Introduction
In geometrical optics, light in a medium of refractive index n ( r ) n(\mathbf{r}) n ( r ) [2 , 3 , 4 , 5 , 6 ] . In the language of semiclassical dynamics it is an anomalous-velocity term generated by the Berry curvature in momentum space [7 ] .
The effect is described in the literature in two equivalent forms. In the first [3 , 6 ] , the equations of motion contain the Berry curvature explicitly. In the second, the helicity is built into the Hamiltonian through the Berry connection, as a shift of the position at which the refractive index is evaluated. In this paper we derive the helicity-dependent ray equations from Maxwell’s equations, show how the two forms are related, make the gauge dependence explicit, state the approximations involved, and set up the equations for the numerical propagation of rays in a medium with a spatially varying refractive index. The effect was predicted by Liberman and Zel’dovich [2 ] , derived for a smoothly inhomogeneous medium by Bliokh and Bliokh [3 , 4 ] , and described as a Hall effect of light, with the semiclassical equations of motion of a wave packet, by Onoda, Murakami and Nagaosa [5 ] ; reviews are given in Refs. [8 , 9 ] . The helicity-dependent deflection was observed for helical rays in a glass cylinder [6 ] , where the displacement, of the order of the wavelength per turn of the direction of propagation, accumulates over many turns. The associated Berry phase, the modern form of the Rytov-Vladimirskii law of rotation of the polarization plane along a curved ray [16 , 17 , 18 ] , was measured in helically coiled optical fibers [19 ] . At a sharp interface the corresponding effect is the Imbert-Fedorov shift [10 , 11 ] .
The same ray equations describe light in gravitational fields [12 , 13 , 14 ] and, for light carrying orbital angular momentum, in an optical waveguide [15 ] ; numerical integrations of the helicity-dependent ray equations have been reported for the Schwarzschild spacetime [13 ] and for such a waveguide [15 ] . We are not aware of a numerical propagation for a general three-dimensional gradient-index profile.
The new elements of this paper are the following. (i) The parallel-transport law of the polarization and the helicity-dependent ray equations are derived from the first-order Maxwell equations for a medium in which both the permittivity and the permeability vary; the polarization transport and the ray equations depend only on the refractive index, while the impedance enters only on the amplitude. (ii) The relation between the shifted-argument (Berry-connection) Hamiltonian and the Berry-curvature equations of motion is made explicit, including the gauge dependence of the canonical coordinate and the identification of the gauge-invariant physical position of the ray. (iii) The exact conservation law of the total angular momentum is given in the form used for numerical integration, together with a set of analytic tests. (iv) The ray equations are integrated numerically for a three-dimensional gradient-index profile. The equivalent formulations (Lagrangian of the ray path, non-canonical brackets, and frequency as the Hamiltonian) and the detailed comparison with Bliokh’s review [8 ] are collected in the appendices.
The paper is organized as follows. In Sec. II we write the eikonal expansion of Maxwell’s equations and show that the polarization of the first-order field is parallel transported along the ray. Section III defines the circular polarization basis, the Berry phase, the Berry connection and the Berry curvature. Section IV gives the ray action and the Hamiltonian including the Magnus term, derives the equations of motion in terms of the physical position, and gives the conservation law of the total angular momentum. Section V estimates the size of the effect and discusses the validity of the description, Sec. VI treats a planar interface as a check, and Sec. VII presents the numerical propagation and the tests of the numerical calculation. Section VIII summarizes the results. Appendix A gives the details of the first-order solvability condition, Appendix B the explicit Berry connections and the Frenet-Serret construction, Appendix C the centroid of a wave packet, Appendix D the equivalent formulations of the ray dynamics, and Appendix E the comparison with Bliokh’s review and the analytic results for a helical ray.
Maxwell’s equations and the eikonal expansion
The equations of this section are written in SI units. The ray equations that follow involve only the dimensionless optical momentum p \mathbf{p} p k 0 = ω / c k_0 = \omega/c k 0 = ω / c
Maxwell’s equations
Consider a monochromatic electromagnetic wave with time dependence e − i ω t e^{-i\omega t} e − iω t ϵ ( r ) \epsilon(\mathbf{r}) ϵ ( r ) μ ( r ) \mu(\mathbf{r}) μ ( r )
∇ × E ( r ) = i ω μ ( r ) H ( r ) , (1) \nabla\times\mathbf{E}(\mathbf{r}) = i\omega\mu(\mathbf{r}) \mathbf{H}(\mathbf{r}), \tag*{(1)} ∇ × E ( r ) = iω μ ( r ) H ( r ) , (1) ∇ × H ( r ) = − i ω ϵ ( r ) E ( r ) . (2) \nabla\times\mathbf{H}(\mathbf{r}) = -i\omega\epsilon(\mathbf{r}) \mathbf{E}(\mathbf{r}). \tag*{(2)} ∇ × H ( r ) = − iω ϵ ( r ) E ( r ) . (2) The local refractive index and the local impedance are
n ( r ) = ϵ ( r ) μ ( r ) ϵ 0 μ 0 = c ϵ ( r ) μ ( r ) , (3) n(\mathbf{r}) = \sqrt{\frac{\epsilon(\mathbf{r})\mu(\mathbf{r})}{\epsilon_0\mu_0}} = c\sqrt{\epsilon(\mathbf{r})\mu(\mathbf{r})}, \tag*{(3)} n ( r ) = ϵ 0 μ 0 ϵ ( r ) μ ( r ) = c ϵ ( r ) μ ( r ) , (3) Z ( r ) = μ ( r ) ϵ ( r ) , (4) Z(\mathbf{r}) = \sqrt{\frac{\mu(\mathbf{r})}{\epsilon(\mathbf{r})}}, \tag*{(4)} Z ( r ) = ϵ ( r ) μ ( r ) , (4) where c = 1 / ϵ 0 μ 0 c = 1/\sqrt{\epsilon_0\mu_0} c = 1/ ϵ 0 μ 0 k 0 = 2 π / λ 0 = ω / c k_0 = 2\pi/\lambda_0 = \omega/c k 0 = 2 π / λ 0 = ω / c λ 0 \lambda_0 λ 0
Eikonal expansion
To study the short-wavelength limit (k 0 → ∞ k_0 \to\infty k 0 → ∞ [21 ] , we separate the rapidly varying phase (the eikonal ψ ( r ) \psi(\mathbf{r}) ψ ( r ) E ( r ) \mathbf{E}(\mathbf{r}) E ( r ) H ( r ) \mathbf{H}(\mathbf{r}) H ( r )
E ( r ) = E ( r ) e i k 0 ψ ( r ) , (5) \mathcal{E}(\mathbf{r}) = \mathbf{E}(\mathbf{r}) e^{ik_0\psi(\mathbf{r})}, \tag*{(5)} E ( r ) = E ( r ) e i k 0 ψ ( r ) , (5) H ( r ) = H ( r ) e i k 0 ψ ( r ) . (6) \mathcal{H}(\mathbf{r}) = \mathbf{H}(\mathbf{r}) e^{ik_0\psi(\mathbf{r})}. \tag*{(6)} H ( r ) = H ( r ) e i k 0 ψ ( r ) . (6) We define the optical momentum
p = ∇ ψ . (7) \mathbf{p} = \nabla\psi. \tag*{(7)} p = ∇ ψ . (7) Since ∇ × ( V e i k 0 ψ ) = ( ∇ × V + i k 0 p × V ) e i k 0 ψ \nabla\times(\mathbf{V}e^{ik_0\psi}) = (\nabla\times\mathbf{V} + ik_0\mathbf{p} \times\mathbf{V})e^{ik_0\psi} ∇ × ( V e i k 0 ψ ) = ( ∇ × V + i k 0 p × V ) e i k 0 ψ V \mathbf{V} V
∇ × E + i k 0 p × E = i k 0 c μ H , (8) \nabla\times\mathbf{E} + ik_0\mathbf{p} \times\mathbf{E} = ik_0 c\mu\mathbf{H}, \tag*{(8)} ∇ × E + i k 0 p × E = i k 0 c μ H , (8) ∇ × H + i k 0 p × H = − i k 0 c ϵ E . (9) \nabla\times\mathbf{H} + ik_0\mathbf{p} \times\mathbf{H} = -ik_0 c\epsilon\mathbf{E}. \tag*{(9)} ∇ × H + i k 0 p × H = − i k 0 cϵ E . (9) We expand the amplitudes in powers of 1 / ( i k 0 ) 1/(ik_0) 1/ ( i k 0 )
E = ∑ l = 0 ∞ 1 ( i k 0 ) l E l , H = ∑ l = 0 ∞ 1 ( i k 0 ) l H l . (10) \mathbf{E} = \sum_{l=0}^{\infty} \frac{1}{(ik_0)^l}\mathbf{E}_l,\qquad
\mathbf{H} = \sum_{l=0}^{\infty} \frac{1}{(ik_0)^l}\mathbf{H}_l. \tag*{(10)} E = l = 0 ∑ ∞ ( i k 0 ) l 1 E l , H = l = 0 ∑ ∞ ( i k 0 ) l 1 H l . (10) Inserting the series into Eqs. (8) and (9) and collecting the coefficients of each power of i k 0 ik_0 i k 0
p × E 0 − c μ H 0 = 0 , (11) \mathbf{p} \times\mathbf{E}_0 - c\mu\mathbf{H}_0 = 0, \tag*{(11)} p × E 0 − c μ H 0 = 0 , (11) p × H 0 + c ϵ E 0 = 0 , (12) \mathbf{p} \times\mathbf{H}_0 + c\epsilon\mathbf{E}_0 = 0, \tag*{(12)} p × H 0 + cϵ E 0 = 0 , (12) and, for l ≥ 1 l \ge1 l ≥ 1
∇ × E l − 1 + p × E l − c μ H l = 0 , (13) \nabla\times\mathbf{E}_{l-1} + \mathbf{p} \times\mathbf{E}_l - c\mu\mathbf{H}_l = 0, \tag*{(13)} ∇ × E l − 1 + p × E l − c μ H l = 0 , (13) ∇ × H l − 1 + p × H l + c ϵ E l = 0. (14) \nabla\times\mathbf{H}_{l-1} + \mathbf{p} \times\mathbf{H}_l + c\epsilon\mathbf{E}_l = 0. \tag*{(14)} ∇ × H l − 1 + p × H l + cϵ E l = 0. (14) Section II C solves the zeroth-order equations; Sec. II D treats the case l = 1 l=1 l = 1
Zeroth order
Eliminating H 0 \mathbf{H}_0 H 0
p × [ p × E 0 ] + n 2 E 0 = 0 , (15) \mathbf{p} \times[\mathbf{p} \times\mathbf{E}_0] + n^2\mathbf{E}_0 = 0, \tag*{(15)} p × [ p × E 0 ] + n 2 E 0 = 0 , (15) with n n n (3) . (15) has nontrivial solutions if and only if
det [ p j p j ′ + ( n 2 − p 2 ) δ j j ′ ] = n 2 ( p 2 − n 2 ) 2 = 0 , (16) \det\left[p_jp_{j'} + (n^2-p^2)\delta_{jj'}\right] = n^2\left(p^2-n^2\right)^2 = 0, \tag*{(16)} det [ p j p j ′ + ( n 2 − p 2 ) δ j j ′ ] = n 2 ( p 2 − n 2 ) 2 = 0 , (16) (the matrix has the eigenvalue n 2 n^2 n 2 p \mathbf{p} p n 2 − p 2 n^2-p^2 n 2 − p 2 p \mathbf{p} p
p = ∣ p ∣ = n ( r ) , (17) p = |\mathbf{p}| = n(\mathbf{r}), \tag*{(17)} p = ∣ p ∣ = n ( r ) , (17) since p \mathbf{p} p n n n p ⋅ E 0 = p ⋅ H 0 = 0 \mathbf{p}\cdot\mathbf{E}_0 = \mathbf{p}\cdot\mathbf{H}_0 = 0 p ⋅ E 0 = p ⋅ H 0 = 0
H 0 = 1 c μ p × E 0 . (18) \mathbf{H}_0 = \frac{1}{c\mu}\mathbf{p} \times\mathbf{E}_0. \tag*{(18)} H 0 = c μ 1 p × E 0 . (18) that is, the zeroth-order fields are transverse. In the sense of the bilinear scalar product, { E 0 , H 0 , p } \{\mathbf{E}_0,\mathbf{H}_0,\mathbf{p}\} { E 0 , H 0 , p } p \mathbf{p} p r ( ℓ ) \mathbf{r}(\ell) r ( ℓ ) p ^ = p / p = d r / d ℓ \hat{\mathbf{p}}=\mathbf{p}/p=d\mathbf{r}/d\ell p ^ = p / p = d r / d ℓ ℓ \ell ℓ (17) implies ψ ( r ( ℓ ) ) = ∫ 0 ℓ n d ℓ ′ \psi(\mathbf{r}(\ell))=\int_0^\ell n\,d\ell' ψ ( r ( ℓ )) = ∫ 0 ℓ n d ℓ ′ d ( n p ^ ) / d ℓ = ∇ n d(n\hat{\mathbf{p}})/d\ell=\nabla n d ( n p ^ ) / d ℓ = ∇ n
d r d ℓ = p ^ , d p d ℓ = ∇ n . (19) \frac{d\mathbf{r}}{d\ell}=\hat{\mathbf{p}},\qquad\frac{d\mathbf{p}}{d\ell}=\nabla n. \tag*{(19)} d ℓ d r = p ^ , d ℓ d p = ∇ n . (19) Here r \mathbf{r} r r ( ℓ ) \mathbf{r}(\ell) r ( ℓ ) ℓ \ell ℓ 1 / k 0 1/k_0 1/ k 0 R ( ℓ ) \mathbf{R}(\ell) R ( ℓ ) r ( ℓ ) \mathbf{r}(\ell) r ( ℓ ) R \mathbf{R} R
First order and the solvability condition
For l = 1 l=1 l = 1 (13) and (14) read
p × E 1 − c μ H 1 = − ∇ × E 0 , (20) \mathbf{p}\times\mathbf{E}_1-c\mu\mathbf{H}_1=-\nabla\times\mathbf{E}_0, \tag*{(20)} p × E 1 − c μ H 1 = − ∇ × E 0 , (20) p × H 1 + c ϵ E 1 = − ∇ × H 0 . (21) \mathbf{p}\times\mathbf{H}_1+c\epsilon\mathbf{E}_1=-\nabla\times\mathbf{H}_0. \tag*{(21)} p × H 1 + cϵ E 1 = − ∇ × H 0 . (21) This is an inhomogeneous linear system whose homogeneous part has vanishing determinant, so it has a solution only if the inhomogeneous terms satisfy a condition. Solving Eq. (20) for H 1 \mathbf{H}_1 H 1
H 1 = 1 c μ [ p × E 1 + ∇ × E 0 ] , (22) \mathbf{H}_1=\frac{1}{c\mu}\left[\mathbf{p}\times\mathbf{E}_1+\nabla\times\mathbf{E}_0\right], \tag*{(22)} H 1 = c μ 1 [ p × E 1 + ∇ × E 0 ] , (22) and inserting the result into Eq. (21) , multiplied by c μ c\mu c μ
p × [ p × E 1 ] + n 2 E 1 = C , (23) \mathbf{p}\times[\mathbf{p}\times\mathbf{E}_1]+n^2\mathbf{E}_1=\mathbf{C}, \tag*{(23)} p × [ p × E 1 ] + n 2 E 1 = C , (23) where
C = − c μ ∇ × H 0 − p × ( ∇ × E 0 ) . (24) \mathbf{C}=-c\mu\nabla\times\mathbf{H}_0-\mathbf{p}\times(\nabla\times\mathbf{E}_0). \tag*{(24)} C = − c μ ∇ × H 0 − p × ( ∇ × E 0 ) . (24) On the surface p = n p=n p = n (23) is
M E 1 = p ( p ⋅ E 1 ) = C , M j j ′ = p j p j ′ + ( n 2 − p 2 ) δ j j ′ , (25) \mathcal{M}\mathbf{E}_1=\mathbf{p}(\mathbf{p}\cdot\mathbf{E}_1)=\mathbf{C},\qquad\mathcal{M}_{jj'}=p_jp_{j'}+(n^2-p^2)\delta_{jj'}, \tag*{(25)} M E 1 = p ( p ⋅ E 1 ) = C , M j j ′ = p j p j ′ + ( n 2 − p 2 ) δ j j ′ , (25) because M \mathcal{M} M p = n p=n p = n (25) can be solved if and only if
C is collinear with p , (26) \mathbf{C}\text{ is collinear with }\mathbf{p}, \tag*{(26)} C is collinear with p , (26) i.e., the components of C \mathbf{C} C p \mathbf{p} p Δ = det M \Delta=\det\mathcal{M} Δ = det M Δ j ′ ′ \Delta_{j''} Δ j ′′ M \mathcal{M} M j ′ ′ j'' j ′′ C \mathbf{C} C Δ j ′ ′ / Δ = [ n 2 C j ′ ′ − p j ′ ′ ( p ⋅ C ) ] / [ n 2 ( n 2 − p 2 ) ] \Delta_{j''}/\Delta=[n^2C_{j''}-p_{j''}(\mathbf{p}\cdot\mathbf{C})]/[n^2(n^2-p^2)] Δ j ′′ /Δ = [ n 2 C j ′′ − p j ′′ ( p ⋅ C )] / [ n 2 ( n 2 − p 2 )] p → n p\to n p → n p = n p=n p = n (26) .)
We evaluate C \mathbf{C} C
∇ × X = p ^ × d X d ℓ , (27) \nabla\times\mathbf{X}=\hat{\mathbf{p}}\times\frac{d\mathbf{X}}{d\ell}, \tag*{(27)} ∇ × X = p ^ × d ℓ d X , (27) which neglects the derivatives transverse to the ray (the spreading of the ray bundle). With this approximation both terms of C \mathbf{C} C p ^ \hat{\mathbf{p}} p ^ p ⋅ C = 0 \mathbf{p}\cdot\mathbf{C}=0 p ⋅ C = 0 C = 0 \mathbf{C}=0 C = 0
To find C \mathbf{C} C ν ^ \hat{\boldsymbol{\nu}} ν ^ β ^ = p ^ × ν ^ \hat{\boldsymbol{\beta}}=\hat{\mathbf{p}}\times\hat{\boldsymbol{\nu}} β ^ = p ^ × ν ^ { p ^ , ν ^ , β ^ } \{\hat{\mathbf{p}},\hat{\boldsymbol{\nu}},\hat{\boldsymbol{\beta}}\} { p ^ , ν ^ , β ^ } [1 ] ,
d p ^ d ℓ = K ν ^ , d ν ^ d ℓ = − K p ^ + T β ^ , d β ^ d ℓ = − T ν ^ , (28) \frac{d\hat{\mathbf{p}}}{d\ell}=\mathcal{K}\hat{\boldsymbol{\nu}},\qquad\frac{d\hat{\boldsymbol{\nu}}}{d\ell}=-\mathcal{K}\hat{\mathbf{p}}+\mathcal{T}\hat{\boldsymbol{\beta}},\qquad\frac{d\hat{\boldsymbol{\beta}}}{d\ell}=-\mathcal{T}\hat{\boldsymbol{\nu}}, \tag*{(28)} d ℓ d p ^ = K ν ^ , d ℓ d ν ^ = − K p ^ + T β ^ , d ℓ d β ^ = − T ν ^ , (28) where K = ∣ d p ^ / d ℓ ∣ \mathcal{K}=|d\hat{\mathbf{p}}/d\ell| K = ∣ d p ^ / d ℓ ∣ T = − ν ^ ⋅ d β ^ / d ℓ \mathcal{T}=-\hat{\boldsymbol{\nu}}\cdot d\hat{\boldsymbol{\beta}}/d\ell T = − ν ^ ⋅ d β ^ / d ℓ
E 0 = E ν ν ^ + E β β ^ , (29) \mathbf{E}_0=E_\nu\hat{\boldsymbol{\nu}}+E_\beta\hat{\boldsymbol{\beta}}, \tag*{(29)} E 0 = E ν ν ^ + E β β ^ , (29) with complex components E ν E_\nu E ν E β E_\beta E β (18) gives
H 0 = Y [ − E β ν ^ + E ν β ^ ] , Y = 1 Z = ϵ μ . (30) \mathbf{H}_0=\mathcal{Y}\left[-E_\beta\hat{\boldsymbol{\nu}}+E_\nu\hat{\boldsymbol{\beta}}\right],\qquad\mathcal{Y}=\frac{1}{\mathcal{Z}}=\sqrt{\frac{\epsilon}{\mu}}. \tag*{(30)} H 0 = Y [ − E β ν ^ + E ν β ^ ] , Y = Z 1 = μ ϵ . (30) Evaluating C \mathbf{C} C (27) –(30) (Appendix A) gives C = 2 n [ D ν ν ^ + D β β ^ ] + n ( d ln Y / d ℓ ) E 0 \mathbf{C}=2n[D_\nu\hat{\boldsymbol{\nu}}+D_\beta\hat{\boldsymbol{\beta}}]+n(d\ln\mathcal{Y}/d\ell)\mathbf{E}_0 C = 2 n [ D ν ν ^ + D β β ^ ] + n ( d ln Y / d ℓ ) E 0 D ν D_\nu D ν D β D_\beta D β E 0 \mathbf{E}_0 E 0 (83) ]. The condition C = 0 \mathbf{C}=0 C = 0
( d E 0 d ℓ ) ⊥ = − 1 2 d ln Y d ℓ E 0 , (31) \left(\frac{d\mathbf{E}_0}{d\ell}\right)_{\perp}=-\frac{1}{2}\frac{d\ln\mathcal{Y}}{d\ell}\mathbf{E}_0, \tag*{(31)} ( d ℓ d E 0 ) ⊥ = − 2 1 d ℓ d ln Y E 0 , (31) where ( ⋅ ) ⊥ (\cdot)_{\perp} ( ⋅ ) ⊥ p ^ \hat{\mathbf{p}} p ^ E 0 \mathbf{E}_0 E 0 D ν = D β = 0 D_\nu=D_\beta=0 D ν = D β = 0 E 0 \mathbf{E}_0 E 0 ν ^ \hat{\boldsymbol{\nu}} ν ^ β ^ \hat{\boldsymbol{\beta}} β ^ T \mathcal{T} T ∫ T d ℓ \int\mathcal{T}\,d\ell ∫ T d ℓ (31) gives the amplitude ∝ Z 1 / 2 \propto\mathcal{Z}^{1/2} ∝ Z 1/2 ∣ E 0 ∣ 2 / Z |\mathbf{E}_0|^2/\mathcal{Z} ∣ E 0 ∣ 2 / Z
Helicity states, Berry phase and Berry curvature
Circular polarization basis and uncoupled helicity states
We use the local circular polarization vectors
e ^ σ = 1 2 [ ν ^ + i σ β ^ ] , σ = ± 1. (32) \hat{\mathbf{e}}_\sigma=\frac{1}{\sqrt{2}}\left[\hat{\boldsymbol{\nu}}+i\sigma\hat{\boldsymbol{\beta}}\right],\qquad\sigma=\pm1. \tag*{(32)} e ^ σ = 2 1 [ ν ^ + iσ β ^ ] , σ = ± 1. (32) With the time dependence e − i ω t e^{-i\omega t} e − iω t e ^ σ \hat{\mathbf{e}}_\sigma e ^ σ ν ^ cos ω t + σ β ^ sin ω t \hat{\boldsymbol{\nu}}\cos\omega t+\sigma\hat{\boldsymbol{\beta}}\sin\omega t ν ^ cos ω t + σ β ^ sin ω t ν ^ × β ^ = p ^ \hat{\boldsymbol{\nu}}\times\hat{\boldsymbol{\beta}}=\hat{\mathbf{p}} ν ^ × β ^ = p ^ σ = + 1 \sigma=+1 σ = + 1 σ = + 1 \sigma=+1 σ = + 1 p ^ \hat{\mathbf{p}} p ^ σ = + 1 \sigma=+1 σ = + 1 σ = − 1 \sigma=-1 σ = − 1 E ν = E σ / 2 \mathcal{E}_{\nu}=\mathcal{E}_{\sigma}/\sqrt{2} E ν = E σ / 2 E β = i σ E σ / 2 \mathcal{E}_{\beta}=i\sigma\mathcal{E}_{\sigma}/\sqrt{2} E β = iσ E σ / 2
E 0 = E σ e ^ σ , H 0 = − i σ Y E σ e ^ σ , (33) \mathbf{E}_0=\mathcal{E}_{\sigma}\hat{\mathbf{e}}_{\sigma},\qquad\mathbf{H}_0=-i\sigma\mathcal{Y}\mathcal{E}_{\sigma}\hat{\mathbf{e}}_{\sigma}, \tag*{(33)} E 0 = E σ e ^ σ , H 0 = − iσ Y E σ e ^ σ , (33) where we used Eq. (30) . Then D β = i σ D ν D_{\beta}=i\sigma D_{\nu} D β = iσ D ν (31) reduce to a single equation,
d E σ d ℓ − i σ T E σ + 1 2 d ln Y d ℓ E σ = 0 , (34) \frac{d\mathcal{E}_{\sigma}}{d\ell}-i\sigma\mathcal{T}\mathcal{E}_{\sigma}+\frac{1}{2}\frac{d\ln\mathcal{Y}}{d\ell}\mathcal{E}_{\sigma}=0, \tag*{(34)} d ℓ d E σ − iσ T E σ + 2 1 d ℓ d ln Y E σ = 0 , (34) with the solution
E σ ( ℓ ) = [ Y ( 0 ) Y ( ℓ ) ] 1 / 2 exp ( i σ ∫ 0 ℓ T ( ℓ ′ ) d ℓ ′ ) E σ ( 0 ) . (35) \mathcal{E}_{\sigma}(\ell)=\left[\frac{\mathcal{Y}(0)}{\mathcal{Y}(\ell)}\right]^{1/2}\exp\left(i\sigma\int_{0}^{\ell}\mathcal{T}(\ell')\,d\ell'\right)\mathcal{E}_{\sigma}(0). \tag*{(35)} E σ ( ℓ ) = [ Y ( ℓ ) Y ( 0 ) ] 1/2 exp ( iσ ∫ 0 ℓ T ( ℓ ′ ) d ℓ ′ ) E σ ( 0 ) . (35) The two helicity states are therefore uncoupled: the transverse part of d e ^ σ / d ℓ d\hat{\mathbf{e}}_{\sigma}/d\ell d e ^ σ / d ℓ e ^ σ \hat{\mathbf{e}}_{\sigma} e ^ σ (90) of Appendix B], and the helicity is not changed by propagation to this order. A linearly polarized wave is a superposition of the two helicity states, whose phases ± ∫ T d ℓ \pm\int\mathcal{T}\,d\ell ± ∫ T d ℓ ∫ T d ℓ \int\mathcal{T}\,d\ell ∫ T d ℓ
Berry phase and Berry connection
The solution (35) contains the phase
φ σ ( ℓ ) = σ ∫ 0 ℓ T ( ℓ ′ ) d ℓ ′ , (36) \varphi_{\sigma}(\ell)=\sigma\int_{0}^{\ell}\mathcal{T}(\ell')\,d\ell', \tag*{(36)} φ σ ( ℓ ) = σ ∫ 0 ℓ T ( ℓ ′ ) d ℓ ′ , (36) which is defined with respect to the Frenet basis of the ray. The same phase can be expressed through the momentum p ( ℓ ) \mathbf{p}(\ell) p ( ℓ ) e ^ σ ( p ) \hat{\mathbf{e}}_{\sigma}(\mathbf{p}) e ^ σ ( p ) p \mathbf{p} p
A σ ( p ) = i e ^ σ ∗ ⋅ ∇ p e ^ σ = i ⟨ e ^ σ ∣ ∇ p ∣ e ^ σ ⟩ , (37) \mathbf{A}_{\sigma}(\mathbf{p})=i\hat{\mathbf{e}}_{\sigma}^{*}\cdot\nabla_{\mathbf{p}}\hat{\mathbf{e}}_{\sigma}=i\langle\hat{\mathbf{e}}_{\sigma}|\nabla_{\mathbf{p}}|\hat{\mathbf{e}}_{\sigma}\rangle, \tag*{(37)} A σ ( p ) = i e ^ σ ∗ ⋅ ∇ p e ^ σ = i ⟨ e ^ σ ∣ ∇ p ∣ e ^ σ ⟩ , (37) which is real because e ^ σ ∗ ⋅ e ^ σ = 1 \hat{\mathbf{e}}_{\sigma}^{*}\cdot\hat{\mathbf{e}}_{\sigma}=1 e ^ σ ∗ ⋅ e ^ σ = 1 E 0 = E σ e ^ σ ( p ( ℓ ) ) \mathbf{E}_0=\mathcal{E}_{\sigma}\hat{\mathbf{e}}_{\sigma}(\mathbf{p}(\ell)) E 0 = E σ e ^ σ ( p ( ℓ )) (31) on e ^ σ ∗ \hat{\mathbf{e}}_{\sigma}^{*} e ^ σ ∗
d E σ d ℓ + ⟨ e ^ σ ∣ d d ℓ e ^ σ ⟩ E σ + 1 2 d ln Y d ℓ E σ = 0 , \frac{d\mathcal{E}_{\sigma}}{d\ell}+\left\langle\hat{\mathbf{e}}_{\sigma}\left|\frac{d}{d\ell}\hat{\mathbf{e}}_{\sigma}\right.\right\rangle\mathcal{E}_{\sigma}+\frac{1}{2}\frac{d\ln\mathcal{Y}}{d\ell}\mathcal{E}_{\sigma}=0, d ℓ d E σ + ⟨ e ^ σ d ℓ d e ^ σ ⟩ E σ + 2 1 d ℓ d ln Y E σ = 0 , ⟨ e ^ σ ∣ d d ℓ e ^ σ ⟩ = − i A σ ⋅ d p d ℓ , (38) \left\langle\hat{\mathbf{e}}_{\sigma}\left|\frac{d}{d\ell}\hat{\mathbf{e}}_{\sigma}\right.\right\rangle=-i\mathbf{A}_{\sigma}\cdot\frac{d\mathbf{p}}{d\ell}, \tag*{(38)} ⟨ e ^ σ d ℓ d e ^ σ ⟩ = − i A σ ⋅ d ℓ d p , (38) so that
E σ ( ℓ ) = [ Y ( 0 ) Y ( ℓ ) ] 1 / 2 e i φ σ ( ℓ ) E σ ( 0 ) , (39) \mathcal{E}_{\sigma}(\ell)=\left[\frac{\mathcal{Y}(0)}{\mathcal{Y}(\ell)}\right]^{1/2}e^{i\varphi_{\sigma}(\ell)}\mathcal{E}_{\sigma}(0), \tag*{(39)} E σ ( ℓ ) = [ Y ( ℓ ) Y ( 0 ) ] 1/2 e i φ σ ( ℓ ) E σ ( 0 ) , (39) φ σ ( ℓ ) = ∫ A σ ( p ) ⋅ d p . (40) \varphi_{\sigma}(\ell)=\int\mathbf{A}_{\sigma}(\mathbf{p})\cdot d\mathbf{p}. \tag*{(40)} φ σ ( ℓ ) = ∫ A σ ( p ) ⋅ d p . (40) This is the Berry phase of the helicity state, in the gauge in which e ^ σ \hat{\mathbf{e}}_{\sigma} e ^ σ e ^ σ = ( ν ^ + i σ β ^ ) / 2 \hat{\mathbf{e}}_{\sigma}=(\hat{\boldsymbol{\nu}}+i\sigma\hat{\boldsymbol{\beta}})/\sqrt{2} e ^ σ = ( ν ^ + iσ β ^ ) / 2 A σ ⋅ d p / d ℓ = σ T \mathbf{A}_{\sigma}\cdot d\mathbf{p}/d\ell=\sigma\mathcal{T} A σ ⋅ d p / d ℓ = σ T (36) .
The fields are
E ( r ) = E r e ^ σ ( r ) e i k 0 ψ ( r ) + i φ σ ( r ) , (41) \mathbf{E}(\mathbf{r})=\mathcal{E}_{r}\hat{\mathbf{e}}_{\sigma}(\mathbf{r})e^{ik_{0}\psi(\mathbf{r})+i\varphi_{\sigma}(\mathbf{r})}, \tag*{(41)} E ( r ) = E r e ^ σ ( r ) e i k 0 ψ ( r ) + i φ σ ( r ) , (41) H ( r ) = − i σ E r Y ( r ) e ^ σ ( r ) e i k 0 ψ ( r ) + i φ σ ( r ) , (42) \mathbf{H}(\mathbf{r})=-i\sigma\mathcal{E}_{r}\mathcal{Y}(\mathbf{r})\hat{\mathbf{e}}_{\sigma}(\mathbf{r})e^{ik_{0}\psi(\mathbf{r})+i\varphi_{\sigma}(\mathbf{r})}, \tag*{(42)} H ( r ) = − iσ E r Y ( r ) e ^ σ ( r ) e i k 0 ψ ( r ) + i φ σ ( r ) , (42) where E r ( r ) ∝ Y − 1 / 2 \mathcal{E}_{r}(\mathbf{r})\propto\mathcal{Y}^{-1/2} E r ( r ) ∝ Y − 1/2 ψ ( r ( ℓ ) ) = ∫ 0 ℓ n d ℓ ′ \psi(\mathbf{r}(\ell))=\int_{0}^{\ell}n\,d\ell' ψ ( r ( ℓ )) = ∫ 0 ℓ n d ℓ ′
The Berry curvature of the helicity state σ \sigma σ e ^ σ ( p ) \hat{\mathbf{e}}_{\sigma}(\mathbf{p}) e ^ σ ( p )
Ω σ ( p ) = ∇ p × A σ ( p ) = − σ p p 3 , (43) \boldsymbol{\Omega}_{\sigma}(\mathbf{p})=\nabla_{\mathbf{p}}\times\mathbf{A}_{\sigma}(\mathbf{p})=-\sigma\frac{\mathbf{p}}{p^{3}}, \tag*{(43)} Ω σ ( p ) = ∇ p × A σ ( p ) = − σ p 3 p , (43) the field of a magnetic monopole of strength − σ -\sigma − σ p = 0 \mathbf{p}=0 p = 0 e ^ σ → e i χ ( p ) e ^ σ \hat{\mathbf{e}}_{\sigma}\rightarrow e^{i\chi(\mathbf{p})}\hat{\mathbf{e}}_{\sigma} e ^ σ → e i χ ( p ) e ^ σ e ^ σ \hat{\mathbf{e}}_{\sigma} e ^ σ
A σ ⟶ A σ − σ ∇ p χ , φ σ ⟶ φ σ − σ Δ χ , (44) \mathbf{A}_{\sigma}\longrightarrow\mathbf{A}_{\sigma}-\sigma\nabla_{\mathbf{p}}\chi,\qquad\varphi_{\sigma}\longrightarrow\varphi_{\sigma}-\sigma\Delta\chi, \tag*{(44)} A σ ⟶ A σ − σ ∇ p χ , φ σ ⟶ φ σ − σ Δ χ , (44) where Δ χ \Delta\chi Δ χ χ \chi χ p \mathbf{p} p Ω σ \boldsymbol{\Omega}_{\sigma} Ω σ − σ -\sigma − σ p ^ \hat{\mathbf{p}} p ^ 2 π 2\pi 2 π Ω σ \boldsymbol{\Omega}_{\sigma} Ω σ
Ray action, Hamiltonian and equations of motion
Optical path length with the Berry phase
The phase of the field in Eq. (41) is k 0 ψ + φ σ k_{0}\psi+\varphi_{\sigma} k 0 ψ + φ σ k 0 k_{0} k 0 r i \mathbf{r}_{i} r i r f \mathbf{r}_{f} r f
L σ = ∫ 0 ℓ [ n ( r ( ℓ ′ ) ) + 1 k 0 A σ ( p ) ⋅ d p d ℓ ′ ] d ℓ ′ , (45) \mathcal{L}_{\sigma}=\int_{0}^{\ell}\left[n(\mathbf{r}(\ell'))+\frac{1}{k_{0}}\mathbf{A}_{\sigma}(\mathbf{p})\cdot\frac{d\mathbf{p}}{d\ell'}\right]d\ell', \tag*{(45)} L σ = ∫ 0 ℓ [ n ( r ( ℓ ′ )) + k 0 1 A σ ( p ) ⋅ d ℓ ′ d p ] d ℓ ′ , (45) where the first term is the usual optical path and the second is the spin-dependent Berry phase accumulated along the trajectory, divided by k 0 k_{0} k 0 L σ \mathcal{L}_{\sigma} L σ δ L σ = 0 \delta\mathcal{L}_{\sigma}=0 δ L σ = 0 1 / k 0 1/k_{0} 1/ k 0
Phase-space action
Equation (45) can be written as a phase-space action in which the momentum p \mathbf{p} p R ( ℓ ) \mathbf{R}(\ell) R ( ℓ )
S = ∫ d ℓ [ p ⋅ d R d ℓ + 1 k 0 A σ ( p ) ⋅ d p d ℓ − ( ∣ p ∣ − n ( R ) ) ] . (46) S=\int d\ell\left[\mathbf{p}\cdot\frac{d\mathbf{R}}{d\ell}+\frac{1}{k_0}\mathbf{A}_{\sigma}(\mathbf{p})\cdot\frac{d\mathbf{p}}{d\ell}-(|\mathbf{p}|-n(\mathbf{R}))\right]. \tag*{(46)} S = ∫ d ℓ [ p ⋅ d ℓ d R + k 0 1 A σ ( p ) ⋅ d ℓ d p − ( ∣ p ∣ − n ( R )) ] . (46) (Eliminating p \mathbf{p} p d R / d ℓ = p ^ d\mathbf{R}/d\ell=\hat{\mathbf{p}} d R / d ℓ = p ^ ∣ p ∣ = n |\mathbf{p}|=n ∣ p ∣ = n (45) ). Varying p \mathbf{p} p R \mathbf{R} R
∂ A j ∂ p i − ∂ A i ∂ p j = ϵ i j l Ω l , (47) \frac{\partial A_j}{\partial p_i}-\frac{\partial A_i}{\partial p_j}=\epsilon_{ijl}\Omega_l, \tag*{(47)} ∂ p i ∂ A j − ∂ p j ∂ A i = ϵ ij l Ω l , (47) which follows from Ω l = ϵ l m n ∂ p m A n \Omega_l=\epsilon_{lmn}\partial_{p_m}A_n Ω l = ϵ l mn ∂ p m A n ϵ i j l ϵ l m n = δ i m δ j n − δ i n δ j m \epsilon_{ijl}\epsilon_{lmn}=\delta_{im}\delta_{jn}-\delta_{in}\delta_{jm} ϵ ij l ϵ l mn = δ im δ j n − δ in δ j m A = A σ \mathbf{A}=\mathbf{A}_{\sigma} A = A σ Ω = Ω σ \mathbf{\Omega}=\mathbf{\Omega}_{\sigma} Ω = Ω σ
d R d ℓ = p ^ − 1 k 0 d p d ℓ × Ω σ ( p ) , (48) \frac{d\mathbf{R}}{d\ell}=\hat{\mathbf{p}}-\frac{1}{k_0}\frac{d\mathbf{p}}{d\ell}\times\mathbf{\Omega}_{\sigma}(\mathbf{p}), \tag*{(48)} d ℓ d R = p ^ − k 0 1 d ℓ d p × Ω σ ( p ) , (48) d p d ℓ = ∇ n ( R ) . (49) \frac{d\mathbf{p}}{d\ell}=\nabla n(\mathbf{R}). \tag*{(49)} d ℓ d p = ∇ n ( R ) . (49) Inserting Eq. (49) into Eq. (48) and using Eq. (43) ,
d R d ℓ = p ^ − 1 k 0 ∇ n ( R ) × Ω σ ( p ) = p ∣ p ∣ + σ k 0 ∇ n ( R ) × p p 3 , (50) \frac{d\mathbf{R}}{d\ell}=\hat{\mathbf{p}}-\frac{1}{k_0}\nabla n(\mathbf{R})\times\mathbf{\Omega}_{\sigma}(\mathbf{p})=\frac{\mathbf{p}}{|\mathbf{p}|}+\frac{\sigma}{k_0}\nabla n(\mathbf{R})\times\frac{\mathbf{p}}{p^3}, \tag*{(50)} d ℓ d R = p ^ − k 0 1 ∇ n ( R ) × Ω σ ( p ) = ∣ p ∣ p + k 0 σ ∇ n ( R ) × p 3 p , (50) d p d ℓ = ∇ n ( R ) . (51) \frac{d\mathbf{p}}{d\ell}=\nabla n(\mathbf{R}). \tag*{(51)} d ℓ d p = ∇ n ( R ) . (51) These are the helicity-dependent ray equations in terms of the physical position R \mathbf{R} R p \mathbf{p} p ∣ p ∣ − n ( R ) |\mathbf{p}|-n(\mathbf{R}) ∣ p ∣ − n ( R ) d d ℓ ( ∣ p ∣ − n ( R ) ) = − ∇ n ⋅ ( ∇ n × Ω σ ) / k 0 = 0 \frac{d}{d\ell}(|\mathbf{p}|-n(\mathbf{R}))=-\nabla n\cdot(\nabla n\times\mathbf{\Omega}_{\sigma})/k_0=0 d ℓ d ( ∣ p ∣ − n ( R )) = − ∇ n ⋅ ( ∇ n × Ω σ ) / k 0 = 0 ∣ p ∣ = n ( R ) |\mathbf{p}|=n(\mathbf{R}) ∣ p ∣ = n ( R ) (50) is
σ k 0 n 2 ∇ n × p ^ . (52) \frac{\sigma}{k_0n^2}\nabla n\times\hat{\mathbf{p}}. \tag*{(52)} k 0 n 2 σ ∇ n × p ^ . (52) It is the optical Magnus term: it is transverse to both ∇ n \nabla n ∇ n p ^ \hat{\mathbf{p}} p ^ λ 0 \lambda_0 λ 0 ∇ n − ( p ^ ⋅ ∇ n ) p ^ = n K ν ^ \nabla n-(\hat{\mathbf{p}}\cdot\nabla n)\hat{\mathbf{p}}=nK\hat{\boldsymbol{\nu}} ∇ n − ( p ^ ⋅ ∇ n ) p ^ = n K ν ^
σ k 0 n 2 ∇ n × p ^ = − σ K k 0 n β ^ . (53) \frac{\sigma}{k_0n^2}\nabla n\times\hat{\mathbf{p}}=-\frac{\sigma K}{k_0n}\hat{\boldsymbol{\beta}}. \tag*{(53)} k 0 n 2 σ ∇ n × p ^ = − k 0 n σ K β ^ . (53) The spin-induced deflection is therefore determined by the curvature of the ray alone and is directed along the binormal, perpendicular to the local plane of the ray [8 ] .
Canonical Hamiltonian
Define the canonical coordinate r \mathbf{r} r
R = r + 1 k 0 A σ ( p ) . (54) \mathbf{R}=\mathbf{r}+\frac{1}{k_0}\mathbf{A}_{\sigma}(\mathbf{p}). \tag*{(54)} R = r + k 0 1 A σ ( p ) . (54) Then p ⋅ R ′ + k 0 − 1 A σ ⋅ p ′ = p ⋅ r ′ + k 0 − 1 d ( p ⋅ A σ ) / d ℓ \mathbf{p}\cdot\mathbf{R}'+k_0^{-1}\mathbf{A}_{\sigma}\cdot\mathbf{p}'=\mathbf{p}\cdot\mathbf{r}'+k_0^{-1}d(\mathbf{p}\cdot\mathbf{A}_{\sigma})/d\ell p ⋅ R ′ + k 0 − 1 A σ ⋅ p ′ = p ⋅ r ′ + k 0 − 1 d ( p ⋅ A σ ) / d ℓ ∫ d ℓ [ p ⋅ r ′ − H σ ] \int d\ell\,[\mathbf{p}\cdot\mathbf{r}'-\mathcal{H}_{\sigma}] ∫ d ℓ [ p ⋅ r ′ − H σ ]
H σ ( r , p ) = ∣ p ∣ − n ( r + 1 k 0 A σ ( p ) ) ≃ ∣ p ∣ − n ( r ) − 1 k 0 ∇ n ( r ) ⋅ A σ ( p ) , (55) \mathcal{H}_{\sigma}(\mathbf{r},\mathbf{p})=|\mathbf{p}|-n\left(\mathbf{r}+\frac{1}{k_0}\mathbf{A}_{\sigma}(\mathbf{p})\right)
\simeq|\mathbf{p}|-n(\mathbf{r})-\frac{1}{k_0}\nabla n(\mathbf{r})\cdot\mathbf{A}_{\sigma}(\mathbf{p}), \tag*{(55)} H σ ( r , p ) = ∣ p ∣ − n ( r + k 0 1 A σ ( p ) ) ≃ ∣ p ∣ − n ( r ) − k 0 1 ∇ n ( r ) ⋅ A σ ( p ) , (55) where ( r , p ) (\mathbf{r},\mathbf{p}) ( r , p ) { r i , p j } = δ i j \{r_i,p_j\}=\delta_{ij} { r i , p j } = δ ij A σ / k 0 \mathbf{A}_{\sigma}/k_0 A σ / k 0 H σ \mathcal{H}_{\sigma} H σ 1 / k 0 2 1/k_0^2 1/ k 0 2 ∇ n = d p / d ℓ \nabla n=d\mathbf{p}/d\ell ∇ n = d p / d ℓ − k 0 − 1 A σ ⋅ d p / d ℓ -k_0^{-1}\mathbf{A}_{\sigma}\cdot d\mathbf{p}/d\ell − k 0 − 1 A σ ⋅ d p / d ℓ [8 ] [Eq. (3.11)], as discussed in Appendix § 4.5 1.
We derive Hamilton's equations from the first form of Eq. (55) . With ∂ j = ∂ / ∂ R j \partial_j=\partial/\partial R_j ∂ j = ∂ / ∂ R j
d r i d ℓ = ∂ H σ ∂ p i = p ^ i − 1 k 0 ∂ j n ( R ) ∂ A σ , j ∂ p i , (56) \frac{d r_i}{d\ell}=\frac{\partial\mathcal{H}_{\sigma}}{\partial p_i}=\hat{p}_i-\frac{1}{k_0}\partial_j n(\mathbf{R})\frac{\partial A_{\sigma,j}}{\partial p_i}, \tag*{(56)} d ℓ d r i = ∂ p i ∂ H σ = p ^ i − k 0 1 ∂ j n ( R ) ∂ p i ∂ A σ , j , (56) d p i d ℓ = − ∂ H σ ∂ r i = ∂ i n ( R ) , (57) \frac{d p_i}{d\ell}=-\frac{\partial\mathcal{H}_{\sigma}}{\partial r_i}=\partial_i n(\mathbf{R}), \tag*{(57)} d ℓ d p i = − ∂ r i ∂ H σ = ∂ i n ( R ) , (57) where we used ∂ R j / ∂ r i = δ i j \partial R_j/\partial r_i=\delta_{ij} ∂ R j / ∂ r i = δ ij R \mathbf{R} R ∂ A σ , j / ∂ p i \partial A_{\sigma,j}/\partial p_i ∂ A σ , j / ∂ p i (56) is not gauge invariant. For the physical coordinate, d R i / d ℓ = d r i / d ℓ + k 0 − 1 ( ∂ A σ , i / ∂ p j ) d p j / d ℓ dR_i/d\ell=dr_i/d\ell+k_0^{-1}(\partial A_{\sigma,i}/\partial p_j)dp_j/d\ell d R i / d ℓ = d r i / d ℓ + k 0 − 1 ( ∂ A σ , i / ∂ p j ) d p j / d ℓ
d R i d ℓ = p ^ i + 1 k 0 ∂ j n ( R ) ( ∂ A σ , i ∂ p j − ∂ A σ , j ∂ p i ) = p ^ i − 1 k 0 [ ∇ n ( R ) × Ω σ ] i , (58) \begin{aligned}
\frac{dR_i}{d\ell}=\hat{p}_i+\frac{1}{k_0}\partial_j n(\mathbf{R})\left(\frac{\partial A_{\sigma,i}}{\partial p_j}-\frac{\partial A_{\sigma,j}}{\partial p_i}\right) \\
&=\hat{p}_i-\frac{1}{k_0}\left[\nabla n(\mathbf{R})\times\mathbf{\Omega}_{\sigma}\right]_i, \tag*{(58)}
\end{aligned} d ℓ d R i = p ^ i + k 0 1 ∂ j n ( R ) ( ∂ p j ∂ A σ , i − ∂ p i ∂ A σ , j ) = p ^ i − k 0 1 [ ∇ n ( R ) × Ω σ ] i , (58) which reproduces Eqs. (50) and (51) . The term of Eq. (56) , − k 0 − 1 ( ∇ n ⋅ ∇ p ) A σ -k_0^{-1}(\nabla n\cdot\nabla_{\mathbf{p}})\mathbf{A}_{\sigma} − k 0 − 1 ( ∇ n ⋅ ∇ p ) A σ R \mathbf{R} R d r / d ℓ d\mathbf{r}/d\ell d r / d ℓ d R / d ℓ d\mathbf{R}/d\ell d R / d ℓ R \mathbf{R} R r \mathbf{r} r r \mathbf{r} r
Why R \mathbf{R} R r \mathbf{r} r
In Secs. II and III, r \mathbf{r} r ψ ( r ) \psi(\mathbf{r}) ψ ( r ) n ( r ) n(\mathbf{r}) n ( r ) r ( ℓ ) \mathbf{r}(\ell) r ( ℓ ) (19) . To this order there is only one candidate for the position of the ray. At first order in 1 / k 0 1/k_0 1/ k 0 R ( ℓ ) \mathbf{R}(\ell) R ( ℓ ) r ( ℓ ) = R ( ℓ ) − A σ ( p ) / k 0 \mathbf{r}(\ell)=\mathbf{R}(\ell)-\mathbf{A}_{\sigma}(\mathbf{p})/k_0 r ( ℓ ) = R ( ℓ ) − A σ ( p ) / k 0 ∣ A σ ∣ / k 0 ∼ 1 / ( k 0 n ) |\mathbf{A}_{\sigma}|/k_0 \sim1/(k_0n) ∣ A σ ∣/ k 0 ∼ 1/ ( k 0 n ) λ / 2 π \lambda/2\pi λ /2 π R \mathbf{R} R
(i) Gauge invariance. The connection A σ \mathbf{A}_{\sigma} A σ e ^ σ ( p ) \hat{\mathbf{e}}_{\sigma}(\mathbf{p}) e ^ σ ( p ) (87) and (44) ]. Under a change of this choice, r → r + ( σ / k 0 ) ∇ p χ \mathbf{r} \to\mathbf{r}+(\sigma/k_0)\nabla_{\mathbf{p}}\chi r → r + ( σ / k 0 ) ∇ p χ R \mathbf{R} R a ^ \hat{\mathbf{a}} a ^ σ ∇ p ϕ a ^ \sigma\nabla_{\mathbf{p}}\phi_{\hat{\mathbf{a}}} σ ∇ p ϕ a ^ ( σ / k 0 ) ∇ p ϕ a ^ (\sigma/k_0)\nabla_{\mathbf{p}}\phi_{\hat{\mathbf{a}}} ( σ / k 0 ) ∇ p ϕ a ^ 1 / ( k 0 p sin ϑ a ^ ) 1/(k_0p\sin\vartheta_{\hat{\mathbf{a}}}) 1/ ( k 0 p sin ϑ a ^ ) ϑ a ^ \vartheta_{\hat{\mathbf{a}}} ϑ a ^ p \mathbf{p} p a ^ \hat{\mathbf{a}} a ^ (73) . A measured position of the light cannot depend on how the phase of the polarization basis vector is chosen. Therefore r \mathbf{r} r R \mathbf{R} R
(ii) Equations of motion and conservation laws. The velocity d R / d ℓ d\mathbf{R}/d\ell d R / d ℓ (50) , contains only the gauge-invariant curvature Ω σ \boldsymbol{\Omega}_{\sigma} Ω σ d r / d ℓ d\mathbf{r}/d\ell d r / d ℓ (56) , contains the gauge-dependent term ∂ A σ , j / ∂ p i \partial A_{\sigma,j}/\partial p_i ∂ A σ , j / ∂ p i R × p \mathbf{R}\times\mathbf{p} R × p R = r + A σ ( p ) / k 0 \mathbf{R}=\mathbf{r}+\mathbf{A}_{\sigma}(\mathbf{p})/k_0 R = r + A σ ( p ) / k 0 (54) gives the same conserved quantity in the canonical variables,
J = r × p + 1 k 0 A σ ( p ) × p + σ k 0 p ^ . (59) \mathbf{J}=\mathbf{r}\times\mathbf{p}+\frac{1}{k_0}\mathbf{A}_{\sigma}(\mathbf{p})\times\mathbf{p}+\frac{\sigma}{k_0}\hat{\mathbf{p}}. \tag*{(59)} J = r × p + k 0 1 A σ ( p ) × p + k 0 σ p ^ . (59) Some of the terms in this equation which are not gauge invariant. Under a change of gauge, A σ → A σ − σ ∇ p χ \mathbf{A}_{\sigma}\to\mathbf{A}_{\sigma}-\sigma\nabla_{\mathbf{p}}\chi A σ → A σ − σ ∇ p χ r → r + ( σ / k 0 ) ∇ p χ \mathbf{r}\to\mathbf{r}+(\sigma/k_0)\nabla_{\mathbf{p}}\chi r → r + ( σ / k 0 ) ∇ p χ (59) then changes by + ( σ / k 0 ) ∇ p χ × p +(\sigma/k_0)\nabla_{\mathbf{p}}\chi\times\mathbf{p} + ( σ / k 0 ) ∇ p χ × p − ( σ / k 0 ) ∇ p χ × p -(\sigma/k_0)\nabla_{\mathbf{p}}\chi\times\mathbf{p} − ( σ / k 0 ) ∇ p χ × p J \mathbf{J} J r × p \mathbf{r}\times\mathbf{p} r × p [20 ] .
(iii) Centroid of a wave packet. For a wave packet of helicity σ \sigma σ n n n R = r + A σ / k 0 \mathbf{R}=\mathbf{r}+\mathbf{A}_{\sigma}/k_0 R = r + A σ / k 0 r \mathbf{r} r A σ / k 0 \mathbf{A}_{\sigma}/k_0 A σ / k 0 [3 , 5 , 7 ] and in Ref. [8 ] , where the ray is the trajectory of the centre of gravity of a paraxial beam.
Notation. The symbol r \mathbf{r} r ψ \psi ψ n n n § 7 is R ( ℓ ) \mathbf{R}(\ell) R ( ℓ ) x ( ℓ ) x(\ell) x ( ℓ ) y ( ℓ ) y(\ell) y ( ℓ ) z ( ℓ ) z(\ell) z ( ℓ ) r \mathbf{r} r q \mathbf{q} q [8 ] uses r \mathbf{r} r R \mathbf{R} R
Check with the expanded Hamiltonian
If one starts from the expanded form of Eq. (55) , Hamilton’s equations are
d r i d ℓ = p ^ i − 1 k 0 ∂ j n ( r ) ∂ A σ , j ∂ p i , (60) \frac{d r_i}{d\ell}=\hat{p}_i-\frac{1}{k_0}\partial_j n(\mathbf{r})\frac{\partial A_{\sigma,j}}{\partial p_i}, \tag*{(60)} d ℓ d r i = p ^ i − k 0 1 ∂ j n ( r ) ∂ p i ∂ A σ , j , (60) d p i d ℓ = ∂ i n ( r ) + 1 k 0 ∂ i ∂ j n ( r ) A σ , j . (61) \frac{d p_i}{d\ell}=\partial_i n(\mathbf{r})+\frac{1}{k_0}\partial_i\partial_j n(\mathbf{r})A_{\sigma,j}. \tag*{(61)} d ℓ d p i = ∂ i n ( r ) + k 0 1 ∂ i ∂ j n ( r ) A σ , j . (61) These differ from Eqs. (56) and (57) at order 1 / k 0 2 1/k_0^2 1/ k 0 2 ∇ n ( R ) = ∇ n ( r ) + k 0 − 1 ( A σ ⋅ ∇ ) ∇ n + O ( k 0 − 2 ) \nabla n(\mathbf{R})=\nabla n(\mathbf{r})+k_0^{-1}(\mathbf{A}_{\sigma}\cdot\nabla)\nabla n+O(k_0^{-2}) ∇ n ( R ) = ∇ n ( r ) + k 0 − 1 ( A σ ⋅ ∇ ) ∇ n + O ( k 0 − 2 ) d p / d ℓ = ∇ n ( R ) + O ( k 0 − 2 ) d\mathbf{p}/d\ell=\nabla n(\mathbf{R})+O(k_0^{-2}) d p / d ℓ = ∇ n ( R ) + O ( k 0 − 2 ) (50) is obtained with an error of order k 0 − 2 k_0^{-2} k 0 − 2 1 / k 0 1/k_0 1/ k 0
Gauge dependence
A change of gauge, [see Eq. (44) ],
A σ → A σ ′ = A σ − σ ∇ p χ ( p ) (62) \mathbf{A}_{\sigma}\to\mathbf{A}'_{\sigma}=\mathbf{A}_{\sigma}-\sigma\nabla_{\mathbf{p}}\chi(\mathbf{p}) \tag*{(62)} A σ → A σ ′ = A σ − σ ∇ p χ ( p ) (62) does not change Ω σ \boldsymbol{\Omega}_{\sigma} Ω σ n n n
r ′ = r + σ k 0 ∇ p χ ( p ) , p ′ = p . (63) \mathbf{r}'=\mathbf{r}+\frac{\sigma}{k_0}\nabla_{\mathbf{p}}\chi(\mathbf{p}),\qquad\mathbf{p}'=\mathbf{p}. \tag*{(63)} r ′ = r + k 0 σ ∇ p χ ( p ) , p ′ = p . (63) This is a canonical transformation, generated by F ( r , p ′ ) = r ⋅ p ′ + ( σ / k 0 ) χ ( p ′ ) F(\mathbf{r},\mathbf{p}')=\mathbf{r}\cdot\mathbf{p}'+(\sigma/k_0)\chi(\mathbf{p}') F ( r , p ′ ) = r ⋅ p ′ + ( σ / k 0 ) χ ( p ′ ) r ′ = ∂ F / ∂ p ′ \mathbf{r}'=\partial F/\partial\mathbf{p}' r ′ = ∂ F / ∂ p ′ p = ∂ F / ∂ r \mathbf{p}=\partial F/\partial\mathbf{r} p = ∂ F / ∂ r H σ ( r ; p ; A σ ) = H σ ( r ′ ; p ′ ; A σ ′ ) \mathcal{H}_{\sigma}(\mathbf{r};\mathbf{p};\mathbf{A}_{\sigma})=\mathcal{H}_{\sigma}(\mathbf{r}';\mathbf{p}';\mathbf{A}'_{\sigma}) H σ ( r ; p ; A σ ) = H σ ( r ′ ; p ′ ; A σ ′ ) r + A σ / k 0 = r ′ + A σ ′ / k 0 \mathbf{r}+\mathbf{A}_{\sigma}/k_0=\mathbf{r}'+\mathbf{A}'_{\sigma}/k_0 r + A σ / k 0 = r ′ + A σ ′ / k 0 R \mathbf{R} R − k 0 − 1 ∇ n ⋅ A σ -k_0^{-1}\nabla n\cdot\mathbf{A}_{\sigma} − k 0 − 1 ∇ n ⋅ A σ + ( σ / k 0 ) ∇ n ⋅ ∇ p χ +(\sigma/k_0)\nabla n\cdot\nabla_{\mathbf{p}}\chi + ( σ / k 0 ) ∇ n ⋅ ∇ p χ r \mathbf{r} r
Conservation of angular momentum
The displacement of the ray is connected to the conservation of the total angular momentum. To obtain a formula for the conserved angular momentum, we start from the orbital angular momentum of the ray, R × p \mathbf{R}\times\mathbf{p} R × p (50) and (51) . Since p ^ × p = 0 \hat{\mathbf{p}}\times\mathbf{p}=0 p ^ × p = 0
d d ℓ ( R × p ) = d R d ℓ × p + R × d p d ℓ = σ k 0 p 3 ( ∇ n × p ) × p + R × ∇ n . (64) \frac{d}{d\ell}(\mathbf{R}\times\mathbf{p})=\frac{d\mathbf{R}}{d\ell}\times\mathbf{p}+\mathbf{R}\times\frac{d\mathbf{p}}{d\ell}=\frac{\sigma}{k_0p^3}(\nabla n\times\mathbf{p})\times\mathbf{p}+\mathbf{R}\times\nabla n. \tag*{(64)} d ℓ d ( R × p ) = d ℓ d R × p + R × d ℓ d p = k 0 p 3 σ ( ∇ n × p ) × p + R × ∇ n . (64) For any b \mathbf{b} b ( b × p ) × p = ( b ⋅ p ) p − p 2 b (\mathbf{b}\times\mathbf{p})\times\mathbf{p}=(\mathbf{b}\cdot\mathbf{p})\mathbf{p}-p^2\mathbf{b} ( b × p ) × p = ( b ⋅ p ) p − p 2 b − ( σ / k 0 p ) ∇ ⊥ n -(\sigma/k_0p)\nabla_{\perp}n − ( σ / k 0 p ) ∇ ⊥ n ∇ ⊥ n = ∇ n − p ^ ( p ^ ⋅ ∇ n ) \nabla_{\perp}n=\nabla n-\hat{\mathbf{p}}(\hat{\mathbf{p}}\cdot\nabla n) ∇ ⊥ n = ∇ n − p ^ ( p ^ ⋅ ∇ n ) ∇ n \nabla n ∇ n p ^ \hat{\mathbf{p}} p ^
d p ^ d ℓ = 1 p [ d p d ℓ − p ^ ( p ^ ⋅ d p d ℓ ) ] = ∇ n p . \frac{d\hat{\mathbf{p}}}{d\ell}=\frac{1}{p}\left[\frac{d\mathbf{p}}{d\ell}-\hat{\mathbf{p}}\left(\hat{\mathbf{p}}\cdot\frac{d\mathbf{p}}{d\ell}\right)\right]=\frac{\nabla n}{p}. d ℓ d p ^ = p 1 [ d ℓ d p − p ^ ( p ^ ⋅ d ℓ d p ) ] = p ∇ n . The anomalous (Magnus) velocity therefore changes the orbital angular momentum by a total derivative, − ( σ / k 0 ) d p ^ / d ℓ -(\sigma/k_0)d\hat{\mathbf{p}}/d\ell − ( σ / k 0 ) d p ^ / d ℓ d [ R × p + ( σ / k 0 ) p ^ ] / d ℓ = R × ∇ n d[\mathbf{R}\times\mathbf{p}+(\sigma/k_0)\hat{\mathbf{p}}]/d\ell=\mathbf{R}\times\nabla n d [ R × p + ( σ / k 0 ) p ^ ] / d ℓ = R × ∇ n
J = R × p + σ k 0 p ^ , (65) \mathbf{J}=\mathbf{R}\times\mathbf{p}+\frac{\sigma}{k_0}\hat{\mathbf{p}}, \tag*{(65)} J = R × p + k 0 σ p ^ , (65) as the sum of the orbital angular momentum and the spin angular momentum ( σ / k 0 ) p ^ (\sigma/k_0)\hat{\mathbf{p}} ( σ / k 0 ) p ^
d J d ℓ = R × ∇ n ( R ) . (66) \frac{d\mathbf{J}}{d\ell}=\mathbf{R}\times\nabla n(\mathbf{R}). \tag*{(66)} d ℓ d J = R × ∇ n ( R ) . (66) Multiplied by ℏ k 0 \hbar k_0 ℏ k 0 ℏ k 0 J = R × ℏ k + ℏ σ k ^ \hbar k_0\mathbf{J}=\mathbf{R}\times\hbar\mathbf{k}+\hbar\sigma\hat{\mathbf{k}} ℏ k 0 J = R × ℏ k + ℏ σ k ^ ℏ σ k ^ \hbar\sigma\hat{\mathbf{k}} ℏ σ k ^ σ \sigma σ § 6 .
Hence J \mathbf{J} J n = n ( ∣ R ∣ ) n=n(|\mathbf{R}|) n = n ( ∣ R ∣ ) J \mathbf{J} J [8 ] [Eq. (3.42)], J = r × p + λ 0 s \mathbf{J}=\mathbf{r}\times\mathbf{p}+\lambda_0\mathbf{s} J = r × p + λ 0 s R \mathbf{R} R t = d R / d ℓ \mathbf{t}=d\mathbf{R}/d\ell t = d R / d ℓ p ^ \hat{\mathbf{p}} p ^ 1 / k 0 1/k_0 1/ k 0 § 6 is the case of the axis along the normal to the interface.
Size of the Magnus Effect and Validity
Integrating the Magnus term of Eq. (50) along a ray and using ∇ n d ℓ = d p \nabla n\,d\ell=d\mathbf{p} ∇ n d ℓ = d p
Δ R σ , M = − 1 k 0 ∫ d p × Ω σ ( p ) = σ k 0 ∫ d p × p p 3 , (67) \Delta\mathbf{R}_{\sigma,\mathrm{M}}=-\frac{1}{k_0}\int d\mathbf{p}\times\boldsymbol{\Omega}_{\sigma}(\mathbf{p})=\frac{\sigma}{k_0}\int d\mathbf{p}\times\frac{\mathbf{p}}{p^3}, \tag*{(67)} Δ R σ , M = − k 0 1 ∫ d p × Ω σ ( p ) = k 0 σ ∫ d p × p 3 p , (67) where the integral is along the path of p \mathbf{p} p n ( r ) n(r) n ( r ) ∫ p ^ d ℓ \int\hat{\mathbf{p}}\,d\ell ∫ p ^ d ℓ p \mathbf{p} p Δ ϕ σ \Delta\phi_\sigma Δ ϕ σ σ \sigma σ n n n ∣ Δ R σ , M ∣ ≃ Δ ϕ σ / ( k 0 n ) = λ Δ ϕ σ / 2 π |\Delta\mathbf{R}_{\sigma,\mathrm{M}}|\simeq\Delta\phi_\sigma/(k_0n)=\lambda\Delta\phi_\sigma/2\pi ∣Δ R σ , M ∣ ≃ Δ ϕ σ / ( k 0 n ) = λ Δ ϕ σ /2 π λ = λ 0 / n \lambda=\lambda_0/n λ = λ 0 / n σ \sigma σ ∇ n ( R ) \nabla n(\mathbf{R}) ∇ n ( R ) p \mathbf{p} p
Δ ϕ σ = Δ ϕ 0 + O ( 1 / k 0 ) , (68) \Delta\phi_\sigma=\Delta\phi_0+O(1/k_0), \tag*{(68)} Δ ϕ σ = Δ ϕ 0 + O ( 1/ k 0 ) , (68) where Δ ϕ 0 \Delta\phi_0 Δ ϕ 0 σ = 0 \sigma=0 σ = 0 § 7 A). The correction is small, of relative order 1 / ( k 0 L ) 1/(k_0L) 1/ ( k 0 L ) Δ R σ , M \Delta\mathbf{R}_{\sigma,\mathrm{M}} Δ R σ , M 1 / k 0 1/k_0 1/ k 0 Δ R σ , M \Delta\mathbf{R}_{\sigma,\mathrm{M}} Δ R σ , M 1 / k 0 2 1/k_0^2 1/ k 0 2 Δ ϕ σ \Delta\phi_\sigma Δ ϕ σ Δ ϕ 0 \Delta\phi_0 Δ ϕ 0
Equation (67) is, however, only the direct contribution of the anomalous-velocity term. To first order in 1 / k 0 1/k_0 1/ k 0 δ R \delta\mathbf{R} δ R σ \sigma σ σ = 0 \sigma=0 σ = 0 σ = 0 \sigma=0 σ = 0
d δ R d ℓ = δ p ⊥ n + σ k 0 n 2 ∇ n × p ^ , d δ p d ℓ = ( δ R ⋅ ∇ ) ∇ n , (69) \frac{d\delta\mathbf{R}}{d\ell}=\frac{\delta\mathbf{p}_{\perp}}{n}+\frac{\sigma}{k_0n^2}\nabla n\times\hat{\mathbf{p}},
\qquad
\frac{d\delta\mathbf{p}}{d\ell}=(\delta\mathbf{R}\cdot\nabla)\nabla n, \tag*{(69)} d ℓ d δ R = n δ p ⊥ + k 0 n 2 σ ∇ n × p ^ , d ℓ d δ p = ( δ R ⋅ ∇ ) ∇ n , (69) where δ p ⊥ \delta\mathbf{p}_{\perp} δ p ⊥ δ p \delta\mathbf{p} δ p p ^ \hat{\mathbf{p}} p ^ [8 ] ). The first term of Eq. (69) is the response of the ray to its displacement in the inhomogeneous medium. Equation (67) is obtained by dropping it, i.e., by setting δ p = 0 \delta\mathbf{p}=0 δ p = 0 [8 ] ]. The response vanishes if n n n δ R \delta\mathbf{R} δ R § 6 , but not in general: it changes the displacement of a helical ray by a factor 1 / 2 1/2 1/2 § 7 D.
The following remarks concern the validity of the description.
The ray equations are derived to first order in 1 / ( k 0 L ) 1/(k_0L) 1/ ( k 0 L ) L L L n n n ϵ \epsilon ϵ μ \mu μ n ( r + A σ / k 0 ) n(\mathbf{r}+\mathbf{A}_\sigma/k_0) n ( r + A σ / k 0 )
The shift A σ / k 0 \mathbf{A}_\sigma/k_0 A σ / k 0 R \mathbf{R} R R \mathbf{R} R [3 , 5 , 7 ] ; a short derivation for a locally homogeneous region is given in Appendix C.
The sign of the Magnus term depends on the convention for the helicity σ \sigma σ σ = + 1 \sigma=+1 σ = + 1
The polarization analysis neglects the transverse derivatives of the amplitudes (see Sec. II D) and assumes ϵ , μ \epsilon,\mu ϵ , μ 1 / ( k 0 L ) 1/(k_0L) 1/ ( k 0 L )
For linear or elliptical polarization the description requires the superposition of the two helicity rays (spin Hall splitting).
Example: Refraction at a Planar Interface
As a check, consider a refractive index that depends on z z z
n ( z ) = { n 1 , z < − a , n ~ ( z ) , − a ≤ z ≤ a , n 2 , z > a , (70) n(z)=\begin{cases}
n_1, & z<-a,\\
\tilde{n}(z), & -a\le z\le a,\\
n_2, & z>a,
\end{cases} \tag*{(70)} n ( z ) = ⎩ ⎨ ⎧ n 1 , n ~ ( z ) , n 2 , z < − a , − a ≤ z ≤ a , z > a , (70) where a > 0 a>0 a > 0 n ~ ( z ) \tilde{n}(z) n ~ ( z ) n ~ ( − a ) = n 1 \tilde{n}(-a)=n_1 n ~ ( − a ) = n 1 n ~ ( a ) = n 2 \tilde{n}(a)=n_2 n ~ ( a ) = n 2 n ~ ( z ) = 1 2 ( n 1 + n 2 ) − 1 2 ( n 1 − n 2 ) z / a \tilde{n}(z)=\frac{1}{2}(n_1+n_2)-\frac{1}{2}(n_1-n_2)z/a n ~ ( z ) = 2 1 ( n 1 + n 2 ) − 2 1 ( n 1 − n 2 ) z / a x x x z z z z < − a z<-a z < − a θ i \theta_i θ i p x = n 1 sin θ i p_x=n_1\sin\theta_i p x = n 1 sin θ i
p = ( s , 0 , n 2 − s 2 ) , where s ≡ n 1 sin θ i = n sin θ . (71) \mathbf{p}=\left(s,0,\sqrt{n^2-s^2}\right),\qquad\text{where }s\equiv n_1\sin\theta_i=n\sin\theta. \tag*{(71)} p = ( s , 0 , n 2 − s 2 ) , where s ≡ n 1 sin θ i = n sin θ . (71) Here θ \theta θ z z z ∇ n = n ′ ( z ) z ^ \nabla n=n'(z)\hat{\mathbf{z}} ∇ n = n ′ ( z ) z ^ z ^ × p ^ = sin θ y ^ \hat{\mathbf{z}}\times\hat{\mathbf{p}}=\sin\theta\hat{\mathbf{y}} z ^ × p ^ = sin θ y ^ (50) is along y ^ \hat{\mathbf{y}} y ^
d Y d ℓ = σ n ′ ( z ) sin θ k 0 n 2 , (72) \frac{dY}{d\ell}=\frac{\sigma n'(z)\sin\theta}{k_0n^2}, \tag*{(72)} d ℓ d Y = k 0 n 2 σ n ′ ( z ) sin θ , (72) where Y Y Y y y y R \mathbf{R} R z z z (50) gives d z / d ℓ = cos θ dz/d\ell=\cos\theta d z / d ℓ = cos θ d Y = σ tan θ d n / ( k 0 n 2 ) dY=\sigma\tan\theta\,dn/(k_0n^2) d Y = σ tan θ d n / ( k 0 n 2 ) tan θ = s / n 2 − s 2 \tan\theta=s/\sqrt{n^2-s^2} tan θ = s / n 2 − s 2
Δ Y = σ k 0 ∫ n 1 n 2 s d n n 2 n 2 − s 2 = σ ( cos θ t − cos θ i ) k 0 n 1 sin θ i , (73) \Delta Y=\frac{\sigma}{k_0}\int_{n_1}^{n_2}\frac{s\,dn}{n^2\sqrt{n^2-s^2}}=\frac{\sigma(\cos\theta_t-\cos\theta_i)}{k_0n_1\sin\theta_i}, \tag*{(73)} Δ Y = k 0 σ ∫ n 1 n 2 n 2 n 2 − s 2 s d n = k 0 n 1 sin θ i σ ( cos θ t − cos θ i ) , (73) where we have used ∫ d n n 2 n 2 − s 2 = n 2 − s 2 n 2 s \int\frac{dn}{n^2\sqrt{n^2-s^2}}=\frac{\sqrt{n^2-s^2}}{n^2s} ∫ n 2 n 2 − s 2 d n = n 2 s n 2 − s 2 s = n 1 sin θ i s=n_1\sin\theta_i s = n 1 sin θ i sin θ t = ( n 1 / n 2 ) sin θ i \sin\theta_t=(n_1/n_2)\sin\theta_i sin θ t = ( n 1 / n 2 ) sin θ i n ~ ( z ) \tilde{n}(z) n ~ ( z ) a a a p \mathbf{p} p x x x z z z n n n y y y y y y (69) vanishes], so that Eq. (73) is exact within Eqs. (50) and (51) . For example, for n 1 = 1 n_1=1 n 1 = 1 n 2 = 1.5 n_2=1.5 n 2 = 1.5 θ i = 60 ∘ \theta_i=60^\circ θ i = 6 0 ∘ Δ Y = 0.366 σ / k 0 = 0.058 σ λ 0 \Delta Y=0.366\,\sigma/k_0=0.058\,\sigma\lambda_0 Δ Y = 0.366 σ / k 0 = 0.058 σ λ 0
Equation (73) coincides with what follows from the conservation of the component of the total angular momentum along the normal to the interface, J z = ( R × ℏ k ) z + ℏ σ cos θ J_z=(\mathbf{R}\times\hbar\mathbf{k})_z+\hbar\sigma\cos\theta J z = ( R × ℏ k ) z + ℏ σ cos θ σ \sigma σ p ^ \hat{\mathbf{p}} p ^ k y = 0 k_y=0 k y = 0 k x = k 0 n 1 sin θ i k_x=k_0n_1\sin\theta_i k x = k 0 n 1 sin θ i J z / ℏ = − Y k x + σ cos θ J_z/\hbar=-Yk_x+\sigma\cos\theta J z /ℏ = − Y k x + σ cos θ Δ Y = σ Δ cos θ / k x \Delta Y=\sigma\Delta\cos\theta/k_x Δ Y = σ Δ cos θ / k x (73) . This confirms the sign convention of Eq. (43) with σ = + 1 \sigma=+1 σ = + 1
The Berry phase is gauge dependent and is not an observable by itself. In the gauge (B3) with a ^ = y ^ \hat{\mathbf{a}}=\hat{\mathbf{y}} a ^ = y ^ A ~ σ = σ ( − p z , 0 , p x ) / p 2 \tilde{\mathbf{A}}_\sigma=\sigma(-p_z,0,p_x)/p^2 A ~ σ = σ ( − p z , 0 , p x ) / p 2 d p = ( 0 , 0 , d p z ) d\mathbf{p}=(0,0,dp_z) d p = ( 0 , 0 , d p z )
∫ A ~ σ ⋅ d p = σ ∫ n 1 n 2 s d n n n 2 − s 2 = σ [ arccos ( s / n ) ] n 1 n 2 = σ ( θ i − θ t ) . (74) \int\tilde{\mathbf{A}}_\sigma\cdot d\mathbf{p}=\sigma\int_{n_1}^{n_2}\frac{s\,dn}{n\sqrt{n^2-s^2}}=\sigma[\arccos(s/n)]_{n_1}^{n_2}=\sigma(\theta_i-\theta_t). \tag*{(74)} ∫ A ~ σ ⋅ d p = σ ∫ n 1 n 2 n n 2 − s 2 s d n = σ [ arccos ( s / n ) ] n 1 n 2 = σ ( θ i − θ t ) . (74) This is the gauge term − σ Δ ϕ a ^ -\sigma\Delta\phi_{\hat{\mathbf{a}}} − σ Δ ϕ a ^ (44) : Δ ϕ a ^ = θ t − θ i \Delta\phi_{\hat{\mathbf{a}}}=\theta_t-\theta_i Δ ϕ a ^ = θ t − θ i p \mathbf{p} p a ^ \hat{\mathbf{a}} a ^ a ^ \hat{\mathbf{a}} a ^ p ⋅ a ^ = 0 \mathbf{p}\cdot\hat{\mathbf{a}}=0 p ⋅ a ^ = 0 A σ = 0 \mathbf{A}_\sigma=0 A σ = 0 (36) is also zero. The observable effect is the displacement (73) , which depends only on the curvature.
The spin Hall effect in a gradient-index medium and the transverse Imbert-Fedorov shift of a beam reflected or transmitted at a plane interface are the weak- and strong-scattering limits of the same polarization transport [8 ] . For a low contrast the Imbert-Fedorov shift corresponds to the ray equations (50) ; for a general contrast the evolution at a sharp interface is not adiabatic and the shift cannot be described by the Berry-phase formalism [10 , 11 ] . Equation (73) applies to a graded interface whose width is large compared with the wavelength. For a small change δ θ = θ t − θ i \delta\theta=\theta_t-\theta_i δ θ = θ t − θ i Δ Y ≃ − σ δ θ / ( k 0 n 1 ) \Delta Y\simeq-\sigma\,\delta\theta/(k_0n_1) Δ Y ≃ − σ δ θ / ( k 0 n 1 ) 2 π 2\pi 2 π § 5 .
Numerical Propagation of Rays
Equations to be integrated
The ray of helicity σ \sigma σ (50) and (51) in the variables ( R , p ) (\mathbf{R},\mathbf{p}) ( R , p )
d R d ℓ = p ∣ p ∣ + σ k 0 ∇ n ( R ) × p ∣ p ∣ 3 , (75) \frac{d\mathbf{R}}{d\ell}=\frac{\mathbf{p}}{|\mathbf{p}|}+\frac{\sigma}{k_0}\frac{\nabla n(\mathbf{R})\times\mathbf{p}}{|\mathbf{p}|^3}, \tag*{(75)} d ℓ d R = ∣ p ∣ p + k 0 σ ∣ p ∣ 3 ∇ n ( R ) × p , (75) d p d ℓ = ∇ n ( R ) , (76) \frac{d\mathbf{p}}{d\ell}=\nabla n(\mathbf{R}), \tag*{(76)} d ℓ d p = ∇ n ( R ) , (76) with the initial conditions R ( 0 ) = R 0 \mathbf{R}(0)=\mathbf{R}_0 R ( 0 ) = R 0 p ( 0 ) = n ( R 0 ) p ^ 0 \mathbf{p}(0)=n(\mathbf{R}_0)\hat{\mathbf{p}}_0 p ( 0 ) = n ( R 0 ) p ^ 0 ∣ p ∣ = n ( R ) |\mathbf{p}|=n(\mathbf{R}) ∣ p ∣ = n ( R ) (75) and (76) , ∣ p ∣ − n ( R ) |\mathbf{p}|-n(\mathbf{R}) ∣ p ∣ − n ( R ) R ( ℓ ) = ( X ( ℓ ) , Y ( ℓ ) , Z ( ℓ ) ) \mathbf{R}(\ell)=(X(\ell),Y(\ell),Z(\ell)) R ( ℓ ) = ( X ( ℓ ) , Y ( ℓ ) , Z ( ℓ )) r \mathbf{r} r σ = 0 \sigma=0 σ = 0 (75) and (76) switches the Magnus term off and gives the reference ray: the ray of ordinary geometrical optics, Eq. (19) . No light has σ = 0 \sigma=0 σ = 0 S 3 = 0 S_3=0 S 3 = 0 [8 ] , whose two circular components, σ = ± 1 \sigma=\pm1 σ = ± 1 σ = + 1 \sigma=+1 σ = + 1 σ = − 1 \sigma=-1 σ = − 1
Medium with a localized refractive-index profile
As an example, we consider the index
n ( x , y , z ) = n 0 + Δ n exp [ − x 2 + z 2 2 w 1 2 − y 2 2 w 2 2 ] , (77) n(x,y,z)=n_0+\Delta n\exp\left[-\frac{x^2+z^2}{2w_1^2}-\frac{y^2}{2w_2^2}\right], \tag*{(77)} n ( x , y , z ) = n 0 + Δ n exp [ − 2 w 1 2 x 2 + z 2 − 2 w 2 2 y 2 ] , (77) with n 0 = 1 n_0=1 n 0 = 1 Δ n = 2 \Delta n=2 Δ n = 2 w 1 = 8 w_1=8 w 1 = 8 w 2 = 4 w_2=4 w 2 = 4 k 0 k_0 k 0
∇ n = − ( n − n 0 ) ( x w 1 2 , y w 2 2 , z w 1 2 ) . (78) \nabla n=-(n-n_0)\left(\frac{x}{w_1^2},\frac{y}{w_2^2},\frac{z}{w_1^2}\right). \tag*{(78)} ∇ n = − ( n − n 0 ) ( w 1 2 x , w 2 2 y , w 1 2 z ) . (78) Tests of the numerical calculation
The following properties of Eqs. (75) and (76) can be used to test an integration; the first two are exact, and the others reproduce the analytic results of Sec. VI and Appendix E2.
Conservation of ∣ p ∣ − n ( R ) |\mathbf{p}|-n(\mathbf{R}) ∣ p ∣ − n ( R )
Conservation of angular momentum. The profile (77) is invariant under rotations about the y y y (65) , the component
J y = Z p x − X p z + σ k 0 p y ∣ p ∣ (79) J_y=Zp_x-Xp_z+\frac{\sigma}{k_0}\frac{p_y}{|\mathbf{p}|} \tag*{(79)} J y = Z p x − X p z + k 0 σ ∣ p ∣ p y (79) is exactly conserved. For a spherically symmetric profile the full vector (65) is conserved. The spin term is of order 1 / k 0 1/k_0 1/ k 0 (79) with the spin term omitted fails at this order.
Helical ray. For a cylindrically symmetric profile n ( ρ ) n(\rho) n ( ρ ) n ( ρ ) = n a ( 1 − ρ 2 / 2 Λ 2 ) n(\rho)=n_a(1-\rho^2/2\Lambda^2) n ( ρ ) = n a ( 1 − ρ 2 /2 Λ 2 ) n a n_a n a Λ \Lambda Λ sin 2 θ = ( ρ 2 / Λ 2 ) / ( 1 − ρ 2 / 2 Λ 2 ) \sin^2\theta=(\rho^2/\Lambda^2)/(1-\rho^2/2\Lambda^2) sin 2 θ = ( ρ 2 / Λ 2 ) / ( 1 − ρ 2 /2 Λ 2 ) R 0 = ( ρ , 0 , 0 ) \mathbf{R}_0=(\rho,0,0) R 0 = ( ρ , 0 , 0 ) σ = 0 \sigma=0 σ = 0 p 0 = n ( ρ ) ( 0 , sin θ , cos θ ) \mathbf{p}_0=n(\rho)(0,\sin\theta,\cos\theta) p 0 = n ( ρ ) ( 0 , sin θ , cos θ ) ρ \rho ρ Z Z Z σ = ± 1 \sigma=\pm1 σ = ± 1 ρ \rho ρ p 0 = n ( ρ ) ( 0 , sin θ p , cos θ p ) \mathbf{p}_0=n(\rho)(0,\sin\theta_p,\cos\theta_p) p 0 = n ( ρ ) ( 0 , sin θ p , cos θ p ) θ p ≃ θ − σ K / ( 2 k 0 n ) \theta_p\simeq\theta-\sigma K/(2k_0n) θ p ≃ θ − σ K / ( 2 k 0 n ) σ = 0 \sigma=0 σ = 0 ℓ = 2 π ρ / sin θ \ell=2\pi\rho/\sin\theta ℓ = 2 π ρ / sin θ δ R 0 \delta\mathbf{R}_0 δ R 0 (116) , where β ^ = ( 0 , − cos θ , sin θ ) \hat{\boldsymbol{\beta}}=(0,-\cos\theta,\sin\theta) β ^ = ( 0 , − cos θ , sin θ ) σ = + 1 \sigma=+1 σ = + 1 σ = − 1 \sigma=-1 σ = − 1 λ sin θ \lambda\sin\theta λ sin θ p 0 \mathbf{p}_0 p 0 1 / k 0 1/k_0 1/ k 0
Graded interface. For n = n ( z ) n=n(z) n = n ( z ) Δ Y \Delta Y Δ Y (73) is independent of the profile of the interface, and tests the integration at oblique incidence.
Test 3 is a first-order result, reproduced by the numerical solution up to terms of order k 0 − 2 k_0^{-2} k 0 − 2 k 0 ρ k_0\rho k 0 ρ σ = ± 1 \sigma=\pm1 σ = ± 1
Results for the Gaussian profile
We now present the numerical results of the integration of the equations of motion, Eqs. (75) and (76) , for the profile (77): rays for σ = ± 1 \sigma=\pm1 σ = ± 1 X ( 0 ) = − 30 X(0)=-30 X ( 0 ) = − 30 Y ( 0 ) = 3 Y(0)=3 Y ( 0 ) = 3 Z ( 0 ) = 0 Z(0)=0 Z ( 0 ) = 0 p x ( 0 ) = n ( − 30 , 3 , 0 ) p_x(0)=n(-30,3,0) p x ( 0 ) = n ( − 30 , 3 , 0 ) p y ( 0 ) = 0 p_y(0)=0 p y ( 0 ) = 0 p z ( 0 ) = 0 p_z(0)=0 p z ( 0 ) = 0 k 0 = 10 k_0=10 k 0 = 10 Figure 1 –Figure 5 .
Figure 1. Parametric plot of the ray path R ( ℓ ) \mathbf{R}(\ell) R ( ℓ ) σ = + 1 \sigma=+1 σ = + 1 σ = − 1 \sigma=-1 σ = − 1 z z z
Figure 5. The torsion of the ray T = − ν ^ ⋅ d β ^ / d ℓ \mathcal{T}=-\hat{\nu}\cdot d\hat{\beta}/d\ell T = − ν ^ ⋅ d β ^ / d ℓ ℓ \ell ℓ
Figures 1 and 2 show the ray path R ( ℓ ) \mathbf{R}(\ell) R ( ℓ ) (77) . Figure 1 plots the rays versus ℓ \ell ℓ σ = ± 1 \sigma=\pm1 σ = ± 1 R ( ℓ ) = ( X ( ℓ ) , Y ( ℓ ) , Z ( ℓ ) ) \mathbf{R}(\ell)=(X(\ell),Y(\ell),Z(\ell)) R ( ℓ ) = ( X ( ℓ ) , Y ( ℓ ) , Z ( ℓ )) ℓ \ell ℓ σ = + 1 \sigma=+1 σ = + 1 Z ( ℓ ) Z(\ell) Z ( ℓ ) x x x y y y x x x y y y x x x y y y
Figure 3 shows the momentum p ( ℓ ) \mathbf{p}(\ell) p ( ℓ ) σ = ± 1 \sigma=\pm1 σ = ± 1 x x x y y y p ( ℓ ) \mathbf{p}(\ell) p ( ℓ ) z z z σ = ± 1 \sigma=\pm1 σ = ± 1 p z ( ℓ ) p_z(\ell) p z ( ℓ )
We checked that for k 0 = 20 k_0=20 k 0 = 20 k 0 = 10 k_0=10 k 0 = 10 Z ( ℓ ) Z(\ell) Z ( ℓ ) p z ( ℓ ) p_z(\ell) p z ( ℓ ) 1 / 2 1/2 1/2 k 0 k_0 k 0 ζ \zeta ζ Z ( ℓ ) Z(\ell) Z ( ℓ ) p z ( ℓ ) p_z(\ell) p z ( ℓ ) 1 / ζ 1/\zeta 1/ ζ
Figure 2. The ray path X ( ℓ ) X(\ell) X ( ℓ ) Y ( ℓ ) Y(\ell) Y ( ℓ ) Z ( ℓ ) Z(\ell) Z ( ℓ ) σ = + 1 \sigma= +1 σ = + 1 Z ( ℓ ) Z(\ell) Z ( ℓ ) σ = − 1 \sigma= -1 σ = − 1 X X X Y Y Y Z Z Z
Figure 3. Parametric plot of the momentum, i.e. the gradient of the eikonal p ( ℓ ) = ∇ ψ ( ℓ ) \mathbf{p}(\ell) = \nabla\psi(\ell) p ( ℓ ) = ∇ ψ ( ℓ ) σ = + 1 \sigma= +1 σ = + 1 σ = − 1 \sigma= -1 σ = − 1 p z p_z p z
Figure 4 shows the ray curvature K ( ℓ ) \mathcal{K}(\ell) K ( ℓ ) ℓ \ell ℓ Figure 5 shows the ray torsion T σ ( ℓ ) \mathcal{T}_{\sigma}(\ell) T σ ( ℓ ) Figure 4 is the same for the left- and right-polarized rays. It has three peaks, at ℓ = 16.9 \ell= 16.9 ℓ = 16.9 33.1 33.1 33.1 48 48 48 ℓ = 23.2 \ell= 23.2 ℓ = 23.2 44.0 44.0 44.0 Figure 5 satisfies the equality T − 1 ( ℓ ) = − T 1 ( ℓ ) \mathcal{T}_{-1}(\ell) = -\mathcal{T}_{1}(\ell) T − 1 ( ℓ ) = − T 1 ( ℓ ) T 1 ( ℓ ) \mathcal{T}_{1}(\ell) T 1 ( ℓ ) ℓ = 23.2 \ell= 23.2 ℓ = 23.2 44.0 44.0 44.0
Figure 4. The ray curvature K = ∣ d p ^ / d ℓ ∣ \mathcal{K} = |d\hat{\mathbf{p}}/d\ell| K = ∣ d p ^ / d ℓ ∣ ℓ \ell ℓ
The out-of-plane displacement has a simple origin. For σ = 0 \sigma= 0 σ = 0 § 7 A) the ray stays in the plane z = 0 z = 0 z = 0 p \mathbf{p} p ∇ n \nabla n ∇ n ∇ n × p \nabla n \times\mathbf{p} ∇ n × p z ^ \hat{\mathbf{z}} z ^ z → − z z \to-z z → − z σ → − σ \sigma\to-\sigma σ → − σ X X X Y Y Y p x p_x p x p y p_y p y σ = ± 1 \sigma= \pm1 σ = ± 1 Z Z Z p z p_z p z 1 / k 0 1/k_0 1/ k 0 (69) reduces, along the σ = 0 \sigma= 0 σ = 0
d δ Z d ℓ = δ p z n − σ k 0 n d φ p d ℓ , d δ p z d ℓ = − n − n 0 w 1 2 δ Z (80) \frac{d\,\delta Z}{d\ell} = \frac{\delta p_z}{n} - \frac{\sigma}{k_0 n}\frac{d\varphi_p}{d\ell}, \qquad\frac{d\,\delta p_z}{d\ell} = -\frac{n-n_0}{w_1^2}\delta Z \tag*{(80)} d ℓ d δ Z = n δ p z − k 0 n σ d ℓ d φ p , d ℓ d δ p z = − w 1 2 n − n 0 δ Z (80) where φ p \varphi_p φ p ( p x , p y ) (p_x,p_y) ( p x , p y ) z = 0 z = 0 z = 0 n n n k 0 = 10 k_0 = 10 k 0 = 10 (67) ] gives max ∣ Z ∣ ≃ 0.023 \max|Z| \simeq0.023 max ∣ Z ∣ ≃ 0.023 Z ( ℓ = 140 ) ≃ + 0.004 Z(\ell= 140) \simeq+0.004 Z ( ℓ = 140 ) ≃ + 0.004 max ∣ Z ∣ ≃ 0.032 \max|Z| \simeq0.032 max ∣ Z ∣ ≃ 0.032 Z ( 140 ) ≃ − 0.013 Z(140) \simeq-0.013 Z ( 140 ) ≃ − 0.013 (80) reproduces the full solution to better than 10 − 4 10^{-4} 1 0 − 4
The torsion peaks of Figure 5 are located at the zeros of the curvature, i.e., at the inflection points of the ray. The σ = 0 \sigma= 0 σ = 0 T = 0 \mathcal{T} = 0 T = 0 σ = ± 1 \sigma= \pm1 σ = ± 1 ν ^ \hat{\nu} ν ^ β ^ \hat{\beta} β ^ ± π \pm\pi ± π k 0 k_0 k 0 ∫ T d ℓ ≃ 3.06 \int\mathcal{T}\,d\ell\simeq3.06 ∫ T d ℓ ≃ 3.06 3.03 3.03 3.03 k 0 = 10 k_0 = 10 k 0 = 10 ∫ T d ℓ \int\mathcal{T}\,d\ell ∫ T d ℓ π \pi π
Figure 6 shows the change of the component J y J_y J y (79) along the ray (solid line), which vanishes to within the numerical accuracy (∼ 10 − 13 \sim10^{-13} ∼ 1 0 − 13 Z p x − X p z Zp_x - Xp_z Z p x − X p z ± 0.03 \pm0.03 ± 0.03 ( σ / k 0 ) p y / ∣ p ∣ (\sigma/k_0)p_y/|\mathbf{p}| ( σ / k 0 ) p y /∣ p ∣ ∣ p ∣ − n ( R ) |\mathbf{p}|-n(\mathbf{R}) ∣ p ∣ − n ( R ) 10 − 12 10^{-12} 1 0 − 12
Figure 6. Change Δ J y ( ℓ ) = J y ( ℓ ) − J y ( 0 ) \Delta J_y(\ell)=J_y(\ell)-J_y(0) Δ J y ( ℓ ) = J y ( ℓ ) − J y ( 0 ) (79) (solid line, zero within the numerical accuracy) and of its orbital part Z p x − X p z Zp_x-Xp_z Z p x − X p z σ = + 1 \sigma=+1 σ = + 1
Summary and Conclusions
Starting from Maxwell’s equations in an inhomogeneous medium and the eikonal expansion to first order in 1 / k 0 1/k_0 1/ k 0 φ σ = ∫ A σ ⋅ d p \varphi_\sigma=\int\mathbf{A}_\sigma\cdot d\mathbf{p} φ σ = ∫ A σ ⋅ d p A σ \mathbf{A}_\sigma A σ Ω σ = − σ p / p 3 \boldsymbol{\Omega}_\sigma=-\sigma\mathbf{p}/p^3 Ω σ = − σ p / p 3 H σ = ∣ p ∣ − n ( r + A σ / k 0 ) ≃ ∣ p ∣ − n ( r ) − k 0 − 1 ∇ n ⋅ A σ \mathcal{H}_\sigma=|\mathbf{p}|-n(\mathbf{r}+\mathbf{A}_\sigma/k_0)\simeq|\mathbf{p}|-n(\mathbf{r})-k_0^{-1}\nabla n\cdot\mathbf{A}_\sigma H σ = ∣ p ∣ − n ( r + A σ / k 0 ) ≃ ∣ p ∣ − n ( r ) − k 0 − 1 ∇ n ⋅ A σ R = r + A σ / k 0 \mathbf{R}=\mathbf{r}+\mathbf{A}_\sigma/k_0 R = r + A σ / k 0
d R d ℓ = p ^ + σ k 0 ∇ n ( R ) × p p 3 , d p d ℓ = ∇ n ( R ) (81) \frac{d\mathbf{R}}{d\ell}=\hat{\mathbf{p}}+\frac{\sigma}{k_0}\nabla n(\mathbf{R})\times\frac{\mathbf{p}}{p^3},\qquad\frac{d\mathbf{p}}{d\ell}=\nabla n(\mathbf{R}) \tag*{(81)} d ℓ d R = p ^ + k 0 σ ∇ n ( R ) × p 3 p , d ℓ d p = ∇ n ( R ) (81) which contain only the gauge-invariant curvature. The second term of the first equation is the optical Magnus term, transverse to ∇ n \nabla n ∇ n 1 / k 0 1/k_0 1/ k 0 ω ~ = c ~ k / n \tilde{\omega}=\tilde{c}k/n ω ~ = c ~ k / n § 12.3 ). For refraction at a planar interface the Magnus term gives the displacement (73), which agrees with the conservation of angular momentum about the normal to the interface (§ 6 ). The equations are in a form suitable for numerical propagation (§ 7 ). The Magnus term is the helicity-dependent transverse deflection; it scales as 1 / k 0 1/k_0 1/ k 0 k 0 → ∞ k_0\to\infty k 0 → ∞
The Berry phase and the spin Hall effect are consequences of the same spin-orbit term, but they differ in their dependence on the frame: the Berry phase of an open path depends on the gauge (Sec. III C, Appendix B and the interface example), whereas the displacement of the ray is gauge invariant and is determined by the curvature of the ray, Eq. (53) . The two helicity states are uncoupled to first order; an elliptically polarized beam splits upon propagation into two overlapping but displaced circularly polarized beams [8 ] , each following Eqs. (50) and (51) . The Berry-phase description requires a smooth medium; sharp interfaces give the non-adiabatic Imbert-Fedorov shift, which is not contained in the equations of this paper.
The equations agree with those of Bliokh’s review [8 ] , including the sign of the Magnus term (Appendix § 4.5 1). They reproduce the known analytic results: the deflection along the binormal determined by the curvature of the ray, Eq. (53) ; the shift of a helical ray, Eq. (116) ; the conservation of the total angular momentum, Eq. (65) ; and the displacement at a graded interface, Eq. (73) , which agrees with the low-contrast limit of the Imbert-Fedorov shift (§ 6 ). Experiments confirm the ray equations for helical rays [6 , 9 ] , and the same equations have been integrated numerically for light in the Schwarzschild spacetime [13 ] and for vortex light in a waveguide [15 ] . We did not find published numerical propagation for a general three-dimensional gradient-index profile; the exact invariants and the analytic tests of § 7 C are provided for such a calculation.
First-order solvability condition
Here we evaluate the vector C \mathcal{C} C (24) for the zeroth-order fields (29) and (30), using Eq. (27) and the Frenet-Serret formulas (28) . Using Eqs. (28) one finds
d E 0 d ℓ = D ν ν ^ + D β β ^ − κ E ν p ^ , (82) \frac{d\mathbf{E}_0}{d\ell}=D_{\nu}\hat{\nu}+D_{\beta}\hat{\beta}-\kappa E_{\nu}\hat{p}, \tag*{(82)} d ℓ d E 0 = D ν ν ^ + D β β ^ − κ E ν p ^ , (82) with
D ν = d E ν d ℓ − T E β , D β = d E β d ℓ + T E ν . (83) D_{\nu}=\frac{dE_{\nu}}{d\ell}-T E_{\beta},\qquad D_{\beta}=\frac{dE_{\beta}}{d\ell}+T E_{\nu}. \tag*{(83)} D ν = d ℓ d E ν − T E β , D β = d ℓ d E β + T E ν . (83) The curls follow from Eq. (27) ,
∇ × E 0 = − D β ν ^ + D ν β ^ , (84) \nabla\times\mathbf{E}_0=-D_{\beta}\hat{\nu}+D_{\nu}\hat{\beta}, \tag*{(84)} ∇ × E 0 = − D β ν ^ + D ν β ^ , (84) c μ ∇ × H 0 = − n D ν ν ^ − n D β β ^ − n d ln Y d ℓ E 0 , (85) c\mu\nabla\times\mathbf{H}_0=-nD_{\nu}\hat{\nu}-nD_{\beta}\hat{\beta}-n\frac{d\ln\mathcal{Y}}{d\ell}\mathbf{E}_0, \tag*{(85)} c μ ∇ × H 0 = − n D ν ν ^ − n D β β ^ − n d ℓ d ln Y E 0 , (85) where we used c μ Y = n c\mu\mathcal{Y}=n c μ Y = n (85) comes from the variation of the impedance along the ray. Here H 0 = Y p ^ × E 0 \mathbf{H}_0=\mathcal{Y}\hat{p}\times\mathbf{E}_0 H 0 = Y p ^ × E 0 ∇ × H 0 = p ^ × d H 0 / d ℓ \nabla\times\mathbf{H}_0=\hat{p}\times d\mathbf{H}_0/d\ell ∇ × H 0 = p ^ × d H 0 / d ℓ Y \mathcal{Y} Y p ^ × ν ^ = β ^ \hat{p}\times\hat{\nu}=\hat{\beta} p ^ × ν ^ = β ^ p ^ × β ^ = − ν ^ \hat{p}\times\hat{\beta}=-\hat{\nu} p ^ × β ^ = − ν ^ (84) gives p × ( ∇ × E 0 ) = − n ( D ν ν ^ + D β β ^ ) \mathbf{p}\times(\nabla\times\mathbf{E}_0)=-n(D_{\nu}\hat{\nu}+D_{\beta}\hat{\beta}) p × ( ∇ × E 0 ) = − n ( D ν ν ^ + D β β ^ ) − c μ ∇ × H 0 = n ( D ν ν ^ + D β β ^ ) + n ( d ln Y / d ℓ ) E 0 -c\mu\nabla\times\mathbf{H}_0=n(D_{\nu}\hat{\nu}+D_{\beta}\hat{\beta})+n(d\ln\mathcal{Y}/d\ell)\mathbf{E}_0 − c μ ∇ × H 0 = n ( D ν ν ^ + D β β ^ ) + n ( d ln Y / d ℓ ) E 0
C = n [ 2 D ν ν ^ + 2 D β β ^ ] + n d ln Y d ℓ E 0 , (86) \mathcal{C}=n\left[2D_{\nu}\hat{\nu}+2D_{\beta}\hat{\beta}\right]+n\frac{d\ln\mathcal{Y}}{d\ell}\mathbf{E}_0, \tag*{(86)} C = n [ 2 D ν ν ^ + 2 D β β ^ ] + n d ℓ d ln Y E 0 , (86) Setting C = 0 \mathcal{C}=0 C = 0 (31) of the main text.
Explicit Berry connections and the Frenet-Serret construction
Berry connection and curvature as functions of p
To obtain A σ ( p ) \mathbf{A}_{\sigma}(\mathbf{p}) A σ ( p ) a ^ \hat{a} a ^ p ^ \hat{p} p ^
e ^ 1 ( p ) = a ^ × p ∣ a ^ × p ∣ , e ^ 2 ( p ) = p ^ × e ^ 1 ( p ) , \hat{e}_1(\mathbf{p})=\frac{\hat{a}\times\mathbf{p}}{\lvert\hat{a}\times\mathbf{p}\rvert},\qquad\hat{e}_2(\mathbf{p})=\hat{p}\times\hat{e}_1(\mathbf{p}), e ^ 1 ( p ) = ∣ a ^ × p ∣ a ^ × p , e ^ 2 ( p ) = p ^ × e ^ 1 ( p ) , e ^ σ ( p ) = 1 2 [ e ^ 1 + i σ e ^ 2 ] . (87) \hat{e}_{\sigma}(\mathbf{p})=\frac{1}{\sqrt{2}}\left[\hat{e}_1+i\sigma\hat{e}_2\right]. \tag*{(87)} e ^ σ ( p ) = 2 1 [ e ^ 1 + iσ e ^ 2 ] . (87) These are not the Frenet normal and binormal: they depend on p \mathbf{p} p (37) and the orthonormality of e ^ 1 \hat{e}_1 e ^ 1 e ^ 2 \hat{e}_2 e ^ 2 A σ = σ e ^ 2 ⋅ ∇ p e ^ 1 \mathbf{A}_{\sigma}=\sigma\hat{e}_2\cdot\nabla_p\hat{e}_1 A σ = σ e ^ 2 ⋅ ∇ p e ^ 1
A σ ( p ) = − σ ( p ⋅ a ^ ) [ p × a ^ ] p ∣ p × a ^ ∣ 2 . (88) \mathbf{A}_{\sigma}(\mathbf{p})=-\sigma\frac{(\mathbf{p}\cdot\hat{a})[\mathbf{p}\times\hat{a}]}{p\lvert\mathbf{p}\times\hat{a}\rvert^2}. \tag*{(88)} A σ ( p ) = − σ p ∣ p × a ^ ∣ 2 ( p ⋅ a ^ ) [ p × a ^ ] . (88) For a ^ = z ^ \hat{a}=\hat{z} a ^ = z ^ ( p , ϑ , ϕ ) (p,\vartheta,\phi) ( p , ϑ , ϕ ) p \mathbf{p} p (88) is A σ = σ cos ϑ ϕ ^ / ( p sin ϑ ) \mathbf{A}_{\sigma}=\sigma\cos\vartheta\,\hat{\phi}/(p\sin\vartheta) A σ = σ cos ϑ ϕ ^ / ( p sin ϑ ) (89) below is A ~ σ = − σ ( 1 − cos ϑ ) ϕ ^ / ( p sin ϑ ) \tilde{\mathbf{A}}_{\sigma}=-\sigma(1-\cos\vartheta)\hat{\phi}/(p\sin\vartheta) A ~ σ = − σ ( 1 − cos ϑ ) ϕ ^ / ( p sin ϑ ) ( p sin ϑ ) − 1 ∂ ϑ ( sin ϑ A ϕ ) (p\sin\vartheta)^{-1}\partial_{\vartheta}(\sin\vartheta A_{\phi}) ( p sin ϑ ) − 1 ∂ ϑ ( sin ϑ A ϕ ) − σ / p 2 -\sigma/p^2 − σ / p 2 A σ − A ~ σ = σ ϕ ^ / ( p sin ϑ ) = σ ∇ p ϕ \mathbf{A}_{\sigma}-\tilde{\mathbf{A}}_{\sigma}=\sigma\hat{\phi}/(p\sin\vartheta)=\sigma\nabla_p\phi A σ − A ~ σ = σ ϕ ^ / ( p sin ϑ ) = σ ∇ p ϕ ∇ p × A σ = − σ p / p 3 \nabla_p\times\mathbf{A}_{\sigma}=-\sigma\mathbf{p}/p^3 ∇ p × A σ = − σ p / p 3 a ^ \hat{a} a ^ a ^ \hat{a} a ^ ∇ p χ ( p ) \nabla_p\chi(\mathbf{p}) ∇ p χ ( p )
A ~ σ ( p ) = σ p × a ^ p ( p + p ⋅ a ^ ) , (89) \tilde{\mathbf{A}}_{\sigma}(\mathbf{p})=\sigma\frac{\mathbf{p}\times\hat{a}}{p(p+\mathbf{p}\cdot\hat{a})}, \tag*{(89)} A ~ σ ( p ) = σ p ( p + p ⋅ a ^ ) p × a ^ , (89) with the same curvature, ∇ p × A ~ σ = Ω σ \nabla_p\times\tilde{\mathbf{A}}_{\sigma}=\boldsymbol{\Omega}_{\sigma} ∇ p × A ~ σ = Ω σ A ~ σ = A σ − σ ∇ p ϕ a ^ \tilde{\mathbf{A}}_{\sigma}=\mathbf{A}_{\sigma}-\sigma\nabla_p\phi_{\hat{a}} A ~ σ = A σ − σ ∇ p ϕ a ^ ϕ a ^ \phi_{\hat{a}} ϕ a ^ p \mathbf{p} p a ^ \hat{a} a ^ p ∥ ± a ^ \mathbf{p}\parallel\pm\hat{a} p ∥ ± a ^ p ∥ − a ^ \mathbf{p}\parallel-\hat{a} p ∥ − a ^ p \mathbf{p} p − 1 -1 − 1 p \mathbf{p} p
Helicity states in the Frenet basis
Using Eqs. (28) and (32) ,
d e ^ σ d ℓ = 1 2 [ − κ p ^ + T β ^ − i σ T ν ^ ] = − i σ T e ^ σ − κ 2 p ^ , (90) \frac{d\hat{e}_{\sigma}}{d\ell}=\frac{1}{\sqrt{2}}\left[-\kappa\hat{p}+T\hat{\beta}-i\sigma T\hat{\nu}\right]=-i\sigma T\hat{e}_{\sigma}-\frac{\kappa}{\sqrt{2}}\hat{p}, \tag*{(90)} d ℓ d e ^ σ = 2 1 [ − κ p ^ + T β ^ − iσ T ν ^ ] = − iσ T e ^ σ − 2 κ p ^ , (90) because T β ^ − i σ T ν ^ = − i σ T ( ν ^ + i σ β ^ ) T\hat{\beta}-i\sigma T\hat{\nu}=-i\sigma T(\hat{\nu}+i\sigma\hat{\beta}) T β ^ − iσ T ν ^ = − iσ T ( ν ^ + iσ β ^ ) d e ^ σ / d ℓ d\hat{e}_{\sigma}/d\ell d e ^ σ / d ℓ e ^ σ \hat{e}_{\sigma} e ^ σ ⟨ e ^ σ ∣ d e ^ σ / d ℓ ⟩ = − i σ T \langle\hat{e}_{\sigma}|d\hat{e}_{\sigma}/d\ell\rangle=-i\sigma T ⟨ e ^ σ ∣ d e ^ σ / d ℓ ⟩ = − iσ T (37) , in the Frenet gauge,
A σ ⋅ d p d ℓ = i ⟨ e ^ σ | d d ℓ | e ^ σ ⟩ = σ T . (91) \mathbf{A}_{\sigma}\cdot\frac{d\mathbf{p}}{d\ell}=i\left\langle\hat{e}_{\sigma}\middle|\frac{d}{d\ell}\middle|\hat{e}_{\sigma}\right\rangle=\sigma T. \tag*{(91)} A σ ⋅ d ℓ d p = i ⟨ e ^ σ d ℓ d e ^ σ ⟩ = σ T . (91) Relation between the Frenet gauge and the gauge of (87)
The Frenet frame { ν ^ , β ^ } \{\hat{\nu},\hat{\beta}\} { ν ^ , β ^ } { e ^ 1 , e ^ 2 } \{\hat{e}_1,\hat{e}_2\} { e ^ 1 , e ^ 2 } (87) differ by a rotation by the angle χ ( ℓ ) \chi(\ell) χ ( ℓ ) p ^ \hat{p} p ^ ν ^ = cos χ e ^ 1 + sin χ e ^ 2 \hat{\nu}=\cos\chi\,\hat{e}_1+\sin\chi\,\hat{e}_2 ν ^ = cos χ e ^ 1 + sin χ e ^ 2 β ^ = − sin χ e ^ 1 + cos χ e ^ 2 \hat{\beta}=-\sin\chi\,\hat{e}_1+\cos\chi\,\hat{e}_2 β ^ = − sin χ e ^ 1 + cos χ e ^ 2 e ^ σ F = e − i σ χ e ^ σ ( a ) \hat{e}_{\sigma}^{F}=e^{-i\sigma\chi}\hat{e}_{\sigma}^{(a)} e ^ σ F = e − iσ χ e ^ σ ( a )
σ T = A σ ( a ) ⋅ d p d ℓ + σ d χ d ℓ , ∫ A σ ( a ) ⋅ d p = σ ∫ T d ℓ − σ Δ χ . (92) \sigma T=\mathbf{A}_{\sigma}^{(a)}\cdot\frac{d\mathbf{p}}{d\ell}+\sigma\frac{d\chi}{d\ell},\qquad\int\mathbf{A}_{\sigma}^{(a)}\cdot d\mathbf{p}=\sigma\int T\,d\ell-\sigma\Delta\chi. \tag*{(92)} σ T = A σ ( a ) ⋅ d ℓ d p + σ d ℓ d χ , ∫ A σ ( a ) ⋅ d p = σ ∫ T d ℓ − σ Δ χ . (92) The Berry phase of Eq. (40) in the gauge ( a ) (a) ( a ) Δ χ \Delta\chi Δ χ (44) .
Centroid of a wave packet
Consider a wave packet of helicity σ \sigma σ n n n
E ( r ) = ∫ d 3 k a ( k ) e ^ σ ( k ) e i k ⋅ r , a ( k ) = ∣ a ( k ) ∣ e − i γ ( k ) . (93) \mathcal{E}(\mathbf{r})=\int d^3k\,a(\mathbf{k})\hat{e}_{\sigma}(\mathbf{k})e^{i\mathbf{k}\cdot\mathbf{r}},\qquad a(\mathbf{k})=\lvert a(\mathbf{k})\rvert e^{-i\gamma(\mathbf{k})}. \tag*{(93)} E ( r ) = ∫ d 3 k a ( k ) e ^ σ ( k ) e i k ⋅ r , a ( k ) = ∣ a ( k )∣ e − iγ ( k ) . (93) with ∣ a ( k ) ∣ \lvert a(\mathbf{k})\rvert ∣ a ( k )∣ k ˉ = k 0 p ^ \bar{\mathbf{k}}=k_0\hat{p} k ˉ = k 0 p ^ e ^ σ ( k ) \hat{e}_{\sigma}(\mathbf{k}) e ^ σ ( k ) k \mathbf{k} k r e i k ⋅ r = − i ∇ k e i k ⋅ r r e^{i\mathbf{k}\cdot\mathbf{r}}=-i\nabla_k e^{i\mathbf{k}\cdot\mathbf{r}} r e i k ⋅ r = − i ∇ k e i k ⋅ r ∣ E ∣ 2 |\mathcal{E}|^2 ∣ E ∣ 2
⟨ r ⟩ = ∫ d 3 k ( a e ^ σ ) ∗ ⋅ i ∇ k ( a e ^ σ ) ∫ d 3 k ∣ a ∣ 2 = ∫ d 3 k ∣ a ∣ 2 [ ∇ k γ + A ~ σ ( k ) ] ∫ d 3 k ∣ a ∣ 2 ≃ ∇ k γ ( k ˉ ) + 1 k 0 A σ ( p ˉ ) , (94) \langle\mathbf{r}\rangle= \frac{\int d^3k\,(\mathbf{a}\hat{\mathbf{e}}_\sigma)^* \cdot i\nabla_k(\mathbf{a}\hat{\mathbf{e}}_\sigma)}{\int d^3k\,|\mathbf{a}|^2}
= \frac{\int d^3k\,|\mathbf{a}|^2\left[\nabla_k\gamma+\widetilde{\mathbf{A}}_\sigma(\mathbf{k})\right]}{\int d^3k\,|\mathbf{a}|^2}
\simeq\nabla_k\gamma(\bar{\mathbf{k}})+\frac{1}{k_0}\mathbf{A}_\sigma(\bar{\mathbf{p}}), \tag*{(94)} ⟨ r ⟩ = ∫ d 3 k ∣ a ∣ 2 ∫ d 3 k ( a e ^ σ ) ∗ ⋅ i ∇ k ( a e ^ σ ) = ∫ d 3 k ∣ a ∣ 2 ∫ d 3 k ∣ a ∣ 2 [ ∇ k γ + A σ ( k ) ] ≃ ∇ k γ ( k ˉ ) + k 0 1 A σ ( p ˉ ) , (94) where A ~ σ ( k ) = i e ^ σ ∗ ⋅ ∇ k e ^ σ = A σ ( p ) / k 0 \widetilde{\mathbf{A}}_\sigma(\mathbf{k})=i\hat{\mathbf{e}}_\sigma^*\cdot\nabla_k\hat{\mathbf{e}}_\sigma=\mathbf{A}_\sigma(\mathbf{p})/k_0 A σ ( k ) = i e ^ σ ∗ ⋅ ∇ k e ^ σ = A σ ( p ) / k 0 (107) ], and the term i ∣ a ∣ ∇ k ∣ a ∣ = i 2 ∇ k ∣ a ∣ 2 i|\mathbf{a}|\nabla_k|\mathbf{a}|=\frac{i}{2}\nabla_k|\mathbf{a}|^2 i ∣ a ∣ ∇ k ∣ a ∣ = 2 i ∇ k ∣ a ∣ 2 ∇ k γ \nabla_k\gamma ∇ k γ k \mathbf{k} k r \mathbf{r} r k \mathbf{k} k R = r + A σ / k 0 \mathbf{R}=\mathbf{r}+\mathbf{A}_\sigma/k_0 R = r + A σ / k 0 e ^ σ → e i σ χ e ^ σ \hat{\mathbf{e}}_\sigma\to e^{i\sigma\chi}\hat{\mathbf{e}}_\sigma e ^ σ → e iσ χ e ^ σ a → e − i σ χ a \mathbf{a}\to e^{-i\sigma\chi}\mathbf{a} a → e − iσ χ a ∇ k γ → ∇ k γ + σ ∇ k χ \nabla_k\gamma\to\nabla_k\gamma+\sigma\nabla_k\chi ∇ k γ → ∇ k γ + σ ∇ k χ A ~ σ → A ~ σ − σ ∇ k χ \widetilde{\mathbf{A}}_\sigma\to\widetilde{\mathbf{A}}_\sigma-\sigma\nabla_k\chi A σ → A σ − σ ∇ k χ H 0 ∝ e ^ σ \mathbf{H}_0\propto\hat{\mathbf{e}}_\sigma H 0 ∝ e ^ σ (33) ], the magnetic and total energy densities have the same centroid. This is the argument of the semiclassical wave-packet theory [3 , 5 , 7 ] , and it agrees with the review [8 ] , where the ray is the trajectory of the centre of gravity of a paraxial beam.
Eliminating p \mathbf{p} p
p = n ( R ) R ˙ ∣ R ˙ ∣ + O ( k 0 − 1 ) , (95) \mathbf{p}=n(\mathbf{R})\frac{\dot{\mathbf{R}}}{|\dot{\mathbf{R}}|}+O(k_0^{-1}), \tag*{(95)} p = n ( R ) ∣ R ˙ ∣ R ˙ + O ( k 0 − 1 ) , (95) where the dot is the derivative with respect to an arbitrary parameter, gives the Lagrangian of the ray path,
L ( R , R ˙ , R ¨ ) = n ( R ) ∣ R ˙ ∣ + 1 k 0 R ¨ ⋅ A σ ( R ˙ ) . (96) \mathcal{L}(\mathbf{R},\dot{\mathbf{R}},\ddot{\mathbf{R}})=n(\mathbf{R})|\dot{\mathbf{R}}|+\frac{1}{k_0}\ddot{\mathbf{R}}\cdot\mathbf{A}_\sigma(\dot{\mathbf{R}}). \tag*{(96)} L ( R , R ˙ , R ¨ ) = n ( R ) ∣ R ˙ ∣ + k 0 1 R ¨ ⋅ A σ ( R ˙ ) . (96) To obtain the second term from k 0 − 1 A σ ( p ) ⋅ p ˙ k_0^{-1}\mathbf{A}_\sigma(\mathbf{p})\cdot\dot{\mathbf{p}} k 0 − 1 A σ ( p ) ⋅ p ˙ p ˙ = n R ¨ / ∣ R ˙ ∣ − n ( R ˙ ⋅ R ¨ ) R ˙ / ∣ R ˙ ∣ 3 + ( ∇ n ⋅ R ˙ ) R ˙ / ∣ R ˙ ∣ \dot{\mathbf{p}}=n\ddot{\mathbf{R}}/|\dot{\mathbf{R}}|-n(\dot{\mathbf{R}}\cdot\ddot{\mathbf{R}})\dot{\mathbf{R}}/|\dot{\mathbf{R}}|^3+(\nabla n\cdot\dot{\mathbf{R}})\dot{\mathbf{R}}/|\dot{\mathbf{R}}| p ˙ = n R ¨ /∣ R ˙ ∣ − n ( R ˙ ⋅ R ¨ ) R ˙ /∣ R ˙ ∣ 3 + ( ∇ n ⋅ R ˙ ) R ˙ /∣ R ˙ ∣ R ˙ \dot{\mathbf{R}} R ˙ A σ \mathbf{A}_\sigma A σ A σ \mathbf{A}_\sigma A σ − 1 -1 − 1 A σ ( p ) = ( ∣ R ˙ ∣ / n ) A σ ( R ˙ ) \mathbf{A}_\sigma(\mathbf{p})=(|\dot{\mathbf{R}}|/n)\mathbf{A}_\sigma(\dot{\mathbf{R}}) A σ ( p ) = ( ∣ R ˙ ∣/ n ) A σ ( R ˙ ) R ¨ \ddot{\mathbf{R}} R ¨ ∂ L / ∂ R − d / d ℓ ∂ L / ∂ R ˙ + d 2 / d ℓ 2 ∂ L / ∂ R ¨ = 0 \partial\mathcal{L}/\partial\mathbf{R}-d/d\ell\,\partial\mathcal{L}/\partial\dot{\mathbf{R}}+d^2/d\ell^2\,\partial\mathcal{L}/\partial\ddot{\mathbf{R}}=0 ∂ L / ∂ R − d / d ℓ ∂ L / ∂ R ˙ + d 2 / d ℓ 2 ∂ L / ∂ R ¨ = 0 ( a ⋅ ∇ R ˙ ) B − ∇ R ˙ ( a ⋅ B ) = − a × ( ∇ R ˙ × B ) (\mathbf{a}\cdot\nabla_{\dot{\mathbf{R}}})\mathbf{B}-\nabla_{\dot{\mathbf{R}}}(\mathbf{a}\cdot\mathbf{B})=-\mathbf{a}\times(\nabla_{\dot{\mathbf{R}}}\times\mathbf{B}) ( a ⋅ ∇ R ˙ ) B − ∇ R ˙ ( a ⋅ B ) = − a × ( ∇ R ˙ × B ) a \mathbf{a} a − k 0 − 1 d / d ℓ [ R ¨ × Ω σ ( R ˙ ) ] -k_0^{-1}d/d\ell[\ddot{\mathbf{R}}\times\boldsymbol{\Omega}_\sigma(\dot{\mathbf{R}})] − k 0 − 1 d / d ℓ [ R ¨ × Ω σ ( R ˙ )]
Ω σ ( R ˙ ) = ∇ R ˙ × A σ ( R ˙ ) = − σ R ˙ ∣ R ˙ ∣ 3 . \boldsymbol{\Omega}_\sigma(\dot{\mathbf{R}})=\nabla_{\dot{\mathbf{R}}}\times\mathbf{A}_\sigma(\dot{\mathbf{R}})=-\sigma\frac{\dot{\mathbf{R}}}{|\dot{\mathbf{R}}|^3}. Ω σ ( R ˙ ) = ∇ R ˙ × A σ ( R ˙ ) = − σ ∣ R ˙ ∣ 3 R ˙ . The equation becomes
∇ n ∣ R ˙ ∣ − d d ℓ ( n R ˙ ∣ R ˙ ∣ ) − 1 k 0 d d ℓ [ R ¨ × Ω σ ( R ˙ ) ] = 0. (97) \nabla n|\dot{\mathbf{R}}|-\frac{d}{d\ell}\left(n\frac{\dot{\mathbf{R}}}{|\dot{\mathbf{R}}|}\right)-\frac{1}{k_0}\frac{d}{d\ell}\left[\ddot{\mathbf{R}}\times\boldsymbol{\Omega}_\sigma(\dot{\mathbf{R}})\right]=0. \tag*{(97)} ∇ n ∣ R ˙ ∣ − d ℓ d ( n ∣ R ˙ ∣ R ˙ ) − k 0 1 d ℓ d [ R ¨ × Ω σ ( R ˙ ) ] = 0. (97) Choosing the arclength as the parameter after the variation, ∣ R ˙ ∣ = 1 |\dot{\mathbf{R}}|=1 ∣ R ˙ ∣ = 1 R ˙ = t ^ \dot{\mathbf{R}}=\hat{\mathbf{t}} R ˙ = t ^ R ¨ \ddot{\mathbf{R}} R ¨ R ¨ = ∇ ⊥ ln n \ddot{\mathbf{R}}=\nabla_\perp\ln n R ¨ = ∇ ⊥ ln n ∇ ln n \nabla\ln n ∇ ln n
d d ℓ ( n t ^ ) = ∇ n + σ k 0 d d ℓ [ ∇ ln n × t ^ ] . (98) \frac{d}{d\ell}(n\hat{\mathbf{t}})=\nabla n+\frac{\sigma}{k_0}\frac{d}{d\ell}\left[\nabla\ln n\times\hat{\mathbf{t}}\right]. \tag*{(98)} d ℓ d ( n t ^ ) = ∇ n + k 0 σ d ℓ d [ ∇ ln n × t ^ ] . (98) where ∇ ⊥ ln n × t ^ = ∇ ln n × t ^ \nabla_\perp\ln n\times\hat{\mathbf{t}}=\nabla\ln n\times\hat{\mathbf{t}} ∇ ⊥ ln n × t ^ = ∇ ln n × t ^ t = d R / d ℓ = p ^ + δ \mathbf{t}=d\mathbf{R}/d\ell=\hat{\mathbf{p}}+\boldsymbol{\delta} t = d R / d ℓ = p ^ + δ δ = ( σ / k 0 n 2 ) ∇ n × p ^ \boldsymbol{\delta}=(\sigma/k_0n^2)\nabla n\times\hat{\mathbf{p}} δ = ( σ / k 0 n 2 ) ∇ n × p ^ p ^ \hat{\mathbf{p}} p ^ p = n ( t ^ − δ ) \mathbf{p}=n(\hat{\mathbf{t}}-\boldsymbol{\delta}) p = n ( t ^ − δ ) d p / d ℓ = ∇ n d\mathbf{p}/d\ell=\nabla n d p / d ℓ = ∇ n d ( n t ^ ) / d ℓ = ∇ n + d ( n δ ) / d ℓ d(n\hat{\mathbf{t}})/d\ell=\nabla n+d(n\boldsymbol{\delta})/d\ell d ( n t ^ ) / d ℓ = ∇ n + d ( n δ ) / d ℓ 1 / k 0 1/k_0 1/ k 0
The ray equations follow from the Hamiltonian H = ∣ p ∣ − n ( R ) \mathcal{H}=|\mathbf{p}|-n(\mathbf{R}) H = ∣ p ∣ − n ( R ) ( R , p ) (\mathbf{R},\mathbf{p}) ( R , p ) { r i , p j } = δ i j \{r_i,p_j\}=\delta_{ij} { r i , p j } = δ ij
{ R i , p j } = δ i j , { R i , R j } = 1 k 0 ϵ i j l Ω σ , l ( p ) , (99) \{R_i,p_j\}=\delta_{ij},\qquad\{R_i,R_j\}=\frac{1}{k_0}\epsilon_{ijl}\Omega_{\sigma,l}(\mathbf{p}), \tag*{(99)} { R i , p j } = δ ij , { R i , R j } = k 0 1 ϵ ij l Ω σ , l ( p ) , (99) where the second relation follows from { R i , R j } = k 0 − 1 ( ∂ p i A σ , j − ∂ p j A σ , i ) \{R_i,R_j\}=k_0^{-1}(\partial_{p_i}A_{\sigma,j}-\partial_{p_j}A_{\sigma,i}) { R i , R j } = k 0 − 1 ( ∂ p i A σ , j − ∂ p j A σ , i )
d R i d ℓ = { R i , H } = p ^ i + { R i , R j } ∂ H ∂ R j , (100) \frac{dR_i}{d\ell}=\{R_i,\mathcal{H}\}=\hat{p}_i+\{R_i,R_j\}\frac{\partial\mathcal{H}}{\partial R_j}, \tag*{(100)} d ℓ d R i = { R i , H } = p ^ i + { R i , R j } ∂ R j ∂ H , (100) d p i d ℓ = { p i , H } = ∂ i n ( R ) , (101) \frac{dp_i}{d\ell}=\{p_i,\mathcal{H}\}=\partial_i n(\mathbf{R}), \tag*{(101)} d ℓ d p i = { p i , H } = ∂ i n ( R ) , (101) which reproduce Eqs. (50) and (51) because ∂ H / ∂ R j = − ∂ j n \partial\mathcal{H}/\partial R_j=-\partial_j n ∂ H / ∂ R j = − ∂ j n [ r i , p j ] = i δ i j / k 0 [r_i,p_j]=i\delta_{ij}/k_0 [ r i , p j ] = i δ ij / k 0 [ r i , k j ] = i δ i j [r_i,k_j]=i\delta_{ij} [ r i , k j ] = i δ ij p = k / k 0 \mathbf{p}=\mathbf{k}/k_0 p = k / k 0 1 / k 0 1/k_0 1/ k 0 [ A , B ] = ( i / k 0 ) { A , B } [A,B]=(i/k_0)\{A,B\} [ A , B ] = ( i / k 0 ) { A , B } [ R i , R j ] = i ϵ i j l Ω σ , l / k 0 2 = − i σ ϵ i j l k l / k 3 [R_i,R_j]=i\epsilon_{ijl}\Omega_{\sigma,l}/k_0^2=-i\sigma\epsilon_{ijl}k_l/k^3 [ R i , R j ] = i ϵ ij l Ω σ , l / k 0 2 = − iσ ϵ ij l k l / k 3
Frequency as the Hamiltonian
The ray Hamiltonian uses the arclength ℓ \ell ℓ
H ( r , k ) = ℏ ω ( r , k ) = ℏ c ∣ k ∣ n ( r ) , (102) H(\mathbf{r},\mathbf{k})=\hbar\omega(\mathbf{r},\mathbf{k})=\frac{\hbar c|\mathbf{k}|}{n(\mathbf{r})}, \tag*{(102)} H ( r , k ) = ℏ ω ( r , k ) = n ( r ) ℏ c ∣ k ∣ , (102) with canonical momentum ℏ k \hbar\mathbf{k} ℏ k r ˙ = ∂ ω / ∂ k \dot{\mathbf{r}}=\partial\omega/\partial\mathbf{k} r ˙ = ∂ ω / ∂ k k = − ∂ ω / ∂ r \mathbf{k}=-\partial\omega/\partial\mathbf{r} k = − ∂ ω / ∂ r ℏ \hbar ℏ ω \omega ω { r i , k j } = δ i j \{r_i,k_j\}=\delta_{ij} { r i , k j } = δ ij
Same constraint surface and reparametrization
Fix the frequency ω 0 = c k 0 \omega_0 = ck_0 ω 0 = c k 0 k = k 0 p \mathbf{k} = k_0\mathbf{p} k = k 0 p ω ( r , k ) = ω 0 \omega(\mathbf{r},\mathbf{k}) = \omega_0 ω ( r , k ) = ω 0 c k 0 ∣ p ∣ / n = c k 0 ck_0|\mathbf{p}|/n = ck_0 c k 0 ∣ p ∣/ n = c k 0 ∣ p ∣ = n ( r ) |\mathbf{p}| = n(\mathbf{r}) ∣ p ∣ = n ( r ) (17) , which is the constraint H 0 = ∣ p ∣ − n = 0 \mathcal{H}_0 = |\mathbf{p}| - n = 0 H 0 = ∣ p ∣ − n = 0
ω − ω 0 = ω 0 ( ∣ p ∣ n − 1 ) = ω 0 n ( r ) H 0 ( r , p ) . (103) \omega-\omega_0=\omega_0\left(\frac{|\mathbf{p}|}{n}-1\right)=\frac{\omega_0}{n(\mathbf{r})}\mathcal{H}_0(\mathbf{r},\mathbf{p}). \tag*{(103)} ω − ω 0 = ω 0 ( n ∣ p ∣ − 1 ) = n ( r ) ω 0 H 0 ( r , p ) . (103) In Eq. (102) the frequency is the value of the Hamiltonian (an integral of the motion, since ω \omega ω k 0 k_0 k 0 ω 0 / n \omega_0/n ω 0 / n ∂ / ∂ k = k 0 − 1 ∂ / ∂ p \partial/\partial\mathbf{k}=k_0^{-1}\partial/\partial\mathbf{p} ∂ / ∂ k = k 0 − 1 ∂ / ∂ p ω = ω 0 ∣ p ∣ / n \omega=\omega_0|\mathbf{p}|/n ω = ω 0 ∣ p ∣/ n
d r d t = ∂ ω ∂ k = 1 k 0 ω 0 n p ^ = c n p ^ , (104) \frac{d\mathbf{r}}{dt}=\frac{\partial\omega}{\partial\mathbf{k}}=\frac{1}{k_0}\frac{\omega_0}{n}\widehat{\mathbf{p}}=\frac{c}{n}\widehat{\mathbf{p}}, \tag*{(104)} d t d r = ∂ k ∂ ω = k 0 1 n ω 0 p = n c p , (104) d p d t = − 1 k 0 ∂ ω ∂ r = c ∣ p ∣ n 2 ∇ n = c n ∇ n , (105) \frac{d\mathbf{p}}{dt}=-\frac{1}{k_0}\frac{\partial\omega}{\partial\mathbf{r}}=\frac{c|\mathbf{p}|}{n^2}\nabla n=\frac{c}{n}\nabla n, \tag*{(105)} d t d p = − k 0 1 ∂ r ∂ ω = n 2 c ∣ p ∣ ∇ n = n c ∇ n , (105) where the last step uses ∣ p ∣ = n |\mathbf{p}|=n ∣ p ∣ = n (19) , these are the equations of Eq. (19) multiplied by c / n c/n c / n
d ℓ = c n d t . (106) d\ell=\frac{c}{n}\,dt. \tag*{(106)} d ℓ = n c d t . (106) Berry-connection Hamiltonian in the time parametrization
The shifted-argument Hamiltonian (55) has a counterpart for Eq. (102) . Define the Berry connection in k \mathbf{k} k
A ~ σ ( k ) = 1 k 0 A σ ( k k 0 ) , F σ ( k ) = ∇ k × A ~ σ = − σ k k 3 , (107) \widetilde{\mathbf{A}}_\sigma(\mathbf{k})=\frac{1}{k_0}\mathbf{A}_\sigma\left(\frac{\mathbf{k}}{k_0}\right),\qquad
\mathbf{F}_\sigma(\mathbf{k})=\nabla_{\mathbf{k}}\times\widetilde{\mathbf{A}}_\sigma=-\sigma\frac{\mathbf{k}}{k^3}, \tag*{(107)} A σ ( k ) = k 0 1 A σ ( k 0 k ) , F σ ( k ) = ∇ k × A σ = − σ k 3 k , (107) where the second relation follows from Eq. (43) and ∇ k = k 0 − 1 ∇ p \nabla_{\mathbf{k}}=k_0^{-1}\nabla_{\mathbf{p}} ∇ k = k 0 − 1 ∇ p A ~ σ \widetilde{\mathbf{A}}_\sigma A σ R = r + A ~ σ ( k ) \mathbf{R}=\mathbf{r}+\widetilde{\mathbf{A}}_\sigma(\mathbf{k}) R = r + A σ ( k ) (54) . The helicity-dependent Hamiltonian is
H σ ( r , k ) = ℏ c ∣ k ∣ n ( r + A ~ σ ( k ) ) , ω σ = c ∣ k ∣ n ( R ) . (108) \mathcal{H}_\sigma(\mathbf{r},\mathbf{k})=\frac{\hbar c|\mathbf{k}|}{n(\mathbf{r}+\widetilde{\mathbf{A}}_\sigma(\mathbf{k}))},\qquad
\omega_\sigma=\frac{c|\mathbf{k}|}{n(\mathbf{R})}. \tag*{(108)} H σ ( r , k ) = n ( r + A σ ( k )) ℏ c ∣ k ∣ , ω σ = n ( R ) c ∣ k ∣ . (108) Hamilton’s equations are
r ˙ i = ∂ ω σ ∂ k i = c k ^ i n − c k n 2 ∂ j n ∂ A ~ σ , j ∂ k i , (109) \dot{r}_i=\frac{\partial\omega_\sigma}{\partial k_i}=\frac{c\widehat{k}_i}{n}-\frac{ck}{n^2}\partial_j n\frac{\partial\widetilde{A}_{\sigma,j}}{\partial k_i}, \tag*{(109)} r ˙ i = ∂ k i ∂ ω σ = n c k i − n 2 c k ∂ j n ∂ k i ∂ A σ , j , (109) k ˙ i = − ∂ ω σ ∂ r i = c k n 2 ∂ i n , (110) \dot{k}_i=-\frac{\partial\omega_\sigma}{\partial r_i}=\frac{ck}{n^2}\partial_i n, \tag*{(110)} k ˙ i = − ∂ r i ∂ ω σ = n 2 c k ∂ i n , (110) with n n n R \mathbf{R} R R ˙ i = r ˙ i + ( ∂ A ~ σ , i / ∂ k j ) k ˙ j \dot{R}_i=\dot{r}_i+(\partial\widetilde{A}_{\sigma,i}/\partial k_j)\dot{k}_j R ˙ i = r ˙ i + ( ∂ A σ , i / ∂ k j ) k ˙ j ∂ k i A ~ σ , j − ∂ k j A ~ σ , i = ϵ i j l F σ , l \partial_{k_i}\widetilde{A}_{\sigma,j}-\partial_{k_j}\widetilde{A}_{\sigma,i}=\epsilon_{ijl}F_{\sigma,l} ∂ k i A σ , j − ∂ k j A σ , i = ϵ ij l F σ , l (47) ] gives
R ˙ = c n k ^ − k ˙ × F σ ( k ) = c n k ^ + σ k ˙ × k k 3 , k ˙ = c k n 2 ∇ n . (111) \dot{\mathbf{R}}=\frac{c}{n}\widehat{\mathbf{k}}-\dot{\mathbf{k}}\times\mathbf{F}_\sigma(\mathbf{k})=\frac{c}{n}\widehat{\mathbf{k}}+\sigma\dot{\mathbf{k}}\times\frac{\mathbf{k}}{k^3},\qquad
\dot{\mathbf{k}}=\frac{ck}{n^2}\nabla n. \tag*{(111)} R ˙ = n c k − k ˙ × F σ ( k ) = n c k + σ k ˙ × k 3 k , k ˙ = n 2 c k ∇ n . (111) This is the time-parametrized formulation with the Berry curvature F σ \mathbf{F}_\sigma F σ − k ˙ × F σ -\dot{\mathbf{k}}\times\mathbf{F}_\sigma − k ˙ × F σ [3 , 6 , 7 ] . Multiplying Eq. (111) by d t / d ℓ = n / c dt/d\ell=n/c d t / d ℓ = n / c k ˙ = ( c k / n 2 ) ∇ n \dot{\mathbf{k}}=(ck/n^2)\nabla n k ˙ = ( c k / n 2 ) ∇ n k = k 0 n \mathbf{k}=k_0n k = k 0 n σ ( ∇ n × k ^ ) / ( n k ) = σ ∇ n × p ^ / ( k 0 n 2 ) \sigma(\nabla n\times\widehat{\mathbf{k}})/(nk)=\sigma\nabla n\times\widehat{\mathbf{p}}/(k_0n^2) σ ( ∇ n × k ) / ( nk ) = σ ∇ n × p / ( k 0 n 2 ) (111) reproduces Eq. (50) on the surface p = n p=n p = n
Lagrangian with the Berry term
The Lagrangian of the Hamiltonian in Eq. (108) is L = k ⋅ r ˙ − ω σ L=\mathbf{k}\cdot\dot{\mathbf{r}}-\omega_\sigma L = k ⋅ r ˙ − ω σ r ˙ = R ˙ − ( k ˙ ⋅ ∇ k ) A ~ σ \dot{\mathbf{r}}=\dot{\mathbf{R}}-(\dot{\mathbf{k}}\cdot\nabla_{\mathbf{k}})\widetilde{\mathbf{A}}_\sigma r ˙ = R ˙ − ( k ˙ ⋅ ∇ k ) A σ k ⋅ ( k ˙ ⋅ ∇ k ) A ~ σ = d ( k ⋅ A ~ σ ) / d t − A ~ σ ⋅ k ˙ \mathbf{k}\cdot(\dot{\mathbf{k}}\cdot\nabla_{\mathbf{k}})\widetilde{\mathbf{A}}_\sigma=d(\mathbf{k}\cdot\widetilde{\mathbf{A}}_\sigma)/dt-\widetilde{\mathbf{A}}_\sigma\cdot\dot{\mathbf{k}} k ⋅ ( k ˙ ⋅ ∇ k ) A σ = d ( k ⋅ A σ ) / d t − A σ ⋅ k ˙
L = k ⋅ R ˙ + A ~ σ ( k ) ⋅ k ˙ − ω ( R , k ) . (112) L=\mathbf{k}\cdot\dot{\mathbf{R}}+\widetilde{\mathbf{A}}_\sigma(\mathbf{k})\cdot\dot{\mathbf{k}}-\omega(\mathbf{R},\mathbf{k}). \tag*{(112)} L = k ⋅ R ˙ + A σ ( k ) ⋅ k ˙ − ω ( R , k ) . (112) The term A ~ σ ⋅ k ˙ \widetilde{\mathbf{A}}_\sigma\cdot\dot{\mathbf{k}} A σ ⋅ k ˙ k \mathbf{k} k (111) . This is the form of the semiclassical Lagrangian of Sundaram and Niu [7 ] . It is equivalent to the action (46): ∫ ( k ⋅ d R + A ~ σ ⋅ d k ) = k 0 ∫ [ p ⋅ d R + k 0 − 1 A σ ⋅ d p ] \int(\mathbf{k}\cdot d\mathbf{R}+\widetilde{\mathbf{A}}_\sigma\cdot d\mathbf{k})=k_0\int[\mathbf{p}\cdot d\mathbf{R}+k_0^{-1}\mathbf{A}_\sigma\cdot d\mathbf{p}] ∫ ( k ⋅ d R + A σ ⋅ d k ) = k 0 ∫ [ p ⋅ d R + k 0 − 1 A σ ⋅ d p ]
Summary of the correspondence
Hamiltonian: H σ = ∣ p ∣ − n ( R ) \mathcal{H}_\sigma=|\mathbf{p}|-n(\mathbf{R}) H σ = ∣ p ∣ − n ( R ) H σ / ℏ = c k / n ( R ) \mathcal{H}_\sigma/\hbar=ck/n(\mathbf{R}) H σ /ℏ = c k / n ( R )
Variables: ( r , p ) (\mathbf{r},\mathbf{p}) ( r , p ) ( r , k ) (\mathbf{r},\mathbf{k}) ( r , k ) k = k 0 p \mathbf{k}=k_0\mathbf{p} k = k 0 p
Evolution parameter: ℓ \ell ℓ t t t d ℓ = ( c / n ) d t d\ell=(c/n)\,dt d ℓ = ( c / n ) d t
Shift of the argument of n n n A σ ( p ) / k 0 \mathbf{A}_\sigma(\mathbf{p})/k_0 A σ ( p ) / k 0 A ~ σ ( k ) \widetilde{\mathbf{A}}_\sigma(\mathbf{k}) A σ ( k )
Curvature: Ω σ = − σ p / p 3 \boldsymbol{\Omega}_\sigma=-\sigma\mathbf{p}/p^3 Ω σ = − σ p / p 3 F σ = − σ k / k 3 \mathbf{F}_\sigma=-\sigma\mathbf{k}/k^3 F σ = − σ k / k 3
Frequency: the fixed parameter k 0 k_0 k 0 H / ℏ H/\hbar H /ℏ
Comparison with Bliokh’s review and the helical ray
Comparison of conventions and equations
The ray equations (50) and (51) have exactly the form given in Bliokh’s review [8 ] . There they are obtained from the Coriolis term that appears in the Maxwell equations written in a frame rotating with the ray, and Eq. (3.40) of the review reads
p ˙ = ∇ n and r ˙ = p p + λ 0 p 3 s p ˙ × p \dot{\mathbf{p}}=\nabla n \quad\text{and}\quad\dot{\mathbf{r}}=\frac{\mathbf{p}}{p}+\frac{\lambda_0}{p^3}s\,\dot{\mathbf{p}}\times\mathbf{p} p ˙ = ∇ n and r ˙ = p p + p 3 λ 0 s p ˙ × p where λ ˉ 0 = 1 / k 0 \bar{\lambda}_0=1/k_0 λ ˉ 0 = 1/ k 0 s = ± 1 s=\pm1 s = ± 1 s = σ s=\sigma s = σ p ˙ = ∇ n \dot{\mathbf{p}}=\nabla n p ˙ = ∇ n (50) and (51) , including the sign of the Magnus term. The two derivations are therefore consistent. The notation corresponds as follows.
The circular polarization vector of the review, ξ = ( v + i w ) / 2 \boldsymbol{\xi}=(\mathbf{v}+i\mathbf{w})/\sqrt{2} ξ = ( v + i w ) / 2 ( v , w , t ) (\mathbf{v},\mathbf{w},\mathbf{t}) ( v , w , t ) e ^ σ \hat{\mathbf{e}}_\sigma e ^ σ (32) with σ = + 1 \sigma=+1 σ = + 1 ( v , w , t ) ↔ ( ν ^ , β ^ , p ^ ) (\mathbf{v},\mathbf{w},\mathbf{t})\leftrightarrow(\hat{\boldsymbol{\nu}},\hat{\boldsymbol{\beta}},\hat{\mathbf{p}}) ( v , w , t ) ↔ ( ν ^ , β ^ , p ^ )
The Berry connection of the review is A B = − i ξ ∗ ⋅ ∇ p ξ \mathbf{A}^{\mathrm{B}}=-i\boldsymbol{\xi}^{*}\cdot\nabla_{\mathbf{p}}\boldsymbol{\xi} A B = − i ξ ∗ ⋅ ∇ p ξ F = p / p 3 \mathbf{F}=\mathbf{p}/p^3 F = p / p 3 s s s A σ = − σ A B \mathbf{A}_{\sigma}=-\sigma\mathbf{A}^{\mathrm{B}} A σ = − σ A B Ω σ = − σ F \boldsymbol{\Omega}_{\sigma}=-\sigma\mathbf{F} Ω σ = − σ F
The spin-orbit Lagrangian − λ ˉ 0 s A B ⋅ p ˙ -\bar{\lambda}_0s\mathbf{A}^{\mathrm{B}}\cdot\dot{\mathbf{p}} − λ ˉ 0 s A B ⋅ p ˙ + k 0 − 1 A σ ⋅ d p / d ℓ +k_0^{-1}\mathbf{A}_{\sigma}\cdot d\mathbf{p}/d\ell + k 0 − 1 A σ ⋅ d p / d ℓ r \mathbf{r} r p \mathbf{p} p (50) and (51) , as in the review. The coordinate r \mathbf{r} r R \mathbf{R} R
The phase of the review is Φ = Φ 0 − s Φ B \Phi=\Phi_0-s\Phi_{\mathrm{B}} Φ = Φ 0 − s Φ B Φ B = ∫ A B ⋅ d p = − ∫ T d ℓ \Phi_{\mathrm{B}}=\int\mathbf{A}^{\mathrm{B}}\cdot d\mathbf{p}=-\int\mathcal{T}\,d\ell Φ B = ∫ A B ⋅ d p = − ∫ T d ℓ φ σ = ∫ A σ ⋅ d p = σ ∫ T d ℓ \varphi_{\sigma}=\int\mathbf{A}_{\sigma}\cdot d\mathbf{p}=\sigma\int\mathcal{T}\,d\ell φ σ = ∫ A σ ⋅ d p = σ ∫ T d ℓ (40) and (36) . The review also gives the parallel-transport law for the polarization vector, e ˙ = − ( e ⋅ t ˙ ) t \dot{\mathbf{e}}=-(\mathbf{e}\cdot\dot{\mathbf{t}})\mathbf{t} e ˙ = − ( e ⋅ t ˙ ) t (31) for a constant impedance.
The Hamiltonian of the review is H = p − n − λ ˉ 0 s Ω ∥ \mathcal{H}=p-n-\bar{\lambda}_0s\Omega_{\parallel} H = p − n − λ ˉ 0 s Ω ∥ Ω ∥ \Omega_{\parallel} Ω ∥ t \mathbf{t} t − k 0 − 1 A σ ⋅ d p / d ℓ -k_0^{-1}\mathbf{A}_{\sigma}\cdot d\mathbf{p}/d\ell − k 0 − 1 A σ ⋅ d p / d ℓ − ( σ / k 0 ) T -(\sigma/k_0)\mathcal{T} − ( σ / k 0 ) T (91) , which is the same term.
Two differences should be noted. The review assumes a nonmagnetic medium (n 2 = ϵ / ϵ 0 n^2=\epsilon/\epsilon_0 n 2 = ϵ / ϵ 0 ϵ ( r ) \epsilon(\mathbf{r}) ϵ ( r ) μ ( r ) \mu(\mathbf{r}) μ ( r ) n n n Z 1 / 2 Z^{1/2} Z 1/2 (35) . Second, the review describes the rays as the trajectories of the center of gravity of a paraxial wave packet or beam, which supports the identification of R \mathbf{R} R
Helical ray in a cylindrically symmetric medium
For n = n ( ρ ) n=n(\rho) n = n ( ρ ) ( ρ , Φ , Z ) (\rho,\Phi,Z) ( ρ , Φ , Z ) ρ \rho ρ p ^ = sin θ e Φ + cos θ e Z \hat{\mathbf{p}}=\sin\theta\,\mathbf{e}_{\Phi}+\cos\theta\,\mathbf{e}_{Z} p ^ = sin θ e Φ + cos θ e Z p = n p ^ \mathbf{p}=n\hat{\mathbf{p}} p = n p ^
sin 2 θ = − ρ n ′ ( ρ ) n ( ρ ) (113) \sin^2\theta=-\frac{\rho n'(\rho)}{n(\rho)} \tag*{(113)} sin 2 θ = − n ( ρ ) ρ n ′ ( ρ ) (113) (Eq. (4.3) of the review [[8 ] ). The curvature and the torsion of the helix are K = sin 2 θ / ρ \mathcal{K}=\sin^2\theta/\rho K = sin 2 θ / ρ T = sin θ cos θ / ρ \mathcal{T}=\sin\theta\cos\theta/\rho T = sin θ cos θ / ρ ν ^ = − e ρ \hat{\boldsymbol{\nu}}=-\mathbf{e}_{\rho} ν ^ = − e ρ β ^ = − cos θ e Φ + sin θ e Z \hat{\boldsymbol{\beta}}=-\cos\theta\,\mathbf{e}_{\Phi}+\sin\theta\,\mathbf{e}_{Z} β ^ = − cos θ e Φ + sin θ e Z ∇ n = n ′ e ρ \nabla n=n'\mathbf{e}_{\rho} ∇ n = n ′ e ρ (53)
d R M d ℓ = − σ sin 2 θ k 0 n ρ β ^ (114) \frac{d\mathbf{R}_{\mathrm{M}}}{d\ell}=-\sigma\frac{\sin^2\theta}{k_0 n\rho}\,\hat{\boldsymbol{\beta}} \tag*{(114)} d ℓ d R M = − σ k 0 n ρ sin 2 θ β ^ (114) which is Eq. (4.13) of the review [[8 ] (the instantaneous Magnus velocity).
Integrating the velocity in Eq. (114) at fixed p \mathbf{p} p ℓ = 2 π ρ / sin θ \ell=2\pi\rho/\sin\theta ℓ = 2 π ρ / sin θ δ p = 0 \delta\mathbf{p}=0 δ p = 0 − σ λ sin θ β ^ -\sigma\lambda\sin\theta\,\hat{\boldsymbol{\beta}} − σ λ sin θ β ^ λ = 2 π / [ k 0 n ( ρ ) ] \lambda=2\pi/[k_0n(\rho)] λ = 2 π / [ k 0 n ( ρ )] d p / d ℓ = ∇ n ( R ) d\mathbf{p}/d\ell=\nabla n(\mathbf{R}) d p / d ℓ = ∇ n ( R ) (51) : R ′ = p ^ − σ κ β ^ / ( k 0 n ) \mathbf{R}'=\hat{\mathbf{p}}-\sigma\kappa\hat{\boldsymbol{\beta}}/(k_0n) R ′ = p ^ − σ κ β ^ / ( k 0 n ) Φ ′ = [ sin θ p + σ K cos θ p / ( k 0 n ) ] / ρ \Phi'=[\sin\theta_{\mathrm{p}}+\sigma\mathcal{K}\cos\theta_{\mathrm{p}}/(k_0n)]/\rho Φ ′ = [ sin θ p + σ K cos θ p / ( k 0 n )] / ρ Φ ′ = − n ′ / [ n sin θ p ] \Phi'=-n'/[n\sin\theta_{\mathrm{p}}] Φ ′ = − n ′ / [ n sin θ p ] θ p \theta_{\mathrm{p}} θ p ρ \rho ρ
sin 2 θ p = sin 2 θ − σ sin 2 θ k 0 n ρ = sin θ p cos θ p ⇒ θ p ≃ θ − σ K 2 k 0 n (115) \sin^2\theta_{\mathrm{p}}=\sin^2\theta-\sigma\frac{\sin^2\theta}{k_0n\rho}
=\sin\theta_{\mathrm{p}}\cos\theta_{\mathrm{p}}
\Rightarrow
\theta_{\mathrm{p}}\simeq\theta-\frac{\sigma\mathcal{K}}{2k_0n} \tag*{(115)} sin 2 θ p = sin 2 θ − σ k 0 n ρ sin 2 θ = sin θ p cos θ p ⇒ θ p ≃ θ − 2 k 0 n σ K (115) while the tangent of the ray is at the angle θ + σ K / ( 2 k 0 n ) \theta+\sigma\mathcal{K}/(2k_0n) θ + σ K / ( 2 k 0 n ) σ = 0 \sigma=0 σ = 0 σ K / ( k 0 n ) \sigma\mathcal{K}/(k_0n) σ K / ( k 0 n ) p \mathbf{p} p
δ R 0 = − σ λ 2 sin θ β ^ , λ = 2 π k 0 n ( ρ ) (116) \delta\mathbf{R}_0=-\sigma\frac{\lambda}{2}\sin\theta\,\hat{\boldsymbol{\beta}},\qquad\lambda=\frac{2\pi}{k_0n(\rho)} \tag*{(116)} δ R 0 = − σ 2 λ sin θ β ^ , λ = k 0 n ( ρ ) 2 π (116) and the splitting between σ = + 1 \sigma=+1 σ = + 1 σ = − 1 \sigma=-1 σ = − 1 λ sin θ \lambda\sin\theta λ sin θ (75) and (76) numerically for n ( ρ ) = n a ( 1 − ρ 2 / 2 Λ 2 ) n(\rho)=n_a(1-\rho^2/2\Lambda^2) n ( ρ ) = n a ( 1 − ρ 2 /2 Λ 2 ) ρ / Λ = 0.5 \rho/\Lambda=0.5 ρ /Λ = 0.5 k 0 = 400 k_0=400 k 0 = 400 Λ \Lambda Λ (116) is 1.000 to the value of the review 0.500, in the same direction. If both rays are started with the same p 0 \mathbf{p}_0 p 0 σ ∫ T d ℓ = 2 π σ cos θ \sigma\int\mathcal{T}\,d\ell=2\pi\sigma\cos\theta σ ∫ T d ℓ = 2 π σ cos θ 2 π 2\pi 2 π − σ Θ -\sigma\Theta − σ Θ Θ = 2 π ( 1 − cos θ ) \Theta=2\pi(1-\cos\theta) Θ = 2 π ( 1 − cos θ ) (44) and with Eq. (4.11) of the review. The shift accumulates linearly with the number of periods.
[20 ] For example, in the gauge (B3), using ( p × a ^ ) × p = a ^ − ( p ⋅ a ^ ) p (\mathbf{p} \times\hat{\mathbf{a}}) \times\mathbf{p} = \hat{\mathbf{a}} - (\mathbf{p} \cdot\hat{\mathbf{a}})\mathbf{p} ( p × a ^ ) × p = a ^ − ( p ⋅ a ^ ) p
J = r × p + σ k 0 a ^ + p ^ 1 + p ^ ⋅ a ^ , (117) \mathbf{J} = \mathbf{r} \times\mathbf{p} + \frac{\sigma}{k_0}\frac{\hat{\mathbf{a}}+\hat{\mathbf{p}}}{1+\hat{\mathbf{p}}\cdot\hat{\mathbf{a}}}, \tag*{(117)} J = r × p + k 0 σ 1 + p ^ ⋅ a ^ a ^ + p ^ , (117) which is singular only on the Dirac string p ^ = − a ^ \hat{\mathbf{p}}=-\hat{\mathbf{a}} p ^ = − a ^ a ^ \hat{\mathbf{a}} a ^ ( r × p ) ⋅ a ^ + σ / k 0 (\mathbf{r}\times\mathbf{p})\cdot\hat{\mathbf{a}}+\sigma/k_0 ( r × p ) ⋅ a ^ + σ / k 0 a ^ \hat{\mathbf{a}} a ^ J ⋅ a ^ \mathbf{J}\cdot\hat{\mathbf{a}} J ⋅ a ^ σ / k 0 \sigma/k_0 σ / k 0
[22] The same result follows from the symmetry of the problem. With the non-canonical brackets, Eq. (99) , { R i , R j } = k 0 − 1 ϵ i j l Ω σ , l \{R_i,R_j\}=k_0^{-1}\epsilon_{ijl}\Omega_{\sigma,l} { R i , R j } = k 0 − 1 ϵ ij l Ω σ , l { R i , p j } = δ i j \{R_i,p_j\}=\delta_{ij} { R i , p j } = δ ij { p i , p j } = 0 \{p_i,p_j\}=0 { p i , p j } = 0 R \mathbf{R} R { ( R × p ) i , R j } = ϵ i j l R l + k 0 − 1 ( p j Ω σ , i − δ i j p ⋅ Ω σ ) \{(\mathbf{R}\times\mathbf{p})_i,R_j\}=\epsilon_{ijl}R_l+k_0^{-1}(p_j\Omega_{\sigma,i}-\delta_{ij}\mathbf{p}\cdot\boldsymbol{\Omega}_\sigma) {( R × p ) i , R j } = ϵ ij l R l + k 0 − 1 ( p j Ω σ , i − δ ij p ⋅ Ω σ ) { ( σ / k 0 ) p ^ i , R j } = − ( σ / k 0 ) ( δ i j − p ^ i p ^ j ) / p \{(\sigma/k_0)\hat{p}_i,R_j\}=-(\sigma/k_0)(\delta_{ij}-\hat{p}_i\hat{p}_j)/p {( σ / k 0 ) p ^ i , R j } = − ( σ / k 0 ) ( δ ij − p ^ i p ^ j ) / p Ω σ = − σ p / p 3 \boldsymbol{\Omega}_\sigma=-\sigma\mathbf{p}/p^3 Ω σ = − σ p / p 3
{ J i , R j } = ϵ i j l R l , { J i , p j } = ϵ i j l p l , \{J_i,R_j\}=\epsilon_{ijl}R_l,\qquad\{J_i,p_j\}=\epsilon_{ijl}p_l, { J i , R j } = ϵ ij l R l , { J i , p j } = ϵ ij l p l , so J \mathbf{J} J ( R , p ) (\mathbf{R},\mathbf{p}) ( R , p ) d J / d ℓ = { J , H } d\mathbf{J}/d\ell=\{\mathbf{J},\mathcal{H}\} d J / d ℓ = { J , H } H = ∣ p ∣ − n ( R ) \mathcal{H}=|\mathbf{p}|-n(\mathbf{R}) H = ∣ p ∣ − n ( R )
