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Research ArticleModulation Transfer Function of a Gaussian Beam
Based onthe Generalized Modified Atmospheric Spectrum
Chao Gao and Xiaofeng Li
School of Astronautics and Aeronautics, University of Electronic
Science and Technology of China, 2006 Xiyuan Ave,West Hi-Tech Zone,
Chengdu 611731, China
Correspondence should be addressed to Xiaofeng Li;
[email protected]
Received 26 May 2016; Accepted 3 August 2016
Academic Editor: Sulaiman Wadi Harun
Copyright © 2016 C. Gao and X. Li. This is an open access
article distributed under the Creative Commons Attribution
License,which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly
cited.
This paper investigates the modulation transfer function of a
Gaussian beam propagating through a horizontal path in
weak-fluctuation non-Kolmogorov turbulence. Mathematical
expressions are obtained based on the generalized modified
atmosphericspectrum, which includes the spectral power law value of
non-Kolmogorov turbulence, the finite inner and outer scales
ofturbulence, and other optical parameters of the Gaussian beam.
The numerical results indicate that the atmospheric turbulencewould
produce less negative effects on the wireless optical communication
system with an increase in the inner scale of
turbulence.Additionally, the increased outer scale of turbulence
makes a Gaussian beam influenced more seriously by the
atmosphericturbulence.
1. Introduction
Optical wireless communication technology has drawnmuchattention
for its significant technological challenges andprospective
applications. It uses beams of laser propagatingin the atmosphere
to wirelessly transmit data at high speed.However, the atmosphere
is full of numerous turbulenceeddies, which has great degrading
impacts on the perfor-mance of the communication system. The
degrading effectsof atmospheric turbulence on the communication
systemcan be characterized statistically by the modulation
trans-fer function (MTF) [1]. In the past few decades, variouspower
spectrum models of refractive index have been pro-posed to analyze
the MTF for different situations. Generallyspeaking, these
turbulence power spectrum models can beclassified into two typical
categories: Kolmogorov and non-Kolmogorovmodels.The former have a
fixed power law valueof 11/3, while the latter allow the power law
value to vary inthe range from three to four. Most non-Kolmogorov
modelscan be generalized from their corresponding Kolmogorovmodels,
and thus the Kolmogorov models can be regardedas specific cases of
the non-Kolmogorov models [2]. Amongthese models, the generalized
modified atmospheric spec-trum not only considers the variable
spectral power law valuebetween the ranges from 3 to 4, but also
takes the finite
inner and outer scales of turbulence into account [3].
Besides,the generalized modified atmospheric spectrum features
thesmall rise at a high wavenumber, which is clearly seen
intemperature data recorded by sensors.These properties makethe
generalized modified atmospheric spectrum suitable andunique in the
investigation of theMTF for plane and sphericalwaves [4].
In this study, the generalized modified atmospheric spec-trum is
used to investigate the MTF of a Gaussian beamin non-Kolmogorov
turbulence along a horizontal path. TheGaussian beam, whose
transverse electric field and intensityare normally distributed, is
a typical kind of electromagneticwave [5]. The rest of the paper is
organized as follows.Section 2 introduces the generalized modified
atmosphericspectrum and the MTF of a Gaussian beam. Section
3presents a detailed expression reduction.The influences of
theinner and outer scales of turbulence on theMTFof aGaussianbeam
are analyzed in Section 4, followed by conclusions inSection 5.
2. Theoretical Models
2.1. Generalized Modified Atmospheric Spectrum. The gener-alized
modified atmospheric spectrum takes the form [3]
Φ𝑛(𝜅) = 𝐴 (𝛼) 𝐶
2
𝑛𝜅−𝛼
𝑓 (𝜅, 𝛼) , (1)
Hindawi Publishing CorporationInternational Journal of
OpticsVolume 2016, Article ID 2613816, 8
pageshttp://dx.doi.org/10.1155/2016/2613816
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2 International Journal of Optics
where 𝜅 ∈ [0, +∞) is the angular wavenumber of the turbu-lence
scale, 𝛼 ∈ (3, 4) is the spectral power law value, 𝐶2
𝑛is
the generalized atmospheric structure parameter, 𝑙0≥ 0 is
the inner scale of turbulence, and 𝐿0≥ 𝑙0is the outer scale
of
turbulence. 𝐴(𝛼) in (1) is a function related to 𝛼:
𝐴 (𝛼) =Γ (𝛼 − 1)
4𝜋2sin 𝜋 (𝛼 − 3)
2, (2)
where Γ(𝑥) is the gamma function.For the convenience of
mathematical analysis, let
𝐶𝛼=3 − 𝛼
3Γ (
3 − 𝛼
2) + 𝑎
4 − 𝛼
3Γ (
4 − 𝛼
2)
− 𝑏3 + 𝛽 − 𝛼
3Γ(
3 + 𝛽 − 𝛼
2) ,
(3)
where the constant coefficients 𝑎, 𝑏, and 𝛽 in (3) are
usuallyset as
𝑎 = 1.802,
𝑏 = 0.254,
𝛽 =7
6.
(4)
It must be pointed out that the values of these coefficientsare
based on the experiments for the classic Kolmogorovturbulence but
are widely used for theoretical analyses ofnon-Kolmogorov
turbulence [1, 4]. Nevertheless, 𝑓(𝜅, 𝛼) in(1) takes the form
𝑓 (𝜅, 𝛼) = exp(−𝜅2
𝜅2
𝑙
) × (1 − exp(−𝜅2
𝜅2
0
))
× (1 + 𝑎(𝜅
𝜅𝑙
) − 𝑏(𝜅
𝜅𝑙
)
𝛽
)
=
3
∑
𝑖=1
2
∑
𝑗=1
(−1)𝑗−1
𝑐𝑖𝜅𝑝𝑖 exp (−𝑑2
𝑗𝜅2
) ,
(5)
where
𝜅0=4𝜋
𝐿0
,
𝜅𝑙=(𝜋𝐴 (𝛼) 𝐶
𝛼)1/(𝛼−5)
𝑙0
.
(6)
And the coefficients are 𝑐1= 1, 𝑐2= 𝑎/𝜅
𝑙, 𝑐3= −𝑏/𝜅
𝛽
𝑙, 𝑝1= 0,
𝑝2= 1, 𝑝
3= 𝛽, 𝑑
1= √1/𝜅
2
𝑙, and 𝑑
2= √1/𝜅
2
𝑙+ 1/𝜅2
0.
2.2.MTF of aGaussian Beam. TheMTF is relative to
thewavestructure function (WSF). Based on theRytov
approximation,the WSF of Gaussian beam takes the simple form
[1]
𝐷(𝜌) = 8𝐿𝑘2
𝜋2
∫
1
0
d𝜉∫+∞
0
d𝜅
× 𝜅Φ𝑛(𝜅) exp(−𝐿Λ𝜅
2
𝜉2
𝑘)
× (𝐼0(Λ𝜌𝜅𝜉) − 𝐽
0(𝜌𝜅 (1 − Θ𝜉))) ,
(7)
where 𝜌 is the scalar separation between two observationpoints
and 𝐿 is the propagation optical path length. 𝑘 in (7) isthe
angular wavenumber of Gaussian beam wave
𝑘 =2𝜋
𝜆, (8)
where 𝜆 is the wavelength of Gaussian beam. BothΛ andΘ in(7) are
optical parameters of theGaussian beamat the receiver
Θ =Θ0
Θ2
0+ Λ2
0
,
Θ = 1 − Θ,
Λ =Λ0
Θ2
0+ Λ2
0
,
(9)
where Θ0is the curvature parameter of Gaussian beam at
transmitter and Λ0is the Fresnel ratio of Gaussian beam at
transmitter
Θ0= 1 −
𝐿
𝑅0
,
Λ0=
2𝐿
𝑘𝑊0
.
(10)
In (10), 𝑅0is the phase front radius of Gaussian beam
at transmitter, and 𝑊0is the radius of Gaussian beam at
transmitter. 𝐼0(𝑥) in (7) is the modified Bessel function of
the first kind with zero order, and 𝐽0(𝑥) in (7) is the
Bessel
function of the first kind with zero order [6]
𝐼0(𝑥) =
+∞
∑
𝑛=0
1
(𝑛!)2(𝑥
2)
2𝑛
,
𝐽0(𝑥) =
+∞
∑
𝑛=0
(−1)𝑛
(𝑛!)2(𝑥
2)
2𝑛
.
(11)
The atmospheric turbulence MTF takes the form [1]
MTF (𝜇) = exp (−12𝐷 (𝜇𝑑)) , (12)
where 𝜇 is the normalized spatial frequency and 𝑑 is thereceiver
aperture diameter. It is clear that the value range ofthe MTF is
the interval from 0 to 1.
3. Expression Reduction
The calculation equation (7) will spend too much timebecause of
its improper iterated integral. As an alternative, theclosed-form
expression of (7) can replace the improper iter-ated integral with
special functions, which has corresponding
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International Journal of Optics 3
packages in frequently used software. This section
mainlydiscusses the reduction of (7).
Substituting (1) into (7), it follows that
𝐷(𝜌) = 8𝐴 (𝛼) 𝐶2
𝑛𝐿𝑘2
𝜋2
∫
1
0
d𝜉 ∫+∞
0
d𝜅
× 𝜅1−𝛼
𝑓 (𝜅, 𝛼) exp(−𝐿Λ𝜅2
𝜉2
𝑘)
× (𝐼0(Λ𝜌𝜅𝜉) − 𝐽
0(𝜌𝜅 (1 − Θ𝜉))) .
(13)
For mathematical convenience, let
𝐷𝐼= ∫
1
0
d𝜉 ∫+∞
0
d𝜅 × 𝜅1−𝛼𝑓 (𝜅, 𝛼) exp(−𝐿Λ𝜅2
𝜉2
𝑘)
× (𝐼0(Λ𝜌𝜅𝜉) − 1) ,
𝐷𝐽= ∫
1
0
d𝜉 ∫+∞
0
d𝜅 × 𝜅1−𝛼𝑓 (𝜅, 𝛼) exp(−𝐿Λ𝜅2
𝜉2
𝑘)
× (𝐽0(𝜌𝜅 (1 − Θ𝜉)) − 1) .
(14)
Thus, (13) can be presented by
𝐷(𝜌) = 8𝐴 (𝛼) 𝐶2
𝑛𝐿𝑘2
𝜋2
× (𝐷𝐼− 𝐷𝐽) . (15)
3.1. Reduction of 𝐷𝐼. Substituting (11) into (14), 𝐷
𝐼is rewrit-
ten as
𝐷𝐼= ∫
1
0
d𝜉∫+∞
0
d𝜅 ×+∞
∑
𝑛=1
1
(𝑛!)2(Λ𝜌𝜅𝜉
2)
2𝑛
× 𝜅1−𝛼
𝑓 (𝜅, 𝛼) exp(−𝐿Λ𝜅2
𝜉2
𝑘) .
(16)
In most situations, 𝜌 ≪ 1. This is because MTF will
quicklyconverge to zero when 𝜌 approaches one; that is, MTF
issignificantly larger than zero when 𝜌 approaches zero. Thus,(16)
could be approximated by the simpler expression
𝐷𝐼≈Λ2
𝜌2
4∫
1
0
𝜉2d𝜉∫+∞
0
d𝜅
× 𝜅3−𝛼
𝑓 (𝜅, 𝛼) exp(−𝐿Λ𝜅2
𝜉2
𝑘) .
(17)
Consider the iterated integral in (17). According to (5),
thereis
𝜅3−𝛼
𝑓 (𝜅, 𝛼) exp(−𝐿Λ𝜅2
𝜉2
𝑘)
=
3
∑
𝑖=1
2
∑
𝑗=1
(−1)𝑗−1
𝑐𝑖𝜅3−𝛼+𝑝𝑖
× exp(−(𝑑2𝑗+𝐿Λ𝜉2
𝑘) 𝜅2
) .
(18)
Based on the equation for 𝑢 > −1 and V > 0 [7],
∫
+∞
0
𝑥𝑢 exp (−V𝑥2) d𝑥 = 1
2V−(𝑢+1)/2Γ (
𝑢 + 1
2) , (19)
we can get
∫
+∞
0
𝜅3−𝛼
𝑓 (𝜅, 𝛼) exp(−𝐿Λ𝜅2
𝜉2
𝑘) d𝜅
=
3
∑
𝑖=1
2
∑
𝑗=1
(−1)𝑗−1
𝑐𝑖
2(𝑑2
𝑗+𝐿Λ𝜉2
𝑘)
−(4−𝛼+𝑝𝑖)/2
× Γ (4 − 𝛼 + 𝑝
𝑖
2) .
(20)
Without loss of generality, the integrand in (17) takes the
form
𝜉𝑛
∫
+∞
0
𝜅𝑝
𝑓 (𝜅, 𝛼) exp(−𝐿Λ𝜅2
𝜉2
𝑘) d𝜅
=
3
∑
𝑖=1
2
∑
𝑗=1
(−1)𝑗−1
𝑐𝑖
2Γ (
1 + 𝑝 + 𝑝𝑖
2)
× (𝐿Λ
𝑘)
−(1+𝑝+𝑝𝑖)/2
𝜉𝑛
(
𝑘𝑑2
𝑗
𝐿Λ+ 𝜉2
)
−(1+𝑝+𝑝𝑖)/2
,
(21)
where 𝑛 = 2 and 𝑝 = 3 − 𝛼. Based on the equation for 𝑢 > 0and
𝑤 > 0 [7],
∫
1
0
𝑥𝑢−1
(𝑤2
+ 𝑥2
)Vd𝑥
=1
𝑢𝑤2V2𝐹1(−V,
𝑢
2;𝑢 + 2
2; −
1
𝑤2) ,
(22)
we can get
∫
1
0
𝜉𝑛d𝜉∫+∞
0
d𝜅 × 𝜅𝑝𝑓 (𝜅, 𝛼) exp(−𝐿Λ𝜅2
𝜉2
𝑘)
=
3
∑
𝑖=1
2
∑
𝑗=1
(−1)𝑗−1
𝑐𝑖
2𝑑1+𝑝+𝑝𝑖
𝑗(𝑛 + 1)
× Γ (1 + 𝑝 + 𝑝
𝑖
2)
×2𝐹1(1 + 𝑝 + 𝑝
𝑖
2,𝑛 + 1
2;𝑛 + 3
2; −
𝐿Λ
𝑘𝑑2
𝑗
) ,
(23)
where2𝐹1(𝑎, 𝑏; 𝑐; 𝑧) is the Gaussian hypergeometric function
[6]. Thus,𝐷𝐼can be computed by (17) and (23) with 𝑛 = 2.
3.2. Reduction of 𝐷𝐽. Following similar procedures as pre-
sented in Section 3.1,𝐷𝐽in (14) is rewritten as
𝐷𝐽≈ −
𝜌2
4∫
1
0
(1 − Θ𝜉)2
d𝜉∫+∞
0
d𝜅
× 𝜅3−𝛼
𝑓 (𝜅, 𝛼) exp(−𝐿Λ𝜅2
𝜉2
𝑘) .
(24)
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4 International Journal of Optics
𝛼 = 3.5
𝛼 = 3.6
𝛼 = 3.7
0.2 0.4 0.6 0.8 10𝜇
0
0.2
0.4
0.6
0.8
1MTF
(a) Convergent Θ0= 0.5
𝛼 = 3.5
𝛼 = 3.6
𝛼 = 3.7
0.2 0.4 0.6 0.8 10𝜇
0
0.2
0.4
0.6
0.8
1
MTF
(b) Collimated Θ0= 1
𝛼 = 3.5
𝛼 = 3.6
𝛼 = 3.7
0.2 0.4 0.6 0.8 10𝜇
0
0.2
0.4
0.6
0.8
1
MTF
(c) Divergent Θ0= 2
Figure 1: Effects of spectral power law value on MTF for
different types of Gaussian beams.
Expanding (24) by the binomial theorem, it follows that
𝐷𝐽= −
Θ2
𝜌2
4∫
1
0
𝜉2d𝜉∫+∞
0
d𝜅
× 𝜅3−𝛼
𝑓 (𝜅, 𝛼) exp(−𝐿Λ𝜅2
𝜉2
𝑘)
+Θ𝜌2
2∫
1
0
𝜉 d𝜉∫+∞
0
d𝜅
× 𝜅3−𝛼
𝑓 (𝜅, 𝛼) exp(−𝐿Λ𝜅2
𝜉2
𝑘)
−𝜌2
4∫
1
0
d𝜉∫+∞
0
d𝜅
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International Journal of Optics 5
0.2 0.4 0.6 0.8 10𝜇
0
0.2
0.4
0.6
0.8
1MTF
l0 = 0.001ml0 = 0.01ml0 = 0.1m
(a) Convergent Θ0= 0.5
0 0.2 0.4 0.6 0.8 1𝜇
0
0.2
0.4
0.6
0.8
1
MTF
l0 = 0.001ml0 = 0.01ml0 = 0.1m
(b) Collimated Θ0= 1
0.2 0.4 0.6 0.8 10𝜇
0
0.2
0.4
0.6
0.8
1
MTF
l0 = 0.001ml0 = 0.01ml0 = 0.1m
(c) Divergent Θ0= 2
Figure 2: Effects of inner scale on MTF for different types of
Gaussian beams.
× 𝜅3−𝛼
𝑓 (𝜅, 𝛼) exp(−𝐿Λ𝜅2
𝜉2
𝑘) .
(25)
Thus,𝐷𝐽could be computed by (25) and (23) with 𝑛 = 0, 1, 2.
4. Numerical Simulations
The following simulations are conducted by the Gaussianbeam with
these settings: 𝜆 = 1.55 × 10−6m, 𝐿 = 1000m,𝑘 ≈ 4.0537 × 10
6 rad/m, 𝐶2𝑛= 1.7 × 10
−14m3−𝛼,𝑊0= 0.1m,
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6 International Journal of Optics
0.2 0.4 0.6 0.8 10𝜇
0
0.2
0.4
0.6
0.8
1MTF
L0 = 1mL0 = 5mL0 = 25m
(a) Convergent Θ0= 0.5
0.2 0.4 0.6 0.8 10𝜇
0
0.2
0.4
0.6
0.8
1
MTF
L0 = 1mL0 = 5mL0 = 25m
(b) Collimated Θ0= 1
0.2 0.4 0.6 0.8 10𝜇
0
0.2
0.4
0.6
0.8
1
MTF
L0 = 1mL0 = 5mL0 = 25m
(c) Divergent Θ0= 2
Figure 3: Effects of outer scale on MTF for different types of
Gaussian beams.
Λ0≈ 0.0493, and 𝑑 = 0.1m. Of course, other values can also
be chosen.Figure 1 depicts the effects of spectral power law
value
on MTF for different types of Gaussian beams. In
thiscalculation, the inner and outer scales of turbulence are setas
𝑙0= 0.01m and 𝐿
0= 5m, respectively. As shown in
Figure 1(a), the atmospheric turbulence apparently produces
more effects on the propagation of the convergent Gaussianbeam
(Θ
0= 0.5) with an increase in the normalized
spatial frequency 𝜇, which acts in accordance with commonsense.
Besides, from Figure 1(a), it can be found that thenon-Kolmogorov
atmospheric turbulence would bring moreeffects on the wireless
optical communication system whenthe spectral power law value 𝛼
decreases. The same trends
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International Journal of Optics 7
0.2 0.4 0.6 0.8 10
𝜇
0
0.2
0.4
0.6
0.8
1MTF
𝜆 = 800 nm𝜆 = 1060nm𝜆 = 1550nm
(a) Convergent Θ0= 0.5
0.2 0.4 0.6 0.8 10
𝜇
0
0.2
0.4
0.6
0.8
1
MTF
𝜆 = 800 nm𝜆 = 1060nm𝜆 = 1550nm
(b) Collimated Θ0= 1
𝜆 = 800 nm𝜆 = 1060nm𝜆 = 1550nm
0.2 0.4 0.6 0.8 10
𝜇
0
0.2
0.4
0.6
0.8
1
MTF
(c) Divergent Θ0= 2
Figure 4: Effects of wavelength on MTF for different types of
Gaussian beams.
are obtained for the collimated Gaussian beam (Θ0= 1) in
Figure 1(b) and the divergent Gaussian beam (Θ0= 2) in
Figure 1(c).To analyze the effects of the turbulence inner scale
on
MTF, the spectral power law value and the outer scale
ofturbulence are fixed to constant values as 𝛼 = 3.6 and𝐿0= 5m.
Several inner scales of turbulence are used, and
calculation results are depicted in Figure 2 for different
types
of Gaussian beams. It can be seen that, with an increase in
theinner scale of turbulence, the value of MTF also increases.This
can be physically explained by the change of inertialsubrange of
turbulence. When the inner scale of turbulenceincreases, the
frequency’s upper bound of inertial subrangewould move to a lower
position, and thus the atmosphericturbulence would bring less
effects on the propagation of theGaussian beam.
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8 International Journal of Optics
The influences of outer scale of turbulence on MTF aredepicted
in Figure 3 for different types ofGaussian beams. Forthe real
atmospheric turbulence, the outer scale of turbulenceis usually in
the order of meters. Hence, it is set to 1m, 5m,and 25m,
respectively. The spectral power law value and theinner scale of
turbulence are set to 𝛼 = 3.6 and 𝑙
0= 0.01m as
example. It can be seen that, with an increase in the outer
scaleof turbulence, the value ofMTFdecreases and thus the qualityof
theGaussian beam is degraded severely by the atmosphericturbulence.
This is because WSF is mostly influenced by thelarge-scale
turbulence eddies, which are relevant to the low-frequency part of
the atmospheric turbulence spectrum. Alarger turbulence outer scale
would lead to a larger range ofinertial subrange.
For further discussions and analyses, the inner and outerscales
of turbulence are assigned to constant values 𝑙
0=
0.01m and 𝐿0= 5m, respectively. The spectral power law
value still uses the default value 𝛼 = 3.6. Some typical
valuesof wavelength in the near infrared region, 𝜆 = 850 nm,𝜆 =
1060 nm, and 𝜆 = 1550 nm, are investigated in thissimulation.
Figure 4 depicts MTF for different Gaussianbeams as a function of 𝜇
with different 𝜆. It is obvious thatthe value of MTF increases with
an increase in 𝜆 for certaintype of Gaussian beam if other optical
parameters are fixed.This phenomenon may be caused by the fact that
the largerthe beam wavelength, the more pronounced the
diffraction.Thus, a laser beam with larger wavelength can be less
affectedby turbulence eddies.
5. Conclusions
In this paper, a theoretical expression of the MTF isderived for
a Gaussian beam propagating through the non-Kolmogorov atmospheric
turbulence along a horizontal path.This expression contains a
variable spectral power lawvalue, finite inner and outer scales of
turbulence, and otherimportant optical parameters of a Gaussian
beam. Numericalsimulations indicate that the atmospheric turbulence
wouldproduce less degrading effects on the wireless optical
com-munication system with an increase in the spectral powerlaw
value. The decreased inner scale of turbulence makes aGaussian beam
influencedmore seriously by the atmosphericturbulence. With an
increase in the outer scale of turbulence,the quality of a Gaussian
beam is degraded more severelyby the atmospheric turbulence. A
laser beam with largerwavelength can be less affected by turbulence
eddies.
Competing Interests
The authors declare that there is no conflict of
interestsregarding the publication of this paper.
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