## Multi-transmitter aperture synthesis |

Optics Express, Vol. 18, Issue 24, pp. 24937-24945 (2010)

http://dx.doi.org/10.1364/OE.18.024937

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### Abstract

Multi-transmitter aperture synthesis is a method in which multiple transmitters can be used to improve resolution and contrast of distributed aperture systems. Such a system utilizes multiple transmitter locations to interrogate a target from multiple look angles thus increasing the angular spectrum content captured by the receiver aperture array. Furthermore, such a system can improve the contrast of sparsely populated receiver arrays by capturing field data in the region between sub-apertures by utilizing multiple transmitter locations. This paper discusses the theory behind multi-transmitter aperture synthesis and provides experimental verification that imagery captured using multiple transmitters will provide increased resolution.

© 2010 OSA

## 1. Introduction

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

2. S. T. Thurman and J. R. Fienup, “Phase-error correction in digital holography,” J. Opt. Soc. Am. A **25**(4), 983–994 (2008). [CrossRef]

4. N. J. Miller, M. P. Dierking, and B. D. Duncan, “Optical sparse aperture imaging,” Appl. Opt. **46**(23), 5933–5943 (2007). [CrossRef] [PubMed]

5. A. J. Stokes, B. D. Duncan, and M. P. Dierking, “Improving mid-frequency contrast in sparse aperture optical imaging systems based upon the Golay-9 array,” Opt. Express **18**(5), 4417–4427 (2010). [CrossRef] [PubMed]

6. 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]

7. J. W. Stafford, B. D. Duncan, and M. P. Dierking, “Experimental demonstration of a stripmap holographic aperture ladar system,” Appl. Opt. **49**(12), 2262-2270 (2010). [CrossRef] [PubMed]

8. B. D. Duncan and M. P. Dierking, “Holographic aperture ladar,” Appl. Opt. **48**(6), 1168–1177 (2009). [CrossRef]

## 2. Theory

*(x*will yield a target plane field

_{n},y_{n})*U*given bywhere

_{T}(x_{T},y_{T})*z*is the propagation distance between the transmitter and target planes,

*λ*is the wavelength of the source,

*U(x-x*. Note that the transmitted field is written as a function of transmitter location coordinates

_{n},y-y_{n})*(x*and is defined to be identical between all transmitter locations. By rearranging the terms in Eq. (1) the target plane field can then be written aswhere

_{n},y_{n})*δ()*is the dirac delta function. Furthermore, the transmitter location-dependent tilt term can be factored out of Eq. (2) and the diffracted field can then be rewritten such thatwhere

*U*is the general field given by far-field diffraction from the identical transmitters:The reflected field in the target plane

_{T}’(x_{T},y_{T})*U*is written as the product of the illuminating field and the reflection coefficient of the scene such thatwhere

_{refl}(x_{T},y_{T})*r(x*describes the reflection coefficient of the target scene. The field measured at the receiver plane

_{T},y_{T})*U*is found by multiplying the receiver pupil function by the Fraunhofer diffraction pattern of the reflected fieldwhere

_{R}(x,y)*P(x,y)*is the pupil function of the receiver array. The expression for the received field can be simplified through use of the convolution theorem. The resulting expression iswhere

*P(x,y)*can be used to receive a large area of the backscattered wavefront by illuminating the target from multiple transmitter positions. The large collection area is in contrast to the relatively small receiver area of an individual sub-aperture. In short, a translated transmitter creates a tilt phase in the target plane which itself results in a shifted field across the static receiver array. This process is equivalent to translating the sub-apertures and leads to the aforementioned aperture gain across the receiver array.

## 3. Experiment

### 3.1 Hardware

*U*is imaged onto a

_{t}(x,y)*Lumenera 120m*camera and mixed with a tilted plane wave LO field described by

*U*. The resultant fringes are captured by the camera and can be digitally processed using standard digital holographic techniques so that the field

_{LO}(x,y)*U*can be recovered [6

_{t}(x,y)6. 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]

*(Δx,Δy)*and the initial transmitter aperture location is defined to be location

*(0,0)*. The transmitter locations were based on shifts equal to half of the aperture-to-aperture separation of the fixed array, thus maximizing the overlap in between subsequent captures of the received field.

*U*. A transmitter is moved to each of the locations to gather the coherent pupil plane information for each transmitter location separately. For each realization the aperture function,

_{t}(x,y)*P(x,y)*, and the transmitter locations

*(x*are used to place the reconstructed field in the combined coherent pupil plane.

_{n},y_{n})### 3.2 Processing and results

2. S. T. Thurman and J. R. Fienup, “Phase-error correction in digital holography,” J. Opt. Soc. Am. A **25**(4), 983–994 (2008). [CrossRef]

6. 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]

*S*used here is given bywhere

_{A}*N*is the total number of images captured and

*I*is the

_{n}(x,y)*n*image. This method allows for the individual sub-aperture images to be sharpened based on the incoherent, speckle-averaged image [6

^{th}**18**(10), 10334–10342 (2010). [CrossRef] [PubMed]

## 4. Conclusions

## References and links

1. | J. C. Marron and R. L. Kendrick, “Distributed Aperture Active Imaging,” Proc. SPIE |

2. | S. T. Thurman and J. R. Fienup, “Phase-error correction in digital holography,” J. Opt. Soc. Am. A |

3. | J. C. Marron, and R. L. Kendrick, “Multi-Aperture 3D Imaging Systems,” Aerospace Conference, 2008 IEEE (2008). |

4. | N. J. Miller, M. P. Dierking, and B. D. Duncan, “Optical sparse aperture imaging,” Appl. Opt. |

5. | A. J. Stokes, B. D. Duncan, and M. P. Dierking, “Improving mid-frequency contrast in sparse aperture optical imaging systems based upon the Golay-9 array,” Opt. Express |

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

7. | J. W. Stafford, B. D. Duncan, and M. P. Dierking, “Experimental demonstration of a stripmap holographic aperture ladar system,” Appl. Opt. |

8. | B. D. Duncan and M. P. Dierking, “Holographic aperture ladar,” Appl. Opt. |

9. | C. V. Jakowatz, D. E. Wahl, P. H. Eichel, D. C. Ghiglia, and P. A. Thompson, |

**OCIS Codes**

(100.3010) Image processing : Image reconstruction techniques

(090.1995) Holography : Digital holography

**ToC Category:**

Image Processing

**History**

Original Manuscript: August 25, 2010

Revised Manuscript: November 5, 2010

Manuscript Accepted: November 8, 2010

Published: November 15, 2010

**Citation**

David J. Rabb, Douglas F. Jameson, Jason W. Stafford, and Andrew J. Stokes, "Multi-transmitter aperture synthesis," Opt. Express **18**, 24937-24945 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-24-24937

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### References

- J. C. Marron and R. L. Kendrick, “Distributed Aperture Active Imaging,” Proc. SPIE 6550, 65500A (2007). [CrossRef]
- S. T. Thurman and J. R. Fienup, “Phase-error correction in digital holography,” J. Opt. Soc. Am. A 25(4), 983–994 (2008). [CrossRef]
- J. C. Marron, and R. L. Kendrick, “Multi-Aperture 3D Imaging Systems,” Aerospace Conference, 2008 IEEE (2008).
- N. J. Miller, M. P. Dierking, and B. D. Duncan, “Optical sparse aperture imaging,” Appl. Opt. 46(23), 5933–5943 (2007). [CrossRef] [PubMed]
- A. J. Stokes, B. D. Duncan, and M. P. Dierking, “Improving mid-frequency contrast in sparse aperture optical imaging systems based upon the Golay-9 array,” Opt. Express 18(5), 4417–4427 (2010). [CrossRef] [PubMed]
- 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]
- J. W. Stafford, B. D. Duncan, and M. P. Dierking, “Experimental demonstration of a stripmap holographic aperture ladar system,” Appl. Opt. 49(12), 2262-2270 (2010). [CrossRef] [PubMed]
- B. D. Duncan and M. P. Dierking, “Holographic aperture ladar,” Appl. Opt. 48(6), 1168–1177 (2009). [CrossRef]
- C. V. Jakowatz, D. E. Wahl, P. H. Eichel, D. C. Ghiglia, and P. A. Thompson, Spotlight-mode synthetic aperture radar: a signal processing approach (Springer, 1996), Chap. 2.

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