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 [1–6], Zernike polynomials method [7] and correlation matrix methods [8–10]. 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 [2–6], 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]

In the standard FFT-based methods, the zero frequency term at the origin of ${\Phi }_n(m^{\text{'}},n^{\text{'}})$is set to zero, i.e.,${\Phi }_n(0,0)=0$, 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 N_zx$\times$N_zy rectangular region centered at the origin of ${\Phi }_n(m^{\text{'}},n^{\text{'}})$should be set to zero

For rectangular phase screen (N_x /N_y = 2, 4, 8, 16,$\cdots$), I suggest

After setting more low frequency terms of ${\Phi }_n(m^{\text{'}},n^{\text{'}})$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 D_x and D_y 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 $1/2\times 1/2$of the initial FFT-based phase screen.

3. Compensation of the low frequency components

Let ${\varphi }_{low}$ denote the low frequency compensating phase screen, then the final compensated phase screen can be represented by

It is reasonable to assume ${\varphi }_{FFT}$and ${\varphi }_{low}$ are mutually independent random variables with zero-mean, thus the autocorrelation function of the final compensated phase screen is given by

The theoretical phase autocorrelation function is given by [10]

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

Here the low frequency terms of ${\Phi }_n(m^{\text{'}},n^{\text{'}})$,$\left|m^{\text{'}}\right|\le (N_{zx}-1)/2$and$\left|n^{\text{'}}\right|\le (N_{zy}-1)/2$, 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 ${\varphi }_{low}$ is obtained

The low frequency compensating phase screen ${\varphi }_{low}$is 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 ${\varphi }_{low}$ can be obtained from a low resolution phase screen ${\varphi }_{low,low}$ through an interpolation operation.

In the x and y directions of the low resolution phase screen ${\varphi }_{low,low}$, assuming the spatial sampling intervals are both $q\Delta$, where q is an integer power of two, the numbers of points are $N_{lx}=N_x/(2q)+1$ and$N_{ly}=N_y/(2q)+1$, the sizes are$D_{lx}=D_x/2+q\Delta$ and$D_{ly}=D_y/2+q\Delta$, respectively. Then the autocorrelation function of the low resolution phase screen is given by

The low resolution phase screen ${\varphi }_{low,low}$with autocorrelation $B_{{\varphi }_{low,low}}$can be generated by the correlation matrix method [8,9], which is described below

Note that part of eigenvalues${\lambda }_j$ may be negative, so$\text{Re(}\cdot \text{)}$is used to avoid the phase screen ${\phi }_{low,low}$ being complex values. This operation will induce some simulation errors.

The phase autocorrelation function of any two points $(m_1,n_1)$and $(m_2,n_2)$ in the actual generated low resolution phase screen is given by

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

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

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

The error induced by the negative eigenvalues of matrix $B_{{\phi }_{low,low}}$can be defined as

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. N_zx = N_zy = 1, then the maximum error is about 6%. By setting more low frequency terms of ${\Phi }_n(m^{\text{'}},n^{\text{'}})$ to zero, the negative eigenvalues can be eliminated gradually, and the errors decrease correspondingly, when N_zx = N_zy = 3, the errors are less than 0.1%. For rectangular screen, when $N_{zx}=3N_x/N_y+1$ and $N_{zy}=3$, the errors are less than 0.2%. The parameters N_x, N_y, N_lx and N_ly also have some influence on the errors, with the increasing of N_x, N_y, N_lx and N_ly, the errors increase slightly.

 

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

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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, D_x = D_y = 1 m, N_x = N_y = 256, N_lx = N_ly = 9, N_zx = N_zy = 3, L₀ = 3 m, averaged over 10⁵ screens.

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The simulation errors are shown in Fig. 3a for square screen. When N_l = N_lx = N_ly = 9 and N_z = N_zx = N_zy = 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 N_z or N_l. Increasing N_z, the errors induced by the negative eigenvalues of matrix B_low,low will decrease, N_z = 5 is adequate to completely eliminate the negative eigenvalues of matrix B_low,low. Increasing N_l, the interpolation errors will decrease. In order to avoid large interpolation errors, the condition of $N_l\ge 4(N_z-1)+1$ 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, D_x = D_y = 1 m, N_x = N_y = 256 and N_zx = N_zy = 3, averaged over 5$\times$10⁶ screens. (b) Rectangular screen. Solid line: D_x = 32 m, D_y = 1 m, N_x = 2048_, N_y = 64, N_lx = 257, N_ly = 9, N_zx = 97, N_zy = 3. Dashed line: D_x = 128 m, D_y = 1 m, N_x = 8192_, N_y = 64, N_lx = 513, N_ly = 5, N_zx = 385, N_zy = 3, averaged over 10⁶ screens.

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For rectangular screen (N_x > N_y ), the conditions of $N_{lx}\ge 8N_x/N_y+1$ and $N_{ly}\ge 9$ should be satisfied as for as possible. Under the current personal computer capacity, ${\phi }_{low,low}$with $257\times 9$ points can be generated easily, then rectangular screen with aspect ratio of $32$can 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 ${\varphi }_{low,low}$is 513$\times$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 N_zx, the interpolation errors in the x direction will decrease, but at the same time, the errors induced by the negative eigenvalues of matrix $B_{low,low}$will increase. There exists an optimum N_zx to minimize the total errors.

The time of generating one compensated screen with N_x = N_y = 1024 and N_lx = N_ly = 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 $1/2\times 1/2$of 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.