1. Introduction

Historically, classical holography started around 65 years ago when Dennis Gabor reported on a method to avoid spherical aberration and to improve image quality in electron microscopy by sparing lenses in the experimental setup [1]. The holographic principle proposed by Gabor is the simplest holographic layout ever implemented and its application to the visible (optical) range was later performed by Rogers [2]. The progress in holography continued with the insertion of an external reference beam at both off-axis [3–6] and on-axis [7] configurations and with the first evidences of image formation with digitally reconstructed holograms [8, 9].

Since then, digital holographic microscopy (DHM) has becoming in a powerful and versatile tool in many significant fields of Biophotonics, Life Sciences and Medicine [10–13]. DHM takes the advantages of digital holography concerning numerical manipulation of the complex object wavefront and improves classical microscopy in some aspects such as, for instance, by avoiding both the limited depth of focus in high NA lenses and the high magnification ratios needed in conventional optical microscope imaging. As a result, DHM combines high-quality imaging provided by microscopy, whole-object wave front recovery provided by holography, and numerical processing capabilities provided by computers [14–17]. DHM allows visualization of phase samples using a non-invasive (no need for stained samples), full-field (non-scanning), real-time (on-line control), non-contact (no sample damage) and static (no moving components) operating principle [18].

Because of its interferometric underlying principle, DHM has been implemented using different classical interferometric configurations. Thus, one can find Mach-Zehnder [19, 20], Michelson [21], Linnik [22], Mirau [23], Twyman-Green [24] and common-path [25, 26] interferometric architectures as background layouts. Among them, Mach-Zehnder architecture is by far the most used one in DHM practice for transmissive configuration. However, common-path interferometric (CPI) configuration provides significant advantages over all the configurations such as robustness (can be assembled in a more compact way), simplicity (require fewer optical elements) and stability (relatively insensitive to vibrations). The key point in CPI configuration is the transmission (somehow but in parallel) of the imaging and reference beams through the same microscope lens. Essentially, one can distinguish between three general types of CPI layouts. On one hand, the reference beam can be synthesized from the imaging beam after passing the microscope lens; we will name these types of setups as reference-generation (R-G) CPI. On the other hand and assuming that the sample is sparse, the surroundings of the inspected sample area can act in good approximation as a clear region for reference beam transmission; we will refer to those setups as self-interference (S-I) CPI. And finally, the third type of CPI implies the independent transmission of both the reference and the imaging beams through the imaging system and their a posteriori overlapping at the recording plane; we have named these layouts as spatially-multiplexed (S-M) CPI.

Concerning R-G CPI, the interferometric setup is fully assembled after passing the microscope lens, that is, at the image space, thus reducing the sensitivity of the system to vibrations and/or thermal changes in comparison to regular interferometers. Essentially, the transmitted imaging beam is split into two optical arms and one of them is spatially filtered using a pinhole mask at an intermediate Fourier plane. As a result, the DC term is transmitted while the sample’s information is blocked, or in other words, a reference beam is generated by spatial filtering. Owing to its interferometric nature, CPI has been implemented in Mach-Zehnder [27, 28] and Michelson [29–31] architectures used in off-axis [25, 27, 28] as well as on-axis [26, 29, 31] recording geometries and in combination with digital spatial filtering and phase-shifting strategy, respectively, for retrieving the sample’s complex amplitude distribution. The main advantage of R-G CPI is that the field of view (FOV) is fully preserved while their strong drawback is the need to implement a relatively complex opto-mechanical stage at the exit port of the microscope.

S-I CPI produces the overlapping of the imaging beam with a shifted version of itself because there is a blank or clear region without sample information in the imaging beam which is considered to not alter the light passing through it [32–36]. To allow this, the sample must be a sparse sample (as it happen in many cases when using biosamples) and, again, different interferometric architectures such as CPI [32, 33], Michelson [34], using a Lloyd’s mirror [35] or a lateral shearing plate [36] have been reported for producing the overlapping of both areas. S-I CPI can be easily assembled (less optical elements, more compact and simple configuration, etc) than R-G CPI but its applicability is restricted to sparse samples.

And finally, S-M CPI is based on the transmission in parallel of the two separated interferometric beams by using a specially defined input plane spatial distribution [37–40]. On one hand, the imaging beam containing information of the sample to be inspected is conventionally imaged either in transmission [37–39] or in reflection [40] modes. And on the other hand, the reference beam is transmitted by the specific constraints of the input plane design: a black region at the input plane in side-by-side configuration with the sample [37–39] or by back-reflection at the tilted coverslip of a specially designed chamber [40] for transmissive and reflective configurations, respectively. After that, both beams are overlapped allowing holographic recording by using minimal elements (diffraction gratings and tube lenses). S-M CPI allows reference beam transmission for all sample cases by minimal modifications in the setup with the inconvenience of a specific input plane spatial distribution design yielding a multiplexing in the FOV and a specific chamber design for transmissive and reflective configurations, respectively.

Similarly to other approaches where regular microscopes have been equipped with coherence sensing capabilities [41, 42], this manuscript exploits the advantages of our previously reported S-M CPI method [37–39] concerning its simplicity, robustness and easiness of integration to convert a commercially available standard microscope into a DHM with only minimal modifications. By replacing the broadband light source by a laser diode, by leaving a clear region at the input plane for reference beam transmission and by properly placing a one-dimensional (1D) diffraction grating in the microscope embodiment, it is possible to implement off-axis holographic recording in a commercial microscope. We have named the method as spatially-multiplexed interferometric microscopy (SMIM) since the input plane spatial distribution is divided into, at least, two useful regions in side-by-side configuration: one containing the sample and another clear region for reference beam transmission. This spatial multiplexing restricts the FOV provided by the microscope lens but enables phase information availability.

The paper is organized as follows. Section 2 provides both a graphical interpretation and the theoretical framework of the updated microscope concerning FOV limitation. Section 3 validates the SMIM first with synthetic sample (USAF – United State Air Force resolution test target) for calibration purposes and second with complex biosamples (red blood cells and sperm cells). Finally, Section 4 concludes the paper.

2. Theoretical analysis

2.1 Qualitative SMIM description

Our proposed SMIM is presented in Fig. 1. Essentially, SMIM is implemented using the embodiment of a BX60 Olympus microscope with three minimal changes. From down to up in Fig. 1, the first one involves the use of a coherent light source for interferometric recording. The second defines a specific spatial multiplexing at the input plane for allowing reference and imaging beam transmission in common-path configuration. And the third one performs the insertion of a 1D diffraction grating for mixing both interferometric beams at the recording plane. For the sake of simplicity in the descriptions, the grating is not included in Fig. 1 but only the place where it is inserted.

 

Fig. 1 Picture of the experimental layout (left) and scheme (right) of the proposed SMIM where the main components of the proposed SMIM can be identified at both the picture and the scheme.

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Concerning the inclusion of coherent light illumination in the layout and instead of using the microscope illumination light path, we have externally inserted a commercial grade laser diode just below the microscope’s XY translation stage (see picture in front view at Fig. 1). The laser diode provides spherical divergent coherent illumination onto the input plane which has been divided into three spatial regions with the same size. Those three regions are identified as R-O-X in Fig. 1 incoming from the Reference, the Object and the X-blocking areas, respectively. The O region is centered on axis while the R and X ones are at both sides of the O region. Such spatially multiplexed input plane is imaged and magnified by the microscope system consisting in an infinity corrected objective and a tube lens. Then and in absence of the grating, the output distribution at the CCD plane will be composed by three spatial regions named as R’-O’-X’ incoming from the R-O-X images. Since the CCD is centered with the optical axis of the microscope embodiment, only the O’ region will fall in its sensitive area. Note that this fact can be easily accomplished by selecting the CCD dimensions and/or the magnification between input and output planes. In this configuration, the proposed system acts as a regular microscope working under coherent illumination but with a useful FOV reduced by a factor of 3 according to the implemented spatial multiplexing.

Then, a 1D diffraction grating is inserted in the analyzer insertion slot just before the tube lens included in the observation tube of the microscope. Because of the infinity corrected configuration, the grating will not result in any additional restriction due to field or aperture diaphragm effects. For simplicity, we will consider a sinusoidal grating so there will be 3 shifted replicas of the output plane spatial distribution. This situation is presented in Fig. 2(a). By properly selecting the grating period (see Section 2.2), it is possible to allow a displacement equal to one third of the FOV at the image space. This fact means that the 3 spatially multiplexed regions R’-O’-X’ will perfectly overlap one to each other at the recording plane. In particular and at the CCD position [see Fig. 2(a)], the O’ region incoming from the 0th grating order will overlap with the R’ region incoming from the −1st grating order (the right one in dark red color) and with the X’ region incoming from the + 1st grating order (the left one in black color). Since the X’ region contains no light (it is blocked at the input plane), the CCD will record an off-axis hologram incoming from the addition of the imaging beam (O’) and a tilted reference beam (R’) is arriving at the CCD with a specific off-axis propagation angle. Such off-axis hologram can be then digitally processed with regular numerical operations (Fourier transform, Fourier filtering and inverse Fourier transform) to recover the complex amplitude distribution of the imaged sample.

 

Fig. 2 Two proposed SMIM configurations where the input plane is divided into: (a) 3 regions and the CCD is centered on-axis, and (b) 2 regions and the CCD is laterally shifted from the optical axis.

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The position of the grating regarding the position of the Fourier plane provided by the microscope lens is crucial for properly selection of the diffraction grating’s period. There are two main scenarios to be considered (see [37–39]): the grating placed just at or in the proximity of the Fourier plane or the grating placed far from it. For the grating placed in the Fourier plane, the reference (R’) and imaging (O’) beams follow the same optical path with the same propagation angle and will interfere in on-axis mode at the recording plane. Thus, phase-shifting algorithm will be needed to recover the complex amplitude distribution of O’. Moreover, in this case, a too low basic grating frequency will not produce a correct overlapping of the regions R’ and O’ (probably, O’ will interfere with a shifted version of O’), while a too high grating frequency will diffract R’ outside the tube lens aperture preventing the holographic recording. The second scenario defined by the grating placed far from the Fourier plane (our implementation case due to the position of the analyzer insertion slot in the microscope embodiment) takes the advantage of the off-axis holographic recording concerning whole sample information retrieval in a single hologram acquisition. However, two parameters should be taken into account: the axial distance between the grating’s position plane from the Fourier plane and the grating basic frequency. The former is depending on the objective lens selected for imaging since the focal length changes (and thus the Fourier plane position) but we will assume such distance big enough to introduce significant variations. The latter defines the shift of the replicas in the recording plane and becomes crucial because a too low basic grating frequency will produce overlapping of the hologram orders at the Fourier domain while a too high grating frequency will derive in aliasing problems, both cases preventing an accurate retrieval of the complex amplitude distribution of the imaged sample.

Aside of a specific design of the chamber containing the sample, the only payment is performed in the FOV since the input plane must be divided into 3 regions because of the spatial multiplexing (FOV restricted by a factor of 3). However, an additional implementation can be performed allowing a FOV reduction factor of only 2. Figure 2(b) depicts this configuration which is essentially the background setup included in [37–39]. In that second case, only 2 regions are multiplexed at the input plane (no X region is needed) but as trade-off the CCD must be laterally shifted to an off-axis position to record the hologram. In this manuscript we have implemented and validated the configuration included in case (a).

2.2 Mathematical analysis of SMIM

Let us suppose that our SMIM microscope included in Fig. 2(a) can be represented by the scheme depicted in Fig. 3. A spherical wave is originated at a point source S at a distance a in front of the objective and the input plane coincides with the objective’s front focal plane due to the infinity-corrected configuration. The objective is represented by a thin lens of focal length f’₁ and the tube lens by another one of focal length f’₂. Just in front of the tube lens, we place the 1D grating with spatial frequency of N line pairs/mm (lp/mm). In absence of the grating, the layout provides an image of the object at the back focal plane of the tube lens (CCD plane). In principle, the FOV size at the input plane is such that its image exceeds the size of the CCD detector. The inclusion of the sinusoidal grating generates 3 replicas of the output distribution. The condition to optimize the holographic recording is essentially a function of both the CCD characteristics (pixel size and number of pixels) and the grating’s basic frequency. If the frequency of the grating is appropriately chosen, only one diffracted order will impinge on the CCD for the reference beam (the −1st order at Fig. 3). Then, the N value must be selected taking into account that it should be enough to redirect the reference beam towards the CCD and at the same time it must be able to separate the different hologram orders when performing Fourier transform at the recorded hologram.

 

Fig. 3 Optical diagram and ray tracing scheme for the theoretical analysis of the proposed SMIM.

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Let us suppose that the width of the CCD detector is 2z (so it extends from -z to + z at the recording plane) and that the R region extends from point R₁ to point R₂ at the input plane being R₀ its central point. For the sake of simplicity in the calculations but without lack of generality, we have considered that a = −2f’₁ and, so, a’ = 2f’₁. In the following, we will consider as positives the angles with the axis when bringing the ray to coincide with the axis through the shortest angular way we move counterclockwise. Let us analyze the trajectory of a ray starting at S and passing through R₀ (solid thick red line). Such a ray will form an angle θ₀ which is a function of the distance a of the point source and the R₀ height (named as r₀ using the lower case letters), that is: tanθ₀ = r₀/f’₁ > 0, according our sign convention. After passing the objective lens, the refracted ray will cross the optical axis at S’ point (image of S through the microscope lens) and will arrive to the grating with the same angle θ₀. After grating’s diffraction, we want that the −1 diffraction order will redirect our ray to the back focal point of the tube lens or, in other words, that the ray passing through the central reference region will cross the optical axis at the center of the CCD sensor area. To satisfy this condition, our ray must be diffracted parallel (θ’₀ = 0°) to the optical axis by the 1D grating, that is:

In a first approximation, taking into account that we are working with small angles, we can assume that tanθ₀ ≅ sinθ₀. So then, we can write that r₀ = Nλf’₁.

We want to determine now the positions of the points R₁ and R₂. The rays coming from R₁ impinge on the grating with angle θ₁ > 0. For R₁, the ray at the image space must reach the point placed at + z in the back focal plane of the tube lens in order to cover all the CCD sensitive area. Let us call θ’₁ the inclination of that ray at the image space. This angle is negative according to our notation and it can be calculated as tanθ’₁ = -z/f’₂. By applying the grating equation, one arrives at

A similar procedure can be applied for the R₂ point arriving to a similar expression:

The reference zone at the input plane extends from R₁ to R₂ and its width is given by r₂ – r₁ = 2zf’₁/f’₂ = 2z/M, being M the magnification layout. This width equals the part of the O region that is imaged on the CCD detector, as could be expected. Theoretically, the largest object that can be imaged will be confined to the interval [-r₁, r₁]. At the image plane, the O’ region incoming from the zero order term extends in the interval [r’₁ = -Mr₁, r’₂ = + Mr₁]. In the case that r’₁ > z, only the central part of the object in the interval [y₁ = -z/M, y₂ = + z/M] can be used for recording the hologram. On the other hand, the spatial frequency of the grating (N) must be appropriate to avoid the overlapping of the O’ regions incoming from the 0th order of the diffraction grating (direct imaging) and its diffracted versions incoming from the ± 1st diffraction grating orders (shifted replicas of O’ at the CCD plane).

The center of the input plane spatial distribution at the image space provided by the −1 diffraction order is shown in Eq. (1) assuming that θ₀ = 0. This situation is depicted by the brown dotted line at Fig. 3. If we name θ’ as the angle provided by the ± 1 diffraction orders of the grating with the optical axis (sinθ’ = ± Nλ) and assuming that the grating is close to the tube lens, we can calculate the shift at the recording plane provided by both ± 1 diffraction grating orders as: h_c = ± f’₂ tanθ’, where both θ’ and h_c can be identified from Fig. 3. We will pay attention to the −1 diffraction order of the grating since it is the one providing the reference beam. Thus, for a total object extension of 2y at the input plane, the zero order image extends in the interval [-My, + My] at the output plane while the −1 diffraction order do it from [h_c - My, h_c + My]. Notice that My = 2z for the ideal case represented in Fig. 2(a). For a CCD capable of recording the whole region named as O’ incoming from the zero diffraction order of the grating, the proper separation to avoid the overlapping of the images provided by the grating replicas and to allow presence of reference beam in the whole CCD sensitive area must verify that h_c = 2z = 2My. However, for a CCD size smaller than the extent of the zero order image, the minimum separation is given by h_{c, min} = z + My = f’₂ tanθ’_min ≅ f’₂ N_min λ. Then, the grating’s spatial frequency determines the value of h_{c, min} which also determines the maximum size of the object without generating overlapping of the images at the CCD area.

Conversely, we can calculate the minimum value of the grating’s spatial frequency to avoid overlapping for a given object size. For instance, for the case when the O’ region is perfectly adjusted with the CCD size (2z = 2My), the minimum separation of the replicas incoming from the zero and −1 diffraction orders must be

Notice that in the last case, it is now the size of the O region which determines the beginning of the reference R region and permits the ideal case shown in Fig. 2(a).

Finally, the pixel size of the CCD sensor and the NA of the objective lens also restrict the maximum and minimum angle between the interferometric beams in order to avoid aliasing errors from overlapping frequencies in the Fourier plane [43]. In our setup, this issue is optimized by providing diffraction along the oblique direction at the Fourier plane.

3. Experimental validation of the proposed layout

In this section we present the experimental validation of SMIM. We have used a commercial BX60 Olympus microscope for implementing the modifications involved in SMIM. We have used a DVD laser diode source (650 nm) as coherent light source which is placed just below the manual XY translation stage of the microscope (see Fig. 1). As optics, we have used 3 different microscope lenses (UMPlanFl) all of them infinity corrected ones: 5X/0.15NA, 10X/0.30NA and 20X/0.46NA. A Ronchi ruled grating (80 lp/mm period) and a commercial grade CCD camera (Basler A312f, 582x782 pixels, 8.3 μm pixel size, 12 bits/ pixel) are used as 1D diffraction grating and imaging device, respectively. Note as the use of a Ronchi grating instead of a sinusoidal one does not introduce additional problems because of its high diffraction orders since even orders have zero diffraction efficiency while odd orders provide replicas outside the CCD sensor area due to their high diffraction angle.

3.1 System validation from a mathematical point of view

First, experimental validation that the CCD camera is properly placed so that the lateral magnification between input and output planes equals the nominal magnification stated in the microscope lenses is provided. The 5X objective images the Element 4 of Group 1 (from now on named as G1-E4) of the USAF test (period of 62.5 μm or spatial frequency of 16 lp/mm). Such image has a size equal to 38 pixels or, equivalently, 315.4 μm, thus defining a lateral magnification value of M = 5.05 which is in good agreement with the nominal one.

In addition, since the CCD width (2z) is 6.5 mm on its largest direction (782 pixels), it extends from [-3.25, + 3.25] mm at the recording plane. Considering the illumination wavelength (λ = 650 nm) and the tube lens focal length (f’₂ = 180 mm according to the microscope specifications), the FOV of the O and R regions at the input plane are [-0.65, + 0.65] mm and [ + 0.65, + 1.95] mm, respectively. Note as, although the useful FOV is reduced by a factor of 3, it still provides a reasonable value (1.3 mm for the largest CCD direction). Moreover, the minimum spatial frequency of the grating is N_min ≅ 56 lp/mm according to Eq. (7). For the experiments, our grating has 80 lp/mm defining a reference region R equal to [r₁, r₂] = [1.22, 2.53] mm. Theoretically, the object region O can then extend in the interval [-1.22, + 1.22] mm but its image on the CCD plane will extend from [-6.16, + 6.16] mm according to M = 5.05. This means that part of that image will lie outside the CCD sensor area and only its central part, the interval [-0.69M, + 0.69M] = [-3.25, + 3.25] mm, will be useful for recording of the hologram. According to Eq. (6), the minimum separation of the replicas provided by the zero and −1 diffraction order images is h_c,min = 9.37 mm and the −1 diffraction order replica extends from [ + 3.25, + 15.48] mm, so it is non-overlapping with the zero order one over the CCD sensor area (there is a superposition from [ + 3.25, + 6.16] mm outside the CCD area).

3.2 System calibration and involved digital image processing using a resolution target

In this subsection, we will present the experimental results provided by the proposed SMIM for the 3 available microscope lenses when using a USAF resolution test target as input object. Obviously, our calibration experiment does not involve placing the USAF into the microscope chamber specifically designed for the input plane spatial multiplexing. Nevertheless, we have used the USAF largest clear area (R region) at one side of the maximum resolution region of the USAF (O region) to transmit the reference beam and manually block the other side (X region).

Figure 4 includes the results obtained when using the 5X objective. The hologram is included in (a), (b), an averaged plot of the normalized intensity of the interferometric fringes of the region marked with a solid line white rectangle in (a) is included in (b), the Fourier transform of (a) is shown in (c) where the diffraction order to be spatially filtered is enhanced with a white ellipse, and the final complex retrieved image in (d). As general comments, we can see as the insets in (a) and (d) has and has not, respectively, interferometric fringes, and, through (b), that the interferometric fringes have a peak-to-valley modulation of approximately the 60% of the total dynamic range of the camera sensor. Notice also that in order to enhance image contrast, the image included in (c) is the intensity represented in log scale and that the DC term of the Fourier spectrum is blocked down. Also, the aperture shape of the objective lens in the Fourier domain is elliptical [white ellipse in (c)] as consequence of a different scale factor incoming from a non-square image (582x782 pixels) when performing digital fast Fourier transform operation. According to the experimental images, the FOV is approximately [-0.6, + 0.6] mm, values that are in good agreement with the theoretical ones previously presented.

 

Fig. 4 SMIM with a 5X/0.15NA objective lens and a USAF resolution target: (a) the hologram, (b) an averaged plot of the interferometric fringes included in the white rectangle of (a), (c) the Fourier transform of (a), and (d) recovered image.

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For completeness, we have also included an additional experiment showing what happen when the input plane spatial multiplexing is not properly matched. In this case, we have rotated the USAF test in such a way that the reference beam does not pass through a clear region of the test. Then, overlapping of object regions will be produced when performing the holographic recording since the reference beam contains input object information. Figure 5 includes the experimental results where, although the object information incoming from the reference beam is difficult to see in the hologram because of the orders grating efficiency, in the obtained reconstruction after the Fourier filtering it is more than evident. Thus, it is not possible to retrieve a non-distorted image of the object region due to the replicas overlapping at the recording plane.

 

Fig. 5 Non-proper adjustment of the input plane spatial multiplexing in SMIM: (a) hologram, (b) Fourier transform of (a), and (c) retrieved image showing the overlapping of the replicas.

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Finally, the SMIM concept has also been validated for the 10X and 20X microscope lenses. Figure 6 shows the experimental results for both objectives where the orientation of the USAF is properly adjusted for each magnification to allow clear reference beam transmission.

 

Fig. 6 SMIM with a 10X/0.30NA (upper row) and 20X/0.46NA (lower row) objective lenses and a USAF resolution target: (a) and (d) the holograms, (b) and (e) the Fourier transforms of (a) and (d), and (c) and (f) the recovered images for 10X and 20X, respectively.

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We want to strength here that because of the fact that the 1D grating is inserted after the microscope lens and before the tube lens, that is, in the optical path defined by the infinity corrected imaging configuration, the grating is the same one for all the objectives. In other words, the grating’s basic frequency does not depend on the magnification of the objective. So, there is no need to change gratings as a function of the lens used for imaging.

In addition, we have computed two quantitative values in order to evaluate spatial noise and image quality provided by the proposed method. On one hand, signal to noise ratio (SNR) is computed as the ratio of the signal mean to the standard deviation (STD) of the noise. To allow SNR calculation, we have defined a binary mask to separate the contribution of the signal (those points at the image corresponding with squares and vertical bars of the USAF test) and the background (those points at the image that are not considered as signal). Such binary mask can be easily obtained by thresholding over the intensity of the final retrieved images. In this case, the background is rendered with zeros and the image details with ones. And an inverse mask is the applied for calculating the STD of the noise associated with the background. And on the other hand, STD of the phase distribution provides a direct value on the phase stability provided by the method. STD is computed over a clear region in the phase distribution of the retrieved images. Results are summarized in Table 1.

Table 1. SNR and STD analysis

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3.3 Experimental results with biosamples

And finally, we have tested the proposed SMIM concept using biological samples (sperm cells and human erythrocytes). Since we only want to perform a proof of principle validation, we have not built the specially designed chamber for the input plane spatial multiplexing. Notice that such specific chamber should be designed as a function of the microscope lens magnification to be used since the FOV varies when changing between different objectives and, consequently, the width of the O region (from -R₁ to + R₁ according to Fig. 3) must be adapted to them in order to adjust its image into the CCD sensitive area (from -z to + z according to Fig. 3). Instead of this, we have followed two different strategies depending on the type of biosample we have used in this validation. For the sperm cells biosample, we have utilized a counting chamber used for sperm characterization and to have a thickness of 20 μm. Because of both the sperm cell dimensions (head width of 6x9 μm, total length of 55 μm, and a tail’s width of 2 μm on the head side and below 1 μm on the end, approximately) and its preparation (unstained sample which is dried up for fixing the sperm cells), the cells will be deposited at both sides of the counting chamber after dying, thus allowing two imaging planes. On the other hand, the red blood cells (RBCs) are stained and placed in a conventional microscope slide.

Whatever the case (the counting chamber for the sperm cells or the microscope slide for the RBCs), the biosample is arranged in such a way that it produces a clear-cut, sudden and abrupt change from the O to the R region ( + R₁ point). Then, the separation between these two regions is perfectly defined and valid for all the microscope lenses included in the experimental validation. Now, it only remains to properly adjust the width of the O region to the FOV of each objective. This is accomplished by externally adding a blocking strip of black thin paper that disables light transmission below the -R₁ point. Thus, the input plane spatial multiplexing (R-O-X regions) are properly matched for each objective. Finally, because of its low magnification, the 5X/0.15NA lens has not been included in the biosample experimental validation.

Figure 7 presents the images provided by the 10X/0.30NA objective when imaging swine sperm cells. Essentially, the sperm sample is a phase sample and, thus, it is better to look at the phase distribution in order to visualize and characterize the sample. Figure 7(a) shows the recorded off-axis hologram while 7(b) magnifies the area marked with a solid white line rectangle in 7(a). The direct result after digital image processing is included in Fig. 7(c) where the unwrapped phase distribution is included. We can see that some of the sperm cells that are almost invisible under conventional intensity imaging mode [see Fig. 7(b)] are now perfectly identified because of the holographic recording. Moreover, the accessibility to the phase distribution allows additional capabilities concerning three-dimensional (3D) quantitative phase imaging and numerical focusing to different sample planes in order to characterize cells outside the depth of field provided by the objective lens. We have implemented these two imaging modalities and the results are included at Figs. 7(d)–7(f). Figure 7(d) depicts the 3D plot of the area marked with a solid white rectangle in 7(c) while the gray color scale bar represents optical phase in radians. Then, the recovered complex amplitude distribution is digitally propagated using the angular spectrum method to the plane where the other sperm cells are in focus, that is, to the other side of the counting chamber. The new results are included in Figs. 7(e) and 7(f) where the retrieved unwrapped phase distribution and the 3D quantitative phase imaging of the same area than in Fig. 7(d), respectively, are presented. We can see that the cells that were in focus in Figs. 7(c) and 7(d) appear now a bit misfocused in Figs. 7(e) and 7(f) while the defocused ones are now in focus.

 

Fig. 7 SMIM using a 10X/0.30NA objective and a swine sperm sample. (a)-(b) the recorded hologram and a magnified area of it, (c)-(d) the 2D retrieved unwrapped phase distribution and its 3D plot of the area marked with a solid line white rectangle in (c), and (e)-(f) the same as in (c)-(d) but after numerical propagation to focus at the plane where different sperm cells are contained. Lateral gray scale bars in (d) and (f) represents optical phase in radians.

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The experimental results for the 20X/0.46NA microscope objective when using both biosamples (RBCs and sperm cells) as input objects are included in Fig. 8. Images in cases (a) and (b) show the recorded holograms, (c) and (d) depict the retrieved phase distributions, and (e) and (f) present a 3D plot of the unwrapped phase distributions marked with a solid line white rectangle in (c) and (d) cases for the RBCs and swine sperm cells samples, respectively. We can see as wide-field holographic imaging provided by SMIM allows the characterization (qualitative and quantitative phase imaging) of biological samples which are almost invisible under conventional intensity imaging modes.

 

Fig. 8 SMIM using a 20X/0.46NA lens for RBCs and swine sperm cells: (a)-(b) are the recorded holograms, (c)-(d) are the retrieved phase distributions, and (e)-(f) are the 3D plots of the unwrapped phase distributions of the areas marked with a solid line white rectangle in (c)-(d), respectively. Lateral gray scale bar represents optical phase in radians.

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Finally, we have assembled a conventional Mach-Zehnder interferometric configuration at the lab for comparison between quantitative phase values provided by SMIM [Fig. 8(f)] and obtained with conventional DHM (Fig. 9). For the DHM layout, we have used a He-Ne laser as illumination source, a 0.42NA Mitutoyo infinity-corrected long-working distance objective as imaging lens, and a group of 4 sperm cells as input object. The results from DHM are presented in Fig. 9 including 2D as well as 3D plots of the unwrapped phase distributions. Although the cells are not the same ones in both images, they come from the same swine sperm biosample. As one can see, the phase step between the background and the higher part of the sperm’s heads is around 3 rads in both cases. This fact shows a high degree of correlation between the unwrapped phase distributions provided by both methods. Note that the background phase value of both images has been equalized to zero value for direct comparison. Concerning phase stability, we have computed the STD value of the background resulting in 0.22 rad and 0.1 rad for the SMIM and DHM configurations, respectively. Although a bit noisier maybe due to the presence of less optical components in the DHM layout, the quantitative phase information provided by the proposed approach perfectly matches the one provided by conventional holographic methods.

 

Fig. 9 (a) 2D and (b) 3D unwrapped phase distribution plots of a different group of sperm cells obtained with conventional DHM. Gray level scale represents optical phase in radians.

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4. Conclusions

Our reported CPI architecture by input plane spatial multiplexing previously validated as on-the-table demonstrator at the lab [37–39] has now been successfully implemented in a regular microscope. The proposed SMIM allows holographic imaging by implementing three small modifications to a regular microscope: a coherent illumination source, a 1D diffraction grating and a specific input plane spatial distribution. With only these simple changes, holographic imaging has been validated using a synthetic sample (USAF resolution test) as well as biological samples (RBCs and sperm cells).

Due to the microscope infinity corrected configuration, the 1D diffraction grating can be preserved for all the microscope lenses without penalizing the holographic recording. However, changing the objective lens will modify the axial distance between Fourier and grating planes (see [37–39]) and, consequently, the fringe period. Nevertheless, this variation can be assumed as tolerable because, approximately, it supposes a change of 1-2 cm in the Fourier plane position (due to the objective exchange) in a separation distance of 14 cm (the axial distance between the back plane of the objective wheel and the grating’s insertion slot). In any case, the small change in the period of the interferometric fringes will be a bit higher for low magnification objectives (shorter axial separation) than for high magnification ones (higher axial separation distance). Thus, the order’s separation in the Fourier domain will be slightly different depending on the objective choice: high magnification lenses will provide higher separation than low magnification ones. Note that low magnification lenses are associated with low NA values, so the diffraction order separation needed to allow non-overlapping among them is lower than high magnification lenses (or objectives with high NA values); then, the same diffraction grating can be used for the whole set of objectives.

The strong point of SMIM comes from the CPI architecture implemented using the embodiment of a regular microscope. Thus, imaging and reference beams follow nearly the same optical path providing two main advantages. First, the instabilities of the system, due to mechanical or thermal changes on both optical paths, do not affect the obtained results leading in phase disturbance reduction. And second, low coherence length light sources are suitable to be used as illumination, thus reducing even more spatial phase noise. As main drawback, the FOV of the objective lens is penalized because of the spatial multiplexing at the input plane to allow reference beam transmission. We have experimentally validated a configuration in which the FOV is restricted to one third of the initial one and proposed [see Fig. 2(b)] an additional implementation with a lower (one half) FOV limitation.

This new configuration of holographic microscopy could be specially useful in, for instance, microfluidic applications where the inspected sample is force to flow through a narrow channel which size that can be adapted to one third (or one half, see Fig. 2) of the FOV provided by the microscope lens and the rest can be used as reference region.