Abstract

We show, that by performing a simultaneous analysis of the angular dependencies of the ± first and the zeroth diffraction orders of mixed holographic gratings, each of the relevant parameters can be obtained: the strength of the phase grating and the amplitude grating, respectively, as well as a potential phase between them. Experiments on a pure lithium niobate crystal are used to demonstrate the applicability of the analysis.

© 2008 Optical Society of America

Full Article  |  PDF Article

Corrections

Martin Fally, Mostafa A. Ellabban, and Irena Drevenšek-Olenik, "Out-of-phase mixed holographic gratings : a quantative analysis: erratum," Opt. Express 17, 23350-23350 (2009)
https://www.osapublishing.org/oe/abstract.cfm?uri=oe-17-25-23350

References

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  1. L. Carretero, R. F. Madrigal, A. Fimia, S. Blaya, and A. Beléndez, "Study of angular responses of mixed amplitude-phase holographic gratings: shifted Borrmann effect," Opt. Lett. 26, 786-788 (2001).
    [CrossRef]
  2. C. Neipp, C. Pascual, and A. Beléndez, "Mixed phase-amplitude holographic gratings recorded in bleached silver halide materials," J. Phys. D Appl. Phys. 35, 957-967 (2002).
    [CrossRef]
  3. C. Neipp, I. Pascual, and A. Beléndez, "Experimental evidence of mixed gratings with a phase difference between the phase and amplitude grating in volume holograms," Opt. Express 10, 1374-1383 (2002).
    [PubMed]
  4. M. A. Ellabban, M. Fally, R. A. Rupp, and L. Kovács, "Light-induced phase and amplitude gratings in centrosymmetric Gadolinium Gallium garnet doped with Calcium," Opt. Express 14, 593-602 (2006).
    [CrossRef] [PubMed]
  5. A. S. Shcheulin, A. V. Veniaminov, Y. L. Korzinin, A. E. Angervaks, and A. I. Ryskin, "A Highly Stable Holographic Medium Based on CaF2 :Na Crystals with Colloidal Color Centers: III. Properties of Holograms," Opt. Spectrosc. (USSR) 103, 655-659 (2007).
    [CrossRef]
  6. H. Kogelnik, "Coupled Wave Theory for Thick Hologram Gratings," AT&T Tech. J. 48, 2909-2947 (1969).
  7. E. Guibelalde, "Coupled wave analysis for out-of-phase mixed thick hologram gratings," Opt. Quantum Electron. 16, 173-178 (1984).
    [CrossRef]
  8. C. Neipp, I. Pascual, and A. Beléndez, "Theoretical and experimental analysis of overmodulation effects in volume holograms recorded on BB-640 emulsions," J. Opt. A-Pure Appl. Opt. 3, 504-513 (2001).
    [CrossRef]
  9. S. Gallego, M. Ortuño, C. Neipp, C. García, A. Beléndez, and I. Pascual, "Overmodulation effects in volume holograms recorded on photopolymers," Opt. Commun. 215, 263-269 (2003).
    [CrossRef]
  10. I. Drevensek-Olenik, M. Fally, and M. Ellabban, "Optical anisotropy of holographic polymer-dispersed liquid crystal transmission gratings," Phys. Rev. E 74, 021707 (2006).
    [CrossRef]
  11. M. Fally, I. Drevensek-Olenik, M. A. Ellabban, K. P. Pranzas, and J. Vollbrandt, "Colossal light-induced refractive-index modulation for neutrons in holographic polymer-dispersed liquid crystals," Phys. Rev. Lett. 97, 167803 (2006).
    [CrossRef] [PubMed]
  12. G. Montemezzani and M. Zgonik, "Light diffraction at mixed phase and absorption gratings in anisotropic media for arbitrary geometries," Phys. Rev. E 55, 1035-1047 (1997).
    [CrossRef]
  13. M. G. Moharam and T. K. Gaylord, "Rigorous coupled-wave analysis of planar-grating diffraction," J. Opt. Soc. Am. 71, 811-818 (1981).
    [CrossRef]
  14. M. A. Ellabban, M. Bichler, M. Fally, and I. Drevensek Olenik, "Role of optical extinction in holographic polymer-dispersed liquid crystals," in Liquid Crystals and Applications in Optics, M. Glogarova, P. Palffy-Muhoray, and M. Copic, eds., Proc. SPIE 6587, 65871J (2007).
    [CrossRef]
  15. L. De Sio, R. Caputo, A. De Luca, A. Veltri, C. Umeton, and A. V. Sukhov, "In situ optical control and stabilization of the curing process of holographic gratings with a nematic film-polymer-slice sequence structure," Appl. Opt. 45, 3721-3727 (2006).
    [CrossRef]
  16. N. Uchida, "Calculation of diffraction efficiency in hologram gratings attenuated along the direction perpendicular to the grating vector," J. Opt. Soc. Am. 63, 280-287 (1973).
    [CrossRef]
  17. B. W. Batterman and H. Cole, "Dynamical Diffraction of X Rays by Perfect Crystals," Rev. Mod. Phys. 36, 681-717 (1964).
    [CrossRef]
  18. K. Sutter and P. Günter, "Photorefractive gratings in the organic crystal 2-cyclooctylamino-5-nitropyridine doped with 7,7,8,8-tetracyanoquinodimethane," J. Opt. Soc. Am. B 7, 2274-2278 (1990).
    [CrossRef]
  19. F. Kahmann, "Separate and simultaneous investigation of absorption gratings and refractive-index gratings by beam-coupling analysis," J. Opt. Soc. Am. A 10, 1562-1569 (1993).
    [CrossRef]
  20. M. Fally, "Separate and simultaneous investigation of absorption gratings and refractive-index gratings by beamcoupling analysis: comment," J. Opt. Soc. Am. A 23, 2662-2663 (2006).
    [CrossRef]

2007 (1)

A. S. Shcheulin, A. V. Veniaminov, Y. L. Korzinin, A. E. Angervaks, and A. I. Ryskin, "A Highly Stable Holographic Medium Based on CaF2 :Na Crystals with Colloidal Color Centers: III. Properties of Holograms," Opt. Spectrosc. (USSR) 103, 655-659 (2007).
[CrossRef]

2006 (5)

2003 (1)

S. Gallego, M. Ortuño, C. Neipp, C. García, A. Beléndez, and I. Pascual, "Overmodulation effects in volume holograms recorded on photopolymers," Opt. Commun. 215, 263-269 (2003).
[CrossRef]

2002 (2)

C. Neipp, C. Pascual, and A. Beléndez, "Mixed phase-amplitude holographic gratings recorded in bleached silver halide materials," J. Phys. D Appl. Phys. 35, 957-967 (2002).
[CrossRef]

C. Neipp, I. Pascual, and A. Beléndez, "Experimental evidence of mixed gratings with a phase difference between the phase and amplitude grating in volume holograms," Opt. Express 10, 1374-1383 (2002).
[PubMed]

2001 (2)

L. Carretero, R. F. Madrigal, A. Fimia, S. Blaya, and A. Beléndez, "Study of angular responses of mixed amplitude-phase holographic gratings: shifted Borrmann effect," Opt. Lett. 26, 786-788 (2001).
[CrossRef]

C. Neipp, I. Pascual, and A. Beléndez, "Theoretical and experimental analysis of overmodulation effects in volume holograms recorded on BB-640 emulsions," J. Opt. A-Pure Appl. Opt. 3, 504-513 (2001).
[CrossRef]

1997 (1)

G. Montemezzani and M. Zgonik, "Light diffraction at mixed phase and absorption gratings in anisotropic media for arbitrary geometries," Phys. Rev. E 55, 1035-1047 (1997).
[CrossRef]

1993 (1)

1990 (1)

1984 (1)

E. Guibelalde, "Coupled wave analysis for out-of-phase mixed thick hologram gratings," Opt. Quantum Electron. 16, 173-178 (1984).
[CrossRef]

1981 (1)

1973 (1)

1969 (1)

H. Kogelnik, "Coupled Wave Theory for Thick Hologram Gratings," AT&T Tech. J. 48, 2909-2947 (1969).

1964 (1)

B. W. Batterman and H. Cole, "Dynamical Diffraction of X Rays by Perfect Crystals," Rev. Mod. Phys. 36, 681-717 (1964).
[CrossRef]

Angervaks, A. E.

A. S. Shcheulin, A. V. Veniaminov, Y. L. Korzinin, A. E. Angervaks, and A. I. Ryskin, "A Highly Stable Holographic Medium Based on CaF2 :Na Crystals with Colloidal Color Centers: III. Properties of Holograms," Opt. Spectrosc. (USSR) 103, 655-659 (2007).
[CrossRef]

Batterman, B. W.

B. W. Batterman and H. Cole, "Dynamical Diffraction of X Rays by Perfect Crystals," Rev. Mod. Phys. 36, 681-717 (1964).
[CrossRef]

Beléndez, A.

S. Gallego, M. Ortuño, C. Neipp, C. García, A. Beléndez, and I. Pascual, "Overmodulation effects in volume holograms recorded on photopolymers," Opt. Commun. 215, 263-269 (2003).
[CrossRef]

C. Neipp, C. Pascual, and A. Beléndez, "Mixed phase-amplitude holographic gratings recorded in bleached silver halide materials," J. Phys. D Appl. Phys. 35, 957-967 (2002).
[CrossRef]

C. Neipp, I. Pascual, and A. Beléndez, "Experimental evidence of mixed gratings with a phase difference between the phase and amplitude grating in volume holograms," Opt. Express 10, 1374-1383 (2002).
[PubMed]

L. Carretero, R. F. Madrigal, A. Fimia, S. Blaya, and A. Beléndez, "Study of angular responses of mixed amplitude-phase holographic gratings: shifted Borrmann effect," Opt. Lett. 26, 786-788 (2001).
[CrossRef]

C. Neipp, I. Pascual, and A. Beléndez, "Theoretical and experimental analysis of overmodulation effects in volume holograms recorded on BB-640 emulsions," J. Opt. A-Pure Appl. Opt. 3, 504-513 (2001).
[CrossRef]

Blaya, S.

Caputo, R.

Carretero, L.

Cole, H.

B. W. Batterman and H. Cole, "Dynamical Diffraction of X Rays by Perfect Crystals," Rev. Mod. Phys. 36, 681-717 (1964).
[CrossRef]

De Luca, A.

De Sio, L.

Drevensek-Olenik, I.

M. Fally, I. Drevensek-Olenik, M. A. Ellabban, K. P. Pranzas, and J. Vollbrandt, "Colossal light-induced refractive-index modulation for neutrons in holographic polymer-dispersed liquid crystals," Phys. Rev. Lett. 97, 167803 (2006).
[CrossRef] [PubMed]

I. Drevensek-Olenik, M. Fally, and M. Ellabban, "Optical anisotropy of holographic polymer-dispersed liquid crystal transmission gratings," Phys. Rev. E 74, 021707 (2006).
[CrossRef]

Ellabban, M.

I. Drevensek-Olenik, M. Fally, and M. Ellabban, "Optical anisotropy of holographic polymer-dispersed liquid crystal transmission gratings," Phys. Rev. E 74, 021707 (2006).
[CrossRef]

Ellabban, M. A.

M. Fally, I. Drevensek-Olenik, M. A. Ellabban, K. P. Pranzas, and J. Vollbrandt, "Colossal light-induced refractive-index modulation for neutrons in holographic polymer-dispersed liquid crystals," Phys. Rev. Lett. 97, 167803 (2006).
[CrossRef] [PubMed]

M. A. Ellabban, M. Fally, R. A. Rupp, and L. Kovács, "Light-induced phase and amplitude gratings in centrosymmetric Gadolinium Gallium garnet doped with Calcium," Opt. Express 14, 593-602 (2006).
[CrossRef] [PubMed]

Fally, M.

M. A. Ellabban, M. Fally, R. A. Rupp, and L. Kovács, "Light-induced phase and amplitude gratings in centrosymmetric Gadolinium Gallium garnet doped with Calcium," Opt. Express 14, 593-602 (2006).
[CrossRef] [PubMed]

M. Fally, "Separate and simultaneous investigation of absorption gratings and refractive-index gratings by beamcoupling analysis: comment," J. Opt. Soc. Am. A 23, 2662-2663 (2006).
[CrossRef]

M. Fally, I. Drevensek-Olenik, M. A. Ellabban, K. P. Pranzas, and J. Vollbrandt, "Colossal light-induced refractive-index modulation for neutrons in holographic polymer-dispersed liquid crystals," Phys. Rev. Lett. 97, 167803 (2006).
[CrossRef] [PubMed]

I. Drevensek-Olenik, M. Fally, and M. Ellabban, "Optical anisotropy of holographic polymer-dispersed liquid crystal transmission gratings," Phys. Rev. E 74, 021707 (2006).
[CrossRef]

Fimia, A.

Gallego, S.

S. Gallego, M. Ortuño, C. Neipp, C. García, A. Beléndez, and I. Pascual, "Overmodulation effects in volume holograms recorded on photopolymers," Opt. Commun. 215, 263-269 (2003).
[CrossRef]

García, C.

S. Gallego, M. Ortuño, C. Neipp, C. García, A. Beléndez, and I. Pascual, "Overmodulation effects in volume holograms recorded on photopolymers," Opt. Commun. 215, 263-269 (2003).
[CrossRef]

Gaylord, T. K.

Guibelalde, E.

E. Guibelalde, "Coupled wave analysis for out-of-phase mixed thick hologram gratings," Opt. Quantum Electron. 16, 173-178 (1984).
[CrossRef]

Günter, P.

Kahmann, F.

Kogelnik, H.

H. Kogelnik, "Coupled Wave Theory for Thick Hologram Gratings," AT&T Tech. J. 48, 2909-2947 (1969).

Korzinin, Y. L.

A. S. Shcheulin, A. V. Veniaminov, Y. L. Korzinin, A. E. Angervaks, and A. I. Ryskin, "A Highly Stable Holographic Medium Based on CaF2 :Na Crystals with Colloidal Color Centers: III. Properties of Holograms," Opt. Spectrosc. (USSR) 103, 655-659 (2007).
[CrossRef]

Kovács, L.

Madrigal, R. F.

Moharam, M. G.

Montemezzani, G.

G. Montemezzani and M. Zgonik, "Light diffraction at mixed phase and absorption gratings in anisotropic media for arbitrary geometries," Phys. Rev. E 55, 1035-1047 (1997).
[CrossRef]

Neipp, C.

S. Gallego, M. Ortuño, C. Neipp, C. García, A. Beléndez, and I. Pascual, "Overmodulation effects in volume holograms recorded on photopolymers," Opt. Commun. 215, 263-269 (2003).
[CrossRef]

C. Neipp, C. Pascual, and A. Beléndez, "Mixed phase-amplitude holographic gratings recorded in bleached silver halide materials," J. Phys. D Appl. Phys. 35, 957-967 (2002).
[CrossRef]

C. Neipp, I. Pascual, and A. Beléndez, "Experimental evidence of mixed gratings with a phase difference between the phase and amplitude grating in volume holograms," Opt. Express 10, 1374-1383 (2002).
[PubMed]

C. Neipp, I. Pascual, and A. Beléndez, "Theoretical and experimental analysis of overmodulation effects in volume holograms recorded on BB-640 emulsions," J. Opt. A-Pure Appl. Opt. 3, 504-513 (2001).
[CrossRef]

Ortuño, M.

S. Gallego, M. Ortuño, C. Neipp, C. García, A. Beléndez, and I. Pascual, "Overmodulation effects in volume holograms recorded on photopolymers," Opt. Commun. 215, 263-269 (2003).
[CrossRef]

Pascual, C.

C. Neipp, C. Pascual, and A. Beléndez, "Mixed phase-amplitude holographic gratings recorded in bleached silver halide materials," J. Phys. D Appl. Phys. 35, 957-967 (2002).
[CrossRef]

Pascual, I.

S. Gallego, M. Ortuño, C. Neipp, C. García, A. Beléndez, and I. Pascual, "Overmodulation effects in volume holograms recorded on photopolymers," Opt. Commun. 215, 263-269 (2003).
[CrossRef]

C. Neipp, I. Pascual, and A. Beléndez, "Experimental evidence of mixed gratings with a phase difference between the phase and amplitude grating in volume holograms," Opt. Express 10, 1374-1383 (2002).
[PubMed]

C. Neipp, I. Pascual, and A. Beléndez, "Theoretical and experimental analysis of overmodulation effects in volume holograms recorded on BB-640 emulsions," J. Opt. A-Pure Appl. Opt. 3, 504-513 (2001).
[CrossRef]

Pranzas, K. P.

M. Fally, I. Drevensek-Olenik, M. A. Ellabban, K. P. Pranzas, and J. Vollbrandt, "Colossal light-induced refractive-index modulation for neutrons in holographic polymer-dispersed liquid crystals," Phys. Rev. Lett. 97, 167803 (2006).
[CrossRef] [PubMed]

Rupp, R. A.

Ryskin, A. I.

A. S. Shcheulin, A. V. Veniaminov, Y. L. Korzinin, A. E. Angervaks, and A. I. Ryskin, "A Highly Stable Holographic Medium Based on CaF2 :Na Crystals with Colloidal Color Centers: III. Properties of Holograms," Opt. Spectrosc. (USSR) 103, 655-659 (2007).
[CrossRef]

Shcheulin, A. S.

A. S. Shcheulin, A. V. Veniaminov, Y. L. Korzinin, A. E. Angervaks, and A. I. Ryskin, "A Highly Stable Holographic Medium Based on CaF2 :Na Crystals with Colloidal Color Centers: III. Properties of Holograms," Opt. Spectrosc. (USSR) 103, 655-659 (2007).
[CrossRef]

Sukhov, A. V.

Sutter, K.

Uchida, N.

Umeton, C.

Veltri, A.

Veniaminov, A. V.

A. S. Shcheulin, A. V. Veniaminov, Y. L. Korzinin, A. E. Angervaks, and A. I. Ryskin, "A Highly Stable Holographic Medium Based on CaF2 :Na Crystals with Colloidal Color Centers: III. Properties of Holograms," Opt. Spectrosc. (USSR) 103, 655-659 (2007).
[CrossRef]

Vollbrandt, J.

M. Fally, I. Drevensek-Olenik, M. A. Ellabban, K. P. Pranzas, and J. Vollbrandt, "Colossal light-induced refractive-index modulation for neutrons in holographic polymer-dispersed liquid crystals," Phys. Rev. Lett. 97, 167803 (2006).
[CrossRef] [PubMed]

Zgonik, M.

G. Montemezzani and M. Zgonik, "Light diffraction at mixed phase and absorption gratings in anisotropic media for arbitrary geometries," Phys. Rev. E 55, 1035-1047 (1997).
[CrossRef]

Appl. Opt. (1)

AT&T Tech. J. (1)

H. Kogelnik, "Coupled Wave Theory for Thick Hologram Gratings," AT&T Tech. J. 48, 2909-2947 (1969).

J. Opt. A-Pure Appl. Opt. (1)

C. Neipp, I. Pascual, and A. Beléndez, "Theoretical and experimental analysis of overmodulation effects in volume holograms recorded on BB-640 emulsions," J. Opt. A-Pure Appl. Opt. 3, 504-513 (2001).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (1)

J. Phys. D Appl. Phys. (1)

C. Neipp, C. Pascual, and A. Beléndez, "Mixed phase-amplitude holographic gratings recorded in bleached silver halide materials," J. Phys. D Appl. Phys. 35, 957-967 (2002).
[CrossRef]

Opt. Commun. (1)

S. Gallego, M. Ortuño, C. Neipp, C. García, A. Beléndez, and I. Pascual, "Overmodulation effects in volume holograms recorded on photopolymers," Opt. Commun. 215, 263-269 (2003).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Opt. Quantum Electron. (1)

E. Guibelalde, "Coupled wave analysis for out-of-phase mixed thick hologram gratings," Opt. Quantum Electron. 16, 173-178 (1984).
[CrossRef]

Opt. Spectrosc. (USSR) (1)

A. S. Shcheulin, A. V. Veniaminov, Y. L. Korzinin, A. E. Angervaks, and A. I. Ryskin, "A Highly Stable Holographic Medium Based on CaF2 :Na Crystals with Colloidal Color Centers: III. Properties of Holograms," Opt. Spectrosc. (USSR) 103, 655-659 (2007).
[CrossRef]

Phys. Rev. E (2)

I. Drevensek-Olenik, M. Fally, and M. Ellabban, "Optical anisotropy of holographic polymer-dispersed liquid crystal transmission gratings," Phys. Rev. E 74, 021707 (2006).
[CrossRef]

G. Montemezzani and M. Zgonik, "Light diffraction at mixed phase and absorption gratings in anisotropic media for arbitrary geometries," Phys. Rev. E 55, 1035-1047 (1997).
[CrossRef]

Phys. Rev. Lett. (1)

M. Fally, I. Drevensek-Olenik, M. A. Ellabban, K. P. Pranzas, and J. Vollbrandt, "Colossal light-induced refractive-index modulation for neutrons in holographic polymer-dispersed liquid crystals," Phys. Rev. Lett. 97, 167803 (2006).
[CrossRef] [PubMed]

Rev. Mod. Phys. (1)

B. W. Batterman and H. Cole, "Dynamical Diffraction of X Rays by Perfect Crystals," Rev. Mod. Phys. 36, 681-717 (1964).
[CrossRef]

Other (1)

M. A. Ellabban, M. Bichler, M. Fally, and I. Drevensek Olenik, "Role of optical extinction in holographic polymer-dispersed liquid crystals," in Liquid Crystals and Applications in Optics, M. Glogarova, P. Palffy-Muhoray, and M. Copic, eds., Proc. SPIE 6587, 65871J (2007).
[CrossRef]

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Figures (4)

Fig. 1.
Fig. 1.

Schematic of the setup. The angular dependence of the zeroth η 0(Θ) and ± first diffraction orders η ±1(Θ) from a mixed grating is measured by rotating the sample around an axis perpendicular to the grating vector K⃗. Θ denotes angles outside the medium. Note, that we are within the thick grating regime, where only two beams are propagating simultaneously: the zero order together with either the -1 st or the +1 st order.

Fig. 2.
Fig. 2.

Angular dependence of the zeroth and first orders diffraction efficiencies for increasing grating strength of the phase grating contribution. For all graphs: α 0=2.5×104,α 0 d=1, κ 2=α 0/2. Red thin lines: κ 1=κ 2/4, blue lines: κ 1=κ 2 and green thick lines:κ 1=4κ 2 with φ=0 (a), φ=π/4 (b), φ=π/2 (c), and φ=3π/4. The dash-dot line indicates the mean absorption curve A(θ). θ=0,±θ B are marked by vertical lines. Note, that for φ=π/2,κ 1=κ2 the minus first order Bragg peak completely disappears and the zero order peaks, too (shown in (c), blue lines)

Fig. 3.
Fig. 3.

Angular dependence of the diffraction efficiencies for the zero and ± first orders of a grating recorded in a pure LiNbO3 sample. The unequal values of the first order diffraction efficiencies are an impressive signature for the existence of mixed phase and amplitude gratings that are out of phase. The zero order diffraction efficiency at the Bragg angles show only a slight asymmetry with respect to θB because the diffraction efficiencies are small (<1%) and the phase grating is by far dominating. The solid lines show a simultaneous fit to η ±1 and η 0. The dashed-dot line indicates the mean absorption curve.

Fig. 4.
Fig. 4.

Zeroth and ± first diffraction orders of a strongly overmodulated grating in HPDLC at 63° Celsius. Λ=1.2µm, λ=543 nm, d=30 µm (same sample as used for the investigations in Ref. [10]. The lower graphs show a simulation according to Eq. 1 and Eq. 2. Note, that here the mean extinction is already rather high. We do not expect that a fit could be successful for at least three reasons: (1) as in HPDLCs anisotropic gratings are formed, for the basic equations the full theory of Montemezzani and Zgonik should be employed [12]. (2) It can be noticed, that around θ=0 more than two waves are propagating in the medium. Therefore, also the two-wave coupling theory is not fully applicable. Instead a rigorous coupled wave analysis should be performed [13]. (3) The gratings are expected to be inhomogeneous and non-sinusoidal [11], thus not completely fulfilling the requirements

Equations (10)

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

η ± 1 ( θ ) = A ( θ ) κ 1 2 + κ 2 2 ± 2 κ 1 κ 2 sin φ 2 z ( cosh [ z 1 2 D cos ψ ] cos [ z 1 2 D sin ψ ] )
η 0 ( θ ) = A ( θ ) ( ( z + ϑ 2 ) cosh [ z 1 2 D cos ψ ] + ( z ϑ 2 ) cos [ z 1 2 D sin ψ ]
+ 2 cos φ cos φ ϑ z 1 2 { sin ψ sinh [ z 1 2 D cos ψ ] cos ψ sin [ z 1 2 D sin ψ ] } )
ϑ = K ( sin θ sin θ B )
z = { [ ϑ 2 + 4 ( κ 1 2 κ 2 2 ) ] 2 + [ 8 κ 1 κ 2 cos φ ] 2 } 1 2
2 ψ = arccos ( ϑ 2 + 4 ( κ 1 2 κ 2 2 ) z ) .
η ± 1 ( θ ) = A ( θ ) z 1 4 ( κ 1 ± κ 2 ) 2 sin 2 ( z 1 2 D 2 ) =
  = A ( θ ) z 1 r ± 1 4 ( κ 1 2 κ 2 2 ) sin 2 ( z 1 2 D 2 )
η 0 = A ( θ ) z 1 [ ϑ 2 + 4 ( κ 1 2 κ 2 2 ) cos 2 ( z 1 2 D 2 ) ]
z = ϑ 2 + 4 ( κ 1 2 κ 2 2 ) .

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