Phase Space Imaging

3D imaging in volumetric scattering media using phase-space measurements
H. Liu, E. Jonas, L. Tian, J. Zhong, B. Recht, L. Waller
Opt. Express 23, 14461-14471 (2015).

We demonstrate the use of phase-space imaging for 3D localization of multiple point sources inside scattering material. The effect of scattering is to spread angular (spatial frequency) information, which can be measured by phase space imaging. We derive a multi-slice forward model for homogeneous volumetric scattering, then develop a reconstruction algorithm that exploits sparsity in order to further constrain the problem. By using 4D measurements for 3D reconstruction, the dimensionality mismatch provides significant robustness to multiple scattering, with either static or dynamic diffusers. Experimentally, our high-resolution 4D phase-space data is collected by a spectrogram setup, with results successfully recovering the 3D positions of multiple LEDs embedded in turbid scattering media.


3D intensity and phase imaging from light field measurements in an LED array microscope
Lei Tian, L. Waller
Optica 2, 104-111 (2015).

Realizing high resolution across large volumes is challenging for 3D imaging techniques with high-speed acquisition. Here, we describe a new method for 3D intensity and phase recovery from 4D light field measurements, achieving enhanced resolution via Fourier Ptychography. Starting from geometric optics light field refocusing, we incorporate phase retrieval and correct diffraction artifacts. Further, we incorporate dark-field images to achieve lateral resolution beyond the diffraction limit of the objective (5x larger NA) and axial resolution better than the depth of field, using a low magnification objective with a large field of view. Our iterative reconstruction algorithm uses a multi-slice coherent model to estimate the 3D complex transmittance function of the sample at multiple depths, without any weak or single-scattering approximations. Data is captured by an LED array microscope with computational illumination, which enables rapid scanning of angles for fast acquisition. We demonstrate the method with thick biological samples in a modified commercial microscope, indicating the technique’s versatility for a wide range of applications.


3D differential phase contrast microscopy with computational illumination using an LED array
Lei Tian, J. Wang, L. Waller
Optics Letters 39, 1326 – 1329 (2014).

We demonstrate 3D differential phase-contrast (DPC) microscopy, based on computational illumination with a programmable LED array. By capturing intensity images with various illumination angles generated by sequentially patterning an LED array source, we digitally refocus images through various depths via light field processing. The intensity differences from images taken at complementary illumination angles are then used to generate DPC images, which are related to the gradient of phase. The proposed method achieves 3D DPC with simple, inexpensive optics and no moving parts. We experimentally demonstrate our method by imaging a camel hair sample in 3D.


Wigner function measurement using a lens let array
Lei Tian, Z. Zhang, Jon. C. Petruccelli, G. Barbastathis
Optics Express 21, 10511-10525 (2013).

Geometrical–optical arguments have traditionally been used to explain how a lenslet array measures the distribution of light jointly over space and spatial frequency. Here, we rigorously derive the connection between the intensity measured by a lenslet array and wave–optical representations of such light distributions for partially coherent optical beams by using the Wigner distribution function (WDF). It is shown that the action of the lenslet array is to sample a smoothed version of the beam’s WDF (SWDF). We consider the effect of lenslet geometry and coherence properties of the beam on this measurement, and we derive an expression for cross–talk between lenslets that corrupts the measurement. Conditions for a high fidelity measurement of the SWDF and the discrepancies between the measured SWDF and the WDF are investigated for a Schell–model beam.


Experimental compressive phase space tomography
Lei Tian, J. Lee, S. B. Oh, G. Barbastathis
Optics Express 20, 8296-8308 (2012).
*Highlighted in the OSA Spotlight on Optics

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.