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Propagation of a twist Gaussian–Schell model beam in non-Kolmogorov turbulence
Optics Communications 324 (2014) 108–113
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Propagation of a twist Gaussian–Schell model beam in non-Kolmogorov turbulence Yan Cui, Fei Wang n, Yangjian Cai School of Physical Science and Technology & Collaborative Innovation Center of Suzhou Nano Science and Technology, Soochow University, Suzhou 215006, China
art ic l e i nf o
a b s t r a c t
Article history: Received 20 January 2014 Received in revised form 9 March 2014 Accepted 14 March 2014 Available online 27 March 2014
Based on the extended Huygens–Fresnel integral, an analytical formula for the cross-spectral density (CSD) function of a twist Gaussian–Schell model (TGSM) beam propagating through non-Kolmogorov turbulence is derived. The statistical properties, such as the beam width, the radius of curvature, the transverse coherence length and the twist factor of the TGSM beam in non-Kolmogorov turbulence, are studied numerically. It is found that the TGSM beam is less affected than a GSM beam without twist phase both in non-Kolmogorov turbulence and in Kolmogorov turbulence. Our results will be useful in free-space communication. & 2014 Elsevier B.V. All rights reserved.
Keywords: Turbulent atmosphere Propagation Twist factor Partially coherent beam
1. Introduction In the past decades, partially coherent beams have been extensively investigated due to their wide applications in optical image, inertial confinement fusion, free-space communications and optical trappings [1–4]. Gaussian–Schell-model (GSM) beam whose spectral degree of coherence and intensity distribution satisfy Gaussian distribution is a typical partially coherent beam, and has been studied extensively [5–8]. A more general GSM beam can possess a twist phase, which differs in many aspects from the customary quadratic phase factor. In 1993, Simon and Mukunda first introduced the GSM beam with twist phase named as a twist Gaussian Schell-model (TGSM) beam [9,10]. Unlike the usual phase curvature, the twist phase is bounded by strength due to the fact that the cross-spectral density (CSD) function must be non-negative definite, and it is absent in a coherent Gaussian beam [9]. Since then, various properties, such as the spectral shift, the paraxial propagation, the orbital angular momentum, the second-harmonic generation and the ghost imaging of the TGSM beam, have been investigated [11– 20]. Recently, Zhao et al. investigated the radiation force of the TGSM beam on a Rayleigh particle [21]. The second-order statistical properties and the scintillation index of the TGSM beam in Kolmogorov turbulence have been studied in Refs. [22] and [23]. It was shown in Ref. [24] that the TGSM beam could serve as illumination that might produce images with a resolution n
Corresponding author. E-mail addresses: [email protected] (F. Wang), [email protected] (Y. Cai). http://dx.doi.org/10.1016/j.optcom.2014.03.030 0030-4018/& 2014 Elsevier B.V. All rights reserved.
overcoming the Rayleigh limit. More recently, the orbital angular moment of the TGSM beam propagating through an astigmatic ABCD optical system with loss or gain was explored [25]. On the other hand, considerable attention is being paid to the propagation of laser beams through atmospheric turbulence due to their wide applications in free-space communications, remote sensing, imaging and radar system [26–39]. The Kolmogorov model has been adopted to study the effect of atmospheric turbulence in free-space communication and has shown good agreement with the experimental results [40]. The suitable condition for the Kolmogorov model is that the atmospheric turbulence is homogeneous. However, some experimental results have shown that there exists great deviation from the Kolmogorov model's prediction in some portions of atmosphere [41–43]. The reason is that the atmosphere is extremely stable, and the turbulence is no longer homogeneous in three dimensions since the vertical component is suppressed [44]. Toselli et al. introduced the non-Kolmogorov model, in which a generalized exponent factor was used instead of the standard exponent value 11/3 in the Kolmogorov model [45,46]. Recently, the behaviors of various laser beams, such as Hermite–Gaussian beam, elegant Laguerre– Gaussian beam, laser beam array and GSM beam propagating through non-Kolmogorov turbulence, have been explored [47–55], and some interesting and useful results have been found. In this paper, our aim is to investigate the statistical properties of the TGSM beam in non-Kolmogorov turbulence. An analytical expression for the CSD function of the TGSM beam propagating through non-Kolmogorov turbulence is derived, and numerical results are illustrated.
Y. Cui et al. / Optics Communications 324 (2014) 108–113
2. Cross-spectral density function of the TGSM beam in nonKolmogorov turbulence The CSD function of the TGSM beam in the source plane (z¼0) is expressed as [9,18] " # r 2 þ r 2 jr 1 r 2 j2 ikμ Wðr1 ; r2 ; 0Þ ¼ exp 1 2 2 exp 0 ðr1 r2 ÞJðr1 þ r2 ÞT ; 2 2 2sg 4sI0 ð1Þ where r1 ðx1 ; y1 Þ and r2 ðx2 ; y2 Þ denote two arbitrary position vectors in the source plane. sI0 and sg represent the transverse beam width and the transverse coherence length, respectively. The superscript T in Eq. (1) represents the transpose symbol, J is an 0 1 anti-symmetric matrix given by J ¼ , and k ¼ 2π=λ is the 1 0 wave number with λ being the wavelength. μ0 is a scalar realvalued twist factor with an inverse length dimension, limited by h i1 2 the double inequality 0 r μ20 r k s4g due to the non-negativity requirement of Eq. (1). Under the condition of μ0 ¼ 0, Eq. (1) reduces to a CSD function of GSM beam without twist phase [1]. Due to the existence of the term ðr1 r2 ÞJðr1 þ r2 ÞT ¼ x1 y2 x2 y1 in Eq. (1), the two-dimensional CSD function cannot be split into a product of two one-dimensional cross-spectral density functions. Within the validity of the paraxial approximation, propagation of the CSD function of a partially coherent beam in turbulent atmosphere can be studied with the help of the following extended Huygens–Fresnel integral [49,53]: Z 1 Z 1 1 W o ðρ1 ; ρ2 ; zÞ ¼ 2 Wðr1 ; r2 ; 0Þ 2 λ z 1 1 ik ik exp ðr1 ρ1 Þ2 þ ðr2 ρ2 Þ2 2z 2z 2
2
〈exp½ψ n ðr1 ; ρ1 ; zÞ þ ψðr2 ; ρ2 ; zÞ〉m d r1 d r2 ;
ð2Þ
where ρ1 ðξ1 ; η1 Þ; ρ2 ðξ2 ; η2 Þ denote the two arbitrary position vectors in the output plane. ψðr; ρ; zÞ denotes the random part of the complex phase of a spherical wave propagating in turbulent medium from the point ðr; 0Þ to ðρ; zÞ. The angle brackets with subscript “m” represent the ensemble average over the medium, and the term 〈exp½ψ n ðr1 ; ρ1 ; zÞ þ ψðr2 ; ρ2 ; zÞ〉m is expressed as 〈exp½ψ n ðr1 ; ρ1 ; zÞ þ ψ ðr2 ; ρ2 ; zÞ〉m ( 2
¼ exp 4π 2 k z
Z
Z
1
dξ 0
0
1
)
1 J 0 ðκ ξðr1 r2 Þ þ ð1 ξÞðρ1 ρ2 ÞÞ Φn ðκÞκ dκ ;
ð3Þ where Φn ðκÞ stands for the spatial power spectrum of the refractive-index fluctuation of the atmospheric turbulence with κ being the two-dimensional spatial frequency. J 0 is the first-kind of Bessel function of order 0. For the convenience of integration, J 0 is expanded and approximated as follows [56]: 1 J 0 ðκ ξðr1 r2 Þ þ ð1 ξÞðρ1 ρ2 ÞÞ 1 ðκ ξðr1 r2 Þ þ ð1 ξÞðρ1 ρ2 ÞÞ2 : 4 ð4Þ Above approximation is valid only when κ ξðr1 r2 Þ þ ð1 ξÞ È ðρ1 ρ2 Þj1, and this approximation has been used to treat the propagation of a partially coherent beam both in Kolmogorov turbulence [56] and in non-Kolmogorov turbulence [53]. We assume that the turbulence obeys the non-Kolmogorov statistics and the power spectrum Φn ðκÞ has the van Karman form, in which the slope 11/3 is generalized to an arbitrary parameter α
109
[53,57], i.e., 2 exp½ κ 2 =κ 2 m Φn ðκÞ ¼ AðαÞC~ n ; ðκ2 þ κ 20 Þα=2
0 o κ o 1;
3 oα o 4;
ð5Þ
where κ 0 ¼ 2π=L0 with L0 being the outer scale of turbulence, κm ¼ cðαÞ=l0 with l0 being the inner scale of turbulence, and 2πΓð5 α=2ÞAðαÞ 1=ðα 5Þ cðαÞ ¼ : ð6Þ 3 The term C~ n in Eq. (5) is a generalized refractive-index structure parameter with units m3 α , and απ
1 : ð7Þ AðαÞ ¼ 2 Γðα 1Þ cos 2 4π 2
Here Γð U Þ represents the Gamma function. Substituting Eqs. (4) and (5) into Eq. (3), we obtain (after integration over ξ; κ) [53] 〈exp½ψ n ðr1 ; ρ1 ; zÞ þ ψðr2 ; ρ2 ; zÞ〉m ( ) 2 i π 2 k Tzh ðr1 r2 Þ2 þ ðr1 r2 Þ U ðρ1 ρ2 Þ þ ðρ1 ρ2 Þ2 ; ¼ exp 3 ð8Þ with T¼
AðαÞ ~ 2 2 α C βκ m expðκ 20 =κ 2m ÞΓ 1 ð2 α=2; κ 20 =κ2m Þ 2κ 40 α ; 2ðα 2Þ n
ð9Þ
where β ¼ 2κ 20 2κ 2m þ ακ2m , and Γ 1 is the incomplete Gamma function. In the case of α ¼ 11=3, the power spectrum Φn ðκÞ reduces to the van Karman spectrum with Kolmogorov statistics, given by [26] κ2 ð10Þ Φn ðκÞ ¼ 0:033C 2n ðκ 2 þ κ20 Þ 11=6 exp 2 ; κm where C 2n is a constant structure parameter with unit m 2=3 . If the outer scale of turbulence L0 tends to infinity, the power spectrum Φn ðκÞ further reduces to the Tatarskii spectrum with Kolmogorov statistics κ2 ð11Þ Φn ðκÞ ¼ 0:033C 2n κ 11=3 exp 2 : κm The power spectrum of Eq. (11) has been used in many literatures to explore the statistical properties of various laser beams propagating through Kolmogorov turbulence. The statistical properties of the TGSM beam in Kolmogorov turbulence can be found in Ref. [22]. Eqs. (1), (2) and (8) provide the theoretical basis for studying the propagation of the TGSM beam in non-Kolmogorov turbulence. Substituting Eqs. (1) and (8) into Eq. (2), we obtain (after integration) the following expression for the CSD function of the TGSM beam in non-Kolmogorov turbulence:\scale87%{
ρ2 þ ρ22 1 W o ðρ1 ; ρ2 ; zÞ ¼ ΔðzÞ exp 4s12 ΔðzÞ I0
"
! # 2 2 1 π 2 k Tz 2 π 4 k T 2 z4 2 1 þ exp ρ Þ ðρ þ 1 2 3 ΔðzÞ 2s2g ΔðzÞ 18ΔðzÞs2I0 exp
with
"
ΔðzÞ ¼ 1 þ μ20 þ
! ik ρ21 ρ22 iμ0 k exp ðρ1 ρ2 ÞJðρ1 þ ρ2 ÞT ; 2ΔðzÞ 2R1 ðzÞ
1 4k s4I0 2
þ
1
1
þ 2
2 k s2I0 sg
2
2π 2 k Tz 3
!# z2 ;
ð12Þ
110
Y. Cui et al. / Optics Communications 324 (2014) 108–113
RðzÞ ¼ z þ
s2I0 z π 2 Tz4 =3 ðΔðzÞ 1Þs2I0 þ π 2 Tz3 =3
ð15Þ
turbulence with α ¼ 11=3 for different values of the twist factor μ0 and the transverse coherence length sg . One finds from Fig. 1 that the beam width of the TGSM beam increases as the propagation distance increases both in non-Kolmogorov turbulence and in Kolmogorov turbulence. The beam exhibits the different spreading speeds for different values of the twist factor μ0 and the transverse coherence length sg . Here the spreading speed is defined as the speed of increase in the beam width. The spreading speed of the TGSM beam increases as the value of the twist factor increases or the transverse coherence length sg decreases both in nonKolmogorov turbulence and in Kolmogorov turbulence. To show the advantage of the TGSM beam over the GSM beam quantitatively, we adopt the parameter ΔsI ðzÞ named the deviation percentage of the beam width introduced in Ref. [22] to show the difference between the beam width ofthe TGSM beam or GSM beam in turbulent atmosphere and that of the TGSM or GSM beam in free space. ΔsI ðzÞ is defined as [22]
ð16Þ
ΔsI ðzÞ ¼
Eq. (12) provides a convenient way for studying the statistical properties of the TGSM beam in non-Kolmogorov turbulence. Comparing Eqs. (1) and (12), we can obtain the following expressions for the transverse beam width, the transverse coherence length, the radius of curvature and the twisted factor in nonKolmogorov turbulence, respectively: pffiffiffiffiffiffiffiffiffi sI ðzÞ ¼ sI0 ΔðzÞ; ð13Þ
sg ðzÞ ¼ sg RðzÞ ¼ z þ
! 1=2 2 2 2s2g π 2 k Tz 2s2g π 4 k T 2 z4 1 2 þ ; 1þ ΔðzÞ ΔðzÞ 3 18ΔðzÞs2I0
s2I0 z π 2 Tz4 =3 ; ðΔðzÞ 1Þs2I0 þ π 2 Tz3 =3
μðzÞ ¼ μ0 =ΔðzÞ:
ð14Þ
3. Numerical results In this section, we study numerically the statistical properties of the TGSM beam in non-Kolmogorov turbulence by applying Eqs. (13) and (16). The parameters in the following calculations are chosen as sI0 ¼ 0:01 m, l0 ¼ 0:01 m, L0 ¼ 1 m, λ ¼ 1550 nm, and the other parameters will be specified in numerical examples. We calculate in Fig. 1(a) and (c) the beam width sI ðzÞ of the TGSM beam versus the propagation distance z in non-Kolmogorov turbulence with α ¼ 3:2 for different values of the twist factor μ0 and the transverse coherence length sg . For the convenience of comparison, we calculate in Fig. 1(b) and (d) the beam width sI ðzÞ of the TGSM beam versus the propagation distance z in Kolmogorov
sI ðzÞtur sI ðzÞf ree : sI ðzÞf ree
ð17Þ
The subscripts “tur” or “free” in Eq. (17) represent the beam 2 width in turbulence or in free space (C~ n ¼ 0) on propagation, respectively. One sees that the beam with smaller value of ΔsI ðzÞ is less affected by the turbulence. We calculate in Fig. 2 the deviation percentage of the beam width of the TGSM beam versus the propagation distance z in non-Kolmogorov turbulence with α ¼ 3:2 or in Kolmogorov turbulence with α ¼ 11=3 for different values of the twist factor μ0 and the transverse coherence length sg . One finds from Fig. 2 that the deviation percentage of the beam width of the TGSM beam with larger value of μ0 or smaller value of sg is always smaller than that of the TGSM beam with smaller value of μ0 or larger value of sg , which means that the TGSM beam with larger value of μ0 or smaller value of sg is less affected by turbulence both in non-Kolmogorov turbulence and in Kolmogorov turbulence.
Fig. 1. Beam width sI ðzÞ of the TGSM beam versus the propagation distance z in non-Kolmogorov turbulence with α ¼ 3:2 or in Kolmogorov turbulence with α ¼ 11=3 for different values of the twist factor μ0 and the transverse coherence length sg .
Y. Cui et al. / Optics Communications 324 (2014) 108–113
111
Fig. 2. Deviation percentage of the beam width of the TGSM beam versus the propagation distance z in non-Kolmogorov turbulence with α ¼ 3:2 or in Kolmogorov turbulence with α ¼ 11=3 for different values of the twist factor μ0 and the transverse coherence length sg .
Fig. 3. Radius of curvature of the TGSM beam versus the propagation distance z in non-Kolmogorov turbulence with α ¼ 3:2 or in Kolmogorov turbulence with α ¼ 11=3 for 2 different values of μ0 , C~ n and sg .
Fig. 3 shows the calculation of the radius of curvature of the TGSM beam versus the propagation distance z in non-Kolmogorov turbulence with α ¼ 3:2 or in Kolmogorov turbulence with 2 α ¼ 11=3 for different values of the twist factor μ0 , C~ n and the transverse coherence length sg . It is clear from Fig. 3 that the radius of curvature of the TGSM beam on propagation in free space 2 (C~ n ¼ 0) or in turbulence initially displays a downward trend in the
near field, but after reaching a dip, starts to increase. The difference between the radius of curvature of the TGSM beam in free space and that in Kolmogorov turbulence or non-Kolmogorov turbulence is smaller than the difference between the radius of curvature of the GSM beam (μ0 ¼ 0) in free space and that in Kolmogorov turbulence or non-Kolmogorov turbulence, which means that the TGSM beam is less affected by turbulence than
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Y. Cui et al. / Optics Communications 324 (2014) 108–113
Fig. 4. Transverse coherence length of the TGSM beam versus the propagation distance z in non-Kolmogorov turbulence with α ¼ 3:2 or in Kolmogorov turbulence with α ¼ 11=3 for different values of the twist factor μ0 . The initial transverse coherence length sg is 10 mm.
Fig. 5. Twist factor of the TGSM beam versus the propagation distance z in non-Kolmogorov turbulence with α ¼ 3:2 or in Kolmogorov turbulence with α ¼ 11=3. The initial transverse coherence length sg is 10 mm.
the GSM beam from the aspect of the radius of curvature. Furthermore, one finds from Fig. 3 that the TGSM beam with lower coherence is less affected by the turbulence than that with higher coherence. Now we study the evolution properties of the transverse coherence length and twist factor of the TGSM beam in turbulence. Calculation of transverse coherence length of the TGSM beam versus the propagation distance z in non-Kolmogorov turbulence with α ¼ 3:2 or in Kolmogorov turbulence with α ¼ 11=3 for different values of the twist factor μ0 is shown in Fig. 4. One finds from Fig. 4 that the transverse coherence length first increases in the near field, but after reaching a maximum value, starts to decrease both in non-Kolmogorov turbulence and in Kolmogorov turbulence. In the near field, the transverse coherence length increases more rapidly as the twist factor increases, while its value almost does not depend on the twist factor in the far field. This phenomenon can be explained by the fact that the evolution properties of transverse coherence length are mainly determined by the beam diffraction and the turbulence together. In the near field, the effect of turbulence is small, and the beam diffraction plays a dominant role; thus the coherence length increases as the propagation distance increases, which is similar to its behavior in free space. However, when the propagation distance is long enough, the effect of turbulence overpasses the effect of beam diffraction, and plays a dominant role. The effect of turbulence is to reduce the coherence of the beam. Thus, the coherence length decreases on propagation when the propagation distance is long enough. Fig. 5 shows the calculation of the twist factor of the TGSM beam versus the propagation distance z in nonKolmogorov turbulence with α ¼ 3:2 or in Kolmogorov turbulence with α ¼ 11=3. One finds from Fig. 5 that the twist factor of the TGSM beam decreases on propagation, and approaches to zero in the far field both in non-Kolmogorov turbulence and in Kolmogorov turbulence.
4. Conclusion In conclusion, we have derived the explicit expression for the CSD function of the TGSM beam on propagation in nonKolmogorov turbulence based on the extended Huygens–Fresnel principle under the condition of strong turbulence. The derived formula can be reduced to the propagation formula for the TGSM beam in Kolmogorov turbulence under certain condition. With the help of the derived formula, we have explored the statistical properties of the TGSM beam both in non-Kolmogorov turbulence and in Kolmogorov turbulence. We have found that the TGSM beam is less affected by turbulence than the GSM beam both in non-Kolmogorov turbulence and in Kolmogorov turbulence, which will be useful in free-space communications.
Acknowledgments This work is supported by the National Natural Science Foundation of China under Grant nos. 11274005 and 11104195, the Huo Ying Dong Education Foundation of China under Grant no. 121009, the Key Project of Chinese Ministry of Education under Grant no. 210081, the Universities Natural Science Research Project of Jiangsu Province under Grant 11KJB140007, the project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions, and the project sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry. References [1] L. Mandel, E. Wolf, Optical Coherence and Quantum OpticsCambridge University Press, New York, 1995. [2] Y. Cai, S.Y. Zhu, Phys. Rev. E 71 (2005) 056607. [3] M. Alavinejad, B. Ghafary, Opt. Lasers Eng. 46 (2008) 357. [4] Z. Qin, R. Tao, P. Zhou, X. Xu, Z. Liu, Opt. Laser Technol. 56 (2014) 182.
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Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Propagation of a twist Gaussian–Schell model beam in non-Kolmogorov turbulence Yan Cui, Fei Wang n, Yangjian Cai School of Physical Science and Technology & Collaborative Innovation Center of Suzhou Nano Science and Technology, Soochow University, Suzhou 215006, China
art ic l e i nf o
a b s t r a c t
Article history: Received 20 January 2014 Received in revised form 9 March 2014 Accepted 14 March 2014 Available online 27 March 2014
Based on the extended Huygens–Fresnel integral, an analytical formula for the cross-spectral density (CSD) function of a twist Gaussian–Schell model (TGSM) beam propagating through non-Kolmogorov turbulence is derived. The statistical properties, such as the beam width, the radius of curvature, the transverse coherence length and the twist factor of the TGSM beam in non-Kolmogorov turbulence, are studied numerically. It is found that the TGSM beam is less affected than a GSM beam without twist phase both in non-Kolmogorov turbulence and in Kolmogorov turbulence. Our results will be useful in free-space communication. & 2014 Elsevier B.V. All rights reserved.
Keywords: Turbulent atmosphere Propagation Twist factor Partially coherent beam
1. Introduction In the past decades, partially coherent beams have been extensively investigated due to their wide applications in optical image, inertial confinement fusion, free-space communications and optical trappings [1–4]. Gaussian–Schell-model (GSM) beam whose spectral degree of coherence and intensity distribution satisfy Gaussian distribution is a typical partially coherent beam, and has been studied extensively [5–8]. A more general GSM beam can possess a twist phase, which differs in many aspects from the customary quadratic phase factor. In 1993, Simon and Mukunda first introduced the GSM beam with twist phase named as a twist Gaussian Schell-model (TGSM) beam [9,10]. Unlike the usual phase curvature, the twist phase is bounded by strength due to the fact that the cross-spectral density (CSD) function must be non-negative definite, and it is absent in a coherent Gaussian beam [9]. Since then, various properties, such as the spectral shift, the paraxial propagation, the orbital angular momentum, the second-harmonic generation and the ghost imaging of the TGSM beam, have been investigated [11– 20]. Recently, Zhao et al. investigated the radiation force of the TGSM beam on a Rayleigh particle [21]. The second-order statistical properties and the scintillation index of the TGSM beam in Kolmogorov turbulence have been studied in Refs. [22] and [23]. It was shown in Ref. [24] that the TGSM beam could serve as illumination that might produce images with a resolution n
Corresponding author. E-mail addresses: [email protected] (F. Wang), [email protected] (Y. Cai). http://dx.doi.org/10.1016/j.optcom.2014.03.030 0030-4018/& 2014 Elsevier B.V. All rights reserved.
overcoming the Rayleigh limit. More recently, the orbital angular moment of the TGSM beam propagating through an astigmatic ABCD optical system with loss or gain was explored [25]. On the other hand, considerable attention is being paid to the propagation of laser beams through atmospheric turbulence due to their wide applications in free-space communications, remote sensing, imaging and radar system [26–39]. The Kolmogorov model has been adopted to study the effect of atmospheric turbulence in free-space communication and has shown good agreement with the experimental results [40]. The suitable condition for the Kolmogorov model is that the atmospheric turbulence is homogeneous. However, some experimental results have shown that there exists great deviation from the Kolmogorov model's prediction in some portions of atmosphere [41–43]. The reason is that the atmosphere is extremely stable, and the turbulence is no longer homogeneous in three dimensions since the vertical component is suppressed [44]. Toselli et al. introduced the non-Kolmogorov model, in which a generalized exponent factor was used instead of the standard exponent value 11/3 in the Kolmogorov model [45,46]. Recently, the behaviors of various laser beams, such as Hermite–Gaussian beam, elegant Laguerre– Gaussian beam, laser beam array and GSM beam propagating through non-Kolmogorov turbulence, have been explored [47–55], and some interesting and useful results have been found. In this paper, our aim is to investigate the statistical properties of the TGSM beam in non-Kolmogorov turbulence. An analytical expression for the CSD function of the TGSM beam propagating through non-Kolmogorov turbulence is derived, and numerical results are illustrated.
Y. Cui et al. / Optics Communications 324 (2014) 108–113
2. Cross-spectral density function of the TGSM beam in nonKolmogorov turbulence The CSD function of the TGSM beam in the source plane (z¼0) is expressed as [9,18] " # r 2 þ r 2 jr 1 r 2 j2 ikμ Wðr1 ; r2 ; 0Þ ¼ exp 1 2 2 exp 0 ðr1 r2 ÞJðr1 þ r2 ÞT ; 2 2 2sg 4sI0 ð1Þ where r1 ðx1 ; y1 Þ and r2 ðx2 ; y2 Þ denote two arbitrary position vectors in the source plane. sI0 and sg represent the transverse beam width and the transverse coherence length, respectively. The superscript T in Eq. (1) represents the transpose symbol, J is an 0 1 anti-symmetric matrix given by J ¼ , and k ¼ 2π=λ is the 1 0 wave number with λ being the wavelength. μ0 is a scalar realvalued twist factor with an inverse length dimension, limited by h i1 2 the double inequality 0 r μ20 r k s4g due to the non-negativity requirement of Eq. (1). Under the condition of μ0 ¼ 0, Eq. (1) reduces to a CSD function of GSM beam without twist phase [1]. Due to the existence of the term ðr1 r2 ÞJðr1 þ r2 ÞT ¼ x1 y2 x2 y1 in Eq. (1), the two-dimensional CSD function cannot be split into a product of two one-dimensional cross-spectral density functions. Within the validity of the paraxial approximation, propagation of the CSD function of a partially coherent beam in turbulent atmosphere can be studied with the help of the following extended Huygens–Fresnel integral [49,53]: Z 1 Z 1 1 W o ðρ1 ; ρ2 ; zÞ ¼ 2 Wðr1 ; r2 ; 0Þ 2 λ z 1 1 ik ik exp ðr1 ρ1 Þ2 þ ðr2 ρ2 Þ2 2z 2z 2
2
〈exp½ψ n ðr1 ; ρ1 ; zÞ þ ψðr2 ; ρ2 ; zÞ〉m d r1 d r2 ;
ð2Þ
where ρ1 ðξ1 ; η1 Þ; ρ2 ðξ2 ; η2 Þ denote the two arbitrary position vectors in the output plane. ψðr; ρ; zÞ denotes the random part of the complex phase of a spherical wave propagating in turbulent medium from the point ðr; 0Þ to ðρ; zÞ. The angle brackets with subscript “m” represent the ensemble average over the medium, and the term 〈exp½ψ n ðr1 ; ρ1 ; zÞ þ ψðr2 ; ρ2 ; zÞ〉m is expressed as 〈exp½ψ n ðr1 ; ρ1 ; zÞ þ ψ ðr2 ; ρ2 ; zÞ〉m ( 2
¼ exp 4π 2 k z
Z
Z
1
dξ 0
0
1
)
1 J 0 ðκ ξðr1 r2 Þ þ ð1 ξÞðρ1 ρ2 ÞÞ Φn ðκÞκ dκ ;
ð3Þ where Φn ðκÞ stands for the spatial power spectrum of the refractive-index fluctuation of the atmospheric turbulence with κ being the two-dimensional spatial frequency. J 0 is the first-kind of Bessel function of order 0. For the convenience of integration, J 0 is expanded and approximated as follows [56]: 1 J 0 ðκ ξðr1 r2 Þ þ ð1 ξÞðρ1 ρ2 ÞÞ 1 ðκ ξðr1 r2 Þ þ ð1 ξÞðρ1 ρ2 ÞÞ2 : 4 ð4Þ Above approximation is valid only when κ ξðr1 r2 Þ þ ð1 ξÞ È ðρ1 ρ2 Þj1, and this approximation has been used to treat the propagation of a partially coherent beam both in Kolmogorov turbulence [56] and in non-Kolmogorov turbulence [53]. We assume that the turbulence obeys the non-Kolmogorov statistics and the power spectrum Φn ðκÞ has the van Karman form, in which the slope 11/3 is generalized to an arbitrary parameter α
109
[53,57], i.e., 2 exp½ κ 2 =κ 2 m Φn ðκÞ ¼ AðαÞC~ n ; ðκ2 þ κ 20 Þα=2
0 o κ o 1;
3 oα o 4;
ð5Þ
where κ 0 ¼ 2π=L0 with L0 being the outer scale of turbulence, κm ¼ cðαÞ=l0 with l0 being the inner scale of turbulence, and 2πΓð5 α=2ÞAðαÞ 1=ðα 5Þ cðαÞ ¼ : ð6Þ 3 The term C~ n in Eq. (5) is a generalized refractive-index structure parameter with units m3 α , and απ
1 : ð7Þ AðαÞ ¼ 2 Γðα 1Þ cos 2 4π 2
Here Γð U Þ represents the Gamma function. Substituting Eqs. (4) and (5) into Eq. (3), we obtain (after integration over ξ; κ) [53] 〈exp½ψ n ðr1 ; ρ1 ; zÞ þ ψðr2 ; ρ2 ; zÞ〉m ( ) 2 i π 2 k Tzh ðr1 r2 Þ2 þ ðr1 r2 Þ U ðρ1 ρ2 Þ þ ðρ1 ρ2 Þ2 ; ¼ exp 3 ð8Þ with T¼
AðαÞ ~ 2 2 α C βκ m expðκ 20 =κ 2m ÞΓ 1 ð2 α=2; κ 20 =κ2m Þ 2κ 40 α ; 2ðα 2Þ n
ð9Þ
where β ¼ 2κ 20 2κ 2m þ ακ2m , and Γ 1 is the incomplete Gamma function. In the case of α ¼ 11=3, the power spectrum Φn ðκÞ reduces to the van Karman spectrum with Kolmogorov statistics, given by [26] κ2 ð10Þ Φn ðκÞ ¼ 0:033C 2n ðκ 2 þ κ20 Þ 11=6 exp 2 ; κm where C 2n is a constant structure parameter with unit m 2=3 . If the outer scale of turbulence L0 tends to infinity, the power spectrum Φn ðκÞ further reduces to the Tatarskii spectrum with Kolmogorov statistics κ2 ð11Þ Φn ðκÞ ¼ 0:033C 2n κ 11=3 exp 2 : κm The power spectrum of Eq. (11) has been used in many literatures to explore the statistical properties of various laser beams propagating through Kolmogorov turbulence. The statistical properties of the TGSM beam in Kolmogorov turbulence can be found in Ref. [22]. Eqs. (1), (2) and (8) provide the theoretical basis for studying the propagation of the TGSM beam in non-Kolmogorov turbulence. Substituting Eqs. (1) and (8) into Eq. (2), we obtain (after integration) the following expression for the CSD function of the TGSM beam in non-Kolmogorov turbulence:\scale87%{
ρ2 þ ρ22 1 W o ðρ1 ; ρ2 ; zÞ ¼ ΔðzÞ exp 4s12 ΔðzÞ I0
"
! # 2 2 1 π 2 k Tz 2 π 4 k T 2 z4 2 1 þ exp ρ Þ ðρ þ 1 2 3 ΔðzÞ 2s2g ΔðzÞ 18ΔðzÞs2I0 exp
with
"
ΔðzÞ ¼ 1 þ μ20 þ
! ik ρ21 ρ22 iμ0 k exp ðρ1 ρ2 ÞJðρ1 þ ρ2 ÞT ; 2ΔðzÞ 2R1 ðzÞ
1 4k s4I0 2
þ
1
1
þ 2
2 k s2I0 sg
2
2π 2 k Tz 3
!# z2 ;
ð12Þ
110
Y. Cui et al. / Optics Communications 324 (2014) 108–113
RðzÞ ¼ z þ
s2I0 z π 2 Tz4 =3 ðΔðzÞ 1Þs2I0 þ π 2 Tz3 =3
ð15Þ
turbulence with α ¼ 11=3 for different values of the twist factor μ0 and the transverse coherence length sg . One finds from Fig. 1 that the beam width of the TGSM beam increases as the propagation distance increases both in non-Kolmogorov turbulence and in Kolmogorov turbulence. The beam exhibits the different spreading speeds for different values of the twist factor μ0 and the transverse coherence length sg . Here the spreading speed is defined as the speed of increase in the beam width. The spreading speed of the TGSM beam increases as the value of the twist factor increases or the transverse coherence length sg decreases both in nonKolmogorov turbulence and in Kolmogorov turbulence. To show the advantage of the TGSM beam over the GSM beam quantitatively, we adopt the parameter ΔsI ðzÞ named the deviation percentage of the beam width introduced in Ref. [22] to show the difference between the beam width ofthe TGSM beam or GSM beam in turbulent atmosphere and that of the TGSM or GSM beam in free space. ΔsI ðzÞ is defined as [22]
ð16Þ
ΔsI ðzÞ ¼
Eq. (12) provides a convenient way for studying the statistical properties of the TGSM beam in non-Kolmogorov turbulence. Comparing Eqs. (1) and (12), we can obtain the following expressions for the transverse beam width, the transverse coherence length, the radius of curvature and the twisted factor in nonKolmogorov turbulence, respectively: pffiffiffiffiffiffiffiffiffi sI ðzÞ ¼ sI0 ΔðzÞ; ð13Þ
sg ðzÞ ¼ sg RðzÞ ¼ z þ
! 1=2 2 2 2s2g π 2 k Tz 2s2g π 4 k T 2 z4 1 2 þ ; 1þ ΔðzÞ ΔðzÞ 3 18ΔðzÞs2I0
s2I0 z π 2 Tz4 =3 ; ðΔðzÞ 1Þs2I0 þ π 2 Tz3 =3
μðzÞ ¼ μ0 =ΔðzÞ:
ð14Þ
3. Numerical results In this section, we study numerically the statistical properties of the TGSM beam in non-Kolmogorov turbulence by applying Eqs. (13) and (16). The parameters in the following calculations are chosen as sI0 ¼ 0:01 m, l0 ¼ 0:01 m, L0 ¼ 1 m, λ ¼ 1550 nm, and the other parameters will be specified in numerical examples. We calculate in Fig. 1(a) and (c) the beam width sI ðzÞ of the TGSM beam versus the propagation distance z in non-Kolmogorov turbulence with α ¼ 3:2 for different values of the twist factor μ0 and the transverse coherence length sg . For the convenience of comparison, we calculate in Fig. 1(b) and (d) the beam width sI ðzÞ of the TGSM beam versus the propagation distance z in Kolmogorov
sI ðzÞtur sI ðzÞf ree : sI ðzÞf ree
ð17Þ
The subscripts “tur” or “free” in Eq. (17) represent the beam 2 width in turbulence or in free space (C~ n ¼ 0) on propagation, respectively. One sees that the beam with smaller value of ΔsI ðzÞ is less affected by the turbulence. We calculate in Fig. 2 the deviation percentage of the beam width of the TGSM beam versus the propagation distance z in non-Kolmogorov turbulence with α ¼ 3:2 or in Kolmogorov turbulence with α ¼ 11=3 for different values of the twist factor μ0 and the transverse coherence length sg . One finds from Fig. 2 that the deviation percentage of the beam width of the TGSM beam with larger value of μ0 or smaller value of sg is always smaller than that of the TGSM beam with smaller value of μ0 or larger value of sg , which means that the TGSM beam with larger value of μ0 or smaller value of sg is less affected by turbulence both in non-Kolmogorov turbulence and in Kolmogorov turbulence.
Fig. 1. Beam width sI ðzÞ of the TGSM beam versus the propagation distance z in non-Kolmogorov turbulence with α ¼ 3:2 or in Kolmogorov turbulence with α ¼ 11=3 for different values of the twist factor μ0 and the transverse coherence length sg .
Y. Cui et al. / Optics Communications 324 (2014) 108–113
111
Fig. 2. Deviation percentage of the beam width of the TGSM beam versus the propagation distance z in non-Kolmogorov turbulence with α ¼ 3:2 or in Kolmogorov turbulence with α ¼ 11=3 for different values of the twist factor μ0 and the transverse coherence length sg .
Fig. 3. Radius of curvature of the TGSM beam versus the propagation distance z in non-Kolmogorov turbulence with α ¼ 3:2 or in Kolmogorov turbulence with α ¼ 11=3 for 2 different values of μ0 , C~ n and sg .
Fig. 3 shows the calculation of the radius of curvature of the TGSM beam versus the propagation distance z in non-Kolmogorov turbulence with α ¼ 3:2 or in Kolmogorov turbulence with 2 α ¼ 11=3 for different values of the twist factor μ0 , C~ n and the transverse coherence length sg . It is clear from Fig. 3 that the radius of curvature of the TGSM beam on propagation in free space 2 (C~ n ¼ 0) or in turbulence initially displays a downward trend in the
near field, but after reaching a dip, starts to increase. The difference between the radius of curvature of the TGSM beam in free space and that in Kolmogorov turbulence or non-Kolmogorov turbulence is smaller than the difference between the radius of curvature of the GSM beam (μ0 ¼ 0) in free space and that in Kolmogorov turbulence or non-Kolmogorov turbulence, which means that the TGSM beam is less affected by turbulence than
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Fig. 4. Transverse coherence length of the TGSM beam versus the propagation distance z in non-Kolmogorov turbulence with α ¼ 3:2 or in Kolmogorov turbulence with α ¼ 11=3 for different values of the twist factor μ0 . The initial transverse coherence length sg is 10 mm.
Fig. 5. Twist factor of the TGSM beam versus the propagation distance z in non-Kolmogorov turbulence with α ¼ 3:2 or in Kolmogorov turbulence with α ¼ 11=3. The initial transverse coherence length sg is 10 mm.
the GSM beam from the aspect of the radius of curvature. Furthermore, one finds from Fig. 3 that the TGSM beam with lower coherence is less affected by the turbulence than that with higher coherence. Now we study the evolution properties of the transverse coherence length and twist factor of the TGSM beam in turbulence. Calculation of transverse coherence length of the TGSM beam versus the propagation distance z in non-Kolmogorov turbulence with α ¼ 3:2 or in Kolmogorov turbulence with α ¼ 11=3 for different values of the twist factor μ0 is shown in Fig. 4. One finds from Fig. 4 that the transverse coherence length first increases in the near field, but after reaching a maximum value, starts to decrease both in non-Kolmogorov turbulence and in Kolmogorov turbulence. In the near field, the transverse coherence length increases more rapidly as the twist factor increases, while its value almost does not depend on the twist factor in the far field. This phenomenon can be explained by the fact that the evolution properties of transverse coherence length are mainly determined by the beam diffraction and the turbulence together. In the near field, the effect of turbulence is small, and the beam diffraction plays a dominant role; thus the coherence length increases as the propagation distance increases, which is similar to its behavior in free space. However, when the propagation distance is long enough, the effect of turbulence overpasses the effect of beam diffraction, and plays a dominant role. The effect of turbulence is to reduce the coherence of the beam. Thus, the coherence length decreases on propagation when the propagation distance is long enough. Fig. 5 shows the calculation of the twist factor of the TGSM beam versus the propagation distance z in nonKolmogorov turbulence with α ¼ 3:2 or in Kolmogorov turbulence with α ¼ 11=3. One finds from Fig. 5 that the twist factor of the TGSM beam decreases on propagation, and approaches to zero in the far field both in non-Kolmogorov turbulence and in Kolmogorov turbulence.
4. Conclusion In conclusion, we have derived the explicit expression for the CSD function of the TGSM beam on propagation in nonKolmogorov turbulence based on the extended Huygens–Fresnel principle under the condition of strong turbulence. The derived formula can be reduced to the propagation formula for the TGSM beam in Kolmogorov turbulence under certain condition. With the help of the derived formula, we have explored the statistical properties of the TGSM beam both in non-Kolmogorov turbulence and in Kolmogorov turbulence. We have found that the TGSM beam is less affected by turbulence than the GSM beam both in non-Kolmogorov turbulence and in Kolmogorov turbulence, which will be useful in free-space communications.
Acknowledgments This work is supported by the National Natural Science Foundation of China under Grant nos. 11274005 and 11104195, the Huo Ying Dong Education Foundation of China under Grant no. 121009, the Key Project of Chinese Ministry of Education under Grant no. 210081, the Universities Natural Science Research Project of Jiangsu Province under Grant 11KJB140007, the project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions, and the project sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry. References [1] L. Mandel, E. Wolf, Optical Coherence and Quantum OpticsCambridge University Press, New York, 1995. [2] Y. Cai, S.Y. Zhu, Phys. Rev. E 71 (2005) 056607. [3] M. Alavinejad, B. Ghafary, Opt. Lasers Eng. 46 (2008) 357. [4] Z. Qin, R. Tao, P. Zhou, X. Xu, Z. Liu, Opt. Laser Technol. 56 (2014) 182.
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