Abstract

The microscopic physical origins of photoinduced anisotropy in optical fibers are discussed, and the perturbations to the dielectric tensor resulting from interference of two arbitrarily polarized waves are derived. Both single- and two-photon absorption processes are treated, making the formulation valid for filter formation both by UV side writing and by axial exposure to copropagating or counterpropagating blue-green guided modes. In addition to elucidating the dielectric tensor for various types of known polarization convertor, the treatment predicts the formation of a number of unusual polarization convertors and filters. These predictions underline the universal result that unusual and complicated periodic forms of dielectric tensor—unlike anything encountered in crystals—can arise in amorphous media such as glass and polymers on exposure to light.

© 1994 Optical Society of America

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References

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  1. K. O. Hill, Y. Fuji, D. C. Johnson, and B. S. Kawasaki, “Photosensitivity in optical fiber waveguides: application to reflection filter fabrication,” Appl. Phys. Lett. 32, 647 (1978).
    [Crossref]
  2. G. Meltz, W. W. Morey, and W. H. Glenn, “Formation of Bragg gratings in optical fibers by a transverse holographic method,” Opt. Lett. 14, 823 (1989).
    [Crossref] [PubMed]
  3. D. P. Hand, P. St, and J. Russell, “Single mode fiber grating written into Sagnac loop using photosensitive fiber: Transmission filters,” presented at the Seventh International Conference on Integrated Optics and Optical Communication, IOOC ’89, Kobe, Japan, 1989, paper 21C3-4.
  4. J.-L. Archambault, L. Reekie, P. St, and J. Russell, “100% reflectivity Bragg reflectors produced in optical fibres by single excimer laser pulses,” Electron. Lett. 26, 453 (1993).
    [Crossref]
  5. M. Parent, J. Bures, S. Lacroix, and J. Lapierre, “Propriétés de polarization des réflecteurs de Bragg induits par photosensibilité dans les fibres optiques monomodes,” Appl. Opt. 24, 354 (1985).
    [Crossref]
  6. F. Ouellette, D. Gagnon, and M. Poirer, “Permanent photo-induced birefringence in a Ge-doped fiber,” Appl. Phys. Lett. 58, 1813 (1991).
    [Crossref]
  7. S. Bardal, A. Kamal, P. St, and J. Russell, “Photo-induced birefringence in optical fibers: a comparative study of Lo-Bi and Hi-Bi fibers,” Opt. Lett. 17, 411 (1992).
    [Crossref] [PubMed]
  8. P. St, J. Russell, and D. P. Hand, “Rocking filter formation in photosensitive high birefringence optical fibres,” Electron. Lett. 26, 1846 (1990).
    [Crossref]
  9. S. E. Kanellopoulos, L. C. G. Valente, V. A. Handerek, and A. J. Rogers, “Photorefractive polarization couplers in elliptical core fibers,” IEEE Photon. Technol. Lett. 3, 806 (1991).
    [Crossref]
  10. K. O. Hill, F. Bilodeau, B. Malo, and D. C. Johnson, “Birefringent photosensitivity in monomode optical fiber: application to external writing of rocking filters,” Electron. Lett. 27, 1548 (1991).
    [Crossref]
  11. A. Kamal, J.-L. Archambault, P. St, J. Russell, V. A. Handerek, and A. J. Rogers, “Holographically written reflective polarization filter in single-mode optical fibers,” Opt. Lett. 17, 1189 (1992).
    [Crossref] [PubMed]
  12. H. Fritzche, Department of Physics, University of Chicago, Chicago, Illinois (personal communication, 1992).
  13. R. M. Atkins, “Measurement of the ultraviolet absorption spectrum of optical fibers,” Opt. Lett. 17, 469 (1992).
    [Crossref] [PubMed]
  14. A. Kamal, R. W. Terhune, and D. A. Weinberger, “Dynamics of self-organized χ(2)gratings in optical fibers,” in International Workshop on Photoinduced Self-Organization Effects in Optical Fiber, F. Ouellette, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1516, 137 (1991).
    [Crossref]
  15. R. M. Atkins, V. Mizrahi, and T. Erdogen, “248 nm induced vacuum UV spectral changes in optical fibre preform cores: support for a colour centre model of photosensitivity,” Elec. Lett. 29, 385 (1993).
    [Crossref]
  16. J. Lauzon, D. Gagnon, S. LaRochelle, A. Blouin, and F. Ouellette, “Dynamic polarization coupling in elliptical core photosensitive optical fiber,” Opt. Lett. 17, 1664 (1992).
    [Crossref] [PubMed]
  17. While we were responding to reviewers’ suggestions on this paper, another set of measurements [J. Albert, B. Malo, D. C. Johnson, F. Bilodeau, K. O. Hill, J. L. Brebner, and G. Kajrys, “Dichroism in the absorption spectrum of photobleached ion-implanted silica,” Opt. Lett. 18, 1126 (1993)] appeared in the literature, corroborating the preferential depletion picture. In these measurements a dichroism in the absorption spectrum of ion-implanted silica was observed after exposure to a linearly polarized UV beam. The dichroism was such that it suggested a depletion of the UV-absorbing defect centers along the polarization state of the pump UV beam.
    [Crossref] [PubMed]
  18. P. St, J. Russell, and R. Ulrich, “Grating-fiber coupler as a high-resolution spectrometer,” Opt. Lett. 10, 291 (1986).
  19. S. An and J. E. Sipe, “Polarization aspects of two-photon photosensitivity in birefringent optical fibers,” Opt. Lett. 17, 490 (1992).
    [Crossref] [PubMed]
  20. V. Evtuhov and A. E. Siegman, “A ‘twisted mode’ technique for obtaining axially uniform energy density in a laser cavity,” Appl. Opt. 4, 142 (1965).
    [Crossref]
  21. F. Glocking, The Chemistry of Germanium (Academic, London, 1969), Chap. 1, p. 13.

1993 (3)

1992 (5)

1991 (3)

S. E. Kanellopoulos, L. C. G. Valente, V. A. Handerek, and A. J. Rogers, “Photorefractive polarization couplers in elliptical core fibers,” IEEE Photon. Technol. Lett. 3, 806 (1991).
[Crossref]

K. O. Hill, F. Bilodeau, B. Malo, and D. C. Johnson, “Birefringent photosensitivity in monomode optical fiber: application to external writing of rocking filters,” Electron. Lett. 27, 1548 (1991).
[Crossref]

F. Ouellette, D. Gagnon, and M. Poirer, “Permanent photo-induced birefringence in a Ge-doped fiber,” Appl. Phys. Lett. 58, 1813 (1991).
[Crossref]

1990 (1)

P. St, J. Russell, and D. P. Hand, “Rocking filter formation in photosensitive high birefringence optical fibres,” Electron. Lett. 26, 1846 (1990).
[Crossref]

1989 (1)

1986 (1)

1985 (1)

1978 (1)

K. O. Hill, Y. Fuji, D. C. Johnson, and B. S. Kawasaki, “Photosensitivity in optical fiber waveguides: application to reflection filter fabrication,” Appl. Phys. Lett. 32, 647 (1978).
[Crossref]

1965 (1)

Albert, J.

An, S.

Archambault, J.-L.

J.-L. Archambault, L. Reekie, P. St, and J. Russell, “100% reflectivity Bragg reflectors produced in optical fibres by single excimer laser pulses,” Electron. Lett. 26, 453 (1993).
[Crossref]

A. Kamal, J.-L. Archambault, P. St, J. Russell, V. A. Handerek, and A. J. Rogers, “Holographically written reflective polarization filter in single-mode optical fibers,” Opt. Lett. 17, 1189 (1992).
[Crossref] [PubMed]

Atkins, R. M.

R. M. Atkins, V. Mizrahi, and T. Erdogen, “248 nm induced vacuum UV spectral changes in optical fibre preform cores: support for a colour centre model of photosensitivity,” Elec. Lett. 29, 385 (1993).
[Crossref]

R. M. Atkins, “Measurement of the ultraviolet absorption spectrum of optical fibers,” Opt. Lett. 17, 469 (1992).
[Crossref] [PubMed]

Bardal, S.

Bilodeau, F.

Blouin, A.

Brebner, J. L.

Bures, J.

Erdogen, T.

R. M. Atkins, V. Mizrahi, and T. Erdogen, “248 nm induced vacuum UV spectral changes in optical fibre preform cores: support for a colour centre model of photosensitivity,” Elec. Lett. 29, 385 (1993).
[Crossref]

Evtuhov, V.

Fritzche, H.

H. Fritzche, Department of Physics, University of Chicago, Chicago, Illinois (personal communication, 1992).

Fuji, Y.

K. O. Hill, Y. Fuji, D. C. Johnson, and B. S. Kawasaki, “Photosensitivity in optical fiber waveguides: application to reflection filter fabrication,” Appl. Phys. Lett. 32, 647 (1978).
[Crossref]

Gagnon, D.

J. Lauzon, D. Gagnon, S. LaRochelle, A. Blouin, and F. Ouellette, “Dynamic polarization coupling in elliptical core photosensitive optical fiber,” Opt. Lett. 17, 1664 (1992).
[Crossref] [PubMed]

F. Ouellette, D. Gagnon, and M. Poirer, “Permanent photo-induced birefringence in a Ge-doped fiber,” Appl. Phys. Lett. 58, 1813 (1991).
[Crossref]

Glenn, W. H.

Glocking, F.

F. Glocking, The Chemistry of Germanium (Academic, London, 1969), Chap. 1, p. 13.

Hand, D. P.

P. St, J. Russell, and D. P. Hand, “Rocking filter formation in photosensitive high birefringence optical fibres,” Electron. Lett. 26, 1846 (1990).
[Crossref]

D. P. Hand, P. St, and J. Russell, “Single mode fiber grating written into Sagnac loop using photosensitive fiber: Transmission filters,” presented at the Seventh International Conference on Integrated Optics and Optical Communication, IOOC ’89, Kobe, Japan, 1989, paper 21C3-4.

Handerek, V. A.

A. Kamal, J.-L. Archambault, P. St, J. Russell, V. A. Handerek, and A. J. Rogers, “Holographically written reflective polarization filter in single-mode optical fibers,” Opt. Lett. 17, 1189 (1992).
[Crossref] [PubMed]

S. E. Kanellopoulos, L. C. G. Valente, V. A. Handerek, and A. J. Rogers, “Photorefractive polarization couplers in elliptical core fibers,” IEEE Photon. Technol. Lett. 3, 806 (1991).
[Crossref]

Hill, K. O.

Johnson, D. C.

Kajrys, G.

Kamal, A.

Kanellopoulos, S. E.

S. E. Kanellopoulos, L. C. G. Valente, V. A. Handerek, and A. J. Rogers, “Photorefractive polarization couplers in elliptical core fibers,” IEEE Photon. Technol. Lett. 3, 806 (1991).
[Crossref]

Kawasaki, B. S.

K. O. Hill, Y. Fuji, D. C. Johnson, and B. S. Kawasaki, “Photosensitivity in optical fiber waveguides: application to reflection filter fabrication,” Appl. Phys. Lett. 32, 647 (1978).
[Crossref]

Lacroix, S.

Lapierre, J.

LaRochelle, S.

Lauzon, J.

Malo, B.

Meltz, G.

Mizrahi, V.

R. M. Atkins, V. Mizrahi, and T. Erdogen, “248 nm induced vacuum UV spectral changes in optical fibre preform cores: support for a colour centre model of photosensitivity,” Elec. Lett. 29, 385 (1993).
[Crossref]

Morey, W. W.

Ouellette, F.

J. Lauzon, D. Gagnon, S. LaRochelle, A. Blouin, and F. Ouellette, “Dynamic polarization coupling in elliptical core photosensitive optical fiber,” Opt. Lett. 17, 1664 (1992).
[Crossref] [PubMed]

F. Ouellette, D. Gagnon, and M. Poirer, “Permanent photo-induced birefringence in a Ge-doped fiber,” Appl. Phys. Lett. 58, 1813 (1991).
[Crossref]

Parent, M.

Poirer, M.

F. Ouellette, D. Gagnon, and M. Poirer, “Permanent photo-induced birefringence in a Ge-doped fiber,” Appl. Phys. Lett. 58, 1813 (1991).
[Crossref]

Reekie, L.

J.-L. Archambault, L. Reekie, P. St, and J. Russell, “100% reflectivity Bragg reflectors produced in optical fibres by single excimer laser pulses,” Electron. Lett. 26, 453 (1993).
[Crossref]

Rogers, A. J.

A. Kamal, J.-L. Archambault, P. St, J. Russell, V. A. Handerek, and A. J. Rogers, “Holographically written reflective polarization filter in single-mode optical fibers,” Opt. Lett. 17, 1189 (1992).
[Crossref] [PubMed]

S. E. Kanellopoulos, L. C. G. Valente, V. A. Handerek, and A. J. Rogers, “Photorefractive polarization couplers in elliptical core fibers,” IEEE Photon. Technol. Lett. 3, 806 (1991).
[Crossref]

Russell, J.

J.-L. Archambault, L. Reekie, P. St, and J. Russell, “100% reflectivity Bragg reflectors produced in optical fibres by single excimer laser pulses,” Electron. Lett. 26, 453 (1993).
[Crossref]

A. Kamal, J.-L. Archambault, P. St, J. Russell, V. A. Handerek, and A. J. Rogers, “Holographically written reflective polarization filter in single-mode optical fibers,” Opt. Lett. 17, 1189 (1992).
[Crossref] [PubMed]

S. Bardal, A. Kamal, P. St, and J. Russell, “Photo-induced birefringence in optical fibers: a comparative study of Lo-Bi and Hi-Bi fibers,” Opt. Lett. 17, 411 (1992).
[Crossref] [PubMed]

P. St, J. Russell, and D. P. Hand, “Rocking filter formation in photosensitive high birefringence optical fibres,” Electron. Lett. 26, 1846 (1990).
[Crossref]

P. St, J. Russell, and R. Ulrich, “Grating-fiber coupler as a high-resolution spectrometer,” Opt. Lett. 10, 291 (1986).

D. P. Hand, P. St, and J. Russell, “Single mode fiber grating written into Sagnac loop using photosensitive fiber: Transmission filters,” presented at the Seventh International Conference on Integrated Optics and Optical Communication, IOOC ’89, Kobe, Japan, 1989, paper 21C3-4.

Siegman, A. E.

Sipe, J. E.

St, P.

J.-L. Archambault, L. Reekie, P. St, and J. Russell, “100% reflectivity Bragg reflectors produced in optical fibres by single excimer laser pulses,” Electron. Lett. 26, 453 (1993).
[Crossref]

A. Kamal, J.-L. Archambault, P. St, J. Russell, V. A. Handerek, and A. J. Rogers, “Holographically written reflective polarization filter in single-mode optical fibers,” Opt. Lett. 17, 1189 (1992).
[Crossref] [PubMed]

S. Bardal, A. Kamal, P. St, and J. Russell, “Photo-induced birefringence in optical fibers: a comparative study of Lo-Bi and Hi-Bi fibers,” Opt. Lett. 17, 411 (1992).
[Crossref] [PubMed]

P. St, J. Russell, and D. P. Hand, “Rocking filter formation in photosensitive high birefringence optical fibres,” Electron. Lett. 26, 1846 (1990).
[Crossref]

P. St, J. Russell, and R. Ulrich, “Grating-fiber coupler as a high-resolution spectrometer,” Opt. Lett. 10, 291 (1986).

D. P. Hand, P. St, and J. Russell, “Single mode fiber grating written into Sagnac loop using photosensitive fiber: Transmission filters,” presented at the Seventh International Conference on Integrated Optics and Optical Communication, IOOC ’89, Kobe, Japan, 1989, paper 21C3-4.

Terhune, R. W.

A. Kamal, R. W. Terhune, and D. A. Weinberger, “Dynamics of self-organized χ(2)gratings in optical fibers,” in International Workshop on Photoinduced Self-Organization Effects in Optical Fiber, F. Ouellette, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1516, 137 (1991).
[Crossref]

Ulrich, R.

Valente, L. C. G.

S. E. Kanellopoulos, L. C. G. Valente, V. A. Handerek, and A. J. Rogers, “Photorefractive polarization couplers in elliptical core fibers,” IEEE Photon. Technol. Lett. 3, 806 (1991).
[Crossref]

Weinberger, D. A.

A. Kamal, R. W. Terhune, and D. A. Weinberger, “Dynamics of self-organized χ(2)gratings in optical fibers,” in International Workshop on Photoinduced Self-Organization Effects in Optical Fiber, F. Ouellette, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1516, 137 (1991).
[Crossref]

Appl. Opt. (2)

Appl. Phys. Lett. (2)

F. Ouellette, D. Gagnon, and M. Poirer, “Permanent photo-induced birefringence in a Ge-doped fiber,” Appl. Phys. Lett. 58, 1813 (1991).
[Crossref]

K. O. Hill, Y. Fuji, D. C. Johnson, and B. S. Kawasaki, “Photosensitivity in optical fiber waveguides: application to reflection filter fabrication,” Appl. Phys. Lett. 32, 647 (1978).
[Crossref]

Elec. Lett. (1)

R. M. Atkins, V. Mizrahi, and T. Erdogen, “248 nm induced vacuum UV spectral changes in optical fibre preform cores: support for a colour centre model of photosensitivity,” Elec. Lett. 29, 385 (1993).
[Crossref]

Electron. Lett. (3)

P. St, J. Russell, and D. P. Hand, “Rocking filter formation in photosensitive high birefringence optical fibres,” Electron. Lett. 26, 1846 (1990).
[Crossref]

K. O. Hill, F. Bilodeau, B. Malo, and D. C. Johnson, “Birefringent photosensitivity in monomode optical fiber: application to external writing of rocking filters,” Electron. Lett. 27, 1548 (1991).
[Crossref]

J.-L. Archambault, L. Reekie, P. St, and J. Russell, “100% reflectivity Bragg reflectors produced in optical fibres by single excimer laser pulses,” Electron. Lett. 26, 453 (1993).
[Crossref]

IEEE Photon. Technol. Lett. (1)

S. E. Kanellopoulos, L. C. G. Valente, V. A. Handerek, and A. J. Rogers, “Photorefractive polarization couplers in elliptical core fibers,” IEEE Photon. Technol. Lett. 3, 806 (1991).
[Crossref]

Opt. Lett. (8)

A. Kamal, J.-L. Archambault, P. St, J. Russell, V. A. Handerek, and A. J. Rogers, “Holographically written reflective polarization filter in single-mode optical fibers,” Opt. Lett. 17, 1189 (1992).
[Crossref] [PubMed]

G. Meltz, W. W. Morey, and W. H. Glenn, “Formation of Bragg gratings in optical fibers by a transverse holographic method,” Opt. Lett. 14, 823 (1989).
[Crossref] [PubMed]

S. Bardal, A. Kamal, P. St, and J. Russell, “Photo-induced birefringence in optical fibers: a comparative study of Lo-Bi and Hi-Bi fibers,” Opt. Lett. 17, 411 (1992).
[Crossref] [PubMed]

J. Lauzon, D. Gagnon, S. LaRochelle, A. Blouin, and F. Ouellette, “Dynamic polarization coupling in elliptical core photosensitive optical fiber,” Opt. Lett. 17, 1664 (1992).
[Crossref] [PubMed]

While we were responding to reviewers’ suggestions on this paper, another set of measurements [J. Albert, B. Malo, D. C. Johnson, F. Bilodeau, K. O. Hill, J. L. Brebner, and G. Kajrys, “Dichroism in the absorption spectrum of photobleached ion-implanted silica,” Opt. Lett. 18, 1126 (1993)] appeared in the literature, corroborating the preferential depletion picture. In these measurements a dichroism in the absorption spectrum of ion-implanted silica was observed after exposure to a linearly polarized UV beam. The dichroism was such that it suggested a depletion of the UV-absorbing defect centers along the polarization state of the pump UV beam.
[Crossref] [PubMed]

P. St, J. Russell, and R. Ulrich, “Grating-fiber coupler as a high-resolution spectrometer,” Opt. Lett. 10, 291 (1986).

S. An and J. E. Sipe, “Polarization aspects of two-photon photosensitivity in birefringent optical fibers,” Opt. Lett. 17, 490 (1992).
[Crossref] [PubMed]

R. M. Atkins, “Measurement of the ultraviolet absorption spectrum of optical fibers,” Opt. Lett. 17, 469 (1992).
[Crossref] [PubMed]

Other (4)

A. Kamal, R. W. Terhune, and D. A. Weinberger, “Dynamics of self-organized χ(2)gratings in optical fibers,” in International Workshop on Photoinduced Self-Organization Effects in Optical Fiber, F. Ouellette, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1516, 137 (1991).
[Crossref]

F. Glocking, The Chemistry of Germanium (Academic, London, 1969), Chap. 1, p. 13.

D. P. Hand, P. St, and J. Russell, “Single mode fiber grating written into Sagnac loop using photosensitive fiber: Transmission filters,” presented at the Seventh International Conference on Integrated Optics and Optical Communication, IOOC ’89, Kobe, Japan, 1989, paper 21C3-4.

H. Fritzche, Department of Physics, University of Chicago, Chicago, Illinois (personal communication, 1992).

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Figures (3)

Fig. 1
Fig. 1

Geometry of interferometric exposure to two arbitrarily polarized UV beams. E1 and E2 represent two arbitrary polarization states. The propagation vectors k1 and k2 lie in the yz plane and both make an angle α with the y axis.

Fig. 2
Fig. 2

Single-beam writing geometry. E1 represents a linear polarization state, which lies in the xy plane. K1 is also in the xy plane and makes an angle ϕo with the x axis.

Fig. 3
Fig. 3

Isotropic and anisotropic grating couplers. Grating couplers are conventionally formed by etching of a corrugation into the side of a fiber core (for simplicity an overlay of cladding glass is assumed in the figure). Phase-matching along the z direction (construction line CC′) determines the radiation angle ψ,. As the wavelength of the incident TE polarized light is tuned, ψ steers from +90° to −90°. For TM polarization the phase-matching condition is unchanged, but the radiated intensity is zero at ψ = 0°. The anisotropic tap shows the opposite effect—the radiated intensity is maximum at ψ = 0° and zero at ψ = 90°.

Tables (2)

Tables Icon

Table 1 Rocking Filtersa

Tables Icon

Table 2 Chiral Filtersa

Equations (38)

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N o = ϕ = 0 2 π θ = 0 π N ( θ , ϕ ) ( sin θ ) d θ d ϕ .
u ^ = ( u x , u y , u z ) = ( sin θ cos ϕ , sin θ sin ϕ , cos θ ) ,
Δ χ k l = γ ( 1 ) ϕ = 0 2 π θ = 0 π Δ N ( θ , ϕ ) u k u l sin θ d θ d ϕ .
E = 1 2 ( E 1 + E 2 ) exp ( j ω t ) + c . c . = 1 2 k = x , y , z k ^ ( E 1 k + E 2 k ) exp ( j ω t ) + c . c . = 1 2 k = x , y , z k ^ E k exp ( j ω t ) + c . c .
H int = μ · E = μ 0 2 k = x , y , z u k E k exp ( j ω t ) + c . c . = H exp ( j ω t ) + H * exp ( - j ω t ) ,
w = 1 2 f H i 2 g ( ν - ν 0 ) ,
g ( ν - ν 0 ) d v = 1.
w ( θ , ϕ ) = μ f i 2 4 2 g ( ν - ν 0 ) u k u l E k E l * ,
Δ N ( θ , ϕ ) = - N o 4 π w ( θ , ϕ ) Δ t .
Δ χ i j = C o Q i j k l E k E l * ,
C o = - ( μ f i 4 ) 2 N o γ ( 1 ) Δ t π g ( ν - ν o )
Q i j k l = ϕ = 0 2 π θ = 0 π u i u j u k u l ( sin θ ) d θ d ϕ
Q i i i i = Q 4 = 4 π / 5 , Q i i j j = Q i j i j = Q i j j i = Q 22 = 4 π / 15 ,             i j ,
Δ χ x x = C o ( Q x x x x E x 2 + Q x x y y E y 2 + Q x x z z E z 2 ) = 4 π C o 15 ( 3 E x 2 + E y 2 + E z 2 ) Δ χ x y = C o Q x y x y E x E y * + c . c . = 4 π C o 15 E x E y * + c . c .
H int = α : EE = ( α 0 / 4 ) u k u l E k E l exp ( j 2 ω t ) + c . c . = H exp ( j 2 ω t ) + H * exp ( - j 2 ω t ) .
w ( θ , ϕ ) = H 2 ( 1 / 2 ) g ( ν - ν 0 ) = ( α f i 4 ) 2 g ( ν - ν 0 ) u k u l u m u n E k E l * E m E n * ,
α f i = m μ f m μ m i ( U m - U i - U o ) ,
Δ χ i j = C 1 Q i j k l m n E k E l * E m E n * ,
C 1 = C o ( α f i / 2 μ f i ) 2 ,
Q i j k l m n = ϕ = 0 2 π θ = 0 π u i u j u k u l u m u n sin θ d θ d ϕ
Q 6 = 4 π / 7 , Q 42 = 4 π / 35 , Q 222 = 4 π / 105 ,
Δ χ x x = C 1 { Q 6 E x 4 + Q 42 [ E y 4 + 4 E x 2 E y 2 + ( E x 2 E y * 2 + c . c . ) ] } , Δ χ x y = 2 C 1 Q 42 ( E x 2 + E y 2 ) ( E x E y * + c . c . ) ,
E x = E 1 sin ϕ o exp ( - j k 1 · r ) , E y = - E 1 cos ϕ o exp ( - j k 1 · r ) .
[ Δ χ ] = Δ χ o × [ E 1 2 ( 1 + 2 sin 2 ϕ o ) - E 1 2 sin ( 2 ϕ o ) 0 - E 1 2 sin ( 2 ϕ o ) - E 1 2 ( 1 + 2 cos 2 ϕ o ) 0 0 0 E 1 2 ] ,
Δ χ o = ( 4 π / 15 ) C o .
[ Δ χ ] = Δ χ o [ E 1 2 0 0 0 3 E 1 2 0 0 0 E 1 2 ] .
E x = E 1 exp ( - j k 1 · r ) , E y = - E 2 sin α exp ( - j k 2 · r ) , E z = E 2 cos α exp ( - j k 2 · r ) ,
[ Δ χ ] = Δ χ 0 [ 3 E 1 2 + E 2 2 - 2 E 1 E 2 sin α cos ( K z ) 2 E 1 E 2 cos α cos ( K z ) - 2 E 1 E 2 sin α cos ( K z ) E 1 2 + E 2 2 ( 1 + 2 sin 2 α ) - E 2 2 sin ( 2 α ) 2 E 1 E 2 cos α cos ( K z ) - E 2 2 sin ( 2 α ) E 1 2 + E 2 2 ( 1 + 2 cos 2 α ) ] ,
K = ( k 1 - k 2 ) · z ^ .
κ = k o Δ χ x y 8 [ ( x x + Δ χ x x ) ( y y + Δ χ y y ) ] 1 / 2 ,
E x = E 1 exp ( - j β x z ) ,             E y = E 2 exp ( j β y z ) ,
[ Δ χ ] = [ Δ 11 + Δ m cos ( 2 K z ) Δ 12 cos ( K z ) 0 Δ 12 cos ( K z ) Δ 22 + Δ m cos ( 2 K z ) 0 0 0 Δ 33 ] ,
Δ 11 = Δ χ 1 [ 5 E 1 4 + E 2 2 ( E 2 2 + 4 E 1 2 ) ] , Δ 22 = Δ χ 1 [ 5 E 2 4 + E 1 2 ( E 1 2 + 4 E 2 2 ) ] , Δ 12 = Δ χ 1 ( 4 E o 2 E 1 E 2 ) , Δ m = Δ χ 1 ( 2 E 1 2 E 2 2 ) , Δ 33 = Δ χ 1 { 2 3 E 1 2 E 2 2 [ 2 + cos ( 2 K z ) ] + E 1 4 + E 2 4 } , Δ χ 1 = 4 π C 1 / 35 , E o 2 = E 1 2 + E 2 2 .
E x = ( 1 / 2 ) [ E 1 exp ( - j k 2 · r ) + E 2 exp ( - j k 2 · r ) ] , E y = j ( sin α / 2 ) [ E 1 exp ( - j k 1 · r ) - s E 2 exp ( - j k 2 · r ) ] , E z = j ( cos α / 2 ) [ E 1 exp ( - j k 1 · r ) - s E 2 exp ( - j k 2 · r ) ] ,
Δ χ x x = Δ χ o { 2 E o 2 + E 1 E 2 [ 3 + s cos ( 2 α ) ] cos ( K z ) } , Δ χ y y = Δ χ o [ E o 2 ( 1 + sin 2 α ) + E 1 E 2 [ 1 + s - 4 s sin 2 α ) ] cos ( K z ) } , Δ χ z z = Δ χ o [ E o 2 ( 1 + cos 2 α ) + E 1 E 2 [ 1 + 3 s - 4 s sin 2 α ) cos ( K z ) ] , Δ χ x y = Δ χ y x = Δ χ o E 1 E 2 ( sin α ) ( 1 + s ) sin ( K z ) , Δ χ x z = Δ χ z x = Δ χ o E 1 E 2 ( cos α ) ( 1 - s ) sin ( K z ) , Δ χ y z = Δ χ z y = Δ χ o 2 { E 1 2 - E 2 2 } sin ( 2 α ) .
E x = ( 1 / 2 ) [ E 1 exp ( - j β R z ) + E 2 exp ( - j β L z ) ] , E y = ( j / 2 ) [ E 1 exp ( - j β R z ) - E 2 exp ( - j β L z ) ] ,
[ Δ χ ] = [ Δ 11 + Δ 1 cos ( K z ) Δ 1 sin ( K z ) 0 Δ 1 sin ( K z ) Δ 11 - Δ 1 cos ( K z ) 0 0 0 Δ 33 ] ,
Δ 11 = Δ χ 1 ( 2 E o 4 + 4 E 1 2 E 2 2 ) , Δ 1 = Δ χ 1 4 E o 2 E 1 E 2 , Δ 33 = Δ χ 1 3 ( 2 E o 4 - 2 E 1 2 E 2 2 ) , K = β R - β L .

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