Total internal reflection of orbital angular momentum beams

Simple optical reflection at planar interfaces still offers new surprises. More than 60 years ago, Goos and Hänchen found that a realistic optical beam experiences an in-plane displacement, with respect to the geometric optics expected path, under total internal reflection (TIR) [1]. This is caused by the fact that each of the plane-wave components, which constitute the beam, picks up a slightly different reflection coefficient. In other words, even if a beam has a flat wavefront (which is approximately true for well-collimated beams), beams are not plane waves and have a finite wavevector spread that needs to be taken into account during reflection. Nevertheless, beams are the best approximation of a geometric-optics ray, and it turns out that beam shifts are the first-order diffractive corrections to geometric optics. Beam shifts are polarization dependent. In-plane Goos–Hänchen shifts appear for p and s linear polarization while the (transverse) Imbert–Fedorov shifts appear for circular polarization [2, 3]. In recent years, the study of beam shift phenomena has received a strong boost from (amongst other effects) the discovery of the spin-Hall effect of light [4–6] and angular shifts [7–9]. In the meantime, shifts have been discovered not only for light, but also for matter waves [10, 11].

Beam shifts occurring under partial (external) reflection, such as angular beam shifts [9], are known to be sensitive to the spatial structure of the beam [12]. In particular, these shifts are sensitive to the beam's orbital angular momentum [13], which appears, for instance, in Laguerre–Gauss (LG) laser modes. We have recently shown that one can describe the diffractive corrections appearing upon reflection also as a change in the transverse mode spectrum of the beam; notably a pure LG input mode was found to acquire sidebands [14] upon external reflection. These results were obtained by external reflection of the input beam. How does this change if we investigate total internal reflection? Specifically, how does the orbital angular momentum influence beam shifts, and how is the OAM spectrum modified by TIR? This has not been studied yet and we shed light on this by theory and experiment.

We start with the general case of optical reflection, i.e. we do not yet specialize to the TIR case. Consider an incoming paraxial, monochromatic and homogeneously polarized (λ = 1,2 ≡ p,s), but otherwise arbitrary, optical field ${\mathbf{U}}^{\mathrm{i}}(x,y,z)={\sum }_{\lambda }{U}^{\mathrm{i}}(x,y,z){a}_{\lambda }{\hat {\mathbf{x}}}_{\lambda }^{\mathrm{i}}$ propagating along ${\hat {\mathbf{x}}}_{3}^{\mathrm{i}}$ (z coordinate), where (a₁,a₂) is the polarization Jones vector of the incoming beam. We use dimensionless quantities in units of 1/k₀, where k₀ is the wavevector. The coordinate systems and their unit vectors ${\hat {\mathbf{x}}}_{\lambda }^{\mathrm{i},\mathrm{r}}$ are attached to the incoming (i) and reflected (r) beam, respectively. After reflection at a dielectric interface, the polarization and spatial degree of freedom are coupled by the Fresnel reflection coefficients r_p,s as [12]

$$\begin{eqnarray} {\mathbf{U}}^{\mathrm{r}}(x,y,z)&=&\mathop{\sum }\limits _{\lambda }{a}_{\lambda }{r}_{\lambda }U(-x+{X}_{\lambda },y-{Y}_{\lambda },z)\hat {\mathbf{x}}_{\lambda }^{\mathrm{r}}\nonumber\\ &\equiv &\mathop{\sum }\limits _{\lambda }{a}_{\lambda }{r}_{\lambda }\mathbf{U}_{\lambda }^{\mathrm{r}}. \end{eqnarray} \tag{ 1 }$$

X_λ and Y_λ are the polarization-dependent dimensionless beam shifts (θ is the angle of incidence):

$$\begin{eqnarray} &&{X}_{1}=-\mathrm{i}\hspace{0.167em} {\partial }_{\theta }\left [\ln {r}_{1}(\theta )\right ],\tqs \nonumber\\ &&{Y}_{1}=\mathrm{i}\frac{{a}_{2}}{{a}_{1}}\left (1+\frac{{r}_{2}}{{r}_{1}}\right )\cot \theta \end{eqnarray} \tag{ 2a }$$

$$\begin{eqnarray} &&{X}_{2}=-\mathrm{i}\hspace{0.167em} {\partial }_{\theta }\left [\ln {r}_{2}(\theta )\right ],\tqs \nonumber\\ &&{Y}_{2}=-\mathrm{i}\frac{{a}_{1}}{{a}_{2}}\left (1+\frac{{r}_{1}}{{r}_{2}}\right )\cot \theta . \end{eqnarray} \tag{ 2b }$$

Their real parts yield the spatial beam shifts, and their imaginary parts the angular beam shifts. They can appear as longitudinal Goos–Hänchen type shifts [1] X_λ (along $\hat {x}$ coordinate), or as transverse Imbert–Fedorov type shifts [3, 4] Y_λ along $\hat {y}$. Transverse shifts Y_λ require that both a₁ and a₂ are finite, such as present in circularly polarized light; this is not necessary for the longitudinal shifts X_λ. To make the step from these dimensionless shifts to observable shifts, we have to find the centroid of the reflected beam:

$$\begin{eqnarray} &&\langle \mathbf{R}\rangle (z)=\mathop{\sum }\limits _{\lambda }{w}_{\lambda }\frac{\int \mathbf{R}\left \vert U(-x+{X}_{\lambda },y-{Y}_{\lambda },z)\right \vert ^{2}\mathrm{d}x \hspace{0.167em} \mathrm{d}y}{\int \left \vert U(-x+{X}_{\lambda },y-{Y}_{\lambda },z)\right \vert ^{2}\mathrm{d}x \hspace{0.167em} \mathrm{d}y}, \end{eqnarray} \tag{ 3 }$$

where w_λ = |r_λa_λ|²/∑_ν|r_νa_ν|² is the fraction of the reflected intensity with polarization λ, and $\mathbf{R}=x{\hat {\mathbf{x}}}_{1}^{\mathrm{r}}+y{\hat {\mathbf{x}}}_{2}^{\mathrm{r}}$. The shift of the centroid depends on the structure of the field, while the dimensionless beam shifts (equation (2)) are independent of the exact form of U. Equation (3) can be calculated straightforwardly by first-order Taylor expansion around zero shift (X_λ = Y_λ = 0). We obtain for the centroid, which is the expectation value for the 2D position vector R at distance z from the beam waist,

$$\begin{eqnarray} &&\langle \mathbf{R}\rangle (z)=\mathop{\sum }\limits _{\lambda }{w}_{\lambda }\left [\mathrm{Re}\left (\begin{array}{@{}c@{}} \displaystyle {X}_{\lambda }\\ \displaystyle {Y}_{\lambda } \end{array}\right )+M(z)\hspace{0.167em} \mathrm{Im}\left (\begin{array}{@{}c@{}} \displaystyle {X}_{\lambda }\\ \displaystyle {Y}_{\lambda } \end{array}\right )\right ], \end{eqnarray} \tag{ 4 }$$

where M(z) is a polarization-independent 2 × 2 matrix depending on the transverse mode of the field [15]:

$$\begin{eqnarray} &&{M}_{i j}(z)=2\frac{\mathrm{Im}\int {U}^{\ast }({x}_{i}\frac{\partial }{\partial {x}_{j}})U \hspace{0.167em} \mathrm{d}x \hspace{0.167em} \mathrm{d}y}{\int {U}^{\ast }U \hspace{0.167em} \mathrm{d}x \hspace{0.167em} \mathrm{d}y}, \end{eqnarray} \tag{ 5 }$$

with i,j = 1,2,x₁ ≡ x, and x₂ ≡ y. The z-dependent diagonal elements of M(z) describe how angular shifts influence the apparent position 〈R〉(z), and the off-diagonal elements effectively mix transverse angular shift into the longitudinal spatial shift, and the longitudinal angular shift into the transverse spatial shift. Evaluating (equation (3)) for Laguerre–Gauss beams with p = 0 and an OAM of ℓħ, and using the dimensionless Rayleigh range $\Lambda =2/{\theta }_{0}^{2}$, we obtain [12]:

$$\begin{eqnarray} \langle \mathbf{R}\rangle (z)&=&\mathop{\sum }\limits _{\lambda }{w}_{\lambda }\left [\mathrm{Re}\left (\begin{array}{@{}c@{}} \displaystyle {X}_{\lambda }\\ \displaystyle {Y}_{\lambda } \end{array}\right )\right .\nonumber\\ &&\left .+~\left (\begin{array}{@{}cc@{}} \displaystyle \frac{z}{\Lambda }(1+\vert \ell \vert )&\displaystyle -\ell \\ \displaystyle \ell &\displaystyle \frac{z}{\Lambda }(1+\vert \ell \vert ) \end{array}\right )\mathrm{Im}\left (\begin{array}{@{}c@{}} \displaystyle {X}_{\lambda }\\ \displaystyle {Y}_{\lambda } \end{array}\right )\right ]. \end{eqnarray} \tag{ 6 }$$

Finally, we specialize to the case of total internal reflection, where the Fresnel coefficients are imaginary. Hence, the dimensionless beam shifts X_λ and Y_λ (equations (2)) become real, which means that mixing via M(z) does not occur; therefore the occurring shifts are expected to be of purely spatial nature and independent of the orbital angular momentum ℓ.

This brings us to our first experiment as shown in figure 1, where we investigate if the OAM mode influences spatial Goos–Hänchen and Imbert–Fedorov shifts under TIR, and if angular shifts disappear as expected. The OAM beam is prepared with a custom HeNe laser and mode conversion. In the laser cavity, a wire (40 μm diameter) is introduced to enforce a Hermite–Gaussian (HG_nm) fundamental mode with m = 0. This mode is sent through an astigmatic mode converter [13] consisting of two cylindrical lenses. The nodal line of the HG mode is oriented at 45° relative to the common axis of the mode converter, such that a HG_n0 mode is transformed into an LG_ℓp mode with ℓ = n and p = 0. Additional lenses (L1 and L2) are used to ensure mode matching; the final beam after L2 has a beam waist of 775 μm. After polarization modulation with a liquid-crystal variable retarder (LCVR), this beam is then reflected internally at the hypotenuse of a 45°–90°–45° BK7 prism. We measure the polarization-differential beam displacement using a quadrant detector (which is binned to act effectively as a split detector), and lock-in techniques (for details, see [12]). Figure 2 demonstrates that our theoretical expectations were accurate: under TIR, only spatial shifts, here in the polarization-differential form Δ_GH ≡ Re[X₁ − X₂] and Δ_IF ≡ Re[Y₁ − Y₂], occur, and are independent of the OAM ℓ. The angular shifts Θ_GH ≡ Im[X₁ − X₂] and Θ_IF ≡ Im[Y₁ − Y₂] are identically zero. The shifts are also independent of the collimation properties of the beam as determined by the beam opening angle θ₀ (data not shown), which we expect from the discussion above.

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We now come to the second experiment, which addresses changes in the OAM spectrum upon total internal reflection, following [14]. We start by now evaluating equation (1) directly. A first-order Taylor expansion around zero shift (X_λ = Y_λ = 0) results in

$$\begin{eqnarray} \fl U(-x+{X}_{\lambda },y-{Y}_{\lambda },z)&\simeq &U(-x,y,z)\nonumber\\ &&+~{\mathbf{R}}_{\lambda }\boldsymbol{ \cdot }\frac{\partial }{\partial \mathbf{R}}U(-x,y,z), \end{eqnarray} \tag{ 7 }$$

with R = (x,y) and R_λ = (X_λ,Y_λ). We substitute U for the well-known normalized Laguerre–Gauss functions, $U\rightarrow {U}_{p,\ell }\rightarrow {\mathrm{LG}}_{p}^{\ell }$, where ℓ and p are the azimuthal and radial-mode indices, respectively. The spatial Fresnel coefficients ${c}_{\ell ,{\ell }^{\prime},p,{p}^{\prime}}^{\lambda }$ describe the scattering amplitude for an incoming ${\mathrm{LG}}_{p}^{\ell }$ mode into the ${\mathrm{LG}}_{{p}^{\prime}}^{{\ell }^{\prime}}$ output channel. These coefficients are obtained by OAM decomposition of the shifted reflected beam from equation (7)

$$\begin{eqnarray} &&{c}_{\ell ,{\ell }^{\prime},p,{p}^{\prime}}^{\lambda }=\int {\mathrm{d}}^{2}R \hspace{0.167em} {\mathrm{LG}}_{{p}^{\prime}}^{{\ell }^{\prime}\ast }(\mathbf{R}){U}_{p,\ell }(-x+{X}_{\lambda },y-{Y}_{\lambda },z). \end{eqnarray} \tag{ 8 }$$

If we consider only the OAM part of the Laguerre–Gauss modes (by setting p = p' = 0), we find the following simple coefficients (upper and lower signs refer to the case $\ell \ge 0$, and ℓ < 0, respectively):

$$\begin{eqnarray} {c}_{\ell ,{\ell }^{\prime}}^{\lambda }=\left \{ \begin{array}{@{}ll@{}} \displaystyle \pm {Z}_{\lambda }^{\pm }\sqrt{\vert \ell \vert +1}&\displaystyle \tqs \text{for}~{\ell }^{\prime}=-\ell \mp 1\\ \displaystyle \mp {Z}_{\lambda }^{\mp }\sqrt{\vert \ell \vert }&\displaystyle \tqs \text{for}~{\ell }^{\prime}=-\ell \pm 1\\ \displaystyle (-1)^{\ell }&\displaystyle \tqs \text{for}~{\ell }^{\prime}=-\ell \\ \displaystyle 0&\displaystyle \tqs \text{otherwise}. \end{array}\right . \end{eqnarray} \tag{ 9 }$$

We see that (in our first-order approximation, see equation (7)) these coefficients couple 'neighboring' OAM modes with ℓ' =− ℓ ± 1, where the minus sign stems from image reversal upon reflection. In other words, the OAM spectrum acquires sidebands upon reflection. The complex-valued parameters

$$\begin{eqnarray} &&{Z}_{\lambda }^{\pm }=\frac{{\theta }_{0}}{{2}^{3/2}}(-1)^{\ell }\left ({X}_{\lambda }\pm \mathrm{i}\hspace{0.167em} {Y}_{\lambda }\right ) \end{eqnarray} \tag{ 10 }$$

combine all dimensionless shifts. The intensity which appears in a specific sideband after reflection is ${C}_{\ell ,{\ell }^{\prime}}^{\lambda }=\vert {c}_{\ell ,{\ell }^{\prime}}^{\lambda }\vert ^{2}$; it depends on the strength of the shifts via equation (10), further it is proportional to ℓ and to the square of the beam opening angle ${\theta }_{0}^{2}$. In contrast to the previous case of beam shifts (equation (6)), the OAM sideband method does not discriminate real (spatial) and imaginary (angular) components of the dimensionless shifts X_λ and Y_λ, which is surprising.

To demonstrate experimentally the appearance of OAM sidebands under total internal reflection, we check these dependences by varying ℓ and θ₀. The setup is shown in figure 3, where we use again total internal reflection in a 45°–90°–45° prism (BK7). We collimate a fiber-coupled 635 nm diode laser with a 20 ×  objective, then use a spatial light modulator to imprint the incoming beam helical phase exp(iℓϕ). This method produces, in terms of Laguerre–Gauss modes, a superposition of modes with the same azimuthal index ℓ, but with many radial modes of different p; this superposition depends on the magnitude of ℓ. This OAM beam is then reflected internally at the prism hypotenuse. We analyze the reflected beam by its OAM spectrum with another combination of SLM and a single-mode fiber, which in turn is connected to a photo-diode. We align the setup for best mode matching between the single-mode fibers of the laser and the detector for the case of ℓ = ℓ' = 0. To vary θ₀, we introduce two microscope objectives (10 × , 0.25 NA, underfilled aperture). To compensate for small residual alignment errors, we use polarization modulation by λ/2 wave plates on rotation stages, and determine the total polarization-differential OAM sideband intensity I_pd(ℓ), see [14]. To obtain a theoretical prediction, we use numerical modeling of the experiment: this is required because (i) our setup has a transmission which depends strongly on the selected OAM mode (i.e., transmission for ℓ' ≡ ℓ, see supplementary information [14]); and (ii) the radial-mode superposition as produced by the SLMs has to be taken into account. In any case, the OAM part of the modes is well defined throughout our setup.

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In figure 4(a) we compare the OAM sideband intensity from experiment and theory at an angle of 45°, for a collimated (θ₀ = 0.0002) and a focused (θ₀ = 0.05) beam. We can confirm that the sideband intensity depends on the input OAM ℓ and on the beam opening angle θ₀, and agrees with the simulated data. Figure 4(b) shows the theoretically calculated total sideband intensity for an incoming ℓ = 4 OAM beam over a larger range of incident angles. In the case of a focused beam (θ₀ = 0.05), the sideband intensity exceeds 1% for the s polarization over the whole range of total internal reflection. This has to be taken into account if total internal reflection is used in, e.g., beam steering applications.

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We compare now our two experiments: in the first case, we found that in total internal reflection, beam shifts are independent of the OAM of the beam, and independent of beam focusing. However, if we measure the OAM sidebands appearing during total reflection, both properties influence the result (i.e., the sideband strength). As is obvious from the similarity of the experiments, both effects describe the same underlying physical phenomenon: diffractive corrections to geometric optics for total internal reflection. We want to explore this relation briefly theoretically. We start by writing the spatial part (for polarization λ) of the reflected field in a quantum-like notation based on the spatial Fresnel coefficients (we use their full form, see [14], for ℓ > 0):

$$\begin{eqnarray} \fl \vert \mathrm{out}\rangle _{\lambda }&=&\mathop{\sum }\limits _{{\ell }^{\prime},{p}^{\prime}}{a}_{\lambda }{r}_{\lambda }c_{\ell ,{\ell }^{\prime},p,{p}^{\prime}}^{\lambda }\vert \lambda \rangle \vert {\ell }^{\prime},{p}^{\prime}\rangle ={a}_{\lambda }(-1)^{\ell }\vert -\ell ,p\rangle \nonumber\\ &&+~{a}_{\lambda }{Z}_{\lambda }^{+}\sqrt{\ell +p+1}\vert -\ell -1,p\rangle \nonumber\\ &&+~{a}_{\lambda }{Z}_{\lambda }^{+}\sqrt{p}\vert -\ell -1,p-1\rangle \nonumber\\ &&-~{a}_{\lambda }{Z}_{\lambda }^{-}\sqrt{\ell +p}\vert -\ell +1,p\rangle \nonumber\\ &&-~{a}_{\lambda }{Z}_{\lambda }^{-}\sqrt{p+1}\vert -\ell +1,p+1\rangle . \end{eqnarray} \tag{ 11 }$$

The real space representation of this leads to the centroid as follows:

$$\begin{eqnarray} &&\langle \mathbf{R}\rangle (z)=\frac{\langle \mathrm{out}\vert \mathbf{R}\vert \mathrm{out}\rangle }{\langle \mathrm{out}\vert \mathrm{out}\rangle }. \end{eqnarray} \tag{ 12 }$$

To be able to compare this to equation (6), we need to find the sidebands for pure azimuthal LG input modes with p = 0. Equation (12) can easily be evaluated by using the known orthogonality relations of the LG modes, and by recognizing that the position operator R can be written as

$$\begin{eqnarray} \mathbf{R}&=&r\left ({\hat {\mathbf{x}}}_{1}^{\mathrm{r}}\cos \phi +{\hat {\mathbf{x}}}_{2}^{\mathrm{r}}\sin \phi \right )\nonumber\\ &=&\frac{r}{2}[{\mathrm{e}}^{\mathrm{i}\phi }({\hat {\mathbf{x}}}_{1}^{\mathrm{r}}-\mathrm{i}{\hat {\mathbf{x}}}_{2}^{\mathrm{r}})+{\mathrm{e}}^{-\mathrm{i}\phi }({\hat {\mathbf{x}}}_{1}^{\mathrm{r}}+\mathrm{i}{\hat {\mathbf{x}}}_{2}^{\mathrm{r}})]. \end{eqnarray} \tag{ 13 }$$

We see that this operator couples modes with Δℓ =± 1 in 〈out|R|out〉. Straightforward evaluation of equation (12) leads then exactly to equation (6), i.e., the dependency of ℓ and θ₀ (which was implicit in the coefficients ${c}_{\ell ,{\ell }^{\prime},p,{p}^{\prime}}^{\lambda }$) disappears for total internal reflection.

Can we give an intuitive explanation why θ₀-dependent OAM sidebands occur for TIR, while angular beam shifts are absent? In short, the measurement method is different. Spatial beam shifts are absolute, they do not depend on the beam waist. The OAM spectrum of a displaced beam, however, depends on the ratio of the displacement to the beam waist [16, 17]. If the beam is collimated, the Goos–Hänchen and Imbert–Fedorov shifts are usually very small compared to the beam waist, and the OAM spectrum is not modified; if the beam is focused, appreciable sidebands appear, in agreement with our experimental results in figure 4(a).

In conclusion, we have studied the behavior of OAM beams under total internal reflection. We have investigated this by two complementary methods: firstly, by the analysis of optical beam shifts, and secondly by the observation of modifications in the OAM spectrum of a probe beam by the spatial Fresnel coefficients. In the first case, we found that OAM does not modify the spatial beam shifts under total reflection. In the other case, the opposite is true: the OAM spectrum of a beam is modified under TIR, and the strength of these modifications increase with the OAM of the beam as well as its focusing. To resolve this issue, we have shown how to derive the beam shifts from the spatial Fresnel coefficients in the short didactical discussion at the top of this section.

Very recently, a third method to study such diffractive corrections was found: it turns out that physical reflection induces a splitting of a high-order vortex into spatially separated first-order vortices, and the splitting is characteristic for a given experimental condition [18]. Based upon our experimental result here, we would expect that such splitting will occur also under TIR: vortex splitting can be explained by coherent background fields [19], and the OAM sidebands, as found here, are of course an example of such a coherent background field.

We acknowledge financial support by NWO and the EU STREP program 255914 (PHORBITECH).