Abstract

Analytical expressions for the mode expansion for Gaussian Schell-model (GSM) beams with partially correlated modes are derived on the basis of the partial-coherence theory and the M2-factor concept. It is shown that our results have general characteristics and are valid for partially coherent laser light, which can be expressed as GSM beams. Moreover, owing to the cross correlation of modes, some modifications to the previous theory have to be made.

© 1999 Optical Society of America

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  1. A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
    [CrossRef]
  2. A. E. Siegman, “How to (maybe) measure laser beam quality,” in DPSS Lasers: Application and Issues, M. W. Dowley, ed., Vol. 17 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1998), 184–199.
  3. H. Weber, ed., Special issue on laser beam quality, Opt. Quantum Electron.24, 861–1135 (1992).
    [CrossRef]
  4. K. M. Du, G. Herziger, P. Loosen, F. Rühl, “Coherence and intensity moments of laser light,” Opt. Quantum Electron. 24, 1081–1093 (1992).
    [CrossRef]
  5. K. M. Du, G. Herziger, P. Loosen, F. Rühl, “Computation of the statistical properties of laser light,” Opt. Quantum Electron. 24, 1095–1108 (1992).
    [CrossRef]
  6. K. M. Du, G. Herziger, P. Loosen, F. Rühl, “Measurement of the mode coherence coefficients,” Opt. Quantum Electron. 24, 1119–1127 (1992).
    [CrossRef]
  7. K. M. Du, “Kohörenz, Polarization und Fluktuation von Laserstrahlung,” Ph.D. dissertation (Verlag der Augustinus-Buchhandlung, Aachen, Germany1992).
  8. H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24, 1027–1049 (1992).
    [CrossRef]
  9. A. Starikov, E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and their radiation fields,” J. Opt. Soc. Am. 72, 923–928 (1982).
    [CrossRef]
  10. A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
    [CrossRef]
  11. J. Turunen, A. T. Friberg, “Matrix representation of Gaussian Schell-model beams in optical systems,” Opt. Laser Technol. 18, 259–267 (1986).
    [CrossRef]
  12. A. T. Friberg, J. Turunen, “Imaging of Gaussian Schell-model sources,” J. Opt. Soc. Am. A 5, 713–719 (1988).
    [CrossRef]
  13. F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
    [CrossRef]
  14. R. Gase, “The multimode laser radiation as a Gaussian Schell-model beams,” J. Mod. Opt. 38, 1107–1115 (1991).
    [CrossRef]
  15. B. Lü, B. Zhang, B. Cai, C. Yang, “A simple method for estimating the number of effectively oscillating modes and weighting factors of mixed-mode laser beams,” Opt. Commun. 101, 49–52 (1993).
    [CrossRef]
  16. B. Zhang, B. Lü, “Transformation of Gaussian Schell-model beams and their coherent-mode representation,” J. Opt. (Paris) 27, 99–103 (1996).
    [CrossRef]
  17. L. Mandel, E. Wolf, “Spectral coherence and the concept of cross-spectral purity,” J. Opt. Soc. Am. 66, 529–535 (1976).
    [CrossRef]
  18. T. Shirai, T. Asakura, “Spatial coherence of light generated from a partially coherent source and its control using a source filter,” Optik (Stuttgart) 94, 1–17 (1993).
  19. L. Mandel, E. Wolf, eds., Coherence and Quantum Optics (Plenum, New York, 1973).
  20. H. Gamo, “Matrix Treatment of Partial Coherence,” Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Vol. III, pp. 187–247.
  21. A. Erdelyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954).
  22. I. S. Gradshtein, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).
  23. F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
    [CrossRef]
  24. L. W. Caperson, A. A. Tovar, “Hermite-sinusoidal-Gaussian beams in complex optical systems,” J. Opt. Soc. Am. A 15, 954–961 (1998).
    [CrossRef]
  25. B. Lü, B. Zhang, S. Luo, “Far-field intensity distribution, M2 factor, and propagation of flattened Gaussian beams,” Appl. Opt. 38, 4581–4584 (1999).
    [CrossRef]
  26. B. Lü, B. Zhang, H. Ma, “Beam-propagation factor and mode-coherence coefficients of hyperbolic-cosine–Gaussian beams,” Opt. Lett. 24, 640–642 (1999).
    [CrossRef]

1999 (2)

1998 (1)

1996 (1)

B. Zhang, B. Lü, “Transformation of Gaussian Schell-model beams and their coherent-mode representation,” J. Opt. (Paris) 27, 99–103 (1996).
[CrossRef]

1994 (1)

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
[CrossRef]

1993 (2)

B. Lü, B. Zhang, B. Cai, C. Yang, “A simple method for estimating the number of effectively oscillating modes and weighting factors of mixed-mode laser beams,” Opt. Commun. 101, 49–52 (1993).
[CrossRef]

T. Shirai, T. Asakura, “Spatial coherence of light generated from a partially coherent source and its control using a source filter,” Optik (Stuttgart) 94, 1–17 (1993).

1992 (4)

K. M. Du, G. Herziger, P. Loosen, F. Rühl, “Coherence and intensity moments of laser light,” Opt. Quantum Electron. 24, 1081–1093 (1992).
[CrossRef]

K. M. Du, G. Herziger, P. Loosen, F. Rühl, “Computation of the statistical properties of laser light,” Opt. Quantum Electron. 24, 1095–1108 (1992).
[CrossRef]

K. M. Du, G. Herziger, P. Loosen, F. Rühl, “Measurement of the mode coherence coefficients,” Opt. Quantum Electron. 24, 1119–1127 (1992).
[CrossRef]

H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24, 1027–1049 (1992).
[CrossRef]

1991 (1)

R. Gase, “The multimode laser radiation as a Gaussian Schell-model beams,” J. Mod. Opt. 38, 1107–1115 (1991).
[CrossRef]

1988 (1)

1986 (1)

J. Turunen, A. T. Friberg, “Matrix representation of Gaussian Schell-model beams in optical systems,” Opt. Laser Technol. 18, 259–267 (1986).
[CrossRef]

1982 (2)

A. Starikov, E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and their radiation fields,” J. Opt. Soc. Am. 72, 923–928 (1982).
[CrossRef]

A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[CrossRef]

1980 (1)

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

1976 (1)

Asakura, T.

T. Shirai, T. Asakura, “Spatial coherence of light generated from a partially coherent source and its control using a source filter,” Optik (Stuttgart) 94, 1–17 (1993).

Cai, B.

B. Lü, B. Zhang, B. Cai, C. Yang, “A simple method for estimating the number of effectively oscillating modes and weighting factors of mixed-mode laser beams,” Opt. Commun. 101, 49–52 (1993).
[CrossRef]

Caperson, L. W.

Du, K. M.

K. M. Du, G. Herziger, P. Loosen, F. Rühl, “Coherence and intensity moments of laser light,” Opt. Quantum Electron. 24, 1081–1093 (1992).
[CrossRef]

K. M. Du, G. Herziger, P. Loosen, F. Rühl, “Computation of the statistical properties of laser light,” Opt. Quantum Electron. 24, 1095–1108 (1992).
[CrossRef]

K. M. Du, G. Herziger, P. Loosen, F. Rühl, “Measurement of the mode coherence coefficients,” Opt. Quantum Electron. 24, 1119–1127 (1992).
[CrossRef]

K. M. Du, “Kohörenz, Polarization und Fluktuation von Laserstrahlung,” Ph.D. dissertation (Verlag der Augustinus-Buchhandlung, Aachen, Germany1992).

Erdelyi, A.

A. Erdelyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954).

Friberg, A. T.

A. T. Friberg, J. Turunen, “Imaging of Gaussian Schell-model sources,” J. Opt. Soc. Am. A 5, 713–719 (1988).
[CrossRef]

J. Turunen, A. T. Friberg, “Matrix representation of Gaussian Schell-model beams in optical systems,” Opt. Laser Technol. 18, 259–267 (1986).
[CrossRef]

A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[CrossRef]

Gamo, H.

H. Gamo, “Matrix Treatment of Partial Coherence,” Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Vol. III, pp. 187–247.

Gase, R.

R. Gase, “The multimode laser radiation as a Gaussian Schell-model beams,” J. Mod. Opt. 38, 1107–1115 (1991).
[CrossRef]

Gori, F.

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
[CrossRef]

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

Gradshtein, I. S.

I. S. Gradshtein, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

Herziger, G.

K. M. Du, G. Herziger, P. Loosen, F. Rühl, “Measurement of the mode coherence coefficients,” Opt. Quantum Electron. 24, 1119–1127 (1992).
[CrossRef]

K. M. Du, G. Herziger, P. Loosen, F. Rühl, “Computation of the statistical properties of laser light,” Opt. Quantum Electron. 24, 1095–1108 (1992).
[CrossRef]

K. M. Du, G. Herziger, P. Loosen, F. Rühl, “Coherence and intensity moments of laser light,” Opt. Quantum Electron. 24, 1081–1093 (1992).
[CrossRef]

Loosen, P.

K. M. Du, G. Herziger, P. Loosen, F. Rühl, “Coherence and intensity moments of laser light,” Opt. Quantum Electron. 24, 1081–1093 (1992).
[CrossRef]

K. M. Du, G. Herziger, P. Loosen, F. Rühl, “Computation of the statistical properties of laser light,” Opt. Quantum Electron. 24, 1095–1108 (1992).
[CrossRef]

K. M. Du, G. Herziger, P. Loosen, F. Rühl, “Measurement of the mode coherence coefficients,” Opt. Quantum Electron. 24, 1119–1127 (1992).
[CrossRef]

Lü, B.

B. Lü, B. Zhang, S. Luo, “Far-field intensity distribution, M2 factor, and propagation of flattened Gaussian beams,” Appl. Opt. 38, 4581–4584 (1999).
[CrossRef]

B. Lü, B. Zhang, H. Ma, “Beam-propagation factor and mode-coherence coefficients of hyperbolic-cosine–Gaussian beams,” Opt. Lett. 24, 640–642 (1999).
[CrossRef]

B. Zhang, B. Lü, “Transformation of Gaussian Schell-model beams and their coherent-mode representation,” J. Opt. (Paris) 27, 99–103 (1996).
[CrossRef]

B. Lü, B. Zhang, B. Cai, C. Yang, “A simple method for estimating the number of effectively oscillating modes and weighting factors of mixed-mode laser beams,” Opt. Commun. 101, 49–52 (1993).
[CrossRef]

Luo, S.

Ma, H.

Magnus, W.

A. Erdelyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954).

Mandel, L.

Oberhettinger, F.

A. Erdelyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954).

Rühl, F.

K. M. Du, G. Herziger, P. Loosen, F. Rühl, “Measurement of the mode coherence coefficients,” Opt. Quantum Electron. 24, 1119–1127 (1992).
[CrossRef]

K. M. Du, G. Herziger, P. Loosen, F. Rühl, “Computation of the statistical properties of laser light,” Opt. Quantum Electron. 24, 1095–1108 (1992).
[CrossRef]

K. M. Du, G. Herziger, P. Loosen, F. Rühl, “Coherence and intensity moments of laser light,” Opt. Quantum Electron. 24, 1081–1093 (1992).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshtein, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

Shirai, T.

T. Shirai, T. Asakura, “Spatial coherence of light generated from a partially coherent source and its control using a source filter,” Optik (Stuttgart) 94, 1–17 (1993).

Siegman, A. E.

A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
[CrossRef]

A. E. Siegman, “How to (maybe) measure laser beam quality,” in DPSS Lasers: Application and Issues, M. W. Dowley, ed., Vol. 17 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1998), 184–199.

Starikov, A.

Sudol, R. J.

A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[CrossRef]

Tovar, A. A.

Tricomi, F. G.

A. Erdelyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954).

Turunen, J.

A. T. Friberg, J. Turunen, “Imaging of Gaussian Schell-model sources,” J. Opt. Soc. Am. A 5, 713–719 (1988).
[CrossRef]

J. Turunen, A. T. Friberg, “Matrix representation of Gaussian Schell-model beams in optical systems,” Opt. Laser Technol. 18, 259–267 (1986).
[CrossRef]

Weber, H.

H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24, 1027–1049 (1992).
[CrossRef]

Wolf, E.

Yang, C.

B. Lü, B. Zhang, B. Cai, C. Yang, “A simple method for estimating the number of effectively oscillating modes and weighting factors of mixed-mode laser beams,” Opt. Commun. 101, 49–52 (1993).
[CrossRef]

Zhang, B.

B. Lü, B. Zhang, S. Luo, “Far-field intensity distribution, M2 factor, and propagation of flattened Gaussian beams,” Appl. Opt. 38, 4581–4584 (1999).
[CrossRef]

B. Lü, B. Zhang, H. Ma, “Beam-propagation factor and mode-coherence coefficients of hyperbolic-cosine–Gaussian beams,” Opt. Lett. 24, 640–642 (1999).
[CrossRef]

B. Zhang, B. Lü, “Transformation of Gaussian Schell-model beams and their coherent-mode representation,” J. Opt. (Paris) 27, 99–103 (1996).
[CrossRef]

B. Lü, B. Zhang, B. Cai, C. Yang, “A simple method for estimating the number of effectively oscillating modes and weighting factors of mixed-mode laser beams,” Opt. Commun. 101, 49–52 (1993).
[CrossRef]

Appl. Opt. (1)

J. Mod. Opt. (1)

R. Gase, “The multimode laser radiation as a Gaussian Schell-model beams,” J. Mod. Opt. 38, 1107–1115 (1991).
[CrossRef]

J. Opt. (Paris) (1)

B. Zhang, B. Lü, “Transformation of Gaussian Schell-model beams and their coherent-mode representation,” J. Opt. (Paris) 27, 99–103 (1996).
[CrossRef]

J. Opt. Soc. Am. (2)

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

Opt. Laser Technol. (1)

J. Turunen, A. T. Friberg, “Matrix representation of Gaussian Schell-model beams in optical systems,” Opt. Laser Technol. 18, 259–267 (1986).
[CrossRef]

Opt. Commun. (4)

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[CrossRef]

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
[CrossRef]

B. Lü, B. Zhang, B. Cai, C. Yang, “A simple method for estimating the number of effectively oscillating modes and weighting factors of mixed-mode laser beams,” Opt. Commun. 101, 49–52 (1993).
[CrossRef]

Opt. Lett. (1)

Opt. Quantum Electron. (4)

K. M. Du, G. Herziger, P. Loosen, F. Rühl, “Coherence and intensity moments of laser light,” Opt. Quantum Electron. 24, 1081–1093 (1992).
[CrossRef]

K. M. Du, G. Herziger, P. Loosen, F. Rühl, “Computation of the statistical properties of laser light,” Opt. Quantum Electron. 24, 1095–1108 (1992).
[CrossRef]

K. M. Du, G. Herziger, P. Loosen, F. Rühl, “Measurement of the mode coherence coefficients,” Opt. Quantum Electron. 24, 1119–1127 (1992).
[CrossRef]

H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24, 1027–1049 (1992).
[CrossRef]

Optik (Stuttgart) (1)

T. Shirai, T. Asakura, “Spatial coherence of light generated from a partially coherent source and its control using a source filter,” Optik (Stuttgart) 94, 1–17 (1993).

Other (8)

L. Mandel, E. Wolf, eds., Coherence and Quantum Optics (Plenum, New York, 1973).

H. Gamo, “Matrix Treatment of Partial Coherence,” Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Vol. III, pp. 187–247.

A. Erdelyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954).

I. S. Gradshtein, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

K. M. Du, “Kohörenz, Polarization und Fluktuation von Laserstrahlung,” Ph.D. dissertation (Verlag der Augustinus-Buchhandlung, Aachen, Germany1992).

A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
[CrossRef]

A. E. Siegman, “How to (maybe) measure laser beam quality,” in DPSS Lasers: Application and Issues, M. W. Dowley, ed., Vol. 17 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1998), 184–199.

H. Weber, ed., Special issue on laser beam quality, Opt. Quantum Electron.24, 861–1135 (1992).
[CrossRef]

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Tables (1)

Tables Icon

Table 1 Some Numerical Examples of the Mode Indices and Weighting Factors Calculated from the Given M2 for GSM Beams

Equations (62)

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W(r1, θ1, r2, θ2, z)
=ppllλppllϕpl(r1, θ1, z)ϕpl*(r2, θ2, z),
ϕpl(r1, θ1)
=upl exp[-(α2/2)r12](αr1)lLpl(α2r12)exp(-ilθ1),
ϕpl(r2, θ2)
=upl exp[-(α2/2)r22](αr2)lLpl(α2r22)exp(-ilθ2),
α=2/woh,
W(r1, θ1, r2, θ2, 0)
=I0 exp-r12+r22w02×exp-r12+r22-2r1r2 cos(θ1-θ2)2σ02,
λppll=0+0+02π02πϕpl*(r1, θ1)×W(r1, θ1, r2, θ2, 0)ϕpl(r2, θ2)×r1r2dθ1dθ2dr1dr2.
02π expr1r2 cos(θ1-θ2)σ02exp(-ilθ1)dθ1
=2πilJl-i r1r2σ02exp(-ilθ2),
02π exp[-i(l-l)θ2]dθ2
=2π0l=lll ,
0+xl exp(-βx2)Lpl(νx2)Jl(xy)xdx
=2-l-1β-p-l-1(β-ν)pyl exp-y24β
× Lplνy24β(ν-β),
0+tlLpl(λt)Lpl(kt)exp(-qt)dt
=Γ(p+p+l+1)p!p! (q-λ)p(q-k)pqp+p+l+1×2F1-p,-p;-p-p-l; q(q-λ-k)(q-λ)(q-k),
λppll=λ0c2p+l Γ(p+p+l+1)[p!p!Γ(p+l+1)Γ(p+l+1)]1/2×[(a/b)-1]p(a-1)pap+p+l+1×2F1-p,-p;-p-p-l; a(a-b-1)(a-b)(a-1),
a=4σ04(1/w02+1/2σ02+α2/2)2-14σ04α2(1/w02+1/2σ02+α2/2),
1b=4σ041w02+12σ022-α44,
c=[(1/w02+1/2σ02)2-α4/4]1/21/w02+1/2σ02+α2/2,
λ0=I0π1/w02+1/2σ02+α2/21/2,
2F1-p,-p;-p-p-l; a(a-b-1)(a-b)(a-1)
=k=0min(p, p) (p+p+l-k)!(p-k)!(p-k)!(p+p+l)!
× a(-a+b+1)(a-b)(a-1)kk!,
λppll=λ0c2p+l (p!p!)1/2[(p+l)!(p+l)!]1/2ap+p+l+1×k=0min(p, p) (p+p+l-k)!(p-k)!(p-k)!× [(a/b)(-a+b+1)]kk!×ab-1p-k(a-1)p-k.
λppll=λ01/2σ021/w02+1/2σ02+α2/22p+l,
σr2=02π0+r3I(r)drdθ02π0+I(r)rdrdθ,
I(r)=W(r, θ, r, θ).
(p+1)Lp+1l(z)-(2p+l+1-z)Lpl(z)
+(p+l)Lp-1l(z)=0,
0+zl exp(-z)Lpl(z)Lpl(z)dz
=[Γ(p+l+1)]/p!p=p0pp
σr2=1α2 Σ(2p+l+1)λppll-Σ[(p+1)(p+l+1)]1/2λp,p+1,llΣλppll.
σk2=α2 Σ(2p+l+1)λppll+Σ[(p+1)(p+l+1)]1/2λp,p+1,llΣλppll.
M2=σrσk={[Σ(2p+l+1)λppll]2-[Σ(p+1)(p+l+1)λp,p+1,ll]2}1/2Σλppll.
M2=Σ(2p+l+1)λppllΣλppll.
M2=1+w0σ021/2.
λppll=λ0c2p+l p!(p+l)!a2p+l+1k=0min(p, p)(2p+l-k)!(p-k)!2× [(a/b)(-a+b+1)]kk!×ab-1p-k(a-1)p-k,
λp,p+1,ll=λ0c2p+l (p+1)1/2p!(p+l+1)1/2(p+l)!a2p+l+2×k=0min(p, p) (2p+l+1-k)!(p-k)!(p-k+1)!×[(a/b)(-a+b+1)]kk!× ab-1p-k(a-1)p-k+1,
a=4(M2)2+[(b-1)/2b][(M2)2-1]2+[(M2)2+1]{4(M2)2+[(b-1)/b][(M2)2-1]2}1/24(M2)2+[(b-1)/b][(M2)2-1]2+[(M2)2+1]{4(M2)2+[(b-1)/b][(M2)2-1]2}1/2,
c=(M2)2-1[(M2)2+1]+{4(M2)2+[(b-1)/b][(M2)2-1]2}1/2,
Cppll=λppllΣλppll,
Cp,p+1,ll=λp,p+1,llΣλppll.
woh2=λLπ [g1g2(1-g1g2)]1/2|g1+g2-2g1g2|,
ACBD
W(r1, θ1, r2, θ2, z)
=k2B202π02π0+0+W(r1, θ1, r2, θ2, 0)×exp-ik2B {A(r12-r22)-2[r1r1 cos(θ1-θ1)-r2r2 cos(θ2-θ2)]+D(r12-r22)}r1r2dr1dr2dθ1dθ2.
W(r1, θ1, r2, θ2, z)
=ppllλppllϕpl(r1, θ1)ϕpl*(r2, θ2),
λppll=λppll
ϕpl(r1, θ1)=α2p!π(p+l)!1/2exp-α22 r12(αr1)l×Lpl(α2r12)exp-ilθ1+k2R+ξp,
ϕpl(r2, θ2)=α2p!π(p+l)!exp-α22 r22(αr2)l×Lpl(α2r22)×exp-ilθ2+k2R+ξp,
α2=α2A2+B2α4/k2;
R=A2+B2α4/k2AC+BDα4/k2,
ξp=2p tan-1kABα2+pπ,
ξp=2p tan-1kABα2+pπ.
q-1=-i α2k,q-1=1R-i α2k
q=Aq+BCq+D.
β=1+w02σ02-1/2=1+w2σ2-1/2,

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