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Cited by 217Orbital Angular Momentum in OpticsQuantum and Classical ElectrodynamicsMetamaterials and Metasurfaces Applications

Geometrodynamics of polarized light: Berry phase and spin Hall effect in a gradient-index medium

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Bliokh, Konstantin Y·2009·Journal of Optics A: Pure and Applied Optics
DOI 10.1088/1464-4258/11/9/094009
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Abstract

We review the geometrical-optics evolution of an electromagnetic wave propagating along a curved ray trajectory in a gradient-index dielectric medium. A Coriolis-type term appears in Maxwell equations under transition to the rotating coordinate system accompanying the ray. This term describes the spin-orbit coupling of light which consists of (i) the Berry phase responsible for a trajectory-dependent polarization variations and (ii) the spin Hall effect representing polarization-dependent trajectory perturbations. These mutual phenomena are described within universal geometrical structures underlying the problem and are explained by the dynamics of the intrinsic angular momentum carried by the wave. Such close geometro-dynamical interrelations illuminate a dual physical nature of the phenomena.

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Written from the paper’s full text.

TL;DR

This review develops a unified geometrical-optics description of spin-orbit coupling of light in a gradient-index dielectric, where the Berry phase and spin Hall effect are mutual consequences of a Coriolis-type term in the ray-accompanying frame. It shows that polarization evolution and polarization-dependent ray deflection are governed by Berry connection/curvature and by conservation of total angular momentum.

Key findings

  • The Berry phase and spin Hall effect of light are two sides of the same spin-orbit interaction: the Berry phase describes the trajectory’s effect on polarization, while the spin Hall effect describes polarization’s reaction on the ray trajectory; both follow from a single spin-orbit Lagrangian.
  • The Berry curvature takes the form of a topological monopole in momentum space and equals \(F = t\) on the unit \(t\)-sphere, indicating unit Gaussian curvature; the Berry phase for a cyclic evolution equals the solid angle \(\Theta\) cut by the loop on the \(t\)-sphere, up to a \(2\pi\) frame-rotation difference.
  • The spin Hall effect adds a helicity-dependent topological term to the ray equations, producing a transverse deflection orthogonal to both the wave momentum \(p\) and the inhomogeneity gradient \(\nabla n\); its magnitude is estimated as the wavelength \(\lambda\) for a complete \(2\pi\) turn of the propagation direction, and the splitting between right- and left-hand circularly polarized beams is \(\Delta = 2\delta r_0\).
  • For a helical ray in a cylindrically symmetric medium, the Berry phase grows linearly with propagation distance and rotates the polarization ellipse uniformly; the spin Hall displacement is directed along the binormal and accumulates over repeated cyclical trajectories.

Methods

  • Theoretical review using short-wavelength geometrical optics for a monochromatic electromagnetic wave in a smoothly inhomogeneous isotropic dielectric, with small parameter \(\mu\) comparing wavelength to medium inhomogeneity scale.
  • Maxwell equations are projected onto the plane transverse to the ray, transformed to a ray-accompanying rotating coordinate frame, and diagonalized in the circular-polarization basis; the resulting Coriolis term is treated to first order in \(\mu\).
  • The analysis uses Berry connection/curvature, Jones-vector and Stokes/Poincaré representations, Euler-Lagrange ray equations, and a worked example of a helical ray in a cylindrical gradient-index medium.

Limitations

  • The paper states that it considers only the spin-redirection Rytov–Vladimirskii–Berry phase in inhomogeneous media and does not cover generalizations to step-index interfaces, anisotropic media, or higher-order states of light carrying orbital angular momentum.
  • It notes that geometrical optics is rigorously inapplicable in single-mode waveguides, although the polarization evolution and Berry phase of waveguide modes obey the same geometrical formalism.
  • It states that the spin Hall displacement is typically much smaller than the width of a paraxial optical beam, though observable because beam centroids can be measured with subwavelength accuracy; it also notes that the Imbert–Fedorov shift cannot generally be described within the Berry-phase formalism because of non-adiabatic evolution at sharp interfaces.

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Crossref · 10.1088/1464-4258/11/9/094009

Cites

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Claims the paper makes

0 of 184 cited sources checked · 0 supported · 0 flagged

  • Findingnone given
    § Geometrodynamics of polarized light: Berry phase and spin Hall effect in a gradient-index medium · p. 1

    A Coriolis-type term appears in Maxwell equations under transition to the rotating coordinate system accompanying the ray.

    “A Coriolis-type term appears in Maxwell equations under transition to the rotating coordinate system accompanying the ray.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The inhomogeneity of the medium is smooth, i.e., its characteristic scale L is large compared with the wavelength.
    • The authors state this as an assumption: Higher-order terms proportional to Ω^2 and Ω dot are neglected.
  • Findingnone given
    § 3.3. Berry connection and curvature · p. 6

    The spin-orbit Lagrangian brings about an additional polarization-dependent wave phase.

    “The spin-orbit Lagrangian (3.12) brings about an additional polarization-dependent wave phase.”
    Quote in the paperSubstantial claim
  • Findingnone given
    § 3.2. Coriolis term in the co-moving frame · p. 6

    The spin-orbit Lagrangian is proportional to the longitudinal component of the angular velocity of the co-moving coordinate frame.

    “The spin-orbit Lagrangian is proportional to the longitudinal component of the angular velocity of the co-moving coordinate frame, which is”
    Quote in the paperSubstantial claim
  • Findingreasoning
    § 3.2. Coriolis term in the co-moving frame · p. 4

    The second term in Eq. (3.6) describes the Coriolis effect caused by the rotation of the ray coordinate frame.

    “Here the second term describes the Coriolis effect caused by the rotation of the ray coordinate frame.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: Higher-order terms proportional to Ω^2 and Ω dot are neglected.
  • Findingreasoning
    § 3.4. Berry phase and polarization evolution · p. 9

    The angle of the resulting rotation of e equals the Berry phase Φ_B.

    “The angle of the resulting rotation of e equals the Berry phase B Φ .”
    Quote in the paperSubstantial claim
  • May combine two claims. 44
  • The quote only partly states this claim. 5
  • Classified as probably not a substantial claim. 18

  • May combine two claims.

    Findingreasoning
    § 3.2. Coriolis term in the co-moving frame · p. 5

    According to Eq. (3.10), in the first approximation of geometrical optics, the two circular polarization states evolve independently, and the zero-order polarization degeneracy is lifted by the Coriolis term.

    “According to Eq. (3.10), in the first approximation of the geometrical optics, these two states evolve independently, and the zero-order polarization degeneracy is lifted by the Coriolis term.”
    Quote in the paperSubstantial claim

  • May combine two claims. The quote only partly states this claim.

    Findingnone given
    § 3.2. Coriolis term in the co-moving frame · p. 4

    This Coriolis term is small: Ω ~ μ/λ0, but should be taken into account in the first-order approximation in μ.

    “This term is small: 0 ~ µ Ω Ż , but should be taken into account in the first-order approximation in µ .”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingnone given
    § 3.4. Berry phase and polarization evolution · p. 11

    The third component of the Stokes vector indicates the wave helicity and represents a continuous analogue of the quantized helicity s=±1.

    “Its third component, [ ] 2 2 3 1,1 S e e + -= -∈ -, indicates the wave helicity and represents a continuous analogue of the quantized helicity 1 s = ± .”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § 3.4. Berry phase and polarization evolution · p. 9

    The Berry phase is equal to the contour integral of the effective vector-potential A or to the flux of the 'magnetic field' F through this contour.

    “It is equal to the contour integral of the effective vector-potential A or to the flux of the 'magnetic field' F through this contour.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § 3.4. Berry phase and polarization evolution · p. 9

    Equation (3.32) is independent of the local coordinate frame and describes the polarization dynamics in a universal way.

    “In contrast to the previous Eqs. (3.25) -(3.29), equation (3.32) is independent of the local coordinate frame ( ) , , v w t and describes the polarization dynamics in a universal way.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingreasoning
    § 3.4. Berry phase and polarization evolution · p. 9

    Equation (3.32) is non-integrable, i.e., its solution e(l) is non-local due to the non-integrability of the Berry phase.

    “Equation (3.32) is non-integrable , i.e., its solution ( ) l e is non-local due to the non-integrability of the Berry phase (3.25) and (3.26).”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data
    § 3.3. Berry connection and curvature · p. 7

    On the surface of the unit t-sphere (i.e., at p=1), the Berry curvature equals F=t, indicating the unit Gaussian curvature of the surface.

    “On the surface of the unit t -sphere (i.e., at 1 p = ), the Berry curvature equals = F t indicating the unit Gaussian curvature of the surface.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § 4.4. Spin Hall effect · p. 15

    The displacement δr due to the spin Hall effect is directed locally orthogonal to the inhomogeneity, with δp ≡ 0.

    “Therefore, displacement δ r is directed locally orthogonal to inhomogeneity, 0 δ ≡ p , and we can use Eqs. (3.45) and (3.46) for calculation of δ r .”
    Quote in the paperSubstantial claim

  • The quote only partly states this claim.

    Findingreasoning
    § 4.3. Evolution of polarization and the Berry phase · p. 15

    In the Stokes-vector formalism, the Stokes vector precesses uniformly with angular velocity (4.9).

    “In the Stokes-vector formalism, Eq. (3.37), the Stokes vector precesses uniformly with angular velocity, Fig. 3:”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § 4.3. Evolution of polarization and the Berry phase · p. 15

    The Berry phase grows linearly with the propagation distance l, and the polarization ellipse experiences a uniform rotation with angular velocity (4.7) in the ray reference frame.

    “Thus, the Berry phase grows linearly with the propagation distance l , and polarization ellipse experience a uniform rotation with angular velocity (4.7) in the ray reference frame, see Fig. 1.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § 3.5. Ray equations and spin Hall effect · p. 12

    The spin-dependent term of Eqs. (3.40) is proportional to the curvature of the ray and directed orthogonally to the local propagation plane.

    “Indeed, the spin-dependent term of Eqs. (3.40) is proportional to the curvature of the ray and directed orthogonally to the local propagation plane (see Section A.2).”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § 3.5. Ray equations and spin Hall effect · p. 12

    Equations (3.40) depend on the Berry curvature F and therefore are gauge-invariant, i.e., independent of the coordinate frame.

    “Equations (3.40) depend on the Berry curvature F and therefore are gauge-invariant, i.e., independent on the coordinate frame.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Findingcitation
    § 3.5. Ray equations and spin Hall effect · p. 14

    A number of experiments observed spin-orbit coupling and polarization transport of light in complex speckle fields of multimode optical fibers.

    “There had been a number of remarkable experiment [120 -123] which observed the spin-orbit coupling and polarization transport of light in complex speckle fields of multimode optical fibers (see also [24]),”
    Quote in the paperNot a substantial claim0 of 4 sources checked
    • cites [120] source

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    • cites [121] source

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    • cites [122] source

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    • cites [123] source

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  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Findingnone given
    § 4.4. Spin Hall effect · p. 16

    The curvature of the helical ray is K = ρ^{-1} sin^{-1}θ, and Eq. (4.14) agrees with Eq. (A12).

    “Note that 1 2 1 sin K ρ ϑ --= is the curvature of the helical ray, and Eq. (4.14) agrees with Eq. (A12).”
    Quote in the paperNot a substantial claim

  • The quote only partly states this claim.

    Findingreasoning
    § 4.3. Evolution of polarization and the Berry phase · p. 15

    The Berry phase per one period of the helix equals Eq. (4.11), where Θ is the corresponding area on the t-sphere.

    “The Berry phase (the angle of rotation of the polarization ellipse) per one period of the helix, 2 / sin l π ρ ϑ = , equals”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingcitation
    § 3.5. Ray equations and spin Hall effect · p. 14

    The spin Hall effect of light and the ray equations (3.40) were verified by direct measurements for helical trajectories as in Fig. 4.

    “Recently, the spin Hall effect of light and the ray equations (3.40) were verified by direct measurements for a helical trajectories as in Fig. 4 [35].”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [35] source

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  • May combine two claims.

    Findingreasoning
    § 3.5. Ray equations and spin Hall effect · p. 12

    The spin-orbit term in Eqs. (3.40) is proportional to ħ0 and can be treated as a 'semiclassical' correction to the ray equations.

    “The spin-orbit term in Eqs. (3.40) is proportional to 0 Ż , and, thus, can be treated as a ' semiclassical ' correction to the ray equations.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § 4.4. Spin Hall effect · p. 16

    The shift per one period of the helix equals Eq. (4.15), and the splitting between the centers of right-hand and left-hand polarized beams is Δ = 2δr.

    “The shift per one period of helix, 2 / sin l π ρ ϑ = , equals”
    • Also stated in: § 4.4. Spin Hall effect · p. 16 · plain
      “whereas the splitting between the centers of right-hand, 1 s = , and left-hand, 1 s = -, polarized beams is, evidently, 0 2 δ ∆ = r , see Fig. 4.”
    Quote in the paperSubstantial claim

  • May combine two claims. The quote only partly states this claim.

    Findingreasoning
    § A.2. Frenet frame · p. 20

    The torsion characterizes local rotation of the Frenet frame about t, and the Berry phase in this frame is given by Eq. (A11).

    “Thus, the torsion characterizes local rotation of this frame about t , and the Berry phase in this frame is”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingreasoning
    § A.1. Euler-angles frame · p. 20

    The Berry curvature is the monopole (3.20), whereas the Berry connection becomes Eq. (3.35) after gauge transformation (3.23) with α = -φ.

    “Here the Berry curvature is the monopole (3.20), whereas the Berry connection becomes Eq. (3.35) after gauge transformation (3.23) with α = -φ .”
    Quote in the paperNot a substantial claim

  • Classified as probably not a substantial claim.

    Findingcitation
    § A.2. Frenet frame · p. 20

    The topological term in equations (3.40) can be written as Eq. (A12).

    “Indeed, the topological term in equations (3.40) can be written as [35]”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [35] source

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisenone given
    § 2. Geometrical optics of scalar waves · p. 3

    The central wave phase in geometrical optics is given by the WKB integral along the ray.

    “In particular, the central wave phase 0 Φ ( 0 i e ψ Φ ∝ ) in this approximation is given by the WKB integral along the ray:”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Premisecitation
    § 1. Introduction · p. 1

    In the short-wavelength (semiclassical) approximation, the spin-orbit coupling can be described within the Berry-phase formalism, which has a universal geometrical character and can be applied to a diversity of quantum and classical wave systems.

    “In the short-wavelength (semiclassical) approximation, the spin-orbit coupling can be described within the Berry-phase formalism, which has a universal geometrical character and can be applied to a diversity of quantum and classical wave systems [1 -3].”
    Quote in the paperSubstantial claim0 of 3 sources checked
    • cites [1] Shapere A, 1989, Geometric Phases in Physics

      Not checked in the free run.

    • cites [2] Markovski B, 1989, Topological Phases in Quantum Theory

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    • cites [3] Bohm A, 2003, Geometrical Phase in Quantum Systems source

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  • May combine two claims.

    Premisereasoning
    § 2. Geometrical optics of scalar waves · p. 3

    The rays are curvilinear because of the wave refraction in the medium, and the gradient n ∇ plays the role of an 'external force' affecting the motion of the particle.

    “The rays are curvilinear because of the wave refraction in the medium, and the gradient n ∇ plays the role of an 'external force' affecting the motion of the particle.”
    Quote in the paperNot a substantial claim

  • Classified as probably not a substantial claim.

    Premisecitation
    § 1. Introduction · p. 1

    This polarization-dependent shift is known as the spin Hall effect of light or optical Magnus effect.

    “This effect is known as the spin Hall effect of light or optical Magnus effect [24 -35]”
    Quote in the paperSubstantial claim0 of 10 sources checked
    • cites [24] Pancharatnam S, 1975, Collected Works

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    • cites [25] source

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  • The quote only partly states this claim.

    Interpretationreasoning
    § Geometrodynamics of polarized light: Berry phase and spin Hall effect in a gradient-index medium · p. 1

    The spin-orbit coupling of light consists of the Berry phase, which is responsible for trajectory-dependent polarization variations.

    “This term describes the spin-orbit coupling of light which consists of (i) the Berry phase responsible for a trajectory-dependent polarization variations and (ii) the spin Hall effect representing polarizationdependent trajectory perturbations.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Premisereasoning
    § 3.1. Maxwell equations and polarization orthogonality · p. 3

    In the zero-order approximation in μ, Maxwell equations result in the scalar Hamiltonian and the ray equations, but polarization is indeterminate in this approximation.

    “In the zero -order approximation in µ , Eq. (3.1) results in the scalar Hamiltonian and the ray equations (2.3) -(2.6). However, polarization is indeterminate in this approximation.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning
    § 1. Introduction · p. 2

    The Berry phase and spin Hall effect are closely related to the dynamics of the spin angular momentum carried by the wave and can be described in terms of the Coriolis effect or the angular Doppler effect.

    “It will be shown that the Berry phase and spin Hall effect are closely related to the dynamics of the spin angular momentum carried by the wave and can be described in terms of the Coriolis effect or the angular Doppler effect.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S4 (based).

    Premisenone given
    § 3. Geometrical optics of vector waves: spin-orbit geometrodynamics · p. 3

    Evolution of vector waves involves an additional internal degree of freedom - polarization (spin).

    “Evolution of vector waves, e.g., electromagnetic or elastic ones, requires more sophisticated approaches, since it involves an additional, internal degree of freedom -polarization (spin).”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S2 (checkable).

    Interpretationnone given
    § Geometrodynamics of polarized light: Berry phase and spin Hall effect in a gradient-index medium · p. 1

    These mutual phenomena are described within universal geometrical structures underlying the problem.

    “These mutual phenomena are described within universal geometrical structures underlying the problem”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S2 (checkable), S3 (load-bearing).

    Interpretationreasoning
    § Geometrodynamics of polarized light: Berry phase and spin Hall effect in a gradient-index medium · p. 1

    Such close geometro-dynamical interrelations illuminate a dual physical nature of the phenomena.

    “Such close geometro-dynamical interrelations illuminate a dual physical nature of the phenomena.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Premisenone given
    § 3.1. Maxwell equations and polarization orthogonality · p. 3

    The polarization term in Maxwell equations is small in a gradient-index medium (of the order of μ) and can be treated perturbatively within the geometrical optics approach.

    “This polarization term is small in a gradient-index medium (of the order of µ ) and can be treated perturbatively within the geometrical optics approach.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation
    § 1. Introduction · p. 1

    In a more general context, the Berry phase and the accompanying geometric reaction forces arise from a coupling between the fast and slow degrees of freedom.

    “In a more general context, the Berry phase and the accompanying geometric reaction forces arise from a coupling between the fast and slow degrees of freedom [44]”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [44] source

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    Premisecitation
    § 3.4. Berry phase and polarization evolution · p. 10

    The polarization rotation corresponding to the Berry phase has been measured experimentally in single-mode optical fibers coiled in a helix and for helical rays in a multimode rectilinear waveguide.

    “The polarization rotation corresponding to the Berry phase (3.33) and (3.34) has been measured experimentally in single-mode optical fibers coiled in a helix similar to Fig. 1 [10,11] and for helical rays in a multimode rectilinear waveguide [20,35], cf. Section 4.”
    Quote in the paperNot a substantial claim0 of 4 sources checked
    • cites [10] Rytov S M, 1938, Dokl. Akad. Nauk. SSSR

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    • cites [11] Vladimirskii V V, 1941, Dokl. Akad. Nauk. SSSR

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    • cites [35] source

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    Interpretationcitation
    § 3.4. Berry phase and polarization evolution · p. 11

    This evidences an adiabatic regime of the wave evolution, i.e., independence of the right- and left-hand circular polarizations.

    “This evidences adiabatic regime of the wave evolution [4,12,14,17], i.e., independence of the right- and left-hand circular polarizations.”
    Quote in the paperSubstantial claim0 of 4 sources checked
    • cites [4] Vinitskii S I, 1990, Usp. Fiz. Nauk source

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    • cites [12] source

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  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation
    § 3.4. Berry phase and polarization evolution · p. 9

    The Berry phase is analogous to the Aharonov-Bohm (or Dirac) phase in quantum mechanics.

    “At the same time, it shows that the Berry phase is analogous to the Aharonov -Bohm (or Dirac) phase in quantum mechanics [94 -96].”
    Quote in the paperNot a substantial claim0 of 3 sources checked
    • cites [94] source

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    • cites [95] source

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    • cites [96] Landau L D, 1975, The Classical Theory of Fields

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  • May combine two claims.

    Interpretationreasoning
    § 3.5. Ray equations and spin Hall effect · p. 13

    The spin Hall effect in a gradient-index medium and the Imbert-Fedorov shift represent, respectively, the weak- and strong-scattering limits of the polarization transport induced by the spin-orbit interaction of light.

    “In fact, the spin Hall effect in a gradient-index medium and the Imbert-Fedorov shift represent, respectively, the weak- and strong-scattering limits of the polarization transport induced by the spin-orbit interaction of light.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning
    § 5. Concluding remarks · p. 16

    The spin Hall effect represents a real deflection of the ray trajectory, which is independent of the observation frame.

    “In contrast, the spin Hall effect represents a real deflection of the ray trajectory, which is independent on the observation frame.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning
    § 3.5. Ray equations and spin Hall effect · p. 14

    All those experiments, apart from numerical simulations in [123], dealt with step-index fibers, so their results should rather be associated with Imbert-Fedorov shifts upon reflection at sharp inhomogeneities.

    “but all of them (apart from numerical simulations in [123]) dealt with step-index index fibers, so that these results should rather be associated with the Imbert-Fedorov shifts upon the reflection at sharp inhomogeneities.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [123] source

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  • Classified as probably not a substantial claim.

    Interpretationnone given
    § 3.5. Ray equations and spin Hall effect · p. 13

    J is associated with the total angular momentum of light, consisting of the extrinsic orbital angular momentum L = r × p and the intrinsic spin angular momentum Σ.

    “Here J is obviously associated with the total angular momentum of light, consisting of the extrinsic orbital angular momentum = × L r p and the intrinsic spin angular momentum Σ .”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Premisecitation
    § 3.4. Berry phase and polarization evolution · p. 11

    The pseudospin S completely describes the polarization state of the wave, while the real spin angular momentum s=Σtħ does not.

    “It is worth remarking that the pseudospin S arrowrightnosp completely describes the polarization state of the wave, while the real spin angular momentum 0 s = Σ t Ż does not [97].”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [97] source

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  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

    Interpretationnone given
    § 3.5. Ray equations and spin Hall effect · p. 14

    Propagation of light in optical fibers does not allow one to observe geometrical-optics ray trajectories hidden in the complex speckle patterns of the interfering modes.

    “In addition, propagation of light in optical fibers does not allow one to observe the geometricaloptics ray trajectories hidden in the complex speckle patterns of the interfering modes.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning
    § 3.5. Ray equations and spin Hall effect · p. 13

    The ray displacement takes the form of a nonlocal contour integral in momentum space, which can grow unlimitedly with the length of the trajectory.

    “Thus, akin to the Berry phase (3.26) and (3.33), the ray displacement takes the form of a nonlocal contour integral in momentum space, which can grow unlimitedly with the length of the trajectory.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationnone given
    § 5. Concluding remarks · p. 16

    The Berry curvature appears in momentum rather than coordinate space and takes the form of the topological monopole as in the case of massless spinning particles.

    “Important to note that the Berry curvature appears in momentum rather than coordinate space and takes the form of the topological monopole as in the case of massless spinning particles.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning
    § 3.5. Ray equations and spin Hall effect · p. 12

    In the spin Hall effect, the Berry curvature manifests itself dynamically - the polarization term in Eq. (3.40) represents a 'Lorentz force' from the 'magnetic monopole'.

    “In the spin Hall effect, the Berry curvature manifests itself dynamically - the polarization term in Eq. (3.40) represents a ' Lorentz force ' from the 'magnetic monopole' (3.20).”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisereasoning
    § 3.5. Ray equations and spin Hall effect · p. 13

    The Imbert-Fedorov shift is a polarization-dependent displacement of a wave beam reflected or transmitted at a plane dielectric interface.

    “The latter effect is a similar polarization-dependent displacement of a wave beam reflected or transmitted at a plane dielectric interface.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S4 (based).

    Interpretationnone given
    § 5. Concluding remarks · p. 16

    In the zero-order scalar approximation, light propagates along a smooth trajectory which ensures the adiabatic regime of the evolution.

    “In the zero-order (scalar) approximation the light propagates along a smooth trajectory which ensures the adiabatic regime of the evolution.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationnone given
    § 5. Concluding remarks · p. 16

    The first-order corrections involve the wave polarization and bring about two mutual phenomena: the Berry phase and the spin Hall effect of light.

    “The first-order corrections involve the wave polarization and bring about two mutual phenomena -the Berry phase and the spin Hall effect of light.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning
    § 4.3. Evolution of polarization and the Berry phase · p. 15

    Since the Euler-angles and Frenet frames coincide, the Berry-phase expressions are equivalent as well.

    “Since the Euler-angles and Frenet frames coincide with each other, the Berry-phase expressions are equivalent as well.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning
    § 5. Concluding remarks · p. 16

    The Berry phase can be associated with the inertia of the wave field which remains locally non-rotating, and can be eliminated in the proper inertial coordinate frame.

    “The Berry phase can be associated with the inertia of the wave field which remains locally non-rotating, and can be eliminated in the proper inertial coordinate frame (see Section A3).”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Premisecitation
    § 3.4. Berry phase and polarization evolution · p. 11

    The Jones and Stokes representations characterize the polarization state of light by the complex two-component vector ψ and the real three-component vector S, respectively; in the generic case, they obey SU(2) and SO(3) evolutions and correspond to the Schrödinger and Heisenberg representations of spin in quantum mechanics.

    “The Jones and Stokes representations characterize the polarization state of light, respectively, by the complex two-component vector ψ and the real three-component vector S arrowrightnosp . In the generic case, they obey ( ) SU 2 and ( ) SO 3 evolutions and correspond, respectively, to the Schrödinger and Heisenberg representations of spin in quantum mechanics [34,98].”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites [34] source

      Not checked in the free run.

    • cites [98] source

      Not checked in the free run.

  • May combine two claims.

    Interpretationreasoning
    § 5. Concluding remarks · p. 16

    The Berry phase represents the effect of the trajectory on the polarization of light, while the spin Hall effect is the inverse effect which accounts for the reaction of polarization on the trajectory of propagation.

    “The Berry phase represents the effect of the trajectory on the polarization of light, while the spin Hall effect is the inverse effect which accounts for the reaction of polarization on the trajectory of propagation.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning
    § 3.5. Ray equations and spin Hall effect · p. 11

    The spin Hall effect can be treated as an effective circular birefringence of the inhomogeneous medium, but in fact it arises because of the intrinsic property (spin) of light coupled to a curved ray trajectory.

    “It can be treated as an effective circular birefringence of the inhomogeneous medium [25,26], but, in fact, it arises because of the intrinsic property (spin) of light coupled to a curved ray trajectory.”
    Quote in the paperSubstantial claim0 of 2 sources checked
    • cites [25] source

      Not checked in the free run.

    • cites [26] source

      Not checked in the free run.

  • May combine two claims.

    Interpretationreasoning
    § A.2. Frenet frame · p. 20

    While the Berry phase is an artifact of a particular rotating ray-accompanying coordinate frame, the spin Hall effect is a Coriolis-type transverse deflection induced by a real bending of spinning light.

    “Thus, while the Berry phase is an artifact of a particular rotating ray-accompanying coordinate frame, the spinHall effect is a Coriolis-type transverse deflection induced by a real bending of spinning light.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationown data
    § A.3. Parallel-transport frame · p. 21

    The parallel-transport frame is non-local: equation (A14) contains the phase which depends on the integral evolution of t.

    “However, the parallel-transport frame is non-local -equation (A14) contains the phase which depends on the integral evolution of t .”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationnone given
    § 5. Concluding remarks · p. 17

    For higher-order beams with vortices, the intrinsic orbital angular momentum behaves, in the geometrical optics approximation, very much like spin and the vortex charge determines spin-like helicity of the beam.

    “In this case the intrinsic orbital angular momentum behaves, in the geometrical optics approximation, very much like spin and the vortex charge determines spin-like helicity of the beam.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning
    § A.3. Parallel-transport frame · p. 21

    Although the rotation of the ray-accompanying frame about t can be eliminated, rotation associated with the ray bending and the ray curvature is unremovable.

    “Thus, although the rotation of the ray-accompanying frame about t can be eliminated, rotation associated with the ray bending and the ray curvature is unremovable.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Interpretationcitation
    § 5. Concluding remarks · p. 17

    A number of effects related to the spin-orbit coupling of light take place in multimode optical fibers.

    “A number of effects related to the spin-orbit coupling of light take place in multimode optical fibers [24,120 -123,141 -144].”
    Quote in the paperNot a substantial claim0 of 9 sources checked
    • cites [24] Pancharatnam S, 1975, Collected Works

      Not checked in the free run.

    • cites [120] source

      Not checked in the free run.

    • cites [121] source

      Not checked in the free run.

    • cites [122] source

      Not checked in the free run.

    • cites [123] source

      Not checked in the free run.

    • cites [141] Kravtsov Y A, 1996, Phys.-Usp. source

      Not checked in the free run.

    • cites [142] source

      Not checked in the free run.

    • cites [143] source

      Not checked in the free run.

    • cites [144] source

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Interpretationnone given
    § A.2. Frenet frame · p. 20

    The Frenet basis vectors characterize local properties of the curve and are independent of the laboratory coordinate frame.

    “These basis vectors characterize local properties of the curve and are independent on the laboratory coordinate frame.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S4 (based).

    Interpretationnone given
    § 5. Concluding remarks · p. 17

    The Berry phase is the phase of the central plane wave in the packet.

    “The Berry phase is the phase of the central plane wave in the packet.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning
    § 5. Concluding remarks · p. 17

    The spin-orbit-type interaction occurs between the intrinsic and extrinsic orbital angular momentums of light beams, which induces the corresponding Berry phase and orbital Hall effect.

    “Thus, the spin-orbit-type interaction occurs between the intrinsic and extrinsic orbital angular momentums of light beams, which induces the corresponding Berry phase and orbital Hall effect.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning
    § A.2. Frenet frame · p. 20

    The spin-induced ray deflection is determined solely by the ray curvature and is directed along the binormal, i.e., orthogonal to the local propagation plane (t,n).

    “This shows that the spin-induced ray deflection is determined solely by the ray curvature and is directed along the binormal, i.e., orthogonal to the local propagation plane ( , ) t n .”
    Quote in the paperSubstantial claim
62 of 62

  • The authors state this as an assumption.

    The inhomogeneity of the medium is smooth, i.e., its characteristic scale L is large compared with the wavelength.

    “We assume that inhomogeneity of the medium is smooth, i.e., its characteristic scale, L , is large as compared with the wavelength:”

  • The authors state this as an assumption.

    The dielectric constant of the medium is real.

    “where 2 n ε = is the dielectric constant of the medium, which is assumed to be real.”

  • The authors state this as an assumption.

    Higher-order terms proportional to Ω^2 and Ω dot are neglected.

    “Neglecting higher-order terms proportional to 2 Ω and Ω ɺ , we arrive at”

  • The authors state this as an assumption.

    The wave beams or packets are Gaussian-type and do not bear an intrinsic orbital angular momentum.

    “We imply Gaussian-type wave beams or packets which do not bear an intrinsic orbital angular momentum.”

  • The paper relies on this without support.

    The position of the center of gravity of the beam can be measured with subwavelength accuracy.

    “since the position of center of gravity of the beam can be measured with a subwavelength accuracy.”
5 of 5

Claims other papers make about this one

Each is checked against this paper’s own text: does it say what the citing paper says it does?

  • The Berry connection and Berry curvature immediately appear in all intrinsic SOI phenomena, such as the spin-Hall effect.

    “Therefore, the Berry connection and Berry curvature immediately appear in all intrinsic SOI phenomena, such as the spin-Hall effect.”

    From Spin–orbit interactions of light (2015) · cites this paper as [7]

    The Concluding remarks (p. 16) describe the Berry phase and spin-Hall effect of light as manifestations of the Berry connection and curvature, supporting the spin-Hall example. But the claim extends this to all intrinsic spin-orbit-interaction phenomena: Section 3.5 (p. 13) says a related spin-orbit-induced shift cannot, in general, be described within the Berry-phase formalism. (LLM, conf=0.87)

    Experimental — accuracy not yet measured

  • The intrinsic SOI of light originate from fundamental properties of the Maxwell equations and are analogous to the spin-orbit interactions of relativistic quantum particles and electrons in solids.

    “The intrinsic SOI of light originate from fundamental properties of the Maxwell equations 7,8 and are analogous to the spin-orbit interactions of relativistic quantum particles 2,9,10 and electrons in solids 11,12 .”

    From Spin–orbit interactions of light (2015) · cites this paper as [7]

  • In a gradient-index medium, the trajectory of light is described by equations of motion whose last term describes the transverse spin-dependent displacement (spin-Hall effect of light).

    “The last term in Eq. (1) describes the transverse spin-dependent displacement of the trajectory, i.e., the spin-Hall effect of light (Fig. 1a).”

    From Spin–orbit interactions of light (2015) · cites this paper as [7]

  • The helicity-dependent term in the equations of motion can be considered as a Lorentz force produced by the Berry curvature acting in momentum space.

    “Later, it was shown that the helicity-dependent term in Eq. (1) can be considered as a 'Lorentz force' produced by the Berry curvature F σ P ( ) acting in momentum space 7,13,14,17 (Box 2).”

    From Spin–orbit interactions of light (2015) · cites this paper as [7]

Cites

Flagged first, then by citations

3 of the 162 references couldn’t be matched to a known paper.

Research works printed but not linked: 14

  1. Rytov S M, 1938, Dokl. Akad. Nauk. SSSR, 18, 263
  2. Vladimirskii V V, 1941, Dokl. Akad. Nauk. SSSR, 31, 222
  3. Pancharatnam S, 1975, Collected Works
  4. Berry M V, 1989, Geometric Phases in Physics, 7
  5. Horváthy P A, 2006, SIGMA, 2, 090
  6. Dandoloff R, 1989, J. Physique, 22, L461
  7. Fedorov F I, 1955, Dokl. Akad. Nauk SSSR, 105, 465
  8. Punko N N, 1984, JETP Lett., 39, 20
  9. Fedoseev V G, 1985, Opt. Spectrosc. (USSR), 58, 296
  10. Volyar A V, 1992, Ukr. Fiz. Zh., 37, 1468
  11. Volyar A V, 1998, Opt. Atmos. Okeana, 11, 1199
  12. Volyar A V, 1999, Opt. Spektrosc., 86, 593
  13. Baranova N B, 1994, JETP Lett., 59, 232
  14. Zel’dovich B Y, 1994, JETP Lett., 59, 766

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Authors
Bliokh, K. Y.
Affiliations
Australian National University (AU)
Corresponding author
Bliokh, Konstantin Y
ORCID
Bliokh, Konstantin Y
Venue
Journal of Optics A: Pure and Applied Optics
Publisher
IOP Publishing
Published
2009-08-04
Volume
11
Issue
9
Pages
094009
ISSN
  • 1464-4258 (ISSN-L)
  • 1741-3567
DOI
10.1088/1464-4258/11/9/094009
OpenAlex
W2062673708
Semantic Scholar
9c34209b9b5360bc9bb49a558828f9a34020ee7f
Cited by
217
References
162
Citations by year
2026: 7 · 2025: 17 · 2024: 18 · 2023: 35 · 2022: 21 · 2021: 28 · 2020: 8 · 2019: 16 · 2018: 8 · 2017: 5 · 2016: 8 · 2015: 9 · 2014: 2 · 2013: 10 · 2012: 7
FWCI
4.64
Type
journal article
Language
en
Open access
Locations
  • Journal of Optics A Pure and Applied Optics (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • ANU Open Research (Australian National University) (submittedVersion)
  • arXiv (Cornell University)
Funders
Australian Research Council
Topics
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Orbital Angular Momentum in Optics
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum and Classical Electrodynamics
  • Physical Sciences › Materials Science › Electronic, Optical and Magnetic Materials › Metamaterials and Metasurfaces Applications
Keywords
Berry phase, spin Hall effect, geometrical optics, polarized light, rotating coordinate system, polarization evolution

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Claims read by deepseek-v4.1-flash under definition version 3 (extract-v3) on 2026-10-08.

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