The resolution of lensless imaging devices is limited by the pixel size of the camera chip ($\sim \mathrm{several}$ micrometers) and the reconstruction algorithm. Indeed, Claus et al. [1] showed that, since the real resolution limitation comes from the numerical aperture (NA) of the detector subtending the object, it can be taken into account in the reconstruction algorithm of the holograms to show object details smaller than the detector pixel size. To increase this resolution, several lensless techniques in microscopy have been proposed. Subpixel perspective sweeping microscopy uses images taken at several illumination angles. Between two illumination angles, the shadows of the sample move over a subpixel distance. Then a highly resolved image is reconstructed numerically. This approach is particularly interesting for samples with a high confluence. A resolution of 660 nm with a 2.2 μm pixel size over a $24{\mathrm{mm}}^2$ field of view (FOV) has been reported with an on-chip device [2,3]. However, in this technique, the illumination is on top of the sample which obstructs its view. Another way to increase the resolution is to use optofluidics. Cui et al. [4] and Zheng et al. [5] used a flow of objects in a microfluidic channel with submicrometer holes placed along the channel. Several projection images are taken, as the object moves with the flow, with a complementary metal oxide semi-conductor camera of 9.9 μm pixel size. Then a highly resolved image is reconstructed numerically. A resolution of 750 nm has been reported.

Finally, in digital inline holography, the increase in resolution has been investigated by using several subpixel shifted holograms on the camera to retrieve a highly resolved hologram processed in a pixel super-resolution algorithm [6–11]. This technique has been proposed with incoherent illumination to create speckle free images; however, the compactness of the imager is then compromised, since a rather large distance (several centimeters) is needed between the source and the sample to obtain enough spatial coherence. The subpixel shifts are induced by changing either the wavelength of the source by several tens of nanometers [8] or the position of the source on top of the sample [6,7,9–11].

In this Letter, we present a compact digital lensless inline holographic imager where subpixel shifts are created by changing the injection current of a vertical-cavity surface-emitting laser (VCSEL). The final reconstructed image has a lateral resolution smaller than the pixel size.

A compact lensless imager performing phase retrieval has already been presented by the authors [12], but it had a resolution limited by the pixel size of the camera. Here a side illumination system is constructed with a prism onto which a volume phase grating is recorded in a 70 μm thick photopolymer (Covestro BAYFOL HX) whose refractive index is similar to that of the prism (1.485 versus 1.52, respectively). The beam from the VCSEL is reflected by total internal reflection (TIR) at the photopolymer-air interface. Indeed, the critical angle at the interface between the photopolymer and the prism is $\sim 80°$, and the beam’s incident angle is $\sim 60°$ with an angular spread of the VCSEL of $\sim 8°$. The TIR occurs at the photopolymer-air interface, since the critical angle at the interface between the photopolymer and air is $\sim 42°$. This ensures that the zero order of the incident beam does not go to the camera. The signal beam coming from the top is transmitted at the photopolymer-air interface due to the 60 deg prism. The recording scheme is shown in Fig. 1. The signal beam has been chosen to be collimated in order to obtain a collimated diffracted beam while using the VCSEL readout depicted in Fig. 2. The recording process is detailed in [12].

 

Fig. 1. Side-view hologram grating recording. The two beams interfere in the photopolymer, inducing a phase hologram grating.

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Fig. 2. Compact lensless subpixel resolution large FOV microscope sketch.

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Moreover, in order to better control the signal beam direction (to avoid refraction at the output of the prism), a recording prism, with the same characteristics of the prism, is added on top of the prism. Index matching oil is added between the two prisms. This is shown in Fig. 1.

The volume hologram is illuminated by a VCSEL (Vixar 680S, NA0.21), which produces a collimated diffracted beam illuminating the sample. Figure 2 is a sketch of the device. The employed monochrome camera has a 5.2 μm pixel size (Thorlabs DCC1545M).

When the injection current of the light source is changed, it induces a wavelength change of the light. This is shown in Fig. 3 which shows the wavelength shift with respect to the VCSEL driving current.

 

Fig. 3. Wavelength shift versus driving current of the VCSEL.

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The change of wavelength results in a slight change in the diffracted angle from the grating. The volume grating has been recorded with a collimated beam and, since the wavelength change is very small (0.044%), the diffraction results in a small angular change with negligible deviation from a plane wave. This creates a shift of the hologram on the camera, which depends on the distance between the sample and the camera. Figure 4 illustrates this situation. The digital hologram depends also on the wavelength; however, since the maximum wavelength change is smaller than 0.5 nm, this change is negligible in the reconstruction process.

 

Fig. 4. Sketch of the subpixel shift created by the wavelength selective diffraction which changes the illumination angle.

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The shift of each hologram is estimated using the hologram taken with the lowest driving current as reference. The algorithm [13] first estimates roughly the shift by cross-correlation and fast Fourier transform and then refines it using upsampling of the discrete Fourier transform in the neighborhood of the first estimation.

Table 1 shows the estimated hologram shifts, obtained with the algorithm above, in horizontal and vertical directions at the camera plane with respect to the wavelength shifts for a sample-camera distance of $\sim 7\mathrm{mm}$. This distance has been chosen to obtain a compact device and to provide a maximum shift of one full pixel at the same time.

Table 1. Hologram Shifts for Different VCSEL Driving Currents for a Sample-Camera Distance of $\sim 7\mathrm{mm}$

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The change of the illumination angle induces a shift in the Fourier domain; however, since the maximum angular shift is $\sim 0.04°$, this change is negligible in the reconstruction process.

Ten digital inline holograms are recorded, corresponding to one per driving current of the VCSEL. The stack of holograms is then processed in a pixel super-resolution algorithm [14] to reconstruct a highly resolved hologram that is then backpropagated [15].

A comparison between the resolution of the amplitude reconstructed image obtained with 10 subpixel shifted holograms and the resolution obtained with 1 hologram taken with a high dynamic range (HDR) is shown in Fig. 5. The HDR image uses as many holograms as the one reconstructed with the proposed method, which makes the contrast (hence, the resolution) comparison more correct.

 

Fig. 5. (a) Reconstructed amplitude using 10 holograms (full FOV). (b) Crop of image (a) (FOV $\sim 0.1{\mathrm{mm}}^2$). (c) Reconstructed amplitude using 1 hologram (crop of $\sim 0.1{\mathrm{mm}}^2$). (d) Reconstructed amplitude from an HDR image using 10 holograms with different exposure times (crop of $\sim 0.1{\mathrm{mm}}^2$).

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A resolution of $\sim 2.76\mathrm{\mu m}$ over a FOV of $\sim 28{\mathrm{mm}}^2$ is demonstrated using a camera with pixels of 5.2 μm. The resolution is improved approximatively by a factor of two. Increasing the sampling rate, therefore, the number of images would result in a higher resolution gain. Unfortunately, this would also result in a longer acquisition time. In order to further increase the resolution without increasing the number of samples, in the future we will consider the introduction of a regularizer, e.g., total variation [16], in the current super-resolution algorithm. Ultimately, the improvement in resolution is limited by the NA of the detector subtending the object and by noise. The gray scale corresponds to intensity values.

The noisy background that can be observed in the reconstructed image is mainly due to two sources. The first one is the nonuniformity of the diffracted beam itself, especially because of the substrate of the photopolymer used to fabricate the gratings. The second is the twin image that creates unwanted diffraction patterns around the features. Indeed, in inline digital holography, the real and virtual images are superimposed with the DC term in the spectrum of the hologram. Therefore, one image (real or virtual) cannot be isolated before backpropagation. It is a well-known problem in inline holography, and it could be tackled using more image acquisition and processing [17–22]. This is out of the scope of this Letter, which has the objective of demonstrating pixel super-resolution. However, if the presented technique is combined with phase retrieval, the twin image problem can be solved in the phase retrieval process [17,20].

Two USAF 1951 test targets were superimposed and imaged simultaneously. One was situated at $\sim 7\mathrm{mm}$ from the camera, and the second one was situated at $\sim 8\mathrm{mm}$ from the camera. Since the samples are highly absorbing, the smallest features were placed at different locations on the camera in $x$ and $y$ directions to avoid the overlapping of large absorbing features. Figure 6 shows a sketch of the experiment.

 

Fig. 6. Sketch of the two superimposed sample experiment. Sample 2 is placed on top of sample 1, and its smallest features are placed in order to avoid overlapping with large absorbing features from sample 1.

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A comparison between the resolution of the reconstructed amplitude obtained with 10 subpixel shifted holograms for both depth planes is shown in Fig. 7. Since the subpixel shifts depend on the distance between the sample and the camera, two sets of shifts were computed, one per object-camera distance. The two reconstructions of the hologram were carried out separately. Finally, the reconstructed holograms were propagated with the corresponding object-camera distance.

 

Fig. 7. (a) Reconstructed amplitude using 10 holograms of the test target situated at $\sim 7\mathrm{mm}$ from the camera. (b) Reconstructed amplitude using 10 holograms of the test target situated at $\sim 8\mathrm{mm}$ from the camera.

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The same resolution is obtained for both targets. This resolution corresponds to the one obtained with the same number of holograms and only one sample between the camera and the prism (see Fig. 5). This shows that the same resolution increase can be obtained at different depths of a sample.

The presented device offers a large FOV of $\sim 28{\mathrm{mm}}^2$ and a resolution of $\sim 2.76\mathrm{\mu m}$ (four more groups are resolved compare to a reconstruction with only one hologram on 1951 USAF test target). The increase of resolution is obtained without any mechanical shifts and by using a cheap and efficient light source. This technique could be combined with digital off-axis holography design, such as in [23], to also obtain the phase of the sample with high resolution.