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The effect of non-Kolmogorov turbulence on the propagation of cosh-Gaussian beam
Optics Communications 283 (2010) 3398–3403
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Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m
The effect of non-Kolmogorov turbulence on the propagation of cosh-Gaussian beam☆ Xiuxiang Chu a,b,⁎, Chunhong Qiao b, Xiaoxing Feng b a b
School of Sciences, Zhejiang Forestry University, Lin'an 311300, China Key Laboratory of Atmospheric Composition and Optical Radiation, Chinese Academy of Sciences, Hefei 230031, China
a r t i c l e
i n f o
Article history: Received 2 February 2010 Received in revised form 22 April 2010 Accepted 23 April 2010
a b s t r a c t The propagation of cosh-Gaussian beam in turbulence with different power spectral density of refractive index is investigated. By using the expansion of mutual coherence function in Taylor series, analytical expression for the average intensity is presented. With the help of this expression, the intensity profiles with different parameters are analyzed. Relative errors of the analytical expression are studied, and the effects of power spectral density on beam spreading and evolvement are discussed in details. It shows that the analytical expression should provide reasonable approximations to study the propagation of cosh-Gaussian in non-Obukhov–Kolmogorov turbulence. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Obukhov–Kolmogorov (OK) turbulence model has been widely used and accepted in atmospheric and adaptive optics. Predictions based on the model have been found to be in good agreement with the results in many applications. However, there are some results that deviated from the OK theory. Some experimental data deviated from the OK theory has been reported [1–4]. Further development of turbulence theory has also shown that there exists turbulence in atmosphere that does not obey OK theory [5]. Over the past decades, the propagation and compensation of beams in non-Obukhov– Kolmogorov (NOK) turbulence have attracted more and more attention. Some results have been published [6–14]. As we know, the effects of turbulence on different types of beams are different, and system performance can be improved by choosing a
suitable beam model. The propagation of Gaussian beam in NOK turbulence has been studied [11]. But, to date, the propagation of other type of beams in NOK turbulence has not been taken into account, even though the propagations of various types of laser beams in OK turbulence have been widely studied [15–21]. Therefore, we focus our attention on the properties of various types of beams in NOK turbulence. Cosh-Gaussian beams are the exact solution of a paraxial wave equation [22,23], and can be obtained by superposition of four fundamental Gaussian beams [23]. Therefore, the profiles of these beams are diverse that can be used in the space diversity applications in free-space optic (FSO) systems [15,19]. The propagation of coshGaussian beam in vacuum and in Kolmogorov turbulence has been studied [15,19]. In the present paper, we choose a cosh-Gaussian beam as an example to illustrate the propagation of beam with different power spectral density of refractive index.
2. Theoretical formulation Considering an optical source in the z = 0 plane as seen in Fig. 1, and incident optical field of cosh-Gaussian beam is [22] x2 + y2 uðx0 ; y0 ; 0Þ = cosh ðΩ0 x0 Þ cosh ðΩ0 y0 Þ exp − 0 2 0 w0
!
ik 2 2 x0 + y0 ; exp − 2R0
ð1Þ
☆ The project was supported by Open Research Fund of the Key Laboratory of Atmospheric Composition and Optical Radiation, Chinese Academy of Sciences (JJ0903) and the program for talents in Zhejiang Forestry University. ⁎ Corresponding author. School of Sciences, Zhejiang Forestry University, Lin'an 311300, China. E-mail address: [email protected] (X. Chu). 0030-4018/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2010.04.092
X. Chu et al. / Optics Communications 283 (2010) 3398–3403
3399
Fig. 1. Propagation geometry.
where (x0, y0) is the transverse coordinates at the input plane, Ω0 which has the units of m− 1 is the parameter associated with the cosh part and determined the separation distance between each beamlet (see Section 3.1), w0 is the 1/e waist width of Gaussian amplitude distribution, R0 is the phase front radius of curvature, and k = 2π / λ (λ is the wavelength) is the wave number. If we set p1 = (x01 + x02) / 2, q1 = (y01 + y02) / 2, p2 = x01 − x02, and q2 = y01 − y02 [(x01, y01) and (x02, y02) are the coordinates of two points at z = 0 plane], the average intensity at the z-plane can be expressed as in Eq. (1) by using the extended Huygens–Fresnel principle [24] Iðx; y; zÞ =
k 2πz
2
∞
∞
∫−∞ ∫−∞ Ηðp2 ; q2 ÞMðp2 ; q2 Þ exp
ik ðxp2 + yq2 Þ dp2 dq2 ; z
ð2Þ
where ∞ ∞ p q p q ik ðp1 p2 + q1 q2 Þ dp1 q1 ; Hðp2 ; q2 Þ = ∫−∞ ∫−∞ u0 p1 + 2 ; q1 + 2 u0 p1 − 2 ; q1 − 2 exp z 2 2 2 2
ð3Þ
Mðp2 ; q2 Þ = exp ½−0:5Dw ðp2 ; q2 Þ:
ð4Þ
and
Here, M(p2, q2) is the mutual coherence function and Dw is the wave structure function. The 3D power spectrum of arbitrary refractive index can be defined as [6] Φn ðα; K; zÞ = AðαÞβðα; zÞK
−α
;
ð5Þ
where α is the spectral exponent, K is the spatial wavenumber, AðαÞ = sin ½ðα−3Þπ = 2
Γðα−1Þ 4π2
ð6Þ
is a constant that maintains consistency between the index structure function and its power spectrum [A(11/3) = 0.033] [6,13], β(α, z) is the general refractive index structure constant [β(11/3, z) ≡ C2n(z)] and has the units of m3 − α [6,13]. In Eq. (6) Г denotes Euler's gamma function. If we assume 3 b α b 4, Dw can be derived from Eq. (5) as [6] α−2 ρ Dw ðα; ρ; zÞ = 2 ð3 b α b 4Þ; ð7Þ ρ0 ðα; zÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where ρ = p22 + q22 is the distance between the two points (x01, y01) and (x02, y02), and ρ0(α, z) [ρ0(11/3, z) = (0.545C2nk2z)− 3/5] is a spherical wave coherence length that corresponds to Φn(α, K, z) [6]. Following Ref. [6], the expression for the coherence length is " #− 1 22−α π2 k2 αβðαÞzAðαÞΓð−α =2Þ α−2 : ρ0 ðα; zÞ = ðα−1ÞΓðα=2Þ
ð8Þ
Considering cosh (x) = [exp (x) + exp (− x)] / 2, and substituting Eqs. (1), (3), (4) and (6) into Eq. (2), we can obtain ( 1 1 1 1 1 γ2 h i Iðx′; y′; z; δÞ = 2 + ð−1Þ ∑ ∑ ∑ ∑ exp 4 8τ23 m = 0 n = 1 i = 0 j = 0
+ j
m + n
+ ð−1Þ
i
) ∞ ∫0
" ! # pffiffiffi ! τ2 2 2 2−δ 2 exp − 2 −1 t J0 exp −t bt tdt; τ3 τ3
ð9Þ
where δ = 4 − α (0 b δ b 1), J0 (x) is the Bessel function of the first kind of order zero, and b=
1 4
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ffi ½4x′ + ð−1Þm γðτ1 −iτ2 Þ + ð−1Þn γðτ1 + iτ2 Þ2 + 4y′ + ð−1Þi γðτ1 −iτ2 Þ + ð−1Þj γðτ1 + iτ2 Þ :
ð10Þ
In Eqs.pffiffiffi (9) and (10), the non-dimensional parameters are defined as x′ = x / w0, y′ = y / w0, γ = Ω0w0, τ1 = 1 − z / R0, τ2 = 2z / (kw20), qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 τ3 = 2 2z = ½kw0 ρ0 ðα; zÞ and τ = τ1 + τ2 + τ23 . From Eqs. (9) and (10) we can see that the average intensity distribution is only determined by γ, τ1, τ2 and τ3.
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X. Chu et al. / Optics Communications 283 (2010) 3398–3403
Because Eq. (9) cannot be solved analytically and exp (−t2 − δ) can be expanded in Taylor series about δ= 0. Using the series representation of Bessel function and formula 4.335 in Ref. [25] with integration by parts, the approximate solution for Eq. (9) in third-order expansion can be expressed as 3
Ia ðx′; y′; z; δÞ = ∑ Il ðx′; y′; z; δÞ;
ð11Þ
l=0
where I0 ðx′; y′; z; δÞ =
1 1 1 1 1 2 2 ∑ ∑ ∑ ∑ exp − b ; 16τ2 m = 0 n = 1 i = 0 j = 1 τ2 1
1
1
1
∞
s
I1 ðx′; y′; z; δÞ = ∑ ∑ ∑ ∑ ∑ ð−1Þ m=0 n=1 i=0 j=1 s=0
s−4
ð12Þ
2s
2 δb A1 ð2Þ; s!2 τ23 τ2s
Fig. 2. Intensity profiles of Cosh-Gaussian beam with different γ.
ð13Þ
X. Chu et al. / Optics Communications 283 (2010) 3398–3403 1
1
1
1
∞
s
I2 ðx′; y′; z; δÞ = ∑ ∑ ∑ ∑ ∑ ð−1Þ m=0 n=1 i=0 j=1 s=0
3401
2s−4 δ2 b2s ½A2 ð3Þ−A2 ð2Þ; s!2 τ23 τ2s
ð14Þ
2s−4 δ3 b2s ½A3 ð2Þ−3A3 ð3Þ + A3 ð4Þ: s!2 τ23 τ2s
ð15Þ
and 1
1
1
1
∞
s
I3 ðx′; y′; z; δÞ = ∑ ∑ ∑ ∑ ∑ ð−1Þ m=0 n=1 i=0 j=1 s=0
In the present paper, 〈Ia(x′, y′, z, δ)〉 denotes the approximation of average intensity [see Eq. (11)], and 〈I(x′, y′, z, δ)〉 denotes the average intensity numerically calculated from Eq. (9). In Eqs. (13)–(15) A1 ðβÞ =
" ! # 1 τ3 2β τ2 Γðβ + sÞ ln 32 + ψðβ + sÞ ; 2 τ τ
ð16Þ
A2 ðβÞ =
(" ) ! #2 2 1 τ3 2β τ ð1Þ Γðβ + sÞ ln 32 + ψðβ + sÞ + ψ ðβ + sÞ ; 8 τ τ
ð17Þ
A3 ðβÞ =
1 τ3 48 τ2
and 2
!β
" Γðs + βÞ ln
2
τ3 τ2
!
#3 + ψðs + βÞ
" + 3 ln
2
τ3 τ2
!
# ð1Þ
+ ψðs + βÞ ψ
ð2Þ
ðs + βÞ + ψ
ðs + βÞ;
ð18Þ
where ψ(x) = d[ln Γ(x)] / dx is the psi function, and ψ(n)(x) = dnψ(x) / dxn is poly-gamma function. It should be noted that more higher-order expression could be derived with the same way. Calculation also shows that the infinite sum occurring in Eqs. (13)–(15) converges rapidly.
Fig. 3. The variations of relative error with different parameters.
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X. Chu et al. / Optics Communications 283 (2010) 3398–3403
(see Fig. 3a–c). For small τ3, the relative error is small. With the increase of τ3, the relative errors become large. The effect of γ on relative error is small against variations in δ and τ3.
3. Numerical calculation 3.1. The intensity distribution of cosh-Gaussian beam A cosh-Gaussian beam can be produced simply by superposition of four decentered Gaussian beams with the same waist width and in phase, whose centers are located at the positions (γ / 2, γ / 2), (γ / 2, −γ / 2), (−γ / 2, −γ / 2) and (− γ / 2, γ / 2) [23]. Its intensity profiles are plotted in Fig. 2 with different γ. The normalized intensity is defined as IN(x0, y0) = I(x0, y0) / max [I(x0, y0)]. When γ is small (for example γ = 1), the intensity profile has a Gaussian shape (see Fig. 2a). Fig. 2b shows that the intensity distribution has a flattened Gaussian shape if γ = 1.4. With increasing γ, intensity distribution gradually becomes four separate Gaussian shapes (see Fig. 2c). Cross section of intensity profiles corresponding to Fig. 2a, b and c is plotted in Fig. 2d.
3.3. Intensity profiles of cosh-Gaussian in NOK turbulence For NOK turbulence (α ≠ 11/3 or δ ≠ 1/3), β(α) ≠ C2n. The choice of β(α) for comparing intensity distributions will influence the comparison. These choices include 1) zβ(α) = constant, namely, power spectrum ispconstant at a specific wavenumber, which is selected as ffiffiffiffiffiffi 1 m− 1 or λz in Refs. [6] and [13], respectively, 2) power of the spectrum in a finite bandwidth is a constant [6], 3) β(α) = constant [11]. In present paper the choice of β(α) is the same as in Ref. [11], that is β(α) = constant. The normalized average intensity is defined as In ðx′; y′; z; δÞ =
Ia ðx′; y′; z; δÞ ½Ia ðx′; y′; z; 1 = 3Þ: max
ð20Þ
3.2. The relative error due to the expansion To illuminate the validity of the approximate expression for average intensity, the relative errors of the on-axis intensity is defined as σ = 1−Ia ð0; 0; z; δÞ = Ið0; 0; z; δÞ
ð19Þ
and shown in Fig. 3. The parameters are selected as τ1 = 1 (collimated beam), w0 = 1 m, z = 10 km and λ = 1.06 μm. It can be seen from the graphs that Eqs. (11)–(18) should provide reasonable approximations. These relative errors are small with small δ, and increase with δ
Fig. 4. The cross section of average intensity profiles with γ = 1.
The parameters are selected as w0 = 1 m, β(α) = 10− 14 m3 − α and λ = 1.06 μm. The cross section of intensity profiles with γ = 1 and different propagation distance is plotted in Fig. 4. Comparison with Fig. 2d shows that the beam spreading is small with z = 10 km. With increasing propagation distance, beam spreading increases. The speed of beam spreading with δ = 1/3 (or α = 11/3) is smaller than that with δ = 1/2 (α = 7/2) and δ = 2/3 (α = 10/3). For exposing the effects of γ on beam spreading, the intensity profiles with γ = 1.44 and γ = 2 are plotted in Figs. 5 and 6. It can be seen from the graphs that the effects
Fig. 5. The cross section of average intensity profiles with γ = 1.4.
X. Chu et al. / Optics Communications 283 (2010) 3398–3403
3403
hollow beam into Gaussian shape is slower with δ = 1/3 than that with δ = 1/2 and δ = 2/3. The effects of δ on beam spreading are in agreement with its effects on the evolvement of beam shape. Appendix A The relation between wave structure function and power spectrum is Kρz′ 2 2 z ∞ Dw ðα; ρ; zÞ = 8π k ∫0 ∫0 Φn ðα; K; z′ÞK 1−J0 dKdz′: z
ðA1Þ
For 3D power spectrum of arbitrary refractive index with the form Φn ðα; K; zÞ = AðαÞβðα; zÞK
−α
:
ðA2Þ
Eq. (A1) converge on [0, ∞) only when 2 b α b 4. Considering 3 b α b 5 AðαÞ = sin ½ðα−3Þπ = 2
Γðα−1Þ ; 4π2
ðA3Þ
therefore, alpha range values in Eq. (7) are from 3 to 4. Assuming β(α, z) is independent on z and performing the integrals in Eq. (A1) we can obtain α−2 ρ Dw ðα; ρ; zÞ = 2 ð3 b α b 4Þ; ρ0 ðα; zÞ
ðA4Þ
where " #− 1 22−α π2 k2 αβðαÞzAðαÞΓð−α=2Þ α−2 : ρ0 ðα; zÞ = ðα−1ÞΓðα= 2Þ
ðA5Þ
References Fig. 6. The cross section of average intensity profiles with γ = 2.
of δ and z with γ = 1.44 and γ = 2 are in agreement with γ = 1. For the case of γ = 1.44 (see Fig. 5), the flat top evolves into Gaussian shape gradually with the increase of propagation distance. The speed of evolvement with δ = 1/3 is slower than that with δ = 1/2 and δ = 2/3. Fig. 6 shows that the central dip of the hollow shape gradually disappears, and becomes Gaussian shape with the increase of propagation distance. The effects of δ on the speed of evolvement are accordance with the case of γ = 1.44. 4. Conclusion Analytically approximate expression for average intensity of coshGaussian beam in NOK turbulence is presented. Relative errors of the expression are studied. It shows that the expression should provide reasonable approximations. With the help of the expression, the average intensity profiles with different parameters are discussed. From the evolvement of intensity profiles we can see that the effects of δ and z on intensity distribution with γ = 1 are in accordance with that γ = 1.4 and γ = 2. The speed of the evolvement from flat top and
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R.G. Buser, J. Opt. Soc. Am. 61 (1971) 488. D. Dayton, B. Pierson, B. Spielbusch, J. Gonglewski, Opt. Lett. 17 (1992) 1737. E. Golbraikh, S.S. Moiseev, Phys. Lett. A 305 (2002) 173. C. Rao, W. Jiang, N. Ling, J. Mod. Opt. 47 (2000) 1111. E. Golbraikh, N.S. Kopeika, Appl. Opt. 43 (2004) 6151. B.E. Stribling, B.M. Welsh, M.C. Roggemann, Proc. SPIE 2471 (1995) 181. G.D. Boreman, C. Dainty, J. Opt. Soc. Am. A 13 (1996) 517. C. Rao, W. Jiang, N. Ling, Appl. Opt. 40 (2001) 3441. E. Golbraikh, H. Branover, N.S. Kopeika, A. Zilberman, Nonlinear Proc. in Geoph. 13 (2006) 297. I. Toselli, L.C. Andrews, R.L. Phillips, V. Ferrero, Proc. SPIE 6551 (2007) 65510E-1. I. Toselli, L.C. Andrews, R.L. Phillips, V. Ferrero, Opt. Eng. 47 (2) (2008) 026003. D.G. Pérez, L. Zunino, Opt. Lett. 33 (2008) 572. A. Zilberman, E. Golbraikh, N.S. Kopeika, Appl. Opt. 47 (2008) 6385. N.S. Kopeika, A. Zilberman, E. Golbraikh, Proc. SPIE 7463 (2009) 746307-1. H.T. Eyyuboğlu, Y. Baykal, Opt. Express 12 (2004) 4659. Y. Cai, S. He, Opt. Express 14 (2006) 1353. H.T. Eyyuboğlu, C. Arpali, Y. Baykal, Opt. Express 14 (2006) 4196. X. Chu, Y. Ni, G. Zhou, Opt. Commun. 274 (2007) 274. X. Chu, Opt. Express 24 (2007) 17613. X. Chu, Z. Liu, Y. Wu, J. Opt. Soc. Am. A 25 (2008) 74. Y. Zhu, D. Zhao, X. Du, Opt. Express 16 (2008) 18437. A.A. Tovar, L.W. Casperson, J. Opt. Soc. Am. A 15 (1998) 2425. Y. Zhang, Y. Song, Z. Chen, J. Ji, Z. Shi, Opt. Lett. 32 (2007) 292. H.T. Yura, S.G. Hanson, J. Opt. Soc. Am. A 4 (1987) 1931. I.S. Gradysteyn, I.M. Ryzhik, Tables of Integrals, Series and Products, Academic Press, New York, 1980.
Contents lists available at ScienceDirect
Optics Communications j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / o p t c o m
The effect of non-Kolmogorov turbulence on the propagation of cosh-Gaussian beam☆ Xiuxiang Chu a,b,⁎, Chunhong Qiao b, Xiaoxing Feng b a b
School of Sciences, Zhejiang Forestry University, Lin'an 311300, China Key Laboratory of Atmospheric Composition and Optical Radiation, Chinese Academy of Sciences, Hefei 230031, China
a r t i c l e
i n f o
Article history: Received 2 February 2010 Received in revised form 22 April 2010 Accepted 23 April 2010
a b s t r a c t The propagation of cosh-Gaussian beam in turbulence with different power spectral density of refractive index is investigated. By using the expansion of mutual coherence function in Taylor series, analytical expression for the average intensity is presented. With the help of this expression, the intensity profiles with different parameters are analyzed. Relative errors of the analytical expression are studied, and the effects of power spectral density on beam spreading and evolvement are discussed in details. It shows that the analytical expression should provide reasonable approximations to study the propagation of cosh-Gaussian in non-Obukhov–Kolmogorov turbulence. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Obukhov–Kolmogorov (OK) turbulence model has been widely used and accepted in atmospheric and adaptive optics. Predictions based on the model have been found to be in good agreement with the results in many applications. However, there are some results that deviated from the OK theory. Some experimental data deviated from the OK theory has been reported [1–4]. Further development of turbulence theory has also shown that there exists turbulence in atmosphere that does not obey OK theory [5]. Over the past decades, the propagation and compensation of beams in non-Obukhov– Kolmogorov (NOK) turbulence have attracted more and more attention. Some results have been published [6–14]. As we know, the effects of turbulence on different types of beams are different, and system performance can be improved by choosing a
suitable beam model. The propagation of Gaussian beam in NOK turbulence has been studied [11]. But, to date, the propagation of other type of beams in NOK turbulence has not been taken into account, even though the propagations of various types of laser beams in OK turbulence have been widely studied [15–21]. Therefore, we focus our attention on the properties of various types of beams in NOK turbulence. Cosh-Gaussian beams are the exact solution of a paraxial wave equation [22,23], and can be obtained by superposition of four fundamental Gaussian beams [23]. Therefore, the profiles of these beams are diverse that can be used in the space diversity applications in free-space optic (FSO) systems [15,19]. The propagation of coshGaussian beam in vacuum and in Kolmogorov turbulence has been studied [15,19]. In the present paper, we choose a cosh-Gaussian beam as an example to illustrate the propagation of beam with different power spectral density of refractive index.
2. Theoretical formulation Considering an optical source in the z = 0 plane as seen in Fig. 1, and incident optical field of cosh-Gaussian beam is [22] x2 + y2 uðx0 ; y0 ; 0Þ = cosh ðΩ0 x0 Þ cosh ðΩ0 y0 Þ exp − 0 2 0 w0
!
ik 2 2 x0 + y0 ; exp − 2R0
ð1Þ
☆ The project was supported by Open Research Fund of the Key Laboratory of Atmospheric Composition and Optical Radiation, Chinese Academy of Sciences (JJ0903) and the program for talents in Zhejiang Forestry University. ⁎ Corresponding author. School of Sciences, Zhejiang Forestry University, Lin'an 311300, China. E-mail address: [email protected] (X. Chu). 0030-4018/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2010.04.092
X. Chu et al. / Optics Communications 283 (2010) 3398–3403
3399
Fig. 1. Propagation geometry.
where (x0, y0) is the transverse coordinates at the input plane, Ω0 which has the units of m− 1 is the parameter associated with the cosh part and determined the separation distance between each beamlet (see Section 3.1), w0 is the 1/e waist width of Gaussian amplitude distribution, R0 is the phase front radius of curvature, and k = 2π / λ (λ is the wavelength) is the wave number. If we set p1 = (x01 + x02) / 2, q1 = (y01 + y02) / 2, p2 = x01 − x02, and q2 = y01 − y02 [(x01, y01) and (x02, y02) are the coordinates of two points at z = 0 plane], the average intensity at the z-plane can be expressed as in Eq. (1) by using the extended Huygens–Fresnel principle [24] Iðx; y; zÞ =
k 2πz
2
∞
∞
∫−∞ ∫−∞ Ηðp2 ; q2 ÞMðp2 ; q2 Þ exp
ik ðxp2 + yq2 Þ dp2 dq2 ; z
ð2Þ
where ∞ ∞ p q p q ik ðp1 p2 + q1 q2 Þ dp1 q1 ; Hðp2 ; q2 Þ = ∫−∞ ∫−∞ u0 p1 + 2 ; q1 + 2 u0 p1 − 2 ; q1 − 2 exp z 2 2 2 2
ð3Þ
Mðp2 ; q2 Þ = exp ½−0:5Dw ðp2 ; q2 Þ:
ð4Þ
and
Here, M(p2, q2) is the mutual coherence function and Dw is the wave structure function. The 3D power spectrum of arbitrary refractive index can be defined as [6] Φn ðα; K; zÞ = AðαÞβðα; zÞK
−α
;
ð5Þ
where α is the spectral exponent, K is the spatial wavenumber, AðαÞ = sin ½ðα−3Þπ = 2
Γðα−1Þ 4π2
ð6Þ
is a constant that maintains consistency between the index structure function and its power spectrum [A(11/3) = 0.033] [6,13], β(α, z) is the general refractive index structure constant [β(11/3, z) ≡ C2n(z)] and has the units of m3 − α [6,13]. In Eq. (6) Г denotes Euler's gamma function. If we assume 3 b α b 4, Dw can be derived from Eq. (5) as [6] α−2 ρ Dw ðα; ρ; zÞ = 2 ð3 b α b 4Þ; ð7Þ ρ0 ðα; zÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where ρ = p22 + q22 is the distance between the two points (x01, y01) and (x02, y02), and ρ0(α, z) [ρ0(11/3, z) = (0.545C2nk2z)− 3/5] is a spherical wave coherence length that corresponds to Φn(α, K, z) [6]. Following Ref. [6], the expression for the coherence length is " #− 1 22−α π2 k2 αβðαÞzAðαÞΓð−α =2Þ α−2 : ρ0 ðα; zÞ = ðα−1ÞΓðα=2Þ
ð8Þ
Considering cosh (x) = [exp (x) + exp (− x)] / 2, and substituting Eqs. (1), (3), (4) and (6) into Eq. (2), we can obtain ( 1 1 1 1 1 γ2 h i Iðx′; y′; z; δÞ = 2 + ð−1Þ ∑ ∑ ∑ ∑ exp 4 8τ23 m = 0 n = 1 i = 0 j = 0
+ j
m + n
+ ð−1Þ
i
) ∞ ∫0
" ! # pffiffiffi ! τ2 2 2 2−δ 2 exp − 2 −1 t J0 exp −t bt tdt; τ3 τ3
ð9Þ
where δ = 4 − α (0 b δ b 1), J0 (x) is the Bessel function of the first kind of order zero, and b=
1 4
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2ffi ½4x′ + ð−1Þm γðτ1 −iτ2 Þ + ð−1Þn γðτ1 + iτ2 Þ2 + 4y′ + ð−1Þi γðτ1 −iτ2 Þ + ð−1Þj γðτ1 + iτ2 Þ :
ð10Þ
In Eqs.pffiffiffi (9) and (10), the non-dimensional parameters are defined as x′ = x / w0, y′ = y / w0, γ = Ω0w0, τ1 = 1 − z / R0, τ2 = 2z / (kw20), qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 τ3 = 2 2z = ½kw0 ρ0 ðα; zÞ and τ = τ1 + τ2 + τ23 . From Eqs. (9) and (10) we can see that the average intensity distribution is only determined by γ, τ1, τ2 and τ3.
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Because Eq. (9) cannot be solved analytically and exp (−t2 − δ) can be expanded in Taylor series about δ= 0. Using the series representation of Bessel function and formula 4.335 in Ref. [25] with integration by parts, the approximate solution for Eq. (9) in third-order expansion can be expressed as 3
Ia ðx′; y′; z; δÞ = ∑ Il ðx′; y′; z; δÞ;
ð11Þ
l=0
where I0 ðx′; y′; z; δÞ =
1 1 1 1 1 2 2 ∑ ∑ ∑ ∑ exp − b ; 16τ2 m = 0 n = 1 i = 0 j = 1 τ2 1
1
1
1
∞
s
I1 ðx′; y′; z; δÞ = ∑ ∑ ∑ ∑ ∑ ð−1Þ m=0 n=1 i=0 j=1 s=0
s−4
ð12Þ
2s
2 δb A1 ð2Þ; s!2 τ23 τ2s
Fig. 2. Intensity profiles of Cosh-Gaussian beam with different γ.
ð13Þ
X. Chu et al. / Optics Communications 283 (2010) 3398–3403 1
1
1
1
∞
s
I2 ðx′; y′; z; δÞ = ∑ ∑ ∑ ∑ ∑ ð−1Þ m=0 n=1 i=0 j=1 s=0
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2s−4 δ2 b2s ½A2 ð3Þ−A2 ð2Þ; s!2 τ23 τ2s
ð14Þ
2s−4 δ3 b2s ½A3 ð2Þ−3A3 ð3Þ + A3 ð4Þ: s!2 τ23 τ2s
ð15Þ
and 1
1
1
1
∞
s
I3 ðx′; y′; z; δÞ = ∑ ∑ ∑ ∑ ∑ ð−1Þ m=0 n=1 i=0 j=1 s=0
In the present paper, 〈Ia(x′, y′, z, δ)〉 denotes the approximation of average intensity [see Eq. (11)], and 〈I(x′, y′, z, δ)〉 denotes the average intensity numerically calculated from Eq. (9). In Eqs. (13)–(15) A1 ðβÞ =
" ! # 1 τ3 2β τ2 Γðβ + sÞ ln 32 + ψðβ + sÞ ; 2 τ τ
ð16Þ
A2 ðβÞ =
(" ) ! #2 2 1 τ3 2β τ ð1Þ Γðβ + sÞ ln 32 + ψðβ + sÞ + ψ ðβ + sÞ ; 8 τ τ
ð17Þ
A3 ðβÞ =
1 τ3 48 τ2
and 2
!β
" Γðs + βÞ ln
2
τ3 τ2
!
#3 + ψðs + βÞ
" + 3 ln
2
τ3 τ2
!
# ð1Þ
+ ψðs + βÞ ψ
ð2Þ
ðs + βÞ + ψ
ðs + βÞ;
ð18Þ
where ψ(x) = d[ln Γ(x)] / dx is the psi function, and ψ(n)(x) = dnψ(x) / dxn is poly-gamma function. It should be noted that more higher-order expression could be derived with the same way. Calculation also shows that the infinite sum occurring in Eqs. (13)–(15) converges rapidly.
Fig. 3. The variations of relative error with different parameters.
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X. Chu et al. / Optics Communications 283 (2010) 3398–3403
(see Fig. 3a–c). For small τ3, the relative error is small. With the increase of τ3, the relative errors become large. The effect of γ on relative error is small against variations in δ and τ3.
3. Numerical calculation 3.1. The intensity distribution of cosh-Gaussian beam A cosh-Gaussian beam can be produced simply by superposition of four decentered Gaussian beams with the same waist width and in phase, whose centers are located at the positions (γ / 2, γ / 2), (γ / 2, −γ / 2), (−γ / 2, −γ / 2) and (− γ / 2, γ / 2) [23]. Its intensity profiles are plotted in Fig. 2 with different γ. The normalized intensity is defined as IN(x0, y0) = I(x0, y0) / max [I(x0, y0)]. When γ is small (for example γ = 1), the intensity profile has a Gaussian shape (see Fig. 2a). Fig. 2b shows that the intensity distribution has a flattened Gaussian shape if γ = 1.4. With increasing γ, intensity distribution gradually becomes four separate Gaussian shapes (see Fig. 2c). Cross section of intensity profiles corresponding to Fig. 2a, b and c is plotted in Fig. 2d.
3.3. Intensity profiles of cosh-Gaussian in NOK turbulence For NOK turbulence (α ≠ 11/3 or δ ≠ 1/3), β(α) ≠ C2n. The choice of β(α) for comparing intensity distributions will influence the comparison. These choices include 1) zβ(α) = constant, namely, power spectrum ispconstant at a specific wavenumber, which is selected as ffiffiffiffiffiffi 1 m− 1 or λz in Refs. [6] and [13], respectively, 2) power of the spectrum in a finite bandwidth is a constant [6], 3) β(α) = constant [11]. In present paper the choice of β(α) is the same as in Ref. [11], that is β(α) = constant. The normalized average intensity is defined as In ðx′; y′; z; δÞ =
Ia ðx′; y′; z; δÞ ½Ia ðx′; y′; z; 1 = 3Þ: max
ð20Þ
3.2. The relative error due to the expansion To illuminate the validity of the approximate expression for average intensity, the relative errors of the on-axis intensity is defined as σ = 1−Ia ð0; 0; z; δÞ = Ið0; 0; z; δÞ
ð19Þ
and shown in Fig. 3. The parameters are selected as τ1 = 1 (collimated beam), w0 = 1 m, z = 10 km and λ = 1.06 μm. It can be seen from the graphs that Eqs. (11)–(18) should provide reasonable approximations. These relative errors are small with small δ, and increase with δ
Fig. 4. The cross section of average intensity profiles with γ = 1.
The parameters are selected as w0 = 1 m, β(α) = 10− 14 m3 − α and λ = 1.06 μm. The cross section of intensity profiles with γ = 1 and different propagation distance is plotted in Fig. 4. Comparison with Fig. 2d shows that the beam spreading is small with z = 10 km. With increasing propagation distance, beam spreading increases. The speed of beam spreading with δ = 1/3 (or α = 11/3) is smaller than that with δ = 1/2 (α = 7/2) and δ = 2/3 (α = 10/3). For exposing the effects of γ on beam spreading, the intensity profiles with γ = 1.44 and γ = 2 are plotted in Figs. 5 and 6. It can be seen from the graphs that the effects
Fig. 5. The cross section of average intensity profiles with γ = 1.4.
X. Chu et al. / Optics Communications 283 (2010) 3398–3403
3403
hollow beam into Gaussian shape is slower with δ = 1/3 than that with δ = 1/2 and δ = 2/3. The effects of δ on beam spreading are in agreement with its effects on the evolvement of beam shape. Appendix A The relation between wave structure function and power spectrum is Kρz′ 2 2 z ∞ Dw ðα; ρ; zÞ = 8π k ∫0 ∫0 Φn ðα; K; z′ÞK 1−J0 dKdz′: z
ðA1Þ
For 3D power spectrum of arbitrary refractive index with the form Φn ðα; K; zÞ = AðαÞβðα; zÞK
−α
:
ðA2Þ
Eq. (A1) converge on [0, ∞) only when 2 b α b 4. Considering 3 b α b 5 AðαÞ = sin ½ðα−3Þπ = 2
Γðα−1Þ ; 4π2
ðA3Þ
therefore, alpha range values in Eq. (7) are from 3 to 4. Assuming β(α, z) is independent on z and performing the integrals in Eq. (A1) we can obtain α−2 ρ Dw ðα; ρ; zÞ = 2 ð3 b α b 4Þ; ρ0 ðα; zÞ
ðA4Þ
where " #− 1 22−α π2 k2 αβðαÞzAðαÞΓð−α=2Þ α−2 : ρ0 ðα; zÞ = ðα−1ÞΓðα= 2Þ
ðA5Þ
References Fig. 6. The cross section of average intensity profiles with γ = 2.
of δ and z with γ = 1.44 and γ = 2 are in agreement with γ = 1. For the case of γ = 1.44 (see Fig. 5), the flat top evolves into Gaussian shape gradually with the increase of propagation distance. The speed of evolvement with δ = 1/3 is slower than that with δ = 1/2 and δ = 2/3. Fig. 6 shows that the central dip of the hollow shape gradually disappears, and becomes Gaussian shape with the increase of propagation distance. The effects of δ on the speed of evolvement are accordance with the case of γ = 1.44. 4. Conclusion Analytically approximate expression for average intensity of coshGaussian beam in NOK turbulence is presented. Relative errors of the expression are studied. It shows that the expression should provide reasonable approximations. With the help of the expression, the average intensity profiles with different parameters are discussed. From the evolvement of intensity profiles we can see that the effects of δ and z on intensity distribution with γ = 1 are in accordance with that γ = 1.4 and γ = 2. The speed of the evolvement from flat top and
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