Influence of atmospheric turbulence on the propagation of superimposed partially coherent Hermite–Gaussian beams

Optics Communications 282 (2009) 7–13

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Influence of atmospheric turbulence on the propagation of superimposed partially coherent Hermite–Gaussian beams Xiaoqing Li, Xiaowen Chen, Xiaoling Ji * Department of Physics, Sichuan Normal University, Chengdu 610068, China

a r t i c l e

i n f o

Article history: Received 4 June 2008 Received in revised form 16 September 2008 Accepted 20 September 2008

Keywords: Superimposed partially coherent Hermite– Gaussian beams Atmospheric turbulence Propagation Beam power focusability Maximum intensity

a b s t r a c t The influence of atmospheric turbulence on the propagation of superimposed partially coherent Hermite–Gaussian (H–G) beams is studied in detail. The closed-form propagation equation of superimposed partially coherent H–G beams through atmospheric turbulence is derived. It is shown that the turbulence accelerates the evolution of three stages which superimposed partially coherent H–G beams undergo. The turbulence results in a beam spreading and a decrease of the maximum intensity. However, the larger the beam number M, the beam order m, the separate distance xd, and the smaller the beam correlation length r0 are, the less the power focusability of superimposed partially coherent H–G beams is affected by the turbulence. Specially, superimposed partially coherent H–G beams are less sensitive to turbulence than superimposed fully ones, and than partially coherent H–G beams if the beam power focusability and the maximum intensity are taken as beam criterions. However, the maximum intensity of superimposed partially coherent H–G beams is less sensitive or more sensitive to turbulence than that of superimposed Gaussian Schell-model (GSM) beams depending on r0. Ó 2008 Elsevier B.V. All rights reserved.

1. Introduction In the past decades, the off-axis superposition of beams has been attracted much attention because of their wide applications in high-power system, inertial confinement fusion and high-energy weapons, and so on [1–4]. Up to now, theoretical and numerical analysis for coherent and incoherent beam superposition with linear, rectangular and radial array geometries for coherent off-axis Gaussian beams, Hermite–Gaussian (H–G) beams, etc have been performed [4–7]. On the other hand, the propagation of laser beams through atmospheric turbulence is a subject of considerable importance [8]. Much work has been carried out concerning the spreading of laser beams in atmospheric turbulence, and it was demonstrated that partially coherent beams are less sensitive to the effects of turbulence than fully coherent ones [9–12]. Recently, we studied the propagation properties of coherent off-axis Gaussian beams and H–G beams through atmospheric turbulence, respectively [13,14], in which the incoherent beam superposition is considered. In practice, particularly in high-power technology, a partially coherent laser source is often encountered. In 2003, Li and Lü studied characteristics of the off-axis Gaussian Schell-model (GSM) beams for the superposition of the cross-spectral density function in free space by using the Wigner distribution function [15]. Recently, the propagation of GSM array beams for the superposition of the intensity through atmospheric turbulence was investigated by Cai * Corresponding author. E-mail address: [email protected] (X. Ji). 0030-4018/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2008.09.063

and Lin [16]. Very recently, we studied two types of the superposition, i.e., the superposition of the intensity and the superposition of the cross-spectral density function of off-axis GSM beams propagating through atmospheric turbulence [17]. However, in practice, there are certain scenarios to minimize excitation of nonlinearities within the crystal of a laser, or when the received spot needs to have a multiple spot pattern. In addition, the conversion efficiency, power out/power in, is greater for higher order modes. In those cases, the optical field is modeled mathematically as a multimode beam with Hermite and Laguerre polynomials in rectangular (CO2 laser) and cylindrical (HeNe laser) coordinates, respectively. The aim of this paper is to perform a detailed study of the propagation of superimposed partially coherent H–G beams through atmospheric turbulence, in which the superposition of the crossspectral density function of off-axis partially coherent H–G beams is considered. The propagation properties of partially coherent H–G beams, superimposed GSM beams and superimposed fully coherent H–G beams through atmospheric turbulence can be treated as special cases of our general result. In comparison with the behavior of partially coherent H–G beams, superimposed GSM beams and superimposed fully coherent H–G beams in turbulence, some interesting results are obtained.

2. Propagation equation As shown in Fig. 1, one-dimensional (1D) array beam in rectangular symmetry consists of M individual off-axis partially coherent

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X. Li et al. / Optics Communications 282 (2009) 7–13

H–G beams positioned at the source plane z = 0, the waist width of the corresponding TEM00 mode is w0, and the separate distance is xd. For definiteness let M = odd numbers. We assume that individual off-axis partially coherent H–G beams are correlated and are perfect replicas of each other. Thus, the cross-spectral density function of the superimposed partially coherent H–G beam at the source plane z = 0 is expressed as [15,18]

and

2m=2 Hm

   X m  m xþy pffiffiffi ¼ Hmn ðyÞHn ðxÞ n 2 n¼0

ð7Þ

after very tedious integral calculations, we obtain the average intensity of the superimposed partially coherent beam propagating through atmospheric turbulence at the plane z, which is given by

W 0 ðx01 ; x02 ; z ¼ 0Þ

"pffiffiffi # "pffiffiffi # 2ðx01  pxd Þ 2ðx02  qxd Þ H m w20 w20 p¼M1 q¼M1 2 2 ( ) ðx0  pxd Þ2 þ ðx02  qxd Þ2  exp  1 w20 ( ) ½ðx0  pxd Þ  ðx02  qxd Þ2  exp  1 ; 2r20 M1

¼

2 X

M1

2 X

y

Hm

x

xd

ð1Þ

Fig. 1. A schematic diagram of the one-dimensional beam array.

where r0 is the coherence length of individual off-axis partially coherent H–G beams at the plane z = 0, Hj (j = m, n) denotes the jth order Hermite polynomial. Based on the extended Huygens–Fresnel principle, the crossspectral density function of the superimposed partially coherent beam represented by Eq. (1) propagating through atmospheric turbulence reads as [8]

Z

k 2pz

1

1

Z

1

z=6km

1

z=3km

0.8

dx01 dx02 W 0 ðx01 ; x02 ; z ¼ 0Þ

1   ik  01 0 0 2 2  exp ðx2  x02 2 Þ  2ðx1 x1  x2 x2 Þ þ ðx1  x2 Þ 2z



 < exp½wðx01 ; x1 Þ þ w ðx02 ; x2 Þ>m ;

/Imax

Wðx1 ; x2 ; zÞ ¼

1.2

ð2Þ

ð3Þ

q0

z=0

z=12km 0.4 0.2

where k is the wave number related to the wavelength k by k = 2p/ k, w(x0 , x) represents the random part of the complex phase of a spherical wave due to the turbulence. him denotes average over the ensemble of the turbulent medium statistics, and is given by [19]

hexp½wðx01 ; x1 Þ þ w ðx02 ; x2 Þim " # ðx01  x02 Þ2 þ ðx01  x02 Þðx1  x2 Þ þ ðx1  x2 Þ2 ; ffi exp  2

0.6

0 -20

-10

0

10

20

x / w0 Fig. 2. Normalized average intensity distributions hI(x, z)i/Imax at different propagation distance z = 0, 3, 6 and 12 km. The calculation parameters are C 2n ¼ 5  1015 m2=3 , r0 = 0.05 m, M = 7, m = 2 and xd = 0.1 m.

with

q0 ¼ ð0:545C 2n k2 zÞ3=5 ;

ð4Þ

Z

1

pffiffiffiffi exp½ðx  yÞ Hn ðaxÞdx ¼ pð1  a2 Þn=2 Hn 2

1

Z

"

#

ay ; ð1  a2 Þ1=2

ð5Þ

1.2

C n2 = 5 × 10 −15 m -2/3

C n2 = 5 × 10 −16 m -2/3

1 0.8

/Imax

where q0 is the spatial coherence radius of a spherical wave propagating in turbulence, C 2n is the refraction index structure constant which describes how strong the turbulence is. It is noted that a quadratic approximation of Rytov’s phase structure function is used in Eq. (3) to obtain an analytical result. This approximation has been shown to be a good approximation in practical applications [20]. On substituting from Eq. (1) into Eq. (2), letting x1 = x2 = x, and recalling the formulae [21]

C n2 = 3 × 10 −14 m -2/3

C n2 = 0

0.6 0.4 0.2

1

2

exp½ðx  yÞ Hm ðaxÞHn ðbxÞdx 1 ½n=2 pffiffiffiffi X ¼ ðb=aÞn p

minðm;n2kÞ X 

m



n! 1 Þk 2l ð1  k!ðn  2k  lÞ! l ðb=aÞ2 k¼0 l¼0 " # mþn ay 2 2 kl ; ð6Þ  ð1  a Þ  Hmþn2k2l ð1  a2 Þ1=2

0 -20

-10

0

10

20

x / w0 Fig. 3. Normalized average intensity distributions hI(x, z)i/Imax for different values of the refraction index structure constant C 2n ¼ 0, 5  1016 m2/3, 5  1015 m2/3 and 3  1014 m2/3. The calculation parameters are z = 6 km, r0 = 0.05 m, M = 7, m = 2 and xd = 0.1 m.

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X. Li et al. / Optics Communications 282 (2009) 7–13

hIðx;zÞi ¼ Wðx; x;zÞ M1 2

M1 2

X

¼

X

k 2 1 2 2 2zAD w0 A M1

p¼M1 q¼ 2

(

2

 exp 

n¼0 r¼0



2

Rk 2 ikE ik x þ ð 2  2Þðp  qÞxd  RC x þ S T; ð8Þ 2z 4z2 zD

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 ik A¼ þ B ; 2z w20 2

A

þ 2

E¼1

B¼

S¼

ð10Þ

ð11Þ

; D2

2A2

ð12Þ

;

! 2 1 2 ik 2 k 1 ðp  qÞ2 x2d þ RC þ ðp  q2 Þx2d  4 2z 4z2 D2 q20 

ikEC 2zD2

ðp  qÞxd ;

1

r20

þ

2

q20

ð14Þ

ð15Þ

;

ik 2 px  ðp  qÞxd ; z d q20 1 F ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : A w20 A2  2

C¼

ð16Þ ð17Þ

Eq. (8) is the closed-form propagation equation of superimposed partially coherent H–G beams through atmospheric turbulence, which is the main result obtained in this paper. From Eq. (8) it can be seen that the average intensity depends on the refraction index structure constant C 2n , beam correlation length r0, beam number M, beam order m, separate distance xd and propagation distance z.

ð13Þ

a a

l

with

E2

B

n !r

2 2  1 2 2 w20 F 2 B2 w0 D " #   ik ikEx þ ikðp  qÞxd  ECz ;  Hmn  Fx þ FC Hmþn2r2l pffiffiffi 2 z 2ðD w20  2Þ1=2 z

ð9Þ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 ik B2 þ B þ  2; D¼ 2 2z 4A w0 2 1

l¼0

 n m n! w0 2 2 þl pffiffiffi FB r!ðn  2r  lÞ! 2 !mþn rl 2

 1

)



where

R¼

m ½n=2 X X minðm;n2rÞ X  m  m 

T¼

!m=2

1.2

1.2 1

σ 0 = 0.03m

/Imax

0.8

/Imax

1

M=5 M=1

0.6

0.8

σ 0 = 0.01m

0.6

σ0 → ∞ 0.4

M=7 0.4 0.2

0.2 0 -15

0 -15

-10

-5

0

5

10

-5

0

5

10

15

x / w0

15

x / w0

b

-10

b

1.2

1.2 1

1

M=5 /Imax

0.8

/Imax

0.8 0.6

M=1

M=7

0.4

σ0 → ∞

0.6

0.4

0.2

0.2 0 -50

σ 0 = 0.03m

σ 0 = 0.01m

-25

0

25

50

x / w0 Fig. 4. Normalized average intensity distributions hI(x, z)i/Imax for different values of the beam number M = 1, 5 and 7. The calculation parameters are z = 18 km, r0 = 0.04 m, m = 2 and xd = 0.1 m. (a) C 2n ¼ 0, (b) C 2n ¼ 1014 m2=3 .

0 -40

-20

0

20

40

x / w0 Fig. 5. Normalized average intensity distributions hI(x, z)i/Imax for different values of the beam correlation length r0 = 0.01 m, 0.03 m and 1. The calculation parameters are z = 18 km, M = 5, m = 2 and xd = 0.1 m. (a) C 2n ¼ 0, (b) C 2n ¼ 1014 m2=3 .

10

X. Li et al. / Optics Communications 282 (2009) 7–13

Some interesting results can be treated as special cases of Eq. (8), which are given as follows:

(a) For m = 0, Eq. (8) reduces to

(a) For M = 1, Eq. (8) reduces to the average intensity of partially coherent H–G beams propagating atmospheric turbulence, i.e.,

hIðx; zÞi ¼

k 2 1 2zAD w20 A2

!m2

2

exp 

!

Rk 2 x T1; 4z2

ð18Þ

where

T1 ¼

m ½n=2 X minðm;n2rÞ X X m n¼0 r¼0

 1

!

m

n

l¼0

l

!r

2 w20 F 2 B2

!

 1

2

 n m n! w0 2 2 þl pffiffiffi FB r!ðn  2r  lÞ! 2 mþn ! 2 rl

w20 D2

" #   ik ikEx :  Hmn  Fx Hmþn2r2l pffiffiffi 2 z 2ðD w20  2Þ1=2 z

ð19Þ

For C 2n ¼ 0 case (without turbulence), Eq. (18) in this paper is essentially consistent with Eq. (4) in Ref. [18] if the corresponding intensity of partially H–G beams propagating in free space is considered.

2 X

M1

2 X

ð20Þ

Eq. (20) is the average intensity of superimposed GSM beams through atmospheric turbulence. It is noted that, for C 2n ¼ 0 (i.e., q2 0 ¼ 0) Eq. (8) reduces to the average intensity of superimposed partially coherent H–G beams propagating in free space; while for r0 ? 1 Eq. (8) reduces to the average intensity of superimposed fully coherent H–G beams propagating through atmospheric turbulence. The analytical expressions are omitted here. 3. Numerical calculation results and analysis Numerical calculations were performed to illustrate the influence of turbulence on the propagation of superimposed partially coherent H–G beams. The calculation parameters are k = 107 m1 and w0 = 0.05 m. Fig. 2 gives the normalized average intensity distributions hI(x, z)i/Imax of superimposed partially coherent H–G beams in turbulence C 2n ¼ 5  1015 m2=3 at different propagation distance z = 0, 3, 6 and 12 km, respectively, where Imax denotes the maximum

a a

( 2 k Rk exp  2 x2 2zAD 4z p¼M1 q¼M1 2 2  

 ikE ik  2 ðp  qÞxd  RC þ xþS ; 2 2z zD M1

hIðx; zÞi ¼

1.2

1.2 1

m=6

1

xd =2 w0

/Imax

0.8

/Imax

0.8 0.6

m=12 m=0

0.6

xd =3 w0

xd =1 w0

0.4

0.4 0.2 0.2 0 -20

-10

0

10

0 -15

20

-10

-5

b

b

1.2

/Imax

/Imax

10

15

1.2 xd =2 w0

0.8

0.8

m=6

m=12

m=0

0.4

0.6

xd =1 w0

xd =3 w0

0.4 0.2

0.2 0 -50

5

1

1

0.6

0

x / w0

x / w0

-25

0

25

50

x / w0 Fig. 6. Normalized average intensity distributions hI(x, z)i/Imax for different values of beam order m = 0, 6 and 12. The calculation parameters are z = 18 km, r0 = 0.04 m, M = 7, and xd = 0.1 m. (a) C 2n ¼ 0, (b) C 2n ¼ 1014 m2=3 .

0 -60

-40

-20

0

20

40

60

x / w0 Fig. 7. Normalized average intensity distributions hI(x, z)i/Imax for different values of the relative separate distance xd/w0 = 1, 2 and 3. The calculation parameters are z = 18 km, r0 = 0.04 m, M = 5 and m = 2. (a) C 2n ¼ 0, (b) C 2n ¼ 2  1014 m2=3 .

11

X. Li et al. / Optics Communications 282 (2009) 7–13

intensity. From Fig. 2 it can be seen that superimposed partially coherent H–G beams propagating through atmospheric turbulence undergo three stages of evolution with increasing propagation distance z, i.e., at first their beam profile is the fringe pattern which is similar to the initial one (see z = 3 km), and then it becomes a flattopped profile with increasing z (see z = 6 km), at last it turns into a Gaussian-like profile in the far field (see z = 12 km). The normalized average intensity distributions hI(x, z)i/Imax of superimposed partially coherent H–G beams for different values of C 2n at the plane z = 6 km are depicted in Fig. 3. Fig. 3 shows that for C 2n ¼ 0 (in free space) the beam profile is the fringe pattern, but the oscillatory behavior becomes weaker with increasing turbulence. The beam profile becomes a flat-topped profile for C 2n ¼ 5  1015 m2=3 , at last it turns into a Gaussian-like profile when the turbulence is strong enough (e.g., C 2n ¼ 3  1014 m2=3 ). The normalized average intensity distributions hI(x, z)i/Imax of superimposed partially coherent H–G beams with different values of beam number M in free space and in turbulence at the plane z = 18 km are plotted in Fig. 4a and b, respectively. As can be seen, in free space the beam spreading increases with increasing M (see Fig. 4a); but the curves of hI(x, z)i/Imax are close to each other for different values of M, and become Gaussian-like profile due to turbulence (see Fig. 4b, C 2n ¼ 1014 m2=3 ). It means that beam power focusability of superimposed partially coherent H–G beams with larger M is less sensitive to turbulence than those with smaller M. The normalized average intensity distributions hI(x, z)i/Imax of superimposed partially coherent H–G beams with and without tur-

a

bulence for different values of the beam correlation length r0, beam order m and separate distance xd at the plane z = 18 km are plotted in Figs. 5–7, respectively. It can be seen that in free space the beam spreading increases with decreasing r0 (see Fig. 5a), with increasing m (see Fig. 6a), and with increasing xd (see Fig. 7a). However, in turbulence (C 2n ¼ 1014 m2=3 ) the curves of hI(x, z)i/Imax are very close to each other for different values of r0 (see Fig. 5b), for different values of m (see Fig. 6b), and for different values of xd (see Fig. 7b). The normalized average intensity distributions hI(x, z)i/I0max of superimposed partially coherent H–G beams with and without turbulence for M = 1 (i.e., a partially coherent H–G beam) and M = 11 (i.e., a superimposed partially coherent H–G beam) are shown in Fig. 8a and b, respectively, where I0max is the maximum intensity of the corresponding beam in free space. The solid and dashed curves represent hI(x, z)i/I0max in free space and in turbulence (C 2n ¼ 1014 m2=3 ), respectively. It is clear that the turbulence results in a beam spreading and a decrease of the maximum intensity both for M = 1 and 11. However, the decrement in the maximum intensity is larger for M = 1 than that for M = 11 due to turbulence, e.g., the maximum intensity is 0.261 and 0.522 times in turbulence as that in free space for M = 1 and 11, respectively. The normalized average intensity distributions hI(x, z)i/I0max of superimposed partially coherent H–G beams with and without turbulence for r0 ? 1 (i.e., a superimposed fully coherent H–G beam) and r0 = 0.03 m (i.e., a superimposed partially coherent H–G beam) are given in Fig. 9a and b, respectively. As can be seen that the decrement in the maximum intensity is larger for r0 ? 1 than

a

1.2

1.2 1

1

C n2 = 0

0.8

0.6

/I0max

/I0max

C n2 = 0

C n2 = 10 −14 m -2/3

0.4

0.8 0.6

C n2 = 10 −14 m -2/3

0.4 0.2

0.2

0 -20

-10

0

10

0 -25

20

-20

-15

-10

-5

x / w0

b

b

1.2

0.8 0.6

C n2 = 10 −14 m -2/3

0.6

0.2

0.2

-10

0

10

20

30

x / w0 Fig. 8. Normalized average intensity distributions hI(x, z)i/I0max with turbulence (C 2n ¼ 1014 m2=3 ) and without turbulence (C 2n ¼ 0). The calculation parameters are z = 12 km, r0 = 0.04 m, xd = 0.1 m and m = 2. (a) M = 1, (b) M = 11.

15

20

25

20

25

C n2 = 0

0.8

0.4

-20

10

1.2

0.4

0 -30

5

1

C n2 = 0

/I0max

/I0max

1

0

x / w0

0 -25

C n2 = 10 −14 m -2/3

-20

-15

-10

-5

0

5

10

15

x / w0 Fig. 9. Normalized average intensity distributions hI(x, z)i/I0max with turbulence (C 2n ¼ 1014 m2=3 ) and without turbulence (C 2n ¼ 0). The calculation parameters are z = 12 km, M = 7, m = 2, and xd = 0.1 m. (a) r0 ? 1, (b) r0 = 0.03 m.

12

X. Li et al. / Optics Communications 282 (2009) 7–13

that for r0 = 0.03 m because of turbulence, e.g., the maximum intensity is 0.266 and 0.474 times in turbulence as that in free space for r0 ? 1 and 0.03 m, respectively. It implies that the maximum intensity of superimposed partially coherent H–G beams is less sensitive to turbulence than that of partially coherent H–G beams, and than that of superimposed fully coherent H–G beams. It is noted that the maximum intensity of superimposed partially coherent H–G beam is less sensitive or more sensitive to turbulence than that of superimposed GSM beams depending on the beam correlation length r0. Several numerical examples are given in Fig. 10a–d, respectively. From Fig. 10a and b it can be seen that for r0 = 0.01 m the decrement in the maximum intensity is larger for m = 0 (i.e., a superimposed GSM beam) than that for m = 16 (i.e., a superimposed partially coherent H–G beam) due to turbulence, e.g., the maximum intensity is 0.388 and 0.563 times in turbulence as that in free space for m = 0 and 16, respectively. On the other hand, Fig. 10c and d indicate that for r0 = 0.3 m the decrement in the maximum intensity is smaller for m = 0 than that for m = 16 due to turbulence, e.g., the maximum intensity is 0.257 and 0.180 times in turbulence as that in free space for m = 0 and 16, respectively. It means that the maximum intensity of superimposed partially coherent H–G beams is less sensitive to turbulence than that of superimposed GSM beams when the value of r0 is

a

small. Conversely, the opposite outcome will appear when the value of r0 is large. 4. Conclusions The influence of turbulence on the propagation of superimposed partially coherent H–G beams has been studied in detail, in which the superposition of the cross-spectral density function of off-axis partially coherent H–G beams is considered. The closed-form propagation equation of superimposed partially coherent H–G beams through atmospheric turbulence has been derived. It has been shown that superimposed partially coherent H–G beams propagating through atmospheric turbulence undergo three stages of evolution with increasing propagation distance, and turbulence accelerates the evolution of three stages which multi-Gaussian beams undergo. The turbulence results in a beam spreading and a decrease of the maximum intensity of superimposed partially coherent H–G beams. However, the power focusability of superimposed partially coherent H–G beams with larger M, m, xd and smaller r0 is less sensitive to turbulence than those with smaller M, m, xd and larger r0. If the beam power focusability and the maximum intensity are taken as beam criterions, superimposed partially coherent H–G beams are less sensitive to turbulence than

b

1.2

1

1

Cn2 = 0

0.8

/I0max

/I0max

0.8

0.6

1.2

Cn2 = 10−14 m -2/3

0.4

0.2

0.2

-10

0

10

Cn2 = 10 −14 m -2/3

0.6

0.4

0 -20

C n2 = 0

0 -20

20

-10

x / w0

c

0

10

20

x / w0

d

1.2

1

1.2

1 C n2 = 0

/I0max

/I0max

0.6

0.4

C n2 = 10 −14 m -2/3

0.6

Cn2 = 10−14 m-2/3

0.4

0.2

0.2

0 -20

C n2 = 0

0.8

0.8

-10

0

x / w0

10

20

0 -20

-10

0

10

20

x / w0

Fig. 10. Normalized average intensity distributions hI(x, z)i/I0max with turbulence (C 2n ¼ 1014 m2=3 ) and without turbulence (C 2n ¼ 0). The calculation parameters are z = 12 km, M = 3 and xd = 0.1 m. (a) r0 = 0.01 m, m = 0, (b) r0 = 0.01 m, m = 16, (c) r0 = 0.3 m, m = 0, (d) r0 = 0.3 m, m = 16.

X. Li et al. / Optics Communications 282 (2009) 7–13

superimposed fully ones, and than partially coherent H–G beams. However, the maximum intensity of superimposed partially coherent H–G beams is less sensitive or more sensitive to turbulence than that of superimposed GSM beams depending on r0. The results obtained in this paper would be useful for studying superimposed partially coherent higher order mode laser beams in passage through atmospheric turbulence. Acknowledgment This work was supported by the National Natural Science Foundation of China under Grant 60778048. References [1] N. NIishi, T. Jitduno, K. Tsubakimoto, S. Matsuoka, N. Miyanaga, M. Nakatsuka, Opt. Rew. 7 (2000) 216.

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