Abstract

The principal focus of my report is on the theoretical study of the properties of spectral coefficients in a complex wave-number plane. The basic results of the study are described, and their application to the synthesis of a rugate filter and to inhomogeneous layer recognition problems are considered. General results concerning the existence of solutions to synthesis problems are also presented. The close analogy between synthesis problems in thin-film optics and optimal control problems is outlined, and some applications of Pontryagin's maximum principle are considered.

© 1993 Optical Society of America

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References

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  1. L. D. Landau, E. M. Lifshiz, Electrodynamics of Continuous Media (Pergamon, London, Paris, 1958).
  2. H. M. Nussenzveig, Causality and Dispersion Relations (Academic, New York, London, 1972).
  3. H. Kaiser, “Analysis of the transfer coefficients in layered media,” in Optical Interference Coatings, Vol. 15 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D. C., 1992), pp. 72–74.
  4. H. Kaiser, “Principles of thin film systems design,” in Thin Films for Optical Systems, K. H. Guenther, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1782 (to be published).
  5. A. V. Tikhonravov, “General layered media properties in connection with coating design and investigation techniques,” in Optical Interference Coatings, Vol. 15 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 70–71.
  6. V. G. Boltyanskii, Mathematical Methods of Optimal Control (Holt, Rinehart & Winston, New York, 1971).
  7. Sh. A. Furman, A. V. Tikhonravov, Basics of Optics of Multilayer Systems (Editions Frontieres, Gif-sur-Yvette, France, 1992).
  8. A. V. Tikhonravov, “About the limit accuracy of the synthesis problem solution,” Zh. Vychisl. Mat. Mat. Fiz. 22, 1421–1433 (1982).
  9. A. V. Tikhonravov, “Amplitude and phase properties of layered media spectral characteristics,” Zh. Vychisl. Mat. Mat. Fiz. 25, 442–450 (1985).
  10. J. A. Dobrowolski, P. Lowe, “Optical thin film synthesis program based on the use of Fourier transforms,” Appl. Opt. 17, 3039–3050 (1978).
    [CrossRef] [PubMed]
  11. P. G. Verly, J. A. Dobrowolski, “Iterative correction process for optical thin film synthesis with Fourier-transform method,” Appl. Opt. 29, 3672–3684 (1990).
    [CrossRef] [PubMed]
  12. B. G. Bovard, “Rugate filter design: the modified Fourier transform technique,” Appl. Opt. 30, 24–30 (1990).
    [CrossRef]
  13. A. V. Tikhonravov, “A stable method for solving layered media synthesis problems,” Dokl. Akad. Nauk SSSR 283, 582–585 (1985).
  14. A. V. Tikhonravov, “Synthesis of layered media with pre-set amplitude-phase properties,” Zh. Vychisl. Mat. Mat. Fiz. 25, 1674–1688 (1985).
  15. A. V. Tikhonravov, A. D. Poezd, I. V. Zuev, “On the applicability of the Kramers–Kronig relation to the determination of layered media optical parameters,” in Optical Interference Coatings, Vol. 15 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 304–306.
  16. S. Lang, Complex Analysis (Addison-Wesley, Reading, Mass., 1977), p. 253.
  17. V. B. Glasko Yu, I. Khudak, “Additive representations of layered media characteristics and uniqueness of inverse problems,” Zh. Vychisl. Mat. Mat. Fiz. 20, 482–490 (1980).
  18. Ye. Ya. Khruslov, “One-dimensional inverse problems in electrodynamics,” Zh. Vychisl. Mat. Mat. Fiz. 25, 548–561 (1985).
  19. P. Grosse, V. Offerman, “Analysis or reflectance data using the Kramers–Kronig relations,” Appl. Phys. A 52, 138–144 (1991).
    [CrossRef]
  20. I. V. Zuev, A. V. Tikhonravov, “On the uniqueness of layered media parameters determination basing on reflectance measurement,” Zh. Vychisl. Mat. Mat. Fiz. 33, 428–438 (1993).
  21. A. N. Tikhonov, V. Ya. Arsenin, Solutions of Ill-Posed Problems (Winston Wiley, New York, 1977).
  22. L. Li, J. A. Dobrowolski, “Computation speeds of different optical thin-film synthesis methods,” Appl. Opt. 31, 3790–3799 (1992).
    [CrossRef] [PubMed]
  23. J. A. Dobrowolski, “Computer design of optical coatings,” Thin Solid Films. 163, 97–110 (1988).
    [CrossRef]
  24. A. V. Tikhonravov, “On the optimal control theory problems connected with the layered media synthesis,” Dif. Urav. 21, 1516–1523 (1985).
  25. A. V. Tikhonravov, “On the optimality of thin-film optical coating design,” in Optical Thin Films and Applications, R. Herrmann, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1270, 28–35 (1990).
  26. P. Baumeister, “Detailed knowledge of optical coating design techniques may be superfluous to produce usable coatings,” in Optical Interference Coatings, Vol. 15 of OSA 1992 Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 8–10.
  27. A. N. Baskakov, A. V. Tikhonravov, “Synthesis of two-component optical coatings,” Opt. Spectrosc. (USSR) 56, 559–562 (1984).
  28. A. V. Tikhonravov, M. K. Trubetskov, “Thin film coating design using second-order optimization methods,” in Thin Films for Optical Systems, K. H. Guenther, ed., Proc. Soc. Photo-Opt. Instrum. Eng. (to be published).
  29. B. Sullivan, J. Dobrowolski, “Deposition error compensation for optical multilayer coatings. 1. Theoretical description,” Appl. Opt. 31, 3821–3835 (1992).
    [CrossRef] [PubMed]
  30. B. T. Sullivan, J. A. Dobrowolski, “Deposition error compensation for optical multilayer coatings. II. Experimental results—sputtering system,” Appl. Opt. 32, 2351–2360 (1993).
    [CrossRef] [PubMed]

1993 (2)

I. V. Zuev, A. V. Tikhonravov, “On the uniqueness of layered media parameters determination basing on reflectance measurement,” Zh. Vychisl. Mat. Mat. Fiz. 33, 428–438 (1993).

B. T. Sullivan, J. A. Dobrowolski, “Deposition error compensation for optical multilayer coatings. II. Experimental results—sputtering system,” Appl. Opt. 32, 2351–2360 (1993).
[CrossRef] [PubMed]

1992 (2)

1991 (1)

P. Grosse, V. Offerman, “Analysis or reflectance data using the Kramers–Kronig relations,” Appl. Phys. A 52, 138–144 (1991).
[CrossRef]

1990 (2)

1988 (1)

J. A. Dobrowolski, “Computer design of optical coatings,” Thin Solid Films. 163, 97–110 (1988).
[CrossRef]

1985 (5)

A. V. Tikhonravov, “On the optimal control theory problems connected with the layered media synthesis,” Dif. Urav. 21, 1516–1523 (1985).

A. V. Tikhonravov, “A stable method for solving layered media synthesis problems,” Dokl. Akad. Nauk SSSR 283, 582–585 (1985).

A. V. Tikhonravov, “Synthesis of layered media with pre-set amplitude-phase properties,” Zh. Vychisl. Mat. Mat. Fiz. 25, 1674–1688 (1985).

A. V. Tikhonravov, “Amplitude and phase properties of layered media spectral characteristics,” Zh. Vychisl. Mat. Mat. Fiz. 25, 442–450 (1985).

Ye. Ya. Khruslov, “One-dimensional inverse problems in electrodynamics,” Zh. Vychisl. Mat. Mat. Fiz. 25, 548–561 (1985).

1984 (1)

A. N. Baskakov, A. V. Tikhonravov, “Synthesis of two-component optical coatings,” Opt. Spectrosc. (USSR) 56, 559–562 (1984).

1982 (1)

A. V. Tikhonravov, “About the limit accuracy of the synthesis problem solution,” Zh. Vychisl. Mat. Mat. Fiz. 22, 1421–1433 (1982).

1980 (1)

V. B. Glasko Yu, I. Khudak, “Additive representations of layered media characteristics and uniqueness of inverse problems,” Zh. Vychisl. Mat. Mat. Fiz. 20, 482–490 (1980).

1978 (1)

Arsenin, V. Ya.

A. N. Tikhonov, V. Ya. Arsenin, Solutions of Ill-Posed Problems (Winston Wiley, New York, 1977).

Baskakov, A. N.

A. N. Baskakov, A. V. Tikhonravov, “Synthesis of two-component optical coatings,” Opt. Spectrosc. (USSR) 56, 559–562 (1984).

Baumeister, P.

P. Baumeister, “Detailed knowledge of optical coating design techniques may be superfluous to produce usable coatings,” in Optical Interference Coatings, Vol. 15 of OSA 1992 Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 8–10.

Boltyanskii, V. G.

V. G. Boltyanskii, Mathematical Methods of Optimal Control (Holt, Rinehart & Winston, New York, 1971).

Bovard, B. G.

B. G. Bovard, “Rugate filter design: the modified Fourier transform technique,” Appl. Opt. 30, 24–30 (1990).
[CrossRef]

Dobrowolski, J.

Dobrowolski, J. A.

Furman, Sh. A.

Sh. A. Furman, A. V. Tikhonravov, Basics of Optics of Multilayer Systems (Editions Frontieres, Gif-sur-Yvette, France, 1992).

Glasko Yu, V. B.

V. B. Glasko Yu, I. Khudak, “Additive representations of layered media characteristics and uniqueness of inverse problems,” Zh. Vychisl. Mat. Mat. Fiz. 20, 482–490 (1980).

Grosse, P.

P. Grosse, V. Offerman, “Analysis or reflectance data using the Kramers–Kronig relations,” Appl. Phys. A 52, 138–144 (1991).
[CrossRef]

Kaiser, H.

H. Kaiser, “Analysis of the transfer coefficients in layered media,” in Optical Interference Coatings, Vol. 15 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D. C., 1992), pp. 72–74.

H. Kaiser, “Principles of thin film systems design,” in Thin Films for Optical Systems, K. H. Guenther, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1782 (to be published).

Khruslov, Ye. Ya.

Ye. Ya. Khruslov, “One-dimensional inverse problems in electrodynamics,” Zh. Vychisl. Mat. Mat. Fiz. 25, 548–561 (1985).

Khudak, I.

V. B. Glasko Yu, I. Khudak, “Additive representations of layered media characteristics and uniqueness of inverse problems,” Zh. Vychisl. Mat. Mat. Fiz. 20, 482–490 (1980).

Landau, L. D.

L. D. Landau, E. M. Lifshiz, Electrodynamics of Continuous Media (Pergamon, London, Paris, 1958).

Lang, S.

S. Lang, Complex Analysis (Addison-Wesley, Reading, Mass., 1977), p. 253.

Li, L.

Lifshiz, E. M.

L. D. Landau, E. M. Lifshiz, Electrodynamics of Continuous Media (Pergamon, London, Paris, 1958).

Lowe, P.

Nussenzveig, H. M.

H. M. Nussenzveig, Causality and Dispersion Relations (Academic, New York, London, 1972).

Offerman, V.

P. Grosse, V. Offerman, “Analysis or reflectance data using the Kramers–Kronig relations,” Appl. Phys. A 52, 138–144 (1991).
[CrossRef]

Poezd, A. D.

A. V. Tikhonravov, A. D. Poezd, I. V. Zuev, “On the applicability of the Kramers–Kronig relation to the determination of layered media optical parameters,” in Optical Interference Coatings, Vol. 15 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 304–306.

Sullivan, B.

Sullivan, B. T.

Tikhonov, A. N.

A. N. Tikhonov, V. Ya. Arsenin, Solutions of Ill-Posed Problems (Winston Wiley, New York, 1977).

Tikhonravov, A. V.

I. V. Zuev, A. V. Tikhonravov, “On the uniqueness of layered media parameters determination basing on reflectance measurement,” Zh. Vychisl. Mat. Mat. Fiz. 33, 428–438 (1993).

A. V. Tikhonravov, “Amplitude and phase properties of layered media spectral characteristics,” Zh. Vychisl. Mat. Mat. Fiz. 25, 442–450 (1985).

A. V. Tikhonravov, “On the optimal control theory problems connected with the layered media synthesis,” Dif. Urav. 21, 1516–1523 (1985).

A. V. Tikhonravov, “A stable method for solving layered media synthesis problems,” Dokl. Akad. Nauk SSSR 283, 582–585 (1985).

A. V. Tikhonravov, “Synthesis of layered media with pre-set amplitude-phase properties,” Zh. Vychisl. Mat. Mat. Fiz. 25, 1674–1688 (1985).

A. N. Baskakov, A. V. Tikhonravov, “Synthesis of two-component optical coatings,” Opt. Spectrosc. (USSR) 56, 559–562 (1984).

A. V. Tikhonravov, “About the limit accuracy of the synthesis problem solution,” Zh. Vychisl. Mat. Mat. Fiz. 22, 1421–1433 (1982).

Sh. A. Furman, A. V. Tikhonravov, Basics of Optics of Multilayer Systems (Editions Frontieres, Gif-sur-Yvette, France, 1992).

A. V. Tikhonravov, “General layered media properties in connection with coating design and investigation techniques,” in Optical Interference Coatings, Vol. 15 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 70–71.

A. V. Tikhonravov, M. K. Trubetskov, “Thin film coating design using second-order optimization methods,” in Thin Films for Optical Systems, K. H. Guenther, ed., Proc. Soc. Photo-Opt. Instrum. Eng. (to be published).

A. V. Tikhonravov, “On the optimality of thin-film optical coating design,” in Optical Thin Films and Applications, R. Herrmann, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1270, 28–35 (1990).

A. V. Tikhonravov, A. D. Poezd, I. V. Zuev, “On the applicability of the Kramers–Kronig relation to the determination of layered media optical parameters,” in Optical Interference Coatings, Vol. 15 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 304–306.

Trubetskov, M. K.

A. V. Tikhonravov, M. K. Trubetskov, “Thin film coating design using second-order optimization methods,” in Thin Films for Optical Systems, K. H. Guenther, ed., Proc. Soc. Photo-Opt. Instrum. Eng. (to be published).

Verly, P. G.

Zuev, I. V.

I. V. Zuev, A. V. Tikhonravov, “On the uniqueness of layered media parameters determination basing on reflectance measurement,” Zh. Vychisl. Mat. Mat. Fiz. 33, 428–438 (1993).

A. V. Tikhonravov, A. D. Poezd, I. V. Zuev, “On the applicability of the Kramers–Kronig relation to the determination of layered media optical parameters,” in Optical Interference Coatings, Vol. 15 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 304–306.

Appl. Opt. (6)

Appl. Phys. A (1)

P. Grosse, V. Offerman, “Analysis or reflectance data using the Kramers–Kronig relations,” Appl. Phys. A 52, 138–144 (1991).
[CrossRef]

Dif. Urav. (1)

A. V. Tikhonravov, “On the optimal control theory problems connected with the layered media synthesis,” Dif. Urav. 21, 1516–1523 (1985).

Dokl. Akad. Nauk SSSR (1)

A. V. Tikhonravov, “A stable method for solving layered media synthesis problems,” Dokl. Akad. Nauk SSSR 283, 582–585 (1985).

Opt. Spectrosc. (USSR) (1)

A. N. Baskakov, A. V. Tikhonravov, “Synthesis of two-component optical coatings,” Opt. Spectrosc. (USSR) 56, 559–562 (1984).

Thin Solid Films. (1)

J. A. Dobrowolski, “Computer design of optical coatings,” Thin Solid Films. 163, 97–110 (1988).
[CrossRef]

Zh. Vychisl. Mat. Mat. Fiz. (6)

I. V. Zuev, A. V. Tikhonravov, “On the uniqueness of layered media parameters determination basing on reflectance measurement,” Zh. Vychisl. Mat. Mat. Fiz. 33, 428–438 (1993).

A. V. Tikhonravov, “About the limit accuracy of the synthesis problem solution,” Zh. Vychisl. Mat. Mat. Fiz. 22, 1421–1433 (1982).

A. V. Tikhonravov, “Amplitude and phase properties of layered media spectral characteristics,” Zh. Vychisl. Mat. Mat. Fiz. 25, 442–450 (1985).

V. B. Glasko Yu, I. Khudak, “Additive representations of layered media characteristics and uniqueness of inverse problems,” Zh. Vychisl. Mat. Mat. Fiz. 20, 482–490 (1980).

Ye. Ya. Khruslov, “One-dimensional inverse problems in electrodynamics,” Zh. Vychisl. Mat. Mat. Fiz. 25, 548–561 (1985).

A. V. Tikhonravov, “Synthesis of layered media with pre-set amplitude-phase properties,” Zh. Vychisl. Mat. Mat. Fiz. 25, 1674–1688 (1985).

Other (13)

A. V. Tikhonravov, A. D. Poezd, I. V. Zuev, “On the applicability of the Kramers–Kronig relation to the determination of layered media optical parameters,” in Optical Interference Coatings, Vol. 15 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 304–306.

S. Lang, Complex Analysis (Addison-Wesley, Reading, Mass., 1977), p. 253.

L. D. Landau, E. M. Lifshiz, Electrodynamics of Continuous Media (Pergamon, London, Paris, 1958).

H. M. Nussenzveig, Causality and Dispersion Relations (Academic, New York, London, 1972).

H. Kaiser, “Analysis of the transfer coefficients in layered media,” in Optical Interference Coatings, Vol. 15 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D. C., 1992), pp. 72–74.

H. Kaiser, “Principles of thin film systems design,” in Thin Films for Optical Systems, K. H. Guenther, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1782 (to be published).

A. V. Tikhonravov, “General layered media properties in connection with coating design and investigation techniques,” in Optical Interference Coatings, Vol. 15 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 70–71.

V. G. Boltyanskii, Mathematical Methods of Optimal Control (Holt, Rinehart & Winston, New York, 1971).

Sh. A. Furman, A. V. Tikhonravov, Basics of Optics of Multilayer Systems (Editions Frontieres, Gif-sur-Yvette, France, 1992).

A. N. Tikhonov, V. Ya. Arsenin, Solutions of Ill-Posed Problems (Winston Wiley, New York, 1977).

A. V. Tikhonravov, M. K. Trubetskov, “Thin film coating design using second-order optimization methods,” in Thin Films for Optical Systems, K. H. Guenther, ed., Proc. Soc. Photo-Opt. Instrum. Eng. (to be published).

A. V. Tikhonravov, “On the optimality of thin-film optical coating design,” in Optical Thin Films and Applications, R. Herrmann, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1270, 28–35 (1990).

P. Baumeister, “Detailed knowledge of optical coating design techniques may be superfluous to produce usable coatings,” in Optical Interference Coatings, Vol. 15 of OSA 1992 Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 8–10.

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Figures (8)

Fig. 1
Fig. 1

Optical coating model.

Fig. 2
Fig. 2

Relationship between analysis and synthesis.

Fig. 3
Fig. 3

Complex wave-number plane, ν = k + iσ. Two pairs of amplitude reflectance zeros (○) are shown.

Fig. 4
Fig. 4

Transferring a pair of amplitude reflectance zeros from one half-plane to the other causes a change in the phase shift on reflection but no changes in the intensity reflectance.

Fig. 5
Fig. 5

Plot of the ψ(k) function, which gives the main contribution to the phase factor ϕ(k) for k close to the zero ν0 = k0 + iσ0, with σ0 ≈ 0: for curve (2) the amplitude reflectance zero is closer to the real axis than for curve (1).

Fig. 6
Fig. 6

Plots on the refractive-index profile: (a) no zero transformations, (b) after one zero transformation, (c) after two zero transformations. The zero transformations corresponding to (b) and (c) are shown in (d). The refractive indices of the substrate and the incident medium are 1.52 and 1.00, respectively.

Fig. 7
Fig. 7

This figure illustrates the maximum principle. When b0(z) is positive, then the optimal value of ∊ is ∊max; when b0(z) is negative, the optimal value of ∊ is ∊min.

Fig. 8
Fig. 8

Optimal 14-layer antireflection coating from Ref. 26, where (a) is the reflectance of the coating, (b) is the refractive-index profile, and (c) is the b0(z) function from the maximum principle. There is an exact agreement between the expected refractive-index profile and the profile of the actual design.

Equations (39)

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

n s = s 1 / 2 , n a = a 1 / 2 , n ( z ) = 1 / 2 ( z ) .
d u / d z = i k υ , d υ / d z = i k ( z ) u ,
u ( 0 , k ) = 1 , υ ( 0 , k ) = n s .
r ( k ) = n a u ( z a , k ) υ ( z a , k ) n a u ( z a , k ) + υ ( z a , k ) , t ( k ) = 2 n a n a u ( z a , k ) + υ ( z a , k ) .
R ( k ) = | r ( k ) | 2 , T ( k ) = ( n s / n a ) | t ( k ) | 2 .
x = 0 z n ( z ) d z
d u / d x = ikn 1 ( x ) υ , d υ / d x = ikn ( x ) u .
r ( k ) = n a u ( x a , k ) υ ( x a , k ) n a u ( x a , k ) + υ ( x a , k ) , t ( k ) = 2 n a n a u ( x a , k ) + υ ( x a , k ) ,
x a = 0 z a n ( z ) d z .
f 1 ( ν ) = ( n a n s ) 1 / 2 1 t ( ν ) , f 2 ( ν ) = ( n a n s ) 1 / 2 r ( ν ) t ( ν )
f 1 ( k ) = exp ( i k x a ) + x a x a F 1 ( t ) exp ( ikt ) d t ,
f 2 ( k ) = x a x a F 2 ( t ) exp ( ikt ) d t ,
r ( k ) 0 at k .
Q ( k ) = | f 2 ( k ) | = [ R ( k ) / T ( k ) ] 1 / 2 .
f 2 ( k ) = Q ( k ) exp [ i ϕ ( k ) ] = 1 2 n ( t ) n ( t ) exp ( ikt ) d t .
r ( ν ) = r ( ν ) ( ν ν 0 * ) ( ν + ν 0 ) ( ν + ν 0 * ) ( ν ν 0 ) .
| r ( k ) | = | r ( k ) | ,
ñ ( x ) = n ( x ) [ 1 + 2 σ 0 0 x | u ( x , ν 0 ) | 2 n ( x ) n s 1 d x 1 + 2 σ 0 0 x | υ ( x , ν 0 ) | 2 n 1 ( x ) n s 1 d x ] 2 .
( 1 ν ν i ) ,
f 2 ( ν ) = ( 1 ν ν 0 ) ( 1 + ν ν 0 * ) = ( 1 + 2 i ν σ 0 ν 2 k 0 2 + σ 0 2 ) ·
1 + 2 i k σ 0 k 2 k 0 2 + σ 0 2 = ρ ( k ) exp [ i ψ ( k ) ] ,
ρ ( k ) = [ ( 1 k 2 k 0 2 + σ 0 2 ) 2 + 4 k 2 σ 0 2 ( k 0 2 + σ 0 2 ) 2 ] 1 / 2 , ψ ( k ) = A tan 2 σ 0 k k 0 2 + σ 0 2 k 2 .
arg r ( k ) = k π 0 log R ( ξ ) ξ 2 k 2 d ξ .
max k 1 k k 2 | T ( k ) T ( k ) | δ .
min ( z ) max .
F [ ( z ) ] = { k j } w j [ T ( k j ) T ( k j ) ] 2 ,
d Y / d z = f [ Y ( z ) , ( z ) , k ] ,
Y ( 0 ) = Y 0 .
F [ ( z ) ] = { k j } Φ [ Y ( z a , k j ) ] .
d Y / d t = f [ Y ( t ) , u ( t ) ] .
Y ( 0 ) = Y 0 .
F = Φ [ Y ( T ) ] .
u 1 u ( t ) u 2 .
d ϕ / d z = i k ( z ) ψ , d ψ / d z = i k ϕ .
ϕ ( z a ) = 2 w j ( T T ) r ( 1 r * ) t * , ψ ( z a ) = 2 n a 1 w j ( T T ) r ( 1 + r * ) t * .
H ( u , υ , ϕ , ψ , ) = { k j } i k j [ υ ( j ) ( z ) ϕ ( j ) ( z ) + ( z ) u ( j ) ( z ) ψ ( j ) ( z ) ] .
Re H [ u 0 ( z ) , υ 0 ( z ) , ϕ 0 ( z ) , ψ 0 ( z ) , ] Re H [ u 0 ( z ) , υ 0 ( z ) , ϕ 0 ( z ) , ψ 0 ( z ) , 0 ( z ) ] .
a 0 ( z ) = Re { k j } i k j υ 0 ( j ) ( z ) ϕ 0 ( j ) ( z ) , b 0 ( z ) = Re { k j } i k j u 0 ( j ) ( z ) ψ 0 ( j ) ( z ) .
a 0 ( z ) + b 0 ( z ) a 0 ( z ) + 0 ( z ) b 0 ( z ) .

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