Goos–Hänchen and Imbert–Fedorov beam shifts: an overview

First, we consider a monochromatic light beam propagating along the z-axis in free space, which is characterised by the frequency ω, the wavenumber k = ω, and the complex electric field E(r). The actual real-valued field is E(r,t) =  Re[E(r) e^−iωt] and throughout the paper we use units ħ = c = 1. The complex field can equivalently be characterized by its Fourier spectrum $\tilde {\mathbf{E}}(\mathbf{k})$:

$$\begin{eqnarray} &&\mathbf{E}(\mathbf{r})\propto \int \tilde {\mathbf{E}}(\mathbf{k})\hspace{0.167em} {\mathrm{e}}^{\mathrm{i}\mathbf{k}\boldsymbol{\cdot }\mathbf{r}}\hspace{0.167em} {\mathrm{d}}^{2}{\mathbf{k}}_{\perp }. \end{eqnarray} \tag{ 2.1 }$$

Here d²k_⊥ = dk_x dk_y, ${k}_{z}({\mathbf{k}}_{\perp })=\sqrt{{k}^{2}-{k}_{x}^{2}-{k}_{y}^{2}}$, and we neglect the evanescent waves [55]. The plane-wave Fourier amplitudes $\tilde {\mathbf{E}}(\mathbf{k})$ determine the momentum representation of the field. We will also use the unit polarization vectors of the fields: e = E/E and $\tilde {\mathbf{e}}=\tilde {\mathbf{E}}/\tilde {E}$.

From the Maxwell equation ∇⋅E = 0 in a homogeneous isotropic medium, it follows that

$$\begin{eqnarray} &&\mathbf{k}\boldsymbol{ \cdot }\tilde {\mathbf{E}}=0. \end{eqnarray} \tag{ 2.2 }$$

This is the transversality condition which compels the electric field of a plane wave, $\tilde {\mathbf{E}}$, to be orthogonal to its wavevector k. This constraint couples the polarization and momentum of light, which is a typical feature of the spin–orbit interaction. The condition (2.2) has a natural geometrical representation: the wave electric field must be tangent to the surface of the unit sphere of directions, S² = {κ}, κ = k/k, in momentum k-space, figure 2.

Alongside the 'absolute' field vectors E and $\tilde {\mathbf{E}}$, which are independent of the choice of the coordinate frame, we introduce 'vectors of components', |E)_A and $\vert \tilde {\mathbf{E}})_{A}$, i.e., field components in the given coordinate frame A. For instance, in the global Cartesian frame (x,y,z), the laboratory frame, we have

$$\begin{eqnarray} &&\mathbf{E}={E}_{x}{\mathbf{u}}_{x}+{E}_{y}{\mathbf{u}}_{y}+{E}_{z}{\mathbf{u}}_{z},\tqs \vert \mathbf{E})_{\mathrm{L}}=\left (\begin{array}{@{}c@{}} \displaystyle {E}_{x}\\ \displaystyle {E}_{y}\\ \displaystyle {E}_{z} \end{array}\right ), \end{eqnarray} \tag{ 2.3 }$$

where u_α stands for the unit basis vector of the corresponding α-axis. We will also use the transverse vector components considered with respect to the z-axis of the frame in use and denoted by the '⊥' subscript: |E)_⊥L = (E_x,E_y)^T, r_⊥ = xu_x + yu_y, etc.

Components of the same vector in different bases are related by a unitary transformation which links the basic vectors of the frames:

$$\begin{eqnarray} &&\left (\begin{array}{@{}c@{}} \displaystyle {E}_{x}^{\prime}\\ \displaystyle {E}_{y}^{\prime}\\ \displaystyle {E}_{z}^{\prime} \end{array}\right )=\hat {U}\left (\begin{array}{@{}c@{}} \displaystyle {E}_{x}\\ \displaystyle {E}_{y}\\ \displaystyle {E}_{z} \end{array}\right ),\tqs \left (\begin{array}{@{}c@{}} \displaystyle {\mathbf{u}}_{x}^{\prime}\\ \displaystyle {{\mathbf{u}}^{\prime}}_{y}\\ \displaystyle {{\mathbf{u}}^{\prime}}_{z} \end{array}\right )=\hat {U}\left (\begin{array}{@{}c@{}} \displaystyle {\mathbf{u}}_{x}\\ \displaystyle {\mathbf{u}}_{y}\\ \displaystyle {\mathbf{u}}_{z} \end{array}\right ). \end{eqnarray} \tag{ 2.4 }$$

For instance, in the 'laboratory circular basis' $({\mathbf{u}}_{0}^{+},{\mathbf{u}}_{0}^{-},{\mathbf{u}}_{z})$ of the circular polarizations in the (x,y) plane, with ${\mathbf{u}}_{0}^{\pm }=({\mathbf{u}}_{x}\pm \mathrm{i}{\mathbf{u}}_{y})/\sqrt{2}$, one has ${E}_{0}^{\pm }=({E}_{x}\mp \mathrm{i}{E}_{y})/\sqrt{2}$, i.e., 

$$\begin{eqnarray} \vert \mathbf{E})_{\mathrm{C}}=\hat {V}\vert \mathbf{E})_{\mathrm{L}},\tqs \hat {V}=\frac{1}{\sqrt{2}}\left (\begin{array}{@{}ccc@{}} \displaystyle 1&\displaystyle -\mathrm{i}&\displaystyle 0\\ \displaystyle 1&\displaystyle \mathrm{i}&\displaystyle 0\\ \displaystyle 0&\displaystyle 0&\displaystyle \sqrt{2} \end{array}\right ). \end{eqnarray} \tag{ 2.5 }$$

For the wave propagating along the z-axis, k = ku_z, the polarization vectors of right- and left-circularly polarized fields (σ =± 1) are given by:

$$\begin{eqnarray} &&\begin{array}{@{}c@{}} \displaystyle \vert {\tilde {\mathbf{E}}}^{\sigma })_{\mathrm{C}}\propto \vert { \boldsymbol {\mathcal {e}} }^{\sigma }),\tqs \vert { \boldsymbol {\mathcal {e}} }^{+})=(1,0,0)^{\mathrm{T}},\\ \displaystyle \vert { \boldsymbol {\mathcal {e}} }^{-})=(0,1,0)^{\mathrm{T}}. \end{array} \end{eqnarray} \tag{ 2.6 }$$

Here σ represents the helicity quantum number, which determines the spin states of photons. Polarizations vectors equation (2.6) are eigenvectors of the diagonal helicity operator $\hat {\sigma }$:

$$\begin{eqnarray} &&\begin{array}{@{}c@{}} \displaystyle \hat {\sigma }=\mathrm{diag}(1,-1,0),\tqs \hat {\sigma }\vert { \boldsymbol {\mathcal {e}} }^{\sigma })=\sigma \vert { \boldsymbol {\mathcal {e}} }^{\sigma }),\\ \displaystyle \sigma =\pm 1. \end{array} \end{eqnarray} \tag{ 2.7 }$$

The third eigenvector of $\hat {\sigma }$, |ℯ_z) = (0,0,1)^T, with eigenvalue σ = 0, corresponds to the longitudinal polarization prohibited by equation (2.2).

Since the transversality condition (2.2) attaches the generic wave electric field $\tilde {\mathbf{E}}$ to the surface of the κ-sphere, it is natural to describe the polarization evolution in the spherical coordinates (θ,ϕ,k) in momentum space (see figure 2). Their basic vectors are

$$\begin{eqnarray} &&\begin{array}{@{}c@{}} \displaystyle {\mathbf{u}}_{\theta }(\mathbf{k})={\mathbf{u}}_{\phi }(\mathbf{k})\times \boldsymbol{\kappa},\tqs {\mathbf{u}}_{\phi }(\mathbf{k})=\frac{{\mathbf{u}}_{z}\times \boldsymbol{\kappa}}{\vert {\mathbf{u}}_{z}\times \boldsymbol{\kappa}\vert },\\ \displaystyle {\mathbf{u}}_{k}(\mathbf{k})=\boldsymbol{\kappa}. \end{array} \end{eqnarray} \tag{ 2.8 }$$

Transformation between the laboratory and spherical frames can be represented by the product of two rotations through the angles θ and ϕ, ${\hat {U}}_{\mathrm{S}}={\hat {R}}_{y}(\theta ){\hat {R}}_{z}(\phi )$:

$$\begin{eqnarray} &&\begin{array}{@{}c@{}} \displaystyle \vert \tilde {\mathbf{E}})_{\mathrm{S}}={\hat {U}}_{\mathrm{S}}(\mathbf{k})\vert \tilde {\mathbf{E}})_{\mathrm{L}},\\ \displaystyle {\hat {U}}_{\mathrm{S}}=\left (\begin{array}{@{}ccc@{}} \displaystyle \cos \theta \cos \phi &\displaystyle \cos \theta \sin \phi &\displaystyle -\sin \theta \\ \displaystyle -\sin \phi &\displaystyle \cos \phi &\displaystyle 0\\ \displaystyle \sin \theta \cos \phi &\displaystyle \sin \theta \sin \phi &\displaystyle \cos \theta \end{array}\right ). \end{array} \end{eqnarray} \tag{ 2.9 }$$

Here we introduced the operators of SO(3) rotations: ${\hat {R}}_{\alpha }(\delta )=\exp (\mathrm{i}\delta {\hat {S}}_{\alpha })$, α = x,y,z, where $({\hat {S}}_{\alpha })_{i j}=-\mathrm{i}{\varepsilon }_{\alpha i j}$ (ε_αij is the Levi-Civita symbol) are the generators of rotations, i.e., spin-1 matrices. In spherical coordinates, the transversality condition (2.2) becomes particularly simple: ${\tilde {E}}_{k}=0$, so that the field is two-component: $\vert \tilde {\mathbf{E}})_{\mathrm{S}}=({\tilde {E}}_{\theta },{\tilde {E}}_{\phi },0)^{\mathrm{T}}$.

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Spherical coordinates provide a description of a plane wave in the field spectrum in the local basis attached to its k-vector. Akin to the laboratory circular-polarization basis ${\mathbf{u}}_{0}^{\sigma }$, the circular polarizations defined with respect to the spherical vectors form the helicity basis [82, 83]:

$$\begin{eqnarray} &&{\mathbf{u}}^{\sigma }=\frac{{\mathbf{u}}_{\theta }+\mathrm{i}\sigma {\mathbf{u}}_{\phi }}{\sqrt{2}},\tqs \sigma =\pm 1. \end{eqnarray} \tag{ 2.10 }$$

This basis has a polar singularity at θ = 0. In the paraxial limit θ → 0 (propagation along the z-axis), the basic vectors u^σ tend to the laboratory circular-polarization basis with an additional azimuthal rotation: ${\mathbf{u}}^{\sigma }\rightarrow {\mathbf{u}}_{0}^{\sigma }\hspace{0.167em} {\mathrm{e}}^{-\mathrm{i}\sigma \phi }$.⁵ Transformation from the laboratory circular frame (2.5) to the helicity basis (2.10) is given by the unitary matrix ${\hat {U}}_{\mathrm{H}}(\mathbf{k})=\hat {V}{\hat {R}}_{y}(\theta ){\hat {R}}_{z}(\phi ){\hat {V}}^{\dagger }$.

Any polarization state is defined up to a common phase. In the case of circular polarization, the phase can be induced by a rotation of the coordinate frame about the k-vector. For instance, if a plane wave propagates along the z-axis, rotation of the coordinates (x,y) by the angle −Φ, $\vert {\mathbf{E}}^{\prime})_{\mathrm{L}}={\hat {R}}_{z}(-\Phi )\vert \mathbf{E})_{\mathrm{L}}$, induces the geometric phase σΦ in the circular field components:

$$\begin{eqnarray} &&\begin{array}{@{}c@{}} \displaystyle {\mathbf{u}}_{0}^{{\sigma }^{\prime}}={\mathrm{e}}^{\mathrm{i}\sigma \Phi }\hspace{0.167em} {\mathbf{u}}_{0}^{\sigma },\tqs \vert {\mathbf{E}}^{\prime})_{\mathrm{C}}=\exp (-\mathrm{i}\hat {\sigma }\Phi )\vert \mathbf{E})_{\mathrm{C}},\\ \displaystyle {\mathrm{e}}^{-\mathrm{i}\hat {\sigma }\Phi }=\left (\begin{array}{@{}ccc@{}} \displaystyle {\mathrm{e}}^{-\mathrm{i}\Phi }&\displaystyle 0&\displaystyle 0\\ \displaystyle 0&\displaystyle {\mathrm{e}}^{\mathrm{i}\Phi }&\displaystyle 0\\ \displaystyle 0&\displaystyle 0&\displaystyle 1 \end{array}\right ). \end{array} \end{eqnarray} \tag{ 2.11 }$$

This is the simplest 2D example of the spin-rotation coupling [84–87].

In the general 3D case, polarizations of different waves are defined in the different planes orthogonal to the wavevectors k, i.e., tangent to the surface of the κ-sphere, figure 2. Hence, to compare the phases and polarizations of plane waves with different k-vectors, one has to transport their electric fields over the surface of sphere in order to bring them to the same plane. The geometric parallel transport of vectors tangent to the sphere is defined in such a way that the vector does not experience local rotation about the normal κ-vector. However, the parallel transport frame cannot be defined globally on the sphere, and any globally-defined coordinate system on the sphere inevitably experiences local rotations of its axes about the κ-vector and induces geometric phases for circularly polarized modes. In particular, for the spherical coordinates (2.8) and helicity basic vectors (2.10), the corresponding differential angle of rotation and the geometric phase reads [88–91]:

$$\begin{eqnarray} &&-\sigma \hspace{0.167em} \mathrm{d}\Phi =\mathrm{i}{\mathbf{u}}^{\sigma \ast }\boldsymbol{\cdot }\mathrm{d}{\mathbf{u}}^{\sigma }=\mathrm{i}\mathop{\sum }\limits _{j}\left [{\mathbf{u}}^{\sigma \ast }(\mathbf{k})\boldsymbol{ \cdot }\left (\frac{\partial }{\partial {k}_{j}}\right ){\mathbf{u}}^{\sigma }(\mathbf{k})\right ]\hspace{0.167em} \mathrm{d}{k}_{j}.\nonumber\\ \end{eqnarray} \tag{ 2.12 }$$

For the evolution of the field along a contour C between k and k' in momentum space (owing to the transversality, it can always be projected onto the κ-sphere), integration of equation (2.12) results in the accumulation of the geometric phase (2.11):

$$\begin{eqnarray} &&\begin{array}{@{}c@{}} \displaystyle \vert {\tilde {\mathbf{E}}}^{\prime})_{\mathrm{H}}=\exp (-\mathrm{i}\hat {\sigma }{\Phi }_{\mathrm{B}})\vert \tilde {\mathbf{E}})_{\mathrm{H}},\\ \displaystyle {\Phi }_{\mathrm{B}}(C)=\int _{C}{\mathbf{A}}_{\mathrm{B}}(\mathbf{k})\boldsymbol{ \cdot }\mathrm{d}\mathbf{k}=-\int \cos \theta \hspace{0.167em} \mathrm{d}\phi , \end{array} \end{eqnarray} \tag{ 2.13 }$$

$$\begin{eqnarray} &&{\mathbf{A}}_{\mathrm{B}}=-\mathrm{i}{\sigma }^{-1}{\mathbf{u}}^{\sigma \ast }\boldsymbol{\cdot }\left (\frac{\partial }{\partial \mathbf{k}}\right ){\mathbf{u}}^{\sigma }=-{k}^{-1}\cot \theta \hspace{0.167em} {\mathbf{u}}_{\phi }. \end{eqnarray} \tag{ 2.14 }$$

Here Φ_B is the Berry geometric phase described by the momentum space contour integral of the Berry connection A_B(k) which was calculated from equations (2.8), (2.10), and (2.12). It follows from equation (2.13) that Berry phase difference between the waves with a fixed polar angle θ = const distributed in the range of azimuthal angles (0,ϕ) is equal to

$$\begin{eqnarray} &&{\Phi }_{\mathrm{B}}=-\phi \cos \theta . \end{eqnarray} \tag{ 2.15 }$$

An example of the Berry phase Φ_B =− 2πcosθ caused by the parallel transport of the field along a closed contour of evolution C = {θ = const,ϕ∈(0,2π)} is shown in figure 2.

Berry phase (2.13) has been extensively studied in various optical systems (for reviews, see [89–93]). It is associated with the variations of the k-vector and becomes significant in globally nonparaxial optical systems involving 3D distributions of the wavevectors. One can distinguish two typical situations: (i) successive nonplanar variations of the direction of propagation of a polarized wave (in multiple reflections, refractions, curved optical fibres, etc) [89–93]; and (ii) simultaneous interference involving multiple polarized waves with different k-vectors (e.g., tightly focused or scattered nonparaxial fields) [83, 87]. As we will see, the Berry phase of the second type underlies the spin Hall effect of light, i.e., the IF shifts of polarized beams.

The Berry phase also allows simple dynamical interpretations. Upon the evolution of the k-vector, the electric field $\tilde {\mathbf{E}}(\mathbf{k})$ possesses a sort of inertia and remains locally non-rotating about k. Observation in a globally-defined spherical frame produces a Coriolis effect caused by local rotations of the basis. This is expressed in the simple 'dynamical' formulae for the Berry phase [87, 91, 94]:

$$\begin{eqnarray} &&\sigma {\Phi }_{\mathrm{B}}=-\sigma \int \boldsymbol{\kappa}\boldsymbol{ \cdot }{\boldsymbol{\Omega}}_{\tau }\hspace{0.167em} \mathrm{d}\tau . \end{eqnarray} \tag{ 2.16 }$$

Here τ is a parameter underlying the evolution of waves in momentum space (this can be time, a coordinate in real space, etc) and Ω_τ is the angular velocity of rotation of the coordinate frame defined with respect to this parameter τ. In particular, waves with fixed polar angle θ = const but different azimuthal angles ϕ on the κ-sphere are related via rotation by the angle ϕ about the z-axis, i.e., Ω_ϕ = u_z and Φ_B =− ∫κ ⋅u_z dϕ =− ∫cosθ dϕ [87, 91, 94], cf equation (2.13). Equation (2.16) represents the Berry phase as a result of the spin-rotation coupling between the spin AM σκ and the rotation Ω_τ. In this spirit, it is completely analogous to the Coriolis effect [84–87] and the rotational Doppler effect for waves carrying intrinsic AM [95–98].

While the Berry phase describes the spin–orbit interaction phenomena on the level of geometric interference of plane waves forming the field, the coordinate and AM represent integral dynamical characteristics of the localized wavepackets or beams. They are naturally described within a quantum-like operator formalism.

The operators for the coordinate, momentum, and energy of a photon are written in the momentum representation as:

$$\begin{eqnarray} &&\hat {\mathbf{r}}=\mathrm{i}\frac{\partial }{\partial \mathbf{k}},\tqs \hat {\mathbf{p}}=\mathbf{k},\tqs \hat {w}=\omega , \end{eqnarray} \tag{ 2.17 }$$

whereas the canonical orbital and spin AM operators are known to be [99]

$$\begin{eqnarray} &&\hat {\mathbf{L}}=-\mathrm{i}\left (\mathbf{k}\times \frac{\partial }{\partial \mathbf{k}}\right ),\tqs ({\hat {S}}_{\alpha })_{i j}=-\mathrm{i}{\varepsilon }_{\alpha i j}. \end{eqnarray} \tag{ 2.18 }$$

The orbital and spin AM, $\hat {\mathbf{L}}$ and $\hat {\mathbf{S}}$, represent differential and matrix operators that act in Hilbert-momentum and vector-polarization spaces, respectively⁶. The spin operator $\hat {\mathbf{S}}$ consists of the generators of SO(3) rotations that act on the components of the field in the laboratory frame, $\vert \tilde {\mathbf{E}})_{\mathrm{L}}$.

The amounts of energy, momentum, and AM carried by a light beam propagating along the z-axis are characterized by the linear densities of these quantities per unit z-length [58, 101]. In this manner, the normalized linear densities of the energy, 〈W〉, momentum, 〈P〉, orbital AM, 〈L〉, and spin AM, 〈S〉, as well as the coordinates of the field centroid, 〈R〉, can be calculated as expectation values of the operators (2.17) and (2.18):

$$\begin{eqnarray} &&\begin{array}{@{}c@{}} \displaystyle \langle \mathbf{R}\rangle ={N}^{-1}\langle \tilde {\mathbf{E}}\vert \hat {\mathbf{r}}\vert \tilde {\mathbf{E}}\rangle _{\mathrm{L}},\tqs \langle \mathbf{P}\rangle ={N}^{-1}\langle \tilde {\mathbf{E}}\vert \mathbf{k}\vert \tilde {\mathbf{E}}\rangle _{\mathrm{L}},\\ \displaystyle \langle W\rangle ={N}^{-1}\langle \tilde {\mathbf{E}}\vert \hat {w}\vert \tilde {\mathbf{E}}\rangle _{\mathrm{L}}.\\ \displaystyle \end{array} \end{eqnarray} \tag{ 2.19 }$$

$$\begin{eqnarray} &&\langle \mathbf{L}\rangle ={N}^{-1}\langle \tilde {\mathbf{E}}\vert \hat {\mathbf{L}}\vert \tilde {\mathbf{E}}\rangle _{\mathrm{ L}},\tqs \langle \mathbf{S}\rangle ={N}^{-1}\langle \tilde {\mathbf{E}}\vert \hat {\mathbf{S}}\vert \tilde {\mathbf{E}}\rangle _{\mathrm{ L}}.\nonumber\\ \end{eqnarray} \tag{ 2.20 }$$

Here $N=\langle \tilde {\mathbf{E}}\vert \tilde {\mathbf{E}}\rangle _{\mathrm{L}}$ is the norm (the number of photons per unit z-length), whereas the state vector $\vert \tilde {\mathbf{E}}\rangle _{\mathrm{L}}=\vert \tilde {\mathbf{E}}({\mathbf{k}}_{\perp }))_{\mathrm{L}}\exp [\mathrm{i}{k}_{z}({\mathbf{k}}_{\perp })z]$ is proportional to the field polarization vector $\vert \tilde {\mathbf{E}})_{\mathrm{L}}$ but is also considered as a vector in the Hilbert space [55, 83]. Correspondingly, the 'bra-ket' inner product implies the scalar product of complex polarizations together with the integration over the momentum space: $\langle \tilde {\mathbf{E}}\vert {\tilde {\mathbf{E}}}^{\prime}\rangle _{\mathrm{L}}=\int (\tilde {\mathbf{E}}\vert {\tilde {\mathbf{E}}}^{\prime})_{\mathrm{L}}\hspace{0.167em} {\mathrm{d}}^{2}{\mathbf{k}}_{\perp }$.

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While the spin AM 〈S〉 is purely intrinsic (i.e., independent of the origin), the orbital AM of light can be divided into extrinsic (origin-dependent) and intrinsic contributions, 〈L^ext〉 and 〈L^int〉 [83, 102]. The extrinsic contribution is determined by the motion of the centroid of the field and is given by the 'mechanical' cross-product of the central coordinate and momentum [40, 41, 50–52, 61–63, 83]:

$$\begin{eqnarray} &&\langle {\mathbf{L}}^{\mathrm{ext}}\rangle =\langle \mathbf{R}\rangle \times \langle \mathbf{P}\rangle ,\tqs \langle {\mathbf{L}}^{\mathrm{int}}\rangle =\langle \mathbf{L}\rangle -\langle {\mathbf{L}}^{\mathrm{ext}}\rangle .\nonumber\\ \end{eqnarray} \tag{ 2.21 }$$

Thus, there are three types of the AM of light: (i) spin, (ii) intrinsic orbital, and (iii) extrinsic orbital AM. For the locally paraxial fields they are associated, respectively, with circular polarization, optical vortices, and transverse beam shifts, figure 3. If the medium possesses rotational symmetry about a certain axis, the corresponding component of the total AM, 〈L〉 + 〈S〉 = 〈L^int〉 + 〈L^ext〉 + 〈S〉, is conserved upon evolution of light. However, mutual conversion between different parts of the AM is possible, which signals the spin–orbit (or orbit–orbit) interaction of light. In a similar way, if the medium is stationary and possesses translational symmetry about a certain axis, the energy 〈W〉 and corresponding component of the momentum 〈P〉 must be conserved upon light evolution.

Considering paraxial beams propagating along the z-axis, θ ≪ 1, we neglect small longitudinal z-components of the field and deal with the transverse (x,y)-components denoted by $\vert \tilde {\mathbf{E}})_{\perp }$. As an example, let us consider paraxial circularly polarized Laguerre–Gaussian vortex beams with the azimuthal quantum number ℓ = 0, ±1, ±2, ... and radial quantum number p = 0 [58]. The electric field in the laboratory basis of circular polarizations can be written as:

$$\begin{eqnarray} &&\begin{array}{@{}c@{}} \displaystyle \vert {\tilde {\mathbf{E}}}_{\ell }^{\sigma }(\mathbf{k})\rangle _{\perp \mathrm{C}}=\vert { \boldsymbol {\mathcal {e}} }^{\sigma })_{\perp }\vert {\tilde {E}}_{\ell }(\mathbf{k})\rangle ,\\ \displaystyle \vert {\tilde {E}}_{\ell }(\mathbf{k})\rangle ={\theta }^{\vert \ell \vert }G(\theta )\hspace{0.167em} {\mathrm{e}}^{\mathrm{i}\ell \phi +\mathrm{i}k(1-{\theta }^{2}/2)\hspace{0.167em} z}, \end{array} \end{eqnarray} \tag{ 2.22 }$$

where G(θ) ∝ exp[ − (kw₀)²θ²/4] is the Gaussian envelope function with w₀ ≫ k⁻¹ being the beam waist. The azimuthal phase factor exp(iℓϕ) in equation (2.22) represents the optical vortex of charge ℓ. Note that the spin and orbital degrees of freedom are factorized here into the constant polarization vector |ℯ^σ) in the real space and scalar field $\vert {\tilde {E}}_{\ell }(\mathbf{k})\rangle$ as a vector in the Hilbert space. Substituting field (2.22) into equations (2.19) and (2.20), we obtain the expectation values:

$$\begin{eqnarray} &&\begin{array}{@{}c@{}} \displaystyle \langle \mathbf{R}\rangle =z\mathbf{K},\tqs \langle \mathbf{P}\rangle =k\mathbf{K},\tqs \langle W\rangle =\omega ,\\ \displaystyle \langle \mathbf{L}\rangle =\ell \hspace{0.167em} \mathbf{K},\tqs \langle \mathbf{S}\rangle =\sigma \mathbf{K} \end{array} \end{eqnarray} \tag{ 2.23 }$$

where K = u_z is the direction of propagation of the beam and we took into account that the spin operator in the basis of circular polarizations (2.5) is $(\hat {\mathbf{S}})_{\mathrm{C}}=\hat {V}\hat {\mathbf{S}}{\hat {V}}^{\dagger }$. Equations (2.23) represent well-known results for the intrinsic orbital and spin AM of the paraxial vortex beams [57, 58] (see figure 3). It is easy to see that beams (2.22) are eigenmodes of the ${\hat {L}}_{z}$ and $({\hat {S}}_{z})_{\mathrm{C}}$ operators with the discrete eigenvalues ℓ and σ. Indeed, these operators read

$$\begin{eqnarray} &&{\hat {L}}_{z}=-\mathrm{i}\frac{\partial }{\partial \phi },\tqs ({\hat {S}}_{z})_{\mathrm{C}}=\hat {V}{\hat {S}}_{z}{\hat {V}}^{\dagger }=\hat {\sigma }, \end{eqnarray} \tag{ 2.24 }$$

so that vortices exp(iℓϕ) and polarizations |e^σ), equations (2.6) and (2.7), represent their eigenmodes. Obviously, the extrinsic AM (2.21) vanishes for the beam (2.22) and (2.23): 〈L^ext〉 = 0. However, mutually orthogonal transverse shift and tilt of the beam would generate nonzero extrinsic AM 〈L^ext〉 = 〈R〉 × 〈P〉 ≠ 0.

Note that transverse part of the helicity operator (2.7) represents the Pauli matrix, ${\hat {\sigma }}_{\perp }=\mathrm{diag}(1,-1)={\hat {\sigma }}_{3}$, with the corresponding eigenvectors of circular polarizations: |ℯ⁺)_⊥ = (1,0)^T and |ℯ⁻)_⊥ = (0,1)^T. An arbitrary uniform polarization state of the paraxial field is described by the complex unit Jones vector |e)_⊥C = (e⁺,e⁻)^T, |e⁺|² + |e⁻|² = 1, obeying SU(2) symmetry of a two-level system. For the beam with such polarization, the helicity σ in equations (2.23) is replaced by the mean helicity $\bar {\sigma }=(\mathbf{e}\vert {\hat {\sigma }}_{3}\vert \mathbf{e})_{\perp \mathrm{C}}=\vert {e}^{+}\vert ^{2}-\vert {e}^{-}\vert ^{2}$. Thus, the Pauli matrix ${\hat {\sigma }}_{3}$ underpins the degree of the circular polarization responsible for the spin AM carried by the field. The complete characterization of the polarization requires the use of all the Pauli matrices $\hat {\vec{\sigma }}=({\hat {\sigma }}_{1},{\hat {\sigma }}_{2},{\hat {\sigma }}_{3})$, where ${\hat {\sigma }}_{1}$ and ${\hat {\sigma }}_{2}$ are unrelated to the AM [103]. A uniform polarization state is completely characterized by the expectation values

$$\begin{eqnarray} &&\vec{\frak{S}}=(\tilde {\mathbf{e}}\vert \hat {\vec{\sigma }}\vert \tilde {\mathbf{e}})_{\perp \mathrm{C}}, \end{eqnarray} \tag{ 2.25 }$$

which form the normalized Stokes vector [104, 105]. It can be considered as a pseudo-spin which represents the SU(2) polarization state on the abstract SO(3) Poincaré sphere—an analogue of the Bloch sphere. The pure helicity states |ℯ^σ)_⊥ correspond to the poles of the Poincaré sphere, and the third component of the Stokes vectors determines the mean helicity: ${\frak{S}}_{3}=\bar {\sigma }$. Note also that in the basis of linear polarizations, the Stokes vector components are determined by equation (2.25) with the permuted Pauli matrices $\hat {{\vec{\sigma }}^{\prime}}=({\hat {\sigma }}_{3},{\hat {\sigma }}_{1},{\hat {\sigma }}_{2})$:

$$\begin{eqnarray} &&\vec{\frak{S}}=(\tilde {\mathbf{e}}\vert \hat {{\vec{\sigma }}^{\prime}}\vert \tilde {\mathbf{e}})_{\perp \mathrm{L}}. \end{eqnarray} \tag{ 2.26 }$$

In this section we examine reflection and transmission of polarized Gaussian-type beams without intrinsic orbital AM, i.e., without vortex: ℓ = 0. The geometry of the problem is depicted in figures 1 and 4. A paraxial optical beam propagates in the (x,z) plane at an angle ϑ to the z-axis and undergoes partial reflection and refraction at the plane interface z = 0 separating two non-absorbing dielectric media. The interface is characterized by the relative refractive index of the second medium, n, and its relative impedance ζ.

In the geometrical-optics description, the beams are associated with their central plane waves, which have wavevectors ${\mathbf{k}}_{\mathrm{c}}^{a}$. Hereafter index a = i,r,t denotes incident, reflected, and transmitted beams, respectively, and the index i is omitted in the explicit expressions: ${\mathbf{k}}_{\mathrm{c}}^{\mathrm{i}}\equiv {\mathbf{k}}_{\mathrm{c}}$, etc. The wavenumbers in the three beams are k^r = kⁱ ≡ k and k^t = nk. From Snell's law (which represents conservation of the tangent momentum components), it follows that the central wavevectors of all the three beams lie in the same (x,z) plane: ${\mathbf{k}}_{\mathrm{c}}^{a}={k}^{a}(\sin {\vartheta }^{a},0,\cos {\vartheta }^{a})$ and form angles [1]

$$\begin{eqnarray} &&\begin{array}{@{}c@{}} \displaystyle {\vartheta }^{\mathrm{i}}=\vartheta ,\tqs {\vartheta }^{\mathrm{r}}=\pi -\vartheta ,\\ \displaystyle {\vartheta }^{\mathrm{t}}={\sin }^{-1}({n}^{-1}\sin \vartheta )\equiv {\vartheta }^{\prime}, \end{array} \end{eqnarray} \tag{ 3.1 }$$

with the z-axis. It is natural to introduce coordinate frames of individual beams (X^a,y,Z^a) with the Z^a-axes attached to their directions of propagation: ${\mathbf{k}}_{\mathrm{c}}^{a}={k}^{a}{\mathbf{u}}_{Z}^{a}$ (see figure 4). The beam frames are obtained via rotation of the laboratory coordinate frame (x,y,z) by the angle ϑ^a about the y-axis:

$$\begin{eqnarray} &&\begin{array}{@{}c@{}} \displaystyle \left (\begin{array}{@{}c@{}} \displaystyle {\mathbf{u}}_{X}^{a}\\ \displaystyle {\mathbf{u}}_{y}\\ \displaystyle {\mathbf{u}}_{Z}^{a} \end{array}\right )={\hat {R}}_{y}({\vartheta }^{a})\left (\begin{array}{@{}c@{}} \displaystyle {\mathbf{u}}_{x}\\ \displaystyle {\mathbf{u}}_{y}\\ \displaystyle {\mathbf{u}}_{z} \end{array}\right ),\\ \displaystyle {\hat {R}}_{y}(\vartheta )=\left (\begin{array}{@{}ccc@{}} \displaystyle \cos \vartheta &\displaystyle 0&\displaystyle -\sin \vartheta \\ \displaystyle 0&\displaystyle 1&\displaystyle 0\\ \displaystyle \sin \vartheta &\displaystyle 0&\displaystyle \cos \vartheta \end{array}\right ). \end{array} \end{eqnarray} \tag{ 3.2 }$$

The electric field of the incident central plane wave can be written as E_⊥ = e_Xu_X + e_yu_y, where we assumed normalization |E_⊥|² = |e_X|² + |e_y|² = 1. The field components form the effective normalized Jones vector in the beam frame, |E)_⊥B = |e)_⊥B = (e_X,e_y)^T, and for the central waves the X–y basis coincides with the basis of TM (p) and TE (s) modes with respect to the interface. The fields of the secondary beams are determined by the Fresnel reflection and transmission coefficients, R_p,s(ϑ) and T_p,s(ϑ) for the p and s modes [1]. Namely, implying natural transitions between the corresponding beam coordinate frames $({\mathbf{u}}_{X}^{a},{\mathbf{u}}_{y},{\mathbf{u}}_{Z}^{a})$, the transverse electric fields of the secondary beams are obtained by application of the effective Fresnel Jones matrix ${\hat {F}}^{a}$:

$$\begin{eqnarray} \vert {\mathbf{E}}^{a})_{\perp \mathrm{B}}={\hat {F}}^{a}\vert \mathbf{E})_{\perp \mathrm{B}},\tqs {\hat {F}}^{a}=\left (\begin{array}{@{}cc@{}} \displaystyle {f}_{\mathrm{ p}}^{a}&\displaystyle 0\\ \displaystyle 0&\displaystyle {f}_{\mathrm{ s}}^{a} \end{array}\right ), \end{eqnarray} \tag{ 3.3 }$$

where we denoted f^i,r,t ≡ 1,R,T. The normalized Jones vectors for the beams take the form:

$$\begin{eqnarray} &&\vert {\mathbf{e}}^{a})_{\perp \mathrm{B}}={Q}^{{a}^{-1}}\vert {\mathbf{E}}^{a})_{\perp \mathrm{B}},\tqs {Q}^{{a}^{2}}=({\mathbf{E}}^{a}\vert {\mathbf{E}}^{a})_{\perp \mathrm{B}}. \end{eqnarray} \tag{ 3.4 }$$

Here ${Q}^{a}=\sqrt{\vert {f}_{\mathrm{p}}^{a}\vert ^{2}\vert {e}_{X}\vert ^{2}+\vert {f}_{\mathrm{s}}^{a}\vert ^{2}\vert {e}_{y}\vert ^{2}}$ are the amplitude coefficients of the fields, whereas their energy coefficients, satisfying the conservation law, are given by [1]:

$$\begin{eqnarray} &&\begin{array}{@{}c@{}} \displaystyle {\mathsf{Q}}^{a}={\zeta }^{a}\left \vert \frac{\cos {\vartheta }^{a}}{\cos \vartheta }\right \vert {Q}^{{a}^{2}},\\ \displaystyle {\mathsf{Q}}^{\mathrm{r}}+{\mathsf{Q}}^{\mathrm{t}}=\mathsf{Q}=1. \end{array} \end{eqnarray} \tag{ 3.5 }$$

Here ζ^a is the relative impedance of the medium, i.e., ζⁱ = ζ^r = 1 and ζ^t = ζ.

Equations (3.1)–(3.5) describe the Snell–Fresnel reflection and transmission of the central plane wave in the beam.

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Now, let us examine real beams possessing finite Fourier spectra with wavevectors k^a narrowly distributed around the central wavevectors ${\mathbf{k}}_{\mathrm{c}}^{a}$. For a monochromatic field, $\vert {\mathbf{k}}^{a}\vert =\vert {\mathbf{k}}_{\mathrm{c}}^{a}\vert ={k}^{a}$, and the wavevectors can be characterized by small orthogonal deflections χ^a:

$$\begin{eqnarray} &&\begin{array}{@{}c@{}} \displaystyle {\mathbf{k}}^{a}={\mathbf{k}}_{\mathrm{c}}^{a}+{\boldsymbol{\chi}}^{a},\tqs \vert {\boldsymbol{\chi}}^{a}\vert \ll {k}^{a},\\ \displaystyle {\boldsymbol{\chi}}^{a}\simeq {k}^{a}({\mu }^{a}{\mathbf{u}}_{X}^{a}+{\nu }^{a}{\mathbf{u}}_{y}). \end{array} \end{eqnarray} \tag{ 3.6 }$$

Thus, μ and ν specify the in-plane and out-of-plane deflections of non-central wavevectors, as shown in figure 5. First, Snell's law provides a connection between the components of the wavevectors (3.6):

$$\begin{eqnarray} &&\begin{array}{@{}c@{}} \displaystyle {\mu }^{a}=\frac{k}{{k}^{a}}{\gamma }^{a}\mu ,\tqs {\nu }^{a}=\frac{k}{{k}^{a}}\nu ,\\ \displaystyle {\gamma }^{a}=\frac{\cos \vartheta }{\cos {\vartheta }^{a}}. \end{array} \end{eqnarray} \tag{ 3.7 }$$

Here we introduced the coefficients of the elliptical deformations of the beams: γⁱ = 1, γ^r =− 1, and γ^t = cosϑ/cosϑ'. Second, the field components are related by the Fresnel coefficients, which are written for TM and TE plane waves, i.e., waves with the electric field being parallel and orthogonal to their planes of incidence. However, the plane of incidence of a wave with ν ≠ 0 does not coincide with the (x,z) plane, and, hence, the basis of p and s modes differs from the natural beam X–y basis (see figure 5). Indeed, the TM and TE modes with respect to the interface z = 0 are associated with polar and azimuthal polarizations along the basic vectors u_θ and u_ϕ of the global spherical coordinate frame, equations (2.8) and (2.9) (cf [74]). Thus, the deceptively simple problem of the reflection or refraction of a wave beam essentially involves three coordinate frames for its adequate description: (i) the laboratory frame, (ii) the beam frames, and (iii) the spherical frame of the TM and TE modes.

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Let the beam spectra be given in the (X^a,y,Z^a) coordinates as $\vert {\tilde {\mathbf{E}}}^{a}({\mathbf{k}}^{a}))_{\mathrm{B}}$. Transition to the laboratory frame (x,y,z) is realized by inverse rotations (3.2) ${\hat {R}}_{y}(-{\vartheta }^{a})$, $\vert {\tilde {\mathbf{E}}}^{a})_{\mathrm{L}}={\hat {R}}_{y}(-{\vartheta }^{a})\vert {\tilde {\mathbf{E}}}^{a})_{\mathrm{B}}$, whereas the next transitions to the global spherical coordinates are accomplished with the help of the transformation (2.9) ${\hat {U}}_{\mathrm{S}}({\mathbf{k}}^{a})={\hat {R}}_{y}({\theta }^{a}){\hat {R}}_{z}({\phi }^{a})$ : $\vert {\tilde {\mathbf{E}}}^{a})_{\mathrm{S}}={\hat {U}}_{\mathrm{S}}({\mathbf{k}}^{a})\vert {\tilde {\mathbf{E}}}^{a})_{\mathrm{L}}$. Here the spherical angles (θ^a,ϕ^a) determine the direction of k^a. The resulting rotational transformation to the basis of p and s modes is

$$\begin{eqnarray} &&\begin{array}{@{}c@{}} \displaystyle \vert {\tilde {\mathbf{E}}}^{a})_{\mathrm{S}}={\hat {U}}_{\vartheta }({\vartheta }^{a},{\mathbf{k}}^{a})\vert {\tilde {\mathbf{E}}}^{a}({\mathbf{k}}^{a}))_{\mathrm{B}},\\ \displaystyle {\hat {U}}_{\vartheta }({\vartheta }^{a},{\mathbf{k}}^{a})={\hat {R}}_{y}({\theta }^{a}){\hat {R}}_{z}({\phi }^{a}){\hat {R}}_{y}(-{\vartheta }^{a}). \end{array} \end{eqnarray} \tag{ 3.8 }$$

For the central plane wave ${\mathbf{k}}^{a}={\mathbf{k}}_{\mathrm{c}}^{a}$, we have (θ^a,ϕ^a) = (ϑ^a,0) and ${\hat {U}}_{\vartheta }({\vartheta }^{a},{\mathbf{k}}_{c}^{a})=\hat {1}$. For a non-central wave (3.6), small wavevector deflections μ^a and ν^a induce changes in θ^a and ϕ^a, respectively:

$$\begin{eqnarray} &&{\theta }^{a}\simeq {\vartheta }^{a}+{\mu }^{a},\tqs {\phi }^{a}\simeq \frac{{\nu }^{a}}{\sin {\vartheta }^{a}}. \end{eqnarray} \tag{ 3.9 }$$

Thus, the in-plane deflection μ changes the angle of incidence, whereas the out-of-plane deflection ν turns the plane of incidence by the angle v/sinϑ about the z axis (figure 5). Substituting equation (3.9) into (3.8), we obtain the rotation matrix ${\hat {U}}_{\vartheta }$ in the linear approximation in χ^a:

$$\begin{eqnarray} &&\fl {\hat {U}}_{\vartheta }({\vartheta }^{a},{\boldsymbol{\chi}}^{a})\simeq {\hat {R}}_{y}({\vartheta }^{a}+{\mu }^{a}){\hat {R}}_{z}({\nu }^{a}/\mathrm{sin}~{\vartheta }^{a}){\hat {R}}_{y}(-{\vartheta }^{a})\nonumber\\ \simeq \left (\begin{array}{@{}ccc@{}} \displaystyle 1&\displaystyle {\nu }^{a}\cot {\vartheta }^{a}&\displaystyle -{\mu }^{a}\\ \displaystyle -{\nu }^{a}\cot {\vartheta }^{a}&\displaystyle 1&\displaystyle -{\nu }^{a}\\ \displaystyle {\mu }^{a}&\displaystyle {\nu }^{a}&\displaystyle 1 \end{array}\right ). \end{eqnarray} \tag{ 3.10 }$$

In fact, only the transverse components of the fields are essential for the paraxial problem under consideration, which are described by the 2×2 upper left sector of equation (3.10). In the basis of linear and circular polarizations it yields:

$$\begin{eqnarray} \begin{array}{@{}l@{}} \displaystyle \fl {\hat {U}{}_{\vartheta }}_{\perp }\simeq \left (\begin{array}{@{}cc@{}} \displaystyle 1&\displaystyle -{\Phi }_{\mathrm{B}}^{a}\\ \displaystyle {\Phi }_{\mathrm{B}}^{a}&\displaystyle 1 \end{array}\right ),\\ \displaystyle \fl {\hat {V}}_{\perp }{\hat {U}{}_{\vartheta }}_{\perp }{\hat {V}}_{\perp }^{\dagger }\simeq \left (\begin{array}{@{}cc@{}} \displaystyle \exp (-\mathrm{i}{\Phi }_{\mathrm{B}}^{a})&\displaystyle 0\\ \displaystyle 0&\displaystyle \exp (\mathrm{i}{\Phi }_{\mathrm{B}}^{a}) \end{array}\right )=\exp (-\mathrm{i}{\hat {\sigma }}_{3}{\Phi }_{\mathrm{B}}^{a}). \end{array} \end{eqnarray} \tag{ 3.11 }$$

Here the basis of circular polarizations corresponds to the helicity basis (2.10), and

$$\begin{eqnarray} &&{\Phi }_{\mathrm{B}}^{a}=-{\nu }^{a}\cot {\vartheta }^{a}=-{\phi }^{a}\cos {\vartheta }^{a} \end{eqnarray} \tag{ 3.12 }$$

is the Berry geometric phase (2.13)–(2.16) induced by the azimuthal rotation of the plane of incidence by the angle (3.9) ϕ^a. The geometric phase matrix (3.11) represents the effective Jones matrix of the spin–orbit interaction caused by the transition from the beam coordinate frame to the global spherical frame. It is this free space rotational transformation that results in the transverse IF shifts, i.e., the spin Hall effect of light.

After transforming the fields to the global spherical basis of p and s modes, one can connect them via the Fresnel boundary conditions (3.3):

$$\begin{eqnarray} &&\begin{array}{@{}l@{}} \displaystyle \fl \vert {\tilde {\mathbf{E}}}^{a})_{\perp \mathrm{S}}={\hat {F}}^{a}\vert \tilde {\mathbf{E}})_{\perp \mathrm{S}},\\ \displaystyle \fl {\hat {F}}^{a}\simeq \left (\begin{array}{@{}cc@{}} \displaystyle {f}_{\mathrm{p}}^{a}\left (1+\mu \hspace{0.167em} \frac{\partial \ln {f}_{\mathrm{p}}^{a}}{\partial \vartheta }\right )&\displaystyle 0\\ \displaystyle 0&\displaystyle {f}_{\mathrm{s}}^{a}\left (1+\mu \hspace{0.167em} \frac{\partial \ln {f}_{\mathrm{s}}^{a}}{\partial \vartheta }\right ) \end{array}\right ). \end{array} \end{eqnarray} \tag{ 3.13 }$$

Here we took into account variations in the angle of incidence, equation (3.9): f(ϑ + μ) ≃ f(ϑ) + μ f'(ϑ). Corrections from the gradients of the Fresnel coefficients in equation (3.13) are responsible for the in-plane GH shifts of the beams.

Together, equations (3.6)–(3.13) yield the complete transformation and Jones matrix relating the incident and secondary fields in the beam coordinate frames: ${\hat {T}}_{\perp }^{a}={\hat {U}}_{\vartheta \perp }^{\dagger }({\vartheta }^{a},{\boldsymbol{\chi}}^{a}){\hat {F}}^{a}(\vartheta ,\boldsymbol{\chi}){\hat {U}}_{\vartheta \perp }(\vartheta ,\boldsymbol{\chi})$,

$$\begin{eqnarray} &&\begin{array}{@{}c@{}} \displaystyle \vert {\tilde {\mathbf{E}}}^{a})_{\perp \mathrm{B}}={\hat {T}}_{\perp }^{a}\vert \tilde {\mathbf{E}})_{\perp \mathrm{B}},\\ \displaystyle {\hat {T}}_{\perp }^{a}\simeq \left (\begin{array}{@{}cc@{}} \displaystyle {f}_{\mathrm{p}}^{a}(1+\mu \hspace{0.167em} {\mathcal{X}}_{\mathrm{p}}^{a})&\displaystyle {f}_{\mathrm{p}}^{a}\nu {\mathcal{Y}}_{\mathrm{p}}^{a}\\ \displaystyle -{f}_{\mathrm{s}}^{a}\nu {\mathcal{Y}}_{\mathrm{s}}^{a}&\displaystyle {f}_{\mathrm{s}}^{a}(1+\mu \hspace{0.167em} {\mathcal{X}}_{\mathrm{s}}^{a}) \end{array}\right ). \end{array} \end{eqnarray} \tag{ 3.14 }$$

Here we introduced the quantities

$$\begin{eqnarray} &&\begin{array}{@{}c@{}} \displaystyle {\mathcal{X}}_{\mathrm{p,s}}^{a}=\frac{\partial \ln {f}_{\mathrm{p,s}}^{a}}{\partial \vartheta },\\ \displaystyle {\mathcal{Y}}_{\mathrm{p,s}}^{a}=\left (1-{\gamma }^{{a}^{-1}}\frac{{f}_{\mathrm{s,p}}^{a}}{{f}_{\mathrm{p,s}}^{a}}\right )\cot \vartheta . \end{array} \end{eqnarray} \tag{ 3.15 }$$

Equations (3.14) and (3.15) are the central results in this work; they describe field transformations upon the reflection and refraction of a paraxial optical beam at a plane dielectric interface [52, 54, 56, 70, 73]. The Jones matrix ${\hat {T}}_{\perp }^{a}$ characterizes the effective spatial dispersion of the interface through μ- and ν-dependent terms. Importantly, the beam interaction with more complex interfaces (metallic, multilayer, etc) can be described by the same equations with the corresponding transmission and reflection coefficients. While the 'spin–orbit transformation' (3.11) is diagonal in the basis of circular polarizations, the Fresnel boundary conditions (3.13) are naturally diagonal in the basis of linear polarizations. Therefore, there is no global polarization basis that would diagonalize the whole transformation (3.14).

To calculate the shifts explicitly, let us take an incident Gaussian beam at its waist Z = 0: $\vert \tilde {\mathbf{E}})_{\perp \mathrm{B}}=\vert \mathbf{e})_{\perp \mathrm{B}}G(\mu ,\nu )$, where |e)_⊥B = (e_X,e_y)^T is the Jones vector of the central plane wave and $G(\mu ,\nu )=\frac{{w}_{0}^{2}}{2 \pi }\exp [-(k{w}_{0})^{2}\frac{{\mu }^{2}+{\nu }^{2}}{4}]$ is the normalized Gaussian envelope with the waist w₀. By applying the Jones matrix (3.14) we obtain the fields of the secondary beams and calculate the expectation values of their transverse momenta and coordinates using equations (2.19):

$$\begin{eqnarray} &&\begin{array}{@{}c@{}} \displaystyle \langle {X}^{a},{Y}^{a}\rangle ={Q}^{{a}^{-2}}\left \langle {\tilde {\mathbf{E}}}^{a}\left \vert \mathrm{i}\frac{\partial }{\partial {k}_{X,y}^{a}}\right \vert {\tilde {\mathbf{E}}}^{a}\right \rangle _{\mathrm{B}\perp },\\ \displaystyle \langle {P}_{X,y}^{a}\rangle ={Q}^{{a}^{-2}}\langle {\tilde {\mathbf{E}}}^{a}\vert {k}_{X,y}^{a}\vert {\tilde {\mathbf{E}}}^{a}\rangle _{\mathrm{B}\perp }. \end{array} \end{eqnarray} \tag{ 3.16 }$$

Here the integration is taken over the transverse momentum components ${k}_{X}^{a}={\gamma }^{a}k \mu$ and ${k}_{y}^{a}=k \nu$, and we took into account that the norms are equal to ${N}^{a}=\langle {\tilde {\mathbf{E}}}_{\perp }^{a}\vert {\tilde {\mathbf{E}}}_{\perp }^{a}\rangle ={Q}^{{a}^{2}}$.

In the case of partial reflection and transmission, sinϑ < n, the Fresnel coefficients ${f}_{\mathrm{p,s}}^{a}$ are real, and equations (3.14)–(3.16) at Z^a = 0 result in

$$\begin{eqnarray} &&\langle {X}^{a}\rangle =0,\tqs \langle {P}_{X}^{a}\rangle =\frac{{\gamma }^{a}}{k{w}_{0}^{2}}\hspace{0.167em} \frac{\mathrm{d}\ln {\mathsf{Q}}^{a}}{\mathrm{d}\vartheta }, \end{eqnarray} \tag{ 3.17 }$$

$$\begin{eqnarray} &&\begin{array}{@{}c@{}} \displaystyle \langle {Y}^{a}\rangle =-\frac{\bar {\sigma }}{2 k}\frac{{f}_{\mathrm{p}}^{{a}^{2}}{\mathcal{Y}}_{\mathrm{ p}}^{a}+{f}_{\mathrm{ s}}^{{a}^{2}}{\mathcal{Y}}_{\mathrm{ s}}^{a}}{{Q}^{{a}^{2}}},\\ \displaystyle \langle {P}_{y}^{a}\rangle =\frac{\bar {\rho }}{k{w}_{0}^{2}}\frac{{f}_{\mathrm{p}}^{{a}^{2}}{\mathcal{Y}}_{\mathrm{ p}}^{a}-{f}_{\mathrm{ s}}^{{a}^{2}}{\mathcal{Y}}_{\mathrm{ s}}^{a}}{{Q}^{{a}^{2}}}. \end{array} \end{eqnarray} \tag{ 3.18 }$$

Here $\bar {\sigma }=(\mathbf{e}\vert {\hat {\sigma }}_{2}\vert \mathbf{e})_{\perp \mathrm{B}}=2 \hspace{0.167em} \mathrm{Im}({e}_{X}^{\ast }{e}_{y})$ is the average helicity of the incident beam (the Stokes parameter S₃) and $\bar {\rho }=(\mathbf{e}\vert {\hat {\sigma }}_{1}\vert \mathbf{e})_{\perp \mathrm{B}}=2 \hspace{0.167em} \mathrm{Re}({e}_{X}^{\ast }{e}_{y})$ is the degree of linear polarization inclined at π/4 angle (the Stokes parameter S₂), see equation (2.26).

In the case of total internal reflection, sinϑ > n, there is no transmitted propagating wave, ${f}_{\mathrm{p,s}}^{\mathrm{t}}=0$, while the reflection coefficients are complex: ${f}_{\mathrm{p,s}}^{\mathrm{r}}=\exp (\mathrm{i}{\delta }_{\mathrm{p,s}})$, with real phases δ_p,s. In this case equations (3.14)–(3.16) bring about

$$\begin{eqnarray} &&\begin{array}{@{}c@{}} \displaystyle \langle {X}^{\mathrm{tot}\hspace{0.167em} \mathrm{r}}\rangle =\frac{1}{k}(\vert {e}_{X}\vert ^{2}\hspace{0.167em} \mathrm{Im}{\mathcal{X}}_{\mathrm{ p}}^{\mathrm{r}}+\vert {e}_{y}\vert ^{2}\hspace{0.167em} \mathrm{Im}{\mathcal{X}}_{\mathrm{ s}}^{\mathrm{r}}), \\ \displaystyle \langle {P}_{X}^{\mathrm{tot}\hspace{0.167em} \mathrm{r}}\rangle =0, \end{array} \end{eqnarray} \tag{ 3.19 }$$

$$\begin{eqnarray} &&\begin{array}{@{}c@{}} \displaystyle \langle {Y}^{\mathrm{tot}\hspace{0.167em} \mathrm{r}}\rangle =-\frac{1}{2 k}[\bar {\sigma }\hspace{0.167em} \mathrm{Re}({\mathcal{Y}}_{\mathrm{p}}^{\mathrm{r}}+{\mathcal{Y}}_{\mathrm{ s}}^{\mathrm{r}})+\bar {\rho }\hspace{0.167em} \mathrm{Im}({\mathcal{Y}}_{\mathrm{ p}}^{\mathrm{r}}-{\mathcal{Y}}_{\mathrm{ s}}^{\mathrm{r}})],\\ \displaystyle \langle {P}_{y}^{\mathrm{tot}\hspace{0.167em} \mathrm{r}}\rangle =0. \end{array} \end{eqnarray} \tag{ 3.20 }$$

The centroid displacement 〈X^a〉 and momentum shift (deflection) $\langle {P}_{X}^{a}\rangle$ represent the spatial and angular Goos–Hänchen shifts, whereas displacement 〈Y^a〉 and deflection $\langle {P}_{y}^{a}\rangle$ represent the spatial and angular Imbert–Fedorov shifts, respectively, see figure 1. The angular shifts depend on the beam waist and vanish in the paraxial limit w₀ → ∞, whereas the spatial shifts are independent of the beam profile. Nonetheless, both spatial and angular shifts have a common origin: the in-plane momentum gradients of the Fresnel coefficients ${\mathcal{X}}_{\mathrm{p,s}}^{a}$ for the GH effects (3.17) and (3.19) and the geometric spin–orbit terms ${\mathcal{Y}}_{\mathrm{p,s}}^{a}$ for the IF effects (3.18) and (3.20). The latter can be associated with the transverse momentum gradients of the Berry phases across the beams: $\partial \hspace{0.167em} {\Phi }_{\mathrm{B}}^{a}/\partial {\nu }^{a}=-\cot {\vartheta }^{a}$. The IF effects show divergence at ϑ → 0 because the approximate transition (3.9) and (3.10) to the spherical coordinates, becomes singular at normal incidence; in reality, of course, the transverse shift vanishes at ϑ = 0. Away from the beam waist, at Z^a ≠ 0, the real-space shifts grow according to the angular deviation: $\langle {X}^{a}\rangle ({Z}^{a})=\langle {X}^{a}\rangle +\langle {P}_{X}^{a}\rangle {Z}^{a}/{k}^{a}$, $\langle {Y}^{a}\rangle ({Z}^{a})=\langle {Y}^{a}\rangle +\langle {P}_{y}^{a}\rangle {Z}^{a}/{k}^{a}$, figure 1, which is used to measure small deviation effects [14, 53, 54, 56, 68, 69].

The widely known spatial GH shift 〈X^a〉 is caused by the angular gradient of the phase of complex reflection or refraction coefficients and appears, e.g., upon the total internal reflection or interaction with complex dispersive interfaces [2–5, 15–28]. In turn, the angular GH shift $\langle {P}_{X}^{a}\rangle$ is caused by the angular gradients of the amplitude of the Fresnel coefficients and can be observed in the case of partial reflection and refraction with real coefficients [9–14]. Thus, both GH effects are intimately related to the spatial dispersion of the scattering coefficients. The eigenmodes of the GH shifts are p and s linearly polarized waves, but polarization is not crucial for the existence of these phenomena, as it can also occur for scalar waves.

The IF shifts arise essentially owing to the intrinsic polarization properties of light. In the partial reflection/transmission case, the eigenmodes of the spatial shift 〈Y^a〉 are circularly polarized waves $\bar {\sigma }=\pm 1$, the shift is proportional to the helicity of the incident wave, and, thus, represents the spin Hall effect of light, see figure 6. This transverse shift has been discussed and studied for many years, both theoretically and experimentally [6–8, 29–53]. Strikingly, while the clear theoretical explanation of the GH effect was given by Artmann the year after its discovery [3], the accurate formulae equation (3.18) for the IF shift were derived and verified experimentally only recently [51–54] (although a formula equivalent to equation (3.20) appeared first in an early paper by Schilling [7]). This could be explained by the fact that while the GH shifts can be fully explained within the 2D geometry of the plane of incidence, the transverse IF effect essentially involves a 3D description and requires an accurate characterization of the polarization of the incident field [52]. The angular IF shift $\langle {P}_{y}^{a}\rangle$ was revealed only recently [52–56, 63, 64]. Despite the fact that the eigenmodes of this effect are linearly polarized waves with $\bar {\rho }=\pm 1$, it originates from the same geometric phase terms as the spatial transverse shift.

Importantly, the spatial IF shift is intimately related to the balance and conservation of the AM of light [40, 41, 50–52]. Indeed, the rotational symmetry of the medium about the z-axis implies that the z-component of the total AM must be conserved upon reflection and refraction. Using equation (2.23) for paraxial beams, the spin AM per photon in the three beams can be written as $\langle {\mathbf{S}}^{a}\rangle ={\bar {\sigma }}^{a}{\mathbf{u}}_{Z}^{a}$, $\langle {S}_{z}^{a}\rangle ={\bar {\sigma }}^{a}\cos {\vartheta }^{a}$. From the Fresnel boundary conditions (3.3) and (3.4), one can determine the mean helicities ${\bar {\sigma }}^{a}=({\mathbf{e}}^{a}\vert {\hat {\sigma }}_{2}\vert {\mathbf{e}}^{a})_{\perp \mathrm{B}}$, which yield the spin AM for the cases of partial reflection/transmission and for total internal reflection:

$$\begin{eqnarray} &&\begin{array}{@{}c@{}} \displaystyle \langle {S}_{z}^{a}\rangle =\bar {\sigma }\cos {\vartheta }^{a}\frac{{f}_{\mathrm{p}}^{a}{f}_{\mathrm{ s}}^{a}}{{Q{}^{a}}^{2}},\\ \displaystyle \langle {S}_{z}^{\mathrm{tot}\hspace{0.167em} \mathrm{r}}\rangle =-(\bar {\sigma }\cos \delta +\bar {\rho }\sin \delta )\cos \vartheta , \end{array} \end{eqnarray} \tag{ 3.21 }$$

where δ = δ_s − δ_p. The transverse coordinate shifts 〈Y^a〉, equations (3.18) and (3.20), together with the momentum component $\langle {P}_{x}^{a}\rangle \simeq k\sin \vartheta$ generate the extrinsic orbital AM (2.21):

$$\begin{eqnarray} &&\langle {L}_{z}^{\mathrm{ext}\hspace{0.167em} a}\rangle =[\langle {\mathbf{R}}^{a}\rangle \times \langle {\mathbf{P}}^{a}\rangle ]_{z}=-\langle {Y}^{a}\rangle k\sin \vartheta . \end{eqnarray} \tag{ 3.22 }$$

Since the number of photons in the beams is proportional to the energy Q^a, we write the conservation of the z-component of the total AM in the scattering as

$$\begin{eqnarray} &&{\mathsf{Q}}^{\mathrm{r}}[\langle {S}_{z}^{\mathrm{r}}\rangle -\langle {Y}^{\mathrm{r}}\rangle k\sin \vartheta ]+{\mathsf{Q}}^{\mathrm{t}}[\langle {S}_{z}^{\mathrm{t}}\rangle -\langle {Y}^{\mathrm{t}}\rangle k\sin \vartheta ]=\langle {S}_{z}\rangle .\nonumber\\ \end{eqnarray} \tag{ 3.23 }$$

Here we took into account that $\langle {L}_{z}^{\mathrm{ext}}\rangle =\langle Y\rangle =0$ and Q = 1 for the incident beam. Importantly, this equation is not satisfied with 〈Y^r〉 = 〈Y^t〉 = 0. Changes in the spin AM (3.21) which occur upon Snell–Fresnel reflection/refraction of the wave beam must be compensated by the nonzero extrinsic orbital AM (3.22) of the secondary beams. Substituting equations (3.18), (3.20), and (3.21) into (3.23) and using relations (3.1)–(3.7) and (3.15), one can verify that equation (3.23) is fulfilled identically. Thus, the nonzero transverse shifts of the reflected and refracted beams ensure the conservation of the AM in the problem.

In the case of total internal reflection, there is no transmitted beam, Q^t = 0, Q^r = 1, and the shift can be derived solely from the AM conservation:

$$\begin{eqnarray} &&\langle {Y}^{\mathrm{tot}\hspace{0.167em} \mathrm{r}}\rangle =\frac{\langle {S}_{z}^{\mathrm{tot}\hspace{0.167em} \mathrm{r}}\rangle -\langle {S}_{z}\rangle }{k\sin \vartheta }, \end{eqnarray} \tag{ 3.24 }$$

which, together with equation (3.21), immediately yields equation (3.20). This shows that this IF shift is very robust and is practically independent of the shape and fine details of the incident beam field. In other cases, the AM conservation imposes a constraint on the beam shifts but does not fix their values.

In a similar manner, one can verify that the angular shifts obey a constraint following from the conservation of the linear momentum [63, 106]. The balance of the tangential components of the beam momenta yields:

$$\begin{eqnarray} &&\begin{array}{@{}c@{}} \displaystyle -{\mathsf{Q}}^{\mathrm{r}}\langle {P}_{X}^{\mathrm{r}}\rangle \cos \vartheta +{\mathsf{Q}}^{\mathrm{t}}\langle {P}_{X}^{\mathrm{t}}\rangle \cos {\vartheta }^{\prime}=0,\\ \displaystyle {\mathsf{Q}}^{\mathrm{r}}\langle {P}_{y}^{\mathrm{r}}\rangle +{\mathsf{Q}}^{\mathrm{t}}\langle {P}_{y}^{\mathrm{t}}\rangle =0. \end{array} \end{eqnarray} \tag{ 3.25 }$$

Here we took into account the fact that the x-component of the momentum is $\langle {P}_{x}^{a}\rangle =\langle {P}_{X}^{a}\rangle \cos {\vartheta }^{a}+\langle {P}_{Z}^{a}\rangle \sin {\vartheta }^{a}$ and the balance of $\langle {P}_{Z}^{a}\rangle \sin {\vartheta }^{a}={k}^{a}\sin {\vartheta }^{a}=k\sin \vartheta$ is guaranteed by Snell's law (3.1). Using equations (3.1)–(3.7) and (3.15), one can readily show that equations (3.17) and (3.18) fulfil equations (3.25) identically. Note, however, that the linear momentum conservation (3.25) could be satisfied without angular shifts, i.e., at $\langle {P}_{X,y}^{\mathrm{r}}\rangle =\langle {P}_{X,y}^{\mathrm{t}}\rangle =0$ (which is achieved for special beam models [52]), while the AM balance (3.23) essentially requires at least one transverse shift, 〈Y^r〉 or 〈Y^t〉, to be nonzero. In the case of total internal reflection (Q^t = 0,Q^r = 1), the momentum conservation (3.25) results in vanishing of angular shifts, $\langle {P}_{X,y}^{\mathrm{tot}\hspace{0.167em} \mathrm{r}}\rangle =0$, in agreement with (3.19) and (3.20).

Thus, the 'microscopic' plane-wave interference resulting in the shifts (3.17)–(3.20) and 'macroscopic' arguments of the conservation laws (3.23)–(3.25) are in complete agreement with each other. This demonstrates a remarkable interplay of the 'geometric' and 'dynamical' mechanisms in the wave interaction phenomena.

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The GH and IF shifts of the beam centroids have a typical magnitude k⁻¹, i.e., a fraction of the wavelength. Measurement of such shifts is a challenging problem for experimentalists. However, there is a method of 'quantum weak measurements' [108–111] which allows significant magnification of the beam shifts and measurements of the elements of the interface Jones matrix (3.14). Roughly speaking, the method consists in the use of two almost orthogonal polarizers: one pre-selecting the incident beam state and the other post-selecting reflected or refracted beam states after the interface. Depending on the mutual orientation of the two polarizers, one can increase the shift of the post-selected beam up to the order of the beam width. The weak-measurement technique was used in experiments [53, 56, 68, 69, 112–115] measuring the IF effect, whereas its detailed theoretical analysis in the beam-shift problem is given in [53, 54, 116, 117]. In fact, the weakly measured beam-shift yields the corresponding element of the interface Jones matrix (3.14) which is interpreted as an effective Hamiltonian of the polarization-momentum coupling. A predecessor of the weak-measurement technique was the idea to measure the splitting of the beam intensity in the crossed input and output polarizers [38, 51].

The shifts of the beam centroid are the first moments of the intensity distributions, so that they describe only the simplest first-order deformations of the beam (figure 1). At the same time, the reflected and transmitted beams can undergo fine higher-order deformation (aberration) effects, which are not accounted in the beam shifts (see, e.g., the vortex beam deformations in figure 6). Such deformations can be described, e.g., via expanding the beam intensity as a superposition of the orbital AM vortex eigenmodes taken with respect to the geometrical-optics axis. In this manner, the shift and extrinsic orbital AM of the secondary beams arises from the superposition of vortex modes with ℓ differing by ±1. Calculations and measurements of the orbital AM spectrum of the reflected beam were presented recently [118]. In the case of the incident vortex beam, the shift effects are accompanied by the shift and fine splitting of the charge-ℓ vortex into a constellation of the |ℓ| unit strength vortices. Such a constellation characterizes the beam aberrations and properties of the interface up to the |ℓ| order [119].

The beam reflection from a plane dielectric interface has two peculiar points: the Brewster angle of incidence and the critical angle upon the reflection from a less dense medium. The beam shifts can be significantly enhanced in the vicinity of these angles (the spin Hall effect diverges at the Brewster angle, while the GH shift diverges at the critical incidence), and the problem of the beam reflection can be challenging for the accurate theoretical treatment [74]. The beam shifts are well studied, both theoretically and experimentally, in the vicinity of the Brewster angle [11, 14, 60, 68, 72, 112, 120, 121]. At the same time, the situation is more complicated for near-critical incidence, where the theoretical and experimental results are still far from good agreement. Indeed, see [122–124] for the GH effect, and we have found that the previously reported measurements of the IF shift near the critical angle [35, 38, 44, 47, 49] have values about 1.5 times larger than those predicted by the Schilling formulae (3.20). The plane-wave Fourier analysis of the beam reflection becomes quite complicated near the critical angle [74], and perhaps one should apply real-space boundary conditions at the interface to find the actual reflected beam transformations [125]. It is also possible that the transverse energy-flow effect in the evanescent transmitted field, which was originally discovered by Fedorov and Imbert [6, 8], still contributes the IF shift near the critical angle.

The beam-shift effects can also be sensitive to the shape of the incident beam. In addition to the Gaussian and Laguerre–Gaussian beams, the beam shifts were considered for the following types of beams: paraxial Bessel [81], nonparaxial Bessel [126], Laguerre–Gaussian with higher radial order [127], Hermite–Gaussian [128, 129], and for the dipole radiation [74]. General aspects of the shape-dependent and shape-independent beam shifts are analysed in [130]. Another important characteristic of the incident beam is the degree of its spatial coherence. Dependence of different beam shifts on this characteristic caused a theoretical controversy [131–134] which was recently resolved by experimental measurements [135, 136].

Apparently, the most popular extensions of the beam-shift problems are generalizations of these effects to the cases of various complex media. These include metals [26, 137, 138], dissipative media [15, 17, 55, 66, 67, 120, 139, 140], semiconductors [66, 67], various anisotropic (including chiral) media [69, 141–145], multilayered structures [48, 146–150], metamaterials [23, 70, 141, 147, 151, 152], photonic crystals [24, 148, 150, 153], thin films [48, 113, 114], resonators [16, 44, 154], plasmonic systems [19, 23, 25, 87, 115, 155], curved interfaces (lenses) [13, 87, 154, 156–158], nonlinear media [18, 67, 159], etc. In addition, the GH shift is studied for matter waves in various quantum and condensed-matter systems [27, 28, 160].

It is important to mention effects related to the beam shifts that occur upon reflection/refraction at sharp interfaces.

First, propagation of light in a gradient-index medium exhibits an AM-dependent transverse transport of the beam [43, 50, 61, 161–167]. This includes the spin and orbital Hall effects (also referred to as optical Magnus effect [43, 168]). The Hall effects in a gradient-index medium can be derived as a limiting case of the spatial IF shifts when the beam is transmitted through multiple dielectric interfaces with low contrasts [43, 50, 61]. This effect is also intimately related to the AM conservation, and is described by the action of the 'geometric Berry force' in the momentum space. (Note that the IF shift at a sharp interface can be attributed to the action of the so-called 'Abraham force' related to the difference between the Abraham and Minkowski momenta of photons in a dielectric medium [169].)

Another related beam-shift effect is the geometric Hall effect of light [170–172], which occurs upon observation of a beam of light in a tilted cross-section. This effect can be observed even in free space via measurements of the energy flux density through the tilted cross-section of the beam. The geometric Hall beam shifts are also intimately related to the transformations of the AM and are similar to the orbital Hall effect shift (4.2)—both phenomena have a characteristic ℓtanϑ/2k dependence, which originates from the spatial tilt rather than gradients in the momentum Fourier space. In the space–time domain, an entirely analogous beam-shift occurs upon observation of the beam in the relativistic moving frame; this is the relativistic Hall effect [173].