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We experimentally demonstrated single-shot phase imaging with a coded aperture (SPICA), which connects digital holography and coherent diffractive imaging based on compressive sensing to realize the advantages of both methods simultaneously. SPICA allows the observation of a complex field with a simple, single-shot optical setup that does not need reference light and does not suffer from losses associated with the field-of-view and spatial resolution. Experiments showed the promising capabilities of SPICA for single-shot holographic imaging.
© 2015 Optical Society of America
1. Introduction
Phase imaging allows observation of not only the amplitude but also the phase of the light wave distribution of an object. In particular, quantitative phase imaging has been widely used in various fields, such as biomedicine and industry [1]. It can be used to image label-free specimens with a low light absorption and/or a three-dimensional shape. Phase imaging has become an attractive research topic in optics and photonics.
Digital holography (DH) is a representative method for quantitative phase imaging [2]. Several DH methods for multidimensional imaging from a single hologram, such as three-dimensional and/or spectral imaging, have been reported [3, 4]. To remove the autocorrelation and the conjugation signals of an object wave, off-axis holography and phase-shifting holography have been proposed [5, 6]. The former approach achieves single-shot phase imaging but it has a limited spatial frequency bandwidth (SFB). The latter does not restrict the SFB, but it re-quires multiple measurements. Furthermore, both approaches need reference light to determine the interference quantitatively. Introducing reference light makes the optical set-ups larger and more complex.
Coherent diffractive imaging (CDI) is also an established method for quantitative phase imaging [7]. CDI can realize single-shot phase imaging without reference light, and therefore, its optical set-up can be simplified compared with DH. However, the field-of-view (FOV) of CDI is severely limited for phase retrieval (PR) from a single intensity image of a diffraction pattern [8]. Multi-shot CDI approaches, such as ptychography and the transport of intensity equation, have been proposed to extend the FOV [9, 10]. In these multi-shot approaches, the imaging conditions, including the FOV and the focusing distance, are sequentially scanned. Both single-shot and multi-shot CDI have been used particularly in the X-ray regime, where focusing lenses are difficult to fabricate [11–13]. In the visible regime, Fourier ptychography, which can be regarded as a type of multi-shot CDI, has become a hot topic recently [14,15].
Recently, we have proposed and numerically demonstrated single-shot phase imaging with a coded aperture (SPICA), which combines DH and CDI to achieve their respective advantages simultaneously [16, 17]. SPICA can realize single-shot phase imaging without reference light and alleviates the limitations of the SFB and the FOV in current methods for quantitative phase imaging. In the work described in this paper, we experimentally demonstrated SPICA.
2. Single-shot phase imaging with a coded aperture (SPICA)
The optical set-up of SPICA used for experiments described in this paper is shown in Fig. 1. SPICA connects DH and CDI with a coded aperture (CA), which is composed of random pin-holes on a mask. The complex field of the object illuminated with a collimated beam propagates onto the CA. The CA sieves the propagating field. The sieved field propagates onto the image sensor. The image sensor measures a single intensity image of the propagating field. This imaging process is written as
where is the vectored captured intensity image, is a Toeplitz matrix for the Fresnel propagation at distance z [18], is a diagonal matrix to sieve the complex field with the CA, and is the vectored complex field of the object. Here, zs is the distance between the CA and the image sensor, zo is the distance between the object and the CA, and Nx and Ny are the numbers of elements along the x- and y-axes, respectively, as shown in Fig. 1.As indicated in the bottom part of Fig. 1, the sieved field just behind the CA can be reconstructed by a PR method with the CA support because the number of transmissive pixels on the CA is sufficiently smaller than the pixel count of the captured intensity image [8]. The object field is retrieved from the reconstructed sieved field with a compressive sensing algorithm with a sparsity constraint [19,20]. Compressive sensing is a powerful framework for observing large object data from a number of measurements smaller than that stipulated by the sampling theorem [21,22], and it has been recently introduced to DH [23–26]. Theories of SPICA, including details of imaging models and conditions, were derived in [16,17].
3. Experiments
In the experiments, we used several objects to show the imaging performance of SPICA. As shown in Fig. 1, the CA was implemented with a spatial light modulator (SLM, LC2002, manufactured by Holoeye, pixel count: 600 × 800 pixels, pixel pitch: 32 μm × 32 μm) located between two perpendicularly arranged polarizers and capable of pixelwise intensity modulation. The image sensor (CoolSNAP MYO, manufactured by Photometrics, pixel count: 1460 × 1940, pixel pitch: 4.54 μm × 4.54 μm) was located 8 cm (zs) from the SLM. The objects were illuminated with a collimated beam from a laser diode (L658P050, manufactured by Thorlabs, wavelength: 658 nm). The hybrid input-output (HIO) algorithm was used to retrieve the complex field just behind the CA [8]. The two-step iterative shrinkage/thresholding (TwIST) algorithm with two-dimensional total variation was chosen to reconstruct the object field from the complex field retrieved with HIO [27, 28]. Details of this reconstruction process are shown in [16,17].
3.1. Wires
In the first experiment, the object was composed of two wires with a width of 30 μm located 13 cm (zo1) and 20 cm (zo2) away from the CA, respectively. These wires were arranged perpendicularly to each other. Figure 2(a) shows the CA pattern, which corresponds to 105 × 105 pixels on the SLM. The transmission factor, which is the ratio of the total area of the pinholes to the whole CA area, was 30 %. This factor was derived from [16], and its inversion corresponds to the compression factor in the literature on compressive sensing. The captured intensity image is shown in Fig. 2(b). The pixel count was 800×800 (Nx × Ny) pixels. The width of the whole CA was 93 % of that of the captured image. From Fig. 2(b), the amplitude and phase of the retrieved complex field just behind the CA are shown in Figs. 2(c) and 2(d). From Figs. 2(c) and 2(d), the amplitude and phase of the reconstructed object at zo1 and zo2 are shown in Figs. 2(e)–2(h), respectively. The two wires were reconstructed correctly. The pixel counts of the reconstructions were the same as that of the captured image, confirming that the FOV of SPICA was as predicted by theory.
Fig. 2 Experimental results with two wires. (a) The CA pattern and (b) the captured image. (c) The amplitude and (d) the phase of the retrieved field on the CA. (e) The amplitude and (f) the phase of the reconstructed field at zo1. (g) The amplitude and (h) the phase of the reconstructed field at zo2. Phases are normalized in the interval [−π,π]. Black lines in Figs. (e) and (g) show the positions of the line profiles in Fig. 3.
Line profiles of the reconstructed wires are shown in Fig. 3. The positions of the profiles are indicated with black lines in Figs. 2(e) and 2(g). The full widths at half maximum (FWHMs) of the profiles of zo1 and zo2 are 59.0 μm and 72.7 μm, respectively. The theoretical size, δ, of the point spread function (PSF) of SPICA is approximately given by
where λ is the wavelength of illumination, and w is the width of the whole CA, not the width of the individual pinholes [16]. In the case of the experiment, the widths of the PSFs at zo1 and zo2 are calculated to be 25.5 μm and 39.2 μm from Eq. (2). Then the estimated width of the wires in the reconstructions are 55.5 μm and 69.2 μm at zo1 and zo2, respectively, because the PSF in Eq. (2) is convoluted with the images of the wires, whose widths are 30 μm. These estimated widths are consistent with the measured FWHMs, confirming that the spatial resolution and the SFB of SPICA were as predicted by theory.3.2. A convex lens
In the second experiment, a convex lens with a focal length of 80 cm was used as the object. It was located 11 cm (zo) from the CA. The other parameters and configurations were the same as those in the first experiment. Figure 4(a) is the captured intensity image. The retrieved amplitude and phase just behind the CA are as shown in Figs. 4(b) and 4(c). The reconstructed amplitude and phase on the object plane are shown in Figs. 4(d) and 4(e), respectively. The phase map after an unwrapping process is three-dimensionally plotted in Fig. 5(a) [29]. Its line profile is shown in Fig. 5(b), where the theoretical phase delay of a lens with a focal length of 80 cm is also plotted as a reference. The measured and reference phases were in good agreement, and the root mean square error between them was 0.4 rad. This result indicates the quantitative phase imaging capability of SPICA.
Fig. 4 Experimental results obtained with a lens. (a) The captured image. (b) The amplitude and (c) the phase of the retrieved field on the CA. (d) The amplitude and (e) the phase of the reconstructed field.
Fig. 5 Unwrapped phase maps of the reconstructed lens. (a) The three-dimensional map and (b) the line profile.
3.3. Oil bubbles
In the third experiment, the object was oil bubbles [30], prepared by shaking a mixture of oil and water, which was then placed on a slide glass. The slide glass was located 8 cm (zo) from the CA. The other parameters and configurations were the same as those in the first experiment. Figure 6(a) shows the captured intensity image. The amplitude and phase just behind the CA are shown in Figs. 6(b) and 6(c). The reconstructed amplitude and phase of the object field were as shown in Figs. 6(d) and 6(e), respectively. The unwrapped phase image in Fig. 6(e) is shown in Fig. 6(f). Figure 6(g) is an image of the region surrounded by the square in Fig. 6(b), captured with a microscope (IX71, manufactured by Olympus, 5× objective lens). The reconstruction results of SPICA were in agreement with the microscope image, which has a higher resolution than the SPICA image because the former has a higher and numerical aperture than the latter (0.15 and 0.02, respectively).
Fig. 6 Experimental results with mixture of oil and water. (a) The captured image. (b) The amplitude and (c) the phase of the retrieved field on the CA. (d) The amplitude and (e) the phase of the reconstructed field. (f) The unwrapped phase of (e). (g) Image of the region surrounded by the square in Fig. (d) observed with a microscope, where the scale bar is 200 μm.
4. Conclusions
In this paper, we experimentally implemented and demonstrated SPICA. SPICA realizes reference light-free single-shot holographic imaging without losses associated with the FOV and SFB. In the experiments, a large FOV and SFB were verified with thin wires, and the ability to perform quantitative phase acquisition was confirmed with a convex lens. An image of oil bubbles captured by SPICA was also consistent with that captured with a microscope.
The experimental results supported the promising imaging capabilities introduced in our previous work, where the imaging conditions, such as the transmission factor and the alignment error of the CA, the sparsity of objects, and the measurement noise, also have been discussed [16,17]. These results are readily applicable to various spectral regions because SPICA is composed of only a coherent light source, a CA, and an image sensor. Future work will include extending the system to single-shot multidimensional complex field acquisition and an implementation of a CA with high resolution and high precision for practical applications.
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