Open Access
Add to BibSonomy
Abstract
This paper proposes one-shot synthetic aperture digital holographic microscopy using a combination of angular-multiplexing and coherence gating. The proposed angular-multiplexing technique uses multiple noncoplanar incident beams into the synthetic aperture to create tight packed passbands so as to extend spatial frequency spectrum. Coherence gating is performed to prevent the self-interference among the multiple beams. Based on the design guideline proposed herein, a phase-only spatial light modulator is employed as an adjustable blazed grating to split multiple noncoplanar beams and perform angular-multiplexing, and then using coherence gating based on low-coherence-light, superresolution imaging is achieved after one-shot acquisition.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Digital holography [1–4] refers to the acquisition of holographic interferograms of object beams and reference beams on CCD/CMOS image sensors and numerical reconstruction of the three-dimensional information of the object wavefront. This imaging method is extremely beneficial and has great application potential for conducting precise analyses and examinations of microscopic samples with amplitude and/or phase. To determine the fine structures of the sample, digital holography microscopy (DHM) can be employed [5–7]. In DHM, a microscope objective is used to enlarge samples, thereby increasing the spatial resolution of an obtained image. However, similar to traditional optical imaging systems, DHM can achieve a spatial resolution that is limited by the aperture size of its microscope objective. To overcome the limited spatial resolution imposed by the aperture of a microscope objective under normal incidence of light, researchers often introduce synthetic aperture (SA) techniques to acquire additional high-frequency information [8–14]. The SA techniques include image sensor shifting, sample rotation, beam scanning, and diffraction grating usage [15–18]. The positions and directions of an object acquired using such methods contain high-frequency information and the frequency spectrum can be extended using passband superposition and then employed to reconstruct superresolution imaging. One of the most commonly employed techniques is to insert a galvo mirror into the experiment framework to alter the angle of incidence when scanning a sample [19, 20]. Beam scanning with a galvo mirror must often be time-shared or use time-multiplexing to record numerous holograms for subsequent spectrum synthesis. However, when holograms are being recorded, if the beams or galvo mirror are unstable or if environmental factors interfere, the quality of the holograms is reduced, which would consequently affect the spectrum synthesis and image reconstruction. One feasible method of overcoming the aforementioned problems, including time-consuming multiple exposures and disturbances in recordings and mechanisms, is to employ an SA with angular-multiplexing to reduce the recording period, even to the degree of that only one-shot is required.
Laser pulses were previously employed in angular-multiplexing SA digital holography [21, 22]. Ultrashort pulsed laser was used as light source and split into three incident beams with different angles of incidence; in one exposure cycle, angular-multiplexing holography is performed. However, this technique requires a rather complicated collimated experiment structure to split and record only three light beams. Composed of complicated lenses and beam splitters, this technique also acutely reduces the degrees of freedom and flexibility in optical system arrangement. Consequently, the feasibility of recording more light beams and the convenience of practical operations is limited. These limitations make the SA unfavorable for application to one-shot superresolution imaging. By contrast, to simplify the structure of an optical imaging system, computer-generated holograms or grating structures with a spatial light modulator (SLM) can be applied to split light beams and perform angular-multiplexing [23–25]. To produce angular-multiplexing holograms, polarization coding of incident beams from a continued-wave or pulsed laser source [26, 27] is then required to produce orthogonal double-polarized object beams that enable SA imaging. Simultaneously, the self-interference effect between double beams, which is caused by simultaneous exposure, must be avoided. Using the said orthogonal-polarization coding, only two orthogonally polarized beams can be exposure on the same plane; simultaneous recording of multiple beams cannot be achieved. In addition, the holograms recorded using orthogonal polarization multiplexing may be phase-shifted because of anisotropy in observation and a polarization-sensitive medium. Therefore, this paper proposes one-shot SA digital holography using noncoplanar multibeams angular-multiplexing and coherence gating. Through theoretical analysis, the design guideline for angular-multiplexing of each beam could be determined. According to the design guideline, experiments were conducted for verification. In the experiment, a phase-only SLM was used as an adjustable blazed grating and noncoplanar multibeams splitting and angular-multiplexing were performed. In the proposed technique, multiple incident beams pass through the SA, produce the tight packed passbands, and extend the spectrum coverage to achieve superresolution imaging in a single exposure.
2. Principle
The concept of one-shot SA digital holography is illustrated in Fig. 1. The proposed system designs multiple noncoplanar incident beams that arrive at the object from different spatial angles. Incident beams are split using a diffractive optical element and become noncoplanar beams that follow the designed split angles and arrive at the object plane. Using incident beams with different angles and directions, one-shot SA imaging is achieved. Low-coherence light, with the characteristics of coherence gating, is utilized to prevent self-interference among the spilt light beams. In practice, the proposed one-shot SA method utilizes the design guideline regarding how the angles of object incident beams and reference beams can be determined to enable angular-multiplexing: (1) self-interference should be avoided in the split beams formed from incident beams passing through the diffractive optical element and arriving at the hologram plane; and (2) the interference angles of the object beams and reference beams should be designed in such a way that their passbands do not overlap.
Fig. 1 The concept of the angular-multiplexing and coherence gating in the non-coplanar configuration
Consider an incident optical field of noncoplanar multiple beams, represented as a complex number in the following equation [28]:
where is the amplitude of optical field and is wave vector, and is the position vector. The angle between incident beam j and the z-axis is , is the initial phase (phase-shifted) of light beam j, is the polarized direction of light beam j, and n is the number of incident beams. The noncoplanar multiple beams penetrate the object and interfere with each of their corresponded reference beams, and the object wavefront is recorded on the hologram plane. At this moment, the interferences produced by the object and reference beams, self-interference among the noncoplanar multiple beams may also occur. Therefore, the optical field intensity distribution of the multiple incident beams recorded on the hologram plane can be represented aswhere and are any two light beams and is the inner product of any two light beams in the polarization direction. In Eq. (2), the first term is a direct-current term, whereas the second is a self-interference term. If the spectrum of the incident light source has a Gaussian distribution, the interference contrast among light beams, , can be written as [29]where h is the optical path difference (OPD) between any two beams, λ and ∆λ are an central wavelength and spectral bandwidth of the incident beam, respectively. In this study, h was sufficiently long to avoid self-interference among the multiple beams. Similarly, on the reference beam’s side, self-interference among the multiple reference beams should be avoided. Therefore, the reference beams were also designed with an OPD that met the requirements in Eqs. (2) and (3). Consider a hologram recording structure composed of two 4-f imaging systems, as illustrated in Fig. 2. The OPD between any two beams with different split angles can be calculated to avoid self-interference between the beams. Although it would be difficult to consider the actual OPD of a commercial objective within the compound lenses (refractive index, lens shapes etc.), the OPD in the air (up to about 1 mm) is larger enough to ignore the OPD in the lens and thus the dominant OPD in the air between MO1 and MO2 can be derived. Given that the optical path length of an incident beam is L, the OPD of any two splitting beams in the 4-f imaging system can be written asusing Eq. (4), the angular-multiplexing design based on the coherence length of the light source is implemented in an actual system. Consider that the OPD going through L1 and L2 is small enough than the OPD induced by MO1 and MO2 having the same focal length (f2). The OPD of any two split beams in the 4f system can be further simplified and rewritten as which can be the approximate OPD calculation for design guidance. To avoid self-interference, the optical path lengths of different light beams with different split angles are altered to implement coherence gating to filter out beams with OPD greater than the coherence length () for the hologram recording. In this manner, the object and reference beams meet the requirement that incident and reference beam interferences are produced that can be recorded as required hologram. Additionally, unwanted noise from the interference terms that produce self-interference among beams is avoided. Thus, the design guideline (1) of the proposed system is determined. The incident angle of object beam must be satisfied with the condition:In other words, self-interference is avoided by setting conditions for the angle of incident object beams that ensure the OPD of any two light beams is longer than the coherence length. Also, the multibeams noncoplanar design adopts tight packed passband spectrum stacking to widen the spectrum, thereby preventing the spectrum of each beam from stacking in holograms. Therefore, the angle between the object beam’s angle of incidence θj and the reference beam’s direction angle θR—that is, θR + θj—should also satisfy the condition that the passbands of the two interference terms be spatially separated [30, 31]. The passband of an interference term is determined by the cutoff frequency of the imaging system [28]. Also, the maximum coverage of the multipassbands that a digital hologram records should not exceed the maximum bandwidth limit determined [28] by the sampling size that the digital hologram records. From the aforementioned conditions, the system’s design guideline (2) can be stated: the reference light beam’s angle of incidence must satisfy
where n is the refractive index of the environmental medium, D is the diameter of the aperture of objective lens MO2, and ∆x is the sampling point size of the digital hologram in which information is recorded.3. Simulations
To verify the feasibility and effect of the one-shot SA digital holography using noncoplanar multiple beams, angular-multiplexing, and coherence gating, this study conducted computer simulation. According to Eqs. (1)-(6), five object beams and five reference beams in two 4-f imaging systems interfering with each other simultaneously were simulated. Consider that the central wavelength of light beam adopted for coherence gating was 405 nm with coherence length approximating 120 μm, the pixel number was 1024 × 1024 pixels, the pixel size was 5.2 μm, and the magnification of the imaging system was M = 100 × . Firstly, following the design guidelines as shown in Eqs. (5) and (6), five object beams entered the imaging system, and their angles of incidence were 0.55°, −0.65° between the optical axis and the x- axis, and 0.60°, −0.70° between the optical axis and the y-axis, respectively. On the other hand, the angles of five reference beams were −0.55°, 0.65° between the optical axis and the x- axis, and −0.60°, 0.70° between the optical axis and the y-axis, respectively. Then, based on the Fresnel diffraction approximation [28], the propagating light fields of object beams were calculated to generate the interference fringes with reference beams under low coherence effect from statistics optics [29]. The OPDs of each beam were calculated from Eq. (4), respectively, to ensure the required conditions that five angles of incidence for the five object beams and their OPDs were designed to allow only beams that, after passing through the object, would interfere with their corresponding reference beams. The results were recorded in a digital hologram, which subsequently underwent spectrum selection and synthesis before image reconstruction. Finally, the SA approach was employed for spectrum synthesis to determine the horizontal resolution of the reconstructed image. The simulation results are presented in Fig. 3. The testing samples were complex images composed of three line pairs in different width, from top to bottom, of 1.6, 1.0, and 0.5 μm, respectively. Each of the object beams interfered with the five corresponding reference beams of the same angles, and the results were captured in a hologram. Figure 3(a) presents a hologram recording of five object and five reference beams interfered. Figure 3(b) displays the spectra from the hologram in Fig. 3(a) transformed into the Fourier plane. Examination of the spectra revealed that they could be successfully separated without self-interference. As shown in the figure, the interference terms in the spectra were extracted and then their passbands were synthesized for image reconstruction. The passbands obtained from different angles in Fig. 3(b) were used for SA spectrum synthesis as in Fig. 3(f) and for superresolution image reconstruction [32], as illustrated in Figs. 3(f)-3(h). Figures 3(e) and 3(h) present phase images reconstructed for the 0.5 μm linewidth before and after SA spectrum synthesis were performed. Figures 3(i) and 3(j) plot the cross-section of phase images along the x-axis and y-axis. The results reveal that after one-shot SA spectrum synthesis was performed, the horizontal resolution of the reconstructed images in both the x-direction and y-direction was clearly higher.
Fig. 3 Simulations of synthetic aperture recording and reconstruction by single shot angular multiplexing method, (a) and (b) Recorded hologram and its spectrum, (c) spectrum of normal aperture reconstruction, (d) and (e) reconstructed amplitude and phase by normal aperture. (f)Spectrum of synthetic aperture reconstruction, (g) and (h) reconstructed amplitude and phase by synthetic aperture. (i) and (j) is the cross section of x and y directions both in (e) and (h).
4. Experiments
Figure 4(a) displays the experiment framework used for one-shot SA-DHM. The light source employed in this experiment was a diode laser with the central wavelength of 405 nm and coherence length approximating 120 μm. The source beam entered a spatial filter and was expanded into a collimated beam, after which the beam passed through a beam splitter and was split in two; an incidence beam entering the object and a reference beam. Liquid crystal on silicon (LCoS) was employed as a spatial light modulator (SLM). After inputting a series of blazed gratings that were controlled using software, the SLM became a multiplexing element that could split light into different angles [Fig. 4(b)]. At the object end, the beams entered SLM1 [blazed grating shown in Fig. 4(b)]. The reflected light produced four beams with the designed split angles. SLM1 was placed on the front focal plane in the first 4f imaging system (lens L1: f = 150 mm; microscope MO1: 100 × , NA = 0.9), whereas the sample (S) was placed on the back focal plane of this 4f system, enabling the incident beam that had been split by angular-multiplexing to enter the sample. The object beams that exited the sample went through another 4f imaging system (lens L2: f = 150 mm; microscope MO2: 100 × , NA = 0.9) for enlargement and interfered with the reference beams on the image sensor, with the interference captured in a digital hologram. The beam route at the reference end adopted an optical framework similar to that at the object end. SLM2 in the reference end contained two pairs of 4f imaging systems (lenses L3, 4: f = 150 mm; microscope MO3, 4: 40 × , NA = 0.65). The blazed grating controlled by a computer [Fig. 4(c)] produced the required split reference beams. In the one-shot holographic recording procedure with a digital camera (uEye UI-1545), the exposure time can be set up to 35 μs at the power density of 0.1 mW/cm2 for the hologram recording, which the imaging speed depended on the sensitivity of image sensor and the incident power density of object and reference beams.
Fig. 4 Experiments of one-shot SA-DHM. (a)Experimental setup, (b) and (c) Blazing grating for SLM in object and reference arms, respectively. SF: spatial filter, M: mirror, P: Polarizer, BS: Beam spliter, MO: Microscopic objective, L: Lens, S: sample.
The testing samples used in this experiment were Extreme Siemens Star Resolution Target [33–35], which enabled the observation of spatial frequency responses from all directions, enabling a comprehensive analysis of system resolution and image quality. In the resolution test, normal aperture DHM was employed to record and reconstruct testing samples as standards for comparison. The experiment results are presented in Fig. 5. Figure 5(a) displays a hologram (left) and the hologram’s frequency spectrum (right), which shows that only one interference term is present in the spectrum. The reconstructed amplitude and phase image of the interference term are shown in Figs. 5(b) and 5(c), respectively. The phase sensitivity in the image is approximately 8.9°. The horizontal resolution was determined through analyzing the tightest line pairs with clear resolution in the amplitude image; the result is shown in Fig. 5(d). The blue circle in Fig. 5(d) is located in a region wherein the spokes from all directions are clearly outlined. Figure 5(e) plots the sectional view of the spokes, which have a linewidth of approximately 346 nm. The amplitudes of the sectional view indicate that the line pairs can be clearly separated. By contrast, the inner red circle in Fig. 5(d) corresponds to the narrower spokes; as shown in Fig. 5(f), their linewidth is approximately 312 nm. However, some of the line pairs, as indicated by the dashed ellipse, are not clear to be resolved. Therefore, the normal aperture DHM system’s full resolution is approximately 346 nm; this experimental result matches the result calculated from optical imaging theory.
Fig. 5 Experiments results of normal aperture recording. (a) Hologram and its spectrum, (b) and (c) reconstructed amplitude and phase, (d) magnification image at the center of (b), (e) and (f) is the cross-section of (d) at line pair of 346nm and 312 nm.
Figure 6 presents the SA records and reconstructions acquired using the one-shot SA-DHM system proposed in this study. Figure 6(a) shows a digital hologram and the spectrum distribution with multibeams recorded simultaneously. The spectrum distribution reveals that a total of five beams, which incident at different angles, could be simultaneously recorded in one hologram. Because of the multiple diffraction term from the periodic structure of LCoS device [36], some undesired terms were also found in spectrum of Fig. 6(a), which might interfere the desired passbands and degrade the reconstructed image quality. In the reconstruction process, these five interference terms were individually circled to reconstruct SA images as the procedure in the simulation section.”. The reconstructed amplitudes and phase images are shown in Figs. 6(b) and 6(c), respectively. The phase sensitivity was approximately 14.9°. The horizontal resolution was again determined by the linewidth of the tightest fringes in the amplitude image, and the results are presented in Figs. 6(d). The blue circle in Fig. 6(d) corresponds to the sectional view in Fig. 6(e), which reveals that all the spokes were resolved. At this position, the line pairs with clear resolution are approximately 290 nm. However, if thinner line pairs were observed in the sectional view further toward the center—as indicated by the red circle in Fig. 6(d), which contains a linewidth of 230 nm—some of line pairs in certain regions can be resolved, as shown by the dotted ellipse in the sectional view [Fig. 6(f)]. Therefore, even though the impact of aberrations was not eliminated and deconvolution analysis was not performed [36], the one-shot SA-DHM proposed in this study achieved a spatial resolution of 230 nm. Compared with that achieved using the normal aperture method, the resolution achieved with the proposed method was approximately 1.5 times higher. A possible cause of parts of the high-frequency spatial information not resolved is likely to be the increase in high-frequency noise as the one-shot SA passband frequencies were synthesized and the image reconstructed.
Fig. 6 Experiments results of one-shot synthetic aperture recording. (a) Hologram and its spectrum, (b) and (c) reconstructed amplitude and phase, (d) magnification image at the center of (b), (e) and (f) is the cross-section of (d) at line pair of 290 and 230 nm.
Therefore, to analyze the impact of high-frequency noise on image reconstruction, this study first employed hologram recording without test samples to determine how the number of one-shot multibeams interference terms affected phase noise, which subsequently affected the system’s phase sensitivity. The result is illustrated in Fig. 7. The passband frequency spectra of each interference term in a single hologram were synthesized one by one, and image reconstruction was then performed to observe the change in phase sensitivity on the reconstructed plane. The change in phase sensitivity is plotted in Fig. 7. Figure 7(a) reveals that when the number of interference terms increased, the phase sensitivity worsened from the original 8.9° to 14.9°. This may have been because high-frequency noise of the image sensor, such as shot noise [37], was being repeatedly stacked into the synthesized spectrum following stacking of passband frequencies at different positions. Consequently, the noise in the reconstructed image was increased and the corresponding phase sensitivity decreased. When the beam was split, the exposure intensity was also reduced among the multibeams, resulting in an increase in the shot noise of the image sensor and consequently leading to the reduction of the phase sensitivity for DHM reconstruction. Given this understanding, multiple holograms were adopted to decrease the high-frequency noise. The frequencies at the same locations were stacked and then averaged over time to reduce the noise. To acquire multiple holograms, another experiment was conducted: the blazed grating in an SLM was rotated to different angles to produce five beams with different directions. Starting from 0°, a hologram was obtained when the blazed grating was rotated by 15°; a total of six holograms were produced to meet the goal of acquiring multiple holograms and averaging them over time. The results, presented in Fig. 7(b), reveal that the phase sensitivity of a reconstructed plane improves as the number of holograms increases, reaching approximately 12.3°. The aforementioned method, adopting multiple holograms and averaging them over time, was incorporated into this study’s experiment framework, and samples were analyzed using Extreme Siemen's Star Resolution Target. The reconstructed amplitudes and phase images from a total of six holograms and spectra (including 30 passbands) were stacked and are shown in Figs. 8(a) and 8(b), where the phase sensibility was approximately 12°. The sectional view of the amplitude was obtained from the line pairs in Fig. 8(a) around 200 nm. As Fig. 8(c) shows, most of the line pairs can be resolved, only some of the line pairs were blurred as indicated by the dotted ellipse. The experiment results indicated that when only six holograms were employed, noise could be reduced and the resolution of an image reconstruction was up to 1.72 times higher than that of the images formed using conventional-incidence normal aperture DHM imaging.
Fig. 7 (a) phase sensitivity variation according to number of interference term in single hologram, (b) enhancement of phase sensitivity by numbers of time average.
Fig. 8 Synthetic aperture reconstructed image after time average, (a) amplitude, (b) phase image, and (c) cross section of (a) at line pair of 200 nm.
5. Conclusions
This study has proposed and experimentally demonstrated a novel one-shot SA-DHM method that employs multibeams angular-multiplexing and coherence gating. Additionally, this study provides a useful design guideline for performing noncoplanar object and reference beam angular-multiplexing and coherence gating to effectively obtain one-shot digital holograms. The proposed method employs software-controlled SLMs to incorporate blazed gratings that split multiple beams and perform angular-multiplexing. Without requiring the use of a complicated optical experiment framework and algorithm, this method hugely increased the degrees of freedom and flexibility of the system, thereby increasing its practical operation convenience. The results demonstrated that compared with the normal aperture DHM, the proposed method achieved a lateral resolution under single exposure that was approximately 1.5 times higher. If multiple holograms were averaged by space, the system’s lateral resolution was improved approximately to 1.7 times higher. By combining the tomographic imaging technique [38], the proposed one-shot SA-DHM has potential to observe and track 3D moving biological specimen.
6. Acknowledgment
This work was financially supported by the Ministry of Science and Technology, Taiwan (MOST 105-2221-E-003-015-MY3 and 104-2221-E-034-010-MY3).
References and links
1. M. K. Kim, Digital Holographic Microscopy: Principles, Techniques, and Applications (Springer, 2011).
2. E. Cuche, F. Bevilacqua, and C. Depeursinge, “Digital holography for quantitative phase-contrast imaging,” Opt. Lett. 24(5), 291–293 (1999). [CrossRef] [PubMed]
3. W. Osten, A. Faridian, P. Gao, K. Körner, D. Naik, G. Pedrini, A. K. Singh, M. Takeda, and M. Wilke, “Recent advances in digital holography [invited],” Appl. Opt. 53(27), G44–G63 (2014). [CrossRef] [PubMed]
4. A. Vijayakumar, Y. Kashter, R. Kelner, and J. Rosen, “Coded aperture correlation holography-a new type of incoherent digital holograms,” Opt. Express 24(11), 12430–12441 (2016). [CrossRef] [PubMed]
5. 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(34), 6994–7001 (1999). [CrossRef] [PubMed]
6. X.-J. Lai, H. Y. Tu, Y. C. Lin, and C. J. Cheng, “Coded aperture structured illumination digital holographic microscopy for superresolution imaging,” Opt. Lett. 43(5), 1143–1146 (2018). [CrossRef] [PubMed]
7. B. Rappaz, I. Moon, F. Yi, B. Javidi, P. Marquet, and G. Turcatti, “Automated multi-parameter measurement of cardiomyocytes dynamics with digital holographic microscopy,” Opt. Express 23(10), 13333–13347 (2015). [CrossRef] [PubMed]
8. M. Karaman, P.-C. Li, and M. O’Donnell, “Synthetic aperture imaging for small scale systems,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 42(3), 429–442 (1995). [CrossRef]
9. M. Bashkansky, R. L. Lucke, E. Funk, L. J. Rickard, and J. Reintjes, “Two-dimensional synthetic aperture imaging in the optical domain,” Opt. Lett. 27(22), 1983–1985 (2002). [CrossRef] [PubMed]
10. A. Stern and B. Javidi, “3-D computational synthetic aperture integral imaging (COMPSAII),” Opt. Express 11(19), 2446–2451 (2003). [CrossRef] [PubMed]
11. M. Kim, Y. Choi, C. Fang-Yen, Y. Sung, R. R. Dasari, M. S. Feld, and W. Choi, “High-speed synthetic aperture microscopy for live cell imaging,” Opt. Lett. 36(2), 148–150 (2011). [CrossRef] [PubMed]
12. G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nat. Photonics 7(9), 739–745 (2013). [CrossRef] [PubMed]
13. Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marquet, and C. Depeursinge, “Marker-free phase nanoscopy,” Nat. Photonics 7(2), 113–117 (2013). [CrossRef]
14. W. Luo, A. Greenbaum, Y. B. Zhang, and A. Ozcan, “Synthetic aperture-based on-chip microscopy,” Light Sci. Appl. 4(3), e261 (2015). [CrossRef]
15. H. E. de Bruijn, R. P. H. Kooyman, and J. Greve, “Surface plasmon resonance microscopy: improvement of the resolution by rotation of the object,” Appl. Opt. 32(13), 2426–2430 (1993). [CrossRef] [PubMed]
16. J. H. Massig, “Digital off-axis holography with a synthetic aperture,” Opt. Lett. 27(24), 2179–2181 (2002). [CrossRef] [PubMed]
17. C. Liu, Z. G. Liu, F. Bo, Y. Wang, and J. Q. Zhu, “Super-resolution digital holographic imaging method,” Appl. Phys. Lett. 81(17), 3143–3145 (2002). [CrossRef]
18. S. A. Alexandrov, T. R. Hillman, T. Gutzler, and D. D. Sampson, “Synthetic aperture fourier holographic optical microscopy,” Phys. Rev. Lett. 97(16), 168102 (2006). [CrossRef] [PubMed]
19. V. Lauer, “New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope,” J. Microsc. 205(2), 165–176 (2002). [CrossRef] [PubMed]
20. X. J. Lai, H. Y. Tu, C. H. Wu, Y. C. Lin, and C. J. Cheng, “Resolution enhancement of spectrum normalization in synthetic aperture digital holographic microscopy,” Appl. Opt. 54(1), A51–A58 (2015). [CrossRef] [PubMed]
21. C. Yuan, H. Zhai, and H. Liu, “Angular multiplexing in pulsed digital holography for aperture synthesis,” Opt. Lett. 33(20), 2356–2358 (2008). [CrossRef] [PubMed]
22. M. Paturzo, P. Memmolo, A. Tulino, A. Finizio, and P. Ferraro, “Investigation of angular multiplexing and de-multiplexing of digital holograms recorded in microscope configuration,” Opt. Express 17(11), 8709–8718 (2009). [CrossRef] [PubMed]
23. Y. Kuroiwa, N. Takeshima, Y. Narita, S. Tanaka, and K. Hirao, “Arbitrary micropatterning method in femtosecond laser microprocessing using diffractive optical elements,” Opt. Express 12(9), 1908–1915 (2004). [CrossRef] [PubMed]
24. A. Hussain, J. L. Martínez, A. Lizana, and J. Campos, “Super resolution imaging achieved by using on-axis interferometry based on a Spatial Light Modulator,” Opt. Express 21(8), 9615–9623 (2013). [CrossRef] [PubMed]
25. S. Hasegawa and Y. Hayasaki, “Polarization distribution control of parallel femtosecond pulses with spatial light modulators,” Opt. Express 21(11), 12987–12995 (2013). [CrossRef] [PubMed]
26. X. J. Lai, C. J. Cheng, Y. C. Lin, and H. Y. Tu, “Angular- and polarization-multiplexing with spatial light modulators for resolution enhancement in digital holographic microscopy,” J. Opt. 19(5), 055607 (2017). [CrossRef]
27. Y. C. Lin, C. J. Cheng, and L. C. Lin, “Tunable time-resolved tick-tock pulsed digital holographic microscopy for ultrafast events,” Opt. Lett. 42(11), 2082–2085 (2017). [CrossRef] [PubMed]
28. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company Publishers, 2005).
29. J. W. Goodman, Statistical Optics (Wiley, 2000).
30. U. Schnars and W. P. O. Jüptner, Digital Holography, (Springer US, 2005).
31. N. Pavillon, C. S. Seelamantula, J. Kühn, M. Unser, and C. Depeursinge, “Suppression of the zero-order term in off-axis digital holography through nonlinear filtering,” Appl. Opt. 48(34), H186–H195 (2009). [CrossRef] [PubMed]
32. J. P. Wilde, J. W. Goodman, Y. C. Eldar, and Y. Takashima, “Coherent superresolution imaging via grating-based illumination,” Appl. Opt. 56(1), A79–A88 (2017). [CrossRef]
33. Y. Takahashi, A. Suzuki, S. Furutaku, K. Yamauchi, and T. Ishikawa, “High-resolution and high-sensitivity phase-contrast imaging by focused hard x-ray ptychography with a spatial filter,” Appl. Phys. Lett. 102(9), 094102 (2013). [CrossRef]
34. S. B. Mehta and R. Oldenbourg, “Image simulation for biological microscopy: microlith,” Biomed. Opt. Express 5(6), 1822–1838 (2014). [CrossRef] [PubMed]
35. R. Horstmeyer, R. Heintzmann, G. Popescu, L. Waller, and C. Yang, “Standardizing the resolution claims for coherent microscopy,” Nat. Photonics 10(2), 68–71 (2016). [CrossRef]
36. J. Li, Y. C. Lin, H. Y. Tu, J. Gui, C. Li, Y. Lou, and C. J. Cheng, “Image formation of holographic three-dimensional display based on spatial light modulator in paraxial optical systems,” J. Micro/Nanolith. MEMS MOEMS. 14(4), 041303 (2015). [CrossRef]
37. Y. L. Lee, Y. C. Lin, H. Y. Tu, and C. J. Cheng, “Phase measurement accuracy in digital holographic microscopy using a wavelength-stabilized laser diode,” J. Opt. 15(2), 025403 (2013). [CrossRef]
38. B. Vinoth, X.-J. Lai, Y.-C. Lin, H.-Y. Tu, and C.-J. Cheng, “Integrated dual-tomography for refractive index analysis of free-floating single living cell with isotropic superresolution,” Sci. Rep. 8(1), 5943 (2018). [CrossRef] [PubMed]
