Generalized matrix transformation formalism for reflection and transmission of complex optical waves at a plane dielectric interface

Abstract

We describe a generalized formalism, addressing the fundamental problem of reflection and transmission of complex optical waves at a plane dielectric interface. Our formalism involves the application of generalized operator matrices to the incident constituent plane wave fields to obtain the reflected and transmitted constituent plane wave fields. We derive these matrices and describe the complete formalism by implementing these matrices. This formalism, though physically equivalent to Fresnel formalism, has greater mathematical elegance and computational efficiency as compared to the latter. We utilize exact 3D expressions of the constituent plane wavevectors and electric fields of the incident, reflected and transmitted waves, which enable us to seamlessly analyse plane waves, paraxial and non-paraxial beams, highly diverging and tightly focused beam-fields as well as waves of miscellaneous wavefront-shapes and properties using the single formalism. The exact electric field expressions automatically include the geometric phase information; while we retain the wavefront curvature information by using appropriate multiplicative factors. We demonstrate our formalism by obtaining the reflected and transmitted fields in a simulated Gaussian beam model; in particular, by exploring the existence and nature of phase vortex in the longitudinal electric field component of the reflected beam -thus showing spin-to-orbital angular momentum conversion. Finally, we briefly discuss how our generalized formalism is capable of analysing the reflection-transmission problem of a very large class of complex optical waves -by referring to some novel works from the current literature as exemplary cases.

Observation of Polarization Singularities in a Brewster-Reflected Paraxial Beam

“…We define (i) the initial input collimated beam (before the lens L 1 ) in the I coordinate system; and (ii) analyse the final output collimated beam (after the lens L 2 ) in the R coordinate system. The incident diverging beam after L 1 and the reflected diverging beam before L 2 are analysed in the simulation [60] (including the lens transformations); whose mathematical details are not required here. The mathematical forms of only the (i) initial and (ii) final collimated beams are relevant for the purpose of the present paper.…”

“…The fundamental phase and polarization (lemons, node) singularity patterns, simulated based on our formalism of Ref. [60] applied to a Brewster-reflected post-paraxial optical beam, are also experimentally observed [82] To observe the underlying OAM structure in the E ±σ ± fields [Eq. (8)] for the θ E = 0 • case, we first pass the output beam through an appropriately oriented QWP to transform the E beam-field to where,d ± = (x ±ŷ)/ √ 2.…”

“…1] based on our formalism developed in Ref. [60]. In a medium of refractive index n 1 , an initial collimated Gaussian beam is converted to a weakly-diverging post-paraxial Gaussian beam by passing it through a concave lens L 1 of focal length F 1 = −O F O I .…”

“…(topological indices respectively +1/2, −1/2, −1/2, +1, +1, −1) [1,6,7,[9][10][11][12][13][14][15][16][17][18][19]. Standard methods of generating vortex beams with embedded polarization singularities [1,17,[45][46][47][48][49][50][51][52][53][54] thus involve, in some form or the other, the inherent inhomogeneous polarization of the considered beam-field [55][56][57][58][59][60][61].…”

“…A classic case of inhomogeneous polarization in the beam cross-section occurs for the reflection and transmission of a composite optical wave, where, the constituent plane waves reflect and transmit differently due to differences in Fresnel coefficients [62][63][64]; thus generating complicated reflected and transmitted field profiles [55][56][57][58][59][60][65][66][67][68][69][70][71][72]. This effect is especially unique when a constituent wave with transverse magnetic (TM) polarization is incident at Brewster angle (θ B ) [62][63][64], because this constituent wave is not reflected at all.…”