## Non-iterative aberration correction of a multiple transmitter system |

Optics Express, Vol. 19, Issue 25, pp. 25048-25056 (2011)

http://dx.doi.org/10.1364/OE.19.025048

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### Abstract

Multi-transmitter aperture synthesis provides aperture gain and improves effective aperture fill factor by shifting the received speckle field through the use of multiple transmitter locations. It is proposed that by utilizing methods based on shearing interferometry some low-order aberrations, such as defocus, can be found directly rather than through iterative algorithms. The current work describes the theory behind multi-transmitter aberration correction and describes experiments used to validate this method. Experimental results are shown which demonstrate the ability of such a sensor to solve directly for defocus and toric curvature in the captured field values.

© 2011 OSA

## 1. Introduction

1. J. C. Marron and R. L. Kendrick, “Distributed aperture active imaging,” Proc. SPIE **6550**, 65500A (2007). [CrossRef]

2. D. J. Rabb, D. F. Jameson, A. J. Stokes, and J. W. Stafford, “Distributed aperture synthesis,” Opt. Express **18**(10), 10334–10342 (2010). [CrossRef] [PubMed]

3. D. J. Rabb, D. F. Jameson, J. W. Stafford, and A. J. Stokes, “Multi-transmitter aperture synthesis,” Opt. Express **18**(24), 24937–24945 (2010). [CrossRef] [PubMed]

4. J. R. Fienup and J. J. Miller, “Aberration correction by maximizing generalized sharpness metrics,” J. Opt. Soc. Am. A **20**(4), 609–620 (2003). [CrossRef] [PubMed]

3. D. J. Rabb, D. F. Jameson, J. W. Stafford, and A. J. Stokes, “Multi-transmitter aperture synthesis,” Opt. Express **18**(24), 24937–24945 (2010). [CrossRef] [PubMed]

## 2. Theory

*x*,

_{T}*y*) which are a function of the transmitter location.

_{T}*U*(

_{d}*x*,

*y*) can be written aswhere

*P*(

*x*,

*y*) is the pupil function,

*U*(

_{b}*x-x*,

_{T}*y-y*) is the backscattered field and

_{T}*ϕ*(

_{e}*x*,

*y*) is the phase error across the pupil. The shift due to transmitter locations is given by x

_{T}and y

_{T}. Note that for a given aperture the phase error value is static and doesn’t shift with backscattered field. The phase-front detected by the holographic receiver,

*U*(

_{d}*x*,

*y*), can be described by taking the argument of Eq. (1) such thatwhere

*W*(

_{e}*x*,

*y*) is the wavefront error and

*W*(

_{b}*x*,

*y*) is the wavefront of the backscattered field. Therefore the detected optical wavefront is the sum of the hardware wavefront error and the transmitter-location-translated, backscattered wavefront. From this point on the amplitude terms will be dropped for convenience. The captured fields must be registered in a common, digital pupil plane by shifting the captured data by the known transmitter location so that the individually captured, and now registered, wavefronts are described bywhere it can be observed that the wavefront error term now shifts with the registered field. An estimate of the “shear” between two overlapped wavefronts estimates can be found by finding the difference in the captured, and registered, wavefronts Δ

*W*(

*x*,

*y*) given bywhere (

*x*,

_{T0}*y*) and (

_{T0}*x*,

_{T1}*y*) represent the effective transmitter shift of each of the captured fields. Note that the receivers measure a common backscattered field over any overlap area and that the difference will represent only the difference in error terms such that

_{T1}*W*(

_{e}*x*,

*y*) will be modeled using the bivariate expansion given by

_{T0},y

_{T0}) and (x

_{T1},y

_{T1}).

*a*,

_{11}*a*and

_{02}*a*from Eq. (7) can be used to calculate these wavefront terms of interest. Solutions for them can be found by isolating the first-order terms in the difference equation and rewriting the first-order terms Δ

_{20}*W*’(

*x*,

*y*) as

*a*,

_{11}*a*and

_{02}*a*, one more shear measurement is required, leading to an over determined system of equations from which the parameter values are calculated.

_{20}*x*and

*y*directions respectively, are shown with the pupil planes registered using the known shifts corresponding to the transmitter locations.

*x*,

_{T0}*y*,

_{T0}, y_{T1}*x*are all zero. The resulting equations for the shear arising from the transmitters at (0,0) and (

_{T2}*x*,0), Δ

_{T1}*W*(

_{01}*x*,

*y*), and the shear arising from transmitters at (0,0) and (0,

*y*), Δ

_{T2}*W*(

_{02}*x*,

*y*), are

*x*tilt, γ

_{01x}, and

*y*tilt, γ

_{01y}, of Δ

*W*(

_{01}*x*,

*y*) and the

*x*tilt, γ

_{02x}, and

*y*tilt, γ

_{02y}, of Δ

*W*(

_{02}*x*,

*y*) can then be used to calculate

*a*,

_{11}*a*and

_{02}*a*

_{20}*a*are simply averaged in Eq. (11). Using the known transmitter locations as well as the measured tilts to calculate the coefficients yields the wavefront error which is used to “flatten” the pupil fields for synthesis in a common pupil plane. Note that the measurement is independent of the target being imaged (as long as there is sufficient backscatter to close the link-budget).

_{11}## 3. Experiment

### 3.1 Hardware

*Z*in front of the imaging lens.

_{0}is chosen to be at the origin (

*x*and

_{T0}*y*are both zero) in describing the transmitter positions, then

_{T0}*x*= 16 mm,

_{T1}*y*= 0,

_{T1}*x*= 0,

_{T2}*y*= 16 mm,

_{T2}*x*= 16 mm, and

_{T3}*y*= 16 mm. The additional transmitter location relative to Fig. 2 results in additional

_{T3}*x*and

*y*shear, providing redundant calculations to those shown in Eq. (11-13), which are averaged to increase accuracy of the estimates. Equations (14-16) show the implementation of Eq. (11-13) for this particular setup, accounting for the averaging of the redundant measurements and the specific transmitter locations.

### 3.2 Processing and Results

7. M. Guizar-Sicairos, S. T. Thurman, and J. R. Fienup, “Efficient subpixel image registration algorithms,” Opt. Lett. **33**(2), 156–158 (2008). [CrossRef] [PubMed]

8. M. Guizar, “Efficient subpixel image registration by cross-correlation,” http://www.mathworks.com/matlabcentral/fileexchange/18401-efficient-subpixel-image-registration-by-cross-correlation

*a*,

_{11}*a*,

_{02}*a*can be found. The aberrations are corrected by subtracting the aberrated wavefront from each of the detected fields

_{20}*U*(

_{d}*x*,

*y*).

*Z*values between 3.5 and 7.5 meters. Single aperture data taken at best focus (

*Z*= 6.97 m) is shown in Fig. 6(a) , the residual, single aperture, wavefront error (0.19 waves peak to valley) captured through the current method is shown in Fig. 6(b), and synthesized, higher resolution, data is shown in Fig. 6(c). Figure 6(d) show a single aperture image taken at a distance of

*Z*= 3.87 m, Fig. 6(e) shows the solved wavefront error (5.02 waves peak to valley), and Fig. 6(f) shows the final, synthesized image. The final, synthesized image is larger than the others of the set because of the shorter range. Note that the target is originally severely blurred, to the point of being indiscernible and that the image has been synthesized without knowledge of target range or of any other target information.

*Z*. Toric curvature is added to the system by rotating a 1000 mm focal length cylindrical lens relative to another −1000 mm focal length cylindrical lens. The two lenses are closely spaced with curved surfaces facing one another, so as to minimize aberrations when the two are aligned. Figure 6(g) shows an image (target at 6.97 m) taken through a single aperture with a relative rotation of 1.5 degrees. The image is barely perceptible due to the wavefront error (2.22 waves peak to valley) found by the system and shown in Fig. 6(h), however it is easily corrected and the resulting synthesized image is shown in Fig. 6(i). A relative rotation of 6 degrees is shown in Fig. 6(j-l) (target at 6.97 m, 8.79 waves peak to valley wavefront error), however here the matched pair has been rotated 45 degrees with respect to the optical system as compared with the results shown in Fig. 6(g-i). Figure 7(b) shows the relative toric curvature found through both OSLO simulations and experiment as a function of the rotation between the cylindrical lenses, where the target was placed at 6.97 m.

## 4. Conclusion

## References and links

1. | J. C. Marron and R. L. Kendrick, “Distributed aperture active imaging,” Proc. SPIE |

2. | D. J. Rabb, D. F. Jameson, A. J. Stokes, and J. W. Stafford, “Distributed aperture synthesis,” Opt. Express |

3. | D. J. Rabb, D. F. Jameson, J. W. Stafford, and A. J. Stokes, “Multi-transmitter aperture synthesis,” Opt. Express |

4. | J. R. Fienup and J. J. Miller, “Aberration correction by maximizing generalized sharpness metrics,” J. Opt. Soc. Am. A |

5. | R. A. Hutchin, “Sheared coherent interferometric photography, a technique for lensless imaging,” in Digital Image Recover and Synthesis, P. S. Idell, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 2029, 161–168 (1993). |

6. | T. Kreis, |

7. | M. Guizar-Sicairos, S. T. Thurman, and J. R. Fienup, “Efficient subpixel image registration algorithms,” Opt. Lett. |

8. | M. Guizar, “Efficient subpixel image registration by cross-correlation,” http://www.mathworks.com/matlabcentral/fileexchange/18401-efficient-subpixel-image-registration-by-cross-correlation |

**OCIS Codes**

(100.3010) Image processing : Image reconstruction techniques

(090.1995) Holography : Digital holography

**ToC Category:**

Image Processing

**History**

Original Manuscript: September 2, 2011

Revised Manuscript: September 28, 2011

Manuscript Accepted: October 4, 2011

Published: November 22, 2011

**Citation**

David J. Rabb, Jason W. Stafford, and Douglas F. Jameson, "Non-iterative aberration correction of a multiple transmitter system," Opt. Express **19**, 25048-25056 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-25-25048

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### References

- J. C. Marron and R. L. Kendrick, “Distributed aperture active imaging,” Proc. SPIE6550, 65500A (2007). [CrossRef]
- D. J. Rabb, D. F. Jameson, A. J. Stokes, and J. W. Stafford, “Distributed aperture synthesis,” Opt. Express18(10), 10334–10342 (2010). [CrossRef] [PubMed]
- D. J. Rabb, D. F. Jameson, J. W. Stafford, and A. J. Stokes, “Multi-transmitter aperture synthesis,” Opt. Express18(24), 24937–24945 (2010). [CrossRef] [PubMed]
- J. R. Fienup and J. J. Miller, “Aberration correction by maximizing generalized sharpness metrics,” J. Opt. Soc. Am. A20(4), 609–620 (2003). [CrossRef] [PubMed]
- R. A. Hutchin, “Sheared coherent interferometric photography, a technique for lensless imaging,” in Digital Image Recover and Synthesis, P. S. Idell, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 2029, 161–168 (1993).
- T. Kreis, Handbook of Holographic Interferometry: Optical and Digital Methods (Wiley, 2005).
- M. Guizar-Sicairos, S. T. Thurman, and J. R. Fienup, “Efficient subpixel image registration algorithms,” Opt. Lett.33(2), 156–158 (2008). [CrossRef] [PubMed]
- M. Guizar, “Efficient subpixel image registration by cross-correlation,” http://www.mathworks.com/matlabcentral/fileexchange/18401-efficient-subpixel-image-registration-by-cross-correlation

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