Abstract

Standard FFT-based turbulent phase screen generation method has very large errors due to the undersampling of the low frequency components. Subharmonic methods are the main low frequency components compensating methods to improve the accuracy, but the residual errors are still large. In this paper I propose a new low frequency components compensating method, which is based on the correlation matrix phase screen generation methods. Using this method, the low frequency components can be compensated accurately, both of the accuracy and speed are superior to those of the subharmonic methods.

© 2011 OSA

1. Introduction

The simulation of Kolmogorov turbulent phase screen is important in the study of light propagation though the turbulent atmosphere. Several methods are available for generating this kind of phase screen, e.g., fast Fourier transformed (FFT) based method [16], Zernike polynomials method [7] and correlation matrix methods [810]. The most popular method among them is the FFT-based method proposed by McGlamery [1]. This method has the advantages of much faster speed and smaller memory requirements than other methods, is very suitable for generating large phase screen. But this method suffers from a well-known drawback of the undersampling of the low spatial frequency components.

The main solutions for compensating the low frequency components are subharmonic methods. Over the past two decades, several subharmonic methods have been proposed to improve the simulation accuracy [26], but the residual errors are still large. Because in the processes of generating the initial FFT-based phase screen and the subharmonic phase screen, the turbulence power spectrum is discretely sampled, then the simulated phase screen has the ladder-like power spectrum, which does not agree with the theoretical continuous power spectrum. This disagreement in the low frequency region leads to larger errors.

In this paper I propose a new low frequency components compensating method, which is based on the correlation matrix methods instead of subharmonic methods.

2. FFT-based turbulent phase screen

The standard FFT-based phase screen can be generated by the following equations [4,5]

φFFT(m,n)=m'=Nx/2Nx/21n'=Ny/2Ny/21h(m',n')Φn(m',n')exp[i2π(m'mNx+n'nNy)]
Φn(m',n')=0.490r05/3[(m'Δkx)2+(n'Δky)2+(2π/L0)2]11/6ΔkxΔky
where ϕFFT is the FFT-based phase screen, m,m'=Nx/2,,Nx/21 and n,n'=Ny/2, ,Ny/21 are integer indices, Δkx=2π/Dx and Δky=2π/Dy are the sampling intervals in power spectrum domain, Dx and Dyare the FFT-based phase screen sizes in the x and y directions each withNxand Ny points, respectively,Φn(m',n')is the discrete turbulence power spectrum,h(m',n')is the Hermitian complex Gaussian noise with zero-mean and unit-variance, L0 is the outer scale of turbulence, r0 is the Fried parameter. The phase screen is assumed to be sampled at a constant spatial intervalΔ = Dx /Nx = Dy /Ny. Equation (1) can be implemented by means of FFT.

In the standard FFT-based methods, the zero frequency term at the origin of Φn(m',n')is set to zero, i.e.,Φn(0,0)=0, to remove the direct component of the phase screen, whereas in my method, in order to accurately generate the low frequency compensating phase screen, more low frequency terms in the Nzx×Nzy rectangular region centered at the origin of Φn(m',n')should be set to zero

Φn(m',n')=0for|m'|(Nzx1)/2and|n'|(Nzy1)/2
where Nzx and Nzy are the numbers of points being set to zero in the x and y directions, respectively, and only odd numbers are allowable. For square phase screen, I suggest

Nzx=Nzy=3

For rectangular phase screen (Nx /Ny = 2, 4, 8, 16,), I suggest

{Nzx=3Nx/Ny+1Nzy=3

After setting more low frequency terms of Φn(m',n')to zero, the FFT-based phase screen has a greater deficiency of low frequency components than the standard FFT-based phase screen, but those low frequency components will be contained within the compensating phase screen automatically.

The FFT-based phase screen has several drawbacks. The first is the undersampling of the low spatial frequency components. For a given turbulent outer scale, the loss degree of the low frequency components decreases with the increasing of the screen size. The second drawback is the anisotropy of the FFT-based phase screen. Not to mention the rectangular screen, even for the square screen, the loss degree of the low frequency components in the diagonal direction is less than that in the x or y direction, naturally, the statistical properties of the FFT-based phase screen are anisotropic. The third drawback is the periodicity of the phase screen. This is due to the periodicity of the FFT algorithm, the periods are Dx and Dy in the x and y directions, respectively. To eliminate the influence of the periodicity, the size of the FFT-based phase screen should be much larger than the required size.

The following steps can accurately compensate the low frequency components and the anisotropy in the region of 1/2×1/2of the initial FFT-based phase screen.

3. Compensation of the low frequency components

Let ϕlow denote the low frequency compensating phase screen, then the final compensated phase screen can be represented by

ϕ=ϕFFT+ϕlow

It is reasonable to assume ϕFFTand ϕlow are mutually independent random variables with zero-mean, thus the autocorrelation function of the final compensated phase screen is given by

Bϕ(r)=<ϕ(s1)ϕ(s2)>=<[ϕFFT(s1)+ϕlow(s1)][ϕFFT(s2)+ϕlow(s2)]>=BϕFFT(r)+Bϕlow(r)
where BϕFFT(r) is the autocorrelation function of the FFT-based phase screen, Bϕlow(r) is the autocorrelation function of the low frequency compensating phase screen, r=|s1s2|is the separation distance between the two points s1 and s2, <>denotes ensemble average.

The theoretical phase autocorrelation function is given by [10]

Bϕ(r)=(L0/r0)5/3Γ(11/6)25/6π8/3[245Γ(6/5)]5/6(2πrL0)5/6K5/6(2πrL0)
whereK5/6()is the modified Bessel function of the third kind, and Γ()is the gamma function.

The autocorrelation function of the FFT-based phase screen is given by [4]

BϕFFT(m2+n2Δ)=m'=Nx/2Nx/21n'=Ny/2Ny/21Φn(m',n')exp[i2π(m'mNx+n'nNy)]

Here the low frequency terms of Φn(m',n'),|m'|(Nzx1)/2and|n'|(Nzy1)/2, are also set to zero to be consistent with that in Eq. (1), and Eq. (9) can be calculated using FFT.

Then the autocorrelation function of ϕlow is obtained

Bϕlow(m2+n2Δ)=Bϕ(m2+n2Δ)BϕFFT(m2+n2Δ)

The low frequency compensating phase screen ϕlowis mainly composed of the low frequency components that are missing in the FFT-based phase screen, and the high frequency components are very little, so ϕlow can be obtained from a low resolution phase screen ϕlow,low through an interpolation operation.

In the x and y directions of the low resolution phase screen ϕlow,low, assuming the spatial sampling intervals are both qΔ, where q is an integer power of two, the numbers of points are Nlx=Nx/(2q)+1 andNly=Ny/(2q)+1, the sizes areDlx=Dx/2+qΔ andDly=Dy/2+qΔ, respectively. Then the autocorrelation function of the low resolution phase screen is given by

Bφlow,low[(m11)Nly+n1,(m21)Nly+n2]=Bφlow[(m1m2)2+(n1n2)2qΔ]
where m1,m2=1,2,,Nlxandn1,n2=1,2,,Nly,(m1,n1)and (m2,n2)are any two points in the phase screenϕlow,low. The dimensions of Bϕlow,lowareNlxNly×NlxNly.

The low resolution phase screen ϕlow,lowwith autocorrelation Bϕlow,lowcan be generated by the correlation matrix method [8,9], which is described below

ψlow,low=URe(L)X
whereUdenotes a matrix whose columns Ujare the orthonormal eigenvectors of matrix Bφlow,low,Lis a diagonal matrix whose diagonal elementsλjare the eigenvalues corresponding to the eigenvectors Uj, Uand L are obtained by singular value decomposition operation of Bφlow,low, Xis a column vector whose elements are independent Gaussian random variables with zero-mean and unit-variance, Re()denotes the real part.ψlow,lowgiven by Eq. (12) is a column vector of length NlxNly, The two-dimensional low resolution phase screen ϕlow,lowcan be obtained by rearranging ψlow,lowinto a two-dimensional matrix of dimensionsNlx×Nly.

Note that part of eigenvaluesλj may be negative, soRe()is used to avoid the phase screen φlow,low being complex values. This operation will induce some simulation errors.

The phase autocorrelation function of any two points (m1,n1)and (m2,n2) in the actual generated low resolution phase screen is given by

Bϕlow,lowactual(p1,p2)=<ψlow,low(p1)ψlow,lowT(p2)>=j=1NlxNlyU(p1,j)[Re(λj)]2U(j,p2)
wherep1=(m11)Nly+n1,p2=(m21)Nly+n2, the superscript T denotes matrix transpose.

Phase structure function is often used to evaluate the simulation accuracy. The phase structure function of the actual generated low resolution phase screen is given by

Dϕlow,lowactual(p1,p2)=<[ϕlow,low(m1,n1)ϕlow,low(m2,n2)]2>=Bϕlow,lowactual(p1,p1)+Bϕlow,lowactual(p2,p2)2Bϕlow,lowactual(p1,p2)

The expected structure function of the low resolution phase screen is given by

Dϕlow,lowexp(p1,p2)=2[Bϕlow,low(1,1)Bϕlow,low(p1,p2)]

The expected structure function of the final compensated phase screen is given by [10]

Dϕexp(r)=(L0r0)5/321/6Γ(11/6)π8/3[245Γ(65)]5/6[Γ(6/5)21/6(2πrL0)5/6K5/6(2πrL0)]

The error induced by the negative eigenvalues of matrix Bφlow,lowcan be defined as

ERRSVD=max(|Dϕlow,lowactualDϕlow,lowexp|/Dϕexp)
where max(.) denotes the maximum value in the errors matrix.

This kind of errors is shown in Fig. 1 . For square screen, if the initial FFT-based phase screen is generated by the standard FFT-based method, i.e. Nzx = Nzy = 1, then the maximum error is about 6%. By setting more low frequency terms of Φn(m',n') to zero, the negative eigenvalues can be eliminated gradually, and the errors decrease correspondingly, when Nzx = Nzy = 3, the errors are less than 0.1%. For rectangular screen, when Nzx=3Nx/Ny+1 and Nzy=3, the errors are less than 0.2%. The parameters Nx, Ny, Nlx and Nly also have some influence on the errors, with the increasing of Nx, Ny, Nlx and Nly, the errors increase slightly.

 

Fig. 1 Errors induced by the negative eigenvalues of matrix Blow,low as a function of L0/Dy.

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4. Simulation results

In the following simulation, r0 is 0.2 m, interpolation method is spline. Spline interpolation has both higher accuracy and faster speed than cubic interpolation on my computer. Figure 2 shows the phase structure functions of the simulated phase screen. The phase structure functions of the final compensated phase screen agree very well with the theoretical phase structure functions, cannot distinguish them by naked eyes. The anisotropy of the FFT-based phase screen, which also exists in the standard FFT-based phase screen, is also eliminated.

 

Fig. 2 Phase structure functions of the final compensated phase screen, the initial FFT-based phase screen and the low frequency compensating phase screen. Square screen, Dx = Dy = 1 m, Nx = Ny = 256, Nlx = Nly = 9, Nzx = Nzy = 3, L0 = 3 m, averaged over 105 screens.

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The simulation errors are shown in Fig. 3a for square screen. When Nl = Nlx = Nly = 9 and Nz = Nzx = Nzy = 3, the maximum relative error is on the order of 0.1% in the low frequency region (large spatial distance r), whereas that of the weighted subharmonic method is about 1% [5]. The low frequency errors can be further reduced easily by increasing Nz or Nl. Increasing Nz, the errors induced by the negative eigenvalues of matrix Blow,low will decrease, Nz = 5 is adequate to completely eliminate the negative eigenvalues of matrix Blow,low. Increasing Nl, the interpolation errors will decrease. In order to avoid large interpolation errors, the condition of Nl4(Nz1)+1 should be satisfied. In the high frequency region, larger relative errors occur due to the undersampling of the high frequency components. This is less important because the energy loss of high frequency is very little.

 

Fig. 3 Relative errors of the phase structure functions on the edge line of the final compensated phase screen. (a) Square screen, Dx = Dy = 1 m, Nx = Ny = 256 and Nzx = Nzy = 3, averaged over 5×106 screens. (b) Rectangular screen. Solid line: Dx = 32 m, Dy = 1 m, Nx = 2048, Ny = 64, Nlx = 257, Nly = 9, Nzx = 97, Nzy = 3. Dashed line: Dx = 128 m, Dy = 1 m, Nx = 8192, Ny = 64, Nlx = 513, Nly = 5, Nzx = 385, Nzy = 3, averaged over 106 screens.

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For rectangular screen (Nx > Ny ), the conditions of Nlx8Nx/Ny+1 and Nly9 should be satisfied as for as possible. Under the current personal computer capacity, φlow,lowwith 257×9 points can be generated easily, then rectangular screen with aspect ratio of 32can be obtained, and the low frequency errors are about 0.2%, slightly larger than those of the square screen, as shown in Fig. 3b. If the number of points in ϕlow,lowis 513×5, then rectangular screen with aspect ratio of 128 can be obtained, but the low frequency errors increase to the order of 1%. These larger errors are dominanted by the interpolation errors, and the errors in the x direction are larger than those in the y direction. By decreasing Nzx, the interpolation errors in the x direction will decrease, but at the same time, the errors induced by the negative eigenvalues of matrix Blow,lowwill increase. There exists an optimum Nzx to minimize the total errors.

The time of generating one compensated screen with Nx = Ny = 1024 and Nlx = Nly = 9 was about one second on my lenovo computer running Matlab R2010a with 2.0GHz Pentium dual core processor and 2GB memory. Typically 34.6% of the time was spent for the preparing work of the initial FFT-based screen, 30.8% for an FFT, 18.9% for the preparing work of the compensating phase screen, and 15.7% for a spline interpolation. Take no account of the preparing time, the execution time increases about 51% compared to the standard FFT-based phase screen, whereas that of the subharmonic methods is larger than 200% [5].

5. Conclusion

A new method is proposed to compensate the low frequency components of the FFT-based turbulent phase screen. Using this method, the low frequency components in the region of 1/2×1/2of the initial FFT-based phase screen can be compensated perfectly for both square and rectangular phase screens. Compared to the subharmonic compensating methods, this method has higher accuracy, faster speed and lower complexity. For the square screen, the maximum low frequency error is on the order of 0.1%, about an order of magnitude less than that of the weighted subharmonic method, and it can be further reduced easily. The execution time of this low frequency compensation, which is dominanted by the time of the spline interpolation, is about 1/4 that of the subharmonic methods. Finally, this low frequency compensation method is simple and easy to understand, whereas in the weighted subharmonic methods, the calculation of the subharmonic weight coefficients is a troublesome work.

Acknowledgments

This research was supported by Chongqing Natural Science Foundation Project of CSTC2008BB2414 and Chongqing Municipal Education Commission Science and Technology Project of KJ080513.

References

1. B. L. McGlamery, “Computer simulation studies of compensation of turbulence degraded images,” Proc. SPIE 74, 225–233 (1976).

2. B. J. Herman and L. A. Strugala, “Method for inclusion of low-frequency contributions in numerical representation of atmospheric turbulence,” Proc. SPIE 1221, 183–192 (1990). [CrossRef]  

3. R. G. Lane, A. Glindemann, and J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2(3), 209–224 (1992). [CrossRef]  

4. E. M. Johansson and D. T. Gavel, “Simulation of stellar speckle imaging,” Proc. SPIE 2200, 372–383 (1994). [CrossRef]  

5. G. Sedmak, “Implementation of fast-Fourier-transform-based simulations of extra-large atmospheric phase and scintillation screens,” Appl. Opt. 43(23), 4527–4538 (2004). [CrossRef]   [PubMed]  

6. J. Recolons and F. Dios, “Accurate calculation of phase screens for the modelling of laser beam propagation through atmospheric turbulence,” Proc. SPIE 5891, 589107, 589107-12 (2005). [CrossRef]  

7. N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29(10), 1174–1180 (1990). [CrossRef]  

8. K. A. Winick, “Atmospheric turbulence-induced signal fades on optical heterodyne communication links,” Appl. Opt. 25(11), 1817–1825 (1986). [CrossRef]   [PubMed]  

9. C. M. Harding, R. A. Johnston, and R. G. Lane, “Fast simulation of a kolmogorov phase screen,” Appl. Opt. 38(11), 2161–2170 (1999). [CrossRef]   [PubMed]  

10. F. Assémat, R. W. Wilson, and E. Gendron, “Method for simulating infinitely long and non stationary phase screens with optimized memory storage,” Opt. Express 14(3), 988–999 (2006). [CrossRef]  

References

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  1. B. L. McGlamery, “Computer simulation studies of compensation of turbulence degraded images,” Proc. SPIE 74, 225–233 (1976).
  2. B. J. Herman and L. A. Strugala, “Method for inclusion of low-frequency contributions in numerical representation of atmospheric turbulence,” Proc. SPIE 1221, 183–192 (1990).
    [CrossRef]
  3. R. G. Lane, A. Glindemann, and J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2(3), 209–224 (1992).
    [CrossRef]
  4. E. M. Johansson and D. T. Gavel, “Simulation of stellar speckle imaging,” Proc. SPIE 2200, 372–383 (1994).
    [CrossRef]
  5. G. Sedmak, “Implementation of fast-Fourier-transform-based simulations of extra-large atmospheric phase and scintillation screens,” Appl. Opt. 43(23), 4527–4538 (2004).
    [CrossRef] [PubMed]
  6. J. Recolons and F. Dios, “Accurate calculation of phase screens for the modelling of laser beam propagation through atmospheric turbulence,” Proc. SPIE 5891, 589107, 589107-12 (2005).
    [CrossRef]
  7. N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29(10), 1174–1180 (1990).
    [CrossRef]
  8. K. A. Winick, “Atmospheric turbulence-induced signal fades on optical heterodyne communication links,” Appl. Opt. 25(11), 1817–1825 (1986).
    [CrossRef] [PubMed]
  9. C. M. Harding, R. A. Johnston, and R. G. Lane, “Fast simulation of a kolmogorov phase screen,” Appl. Opt. 38(11), 2161–2170 (1999).
    [CrossRef] [PubMed]
  10. F. Assémat, R. W. Wilson, and E. Gendron, “Method for simulating infinitely long and non stationary phase screens with optimized memory storage,” Opt. Express 14(3), 988–999 (2006).
    [CrossRef]

2006 (1)

2005 (1)

J. Recolons and F. Dios, “Accurate calculation of phase screens for the modelling of laser beam propagation through atmospheric turbulence,” Proc. SPIE 5891, 589107, 589107-12 (2005).
[CrossRef]

2004 (1)

1999 (1)

1994 (1)

E. M. Johansson and D. T. Gavel, “Simulation of stellar speckle imaging,” Proc. SPIE 2200, 372–383 (1994).
[CrossRef]

1992 (1)

R. G. Lane, A. Glindemann, and J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2(3), 209–224 (1992).
[CrossRef]

1990 (2)

B. J. Herman and L. A. Strugala, “Method for inclusion of low-frequency contributions in numerical representation of atmospheric turbulence,” Proc. SPIE 1221, 183–192 (1990).
[CrossRef]

N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29(10), 1174–1180 (1990).
[CrossRef]

1986 (1)

1976 (1)

B. L. McGlamery, “Computer simulation studies of compensation of turbulence degraded images,” Proc. SPIE 74, 225–233 (1976).

Assémat, F.

Dainty, J. C.

R. G. Lane, A. Glindemann, and J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2(3), 209–224 (1992).
[CrossRef]

Dios, F.

J. Recolons and F. Dios, “Accurate calculation of phase screens for the modelling of laser beam propagation through atmospheric turbulence,” Proc. SPIE 5891, 589107, 589107-12 (2005).
[CrossRef]

Gavel, D. T.

E. M. Johansson and D. T. Gavel, “Simulation of stellar speckle imaging,” Proc. SPIE 2200, 372–383 (1994).
[CrossRef]

Gendron, E.

Glindemann, A.

R. G. Lane, A. Glindemann, and J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2(3), 209–224 (1992).
[CrossRef]

Harding, C. M.

Herman, B. J.

B. J. Herman and L. A. Strugala, “Method for inclusion of low-frequency contributions in numerical representation of atmospheric turbulence,” Proc. SPIE 1221, 183–192 (1990).
[CrossRef]

Johansson, E. M.

E. M. Johansson and D. T. Gavel, “Simulation of stellar speckle imaging,” Proc. SPIE 2200, 372–383 (1994).
[CrossRef]

Johnston, R. A.

Lane, R. G.

C. M. Harding, R. A. Johnston, and R. G. Lane, “Fast simulation of a kolmogorov phase screen,” Appl. Opt. 38(11), 2161–2170 (1999).
[CrossRef] [PubMed]

R. G. Lane, A. Glindemann, and J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2(3), 209–224 (1992).
[CrossRef]

McGlamery, B. L.

B. L. McGlamery, “Computer simulation studies of compensation of turbulence degraded images,” Proc. SPIE 74, 225–233 (1976).

Recolons, J.

J. Recolons and F. Dios, “Accurate calculation of phase screens for the modelling of laser beam propagation through atmospheric turbulence,” Proc. SPIE 5891, 589107, 589107-12 (2005).
[CrossRef]

Roddier, N.

N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29(10), 1174–1180 (1990).
[CrossRef]

Sedmak, G.

Strugala, L. A.

B. J. Herman and L. A. Strugala, “Method for inclusion of low-frequency contributions in numerical representation of atmospheric turbulence,” Proc. SPIE 1221, 183–192 (1990).
[CrossRef]

Wilson, R. W.

Winick, K. A.

Appl. Opt. (3)

Opt. Eng. (1)

N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29(10), 1174–1180 (1990).
[CrossRef]

Opt. Express (1)

Proc. SPIE (4)

E. M. Johansson and D. T. Gavel, “Simulation of stellar speckle imaging,” Proc. SPIE 2200, 372–383 (1994).
[CrossRef]

J. Recolons and F. Dios, “Accurate calculation of phase screens for the modelling of laser beam propagation through atmospheric turbulence,” Proc. SPIE 5891, 589107, 589107-12 (2005).
[CrossRef]

B. L. McGlamery, “Computer simulation studies of compensation of turbulence degraded images,” Proc. SPIE 74, 225–233 (1976).

B. J. Herman and L. A. Strugala, “Method for inclusion of low-frequency contributions in numerical representation of atmospheric turbulence,” Proc. SPIE 1221, 183–192 (1990).
[CrossRef]

Waves Random Media (1)

R. G. Lane, A. Glindemann, and J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2(3), 209–224 (1992).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Errors induced by the negative eigenvalues of matrix Blow,low as a function of L0/Dy.

Fig. 2
Fig. 2

Phase structure functions of the final compensated phase screen, the initial FFT-based phase screen and the low frequency compensating phase screen. Square screen, Dx = Dy = 1 m, Nx = Ny = 256, Nlx = Nly = 9, Nzx = Nzy = 3, L0 = 3 m, averaged over 105 screens.

Fig. 3
Fig. 3

Relative errors of the phase structure functions on the edge line of the final compensated phase screen. (a) Square screen, Dx = Dy = 1 m, Nx = Ny = 256 and Nzx = Nzy = 3, averaged over 5 × 106 screens. (b) Rectangular screen. Solid line: Dx = 32 m, Dy = 1 m, Nx = 2048, Ny = 64, Nlx = 257, Nly = 9, Nzx = 97, Nzy = 3. Dashed line: Dx = 128 m, Dy = 1 m, Nx = 8192, Ny = 64, Nlx = 513, Nly = 5, Nzx = 385, Nzy = 3, averaged over 106 screens.

Equations (17)

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φ FFT (m,n)= m ' = N x /2 N x /21 n ' = N y /2 N y /21 h( m ' , n ' ) Φ n ( m ' , n ' ) exp[i2π( m ' m N x + n ' n N y )]
Φ n ( m ' , n ' )=0.490 r 0 5/3 [ ( m ' Δ k x ) 2 + ( n ' Δ k y ) 2 + (2π/ L 0 ) 2 ] 11/6 Δ k x Δ k y
Φ n ( m ' , n ' )=0for| m ' |( N zx 1)/2and| n ' |( N zy 1)/2
N zx = N zy =3
{ N zx =3 N x / N y +1 N zy =3
ϕ= ϕ FFT + ϕ low
B ϕ (r)=<ϕ( s 1 )ϕ( s 2 )>=<[ ϕ FFT ( s 1 )+ ϕ low ( s 1 )][ ϕ FFT ( s 2 )+ ϕ low ( s 2 )]> = B ϕ FFT (r)+ B ϕ low (r)
B ϕ (r)= ( L 0 / r 0 ) 5/3 Γ(11/6) 2 5/6 π 8/3 [ 24 5 Γ(6/5)] 5/6 ( 2πr L 0 ) 5/6 K 5/6 ( 2πr L 0 )
B ϕ FFT ( m 2 + n 2 Δ)= m ' = N x /2 N x /21 n ' = N y /2 N y /21 Φ n ( m ' , n ' ) exp[i2π( m ' m N x + n ' n N y )]
B ϕ low ( m 2 + n 2 Δ)= B ϕ ( m 2 + n 2 Δ) B ϕ FFT ( m 2 + n 2 Δ)
B φ low,low [( m 1 1) N ly + n 1 ,( m 2 1) N ly + n 2 ]= B φ low [ ( m 1 m 2 ) 2 + ( n 1 n 2 ) 2 qΔ]
ψ low,low =URe( L )X
B ϕ low,low actual ( p 1 , p 2 )=< ψ low,low ( p 1 ) ψ low,low T ( p 2 )>= j=1 N lx N ly U( p 1 ,j) [Re( λ j )] 2 U(j, p 2 )
D ϕ low,low actual ( p 1 , p 2 )=< [ ϕ low,low ( m 1 , n 1 ) ϕ low,low ( m 2 , n 2 )] 2 > = B ϕ low,low actual ( p 1 , p 1 )+ B ϕ low,low actual ( p 2 , p 2 )2 B ϕ low,low actual ( p 1 , p 2 )
D ϕ low,low exp ( p 1 , p 2 )=2[ B ϕ low,low (1,1) B ϕ low,low ( p 1 , p 2 )]
D ϕ exp (r)= ( L 0 r 0 ) 5/3 2 1/6 Γ(11/6) π 8/3 [ 24 5 Γ( 6 5 ) ] 5/6 [ Γ(6/5) 2 1/6 ( 2πr L 0 ) 5/6 K 5/6 ( 2πr L 0 ) ]
ERRSVD=max(| D ϕ low,low actual D ϕ low,low exp |/ D ϕ exp )

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