## Atmospheric turbulence correction using digital holographic detection: experimental results

Optics Express, Vol. 17, Issue 14, pp. 11638-11651 (2009)

http://dx.doi.org/10.1364/OE.17.011638

Acrobat PDF (9137 KB)

### Abstract

The performance of long distance imaging systems is typically degraded by phase errors imparted by atmospheric turbulence. In this paper we apply coherent imaging methods to determine, and remove, these phase errors by digitally processing coherent recordings of the image data. In this manner we are able to remove the effects of atmospheric turbulence without needing a conventional adaptive optical system. Digital holographic detection is used to record the coherent, complex-valued, optical field for a series of atmospheric and object realizations. Correction of atmospheric phase errors is then based on maximizing an image sharpness metric to determine the aberrations present and correct the underlying image. Experimental results that demonstrate image recovery in the presence of turbulence are presented. Results obtained with severe turbulence that gives rise to anisoplanatism are also presented.

© 2009 Optical Society of America

## 1. Introduction

1. J. W. Goodman, D. W. Jackson, M. Lehmann, and J. Knotts, “Experiments in Long-Distance Holographic Imagery,” Appl. Opt. **8**, 1581–1586 (1969).
[CrossRef] [PubMed]

2. J. W. Goodman and R.W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. **11**, 77–79 (1967).
[CrossRef]

3. R. A. Muller and A. Buffington, “Real-time correction of atmospherically degraded telescope images through image sharpening,” J. Opt. Soc. Am. **64**, 1200–1210 (1974).
[CrossRef]

5. J. R. Fienup and J. J. Miller, “Aberration Correction by Maximizing Generalized Image Sharpness Metrics,” J. Opt. Soc. Am. A **20**, 609–620 (2003).
[CrossRef]

## 2. Digital holographic recording

6. The U.S. Department of Commerce Table Mountain Field Site and Radio Quiet Zone. http://www.its.bldrdoc.gov/table_mountain/.

*f*(

*x*) and the reference point as

*g*(

*x*) with their corresponding Fourier transforms given by

*F*(

*u*) and

*G*(

*u*) respectively. The intensity recorded by the detector array can be written as

*x=b*, it follows that the Fourier transform of the intensity pattern is given by

*b*. These images are complex-valued and by extracting one of them the complex-valued representation of the object field is obtained. In the experiments reported here we digitally correct for instrumentation and atmospheric phase aberrations. The instrumentation phase aberrations in our system included several waves of defocus. Thus prior to computing the digital Fourier transform of the recorded intensity data, we typically apply a quadratic phase term to the pupil intensity data. This produces a better focused image term in the Fourier transform data shown on the right in Fig. 3(B). Notice, however, that the conjugate image on the left of Fig. 3(B) was more severely defocused by adding the quadratic term. With the defocus removed the image is localized and we are readily able to extract the complex-valued, optical field data from the Fourier transform by simply extracting the desired region. The result of this extraction is shown in Fig. 3(C). Note that the sum of the squared magnitudes of the 16 complex-valued realizations is shown. In this example the object is the 1951 USAF resolution test chart. The realizations result from changes in the atmosphere and changes in the underlying speckle caused by slight object motion. Note that for this image we have not yet corrected for atmospheric turbulence, or fully corrected for instrumentation errors.

## 3. Aberration correction using image sharpness

9. J. Marron and G. M. Morris, “Image-plane speckle from rotating, rough objects,” J. Opt. Soc. Am. A **2**, 1395–1402 (1985).
[CrossRef]

*t*(

*x,y*) multiplied by a random phase function

*ϕ*(

*x,y*). At this point we take the atmosphere to be static and thus the ensemble of realizations corresponds to realizations of the object’s random phase function

*ϕ*(

*x,y*). Note that

*t*(

*x,y*) includes both the object’s underlying reflectivity and any variation that might be caused by the scintillation and wander of the illuminating beam. We can thus write the image intensity as

*V*(

*m,n*)=<

*I*(

*m,n*)>. Following Refs. 4, 11 and 12 we can write the speckled image intensity as the product of the underlying intensity image

*V*(

*m,n*) multiplied by a speckle noise function or

*I*(

_{s}*m,n*) is a spatially stationary negative exponential random variable with mean value equal to unity. The relationship in Eq. (6) is referred to as the multiplicative speckle noise model because it shows that the intensity of the speckled image is given by the product of the underlying image intensity multiplied by spatially uniform speckle noise function. Note that this model does have limitations that are discussed in Ref. 12. From Eq. (6) it follows that the

*r*moment of the speckled image intensity is given by

^{th}13. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. **66**, 207–211 (1976).
[CrossRef]

3. R. A. Muller and A. Buffington, “Real-time correction of atmospherically degraded telescope images through image sharpening,” J. Opt. Soc. Am. **64**, 1200–1210 (1974).
[CrossRef]

*β*is dependent on the nature of the image. For images with bright spots or glints, values of

*β*>2 are generally more suitable, whereas for images without bright points a value of 1.0<

*β*<2.0 is generally more suitable [5

5. J. R. Fienup and J. J. Miller, “Aberration Correction by Maximizing Generalized Image Sharpness Metrics,” J. Opt. Soc. Am. A **20**, 609–620 (2003).
[CrossRef]

*β*=1.2. An optimization algorithm, such as a quasi-Newton line search, is then used to determine a set of polynomial coefficients that maximize the sharpness. Finally the output image and phase aberration masks are generated.

^{th}order Zernike polynomial. The actual 48

^{th}polynomial is shown in Fig. 4(B) and it is apparent that the spatial features of this polynomial are not fine enough to negate the phase features of the input image, which are approximately the speckle size in Fig. 4(A).

*β*in Eq. (9) that is too large. For this reason we often use values of

*β*that are in the range of 1<

*β*<2. In practice one should evaluate the value of

*β*with respect to the specific imaging task as discussed in Ref. 5 and if the processing appears to generate artificial point like features, the value of

*β,*or the number of Zernike polynomials, should be reduced.

## 4. Experiment description

6. The U.S. Department of Commerce Table Mountain Field Site and Radio Quiet Zone. http://www.its.bldrdoc.gov/table_mountain/.

*R*=100 meters. The beam height was 1.2 meters along the path and the terrain was a uniform grassy desert. Based on the uniformity of the path we operated under the assumption that the level of refractive turbulence quantified by

*C*was constant along the path. In this case the values of Fried’s parameter,

_{n}^{2}*r*

_{0}, and the isoplanatic angle,

*α*

_{0}, are defined by algebraic formulas rather than path integrals and are given by

*λ*=532 nm. The coherence length was significantly longer than the 200 m roundtrip path length and thus the return light and reference beam interfered coherently. The target was flood illuminated with a laser beam having a power level of roughly 200 mW.

*D*=48.3 mm. A simple telescope consisting of a set of doublets was used to reimage the entrance pupil onto a two-dimensional CCD detector array. A small portion of the laser transmitter was coupled into a fiber and was used as the reference beam, or local oscillator. A beamsplitter was used to mix the target return and the local oscillator on the 2D detector array creating the digital holographic interference pattern shown, for example, in Fig. 2. Image formation was performed using a 1024×1024 pixel region of the detector array and the pixel spacing of the detector array was 6.7 microns. Integration times for the exposures of the detector array were typically 0.5 msec. This value was chosen to give sufficient signal level, while also corresponding to a quasi-static realization of the atmosphere. The CCD’s quantum efficiency was roughly 0.4.

*D/λ R*=0.907 lp/mm it follows that the MTF for our system is near zero for element 6 and increases somewhat for group -1, element 5.

*C*and

^{2}_{n}*r*

_{0}values. The scintillometer determines the level of turbulence over 60 second time intervals and reports values for each such interval. Figs. 5 and 6 contain examples of

*C*and

^{2}_{n}*r*

_{0}values for the Table Mountain test site on the March 6, 2008. Note that the turbulence levels were relatively strong during the early afternoon and improved as sunset approached. For the results discussed below we selected three data sets with atmospheric parameters given in Table 1. The corresponding values of

*D/r*

_{0}were

*D/r*

_{0}=0.4,

*D/r*

_{0}=1.6 and

*D/r*

_{0}=5.9.

## 5. Results

^{th}, 2008. The level of atmospheric turbulence increases in going from top to bottom.

*D/r*

_{0}<1 which is the case in Fig. 7(A).

*D/r*

_{0}increases, a single fixed correction does not correct for the turbulence as shown in Fig. 7(C). However, when we correct for each image realization, the quality of the image is improved substantially with Figs. 7(E) and (F) showing marked improvement over the single screen corrections. Image details approaching the diffraction limit, for example group -1 element 4 in Fig. 7(E), are clearly visible.

*D/r*

_{0}is approximately 5.9. Using the equations derived in Ref. 13 we find that the residual rms wavefront error after removing 48 Zernikes is approximately 0.11 waves. Thus improvement could be obtained by using more Zernike polynomials, however, anisoplanatism (aberration variation as a function of field angle) also degrades the imaging performance in this case.

*I*-

_{max}*I*)/(

_{min}*I*+

_{max}*I*), for the bars in the lower right corner are: 0.26, 0.43, 0.61 for images (A), (B), and (C), respectively.

_{min}## 6. Summary

*D*/

*r*

_{0}=0.4 to

*D/r*

_{0}=5.9 and isoplanatic patch sizes that covered the entire object to the case where there are approximately seven isoplanatic patches across the object. In all conditions the aberrations due to atmospheric turbulence were corrected digitally by maximizing an image sharpness criterion. For the case where the isoplanatic patch is the same size as the target, the result is essentially diffraction limited. For the severe turbulence case a reasonable image is recovered but it is not diffraction limited. We also demonstrated that by isolating a small section of an image corresponding to an isoplanatic patch we are able to obtain good atmospheric correction.

## References and links

1. | J. W. Goodman, D. W. Jackson, M. Lehmann, and J. Knotts, “Experiments in Long-Distance Holographic Imagery,” Appl. Opt. |

2. | J. W. Goodman and R.W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. |

3. | R. A. Muller and A. Buffington, “Real-time correction of atmospherically degraded telescope images through image sharpening,” J. Opt. Soc. Am. |

4. | R. G. Paxman and J. C. Marron, “Aberration Correction of Speckled Imagery With an Image Sharpness Criterion,” In Proc. of the SPIE Conference on Statistical Optics, 976, San Diego, CA, August (1988). |

5. | J. R. Fienup and J. J. Miller, “Aberration Correction by Maximizing Generalized Image Sharpness Metrics,” J. Opt. Soc. Am. A |

6. | The U.S. Department of Commerce Table Mountain Field Site and Radio Quiet Zone. http://www.its.bldrdoc.gov/table_mountain/. |

7. | M. C. Roggeman and B. Welsh, |

8. | R. R. Beland, “Propagation through atmospheric optical turbulence,” in |

9. | J. Marron and G. M. Morris, “Image-plane speckle from rotating, rough objects,” J. Opt. Soc. Am. A |

10. | J. C. Dainty, “Stellar speckle interferometry,” in |

11. | G. April and H. H. Arsenault, “Nonstationary image-plane speckle statistics,” J. Opt. Soc. Am. A |

12. | M. Tur, K. C. Chin, and J. W. Goodman, “When is speckle noise multiplicative?,” Appl. Opt. |

13. | R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. |

**OCIS Codes**

(110.0115) Imaging systems : Imaging through turbulent media

(090.1995) Holography : Digital holography

(110.3010) Imaging systems : Image reconstruction techniques

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: April 6, 2009

Revised Manuscript: May 26, 2009

Manuscript Accepted: June 8, 2009

Published: June 26, 2009

**Citation**

Joseph C. Marron, Richard L. Kendrick, Nathan Seldomridge, Taylor D. Grow, and Thomas A. Höft, "Atmospheric turbulence correction using digital holographic detection: experimental results," Opt. Express **17**, 11638-11651 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-14-11638

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### References

- J. W. Goodman, D. W. Jackson, M. Lehmann, and J. Knotts, "Experiments in Long-Distance Holographic Imagery," Appl. Opt. 8, 1581-1586 (1969). [CrossRef] [PubMed]
- J. W. Goodman and R.W. Lawrence, "Digital image formation from electronically detected holograms," Appl. Phys. Lett. 11, 77-79 (1967). [CrossRef]
- R. A. Muller and A. Buffington, "Real-time correction of atmospherically degraded telescope images through image sharpening," J. Opt. Soc. Am. 64, 1200-1210 (1974). [CrossRef]
- R. G. Paxman and J. C. Marron, "Aberration Correction of Speckled Imagery With an Image Sharpness Criterion," In Proc. of the SPIE Conference on Statistical Optics, 976, San Diego, CA, August (1988).
- J. R. Fienup and J. J. Miller, "Aberration Correction by Maximizing Generalized Image Sharpness Metrics," J. Opt. Soc. Am. A 20, 609-620 (2003). [CrossRef]
- TheU.S. Department of Commerce Table Mountain Field Site and Radio Quiet Zone. http://www.its.bldrdoc.gov/table_mountain/.
- M. C. Roggeman and B. Welsh, Imaging Through Turbulence (CRC, New York, N.Y. 1996).
- R. R. Beland, "Propagation through atmospheric optical turbulence," in The Infrared and Electro-Optical Handbook, Vol. 2: Atmospheric Propagation of Radiation, F. G. Smith, ed. (SPIE Optical Engineering Press: Bellingham, Wash., 1993), pp. 157-232.
- J. Marron and G. M. Morris, "Image-plane speckle from rotating, rough objects," J. Opt. Soc. Am. A 2, 1395-1402 (1985). [CrossRef]
- J. C. Dainty, "Stellar speckle interferometry," in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. Springer-Verlag, Berlin, (1983), pp. 256-320.
- G. April and H. H. Arsenault, "Nonstationary image-plane speckle statistics," J. Opt. Soc. Am. A 1, 738-741 (1984). [CrossRef]
- M. Tur, K. C. Chin, and J. W. Goodman, "When is speckle noise multiplicative?," Appl. Opt. 21, 1157-1159 (1982). [CrossRef] [PubMed]
- R. J. Noll, "Zernike polynomials and atmospheric turbulence," J. Opt. Soc. Am. 66, 207- 211 (1976). [CrossRef]

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