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Abstract
We propose a new, to the best of our knowledge, method of incoherent optical frequency selection called three-pack frequency-selective incoherent holography. Compressed holography is reconstructed using phase shift intercepts and spatial transfer function convolution in the form of separation without loss of magnification or resolution. The frequency-selective reconstruction process removes the conjugate and DC terms along with the interception of the object wave. This work attempts three-dimensional reconstruction and selected-frequency phase extraction of axial slices in submicron steps, and the experimental results show the potential of the proposed method in areas such as compressed holography, extended field of view, and slice tomography.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The simultaneous capture of several complex wavefronts by a single CCD is known as multiplexed holography. Each wavefront is holographically encoded with a separate message. These intricate wavefronts are compressed, and all of them are captured simultaneously. Multiplexing with microscopic amplification allows digital microscopy to have significantly improved performance in terms of wavefront acquisition and field of view (FOV). Multiplexed micro-holography can therefore be used for a variety of purposes. For instance, non-scanning holographic lamination [1–4], depth-of-field multiplexing [5–9], and field-of-view multiplexing [10–16]. Holography is a method of recording and reconstructing the amplitude and phase of wave fields, invented by Dennis Gabor in 1948 [17]. During this period, researchers used wet chemical and thin film materials for recording holograms. It was not until 1967 that the concept of digital holography was first introduced by J. W. Goodman and R. Lawrence [18]. It is through digital processing that objects are digitally reconstructed. From then on, the era of digital holography began.
The laser has strong coherence for the development of holography, and the interferometer makes it simple to create holograms. Coherent optical is the first area where multiplexed holography initially debuted. Frenklach et al. [19] introduced an interferometric technique in 2014, referred to as interferometry with a triple imaging area, in which the camera simultaneously captures four holograms, tripling the amount of quantitative data it gathers during a single exposure. Six off-axis holograms were combined into one multiplexed off-axis hologram by Rubin et al. in 2016 [20] without sacrificing resolution or magnification. Additionally, it is possible to recreate it using spatial filtering. The use of spatial bandwidth is significantly improved. In 2017, Dardikman et al. [21] positioned two pairs of complicated wave fronts with conjugates in the area inhabited by the DC term. In one pass, eight-pack holograms (8 PH) were reused. It is necessary to acquire two phase-shifted holograms in order to eliminate the DC term. W. Zhang et al. [22] produced the first two-pack multiplexed frequency-selective (TPFS) holography in the area of incoherent light in 2022. Longitudinal separation of on-axis overlapping spectra was achieved by combining the phase shift method of fresnel correlation holography. Without sacrificing resolution, TPFS doubled FOV. The phase of 3D objects may be reconstructed more effectively by using multiplexed holographic phase extraction methods. Currently, a maximum of eight holograms can be extended from the coherent optical multiplexing that is experimentally feasible. The hologram's frequency domain reveals that the spectrum of the eight off-axis wave fronts already fills the two-dimensional frequency domain when the cutoff angular frequency necessary for the multiplexing method is four times the sample's maximum angular frequency. If there are more off-axis holographic waves in this situation, spatial filtering in the two-dimensional frequency domain will not be able to pick the spectrum of extra conjugate terms. In the coherent optical domain, this restricts the number of multiplexed holographic envelopes. Fresnel Incoherent Correlation Holography (FINCH) [23], a self-interference reference holography that defies the Lagrangian invariance requirement of conventional optical systems, was proposed by Rosen et al. in 2007. Compared to typical coherent imaging systems with similar numerical apertures (NA), FINCH's optical transfer function (OTF) is twice as broad and reacts uniformly to all OTF frequencies [24,25]. Because of this, FINCH has a resolution that is around two and one and a half times greater than analogous coherent and incoherent imaging systems, respectively. Hence, in incoherent multiplexed frequency-selective holography, the self-interference holography principle of FINCH is necessary. In addition, the theory of multiplexed frequency-selective holography provides a good foundation for the advancement of frequency-selective slice laminography and holographic FOV expansions in the incoherent field.
In this paper, we extend the concept of frequency selection experimentally to the field of incoherent three-pack holography (3 PH). This work not only realized the incoherent three-pack object wave frequency-selective reconstruction. It is also the first demonstration of incoherent optical frequency-selective slice tomography. We then present the design, experimental studies, and results of a triple-beam frequency-selective Fresnel incoherent correlated holography (TBFS-FINCH) structure.
2. Methodology
In triple-beam frequency-selective Fresnel incoherent correlated holography (TBFS-FINCH), three object waves leaving the traditional microscope configuration are introduced into the SLM loaded with a phase mask (PM). Due to the interferometric properties of the single FINCH, the three beams containing different specimen information interfere with each other to form a mixed hologram. In the proposed system, there are two cases of mixed-stack holograms. The first is a triple-wave interference hologram, which is formed when the optical path difference (OPD) between any two of the three branches of the object wave is less than the temporal coherence length (TCL). If the OPD between the object wave is larger than the TCL, it is the second case, where the intensities of the three holograms are superimposed and do not contain inter-correlation terms. In the following, we discuss the frequency selection options for the first scenario. This is because the three-branch interference already includes the case of holographic superposition. The multiplexed holography received by the camera evolves into an intensity superposition as soon as the mixed interferometric correlation term is zero. So the reconstruction for the second scenario can be included in the resolution for the first scenario. According to Fresnel diffraction theory, the complex amplitude distribution of the three object waves reaching the sensor surface through the imaging system can be represented separately. We start with the example of specimen O.
Based on the first branch in Fig. 1, in the case of a point object with the amplitude $\sqrt {{I_s}}$ located at $({\overline r _s},{z_s}) = ({x_s},{y_s},{z_s})$, the light diffracted from the object reaches the left surface of the microscope objective (MO) with complex amplitude is expressed as:
where $L(\overline s /z) = \exp [i2\pi {(\lambda z)^{ - 1}}({s_x}x + {s_y}y)]$ and $Q(a) = \exp [i\pi a{\lambda ^{ - 1}}({x^2} + {y^2})]$ are the linear and the quadratic phase functions, respectively, in which $\lambda$ is defined as the central wavelength. ${C_1}$ is a complex valued constant dependent on the position of the point source. The complex amplitude immediately after MO is given as $\tau (x,y,{r_s},{z_s}) \cdot Q( - 1/{f_{mo1}})$. The diffractive optical element $r(x,y) = Q( - {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 a}} \right.}\!\lower0.7ex\hbox{$a$}}){e^{i\theta }}$ is shown on a spatial light modulator (SLM) oriented at a small angle with respect to the optical axis. The SLM is polarization sensitive, and the beams incident on the SLM need to be modulated by the linear polarizer $L{P_1}$. For the first branch, LP1 leaves half of the incident light unmodulated by the SLM. At this point, the SLM is equivalent to a reflector, and the complex amplitude arriving at the sensor surface is
Fig. 1. TBFS-FINCH optical multiplexing system. BF, bandpass filter; $L{P_1}$, linear polarizer 1; $L{P_2}$, linear polarizer 2; BS, beam splitter; MO, microscope objective, ${\times}$ 4, 0.1-NA, ${f_{mo1}}$=7 mm; HWP, half-wave plate; M, mirror; SLM, spatial light modulator; CCD, charge coupled device.
$L{P_1}$ similarly causes half of the incident beam to be modulated by the PM of the SLM, and the modulated wave at the same position is
Obviously, the triple multiplexed hologram is ${H_{mn}} = \sum\limits_{k = 1}^k {{H_k}({r_0};{z_s})}$. (m,n) are the markers of the holograms recorded in the different cases, nothing more, and not parameters.
Using demodulation (i.e., filtering in the Fourier domain), the cross-referenced holograms cannot be effectively separated, so that single phase recovery is in principle impossible. The valuable Fourier terms ${D_R}(x,y;\overline {{r_s}} ,{z_s}) \cdot D_{_O}^{\dagger} (x,y;{\overline r _{s,\theta }},{z_s})$ and $D_{_R}^{\dagger} (x,y;\overline {{r_s}} ,{z_s}) \cdot {D_O}(x,y;{\overline r _{s,\theta }},{z_s})$ are modulated at the spatial carrier frequency of Fig. 2(d) and are concentrated in the centre of the two-dimensional frequency domain. The symbol ${\dagger}$ here stands for complex conjugate. The analysis is performed with respect to the first branch. ${D_O}(x,y;{\overline r _{s,\theta }},{z_s})$ the complex amplitude equation generated by the imaging system for the point object, is available for processing. For the remaining arbitrary branches also the valuable self-referenced signal waves are processed.
Fig. 2. (a) A hologram of TPFS-FINCH. (c) A hologram of TBFS-FINCH. (b) and (d) are the spectrum of (a) and (c). (e) is a reconstruction of the angular spectrum of (a). (f) is a reconstruction of the angular spectrum of (c). (f) is a reconstruction of the power spectrum of (c). (g) and (h) are the intensity distributions of the corresponding colour paths in (e) and (f), respectively.
In order to implement triple-beam frequency-selective FINCH (TBFS-FINCH), the corresponding experimental steps are as follows:
- (1) The specimens are placed at symbols O,S,A in Fig. 1(a) and the fast axis of the HWP is adjusted to be parallel to the direction of polarization of the object wave light.
- (2) Figure 1(b) is loaded onto the SLM. With the HWP placed on the first branch only, the intensity distribution acquired on the CCD is ${H_{11}}$. By placing the HWP on the second branch only, the intensity distribution acquired on the CCD is ${H_{12}}$. Applying the HWP on the third branch only, the intensity distribution acquired on the CCD is ${H_{13}}$. When placing two HWPs on the first and second branches respectively, the intensity distribution acquired on the CCD is ${H_{14}}$. The two HWPs are placed in the second and third branches and the intensity distribution acquired on the CCD is ${H_{15}}$. The two HWPs are placed in the first and third branches and the intensity distribution acquired on the CCD is ${H_{16}}$.
- (3) Load Fig. 1(c) onto the SLM. Repeat the recording sequence from step (2) (the PM of the SLM has been replaced). The intensity distributions acquired by the CCD are: ${H_{21}}$, ${H_{22}}$, ${H_{23}}$, ${H_{24}}$, ${H_{25}}$, ${H_{26}}$.
In post-processing, a Fresnel modulation and a straight reference wave are used to remove defects in the optical path. The redundant terms in Eq. (4) are eliminated by a triple frequency selection scheme. The frequency selection equation is shown below.
3. Experiments
The principle of the TBFS-FINCH was demonstrated experimentally using the digital microholographic setup shown in Fig. 1(a). The xenon lamp (CET-TCX250, 250W) is mounted on the left side of the entire illumination channel. Narrow bandwidth filter $\varDelta \lambda$ = 20 nm with a central wavelength of 633 nm. The plano-convex lens (CL) allows the conversion of point light sources into parallel beams. The effective focal length of the biconvex lens L is 50.8 mm and the light from the incoherent light source is focused by the lens L onto the object in the critical illumination configuration. Three pairs of identical Mo and USAF 1951 charts were placed in three branches in order to strictly magnify the information of the specimen under test by a factor of four. The light aggregated by the BS is polarized by the linear polarizer $L{P_1}$ oriented to an angle of ${45^\circ }$ with respect to the active axis of SLM (FSLM-2K70-VIS, CAS Microstar, 1920 ${\times}$ 1080 pixels, 8 $\mu m$ pixel pitch, phase-only modulation). The mask, calculated using the Fresnel diffraction principle, is displayed on the SLM in 1920 ${\times}$ 1080 pixels. About half of the light intensity is modulated by the mask, the remaining beam is unaffected and reflected. As for the phase shift, the frequency selector combined with the filtering algorithm uses a two-step phase shift, corresponding to a phase shift difference of $\pi$ between the two PMs displayed in the SLM to eliminate twin images and zero-order terms. In order to enable only similarly oriented components of the modulated and unmodulated waves to propagate and cause interference at the sensor plane (MER-502-79U3M/C, 2448 ${\times}$ 2048 pixels, 3.45$\mu m$ pixel size, 79 fps capturing speed), a second polarizer $L{P_2}$, positioned after the SLM, is orientated at 45 degrees with regard to the active axis of the SLM. The distance ${z_s}$ is variable, and the distance ${z_h}$ = 15 cm between the SLM and the camera is aligned with the target so that elements 3,4 and 5 of group 2 of the USAF charts can be imaged onto the camera.
To further experimentally analyse the axial resolution of the TBFS in real scenarios, three planar objects were constructed at same relative distances using USAF resolution charts mounted on channels 1,2 and 3 respectively.
The two-pack incoherent frequency-selective holography technique has been demonstrated in a previous study [22]. This technique was shown to reconstruct the object waves of both branches without loss of resolution. To demonstrate that the 3 PH and 2 PH in the incoherent domain are equally capable of frequency selection. The effect of mixing and superimposing holographic experiments of elements 3 (5.04 lp/mm), 4 (5.66 lp/mm) and 5 (6.35 lp/mm) of group 2 of the USAF chart is shown in Fig. 2. Figure 2(e) is a reconstruction of the angular spectrum of Fig. 3(a). The holograms of the overlapping numbers 3 and 5 were separated without eliminating the coherent background. Figure 2(g) shows the intensity distribution of the corresponding path in (e). The background intensity is around 80 arbitrary unit (a.u.) and the intensity of the figures is around 90 a.u. This is a capability that the two-pack frequency-selective-FINCH (TPFS-FINCH) setup possesses and has been demonstrated previously in the literature. The hologram of the experiment we performed with the setup TBFS-FINCH according to Fig. 1 is shown in Fig. 2(c). The background Fresnel diffraction ring in this hologram is still visible, indicating that the hologram is a superposition of three self-interfering holograms. Similarly, 3 PH is available for spectral selection without loss of resolution. Figure 2(f) shows the power spectrum reconstructed by stripping elements 3, 4, and 5 of group 2 of the USAF chart from the corresponding branches. Figure 2(h) presents the intensity distribution of the three colour paths in (f). Since the graph (f) is reconstructed as a power spectrum, i.e., as the square of the inverse Fourier transform, the background intensity is essentially zero. It is easy to discern the contour information of the object. Figure 2(f) also demonstrates that the TBFS-FINCH has a triple field of view, which is advantageous in situations where detector resolution is insufficient. In the frequency selection scheme for axial holographic superposition, the ratio of the cut-off corner frequency to the maximum corner frequency is 1. The calculation of the wavefront acquisition efficiency for intensity superposition shows that the efficiency of arbitrary n-branch multiplexed holographic frequency selection is n/8. For a on-axis FINCH, a single acquisition of an object wavefront typically requires a three-step phase shift technique to eliminate the conjugate and DC terms. The acquisition of n wavefronts requires 3n holograms. For the setup described in this paper, the improvement in wavefront acquisition efficiency is a factor of 9/8, breaking the limit of less than 1 for the on-axis holographic frequency selection technique.
Fig. 3. (a) is a hologram of TBFS-FINCH, containing paramecium and water cotton cell. (c), (d), (e) are three-branch inverse Fourier reconstructions of (a). (f)-(h) are the intensity figures of (c)-(e). (b) is the intensity distribution of (f)-(h) corresponding to the colour path.
The TBFS-FINCH setup also has excellent capabilities in biomicroscopic imaging. Many new optical systems [28–30] use microlens arrays to split a complete laser wavefront into many tiny parts in space, effectively superimposing a number of images on the camera target surface. Due to the limitations of the imaging equipment’s field of view, there are multiple biological cell images that are multiplexed on a single camera. The ability to actively select the desired cells for individual analysis is extremely beneficial in this case to exclude extraneous interference factors. In Fig. 3(a), we have taken multiplexed holograms of a water cotton cell and two paramecium of different sizes using a frequency-selective setup. Figure 3(c),(d),(e) shows the inverse Fourier patterns reconstructed by frequency selection from Eqs. (6)-(8) in Section 2, while also feeding back the object wave information of the three branches. The results show that the three spectral stripping reconstructions of the triplexed holograms have been successfully performed. (f)-(h) are intensity figures corresponding to (c)-(e). The intensity distributions of the selected lines S1, S2, and S3 in (f)-(h) are shown in Fig. 3(b). It is demonstrated that the reconstructed cell images after frequency selection can be highlighted in the complex surrounding environment. The incoherent optical holographic frequency selection technique reconstructs the cellular hologram without destroying the cellular structure. It is valuable for studying the parameters of cellular physiological activity.
Figure 4 provides the average cross section of a 300 lp/mm grating based on the reconstruction of the hologram obtained with the TBFS setup. The grating visibility is 0.84, and the frequency selective setup still provides excellent lateral resolution compared to the grating visibility of conventional imaging, FINCH, and COACH in the literature [23,24,31].
Fig. 4. (a) a hologram of TBFS-FINCH. (b) The reconstruction of grating. (c) Average cross-sections of the 300 lp/mm grating.
It is noticeable that the TFBS-FINCH setup demonstrates notable slicing capabilities, which we demonstrate with three paramecium specimens. As shown in Fig. 5, with the first two branches being the front and back of the 3D object and the third being the internal section of the 3D object, the specimens are placed on the stepper motor and moved in 0.3 mm steps. The stripped spectrum inverse Fourier transform (IFT) reconstruction of multiplexed holograms using a frequency selection algorithm is used in the experiment. As can be seen from the reconstructed figure, as the focus position of the MO is moved axially over the specimen, the reconstructed section also corresponds to a slice of the corresponding step inside the 3D object. An intensity distribution was made for the red line cross section in Fig. 5, indicating that the internal refractive index and profile of the object are different at different locations in the axial direction, as shown in Fig. 6.
Fig. 5. Experimental comparison of the results of the third path reconstruction when the position of the third branch relative to the focal point of the MO ($\Delta d$) is varied from 0 to 1.8 mm in steps of 0.3 mm.
Fig. 6. Intensity distribution of the red line segments corresponding to the step IFT reconstruction in Fig. 5.
To test the numerical focusing capability of the setup, we made the aerosol generator spray alumina particles so that they spread around the sample location in the three branches. The multiplexed aerosol hologram captured on the CCD is shown in 7(a). The ideal plane for focused reconstruction can be determined using an autofocus technique based on weighted spectral analysis [32] for the microscopes discussed here. The phase variation brought on by aberrations during phase extraction is not insignificant. Here, we employ the double exposure method to solely display the object’s phase distribution. The reconstructed phase has been unwrapped, and the background phase noise and Fresnel mask geometry tilt-related phases have been effectively eliminated, as can be seen in Fig. 7(e). The phase distribution was converted to thickness distribution using $d = {\raise0.7ex\hbox{${\Delta \phi }$} \!\mathord{\left/ {\vphantom {{\Delta \phi } {[({n_a} - {n_r})k]}}} \right.}\!\lower0.7ex\hbox{${[({n_a} - {n_r})k]}$}}$, where ${n_a}$=1.7659 and ${n_r}$=1.0003 [33] are the constant mean refractive indices of the monochromatic light at 633 nm and the aluminium oxide aerosol's surrounding medium (in this example, air), respectively.
Fig. 7. (a) a aerosol hologram of TBFS-FINCH. (b), (c), (d) are three-branch reconstructions of (a). (e) is the unwrapped phase of (c). (f) Three-dimensional thickness reconstruction of partially phased. (g) Corresponds to the intensity distribution of the dashed line. (f) Local magnification curve in (g).
Figures 7(b)-(d) are based on frequency-selected reconstructions under TBFS-FINCH, representing aerosol lateral cross sections at different axial distances from MO. This has enlightening implications for the determination of aerosol concentrations within a certain space. Figure 7(f) shows the aerosol thickness of a partial phase conversion. The holographic technique can record not only the amplitude of the aerosol particles but also the phase diagram of the particles. The average intensity of the red curve in Fig. 7(g) is approximately 160 a.u. It indicates that the unmodulated background in the reconstructed map of (c) is still present. The average intensity after phase unwrapping is approximately 35 a.u., eliminating the background noise, as shown in Fig. 7(h). We expanded the true phase of the aerosol particles using a double-exposure unwrapping algorithm. The thickness converted by the particle phase reveals the exact size of the particles, which range from approximately 1 $\mu m$ to 10 $\mu m$. Figure 7 also confirms that the TBFS setup has the capability to quantitatively visualize phase. Clearly, technology that can accurately record the phase of an object has important implications for research in the field of holographic tomography.
4. Discussion and summary
In this study, we report on the implementation of incoherent multiplexed frequency-selective holography using a triple-beam setup. Our primary contributions in this area are the first demonstrations of frequency-selective reconstruction of incoherent three-packet object waves and optical frequency-selective slice tomography. It is not a conventional multi-camera independent exposure phase shift reconstruction [34–36] or inverse filtered projection laminar reconstruction [37,38]. The reflective SLM can be reloaded twice as needed with the phase mask without moving the camera. The proposed TBFS technique is based on only a few components in a simple and static configuration, allowing it to be implemented as a stand-alone module for fast and accurate quantitative phase imaging (QPI). Compared to other multiplexed imaging techniques, TBFS not only improves efficiency by reducing resolution, but also extends FOV.
Coherent optical frequency-selective holography [19,20,21] is being studied and cited by an increasing number of researchers due to its vastly improved performance in terms of wavefront acquisition efficiency and FOV. So far, the number of multiplexed waves of coherent light has been experimentally achieved up to 8 PH. Coherent optical multiplexing differs from TBFS in theory in that it employs a dual image area interferometry technique. Using this technique, two interferograms are projected onto the camera, each of which captures a distinct area of the sample or a complicated wave front for a separate sample. Coherent optical multiplexing allows for the reconstruction and splicing of two or more FOVs using spatial filters since the FOVs do not overlap in the spatial frequency domain. Equations (6)-(8) demonstrate how the TBFS technique used in this study vertically shears the matching spectrum using a spatial transfer function with phase shift. Compared to the TPFS scheme [22], the TBFS scheme deterministically achieves an extension from two-branch multiplexed frequency selection to three-branch. It illustrates the theoretical possibility of extending the pack number from two to infinite dimensions for incoherent optical frequency selection.
Depending on the form of spectral filtering, the wavefront acquisition efficiency of TBFS is lower than that of the coherent light multiplexing holographic frequency selection scheme. Since the temporal coherence length of incoherent light is very small, in the sub-micron order, it is difficult to achieve off-axis holography. Therefore, incoherent holography is mainly used in on-axis holography, which requires the use of phase shift techniques. For a single FINCH, wavefront acquisition usually requires a three-step phase shift to remove the conjugate and zero-level terms. The TBFS scheme improves the wavefront acquisition efficiency by 3n/8 compared to conventional on-axis holography. n is the number of object waves to be acquired. Clearly, the larger the value of n, the more efficient the incoherent frequency selection, i.e., the higher the reconstruction time resolution. Multiplexed frequency selection offers more options for the development of holographic encrypted transmissions, quantum encryption [39,40] and so on.
The TBFS technique in this paper shows the capability of QPI on the micrometer scale in aerosol separation reconstruction. The setup has a single-channel multiplexing feature, a feature that reduces phase fluctuations that may originate from ambient noise [41,42]. This opens up the option of using intensity superposition states to extract phases in the field of remote sensing detection. Multi-pack holography enables the compressed transmission of many layers of information, which is useful when detector resolution is insufficient. This implies that it is unnecessary to factor each subwavefront’s capture interval into the camera’s frame rate. Our research supports the premise that several camera exposures are no longer necessary for QPI based on TBFS. Comparing the precision and stability of the TBFS imaging method to that of traditional multiplexed holography [22–25], we discovered that there was no degradation (see Fig. 2). Our study paves the way for the development of wide-field super-resolution TBFS-FINCH for biological or metrological imaging, as proven by the USAF1951 resolution charts, paramecium sections, and aerosol dynamics frequency-selective reconstructions.
Funding
National Natural Science Foundation of China (41371336, 52177137).
Acknowledgment
We thank the anonymous reviewers for their comments on this paper.
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.
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