Nanocomposite gradient-index variable-focus optic

Abstract

An optic configured for variable wavefront shaping of electromagnetic radiation comprises first and second optical elements each including a solidified heterogeneous coalescence of nanocomposite material providing respective first and second complex dielectric-function gradients. The first and second optical elements are arranged in tandem along an optical axis and together provide wavefront shaping that varies in dependence on a displacement of the first optical element relative to the second optical element.

Description

Docket No. NVX21306CON1CIPPCT NANOCOMPOSITE GRADIENT-INDEX VARIABLE-FOCUS OPTICCROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application claims priority to U.S. Continuation-in-Part Patent ApplicationSerial Number 18 / 341,521 filed 26 June 2023 entitled NANOCOMPOSITE GRADIENT-INDEX VARIABLE-FOCUS OPTIC, the entirety of which is hereby incorporated herein byreference for all purposes.TECHNICAL FIELD

[0002] This disclosure relates generally to optical elements and more particularly topaired optical elements that, when translated relative each other, vary the phase of thewavefront incident thereon.SUMMARY

[0003] This disclosure is directed to composite, variable dielectric-property optics. Inone aspect, a composite, gradient refractive-index optic comprises first and secondoptical elements each including a non-homogeneous material and having an opticalfunction such that when arranged in tandem along the optical axis, the shape ordirection of the transmitted electromagnetic wavefront varies as a function of therelative displacement between the elements.

[0004] The optical functions may include freeform, anamorphic, or non-axisymmetricoptical functions, for instance. The variable dielectric-property optic may includerefraction, permittivity, and permeability, which are related to one another. A gradientrefractive index (GRIN) optic is one example, where a changing refractive index canchange the direction of an optical wavefront. This approach applies to wavefrontsbeyond the visible wavelength range, extending to the infrared (IR), radio-frequency(RF), and millimeter wave (MW) domains. GRIN optics may include composite materialswith particle sizes much smaller than the wavelengths to be refracted. Compositematerials may include inkjet-printed nanocomposites deposited with a concentrationgradient within the optical elements formed.Docket No. NVX21306CON1CIPPCT

[0005] This Summary is provided to introduce in simplified form a selection ofconcepts that are further described in the Detailed Description. This Summary is notintended to represent key or essential features of the claimed subject matter, nor tolimit the scope of the claimed subject matter. Neither is the claimed subject matterlimited to implementations that solve any disadvantage noted in any part of thisdisclosure.BRIEF DESCRIPTION OF THE DRAWINGS

[0006] FIG. 1A is a perspective view illustrating a nanocomposite-ink gradientrefractive-index with variable focus optic comprising a first optical element, a secondoptical element, each the optical elements comprised of a cured nanocomposite inkwherein the first and second optical element have a cubic volumetric gradientrefractive-index such that when arranged in tandem along an optical axis the opticalpower varies based on linear translation with respect to another.

[0007] FIG. 1B is a perspective view of the variable focus optic shown in FIG. 1Awherein the first optical element and the second optical element are tandemly arranged.

[0008] FIG. 2A is a plan view of the variable focus optic as shown in FIG. 1A and FIG. 1Billustrating a neutral alignment position with an offset of zero.

[0009] FIG. 2B is a cross-section view of the neutrally aligned variable focus optic asshown in FIG. 2A, further illustrating optical ray propagation.

[0010] FIG. 2C is a plan view of the variable focus optic as shown in FIG. 1A and FIG. 1Billustrating a positive alignment position with a positive offset.

[0011] FIG. 2D is a cross-section view of the positive aligned variable focus optic asshown in FIG. 2C further illustrating optical ray propagation.

[0012] FIG. 2E is a plan view of the variable focus optic as shown in FIG. 1A and FIG. 1Billustrating a negative alignment position with a negative offset.

[0013] FIG. 2F is a cross-section view of the negative aligned variable focus opticshown in FIG. 2E further illustrating optical ray propagation.

[0014] FIG. 3 is a cross-section view of the variable focus optic further comprising anintermediate layer between the first optical element and the second optical element;Docket No. NVX21306CON1CIPPCT

[0015] FIG. 4 is a plan view of a variable focus optic further comprising an alignmentmark for rotational.

[0016] FIG. 5 is partial perspective view of the sidewall of a variable focus optic,wherein the alignment feature is scaled for a linear encoder.

[0017] FIG. 6 is a cross-section view of a variable focus optic with an alignment markfollowing an aspheric contour to guide post-processing.

[0018] FIG. 7 is a cross-section view of a variable focus optic, wherein the first opticalelement and second optical element have a cubic free form surface.

[0019] FIG. 8A is a plan view of a variable focus optic with an array of cubic volumetricrefractive gradient profiles.

[0020] FIG. 8B is a perspective view of the variable focus optic with an array of cubicvolumetric refractive gradient profiles as shown in FIG. 8A wherein the first opticalelement and the second optical element are tandemly arranged.

[0021] FIG. 9 shows aspects of an example method of manufacture of a nanocompositeink-based optic with a complex dielectric-function gradient and variable focus.

[0022] FIG. 10 shows aspects of (a) a variable-thickness surface-figured homogeneous-index Alvarez-lens elements, and (b) an example GRIN representation of a plano-planoAlvarez-lens elements with a uniform thickness, in which the plano-plano GRINelements are designed to have the same optical path difference (OPD) as the surface-figured homogeneous-index Alvarez-lens elements of panel (a).

[0023] FIG. 11 shows example aspects of configurations A and B of Table 1 herein,illustrating one element of a pair of opposing GRIN optical elements.

[0024] FIG. 12 shows aspects of a conjugated GRIN devices comprised of plano-planoGRIN optical elements, wherein the rotational shift of the plano-plano GRIN opticalelements cause the angle of deflection of the beam to change in azimuth or elevation asa function of the relative angle of rotation between the two.

[0025] FIG. 13 shows aspects of a GRIN phase plate optical element with a refractiveindex distribution that implements a parabolic wedge index function across the device,such that when translated linearly relative to a complementary GRIN phase plate opticalelement it creates a variable cylindrical wedge function.Docket No. NVX21306CON1CIPPCT

[0026] FIG. 14 shows aspects of an example of a GRIN phase plate with an indexdistribution that implements a helical phase function, wherein the index of refractiongradient changes radially as a function of the angle α.

[0027] FIG. 15 shows aspects of a GRIN phase plate optical element implemented witha refractive index profile that implements a saddle function.

[0028] FIG. 16 shows (a) aspects of an example zoom lens comprising two conjugatedGRIN devices that are positioned relative to one another along the optical axis and arepositioned relative to one another along the optical axis, between surface figuredhomogeneous index lenses; and (b), an aspect of an afocal arrangement of twoconjugated GRIN devices, positioned between homogeneous index lenses, wherein thesecond conjugated GRIN device is positioned at the focal plane of the first conjugatedGRIN device.

[0029] FIG. 17 shows aspects of an example optic configured for variable wavefrontshaping of electromagnetic radiation.

[0030] FIG. 18 shows aspects of an example of a GRIN phase plate in which thecomplex dielectric-function gradients change in three coordinate dimensions.

[0031] FIG. 19 shows aspects of a GRIN phase plate optical element implemented witha refractive index profile that implements a cubic index function, where the GRIN phaseplate optical element is designed to change the focus of a beam when translated relativeto a complementary GRIN phase plate optical element.

[0032] FIG. 20 shows aspects of (a) a surface-figured optical element composed of twoconjugate parts designed to provide variable optical power as a function of translationas a result of the combined surface shapes; (b) an example equivalent GRIN optic in twoopposing plano-plano parts designed to modulate incident wavefronts to cause variableoptical power as a function of translation of the combined GRIN profiles; and (c) anexample GRIN optic in two opposing plano-plano parts designed to modulate incidentwavefronts to cause deflection of an incident beam as a function of translation of thecombined GRIN optical elements.

[0033] FIG. 21 shows aspects of a GRIN phase plate optical element with a refractiveindex distribution that implements a spiral phase function across the device.

[0034] FIG. 22 shows aspects of an example of a GRIN optical element.Docket No. NVX21306CON1CIPPCT

[0035] FIG. 23 is a graph of example first and second volume-fraction profiles of aGRIN optic, each plotted as a function of a coordinate r.

[0036] FIG. 24 shows aspects of an example apparatus configured for additivemanufacture of a GRIN optic.DETAILED DESCRIPTIONI.

[0037] A zoom optic or variable-focus optic has an effective focal length or power thatcan be manipulated to change magnification. The most prevalent type of zoom lenscomprise a grouping of optical elements situated along an optical axis wherein changein effective focal length is accomplished by movement of one or more of the opticalelements along the optical axis. Other systems include optical elements wherein surfacecurvature or shape can be altered mechanically or by some other stimulus. Thisdisclosure relates to another approach.

[0038] This disclosure is directed to a nanocomposite refractive gradient variablefocus optic. In one aspect the nanocomposite-ink refractive gradient optic with variablefocus optic comprising a first optical element, a second optical element, each the opticalelements comprised of a cured nanocomposite ink wherein the first and second opticalelement have a cubic volumetric gradient refractive-index such that when arranged intandem along an optical axis the optical power varies based on linear translation withrespect to another.

[0039] Referring now to the drawings, wherein like components are designated by likereference numerals. Figures are characterized by mutually perpendicular axis inCartesian coordinates allow other coordinate systems to be used. Methods ofmanufacture and various embodiments of the disclosure are described furtherhereinafter.

[0040] Referring to FIG. 1A and FIG. 1B, a nanocomposite-ink gradient refractive-indexoptic with variable focus optic 10, also referred herein as variable focus optic, comprisestwo optical elements. The optical elements are normally situated in tandemarrangement as that shown in FIG. 1B, drawn side-by-side for illustrative purposes inFIG. 1A. Variable focus optic 10 has a first optic element 14 with a first surface 16, aDocket No. NVX21306CON1CIPPCTsecond surface 18, and a cured nanocomposite ink 20 within and a second optic element24 with a first surface 26 and a second surface 28 and a cured nanocomposite ink 30 arealigned in tandem on an optical-axis 19. Here, the first and the second surface of each ofthe optical elements are planar, although the surfaces can be figured into any curvatureincluding symmetric positive, symmetric negative, cylindrical, and freeform shapes. Thecured nanocomposite ink comprises of an organic matrix with a nanoparticle dispersedwithin.

[0041] The cubic volumetric gradient refractive-index is achieved by depositing andcuring one or more types of the nanocomposite inks. The optical properties of theorganic matrix, the nanoparticles, and the nanoparticle concentration determine therefractive-index in any particular area. The cured nanocomposite inks comprising thenanoparticles dispersed within the organic matrix can be composed of variousmaterials. The organic matrix of the nanocomposite ink is a curable resin opticallytransmissive for those wavelengths of the optical elements intended use. Within thisdisclosure, nanocomposite inks can also include the organic matrix withoutnanoparticles, also referred to as neat organic matrix. The organic matrix can be curedby photo exposure, thermal processes chemical process, and combinations thereof.Non-limiting examples of organic-matrix materials include polyacrylate, hexanedioldiacrylate (HDODA), polymethyl methacrylate (PMMA), diethylene glycol diacrylate(DEGDA), neopentyl glycol diacrylate (NPGDA), tricyclododecane dimethanol diacrylate(TCDDMDA), bisphenol A novolac epoxy dissolved in organic solvent (SU-8), and othersuch materials.

[0042] The nanoparticles dispersed within the organic matrix can be any material ornanostructure that is sufficiently small to not scatter light of wavelengths intended to beused with the optical element. The nanoparticles can comprise one or more metal,dielectric, semiconductor, or organic materials.

[0043] Nonlimiting examples of nanoparticles include beryllium oxide (BeO), bariumtitanate (BaTiO3), aluminum oxide (Al2O3), silicon carbide (SiC), zinc oxide (ZnO), silicondioxide (SiO2), hollow silicon dioxide nanospheres (hollow SiO2) zinc sulfide (ZnS),zirconium oxide (ZrO2), yttrium orthovanadate (YVO4), titanium oxide (TiO2), coppersulfide (CuS2), cadmium selenide (CdSe), lead sulfide (PbS), tellurium oxide (TeO2),magnesium oxide (MgO), aluminum nitride (AIN), LaF3, GaSbO, nano-diamond, ThF4,Docket No. NVX21306CON1CIPPCTHfO2-Y2O3, Yb2O3, Dy2O3, ZrO2-Y2O3, Si3N4, Y2O3, KBr, Ta2O5, HfO2, AlGaP, SiGe, GaAs, Au,LiF, and molybdenum disulfide (MoS2) including those with core, core-shell, core-shell-ligand, and hollow architectures.

[0044] The nanocomposite inks can be formulated by the nanoparticles type or type,the organic matrix, organic-matrix type, concentration of nanoparticles, andcombinations thereof. The refractive-index of the nanocomposite inks is influenced bythe formulation. An approximation of the optical properties can be calculated based onthe linear summation of the optical properties calculated for the proportionate volumepercentage of the organic-matrix materials and the optical properties calculated for thevolume percentage of the nanoparticles, although direct measurement is a preferredmethod of determining the refractive index for any given nanocomposite-ink formation.For a nanocomposite ink with one nanoparticle type, the refractive index is given by thefollowing equation,where ^^^^^^ ^^^ is the effective index of the nanocomposite ink, ^^%ே^^ ^^^ is the percentvolume of the nanoparticles, ^^ே^^ ^^^ is the refractive-index of the nanoparticles,^^%ைெ^ ^^^ is the percent volume of the organic matrix, and ^^ைெ^ ^^^ is the refractive-indexof the organic matrix. Additional nanoparticles types can be added and percent volumeand refractive-index included in the equation. For instance, nanocomposite ink withnanoparticles that have a high-index relative to the organic matrix will have arefractive-index that increases in proportion to the volume of nanoparticles relative tothat matrix host material increases. Likewise, a nanocomposite ink with a low-indexnanoparticle, for instance a hollow Buckminsterfullerene or a hollow nanosphere,comprised mostly of air, which has an optical refractive index (n) of n = 1, results in ananocomposite ink with a refractive index lower than the organic matrix, whichdecreases closer to n = 1 as the percentage of nanoparticles in the compositionincreases.

[0045] Using one or more of nanocomposite inks, each formulated with differentcompositions, the deposition of droplets of various nanocomposite inks, allows solidswith volumetrically varying complex dielectric functions to be fabricated, with allowsthe materials to exhibit first- and higher-order complex optical properties. Theseproperties can include the index of refraction, third-order susceptibility, or otherDocket No. NVX21306CON1CIPPCTnonlinear optical effects. One method of manufacturing the optical element of thisdisclosure is inkjet printing, described in detail further below.

[0046] Inkjet printing the nanocomposite ink allows materials with complex opticaleffects to be formed that can vary throughout their volume. These effects include thefirst order complex refractive index and higher order nonlinear effects such as the realand imaginary parts of the third-order susceptibility and the nonlinear refractive indexand absorption coefficients.

[0047] To manufacture a volumetric gradient refractive-index at least two of thenanocomposite inks must be used, although additional optical inks, including opticalinks without nanoparticles. These inks can be printed individually or can be mixedduring the printing process to yield optical properties that differ from that of thedroplets themselves. One of the nanocomposite inks printed must have an optical indexat least as low as that required by the gradient optical profile and the othernanocomposite ink must have an optical index as least as high as the highest requiredby the gradient optical index profile. Intermediate values can be obtained by controlleddeposition techniques including nanoparticle diffusion control and advective mixing.Such printing apparatus and printing techniques are described in U.S. Pat. App. No.14 / 863,297, assigned to the assignee of this disclosure, and hereby incorporated byreference in its entirety.

[0048] One method of manufacturing the nanocomposite-ink gradient complex opticalindex solid includes the steps of having or providing a nanocomposite-ink printingapparatus with a nanocomposite ink comprising of an organic matrix with ananoparticle dispersed within. Depositing and forming a first optic element having afirst surface and a second surface with a cubic volumetric gradient optical index.Depositing and forming a second optic element having a first and a second surface witha cubic volumetric gradient refractive-index.

[0049] The variable focus optic can be printed separately as shown in FIG. 1A or intandem arrangement as shown in FIG. 1B. The printing process can include additionalprocess steps and features. For instance, additional optical elements, alignmentfeatures, and sacrificial areas can be printed. Alignment features can be depositedwithin the optical elements, on surfaces, and combinations thereof. For instanceDocket No. NVX21306CON1CIPPCTalignment features can be printed to aid in rotational alignment, inform post-processsurface figuring, and as a guide for cleaving.

[0050] First optical element 14 and second optical element 24 have a cubic volumetricgradient refractive-index profile wherein the z-axis integrated nanocomposite-inkprofile through both the first optical element and the second optical element has at leastan approximate parabolic refractive-index profile. The parabolic refractive-index profilechanges as a function of linear translation between first optic 14 and second optic 24. Insome embodiments the parabolic profile has a symmetric change as a function oftranslation in the x-axis. Such embodiments have a cubic refractive-index profiledescribed bywhere A, B, C, D, E, and F are constants that can be optimized to obtain a desired profile.The cubic concentration profile of the first optical element has a cubic term that is theopposite in sign of the second optical element such that the cubic term is eliminated insummation of each of the cubic concentration profiles and the summed concentrationprofile has a parabolic term െ2 ^^ ^^^ ^^ଶ ^ ^^ଶ^, where offset δ is the linear offset from theoptical axis of each the optical element. In other embodiments the parabolic refractive-index profile has a cylindrical power change when translated in the x-axis or the y-axis.Such embodiments have a cubic concentration profile described by:where both cubic terms are eliminated in summation. Equal spatial translation in boththe x-axis and y-axis causes a symmetric power change.

[0051] The focal length of the variable power optic is inversely proportional to theoffset and thickness of the optical elements although the thickness of the opticalelements and spatial separation between the two optical elements must remainsufficiently thin such that the thin lens approximation remains accurate. For planaroptics with approximately the same magnitude coefficients, thickness and offset of zero,the focal length is infinite and therefore the optic has zero power. For a positive offset,the power increases, for a negative offset power decreases.Docket No. NVX21306CON1CIPPCT

[0052] Referring to FIG. 2A and FIG. 2B, a neutrally aligned variable power optic 40Awith a first optical element 42 and a second optical element 44. First optical element 42has a first surface 46 and a second surface 48 with a cubic volumetric nanoparticlegradient refractive profile. Second-optical element 44 has a first surface 50 and a secondsurface 52 with a cubic volumetric nanoparticle gradient refractive profile. First opticalelement 42 is aligned in tandem with second optical element 44, here with noorthogonal offset. As aligned the concentration of nanoparticles as integrated along thez-axis through both the first optical element and the second optical element is constantexemplified by the uniform concentration as illustrated in FIG. 2A.

[0053] An on-axis ray 57A, a marginal ray 56A, and a marginal ray 58A propagate inparallel to neutrally aligned optical elements 42 and 44. The rays enter at an orthogonalangle to first surface 50, continuously refract through first optical element 42 exitsecond surface 52 into an air gap 54 at an angle oblique with the optical axis. The raysrefract at first surface 46 and continuously refract through first optical element 42 suchthat the rays exit orthogonal to second surface 48 and parallel to the optical axis.

[0054] Referring to FIG. 2C and FIG. 2D a positively aligned variable power optic 40Bhas that shown in Figs. 2A and 2B, except here, first optic 42 has a positive offset δ inrelation to the origin (x^ = 0, y^ = 0). Aligned the positive offset the concentration ofnanoparticles as integrated along the z-axis through both the first optical element andthe second optical element has a positive parabolic shape exemplified by the plan viewillustration shown in FIG. 2C.

[0055] An on-axis ray 57B, a marginal ray 56B, and a marginal ray 58B propagate inparallel to positively aligned optical elements 42 and 44. As before all the optical raysenter at an orthogonal angle to first surface 50. Here, optical ray 56B refracts littlethough second optical element 44, exits, then refracts towards the optical-axis throughfirst optical element 42 towards the larger concentration of nanoparticles, and exits firstsecond surface 48 converging towards a focal spot. On axis ray 57B continuouslyrefracts through second optical element 50 in the positive x-direction, refracts throughfirst optical element 52 in the negative x-direction towards the optical axis and exitsabout parallel with the optical-axis, although some parallax may occur due to theasymmetry. Marginal ray 58B refracts continuously through second optical element 44towards the larger concentration of nanoparticles and exits second surface 52 towardsDocket No. NVX21306CON1CIPPCTfirst optical element 42. Marginal ray 58B refracts little through first optical element 42and exits first surface 48 towards the focal spot.

[0056] Referring to FIG. 2E and FIG. 2F, a negatively aligned variable power optic 40Chas that shown in Figs. 2A and 2B, except here, first optic 42 has a negative offset +δ inrelation to the origin (x^ = 0, y^ = 0). Aligned the negative offset the concentration ofnanoparticles as integrated along the z-axis through both the first optical element andthe second optical element has a positive parabolic shape exemplified by the plan viewillustration shown in FIG. 2E.

[0057] An on-axis ray 57C, a marginal ray 56C, and a marginal ray 58C propagate inparallel to positively aligned optical elements 42 and 44. As before all the optical raysenter at an orthogonal angle to first surface 50. Here, optical ray 56C refracts littlethough second optical element 44, exits, then refracts away from the optical-axisthrough first optical element 42 towards the larger concentration of nanoparticles, hereshifted away from the optical axis. Optical ray 56C exits first second surface 48diverging from the optical-axis. On-axis ray 57C continuously refracts through secondoptical element 50 in the positive x-direction, refracts through first optical element 42in the negative x-direction towards the optical axis and exits about parallel with theoptical-axis, although some parallax may occur due to the asymmetry. Marginal ray 58Crefracts continuously through second optical element 44 towards the largerconcentration of nanoparticles, here diverging from the optical-axis, and exits secondsurface 52 towards first optical element 42. Marginal ray 58B refracts little through firstoptical element 42 and exits first surface 48 diverging from the optical-axis.

[0058] Referring to FIG. 3, a variable focus optic 100 has that shown in FIG. 1B, furthercomprising a sacrificial layer 102. When printed in tandem arrangement, sacrificiallayer 102 can be deposited between first optical element 42 and second optical element44. The sacrificial layer facilitates cleave between the two optical elements.Alternatively, an elastomeric layer can be deposited between the optical elementsallowing movement between the two-optical elements and reducing surface refractionat facing surfaces.

[0059] Referring to FIG. 4, an optical element 120 has a first surface 122 with analignment feature 126, alignment feature 128, and alignment feature 130. Here theDocket No. NVX21306CON1CIPPCTalignment features are on first surface 122, distributed on the perimeter of opticalelement 120 to facilitate rotational alignment to another optical element.

[0060] Referring to FIG. 5, an optical element 130 is shown with a first surface 132, asecond surface 124, and alignment feature 126. Here, alignment feature 126 positionedon an outer sidewall 129 are spaced scales. Various types of scales can be depositedinducing optical, magnetic, capacitive and inductive. During the printing process thescales can be deposited on the outer sidewall to pair with a sensor thereby forming anencoder for position feedback. The variable focus optic can be paired with a lineartranslation stage, such as a MEMS stage, and the encoder can provide direct positionalfeedback.

[0061] Referring to FIG. 6, an optical element in process 150 has a first surface 152, asecond surface 154, and an alignment feature 156. Here alignment feature 156 ispositioned with the volume of the optical element along an aspheric contour 158.Alignment feature 156 provides positional feedback to inform post-process surfacefiguring. For instance a single-point diamond turning head 160 can either use thealignment feature to setup a CNC tool or if equipped with optical recognition can followalignment feature 156 to form the aspheric contour 158.

[0062] Referring to FIG. 7, an optical element 180 has a first optical element 182 and asecond optical element 184 each with a cubic volumetric gradient refractive-index.Here, first optical element has a first surface 186 that is planar and a second surface 188that has a cubic contour. Second optical element 184 has a first surface 192 that has acubic contour and a second surface 190 that is planar.

[0063] Referring to FIG. 8A and FIG. 8B, a variable focus optic with an array of cubicvolumetric refractive gradient profiles has a first optical element array 202 and asecond optical element array 204. First optical element array 202 has a first surface 206and a second surface 208 with a plurality of cubic volumetric refractive gradientsbetween. Second optical element array 204 has a first surface 210 and a second surface212 with a plurality of cubic volumetric refractive gradients between. Here, the cubicvolumetric refractive gradients is a four-by-four array.

[0064] The first optical element’s cubic volumetric refractive gradients are paired withthe second optical element’s cubic refractive gradients wherein each pair have anDocket No. NVX21306CON1CIPPCToptical power that varies on linear translation. By way of example, an exemplary cubicvolumetric refractive gradient 216 and 218 has a power when tandemly arranged suchas that shown in FIG. 8B. Each of the paired volumetric refractive gradients can have thesame power or the power can vary across the array.

[0065] Here, each of the cubic volumetric refractive gradients have a square opticalshape to increase the fill factor. In other embodiments the optical aperture can becircular. Carbon, metal, or other opaque inks can be used to separate isolate each of thepaired cubic volumetric refractive gradients to reduce or eliminate crosstalk duringlinear translation. As noted above, the area between the first optical element and secondoptical element filled and surfaces can shaped.

[0066] Some implementations will now be summarized. The first implementation is amethod of manufacturing a nanocomposite-ink gradient complex optical index opticwith variable focus comprising: (a) having or providing a nanocomposite-ink printingapparatus with a nanocomposite ink comprising of an organic matrix with ananoparticle dispersed within; (b) depositing and forming a first optic element having afirst surface and a second surface with a cubic volumetric gradient optical index; and (c)depositing and forming a second optic element having a first and a second surface witha cubic volumetric gradient optical index. Here the first optical element and the secondoptical element each comprise a cured nanocomposite ink with an organic matrix and ananoparticle dispersed within; and the first and the second optical element arearranged in tandem along on an optical axis have an optical power that varies based onlinear translation between the first and the second optical element orthogonal to theoptical axis.

[0067] The second implementation is a method according to the first implementation,wherein the first optical element and the second optical element are printed in atandem arrangement. The third implementation is a method according to the secondimplementation, further comprising the step of depositing a sacrificial layer betweenthe first optical element and the second optical element, the sacrificial layer facilitatingpost fabrication physical separation of the first optical element and the second opticalelement. The fourth implementation is a method according to the secondimplementation, further comprising the step of depositing an elastomeric layer betweenthe first optical element and the second optical element. The fifth implementation is aDocket No. NVX21306CON1CIPPCTmethod according to the first implementation, further comprising the step of depositingan alignment feature on the first optical element, the second optical element, or both.The sixth implementation is a method according to the fifth implementation, where thealignment features are provided to inform post-process surface figuring. The seventhimplementation is a method according to the fifth implementation, wherein thealignment features are provided as a guide for cleaving, sawing, or otherwise physicallyseparating the first optical element from the second optical element. The eighthimplementation is a method according to the fifth implementation, wherein thealignment features are provided for aligning the first and the second optical elementduring assembly of an optical system. The ninth implementation is a method accordingto the first implementation, wherein the first optical element and the second opticalelement are composed of an array of optical elements, each composed of a cubicvolumetric refractive gradient profile. The tenth implementation is a method accordingto the ninth implementation, wherein one or more array of optical elements are printedin square, hexagonal, or other close packed geometry chosen to eliminate theproportion of light striking a non-index modulated portion of the element.

[0068] The eleventh implementation is a nanocomposite-ink gradient refractive-indexoptic with variable focus comprising: a first volumetric gradient refractive-index opticalelement having a first surface and a second surface; and a second volumetric gradientrefractive-index optical element having a first surface and a second surface, wherein thefirst and the second optical element are arranged in tandem along on an optical axis andhave an optical power that varies based on linear translation between the first and thesecond optical element orthogonal to the optical axis.

[0069] The twelfth implementation is an optic according to the eleventhimplementation, wherein the first and the second optical element have each have atleast one planar surface. The thirteenth implementation is an optic according to theeleventh implementation, wherein the first optical element and the second opticalelements are formed using a nanocomposite ink having a common organic matrixmaterial. The fourteenth implementation is an optic according to the eleventhimplementation, wherein the gradient pattern of one or more optical inks is chosen tominimize geometric aberrations from the first optical element or the second opticalelements surfaces. The fifteenth implementation is an optic according to the eleventhDocket No. NVX21306CON1CIPPCTimplementation, wherein the volumetric gradient refractive-index minimizes chromaticaberration. The sixteenth implementation is an optic according to the eleventhimplementation, wherein the gradient pattern is chosen to minimize geometricaberration. The seventeenth implementation is an optic according to the eleventhimplementation, further comprising a means of translating the first optic, the secondoptic, or combinations thereof. The eighteenth implementation is an optic according tothe seventeenth implementation, wherein the means of translating includes manualmechanisms, motorized mechanisms, and combinations thereof. The nineteenthimplementation is an optic according to the seventeenth implementation, wherein themeans of translation is a microelectromechanical system. The twentiethimplementation is an optic according to the first implementation, wherein the firstoptical element and the second optical element are made from different nanoparticlematerials to correct chromatic aberration. The twenty-first implementation is an opticaccording to the first implementation, further comprising an intermediary layerbetween the first and the second optical elements. The twenty-second implementationis an optic according to the twenty-first implementation, wherein the intermediarylayer has a gradient refractive index. The twenty-second implementation is an opticaccording to the twenty-first implementation, wherein the intermediary layer correctschromatic aberration.

[0070] From the description herein one skilled in the art can manufacture thedisclosed apparatus and practice the disclosed methods. While this disclosure has beendescribed in terms of particular embodiments and examples, other embodiments andexamples can be implemented without departing from the intended spirit or scope. Thisdisclosure is not limited to the illustrated embodiments but only by the claimsappended hereto.II.

[0071] FIG. 9 shows aspects of an example method 15 of manufacture of ananocomposite ink-based optic with a complex dielectric-function gradient and variablefocus. In some examples the complex optical index is a freeform function with no axis ofsymmetry.

[0072] At 17 of method 15 is provided a nanocomposite-ink printing apparatus with ananocomposite ink including an organic matrix with a nanoparticle dispersed within theDocket No. NVX21306CON1CIPPCTorganic matrix. At 19 a first optical element having a first surface and a second surface isdeposited and formed. The first optical element has a gradient optical index betweenthe first and second surfaces. At 21 a second optical element having a third surface anda fourth surface is deposited and formed. The second optical element has a gradientoptical index between the third and fourth surfaces. In some examples, here and / or instep 19 above, one or more layers of deposited materials may be cured beforeadditional layers are deposited.

[0073] In method 15 the first optical element and the second optical element eachcomprise a cured nanocomposite ink with an organic matrix and a nanoparticledispersed within the organic matrix. The first and the second optical elements arearranged in tandem along on an optical axis and have an optical power that variesaccording to a translation between the first and second optical elements. Thetranslation can be a circular translation orthogonal to the optical axis, in someexamples. In some examples the optical axis is not necessarily perpendicular to the firstoptical element. In other examples the optical axis may be perpendicular to the firstoptical element.

[0074] At optional step 23, a third, fourth optical element, etc., may be deposited andformed by analogous processing until the final desired optical element is deposited andformed. In some examples, a third optical element is configured to cause a focal pointcreated by the first and second optical elements to be minimized over a range oftranslations. In some examples, a third optical element is configured to co-locate anoptical axis of the first optical element to an optical axis of the second optical elementover an operating range of the translation. In method 15 the nanocomposite ink of thefirst optical element and the nanocomposite ink of the second optical element may beselected such that a slope of refractive index with respect to wavelength of a highestaverage refractive index nanocomposite ink and slope with respect to wavelength of alowest average refractive index ink are parallel to 1% or better.III.

[0075] ‘Freeforms’, as used herein, are optical shapes or optical surfaces designed withlittle or no symmetry. The added degrees of freedom afforded by freeforms makes themuseful for implementing non-radially symmetric phase plates—e.g., for off-axis opticalsystems where axial symmetry is broken. Interest in freeform optics is driven in part byDocket No. NVX21306CON1CIPPCTpotential applications in near-eye displays and compact optical systems for medical,military, and mobile imaging, as well as illumination devices with constraints on sizeand weight.

[0076] Cubic-phase optical elements are freeforms of particular interest. Cubic lenseshave been shown to exhibit increased depth of focus. A variant of cubic-phase optics is afocus-correcting lens called an ‘Alvarez’, ‘Lohmann’, or ‘Alvarez-Lohmann’ lens. Despiteits straightforward operating principle, the Alvarez-Lohmann lens can be impractical toimplement due to the aberrated optical performance of conventionally manufacturedoptical elements. In particular, the manufacture and metrology of transmissive cubicsurfaces with the necessary precision remains a time-consuming and often expensiveprocess. Moreover, as existing refractive freeforms are made using homogeneousmaterials, complex chromatic aberrations result from the dispersion introduced at theirshaped surfaces.

[0077] In U.S. Patent Application 14 / 970,378 (incorporated by reference herein for allpurposes), Vadient described a gradient refractive-index optic fabricated via inkjetprinting of nanocomposite materials, where nanocomposite optical elements aretranslated relative to one another, in a direction orthogonal to the optical axis, to changethe focus. In the present disclosure, the nanocomposite optical elements are generalizedto include a broader class of optical elements, where the displacements (between andamong the optical elements) may be rotational as well as translational, and where thecombined optical functions extend beyond optical power. Furthermore, pairs of opticsof this kind may be included in an optical system to allow for cascaded optical effectssuch as zoom lenses (FIG. 16), or beam scanning (FIG. 12). In some examples the devicesmay be optimized for the part of the electromagnetic spectrum other than visible byforming permittivity or permeability index gradients. More general freeform gradient-index (GRIN) optics, in sum, offer a method of addressing the limitations of theconventional Alvarez-Lohmann approach and providing novel and non-obviousextensions directed to specific applications.

[0078] GRIN offers added degrees of freedom in the form of dimensionally varyingindex gradients, which may be used to reduce the size and weight of optical systems.However, confined by the available fabrication processes, GRIN-lens configurationshave historically been limited to shallow, radially-symmetric index gradient profiles andDocket No. NVX21306CON1CIPPCTto optics of relatively small size. Further, the dispersion properties of the GRINstructures have been limited by process-compatible materials. Recent advancements inthe manufacturing of gradient-index (GRIN) media now make possible refractive-indexdistributions that can vary arbitrarily in up to three spatial dimensions, ^^( ^^, ^^, ^^).

[0079] Additive manufacture of optics has also been explored. One attribute of additivemanufacture is that designs can be readily fabricated, on demand, directly fromsoftware design tools, eliminating the cost and lead times associated withmanufacturing conventional freeform optics. It has recently been demonstrated thatdrop-on-demand, inkjet-print, additive manufacture can be used to create volumetricindex gradients within an optical element. In addition to enabling fabrication of radially-symmetric spherical GRIN elements, inkjet-print manufacture enables fabrication ofcomplex, radially symmetric, aspheric index gradients, as well as three-dimensional(3D) aspheric index gradients, where the high-order gradient profiles vary axially asfunction of position on the optical axis. Inkjet-print manufacture is also well-suited formanufacturing freeform index gradients.

[0080] In addition to these advantages, freeform GRIN optics offer a method oftransforming optical phase. In plano-plano freeform GRIN phase plates, the refractiveindex gradient controls optical path differences (OPD), as opposed to the summation offigure thicknesses (where the OPD is controlled by the shapes of the homogeneousfreeforms). While there are similarities between freeform GRIN and freeform surfaces, adistinct benefit of a plane-parallel freeform GRIN is that it provides degrees of freedomin three dimensions, n(x, y, z), whereas freeform surfaces provide degrees of freedom inonly two dimensions, ^^( ^^, ^^). Also, with plane-parallel freeform GRIN, pairs of devicescan be brought into closer proximity to one another, such that the pair better representsa ‘thin lens’ approximation. With 3D freeform GRIN optics, it is possible to implementarbitrary volumetric gradient profiles in which there are no axes of symmetry. Theability to create polynomial terms with longitudinal (i.e., z direction) variations in thegradient index profiles provides extra degrees of freedom not available with surface-figured freeforms. This can be used to correct for axis deviation, reduce aberrations,accommodate non-paraxial rays, and otherwise improve the performance of an AlvarezLohmann-type lens. To maximize the degrees of freedom, it is possible to polish ofDocket No. NVX21306CON1CIPPCTmachine surface figures on 3D freeform GRIN optical materials, thereby combining bothapproaches.

[0081] Significantly, the ability to engineer multiple nanocomposite optical materialsusing multi-constituent blends, and to then mix the multiple multiple-constituentoptical feedstocks when print composing substrates, makes it possible to independentlyspecify the index gradient and the dispersion of additively manufactured GRIN opticalmaterials, allowing for fabrication of achromatic singlet lenses and achromatic freeformrefractive optics.

[0082] Several of the degrees of freedom afforded by the inkjet-print additivemanufacture of freeform GRIN lenses are demonstrated in the optimization, fabrication,and metrology of planar Alvarez-Lohmann lenses, demonstrated and environmentallytested for used in accommodating vision in respiratory masks. In particular, theAlvarez-Lohmann class of lenses offers a method of performing positive and negativediopter adjustments for vision accommodation in respirator masks.

[0083] As shown by schematic example in FIG. 10, the Alvarez-Lohmann lens is acomposite lens comprising two, spatially separate variable phase plate elements that,combined, make an effective lens. Panel (a) of FIG. 10 provides a schematic illustrationof a common implementation of an Alvarez-Lohmann lens, with variable-thickness,surface-figured Alvarez-lens elements. In its most common implementation both lenscomponents are plano-freeform elements with surface geometries described by a cubicpolynomial equation; the first element features the negative function of the second f1(x,y) = – f2(x, y). With translation between the upper and lower parts of the device, thesurface functions produce quadratic wavefront changes.

[0084] In a traditional Alvarez-Lohmann lens, the two variable phase plates are alignedalong the optical axis, with the cubic surfaces inverted with respect to each other. Whenlaterally aligned, the effective cumulative thickness introduces a phase delayproportional to the combined thickness of the shaped elements. The structure isdesigned such that at one value of lateral shift between the two parts, δ = δ0, the deviceacts as a neutral optical element that does not change the wavefront of a propagatingplane wave, owing to the cancellation of the cumulative phase delays introduced by thesurface profiles of one element by those of other element. Thus, the system has aninfinite focal length. The optical power of the device changes, however, proportionallyDocket No. NVX21306CON1CIPPCTwith the lateral shift δ. In performing a transverse shift of the surfaces relative to oneanother there is no longer perfect cancellation of the variable phase delays of the twoelements. In this case, the component can be described by a phase function distributioncorresponding to a simple delay on the incident illuminating wavefront, which isproportional to the effective thickness of the composite surfaces. The differential of thetwo cubic profiles results in a composite surface with a spherical thickness variationinducing a quadratic variation in the wavefront, such that depending on the direction ofthe shift, the composite thickness is equivalent to a converging or diverging lens of acertain focal length.

[0085] Alvarez and similarly Lohmann used this approach to generate varifocal lenses.The difference between Alvarez and Lohmann lens functions is the shape of the cubicsurface; the relation between the Alvarez and the Lohmann descriptions is a 45-degreerotation and a scale factor of √2. In these designs, each lens has a cubic-type surfaceprofile, which may be represented by:z1(x, y) = A(x3 / 3 + xy2) + Fx + E, (4) z2(x, y) = –A(x3 / 3 + xy2) – Fx + E, (5)where x and y are in-plane transverse coordinates normal to z, representing position,and A, D, and E are design variables.

[0086] While their values can highly influence wavefront error, the F and E coefficientshave no direct effect on the focal length. The coefficient E is a constant representing thelens element thickness at δ0; it can be adjusted to reduce overall lens thickness.Although a finite gap is used to prevent collisions between the surfaces as they areshifted, assuming that there is no gap between the elements, when aligned at δ0, thecombined thickness of the two-element system is z = z1 + z2 = 2 E, the equivalent of aparallel plate. Meanwhile, the coefficient F defines the tilt term of the freeform surfacethat affects the slope of the prism along the x direction of the cubic profile; it may beused as a compensating term to reduce the effects on sag that result from increasing thecoefficient A. The coefficient A is the area scale that represents the rate of lens powervariation with lens movement in the x direction.

[0087] Working within the thin-lens and paraxial approximations, one can vary thefocal length of the composite pair by transversely shifting the plates relative to oneDocket No. NVX21306CON1CIPPCTanother. The optical power varies linearly with δ, and both positive and negative powercan be obtained by altering the direction of shear (i.e., the sign of δ). When the firstelement moves a displacement δ and the second moves −δ along the x direction, thecombined thickness, z, has a parabolic term −2 A^ δ(x2 + y2). The parabolic term isequivalent to a standard axially symmetric constant curvature term. For a determinedlateral displacement range, the value of A affects the sag and curvature of the object-conjugate plates. The effective sag of the composite-surfaced pair gives rise to a focalpoint. In order to relate the spherical profile to focal length, a first-order paraxialapproximation is assumed. From the sag equation of a surface, the radius of curvature, r,is given by:R= 4 r^δ. (6)The effective focal length, f, of the combination, can then be calculated asf= [4 A^δ(n – 1)]– 1, (7)where n is the refractive index of the lens material. The refractive index contributes tothe optical path difference, and this generally forces both components to be the samematerial if they are identical cubic surfaces. However, if both the coefficient A and n arechanged proportionally, it is possible to maintain purely quadratic phase variations.

[0088] The performance limitations of prior Alvarez-Lohmann lens implementationsstem from the deviation of practically manufactured lenses from their ‘thin-lens’approximations, exacerbated by the air gap between the pair. A ‘thin-lens’approximation allows the Alvarez-Lohmann lens to be represented by two cubicfunctions superimposed on top of one another, such that the coordinates at which theray of light exits the first element and the coordinates at which it intersects the secondelement are the same. However, if the phase plates have a finite thickness and areseparated—as is the case with homogeneous index surfaces—there can be a significantdifference in the ray coordinates relative to the theoretical, causing Alvarez-Lohmannlens implementations to deviate significantly from ideal performance.

[0089] To obtain a large range of lens powers with minimal lateral translation, thevalue of A needs to be large, which, in turn, increases the lens sag. This causes the lensesto be increasingly thick away from the optical center. Obtaining a large optical poweradjustment range in a compact configuration is also challenging. The change in opticalDocket No. NVX21306CON1CIPPCTpower is proportional to the product of A and the lateral displacement, δ; so, for acompact implementation, δmax is made small, which requires that A be large. However,increasing the value of A increases curvature, which in turn increases the sag and causeslarger amounts of aberrations from the combined elements. Since the elements are non-rotationally symmetric freeforms, aberrations specific to off-axis propagation, includingcoma and astigmatism may also degrade performance.

[0090] Higher-order surface terms may be used to partially compensate for theseaberrations; however, realizing precise high-order freeform surfaces is constrained byexisting process limitations, and the correction is difficult to maintain throughout thepower range. Moreover, as prior Alvarez-Lohmann lenses have been realized usinghomogeneous optical media (e.g., glass or plastics), chromatic aberrations present agenuine limitation. Combined with the non-superposition of the dual optical functionsresulting from lens thickness, optical axis dislocation, and non-paraxial behavior of gazeangles, practical Alvarez-Lohmann lens implementations deviate from their ideals.IV.

[0091] To address these issues and to provide other advantages, this disclosurepresents a custom additive manufacture technology platform, ‘Variable Index ofRefractive Gradient Optics’ (VIRGOTM), which uses inkjet-printhead deposition ofspecific compositions of optical nanocomposite feedstock.

[0092] The properties of the printable primary ‘optical ink’ feedstocks play animportant role in inkjet-constructed optical elements. The nanocomposite opticalfeedstocks are formulated by embedding one, or more, non-scattering organic orceramic nanoparticles in one or more low-viscosity, optical-grade, photocurablemonomers. Each nanoparticle is small (e.g., < ~10 nm, less than 1 / 30th the wavelengthof light passing through the optic) and is chemically coated to eliminate agglomeration,such that Rayleigh and Mie scattering are insignificant. The optical inks are formulatedwith the rheological properties necessary for reliable inkjet printhead deposition.

[0093] At minimum, using inkjet print fabrication, two optical inks are used to create aGRIN element, a ‘high index’ optical ink, nhigh, and a ‘low index’ optical ink, nlow. Thedifference in the index values of the two primary inks is the refractive index contrast,Δn. A unique feature of multi-constituent nanocomposite optical inks is that it isDocket No. NVX21306CON1CIPPCTpossible to precisely tailor the refractive index spectra of each optical ink relative to theother(s), to control primary and secondary color. The refractive-index spectralproperties of the primary optical inks are a linear function of the volume fraction of theconstituent properties of the inks. The dispersion of homogeneous materials over the‘d;F;C’ spectrum is commonly characterized by the Abbe number, ν, and partialdispersion, P, asν = (nd – 1) / (NF – nC), (8) Pd, F = (nF – nd) / (nF – nC). (9)For homogeneous materials with normal dispersion, ν is always positive and the valueof P is around 0.7 and is bounded between zero and one. The partial dispersion valuesof optical materials strongly trend linearly with their dispersion values.

[0094] For GRIN optics, the GRIN Abbe number (ν GRIN) and partial dispersion (Pd, F GRIN)are defined asνGRIN= (Δnd) / ΔnF– ΔnC) (10) Pd, F_GRIN = (ΔnF – Δnd) / (ΔnF – ΔnC), (11)so their values are no longer constrained. Defined by the difference in the refractive-index spectra of the high- and low-index primary optical inks used to print thegradients, the index gradients can have positive, zero, or negative νGRIN values, and whenmore than three constituents are used to synthesize the feedstock, the dispersion can bedefined independently from the index gradient.

[0095] Independent control of dispersion relative to the index gradient allows forachromatic singlet GRIN lenses to be fabricated. For example, even for a binary primaryoptical ink pair, if the slope of the high-index optical ink, nF(high) – nC(high), is matchedto the slope of the low-index optical ink, nF(low) – nC (low), then ΔnF = Δnc and the GRINlens is achromatic. Moreover, by introducing sufficient additional constituents into thefeedstock composition, it is possible to relax the dependence of the PGRIN values relativeto the ^^ values, making possible a wide range of anomalous partial dispersion GRINmaterials not available in standard glass or plastic materials. Achieving dispersion andpartial-dispersion values independent of the index gradient is desirable for freeformrefractive optical elements, due to the complexity of dispersion attributable to freeformsurfaces and gradient index functions.Docket No. NVX21306CON1CIPPCT

[0096] Increasing the number of primary optical inks expands the degrees of freedom.The number of primary optical inks that can be printed concurrently is limited by thenumber of available printheads. For the simple case of a binary ink set, at least twoprintheads are required. Constructing each layer of the refractive index gradientrequires separate bitmaps for each printhead. For each pass of the printhead over thesubstrate, the bitmaps define the drop density patterns of each primary optical ink.

[0097] Using inkjet print deposition, ‘print composition’ is used to define theintermediate refractive index values of the gradient profiles. With spatial dropletconcentrations defined by the bitmaps, local mixing and inter-diffusion of the co-deposited primary optical inks causes the spatial localities to assume refractive indexspectra that are the weighted average of the spectral properties of the constituent inks.A simple binary linear composition model allows the substrate index value to beapproximated at each wavelength as a function of two primary optical inks, nlow(λ) andnhigh(λ), aswhere ^^^ ^^, ^^^^^௪ and ^^^ ^^, ^^^ ^^^^ are the local volume concentrations of the primaryoptical inks defined by index nlow and index nhigh, respectively, at location (x, y), and

[0098] In the graphics-print industry, halftoning algorithms are used to determine theplacement of the optical ink droplets such that the reflective properties of the substratecreate grayscale images, using only black droplets. To create refractive gradient indexprofile bitmaps using optical ink pairs, a comparable process is used. To accommodatethree-dimensional refractive index volumes (i.e., n(x, y, z)), the halftoning algorithmsquantize the refractive index profile designs in three dimensions. The residuals fromquantization are distributed to neighboring pixels that have not yet been processed.This method can also be extended to three-level or multi-level halftoning, so thatseveral primary inks can be printed concurrently. Printing multiple inks reduces thelevels of quantization, allowing for more precise control over gradient index patterns.Concurrent printing of multiple inks also provides degrees of freedom for controllingdispersion and secondary color.Docket No. NVX21306CON1CIPPCT

[0099] The stack of print maps, one for each printhead each layer, are uploaded to theprinter when fabricating the optic. Using industrial printers, the drop placement of theoptical inks can be controlled to better than one micron precision, but after inter-diffusion of the concentrations of optical inks and polymerization, it is possible tofabricate complex sub-wavelength smooth gradient profiles that precisely match thedesign intent.

[0100] Planar freeform-GRIN elements can be printed with better than λ632nm / 6 flatsurfaces, without post-processing. Due to the benefits of the inter-dispersed ceramicnanoparticles that are tightly crosslinked within the polymer matrix, the nanocompositeoptical materials are sufficiently strong and hard that, using industry standardprocesses, the surfaces may be polished or shaped to high precision industry standards.The nanocomposite materials are also non-hydroscopic and have lower temperaturesensitivity than plastic materials.V.

[0101] Inkjet printing of nanocomposite GRIN materials allows for performanceoptimization of Alvarez Lohmann-type lenses not available with surface-shapedhomogeneous materials. Panel (b) of FIG. 10 shows an example GRIN representation ofa plano-plano Alvarez-lens elements with a uniform thickness, in which the plano-planoGRIN elements are designed to have the same optical path difference (OPD) as thesurface-figured homogeneous-index Alvarez-lens elements of panel (a). In a plano-planofreeform GRIN Alvarez-Lohmann lens, optical power is generated in a manneranalogous to that of surface-figured homogeneous index elements. The OPD isgenerated across the GRIN phase plates not by summing the thicknesses attributable tothe surface shapes of each element, however, but by controlling Δn and the elementthickness t.

[0102] Assuming no variation of the refractive index perpendicular to the optical (i.e.,z) axis, the generally cubic Alvarez-Lohmann phase profiles may be represented forGRIN implementations as:n1(x, y) = n00 + k(x3 / 3 + xy2+ Rx +E), (13) n2(x, y) = n00– k(x3 / 3 + xy2– Rx + E), (14)^Docket No. NVX21306CON1CIPPCTwhere ni(x, y) is the refractive index at point x, y on the lens, n00 is the base index (valueof the lowest index optical ink), the index profile coefficient k is a scaling factor thatdescribes the magnitude of the refractive index term, the coefficient R is a field constantthat describes the tilt term, and E is a scalar constant. These equations do not take intoaccount any axial (z-direction) variations of the refractive-index profiles. However, theability to change the gradient profiles expressed in eqs 13 and 14, by adding higherorder polynomial terms that vary along the axial dimension, or adding aberration orperformance enhancing optical functions, allows for the devices to be optimized forspecific applications. For example, aberrations can be corrected, axis-offsets may beaccommodated, off axis implementations are made possible, profiles can be optimizedfor gaze angles, and the profiles can compensate for offsets in magnification.

[0103] The similarity between eqs 4 and 5 and eqs 13 and 14 is apparent. The indexprofile coefficient k of eq 13 is equivalent to coefficient A of eq 4, used to describe themagnitude of lens-power variation for a surface-figured Alvarez-Lohmann lens. Thecoefficient R of eq 13, when scaled by the coefficient k (i.e., R^ / k) is functionallyequivalent to coefficient F of eq 4, which governs the amount by which the polynomialfunctions are tilted. The similarity shows that the OPD determines the optical functions,and that there is a GRIN profile that can be derived for freeform surface profiles thatallow GRIN analogs of ‘Alvarez-Lohmann’ type elements to be realized.

[0104] For a thin Alvarez-Lohmann component, the variation in OPD within the clearaperture of a single thin plano-plano lens plate, is given by:OPD(x, y) = t [n(x, y) – n00], (15)where n00 is the base refractive index value at which point OPD = 0. The OPD of thecombined elements is obtained by summing the index maps of eq 14, calculated with xvaried with a positive and negative translational shift, in the z-direction ±δ , relative toone another other, with a combined thickness of 2 t.VI.

[0105] Naturally, the eye is a compound optical system, and an eye-lens-object opticalsystem is set according to the wearer’s visual performance and the characteristic of thelens assembly, as defined by dioptric power, astigmatism, and visual axis. For use inDocket No. NVX21306CON1CIPPCTvision-corrective optics, the optical power of the Alvarez-Lohmann lens is modeled asthough it were a radial lens fitting within the eyebox, with a focal length, fGRIN:where Δn is the maximum index change across the eyebox, r is half the width of theeyebox, t is the thickness of the lens. The focal length is inversely proportional todiopters according to the relation D = 1000 / fGRIN; thus:D= (2000 t^Δn) / r2. (17)

[0106] There are different combinations of t and Δn that grant the same powervariation, and increasing either t or Δn enables larger values of the coefficient k, whichenables high optical power to be achieved. However, Δn is practically limited by theproperties of the available optical feedstock. With materials optimized for inkjetprinting, which are constrained by the printhead-compatible rheological properties ofthe optical inks, Δn is typically between 0.06 and 0.12 in the visible wavelength range.For some industrial printheads, Δn values larger than 0.28 are possible.

[0107] Within the constraints of the application, the Alvarez-Lohmann cubic profilesmay be optimized by varying the coefficients k and R, which may be assumed equal andopposite for each element. To select the ideal scaling factor k and field constant, thecontributions of the Seidel aberrations and the overall RMS wavefront error may beevaluated at various positions in the eyebox.Field tilt constant, R 0 68 25 25 38 38 56 38 38R / k (10 ) 0.00 59.03 7.51 8.42 22.22 22.22 56.74 7.51 11.14Optical power (±D) 5 5 5 5 5 5 5 5 9GRIN Δn 0.089 0.089 0.06 0.06 0.06 0.06 0.06 0.06 0.12Thickness (z, mm) 5 1.40 0.5 0.84 0.98 1.22 1.26 1.4 1.1Lateral Shift (Δδ, mm) 7.5 7.5 7.5 5 7.5 6 10 7.5 6Lens Width (mm) 25 25 20 20 25 25 30 25 25Eyebox width (mm) 10 10 5 10 10 10 10 11 11Width / Eyebox Ratio 2.5 2.5 4 2 2.5 2.5 3 2.5 2.50.004WFE – eyebox center 0.0012 0.0004 0.0011 0.0009 0.0014 0.0008 0.0010 0.003940.020WFE–eyebox edge x (left) 0.0068 0.0009 0.0067 0.0084 0.0076 0.0049 0.0030 0.02542WFE–eyebox edge y (top) 0.040 0.0127 0.0008 0.0065 0.0051 0.0077 0.0049 0.0130 0.0234Docket No. NVX21306CON1CIPPCT 5

[0108] A summary of nine different configurations selected from a configuration-trade study is shown in Table 1. FIG. 11 shows example aspects of configurations A andB of Table 1 herein, illustrating one element of a pair of opposing GRIN optical elements.Panels (a) and (b) show gradient-index maps; panels (c) and (d) show x-axis crosssections of the gradient index through the center; and panels (e) and (f) show theoptical power as a function of lateral translation when one element is translated againsta duplicate of itself inverted in the y direction. The examples show how the variables ofthe Alvarez- Lohmann lens may be changed to change the optical performance: in thiscase optical power. Configuration B shows an example of how the Alvarez-Lohmannfunction can be tilted relative to the optical axis so that the full value range of indexvalues can be used and the optical power can be increased in a thinner configuration.

[0109] Configurations A and B are implemented with Δn = 0.089 and differ from eachother by the value of coefficient R, which describes the tilt term. The thickness of theelements is minimized when Δn is maximized and the tilt term is optimized.Configuration A was implemented without tilt of the polynomial function and as a resultwas 3.57 times thicker than Configuration B. As shown in Table 1, due to it beingthinner, Configuration B had lower WFE across the eyebox than did Configuration A.

[0110] Further with respect to FIG. 11, the equation for the cubic Alvarez lens isz(x,^y)^=^(ax3^+^by3^+^cx2y^+^dxy2^+^ex2^+^fy2^+^gx^+^hy^+^i), (18)where z(x,^y) is the surface height of the lens at position (x,^y), and a,^b,^c,^d,^e,^f,^g,^h,^aId iare coefficients that control the shape of the lens. The cubic Alvarez lens is capable ofproviding a larger change in focal length compared to other varifocal lenses, making ituseful in a variety of optical applications.

[0111] Configurations C through G may be all configured to achieve the same opticalpower (±5 D) and may be configured with Δn = 0.06. The progression of these lensdesigns decreased the coefficient k values, which influences the term for the rate ofrefractive index power, and increased the coefficient R values, which influences theterm for tilt. The effects of varying the two coefficient values on the WFE, requiredlateral shift, width / eyebox ratio, and the element thickness are shown in Table 1.Configuration C could not achieve sufficient eyebox size. As can be seen in Table 1, theDocket No. NVX21306CON1CIPPCTWFE is lowest for Configuration G, which has the lowest coefficient k value and thelargest width / eyebox ratio. This is not surprising, as introduced above, as minimizingthe rate of optical power generation as a function or shift is expected to reduceaberrations. The trends show the importance of using the tilt term to minimize Δn andto optimize performance.

[0112] Configurations E and F may be optimized with the same R and k coefficientsand may differ only in the lens element thickness. By increasing the thickness, a smallerlateral shift was required, but the WFE increased. This is also expected, as increasingthickness from the infinitely thin ideal increases aberrations.

[0113] Configurations H and I differ primarily on the required Δn value;Configuration H was implemented with Δn = 0.06 and Configuration I was implementedwith Δn = 0.12. As expected, a larger Δn value achieves a larger OPD, which allows forConfiguration I to achieve an optical power that is twice that of Configuration H, in aconfiguration 22% thinner and with a smaller δ.

[0114] Both Δn values are compatible with the print equipment. Nevertheless,Configuration H was chosen for implementation in the vision correction insert, as theavailable Δn = 0.06 ink set met the specification, and it allows for more control overdispersion.

[0115] U.S. Patent Number 9,855,752 B2 is incorporated by reference herein for allpurposes. As noted therein, to create the bitmaps that are communicated to eachprintheads during fabrication, the configurations are first reduced to refractive indexvalue volumes, n(x, y, z). Using the three-dimensional halftoning algorithms thataccommodated the ink-diffusion characteristics, the gradient index profiles may betranslated to bitmap-print patterns, which determined the spatial density patterns forthe droplets from optical primary ink. As a simple binary-ink pair was used to fabricatethe lens elements in this effort, two bitmaps may be required: one for the ‘high index’and one for the ‘low index’ optical ink. As for simplicity, the cubic polynomials were notvaried as a function of the axial coordinate, so that the same two bitmaps may be usedto fabricate each layer of the device.VII.Docket No. NVX21306CON1CIPPCT

[0116] GRIN lens configurations can be optimized for operation in the visible,infrared, terahertz, and RF portions of the electromagnetic spectrum. The interaction oflight with solids takes place through different mechanisms, depending on the type ofmaterial and the range of wavelength investigated. Insulators or dielectrics are typicallytransparent to visible light while most semiconductors are opaque to visible light yettransparent to infrared radiations; in contrast metallic solids appear shiny because theyreflect all wavelength up to the ultraviolet region. The optical properties of a soliddepends on its chemical composition and its structural properties and vary for everymaterial.

[0117] When electromagnetic radiation impinges upon a material it interacts bypolarizing the molecular units, producing oscillating dipole moments. This interactionresults in several observable optical phenomena such as reflection, transmission,absorption, or scattering. The classical model of light propagation assumes that theoscillating electric field can interact with several different types of dipole oscillatorswithin the material. Different dipoles are usually accessed by light wave from differentfrequency range depending on their mass.

[0118] The propagation of light through materials is described by a wave equationsimilar to the one that describes light travel in a vacuum (free space). The refractiveindex, n, describes how matter affects light speed: through the electric permittivity εand the magnetic permeability μ.

[0119] The refractive index (RI) is the measure of how light propagates through amaterial. The refractive index is a wavelength-dependent quantity and is a complexquantity. The complex RI is usually expressed aswhere η is the real portion of the RI also defined as the ratio of the wave velocity invacuum to the velocity in the medium, η =^c^v, and k is the extinction coefficient, whichis directly related to the absorption coefficient.; ε0 = permittivity of free space, ε =permittivity, εr = relative permittivity, µ0 = permeability of free space, μr = relativepermeability. Relative permittivity can be expressed as εr^=^ε^ / ^ε0.

[0120] The real part of the RI describes the change in velocity or wavelength of awave propagating from a vacuum into a medium, defining the extent the light rays canDocket No. NVX21306CON1CIPPCTbend, or refract, when passing from one medium to another, while the imaginary part isa measure of the dissipation rate of the wave in the medium. For the optical domain,n= (^ / ^0)1 / 2 = (^r)1 / 2. (20)

[0121] The relative permittivity is sometimes referred to as dielectric constant. Thedielectric constant is the characteristic of an insulating material or a dielectric whichrepresents its ability to store electrical energy in an electrical field. It shows how easilya material tends to be polarized when placed in an external electric field.

[0122] The dielectric constant k of a material is the ratio of its permittivity ε to thepermittivity of vacuum εo, so k = ε / εo. The dielectric constant is therefore also known asthe relative permittivity of the material. A low-k dielectric is a dielectric that has a lowpermittivity, or low ability to polarize and hold charge. Low-k dielectrics are very goodinsulators for isolating signal-carrying conductors from each other. A high-k dielectric,on the other hand, has a high permittivity. Because high-k dielectrics are good at holdingcharge, they are the preferred dielectric for capacitors.

[0123] The relationship between the complex relative permittivity, also known as thecomplex dielectric constant, and the complex refractive index is given by^r + i^^img = (n + i^k)2. (21)The relationships between the real and imaginary parts are^r = n2 – k2 (22)and^img = 2 n^k. (23)The dielectric constant of a material and its refractive index are closely linked by theequationκ=^ ^ / ^^0^= n2 , (24)which can be applied to the static dielectric constants of non-polar materials, or to thehigh-frequency dielectric constants of any dielectric. As the refractive index n iscomplex quantity, ^r must also be complex since n = (^r)0.5. The dielectric constant canvary significantly with frequency, n = n(ω) and k = k(ω), so that ^ and ^0 are frequencyDocket No. NVX21306CON1CIPPCTdependent, εr(ω) = relative permittivity or complex frequency dependent dielectricconstant.

[0124] By manipulating the permittivity and permeability of the nanocompositematerials, the refractive index of wavefronts in the radio-frequency (RF) and millimeter(MM) wave spectral ranges can be manipulated. The index of refraction is complex, andthe GRIN functions can include variation in either the real or imaginary part of the indexof refraction function. The imaginary part of the refractive index is the extinctioncoefficient in the material—a measure of how much light is being absorbed at a givenwavelength. In the optical domain, this governs how much the material absorbs light,and in the RF and MM wavelength domains, it determines the losses of the materials.VIII.

[0125] In one example, the ‘Alvarez Lohmann-like’ (‘AL-like’) tunable GRIN opticaldevices are generalized by deriving a GRIN profile, hence conjugating optical elementprofiles, from the phase of the wavefront the combined optical elements are intended toproduce as they are translated relative to one another laterally or rotationally.

[0126] The approach to combining the prescriptions of two or more GRIN opticalelements with high-order polynomial functions to create combined variable lower orderoptical functions can be extended to a wide range of optical elements that implement awide variety of optical functions. For example, conjugate fourth order optical elementscan be combined to create cubic functions, and pairs of second order optical elementscan be combined to create linear functions. It is also possible to conjugate multipleorder polynomial profiles optical that combined create a lower-order polynomialoptical functions. These devices can be used to provide variable optical power, to steerlaser beams in multiple directions, to implement phased array optical devices, or tootherwise implement variable optical functions on waveforms.

[0127] In general, functions with odd-order terms tend to be asymmetric and causebeam steering, while functions with even-order terms tend to be symmetric and cause achange of focus. However, this is not a hard and fast rule and there are exceptions.

[0128] Inkjet print additive manufacturing allows for the GRIN polynomial functionsimplemented on the optical elements to be optimized in three dimensions. Thisprovides more degrees of freedom than is provided by surface profiling only. ThisDocket No. NVX21306CON1CIPPCTallows, for example, for implementation of plano-plano optical elements that compriseof three-dimensional GRIN functions that reduce aberrations, align optical axis, improvefield of view, correct gaze angles, and eliminate distortions in ways that surface-figuredoptical devices cannot provide.

[0129] The OPD imparted by propagation of a wave through the local thickness of thehomogeneous index device with a variable surface profile is OPD = (n – 1) z(x, y), suchthat when applying a directional shift, δ, in the z direction, results in a wavefrontdeformationW(x, y, δ) = (n – 1)[z (x + δ, y) – z (x – δ, y)]. (25)

[0130] A comparable plano-plano GRIN device, composed of elements of equalthickness, t, can be designed the same optical path difference, OPD = (t)[n(x, y) – n0] =(t) nr(x, y), where the relative refractive index, nr = n – n0, and n0 is the base refractiveindex value at which point OPD = 0. When applying a directional shift, δ, in the zdirection, results in a wavefront deformation isW(x, y, δ) = t [nr (x + δ, y) – nr (x – δ, y)]. (26)Notice that for no shift, δ = 0, the contribution of the two elements cancel each other out,and no wavefront aberration is introduced.

[0131] In this way, for a fixed thickness device, the relative index, nr(x, y) may besubstituted for variable z(x, y), such that plano-plano variable GRIN patterned devicesmay be substituted to achieve the same spatially resolved OPD as is obtained with avariable surface figure thickness devices. However, it should be noted, that a plano-plano GRIN element can further ontrol OPD by changing the index profile in three-dimensions n(x, y, z) which has more degrees of freedom than a surface-figuredhomogeneous-index optical element, where in the OPD is varied with only twodimensions. Even more degrees of freeform can be realized when freeform surfaceprofiles are implemented on GRIN materials comprised of three-dimensional indexgradients.

[0132] For simplicity and without limitations, it is assumed here that the polynomialfunction is freeform in the x and y planes and is constant in the z direction. Generalizingthe Alvarez Lohmann type lens, it is possible for two elements with other like cubicfunctions to combine to create a variable power spheric lens. The approach herein canDocket No. NVX21306CON1CIPPCTbe extended to a broader range of conjugated GRIN optical functions, which that whenlinearly, angularly, or rotationally translated may be used to achieve variable opticalpower, beam steering, or other wavefront manipulations.

[0133] The change in optical phase of a plane wavefront passing through a surfacefigured homogenous index device is,Δς^(x, y, δ) = h^(x, y, δ) * (n0 – 1) 2π / λ,^ (27)where h is the total thickness of the device, which is assumed to be small to meet thin-lens approximations, and n0 is the base index.

[0134] The device acts as a neutral optical when the lateral shift between elements iszero—i.e., δ0 = 0, such that for a given wavelength the device will add a constant phaseto the incident wavefrontΔς (x, y, 0) = z2(x, y) – z1(x, y) = ς0. (28)The functional surface profiles, z1 and z2, satisfyh0 = h(x, y, 0) = z2(x, y) – z1(x, y) = λ / (2π(n0 – 1)) ς0. (29)The two halves of the device must therefore have the same profile shape, offset by theequivalent neutral optical path length OPL, which may be written asz(x, y) = z2(x, y) – z0 – h0 / 2 = z1(x, y) – z0 + h0 / 2, (30)in which h0 > max[z(x, y)] − min[z(x, y)].

[0135] For small shifts δ, the derivative of the profile x may be approximated asfollows:δz / δx = [[z(x^+ δ / 2, y) – z(x – δ / 2, y)] / δ ] = [λ / (2πδ(n0 – 1)] Δς(x, y, δ). (31)Equating the change in phase introduced by the device to the target phase, ϕ’, requiredto produce at a specified working shift, δ = δW, between the two halves of the device, onemay write ϕ’ (x, y) = Δϕ(x, y, δW). The function z(x, y), which defines the profile of ageneralized conjugating optical element for an arbitrary target phase ϕT at a workingshift may then be represented asz(x, y) = [λ / (2πδ(n0 – 1)] ʃ ϕT (x, y) dx, (32)Docket No. NVX21306CON1CIPPCTwhere ϕT(x, y) is the aberration function. Eq 32 applies to small shifts in the x direction,but a series expansion may be applied for large phase angles. A double integral wouldbe required for translation in both the x and y coordinates.

[0136] The equivalent of eq 32 for a plano-plano GRIN device isThe equation relates the refractive index profile of the optical element to its transverseaberration function, which describes the deviation of the wavefront from a perfectplanar wavefront. The integral over the transverse aberration function represents thetotal phase shift experienced by the light as it passes through the optical element. Byadjusting the transverse aberration function, the refractive index profile of the opticalelement can be tailored to achieve specific optical functions, such as focusing, imaging,and beam shaping.

[0137] The approach can be extended to other optical functions. The structure of ageneric conjugated optical device is shown by example in FIG. 10 (a), wherein the twoconjugate optical elements are defined by their surface profiles, z1 and z2, respectively,and are shifted against each other in the x direction along the internal surface z0.

[0138] As discussed above, for a homogeneous refractive device, the optical pathlengths across the device, OPL(x, y), are created by the summation of the thicknessprofiles of the two elements of the conjugate pair as they are translated relative to oneanother. It is understood that for any device with a homogeneous index n and a variablethickness z(x, y), a plano-plano GRIN device with an equivalent OPL(x, y) can be createdwith a device with a uniform thickness, t , implemented with a variable index profiledefined by n(x, y). FIG. 10 (b) shows two plano-plano GRIN elements that implement thesame OPL(x, y) pattern as the surface figured elements shown in FIG. 10 (a).

[0139] For a given output optical wavefront, a general two-part GRIN device can bemade such that at one value of the shift, δ = δ0, between the two GRIN parts, the deviceacts as a neutral optical element that does not change the wavefront of a propagatingplane wave; however, the primary optical property of the device changescommensurate with δ. The devices may be tuned by translation of the two parts (e.g.,halves) in the x direction as was discussed above. However, shifts in y direction, both xand y^directions,^or rotational angle, a are also envisaged.Docket No. NVX21306CON1CIPPCT

[0140] A simple device is a linear optical wedge. The wedge-shaped element can beused in various optical applications such as beam steering, wavefront correction, andoptical metrology. By controlling the thickness and apex angle of the wedge, therefractive index profile can be tuned to achieve specific optical properties, such asdeflection angle, wavefront curvature, and beam focusing.

[0141] A nonlimiting example of a linear wedge that can be introduced into a GRINdevice has a distributionn(x, y) =(n0+n1) / 2+(x∙(n1 – n0)) / (2∙ xmax), (34)where n1 and n0 represent the high and low refractive, index values respectively andxmax is the maximum value of x. An example of a GRIN lens implementing a linear wedgefunction is shown in FIG. 12 at (a). More generally, FIG. 12 shows aspects of aconjugated GRIN devices comprised of plano-plano GRIN optical elements, wherein therotational shift of the plano-plano GRIN optical elements cause the angle of deflection ofthe beam to change in azimuth or elevation as a function of the relative angle of rotationbetween the two. In the exemplary GRIN lens of FIG. 12 at (a), n0 = 1.4, n1 = 1.6, and xmax= 10 mm.

[0142] If a linear wedge function is rotated relative to its complement, they willcounteract each other and, ideally, result in a net zero optical power if they are perfectlyaligned and have identical properties. However, the wedge angle, in conjunction withthe prism's index of refraction, causes light entering the prism to be deviated by a smallangle. If the prism is rotated, the direction of deviation also rotates. As a pair ofcomplementary wedge prisms are rotated relative to each other, the deviationintroduced by the first wedge can be compensated by the second wedge to a varyingdegree depending on the relative angle between them. The aspect shown in FIG. 12 at(b) shows that at certain relative orientations, the two wedges can perfectly cancel eachother's deviation, resulting in a collimated beam with no net deviation. Here θ is therelative rotational translation between the two elements, where either or both mayrotate. The phase can introduce change in both the azimuth and the elevational beam-pointing angle. The change in beam steering is not necessarily linear in the azimuth orelevational angle; it may sweep out arcs, circles, etc. The aspect shown in FIG. 12 at (b)shows that at other orientations, the two wedges can add up their deviations, resultingin a beam that is steered to a different direction. The aspect shown in FIG. 12 at (c)Docket No. NVX21306CON1CIPPCTshows that it is possible to design the conjugate GRIN device such that as a function thedirection of the rotation, the phase change is linear with respect to the angle of rotation,as a function of rotational direction.

[0143] Odd functions are symmetric about the origin, meaning f (–x) = –f (x). Whenone has a pair of odd-order surfaces (like cubic, quintic, etc.) and they are translatedalong the axis relative to each other, their effects tend to cancel out. This cancellationresults in an optical system that behaves like a lower order system. Even functions aresymmetric about the y-axis, meaning f(–x) = f(x). When one has a pair of even-ordersurfaces (like quadratic, quartic, etc.) and they are translated along the axis relative toeach other, their effects do not cancel out. Instead, they can add up or produce adifferent functionality depending on the specifics of the translation. Periodic patterns ofwedges can be made into gratings, including diffraction gratings.

[0144] A cylindrical wedge introduces a phaseϕwedge(x, y) = α x (n0 – 1) * 2π^ / λ (35)for a wedge oriented in the x direction, where α is the angle of the wedge. Using a wedgefunction, gratings can be created that impart periodic tunable blaze angles. A cylindricalwedge, when linearly translated with its complement, will produce a net effect similarto a cylindrical lens. A cylindrical wedge refracts light in one direction only, while acylindrical lens refracts light in one direction while leaving light in the orthogonaldirection unchanged. When one has a pair of cylindrical wedges and slides them in thedirection orthogonal to the refraction, the net effect is a lens-like behavior, similar to acylindrical lens. The reason for this is that the refraction of light by a wedge is linearlydependent on the position across the wedge. By sliding two wedges against each other,one effectively creates a position-dependent phase delay that varies quadratically withposition – this is the same kind of phase delay induced by a lens.

[0145] The phase change of eq 33 may be induced by a device with a parabolic GRINprofile for each of the conjugate optical elements ofnr,wedge (x, y) = nlow + (nhi^^– nlow) / 2 + [(nhi – nlow) / (δmax] (α^x2), (36)where nlow is the base refractive index, nhi and nlow represent the high and lowrefractive, index values, δmax is the maximum value of the ranslation, α is a coefficientthat determines the strength of the wedge component. FIG. 13 shows aspects of a GRINDocket No. NVX21306CON1CIPPCTphase plate optical element with a refractive index distribution that implements aparabolic wedge index function across the device, such that when translated linearlyrelative to a complementary GRIN phase plate optical element it creates a variablecylindrical wedge function. More particularly, FIG. 13 shows an example of a GRIN lenswith a parabolic wedge function created using α = 1, nlow =1.41, (nhi – nlow) = 0.18, andδmax = 10 mm.

[0146] Whereas the cubic phase plates of an Alvarez lens, when combined in aconjugating GRIN device, creates a parabolic lens function, it is also possible to design aconjugate GRIN device that forms a spherical lens. The target phase of a sphericalsurface homogeneous index lens has a target wavefront ofΦsphere (x, y) = [ R2 – (x2 – y2)]1 / 2 + R ] * (n – 1) * 2^π / λ. (37)which represents the phase shift of light passing through a spherical surface with radiusR and refractive index n. The first term, [ R2 – (x2 – y2)]1 / 2 + R, represents the distancefrom the center of the spherical surface to the point (x, y) in the image plane taking intoaccount the curvature of the surface. The second term, (n – 1) * 2π / λ, represents thephase shift of the light passing through the surface due to the difference in refractiveindex between the surface and the surrounding medium.

[0147] To create the phase plate optical elements, the phase delay function is dividedbetween the two phase plates. Assuming that they are identical, each phase plate wouldneed to impart half the target phase delay, so the phase function for each would beΦ_sphere,^plate (x, y) = [ R2 – (x2 + y2)]1 / 2 + R ) * (n – 1) * π / λ, (38)where, n is the refractive index of the homogeneous index phase plate material, R is theradius of curvature of the spherical wavefront, and λ is the wavelength of the light.In order to find the index distribution n(x, y) that achieves this, the relationship betweenrefractive index and optical path length OPL (x, y) is considered. For a given ray, the OPLis the product of the refractive index and the physical path length.

[0148] A plano-plano GRIN lens doesn’t have a curved surface, so R doesn’tcorrespond to a physical feature of the lens. Instead, it’s a parameter in the refractiveindex distribution that creates an equivalent optical effect to the original lens. The phasedelay φ of a wavefront after passing through a homogeneous lens with index n andradius R is given by φ = 2^π / λ * (n – 1) * R. To create the same phase delay with a GRINDocket No. NVX21306CON1CIPPCTlens, one may consider a spheric index profile. A common form for a GRIN lens indexprofile is nGRIN(x, y) = n0 + α * (x2 + y2) where n0 is the index at the center of the lens, α isthe gradient constant, and x and y are the transverse coordinates. The sign of αdetermines if the lens converges light (is a positive lens) or diverges light (is a negativelens). The behavior is reversed for α < 0 compared to α > 0.

[0149] The refractive index distribution n(x, y, z) of the GRIN lens can be found bysolving the eikonal equation, which connects the refractive index distribution to thephase of a wavefront. In the paraxial approximation, the eikonal equation can be writtenas Φ_plate(x, y) = ∫ n(x, y, z) ds, where the integral is along the optical path. For a plano-plano GRIN lens with a thickness t, with no index variation along the z axis, n(x, y) onemay assume that the rays travel straight through the element, the differential lengthelement can be assumed to be dz. This equation can be rearranged to solve for therefractive index distribution: n(x, y) = Φ_plate(x, y) / t. This results in a conjugate opticalelement that may have a (x, y) coordinate distributionwhere no is the low index and n1 is the high index, and R is the radial term.

[0150] An example of a phase plate that can be translated rotationally is a spiralphase plate. A spiral phase plate (SPP) can be defined by a target wavefront ofΦSPP (x, y) = atan (y / x) * [[h*(n – 1)] / λ]. (40)The atan(y / x) term generates an azimuthal phase ramp that increases linearly with theangle from the x-axis, giving the optical field a helical or spiral wavefront.

[0151] Applying eq 32 leads to a surface profile for each of the homogeneous indexoptical elements ofzSPP (x, y) = [h / δW][x *atan(y / x) – y / 2 ln(x2 + y2)]α, (41)where h defines the phase per turn, and α is the angle. In this case, the term x * atan(y / x) – y / 2 * ln(x^2 + y^2) gives the phase of the optical vortex, which increases linearlywith the azimuthal angle and logarithmically with the radial distance from the origin.The parameter α allows for adjustment of the topological charge of the optical vortex,and h / δW controls the overall phase ramp.Docket No. NVX21306CON1CIPPCT

[0152] The GRIN equivalent is derived by eq 33 by creating the equivalent equationfor the optical path difference.nSPP(x, y) = ^^^ + ^ ^^^ െ ^^^^ * [h / δW] * [x * atan(y / x) – y / 2 * ln(x^2 + y^2)] * α / t.(42)A helical phase distribution is a spiral phase plate, which introduces a phase delay thatincreases azimuthally around the center of the plate. When a helical phase distributionis counter-rotated with its complement, the resulting phase delay across the beam caneither add up or cancel out, depending on the rotation angle and the specific values ofthe topological charges. If the two phase plates have equal and opposite topologicalcharges and are perfectly aligned, the phase delay introduced by one plate will beexactly canceled out by the other, resulting in a collimated beam. If they are slightlymisaligned, the combination can act like a lens, causing the beam to focus or defocusdepending on the direction and degree of misalignment. If the two phase plates havedifferent topological charges, or if they are rotated relative to each other by an angleother than 180 degrees, the phase delay will not cancel out perfectly. This results in anet phase gradient across the beam, which causes the beam to be steered or deflected ina particular direction. The direction and amount of deflection will depend on thespecific values of the topological charges and the rotation angle.

[0153] For a helical profiles to form a varifocal lenses, when rotated relative to itscomplement, in the thin-lens approximation, a rotation angle-dependent refractionpower D( ^^) can be formulated in dependence on the azimuthal change of curvature of afirst surface profile C1(α) and a second GRIN profile C2(α)D( ^^) = (nL – n0) / no [C1(α) – C2(α)], (43)where nL is the refraction index of the lens body, n0 is the based refraction index of theGRIN element, α is the azimuth, φ is the rotation angle, and C = 1 ∕ R, where R is theradius of the profile. Choosing a linear dependence of the surface curvature on theazimuth α, the curvature C1(α) of the first lens body can be described as a function of α,C1(α) = C10 + j α, where j is the linear factor of the curvature distribution, and C10 is thecurvature at α = 0.

[0154] For an equivalent GRIN expression, one may substitute for the first surface,D1( ^^) = no+ (n1– n0)* [C10(α) + j1α] / [C10(α) + j12 π]. (44)Docket No. NVX21306CON1CIPPCT

[0155] Rotating the second lens body, the respective opposing curvature alsodepends on the rotation angle φ. The equation of the curvature distribution C2(α),where C2 = 1 ∕ R2, can be described in dependence on the rotation angle φ, where C2(α) =C20 + j2(α – ^^) for φ ≤ α ≤ 2^π, and C2(α) = C20 + j2(α – ^^ + 2 π) for 0 ≤ α ≤ φ, and C20 is thecurvature at α = 0, where j2 is the linear factor of the curvature distribution. Substitutingfor the second surfaceD2( ^^) = no + (n1 – n0)* [C20(α) + j2(α – ^^)] / [C20(α) + j2(α – ^^)] for φ ≤ α ≤ 2^π (45)andD2( ^^) = no + (n1 – n0)Equating, for convenience, the linear factors j1 = j2 = j,D ( ^^) = no+ (n1– n0)( C10– C20+ j ^^) / ( C10– C20+ j 2^π) for φ ≤ α ≤ 2^π, (47) D( ^^) = no + (n1 – n0)( C10 – C20 + j ( ^^ – 2^π)) / ( C10 – C20+ j ^^) for 0 ≤ α ≤ ^^.(48)The above equation describes the angle-dependent refraction power of a bifocalrotation optic. For every rotation φ ≠ 0 there exist two lens sectors providing a certainrefractive power.

[0156] The curvature varies such that lens sectors of two opposing lens bodies willresult in the same refraction power over the whole azimuth range in the initial state. Arotation of one of the lens bodies by an angle φ around the optical axis will change thetwo opposing curvatures, resulting in a change of refraction power. Two sectors withdifferent tunable optical refraction powers are formed by mutual rotation, thusresulting in a tunable bifocal optics.

[0157] FIG. 14 shows aspects of an example of a GRIN phase plate with an indexdistribution that implements a helical phase function, wherein the index of refractiongradient changes radially as a function of the angle α. The index scale shows the rangeof refractive index values; showing a range from a low refractive index value of about1.4 to a high refractive index value of about 1.6. Proceeding counterclockwise, at 0o theradial gradient distribution changes from a convex index function to a concave indexfunction. Proceeding counterclockwise the second derivative of the radial indexDocket No. NVX21306CON1CIPPCTgradient becomes less positive, gradually flattens, and reaches a minimum at 180°. At180°, the radial gradient distribution switches from concave to convex, and proceedingcounterclockwise, the second derivative of the radial gradient distribution graduallyincreases until it reaches 0o where the second derivative is at its maximum negativevalue.

[0158] Toroidal optical elements may also be used in conjugated GRIN devices.Toroidal lenses are anamorphic optical elements, which primarily have a transmissionfunction corresponding to that of two crossed cylindrical lenses, which may havedifferent focal lengths. A toroidal lens may be thought of as a combination of twocylindrical lenses oriented at right angles to each other (crossed), and these cylindricallenses may have different focal lengths. Both, cylindrical lenses and saddle lenses are asub-group of toroidal lenses. A saddle lens has the transmission function of two crossedcylindrical lenses with opposite optical powers. The transmission function of acylindrical lens may be obtained from that of a saddle lens by combining it with anadjacent spherical lens. A general toroidal lens may be assumed to be composed of arotationally symmetric lens, and a saddle lens.

[0159] Assuming a thin lens approximation, the surface profile (height) a semi-planartoroidal lens is represented in cartesian (x, y) coordinates byz(x, y) = [Fxx2 + Fyy2] / [2(n^– 1)], (49)where n^is the refractive index of the lens material. Fx and Fy are the cylindrical opticalpowers of the lens in the xz- and yz- planes, respectively, and may be positive (convexlens), or negative (concave lens). In a thin lens approximation, the correspondingtransmission function for light with a wavelengthTt = exp[–i π / λ(Fxx2 + Fyy2)]. (50)This transmission function of a semi-planar toroidal lens may be factorizedTt = exp[–i^π^ / λ (Fxx + Fyy) / 2](x2 + y2) * exp[–i^π (Fxx – Fyy) / 2](x2 – y2) = T1 * T2.(51)

[0160] There, the first factor T1 corresponds to the transmission function of aparabolic (or a general spherical) lens with an optical power of (Fx + Fy) / 2 and thesecond factor T2 to that of a quadrupole saddle lens with a ‘quadrupole’ optical power of(Fx – Fy) / 2, whose transmission function corresponds to that of two crossed cylindricalDocket No. NVX21306CON1CIPPCTlenses with opposite optical powers of ±(Fx – Fy) / 2. Thus, the general toroidaltransmission functions include the cases of pure spherical (parabolic) lenses if Fx = Fy, ofpure saddle lenses if Fx = –Fy, and of cylindrical lenses if either Fx = 0 or Fy = 0.

[0161] Assuming a thin lens approximation, the GRIN profile of n(x, y) of a semi-planar toroidal GRIN lens is represented in cartesian (x, y) coordinates by:n(x, y) = n0 – π / λ t (Fxx + Fyy) / 2 * (x2 + y2) – π / λ^t (Fxx – Fyy) / 2 * (x2 – y2), (52)where the device thickness is given by t. As above, Fx and Fy are the cylindrical opticalpowers of the lens in the xz-plane and yz-plane, respectively, and may be positive(convex lens), or negative (concave lens). Only quadratic terms are considered here,although correction terms of different order may be present in practicalimplementations, and it is assumed that the optical axes of the lens are aligned parallelto the x-axis and y-axis.

[0162] A single toroidal lens (e.g., a saddle lens or a cylindrical lens) typically cannotbe used for imaging, since it affects the focal length in two orthogonal planes in adifferent way. However, this issue can be resolved by using two toroidal lenses (or twosets of combined, tunable toroidal lenses) placed at different positions within an opticalsystem.

[0163] FIG. 15 shows aspects of a GRIN phase plate optical element implementedwith a refractive index profile that implements a saddle function. A tunable saddle lenscan be constructed by different methods. One of them is to just stack two individualsaddle lenses with equal quadrupole optical powers into a single, combined conjugatingGRIN device, a ‘conjugated-saddle lens’. A second method to realize a tunable saddlelens is to just combine two cylindrical lenses with opposite optical powers in a scissorarrangement, i.e., with a variable angle between the cylinder axes. More generally, anycombination of two mutually rotatable cylindrical (or toroidal) lenses in combinationwith adequately chosen spherical lenses (which correct for additionally appearingspherical lens terms) can be used as a tunable combi-saddle lens.

[0164] Like an Alvarez lens, the insertion of two tandem-saddle lens lenses in anoptical setups allows one to construct a zoom system, which acts as an afocal telescope,whose angular magnification can be continuously tuned by a rotation of the individualsaddle lens elements around the optical axis. The working principle of a saddle lensDocket No. NVX21306CON1CIPPCTtelescope is based on the fact that a convolution of an input image with the transmissionfunction of a saddle lens yields a Fourier transform of the input image, which is scaledby an amount that depends on the adjusted quadrupole optical power of the saddle lens.

[0165] When two saddle lenses are stacked—i.e., one mounted directly behind theother, and (in a thin-lens approximation), the corresponding ‘conjugated-saddle’ lenshas a transmission function of a single saddle lens, but with a different quadrupoleoptical power that can be tuned by rotating one element with respect to the otheraround the optical axis.

[0166] Zooming can also be achieved by changing the mutual rotation angle of a set offour rotationally asymmetric lenses, namely of four cylindrical lenses, or of four saddlelenses. For example, the insertion of two conjugated-saddle devices in certain opticalsetups allows one to construct a zoom system, which may act as an afocal telescope,whose angular magnification can be continuously tuned by a rotation of the individualsaddle lens elements around the optical axis. FIG. 16 shows, at (a), aspects of anexample zoom lens comprising two conjugated GRIN devices that are positionedrelative to one another along the optical axis and are positioned relative to one anotheralong the optical axis, between surface figured homogeneous index lenses. Eachconjugated GRIN device is comprised of a pair of GRIN optical elements that arepositioned relative to one another along the optical axis, such that the optical Fouriertransform of the first tunable conjugated GRIN device is projected onto the plane of thesecond tunable conjugated GRIN device. In the aspect shown the second tunableconjugated GRIN device is positioned after the focal point of the first tunableconjugated GRIN device. When the GRIN optical elements are translated linearly orrotationally relative to one each other, they change the focal length of the associatedconjugated GRIN device, resulting in a change in magnification of the zoom lens; at (b)an aspect of an afocal arrangement of two conjugated GRIN devices, positioned betweenhomogeneous index lenses, wherein the second conjugated GRIN device is positioned atthe focal plane of the first conjugated GRIN device. In other words, panel (b) of FIG. 16shows an example of two conjugated-saddle devices, configured between twohomogeneous index lenses, to form a zoom telescope, which have stable image andobject planes as a function of magnification.Docket No. NVX21306CON1CIPPCT

[0167] These devices can be applied to any portion of the electromagnetic (EM)spectrum including the optical, radio-frequency (RF) or millimeter wavelength regions.Hereinafter, nanocomposites refer to materials which sized at < λ / 10.IX.

[0168] FIG. 17 shows aspects of an example optic 25 configured for variablewavefront shaping of electromagnetic (EM) radiation. The wavelength band of the EMradiation is not particularly limited in this disclosure; the EM radiation may comprisevisible, near-infrared, infrared, millimeter-wave, or radio-frequency radiation, forinstance. Optic 25 comprises a first optical element 27A including a solidifiedheterogeneous coalescence of nanocomposite material providing a first complexdielectric-function gradient. Optic 25 also comprises a second optical element 27Bincluding a solidified heterogeneous coalescence of nanocomposite material providing asecond complex-dielectric function gradient. Any, some, or all of the optical elements ofoptic 25 may comprise nanoparticles embedded in a cured polymer. Such nanoparticlesmay include oxide, semiconductor, fluoride, metal, hexaferrite, chalcogenide, ferrite,carbon, and / or hexaferrite, for example—e.g., materials with hollow cores or configuredin core-shell architectures. Any, some, or all of the optical elements may be fabricatedvia inkjet-print fabrication, though other modes of fabrication are also envisaged. Insome examples the nanocomposite materials are formulated for achromatic orapochromatic performance. In other examples, the wavelength-dispersive properties ofthe nanocomposite materials may impart a wavelength dependence to the variablewavefront shaping.

[0169] The first and / or second complex dielectric-function gradient may be afreeform gradient, a non-radially symmetric gradient, a non-axially symmetric gradient,or an anamorphic gradient, for instance. In some examples the first and / or secondcomplex dielectric-function gradient comprises a permittivity or permeability gradient.Generally speaking, the first or second complex dielectric-function may vary radially(perpendicular to optical axis A), and / or axially (in the z direction, along A). In someexamples the complex dielectric function characterizing the first and second opticalelements may vary in three dimensions. In some examples the first and / or secondcomplex dielectric-function gradient may be modified by laser radiation. The firstand / or second complex dielectric-function gradient may comprise a real part usable toDocket No. NVX21306CON1CIPPCTmanipulate the EM radiation received into the first optical element. In some examplesthe first and / or second complex dielectric-function gradient may also comprise animaginary part.

[0170] As shown in FIG. 17, first optical element 27A and second optical element 27Bare arranged in tandem along optical axis A; together they provide wavefront shapingthat varies according to the displacement of the first optical element relative to thesecond optical element. In some examples, displacement of first optical element 27Arelative to second optical element 27B changes the focal length of optic 25. In someexamples the displacement changes the direction of the beam exiting the second opticalelement relative to the direction of the beam entering the first optical element. In someexamples the displacement imparts the effect of a variable wedge function on theincident EM radiation. In some examples the displacement reproduces the effect of avariable phase plate on the EM radiation. In some examples the displacementreproduces the effect of a variable blazed grating on the EM radiation.

[0171] In order to effect the displacement of first optical element 27A relative tosecond optical element 27B, optic 25 includes an actuator 29. In some examples theactuator includes a piezoelectric motor configured to translate the first or secondoptical element. In some examples the actuator includes an integrated micro-mechanical actuator configured to translate or rotate the first or second optical element.In some examples the actuator is configured to rotate the first or second optical elementabout optical axis A.

[0172] Optic 25 of FIG. 17 includes an anti-reflective coating 31 arranged on the firstand / or second optical element. Optic 25 may be configured to transmit EM radiationonly through an area of overlap between the first and second optical elements. In theexample shown in FIG. 17, optic 25 includes at least one opaque baffle 33 arrangedbetween first optical element 27A and second optical element 27B.

[0173] The range of applications of optic 25 is not particularly limited. The optic maybe arranged in a vision-correcting device, an optical scanner, a variable-magnificationtelescope, a variable-magnification microscope, for instance, or in a head-up displayconfigured for virtual- or augmented-reality applications. In some examples, the optic isconfigured to emit a light field or hologram—e.g., the optic may be a light-field orcomputational-imaging optic. In some examples optic 25 may be arranged in anDocket No. NVX21306CON1CIPPCTantenna. To support these applications, among others, first optical element 27A and / orsecond optical element 27B may be integrated with one or more structural elements tofacilitate mounting. In some examples the first and second optical elements may beconfigured for dynamic illumination.

[0174] In some examples, each optical element in optic 25 may be configured tomodel a conventional spherical lens. In other examples, the first and / or second complexdielectric-function gradient may Fresnel implementations of the desired complex orfreeform dielectric-function gradient. In some examples the first and / or secondcomplex dielectric-function gradient may comprise a segmented implementation of adesired complex or freeform dielectric-function gradient. In some examples the firstand / or second complex dielectric-function gradient may be a gradient of a function ofpolynomial terms higher than third-order, to reduce aberrations or otherwise improveoptical performance quality.

[0175] The number of optical elements in optic 25 is not particularly limited. In theexample illustrated in FIG. 17, optic 25 includes a third optical element 27C including asolidified heterogeneous coalescence of nanocomposite material providing a thirdcomplex dielectric-function gradient arranged in the optical path of the first and secondoptical elements, and configured to correct for aberrations. More generally, the first andsecond optical elements, etc., may be arranged in an array of analogously configuredoptical elements. In such an array the complex dielectric-function gradient may vary independence on the position of each optical element in the array. Alternatively, or inaddition, the size of each optical element may vary in dependence on the position of thatoptical element in the array. Alternatively, or in addition, the orientation of each opticalelement may vary in dependence on the position of that optical element in the array. Insome examples the first and second optical elements may be tiled in a square,hexagonal, triangular, circumscribed circular, or chirped lattice configuration. In someexamples the first and second optical elements may be tiled in a cubic, square,hexagonal, triangular, circular, or chirped packing.

[0176] FIG. 16 shows at (a), aspects of an example optical system 39 configured forvariable focus. The system comprises a first optic 25A including first and secondgradient complex dielectric-function optical elements 27F and 27G, arranged in tandemalong an optical axis, which together provide an optical power that varies according to aDocket No. NVX21306CON1CIPPCTdisplacement of the first optical element relative to the second optical element. Thesystem further comprises a second optic 25B including third and fourth gradientcomplex dielectric-function optical elements 27H and 27I, arranged in tandem along anoptical axis, which together provide an optical power that varies according to adisplacement of the third optical element relative to the fourth optical element. In thisexample the focal lengths of the first and second optics are adjustable relative to eachother.

[0177] In some examples, system 39 may include at least one additional lens element(not shown in FIG. 16 (a)) arranged between the first optic 25A and second optic 25B. Insome examples the dispersive properties of the first and second optics are matched toachieve achromatic performance. In system 37 a collimator lens system 41 is arrangedbetween the source of the EM radiation and first optic 25A.

[0178] FIG. 16 (b) shows aspects of two example GRIN optics, which have an angularmagnification that changes with rotation of each element relative to its conjugate opticalelement. When configured in a pair and configured with an additional optic, a zoom lensis formed. Setups allows one to construct a zoom system, which acts as an afocaltelescope, whose angular magnification can be continuously tuned by a rotation of theindividual saddle lens elements around the optical axis.

[0179] Additional support for the examples above is provided in FIGS. 17 through 20.FIG. 18 shows aspects of an example of a GRIN phase plate in which the complexdielectric-function gradients change in three coordinate dimensions. FIG. 12 (a) showsan example of a beam-steering conjugated GRIN device effected by a pair of GRIN phaseplate optical elements, wherein the rotational displacement of the GRIN phase-platedelements relative to its complement causes beam steering. FIG. 12 (b) shows aspects ofa rotational shift of an example pair of plano-plano GRIN phase plate optical elements—viz., the accumulated phase achieved by the combined index of refraction profiles of thepair as they are rotated relative to one another, cause the angle of deflection to changeas a function of the angle of rotation. In the aspect shown, at one angle the accumulatedphase is uniform across the conjugated GRIN device is uniform such that the beam is notdeflected. The illustration shows rotational phase functions, which, as they shift relativeto one another, cause the beam to deflect. When there is no shift of the elements relativeto each otherthe beam is undeflected from the incident orientation. TheDocket No. NVX21306CON1CIPPCTazimuthal angle φ′ of the beam may be swept (dashed line) by co-rotation of the prismpairs. In the aspect shown in FIG. 12 (c) the angle of deviation is linear with respect tothe angular displacement between the complementary pair and the direction of theangle of the deflection is dependent on the direction of the relative rotation of the pair.X.

[0180] This section provides additional development of GRIN formulas and detailslinear-translation and rotational-translation GRIN distributions that can be used,including quadrupole, torroids, etc., for both beam steering and varifocus applications.

[0181] The Alvarez lens is typically configured with a cubic function given byn(x,^y)^=^n0^+^a1^x^+^a2^y^+^a3(x3^–^3^x^y2), (53)where n(x,^y) is the refractive index of the lens at a given point (x,^y), n0 is the averagerefractive index, and a1,^a2,^and a3 are coefficients that determine the shape of the lens.The cubic function has a saddle shape, which allows for the lens to perform bothpositive and negative refraction.

[0182] When a GRIN Alvarez lens with the function n(x,^y)^=^n0^+^a1^x^+^a2^y^+^a3(x3^–^3^x^ y2) is translated linearly with its opposite, the combined optical function can bedescribed by the equationn_comb(x,^y)^=^n0^+^a1^x^+^a2^y^+^a3(x3^–^3^x^y2)+(d2 / ^2)(∂2n / ∂x2^+^∂2n / ∂y2),^ (54)where ∂2n / ∂x2 and ∂2n / ∂y2 are the second partial derivatives of n(x,^y) with respect to xand y, respectively.

[0183] The optical power of the combined lens can be calculated by taking thegradient of the refractive index n_comb(x, y),P(x,^y)^=^–(1 / n) ∇n_comb(x,^y), (55)where n is the refractive index of the surrounding medium and the gradient operator ∇is given by ∇ = (∂ / ∂x) i + (∂ / ∂y) j. The optical power describes the direction andmagnitude of the light's path as it passes through the lens and can be used to determinethe focal length and other optical properties of the lens.Docket No. NVX21306CON1CIPPCT

[0184] FIG. 19 shows aspects of a GRIN phase plate optical element implementedwith a refractive index profile that implements a cubic index function, where the GRINphase plate optical element is designed to change the focus of a beam when translatedrelative to a complementary GRIN phase plate optical element.. Here the Alvarez lensequation is expressed as a fourth-order polynomial equation that describes thevariation of the refractive index along the x and y axes of a lens. The general form of theequation isn(x,^y)^=^n0^+^a1x^+^a2y^+^a3(x2^+^y2)^+^a4(x2–^y2)^+^a5(2xy)^+^a6x(x2–^3y2)^+^a7y(3x2–^y2)^+^ a8x2y^+^a9xy2.^(56)

[0185] This equation has nine coefficients (a1 to a9) and includes terms up to fourthorder. The x2 – y2 term gives it the shape characteristic of Alvarez lenses, and theadditional terms provide further control over the lens's refractive properties. Thehigher-order terms, such as x3,^y3,^x2y, and x^y2, can introduce additional complexity andcurvature to the GRIN profile, allowing for greater control over the lens's opticalproperties, which can be used to remove aberrations. Although not shown, the use ofthree-dimensional index variation, n(x, y,z) can be used to further reduce aberrationsand better accommodate designs which deviate from the paraxial approximation.

[0186] Returning briefly to FIG. 15, this drawing shows aspects of a general saddlelens. An example of a saddle function is f(x,^y)^=^x2^–^y2. A plan-plano GRIN lens with arefractive index distribution shaped like a saddle function is a type of lens where theindex gradient curvature is positive in one direction and negative in the orthogonaldirection. This is also called a hyperbolic paraboloid shape. A GRIN saddle lens is givenby an index distribution n(x, y) = n0 + a * (x2 – y2), where n0 is the base refractive index, ais a constant that determines the strength of the saddle lens effect, and x and y are thecoordinates in the plane of the lens.

[0187] If two lenses with opposite refractive index distributions shaped like a saddlefunction are rotated relative to each other, the resulting lens can be modeledmathematically by multiplying the transmission functions of the two lenses in thefrequency domain.

[0188] The transmission function of a GRIN lens is given by the following equation,T(x, y) = exp[i^ϕ(x, y)] = exp[i * 2^π * a * (x2– y2) * t / λ], (57)Docket No. NVX21306CON1CIPPCTwhere i is the imaginary unit, t is the thickness of the device, and a is a constant thatdetermines the strength of the lens. The coefficient a is the same for both the x and ydirections in the quadratic term, resulting in a saddle shape where the curvature in the xand y directions is equal and opposite. If the coefficients differ, an elliptical rather than asaddle shape is obtained.

[0189] Assuming that the refractive index distribution of the first lens is given by f(x,^y) and that of the second lens is given by –f(x,^y), the combined transmission functioncan be written asT^(x,^y,^θ)^=^exp[i^*^2π^*^((ax^*^(x^cosθ^+^y^sinθ)2 –^ay^*^(y^cosθ –^x^sinθ)2)^*^t)^ / ^λ]. (58)The actual shape and properties of the combined lens will depend on the specificparameters of the two lenses used and the angle at which they are rotated relative toeach other.

[0190] Rotational translation will now be described. The mathematical description ofa change in optical power with rotation as a change in focal length with rotation withrespect to the inverse of the lens function can be expressed as:f(θ)^=^1^ / ^D(θ), (59)where D(θ) is the distribution function of the GRIN lens and f(θ) is the focal length atangle θ. If the optical power changes linearly with respect to rotation, then the focallength will also change linearly with respect to rotation, as expressed by:f(θ)^=^aθ^+^b,^ (60)where a and b are constants that depend on the specific GRIN lens design. The inversefunction of this equation can be used to determine the distribution function D(θ) thatwould produce a linear change in focal length with respect to rotation.

[0191] To make a cubic Alvarez-like lens varifocal with rotation, one may add higher-order terms to the refractive index distribution equation that depend on the rotationangle. One way to do this is to introduce a parameter that controls the variation of therefractive index distribution with the rotation angle. This can be done by adding a termproportional to the sine of the rotation angle to the refractive index distributionequation.

[0192] A possible equation for a varifocal cubic Alvarez lens with rotation isDocket No. NVX21306CON1CIPPCT n(x,^y,^θ)^=^n0^+^a1^x^+^a2^y^+^a3(x3^–^3^x^y2)^+^a4^sin(θ)(x3^–^3^x^y2), (61)where θ is the rotation angle, and a4^ is a parameter that controls the variation of therefractive index distribution with the rotation angle. The coefficients a3 and a4 introduceangular dependence in the refractive index distribution. To achieve a change of focuswith rotation, one may set non-zero values for a3 and / or a4. These coefficientsdetermine the strength and nature of the angular variation. The term a3(x3 – 3 x y2)represents the cubic component of the refractive index distribution, which gives thelens an Alvarez-like shape. The sin(θ) term associated with a4 allows for an angularmodulation of the refractive index. The term a4 sin(θ)(x3 – 3 x y2) introduces asinusoidal variation in the refractive index distribution as a function of the rotationangle θ, further enhancing the varifocal properties of the lens. This equation has thesame cubic dependence on x and y as the original cubic Alvarez equation, but it alsoincludes a sinusoidal dependence on the rotation angle θ.

[0193] When the lens is rotated relative to its complement, the sinusoidal term in therefractive index distribution equation produces a shift in the focal length of the lens thatdepends on the rotation angle. As the lens rotates, the focal length changes in a varifocalmanner, allowing the lens to focus at a different distances depending on the rotationangle.Δf(θ)^=^–(1 / 2)^k^(Δn)a4^sin(2θ), (62)where Δf(θ) is the change in focal length as a function of rotation angle θ, Δn is thedifference in refractive index between the GRIN lens, a4 is the coefficient of the sin(θ)term in the GRIN equation, and k is a constant that depends on the specific GRINdistribution. The change in focal length is determined by several factors, including thecoefficient a4, which controls the extent of the varifocal effect, the change in refractiveindex denoted by Δn, and the rotation angle θ. The term (1 / 2)k is a constant thatdepends on the specific design of the lens. The sin(2 θ) term indicates that the change infocal length is sinusoidal with respect to the rotation angle, with a maximum changeoccurring at θ = π / 4 and 3π / 4, and a minimum change at θ = 0, π / 2, and π.

[0194] When rotated relative to a negative version of itself, the following phaseequation may be used:f(θ)^=^–λ / 2 π^n0(∆n / λ)^R(θ),^ (63)Docket No. NVX21306CON1CIPPCTwhere λ is the wavelength of light, ∆n is the difference between the maximum andminimum refractive index. The term –λ / 2π n0(Δn^ / λ) is a constant that relates to therefractive index difference Δn and the wavelength λ. The function R(θ) represents therotation function or the angular dependence of the phase delay. The specific form of therotation function will depend on the optical system and the desired properties. It can bea simple trigonometric function or a more complex function depending on the specificsituation. The term rotation-dependent term R(θ) may be given by:R(θ)^=^[a12^+^a22^+^(a3^+^2a4)2]^sin2(θ / 2)^+^4a^42^cos4(θ / 2)^+^4a3^a4^sin2(θ / 2)^cos2(θ / 2).^(64)

[0195] There are various n(x, y) distributions that can allow for a linear change inoptical power as a function of rotation, some of which are:n(x,^y)^=^n0^+^a1^x^+^a2^y^+^a3^cos(2θ), (65)where the coefficient a3 cos(2θ) is responsible for introducing angular-dependentvariations in the refractive index. The value of a3 determines the amplitude of therefractive index modulation caused by the cos(2θ) term. By adjusting a3, one maycontrol the extent of the focal length variation; orn(r,^θ)^=^n0^+^r(a1^cosθ^+^a2^sinθ)^+^a3^sin(2θ)),^ (66)where the coefficient a3 introduces angular-dependent variations in the refractiveindex, specifically a sinusoidal modulation with a period of 2θ; orn(x,^y)^=^n0^+^a1x^+^a2y^+^a3^sin(θ)^+^a4^cos(θ), (67)where the coefficients a3 and a4 contribute to the angular-dependent variations in therefractive index. The a3 sin(θ) term and the a4 cos(θ) term introduce sinusoidalmodulations in the refractive index as a function of the rotation angle θ. To achievevarifocality, non-zero values for both a3 and a4 are typically used. The specific valuesand the ratio between these coefficients would determine the magnitude and nature ofthe focal length variations.

[0196] Adjusting these coefficients would allow control over the extent of the focallength change as the lens is rotated; or,^Docket No. NVX21306CON1CIPPCTwhere An, Bn(r), and Cn(r) are coefficients that depend on the particular GRIN function;orn(r,^θ)^=^n0^+^a1^r^cos(θ)^+^a2^r^sin(θ)^+^a3^sin(θ)^+^a4^cos(θ). (69)

[0197] Quadrupole optical power implies that the optical power of the lens changesquadratically with respect to the transverse position of the beam. However, linearchange in focus with respect to rotation can be achieved using various higher orderGRIN distributions that may not exhibit quadrupole optical power.

[0198] A quadrupole optical power GRIN lens has an asymmetric structure andinduces a linear change in focus as it is rotated relative to its inverse. To achieve aQuadrupole GRIN lens with a linear change in optical power when rotated, a suitablevalue of k can be chosen, and the power-law exponent p can be adjusted accordingly toachieve the desired index profile. The rotational symmetry of the lens can then beexploited to achieve the desired linear change in optical power.

[0199] One may calculate the optical power of the lens at a given angle θ as the sum ofthe paraxial optical power and the non-paraxial contributions, which can be expressedas:P(θ)^=^P0^+^P2(θ),^ (70)where P0 is the paraxial optical power, given by:P0^=^(n0^–^1)^ / ^f0, (71)and f0 is the focal length of the lens in the absence of any rotation. The non-paraxialcontributions to the optical power can be expressed asP2(θ)^=^^dxdy, (72)where the integral is taken over the entire lens aperture.

[0200] Substituting the expression for the refractive index distribution and simplifythe expression by transforming to polar coordinates, the total optical power of the lensas a function of angle θ can be expressed as:P(θ)^=^ (n0^–^1)^ / ^ f0 +^πR5(n0^–^1)^ [a1^cos(θ)^+^a2^sin(θ)^+^a3^cos(2θ)^+^a4^sin(2θ)]^ / ^5.^(73)Docket No. NVX21306CON1CIPPCTThis expression shows that the optical power of the lens varies linearly with angle, witha slope determined by the quadrupole refractive index gradient coefficients a3^and^a4.

[0201] The optical power of the lens is given by the second derivative of therefractive index with respect to the radial distance r from the optical axis, i.e.,P(r)^=^–(1 / n)(d2n / dr2),^ (74)where n is the refractive index.Δf(θ)^=^–(1 / n0)d2n / dr2=^–(1 / n0)[(6a3x^–^6a3y2^+^2a4)^cos2θ^–^12a3^x^y^sinθcosθ^+^(–6a3^ x2^+^6a3^y^+^2a4)^sin2θ], (75)where x and y are the radial distances from the optical axis.

[0202] In polar coordinates:n(r,^θ)^=^n0^+^a1r^cos(θ)^+^a2r^sin(θ) +^a3r3(^cos(θ)3^–^3^cos(θ)^sin(θ)2)^+^a4r2(^sin(θ)2^–^cos(θ)2). (76)And the change in focus as a function of rotation θ is given by:Δf^=^–λ / (2π)^*^(nL^–^n0)^*^D(θ),^ (77)where: D(θ) = [n0C1(a) – nL(C1(a) cos(θ) + C2(a) sin(θ))] / [nL – n0], and^C1(a)^=^1^+^a1^+^3a3^cos(θ)2^+^2a4^sin(θ)^cos(θ)^C2(a) =^a2^+^3a3^sin(θ)^cos(θ)^–^2a4^cos(θ)2 . ^

[0203] The equation for a Quadrupole GRIN lens that has a linear change in opticalpower when rotated relative to itself can be expressed asP(z)^=^P0^+^k^z, (78)where P(z) is the optical power at a distance z along the optical axis, P0 is the initialoptical power at z = 0, k is the rate of change of optical power per unit distance, and z isthe distance along the optical axis.

[0204] One example of a GRIN distribution that can be used to achieve a QuadrupoleGRIN lens with a linear change in optical power when rotated is a power-lawdistribution. In this case, the index profile is given byn(r)^=^n0^[1^+^(k / d)^(r / d)p], (79)where n(r) is the refractive index at a distance r from the center of the lens, n0 is therefractive index at the center of the lens, d is the diameter of the lens, k is the rate ofDocket No. NVX21306CON1CIPPCTchange of optical power per unit distance. P(z) represents the optical power of the lensas a function of the distance along the optical axis, as given in the equation P(z) = P0 +kz), where P0 is the optical power at the center of the lens and k is the rate of change ofoptical power per unit distance, and p is a power-law exponent that controls the shapeof the index profile.

[0205] Quadrupole optical power specifically describes the variation in opticalpower that arises due to changes in the second-order terms of the refractive indexdistribution. An example of a GRIN saddle function with a quadrupole optical powerthat can function as a toroidal lens isn(x,^y)^=^n0^+^a1x^+^a2y^+^a3(x3^–^3xy2)^+^a4(x2^–^y2),^ (80)where n0 is the refractive index at the center, a1 and a2 are the linear refractive indexcoefficients, a3 and^a4 are the higher-order refractive index coefficients, and x and y arethe coordinates in the transverse plane. When this lens is rotated relative to itscomplement, it causes a change in focus that varies linearly with the rotation angle θ.

[0206] By choosing the coefficients a1 and a2 to be equal in magnitude and opposite insign, so that the linear variation of the index along the x direction cancels out the linearvariation along the y direction, results in a purely quadratic variation along the diagonaldirection.Δf(θ)^=^2πkL2^[n02^+^(2 / 5)a12^+^(2 / 5)a22^+^(8 / 35)a32^+^(4 / 35)(a1a2^+^a2a1)^cos(2θ)^+^(2 / 35)(a1a3^–^a3a1)^cos(3θ)^+^(2 / 35)(a2a3^–^a3a2)^sin(3θ)],^ (81)where θ is the angle of rotation of the lens, L is the length of the lens, k is a constantrelated to the material properties of the lens, and the coefficients a1,^a2, and a3 describethe variation of the refractive index along the x, y, and z axes of the lens, respectively.The coefficient a4, which describes the variation of the refractive index along thediagonal of the lens, is not included in this equation because it does not affect thechange in focal length due to rotation.

[0207] A cylindrical Quadrupole GRIN lens that has a linear change in optical poweralong the optical axis when rotated relative to itself can be designed using a specificrefractive index distribution. The refractive index distribution that achieves thisbehavior is called a parabolic refractive index distribution. The parabolic refractiveindex distribution is given byDocket No. NVX21306CON1CIPPCT n(r)^=^n0^+^Ar2, (82)where n(r) is the refractive index at a radial distance r from the center of the lens, n0 isthe refractive index at the center of the lens, A is a constant that determines the strengthof the refractive index gradient, and r is the radial distance from the center of the lens.

[0208] For a cylindrical lens, the optical axis is the axis perpendicular to thecylindrical surface, so the linear change in optical power must be along this axis.Therefore, the parabolic refractive index distribution must be along the axial direction.This can be achieved by varying the refractive index as a function of the distance alongthe axial direction.

[0209] The refractive index distribution for a cylindrical Quadrupole GRIN lens with aparabolic refractive index distribution along the axial direction can be expressed as:n(x,^y)^=^n0^+^A(x2^+^y2),^ (83)where n(x, y) is the refractive index at a point (x, y) in the lens, n0 is the refractive indexat the center of the lens, A is a constant that determines the strength of the refractiveindex gradient, x and y are the coordinates of the point in the plane perpendicular to theoptical axis. This GRIN distribution has a linear change in optical power along theoptical axis when rotated relative to itself. Therefore, it should result in a linear changein focal length with rotation. The change in focal length Δf with rotation angle θ can becalculated using the following equation:Δf^=^(n0 / 2A)^sin(2θ), (84)where n0^ is the refractive index at the lens center and A is a constant related to thestrength of the Quadrupole term,

[0210] A GRIN toroidal lens is a type of GRIN lens that has a refractive index thatradially from the center of the toroid to its outer edge, which allows the lens to focuslight in a unique way. The mathematical function that describes the refractive index of aGRIN toroidal lens is typically a variation of the power-law GRIN equation, such aswhere n(r) is the refractive index at a radial distance r from the center of the toroid, n0is the refractive index at the center of the toroid (i.e., the hole), a, b, and c are constantsthat determine the refractive index profile, and R is the radius of the toroid.Docket No. NVX21306CON1CIPPCT

[0211] When two GRIN lenses are rotated relative to each other, they can function astoroidal lenses and cause a change in optical power. The specific change in opticalpower will depend on the exact parameters of the GRIN function and the angle ofrotation. When these lenses are placed in close proximity, the refractive indexdistribution of one lens is the mirror image of the other lens. This creates an indexprofile that varies linearly with distance from the center of the lens in both the x-y andy-z planes. This linear variation in refractive index is similar to the surface shape of atoroidal lens and can therefore produce similar optical effects. One example of an indexdistribution that could be used to achieve this isn(r,^z)^=^n0^+^(n2^–^n0)^*^(r / R)2^*^(1^+^a1^*^cos(2θ)^+^a2^*^cos(4θ))^*^exp(–z2^ / ^(2^*^L2)).^ (86)

[0212] Here, n0 is the refractive index at the center of the lens, n2 is the refractiveindex at the outer edge of the lens (at a radius R), r is the radial distance from the centerof the lens, z is the axial distance along the optical axis, θ is the azimuthal angle, a1 anda2 are coefficients that control the degree of toroidicity in the lens, and L is acharacteristic length scale that controls the axial variation of the index distribution. Byrotating this lens relative to its inverse, the toroidal shape of the lens will cause a linearchange in optical power along the optical axis.

[0213] The primary difference between a quadrupole lens and a toroidal lens whenthey are rotated relative to their inverse is in the direction of their power axis. In aquadrupole lens, the power axis is perpendicular to the optical axis, while in a toroidallens, the power axis is parallel to the optical axis. As a result, when rotated relative totheir complement, a quadrupole lens has a linear change in optical power, while atoroidal lens has a sinusoidal change in optical power. Additionally, the shape of the lenssurface in a toroidal lens is different from that in a quadrupole lens, which affects otherproperties of the lens such as aberrations.

[0214] In general, the idea is to have two GRIN lenses with different saddle functionssuch that when they are rotated relative to each other, the resulting optical powerdistribution approximates that of a toroid. One possible mathematical approach is toconsider the two saddle functions as perturbations of a toroidal function. The toroidalfunction can be expressed asDocket No. NVX21306CON1CIPPCT n(φ)^=^n0^+^Δn^cos(φ),^ (87)where φ is the azimuthal angle, n0 is the refractive index in the center of the toroid, andΔn is the index change due to the toroidal shape. The saddle functions can then beexpressed as perturbations of this function, for example,n1(φ)^=^n0^+^Δn^cos(φ)^+^a^cos(2φ)^n2(φ)^=^n0^+^Δn^cos(φ)^+^b^sin(2φ), (88)where a and b are coefficients that determine the strength and orientation of the saddlefunctions.

[0215] When these two saddle functions are combined, the resulting refractive indexdistribution is:n(φ)^=^n0^+^Δn^cos(φ)^+^a^cos(2φ)^+^b^sin(2(φ+θ)),^ (89)where θ is the relative rotation angle between the two lenses. This distribution can beapproximated as a toroid when a and b are chosen appropriately and θ is small enough.

[0216] One example of a GRIN expression for a toroidal lens that can change theoptical power when rotated relative to its opposite isn(r,^θ)^=^n0^+^a1r^cos(θ)^+^a2r^sin(θ)^+^a3r2^cos(2θ)^+^a4r2^sin(2θ), (90)where n0 is the refractive index at the center of the toroid, r is the radial distance fromthe center of the toroid to a point in the lens, θ is the azimuthal angle measured from areference direction, a1 and a2 are coefficients that control the linear variation inrefractive index in the x and y directions, respectively, and a3 and a4 are coefficients thatcontrol the quadrupole variation in refractive index in the x and y directions,respectively.

[0217] The linear variation in refractive index in the x and y directions (a1 and a2terms) causes a linear change in the focal length of the lens as it is rotated relative to itsinverse. The quadrupole variation in refractive index in the x and y directions (a3 and a4terms) can help to reduce aberrations and improve the quality of the beam steering.

[0218] When a cylindrical Quadrupole GRIN lensn(x,^y)^=^n0^+^A(x2^+^y2)^ (91)is rotated relative to its inverse, it creates a varifocal lens where the focal lengthchanges parabolically with rotation.Docket No. NVX21306CON1CIPPCT

[0219] To make a zoom lens using rotationally variant GRIN lenses, one could use acombination of toroidal GRIN lenses and quadrupole GRIN lenses. The toroidal GRINlenses could be used to adjust the focal length of the system, while the quadrupole GRINlenses could be used to adjust the optical power and correct for aberrations. To achievezooming, the relative orientations of the lenses could be adjusted, either mechanicallyor electronically, to change the effective focal length of the system. By adjusting therelative orientations of the toroidal and quadrupole GRIN lenses, the user could controlboth the focal length and optical power of the lens system, allowing for a zooming effect.One approach is to use a series of GRIN lenses with different optical powers that can beselectively combined to provide the desired zoom range. For example, one may use atoroidal GRIN lens with a linear change in focus as a function of rotation to provide asmall amount of zoom (e.g.,^1.2x to 1.5x), and then use a series of quadrupole GRINlenses with progressively larger optical powers to provide additional zoom (e.g., 2x to5x).

[0220] Beam-steering will now be described. A beam-steering lens uses a GRINdistribution that causes a linear change in optical power with rotation. One suchdistribution isn(x,^y)^=^n0^+^k(x2^–^y2), where k is a constant. (92)

[0221] The main difference between the two GRIN distributions is that theQuadrupole GRIN distributionn(x,^y)^=^n0^+^A(x2^+^y2)^ (93)creates a varifocal lens with a parabolic change in focal length, while the GRINdistributionn(x,^y)^=^n0^+^k(x2^–^y2)^ (94)creates a beam steering lens with a linear change in optical power. The change in focallength Δf as a function of the rotation angle θ can be expressed as:Δf(θ)^=^2n^f(0) sin2(θ / 2) for the varifocal lens, (95)where n is the refractive index at the center of the lens, f(0) is the focal length at θ = 0,and θ is the rotation angle, andΔf(θ)^=^n^f(0)^sin(2θ) for the beam steering lens, (96)Docket No. NVX21306CON1CIPPCTwhere n is the refractive index of the lens and f(0) is the focal length when it is notrotated, i.e., when θ = 0. It represents the focal length of the lens when it is not rotatedand serves as a reference point for calculating the change in focal length with rotationangle.

[0222] The difference between the two GRIN distributions is in the power of the sinefunction. The varifocal lens has a sine squared function which results in a parabolicchange in focal length, while the beam steering lens has a sine function which results ina linear change in focal length.

[0223] For a varifocal lens with a GRIN distribution that varies linearly with radius,the phase change is proportional to the angle of rotation. For a beam steering lens witha GRIN distribution that varies sinusoidally with radius, the phase change isproportional to the square of the angle of rotation. In both cases, the phase change isrelated to the change in the OPL created by the lens as it rotates. This change in opticalpath length causes a change in the focal length or direction of the beam, depending onthe application.

[0224] The OPL created by a GRIN lens is given by the integral of the refractive indexalong the path of light passing through the lens. If the optical path length created by thefirst GRIN lens is denoted OPL1 and the optical path length created by the second GRINlens as OPL2, then the total optical path length difference between the two lenses as theyrotate relative to one another can be expressed as:OPL^=^OPL2^–^OPL1. (97)This optical path length difference will cause a phase difference between the two lenses,which can result in various optical effects such as beam steering, focusing, or varifocalcapabilities.

[0225] To make a lens that beam steers with rotation, an asymmetry is introduced inthe lens design. One way to do this is to add a term that varies with the angle of rotation.This can be achieved by replacing x and y in the lens equation with expressionsinvolving sin(θ) and cos(θ), respectively. For example, one may modify the GRIN lensequation used for linear shiftn(x,^y)^=^n0^+^a1x^+^a2y^+^a3(x3^–^3xy2)^ (98)Docket No. NVX21306CON1CIPPCTto make it beam steer with rotation by replacing x with r sin(θ) and y with r cos(θ),where r is the radial distance from the center of the lens and θ is the angle of rotation.n(r, θ) = n0 + a1 r sin(θ) + a2 r cos(θ) + a3(r3 sin3(θ) – 3r2 sin(θ) cos2(θ)). (99)Now, when the lens is rotated, the sin(θ) and cos(θ) terms will cause the refractiveindex to vary with the angle of rotation, resulting in beam steering.

[0226] There are several high-order GRIN functions that can perform beam steeringwhen rotated relative to one another. Some examples include cubic and quartic GRINdistributions, which produce lenses that have different refractive indices along the x-and y-axes, allowing for astigmatism correction and beam steering. By rotating twolenses with the same cubic or quartic GRIN distribution, but with opposite signs,relative to each other, the combined lens can produce beam steering.

[0227] Spiral phase plates (SPP) introduce a spiral phase shift to the wavefront of abeam, which can produce a vortex beam or other types of structured light. By rotatingtwo spiral phase plates with opposite handedness relative to each other, the combinedsystem can produce beam steering. Axicon lenses are conical lenses that produce aBessel beam or other types of non-diffracting beams. By rotating two axicon lenses withthe same refractive index profile but opposite orientations relative to each other, thecombined system can produce beam steering.

[0228] There are also other high-order GRIN functions that can also be used for beamsteering, but the specific function that is optimal for a given application depends onseveral factors, including the desired level of steering, the size and shape of the beam,the wavelength of the light, and other optical parameters.

[0229] The combined function of two lenses with the same GRIN distribution,complementary to one another, as it relates to beam steering, can be described asfollows. Let the GRIN distribution for the lenses be given by:n(r,^θ)^=^n0^+^a1r^cos(θ)^+^a2r^sin(θ)^+^a3r2^+^a4r3^cos(θ)^+^a4r3^((1 / 3)^sin(3θ)^–^(1 / 2)^sin(θ))^+^…^ (100),where n0 is the refractive index at the center of the lens, a1,^ a2,^ a3,^ a4,^ ... are thecoefficients of the GRIN distribution.Docket No. NVX21306CON1CIPPCTIf the two lenses are rotated relative to one another by an angle θ, and the beam isincident along the z-axis, then the resulting phase shift of the beam at the output planecan be expressed as:φ(x,^y)^=^k^[n(x,^y)^–^n0]^t ( 101)where k is the wavevector of the light, and t is the thickness of the lenses.

[0230] Assuming that the beam is a plane wave incident along the z-axis, the outputfield at the plane z = t can be obtained by taking the Fourier transform of the phase shift:E(x,^y,^L)^=^F^{exp[i^φ(x,^y)]}, (102)where F{ } denotes the Fourier transform.

[0231] The resulting beam will be steered in the direction perpendicular to the axis ofrotation of the lenses, with the amount of steering depending on the angle of rotationand the coefficients of the GRIN distribution. The exact form of the function will dependon the specific GRIN distribution used.

[0232] FIG. 12 shows aspects of a GRIN phase plates that operate similarly to Risleyedge prims elements. The refractive index profile of a Risley prism can be approximatedby a GRIN equation with only linear terms, such as:^n(x)^=^n0^+^a1x. A GRIN phase platewith this distribution is shown in FIG. 12 (a). When this device is rotated with itscomplement, it will induce beam steering. The steering effect will be linearlyproportional to the rotation angle and the coefficient a1. The amount of beam steeringwill depend on the gradient of the refractive index distribution and the rotation angle.

[0233] The refractive index distribution of a GRIN lens that can act as an equivalentdevice to a Risley prism will have a sinusoidal variation, with a period that is equal tothe pitch of the prism. Mathematically, the refractive index distribution can beexpressed asn(x,^y)^=^n0^+^Δn^sin(2πx / p), (103)where n0 is the average refractive index of the lens, Δn is the amplitude of the refractiveindex variation, x and y are the Cartesian coordinates in the transverse plane, and p isthe pitch of the prism. In this device, the refractive index varies sinusoidally along the x-axis with a period of p. When this device is rotated with its complement, it will inducebeam steering by changing the propagation direction of the incident light based on theDocket No. NVX21306CON1CIPPCTsinusoidal refractive index variation. The amount of beam steering will depend on therotation angle and the amplitude of the refractive index variation (Δn).

[0234] Assuming that the two lenses are identical and rotate at different rates, thenthe angular deviation of the collimated light will change over time as the two lensesrotate relative to each other. The direction and magnitude of the angular deviation willdepend on the relative orientation of the two lenses at any given time, which in turndepends on their respective rotation angles and rates.

[0235] It is possible that the two lenses could be designed such that their combinedeffect on the light results in a desired change in focal length or other optical property.However, designing such a system would require careful consideration of the individuallens properties, as well as the relative rates and orientations of their rotation.

[0236] One possible GRIN equation that can function like a beam scanner withoutabrupt changes is:n(x,^y)^=^n0^+^Δn^cos(2π(x / p^+^φ(y))), (104)where n0 is the refractive index at the center of the GRIN element, Δn is the maximumrefractive index variation from n0, p is the period of the refractive index variation, andφ(y) is a smooth function that varies the phase of the refractive index variation alongthe y-axis. By varying the function φ(y), the scanning pattern can be controlled withoutabrupt changes in phase. This type of GRIN element is known as a smoothly varyingphase plate.

[0237] When two lenses with the same quartic GRIN distribution, but with oppositesigns, are rotated relative to each other by an angle θ, the resulting combined lens willhave a refractive index distribution that can cause beam steering. An example of aquartic GRIN distribution that can be used for beam steering similarly is:n(x,^y)^=^n0^+^a1x^+^a2y^+^a3x2^+^a4y2^+^a5x3^+^a6y3^+^a7x4^+^a8y4, (105)where n0 is the refractive index at the center, a1 and a2 are linear coefficients, a3 and a4are quadratic coefficients, a5^ and a6 are cubic coefficients, and a7 and a8^ are quarticcoefficients. By adjusting the coefficients in the quartic distribution, the amount anddirection of beam steering can be precisely controlled.Docket No. NVX21306CON1CIPPCTThe quartic GRIN distribution, when scanned relative to its complement, can effectivelychange the effective focal length of the GRIN element. This allows for the ability todynamically focus or defocus the beam as it is scanned. By adjusting the scanningparameters and the coefficients of the quartic distribution, the focal length can bemodified to achieve different beam steering effects.

[0238] One possible equation to describe the combined refractive index distributionis:n(x,^y,^θ)^=^n0^+^a1x^+^a2y^+^a3(x2^+^y2)^+^a4(x3^+^y3)^–^a4[(x^cosθ^+^y^sinθ)3^+^(–x^sinθ^+^y^cosθ)3],^ (106)where n0 is the background refractive index, a1 to a4 are coefficients that determine thestrength of the GRIN distribution, and θ is the rotation angle between the two lenses.

[0239] The presence of the cubic terms in the refractive index distribution introducesa nonlinear phase modulation across the GRIN element. This nonlinearity allows forprecise control over the phase profile of the transmitted light, resulting in controlleddeflection or steering of the beam. By adjusting the coefficients in the distribution, thebeam can be steered in a desired direction with high precision. The additional terminvolving the angle θ introduces a dependence on the rotation angle, allowing forvariable steering angles. By rotating the GRIN element and its complementarycounterpart, the combined effect of the refractive index distribution and the rotationenables the beam to be steered in different directions as the rotation angle changes.This provides flexibility in controlling the steering angle of the beam. The combinationof the quadratic and cubic terms in the refractive index distribution allows for morecomplex beam manipulation. The quadratic term contributes to focusing and defocusingeffects, while the cubic term introduces asymmetry and non-uniform steering. Thisenables more versatile beam shaping and steering capabilities, allowing for customizedbeam trajectories and profiles.

[0240] An example of a higher-order equation that could function like beam steeringdevice is:n(x,^y)^=^n0^+^Δn^[^sin(2πx / p)^+^b1sin(4πx / p)^+^b2sin(6πx / p)^+^...],^ (107)Docket No. NVX21306CON1CIPPCTwhere b1, b2, etc. are coefficients that determine the strength of the higher-order terms.The additional terms can help smooth out the phase transition between the two lenses,resulting in a more gradual change in phase as the beam is scanned.

[0241] Another phase distribution that can be used for beam steeringφ(r,θ)^=^mθ^+^l(r)exp(imθ),^^^^^^^^^^^^^^^^^^^^^^^^^^(108)where m is the topological charge (the number of spiral arms or lobes in the resultingphase pattern.), l(R) is the radial distribution function, and R and θ are the polarcoordinates in the transverse plane. The radially symmetric refractive indexdistribution where the refractive index depends only on the radial distance from thecenter, R. The equation, with its radial symmetry, is particularly suitable for shaping thewavefront of a beam with radial symmetry, such as cylindrical beams. It can introduce acontrolled phase variation as a function of the radial position, allowing for wavefrontmanipulation. The linear phase term mθ introduces a constant angular shift to the beamas it propagates. This angular shift can be used to control the direction or steering angleof the beam. By adjusting the value of m, one may change the amount of beamdeflection, allowing for precise control over the steering direction. The radial phaseterm introduces a radial variation in the phase of the beam. This can be utilized to shapethe intensity profile of the beam, such as creating focused spots or controlling the beamprofile. By appropriately choosing the function l(R), one may achieve specific intensitydistributions or tailor the beam shape according to the application requirements.

[0242] The index distribution that can create that phase pattern isn(r)^=^n0^+^(n1 –^n0) l(r) exp(i^m^θ),^^^^^^^^^^^(109)where n0 and n1 are the refractive indices of the ambient medium and the vortexmaterial, respectively. The refractive index varies as a function of both the radialdistance R and the azimuthal angle θ. The terms describe the amplitude of eachsinusoidal component with different azimuthal modes, and φm represents the phaseoffset for each mode. The equation, with its azimuthal symmetry, can be used for beamsteering applications. By adjusting the amplitudes and phases of the different azimuthalmodes, the direction and intensity profile of the transmitted beam can be controlled,enabling beam steering functionalities.Docket No. NVX21306CON1CIPPCT

[0243] To describe the refractive index variation in a GRIN spiral phase plate, thefollowing equation may be used:where n0 is the background refractive index, r and θ are the radial and azimuthalcoordinates in polar coordinates, R is the radius of the spiral phase plate, am and φm arethe amplitude and phase of the m-th order spiral term, respectively. This equationdescribes the refractive index variation in a GRIN spiral phase plate using a Fourierseries expansion, where n0 is the average refractive index, am are the Fouriercoefficients that determine the amplitude of the refractive index variation at eachharmonic frequency, R is the radius of the spiral, θ is the azimuthal angle, and φm is thephase angle for each harmonic component.

[0244] The key feature of this expression is the presence of different azimuthalharmonics with odd values of m (1, 3, 5, ...). Each term contributes to the overallrefractive index distribution, resulting in a specific phase modulation across the beam.The amplitude (am) determines the strength of each harmonic component, while thephase shift (φm) determines the offset or rotation of the phase pattern for eachharmonic.

[0245] The first coefficient a1 determines the magnitude of the linear phase gradientacross the plate and does not affect the optical vortex properties of the plate. Thesecond coefficient a3 controls the strength of the optical vortex and the direction ofrotation of the helical wavefront. The higher order coefficients a5, a7, etc. introduceadditional spiral components to the phase distribution. The exact values of the amcoefficients depend on the design requirements of the spiral phase plate, such as thedesired optical vortex charge, the radius of the spiral structure, and the wavelength ofthe incident light.

[0246] The mathematical function that describes the two spiral phase plates, oneopposite the other, rotated relative to one another can be written as follows:Φ(x,^y,^θ)^=^mφ(x,^y)^–^mφ(x^cosθ^+^y^sinθ,^y^cosθ^–^x^sinθ), (111)where y,^ θ)^ is the combined phase function of the two spiral phase plates atposition (x, y) and angle θ,^φ(x,^y) is the individual phase function of each spiral phaseDocket No. NVX21306CON1CIPPCTplate, m is the topological charge of the spiral phase plate, and θ is the rotation anglebetween the two plates.

[0247] To further illustrate some of the features above, FIG. 20 shows aspects of (a) asurface-figured optical element composed of two conjugate parts designed to providevariable optical power as a function of translation as a result of the combined surfaceshapes; (b) an example equivalent GRIN optic in two opposing plano-plano partsdesigned to modulate incident wavefronts to cause variable optical power as a functionof translation of the combined GRIN profiles; and (c) an example GRIN optic in twoopposing plano-plano parts designed to modulate incident wavefronts to causedeflection of an incident beam as a function of translation of the combined GRIN opticalelements.

[0248] FIG. 21 shows aspects of a GRIN phase plate optical element with a refractiveindex distribution that implements a spiral phase function across the device, such thatwhen translated rotationally relative to a complementary GRIN phase plate opticalelement, the helical phase profile that changes phase with the relative rotation angle,may reverse the handiness of the resulting vortex.XI.

[0249] To summarize, one aspect of this disclosure is directed to a device to variablymanipulate electromagnetic waveforms using two freeform elements, which, whentranslated relative to one another, either laterally or rotationally, result in a variableoptical function. The device media have complex dielectric properties that varythroughout the device. In one aspect, the refractive index, permeability, or permittivitymay be varied by forming the media with three-dimensional freeform compositionalpatterns in which the patterns have no axis of symmetry. The compositional materialpatterns result in complex dielectric functions that change the phase delays oftransmitted wavefronts. The combined phased delays of the elements can change theoptical functions of the combined elements. The optical functions may include opticalpower, beam steering, or other wavefront manipulation.

[0250] FIG. 22 shows aspects of an example gradient refractive index (GRIN) optic102. In the illustrated example, optic 102 takes the form of a disc symmetric aboutoptical axis A. Optic 102 comprises first material 104A and second material 104B andDocket No. NVX21306CON1CIPPCToptionally may comprise additional materials (vide^ infra). The first and secondmaterials are distributed inhomogeneously within optic 102. The first material isdistributed according to a first volume-fraction profile x1(r), and the second material isdistributed according to a second volume-fraction profile x2(r). The graph of FIG. 23shows example first and second volume-fraction profiles, each plotted as a function of acoordinate r. Generally speaking, r is a geometric coordinate of optic 102. In theillustrated example r corresponds to radial distance from axis A.^ In other examples rmay correspond to a different geometric coordinate or to a linear combination ofgeometric coordinates. In some examples a GRIN optic may have more or less symmetrythan optic 102 and a different overall shape.

[0251] First material 104A has a first refractive index n1(λ), and second material104B has a second refractive index n2(λ), where λ denotes wavelength. In the examplesherein the first refractive index is greater than the second refractive index for λB < λ < λR.The difference in the refractive index of the first and second materials is denoted Δn(λ)= n1(λ) – n2(λ). To a good approximation, the observed refractive index n in radiallysymmetric optic 102, at any value of r, is a linear combination of n1(λ) and n2(λ)weighted according to the respective volume fractions of the first and second materialsat that same r:n(λ, r) = x1(r) n1(λ) + x2(r) n2(λ). (112)

[0252] For an optic including a third material, etc., the weighted sum is extendedaccordingly. In combination with n1(λ) and n2(λ), the first and second volume-fractionprofiles define a gradient in the observed refractive index of the optic. Thus, for a givenwavelength λ (or sufficiently narrow range of wavelengths), it is possible to engineer adesired refractive-index gradient by appropriate material selection and control over thefirst and second volume-fraction profiles x1(r) and x2(r).

[0253] In optic 102 of FIG. 22, the observed refractive index decreases withincreasing distance r from optical axis A, thereby defining a radial component of therefractive-index gradient. In more particular examples, the radial component may besuch that the refractive index changes as a function of one or more terms of rx, forexample where x ≥ 2. In other examples, the radial component may have a morecomplex refractive-index distribution. For some radially symmetric optics the radialcomponent may be a superposition of radial components—e.g.,^Docket No. NVX21306CON1CIPPCT^ n(λ, r) = n0 + Δn(a2 r2 + a4 r4 + a6 r6 + …), (113)where coefficients ax weight corresponding radial powers rx, and where n0 is therefractive index at the center of the optic. In some examples the radial component mayvary as a function of depth z along the optical axis. The refractive index along z may alsovary as a function of one or more terms of zx, for example where x ≥ 2. In the case wherethe refractive index varies with both r and z, the refractive index at any location may berepresented asn(λ, r, z) = n0 + Δn(a2 r2 + a4 r4 + a6 r6 + b1 z, b1 z2 + a2 b1 r2 z + a2 b2 r2 z2 +…),(114)where coefficients bx weight corresponding depth powers zx.

[0254] Thus, the observed refractive-index may vary in directions perpendicularand / or parallel to the optical axis. GRIN optics having refractive-index profiles of lowersymmetry are also envisaged. In particular, an optic consonant with this disclosure mayhave a refractive index profile with no translational or rotational symmetry about axesnormal to a mean plane. An optic consonant with this disclosure may have a surfaceprofile with no translational or rotational symmetry about axes normal to a mean plane.Equally envisaged are freeform optics where the refractive-index gradient isasymmetric about the optical axis, as can be described by more complex polynomialrepresentations. In optic 102, however, optical power derives from the controlledgradient of the observed refractive index in the radial direction, ∂n / ∂r. As describedhereinafter, one way to exert such control is to form optic 102 from a cured coalescenceof ‘ink’ droplets providing the controlled volume fractions of the first and secondmaterials. Such an optic can be engineered to provide optical power—e.g., convergentfocus of light rays passing through the optic. In such examples, the refractive-indexgradient provides a function analogous to the gradient entry and / or exit surface anglesof a conventional spherical lens. Accordingly, optic 102 can be engineered to provideoptical power despite having no curvature on the entry or exit faces. Nevertheless, optic102 optionally may include at least one curved surface for additional optical power. Theskilled reader will note that a ‘gradient’ defined as a scalar departs somewhat fromstandard usage; the direction of the gradient is assumed to be the direction of greatestchange unless otherwise stated.Docket No. NVX21306CON1CIPPCT

[0255] A practical way to realize optical materials with refractive indices amenable tothe approach herein is to base each material on a polymer species or mixture ofpolymer species. A polymer-based material can be deposited in a controlled manner inthe form of liquid droplets, which coalesce and subsequently solidify in a desired shape(vide^ infra). Accordingly, first material 104A and second material 104B of optic 102(and a third material, etc., in examples in which additional materials are incorporated)may each include at least one polymer species. The term ‘matrix’ refers herein to the atleast one polymer species on which a material is based. In examples in which asubstantially transparent optic is desired, each polymer species may be an opticallytransparent polymer species. Suitable polymer species include propylene carbonate(PC), di(ethylene glycol) diacrylate (DEGDA), fluoroethylene glycol diacrylates (FEGDA,FEGDA(2)), neopentyl glycol diacrylate (NPGDA), 2-hydroxyethylmethacrylate (HEMA)and hexanediol diacrylate (HDDA or HDODA) polymers, bisphenol A novolak epoxy(SU8), polyacrylate (PA), polymethyl methacrylate (PMMA), polystyrene,polydiacetylene (PDA), poly(ethylene glycol diacrylate (PEGDA), and poly[(2, 3, 4, 4, 5,5-hexafluorotetrahydrofuran-2, 3-diyl)(1, 1, 2, 2-tetrafluoroethyl-ene)] (CYTOP)). Otherpolymer species providing desired physicochemical properties may also be used.

[0256] In some examples, one or more nanoparticle species may be dispersed in amatrix in order to modify the wavelength-dependent refractive index of the matrix.Accordingly, first material 104A and / or second material 104B of optic 102 (and / or athird material, etc., in examples in which additional materials are incorporated) may becomposite materials of fixed composition. More particularly, each material may includeat least one nanoparticle species dispersed in a matrix. The term ‘nanocompositematerial’ refers herein to a dispersion of at least one nanoparticle species in a matrix. Inexamples in which a substantially non-scattering optic is desired, an averagenanoparticle size may be selected for each nanoparticle species such that the size is toosmall to effect significant Rayleigh or Mie scattering in optic 102. Accordingly, theselected average size may depend on the wavelength band of interest. For non-scattering optics engineered for the visible wavelengths, the selected average size maybe less than 50 nanometers (nm), for example. Further, the coefficient of extinction,combining absorbance and reflection, of a nanocomposite material may be 10% orlower, preferably 1% or lower, over the band of interest.Docket No. NVX21306CON1CIPPCT

[0257] Nanoparticle species suitable for modifying the refractive index of a matrixinclude various metal, metal oxide, chalcogenide, and semiconductor nanoparticles.More particular examples include zinc sulfide (ZnS), zirconium dioxide (ZrO2), bariumtitanate (BTO), bismuth germanate (BGO), nano-diamond (NanoD), zinc oxide (ZnO),beryllium oxide (BeO), magnesium oxide (MgO), aluminum nitride (AlN), wurtzite AlN(w-AlN), titanium dioxide (TiO2), tellurium dioxide (TeO2), aluminum oxide imide(Al2O3HN), molybdenum trioxide (MoO3), aluminum-doped ZnO (AZO), germanium-doped silicon (SiGe), silicon dioxide (SiO2), and lithium fluoride (LiF) nanoparticles,hollow SiO2 nanospheres (h-SiO2), and shelled variants of any of the foregoingnanoparticles supporting ZrO2, MgO, SiO2, ZnO, or other shells, including those thatcause the nanoparticles to be more or less reactive with the matrix. Other nanoparticlespecies providing desired physicochemical properties may also be used. For somenanoparticle species, nanoparticle stability and / or dispersability in a matrix can beenhanced by chemical modification of the surface of each nanoparticle. For instance, thenanoparticles may be surface-functionalized by a suitable ligand—e.g., acrylic acid,phosphonic acid, or a silane—that provides chemical compatibility dispersability withthe matrix, thereby enhancing optical clarity. Ligands may be selected to covalentlybond to the surface of the nanocrystal via an ‘anchor’ moiety and / or repel each othervia a ‘buoy’ moiety, thereby discouraging aggregation. In some examples, a distal site ona ligand may bond covalently to a monomer of the matrix so that dispersability ismaintained during polymerization.

[0258] FIG. 24 shows aspects of a non-limiting example apparatus configured foradditive manufacture of an article. Additional details are found in U.S. PatentApplications 16 / 224,512 entitled NANOCOMPOSITE OPTICAL-DEVICE WITHINTEGRATED CONDUCTIVE PATHS and 16 / 507,658 entitled PRINTED CIRCUIT BOARDWITH INTEGRATED OPTICAL WAVEGUIDES; FUNCTIONALLY GRADED POLYMERMATRIX NON-COMPOSITES BY SOLID FREEFORM FABRICATION, Solid Freeform (SFF)Symposium (2003); and POLYMER MATRIX NANOCOMPOSITES BY INK-JET PRINTING,Solid Freeform (SFF) Symposium (2005), which are hereby incorporated herein byreference for all purposes. Nevertheless, various other deposition methods andapparatuses are also applicable to the approach herein.Docket No. NVX21306CON1CIPPCT

[0259] Apparatus 906 of FIG. 24 includes reservoir 908A holding a first ink andreservoir 908B holding a second ink. The first ink is a liquid precursor of first material104A, in which the one or more polymer species takes resinous form, is not curedand / or not cross-linked. Likewise the second ink is a liquid precursor of secondmaterial 104B, in which the one or more polymer species takes resinous form, is notcured and / or not cross-linked. Reservoirs 908A and 908B are coupled fluidically toprint heads 910A and 910B, respectively. Each print head is configured to discharge thecorresponding ink with high spatial accuracy onto optic 902, arranged on platen 912.More particularly, each print head is configured to add individual voxels of ink to theoptic. In examples in which a third, etc., material is used, the apparatus may include aseparate reservoir and print head for additional, corresponding inks. In these and otherexamples, both the order of deposition of the ink droplets and the location of eachdroplet may be controlled to high precision.

[0260] Platen 912 is coupled mechanically to translational stage 914. Thetranslational stage is configured to adjust the displacement of the platen along each ofthe three Cartesian axes. In other examples, displacement along any, some, or all of theCartesian axes may be adjusted by movement of the print heads instead of, or inaddition to, the platen. In still other examples, a translational stage may adjust therelative displacement of the platen and print heads along two Cartesian axes, and arotational stage (not shown in the drawings) may be used to adjust the azimuth of voxeldeposition in the plane orthogonal to the two Cartesian axes. In every case, theadjustment is controlled (e.g., servomechanically), pursuant to control signals fromcontroller 915. More particularly, the controller may be configured to transmit, to thetranslational stage and to the first and second print heads, signal defining the first andsecond volume-fraction profiles, for each of a plurality of voxel-thick layers of the optic.The controller may compute these patterns by parsing a 3D digital model of the optic tobe fabricated and returning the intersection of the 3D digital model with a series ofcutting planes corresponding to the plurality of layers.

[0261] Continuing in FIG. 24, apparatus 906 includes a directed optical emitter 916and a diffuse optical emitter 918. The optical emitters may comprise lasers or lamps ofany emission profile suitable for curing the inks. The displacement of the opticalemitters relative to platen 912 may be controlled in the same manner as theDocket No. NVX21306CON1CIPPCTdisplacement of the print heads relative to the platen. The directed optical emitter maybe used for selective, localized curing of certain regions of voxels, and the diffuse opticalemitter may be used to cure larger regions of the optic. In examples in which one of theinks is thermally curable, a heat emitter may be included.

[0262] This disclosure is presented by way of example and with reference to theattached drawing figures. Components, process steps, and other elements that may besubstantially the same in one or more of the figures are identified coordinately anddescribed with minimal repetition. It will be noted, however, that elements identifiedcoordinately may also differ to some degree. It will be further noted that the figures areschematic and generally not drawn to scale. Rather, the various drawing scales, aspectratios, and numbers of components shown in the figures may be purposely distorted tomake certain features or relationships easier to see.

[0263] It will be understood that the configurations and / or approaches describedherein are exemplary in nature, and that these specific examples are not to beconsidered in a limiting sense, because numerous variations are possible. The specificroutines or methods described herein may represent one or more of any number ofprocessing strategies. As such, various acts illustrated and / or described may beconducted in the sequence illustrated and / or described, in other sequences, in parallel,or omitted. Likewise, the order of the above-described processes may be changed.

[0264] The subject matter of the present disclosure includes all novel and non-obvious combinations and sub-combinations of the various processes, systems andconfigurations, and other features, functions, acts, and / or properties disclosed herein,as well as any and all equivalents thereof.

Claims

Docket No. NVX21306CON1CIPPCT CLAIMS:

1. An optic configured for variable wavefront shaping of electromagnetic radiation,the optic comprising:a first optical element including a solidified heterogeneous coalescence ofnanocomposite material providing a first complex dielectric-function gradient;a second optical element including solidified heterogeneous coalescence ofnanocomposite material providing a second complex dielectric-function gradient,wherein the first and second optical elements are arranged in tandem along an opticalaxis and together provide wavefront shaping that varies in dependence on adisplacement of the first optical element relative to the second optical element.

2. The optic of claim 1 wherein the first and / or second complex dielectric-functiongradient is a freeform gradient, a non-radially symmetric gradient, a non-axiallysymmetric gradient, or an anamorphic gradient.

3. The optic of claim 1 wherein the first or second complex dielectric-function variesradially and axially, relative to the optical axis.

4. The optic of claim 1 wherein the first and / or second complex dielectric-functiongradient is modified by laser radiation.

5. The optic of claim 1 wherein the displacement changes a focal length of the optic.

6. The optic of claim 1 wherein the displacement changes a direction of a beamexiting the second optical element relative to the direction of the beam entering the firstoptical element.

7. The optic of claim 1 wherein the displacement imparts an effect of a variablewedge function on the electromagnetic radiation.

8. The optic of claim 1 wherein the displacement reproduces an effect of a variablephase plate on the electromagnetic radiation.Docket No. NVX21306CON1CIPPCT 9. The optic of claim 1 wherein the displacement reproduces an effect of a variableblazed grating on the electromagnetic radiation.

10. The optic of claim 1 wherein the electromagnetic radiation comprises near-infrared, infrared, millimeter-wave, or radio-frequency radiation.

11. The optic of claim 1 wherein the optic is arranged in a vision-correcting device,optical scanner, variable-magnification telescope, or variable-magnification microscope.

12. The optic of claim 1 wherein the first and second optical elements are configuredfor time varying spatial radiance.

13. The optic of claim 1 wherein the optic is arranged in an antenna.

14. The optic of claim 1 wherein dispersive properties of the nanocompositematerials of the first and second optical elements impart a wavelength dependence tothe variable wavefront shaping.

15. The optic of claim 1 further comprising an anti-reflective coating arranged on thefirst and / or second optical element.

16. The optic of claim 1 further comprising a beam deflector configured to extractoptical power from the optic.

17. The optic of claim 1 wherein the first and / or second complex dielectric-functiongradient comprises a Fresnel implementation of a complex dielectric-function gradient.

18. The optic of claim 1 wherein the first and / or second complex dielectric-functiongradient comprises a segmented freeform implementation of a complex dielectric-function gradient.

19. The optic of claim 1 wherein the optic is configured to emit a light field orhologram.Docket No. NVX21306CON1CIPPCT 20. The optic of claim 1 further comprising at least one opaque baffle arrangedbetween the first and second optical elements.

21. The optic of claim 1 further configured to transmit the electromagnetic radiationonly through an area of overlap between the first and second optical elements.

22. The optic of claim 1 wherein the first and second optical elements are arrangedin an array of analogously configured optical elements, and wherein the complexdielectric-function gradient varies in dependence on a position of each optical elementin the array.

23. A system of optics configured for variable focus, the system comprising:a first optic including first and second gradient complex dielectric-function opticalelements arranged in tandem along an optical axis, which together provide anoptical power that varies according to a displacement of the first optical elementrelative to the second optical element; anda second optic including third and fourth gradient complex dielectric-function opticalelements arranged in tandem along an optical axis, which together provide anoptical power that varies according to a displacement of the third optical elementrelative to the fourth optical element,wherein focal lengths of the first and second optics are adjustable relative to eachother.

24. The system of claim 23 further comprising at least one additional lens elementarranged between the first and second optics.

25. The system of claim 23 wherein dispersive properties of the first and secondoptics are matched to achieve achromatic performance.

26. A method of manufacture of a nanocomposite ink-based optic with a complexdielectric-function gradient and variable focus, the method comprising:having or providing a nanocomposite-ink printing apparatus with a nanocomposite inkincluding an organic matrix with a nanoparticle dispersed within the organic matrix;Docket No. NVX21306CON1CIPPCTdepositing and forming a first optical element having a first surface and a secondsurface with a gradient optical index therebetween;depositing and forming a second optical element having a third surface and a fourthsurface with a gradient optical index therebetween, the first optical element and thesecond optical element each comprising a cured nanocomposite ink with an organicmatrix and a nanoparticle dispersed within the organic matrix,wherein the first and the second optical elements are arranged in tandem along on anoptical axis and have an optical power that varies according to a translation betweenthe first and second optical elements.

27. The method of claim 26 further comprising depositing and forming a thirdoptical element configured to co-locate an optical axis of the first optical element to anoptical axis of the second optical element over an operating range of the translation.

28. The method of claim 26, wherein the nanocomposite ink of the first opticalelement and the nanocomposite ink of the second optical element are selected such thata slope of refractive index with respect to wavelength of a highest average refractiveindex nanocomposite ink and slope with respect to wavelength of a lowest averagerefractive index ink are parallel to 1% or better.

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