Abstract

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.

© 1997 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–935 (1985).
    [CrossRef]
  2. M. G. Moharam, E. B. Grann, D. A. Pommet, T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995).
    [CrossRef]
  3. M. G. Moharam, E. B. Grann, D. A. Pommet, 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, 1077–1083 (1995).
    [CrossRef]
  4. P. Lalanne, G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
    [CrossRef]
  5. L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
    [CrossRef]
  6. L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “Finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
    [CrossRef]
  7. L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40, 553–573 (1993).
    [CrossRef]
  8. L. Li, “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 10, 2581–2591 (1993).
    [CrossRef]
  9. L. Li, “Bremmer series, R-matrix propagation algorithm, and numerical modeling of diffraction gratings,” J. Opt. Soc. Am. A 11, 2829–2836 (1994).
    [CrossRef]
  10. J. M. Elson, P. Tran, “Dispersion in photonic media and diffraction from gratings: a different modal expansion for the R-matrix propagation technique,” J. Opt. Soc. Am. A 12, 1765–1771 (1995).
    [CrossRef]
  11. J. B. Pendry, “Photonic band structure,” J. Mod. Opt. 41, 209–229 (1994).
    [CrossRef]
  12. K. Knop, “Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,” J. Opt. Soc. Am. 68, 1206–1210 (1978).
    [CrossRef]
  13. S. Peng, G. M. Morris, “Efficient implementation of rigorous coupled-wave analysis for surface-relief gratings,” J. Opt. Soc. Am. A 12, 1087–1096 (1995).
    [CrossRef]
  14. S. T. Han, Y. T. Tsao, R. M. Walser, M. F. Becker, “Electromagnetic scattering of two-dimensional surface-relief gratings,” Appl. Opt. 31, 2343–2352 (1992).
    [CrossRef] [PubMed]
  15. E. Noponen, J. Turunen, “Eigenmode method for electromagnetic synthesis of diffractive elements with three-dimensional profiles,” J. Opt. Soc. Am. A 11, 2494–2502 (1994).
    [CrossRef]
  16. P. St. Russell, “Power conservation and field structures in uniform dielectric gratings,” J. Opt. Soc. Am. A 1, 293–299 (1984).
    [CrossRef]
  17. A. Zylberberg, E. Marom, “Rigorous coupled-wave analysis of pure reflection gratings,” J. Opt. Soc. Am. 73, 392–398 (1983).
    [CrossRef]
  18. M. G. Moharam, T. K. Gaylord, “Comments on analyses of reflection gratings,” J. Opt. Soc. Am. 73, 399–401 (1983).
    [CrossRef]
  19. B. Hecht, H. Heinzelmann, D. W. Pohl, “Combined aperture SNON/PSTM: best of both worlds,” Ultramicroscopy 57, 228–230 (1995).
    [CrossRef]
  20. H. Kogelnik, “Theory of optical waveguides,” in Guided Wave Optics, T. Tamir, ed., Vol. 26 of Electronics and Photonics (Springer-Verlag, New York, 1988).
  21. W. H. Steel, Interferometry (Cambridge U. Press, Cambridge, 1983).
  22. H. Haus, Waves and Fields in Opto-Electronics (Wiley, New York, 1991).
  23. D. M. Pai, K. A. Awada, “Analysis of dielectric gratings of arbitrary profiles and thicknesses,” J. Opt. Soc. Am. A 8, 755–762 (1991).
    [CrossRef]
  24. H. Bremmer, “The WKB approximation as the first term of a geometric series,” Commun. Pure Appl. Math. 4, 105–115 (1951).
    [CrossRef]
  25. M. Nevière, F. Montiel, “Bragg–Fresnel gratings, zone plates and periodic stratified media: differential formalism revisited,” in Application and Theory of Periodic Structures, T. Janson, N. C. Gallagher, eds., Proc. SPIE2532, 56 (1995).
    [CrossRef]
  26. L. F. DeSandre, J. M. Elson, “Extinction-theorem analysis of diffraction anomalies in overcoated gratings,” J. Opt. Soc. Am. A 8, 763–777 (1991).
    [CrossRef]
  27. F. Montiel, M. Neviere, “Differential theory of gratings: extension to deep gratings of arbitrary profile and permittivity through the R-matrix propagation algorithm,” J. Opt. Soc. Am. A 11, 3241–3250 (1994).
    [CrossRef]
  28. L. Li, C. W. Haggans, “Convergence of the coupled-wave method for metallic lamellar diffraction gratings,” J. Opt. Soc. Am. A 10, 1184–1189 (1993).
    [CrossRef]
  29. J. M. Miller, J. Turunen, E. Noponen, A. Vasara, M. R. Taghizadeh, “Rigorous modal theory for multiply grooved lamellar gratings,” Opt. Commun. 111, 526–535 (1994).
    [CrossRef]
  30. L. C. Botten, M. S. Craig, R. C. McPhedran, “Highly conducting lamellar diffraction gratings,” Opt. Acta 28, 1103–1106 (1981).
    [CrossRef]
  31. G. Tayeb, R. Petit, “On the numerical study of deep conducting lamellar diffraction gratings,” Opt. Acta 31, 1361–1365 (1984).
    [CrossRef]
  32. R. H. Morf, “Exponentially convergent and numerically efficient solution of Maxwell's equations for lamellar gratings,” J. Opt. Soc. Am. A 12, 1043–1056 (1995).
    [CrossRef]

1996 (1)

1995 (6)

1994 (5)

1993 (3)

1992 (1)

1991 (2)

1985 (1)

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–935 (1985).
[CrossRef]

1984 (2)

G. Tayeb, R. Petit, “On the numerical study of deep conducting lamellar diffraction gratings,” Opt. Acta 31, 1361–1365 (1984).
[CrossRef]

P. St. Russell, “Power conservation and field structures in uniform dielectric gratings,” J. Opt. Soc. Am. A 1, 293–299 (1984).
[CrossRef]

1983 (2)

1981 (3)

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “Finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, “Highly conducting lamellar diffraction gratings,” Opt. Acta 28, 1103–1106 (1981).
[CrossRef]

1978 (1)

1951 (1)

H. Bremmer, “The WKB approximation as the first term of a geometric series,” Commun. Pure Appl. Math. 4, 105–115 (1951).
[CrossRef]

Adams, J. L.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “Finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

Andrewartha, J. R.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “Finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

Awada, K. A.

Becker, M. F.

Botten, L. C.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “Finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, “Highly conducting lamellar diffraction gratings,” Opt. Acta 28, 1103–1106 (1981).
[CrossRef]

Bremmer, H.

H. Bremmer, “The WKB approximation as the first term of a geometric series,” Commun. Pure Appl. Math. 4, 105–115 (1951).
[CrossRef]

Craig, M. S.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “Finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, “Highly conducting lamellar diffraction gratings,” Opt. Acta 28, 1103–1106 (1981).
[CrossRef]

DeSandre, L. F.

Elson, J. M.

Gaylord, T. K.

Grann, E. B.

Haggans, C. W.

Han, S. T.

Haus, H.

H. Haus, Waves and Fields in Opto-Electronics (Wiley, New York, 1991).

Hecht, B.

B. Hecht, H. Heinzelmann, D. W. Pohl, “Combined aperture SNON/PSTM: best of both worlds,” Ultramicroscopy 57, 228–230 (1995).
[CrossRef]

Heinzelmann, H.

B. Hecht, H. Heinzelmann, D. W. Pohl, “Combined aperture SNON/PSTM: best of both worlds,” Ultramicroscopy 57, 228–230 (1995).
[CrossRef]

Knop, K.

Kogelnik, H.

H. Kogelnik, “Theory of optical waveguides,” in Guided Wave Optics, T. Tamir, ed., Vol. 26 of Electronics and Photonics (Springer-Verlag, New York, 1988).

Lalanne, P.

Li, L.

Marom, E.

McPhedran, R. C.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “Finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, “Highly conducting lamellar diffraction gratings,” Opt. Acta 28, 1103–1106 (1981).
[CrossRef]

Miller, J. M.

J. M. Miller, J. Turunen, E. Noponen, A. Vasara, M. R. Taghizadeh, “Rigorous modal theory for multiply grooved lamellar gratings,” Opt. Commun. 111, 526–535 (1994).
[CrossRef]

Moharam, M. G.

Montiel, F.

F. Montiel, M. Neviere, “Differential theory of gratings: extension to deep gratings of arbitrary profile and permittivity through the R-matrix propagation algorithm,” J. Opt. Soc. Am. A 11, 3241–3250 (1994).
[CrossRef]

M. Nevière, F. Montiel, “Bragg–Fresnel gratings, zone plates and periodic stratified media: differential formalism revisited,” in Application and Theory of Periodic Structures, T. Janson, N. C. Gallagher, eds., Proc. SPIE2532, 56 (1995).
[CrossRef]

Morf, R. H.

Morris, G. M.

Neviere, M.

Nevière, M.

M. Nevière, F. Montiel, “Bragg–Fresnel gratings, zone plates and periodic stratified media: differential formalism revisited,” in Application and Theory of Periodic Structures, T. Janson, N. C. Gallagher, eds., Proc. SPIE2532, 56 (1995).
[CrossRef]

Noponen, E.

J. M. Miller, J. Turunen, E. Noponen, A. Vasara, M. R. Taghizadeh, “Rigorous modal theory for multiply grooved lamellar gratings,” Opt. Commun. 111, 526–535 (1994).
[CrossRef]

E. Noponen, J. Turunen, “Eigenmode method for electromagnetic synthesis of diffractive elements with three-dimensional profiles,” J. Opt. Soc. Am. A 11, 2494–2502 (1994).
[CrossRef]

Pai, D. M.

Pendry, J. B.

J. B. Pendry, “Photonic band structure,” J. Mod. Opt. 41, 209–229 (1994).
[CrossRef]

Peng, S.

Petit, R.

G. Tayeb, R. Petit, “On the numerical study of deep conducting lamellar diffraction gratings,” Opt. Acta 31, 1361–1365 (1984).
[CrossRef]

Pohl, D. W.

B. Hecht, H. Heinzelmann, D. W. Pohl, “Combined aperture SNON/PSTM: best of both worlds,” Ultramicroscopy 57, 228–230 (1995).
[CrossRef]

Pommet, D. A.

Russell, P. St.

Steel, W. H.

W. H. Steel, Interferometry (Cambridge U. Press, Cambridge, 1983).

Taghizadeh, M. R.

J. M. Miller, J. Turunen, E. Noponen, A. Vasara, M. R. Taghizadeh, “Rigorous modal theory for multiply grooved lamellar gratings,” Opt. Commun. 111, 526–535 (1994).
[CrossRef]

Tayeb, G.

G. Tayeb, R. Petit, “On the numerical study of deep conducting lamellar diffraction gratings,” Opt. Acta 31, 1361–1365 (1984).
[CrossRef]

Tran, P.

Tsao, Y. T.

Turunen, J.

E. Noponen, J. Turunen, “Eigenmode method for electromagnetic synthesis of diffractive elements with three-dimensional profiles,” J. Opt. Soc. Am. A 11, 2494–2502 (1994).
[CrossRef]

J. M. Miller, J. Turunen, E. Noponen, A. Vasara, M. R. Taghizadeh, “Rigorous modal theory for multiply grooved lamellar gratings,” Opt. Commun. 111, 526–535 (1994).
[CrossRef]

Vasara, A.

J. M. Miller, J. Turunen, E. Noponen, A. Vasara, M. R. Taghizadeh, “Rigorous modal theory for multiply grooved lamellar gratings,” Opt. Commun. 111, 526–535 (1994).
[CrossRef]

Walser, R. M.

Zylberberg, A.

Appl. Opt. (1)

Commun. Pure Appl. Math. (1)

H. Bremmer, “The WKB approximation as the first term of a geometric series,” Commun. Pure Appl. Math. 4, 105–115 (1951).
[CrossRef]

J. Mod. Opt. (2)

L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40, 553–573 (1993).
[CrossRef]

J. B. Pendry, “Photonic band structure,” J. Mod. Opt. 41, 209–229 (1994).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (14)

E. Noponen, J. Turunen, “Eigenmode method for electromagnetic synthesis of diffractive elements with three-dimensional profiles,” J. Opt. Soc. Am. A 11, 2494–2502 (1994).
[CrossRef]

L. Li, “Bremmer series, R-matrix propagation algorithm, and numerical modeling of diffraction gratings,” J. Opt. Soc. Am. A 11, 2829–2836 (1994).
[CrossRef]

F. Montiel, M. Neviere, “Differential theory of gratings: extension to deep gratings of arbitrary profile and permittivity through the R-matrix propagation algorithm,” J. Opt. Soc. Am. A 11, 3241–3250 (1994).
[CrossRef]

P. St. Russell, “Power conservation and field structures in uniform dielectric gratings,” J. Opt. Soc. Am. A 1, 293–299 (1984).
[CrossRef]

D. M. Pai, K. A. Awada, “Analysis of dielectric gratings of arbitrary profiles and thicknesses,” J. Opt. Soc. Am. A 8, 755–762 (1991).
[CrossRef]

L. F. DeSandre, J. M. Elson, “Extinction-theorem analysis of diffraction anomalies in overcoated gratings,” J. Opt. Soc. Am. A 8, 763–777 (1991).
[CrossRef]

L. Li, C. W. Haggans, “Convergence of the coupled-wave method for metallic lamellar diffraction gratings,” J. Opt. Soc. Am. A 10, 1184–1189 (1993).
[CrossRef]

L. Li, “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 10, 2581–2591 (1993).
[CrossRef]

P. Lalanne, G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
[CrossRef]

R. H. Morf, “Exponentially convergent and numerically efficient solution of Maxwell's equations for lamellar gratings,” J. Opt. Soc. Am. A 12, 1043–1056 (1995).
[CrossRef]

M. G. Moharam, E. B. Grann, D. A. Pommet, T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995).
[CrossRef]

M. G. Moharam, E. B. Grann, D. A. Pommet, 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, 1077–1083 (1995).
[CrossRef]

S. Peng, G. M. Morris, “Efficient implementation of rigorous coupled-wave analysis for surface-relief gratings,” J. Opt. Soc. Am. A 12, 1087–1096 (1995).
[CrossRef]

J. M. Elson, P. Tran, “Dispersion in photonic media and diffraction from gratings: a different modal expansion for the R-matrix propagation technique,” J. Opt. Soc. Am. A 12, 1765–1771 (1995).
[CrossRef]

Opt. Acta (4)

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “Finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, “Highly conducting lamellar diffraction gratings,” Opt. Acta 28, 1103–1106 (1981).
[CrossRef]

G. Tayeb, R. Petit, “On the numerical study of deep conducting lamellar diffraction gratings,” Opt. Acta 31, 1361–1365 (1984).
[CrossRef]

Opt. Commun. (1)

J. M. Miller, J. Turunen, E. Noponen, A. Vasara, M. R. Taghizadeh, “Rigorous modal theory for multiply grooved lamellar gratings,” Opt. Commun. 111, 526–535 (1994).
[CrossRef]

Proc. IEEE (1)

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–935 (1985).
[CrossRef]

Ultramicroscopy (1)

B. Hecht, H. Heinzelmann, D. W. Pohl, “Combined aperture SNON/PSTM: best of both worlds,” Ultramicroscopy 57, 228–230 (1995).
[CrossRef]

Other (4)

H. Kogelnik, “Theory of optical waveguides,” in Guided Wave Optics, T. Tamir, ed., Vol. 26 of Electronics and Photonics (Springer-Verlag, New York, 1988).

W. H. Steel, Interferometry (Cambridge U. Press, Cambridge, 1983).

H. Haus, Waves and Fields in Opto-Electronics (Wiley, New York, 1991).

M. Nevière, F. Montiel, “Bragg–Fresnel gratings, zone plates and periodic stratified media: differential formalism revisited,” in Application and Theory of Periodic Structures, T. Janson, N. C. Gallagher, eds., Proc. SPIE2532, 56 (1995).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

Reflection and transmission of light at the interface of (a) an infinitely thick homogeneous layer with complex permittivities 2, μ2; and (b) an infinitely thick lamellar grating with complex permittivities 2(x), μ2(x). Here a0 is the incident amplitude of the relevant field (a0=Ey0 for TE polarization, and a0=Hy0 for TM polarization). See text for further explanation of notation.

Fig. 2
Fig. 2

Reflection and transmission of light at the interface of (a) an homogeneous layer with complex permittivities 2, μ2 and thickness d; and (b) a lamellar grating with complex permittivities 2(x), μ2(x) and thickness d. Here a0 is the incident amplitude of the relevant field (a0=Ey0 for TE polarization, and a0 =Hy0 for TM polarization). See text for further explanation of notation.

Fig. 3
Fig. 3

Incoming and outgoing (vector) fields at the interfaces of a lamellar grating. Here ai and bi are vectors whose elements are the mode amplitudes of the incoming and the outgoing fields, respectively. These vectors are related by means of matrix equations detailed in the text.

Fig. 4
Fig. 4

Diffraction from a surface grating is modeled as diffraction from a stack of lamellar gratings. The surface-relief grating is divided into M layers, with each being modeled as a lamellar grating [i.e., (x){1(x), 2(x)·M(x)}]. One computes diffraction efficiencies for the entire structure by recursively calculating the reflection matrix Γi, using the reflection matrix from the adjacent layer Γi-1. The recursive algorithm for this procedure is detailed in the text.

Tables (4)

Tables Icon

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

Tables Icon

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

Tables Icon

Table 3 Simplification of Table 1 when Regions 1 and 3 in Fig. 2 are Homogeneous with σ1, Constant Permittivitya

Tables Icon

Table 4 Comparison of Diffraction Efficiencies Computed by Use of Coupled-Wave (CW) and Hybrid Coupled-Wave (HCW) Models Described in the Texta

Equations (56)

Equations on this page are rendered with MathJax. Learn more.

lϕl(x)(al+bl)=qψq(x)tq,
l 1σ1(x)ϕl(x)(k1z)l(al-bl)
=k 1σ2(x)ψq(x)(k2z)qtq,
σi(x)μi(x)for TE polarization,
σi(x)i(x)for TM polarization.
Einc=ES+EL,
Hinc=HS+HL,
(Σ1-1)ql=ψq|1σ1|ϕlσ2σ1,
(Σ2-1)ql=ψq|1σ2|ϕlσ1σ2,
(K1z)ll=(k1z)lδll,
(K2z)qq=(k2z)qδqq.
Σ2-1·(a+b)=t,
Σ1-1K1z·(a-b)=K2z·t,
b=1Σ1-1K1z+K2zΣ2-1·[Σ1-1K1z-K2zΣ2-1]·aΓ·a,
t=Σ2-1 2Σ1-1K1z+K2zΣ2-1Σ1-1K1z·aT·a.
Sx=12Λ0Λdx(E×H*+E*×H),
S1z=12l(al*-bl*)(k1z)l(al+bl)+l(al-bl)(k1z)l(al*-bl*)=a·K1z+K1z2·aIncident Power-a·Γ·K1z+K1z2·ΓaReflected Power+aΓ·K1z-K1z2·a+a·K1z-K1z2·Γ·a,Mode Conversion/Tunneling Power S2z=a·T·K2z+K2z2·T·a.
S1z=S2z.
Γ=r1+t1t2r2 exp(-2jϕ)1-r22 exp(-2jϕ),
t=t1t2 exp(-2jϕ)1-r22 exp(-jϕ).
qψq(x)(aq+bq)=lϕl(x)tl,
q 1σ2(x)ψq(x)(k2z)q(aq-bq)
=l 1σ1(x)ϕl(x)(k1z)ltl,
Γ2=-Σ2-1Γ1Σ2,
T2=Σ2T1K1z-1Σ1K2z.
b1=Γ1a1+T2a2,b3=Γ2a3+T1a4,
b2=T1a1+Γ2a2,b4=T2a3+Γ1a4.
a2=exp(-jK2zh)b3P0b3
a3=exp(-jK2zh)b2P0b2.
b1=Γ1+T2P0 11-Γ2P0Γ2P0Γ2P0T1a1=Γ1-(I-Γ1)P˜0 11-Γ1P˜0Γ1P˜0Γ1P˜0(I+Γ1)a1Γa¯1,
b4=T2P0 11-Γ2P0Γ2P0T1a1=(I-Γ1)P˜0 11-Γ1P˜0Γ1P˜0(I+Γ1)a1T˜a1,
P˜0=Σ2P0Σ2-1.
S1z=(a1-b1)K1z(a1+b1)+adj=(b2-a2)K2z(a2+b2)+adj,
S3z=b4K3zb4+adj=(a3-b3)K3z(a3+b3)+adj,
S1z=S3z.
b1=γiai+tia2,b3=Γ˜i-1+a3,
b2=tia1+γia2,b4=T˜i-1+a3,
a2=b3,a3=b2.
Γ˜i+=γi+tiΓ˜i-1+ 11-γiΓ˜i-1+ti,
T˜i+=T˜i-1+ 11-γiΓ˜i-1+ti.
(σˆ2)nmϕn|σ2(x)|ϕm=σ1Σ2Σ1-1,
(χˆ2)nm=ϕn|1σ2(x)|ϕm=1σ1Σ1Σ2-1,
HΣ2-σˆ2G 1σˆ2G+k02EΣ2=Σ2K2z2,
HΣ2[-χ2-1GE-1G+k02χˆ-1]Σ2=Σ2K2z2.
Σ2Σ2=σˆ2χˆ2-1,
K2zexact=F(K2zrcw),
K˜2z=Σ2rcwK2zexact(Σ2rcw)-1,
F(x, z)=F(x)exp(-jkzz),
σ(x) ddx1σ(x)ddx+k02(x)Fl(x)=kz2Fl(x).
{F(x), σi(x)}{Ey, μi(x)}for TE polarization,
{F(x), σi(x)}{Hy, i(x)}for TM polarization,
σ¯(x) ddx1σ¯(x)ddx+k02¯(x)Fl(x)=(kz)2Fl(x),
Fl| 1σ(x)|Fl0Λdx 1σ(x)F¯l(x)Fl(x)=δll,
h(x)=l=-Fl| 1σ(x)|hFl(x),
(Σ1)lq=ϕl| 1σ2|ψq,
(Σ2)lq=ϕl| 1σ1|ψq.

Metrics