Theoretical and experimental studies of polarization fluctuations over atmospheric turbulent channels for wireless optical communication systems Jiankun Zhang,1 Shengli Ding,1 Huili Zhai,1 and Anhong Dang1,* 1
State Key Laboratory of Advanced Optical Communication Systems & Networks, Department of Electronics, Peking University, Beijing 100871, China *[email protected]
Abstract: In wireless optical communications (WOC), polarization multiplexing systems and coherent polarization systems have excellent performance and wide applications, while its state of polarization affected by atmospheric turbulence is not clearly understood. This paper focuses on the polarization fluctuations caused by atmospheric turbulence in a WOC link. Firstly, the relationship between the polarization fluctuations and the index of refraction structure parameter is introduced and the distribution of received polarization angle is obtained through theoretical derivations. Then, turbulent conditions are adjusted and measured elaborately in a wide range of scintillation indexes (SI). As a result, the root-mean-square (RMS) variation and probability distribution function (PDF) of polarization angle conforms closely to that of theoretical model. ©2014 Optical Society of America OCIS codes: (060.2605) Free-space optical communication; (260.5430) Polarization.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9.
10. 11. 12. 13. 14. 15.
V. Chan, “Free-space optical communications,” J. Lightwave Technol. 24(12), 4750–4762 (2006). L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005). M. Niu, X. Song, J. Cheng, and J. F. Holzman, “Performance analysis of coherent wireless optical communications with atmospheric turbulence,” Opt. Express 20(6), 6515–6520 (2012). A. K. Majumdar, “Free-space laser communication performance in the atmospheric channel,” J. Opt. Fiber Commun. Rep. 2(4), 345–396 (2005). J. C. Ricklin and F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: Implications for free-space laser communication,” J. Opt. Soc. Am. A 19(9), 1794–1802 (2002). X. Ji and G. Ji, “Effect of turbulence on the beam quality of apertured partially coherent beams,” J. Opt. Soc. Am. A 25(6), 1246–1252 (2008). M. L. Wesely, “The combined effect of temperature and humidity fluctuations on refractive index,” J. Appl. Meteorol. 15(1), 43–49 (1976). N. Cvijetic, D. Qian, J. Yu, Y. Huang, and T. Wang, “Polarization-multiplexed optical wireless transmission with coherent detection,” J. Lightwave Technol. 28(8), 1218–1227 (2010). H. Huang, G. Xie, Y. Yan, N. Ahmed, Y. Ren, Y. Yue, D. Rogawski, M. J. Willner, B. I. Erkmen, K. M. Birnbaum, S. J. Dolinar, M. P. J. Lavery, M. J. Padgett, M. Tur, and A. E. Willner, “100 Tbit/s free-space data link enabled by three-dimensional multiplexing of orbital angular momentum, polarization, and wavelength,” Opt. Lett. 39(2), 197–200 (2014). G. Xie, F. Wang, A. Dang, and H. Guo, “A novel polarization-multiplexing system for free-space optical links,” Photon. Technol. Lett. 23(20), 1484–1486 (2011). Y. Han and G. Li, “Coherent optical communication using polarization multiple-input-multiple-output,” Opt. Express 13(19), 7527–7534 (2005). X. Tang, Z. Xu, and Z. Ghassemlooy, “Coherent polarization modulated transmission through MIMO atmospheric optical turbulence channel,” J. Lightwave Technol. 31(20), 3221–3228 (2013). X. Zhao, Y. Yao, Y. Sun, and C. Liu, “Circle polarization shift keying with direct detection for free-space optical communication,” J. Opt. Commun. Netw. 1(4), 307–312 (2009). Z. Ghassemlooy, X. Tang, and S. Rajbhandari, “Experimental investigation of polarization modulated free space optical communication with direct detection in a turbulence channel,” IET Commun. 6(11), 1489–1494 (2012). J. W. Strohbehn and S. F. Clifford, “Polarization and angle-of -arrival fluctuations for a plane wave propagated through a turbulent medium,” IEEE Trans. Antenn. Propag. 15(3), 416–421 (1967).
#224168 - $15.00 USD Received 30 Sep 2014; revised 26 Nov 2014; accepted 10 Dec 2014; published 24 Dec 2014 (C) 2014 OSA 3 November 2014 | Vol. 22, No. 22 | DOI:10.1364/OE.22.032482 | OPTICS EXPRESS 32482
16. E. Collett and R. Alferness, “Depolarization of a laser beam in a turbulent medium,” J. Opt. Soc. Am. 62(4), 529–533 (1972). 17. D. H. Höhn, “Depolarization of a laser beam at 6328 Ǻ due to atmospheric transmission,” Appl. Opt. 8(2), 367– 369 (1969). 18. M. Toyoshima, H. Takenaka, Y. Shoji, Y. Takayama, Y. Koyama, and H. Kunimori, “Polarization measurements through space-to-ground atmospheric propagation paths by using a highly polarized laser source in space,” Opt. Express 17(25), 22333–22340 (2009). 19. D. F. James, “Change of polarization of light beams on propagation in free space,” J. Opt. Soc. Am. 11(5), 1641–1643 (1994). 20. X. Ji and X. Chen, “Changes in the polarization, the coherence and the spectrum of partially coherent electromagnetic Hermite–Gaussian beams in turbulence,” Opt. Laser Technol. 41(2), 165–171 (2009). 21. F. Kashani, M. Alavinejad, and B. Ghafary, “Polarization characteristics of aberrated partially coherent flattopped beam propagating through turbulent atmosphere,” Opt. Commun. 282(20), 4029–4034 (2009). 22. Y. Wang, Y. Zhang, J. Wang, and J. Jia, “Degree of polarization for quantum light field propagating through non-Kolmogorov turbulence,” Opt. Laser Technol. 43(4), 776–780 (2011). 23. O. Korotkova and N. Farwell, “Effect of oceanic turbulence on polarization of stochastic beams,” Opt. Commun. 284(7), 1740–1746 (2011). 24. X. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun. 50(8), 1293–1300 (2002). 25. Y. Han, A. Dang, Y. Ren, J. Tang, and H. Guo, “Theoretical and experimental studies of turbo product code with time diversity in free space optical communication,” Opt. Express 18(26), 26978–26988 (2010). 26. M. Al-Habash, C. Andrews, and R. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40(8), 1554–1562 (2001). 27. A. Dang, “A closed-form solution of the bit-error rate for optical wireless communication systems over atmospheric turbulence channels,” Opt. Express 19(4), 3494–3502 (2011). 28. A. Mostafa and S. Hranilovic, “Channel measurement and Markov modeling of an urban free-space optical link,” J. Opt. Commun. Netw. 4(10), 836–846 (2012). 29. F. S. Vetelino, C. Young, L. Andrews, and J. Recolons, “Aperture averaging effects on the probability density of irradiance fluctuations in moderate-to-strong turbulence,” Appl. Opt. 46(11), 2099–2108 (2007).
1. Introduction Wireless optical communication (WOC) is a promising candidate for data transmission due to its flexibility, wide-bandwidth and license-free [1]. However, the random variation of the refractive index (RI) caused by atmospheric turbulence can degrade the performance of WOC system severely. When transmitting through atmosphere, WOC signals suffer from intensity scintillation, phase fluctuation, as well as polarization alteration caused by turbulence [2–6]. The variation of RI is caused by the random variation of the temperature [7]. The longitudinal wind velocity associated with the turbulent atmosphere fluctuates randomly about its mean value, resulting in the changing of RI. When polarized light passes through the turbulent atmosphere, intensities in the two orthogonal polarizations are impacted differently, leading to the alteration of the direction and degree of polarization. In WOC systems, polarization techniques are widely applied, such as polarizationmultiplexing systems [8–11], coherent polarization shift keying (POLSK) systems [12,13], and so on. However, these systems are quite sensitive to polarization directions, requiring polarization agreement in both transmitting and receiving ends. Polarization fluctuations may cause a crosstalk between the signals in different polarizations, resulting in the degradation of optical signal-to-noise ratio (OSNR) and bit error rate (BER) performance [14]. Polarization fluctuations may also increase the complexity of polarization control feedback in the systems. Hence an applicable model for polarization fluctuations is important and necessary for performance estimations. Up to now, many researchers have studied the polarization fluctuations caused by atmospheric turbulence [15–23]. However, their results are limited for some specific conditions to some extent. For example, an early theoretical model proposed in [15,16] is based on slight RI variations, and it shows two orders of magnitude smaller than experimental results presented in [17]. A space-to-ground propagation was carried in [18], but the turbulent parameters of which is unknown. Hence the measurement results are not compared with any theory. Recently, polarization fluctuations for partially coherent beams were studied [19–21], and polarization in non-Kolmogorov turbulence [22] and oceanic turbulence [23] was researched. However, a widely accepted model is not yet proposed to describe turbulenceinduced polarization fluctuations. Moreover, most of the accomplished experiments are #224168 - $15.00 USD Received 30 Sep 2014; revised 26 Nov 2014; accepted 10 Dec 2014; published 24 Dec 2014 (C) 2014 OSA 3 November 2014 | Vol. 22, No. 22 | DOI:10.1364/OE.22.032482 | OPTICS EXPRESS 32483
measured under a single turbulent condition, and the distribution of the polarization is seldom discussed. This paper presents a novel and widely applicable theory to evaluate polarization fluctuations in turbulence. Firstly, the relationship between the polarization fluctuations and the index of refraction structure parameter is introduced and the probability distribution function (PDF) of received polarization angle is obtained through theoretical derivations. Then, a related experiment is carried out to measure the distribution of the polarization angle. In the experiment, different turbulent conditions are realized to confirm our theory in a wide range. Finally, the experimental results are compared with the theoretical conclusions. 2. Theoretical model of turbulence on polarization Suppose the incident light is polarized along the z-axis and propagates along the x-axis. The time-varying light-field can be expressed as E0 (t ) = A0 exp[j(ω0 t + S0 )]xˆ ,
(1)
where A0, ω0 and S0 denote the amplitude, frequency and phase of transmitting wave respectively. When propagating through turbulence, polarization alteration leads to the amplitude variation both in x and y directions. Then the receiving light-field is E = ( Ax x + Ay y ) exp{ j[ω0 t + S (t )]},
(2)
where Ax and Ay are the polarization components of x and y direction separately, which satisfy A=
2
Ax + Ay
2
; S(t) is the phase fluctuations caused by turbulence. The polarization
angle is defined as φ = arctan(Ay / Ax), with Ax = Re{Aejφ} and Ay = Im{Aejφ}. When the polarization angle is quite small, there is φ ≈M where M = |Ey| / |E0| is the polarizability. According to the perturbation theory, M is given by [14] M2 =
4 π Lσ n , k 2l 3
(3)
where σn is the root mean square of refractive index and satisfies σ n = Δn 2
1/ 2
; l is the scale
factor; k = 2π / λ is the wavenumber, where λ is the wavelength; L is propagation length. Then the root mean square σφ of the polarization angle φ caused by atmospheric turbulence is [17]
σϕ =
σ n λ L1/ 2 . 2π 3/ 4 l 3/ 2
(4)
For experiments, Δn is usually difficult to measure directly. Since the atmospheric structure function Dn (r ) = [n(r1 ) − n(r2 )]2 = [n(r1 )]2 + [n(r2 )]2 − 2 n(r1 )n(r2 ) ([2], Sec 3.2), we can get Dn (0) = 2 n 2 − 2 n
2
= 2Cn2 d 02/3 in the correlation range. By substituting
2
n ≈ Cn2 d 02/3 [16] and d 0 ≈ (λ L)1/ 2 [24,25], where d0 is the correlation length, σ n2 can be approximated as
σ n2 = 0.5Cn2 λ 1/3 L1/3 ,
(5)
where Cn2 is the index of refraction structure parameter which can be easily measured and compared with the theoretical results. Thus the RMS of depolarization can be written as
σϕ =
(Cn2 )1/ 2 λ 7/ 6 L2/3 2 2π 3/ 4 l 3/ 2
.
(6)
#224168 - $15.00 USD Received 30 Sep 2014; revised 26 Nov 2014; accepted 10 Dec 2014; published 24 Dec 2014 (C) 2014 OSA 3 November 2014 | Vol. 22, No. 22 | DOI:10.1364/OE.22.032482 | OPTICS EXPRESS 32484
The PDF of polarization angle Δϕ after atmospheric turbulence can be deduced according to Rytov theory. The perturbation decomposition method applying to wave equation converts its free space solution into perturbation multiplication. We have ψ = ψ 0 + ψ 1 when the higher order terms of ψ (r ) can be neglected, where ψ 0 is the certain solution and ψ 1 is the perturbation term. The formal solution of scalar wave equation can be expressed as u = exp (ψ 0 + ψ 1 ) which satisfies Riccati equation ([2], Sec 5.3)
∇ 2ψ (r ) + [∇ψ (r ) ] + k 2 n 2 (r ) = 0. (7) In the Born approximation, any component of the propagating light field in turbulent medium can be expressed as the sum of free space perturbation solutions. If we put Ax and Ay together to form complex amplitude as A = Ax + jAy = Ae jϕ , where its real and image parts denote the x and y components respectively, then the complex amplitude without disturbance is A0 = A0 accordingly. Combining with Eq. (1),(2), we can get 2
ψ 1 =ψ -ψ 0 =ln AA + jϕ + j( S − S0 ). 0
(8)
With the help of scalar diffraction theory and forward scattering approximation, we suppose the wave scattering angle caused by inhomogeneity of refractivity satisfies θ0 = λ/l0
State Key Laboratory of Advanced Optical Communication Systems & Networks, Department of Electronics, Peking University, Beijing 100871, China *[email protected]
Abstract: In wireless optical communications (WOC), polarization multiplexing systems and coherent polarization systems have excellent performance and wide applications, while its state of polarization affected by atmospheric turbulence is not clearly understood. This paper focuses on the polarization fluctuations caused by atmospheric turbulence in a WOC link. Firstly, the relationship between the polarization fluctuations and the index of refraction structure parameter is introduced and the distribution of received polarization angle is obtained through theoretical derivations. Then, turbulent conditions are adjusted and measured elaborately in a wide range of scintillation indexes (SI). As a result, the root-mean-square (RMS) variation and probability distribution function (PDF) of polarization angle conforms closely to that of theoretical model. ©2014 Optical Society of America OCIS codes: (060.2605) Free-space optical communication; (260.5430) Polarization.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9.
10. 11. 12. 13. 14. 15.
V. Chan, “Free-space optical communications,” J. Lightwave Technol. 24(12), 4750–4762 (2006). L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005). M. Niu, X. Song, J. Cheng, and J. F. Holzman, “Performance analysis of coherent wireless optical communications with atmospheric turbulence,” Opt. Express 20(6), 6515–6520 (2012). A. K. Majumdar, “Free-space laser communication performance in the atmospheric channel,” J. Opt. Fiber Commun. Rep. 2(4), 345–396 (2005). J. C. Ricklin and F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: Implications for free-space laser communication,” J. Opt. Soc. Am. A 19(9), 1794–1802 (2002). X. Ji and G. Ji, “Effect of turbulence on the beam quality of apertured partially coherent beams,” J. Opt. Soc. Am. A 25(6), 1246–1252 (2008). M. L. Wesely, “The combined effect of temperature and humidity fluctuations on refractive index,” J. Appl. Meteorol. 15(1), 43–49 (1976). N. Cvijetic, D. Qian, J. Yu, Y. Huang, and T. Wang, “Polarization-multiplexed optical wireless transmission with coherent detection,” J. Lightwave Technol. 28(8), 1218–1227 (2010). H. Huang, G. Xie, Y. Yan, N. Ahmed, Y. Ren, Y. Yue, D. Rogawski, M. J. Willner, B. I. Erkmen, K. M. Birnbaum, S. J. Dolinar, M. P. J. Lavery, M. J. Padgett, M. Tur, and A. E. Willner, “100 Tbit/s free-space data link enabled by three-dimensional multiplexing of orbital angular momentum, polarization, and wavelength,” Opt. Lett. 39(2), 197–200 (2014). G. Xie, F. Wang, A. Dang, and H. Guo, “A novel polarization-multiplexing system for free-space optical links,” Photon. Technol. Lett. 23(20), 1484–1486 (2011). Y. Han and G. Li, “Coherent optical communication using polarization multiple-input-multiple-output,” Opt. Express 13(19), 7527–7534 (2005). X. Tang, Z. Xu, and Z. Ghassemlooy, “Coherent polarization modulated transmission through MIMO atmospheric optical turbulence channel,” J. Lightwave Technol. 31(20), 3221–3228 (2013). X. Zhao, Y. Yao, Y. Sun, and C. Liu, “Circle polarization shift keying with direct detection for free-space optical communication,” J. Opt. Commun. Netw. 1(4), 307–312 (2009). Z. Ghassemlooy, X. Tang, and S. Rajbhandari, “Experimental investigation of polarization modulated free space optical communication with direct detection in a turbulence channel,” IET Commun. 6(11), 1489–1494 (2012). J. W. Strohbehn and S. F. Clifford, “Polarization and angle-of -arrival fluctuations for a plane wave propagated through a turbulent medium,” IEEE Trans. Antenn. Propag. 15(3), 416–421 (1967).
#224168 - $15.00 USD Received 30 Sep 2014; revised 26 Nov 2014; accepted 10 Dec 2014; published 24 Dec 2014 (C) 2014 OSA 3 November 2014 | Vol. 22, No. 22 | DOI:10.1364/OE.22.032482 | OPTICS EXPRESS 32482
16. E. Collett and R. Alferness, “Depolarization of a laser beam in a turbulent medium,” J. Opt. Soc. Am. 62(4), 529–533 (1972). 17. D. H. Höhn, “Depolarization of a laser beam at 6328 Ǻ due to atmospheric transmission,” Appl. Opt. 8(2), 367– 369 (1969). 18. M. Toyoshima, H. Takenaka, Y. Shoji, Y. Takayama, Y. Koyama, and H. Kunimori, “Polarization measurements through space-to-ground atmospheric propagation paths by using a highly polarized laser source in space,” Opt. Express 17(25), 22333–22340 (2009). 19. D. F. James, “Change of polarization of light beams on propagation in free space,” J. Opt. Soc. Am. 11(5), 1641–1643 (1994). 20. X. Ji and X. Chen, “Changes in the polarization, the coherence and the spectrum of partially coherent electromagnetic Hermite–Gaussian beams in turbulence,” Opt. Laser Technol. 41(2), 165–171 (2009). 21. F. Kashani, M. Alavinejad, and B. Ghafary, “Polarization characteristics of aberrated partially coherent flattopped beam propagating through turbulent atmosphere,” Opt. Commun. 282(20), 4029–4034 (2009). 22. Y. Wang, Y. Zhang, J. Wang, and J. Jia, “Degree of polarization for quantum light field propagating through non-Kolmogorov turbulence,” Opt. Laser Technol. 43(4), 776–780 (2011). 23. O. Korotkova and N. Farwell, “Effect of oceanic turbulence on polarization of stochastic beams,” Opt. Commun. 284(7), 1740–1746 (2011). 24. X. Zhu and J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun. 50(8), 1293–1300 (2002). 25. Y. Han, A. Dang, Y. Ren, J. Tang, and H. Guo, “Theoretical and experimental studies of turbo product code with time diversity in free space optical communication,” Opt. Express 18(26), 26978–26988 (2010). 26. M. Al-Habash, C. Andrews, and R. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40(8), 1554–1562 (2001). 27. A. Dang, “A closed-form solution of the bit-error rate for optical wireless communication systems over atmospheric turbulence channels,” Opt. Express 19(4), 3494–3502 (2011). 28. A. Mostafa and S. Hranilovic, “Channel measurement and Markov modeling of an urban free-space optical link,” J. Opt. Commun. Netw. 4(10), 836–846 (2012). 29. F. S. Vetelino, C. Young, L. Andrews, and J. Recolons, “Aperture averaging effects on the probability density of irradiance fluctuations in moderate-to-strong turbulence,” Appl. Opt. 46(11), 2099–2108 (2007).
1. Introduction Wireless optical communication (WOC) is a promising candidate for data transmission due to its flexibility, wide-bandwidth and license-free [1]. However, the random variation of the refractive index (RI) caused by atmospheric turbulence can degrade the performance of WOC system severely. When transmitting through atmosphere, WOC signals suffer from intensity scintillation, phase fluctuation, as well as polarization alteration caused by turbulence [2–6]. The variation of RI is caused by the random variation of the temperature [7]. The longitudinal wind velocity associated with the turbulent atmosphere fluctuates randomly about its mean value, resulting in the changing of RI. When polarized light passes through the turbulent atmosphere, intensities in the two orthogonal polarizations are impacted differently, leading to the alteration of the direction and degree of polarization. In WOC systems, polarization techniques are widely applied, such as polarizationmultiplexing systems [8–11], coherent polarization shift keying (POLSK) systems [12,13], and so on. However, these systems are quite sensitive to polarization directions, requiring polarization agreement in both transmitting and receiving ends. Polarization fluctuations may cause a crosstalk between the signals in different polarizations, resulting in the degradation of optical signal-to-noise ratio (OSNR) and bit error rate (BER) performance [14]. Polarization fluctuations may also increase the complexity of polarization control feedback in the systems. Hence an applicable model for polarization fluctuations is important and necessary for performance estimations. Up to now, many researchers have studied the polarization fluctuations caused by atmospheric turbulence [15–23]. However, their results are limited for some specific conditions to some extent. For example, an early theoretical model proposed in [15,16] is based on slight RI variations, and it shows two orders of magnitude smaller than experimental results presented in [17]. A space-to-ground propagation was carried in [18], but the turbulent parameters of which is unknown. Hence the measurement results are not compared with any theory. Recently, polarization fluctuations for partially coherent beams were studied [19–21], and polarization in non-Kolmogorov turbulence [22] and oceanic turbulence [23] was researched. However, a widely accepted model is not yet proposed to describe turbulenceinduced polarization fluctuations. Moreover, most of the accomplished experiments are #224168 - $15.00 USD Received 30 Sep 2014; revised 26 Nov 2014; accepted 10 Dec 2014; published 24 Dec 2014 (C) 2014 OSA 3 November 2014 | Vol. 22, No. 22 | DOI:10.1364/OE.22.032482 | OPTICS EXPRESS 32483
measured under a single turbulent condition, and the distribution of the polarization is seldom discussed. This paper presents a novel and widely applicable theory to evaluate polarization fluctuations in turbulence. Firstly, the relationship between the polarization fluctuations and the index of refraction structure parameter is introduced and the probability distribution function (PDF) of received polarization angle is obtained through theoretical derivations. Then, a related experiment is carried out to measure the distribution of the polarization angle. In the experiment, different turbulent conditions are realized to confirm our theory in a wide range. Finally, the experimental results are compared with the theoretical conclusions. 2. Theoretical model of turbulence on polarization Suppose the incident light is polarized along the z-axis and propagates along the x-axis. The time-varying light-field can be expressed as E0 (t ) = A0 exp[j(ω0 t + S0 )]xˆ ,
(1)
where A0, ω0 and S0 denote the amplitude, frequency and phase of transmitting wave respectively. When propagating through turbulence, polarization alteration leads to the amplitude variation both in x and y directions. Then the receiving light-field is E = ( Ax x + Ay y ) exp{ j[ω0 t + S (t )]},
(2)
where Ax and Ay are the polarization components of x and y direction separately, which satisfy A=
2
Ax + Ay
2
; S(t) is the phase fluctuations caused by turbulence. The polarization
angle is defined as φ = arctan(Ay / Ax), with Ax = Re{Aejφ} and Ay = Im{Aejφ}. When the polarization angle is quite small, there is φ ≈M where M = |Ey| / |E0| is the polarizability. According to the perturbation theory, M is given by [14] M2 =
4 π Lσ n , k 2l 3
(3)
where σn is the root mean square of refractive index and satisfies σ n = Δn 2
1/ 2
; l is the scale
factor; k = 2π / λ is the wavenumber, where λ is the wavelength; L is propagation length. Then the root mean square σφ of the polarization angle φ caused by atmospheric turbulence is [17]
σϕ =
σ n λ L1/ 2 . 2π 3/ 4 l 3/ 2
(4)
For experiments, Δn is usually difficult to measure directly. Since the atmospheric structure function Dn (r ) = [n(r1 ) − n(r2 )]2 = [n(r1 )]2 + [n(r2 )]2 − 2 n(r1 )n(r2 ) ([2], Sec 3.2), we can get Dn (0) = 2 n 2 − 2 n
2
= 2Cn2 d 02/3 in the correlation range. By substituting
2
n ≈ Cn2 d 02/3 [16] and d 0 ≈ (λ L)1/ 2 [24,25], where d0 is the correlation length, σ n2 can be approximated as
σ n2 = 0.5Cn2 λ 1/3 L1/3 ,
(5)
where Cn2 is the index of refraction structure parameter which can be easily measured and compared with the theoretical results. Thus the RMS of depolarization can be written as
σϕ =
(Cn2 )1/ 2 λ 7/ 6 L2/3 2 2π 3/ 4 l 3/ 2
.
(6)
#224168 - $15.00 USD Received 30 Sep 2014; revised 26 Nov 2014; accepted 10 Dec 2014; published 24 Dec 2014 (C) 2014 OSA 3 November 2014 | Vol. 22, No. 22 | DOI:10.1364/OE.22.032482 | OPTICS EXPRESS 32484
The PDF of polarization angle Δϕ after atmospheric turbulence can be deduced according to Rytov theory. The perturbation decomposition method applying to wave equation converts its free space solution into perturbation multiplication. We have ψ = ψ 0 + ψ 1 when the higher order terms of ψ (r ) can be neglected, where ψ 0 is the certain solution and ψ 1 is the perturbation term. The formal solution of scalar wave equation can be expressed as u = exp (ψ 0 + ψ 1 ) which satisfies Riccati equation ([2], Sec 5.3)
∇ 2ψ (r ) + [∇ψ (r ) ] + k 2 n 2 (r ) = 0. (7) In the Born approximation, any component of the propagating light field in turbulent medium can be expressed as the sum of free space perturbation solutions. If we put Ax and Ay together to form complex amplitude as A = Ax + jAy = Ae jϕ , where its real and image parts denote the x and y components respectively, then the complex amplitude without disturbance is A0 = A0 accordingly. Combining with Eq. (1),(2), we can get 2
ψ 1 =ψ -ψ 0 =ln AA + jϕ + j( S − S0 ). 0
(8)
With the help of scalar diffraction theory and forward scattering approximation, we suppose the wave scattering angle caused by inhomogeneity of refractivity satisfies θ0 = λ/l0
