Interferometric displacement measurement system and methods

Abstract

A displacement measurement interferometer has a beam coupler to decollimate source radiation; beam-splitting optics that (a) output the source radiation toward a reflective target and (b) split reflected radiation into a single-pass optical beam and a multi-pass optical beam; and reflective optics to retroreflect the multi-pass optical beam through the beam-splitting optics to the reflective target. A method for measuring displacement of an object includes: reflecting non-collimated source radiation off a surface of the object; splitting reflected radiation from the surface of the object into a single-pass optical beam and a multi-pass optical beam; retroreflecting the multi-pass optical beam to the surface; and determining the displacement from an interference signal between the single-pass optical beam and the multi-pass optical beam after the single-pass optical beam reflects off the surface and after the multi-pass optical beam reflects off the surface at least twice.

Description

PATENT Attorney Docket No: TUOE. P2012WO / 0065313INTERFEROMETRIC DISPLACEMENT MEASUREMENT SYSTEM AND METHODS

[0001] This application claims priority to US Provisional Application No.63 / 730,587, titled “Interferometric Displacement Measurement System”, filed 11 December 2024 and incorporated herein by reference.BACKGROUND

[0002] Interferometry used to determine displacement of an object includes the use of collimated laser radiation. Displacement measuring interferometers are widely used m the field of mechatronic motion systems and measurement machines from atomic force microscopes to high precision planar stages. Some of the many advantages of these interferometers are the large measurement stroke, relatively fast sampling, non-contact measuring, and low measurement uncertainty in the range of (sub-)nanometers. [1] [2], Within the semiconductor industry, there is a trend towards complex 3D integration schemes to increase functionality and decrease footprint, cost and power usage of chips [3], One of these methods includes stacking integrated circuits (both die to die, die to wafer, and wafer to wafer). Planar stages could be used to align the chips in three degrees of freedom with respect to each other. Tire rotation is typically aligned by a second stage because of the limited angle acceptance of displacement measurement interferometers, increasing system cost and complexity.SUMMARY

[0003] Embodiments disclosed herein include displacement measuring interferometers, and associated displacement measurement methods, with an increased angle of acceptance (e.g., more than 10x) without incurring tire costs of previous approaches, such as increased size, cost, and system complexity. Increasing the angle of acceptance of a displacement measuring interferometer (“DMI”) as disclosed hereinbelow is valuable for at least two reasons. First, it increases the DMI's alignment tolerance. Second, it increases the DMI's range of applications by enabling it to measure displacement of targets whose linear motion (displacement) is accompanied by rotational motion (tip and tilt) that changes the direction of light reflected by the target.

[0004] The rotational range of many high-end motion platforms is limited by the use of plane mirror interferometers, for example, in stacking integrated circuits. Embodiments disclosed herein increase the angle acceptance of the DMI, for example, with a measurement 1LEGAL\81463546\914PATENT Attorney Docket No: TUOE. P2012WO / 0065313 stroke of about 0.5 m, as typical for motion stages in the semiconductor industry.Embodiments described below may use a diverging laser beam to significantly increase the target mirror range of rotation. Further, by interfering the beams from a single path and a double path on a detector, for example, the displacement of the target can be measured while eliminating first-order wavefront tilt.

[0005] Embodiments disclosed herein include two interferometer layouts that apply a diverging beam. A model based on Gaussian beam theory’ is developed as a design tool to characterize these embodiments based on key performance indicators. A prototype of the most promising layout is also described, demonstrating that the angle acceptance increases thirty-twofold from 0.5 mrad to 16 mrad, while reaching a detected contrast of at least 0.5 and a signal-to-noise ratio of at least 10 at a rotation of ±16 mrad and a distance of 0.5 m.

[0006] In one embodiment, a displacement measurement interferometer has beam coupler to decollimate source radiation: beam-splitting optics that (a) output the source radiation toward a reflective target and (b) split reflected radiation into a single-pass optical beam and a multi-pass optical beam; and reflective optics to retroreflect the multi-pass optical beam through the beam-splitting optics to the reflective target.

[0007] In an embodiment, a method is provided for measuring displacement of an object, including: reflecting non-collimated source radiation off a surface of the object; splitting reflected radiation from the surface of the object into a single-pass optical beam and a multi-pass optical beam; retroreflecting the multi-pass optical beam to tire surface; and determining the displacement from an interference signal between the single-pass optical beam and the multi-pass optical beam after the single-pass optical beam reflects off the surface and after the multi-pass optical beam reflects off the surface at least twice. In an embodiment, the step of retroreflecting tire multi-pass optical beam to the surface includes reflecting the multi-pass optical beam off the surface.BRIEF DESCRIPTION OF THE FIGURES

[0008] In the drawings, identical reference numbers identify similar elements or acts. The sizes and relative positions of elements in the drawings are not necessarily drawn to scale. For example, the shapes of various elements and angles are not drawn to scale, and some of these elements are arbitrarily enlarged and positioned to improve drawing legibility. Further, the particular shapes of the elements as drawn are not intended to convey any information regarding the actual shape of the particular elements, and have been solely selected for ease of recognition in the drawings.2LEGAL\81463546\914PATENT Attorney Docket No: TUOE. P2012WO / 0065313 i

[0009] FIG. 1 shows one displacement measurement interferometer with a single-pass optical beam, in an embodiment.

[0010] FIG. 2 shows the displacement measurement interferometer in FIG. 1 but with a multi-pass optical beam, in an embodiment.

[0011] FIG. 3 shows one displacement measurement interferometer with a single-pass optical beam, in an embodiment.

[0012] FIG. 4 shows the displacement measurement interferometer in FIG. 3 but with a multi-pass optical beam, in an embodiment.

[0013] FIG. 5 and FIG. 6 show the displacement measurement interferometer in FIG. 3 in different operational displacements of the reflective target, to illustrate behavior of the single-pass optical beam.

[0014] FIG. 7 and FIG. 8 show the displacement measurement interferometer in FIG. 3 in different operational displacements of the reflective target, to illustrate behavior of the single-pass optical beam but just showing a primary ray path,

[0015] FIG. 9 and FIG. 10 show' the displacement measurement interferometer in FIG. 4 in different operational displacements of the reflective target, to illustrate behavior of the multi-pass optical beam.

[0016] FIG. 11 and FIG. 12 show the displacement measurement interferometer in FIG. 4 in different operational displacements of the reflective target, to illustrate behavior of the multi-pass optical beam but just showing a primary ray path.

[0017] FIG. 13 is a flowchart illustrating a method for measuring displacement of an object, in an embodiment.

[0018] FIG. 14 illustrates a comparison between embodiments of a single-pass DMI, a double-pass DMI, a beam steering DMI and the diverging beam DMI for a tilted measurement mirror.

[0019] FIG. 15 show's the interference patern of two optical beams of equal curvature, but interfering under an angle.

[0020] FIG. 16 shows the interference pattern of two beams with different radius but equal angle.

[0021] FIG. 17 shows, for embodiments of a DMI, the relative phase difference between the first and second pass for different optical path lengths and different aperture sizes.

[0022] FIG. 18 depicts an embodiment that reflects a portion of the first pass onto the detector by means of a polarizing beamsplitter and a quarter wave plate3LEGAL\81463546\914PATENT Attorney Docket No: TUOE. P2012WO / 0065313

[0023] FIG. 19 shows optical efficiency of the multiple beam passes in an embodiment of a DMI.

[0024] FIG. 20 is a schematic of a non-polarization splitting embodiment of a DMI.

[0025] FIG. 21 shows an example optical efficiency of the two beam passes in an embodiment of a DMI.

[0026] FIG. 22 shows elliptical beam intensity profiles associated with an embodiment of a DMI,

[0027] FIG. 23 shows power on the detector and the power ratio for the two passes through an embodiment of a DMI.

[0028] FIG. 24 shows detected contrast and signal-to-noise ratio for an embodiment of a DMI

[0029] FIG. 25 is a photograph of an example non-polarization splitting test setup of a diverging beam DMI.

[0030] FIG. 26 shows key performance indicators for an embodiment of a diverging beam DMI.

[0031] FIG. 27 shows the measured, low-pass filtered, and noise, voltages for an embodiment of a DMI.

[0032] FIG. 28 shows measurement noise and periodic error for an embodiment of a diverging beam DMI.

[0033] FIG. 29 show's power on the detector and the power ratio for an embodiment of a DMI.

[0034] FIG. 30 shows modelled key performance indicators: detected contrast and signal-to-noise ratio for an embodiment of a DMI.

[0035] FIG. 31 shows power on the detector and the power ratio for the two passes for an embodiment of a DMI.

[0036] FIG. 32 show's detected contrast and signal-to-noise ratio of an embodiment of a DMI.

[0037] FIG. 33 show's power on the detector for the three passes, for an embodiment of a DMI.

[0038] FIG. 34 shows power ratio on the detector between the three passes, for an embodiment of a DMI.

[0039] FIG. 35 shows modelled key performance indicators: detected contrast and signal-to-noise ratio of an embodiment.4LEGAL\81463546\914PATENT Attorney Docket No: TUOE. P2012WO / 0065313 i

[0040] FIG. 36 shows the periodic error and periodic error offset caused by the third pass of an embodiment.

[0041] FIG. 37 is a plot of beam width for a circular Gaussian beam.

[0042] FIG. 38 is a plot of beam radius of curvature for a circular Gaussian beam.

[0043] FIG. 39 is a plot of the Gouy phase for a circular Gaussian beam along the optical axis.

[0044] FIG. 40 is a plot showing a motion profile of the piezo stepper.

[0045] FIG. 41 shows the FFT of a measured signal withou t and with a band-stop filter for an embodiment of a DMI.

[0046] FIG. 42. shows the FFT of the noise without and with a band-stop filter, for an embodiment of a DMI.

[0047] FIG. 43 show's typical measured data without and with a band-stop filter at the resonant frequency of the piezo - measurement mirror assembly, for an embodiment of a DMI.DETAILED DESCRIPTION OF THE EMBODIMENTS

[0048] In the following description, certain specific details are set forth in order to provide a thorough understanding of various disclosed embodiments. However, one skilled in the relevant art will recognize that embodiments may be practiced without one or more of these specific details, or with other methods, components, materials, etc.

[0049] Unless the context requires otherwise, throughout the specification and claims which follow, the word “comprise'’ and variations thereof, such as, “comprises” and “comprising” are to be construed in an open, inclusive sense that is as “including, but not limited to.”

[0050] Reference throughout this specification to “one implementation” or “an implementation” or “one embodiment” or “an embodiment” means that a particular feature, structure or characteristic described in connection with the embodiment is included in at least one implementation or embodiment. Thus, the appearances of the phrases “one implementation” or “an implementation” or “in one embodiment” or “in an embodiment” in various places throughout this specification are not necessarily all referring to the same implementation or embodiment. Furthermore, the particular features, structures, or characteristics may be combined in any suitable manner in one or more implementations or one or more embodiments.5LEGAL\81463546\914PATENT Attorney Docket No: TUOE. P2012WO / 0065313

[0051] As used in this specification and the appended claims, the singular forms “a,” “an,” and “the” include plural referents unless the content clearly dictates otherwise. It should also be noted that the term “or” is generally employed in its sense including “and / or” unless the content clearly dictates otherwise.

[0052] Regarding instances of the terms “and / or” and “at least one of,” for example, in the cases of “A and / or B,” “at least one of A and B,” and “at least one of A or B,” such phrasing encompasses the selection of (i) A only, or (ii) B only, or (iii) both A and B. In the cases of “A, B, and / or C, ” “at least one of A, B, and C,” and “at least one of A, B, or C,” such phrasing encompasses the selection of (i) A only, or (ii) B only, or (iii) C only, or (iv) A and B only, or (v) A and C only, or (vi) B and C only, or (v ii) each of A and B and C. This may be extended for as many items as are listed.

[0053] In the embodiment shown in FIG. 1 and FIG. 2, a displacement measurement interferometer 10 is shown. Specifically, in FIG. 1, a single-pass optical beam 12 is shown within displacement measurement interferometer 10; while in FIG. 2, a multi-pass optical beam 14 is shown within displacement measurement interferometer 10. Both the single-pass optical beam 12 and the multi-pass optical beam 14 derive from source radiation 16 generated by a source 18 (e.g., a laser) through a coupler 20 (e.g., a lens, aperture or source exit pupil). Beam-splitting optics 22 outputs source radiation 16 toward a reflective target 24 (e.g., a plane mirror mounted to tire measurement head) and then splits radiation, reflected by reflective target 24, into the single-pass optical beam 12 and the multi-pass optical beam 14. Reflective optics 26 retroreflects the multi-pass optical beam 14 through the beam-splitting optics 22 to the reflective target 24.

[0054] Displacement measurement interferometer 10 may further include a beamsplitter 28 that separates (a) source radiation 16 from (b) both the single-pass optical beam 12 and the multi-pass optical beam 14 output to a detector 30. Beamsplitter 28 is for example a 50 / 50 transmission / reflection splitter. Coupler 20 decollimates source radiation 16 so that both single-pass optical beam 12 and multi-pass optical beam 14 are non-collimated, e.g., diverging. Coupler 20 is, for example, a positive or negative lens that decollimates source radiation 16. Coupler 20 may alternatively be an aperture stop or an exit aperture of source 18, e.g., when source 18 generates a diverging or converging beam of source radiation 16. Coupler 20 may comprise one or more, or a set of, lenses or reflective components to provide like function, as a matter of design choice.

[0055] An important use of displacement measurement interferometer 10 is to measure relevant displacement between a measurement head and object upon which6LEGAL\81463546\914PATENT Attorney Docket No: TUOE. P2012WO / 0065313 reflective target 24 is mounted, so that an interference pattern generated at detector 30 can be used to determine displacement of the object. Unlike a traditional interferometer, displacement measurement interferometer 10 advantageously uses de collimated source radiation 16 so that some portion of that beam will always re-enter a center of reflective target 24 despite enlarged rotations of reflective target 24. An aperture 32 adjacent beam-splitting optics 22 ensures, for single-pass optical beam 12, that only its orthogonally reflected radiation from reflective target 24 passes through aperture 32. This eliminates scaling errors since the portion of single-pass optical beam arriving at detector 30 is the portion that orthogonally touched reflective target 24 (that is, normally-incident on, and hence retroreflected by, reflective target 24).

[0056] Accordingly, multi-pass optical beam 14 becomes a reference used to interfere with single-pass optical beam 12. Reflective optics 26 reflects part of multi-pass optical beam 14 back to reflective target 24 for a second pass. The second pass by multi-pass optical beam 14 interferes with the first pass by single-pass optical beam 12 to generate a signal proportional to displacement of reflective target 24 (and of the underlying measurement head upon which reflective target 24 is mounted). Additional passes and reflections off reflective target 24 by multi-pass optical beam 14 are also possible, including a 3rdpass, a 4s!lpass, and so on. Second and third passes by multi-pass optical beam 14 are particularly important, as described hereinbelow'.

[0057] Displacement measurement interferometer 10 may include one or more lenses 34 used to modify multi-pass optical beam into a beam of desired divergence, for example to focus beam 14 onto reflective optics 26. Reflective optics 26 may be a retroreflector.[00581 In an embodiment, beam-splitting optics 22 includes polarizing optics that (a) output source radiation 16 with a first polarization toward reflective target 24, (b) polarize reflected radiation from reflective target 24 with a second polarization, and (c) split the reflected radiation, based on polarization, into single-pass optical beam 12 and the multi-pass optical beam 14. For example, the first polarization may be elliptical polarization and the second polarization may be linear polarization. Beam-splitting optics 22 may further include a polarizing beamsplitter, and may also include a wave plate 36 as shown (e.g., a quarter¬ wave plate).

[0059] In an example of operation for the single-pass optical beam 12 in FIG. 1, the beam-splitting optics 22 are oriented to reflect all incoming radiation from source 18 so that a diverging beam travels through the quarter wave plate 36 to convert the diverging beam to an elliptical polarization state, which then strikes reflective target 24. Electromagnetic radiation 7LEGAL\81463546\914PATENT Attorney Docket No: TUOE. P2012WO / OQ65313 reflected from reflective target 24 hits aperture 32 such that a small portion of that reflected radiation continues to the quarter wave plate 36; this converts the elliptical polarization state to a linear polarization state again. The quarter wave plate 36 is aligned, in this example, such that after two passes the polarization state has rotated more than 90 degrees (or less than 90 degrees). Therefore, upon hitting beamsplitter optics 22, a portion reflects towards beamsplitter 28, as single-pass optical beam 12, which in turn reflects towards detector 30.

[0060] Continuing with the example but for operation of multi-pass optical beam 14 in FIG. 2, part of the diverging beam that travels to the reflective target 24 on the first pass is not reflected towards detector 30, but instead is transmitted by beam-splitting optics 22. This effectively originates multi-pass optical beam 14. From beam-splitting optics 22, multi-pass optical beam 14 travels left towards lens 34, which focusses multi-pass optical beam 14 onto reflective optics 26 positioned at the focal point of lens 34. This arrangement retroreflects multi-pass optical beam 14, which travels right again through beam-splitting optics 22 at the same angle as before, and through lens 34. Multi-pass optical beam 14 travels through quarter wave plate 36 and converts to elliptical polarization, then passes through aperture 32 and travels to reflective target 24. After reflecting off of reflective target 24, multi-pass optical beam 14 travels left again towards aperture 32. Part of multi-pass optical beam 14 passes through aperture 32 and travels to quarter wave plate 36, which converts multi-pass optical beam 14 to a linear polarization state so that most reflects down toward beamsplitter 28. Single-pass optical beam 12 and multi-pass optical beam 14 then combine to interfere at detector 30.

[0061] By generating multi-pass optical beam 14 this way, when single-pass optical beam 12 and multi-pass optical beam 14 pass through aperture 32 to detector 30 they are not tilted with respect to each other; though each are tilted relative to the optical axis with an angle equal to the angle of reflective target 24. Since the tilt of beams 12 and 14 are equal, contrast at detector 30 is not diminished. Experimentation has shown that this configuration creates high detected contrast when the beam waist, aperture 32, and nodal point of reflective optics 26 are collocated.

[0062] In the embodiment shown in FIG. 3 and FIG. 4, a displacement measurement interferometer 100 is shown. Specifically, in FIG. 3, a single-pass optical beam 112 is shown within displacement measurement interferometer 100; while in FIG. 4, a multi-pass optical beam 114 is shown within displacement measurement interferometer 100. Both the single¬ pass optical beam 112 and the multi-pass optical beam 114 derive from source radiation 116 generated by a source 118 (e.g., a laser) through a coupler 120 (e.g., a lens, aperture or source 8LEGAL\81463546\914PATENT Attorney Docket No: TUOE. P2012WO / 0065313 i exit pupil). Beam-splitting optics 122 outputs source radiation 116 towards a reflective target 124 (e.g., a plane mirror mounted to the measurement head) and then splits reflected radiation, from the reflective target 124, into the single-pass optical beam 112 and the multipass optical beam 114. Reflective optics 126 retroreflects the multi-pass optical beam 114 through the beam-splitting optics 122 to the reflective target 124.

[0063] Displacement measurement interferometer 100 may further include a beamsplitter 128 that separates (a) source radiation 116 from (b) both the single-pass optical beam 112 and the multi-pass optical beam 114 output to a detector 130. Beamsplitter 128 is for example a 50 / 50 transmission / reflection splitter. Coupler 120 decollimates source radiation 116 so that both single-pass optical beam 112 and multi-pass optical beam 114 are non-collimated, e.g., diverging. Coupler 120 is, for example, a positive or negative lens; although coupler 120 may alternatively be an aperture stop or an exit aperture of source 118 when source 118 generates a diverging or converging beam of source radiation 116. Coupler 12.0 may comprise one or more, or a set of, lenses or reflective components to provide like function, as a matter of design choice. A fold mirror Ml may be used as a matter of design choice, to redirect source radiation 116, as shown in FIG. 3 and FIG. 4. A polarizing filter PF may also be included.

[0064] An important use of di splacement measurement interferometer 100 is to measure relevant displacement between a measurement head and object upon which reflective target 124 is mounted, so that an interference pattern generated at detector 130 may be used to determine displacement of the object. Unlike a traditional interferometer, displacement measurement interferometer 100 advantageously uses decollimated source radiation 116 so that some portion of that beam strikes reflective target 124 and re-enters interferometer 100 despite enlarged rotations of reflective target 124 (and with sufficiently high power to be detected). An aperture 132 adjacent detector 130 ensures, for single-pass optical beam 112, that only its orthogonally reflected radiation from reflective target 124 passes through aperture 132. This eliminates scaling errors since the portion of single-pass optical beam 112 arriving at detector 130 is the portion that orthogonally touched reflective target 124 (that is, normally-incident on, and hence retroreflected by, reflective target 124).[00651 Accordingly, multi-pass optical beam 114 becomes a reference used to interfere with single-pass optical beam 112. Reflective optics 126 reflects part of multi-pass optical beam 114 back to reflective target 124 for a second pass (or more). The second pass by multi-pass optical beam 114 interferes with the first pass by single-pass optical beam 112 to generate a signal proportional to displacement of reflective target 124 (and of the9LEGAJ,\81463546\914PATENT Attorney Docket No: TUOE. P2012WO / OQ65313 underlying measurement head upon which reflective target 124 is mounted). Additional passes and reflections off reflective target 124 by multi-pass optical beam 114 are also possible, including a 3rdpass, a 4thpass, and so on. Second and third passes by multi-pass optical beam 114 are particularly important, as described hereinbelow,

[0066] In embodiments, displacement measurement interferometer 100 includes one or more lenses 134 used to modify multi-pass optical beam 114 into a beam of desired divergence, for example to direct beam 114 onto reflective optics 126. Reflective optics 126 may be a retroreflector,[00671 In an embodiment, beam-splitting optics 122 includes polarizing optics that (a) output source radiation 116 with a first polarization toward reflective target 124, (b) polarize reflected radiation from reflective target 124 with a second polarization and (c) split the reflected radiation, based on polarization, into single-pass optical beam 112 and the multi¬ pass optical beam 114. For example, the first polarization may be elliptical polarization and the second polarization may be linear polarization. Beam-splitting optics 122 may further include a polarizing beamsplitter, and may also include a wave plate 136 as shown (e.g., a quarter-wave plate).

[0068] In an example of operation for the single-pass optical beam 112 in FIG. 3, beam-splitting optics 12.2. are oriented to reflect all incoming radiation from source 118 so that a diverging beam travels through the quarter wave plate 136 to convert the diverging beam to an elliptical polarization state, which then strikes reflective target 124.Electromagnetic radiation reflected from reflective target 124 again passes through wave plate 136, converting the elliptical polarization state to a linear polarization state, and reflects off beamsplitter 128 to aperture 132, such that a small portion of that reflected radiation continues to detector 130. The quarter wave plate 136 is aligned, in this example, such that after two passes the polarization state has rotated more than 90 degrees (or less than 90 degrees).

[0069] Continuing with the example but for operation of multi-pass optical beam 114 in FIG. 4, part of the diverging beam that, travels to the reflective target 124 on the first pass is not reflected towards detector 130, but instead passes straight through beamsplitter 128 and beam-splitting optics 122. This effectively originates multi-pass optical beam 114. From beam-splitting optics 122, multi-pass optical beam 114 travels left toward lens 134, which focusses multi-pass optical beam 114 onto reflective optics 12.6 positioned at the focal point of lens 134. This arrangement retroreflects multi-pass optical beam 114, which travels right again through lens 134, beam-splitting optics 122 and beamsplitter 128 at the same angle as 10LEGAL\81463546\914PATENT Attorney Docket No: TUOE. P2012WO / 0065313 before. Multi-pass optical beam 114 travels through quarter wave plate 136 and converts to elliptical polarization, and travels to reflective target 124. After reflecting off of reflective target 124, multi-pass optical beam 114 travels left again through quarter wave plate 136, which converts multi-pass optical beam 114 to a linear polarization state and then to beamsplitter 128 so that some radiation reflects down toward to aperture 132. Part of multi¬ pass optical beam 114 passes through aperture 132. Single-pass optical beam 112 and multi¬ pass optical beam 114 then combine to interfere at detector 130.

[0070] By generating multi-pass optical beam 114 this way, when single-pass optical beam 112 and multi-pass optical beam 114 pass through aperture 132 to detector 130 they are not tilted with respect to each other; though each are tilted relative to the optical axis with an angle equal to the angle of reflective target 124. Since the tilt of beams 112 and 114 are equal, contrast at detector 130 is not diminished. Experimentation has shown that this configuration creates high detected contrast when the beam waist, aperture 132, and nodal point of reflective optics 126 are collocated, FIG, 5 and FIG, 6 illustrate a displacement measurement interferometer 500, which is an example of interferometer 100 in which reflective optics 126 and lens 134 form a cat’s eye retroreflector. Additionally, FIG. 5 shows reflective target 124 in a first position while FIG. 6 shows reflective target 124 in a different position, thus illustrating how single-pass optical beam 112 varies with position and rotation of target 124. Similarly FIG. 7 and FIG. 8 show a primary ray path within single-pass optical beam 112 for those two positions. As shown in these figures, a lens 135 may be used to decollimate (diverge or converge) source radiation 116.

[0071] FIG. 9 and FIG. 10 illustrate displacement measurement interferometer 500 with the specific configuration that reflective optics 126 and lens 134 form a cat’s eye retroreflector. Additionally, FIG. 9 show's reflective target 12.4 in a first position while FIG. 10 show's reflective target 124 in a different position, thus illustrating how multi-pass optical beam 114 varies with position. Similarly FIG. 11 and FIG. 12 show' a primary ray path within multi-pass optical beam 114 for those two positions.

[0072] FIG. 13 is a flowchart illustrating a method 1300 for measuring displacement of an object. Method 1300 includes at least one of steps 1302, 1304, and 1306 and may be implemented by interferometer 10 (FIG. 1) or interferometer 100 (FIGs. 3, 4). In step 1302, non-collimated radiation is reflected off a surface of the object. In an example of step 1302, and shown in FIG. 1, and after non-collimated radiation 16 passes through beamspliter 28, beamsplitter optics 22, wave plate 36 and aperture 32, non-collimated source radiation 16 is directed to reflect off of reflective target 124. In another example of step 1302, single-pass 11LEGAL\81463546\914PATENT Attorney Docket No: TUOE. P2012WO / 0065313 optical beam 112 is directed to reflect off of reflective target 124 of interferometer 100 (FIG.4).[0073 j In step 1304, reflected radiation from the surface of the object is split into a single-pass optical beam and a multi-pass optical beam. In an example of step 1304, and shown in FIG. 1, radiation reflected by target 24 passes through aperture stop 32, through wave plate 36, and enters beamsplitter optics 22. There, at beamsplitter optics 22, single-pass optical beam 12. reflects down through beamsplitter 28 and reflects to detector 30. At the same time, as shown in FIG. 2, multi-pass optical beam 14 passes through beamspliter optics 22, to lens 34 and reflective optics 26. In another example of step 1304, and reviewing FIG. 3, reflected radiation off of target 124 passes through wave plate 136 and enters beamspliter 128. There, single-pass optical beam 112 reflects to and through aperture 132 to reach detector 130. At the same time, and shown in FIG. 4, multi-pass optical beam 114 passes through beamsplitter 128, beamsplitter optics 122 and lens 134 to retroreflect off of reflective optics 12.6 and travel back to reflective target 12.4,[00741 In step 1306, the multi-pass optical beam is retroreflected to the surface. In an example of step 1306, multi-pass optical beam 14 retroreflects off of reflective optics 26 and travels back through lens 34, beamsplitter optics 22, wave plate 36, aperture 32, and finally to reflective target 2.4, as shown in FIG. 2. In another example of step 1306, multi-pass optical beam 114 retroreflects off of reflective optics 126 and travels back through lens 134, beamsplitter optics 12.2, wave plate 136, aperture 132, and finally to reflective target 124, as shown in FIG. 4.

[0075] Accordingly, it should be clear that the multi-pass optical beam touches the reflective target at least twice; but it also continues to travel back and forth between the reflective target and the retroreflector to touch reflective target additional times, which may be used depending on measurement goals. Thus, as shown in FIG, 2 and FIG, 4, step 1306 includes the step of reflecting multi-pass optical beam 14, 114 off the surface of reflective target 24, 124 respectively.

[0076] In step 1308, displacement of the object is determined from an interference signal between the single-pass optical beam and the multi-pass optical beam after the single¬ pass optical beam reflects off the surface of the object and after the multi-pass optical beam reflects off the surface of the object at least twice. In an example of step 1308, and shown in FIG. I and FIG. 2, single-pass optical beam 12 and multi-pass optical beam 14 combine at detector 30. In another example of step 1308, and viewing FIG. 3 and FIG. 4, single-pass optical beam 112 and multi-pass optical beam 114 combine at detector 130.12LEGAL\81463546\914PATENT Attorney Docket No: TUOE. P2012WO / 0065313

[0077] Step 1304 may include spliting based on polarization. Tins step may further include rotating polarization of the multi-pass optical beam. Step 1302 may include the step of diverging source radiation, such as by use of a coupler 20 or 120.In step 1306, the method may further include retroreflecting the multi-pass optical beam by focusing the multi-pass optical beam to a reflective optical element.

[0078] Before determining displacement in step 1308, the method may include (a) reflecting the multi-pass optical beam off the surface and outputting the single-pass optical beam and the multi-pass optical beam to the detector with (a) comparably equal optical powers, (b) negligible tilt relative to each other and (c) a propagation angle either proportional to, or equal to, twice the tilt angle of the surface. ’‘Comparably” may for example mean having optical powers that ratio near 1 without degrading interference contrast needed for desired displacement measurement accuracy. ‘" Negligible” may for example mean relative angular deviation between interfering beams is sufficiently small so that resulting fringe spacing exceeds the detector aperture and prevents significant contrast loss,

[0079] Again before step 1308, the method may include stopping down the single¬ pass optical beam and the multi-pass optical beam.Additional Detailed Overview of DMI Embodiments

[0080] As noted above, DMI embodiments disclosed herein may use a diverging beam incident on a target mirror mounted on an object affixed to a measurement head (MH). Iliis provides a larger angle of acceptance for displacement measurement. We have seen for example a measurement stroke of 0.5 m (x) and ±20 mrad (0), using an eye-safe laser source, with a periodic error of similar magnitude to conventional DMIs, which is at least below 4 nm. Detailed techniques that enable diverging beam DMIs are discussed below, with two optical layouts presented and compared based on optical efficiency and inherent periodic error. A model based on Gaussian beam theory is used to characterize modeling and to optimize an optical layout. Evaluation is based on detected contrast, signal-to-noise ratio and periodic error. A prototype achieved an increase in the angle acceptance of more than an order of magnitude, from 0.5 to 16 mrad, compared to a conventional DMI.

[0081] Traditional approaches for increasing tire angle acceptance of con ventional DMIs, either severely limit movements orthogonal to the direction of propagation of the measurement beam, or increase the size of the optics. A diverging beam DMI described herein can be used to increase the angle acceptance while allowing transverse movement, all within a similar form factor.LEGAL\81463546\914PATENT Attorney Docket No: TUOE. P2012WO / 00653131

[0082] FIG. 14 shows a comparison between a single-pass DMI, a double-pass DMT, a beam steering DMI and the diverging beam DMI for a tilted measurement mirror (MM). The single-pass DMI introduces a wave front tilt and a beam walk-off of the measurement beam with respect to the reference beam [5], The double-pass DMI will only introduce a beam walk-off of the measurement beam with respect to the reference beam [5], while the beam steering concept doesn’t introduce any wavefront tilt or beam walk-off [4]. This is accomplished by steering the beam before entry into the MH to give the reference beam and the measurement beam the same angle.

[0083] In FIG. 14, rotation of the planar stage leads to: both beam w'aik-off and wavefront tilt for a single pass DMI, beam walk-off and no wavefront tilt for a double pass DMI, no beam walk-off or wavefront tilt is present for a beam steering DMI and the diverging beam DMI presented in embodiments.

[0084] The diverging beam DMI shows that, when the tilt angle of the MM is smaller than the divergence angle of the DMI, there is a ray orthogonal to the MM that traces back to the optical axis at the beam waist. However, simply increasing the divergence of the beam will not lead to desirable results as the first pass will have wavefront tilt of the same angle as the MM with respect to a static non-tilted reference beam, as well as a difference in radius of curvature. These effects lower the interference contrast on the detector. To be able to implement a diverging beam to increase the angle acceptance of a DMI these challenges have to be solved.

[0085] The detected contrast is decreased when more than one interference fringe falls on the detector area, even though the fringe contrast remains the same. When two beams interfere at an angle, an interference fringe pattern emerges. This effect is visualized in FIG. 15, in which Interference fringe pattern introduced by a wavefront tilt of themeasurement beam with respect to the reference beam with periodicity A = leads to a decrease of detected contrast when A < Ddet. Hie distance between two fringes decreases when the angle increases, leading to more fringes on the detector and a decrease in detected contrast.

[0086] To minimize wavefront tilt between the two interfering beams, a second pass to the mirror with the same angle as the first pass may be used instead of a static, non-tilted, reference beam. The phase change of the first pass to the MM is proportional to twice the displacement of the MM (A0Xoc 2Az), while the phase change of the second pass isLEGAL\81463546\914PATENT Attorney Docket No: TUOE. P2012WO / 0065313 proportional to four times this displacement (A02K4Az). Interference between these two beams will result in a phase change corresponding to twice the displacement (A0 x 2 Az),

[0087] To return the second pass to the MM at the same angle as the first pass, a retroreflective component may be used. Especially suited is a Cat’s Eye Retroreflector (CER), which includes a lens and a mirror placed at the back focal plane of the lens. The CER does not introduce beam walk-off between the first and second pass for the orthogonal ray, when the nodal point (front focal point) of the CER is placed on the optical axis at the beam waist. Using a CER ensures that the ray orthogonal to the MM will be reflected with the same angle as the first pass beam, leading to no first principle wavefront tilt and beam walk-off of the two passes with respect to each other and minimal scaling errors.

[0088] Tire detected contrast can be decreased by the fringe pattern that results from a difference in radius of curvature between the two interfering beams, which is present as a result of the divergence. This interference pattern is created as a result of the phase difference between the two beams. When the phase difference between the two beams is equal to ~A + nA, with n E and A the wavelength in nm, the beams will be in phase and will create constructive interference, while the beams create destructive interference when the phase difference is equal to A + nA. The interference results in a fringe pattern of alternating bright and dark rings. When multiple fringes fall on the detector the detected contrast is decreased. However, by limiting the size of the aperture of the detector to be smaller than the radius of the first fringe, the detected contrast can be increased. This corresponds to sampling the part of the interference pattern where the phase difference between the two wavefronts is significantly smaller than A / 4.

[0089] FIG. 16 shows the resulting interference pattern when two beams with different radii of curvature interfere and defines parameters to geometrically approximate the relative phase difference between the two wavefronts at the radius of an aperture (Aw = wq — w2) with:Wj = Rf ~ Ricos(arcsm(~^-'))R;Where rAP. is the aperture size m m and R^ is the radius of curvature in m for pass i. In FIG. 16, an interference pattern created by a difference in radius of curvature between the reference beam and measurement beam leads to a decrease in detected contrast when rAP.: ^fringe-15LEGAL\81463546\914PATENT Attorney Docket No: TUOE. P2012WO / 0065313

[0090] The relative phase difference becomes larger tor smaller optical path lengths. The aperture size should be chosen, such that the relative phase difference remains smaller than 2 / 4 for the entire measurement stroke. FIG. 17 shows the relative phase difference between the first and second pass (Aw) for different optical path lengths (7?2= 2 / ?3) and different aperture sizes. For an aperture radius of 0.5 mm the relative phase difference remains below 2 / 4 over the measurement stroke.

[0091] It is clear that there are aperture sizes for which the relative phase difference is sufficiently small over the entire measurement stroke. Decreasing the aperture size, unfortunately also decreases the intensity that falls on the detector lowering the Signal-to-noise ratio (SNR). Thus, there is an engineering optimum in aperture size for contrast and SNR. It can be concluded that, in embodiments, an aperture size of 0,5 mm is the largest radius that leads to a difference in curvature of at most 2 / 4 over the entire m easurement stroke for the case where R2~ 2R, In any event, optimum aperture size depends on desired displacement range and wavelength and is not a fixed value.

[0092] By replacing the static reference beam with a second pass, it is no longer possible to use a heterodyne laser source that separates the measurement and reference beam based on polarization, and thus limits the architecture to homodyne interferometry.By introducing an aperture and a diverging beam the total power that falls through the aperture onto the detector and, thus, the optical efficiency of this design is decreased compared to a conventional DM1. The second pass has a lower detected power as it traverses a longer optical path and diverges even more than the first pass. For optimal contrast at the detector the beam power of the reference and measurement beam should be equal. The splitting ratio between the first and second pass may be such that a power ratio (PJ / PJ) on the detector is close to 1 or equals 1 to maximize the detected contrast.

[0093] New coherent laser diode sources entering the market offer increased power, and more precise modulation capabilities. These advancements are particularly beneficial for a diverging beam DMI that limits the design to homodyne interferometry, where directional measurements rely on either quadrature detection or the modulation of the source. The increase in available source power will lead to an increase in SNR and can account for the lower optical efficiency that is inherent in the diverging beam DMI.

[0094] We now present two layouts that increase the angle acceptance of a conventional DMI by means of a diverging beam. Both layouts increase contrast on tire detector using a C-ER as the retroreflective element to create a second pass (reference beam)LEGAL.\81463546\914PATENT Attorney Docket No: TUOE. P2012WO / 0065313 i with the same angle as the first pass (measurement beam) and using an aperture to limit the relative spatial phase difference at the detector between the two passes. Whereas in a conventional DM1 a single beamsplitter may be used to split the source beam into a reference and measurement beam, the diverging beam DMI uses an additional beamsplitter and the polarization of the laser beam to create a second (or higher order) pass to the MM.

[0095] lire two layouts differ m the placement of the additional beamsplitter and in how the passes are directed to the detector. The first layout has the additional beamsplitter placed before the polarizing beamsplitter and reflects a portion of the light onto the detector based on the polarization direction of the beam, which is manipulated by means of a Quarter Wave Plate (QWP). lire second layout uses a non-polarizing beamsplitter after the polarizing beamsplitter to reflect a portion of the light onto the detector. In both layouts the rest of the light (orthogonal ray) is reflected by the CER for a second pass at the same angle. The best perfonning layout is selected by comparing the optical efficiency and inherent periodic error.

[0096] Layout 1: Polarization Splitting. The first layout involves polarization splitting and makes use of a QWP aligned with its fast axis at an angle to the horizontal axis. After returning through the QWP, the polarization direction of the beam has been rotated. The polarizing beam splitter reflects the horizontally polarized portion of the beam onto the detector. The vertically polarized portion is transmitted into the CER and then reflected for a second pass. After reflecting on the MM, the second pass is returned into the polarizing beam splitter at the polarization angle imposed by the QWP, which will again reflect tire horizontally polarized portion of the beam into the detector, leading to the interferometric signal. However, the remaining vertically polarized portion of the beam will be transmitted into the CER, leading to higher order passes. This layout is henceforth called the Polarization Splitting (PS) layout and is depicted in FIG. 18. The layout of FIG. 18 reflects a portion of the first pass onto the detector by means of a polarizing beamsplitter and a quarter wave plate with its fast axis at an angle to the horizontal axis. A cat’s eye retroreflector returns the second pass at the same angle as the first pass.

[0097] Higher order passes are undesirable as they will manifest as periodic errors on the interferometric signal, even though they do contain information about the displacement of the MM. In particular they will manifest as frequencies at multiples of the measured frequency on the signal, where the phase and amplitude is dependent on the rotation and distance of the MM. Atypical signal for 3 passes is displayed in the equations below [6],17LEGAL\81463546\914PATENT Attorney Docket No: TUOE. P2012WO / 0065313 KZ) ■■■' + \E22 I + 1^33 I +2 |£12|COS( - ™) -4- A2TTAZ-7-J2 |E23|cos(~“--y-^) +2JTAZ2 |E13|C0s(____^13)AIn the above equations, / is the irradiance field in W / m2, | is the DC magnitude of the electric field in N / C,is the AC magnitude of the electric field between field i and j in N / C and Az{;- is the difference in optical path length between the field i and j. Note that the optical path length of Az13~ 2 zi2= 2Az23. The third pass will, therefore, result in a sinusoidal signal with twice the frequency as the interference signal of the first and second passes.

[0098] FIG. 19 show's optical efficiency of the multiple beam passes assuming a collimated beam and no power loss due to the aperture in the polarization splitting layout. Above a transmittance ratio of 0.62 the power of the second pass and first pass may be matched for various divergence angles.[00991 A desirable interferometer will have a power ratio of 1 between the reference and measurement beam. In the case of a diverging beam, however, the second pass has had a longer optical path to diverge its power over a larger beam width. Inherently it is beneficial for a diverging beam DMI to ensure that the second pass has a higher undiverged power than the first pass to ensure that the powers that fall on the aperture in front of the detector for the two passes are equal. The relation between the optical efficiency of each of the passes and the transmittance ratio (based on the QWP angle) is visualized in FIG. 19 without taking the divergence of the passes into account yet. For transmittance ratios higher than approximately 0.62. the power of tire second pass is higher than that of the first pass. The exact optimal transmittance ratio, at which the power ratio is 1, is dependent on the amount of divergence of the laser and may be optimized based on the other design parameters.

[0100] lire second layout involves non-polarization splitting and makes use of a non¬ polarizing beamsplitter (BS) to reflect a portion of the first pass onto the detector. The QWP in this layout is aligned with its fast axis at 45° to the horizontal axis, such that the polarization direction of the returning beam is rotated 90". The polarizing beamsplitter (PBS) transmits all the light that is not reflected in the BS into the CER, which reflects the second pass onto the MM. Tire BS reflects a portion of the second pass onto the detector, creating the interferometric signal. The remaining portion will get reflected back to the source and thus 18LEGAL\81463546\914PATENT Attorney Docket No: TUOE. P2012WO / 0065313 i will not interfere on the detector. Since the first and second passes are now orthogonally polarized with respect to each other, a polarizing filter is placed before the detector.

[0101] This layout will henceforth be called the Non-Polarization Splitting (NPS) layout and is presented in FIG. 20. This layout uses a quarter waveplate at 45° to create a second pass to the measurement mirror. A non-polarizing beamsplitter is used to reflect a portion of the first and second passes onto the detector. A cat’s eye retroreflector returns the second pass at the same angle as the first pass. A polarizer is placed in front of the detector to create interference between the two orthogonally polarized passes.

[0102] As previously stated, it is beneficial for an interferometer when the power of the two passes are equal. However, the second pass has traversed a longer optical path and has diverged more than the first pass. 'The first pass will thus have a higher power. By rotating the polarization filter, the power between the first and second pass may be matched to ensure a power ratio of 1. The remaining power is dissipated in the polarizer, so this method can increase the fringe contrast at the cost of the optical efficiency of the layout. FIG. 21 shows the resulting optical efficiency at the baseline for a polarizer angle of 45° and at a polarizer angle of 17.5°, which is a realistic polarizer angle for a diverged beam to match the power of the two passes.

[0103] FIG. 21 shows an example optical efficiency of the two beam passes assuming a collimated beam and no power loss due to the aperture and a polarizer at 45° & 17.5° in an embodiment of the non -polarization splitting layout. For the polarizer angle at 17.5° and a transmittance ratio above 0.6, the power of the second pass and first pass may be matched for various divergence angles.

[0104] The PS layout has a bigger optical efficiency, which will result in a higher SNR compared to the NPS layout. However, the PS layout has a larger inherent periodic error as a result of the higher order passes. Tire periodic errors may be compensated by real time compensation algorithms [7], but these are not directly able to compensate with non-constant velocity or amplitude varying periodic errors that will be present in the PS layout. The NPS layout, however, has no first principle periodic errors. For DMIs a lower SNR is more easily solved by, for example, increasing the source power or decreasing noise sources present, in the electronics than it is to account and compensate for periodic errors. Thus, subsequent sections investigate the NPS layout in more detail.

[0105] A model based on Gaussian beam theory? was created to characterize the performance of the diverging beam DMI with available optics and as a design tool toLEGAL\81463546\914PATENT Attorney Docket No: TUOE. P2012WO / 0065313 optimize parameters for future designs. Herein, the following Key Performance Indicators (KPIs) are used to evaluate the performance of a specific design: Detected contrast and Signal-to-noise ratio (SNR). The KPIs of the NPS layout are calculated by using the electric field of each of the passes to calculate the power on the detector within the range of distances and rotations of the MM. This may be done by taking the definition of the electric field of a Gaussian beam in Cartesian coordinates as displayed in the expression below' for E (%, y, z) [8], Cartesian coordinates are used to accommodate integrating over the aperture, which is a circle at an apparent offset to the beam center. 'The z-direction has been assumed to coincide with the optical axis of the system, hence the electric field E is given by:E(x,y,z) ~,•2.2| £01 exp.v / wx(z)wy(z) / 4>0,y(Z).kx'2,ky'2“P l"12 -+

[0106] In which i is the imaginary unit, k is the wavenumber which is given by: k ™ in m~1and £0is the peak magnitude of the electric field in Fm"1which can be defined by the power of the laser beam as |Fnl = 2 i - ■■■ —. with e being the permittivity of thepropagation medium in A2s4kg~lm~3(air in this case), v the speed of light in air in ms" 1, Pothe source power in W and wO x / >, is the beam waist diameter in m. Other parameters defining the electric field in £(%, y, z) are the beam radii, which are defined as follows: / 1(Z - Zo.^ / y) \Wx / y(z) = WOiX / y1 +\nwL / y )where Zo.x / yare iebeam waist offsets with respect to the origin. E x, y, z) is also dependent on the Gouy phase ( o,x / y)-- which is given as:The beam radius (Rx / y) is given as:«x / y(z) = U - zox)

[0107] When assuming no loss of power in the optic components, the electric field at the detector will include the sum of the electric field of the first pass ( and that of theLEGAL\81463546\914PATENT Attorney Docket No: TUOE. P2012WO / 0065313 second pass (E'2) and can be defined as a function of the beamsplitter transmission ratio (^BS), the diverging beam electric field of the laser Elaser(x, y, z)) and the polarizer angle 9polEi (x, y, z) = Elaser(% - %3, y - y\, z, )£’2 (%, y, z) = Elaser(x - x2, y - y2, z2)^Bs]cos($pof)EDe£(x,y,z) = EiC^y.z) + Ez(x,y,z)

[0108] Tilt of the mirror causes the beam axis to shift relative to the aperture center, leading to an apparent detector offset. Hie detector offsets xnand ynfor pass n are thus dependent on the rotation angle (a) and distance (z) of the MM xn= zsin(2a)). For larger MM angles and larger distances, the detector will have a larger apparent offset with respect to the beam axis. The detector offset with respect to the Gaussian beam can be seen in FIG. 22 for a MM distance of 0.3 m and a rotation of 10 mrad.

[0109] FIG. 22 shows elliptical beam intensity profiles at z = 0.3 m for a measurement mirror angle of a = 10 mrad. The detector position is offset with respect to the beam axis by the rotation and distance of the measurement mirror.

[0110] FIG. 23 shows power on the detector and tire power ratio for the two passes. The power ratio of both passes is Pined to 1 with the polarizer angle for the smallest distances to the measurement mirror, where the contrast is lowest.

[0111] FIG. 24 shows modelled key performance indicators: detected contrast and Signal -to-noise ratio show acceptable performance for a photodiode (Thorlabs SM05PD1A) and variable high speed current amplifier (Femto DHPCA-100) combination for a sample frequency (fs) of 20 kHz over the entire measurement stroke.

[0112] FIG. 25 is a photograph of an example non-polarization splitting layout of the diverging beam DMI including a stabilized HeNe laser source (Thorlabs HRS015B), an optical isolator (Faraday rotator and two polarizing filters), a folding mirror (Ml), a plano¬ convex cylindrical lens with f = 50 mm (LI), a polarizing beam splitter (PBS), a 30:70 non¬ polarizing beam splitter (BS), a quarter wave plate (QWP), a 2 meh measurement mirror (MM) mounted on a linear piezo stepper stage (PI P-753.3CD), a Cat’s Eye Retrorelector (CER) including a plano-convex lens with / = 100 mm and a mirror, a detector assembly (DET) including a polarizing filter (PF), an aperture rAP~ 0.5 mm (AP) and a photodiode (Thorlabs SM05PD1 A), a switchable transimpedance amplifier (AMP)(Femto DHPCA-100) and an analogue to digital converter (ADC)(Picoscope 5442D).21LEGAL\81463546\914PATENT Attorney Docket No: TUOE. P2012WO / 0065313 i

[0113] With all the parameters for the electric field defined, the irradiance can be; £’; y.. y'j i 2calculated with the following relation / (%, y, z)=inand the waveimpedance riair= « 377!npp Here is the permeability of the medium in-vcadrVsA~1m~. Finally, the irradiance may be used to calculate the power that falls through the aperture on the detector by taking the integral over the aperture area within the irradiance field at a certain z-coordinate with the following equation:f JrAP~x2frAPP(z) = I \ - I / (x,y, z)dxdyJ- / 'AP-*2‘' ~rAPWhen the beam waist, nodal point of the CER and aperture in front of the detector are not perfectly aligned, an additional detector offset will be present for the second pass, which leads to wavefront tilt and / or beam walk-off between the two passes. For now, this model assumes perfect alignment and paraxiallity of all the optical components. With the above expresso for P(z), the resulting power for any electric field that falls on the detector through tlie aperture can be calculated. The beam waist in x and y-direction (w0 x / y) dictates the divergence of the beam and is set to create a divergence of 20 mrad in x-direction and 1,3 mrad in y-direction, which is a common divergence angle tor typical HeNe laser sources used in interferometry. FIG. 22 displays the irradiance field in x- and y-direction and the aperture integration bounds. This displays that only a part of the Gaussian beam falls on the detector to minimize the effects of the difference in radius of curvature.Table 1: Input parameters of the simulation corresponding to the optics used in FIG. 25. Parameter Meaning Value Unit2 Wavelength 633 nmPoSource power 1.4 mWdiv.x Divergence angle 20 mradrjssTransmittance factor 0.70poiPolarizer angle 17.5rAPAperture radius 0.5 mmfsSample freq. 20 kHz

[0114] The model is used to simulate the power of both passes Pj), the power ratio p(-p), detected con trast and SNR at a discrete number of rotations and distances within the measurement stroke range. The simulation inputs are displayed in Table 1. The wavelengthLEGAL\81463546\914PATENT Attorney Docket No: TUOE. P2012WO / 00653131 and source power are determined by the available stabilized HeNe laser source. The polarizer angle is chosen by optimizing for a power ratio of 1 at distances of d — 0.1 m. The aperture radius is based on the calculations as described herein. The sampling frequency is typical for metrology inputs into the control loop of motion systems and gives the bandwidth used to calculate the noise floor of the amplifier and photodiode combination. The Noise Equivalent Power (NEP) of the photodiode gives the following noise current Iphotodiode ~ NEP X)^ fs& the input noise current density Jnoise) of the amplifier gives the following amplifier noise current IAMP— Jnoise Ts- Note that the input noise current density is dependent on the selected transimpedance gam.

[0115] The simulation outcomes and KPIs are displayed in FIGs. 23 and 24. The optimal and minimal positions for each of the model outcomes are summarized in Table 2. Note that the detected contrast is expected to be in the range of [0.74 ■■■■ 0.98] and that a contrast of 0.5 is deemed sufficient for phase retrieval that is required to obtain displacement information. Tills leads to the conclusion that an aperture radius of rAP= 0.5 mm leads to a sufficient contrast on the detector.Table 2: Summary of the simulation results at the optimum and minimum positions along the measurement stroke.Parameter Value Unit Distance [m] Rotation [mrad] max 1200 nW 0.1 01p mi.n 1.8 nW 0.5 ± 20P. 1 •■■■ 0.1

[0020] R,JmaxPz 0.76 - 0.5

[0020] pminr ^max 0.98 - 0.5

[0020] 0.74 - 0.1

[0020] SNRmax1692 ■■■■ 0.1 0SN 79 - 0.5 ± 20

[0116] Although the small aperture radius leads to sufficient contrast, the low power that falls through the aperture on the detector at large distances and rotations is noteworthy. Ilns low power leads to a low SNR. While a SNR below 100 is relatively low for a DM1, for the prototype it is high enough to be able to proof that the angle acceptance is increased. Also note the large difference in power between the maximum and minimum power, which necessitates the use of a gain-switchable transimpedance amplifier. Switching the gain to 23LEGAJ,\81463546\914PATENT Attorney Docket No: TUOE. P2012WO / 0065313 i maximize the measured voltage will lead to a jump in the SNR as the signal is amplified more than the noise is increased. A continuously variable gain would be beneficial to maximize the peak measured voltage and thus improve the SNR.Oil 7] By choosing better parameters for a final design, the SNR can be increased to decrease the measurement uncertainty of the diverging beam DMI further, while keeping the same contrast on the detector. By for example, switching to a wavelength of A = 1550 nm, the aperture radius can be increased, since the difference in radius of curvature will decrease. Effectively more power will fall on the detector. Another improvement is increasing the source power to Po= 80 mW, which is still deemed eye safe in class 3B at 1550 nm according to the NEN-EN-IEC 60825 standard and will improve the SNR by about 80 fold as the SNR is not shot noise dominated. Tire model may be used as a design tool to simulate the performance increases for changes in the design parameters.

[0118] Even for the suboptimal input parameters given in Table 1, the results of the simulation predict satisfactory performance of the prototype, with a detected contrast bigger than 0.5 and a SNR above 20 over the entire measurement stroke, the prototype will meet the requirements specified to prove increased angle acceptance.

[0119] The outcome of the Gaussian beam model demonstrates that the inputs as given in Table 1 will highly likely lead to a functioning diverging beam DMI. Therefore, a prototype was built as shown in FIG. 25.

[0120] The MM is placed at a number of distances from the measurement head in the range of d = 0.1 m to d = 0.5 m and rotated to a number of rotations within the range of 0 = +20 mrad. To create at least two fringes on the detector for homodyne detection in DMIs the target mirror should move at least one wavelength. At each of the discrete measurement points the mirror is displaced by two wavelengths forward with a constant velocity, before moving back in a saw-tooth profile at a frequency of fdrive ~ 1-5 wavelengths / s with smoothed transitions to minimize exciting dynamics while measuring the incident power on the detector.

[0121] FIG. 26 shows key performance indicators of the interferometer prototype for the parameters given in Table 1. The prototype performs above the acceptable levels within the rotation range of + 16 mrad.

[0122] A typical measured response can be seen in FIG. 27. FIG. 27 shows the measured, low-pass filtered= 20 hz) and noise Vmeas— VLP) voltages at a distance of z ~ 0.3 m and rotation of 9 = 10 mrad for a constant velocity movement of « 2 wavelengthsLEGAL\81463546\914PATENT Attorney Docket No: TUOE. P2012WO / 0065313 i of the measurement mirror. This signal is filtered using a 4th order low-pass filter at fLP~ 20 Hz, which is higher than the nyquist frequency fnyc. = 1 Ofdrtve)-difference between the measured and low-pass filtered voltage may be used to determine the noise present in the measured signal. To limit the contribution of motion stage dynamics to the SNR, additionally, the eigenfrequency of the motion stage is determined and a narrow' band-stop filter is implemented on the measured voltage at this frequency.[01231 To find the contrast on the detector the maximum and minimum voltage of the filtered signal are used in the following equation [5]:„ _i / max —i / mt ■n^fringe “lmax ’sminWith / the detected power in W. The SNR is given by the Root Mean Square (RMS) amplitude of the measured voltage divided by the RMS amplitude of the noise (SNR = A- To determine the periodic error, a 4th order sum of sines is fitted to the filtered signal, noisewhich is of the following form:2nz 4nzVfit— AiSin (—; — F 02) + fi2sln(“~5 — r $2) +A A6TCZ STCZA3sin (-— + 03) + A4sin (—7— + 04)a a(’A- 4-A-j +Aj.\- 1 & the periodic errordisplacement can be found with: Aze=

[0124] Additionally, the displacement noise expressed in nm can be obtained by converting the measured RMS voltage noise to an equivalent displacement noise using the slope of the interferometric si "g"nal. ' '. Ax / in n -■■m■■■■:I'noise.rms&no ise,rms (AVII Ax IHere Vnoise,rms is the RMS noise voltage in V, and ~is the voltage sensitivity of the interfer¬ ometer with respect to displacement. By using the phase relation to displacement of this25LEGAL\81463546\914PATENT Attorney Docket No: TUOE. P2012WO / 0665313 i interferometer layout (A< > oc 2Az), the slope can be expressed using the peak-to-peak signal voltage Vppand the wavelength z in the following way:^noise,rms2nV

[0125] The previously discussed KPIs are calculated for the measured signal at each of the discrete measurement points. By plotting the KPIs over the entire measurement grid the performance of the diverging beam DMI prototype can be visualized. This is done in FIG. 26 for the contrast and SNR and in FIG. 28 for the measurement noise and periodic error. FIG. 28 shows measurement noise and periodic error of the interferometer prototype for the parameters given in Table 1. The prototype performs above the acceptable levels within the rotation range of + 16 mrad. A numerical representation of these results is given in Table 3.Table 3: Summary of the prototype results at the optimum and minimum positions along the measurement stroke.Parameter Value Unit Distance [m Rotation [mrad] (‘max 0-83 ■■■■ 0.3 -2, [2 - 10]0.28 - 0.1 ±20SNR:!:y:iX50 - 0.1 0S^Rmin1- 0.5 ±20p '■'noise,mi ■.n G G nm 0.1 07 fi nm 0.5 -20ecp er t -o a,. m i ■n 095 nm 0.1, 0.2, 0.4 2, -8, -4p ^peri ■od j, max 7 * nm 0.4 -20

[0126] It is noticeable that the detected contrast is quite flat over a large range of angles. A decrease in detected contrast can be noticed at small measurement distances, which can also be observed in the Gaussian beam model in FIG. 24 and is expected to be a result of the difference in radius of curvature as discussed herein. However, the contrast at the detector also seems to decrease for larger rotations, this effect is not present in the model and can be explained by the assumption of perfect alignment of the beam waist, CER nodal point and aperture in the model. In the prototype small misalignments are bound to be present and these will lead to wavefront tilt and / or beam walk-off between the first and second passes depending on the rotation and distance of the MM.LEGAL\81463546\914PATENT Attorney Docket No: TUOE. P2012WO / 0065313

[0127] Furthermore, it cars be noticed that the SNR decreases for larger angles and larger distances and jumps up whenever the transimpedance gain is switched to a higher amplification, as the signal is then set closer to the limit of the detected voltage VDet= 1 V. Iliis effect is also predicted in the Gaussian beam model and can be seen in FIG. 24 as well. However, the magnitude of the SNR is significantly lower in the prototype compared to the simulation, which is expected to be a result of other sources of noise, which have not been taken into account in the Gaussian beam model. These noise sources include dynamic vibrations in the CER, unmodelled electronic noise / non-optimized electronics in the setup or environmental drift, such as in pressure, temperature and humidity'.[01281 When evaluating FIG. 28, it can be seen that within the range of rotations [-16 16] mrad the measurement noise remains below 6 nm. For this prototype this is an acceptable range, however, for many' interferometric applications the measurement noise should be lower. Whenever the SNR increases either the RAIS noise voltage wili decrease or the slope of the measured voltage over the stage displacement is increased, leading to lower measurement noise. For next versions it is thus beneficial if the SNR is improved to decrease the measurement noise.

[0129] In addition, the periodic error remains at a magnitude comparable to conventional DMls over the entire measurement stroke, increasing slightly for larger angles. This could either be a physical effect in the DMI, where ghost reflections increase m magnitude for larger angles of the beam. Or it can be atributed to the fact that the SNR decreases at those rotations as well, which increases the confidence interval of the fitting algorithm. By evaluating the periodic error after improving the SNR at larger rotations, it can be determined which effect is dominant.

[0130] Upon comparing the simulation results to the prototype it can be stated that the quantitative behavior of the detected contrast and the qualitative behavior of the SNR of the diverging beam DMI are predicted correctly by the model. The prototype, thus, partly validates the model, which can be used as a design tool to improve the KPIs of the diverging beam DMI for future designs.

[0131] It can be concluded from the results that the diverging beam concept reaches the performance goals for a large portion of the measurement stroke. However, at rotations higher than 16 mrad, the detected contrast is below 0.5, the SNR is below 10 and the periodic error is higher than 4 nm at some distances. Hie diverging beam DMI prototype fulfils the requirements for proving the increased angle acceptance for a measurement stroke of 0.5 m and rotations of up to 16 mrad. This validates that the use of a diverging beam together with 27LEGAL\81463546\914PATENT Attorney Docket No: TUOE. P2012WO / 0065313 the optical layout as discussed herein is a viable technique to significantly increase the angle acceptance of DMIs.

[0132] It can also be concluded from the results that the performance of the diverging beam DMT can be improved by increasing the SNR, which can be done by increasing the laser wavelength to, for example, 1550 nm, which leads to a smaller difference in radius of curvature and allows the size of the aperture to increase while reaching the same contrast. A 1550 nm source is a good candidate due to its availability and power levels up to 80 mW (within class 3B according to NEN-EN-IEC 60825). Tire power that falls on the detector through the aperture will increase exponentially with an increasing aperture radius. However, by increasing the wavelength the interpolation distance within one wavelength increases linearly. On the basis of first principle scaling laws it is expected that the combined result of these effects will lead to an increase in the SNR by two orders of magnitude.

[0133] Increasing the angle acceptance of displacement measuring interferometers (DMIs) is beneficial for increasing the angular range of motion systems, which is useful in, for example, stacking chiplets and can enable the use of plane mirror DMIs for new' applications. While other techniques exist for enlarging the angle acceptance of DMIs, these solutions either require increasing the size of the optics or add complexity and costs with active elements.

[0134] Embodiments disclosed herein thus includes a novel passive method of increasing the angle acceptance of DMIs by means of a diverging laser beam. The effects of wavefront tilt are minimized by measuring the interference between a first and a second pass to the measurement mirror instead of a static non-tilted reference beam. Two optical layouts that incorporate a diverging beam have been introduced. One layout is selected by comparing the optical efficiency and inherent periodic error of the two layouts and worked out in more detail. A model based on Gaussian beam was created to predict the performance of this layout, based on key performance indicators and as a design tool. A prototype of the diverging beam DMI was constructed.

[0135] The prototype reaches exemplary' goals of a detected contrast higher than 0.5, a signal -to-noise ratio of at least 10 and a periodic error of at most.4 nm, while using an eye safe laser source, for a measurement stroke of 0.5 m and a rotation range of ±16 mrad, which is an increase of 32 times the angle acceptance of a conventional DMI. By increasing the wavelength to 1550 nm the signal-to-noise ratio and angle acceptance of the prototype is expected to increase even further. Based on these results it can be concluded that the use of aLEGAL\81463546\914PATENT Attorney Docket No: TUOE. P2012WO / 00653131 diverging beam is a viable technique to increase the angle acceptance of DMIs by at least an order of magnitude.

[0136] The model has been used as a design tool to find design parameters for a diverging beam DMI at a wavelength of A ~ 1550 nm. These are presented in Table 4:Table 4: Modelled best parameters for a diverging beam DMI. Parameter Meaning Best value UnitWavelength 1550 nmP0Source power 80Divergence angle 20 mradies Transmittance factor 0.7pol Polarizer angle 17.5rAP Aperture radius 0.8 mmSample freq. 20 kHz

[0137] The outputs of a simulation of the model as described herein with the inputs as shown in Table 4 are shown m FIGs. 29 and 30. FIG. 29 show's power on the detector and the power ratio for the two passes. The power ratio of both passes is tuned to 1 with the polarizer angle for the smallest distances to the measurement mirror, where the contrast is lowest.

[0138] FIG. 30 shows modelled key performance indicators: detected contrast and Signal-to-noise ratio show acceptable performance for a photodiode (Thorlabs SM05PD1A) and variable high speed current amplifier (Femto DHPCA-100) combination for a sample frequency (fs) of 20 kHz over the entire measurement stroke. A summary of the results is given in Table 5.Table 5: Summary of the simulation results for inputs at optimum and minimum positions along the measurement stroke.Parameter Value Unit Distance [m' Rotation [mrad] - p max 142 μW 0.1 0JP m n 2.4 μW 0.5 ± 201.05 - 0.1

[0020] p-t1maxp20.78 •■■■ 0.5

[0020] r '-•max 0.98 - 0.5

[0020] ' C“•mi ■n 0.71 - 0.1

[0020] SNRmax12893 •■■■ 0.1 0SNRmm1161 - 0.5 ± 2029LEGAL\81463546\914PATENT Attorney Docket No: TUOE. P2012WO / 0065313

[0139] The results show a great improvement in signal -to-noise ratio, while reaching a similar contrast as the results for the Simulation at a wavelength of A = 633 nm. This increase is a result of both the increase in source power and the increase in aperture radius, which is possible as a result of the decrease in radius of curvature at this larger wavelength.

[0140] The outputs of a simulation of the model are now described with the inputs as shown in Table 1 for a slight misalignment of 1 mm in the beam waist, cat’s eye retroreflector and detector positions are shown in 31 and 32. FIG. 31 show's power on the detector and the power ratio for the two passes. Ihe power ratio of both passes is tuned to 1 with the polarizer angle for the smallest distances to the measurement mirror, where the contrast is lowest.

[0141] FIG. 32. shows modelled key performance indicators: detected contrast and Signal-to-noise ratio show acceptable performance for a photodiode (Thorlabs SM05PD1 A) and variable high speed current amplifier (Femto DHPCA-100) combination for a sample frequency ( / j) of 20 kHz over the entire measurement stroke.

[0142] Table 6 is a summary of the simulation results for inputs with a slight misalignment of 1 mm of the beam waist, cat’s eye retroreflector nodal point and the detector positions at the optimum and minimum positions along the measurement stroke.Table 6Parameter Value Unit Distance [mj Rotation [mradj Pmax1100 nW 0.1 0Pmin1-7 nlV 0.5 + 2.0Pz- 1.16 - 0.1 ± 20p Im axP- 0.78 - 0.5

[0020] P J,-mi ■nCmax0.98 - 0.5 0Cmm0.39 — 0.14 20SNR^ax1579 - 0.1 0SNRmin69 •■■■ 0.5 ± 2.0

[0143] The results show that there is a dependency of both the rotation and distance to the MM introduced by the misalignment. This effect is also present in the prototype as described in 5. This is a strong indicator that the alignment of the prototype can be perfected to increase the performance of the diverging beam DMI.LEGAL\81463546\914PATENT Attorney Docket No: TUOE. P2012WO / 00653131

[0144] The model cars also be used to simulate the performance of the polarization splitting layout. The input parameters are chosen to be the same as those for the non¬ polarization layout, which are presented in Table 1. The results of the simulation for the polarization spliting layout are shown in FIGs. 33-36.

[0145] FIG. 33 shows power on the detector for the three passes. FIG. 34 shows power ratio on the detector between the three passes. FIG. 35 shows modelled key performance indicators: detected contrast and Signal -to-noise ratio show acceptable performance for a photodiode (Thorlabs SM05PD1A) and variable high speed current amplifier (Femto DHPCA-100) combination for a sample frequency (fs) of 20 kHz over the entire measurement stroke. FIG. 36 shows the periodic error and periodic error offset caused by the third pass.

[0146] Table 7 summari zes of the simulation results for inputs of Table 1 at the optimum and minimum positions along the measurement stroke.Table 7Parameter Value Unit Distance [m] Rotation [mrad] P„ax6700 nF / 0.1 0Pmin10 nW 0.5 ± 20P21.21 - 0.1 ± 20ft1maxP20.83 - 0.5 0Pi1mt ■ nP30.064 - 0.1 ± 201maxP30.036 - 0.5 0Pi ■P30.053 •■■■ 0.1 + 20p 2 maxP30.044 - 0.5

[0020] P? z 'mt •nCmax0.96 - 0.5 0Cmin0.68 - 0.12 ± 20SNRmax5202 — 0.1 0SNRm(n76 — 0.5 + 20 Periodicerrormin6.9 nm 0.18 + 20 Periodicerrormax8.8 nm 0.5 0LEGAL\81463546\914PATENT Attorney Docket No: TUOE. P2012WO / OQ65313

[0147] The polarization spliting layout has a deal more outputs as there is another modelled pass to investigate. Additionally to the Non-Polarization Splitting (NPS) layout, the beam power for pass 3, power ratios (P3 / Pi & P3 / P2), the periodic error and the periodic error offset are modelled. The main takeaways from these results:• Even though the optical efficiency is higher than the NPS layout, the SNR for large angles is not higher (as a result of the higher order passes influencing the AC- magnitude).• The power ratio between the third and first pass is about 5%, which is a significant portion of the measured signal and will lead to significant periodic errors.® It is more difficult to tune the power ratio to 1, as the shape is dependent on the angle.This leads to a dependency on the angle for the contrast as well.• There is a significant periodic error present as a result of the higher order passes. This periodic error doesn’t only have an AC -component, but also has an offset. This is measured as an additional optical path distance, which is not necessarily undesirable as it manifests as a constant angle offset in the measured phase. It is, however, undesirable for the periodic error offset to change over the measurement stroke, which is the case here. This manifests itself as an additional measured displacement that is not made by the measurement mirror. Both the AC- and DC-component of the periodic errors are significant and will result in higher than acceptable measurement uncertainty'.• The NPS layout will, thus, have a lower measurement uncertainty than the PS layout.

[0148] To model the electric field of a Gaussian beam as a function of the Cartesian coordinates (x,y,z) a couple of defining relations for the Gaussian beam have to be described. A Gaussian beam can be described in full with the beam width, the radius of curvature and the Gouy phase as a function of the z-coordinate. The equations for the beam w idths (in x- and y-di recti on) are given by the following equations.i fA(z - zx)\2[A(z - zy)\2wr(z) = wO y11 + — — 5 — and wv(z) = wO v1 + — — 5 —.V ' y Vnwo,y /

[0149] In the above equations, wO- is the beam width in direction I in [m], A is the wavelength of light in [m] and z.{ is the offset of the beam waist in direction i w.r.t. z = 0 in [m]. When assuming that ivO x& wOj, are equal the laser beam will be circular. If they differ the resulting beam will be elliptical. Figure 24 shows the evolution of the beam width over 32LEGAL\81463546\914PATENT Attorney Docket No: TUOE. P2012WO / 0665313 the optical axis, specifically around the beam waist (with zt= 0). It follows a hyperbolic relation with the distance along the optical axis. Also depicted is the Rayleigh length, which is given by zR i= —A- The Rayleigh length is the position along the optical axis where the cross-sectional area of the beam beam is doubled wizR i') = V2w0>J. Only when the distance along the optical axis is much larger than the Rayleigh length (zf» zR i) one is able to use geometrical optics to perform calculations.

[0150] FIG. 37 is a plot of beam width for a circular Gaussian beam. Geometrical optics are displayed in FIG. 37 by the dashed black line. It can clearly be seen that the Gaussian beam width deviates most around the beam waist. Further away from the Rayleigh length the beam widths start to coincide.

[0151] The curvature of radius of the beam changes with the distance along tire optical axis as well. This relation in rectangular coordinates is given by:and Ry(2) a(z - zx)In the above equation, R;(z) is the radius of curvature in [mJ. A positive radius of curvature dictates that the curvature is leaning back w.r.t. the focal point of the beam. It is expected that the radius of curvature will asymptotically approach infinity near the focal point of the beam as a result of the z in the denominator. This means that near the focal point of the beam the wavefront will approach a planar wave. The farther away from the focal point the beam gets the larger the radius of curvature will become. This relation for both negative and positive distances along the optical axis can be seen in FIG, 38, which is a plot of beam radius of curvature for a circular Gaussian beam.

[0152] The final relation that is needed to model the behavior of a Gaussian beam is the Gouy phase. This relation is given as follows: / A(z ■■■■ zx)\ / T(z - Zy)\<pr.y= arctan - - — and <»0 y= arctan - - —.\ J \!;- / In the above expressions,O.» is the Gouy phase in [rad], The Gouy phase is the result of the different path the light takes in the Gaussian model w.r.t. geometrical optics. In effect the phase of the beam is shifted by TI [rad] after the focal point. FIG. 39 is a plot of the Gouy phase for a circular Gaussian beam along the optical axis.LEGAL\81463546\914PATENT Attorney Docket No: TUOE. P2012WO / 0065313

[0153] FIG. 40 is a plot showing amotion profile of the piezo stepper. Note that the sawtooth has been smoothed around the transitions to minimize exciting the dynamics of the system and that the drive frequency is approximately 1.5 Hz (2 wavelengths each 1.3 seconds).

[0154] To find the signal -to-noise ratio, both the AC-ampiitude and the noise of the measured signal have to be found. This is done by separating the signal based on the frequency. The displacement imposed by the linear piezo stepper has a frequency of approximately 2 Hz. Tire AC-amplitude of the signal can be determined by finding the minimum and maximum voltage of the measured signal after implementation of a low pass filter at 20 Hz (Nyquist frequency). The noise can be defined as all contributions to tire measured signal above this frequency. The frequency response function of the measured data and of the noise band can be seen in FIGs 41 and 42, respectively.

[0155] FIG. 41 shows the FFT of a measured signal without and with a band-stop filter at a distance of d = 0.3 m and a rotation of a = 0 mrad. Note the resonant frequency at fres == 362 Hz. FIG. 42 shows the FFT of the noise without and with a band-stop filter at a distance of d = 0.3 m and a rotation of a — 0 mrad. Note the resonant frequency at fres= 362 Hz.

[0156] However, when inspecting the frequency response function of the measured data a spike can be seen around 362 Hz, whereas a constant noise band is expected as a result of electronic noise sources. By changing the mass of the measurement mirror a shift in this frequency can be observed. It is thus highly likely that this spike is a result of the eigenfrequency of the linear piezo stage - measurement mirror combination. Even after tuning a notch filter on this particular frequency m the controller of the linear piezo stepper, the contribution of this eigenfrequency can be observed in the measured signal. Since the inherent signal -to-noise ratio of this particular optical concept is of interest, the dynamic behavior can be left out of scope. To this end a band-stop filter at the eigenfrequency of the piezo stepper is implemented on the measured signal. This effectively decreases the contribution of the dynamics in the piezo stepper to the signal -to-noise ratio.

[0157] The filtered signal and difference in noise band can be seen in FIG. 43.FIG. 43 shows The typical measured data without and with a band-stop filter at the resonant frequency of the piezo - measurement mirror assembly.

[0158] After the implementation of the bandstop, the noise band is much more constant. The parts of the signal where the noise is temporarily higher without the bandstop isLEGAL\81463546\914PATENT Attorney Docket No: TUOE. P2012WO / 0065313 i likely where the controller imposes a larger control effort and thus excites the dynamics of the stage. By using this band-stop filtering technique the signal-to-noise ratio gives abetter insight into the inherent performance of the optical system, but it overestimates the signal-to- noise ratio slightly, as all other noise contributions around the bandstop frequency are no longer taken into account.

[0159] In additional embodiments, a displacement measurement interferometer includesfirst optics to decollimate source radiation. Tire first optics are, for example, a light source, such as a laser, that itself outputs decollimated, e.g., diverging radiation. The displacement measurement interferometer may also include beam-splitting optics that (a) output the source radiation toward a reflective target and (b) split the reflected radiation into a first optical beam and a second optical beam. Hie displacement measurement interferometer may also includesecond optics to retroreflect the second optical beam toward the reflective target, such that the second optical beam undergoes one or more additional reflection(s) off the reflective target compared to the first optical beam.

[0160] The beam-splitting optics may include polarizing optics that (a) output the source radiation as a first polarization type, such as elliptical, toward the reflective target, (b) polarize reflected radiation from the reflective target as a second type polarization, such as linear, and (c) split the reflected radiation, based on polarization, into the first optical beam and the second optical beam.

[0161] Tire displacement measurement interferometer may further include a first beamsplitter that separates (a) the source radiation from (b) both the first optical beam and the second optical beam output to a detector. Tire polarizing optics may include a polarizing beamsplitter and / or a wave plate, such as a quarter wave plate.

[0162] The displacement measurement interferometer may include a non-polarizing beamsplitter that outputs the first optical beam and the second optical beam to a detector. A detector and an aperture stop can exist between the detector and the beam-splitting optics. The detector may be configured or selected to limit the part of the light used for detection, such that no aperture stop is needed

[0163] The first optical beam and tire second optical beam are for example orthogonally polarized with respect to each other when output to the detector. A polarizing filter may be placed between the beam-splitting optics and the detector, to create interference between the first optical beam and the second optical beam.35LEGAL\81463546\914PATENT Attorney Docket No: TUOE. P2012WO / OQ65313 i

[0164] The second optics may include a retroreflective element, which may further include a reflective optical element and focusing optics, where the reflective element is positioned at or near a focal area of the focusing optics. A beam waist of the decollimated source radiation is most often optically co-located with a nodal point of the retroreflective optical element and the aperture stop, when present. The aperture stop can be a separate element or a part of another component of the displacement measurement interferometer, such as a detector aperture.

[0165] Tire displacement measurement interferometer may include a detector exposed to the first and second optical beams output from the displacement measurement interferometer. In this case, the first optical beam and the second optical beam, when incident on the detector, have (a) negligible wavefront tilt with respect to each other and (b) propagation angle, relative to tin optical axis of the interferometer, equal to tilt of the reflective target. This wavefront tilt is usual non-existent or small that tire interferogram produced by interfering the first and second beams on the detector keeps producing a practical contrast over the detector area within the angular range of the reflective target.

[0166] In embodiments, interference between the first optical beam and the second optical beam output from the displacement measurement interferometer is detectable as a signal proportional to displacement of the reflective target. The interference between the first optical beam and the second optical beam output from the displacement measurement interferometer may include interference between optical radiation reflected by tire reflective target once and optical radiation reflected by the reflective target twice. Or, this interference may be between optical radiation reflected off the reflective target once and optical radiation reflected off the reflective target three times. Additional reflections also may be used.

[0167] In one embodiment, the first optical beam is comparable (e.g., equal) in optical power to the second optical beam when interfered together as output to a detector.

[0168] In other embodiments, a method for measuring displacement of an object is provided, including: reflecting non-collimated source radiation off a surface of the object; splitting reflected radiation from the surface of the object into a first optical beam and a second optical beam; retroreflecting the second optical beam to the surface of the object; reflecting the second optical beam off the surface of the object; and determining the displacement from an interference signal between the first optical beam and the second optical beam after the second optical beam has reflected off the surface of the object.

[0169] Such splitting may include splitting based on polarization. The method may further include rotating polarization of the second optical beam. Reflecting non-collimated 36LEGAL\81463546\914PATENT Attorney Docket No: TUOE. P2012WO / 0065313 i source radiation off a surface of the object may include diverging the radiation.Retroreflecting may include focusing the second optical beam at or near a reflective optical element.

[0170] The method may further include, after reflecting the second optical beam off die surface of the object and before determining the displacement from the interference signal, outputting the first optical beam and the second optical beam (a) with comparable, preferably equal, optical powers, (b) with negligible tilt relative to each other and (c) with a propagation angle equal to tilt of the surface. The method may further include stopping down the first optical beam and the second optical beam before determining the displacement from the interference signal.Combinations of Features

[0171] Features described above, as well as those claimed below', may be combined in various ways without departing from the scope hereof. The following enumerated examples illustrate some possible, non-limiting combinations,

[0172] Embodiment 1. A displacement measurement interferometer, comprising: beam coupler to decollimate source radiation; beam-splitting optics that (a) output the source radiation toward a reflective target and (b) split reflected radiation into a single-pass optical beam and a multi-pass optical beam; and reflective optics to retroreflect the multi-pass optical beam through the beam-splitting optics to the reflective target.

[0173] Embodiment 2. The displacement measurement interferometer of embodiment 1, the beam-splitting optics comprising polarizing optics that (a) output the source radiation with a first polarization toward the reflective target, (b) polarize reflected radiation from the reflective target with a second polarization and (c) split the reflected radiation, based on polarization, into the single-pass optical beam and the multi-pass optical beam.

[0174] Embodiment 3. Tire displacement measurement interferometer of embodiment 2, wherein the first polarization is elliptical polarization and the second polarization is linear polarization.

[0175] Embodiment 4. The displacement measurement interferometer of either one of embodiments 2 or 3, wherein the polarizing optics comprise a polarizing beamsplitter.

[0176] Embodiment 5. The displacement measurement interferometer of embodiment 4, wherein the polarizing optics further comprise a wave plate.

[0177] Embodiment 6. The displacement measurement interferometer of embodiment 5, wherein the wave plate is a quarter wave plate.37LEGAL\81463546\914PATENT Attorney Docket No: TUOE. P2012WO / 0665313

[0178] Embodiment 7. The displacement measurement interferometer of any one of embodiments 1-6, further comprising a first beamsplitter that separates (a) the source radiation from (b) both the single-pass optical beam and tire multi-pass optical beam output to a detector.

[0179] Embodiment 8. The displacement measurement interferometer of any one of embodiments 1-7, further comprising an aperture stop between a detector and beam-splitting optics, to transmit part of reflected radiation orthogonal to the reflective target.

[0180] Embodiment 9. The displacement measurement interferometer of any one of embodiments 1-7, further comprising a non-polarizing beamspliter that outputs the single¬ pass optical beam and the multi-pass optical beam to a detector.

[0181] Embodiment 10. The displacement measurement interferometer of embodiment 9, further comprising an aperture stop between the detector and the non¬ polarization beamsplitter, to transmit part of reflected radiation orthogonal to the reflective target.

[0182] Embodiment 11. The displacement measurement interferometer of any one of embodiments 1, 9 to 10, the single-pass optical beam and the multi-pass optical beam being orthogonally polarized with respect to each other when output to a detector, and further comprising a polarizing filter between the detector and the beam-splitting optics, to create interference between the single-pass optical beam and the multi-pass optical beam.

[0183] Embodiment 12. The displacement measurement interferometer of any one of embodiments Ito 11, wherein the reflective optics compose one of (a) a retroreflector and (b) a reflective optical element and focusing optics, the reflective optical element being positioned at or near a focal point of the focusing optics.

[0184] Embodiment 13. lire displacement measurement interferometer of embodiment 11, a beam waist of the decollimated source radiation being optically co-located with a nodal point of the reflective optics.

[0185] Embodiment 14. The displacement measurement interferometer of any one of embodiments 1—12, further comprising a detector exposed to the single-pass and multi-pass optical beams output from the displacement measurement interferometer, wherein the single¬ pass optical beam and the multi-pass optical beam, when incident on the detector, have (a) negligible wavefront tilt with respect to each other and (b) propagation angle, relative to an optical axis of the interferometer, equal to tilt of the reflective target.

[0186] Embodiment 15. The displacement measurement interferometer of any one of embodiments 1—14, wherein interference between the single-pass optical beam and the multi- 38LEGAL\81463546\914PATENT Attorney Docket No: TUOE. P2012WO / 0665313 i pass optical beam output from the displacement measurement interferometer is detectable as a signal proportional to displacement of the reflective target.

[0187] Embodiment 16. lire displacement measurement interferometer of embodiment 15, wherein the interference between the single-pass optical beam and the multi¬ pass optical beam output from the displacement measurement interferometer comprises interference between optical radiation reflected by the reflective target once and optical radiation reflected by the reflective target twice.

[0188] Embodiment 17. The displacement measurement interferometer of embodiment 15, wherein the interference between the single-pass optical beam and the multi¬ pass optical beam output from the displacement measurement interferometer comprises interference between optical radiation reflected off the reflective target once and optical radiation reflected off the reflective target three times.

[0189] Embodiment 18. The displacement measurement interferometer of embodiment 15, the single-pass optical beam being similar in optical power to the multi-pass optical beam when interfered together as output to a detector.

[0190] Embodiment 19. A method for measuring displacement of an object, comprising: reflecting non-collimated source radiation off a surface of the object; splitting reflected radiation from the surface of the object into a single-pass optical beam and a multi¬ pass optical beam; retroreflecting the multi-pass optical beam to the surface; reflecting the multi-pass optical beam off the surface; and determining the displacement from an interference signal between the single-pass optical beam and the multi-pass optical beam after the single-pass optical beam reflects off the surface and after the multi-pass optical beam reflects off the surface at least twice.

[0191] Embodiment 20. lire method of embodiment 19, wherein splitting comprises splitting based on polarization.

[0192] Embodiment 21. The method of embodiment 20, further comprising rotating polarization of the multi-pass optical beam.

[0193] Embodiment 22. The method of any one of embodiments 19-21, wherein reflecting non-collimated source radiation off a surface of the object comprises diverging the radiation

[0194] Embodiment 23. lire method of any one of embodiments 19-22, wherein retroreflecting comprises focusing the multi-pass optical beam to a reflective optical element.

[0195] Embodiment 24. The method of any one of embodiments 19-23, further comprising, after reflecting the multi-pass optical beam off the surface and before39LEGAL\81463546\914PATENT Attorney Docket No: TUOE. P2012WO / 00653131 determining the displacement, outputting the single-pass optical beam and the multi-pass optical beam with (a) comparably equal optical powers, (b) negligible tilt relative to each other and (c) a propagation angle equal to tilt of tire surface.

[0196] Embodiment 25. The method of any one of embodiments 19-2.4, further comprising stopping down the single-pass optical beam and the multi-pass optical beam before determining the displacement from the interference signal.

[0197] Changes may be made in the above methods and systems without departing from the scope hereof. It should thus be noted that the matter contained in the above description or shown in the accompanying drawings should be interpreted as illustrative and not in a limiting sense. The following claims are intended to cover all generic and specific features described herein, as well as all statements of the scope of the present method and system, which, as a matter of language, might be said to fall therebetween.References[1] S. Yang and G. Zhang, “A review of interferometry for geometric measurement,” Measurement Science and Technology, vol. 29, no. 10, p. 102001, Sep. 2018, doi:10.1088 / 1361 -6501 / aad732.[2] S. J. A. G. Cosijns, “Displacement laser interferometry with sub-nanometer uncertainty,” Phd Thesis 1 (Research TU / e / Graduation TU / e), Mechanical Engineering; Technische Universiteit Eindhoven, 2004. doi: 10.6100 / IR579456.[3] J. H. Lan, 3D IC integration and packaging. McGraw-Hill Education, 2016.[4] K. Looman, L, Cacace, J. P. M. B. (Hans). Vermeulen, and R. Hendrix, “Beam steering interferometry for enlarged angle acceptance: Analysis of measurement uncertainty,” in ASPE proceedings of the 38 th annual meeting, Nov. 2023, pp. 445-449.[5] J. D. Ellis, Field guide to displacement measuring interferometry, in SPIE field guides.SPIE, 2014.[6] T. L. Schmitz and J. F. Beckwith, “An investigation of two unexplored periodic error sources in differential-path interferometry,” Precision Engineering, vol. 27, no. 3, pp. 311-322, 2003, doi: httgs_: / Al roEv / l_0 J.01.6 / S0J.4i.-6359(03)00036"9.[7] C. Lu, E. D. Bumham-Fay, J. D. Ellis, T, L. Schmitz, and J. A. Tarbutton, “Periodic error compensation in fiber-coupled heterodyne interferometry,” Procedia Manufacturing, vol.10, pp. 674-682, 2017, doi: https: / / doi. org / 10.1016 / j.promt g.2017.07.015.[8] M. Bom et al.. Principles of optics: Electromagnetic theory of propagation, interference and diffraction of light, 7th ed. Cambridge University Press, 1999.40LEGAL\81463546\914

Claims

PATENT Attorney Docket No: TUOE.P2012WO / 00653131 CLAIMSWhat is claimed is:

1. A displacement measurement interferometer, comprising:beam coupler to decollimate source radiation;beam-splitting optics that (a) output the source radiation toward a reflective target and (b) split reflected radiation into a single-pass optical beam and a multi-pass optical beam; andreflective optics to retroreflect the multi-pass optical beam through the beam -splitting optics to the reflective target.

2. The displacement measurement interferometer of claim 1, the beam-splitting optics comprising polarizing optics that (a) output the source radiation with a first polarization toward the reflective target, (b) polarize reflected radiation from the reflective target with a second polarization and (c) split the reflected radiation, based on polarization, into the single-pass optical beam and the multi-pass optical beam.

3. The displacement measurement interferometer of claim 2, wherein the first polarization is elliptical polarization and the second polarization is linear polarization.

4. The displacement measurement interferometer of either one of claims 2 or 3, wherein the polarizing optics comprise a polarizing beamsplitter.

5. The displacement measurement interferometer of claim 4, wherein the polarizing optics further comprise a wave plate.

6. The displacement measurement interferometer of claim 5, wherein the wave plate is a quarter wave plate.

7. The displacement measurement interferometer of any one of claims 1-6, further comprising a first beamsplitter that separates (a) the source radiation from (b) both the single-pass optical beam and the multi-pass optical beam output to a detector.

8. The displacement measurement interferometer of any one of claims 1—7, further comprising an aperture stop between a detector and beam-splitting optics, to transmit part of reflected radiation orthogonal to the reflective target.41LEGAL\81463546\914PATENT Attorney Docket No: TUOE.P2012WO / 00653131 9. The displacement measurement interferometer of any one of claims 1-7, further comprising a non-polarizing beamsplitter that outputs the single-pass optical beam and the multi-pass optical beam to a detector.

10. The displacement measurement interferometer of claim 9, further comprising an aperture stop between the detector and the non-polarizing beamsplitter, to transmit part of reflected radiation orthogonal to the reflective target.

11. The displacement measurement interferometer of any one of claims 1, 9 to 10, the single¬ pass optical beam and the multi-pass optical beam being orthogonally polarized with respect to each other when output to a detector, and further comprising a polarizing filter between the detector and the beam-splitting optics, to create interference between the single-pass optical beam and the multi-pass optical beam.

12. The displacement measurem ent interferometer of any one of claim s 1-11, wherein the reflective optics comprise one of (a) a retroreflector and (b) a reflective optical element and focusing optics, the reflective optical element being positioned at or near a focal point of the focusing optics.

13. The displacement measurement interferometer of claim 11, a beam waist of the decollimated source radiation being optically co-located with a nodal point of the reflective optics.

14. The displacement measurement interferometer of any one of claims 1-12, further comprising a detector exposed to the single-pass and multi-pass optical beams output from the displacement measurement interferometer, wherein the single-pass optical beam and the multi-pass optical beam, when incident on the detector, have (a) negligible wavefront tilt with respect to each other and (b) propagation angle, relative to an optical axis of the interferometer, equal to tilt of the reflective target.

15. The displacement measurement interferometer of any one of claims 1-14, wherein interference between the single-pass optical beam and the multi-pass optical beam output from the displacement measurement interferometer is detectable as a signal proportional to displacement of the reflective target.42LEGAL\81463546\914PATENT Attorney Docket No: TUOE.P2012WO / 00653131 16. The displacement measurement interferometer of claim 15, wherein the interference between the single-pass optical beam and the multi-pass optical beam output from the displacement measurement interferometer comprises interference between optical radiation reflected by the reflective target once and optical radiation reflected by the reflective target twice.

17. The displacement measurement interferometer of claim 15, wherein the interference between the single-pass optical beam and the multi-pass optical beam output from the displacement measurement interferometer comprises interference between optical radiation reflected off the reflective target once and optical radiation reflected off the reflective target three times.

18. The displacement measurement interferometer of claim 15, the single-pass optical beam being comparable in optical power to the multi-pass optical beam when interfered together as output to a detector.

19. A method for measuring displacement of an object, comprising:reflecting non-collimated source radiation off a. surface of the object;splitting reflected radiation from the surface of the object into a single-pass optical beam and a multi-pass optical beam;retroreflecting the multi-pass optical beam to the surface; anddetermining the displacement from an interference signal between the single-pass optical beam and the multi-pass optical beam after the single-pass optical beam reflects off the surface and after the multi-pass optical beam reflects off the surface at least twice.

20. The method of claim 19, wherein spliting comprises splitting based on polarization.

21. The method of claim 20, further comprising rotating polarization of the multi-pass optical beam.

22. The method of any one of claims 19-21, wherein reflecting non-collimated source radiation off a surface of the object comprises diverging the radiation43LEGAL\81463546\914PATENT Attorney Docket No: TUOE.P2012WO / 00653131 23. The method of any one of claims 19-22, wherein retroreflecting comprises focusing the multi-pass optical beam to a reflective optical element.

24. The method of any one of claims 19—23, further comprising, after reflecting the multi¬ pass optical beam off the surface and before determining the displacement, outputting the single-pass optical beam and the multi-pass optical beam with (a) comparably equal optical powers, (b) negligible tilt relative to each other and (c) a propagation angle equal to tilt of the surface.

25. The method of any one of claims 19 -24, further comprising stopping down the single- pass optical beam and the multi-pass optical beam before determining the displacement from the interference signal.

26. The method of any one of claims 19-25, wherein retroreflecting the multi-pass optical beam to the surface comprises reflecting the multi-pass optical beam off the surface.LEGAL\81463546\914

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