Atmosphere turbulence MTF models in moderate-to-strong anisotropic turbulence

Optik 130 (2017) 68–75

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Original research article

Atmosphere turbulence MTF models in moderate-to-strong anisotropic turbulence Linyan Cui School of Astronautics, Beihang University, Beijing 100191, China

a r t i c l e

i n f o

Article history: Received 8 September 2016 Accepted 6 November 2016 Keywords: Modulation transfer function Anisotropic turbulence Non-Kolmogorov turbulence

a b s t r a c t In this study, based on the extended Rytov approximation theory and the modified anisotropic non-Kolmogorov turbulence refractive-index fluctuations spectrum, new analytic expressions of the atmospheric turbulence modulation transfer function (MTF) are derived for optical plane and spherical waves propagating through moderate-to-strong anisotropic non-Kolmogorov turbulence. The anisotropic factor which parameterizes the asymmetric turbulence cells or eddies in the horizontal and vertical directions is introduced. When the anisotropic factor equals one, the models derived in this work can reduce correctly to the previously published results in moderate-to-strong isotropic turbulence. In addition, a wide range of turbulence strengths (weak, moderate-to-strong, and strong turbulence) is considered in this work, and it has good consistency with the previous models which were developed in weak anisotropic non-Kolmogorov turbulence in the weak turbulence regime. © 2016 Elsevier GmbH. All rights reserved.

1. Introduction Atmospheric turbulence modulation transfer function (MTF) describes the degrading effects of atmosphere turbulence on the long-range optical imaging system. For isotropic non-Kolmogorov turbulence which assumes that the sizes of turbulence cells or eddies are the same in horizontal and vertical directions, based on the Rytov and extended Rytov approximation theories, the analytic expressions for the atmospheric turbulence MTF models have been developed for optical waves propagating through weak and moderate-to-strong isotropic non-Kolmogorov turbulence [1–4]. In the investigations, the general spectral power law in the range 3–4 was considered (it is set to the classic value of 11/3 for the isotropic Kolmogorov turbulence). However, experiments and theoretical investigations have shown that the atmosphere turbulence also exhibits anisotropic property [5–12]. Compared with the isotropic turbulence, the sizes of anisotropic turbulence cells or eddies are no longer the same in horizontal and vertical directions. Commonly, a horizontal turbulence cell or eddy scale is bigger than the vertical one in anisotropic turbulence, and it may lead to different statistical properties on optical waves’ propagation. For weak anisotropic non-Kolmogorov turbulence, theoretical expressions for the atmospheric turbulence MTF models have been derived [13–15]. When the anisotropic factors equal one, they have good consistency with those derived in weak isotropic non-Kolmogorov turbulence. As turbulence strength continues to increase up to moderate-to-strong or strong regime, the derived turbulence MTF models in weak anisotropic turbulence may not be applicable. To investigate theoretically the optical waves’ propagation through moderate-to-strong or strong anisotropic non-Kolmogorov turbulence, the modified anisotropic non-Kolmogorov turbulence spectral model [16] which modified the conventional anisotropic

E-mail address: [email protected] http://dx.doi.org/10.1016/j.ijleo.2016.11.012 0030-4026/© 2016 Elsevier GmbH. All rights reserved.

L. Cui / Optik 130 (2017) 68–75

69

non-Kolmogorov turbulence spectrum [17] with an amplitude spatial filter function was derived with the extended Rytov approximation theory. In this study, within the extended Rytov approximation theory framework, the modified anisotropic non-Kolmogorov spectral model is applied to investigate theoretically the atmospheric turbulence MTF for optical plane and spherical waves propagating through moderate-to-strong anisotropic non-Kolmogorov turbulence. The anisotropic factor and a wide range of turbulence strength (weak, moderate-to-strong, and strong turbulence regimes) are considered. Numerical calculations are conducted to analyze the anisotropic factor and turbulence strength’s influences on the derived turbulence MTF models. 2. Modified turbulence refractive-index fluctuations spectrum for moderate-to-strong anisotropic non-Kolmogorov turbulence The anisotropic non-Kolmogorov turbulence refractive-index fluctuations spectrum over the inertial subrange takes the form as [17]:





˚n (, ˛, ς ) = A (˛) · Cˆ n2 · ς 2 · z2 + ς 2 x2 + y2 A (˛) =

1  (˛ − 1) cos 42

− ˛ 

2 , 1/L0 <  < 1/l0 ,



3<˛<4 .

(1)

 ˛ 

(2)

2

in which,  is the wavenumber related to the turbulence cell or eddy size, and  =







z2 + ς 2 x2 + y2 . x , y , and z are

the components of  in the x, y, and z directions. ˛ is the general spectral power law value, Cˆ n2 = ˇCn2 is the generalized structure parameter with unit [m3−˛ ], ˇ is a dimensional constant with unit [m11/3−˛ ], and  (·) is the gamma function. ς is the anisotropic factor which parameterizes the asymmetry of turbulence cells or eddies in horizontal and vertical directions. If the turbulence cells or eddies are anisotropic, the horizontal turbulence outer cell or eddy is larger than the vertical one, ς is bigger than one. l0 and L0 are the turbulence inner and outer scales, respectively. By invoking the Markov approximation which assumes that the index of refraction is delta-correlated at any pair of points located along the direction of propagation, z in Eq. (1) can be ignored. At this time, Eq. (1) becomes [17]: ˚n (, ˛, ς ) = A (˛) · Cˆ n2 · ς 2−˛ · −˛ ,  =



x2 + y2 .

(3)

For moderate-to-strong anisotropic non-Kolmogorov turbulence, according to the extended Rytov approximation theory, the modified anisotropic turbulence refractive-index fluctuations spectral model can be expressed as [16]:



˚n1 (, ˛, ς ) = ˚n (, ˛, ς ) G (, ˛, ς ) , 1/L0 <  < 1/l0 ,



3<˛<4 .

G (, ˛, ς ) = GX (, ˛, ς ) + GY (, ˛, ς ) ,



GX (, ˛, ) = exp −

(5)



2 X2

, GY (, ˛, ) =

(˛, )

(4)



˛

2 + Y2 (˛, )

˛/2 .

(6)

where, G (, ˛, ς ) is the amplitude spatial filter function, and it can be expressed as the sum of large-scale filter GX (, ˛, ς ) and small-scale filter GY (, ˛, ς ). X (˛, ) and Y (˛, ) are large-scale (or refractive) and small-scale (or diffractive) cutoff spatial frequencies, respectively. Appendix A gives the expressions of these two parameters. 3. Atmospheric turbulence MTF in moderate-to-strong anisotropic turbulence Following the analysis of Hufnagel and Stanley [18] and Fried [19], the long-exposure atmospheric turbulence MTF takes the form as:



1 MTFturb () = exp − Dω ( F) 2



(7)

where is the separation between points in the image plane transverse to the direction,  is the spatial frequency measured in cycles per unit length, and F is the focal length. Dω ( F) represents the wave structure function, and it is the sum of logamplitude and phase structure functions. For moderate-to-strong anisotropic non-Kolmogorov turbulence, the extended Rytov approximation theory will be adopted, and the modified anisotropic non-Kolmogorov turbulence refractive-index fluctuations spectrum is assumed to be valid in the analysis. At this time, the wave structure functions for plane and spherical waves take the forms as

L 2 2

Dωp ( , ˛, ς ) = 8 k

∞ [1 − J0 ( )] ˚n1 (, ˛, ς ) d.

dz 0

0

(8)

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L. Cui / Optik 130 (2017) 68–75

L 2 2

Dωs ( , ˛, ς ) = 8 k

∞





1 − J0  z/L

d 0



˚n1 (, ˛, ς ) d.

(9)

0

where J0 is Bessel function of the first kind and zero order and L is the optical path length. The detailed derivations for Eqs. (8) and (9) will be given as follows. First, expanding J0 in the form of Maclaurin series [20]: J0 (x) =

∞ n=0

x 2n

(−1)n · n! ·  (n + 1)

2

,

(10)

Eqs. (8) and (9) become

L 2 2

Dωp ( , ˛, ς ) = 8 k

dz

∞ ∞

0

0



 ∞ ∞

L

Dωs ( , ˛, ς ) = 82 k2

n=1

d 0

0

(−1)n−1 · n! ·  (n + 1)

(−1)n−1 · n! ·  (n + 1)

n=1

 2n 2

˚n1 (, ˛, ς ) d.

 z 2n 2L

(11)

˚n1 (, ˛, ς ) d.

(12)

Second, using the definitions of gamma function  (·), hypergeometric function 1 F 1 (·), and generalized hypergeometric function 2 F 2 (·) [20]:

∞ x−1 · e− d

 (x) =

( > 0, x > 0) ,

(13)

0

1F 1

(a; b; z) =

∞ (a)n · z n n=0

2F 2

(a, b; c, d; z) =

(b)n · n!

,

(14)

∞ (a)n (b)n · z n n=0

(15)

(c)n (d)n · n!

the analytic expressions of the wave structure function for plane and spherical waves propagating through moderate-tostrong anisotropic non-Kolmogorov turbulence are finally obtained. Dωp ( , ˛, ς ) =





× 1 − 1F 1

2 4 R(pl) (˛, ς )

ˇ1 (˛)



˛ 1− 2



k 2 X(pl) (˛, ς ) ˛ 1 − ; 1; − 2 4L



˛

X(pl)2 (˛, ς ) 1−



−





1

 ˛/2



k 2 4L

˛ − 1 2

(16)

˛    k 2 Y (pl) (˛, ς ) ˛ 2 , − Y (pl) (˛, ς ) 1 − 1 F 1 1 − ; 1; − 2 4L 1−

Dωs ( , ˛, ς ) =

× 1 − 2F 2



2 4 R(sp) (˛, ς )

ˇ2 (˛)



˛ 1− 2





˛

X(sp)2 (˛, ς ) 1−

k 2 X(sp) (˛, ς ) ˛ 1 3 1 − , ; 1, ; − 2 2 2 4L



−

1

 



(˛ − 1)  ˛/2

k 2 4L

˛ − 1 2

(17)

˛    2 ς k

(˛, ) 1 ˛ 3 Y (sp) . − Y (sp)2 (˛, ς ) 1 − 2 F 2 1 − , ; 1, ; − 2 2 2 4L 1−

2 2 R(pl) (˛, ς ) and R(sp) (˛, ς ) are the irradiance scintillation indexes for plane and spherical wave propagating through 2 2 weak anisotropic non-Kolmogorov turbulence. The expressions of R(pl) (˛, ς ), X(pl) (˛, ς ), Y (pl) (˛, ς ), ˇ1 (˛), R(sp) (˛, ς ),

X(sp) (˛, ς ), Y (sp) (˛, ς ) and ˇ2 (˛) are exhibited in Appendix A.

L. Cui / Optik 130 (2017) 68–75

71

In Eq. (13) and (14), the hypergeometric function 1 F 1 (·) and generalized hypergeometric function 2 F 2 (·) are included. They make the calculation of wave structure function very complex. Therefore, the following approximation is adopted for ease of calculation [21]:

˛ ; 1; −x 2

1 − 1F 1 1 −

1 − 2F 2



≈

⎧ ⎨  ˛

1−

˛ 1 3 1 − , ; 1, ; −x 2 2 2



x

2

 2

1+

⎩

⎧

2 − ˛ ⎨

≈

x

6



(˛ − 2)  ˛/2



⎫˛/2−2 2  ˛−4 ⎬ x

,

⎭



1+

⎩

6





(18)

⎫˛/2−2 2  ˛−4 ⎬

(˛ − 1) (˛ − 2)  ˛/2

x

⎭

,

(19)

At this time, the wave structure functions for plane and spherical wave in moderate-to-strong anisotropic nonKolmogorov turbulence can be approximately expressed with more convenient forms for calculations 2 R(pl) (˛, ς )

Dωp ( , ˛, ς ) =

⎧ ⎪ ⎪ ⎪ ⎨

ˇ1 (˛)

⎧ ⎪ ⎪ ⎨

2−˛/2

X(pl) (˛, ς )

⎪ ⎪ ⎪ ⎩ −





2−˛/2 − Y (pl)

⎧ ⎪ ⎪ ⎪ ⎨

⎪ ⎪ ⎩

⎪ ⎪ ⎪ ⎩

6

k 2 4L

1  3

1+

⎪ ⎪ ⎩

˛ − 4 k 2 X(pl)



˛ 2

(20)





 2

R(sp)

⎪ ⎪ ⎩

4L

⎪ ⎪ ⎭

⎪ ⎪ ⎪ ⎭

(˛, ς ) k 2 L

6





˛ − 4 k 2 X(sp)

k 2 4L

(˛ − 1) (˛ − 2)  ˛/2



⎪ ⎪ ⎭

˛/2−2

(21) 2−˛/2

− Y (sp) (˛, ς )

 2 

⎫˛/2−2 ⎪ ⎪ ⎬ (˛, ς )

4L

(˛ − 1) (˛ − 2)  ˛/2



,

ˇ2 (˛)

 6

⎫

 2

 

⎫

˛/2−2 ⎪ ⎪ ⎪  2 ⎪ ⎪ ⎬ ⎬ ˛ − 4 k 2 ς (˛, ) Y (pl)



1+

⎪ ⎪ ⎭

4L

(˛ − 2)  ˛/2

2−

⎫˛/2−2 ⎪ ⎪ ⎬ (˛, ς )

˛/2−2

2

1+

⎪ ⎪ ⎩

(2 − ˛) (˛ − 1)  ˛/2

⎧ ⎪ ⎪ ⎨





⎧ ⎪ ⎪ ⎨

2−˛/2

L

(˛ − 2)  ˛/2



⎧ ⎪ ⎪ ⎨

X(sp) (˛, ς )

−

2

1+

1 − ˛/2

Dωs ( , ˛, ς ) =

 k 2  2



(˛, ς )

˛ 2

2−



1

 ˛/2



˛ − 4 k 2 Y (sp) 4L

⎫˛/2−2 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎬ ς (˛, ) ⎪ ⎪ ⎭

⎪ ⎪ ⎪ ⎭

.

Substituting Eqs. (18) and (19) into Eq. (7), the analytic expressions of atmospheric turbulence MTF for plane and spherical wave propagating through moderate-to-strong anisotropic non-Kolmogorov turbulence are obtained





1 MTFturb(pl) (u, ˛, ς ) = exp − Dωp (uD, ˛, ς ) , 2



(3 < ˛ < 4) .

(22)

(3 < ˛ < 4) .

(23)



1 MTFturb(sp) (u, ˛, ς ) = exp − Dωs (uD, ˛, ς ) , 2

u = F/D denotes the normalized spatial frequency, and it is between 0 and 1. D is the receiver aperture diameter.

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L. Cui / Optik 130 (2017) 68–75

(b)

(a)

(c)

Fig. 1. Comparison between the results derived in this work and those in [13] (˛ = 10/3). (a): weak turbulence ( R2 = 0.1); (b) moderate-to-strong turbulence ( R2 = 4); (c): strong turbulence ( R2 = 50).

In the special case of weak non-Kolmogorov turbulence, the wave structure functions of plane and spherical waves (Eqs. (18) and (19)) become



4−˛

Dωp ( , ˛, ς ) = −2

2 2

 k A (˛) Cˆ n2 ς 2−˛ L





4−˛

−2

2 2

(24)

 ˛/2

 1 Dωs ( , ˛, ς ) = ˛−1

  ˛−2 ,

 1 − ˛/2

 k A (˛) · Cˆ n2 ς 2−˛ L

 1 − ˛/2





 ˛/2



 ˛−2

.

(25)

at this time, the anisotropic turbulence MTF models of plane and spherical waves derived in this work are consistent with that in [13]. 4. Numerical calculations In this section, calculations will be performed to analyze the atmospheric turbulence MTF as a function of normalized spatial frequency for optical plane and spherical waves propagating through moderate-to-strong anisotropic non-Kolmogorov turbulence. Influences of anisotropic factor, turbulence strength and general spectral power law values on the final results will be analyzed, and the parameters are set as: L = 3 km, D = 30 mm, = 0.55 ␮m. Figs. 1 and 2 show the comparisons between the results derived in this work and those in [13] which focused on the weak anisotropic turbulence and were derived with Rytov theory. Two different ˛ values of ˛ = 10/3 and ˛ = 3.8 are adopted 2 = 0.2), moderatefor theoretical analyses purpose and other values in the range 3–4 can also be chosen. The weak ( R(pl) 2 2 to-strong ( R(pl) = 4), and strong ( R(pl) = 50) turbulence strengths are analyzed, respectively. Different anisotropic factor

values of  = 1 (for isotropic turbulence), = 2,  = 3, and  = 4 are adopted for each figure. Note that these parameters are set as examples for theoretical investigations and other values can also be chosen according to the specific atmosphere turbulence scene if they can be measured by equipment.

L. Cui / Optik 130 (2017) 68–75

(a)

(b)

73

(c)

Fig. 2. Comparison between the results derived in this work and those in [13] (˛ = 3.8). (a): weak turbulence ( R2 = 0.1); (b) moderate-to-strong turbulence ( R2 = 4); (c): strong turbulence ( R2 = 50).

As shown, for weak non-Kolmogorov turbulence, the atmospheric turbulence MTF models derived in this work are basically consistent with those derived with the Rytov theory. When turbulence strength continues to increase up to 2 2 moderate-to-strong ( R(pl) = 4) or strong ( R(pl) = 50) atmosphere turbulence regime, the anisotropic turbulence produces less effects on the optical imaging system compared with those derived with Rytov theory. That is because with the increased turbulence strength, the multiple scattering of optical wave, which causes the optical wave to become increasingly less coherent as propagates through the atmospheric turbulence, weaken the turbulence effects. To quantitatively compare the discrepancy between the MTF models derived in this work and those in [13], the maximum percentage differences between our results and those in [13] are calculated and listed in Table 1. For weak turbulence, the maximum percentage differences are less than 1%, and they increase rapidly for moderate-to-strong and strong turbulence. From Table 1, it can also be seen that with the increase of anisotropic factor, the maximum percentage difference becomes smaller and smaller especially for the spherical wave. This phenomenon can be explained from this physical point of view: when the anisotropic factor increases, the anisotropic turbulence cells or eddies act as lenses with a higher radius of curvature, and they will make the focusing effects of turbulence cells or eddies on the optical waves’ propagation exhibit more and more obvious [17]. That is, when an optical beam propagates along the short axis of anisotropic turbulence cells or eddies, it will be less deviated from the direction of propagation. At this time, the multiple scattering effects which are produced by the anisotropic turbulence cells or eddies and alleviate the coherence property of optical wave will become weaker and weaker. The anisotropic turbulence with a higher anisotropic factor will produce more effects on the optical waves’ propagation through moderate-to-strong and strong turbulence compared with the case with a lower anisotropic factor. Therefore, the results in this work will be closer to those obtained with Rytov theory with the increased anisotropic factor values. 5. Conclusions and discussions In this study, based on the extended Rytov approximation theory and the modified anisotropic non-Kolmogorov turbulence spectrum, new analytic expressions for the turbulence MTF models have been derived for optical plane and spherical

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L. Cui / Optik 130 (2017) 68–75

Table 1 The maximum percentage difference between the results derived in this work and those in [13]. ˛ = 10/3

˛ = 3.8

Plane wave

Spherical wave

Plane wave

Spherical wave

R2 = 0.1

   

=1 =2 =3 =4

0.73% 0.27% 0.15% 0.10%

0.16% (6.05e-2)% (3.50e-2)% (2.38e-2)%

0.90% 0.14% (5.76e-2)% (3.20e-2)%

(4.75e-2)% (5.62e-3)% (1.98e-3)% (1.03e-3)%

R2 = 4

   

=1 =2 =3 =4

93.67% 44.14% 19.28% 10.05%

38.32% 8.04% 3.05% 1.62%

98.88% 47.98% 16.87% 7.03%

45.00% 5.67% 1.36% 0.49%

R2 = 50

   

=1 =2 =3 =4

101% 100.5% 100.1% 99.93%

101% 99.87% 94.76% 79.53%

101% 100% 99.99% 99.08%

101% 99.06% 77.88% 46.77%

waves propagating through moderate-to-strong (or strong) anisotropic non-Kolmogorov turbulence. The anisotropic factor, the general spectral power law value, and a more wide range of turbulence strength (weak, moderate-to-strong, and strong turbulence) have been considered. Calculations show that in the weak anisotropic turbulence regime, the results derived in this work have good consistency with those in [13] which were focused on the weak anisotropic non-Kolmogorov turbulence and derived with classic Rytov approximation theory. With the increase of turbulence strength, the discrepancy between the MTF models obtained in this work and those in [13] appears and exhibits more and more obvious. In the investigations, the circular symmetry assumption of turbulence cells or eddies in the orthogonal xy-plane was adopted for theoretical derivation convenience purpose. In fact, the asymmetry property and the finite turbulence inner and outer scales should be considered theoretically. They will be investigated in the future work. Acknowledgments This work is partly supported by the National Natural Science Foundation of China (61405004), the China Scholarship Council (201506025046), and the Aeronautical Science Foundation of China (20150151001). Appendix A. For plane and spherical waves, X (˛, ς ) and Y (˛, ς ) separately take the forms as [16]: 2 X(pl) (˛, ς ) =

k

(˛, ς ) , L X(pl)

Y2 (pl) (˛, ς ) =

k

Y (pl) (˛, ς ) L

(A.1)

2 X(sp) (˛, ς ) =

k

X(sp) (˛, ς ) , L

Y2 (sp) (˛, ς ) =

k

Y (sp) (˛, ς ) L

(A.2)

X(pl) (˛, ς ), Y (pl) (˛, ς ), X(sp) (˛, ς ) and Y (sp) (˛, ς ) in Eqs. (A.1) and (A.2) can be expressed as

 1.47ˇ1 (˛)



X(pl) (˛, ς ) =

2  6−˛



2 3 − ˛/2



Y (pl) (˛, ς ) =

0.51 (˛ − 2) ˇ1 (˛) 8

 7.35ˇ2 (˛)



X(sp) (˛, ς ) =



Y (sp) (˛, ς ) =

2

2−˛



0.51 (˛ − 2) ˇ2 (˛) 8

,

(A.3)



4

˛−2 1 + fY (pl) (˛) R(pl) (˛, ς ) ,

2  6−˛

 3 − ˛/2

−1

4

˛−2 1 + fX(pl) (˛) R(pl) (˛, ς )

−1

4 ˛−2

1 + fX(sp) (˛) R(sp) (˛, ς ) 2

2−˛

(A.4)

4

,

(A.5)



˛−2 1 + fY (sp) (˛) R(sp) (˛, ς ) ,

(A.6)

2 2 where, fX(pl) (˛), fY (pl) (˛), fX(sp) (˛), fY (sp) (˛), R(pl) (˛, ς ), R(sp) (˛, ς ), ˇ1 (˛) and ˇ2 (˛)in Eqs. (A.3)–(A.6) are given as

fX(pl) (˛) =

r

1

(˛) I1 (˛) 0.98

2  ˛−6

, fY (pl) (˛) =

2

ln 2  2−˛

0.51

.

(A.7)

L. Cui / Optik 130 (2017) 68–75

fX(sp) (˛) =

r1 (˛) =

r2 (˛) =

I1 (˛) =

2

r

2

(˛) I2 (˛) 0.98

(3−˛)(˛−10) ˛−2



2

(3−˛)(˛−10) ˛−2

˛−2 2F 1

 6−˛ ˛−2

, fY (sp) (˛) =



−

˛−2

2  ˛−6



 1 − ˛/2



 ˛/2



−



 ˛/2

, ˛ − 3; ˛ − 2; ˛−3

0.51

  ˛−6 ˛−2



 1 − ˛/2

2

ln 2  2−˛



  ˛−6 ˛−2



˛−2 ˛−1



.

(A.8)

6 − ˛ ˛−2

ˇ1 (˛)

6 − ˛ ˛−2



75

ˇ2 (˛)

 8−2˛ ˛−2

,

(A.9)

 8−2˛ ˛−2

,

(A.10)

6−˛

, I2 (˛) =

(˛ − 1) ˛−2  2 (˛ − 3) .  (2˛ − 6)

2 R(pl) (˛, ς ) = ˇ1 (˛) A (˛) Cˆ n2 ς 2−˛ 2 k3−˛/2 L˛/2 , ˇ1 (˛) = 4

˛

2 R(sp) (˛, ς ) = ˇ2 (˛) A (˛) Cˆ n2 ς 2−˛ 2 k3−˛/2 L˛/2 , ˇ2 (˛) = −4

−

2 1−

(A.11)

sin ˛ 2

˛  4

 sin

.

˛   4

(A.12)

 2



˛/2

 (˛)

.

(A.13)

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