Abstract

A generalized exponential spectrum model is derived, which considers finite turbulence inner and outer scales and has a general spectral power law value between the range 3 to 5 instead of standard power law value 11/3. Based on this generalized spectrum model, a new generalized long exposure turbulence modulation transfer function (MTF) is obtained for optical plane and spherical wave propagating through horizontal path in weak fluctuation turbulence. When the inner scale and outer scale are set to zero and infinite, respectively, the new generalized MTF is reduced to the classical generalized MTF derived from the non-Kolmogorov spectrum.

© 2010 OSA

1. Introduction

Atmospheric turbulence has a significance degrading impact on the quality of imaging system, and this can be described by the MTF. Traditionally, in imaging systems, the long exposure MTF and the short-exposure MTF are derived with the assumption that the turbulence is Kolmogorov type and the Kolmogorov spectrum is applied for mathematical simplicity [1]. However, the Kolmogorov spectrum is only effective in inertial subrange. When it is used to estimate the MTF of the imaging system, it is extended to all ranges by assuming the inner scale size is zero and the outer scale is infinite. This assumption may lead to divergent integrals in some cases [2]. So some turbulence spectrums with specific inner and outer scale were used to estimate the MTF [2].

In the past decade, both the experimental data and theoretical investigations [36] have exhibited non-Kolmogorov turbulence case in certain portions of the atmosphere. The non-Kolmogorov spectrum with variable power law value between the range 3 to 5 instead of standard power law value 11/3 for Kolmogorov turbulence and a more general amplitude factor instead of constant value 0.033 has been proposed, and applied to obtain the generalized MTF [7,8]. Still it has the same problem as the Kolmogorov spectrum.

In this study, the exponential spectrum model (with both finite inner scale and outer scale) [2] is generalized to apply in non-Kolmogorov atmospheric turbulence. Based on the new generalized exponential spectrum model, a new generalized long exposure turbulence MTF is derived for optical plane and spherical wave propagating through horizontal path in weak fluctuation non-Kolmogorov turbulence. And then the influences of turbulence inner scale, outer scale and spectral power law’s variations on the atmospheric turbulence MTF are analyzed.

2. Generalized exponential spectrum

2.1 Exponential spectrum

Exponential spectrum is a turbulence spectrum model which considers the influence of finite inner and outer scales, and is given by [2]

Φn(κ)=0.033Cn2κ11/3exp(κ2κl2)[1exp(κ2κ02)](0k<)
Where Cn2 represents the refractive-index structure parameter for Kolmogorov turbulence and has the unit of m2/3, κ is the spatial wave number, κl=5.92/l0,κ0=C0/L0, l0and L0 are the turbulence inner and outer scale, C0is chosen differently depending on the application, and because the outer scale itself is not well defined, it is difficult to proclaim any particular constant C0 with the outer scale parameter κ0. In this study, we set C0=4πjust as in [2].

In order to include both inner scale and outer scale’s effects, the generalized spectrum recently adopted has the form as [8,9]

Φn(κ,α)=A(α)Cn2^F(k,l0,L0,α)(0κ<,3<α<5)
Where A(α) is a constant which maintains consistency between the refractive index structure function and its power spectrum, Cn2^is the generalized refractive-index structure parameter with unitm3α, F(k,l0,L0,α) is the function which includes the influence of finite inner and/or outer scale, and has different forms for different spectrums [8,9].

To generalize the exponential spectrum model to apply in non-Kolmogorov atmospheric turbulence, the exponential spectrum model should take the form as

Φn(κ,α,l0,L0)=A^(α)Cn2^καf(k,l0,L0,α)(0κ<,3<α<5)
f(κ,l0,L0,α)=exp(κ2κl2)[1exp(κ2κ02)]
Where A^(α) has the same meaning asA(α),κl=c(α)/l0,κ0=C0/L0, c(α) is the scaling constant. In the next section, the expression forms of A^(α) and c(α) would be derived.

2.2 Refractive-index structure function for generalized exponential spectrum model

One important parameter to infer the generalized exponential spectrum is the refractive-index structure functionDn(R,α), which describes the behavior of the correlations of turbulence refractive index filed fluctuations between two given points separated by a distance R. For Kolmogorov turbulence, the relationship between Dn(R) and Φn(κ)can be described as [1]

Dn(R)=8π0κ2Φn(k)(1sinκRκR)dκ
For non-Komogorov turbulence, the relationship between Dn(R) and Φn(κ) is
Dn(R,α)=8π0κ2Φn(k,α)(1sinκRκR)dκ
Substituting Eq. (3) into Eq. (6), and set L0 to infinite (this will be explained later), the following expression is derived:
Dn(R,α)=8π0κ2αA^(α)Cn2^exp(κ2κl2)(1sinκRκR)dκ
Expanding 1sinκRκR with Maclaurin series [10]:
1sinκRκR=n=1(1)n1(2n+1)!κ2nR2n
and inserting it into Eq. (7), then interchanging the order of series summation and integration, as a result, Eq. (7) becomes:

Dn1(R,α)=8πA^(α)Cn2^n=1(1)n1(2n+1)!R2n0κ2α+2nexp(κ2κl2)dκ

In view of the definition of gamma function Γ(x) and the hypergeometric function F11(a;b;z) [10]:

Γ(x)=0κx1eκdκ(κ>0,x>0),F11(a;b;z)=n=0(a)nzn(b)nn!
Where (a)n is the Pochhammer Symbol and has the form
(a)n=Γ(a+n)Γ(a)=a(a+1)(a+n1)
The integration and series summation in Eq. (9) is solved, as a result, the refractive-index structure function can be expressed as:
Dn(R,α)=4πA^(α)Cn2^κl3α{Γ(α2+32)[1F11(α2+32;32;R2κl24)]}
For statistically homogeneous and isotropic fluctuation non-Komogorov atmospheric turbulence, the related refractive-index structure function is given by [2,9]
Dn(R,α)={Cn2^l0α5R20Rl0Cn2^Rα3l0RL0
It needs to say that because the random field of index refractive fluctuation is nonisotropic for scale sizes larger than outer scaleL0, no general description of the refractive-index structure function can be predicted forR>L0. So, in the derivation of refractive-index structure function for generalized modified atmospheric spectrum, L0 is set to infinite for calculation purpose [2,9].

Using Eq. (12) and Eq. (13), the unknown A^(α) and c(α) in Eq. (3) will be derived next.

2.2.1 Expression derivation of A^(α)

Whenl0RL0 thenR2κl24=R2c2(α)4l021, the F11(a;b;x) in the Eq. (12) can be approximately expanded with big arguments and is given by [10]

F11(a;b;x)Γ(b)Γ(ba)xa(x1)
Substituting Eq. (14) into Eq. (12), then Dn(R,α) (see Eq. (12)) becomes
Dn(R,α)4πA^(α)Cn2^Γ(α2+32)Γ(32)Γ(α2)(12)α3(R)α3(l0RL0)
Using Eq. (15) and Eq. (13), and considering the properties of gamma function [10]
Γ(α+1)=αΓ(α), Γ(1α)Γ(α)=πsin(πα), Γ(α)Γ(α+12)=212απΓ(2α)
As a result, A^(α)can be expressed with the form
A^(α)=Γ(α1)4π2sin[(α3)π2]
It has the same expression as A(α) in non-Kolmogorov spectrum model [7,8].

2.2.2 Expression derivation of c(α)

When0Rl0 thenR2κl24=R2c2(α)4l021, the F11(a;b;x) in the Eq. (12) can be approximately expanded with small arguments and is given by [10]

F11(a;b;x)n=01(a)nzn(b)nn!=1+abx(x1)
Substituting Eq. (18) into Eq. (12), then Dn(R,α) becomes
Dn(R,α)πA^(α)Cn2^κl5αR2[Γ(α2+32)(3α3)](0Rl0)
Using Eq. (19) and Eq. (13), the expression of c(α)can be derived:

c(α)={πA^(α)[Γ(α2+32)(3α3)]}1α5

2.3 Generalized exponential spectrum model

Substituting Eq. (17) and Eq. (20) into Eq. (3), we can obtain the expression of the generalized exponential spectrum model. The curves of the A^(α) and c(α) as the function of α are plotted and shown in Fig. 1(a) and Fig. 1(b). Whenα=11/3, A^(11/3)=0.033 and c(11/3)5.92, Eq. (3) is reduced to the exponential spectrum.

 

Fig. 1 A(α) and c(α) as a function of α. (a): A(α), (b): c(α)

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When inner scale is set to zero and outer scale is set to infinite, the generalized exponential spectrum becomes

Φn(κ,α)=A^(α)Cn2^κα(0κ<,3<α<5)
Where A^(α) has the same expression as A(α) in non-Kolmogorov spectrum model, and that means the generalized exponential spectrum can be reduced to the classical non-Kolmogorov spectrum with the particular case of zero inner scale and infinite outer scale.

3. Generalized atmospheric turbulence MTF based on the generalized exponential spectrum

In this section, a new generalized atmospheric turbulence MTF model is derived with the generalized exponential spectrum for plane and spherical wave propagating through horizontal path in weak fluctuation non-Kolmogorov turbulence.

Following the analysis of Hufnagel [11] and Fried [12], the long-exposure weak fluctuation atmospheric turbulence MTF can be deduced from the formulation of the mutual coherence function (MCF) in the receiver aperture plane, and it is the product of the MCF evaluated atρ=λFν. For the case of plane/spherical wave, the long exposure weak fluctuation turbulence MTF in the focal plane of the receiver is given by [12]

MTFturb(ν)=exp[12Dω(λFν)]
Where ρ is the separation between points in the image plane transverse to the direction, ν is the spatial frequency measured in cycles per unit length, F is focal length. Dω(λFν) represents the wave structure function for plane/spherical wave, and it is the sum of the log-amplitude structure function and the phase structure function.

For weak, homogeneous, and isotropic turbulence, the method of small perturbations can be used to solve the wave propagation problem. The solution yields the wave structure function for plane and spherical waves from point object within the atmospheric turbulence [1]

Dωp(ρ)=8π2k20Ldz0[1J0(κρ)]Φn(κ,z)κdκ
Dωs(ρ)=8π2k20Ldz0[1J0(κρz/L)]Φn(κ,z)κdκ
WhereJ0is Bessel function of the first kind and zero order. L is the optical path. Dωp(ρ)andDωs(ρ) represents plane and spherical wave structure function, respectively.It is noted that in the derivation of Eq. (22), Eq. (23) and Eq. (24), no particular assumption of Kolmogorov turbulence is needed, so the relationships betweenMTF(), Dω()andΦn() can still be applied in the non-Kolmogorov turbulence case.

In the next section, the expression forms of plane and spherical wave structure functions for non-Kolmogorov turbulence case will be derived, and subsequently the generalized MTF can be obtained. It is noted that in the following calculation, the range of α is further restricted to the range 3 to 4 just as [7,8,13].

3.1 Plane wave structure function for non-Kolmogorov turbulence

For generalized exponential spectrum, Eq. (23) can be written in the form as

Dωp(ρ,α,l0,L0)=8π2k20Ldz0[1J0(κρ)]Φn(κ,α,l0,L0)κdκ(3<α<4)
Substituting Eq. (3) into Eq. (25), and expanding J0 in the form of Maclaurin series [10]
J0(x)=n=0(1)nn!Γ(n+1)(x2)2n
Where, Γ(n+1)in the above equation can be expressed withΓ(n+1)=n!=(1)n.

The wave structure function for plane wave becomes

Dωp(ρ,α,l0,L0)=8π2k2A^(α)Cn2^0L{n=1(1)n1n!(1)n(ρ2)2n0κ2nα+1f(κ,l0,L0,α)dκ}dz
Interchanging the order of series summation and integration, and making integration with respect to z, as a result, Eq. (27) is expressed as

Dωp(ρ,α,l0,L0)=8π2k2A^(α)Cn2^Ln=1(1)n1n!(1)n(ρ24)n0κ2nα+1f(κ,l0,L0,α)dκ

To calculate for simplicity, f(κ,l0,L0,α) is divided into two parts

f1(κ,l0,L0,α)=exp(κ2κl2), f2(κ,l0,L0,α)=exp[κ2(1κ02+1κl2)]
Then using the Eq. (10) for the calculation of integration and series summation, the plane wave structure function for non-Kolmogorov turbulence is finally derived
Dωp(ρ,α,l0,L0)=Dωp1(ρ,α,l0,L0)+Dωp2(ρ,α,l0,L0)
Where Dωp1(ρ,α,l0,L0) and Dωp2(ρ,α,l0,L0) are obtained from f1(κ,l0,L0,α) and f2(κ,l0,L0,α) separately, and expressed with the forms as
Dωp1(ρ,α,l0,L0)=8π2k2A^(α)Cn2^Lκl2α{12Γ(α2+1)[1F11(α2+1;1;ρ2κl24)]}
Dωp2(ρ,α,l0,L0)=8π2k2A^(α)Cn2^L{12κ2α21Γ(α2+1)[1F11(α2+1;1;ρ24κ2)]}
Where, κ2=1κ02+1κl2.

Hitherto, the wave structure function for plane wave propagating through horizontal path has been obtained, and it contains finite inner and outer scale. To compare with the result derived from the non-Kolmogorov spectrum in which the inner scale is set to zero and outer scale is set to infinite for calculation purpose, we discuss the particular case of L0 and l00.WhenL0andl00, thenκ0=C0L00andκl=c(α)l0, soκ2=1κ02+1κl2, ρ24κ20, ρ2κl24. The F11(a;b;z)in Eq. (31) and Eq. (32) can be approximately expanded with big arguments (see Eq. (14)) and small arguments (see Eq. (18)), respectively. So the wave structure function can be approximately expressed as

Dωp1(ρ,α,l0,L0)24απ2k2A^(α)Cn2^LΓ(α/2+1)Γ(α/2)ρα2(3<α<4)
Dωp2(ρ,α,l0,L0)0(3<α<4)
At this time, the Dωp(ρ,α,l0,L0) can be expressed with Eq. (33) which has the same form as the case derived from the non-Kolmogorov spectrum [13], and this means the plane wave structure function derived in this study can be reduced to the classical result under the special condition of zero inner scale and infinite outer scale.

3.2 Spherical wave structure function for non-Kolmogorov turbulence

For generalized exponential spectrum, Eq. (24) can be expressed as

Dωs(ρ,α,l0,L0)=8π2k20Ldz0[1J0(κρz/L)]Φn(κ,α,l0,L0)κdκ(3<α<4)
Substituting Eq. (3) into Eq. (35), and expanding J0 in the form of Maclaurin series just as the Eq. (26), the wave structure function for spherical wave becomes
Dωs(ρ,α,l0,L0)=8π2k2A^(α)Cn2^0L{n=1(1)n1n!(1)n(ρz2L)2n0κ2nα+1f(κ,l0,L0,α)dκ}dz
Interchanging the order of series summation and integration, and making the integrating with respect to z, Eq. (36) becomes
Dωs(ρ,α,l0,L0)=8π2k2A^(α)Cn2^Ln=1{(1)n1n!(1)n12n+1(ρ2)2n0κ2nα+1f(κ,l0,L0,α)dκ}
Where, 12n+1 in the above equation can be expressed with the form12n+1=(1/2)n(3/2)n.

To calculate for simplicity, f(κ,l0,L0,α) is expressed with the sum of two parts just as Eq. (29), then considering the definition of Γ(x)function (see Eq. (10)) and the generalized hypergeometric function F22(a,b;c,d;z) which is defined as [10]:

F22(a,b;c,d;z)=n=0(a)n(c)nzn(b)n(d)nn!
As a result, the spherical wave structure function for non-Kolmogorov turbulence can be expressed as
Dωs(ρ,α,l0,L0)=Dωs1(ρ,α,l0,L0)+Dωs2(ρ,α,l0,L0)
Where Dωs1(ρ,α,l0,L0) and Dωs2(ρ,α,l0,L0) are obtained from the part of f1(κ,l0,L0,α) and f2(κ,l0,L0,α) separately, and expressed with the forms as
Dωs1(ρ,α,l0,L0)=8π2k2A^(α)Cn2^Lκl2α{12Γ(α2+1)[1F22(α2+1,12;1,32;ρ2κl24)]}
Dωs2(ρ,α,l0,L0)=8π2k2A^(α)Cn2^L{12κ2α21Γ(α2+1)[1F22(α2+1,12;1,32;ρ24κ2)]}
Hitherto, the wave structure function for spherical wave propagating through horizontal path has been obtained, and it contains finite inner and outer scale. To compare with the result derived from the non-Kolmogorov spectrum in which the inner scale is set to zero and outer scale is set to infinite for calculation purpose, we discuss the particular case of L0 and l00.

Similar to the discussion in section 3.2, here it needs to consider the approximate expansions of F22(a,b;c,d;z)with big arguments and small arguments, and they are given by [10]:

F22(a,b;c,d;x)Γ(c)Γ(d)Γ(ba)Γ(b)Γ(ca)Γ(da)xa+Γ(c)Γ(d)Γ(ab)Γ(a)Γ(cb)Γ(db)xb(x1)
F22(a,b;c,d;z)n=01(a)n(c)nzn(b)n(d)nn!=1+abcdx(x1)
Substituting Eq. (42) and Eq. (43) into the Eq. (40) and Eq. (41), the spherical wave structure function for non-Kolmogorov turbulence can be approximately expressed as
Dωs1(ρ,α,l0,L0)1α1(24απ2k2A^(α)Cn2^LΓ(α/2+1)Γ(α/2)ρα2)(3<α<4)
Dωs2(ρ,α,l0,L0)0(3<α<4)
At this time, the Dωs(ρ,α,l0,L0) can be expressed with Eq. (44) which has the same form as the case derived from the non-Kolmogorov spectrum [13], and this means the spherical wave structure function derived in this study can also be reduced to the classical result under the special condition of zero inner scale and infinite outer scale.

3.3 Generalized atmospheric turbulence MTF for plane wave and spherical wave

Taking into account Eq. (30) and Eq. (39) for plane and spherical wave structure function for wave propagating through horizontal path in weak fluctuation non-Kolmogorov turbulence, the generalized form of the atmospheric turbulence MTF can be expressed as

MTFturb(pl)(u,α,l0,L0)=exp[12Dωp(uD,α,l0,L0)](3<α<4)
MTFturb(sp)(u,α,l0,L0)=exp[12Dωs(uD,α,l0,L0)](3<α<4)
WhereMTFturb(pl)(u,α,l0,L0) and MTFturb(sp)(u,α,l0,L0) represent the MTF for plane wave and spherical wave propagating through horizontal path in weak non-Kolmogorov atmospheric turbulence, respectively. u=λFν/D denotes the normalized spatial frequency, and it is between 0 and 1, D is the receiver aperture diameter.

In practice, if we further consider the influence of receiver, atmospheric particulates (aerosol, dust, etc.), the total long-exposure MTF of propagation path up to detector of imaging system can be described as the product [7]

MTFtotal(u)=2π[cos1uu1u2]×MTFturb(u)×MTFaersol(u)
Where 2π[cos1uu1u2] is the MTF of the receiver optics, MTFaersol(u) represents the MTF caused by atmospheric particulates (aerosol, dust, etc.) scattering.In this study, we focus on the influence of atmospheric turbulence on the imaging system, that isMTFturb(u).

4. Numerical results

In this section, simulations are conducted to analyze the generalized atmospheric turbulence MTF as a function of normalized spatial frequency. Because the MTF contains three changeable parameters: α, l0 (turbulence inner scale) and L0 (turbulence outer scale), in order to avoid the mutual interferences between parameters, in the following simulations, we fix two of the three parameters at a time and analyze only one parameter’s influence on the MTF. All the experiments are conducted for plane wave and spherical wave propagating in horizontal path with the settings:

Cn2^=1.6×1014m3α,λ=1.55μm,L=1000m,D=0.1m

4.1 Effect of outer scale’s variation on the MTF

In the first simulation experiment, we fix α and l0 to constant values, and choose different turbulence outer scale sizes to analyze the influence on MTF. To well compare with the classical MTF derived from the non-Kolmogorov spectrum in which l0 equals zero and L0 is infinite for calculation purpose, l0 is set to a very small value of 0.01×(λL)1/2 and α is set to a constant value of 11/3. Then outer scale L0 is set to5m, 50m and500m, respectively, and the corresponding atmospheric turbulence MTFs are plotted in Fig. 2 . To further analyze the impact of L0’s variation on MTF, we also exhibit the classical generalized MTF derived from non-Kolmogorov spectrum.

 

Fig. 2 MTF as a function of normalized spatial frequency with different outer scale values. (a): plane wave. (b): spherical wave.

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As shown in Fig. 2, with the increase of the turbulent outer scale, the value of MTF decreases (it means the quality of imaging system is degraded more severely by the turbulence), and especially when L0 is set to a very big value (here it equals500m), at this time, the result is very close to the classical non-Kolmogorov case (with infinite outer scale). This can be explained directly from the function f(κ,α,l0,L0) in Eq. (4). WhenL0increases, f(κ,α,l0,L0)raises, that makes the plane and spherical wave structure function in Eq. (25) and Eq. (35) increases, so the final MTF decreases.

This can also be interpreted from another point: big turbulent cells (>(λL)1/2) mainly bring the variation of wave phase [14], and it can be expressed with phase structure function which is the dominant component in the wave structure function [15]. So when theL0 is assumed high value the wave meets a major number of large-scale turbulent cells along its propagation length and these cells leads to higher phase wave structure function with respect to the case of low outer scale value (where more large scale cells are cut out) [9], and this makes the plane and spherical wave structure function increases, subsequently the final MTF decreases.

4.2 Effect of inner scale’s variation on MTF

To analyze turbulence inner scale’s influence on MTF, α and L0 are fixed to constant values, and different inner scale sizes are chosen. In order to alleviate the other two parameters’ interference, α is chosen to be 11/3 and L0 is fixed to a very big value of 5000 m. Then the inner scale l0 is set to0.1×(λL)1/2, (λL)1/2and10×(λL)1/2, respectively, and the corresponding MTFs as a function of normalized spatial frequency are plotted in Fig. 3 . To further analyze this simulation, we also exhibit the classical generalized MTF derived from conventional non-Kolmogorov spectrum.

 

Fig. 3 MTF as a function of normalized spatial frequency with different inner scale values. (a): plane wave. (b): spherical wave

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From Fig. 3, we can infer that when the inner scale is much smaller than the first Fresnel zone, that isl0<<(λL)1/2, its influence on the final form of the MTF can be ignored compared with the classical result. That is because the log-amplitude structure function has a correlation distance of the order of (λL)1/2, and it is primarily affected by the turbulence cells with the size of (λL)1/2 [14]. Whenl0<<(λL)1/2, the number of turbulent cells with the size of (λL)1/2 almost keeps parallel with the classical case which assumes zero inner scale, therefore, log-amplitude structure function will not be changed. Meanwhile, the outer scale L0 is fixed to a very big value in this simulation, which makes the phase structure function nearly the same as the classical case which takes infinite outer scale. All these make the wave structure function and the MTF unchanged compared with the result derived from non-Kolmogorov spectrum

However, when the finite inner scale is much larger than the Fresnel zone, that is l0(λL)1/2, the log-amplitude structure function is nearly ignorable and the number of big turbulent cells (>(λL)1/2) that leads to high phase structure function becomes less. This makes the wave structure function decreases and the MTF increases just as shown in Fig. 3. At this time, the influence of finite inner scale on the MTF should be considered theoretically. However, in practice, the inner scale is commonly in the unit of millimeter and does not satisfyl0(λL)1/2.

4.3 Effect of α’s variation on MTF

To analyze α’s influence on MTF, l0is fixed to a constant value of 0.1×(λL)1/2 and L0 is set to5000m. So, we can only focus on the influence of α’s changes on MTF. In this simulations, α is set to 10/3, 11/3 and 3.9 respectively, and the corresponding MTFs as a function of normalized spatial frequency are plotted in Fig. 4 , from which we can infer that various α brings different impact on the form of MTF just as in [7,8].

 

Fig. 4 MTF as a function of normalized spatial frequency with different α values. (a): plane wave. (b): spherical wave

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As shown in Fig. 4, for helical turbulence, α=10/3, compared with the Kolmogorov case, the turbulence brings more degrading on the imaging in the low frequency and less degrading in the high frequency, and this is consistent with [7], although the latter describes the condition of slant path propagation. While, for the power law value higher than the Kolmogorov case, in this simulation, α=3.9>11/3, the opposite trend appears just as [7].

5. Conclusions

In this study, a theoretical generalized exponential spectrum to apply in non-Kolmogorov atmospheric turbulence is derived, which considers finite inner scale, finite outer scale and general power law values. Base on this atmospheric spectrum model, a new generalized atmospheric turbulence long exposure MTF model is developed for plane wave and spherical wave propagating through non-Kolmogorov atmospheric turbulence medium for a horizontal path. When the inner scale and outer scale are set to zero and infinite, respectively, the new generalized MTF is reduced to the classical generalized MTF derived from the non-Kolmogorov spectrum.

Simulation results show that finite outer and inner scale does introduce influences on the MTF for plane wave and spherical wave. The former alleviates the influence of turbulence on the imaging system especially in high spatial frequency than the case derived from non-Kolmogorov spectrum, and the inner scale with very small value (here it is set tol0(λL)1/2) has ignorable influence on the MTF compared with the classical result, but as it continues to increase, the effect on the propagating needs to be considered. Different power law values also produces different impacts on imaging system, and this conclusion is consist with [7]. The results will help to better study the effects of turbulence on the optical plane wave and spherical wave propagating through horizontal path.

Acknowledgments

This work is partly supported by the United Fund Foundation of the Civil Aviation and the National Natural Science Foundation of China (No. 60832011), Aeronautical Science Foundation of China (20080151009).

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5. M. S. Belen’kii, E. Cuellar, K. A. Hughes, and V. A. Rye, “Experimental study of spatial structure of turbulence at Maui Space Surveillance Site (MSSS),” Proc. SPIE 6304, 63040U (2006). [CrossRef]  

6. A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov trubulence,” Atmos. Res. 88(1), 66–77 (2008). [CrossRef]  

7. A. Zilberman, and N. S. Kopeika, “Slant-path generalized atmospheric MTF,” in Proceedings of IEEE 25th Convention of Electrical and Electronics Engineers (Institute of Electrical and Electronics Engineers, Israel, 2008), pp. 217–221.

8. N. S. Kopeika, A. Zilberman, and E. Golbraikh, “Generalized atmospheric turbulence: implications regarding imaging and communications,” Proc. SPIE 7588, 758808 (2010). [CrossRef]  

9. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of Arrival Fluctuations for Free Space Laser Beam Propagation through non Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007). [CrossRef]  

10. L. C. Andrews, Special Functions of Mathematics for Engineers, 2nd ed. (SPIE Optical Engineering Press, Bellingham, Wash., 1998).

11. R. E. Hufnagel and N. R. Stanley, “Modulation Transfer Function associated with Image Transmission through Turbulent Media,” J. Opt. Soc. Am. 54(1), 52–61 (1964). [CrossRef]  

12. D. L. Fried, “Optical Resolution Through a Randomly Inhomogeneous Medium for Very Long and very Short Exposures,” J. Opt. Soc. Am. 56(10), 1372–1379 (1966). [CrossRef]  

13. B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical Propagation in non-Kolmogorov Atmospheric Turbulence,” Proc. SPIE 2471, 181–196 (1995). [CrossRef]  

14. A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Propagation of electromagnetic waves in Kolmogorov and non-Kolmogorov atmospheric turbulence: three-layer altitude model,” Appl. Opt. 47(34), 6385–6391 (2008). [CrossRef]   [PubMed]  

15. W. B. Miller, J. C. Ricklin, and L. C. Andrews, “Log-amplitude variance and wave structure function: a new perspective for Gaussian beams,” J. Opt. Soc. Am. A 10(4), 661–672 (1993). [CrossRef]  

References

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  • |

  1. V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation, (trans.for NOAA by Israel Program for Scientific Translations, Jerusalem, 1971).
  2. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media. (SPIE Optical Engineering Press, Bellingham, 2005).
  3. D. T. Kyrazis, J. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55 (1994).
    [CrossRef]
  4. M. S. Belen’kii, S. J. Karis, J. M. Brown, and R. Q. Fugate, “Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE 3126, 113–123 (1997).
    [CrossRef]
  5. M. S. Belen’kii, E. Cuellar, K. A. Hughes, and V. A. Rye, “Experimental study of spatial structure of turbulence at Maui Space Surveillance Site (MSSS),” Proc. SPIE 6304, 63040U (2006).
    [CrossRef]
  6. A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov trubulence,” Atmos. Res. 88(1), 66–77 (2008).
    [CrossRef]
  7. A. Zilberman, and N. S. Kopeika, “Slant-path generalized atmospheric MTF,” in Proceedings of IEEE 25th Convention of Electrical and Electronics Engineers (Institute of Electrical and Electronics Engineers, Israel, 2008), pp. 217–221.
  8. N. S. Kopeika, A. Zilberman, and E. Golbraikh, “Generalized atmospheric turbulence: implications regarding imaging and communications,” Proc. SPIE 7588, 758808 (2010).
    [CrossRef]
  9. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of Arrival Fluctuations for Free Space Laser Beam Propagation through non Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
    [CrossRef]
  10. L. C. Andrews, Special Functions of Mathematics for Engineers, 2nd ed. (SPIE Optical Engineering Press, Bellingham, Wash., 1998).
  11. R. E. Hufnagel and N. R. Stanley, “Modulation Transfer Function associated with Image Transmission through Turbulent Media,” J. Opt. Soc. Am. 54(1), 52–61 (1964).
    [CrossRef]
  12. D. L. Fried, “Optical Resolution Through a Randomly Inhomogeneous Medium for Very Long and very Short Exposures,” J. Opt. Soc. Am. 56(10), 1372–1379 (1966).
    [CrossRef]
  13. B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical Propagation in non-Kolmogorov Atmospheric Turbulence,” Proc. SPIE 2471, 181–196 (1995).
    [CrossRef]
  14. A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Propagation of electromagnetic waves in Kolmogorov and non-Kolmogorov atmospheric turbulence: three-layer altitude model,” Appl. Opt. 47(34), 6385–6391 (2008).
    [CrossRef] [PubMed]
  15. W. B. Miller, J. C. Ricklin, and L. C. Andrews, “Log-amplitude variance and wave structure function: a new perspective for Gaussian beams,” J. Opt. Soc. Am. A 10(4), 661–672 (1993).
    [CrossRef]

2010 (1)

N. S. Kopeika, A. Zilberman, and E. Golbraikh, “Generalized atmospheric turbulence: implications regarding imaging and communications,” Proc. SPIE 7588, 758808 (2010).
[CrossRef]

2008 (2)

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov trubulence,” Atmos. Res. 88(1), 66–77 (2008).
[CrossRef]

A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Propagation of electromagnetic waves in Kolmogorov and non-Kolmogorov atmospheric turbulence: three-layer altitude model,” Appl. Opt. 47(34), 6385–6391 (2008).
[CrossRef] [PubMed]

2007 (1)

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of Arrival Fluctuations for Free Space Laser Beam Propagation through non Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[CrossRef]

2006 (1)

M. S. Belen’kii, E. Cuellar, K. A. Hughes, and V. A. Rye, “Experimental study of spatial structure of turbulence at Maui Space Surveillance Site (MSSS),” Proc. SPIE 6304, 63040U (2006).
[CrossRef]

1997 (1)

M. S. Belen’kii, S. J. Karis, J. M. Brown, and R. Q. Fugate, “Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE 3126, 113–123 (1997).
[CrossRef]

1995 (1)

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical Propagation in non-Kolmogorov Atmospheric Turbulence,” Proc. SPIE 2471, 181–196 (1995).
[CrossRef]

1994 (1)

D. T. Kyrazis, J. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55 (1994).
[CrossRef]

1993 (1)

1966 (1)

1964 (1)

Andrews, L. C.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of Arrival Fluctuations for Free Space Laser Beam Propagation through non Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[CrossRef]

W. B. Miller, J. C. Ricklin, and L. C. Andrews, “Log-amplitude variance and wave structure function: a new perspective for Gaussian beams,” J. Opt. Soc. Am. A 10(4), 661–672 (1993).
[CrossRef]

Belen’kii, M. S.

M. S. Belen’kii, E. Cuellar, K. A. Hughes, and V. A. Rye, “Experimental study of spatial structure of turbulence at Maui Space Surveillance Site (MSSS),” Proc. SPIE 6304, 63040U (2006).
[CrossRef]

M. S. Belen’kii, S. J. Karis, J. M. Brown, and R. Q. Fugate, “Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE 3126, 113–123 (1997).
[CrossRef]

Bishop, K. P.

D. T. Kyrazis, J. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55 (1994).
[CrossRef]

Brown, J. M.

M. S. Belen’kii, S. J. Karis, J. M. Brown, and R. Q. Fugate, “Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE 3126, 113–123 (1997).
[CrossRef]

Cuellar, E.

M. S. Belen’kii, E. Cuellar, K. A. Hughes, and V. A. Rye, “Experimental study of spatial structure of turbulence at Maui Space Surveillance Site (MSSS),” Proc. SPIE 6304, 63040U (2006).
[CrossRef]

Ferrero, V.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of Arrival Fluctuations for Free Space Laser Beam Propagation through non Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[CrossRef]

Fried, D. L.

Fugate, R. Q.

M. S. Belen’kii, S. J. Karis, J. M. Brown, and R. Q. Fugate, “Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE 3126, 113–123 (1997).
[CrossRef]

Golbraikh, E.

N. S. Kopeika, A. Zilberman, and E. Golbraikh, “Generalized atmospheric turbulence: implications regarding imaging and communications,” Proc. SPIE 7588, 758808 (2010).
[CrossRef]

A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Propagation of electromagnetic waves in Kolmogorov and non-Kolmogorov atmospheric turbulence: three-layer altitude model,” Appl. Opt. 47(34), 6385–6391 (2008).
[CrossRef] [PubMed]

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov trubulence,” Atmos. Res. 88(1), 66–77 (2008).
[CrossRef]

Hufnagel, R. E.

Hughes, K. A.

M. S. Belen’kii, E. Cuellar, K. A. Hughes, and V. A. Rye, “Experimental study of spatial structure of turbulence at Maui Space Surveillance Site (MSSS),” Proc. SPIE 6304, 63040U (2006).
[CrossRef]

Karis, S. J.

M. S. Belen’kii, S. J. Karis, J. M. Brown, and R. Q. Fugate, “Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE 3126, 113–123 (1997).
[CrossRef]

Keating, D. B.

D. T. Kyrazis, J. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55 (1994).
[CrossRef]

Kopeika, N. S.

N. S. Kopeika, A. Zilberman, and E. Golbraikh, “Generalized atmospheric turbulence: implications regarding imaging and communications,” Proc. SPIE 7588, 758808 (2010).
[CrossRef]

A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Propagation of electromagnetic waves in Kolmogorov and non-Kolmogorov atmospheric turbulence: three-layer altitude model,” Appl. Opt. 47(34), 6385–6391 (2008).
[CrossRef] [PubMed]

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov trubulence,” Atmos. Res. 88(1), 66–77 (2008).
[CrossRef]

Kupershmidt, I.

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov trubulence,” Atmos. Res. 88(1), 66–77 (2008).
[CrossRef]

Kyrazis, D. T.

D. T. Kyrazis, J. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55 (1994).
[CrossRef]

Miller, W. B.

Phillips, R. L.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of Arrival Fluctuations for Free Space Laser Beam Propagation through non Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[CrossRef]

Preble, A. J.

D. T. Kyrazis, J. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55 (1994).
[CrossRef]

Ricklin, J. C.

Roggemann, M. C.

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical Propagation in non-Kolmogorov Atmospheric Turbulence,” Proc. SPIE 2471, 181–196 (1995).
[CrossRef]

Rye, V. A.

M. S. Belen’kii, E. Cuellar, K. A. Hughes, and V. A. Rye, “Experimental study of spatial structure of turbulence at Maui Space Surveillance Site (MSSS),” Proc. SPIE 6304, 63040U (2006).
[CrossRef]

Shtemler, Y.

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov trubulence,” Atmos. Res. 88(1), 66–77 (2008).
[CrossRef]

Stanley, N. R.

Stribling, B. E.

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical Propagation in non-Kolmogorov Atmospheric Turbulence,” Proc. SPIE 2471, 181–196 (1995).
[CrossRef]

Toselli, I.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of Arrival Fluctuations for Free Space Laser Beam Propagation through non Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[CrossRef]

Virtser, A.

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov trubulence,” Atmos. Res. 88(1), 66–77 (2008).
[CrossRef]

Welsh, B. M.

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical Propagation in non-Kolmogorov Atmospheric Turbulence,” Proc. SPIE 2471, 181–196 (1995).
[CrossRef]

Wissler, J.

D. T. Kyrazis, J. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55 (1994).
[CrossRef]

Zilberman, A.

N. S. Kopeika, A. Zilberman, and E. Golbraikh, “Generalized atmospheric turbulence: implications regarding imaging and communications,” Proc. SPIE 7588, 758808 (2010).
[CrossRef]

A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Propagation of electromagnetic waves in Kolmogorov and non-Kolmogorov atmospheric turbulence: three-layer altitude model,” Appl. Opt. 47(34), 6385–6391 (2008).
[CrossRef] [PubMed]

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov trubulence,” Atmos. Res. 88(1), 66–77 (2008).
[CrossRef]

Appl. Opt. (1)

Atmos. Res. (1)

A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov trubulence,” Atmos. Res. 88(1), 66–77 (2008).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (1)

Proc. SPIE (6)

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical Propagation in non-Kolmogorov Atmospheric Turbulence,” Proc. SPIE 2471, 181–196 (1995).
[CrossRef]

N. S. Kopeika, A. Zilberman, and E. Golbraikh, “Generalized atmospheric turbulence: implications regarding imaging and communications,” Proc. SPIE 7588, 758808 (2010).
[CrossRef]

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of Arrival Fluctuations for Free Space Laser Beam Propagation through non Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007).
[CrossRef]

D. T. Kyrazis, J. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55 (1994).
[CrossRef]

M. S. Belen’kii, S. J. Karis, J. M. Brown, and R. Q. Fugate, “Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE 3126, 113–123 (1997).
[CrossRef]

M. S. Belen’kii, E. Cuellar, K. A. Hughes, and V. A. Rye, “Experimental study of spatial structure of turbulence at Maui Space Surveillance Site (MSSS),” Proc. SPIE 6304, 63040U (2006).
[CrossRef]

Other (4)

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation, (trans.for NOAA by Israel Program for Scientific Translations, Jerusalem, 1971).

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media. (SPIE Optical Engineering Press, Bellingham, 2005).

L. C. Andrews, Special Functions of Mathematics for Engineers, 2nd ed. (SPIE Optical Engineering Press, Bellingham, Wash., 1998).

A. Zilberman, and N. S. Kopeika, “Slant-path generalized atmospheric MTF,” in Proceedings of IEEE 25th Convention of Electrical and Electronics Engineers (Institute of Electrical and Electronics Engineers, Israel, 2008), pp. 217–221.

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Figures (4)

Fig. 1
Fig. 1

A ( α ) and c ( α ) as a function of α. (a): A ( α ) , (b): c ( α )

Fig. 2
Fig. 2

MTF as a function of normalized spatial frequency with different outer scale values. (a): plane wave. (b): spherical wave.

Fig. 3
Fig. 3

MTF as a function of normalized spatial frequency with different inner scale values. (a): plane wave. (b): spherical wave

Fig. 4
Fig. 4

MTF as a function of normalized spatial frequency with different α values. (a): plane wave. (b): spherical wave

Equations (55)

Equations on this page are rendered with MathJax. Learn more.

Φ n ( κ ) = 0.033 C n 2 κ 11 / 3 exp ( κ 2 κ l 2 ) [ 1 exp ( κ 2 κ 0 2 ) ] ( 0 k < )
Φ n ( κ , α ) = A ( α ) C n 2 ^ F ( k , l 0 , L 0 , α ) ( 0 κ < , 3 < α < 5 )
Φ n ( κ , α , l 0 , L 0 ) = A ^ ( α ) C n 2 ^ κ α f ( k , l 0 , L 0 , α ) ( 0 κ < , 3 < α < 5 )
f ( κ , l 0 , L 0 , α ) = exp ( κ 2 κ l 2 ) [ 1 exp ( κ 2 κ 0 2 ) ]
D n ( R ) = 8 π 0 κ 2 Φ n ( k ) ( 1 sin κ R κ R ) d κ
D n ( R , α ) = 8 π 0 κ 2 Φ n ( k , α ) ( 1 sin κ R κ R ) d κ
D n ( R , α ) = 8 π 0 κ 2 α A ^ ( α ) C n 2 ^ exp ( κ 2 κ l 2 ) ( 1 sin κ R κ R ) d κ
1 sin κ R κ R = n = 1 ( 1 ) n 1 ( 2 n + 1 ) ! κ 2 n R 2 n
D n 1 ( R , α ) = 8 π A ^ ( α ) C n 2 ^ n = 1 ( 1 ) n 1 ( 2 n + 1 ) ! R 2 n 0 κ 2 α + 2 n exp ( κ 2 κ l 2 ) d κ
Γ ( x ) = 0 κ x 1 e κ d κ ( κ > 0 , x > 0 ) , F 1 1 ( a ; b ; z ) = n = 0 ( a ) n z n ( b ) n n !
( a ) n = Γ ( a + n ) Γ ( a ) = a ( a + 1 ) ( a + n 1 )
D n ( R , α ) = 4 π A ^ ( α ) C n 2 ^ κ l 3 α { Γ ( α 2 + 3 2 ) [ 1 F 1 1 ( α 2 + 3 2 ; 3 2 ; R 2 κ l 2 4 ) ] }
D n ( R , α ) = { C n 2 ^ l 0 α 5 R 2 0 R l 0 C n 2 ^ R α 3 l 0 R L 0
F 1 1 ( a ; b ; x ) Γ ( b ) Γ ( b a ) x a ( x 1 )
D n ( R , α ) 4 π A ^ ( α ) C n 2 ^ Γ ( α 2 + 3 2 ) Γ ( 3 2 ) Γ ( α 2 ) ( 1 2 ) α 3 ( R ) α 3 ( l 0 R L 0 )
Γ ( α + 1 ) = α Γ ( α )
Γ ( 1 α ) Γ ( α ) = π sin ( π α )
Γ ( α ) Γ ( α + 1 2 ) = 2 1 2 α π Γ ( 2 α )
A ^ ( α ) = Γ ( α 1 ) 4 π 2 sin [ ( α 3 ) π 2 ]
F 1 1 ( a ; b ; x ) n = 0 1 ( a ) n z n ( b ) n n ! = 1 + a b x ( x 1 )
D n ( R , α ) π A ^ ( α ) C n 2 ^ κ l 5 α R 2 [ Γ ( α 2 + 3 2 ) ( 3 α 3 ) ] ( 0 R l 0 )
c ( α ) = { π A ^ ( α ) [ Γ ( α 2 + 3 2 ) ( 3 α 3 ) ] } 1 α 5
Φ n ( κ , α ) = A ^ ( α ) C n 2 ^ κ α ( 0 κ < , 3 < α < 5 )
M T F t u r b ( ν ) = exp [ 1 2 D ω ( λ F ν ) ]
D ω p ( ρ ) = 8 π 2 k 2 0 L d z 0 [ 1 J 0 ( κ ρ ) ] Φ n ( κ , z ) κ d κ
D ω s ( ρ ) = 8 π 2 k 2 0 L d z 0 [ 1 J 0 ( κ ρ z / L ) ] Φ n ( κ , z ) κ d κ
D ω p ( ρ , α , l 0 , L 0 ) = 8 π 2 k 2 0 L d z 0 [ 1 J 0 ( κ ρ ) ] Φ n ( κ , α , l 0 , L 0 ) κ d κ ( 3 < α < 4 )
J 0 ( x ) = n = 0 ( 1 ) n n ! Γ ( n + 1 ) ( x 2 ) 2 n
D ω p ( ρ , α , l 0 , L 0 ) = 8 π 2 k 2 A ^ ( α ) C n 2 ^ 0 L { n = 1 ( 1 ) n 1 n ! ( 1 ) n ( ρ 2 ) 2 n 0 κ 2 n α + 1 f ( κ , l 0 , L 0 , α ) d κ } d z
D ω p ( ρ , α , l 0 , L 0 ) = 8 π 2 k 2 A ^ ( α ) C n 2 ^ L n = 1 ( 1 ) n 1 n ! ( 1 ) n ( ρ 2 4 ) n 0 κ 2 n α + 1 f ( κ , l 0 , L 0 , α ) d κ
f 1 ( κ , l 0 , L 0 , α ) = exp ( κ 2 κ l 2 )
f 2 ( κ , l 0 , L 0 , α ) = exp [ κ 2 ( 1 κ 0 2 + 1 κ l 2 ) ]
D ω p ( ρ , α , l 0 , L 0 ) = D ω p 1 ( ρ , α , l 0 , L 0 ) + D ω p 2 ( ρ , α , l 0 , L 0 )
D ω p 1 ( ρ , α , l 0 , L 0 ) = 8 π 2 k 2 A ^ ( α ) C n 2 ^ L κ l 2 α { 1 2 Γ ( α 2 + 1 ) [ 1 F 1 1 ( α 2 + 1 ; 1 ; ρ 2 κ l 2 4 ) ] }
D ω p 2 ( ρ , α , l 0 , L 0 ) = 8 π 2 k 2 A ^ ( α ) C n 2 ^ L { 1 2 κ 2 α 2 1 Γ ( α 2 + 1 ) [ 1 F 1 1 ( α 2 + 1 ; 1 ; ρ 2 4 κ 2 ) ] }
D ω p 1 ( ρ , α , l 0 , L 0 ) 2 4 α π 2 k 2 A ^ ( α ) C n 2 ^ L Γ ( α / 2 + 1 ) Γ ( α / 2 ) ρ α 2 ( 3 < α < 4 )
D ω p 2 ( ρ , α , l 0 , L 0 ) 0 ( 3 < α < 4 )
D ω s ( ρ , α , l 0 , L 0 ) = 8 π 2 k 2 0 L d z 0 [ 1 J 0 ( κ ρ z / L ) ] Φ n ( κ , α , l 0 , L 0 ) κ d κ ( 3 < α < 4 )
D ω s ( ρ , α , l 0 , L 0 ) = 8 π 2 k 2 A ^ ( α ) C n 2 ^ 0 L { n = 1 ( 1 ) n 1 n ! ( 1 ) n ( ρ z 2 L ) 2 n 0 κ 2 n α + 1 f ( κ , l 0 , L 0 , α ) d κ } d z
D ω s ( ρ , α , l 0 , L 0 ) = 8 π 2 k 2 A ^ ( α ) C n 2 ^ L n = 1 { ( 1 ) n 1 n ! ( 1 ) n 1 2 n + 1 ( ρ 2 ) 2 n 0 κ 2 n α + 1 f ( κ , l 0 , L 0 , α ) d κ }
F 2 2 ( a , b ; c , d ; z ) = n = 0 ( a ) n ( c ) n z n ( b ) n ( d ) n n !
D ω s ( ρ , α , l 0 , L 0 ) = D ω s 1 ( ρ , α , l 0 , L 0 ) + D ω s 2 ( ρ , α , l 0 , L 0 )
D ω s 1 ( ρ , α , l 0 , L 0 ) = 8 π 2 k 2 A ^ ( α ) C n 2 ^ L κ l 2 α { 1 2 Γ ( α 2 + 1 ) [ 1 F 2 2 ( α 2 + 1 , 1 2 ; 1 , 3 2 ; ρ 2 κ l 2 4 ) ] }
D ω s 2 ( ρ , α , l 0 , L 0 ) = 8 π 2 k 2 A ^ ( α ) C n 2 ^ L { 1 2 κ 2 α 2 1 Γ ( α 2 + 1 ) [ 1 F 2 2 ( α 2 + 1 , 1 2 ; 1 , 3 2 ; ρ 2 4 κ 2 ) ] }
F 2 2 ( a , b ; c , d ; x ) Γ ( c ) Γ ( d ) Γ ( b a ) Γ ( b ) Γ ( c a ) Γ ( d a ) x a + Γ ( c ) Γ ( d ) Γ ( a b ) Γ ( a ) Γ ( c b ) Γ ( d b ) x b ( x 1 )
F 2 2 ( a , b ; c , d ; z ) n = 0 1 ( a ) n ( c ) n z n ( b ) n ( d ) n n ! = 1 + a b c d x ( x 1 )
D ω s 1 ( ρ , α , l 0 , L 0 ) 1 α 1 ( 2 4 α π 2 k 2 A ^ ( α ) C n 2 ^ L Γ ( α / 2 + 1 ) Γ ( α / 2 ) ρ α 2 ) ( 3 < α < 4 )
D ω s 2 ( ρ , α , l 0 , L 0 ) 0 ( 3 < α < 4 )
M T F t u r b ( p l ) ( u , α , l 0 , L 0 ) = exp [ 1 2 D ω p ( u D , α , l 0 , L 0 ) ] ( 3 < α < 4 )
M T F t u r b ( s p ) ( u , α , l 0 , L 0 ) = exp [ 1 2 D ω s ( u D , α , l 0 , L 0 ) ] ( 3 < α < 4 )
M T F t o t a l ( u ) = 2 π [ cos 1 u u 1 u 2 ] × M T F t u r b ( u ) × M T F a e r s o l ( u )
C n 2 ^ = 1.6 × 10 14 m 3 α
λ = 1.55 μ m
L = 1000 m
D = 0.1 m

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