Abstract

Log-amplitude and phase-correlation and structure functions of higher-order annular laser beams in a turbulent atmosphere are derived. A higher-order annular beam source is defined as the superposition of two different higher-order Hermite–Gaussian beams. A special case of such an excitation is the annular Gaussian beam in which two beams operate at fundamental modes of different Gaussian beam sizes, yielding a doughnut-shaped (annular) beam when the second beam is subtracted from the first beam. Our formulation utilizes Rytov approximation, which makes it applicable in the weak-turbulence regime, especially for log-amplitude fluctuations. Limiting cases of our formulations correctly match with known higher-order-mode solutions that in turn reduce to the Gaussian-beam-wave (TEM00-mode) results. Our results can be applied to determine the scintillation index and the phase fluctuations in free-space optics links under higher-order annular laser beam excitation. Except for the numerical evaluation of a specific example covering an annular Gaussian beam, the results in general are left in integral form and need to be numerically evaluated in detail to obtain quantitative results.

© 2005 Optical Society of America

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References

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    [CrossRef]
  14. Y. Baykal, “Correlation and structure functions for multimode-laser-beam incidence in atmospheric turbulence,” J. Opt. Soc. Am. A 4, 817–819 (1987).
    [CrossRef]
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  16. I. S. Gradysteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1980).

2004

2000

M. Ciofini, A. Lapucci, “Guided Talbot resonators for annular laser sources,” J. Opt. A Pure Appl. Opt. 2, 223–227 (2000).
[CrossRef]

1999

1997

1996

G. Huyet, C. Mathis, J. R. Tredicce, “Dynamics of annular lasers,” Opt. Commun. 127, 257–262 (1996).
[CrossRef]

1994

1993

1987

1971

P. K. Katti, B. N. Gupta, K. Singh, “Effect of atmospheric turbulence on the far field diffraction of an annular aperture,” J. Phys. D 4, 666–671 (1971).
[CrossRef]

1970

A. K. Aggarwal, R. S. Sirohi, “The fluctuations of intensity at the focus of an annular aperture,” Opt. Acta 17, 623–629 (1970).
[CrossRef]

1969

A. Ishimaru, “Fluctuations in the parameters of spherical waves propagating in a turbulent atmosphere,” Radio Sci. 4, 295–305 (1969).
[CrossRef]

Aggarwal, A. K.

A. K. Aggarwal, R. S. Sirohi, “The fluctuations of intensity at the focus of an annular aperture,” Opt. Acta 17, 623–629 (1970).
[CrossRef]

Andrews, L. C.

F. E. S. Vetelino, L. C. Andrews, “Annular Gaussian beams in turbulent media,” in Proceedings of Free-Space Laser Communication and Active Laser Illumination III, D. G. Voelz, J. C. Ricklin, eds., Proc. SPIE5160, 86–97 (2004).

Baykal, Y.

Bochum, H.

Buchter, S. C.

A. Shevchenko, S. C. Buchter, N. V. Tabiryan, M. Kaivola, “Creation of a hollow laser beam using self-phase modulation in a nematic liquid crystal,” Opt. Commun. 232, 77–82 (2004).
[CrossRef]

Chaloupka, J. L.

Ciofini, M.

M. Ciofini, A. Lapucci, “Guided Talbot resonators for annular laser sources,” J. Opt. A Pure Appl. Opt. 2, 223–227 (2000).
[CrossRef]

A. Lapucci, M. Ciofini, “Extraction of high-quality beams from narrow annular laser sources,” Appl. Opt. 38, 4552–4557 (1999).
[CrossRef]

Ehrlichmann, D.

Gradysteyn, I. S.

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

Gupta, B. N.

P. K. Katti, B. N. Gupta, K. Singh, “Effect of atmospheric turbulence on the far field diffraction of an annular aperture,” J. Phys. D 4, 666–671 (1971).
[CrossRef]

Habich, U.

Huyet, G.

G. Huyet, C. Mathis, J. R. Tredicce, “Dynamics of annular lasers,” Opt. Commun. 127, 257–262 (1996).
[CrossRef]

Ishimaru, A.

A. Ishimaru, “Fluctuations in the parameters of spherical waves propagating in a turbulent atmosphere,” Radio Sci. 4, 295–305 (1969).
[CrossRef]

Kaivola, M.

A. Shevchenko, S. C. Buchter, N. V. Tabiryan, M. Kaivola, “Creation of a hollow laser beam using self-phase modulation in a nematic liquid crystal,” Opt. Commun. 232, 77–82 (2004).
[CrossRef]

Katti, P. K.

P. K. Katti, B. N. Gupta, K. Singh, “Effect of atmospheric turbulence on the far field diffraction of an annular aperture,” J. Phys. D 4, 666–671 (1971).
[CrossRef]

Lapucci, A.

M. Ciofini, A. Lapucci, “Guided Talbot resonators for annular laser sources,” J. Opt. A Pure Appl. Opt. 2, 223–227 (2000).
[CrossRef]

A. Lapucci, M. Ciofini, “Extraction of high-quality beams from narrow annular laser sources,” Appl. Opt. 38, 4552–4557 (1999).
[CrossRef]

Masamori, E.

Mathis, C.

G. Huyet, C. Mathis, J. R. Tredicce, “Dynamics of annular lasers,” Opt. Commun. 127, 257–262 (1996).
[CrossRef]

Meyerhofer, D. D.

Nishimae, J.

Peatross, J.

Plum, H. D.

Ryzhik, I. M.

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

Shevchenko, A.

A. Shevchenko, S. C. Buchter, N. V. Tabiryan, M. Kaivola, “Creation of a hollow laser beam using self-phase modulation in a nematic liquid crystal,” Opt. Commun. 232, 77–82 (2004).
[CrossRef]

Singh, K.

P. K. Katti, B. N. Gupta, K. Singh, “Effect of atmospheric turbulence on the far field diffraction of an annular aperture,” J. Phys. D 4, 666–671 (1971).
[CrossRef]

Sirohi, R. S.

A. K. Aggarwal, R. S. Sirohi, “The fluctuations of intensity at the focus of an annular aperture,” Opt. Acta 17, 623–629 (1970).
[CrossRef]

Tabiryan, N. V.

A. Shevchenko, S. C. Buchter, N. V. Tabiryan, M. Kaivola, “Creation of a hollow laser beam using self-phase modulation in a nematic liquid crystal,” Opt. Commun. 232, 77–82 (2004).
[CrossRef]

Tamida, T.

Tredicce, J. R.

G. Huyet, C. Mathis, J. R. Tredicce, “Dynamics of annular lasers,” Opt. Commun. 127, 257–262 (1996).
[CrossRef]

Vetelino, F. E. S.

F. E. S. Vetelino, L. C. Andrews, “Annular Gaussian beams in turbulent media,” in Proceedings of Free-Space Laser Communication and Active Laser Illumination III, D. G. Voelz, J. C. Ricklin, eds., Proc. SPIE5160, 86–97 (2004).

Appl. Opt.

J. Opt. A Pure Appl. Opt.

M. Ciofini, A. Lapucci, “Guided Talbot resonators for annular laser sources,” J. Opt. A Pure Appl. Opt. 2, 223–227 (2000).
[CrossRef]

J. Opt. Soc. Am. A

J. Phys. D

P. K. Katti, B. N. Gupta, K. Singh, “Effect of atmospheric turbulence on the far field diffraction of an annular aperture,” J. Phys. D 4, 666–671 (1971).
[CrossRef]

Opt. Acta

A. K. Aggarwal, R. S. Sirohi, “The fluctuations of intensity at the focus of an annular aperture,” Opt. Acta 17, 623–629 (1970).
[CrossRef]

Opt. Commun.

G. Huyet, C. Mathis, J. R. Tredicce, “Dynamics of annular lasers,” Opt. Commun. 127, 257–262 (1996).
[CrossRef]

A. Shevchenko, S. C. Buchter, N. V. Tabiryan, M. Kaivola, “Creation of a hollow laser beam using self-phase modulation in a nematic liquid crystal,” Opt. Commun. 232, 77–82 (2004).
[CrossRef]

Opt. Express

Opt. Lett.

Radio Sci.

A. Ishimaru, “Fluctuations in the parameters of spherical waves propagating in a turbulent atmosphere,” Radio Sci. 4, 295–305 (1969).
[CrossRef]

Other

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

F. E. S. Vetelino, L. C. Andrews, “Annular Gaussian beams in turbulent media,” in Proceedings of Free-Space Laser Communication and Active Laser Illumination III, D. G. Voelz, J. C. Ricklin, eds., Proc. SPIE5160, 86–97 (2004).

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Figures (2)

Fig. 1
Fig. 1

On-axis scintillation index in turbulence versus link length for an annular Gaussian beam formed by superposition of a first plane-wave beam ( α s 1 = ) and a second beam with Gaussian beam size α s 2 = 5   cm .

Fig. 2
Fig. 2

Same as Fig. 1 but the second beam is of size α s 2 = 2.5   cm .

Equations (50)

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u n 1 , m 1 ( s x ,   s y ,   z = 0 )
= A n 1 , m 1 H n 1 ( a x 1 s x + b x 1 ) H m 1 ( a y 1 s y + b y 1 ) × exp [ - k / 2 ( α x 1 s x 2 + α y 1 s y 2 ) ] exp ( - i ϕ 1 ) ,
α x 1 = 1 k α sx 1 2 + i F x 1 ,
α y 1 = 1 k α sy 1 2 + i F y 1 ,
u n 1 , m 1 FS ( p ,   z ) = k   exp ( ikz ) / ( 2 π iz ) × - d s x - d s y u n 1 , m 1 ( s x ,   s y ,   z = 0 ) × exp { [ ik / ( 2 z ) ] [ ( s x - p x ) 2 + ( s y - p y ) 2 ] } ,
u n 1 , m 1 FS ( p ,   z ) = A n 1 , m 1 exp ( ikz ) × exp ( - i ϕ 1 ) ( 1 - 2 ia x 1 2 zA x 1 / k ) n 1 / 2 × ( 1 - 2 ia y 1 2 zA y 1 / k ) m 1 / 2 A x 1 1 / 2 A y 1 1 / 2 × exp ( - k α x 1 A x 1 p x 2 / 2 ) × exp ( - k α y 1 A y 1 p y 2 / 2 ) H n 1 ( × β 2 x 1 p x + β 1 x 1 ) H m 1 ( β 2 y 1 p y + β 1 y 1 ) ,
A x 1 = 1 / ( 1 + i α x 1 z ) ,
A y 1 = 1 / ( 1 + i α y 1 z ) ;
β 2 x 1 = a x 1 A x 1 1 / 2 / { 1 - iz [ ( 2 a x 1 2 / k ) - α x 1 ] } 1 / 2 ,
β 1 x 1 = β 2 x 1 [ kb x 1 ( 1 + i α x 1 z ) ] / ( ka x 1 ) .
B χ ( p 1 ,   p 2 ,   L )
= π   Re 0 L d η 0 κ d κ 0 2 π d θ [ G 11 ( p 1 ,   p 2 ,   η ,   κ ,   θ ,   L ) + G 21 ( p 1 ,   p 2 ,   η ,   κ ,   θ ,   L ) ] Φ n ( κ ) ,
G 11 ( p 1 ,   p 2 ,   η ,   κ ,   θ ,   L ) = { - k 2 / [ E 1 ( p 1 ,   0 ) E 1 ( p 2 ,   0 ) ] } × exp ( b 3 x 1 κ 2 cos 2   θ + b 3 y 1 κ 2 sin 2   θ ) × exp [ i γ x 1 ( p x 1 - p x 2 ) κ   cos   θ   + i γ y 1 ( p y 1 - p y 2 ) κ   sin   θ ] × E 1 ( p 1 ,   κ ) E 1 ( p 2 ,   - κ ) ,
G 21 ( p 1 ,   p 2 ,   η ,   κ ,   θ ,   L )
= { k 2 / [ E 1 ( p 1 ,   0 ) E 1 * ( p 2 ,   0 ) ] } × exp ( ζ 3 x 1 κ 2 cos 2   θ + ζ 3 y 1 κ 2 sin 2   θ ) × exp [ i ( γ x 1 p x 1 - γ x 1 * p x 2 ) κ   cos   θ   + i ( γ y 1 p y 1 - γ y 1 * p y 2 ) κ   sin   θ ] × E 1 ( p 1 ,   κ ) E 1 * ( p 2 ,   κ ) ;
E 1 ( p 1 ,   κ )
= H n 1 ( g 4 x 1 p x 1 + g 5 x 1 κ   cos   θ + g 7 x 1 ) × H m 1 ( g 4 y 1 p y 1 + g 5 y 1 κ   sin   θ + g 7 y 1 ) ,
E 1 ( p 2 ,   κ )
= H n 1 ( g 4 x 1 p x 2 + g 5 x 1 κ   cos   θ + g 7 x 1 ) × H m 1 ( g 4 y 1 p y 2 + g 5 y 1 κ   sin   θ + g 7 y 1 ) ;
γ x 1 = ( 1 + i α x 1 η ) / ( 1 + i α x 1 L ) ;
b 3 x 1 = i γ x 1 ( η - L ) / k ;
g 4 x 1 = a x 1 / ( ( 1 + i α x 1 L ) { 1 - iL [ ( 2 a x 1 2 / k ) - α x 1 ] } ) 1 / 2 ,
g 5 x 1 = g 4 x 1 ( η - L ) / k ;
g 7 x 1 = g 4 x 1 b x 1 ( 1 + i α x 1 L ) / a x 1 ,
ζ 3 x 1 = [ Im ( γ x 1 ) ] ( L - η ) / k .
D χ ( p 1 ,   p 2 ,   L )
= π   Re 0 L d η 0 κ d κ 0 2 π d θ [ G 11 ( p 1 ,   p 1 ,   η ,   κ ,   θ ,   L ) + G 21 ( p 1 ,   p 1 ,   η ,   κ ,   θ ,   L ) + G 11 ( p 2 ,   p 2 ,   η ,   κ ,   θ ,   L ) + G 21 ( p 2 ,   p 2 ,   η ,   κ ,   θ ,   L ) - 2 G 11 ( p 1 ,   p 2 ,   η ,   κ ,   θ ,   L ) - 2 G 21 ( p 1 ,   p 2 ,   η ,   κ ,   θ ,   L ) ] Φ n ( κ ) .
B S ( p 1 ,   p 2 ,   L )
= - π   Re 0 L d η 0 κ d κ 0 2 π d θ [ G 11 ( p 1 ,   p 2 ,   η ,   κ ,   θ ,   L ) - G 21 ( p 1 ,   p 2 ,   η ,   κ ,   θ ,   L ) ] Φ n ( κ ) .
D S ( p 1 ,   p 2 ,   L )
= - π   Re 0 L d η 0 κ d κ 0 2 π d θ [ G 11 ( p 1 ,   p 1 ,   η ,   κ ,   θ ,   L ) - G 21 ( p 1 ,   p 1 ,   η ,   κ ,   θ ,   L ) + G 11 ( p 2 ,   p 2 ,   η ,   κ ,   θ ,   L ) - G 21 ( p 2 ,   p 2 ,   η ,   κ ,   θ ,   L ) - 2 G 11 ( p 1 ,   p 2 ,   η ,   κ ,   θ ,   L ) + 2 G 21 ( p 1 ,   p 2 ,   η ,   κ ,   θ ,   L ) ] Φ n ( κ ) .
D ψ = D χ + D S .
u HOA ( s x ,   s y ,   z = 0 ) = A n 1 , m 1 H n 1 ( a x 1 s x + b x 1 ) H m 1 ( a y 1 s y + b y 1 ) × exp [ - k / 2 ( α x 1 s x 2 + α y 1 s y 2 ) ] exp ( - i ϕ 1 )   + A n 2 , m 2 H n 2 ( a x 2 s x + b x 2 ) H m 2 ( a y 2 s y + b y 2 ) × exp [ - k / 2 ( α x 2 s x 2 + α y 2 s y 2 ) ] exp ( - i ϕ 2 ) = u n 1 , m 1 ( s x ,   s y ,   z = 0 ) + u n 2 , m 2 ( s x ,   s y ,   z = 0 ) ,
u FS , HOA ( p ,   z ) = u n 1 , m 1 FS ( p ,   z ) + u n 2 , m 2 FS ( p ,   z ) ,
H HOA ( p x ,   p y ,   L ,   κ x ,   κ y ,   z ) = H 1 HOA ( p x ,   p y ,   L ,   κ x ,   κ y ,   z )
+ H 2 HOA ( p x ,   p y ,   L ,   κ x ,   κ y ,   z ) ,
H 1 HOA ( p ) = { u n 1 , m 1 FS ( p ,   L ) / [ u n 1 , m 1 FS ( p ,   L ) + u n 2 , m 2 FS ( p ,   L ) ] } H 1 ( p ) ,
H 2 HOA ( p ) = { u n 2 , m 2 FS ( p ,   L ) / [ u n 1 , m 1 FS ( p ,   L ) + u n 2 , m 2 FS ( p ,   L ) ] } H 2 ( p ) .
B χ HOA ( p 1 ,   p 2 ,   L ) = π   Re 0 L d η 0 κ d κ 0 2 π d θ × [ G 1 A ( p 1 ,   p 2 ,   η ,   κ ,   θ ,   L ) + G 2 A ( p 1 ,   p 2 ,   η ,   κ ,   θ ,   L ) ] Φ n ( κ ) ,
G 1 A ( p 1 ,   p 2 ) = { 1 / [ u n 1 , m 1 FS ( p 1 ,   L ) + u n 2 , m 2 FS ( p 1 ,   L ) ] } × { 1 / [ u n 1 , m 1 FS ( p 2 ,   L ) + u n 2 , m 2 FS ( p 2 ,   L ) ] } × [ u n 1 , m 1 FS ( p 1 ,   L ) u n 1 , m 1 FS ( p 2 ,   L ) G 11 + u n 1 , m 1 FS ( p 1 ,   L ) u n 2 , m 2 FS ( p 2 ,   L ) G 112 + u n 2 , m 2 FS ( p 1 ,   L ) u n 1 , m 1 FS ( p 2 ,   L ) G 121 + u n 2 , m 2 FS ( p 1 ,   L ) u n 2 , m 2 FS ( p 2 ,   L ) G 12 ] ,
G 2 A ( p 1 ,   p 2 ) = { 1 / [ u n 1 , m 1 FS ( p 1 ,   L ) + u n 2 , m 2 FS ( p 1 ,   L ) ] } × { 1 / [ u n 1 , m 1 FS ( p 2 ,   L ) + u n 2 , m 2 FS ( p 2 ,   L ) ] } * × { u n 1 , m 1 FS ( p 1 ,   L ) [ u n 1 , m 1 FS ( p 2 ,   L ) ] * G 21 + u n 1 , m 1 FS ( p 1 ,   L ) [ u n 2 , m 2 FS ( p 2 ,   L ) ] * G 212 + u n 2 , m 2 FS ( p 1 ,   L ) [ u n 1 , m 1 FS ( p 2 ,   L ) ] * G 221 + u n 2 , m 2 FS ( p 1 ,   L ) [ u n 2 , m 2 FS ( p 2 ,   L ) ] * G 22 } ;
G 112 = { - k 2 / [ E 1 ( p 1 ,   0 ) E 2 ( p 2 ,   0 ) ] } exp { [ ( b 3 x 1 / 2 ) + ( b 3 x 2 / 2 ) ] κ 2 cos 2   θ + [ ( b 3 y 1 / 2 ) + ( b 3 y 2 / 2 ) ] κ 2 sin 2   θ } × exp [ i ( γ x 1 p x 1 - γ x 2 p x 2 ) κ   cos   θ + i ( γ y 1 p y 1 - γ y 2 p y 2 ) κ   sin   θ ] × E 1 ( p 1 ,   κ ) E 2 ( p 2 ,   - κ ) ,
G 121 = { - k 2 / [ E 2 ( p 1 ,   0 ) E 1 ( p 2 ,   0 ) ] } exp { [ ( b 3 x 1 / 2 ) + ( b 3 x 2 / 2 ) ] κ 2 cos 2   θ + [ ( b 3 y 1 / 2 ) + ( b 3 y 2 / 2 ) ] κ 2 sin 2   θ } × exp [ i ( γ x 2 p x 1 - γ x 1 p x 2 ) κ   cos   θ + i ( γ y 2 p y 1 - γ y 1 p y 2 ) κ   sin   θ ] E 2 ( p 1 ,   κ ) × E 1 ( p 2 ,   - κ ) ,
G 212 = { k 2 / [ E 1 ( p 1 ,   0 ) E 2 * ( p 2 ,   0 ) ] } exp { [ ( ζ 3 x 1 / 2 ) + ( ζ 3 x 2 / 2 ) ] κ 2 cos 2   θ + [ ( ζ 3 y 1 / 2 ) + ( ζ 3 y 2 / 2 ) ] κ 2 sin 2   θ } × exp [ i ( γ x 1 p x 1 - γ x 2 * p x 2 ) κ   cos   θ + i ( γ y 1 p y 1 - γ y 2 * p y 2 ) κ   sin   θ ] × E 1 ( p 1 ,   κ ) E 2 * ( p 2 ,   - κ ) ,
G 221 = { k 2 / [ E 2 ( p 1 ,   0 ) E 1 * ( p 2 ,   0 ) ] } exp { [ ( ζ 3 x 1 / 2 ) + ( ζ 3 x 2 / 2 ) ] κ 2 cos 2   θ + [ ( ζ 3 y 1 / 2 ) + ( ζ 3 y 2 / 2 ) ] κ 2 sin 2   θ } × exp [ i ( γ x 2 p x 1 - γ x 1 * p x 2 ) κ   cos   θ + i ( γ y 2 p y 1 - γ y 1 * p y 2 ) κ   sin   θ ] × E 2 ( p 1 ,   κ ) E 1 * ( p 2 ,   - κ ) .
G 12 ( p 1 ,   p 2 ,   η ,   κ ,   θ ,   L ) = { - k 2 / [ E 2 ( p 1 ,   0 ) E 2 ( p 2 ,   0 ) ] } × exp ( b 3 x 2 κ 2 cos 2   θ + b 3 y 2 κ 2 sin 2   θ ) × exp [ i γ x 2 ( p x 1 - p x 2 ) κ   cos   θ   + i γ y 2 ( p y 1 - p y 2 ) κ   sin   θ ] × E 2 ( p 1 ,   κ ) E 2 ( p 2 ,   - κ ) ,
G 22 ( p 1 ,   p 2 ,   η ,   κ ,   θ ,   L ) = { k 2 / [ E 2 ( p 1 ,   0 ) E 2 * ( p 2 ,   0 ) ] } × exp ( ζ 3 x 2 κ 2 cos 2   θ + ζ 3 y 2 κ 2 sin 2   θ ) × exp [ i ( γ x 2 p x 1 - γ x 2 * p x 2 ) κ   cos   θ   + i ( γ y 2 p y 1 - γ y 2 * p y 2 ) κ   sin   θ ] × E 2 ( p 1 ,   κ ) E 2 * ( p 2 ,   κ ) .
D χ HOA ( p 1 ,   p 2 ,   L ) = π   Re 0 L d η 0 κ d κ 0 2 π d θ × [ G 1 A ( p 1 ,   p 1 ,   η ,   κ ,   θ ,   L )   + G 2 A ( p 1 ,   p 1 ,   η ,   κ ,   θ ,   L )   + G 1 A ( p 2 ,   p 2 ,   η ,   κ ,   θ ,   L )   + G 2 A ( p 2 ,   p 2 ,   η ,   κ ,   θ ,   L )   - 2 G 1 A ( p 1 ,   p 2 ,   η ,   κ ,   θ ,   L )   - 2 G 2 A ( p 1 ,   p 2 ,   η ,   κ ,   θ ,   L ) ] Φ n ( κ ) ,
B S HOA ( p 1 ,   p 2 ,   L ) = - π   Re 0 L d η 0 κ d κ 0 2 π d θ × [ G 1 A ( p 1 ,   p 2 ,   η ,   κ ,   θ ,   L ) - G 2 A ( p 1 ,   p 2 ,   η ,   κ ,   θ ,   L ) ] Φ n ( κ ) .
D S HOA ( p 1 ,   p 2 ,   L ) = - π   Re 0 L d η 0 κ d κ 0 2 π d θ × [ G 1 A ( p 1 ,   p 1 ,   η ,   κ ,   θ ,   L )   - G 2 A ( p 1 ,   p 1 ,   η ,   κ ,   θ ,   L )   + G 1 A ( p 2 ,   p 2 ,   η ,   κ ,   θ ,   L )   - G 2 A ( p 2 ,   p 2 ,   η ,   κ ,   θ ,   L )   - 2 G 1 A ( p 1 ,   p 2 ,   η ,   κ ,   θ ,   L )   + 2 G 2 A ( p 1 ,   p 2 ,   η ,   κ ,   θ ,   L ) ] Φ n ( κ ) ,

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