Abstract

The propagation of electromagnetic radiation in periodically stratified media is considered. Media of finite, semi-infinite, and infinite extent are treated. A diagonalization of the unit cell translation operator is used to obtain exact solutions for the Bloch waves, the dispersion relations, and the band structure of the medium. Some new phenomena with applications to integrated optics and laser technology are presented.

© 1977 Optical Society of America

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References

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  1. F. Abeles, Ann. Phys. (Paris) 5, 596 (1950);Ann. Phys. (Paris) 5, 706 (1950).
  2. A. Ashkin and A. Yariv, Bell Labs. Tech. Memo MM-61-124-46 (13November1961) (unpublished).
  3. N. Bloembergen and A. J. Sievers, Appl. Phys. Lett. 17, 483 (1970).
    [Crossref]
  4. C. L. Tang and P. P. Bey, IEEE J. Quantum Electron. QE-9, 9 (1973).
    [Crossref]
  5. S. M. Rytov, Zh. Eksp. Teor. Fiz. 29, 605 (1955)[Sov. Phys. -JETP 2, 466 (1956)].
  6. J. P. van der Ziel, M. Illegems, and R. M. Mikulyak, Appl. Phys. Lett. 28, 735 (1976).
    [Crossref]
  7. A. Y. Cho and J. R. Arthur, Progress in Solid State Chemistry, Vol. 10 (Pergamon, New York, 1975), Part 3, pp. 157157–191.
    [Crossref]
  8. A. Yariv, Appl. Phys. Lett. 25, 105 (1974).
    [Crossref]
  9. F. Bloch, Z. Phys. 52, 555 (1928).
  10. K. Aiki, M. Nakamura, J. Umeda, A. Yariv, A. Katzir, and H. W. Yen, Appl. Phys. Lett. 27, 145 (1975).
    [Crossref]
  11. H. C. Casey, S. Somekh, and M. Ilegems, Appl. Phys. Lett. 27, 142 (1975).
    [Crossref]
  12. F. K. Reinhart, R. A. Logan, and C. V. Shank, Appl. Phys. Lett. 27, 45 (1975).
    [Crossref]
  13. A. Yariv, IEEE J. Quant. Electron. QE-9, 919 (1973).
    [Crossref]
  14. M. Born and E. Wolf, Principles of Optics (Macmillan, New York, 1964), p. 67.
  15. H. Yajima, Proceedings of the Symposium on Optical and Acoustical Micro-Electronics, New York, April1974) (unpublished).
  16. L. B. Stotts, Opt. Commun. 17, 133 (1976).
    [Crossref]
  17. S. Somekh, E. Garmire, A. Yariv, H. Garvin, and R. Hunsperger, Appl. Phys. Lett. 22, 46 (1973).
    [Crossref]
  18. A. B. Buckman, J. Opt. Soc. Am. 66, 30 (1976).
    [Crossref]
  19. E. A. Ash, “Grating Surface Waveguides,” presented at International Microwave Symposium, Newport Beach, Calif., May 1970 (unpublished).
  20. A. Yariv, Quantum Electronics (Wiley, New York, 1975).
  21. P. K. Tien, Appl. Opt. 10, 2395 (1971).
    [Crossref] [PubMed]
  22. D. Kossel, “Analogies between Thin-Film Optics and Electron-Band Theory of Solids,” J. Opt. Soc. Am. 56, 1434 (1966).
  23. J. A. Arnaud and A. A. M. Saleh, Appl. Opt. 13, 2343 (1974).
    [Crossref] [PubMed]
  24. W. Shockley, Phys. Rev. 56, 317 (1939).
    [Crossref]
  25. J. P. van der Ziel and M. Ilegems, Appl. Phys. Lett. 28, 437 (1976).
    [Crossref]

1976 (4)

L. B. Stotts, Opt. Commun. 17, 133 (1976).
[Crossref]

J. P. van der Ziel, M. Illegems, and R. M. Mikulyak, Appl. Phys. Lett. 28, 735 (1976).
[Crossref]

J. P. van der Ziel and M. Ilegems, Appl. Phys. Lett. 28, 437 (1976).
[Crossref]

A. B. Buckman, J. Opt. Soc. Am. 66, 30 (1976).
[Crossref]

1975 (3)

K. Aiki, M. Nakamura, J. Umeda, A. Yariv, A. Katzir, and H. W. Yen, Appl. Phys. Lett. 27, 145 (1975).
[Crossref]

H. C. Casey, S. Somekh, and M. Ilegems, Appl. Phys. Lett. 27, 142 (1975).
[Crossref]

F. K. Reinhart, R. A. Logan, and C. V. Shank, Appl. Phys. Lett. 27, 45 (1975).
[Crossref]

1974 (2)

1973 (3)

S. Somekh, E. Garmire, A. Yariv, H. Garvin, and R. Hunsperger, Appl. Phys. Lett. 22, 46 (1973).
[Crossref]

A. Yariv, IEEE J. Quant. Electron. QE-9, 919 (1973).
[Crossref]

C. L. Tang and P. P. Bey, IEEE J. Quantum Electron. QE-9, 9 (1973).
[Crossref]

1971 (1)

1970 (1)

N. Bloembergen and A. J. Sievers, Appl. Phys. Lett. 17, 483 (1970).
[Crossref]

1966 (1)

D. Kossel, “Analogies between Thin-Film Optics and Electron-Band Theory of Solids,” J. Opt. Soc. Am. 56, 1434 (1966).

1955 (1)

S. M. Rytov, Zh. Eksp. Teor. Fiz. 29, 605 (1955)[Sov. Phys. -JETP 2, 466 (1956)].

1950 (1)

F. Abeles, Ann. Phys. (Paris) 5, 596 (1950);Ann. Phys. (Paris) 5, 706 (1950).

1939 (1)

W. Shockley, Phys. Rev. 56, 317 (1939).
[Crossref]

1928 (1)

F. Bloch, Z. Phys. 52, 555 (1928).

Abeles, F.

F. Abeles, Ann. Phys. (Paris) 5, 596 (1950);Ann. Phys. (Paris) 5, 706 (1950).

Aiki, K.

K. Aiki, M. Nakamura, J. Umeda, A. Yariv, A. Katzir, and H. W. Yen, Appl. Phys. Lett. 27, 145 (1975).
[Crossref]

Arnaud, J. A.

Arthur, J. R.

A. Y. Cho and J. R. Arthur, Progress in Solid State Chemistry, Vol. 10 (Pergamon, New York, 1975), Part 3, pp. 157157–191.
[Crossref]

Ash, E. A.

E. A. Ash, “Grating Surface Waveguides,” presented at International Microwave Symposium, Newport Beach, Calif., May 1970 (unpublished).

Ashkin, A.

A. Ashkin and A. Yariv, Bell Labs. Tech. Memo MM-61-124-46 (13November1961) (unpublished).

Bey, P. P.

C. L. Tang and P. P. Bey, IEEE J. Quantum Electron. QE-9, 9 (1973).
[Crossref]

Bloch, F.

F. Bloch, Z. Phys. 52, 555 (1928).

Bloembergen, N.

N. Bloembergen and A. J. Sievers, Appl. Phys. Lett. 17, 483 (1970).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics (Macmillan, New York, 1964), p. 67.

Buckman, A. B.

Casey, H. C.

H. C. Casey, S. Somekh, and M. Ilegems, Appl. Phys. Lett. 27, 142 (1975).
[Crossref]

Cho, A. Y.

A. Y. Cho and J. R. Arthur, Progress in Solid State Chemistry, Vol. 10 (Pergamon, New York, 1975), Part 3, pp. 157157–191.
[Crossref]

Garmire, E.

S. Somekh, E. Garmire, A. Yariv, H. Garvin, and R. Hunsperger, Appl. Phys. Lett. 22, 46 (1973).
[Crossref]

Garvin, H.

S. Somekh, E. Garmire, A. Yariv, H. Garvin, and R. Hunsperger, Appl. Phys. Lett. 22, 46 (1973).
[Crossref]

Hunsperger, R.

S. Somekh, E. Garmire, A. Yariv, H. Garvin, and R. Hunsperger, Appl. Phys. Lett. 22, 46 (1973).
[Crossref]

Ilegems, M.

J. P. van der Ziel and M. Ilegems, Appl. Phys. Lett. 28, 437 (1976).
[Crossref]

H. C. Casey, S. Somekh, and M. Ilegems, Appl. Phys. Lett. 27, 142 (1975).
[Crossref]

Illegems, M.

J. P. van der Ziel, M. Illegems, and R. M. Mikulyak, Appl. Phys. Lett. 28, 735 (1976).
[Crossref]

Katzir, A.

K. Aiki, M. Nakamura, J. Umeda, A. Yariv, A. Katzir, and H. W. Yen, Appl. Phys. Lett. 27, 145 (1975).
[Crossref]

Kossel, D.

D. Kossel, “Analogies between Thin-Film Optics and Electron-Band Theory of Solids,” J. Opt. Soc. Am. 56, 1434 (1966).

Logan, R. A.

F. K. Reinhart, R. A. Logan, and C. V. Shank, Appl. Phys. Lett. 27, 45 (1975).
[Crossref]

Mikulyak, R. M.

J. P. van der Ziel, M. Illegems, and R. M. Mikulyak, Appl. Phys. Lett. 28, 735 (1976).
[Crossref]

Nakamura, M.

K. Aiki, M. Nakamura, J. Umeda, A. Yariv, A. Katzir, and H. W. Yen, Appl. Phys. Lett. 27, 145 (1975).
[Crossref]

Reinhart, F. K.

F. K. Reinhart, R. A. Logan, and C. V. Shank, Appl. Phys. Lett. 27, 45 (1975).
[Crossref]

Rytov, S. M.

S. M. Rytov, Zh. Eksp. Teor. Fiz. 29, 605 (1955)[Sov. Phys. -JETP 2, 466 (1956)].

Saleh, A. A. M.

Shank, C. V.

F. K. Reinhart, R. A. Logan, and C. V. Shank, Appl. Phys. Lett. 27, 45 (1975).
[Crossref]

Shockley, W.

W. Shockley, Phys. Rev. 56, 317 (1939).
[Crossref]

Sievers, A. J.

N. Bloembergen and A. J. Sievers, Appl. Phys. Lett. 17, 483 (1970).
[Crossref]

Somekh, S.

H. C. Casey, S. Somekh, and M. Ilegems, Appl. Phys. Lett. 27, 142 (1975).
[Crossref]

S. Somekh, E. Garmire, A. Yariv, H. Garvin, and R. Hunsperger, Appl. Phys. Lett. 22, 46 (1973).
[Crossref]

Stotts, L. B.

L. B. Stotts, Opt. Commun. 17, 133 (1976).
[Crossref]

Tang, C. L.

C. L. Tang and P. P. Bey, IEEE J. Quantum Electron. QE-9, 9 (1973).
[Crossref]

Tien, P. K.

Umeda, J.

K. Aiki, M. Nakamura, J. Umeda, A. Yariv, A. Katzir, and H. W. Yen, Appl. Phys. Lett. 27, 145 (1975).
[Crossref]

van der Ziel, J. P.

J. P. van der Ziel, M. Illegems, and R. M. Mikulyak, Appl. Phys. Lett. 28, 735 (1976).
[Crossref]

J. P. van der Ziel and M. Ilegems, Appl. Phys. Lett. 28, 437 (1976).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Macmillan, New York, 1964), p. 67.

Yajima, H.

H. Yajima, Proceedings of the Symposium on Optical and Acoustical Micro-Electronics, New York, April1974) (unpublished).

Yariv, A.

K. Aiki, M. Nakamura, J. Umeda, A. Yariv, A. Katzir, and H. W. Yen, Appl. Phys. Lett. 27, 145 (1975).
[Crossref]

A. Yariv, Appl. Phys. Lett. 25, 105 (1974).
[Crossref]

S. Somekh, E. Garmire, A. Yariv, H. Garvin, and R. Hunsperger, Appl. Phys. Lett. 22, 46 (1973).
[Crossref]

A. Yariv, IEEE J. Quant. Electron. QE-9, 919 (1973).
[Crossref]

A. Ashkin and A. Yariv, Bell Labs. Tech. Memo MM-61-124-46 (13November1961) (unpublished).

A. Yariv, Quantum Electronics (Wiley, New York, 1975).

Yen, H. W.

K. Aiki, M. Nakamura, J. Umeda, A. Yariv, A. Katzir, and H. W. Yen, Appl. Phys. Lett. 27, 145 (1975).
[Crossref]

Ann. Phys. (Paris) (1)

F. Abeles, Ann. Phys. (Paris) 5, 596 (1950);Ann. Phys. (Paris) 5, 706 (1950).

Appl. Opt. (2)

Appl. Phys. Lett. (8)

J. P. van der Ziel and M. Ilegems, Appl. Phys. Lett. 28, 437 (1976).
[Crossref]

N. Bloembergen and A. J. Sievers, Appl. Phys. Lett. 17, 483 (1970).
[Crossref]

J. P. van der Ziel, M. Illegems, and R. M. Mikulyak, Appl. Phys. Lett. 28, 735 (1976).
[Crossref]

A. Yariv, Appl. Phys. Lett. 25, 105 (1974).
[Crossref]

K. Aiki, M. Nakamura, J. Umeda, A. Yariv, A. Katzir, and H. W. Yen, Appl. Phys. Lett. 27, 145 (1975).
[Crossref]

H. C. Casey, S. Somekh, and M. Ilegems, Appl. Phys. Lett. 27, 142 (1975).
[Crossref]

F. K. Reinhart, R. A. Logan, and C. V. Shank, Appl. Phys. Lett. 27, 45 (1975).
[Crossref]

S. Somekh, E. Garmire, A. Yariv, H. Garvin, and R. Hunsperger, Appl. Phys. Lett. 22, 46 (1973).
[Crossref]

IEEE J. Quant. Electron. (1)

A. Yariv, IEEE J. Quant. Electron. QE-9, 919 (1973).
[Crossref]

IEEE J. Quantum Electron. (1)

C. L. Tang and P. P. Bey, IEEE J. Quantum Electron. QE-9, 9 (1973).
[Crossref]

J. Opt. Soc. Am. (2)

D. Kossel, “Analogies between Thin-Film Optics and Electron-Band Theory of Solids,” J. Opt. Soc. Am. 56, 1434 (1966).

A. B. Buckman, J. Opt. Soc. Am. 66, 30 (1976).
[Crossref]

Opt. Commun. (1)

L. B. Stotts, Opt. Commun. 17, 133 (1976).
[Crossref]

Phys. Rev. (1)

W. Shockley, Phys. Rev. 56, 317 (1939).
[Crossref]

Z. Phys. (1)

F. Bloch, Z. Phys. 52, 555 (1928).

Zh. Eksp. Teor. Fiz. (1)

S. M. Rytov, Zh. Eksp. Teor. Fiz. 29, 605 (1955)[Sov. Phys. -JETP 2, 466 (1956)].

Other (6)

A. Ashkin and A. Yariv, Bell Labs. Tech. Memo MM-61-124-46 (13November1961) (unpublished).

A. Y. Cho and J. R. Arthur, Progress in Solid State Chemistry, Vol. 10 (Pergamon, New York, 1975), Part 3, pp. 157157–191.
[Crossref]

E. A. Ash, “Grating Surface Waveguides,” presented at International Microwave Symposium, Newport Beach, Calif., May 1970 (unpublished).

A. Yariv, Quantum Electronics (Wiley, New York, 1975).

M. Born and E. Wolf, Principles of Optics (Macmillan, New York, 1964), p. 67.

H. Yajima, Proceedings of the Symposium on Optical and Acoustical Micro-Electronics, New York, April1974) (unpublished).

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Figures (23)

FIG. 1
FIG. 1

Portion of a typical periodic stratified medium.

FIG. 2
FIG. 2

Plane wave amplitudes associated with the nth unit cell and its neighboring cells.

FIG. 3
FIG. 3

TE waves (E perpendicular to the direction of periodicity) band structure in the ωβ plane. The dark zones are the allowed bands.

FIG. 4
FIG. 4

TM waves (H perpendicular to the direction of periodicity) band structure in the ωβ plane. The dashed line is β = (ω/c)n2 sinθB. The dark zones are the allowed bands.

FIG. 5
FIG. 5

Dispersion relation between ω and K when β = 0 (normal incidence). Dotted curves are the imaginary part of K in arbitrary scales.

FIG. 6
FIG. 6

Geometry of a typical N-period Bragg reflector.

FIG. 7
FIG. 7

TE waves reflectivity spectrum of a 15-period Bragg reflector at various angles of incidence.

FIG. 8
FIG. 8

TM waves reflectivity spectrum of a 15-period Bragg reflector at various angles of incidence.

FIG. 9
FIG. 9

Section view of a typical N-channel symmetric waveguide.

FIG. 10
FIG. 10

Dispersion curves for the confined modes of a typical single channel waveguide (N = 1).

FIG. 11
FIG. 11

Dispersion curves for the confined modes of a typical two-channel waveguide (N = 2). Note the splitting in comparison with Fig. 10.

FIG. 12
FIG. 12

β vs separation for two asymmetric multichannel waveguides with N = 2 (upper diagram) and N = 5 (lower diagram) at ω = 3 4 π c / a .. The dark zones are the allowed bands. Dashed curves are the band edges. The inset shows the refractive index profile.

FIG. 13
FIG. 13

β vs separation for two asymmetric multichannel waveguides with N = 2 (upper diagram) and N = 5 (lower diagram) at ω = πc/a. The dark zones are the allowed bands. Dashed curves are the band edges. The inset shows the refractive index profile.

FIG. 14
FIG. 14

Transverse field distribution for the confined modes in the first band of a two-channel waveguide.

FIG. 15
FIG. 15

Transverse field distribution for the confined modes in the first band of a five-channel waveguide.

FIG. 16
FIG. 16

Transverse field distribution for the TE00 mode of a two-channel waveguide at various na’s.

FIG. 17
FIG. 17

Transverse field distribution for the TE00 mode of a five-channel waveguide at various na’s.

FIG. 18
FIG. 18

Bragg reflection (slab) waveguide (∂/∂y = 0).

FIG. 19
FIG. 19

Transverse field distribution of the fundamental modes of a typical Bragg reflection (slab) waveguide.

FIG. 20
FIG. 20

Transverse field distribution of the fundamental mode of a typical Bragg reflection waveguide with air as the guiding channel.

FIG. 21
FIG. 21

Semi-infinite periodic stratified medium.

FIG. 22
FIG. 22

Transverse field distribution for a typical fundamental surface mode guided by the surface of a semi-infinite periodic stratified medium.

FIG. 23
FIG. 23

Transverse field distribution for a typical higher-order surface mode guided by a semi-infinite peridic stratified medium.

Equations (96)

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n ( x ) = { n 2 , 0 < x < b , n 1 , b < x < Λ ,
n ( x + Λ ) = n ( x ) ,
E ( x , z ) = E ( x ) e i β z .
( a n ( α ) b n ( α ) ) .
E ( x , z ) = ( a n ( α ) e i k α x ( x n Λ ) + b n ( α ) e i k α x ( x n Λ ) ) e i β z ,
k α x = { [ ( ω / c ) n α ] 2 β 2 } 1 / 2 , α = 1 , 2 .
a n 1 + b n 1 = e i k 2 x Λ c n + e i k 2 x Λ d n , i k 1 x ( a n 1 b n 1 ) = i k 2 x ( e i k 2 x Λ c n e i k 2 x Λ d n ) , e i k 2 x a c n + e i k 2 x a d n = e i k 1 x a a n + e i k 1 x a b n , i k 2 x ( e i k 2 x a c n e i k 2 x a d n ) = i k 1 x ( e i k 1 x a a n e i k 1 x a b n ) .
( 1 1 1 1 ) ( a n 1 b n 1 ) = ( e i k 2 x Λ e i k 2 x Λ k 2 x k 1 x e i k 2 x Λ k 2 x k 1 x e i k 2 x Λ ) ( c n d n ) ,
( e i k 2 x a e i k 2 x a e i k 2 x a e i k 2 x a ) ( c n d n ) = ( e i k 1 x a e i k 1 x a k 1 x k 2 x e i k 1 x a k 1 x k 2 x e i k 1 x a ) ( a n b n ) ,
a n a n ( 1 ) , b n b n ( 1 ) , c n a n ( 2 ) , d n b n ( 2 ) .
( c n d n ) ,
( a n 1 b n 1 ) = ( A B C D ) ( a n b n )
A = e i k 1 x a [ cos k 2 x b 1 2 i ( k 2 x k 1 x + k 1 x k 2 x ) sin k 2 x b ] ,
B = e i k 1 x a [ 1 2 i ( k 2 x k 1 x k 1 x k 2 x ) sin k 2 x b ] ,
C = e i k 1 x a [ 1 2 i ( k 2 x k 1 x k 1 x k 2 x ) sin k 2 x b ] ,
D = e i k 1 x a [ cos k 2 x b + 1 2 i ( k 2 x k 1 x + k 1 x k 2 x ) sin k 2 x b ] ,
A D B C = 1 .
( c n 1 d n 1 ) to ( c n d n )
A TM = e i k 1 x a [ cos k 2 x b 1 2 i ( n 2 2 k 1 x n 1 2 k 2 x + n 1 2 k 2 x n 2 2 k 1 x ) sin k 2 x b ] ,
B TM = e i k 1 x a [ 1 2 i ( n 2 2 k 1 x n 1 2 k 2 x n 1 2 k 2 x n 2 2 k 1 x ) sin k 2 x b ] ,
C TM = e i k 1 x a [ 1 2 i ( n 2 2 k 1 x n 1 2 k 2 x n 1 2 k 2 x n 2 2 k 1 x ) sin k 2 x b ] ,
D TM = e i k 1 x a [ cos k 2 x b + 1 2 i ( n 2 2 k 1 x n 1 2 k 2 x + n 1 2 k 2 x n 2 2 k 1 x ) sin k 2 x b ] .
( a n b n ) = ( A B C D ) n ( a 0 b 0 ) .
( a n b n ) = ( D B C A ) n ( a 0 b 0 ) .
T E ( x ) = E ( T 1 x ) = E ( x l Λ ) .
E K ( x , z ) = E K ( x ) e i K x e i β z ,
E K ( x + Λ ) = E K ( x ) .
( a n b n ) = e i K Λ ( a n 1 b n 1 ) .
( A B C D ) ( a n b n ) = e i K Λ ( a n b n ) .
e i K Λ = 1 2 ( A + D ) ± { [ 1 2 ( A + D ) ] 2 1 } 1 / 2 .
( a 0 b 0 ) = ( B e i K Λ A )
K ( β , ω ) = ( 1 / Λ ) cos 1 [ 1 2 ( A + D ) ] .
E K ( x ) e i K x = [ ( a 0 e i k 1 x ( x n Λ ) + b 0 e i k 1 x ( x n Λ ) ) e i K ( x n Λ ) ] e i K x ,
r N = ( b 0 / a 0 ) b N = 0 .
( a 0 b 0 ) = ( A B C D ) N ( a N b N ) .
( A B C D ) N = ( A U N 1 U N 2 B U N 1 C U N 1 D U N 1 U N 2 ) ,
U N = sin ( N + 1 ) K Λ / sin K Λ ,
r N = C U N 1 / ( A U N 1 U N 2 ) .
| r N | 2 = | C | 2 | C | 2 + ( sin K Λ / sin N K Λ ) 2 .
| r 1 | 2 = | C | 2 / ( | C | 2 + 1 )
| C | 2 = | r 1 | 2 / ( 1 | r 1 | 2 ) .
| r N | 2 = | C | 2 / [ | C | 2 + ( 1 / N ) 2 ] .
K Λ = m π + i K i Λ .
| r N | 2 = | C | 2 | C | 2 + ( sinh K i Λ / sinh N K i Λ ) 2 .
n ( x , z ) = { n 2 , m Λ x m Λ + b ( m = 0 , 1 , 2 , , N 1 ) , n 1 , otherwise ,
n 1 < n 2 .
T = ( A B C D ) ,
A = e q a [ cos p b 1 2 ( p / q q / p ) sin p b ] , B = e q a [ 1 2 ( p / q + q / p ) sin p b ] , C = e q a [ 1 2 ( p / q + q / p ) sin p b ] , D = e q a [ cos p b + 1 2 ( p / q q / p ) sin p b ] ,
q = { β 2 [ ( ω / c ) n 1 ] 2 } 1 / 2 = i k 1 x , p = { [ ( ω / c ) n 2 ] 2 β 2 } 1 / 2 = k 2 x .
A ( sin N K Λ sin K Λ ) ( sin ( N 1 ) K Λ sin K Λ ) = 0 .
n ( x , z ) = { n a , x < 0 , n 2 , m Λ x < m Λ + b ( m = 0 , 1 , 2 , , N 1 ) , n 1 , otherwise .
( a 0 b 0 ) to ( a N b N )
M = ( 1 2 ( 1 + q / q a ) 1 2 ( 1 q / q a ) 1 2 ( 1 q / q a ) 1 2 ( 1 + q / q a ) ) ( A B C D ) N ,
q a = { β 2 [ ( ω / c ) n a ] 2 } 1 / 2 .
( A + q a q q a + q C ) sin N K Λ sin K Λ sin ( N 1 ) K Λ sin K Λ = 0 .
n a = n a ( E dc = 0 ) + α E dc ,
n a ( E dc = 0 ) = n 1 .
β min < β < β max ,
β max = ( ω / c ) n 2 ,
β min = max ( ω c n a , ω c n s ) ,
2 E y z 2 + 2 E y x 2 + ω 2 c 2 n 2 ( x ) E y = 0 .
2 E ( x ) x 2 + ( ω 2 c 2 n 2 ( x ) β 2 ) E ( x ) = 0 .
E ( x ) = { ( i ) e q a ( x + t ) , x < t , ( ii ) c 1 cos ( k g x ) + c 2 sin ( k g x ) , t x < 0 , ( iii ) E K ( x ) e i K x , 0 x ,
q a = [ β 2 ( ω c n a ) 2 ] 1 / 2 , k g = [ ( ω c n g ) 2 β 2 ] 1 / 2 .
k g ( q a cos k g t k g sin k g t q a sin k g t + k g cos k g t ) = i k 1 x e i K Λ A B e i K Λ A + B .
i k 1 x ( e i K Λ A B e i K Λ A + B ) = { k a tan ( 1 2 k a t ) , even TE modes , k a cot ( 1 2 k a t ) , odd TE modes ,
k a = { [ ( ω / c ) n a ] 2 β 2 } 1 / 2 .
Λ 2 / t 2 = l / s , l = 1 , 2 , 3 ,
n ( x , z ) = { n a , x 0 , n 2 , m Λ x < m Λ + b , n 1 , m Λ + b x < ( m + 1 ) Λ ( m = 0 , 1 , 2 , ) .
E ( x ) = { α e q a x , x 0 , E K ( x ) e i K x , x 0 ,
q a = { β 2 [ ( ω / c ) n a ] 2 } 1 / 2 ,
q a = q ( e i K Λ A B ) / ( e i K Λ A + B ) .
energy in the first period energy in the whole semi - infinite periodic structrue = ( 1 e 2 K i Λ ) ,
n ( x ) = { n 1 , x 0 < x < x 1 , n 2 , x 1 < x < x 2 , n M , x M 1 < x < x M ,
n ( x + Λ ) = n ( x ) , Λ = x M x 0 .
t m = x m x m 1 .
E ( x ) = a n ( m ) e i k m x ( x n Λ ) + b n ( m ) e i k m x ( x n Λ ) .
( a n 1 ( m ) e i k m x x m b n 1 ( m ) e i k m x x m ) = T ( m ) ( a n ( m ) e i k m x x m b n ( m ) e i k m x x m ) .
T ( m ) = 1 2 M α = m + 1 M + m ( ( 1 + C α ) e i k α x t α ( 1 C α ) e i k α x t α ( 1 C α ) e i k α x t α ( 1 + C α ) e i k α x t α ) ,
C α = { k α x / k ( α 1 ) x , TE waves , n α 2 k ( α 1 ) x / n ( α 1 ) 2 k α x , TM waves .
C M + α = C α , t M + α = t α
α = 1 M C α = 1 , α = 1 M t α = Λ .
C α = { k α x μ α 1 / k ( α 1 ) x μ α , TE waves , n α 2 k ( α 1 ) x μ α 1 / n α 1 2 k α x μ α , TM waves .
( A B C D ) N = ( A U N 1 U N 2 B U N 1 C U N 1 D U N 1 U N 2 ) ,
U N = sin ( N + 1 ) K Λ / sin K Λ ,
K Λ = cos 1 [ 1 2 ( A + D ) ]
( A B C D ) V ± = e ± i K Λ V ± .
e ± i K Λ = [ 1 2 ( A + D ) ] ± { [ 1 2 ( A + D ) ] 2 1 } 1 / 2 ,
V ± = ( α ± β ± )
α ± = B [ B 2 + ( e ± i K Λ A ) 2 ] 1 / 2 , β ± = e ± i K Λ A [ B 2 + ( e ± i K Λ A ) 2 ] 1 / 2 .
{ M ( A B C D ) M 1 } N = M ( A B C D ) N M 1 ,
M ( A B C D ) M 1 = ( e i K Λ 0 0 e i K Λ ) ,
( A B C D ) N = M 1 ( e i N K Λ 0 0 e i N K Λ ) M .
M 1 = 1 ( α + β α β + ) 1 / 2 ( α + α β + β ) ,
M 1 = 1 ( α + β α β + ) 1 / 2 ( β α β + α + ) .
( A B C D ) N = 1 α + β α β + ( α + α β + β ) × ( e i N K Λ 0 0 e i N K Λ ) ( β α β + α + ) = ( A sin N K Λ sin ( N 1 ) K Λ sin K Λ B sin N K Λ sin K Λ C sin N K Λ sin K Λ D sin N K Λ sin ( N 1 ) K Λ sin K Λ ) .