Abstract

A mode coupling approach based on the modal theory of coherence is suggested for the study of partially coherent beams in atmospheric turbulence. An approximate expression is derived for the mode power coupling coefficients, and some specific cases are studied using numerical methods. Several general results derived from the properties of the coupling coefficients are also presented.

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References

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  1. R. L. Fante, "Wave propagation in random media: a systems approach," in Progress in Optics, Vol. XXII, E.Wolf, ed. (Elsevier, 1985), pp. 341-398.
    [CrossRef]
  2. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 1998).
  3. J. C. Ricklin and F. M. Davidson, "Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication," J. Opt. Soc. Am. A 19, 1794-1802 (2002).
    [CrossRef]
  4. G. Gbur and E. Wolf, "Spreading of partially coherent beams in random media," J. Opt. Soc. Am. A 19, 1592-1598 (2002).
    [CrossRef]
  5. S. A. Ponomarenko, J. J. Greffett, and E. Wolf, "The diffusion of partially coherent beams in turbulent media," Opt. Commun. 208, 1-8 (2002).
    [CrossRef]
  6. A. Dogariu and S. Amarande, "Propagation of partially coherent beams: turbulence-induced degradation," Opt. Lett. 28, 10-12 (2003).
    [CrossRef] [PubMed]
  7. T. Shirai, A. Dogariu, and E. Wolf, "Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence," J. Opt. Soc. Am. A 20, 1094-1102 (2003).
    [CrossRef]
  8. E. Wolf, "Unified theory of coherence and polarization of random electromagnetic beams," Phys. Lett. A 312, 263-267 (2003).
    [CrossRef]
  9. E. Wolf, "New theory of partial coherence in the space-frequency domain. Part I: spectra and cross spectra of steady-state sources," J. Opt. Soc. Am. 72, 343-351 (1982).
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  11. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
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  14. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1992).
  15. B. Lu and B. Zhang, "Mode expansion for Gaussian Schell-model beams with partially correlated modes," J. Opt. Soc. Am. A 16, 2453-2458 (1999).
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  16. W. H. Carter, "Spot size and divergence for Hermite Gaussian beams of any order," Appl. Opt. 191027-1029 (1980).
    [CrossRef] [PubMed]
  17. 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]
  18. F. Gori, "Mode propagation of the field generated by Collet-Wolf Schell-model sources," Opt. Commun. 46, 149-154 (1983).
    [CrossRef]

2003 (3)

2002 (4)

G. Gbur and E. Wolf, "Spreading of partially coherent beams in random media," J. Opt. Soc. Am. A 19, 1592-1598 (2002).
[CrossRef]

J. C. Ricklin and F. M. Davidson, "Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication," J. Opt. Soc. Am. A 19, 1794-1802 (2002).
[CrossRef]

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]

S. A. Ponomarenko, J. J. Greffett, and E. Wolf, "The diffusion of partially coherent beams in turbulent media," Opt. Commun. 208, 1-8 (2002).
[CrossRef]

1999 (1)

1986 (2)

1983 (1)

F. Gori, "Mode propagation of the field generated by Collet-Wolf Schell-model sources," Opt. Commun. 46, 149-154 (1983).
[CrossRef]

1982 (1)

1981 (1)

1980 (1)

Amarande, S.

Andrews, L. C.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 1998).

Carter, W. H.

Davidson, F. M.

Dogariu, A.

Fante, R. L.

R. L. Fante, "Wave propagation in random media: a systems approach," in Progress in Optics, Vol. XXII, E.Wolf, ed. (Elsevier, 1985), pp. 341-398.
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1992).

Gbur, G.

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]

Gori, F.

F. Gori, "Mode propagation of the field generated by Collet-Wolf Schell-model sources," Opt. Commun. 46, 149-154 (1983).
[CrossRef]

Greffett, J. J.

S. A. Ponomarenko, J. J. Greffett, and E. Wolf, "The diffusion of partially coherent beams in turbulent media," Opt. Commun. 208, 1-8 (2002).
[CrossRef]

Lu, B.

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]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

Montgomery, W. D.

Phillips, R. L.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 1998).

Ponomarenko, S. A.

S. A. Ponomarenko, J. J. Greffett, and E. Wolf, "The diffusion of partially coherent beams in turbulent media," Opt. Commun. 208, 1-8 (2002).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1992).

Ricklin, J. C.

Shirai, T.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1992).

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1992).

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]

Zhang, B.

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (6)

Opt. Commun. (2)

F. Gori, "Mode propagation of the field generated by Collet-Wolf Schell-model sources," Opt. Commun. 46, 149-154 (1983).
[CrossRef]

S. A. Ponomarenko, J. J. Greffett, and E. Wolf, "The diffusion of partially coherent beams in turbulent media," Opt. Commun. 208, 1-8 (2002).
[CrossRef]

Opt. Eng. (1)

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. Lett. (2)

Phys. Lett. A (1)

E. Wolf, "Unified theory of coherence and polarization of random electromagnetic beams," Phys. Lett. A 312, 263-267 (2003).
[CrossRef]

Other (4)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

R. L. Fante, "Wave propagation in random media: a systems approach," in Progress in Optics, Vol. XXII, E.Wolf, ed. (Elsevier, 1985), pp. 341-398.
[CrossRef]

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 1998).

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1992).

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Figures (8)

Fig. 1
Fig. 1

Illustration of the notation used in the propagation description.

Fig. 2
Fig. 2

Change in the power content of the different modes on propagation in atmospheric turbulence of a beam produced by a TEM00 mode source ( k = 10 7 m 1 , C n 2 = 10 14 m 2 3 , l 0 = 1 cm , w 0 = 1 cm ) .

Fig. 3
Fig. 3

Total power contained up to the mode TEM99 of the beam for the case of a TEM00 source ( k = 10 7 m 1 , C n 2 = 10 14 m 2 3 , l 0 = 1 cm , w 0 = 1 cm ) .

Fig. 4
Fig. 4

Normalized intensity distribution (with respect to free-space propagation) for a TEM00 beam after 1000 m of propagation ( k = 10 7 m 1 , C n 2 = 10 14 m 2 3 , l 0 = 1 cm , w 0 = 1 cm ) .

Fig. 5
Fig. 5

Change in the power content of the different modes on propagation of a doughnut beam ( k = 10 7 m 1 , C n 2 = 10 14 m 2 3 , l 0 = 1 cm , w 0 = 1 cm ) .

Fig. 6
Fig. 6

Doughnut-beam mean intensity distribution after propagation of 1000 m in atmospheric turbulence ( k = 10 7 m 1 , C n 2 = 10 14 m 2 3 , l 0 = 1 cm , w 0 = 1 cm ) .

Fig. 7
Fig. 7

Change in the power content of the different modes on propagation of a GSM beam ( k = 10 7 m 1 , C n 2 = 10 14 m 2 3 , l 0 = 1 cm , w 0 = 1 cm ) .

Fig. 8
Fig. 8

Comparison of beam-spread calculation for free-space propagation and in turbulence using the results from mode expansion coefficients for a GSM beam ( k = 10 7 m 1 , C n 2 = 10 14 m 2 3 , l 0 = 1 cm , w 0 = 1 cm ) .

Equations (52)

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W = W i j ( r 1 , r 2 , ω ) = E i * ( r 1 , ω ) E j ( r 2 , ω ) ,
W ( r 1 , r 2 , ω ) = E * ( r 1 , ω ) E ( r 2 , ω ) .
W ( ρ 1 , ρ 2 , 0 , ω ) = λ l ( ω ) ϕ l * ( ρ 1 , ω ) ϕ l ( ρ 2 , ω ) ,
W ( ρ 1 , ρ 2 , 0 , ω ) ϕ l ( ρ 1 , ω ) d 2 ρ 1 = λ l ( ω ) ϕ l ( ρ 2 , ω ) .
S ( ρ , ω ) = W ( ρ , ρ , 0 , ω ) = l λ l ( ω ) ϕ l * ( ρ , ω ) ϕ l ( ρ , ω ) .
W ( ρ 1 , ρ 2 , z , ω ) = W ( ρ 1 , ρ 2 , 0 , ω ) K 0 ( ρ 1 ρ 1 , ρ 2 ρ 2 , z , ω ) d 2 ρ 1 d 2 ρ 2 ,
K 0 ( ρ 1 ρ 1 , ρ 2 ρ 2 , z , ω ) = G 0 * ( ρ 1 ρ 1 , z , ω ) G 0 ( ρ 2 ρ 2 , z , ω ) ,
G 0 ( ρ ρ , z , ω ) = i k 2 π z exp { i k ρ ρ 2 2 z } ,
W ( ρ 1 , ρ 2 , z , ω ) = l λ l ( ω ) ψ l * ( ρ 1 , z , ω ) ψ l ( ρ 2 , z , ω ) ,
K 0 ( ρ 1 ρ 1 , ρ 2 ρ 2 , z , ω ) = n , m ϕ n ( ρ 1 , ω ) ϕ m * ( ρ 2 , ω ) ψ n * ( ρ 1 , z , ω ) ψ m ( ρ 2 , z , ω ) .
W A T ( ρ 1 , ρ 2 , z , ω ) = W ( ρ 1 , ρ 2 , 0 , ω ) K A T ( ρ 1 , ρ 1 , ρ 2 , ρ 2 , z , ω ) d 2 ρ 1 d 2 ρ 2 ,
K A T ( ρ 1 , ρ 1 , ρ 2 , ρ 2 , z , ω ) = G A T * ( ρ 1 ρ 1 , z , ω ) G A T ( ρ 2 ρ 2 , z , ω ) = G 0 * ( ρ 1 ρ 1 , z , ω ) G 0 ( ρ 2 ρ 2 , z , ω ) × exp [ ψ * ( ρ 1 ρ 1 , z , ω ) + ψ ( ρ 2 ρ 2 , z , ω ) ] = K 0 ( ρ 1 ρ 1 , ρ 2 ρ 2 , z , ω ) × exp [ 1 2 D s p ( ρ 1 ρ 2 , ρ 1 ρ 2 , z , ω ) ] ,
G A T ( ρ ρ , z , ω ) = G 0 ( ρ ρ , z , ω ) exp { Ψ ( ρ ρ , z , ω ) } .
D s p ( ρ 1 ρ 2 , ρ 1 ρ 2 , z , ω ) = 8 π 2 k 2 z 0 1 0 κ Φ n ( κ ) × { 1 J 0 [ ( 1 ξ ) ( ρ 1 ρ 2 ) + ξ ( ρ 1 ρ 2 ) κ ] } d κ d ξ .
K A T ( ρ 1 ρ 1 , ρ 2 ρ 2 , z , ω ) = K 0 ( ρ 1 ρ 1 , ρ 2 ρ 2 , z , ω ) K 0 ( ρ 1 ρ 1 , ρ 2 ρ 2 , z , ω ) × { 1 exp [ 1 2 D s p ( ρ 1 ρ 2 , ρ 1 ρ 2 , z , ω ) ] } .
χ n ( ρ , z , ω ) = l b n l ( z , ω ) ψ l ( ρ , z , ω ) ,
W n ( ρ 1 , ρ 2 , z , ω ) = χ n * ( ρ 1 , z , ω ) χ n ( ρ 2 , z , ω ) = l , m b n l * ( z , ω ) b n m ( z , ω ) ψ l * ( ρ 1 , z , ω ) ψ m ( ρ 2 , z , ω ) = l , m b n l * ( z , ω ) b n m ( z , ω ) ψ l * ( ρ 1 , z , ω ) ψ m ( ρ 2 , z , ω ) = l b n l ( z , ω ) 2 ψ l * ( ρ 1 , z , ω ) ψ l ( ρ 2 , z , ω ) + l m b n l * ( z , ω ) b n m ( z , ω ) ψ l * ( ρ 1 , z , ω ) ψ m ( ρ 2 , z , ω ) .
W A T ( ρ 1 , ρ 2 , z , ω ) = n l b n l ( z , ω ) 2 ψ l * ( ρ 1 , z , ω ) ψ l ( ρ 2 , z , ω ) + n l m b n l * ( z , ω ) b n m ( z , ω ) ψ l * ( ρ 1 , z , ω ) ψ m ( ρ 2 , z , ω ) .
W A T ( ρ , ρ , z , ω ) d 2 ρ = n l b n l ( z , ω ) 2 .
W A T ( d ) ( ρ 1 , ρ 2 , z , ω ) = l [ λ l ( ω ) + n λ n ( ω ) d n l ( z , ω ) ] ψ l * ( ρ 1 , z , ω ) ψ l ( ρ 2 , z , ω ) ,
l [ λ l ( ω ) + n λ n ( ω ) d n l ( z , ω ) ] ψ l * ( ρ 1 , z , ω ) ψ l ( ρ 2 , z , ω ) = l λ l ( ω ) ψ l * ( ρ 1 , z , ω ) ψ l ( ρ 2 , z , ω ) l λ l ( ω ) ϕ l * ( ρ 1 , ω ) ϕ l ( ρ 2 , ω ) × n , m ϕ n ( ρ 1 , ω ) ϕ m * ( ρ 2 , ω ) ψ n * ( ρ 1 , z , ω ) ψ m ( ρ 2 , z , ω ) × { 1 exp [ 1 2 D s p ( ρ 1 ρ 2 , ρ 1 ρ 2 , z , ω ) ] } d 2 ρ 1 d 2 ρ 2 .
d n l ( z , ω ) = ψ n * ( ρ 1 , z , ω ) ψ n ( ρ 2 , z , ω ) ψ l ( ρ 1 , z , ω ) ψ l * ( ρ 2 , z , ω ) × { 1 exp [ 1 2 D Φ ( ρ 1 , ρ 2 , ω ) ] } d 2 ρ 1 d 2 ρ 2 .
l [ λ l ( ω ) + n λ n ( ω ) d n l ( z , ω ) ] = l λ l ( ω ) l [ l λ n ( ω ) d n l ( z , ω ) ] = 0 .
n [ l λ n ( ω ) d n l ( z , ω ) ] = 0
n λ n ( ω ) l d n l ( z , ω ) = 0 .
l d n l ( z , ω ) = 0 .
λ ( z , ω ) = [ I + d ( z , ω ) ] λ ( 0 , ω ) ,
W ( ρ 1 , ρ 2 , z , ω ) = m n β m n ( ω ) ψ m n * ( ρ 1 , z , ω ) ψ m n ( ρ 2 , z , ω ) .
W A T ( d ) ( ρ 1 , ρ 2 , z , ω ) = m n [ β m n ( ω ) + k l β k l ( ω ) d ( m n ) ( k l ) ( z , ω ) ] ψ m n * ( ρ 1 , z , ω ) ψ m n ( ρ 2 , z , ω ) .
d ( m n ) ( k l ) ( z , ω ) = 1 [ w 0 Δ ( z ) ] 2 π 2 m + n + k + l 2 m ! n ! k ! l ! H k ( 2 w 0 Δ ( z ) x 1 ) H l ( 2 w 0 Δ ( z ) y 1 ) × H m ( 2 w 0 Δ ( z ) x 2 ) H n ( 2 w 0 Δ ( z ) y 2 ) H k ( 2 w 0 Δ ( z ) x 2 ) H l ( 2 w 0 Δ ( z ) y 2 ) H m ( 2 w 0 Δ ( z ) x 1 ) H n ( 2 w 0 Δ ( z ) y 1 ) × exp ( x 1 2 + y 1 2 w 0 2 Δ 2 ( z ) ) exp ( x 2 2 + y 2 2 w 0 2 Δ 2 ( z ) ) { 1 exp [ 1 2 D Φ ( ρ 1 , ρ 2 , z , ω ) ] } d x 1 d y 1 d x 2 d y 2 .
Φ n ( κ ) = 0.033 C n 2 exp ( κ 2 κ m 2 ) κ 11 3 ,
D Φ ( ρ 1 , ρ 2 , z , ω ) = D Φ ( ρ = ρ 2 ρ 1 , z , ω ) = 1.093 C n 2 k 2 z l 0 1 3 ρ 2 ( 1 + ρ 2 l 0 2 ) 1 6 .
σ 1 2 = 1.23 C n 2 k 7 6 z 11 6 ,
ρ ( z ) ¯ = [ ρ 2 I ( ρ , z ) d 2 ρ I ( ρ , z ) d 2 ρ ] 1 2 .
ρ ( z ) ¯ = [ 1 2 n m β m n ( ω , z ) ( m + n + 1 ) w 0 2 Δ 2 ( z ) ] 1 2 .
l [ λ l ( ω ) + n λ n ( ω ) d n l ( z , ω ) ] ψ l * ( ρ 1 , z , ω ) ψ l ( ρ 2 , z , ω ) = l λ l ( ω ) ψ l * ( ρ 1 , z , ω ) ψ l ( ρ 2 , z , ω ) l λ l ( ω ) ϕ l * ( ρ 1 , ω ) ϕ l ( ρ 2 , ω ) n , m ϕ n ( ρ 1 , ω ) ϕ m * ( ρ 2 , ω ) ψ n * ( ρ 1 , z , ω ) ψ m ( ρ 2 , z , ω ) × { 1 exp [ 1 2 D Φ ( ρ 1 , ρ 2 , z , ω ) ] } d 2 ρ 1 d 2 ρ 2 = l λ l ( ω ) ψ l * ( ρ 1 , z , ω ) ψ l ( ρ 2 , z , ω ) l n , m λ l ( ω ) δ n l δ m l ψ n * ( ρ 1 , z , ω ) ψ m ( ρ 2 , z , ω ) { 1 exp [ 1 2 D Φ ( ρ 1 , ρ 2 , z , ω ) ] } = l λ l ( ω ) ψ l * ( ρ 1 , z , ω ) ψ l ( ρ 2 , z , ω ) l λ l ( ω ) ψ l * ( ρ 1 , z , ω ) ψ l ( ρ 2 , z , ω ) { 1 exp [ 1 2 D Φ ( ρ 1 , ρ 2 , z , ω ) ] } .
l n λ n ( ω ) d n l ( z , ω ) ψ l * ( ρ 1 , z , ω ) ψ l ( ρ 2 , z , ω )
= l λ n ( ω ) ψ l * ( ρ 1 , z , ω ) ψ l ( ρ 2 , z , ω ) { 1 exp [ 1 2 D Φ ( ρ 1 , ρ 2 , ω ) ] } .
n λ n ( ω ) d n j ( z , ω ) = l λ l ( ω ) ψ l * ( ρ 1 , z , ω ) × ψ l ( ρ 2 , z , ω ) ψ j ( ρ 1 , z , ω ) ψ j * ( ρ 2 , z , ω ) × { 1 exp [ 1 2 D Φ ( ρ 1 , ρ 2 , ω ) ] } d 2 ρ 1 d 2 ρ 2 .
d n l ( z , ω ) = ψ n * ( ρ 1 , z , ω ) ψ n ( ρ 2 , z , ω ) ψ l ( ρ 1 , z , ω ) ψ l * ( ρ 2 , z , ω ) × { 1 exp [ 1 2 D Φ ( ρ 1 , ρ 2 , ω ) ] } d 2 ρ 1 d 2 ρ 2 .
W ( ρ 1 , ρ 2 , ω ) = A ( ω ) exp ( ρ 1 2 + ρ 2 2 4 σ s 2 ( ω ) ) exp ( ρ 1 ρ 2 2 2 σ μ 2 ( ω ) ) .
S ( ρ , ω ) = W ( ρ , ρ , 0 , ω ) = A ( ω ) exp ( ρ 2 2 σ s 2 ( ω ) )
μ ( ρ 1 , ρ 2 , ω ) = W ( ρ 1 , ρ 2 , ω ) S ( ρ 1 , ω ) S ( ρ 2 , ω ) = exp ( ρ 1 ρ 2 2 2 σ μ 2 ( ω ) )
W ( ρ 1 , ρ 2 , ω ) = m n β m n ( ω ) ϕ m n * ( ρ 1 ) ϕ m n ( ρ 2 ) ,
ϕ m n ( ρ ) ϕ m n ( x , y ) = 1 w 0 π 2 m + n 1 m ! n ! × H m ( 2 w 0 x ) H n ( 2 w 0 y ) exp ( x 2 + y 2 w 0 2 ) ,
β m n ( ω ) = A ( ω ) ( π a + b + c ) ( b a + b + c ) m + n ,
a = 1 4 σ s 2 ( ω ) , b = 1 2 σ μ 2 ( ω ) , c = a 2 + 2 a b , w 0 = 1 c .
ψ m n ( ρ , z ) ψ m n ( x , y , z ) = 1 w 0 Δ ( z ) π 2 m + n 1 m ! n ! × H m ( 2 w 0 Δ ( z ) x ) H n ( 2 w 0 Δ ( z ) y ) × exp [ x 2 + y 2 w 0 2 Δ ( z ) 2 ] exp [ i k x 2 + y 2 2 R ( z ) ] × exp [ i ( m + n + 1 ) Φ ( z ) ] ,
Δ ( z ) = 1 + ( z Z R ) 2 ,
R ( z ) = z [ 1 + ( Z R z ) 2 ] ,
Z R = k w 0 2 2 ,
Φ ( z ) = tan 1 [ 2 z ( k w 0 2 ) ] .

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