GRIN-lens-based in-line digital holographic microscopy

Ali Akbar Khorshad; Nicholas Devaney

doi:10.1364/AO.476535

1. INTRODUCTION

Holography was invented by Gabor as a new diffraction-based microscopic principle in 1948 [1]. The original setup for holography introduced by Gabor is shown in [1]. The arrangement is simply composed of an object, positioned near a point source, followed by a photographic plate without any further lenses between the object and plate. The primary spherical waves emanating from the source are partially scattered by the object and interfere on the plate with the coherent secondary waves emitted by the object. Typically, the scattered and un-scattered/direct parts of the beam are referred to as the object and reference waves, respectively. The photographically recorded interference pattern, known as a hologram, provides the whole amplitude and phase information of the three-dimensional object simultaneously in one recording.

The idea of digitally reconstructing images from electronically recorded holograms with a computer using the FFT method was proposed by Goodman et al. [2,3]. This digital image reconstruction method, known as numerical reconstruction, provides the possibility of numerical refocusing. Moreover, using the numerical reconstruction approach, not only the amplitude image but also the phase distribution of the recorded object wave can be digitally calculated from the stored hologram [4]. Due to the development of charge-coupled device (CCD) or complementary metal–oxide semiconductor (CMOS) image sensors and computer technologies, the concept of digital holography has emerged as a popular technology with significant applications in microscopy [5,6], cell biology [7], micro-fluidics [8] and lab-on-chip tomography [9]. Among these applications, the main focus of research is digital holographic microscopy (DHM).

So far, DHM has been utilized in two main optical configurations: in-line [5,7,8] and off-axis [10,11]. In the off-axis geometry, a Michelson or Mach–Zehnder type interferometer is normally required. This commonly used geometry allows in particular for spatial filtering of the zeroth-order diffraction term and “twin image” noise [10]. However, off-axis configurations are not compact and are subject to the effects of vibration. Conversely, the in-line configuration has many advantages, including mechanical stability, compactness, and ease of application. The in-line DHM has been employed in two popular schemes: (I) the sample is placed near an image sensor [6,7], and (II) the sample is positioned close to the point source [5]. The former provides a large field-of-view (FOV) image, at the cost of resolution capability (in this case, the resolution is limited by the pixel size of the image sensor), whereas the latter can theoretically achieve wavelength-scale resolution with a relatively smaller FOV. A theoretical study, in this regard, has been recently reported by Serabyn et al. [11]. They describe these two schemes as a low-magnification (LM) regime, obtained with the object close to a small pixel-size detector array, and a high-magnification (HM) regime, achieved using a large pixel-pitch image sensor positioned relatively far from the object (scheme II above). It is worth noting that thermal issues may arise from the powered detector in the LM case, due to the proximity of the object and detector [12]. In principle, the HM operating mode may be more suited to microscopic imaging where best resolution and thermal isolation are the priorities. However, the HM setup of the in-line DHM typically employs a light-coupling component (composed of a small pinhole, an objective lens, and a micro-mechanical stage/holder for alignment) to provide a high-numerical-aperture (NA) illumination beam, thereby increasing volume and alignment complexity of the holographic system.

Recently, an off-axis DHM approach using radial gradient-index (GRIN) rod lenses was presented to provide ${\sim}{0.95}\;{\unicode{x00B5}{\rm m}}$ resolution microscopy in a compact and robust package, appropriate for the needs of remote deployment [11,12]. Note that software aberration correction has been crucial in obtaining the best resolution reported in [11], using the KOALA acquisition and analysis holography software package developed by Lyncée Tec [13]. GRIN lenses are off-the-shelf products with mm-scale diameters and lengths that eliminate the requirement to use and align focal-plane pinholes. In this work, we utilize a single radial-GRIN lens in the HM operating mode of in-line DHM to develop a compact, stable, and cost-effective holographic imaging system, with resolution comparable to off-axis geometry, suitable for sparsely populated micro-particle imaging for future remote applications. In addition, we replace the GRIN lens with a 1 µm pinhole and use two different pixel-sized detector arrays to compare the resolution of various in-line DHM techniques. In brief, a GRIN-based in-line DHM compared to the conventional pinhole-based version benefits from: (I) simplicity in alignment, (II) no requirement to use a relatively high-NA objective lens to illuminate a pinhole, and (III) providing a higher-NA illumination beam and therefore higher resolution. The overall image quality of the pinhole-based DHM is slightly better than that of GRIN-based, due to the spatial filtering property of the pinhole. A GRIN-based DHM using a single GRIN lens has the advantage of cost effectiveness over the off-axis DHM using a pair of GRIN lenses, especially where the commercial KOALA software package is not available in resource-limited laboratories.

The rest of this paper is organized as follows: in Section 2, we briefly introduce: (I) the theoretical background of holography, (II) image reconstruction approaches with a description of the angular spectrum method (ASM), (III) basic properties of GRIN lenses, and (IV) a simple model to calculate the resolution as a function of the object and detector positions. A schematic diagram of our setup and the experimental results obtained for the resolution measurement and imaging of micrometer particles are given in Section 3. The paper ends with conclusions and suggestions for future work.

2. THEORY

A. Theoretical Background of Holography

In an in-line DHM operating in a HM regime, part of the reference wave is scattered by the object, located near the source, and interferes with the unscattered direct wave on a screen further away from the object. Here, we denote the complex amplitude of the scattered and unscattered parts of the wave by ${E_s}$ and ${E_{\textit{us}}}$, respectively. The intensity of the superimposed waves recorded on the screen is given by

(1)$$\begin{split}{I(\textbf{r})}&={ |{E_{\textit{us}}}(\textbf{r}) + {E_s}(\textbf{r}{{)|}^2}}\\ &={ {E_{\textit{us}}}(\textbf{r})E_{\textit{us}}^*(\textbf{r}) + [{E_{\textit{us}}}(\textbf{r})E_s^*(\textbf{r}) + E_{\textit{us}}^*(\textbf{r}){E_s}(\textbf{r})] + {E_s}(\textbf{r})E_s^*(\textbf{r}),}\end{split}$$

where $*$ denotes the complex conjugate. The first term on the right-hand side of Eq. (1) is the intensity of the unscattered wave passing through the object, the so-called zeroth-order diffraction term. The two central terms in square brackets, linear in the scattered wave, indicate the interference between the unscattered reference wave from the source and the scattered wave from the object. This is termed holographic diffraction and forms the basis of holographic imaging. The last term, which is quadratic in scattered waves, contains the interference between scattered waves. This term is the classical diffraction pattern formed by the light scattered from the object. If the object is small and/or sparse enough to scatter only a small part of the reference wave, the holographic regime is dominant and results in a hologram on the screen. In this case, the classical diffraction term of Eq. (1) can be neglected. Conversely, as more of the illuminating reference wave becomes blocked by larger and/or more dense objects, the scattered wave intensity grows and classical diffraction regime becomes prominent. Note that as there are two terms in the holographic diffraction pattern, inside the square brackets, two images (i.e., real and virtual) will be produced after numerical reconstruction. It is worth mentioning that the zeroth-order or dc diffraction term can be effectively suppressed from Eq. (1) or the numerically reconstructed image by subtracting the hologram’s average intensity from each stored intensity value of the hologram [14].

Fig. 1. Coordinate system for propagation of plane waves and reconstructing the image in the angular spectrum method.

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B. Image Reconstruction Approaches

Based on the Fresnel–Kirchhoff diffraction integral, there are three main approaches to reconstruct the holographic image [15]: (I) Fresnel transform, (II) Huygens convolution, and (III) angular spectrum.

The ASM, or plane wave decomposition, provides the highest degree of accuracy for reconstructing the holographic image. In this approach, schematically shown in Fig. 1, also employed in the work presented here, first the angular spectrum ${A_0}({k_x},{k_y})$ is obtained by taking the Fourier transform of the input wave field ${E_0}({x_0},{y_0})$ (i.e., the hologram) as follows [15,16]:

(2)$$\begin{split}{{A_0}({k_x},{k_y})}&={ {\cal F}\{{E_0}\}}\\ &={ \frac{1}{{2\pi}}\iint _A {E_0}({x_0},{y_0})\exp [- i({k_x}{x_0} + {k_y}{y_0})] {\rm d}{x_0} {\rm d}{y_0},}\end{split}$$

where ${\cal F}$ denotes Fourier transform. The input wave field can be represented as the inverse Fourier transform of the angular spectrum as follows:

(3)$$\begin{split}{{E_0}({x_0},{y_0})}&={ {{\cal F}^{- 1}}\{{A_0}\}}\\ &={ \frac{1}{{2\pi}}\iint _A {A_0}({k_x},{k_y})\exp [i({k_x}{x_0} + {k_y}{y_0})] {\rm d}{k_x} {\rm d}{k_y}.}\end{split}$$

As presented in Fig. 1, Eq. (3) indicates that the input pattern comprises various plane wave components propagating along the wave vector $\textbf{k} = ({k_x},{k_y},{k_z})$, with the complex amplitude of each component given by the angular spectrum ${A_0}({k_x},{k_y})$. Note that the $z$ component of the wave vector is given by ${k_z} = \sqrt {{k^2} - k_x^2 - k_y^2}$. We know that each plane wave component acquires a phase factor $\exp (i{k_z}z)$ after propagation over a distance $z$. Therefore, the reconstructed complex wave field of any plane perpendicular to the $z$ axis is represented by

(4)$$\begin{split}{E(x,y;z) = \frac{1}{{2\pi}}\iint {A_0}({k_x},{k_y})\exp [i({k_x}x + {k_y}y + {k_z}z)] {\rm d}{k_x} {\rm d}{k_y}.}\end{split}$$

Equation (4) is indeed an inverse Fourier transform of ${A_0}({k_x},{k_y})\exp (i{k_z}z)$ and can be rewritten as follows:

(5)$$\begin{split}{E(x,y;z)}&={ {{\cal F}^{- 1}}\{{A_0}({k_x},{k_y})\exp [i\sqrt {{k^2} - k_x^2 - k_y^2} z]\}}\\ &={ {{\cal F}^{- 1}}\{{\rm filter}[{\cal F}\{{E_0}\}]\exp [i\sqrt {{k^2} - k_x^2 - k_y^2} z]\} .}\end{split}$$

As Eq. (5) shows, two FFTs are required to reconstruct the holographic image using the ASM described above. Note that the Fresnel transform (Huygens convolution) method can be directly derived from the ASM by expanding Eq. (5) and taking the paraxial approximation [by expanding Eq. (5) and applying the convolution theorem]. A worthwhile property of the ASM, especially useful for off-axis digital holography, is allowing optional filtering of Eq. (5) in the spectral domain. This spatial frequency filtering removes background noise, i.e., the reconstructed zeroth-order and twin image components [10].

Fig. 2. Configuration used to investigate the effect of source–detector, $d$, and object–detector, $o$, distance on the resolution of the GRIN- or pinhole-based DHM. The number of pixels and pixel size of the detector are $M$ and $q$, respectively.

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C. Basic Properties of GRIN Lenses

Radial-GRIN lenses are composed of many flat disks of a GRIN medium, and therefore their aberrations can be simply controlled in comparison to traditional spherical or aspheric lenses. In addition, a GRIN lens can be used as a point source to produce nearly perfect spherical waves. A GRIN rod lens is an ideal device for wide field microscopy/imaging, due to the nearly diffraction limited performance on the optical axis. GRIN lenses are able to focus collimated light if they have a parabolic refractive index profile as follows [17]:

(6)$$n(r) = {n_{\rm{max}}}\left[1 - \frac{{{{(r\sqrt A)}^2}}}{2}\right],$$

where ${n_{\rm{max}}}$, $\sqrt A$, and $r$ represent the index of refraction on the optical axis, gradient constant, and radial position, respectively.

Radial-GRIN rod lenses are usually specified in terms of their length or pitch. The pitch of a GRIN lens is defined as the fraction of a full sinusoidal period that the ray transverses in the lens. As an example, a GRIN lens with a pitch of 0.25 has a length equal to 1/4 of a sine wave, thus focusing an input collimated beam at the output surface of the lens. The pitch and length of a GRIN rod lens are related as follows [18]:

(7)$$2\pi P = \sqrt A L,$$

where $P$ and $L$ are the pitch and length of the lens, respectively. In this work, a 0.25 pitch GRIN lens with a length of 2.36 mm and gradient constant of $0.664\;{{\rm mm}^{- 1}}$ is used to produce an illumination beam with NA of 0.55 for holographic microscopy.

D. Resolution Model

The configuration of Fig. 2 is considered to calculate the resolution of both GRIN- and pinhole-based setups. Suppose that a GRIN lens or a pinhole illuminates both a detector array (having pixel size of $q$, and pixel count of $M$), at a fixed distance $d$ from the source, and a sparse object at a distance $o$ from the detector plane. The reference and object wavefronts received by the detector have different radii of curvature and consequently have different phases. For an expanding spherical wavefront, the phase, $\phi$, is given by $\phi = kr$, where $r$ is the distance from the wavefront’s point of origin, and $k = 2\pi /\lambda$ is the wavenumber. In the Fresnel (paraxial) approximation, where the spherical wavefront is approximated with a parabolic wavefront, the phase in a plane perpendicular to the optical axis can be written as follows [19]:

(8)$$\phi = \frac{{k{\rho ^2}}}{{2z}},$$

where $z$ is the distance from the origin of the spherical wave, and $\rho$ is the radial distance from the optical axis. The phase difference, $\psi = {\phi _o} - {\phi _d}$, between the object and reference wavefronts at the detector plane is then

(9)$$\psi = \frac{{k{\rho ^2}}}{{2o}} - \frac{{k{\rho ^2}}}{{2d}} = \frac{{\pi {\rho ^2}}}{\lambda}\left(\frac{1}{o} - \frac{1}{d}\right).$$

As Eq. (9) indicates the phase difference depends on the object position. As the object approaches the detector plane, $\psi$ increases and the circular Fresnel fringes resulting from the two parabolic wavefronts become narrower and closely spaced on the detector plane. To resolve these fringes with given detector pixels, the sampling theorem must be satisfied. This theorem implies that at least two sample points are required for each fringe. Therefore, if the maximum spatial frequency of the fringes to be recorded is not limited by the pixel size of the detector array, then the full array size determines the NA of the holographic imaging system. Otherwise, if the fringe width at the periphery of the detector array is smaller than two pixels (i.e., $2q$), under-sampling occurs, and the effective array size would be less than the full size of the detector array. As a result, the effective NA of the imaging system decreases, giving rise to resolution deterioration. To find the effective radius, ${\rho _{\rm{eff}}}$, of the detector for any object position, first, we differentiate Eq. (9) as follows:

(10)$$\frac{{\delta \psi}}{{\delta \rho}} = \frac{{2\pi \rho}}{\lambda}\left(\frac{1}{o} - \frac{1}{d}\right).$$

Then, by setting $\delta \psi = 2\pi$ and $\delta \rho = 2q$, we solve Eq. (10) for the effective radius at which the local fringe spatial period is two pixels:

(11)$$\frac{1}{q} = \frac{{2{\rho _{\textit{eff}}}}}{\lambda}\left(\frac{1}{o} - \frac{1}{d}\right)$$

or

(12)$${\rho _{\rm{eff}}} = \frac{\lambda}{{2q}}\frac{{od}}{{d - o}}.$$

Finally, the effective NA of the holographic imaging system can be written as follows:

(13)$${{\rm NA}_{\rm{eff}}} = \frac{{{\rho _{\rm{eff}}}}}{o} = \frac{\lambda}{{2q}}\frac{d}{{d - o}}.$$

Similar to classical optics, the full width at half maximum of the point spread function has been frequently used to calculate the resolution in DHM systems [5,12]. For a diffraction-limited system, this linear resolution width can be written as follows:

(14)$$R = 1.22\frac{\lambda}{{2{{\rm NA}_{\rm{eff}}}}}.$$

Substituting Eq. (13) into Eq. (14), the resolution of the holographic configuration of Fig. 2 is given by

(15)$$R = 1.22q\frac{{d - o}}{d}.$$

As Eq. (15) implies, the resolution increases (i.e., degrades) as the object is moved closer to the detector array (i.e., $o$ decreases). Therefore, the maximum resolution of the GRIN- or pinhole-based DHM described here can be achieved when the object is positioned near the point source.

Fig. 3. Schematic of our in-line DHM setup using a (a) pinhole and (b) radial-GRIN rod lens. L, BE, MO, MMS, P, MMH, GRIN, S, and D represent diode laser with collimated output, beam expander, microscope objective, micro-mechanical translation stage, pinhole, micro-mechanical holder, GRIN lens, sample, and detector array, respectively.

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3. EXPERIMENT

A. Experimental Setup

Figures 3(a) and 3(b), respectively, show a schematic layout of the pinhole- and GRIN-based in-line DHM we have developed. In the conventional pinhole-based setup [Fig. 3(a)], a collimated blue diode laser beam (450 nm, 40 mW, Z-laser) is expanded to slightly overfill the back aperture of a microscope objective lens (${40} \times$, 0.65 NA, Comar Optics). The microscope objective, mounted on a micro-mechanical translation stage, tightly focuses the laser beam into a pinhole (1 µm aperture, Thorlabs). The pinhole is mounted in a micro-mechanical holder. The coherent nearly perfect spherical waves emanating from the pinhole are partly scattered by the sparse sample and produce a magnified interference pattern (hologram) on a CCD (${1600} \times {1200}\;{\rm pixels}$, 7.4 µm pixel size, QImaging) or CMOS (${3280} \times {2464}\;{\rm pixels}$, 1.12 µm pixel size, Raspberry Pi) image sensor, away from the sample. Note that the configuration with a large pixel-sized CCD (small pixel-pitch CMOS) sensor chip provides the HM (LM) operating regime of in-line DHM, when the sample is positioned near the pinhole (image sensor). In our proposed GRIN-based in-line DHM setup [Fig. 3(b)], the pinhole and light coupling components are simply replaced with a GRIN rod lens (effective focal ${\rm length} = {0.92}\;{\rm mm}$, ${\rm NA} = {0.55}$, ${\rm diameter} = {1}\;{\rm mm}$, Edmund optics) mounted in an opaque disk. In this setup, the CCD array is utilized as the image sensor and the setup operates under HM mode. The recorded holograms of both setups are then transferred to the computer for further analysis and holographic image reconstruction. A MATLAB code based on the ASM was developed to numerically reconstruct the holographic amplitude images. Both the sample and sensor chip in both pinhole- and GRIN-based setups are mounted on translation stages (not shown here) to provide adjustable source–detector and sample–detector distances. The stages are only required to experimentally investigate the theoretical resolution model presented here and also to find the best sample and detector positions resulting in the best image resolution. We use a positive USAF 1951 high-resolution test target (down to 645 lp/mm, Edmund optics) to measure the resolution of both setups. As Fig. 3(b) demonstrates, holographic microscopy using a single GRIN lens practically eliminates the alignment procedure and is compact, stable, and easy to implement.

Fig. 4. Digitally reconstructed holographic amplitude images of a USAF 1951 high-resolution test target obtained from the setup shown in Fig. 3, associated with the normalized intensity profile across the best-resolved element (shown by a solid red line) of each reconstructed image. (a), (b) Best-resolved holographic images produced by the commonly used pinhole-based setup in Fig. 3(a), when the setup operates in the HM and LM regimes, respectively. (c) Best-resolved reconstructed image belonging to the GRIN-based in-line DHM system shown in Fig. 3(b), when the system works in the HM mode. The digitally zoomed regions specified with the green dashed rectangles are shown in the adjacent green solid rectangles.

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B. Resolution Measurement

Figures 4(a)–4(c) show the best-resolved digitally reconstructed holographic amplitude images we obtained with both setups presented in Fig. 3. To evaluate the maximum resolution achieved with each holographic system, we included the average intensity profile across the best-resolved element (shown as a solid red line) of each reconstructed image [Figs. 4(d)–4(f)]. Figures 4(a) and 4(b), respectively, correspond to setup (a) of Fig. 3, when the setup operates under the HM and LM regimes. Figure 4(c) is the reconstructed image produced by the setup (b) of Fig. 3. We changed both the detector and sample positions with respect to the pinhole or GRIN lens to find the positions at which the best resolutions shown in Figs. 4(a)–4(c) are achieved. For the pinhole-based setup, when working in the HM (LM) mode, the pinhole–detector and sample–detector distances were ${\sim}37\;{\rm mm} $ (${\sim}11\;{\rm mm} $) and ${\sim}32\;{\rm mm} $ (${\sim}5.5\;{\rm mm} $), respectively. For the GRIN-based system, the GRIN lens–detector and sample–detector distances were : 30 and ${\sim}28\;{\rm mm} $, respectively. As Figs. 4(a) and 4(d) show, the pinhole-based in-line DHM operating in the HM regime is able to clearly resolve only element 1 of group 8, which corresponds to 1.95 µm resolution. Looking at Figs. 4(b) and 4(e), we observe that element 2 of group 8, corresponding to 1.74 µm resolution, can be clearly resolved with the pinhole-based in-line DHM working in the LM regime. According to Figs. 4(c) and 4(f), we are able to clearly resolve the vertical bars of element 4 that belongs to group 8. Thus, the best resolution for the GRIN-based setup is 1.38 µm. This higher resolving power could be attributed to the larger NA of the GRIN-based compared to the pinhole-based imaging system (0.55 versus 0.45). By comparing Figs. 4(a) and 4(b) with Fig. 4(c), we see that the pinhole-based holographic system provides slightly better image quality compared to the GRIN-based, dues to the spatial filtering capability of the pinhole. The GRIN-based DHM resolution can be improved further using a detector array with a larger diagonal size and/or blue diode laser with a shorter wavelength. Note that the best resolution obtained here is smaller than the pixel size of the detector array, used in the measurement, by a factor of ${\sim}5.4$ (reported as magnification factor). Therefore, it confirms that the limitation of the pixel size can be overcome using the HM regime of the in-line DHM. With the aforementioned distances, the FOV of our GRIN-based DHM would be ${\sim}0.8 \times 0.6\;{{\rm mm}^2}$.

To validate the theoretical resolution model, developed in Subsection 2.D, with the experimental results, the GRIN-based imaging setup was chosen and the GRIN lens to detector distance was fixed at two different distances: 20 and 30 mm. Then, the position of the resolution test target, used as the sample, was changed from close to the GRIN lens to near the detector array. Finally, the setup resolution was measured as a function of the sample to detector distance for both GRIN lens to detector distances, the results of which are shown in Figs. 5(a) and 5(b). It should be mentioned that the procedure for resolution measurement using the Air Force test target was repeated at least at five different locations of the imaging FOV to estimate the error bars within the graphs presented in Fig. 5. According to Fig. 5, we can conclude that: (I) there is a good agreement between the experimental results and theoretical predictions of the resolution model. (II) The best average resolution obtained using both different GRIN lens–detector distances is roughly 1.5 µm; however, the FOV for the configuration with a shorter source–detector distance is ${\sim}1.2 \times 0.9\;{{\rm mm}^2}$. In addition, we repeated this measurement for other source–detector distances of ${\sim}25$ and ${\sim}35\;{\rm mm} $ and achieved the best average resolution of ${\sim}{1.5}\;{\unicode{x00B5}{\rm m}}$ again when the sample was positioned 2 mm away from the GRIN lens (data not shown). (III) The experimental resolution is slightly worse than that predicted by theory. The reasons could be that, first, the zeroth-order and twin image noise are still present in the experimental data. Second, the model developed for resolution calculation is not accurate due to the Fresnel approximation of the phase as well as a first-order analysis to the resolution. (IV) The experimental and theoretical resolutions presented here are in the $x$ direction; we also measured in the $y$ direction with similar conclusions (data not shown).

Fig. 5. Experimental results. Resolution as a function of sample–detector distance when the GRIN lens to detector distance is fixed at (a) 20 mm and (b) 30 mm. The dashed lines show theoretical calculations, while the solid lines represent linear fits to the experimental data. The error bars associated with experimental data are estimated by measuring the resolution at different areas of the FOV.

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Note that a contrast hologram is used as input to the angular spectrum approach to reconstruct the holographic image in this paper. The contrast hologram is obtained by subtraction of the image without a sample from images with the sample [20]. Using the contrast hologram can help to eliminate the intensity variations/imperfections in the primary laser beam profile. While the quality of the reconstructed image could be slightly better with a pinhole-based DHM due to its spatial filtering property, the final resolution obtained is better with the GRIN-based DHM.

C. Micro-Particle Imaging

The final aim of this research is to develop a compact, cost-effective, and stable instrument for imaging/tracking dielectric particles down to 1 µm diameter in a 1 mm scale FOV for remote deployment. To this end, we prepared dilute solutions of polystyrene beads (Polybead sampler kit I, Poly-sciences) with nominal diameters of 3.0 and 2.0 µm on separate clean microscope slides of 1 mm thickness. We opted for the second-best-resolved configuration (i.e., GRIN lens to detector distance of ${\sim}30\;{\rm mm} $) and digitally reconstructed the holographic amplitude images of the polystyrene particles as presented in Figs. 6(a) and 6(b) (Visualization 1 and Visualization 2, respectively). It can be seen in Figs. 6(a) and 6(b), respectively, that 3.0 and 2.0 µm spheres are quite visible and clearly resolved.

Fig. 6. Experimental results for micro-particle imaging using the GRIN-lens-based DHM setup shown in Fig. 3(b). (a) [Visualization 1], (b) [Visualization 2] Clearly resolved reconstructed amplitude images of 3.0 and 2.0 µm polystyrene micro-spheres, respectively. The scale bars indicate a distance of 40 µm. The white arrows denote some particle positions as an example.

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To prepare the visualizations, a minimum edge intensity focus metric recently developed by Tian et al. was employed that brings all particles in the sample volume into focus. Indeed, this technique automatically detects all particles in a selected volume and mainly consists of two steps: (I) first, the holographic amplitude images are reconstructed at successive $z$ planes, i.e., different depth; (II) then, the minimum intensity value at each pixel of the image is recorded, which finally provides a 2D projection of minimum intensity. For further details, refer to [8]. It should be mentioned that the final reconstructed image of each particle is overlapped with a larger, concentric, and less dark image, known as a twin image, which is a common problem in all configurations of in-line DHM. Now it is possible to determine the precise size and position of micro-particles, and these steps will be the subject of future work.

4. CONCLUSION

In summary, we have developed an in-line DHM technique using a single GRIN rod lens. To obtain high-resolution images, not limited by the pixel size of the detector array, our setup operates in the HM regime of in-line DHM, where the object is located near the GRIN lens, and the CCD sensor chip is positioned relatively far from the object. We have investigated, both theoretically and experimentally, the influence of the GRIN lens to detector and object to detector distances on the resolution. As a result of that, an average resolution of ${\sim}{1.5}\;{\unicode{x00B5}{\rm m}}$ within a FOV of ${\sim}0.8 \times 0.6\;{{\rm mm}^2}$ is achieved when the GRIN lens–detector and object–detector distances are set at the optimized values of : 30 and ${\sim}28\;{\rm mm} $, respectively.

Moreover, we developed a conventional pinhole-based DHM setup operating in both HM and LM regimes to compare its resolution with our GRIN-based system. The reconstructed images of a USAF resolution target show that the GRIN-based holographic microscope provides better resolution. Furthermore, this setup is stable and easy to apply, and there is no requirement for alignment. The resolution of the GRIN-based setup can be increased further using a detector array with a larger diagonal size, a blue laser with a shorter wavelength, and an off-the-shelf GRIN lens with NA of 0.58. We show that this holographic microscopy scheme provides clearly resolved dynamic images of 3.0 and 2.0 µm polystyrene beads. Our theoretical and experimental results are in good agreement.

Funding

European Space Agency (contract no.: 4000131569/20/NL/BW); Irish Research eLibrary.

Acknowledgment

The authors acknowledge Stuart A. Harries and Conor McBrierty of University of Galway for their assistance with the machining of mechanical parts. Open access funding provided by Irish Research eLibrary.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

REFERENCES

1. D. Gabor, “A new microscopic principle,” Nature 161, 777–778 (1948). [CrossRef]

2. J. W. Cooley and J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965). [CrossRef]

3. J. W. Goodman and R. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11, 77–79 (1967). [CrossRef]

4. E. Cuche, P. Marquet, and C. Depeursinge, “Simultaneous amplitude-contrast and quantitative phase-contrast microscopy by numerical reconstruction of Fresnel off-axis holograms,” Appl. Opt. 38, 6994–7001 (1999). [CrossRef]

5. J. Garcia-Sucerquia, W. Xu, S. K. Jericho, P. Klages, M. H. Jericho, and H. J. Kreuzer, “Digital in-line holographic microscopy,” Appl. Opt. 45, 836–850 (2006). [CrossRef]

6. D. Tseng, O. Mudanyali, C. Oztoprak, S. O. Isikman, I. Sencan, O. Yaglidere, and A. Ozcan, “Lensfree microscopy on a cellphone,” Lab Chip 10, 1787–1792 (2010). [CrossRef]

7. A. Greenbaum, W. Luo, T.-W. Su, Z. Göröcs, L. Xue, S. O. Isikman, A. F. Coskun, O. Mudanyali, and A. Ozcan, “Imaging without lenses: achievements and remaining challenges of wide-field on-chip microscopy,” Nat. Methods 9, 889–895 (2012). [CrossRef]

8. L. Tian, N. Loomis, J. A. Domínguez-Caballero, and G. Barbastathis, “Quantitative measurement of size and three-dimensional position of fast-moving bubbles in air-water mixture flows using digital holography,” Appl. Opt. 49, 1549–1554 (2010). [CrossRef]

9. S. Seo, T.-W. Su, D. K. Tseng, A. Erlinger, and A. Ozcan, “Lensfree holographic imaging for on-chip cytometry and diagnostics,” Lab Chip 9, 777–787 (2009). [CrossRef]

10. E. Cuche, P. Marquet, and C. Depeursinge, “Spatial filtering for zero-order and twin-image elimination in digital off-axis holography,” Appl. Opt. 39, 4070–4075 (2000). [CrossRef]

11. E. Serabyn, K. Liewer, and J. Wallace, “Resolution optimization of an off-axis lensless digital holographic microscope,” Appl. Opt. 57, A172–A180 (2018). [CrossRef]

12. E. Serabyn, K. Liewer, C. Lindensmith, K. Wallace, and J. Nadeau, “Compact, lensless digital holographic microscope for remote microbiology,” Opt. Express 24, 28540–28548 (2016). [CrossRef]

13. http://www.lynceetec.com/koala-acquisition-analysis/.

14. T. M. Kreis and W. P. Jüptner, “Suppression of the dc term in digital holography,” Opt. Eng. 36, 2357–2360 (1997). [CrossRef]

15. M. K. Kim, “Principles and techniques of digital holographic microscopy,” SPIE Rev. 1, 018005 (2010). [CrossRef]

16. M. K. Kim, L. Yu, and C. J. Mann, “Interference techniques in digital holography,” J. Opt. A 8, S518 (2006). [CrossRef]

17. E. Hecht, Optics (Pearson Education India, 2012).

18. Thorlabs, “GRIN lenses for imaging,” https://www.thorlabs.com.

19. K. Myung, Digital Holographic Microscopy: Principles, Techniques, and Applications, Springer Series in Optical Sciences (Springer, 2011).

20. W. Xu, M. Jericho, I. Meinertzhagen, and H. Kreuzer, “Digital in-line holography of microspheres,” Appl. Opt. 41, 5367–5375 (2002). [CrossRef]

Name	Description
Visualization 1	Video shows digitally reconstructed holographic amplitude images of 3.0 micron polystyrene particles obtained with the GRIN lens-based digital in-line holographic microscope, we have developed.
Visualization 2	Video shows digitally reconstructed holographic amplitude images of 2.0 micron polystyrene particles obtained with the GRIN lens-based digital in-line holographic microscope, we have developed.

GRIN-lens-based in-line digital holographic microscopy

Abstract

1. INTRODUCTION

2. THEORY

A. Theoretical Background of Holography

B. Image Reconstruction Approaches

C. Basic Properties of GRIN Lenses

D. Resolution Model

3. EXPERIMENT

A. Experimental Setup

B. Resolution Measurement

C. Micro-Particle Imaging

4. CONCLUSION

Funding

Acknowledgment

Disclosures

Data availability

REFERENCES

Supplementary Material (2)

Data availability

Cited By

Figures (6)

Equations (15)

Applied Optics