Sumanth Kaushik, "Vector Fresnel equations and Airy formula for one-dimensional multilayer and surface-relief gratings," J. Opt. Soc. Am. A 14, 596-609 (1997)
A simple and intuitive formalism is presented to describe diffraction in multilayered periodic structures. A modal theory of diffraction is used to show how well-known results from scalar analysis (wave propagation in homogeneous layered media) can readily be generalized to vector problems. Specifically, the following results are derived: (1) generalized Fresnel equations appropriate for reflection and transmission from an infinitely thick grating, (2) a generalized equation for power conservation for diffraction gratings, (3) a generalized Airy formula for thin film to describe reflection and transmission of light through a lamellar grating, and (4) a matrix propagation method akin to that used to calculate reflection and transmission of multilayer thin films. Some numerical results are also presented to illustrate the applications of this research and its relationship to previous modal theories.
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Comparison of the Generalized Fresnel Formulas for Light Reflecting from an Infinitely Thick Lamellar Grating [see Fig. 1(b)] with the Classical Fresnel Formulas for Light Reflecting from a Homogeneous Interface [see Fig. 1(a)]a
Interface
Homogeneous
Inhomogeneous
Reflection
Transmission
Power flow
Boldface indicates matrix and vector quantities, and † denotes Hermitian conjugate. See text for detailed description.
Table 2
Comparison of the Generalized Airy Relations for Light Reflecting from a Lamellar Grating [Fig. 2(b)] with Classical Formulas for Light Reflecting from a Homogeneous Slab [see Fig. 2(a)]a
Homogeneous Slab
Lamellar Grating
Reflection
Transmission
Stokes (reciprocity) relations
Boldface indicates matrix and vector quantities. The matrix propagation is given by ; the scalar propagation, by . See text for detailed description.
Table 3
Simplification of Table 1 when Regions 1 and 3 in Fig. 2 are Homogeneous with σ1, Constant Permittivitya
TE Polarization
TM Polarization
The differences in formulas for TE and TM polarization are discussed in Section 5.
Table 4
Comparison of Diffraction Efficiencies Computed by Use of Coupled-Wave (CW) and Hybrid Coupled-Wave (HCW) Models Described in the Texta
The quantities in italics are values that have not converged with respect to the number of orders or modes and therefore could be potentially incorrect.
Ref. 10.
Ref. 8.
Tables (4)
Table 1
Comparison of the Generalized Fresnel Formulas for Light Reflecting from an Infinitely Thick Lamellar Grating [see Fig. 1(b)] with the Classical Fresnel Formulas for Light Reflecting from a Homogeneous Interface [see Fig. 1(a)]a
Interface
Homogeneous
Inhomogeneous
Reflection
Transmission
Power flow
Boldface indicates matrix and vector quantities, and † denotes Hermitian conjugate. See text for detailed description.
Table 2
Comparison of the Generalized Airy Relations for Light Reflecting from a Lamellar Grating [Fig. 2(b)] with Classical Formulas for Light Reflecting from a Homogeneous Slab [see Fig. 2(a)]a
Homogeneous Slab
Lamellar Grating
Reflection
Transmission
Stokes (reciprocity) relations
Boldface indicates matrix and vector quantities. The matrix propagation is given by ; the scalar propagation, by . See text for detailed description.
Table 3
Simplification of Table 1 when Regions 1 and 3 in Fig. 2 are Homogeneous with σ1, Constant Permittivitya
TE Polarization
TM Polarization
The differences in formulas for TE and TM polarization are discussed in Section 5.
Table 4
Comparison of Diffraction Efficiencies Computed by Use of Coupled-Wave (CW) and Hybrid Coupled-Wave (HCW) Models Described in the Texta
The quantities in italics are values that have not converged with respect to the number of orders or modes and therefore could be potentially incorrect.
Ref. 10.
Ref. 8.
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