Abstract

A new numerical method is presented which is suitable for the analysis of scattering by dielectric structures which are periodic in one dimension. This method is based on a rigorous differential equation formulation of the scattering problem. It is particularly appropriate for structures which have periods on the order of the incident wavelength, and which have a depth of several wavelengths or less. Results are presented for periodic dielectric structures for TE polarization. Comparisons with results that have been obtained by other authors, and with a case involving inhomogeneous dielectric for which the exact solution can be found, are made for transmission type dielectric structures. These results establish the accuracy of the method. Further results are presented for reflection structures consisting of arrays of rectangular dielectric cylinders over a ground plane, and for transmission structures consisting of arrays of rectangular cylinders in free space, as well as for transmission structures consisting of semi-infinite dielectric media with sinusoidal or rectangular surface height variation.

© 1978 Optical Society of America

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References

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  1. P. Debye and F. Sears, “On the Scattering of Light by Supersonic Waves,” Proc. Natl. Acad. Sci. (U.S.) 18, 409–414 (1932).
    [CrossRef]
  2. C. Raman and N. Nath, “The Diffraction of Light by Sound Waves of High Frequency,” published in 5 parts, J. Ind. Acad. Sci., Part I,  2, 406 (1935); Part II,  2, 413 (1935); Part III,  3, 75 (1936); Part IV,  3, 119 (1936),; Part V,  3, 459 (1936).
  3. N. Nath, “The Diffraction of Light by High Frequency Sound Waves: Generalized Theory,” Proc. Ind. Acad. Sci. 4, No. 2, 222–242 (1936).
  4. N. Nath, “The Diffraction of Light by Supersonic Waves,” Proc. Ind. Acad. Sci. 8, 499–503 (1938).
  5. R. Aggarwal, “Diffraction of Light by Ultrasonic Waves,” Proc. Ind. Acad. Sci. 31, No. 6, 417–426 (1950).
  6. A. Bhatia and W. Noble, “Diffraction of Light by Ultrasonic Waves,” Parts I and II, Proc. R. Soc. Lon. 220, 356–385 (1953).
    [CrossRef]
  7. P. Phariseau, “On the Diffraction of Light by Progressive Supersonic Waves,” J. Ind. Acad. Sci. 44, No. 4, 165–170 (1956).
  8. C. Burckhardt, “Diffraction of a Plane Wave at a Sinusoidally Stratified Dielectric Grating,” J. Opt. Soc. Am. 56, 1502–1509 (1966).
    [CrossRef]
  9. H. Kogelnik, “Coupled Wave Theory for Thick Hologram Gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
    [CrossRef]
  10. L. Bergstein and D. Kermisch, “Image Storage and Reconstruction in Volume Holography,” in Proceedings of the Symposium on Modern Optics (Polytechnic Press of Polytechnic Institute of Brooklyn, 1967).
  11. D. Kermisch, “Nonuniform sinusoidally modulated dielectric gratings,” J. Opt. Soc. Am. 59, 1409–1414 (1969).
    [CrossRef]
  12. P. Bousquet, “Étude de la Réflexion et de la Transmission de la Lumiére par un Réseau Transparent à Profil Sinusoidal. Extension au cas d’une Surface Irrégulière,” Rev. Opt. 41, No. 6, 277–294 (1962).
  13. J. Pavageau, “Réflexion et Transmission d’une Onde Électromagneétique Plane par un Réseau Transparent à Profil Sinusoidal,” Rev. Opt. 42, No. 9, 451–462 (1963).
  14. E. Ippen, “Diffraction of Light by Acoustic Surface Waves,” Ph.D. dissertation (University of California, Berkeley, 1968).
  15. K. Zaki, “Numerical Methods for the Analysis of Scattering from Nonplanar Periodic Structures,” Ph.D. dissertation (University of California, Berkeley, 1969).
  16. M. Neviere, M. Cadilhac, and R. Petit, “Application des Proprietés des Transformations Conformes à L’étude de Probleme de la Diffraction d’une onde Electromagnétique Plane par un Reseau Infiniment Conducteur,” URSI Symposium, Tbilisi, USSR, 1971; also “Applications of Conformal Mappings to the Diffraction of Electromagnetic Waves by a Grating,” IEEE Trans. Antennas Prop. 21, No. 1, 37–46 (1973).
  17. M. Neviere, R. Petit, and M. Cadilhac, “About the Theory of Optical Grating Coupler-Waveguide Systems,” Opt. Commun. 8, No. 2, 113–117 (1973).
    [CrossRef]
  18. M. Neviere, P. Vincent, and R. Petit, “Sur la Théorie du Réseau Conducteur et sés Applications a’ L’Optique,” Nouv. Rev. Opt. 5, No. 2, 65–77 (1974).
  19. K. K. Mei, “Unimoment Method of Solving Antenna and Scattering Problems,” IEEE Trans. Antennas Propag. 22, 760–766 (1974).
    [CrossRef]
  20. D. E. Tremain, “Analysis of Scattering by Metal, Dielectric, and Inhomogeneous Dielectric Periodic Structures,” Ph.D. dissertation (University of California, Berkeley, 1974).
  21. E. Angel, “Invariant Imbedding, Difference Equations, and Elliptic Boundary Value Problems,” J. Comput. Sys. Sci. 4, 473–491 (1969).
    [CrossRef]
  22. N. Amitay and V. Galindo, “On Energy Conservation and the Method of Moments in Waveguide Discontinuity and Scattering Problems,” URSI Symposium, Washington, D.C., 1969.
  23. D. Tseng, A. Messel, and A. Oliner, “Scattering by a Multimode Corrugated Structure with Application to P-type Wood Anomalies,” URSI Symposium, Stresa, Italy, 1968.
  24. H. Kogelnik, “Reconstructing Response and Efficiency of Hologram Gratings,” in Proceedings of the Symposium on Modern Optics (Polytechnic Press of the Polytechnic Institute of Brooklyn, 1967).

1974 (2)

M. Neviere, P. Vincent, and R. Petit, “Sur la Théorie du Réseau Conducteur et sés Applications a’ L’Optique,” Nouv. Rev. Opt. 5, No. 2, 65–77 (1974).

K. K. Mei, “Unimoment Method of Solving Antenna and Scattering Problems,” IEEE Trans. Antennas Propag. 22, 760–766 (1974).
[CrossRef]

1973 (1)

M. Neviere, R. Petit, and M. Cadilhac, “About the Theory of Optical Grating Coupler-Waveguide Systems,” Opt. Commun. 8, No. 2, 113–117 (1973).
[CrossRef]

1969 (3)

H. Kogelnik, “Coupled Wave Theory for Thick Hologram Gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

D. Kermisch, “Nonuniform sinusoidally modulated dielectric gratings,” J. Opt. Soc. Am. 59, 1409–1414 (1969).
[CrossRef]

E. Angel, “Invariant Imbedding, Difference Equations, and Elliptic Boundary Value Problems,” J. Comput. Sys. Sci. 4, 473–491 (1969).
[CrossRef]

1966 (1)

1963 (1)

J. Pavageau, “Réflexion et Transmission d’une Onde Électromagneétique Plane par un Réseau Transparent à Profil Sinusoidal,” Rev. Opt. 42, No. 9, 451–462 (1963).

1962 (1)

P. Bousquet, “Étude de la Réflexion et de la Transmission de la Lumiére par un Réseau Transparent à Profil Sinusoidal. Extension au cas d’une Surface Irrégulière,” Rev. Opt. 41, No. 6, 277–294 (1962).

1956 (1)

P. Phariseau, “On the Diffraction of Light by Progressive Supersonic Waves,” J. Ind. Acad. Sci. 44, No. 4, 165–170 (1956).

1953 (1)

A. Bhatia and W. Noble, “Diffraction of Light by Ultrasonic Waves,” Parts I and II, Proc. R. Soc. Lon. 220, 356–385 (1953).
[CrossRef]

1950 (1)

R. Aggarwal, “Diffraction of Light by Ultrasonic Waves,” Proc. Ind. Acad. Sci. 31, No. 6, 417–426 (1950).

1938 (1)

N. Nath, “The Diffraction of Light by Supersonic Waves,” Proc. Ind. Acad. Sci. 8, 499–503 (1938).

1936 (1)

N. Nath, “The Diffraction of Light by High Frequency Sound Waves: Generalized Theory,” Proc. Ind. Acad. Sci. 4, No. 2, 222–242 (1936).

1935 (1)

C. Raman and N. Nath, “The Diffraction of Light by Sound Waves of High Frequency,” published in 5 parts, J. Ind. Acad. Sci., Part I,  2, 406 (1935); Part II,  2, 413 (1935); Part III,  3, 75 (1936); Part IV,  3, 119 (1936),; Part V,  3, 459 (1936).

1932 (1)

P. Debye and F. Sears, “On the Scattering of Light by Supersonic Waves,” Proc. Natl. Acad. Sci. (U.S.) 18, 409–414 (1932).
[CrossRef]

Aggarwal, R.

R. Aggarwal, “Diffraction of Light by Ultrasonic Waves,” Proc. Ind. Acad. Sci. 31, No. 6, 417–426 (1950).

Amitay, N.

N. Amitay and V. Galindo, “On Energy Conservation and the Method of Moments in Waveguide Discontinuity and Scattering Problems,” URSI Symposium, Washington, D.C., 1969.

Angel, E.

E. Angel, “Invariant Imbedding, Difference Equations, and Elliptic Boundary Value Problems,” J. Comput. Sys. Sci. 4, 473–491 (1969).
[CrossRef]

Bergstein, L.

L. Bergstein and D. Kermisch, “Image Storage and Reconstruction in Volume Holography,” in Proceedings of the Symposium on Modern Optics (Polytechnic Press of Polytechnic Institute of Brooklyn, 1967).

Bhatia, A.

A. Bhatia and W. Noble, “Diffraction of Light by Ultrasonic Waves,” Parts I and II, Proc. R. Soc. Lon. 220, 356–385 (1953).
[CrossRef]

Bousquet, P.

P. Bousquet, “Étude de la Réflexion et de la Transmission de la Lumiére par un Réseau Transparent à Profil Sinusoidal. Extension au cas d’une Surface Irrégulière,” Rev. Opt. 41, No. 6, 277–294 (1962).

Burckhardt, C.

Cadilhac, M.

M. Neviere, R. Petit, and M. Cadilhac, “About the Theory of Optical Grating Coupler-Waveguide Systems,” Opt. Commun. 8, No. 2, 113–117 (1973).
[CrossRef]

M. Neviere, M. Cadilhac, and R. Petit, “Application des Proprietés des Transformations Conformes à L’étude de Probleme de la Diffraction d’une onde Electromagnétique Plane par un Reseau Infiniment Conducteur,” URSI Symposium, Tbilisi, USSR, 1971; also “Applications of Conformal Mappings to the Diffraction of Electromagnetic Waves by a Grating,” IEEE Trans. Antennas Prop. 21, No. 1, 37–46 (1973).

Debye, P.

P. Debye and F. Sears, “On the Scattering of Light by Supersonic Waves,” Proc. Natl. Acad. Sci. (U.S.) 18, 409–414 (1932).
[CrossRef]

Galindo, V.

N. Amitay and V. Galindo, “On Energy Conservation and the Method of Moments in Waveguide Discontinuity and Scattering Problems,” URSI Symposium, Washington, D.C., 1969.

Ippen, E.

E. Ippen, “Diffraction of Light by Acoustic Surface Waves,” Ph.D. dissertation (University of California, Berkeley, 1968).

Kermisch, D.

D. Kermisch, “Nonuniform sinusoidally modulated dielectric gratings,” J. Opt. Soc. Am. 59, 1409–1414 (1969).
[CrossRef]

L. Bergstein and D. Kermisch, “Image Storage and Reconstruction in Volume Holography,” in Proceedings of the Symposium on Modern Optics (Polytechnic Press of Polytechnic Institute of Brooklyn, 1967).

Kogelnik, H.

H. Kogelnik, “Coupled Wave Theory for Thick Hologram Gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

H. Kogelnik, “Reconstructing Response and Efficiency of Hologram Gratings,” in Proceedings of the Symposium on Modern Optics (Polytechnic Press of the Polytechnic Institute of Brooklyn, 1967).

Mei, K. K.

K. K. Mei, “Unimoment Method of Solving Antenna and Scattering Problems,” IEEE Trans. Antennas Propag. 22, 760–766 (1974).
[CrossRef]

Messel, A.

D. Tseng, A. Messel, and A. Oliner, “Scattering by a Multimode Corrugated Structure with Application to P-type Wood Anomalies,” URSI Symposium, Stresa, Italy, 1968.

Nath, N.

N. Nath, “The Diffraction of Light by Supersonic Waves,” Proc. Ind. Acad. Sci. 8, 499–503 (1938).

N. Nath, “The Diffraction of Light by High Frequency Sound Waves: Generalized Theory,” Proc. Ind. Acad. Sci. 4, No. 2, 222–242 (1936).

C. Raman and N. Nath, “The Diffraction of Light by Sound Waves of High Frequency,” published in 5 parts, J. Ind. Acad. Sci., Part I,  2, 406 (1935); Part II,  2, 413 (1935); Part III,  3, 75 (1936); Part IV,  3, 119 (1936),; Part V,  3, 459 (1936).

Neviere, M.

M. Neviere, P. Vincent, and R. Petit, “Sur la Théorie du Réseau Conducteur et sés Applications a’ L’Optique,” Nouv. Rev. Opt. 5, No. 2, 65–77 (1974).

M. Neviere, R. Petit, and M. Cadilhac, “About the Theory of Optical Grating Coupler-Waveguide Systems,” Opt. Commun. 8, No. 2, 113–117 (1973).
[CrossRef]

M. Neviere, M. Cadilhac, and R. Petit, “Application des Proprietés des Transformations Conformes à L’étude de Probleme de la Diffraction d’une onde Electromagnétique Plane par un Reseau Infiniment Conducteur,” URSI Symposium, Tbilisi, USSR, 1971; also “Applications of Conformal Mappings to the Diffraction of Electromagnetic Waves by a Grating,” IEEE Trans. Antennas Prop. 21, No. 1, 37–46 (1973).

Noble, W.

A. Bhatia and W. Noble, “Diffraction of Light by Ultrasonic Waves,” Parts I and II, Proc. R. Soc. Lon. 220, 356–385 (1953).
[CrossRef]

Oliner, A.

D. Tseng, A. Messel, and A. Oliner, “Scattering by a Multimode Corrugated Structure with Application to P-type Wood Anomalies,” URSI Symposium, Stresa, Italy, 1968.

Pavageau, J.

J. Pavageau, “Réflexion et Transmission d’une Onde Électromagneétique Plane par un Réseau Transparent à Profil Sinusoidal,” Rev. Opt. 42, No. 9, 451–462 (1963).

Petit, R.

M. Neviere, P. Vincent, and R. Petit, “Sur la Théorie du Réseau Conducteur et sés Applications a’ L’Optique,” Nouv. Rev. Opt. 5, No. 2, 65–77 (1974).

M. Neviere, R. Petit, and M. Cadilhac, “About the Theory of Optical Grating Coupler-Waveguide Systems,” Opt. Commun. 8, No. 2, 113–117 (1973).
[CrossRef]

M. Neviere, M. Cadilhac, and R. Petit, “Application des Proprietés des Transformations Conformes à L’étude de Probleme de la Diffraction d’une onde Electromagnétique Plane par un Reseau Infiniment Conducteur,” URSI Symposium, Tbilisi, USSR, 1971; also “Applications of Conformal Mappings to the Diffraction of Electromagnetic Waves by a Grating,” IEEE Trans. Antennas Prop. 21, No. 1, 37–46 (1973).

Phariseau, P.

P. Phariseau, “On the Diffraction of Light by Progressive Supersonic Waves,” J. Ind. Acad. Sci. 44, No. 4, 165–170 (1956).

Raman, C.

C. Raman and N. Nath, “The Diffraction of Light by Sound Waves of High Frequency,” published in 5 parts, J. Ind. Acad. Sci., Part I,  2, 406 (1935); Part II,  2, 413 (1935); Part III,  3, 75 (1936); Part IV,  3, 119 (1936),; Part V,  3, 459 (1936).

Sears, F.

P. Debye and F. Sears, “On the Scattering of Light by Supersonic Waves,” Proc. Natl. Acad. Sci. (U.S.) 18, 409–414 (1932).
[CrossRef]

Tremain, D. E.

D. E. Tremain, “Analysis of Scattering by Metal, Dielectric, and Inhomogeneous Dielectric Periodic Structures,” Ph.D. dissertation (University of California, Berkeley, 1974).

Tseng, D.

D. Tseng, A. Messel, and A. Oliner, “Scattering by a Multimode Corrugated Structure with Application to P-type Wood Anomalies,” URSI Symposium, Stresa, Italy, 1968.

Vincent, P.

M. Neviere, P. Vincent, and R. Petit, “Sur la Théorie du Réseau Conducteur et sés Applications a’ L’Optique,” Nouv. Rev. Opt. 5, No. 2, 65–77 (1974).

Zaki, K.

K. Zaki, “Numerical Methods for the Analysis of Scattering from Nonplanar Periodic Structures,” Ph.D. dissertation (University of California, Berkeley, 1969).

Bell Syst. Tech. J. (1)

H. Kogelnik, “Coupled Wave Theory for Thick Hologram Gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

K. K. Mei, “Unimoment Method of Solving Antenna and Scattering Problems,” IEEE Trans. Antennas Propag. 22, 760–766 (1974).
[CrossRef]

J. Comput. Sys. Sci. (1)

E. Angel, “Invariant Imbedding, Difference Equations, and Elliptic Boundary Value Problems,” J. Comput. Sys. Sci. 4, 473–491 (1969).
[CrossRef]

J. Ind. Acad. Sci. (1)

P. Phariseau, “On the Diffraction of Light by Progressive Supersonic Waves,” J. Ind. Acad. Sci. 44, No. 4, 165–170 (1956).

J. Ind. Acad. Sci., Part I (1)

C. Raman and N. Nath, “The Diffraction of Light by Sound Waves of High Frequency,” published in 5 parts, J. Ind. Acad. Sci., Part I,  2, 406 (1935); Part II,  2, 413 (1935); Part III,  3, 75 (1936); Part IV,  3, 119 (1936),; Part V,  3, 459 (1936).

J. Opt. Soc. Am. (2)

Nouv. Rev. Opt. (1)

M. Neviere, P. Vincent, and R. Petit, “Sur la Théorie du Réseau Conducteur et sés Applications a’ L’Optique,” Nouv. Rev. Opt. 5, No. 2, 65–77 (1974).

Opt. Commun. (1)

M. Neviere, R. Petit, and M. Cadilhac, “About the Theory of Optical Grating Coupler-Waveguide Systems,” Opt. Commun. 8, No. 2, 113–117 (1973).
[CrossRef]

Parts I and II, Proc. R. Soc. Lon. (1)

A. Bhatia and W. Noble, “Diffraction of Light by Ultrasonic Waves,” Parts I and II, Proc. R. Soc. Lon. 220, 356–385 (1953).
[CrossRef]

Proc. Ind. Acad. Sci. (3)

N. Nath, “The Diffraction of Light by High Frequency Sound Waves: Generalized Theory,” Proc. Ind. Acad. Sci. 4, No. 2, 222–242 (1936).

N. Nath, “The Diffraction of Light by Supersonic Waves,” Proc. Ind. Acad. Sci. 8, 499–503 (1938).

R. Aggarwal, “Diffraction of Light by Ultrasonic Waves,” Proc. Ind. Acad. Sci. 31, No. 6, 417–426 (1950).

Proc. Natl. Acad. Sci. (U.S.) (1)

P. Debye and F. Sears, “On the Scattering of Light by Supersonic Waves,” Proc. Natl. Acad. Sci. (U.S.) 18, 409–414 (1932).
[CrossRef]

Rev. Opt. (2)

P. Bousquet, “Étude de la Réflexion et de la Transmission de la Lumiére par un Réseau Transparent à Profil Sinusoidal. Extension au cas d’une Surface Irrégulière,” Rev. Opt. 41, No. 6, 277–294 (1962).

J. Pavageau, “Réflexion et Transmission d’une Onde Électromagneétique Plane par un Réseau Transparent à Profil Sinusoidal,” Rev. Opt. 42, No. 9, 451–462 (1963).

Other (8)

E. Ippen, “Diffraction of Light by Acoustic Surface Waves,” Ph.D. dissertation (University of California, Berkeley, 1968).

K. Zaki, “Numerical Methods for the Analysis of Scattering from Nonplanar Periodic Structures,” Ph.D. dissertation (University of California, Berkeley, 1969).

M. Neviere, M. Cadilhac, and R. Petit, “Application des Proprietés des Transformations Conformes à L’étude de Probleme de la Diffraction d’une onde Electromagnétique Plane par un Reseau Infiniment Conducteur,” URSI Symposium, Tbilisi, USSR, 1971; also “Applications of Conformal Mappings to the Diffraction of Electromagnetic Waves by a Grating,” IEEE Trans. Antennas Prop. 21, No. 1, 37–46 (1973).

L. Bergstein and D. Kermisch, “Image Storage and Reconstruction in Volume Holography,” in Proceedings of the Symposium on Modern Optics (Polytechnic Press of Polytechnic Institute of Brooklyn, 1967).

N. Amitay and V. Galindo, “On Energy Conservation and the Method of Moments in Waveguide Discontinuity and Scattering Problems,” URSI Symposium, Washington, D.C., 1969.

D. Tseng, A. Messel, and A. Oliner, “Scattering by a Multimode Corrugated Structure with Application to P-type Wood Anomalies,” URSI Symposium, Stresa, Italy, 1968.

H. Kogelnik, “Reconstructing Response and Efficiency of Hologram Gratings,” in Proceedings of the Symposium on Modern Optics (Polytechnic Press of the Polytechnic Institute of Brooklyn, 1967).

D. E. Tremain, “Analysis of Scattering by Metal, Dielectric, and Inhomogeneous Dielectric Periodic Structures,” Ph.D. dissertation (University of California, Berkeley, 1974).

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Figures (10)

FIG. 1
FIG. 1

Periodic dielectric structure: (a) Reflection type structure; (b) Transmission type structure.

FIG. 2
FIG. 2

Discretization of region II in defining u(n).

FIG. 3
FIG. 3

Discretization of region II in defining v(n).

FIG. 4
FIG. 4

Comparison of unimoment results with Pavageau’s formulas.

FIG. 5
FIG. 5

Dimensions of the sinusoidal groove for Table II.

FIG. 6
FIG. 6

Dimensions of the periodic dielectric slab of Table III.

FIG. 7
FIG. 7

Relative space harmonic power vs normalized depth for array of rectangular cylinders over a ground plane.

FIG. 8
FIG. 8

Relative space harmonic power vs angle of incidence for array of rectangular cylinders in free space.

FIG. 9
FIG. 9

Relative space harmonics power vs angle of incidence for transmission grating with rectangular groove shape.

FIG. 10
FIG. 10

Relative space harmonics power vs angle of incidence for transmission grating with sinusoidal groove shape.

Tables (3)

Tables Icon

TABLE I Comparison with exact solution for vertically inhomogeneous slab.

Tables Icon

TABLE II Comparison with Zaki’s integral equation approach.

Tables Icon

TABLE III Comparison with the formulas of Bergstein and Kermisch.

Equations (47)

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

ϕ ( x + d , z ) = e - j β 0 d ϕ ( x , z )             for all x , z .
ϕ inc = e - j β 0 x e j γ 0 z ,
ϕ ref = n = - A n e - j β n x e - j γ n z             region I ;
ϕ trans = n = - B n e - j β n x e + j γ n ( z + b )             region III .
β n = β 0 + 2 n π / d ,
β n 2 + γ n 2 = k 1 2 ,
β n 2 + ( γ n ) 2 = k 3 2 , k 3 = ω ( μ 0 ɛ 3 ) 1 / 2 .
[ 2 + k 2 ( x , z ) ] ϕ II = 0 ,             k 2 ( x , z ) = ω 2 μ 0 ɛ ( x , z ) .
ϕ z metal surface = 0.
ϕ inc z = 0 + ϕ ref z = 0 = ϕ II z = 0 ,
z ϕ inc | z = 0 + z ϕ ref | z = 0 = z ϕ II | z = 0 .
ϕ trans z = - b = ϕ II z = - b ,
z ϕ trans | z = - b = z ϕ II | z = - b .
n propagating orders A n 2 γ n / γ 0 = 1
n propagating orders A n 2 γ n γ 0 + n propagating orders B n 2 γ n / γ 0 = 1.
P n r = A n 2 γ n / γ 0 ;             P n t = B n 2 γ n / γ 0 ;             γ n , γ n real .
ϕ ( x , z ) = n = - L L a n ( z ) e - j β n x .
k 2 ( x , z ) = n = - M M b n ( z ) e - j 2 n π x / d ,             - b z 0.
( d 2 / d z 2 ) a ( z ) + D ( z ) a ( z ) = 0 ,
a ( z ) [ a - L , , a + L ] T             ( T transpose )
D m , n ( z ) = b m - n ( z ) - δ m , n β m 2 ,             m , n = - L , , L ,             D [ D m , n ] .
9 ( a )             A m + δ m , 0 = a m ( 0 ) ,             m = - L , , + L ;
9 ( b ) j γ 0 δ m , 0 - j γ m A m = ( / z ) a m ( z ) z = 0 ,             m = - L , , + L .
10 ( a )             a m ( z = - b ) = B m ,
10 ( b )             ( / z ) a m ( z = - b ) = j γ m B m .
a ( z ) = n = - L L [ f n u ( n ) ( z ) + g n v ( n ) ( z ) ] ,             z region II u ( n ) ( u - L ( n ) , u L ( n ) ) T ; v ( n ) ( v - L ( n ) , , v + L ( n ) ) T ,             f n             and             g n are scalars .
( d 2 d z 2 + D ( z ) ) { u ( n ) ( z ) v ( n ) ( z ) } = 0 .
v ( n ) ( 0 ) = u ( n ) ( - b ) = 0 .
u ( n ) ( 0 ) = v ( n ) ( z = - b ) [ 0 1 0 0 ] 1             in             row             n + L + 1             n = - L , , + L .
u ( n ) ( m + 2 ) - D ( m + 1 ) u ( n ) ( m + 1 ) + u ( n ) ( m ) = 0 ,
D ( m ) 2 I - ( Δ z ) 2 D ( m ) , D ( m ) D ( z = row             m ) , u ( m ) u ( z = row m ) .
u ( n ) ( m + 1 ) = R ( m ) u ( n ) ( m ) + s ( m ) ,
R ( m ) = [ D ( m + 1 ) - R ( m + 1 ) ] - 1
R ( q - 1 ) = 0 ,             s ( m ) = 0             for all m .
z f ( z ) z = 0 3 f ( 0 ) - 4 f ( - Δ z ) + f ( - 2 Δ z ) 2 Δ z .
( Δ z ) 2 b - n u L + n ( m + 1 ) ,             n = 1 , , 2 L .
u L ( m + 2 ) + u L ( m ) - [ 2 + ( Δ z ) 2 β L 2 ] u L ( m + 1 ) + ( Δ z ) 2 b 1 u L - 1 ( m + 1 ) + .
A m + δ m , 0 = n = - L L f n u m ( n ) ( z = 0 ) ,
m = - L , , + L B m = n = - L L g n v m ( n ) ( z = - b ) .
28 ( a )             A m + δ m , 0 = f m ,             m = - L , , + L
28 ( b )             B m = g m .
j γ 0 δ m , 0 - j γ m A m = 1 2 Δ z n = - L L { f n [ 3 u m ( n ) ( z = 0 ) - 4 u m ( n ) ( z = - Δ z ) + u m ( n ) ( z = - 2 Δ z ) ] + g n [ - 4 v m ( n ) ( z = - Δ z ) + v m ( n ) ( z = - 2 Δ z ) ] } ,             m = - L , , + L .
j γ m A m + 1 2 Δ z n = - L L { A n 3 u m ( n ) ( 0 ) - 4 u m ( n ) ( - Δ z ) + u m ( n ) ( - 2 Δ z ) ] + B n [ - 4 v m ( n ) ( - Δ z ) + v m ( n ) ( - 2 Δ z ) ] } = - 1 2 Δ z [ 3 u m ( 0 ) ( 0 ) - 4 u m ( 0 ) ( - Δ z ) + u m ( 0 ) ( - 2 Δ z ) ] + j γ 0 δ m , 0 ,             m = - L , , + L .
j γ m B m + 1 2 Δ z n = - L L { A n [ - 4 u m ( n ) ( - b + Δ z ) + u m ( n ) ( - b + 2 Δ z ) ] + B n [ 3 v m ( n ) ( - b ) - 4 v m ( n ) ( - b + Δ z ) + v m ( n ) ( - b + 2 Δ z ) ] } = 1 2 Δ z [ 4 u m ( 0 ) ( - b + Δ z ) - u m ( 0 ) ( - b + 2 Δ z ) ] ,             m = - L , , + L .
a ( z ) = n = - L L f n u ( n ) ( z ) .
j γ m A m + 1 2 Δ z n = - L L A n [ 3 u m ( n ) ( 0 ) - 4 u m ( n ) ( - Δ z ) + u m ( n ) ( - 2 Δ z ) ] = j γ 0 δ m , 0 - 1 2 Δ z [ 3 u m ( 0 ) ( 0 ) - 4 u m ( 0 ) ( - Δ z ) + u m ( 0 ) ( - 2 Δ z ) ] ,             m = - L , , + L .
k 2 ( x , z ) = k 0 2 ɛ ( x , z ) = C 1 2 / ( C 2 + C 3 z ) 4 .