research article modulation transfer function of a gaussian ...research article modulation transfer...

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

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

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


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


𝜉2

𝑘) .

(17)

Consider the iterated integral in (17). According to (5), thereis

𝜅3−𝛼


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


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


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


𝜉2

𝑘) .

(24)


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


𝜉2

𝑘)

+Θ𝜌2

2∫

1

0

𝜉 d𝜉∫+∞

0

d𝜅

× 𝜅3−𝛼


𝜉2

𝑘)

−𝜌2

4∫

1

0

d𝜉∫+∞

0

d𝜅


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


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


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


Figure 2: Effects of inner scale on MTF for different types of Gaussian beams.

× 𝜅3−𝛼


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,


0.2 0.4 0.6 0.8 10𝜇

0

0.2

0.4

0.6

0.8

1MTF

L0 = 1mL0 = 5mL0 = 25m


0.2 0.4 0.6 0.8 10𝜇

0

0.2

0.4

0.6

0.8

1

MTF

L0 = 1mL0 = 5mL0 = 25m


0.2 0.4 0.6 0.8 10𝜇

0

0.2

0.4

0.6

0.8

1

MTF

L0 = 1mL0 = 5mL0 = 25m


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


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


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


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


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.


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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research article modulation transfer function of a gaussian ...research article modulation transfer...

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