## Accurate compensation of the low-frequency components for the FFT-based turbulent phase screen |

Optics Express, Vol. 20, Issue 1, pp. 681-687 (2012)

http://dx.doi.org/10.1364/OE.20.000681

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### 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

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]

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]

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]

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]

## 2. FFT-based turbulent phase screen

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]

*L*

_{0}is the outer scale of turbulence,

*r*

_{0}is the Fried parameter. The phase screen is assumed to be sampled at a constant spatial interval

*D*/

_{x}*N*/

_{x}= D_{y}*N*. Equation (1) can be implemented by means of FFT.

_{y}*N*rectangular region centered at the origin of

_{zx×}N_{zy}*N*and

_{zx}*N*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

_{zy}*N*/

_{x}*N*= 2, 4, 8, 16,

_{y}*D*and

_{x}*D*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.

_{y}## 3. Compensation of the low frequency components

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]

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

*q*is an integer power of two, the numbers of points are

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]

*T*denotes matrix transpose.

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]

*N*=

_{zx}*N*= 1, then the maximum error is about 6%. By setting more low frequency terms of

_{zy}*N*=

_{zx}*N*= 3, the errors are less than 0.1%. For rectangular screen, when

_{zy}*N*,

_{x}*N*,

_{y}*N*and

_{lx}*N*also have some influence on the errors, with the increasing of

_{ly}*N*,

_{x}*N*,

_{y}*N*and

_{lx}*N*, the errors increase slightly.

_{ly}## 4. Simulation results

*r*

_{0}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.

*N*=

_{l}*N*=

_{lx}*N*= 9 and

_{ly}*N*=

_{z}*N*=

_{zx}*N*= 3, the maximum relative error is on the order of 0.1% in the low frequency region (large spatial distance

_{zy}*r*), whereas that of the weighted subharmonic method is about 1% [5

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]

*N*or

_{z}*N*. Increasing

_{l}*N*, the errors induced by the negative eigenvalues of matrix

_{z}*B*will decrease,

_{low,low}*N*= 5 is adequate to completely eliminate the negative eigenvalues of matrix

_{z}*B*. Increasing

_{low,low}*N*, the interpolation errors will decrease. In order to avoid large interpolation errors, the condition of

_{l}*N*>

_{x}*N*), the conditions of

_{y}*N*, the interpolation errors in the x direction will decrease, but at the same time, the errors induced by the negative eigenvalues of matrix

_{zx}*N*to minimize the total errors.

_{zx}*N*1024 and

_{x}= N_{y}=*N*= 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

_{lx}= N_{ly}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]

## 5. Conclusion

## Acknowledgments

## References

1. | B. L. McGlamery, “Computer simulation studies of compensation of turbulence degraded images,” Proc. SPIE |

2. | B. J. Herman and L. A. Strugala, “Method for inclusion of low-frequency contributions in numerical representation of atmospheric turbulence,” Proc. SPIE |

3. | R. G. Lane, A. Glindemann, and J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media |

4. | E. M. Johansson and D. T. Gavel, “Simulation of stellar speckle imaging,” Proc. SPIE |

5. | G. Sedmak, “Implementation of fast-Fourier-transform-based simulations of extra-large atmospheric phase and scintillation screens,” Appl. Opt. |

6. | J. Recolons and F. Dios, “Accurate calculation of phase screens for the modelling of laser beam propagation through atmospheric turbulence,” Proc. SPIE |

7. | N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. |

8. | K. A. Winick, “Atmospheric turbulence-induced signal fades on optical heterodyne communication links,” Appl. Opt. |

9. | C. M. Harding, R. A. Johnston, and R. G. Lane, “Fast simulation of a kolmogorov phase screen,” Appl. Opt. |

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 |

**OCIS Codes**

(010.1290) Atmospheric and oceanic optics : Atmospheric optics

(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence

(350.5030) Other areas of optics : Phase

**ToC Category:**

Atmospheric and Oceanic Optics

**History**

Original Manuscript: October 14, 2011

Manuscript Accepted: November 24, 2011

Published: December 23, 2011

**Citation**

Jingsong Xiang, "Accurate compensation of the low-frequency components for the FFT-based turbulent phase screen," Opt. Express **20**, 681-687 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-1-681

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### References

- B. L. McGlamery, “Computer simulation studies of compensation of turbulence degraded images,” Proc. SPIE74, 225–233 (1976).
- B. J. Herman and L. A. Strugala, “Method for inclusion of low-frequency contributions in numerical representation of atmospheric turbulence,” Proc. SPIE1221, 183–192 (1990). [CrossRef]
- R. G. Lane, A. Glindemann, and J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media2(3), 209–224 (1992). [CrossRef]
- E. M. Johansson and D. T. Gavel, “Simulation of stellar speckle imaging,” Proc. SPIE2200, 372–383 (1994). [CrossRef]
- 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]
- J. Recolons and F. Dios, “Accurate calculation of phase screens for the modelling of laser beam propagation through atmospheric turbulence,” Proc. SPIE5891, 589107, 589107-12 (2005). [CrossRef]
- N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng.29(10), 1174–1180 (1990). [CrossRef]
- K. A. Winick, “Atmospheric turbulence-induced signal fades on optical heterodyne communication links,” Appl. Opt.25(11), 1817–1825 (1986). [CrossRef] [PubMed]
- 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]
- F. Assémat, R. W. Wilson, and E. Gendron, “Method for simulating infinitely long and non stationary phase screens with optimized memory storage,” Opt. Express14(3), 988–999 (2006). [CrossRef]

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