Abstract

The mode theory of transmission volume holograms is developed, taking into account the vector character of the electromagnetic field. In scalar recording media, eigenmodes are twice degenerated with reference to the state of polarization of the reconstructing field. Using this theory, the transformation laws of wave, with arbitrary spatially nonuniform field distribution and the state of polarization by a volume hologram are found. Two construction methods are detailed: (1) the reference beam has a wide angular spectrum, (2) the reference beam consists of two plane orthogonally polarized waves. In both cases the magnitudes of diffraction efficiency and polarization noise are calculated. Expressions obtained permit optimization of construction and reconstruction for both cases. In particular, we found the conditions on the reference beam for full reconstruction of the object field, including the whole structure of polarization.

© 1979 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. Yu. N. Denisiuk, Dokl. Akad. Nauk SSSR 144, 1275 (1962) [Sov. Phys. Dokl. 7, 543 (1962)].
  2. H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).
  3. V. C. Aristov, V. S. Shekhtman, Usp. Fiz. Nauk 104, 51 (1971) [Sov. Phys. Usp. 14, 263 (1971)].
    [Crossref]
  4. V. G. Sidorovich, Zh. Tekh. Fiz. 46, 1306 (1976) [Sov. Phys. Tech. Phys. 21, 742 (1976)].
  5. B. Ya. Zeldovich, V. I. Popovichev, V. V. Ragulsky, F. S. Faizullov, Pisima Zh. Eksp. Teor. Fiz. 15, 160 (1972) [JETP Lett. 15, 109 (1972)].
  6. V. G. Sidorovich, Zh. Tekh. Fiz. 46, 2168 (1976) [Sov. Phys. Tech. Phys. 21, 1270 (1976)].
  7. I. M. Beldyugin, M. G. Galushkin, E. M. Zemskov, V. I. Mandrosov, Kvantovaya Elktron. (Moscow) 3, 2467 (1976) [Sov. J. Quantum Electron. 6, 1349 (1976)].
  8. B. Ya. Zeldovich, V. V. Shkunov, Kvantovaya Elektron. (Moscow) 5, 36 (1978) [Sov. J. Quantum Electron. 8, 15 (1978)].
  9. B. Ya. Zeldovich, V. V. Shkunov, Kvantovaya Elektron. (Moscow) 4, 1090 (1977) [Sov. J. Quantum Electron. 7, 610 (1977)].
  10. B. Ya. Zeldovich, V. V. Shkunov, Kvantovaya Elektron. (Moscow) 4, 2353 (1977) [Sov. J. Quantum Electron. 7, 1345 (1977)].
  11. V. G. Sidorovich, V. V. Shkunov, Opt. Spektrosk. 44, 1001 (1978) [Opt. Spectrosc. 44, 586 (1978)].
  12. B. Ya. Zeldovich, V. V. Shkunov, Zh. Eksp. Teor. Fiz. 75, 428 (1978) [Sov. Phys. JETP 48, 214 (1978)]; V. N. Blaschuk, V. N. Krasheninnikov, N. A. Melnikov, N. F. Pilipetsky, V. V. Ragulsky, V. V. Shkunov, B. Ya. Zeldovich, Opt. Commun. 27, 137 (1978).
    [Crossref]
  13. A. W. Lohmann, Appl. Opt. 4, 1667 (1965).
    [Crossref]
  14. I. A. Deryugin, V. N. Kurashov, Usp. Fiz. Nauk 108, 733 (1972) [Sov. Phys. Usp. 15, 804 (1973)].
    [Crossref]
  15. Sh. D. Kakichashvili, Opt. Spektrosk. 42, 390 (1977) [Opt. Spectrosc. 42, 218 (1977)].
  16. V. N. Kurashov, D. V. Podanchuk, V. G. Chemetis, Kvantovaja Electron. (Moscow) 4, 2157 (1977) [Sov. J. Quantum. Electron. 7, 1234 (1977)].
  17. N. B. Baranova, B. Ya. Zeldovich, V. V. Shkunov, Kvantovaya Elektron. (Moscow) 5, 973 (1978) [Sov. J. Quantum Electron. 8, 559 (1978)].
  18. V. G. Sidorovich, Proceedings of 10th Soviet Holographic School, page 161, Leningrad, 1978.
  19. B. Ya. Zeldovich, V. V. Shkunov, T. V. Yakovleva, Preprint FIAN No. 26, Moscow1979.
  20. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959).
  21. W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958).

1978 (4)

B. Ya. Zeldovich, V. V. Shkunov, Kvantovaya Elektron. (Moscow) 5, 36 (1978) [Sov. J. Quantum Electron. 8, 15 (1978)].

V. G. Sidorovich, V. V. Shkunov, Opt. Spektrosk. 44, 1001 (1978) [Opt. Spectrosc. 44, 586 (1978)].

B. Ya. Zeldovich, V. V. Shkunov, Zh. Eksp. Teor. Fiz. 75, 428 (1978) [Sov. Phys. JETP 48, 214 (1978)]; V. N. Blaschuk, V. N. Krasheninnikov, N. A. Melnikov, N. F. Pilipetsky, V. V. Ragulsky, V. V. Shkunov, B. Ya. Zeldovich, Opt. Commun. 27, 137 (1978).
[Crossref]

N. B. Baranova, B. Ya. Zeldovich, V. V. Shkunov, Kvantovaya Elektron. (Moscow) 5, 973 (1978) [Sov. J. Quantum Electron. 8, 559 (1978)].

1977 (4)

Sh. D. Kakichashvili, Opt. Spektrosk. 42, 390 (1977) [Opt. Spectrosc. 42, 218 (1977)].

V. N. Kurashov, D. V. Podanchuk, V. G. Chemetis, Kvantovaja Electron. (Moscow) 4, 2157 (1977) [Sov. J. Quantum. Electron. 7, 1234 (1977)].

B. Ya. Zeldovich, V. V. Shkunov, Kvantovaya Elektron. (Moscow) 4, 1090 (1977) [Sov. J. Quantum Electron. 7, 610 (1977)].

B. Ya. Zeldovich, V. V. Shkunov, Kvantovaya Elektron. (Moscow) 4, 2353 (1977) [Sov. J. Quantum Electron. 7, 1345 (1977)].

1976 (3)

V. G. Sidorovich, Zh. Tekh. Fiz. 46, 2168 (1976) [Sov. Phys. Tech. Phys. 21, 1270 (1976)].

I. M. Beldyugin, M. G. Galushkin, E. M. Zemskov, V. I. Mandrosov, Kvantovaya Elktron. (Moscow) 3, 2467 (1976) [Sov. J. Quantum Electron. 6, 1349 (1976)].

V. G. Sidorovich, Zh. Tekh. Fiz. 46, 1306 (1976) [Sov. Phys. Tech. Phys. 21, 742 (1976)].

1972 (2)

B. Ya. Zeldovich, V. I. Popovichev, V. V. Ragulsky, F. S. Faizullov, Pisima Zh. Eksp. Teor. Fiz. 15, 160 (1972) [JETP Lett. 15, 109 (1972)].

I. A. Deryugin, V. N. Kurashov, Usp. Fiz. Nauk 108, 733 (1972) [Sov. Phys. Usp. 15, 804 (1973)].
[Crossref]

1971 (1)

V. C. Aristov, V. S. Shekhtman, Usp. Fiz. Nauk 104, 51 (1971) [Sov. Phys. Usp. 14, 263 (1971)].
[Crossref]

1969 (1)

H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).

1965 (1)

1962 (1)

Yu. N. Denisiuk, Dokl. Akad. Nauk SSSR 144, 1275 (1962) [Sov. Phys. Dokl. 7, 543 (1962)].

Aristov, V. C.

V. C. Aristov, V. S. Shekhtman, Usp. Fiz. Nauk 104, 51 (1971) [Sov. Phys. Usp. 14, 263 (1971)].
[Crossref]

Baranova, N. B.

N. B. Baranova, B. Ya. Zeldovich, V. V. Shkunov, Kvantovaya Elektron. (Moscow) 5, 973 (1978) [Sov. J. Quantum Electron. 8, 559 (1978)].

Beldyugin, I. M.

I. M. Beldyugin, M. G. Galushkin, E. M. Zemskov, V. I. Mandrosov, Kvantovaya Elktron. (Moscow) 3, 2467 (1976) [Sov. J. Quantum Electron. 6, 1349 (1976)].

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959).

Chemetis, V. G.

V. N. Kurashov, D. V. Podanchuk, V. G. Chemetis, Kvantovaja Electron. (Moscow) 4, 2157 (1977) [Sov. J. Quantum. Electron. 7, 1234 (1977)].

Davenport, W. B.

W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958).

Denisiuk, Yu. N.

Yu. N. Denisiuk, Dokl. Akad. Nauk SSSR 144, 1275 (1962) [Sov. Phys. Dokl. 7, 543 (1962)].

Deryugin, I. A.

I. A. Deryugin, V. N. Kurashov, Usp. Fiz. Nauk 108, 733 (1972) [Sov. Phys. Usp. 15, 804 (1973)].
[Crossref]

Faizullov, F. S.

B. Ya. Zeldovich, V. I. Popovichev, V. V. Ragulsky, F. S. Faizullov, Pisima Zh. Eksp. Teor. Fiz. 15, 160 (1972) [JETP Lett. 15, 109 (1972)].

Galushkin, M. G.

I. M. Beldyugin, M. G. Galushkin, E. M. Zemskov, V. I. Mandrosov, Kvantovaya Elktron. (Moscow) 3, 2467 (1976) [Sov. J. Quantum Electron. 6, 1349 (1976)].

Kakichashvili, Sh. D.

Sh. D. Kakichashvili, Opt. Spektrosk. 42, 390 (1977) [Opt. Spectrosc. 42, 218 (1977)].

Kogelnik, H.

H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).

Kurashov, V. N.

V. N. Kurashov, D. V. Podanchuk, V. G. Chemetis, Kvantovaja Electron. (Moscow) 4, 2157 (1977) [Sov. J. Quantum. Electron. 7, 1234 (1977)].

I. A. Deryugin, V. N. Kurashov, Usp. Fiz. Nauk 108, 733 (1972) [Sov. Phys. Usp. 15, 804 (1973)].
[Crossref]

Lohmann, A. W.

Mandrosov, V. I.

I. M. Beldyugin, M. G. Galushkin, E. M. Zemskov, V. I. Mandrosov, Kvantovaya Elktron. (Moscow) 3, 2467 (1976) [Sov. J. Quantum Electron. 6, 1349 (1976)].

Podanchuk, D. V.

V. N. Kurashov, D. V. Podanchuk, V. G. Chemetis, Kvantovaja Electron. (Moscow) 4, 2157 (1977) [Sov. J. Quantum. Electron. 7, 1234 (1977)].

Popovichev, V. I.

B. Ya. Zeldovich, V. I. Popovichev, V. V. Ragulsky, F. S. Faizullov, Pisima Zh. Eksp. Teor. Fiz. 15, 160 (1972) [JETP Lett. 15, 109 (1972)].

Ragulsky, V. V.

B. Ya. Zeldovich, V. I. Popovichev, V. V. Ragulsky, F. S. Faizullov, Pisima Zh. Eksp. Teor. Fiz. 15, 160 (1972) [JETP Lett. 15, 109 (1972)].

Root, W. L.

W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958).

Shekhtman, V. S.

V. C. Aristov, V. S. Shekhtman, Usp. Fiz. Nauk 104, 51 (1971) [Sov. Phys. Usp. 14, 263 (1971)].
[Crossref]

Shkunov, V. V.

B. Ya. Zeldovich, V. V. Shkunov, Kvantovaya Elektron. (Moscow) 5, 36 (1978) [Sov. J. Quantum Electron. 8, 15 (1978)].

B. Ya. Zeldovich, V. V. Shkunov, Zh. Eksp. Teor. Fiz. 75, 428 (1978) [Sov. Phys. JETP 48, 214 (1978)]; V. N. Blaschuk, V. N. Krasheninnikov, N. A. Melnikov, N. F. Pilipetsky, V. V. Ragulsky, V. V. Shkunov, B. Ya. Zeldovich, Opt. Commun. 27, 137 (1978).
[Crossref]

N. B. Baranova, B. Ya. Zeldovich, V. V. Shkunov, Kvantovaya Elektron. (Moscow) 5, 973 (1978) [Sov. J. Quantum Electron. 8, 559 (1978)].

V. G. Sidorovich, V. V. Shkunov, Opt. Spektrosk. 44, 1001 (1978) [Opt. Spectrosc. 44, 586 (1978)].

B. Ya. Zeldovich, V. V. Shkunov, Kvantovaya Elektron. (Moscow) 4, 1090 (1977) [Sov. J. Quantum Electron. 7, 610 (1977)].

B. Ya. Zeldovich, V. V. Shkunov, Kvantovaya Elektron. (Moscow) 4, 2353 (1977) [Sov. J. Quantum Electron. 7, 1345 (1977)].

B. Ya. Zeldovich, V. V. Shkunov, T. V. Yakovleva, Preprint FIAN No. 26, Moscow1979.

Sidorovich, V. G.

V. G. Sidorovich, V. V. Shkunov, Opt. Spektrosk. 44, 1001 (1978) [Opt. Spectrosc. 44, 586 (1978)].

V. G. Sidorovich, Zh. Tekh. Fiz. 46, 2168 (1976) [Sov. Phys. Tech. Phys. 21, 1270 (1976)].

V. G. Sidorovich, Zh. Tekh. Fiz. 46, 1306 (1976) [Sov. Phys. Tech. Phys. 21, 742 (1976)].

V. G. Sidorovich, Proceedings of 10th Soviet Holographic School, page 161, Leningrad, 1978.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959).

Yakovleva, T. V.

B. Ya. Zeldovich, V. V. Shkunov, T. V. Yakovleva, Preprint FIAN No. 26, Moscow1979.

Zeldovich, B. Ya.

N. B. Baranova, B. Ya. Zeldovich, V. V. Shkunov, Kvantovaya Elektron. (Moscow) 5, 973 (1978) [Sov. J. Quantum Electron. 8, 559 (1978)].

B. Ya. Zeldovich, V. V. Shkunov, Zh. Eksp. Teor. Fiz. 75, 428 (1978) [Sov. Phys. JETP 48, 214 (1978)]; V. N. Blaschuk, V. N. Krasheninnikov, N. A. Melnikov, N. F. Pilipetsky, V. V. Ragulsky, V. V. Shkunov, B. Ya. Zeldovich, Opt. Commun. 27, 137 (1978).
[Crossref]

B. Ya. Zeldovich, V. V. Shkunov, Kvantovaya Elektron. (Moscow) 5, 36 (1978) [Sov. J. Quantum Electron. 8, 15 (1978)].

B. Ya. Zeldovich, V. V. Shkunov, Kvantovaya Elektron. (Moscow) 4, 1090 (1977) [Sov. J. Quantum Electron. 7, 610 (1977)].

B. Ya. Zeldovich, V. V. Shkunov, Kvantovaya Elektron. (Moscow) 4, 2353 (1977) [Sov. J. Quantum Electron. 7, 1345 (1977)].

B. Ya. Zeldovich, V. I. Popovichev, V. V. Ragulsky, F. S. Faizullov, Pisima Zh. Eksp. Teor. Fiz. 15, 160 (1972) [JETP Lett. 15, 109 (1972)].

B. Ya. Zeldovich, V. V. Shkunov, T. V. Yakovleva, Preprint FIAN No. 26, Moscow1979.

Zemskov, E. M.

I. M. Beldyugin, M. G. Galushkin, E. M. Zemskov, V. I. Mandrosov, Kvantovaya Elktron. (Moscow) 3, 2467 (1976) [Sov. J. Quantum Electron. 6, 1349 (1976)].

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969).

Dokl. Akad. Nauk SSSR (1)

Yu. N. Denisiuk, Dokl. Akad. Nauk SSSR 144, 1275 (1962) [Sov. Phys. Dokl. 7, 543 (1962)].

Kvantovaja Electron. (Moscow) (1)

V. N. Kurashov, D. V. Podanchuk, V. G. Chemetis, Kvantovaja Electron. (Moscow) 4, 2157 (1977) [Sov. J. Quantum. Electron. 7, 1234 (1977)].

Kvantovaya Elektron. (Moscow) (4)

N. B. Baranova, B. Ya. Zeldovich, V. V. Shkunov, Kvantovaya Elektron. (Moscow) 5, 973 (1978) [Sov. J. Quantum Electron. 8, 559 (1978)].

B. Ya. Zeldovich, V. V. Shkunov, Kvantovaya Elektron. (Moscow) 5, 36 (1978) [Sov. J. Quantum Electron. 8, 15 (1978)].

B. Ya. Zeldovich, V. V. Shkunov, Kvantovaya Elektron. (Moscow) 4, 1090 (1977) [Sov. J. Quantum Electron. 7, 610 (1977)].

B. Ya. Zeldovich, V. V. Shkunov, Kvantovaya Elektron. (Moscow) 4, 2353 (1977) [Sov. J. Quantum Electron. 7, 1345 (1977)].

Kvantovaya Elktron. (Moscow) (1)

I. M. Beldyugin, M. G. Galushkin, E. M. Zemskov, V. I. Mandrosov, Kvantovaya Elktron. (Moscow) 3, 2467 (1976) [Sov. J. Quantum Electron. 6, 1349 (1976)].

Opt. Spektrosk. (2)

V. G. Sidorovich, V. V. Shkunov, Opt. Spektrosk. 44, 1001 (1978) [Opt. Spectrosc. 44, 586 (1978)].

Sh. D. Kakichashvili, Opt. Spektrosk. 42, 390 (1977) [Opt. Spectrosc. 42, 218 (1977)].

Pisima Zh. Eksp. Teor. Fiz. (1)

B. Ya. Zeldovich, V. I. Popovichev, V. V. Ragulsky, F. S. Faizullov, Pisima Zh. Eksp. Teor. Fiz. 15, 160 (1972) [JETP Lett. 15, 109 (1972)].

Usp. Fiz. Nauk (2)

V. C. Aristov, V. S. Shekhtman, Usp. Fiz. Nauk 104, 51 (1971) [Sov. Phys. Usp. 14, 263 (1971)].
[Crossref]

I. A. Deryugin, V. N. Kurashov, Usp. Fiz. Nauk 108, 733 (1972) [Sov. Phys. Usp. 15, 804 (1973)].
[Crossref]

Zh. Eksp. Teor. Fiz. (1)

B. Ya. Zeldovich, V. V. Shkunov, Zh. Eksp. Teor. Fiz. 75, 428 (1978) [Sov. Phys. JETP 48, 214 (1978)]; V. N. Blaschuk, V. N. Krasheninnikov, N. A. Melnikov, N. F. Pilipetsky, V. V. Ragulsky, V. V. Shkunov, B. Ya. Zeldovich, Opt. Commun. 27, 137 (1978).
[Crossref]

Zh. Tekh. Fiz. (2)

V. G. Sidorovich, Zh. Tekh. Fiz. 46, 1306 (1976) [Sov. Phys. Tech. Phys. 21, 742 (1976)].

V. G. Sidorovich, Zh. Tekh. Fiz. 46, 2168 (1976) [Sov. Phys. Tech. Phys. 21, 1270 (1976)].

Other (4)

V. G. Sidorovich, Proceedings of 10th Soviet Holographic School, page 161, Leningrad, 1978.

B. Ya. Zeldovich, V. V. Shkunov, T. V. Yakovleva, Preprint FIAN No. 26, Moscow1979.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959).

W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, New York, 1958).

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 (6)

Fig. 1
Fig. 1

Scheme of hologram construction: (a) by a reference wave B(q) with a wide angular spectrum, and (b) by a reference wave in the form of two orthogonally polarized plane waves B1(q1)e1 + B2(q2)e2. In both cases A( q ^) is the object field.

Fig. 2
Fig. 2

Illustration of mode theory for the recording and reconstruction of object field A using reference wave B with a wide angular spectrum. Details are in the text.

Fig. 3
Fig. 3

Propagation constant μi for varying hologram modes (in units x J) vs the polarization degree |ξ| of the constructed field (the reference wave has a wide angular spectrum).

Fig. 4
Fig. 4

Illustration of mode theory for the recording and reconstruction of object field A using plane reference wave B. Details are in the text.

Fig. 5
Fig. 5

(a) Propagation constant μ 1 ( ± ) (in x J units) for modes M 1 ( ± ) vs the parameter Λ = K11[|B1(q1)|2 + K11]−1 that characterizes a part of the object field energy within a constructed field of e1 polarization. (b) Relative weights γ 1 ( ± ) of the object field for modes M 1 ( ± ) vs parameter Λ [see (a)].

Fig. 6
Fig. 6

Structure of the constructed field C(q) consisting of two plane orthogonally polarized waves e1B1 and e2B2 and of an object wave e1A1 + e2A2. The structure of M 1 ( + ) , M 1 ( - ) , M 2 ( + ), and M 2 ( - ) modes is also shown.

Equations (62)

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

E Re ( r , z , t ) = ½ [ E ( r , z ) exp ( - i ω t + i k z ) + C . C . ] ,
z E 0 - i 2 k Δ E 0 = 0 ,
i m ( r , z ) = 0 - 2 i k ( c ω ) 2 G i k l m E k 0 E l 0 * .
G i k l m = - x δ k l δ im ,
z E i - i 2 k Δ E i = G i k l m E k 0 E e 0 * E m .
E 0 ( r , z ) = q C ( q ) exp ( i qr - i q 2 2 k z ) ,
E ( r , z ) = q S ( q , z ) exp ( i qr - i q 2 2 k z ) .
S ( q , z ) = S ( q ) exp ( μ z ) .
μ S i ( q ) = G i k l m { [ - C k ( q ) C i * ( q ) + p C k ( p ) C l ( p ) ] S m ( q ) + C k ( q ) p C l * ( p ) S m ( p ) + q 1 q q 2 q 3 q C k ( q 1 ) C l * ( q 2 ) S m ( q 3 ) exp ( - i q 1 2 - q 2 2 - q 3 2 - q 2 2 k z ) δ ( 2 ) ( q 1 - q 2 + q 3 - q ) } .
k θ 2 x · E 0 2 ,
L ( k θ 2 ) - 1 .
δ ( r , z ) ~ x C ( q 1 ) C * ( q 2 ) exp [ i ( q 1 - q 2 ) r - i q 1 2 - q 2 2 2 k z ] .
μ S i ( q ) = G i k l m { [ I k l - C k ( q ) C l * ( q ) ] S m ( q ) + D l m C k ( q ) } ,
I k l = p C k ( p ) C i * ( p ) = 1 S E k 0 ( r ) E l 0 * ( r ) d 2 r ,
D l m = p C l * ( p ) S m ( p ) = 1 S E l 0 * ( r ) E m ( r ) d 2 r
I ^ = 1 / 2 J [ ( 1 0 0 1 ) + ξ 1 ( 0 1 1 0 ) + ξ 2 ( 0 - i i 0 ) + ξ 3 ( 1 0 0 - 1 ) ] ,
x { [ J - C ( q ) 2 ] S i ( q ) + C l ( q ) D l i } = μ S i ( q ) .
S i ( q ) = C l ( q ) x D l i μ + x [ J - C ( q ) 2 ] ,
D p i = D l i q x C p * ( q ) C l ( q ) μ + x [ J - C ( q ) 2 ] .
C ( q ) = A ( q ) + B ( q ) .
1 S A i * ( r , z ) B k ( r , z ) d 2 r q A i * ( q ) B k ( q ) = 0
S i ( q ) = C l ( q ) D l i x μ + x J ,
D p i = I p l * D l i x μ + x J .
D = 1 S E 0 * ( r ) E ( r ) d 2 r = 0 ;             μ = - x J .
E ( r ) = E 0 ( r ) ;             μ = - 2 x J .
μ = - x [ E ( r ) 2 d 2 r ] - 1 · E ( r ) 2 E 0 ( r ) 2 d 2 r .
I ^ = ( I 11 0 0 I 22 ) = ½ J ( 1 + ξ 0 0 1 - ξ ) ,
e 1 = e x cos Δ + e y · exp ( i δ ) sin Δ ; e 2 = e x sin Δ - e y · exp ( i δ ) cos Δ ,
E 0 ( r , z ) = e 1 E 1 0 ( r , z ) + e 2 E 0 2 ( r , z ) = e 1 [ A 1 ( r , z ) + B 1 ( r , z ) ] + e 2 [ A 2 ( r , z ) + B 2 ( r , z ) ] ,
M 1 , 2 ( r , z ) = ψ 1 , 2 E 1 0 ( r , z ) exp ( μ 1 z ) μ 1 = μ 2 = - 1 / 2 x J ( 3 + ξ ) } ,
M 3 , 4 ( r , z ) = ψ 3 , 4 E 2 0 ( r , z ) exp ( μ 3 z ) μ 3 = μ 4 = - ½ x J ( 3 - ξ ) = - x ( I 11 + 2 I 22 ) .
E ( r , z = 0 ) = α 1 φ 1 B 1 ( r , z = 0 ) + α 2 φ 2 B 2 ( r , z ) .
E ( r , z ) = E 1 0 ( β 11 φ 1 + β 12 φ 2 ) exp ( μ 1 z ) + E 2 0 ( β 21 φ 1 + β 22 φ 2 ) exp ( μ 3 z ) + [ ( B 1 α 1 - E 1 0 β 11 - E 2 0 β 21 ) φ 1 + ( B 2 α 2 - E 1 0 β 12 - E 2 0 β 22 ) φ 2 ] exp ( μ 0 z ) ,
β i k = I i i - 1 α k 1 S E i 0 * B k d 2 r = α k B i * B k d 2 r / E i 0 2 d 2 r .
E ( r , z ) = exp ( - x J z ) { A 1 ( r , z ) f 1 ( z ) + A 2 ( r , z ) f 2 ( z ) + B 1 ( r , z ) [ ψ 1 + f 1 ( z ) ] + B 2 ( r , z ) [ ψ 2 + f 2 ( z ) ] } ,
f i ( z ) = - 2 sinh ( x 2 I i i z ) exp ( - x 2 I i i z ) ( β i 1 φ 1 + β i 2 φ 2 ) .
K i k = 1 s A i * ( r ) A k ( r ) d 2 r .
E 0 ( r , z ) = e 1 [ B 1 ( q 1 ) exp ( i q 1 r - i q 1 2 2 k z ) + A 1 ( r , z ) ] + e 2 [ B 2 ( q 2 ) exp ( i q 2 r - i q 2 2 2 k z ) + A 2 ( r , z ) ] ,
J = K 11 + K 22 + B 1 ( q 1 ) 2 + B 2 ( q 2 ) 2 .
D i k = D i k { x B i ( q i ) 2 μ + x [ J - B i ( q i ) 2 ] + x K i i μ - x J } .
μ i ( ± ) = - x ( J + ½ { K i i ± [ K i i 2 + 4 K i i B i ( q i ) 2 ] 1 / 2 } ) .
M 1 ( ± ) ( r , z ) = [ B 1 ( q 1 ) exp ( i q 1 r - i q 1 2 2 k z ) + γ 1 ( ± ) A 1 ( r , z ) ] exp [ μ 1 ( ± ) z ] ,
γ 1 ( ± ) = { K 11 ± [ K 11 2 + 4 K 11 B 1 ( q 1 ) 2 ] 1 / 2 } / 2 K 11 = μ 1 ( ± ) - x [ J - B 1 ( q 1 ) 2 ] μ 1 ( ± ) + x J .
E ( r , z = 0 ) = α 1 φ 1 B 1 ( q 1 ) exp ( i q 1 r ) + α 2 φ 2 B 2 ( q 2 ) exp ( i q 2 r ) .
E ( r , z ) = exp ( - x J z ) [ A 1 ( r , z ) f 1 ( z ) + A 2 ( r , z ) f 2 ( z ) + B 1 ( q 1 ) g 1 ( z ) exp ( i q 1 r - i q 1 2 2 k z ) + B 2 ( q 2 ) g 2 ( z ) exp ( i q 2 r - i q 2 2 2 k z ) ] ,
f i ( z ) = 2 α i φ i B i ( q i ) 2 ν i sinh ( x ν i z / 2 ) exp ( - x K i i z / 2 ) ,
g i ( z ) = α i φ i [ cosh ( x ν i z / 2 ) + K i i ν i sinh ( x ν i z / 2 ) ] exp ( - x K i i z / 2 ) ,
ν i = [ K i i 2 + 4 K i i B i ( q i ) 2 ] 1 / 2 .
A rec ( r z ) = [ A 1 ( r , z ) f 1 ( z ) + A 2 ( r , z ) f 2 ( z ) ] exp ( - x J z ) ,
A ex ( r , z ) = F ( z ) [ A 1 ( r , z ) e 1 + A 2 ( r , z ) e 2 ] .
f i ( z ) = - 2 α i e i B i 2 d 2 r ( A i 2 + B i 2 ) d 2 r · sinh ( x 2 I i i z ) exp ( - x 2 I i i z ) .
f i ( z ) = α i x B i ( q i ) 2 z · e i .
B 1 2 d 2 r B 2 2 d 2 r B 2 * B 1 d 2 r .
U A = exp [ - ( x + x * ) J z ] i , k = 1 2 ( f i * · f k ) A i * A k d 2 r .
V A = [ E * ( r , z ) · A ( r , z ) ] d 2 r 2 A ( r , z ) 2 d 2 r .
P = U A - V A V A .
f 1 ( z ) = f 1 ( z ) e 1 ,             f 2 ( z ) = f 2 ( z ) e 2
P = f 1 - f 2 2 · A 1 2 d 2 r · A 2 2 d 2 r f 1 · A 1 2 d 2 r + f 2 · A 2 2 d 2 r 2 ,
| A err A cor | 1 , 2 = ( ξ A × ξ B ) · A 2 d 2 r ξ A · A 2 d 2 r + ξ B · B 2 d 2 r · ( 1 ± ξ B · ξ ξ ) - 1 ξ = ξ A · A 2 d 2 r + ξ B · B 2 d 2 r ( A 2 + B 2 ) d 2 r .
η u = U A α 1 B 1 + α 2 B 2 2 d 2 r ;             η v = V A α 1 B 1 + α 2 B 2 2 d 2 r .
η = 4 I A I B ( I A + I B ) 2 | sinh ( x 2 J z ) | 2 exp ( - 3 Re x J z ) ,
η = 4 I B I A + 4 I B | sinh [ x 2 ( I A 2 + 4 I A I B ) 1 / 2 z ] | 2 × exp [ - ( 3 I A + 2 I B ) Re x z ] ,

Metrics