Ptychographic lens-less birefringence microscopy using a mask-modulated polarization image sensor

Birefringence, an inherent characteristic of optically anisotropic materials, is widely utilized in various imaging applications ranging from material characterizations to clinical diagnosis. Polarized light microscopy enables high-resolution, high-contrast imaging of optically anisotropic specimens, but it is associated with mechanical rotations of polarizer/analyzer and relatively complex optical designs. Here, we present a form of lens-less polarization-sensitive microscopy capable of complex and birefringence imaging of transparent objects without an optical lens and any moving parts. Our method exploits an optical mask-modulated polarization image sensor and single-input-state LED illumination design to obtain complex and birefringence images of the object via ptychographic phase retrieval. Using a camera with a pixel size of 3.45 μm, the method achieves birefringence imaging with a half-pitch resolution of 2.46 μm over a 59.74 mm2 field-of-view, which corresponds to a space-bandwidth product of 9.9 megapixels. We demonstrate the high-resolution, large-area, phase and birefringence imaging capability of our method by presenting the phase and birefringence images of various anisotropic objects, including a monosodium urate crystal, and excised mouse eye and heart tissues.


Section 1. PS-PtychoLM forward model
Figure S1.PS-PtychoLM image formation.  , , and  represent the incident light field from the   LED element, complex information of the object, and optical mask, respectively.  ,   , and   denote the distances between the LED array and object, the object to the mask, and the mask to sensor plane, respectively.The spatial coordinate  is omitted for simplicity.
The forward imaging model of PS-PtychoLM was used to predict the manner in which the optical wave passing through the object and mask was recorded at the image sensor (Figure S1).Consider an object illuminated by the  th element in the LED array.In the implementation, the distance between the LED elements and the object was large (~400mm), and thus the illumination light wave from the  th LED element (  ) can be assumed to be a plane wave in the object plane, which is expressed as: where  = (, ) denotes the spatial coordinate,  is the imaginary unit,   = ( , ,  , ) is the incident wave vector, and • is the inner product of vectors.Using the PS-PtychoLM configuration (Figure S1), the incident wave can be re-written as  , = −[ −1 ( , / 0 )],  , = −[ −1 ( , / 0 )], where  is the wave number of light and ( , ,  , ) is the lateral location of the  th LED element.The illumination field then interacts with the object and the exit light field in the object plane can be written as  1 () =   ()⨀(), where () is the object complex information and ⨀ stands for the elementwise multiplication.The exit wave  1 then propagates to the mask plane and the wave incident on the mask is given by  2 () =   1 ⨂ 1 (), where   represents the point spread function (PSF) for freespace propagation over the propagation distance  , and ⨂ denotes a convolution operation.In the implementation, free-space propagation was operated in the Fourier space using the angular spectrum method [1][2][3] .The light field  2 is modulated by the optical mask, resulting in  3 () =  2 ()⨀(), where () is the mask transmission function.The modulated light field  3 further propagates to the sensor plane to result in  4 () =   2 ⨂ 3 ().The intensity image captured by the image sensor with the  th LED illumination (  ) can then be written as:

Section 2. PS-PtychoLM image reconstruction procedure
By sequentially turning on each LED, a set of intensity images   ( = 1, 2, … , ) is acquired and used for PS-PtychoLM image reconstruction.In our demonstration, we recorded 81 intensity images with a size of  ×  by turning on the 81 LEDs centered at the optical axis.In order to reconstruct complex information of object and binary mask, we used the regularized ptychographic iterative engine (rPIE) 4 and the reciprocal-space up-sampling method.A detailed reconstruction procedure is presented as follows.
Step 1) We first initialize the object and mask as constant all-one matrix with a size of( × ) × ( × ), where  represents the up-sampling ratio.Under the  th LED illumination, the field at the sensor plane ( 4 ) can be estimated using the PS-PtychoLM forward model.Since the recorded intensity image (  ) has a size  × , the estimated  4 is down-sampled with a factor of , then the information is updated with the acquired intensity image and then up-sampled again by -times.This process is expressed as follows: where ↑  () and ↓  () denote the N-times nearest-neighbor up-sampling and down-sampling operations, respectively.( × ) represents an  ×  all-one matrix.The convolution procedure in the denominator of Eq. ( 3) using ( × ) enforces the intensity sum of each  ×  segment of the result to be equal to the corresponding pixel of the acquired image 5 .Note that, in our implementation, the  ×  all-one matrix was zero-padded to the size ( × ) × ( × ) and then the convolution operation was performed in the Fourier domain.The value of  was set to 4.
Step 2) The updated light field  4 ′ is then back-propagated to the mask plane as: where () represents complex conjugation.
Step 3) Complex information of the mask and light field above the mask is updated using the rPIE algorithm as: where  and  are the weighting parameters of the rPIE algorithm.We set  = 0.8 and  = 0.5 in the implementation.
Step 4) The updated light field  2 ′ is then back-propagated to the object plane: Step 5) Lastly, the complex information of object is updated with the light field  1 ′ as: The object  and mask  are replaced with the updated object  ′ and mask  ′ before starting the update procedures (steps 1-5) using the LEDs in different positions.One iteration is complete if the update procedure is performed for all the LEDs.When the object and mask converge, the reconstruction is finished, and in our experiment, it generally took approximately 10 iterations to complete.
In our PS-PtychoLM implementation, a binary mask with a pre-defined pattern was employed for robust image reconstruction 6 .Even though the mask pattern is pre-defined and fabricated with high fidelity, it is extremely challenging to align the mask in a designated space in the optical setup.In practice, the actual mask pattern that contributes to the image formation may have a lateral shift and rotation, and it may be a small portion of the designed mask pattern.This uncertainty on the actual mask pattern may result in a complete failure of image recovery.We obtained the mask pattern that contributed to the image formation with the three steps outlined below (Figure S2): Step 1) We first perform a blind recovery using a guessed mask pattern.The objective of this blind recovery is to obtain a rough estimate of the mask pattern.In this step, the complex object information and the mask pattern are initialized by all-one matrix, respectively.Then, one iteration of image recovery is conducted, and then rough estimates of the mask pattern and the object are obtained.
Step 2) We then conduct cross-correlation of the estimated mask pattern obtained from Step 1 and the designed mask pattern to determine the amount of lateral shift and rotation of the actual mask with respect to the designed mask pattern.The resultant information was used to further refine the actual mask pattern that contributes to the image formation.
Step 3) The obtained actual mask pattern is then used as the initial guess for the subsequent iteration of image recovery.In order to evaluate the imaging performance of PS-PtychoLM, we first imaged a USAF 1951 resolution target (Edmund optics, #64-862) (Figure S3a-d).An exemplary raw image captured by the PS-PtychoLM is shown in Figure S3a.After the ptychographic reconstruction with 81 image acquisitions, an amplitude image of the resolution target over the entire FoV of 7.07mm x 8.45mm could be reconstructed (Figure S3b).One can see that features in the resolution target can be clearly visualized.Figure S3c is an enlarged image of the area marked with a blue box in Figure S3b.The element 5 in group 7 (line width = 2.46 µm) could be distinguished.The line profile along the dashed red line in Figure S3c is shown in Figure S3d.Our PS-PtyhcoLM prototype achieved a half-pitch resolution of 2.46 µm for a FoV of 7.07 mm x 8.45 mm, which corresponds to an SBP of 9.9 megapixels.
The phase reconstruction performance was then evaluated using a quantitative phase target (Benchmark, Quantitative phase target).Figure S3e depicts the reconstruction result of the phase target using the PS-PtychoLM platform.The phase distribution along the dashed red circle in Figure S3e is presented in Figure S3f.The target features in the reconstructed area shown in Figure S3e are characterized by a height of 200 nm and a refractive index of 1.52.The theoretical phase value, computed from this information and the LED center wavelength (514.5 nm), is indicated by a red dashed line in figure S3f.The reconstructed phase value agrees with the actual value with an error of 6.15%.We employed a pre-defined binary structure as the modulation mask in our PS-PtychoLM implementation.This mask served as a constraint in the iterative phase retrieval process, which helps, along with the captured images, convergence to the solution.We separately evaluated a random mask with unknown micro-/nano-features to examine if it can be used in our PS-PtychoLM imaging.The random mask has been employed in various forms of lens-less ptychographic microscopes to achieve higher spatial resolution 7,8 .Our results are presented in Figure S4.We considered a birefringence object depicted in Figure S4a as the target and performed numerical simulations with an unknown random diffuser and binary mask.Figure S4b,c present the amplitude and phase distributions of the random diffuser used in the simulation.The feature size of the diffuser was set to be 3.45 μm, and its amplitude and phase values were uniformly distributed in the interval [0, 1] and [-, ], respectively.The reconstructed amplitude images of each polarization channel and birefringence information are shown in Figure S4d,e.The amplitude information of the target could not be reconstructed accurately, and consequently, the reconstructed birefringence map also significantly deviated from the ground truth.This can be explained in part by that the random diffuser significantly scrambles the propagation wavefront of the object exit wave, and thus polarization states in the missing pixels could not be correctly interpolated by Newton's polynomial demosaicing method.
In contrast, the use of a pre-determined binary mask with a mean feature size of 27.6 μm enabled an accurate reconstruction of the object birefringence information.Figure S4f,g are the amplitude and phase of the binary mask used in the simulation, respectively.The amplitude of the mask consisted only of 0 and 1, and the phase was constant across the plane (phase = 0 in this simulation).The reconstructed amplitude images of each polarization channel and birefringence map clearly demonstrate that the binary mask enabled accurate estimation of the polarization information throughout all polarization channels, thus resulting in accurate birefringence map.

Figure S2 .Section 3 .
Figure S2.Procedures to obtain the actual mask pattern in the PS-PtychoLM image reconstruction.

Section 4 .
Figure S4.Comparison of the PS-PtychoLM birefringence reconstruction performance with an unknown random diffuser and pre-defined binary mask.(a) Ground truth birefringence information of a simulation target.(b,f) Amplitude and (c,g) phase information of the random diffuser and binary mask, respectively.(d,h)Reconstructed amplitude information of each polarization channel and (e,i) obtained birefringence map using the random diffuser and binary mask, respectively.