Angular spread of partially coherent Hermite-cosh ... - OSA Publishing

Angular spread of partially coherent Hermite-cosh-Gaussian beams propagating through atmospheric turbulence Ailin Yang1, Entao Zhang1, Xiaoling Ji2 and Baida Lü1 1

Institute of Laser Physics and Chemistry, Sichuan University, Chendu 610064, China 2 Department of Physics, Sichuan Normal University, Chengdu 610068, China [email protected], [email protected]

Abstract: The propagation of partially coherent Hermite-cosh-Gaussian (HChG) beams through atmospheric turbulence is studied in detail. The analytical expression for the angular spread of partially coherent H-ChG beams in turbulence is derived. It is shown that the angular spread of partially coherent H-ChG beams with smaller spatial correlation length σ0, smaller waist width w0, smaller beam parameter Ω0, and larger beam orders m, n is less affected by turbulence than that of partially coherent H-ChG beams with larger σ0, w0, Ω0, and smaller m, n. Under a certain condition partially coherent H-ChG beams may generate the same angular spread as a fully coherent Gaussian beam in free space and also in atmospheric turbulence. The angular spread of partially coherent Hermite-Gaussian (HG), cosh-Gaussian (ChG), Gaussian Schell-model (GSM) beams, and fully coherent H-ChG, H-G, ChG, Gaussian beams is studied and treated as special cases of partially coherent H-ChG beams. The results are interpreted physically. ©2008 Optical Society of America OCIS codes: (050.1970) angular spread; (010.1330) atmospheric turbulence; (030.1640) partially coherent Hermite-cosh-Gaussian (H-ChG) beam.

References and links 1.

E. Collett and E. Wolf, “Is complete spatial coherence necessary for the generation of highly directional light beams?” Opt. Lett. 2, 27-29 (1978). 2. 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). 3. P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29, 256-260 (1979). 4. J. D. Farina, L. M. Narducci, and E. Collett, “Generation of highly directional beams from a globally incoherent source,” Opt. Commun. 32, 203-208 (1980). 5. T. Shirai, A. Dogariu, and E. Wolf, “Directionality of Gaussian Schell-model beams propagating in atmospheric turbulence,” Opt. Lett. 28, 610-612 (2003). 6. 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). 7. L. W. Casperson and A. A. Tovar, “Hermite–sinusoidal-Gaussian beams in complex optical systems,” J. Opt. Soc. Am. A 15, 954-961 (1998). 8. A. A. Tovar and L. W. Casperson, “Production and propagation of Hermite–sinusoidal-Gaussian laser beams,” J. Opt. Soc. Am. A 15, 2425-2432 (1998). 9. 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). 10. H. T. Eyyuboğlu and Y. Baykal. “Analysis of reciprocity of cos-Gaussian and cosh-Gaussian laser beams in a turbulent atmosphere,” Opt. Express 12, 4659-4674 (2004). 11. H. T. Eyyuboğlu, “Propagation of Hermite-cosh-Gaussian laser beams in turbulent atmosphere,” Opt. Commun. 245, 37-47 (2005).

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Received 21 Feb 2008; revised 29 Apr 2008; accepted 7 May 2008; published 23 May 2008

9 June 2008 / Vol. 16, No. 12 / OPTICS EXPRESS 8366

12. H. T. Eyyuboğlu and Y. Baykal, “Average intensity and spreading of cosh-Gaussian laser beams in the turbulent atmosphere,” Appl. Opt. 44, 976-983 (2005). 13. H. T. Eyyuboğlu and Y. Baykal. “Hermite-sine-Gaussian and Hermite-sinh-Gaussian laser beams in turbulent atmosphere,” J. Opt. Soc. Am. A 22, 2709-2718 (2005). 14. Y. Cai and S. He. “Average intensity and spreading of an elliptical Gaussian beam propagating in a turbulent atmosphere,” Opt. Lett. 31, 568-570 (2006). 15. Y. Cai and S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express, 14, 1353-1367 (2006). 16. H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Complex degree of coherence for partially coherent general beams in atmospheric turbulence,” J. Opt. Soc. Am. A 24, 2891-2900 (2007). 17. H. T. Eyyubolu, Y. Baykal, and Y. Cai, “Degree of polarization for partially coherent general beams in turbulent atmosphere,” Appl. Phys. B: Lasers Opt. 89, 91-97 (2007). 18. H. T. Eyyuboğlu and Y. Baykal, “Transmittance of partially coherent cosh-Gaussian, cos-Gaussian and annular beams in turbulence,” Opt. Commun. 278, 17-22 (2007). 19. Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89, 041117 (2006). 20. G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19, 1592-1598 (2002). 21. A. Dogariu and S. Amarande, “Propagation of partially coherent beams: turbulence-induced degradation,” Opt. Lett. 28, 10-12 (2003). 22. M. Zahid and M. S. Zubairy, “Directionality of partially coherent Bessel-Gauss beams,” Opt. Commun. 70, 361-364 (1989). 23. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE Press, 1998). 24. S. Wang, C. Ouyang, and M. A. Plonus, “Optical beam propagation for a partially coherent source in the turbulent atmosphere,” J. Opt. Soc. Am. 69, 1297-1304 (1979). 25. H. T. Yura, "Mutual coherence function of a finite cross section optical beam propagating in a turbulent medium," Appl. Opt. 11, 1399-1406 (1972). 26. V. A. Banakh and V. L. Mironov, “Phase approximation of the Huygens-Kirhhoff method in problems of laser beam propagation in the turbulent atmosphere,” Opt. Lett. 1, 172-174 (1977). 27. V. A. Banakh and V. L. Mironov, “Phase approximation of the Huygens-Kirhhoff method in problems of spase-limited optical-beam propagation in the turbulent atmosphere,” Opt. Lett. 4, 259-261 (1979). 28. C. Leader, “Atmospheric propagation of partially coherent radiation,” J. Opt. Soc. Am. 68, 175-178 (1978).

1.

Introduction

In 1978 Collett and Wolf predicted that partially coherent beams, like Gaussian Schell-model (GSM) beams, may have the same directionality as a fully coherent laser beam [1, 2]. It means that full spatial coherence in free space is not a necessary condition for highly directional light beams. The theoretical prediction was confirmed by the experiments [3, 4]. In 2003 Wolf and his collaborators showed that under a certain condition there exist the equivalent GSM beams in turbulence which may generate the same angular spread as a fully coherent laser beam [5]. Very recently, we have found that, besides the equivalent GSM beams, there also exist the equivalent partially coherent Hermite-Gaussian (H-G) beams which may have the same directionality as a fully coherent laser beam in free space and also in atmospheric turbulence [6]. On the other hand, Hermite-sinusoidal-Gaussian (H-SG) beams are solutions of the paraxial wave equation [7]. H-SG beams cover a broad range of beams such as Hermite sinh-( or cosh-) Gaussian (H-ShG or H-ChG), and Hermite sin- (or cos-) Gaussian (H-SiG or HCoG) beams. The production and propagation of H-SG beams in free space were studied in Ref. [8]. The propagation of laser beams in atmospheric turbulence is a topic that has been of considerable theoretical and practical interest for a long time. The propagation of H-G and Laguerre-Gaussian (L-G) beams, CoG and ChG beams, H-ChG beams, and H-SiG and H-ShG beams etc. through the turbulent atmosphere were investigated by Youngs, Eyyuboğlu, Cai and their coworkers in Refs. [9-15]. However, the studies are limited to the spatially fully coherent case. The complex degree of coherence and degree of polarization for partially

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coherent general beams in atmospheric turbulence, and the influence of turbulence on the propagation of partially coherent ChG, CoG and twisted anisotropic GSM beams was recently studied in Refs. [16-19]. It was shown both theoretically and experimentally that partially coherent beams are less affected by turbulence than the fully coherent laser beams [18, 20, 21]. The aim of this paper is to study the angular spread of partially coherent H-ChG beams propagating through atmospheric turbulence. In comparison with previous work our study is more general. The main results obtained in this paper are illustrated by numerical examples and interpreted physically. 2.

Angular spread of partially coherent Hermite-cosh-Gaussian beams in turbulence

The field distribution of H-ChG beams at the source plane z = 0 in the Cartesian coordinate system reads as [7, 8]

U (ρ, z = 0) = H m (

ρ2 + ρ2 2 2 ρ x )H n ( ρ y ) exp(− x 2 y ) cosh(Ω 0 ρ x + Ω 0 ρ y ) , w0 w0 w0

(1)

where ρ ≡ (ρx, ρy) is the two-dimensional position vector at the source plane z = 0, Hm(·) and Hn(·) denote the mth and nth order Hermite polynomials, w0 is the waist width of the Gaussian part, Ω0 is the beam parameter associated with the cosh part. The fully coherent beam can be extended to the partially coherent one by introducing a Gaussian term of the spectral degree of coherence [22]. The cross spectral density function of partially coherent H-ChG beams at the plane z = 0 is expressed as [7, 8, 22]

W (ρ1 , ρ 2 , z = 0) = H m (

ρ12x + ρ12y 2 2 ) ρ1x ) H n ( ρ1 y ) exp(− w0 w0 w02 ⎡

× cosh(Ω 0 ρ1x + Ω 0 ρ1 y ) exp ⎢ − ⎢⎣

× Hm(

(ρ 1 x − ρ 2 x )2 ⎤ 2σ 0

⎥ ⎥⎦

2

ρ 22x + ρ 22y 2 2 ρ 2x )H n ( ρ 2 y ) exp(− ) w0 w0 w02

× cosh(Ω 0 ρ 2 x

⎡ + Ω 0 ρ 2 y ) exp ⎢− ⎢ ⎣

(ρ

− ρ2y )

2

1y

2σ 0

2

⎤ ⎥, ⎥ ⎦

(2)

where σ0 is the spatial correlation length of the source at the plane z = 0. Using the extended Huygens-Fresnel principle, the cross spectral density function of partially coherent H-ChG beams propagating through atmospheric turbulence is expressed as [23]

k 2 ) ∫∫ d 2 ρ1 ∫∫ d 2 ρ 2W (ρ1 , ρ 2 , z = 0) 2πz ik × exp{ [(ρ1′ − ρ1 ) 2 − (ρ′2 − ρ 2 ) 2 ]}〈exp[ψ (ρ1 , ρ1′ ) +ψ ∗ (ρ 2 , ρ′2 )]〉 m , (3) 2z

W (ρ1′ , ρ′2 , z ) = (

where ρ'≡ (ρ'x, ρ'y) is the two-dimensional position vector at the plane z, k is the wave number related to the wave length λ by k = 2π/λ, ψ(ρ, ρ') represents the random part of the complex #92893 - $15.00 USD

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phase of a spherical wave due to the turbulence, m denotes the average over the ensemble of the turbulent medium statistics, and [24-27] 〈exp[ψ (ρ1 , ρ′1 ) + ψ

≈ exp{−

1

ρ

2 0

∗

(ρ 2 , ρ′2 )]〉 m

2 [(ρ1 − ρ 2 ) 2 + (ρ 1 − ρ 2 ) ⋅ (ρ1′ − ρ′2 ) + (ρ1′ − ρ′2 ) ]} ,

(4)

with

ρ 0 = (0.545C n2 k 2 z )

−3 / 5

,

(5)

where ρ0 is the spatial coherence radius of a spherical wave propagating in turbulence, C n2 is the refraction index structure constant which describes how strong the turbulence is [23]. It is noted that a quadratic approximation of Rytov’s phase structure function was used in Eq. (4) to obtain an analytical result. This approximation has been shown to be a good approximation in practice [24, 28]. To obtain the analytical result, the new variables of integration are introduced as

u=

ρ 2 + ρ1 2

, v = ρ 2 − ρ1 .

(6)

By letting ρ1′ = ρ′2 = ρ′ in Eq. (3), and making use of Eqs. (2), (4) and (6), the intensity of partially coherent H-ChG beams in turbulence at the z plane is given by

I (ρ′, z ) = W (ρ′, ρ ′, z )

=(

k 2 v2 2u 2 ik ik ) ∫∫ d 2 u ∫∫ d 2 v exp(− 2 ) exp(− 2 ) exp(− u ⋅ v ) exp( ρ′ ⋅ v ) w0 ε z z 4πz

× H m[

v v v v 2 2 2 2 (u x − x )]H m [ (u x + x )]H n [ (u y − y )]H n [ (u y + y )] w0 w0 w0 w0 2 2 2 2

× {exp[2Ω 0 (u x + u y )] + exp[−Ω 0 (v x + v y )]

，

+ exp[Ω 0 (v x + v y )] + exp[−2Ω 0 (u x + u y )]}

(7)

where

1

ε

2

=

1 1 1 + + 2. 2 2 2 w0 2σ 0 ρ 0

(8)

The normalized rms beam width is defined as [5]

w( z ) =

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∫ρ

′ 2 I (ρ′, z )d 2 ρ ′

2 ′ ′ ∫ I (ρ , z ) d ρ

.

(9)



By using some integral transform techniques (see Appendix A), we obtain the normalized rms beam width of partially coherent H-ChG beams in turbulence. The final result is arranged as 1

6

16

Q w( z ) = P + 2 z 2 + 4(0.545C n2 k 3 ) 5 z 5 , k

(10)

P = R1 / R0 ,

(11)

Q = R2 / R0 ,

(12)

where

R0 = exp( w02 Ω 20 ) L0m ( − w02 Ω 02 ) L0n ( − w02 Ω 02 ) + 1 ,

(13)

1 + w02 Ω 02 0 1 + 2w02 Ω 02 R1 = w02 {exp(w02 Ω 20 )( Lm (− w02 Ω 02 ) L0n (− w02 Ω 02 ) + 2 2

× [ L0n (− w02 Ω 02 ) L1m −1 (− w02 Ω 02 ) + L0m (− w02 Ω 02 ) L1n −1 (− w02 Ω 02 )] + w02 Ω 20 × [ L0n (− w02 Ω 20 ) L2m − 2 (− w02 Ω 02 ) + L0m (− w02 Ω 02 ) L2n −2 (− w02 Ω 20 )]) + R2 = exp( w02 Ω 20 ){2(

m + n +1 }, 2

(14)

1 1 2 + 2 ) L0m ( − w02 Ω 20 ) L0n ( − w02 Ω 02 ) + 2 [ L1m −1 ( − w02 Ω 20 ) 2 w0 σ 0 w0

× L0n ( − w02 Ω 20 ) + L0m ( − w02 Ω 02 ) L1n −1 (− w02 Ω 02 )]} + 2(

2( m + n ) 1 1 (15) + 2 − Ω 20 ) + 2 w0 σ 0 w02

where Lαj (•) (j=0, 1, 2…) denotes the generalized Laguerre polynomial with indices α , j, P and Q are independent of propagation distance z. The first two terms on the right-hand side of Eq. (10) denote the spread of the beam width in free space due to diffraction, where the first term P is independent of the propagation z, whereas the second term Q z 2 increases 1 distance 6 16 2 2 3 5 5 k2 with z , and the third term 4(0.545C n k ) z indicates how the turbulence affects the beam spread, and increases with z16/5. From Eq. (10) the angular spread of partially coherent H-ChG beams propagating through atmospheric turbulence turns out to be

θ sp ( z ) =

w( z ) z

1

z →∞

=

6

6

Q + 4(0.545C n2 k 3 ) 5 z 5 . 2 k

(16)

Equations (10) and (16) are the main analytical results obtained in this paper. The first term on the right-hand side in Eq. (16) represents the angular spread of partially coherent H-ChG beams in free space which is dependent on the spatial correlation length σ0 and beam parameters w0, Ω0, m, n, but independent of the propagation distance. The second term describes the effect of turbulence on the angular spread, which increases with C n2 and #92893 - $15.00 USD

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propagation distance z, but does not depend on σ0, w0, Ω0 and m, n. It means that the turbulence plays a dominant role in the angular spread as the propagation distance becomes large enough. -2

×10

θsp(mrad)

8 2

-14

-2/3

2

-15

-2/3

2

-15

-2/3

2

-16

-2/3

7

Cn =1×10 m

6

Cn =8×10 m Cn =5×10 m

5

Cn =1×10 m

4

Cn =0

2

3 2 1 0

1

2

3

4

5

6

7

8

σ0 (cm) Fig. 1. Angular spread θsp of a partially coherent H-ChG beam versus the spatial correlation length σ0.The calculation parameters are z = 10km, m = n = 1, w0 = 3cm, Ω0 = 100m-1.

×10

-2

10 9 in turbulence in free space

8

θsp(mrad)

7 6 5 4 3 2 1

1

2

3

4

5

w0 (cm) Fig. 2. Angular spread θsp of a partially coherent H-ChG beam versus the waist width w0. The calculation parameters are z = 10km, m = n = 1, σ0 = 1.732cm, Ω0 = 100m-1, C n2 = 10-14m-2/3.

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-2

×10

1.7 1.6

in turbulence in free space

θsp(mrad)

1.5 1.4 1.3 1.2 1.1 1.0

0

20

40

60

80

100

120

140

-1

Ω0 (m ) Fig. 3. Angular spread θsp of a partially coherent H-ChG beam versus the beam parameter Ω0. The calculation parameters are z = 10km, m = n = 1, w0 = 3cm, σ0 = 4cm, C n2 = 10-15m-2/3. -2

5.5

×10

θsp in turbulence θsp in free space

5.0

θsp(mrad)

4.5 4.0 3.5 3.0 2.5

2

4

6

8

10

12

14

m=n Fig. 4. Angular spread θsp of a partially coherent H-ChG beam versus the beam order m, n. The calculation parameters are z = 10km, w0 =1cm, σ0 = 2 cm, Ω0 = 300m-1, C n2 = 10-14m-2/3.

Numerical examples are given by using Eq. (16) to show the influence of turbulence on the angular spread of partially coherent H-ChG beams, where λ = 1.06μm is kept fixed. The angular spread θsp versus the spatial correlation length σ0 for different values of C n2 is shown in Fig. 1. From Fig. 1 it can be seen that θsp increases with decreasing σ0 and increasing C n2 . In addition, the difference between θsp in turbulence and θsp in free space is smaller for partially coherent H-ChG beams with smaller value of σ0 than for those with larger value of #92893 - $15.00 USD

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σ0. For example, for C n2 = 10-14m-2/3, σ0 = 1 and 6cm, we have θsp|turb/θsp|free = 1.59 and 3.43,

where θsp|turb and θsp|free are the angular spread in turbulence and in free space respectively. Figure 2 gives the angular spread θsp versus the waist width w0. Figure 2 indicates that θsp decreases with increasing w0. Furthermore, the larger the waist width w0 is, the larger the difference between θsp in turbulence and θsp in free space exhibits. For instance, for w0 = 1 and 4cm we have θsp|turb/θsp|free = 1.32 and 2.30, respectively. The angular spread θsp versus the parameter Ω0 is depicted in Fig. 3, which shows that θsp decreases with increasing Ω0, and the difference between θsp in turbulence and θsp in free space increases with increasing Ω0, e.g., for Ω0 = 20 and 120m-1 we have θsp|turb/θsp|free = 1.15 and 1.26 respectively. Figure 4 gives the angular spread θsp versus the beam order m, n. Figure 4 indicates that θsp increases with increasing m, n, and the difference between θsp in turbulence and θsp in free space decreases with increasing m, n, e.g., for m = n = 1 and 15 we have θsp|turb/θsp|free = 1.48 and 1.26, respectively. Therefore, the smaller the spatial correlation length σ0, waist width w0, beam parameter Ω0, and the larger the beam orders m, n are, the less partially coherent H-ChG beams are affected by turbulence. Generally, the angular spread of beams in turbulence is affected by two mechanisms. One is the free-space diffraction, and the other is the atmospheric turbulence [20]. Physically, the existence of any substantial original angular spread reduces the effect of atmospheric turbulence. Figures 1-4 indicate that in free space θsp decreases with increasing σ0, w0, Ω0, and decreasing m, n. Namely, the smaller σ0, w0, Ω0, and larger m, n mean the larger substantial original angular spread. This is the physical reason why the angular spread of partially coherent H-ChG beams with smaller σ0, w0, Ω0, and larger m, n is less affected by turbulence than that of partially coherent H-ChG beams with larger σ0, w0, Ω0, and smaller m, n. 3.

Equivalent beams

Partially coherent H-ChG beams represent a more general type of beams. There are some partially coherent beams and fully coherent beams which can be treated as their special cases. (Ⅰ ) For Ω0 = 0, Eq. (16) reduces to 1

6

6

Q1 θ sp ( z ) = + 4(0.545C n2 k 3 ) 5 z 5 , 2 k

(17)

where

Q1 = 2(

m + n +1 1 + 2 ). w02 σ0

(18)

Equation (17) is the angular spread of partially coherent H-G beams in turbulence, which is in agreement with Eq. (29) in Ref. [6]. (Ⅱ ) For m = n = 0, Eq. (16) reduces to the angular spread of partially coherent ChG beams in turbulence, i.e.,

θ sp ( z ) =

1

6

6

Q2 + 4(0.545C n2 k 3 ) 5 z 5 , k2

(19)

where

Q2 = 2[(

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Ω 20 1 1 + − ) ]. w02 σ 02 1 + exp(w02 Ω 02 )

(20)



(Ⅲ ) For Ω0 = 0 and m = n = 0, Eq. (16) reduces to 1

6

6

Q3 θ sp ( z ) = + 4(0.545Cn2 k 3 ) 5 z 5 , 2 k

(21)

where

Q3 = 2(

1 1 + 2). 2 w0 σ 0

(22)

Equation (21) is the angular spread of GSM beams propagating through atmospheric 6/5 turbulence. It is noted that z occurs in the second term of Eq. (21) because a quadratic approximation of the Rytov’s phase structure function is used in this paper, otherwise z appears as Eq. (14) of Ref. [5]. (Ⅳ ) For σ0 → ∞, Eq. (17) reduces to

θ sp ( z ) =

1

6

6

Q4 + 4(0.545C n2 k 3 ) 5 z 5 , k2

(23)

where

Q4 = exp( w02 Ω 02 )[

2 0 2 Lm ( − w02 Ω 20 ) L0n ( − w02 Ω 20 ) + 2 [ L1m−1 ( − w02 Ω 02 ) L0n ( − w02 Ω 02 ) 2 w0 w0

+ L0m ( − w02 Ω 20 ) L1n−1 ( − w02 Ω 02 )] + 2(

1 2( m + n ) . − Ω 02 ) + 2 w0 w02

(24)

Equation (23) is the the angular spread of fully coherent H-ChG beams in turbulence. (Ⅴ ) For Ω0 = 0 and σ0 → ∞, Eq. (16) reduces to the angular spread of fully coherent H-G beams in turbulence, which is given by

θ sp ( z ) =

1

6

6

Q5 + 4(0.545C n2 k 3 ) 5 z 5 , k2

(25)

where

Q5 = 2(

m + n +1 w02

).

(26)

(Ⅵ ) For m = n = 0 and σ0 → ∞, Eq. (16) reduces to 1

6

6

Q6 θ sp ( z ) = + 4(0.545C n2 k 3 ) 5 z 5 , 2 k

(27)

where

Q6 = 2[(

Ω 02 1 ]. − w02 1 + exp( w02 Ω 02 )

(28)

Equation (27) is the angular spread of fully coherent ChG beams. #92893 - $15.00 USD

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(Ⅶ) For Ω0 = 0, m = n = 0 and σ0 coherent Gaussian beams, i.e.,

→

∞, Eq. (16) reduces to the angular spread of fully 1

6

6

Q7 θ sp ( z ) = + 4(0.545C n2 k 3 ) 5 z 5 , 2 k

(29)

where

Q7 =

2 . w02

(30)

In comparison of Eqs. (16), (17), (19), (21) and (29), we conclude that four partially coherent beams, i.e., partially coherent H-ChG, H-G, ChG and GSM beams will have the same angular spread as a fully coherent Gaussian beam if the condition

Q = Q1 = Q2 = Q3 = Q7

(31)

is fulfilled, provided that the wavelength λ is kept fixed. Such four partially coherent beams are called the equivalent partially coherent H-ChG, H-G, ChG and GSM beams, respectively [1, 2].

40

in turbulence in free space

35

w(z) (cm)

30 25 20 15 a

b

c

d e

10 5 0

0

2

4

6

8

10

z (km) Fig. 5. Normalized rms width w(z) of the four equivalent partially coherent beams and the corresponding fully coherent Gaussian beam versus propagation distance z in free space and in turbulence. a: the corresponding fully coherent Gaussian beam; b: the equivalent GSM beam; c: the equivalent partially coherent H-G beam: d: the equivalent partially coherent H-ChG beam; e: the equivalent partially coherent ChG beam. The calculation parameters are listed in Table 1, and the other parameters are λ = 1.06μm, C n2 = 10-14 m-2/3.

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Table 1. Beam parameters relating to Fig. 5. m

n

w0(cm)

σ0(cm)

Ω0(m-1) 0

a

0

0

1

∞

b

0

0

2.8

1.07

0

c

2

1

3.5

1.22

0

d

2

2

2

1.23

150

e

0

0

3

1.07

100

Figure 5 gives the normalized rms width w(z) of the four equivalent partially coherent beams mentioned above and the corresponding fully coherent Gaussian beam propagating both in free space and in atmospheric turbulence. The calculation parameters are listed in Tab. 1. As can be expected, under the condition (31) the four equivalent partially coherent beams exhibit the same directionality as the corresponding fully coherent Gaussian beam in free space and also in turbulence. Except for the equivalent partially coherent beams, there also exist equivalent fully coherent beams. From a comparison of Eqs. (23), (25), (27) and (29), we see that, three fully coherent beams, i.e., fully coherent H-ChG, H-G and ChG beams will have the same angular spread as a fully coherent Gaussian beam if the condition

Q4 = Q5 = Q6 = Q7

(32)

is satisfied, provided that the wavelength λ is kept fixed. Such three fully coherent beams are referred to as the equivalent fully coherent H-ChG, H-G and ChG beams, respectively [1, 2].

40 in turbulence in free space

35

w(z) (cm)

30 25 20 15 bc

d

a

10 5 0

0

2

4

6

8

10

z (km) Fig. 6. Normalized rms width w(z) of the three fully coherent beams and the corresponding fully coherent Gaussian beam versus propagation distance z. a: the corresponding fully coherent Gaussian beam; b: the equivalent fully coherent ChG beam; c: the equivalent fully coherent H-ChG beam; d: the equivalent fully coherent H-G beams. The calculation parameters are listed in Table 2, and the other parameters are λ = 1.06μm, C n2 = 10-14m-2/3.

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Table 2. Beam parameters relating to Fig. 6. m

n

w0(cm)

Ω0(m-1)

a

0

0

1.01

0

b

0

0

1.00

300

c

2

1

1.20

200

d

3

2

2.47

0

Figure 6 gives the normalized rms width w(z) of the three equivalent fully coherent beams mentioned above and the corresponding fully coherent Gaussian beam propagating both in free space and in atmospheric turbulence. The calculation parameters are compiled in Tab. 2. As can be seen, equivalent fully coherent H-ChG, H-G and ChG beams have also the same directionality as the corresponding fully coherent Gaussian beam both in free space and in turbulence, if the condition (32) is satisfied. The results can be physically explained as follows. In fact, partially coherent H-ChG beams are characterized by four parameters (i.e., σ0, w0, Ω0, m, n), and partially coherent H-G, ChG and GSM beams are characterized by (σ0, w0, m, n), (σ0, w0, Ω0) and (σ0, w0), respectively. From Figs. 1-4 it follows that for partially coherent H-ChG beams the smaller σ0 means the worse spatial coherence, which result in a larger angular spread. The smaller the waist width w0 and the beam parameter Ω0 are, the larger the free-space diffraction spread is; while the smaller the beam orders m, n are, the smaller the free-space diffraction spread is. In addition, Eq. (16) indicates that the angular spread due to turbulence is independent of σ0, w0, Ω0 and m, n. Therefore, both in free space and in turbulence there are two competing mechanisms for partially coherent beams regarding to the directionality, i.e., the spatial coherence and diffraction, which can be balanced by a suitable choice of beam parameters σ0, w0, Ω0 and m, n, whereas for fully coherent beams discussed above, the angular spread resulting from diffraction can be compensated by appropriately adjusting beam parameters w0, Ω0 and m, n to achieve the same directionality as a fully coherent Gaussian beam in free space and also in turbulence. 4.

Concluding remarks

In this paper, the propagation of partially coherent H-ChG beams in turbulence has been studied in detail. The analytical expression for the angular spread of partially coherent H-ChG beams propagating through atmospheric turbulence has been derived by using the extended Huygens-Fresnel principle, quadratic approximation of Rytov’s phase structure function and integral transform techniques. In comparison with previous work our result is more general, because the angular spread of partially coherent H-G, ChG and GSM and fully coherent HChG, H-G, ChG, Gaussian beams can be treated as special cases of Eq. (16). It has been shown that the angular spread of partially coherent H-ChG beams with smaller σ0, w0, Ω0, and larger m, n is less affected by turbulence than that of partially coherent H-ChG beams with larger σ0, w0, Ω0, and smaller m, n. Partially coherent H-ChG, H-G, ChG and GSM beams may have the same angular spread as a fully coherent Gaussian beam in free space and also in atmospheric turbulence if the condition (31) is satisfied. Under the condition (32) there exist the equivalent fully coherent H-ChG, H-G and ChG beams which may have the same directionality as a fully coherent Gaussian beam both in free space and in atmospheric turbulence. The physical interpretation has been given to show the validity of our results. Appendix A: Derivation of Eq. (10) Equation (9) can be written as

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w( z ) =

F , F0

(A1)

where

F0 = ∫ I (ρ′, z ) d 2 ρ ′ ,

(A2)

F = ∫ ρ ′ 2 I (ρ′, z ) d 2 ρ ′ .

(A3)

On substituting Eq. (2) into Eq. (A2) and letting

ρ1 = ρ 2 = ρ , we obtain

F0 = ∫∫ W ( 0) (ρ, ρ, z = 0)d 2 ρ

=

1 m +n−1 2 m! n! w02π [exp( w02 Ω 02 ) L0m ( − w02 Ω 02 ) L0n (− w02 Ω 20 ) + 1] . 2

(A4)

In the derivation of Eq. (A4) the law of conservation of energy was used. Equation (A3) can be rewritten as

F = F1 + F2 + F3 + F4 ,

(A5)

F1 = ∫ ρ ′ 2 I 1 (ρ′, z ) d 2 ρ ′ ,

(A6)

F2 = ∫ ρ ′ 2 I 2 (ρ′, z )d 2 ρ ′ ,

(A7)

F3 = ∫ ρ ′ 2 I 3 (ρ′, z ) d 2 ρ ′ ,

(A8)

F4 = ∫ ρ ′ 2 I 4 (ρ′, z )d 2 ρ ′ ,

(A9)

where

with

I 1 (ρ′, z ) =

1 k 2 ( ) d 2 u ∫∫ d 2 v 4 2πz ∫∫

× H m[

vy vy v v 2 2 2 2 (u x − x )]H n [ (u y − )]H m [ (u x + x )]H n [ (u y + )] w0 w0 w0 w0 2 2 2 2

× exp(− I 2 (ρ′, z ) = × H m[

2u 2 ik ik v2 ) exp( − ) exp(− u ⋅ v ) exp( ρ ′ ⋅ v) exp[2Ω 0 (u x + u y )] , (A10) 2 2 z z w0 ε

1 k 2 ( ) d 2 u ∫∫ d 2 v 4 2πz ∫∫


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× exp(−

v2 2u 2 ik ik − ) exp( ) exp(− u ⋅ v ) exp( ρ′ ⋅ v ) exp[Ω 0 (v x + v y )] . (A11) 2 2 z z w0 ε

I3 = (ρ', z) and I4 = (ρ', z) can be obtained if Ω0 is replaced by -Ω0 in Eqs. (A10) and (A11). Making use of the integral formula ∫x

2

exp(−i 2πxs)dx = −

1 ( 2π ) 2

δ ′′( s) ,

(A12)

the substitution from Eq. (A10) into Eq. (A6) yields

F1 = F11 + F12 ,

(A13)

where

1 z F11 = − ( ) 2 ∫∫ d 2 u ∫∫ d 2 v 4 k × H m[


× exp(−

2u 2 v2 ik ) exp( − ) exp(− u ⋅ v ) exp[2Ω 0 (u x + u y )]δ '' (v x )δ (v y ) , (A14) 2 2 z w0 ε

and

1 z F1 2 = − ( ) 2 ∫∫ d 2 u ∫∫ d 2 v 4 k

× H m[


× exp(−

2u 2 ik v2 ) exp( − ) exp(− u ⋅ v ) exp[2Ω 0 (u x + u y )]δ (v x )δ '' (v y ) , (A15) 2 2 z ε w0

with δ denoting the Dirac delta function , and By virtue of the integral formulae ∫

δ ''

being its second derivative .

，

f ( x)δ ( x )dx = f (0)

(A16)

and ∫ exp[− ( x −

y ) 2 ]H m ( x ) H n ( x) dx = 2 n π m! y n−m Lnn−m ( −2 y 2 ) ,

(A17)

the integration of Eq. (A14) with respect to vy and uy yields

w w2Ω 2 1 z F11 = − ( ) 2 2 n n! π 0 exp( 0 0 ) L0n ( − w02 Ω 02 ) ∫∫ dv x du x 4 k 2 2

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2

v v 2u 2 2 × H m[ (u x − x )]H m [ (u x + x )] exp(− x2 ) w0 w0 2 2 w0 × exp(−

vx

ε

2

2

) exp(−

ik u x v x ) exp(2Ω 0 u x )δ '' (v x ) . z

(A18)

Recalling the integral formulae ∫ exp(− x

2

) H m ( x + y ) H n ( x + z ) dx = 2 n π m! y n −m z n−m Lnn−m ( −2 yz ) ,

(A19)

and ∫

f ( x )δ '' ( x )dx = f '' (0) ,

(A20)

where f is an arbitrary function and f" is its second derivative of f ,and performing the integration of Eq. (A18) with respect to ux and vx, we obtain

1 z 2 F11 = − ( ) 2 2 m + n −1 m! n!πw0 exp(w02 Ω 02 ) L0n (− w02 Ω 20 ) 4 k

× [( −

−(

k 2 w02 k 2 w04 Ω 02 0 k 2 w04 Ω 20 2 2 0 2 2 2 2 L w L w − ) ( − Ω )] − ( − Ω ) − Lm − 2 ( − w02 Ω 02 ) m m 0 0 0 0 2 2 2 2 4z ε 4z z

k 2 w04 Ω 02 1 2 k 2 w02 1 2 2 + ) ( − Ω ) − Lm −1 (− w02 Ω 02 )]. L w m −1 0 0 2z 2 z2 w02

(A21)

F12 can be obtained if m and n are replaced by n and m in Eq. (A21), respectively. Thus, F1 in Eq. (A13) (i.e., Eq. (A6)) can be obtained. Similarly, F2 can be written as

F2 = F21 + F22 ,

(A22)

where

k 2 w02 2 1 z 2 m+ n−1 2 k 2 w02 2 2 − ) + Ω 0 − m( 2 + )] . F21 = − ( ) 2 m!n!πw0 [(− 4 k 4z 2 ε 2 2z 2 w0

(A23)

F22 can be obtained if m and n are replaced by n and m in Eqs. (A23), respectively. In addition, F3 and F4 can be obtained if Ω0 is replaced by -Ω0 in Eqs. (A13) and (A22) respectively. Finally, from Eqs. (A1), (A4), (A5), (A13) and (A22) it turns out that 1

w( z ) = P +

6

16

Q 2 z + 4(0.545C n2 k 3 ) 5 z 5 . k2

(A24)

Acknowledgments This work was supported by the National Natural Science Foundation of China under grant No. 60778048.

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