Abstract

By the Riemann method, a coupled wave model is derived for the ordinary-to-ordinary (OO) and extraordinary-to-extraordinary (EE) Bragg diffraction of a Gaussian beam by overlapping holographic gratings in a uniaxial crystal. The computer simulation is used to discuss the relations among the diffraction efficiency, the index modulation, the wavelength sensitivity, the angular sensitivity, and the the widths of the recording and reading beams. The characteristics of EE and OO diffraction in a uniaxial crystal are found to be remarkably different. The simulation shows that EE diffraction may exhibit far higher diffraction efficiency than does OO diffraction for very low index modulation with the same hologram size, for example, nearly 90% when the size is 8.2×105.

© 2005 Optical Society of America

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References

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  1. R. Müller, M. T. Santos, L. Arizmendi, J. M. Cabrera, “A narrow-band interference filter with photorefractive LiNbO3,” J. Phys. D 27, 241–246 (1994).
    [CrossRef]
  2. S. Breer, K. Buse, “Wavelength demultiplexing with volume phase holograms in photorefractive lithium niobate,” Appl. Phys. B 66, 339–345 (1998).
    [CrossRef]
  3. J. F. Heanue, M. C. Bashaw, A. J. Daiber, R. Snyder, L. Hesselink, “Digital holographic storage system incorporating thermal fixing in lithium niobate,” Opt. Lett. 21, 1615–1617 (1996).
    [CrossRef] [PubMed]
  4. H. Kogelnik, “Coupled wave theory of thick holograms gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
    [CrossRef]
  5. B. Benlarbi, P. St.J. Russell, L. Solymar, “Bragg diffraction of finite beams by gratings: two rival theories,” Appl. Phys. B 28, 63–72 (1982).
    [CrossRef]
  6. B. Benlarbi, P. St.J. Russell, L. Solymar, “Bragg diffraction of Gaussian beams by thick gratings: numerical evaluations by plane-wave decomposition,” Appl. Phys. B 28, 383–390 (1982).
    [CrossRef]
  7. L. Solymar, M. P. Jordan, “Two-dimensional transmission type volume holograms for incident plane waves of arbitrary amplitude distribution,” Opt. Quantum Electron. 9, 437–444 (1977).
    [CrossRef]
  8. R. P. Kenan, “Theory of crossed-beam diffraction gratings,” IEEE J. Quantum Electron. QE-14, 924–930 (1978).
    [CrossRef]
  9. M. G. Moharam, T. K. Gaylord, R. Magnusson, “Diffraction characteristics of three-dimensional crossed-beam volume gratings,” J. Opt. Soc. Am. 70, 437–442 (1979).
    [CrossRef]
  10. M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
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    [CrossRef]
  12. J. M. Jarem, P. P. Banerjee, “Exact, dynamical analysis of the Kukhtarev equations in photorefractive barium titanate using rigorous coupled-wave diffraction theory,” J. Opt. Soc. Am. A 13, 819–831 (1996).
    [CrossRef]
  13. C.-W. Tarn, “Gaussian-beam profile deformation via anisotropic photorefractive gratings formed by diffusive, photovoltaic and drift mechanisms: a system transfer function approach,” Opt. Eng. (Bellingham) 37, 229–236 (1998).
    [CrossRef]
  14. S. Liu, R. Guo, Z. Ling, Photorefractive Nonlinear Optics (Chinese Standard, 1992), pp. 136 (in Chinese).
  15. D. Guo, Mathematical Method of Physics (People’s Education, 1978), pp. 428–433 (in Chinese).
  16. K. Peithmann, K. Buse, E. Krätzig, “Photorefractive properties of highly iron- or copper-doped lithium niobate crystals,” in Advances in Photorefractive Materials, Effects, and Devices, P. E. Andersen, P. M. Johansen, H. C. Pedersen, P M. Petersen, and M. Saffman, eds., Vol. 27 of OSA Trends in Optics and Photonics Series (Optical Society of America, 1999), pp. 50–53 (1999).
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    [CrossRef]

1998 (2)

S. Breer, K. Buse, “Wavelength demultiplexing with volume phase holograms in photorefractive lithium niobate,” Appl. Phys. B 66, 339–345 (1998).
[CrossRef]

C.-W. Tarn, “Gaussian-beam profile deformation via anisotropic photorefractive gratings formed by diffusive, photovoltaic and drift mechanisms: a system transfer function approach,” Opt. Eng. (Bellingham) 37, 229–236 (1998).
[CrossRef]

1996 (3)

1994 (1)

R. Müller, M. T. Santos, L. Arizmendi, J. M. Cabrera, “A narrow-band interference filter with photorefractive LiNbO3,” J. Phys. D 27, 241–246 (1994).
[CrossRef]

1987 (1)

1982 (2)

B. Benlarbi, P. St.J. Russell, L. Solymar, “Bragg diffraction of finite beams by gratings: two rival theories,” Appl. Phys. B 28, 63–72 (1982).
[CrossRef]

B. Benlarbi, P. St.J. Russell, L. Solymar, “Bragg diffraction of Gaussian beams by thick gratings: numerical evaluations by plane-wave decomposition,” Appl. Phys. B 28, 383–390 (1982).
[CrossRef]

1981 (1)

1979 (1)

1978 (1)

R. P. Kenan, “Theory of crossed-beam diffraction gratings,” IEEE J. Quantum Electron. QE-14, 924–930 (1978).
[CrossRef]

1977 (1)

L. Solymar, M. P. Jordan, “Two-dimensional transmission type volume holograms for incident plane waves of arbitrary amplitude distribution,” Opt. Quantum Electron. 9, 437–444 (1977).
[CrossRef]

1969 (1)

H. Kogelnik, “Coupled wave theory of thick holograms gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

Arizmendi, L.

R. Müller, M. T. Santos, L. Arizmendi, J. M. Cabrera, “A narrow-band interference filter with photorefractive LiNbO3,” J. Phys. D 27, 241–246 (1994).
[CrossRef]

Banerjee, P. P.

Bashaw, M. C.

Benlarbi, B.

B. Benlarbi, P. St.J. Russell, L. Solymar, “Bragg diffraction of finite beams by gratings: two rival theories,” Appl. Phys. B 28, 63–72 (1982).
[CrossRef]

B. Benlarbi, P. St.J. Russell, L. Solymar, “Bragg diffraction of Gaussian beams by thick gratings: numerical evaluations by plane-wave decomposition,” Appl. Phys. B 28, 383–390 (1982).
[CrossRef]

Breer, S.

S. Breer, K. Buse, “Wavelength demultiplexing with volume phase holograms in photorefractive lithium niobate,” Appl. Phys. B 66, 339–345 (1998).
[CrossRef]

Buse, K.

S. Breer, K. Buse, “Wavelength demultiplexing with volume phase holograms in photorefractive lithium niobate,” Appl. Phys. B 66, 339–345 (1998).
[CrossRef]

K. Peithmann, K. Buse, E. Krätzig, “Photorefractive properties of highly iron- or copper-doped lithium niobate crystals,” in Advances in Photorefractive Materials, Effects, and Devices, P. E. Andersen, P. M. Johansen, H. C. Pedersen, P M. Petersen, and M. Saffman, eds., Vol. 27 of OSA Trends in Optics and Photonics Series (Optical Society of America, 1999), pp. 50–53 (1999).

Cabrera, J. M.

R. Müller, M. T. Santos, L. Arizmendi, J. M. Cabrera, “A narrow-band interference filter with photorefractive LiNbO3,” J. Phys. D 27, 241–246 (1994).
[CrossRef]

Daiber, A. J.

Gaylord, T. K.

Glytsis, E. N.

Guo, D.

D. Guo, Mathematical Method of Physics (People’s Education, 1978), pp. 428–433 (in Chinese).

Guo, R.

S. Liu, R. Guo, Z. Ling, Photorefractive Nonlinear Optics (Chinese Standard, 1992), pp. 136 (in Chinese).

Heanue, J. F.

Hesselink, L.

Jarem, J. M.

Jordan, M. P.

L. Solymar, M. P. Jordan, “Two-dimensional transmission type volume holograms for incident plane waves of arbitrary amplitude distribution,” Opt. Quantum Electron. 9, 437–444 (1977).
[CrossRef]

Kenan, R. P.

R. P. Kenan, “Theory of crossed-beam diffraction gratings,” IEEE J. Quantum Electron. QE-14, 924–930 (1978).
[CrossRef]

Kogelnik, H.

H. Kogelnik, “Coupled wave theory of thick holograms gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

Krätzig, E.

K. Peithmann, K. Buse, E. Krätzig, “Photorefractive properties of highly iron- or copper-doped lithium niobate crystals,” in Advances in Photorefractive Materials, Effects, and Devices, P. E. Andersen, P. M. Johansen, H. C. Pedersen, P M. Petersen, and M. Saffman, eds., Vol. 27 of OSA Trends in Optics and Photonics Series (Optical Society of America, 1999), pp. 50–53 (1999).

Ling, Z.

S. Liu, R. Guo, Z. Ling, Photorefractive Nonlinear Optics (Chinese Standard, 1992), pp. 136 (in Chinese).

Liu, S.

S. Liu, R. Guo, Z. Ling, Photorefractive Nonlinear Optics (Chinese Standard, 1992), pp. 136 (in Chinese).

Magnusson, R.

Moharam, M. G.

Müller, R.

R. Müller, M. T. Santos, L. Arizmendi, J. M. Cabrera, “A narrow-band interference filter with photorefractive LiNbO3,” J. Phys. D 27, 241–246 (1994).
[CrossRef]

Orlov, S. S.

Peithmann, K.

K. Peithmann, K. Buse, E. Krätzig, “Photorefractive properties of highly iron- or copper-doped lithium niobate crystals,” in Advances in Photorefractive Materials, Effects, and Devices, P. E. Andersen, P. M. Johansen, H. C. Pedersen, P M. Petersen, and M. Saffman, eds., Vol. 27 of OSA Trends in Optics and Photonics Series (Optical Society of America, 1999), pp. 50–53 (1999).

Rakuljic, G. A.

Russell, P. St.J.

B. Benlarbi, P. St.J. Russell, L. Solymar, “Bragg diffraction of Gaussian beams by thick gratings: numerical evaluations by plane-wave decomposition,” Appl. Phys. B 28, 383–390 (1982).
[CrossRef]

B. Benlarbi, P. St.J. Russell, L. Solymar, “Bragg diffraction of finite beams by gratings: two rival theories,” Appl. Phys. B 28, 63–72 (1982).
[CrossRef]

Santos, M. T.

R. Müller, M. T. Santos, L. Arizmendi, J. M. Cabrera, “A narrow-band interference filter with photorefractive LiNbO3,” J. Phys. D 27, 241–246 (1994).
[CrossRef]

Snyder, R.

Solymar, L.

B. Benlarbi, P. St.J. Russell, L. Solymar, “Bragg diffraction of finite beams by gratings: two rival theories,” Appl. Phys. B 28, 63–72 (1982).
[CrossRef]

B. Benlarbi, P. St.J. Russell, L. Solymar, “Bragg diffraction of Gaussian beams by thick gratings: numerical evaluations by plane-wave decomposition,” Appl. Phys. B 28, 383–390 (1982).
[CrossRef]

L. Solymar, M. P. Jordan, “Two-dimensional transmission type volume holograms for incident plane waves of arbitrary amplitude distribution,” Opt. Quantum Electron. 9, 437–444 (1977).
[CrossRef]

Tarn, C.-W.

C.-W. Tarn, “Gaussian-beam profile deformation via anisotropic photorefractive gratings formed by diffusive, photovoltaic and drift mechanisms: a system transfer function approach,” Opt. Eng. (Bellingham) 37, 229–236 (1998).
[CrossRef]

Yariv, K.

Appl. Phys. B (3)

S. Breer, K. Buse, “Wavelength demultiplexing with volume phase holograms in photorefractive lithium niobate,” Appl. Phys. B 66, 339–345 (1998).
[CrossRef]

B. Benlarbi, P. St.J. Russell, L. Solymar, “Bragg diffraction of finite beams by gratings: two rival theories,” Appl. Phys. B 28, 63–72 (1982).
[CrossRef]

B. Benlarbi, P. St.J. Russell, L. Solymar, “Bragg diffraction of Gaussian beams by thick gratings: numerical evaluations by plane-wave decomposition,” Appl. Phys. B 28, 383–390 (1982).
[CrossRef]

Bell Syst. Tech. J. (1)

H. Kogelnik, “Coupled wave theory of thick holograms gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

IEEE J. Quantum Electron. (1)

R. P. Kenan, “Theory of crossed-beam diffraction gratings,” IEEE J. Quantum Electron. QE-14, 924–930 (1978).
[CrossRef]

J. Opt. Soc. Am. (2)

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

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

J. Phys. D (1)

R. Müller, M. T. Santos, L. Arizmendi, J. M. Cabrera, “A narrow-band interference filter with photorefractive LiNbO3,” J. Phys. D 27, 241–246 (1994).
[CrossRef]

Opt. Eng. (Bellingham) (1)

C.-W. Tarn, “Gaussian-beam profile deformation via anisotropic photorefractive gratings formed by diffusive, photovoltaic and drift mechanisms: a system transfer function approach,” Opt. Eng. (Bellingham) 37, 229–236 (1998).
[CrossRef]

Opt. Lett. (1)

Opt. Quantum Electron. (1)

L. Solymar, M. P. Jordan, “Two-dimensional transmission type volume holograms for incident plane waves of arbitrary amplitude distribution,” Opt. Quantum Electron. 9, 437–444 (1977).
[CrossRef]

Other (3)

S. Liu, R. Guo, Z. Ling, Photorefractive Nonlinear Optics (Chinese Standard, 1992), pp. 136 (in Chinese).

D. Guo, Mathematical Method of Physics (People’s Education, 1978), pp. 428–433 (in Chinese).

K. Peithmann, K. Buse, E. Krätzig, “Photorefractive properties of highly iron- or copper-doped lithium niobate crystals,” in Advances in Photorefractive Materials, Effects, and Devices, P. E. Andersen, P. M. Johansen, H. C. Pedersen, P M. Petersen, and M. Saffman, eds., Vol. 27 of OSA Trends in Optics and Photonics Series (Optical Society of America, 1999), pp. 50–53 (1999).

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Figures (11)

Fig. 1
Fig. 1

Geometry of a crossed-beam grating corresponding to (a) OO diffraction and (b) EE diffraction.

Fig. 2
Fig. 2

Boundary of volume grating (OO diffraction).

Fig. 3
Fig. 3

Boundary of volume grating (EE diffraction), corresponding to (top) the first case and (bottom) the second case of the line integral.

Fig. 4
Fig. 4

Relation between the diffraction efficiency and the waist radius of the reading light (OO diffraction). Curves a, b, and c correspond to θ 0 = 10 ° , θ 0 = 15 ° , and θ 0 = 20 ° , respectively.

Fig. 5
Fig. 5

Relation between the diffraction efficiency and the waist radius of the reading light (EE diffraction). Curves a, b, and c correspond to θ 0 = 15 ° , θ 0 = 10 ° , and θ 0 = 20 ° , respectively.

Fig. 6
Fig. 6

Relation between the diffraction efficiency and the waist radius of the reading light (EE diffraction with θ 0 = 15 ° ).

Fig. 7
Fig. 7

Relation between the diffraction efficiency and the refractive index modulation (OO diffraction). Curves a, b, and c correspond to θ 0 = 10 ° , θ 0 = 15 ° , and θ 0 = 20 ° , respectively.

Fig. 8
Fig. 8

Relation between the diffraction efficiency and the refractive index modulation (EE diffraction). Curves a, b, and c correspond to θ 0 = 10 ° , θ 0 = 15 ° , and θ 0 = 20 ° , respectively.

Fig. 9
Fig. 9

Relation between the diffraction efficiency and the refractive index modulation (EE diffraction with θ 0 = 15 ° ).

Fig. 10
Fig. 10

Wavelength sensitivity corresponding to (top) OO diffraction, (middle) EE diffraction and (bottom) EE diffraction with θ 0 = 15 ° . Top, curves a, b, and c correspond to θ 0 = 10 ° , θ 0 = 15 ° , and θ 0 = 20 ° , respectively. Middle, curves a, b, and c correspond to θ 0 = 15 ° , θ 0 = 10 ° , and θ 0 = 20 ° , respectively.

Fig. 11
Fig. 11

Angular sensitivity corresponding to (top) OO diffraction, (middle) EE diffraction and (bottom) EE diffraction with θ 0 = 15 ° . Top, curves a, b, and c correspond to θ 0 = 10 ° , θ 0 = 15 ° , and θ 0 = 20 ° , respectively. Middle, curves a, b, and c correspond to θ 0 = 15 ° , θ 0 = 10 ° , and θ 0 = 20 ° , respectively.

Equations (67)

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E R = e R E R 0 exp ( j K R r ) ,
E S = e S E S 0 exp ( j K S r ) ,
K = 2 π n o λ 0 .
K = 2 π n o n e λ 0 n o 2 + ( n e 2 n o 2 ) sin 2 θ 0 ,
E s c = E 0 [ exp ( j K g r ) + exp ( j K g r ) ] ,
tan β = n e 2 n o 2 2 ( n o 2 cos 2 θ 0 + n e 2 sin 2 θ 0 ) sin 2 ( 90 ° θ 0 ) , or
tan β = n e 2 n o 2 2 ( n o 2 cos 2 θ 0 + n e 2 sin 2 θ 0 ) sin 2 θ 0 .
α 1 = θ 0 + β , α 2 = θ 0 β ,
R = e R R 0 ( x , y , z ) exp ( j K R r ) ,
S = e S S 0 ( x , y , z ) exp [ j ( K R K g ) r ] ,
E = y R 0 ( x , y , z ) exp ( j K R r ) + y S 0 ( x , y , z ) exp [ j ( K R K g ) r ] .
cos θ s = K cos θ K 2 + K g 2 2 K K g sin θ ,
sin θ s = K sin θ K g K 2 + K g 2 2 K K g sin θ .
E = [ R 1 R 0 ( x , y , z ) exp ( j K R r ) + S 1 S 0 ( x , y , z ) exp [ j ( K R K g ) r ] 0 R 2 R 0 ( x , y , z ) exp ( j K R r ) + S 2 S 0 ( x , y , z ) exp [ j ( K R K g ) r ] ] ,
R 1 = n e 2 sin θ n e 4 sin 2 θ + n o 4 cos 2 θ ,
R 2 = n o 2 cos θ n e 4 sin 2 θ + n o 4 cos 2 θ ,
S 1 = n e 2 sin θ s n e 4 sin 2 θ s + n o 4 cos 2 θ s ,
S 2 = n o 2 cos θ s n e 4 sin 2 θ s + n o 4 cos 2 θ s .
2 E ( r ) + ω 2 μ 0 ϵ 0 ϵ r E ( r ) [ E ( r ) ] = 0 .
2 E ( r ) + ω 2 μ 0 ϵ 0 ϵ r E ( r ) = 0 ,
cos θ R 0 x sin θ R 0 z + j κ S 0 = 0 ,
cos θ S 0 x + ( sin θ ϑ ) S 0 z j K g 2 ϑ S 0 + j κ R 0 = 0 .
R 2 2 cos θ R 0 x + R 1 2 sin θ R 0 z + j κ S 0 = 0 ,
S 2 2 cos θ S 0 x + S 1 2 ( ϑ sin θ ) S 0 z j [ K sin θ cos θ ( R 1 2 S 2 2 R 2 2 S 1 2 ) ϑ K g R 1 R 2 S 1 2 ] 2 R 1 R 2 S 0 + j κ R 0 = 0 ,
ϑ = ( 2 K sin θ K g ) K .
κ = π n o 3 r 13 E 0 λ .
κ = π ( n e 4 r 33 R 2 S 2 n o 4 r 13 R 1 S 1 ) E 0 n o 2 + ( n e 2 n o 2 ) sin 2 θ λ n o n e .
r = x sin θ z cos θ ,
s = x ( sin θ ϑ ) + z cos θ ,
r = x R 1 2 sin θ z R 2 2 cos θ ,
s = x S 1 2 ( ϑ sin θ ) + z S 2 2 cos θ .
R ( s , r , y ) s ± j κ 0 S ( s , r , y ) = 0 ,
S ( s , r , y ) r ± j κ 0 R ( s , r , y ) = 0 ,
κ 0 = κ sin 2 θ ϑ cos θ , a = K g ϑ 2 ( sin 2 θ ϑ cos θ ) .
κ 0 = κ cos θ [ R 1 2 S 2 2 sin θ R 2 2 S 1 2 ( ϑ sin θ ) ] ,
a = K sin θ cos θ ( R 1 2 S 2 2 R 2 2 S 1 2 ) ϑ K g R 1 R 2 S 1 2 2 R 1 R 2 cos θ [ R 1 2 S 2 2 sin θ R 2 2 S 1 2 ( ϑ sin θ ) ] .
R ( s , r , y ) = R 0 ( s , r , y ) exp ( jar ) ,
S ( s , r , y ) = S 0 ( s , r , y ) exp ( jar ) .
2 S ( r , s , y ) s r + κ 0 2 S ( r , s , y ) = 0 .
s = f ( r ) = sin ( θ θ 0 ) ϑ cos θ 0 sin ( θ + θ 0 ) r ,
0 r W R sin θ 2 sin θ 0 + W R cos θ 2 cos θ 0 ,
W R cos θ 2 cos θ 0 s W R ( sin θ ϑ ) 2 sin θ 0 ;
s = m ( r ) = sin ( θ + θ 0 ) ϑ cos θ 0 sin ( θ θ 0 ) r ,
W S cos θ 2 cos θ 0 r W S sin θ 2 sin θ 0 ,
0 s W S ( sin θ ϑ ) 2 sin θ 0 + W S cos θ 2 cos θ 0 ;
s = f ( r ) = S 2 2 cos θ sin α 1 S 1 2 ( ϑ sin θ ) cos α 1 R 1 2 sin θ cos α 1 R 2 2 cos θ sin α 1 r ,
W R R 1 2 sin θ cos α 1 sin 2 θ 0 r W R R 2 2 cos θ sin α 1 sin 2 θ 0 ,
W R S 1 2 ( ϑ sin θ ) cos α 1 sin 2 θ 0 W R S 2 2 cos θ sin α 1 sin 2 θ 0 s 0 ;
s = m ( r ) = S 2 2 cos θ sin α 2 + S 1 2 ( ϑ sin θ ) cos α 2 R 1 2 sin θ cos α 2 + R 2 2 cos θ sin α 2 r ,
W S R 1 2 sin θ cos α 2 sin 2 θ 0 W S R 2 2 cos θ sin α 2 sin 2 θ 0 r 0 ,
W S S 1 2 ( ϑ sin θ ) cos α 2 sin 2 θ 0 s W S S 2 2 cos θ sin α 2 sin 2 θ 0 .
S ( r , f ( r ) , y ) = 0 ,
S ( r , f ( r ) , y ) r = j κ 0 R 0 ( r , f ( r ) , y ) exp ( jar ) ,
S ( r , m ( r ) , y ) = 0 ,
S ( r , m ( r ) , y ) r = 0 .
r = g ( u , v ) = u sin ( θ + θ 0 ) + v sin ( θ θ 0 ) sin 2 θ 0 ,
s = h ( u , v ) = u [ sin ( θ θ 0 ) ϑ cos θ 0 ] + v [ sin ( θ + θ 0 ) ϑ cos θ 0 ] sin 2 θ 0 ;
r = g ( u , v ) = [ R 2 2 cos θ sin α 1 R 1 2 sin θ cos α 1 ] u [ R 2 2 cos θ sin α 2 + R 1 2 sin θ cos α 2 ] v sin ( α 1 + α 2 ) ,
s = h ( u , v ) = [ S 1 2 ( ϑ sin θ ) cos α 1 S 2 2 cos θ sin α 1 ] u + [ S 1 2 ( ϑ sin θ ) cos α 2 + S 2 2 cos θ sin α 2 ] v sin ( α 1 + α 2 ) .
S 0 ( W R , v , y ) = j κ 0 exp [ jag ( W R , v ) ] 0 g ( W R , v ) R 0 ( r , f ( r ) , y ) exp ( j a r ) J 0 ( 2 κ 0 [ r g ( W R , v ) ] [ f ( r ) h ( W R , v ) ] ) d r ;
S 0 ( W R , v , y ) = j κ 0 exp [ jag ( W R , v ) ] r A 0 R 0 ( r , f ( r ) , y ) exp ( j a r ) J 0 ( 2 κ 0 ( r g ( W R , v ) ) [ f ( r ) h ( W R , v ) ] ) d r ,
R 2 2 cos θ sin α 1 R 1 2 sin θ cos α 1 R 2 2 cos θ sin α 2 + R 1 2 sin θ cos α 2 W R < v < S 1 2 ( sin θ ϑ ) cos α 1 + S 2 2 cos θ sin α 1 S 1 2 ( ϑ sin θ ) cos α 2 + S 2 2 cos θ sin α 2 W R ,
S 0 ( W R , v , y ) = j κ 0 exp [ jag ( W R , v ) ] f 1 ( h ( W R , v ) ) 0 R 0 ( r , f ( r ) , y ) exp ( j a r ) J 0 ( 2 κ 0 [ r g ( W R , v ) ] [ f ( r ) h ( W R , v ) ] ) d r ,
v < R 2 2 cos θ sin α 1 R 1 2 sin θ cos α 1 R 2 2 cos θ sin α 2 + R 1 2 sin θ cos α 2 W R ,
S 0 ( W R , v , y ) = j κ 0 exp [ jag ( W R , v ) ] f 1 ( h ( W R , v ) ) g ( W R , v ) R 0 ( r , f ( r ) , y ) exp ( j a r ) J 0 ( 2 κ 0 [ r g ( W R , v ) ] [ f ( r ) h ( W R , v ) ] ) d r .
R 0 ( x , y , z ) = exp [ ( x sin θ + z cos θ + W R 2 ) 2 + y 2 w r r 2 ]
η = P d P i = 2 P d α π w r r 2 ,

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