On anomalous optical beam shifts at near-normal incidence

A. Derivation

To exhibit the near-normal Berry phase singularity as a particular example, we consider transmission through a q-plate with topological charge q = 2. Following the Jones matrix formalism in the k-space (see the supplementary material for the detailed derivation), we calculate analytically the GH and IF shifts of the Gaussian beam transmitted through the q-plate when the beam incidence plane is parallel to the waveplate main optic axis, which lies at the interface (Fig. 1). The Gaussian beam angular spectrum is defined by the Fourier spectrum f(μ, ν) = exp[−κ2(μ2 + ν2)/2], where κ=k0w0/2 is the inverse beam divergence. The case of q-plate in a real space is considered in the supplementary material by drawing an analogy between the near-normal and off-center beam incidence.Using this approach, we show that any type of normalized beam shift under near-normal incidence 〈L̃〉 could be defined by the universal formula expressed as the multiplication of three factors: (i) geometric Berry phase singularity factor depending on the beam width and angle of incidence G(κϑ), (ii) polarization-related factor depending on the polarization (Stokes parameters) of the incident beam S, and (iii) material-related factor depending on Fresnel coefficients τ,〈L̃〉=∑i=1,2Gi(κϑ)⋅Si⋅τi.(4)Equation (4) allows us to distinguish the contributions and to evaluate the different cases. For instance, the near-normal geometric Berry phase singularity G(κϑ) could be suppressed by the material-related factor τ (see the supplementary material, Sec. IV). Specific results for the four transmitted beam shifts read (see the supplementary material for the detailed derivation)〈P̃X〉=ΛX(κϑ)⋅S1⋅τ−ϑNt,(5)〈P̃Y〉=ΛY(κϑ)⋅S2⋅τ−ϑNt,(6)〈X̃〉=−γtΛX(κϑ)⋅S1⋅τ×ϑNt,(7)〈Ỹ〉=−ΛY(κϑ)⋅S2⋅τ×ϑNt−ΛSHE(κϑ)⋅S3⋅|t−|2ϑNt,(8)where S1 = |ex|2 − |ey|2, S2=2R[ex*ey], and S3=2I[ex*ey] are the corresponding Stokes parameters of the incident beam, Nt = [τ+ + τ−S1ΛY(κϑ)]/2 is the squared norm of the transmitted beam, t− = te − to, τ+,− = |te|2 ± |to|2, and τ×=2I(teto*) are real coefficients depending on the ordinary to and extraordinary te transmission amplitudes for the plane wave under normal incidence (see the supplementary material). Hereinafter, we use the dimensionless shift values defined as 〈P̃X,Y〉=〈PX,Y〉⋅κ2/k and 〈X̃,Ỹ〉=〈X,Y〉⋅k. The nonlinear geometric resonant factors ΛX, ΛY, and ΛSHE are the following:ΛX(κϑ)=6+4κ2ϑ2+κ4ϑ4e−κ2ϑ2+2κ2ϑ2−6κ4ϑ4,ΛY(κϑ)=6−4κ2ϑ2+κ4ϑ4−2e−κ2ϑ2κ2ϑ2+3κ4ϑ4,ΛSHE(κϑ)=1−e−κ2ϑ2.(9)Note that the squared norm of transmitted beam state Nt(ϑ) plays a role of a slight renormalization, in sharp contrast to large incident angles cases, where it causes singularity in the shifts by approaching zero.12–19,2712. J. B. Götte, W. Löffler, and M. R. Dennis, “Eigenpolarizations for giant transverse optical beam shifts,” Phys. Rev. Lett. 112, 233901 (2014). https://doi.org/10.1103/physrevlett.112.23390113. J. B. Götte and M. R. Dennis, “Limits to superweak amplification of beam shifts,” Opt. Lett. 38, 2295–2297 (2013). https://doi.org/10.1364/ol.38.00229514. A. Aiello, M. Merano, and J. P. Woerdman, “Brewster cross polarization,” Opt. Lett. 34, 1207–1209 (2009). https://doi.org/10.1364/ol.34.00120715. M. Merano, A. Aiello, M. P. Van Exter, and J. P. Woerdman, “Observing angular deviations in the specular reflection of a light beam,” Nat. Photonics 3, 337–340 (2009). https://doi.org/10.1038/nphoton.2009.7516. I. V. Soboleva, V. V. Moskalenko, and A. A. Fedyanin, “Giant Goos-Hänchen effect and Fano resonance at photonic crystal surfaces,” Phys. Rev. Lett. 108, 123901 (2012). https://doi.org/10.1103/physrevlett.108.12390117. L. Salasnich, “Enhancement of four reflection shifts by a three-layer surface-plasmon resonance,” Phys. Rev. A 86, 055801 (2012). https://doi.org/10.1103/physreva.86.05580118. X. Zhou, X. Lin, Z. Xiao, T. Low, A. Alù, B. Zhang, and H. Sun, “Controlling photonic spin Hall effect via exceptional points,” Phys. Rev. B 100, 115429 (2019). https://doi.org/10.1103/physrevb.100.11542919. O. Hosten and P. Kwiat, “Observation of the spin Hall effect of light via weak measurements,” Science 319, 787–790 (2008). https://doi.org/10.1126/science.115269727. Y. Gorodetski, K. Y. Bliokh, B. Stein, C. Genet, N. Shitrit, V. Kleiner, E. Hasman, and T. W. Ebbesen, “Weak measurements of light chirality with a plasmonic slit,” Phys. Rev. Lett. 109, 013901 (2012). https://doi.org/10.1103/PhysRevLett.109.013901 For the reflected beam, we arrive at the same results as in Eqs. (5)(8) with substitutions → , γt → γr, and Nt → Nr. For the q-plates with other nonzero q, the shifts are similar to Eqs. (5)(8), but with different nonlinear functions (9).

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