Optical beam shift as a vectorial pointer of curved-path geodesics: an evolution-operator perspective

Aleksei Zheltikov; Aleksei Zheltikov; Aleksei Zheltikov

doi:10.1364/OE.389165

1. Introduction

Close similarities and multiple parallels between equations of electrodynamics, quantum theory, statistical mechanics, ocean waves, and general relativity have long been inspiring the development of optical analogs of black holes [1], event horizons [2,3], and rogue waves [4,5]. As one important parallel, both optical beams and quantum-mechanical wave packets are subject to a position–momentum uncertainty, setting a fundamental limit on the resolution of position measurements in optics and quantum mechanics [6,7]. Much research in optical physics and quantum metrology is currently focused on high-precision wave-packet position measurements whereby the standard limits on the spatial resolution and information content of such measurements could be overcome. In both areas, solutions to these problems are often multiphoton. Multiphoton processes in optics are paving the ways toward an unprecedented resolution in microscopy [8–13], while many-photon quantum states of light offer much promise for quantum metrology and quantum imaging beyond the standard noise limit [14–17]. As a powerful resource, weak measurements suggest new strategies for a minimally disturbing characterization of and information readout from quantum objects. In optics, small, often subwavelength shifts of optical beams, such as the Goos–Hänchen [18,19] and Imbert–Fedorov [20,21] shifts, offer appealing and practically useful analogs [22–27] of weak measurements [28–31], providing a unique tool to detect and characterize the spin Hall effect of light [22,23].

Here, we focus on yet another important property of weak optical beam shifts. We show that, when set to travel along a curved path, e.g., in a bending-waveguide setting, an optical beam tends to re-adjust its position, shifting away from the center of path curvature. An evolution-operator analysis of this effect shows that its properties can be understood in terms of the modal properties of the field state ket as defined by the symmetry of the beam path and the specific shape of the refractive index profile. We examine the bending-induced beam shift and the related tunneling loss as functions of the index step and the beam diameter for generic refractive index profiles. This analysis shows that the transverse shift of an optical beam propagating along a bending path is highly sensitive to the parameters of the refractive index profile causing this path bending. That the beam shifts in the direction that is exactly opposite to the direction to the center of path curvature indicates that, as a path-curvature pointer, this shift is vectorial in nature, sensing the plane of path curvature and detecting the direction to the center of path curvature.

2. Optical beams on curved paths: scalar wave-equation analysis

We consider a spatially localized mode of an electromagnetic field propagating in a dielectric medium whose refractive index profile, n(x, y, z), is chosen in such a way as to guide the field along a curved path (Fig. 1(a)). We choose the z-axis along the beam path, and assume that, for z < 0, the refractive index profile, n(x, y, z) = n₀(x, y) ≡ n₀(r), is invariant with respect to translations along z. In the xy-plane (the inset in Fig. 1(a)), the transverse profile of the refractive index is assumed to be axially symmetric relative to the z-axis, n₀(r) = n₀(r), monotonically decreasing with r = (x² + y²)^1/2 on a spatial scale ρ from its maximum value, n₀(r = 0) = n₁, to its lower, perhaps asymptotic bound n₂ for r ≥ ρ.

Fig. 1. (a) An optical beam traveling along a curved path in a medium with a bending refractive index profile (shown by color coding). Notation is as explained in the text. Also shown is the ray-optic representation of the mode (purple line). A cross-sectional view of the refractive index profile is shown in the inset. (b) An x-cut of the potential V(r) induced by a gradient index profile for the rectilinear (z < 0, dashed line) and bending (z > 0, solid line) segments of the beam path. Also shown is the energy eigenvalue, E = n₁ – β/k₀: (solid horizontal line) E > V and (dotted horizontal line) E < V.

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At z = 0, the refractive-index profile starts to bend, causing the beam to propagate along a curved path with a constant curvature of radius R (Fig. 1(a)). Such index settings are routinely found in bending optical fibers [32,33]. Less routine examples of curved-path electromagnetic field propagation induced by a spatially non-uniform refractivity include bending laser beams in pre-ionized or pre-heated atmosphere [34], as well as curved-path propagation of microwaves in the ionosphere [35].

In a standard approximation, the transverse Cartesian components of the electric and magnetic fields in a curved-path optical beam, Ψ(x, y, z) = ψ(x, y)exp(iβz), are found by solving a scalar paraxial-approximation version of the reduced wave equation,

(1)$$\nabla _ \bot ^2\psi + [{k_0^2n_{}^2(\textbf{r} )- \beta_{}^2} ]\psi = 0,$$

where k₀ = ω/c = 2π/λ, ω is the frequency, λ is the wavelength, β is the propagation constant, $\nabla _ \bot ^2 = {{{\partial ^2}} \mathord{\left/ {\vphantom {{{\partial^2}} {\partial {x^2}}}} \right.} {\partial {x^2}}} + {{{\partial ^2}} \mathord{\left/ {\vphantom {{{\partial^2}} {\partial {y^2}}}} \right.} {\partial {y^2}}}$ is the transverse Laplacian, and c is the speed of light in vacuum. This approximation has been shown to be adequate for a vast class of optical problems and a broad variety of physical settings [6,32].

For R << ρ, the effective index profile n(r) in Eq. (1) is given by [33]

(2)$${n^2}(\textbf{r} )= \left\{ {\begin{array}{c} {n_0^2(r ),z < 0}\\ {n_0^2(r )+ 2n_1^2\frac{r}{R}\cos \varphi ,z \ge 0} \end{array}} \right. .$$

Here, z is the coordinate along the beam path, while r and φ are the polar coordinates in the plane perpendicular to the z-axis. With n(r) as defined by Eq. (2), the lowest-β variational solution to Eq. (1) in the class of Gaussian functions is [32,33]

(3)$$\Phi (\textbf{r} )\approx \left( {1 + \frac{\delta }{{r_0^2}}r\cos \varphi } \right)\exp \left( { - \frac{{{r^2}}}{{2r_0^2}}} \right),$$

where

(4)$$\delta = \left\{ {\begin{array}{c} {0,z < 0}\\ {\frac{{{v^2}r_0^4}}{{2\Delta R{\rho^2}}},z \ge 0} \end{array}} \right.,$$

$\Delta = {{({n_1^2 - n_2^2} )} \mathord{\left/ {\vphantom {{({n_1^2 - n_2^2} )} {({2n_1^2} )}}} \right.} {({2n_1^2} )}}$, $v = {k_0}\rho {({n_1^2 - n_2^2} )^{{1 \mathord{\left/ {\vphantom {1 2}} \right.} 2}}}$, and r₀ is the beam radius, defined as a solution to ${{\partial {\beta ^2}} \mathord{\left/ {\vphantom {{\partial {\beta^2}} {\partial {r_0}}}} \right.} {\partial {r_0}}} = 0$. Thus, while ψ(r) is defined as the exact solution to Eq. (1), Φ(r) stands for the lowest-β variational approximation of ψ(r).

The bending-induced change in the refractive index, $\Delta n(\textbf{r} )= n(\textbf{r} )- {n_0}(r )\approx n_1^{}({{r \mathord{\left/ {\vphantom {r R}} \right.} R}} )\cos \varphi$, thus shifts the maximum of the transverse field profile. This shift is maximal in the plane of n(x, y, z) curvature, i.e., for φ = 0, where the solution for ψ can be approximated, to the first order in δ/r₀ as

(5)$$\Phi ({r,\varphi = 0} )= \exp \left[ { - \frac{{{{({r - \delta } )}^2}}}{{2r_0^2}}} \right].$$

3. Hamiltonian and time-dependent Schrödinger equation

In the paraxial approximation, β ≈ k₀n₁, equations for the field components satisfying the scalar Helmholtz equation can be reduced [6,36,37] to a canonical form of the time-dependent Schrödinger equation (TDSE),

(6)$$\frac{i}{{{k_0}}}\frac{{\partial \Psi }}{{\partial z}} = \hat{H}\Psi ,$$

with a Hamiltonian

(7)$$\hat{H} ={-} \frac{{\nabla _ \bot ^2}}{{2k_0^2{n_1}}} + [{{n_1} - n(\textbf{r} )} ].$$

The transverse field profiles ψ = ψ(x, y) and the propagation constants β can then be found as the eigenfunctions and eigenvalues of

(8)$$\hat{H}\psi ={-} \frac{\beta }{{{k_0}}}\psi .$$

The TDSE formalism for a quantum particle of mass m₀ and energy ħω in the presence of a potential V(r) is thus recovered via the replacement

(9)$$z \to t,\,\beta \to - \omega ,\,{k_0} \to 1/\hbar ,\,{n_1} - n({\bf r}) \to V({\bf r}),\,{n_1} \to {m_0},$$

where t is the time and ħ is the Planck constant.

4. Path bending and tunneling

Within the rectilinear segment of the beam path (z < 0, Fig. 1(a)), an axially symmetric refractive-index profile n₀(r) (the inset in Fig. 1(a)) acts, in accordance with Eq. (9), as a binding potential V₀(r) = n₁ − n₀(r) (dashed line in Fig. 1(b)), confining the field within a stationary bound state with an energy eigenvalue E = n₁ – β/k₀ (horizontal line in Fig. 1(b)) – an optical analogue of a quantum bound state with a stationary energy eigenvalue ħω [see Eq. (9)]. Within the z > 0 segment of the beam path, however, the bending-induced change in the refractive index Δn(r) gives rise to an r-dependent shift of the potential, $\Delta V(\textbf{r} )={-} n_1^{}({{r \mathord{\left/ {\vphantom {r R}} \right.} R}} )\cos \varphi$. For φ falling within the range from −π/2 to +π/2, this shift lowers the binding potential [in an x-cut of V(r) shown in Fig. 1(b), (x) = rcosφ, with the right (x > 0) and left (x < 0) branches of V(x) corresponding to φ = 0 and φ = π, respectively]. As a result, a rectangular, axially symmetric potential well V₀(r) = n₁ − n₀(r) (dashed line in Fig. 1(b)) is transformed into a finite-width potential barrier (solid line in Fig. 1(b)). Now, the state of the field prepared in a bound state of V₀(r) with a stationary energy E = n₁ – β/k₀ < n₁ + n₂ (Fig. 1(b)) can tunnel through the region of evanescence, n₁ –β/k₀ < V(r) (the dashed section of the –β/k₀ line in Fig. 1(b)), becoming radiative again for r > r_c, where r_c is the caustic radius [32,38,39] (Figs. 1(a) and 1(b)).

5. Evolution operator

As a central methodological step of our analysis, we follow the guidance of variable replacement as suggested by Eq. (10) to define an evolution operator [6,36,37],

(10)$$\hat{U}({z,{z_0}} )= \Theta ({z - {z_0}} )\exp [{ - i{k_0}\hat{H}({z - {z_0}} )} ],$$

where Θ(ξ) is the Heaviside step function.

The coordinate-dependent function Ψ(x, y, z) =Ψ(r, z) can now be viewed as the position-representation projection, Ψ(r, z) = $\left\langle {\textbf{r}} \mathrel{|{\vphantom {\textbf{r} {\boldsymbol{\alpha };z}}}} {{\boldsymbol{\alpha };z}} \right\rangle$, of a state ket |α; z>, which provides a representation-nonspecific description of field evolution. The solution to the field evolution equation can then be written as

(11)$$|{\boldsymbol{\alpha };z} \rangle = \hat{U}({z,{z_0}} )|{\boldsymbol{\alpha };{z_0}} \rangle .$$

The evolution of the spatial field profile, Ψ(r, z), is then found as

(12)$$\Psi ({\textbf{r},z} )\, = \,\left\langle {\textbf{r}} \mathrel{|{\vphantom {\textbf{r} {\boldsymbol{\alpha };z}}}} {{\boldsymbol{\alpha };z}} \right\rangle = \sum\limits_q {\left\langle {\textbf{r}} \mathrel{|{\vphantom {\textbf{r} q}} } {q} \right\rangle } \left\langle {q} \mathrel{|{\vphantom {q {\boldsymbol{\alpha };{z_0}}}} } {{\boldsymbol{\alpha };{z_0}}} \right\rangle \exp [{ - i{\beta_q}({z - {z_0}} )} ],$$

where $|q \rangle$ is an eigenket of $\hat{H}$ with an eigenvalue β_q, $\hat{H}|q \rangle ={-} ({{{{\beta_q}} \mathord{\left/ {\vphantom {{{\beta_q}} {{k_0}}}} \right.} {{k_0}}}} )|q \rangle$.

Close similarities between the equations describing electromagnetic waves and quantum-mechanical wave functions can be appreciated already by observing that, with k₀ → 1/ħ, $n_1^2 - {({{\beta \mathord{\left/ {\vphantom {\beta {{k_0}}}} \right.} {{k_0}}}} )^2} \to 2E$, and $n_1^2 - {n^2}(\textbf{r} )\to 2V(\textbf{r} )$, Eq. (1) takes the form of the stationary Schrödinger equation [6,36,37]. It is, however, the evolution-operator formalism [Eqs. (10)–(12) above] that proves to be especially beneficial for our treatment here, as it offers deeper insights into the universal aspects of wave dynamics behind the path-curvature-sensing ability of optical beams.

It is instructive to represent the evolution of the spatial field profile [Eq. (12)] as

(13)$$\Psi ({\textbf{r},z} )= \int {K({\textbf{r},\textbf{r}^{\prime};z - {z_0}} )} \Psi ({\textbf{r}^{\prime},{z_0}} )d\textbf{r}^{\prime},$$

where

(14)$$K({\textbf{r},\textbf{r}^{\prime};z - {z_0}} )= \sum\limits_q {\left\langle {\textbf{r}} \mathrel{|{\vphantom {\textbf{r} q}}} {q} \right\rangle } \left\langle {q} \mathrel{|{\vphantom {q {\textbf{r}^{\prime}}}} } {{\textbf{r}^{\prime}}} \right\rangle \exp [{i{\beta_q}({z - {z_0}} )} ],$$

is the propagator, which can expressed as Feynman’s path integral [40], i.e., a sum of ∼exp(iS_j/ħ) terms taken over the entire manifold of paths leading from (x₀, y₀, z₀) to (x, y, z), with S_j being the action for the jth path in this manifold. This path-integral formalism provides a powerful tool for the analysis of a vast class of problems in electrodynamics [41–43].

With the imaginary path defined as ζ = −i(z − z₀), we arrive at the Feynman − Kac-type relation [43] for the lowest propagation constant $\bar{\beta }$:

(15)$$\bar{\beta } ={-} \mathop {\lim }\limits_{\zeta \to \infty } \frac{1}{\zeta }\ln \textrm{Tr}K({\textbf{r},\textbf{r}^{\prime};i\zeta } ),$$

where $\textrm{Tr}K({\textbf{r},\textbf{r}^{\prime};z - {z_0}} )\equiv \int {K({\textbf{q},\textbf{q};z - {z_0}} )d\textbf{q}}$.

6. Local propagation constant and symmetry arguments

It is instructive to express the Hamiltonian (7) as $\hat{H} \approx {\hat{H}_0} + \hat{W}$, with

(16)$$\hat{H} ={-} \frac{{\nabla _ \bot ^2}}{{2k_0^2{n_1}}} + [{{n_1} - {n_0}(\textbf{r} )} ]$$

and

(17)$$\hat{W} = \left\{ {\begin{array}{c} {0,z < 0}\\ { - n_1^{}\frac{r}{R}\cos \varphi ,z \ge 0} \end{array}} \right..$$

If the initial field is prepared in one of the eigenmodes of ${\hat{H}_0}$, the initial state ket is one of the eigenkets of ${\hat{H}_0}$, $|{\boldsymbol{\alpha };{z_0}} \rangle = |{{q_0}} \rangle$, with a stationary propagation constant β_q₀, defined as an eigenvalue of ${\hat{H}_0}$. Equation (11) then gives for the rectilinear segment of the beam path, $|{\boldsymbol{\alpha };z < 0} \rangle = \exp [ - i{k_0}\hat{H}({z - {z_0}} )]|{\boldsymbol{\alpha };{z_0}} \rangle = \exp [i{\beta _{q0}}({z - {z_0}} )]|{\boldsymbol{\alpha };{z_0}} \rangle$. In full analogy with the textbook result of quantum dynamics, once prepared as an eigenket of a z-independent Hamiltonian (an analog of a time-independent Hamiltonian in quantum dynamics), a state ket $|{\boldsymbol{\alpha };{z_0}} \rangle$ remains unchanged in its norm at any z < 0, changing only its phase.

For a state ket $|{\tilde{\boldsymbol{\alpha }};{z_0}} \rangle$ such that $\left|{\left\langle {\textbf{r}} \mathrel{|{\vphantom {\textbf{r} {\tilde{\boldsymbol{\alpha }};{z_0}}}}} {{\tilde{\boldsymbol{\alpha }};{z_0}}} \right\rangle } \right|$= Φ(r), Eq. (12) yields for z < 0

(18)$$|{\tilde{\boldsymbol{\alpha }};z < 0} \rangle \approx \exp [ - i{k_0}{\hat{H}_0}({z - {z_0}} )]|{\tilde{\boldsymbol{\alpha }};{z_0}} \rangle \approx \exp [i{\beta _0}({z - {z_0}} )]|{\tilde{\boldsymbol{\alpha }};{z_0}} \rangle ,$$

where β₀ ≈ $\bar{\beta }$, with $\bar{\beta }$ as defined by Eq. (15) for the propagator ${K_0}({\textbf{r},\textbf{r}^{\prime};z - {z_0}} )= \sum\nolimits_{{q_0}} \left\langle {\textbf{r}} \mathrel{|{\vphantom {\textbf{r} {{q_0}}}}} {{{q_0}}} \right\rangle$$\left\langle {{{q_0}}} \mathrel{|{\vphantom {{{q_0}} {\textbf{r}^{\prime}}}}} {{\textbf{r}^{\prime}}} \right\rangle \exp [{i{\beta_{q0}}({z - {z_0}} )} ]$.

As the potential changes at z = 0 [Eq. (14)], the evolution equation becomes

(19)$$\left\langle \textbf{r} \right.\left| {{\tilde{\boldsymbol{\alpha} }};z \ge 0} \right\rangle \approx \left\langle \textbf{r} \right|\exp [ - i{k_0}\hat{H}\left( {z - {z_0}} \right)]\left| {{\tilde{\boldsymbol{\alpha} }};{z_0} \ge 0} \right\rangle \approx \exp [i\beta '\left( {z - {z_0}} \right)]\left\langle \textbf{r} \right.\left| {{\tilde{\boldsymbol{\alpha} }};{z_0} \ge 0} \right\rangle.$$

Expressing $\left\langle {\textbf{r}} \mathrel{|{\vphantom {\textbf{r} {\tilde{\boldsymbol{\alpha }};z \ge 0}}}} {{\tilde{\boldsymbol{\alpha }};z \ge 0}} \right\rangle$= Φ_z_≥0(r)exp(iβ′z), expanding $\exp ({ - i{k_0}\hat{H}\xi } )=$ $= 1 - i{k_0}[{{{\hat{H}}_0} - {n_1}({{r \mathord{\left/ {\vphantom {r R}} \right.} R}} )\cos \varphi } ]\xi + \ldots$, discarding the terms smaller than r/R, we find

(20)$$\beta {\prime} \approx {\beta _0}[1\, - \,({r/R} )\cos\varphi ]\textrm{ } \approx {\beta _0}R/(R\, + \,r\cos\varphi ).$$

Unlike β₀ in Eq. (18), the local propagation constant β′ depends on both r and φ. That the true, i.e., r- and φ-independent, propagation constant does not exist for z ≥ 0 reflects the symmetry of the beam path. Unlike the z < 0 segment, where the translational symmetry of n(r) allows the z dependence of the field state kets to be isolated in the form of an exp(iβ₀z) multiplier, the azimuthal symmetry of n(r) for z ≥ 0 dictates a very different, exp(iɛθ) modal dependence with a new mode constant ɛ and the angle θ measured relative to the Ox-axis in the xz-plane as shown in Fig. 1(a). Expressing z through θ, z = (R + rcosφ)θ, we find that, with β′ as defined by Eq. (20), the phase factor exp(iβ′z) of the state ket $\left\langle \textbf{r} \right.|{\tilde{\boldsymbol{\alpha }};z \ge 0} \rangle$ can be written as exp(iβ′z) = exp(iɛθ), with ɛ = βR. We see that, since the refractive index profile features no translational symmetry for z ≥ 0, the field state ket is stripped of its ∼exp(iβ₀z) phase multiplier. Instead, the azimuthal symmetry of n(r) dictates an ∼exp(iɛθ) modal dependence. Our result expressed by Eq. (20) is fully consistent with these symmetry arguments.

7. Azimuthal ray invariant and bending-induced tunneling

As a signature of a new, azimuthal symmetry that the refractive index profile assumes for z ≥ 0, the beam acquires an azimuthal ray invariant [32], as dictated by Noether’s theorem [44], l₀ = [r/(R + r)]n(r)cosθ_φ, with θ_φ being the angle between the tangent to the ray trajectory (purple line in Fig. 1(a)) and the beam axis (dotted line in Fig. 1(a)). Since the field is confined within a guided mode of n(r) for z ≤ 0, its phase evolves as exp(iβ₀z) all the way up to z = 0, with n₂ ≤ β₀ ≤ n₁. Invoking the continuity of ray invariants at z = 0, we can relate l₀ to β₀ to find that n₂(R − ρ)/(R + ρ) ≤ l₀ ≤ n₁, in full analogy with mode analysis in bending waveguides [32]. The fields that become evanescent for r > R + ρ (dotted segments in Figs. 1(a) and 1(b)) start radiating again beyond the caustic at r_c = (R + ρ)l₀/n₂ (wavy lines in Fig. 1(a)). Because l₀ is bounded from above, the caustic occurs at finite r (Fig. 1(b)), r_c ≤ (R + ρ)n₁/n₂, proving that each ray for z > 0 is leaky.

To calculate the evanescent-to-radiation-field transmission coefficient at the caustic surface, r = r_c, we follow the generic recipes for the analysis of tunneling in guided-wave optics [38,39,45], arriving at

(21)$$T \approx {T_0}\exp \left[ { - 2{k_0}\int\limits_\rho^{{r_c}} {{{\left( {l_0^2\frac{{{\rho^2}}}{{{r^2}}} - n_2^2} \right)}^{{1 \mathord{\left/ {\vphantom {1 2}} \right.} 2}}}dr} } \right],$$

where ${T_0} \approx 4{({n_1^2 - l_0^2} )^{{1 \mathord{\left/ {\vphantom {1 2}} \right.} 2}}}{({l_0^2 - n_2^2} )^{{1 \mathord{\left/ {\vphantom {1 2}} \right.} 2}}}{({n_1^2 - n_2^2} )^{ - 1}}$.

Equation (21) is seen to feature a signature tunneling exponential. With the refractive index profile given, the transmission T is fully controlled by the azimuthal ray invariant l₀.

For a quantitative analysis of the bending-induced transition loss κ from a spatially localized mode, we assume that the field in the rectilinear segment of the beam path is prepared in the $\left\langle {\textbf{r}} \mathrel{|{\vphantom {\textbf{r} {\tilde{\boldsymbol{\alpha }};z < 0}}} } {{\tilde{\boldsymbol{\alpha }};z < 0}} \right\rangle$= Φ_z_<0(r)exp(iβ₀z) state, with Φ(r) as defined by Eqs. (3) and (4) for z < 0. At z = 0, the Hamiltonian changes in a stepwise manner due to an abrupt buildup of $\hat{W}$. The state ket of the field is now $\left\langle {\textbf{r}} \mathrel{|{\vphantom {\textbf{r} {\tilde{\boldsymbol{\alpha }};z > 0}}} } {{\tilde{\boldsymbol{\alpha }};z > 0}} \right\rangle$= Φ_z_>0(r)exp(iβ′z). The efficiency of $\left\langle {\textbf{r}} \mathrel{|{\vphantom {\textbf{r} {\tilde{\boldsymbol{\alpha }};z < 0}}} } {{\tilde{\boldsymbol{\alpha }};z < 0}} \right\rangle$-to-$\left\langle {\textbf{r}} \mathrel{|{\vphantom {\textbf{r} {\tilde{\boldsymbol{\alpha }};z > 0}}} } {{\tilde{\boldsymbol{\alpha }};z > 0}} \right\rangle$ coupling, χ = 1 − κ, is found by following, once again, the quantum-mechanical treatment of the probability of transitions induced by a sudden change in the Hamiltonian $\hat{H} - {\hat{H}_0}$:

(22)$$\chi = \int {{\Phi _{z < 0}}({\textbf{r}^{\prime}} )\Phi _{z \ge 0}^ \ast ({\textbf{r}^{\prime}} )} d\textbf{r}^{\prime}.$$

Using Eqs. (3) and (4) to evaluate this integral, we find for the transition loss: $\kappa = 1 - \exp \left[ { - {{{\delta ^2}} \mathord{\left/ {\vphantom {{{\delta ^2}} {\left( {2r_0^2} \right)}}} \right.} {\left( {2r_0^2} \right)}}} \right]$.

8. Beam shift as a pointer of path curvature

The beam shift is thus seen to provide a pointer for path curvature and bending-induced tunneling radiation loss. That a bending refractive index profile shifts an optical beam in the direction that is exactly opposite to the direction to the center of path curvature (Fig. 2) shows that, as a path-curvature pointer, this shift is vectorial in nature, sensing the plane of path curvature and detecting the direction to the center of path curvature (point O in Fig. 2(b)). To gain deeper insights into these results, we examine specific cases of a Gaussian and stepwise refractive-index profiles, n₀(r) = n_g(r) and n₀(r) = n_s(r). The effective beam size r₀ is r₀ = ρ/(v – 1)^1/2 for n₀(r) = n_g(r) and r₀ = ρ/(2lnv)^1/2 for n₀(r) = n_s(r). The beam shift can thus be expressed as

(23)$${\delta _{g,s}} \approx \textrm{ }[{{F_{g,s}}(v )/({2\Delta } )} ]({{\rho^2}/R} ),$$

Fig. 2. A bending refractive index profile (pink) gives rise to a shift of an optical beam (blue) in the direction opposite to the direction to the center of curvature, providing a vectorial pointer of beam-path curvature.

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with F_g(v) = v²/(v – 1)² for n₀(r) = n_g(r) and F_s(v) = v²/[4(lnv)²] for n₀(r) = n_s(r).

The bending-induced transition loss in the case of δ << r₀ is

(24)$${\kappa _{g,s}} \approx \textrm{ }[{{G_{g,s}}(v )/({8{\Delta ^2}} )} ]{({\rho /R} )^2},$$

with G_g(v) = v⁴/(v – 1)³ for n₀(r) = n_g(r) and G_s(v) = v⁴/[8(lnv)³] for n₀(r) = n_s(r). We note the order of smallness of δ_g,s and κ_g,s in ρ/R: δ_g,s/ρ ∼ O(ρ/R) and κ_g,s ∼ O((δ_g,s/ρ)²) ∼ O((ρ/R)²).

In the limit of low-v index profiles, v ≈ 1 + υ, corresponding to a shallow potential well V(r) = − n₀ (r) that can support only one or a few bound states, we approximate lnv as lnv ≈ υ to derive

(25)$${\delta _{g,s}}/\rho \approx \textrm{ }[/({2\Delta } )]({\rho /R} ){\upsilon ^{ - 2}}$$

And

(26)$${\kappa _{g,s}} \approx \textrm{ }[/({8{\Delta ^2}} )]{({\rho /R} )^2}{\upsilon ^{ - 3}},$$

with C_g = 1 and C_s = 1/2.

In the opposite limit of v >> 1, corresponding to a deep potential well V(r) = − n₀ (r) that supports a large manifold of bound states, we find that, as long as δ << r₀, the expressions for δ_g/ρ and κ_g are especially simple and instructive: δ_g/ρ ≈ (2Δ)⁻¹(ρ/R) and κ_g ≈ v/(8Δ²)(ρ/R)² ≈ (δ_g/ρ)²/2 . With 2δ_g/ρ being simply κ_g^1/2, this generic case offers an especially clear illustration of the ability of the beam shift as a pointer of bending-induced tunneling. That δ_g/ρ is simply (2Δ)⁻¹(ρ/R), on the other hand, clearly shows that lower-Δ and larger-ρ index profiles, e.g., large-core high-index-step waveguides, will enhance the sensitivity of path-curvature detection.

9. Conclusion

To summarize, we have shown that, when set to travel along a curved path, an optical beam tends to re-adjust its position, shifting away from the center of path curvature. An evolution-operator analysis of this effect shows that its properties can be understood in terms of the modal properties of the field state ket as defined by the symmetry of the beam path and the specific shape of the refractive index profile. The bending-induced beam shift and the related tunneling loss have been examined as functions of the index step and the beam diameter for generic refractive index profiles, showing that this beam shift can serve as a highly sensitive vectorial pointer for curved-path geodesics and bending-induced optical tunneling.

Funding

Russian Foundation for Basic Research (18-29-20031, 19-02-00473); Russian Science Foundation (20-12-00088- multidecade nonlinear optics); Welch Foundation (A-1801-20180324).

Acknowledgments

Useful discussions with I.V. Fedotov and A.A. Voronin are gratefully acknowledged.

Disclosures

The author declares no conflicts of interest.

References

1. W. G. Unruh and R. Schützhold, “On Slow Light as a Black Hole Analogue,” Phys. Rev. D 68(2), 024008 (2003). [CrossRef]

2. T. G. Philbin, C. Kuklewicz, S. Robertson, S. Hill, F. König, and U. Leonhardt, “Fiber-optical analog of the event horizon,” Science 319(5868), 1367–1370 (2008). [CrossRef]

3. K. E. Webb, M. Erkintalo, Y. Xu, N. G. R. Broderick, J. M. Dudley, G. Genty, and S. G. Murdoch, “Nonlinear optics of fibre event horizons,” Nat. Commun. 5(1), 4969 (2014). [CrossRef]

4. D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450(7172), 1054–1057 (2007). [CrossRef]

5. J. M. Dudley, F. Dias, M. Erkintalo, and G. Genty, “Instabilities, breathers and rogue waves in optics,” Nat. Photonics 8(10), 755–764 (2014). [CrossRef]

6. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, 1972).

7. A. M. Zheltikov, “Optical phase-space modes, self-focusing, and the wavelength as tunable ħ,” Phys. Scr. 90(12), 128003 (2015). [CrossRef]

8. W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248(4951), 73–76 (1990). [CrossRef]

9. W. R. Zipfel, R. M. Williams, and W. W. Webb, “Nonlinear magic: multiphoton microscopy in the biosciences,” Nat. Biotechnol. 21(11), 1369–1377 (2003). [CrossRef]

10. F. Helmchen and W. Denk, “Deep tissue two-photon microscopy,” Nat. Methods 2(12), 932–940 (2005). [CrossRef]

11. C. Freudiger, W. Min, B. Saar, S. Lu, G. Holtom, C. He, J. Tsai, J. Kang, and X. Sunney Xie, “Label-Free Biomedical Imaging with High Sensitivity by Stimulated Raman Scattering Microscopy,” Science 322(5909), 1857–1861 (2008). [CrossRef]

12. V. Westphal, S. O. Rizzoli, M. A. Lauterbach, D. Kamin, R. Jahn, and S. W. Hell, “Video-Rate Far-Field Optical Nanoscopy Dissects Synaptic Vesicle Movement,” Science 320(5873), 246–249 (2008). [CrossRef]

13. S. W. Hell, “Microscopy and its focal switch,” Nat. Methods 6(1), 24–32 (2009). [CrossRef]

14. V. Boyer, A. M. Marino, R. C. Pooser, and P. D. Lett, “Entangled images from four-wave mixing,” Science 321(5888), 544–547 (2008). [CrossRef]

15. G. Brida, M. Genovese, and I. R. Berchera, “Experimental realization of sub-shot-noise quantum imaging,” Nat. Photonics 4(4), 227–230 (2010). [CrossRef]

16. T. Ono, R. Okamoto, and S. Takeuchi, “An entanglement-enhanced microscope,” Nat. Commun. 4(1), 2426 (2013). [CrossRef]

17. G. B. Lemos, V. Borish, G. D. Cole, S. Ramelow, R. Lapkiewicz, and A. Zeilinger, “Quantum imaging with undetected photons,” Nature 512(7515), 409–412 (2014). [CrossRef]

18. F. Goos and H. Hänchen, “Ein neuer und fundamentaler Versuch zur total reflexion,” Ann. Phys. 436(7-8), 333–346 (1947). [CrossRef]

19. F. Goos and H. Hänchen, “Neumessung des strahlversetzungseffektes bei total reflexion,” Ann. Phys. 440(3-5), 251–252 (1949). [CrossRef]

20. F. I. Fedorov, “K teorii polnovo otrazenija,” Dokl. Akad. Nauk SSSR 105, 465–468 (1955).

21. C. Imbert, “Calculation and experimental proof of the transverse shift induced by total internal reflection of a circularly polarized light beam,” Phys. Rev. D 5(4), 787–796 (1972). [CrossRef]

22. M. Onoda, S. Murakami, and N. Nagaosa, “Hall effect of light,” Phys. Rev. Lett. 93(8), 083901 (2004). [CrossRef]

23. O. Hosten and P. Kwiat, “Observation of the spin-Hall effect of light via weak measurements,” Science 319(5864), 787–790 (2008). [CrossRef]

24. A. Aiello and J. P. Woerdman, “Role of beam propagation in Goos–Hänchen and Imbert–Fedorov shifts,” Opt. Lett. 33(13), 1437–1439 (2008). [CrossRef]

25. G. Jayaswal, G. Mistura, and M. Merano, “Weak measurement of the Goos–Hänchen shift,” Opt. Lett. 38(8), 1232–1234 (2013). [CrossRef]

26. G. Jayaswal, G. Mistura, and M. Merano, “Observation of the Imbert–Fedorov effect via weak value amplification,” Opt. Lett. 39(8), 2266–2269 (2014). [CrossRef]

27. F. Toppel, M. Ornigotti, and A. Aiello, “Goos–Hanchen and Imbert–Fedorov shifts from a quantum-mechanical perspective,” New J. Phys. 15(11), 113059 (2013). [CrossRef]

28. Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60(14), 1351–1354 (1988). [CrossRef]

29. N. W. M. Ritchie, J. G. Story, and R. G. Hulet, “Realization of a measurement of a ‘‘weak value’’,” Phys. Rev. Lett. 66(9), 1107–1110 (1991). [CrossRef]

30. S. Kocis, B. Braverman, S. Ravets, M. J. Stevens, R. P. Mirin, L. K. Shalm, and A. Steinberg, “Observing the Average Trajectories of Single Photons in a Two-Slit Interferometer,” Science 332(6034), 1170–1173 (2011). [CrossRef]

31. C. R. Leavens, “Weak Measurements from the point of view of Bohmian Mechanics,” Found. Phys. 35(3), 469–491 (2005). [CrossRef]

32. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

33. D. Marcuse, “Field deformation and loss caused by curvature of optical fibers,” J. Opt. Soc. Am. 66(4), 311–320 (1976). [CrossRef]

34. P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324(5924), 229–232 (2009). [CrossRef]

35. A. M. Zheltikov, M. N. Shneider, A. A. Voronin, A. V. Sokolov, and M. O. Scully, “Remote steering of laser beams by radar- and laser-induced refractive-index gradients in the atmosphere,” Laser Phys. Lett. 9(1), 68–72 (2012). [CrossRef]

36. D. Gloge and D. Marcuse, “Formal Quantum Theory of Light Rays,” J. Opt. Soc. Am. 59(12), 1629–1631 (1969). [CrossRef]

37. D. Stoler, “Operator methods in physical optics,” J. Opt. Soc. Am. 71(3), 334–341 (1981). [CrossRef]

38. A. W. Snyder and J. D. Love, “Tunnelling leaky modes on optical waveguides,” Opt. Commun. 12(3), 326–328 (1974). [CrossRef]

39. A. W. Snyder and D. J. Mitchell, “Leaky rays on circular optical fibers,” J. Opt. Soc. Am. 64(5), 599–607 (1974). [CrossRef]

40. R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, 1965).

41. G. Eichmann, “Quasi-Geometric Optics of Media with Inhomogeneous Index of Refraction,” J. Opt. Soc. Am. 61(2), 161–168 (1971). [CrossRef]

42. M. Eve, “The use of path integrals in guided wave theory,” Proc. Royal Soc. A 347, 405–417 (1976).

43. L. S. Schulman, Techniques and Applications of Path Integration (John Wiley & Sons, 1981).

44. J. W. Blaker and M. A. Tavel, “The application of Noether’s theorem to optical systems,” Am. J. Phys. 42(10), 857–861 (1974). [CrossRef]

45. J. D. Love and C. Winkler, “A universal tunneling coefficient for step- and graded-index multimode fibres,” Opt. Quantum Electron. 10(4), 341–351 (1978). [CrossRef]