Abstract

Propagation of a dark hollow beam (DHB) of circular, elliptical or rectangular symmetry in a turbulent atmosphere is investigated. Analytical formulas for the average intensity of various DHBs propagating in a turbulent atmosphere are derived in a tensor form based on the extended Huygens-Fresnel integral. The intensity and spreading properties of the DHBs in a turbulent atmosphere are studied numerically. It is found that after a long propagation distance a dark hollow beam of circular or non-circular eventually becomes a circular Gaussian beam (without dark hollow) in a turbulent atmosphere, which is much different from its propagation properties in free space. The conversion from a DHB to a circular Gaussian beam becomes quicker and the beam spot in the far field spreads more rapidly for a larger structure constant, a shorter wavelength, a lower beam order and a smaller waist size of the initial beam.

© 2006 Optical Society of America

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References

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2005

2004

2003

2002

C. Y. Young, Y. V. Gilchrest, and B. R. Macon, "Turbulence-induced beam spreading of higher-order mode optical waves," Opt. Eng. 41, 1097-1103 (2002).
[CrossRef]

K. Zhu, H. Tang, X. Sun, X. Wang, and T. Liu, ‘‘Flattened multi-Gaussian light beams with an axial shadow generated through superposing Gaussian beams, ’’Opt. Commun. 207, 29-34 (2002).
[CrossRef]

Q. Lin and Y. Cai, "Tensor ABCD law for partially coherent twisted anisotropic Gaussian Schell-model beams," Opt. Lett. 27, 216-218 (2002).
[CrossRef]

2000

J. Arlt and K. Dholakia, ‘‘Generation of high-order Bessel beams by use of an axicon,’’Opt. Commun. 177, 297-301 (2000).
[CrossRef]

1997

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, ‘‘Novel optical trap of atoms with a doughnut beam,’’Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

1991

J. Alda, S. Wang, and E. Bernabeu, "Analytical expression for the complex radius of curvature tensor Q for generalized Gaussian beams," Opt. Commun. 80, 350-352 (1991).
[CrossRef]

1990

Q. Lin, S. Wang, J. Alda, and E. Bernabeu, "Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law," Optik 85, 67-72 (1990).

1980

1979

1978

1970

J. A. Arnaud, "Nonorthogonal optical waveguides and resonators," Bell Syst. Tech. J. 49, 2311-2348 (1970).

1967

Z. I. Feizulin and Y. A. Kravtsov, "Broadening of a laser beam in a turbulent medium," Radiophys. Quantum Electron. 10, 33-35 (1967).
[CrossRef]

Alda, J.

J. Alda, S. Wang, and E. Bernabeu, "Analytical expression for the complex radius of curvature tensor Q for generalized Gaussian beams," Opt. Commun. 80, 350-352 (1991).
[CrossRef]

Q. Lin, S. Wang, J. Alda, and E. Bernabeu, "Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law," Optik 85, 67-72 (1990).

Arlt, J.

J. Arlt and K. Dholakia, ‘‘Generation of high-order Bessel beams by use of an axicon,’’Opt. Commun. 177, 297-301 (2000).
[CrossRef]

Arnaud, J. A.

J. A. Arnaud, "Nonorthogonal optical waveguides and resonators," Bell Syst. Tech. J. 49, 2311-2348 (1970).

Baykal, Y.

Bernabeu, E.

J. Alda, S. Wang, and E. Bernabeu, "Analytical expression for the complex radius of curvature tensor Q for generalized Gaussian beams," Opt. Commun. 80, 350-352 (1991).
[CrossRef]

Q. Lin, S. Wang, J. Alda, and E. Bernabeu, "Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law," Optik 85, 67-72 (1990).

Cai, Y.

Carter, W. H.

Davis, C. C.

C. C. Davis, I. I. Smolyaninov, and S. D. Milner, "Flexible optical wireless link and networks," IEEE Commun. Mag. 41, 51-57 (2003).
[CrossRef]

Deng, D.

Dholakia, K.

J. Arlt and K. Dholakia, ‘‘Generation of high-order Bessel beams by use of an axicon,’’Opt. Commun. 177, 297-301 (2000).
[CrossRef]

Dogariu, A.

Dolezal, F.

H. Izadpanah, T. Elbatt, V. Kukshya, F. Dolezal, and B. K. Ryu, "High-availability free space optical and RF hybrid wireless networks," IEEE Wireless Commun. 10, 45-55 (2003).
[CrossRef]

Elbatt, T.

H. Izadpanah, T. Elbatt, V. Kukshya, F. Dolezal, and B. K. Ryu, "High-availability free space optical and RF hybrid wireless networks," IEEE Wireless Commun. 10, 45-55 (2003).
[CrossRef]

Eyyuboðlu, H. T.

Fan, Z.

Feizulin, Z. I.

Z. I. Feizulin and Y. A. Kravtsov, "Broadening of a laser beam in a turbulent medium," Radiophys. Quantum Electron. 10, 33-35 (1967).
[CrossRef]

Fu, X.

Gan, X.

Ganic, D.

Gilchrest, Y. V.

C. Y. Young, Y. V. Gilchrest, and B. R. Macon, "Turbulence-induced beam spreading of higher-order mode optical waves," Opt. Eng. 41, 1097-1103 (2002).
[CrossRef]

Gu, M.

Hirano, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, ‘‘Novel optical trap of atoms with a doughnut beam,’’Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

Izadpanah, H.

H. Izadpanah, T. Elbatt, V. Kukshya, F. Dolezal, and B. K. Ryu, "High-availability free space optical and RF hybrid wireless networks," IEEE Wireless Commun. 10, 45-55 (2003).
[CrossRef]

Kravtsov, Y. A.

Z. I. Feizulin and Y. A. Kravtsov, "Broadening of a laser beam in a turbulent medium," Radiophys. Quantum Electron. 10, 33-35 (1967).
[CrossRef]

Kuga, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, ‘‘Novel optical trap of atoms with a doughnut beam,’’Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

Kukshya, V.

H. Izadpanah, T. Elbatt, V. Kukshya, F. Dolezal, and B. K. Ryu, "High-availability free space optical and RF hybrid wireless networks," IEEE Wireless Commun. 10, 45-55 (2003).
[CrossRef]

Leader, J. C.

Lin, Q.

Liu, T.

K. Zhu, H. Tang, X. Sun, X. Wang, and T. Liu, ‘‘Flattened multi-Gaussian light beams with an axial shadow generated through superposing Gaussian beams, ’’Opt. Commun. 207, 29-34 (2002).
[CrossRef]

Lu, X.

Macon, B. R.

C. Y. Young, Y. V. Gilchrest, and B. R. Macon, "Turbulence-induced beam spreading of higher-order mode optical waves," Opt. Eng. 41, 1097-1103 (2002).
[CrossRef]

Mei, Z.

Milner, S. D.

C. C. Davis, I. I. Smolyaninov, and S. D. Milner, "Flexible optical wireless link and networks," IEEE Commun. Mag. 41, 51-57 (2003).
[CrossRef]

Plonus, M. A.

Ryu, B. K.

H. Izadpanah, T. Elbatt, V. Kukshya, F. Dolezal, and B. K. Ryu, "High-availability free space optical and RF hybrid wireless networks," IEEE Wireless Commun. 10, 45-55 (2003).
[CrossRef]

Sasada, H.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, ‘‘Novel optical trap of atoms with a doughnut beam,’’Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

Shao, J.

Shimizu, Y.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, ‘‘Novel optical trap of atoms with a doughnut beam,’’Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

Shiokawa, N.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, ‘‘Novel optical trap of atoms with a doughnut beam,’’Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

Shirai, T.

Smolyaninov, I. I.

C. C. Davis, I. I. Smolyaninov, and S. D. Milner, "Flexible optical wireless link and networks," IEEE Commun. Mag. 41, 51-57 (2003).
[CrossRef]

Sun, X.

K. Zhu, H. Tang, X. Sun, X. Wang, and T. Liu, ‘‘Flattened multi-Gaussian light beams with an axial shadow generated through superposing Gaussian beams, ’’Opt. Commun. 207, 29-34 (2002).
[CrossRef]

Tang, H.

K. Zhu, H. Tang, X. Sun, X. Wang, and T. Liu, ‘‘Flattened multi-Gaussian light beams with an axial shadow generated through superposing Gaussian beams, ’’Opt. Commun. 207, 29-34 (2002).
[CrossRef]

Torii, Y.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, ‘‘Novel optical trap of atoms with a doughnut beam,’’Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

Wang, S.

J. Alda, S. Wang, and E. Bernabeu, "Analytical expression for the complex radius of curvature tensor Q for generalized Gaussian beams," Opt. Commun. 80, 350-352 (1991).
[CrossRef]

Q. Lin, S. Wang, J. Alda, and E. Bernabeu, "Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law," Optik 85, 67-72 (1990).

Wang, S. C. H.

Wang, X.

K. Zhu, H. Tang, X. Sun, X. Wang, and T. Liu, ‘‘Flattened multi-Gaussian light beams with an axial shadow generated through superposing Gaussian beams, ’’Opt. Commun. 207, 29-34 (2002).
[CrossRef]

Wei, C.

Wolf, E.

Young, C. Y.

C. Y. Young, Y. V. Gilchrest, and B. R. Macon, "Turbulence-induced beam spreading of higher-order mode optical waves," Opt. Eng. 41, 1097-1103 (2002).
[CrossRef]

Zhao, D.

Zhu, K.

K. Zhu, H. Tang, X. Sun, X. Wang, and T. Liu, ‘‘Flattened multi-Gaussian light beams with an axial shadow generated through superposing Gaussian beams, ’’Opt. Commun. 207, 29-34 (2002).
[CrossRef]

Appl. Opt.

Bell Syst. Tech. J.

J. A. Arnaud, "Nonorthogonal optical waveguides and resonators," Bell Syst. Tech. J. 49, 2311-2348 (1970).

IEEE Commun. Mag.

C. C. Davis, I. I. Smolyaninov, and S. D. Milner, "Flexible optical wireless link and networks," IEEE Commun. Mag. 41, 51-57 (2003).
[CrossRef]

IEEE Wireless Commun.

H. Izadpanah, T. Elbatt, V. Kukshya, F. Dolezal, and B. K. Ryu, "High-availability free space optical and RF hybrid wireless networks," IEEE Wireless Commun. 10, 45-55 (2003).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

K. Zhu, H. Tang, X. Sun, X. Wang, and T. Liu, ‘‘Flattened multi-Gaussian light beams with an axial shadow generated through superposing Gaussian beams, ’’Opt. Commun. 207, 29-34 (2002).
[CrossRef]

J. Alda, S. Wang, and E. Bernabeu, "Analytical expression for the complex radius of curvature tensor Q for generalized Gaussian beams," Opt. Commun. 80, 350-352 (1991).
[CrossRef]

H. T. Eyyuboðlu, "Propagation of Hermite-cosh-Gaussian laser beams in turbulent atmosphere," Opt. Commun. 245, 37-47 (2005).
[CrossRef]

J. Arlt and K. Dholakia, ‘‘Generation of high-order Bessel beams by use of an axicon,’’Opt. Commun. 177, 297-301 (2000).
[CrossRef]

Opt. Eng.

C. Y. Young, Y. V. Gilchrest, and B. R. Macon, "Turbulence-induced beam spreading of higher-order mode optical waves," Opt. Eng. 41, 1097-1103 (2002).
[CrossRef]

Opt. Express

Opt. Lett.

Optik

Q. Lin, S. Wang, J. Alda, and E. Bernabeu, "Transformation of non-symmetric Gaussian beam into symmetric one by means of tensor ABCD law," Optik 85, 67-72 (1990).

Phys. Rev. Lett.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, ‘‘Novel optical trap of atoms with a doughnut beam,’’Phys. Rev. Lett. 78, 4713-4716 (1997).
[CrossRef]

Radiophys. Quantum Electron.

Z. I. Feizulin and Y. A. Kravtsov, "Broadening of a laser beam in a turbulent medium," Radiophys. Quantum Electron. 10, 33-35 (1967).
[CrossRef]

Other

J. Yin, W. Gao, and Y. Zhu, ‘‘Generation of dark hollow beams and their applications,’’ in Progress in Optics, E. Wolf, ed., (North-Holland, Amsterdam, 2003), Vol. 44, pp. 119-204.
[CrossRef]

J. A. Arnaud, Hamiltonian theory of beam mode propagation, in Progress in Optics, Vol XI, E. Wolf, ed., (North-Holland, Amsterdam, 1973), pp. 247-304.
[CrossRef]

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Figures (9)

Fig. 1.
Fig. 1.

Contour plots of the normalized intensity distribution of a circular DHB for two different N values with p=0.9 and w 0 =2cm. (a) N=3; (b) N=15.

Fig. 2.
Fig. 2.

Contour plots of the normalized intensity distribution of a rectangular DHB for two different sets of (H, N). (a) H=N=5 (b) H=N =15.

Fig. 3.
Fig. 3.

Contour plots of the normalized intensity distributions of an elliptical DHB for two different sets of (w 0x , w 0y , w 0xy ) with N=10 and p=0.9. (a) w 0x = 1cm , w 0y = 2cm, w 0xy = 2cm; (b)w 0x = 2cm, w 0y = 1cm, w 0xy =2cm.

Fig. 4.
Fig. 4.

Cross line (y=0) of the normalized average intensity of a circular DHB at several different propagation distances in a turbulent atmosphere (for two different values of structure constant Cn2) (a) z=0; (b) z=0.5km; (c) z=1.1km; (d) z=1.5km; (e) z=5km; (f) z=10km.

Fig. 5.
Fig. 5.

Cross line (y=0) of the normalized average intensity of a circular DHB for two different λ values at several propagation distances in a turbulent atmosphere.

Fig. 6.
Fig. 6.

Cross line (y=0) of the normalized average intensity of a circular DHB for two different w 0 values at several propagation distances in a turbulent atmosphere.

Fig. 7.
Fig. 7.

Cross line (y=0) of the normalized average intensity of a circular DHB for two different N values at several propagation distances in a turbulent atmosphere.

Fig. 8.
Fig. 8.

Normalized average intensity of an elliptical DHB at several different propagation distances in a turbulent atmosphere. (a) z=0; (b) z=1.1km; (c) z=10km. (d) z=10km (free space).

Fig. 9.
Fig. 9.

Normalized average intensity of a rectangular DHB at z=10km (a) in a turbulent atmosphere with Cn2 = 10-14 m -2/3; (b) in free space

Tables (3)

Tables Icon

Table 1. Difference between the effective beam sizes of a circular DHB at the source plane and output plane (z=10km) for different values of w 0, λ, N, Cn2

Tables Icon

Table 2. Difference between the effective beam sizes of an elliptical DHB at the source plane and output plane (z=10km) in a turbulent atmosphere with different values of w 0x , w 0y , λ, N, Cn2

Tables Icon

Table 3. Difference between the effective beam sizes of a rectangular DHB at the source plane and output plane (z=10km) in a turbulent atmosphere with different values of w 0x , w 0y , λ, N, M, Cn2

Equations (52)

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E N x y 0 = n = 1 N ( 1 ) n 1 N N n [ exp ( n x 2 + n y 2 w 0 2 ) exp ( n x 2 + n y 2 p w 0 2 ) ] ,
E HN x y 0 = h = 1 H n = 1 N ( 1 ) h + n HN H h N n [ exp ( h x 2 w 0 x 2 n y 2 w 0 y 2 ) exp ( h x 2 p w 0 x 2 n y 2 p w 0 y 2 ) ] ,
E N x y 0 = n = 1 N ( 1 ) n 1 N N n [ exp ( n x 2 w 0 x 2 n y 2 w 0 y 2 2 nxy w 0 xy 2 ) exp ( n x 2 p w 0 x 2 n y 2 p w 0 y 2 2 nxy p w 0 xy 2 ) ]
E N r 0 = n = 1 N ( 1 ) n 1 N N n [ exp ( ik 2 r T Q 1 n 1 r ) exp ( ik 2 r T Q 1 np 1 r ) ] ,
Q 1 n 1 = 2 n ik w 0 2 I , Q 1 np 1 = 2 n ipk w 0 2 I ,
Q 1 n 1 = ( 2 n ik w 0 x 2 2 n ik w 0 xy 2 2 n ik w 0 xy 2 2 n ik w 0 y 2 ) , Q 1 np 1 = ( 2 n ipk w 0 x 2 2 n ikp w 0 xy 2 2 n ikp w 0 xy 2 2 n ik pw 0 y 2 ) .
E HN ( x , y , 0 ) = h = 1 H n = 1 N ( 1 ) h + n HN H h N n [ exp ( ik 2 r T Q 1 hn 1 r ) exp ( ik 2 r T Q 1 hnp 1 r ) ] ,
Q 1 hn 1 = ( 2 h ik w 0 x 2 0 0 2 n ik w 0 y 2 ) , Q 1 hnp 1 = ( 2 h ikp w 0 x 2 0 0 2 n ik pw 0 y 2 ) .
E ρ z t = ik 2 πz exp ( ikz ) E r 1 0 exp [ ik 2 z ( r 1 ρ ) 2 + Ψ ( r 1 , ρ ) i 2 πft ] d r 1 ,
I ρ z = k 2 4 π 2 z 2 E r 1 0 E * r 2 0 exp [ ik 2 z ( r 1 ρ ) 2 + ik 2 z ( r 2 ρ ) 2 ]
× exp [ Ψ r 1 ρ + Ψ * r 2 ρ ] d r 1 d r 2 .
exp [ Ψ r 1 ρ + Ψ * r 2 ρ ] = exp [ 0.5 D Ψ ( r 1 r 2 ) ] = exp [ 1 ρ 0 2 ( r 1 r 2 ) 2 ]
I ρ z = k 2 4 π 2 [ det ( B ˜ ) ] 1 2 E r 1 0 E * r 2 0
× exp [ ik 2 ( r ˜ T B ˜ 1 r ˜ 2 r ˜ B ˜ 1 ρ ˜ + ρ ˜ T B ˜ 1 ρ ˜ ) ] exp [ ik 2 r ˜ T P ˜ r ˜ ] d r ˜ ,
B ˜ = ( z I 0 0 z I ) , P ˜ = 2 ik ρ 0 2 ( I I I I ) .
E N r 1 0 E N * r 2 0 = n = 1 N m = 1 N ( 1 ) n + m N 2 N n N m [ exp ( ik 2 r ˜ T Q ˜ 1 nm 1 r ˜ ) exp ( ik 2 r ˜ T Q ˜ 2 nm 1 r ˜ )
exp ( ik 2 r ˜ T Q ˜ 3 nm 1 r ˜ ) + exp ( ik 2 r ˜ T Q ˜ 4 nm 1 r ˜ ) ] ,
Q ˜ 1 nm 1 = ( Q 1 n 1 0 0 ( Q 1 m 1 ) * ) , Q ˜ 2 nm 1 = ( Q 1 n 1 0 0 ( Q 1 mp 1 ) * ) ,
Q ˜ 3 nm 1 = ( Q 1 np 1 0 0 ( Q 1 m 1 ) * ) , Q ˜ 4 nm 1 = ( Q 1 np 1 0 0 ( Q 1 mp 1 ) * ) .
I ρ z = n = 1 N m = 1 N ( 1 ) n + m N 2 N n N m [ det [ S 1 ] 1 2 exp ( ik 2 ρ ˜ T Q ˜ o 1 nm 1 1 ρ ˜
det [ S 2 ] 1 2 exp ( ik 2 ρ ˜ T Q ˜ o 2 nm 1 1 ρ ˜ ) det [ S 3 ] 1 2 exp ( ik 2 ρ ˜ T Q ˜ o 3 nm 1 1 ρ ˜ )
+ det [ S 4 ] 1 2 exp ( ik 2 ρ ˜ T Q ˜ o 4 nm 1 1 ρ ˜ ) ] ,
S i = I ˜ + B ˜ ( Q ˜ inm 1 + P ˜ ) , Q ˜ oinm 1 = [ ( Q ˜ inm 1 + P ˜ ) + B ˜ ] 1 .
E HN r 1 0 E HN * r 2 0 = h = 1 H l = 1 H n = 1 N m = 1 N ( 1 ) h + l + n + m N 2 H 2 H h H l N n N m [ exp ( ik 2 r ˜ T Q 1 hnlm 1 r ˜ )
exp ( ik 2 r 1 T Q 2 hnlm 1 r 1 ) exp ( ik 2 r ˜ T Q 3 hnlm 1 r ˜ ) + exp ( ik 2 r ˜ T Q ˜ 4 hnlm 1 r ˜ ) ] ,
Q ˜ 1 hnlm 1 = ( Q 1 hn 1 0 0 ( Q 1 lm 1 ) * ) , Q ˜ 2 hnlm 1 = ( Q 1 hn 1 0 0 ( Q 1 lmp 1 ) * ) ,
Q ˜ 3 hnlm 1 = ( Q 1 hnp 1 0 0 ( Q 1 lm 1 ) * ) , Q ˜ 4 hnlm 1 = ( Q 1 hnp 1 0 0 ( Q 1 lmp 1 ) * ) .
I ρ z = h = 1 H l = 1 H n = 1 N m = 1 N ( 1 ) h + l + n + m N 2 H h H l N n N m [ det [ S 1 ] 1 2 exp ( ik 2 ρ ˜ T Q ˜ o 1 hnlm 1 1 ρ ˜ )
det [ S 2 ] 1 2 exp ( ik 2 ρ ˜ T Q o 2 hnlm 1 1 ρ ˜ ) det [ S 3 ] 1 2 exp ( ik 2 ρ ˜ T Q o 3 hnlm 1 1 ρ ˜ )
+ det [ S 4 ] 1 2 exp ( ik 2 ρ ˜ T Q ˜ o 4 hnlm 1 1 ρ ˜ ) ] ,
S i = I ˜ + B ˜ ( Q ˜ ihnlm 1 + P ˜ ) , Q ˜ o ihnlm 1 = [ ( Q ˜ ihnlm 1 + P ˜ ) 1 + B ˜ ] 1 , ( i = 1,2,3,4 ) .
I ρ z = k 2 ρ 0 2 w 0 4 k 2 ρ 0 2 w 0 4 + 4 ( ρ 0 2 + 2 w 0 2 ) z 2 exp [ 2 k 2 ρ 0 2 w 0 2 ( ρ x 2 + ρ y 2 ) k 2 ρ 0 2 w 0 4 + 4 ( ρ 0 2 + 2 w 0 2 ) z 2 ] .
W s ( z ) = 2 s 2 I x y z dxdy I x y z dxdy , .
Δ W s ( z ) = W s ( z ) W s ( 0 ) .
I ρ z = k 2 4 π 2 [ det ( B ˜ ) ] 1 2 n = 1 N m = 1 N ( 1 ) n + m N 2 N n N m [ exp ( ik 2 r ˜ T Q ˜ 1 nm 1 r ˜ ) exp ( ik 2 r ˜ T Q ˜ 2 nm 1 r ˜ )
exp ( ik 2 r ˜ T Q ˜ 3 nm 1 r ˜ ) + exp ( ik 2 r ˜ T Q ˜ 4 nm 1 r ˜ ) ] exp [ ik 2 ( r ˜ T B ˜ 1 r ˜ 2 r ˜ B ˜ 1 ρ ˜ + ρ ˜ T B ˜ 1 ρ ˜ ) ] exp [ ik 2 r ˜ T P ˜ r ˜ ] d r ˜
= k 2 4 π 2 [ det ( B ˜ ) ] 1 2 n = 1 N m = 1 N ( 1 ) n + m N 2 N n N m { exp [ ik 2 ρ ˜ T B ˜ 1 ρ ˜ + ik 2 ρ ˜ T B ˜ 1 T ( Q ˜ 1 nm 1 + B ˜ 1 + P ˜ ) B ˜ 1 ρ ˜ ]
× exp [ ik 2 ( Q ˜ 1 nm 1 + B ˜ 1 + P ˜ ) 1 2 r ˜ ( Q ˜ 1 nm 1 + B ˜ 1 + P ˜ ) 1 2 B ˜ 1 ρ ˜ 2 ] d r ˜ exp [ ik 2 ρ ˜ T B ˜ 1 ρ ˜ + ik 2 ρ ˜ T B ˜ 1 T ( Q ˜ 2 nm 1 + B ˜ 1 + P ˜ ) 1 B ˜ 1 ρ ˜ ]
× exp [ ik 2 ( Q ˜ 2 nm 1 + B ˜ 1 + P ˜ ) 1 2 r ˜ ( Q ˜ 2 nm 1 + B ˜ 1 + P ˜ ) 1 2 B ˜ 1 ρ ˜ 2 ] d r ˜ exp [ ik 2 ρ ˜ T B ˜ 1 ρ ˜ + ik 2 ρ ˜ T B ˜ 1 T ( Q ˜ 3 nm 1 + B ˜ 1 + P ˜ ) 1 B ˜ 1 ρ ˜ ]
× exp [ ik 2 ( Q ˜ 3 nm 1 + B ˜ 1 + P ˜ ) 1 2 r ˜ ( Q ˜ 3 nm 1 + B ˜ 1 + P ˜ ) 1 2 B ˜ 1 ρ ˜ 2 ] d r ˜ + exp [ ik 2 ρ ˜ T B ˜ 1 ρ ˜ + ik 2 ρ ˜ T B ˜ 1 T ( Q ˜ 4 nm 1 + B ˜ 1 + P ˜ ) 1 B ˜ 1 ρ ˜ ]
× exp [ ik 2 ( Q ˜ 4 nm 1 + B ˜ 1 + P ˜ ) 1 2 r ˜ ( Q ˜ 4 nm 1 + B ˜ 1 + P ˜ ) 1 2 B ˜ 1 ρ ˜ 2 ] d r ˜ } .
exp ( ax 2 ) d x = π a ,
I ρ z = [ det ( B ˜ ) ] 1 2 n = 1 N m = 1 N ( 1 ) n + m N 2 N n N m
{ [ det ( Q ˜ 1 nm 1 + B ˜ 1 + P ˜ ) ] 1 2 exp [ ik 2 ρ ˜ T B ˜ 1 ρ ˜ + ik 2 ρ ˜ T B ˜ 1 T ( Q ˜ 1 nm 1 + B ˜ 1 + P ˜ ) 1 B ˜ 1 ρ ˜ ]
[ det ( Q ˜ 2 nm 1 + B ˜ 1 + P ˜ ) ] 1 2 exp [ ik 2 ρ ˜ T B ˜ 1 ρ ˜ + ik 2 ρ ˜ T B ˜ 1 T ( Q ˜ 2 nm 1 + B ˜ 1 + P ˜ ) 1 B ˜ 1 ρ ˜ ]
[ det ( Q ˜ 3 nm 1 + B ˜ 1 + P ˜ ) ] 1 2 exp [ ik 2 ρ ˜ T B ˜ 1 ρ ˜ + ik 2 ρ ˜ T B ˜ 1 T ( Q ˜ 3 nm 1 + B ˜ 1 + P ˜ ) 1 B ˜ 1 ρ ˜ ]
+ [ det ( Q ˜ 4 nm 1 + B ˜ 1 + P ˜ ) ] 1 2 exp [ ik 2 ρ ˜ T B ˜ 1 ρ ˜ + ik 2 ρ ˜ T B ˜ 1 T ( Q ˜ 4 nm 1 + B ˜ 1 + P ˜ ) 1 B ˜ 1 ρ ˜ ] } .
[ det ( B ˜ ) ] 1 2 [ det ( Q ˜ 1 nm 1 + B ˜ 1 + P ˜ ) ] 1 2 = [ det ( I ˜ + B ˜ ( Q ˜ 1 nm 1 + P ˜ ) ) ] 1 2 , ( i = 1,2,3,4 )
B ˜ 1 B ˜ 1 T ( Q ˜ inm 1 + B ˜ 1 + P ˜ ) 1 B ˜ 1 = B ˜ 1 B ˜ 1 T ( B ˜ Q ˜ inm 1 + I ˜ + B ˜ P ˜ ) 1
= [ B ˜ 1 ( B ˜ Q ˜ inm 1 + I ˜ + B ˜ P ˜ ) B ˜ 1 T ] ( B ˜ Q ˜ inm 1 + I ˜ + B ˜ P ˜ ) 1
= [ ( Q ˜ inm 1 + P ˜ ) 1 ] 1 [ I ˜ + B ˜ ( Q ˜ inm 1 + P ˜ ) ] 1 = [ ( Q ˜ inm 1 + P ˜ ) 1 + B ˜ ] 1 , ( i = 1,2,3,4 )
S i = I ˜ + B ˜ ( Q ˜ inm 1 + P ˜ ) , Q ˜ oinm 1 = [ ( Q ˜ inm 1 + P ˜ ) 1 + B ˜ ] 1 .

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