Abstract

Polarization, the path traced by light’s electric field vector, appears in all areas of optics. In recent decades, various technologies have enabled the precise control of light’s polarization state, even on a subwavelength scale, at optical frequencies. In this review, we provide a thorough, high-level review of the fundamentals of polarization optics and detail how the Jones calculus, alongside Fourier optics, can be used to analyze, classify, and compare these optical elements. We provide a review of work in this area across multiple technologies and research areas, including recent developments in optical metasurfaces. This review unifies a large body of work on spatially varying polarization optics and may be of interest to both researchers in optics and designers of optical systems more generally.

© 2022 Optical Society of America

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Figures (37)

Figure 1.
Figure 1. (a) Well-formed rhombohedral crystal of “Iceland spar” (i.e., calcite, ${\rm{CaC}}{{\rm{O}}_3}$) is strongly birefringent and exhibits doubly refracted images in transmission, in this case of a square lattice and a single dot. (b) This effect was first formally reported (in Latin) by the Dane Bartholinus in 1670, as exhibited by his own sketch of this effect  [10].
Figure 2.
Figure 2. Polarization ellipse and the Jones vector. The plane wave solution of the wave equation [Eq. (1)] gives an electric field that, in general, traces out an ellipse with time when observed in a fixed plane. Planes of constant phase are perpendicular to the electric field and are separated by the wavelength $\lambda$ along the $\vec k = 2\pi /\lambda$ vector. The shape of the electric field is described by the two-element Jones vector, examples of which are shown at the right.
Figure 3.
Figure 3. Jones matrix polar decomposition. Two broad classes of polarization transformations exist. (a) A unitary transformation (described by a Jones matrix ${\boldsymbol U}$ for which ${{\boldsymbol U}^\dagger}{\boldsymbol U} = {\mathbb I}$) is a lossless transformation in which the components of an incident plane wave along orthogonal eigenvectors are retarded in phase. Physically, wave plates can enact a unitary Jones matrix. (b) A Hermitian transformation is described by a Jones matrix ${\boldsymbol H}$ for which ${{\boldsymbol H}^\dagger} = {\boldsymbol H}$. These attenuate (or, less commonly, amplify) an orthogonal eigenbasis of polarizations. Devices implementing Hermitian Jones matrices are known as diattenuators, of which a polarizer is a limiting case. (c) Any linear polarization transformation given by a $2 \times 2$ Jones matrix ${\boldsymbol J}$ can be written as a cascade of a Hermitian device—a diattenuator—followed by a unitary device—a wave plate. Note that this is true for any ${\boldsymbol J}$ (which may not itself have an orthogonal eigenbasis).
Figure 4.
Figure 4. All plane wave polarization states can be imagined to lie on the surface of a sphere known as the Poincaré sphere [(a), with a view of the northern hemisphere given in (b)]. All linear polarization states lie along its equator and circular polarization states of opposite handedness occupy its poles; everywhere in-between is elliptical. All angles are doubled with respect to the physical polarization ellipse so that orthogonal polarizations lie at diametrically opposite points. The origin corresponds to unpolarized light, while the interior corresponds to partially polarized states (Subsection 2.4c).
Figure 5.
Figure 5. Partially polarized light. Illustrations of the two examples given in Subsection 2.4c of how light may be partially (or un-) polarized. (a) The coherence properties of a light source can be used to deterministically produce partially polarized light. Light can be split (and then recombined) using polarization beam splitters, projecting the incident polarization into orthogonal components. These are then delayed by a path length $\Delta L$ that is much longer than the source’s coherence length, $c\tau$. If the initial polarization is such that equal intensity goes along each path, the resultant light is completely unpolarized. (b) Partially polarized light can also be prepared by combining two (or more) frequencies of completely coherent light in different polarization states. Here the path of the electric field of two slightly different frequencies in $x$ and $y$ linear polarization states is shown after several multiples of the characteristic “beat” time between the two frequencies, $2\pi /({\omega _1} - {\omega _2})$. The electric field traces out a Lissajous figure. At short times, a preferred direction is evident; over these time scales, the light is partially polarized. Over longer times, the electric field fills out a full square. All polarization ellipses are equally represented, and the light is unpolarized. (c) If, on the other hand, the two frequencies have polarization states that are not orthogonal, partially polarized light is produced. Here, $x$ and 45° linearly polarized states of slightly different frequencies are allowed to interfere for $T = 100$ beat periods. There is now a preferred direction of the electric field.
Figure 6.
Figure 6. Retardance and diattenuation space. All unitary and Hermitian Jones matrices can be visualized as lying within a sphere that is referred to as (a) retardance and (b) diattenuation space, respectively. A point in the sphere corresponds to an operator. The line connecting the point to the origin gives the operator’s eigen-polarizations (as the orthogonal Stokes vectors corresponding to that direction on the conventional Poincaré sphere), while its distance from the origin gives its retardance or diattenuation in accordance with Eqs. (92) and (91). For instance, in (a) the surface of the retardance sphere represents all wave plates with half-wave ($\lambda /2$) retardance, while in (b) the surface of the diattenuation sphere represents all perfect polarizers (${\cal D} = 1$).
Figure 7.
Figure 7. Action of a wave plate (unitary) operator visualized on the Poincaré sphere. (a) In general, a wave plate enacts a rotation of the Stokes state-of-polarization (SOP) on the Poincaré sphere. The input state $|{\rm{in}}\rangle$ (whose SOP is represented by a green dot) processes about the eigen-polarization of the wave plate (whose Stokes vector is denoted by a blue arrow) by an angle equal to the retardance ($\pi /3$ shown here), yielding an output $|{\rm{out}}\rangle$ (red dot). In general, output states from the unitary operator for any retardance, given $|{\rm{in}}\rangle$ as an input, are constrained to lie along the purple circle. (b) This readily explains why $x$ polarization is transformed into circular when passed through a wave plate with $\pi /2$ retardance oriented at 45° and (c) why a crystal with optical activity (phase retardance between circular polarizations) rotates the angle of linear polarization.
Figure 8.
Figure 8. Action of a diattenuator visualized with the Poincaré sphere. The effect of a diattenuator (Hermitian) Jones matrix can be visualized by sampling the Poincaré sphere (the space of all possible input polarization states) with a wireframe mesh. A diattenuator in general modifies both the amplitude and polarization state of incoming light. These can be visualized separately by scaling the radius of the wireframe to represent output power as a function of input polarization [(in accordance with Eq. (93)] and by distorting the wireframe to represent output polarization [Eq. (94)]. This is done for a weak diattenuator [(a) ${\cal D} = 0.5$] and a strong one, approaching a perfect polarizer [(b) ${\cal D} = 0.95$], both for $x$-polarized light. In (b), the diattenuator’s tendency to convert all incident polarizations to its preferred one is evident in the sparsity of the output wireframe.
Figure 9.
Figure 9. Two important concepts from Fourier optics. (a) A field $U(x,y)$ is composed of a spectrum of plane waves $A({k_x},{k_y})$—its angular spectrum. Each plane wave in the set can be individually propagated forward in space. (b) Any linear optical system can be abstracted as a black box described by an impulse response function $h({x_2},{y_2};{x_1},{y_1})$, where $h$ describes the field evoked at $({x_2},{y_2})$ at the system’s output in response to a point excitation $({x_1},{y_1})$ at the system’s input. (Only lenses are shown here for illustration, as this formalism is often applied to imaging systems, but any configuration of linear elements, such as sections of free space, lenses, holograms, gratings, and mirrors, can be described in this manner.)
Figure 10.
Figure 10. Geometry of an interference problem between two polarized plane waves.
Figure 11.
Figure 11. Interference of $|x\rangle$ and $|y\rangle$ polarized light. Each circle on the Poincaré sphere denotes a different relative weight between $|x\rangle$ and $|y\rangle$ and corresponds by color to a row in the adjacent table, depicting how the interference between two slightly tilted plane waves appears as a periodic interference pattern of polarization state. One repeating period of this pattern is shown.
Figure 12.
Figure 12. Interference of non-orthogonal polarization states, which induces amplitude modulation.
Figure 13.
Figure 13. Interference of multiple polarized plane waves, equally spaced in angle, with indices lying between $p = m$ and $n$.
Figure 14.
Figure 14. Complex polarization and intensity patterns over one grating period produced by three (left) or four (right) arbitrarily chosen, linearly polarized beams whose diffraction orders and angles of polarization are shown in a table for each case. The polarizations of the plane waves are labeled with blue dots on the Poincaré sphere. In each case, all plane waves have equal weight in the superposition. Each table denotes the polarization state of the assumed diffraction orders—here, $|\theta \rangle$ denotes linearly polarized light with an azimuth angle of $\theta$.
Figure 15.
Figure 15. A hierarchical view of polarization-dependent paraxial diffraction. In the scalar regime (left), a periodic electric field distribution $t(x,y)$ produces discrete orders with scalar weights $\{{a_k}\}$. In the vector regime (middle), the full polarization state $|j(x,y)\rangle$ is allowed to vary periodically with space and produces diffraction orders with characteristic Jones vectors (polarizations) $\{|{j_k}\rangle \}$. Finally, in the matrix regime, the Jones matrix of the grating may vary with space as ${\boldsymbol J}(x,y)$—in this case, the Fourier coefficients $\{{{\boldsymbol J}_k}\}$ are themselves Jones matrix operators, encoding polarization-dependent behaviors. The vector field can be recovered from the matrix description if a particular polarization $|j\rangle$ is incident, and a scalar field from a vector one if analyzed along a particular polarization $|\xi \rangle$.
Figure 16.
Figure 16. A spatially varying Jones matrix described by ${\boldsymbol J}(x,y)$ and its corresponding Jones matrix plane wave spectrum ${\boldsymbol A}({k_x},{k_y})$ (related by Fourier transform ${\cal F}$) constitute a mathematical abstraction neglecting the fact that ${\boldsymbol J}(x,y)$ must be somehow implemented. Different technologies for doing so place different constraints on ${\boldsymbol J}(x,y)$, and the “near-field” produced just after the object, and the far-field behavior contained in ${\boldsymbol A}({k_x},{k_y})$. This physical layer is, thus, a key design consideration.
Figure 17.
Figure 17. Polarization holographic recording. (a) In a photoanisotropic medium, linear polarization induces a local change in refractive index both parallel and perpendicular to its direction and proportional to its intensity. (b) In the general case of elliptically polarized light, a linear photoanisotropic medium feels only the linearly polarized components (denoted in blue), again inducing changes in refractive index in the parallel and perpendicular directions. (c) Full control over the magnitude of $\Delta {n_\parallel}$ and $\Delta {n_ \bot}$ is sufficient to fully control retardance $\delta$ and overall phase shift $\bar \phi$. (d) A spatially varying polarization pattern will induce a spatially varying polarization transfer function in the photoanisotropic film. In this case, the interference of right- and left-circularly polarized light is shown. This particular case induces linear polarization of constant magnitude but periodically changing azimuth, which, in a linear photoanisotropic material, induces a spatially varying wave plate of constant retardance and periodic azimuth. This particular recording scenario, which produces an optic that will produce three diffraction orders regardless of illuminating polarization, has been used in practice in most work in polarization holography.
Figure 18.
Figure 18. Applications of polarization holograms. (a) Spectropolarimetry with a polarization hologram. The full-Stokes vector of a beam can be measured over a band of wavelengths if linear detectors are placed on the diffraction orders of a polarization-insensitive grating and a circular polarization-splitting polarization hologram. Two polarizers on the left two diffraction orders suffice to determine ${S_1}$ and ${S_2}$. Reprinted with permission from [80]. Copyright 2000 Optical Society of America. (b) If one recording beam is passed through a lens, the period of the interference is spatially varying such that upon illumination (after recording), as in (c), the photoanisotropic film will function as an off-axis lens, focusing for one circular polarization and diverging for the other. (b) and (c) reprinted from Opt. Lasers Eng. 44, Ramanujam et al., “Polarisation-sensitive optical elements in azobenzene polyesters and peptides,” pp. 912–915, copyright 2006, with permission from Elsevier [81]. (d) If, in the recording scenario of (b), a transparency is placed (in this case an image of Denis Gabor), a holographic image will be recorded on the two inner diffraction orders. These are sensitive to opposite circular polarization states, and one is inverted with respect to the other, a characteristic of geometric-phase-only devices. Reproduced with permission of Cambridge University Press through PLSclear [70]). (e) This recording can be done twice, reversing the chiralities of the recording beams in-between transparencies. Then, each diffraction order can show two images depending on the sense of circular polarization incident, or both images simultaneously (for linearly polarized illumination). Reprinted with permission from [82]. Copyright 1985 Optical Society of America.
Figure 19.
Figure 19. Computer-generated holograms etched directly into birefringent substrates. (a) Orthogonal linear polarizations can experience two arbitrarily specified phases upon propagation through two birefringent crystals sandwiched together with controllable etch depths. Reprinted with permission from [91]. Copyright 1993 Optical Society of America. (b) If these two etch depths can vary from point-to-point, two patterned birefringent substrates can form a computer-generated hologram that imparts two independent phase profiles on orthogonal linear polarization states. Reprinted with permission from [92]. Copyright 1995 Optical Society of America. (c) Such a hologram implements the function of two non-polarization-sensitive phase holograms placed in a setup in which polarization is split and recombined by polarizing beam splitters. Reprinted with permission from [93]. Copyright 1997 Optical Society of America. (d) In one application, the hologram can be designed to act as a lens with two different focal lengths depending on the polarization of incident light, either imaging a coherent target or producing its Fourier transform when input polarization is switched. (e) The two substrates must be patterned and etched separately in a multi-level fabrication process, then precisely aligned. Reprinted with permission from [92]. Copyright 1995 Optical Society of America.
Figure 20.
Figure 20. Stress-engineered optical elements (SEOs). (a) Example of SEO fabrication. A glass round is held in a mount, and stress is applied through three adjustable set screws, inducing stress birefringence visible in a broadband image viewed through crossed circular polarizers. Reprinted with permission from [105]. Copyright 2007 Optical Society of America. (b) Viewing through crossed circular polarizers with narrowband illumination highlights contours of equal retardance, showing how tailored stress birefringence induces complex patterns of spatially varying birefringence. Near the center of this SEO, the pattern displays azimuthal symmetry with retardance varying with radius and wave plate fast-axis with azimuthal angle. Reprinted by permission from Macmillan Publishers Ltd.: Brown and Beckley, Front. Optoelecton. 6, 89–96 (2013) [104]. Copyright 2013. (c) When placed in the entrance pupil of an imaging system, an SEO modifies the system’s point-spread function in a way that depends on incident polarization. This effect can be used to measure that incident polarization state. Reprinted by permission from Macmillan Publishers Ltd.: Brown and Beckley, Front. Optoelecton. 6, 89–96 (2013) [104]. Copyright 2013. Reprinted with permission from [106]. Copyright 2013. (d) and (e) An application using SEO-enabled polarimetry to determine the orientation, position, and wobble of single molecule fluorophores, described in Subsection 4.4. Reprinted under a Creative Commons Attribution 4.0 International License [107].
Figure 21.
Figure 21. Liquid crystal (LC) devices. (a) A medium of ordered LCs is uniaxially birefringent due to the elongated nature of the LC molecules. At oblique incidence, one polarization eigenmode always experiences ${n_0}$, while the other experiences ${n_0} \le n \le {n_e}$ depending on $\theta$ (and can be computed with aid of the index ellipsoid, pictured here). The retardance experienced varies with $\theta$. (b) A layer of LCs can be sandwiched between substrates that have been mechanically rubbed. Van der Waals forces coax the LCs near the substrate to align with the rubbing directions. In a parallel aligned nematic crystal cell, the rubbing directions for both substrates are parallel so that the cell acts as a retarder. When a voltage is applied, LCs in the center of the cell rotate to align with the E-field, modifying the cell’s retardance. In modern display applications, twisted nematic cells (placed between polarizers) are used instead. Here, lines on substrates denote rubbing directions for LC alignment (not polarizer directions). (c) Two techniques for patterning photoalignment materials with polarized light. Adapted from [119]. (d) A schematic of a typical fabrication procedure for photoaligned LC diffractive optics.
Figure 22.
Figure 22. Optical devices and applications enabled by photoaligned LCs. (a) A geometric phase grating fabricated by illuminating a photoalignment layer with two tilted, circularly polarized beams that create a linearly polarized interference pattern of varying azimuth. LCs align with this patterned photoalignment layer to create a grating that (b) splits circularly polarized incident light into the ${\pm}1$ diffraction orders, with some leakage into the zero order if the cell’s retardance deviates from $\pi$. These can be patterned uniformly over large areas, as shown by a grating fabricated by ImagineOptix, Inc. (a) Escuti and Jones, SID Symp. Dig. Tech. Pap. 37, 1443–1446 (2006) [151]. Copyright Wiley-VCH Verlag GmbH & Co. KGaA. Reproduced with permission. (b) reprinted with permission from [156]. Copyright 2016 Optical Society of America. (c) The LCs can be patterned to realize a lens geometric phase profile. These components can be purchased off-the-shelf from Edmund Optics in ø1” format and $f = 45-100$ mm (example pictured at right). Reprinted with permission from Edmund Optics [157]. (d) Photoaligned holographic LC optics were recently used by Facebook Reality Labs in a virtual reality headset with a sunglasses-like formfactor. The prototype and optical design schematic are shown here (illumination source and screen electronics are located externally). Reprinted under a Creative Commons Attribution International 4.0 License [158]. (e) Systems incorporating several LC polarization gratings have been used for linear-only imaging polarimetry. The example here shows a white light image of a car taken through such a system where green and reddish colors are used to denote angle of linear polarization. Reprinted with permission from [159]. Copyright 2011 Optical Society of America. (f) LC geometric phase polarization holograms have been investigated for use in coronagraphic systems for exoplanet imaging applications. Here, they are referred to as vector apodizing phase plates (“vAPPs”) and are advantageous for creating “cleared out” point-spread functions on both side of a host star where exoplanets can, potentially, be directly imaged. (g) Image of a fabricated vAPP between crossed polarizers. Reprinted with permission from Snik et al., Proc. SPIE 8450 (2012) [160].
Figure 23.
Figure 23. Implementation of spatially varying polarization control with configurations of spatial light modulators (SLMs). (a) A single LC SLM implements a phase profile $\phi (x,y)$ on one linear polarization and a constant phase (disregarded here) on the other, which can be addressed separately by (b) rotating the SLM, or equivalently, using half-wave plates. (c) Two orthogonally oriented SLMs can implement a retarder whose overall phase and retardance can be arbitrarily specified at each point in space. (d) A spatially varying rotation matrix can be realized by sandwiching such an SLM between two quarter-wave plates ($\lambda /4$) oriented at ${\pm}{45^ \circ}$. (e) The operating principle of the system in (a) can be understood with the Poincaré sphere taking incident $x$-polarized light as an example. The cascade of a quarter-wave plate, SLM, and oppositely oriented quarter-wave plate takes the light on a path (in black) that, effectively, amounts to a rotation of the polarization state (in green). (f) Cascade of one rotation unit (as in (d)), two SLMs implementing the phase profiles ${\phi _x}(x,y)$ and ${\phi _y}(x,y)$, rotated 90° with respect to one another, and a final rotation unit to undo the first. Throughout this figure, it is assumed that the SLMs are located optically close to one another (or equivalently, imaged onto each other with systems of lenses) and perfectly aligned.
Figure 24.
Figure 24. Spatially varying polarization transformation with liquid crystal SLMs. (a) Example of a polarization-dependent-phase mask encoded on a LC SLM, viewed between crossed polarizers. Reprinted with permission from [199]. Copyright 2019 Optical Society of America. (b) Typical setup for producing polarization-dependent diffractive optics with an SLM. Incident light passes through a section of the SLM and, by virtue of a reflective $4f$ system and quarter-wave plate, is imprinted with a second polarization-dependent phase profile encoded on the second half of the SLM. Taken together, this setup imparts arbitrary and independent phase profiles on the $x$- and $y$-polarized components of incoming light. Reprinted with permission from [200]. Copyright 2016 Optical Society of America. (c) A setup such as that in (b) can be used to create polarization-sensitive diffractive optics, such as gratings whose orders act as analyzers for particular polarization states, distributing light between orders according to the incident light’s polarization state. Reprinted with permission from [201]. Copyright 2001 Optical Society of America. (d) This concept can be extended to generate gratings whose orders act as analyzers for arbitrarily specified states of polarization. The intensities of the different orders permit reconstruction of incident light’s full-Stokes vector. Reprinted with permission from [200]. Copyright 2016 Optical Society of America. (e) Altogether very similar optical setups (here taken from [202], for sake of example) have been used to realize far-fields with arbitrary polarization state variation [functionally equivalent to (b), with two interactions with a LC SLM]. However, structuring the far-field polarization rather than polarization transfer function represents a subcase, rather than the most general capability of devices such as SLMs to serve as polarization-sensitive optical elements (Subsection 3.3). Reprinted with permission from [202]. Copyright 2018 Optical Society of America.
Figure 25.
Figure 25. A proposed architecture for a lithographically defined polarization grating based on form birefringence capable of splitting $x$- and $y$-polarized light. A combination of a form-birefringent grating and an isotropic material was intended to impose the correct phases on orthogonal polarization states, with ${\rm{Ti}}{{\rm{O}}_2}$ as a proposed dielectric. Given the difficulties associated with non-planar fabrication, this structure was seemingly never realized, but proposals such as these anticipated trends in micro- and nano-optics, especially metasurfaces (Subsection 4.7). Copyright 2000 from “Fourier array illuminators with 100% efficiency: analytical Jones-matrix construction” by Honkanen et al. Reproduced by permission of Taylor and Francis Group, LLC, a division of Informa pic [226].
Figure 26.
Figure 26. Metasurfaces emerged from research in a number of previously disparate fields of optics, as shown in this (very approximate) timeline.
Figure 27.
Figure 27. The origin of form birefringence.
Figure 28.
Figure 28. Pre-metasurface works approximating later work on metasurfaces. (a) Computer-generated hologram comprising pixels with differing form birefringence, etched into silicon. This work notably used a library-based approach to choose structures that best realize a desired pair (${\phi _x}$, ${\phi _y}$). Reprinted with permission from [263]. Copyright 2004 Optical Society of America. (b) A periodic, geometric phase grating that directs $|R\rangle /|L\rangle$ to ${\pm}1$ diffraction orders, including an SEM of the structure as realized in GaAs. Reprinted from Opt. Commun. 209, Hasman et al., “Polarization beam-splitters and optical switches based on space-variant computer-generated subwavelength quasi-periodic structures,” pp. 45–54, copyright 2002 with permission from Elsevier [264]. (c) Discretely turning the grating imparts a geometric phase that varies linearly with orientation angle. (d) This can be used to, for instance, realize a geometric phase lens capable of focusing circularly polarized light. (c) and (d) Reprinted with permission from Hasman et al., Appl. Phys. Lett. 82, 328–330 (2003), AIP Publishing LLC. (e) A geometric phase grating paired with a polarizer and image sensor can be used as a detector of light’s polarization state. Reprinted with permission from [266]. Copyright 2003 Optical Society of America. (f) Example of a device fabricated by Hasman et al. that is, qualitatively speaking, particularly reminiscent of later work on metasurface devices. Reprinted from Prog. Opt. 47, Hasman et al., “Space-variant polarization manipulation,” 215–289, copyright 2005, with permission from Elsevier [267].
Figure 29.
Figure 29. Some typical polarization-sensitive “unit cell” geometries used for metasurfaces. (a) Dielectric pillars with cross sections possessing two axes of mirror symmetry (e.g.,  rectangles, ellipses, or crosses). At each point on a metasurface composed of these, transverse dimensions can be adjusted to affect different phase shifts on light linearly polarized along the structures’ symmetry axes, while the structures can be individually rotated. (b) Gap-plasmon resonator structures, with similarly variable transverse cross sections, for operation in reflection. (c) Freeform structures—the most general case in which all geometrical parameters can be regarded as variable and polarization effects arise as a result of potentially complex modal interference.
Figure 30.
Figure 30. (a) Geometry forward and backward transmission through a planar metasurface device, used for a discussion in Subsection 4.7f. (b) Theoretically, a double-layer metasurface comprising, e.g.,  two layers of pillars whose dimensions and orientations can vary independently at each point of a metasurface is sufficient to realize any unitary Jones matrix, thus filling full retarder space as discussed in Subsection 2.6a.
Figure 31.
Figure 31. (a) Typical “library” of polarization-sensitive metasurface elements, often presented in this format in metasurface works. A polarization-sensitive structure, in this case a rectangular pillar with two orthogonal dimensions, is simulated using, e.g.,  FDTD or RCWA for a variety of different configuration of dimensions $({D_x},{D_y})$ at a given design wavelength (here 532 nm, with the pillars being made of ${\rm{Ti}}{{\rm{O}}_2}$. Phase shift and transmission are shown for $x$-polarized light, with the plots for $y$ being the same given an exchange of the plot axes. This is visualized on the complex plane, showing that geometries exist for phase shifts from 0 to $2\pi$ with relatively high transmission. Figures 1, 2, 3, and S3 reprinted with permission from Mueller et al., Phys. Rev. Lett. 118, 113901 (2017) [290]. Copyright 2017 by the American Physical Society. (b) Dual matrix holography for polarization-dependent amplitude control. As described in Subsection 4.7h, two unitary matrices when added together can realize Hermitian behavior in general, affecting polarization-dependent amplitude control when viewed from the zero-order perspective.
Figure 32.
Figure 32. Examples of propagation (a)–(d) phase-only and (e)–(h) geometric-phase-only metasurface designs and applications. (a) Figures 1, 2, 3, and S3 reprinted with permission from Mueller et al., Phys. Rev. Lett. 118, 113901 (2017) [290]. Copyright 2017 by the American Physical Society. (b) Reprinted by permission from Macmillan Publishers Ltd.: Pors et al., Sci. Rep. 3, 2155 (2013) [280]. Copyright 2013. (c) Reprinted by permission from Macmillan Publishers Ltd.: Arbabi et al., Nat. Nanotechnol. 10, 937–943 (2015) [273]. Copyright 2015. (d) Reprinted with permission from Chen et al., Nano Lett. 14, 225–230 (2014) [299]. Copyright 2014 American Chemical Society. (e) Figures 1, 2, 3, and S3 reprinted with permission from Mueller et al., Phys. Rev. Lett. 118, 113901 (2017) [290]. Copyright 2017 by the American Physical Society. (f) Reprinted by permission from Macmillan Publishers Ltd.: Khorasaninejad and Crozier, Nat. Commun. 5, 5386 (2014) [300]. Copyright 2014. (g) From Lin et al., Science 345, 298–302 (2014) [301]. Reprinted with permission from AAAS. (h) Reprinted under a Creative Commons License [302].
Figure 33.
Figure 33. Combining geometric and propagation phases in a single metasurface. (a) Arbitrary phase (and/or amplitude) profiles can be imparted on an arbitrary basis of elliptical polarization states, so long as the output polarizations are of reversed handedness relative to the input. Figures 1, 2, 3, and S3 reprinted with permission from Mueller et al., Phys. Rev. Lett. 118, 113901 (2017) [290]. Copyright 2017 by the American Physical Society. (b) Conversion of $|R\rangle$ into a vortex beam while leaving $|L\rangle$ unchanged. Reprinted by permission from Macmillan Publishers Ltd.: Arbabi et al., Nat. Nanotechnol. 10, 937–943 (2015) [273]. Copyright 2015. (c) Circular polarization-switchable holograms. (d) Elliptical polarization beam splitting gratings. (c) and (d) Figures 1, 2, 3, and S3 reprinted with permission from Mueller et al., Phys. Rev. Lett. 118, 113901 (2017) [290]. Copyright 2017 by the American Physical Society. (e) Conceptual sketch of a “J-plate,” an arbitrary spin–orbit converter. From Devlin et al., Science 358, 896–901 (2017) [182]. Reprinted with permission from AAAS.
Figure 34.
Figure 34. Summary examples of work using the vector approach to demonstrate metasurface devices producing far-fields or (discrete diffraction orders) whose polarization state and intensity distributions can be controlled through the metasurface’s design. (a) Reprinted with permission from Deng et al., Nano Lett. 18, 2885–2892 (2018) [329]. Copyright 2018 American Chemical Society. (b) Reprinted with permission from Arbabi et al., ACS Photonics 6, 2712–2718 (2019) [330]. Copyright 2019 American Chemical Society. (c) Reprinted under a Creative Commons Attribution 4.0 International License from [331]. (d) Reprinted under a Creative Commons Attribution 4.0 International License from [332]. (e) Reprinted with permission from Wen et al., Nano Lett. 21, 1735–1741 (2021) [333]. Copyright 2021 American Chemical Society. (f) Reprinted with permission from [334]. Copyright 2018 Optical Society of America.
Figure 35.
Figure 35. Examples of metasurface devices designed with the full Jones matrix approach. (a) Metasurfaces can act as diffraction gratings whose orders may have specified polarization sensitivity, emulating through diffraction functions that would ordinarily require bulk birefringent and dichroic optics. (b) One example of these is a 2D periodic grating (shown here made of ${\rm{Ti}}{{\rm{O}}_2}$ pillars for use at visible wavelengths) whose innermost four orders act as analyzers for the set of four non-orthogonal polarization states shown at left. This capability is enabled by the matrix approach described here. (a) and (b) from Rubin et al., Science 365, eaax1839 (2019) [339]. Reprinted with permission from AAAS. (c) Similar ideas can be used to create aperiodic metasurfaces that realize Jones matrix holograms where each point in the far-field can be ascribed its own designer Jones matrix transfer function. One example of this in action is a hologram in which the hologram takes the form of a collection of polarization ellipses, where each polarization state drawn acts as an analyzer for its depicted state. (d) This means that the hologram rearranges light on the basis of its polarization: Each drawing will be bright or dark depending on its state’s projection onto the incident light’s polarization state so that the polarization of incident light can be read out from the hologram by inspection. No external polarization analyzer is present. (c) and (d) from Rubin et al., Sci. Adv. 7, eabg7488 (2021) [340]. Reprinted with permission from AAAS. (e) Similar control can be exerted along the optical axis if a Jones matrix-weighted expansion is carried out in ${k_z}$ using Bessel beams of variable cone angle as a basis. (f) This enables an optical element wherein a custom optical transfer function is created at each point along the optical axis. (g) One example of this is a device that acts as a linear polarizer that virtually “rotates” along the optical axis from 0° to 90°, as shown by the amplitude profile in $z$ for incident $x$- and $y$-polarized light. Reprinted by permission from Macmillan Publishers Ltd.: Dorrah et al., Nat. Photonics 15, 287–296 (2021) [296]. Copyright 2021. All scale bars are 1 µm.
Figure 36.
Figure 36. Polarization imaging using polarization-sensitive metasurfaces. (a) Schematic example of using a spatial multiplexing approach to create a grating that analyzes incoming polarization with respect to $|x\rangle$, $|y\rangle$, ${|45^ \circ}\rangle$, ${|135^ \circ}\rangle$, $|R\rangle$, and $|L\rangle$ simultaneously for independent measurement. (b) Example SEM of a grating interlaced to realize the function in (a). (a) and (b) reprinted with permission from [342]. Copyright 2015 Optical Society of America. (c) Shapes swept out by various sets of four polarization states. (i) and (ii) are not suitable for full-Stokes determination, while (iii), the tetrahedron, represents an optimum choice for full-Stokes polarimetry. (d) One approach to polarization imaging is to use the metasurface as a focal plane element, combining polarization-analyzing gratings (in this case using a spatially multiplexed approach) with focusing power to dedicate a superpixel (e) to the determination of $\vec S$. (d) and (e) reprinted with permission from Arbabi et al., ACS Photonics 5, 3132–3140 (2018) [343]. Copyright 2018 American Chemical Society. (f) In a second approach to metasurface polarimetry, the grating is placed in collimated space, ideally in a pupil plane. Then, if the system’s FOV is limited, several images each analyzed with respect to a particular polarization state are formed, which when registered permit determination of $\vec S$ over a scene. Using this approach, a compact metasurface-based polarization camera has been demonstrated. From Rubin et al., Science 365, eaax1839 (2019) [339]. Reprinted with permission from AAAS.
Figure 37.
Figure 37. Examples of polarization imagery with a metasurface-based polarization camera. Real-world examples of polarization phenomena visualized through a metasurface polarization camera, described in full in [339]. The top row shows the raw sensor exposure in each case, displaying four quadrants corresponding to four polarization channels. When registered, a full-Stokes image can be computed from which the intensity (${S_0}$), azimuth angle, and degree-of-polarization are derived. From Rubin et al., Science 365, eaax1839 (2019) [339]. Reprinted with permission from AAAS.

Tables (4)

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Table 1. Definition of Notation Used Throughout Section 2 and This Review

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Table 2. Various Stokes Vectors and Their Corresponding Jones Vectors

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Table 3. Short-Form Comparison of Technologies Discussed in Section 4

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Table 4. Classes of Spatially Varying Polarization Optics as Defined by the ${\boldsymbol J}$ They Are Capable of Implementinga

Equations (178)

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$${\nabla ^2}\vec E(\vec r,t) = \mu \epsilon \frac{{{\partial ^2}}}{{\partial {t^2}}}\vec E(\vec r,t),$$
$$\vec E(\vec r,t) = {\vec E_0}\cos (\vec k \cdot \vec r - \omega t) = {\vec E_0}{\rm Re} \{{{e^{i(\vec k \cdot \vec r - \omega t)}}}\}.$$
$$\vec E(z,t) = ({E_x}{e^{i{\phi _x}}}\hat x + {E_y}{e^{i{\phi _y}}}\hat y){e^{i(\omega t - kz)}},$$
$$\vec E(z,t) = \left({\begin{array}{*{20}{c}}{{E_x}{e^{i{\phi _x}}}}\\{{E_y}{e^{i{\phi _y}}}}\end{array}} \right){e^{i(\omega t - kz)}}.$$
$$\vec E(z,t) = {E_0}{e^{i{\phi _x}}}\left({\begin{array}{*{20}{c}}{\cos \chi}\\{\sin \chi {e^{{i\phi}}}}\end{array}} \right){e^{i(\omega t - kz)}},$$
$$|j\rangle = \left({\begin{array}{*{20}{c}}{\cos \chi}\\{\sin \chi {e^{{i\phi}}}}\end{array}} \right).$$
$$|E\rangle = \left({\begin{array}{*{20}{c}}{{{\tilde E}_x}}\\{{{\tilde E}_y}}\end{array}} \right).$$
$${\langle E| = (|E\rangle)^\dagger} = \left({\begin{array}{*{20}{c}}{\tilde E_x^*}&{\tilde E_y^*}\end{array}} \right).$$
$$\langle {E_1}|{E_2}\rangle = \tilde E_{x,1}^*{\tilde E_{x,2}} + \tilde E_{y,1}^*{\tilde E_{y,2}}.$$
$$|{\lambda ^ \bot}\rangle = \left({\begin{array}{*{20}{c}}{- \sin \chi}\\{\cos \chi {e^{{i\phi}}}}\end{array}} \right),$$
$$|E^\prime \rangle = {\textbf{J}}|E\rangle ,$$
$${\textbf{J}} = \left({\begin{array}{*{20}{c}}{{{\tilde J}_{11}}}&{{{\tilde J}_{12}}}\\{{{\tilde J}_{21}}}&{{{\tilde J}_{22}}}\end{array}} \right).$$
$$|{j^\prime}\rangle = {\textbf{J}}|j\rangle$$
$$|{j^{\bot ,\prime}}\rangle = {\textbf{J}}|{j^ \bot}\rangle .$$
$${\textbf{A}} = \left({\begin{array}{*{20}{c}}|&|\\{|j\rangle}&{|{j^ \bot}\rangle}\\|&|\end{array}} \right),$$
$${\textbf{B}} = \left({\begin{array}{*{20}{c}}|&|\\{|{j^\prime}\rangle}&{|{j^{\bot ,^\prime}}\rangle}\\|&|\end{array}} \right),$$
$${\textbf{JA}} = {\textbf{B}},$$
$${\textbf{J}} = {\textbf{B}}{{\textbf{A}}^\dagger}$$
$$|{E_2}\rangle \langle {E_1}| = \left({\begin{array}{*{20}{c}}{\tilde E_{x,1}^*{{\tilde E}_{x,2}}}&{\tilde E_{y,1}^*{{\tilde E}_{x,2}}}\\{\tilde E_{x,1}^*{{\tilde E}_{y,2}}}&{\tilde E_{y,1}^*{{\tilde E}_{y,2}}}\end{array}} \right).$$
$$\langle {E_1}|{E_2}\rangle = \tilde E_{x,1}^*{\tilde E_{x,2}} + \tilde E_{y,1}^*{\tilde E_{y,2}}.$$
$$\langle {E_2}|{E_1}\rangle = {\rm{Tr}}(|{E_2}\rangle \langle {E_1}|).$$
$${{\textbf{U}}^\dagger}{\textbf{U}} = {\mathbb I} = \left({\begin{array}{*{20}{c}}1&0\\0&1\end{array}} \right).$$
$${\textbf{U}} = {e^{{i\phi}}}({e^{i\frac{\Delta}{2}}}|j\rangle \langle j| + {e^{- i\frac{\Delta}{2}}}|{j^ \bot}\rangle \langle {j^ \bot}|).$$
$${{\textbf{H}}^\dagger} = {\textbf{H}},$$
$${\textbf{H}} = {e^{- \frac{{{\alpha _0}}}{2}}}({e^{\frac{\alpha}{2}}}|j\rangle \langle j| + {e^{- \frac{\alpha}{2}}}|{j^ \bot}\rangle \langle {j^ \bot}|),$$
$${\cal D} = \frac{{{I_{|j\rangle}} - {I_{|{j^ \bot}\rangle}}}}{{{I_{|j\rangle}} + {I_{|{j^ \bot}\rangle}}}} = \frac{{{e^\alpha} - {e^{- \alpha}}}}{{{e^\alpha} + {e^{- \alpha}}}} = \tanh \,\alpha .$$
$${\textbf{J}} = {\textbf{AD}}{{\textbf{V}}^\dagger},$$
$${\textbf{J}} = ({\textbf{A}}{{\textbf{V}}^\dagger})({\textbf{VD}}{{\textbf{V}}^\dagger}).$$
$${\textbf{J}} = {\textbf{UH}}.$$
$${\textbf{J}} = {{\textbf{H}}^\prime}{{\textbf{U}}^\prime}$$
$$\vec S = \left({\begin{array}{*{20}{c}}{{S_0}}\\{{S_1}}\\{{S_2}}\\{{S_3}}\end{array}} \right) = \left({\begin{array}{*{20}{c}}{{I_{|x\rangle}} + {I_{|y\rangle}}}\\{{I_{|x\rangle}} - {I_{|y\rangle}}}\\{{I_{{{|45}^ \circ}\rangle}} - {I_{{{|135}^ \circ}\rangle}}}\\{{I_{|R\rangle}} - {I_{|L\rangle}}}\end{array}} \right).$$
$$\sqrt {S_1^2 + S_2^2 + S_3^2} \le {S_0}.$$
$$\vec s = \left({\begin{array}{*{20}{c}}{{S_1}/{S_0}}\\{{S_2}/{S_0}}\\{{S_3}/{S_0}}\end{array}} \right) = \left({\begin{array}{*{20}{c}}{{s_1}}\\{{s_2}}\\{{s_3}}\end{array}} \right).$$
$$s_1^2 + s_2^2 + s_3^2 = 1.$$
$$\begin{split}{\vec S}& = \vec X + \vec Y\\[-4pt]&= \frac{1}{2}\left({\begin{array}{*{20}{c}}1\\1\\0\\0\end{array}} \right) + \frac{1}{2}\left({\begin{array}{*{20}{c}}1\\{- 1}\\0\\0\end{array}} \right) = \left({\begin{array}{*{20}{c}}1\\0\\0\\0\end{array}} \right).\end{split}$$
$${\rm{DoP}}(\vec S) = p = \frac{{\sqrt {S_1^2 + S_2^2 + S_3^2}}}{{{S_0}}}.$$
$$\vec S = \left({\begin{array}{*{20}{c}}{{S_0}}\\{{S_1}}\\{{S_2}}\\{{S_3}}\end{array}} \right) = {S_0}\left[{(1 - p)\left({\begin{array}{*{20}{c}}1\\0\\0\\0\end{array}} \right) + p\left({\begin{array}{*{20}{c}}1\\{\frac{{{S_1}}}{{S_1^2 + S_2^2 + S_3^2}}}\\{\frac{{{S_2}}}{{S_1^2 + S_2^2 + S_3^2}}}\\{\frac{{{S_3}}}{{S_1^2 + S_2^2 + S_3^2}}}\end{array}} \right)} \right].$$
$$|E(t)\rangle = \left({\begin{array}{*{20}{c}}{\cos \theta}\\{\sin \theta {e^{i\phi (t)}}}\end{array}} \right).$$
$$\vec S(t) = \left({\begin{array}{*{20}{c}}1\\{\cos 2\theta}\\{\sin 2\theta \cos \phi (t)}\\{\sin 2\theta \sin \phi (t)}\end{array}} \right).$$
$${\langle \vec S(t)\rangle _T} = \left({\begin{array}{*{20}{c}}1\\{\cos 2\theta}\\0\\0\end{array}} \right),$$
$$|E(t)\rangle = |{E_1}(t)\rangle + |{E_2}(t)\rangle = {e^{i{\omega _1}t}}\left({\begin{array}{*{20}{c}}{\cos {\chi _1}}\\{\sin {\chi _1}{e^{i{\phi _1}}}}\end{array}} \right) + {e^{i{\omega _2}t}}\left({\begin{array}{*{20}{c}}{\cos {\chi _2}}\\{\sin {\chi _2}{e^{i{\phi _2}}}}\end{array}} \right).$$
$$\begin{split}{S_1}(t)& ={ (E_1^{(x)}(t{{))}^*}E_1^{(x)}(t) - {{(E_1^{(y)}(t))}^*}E_1^{(y)}(t)}\\&={ ({e^{- i{\omega _1}t}}\cos {\chi _1} + {e^{- i{\omega _2}t}}\cos {\chi _1})({e^{i{\omega _1}t}}\cos {\chi _1} + {e^{i{\omega _2}t}}\cos {\chi _1})}\\&\quad- {({e^{- i{\omega _1}t}}{e^{- i{\phi _1}}}\sin {\chi _1} + {e^{- i{\omega _2}t}}{e^{- i{\phi _2}}}\sin {\chi _1})({e^{i{\omega _1}t}}{e^{i{\phi _1}}}\sin {\chi _1} + {e^{i{\omega _2}t}}{e^{i{\phi _2}}}\sin {\chi _1})}\\&={ (\mathop {\cos}\nolimits^2 {\chi _1} + \mathop {\cos}\nolimits^2 {\chi _2} + 2\cos (({\omega _1} - {\omega _2})t)\cos {\chi _1}\cos {\chi _2})}\\&\quad -{ (\mathop {\sin}\nolimits^2 {\chi _1} + \mathop {\sin}\nolimits^2 {\chi _2} + 2\cos (({\omega _1} - {\omega _2})t + ({\phi _1} - {\phi _2}))\sin {\chi _1}\sin {\chi _2})}.\end{split}$$
$$\langle {S_1}(t)\rangle = (\mathop {\cos}\nolimits^2 {\chi _1} - \mathop {\sin}\nolimits^2 {\chi _1}) + (\mathop {\cos}\nolimits^2 {\chi _2} - \mathop {\sin}\nolimits^2 {\chi _2}) = S_1^{(1)} + S_1^{(2)},$$
$${\rm{DoP}} = \frac{{(S_1^{(1)} + S_1^{(2)}) + (S_2^{(1)} + S_2^{(2)}) + (S_3^{(1)} + S_3^{(2)})}}{{S_0^{(1)} + S_0^{(2)}}} \le 1.$$
$${\textbf{M}} = \left({\begin{array}{*{20}{c}}{{m_{00}}}&\;\;\;{{m_{01}}}&\;\;\;{{m_{02}}}&\;\;\;{{m_{03}}}\\{{m_{10}}}&\;\;\;{{m_{11}}}&\;\;\;{{m_{12}}}&\;\;\;{{m_{13}}}\\{{m_{20}}}&\;\;\;{{m_{21}}}&\;\;\;{{m_{22}}}&\;\;\;{{m_{23}}}\\{{m_{30}}}&\;\;\;{{m_{31}}}&\;\;\;{{m_{32}}}&\;\;\;{{m_{33}}}\end{array}} \right).$$
$${\vec S^{{\rm{out}}}} = {\textbf{M}}{\vec S^{{\rm{in}}}},$$
$$\vec {\cal D} = {\left({\begin{array}{*{20}{c}}{{m_{00}}}&\;\;\;{{m_{01}}}&\;\;\;{{m_{02}}}&\;\;\;{{m_{03}}}\end{array}} \right)^T},$$
$${\cal D} = \frac{{\sqrt {m_{01}^2 + m_{02}^2 + m_{03}^2}}}{{{m_{00}}}}.$$
$${{\textbf{M}}_H} = \left({\begin{array}{*{20}{c}}{{m_{00}}}&\;\;\;{{m_{01}}}&\;\;\;{{m_{02}}}&\;\;\;{{m_{03}}}\\{{m_{01}}}&\;\;\; \cdot &\;\;\; \cdot &\;\;\; \cdot \\{{m_{02}}}&\;\;\; \cdot &\;\;\; \cdot &\;\;\; \cdot \\{{m_{03}}}&\;\;\; \cdot &\;\;\; \cdot &\;\;\; \cdot \end{array}} \right),$$
$${{\textbf{M}}_U} = \left({\begin{array}{*{20}{c}}1&\;\;\;0&\;\;\;0&\;\;\;0\\0&\;\;\; \cdot &\;\;\; \cdot &\;\;\; \cdot \\0&\;\;\; \cdot &\;\;\; \cdot &\;\;\; \cdot \\0&\;\;\; \cdot &\;\;\; \cdot &\;\;\; \cdot \end{array}} \right).$$
$${\textbf{M}} = {{\textbf{M}}_\Delta}{{\textbf{M}}_U}{{\textbf{M}}_H},$$
$${I_{|q\rangle}} = |{{\textbf{A}}_{|q\rangle}}|s\rangle {|^2} = \langle s|{\textbf{A}}_{|q\rangle}^\dagger {{\textbf{A}}_{|q\rangle}}\rangle s.$$
$${I_{|q\rangle}} = \langle s|q\rangle \langle q|s\rangle = |\langle s|q\rangle {|^2},$$
$${S_0} = {I_{|x\rangle}} + {I_{|y\rangle}} = \langle s|x\rangle \langle x|s\rangle + \langle s|y\rangle \langle y|s\rangle = \langle s|\left({\begin{array}{*{20}{c}}1&\;\;\;0\\0&\;\;\;0\end{array}} \right)|s\rangle + \langle s|\left({\begin{array}{*{20}{c}}0&\;\;\;0\\0&\;\;\;1\end{array}} \right)|s\rangle = \langle s|s\rangle .$$
$${S_1} = {I_{|x\rangle}} - {I_{|y\rangle}} = \langle s|x\rangle \langle x|s\rangle - \langle s|y\rangle \langle y|s\rangle = \langle s|\left({\begin{array}{*{20}{c}}1&\;\;\;0\\0&\;\;\;0\end{array}} \right)|s\rangle - \langle s|\left({\begin{array}{*{20}{c}}0&\;\;\;0\\0&\;\;\;1\end{array}} \right)|s\rangle ,$$
$${S_2} = {I_{{{|45}^ \circ}\rangle}} - {I_{{{|135}^ \circ}\rangle}} = \frac{1}{2}\langle s|\left({\begin{array}{*{20}{c}}1&\;\;\;1\\1&\;\;\;1\end{array}} \right)|s\rangle - \frac{1}{2}\langle s|\left({\begin{array}{*{20}{c}}1&\;\;\;{- 1}\\{- 1}&\;\;\;1\end{array}} \right)|s\rangle ,$$
$${S_3} = {I_{|R\rangle}} - {I_{|L\rangle}} = \frac{1}{2}\langle s|\left({\begin{array}{*{20}{c}}1&\;\;\;{- i}\\i&\;\;\;1\end{array}} \right)|s\rangle - \frac{1}{2}\langle s|\left({\begin{array}{*{20}{c}}1&\;\;\;i\\{- i}&\;\;\;1\end{array}} \right)|s\rangle .$$
$$\vec S = \langle s|\vec {\boldsymbol \sigma} |s\rangle ,$$
$$\vec {\boldsymbol \sigma} = {\left({\begin{array}{*{20}{c}}{{{\boldsymbol \sigma} _0}}&\;\;\;{{{\boldsymbol \sigma} _1}}&\;\;\;{{{\boldsymbol \sigma} _2}}&\;\;\;{{{\boldsymbol \sigma} _3}}\end{array}} \right)^T},$$
$${{\boldsymbol \sigma} _0} = {\mathbb I} = \left({\begin{array}{*{20}{c}}1&\;\;\;0\\0&\;\;\;1\end{array}} \right)$$
$${{\boldsymbol \sigma} _1} = \left({\begin{array}{*{20}{c}}1&\;\;\;0\\0&\;\;\;{- 1}\end{array}} \right),\quad {{\boldsymbol \sigma} _2} = \left({\begin{array}{*{20}{c}}0&\;\;\;1\\1&\;\;\;0\end{array}} \right),\quad {{\boldsymbol \sigma} _3} = \left({\begin{array}{*{20}{c}}0&\;\;\;{- i}\\i&\;\;\;0\end{array}} \right).$$
$${\boldsymbol J} = \vec \alpha \cdot \vec {\boldsymbol \sigma} = {a_0}{{\boldsymbol \sigma} _0} + {a_1}{{\boldsymbol \sigma }_1} + {a_2}{{\boldsymbol \sigma} _2} + {a_3}{{\boldsymbol \sigma}_3},$$
$${\alpha _i} = \frac{1}{2}{\rm{Tr}}({\boldsymbol J}{{\boldsymbol \sigma} _i}),$$
$$\vec s \cdot {\vec {\boldsymbol \sigma} ^\prime} = {s_1}{{\boldsymbol \sigma} _1} + {s_2}{{\boldsymbol \sigma} _2} + {s_3}{{\boldsymbol \sigma} _3}$$
$$(\hat s \cdot {\vec {\boldsymbol \sigma} ^\prime})|s\rangle = |s\rangle .$$
$$|\langle p|q\rangle {|^2} = \frac{1}{2}(1 + \hat p \cdot \hat q),$$
$$|t\rangle = {\boldsymbol J}|s\rangle ,$$
$$\vec T = {\boldsymbol M} \vec S.$$
$${T_i} = \langle t|{{\boldsymbol \sigma} _i}|t\rangle = \langle s|{{\boldsymbol J}^\dagger}{{\boldsymbol \sigma} _j}{\boldsymbol J}|s\rangle .$$
$${T_i} = {\rm{Tr}}(|s\rangle \langle s|{{\boldsymbol J}^\dagger}{{\boldsymbol \sigma} _i}{\boldsymbol J}).$$
$$|s\rangle \langle s| = \frac{1}{2}\langle s|s\rangle {\rm{Tr}}({\mathbb I} + \hat s \cdot {\vec {\boldsymbol \sigma} ^\prime}),$$
$$\begin{split}{T_i} &= {\rm{Tr}}(|s\rangle \langle s|{{\boldsymbol J}^\dagger}{{\boldsymbol \sigma} _i}{\boldsymbol J})\\&= \frac{1}{2}\langle s|s\rangle {\rm{Tr}}({{\boldsymbol J}^\dagger}({\mathbb I} + \hat s \cdot {{\vec {\boldsymbol \sigma}}^\prime}){{\boldsymbol \sigma} _i}{\boldsymbol J})\\&= \frac{1}{2}{\rm{Tr}}({{\boldsymbol J}^\dagger}(\vec S \cdot \vec {\boldsymbol \sigma}){{\boldsymbol \sigma} _i}{\boldsymbol J})\\{T_i} &= \frac{1}{2}\sum\limits_{j = 0}^3 {\rm{Tr}}({\boldsymbol J}{{\boldsymbol \sigma} _j}{{\boldsymbol J}^\dagger}{{\boldsymbol \sigma} _i}){S_j},\end{split}$$
$${m_{\textit{ij}}} = \frac{1}{2}{\rm{Tr}}({\boldsymbol J}{{\boldsymbol \sigma} _j}{{\boldsymbol J}^\dagger}{{\boldsymbol \sigma} _i}),$$
$${\boldsymbol \Sigma} = \sum\limits_{i = 0}^3 \sum\limits_{j = 0}^3 {m_{\textit{ij}}}{{\boldsymbol \sigma} _i} \otimes {{\boldsymbol \sigma} _j},$$
$${m_{\textit{ij}}} = {\rm{Tr}}({\boldsymbol \Sigma} ({{\boldsymbol \sigma} _i} \otimes {{\boldsymbol \sigma} _j})).$$
$${\boldsymbol \Sigma} = \sum\limits_{i = 0}^3 {\lambda _i}{\vec V_i} \otimes {\vec V_i},$$
$${e^{\boldsymbol J}} = {\mathbb I} + {\boldsymbol J} + \frac{1}{{2!}}{{\boldsymbol J}^2} + \frac{1}{{3!}}{{\boldsymbol J}^3} + \frac{1}{{4!}}{{\boldsymbol J}^4} + ... = {\mathbb I} + \sum\limits_{n = 1}^\infty \frac{1}{{n!}}{{\boldsymbol J}^n}.$$
$${e^{\boldsymbol A}}{e^{\boldsymbol B}} \ne {e^{\boldsymbol B}}{e^{\boldsymbol A}} \ne {e^{{\boldsymbol A} + {\boldsymbol B}}},$$
$${e^{{\alpha _0}{\mathbb I} + {\boldsymbol J}}} = {e^{{\alpha _0}}}{e^{\boldsymbol J}}$$
$${e^{\frac{{i(\vec \alpha \cdot \vec {\boldsymbol \sigma})}}{2}}} = {e^{i\frac{{{\alpha _0}}}{2}}}{e^{\frac{{i({{\vec \alpha}^\prime} \cdot {{\vec {\boldsymbol \sigma}}^\prime})}}{2}}}.$$
$${e^{\frac{{i({{\vec \alpha}^\prime} \cdot {{\vec {\boldsymbol \sigma}}^\prime})}}{2}}} = {\mathbb I} + \sum\limits_{n = 1}^\infty \frac{1}{{n!}}{\left({\frac{{i({{\vec \alpha}^\prime} \cdot {{\vec {\boldsymbol \sigma}}^\prime})}}{2}} \right)^n}.$$
$$\left({{\mathbb I} + \frac{1}{{2!}}{{\left({\frac{{i(\vec \alpha \cdot \vec {\boldsymbol \sigma})}}{2}} \right)}^2} + \frac{1}{{4!}}{{\left({\frac{{i(\vec \alpha \cdot \vec {\boldsymbol \sigma})}}{2}} \right)}^4} + ...} \right) + \left({\frac{1}{{1!}}\left({\frac{{i(\vec \alpha \cdot \vec {\boldsymbol \sigma})}}{2}} \right) + \frac{1}{{3!}}{{\left({\frac{{i(\vec \alpha \cdot \vec {\boldsymbol \sigma})}}{2}} \right)}^3} + ...} \right).$$
$${\left({\frac{{i(\vec \alpha \cdot {{\vec {\boldsymbol \sigma}}^\prime})}}{2}} \right)^n} = (- {1)^{\frac{n}{2}}}{\left({\frac{{|\vec \alpha |}}{2}} \right)^n}{\mathbb I}.\quad \quad {\rm{(for}} \; n\,{\rm{even),}}$$
$${\left({\frac{{i({{\vec \alpha}^\prime} \cdot {{\vec {\boldsymbol \sigma}}^\prime})}}{2}} \right)^n} = (- {1)^{\frac{{n - 1}}{2}}}{\left({\frac{{|{{\vec \alpha}^\prime}|}}{2}} \right)^{n - 1}}\frac{{i({{\vec \alpha}^\prime} \cdot \vec {\boldsymbol \sigma})}}{2}.\quad \quad {\rm{(for}}\;n\,{\rm{odd)}}$$
$$\left({1 - \frac{1}{{2!}}{{\left({\frac{{|{{\vec \alpha}^\prime}|}}{2}} \right)}^2} + \frac{1}{{4!}}{{\left({\frac{{|{{\vec \alpha}^\prime}|}}{2}} \right)}^4} - \frac{1}{{6!}}{{\left({\frac{{|{{\vec \alpha}^\prime}|}}{2}} \right)}^6} + ...} \right){\mathbb I} = \left({\cos \frac{{|{{\vec \alpha}^\prime}|}}{2}} \right){\mathbb I},$$
$$i\sin \frac{{|{{\vec \alpha}^\prime}|}}{2}({\hat \alpha ^\prime} \cdot \vec {\boldsymbol \sigma}),$$
$${e^{\frac{{i(\vec \alpha \cdot \vec {\boldsymbol \sigma})}}{2}}} = {e^{i\frac{{{\alpha _0}}}{2}}}\left({\cos \frac{{|{{\vec \alpha}^\prime}|}}{2}{\mathbb I} + i\sin \frac{{|{{\vec \alpha}^\prime}|}}{2}({{\hat \alpha}^\prime} \cdot \vec {\boldsymbol \sigma})} \right).$$
$${e^{\frac{{(\vec \beta \cdot \vec {\boldsymbol \sigma})}}{2}}} = {e^{\frac{{{\beta _0}}}{2}}}{e^{\frac{{({{\vec \beta}^\prime} \cdot {{\vec {\boldsymbol \sigma}}^\prime})}}{2}}},$$
$${e^{\frac{{(\vec \beta \cdot \vec {\boldsymbol \sigma})}}{2}}} = {e^{\frac{{{\beta _0}}}{2}}}\left({\cosh \frac{{|{{\vec \beta}^\prime}|}}{2}{\mathbb I} + \sinh \frac{{|{{\vec \beta}^\prime}|}}{2}({{\hat \beta}^\prime} \cdot {{\vec {\boldsymbol \sigma}}^\prime})} \right),$$
$${\boldsymbol J} = {e^{\frac{{i(\vec \alpha \cdot \vec {\sigma})}}{2}}}{e^{\frac{{(\vec \beta \cdot \vec {\sigma})}}{2}}} = {\boldsymbol U}{\boldsymbol H}$$
$${\vec j_{\boldsymbol H}} = \tanh (|{\vec \beta ^\prime}|){\hat \beta ^\prime}.$$
$${\vec j_{\boldsymbol U}} = \sin \left({\frac{{|{{\vec \alpha}^\prime}|}}{2}} \right){\hat \alpha ^\prime}.$$
$$T = \frac{1}{{1 + {\cal D}}}(1 + {\cal D}({\hat \beta ^\prime} \cdot \hat s)),$$
$$\hat t = \frac{{\sqrt {1 - {{\cal D}^2}}}}{{1 + {\cal D}({{\hat \beta}^\prime} \cdot \hat s)}}\hat s + \frac{{{\cal D} + (1 - \sqrt {1 - {{\cal D}^2}})({{\hat \beta}^\prime} \cdot \hat s)}}{{1 + {\cal D}({{\hat \beta}^\prime} \cdot \hat s)}}{\hat \beta ^\prime},$$
$${\vec S^\prime} = {\boldsymbol D}\vec S.$$
$$I = S_0^\prime = \vec D \cdot \vec S.$$
$${I_n} = {\vec D_n} \cdot \vec S.$$
$$\left({\begin{array}{*{20}{c}}{{I_1}}\\{{I_2}}\\ \vdots \\{{I_N}}\end{array}} \right) = \left({\begin{array}{*{20}{c}} - &\;\;\;{{{\vec D}_1}}&\;\;\; - \\ - &\;\;\;{{{\vec D}_2}}&\;\;\; - \\&\;\;\; \vdots &\;\;\;\\ - &\;\;\;{{{\vec D}_N}}&\;\;\; - \end{array}} \right)\left({\begin{array}{*{20}{c}}{{S_0}}\\{{S_1}}\\{{S_2}}\\{{S_3}}\end{array}} \right),$$
$$\vec I = {\boldsymbol A} \vec S,$$
$$\vec S = {{\boldsymbol A}^{- 1}}\vec I.$$
$$\vec S = (({{\boldsymbol A}^T}{\boldsymbol A}{)^{- 1}}{{\boldsymbol A}^T})\vec I.$$
$${\vec S_{{\rm{out}}}} = {\boldsymbol M}{\vec S_{{\rm{in}}}}.$$
$$\left({\begin{array}{*{20}{c}}{- \vec S_{{\rm{out}}}^{(1)} -}\\{- \vec S_{{\rm{out}}}^{(2)} -}\\ \vdots \\{- \vec S_{{\rm{out}}}^{(N)} -}\end{array}} \right) = {\boldsymbol M}\left({\begin{array}{*{20}{c}}{- \vec S_{{\rm{in}}}^{(1)} -}\\{- \vec S_{{\rm{in}}}^{(2)} -}\\ \vdots \\{- \vec S_{{\rm{in}}}^{(N)} -}\end{array}} \right),$$
$${\boldsymbol O} = {\boldsymbol M}{\boldsymbol I},$$
$${\boldsymbol M} = {\boldsymbol O}{{\boldsymbol I}^{- 1}}$$
$${\boldsymbol M} = {\boldsymbol O}({{\boldsymbol I}^T}{({\boldsymbol I}{{\boldsymbol I}^T})^{- 1}})$$
$$({\nabla ^2} + {k^2})U(\vec r) = 0,$$
$$A({k_x},{k_y};z = 0) = \iint _{- \infty}^{+ \infty}U(x,y,z = 0){e^{- i({k_x}x + {k_y}y)}}{\rm{d}}x{\rm{d}}y.$$
$$U(x,y,z = 0) = \iint _{- \infty}^{+ \infty}A({k_x},{k_y}){e^{i({k_x}x + {k_y}y)}}{\rm{d}}{k_x}{\rm{d}}{k_y}$$
$${k_z} = \sqrt {|k{|^2} - (k_x^2 + k_y^2)} = \sqrt {{{\left(\frac{{2\pi}}{\lambda}\right)}^2} - (k_x^2 + k_y^2)} .$$
$$A({k_x},{k_y};z) = A({k_x},{k_y};z = 0){e^{i{k_z}z}}.$$
$$U(x,y,z) = \iint _{- \infty}^{+ \infty}A({k_x},{k_y};z = 0){e^{i(\sqrt {{{(2\pi /\lambda)}^2} - (k_x^2 + k_y^2)})z}}{e^{i({k_x}x + {k_y}y)}}{\rm{d}}{k_x}{\rm{d}}{k_y}.$$
$${U_ +}(x,y,z = 0) = {U_ -}(x,y,z = 0)t(x,y).$$
$${U_2}({x_2},{y_2}) = \iint _{- \infty}^{+ \infty}h({x_2},{y_2};{x_1},{y_1})U({x_1},{y_1}){\rm{d}}{x_1}{\rm{d}}{x_2},$$
$$|{E_1}(\vec r)\rangle = \left({\begin{array}{*{20}{c}}{E_1^x}\\{E_1^y}\end{array}} \right),$$
$$|{E_2}(\vec r)\rangle = \left({\begin{array}{*{20}{c}}{E_2^{{x_2}}}\\{E_2^{{y_2}}}\end{array}} \right).$$
$$\vec E(\vec r) = \left({\begin{array}{*{20}{c}}{{E_x}}\\{{E_y}}\\{{E_z}}\end{array}} \right) = |{E_1}\rangle {e^{i({{\vec k}_1} \cdot \vec r)}} + |{E_2}\rangle {e^{i({{\vec k}_2} \cdot \vec r)}} = \left({\begin{array}{*{20}{c}}{E_1^x{e^{i({{\vec k}_1} \cdot \vec r)}} + E_2^{{x_2}}{e^{i({{\vec k}_2} \cdot \vec r)}}\cos \theta}\\{E_1^y{e^{i({{\vec k}_1} \cdot \vec r)}} + E_2^{{y_2}}{e^{i({{\vec k}_2} \cdot \vec r)}}}\\{E_1^x{e^{i({{\vec k}_1} \cdot \vec r)}} + E_2^{{x_2}}{e^{i({{\vec k}_2} \cdot \vec r)}}\sin \theta}\end{array}} \right).$$
$$\vec E(\vec r) = \left({\begin{array}{*{20}{c}}{E_1^x{e^{i({{\vec k}_1} \cdot \vec r)}} + E_2^x{e^{i({{\vec k}_2} \cdot \vec r)}}}\\{E_1^y{e^{i({{\vec k}_1} \cdot \vec r)}} + E_2^y{e^{i({{\vec k}_2} \cdot \vec r)}}}\\0\end{array}} \right).$$
$$\vec E(\vec r) = \left({\begin{array}{*{20}{c}}{E_1^x + E_2^x{e^{i({k_0}\sin \theta)x}}}\\{E_1^y + E_2^y{e^{i({k_0}\sin \theta)x}}}\\0\end{array}} \right).$$
$$|E(x)\rangle = |{E_1}\rangle + |{E_2}\rangle {e^{i({k_0}\sin \theta)x}}.$$
$$|{E_{{\rm{tot}}}}\rangle (1 - b)|{E_1}\rangle + b|{E_2}\rangle {e^{{i\phi}}},$$
$$|{E_{{\rm{tot}}}}(x)\rangle = \sum\limits_{p = m}^n {w_p}|{E_p}\rangle {e^{ip2\pi x}}.$$
$$|U(x,y)\rangle = \left({\begin{array}{*{20}{c}}{{U_1}(x,y)}\\{{U_2}(x,y)}\end{array}} \right),$$
$${A_1}({k_x},{k_y}) = \iint _{- \infty}^{+ \infty}{U_1}(x,y){e^{i({k_x}x + {k_y}y)}}{\rm{d}}x{\rm{d}}y$$
$${A_2}({k_x},{k_y}) = \iint _{- \infty}^{+ \infty}{U_2}(x,y){e^{i({k_x}x + {k_y}y)}}{\rm{d}}x{\rm{d}}y.$$
$$|A({k_x},{k_y})\rangle = \left({\begin{array}{*{20}{c}}{{A_1}({k_x},{k_y})}\\{{A_2}({k_x},{k_y})}\end{array}} \right) = \iint _{- \infty}^{+ \infty}|U(x,y)\rangle {e^{i({k_x}x + {k_y}y)}}{\rm{d}}x{\rm{d}}y.$$
$${\boldsymbol J}(x,y) = \left({\begin{array}{*{20}{c}}{{J_{11}}(x,y)}&\;\;\;{{J_{12}}(x,y)}\\{{J_{21}}(x,y)}&\;\;\;{{J_{22}}(x,y)}\end{array}} \right),$$
$$|{U_ +}(x,y)\rangle = {\boldsymbol J}(x,y)|{U_ -}(x,y)\rangle .$$
$$|{A_ +}({k_x},{k_y})\rangle = \iint _{- \infty}^{+ \infty}{\boldsymbol J}(x,y)|{U_ -}(x,y)\rangle {e^{i({k_x}x + {k_y}y)}}{\rm{d}}x{\rm{d}}y,$$
$$|{A_ +}({k_x},{k_y})\rangle = \left({\iint _{- \infty}^{+ \infty}{\boldsymbol J}(x,y){e^{i({k_x}x + {k_y}y)}}{\rm{d}}x{\rm{d}}y} \right)|{U_ -}\rangle .$$
$${\boldsymbol A}({k_x},{k_y}) = \iint _{- \infty}^{+ \infty}{\boldsymbol J}(x,y){e^{i({k_x}x + {k_y}y)}}{\rm{d}}x{\rm{d}}y = \left({\begin{array}{*{20}{c}}{{\cal F}\{{J_{11}}(x,y)\}}&\;\;\;{{\cal F}\{{J_{12}}(x,y)\}}\\{{\cal F}\{{J_{21}}(x,y)\}}&\;\;\;{{\cal F}\{{J_{22}}(x,y)\}}\end{array}} \right).$$
$${\boldsymbol J}(x,y) = \sum\limits_{\vec k \in \{\ell \}} {{\boldsymbol J}_k}{e^{- i\vec k \cdot \vec r}}.$$
$$\begin{split}{\Delta {n_\parallel} = {k_\parallel}I}\\{\Delta {n_ \bot} = {k_ \bot}I},\end{split}$$
$${\boldsymbol J} = \left[{\begin{array}{*{20}{c}}{{e^{ik({n_0} + \Delta {n_\parallel})d}}}&\;\;\;0\\0&\;\;\;{{e^{ik({n_0} + \Delta {n_ \bot})d}}}\end{array}} \right],$$
$$\begin{split}{{I_\parallel} = \frac{{{S_0} + ({S_1} + {S_2})}}{2}}\\{{I_ \bot} = \frac{{{S_0} - ({S_1} + {S_2})}}{2}},\end{split}$$
$$\begin{split}{\Delta {n_\parallel} = {k_\parallel}{I_\parallel} + {k_ \bot}{I_ \bot}},\\{\Delta {n_ \bot} = {k_ \bot}{I_\parallel} + {k_\parallel}{I_ \bot}.}\end{split}$$
$$\delta = ({k_\parallel} - {k_ \bot})({I_\parallel} - {I_ \bot})kd = ({k_\parallel} - {k_ \bot})({S_1} + {S_2})kd = ({k_\parallel} - {k_ \bot})({s_1} + {s_2}){S_0}kd.$$
$$\bar \phi = \frac{{({k_\parallel} + {k_ \bot})({I_\parallel} + {I_ \bot})kd}}{2} = \frac{{({k_\parallel} + {k_ \bot}){S_0}kd}}{2},$$
$${\boldsymbol J} = {e^{i\bar \phi}}{e^{i\frac{\delta}{{\sqrt {s_1^2 + s_2^2}}}({s_1} \cdot {\sigma _1} + {s_2} \cdot {\sigma _2})}}.$$
$$|j(x,y)\rangle = \left[{\begin{array}{*{20}{c}}{\cos \frac{{2\pi x}}{\Lambda}}\\{\sin \frac{{2\pi x}}{\Lambda}}\end{array}} \right],$$
$$\vec S(x,y) = {\left[{\begin{array}{*{20}{c}}1&\;\;\;{\mathop {\cos}\nolimits^2 2\frac{{2\pi x}}{\Lambda}}&\;\;\;{\mathop {\sin}\nolimits^2 2\frac{{2\pi x}}{\Lambda}}&\;\;\;0\end{array}} \right]^T}.$$
$${\boldsymbol J}(x,y) \propto {\boldsymbol R}\left({- \frac{{2\pi x}}{\Lambda}} \right)\left[{\begin{array}{*{20}{c}}{{e^{i\frac{\delta}{2}}}}&\;\;\;0\\0&\;\;\;{{e^{- i\frac{\delta}{2}}}}\end{array}} \right]{\boldsymbol R}\left({\frac{{2\pi x}}{\Lambda}} \right),$$
$${\boldsymbol J}(x,y) = \cos \frac{\delta}{2}{\mathbb I} + \left({\frac{i}{2}\sin \frac{\delta}{2}|L\rangle \langle R|} \right){e^{+ i\frac{{2\pi}}{\Lambda}x}} + \left({\frac{i}{2}\sin \frac{\delta}{2}|R\rangle \langle L|} \right){e^{- i\frac{{2\pi}}{\Lambda}x}}.$$
$${\boldsymbol J}(x,y) = \left[{\begin{array}{*{20}{c}}{{e^{i{\phi _x}(x,y)}}}&\;\;\;0\\0&\;\;\;{{e^{i{\phi _y}(x,y)}}}\end{array}} \right],$$
$${\boldsymbol J}(x,y) = {\boldsymbol R}(\theta (x,y))\left[{\begin{array}{*{20}{c}}{{e^{i\frac{\Delta}{2}}}}&\;\;\;0\\0&\;\;\;{{e^{- i\frac{\Delta}{2}}}}\end{array}} \right]{\boldsymbol R}(- \theta (x,y)),$$
$${e^{{im\phi}}},$$
$${\boldsymbol J}(x,y) = \left[{\begin{array}{*{20}{c}}{{e^{i{\phi _x}(x,y)}}}&\;\;\;0\\0&\;\;\;1\end{array}} \right],$$
$${\boldsymbol J}(x,y) = \left[{\begin{array}{*{20}{c}}{{e^{i{\phi _x}(x,y)}}}&\;\;\;0\\0&\;\;\;{{e^{i{\phi _y}(x,y)}}}\end{array}} \right] = {e^{i\frac{{{\phi _x} + {\phi _y}}}{2}}}\left[{\begin{array}{*{20}{c}}{{e^{i\frac{{{\phi _x}(x,y) - {\phi _y}(x,y)}}{2}}}}&\;\;\;0\\0&\;\;\;{{e^{- i\frac{{{\phi _x}(x,y) - {\phi _y}(x,y)}}{2}}}}\end{array}} \right].$$
$${\boldsymbol J}(x) = \left[{\begin{array}{*{20}{c}}{{e^{\textit{ikx}}}}&0\\0&{{e^{\textit{ikx}}}}\end{array}} \right].$$
$${n_{{\rm{eff}},\parallel}} = \sqrt {n_3^2F + n_1^2(1 - F)} .$$
$${n_{{\rm{eff}}, \bot}} = {\left({\frac{1}{{n_1^2}}F + \frac{1}{{n_3^2}}(1 - F)} \right)^{- \frac{1}{2}}}.$$
$${\boldsymbol J}(x,y) = {\boldsymbol R}(\theta (x,y))\left[{\begin{array}{*{20}{c}}{{e^{i{\phi _x}(x,y)}}}&0\\0&{{e^{i{\phi _y}(x,y)}}}\end{array}} \right]{\boldsymbol R}(- \theta (x,y)).$$
$$|{j_{{\rm{top}}}}\rangle = {{\boldsymbol J}_{{\rm{bottom}}}}|{j_{{\rm{bottom}}}}\rangle .$$
$$|j_{{\rm{bottom}}}^\prime \rangle = \left[{\begin{array}{*{20}{c}}{{{(\tilde j_{{\rm{bottom}}}^{(1)})}^*}}\\{{{(- \tilde j_{{\rm{bottom}}}^{(2)})}^*}}\end{array}} \right]\;{\rm{and}}\;|\tilde j_{{\rm{top}}}^\prime \rangle = \left[{\begin{array}{*{20}{c}}{{{(\tilde j_{{\rm{top}}}^{(1)})}^*}}\\{{{(- \tilde j_{{\rm{top}}}^{(2)})}^*}}\end{array}} \right].$$
$$|j_{{\rm{bottom}}}^\prime \rangle = {{\boldsymbol J}_{{\rm{top}}}}|j_{{\rm{top}}}^\prime \rangle ,$$
$$|j_{{\rm{top}}}^\prime \rangle = {\boldsymbol J}_{{\rm{top}}}^{- 1}|j_{{\rm{bottom}}}^\prime \rangle \Leftrightarrow |j_{{\rm{top}}}^\prime \rangle = {\boldsymbol J}_{{\rm{top}}}^\dagger |j_{{\rm{bottom}}}^\prime \rangle ,$$
$$\left[{\begin{array}{*{20}{c}}{\tilde j_{{\rm{top}}}^{(1)}}\\{- \tilde j_{{\rm{top}}}^{(2)}}\end{array}} \right] ={\boldsymbol J}_{{\rm{top}}}^T\left[{\begin{array}{*{20}{c}}{\tilde j_{{\rm{bottom}}}^{(1)}}\\{- \tilde j_{{\rm{bottom}}}^{(2)}}\end{array}} \right].$$
$${{\boldsymbol J}_{{\rm{bottom}}}} = \left[{\begin{array}{*{20}{c}}{\tilde A}&{\tilde B}\\{\tilde C}&{\tilde D}\end{array}} \right],$$
$${J_{{\rm{top}}}} = \left[{\begin{array}{*{20}{c}}{\tilde A}&{- \tilde C}\\{- \tilde B}&{\tilde D}\end{array}} \right],$$
$${{\boldsymbol J}_{{\rm{top}}}} = {{\boldsymbol M}_{\textit{xy}}}{{\boldsymbol J}_{{\rm{bottom}}}}{\boldsymbol M}_{\textit{xy}}^{- 1},$$
$${{\boldsymbol M}_{\textit{xy}}} = \left[{\begin{array}{*{20}{c}}1&0\\0&{- 1}\end{array}} \right].$$
$${{\boldsymbol J}_{{\rm{top}}}} = \left[{\begin{array}{*{20}{c}}{\tilde A}&{- \tilde B}\\{- \tilde C}&{\tilde D}\end{array}} \right].$$
$${\boldsymbol J}_{{\rm{top}}}^T = {{\boldsymbol J}_{{\rm{top}}}},$$
$${{\boldsymbol \Sigma} ^\dagger}{\boldsymbol \Sigma} = (a{{\boldsymbol U}_1} + b{{\boldsymbol U}_2}{)^\dagger}(a{{\boldsymbol U}_1} + b{{\boldsymbol U}_2}) = (|a{|^2} + |b{|^2}){\mathbb I} + {a^*}b{\boldsymbol U}_1^\dagger {{\boldsymbol U}_2} + {b^*}a{\boldsymbol U}_1^\dagger {{\boldsymbol U}_2} \ne {\mathbb I}.$$
$${{\boldsymbol \Sigma} ^T} = (a{{\boldsymbol S}_1} + b{{\boldsymbol S}_2}{)^T} = a{\boldsymbol S}_1^T + b{\boldsymbol S}_2^T = a{{\boldsymbol S}_1} + b{{\boldsymbol S}_2},$$
$$\delta = |{t_x}{e^{i{\phi _x}}} - {e^{i\phi _x^d}}{|^2} + |{t_y}{e^{i{\phi _y}}} - {e^{i\phi _y^d}}{|^2},$$
$${\boldsymbol J}(x,y) = {\boldsymbol R}(\theta)\!\left[\!{\begin{array}{*{20}{c}}{{e^{i{\phi _1}(x,y)}}}&0\\0&{{e^{{\phi _2}(x,y)}}}\end{array}} \!\right]{\boldsymbol R}(- \theta) = {e^{i{\phi _1}(x,y)}}|\theta \rangle \langle \theta | + {e^{i{\phi _2}(x,y)}}\left|\theta + \frac{\pi}{2}\right \rangle \left \langle \theta + \frac{\pi}{2}\right|\!,$$
$${\boldsymbol J}(x,y) = {\boldsymbol R}(\theta (x,y))\left[{\begin{array}{*{20}{c}}{{e^{i\frac{\Delta}{2}}}}&0\\0&{{e^{- i\frac{\Delta}{2}}}}\end{array}} \right]{\boldsymbol R}(- \theta (x,y)),$$
$${\boldsymbol J}(x,y)|j\rangle = {e^{i\phi (x,y)}}|{j^\prime}\rangle \quad {\rm{and}}\quad {\boldsymbol J}(x,y)|{j^ \bot}\rangle = {e^{i{\phi ^ \bot}(x,y)}}|{j^{\bot ,^\prime}}\rangle$$
$$|{j^\prime}\rangle = |(j{)^*}\rangle \quad {\rm{and}}\quad |{j^{\bot ,^\prime}}\rangle = |({j^ \bot}{)^*}\rangle$$
$${\boldsymbol J}(x,y) = \left[{\begin{array}{*{20}{c}}|&|\\{{e^{i\phi (x,y)}}|(j{)^*}\rangle}&{{e^{i{\phi ^ \bot}(x,y)}}|({j^ \bot}{)^*}\rangle}\\|&|\end{array}} \right]{\left[{\begin{array}{*{20}{c}}|&|\\{|j\rangle}&{|{j^ \bot}\rangle}\\|&|\end{array}} \right]^{- 1}},$$
$${\boldsymbol J}(x,y) = {e^{i\phi (x,y)}}|(j{)^*}\rangle \langle j| + {e^{i{\phi ^ \bot}(x,y)}}|({j^ \bot}{)^*}\rangle \langle {j^ \bot}|,$$
$${\boldsymbol A}({k_x},{k_y}) = {{\cal F}_{({k_x},{k_y})}}\{{e^{i{\phi ^ +}(x,y)}}\} |({\lambda ^ +}{)^*}\rangle \langle {\lambda ^ +}| + {{\cal F}_{({k_x},{k_y})}}\{{e^{i{\phi ^ -}(x,y)}}\} |({\lambda ^ -}{)^*}\rangle \langle {\lambda ^ -}|.$$
$$|{j_{{\rm{out}}}}(x,y)\rangle = {\boldsymbol J}(x,y)|{j_{{\rm{in}}}}\rangle ,$$
$$|a({k_x},{k_y})\rangle = {\cal F}\{|{j_{{\rm{out}}}}(x,y)\rangle \} ,$$
$${\boldsymbol J}(x,y) = {\boldsymbol R}(\theta (x,y))\left[{\begin{array}{*{20}{c}}i&0\\0&{- i}\end{array}} \right]{\boldsymbol R}(- \theta (x,y)).$$
$${\cal F}\{{e^{i2\theta (x,y)}}\} |R\rangle \langle L| + {\cal F}\{{e^{- i2\theta (x,y)}}\} |L\rangle \langle R|.$$
$${\boldsymbol J}(x,y) = {\boldsymbol R}(\theta (x,y))\left[{\begin{array}{*{20}{c}}{{e^{i{\phi _x}(x,y)}}}&0\\0&{{e^{i{\phi _y}(x,y)}}}\end{array}} \right]{\boldsymbol R}(- \theta (x,y)).$$