Abstract

By expanding the hard aperture function into a finite sum of complex Gaussian functions, approximate analytical formulas for a decentered Gaussian beam (DEGB) passing through apertured aligned and misaligned paraxial apertured paraxial optical systems are derived in terms of a tensor method. The results obtained by using the approximate analytical expression are in good agreement with those obtained by using the numerical integral calculation. Furthermore, approximate analytical formulas for a decentered elliptical Hermite–Gaussian beam (DEHGB) through apertured paraxial optical systems are derived. As an application example, approximate analytical formulas for a decentered elliptical flattened Gaussian beam through apertured paraxial optical systems are derived. Our results provide a convenient way for studying the propagation and transformation of a DEGB and a DEHGB through apertured paraxial optical systems.

© 2006 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  5. M. L. Luis, M. Y. Omel, and J. J. D. Joris, "Incoherent superposition of off-axis polychromatic Hermite-Gaussian modes," J. Opt. Soc. Am. A. 19, 1572-1582 (2002).
    [CrossRef]
  6. P. J. Cronin, P. Török, P. Varga, and C. Cogswell, "High-aperture diffraction of a scalar, off-axis Gaussian beam," J. Opt. Soc. Am. A. 17, 1556-1564 (2000).
    [CrossRef]
  7. Y. Cai and Q. Lin, "Decentered elliptical Gaussian beam," Appl. Opt. 41, 4336-4340 (2002).
    [CrossRef] [PubMed]
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    [CrossRef]
  9. J. J. Wen and M. A. Breazeale, "A diffraction beam field expressed as the superposition of Gaussian beams," J. Acoust. Soc. Am. 83, 1752-1756 (1988).
    [CrossRef]
  10. D. Ding and X. Liu, "Approximate description for Bessel, Bessel-Gauss, and Gaussian beams with finite aperture," J. Opt. Soc. Am. A , 16, 1286-1293 (1999).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  17. S. Wang and L. Ronchi, "Principles and design of optical array," in Progress in Optics, Vol. 25, E.Wolf, ed. (Elsevier Science, 1988), p. 279.
  18. Y. Cai and Q. Lin, "The elliptical Hermite-Gaussian beam and its propagation through paraxial systems," Opt. Commun. 207, 139-147 (2002).
  19. Y. Cai and Q. Lin, "Light beams with elliptical flat-topped profiles," J. Opt. A 6, 390-395 (2004).
    [CrossRef]

2004

Y. Cai and Q. Lin, "Light beams with elliptical flat-topped profiles," J. Opt. A 6, 390-395 (2004).
[CrossRef]

2003

Y. Cai and Q. Lin, "Decentered elliptical Hermite-Gausian beam," J. Opt. Soc. Am. A. 20, 1111-1119 (2003).
[CrossRef]

2002

M. L. Luis, M. Y. Omel, and J. J. D. Joris, "Incoherent superposition of off-axis polychromatic Hermite-Gaussian modes," J. Opt. Soc. Am. A. 19, 1572-1582 (2002).
[CrossRef]

Y. Cai and Q. Lin, "Propagation of elliptical Gaussian beam through misaligned optical systems in spatial domain and spatialfrequency domain," Opt. Laser Technol. 34, 415-421 (2002).
[CrossRef]

Y. Cai and Q. Lin, "The elliptical Hermite-Gaussian beam and its propagation through paraxial systems," Opt. Commun. 207, 139-147 (2002).

Y. Cai and Q. Lin, "Decentered elliptical Gaussian beam," Appl. Opt. 41, 4336-4340 (2002).
[CrossRef] [PubMed]

2000

P. J. Cronin, P. Török, P. Varga, and C. Cogswell, "High-aperture diffraction of a scalar, off-axis Gaussian beam," J. Opt. Soc. Am. A. 17, 1556-1564 (2000).
[CrossRef]

1999

B. Lü and H. Ma, "Coherent and incoherent combinations of off-axis Gaussian beams with rectangular symmetry," Opt. Commun. 171, 185-194 (1999).
[CrossRef]

D. Ding and X. Liu, "Approximate description for Bessel, Bessel-Gauss, and Gaussian beams with finite aperture," J. Opt. Soc. Am. A , 16, 1286-1293 (1999).
[CrossRef]

1997

1995

1994

1990

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

1988

J. J. Wen and M. A. Breazeale, "A diffraction beam field expressed as the superposition of Gaussian beams," J. Acoust. Soc. Am. 83, 1752-1756 (1988).
[CrossRef]

1973

L. W. Casperson, "Gaussian light beams in inhomogeneous media," Appl. Opt. 12, 2423-2441 (1973).
[CrossRef]

1970

Alda, J.

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

Al-Rashed, A. R.

Arnaud, J. A.

J. A. Arnaud, "Hamiltonian theory of beam mode propagation, inProgress in Optics, Vol 11, E.Wolf, ed. (North-Holland, 1973), pp. 247-304.
[CrossRef]

Bernabeu, E.

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

Breazeale, M. A.

J. J. Wen and M. A. Breazeale, "A diffraction beam field expressed as the superposition of Gaussian beams," J. Acoust. Soc. Am. 83, 1752-1756 (1988).
[CrossRef]

Cai, Y.

Y. Cai and Q. Lin, "Light beams with elliptical flat-topped profiles," J. Opt. A 6, 390-395 (2004).
[CrossRef]

Y. Cai and Q. Lin, "Decentered elliptical Hermite-Gausian beam," J. Opt. Soc. Am. A. 20, 1111-1119 (2003).
[CrossRef]

Y. Cai and Q. Lin, "Propagation of elliptical Gaussian beam through misaligned optical systems in spatial domain and spatialfrequency domain," Opt. Laser Technol. 34, 415-421 (2002).
[CrossRef]

Y. Cai and Q. Lin, "Decentered elliptical Gaussian beam," Appl. Opt. 41, 4336-4340 (2002).
[CrossRef] [PubMed]

Y. Cai and Q. Lin, "The elliptical Hermite-Gaussian beam and its propagation through paraxial systems," Opt. Commun. 207, 139-147 (2002).

Casperson, L. W.

L. W. Casperson, "Gaussian light beams in inhomogeneous media," Appl. Opt. 12, 2423-2441 (1973).
[CrossRef]

Cogswell, C.

P. J. Cronin, P. Török, P. Varga, and C. Cogswell, "High-aperture diffraction of a scalar, off-axis Gaussian beam," J. Opt. Soc. Am. A. 17, 1556-1564 (2000).
[CrossRef]

Collins, S. A.

Cronin, P. J.

P. J. Cronin, P. Török, P. Varga, and C. Cogswell, "High-aperture diffraction of a scalar, off-axis Gaussian beam," J. Opt. Soc. Am. A. 17, 1556-1564 (2000).
[CrossRef]

Ding, D.

Erdelyi, A.

A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).

Joris, J. J. D.

M. L. Luis, M. Y. Omel, and J. J. D. Joris, "Incoherent superposition of off-axis polychromatic Hermite-Gaussian modes," J. Opt. Soc. Am. A. 19, 1572-1582 (2002).
[CrossRef]

Lin, Q.

Y. Cai and Q. Lin, "Light beams with elliptical flat-topped profiles," J. Opt. A 6, 390-395 (2004).
[CrossRef]

Y. Cai and Q. Lin, "Decentered elliptical Hermite-Gausian beam," J. Opt. Soc. Am. A. 20, 1111-1119 (2003).
[CrossRef]

Y. Cai and Q. Lin, "Propagation of elliptical Gaussian beam through misaligned optical systems in spatial domain and spatialfrequency domain," Opt. Laser Technol. 34, 415-421 (2002).
[CrossRef]

Y. Cai and Q. Lin, "Decentered elliptical Gaussian beam," Appl. Opt. 41, 4336-4340 (2002).
[CrossRef] [PubMed]

Y. Cai and Q. Lin, "The elliptical Hermite-Gaussian beam and its propagation through paraxial systems," Opt. Commun. 207, 139-147 (2002).

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

Liu, X.

Lü, B.

B. Lü and H. Ma, "Coherent and incoherent combinations of off-axis Gaussian beams with rectangular symmetry," Opt. Commun. 171, 185-194 (1999).
[CrossRef]

Luis, M. L.

M. L. Luis, M. Y. Omel, and J. J. D. Joris, "Incoherent superposition of off-axis polychromatic Hermite-Gaussian modes," J. Opt. Soc. Am. A. 19, 1572-1582 (2002).
[CrossRef]

Ma, H.

B. Lü and H. Ma, "Coherent and incoherent combinations of off-axis Gaussian beams with rectangular symmetry," Opt. Commun. 171, 185-194 (1999).
[CrossRef]

Magnus, W.

A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).

Nemes, G.

Oberhettinger, F.

A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).

Omel, M. Y.

M. L. Luis, M. Y. Omel, and J. J. D. Joris, "Incoherent superposition of off-axis polychromatic Hermite-Gaussian modes," J. Opt. Soc. Am. A. 19, 1572-1582 (2002).
[CrossRef]

Palma, C.

Ronchi, L.

S. Wang and L. Ronchi, "Principles and design of optical array," in Progress in Optics, Vol. 25, E.Wolf, ed. (Elsevier Science, 1988), p. 279.

Saleh, B. E. A.

Siegman, A. E.

Török, P.

P. J. Cronin, P. Török, P. Varga, and C. Cogswell, "High-aperture diffraction of a scalar, off-axis Gaussian beam," J. Opt. Soc. Am. A. 17, 1556-1564 (2000).
[CrossRef]

Varga, P.

P. J. Cronin, P. Török, P. Varga, and C. Cogswell, "High-aperture diffraction of a scalar, off-axis Gaussian beam," J. Opt. Soc. Am. A. 17, 1556-1564 (2000).
[CrossRef]

Wang, S.

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

S. Wang and L. Ronchi, "Principles and design of optical array," in Progress in Optics, Vol. 25, E.Wolf, ed. (Elsevier Science, 1988), p. 279.

Wen, J. J.

J. J. Wen and M. A. Breazeale, "A diffraction beam field expressed as the superposition of Gaussian beams," J. Acoust. Soc. Am. 83, 1752-1756 (1988).
[CrossRef]

Appl. Opt.

J. Acoust. Soc. Am.

J. J. Wen and M. A. Breazeale, "A diffraction beam field expressed as the superposition of Gaussian beams," J. Acoust. Soc. Am. 83, 1752-1756 (1988).
[CrossRef]

J. Opt. A

Y. Cai and Q. Lin, "Light beams with elliptical flat-topped profiles," J. Opt. A 6, 390-395 (2004).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. A.

Y. Cai and Q. Lin, "Decentered elliptical Hermite-Gausian beam," J. Opt. Soc. Am. A. 20, 1111-1119 (2003).
[CrossRef]

M. L. Luis, M. Y. Omel, and J. J. D. Joris, "Incoherent superposition of off-axis polychromatic Hermite-Gaussian modes," J. Opt. Soc. Am. A. 19, 1572-1582 (2002).
[CrossRef]

P. J. Cronin, P. Török, P. Varga, and C. Cogswell, "High-aperture diffraction of a scalar, off-axis Gaussian beam," J. Opt. Soc. Am. A. 17, 1556-1564 (2000).
[CrossRef]

Opt. Commun.

B. Lü and H. Ma, "Coherent and incoherent combinations of off-axis Gaussian beams with rectangular symmetry," Opt. Commun. 171, 185-194 (1999).
[CrossRef]

Y. Cai and Q. Lin, "The elliptical Hermite-Gaussian beam and its propagation through paraxial systems," Opt. Commun. 207, 139-147 (2002).

Opt. Laser Technol.

Y. Cai and Q. Lin, "Propagation of elliptical Gaussian beam through misaligned optical systems in spatial domain and spatialfrequency domain," Opt. Laser Technol. 34, 415-421 (2002).
[CrossRef]

Optik.

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

Other

A. Erdelyi, W. Magnus, and F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, 1954).

J. A. Arnaud, "Hamiltonian theory of beam mode propagation, inProgress in Optics, Vol 11, E.Wolf, ed. (North-Holland, 1973), pp. 247-304.
[CrossRef]

S. Wang and L. Ronchi, "Principles and design of optical array," in Progress in Optics, Vol. 25, E.Wolf, ed. (Elsevier Science, 1988), p. 279.

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Figures (3)

Fig. 1
Fig. 1

Normalized irradiance distribution of a DEGB and its cross irradiance profile (y = 0) at several propagation distances after passing through a circular aperture located at z = 0: (a) z = 20 mm and (b) z = 500 mm.

Fig. 2
Fig. 2

Normalized irradiance distribution of a DEGB at distance z = 100 mm after passing through a circular aperture for different values of radius located at z = 0: (a) a 1 = 0.5 mm, (b) a 1 = 1 mm, (c) a 1 = 2 mm, (d) a 1 = 10 mm.

Fig. 3
Fig. 3

Normalized irradiance distribution of an EFGB at z = 80 after passing through the circular aperture for different values of radius: (a) a 1 = 0.5 mm, (b) a 1 = 1 mm, (c) a 1 = 2 mm, (d) a 1 = 10 mm.

Equations (53)

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E ( r 1 ) = exp [ i k 2 ( r 1 r 0 ) T Q 1     1 ( r 1 r 0 ) ] ,
Q 1 - 1 = [ q 1 x x - 1 q 1 x y - 1 q 1 x y - 1 q 1 y y - 1 ] .
E 2 ( r 2 ) = i n 1 λ [ det ( B ) ] 1 / 2 exp ( i k l 0 ) E 1 ( r 1 ) H ( r 1 ) × exp ( i k l 1 ) d r 1 ,
l 1 = 1 2 [ r 1 r 2 ] T [ n 1 B 1 A n 1 B 1 n 2 ( C D B 1 A ) n 2 D B 1 ] [ r 1 r 2 ] ,
[ r 2 r 2 ] = [ A B C D ] [ r 1 r 1 ] .
H ( r 1 ) = { 1 , r 1 a 1 0 , r 1 > a 1 ,
H ( r 1 ) = m = 1 M A m exp ( B m a 1     2 r 1 2 ) ,
H ( r ) = m = 1 M A m exp ( i k 2 r T R m r ) ,
R m = 2 B m i k a 1     2 [ 1 0 0 1 ] .
E ( r 2 ) = i λ ( det [ B ] ) 1 / 2 exp ( i k l 0 ) m = 1 M A m exp [ i k 2 ( r 2     T D B 1 r 2 + r 0     T R m r 0 + r 0     T B 1 A r 0 2 r 0     T B 1 r 2 ) ] × exp [ i k 2 ( r 1 r 0 ) T ( Q 1     1 + R m + B 1 A ) ( r 1 r 0 ) ] × exp [ i k ( r 1 r 0 ) T B 1 A r 0 i k ( r 1 r 0 ) T R m r 0 + i k ( r 1 r 0 ) T B 1 r 2 ] d r 1
 = i λ ( det [ B ] ) 1 / 2 exp ( i k l 0 ) m = 1 M A m exp [ i k 2 ( r 2 T D B 1 r 2 + r 0 T R m r 0 + r 0 T B 1 A r 0 2 r 0 T B 1 r 2 ) ] × exp { i k 2 [ r 0 T ( B 1 A ) T + r 0 T R m     T r 2 T B 1 T ] ( Q 1 - 1 + B 1 A + R m ) 1 ( B 1 A r 0 + R m r 0 B 1 r 2 ) } × exp [ i k 2 | ( Q 1 - 1 + B 1 A + R m ) 1 / 2 ( r 1 r 0 ) + ( Q 1 - 1 + B 1 A + R m ) 1 / 2 ( B 1 A r 0 + R m r 0 B 1 r 2 ) | 2 ] d r 1
 =  exp ( i k l 0 ) m = 1 M A m ( det [ A + B Q 1 - 1 + B R m ] ) 1 / 2 exp [ - i k 2 r 2       T Q 2 m - 1 r 2 ] × exp [ i k 2 r 0     T ( B 1 A Q 1 + R m Q 1 + I ) 1 ( B 1 A + R m ) r 0 ] exp [ i k r 0 T ( A Q 1 + B + B R m Q 1 ) 1 r 2 ] ,
Q 2 m           1 = [ C + D ( Q 1     1 + R m ) ] [ A + B ( Q 1     1 + R m ) ] 1 .
exp ( a x 2 ) d x = π / a ,
( B 1 A ) T = B 1 A , ( B 1 ) T = ( C D B 1 A ) ,
( D B 1 ) T = D B 1 .
H ( r 1 ) = { 1 , | x 1 | a 1 , | y 1 | b 1 , 0 , | x 1 | > a 1 , | y 1 | > b 1 ,
H ( x 1 , y 1 ) = m = 1 M A m exp ( B m a 1     2 x 1 2 ) j = 1 J A j exp ( B j b 1     2 y 1 2 ) ,
H ( r 1 ) = m = 1 M j = 1 J A m A j exp ( i k 2 r 1 T R m j r 1 ) ,
R m j = [ 2 B m i k a 1     2 0 0 2 B j i k b 1     2 ] .
E ( r 2 ) = exp ( i k l 0 ) m = 1 M j = 1 J A m A j ( det [ A + B Q 1     1 + B R m j ] ) 1 / 2 exp [ i k 2 r 2     T Q 2 mj            1 r 2 ] × exp [ ik 2 r 0     T ( B 1 A Q 1 + R m j Q 1 + I ) 1 × ( B 1 A + R m ) r 0 ] exp [ i k r 0     T ( A Q 1 + B + B R m j Q 1 ) 1 r 2 ] ,
Q 2 m j             1 = [ C + D ( Q 1     1 + R m j ) ] [ A + B ( Q 1     1 + R m j ) ] 1 .
A = [ 1 0 0 1 ] , B = [ z 0 0 z ] , C = [ 0 0 0 0 ] , D = [ 1 0 0 1 ] ,
E ( r 2 ) = i k 2 π [ det ( B ¯ ) ] 1 / 2 exp ( i k l 0 ) E ( r 1 ) H ( r 1 ) × exp [ i k 2 ( r 1     T B ¯ 1 A ¯ r 1 2 r 1 T B ¯ 1 r 2 + r 2     T D ¯ B ¯ 1 r 2 ) ] × exp [ i k 2 ( r 1     T B ¯ 1 e f + r 2     T B ¯ 1 g h ) ] d r 1 ,
A ¯ = [ a 0 0 a ] , B ¯ = [ b 0 0 b ] , C ¯ = [ c 0 0 c ] , D ¯ = [ d 0 0 d ] ,
( B ¯ 1 A ¯ ) T = B ¯ 1 A ¯ , ( B ¯ 1 ) T = ( C ¯ D ¯ B ¯ 1 A ¯ ) ,
( D ¯ B ¯ 1 ) T = D ¯ B ¯ 1 .
e = 2 ( α T ε x + β T ε x ) ,
f = 2 ( α T ε y + β T ε y ) ,
g = 2 ( b γ T d α T ) ε x + 2 ( b δ T d β T ) ε x ,
h = 2 ( b γ T d α T ) ε y + 2 ( b δ T d β T ) ε y ,
α T = 1 a , β T = 1 b , γ T = c , δ T = ± 1 d .
E ( r 2 ) = exp ( i k l 0 ) m = 1 M A m × ( det [ A ¯ + B ¯ Q 1     1 + B ¯ R m ] ) 1 / 2 × exp [ i k 2 r 2     T B ¯ 1 g h i k 2 r 2     T Q 2 m          1 r 2 ] × exp [ i k 2 r 0     T ( B ¯ 1 A ¯ Q 1 + R m Q 1 + I ) 1 × ( B ¯ 1 A ¯ + R m ) r 0 + i k r 0     T × ( A ¯ Q 1 + B ¯ + B ¯ R m Q 1 ) 1 r 2 ] × exp [ i k 2 r 2     T B ¯ 1 T ( A ¯ + B ¯ Q 1 1 + B ¯ R m ) 1 e f i k 2 r 0     T ( A ¯ Q 1 + B ¯ + B ¯ R m Q 1 ) 1 e f ] × exp [ i k 8 e f     T B ¯ 1 T ( A ¯ + B ¯ Q 1 1 + B ¯ R m ) 1 e f ] ,
Q 2 m           1 = [ C ¯ + D ¯ ( Q 1     1 + R m ) ] [ A ¯ + B ¯ ( Q 1     1 + R m ) ] 1 .
E ( r 2 ) = exp ( i k l 0 ) m = 1 M j = 1 J A m A j × ( det [ A ¯ + B ¯ Q 1     1 + B ¯ R mj ] ) 1 / 2 × exp [ i k 2 r 2     T B ¯ 1 g h i k 2 r 2     T Q 2 m j             1 r 2 ] × exp [ i k 2 r 0     T ( B ¯ 1 A ¯ Q 1 + R m j Q 1 + I ) 1 × ( B ¯ 1 A ¯ + R m j ) r 0 + i k r 0     T × ( A ¯ Q 1 + B ¯ + B ¯ R m j Q 1 ) 1 r 2 ] × exp [ i k 2 r 2     T B ¯ 1 T ( A ¯ + B ¯ Q 1     1 + B ¯ R m j ) 1 e f i k 2 r 0     T ( A ¯ Q 1 + B ¯ + B ¯ R m j Q 1 ) 1 e f ] × exp [ i k 8 e f     T B ¯ 1 T ( A ¯ + B ¯ Q 1     1 + B ¯ R m j ) 1 e f ] ,
Q 2 m j             1 = [ C ¯ + D ¯ ( Q 1     1 + R m j ) ] [ A ¯ + B ¯ ( Q 1     1 + R m j ) ] 1 .
E p ( r 1 ) = exp [ i k 2 ( r 1 r 0 ) T Q e     1 ( r 1 r 0 ) ] × H p [ i k ( r 1 r 0 ) T Q h     1 ( r 1 r 0 ) ] ,
p = 0 , 1 , 2 , 3 ,
exp [ ( a 1 / 2 x a 1 / 2 b ) 2 2 ] H p ( x ) d x = 2 π a × ( 1 2 a ) p / 2 H p [ b 1 2 a ] ,
E 2 p ( r 2 ) = m = 1 M A m ( det [ A + B Q e 1       1 + B R m ] ) 1 / 2 ×     [ 1 2 / det ( B 1 A Q h 1 + Q e 1       1 Q h 1 + R m Q h 1 ) ] p / 2 exp [ i k l 0 ] × exp [ i k 2 r 2     T Q e 2 m           1 r 2 ] × exp [ i k 2 r 0     T ( B 1 A Q e 1 + R m Q e 1 + I ) 1 × ( B 1 A + R m ) r 0 ] H p { [ 1 2 / det × ( B 1 A Q h 1 + Q e 1       1 Q h 1 + R m Q h 1 ) ] 1 / 2 × i k ( r 2 r 0 ) T Q h 2 m               1 ( r 2 r 0 ) } × exp [ i k r 0     T ( A Q e 1 + B + B R m Q e 1 ) 1 r 2 ] ,
Q e 2 m             1 = [ C + D ( Q e 1       1 + R m ) ] [ A + B ( Q e 1       1 + R m ) ] 1 ,
Q h 2 m               1 = [ A + B ( Q e 1       1 + R m ) ] 1 1 T × ( A Q h 1 + B Q e 1       1 Q h 1 + B R m Q h 1 ) 1 .
E 2 p ( r 2 ) = m = 1 M j = 1 J A m A j ( det [ A + B Q e 1       1 + B R m j ] ) 1 / 2 × [ 1 2 / det ( B 1 A Q h 1 + Q e 1       1 Q h 1 + R m j Q h 1 ) ] p / 2 exp [ i k l 0 ] × exp [ i k 2 r 2     T Q e 2 m j - 1 r 2 ] × exp [ i k 2 r 0     T ( B 1 A Q e 1 + R m j Q e 1 + I ) 1 × ( B 1 A + R m j ) r 0 ] H p { [ 1 2 / det × ( B 1 A Q h 1 + Q e 1       1 Q h 1 + R m j Q h 1 ) ] 1 / 2 × i k ( r 2 r 0 ) T Q h 2 m j                  1 ( r 2 r 0 ) } × exp [ i k r 0     T ( A Q e 1 + B + B R m j Q e 1 ) 1 r 2 ] ,
Q e 2 m j                 1 = [ C + D ( Q e 1       1 + R m j ) ] × [ A + B ( Q e 1       1 + R m j ) ] 1 ,
Q h 2 m j                 1 = [ A + B ( Q e 1       1 + R m j ) ] 1 1 T × ( A Q h 1 + B Q e 1       1 Q h 1 + B R m j Q h 1 ) 1 .
E N ( r ) = n = 1 N ( 1 ) n 1 N ( N n ) exp ( i k 2 r T Q 1 n         1 r ) ,
E N ( r ) = n = 1 N ( 1 ) n 1 N ( N n ) × exp [ i k 2 ( r r 0 ) T Q 1 n         1 ( r r 0 ) ] .
E N ( r 2 ) = n = 1 N m = 1 M ( 1 ) n 1 N ( N n ) A m × ( det [ A + B Q l n         1 + B R m ] ) 1 / 2 × exp [ i k 2 r 2     T Q 2 n m               1 r 2 ] × exp [ i k 2 r 0     T ( B 1 A Q 1 n + R m Q 1 n I ) 1 × ( B 1 A + R m ) r 0 ] × exp [ i k r 0     T ( A Q 1 n + B + B R m Q 1 n ) 1 r 2 ] ,
Q 2 n m               1 = [ C + D ( Q 1 n - 1 + R m ) ] [ A + B ( Q 1 n - 1 + R m ) ] 1 .
E N ( r 1 ) = exp [ i k 2 r 1     T Q 1 , N             1 r 1 ] × n = 0 N 1 n ! [ i k 2 r 1     T Q 1 , N             1 r 1 ] n ,
E N ( r 1 ) = exp [ i k 2 ( r 1 r 0 ) T Q 1 , N             1 ( r 1 r 0 ) ] × n = 0 N 1 n ! [ i k 2 ( r 1 r 0 ) T Q 1 , N             1 ( r 1 r 0 ) ] n .
[ ( r 1 r 0 ) T Q 1 , N             1 ( r r 0 ) ] n = ( 2 n ) ! 2 3 n m = 0 n 1 ( n m ) ! ( 2 m ) ! H 2 m [ 2 | Q 1 , N             1 / 2 × ( r 1 r 0 ) | ] ,
E ( r 1 ) = n = 0 N 1 ( 2 n ) ! m = n N 1 2 3 m ( 2 m ) ! ( m n ) ! m ! × exp [ i k 2 ( r 1 r 0 ) T Q 1 , N - 1 ( r 1 r 0 ) ] × H 2 m [ | ( i k Q 1 , N - 1 ) 1 / 2 ( r 1 r 0 ) | ] .

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