1. Introduction

Optical microscopy counts among the most common means of studying structure in a variety of samples ranging from semiconductor microstructures to biological tissues. Microscopy in the visible spectral range represents a highly advanced and well mastered field, which is widely spread in many laboratories worldwide. However, the picture changes considerably with transition to wavelengths above ~1050 nm, where standard silicon-based array detectors have to be replaced by another material – typically InGaAs [1]. Such a detection system is incomparably more expensive than its Si counterpart, partly also due to the high dark current values typical for the infrared (IR) detectors, which necessitate a cooling system implementation and detector temperature stabilization. For these reasons, IR microscopes are less common in scientific laboratories, in spite of a number of interesting materials being opaque in the Si detector spectral range, while being transparent in the near-IR region, such as some metal halide perovskites [2], PbS and PbSe nanostructures, samples on Si wafers, GaAs, CdTe, etc. Moreover, the near-IR region (650-1400 nm) is the spectral range with the lowest absorption in tissue which makes it ideal for use in biology [3].

Standard microscopy acquires an image by a high-quality array detector or by a sequential scanning of the scene of interest by a stabilized detection. In both cases, the number of acquired pixels (datapoints) determines the image resolution. However, the so-called compressed sensing (CS) makes it possible to remove the link between image resolution and the number of pixels of a detector [4]. This is done by rearranging the entire architecture of image acquisition. CS utilizes the fact that a common image is a sparse-like data set in certain bases, e.g., a DCT or a wavelet basis. Sparsity denotes the fact that the vast majority of image components in a certain basis can be neglected and the information can be conserved by using a fraction of the data. This is used in numerous image compression formats, for instance, JPEG.

CS provides a possible solution to the issues of IR microscopy, since the image can be acquired by using a detector as simple as a photodiode. Such imaging is denoted as single-pixel camera (SPC) and it makes it possible to computationally reconstruct an N-pixel image from M measurements of a single-pixel detector, where $M\ll N$ [5]. For each measurement, the image has to be encoded with an uncorrelated pattern, as we will describe in detail below. This is especially appealing in the IR spectral region since the image modulation is incomparably simpler than using an array detector. A differential single-pixel camera is, as we show later, a very useful extension of the SPC concept, which makes it possible, by using a pair of photodiodes, to retain the simplicity of the SPC while considerably improving the quality of the acquired images [6].

The SPC method, as a promising new mode of imaging, has been previously studied with respect to standard and hyperspectral imaging in a broad spectral range by using a variety of approaches to carry out random encoding of an image [7–10]. Implementation in SPC-based fluorescence microscopy in the visible and near-IR spectral range has been previously reported [11,12].

In this article we report on the implementation of the differential single-pixel camera concept in a very simple and robust near-IR microscope (800-1700 nm). We combine previously published ideas of SPC microscopy [6] with the knowledge of the benefits of balanced detection [13] to create an experimental setup, which can be built of parts readily available in most optical laboratories, a digital micromirror device (DMD), and a balanced pair of Ge photodiodes, and can, therefore, be constructed at the extra costs of ~2000€ [14,15]. The microscope can be operated under ambient light in standard laboratory conditions. The microscope detection does not require any temperature stabilization or lock-in detection. The low cost and simplicity of the presented microscope setup can provide access to IR transmission microscopy to a broad range of laboratories. To achieve this goal, the article provides a full description of the used experimental setup, its components, and calibration routines. The setup can, therefore, be assembled and used without deep knowledge in the field of microscopy and CS.

We demonstrate the functionality of the microscope on real samples of CdTe crystals, which are opaque in the visible spectral range, and we compare the images to images obtained by using a standard microscope. At the same time, we prove that the SPC concept cannot be simply replaced by image scanning.

2. Methods

Our aim was to obtain a simple and robust optical setup composed of off-the-shelf components and devices. The optical setup, depicted in Fig. 1, consists of a light source (standard laboratory voltage source 12 V with a halogen light bulb G4) whose light was concentrated by a condenser (achromatic doublet, numerical aperture NA = 0.25) on a sample. The sample was imaged by an objective lens (Olympus Plan Achromat 4x, NA = 0.1) onto a digital micromirror device (DMD, EVM Lightcrafter without the so-called ‘light engine’), where it formed an up-scaled image of the sample. The DMD chip consists of a 2D array (608x684 pixels) of micromirrors, which can be set to a binary position (0 or 1), reflecting the light for each spot into two different directions. The “zero” and “one” images are symmetrical to the incoming beam, separated by ± 12 deg from the incident beam. The input beam from the sample was placed perpendicular to the DMD chip in the horizontal sense. However, in order to detect the two reflected images, we slightly tilted the DMD chip and used an additional rectangular thin mirror to divert the input and output beams on the DMD chip.

 

Fig. 1 Scheme of the experimental setup. Light is propagated from the light source through the sample and then reflected by a DMD in two directions. Both light paths are measured by a balanced pair of photodiodes.

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The detection was carried out by a pair of large-area balanced Ge photodiodes (Newport 2317NF). Owing to the large dimensions of the photodiodes (5 × 5 mm), the detection can be used without any imaging optics beyond the DMD. Nevertheless, with increasing magnification, an improved signal level can be attained by using a pair of lenses (CaF₂, f = 15 mm), each lens refocusing the DMD image onto one photodiode. We closely discuss this in the text below.

The use of a pair of balanced photodiodes enabled us to efficiently cancel the effect of ambient light and to harvest signal from both reflected beams, thus reducing the signal-to-noise ratio by a factor of 2, as we also verified from the experimental data [13]. The signal from the balanced photodiodes was recorded by using a DAQ card (National Instruments, NI USB 6211) and the obtained intensity was used to reconstruct the sample image by using the CS theory, as we describe in the next section.

The spectral response of the microscope is given by the combination of a broad spectrum of light transmitted through the setup (450-2200 nm) with the spectral response of the detector (800-1700 nm, peak efficiency at approx. 1500 nm). The detectable spectral range is, therefore, mainly set by the detector itself.

The presented setup follows the idea of SPC-based microscopy published previously by Radwell et al. [6], where differential detection was implemented on a standard microscopic system by using a DMD applying pairs of a pattern followed by its inverse. The presented setup, on the other hand, puts emphasis on simplicity and robustness. The entire microscope can be created by using three optical elements (condenser, objective, and mirror) and the only unconventional devices used in the setup are a DMD and a balanced pair of photodiodes, which can be purchased for catalogue price not exceeding 2000€. Parallel detection of two inverse patterns on a balanced detector avoids possible interference with oscillatory noise (e.g., power grid frequencies from ambient light) and its higher harmonics.

3. Testing samples

The microscope was initially tested by using a standard USAF 1951 positive resolution target. The microscope performance on real-life samples was evaluated by using a sample of p-type Cl-doped CdTe, which is highly absorbing in the visible spectral range (λ < 860 nm). The sample features crystallographic defects associated with deep levels in the band gap. These defects appear dark in the IR region in contrast to the CdTe crystal, which is transparent.

The sample was previously extensively characterized with respect to its resistivity, Photo-Hall effect spectroscopy, and deep-level activation energies, as can be found in published articles [16,17].

4. Principles of single- and differential single-pixel camera

It was shown in the early works of Donoho [18] and others [19,20] that the measured signal could be sampled well below the Nyquist rate if some simple rules stated by CS are obeyed. CS bypasses the need for precise measurement and allows to directly ‘sense’ the compressed information. This is possible when a random linear combination of single measurements is performed. The SPC experiment, as described by Duarte et al. [5], is based on encoding an image with a series of uncorrelated patterns and detecting the total intensity – see the scheme in Fig. 2(A-B). The image can be subsequently mathematically reconstructed by using knowledge about random patterns and total intensities – see Fig. 2(C).

 

Fig. 2 Principles of the SPC concept with examples of the reconstructed images. (A) Image encoding with random binary masks. (B) Measurement of intensity fluctuations (each datapoint corresponds to a single mask). (C) Reconstruction of USAF 1951 target from the measured data. Blue scale denotes 100 μm. Resolution: 76 × 76 pixels, M/N = 30%. Flat-field correction was applied to the image (see Section 6). (D) Image of the CdTe sample reconstructed for various M/N ratios with finer details emerging with the increasing measurement count. Resolution: 76 × 76 pixels. Note that the same color scheme is used in all presented reconstructions.

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Each uncorrelated pattern (N pixels) can be represented by a N × 1 vector, which is superimposed on an image x (N pixels, 1 × N vector). By generating M different patterns, we gain for each pattern the total intensity y (vector sized 1 × M). This can be expressed in the matrix form as:

Since we aim at significantly reducing the number of measurements M, it holds that $M\ll N$ and the problem in Eq. (1) is heavily underdetermined. However, by using the CS theory, we can still recover the image information based on the so-called regularizing term. The term is a mathematical quantification of an expected image behavior. In our case, we used a commonly employed total variation T, which is a sum of pixel-to-pixel differences in an $n\times n$ image I:

Two examples of reconstructed images are presented in Fig. 2(C) (USAF target) and in Fig. 2(D) (CdTe sample). In principle, Eq. (3) can be solved for a very low M/N ratio, depending on the sparsity of the reconstructed image. A rough reconstruction can already be attained for M/N = 0.05. Nevertheless, we found that the number of measurements M reaching 0.2-0.3 × N is sufficient for a reliable image reconstruction. We also acquired the full number of measurements (M = N, basis scan using Hadamard matrix), as a benchmark of the best attainable reconstruction with the given acquisition noise.

A typical implementation of the SPC approach imposes on an image a binary pattern composed of 0 and 1 pixels, which means that approximately 50% of the incoming light is lost due to the application of the random pattern [5]. Here, we used the so-called digital micromirror device (DMD), which reflects both the “zero” and the “one” pixels in two different directions. By using a pair of balanced photodiodes we were, therefore, able to utilize the signal from both reflected beams and use the difference between the intensities as the signal y. Effectively, this corresponds to the matrix A being filled with + 1 and −1 values. We call this arrangement a differential single-pixel camera.

The differential single-pixel camera (DSPC) features several important advantages. Firstly, it provides an increased signal-to-noise ratio, as stated before. The DSPC is more resistant against fluctuating ambient light, as the fluctuation is cancelled out. Furthermore, the so-called mutual coherence of the sensing matrix A highly improves by switching from the standard SPC to its differential counterpart. Mutual coherence is a measure of the ability of the matrix to provide reconstruction from the compressed measurements, where a lower value ensures a data set reconstruction for a lower M/N ratio.

Finally, SPC measurements are very demanding with respect to the dynamic range of detectors. This arises because the SPC sums up a signal from many random pixels. The useful information is therefore contained in subtle fluctuations around the mean value. The differential detection removes the mean value and shifts the distribution of the signal towards zero. This removes the requirement for a high dynamic range of the signal readout. Nevertheless, it is worth noting that IR detection, including our case, is typically limited by the noise of the detector rather than by the dynamic range of the readout itself.

5. Differential single-pixel camera versus image scanning

In principle, imaging based on a single-pixel detector can be done also by scanning an image pixel by pixel. However, compared to pixel scanning, the DSPC features several advantages. Firstly, every single measurement in DSPC imaging contains information about the whole scene and is, therefore, replaceable by any other measurement. This property is important in applications where noise or outer conditions have a significant influence over the measurement. While in standard image acquisition techniques the noise can corrupt certain image regions, CS can replace the faulty measurements by any other or omit them. Secondly, by using the CS theory, we can attain the same information by decreasing the acquisition time by the factor of 3-5.

Finally, SPC and DSPC techniques are very powerful in cases where the measured values undergo a long-term drift. To demonstrate this, we have carried out a direct comparison between the DSPC and pixel scanning by using the setup in Fig. 1. In one case, we imposed random matrices on the image (DSPC, Fig. 3(A)), while in the other, we acquired the image by using matrices where only one pixel was set to value “1” (pixel scan, Fig. 3(C)). Both measurements were done by using the same intensity of the light illuminating the sample.

 

Fig. 3 USAF 1951 target captured with the same setup via two approaches. (A-B) Measurement using SPC. (C-D) Measurement by a raster scan. (A),(C) Resolution: 76 × 76 pixels. (B),(D) Image binning for resolution:19 × 19 pixels.

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Seemingly, the pixel scanning was heavily penalized by measuring a single pixel at the same light intensity. However, the signal-to-noise ratio (SNR) for the pixel scan and the noise $\epsilon$ is $\epsilon /N$, whereas the measurement employing a random mask features the SNR of $\epsilon /\sqrt{N}$, which arises due to a random sum of pixels. By carrying out the pixel scan at a low resolution (19 × 19 pixels) we were able to reach a SNR comparable to the SPC level, yet the contrast of the acquired image was significantly lower – compare Fig. 3(B) and Fig. 3(D).

The noise during the pixel scan consists of a fine pattern induced by a power grid (50 Hz). Besides, there are prominent dark and bright vertical stripes in the acquired image caused by a significant drift in the detector dark current. The dark current is highly dependent on the detector temperature, which was not stabilized in our measurements.

On the contrary, the detector drift can be easily removed in the DSPC measurements. Here, each measurement is a combination of many random pixels fluctuating around zero. A long-term drift, which corresponds to a temperature change, is extremely unlikely to originate from the measured signal itself. Therefore, we measured short subsets of the whole measurement (96 datapoints) and for each subset the 50% illuminating intensity was obtained by averaging the intensities of two complementary masks. The subsets were then merged according to the 50% intensity level, so that the long-term fluctuations could be eliminated.

In principle, one can improve the SNR during the pixel scan by concentrating the illuminating light into a tiny spot, which will be scanned along the sample – for instance by scanning the sample with a focused laser beam. Nevertheless, this may be a limiting factor for many samples sensitive to light intensity and increased temperature. Moreover, concentrating a high density of an IR light onto a small spot requires either using an IR laser or sophisticated imaging optics to focus the illuminating light source onto the spot.

6. Flat-field correction

The image acquisition can be carried out without any imaging optics beyond the DMD, since the large-area Ge photodiodes in the proximity of the DMD can collect light efficiently. This holds especially for low magnification, because the beam from the objective lens incident on the DMD diverges only slightly. Nevertheless, due to the lower detection efficiency of light on the DMD edge, the recorded image becomes darker on the edge. This effect becomes more pronounced with increasing objective lens magnification.

This effect can be compensated for by creating a flat-field measurement, where no sample is placed in the microscope. Such a measurement leads to a smooth image reflecting only the image edge darkening. The reference image can be then used to correct the reconstructed images for edge darkening and also to calculate the absolute transitivity of the sample.

Moreover, the flat-field image serves as a verification of the position of the detector, where the beams reflected from the DMD should aim at the center of the photodiodes. The detector position was tuned by repeating rapidly four patterns applied on the DMD, where a single quadrant is set to the value “1”, while the rest is set to “0”. When placed in the correct position, the detector shows the same values for all measurements.

7. Microscope calibration and real sample measurements

For the initial acquisitions, we used a USAF 1951 target (see Fig. 2) and a pattern of periodic stripes calibrated on a standard microscope. This allowed us to determine the magnification of the setup. The images were acquired by binning the DMD pixels into the resolution of 76x76 and 114x152. The binning was applied mainly to reduce the acquisition time, which is composed of integration time (0.01 s/measurement, in total around 12 s and 46 s, respectively) and of time for pattern transfer into the DMD. The data transfer was the limiting factor for the acquisition time in our case. This, nevertheless, depends on the used DMD and its internal storage for rapid pattern sequence.

For the 76x76 pixel patterns, our method achieved the physical resolution of 4 and 8$\text{μm}$ per image pixel in the vertical and horizontal direction, respectively. Accordingly, for the 114 × 152 pixel patterns we reached the resolution of 3 and 4$\text{μm}$. The difference between the horizontal and vertical resolution is caused by the aspect ratio of the used DMD, where the micromirrors are arranged in a diamond-like configuration. The resolution can be increased even further by using the native resolution of the DMD and an objective with a high numerical aperture. In our case, the attainable resolution was limited by the used objective lens to approx. 5 μm at the wavelength of 1 μm. To verify the resulting resolution of our setup we extracted a horizontal cut of Fig. 4(A) over the dark spot (defect). Even for the 114 × 152 pixel image, the edges of the spot were step-like, supporting the obtained microscope resolution.

 

Fig. 4 Flat-field correction of an image of the CdTe sample (114 × 152 pixels, M/N = 0.4). (A) Uncorrected image. (B) Flat-field measurement (114 × 152 pixels, M/N = 0.4). (C) Compensated picture. (D) Image of the same sample acquired with a standard microscope with the same magnification; blue scale denotes 100 μm.

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We compared the acquired images of a sample of CdTe with images from a standard microscope in the near-IR range (Olympus IX70, detector based on Sony ICX274AL chip) captured for the same sample – see Fig. 4(D). Clearly, the presented IR microscope can provide the same information about the microstructure of the sample, showing the spherical dark inclusions in the transparent crystal. The quality of the SPC-reconstructed image is to some degree decreased by the flat-field correction, which brings an additional source of noise into the image – compare Fig. 4(A) and Fig. 4(C). This issue can be solved by refocusing the light reflected from the DMD onto the detectors or by correcting the images with a smooth interpolated correction image. At the same time, the SPC image quality can be increased by increasing the number of measurements, which is 2.5 times lower in Fig. 4 compared to standard microscopy (number of pixels).

In contrast to the standard microscope, the SPC image reconstruction can be, to some extent, optimized by tuning the reconstruction parameters in the TVAL3 algorithm – see Fig. 5 [22]. The algorithm can promote either image reconstructions, which faithfully recover the measured total intensities (low-noise measurement, high μ), or it can promote reconstructed images with a low total variation (noisy measurements, low μ),) – see Eq. (3).

 

Fig. 5 Reconstruction of one data set (patterns 114 × 152 pixels, M/N = 0.4) for a range of two TVAL3 parameters μ and maxcnt, demonstrating the effect of the parameters on the resulting image. Flat-field correction was not applied.

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Nevertheless, the optimum conditions of image reconstruction depend mainly on the measurement noise level. Therefore, the reconstruction parameters can be determined in the initial testing measurement and can be retained during the following microscope operation. The operation of the microscope is, therefore, very straightforward and does not require any complex tests of the system.

8. Conclusion

The presented differential single-pixel microscope represents a low-cost alternative to the near-IR microscope. This is achieved by using a simple and robust setup based on standard off-the-shelf optics and devices, using a freely available reconstruction algorithm. The simple construction, calibration, and operation of the microscope enable a broad range of users to build and utilize the setup.

We demonstrate that the microscope can recover the microstructure of a real-life sample equally well as a conventional microscope. Moreover, the presented microscope does not require any temperature stabilization of the detectors and works in ambient light. The cost of the unconventional elements of the setup is 2000€, which can provide access to IR transmission microscopy to a broad range of laboratories. Naturally, an analogous principle can also be extended to other spectral regions, where suitable image encoding is present.