2188
J. Opt. Soc. Am. A / Vol. 30, No. 11 / November 2013
Cui et al.
Analytical expressions for the angle of arrival fluctuations for optical waves’ propagation through moderate-to-strong non-Kolmogorov refractive turbulence Linyan Cui,1,* Bindang Xue,1,2 and Fugen Zhou1,3 1
School of Astronautics, Beihang University, Beijing 100191, China 2 e-mail: [email protected] 3 e-mail: [email protected] *Corresponding author: [email protected]
Received July 8, 2013; revised September 4, 2013; accepted September 8, 2013; posted September 9, 2013 (Doc. ID 193420); published October 3, 2013 The effects of moderate-to-strong non-Kolmogorov turbulence on the angle of arrival (AOA) fluctuations for plane and spherical waves are investigated in detail both analytically and numerically. New analytical expressions for the variance of AOA fluctuations are derived for moderate-to-strong non-Kolmogorov turbulence. The new expressions cover a wider range of non-Kolmogorov turbulence strength and reduce correctly to previously published analytic expressions for the cases of plane and spherical wave propagation through both weak non-Kolmogorov turbulence and moderate-to-strong Kolmogorov turbulence cases. The final results indicate that, as turbulence strength becomes greater, the expressions developed with the Rytov theory deviate from those given in this work. This deviation becomes greater with stronger turbulence, up to moderate-to-strong turbulence strengths. Furthermore, general spectral power law has significant influence on the variance of AOA fluctuations in non-Kolmogorov turbulence. These results are useful for understanding the potential impact of deviations from the standard Kolmogorv spectrum. © 2013 Optical Society of America OCIS codes: (010.1290) Atmospheric optics; (010.1300) Atmospheric propagation; (010.1330) Atmospheric turbulence. http://dx.doi.org/10.1364/JOSAA.30.002188
1. INTRODUCTION Angle of arrival (AOA) fluctuations play an important role in a diverse range of fields, including atmospheric turbulence [1,2], free space optical communication [3], ground-based astronomical observations [4], etc. It involves estimating the variance of AOA fluctuations through integrals of atmospheric turbulence strength along the propagation path. Traditionally, the study is based on the Kolmogorov turbulence, and many methods have been proposed to investigate the variance of AOA fluctuations. These methods can generally be divided into two classes: Rytov theory-based method and extended Rytov theory-based method. The former is suitable for weak Kolmogorov turbulence (characterized by σ 2R ≪ 1, where σ 2R is the Rytov variance [5]), and different atmospheric turbulence spectral models have been adopted to investigate the variance of AOA fluctuations [5–7]. The latter is developed for moderate-to-strong Kolmogorov turbulence (σ 2R ≈ 1), and the effective atmospheric spectral models [8], which modified the conventional spectral models with a filter function, have been developed to analyze the variance of AOA fluctuations [9–11]. The expression derived with the latter method can be reduced to that derived with the former method under the condition of σ 2R ≪ 1 [9–11]. However, the models derived for Kolmogorov turbulence cannot be applied directly in the non-Kolmogorov turbulence case, which has been investigated experimentally [12–15] and theoretically [16,17]. The theoretical atmospheric turbulence spectral models, including the non-Kolmogorov spectrum [18], the generalized von Karman spectrum [19], and the 1084-7529/13/112188-08$15.00/0
generalized exponential spectrum [20], have been proposed and used to investigate the variance of AOA fluctuations for optical waves propagating through weak non-Kolmogorov turbulence [19,21,22]. Compared with other atmospheric turbulence spectral models, the effective non-Kolmogorov spectral model [23], which has been proposed in our previous work, is applicable for moderate-to-strong non-Kolmogorov turbulence, and has been used to analyze the irradiance scintillation for optical waves propagating through moderate to strong non-Kolmogorov turbulence [23]. In this study, the extended Rytov theory is adopted to analyze the AOA fluctuations for optical waves propagating through moderate-to-strong non-Kolmogorov turbulence. Based on this theory, the effective non-Kolmogorov spectral model [23] is assumed to be valid to investigate the variance of AOA fluctuations under moderate-to-strong non-Kolmogorov turbulence. Numerical calculations are conducted to analyze general spectral power law and turbulence strength’s influences on the final expressions.
2. EFFECTIVE NON-KOLMOGOROV SPECTRUM FOR MODERATE-TO-STRONG NON-KOLMOGOROV TURBULENCE The effective non-Kolmogorov turbulence spectrum for moderate-to-strong non-Kolmogorov turbulence takes the form as [23] Φn1 κ; α Φn κ; αGκ; α; 2π∕L0 ≪ κ ≪ 2π∕l0 ; 3 < α < 4:
© 2013 Optical Society of America
(1)
Cui et al.
Vol. 30, No. 11 / November 2013 / J. Opt. Soc. Am. A
Here, Φn κ; α is the non-Kolmogorov turbulence spectrum model derived for weak non-Kolmogorov turbulence, given by [18] Φn κ; α Aα · Cˆ 2n · κ−α ;
Γα − 1 π ; sin α − 3 2 4π 2
(2)
(3)
where α is the spectral power law, Cˆ 2n is the generalized refractive-index structure constant (with units of m3−α , when α 11∕3, with units of m−2∕3 ), κ is the spatial wave number and it denotes the magnitude of the spatial-frequency vector with units of rad/m and is related to the size of turbulence cells, and L0 and l0 are the turbulence outer scale and inner scale, respectively. Γ· is the gamma function. Gκ; α takes the form as Gκ; α GX κ; α GY κ; α;
(4)
where GX κ; α and GY κ; α are the large-scale filter and small-scale filter, respectively,
2
0
Z dξ
∞
0
1 − J 0 κρΦn1 κκdκ;
(7)
1 − J 0 κρξΦn1 κκdκ:
(8)
Z 0
L
Z dz
∞
1 − J 0 κρΦn1 κ; ακdκ:
0
(9)
For analysis purpose, Eq. (9) is expressed as (5)
Dωp ρ; α Dωp l ρ; α Dωp s ρ; α;
Dωp l ρ; α 8π 2 k2
Dωp s ρ; α 8π 2 k2
L 0
Z dz
∞ 0
1 − J 0 κρ
Z 0
L
Z dz
0
∞
(11)
1 − J 0 κρ
× Φn κ; αGY κ; ακdκ:
(12)
Using Weber’s integral and its property [25]
3. VARIANCE OF AOA FLUCTUATIONS FOR MODERATE-TO-STRONG NON-KOLMOGOROV TURBULENCE With the geometrical optics method, the variance of AOA fluctuations of optical wave at the receiver plane can be described in terms of the phase structure function [24]
where Dω D is the phase structure function with the radial distance ρ D. D is the diameter of aperture receiver. k 2π∕λ, λ is the optical wavelength. For moderate-to-strong Kolmogorov turbulence, Dω D takes the form as [8–10]
Z
× Φn κ; αGX κ; ακdκ;
∞ 0
(6)
(10)
where Dωp l ρ; α and Dωp s ρ; α are large-scale and smallscale parts of the variance of AOA fluctuations, respectively,
Z
Dω D ; kD2
1
∞
0
A. Variance of AOA Fluctuations for Plane Wave in Moderate-to-Strong Non-Kolmogorov Turbulence Using Φn1 κ; α for the moderate-to-strong non-Kolmogorov turbulence, Dωp ρ becomes Dωp ρ; α 8π 2 k2
κ X α is a large-scale (or refractive) spatial-frequency cutoff, and κY α is a small-scale (or diffractive) spatial-frequency cutoff. Gκ; α acts like a spatial filter function that permits only low-pass spatial frequencies κ < κX α or high-pass spatial frequencies κ > κ Y α to influence the wave propagation. That is, the small-scale (κ > κY α) and large-scale (κ < κ X α) turbulence eddies’ effects are significant in the analysis of optical waves propagating through moderate-to-strong non-Kolmogorov turbulence, and the intermediate-scale (κX α < κ < κY α) turbulence eddies’ effects can be neglected. The expressions of κX α and κY α have been derived in [23] for plane and spherical waves propagating through moderate-to-strong non-Kolmogorov turbulence and exhibited in Appendix A of this paper.
hβ2a i
Z
Z dz
Here, Dspl ρ and Dssp ρ are the phase structure functions for plane wave and spherical wave, respectively. Φn1 κ is the effective turbulence spectral model for wave propagating through moderate-to-strong Kolmogorov turbulence. J 0 · is the Bessel function of zero order. To consider the influence of general spectral power law values, in the next section, Dspl ρ,Dssp ρ, and Φn1 κ will be replaced by Dspl ρ; α,Dssp ρ; α, and Φn1 κ; α, respectively, and theoretical expressions of variance of AOA fluctuations will be obtained for plane and spherical waves propagating through moderate-to-strong non-Kolmogorov turbulence with horizontal path.
κ GX κ; α exp − 2 ; κX α κα GY κ; α 2 : κ κ2Y αα∕2
L
0
Dωs ρ 8π 2 k2 L
2π∕L0 ≪ κ ≪ 2π∕l0 ; 3 < α < 4;
Aα
Z
Dωp ρ 8π 2 k2
2189
J v ax exp−p2 x2 xμ−1 dx av Γ μν 2 μν a2 ; ; ν 1; − F ν1 μν 2 2 p Γν 1 1 1 4p2 μ ν > 0; Rep2 > 0;
Z 0
∞
(13)
xν1 J ν ax aμ bν−μ dx μ K ab; 2 2 μ1 2 Γμ 1 μ−ν x b 3 a; b > 0; −1 < Reν < Re 2μ ; 2 (14)
2190
J. Opt. Soc. Am. A / Vol. 30, No. 11 / November 2013
Cui et al.
in which J v · is the Bessel function, 1 F 1 a; b; z is the hypergeometric function, and K p z is the modified Bessel function. The analytical expressions of Dωp l ρ; α and Dωp s ρ; α are obtained and take the forms as Dωp l ρ; α
1−α∕2 4σ 2Rpl ηXpl
β1 α
α Γ 1− 2
kρ2 ηXpl α × 1 − 1 F 1 1 − ; 1; − ; 2 4L
Dωp s ρ; α
8σ 2Rpl ηY1−α∕2 pl β1 α
(15)
4 1 − 1 α − 2 Γα∕2
8
J. Opt. Soc. Am. A / Vol. 30, No. 11 / November 2013
Cui et al.
Analytical expressions for the angle of arrival fluctuations for optical waves’ propagation through moderate-to-strong non-Kolmogorov refractive turbulence Linyan Cui,1,* Bindang Xue,1,2 and Fugen Zhou1,3 1
School of Astronautics, Beihang University, Beijing 100191, China 2 e-mail: [email protected] 3 e-mail: [email protected] *Corresponding author: [email protected]
Received July 8, 2013; revised September 4, 2013; accepted September 8, 2013; posted September 9, 2013 (Doc. ID 193420); published October 3, 2013 The effects of moderate-to-strong non-Kolmogorov turbulence on the angle of arrival (AOA) fluctuations for plane and spherical waves are investigated in detail both analytically and numerically. New analytical expressions for the variance of AOA fluctuations are derived for moderate-to-strong non-Kolmogorov turbulence. The new expressions cover a wider range of non-Kolmogorov turbulence strength and reduce correctly to previously published analytic expressions for the cases of plane and spherical wave propagation through both weak non-Kolmogorov turbulence and moderate-to-strong Kolmogorov turbulence cases. The final results indicate that, as turbulence strength becomes greater, the expressions developed with the Rytov theory deviate from those given in this work. This deviation becomes greater with stronger turbulence, up to moderate-to-strong turbulence strengths. Furthermore, general spectral power law has significant influence on the variance of AOA fluctuations in non-Kolmogorov turbulence. These results are useful for understanding the potential impact of deviations from the standard Kolmogorv spectrum. © 2013 Optical Society of America OCIS codes: (010.1290) Atmospheric optics; (010.1300) Atmospheric propagation; (010.1330) Atmospheric turbulence. http://dx.doi.org/10.1364/JOSAA.30.002188
1. INTRODUCTION Angle of arrival (AOA) fluctuations play an important role in a diverse range of fields, including atmospheric turbulence [1,2], free space optical communication [3], ground-based astronomical observations [4], etc. It involves estimating the variance of AOA fluctuations through integrals of atmospheric turbulence strength along the propagation path. Traditionally, the study is based on the Kolmogorov turbulence, and many methods have been proposed to investigate the variance of AOA fluctuations. These methods can generally be divided into two classes: Rytov theory-based method and extended Rytov theory-based method. The former is suitable for weak Kolmogorov turbulence (characterized by σ 2R ≪ 1, where σ 2R is the Rytov variance [5]), and different atmospheric turbulence spectral models have been adopted to investigate the variance of AOA fluctuations [5–7]. The latter is developed for moderate-to-strong Kolmogorov turbulence (σ 2R ≈ 1), and the effective atmospheric spectral models [8], which modified the conventional spectral models with a filter function, have been developed to analyze the variance of AOA fluctuations [9–11]. The expression derived with the latter method can be reduced to that derived with the former method under the condition of σ 2R ≪ 1 [9–11]. However, the models derived for Kolmogorov turbulence cannot be applied directly in the non-Kolmogorov turbulence case, which has been investigated experimentally [12–15] and theoretically [16,17]. The theoretical atmospheric turbulence spectral models, including the non-Kolmogorov spectrum [18], the generalized von Karman spectrum [19], and the 1084-7529/13/112188-08$15.00/0
generalized exponential spectrum [20], have been proposed and used to investigate the variance of AOA fluctuations for optical waves propagating through weak non-Kolmogorov turbulence [19,21,22]. Compared with other atmospheric turbulence spectral models, the effective non-Kolmogorov spectral model [23], which has been proposed in our previous work, is applicable for moderate-to-strong non-Kolmogorov turbulence, and has been used to analyze the irradiance scintillation for optical waves propagating through moderate to strong non-Kolmogorov turbulence [23]. In this study, the extended Rytov theory is adopted to analyze the AOA fluctuations for optical waves propagating through moderate-to-strong non-Kolmogorov turbulence. Based on this theory, the effective non-Kolmogorov spectral model [23] is assumed to be valid to investigate the variance of AOA fluctuations under moderate-to-strong non-Kolmogorov turbulence. Numerical calculations are conducted to analyze general spectral power law and turbulence strength’s influences on the final expressions.
2. EFFECTIVE NON-KOLMOGOROV SPECTRUM FOR MODERATE-TO-STRONG NON-KOLMOGOROV TURBULENCE The effective non-Kolmogorov turbulence spectrum for moderate-to-strong non-Kolmogorov turbulence takes the form as [23] Φn1 κ; α Φn κ; αGκ; α; 2π∕L0 ≪ κ ≪ 2π∕l0 ; 3 < α < 4:
© 2013 Optical Society of America
(1)
Cui et al.
Vol. 30, No. 11 / November 2013 / J. Opt. Soc. Am. A
Here, Φn κ; α is the non-Kolmogorov turbulence spectrum model derived for weak non-Kolmogorov turbulence, given by [18] Φn κ; α Aα · Cˆ 2n · κ−α ;
Γα − 1 π ; sin α − 3 2 4π 2
(2)
(3)
where α is the spectral power law, Cˆ 2n is the generalized refractive-index structure constant (with units of m3−α , when α 11∕3, with units of m−2∕3 ), κ is the spatial wave number and it denotes the magnitude of the spatial-frequency vector with units of rad/m and is related to the size of turbulence cells, and L0 and l0 are the turbulence outer scale and inner scale, respectively. Γ· is the gamma function. Gκ; α takes the form as Gκ; α GX κ; α GY κ; α;
(4)
where GX κ; α and GY κ; α are the large-scale filter and small-scale filter, respectively,
2
0
Z dξ
∞
0
1 − J 0 κρΦn1 κκdκ;
(7)
1 − J 0 κρξΦn1 κκdκ:
(8)
Z 0
L
Z dz
∞
1 − J 0 κρΦn1 κ; ακdκ:
0
(9)
For analysis purpose, Eq. (9) is expressed as (5)
Dωp ρ; α Dωp l ρ; α Dωp s ρ; α;
Dωp l ρ; α 8π 2 k2
Dωp s ρ; α 8π 2 k2
L 0
Z dz
∞ 0
1 − J 0 κρ
Z 0
L
Z dz
0
∞
(11)
1 − J 0 κρ
× Φn κ; αGY κ; ακdκ:
(12)
Using Weber’s integral and its property [25]
3. VARIANCE OF AOA FLUCTUATIONS FOR MODERATE-TO-STRONG NON-KOLMOGOROV TURBULENCE With the geometrical optics method, the variance of AOA fluctuations of optical wave at the receiver plane can be described in terms of the phase structure function [24]
where Dω D is the phase structure function with the radial distance ρ D. D is the diameter of aperture receiver. k 2π∕λ, λ is the optical wavelength. For moderate-to-strong Kolmogorov turbulence, Dω D takes the form as [8–10]
Z
× Φn κ; αGX κ; ακdκ;
∞ 0
(6)
(10)
where Dωp l ρ; α and Dωp s ρ; α are large-scale and smallscale parts of the variance of AOA fluctuations, respectively,
Z
Dω D ; kD2
1
∞
0
A. Variance of AOA Fluctuations for Plane Wave in Moderate-to-Strong Non-Kolmogorov Turbulence Using Φn1 κ; α for the moderate-to-strong non-Kolmogorov turbulence, Dωp ρ becomes Dωp ρ; α 8π 2 k2
κ X α is a large-scale (or refractive) spatial-frequency cutoff, and κY α is a small-scale (or diffractive) spatial-frequency cutoff. Gκ; α acts like a spatial filter function that permits only low-pass spatial frequencies κ < κX α or high-pass spatial frequencies κ > κ Y α to influence the wave propagation. That is, the small-scale (κ > κY α) and large-scale (κ < κ X α) turbulence eddies’ effects are significant in the analysis of optical waves propagating through moderate-to-strong non-Kolmogorov turbulence, and the intermediate-scale (κX α < κ < κY α) turbulence eddies’ effects can be neglected. The expressions of κX α and κY α have been derived in [23] for plane and spherical waves propagating through moderate-to-strong non-Kolmogorov turbulence and exhibited in Appendix A of this paper.
hβ2a i
Z
Z dz
Here, Dspl ρ and Dssp ρ are the phase structure functions for plane wave and spherical wave, respectively. Φn1 κ is the effective turbulence spectral model for wave propagating through moderate-to-strong Kolmogorov turbulence. J 0 · is the Bessel function of zero order. To consider the influence of general spectral power law values, in the next section, Dspl ρ,Dssp ρ, and Φn1 κ will be replaced by Dspl ρ; α,Dssp ρ; α, and Φn1 κ; α, respectively, and theoretical expressions of variance of AOA fluctuations will be obtained for plane and spherical waves propagating through moderate-to-strong non-Kolmogorov turbulence with horizontal path.
κ GX κ; α exp − 2 ; κX α κα GY κ; α 2 : κ κ2Y αα∕2
L
0
Dωs ρ 8π 2 k2 L
2π∕L0 ≪ κ ≪ 2π∕l0 ; 3 < α < 4;
Aα
Z
Dωp ρ 8π 2 k2
2189
J v ax exp−p2 x2 xμ−1 dx av Γ μν 2 μν a2 ; ; ν 1; − F ν1 μν 2 2 p Γν 1 1 1 4p2 μ ν > 0; Rep2 > 0;
Z 0
∞
(13)
xν1 J ν ax aμ bν−μ dx μ K ab; 2 2 μ1 2 Γμ 1 μ−ν x b 3 a; b > 0; −1 < Reν < Re 2μ ; 2 (14)
2190
J. Opt. Soc. Am. A / Vol. 30, No. 11 / November 2013
Cui et al.
in which J v · is the Bessel function, 1 F 1 a; b; z is the hypergeometric function, and K p z is the modified Bessel function. The analytical expressions of Dωp l ρ; α and Dωp s ρ; α are obtained and take the forms as Dωp l ρ; α
1−α∕2 4σ 2Rpl ηXpl
β1 α
α Γ 1− 2
kρ2 ηXpl α × 1 − 1 F 1 1 − ; 1; − ; 2 4L
Dωp s ρ; α
8σ 2Rpl ηY1−α∕2 pl β1 α
(15)
4 1 − 1 α − 2 Γα∕2
8
