1. INTRODUCTION

Off-axis digital holographic microscopy (off-axis DHM) has been of keen interest in applications ranging from particle tracking [1] and microelectromechanical systems characterization [2] to live-cell imaging [3] due to its ability to acquire quantitative phase information and infer the thickness [4] and the refraction index [5] of samples from a single exposure. To recover robust object phase information from the hologram, a conventional off-axis DHM system must overcome parabolic phase aberrations introduced by the microscope objective (MO) lens [6]. Several computational techniques have been proposed to correct some aberrations a posteriori, such as the Zernike polynomial [7] and principal component analysis (PCA) [8] at the expense of additional compute time. Recent advances in telecentric digital holographic microscopy (TDHM) promise a priori phase aberration compensation by utilizing a tube lens to convert the MO-induced spherical wavefront back to a planar wavefront at the camera sensor [9,10]. This optical configuration bypasses the computation cost and image quality (IQ) deterioration [11] through the elimination of numerical calculations of a spherical wave.

As a microscopy technique with a sub-wavelength axial resolution, TDHM is highly sensitive to optical alignment differences [12]. Manipulation of a single optical element may result in changes to multiple control parameters, affecting the final IQ [13,14]. Such a challenge is evident in optimization of the Fourier ${+}{1}$-order position [15,16], a key factor affecting the spatial bandwidth in the filtered hologram and, ultimately [17], the IQ of the resultant image [18]. While a common approach to fine tuning the position of the ${+}{1}$ order is to adjust the near-sensor beam splitter (beam splitter 2 in Fig. 1) in the spatial rotational axes and adjust the preceding mirrors along the optical path [19,20], this method suffers from unwanted complexity in both the optical architecture and the adjustment procedure.

 

Fig. 1. Schematic of TDHM prototype. $\varphi$ represents the object-reference angle (the object and reference beams are aligned in the same horizontal plane), and $\beta$ represents the rotation angle of the camera relative to the optical axis, which will be analyzed in detail in the rest of the laboratory note.

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Using existing mathematical models [21–24], we constructed and validated a set of comprehensive methodologies to control ${+}{1}$-order positioning and filtering of TDHM. This led to a design of a TDHM that isolates the control of the ${+}{1}$-order position into two separate one-dimensional rotations rather than the common optical train. In addition, a relationship was established between camera resolution (in pixels) and the corresponding optimal Fourier filter size. The methods presented in this laboratory note simplify the process of positioning the ${+}{1}$ order and improve the spatial bandwidth, resulting in superior IQ.

2. DESIGN FUNDAMENTAL

A. Microscope Setup

To test and validate the above approaches, a TDHM system was constructed based on the Mach–Zehnder interferometer. As illustrated in Fig. 1, the beam emitted by the laser (CUBE 640-40C, Coherent Inc.) is split by a beam splitter (BS013, Thorlabs Inc.) into the object and reference beams after passing through a beam expander (AC254-030-A-ML, AC254-060-A-ML, Thorlabs Inc.) fitted with a pinhole unit (P50K, Thorlabs Inc.). The reference beam is attenuated by a neutral density (ND) filter (NDC-50C-4M-A, Thorlabs Inc.) and enlarged 10x by a pair of lenses (AC254-030-A-ML, AC254-300-A-ML, Thorlabs Inc.) while the object beam passes through a MO (UPLanSApo, 20x/0.75, Olympus Inc.) and a tube lens (AC508-180-A-ML, Thorlabs Inc.). The holographic interference pattern is generated when the reference and object beams are redirected using another beam splitter (BS031, Thorlabs Inc.), which is held by a manual rotation stage (RP01, Thorlabs Inc.) to superpose at the sensor of the CMOS camera (BFS-U3-120S4M-CS, FLIR). Alignment of the object-reference beam angle is accomplished using this one-degree-of-freedom rotation stage.

On the camera sensor, the intensity projection of the interference between the object wave ${ O}$ and the reference wave ${ R}$ is described by

The terms in Eq. (1) represent the zero order (${| { O} |^2} + {| { R} |^2}$), ${+}{1}$ order (${{ O}^*}{ R}$), and the conjugate ${-}{1}$ order (${{ R}^*}{ O}$). The ${+}{1}$ order is filtered by either a circular or rectangular filter mask to obtain the complex optic field of the imaging samples. The filter size and position directly affect the resolution and IQ of the resulting image [25], as the next two sub-sections attempt to illustrate. The complex field is then reconstructed into the image plane to correct anamorphism by the angular spectrum method [26],

The phase $\Phi_z (x,y)$ represents the optic path length difference between the object beam and the reference beam, which equals the multiplication between the physical propagation distance and the comprehensive refractive index. Combining the above methods, the processing pipeline has been extensively implemented on many DHM systems [15] and is also employed by our TDHM setup, where the processing steps are executed in series to obtain a phase map.

B. Fourier Filter Optimization

The selection and application of an appropriate FFT filter is a critical step in the TDHM processing pipeline. With the zero-order intensity filtered out, the FFT filter mask should maintain good coverage over the high-frequency components of the hologram. Due to TDHM’s use of a tube lens for spherical-to-planar wavefront conversion, the shape of the ${+}{1}$ order would approach a perfect circle (Fig. 2), making it appropriate for the use of a circular filter [25]. Maximizing the filter size allows high spatial bandwidth and the subsequent high lateral resolution in the resulting image [21].

 

Fig. 2. Fourier spectrum of the hologram. The shape and ideal size of diffraction orders are illustrated. For TDHM, the ${+}{1}$ order is near-circular, depending on wavefront curvature. The corresponding FFT filter size is described by $r$, and its position is represented in polar coordinates ($\rho ,\;\theta$) and Cartesian coordinates (${F_x},\;{F_y}$). The size of zero order is double of ${+}{1}$ order [25].

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Expressed below is a set of equations pertinent to the Fourier filter of Fig. 2 (pixel unit in Fourier space):

In Cartesian coordinates, the optimized ${+}{1}$-order position is described as (pixel unit in Fourier space)

It can be reasonably observed from Eq. (7) that the ${+}{1}$-order position with maximum bandwidth is unique to each combination of camera pixel dimensions. It is worth mentioning that the optimized ${+}{1}$ order is not always on the image diagonal as is the case for square sensors. Equations (6) and (7) can otherwise provide the optimized ${+}{1}$-order position for TDHM systems employing non-square sensors. For instance, our camera (BFS-U3-120S4M-CS, FLIR) has a non-square camera resolution of $4000 \times 3000$, and its optimized ${+}{1}$-order position is (1707, 33.04°) (polar coordinates) and (1431, 931) (Cartesian coordinates) in pixel units in Fourier space, which is off the sensor diagonally.

 

Fig. 3. Interference from the difference in beam angles. Two beams meet at the cube and then superpose at the camera sensor, forming beam angle $\varphi$. The phase component can be described by wave vector ${ k}$. The projection of the resulting interference wave vector ${ k}^{\prime}$ should be very close to ${ k}$ since the residual angle $\eta$ is close to 0° due to the small $\varphi ( {\lt} {10}^\circ)$. The hologram (c) is obtained directly from the digital camera, and (d) is the resulting Fourier spectrum, where ${f_x},\;{f_y}$, and $d$ represent the interference fringe sizes, and (${F_x},\;{F_y}$), ($\rho ,\;\theta$) represent the corresponding ${+}{1}$-order position in the Fourier spectrum in Cartesian and polar coordinates, respectively.

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C. Order Position Alignment

This section sets out to detail a simpler method to control the position of the ${+}{1}$ order. Instead of aligning the object and reference beams using a six-degree-of-freedom optical system, we propose a simplified alignment method that utilizes two independent, one-degree rotational parameters (the beam angle and the camera rotation angle) based on Goodman’s derivation of optical interference [21].

The alignment process begins by placing the object and reference beam onto the same plane to eliminate the effect introduced by fringe pattern tilt angle α. As the object and reference beams intersect on the sensor [Figs. 3(a) and 3(b)], the consequent spatial beam angle, $\varphi$, dictates two attributes to the interference fringe pattern: the fringe tilt angle, $\alpha$, and the interference fringe size, $d$ [Fig. 3(c)]. To characterize the relationship between the beam angle and the position of the ${+}{1}$ order, we can first determine how interference patterns on a camera sensor affect the position of ${+}{1}$ order in Fourier space. With a simple geometric relationship [21,22], as shown in Fig. 3, the distance between the interference fringes is described by

Assuming the optic wave of the DHM is propagating in a dielectric medium whose magnetic and electric permittivity is constant throughout the region of propagation and independent of the direction of the polarization, the effect of the medium can be neglected when we only consider the image intensity contrast on the camera sensor. The interference between the object and the reference beam can be represented by

The fringe distance $d$ is exactly the period of this projected intensity distribution [Eq. (11)]. The wave vector ${ k}$ can be described as $2\pi/\lambda$, where $\lambda$ represents the illumination wavelength. Thus, the fringe size is described as [21]

Combined with Eq. (8b), the beam angle can be described as

To summarize this section, an experimental setup and process workflow were introduced, which paved the way for calculation of the optimized ${+}{1}$-order position. Finally, a practical method of controlling the ${+}{1}$-order position by beam angle adjustment and plane (or camera) rotation was added. By doing so, we are able to isolate the alignment parameters to individual optical components.

3. EXPERIMENTAL RESULTS

In this section, three experiments were conducted to verify the control methods detailed in the last section. The precision of ${+}{1}$-order control by either the beam angle or the rotation of the camera (or plane) was examined, and its error range was specified. In addition, the effect of various Fourier filter sizes and camera pixel sizes on TDHM system IQ was investigated.

A. Beam Angle Versus ${+}{1}$-Order Distance

A control experiment was set up to validate Eq. (12), with the object-reference beam angle as the independent variable altered in each experiment trial. To satisfy the requirements, we set fringe pattern tilt angle α to 0 by placing the object beam, reference beam, and camera on the horizontal plane as discussed in Section 2.C. The camera was moved to two positions along the optical axis for each trial to measure the beam angle [Fig. 4(a)]. It should be noted that the size of the fringe pattern size is unaffected by changes in the axial position of the camera (Fig. 5), coinciding with the expression in Eq. (12). A set of data points containing beam angle–fringe size pairs can then be obtained across the experiment to compare with Eq. (12) curve data.

 

Fig. 4. Experiment setting for beam angle. (a) Experimental setup; (b) calculation of the beam angle with respect to alignment error. Setting a line pattern into the object (black dotted) and reference arms (purple solid), respectively. The projected line distances ${d_1}$ and ${d_2}$ can be obtained by moving the camera at a distance $D$ along the optical axis. The alignment error of the objective beam angle is expressed as ${\xi _{\textit{OB}}}$ or ${\xi _{\textit{OB}}}^\prime$ depending on the direction.

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Fig. 5. Scaling of the line pattern. Distance variance between line pattern ${L_1}$ and ${L_2}$ with respect to 5 mm displacement of the camera ($D = {5}\;{\rm mm}$ in Fig. 4), (a) before and (b) after. After the raw hologram is Gaussian-blurred, the line distance can be obtained from the respective intensity plot (row 2). The Gaussian kernel was set to 60 pixels in order to clearly resolve the local minima in the line position. It is noted that, with the displacement of the camera, the distance between L1 and L2 has changed from d1 to d2, but their Fourier spectra remained identical (row 3), illustrating that the fringe pattern size is unaffected by camera position. Indicating that fringe pattern size is not affected by camera displacement. This method is also used to verify the beam angle function in Fig. 6.

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To determine the incoming beam angle, a line target orthogonal to the rotational axis was introduced into the beam paths, as illustrated in Fig. 4(a). Since the beams are collimated, the projection of the line pattern is the same at any point along the path, and only the distance between the reference [Ref in Fig. 4(a)] and object [OB in Fig. 4(a)] beams changes. The beam angle ($\varphi$) can be determined by measuring the distances between the lines (${L_1},\;{L_2}$) at two different camera positions along the optical axis, as shown in Fig. 5, characterized by a simple geometric relationship,

However, we must account for a small angle ($\xi$) between the OB path and the optical axis to compensate for misalignment, as depicted in Fig. 4(b). Accordingly, the beam angle can be described by the displacement of each line (${d_{L1}}$ and ${d_{L2}}$) instead of the distance between lines,

The fringe size $d$ can be obtained via Eq. (8b), which requires selecting the optimized ${+}{1}$ position ${F_x},\;{F_y}$ in the Fourier spectrum (Fig. 5, row 3) in addition to the knowledge of camera resolution (${N_x},\;{N_y}$) and pixel size (${p_x},\;{p_y}$). These values, coupled with their respective beam angle in each trial from Eq. (17), are plotted in comparison to the function curves of Eq. (12) for verification, as illustrated in Fig. 6. The experimental root mean square error (RMSE) for all trials is ${0.131}\;\unicode{x00B5}{\rm m}^2$. The errors are shown in Table 1.

 

Fig. 6. Beam angle versus fringe size between Eq. (12) model curve and experimental data. The gray fitted curve is the result of Eq. (12) with the beam angle–fringe size data points from the experiment in Section 3.A. The red and blue crosses are the data points with the height and width indicating the standard deviation of each data point (${N} = {10}$) by rotating beam splitter 2 clockwise (forward) and counterclockwise (backward), respectively.

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Table 1. Measurement Results with Standard Deviation

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B. Camera Angle Versus Fringe Angle

The above experiment assumes that the object and reference beams interfere on the same plane, where the fringe pattern and corresponding first order would be orthogonal to the field of view, as in Fig. 5. Nevertheless, the optimal ${+}{1}$-order position angle $\theta$ should be adjusted according to Eq. (6), which is conventionally realized by spatially tilting the reference beam and the beam splitter 2 until the angle of the fringes is acceptable [30]. A simpler approach would be to either rotate the object-reference plane or, alternatively, the camera sensor around the optical axis to achieve the optimal order position angle $\theta$, as demonstrated in Fig. 7.

By rotating the camera sensor around the optical axis, achieved by 3D printing multiple camera holders, the corresponding ${+}{1}$-order angle $\theta$ (0°, 22.5°, 45°, 67.5°, and 90°) can be obtained as in Fig. 7(b). When the camera is set at 0° or 90°, rotational errors arise in the first order of the spectrum since the object and reference beams do not perfectly interfere at the same plane, and the object beam is not perfectly parallel to the optical axis. However, the corresponding residual beam angle (${0.2^\circ}\;{\pm}\;{0.03^\circ}$) can be easily obtained by Eq. (13) and compensated by fine tuning the last mirror in the reference path. We then calculated the resulting $+ 1$-order angle, $\theta$, in polar coordinates by Eqs. (6) and (7), which only depends on the camera resolution. The camera rotation angle is subsequently evaluated by Eq. (14).

 

Fig. 7. Optimization of camera rotation. (a) Schematic of the experiment where the camera is rotated around the optical axis; (b) corresponding order positions in Fourier spectrum, where $\beta$ is the camera rotating angle. ${\eta _1}$ and ${\eta _2}$ are typical system errors mainly from the residual beam angles that are calculated as ${0.2^\circ}\;{\pm}\;{0.03^\circ}$.

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Our CMOS camera (BFS-U3-120S4M-CS, FLIR) has a non-square sensor with camera resolution of $4000 \times 3000$, unlike the one used by Carl et al. [31], resulting in an optimal $+1$-order position of (1431, 931) (Cartesian coordinates) in pixel unit in Fourier space [Eq. (7)], and the corresponding filter radius is 1707.2 (pixel unit in Fourier space). Solved with Eq. (14), the optimized camera rotation angle $\beta$ should be 49°, and the corresponding ${+}{1}$-order angle $\theta$ would be 33.04°. The ${+}{1}$ order is located off the diagonal of the spectrum, which is also in line with the results presented in Section 2.B.

C. FFT Filter Versus Image Quality

To verify the proposed optimization methods in the TDHM pipeline, an imaging experiment was set up to examine TDHM IQ with various beam angles and camera pixel size settings. A sample of living HeLa cells was prepared and diluted with Dulbecco’s Phosphate Buffered Saline (PBS, D1408, SIGMA Inc.). 2 ml of the mixture was injected into a Petri dish (catalog#153066, Nunc Cell Culture dishes, Thermo Fisher Scientific Inc.) and left stationary for 10 min before being placed into the sample holder of TDHM without any additional labeling or treatment performed. The HeLa samples were imaged with two different beam angles settings, 2.9° and 6.6°, by an Iris 15 camera [Cam1, (Teledyne Photometrics Inc.)] and a FLIR camera (Cam2, BFS-U3-120S4M-CS, FLIR Inc.), whose pixel sizes are different. We arbitrarily chose an object-reference beam angle of 2.9° on Cam 1 after yielding a usable image on the camera as Fig. 8 (column 2). Its filter size in pixels is then mirrored onto Cam 2, which gives a corresponding beam angle of 6.6° according to Eq. (13) and produced images as shown in Fig. 8 (column 3). The two beam angles were then swapped and applied onto the other camera to test the extremities of the filtering condition, with the results shown in Fig. 8 (column 1) and Fig. 8 (column 3). The refractive index of HeLa cells was assumed to be 1.33, close to that of PBS [32]. After processing the sample through the imaging pipeline described in Section 2, the 3D topographies of the holograms, as well as their height plots, were obtained and rendered by Fiji (Interactive 3D Surface Plot) [33], as depicted in Fig. 8 (row 3).

 

Fig. 8. TDHM result of HeLa cells with different beam angles and camera pixel sizes. Cam 1 is Iris 15, Teledyne Photometrics Inc (pixel size, 4.25 µm); Cam 2 is BFS-U3-120S4M-CS, FLIR Inc (pixel size, 1.85 µm). All scale bars of various drawn lengths in row 1 are of physical length 20 µm, their discrepancy due to pixel sizes difference between the two cameras. The object-reference beam angles of 2.9° and 6.6° are alternatingly applied to Cam 1 and Cam 2. The illumination wavelength is 640 nm. The color scale bars in row 4 are in µm units. The refractive index of the HeLa cells was assumed to be 1.33 [32], as their 3D topographies and the height plot were obtained and plotted in Fiji, shown in row 3. The sizes of all height plots are 13 µm and 1.3 µm for the $x$ and $y$ directions, respectively. The size of the zoomed-in region of interest is $13 \times 13\;{\unicode{x00B5}{\rm m}}$. Note: The fringe pattern in row 1, column 1 of the subfigure is not visible in the hologram since $d = {5.55}\;\unicode{x00B5}{\rm m}$ is less than 2 times the pixel size (8.5 µm) specified in the Nyquist–Shannon sampling theorem.

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According to Eqs. (8c) and (12), the fringe sizes of these holograms are 5.55 µm (6.6°) and 12.75 µm (2.9°), respectively, for the cameras. It is noted that, in one trial where the fringe size was less than the $2 \times$ multiplier to pixel size (4.25 µm) of the Iris 15 camera with 6.6° beam angle in Fig. 8 (column 1), by Nyquist–Shannon sampling theorem, the fringe size is less than the Nyquist rate threshold and, therefore, not sufficient to be sampled by the pixels. This results in the virtual absence of fringe pattern details in Fig. 8 (row1, column 1), whose ${+}\;{1}$ order was too far from the zero order in its Fourier spectrum to produce a usable topographic image. On the other end of the spectrum, the relatively large size of the fringe pattern (12.75 µm) with respect to pixel size (1.85 µm) in the FLIR with a 2.9° beam angle in Fig. 8 (column 4) only permits a small FFT filter not able to fully cover the high-frequency components without risking acquisition of zero-order information, consequently losing high-frequency topography details in comparison with the other two configurations.

Several reports in the literature have explored DHM operation at the diffraction limit [34–36], where its resolution is mainly influenced by MO characteristics and illumination wavelength. However, the employment of an unoptimized Fourier filter inhibits its near-diffraction-limit performance. The above experiment verified the FFT filter mask needs to be sufficiently large to cover high-frequency components of the hologram for high lateral resolution without bleeding into the zero order, and an optimal FFT filter (${+}{1}$-order position) depends on the illumination wavelength and the camera pixel size and camera resolution.

4. DISCUSSION

TDHM improves upon DHM’s aptitude in single-shot quantitative phase imaging by canceling out spherical aberrations in conventional DHM systems. However, its propensity for ${+}{1}$-order position sensitivity opens to high-order, multiplex effects on the IQ by adjustment of isolated optical elements, making conventional alignment difficult. In this laboratory note, we presented a suite of verified methodologies to mathematically locate and mask the optimized ${+}{1}$-order position for a given TDHM system using as few parameters as possible, as well as best practices to control the ${+}{1}$-order position and filtered bandwidth by physical alignment and processing. Its novelty arises from the fact that the alignment task can be approached with one-dimensional readily computable adjustments to the yaw angle of beam splitter 2 and the rotation angle on the optical axis of the camera.

The proposed method is independent of the MO used in the TDHM setup due to the planar wave illumination. Though the optical resolution and the density of TDHM are determined by the properties of MO (magnification and numerical aperture) and the illumination source, the interference pattern would not be influenced. The proposed method sets out to minimize much of the resolution loss during image acquisition and phase signal processing defined by the optical transfer function (OTF).

The method first identifies camera resolution (in pixels) as the only attribute to ${+}{1}$-order position (pixel units in Fourier space). A robust relationship to the optimal ${+}{1}$-order position and circular filter geometry is consequently established, which can be implemented to various camera pixel and sensor shapes. Thirdly, the order positioning alignment of TDHM can be isolated down to two one-dimensional rotational parameters: object-reference beam angle (adjusted by beam splitter 2 Fig. 1) and camera rotation angle (or object-reference plane) (affixed by a 3D printed camera adapter) about the optical axis. Finally, the expression for the interference fringe pattern size can be obtained, depending only on the beam angle and illumination source wavelength.

By using our simplified alignment approach, we verified the order positioning method through control experiments to isolate and calculate the beam angle in each trial. The experimental results confirm the correlation between fringe pattern size and beam angle. Equivalency between the camera rotation angle and object plane rotative angle on the optical axis is confirmed in the experiment, and a more robust ${+}{1}$-order positioning formula is applied in polar coordinates. Lastly, we compared the IQ of images processed with different Fourier filters, demonstrating the merit of large filter size. These experimental validations come together in Fig. 9, where a result phase map is obtained using the computed optimal ${+}{1}$-order position and a maximal Fourier filter size (1707 pixels in radius). Compared to the largest usable filters (1000 pixels in radius) in Fig. 8 (columns 2, 3), the optimal Fourier filter encompasses more high-frequency components, preserving a higher level of detail.

 

Fig. 9. Optimized TDHM result of HeLa cells. (a), (b) According to the proposed methods, the optimized ${+}{1}$-order position ensured the largest FFT filter size. The (c) height map and (d) cross section plot include a high level of detail. The refractive index of the HeLa cell is assumed to be 1.33, close to PBS [32]. The unit of the color scale bar in (c) is µm. The size of the zoomed-in region of interest is $13 \times 13\;{\unicode{x00B5}{\rm m}}$.

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Our comprehensive alignment procedures can be summarized as follows: the camera sensor attributes are first obtained toward the calculation of the optimal ${+}{1}$-order position via camera resolution; theoretical values of the object-reference beam angle and camera rotation angle are calculated; and practical alignment work then constitutes rotating the beam splitter 2 (Fig. 1) and camera one-dimensionally based on the calculated results to achieve the optimal ${+}{1}$-order position. The proposed method could be applied to other off-axis TDHM configurations, such as multiwavelength and multi-angle TDHM, to simplify the alignment process since the relationship between the beam angle and the corresponding Fourier spectrum remains the same for these systems.