Abstract

Using the integral transform technique, the analytical expressions for the mean-squared beam width and the angular spread of the Hermite–Gaussian (H–G) array beam in turbulence are derived for the case of both coherent and incoherent combinations. It is shown that the angular spread of the H–G array beam for the coherent combination may be more or less affected by turbulence than that for the incoherent combination (or a single H–G beam) depending on the beam parameters. For the coherent combination, there exists the oscillatory behavior of the angular spread, and the influence of turbulence on the angular spread is not monotonic versus the beam parameters. In addition, for the coherent combination case, the angular spread of the H–G array beam is less affected by turbulence than that of the Gaussian array beam. On the other hand, it is found that under a certain condition, the H–G array beam may have the same directionality as a single Gaussian beam both in free space and in turbulence if the angular spread is chosen as the characteristic parameter of the beam directionality. The main results are explained physically.

© 2009 Optical Society of America

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References

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2008 (6)

2007 (1)

Y. Cai, Y. Chen, H. T. Eyyuboglu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88, 467-475 (2007).
[CrossRef]

2005 (1)

2003 (3)

2002 (2)

G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19, 1592-1598 (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]

2000 (2)

1998 (2)

1991 (1)

1990 (2)

R. A. Chodzko, J. M. Bernard, and H. Mirels, “Coherent combination of multiline lasers,” Proc. SPIE 1224, 239-253 (1990).
[CrossRef]

A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2-14 (1990).
[CrossRef]

1979 (1)

P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29, 256-260(1979).
[CrossRef]

1978 (1)

E. Wolf and E. Collett, “Partially coherent sources which produce the same far-field intensity distribution as a laser,” Opt. Commun. 25, 293-296 (1978).
[CrossRef]

Andrews, J. R.

Andrews, L. C.

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

Baykal, Y.

Y. Cai, Y. Chen, H. T. Eyyuboglu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88, 467-475 (2007).
[CrossRef]

H. T. Eyyuboglu and Y. Baykal, “Hermite-sine-Gaussian and Hermite-sinH-Gaussian laser beams in turbulent atmosphere,” J. Opt. Soc. Am. A 22, 2709-2718 (2005).
[CrossRef]

Bernard, J. M.

R. A. Chodzko, J. M. Bernard, and H. Mirels, “Coherent combination of multiline lasers,” Proc. SPIE 1224, 239-253 (1990).
[CrossRef]

Bretenaker, F.

Brunel, M.

Cai, Y.

Y. Cai, Y. Chen, H. T. Eyyuboglu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88, 467-475 (2007).
[CrossRef]

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

Chen, X.

X. Ji, X. Chen, and B. Lü, “Spreading and directionality of partially coherent Hermite-Gaussian beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 25, 21-28 (2008).
[CrossRef]

X. Chen and X. Ji, “Directionality of partially coherent annular flat-topped beams propagating through atmospheric turbulence,” Opt. Commun. 281, 4765-4770 (2008).
[CrossRef]

Chen, Y.

Y. Cai, Y. Chen, H. T. Eyyuboglu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88, 467-475 (2007).
[CrossRef]

Chodzko, R. A.

R. A. Chodzko, J. M. Bernard, and H. Mirels, “Coherent combination of multiline lasers,” Proc. SPIE 1224, 239-253 (1990).
[CrossRef]

Clarence, E. C.

Collett, E.

E. Wolf and E. Collett, “Partially coherent sources which produce the same far-field intensity distribution as a laser,” Opt. Commun. 25, 293-296 (1978).
[CrossRef]

De Santis, P.

P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29, 256-260(1979).
[CrossRef]

Dogariu, A.

Du, X.

Eyyuboglu, H. T.

Y. Cai, Y. Chen, H. T. Eyyuboglu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88, 467-475 (2007).
[CrossRef]

H. T. Eyyuboglu and Y. Baykal, “Hermite-sine-Gaussian and Hermite-sinH-Gaussian laser beams in turbulent atmosphere,” J. Opt. Soc. Am. A 22, 2709-2718 (2005).
[CrossRef]

Floch, A. L.

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.

P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29, 256-260(1979).
[CrossRef]

Guattari, G.

P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29, 256-260(1979).
[CrossRef]

Herb, J. J.

Ji, X.

Lin, Q.

Lü, B.

Ma, H.

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]

Marty, J.

Mirels, H.

R. A. Chodzko, J. M. Bernard, and H. Mirels, “Coherent combination of multiline lasers,” Proc. SPIE 1224, 239-253 (1990).
[CrossRef]

Molva, E.

Palma, C.

P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29, 256-260(1979).
[CrossRef]

Phillips, R. L.

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

Schuster, G. L.

Shirai, T.

Siegman, A. E.

A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2-14 (1990).
[CrossRef]

Strohschein, J. D.

Wolf, E.

Yang, A.

Yang, D.

E. Zhang, X. Ji, D. Yang, and B. Lü, “Propagation and far-field beam quality of M×N Hermite-Gaussian beams propagating through atmospheric turbulence,” J. Mod. Opt. 55, 387-400 (2008).
[CrossRef]

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, E.

Zhao, D.

Zhu, Y.

Appl. Opt. (3)

Appl. Phys. B (1)

Y. Cai, Y. Chen, H. T. Eyyuboglu, and Y. Baykal, “Propagation of laser array beams in a turbulent atmosphere,” Appl. Phys. B 88, 467-475 (2007).
[CrossRef]

J. Mod. Opt. (1)

E. Zhang, X. Ji, D. Yang, and B. Lü, “Propagation and far-field beam quality of M×N Hermite-Gaussian beams propagating through atmospheric turbulence,” J. Mod. Opt. 55, 387-400 (2008).
[CrossRef]

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

J. Opt. Soc. Am. B (1)

Opt. Commun. (3)

X. Chen and X. Ji, “Directionality of partially coherent annular flat-topped beams propagating through atmospheric turbulence,” Opt. Commun. 281, 4765-4770 (2008).
[CrossRef]

E. Wolf and E. Collett, “Partially coherent sources which produce the same far-field intensity distribution as a laser,” Opt. Commun. 25, 293-296 (1978).
[CrossRef]

P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29, 256-260(1979).
[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. Express (2)

Opt. Lett. (2)

Proc. SPIE (2)

A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2-14 (1990).
[CrossRef]

R. A. Chodzko, J. M. Bernard, and H. Mirels, “Coherent combination of multiline lasers,” Proc. SPIE 1224, 239-253 (1990).
[CrossRef]

Other (1)

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

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Figures (11)

Fig. 1
Fig. 1

Schematic diagram of a M × N array beam.

Fig. 2
Fig. 2

Angular spread θ sp versus the beam width w 0 for M = N = 11 and 1. The calculation parameters are x d = y d = 0.03 m , m = n = 9 , and z = 20 km . Solid curve, C n 2 = 5 × 10 14 m 2 / 3 ; dashed curve, C n 2 = 0 .

Fig. 3
Fig. 3

Angular spread θ sp versus the beam order m = n for M = N = 7 and 1. The calculation parameters are w 0 = 0.01 m , x d = y = d 0.02 m , and z = 25 km . Solid curve, C n 2 = 5 × 10 14 m 2 / 3 ; dashed curve, C n 2 = 0 .

Fig. 4
Fig. 4

Angular spread θ sp versus the beam separation distance x d = y d for M = N = 7 and 1. The calculation parameters are w 0 = 0.01 m , m = n = 3 , and z = 20 km . Solid curve, C n 2 = 10 14 m 2 / 3 ; dashed curve, C n 2 = 0 .

Fig. 5
Fig. 5

Parameter γ versus the beam width w 0 . The calculation parameters are the same as in Fig. 2.

Fig. 6
Fig. 6

Parameter γ versus the beam order m = n . The calculation parameters are the same as in Fig. 3.

Fig. 7
Fig. 7

Parameter γ versus the beam separation distance x d = y d . The calculation parameters are the same as in Fig. 4.

Fig. 8
Fig. 8

Angular spread θ sp versus the beam width w 0 for m = n = 4 and 0. The calculation parameters are x d = y d = 0.03 m , M = N = 3 , and z = 30 km . Solid curve, C n 2 = 5 × 10 14 m 2 / 3 ; dashed curve, C n 2 = 0 .

Fig. 9
Fig. 9

Angular spread θ sp versus the beam separation distance x d = y d for m = n = 4 and 0. The calculation parameters are w 0 = 0.01 m , M = N = 5 , and z = 20 km . Solid curve, C n 2 = 5 × 10 14 m 2 / 3 ; dashed curve, C n 2 = 0 .

Fig. 10
Fig. 10

Angular spread θ sp versus the beam number M = N for (a)  m = n = 6 and 0 (b)  m = n = 3 and 0. The calculation parameters are w 0 = 0.005 m , x d = y d = 0.01 m , and z = 20 km . Solid curve, C n 2 = 10 14 m 2 / 3 ; dashed curve, C n 2 = 0 .

Fig. 11
Fig. 11

Mean-squared beam width w ( z ) versus propagation distance z. The calculation parameters are listed in Table 1, where Eq. (28) is satisfied. Line a is the H–G array beam, b is the H–G beam, c is the Gaussian array beam, and d is the Gaussian beam. Solid curve, C n 2 = 5 × 10 14 m 2 / 3 ; dashed curve, C n 2 = 0 .

Tables (1)

Tables Icon

Table 1 Beam Parameters Relating to Fig. 11

Equations (40)

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E i j ( x , y , 0 ) = E i ( x , 0 ) E j ( y , 0 ) ,
E i ( x , 0 ) = H m [ 2 ( x i x d ) w 0 ] exp [ ( x i x d ) 2 w 0 2 ] ,
E j ( y , 0 ) = H n [ 2 ( y j x d ) w 0 ] exp [ ( y j y d ) 2 w 0 2 ] ,
i [ M 1 2 , M 1 2 ] , j [ N 1 2 , N 1 2 ] ,
W ( r 1 , r 2 , 0 ) = W ( x 1 , x 2 , 0 ) W ( y 1 , y 2 , 0 ) ,
W ( x 1 , x 2 , 0 ) = i 1 E i 1 ( x 1 , 0 ) i 2 E i 2 * ( x 2 , 0 ) ,
W ( y 1 , y 2 , 0 ) = j 1 E j 1 ( y 1 , 0 ) j 2 E j 2 * ( y 2 , 0 ) .
I ( r , z ) = ( k 2 π z ) 2 d 2 r 1 d 2 r 2 W ( r 1 , r 2 , 0 ) × exp { ( i k 2 z ) [ ( r 1 2 r 2 2 ) 2 r · ( r 1 r 2 ) ] } exp [ ψ * ( r , r 1 , z ) + ψ ( r , r 2 , z ) ] m ,
exp [ ψ * ( r , r 1 , z ) + ψ ( r , r 2 , z ) ] m = exp { 4 π 2 k 2 z 0 1 0 κ Φ n ( κ ) [ 1 J 0 ( κ ξ | r 2 - r 1 | ) ] d κ d ξ } ,
u = ( r 2 + r 1 ) / 2 , v = r 2 r 1 ,
I ( r , z ) = ( k 2 π z ) 2 d 2 u d 2 v W ( u , v , 0 ) exp ( i k z u · v ) exp ( i k z r · v ) × exp { 4 π 2 k 2 z 0 1 0 κ Φ n ( κ ) [ 1 J 0 ( κ ξ v ) ] d κ d ξ } .
w 2 ( z ) = 2 r 2 I ( r , z ) d x d y I ( r , z ) d x d y .
w 2 ( z ) = ( A x + A y ) + B x + B y k 2 z 2 + F z 3 ,
A x = i 1 = M 1 2 M 1 2 i 2 = M 1 2 M 1 2 exp ( D / 2 ) { w 0 2 [ L m ( D ) 2 L m ( 1 ) ( D ) ] + ( i 1 + i 2 ) 2 x d 2 L m ( D ) } / 2 C ,
B x = i 1 = M 1 2 M 1 2 i 2 = M 1 2 M 1 2 2 exp ( D / 2 ) { L m ( D ) 2 L m ( 1 ) ( D ) [ L m ( D ) 4 L m ( 1 ) ( D ) + 4 L m ( 2 ) ( D ) ] D } / w 0 2 C ,
F = 8 3 π 2 0 κ 3 Φ n ( κ ) d κ ,
D = ( i 1 i 2 ) 2 x d 2 / w 0 2 ,
C = i 1 = M 1 2 M 1 2 i 2 = M 1 2 M 1 2 exp ( D / 2 ) L m ( D ) .
θ sp ( z ) lim z w ( z ) z = B / k 2 + F z ,
W i j ( r 1 , r 2 , 0 ) = W i ( x 1 , x 2 , 0 ) W j ( y 1 , y 2 , 0 ) ,
W i ( x 1 , x 2 , 0 ) = E i ( x 1 , 0 ) E i * ( x 2 , 0 ) ,
W j ( y 1 , y 2 , 0 ) = E j ( y 1 , 0 ) E j * ( y 2 , 0 ) ,
I ( r , z ) = i = M 1 2 M 1 2 j = N 1 2 N 1 2 I i j ( r , z ) ,
w 2 ( z ) = [ M 2 1 6 x d 2 + N 2 1 6 y d 2 + ( 1 + n + m ) w 0 2 ] + 4 ( 1 + n + m ) k 2 w 0 2 z 2 + F z 3 .
θ sp ( z ) = 4 ( 1 + n + m ) k 2 w 0 2 + F z .
θ sp ( z ) = 4 k 2 w 0 2 + F z .
Φ n ( κ ) = 0.033 C n 2 ( κ 2 + 1 L 0 2 ) 11 / 6 exp ( κ 2 κ m 2 ) ,
B k 2 = 4 ( 1 + n 1 + m 1 ) k 1 2 w 01 2 = 4 k 2 2 w 0 2 2
w 2 ( z ) = w x 2 ( z ) + w y 2 ( z ) ,
w x 2 ( z ) = 2 F 1 / F 2 ,
F 1 = x 2 I ( r , z ) d x d y ,
F 2 = I ( r , z ) d x d y .
exp ( i 2 π x s ) d x = δ ( s ) ,
F 1 = ( z k ) 2 d u x d u y d v x d v y i 1 = M 1 2 M 1 2 i 2 = M 1 2 M 1 2 H m [ 2 w 0 ( u x 1 2 v x i 1 x d ) ] H m [ 2 w 0 ( u x + 1 2 v x i 2 x d ) ] × exp [ 2 u x 2 + 1 2 v x 2 2 u x ( i 1 + i 2 ) x d + v x ( i 1 i 2 ) x d + ( i 1 2 + i 2 2 ) x d 2 w 0 2 ] × j 1 = N 1 2 N 1 2 j 2 = N 1 2 N 1 2 H n [ 2 w 0 ( u y 1 2 v y j 1 y d ) ] H n [ 2 w 0 ( u y + 1 2 v y j 2 y d ) ] × exp [ 2 u y 2 + 1 2 v y 2 - 2 u y ( j 1 + j 2 ) y d + v y ( j 1 - j 2 ) y d + ( j 1 2 + j 2 2 ) y d 2 w 0 2 ] × exp ( i k z u · v ) exp { 4 π 2 k 2 z 0 1 0 κ Φ n ( κ ) [ 1 J 0 ( κ ξ v ) ] d κ d ξ } δ ( v x ) δ ( v y ) ,
f ( x ) δ ( x ) d x = f ( 0 ) ,
F 1 = w 0 2 2 n π ( z k ) 2 j 1 = N 1 2 N 1 2 j 2 = N 1 2 N 1 2 exp [ ( j 1 j 2 ) 2 y d 2 / 2 w 0 2 ] L n [ ( j 1 j 2 ) 2 y d 2 / w 0 2 ] × d u x d v x i 1 = M 1 2 M 1 2 i 2 = M 1 2 M 1 2 H m [ 2 w 0 ( u x 1 2 v x i 1 x d ) ] H m [ 2 w 0 ( u x + 1 2 v x i 2 x d ) ] × exp ( 2 u x 2 + 1 2 v x 2 2 u x ( i 1 + i 2 ) x d + v x ( i 1 i 2 ) x d + ( i 1 2 + i 2 2 ) x d 2 w 0 2 ) exp ( i k z u · v ) × exp { 4 π 2 k 2 z 0 1 0 κ Φ n ( κ ) [ 1 J 0 ( κ ξ v ) ] d κ d ξ } δ ( v x ) .
F 1 = 2 n + m 1 π w 0 2 C j 1 = N 1 2 N 1 2 j 2 = N 1 2 N 1 2 exp [ ( j 1 j 2 ) 2 y d 2 / 2 w 0 2 ] L n [ ( j 1 j 2 ) 2 y d 2 / w 0 2 ] × ( A x 2 + B x 2 k 2 z 2 + F 4 z 3 ) .
F 2 = 2 n + m 1 π w 0 2 C j 1 = N 1 2 N 1 2 j 2 = N 1 2 N 1 2 exp [ ( j 1 j 2 ) 2 y d 2 / 2 w 0 2 ] L n [ ( j 1 j 2 ) 2 y d 2 / w 0 2 ] .
w x 2 ( z ) = A x + B x k 2 z 2 + F 2 z 3 .
w 2 ( z ) = ( A x + A y ) + B x + B y k 2 z 2 + F z 3 .

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