## Statistical Interpolation Method of Turbulent Phase Screen

Optics Express, Vol. 17, Issue 17, pp. 14649-14664 (2009)

http://dx.doi.org/10.1364/OE.17.014649

Acrobat PDF (328 KB)

### Abstract

A relative displacement between the grid points of optical fields and those of phase screens may occur in the simulation of light propagation through the turbulent atmosphere. A statistical interpolator is proposed to solve this problem in this paper. It is evaluated by the phase structure function and numerical experiments of light propagation through atmospheric turbulence with/without adaptive optics (AO) and it is also compared with the well-known linear interpolator under the same condition. Results of the phase structure function show that the statistical interpolator is more accurate in comparison with the linear one, especially in the high frequency region. More importantly, the long-exposure results of light propagation through the turbulent atmosphere with/without AO also show that the statistical interpolator is more accurate and reliable than the linear one.

© 2009 OSA

## 1. Introduction

1. J. A. Fleck Jr, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. (Berl.) **10**(2), 129–160 (1976). [CrossRef]

6. E. M. Johansson and D. T. Gavel, “Simulation of stellar speckle imaging,” Proc. SPIE **2220**, 372–383 (1994). [CrossRef]

7. V. Sriram and D. Kearney, “An ultra fast Kolmogorov phase screen generator suitable for parallel implementation,” Opt. Express **15**(21), 13709–13714 (2007). [CrossRef] [PubMed]

8. A. Beghi, A. Cenedese, and A. Masiero, “Stochastic realization approach to the efficient simulation of phase screens,” J. Opt. Soc. Am. A **25**(2), 515–525 (2008). [CrossRef]

## 2. Problem statement

### 2.1 One-dimensional relative displacement

9. H. Jakobsson, “Simulations of time series of atmospherically distorted wave fronts,” Appl. Opt. **35**(9), 1561–1565 (1996). [CrossRef] [PubMed]

10. H.-X. Yan, S.-S. Li, and S. Chen, “Numerical simulation investigations of the dynamic control process and frequency response characteristics in an adaptive optics system,” Proc. SPIE **4494**, 156–166 (2002). [CrossRef]

9. H. Jakobsson, “Simulations of time series of atmospherically distorted wave fronts,” Appl. Opt. **35**(9), 1561–1565 (1996). [CrossRef] [PubMed]

12. G. Sedmak, “Performance analysis of and compensation for aspect-ratio effects of fast-fourier-transform-based simulations of large atmospheric wave fronts,” Appl. Opt. **37**(21), 4605–4613 (1998). [CrossRef]

11. B. M. Welsh, “Fourier-series-based atmospheric phase screen generator for simulating anisoplanatic geometries and temporal evolution,” Proc. SPIE **3125**, 327–338 (1997). [CrossRef]

4. F. Dios, J. Recolons, A. Rodríguez, and O. Batet, “Temporal analysis of laser beam propagation in the atmosphere using computer-generated long phase screens,” Opt. Express **16**(3), 2206–2220 (2008). [CrossRef] [PubMed]

13. M. C. Roggemann, B. M. Welsh, D. Montera, and T. A. Rhoadamer, “Method for simulating atmospheric turbulence phase effects for multiple time slices and anisoplanatic conditions,” Appl. Opt. **34**(20), 4037–4051 (1995). [CrossRef] [PubMed]

14. H.-X. Yan, S. Chen, and S.-S. Li, “Turbulent phase screens generated by covariance approach and their application in numerical simulation of atmospheric propagation of laser beam,” Proc. SPIE **6346**, 634628 (2006). [CrossRef]

11. B. M. Welsh, “Fourier-series-based atmospheric phase screen generator for simulating anisoplanatic geometries and temporal evolution,” Proc. SPIE **3125**, 327–338 (1997). [CrossRef]

15. 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] [PubMed]

16. D. L. Fried and T. Clark, “Extruding Kolmogorov-type phase screen ribbons,” J. Opt. Soc. Am. A **25**(2), 463–468 (2008). [CrossRef]

_{0}, the phase perturbations of an optical field can be obtained from the corresponding grid points of a phase screen due to one-to-one correspondence between their grid points. See Fig. 1 (a). If the phase screen shifts by one grid spacing or integer multiples of the grid spacing, the grid points of the optical field are still in one-to-one correspondence with those of the phase screen at the time t = t

_{1}. See Fig. 1 (b). But, if the screen shifts by non-integer multiples of the grid spacing, the grid points of the optical field and those of the screen are staggered at the time t = t

_{2}. See Fig. 1 (c). All grid points of the optical field are among those of the screen. Their grid points are staggered in the direction of the wind. Obviously, the phase perturbations of the optical field induced by atmospheric turbulence cannot be directly obtained from the grid points of the phase screen in this case. This is a one-dimensional relative displacement problem. How to obtain the phase perturbations of the grid points of the optical field is an important problem in the simulation of these scenarios.

### 2.2 Two-dimensional relative displacement

17. C. S. Gardner, B. M. Welsh, and L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE **78**(11), 1721–1743 (1990). [CrossRef]

### 2.3 Solution scheme of the relative displacement problem

*r*.

*k*is the wave number

*λ*is the wavelength.

*L*denotes the thickness of turbulent layer.

## 3. Description of interpolation methods

### 3.1 Linear interpolation

*P*,

_{1}*P*,

_{2}*P*, and

_{3}*P*. See Fig. 3 (b). Then, the phase value of the point

_{4}*P*can be obtained by linear interpolation. It is commonly expressed in the form:where

*P*,

_{1}*P*,

_{2}*P*, and

_{3}*P*are the four corner points of a rectangle or a square and denote the values of these points at the same time (the same below). The interpolation geometry is similar to Fig. 3(a) when

_{4}### 3.2 Statistical interpolation

#### 3.2.1 One-dimensional statistical interpolation

*r*,

*r*and

_{1}*r*are known, the following phase structure functions can be also known from Eq. (1):

_{2}*P*from phase values of the known sample points

*P*,

_{1}*P*. We assume:where

_{2}*a*and

*b*are weight coefficients and satisfy

*a*+

*b*= 1,

*R*is a zero mean Gaussian variable with a variance

*D*,

_{0}*D*and

_{1}*D*are known, we can solve Eq. (3) and obtain:

_{2}*a*, another coefficient

*b*can be obtained by the relation

*a*+

*b*= 1. Thus, by using Eqs. (3)-(5) the phase value of the sample point

*P*can be obtained from the phase values of the known sample points

*P*,

_{1}*P*.

_{2}*α*,

*β*, and

*χ*, having zero mean and equal variances [16

16. D. L. Fried and T. Clark, “Extruding Kolmogorov-type phase screen ribbons,” J. Opt. Soc. Am. A **25**(2), 463–468 (2008). [CrossRef]

*P*coincides with the endpoint

*P*or

_{1}*P*. The coefficient (

_{2}*a*,

*b*) is equal to (1, 0) or (0, 1) and the variance

*P*is the midpoint of

*P*. The coefficient (

_{1}P_{2}*a*,

*b*) is equal to (0.5, 0.5) and

*D*is equal to

_{1}*D*. The variance

_{2}*D*-0.25

_{1}*D*. Obviously, the results of the second case are consistent with those of the known random mid-point displacement method [19

_{0}19. R. G. Lane, A. Glindemann, and J. C. Dainty, “Simulation of Kolmogorov phase screen,” Waves Random Media **2**(3), 209–224 (1992). [CrossRef]

20. 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]

#### 3.2.2 Two-dimensional statistical interpolation

*P*from the values of the known sample points:

*P*,

_{1}*P*,

_{2}*P*, and

_{3}*P*. Similarly to section 3.2.1, we also assume:where

_{4}*a*,

*b*,

*c*and

*d*are weight coefficients and

*a*+

*b*+

*c*+

*d*= 1,

*R*is zero mean Gaussian variables with a variance

*D*,

_{0}*D*,

_{1}*D*,

_{2}*D*,

_{3}*D*and

_{4}*D*are all known.

_{5}*a*,

*b*,

*c*and

*d*, the variance

*P*coincides with the corner point

*P*,

_{1}*P*,

_{2}*P*, or

_{3}*P*. The coefficient (

_{4}*a*,

*b*,

*c*,

*d*) is equal to (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0) or (0, 0, 0, 1) and the variance

*P*. The coefficient (

_{1}P_{2}P_{3}P_{4}*a*,

*b*) is equal to (0.25, 0.25, 0.25, 0.25) and

*D*. The variance

_{1}= D_{2}= D_{3}= D_{4}19. R. G. Lane, A. Glindemann, and J. C. Dainty, “Simulation of Kolmogorov phase screen,” Waves Random Media **2**(3), 209–224 (1992). [CrossRef]

20. 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]

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

20. 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]

11. B. M. Welsh, “Fourier-series-based atmospheric phase screen generator for simulating anisoplanatic geometries and temporal evolution,” Proc. SPIE **3125**, 327–338 (1997). [CrossRef]

## 4. Results and discussions

### 4.1 Comparison of the phase structure function

#### 4.1.1 One-dimensional interpolation

**38**(11), 2161–2170 (1999). [CrossRef]

*r*= 0.20 m. The grid spacing is 0.01 m. The effects of the position of the interpolated point

_{0}*P*on the accuracy of two interpolation methods are also investigated. Assuming

*h*are chosen: 0.3, 0.5, 0.7.The grid points of the base phase screen are 33 × 34 and the grid points of the interpolated phase screen are 33 × 33. A comparison of the relative error is shown in Fig. 4 .

*h*is equal to 0.5. In the low frequency region, there is no substantial difference between the linear interpolator and the statistical interpolator. In addition, a maximum error occurs when the parameter

*h*is equal to 0.5 in the high frequency, especially for the linear interpolator.

#### 4.1.2 Two-dimensional interpolation

*P*is determined by the parameters

*u*and

*v*. Similarly to the case of one-dimensional interpolation, we can assume

*P*can be changed in the square grid

*P*by adjusting the parameter

_{1}P_{2}P_{3}P_{4}*m*and

*n*. On the other hand, the ratio

*l*of the grid spacing of the base screen to that of the interpolated screen also affect the accuracy of the interpolation methods. Being different from the one-dimensional interpolator, the grid spacing of the interpolated screen is generally less than or equal to that of the base screen in the case of two-dimensional interpolation. That is

*l*of the grid spacing affect the accuracy of the interpolation methods.

*r*= 0.14 m. The grid spacing of the base screen is 0.01 m. The grid points of the base screen are 34 × 34 and those of the interpolated screen are 33 × 33.

_{0}*l*on the accuracy of the interpolation method, we change the value of

*l*under the fixed value

*m*and

*n*. The ratio

*l*is chosen as 1.0, 1.2, 1.6 and 2.3 while

*l*in the high frequency region while remains comparatively small in the low frequency region. The ratio

*l*must be equal to 1.0 in order to minimize the relative error in the high frequency region. But, it is generally greater than 1.0 in the simulation of actual scenarios, i.e. the grid spacing of the base screen is greater than that of the interpolated screen. Fortunately, the random mid-point displacement method can be used to solve this problem. By using this method, the local region of the base screen, which is corresponding to the interpolated screen, can be subdivided into a phase screen with higher resolution until the grid spacing of this screen is equal to that of the interpolated screen, i.e.

*m*and

*n*under a fixed value of

### 4.2 Comparison of simulation results of light propagation through turbulent atmosphere

**38**(11), 2161–2170 (1999). [CrossRef]

*P*is a grid point of the low resolution screen. The corresponding point in the high resolution screen (i.e. the base screen) is the grid point

*P*. The phase value of each grid point in the low resolution screen can be obtained by three approaches. The first approach is that the value of the grid point

_{10}*P*in the high resolution screen is directly taken as the phase value of the grid point

_{10}*P*in the low resolution screen. The second approach is that the phase value of the grid point

*P*is obtained by interpolating between two nearest points

*P*and

_{2}*P*or

_{6}*P*and

_{4}*P*. The grid point

_{9}*P*is the interpolated point and it is also the mid-point of the line segment

*P*and

_{2}P_{6}*P*at the same time. This is a case of one-dimensional interpolation. The third approach is that the phase value of the grid point

_{4}P_{9}*P*is obtained by interpolating between four nearest corner points

*P*,

_{1}*P*,

_{3}*P*and

_{5}*P*. The grid point

_{7}*P*is the interpolated point and it is also the center point of the square grid

*P*. This is a case of two-dimensional interpolation. Two interpolators (i.e. the linear interpolator and the statistical interpolator) are used to realize the interpolation process. Obviously, the first approach can provide an accurate result because this approach doesn’t use any interpolation. Thus, this accurate result provided by the first approach will be used as a standard value for comparing to evaluate two interpolation methods. The result provided by the linear interpolator and that provided by the statistical interpolator can be compared to the standard value. A relative error of the long-exposure Strehl ratio is computed. In this way, we can quantitatively estimate the accuracy and reliability of two interpolation methods.

_{1}P_{3}P_{5}P_{7}2. H.-X. Yan, S.-S. Li, D.-L. Zhang, and S. Chen, “Numerical simulation of an adaptive optics system with laser propagation in the atmosphere,” Appl. Opt. **39**(18), 3023–3031 (2000). [CrossRef]

*z*= 100 m. The wavelength is

## 5. Summary

## References and links

1. | J. A. Fleck Jr, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. (Berl.) |

2. | H.-X. Yan, S.-S. Li, D.-L. Zhang, and S. Chen, “Numerical simulation of an adaptive optics system with laser propagation in the atmosphere,” Appl. Opt. |

3. | L. C. Andrews, R. L. Philips, R. J. Sasiela, and R. R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects,” Opt. Eng. |

4. | F. Dios, J. Recolons, A. Rodríguez, and O. Batet, “Temporal analysis of laser beam propagation in the atmosphere using computer-generated long phase screens,” Opt. Express |

5. | H.-X Yan, Han-Ling Wu, Shu-Shan Li and She Chen, “Cone effect in astronomical adaptive optics system investigated by a pure numerical simulation,” Proc. SPIE |

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

7. | V. Sriram and D. Kearney, “An ultra fast Kolmogorov phase screen generator suitable for parallel implementation,” Opt. Express |

8. | A. Beghi, A. Cenedese, and A. Masiero, “Stochastic realization approach to the efficient simulation of phase screens,” J. Opt. Soc. Am. A |

9. | H. Jakobsson, “Simulations of time series of atmospherically distorted wave fronts,” Appl. Opt. |

10. | H.-X. Yan, S.-S. Li, and S. Chen, “Numerical simulation investigations of the dynamic control process and frequency response characteristics in an adaptive optics system,” Proc. SPIE |

11. | B. M. Welsh, “Fourier-series-based atmospheric phase screen generator for simulating anisoplanatic geometries and temporal evolution,” Proc. SPIE |

12. | G. Sedmak, “Performance analysis of and compensation for aspect-ratio effects of fast-fourier-transform-based simulations of large atmospheric wave fronts,” Appl. Opt. |

13. | M. C. Roggemann, B. M. Welsh, D. Montera, and T. A. Rhoadamer, “Method for simulating atmospheric turbulence phase effects for multiple time slices and anisoplanatic conditions,” Appl. Opt. |

14. | H.-X. Yan, S. Chen, and S.-S. Li, “Turbulent phase screens generated by covariance approach and their application in numerical simulation of atmospheric propagation of laser beam,” Proc. SPIE |

15. | F. Assémat, R. W. Wilson, and E. Gendron, “Method for simulating infinitely long and non stationary phase screens with optimized memory storage,” Opt. Express |

16. | D. L. Fried and T. Clark, “Extruding Kolmogorov-type phase screen ribbons,” J. Opt. Soc. Am. A |

17. | C. S. Gardner, B. M. Welsh, and L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE |

18. | B. Formwalt and S. Cain, “Optimized phase screen modeling for optical turbulence,” Appl. Opt. |

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

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

21. | L. C. Andrews, and R. L. Phillips, |

22. | R. K. Tyson, |

**OCIS Codes**

(010.1080) Atmospheric and oceanic optics : Active or adaptive optics

(010.1300) Atmospheric and oceanic optics : Atmospheric propagation

(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence

**ToC Category:**

Atmospheric and Oceanic Optics

**History**

Original Manuscript: May 19, 2009

Revised Manuscript: July 9, 2009

Manuscript Accepted: July 17, 2009

Published: August 4, 2009

**Citation**

Han-Ling Wu, Hai-Xing Yan, Xin-Yang Li, and Shu-Shan Li, "Statistical interpolation method of turbulent phase screen," Opt. Express **17**, 14649-14664 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-17-14649

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### References

- J. A. Fleck, J. R. Morris, and M. D. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. (Berl.) 10(2), 129–160 (1976). [CrossRef]
- H.-X. Yan, S.-S. Li, D.-L. Zhang, and S. Chen, “Numerical simulation of an adaptive optics system with laser propagation in the atmosphere,” Appl. Opt. 39(18), 3023–3031 (2000). [CrossRef]
- L. C. Andrews, R. L. Philips, R. J. Sasiela, and R. R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects,” Opt. Eng. 45(7), 076001–1 (2006). [CrossRef]
- F. Dios, J. Recolons, A. Rodríguez, and O. Batet, “Temporal analysis of laser beam propagation in the atmosphere using computer-generated long phase screens,” Opt. Express 16(3), 2206–2220 (2008). [CrossRef] [PubMed]
- H.-X Yan, Han-Ling Wu, Shu-Shan Li and She Chen, “Cone effect in astronomical adaptive optics system investigated by a pure numerical simulation,” Proc. SPIE 5903, 5903OU1–12 (2005).
- E. M. Johansson and D. T. Gavel, “Simulation of stellar speckle imaging,” Proc. SPIE 2220, 372–383 (1994). [CrossRef]
- V. Sriram and D. Kearney, “An ultra fast Kolmogorov phase screen generator suitable for parallel implementation,” Opt. Express 15(21), 13709–13714 (2007). [CrossRef] [PubMed]
- A. Beghi, A. Cenedese, and A. Masiero, “Stochastic realization approach to the efficient simulation of phase screens,” J. Opt. Soc. Am. A 25(2), 515–525 (2008). [CrossRef]
- H. Jakobsson, “Simulations of time series of atmospherically distorted wave fronts,” Appl. Opt. 35(9), 1561–1565 (1996). [CrossRef] [PubMed]
- H.-X. Yan, S.-S. Li, and S. Chen, “Numerical simulation investigations of the dynamic control process and frequency response characteristics in an adaptive optics system,” Proc. SPIE 4494, 156–166 (2002). [CrossRef]
- B. M. Welsh, “Fourier-series-based atmospheric phase screen generator for simulating anisoplanatic geometries and temporal evolution,” Proc. SPIE 3125, 327–338 (1997). [CrossRef]
- G. Sedmak, “Performance analysis of and compensation for aspect-ratio effects of fast-fourier-transform-based simulations of large atmospheric wave fronts,” Appl. Opt. 37(21), 4605–4613 (1998). [CrossRef]
- M. C. Roggemann, B. M. Welsh, D. Montera, and T. A. Rhoadamer, “Method for simulating atmospheric turbulence phase effects for multiple time slices and anisoplanatic conditions,” Appl. Opt. 34(20), 4037–4051 (1995). [CrossRef] [PubMed]
- H.-X. Yan, S. Chen, and S.-S. Li, “Turbulent phase screens generated by covariance approach and their application in numerical simulation of atmospheric propagation of laser beam,” Proc. SPIE 6346, 634628 (2006). [CrossRef]
- 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] [PubMed]
- D. L. Fried and T. Clark, “Extruding Kolmogorov-type phase screen ribbons,” J. Opt. Soc. Am. A 25(2), 463–468 (2008). [CrossRef]
- C. S. Gardner, B. M. Welsh, and L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78(11), 1721–1743 (1990). [CrossRef]
- B. Formwalt and S. Cain, “Optimized phase screen modeling for optical turbulence,” Appl. Opt. 45(22), 5657–5668 (2006). [CrossRef] [PubMed]
- R. G. Lane, A. Glindemann, and J. C. Dainty, “Simulation of Kolmogorov phase screen,” Waves Random Media 2(3), 209–224 (1992). [CrossRef]
- 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]
- L. C. Andrews, and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE Press, Washington, 2005).
- R. K. Tyson, Principles of Adaptive Optics, 2nd ed. (Academic Press, Boston, 1997).

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