Abstract

We present a theoretical and numerical method for solving problems of inverse scattering in optics given data on the far field, find the scattering object. This method is applied to perfectly conducting diffraction gratings. From the efficiency curve in a Littrow mounting and in the TE case, we derive the shape of the grating surface. Two different cases must be distinguished. The first problem, which we call “reconstruction,” is to compute the profile when the efficiency is experimentally known. In the second one, called “synthesis,” we give a priori an efficiency curve and look for the corresponding grating(s), if it actually exists. We show several theoretical reconstructions for various gratings, and present our first results in the very difficult field of synthesis. The relevance of this method in the domain of electromagnetic optics is then outlined by its application to two other problems.

© 1980 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. Inverse Source Problems in Optics, edited by H. P. Baltes (Springer-Verlag, Heidelberg, 1978).
    [Crossref]
  2. I. J. Wilson, Ph.D. Thesis, University of Tasmania, Hobart, 1977 (unpublished).
  3. A. Roger and D. Maystre, “The perfectly conducting grating from the point of view of inverse diffraction,” Opt. Acta 26, 447–460 (1979).
    [Crossref]
  4. In this grating formula, the incidence angle θ and the diffracted angles θn are defined using, respectively, the counterclockwise sense and the clockwise sense from 0y axis.
  5. R. Petit, “Electromagnetic grating theories: limitations and successes,” Nouv. Rev. Opt. 6, 129–135 (1975).
    [Crossref]
  6. D. Maystre, “A new general integral theory for dielectric coated gratings,” J. Opt. Soc. Am. 68, 490–495 (1978).
    [Crossref]
  7. D. Maystre, “Sur la diffraction d’une onde plane électromagnétique par un réseau métallique,” Opt. Commun. 8, 216–219 (1973).
    [Crossref]
  8. R. Petit and D. Maystre, “Application des lois de l’électromagnétisme à l’étude des réseaux,” Rev. Phys. Ap. 7, 427–441 (1972).
    [Crossref]
  9. D. Maystre, “Sur la diffraction et l’absorption par les réseaux utilisés dans l’infrarouge, le visible et l’ultraviolet,” Thesis, Université Aix-Marseille III, CNRS A.O. 9545, 1974 (unpublished).
  10. Electromagnetic Theory of Gratings, edited by R. Petit (Springer-Verlag, Heidelberg, 1980).
    [Crossref]
  11. C. Miranda, Partial Differential Equations of Elliptic Type (Springer-Verlag, Heidelberg, 1970).
  12. J. C. Guillot and C. H. Wilcox, “Théorie spectrale du laplacien dans des ouverts coniques et cylindriques non bornés,” C.R. Acad. Sci. Paris A 282, 1171–1174 (1976).
  13. L. Schwarz, Théorie des Distributions (Hermann, Paris, 1966).
  14. P. M. Van den Berg, “Diffraction theory of a reflection grating,” Appl. Sci. Res. 24, 261–293 (1971), App. A.
  15. L. Y. Kantorovitch, “On Newton’s method for functional equations,” DAN SSSR 59 (7), 1237–1240 (1948).
  16. Dunford-Schwartz, Linear Operators (Wiley-Interscience, New York, 1966), Chap. II, p. 92.
  17. E. Hille and J. D. Tamarkin, “On the characteristic values of linear integral equations,” Ata Math. 57, 1–75 (1931).
  18. A. Roger, D. Maystre, and M. Cadilhac, “On a problem of inverse scattering in Optics: the dielectric inhomogeneous medium,” J. Opt. 9 (2), 83–90 (1978).
    [Crossref]
  19. M. Bertero, C. de Mol, and G. A. Viano, “Restoration of optical objects using regularization,” Opt. Lett. 3, 51–53 (1978).
    [Crossref] [PubMed]
  20. D. Maystre and P. Vincent, “Diffraction d’une onde électromagnétique plane par un objet cylindrique non infiniment conducteur de section arbitraire,” Opt. Commun. 5, 327–330 (1972).
    [Crossref]
  21. A. Roger and D. Maystre, “Determination of the index profile of a dielectric plate by optical methods,” SPIE 136, 26–28 (1977).
    [Crossref]
  22. A. Roger, “Determination of the index profile of a dielectric plate from scattering data,” Lecture notes in Physics n° 85, Applied Inverse Problems (Springer-Verlag, Heidelberg, 1978).
    [Crossref]
  23. D. Maystre and M. Cadilhac, “A phenomenological theory for gratings, perfect blazing for polarized light in non-zero deviation mounting,” Proceedings of the U.R.S.I Symposium, Munich, 1980 (unpublished).
  24. A. Roger, “Grating profile optimizations by inverse scattering methods,” Opt. Commun. 32, 11–13 (1980).
    [Crossref]
  25. M. Reed and B. Simon, Methods of Modern Mathematical Physics (Academic, New York, 1972).
  26. A. Tikhonov and V. Arsénine, Méthodes de Résolution de Problémes Mal Posés (Editions de Moscou, Moscow, 1976).
  27. K. Miller, “Least squares method for ill posed problems with a prescribed bound,” Siam J. Math. Anal. 1, 52–74 (1970).
    [Crossref]
  28. R. Petit, “Quelques propriétés des réseaux métalliques,” Opt. Acta 14, 301–310 (1967).
    [Crossref]
  29. M. Breidne and D. Maystre, “Equivalence of ruled, holographic and lamellar gratings in constant deviation mountings,” Appl. Opt. (to be published).

1980 (1)

A. Roger, “Grating profile optimizations by inverse scattering methods,” Opt. Commun. 32, 11–13 (1980).
[Crossref]

1979 (1)

A. Roger and D. Maystre, “The perfectly conducting grating from the point of view of inverse diffraction,” Opt. Acta 26, 447–460 (1979).
[Crossref]

1978 (3)

1977 (1)

A. Roger and D. Maystre, “Determination of the index profile of a dielectric plate by optical methods,” SPIE 136, 26–28 (1977).
[Crossref]

1976 (1)

J. C. Guillot and C. H. Wilcox, “Théorie spectrale du laplacien dans des ouverts coniques et cylindriques non bornés,” C.R. Acad. Sci. Paris A 282, 1171–1174 (1976).

1975 (1)

R. Petit, “Electromagnetic grating theories: limitations and successes,” Nouv. Rev. Opt. 6, 129–135 (1975).
[Crossref]

1973 (1)

D. Maystre, “Sur la diffraction d’une onde plane électromagnétique par un réseau métallique,” Opt. Commun. 8, 216–219 (1973).
[Crossref]

1972 (2)

R. Petit and D. Maystre, “Application des lois de l’électromagnétisme à l’étude des réseaux,” Rev. Phys. Ap. 7, 427–441 (1972).
[Crossref]

D. Maystre and P. Vincent, “Diffraction d’une onde électromagnétique plane par un objet cylindrique non infiniment conducteur de section arbitraire,” Opt. Commun. 5, 327–330 (1972).
[Crossref]

1971 (1)

P. M. Van den Berg, “Diffraction theory of a reflection grating,” Appl. Sci. Res. 24, 261–293 (1971), App. A.

1970 (1)

K. Miller, “Least squares method for ill posed problems with a prescribed bound,” Siam J. Math. Anal. 1, 52–74 (1970).
[Crossref]

1967 (1)

R. Petit, “Quelques propriétés des réseaux métalliques,” Opt. Acta 14, 301–310 (1967).
[Crossref]

1948 (1)

L. Y. Kantorovitch, “On Newton’s method for functional equations,” DAN SSSR 59 (7), 1237–1240 (1948).

1931 (1)

E. Hille and J. D. Tamarkin, “On the characteristic values of linear integral equations,” Ata Math. 57, 1–75 (1931).

Arsénine, V.

A. Tikhonov and V. Arsénine, Méthodes de Résolution de Problémes Mal Posés (Editions de Moscou, Moscow, 1976).

Bertero, M.

Breidne, M.

M. Breidne and D. Maystre, “Equivalence of ruled, holographic and lamellar gratings in constant deviation mountings,” Appl. Opt. (to be published).

Cadilhac, M.

A. Roger, D. Maystre, and M. Cadilhac, “On a problem of inverse scattering in Optics: the dielectric inhomogeneous medium,” J. Opt. 9 (2), 83–90 (1978).
[Crossref]

D. Maystre and M. Cadilhac, “A phenomenological theory for gratings, perfect blazing for polarized light in non-zero deviation mounting,” Proceedings of the U.R.S.I Symposium, Munich, 1980 (unpublished).

de Mol, C.

Guillot, J. C.

J. C. Guillot and C. H. Wilcox, “Théorie spectrale du laplacien dans des ouverts coniques et cylindriques non bornés,” C.R. Acad. Sci. Paris A 282, 1171–1174 (1976).

Hille, E.

E. Hille and J. D. Tamarkin, “On the characteristic values of linear integral equations,” Ata Math. 57, 1–75 (1931).

Kantorovitch, L. Y.

L. Y. Kantorovitch, “On Newton’s method for functional equations,” DAN SSSR 59 (7), 1237–1240 (1948).

Maystre, D.

A. Roger and D. Maystre, “The perfectly conducting grating from the point of view of inverse diffraction,” Opt. Acta 26, 447–460 (1979).
[Crossref]

A. Roger, D. Maystre, and M. Cadilhac, “On a problem of inverse scattering in Optics: the dielectric inhomogeneous medium,” J. Opt. 9 (2), 83–90 (1978).
[Crossref]

D. Maystre, “A new general integral theory for dielectric coated gratings,” J. Opt. Soc. Am. 68, 490–495 (1978).
[Crossref]

A. Roger and D. Maystre, “Determination of the index profile of a dielectric plate by optical methods,” SPIE 136, 26–28 (1977).
[Crossref]

D. Maystre, “Sur la diffraction d’une onde plane électromagnétique par un réseau métallique,” Opt. Commun. 8, 216–219 (1973).
[Crossref]

R. Petit and D. Maystre, “Application des lois de l’électromagnétisme à l’étude des réseaux,” Rev. Phys. Ap. 7, 427–441 (1972).
[Crossref]

D. Maystre and P. Vincent, “Diffraction d’une onde électromagnétique plane par un objet cylindrique non infiniment conducteur de section arbitraire,” Opt. Commun. 5, 327–330 (1972).
[Crossref]

D. Maystre, “Sur la diffraction et l’absorption par les réseaux utilisés dans l’infrarouge, le visible et l’ultraviolet,” Thesis, Université Aix-Marseille III, CNRS A.O. 9545, 1974 (unpublished).

M. Breidne and D. Maystre, “Equivalence of ruled, holographic and lamellar gratings in constant deviation mountings,” Appl. Opt. (to be published).

D. Maystre and M. Cadilhac, “A phenomenological theory for gratings, perfect blazing for polarized light in non-zero deviation mounting,” Proceedings of the U.R.S.I Symposium, Munich, 1980 (unpublished).

Miller, K.

K. Miller, “Least squares method for ill posed problems with a prescribed bound,” Siam J. Math. Anal. 1, 52–74 (1970).
[Crossref]

Miranda, C.

C. Miranda, Partial Differential Equations of Elliptic Type (Springer-Verlag, Heidelberg, 1970).

Petit, R.

R. Petit, “Electromagnetic grating theories: limitations and successes,” Nouv. Rev. Opt. 6, 129–135 (1975).
[Crossref]

R. Petit and D. Maystre, “Application des lois de l’électromagnétisme à l’étude des réseaux,” Rev. Phys. Ap. 7, 427–441 (1972).
[Crossref]

R. Petit, “Quelques propriétés des réseaux métalliques,” Opt. Acta 14, 301–310 (1967).
[Crossref]

Reed, M.

M. Reed and B. Simon, Methods of Modern Mathematical Physics (Academic, New York, 1972).

Roger, A.

A. Roger, “Grating profile optimizations by inverse scattering methods,” Opt. Commun. 32, 11–13 (1980).
[Crossref]

A. Roger and D. Maystre, “The perfectly conducting grating from the point of view of inverse diffraction,” Opt. Acta 26, 447–460 (1979).
[Crossref]

A. Roger, D. Maystre, and M. Cadilhac, “On a problem of inverse scattering in Optics: the dielectric inhomogeneous medium,” J. Opt. 9 (2), 83–90 (1978).
[Crossref]

A. Roger and D. Maystre, “Determination of the index profile of a dielectric plate by optical methods,” SPIE 136, 26–28 (1977).
[Crossref]

A. Roger, “Determination of the index profile of a dielectric plate from scattering data,” Lecture notes in Physics n° 85, Applied Inverse Problems (Springer-Verlag, Heidelberg, 1978).
[Crossref]

Schwarz, L.

L. Schwarz, Théorie des Distributions (Hermann, Paris, 1966).

Simon, B.

M. Reed and B. Simon, Methods of Modern Mathematical Physics (Academic, New York, 1972).

Tamarkin, J. D.

E. Hille and J. D. Tamarkin, “On the characteristic values of linear integral equations,” Ata Math. 57, 1–75 (1931).

Tikhonov, A.

A. Tikhonov and V. Arsénine, Méthodes de Résolution de Problémes Mal Posés (Editions de Moscou, Moscow, 1976).

Van den Berg, P. M.

P. M. Van den Berg, “Diffraction theory of a reflection grating,” Appl. Sci. Res. 24, 261–293 (1971), App. A.

Viano, G. A.

Vincent, P.

D. Maystre and P. Vincent, “Diffraction d’une onde électromagnétique plane par un objet cylindrique non infiniment conducteur de section arbitraire,” Opt. Commun. 5, 327–330 (1972).
[Crossref]

Wilcox, C. H.

J. C. Guillot and C. H. Wilcox, “Théorie spectrale du laplacien dans des ouverts coniques et cylindriques non bornés,” C.R. Acad. Sci. Paris A 282, 1171–1174 (1976).

Wilson, I. J.

I. J. Wilson, Ph.D. Thesis, University of Tasmania, Hobart, 1977 (unpublished).

Appl. Sci. Res. (1)

P. M. Van den Berg, “Diffraction theory of a reflection grating,” Appl. Sci. Res. 24, 261–293 (1971), App. A.

Ata Math. (1)

E. Hille and J. D. Tamarkin, “On the characteristic values of linear integral equations,” Ata Math. 57, 1–75 (1931).

C.R. Acad. Sci. Paris A (1)

J. C. Guillot and C. H. Wilcox, “Théorie spectrale du laplacien dans des ouverts coniques et cylindriques non bornés,” C.R. Acad. Sci. Paris A 282, 1171–1174 (1976).

DAN SSSR (1)

L. Y. Kantorovitch, “On Newton’s method for functional equations,” DAN SSSR 59 (7), 1237–1240 (1948).

J. Opt. (1)

A. Roger, D. Maystre, and M. Cadilhac, “On a problem of inverse scattering in Optics: the dielectric inhomogeneous medium,” J. Opt. 9 (2), 83–90 (1978).
[Crossref]

J. Opt. Soc. Am. (1)

Nouv. Rev. Opt. (1)

R. Petit, “Electromagnetic grating theories: limitations and successes,” Nouv. Rev. Opt. 6, 129–135 (1975).
[Crossref]

Opt. Acta (2)

R. Petit, “Quelques propriétés des réseaux métalliques,” Opt. Acta 14, 301–310 (1967).
[Crossref]

A. Roger and D. Maystre, “The perfectly conducting grating from the point of view of inverse diffraction,” Opt. Acta 26, 447–460 (1979).
[Crossref]

Opt. Commun. (3)

D. Maystre, “Sur la diffraction d’une onde plane électromagnétique par un réseau métallique,” Opt. Commun. 8, 216–219 (1973).
[Crossref]

D. Maystre and P. Vincent, “Diffraction d’une onde électromagnétique plane par un objet cylindrique non infiniment conducteur de section arbitraire,” Opt. Commun. 5, 327–330 (1972).
[Crossref]

A. Roger, “Grating profile optimizations by inverse scattering methods,” Opt. Commun. 32, 11–13 (1980).
[Crossref]

Opt. Lett. (1)

Rev. Phys. Ap. (1)

R. Petit and D. Maystre, “Application des lois de l’électromagnétisme à l’étude des réseaux,” Rev. Phys. Ap. 7, 427–441 (1972).
[Crossref]

Siam J. Math. Anal. (1)

K. Miller, “Least squares method for ill posed problems with a prescribed bound,” Siam J. Math. Anal. 1, 52–74 (1970).
[Crossref]

SPIE (1)

A. Roger and D. Maystre, “Determination of the index profile of a dielectric plate by optical methods,” SPIE 136, 26–28 (1977).
[Crossref]

Other (13)

A. Roger, “Determination of the index profile of a dielectric plate from scattering data,” Lecture notes in Physics n° 85, Applied Inverse Problems (Springer-Verlag, Heidelberg, 1978).
[Crossref]

D. Maystre and M. Cadilhac, “A phenomenological theory for gratings, perfect blazing for polarized light in non-zero deviation mounting,” Proceedings of the U.R.S.I Symposium, Munich, 1980 (unpublished).

Dunford-Schwartz, Linear Operators (Wiley-Interscience, New York, 1966), Chap. II, p. 92.

L. Schwarz, Théorie des Distributions (Hermann, Paris, 1966).

D. Maystre, “Sur la diffraction et l’absorption par les réseaux utilisés dans l’infrarouge, le visible et l’ultraviolet,” Thesis, Université Aix-Marseille III, CNRS A.O. 9545, 1974 (unpublished).

Electromagnetic Theory of Gratings, edited by R. Petit (Springer-Verlag, Heidelberg, 1980).
[Crossref]

C. Miranda, Partial Differential Equations of Elliptic Type (Springer-Verlag, Heidelberg, 1970).

In this grating formula, the incidence angle θ and the diffracted angles θn are defined using, respectively, the counterclockwise sense and the clockwise sense from 0y axis.

Inverse Source Problems in Optics, edited by H. P. Baltes (Springer-Verlag, Heidelberg, 1978).
[Crossref]

I. J. Wilson, Ph.D. Thesis, University of Tasmania, Hobart, 1977 (unpublished).

M. Breidne and D. Maystre, “Equivalence of ruled, holographic and lamellar gratings in constant deviation mountings,” Appl. Opt. (to be published).

M. Reed and B. Simon, Methods of Modern Mathematical Physics (Academic, New York, 1972).

A. Tikhonov and V. Arsénine, Méthodes de Résolution de Problémes Mal Posés (Editions de Moscou, Moscow, 1976).

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 (11)

FIG. 1
FIG. 1

Perfectly conducting grating. The profile of the grooves is described by the function y = F(x) of period d. The − 1 order and the incident wave have opposite directions (Littrow mounting).

FIG. 2
FIG. 2

Newton method.

FIG. 3
FIG. 3

Geometrical transformations leaving E(θ) unchanged. (a) Translation in the 0y direction. (b) Translation in the 0x direction. (c) Symmetry about 0y (only when there are two diffracted orders).

FIG. 4
FIG. 4

Schematic representation of the continuity of the operator O and the noncontinuity of the operator O 1.

FIG. 5
FIG. 5

Perfectly conducting cylinder. The profile of the cross section is described by the function 0M = ρ = F(θ).

FIG. 6
FIG. 6

Reconstruction of a sinusoidal profile ( 12 ° θ 19 ° ).1: true profile. 2: computed profile with a random error of 1 % in E(θ). 3: computed profile with a random error of 10%.

FIG. 7
FIG. 7

Reconstruction of a sinusoidal distorted profile ( 12 ° θ 19 ° ). 1: true profile. 2: computed profile with a random error of 1 %. 3: computed profile with a random error of 10%.

FIG. 8
FIG. 8

Reconstruction of a triangular profile ( 12 ° θ 19 ° ). 1: true profile. 2: computed profile with a random error of 1 %. 3: computed profile with a random error of 10%.

FIG. 9
FIG. 9

Reconstruction of a sinusoidal distorted symmetric profile ( 20 ° θ 60 ° ). 1: true profile. 2: computed profile with a random error of 1 %. 3: computed profile with a random error of 10%.

FIG. 10
FIG. 10

Reconstruction of a triangular symmetric profile ( 20 ° θ 60 ° ). 1: true profile. 2: computed profile with a random error from 1 % to 10%.

FIG. 11
FIG. 11

Synthesis of a grating. 1: sinusoidal grating and the corresponding efficiency curve. 2: arbitrary efficiency curve. 3: synthetized grating and the corresponding efficiency curve.

Tables (1)

Tables Icon

TABLE I Singular values of the operators G F and D F in Littrow mounting. The grating profile is F(x) = 0.1 dcos(Kx). The order of multiplicity equal to 2 for each singular value of G F is linked to the symmetry of the profile. The period d is equal to 2π.

Equations (99)

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

sin θ n = sin θ + n λ / d .
2 sin θ = λ / d .
R 1 : N = 2 , 19.47 ° < θ < 90 ° ; R 2 : N = 4 , 11.54 ° < θ < 19.47 ° ; R n : N = 2 n , arcsin ( 1 2 n + 1 ) < θ < arcsin ( 1 2 n 1 ) .
Δ U + k 2 U = 0 if y > F ( x ) ;
lim y F ( x ) U ( x , y ) = 0 ;
U d ( x , y ) = U ( x , y ) U i ( x , y ) ;
Δ G + k 2 G = δ ,
U ( x , y ) = U i ( x , y ) + 0 d G [ x x , y F ( x ) ] ϕ F ( θ , x ) × exp ( i k x sin θ ) d x ,
G ( x 0 , y 0 ) = 1 2 idk + 1 β n exp ( i k β n | y 0 | + i k α n x 0 ) ,
α n = sin θ + n λ / d ,
β n = 1 α n 2 for | α n | 1 = i α n 2 1 for | α n | > 1 .
0 d G F ( θ , x , x ) ϕ F ( θ , x ) d x = exp [ i K cot ( θ ) F ( x ) / 2 ] ,
G F ( θ , x , x ) = sin θ 2 i π n = + 1 β n × exp [ i K ( β n 2 sin θ | F ( x ) F ( x ) | + n ( x x ) ) ] ,
α n = ( 2 n + 1 ) sin θ , K = 2 π / d ;
B ( θ ) = tan θ 2 i π 0 d exp [ i K ( x cot θ 2 F ( x ) ) ] ϕ F ( θ , x ) d x ,
E ( θ ) = | B ( θ ) | 2 .
E = O · F .
O · F 0 = E 0 .
δ E = D F · δ F ,
0 d D ( θ , x ) δ F ( x ) d x = δ E ( θ ) ,
D ( θ , x ) = B ¯ ( θ ) A F ( θ , x ) + B ( θ ) F ( θ , x ) ,
A F ( θ , x ) = ϕ F ( θ , x ) exp ( iKx ) [ tan θ i π 0 d ϕ F ( θ , x ) × G F ( θ , x , x ) d x 1 d exp ( i K cot θ 2 F ( x ) ) ] ,
G F ( θ , x , x ) = 1 2 d sgn [ F ( x ) F ( x ) ] × m exp ( i K β m 2 sin θ | F ( x ) F ( x ) | + imK ( x x ) ) ,
F ( x ) = a 0 + n = 1 a n cos nKx + n = 1 b n sin nKx .
δ F ( x ) = F ( x x 0 ) F ( x ) = x 0 F ( x ) .
F ( x ) = n = 1 a n cos ( nKx ) + n = 2 b n sin ( nKx ) .
j ( 1 , J ) G F ( θ , x j , x ) ϕ F ( θ , x ) d x = exp [ i K cot θ F ( x j ) / 2 ] = ω j .
j ( 1 , J ) G F ( θ , x j , x ) ϕ F ( θ , x ) d x = l γ j l ϕ l .
[ D F * D F + r ( J d 2 d x 2 ) ] δ F = D F * · δ E .
X 1 = 1 d , X 2 n = 2 d cos nKx , X 2 n + 1 = 2 d sin nKx ,
δ f n = δ F , X n , δ g n = D F * · δ E , X n , I n m = X n , [ D F * D F + r ( J d 2 d x 2 ) ] · X m ,
n ( 1 , ) m = 1 I n m δ f m = δ g n .
n ( 1 , M ) m = 1 M I n m δ f m = δ g n .
δ g n = 0 d θ 1 θ 2 D ( θ , x ) X n ( x ) δ E ( θ ) d θ d x ,
I n m = r [ 1 + int 2 ( n / 2 ) K 2 ] δ n m + 0 d 0 d θ 1 θ 2 D ( θ , x ) D ( θ , x ) X n ( x ) X m ( x ) dx d x d θ .
i 4 0 2 π H 0 ( 1 ) [ k R F ( θ , θ ) ] ϕ F ( θ ) d θ = exp [ ikF ( θ ) cos θ ]
R F ( θ , θ ) = [ F 2 ( θ ) + F 2 ( θ ) 2 F ( θ ) F ( θ ) cos ( θ θ ) ] 1 / 2
k = | k | ,
B ( θ ) = i 4 2 π k exp ( i π 4 ) 0 2 π exp [ ikF ( θ ) × cos ( θ θ ) ] ϕ F ( θ ) d θ .
E ( θ ) = 2 π | B ( θ ) | 2 .
δ E = D F · δ F ,
D ( θ , θ ) = 2 π [ B ¯ ( θ ) A F ( θ , θ ) + B ( θ ) F ( θ , θ ) ] , A F ( θ , θ ) = i k ϕ F ( θ ) { cos ( θ θ ) exp [ ikF ( θ ) cos ( θ θ ) ] + p . v . 0 2 π d θ F ( θ ) F ( θ ) cos ( θ θ ) R F ( θ , θ ) × H 1 + ( k R F ( θ , θ ) ϕ F ( θ , θ ) } i k ϕ F ( θ , θ ) { cos θ exp [ ikF ( θ ) cos θ ] + p . v . 0 2 π d θ F ( θ ) F ( θ ) cos ( θ θ ) R F ( θ , θ ) × H 1 + ( k R F ( θ , θ ) ) ϕ F ( θ ) } ,
i 4 0 2 π H ¯ 0 ( 1 ) [ k R F ( θ , θ ) ] ϕ F ( θ , θ ) d θ = exp [ ikF ( θ ) cos ( θ θ ) ] .
E ( θ ) = sin 2 [ A ( h ) cot θ + B ( h ) cot 3 θ ] ,
A ( h ) = 1.570 h 3.225 h 3 + 5.557 h 5 , B ( h ) = 0.5972 h 3 1.125 h 5 ;
G F ( θ = 30 ° )
D F ( 12 ° < θ < 19 ° )
| A ( s , t ) | 2 ds dt < .
t 1 t 2 A ( s , t ) P ( t ) d t = Q ( s )
A * A · P n = a n 2 P n .
N = { n ; a n 0 } , N 2 = { n ; a n = 0 } .
Q n = 1 a n A · P n .
P ( t ) = n N p n P n ( t ) + n N 2 p n P n ( t ) , Q ( s ) = n N q n Q n ( s ) + n N 1 q n Q n ( s ) .
n N a n p n Q n ( s ) = n N q n Q n ( s ) + n N 1 q n Q n ( s ) .
p n = q n / a n ( n N ) .
n N | p n | 2
n N | p n | 2 = n N | q n a n | 2
n N | q n | 2 ,
p n = 0 ( n N 2 ) .
δ P ( t ) = n N δ p n P n ( t ) + n N 2 δ p n P n ( t ) , δ Q ( s ) = n N δ q n Q n ( s ) , δ p n = δ q n / a n ( n N ) .
δ P 2 = n | δ p n | 2 > M , δ Q 2 = n | δ q n | 2 < .
M ( P ) = A · P Q 2 + r R · P 2 r > 0 .
( A * A + r R * R ) · P = A * · Q .
p n = q n a n r + a n 2 · ( n N ) , p n = 0 ( n N 2 ) .
A · P Q = Δ Q .
0 d G F ( θ , x , x ) ϕ F ( θ , x ) d x = exp [ i K cot θ F ( x ) / 2 ] = Q ( x ) ;
G 0 * G 0 ( θ , x , x ) = 0 d G 0 ¯ ( θ , x , x ) · G 0 ( θ , x , x ) d x = d sin 2 θ 4 π 2 n = + 1 | β n | 2 exp [ inK ( x x ) ] ,
a n 2 = a n 1 2 = d 2 sin 2 θ 4 π 2 1 | 1 ( 2 n + 1 ) 2 sin 2 θ | , P n ( x ) = 1 / d exp ( i nKx ) .
a n d 2 π 1 2 n n .
| q n | M 1 / n 3 | p n | = | ϕ n | M 2 / n 2 when n ;
| q n | M 1 / n 2 | ϕ n | M 2 / n .
G F · ϕ F = S ,
S ( x ) = exp [ i K cot θ F ( x ) / 2 ] , B = S 1 | ϕ F ,
S 1 ( x ) = t g θ 2 i π exp [ i K ( x cot θ 2 F ( x ) ) ] .
F 1 F 2 = 0 d F 1 ¯ ( x ) F 2 ( x ) d x .
δ B = δ S 1 ϕ F + S 1 δ ϕ F .
δ S 1 = + i K ( cot θ ) / 2 · S 1 · δ F .
G F · δ ϕ F = δ S δ G F · ϕ F = i K cot θ 2 · S · δ F δ G F · ϕ F .
G F * · ϕ F = S 1 .
S 1 δ ϕ F = G F * · ϕ F δ ϕ F = ϕ F G F · δ ϕ F = i k ( cot θ ) / 2 ϕ F S δ F ϕ F | δ G F · ϕ F .
δ G F ( θ , x , x ) = δ F ( x ) F ( x ) 2 d × m = + exp [ i K ( β m 2 sin θ | F ( x ) F ( x ) | + n ( x x ) ) ] ,
| F ( x ) F ( x ) | = [ F ( x ) F ( x ) ] sgn [ F ( x ) F ( x ) ] , δ | F ( x ) F ( x ) | = δ [ F ( x ) F ( x ) ] sgn [ F ( x ) F ( x ) ] + [ F ( x ) F ( x ) ] δ sgn [ F ( x ) F ( x ) ] .
ϕ F δ G F · ϕ F = 0 d d x ϕ F ¯ ( θ , x ) lim 0 0 d d x × [ δ F ( x ) δ F ( x ) ] ϕ F ( θ , x ) G F ( θ , x , x ) = I 1 I 2 | x x | > ,
G F ( θ , x , x ) = 1 2 d + sgn [ F ( x ) F ( x ) ] × exp [ i K ( β n 2 sin θ | F ( x ) F ( x ) | + n ( x x ) ) ] ,
I 1 = 0 d d x ϕ F ¯ ( θ , x ) δ F ( x ) lim 0 0 d d x ϕ F ( θ , x ) × G F ( θ , x , x ) | x x | > , I 2 = 0 d d x ϕ F ¯ ( θ , x ) lim 0 0 d d x δ F ( x ) ϕ F ( θ , x ) × G F ( θ , x , x ) | x x | > .
I 2 = lim 0 0 d 0 d d x d x ϕ F ¯ ( θ , x ) ϕ F ( θ , x ) × G F ( θ , x , x ) δ F ( x ) | x x | > = 0 d d x ϕ F ( θ , x ) δ F ( x ) lim 0 d x ϕ F ¯ ( θ , x ) × G F ( θ , x , x ) | x x | > .
D F · δ F = 0 d D ( θ , x ) · δ F ( x ) d x ,
D ( θ , x ) = B ¯ ( θ ) · A F ( θ , x ) + B ( θ ) F ( θ , x ) , A F ( θ , x ) = ϕ F ( θ , x ) ( i K cot θ 2 S ¯ 1 ( θ , x ) + p . v . 0 d d x ϕ F ¯ ( θ , x ) G F ( θ , x , x ) ) + ϕ F ¯ ( θ , x ) ( i K cot θ 2 S ( θ , x ) p . v . 0 d d x ϕ F ( θ , x ) G F ( θ , x , x ) ) .
α n = α n 1 , β n = β n 1 .
G F ( θ , x , x ) = exp [ i K ( x x ) ] · G F ( θ , x , x ) ,
G F ( θ , x , x ) = exp [ i K ( x x ) ] · G F ( θ , x , x ) ,
0 d F ( θ , x , x ) exp ( iKx ) exp ( i K x ) ϕ F ( θ , x ) d x = tan θ 2 i π exp [ i K ( x cot θ 2 F ( x ) ) ] .
0 d G F ( θ , x , x ) [ exp ( i K x ) ϕ F ¯ ( θ , x ) ] d x = + tan θ 2 i π exp ( i K cot θ 2 F ( x ) ) .
ϕ F ¯ ( θ , x ) exp ( iKx ) = tan θ 2 i π ϕ F ( θ , x ) .
p . v . 0 d d x ϕ F ¯ ( θ , x ) G F ( θ , x , x ) = tan θ 2 i π exp ( iKx ) p . v . 0 d d x ϕ F ( θ , x ) G F ( θ , x , x ) .
A F ( θ , x ) = ϕ F ( θ , x ) exp ( iKx ) [ 1 d exp ( i K cot θ 2 F ( x ) ) + tan θ i π p . v . d x ϕ F ( θ , x ) G F ( θ , x , x ) ] .
F ( x ) = n = 1 a n cos nKx + n = 2 b n sin nKx .
E 0 ( θ ) + E ( θ ) = 1 .
F ( x ) = n = 1 a n cos nKx = F ( x )