Trojan beam generation waveguide utilizing lagrange points

Abstract

A waveguide for trapping light or electrons utilizing the exotic features of Lagrange points is disclosed. A waveguide formed at a stable Lagrange point, in twisted arrangements, is configured for light or charged particle transmission. The guiding channel for a lightwave is induced by a helical refractive index structure in space while the one for charged particles is generated by twisted static electric field distribution. The resulting Trojan beams can be captured even in defocusing environments at points where the refractive index / potential profile is ordinary. As the mode in this waveguide, these Trojan beams can be trapped at stable Lagrange points along a helical trajectory. This mechanism can be established in a versatile manner through a variety of attractive / repulsive potentials.

Description

TROJAN BEAM GENERATION WAVEGUIDE UTILIZING LAGRANGE POINTS PRIORITY

[0001] This disclosure claims the benefit of and priority to U.S. Provisional Application No. 63 / 528,037, filed July 20, 2023. The contents of that application are hereby incorporated by reference in their entirety. TECHNICAL FIELD

[0002] This disclosure relates to wave propagation and, in particular, to methods, apparatus and systems for generating Trojan optical / particle beams that are transported in a waveguide induced by stable Lagrange points in twisted arrangements. BACKGROUND

[0003] The guided transmission of optical waves is critical to harnessing the power of light for modern communication, information processing, and energy generation systems. Traditionally, guiding lightwaves in structures such as optical fibers is predominantly achieved through the use of total internal reflection (TIR) that occurs when a core of high refractive index is surrounded by a lower-index cladding. In addition to fibers where lightwaves are trapped in a high-index region, waveguiding can also be attained in periodic photonic settings and through other mechanisms such as that of Bragg reflection, evanescent coupling, and bound state in the continuum effects. Similarly, light can propagate in a confined manner in plasmonic or active arrangements that allow for gain guidance.

[0004] Currently, photonic crystal fibers are used in applications over a wide spectrum of fields, ranging from fiber lasers and amplifiers to high-power optical nonlinear sources. Recently, research has occurred aiming to explore how geometric rotation may affect beam propagation in such periodic systems. In this respect, phenomena arising from effects like birefringence and geometric phases have been systematically explored in twisted coreless photonic crystal fibers, liquid crystals, and helical waveguide arrays.

[0005] However, transversely confining light in fully dielectric, non-periodic, and passive configurations still remains a challenge in situations where TIR is not supported, and no other 1 4885-3549-1273.1 065715-000158WOPTmethodology (aside from TIR) has been reported by means of which light can be trapped in a bulk environment, i.e., without using a transverse periodic arrangement.

[0006] Further, the use of electron transport processes within various material systems has utterly revolutionized the technological landscape, ushering in an era of electronic advances. In this regard, there has been a need to devise effective approaches in guiding electrons and, by extension, charged particles such as protons and ions in vacuum. Methods to store charged particles in three- dimensional settings currently rely on static electric fields when used in conjunction with an oscillating electric or a uniform magnetic field, in a way that overcomes the limitations imposed by Earnshaw's theorem. It is also possible to focus charged particles using quadrupole magnetic or electrostatic lenses. Of utmost significance is the principle of strong focusing. Indeed, in facilities like the Large Hadron Collider (LHC), hundreds of quadrupole superconducting magnets are employed to periodically steer and focus protons throughout their journey. However, such arrangements are cumbersome, constrain the degrees of freedom, and are not suitable for guiding charged particles.

[0007] There is therefore a need for an optics waveguide mechanism that does not rely on TIR for operation. There is another need for a waveguide mechanism that may be used for guiding different types of waves including acoustic and charged particle waves. SUMMARY

[0008] One disclosed example is a waveguide system including a beam input generated by a beam source. A waveguide guides the beam input generated by the beam input. The waveguide includes a transmission medium with a twisted potential distribution structure to transport a Trojan beam from the beam input.

[0009] A further implementation of the example system is an embodiment where the waveguide includes a glass cylinder and the transmission medium is polydimethylsiloxane (PDMS). Another implementation is where the example waveguide system includes a current source coupled to the twisted potential distribution structure. The twisted potential distribution structure is a conductive wire. Another implementation is where the conductive wire is shaped in a single helix or a double helix. Another implementation is where the beam input is a Lightwave. The waveguide has a refractive index creating a stable Lagrange point dynamically stabilized by an optical Coriolis force to trap the lightwave as an optical Trojan beam. Another implementation is where the beam 2 4885-3549-1273.1 065715-000158WOPTsource is a laser, and the Trojan beam is guided by the Lagrange point. Another implementation is where the beam source is a stream of electrons generated by an electron gun and the waveguide includes a grounded hollow tube. The twisted potential distribution structure is a helicoidal charged conductive wire with a negative voltage. Another implementation is where the waveguide has a potential distribution generating a stable Lagrange point. Another implementation is where the beam source is an audio wave generator and the waveguide includes a medium with a predetermined ratio of elastic modulus and density. Another implementation is where the beam source is a charged particle emitter.

[0010] Another disclosed example is a method of producing a Trojan beam. An input beam is directed to a transmission medium. A current or voltage is generated through a twisted potential distribution structure in the transmission medium to establish a Lagrange point to produce the Trojan beam from the input beam through the transmission medium.

[0011] A further implementation of the example method is an embodiment where the waveguide includes a glass cylinder and the transmission medium is polydimethylsiloxane (PDMS). Another implementation is where a current source is coupled to the twisted potential distribution structure, and the twisted potential distribution structure is a conductive wire. Another implementation is where the conductive wire is shaped in a single helix or a double helix. Another implementation is where the input beam is a lightwave. The waveguide has a refractive index creating a stable Lagrange point dynamically stabilized by an optical Coriolis force to trap the lightwave as an optical Trojan beam. Another implementation is where the beam input is a laser, and the Trojan beam is guided by the Lagrange point. Another implementation is where the input beam is a stream of electrons generated by an electron gun. The waveguide includes a grounded hollow tube and the twisted potential distribution structure is a helicoidal charged conductive wire with a negative voltage. Another implementation is where the waveguide has a potential distribution generating a stable Lagrange point. Another implementation is where the input beam is an audio wave and the waveguide includes a medium with a predetermined ratio of elastic modulus and density. Another implementation is where the input beam is generated by a charged particle emitter.

[0012] Another disclosed example is an optical waveguide including a waveguide optically coupled to a light beam source generated a light beam. The waveguide includes a light transmitting material and a helical channel embedded in the light transmitting material. A current source is 3 4885-3549-1273.1 065715-000158WOPTcoupled to the helical channel to generate a Lagrange point in the light transmitting medium to generate a Trojan beam from the light beam through the light transmitting material.

[0013] A further implementation of the example optical waveguide is an embodiment where the waveguide includes a glass cylinder and the transmission medium is polydimethylsiloxane (PDMS). Another implementation is where the helical channel includes a wire shaped in a single helix or a double helix. Another implementation is where the waveguide has a refractive index and the Lagrange point is a stable Lagrange point dynamically stabilized by an optical Coriolis force to trap the light beam as an optical Trojan beam. Another implementation is where the light beam source is a laser. Another implementation is where the helical channel is a helical gas channel.

[0014] Another disclosed example is a charged particle waveguide including a cylindrical body having an aperture on one end. The aperture filters a charged particle beam generated from a charged particle source. A helical wire is centered around an axis of the cylindrical body. A voltage source is coupled to the helical wire to establish a Lagrange point in the cylindrical body to generate a Trojan beam from the charged particle beam. The cylindrical body is grounded relative to the voltage source.

[0015] A further implementation of the example charged particle waveguide is an embodiment where the helical wire is shaped in a single helix or a double helix. Another implementation is where the charged particle source is an electron gun and the waveguide includes a grounded hollow tube. A negative voltage is applied via the voltage source. Another implementation is where the electron gun is part of a reflection high-energy electron diffraction gun. Another implementation is where the Lagrange point is a stable Lagrange point generated by the waveguide via a potential distribution. Another implementation is where the charged particle beam is one of a proton beam or an ion beam.

[0016] Another disclosed example is an acoustic waveguide including a gas tube having an input for an acoustic wave. A twisted conductor is embedded in the gas tube. A current source provides current to the twisted conductor to establish a Lagrange point for generating a Trojan beam from the acoustic wave.

[0017] A further implementation of the example acoustic waveguide is an embodiment where the gas tube holds helium. Another implementation is where the acoustic wave is generated by an ultrasonic transducer. 4 4885-3549-1273.1 065715-000158WOPT

[0018] The above summary is not intended to represent each embodiment or every aspect of the present disclosure. Rather, the foregoing summary merely provides an example of some of the novel aspects and features set forth herein. The above features and advantages, and other features and advantages of the present disclosure, will be readily apparent from the following detailed description of representative embodiments and modes for carrying out the present invention, when taken in connection with the accompanying drawings and the appended claims. BRIEF DESCRIPTION OF THE FIGURES

[0019] In order to describe the manner in which the above-recited disclosure and its advantages and features can be obtained, a more particular description of the principles described above will be rendered by reference to specific examples illustrated in the appended drawings. These drawings depict only example aspects of the disclosure, and are therefore not to be considered as limiting of its scope. These principles are described and explained with additional specificity and detail through the use of the following drawings:

[0020] FIG.1A is a perspective view of an example optical waveguide;

[0021] FIG.1B is a close-up view of a Trojan guided beam propagating in the example waveguide in FIG.1A;

[0022] FIG.2A shows a graph of an induced logarithmic defocusing spiraling index potential of the optical waveguide in FIG.1A when viewed within a stationary frame;

[0023] FIG.2B shows a graph of the index potential in FIG.2A in a co-rotating frame;

[0024] FIG.2C shows a graph of numerically obtained intensity distribution for the fundamental Trojan optical mode centered at the Lagrange point in the graphs in FIGs.2A-2B;

[0025] FIG. 3 shows an experimental setup to observe optical Trojan beams from an example optical waveguide;

[0026] FIG.4A shows images of the Trojan beam guiding in the index potential produced from the experimental setup in FIG.3;

[0027] FIG. 4B is a graph showing the dependence of the spot size and ellipticity of the Trojan beam versus the current;

[0028] FIG.5 are images that schematically show the Trojan beam trapping in the index landscape produced by a double-helix current source; 5 4885-3549-1273.1 065715-000158WOPT

[0029] FIG.6 shows a cross-section view of a charged particle waveguide;

[0030] FIG.7A shows another example charged particle waveguide;

[0031] FIG. 7B shows a cross-section view of the example charged particle waveguide in FIG. 7A;

[0032] FIG. 8A is a graph of an induced logarithmic defocusing spiraling index potential of an example charged particle waveguide when viewed within a stationary frame;

[0033] FIG.8B is a graph of the index potential in FIG.7A in a co-rotating frame;

[0034] FIG. 8C is a series of images showing numerically obtained quantum wavefunction probability distributions and phase structures of different Trojan modes;

[0035] FIG.8D is a graph of the helical path of the fundamental quantum eigenmode of an electron in an example charged particle waveguide in the stationary frame;

[0036] FIG. 9A is graph showing the waveguiding action of an example charged particle waveguide;

[0037] FIG.9B is a graph showing the spatial and velocity distributions associated with an input electron in the entrance of a charged particle waveguide;

[0038] FIG. 9C shows histograms of electron spatial distribution and an image of traverse trajectory of an electron in the example waveguide;

[0039] FIG.10 is a table showing example waveguide design parameters for transporting different charged particles;

[0040] FIG.11A is an experimental setup used to observe electron Trojan beams;

[0041] FIG.11B is a close up of the electron beam and test assembly in FIG.11A; and

[0042] FIG.12 is a diagram of an example acoustic waveguide. DETAILED DESCRIPTION

[0043] Unless defined otherwise, technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention belongs. One skilled in the art will recognize many methods and materials similar or equivalent to those described herein, which could be used in the practice of the present invention. Indeed, the present invention is in no way limited to the methods and materials specifically described. 6 4885-3549-1273.1 065715-000158WOPT

[0044] In some embodiments, properties such as dimensions, shapes, relative positions, and so forth, used to describe and claim certain embodiments of the invention are to be understood as being modified by the term “about.”

[0045] The present disclosure relates to an example approach for guiding light in transparent dielectric systems by exploiting the characteristics of Lagrange points. These characteristics can also be applied to charged particle systems such as electron guiding systems. The resulting Trojan beams can be captured even in defocusing environments at points where the refractive index profile (within the stationary frame) is totally ordinary—with no features that could foretell a guiding behavior. In the optical domain, this mechanism can be established in a versatile manner through a variety of attractive / repulsive index potentials. This may also be accomplished in electron setups. A twisted refractive index configuration / potential distribution is first determined. A novel apparatus, method, system for generating Trojan beams in a bulk environment and a special waveguide configuration for trapping light or electrons is disclosed. At least one stable Lagrange point is generated as the guiding channel. An example optical waveguide is induced by a twisted graded refractive index configuration. The embedded helical iron wire with current generates the pertinent index distribution through thermo-optic effects in a transmission medium such as cured PDMS. Besides single wire embedded, multiple wires can be injected in the bulk medium as heating sources for a bunch of guiding channels at different Lagrange points or a single interferometrically combined guiding channel. In general, various mechanisms can be used to generate the desired refractive index profile. For example, the helical iron wire may be substituted by helical gas channels for cooling or, alternatively, the index structure can be generated through direct laser writing.

[0046] A twisted structure induces Coriolis forces for light or charged particles to be stably captured at a pertinent Lagrange point. Pitch and radius of the helical structure as well its transverse profile (refractive index or potential) can be designed to modify the trajectory, modal distribution, and transverse location of the Lagrange point.

[0047] Given that Newton’s third law governing the dynamics of celestial bodies is inconsequential in the example optical setup, multiple Lagrange points can now be simultaneously generated at will in “many-body” configurations, an aspect that is impossible in other settings. In celestial mechanics, Lagrange points represent equilibrium positions whereby the gravitational attraction from two orbiting massive bodies balances the centrifugal force. In this situation, one 7 4885-3549-1273.1 065715-000158WOPTcan identify five Lagrange points designated as ^^^, ^^ଶ, …, ^^ହwith the first three Lagrange points ( ^^^, ^^ଶand ^^ଷ) being by nature unstable, while the remaining two Lagrange points ( ^^ସand ^^ହ) are stable. As a result, a small mass can be forever trapped in the vicinity of the two latter points, such as, for example the Trojan asteroids in the Sun-Jupiter system. Whereas at a first glance, the two- dimensional potential distribution within the co-rotating frame appears to be unstable around the ^^ସ,ହpoints, counterintuitively, the motion of a third body can be dynamically stabilized because of a Coriolis force. In essence, as this body starts to roll down from the ^^ସ,ହpotential hill, the Coriolis force tends to continuously pull it back, thus resulting into a bounded orbit. In recent years, the analog of this process has also been used in atomic physics to produce non-dispersive wavepackets, a prospect enabled by the isomorphism between the Coulomb force and the gravitational attraction.

[0048] FIG.1 is a perspective view of an example optical waveguide 100. The optical waveguide 100 includes glass cylinder 110 supported between two end plates 112 and 114. A twisted wire 120 is attached between the end plates 112 and 114 and extends over the length of the cylinder 110. The glass cylinder 110 is filled with a refractive material 124 such as cured polydimethylsiloxane (PDMS) of refractive index ^^^ൌ 1.46. A current source 130 is coupled to the wire 120. A beam source, such as a laser, provides an optical beam in this example.

[0049] A Trojan beam 140 is excited in the glass cylinder 110 that is approximately 30 cm in length with a radius of ^^ ൌ 1.18 ^^ ^^ that are filled with the cured polydimethylsiloxane (PDMS). To produce optical Lagrange points, the large thermo-optic effect offered by the PDMS ( ^^ ^^ / ^^ ^^ ൌ െ4.5ൈ 10ିସ^^ି^) is exploited. This is achieved by inserting the twisted iron wire 120 (of radius ^^ ൌ 275 ^^ ^^) in the cylinder 110 before the PDMS is cured. Electric current is then passed through the wire 120 that in turn heats the PDMS 124 in order to induce a logarithmic defocusing spiral index potential through the thermo-optic process. The generated Trojan beam 140 is guided by Lagrange points created by the twisted wire 120. Since the twisted iron wire 120 is embedded in the refractive material 124, it creates a helical channel. Thus, the iron wire 120 may be replaced by a single or multiple gas filled helical channels formed in the refractive material.

[0050] An experimental setup may be used to observe optical Trojan beams from the example optical waveguide. A stable Lagrange point is established. FIG.1B shows a close-up view of the light beam 140 in relation to the wire 120. In this example, the current flowing through the wire 120 creates a series of Lagrange points. Thus, FIG.1B displays in a schematic manner the trapping 8 4885-3549-1273.1 065715-000158WOPTof a Trojan beam at a Lagrange point induced by a helicoidal index potential. The Trojan beam 140 is thus guided as will be demonstrated below.

[0051] In corresponding ray dynamics, considering the paraxial approximation, the ray trajectories are described by ^^ଶ^^ / ^^ ^^ଶൌ ∇൫Δ ^^^ ^^, ^^, ^^^൯ / ^^^where Δ ^^^ ^^, ^^, ^^^ represents the helicoidal index profile that is periodic in ^^ and ^^ ^ ^ ^^ ^^^ is the transverse position vector. ^^^is the background refractive index and the total refractive index is ^^ ൌ ^^^^ Δ ^^. In a twisted bulk environment, these two equations of motion can be more conveniently studied within a co-rotating frame ^ ^^ஐబ, ^^ஐబ, ^^ஐబ^ where ^^ஐబൌ ^^, ^ ^^ஐబ, ^^ஐబ൧்ൌ ^^^Ω^^^^^ ^^, ^^^்and the subscript ‘Ω^’ represents the actual units. Here, ^^^Ω^^^^ denotes the rotation matrix:^^^Ω^^^^ ൌ ^ cos^Ω^ ^^^ sin^Ω^ ^^^െsin^Ω^ ^^^ cos^Ω^ ^^^൨ , ^1^and Ω^ൌ 2 ^^ / Λ is thesystem, the spiral index profile can be reduced to an effective index potential Δ ^^ஐబ,^^^൫ ^^ஐబ, ^^ஐబ൯ ൌ ^^^Ωଶ^^ ^^ஐଶ ଶబ ^ ^^ஐబ^ / 2 ^ Δ ^^^ ^^ஐబ, ^^ஐబ^ which now includes centrifugal here,^^ஐబ൯^ / ^^^^ 2^ ^^ ^^ஐబ / ^^ ^^ ൈ Ω^z^^ where the last termunits, with now ^^ஐబൌ ^^ஐబ^^^ஐబ^ ^^ஐబ^^^ஐబ. The Hamiltonian ray dynamics are in every respect equivalent to Newton’s equations of motion with Δ ^^ஐబ,^^^൫ ^^ஐబ, ^^ஐబ൯ playing the role of a potential in which Δ ^^^ ^^ஐబ, ^^ஐబ^ can be either attractiveor defocusing).

[0052] Light propagation dynamics around a stable Lagrange point may be shown in 3-D graphs. FIG. 2A shows a 3-D graph 200 of an induced logarithmic defocusing spiraling index potential from the heat generated from the wire 120 in FIG.1 as shown by a shaded area 210 when viewed within the stationary frame ^ ^^, ^^^. This index profile rotates along ^^ at a constant angular velocity Ω^around the center, C, at coordinate (0,0) represented by a point 224. The corresponding iso- contour lines 212 are also shown. Two Lagrange points 220 and 222 are shown. The Lagrange point 222 ( ^^^^ is a saddle point and hence is unstable. In contrast, the Lagrange point 220 ( ^^^^ exhibits an index where the dynamics can be stabilized through the optical Coriolis force.

[0053] FIG.2B shows a 3-D graph 230 of an index potential when viewed in a co-rotating frame. The index potential is shown as a shaded area 240. Corresponding iso-contour lines 242 are shown with two Lagrange points 250 and 252. The graph 230 displays that in the co-rotating frame ^ ^^, ^^^, 9 4885-3549-1273.1 065715-000158WOPTthe effective index potential 240 now involves centrifugal effects and exhibits two Lagrange points 250 and 252, ^^^and ^^^respectively. The Lagrange point 252 ( ^^^^ is a saddle point and hence is unstable. In contrast, the Lagrange point 250 ( ^^^^ exhibits an index minimum where the dynamics can be stabilized through the optical Coriolis force. In optics, the effective index distribution in the co-rotating coordinate system is ^^ ^^ஐబ,^^^∝ െ ^^ஐబ,^^^.

[0054] FIG.2C shows a graph 260 of numerically obtained intensity distribution shown as an area 270 for the fundamental Trojan optical mode centered at a Lagrange point 272 ( ^^^), corresponding to the stable Lagrange points in FIGs.2A and 2B. In this case, the mode is elliptical in the ^ ^^, ^^^ system.

[0055] In addressing the prospect of trapping light at a Lagrange point, in the stationary coordinate system ( ^^, ^^), the logarithmic defocusing index profile 210 shown in FIG. 2A, is typical to that thermally produced in experiments relating to the example optical waveguide. This long-range potential is spiraling around the center C (224) with a pitch of Λ ൌ 8 ^^ ^^. In this example, theindex contrast induced (with respect to the surface of the metallic wire) is approximately2.25 ൈ 10ିଶ.

[0056] The same index potential, when viewed within the co-rotating frame, is shown in the graph 230 in FIG. 2B along with the corresponding iso-contour lines 242. In this frame, because of centrifugal effects, the effective potential is modified, leading to two Lagrange points 250 and 252 (designated as ^^^and ^^^) where the optical attraction or repulsion is balanced. In this case, only ^^^is stable while ^^^is unstable. The position of the Lagrange point ^^^is also marked in FIG.2A as a point 220 for clarity. As in the case of Trojan asteroids, stability is introduced because of Coriolis forces. In optics, this is possible even though the potential is defocusing (hence, repulsive) and it is produced only by one source, unlike the Sun-Jupiter system. As indicated before, the index landscape around the ^^^point, when viewed within the stationary ( ^^, ^^) system in FIG. 2A, happens to be completely ordinary, with no special features that indicate that light can be trapped.

[0057] To formally assert this possibility within the framework of wave optics, the paraxial wave equation, a reduced form of the Helmholtz electromagnetic problem in weakly guiding arrangements is used ( ^^^ ^^, ^^, ^^^ ൌ ^^^^ Δ ^^^ ^^, ^^, ^^^ where ^^^is the index background while Δ ^^^ ^^, ^^, ^^^ is a small variation Δ ^^^ ^^, ^^, ^^^ ≪ ^^^), that is isomorphic to the Schrödinger equation: ^^ ^^ 1 ^^ଶ ଶ^^ ^^^ 1 ^^ ^^^ ^ ^^^ ^^^ ^^4885-3549-1273.1 065715-000158WOPTwhere ^^ ൌ ^^^^^^ൌ 2 ^^ ^^^ / ^^^is the wavenumber. The coordinates can now be scaled or normalized according to ^^ ൌ ^^ / ^^^^^^^^^ଶ, ^^ ൌ ^^ / ^^^, ^^ ൌ ^^ / ^^^, where ^^^is an arbitrary length scale used for normalization. In this regard, the normalized wave propagation equation shows: ^^^^ ^^^^ ^^ ^1 ^^ଶ ^^ 1 ^^ଶ ^2^^ ^^ଶ ^^2^^ ^^ଶ െ ^^^ ^^, ^^, ^^^ ^^ ൌ 0, ^3^where ^^^ ^^, ^^^ ൌ െ ^^^ଶ^^^^^in the normalized the normalized angular velocity Ω, the paraxial wave propagation equation in the co-rotating frame can be obtained by: ^^^^ ^^^^ ^^ ^ ^^Ω^ ^^ ^^௨ െ ^^ ^^௩^ ^^ ^12^ ^^௨௨ ^ ^^௩௩^ ^^ െ ^^ஐ^ ^^, ^^^ ^^ ൌ 0 . ^4^The second^^^^ ൌ Ω ^^^ ൈ ^ ^^ ^^ (^ ^^ ^^ ൌ ^^ ^^^ ^ ^^ ^^^) akin to that from a magnetic field in the Schrödinger equation, thein the corotating frame can be written as:^12൫ ^^⃗^ ^^ െ ^^^^ ൯ ⋅ ൫ ^^⃗^ ^^ െ ^^^^ ൯ ^ ^^ஐ,^^^൨ ^^ ൌ ^^^^ ^^^^ ^^ ,^5^where ^^⃗^ ^^ describes the^^ஐ,^^^ൌ െΩଶ^^ஐଶ / 2 ^ ^^ஐ^ ^^, ^^^ that nowଶ ଶthe centrifugal potential Ω ^^ஐ / 2. Equation (1) is solved numerically to identify bound modes in the form of ^^ ൌ ^^^ఙ௭^^^^^, ^^^^^^ஏ^௨,௩^, centered at a stable Lagrange point where ^^ denotes their corresponding eigenvalue. FIG.2C depicts the intensity profile of the ground state associated with the ^^^point 250 in FIG. 2B. In this case, the intensity of the fundamental Trojan mode is elliptical, as one could expect given that the index potential 240 around the Lagrange point 250 ( ^^^) has approximately an elliptic paraboloidal dependence.

[0058] To observe Trojan beams, experiments were conducted in PDMS filled cylinders similar to the waveguide in FIG.1A. FIG.3 shows an experimental setup 300 to observe optical Trojan beams from an example waveguide 310 that is constructed in a similar manner with the example waveguide 100 in FIG.1A. The exemplary experiments were conducted with either a single helix wire configuration 311a or a double-helix wire configuration 311b when embedded in a cured PDMS cylinder of the waveguide 310. A laser 312 emitted a beam that was directed toward a first lens 314 with a focal length, F = 25 mm, a second lens 316 with F = 125 mm, and a third lens 318 11 4885-3549-1273.1 065715-000158WOPTwith a focal length Cf. After passing through the lens 314, 316, and 318, the beam is directed via mirrors to one end of the cylinder of the waveguide 310 at the entrance of the waveguide structure (Lagrange point LA). The single-helix wire 311a is connected to a current source 320 at the two terminals. The double-helix wires 311b are wired in series to the current source 320. A camera lens 322 is directed toward the side of the cylinder of the waveguide 310 allowing sideview images to be captured by a CCD camera 324. To enable sideview observations, the PDMS of the test waveguide 310 was mixed with a small amount of TiO2nanoparticles (165 ^^ ^^ in diameter).

[0059] The beam from the waveguide 310 is directed toward a 10ൈ objective lens 326. The output beam from the lends 326 is directed toward a focus lens 328 with a focal length, F =125 mm to a beam splitter 330. The beam splitter 330 directs one part of the beam to an attenuator 332. The other part of the beam is directed toward a lens 334 having a focal length, F = 35 mm and another lens 336 having a focal length, F = 100 mm to a linear polarizer 338. The beam from the attenuator 332 and the linear polarizer 338 are combined by a beam splitter 340. The output is sent through a linear polarizer 342, through a lens 344 having focal length, F =200 mm, and another lens 346 having a focal length, F = 100 mm to a CCD image camera 350.

[0060] In this example, for the single-helix configuration, the experiment was conducted with the waveguide 310 having a single spiral wire 311a positioned 1.85 ^^ ^^ from the center C, having a pitch Λ ൌ 8 ^^ ^^. The beam was injected into the waveguide 310 at the Lagrange point ^^^. For the double-helix experiment, each one of the two spiral wires 311b was positioned at 1.8 mm from the center C and the helix pitch used is Λ= 6.3 cm. In this case, the Lagrange point is located at the center C where the light is in. For both occasions, the elliptical Trojan state is formed when excited with a Gaussian beam of spot size ^^^≃ 110 ^^ ^^. The current for the single helix is 4.0 A while is 3.5 A for the double.

[0061] The optical dynamics of these beams are monitored both at the output facet of the cylinder by the CCD imaging camera 350 as well as from the side using the CCD imaging camera 324. In the experiment with the single spiral iron wire, the Trojan mode is excited by a Gaussian beam with a spot size of ^^^≃ 110 ^^ ^^ from the laser 312. In this example, the laser 312 is a He-Ne laser ( ^^^= 632.8 ^^ ^^), right at the Lagrange point of this arrangement. A DC current ^^ ൌ 4.0 ^^ from the current source 320 is passed through the wire to heat the waveguide 310. In this case, the surface of the wire is estimated to be at 50 ^^ above the outer surface temperature of the test cylinder, and the resulting temperature distribution leads to a spiraling logarithmic index potential 12 4885-3549-1273.1 065715-000158WOPTΔ ^^ ≃ െ2.25 ൈ 10ିଶ^1 െ ^^ ^^ ^ ^^ / ^^^ / ^^ ^^^ ^^ / ^^^^ where ^^ is the distance from the wire center. In this configuration, the Lagrange point ^^^is positioned ~ 430 ^^ ^^ away from the center C of the cylinder.

[0062] The Trojan beam guiding in the index potential produced by a single helical heat source in the experiments were captured in different images shown in FIG.4A. A first image 410 is a side view image of a trapped fundamental Trojan beam 414 in the cylinder in the tested waveguide 310 in FIG.3 with a single spiraling wire. The spiraling wire 412 (Λ= 8 cm), carrying a current I = 4.0 A, is also shown in the image 410. A second image 420 is an intensity profile of the Trojan mode at the output facet of the cylinder, after a distance approximately 30 cm. The mean value of the spot size radius is ~112 μm with an ellipticity of 0.49 (minor / major axis).

[0063] An image 430 shows the numerically obtained wavefront phase structure of the fundamental Trojan state centered around ^^^(depicted as a white dot 432). The presence of the characteristic X-shaped γuv term is evident. An image 440 the experimentally observed wavefront phase of the Trojan state corresponding to the Trojan beam 414 in the image 410 detected via interferometry.

[0064] FIG.4B is a graph 450 that plots ellipicity and spot size (w) against current squared ^^ଶ. A first group of points 452 shows the spot size plotted against current squared and a second group of points 454 shows ellipicity plotted against current squared. The graph 450 shows the dependence of the output mean spot size and ellipticity of the Trojan beam output versus current ^^ଶ.

[0065] The results of experiments demonstrating that the Trojan beam is captured around the Lagrange point ^^^are shown in the image 410. The image 410 demonstrates that the Trojan beam retains its spot size both at the output facet shown in the image 420 and during propagation as shown in the image 410. At the output, the mean value of the spot size radius is ~112 ^^ ^^ with an ellipticity of 0.49.

[0066] FIG.5 are images that schematically show the Trojan beam trapping in the index landscape produced by a double-helix current source. A first image 510 is a side view image of twisted wires 512 and 514 with a trapped fundamental Trojan beam 516 in the cylinder of a waveguide 310 in FIG.3. The image 510 shows that a Trojan optical mode is established when a current ^^ ൌ 3.5 ^^ from the current source 320 is passed through the wires 512 and 514 in series. An elliptical Trojan state is formed when excited with a Gaussian beam of spot size ^^^≃ 110 ^^ ^^ as shown in the image 510. A second image 520 shows an intensity profile of the Trojan mode at the output facet 13 4885-3549-1273.1 065715-000158WOPTof the cylinder, after a distance approximately 30 cm. The beam is trapped around the Lagrange point given that at the output its mean spot size radius is approximately 120 ^^ ^^ with an ellipticity of 0.9 as shown by the image 520. In this case, the Lagrange point is located at the center of this double-helix wire system (pitch Λ ൌ 6.3 ^^ ^^). An image 530 shows an X-shaped interferogram resulting from the corresponding ^^ ^^ ^^ phase structure of the trapped mode of the Trojan mode, indicative of the X-shaped phase term.

[0067] To demonstrate in a definitive manner that what is observed in the experimental setup 300 in FIG.3 is a Trojan beam, the phase structure was experimentally detected as the phase structure is unique to this trapped state. To explain the origin and significance of this phase Ψ^ ^^, ^^^, the effective potential landscape in the vicinity of a Lagrange point ( ^^^, ^^^) is approximated represented by ^^ஐ,^^^ൌ ^^^^௫െ ^ ^^^ଶ^ ^^ െ ^^^^ଶ^ ^^ଶଶ^ ^^ െ ^^^^ଶ^ / 2 where ^^^ଶand ^^ଶଶrepresent local negative of Equation (1)are obtained by: ^^^ ^^, ^^^ ൌ ^^ ^^ି^^௨ି௨^^మ^^ି^^௩ି௩^^మ^^^ఎ^௨ି௨^^^௩ି௩^^^^^ஐ^௨^௩ି௩^௨^^^^ఙ௭^^ି^^^ೌ^௭,^6^9^where ^^,Ψ^^^, ^^^ൌ ^^ ^^ ^ ^^ ^^ ^ ^^ ^^ ^^. The first two terms ( ^^, ^^) are associated with the tilt of the Trojan beam when traversing its helical path while the last term ^^ ^^ ^^ governs the internal energy flow that allows this elliptical state to reorient itself during propagation. The ^^ ^^ ^^ phase is special to these Trojan entities, given that it doesn’t appear in standard dielectric waveguide settings or gain- guiding systems.

[0068] Numerical simulations carried out for the index profiles used in the exemplary experiments as shown in FIG.2B lead to similar results as shown in the image 430 in FIG.4A. To detect the ^^ ^^ ^^ phase, interferometric measurements using a Mach-Zehnder arrangement were performed where the Trojan beam in one of the arms is appropriately tilted so as to remove the effects from 14 4885-3549-1273.1 065715-000158WOPTthe ( ^^ ^^ ^ ^^ ^^) contributions, as shown in the experimental setup 300 in FIG.3. The main body of the Mach-Zehnder interferometry arrangement is between the lens 328 and the lens 344. The light passing through the beam splitter 330 is separated to two arms by 50:50, one of which is attenuated by an attenuator 332 while the other being magnified by 2.9ൈ through a 4f system comprising the lens 334 and the lens 336 to planarize the wave front. The beams along the two arms are recombined at the beam splitter 340 and interfere with each other, whose interference pattern is projected to the CCD camera 350 by a 4f system made of the lens 344 and the lens 346. A polarizer 342 is used to uniformize the polarization states of the two arms, while the other polarizer 338 adjusts the relative light intensity. The resulting X-shaped interferogram (produced by the ^^ ^^ ^^ term) is shown in the image 440 in FIG.4A. This provides evidence that what is observed in the experimental setup 300 is a Trojan beam. The variation of the average beam spot size as a function of current is depicted in the graph 450 in FIG. 4B. Significantly, in optical arrangements, it is possible to excite multiple Lagrange points from several potential sources, something that is not feasible in celestial mechanics because many-body systems are chaotic.

[0069] Equation (5) is mathematically equivalent to a quantum Schrödinger equation for an electron under a twisted electric field in the co-rotating frame. In this respect, similar to the optical Trojan beam, electrons can be captured in a helicoidal electric field distribution by a negative charged metallic wire. Thus, an example charged particle waveguide system 600 is shown in FIG. 6. The system 600 includes a cathode-ray tube (CRT) structure 610. The structure 610 includes a tube section 612 and a conical section 614. An electron gun 620 includes a heater 622, a cathode 624, an anode 626, and a control grid 628. A voltage source 630 is coupled to the cathode 624 and the anode 626. The electron gun 620 emits charged particles such as electrons in the tube section 612. The anode 626 is in the form of a metal plate carrying positive voltage (relative to the cathode 622) with a hole 632 in the center that allows accelerated electrons to pass through. A set of focusing coils 640 surrounds the tube section 612. A set of deflecting coils 642 surrounds the tube section 612. The conical section 614 includes a coating of phosphor on an interior surface 644. A hollow metal tube 650 is suspended in alignment with the tube 612. A helical metal wire 652 is wound in the tube 650. The tube 650 is grounded and the wire 652 is negatively charged via a voltage source 654. In this example, the tube 650 has a diameter of 3 cm and a length of 10 cm. The end of the tube 650 is positioned approximately 5 mm from the interior surface 644 of the structure 610. 15 4885-3549-1273.1 065715-000158WOPT

[0070] In this example, electrons are excited by the electron gun 620 in the cathode-ray tube (CRT) 610. Through the accelerating anode 622 and the focusing and deflecting coils 640 and 642, the high-speed electron beam is deflected to a Lagrange point induced by the negative charged iron wire 652 embedded in the grounded cylindrical hollow tube 650. A negative ^^^^^ 0^ is created by a voltage source 654 connecting the tube 650 at ground and the iron wire 652. This generates a twisted logarithmic electric field. In this exemplary setup, the Lagrange point forms a stable electron waveguide, where electron beams will be stably transported. The pertinent size of guided electron beams can at a sub-nanometer scale under proper design.

[0071] The example Trojan particle beam based waveguide system 600 in FIG.6 or the particle waveguide system 700 in FIG.7A allows the possibility to indefinitely guide and confine charged particles in periodically twisted electrostatic fields by exploiting the counterintuitive attributes of Lagrange points. Similar to celestial mechanics, electrons and protons can be perpetually captured in such arrangements within induced Lagrange points via dynamic stabilization processes facilitated by Coriolis forces. Example electron / proton wave-guiding structures can operate in single-mode formats with very small cross-sections where quantum mechanical effects can be manifested, in a manner that directly emulates the optical fiber paradigm. Here, the problem is treated both quantum-mechanically and classically (up to the relativistic regime). By accounting for Larmor radiation, these arrangements can exhibit exceedingly small losses – thus opening up the possibility for transporting charged particles at high speeds over prolonged distances. The example charged particle guide system introduces new degrees of freedom that does not rely on the well-established strong focusing principle that relies on quadrupole fields. This same strategy can be deployed in other more common settings such as electron microscopies and lithographies, particle accelerators and vacuum electronic and communication systems using high-speed charged particles. Finally, the prospect exists to physically shuttle qubits either between multiplexed traps or nodes of a quantum network where adiabatic means are used to propel particles in the Lagrange waveguides.

[0072] FIG. 7A shows another example waveguide system 700 based on charged-particle beam dynamics in a Lagrange waveguide. The waveguide system 700 includes an outer shield 710 mounted on a base 712. An inner tube 720 includes a twisted wire 722. The inner tube 720 is electrically grounded to form a circuit with a voltage source 726 coupled to the twisted wire 722. An electron gun 730 generates a stream of electrons through an aperture of an anode 732. A 16 4885-3549-1273.1 065715-000158WOPTfocusing coil 740 and a deflecting coil 742 allow the electron beam to be directed to the Lagrange points generated in the inner tube 720.

[0073] The system 700 is set up to observe charged particle Trojan bound states from a beam 750 (bright yellow beam). A stable Lagrange point is established by twisting the electrostatic potential produced by the charged helical metallic wire 722 when kept at a potential V0. FIG.7B is a cross section of the tube 720 and the twisted wire 722. An electron 760 is trapped at a stable Lagrange point and follows a helical trajectory along the z axis. Stability in trapping this charged particle is provided through the Coriolis force Fc represented by an arrow 762 when viewed from the co- rotating frame. R represents the distance between the center represented by an “X” 764 of the grounded metallic tube 720 and the Lagrange point and d is the distance C between the center C and the center of the helical wire 722.

[0074] In the waveguide 700, electrons are emitted from the electron gun 730 after being accelerated by a static electric field. In principle, the electrons can be further accelerated using linear accelerator arrangements before injected in the guiding setup. These high-speed electrons can be confined and guided around stable Lagrange points that are induced by a twisted electric field ^^^ ^^, ^^, ^^^. The twisted electric field configuration is produced by the helical metal wire 722 (of pitch Λ and helical radius ^^) that is kept at a potential ^^^with respect to a surrounding metal cylinder 720 that is grounded. A logarithmic repelling electrostatic potential is induced when the single charged helical metallic wire is axially inserted in a grounded metallic cylinder. COMSOL simulations confirm the theoretical calculation that the potential distribution inside the cylinder is dictated by the radii of the wire and the cylinder tube.

[0075] In the charged particle waveguide systems, the guided electrons will then be transported along a helicoidal trajectory within a small cross-section. A similar platform can be deployed to guide heavier charged particles such as protons, ions, and the like.

[0076] Analysis of the stationary quantum wavefunctions or modes associated with a charged particle like an electron, when trapped and transported within a Lagrange waveguide are as follows. To explore this possibility, the Schrödinger equation is used under the assumption that the particles primarily move along the ^^ axis with paraxial momenta ^^௭ൌ ℏ ^^௭≫ ^^௫, ^^௬. A moving coordinate frame ^^ᇱൌ ^^, ^^ᇱൌ ^^ െ ^^ / ^^௭is used, where ^^௭ൌ ℏ ^^௭ / ^^^represents the dominant ^^ component of the electron’s velocity while ^^^denotes its corresponding mass. 17 4885-3549-1273.1 065715-000158WOPTMoreover, ^^௭≃ 2 ^^ / ^^ௗ^where ^^ௗ^is the electron’s de Broglie wavelength. In this case, the Schrödinger equation takes the form: ^^ ^^௭ ^^ ൌ ^^^′ ^^ (10)where ^^^′ ൌ െ∇ଶ௫,௬ / ^2 ^^௭^ ^ ^ ^^^ / ℏ ^^௭ଶ^ ^^^ ^^, ^^, ^^^ and ^^^ ^^, ^^, ^^^ ൌ ^^ ^^^ ^^, ^^, ^^^ indicates the twisted three-dimensional electrostatic potential in the absolute (stationary) frame. At this point, it is more convenient to consider this problem within the co-rotating frame with normalized coordinates ( ^^, ^^, ^^) of the helix (^ ^^, ^^^்ൌ ^^^Ω ^^^^ ^^, ^^^்) to formally decouple the twisted static electric potential ^^ from ^^, in which case ^^ ൌ ^^^^^, ^^^, ^ ^^, ^^, ^^^ represent the normalized coordinates in the stationary frame and Ω is the normalized spatial angular velocity. Here, ^^^θ^ൌ^cos^ ^^^ sin^ ^^^െsin^ ^^^ cos^ ^^^൨, and the normalized terms ^^ ൌ ^^ / ^^^ ൌ ^^ / ^ ^^ ^^^ଶ^, ^^ ൌ ^^ / ^^^, ^^ ൌ ^^ / ^^^, where^^ ൌ ^^ ൈ ^^Ωis introduced to account for Coriolis effects where ^^ ൌ 2 ^^ ^^^^^^ / Λ with ^^^being the normalization factor in ^^̂ direction and ^^^^ൌ ^^ ^^^ ^ ^^ ^^^ In the rotating frame, the normalized Schrödinger equation now takes the form: ^^^^ ^^^^ ^^ ൌ ^1൫^ ^⃑^ ^^ െ^^^^ ൯ଶ^^^ஐ,^^^^ ^^, ^^^൨ ^^ ^11^where ^ ^⃑^ ൌ െ ^^ ^^^ ^^ / ^^ ^^^^െ ^^ ൌ is an effective potential that now also incorporates thethat the helicoidal metallic wire is kept at a repulsive voltage ^^^^ 0 within a grounded tube of radius ^^ such as the tube 720 in FIG. 7A, the generatedelectrostatic potential is expected to vary in a logarithmic fashion ^^ ൌ ^^^ln^^^′ / ^^^ / ln^^^ / ^^^withinthe stationary coordinate system ( ^^, ^^, ^^) where ^^ is the radius of the wire and ^^′ ൌ^^ ^^ െ ^^ cos^Ω^ ^^^^ଶ^ ^ ^^ െ ^^ sin^Ω^ ^^^^ଶ, Ω^ ൌ 2 ^^ / Λ. To do so, a vector potential^^^^ ൌ^^^^ ൈ ^ ^^ Ωൌ with ^^^ being the normalizationfactor in ^^̂ direction and ^^^^ൌ ^^ ^^^ ^ ^^ ^^^ In the rotating frame, the normalized Schrödinger equation now takes the form: ^^^^ ^^^1 ^^^^ ^ ^^4885-3549-1273.1 065715-000158WOPTwhere ^ ^⃑^^^ൌ െ ^^ ^^^ ^^ / ^^ ^^ െ ^^ ^^^ ^^ / ^^ ^^ is the normalized momentum operator, and ^^ஐ,^^^^ ^^, ^^^ ൌ ^^ ^^^ଶ^^^^^^^^, ^^, ^^ ൌ 0^ / ℏଶെ Ωଶ^^ஐଶ / 2 is an effective potential that now also incorporates the term െΩଶ^^ஐଶ / 2. Given that the helicoidal metallic wire is kept at a repulsive voltage ^^^^ 0 within a grounded tube of radius ^^ such as the tube 720 in FIG. 7A, the generatedelectrostatic potential is expected to vary in a logarithmic fashion ^^ ൌ ^^^ln^^^′ / ^^^ / ln^^^ / ^^^withinthe stationary coordinate system ( ^^, ^^, ^^) where ^^ is the radius of the wire and ^^′ ൌ ^^ ^^ െ ^^ cos^Ω^^^^^ଶ^ ^ ^^ െ ^^ sin^Ω^^^^^ଶ, Ω^ൌ 2 ^^ / Λ.particles at a stable Lagrange point may be shown in relation to an electron based waveguide such as those in FIGs. 6 and 7A. In this example, the electron spiraling around the center of the tube ^^ with a wire pitch of Λ ൌ 6 cm, a radius of wire, ^^ ൌ 275 μm and a tube radius, ^^ ൌ 2 cm at voltage applied to the wire of ^^^ൌ െ2.15 kV. In this example, the injected electrons have a kinetic energy of 30 keV. FIG.8A is a 3D graph 800 that shows an induced logarithmic, defocusing and spiraling electrostatic potential ^^ represented by an area 810 when viewed within the stationary frame at ^^ ൌ 0. This potential profile rotates along z at a constant angular velocity Ω around the center C ^0,0^. The corresponding iso-contour potential lines 812 are also shown. For this scenario, two Lagrange points in positions 814 and 816 are induced (designated as L^and L^) where the electrostatic repulsion balances the centrifugal force, i.e., ∇ ^^ஐ,^^^ൌ 0. In this case, only L^is stable while L^, being a saddle point, is unstable. As previously noted, ^^ஐ,^^^has a maximum at L^.

[0078] FIG.8B shows a 3D graph 820 that shows the corresponding effective potential ^^ஐ,^^^^^^, ^^^as viewed in the co-rotating frame as an area 830. In the co-rotating frame ( ^^, ^^), the effective potential ^^ఆ,^^^now incorporates centrifugal effects and exhibits two Lagrange points at positions 832 and 834 ( ^^^and ^^^respectively). The Lagrange point ^^^is a saddle point and hence is unstable. The Lagrange point ^^^( ^^^, ^^^) exhibits a maximum, around which dynamics are stabilized through Coriolis effects.

[0079] FIG.8C shows a series of images 840, 842, and 844 that are numerically obtained quantum wavefunction probability distributions of different Trojan modes. The image 840 shows the distribution for the fundamental mode, while the images 842 and 844 show the distribution for the next two electron Trojan modes. The quantum states in the images 840, 842, and 844 at the first Lagrange point ^^^correspond to the potential landscapes in the graphs 800 and 820. In this case, 19 4885-3549-1273.1 065715-000158WOPTthe quantum modes are elliptical in the ( ^^ᇱ, ^^ᇱ) system where ^^ᇱൌ ^^ െ ^^^, ^^ᇱൌ ^^ െ ^^^. Images 850, 852, and 854 show respective phase structures associated with these three electron wavefunction states. In all of these states, as shown in the images 850, 852, and 854, the phase profile exhibits an X-shaped pattern. FIG. 8D shows a 3D graph 860 of the twisting path of the probability profile of the electron ground state, shown as a helical volume 862 along the z direction. The graph 860 demonstrates stable propagation of the quantum Trojan fundamental mode as obtained numerically by solving Equation (10). The quantum probability function remains invariant along its helical path.

[0080] To obtain the electron quantum eigenfunctions, Equation (10) is solved numerically. The wavefunction probability distribution corresponding to the ground state is shown in the image 840. In this particular arrangement, the fundamental mode in the image 840 has an elliptical Gaussian- like shape with a mean spot size radius of ~0.3 μm. Similarly, the images 842 and 844 display the wavefunction profiles for the next two modes. In principle, the spot size of the ground state can be further reduced to a few nanometers by either increasing ^^^or by decreasing the de Broglie wavelength. To understand the nature of the trapped quantum wavefunctions, the effective potential is approximately expanded around L^to second order, i.e., ^^ஐ,^^^≅ െ^ ^^^ଶ^^ଶ^ ^^ଶଶ^^ଶ^ / 2, where ^^^ଶ, ^^ଶଶrepresent the corresponding curvatures oflandscape. In this case, it may be shown that the ground state eigenfunction is analytically given by ^^^ൌ ^^ ^^ି^௨మ^^ି^௩మ^^^ఎ௨௩^^^ఉ^. This mode involves a ^^ ^^ phase distribution that is unique to this particular arrangement – a direct byproduct of Coriolis effects. This phase term is manifested in an X-like manner and happens to persist even for high-order states as shown in the images 850, 852, and 854. The theoretical results are confirmed by numerically solving the Schrödinger problem in both the stationary and rotating frames. As shown in the graph 860 in FIG. 8D, the probability profile of the ground state remains invariant (trapped) during propagation while twisting around the center C – a hallmark of the guiding behavior. Finally, Lagrange waveguiding is also possible even in the case where ^^^is attractive ( ^^^^ 0). The dynamics of non-relativistic and relativistic charged particles in a Lagrange waveguide are detailed below. Investigation of a charged particle guiding system under both relativistic and non- relativistic conditions is explained. This classical treatment is imperative to understand how a Lagrange waveguide will respond under incoherent excitation of several modes. As before, the periodic spiraling electric potential distribution ^^^ ^^, ^^, ^^^ ൌ ^^^ ^^, ^^, ^^ ^ Λ^ within the co-rotating 20 4885-3549-1273.1 065715-000158WOPTframe ^ ^^ , ^^ , ^^ ^ with actual units is considered where ^^ ൌ ^^, ^ ^^்ஐబ ஐబ ஐబ ஐబ ஐబ, ^^ஐబ൧ ൌ ^^^Ω ^^^^^^^். In this case, the Newtonian dynamics ^^ஐబ^ ^^^, ^^ஐబ^ ^^^,by: ^^ଶ^^^^ ^^∇௨,௩^ ^^ஐ ,^^^൫ ^^ஐ, ^^ஐ൯^^^ ^^^ ^^ଶൌ െబ బ బ^^ ^ 2^^ ^^ ^^^^ ൈ Ω^ ^^^^ , ^12^ofconfinement is considered around a stable Lagrange point by analyzing the same waveguide configuration. In this case, 30-keV electrons are injected in a 30 cm long Lagrange waveguide similar to that in FIG. 7A after passing through an aperture of radius ^^^ൌ 10 μm. The thermionically emitted electrons from a heated cathode (of temperature ^^ ൌ 1000 K) obey a Maxwell-Boltzmann distribution with a probability density ^^^ ^^^^ ∝ exp ^െ ^^^ଶ / 2 ^^௩ଶ^, where ^^ ൌ ^^, ^^, ^^௩ൌ^^^^^^ / ^^^denotes the velocity variance and ^^^represents the Boltzmann constant.

[0081] graph 900 that shows stable trapping of an electron unfolding around theLagrange point ^^^for the potential landscapes depicted in FIGs.8A and 8B, as viewed within the co-rotating frame. The graph 900 shows the waveguiding action of an example particle waveguide. FIG.9B is a graph 910 that shows spatial and velocity distributions shown as traces 912 and 914 respectively associated with the injected electrons. The distributions are plotted when the aperture at the input of the example Lagrange waveguide has a radius of 10 ^^ ^^ and is centered at the Lagrange point ^^^.

[0082] FIG. 9C shows a histogram 920 of electron spatial distribution after 30 cm in free space ( ^^^ൌ 0). In this case, the electron beam diffracts to a spot size of ~1 mm. In the absence of a repelling potential ( ^^^ൌ 0), the electron beam spreads or diffracts to a spot size of ~1 mm after 30 cm of propagation. FIG.9C also shows a histogram 930 of electron spatial distribution after 30 cm of propagation in a Lagrange waveguide when the repelling potential is ^^^ൌ െ2.15 ^^ ^^. In this latter scenario, the electron beam is stably guided with a mean spot size of ~65 μm via the Lagrange based waveguide. The beam spot size is considerably larger than the mean spot size of the fundamental mode (0.3 μm) since several modes are incoherently excited under these 21 4885-3549-1273.1 065715-000158WOPTconditions. In the example waveguide, the space-charge effect can be ignored even the electron charge current goes up to a μA level.

[0083] FIG.9C shows an image 940 of the transverse trajectory (represented as curve 942) of a high-speed electron (ve=0.999 c) over 100 m. A dashed line 944 depicts the Lagrange point position, located at a radius of ~112 μm from the center. The electron was positioned at a distance ~5 μm away from the Lagrange point. FIG.9C also shows an electron beam transverse distribution histogram 950 after 100 m of propagation in a Lagrange waveguide when the repelling potential is V_0=-80 kV. The electron beam is stably guided with a mean spot size of ~215 μm. A scaling bar 960 in the histograms 920, 930, and 950, corresponds to 300 μm while a scaling bar 962 in the insets of histograms 940 and 960 corresponds to 200 μm.

[0084] This same problem is analyzed in the relativistic regime by solving the equation ofmotion ^^^ௗ^ ^^ ^^ௗ௧ ^ ൌ ^^⃑^ െ ^^^⃑^ ⋅ ^ ^^ ^^^^^ ^^ ^^ / ^^ଶ^ where^^⃑^ ൌ െ ^^∇ ^^. In this case, the Lagrange point islocated same line that connects the center with the helicoidal wire source at a distance ^^ from the center. For a stable Lagrange point L^, this distance is given by ^^ ൌ ^ ^^ െ ^ ^^ଶ^ 4 ^^Θ / ^ ^^^^^Ωଶ^ ^^௭ଶ^^ / 2 where ^^ ൌ 1 / ^1 െ ^^^ଶ / ^^ଶis the Lorentz factor, ^^^is the rest To confirm trapping and guidanceare carried out when ^^^ൌ 0.999 ^^. The trajectory of a high-speed electron ( ^^^ൌ 0.999 ^^) around L^is depicted in the image 840 after 100 m of propagation. In contrast, the histogram 950 shows an electron beam for this samedistance. For the graphs and images in FIGs. 9A-9C, the system parameters were taken to be Λ ൌ6 cm, ^^ ൌ 500 μm, ^^ ൌ 1.8 mm, ^^ ൌ 2 cm, and ^^^ ൌ െ80 kV. Under these conditions, therelativistic Lagrange point is located at a distance of ~112 μm from the center. In obtaining the results of the image 940 and the histogram 950, it is assumed the thermionic electrons emitted from a heated cathode (of temperature ^^ ൌ 1000 K) after acceleration, entered the relativistic Lagrange waveguide through an aperture of radius 10 μm. The pertinent velocity and spatial distributions are depicted in the graph 810 in FIG.8B.

[0085] This same approach can also be used to design a Lagrange waveguide to guide heavier charged particles such as protons and ions. FIG.10 shows a table 1000 that provides waveguide design parameters for transporting protons, Strontium (88Sr+) ions, and Ytterbium ions (171Yb+) after acquiring a terminal velocity ^^௭of ~10଼m / s. The spot size of the quantum ground state 22 4885-3549-1273.1 065715-000158WOPT(fundamental mode) in the Lagrange waveguide is also given for comparison purposes. In this example, the wire radius is .275 mm, the grounded outer tube radius is 2 cm, and the distance between the wire center and C is 1.8 mm.

[0086] FIG.11A shows a test setup 1100 for measuring the Trojan beam generated via a Lagrange point based charged particle waveguide. The test setup 1100 includes a support table 1110 that includes a support surface 1112 that is supported by legs 1114. A scroll pump 1116 is installed under the support surface 1112. The support surface 1112 includes a clamping assembly to hold a test assembly 1120.

[0087] FIG.11B is an isolated view of the test assembly 1120 in FIG.11A. The assembly 1120 includes a reflection high-energy electron diffraction (RHEED) gun 1122, a mounting flange 1124, a guide tube 1126, a turbomolecular pump 1128, an adjustment unit 1130, an actuator 1132, a cylindrical test chamber 1134, a connector 1136 for a helicoil voltage input, a connector 1138 of deflection coils, a pressure gauge 1140 and an end viewport 1142. The RHEED gun 1122 includes an electron gun, while also having a set of deflection coils used to firstly deflect the beam output from the electron gun, and a set of focusing electrodes to focus the beam from the electron gun. In this example, the electron gun in the RHEED gun 1122 is a thermionic emission electron gun that generates an electron beam, but other types of electron guns may be used. The focusing electrodes focus the electron beam to a small spot at a needed distance. The deflection coils deflect the electron beam to change its direction to ensure it moves towards the desired position. In this example, the desired position is an aperture in the adjustment unit 1130. The RHEED gun 1122 is supported by the mounting flange 1124. The mounting flange 1124 is generally circular and is fit into the support surface 1112 of the table 1110. The RHEED gun 1122 emits the electron beam to a guide tube 1126. The guide tube 1126 includes a branch 1150 that is attached to the turbomolecular pump 1128. The turbomolecular pump 1128 along with the scroll pump 1116 are used to pump away gasses in the assembly 1120 to create a vacuum. The scroll pump 1116 vacuums the assembly 1120 from atmospheric pressure to 0.1 mbar and then the turbomolecular 1128 pump from 0.1 mbar to 10ି଼mbar. The other end of the guide tube 1126 is attached to the adjustment unit 1130. The adjustment unit 1130 includes a set of deflection coils, an aperture, and a X-Y stage. In this example, the aperture has a small cross-section of 10 microns in radius. The electron beam is aligned with the aperture by the deflection coil in RHEED gun 1122 to limit the beam size injecting into the waveguide. The set of deflection coils in the adjustment unit 1130 is 23 4885-3549-1273.1 065715-000158WOPTused to deflect the electron beam to a proper tilting angle that matches the excitation requirement of the Trojan beam in the waveguide. The X-Y stage allows movement of the horizontal position of the parts above the stage altogether. Since the test chamber 1334 is attached to the adjustment unit 1130, the X-Y stage allows movement of the test chamber 1134, the pressure gauge 1140 and the viewport 1142, to align the waveguide entrance (the desired Lagrange point) with the deflected electron beam.

[0088] The deflection coil and the X-Y stage of the adjustment unit 1130 direct the electron beam to a test waveguide that is fixed in the test chamber 1134. The test waveguide is a charged particle waveguide designed according to the principles described herein. In this example, the main body of the waveguide, including the metallic helical wire and the outer grounded tube, is assembled in and extends through the test chamber 1134. The exterior of the test chamber 1134 holds the connectors 1136 and 1138. The connectors 1136 and 1138 allow connection of electrical devices inside the chamber 1134 to outer voltage / current sources. The test chamber 1134 includes an aperture attached to the pressure gauge 1140. The pressure gauge 1140 monitors the pressure in the test chamber 1134. The end viewport 1142 has a phosphorus screen. The phosphorus screen generates a light spot when hit by electrons from the waveguide in the test chamber 1134. The view screen of the viewport 1142 is used for viewing the electron beam shape when exiting the waveguide in the test chamber 1134.

[0089] An estimate for the energy loss rate associated with a charged particle when trapped in a Lagrange waveguide may be provided. Given that the particle will be transported along a helical trajectory, it will inevitably radiate energy because of transverse acceleration. In general, in a twisted arrangement, the respective dynamics can be decomposed into a uniform motion ^^௭and a transverse velocity ^^^( ^^^≪ ^^௭) that leads to a centripetal acceleration ^^⃑^^^ൌ െ ^^^ ^^^ / ^^, which is perpendicular toparticle’s total velocity ^^^^ൌ ^^௭^^^ ^ ^^^. In the latter expression, ^^ represents the radius of the helicoidal Lagrange waveguide. In this case, the relativistic radiation loss rate can be obtained from: ଶ^ ଶ^^^^^ ^^ ^^ ^^^^^^ ^^^ଶ^where ^^ ൌ 1 / ^1 െ ^^^ଶis the Lorentz factor and^^^^ ^^ ൌ ^^^ ^^ / ^^ is the velocity ratio. In this regard,theexperienced by a 30-keV electron in a Lagrange waveguide such as the 24 4885-3549-1273.1 065715-000158WOPTwaveguide 700 in FIG. 7A, with Λ ൌ 6 cm and ^^ ൌ 0.74 mm will be 0.27 eV / s, which isnegligible even after a propagation distance of 100,000 km, given that it translates to~4 ൈ 10ି^^ dB / km. In contrast, a 1-TeV proton, moving at a speed ^^^ ൌ ^1 െ 4.4 ൈ 10ି^^ ^^ ina Lagrange waveguide with Λ ൌ 60 cm and ^^ ൌ 260 nm (with wire rotation radius ^^ ൌ1.8 mm, ^^^ൌ 220 kV) is expected to encounter a radiation loss rate of 302 eV / s, which is minute as compared to the energy of the particle. Another source for losses can be manifested when a charged particle is moving close to an imperfectly conducing surface, in which case the drag force leads to Ohmic heating. For the 30 keV electron example considered above, if a copper outer tube has a radius of ^^ ൌ 2 cm, the drag loss rate is predicted to be only 66.3 neV / s. In addition, the thermal noise generated by the twisted wire is also considered and is of no practical significance in this Lagrange waveguide arrangement. Finally, unlike photons for which loss is equivalent to a complete annihilation, in the example waveguides, loss merely means a reduction in the energy of the particles. Consequently, charged particles can still preserve properties like entanglement under appropriate conditions.

[0090] The above estimates indicate that Lagrange waveguides may guide charged particles in accelerator designs or in extremely low-loss electron communication systems that operate in either the classical or quantum domain.

[0091] In the acoustic regime, the propagation of acoustic waves is governed by the wave equation ∇ଶ^^ െ ^1 / ^^^ଶ^ ^^ଶ^^ / ^^ଶ^^ଶൌ 0, where ^^ represents the acoustic pressure and ^^^is the speed of sound in the medium. Within the paraxial approximation, the sound wave equation can be reduced to a form that is mathematically isomorphic to Equation (2). The bulk modulus ^^ and the density ^^ of the medium determine the sound velocity in the fluid ideally, i.e., ^^^ൌ^^^ / ^^. On many occasions the shear modulus ^^ of a solid medium is also involved in determining the sound speed for both transverse and longitudinal sound waves. The elastic modulus and density vary with either temperature or pressure. When an acoustic wave propagates in an ideal gas or air, ^^ ൌ^^^ ^^ where ^^ is the adiabatic index and ^^ stands for the pressure. In this case, the speed of sound in an ideal gas is given by ^^^ൌ^^^ ^^^^^ / ^^ where ^^ is the mass of a single molecule, ^^^is Boltzmann’s constant and ^^ is the absolute temperature. In this regard, an artificial helicoidal temperature distribution similar to the one used in the above optical setup or a graded density distribution will 25 4885-3549-1273.1 065715-000158WOPTprovide a graded “acoustic refractive index,” which may support guiding channels for acoustic Lagrange waves.

[0092] FIG 12 depicts an example of an acoustic Lagrange waveguide system 1200 that includes a gas tube 1210. In this example, the gas tube 1210 contains Helium gas at room temperature. A sound wave source 1220 such as an ultrasonic 3 MHz transducer is embedded in the setup to emit a sound wave beam into the gas tube 1210. A twisted wire 1230 is coupled to a current source 1232. Current passing through the twisted wire 1230 will heat up the gas in the gas tube 1210 above room temperature ^^^with a temperature distribution given by ^^^ ^^, ^^, ^^^ ൌ ^^^^ Δ ^^^ ^^, ^^, ^^^. Hence, the speed of sound, ^^^is given by ^^^ൌ ^^^ / ^^^where ^^^ൌ^^^^ / ^^ ൌ^1 / ^1 ^ Δ ^^ / ^^^^. A first-order Taylor expansion leads to ^^^≃ 1 െ Δ ^^^with Δ ^^^ൌ 0.5 ൈ Δ ^^ / ^^^. In this respect, the heating twisted wire 1230 induces a twisted “acoustic refractive index” profile that is analogous to the optical refractive index landscape, and can therefore induce a Lagrange point for guiding the acoustic wave represented by a line 1236.

[0093] The principles relating to the example charged particle waveguides in FIG.6 and FIG.7A enables charged particle guiding (up to relativistic speeds) in a manner akin to that responsible for light transport in optical fibers. At the induced Lagrange points, the electron / ion beams can be stably captured, because of Coriolis effects, in well-defined quantum states produced by long- range spiraling electrostatic forces. Unlike other approaches relying on quadrupole potential distributions, the example principles provide new degrees of freedom in inducing multiple parallel guiding channels that can be used to enable quantum coupling / splitting arrangements. The principles may be applied to systems transporting accelerated charged particles in vacuum where guiding. Other applications may include shuttling entangled ion qubits over long distances among quantum charge-coupled device systems (where magnetic decoherence could be an issue). Other applications may include optics, electrons, acoustics, microwaves, particle manipulation and detection, quantum processing / computing, free electron lasers, biology, ultracold atoms, TEMS, SEMS, traveling wave tubes and material processing. In various non-limiting aspects, the helical iron wire may be substituted by helical gas channels for cooling or, alternatively, the index structure can be generated through direct laser writing.

[0094] While various embodiments of the present invention have been described above, it should be understood that they have been presented by way of example only, and not limitation. Although the invention has been illustrated and described with respect to one or more implementations, 26 4885-3549-1273.1 065715-000158WOPTequivalent alterations and modifications will occur or be known to others skilled in the art upon the reading and understanding of this specification and the annexed drawings. In addition, while a particular feature of the invention may have been disclosed with respect to only one of several implementations, such feature may be combined with one or more other features of the other implementations as may be desired and advantageous for any given or particular application. Thus, the breadth and scope of the present invention should not be limited by any of the above- described embodiments. Rather, the scope of the invention should be defined in accordance with the following claims and their equivalents. 27 4885-3549-1273.1 065715-000158WOPT

Claims

CLAIMS:

1. A waveguide system, comprising: a beam input generated by a beam source; a waveguide guiding the beam input generated by the beam source, the waveguide including a transmission medium and a twisted potential distribution structure to transport a Trojan beam from the beam input.

2. The system of claim 1, wherein the waveguide includes a glass cylinder and wherein the transmission medium is polydimethylsiloxane (PDMS).

3. The system of claim 2, further comprising a current source coupled to the twisted potential distribution structure, wherein the twisted potential distribution structure is a conductive wire.

4. The system of claim 3, wherein the conductive wire is shaped in a single helix or a double helix.

5. The system of claim 2, wherein the beam input is a lightwave, and wherein the waveguide has a refractive index creating a stable Lagrange point dynamically stabilized by an optical Coriolis force to trap the lightwave as an optical Trojan beam.

6. The system of claim 5, wherein the beam source is a laser, and the Trojan beam is guided by the Lagrange point.

7. The system of claim 1, wherein the beam source is a stream of electrons generated by an electron gun and the waveguide includes a grounded hollow tube and wherein the twisted potential distribution structure is a helicoidal charged conductive wire with a negative voltage.

8. The system of claim 7, wherein the waveguide has a potential distribution generating a stable Lagrange point.

9. The system of claim 1, wherein the beam source is an audio wave generator and the waveguide includes a medium with a predetermined ratio of elastic modulus and density.

10. The system of claim 1, wherein the beam source is a charged particle emitter.

11. A method of producing a Trojan beam, the method comprising: directed an input beam to a transmission medium; and generating a current or voltage through a twisted potential distribution structure in the transmission medium to establish a Lagrange point to produce the Trojan beam from the input beam through the transmission medium. 28 4885-3549-1273.1 065715-000158WOPT12. The method of claim 11, wherein the waveguide includes a glass cylinder and wherein the transmission medium is polydimethylsiloxane (PDMS).

13. The method of claim 12, wherein a current source is coupled to the twisted potential distribution structure, and wherein the twisted potential distribution structure is a conductive wire.

14. The method of claim 13, wherein the conductive wire is shaped in a single helix or a double helix.

15. The method of claim 12, wherein the input beam is a lightwave, and wherein the waveguide has a refractive index creating a stable Lagrange point dynamically stabilized by an optical Coriolis force to trap the lightwave as an optical Trojan beam.

16. The method of claim 15, wherein the beam input is a laser, and the Trojan beam is guided by the Lagrange point.

17. The method of claim 11, wherein the input beam is a stream of electrons generated by an electron gun and the waveguide includes a grounded hollow tube and wherein the twisted potential distribution structure is a helicoidal charged conductive wire with a negative voltage.

18. The method of claim 17, wherein the waveguide has a potential distribution generating a stable Lagrange point.

19. The method of claim 11, wherein the input beam is an audio wave and the waveguide includes a medium with a predetermined ratio of elastic modulus and density.

20. The method of claim 11, wherein the input beam is generated by a charged particle emitter.

21. An optical waveguide comprising: a waveguide optically coupled to a light beam source generated a light beam, the waveguide including a light transmitting material and a helical channel embedded in the light transmitting material; and a current source coupled to the helical channel to generate a Lagrange point in the light transmitting medium to generate a Trojan beam from the light beam through the light transmitting material.

22. The waveguide of claim 21, wherein the waveguide includes a glass cylinder and wherein the light transmitting material is polydimethylsiloxane (PDMS).

23. The waveguide of claim 21, wherein the helical channel includes a wire shaped in a single helix or a double helix. 29 4885-3549-1273.1 065715-000158WOPT24. The waveguide of claim 21, wherein the waveguide has a refractive index and the Lagrange point is a stable Lagrange point dynamically stabilized by an optical Coriolis force to trap the light beam as an optical Trojan beam.

25. The waveguide of claim 21, wherein the light beam source is a laser.

26. The waveguide of claim 21, wherein the helical channel is a helical gas channel.

27. A charged particle waveguide comprising: a cylindrical body having an aperture on one end, the aperture filtering a charged particle beam generated from a charged particle source; a helical wire centered around an axis of the cylindrical body; a voltage source coupled to the helical wire to establish a Lagrange point in the cylindrical body to generate a Trojan beam from the charged particle beam, wherein the cylindrical body is grounded relative to the voltage source.

28. The charged particle waveguide of claim 26, wherein the helical wire is shaped in a single helix or a double helix.

29. The charged particle waveguide of claim 27, wherein the charged particle source is an electron gun and the waveguide includes a grounded hollow tube and wherein a negative voltage is applied via the voltage source.

30. The charged particle waveguide of claim 29, wherein the electron gun is part of a reflection high-energy electron diffraction gun.

31. The charged particle waveguide of claim 27, wherein the Lagrange point is a stable Lagrange point generated by the waveguide via a potential distribution.

32. The charged particle waveguide of claim 27, wherein the charged particle beam is one of a proton beam or an ion beam.

33. An acoustic waveguide comprising: a gas tube including an input for an acoustic wave; a twisted conductor embedded in the gas tube; and a current source providing current to the twisted conductor to establish a Lagrange point for generating a Trojan beam from the acoustic wave.

34. The acoustic waveguide of claim 33, wherein the gas tube holds helium.

35. The acoustic waveguide of claim 33 wherein the acoustic wave is generated by an ultrasonic transducer. 30 4885-3549-1273.1 065715-000158WOPT

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