Abstract

The enhanced transmittance matrix approach developed by Moharam et al. [J. Opt. Soc. Am. A 12, 1077 (1995)] is reformulated in a concise and illuminating form in terms of scattering (reflection and transmission) matrices directly. Two equivalent recursive formulations, corresponding to their full- and partial-solution approaches, are presented and extended to allow simultaneous determination of both reflected and transmitted amplitudes. The relationships between these formulations and the S-matrix algorithm, together with their relative efficiencies and usefulness, are ascertained and compared by means of compact formulas featuring parallel algebraic structures. It is made evident that given the eigenmode solutions, the enhanced approach is the most direct and efficient way for deducing global scattering matrices.

© 2002 Optical Society of America

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References

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  1. D. Y. K. Ko, J. R. Sambles, “Scattering matrix method for propagation of radiation in stratified media: attenuated total reflection studies of liquid crystals,” J. Opt. Soc. Am. A 5, 1863–1866 (1988).
    [CrossRef]
  2. N. P. K. Cotter, T. W. Preist, J. R. Sambles, “Scattering-matrix approach to multilayer diffraction,” J. Opt. Soc. Am. A 12, 1097–1103 (1995).
    [CrossRef]
  3. L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996).
    [CrossRef]
  4. M. G. Moharam, D. A. Pommet, E. B. Grann, T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A 12, 1077–1086 (1995).
    [CrossRef]
  5. Following the common indexing convention for Ssubmatrices, rl,l+1gwould have been denoted by rl,l+2g.Since this matrix as well as Rl+1,l+2glumps all global reflections including those from far beyond layer l+2,the layer indices are simply taken to be two consecutive ones as indicative of the incident boundedness direction.
  6. K. Eidner, “Light propagation in stratified anisotropic media: orthogonality and symmetry properties of the 4×4 matrix formalisms,” J. Opt. Soc. Am. A 6, 1657–1660 (1989).
    [CrossRef]
  7. L. Li, “Multilayer-coated diffraction gratings: differential method of Chandezon et al. revisited,” J. Opt. Soc. Am. A 11, 2816–2828 (1994).
    [CrossRef]

1996 (1)

1995 (2)

1994 (1)

1989 (1)

1988 (1)

Cotter, N. P. K.

Eidner, K.

Gaylord, T. K.

Grann, E. B.

Ko, D. Y. K.

Li, L.

Moharam, M. G.

Pommet, D. A.

Preist, T. W.

Sambles, J. R.

J. Opt. Soc. Am. A (6)

Other (1)

Following the common indexing convention for Ssubmatrices, rl,l+1gwould have been denoted by rl,l+2g.Since this matrix as well as Rl+1,l+2glumps all global reflections including those from far beyond layer l+2,the layer indices are simply taken to be two consecutive ones as indicative of the incident boundedness direction.

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Figures (1)

Fig. 1
Fig. 1

Geometry of multilayered homogeneous or grating structure. There are N layers including the two semi-infinite regions containing the incident (layer 1) and transmitted (layer N) waves. The upper and lower bounding interfaces of each layer l are denoted by Zl> and Zl<, respectively.

Equations (27)

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fl(z)=ΨlPl(z)ul.
Ψl=[Ψl> Ψl<],Pl(z)=Pl>(z)00Pl<(z),
ul=ul>ul<.
fl(z)=ΨlPl(z)Nlcl,Nl=Pl>(-Zl<)00Pl<(-Zl>),
fl(z)=Ψlwl(z).
wl<(Zl>)=rl,l+1gwl>(Zl>),
wl+1>(Zl+1<)=tl,l+1gwl>(Zl>).
wl+1<(Zl+1>)=rl+1,l+2gwl+1>(Zl+1>).
wl+1<(Zl+1<)=Rl+1,l+2gwl+1>(Zl+1<),
Rl+1,l+2g=Pl+1<(Zl+1<-Zl+1>)rl+1,l+2g×Pl+1>(Zl+1>-Zl+1<)
=Pl+1<(-dl+1)rl+1,l+2gPl+1>(dl+1),
Ψl>+Ψl<rl,l+1g=[Ψl+1>+Ψl+1<Rl+1,l+2g]tl,l+1g.
(tl,l+1g)-1rl,l+1g(tl,l+1g)-1=Ψl-1[Ψl+1>+Ψl+1<Rl+1,l+2g].
rl,l+1gtl,l+1g=[-Ψl<[Ψl+1>+Ψl+1<Rl+1,l+2g]]-1[Ψl>],
t1,Ng=tN-1,NgPN-1>(dN-1)t2,3gP2>(d2)t1,2g.
rl,l+1ltl,l+1l=[-Ψl< Ψl+1>]-1[Ψl>].
tl+1,llrl+1,ll=[-Ψl< Ψl+1>]-1[-Ψl+1<].
rl,l+1gtl,l+1g=I0 0I-tl+1,llrl+1,ll[Rl+1,l+2g]-1rl,l+1ltl,l+1l.
rl,l+1g=rl,l+1l+tl+1,llRl+1,l+2gtl,l+1g,
tl,l+1g=[I-rl+1,llRl+1,l+2g]-1tl,l+1l,
tl+1,lgrl+1,lg=[-[Ψl<+Ψl>Rl,l-1g]Ψl+1>]-1[-Ψl+1<].
tl+1,lgrl+1,lg=I0-rl,l+1ltl,l+1l[Rl,l-1g]0I-1tl+1,llrl+1,ll.
wl(Zl>)=Tl+1,lwwl+1(Zl+1<),
Tl+1,lw=[Tl+1,lw> Tl+1,lw<]=Ψl-1Ψl+1,
(tl,l+1g)-1rl,l+1g(tl,l+1g)-1=Tl+1,lw>+Tl+1,lw<Rl+1,l+2g.
rl,l+1gtl,l+1g=[-Tl,l+1w<]IRl+1,l+2g-1[Tl,l+1w>].
Tl,l+1w=[Tl,l+1w> Tl,l+1w<]=Ψl+1-1Ψl.

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