Abstract

We consider wave propagation in longitudinally periodic media near the first- and higher-order Bragg resonances. An extended coupled-waves (ECW) approach is applied to media with single or multiharmonic periodicities and numerically compared to the Floquet theory. The ECW approach predicts coupling coefficients χ that vary as ηN, where η is the magnitude of the perturbation and N is the Bragg order. Explicit expressions are also given for the band-gap width and band-gap shift for all Bragg orders. A number of applications are discussed.

© 1976 Optical Society of America

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  1. L. Brillouin, Wave Propagation in Periodic Structures (Dover, New York, 1956).
  2. R. del Kronig and W. G. Penney, Proc. R. Soc. London Ser. A 130, 499 (1930).
  3. G. Allen, Phys. Rev. 91, 531 (1953).
    [Crossref]
  4. C. Kittel, Introduction to Solid State Physics (Wiley, New York, 1966).
  5. A. Hessel, in Antenna Theory, edited by R. E. Collin and F. J. Zucker (McGraw-Hill, New York, 1969), Pt. 2.
  6. D. J. Mead, J. Sound Vib. 27, 235 (1973).
    [Crossref]
  7. R. M. Bevensee, Electromagnetic Slow Wave Systems (Wiley, New York, 1964).
  8. R. M. White, Proc. IEEE 58, 1238 (1970).
    [Crossref]
  9. C. Elachi and C. Yeh, J. Opt. Soc. Am. 63, 840 (1973).
    [Crossref]
  10. C. Elachi and C. Yeh, J. Appl. Phys. 44, 3146 (1973).
    [Crossref]
  11. A. Yariv, IEEE J. Quantum Electron. 9, 919 (1973).
    [Crossref]
  12. W. S. C. Chang, IEEE Trans. Microwave Theory Tech. 21, 775 (1974).
    [Crossref]
  13. H. Kogelnik and C. V. Shank, J. Appl. Phys. 43, 2327 (1972).
    [Crossref]
  14. S. Wang, Wave Electron. 1, 31 (1974).
  15. S. Wang, IEEE J. Quantum Electron. 10, 413 (1974).
    [Crossref]
  16. C. Elachi, G. Evans, and C. Yeh, J. Opt. Soc. Am. 65, 404 (1975).
    [Crossref]
  17. R. Dingle, A. C. Gossard, and W. Wiegmann, Phys. Rev. Lett. 34, 1327 (1975).
    [Crossref]
  18. T. Tamir, H. C. Wang, and A. A. Oliver, IEEE Trans. Microwave Theory Tech. 12, 324 (1964).
  19. C. Yeh, K. F. Casey, and Z. Kaprielian, IEEE Trans. Microwave Theory Tech. 13, 297 (1965).
    [Crossref]
  20. J. R. Pierce, Almost All About Waves (MIT, Cambridge, 1974).
  21. R. S. Chu and T. Tamir, IEEE Trans. Microwave Theory Tech. 17, 487 (1970).
  22. S. F. Su and T. K. Gaylord, J. Opt. Soc. Am. 65, 59 (1975).
    [Crossref]
  23. E. T. Whitaker and G. N. Watson, A Course of Modern Analysis (Cambridge U. P., Cambridge, 1927).
  24. D. L. Jaggard and G. Evans, Coupled Waves and Floquet Approach to Periodic Structures (, Aug.1975).
  25. A. Yariv and A. Gover, Appl. Phys. Lett. 26, 537 (1975).
    [Crossref]
  26. B. Y. Tong, Phys. Rev. 175, 710 (1961).
    [Crossref]
  27. C. V. Shank and R. V. Schmidt, Appl. Phys. Lett. 23, 154 (1973).
    [Crossref]
  28. J. F. Bjorkholm and V. V. Shank, Appl. Phys. Lett. 20, 306 (1972).
    [Crossref]
  29. C. Elachi, D. L. Jaggard, and C. Yeh, IEEE Trans. Antennas Propag. 23, 352 (1975).
    [Crossref]

1975 (5)

C. Elachi, G. Evans, and C. Yeh, J. Opt. Soc. Am. 65, 404 (1975).
[Crossref]

R. Dingle, A. C. Gossard, and W. Wiegmann, Phys. Rev. Lett. 34, 1327 (1975).
[Crossref]

S. F. Su and T. K. Gaylord, J. Opt. Soc. Am. 65, 59 (1975).
[Crossref]

A. Yariv and A. Gover, Appl. Phys. Lett. 26, 537 (1975).
[Crossref]

C. Elachi, D. L. Jaggard, and C. Yeh, IEEE Trans. Antennas Propag. 23, 352 (1975).
[Crossref]

1974 (3)

S. Wang, Wave Electron. 1, 31 (1974).

S. Wang, IEEE J. Quantum Electron. 10, 413 (1974).
[Crossref]

W. S. C. Chang, IEEE Trans. Microwave Theory Tech. 21, 775 (1974).
[Crossref]

1973 (5)

C. Elachi and C. Yeh, J. Opt. Soc. Am. 63, 840 (1973).
[Crossref]

C. Elachi and C. Yeh, J. Appl. Phys. 44, 3146 (1973).
[Crossref]

A. Yariv, IEEE J. Quantum Electron. 9, 919 (1973).
[Crossref]

D. J. Mead, J. Sound Vib. 27, 235 (1973).
[Crossref]

C. V. Shank and R. V. Schmidt, Appl. Phys. Lett. 23, 154 (1973).
[Crossref]

1972 (2)

J. F. Bjorkholm and V. V. Shank, Appl. Phys. Lett. 20, 306 (1972).
[Crossref]

H. Kogelnik and C. V. Shank, J. Appl. Phys. 43, 2327 (1972).
[Crossref]

1970 (2)

R. M. White, Proc. IEEE 58, 1238 (1970).
[Crossref]

R. S. Chu and T. Tamir, IEEE Trans. Microwave Theory Tech. 17, 487 (1970).

1965 (1)

C. Yeh, K. F. Casey, and Z. Kaprielian, IEEE Trans. Microwave Theory Tech. 13, 297 (1965).
[Crossref]

1964 (1)

T. Tamir, H. C. Wang, and A. A. Oliver, IEEE Trans. Microwave Theory Tech. 12, 324 (1964).

1961 (1)

B. Y. Tong, Phys. Rev. 175, 710 (1961).
[Crossref]

1953 (1)

G. Allen, Phys. Rev. 91, 531 (1953).
[Crossref]

1930 (1)

R. del Kronig and W. G. Penney, Proc. R. Soc. London Ser. A 130, 499 (1930).

Allen, G.

G. Allen, Phys. Rev. 91, 531 (1953).
[Crossref]

Bevensee, R. M.

R. M. Bevensee, Electromagnetic Slow Wave Systems (Wiley, New York, 1964).

Bjorkholm, J. F.

J. F. Bjorkholm and V. V. Shank, Appl. Phys. Lett. 20, 306 (1972).
[Crossref]

Brillouin, L.

L. Brillouin, Wave Propagation in Periodic Structures (Dover, New York, 1956).

Casey, K. F.

C. Yeh, K. F. Casey, and Z. Kaprielian, IEEE Trans. Microwave Theory Tech. 13, 297 (1965).
[Crossref]

Chang, W. S. C.

W. S. C. Chang, IEEE Trans. Microwave Theory Tech. 21, 775 (1974).
[Crossref]

Chu, R. S.

R. S. Chu and T. Tamir, IEEE Trans. Microwave Theory Tech. 17, 487 (1970).

del Kronig, R.

R. del Kronig and W. G. Penney, Proc. R. Soc. London Ser. A 130, 499 (1930).

Dingle, R.

R. Dingle, A. C. Gossard, and W. Wiegmann, Phys. Rev. Lett. 34, 1327 (1975).
[Crossref]

Elachi, C.

C. Elachi, D. L. Jaggard, and C. Yeh, IEEE Trans. Antennas Propag. 23, 352 (1975).
[Crossref]

C. Elachi, G. Evans, and C. Yeh, J. Opt. Soc. Am. 65, 404 (1975).
[Crossref]

C. Elachi and C. Yeh, J. Opt. Soc. Am. 63, 840 (1973).
[Crossref]

C. Elachi and C. Yeh, J. Appl. Phys. 44, 3146 (1973).
[Crossref]

Evans, G.

C. Elachi, G. Evans, and C. Yeh, J. Opt. Soc. Am. 65, 404 (1975).
[Crossref]

D. L. Jaggard and G. Evans, Coupled Waves and Floquet Approach to Periodic Structures (, Aug.1975).

Gaylord, T. K.

Gossard, A. C.

R. Dingle, A. C. Gossard, and W. Wiegmann, Phys. Rev. Lett. 34, 1327 (1975).
[Crossref]

Gover, A.

A. Yariv and A. Gover, Appl. Phys. Lett. 26, 537 (1975).
[Crossref]

Hessel, A.

A. Hessel, in Antenna Theory, edited by R. E. Collin and F. J. Zucker (McGraw-Hill, New York, 1969), Pt. 2.

Jaggard, D. L.

C. Elachi, D. L. Jaggard, and C. Yeh, IEEE Trans. Antennas Propag. 23, 352 (1975).
[Crossref]

D. L. Jaggard and G. Evans, Coupled Waves and Floquet Approach to Periodic Structures (, Aug.1975).

Kaprielian, Z.

C. Yeh, K. F. Casey, and Z. Kaprielian, IEEE Trans. Microwave Theory Tech. 13, 297 (1965).
[Crossref]

Kittel, C.

C. Kittel, Introduction to Solid State Physics (Wiley, New York, 1966).

Kogelnik, H.

H. Kogelnik and C. V. Shank, J. Appl. Phys. 43, 2327 (1972).
[Crossref]

Mead, D. J.

D. J. Mead, J. Sound Vib. 27, 235 (1973).
[Crossref]

Oliver, A. A.

T. Tamir, H. C. Wang, and A. A. Oliver, IEEE Trans. Microwave Theory Tech. 12, 324 (1964).

Penney, W. G.

R. del Kronig and W. G. Penney, Proc. R. Soc. London Ser. A 130, 499 (1930).

Pierce, J. R.

J. R. Pierce, Almost All About Waves (MIT, Cambridge, 1974).

Schmidt, R. V.

C. V. Shank and R. V. Schmidt, Appl. Phys. Lett. 23, 154 (1973).
[Crossref]

Shank, C. V.

C. V. Shank and R. V. Schmidt, Appl. Phys. Lett. 23, 154 (1973).
[Crossref]

H. Kogelnik and C. V. Shank, J. Appl. Phys. 43, 2327 (1972).
[Crossref]

Shank, V. V.

J. F. Bjorkholm and V. V. Shank, Appl. Phys. Lett. 20, 306 (1972).
[Crossref]

Su, S. F.

Tamir, T.

R. S. Chu and T. Tamir, IEEE Trans. Microwave Theory Tech. 17, 487 (1970).

T. Tamir, H. C. Wang, and A. A. Oliver, IEEE Trans. Microwave Theory Tech. 12, 324 (1964).

Tong, B. Y.

B. Y. Tong, Phys. Rev. 175, 710 (1961).
[Crossref]

Wang, H. C.

T. Tamir, H. C. Wang, and A. A. Oliver, IEEE Trans. Microwave Theory Tech. 12, 324 (1964).

Wang, S.

S. Wang, Wave Electron. 1, 31 (1974).

S. Wang, IEEE J. Quantum Electron. 10, 413 (1974).
[Crossref]

Watson, G. N.

E. T. Whitaker and G. N. Watson, A Course of Modern Analysis (Cambridge U. P., Cambridge, 1927).

Whitaker, E. T.

E. T. Whitaker and G. N. Watson, A Course of Modern Analysis (Cambridge U. P., Cambridge, 1927).

White, R. M.

R. M. White, Proc. IEEE 58, 1238 (1970).
[Crossref]

Wiegmann, W.

R. Dingle, A. C. Gossard, and W. Wiegmann, Phys. Rev. Lett. 34, 1327 (1975).
[Crossref]

Yariv, A.

A. Yariv and A. Gover, Appl. Phys. Lett. 26, 537 (1975).
[Crossref]

A. Yariv, IEEE J. Quantum Electron. 9, 919 (1973).
[Crossref]

Yeh, C.

C. Elachi, D. L. Jaggard, and C. Yeh, IEEE Trans. Antennas Propag. 23, 352 (1975).
[Crossref]

C. Elachi, G. Evans, and C. Yeh, J. Opt. Soc. Am. 65, 404 (1975).
[Crossref]

C. Elachi and C. Yeh, J. Opt. Soc. Am. 63, 840 (1973).
[Crossref]

C. Elachi and C. Yeh, J. Appl. Phys. 44, 3146 (1973).
[Crossref]

C. Yeh, K. F. Casey, and Z. Kaprielian, IEEE Trans. Microwave Theory Tech. 13, 297 (1965).
[Crossref]

Appl. Phys. Lett. (3)

A. Yariv and A. Gover, Appl. Phys. Lett. 26, 537 (1975).
[Crossref]

C. V. Shank and R. V. Schmidt, Appl. Phys. Lett. 23, 154 (1973).
[Crossref]

J. F. Bjorkholm and V. V. Shank, Appl. Phys. Lett. 20, 306 (1972).
[Crossref]

IEEE J. Quantum Electron. (2)

A. Yariv, IEEE J. Quantum Electron. 9, 919 (1973).
[Crossref]

S. Wang, IEEE J. Quantum Electron. 10, 413 (1974).
[Crossref]

IEEE Trans. Antennas Propag. (1)

C. Elachi, D. L. Jaggard, and C. Yeh, IEEE Trans. Antennas Propag. 23, 352 (1975).
[Crossref]

IEEE Trans. Microwave Theory Tech. (4)

R. S. Chu and T. Tamir, IEEE Trans. Microwave Theory Tech. 17, 487 (1970).

T. Tamir, H. C. Wang, and A. A. Oliver, IEEE Trans. Microwave Theory Tech. 12, 324 (1964).

C. Yeh, K. F. Casey, and Z. Kaprielian, IEEE Trans. Microwave Theory Tech. 13, 297 (1965).
[Crossref]

W. S. C. Chang, IEEE Trans. Microwave Theory Tech. 21, 775 (1974).
[Crossref]

J. Appl. Phys. (2)

H. Kogelnik and C. V. Shank, J. Appl. Phys. 43, 2327 (1972).
[Crossref]

C. Elachi and C. Yeh, J. Appl. Phys. 44, 3146 (1973).
[Crossref]

J. Opt. Soc. Am. (3)

J. Sound Vib. (1)

D. J. Mead, J. Sound Vib. 27, 235 (1973).
[Crossref]

Phys. Rev. (2)

G. Allen, Phys. Rev. 91, 531 (1953).
[Crossref]

B. Y. Tong, Phys. Rev. 175, 710 (1961).
[Crossref]

Phys. Rev. Lett. (1)

R. Dingle, A. C. Gossard, and W. Wiegmann, Phys. Rev. Lett. 34, 1327 (1975).
[Crossref]

Proc. IEEE (1)

R. M. White, Proc. IEEE 58, 1238 (1970).
[Crossref]

Proc. R. Soc. London Ser. A (1)

R. del Kronig and W. G. Penney, Proc. R. Soc. London Ser. A 130, 499 (1930).

Wave Electron. (1)

S. Wang, Wave Electron. 1, 31 (1974).

Other (7)

J. R. Pierce, Almost All About Waves (MIT, Cambridge, 1974).

L. Brillouin, Wave Propagation in Periodic Structures (Dover, New York, 1956).

C. Kittel, Introduction to Solid State Physics (Wiley, New York, 1966).

A. Hessel, in Antenna Theory, edited by R. E. Collin and F. J. Zucker (McGraw-Hill, New York, 1969), Pt. 2.

R. M. Bevensee, Electromagnetic Slow Wave Systems (Wiley, New York, 1964).

E. T. Whitaker and G. N. Watson, A Course of Modern Analysis (Cambridge U. P., Cambridge, 1927).

D. L. Jaggard and G. Evans, Coupled Waves and Floquet Approach to Periodic Structures (, Aug.1975).

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Figures (9)

FIG. 1
FIG. 1

Brillouin diagram for first three Bragg orders when η = 1.0 using Floquet theory. Dotted lines show imaginary parts of β/K on separate scales. Dashed line represents the unperturbed case.

FIG. 2
FIG. 2

Effect of truncating space harmonics in Floquet theory at first and third Bragg orders when η = 1.0.

FIG. 3
FIG. 3

Coupling diagram of Nth-order Bragg interaction. Dominant cross coupling between F1 and B1 are shown as well as self-coupling for ECW theory.

FIG. 4
FIG. 4

Brillouin diagram at first Bragg order for η = 1.0 (top curve), η = 0.1 (middle curve), and η =0.01 (bottom curve) using Floquet theory. Bottom curve also represents coupled-waves solution for all three cases. Note difference in scale for each case. Imaginary Δβ/K values are the elliptical curves with separate scale.

FIG. 5
FIG. 5

Brillouin diagram at second Bragg order for (a) η = 1.0, (b) η = 0.1, and (c) η = 0.05 showing Floquet and ECW results. Imaginary Δβ/K values are elliptical curves with separate scale. Note the close agreement of theories for η ≤ 0.1.

FIG. 6
FIG. 6

Brillouin diagram at third Bragg order for (a) η = 1.0 and (b) η = 0.5 showing Floquet and ECW results. Imaginary Δβ/K values for the elliptical curves with separate scale.

FIG. 7
FIG. 7

Brillouin diagram at second Bragg order for (a) η f 1 = ( 1 2 ) 1 / 2, ηf2 = 0.25 and (b) η f 1 = ( 1 2 ) 1 / 2, nf2 = −0.25 showing ECW and Floquet results for multiharmonic periodicities. Imaginary Δβ/K values are the elliptical curves with separate scale. Note the dependence of the band gap upon the sign of ηf2.

FIG. 8
FIG. 8

Coupling diagram at fourth-order Bragg interaction for ECW theory. Dominant couplings are shown when (ηf1)4, (ηf2)2, and ηf4 are of the same order of magnitude.

FIG. 9
FIG. 9

Brillouin diagram at fourth Bragg order when ηf1 = 0.5, ηf2 = 0.2, and ηf4 = 0.1 showing ECW and Floquet results. Imaginary Δβ/K values are the elliptical curves with separate scale.

Tables (1)

Tables Icon

TABLE I Summary of the main features of the ECW theory for the first five Bragg orders.

Equations (45)

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λ = 2 Λ / N ,
( z ) = ( 1 + η cos K z ) ,
d 2 E / d z 2 + k 2 ( z ) E = 0 ,
E ( z ) = n = - n = + a n e i ( β + n K ) z .
sin 2 ( π β / K ) = Δ ( 0 ) sin 2 ( π k 1 / 2 / K ) ,
Δ ( 0 ) = det Δ = Hill s determinant ,
Δ m n = { 1 , m = n η 2 k 2 1 / 2 k 2 1 / 2 - m 2 K 2 , m = n ± 1 0 , otherwise .
v av = 1 Λ 0 Λ c d z / [ ( 1 + η cos K z ) ] 1 / 2 ,
v av = c 1 / 2 [ 1 + 3 16 π η 2 + O ( η 4 ) ] .
β β 0 = ± N K / 2             ( N 2 ) .
E ( z ) = n = - 1 N * / 2 ( a n - N - e - i Δ β z + a + n + e i Δ β z ) exp [ i ( 1 - 2 n / N ) β 0 z ] + n = - 1 N * / 2 ( a N - n + e i Δ β z + a - n - e - i Δ β z ) exp - [ i ( 1 - 2 n / N ) β 0 z ] + ( 1 0 ) ( a - N / 2 - e - i Δ β z + a + N / 2 + e i Δ β z ) ,
E ( z ) = n = - 1 N * / 2 { F 1 - 2 n / N ( z ) exp [ i ( 1 - 2 n / N ) β 0 z ] + B 1 - 2 n / N ( z ) exp [ - i ( 1 - 2 n / N ) β 0 z ] } + ( 1 0 ) S ( z ) ,
[ k 2 - ( 1 + 2 N ) 2 β 0 2 ] F 1 + 2 / N + 2 i ( 1 + 2 N ) β 0 F 1 + 2 / N = - η k 2 2 F 1 , [ k 2 - β 0 2 ] F 1 + 2 i β 0 F 1 = - η k 2 2 ( F 1 - 2 / N + F 1 + 2 / N ) , ( 1 0 ) k 2 S = - ( 1 0 ) η k 2 2 ( F 2 / N + B 2 / N ) , [ k 2 - β 0 2 ] B 1 - 2 i β 0 B 1 = - η k 2 2 ( B 1 - 2 / N + B 1 + 2 / N ) , [ k 2 - ( 1 + 2 N ) 2 β 0 2 ] B 1 + 2 / N - 2 i ( 1 + 2 N ) β 0 B 1 + 2 / N = - η k 2 2 B 1 ,
F 1 - 2 / N = det A / det C , B 1 - 2 / N = det G / det C ,
C = | C 1 C C C 2 C 0 0 C C 2 C C C 1 | , A = | A 1 C C C 2 C 0 0 C C 2 C A 2 C C 1 | , G = | G 1 C C C 2 C 0 0 C C 2 C G 2 C C 1 | , C n = k 2 - ( 1 - 2 n N ) 2 β 0 2 ,             G 2 = A 1 = - k 2 η F 1 / 2 , C = η k 2 / 2 ,             G 1 = A 2 = - k 2 η B 1 / 2.
F 1 ( z ) - i δ N F 1 ( z ) = i χ N B 1 ( z ) , - B 1 ( z ) - i δ N B 1 ( z ) = i χ N F 1 ( z ) ,
δ N = k 2 [ 1 - ζ N ( η / 2 ) 2 ( N 2 2 ( N 2 - 1 ) ) ] - β 0 2 2 β 0 , χ N = ( - 1 ) N + 1 k 2 η N 2 N + 1 β 0 1 Π * [ 4 n ( n - N ) / N 2 ] 2 , ζ N = { 0 for N = 1 , 1 otherwise , * f ( n ) = { n = 1 ( N - 1 ) / 2 f ( n ) for N odd n = 1 N / 2 f ( n ) for N even 1 for N = 1 .
e ± i Δ β N z
Δ β N / K = [ ( δ N / K ) 2 - ( χ N / K ) 2 ] 1 / 2 .
( δ N / K ) 2 < ( χ N / K ) 2 .
Im ( Δ β N / K ) max = χ N / K = | η N 2 N + 2 N Π * [ 4 n ( N - n ) / N 2 ] 2 | ,
BGS N = Δ k N 1 / 2 / K δ N = 0 = η 2 ζ N N 3 / 32 ( N 2 - 1 ) ,
W N = | η N 2 N + 1 N Π * [ 4 n ( n - N ) / N 2 ] 2 | = 2 χ N / K .
δ 1 = k 2 - β 0 2 2 β 0 Δ k 1 / 2 ,
χ 1 = η k 2 / 4 β 0 η K / 8 ,
β 0 = K / 2 ,
Δ β 1 K = [ ( Δ k 1 / 2 K ) 2 - ( η / 8 ) 2 ] 1 / 2 .
δ 2 = k 2 ( 1 - η 2 / 6 ) - β 0 2 2 β 0 Δ k 1 / 2 - η 2 K 12 ,
χ 2 = - η 2 k 2 / 8 β 0 - η 2 K / 8 ,
β 0 = K ,
Δ β 2 / K = [ ( Δ k 1 / 2 / K ) 2 - ( Δ k 1 / 2 / K ) 1 6 η 2 - 5 576 η 4 ] 1 / 2 .
( z ) = ( 1 + η n f n cos ( n K z ) ) ,
β N K / 2 k 1 / 2 ,
[ k 2 - ( 1 + 2 N ) 2 β 0 2 ] F 1 + 2 / N + 2 i ( 1 + 2 N ) β 0 F 1 + 2 / N = - k 2 2 η f 1 F 1 , [ k 2 - β 0 2 ] F 1 + 2 i β 0 F 1 = - k 2 2 [ η f 1 ( F 1 - 2 / N + F 1 + 2 / N ) + η f N B 1 ] ( 1 0 ) k 2 S = - ( 1 0 ) k 2 2 η f 1 ( F 2 / N + B 2 / N ) , ( k 2 - β 0 2 ) B 1 - 2 i B 0 B 1 = - k 2 2 [ η f 1 ( B 1 - 2 / N + B 1 + 2 / N ) + η f N F 1 ] , [ k 2 - ( 1 + 2 N ) 2 β 0 2 ] B 1 + 2 / N - 2 i ( 1 + 2 N ) β 0 B 1 + 2 / N = - k 2 2 η f 1 B 1 ,
δ N = k 2 2 β 0 [ 1 - ζ N ( η f 1 2 ) 2 ( N 2 2 ( N 2 - 1 ) ) ] - β 0 2 ,
χ N = k 2 2 β 0 [ η f N 2 + ( - 1 ) N + 1 ( η f 1 2 ) N 1 Π * [ 4 n ( n - N ) / N 2 ] 2 ] .
χ 2 = k 2 2 β 0 ( η f 2 2 - η 2 f 1 2 4 ) .
f 2 = η f 1 2 / 2 ,
Δ β 2 / K = Δ k 1 / 2 / K - η 2 f 1 2 / 8 = δ 2 / K .
η f 2 = ( η f 1 ) 2 / 2 = 1 4
η f 2 = ( η f 1 ) 2 = - 1 4
( k 2 - 9 4 β 0 2 ) F 3 / 2 + 3 i β 0 F 3 / 2 = - k 2 η 2 ( f 1 F 1 ) , ( k 2 - β 0 2 ) F 1 + 2 i β 0 F 1 = - k 2 η 2 [ f 1 ( F 1 / 2 + F 3 / 2 ) + f 2 S + f 4 B 1 ] , ( k 2 - 1 4 β 0 2 ) F 1 / 2 + i β 0 F 1 / 2 = - k 2 η 2 [ f 1 ( F 1 + S ) + f 2 B 1 / 2 ] , k 2 S = - k 2 η 2 [ f 1 ( F 1 / 2 + B 1 / 2 ) + f 2 ( F 1 + B 1 ) ] , ( k 2 - 1 4 β 0 2 ) B 1 / 2 - i β 0 B 1 / 2 = - k 2 η 2 [ f 1 ( B 1 + S ) + f 2 F 1 / 2 ] , ( k 2 - β 0 2 ) B 1 - 2 i β 0 B 1 = - k 2 η 2 [ f 1 ( B 1 / 2 + B 3 / 2 ) + f 2 S + f 4 F 1 ] , ( k 2 - 9 4 β 0 2 ) B 3 / 2 - 3 i β 0 B 3 / 2 = - k 2 η 2 ( f 1 B 1 ) .
δ 4 = k 2 2 β 0 ( 1 - 2 η 2 15 f 1 ) - β 0 2 ,
χ 4 = η k 2 4 β 0 ( - 2 9 η 3 f 1 4 + 10 9 η 2 f 1 2 f 2 - 1 2 η f 2 2 + f 4 ) ,
β 0 = 2 K .