Phase Shifting Holography for THz Near-field/far-field Prediction (2013-10-02)
The following paper Junkin, G. "Phase Shifting Holography for THz Near-Field/Far-Field Prediction", Progress In Electromagnetics Research Letters, Vol. 44, 15-21, 2014 demonstrates how it is possible to recover the near-field amplitude and phase (A) at any point using the power measurements (holograms) of H1 = (A + R1), H2 = (A + R2) and H3 = (A + R3) where R1, R2, R3 are three known reference waves. This paper demonstrates how to choose the correct solution at any point in space (traditionally called the conjugate image problem) based on six possible solutions. There are six possible solutions because each pair of holograms generates two possible complex solutions. There are 3 pairs of holograms since we measure 3 distinct holograms. We suppose that the reference waves have a known (phase & amplitude) relationship between them. The paper demonstrates that the best choice is the have the reference waves phased at 120º with respect to each other. Since there are 3 pairs of holograms, there are three correct solutions (slightly different, because of noise) that can be averaged to improve signal to noise ratio. The performance of the phase-shifter used to produce R1, R2, R3 is a critical parameter that should be calibrated to better than about 0.2º for accurate far-field prediction. The technique could have good data acquisition speed as it avoids switching on/off the reference wave, once the reference wave is known. Unlike Phase Retrieval which requires complete patterns, this method does not have any special sampling requirements (other than Nyquist) and works on a point by point basis. In common with standard complex near-field measurements, these latter features are very convenient for antenna near-field alignment and spatial under-sampling using directive probes.
Please note that holography in itself is not an inverse method. Once the complex field has been recovered, the antenna aperture is produced by inverse propagation. That would be the simple way to do the inverse problem, but only regarding field, not antenna current. Holography is also not a non-linear method, such as Phase Retrieval, there is no cost function, no iteration, no non-linear solver. The method would obviously be quite useless if it didn´t work with noise. The video below demonstrates that it is indeed very noise tolerant. The magical part of the method is not proven analytically in the paper, but a patient and observant reader can appreciate where is the key to the method.
The beauty of using CUDA with the random number generator to simulate holograms in psuedo-realtime was the element that lead to the discovery of the best phase shift of 120º. In near-field/far-field one wishes to avoid asymmetric behaviour in the complex plane because this leads to coherent near-field errors that combine to produce spurious sidelobes. This method is free from this defect. More details can be found in the paper.
Graphics cards have developed into extremely powerful computational devices precisely because of the desire to create and display images quickly. In this application the GPU is used to create holograms, and then using random numbers (https://developer.nvidia.com/curand) simulate noisy measurements, and then subsequently perform image reconstruction using special algorithms. This is all done in device code, with up to 2048X2048 points in the GTX480 card. Please note that because of the YouTube video compression algorithm, at certain points in the video the playback is very pixelated. This has nothing to do with the PSH algorithm.
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