White-light diffraction phase microscopy at doubled space-bandwidth product

White light diffraction microscopy (wDPM) is a quantitative phase imaging method that benefits from both temporal and spatial phase sensitivity, granted, respectively, by the common-path geometry and white light illumination. However, like all off-axis quantitative phase imaging methods, wDPM is characterized by a reduced space-bandwidth product compared to phase shifting approaches. This happens essentially because the ultimate resolution of the image is governed by the period of the interferogram and not just the diffraction limit. As a result, off-axis techniques generates single-shot, i.e., high timebandwidth, phase measurements, at the expense of either spatial resolution or field of view. Here, we show that combining phase-shifting and off-axis, the original space-bandwidth is preserved. Specifically, we developed phase-shifting diffraction phase microscopy with white light, in which we measure and combine two phase shifted interferograms. Due to the white light illumination, the phase images are characterized by low spatial noise, i.e., <1nm pathlength. We illustrate the operation of the instrument with test samples, blood cells, and unlabeled prostate tissue biopsy. © 2016 Optical Society of America OCIS codes: (180.3170) Interference microscopy; (170.0110) Imaging systems; (120.5050) Phase measurement; (999.9999) Quantitative phase imaging. References and links 1. G. Popescu, Quantitative Phase Imaging of Cells and Tissues (McGraw-Hill, 2011). 2. L. Miccio, A. Finizio, R. Puglisi, D. Balduzzi, A. Galli, and P. Ferraro, “Dynamic DIC by digital holography microscopy for enhancing phase-contrast visualization,” Biomed. Opt. Express 2(2), 331–344 (2011). 3. B. Rappaz, P. Marquet, E. Cuche, Y. Emery, C. Depeursinge, and P. Magistretti, “Measurement of the integral refractive index and dynamic cell morphometry of living cells with digital holographic microscopy,” Opt. Express 13(23), 9361–9373 (2005). 4. B. Kemper, A. 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In terms of instrumentation, all QPI approaches rely on interference, because experimentally we only have access to phase differences between two fields, or, equivalently, the phase of cross-correlation functions and not of the fields themselves.Sometimes, in-line holographic setups, e.g., transport of intensity methods are termed "non-interferometric" [23], likely because they do not involve a classical interferometer, in which two beams are physically separated.However, even in this case the phase information is experimentally extracted from the interferogram generated by the incident and scatter light.In order to decouple the phase and amplitude information, several intensity measurements are developed at different phase shifts between the two fields.This class of instruments is referred to as phase-shifting.
The second type of QPI instruments is called off-axis.Unlike phase-shifting methods, the off-axis ones provide single-shot phase information, thus allowing for much faster acquisition rates.However, this high time-bandwidth method comes at the expense of the spacebandwidth product, i.e., in off-axis geometries one has to compromise either spatial resolution or field of view.Slightly off-axis geometries [24] combining off-axis and phase-shifting can provide a trade-off between the time-bandwidth product and space-bandwidth product for accurate QPI instruments.
In this paper we present a new QPI method, referred to as phase shifting diffraction phase microscopy with white light (PSwDPM), which can retrieve the unaltered space-bandwidth product in diffraction phase microscopy (DPM) [25][26][27][28][29].In terms of the advancement with respect to Ref. 24, we would like to point out that our method provides two key improvements.1) the geometry is common path and, as a result, the temporal stability is intrinsically better than in traditional Mach-Zehnder interferometers.2) The white light illumination provides speckle-free images, which otherwise plague the laser methods.Due to these two improvements, our measurements are both temporally and spatially more sensitive.

Theory
The PSwDPM setup is shown in Fig. 1.An inverted microscope is equipped with a halogen lamp, whose light is spatially filtered by a condenser aperture.Thus, the sample is illuminated by a collimated white light beam.The microscope generates a diffraction-limited image at its output port.Precisely at the image plane, we place a diffraction grating, of 33.33 lines/mm.The grating generates diffraction orders, out of which we use two: the 0th and 1st order.Essentially, the grating is used to form a compact Mach-Zehnder type interferometer.In order to covert the 0th order into the reference field of the interferometer, we spatially filter it using Lens L1 and a pinhole.Lens L2 is forming back the image at the CCD plane.The interferogram recorded by the CCD consists of the original image, carried by the 1st order of diffraction, overlapped with the off-axis reference, given by the 0th order.Because of the common-path geometry, DPM is extremely stable, resulting in temporal pathlength noise of <1nm, without active stabilization measurers [26].The measured irradiance distribution has the form ( ) ( ) where I 0 and I 1 are the irradiance of the 0th and 1st order, respectively, and φ is the phase map of interest.The spatial frequency, α = 2π/Λ, is due to the grating, with Λ is the period of the grating.In order to establish the sampling conditions for the interferogram, we perform a spatial frequency domain analysis.Taking the Fourier transform of the measured intensity, we obtain , , where ℑ indicates the Fourier transform operator, and I  is the Fourier transform of I. Using the correlation theorem and the fact that U 0 is uniform after spatial filtering, Eq. ( 2) becomes where ⊗ indicates the convolution operation.Equation ( 3) provides insights into the frequency coverage of the interferogram, as follows.Since U 1 is diffraction limited, it is a band-limited signal, with its maximum frequency governed by the NA of the objective,  has a maximum frequency of 2βNA, we find that the minimum modulation frequency must be α ≥ 3βNA.On the other hand, the maximum frequency of the entire signal is 4βNA.According to the Nyquist theorem, the pixel sampling frequency, k p , must satisfy k p ≥ 8βNA.
In PSwDPM, the correlation term is subtracted out, such that α can be lower, namely, α = βNA, without loss of information.As a result, the highest frequency of the signal is k max ′ = 2βNA, and the new pixel sampling frequency is k p ′ ≥ 4βNA.Thus, the space-bandwidth product is twice longer.An illustration of these sampling conditions can be found in Fig. 2 of Ref [26].Note that, in on-axis methods, the auto correlation term is eliminated via phase shifting and α = 0.In this case, the highest frequency is simply k max ′′ = βNA, and the pixel sampling condition is k p ′′ ≥ 2βNA.Thus, it is apparent that by collecting two phase shifted frames, one gains a factor of 2 in space-bandwidth product, while collecting 4 improves the product by a factor of 4.
Figure 2(a) shows an example of the measured interferogram.Usually, this image is Fourier transformed, which results in a spatial power spectrum as illustrated in Fig. 2(c).The DC and one of the orders are filtered out and the resulting signal Fourier transformed back to the spatial domain.This procedure amounts to a Hilbert transform and yields the complex analytic signal associated with cos[φ(x,y) + αx], the argument of which gives the phase of interest, φ.We show that by shifting the grating by half its period, i.e., by x 0 = π/α, and recording the two respective interferograms, we can eliminate the autocorrelation, i.e., the DC term in Fig. 2(c), thus, extending the frequency coverage of the first order.This result is illustrated in Fig. 2(d).For this proof of principle study, we shifted the grating manually using a precision translation stage.In essence, by acquiring two different interferograms and subtracting them, ( ) ( ) ( )  , we obtain a phase image of higher resolution, as seen by comparing Fig. 2(f) vs. 2e.

Results
In order to demonstrate the resolution gain of PSwDPM, we imaged both test samples and biological samples.Figure 3 shows the comparison between the performance of regular wDPM and PSwDPM when imaging an Airforce target.We perform the side by side comparison as a function of the digital filter (F) applied in the in the spectral domain.Thus, F = 1.0 corresponds to the diameter of the circle in Fig. 2(d) that touches the zero-frequency point.This is the maximum frequency coverage that can be retrieved from the sample and is only available when the DC is eliminated via phase shifting.When the DC order is present, F = 1 results in strong overlap between the frequencies of the first and zeroth order, i.e., aliasing.Note that for all values of F, the spatial phase noise is lower for the phase shifting method (Figs.3(a)-3(h)).These results indicate also that for F ≥ 0.5, the wDPM suffers from aliasing, which becomes severe for F = 1.0.PSwDPM is free of these artifacts for all Fvalues.Figure 3(i) shows the high frequency boost in PSwDPM compared to the artifact-free wDPM (F = 0.3).

Summary and discussion
In summary, we developed PSwDPM to combine phase-shifting and off-axis techniques and overcome the original space-bandwidth product of the image.Of course, we achieve this at the expense of acquisition rate.In contrast to wDPM, PSwDPM is no longer single shot and instead requires the acquisition of two subsequent interferograms.Note that the acquisition rate of PSwDPM is still faster than phase shifting methods, which typically require four frames.Thus, we expect that our new technique will be most practical in applications that require the largest resolution and field of view possible.

Fig. 3 .
Fig. 3. (a-h) Height maps of USAF resolution target with different BPF size, with the color bar indicates height in nm.(i) Angular spectrum Next, we illustrate the increased performance by imaging an unlabeled prostate tissue biopsy.Figures 4 show the comparison between regular wDPM and PSwDPM imaging.Again, the aliasing artifact due to the frequency overlap in the spectrum domain is apparent for wDPM, at F ≥ 0.5.

Fig. 4 .
Fig. 4. Phase maps of an unlabeled prostate biopsy obtained by wDPM and PSwDPM, with the color bar indicating phase in radians.