Maximal breaking of symmetry at critical angles and a closed-form expression for angular deviations of the Snell law

Manoel P. Araújo, Silvânia A. Carvalho, and Stefano De Leo

Phys. Rev. A 90, 033844 – Published 25 September 2014

Abstract

A detailed analysis of the propagation of laser Gaussian beams at critical angles shows under which conditions it is possible to maximize the breaking of symmetry in the angular distribution and for which values of the laser wavelength and beam waist it is possible to find an analytic formula for the maximal angular deviation from the optical path predicted by the Snell law. For beam propagation through $N$ dielectric blocks and for a maximal breaking of symmetry, a closed expression for the Goos-Hänchen shift is obtained. The multiple-peak phenomenon clearly represents additional evidence of the breaking of symmetry in the angular distribution of optical beams. Finally, the laser wavelength and beam-waist conditions to produce focal effects in the outgoing beam are also briefly discussed.

Received 3 July 2014
Revised 20 August 2014

DOI:https://doi.org/10.1103/PhysRevA.90.033844

Authors & Affiliations

Manoel P. Araújo¹, Silvânia A. Carvalho², and Stefano De Leo^2,*

¹Gleb Wataghin Physics Institute, State University of Campinas, Campinas, Brazil
²Department of Applied Mathematics, State University of Campinas, Campinas, Brazil

^*deleo@ime.unicamp.br

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Vol. 90, Iss. 3 — September 2014

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Figure 1
(a) Modeled breaking of symmetry. (b) The breaking of symmetry in the Gaussian angular distribution generates an axial dependence for the peak of the optical beam. This dependence is shown in (c). For the transversal mean value it is possible to obtain an axial linear analytical expression, given in Eq. (10), which is confirmed by the numerical data plotted in (d).
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Figure 2
Geometry of the dielectric block. The nor-mals of the left and right and up and down interfaces and the angular parameters which appear in the transmission coefficient are given in (a). For a symmetric angular distribution the outgoing beam is parallel to the incoming one. The breaking of symmetry generates an angular deviation $α$ of the Snell law, which is drawn in (b) together with the transversal Goos-Hänchen shift. The breaking of symmetry is maximized by building a dielectric structure of $N$ blocks in (c) which in a real optical experiment can be realized by a single elongated prism with sides $N \bar{B C}$ and $\bar{A B}$ .
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Figure 3
Symmetry breaking for $N$ dielectric blocks. The modeled breaking of symmetry discussed in Sec. 2 is now proposed for optical experiments at critical incidence ( $θ_{c} = 0$ , $φ_{c} = π / 4$ ). The plots show that to maximize the breaking of symmetry, we have to decrease the beam waist, increase the blocks number, and use $p$ -polarized waves. For $p$ -polarized waves, an optimal choice to obtain a maximal breaking of symmetry is represented by $N = 50$ and $k w_{0} = 10^{3}$ . To reproduce the maximal symmetry breaking for the other cases, we have to increase the number of blocks.
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Figure 4
Snell's law angular deviation. The axial dependence at the critical angle $θ_{c} = 0$ of the peak and transversal mean value are plotted for a fixed beam waist $w_{0}$ , $k w_{0} = 10^{3}$ , and $b / w_{0} = 10^{2}$ for different block numbers $N$ . The angular deviation of the Snell law is evident in (b) and (d). Observe that the first physical axial points at which we can perform the experimental analysis are given by $z_{out} = z_{in} + N tan φ_{c} \bar{A B}$ (crosses in the plots). From the plots the $N$ amplification of the Goos-Hänchen shift at the critical angle is also clear. The circles represent the points at which the numerical calculation has also been done for BK7 and fused silica blocks (Table 1).
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Figure 5
Multiple-peak phenomenon. For $k w_{0} = 10^{3}$ , $b / w_{0} = 10^{2}$ and for incidence at the critical angle, the outgoing optical beam presents the fascinating phenomenon of multiple peaks. This phenomenon is directly related to the spreading of the optical beam, and it occurs because in the angular distribution the positive angular components are no longer compensated by the negative ones.
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Figure 6
Focal effect. For $k w_{0} = 10^{4}$ , $b / w_{0} = 10$ and for incidence at the critical angle, the multiple-peak phenomenon is no longer so evident. Nevertheless, a new interesting phenomenon appears. Due to the second-order optical phase contribution, in the outgoing beam a focalization effect can be now observed. This effect is clear in (f).
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