Phase Retrieval of Scattered Fields

1. Phase Retrieval of Scattered Fields Greg Hislop and Andrew Hellicar CSIRO, ICT Centre, Sydney greg.hislop@csiro.au

2. Introduction As the frequency of electromagnetic radiation increases so does the difficulty and financial cost of measuring phase information. CSIRO’s Wireless Laboratory’s Imaging Project is developing an all-electronic terahertz imaging system. Measuring phase with such a system would prove extremely difficult so an investigation has been done into the possibility of reconstructing phase from amplitude only measurements. This work considers the retrieval of phase information from unknown scatterers placed in a known plane wave field. To retrieve the phase, the field’s amplitude is measured across two parallel planes and signal processing is applied to reconstruct the phase. Very little literature exists on this problem so two techniques were taken from the related field of phase retrieval for antenna characterisation and adjusted for the problem at hand. These two techniques are the well established method of successive projections and a more recent conjugate gradient approach.

3. Measurement Setup The scenario of interest consists of a known field incident upon an unknown scatterer centered in a hypothetical scatterer’s plane. The scatterer may be compact, as depicted (constrained), or may completely obscure the incident field (unconstrained). The transmitted amplitude is measured across two parallel planes allowing our algorithms to reconstruct the phase.

4. Successive Projections • This is an iterative technique which creates and updates an estimate of the amplitude and phase of the field for constrained scatterers as follows: • Create an estimate of the field and set the estimate’s phase to that which would have been measured at the first plane had no scatterer been present (ie phase of the incident field). • Set the estimate’s amplitude to that measured on the first plane. • Back propagate to the scatterer’s plane and set the estimate’s field terms, outside the scatterer’s physical extent, equal to the incident field. • Propagate to the second measurement plane and change the estimate’s amplitude to the measured amplitude. • Back propagate to the scatterer’s plane as per point 3 and again change the estimate’s terms outside the scatterer’s physical extent to the incident field. • Propagate to the first measurement plane and repeat steps 2-6 until a suitable cost function stabilises.

5. Conjugate Gradients • The method of conjugate gradients operates as follows for constrained scatterers: • Start with an initial estimate of the field at the scatterer’s plane (note the field outside the scatterer’s physical extent does not change and is set to the incident field). • Propagate this field to the two measurement planes. • Evaluate a quadratic cost function (and it’s gradient) relating the estimate's power to the measured power. • Use the gradient and the previous search direction to determine via the method of conjugate gradients a new direction in which to step the field estimate. • Solve for the optimum step distance in the given direction by algebraically minimising the cost function. • Update the estimate of the field at the scatterer’s plane using the distance and direction determined. • Continue steps 2-7 until the cost function stabilises.

6. Unconstrained Scatterers For unconstrained scatterers (scatterers larger then the incident field’s extent), no restraint is available at the scatterer’s plane. This greatly increases the number of local minima making the correct solution hard to find. To cater for unconstrained scatterers our techniques initially include only small spatial frequency terms and then progressively include the higher terms. This allows for false minima avoidance by increasing the ratio of data to unknowns.

7. Testing • Two test scatterers (described below) were used in synthetic experiments, one in a constrained scenario (large measurement plane) and the other unconstrained (measurement plane same size as target). Unbracketed parameters used in constrained case. Bracketed parameters are changes made for unconstrained case. 1λ 3λ 2λ 3λ 2λ 2λ 3λ 3λ σ = 0 S/m εr = 7 (7.5) μr= 1 σ = 0 (50) S/m εr = 3 (4) μr= 1 (1.5) σ = 0 (50) S/m εr = 3 (4) μr= 1 (1.5) σ = 0 S/m εr = 9 (2) μr= 1 σ = 0 S/m εr = 5 μr= 1 σ = 0 S/m εr = 7 (7.5) μr= 1 σ = 0 S/m εr = 5 μr= 1 8λ 18λ

8. Example Reconstructions for the Constrained Scatterer across a Range of Errors

9. Histograms of Average Phase Error at Different SNR for the Constrained Scatterer

10. Example Reconstructions for the Unconstrained Scatterer across a Range of Errors

11. Histograms of Average Phase Error at Different SNR for the Unconstrained Scatterer

12. Comparison Between the Two Techniques • The successive projections performs slightly better then the conjugate gradients technique. • Successive Projections is simpler to implement, faster to run and the input parameters are more logical. • Unlike the successive projections’ cost function, that of the conjugate gradients’ method is guaranteed to decrease monotonically.

13. Conclusions • By using two parallel measurement planes, the phase of a scattered field may be reconstructed using numerical techniques. • The successive projections technique slightly out performs the conjugate gradients technique. • Phase reconstruction is possible for constrained scatterers at significant noise levels. • For unconstrained scatterers reconstruction is possible at more moderate noise levels.