Abstract

Based on dual transverse Pockels effect, complete polarization conversion can be achieved by using only one electro-optic crystal and its two externally applied voltages. The electro-optic phase retardation and the azimuth angle of the field-induced principal dielectric axes of the crystal can be independently and linearly controlled by the amplitude and direction of the applied electric field, and the formulas for this correlation are deduced for arbitrary input and output polarization states. The candidate crystals mainly include the uniaxial crystals of 3m, 6¯2m, and 32 symmetry groups, and the cubic crystals of 4¯3m and 23 symmetry groups. Theoretical analysis demonstrates that one crystal exhibiting both dual transverse Pockels effect and optical activity can also be used for complete polarization converter. The continuous polarization rotation of a linearly polarized light from 0° to 180° has been performed experimentally by use of single lithium niobate crystal with four lateral electrodes. In addition the light beam position- dependent polarization conversion by using a bulk electro-optic crystal is also measured in the ex periment. This new type of polarization converter will have potential applications in many fields due to its simple configuration, explicit control logic of polarization conversion, and lower power consumption.

© 2008 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef] [PubMed]

2007 (1)

2006 (4)

P. Oswald and C. K. Madsen, “Deterministic analysis of endless tuning of polarization controllers,” J. Lightwave Technol. 24, 2932-2939 (2006).
[CrossRef]

A. J. Davidson, S. J. Elston, and E. P. Raynes, “Investigation into chiral active waveplates,” J. Appl. Phys. 99, 093109(2006).
[CrossRef]

S. Haxha, F. Abdel-Malek, and B. M. A. Rahman, “Analysis of polarization conversion in AlGaAs/GaAs electrooptic polarization converter,” Opt. Commun. 262, 47-56 (2006).
[CrossRef]

V. W. S. Chan, “Free-space optical communications,” J. Lightwave Technol. 24, 4750-4762 (2006).
[CrossRef]

2005 (3)

2004 (1)

2003 (3)

2002 (1)

C. Li and T. Yoshino, “Optical voltage sensor based on electrooptic crystal multiplier,” J. Lightwave Technol. 20, 843-849(2002).
[CrossRef]

2001 (1)

C. Li, X. Cui, and T. Yoshino, “Measurement of AC electric power based on dual transverse Pockels effect,” IEEE Trans. Instrum. Meas. 50, 697 (2001).
[CrossRef]

1998 (1)

M. Kurono, “High speed polarization control using a four-electrode LN crystal,” Trans. Inst. Electr. Eng. Jpn. 118-C, 649-655 (1998).

1997 (1)

C. Li and X. Cui, “Pulse-controlled polarization converter and its applications,” Acta Photonica Sin. 26, 929-934 (1997).

1996 (1)

S. Stenholm, “Polarization coding of quantum information,” Opt. Commun. 123, 287-296 (1996).
[CrossRef]

1995 (2)

J. Kobayashi, T. Asahi, M. Ichiki, K. Saito, T. Shimasaki, H. Yoshii, Y. Itagaki, and H. Ikawa, “Optical study on the phase transition of lead lanthanum zirconate titanate PLZT ceramics,” Phys. Rev. B 51, 763-779 (1995).
[CrossRef]

J. S. Kang and D. A. Dunmur, “Electric-filed-induced optical rotation in the pre-transitional isotropic of chiral nematic liquid crystal,” Phys. Rev. E 51, 2129-2136 (1995).
[CrossRef]

1993 (1)

E. Collett, Polarized Light: Fundamentals and Applications (Marcel Dekker, 1993).

1992 (1)

1990 (1)

N. G. Walker and G. R. Walker, “Polarization control for coherent communication,” J. Lightwave Technol. 8, 438-458 (1990).
[CrossRef]

1987 (3)

D. Eimerl, “Crystal symmetry and the electrooptic effect,” IEEE J. Quantum Electron. 23, 2104-2115 (1987).
[CrossRef]

F. Vachss and L. Hesselink, “Measurement of the electrogyratory and electro-optic effects in BSO and BGO,” Opt. Commun. 62, 159-165 (1987).
[CrossRef]

S. Sudo, A. Cordova-Plaza, R. L. Byer, and H. J. Shaw, “MgO: MgO:LiNbO3 single-crystal fiber with magnesium-ion in-diffused cladding,” Opt. Lett. 12, 938-940 (1987).
[CrossRef] [PubMed]

1985 (1)

S. Thaniyavarn, “Wavelength independent, optical damage immune z-propagation LiNbO3 waveguide polarization converter,” Appl. Phys. Lett. 47, 674-677 (1985).
[CrossRef]

1984 (1)

A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley & Sons, 1984).

1970 (1)

D. Kalymnios and F. C. Widdis, “Duplex polarization modulation with cubic crystals,” J. Phys. D 3, 884-888 (1970).
[CrossRef]

1969 (1)

W. J. Tabor and F. S. Chen, “Electromagnetic propagation through materials possessing both Faraday rotation and birefringence: experiments with ytterbium orthoferrite,” J. Appl. Phys. 40, 2760-2765 (1969).
[CrossRef]

1967 (1)

C. L. Hu, “Linear electro-optic retardation schemes for the twenty classes of linear electro-optic crystals and their applications,” J. Appl. Phys. 38, 3275-3284 (1967).
[CrossRef]

1966 (2)

1965 (1)

1964 (1)

1963 (1)

1952 (1)

Abdel-Malek, F.

S. Haxha, F. Abdel-Malek, and B. M. A. Rahman, “Analysis of polarization conversion in AlGaAs/GaAs electrooptic polarization converter,” Opt. Commun. 262, 47-56 (2006).
[CrossRef]

Amano, C.

Asahi, T.

J. Kobayashi, T. Asahi, M. Ichiki, K. Saito, T. Shimasaki, H. Yoshii, Y. Itagaki, and H. Ikawa, “Optical study on the phase transition of lead lanthanum zirconate titanate PLZT ceramics,” Phys. Rev. B 51, 763-779 (1995).
[CrossRef]

Baird, D. H.

Batchman, T. E.

Beyertt, A.

D. Nickel, C. Stolzenburg, A. Beyertt, and A. Giesen, “200 kHz electro-optic switch for ultrafast laser systems,” Rev. Sci. Instrum. 76, 033111 (2005).
[CrossRef]

Bloom, L. R.

Bolla, L.

Buhrer, C. F.

Byer, R. L.

Chan, V. W. S.

Chen, F. S.

W. J. Tabor and F. S. Chen, “Electromagnetic propagation through materials possessing both Faraday rotation and birefringence: experiments with ytterbium orthoferrite,” J. Appl. Phys. 40, 2760-2765 (1969).
[CrossRef]

Chen, L.

Chen, X.

Chen, Y.

Collett, E.

E. Collett, Polarized Light: Fundamentals and Applications (Marcel Dekker, 1993).

Cordova-Plaza, A.

Cui, X.

C. Li, X. Cui, and T. Yoshino, “Measurement of AC electric power based on dual transverse Pockels effect,” IEEE Trans. Instrum. Meas. 50, 697 (2001).
[CrossRef]

C. Li and X. Cui, “Pulse-controlled polarization converter and its applications,” Acta Photonica Sin. 26, 929-934 (1997).

Davidson, A. J.

A. J. Davidson, S. J. Elston, and E. P. Raynes, “Investigation into chiral active waveplates,” J. Appl. Phys. 99, 093109(2006).
[CrossRef]

Dunmur, D. A.

J. S. Kang and D. A. Dunmur, “Electric-filed-induced optical rotation in the pre-transitional isotropic of chiral nematic liquid crystal,” Phys. Rev. E 51, 2129-2136 (1995).
[CrossRef]

Eimerl, D.

D. Eimerl, “Crystal symmetry and the electrooptic effect,” IEEE J. Quantum Electron. 23, 2104-2115 (1987).
[CrossRef]

Elston, S. J.

A. J. Davidson, S. J. Elston, and E. P. Raynes, “Investigation into chiral active waveplates,” J. Appl. Phys. 99, 093109(2006).
[CrossRef]

Giesen, A.

D. Nickel, C. Stolzenburg, A. Beyertt, and A. Giesen, “200 kHz electro-optic switch for ultrafast laser systems,” Rev. Sci. Instrum. 76, 033111 (2005).
[CrossRef]

Hauer, M. C.

Haxha, S.

S. Haxha, F. Abdel-Malek, and B. M. A. Rahman, “Analysis of polarization conversion in AlGaAs/GaAs electrooptic polarization converter,” Opt. Commun. 262, 47-56 (2006).
[CrossRef]

Hesselink, L.

F. Vachss and L. Hesselink, “Measurement of the electrogyratory and electro-optic effects in BSO and BGO,” Opt. Commun. 62, 159-165 (1987).
[CrossRef]

Hirabayashi, K.

Ho, L.

Hu, C. L.

C. L. Hu, “Linear electro-optic retardation schemes for the twenty classes of linear electro-optic crystals and their applications,” J. Appl. Phys. 38, 3275-3284 (1967).
[CrossRef]

Ichiki, M.

J. Kobayashi, T. Asahi, M. Ichiki, K. Saito, T. Shimasaki, H. Yoshii, Y. Itagaki, and H. Ikawa, “Optical study on the phase transition of lead lanthanum zirconate titanate PLZT ceramics,” Phys. Rev. B 51, 763-779 (1995).
[CrossRef]

Ikawa, H.

J. Kobayashi, T. Asahi, M. Ichiki, K. Saito, T. Shimasaki, H. Yoshii, Y. Itagaki, and H. Ikawa, “Optical study on the phase transition of lead lanthanum zirconate titanate PLZT ceramics,” Phys. Rev. B 51, 763-779 (1995).
[CrossRef]

Itagaki, Y.

J. Kobayashi, T. Asahi, M. Ichiki, K. Saito, T. Shimasaki, H. Yoshii, Y. Itagaki, and H. Ikawa, “Optical study on the phase transition of lead lanthanum zirconate titanate PLZT ceramics,” Phys. Rev. B 51, 763-779 (1995).
[CrossRef]

Kalymnios, D.

D. Kalymnios and F. C. Widdis, “Duplex polarization modulation with cubic crystals,” J. Phys. D 3, 884-888 (1970).
[CrossRef]

Kaminow, I. P.

Kang, J. S.

J. S. Kang and D. A. Dunmur, “Electric-filed-induced optical rotation in the pre-transitional isotropic of chiral nematic liquid crystal,” Phys. Rev. E 51, 2129-2136 (1995).
[CrossRef]

Kobayashi, J.

J. Kobayashi, T. Asahi, M. Ichiki, K. Saito, T. Shimasaki, H. Yoshii, Y. Itagaki, and H. Ikawa, “Optical study on the phase transition of lead lanthanum zirconate titanate PLZT ceramics,” Phys. Rev. B 51, 763-779 (1995).
[CrossRef]

Kotlyar, M. V.

Krauss, T. F.

Kurono, M.

M. Kurono, “High speed polarization control using a four-electrode LN crystal,” Trans. Inst. Electr. Eng. Jpn. 118-C, 649-655 (1998).

Li, C.

C. Li and T. Yoshino, “Optical voltage sensor based on electrooptic crystal multiplier,” J. Lightwave Technol. 20, 843-849(2002).
[CrossRef]

C. Li, X. Cui, and T. Yoshino, “Measurement of AC electric power based on dual transverse Pockels effect,” IEEE Trans. Instrum. Meas. 50, 697 (2001).
[CrossRef]

C. Li and X. Cui, “Pulse-controlled polarization converter and its applications,” Acta Photonica Sin. 26, 929-934 (1997).

Madsen, C. K.

Midrio, M.

Nezam, S. M. R. M.

Nickel, D.

D. Nickel, C. Stolzenburg, A. Beyertt, and A. Giesen, “200 kHz electro-optic switch for ultrafast laser systems,” Rev. Sci. Instrum. 76, 033111 (2005).
[CrossRef]

O'Faolain, L.

Oswald, P.

Pan, Z.

Rahman, B. M. A.

S. Haxha, F. Abdel-Malek, and B. M. A. Rahman, “Analysis of polarization conversion in AlGaAs/GaAs electrooptic polarization converter,” Opt. Commun. 262, 47-56 (2006).
[CrossRef]

Ramachandran, G. N.

Ramaseshan, S.

Raynes, E. P.

A. J. Davidson, S. J. Elston, and E. P. Raynes, “Investigation into chiral active waveplates,” J. Appl. Phys. 99, 093109(2006).
[CrossRef]

Saito, K.

J. Kobayashi, T. Asahi, M. Ichiki, K. Saito, T. Shimasaki, H. Yoshii, Y. Itagaki, and H. Ikawa, “Optical study on the phase transition of lead lanthanum zirconate titanate PLZT ceramics,” Phys. Rev. B 51, 763-779 (1995).
[CrossRef]

Shaw, H. J.

She, W.

Shi, J.

Shimasaki, T.

J. Kobayashi, T. Asahi, M. Ichiki, K. Saito, T. Shimasaki, H. Yoshii, Y. Itagaki, and H. Ikawa, “Optical study on the phase transition of lead lanthanum zirconate titanate PLZT ceramics,” Phys. Rev. B 51, 763-779 (1995).
[CrossRef]

Sluss, J. J.

Stenholm, S.

S. Stenholm, “Polarization coding of quantum information,” Opt. Commun. 123, 287-296 (1996).
[CrossRef]

Stolzenburg, C.

D. Nickel, C. Stolzenburg, A. Beyertt, and A. Giesen, “200 kHz electro-optic switch for ultrafast laser systems,” Rev. Sci. Instrum. 76, 033111 (2005).
[CrossRef]

Sudo, S.

Tabor, W. J.

W. J. Tabor and F. S. Chen, “Electromagnetic propagation through materials possessing both Faraday rotation and birefringence: experiments with ytterbium orthoferrite,” J. Appl. Phys. 40, 2760-2765 (1969).
[CrossRef]

Tayag, T. J.

Thaniyavarn, S.

S. Thaniyavarn, “Wavelength independent, optical damage immune z-propagation LiNbO3 waveguide polarization converter,” Appl. Phys. Lett. 47, 674-677 (1985).
[CrossRef]

Turner, E. H.

Vachss, F.

F. Vachss and L. Hesselink, “Measurement of the electrogyratory and electro-optic effects in BSO and BGO,” Opt. Commun. 62, 159-165 (1987).
[CrossRef]

von Willisen, F. K.

Walker, G. R.

N. G. Walker and G. R. Walker, “Polarization control for coherent communication,” J. Lightwave Technol. 8, 438-458 (1990).
[CrossRef]

Walker, N. G.

N. G. Walker and G. R. Walker, “Polarization control for coherent communication,” J. Lightwave Technol. 8, 438-458 (1990).
[CrossRef]

Wang, J.

Widdis, F. C.

D. Kalymnios and F. C. Widdis, “Duplex polarization modulation with cubic crystals,” J. Phys. D 3, 884-888 (1970).
[CrossRef]

Willner, A. E.

Xia, Y.

Yan, L.

Yariv, A.

A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley & Sons, 1984).

Yeh, P.

A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley & Sons, 1984).

Yin, X.

Yoshii, H.

J. Kobayashi, T. Asahi, M. Ichiki, K. Saito, T. Shimasaki, H. Yoshii, Y. Itagaki, and H. Ikawa, “Optical study on the phase transition of lead lanthanum zirconate titanate PLZT ceramics,” Phys. Rev. B 51, 763-779 (1995).
[CrossRef]

Yoshino, T.

C. Li and T. Yoshino, “Optical voltage sensor based on electrooptic crystal multiplier,” J. Lightwave Technol. 20, 843-849(2002).
[CrossRef]

C. Li, X. Cui, and T. Yoshino, “Measurement of AC electric power based on dual transverse Pockels effect,” IEEE Trans. Instrum. Meas. 50, 697 (2001).
[CrossRef]

Zhang, S.

Zucker, J.

Acta Photonica Sin. (1)

C. Li and X. Cui, “Pulse-controlled polarization converter and its applications,” Acta Photonica Sin. 26, 929-934 (1997).

Appl. Opt. (9)

Appl. Phys. Lett. (1)

S. Thaniyavarn, “Wavelength independent, optical damage immune z-propagation LiNbO3 waveguide polarization converter,” Appl. Phys. Lett. 47, 674-677 (1985).
[CrossRef]

IEEE J. Quantum Electron. (1)

D. Eimerl, “Crystal symmetry and the electrooptic effect,” IEEE J. Quantum Electron. 23, 2104-2115 (1987).
[CrossRef]

IEEE Trans. Instrum. Meas. (1)

C. Li, X. Cui, and T. Yoshino, “Measurement of AC electric power based on dual transverse Pockels effect,” IEEE Trans. Instrum. Meas. 50, 697 (2001).
[CrossRef]

J. Appl. Phys. (3)

W. J. Tabor and F. S. Chen, “Electromagnetic propagation through materials possessing both Faraday rotation and birefringence: experiments with ytterbium orthoferrite,” J. Appl. Phys. 40, 2760-2765 (1969).
[CrossRef]

A. J. Davidson, S. J. Elston, and E. P. Raynes, “Investigation into chiral active waveplates,” J. Appl. Phys. 99, 093109(2006).
[CrossRef]

C. L. Hu, “Linear electro-optic retardation schemes for the twenty classes of linear electro-optic crystals and their applications,” J. Appl. Phys. 38, 3275-3284 (1967).
[CrossRef]

J. Lightwave Technol. (6)

J. Opt. Soc. Am. (1)

J. Phys. D (1)

D. Kalymnios and F. C. Widdis, “Duplex polarization modulation with cubic crystals,” J. Phys. D 3, 884-888 (1970).
[CrossRef]

Opt. Commun. (3)

F. Vachss and L. Hesselink, “Measurement of the electrogyratory and electro-optic effects in BSO and BGO,” Opt. Commun. 62, 159-165 (1987).
[CrossRef]

S. Stenholm, “Polarization coding of quantum information,” Opt. Commun. 123, 287-296 (1996).
[CrossRef]

S. Haxha, F. Abdel-Malek, and B. M. A. Rahman, “Analysis of polarization conversion in AlGaAs/GaAs electrooptic polarization converter,” Opt. Commun. 262, 47-56 (2006).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. B (1)

J. Kobayashi, T. Asahi, M. Ichiki, K. Saito, T. Shimasaki, H. Yoshii, Y. Itagaki, and H. Ikawa, “Optical study on the phase transition of lead lanthanum zirconate titanate PLZT ceramics,” Phys. Rev. B 51, 763-779 (1995).
[CrossRef]

Phys. Rev. E (1)

J. S. Kang and D. A. Dunmur, “Electric-filed-induced optical rotation in the pre-transitional isotropic of chiral nematic liquid crystal,” Phys. Rev. E 51, 2129-2136 (1995).
[CrossRef]

Rev. Sci. Instrum. (1)

D. Nickel, C. Stolzenburg, A. Beyertt, and A. Giesen, “200 kHz electro-optic switch for ultrafast laser systems,” Rev. Sci. Instrum. 76, 033111 (2005).
[CrossRef]

Trans. Inst. Electr. Eng. Jpn. (1)

M. Kurono, “High speed polarization control using a four-electrode LN crystal,” Trans. Inst. Electr. Eng. Jpn. 118-C, 649-655 (1998).

Other (2)

E. Collett, Polarized Light: Fundamentals and Applications (Marcel Dekker, 1993).

A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley & Sons, 1984).

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Figures (12)

Fig. 1
Fig. 1

Poincaré sphere illustration of the polarization conversion from point P 1 ( 2 α i , 2 ε i ) to point P 2 ( 2 α o , 2 ε o ) by use of an electro-optic linear birefringent retarder.

Fig. 2
Fig. 2

Poincaré sphere illustration of the polarization conversion from point P 1 ( 2 α i , 2 ε i ) to point P 2 ( 2 α o , 2 ε o ) by use of an electro-optic birefringent retarder with optical activity.

Fig. 3
Fig. 3

Schematic of an electro-optic crystal with dual transverse Pockels effect and its two transversely applied electric fields, where x and y are two electric field-induced principal dielectric axes of the crystal.

Fig. 4
Fig. 4

The equipotential lines distribution of the cross section of a bulk LN crystal with four lateral electrodes and two applied voltages U x = 52.45 V and U y = 202.55 V .

Fig. 5
Fig. 5

Schematic of the polarization conversion from arbitrary polarization state P 1 ( 0.123 , 0.526 ) to 45 ° and 0 ° linear polarization states, i.e., P 2 ( 0 , 1 ) and P 2 ( 1 , 0 ) .

Fig. 6
Fig. 6

Poincaré sphere illustration of the polarization conversion using an electro-optic birefringent retarder with optical activity, where electro-optic retardation and optical activity are separately analyzed.

Fig. 7
Fig. 7

Equipotential lines distribution of the cross section of a bulk LN crystal with four lateral electrodes and two applied voltages U x = U y = 441.5 V .

Fig. 8
Fig. 8

Electric field amplitude distributions on the diagonals of the crystal cross section (a) diagonal a b and (b) diagonal c d .

Fig. 9
Fig. 9

Schematic of experimental setup for the polarization conversion using single LN crystal with four lateral gold electrodes and two applied voltages.

Fig. 10
Fig. 10

Experimental data of the normalized output signals versus the applied voltages of LN crystal at different positions of incident light beam on crystal end surface.

Fig. 11
Fig. 11

Variation of polarization rotation angle as the function of the two voltages applied to the LN crystal.

Fig. 12
Fig. 12

Polarization rotation angle versus the azimuth angle of the applied electric field.

Tables (1)

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Table 1 Electro-Optic Retardation γ and Principal Polarization Azimuth Angle θ for Some Uniaxial Crystals Exhibiting Dual Transverse Pockels Effect

Equations (24)

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J o ( α o , ε o ) = T ( γ , θ ) · J i ( α i , ε i ) ,
S o ( α o , ε o ) = M ( γ , θ ) · S i ( α i , ε i ) ,
a = sin ( γ / 2 ) [ cos ( 2 θ ) sin ( 2 θ ) 0 ] T .
θ = α i + 1 2 arc tan [ cos ( 2 ε i ) cos ( 2 ε o ) sin ( 2 α o 2 α i ) cot ( 2 α o 2 α i ) ] ,
γ = π arc tan ( tan ( 2 ε i ) sin ( 2 θ 2 α i ) ) arc tan ( tan ( 2 ε o ) sin ( 2 α o 2 θ ) ) .
T ( γ , 0 , ρ ) = [ cos Φ 2 i γ Φ sin Φ 2 2 ρ Φ sin Φ 2 2 ρ Φ sin Φ 2 cos Φ 2 + i γ Φ sin Φ 2 ] ,
Φ = 2 [ ρ 2 + ( γ / 2 ) 2 ] 1 / 2 .
T ( γ , θ , ρ ) = R 1 ( θ ) · T ( γ , 0 , ρ ) · R ( θ ) = [ t 11 t 12 t 12 * t 11 * ] ,
t 11 = cos Φ 2 i γ Φ sin Φ 2 cos ( 2 θ ) ,
t 12 = 2 ρ Φ sin Φ 2 i γ Φ sin Φ 2 sin ( 2 θ ) .
a = sin ( Φ / 2 ) Φ / 2 [ γ 2 cos ( 2 θ ) , γ 2 sin ( 2 θ ) , ρ ] T .
tan ( 2 θ ) = cos ( 2 α i ) tan ( 2 ε o ) cos ( 2 α o ) tan ( 2 ε i ) sin ( 2 α o ) tan ( 2 ε i ) sin ( 2 α i ) tan ( 2 ε o ) ,
tan ( 2 ε ) = { cos ( 2 θ 2 α i ) tan ( 2 ε i ) if tan ( 2 ε i ) 0 o r cos ( 2 θ 2 α o ) tan ( 2 ε o ) if tan ( 2 ε o ) 0 ,
cos Φ = sin ( 2 ε i ) sin ( 2 ε o ) cos ( 2 ε i ) cos ( 2 ε o ) cos ( 2 α o 2 α i ) .
γ = 2 ρ tan ( 2 ε ) .
γ = 2 π λ 2 3 n o 3 r 41 E m l ,
θ = 1 2 β = 1 2 arc tan ( E y E x ) ,
γ = π λ n o 3 r 41 l ( E y 2 + 4 E x 2 ) 1 / 2 ,
θ = 1 2 arc tan ( 2 E x E y ) .
γ = 2 π λ n o 3 l r ¯ E m ,
tan ( 2 θ ) = r 22 E x + r 11 E y r 11 E x + r 22 E y ,
tan ( 2 θ ) { r 11 / r 22 for 1 ) E x = 0 , E y 0 ; 2 ) E x < E y 0 r 22 / r 11 for 1 ) E y = 0 , E x 0 ; 2 ) E x > E y 0 ( r 11 + r 22 ) / ( r 11 + r 22 ) for E x = E y 0 .
θ = π / 4 β / 2.
θ = β / 2.

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