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. 2012 Apr 9;20(8):8296-308.
doi: 10.1364/OE.20.008296.

Experimental Compressive Phase Space Tomography

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Free PMC article

Experimental Compressive Phase Space Tomography

Lei Tian et al. Opt Express. .
Free PMC article

Abstract

Phase space tomography estimates correlation functions entirely from snapshots in the evolution of the wave function along a time or space variable. In contrast, traditional interferometric methods require measurement of multiple two-point correlations. However, as in every tomographic formulation, undersampling poses a severe limitation. Here we present the first, to our knowledge, experimental demonstration of compressive reconstruction of the classical optical correlation function, i.e. the mutual intensity function. Our compressive algorithm makes explicit use of the physically justifiable assumption of a low-entropy source (or state.) Since the source was directly accessible in our classical experiment, we were able to compare the compressive estimate of the mutual intensity to an independent ground-truth estimate from the van Cittert-Zernike theorem and verify substantial quantitative improvements in the reconstruction.

Figures

Fig. 1
Fig. 1
(a) Input mutual intensity of a GSMS with paramters σI = 17 and σc = 13, (b) data point locations in the Ambiguity space, mutual intensities estimated by (c) FBP and (d) LRMR methods.
Fig. 2
Fig. 2
The first nine coherent modes of the mutual intensity in Fig. 1(a). (a) Theoretical modes, and (b) LRMR estimates.
Fig. 3
Fig. 3
Eigenvalues of the mutual intensity in Fig. 1(a). (a) Theoretical values, (b) FBP estimates, (c) LRMR estimates, and (d) absolute errors in the LRMR estimates versus mode index.
Fig. 4
Fig. 4
Oversampling rate versus relative MSE of LRMR estimates. The input field is a GSMS with parameters σI = 36 and σc = 18. The noisy data is generated with different SNR from (a) an additive random Gaussian noise model, and (b) a Poisson noise model.
Fig. 5
Fig. 5
Experimental arrangement
Fig. 6
Fig. 6
Intensity measurements at several unequally spaced propagation distances.
Fig. 7
Fig. 7
(a) Real and (b) imaginary parts of the radial slices in Ambiguity space from Fourier transforming the vectors of intensities measured at corresponding propagation distances.
Fig. 8
Fig. 8
Real part of the reconstructed mutual intensity from (a) FBP; (b) LRMR method.
Fig. 9
Fig. 9
Eigenvalues estimated by (a) FBP, and (b) LRMR method.
Fig. 10
Fig. 10
(a) Intensity measured immediately to the right of the illumination slit; (b) real part of van Cittert–Zernike theorem estimated mutual intensity immediately to the right of the object slit; (c) eigenvalues of the mutual intensity in (b); (d) absolute error between the eigenvalues in Fig. 9(b) and 10(c) versus mode index.
Fig. 11
Fig. 11
(a) LRMR estimated coherent modes of the mutual intensities in Fig. 8(b), and (b) coherent modes of the mutual intensities in Fig. 10(b), calculated via use of the van Cittert–Zernike theorem, and assumption of incoherent illumination.

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