Abstract

We propose a phase-shifting interferometry technique using only two in-line phase-shifted self-interference holograms. There is no requirement for additional recording or estimation in the measurement. The proposed technique adopts a mathematical model for self-interference digital holography. The effectiveness of the proposed technique is demonstrated by experiments on incoherent digital holographic microscopy and color-multiplexed fluorescence digital holography with computational coherent superposition. Two-color-multiplexed four-step phase-shifting incoherent digital holography is realized for the first time, to the best of our knowledge, using the proposed technique.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Multidimensional imaging is one of the actively researched themes in both science and industry. Multidimensional information, such as three-dimensional (3D), wavelength, and polarization distributions, has been applied to observe realistic scenes of remote locations, structures with invisible fields of view, such as microscopic and nanoscopic regions, and invisible distributions, such as acoustic and electromagnetic wave distributions in all wavelength bands. The acquisition of 3D information is important, particularly when a person and a machine perceive and observe 3D structures of samples. Color and polarization information is useful for identifying and distinguishing objects. Multiple image sensors, color filters, and an array of polarizers are generally used to obtain multidimensional information.

Using holography [1] and digital holography (DH) [2], multidimensional information is recorded simultaneously on a 2D recording material without any color and polarization filters [37]. Holography exploits phase encoding in the recording of multidimensional information simultaneously on multiplexed holograms and decoding in the reconstruction of object waves at multiple wavelengths and polarization directions separately on the spatial [3] and temporal [4] frequency domains or on the basis of phase-shifting interferometry (PSI) [57]. By using phase encoding in holographic multiwavelength and polarization imaging, absorption by filters is not always needed.

In DH, particularly incoherent DH [8], in-line self-interference optical setups with PSI, such as Fresnel incoherent correlation holography (FINCH) [9], a Michelson interferometer [10] and a single-path phase microscope [11] are adopted frequently. For example, in self-interference incoherent DH, the single-shot techniques of spatial PSI [12,13], off-axis configuration [14], and the use of cross correlation with a point-spread-function (PSF) library [15] have been proposed. By using single-shot PSI, an incoherent hologram with a large space-bandwidth product (SBWP) can be obtained in comparison with an off-axis configuration, and holographic sensing can be conducted for a 3D field whose PSF changes frequently. In single-shot incoherent DH with PSI, the SBWP is improved by reducing the number of phase shifts. In PSI [16], the recording of three or four phase-shifted holograms is generally adopted. However, experts in interferometry and holography have proposed the use of two-step PSI to reduce the number of exposures [17,18]. In the proposals for two-step PSI, Meng’s algorithm requires the recording of the intensity distribution for one of the two waves [17]. Liu and Poon have proposed two-step-only PSI, but it is necessary to estimate the zeroth-order diffraction image using a complex algorithm [18].

In this Letter, we propose two-step PSI for self-interference DH with neither additional recording nor estimation. A mathematical model for self-interference DH is constructed to create the proposed PSI, and the proposed PSI is applied to self-interference incoherent digital holographic microscopy (DHM) systems. The proposed PSI is also applied to color-multiplexed fluorescence DH based on computational coherent superposition (CCS) [19] to reduce the number of recordings. Experimental results show the effectiveness of the proposed two-step PSI and its applicability to CCS with the number of exposures reduced to $ 2N $, where $ N $ is the number of recorded wavelength bands.

Figure 1 illustrates a self-interference incoherent DHM system with FINCH. Using a wide-field inverted optical microscope, a magnified image of the specimens is obtained. The magnified image is introduced to a FINCH system. An object wave of the magnified image passes through relay optics and is linearly polarized by a polarizer. A birefringent material (BM1) is set to generate two object waves whose curvature radii are different and polarization directions are orthogonal. A liquid crystal on silicon spatial light modulator (LCOS-SLM) shifts the phase of one of the two object waves. Another BM (BM2) adjusts the optical-path-length difference, and another polarizer aligns the polarization directions of the two waves. The two waves generated from a point of an object form a Fresnel zone plate (FZP) pattern, and each FZP pattern from each point is incoherently superimposed on the image sensor plane. The recorded image corresponds to a self-interference incoherent hologram, which is generated by the two object waves. A phase shift $ a $ is introduced to the hologram using the LCOS-SLM, and two self-interference phase-shifted incoherent holograms are obtained sequentially. Under the condition of self-interference DH, we express the two phase-shifted holograms as

$${I_1}(x,y) = |A(x,y){|^2}\left({1 + \frac{{2\sqrt M}}{{1 + M}}\cos \phi (x,y)} \right),$$
$$\!\!{I_2}(x,y) = |A(x,y){|^2}\left({1 + \frac{{2\sqrt M}}{{1 + M}}\cos ({\phi (x,y) - \alpha} )} \right),\!$$
where $A(x,y)$ is the amplitude distribution of incident light, $ M $ is the intensity ratio between the two waves, and $\phi (x,y)$ is the phase difference between the two waves. $ M $ can be treated as known by adjusting the axes of the polarizers in Fig. 1; in general, $M = 1$ is the best condition for interferometry. Using Eqs. (1) and (2) and the identity ${{ \cos }^2}\phi + {{ \sin }^2}\phi = 1$, the following equation for $|A(x,y){|^2}$ is derived:
$$p|A(x,y){|^4} + q|A(x,y){|^2} + r = 0,$$
$$p = 2{(1 + M)^2}(1 - \cos \alpha) - 4M{\sin ^2}\alpha ,$$
$$q = - 2{(1 + M)^2}(1 - \cos \alpha)[{{I_1}(x,y) + {I_2}(x,y)} ],$$
$$\begin{split}r = {(1 + M)^2}\big[{{I_1^2}(x,y) + {I_2^2}(x,y) - 2{I_1}(x,y){I_2}(x,y)\cos \alpha} \big].\\\end{split}$$

Using Eqs. (3)–(6), the intensity distribution of the wave $|A(x,y){|^2}$ is derived as

$$|A(x,y){|^2} = \frac{{- q \pm \sqrt {{q^2} - 4pr}}}{{2p}}.$$

When ${M} = 1$, Eq. (7) is simplified as

$$\begin{split}&|A(x,y){|^2} =\\& \frac{{{I_1}(x,y) + {I_2}(x,y) \pm \sqrt {2{I_1}(x,y){I_2}(x,y)({1 + \cos \alpha} )}}}{{1 - \cos \alpha}}.\end{split}$$

Although

 figure: Fig. 1.

Fig. 1. Setup of self-interference incoherent DHM with FINCH.

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two values are obtained from Eq. (7), one of the signs in Eq. (7) gives an incorrect value. We found that the correct sign depends on the phase $\phi$. Figure 2 shows the graph when $|A(x,y){|^2} = 1$, $M = 1$, and $\alpha = \pi /{2}$. The correct sign is minus when ${0}\leqq \phi \lt \pi$ and ${3}\pi /{2}\leqq \phi \lt {2}\pi$ and plus when $\pi \leqq \phi \lt {3}\pi /{2}$. One should determine the sign, and we propose the following criteria. First, we obtain the following two pairs of cosine and sine values from Eqs. (1) and (2) by substituting the two values of Eq. (7):
$${[\cos \phi (x,y)]_ +} = \frac{{(1 + M)[{{I_1}(x,y) - |A(x,y){|^2}_ +} ]}}{{2\sqrt M |A(x,y){|^2}_ +}},$$
$$\begin{split}&{[\sin \phi (x,y)]_ +} = \\&\frac{{(1 + M)\big[{{I_2}(x,y) - {I_1}(x,y)\cos \alpha - |A(x,y){|^2}_ + (1 - \cos \alpha)} \big]}}{{2\sqrt M \sin \alpha |A(x,y){|^2}_ +}},\end{split}$$
$${[\cos \phi (x,y)]_ -} = \frac{{(1 + M)\left[{{I_1}(x,y) - |A(x,y){|^2}_ -} \right]}}{{2\sqrt M |A(x,y){|^2}_ -}},$$
$$\begin{split}&{[\sin \phi (x,y)]_ -} = \\&\frac{{(1 + M)\big[{{I_2}(x,y) - {I_1}(x,y)\cos \alpha - |A(x,y){|^2}_ - (1 - \cos \alpha)} \big]}}{{2\sqrt M \sin \alpha |A(x,y){|^2}_ -}},\end{split}$$
where subscripts ${+}$ and ${-}$ mean the values derived when the signs of $\pm$ in Eq. (7) are ${+}$ and ${-}$, respectively. Using the identity ${{ \cos }^2}\phi + {{ \sin }^2}\phi = 1$, it is clarified that in each pair, one is correct, and the other is incorrect. In a practical case, the sign for which ${{ \cos }^2}\phi + {{ \sin }^2}\phi$ is closer to one should be selected. As an example, the following condition is adopted:
$$|A(x,y){|^2} = \left\{\begin{array}{l}\frac{{- q - \sqrt {{q^2} - 4pr}}}{{2p}},\quad {\rm when}\\\left\{{{{[\cos \phi (x,y)]}_ +}^2 + {{[\sin \phi (x,y)]}_ +}^2} \right\} \ge {\left\{{{{[\cos \phi (x,y)]}_ -}^2 + {{[\sin \phi (x,y)]}_ -}^2} \right\}^{- 1}},\\\frac{{- q + \sqrt {{q^2} - 4pr}}}{{2p}},\quad {\rm when}\\\left\{{{{[\cos \phi (x,y)]}_ +}^2 + {{[\sin \phi (x,y)]}_ +}^2} \right\} \lt {\left\{{{{[\cos \phi (x,y)]}_ -}^2 + {{[\sin \phi (x,y)]}_ -}^2} \right\}^{- 1}}.\end{array} \right.$$

Finally, we obtain the 3D information of the specimens $|A(x,y)|{\exp }[j\phi (x,y)]$ from Eqs. (1), (2), and (13),

$$|A(x,y){|^2}\cos \phi (x,y) = \frac{{(1 + M)\left[{{I_1}(x,y) - |A(x,y){|^2}} \right]}}{{2\sqrt M}},$$
$$\begin{split}&|A(x,y){|^2}\sin \phi (x,y) = \\&\frac{{(1 + M)\big[{{I_2}(x,y) - {I_1}(x,y)\cos \alpha - |A(x,y){|^2}(1 - \cos \alpha)} \big]}}{{2\sqrt M \sin \alpha}},\end{split}$$
where $ j $ is the imaginary unit. Then, a 3D image of the specimens is reconstructed by diffraction integrals. Note that $\phi (x,y)$ in FINCH contains the 3D position of the specimens but no quantitative phase information of the specimens. In contrast, $\phi (x,y)$ in an interferometer for a phase object [11] contains the quantitative phase information.
 figure: Fig. 2.

Fig. 2. Relationship between the two ${({{ \cos }^2}\phi + {{ \sin }^2}\phi)^{1/2}}$ values and the phase $\phi$ in the case where $|A(x,y){|^2} = 1$, $M = 1$, and $\alpha = \pi /{2}$.

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Experiments were conducted to show the effectiveness of the proposed PSI by constructing the system shown in Fig. 1. An optical microscope (IX-73, Olympus) was used. A halogen lamp was used as a light source, and a dichroic mirror was set to obtain a 20 nm wavelength band and central wavelength of 530 nm. A USAF1951 test target was set as a specimen. A complementary metal–oxide–semiconductor (CMOS) image sensor (NEO 5.5, Andor Technology) recorded two phase-shifted incoherent holograms sequentially by using an LCOS-SLM (X10468-01, HAMAMATSU Photonics). The numerical aperture (NA) of the microscope objective was 0.95, and the magnification of the system was 30. We set two crystal lenses (SIGMAKOKI) as BM1 and BM2. The condition $M = 1$ was set. A phase shift $\alpha = \pi /{4}$ at the wavelength $\lambda$ of 530 nm was set. Figure 3 shows the results. A defocused specimen was recorded as incoherent digital holograms. Its numerically focused images were obtained using a single hologram with diffraction integral only (DIO), four-step PSI, and the proposed PSI for comparison. Then, we calculated structural similarity (SSIM) and peak signal-to-noise ratio (PSNR) for the reconstructed images by setting the image obtained by four-step PSI as the reference image. Figure 3 and Table 1 indicate that the quality of the image reconstructed by the proposed PSI is the same as that obtained by four-step PSI and is superior to that obtained from a hologram with DIO.

 figure: Fig. 3.

Fig. 3. Experimental results. (a) One of the recorded incoherent holograms. Images reconstructed by (b) single in-line hologram without PSI, (c) four-step PSI, and (d) the proposed two-step PSI.

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Tables Icon

Table 1. Quantitative Evaluations of the Reconstructed Images

We modified the system shown in Fig. 1 to demonstrate its applicability to the 3D sensing of nanoparticles. A CMOS image sensor (Zyla 4.2, Andor Technology) was used. An oil-immersion microscope objective ($\times 60$, ${\rm NA} = 1.42$) was set, and the relay optics were modified to increase the magnification up to 120. Fluorescent particles with a diameter of 0.2 µm and a peak fluorescence wavelength of 415 nm (F8805, Thermo Fisher Scientific) were distributed in a 3D space. A mercury lamp (U-HGLGPS, Olympus) and a fluorescence mirror unit (U-FUW, Olympus) were set in the DHM to illuminate the particles with ultraviolet excitation light and to introduce fluorescence light without excitation light to the image sensor. $\alpha$ was $\pi /{2}$. Two phase-shifted fluorescence holograms were recorded, and images at multiple depths were reconstructed. For comparison, images at these depths were also obtained from a single hologram with DIO. Figure 4 shows the results. Figures 4(a) and 4(b) indicate that focused images were not obtained from a single hologram owing to the superpositions of undesired-order diffraction images. In contrast, Figs. 4(c) and 4(d) show that sharply focused images were reconstructed by the proposed PSI because undesired images were removed. Then, images at multiple depths were reconstructed successfully (see Visualization 1). The results indicate that it is mandatory to remove the conjugate image to retrieve the 3D information of the specimen. Figures 4(e)–4(g) show 3D submicron resolution. Thus, its applicability to 3D nanoscopy was successfully and experimentally demonstrated.

 figure: Fig. 4.

Fig. 4. Experimental results for fluorescence nanoparticles in a 3D space. Images reconstructed (a), (b) from a single hologram with DIO and (c), (d) by the proposed PSI. Images obtained after numerical refocusing on (a), (c) ${-}{1.9}\;{\rm mm}$ and (b), (d) 1.9 mm depths from the image sensor plane. Arrows indicate focused particles. Plots along (e) $x$-, $y$-, and (f) $z$-axes and (g) $x {\text -} z$ image for the reconstructed particle marked by the violet arrow (Visualization 1).

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Finally, we investigated whether the proposed PSI can be merged into CCS. CCS is a filterless multidimensional imaging technique based on phase encoding and is implemented with $ 2N $ wavelength-multiplexed holograms to obtain $ N $-wavelength object waves, making the best use of the ${2}\pi$ phase ambiguity and exploiting two-step PSI [6]. However, in Ref. [6], it was necessary to record the intensity distribution of one of the two waves at each wavelength. Here, we merge CCS and the proposed PSI to reduce the number of exposures. An $ N $-wavelength-multiplexed phase-shifted self-interference hologram is expressed as

$$\begin{split}&{I_{{\rm WM}}}(x,y;{\alpha _1}, \ldots ,{\alpha _N}) \\&\quad= \sum\limits_{i = 1}^N {\left[{|{A_i}(x,y){|^2}\left({1 + \frac{{2\sqrt {M_i}}}{{1 + {M_i}}}\cos [{\phi _i}(x,y) - {\alpha _i}]} \right)} \right]} ,\end{split}$$
where $ \alpha_{1} $ and $ \alpha_N $ are phase shifts at the wavelengths $ \lambda_{1} $ and $ \lambda_N $, respectively. We assume the case of $N = 2$. When setting $ \alpha_{1} $ as an integral multiple of ${2}\pi$ for three wavelength-multiplexed holograms, ${I_{{\rm WM}}}(x,y;{0},{0})$, ${I_{{\rm WM}}}({x},y;{2}\pi ,{\alpha _{21}})$, and ${I_{{\rm WM}}}({x},y; - 2\pi ,- {\alpha _{21}})$, only the object wave at $ \lambda_{2} $, $|{A_2}(x,y)|{\exp }[j{\phi _2}(x,y)]$, is extracted by the following formula:
$$\begin{split}&|{A_2}(x,y){|^2}{e^{j{\phi _2}(x,y)}} = \\&\frac{{2{I_{{\rm WM}}}(x,y;0,0) - {I_{{\rm WM}}}(x,y;2\pi ,{\alpha _{21}}) - {I_{{\rm WM}}}(x,y; - 2\pi , - {\alpha _{21}})}}{{4(1 - \cos {\alpha _{21}})}}\\ &\quad + j\frac{{{I_{{\rm WM}}}(x,y;2\pi ,{\alpha _{21}}) - {I_{{\rm WM}}}(x,y; - 2\pi , - {\alpha _{21}})}}{{4\sin {\alpha _{21}}}}.\end{split}$$

After that, the $ \lambda_{2} $ component is removed from two wavelength-multiplexed holograms, ${I_{{\rm WM}}}(x,y;{0},{0})$ and ${I_{{\rm WM}}}({x},y;\;{\alpha _{11}},{\alpha _{22}})$, using the derived $|{A_2}(x,y)|$ and ${\phi _2}(x,y)$ and the known $ M_{2} $, $ \alpha_{21} $, and $ \alpha_{22} $. As a result, ${I_1}({x},y)$ and ${I_2}({x},y;\alpha = {\alpha _{11}})$ are numerically obtained, and then the two holograms at $ \lambda_{1} $ are introduced into the proposed PSI to obtain the object wave at $ \lambda_{1} $, $|{A_1}(x,y)|{\exp }[j{\phi _1}(x,y)]$. As described here, only ${2}N$ $ N $-wavelength-multiplexed holograms are required to obtain object waves at $ N $ wavelengths. The applicability to CCS was experimentally investigated with the previously constructed CCS-FINCH system [19]. Figure 1 is the modified version of the system in Ref. [19]. The specification of the system was the same as that in Ref. [19]. Metal complex molecules of europium (${\lambda _1} = 545\;{\rm nm}$) and terbium (${\lambda _2} = 618\;{\rm nm}$) were set as the red and green luminescent specimens. Those full widths at half-maximum were within 10 nm. The specimens were set between a cover glass and a slide glass and were sparsely distributed in a 3D space. The phase shifts at ($ \lambda_{1} $, $ \lambda_{2} $) were (0,0), $ (2\pi ,210\pi /127) $, $ (-2\pi ,-210\pi /127) $, and $ (-\pi /2,-105\pi /254) $. Figure 5 shows the results. Using the combination of the proposed PSI and CCS, we simultaneously retrieved 3D and color information from four wavelength-multiplexed fluorescence holograms.

 figure: Fig. 5.

Fig. 5. Experimental results obtained by the proposed technique and CCS for fluorescent color particles. (a) Photograph of metal complex molecules. (b) One of the recorded wavelength-multiplexed fluorescence holograms. Images numerically focused on (c) 0, (d) 120, and (e) 170 mm depths from the image sensor plane. Arrows indicate the area and the focused particles. Brightness of (b)–(e) is enhanced.

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We have proposed a two-step PSI technique for self-interference DH. Its effectiveness was experimentally demonstrated. It is considered that when the proposed PSI is applied to single-shot CCS [20], the decrease in the spatial sampling density of each hologram is relaxed. The SBWP is 8/3 times that of a single-shot color incoherent DH system with a color filter array [21]. In the proposed PSI, we assume that the intensity distributions of the two waves are the same, although this condition is not always rigorously correct for 3D sensing. However, our experimental results indicate that the proposed PSI works correctly. A detailed analysis will be carried out as future work. The proposed PSI can improve the temporal/spatial information capacity for self-interference DH and will enable 3D sensing with a smaller number of photons by reducing the number of phase shifts.

Funding

Japan Society for the Promotion of Science (18H01456); Precursory Research for Embryonic Science and Technology (JPMJPR15P8, JPMJPR16P8, JPMJPR17P2); Cooperative Research Program of “Network Joint Research Center for Materials and Devices” (20201164).

Disclosures

The authors declare no conflicts of interest.

REFERENCES

1. D. Gabor, Nature 161, 777 (1948). [CrossRef]  

2. J. W. Goodman and R. W. Lawrence, Appl. Phys. Lett. 11, 77 (1967). [CrossRef]  

3. A. W. Lohmann, Appl. Opt. 4, 1667 (1965). [CrossRef]  

4. R. Dandliker, R. Thalmann, and D. Prongue, Opt. Lett. 13, 339 (1988). [CrossRef]  

5. T. Tahara, S. Kikunaga, and Y. Arai, “Digital holography apparatus and digital holography method,” Japanese patent JP6308594 (18 September 2013).

6. T. Tahara, R. Mori, Y. Arai, and Y. Takaki, J. Opt. 17, 125707 (2015). [CrossRef]  

7. T. Tahara, T. Ito, Y. Ichihashi, and R. Oi, Opt. Lett. 45, 2482 (2020). [CrossRef]  

8. J.-P. Liu, T. Tahara, Y. Hayasaki, and T.-C. Poon, Appl. Sci. 8, 143 (2018). [CrossRef]  

9. J. Rosen and G. Brooker, Opt. Lett. 32, 912 (2007). [CrossRef]  

10. M. Imbe, Appl. Opt. 58, A82 (2019). [CrossRef]  

11. W. You, W. Lu, and X. Liu, Opt. Express 28, 34825 (2020). [CrossRef]  

12. T. Tahara, T. Kanno, Y. Arai, and T. Ozawa, J. Opt. 19, 065705 (2017). [CrossRef]  

13. T. Nobukawa, T. Muroi, Y. Katano, N. Kinoshita, and N. Ishii, Opt. Lett. 43, 1698 (2018). [CrossRef]  

14. J. Hong and M. K. Kim, Opt. Lett. 38, 5196 (2013). [CrossRef]  

15. A. Vijayakumar, T. Katkus, S. Lundgaard, D. P. Linklater, E. Ivanova, S. H. Ng, and S. Juodkazis, Opto-Electron. Adv. 3, 200004 (2020). [CrossRef]  

16. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, Appl. Opt. 13, 2693 (1974). [CrossRef]  

17. X. F. Meng, L. Z. Cai, X. F. Xu, X. L. Yang, X. X. Shen, G. Y. Dong, and Y. R. Wang, Opt. Lett. 31, 1414 (2006). [CrossRef]  

18. J.-P. Liu and T.-C. Poon, Opt. Lett. 34, 250 (2009). [CrossRef]  

19. T. Tahara, T. Koujin, A. Matsuda, A. Ishii, T. Ito, Y. Ichihashi, and R. Oi, Appl. Opt. 60, A260 (2021). [CrossRef]  

20. T. Tahara, A. Ishii, T. Ito, Y. Ichihashi, and R. Oi, Appl. Phys. Lett. 117, 031102 (2020). [CrossRef]  

21. T. Tahara, T. Ito, Y. Ichihashi, and R. Oi, J. Opt. 22, 105702 (2020). [CrossRef]  

References

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  1. D. Gabor, Nature 161, 777 (1948).
    [Crossref]
  2. J. W. Goodman and R. W. Lawrence, Appl. Phys. Lett. 11, 77 (1967).
    [Crossref]
  3. A. W. Lohmann, Appl. Opt. 4, 1667 (1965).
    [Crossref]
  4. R. Dandliker, R. Thalmann, and D. Prongue, Opt. Lett. 13, 339 (1988).
    [Crossref]
  5. T. Tahara, S. Kikunaga, and Y. Arai, “Digital holography apparatus and digital holography method,” Japanese patentJP6308594 (18September2013).
  6. T. Tahara, R. Mori, Y. Arai, and Y. Takaki, J. Opt. 17, 125707 (2015).
    [Crossref]
  7. T. Tahara, T. Ito, Y. Ichihashi, and R. Oi, Opt. Lett. 45, 2482 (2020).
    [Crossref]
  8. J.-P. Liu, T. Tahara, Y. Hayasaki, and T.-C. Poon, Appl. Sci. 8, 143 (2018).
    [Crossref]
  9. J. Rosen and G. Brooker, Opt. Lett. 32, 912 (2007).
    [Crossref]
  10. M. Imbe, Appl. Opt. 58, A82 (2019).
    [Crossref]
  11. W. You, W. Lu, and X. Liu, Opt. Express 28, 34825 (2020).
    [Crossref]
  12. T. Tahara, T. Kanno, Y. Arai, and T. Ozawa, J. Opt. 19, 065705 (2017).
    [Crossref]
  13. T. Nobukawa, T. Muroi, Y. Katano, N. Kinoshita, and N. Ishii, Opt. Lett. 43, 1698 (2018).
    [Crossref]
  14. J. Hong and M. K. Kim, Opt. Lett. 38, 5196 (2013).
    [Crossref]
  15. A. Vijayakumar, T. Katkus, S. Lundgaard, D. P. Linklater, E. Ivanova, S. H. Ng, and S. Juodkazis, Opto-Electron. Adv. 3, 200004 (2020).
    [Crossref]
  16. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, Appl. Opt. 13, 2693 (1974).
    [Crossref]
  17. X. F. Meng, L. Z. Cai, X. F. Xu, X. L. Yang, X. X. Shen, G. Y. Dong, and Y. R. Wang, Opt. Lett. 31, 1414 (2006).
    [Crossref]
  18. J.-P. Liu and T.-C. Poon, Opt. Lett. 34, 250 (2009).
    [Crossref]
  19. T. Tahara, T. Koujin, A. Matsuda, A. Ishii, T. Ito, Y. Ichihashi, and R. Oi, Appl. Opt. 60, A260 (2021).
    [Crossref]
  20. T. Tahara, A. Ishii, T. Ito, Y. Ichihashi, and R. Oi, Appl. Phys. Lett. 117, 031102 (2020).
    [Crossref]
  21. T. Tahara, T. Ito, Y. Ichihashi, and R. Oi, J. Opt. 22, 105702 (2020).
    [Crossref]

2021 (1)

2020 (5)

T. Tahara, A. Ishii, T. Ito, Y. Ichihashi, and R. Oi, Appl. Phys. Lett. 117, 031102 (2020).
[Crossref]

T. Tahara, T. Ito, Y. Ichihashi, and R. Oi, J. Opt. 22, 105702 (2020).
[Crossref]

T. Tahara, T. Ito, Y. Ichihashi, and R. Oi, Opt. Lett. 45, 2482 (2020).
[Crossref]

W. You, W. Lu, and X. Liu, Opt. Express 28, 34825 (2020).
[Crossref]

A. Vijayakumar, T. Katkus, S. Lundgaard, D. P. Linklater, E. Ivanova, S. H. Ng, and S. Juodkazis, Opto-Electron. Adv. 3, 200004 (2020).
[Crossref]

2019 (1)

2018 (2)

J.-P. Liu, T. Tahara, Y. Hayasaki, and T.-C. Poon, Appl. Sci. 8, 143 (2018).
[Crossref]

T. Nobukawa, T. Muroi, Y. Katano, N. Kinoshita, and N. Ishii, Opt. Lett. 43, 1698 (2018).
[Crossref]

2017 (1)

T. Tahara, T. Kanno, Y. Arai, and T. Ozawa, J. Opt. 19, 065705 (2017).
[Crossref]

2015 (1)

T. Tahara, R. Mori, Y. Arai, and Y. Takaki, J. Opt. 17, 125707 (2015).
[Crossref]

2013 (1)

2009 (1)

2007 (1)

2006 (1)

1988 (1)

1974 (1)

1967 (1)

J. W. Goodman and R. W. Lawrence, Appl. Phys. Lett. 11, 77 (1967).
[Crossref]

1965 (1)

1948 (1)

D. Gabor, Nature 161, 777 (1948).
[Crossref]

Arai, Y.

T. Tahara, T. Kanno, Y. Arai, and T. Ozawa, J. Opt. 19, 065705 (2017).
[Crossref]

T. Tahara, R. Mori, Y. Arai, and Y. Takaki, J. Opt. 17, 125707 (2015).
[Crossref]

T. Tahara, S. Kikunaga, and Y. Arai, “Digital holography apparatus and digital holography method,” Japanese patentJP6308594 (18September2013).

Brangaccio, D. J.

Brooker, G.

Bruning, J. H.

Cai, L. Z.

Dandliker, R.

Dong, G. Y.

Gabor, D.

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Appl. Opt. (4)

Appl. Phys. Lett. (2)

T. Tahara, A. Ishii, T. Ito, Y. Ichihashi, and R. Oi, Appl. Phys. Lett. 117, 031102 (2020).
[Crossref]

J. W. Goodman and R. W. Lawrence, Appl. Phys. Lett. 11, 77 (1967).
[Crossref]

Appl. Sci. (1)

J.-P. Liu, T. Tahara, Y. Hayasaki, and T.-C. Poon, Appl. Sci. 8, 143 (2018).
[Crossref]

J. Opt. (3)

T. Tahara, R. Mori, Y. Arai, and Y. Takaki, J. Opt. 17, 125707 (2015).
[Crossref]

T. Tahara, T. Kanno, Y. Arai, and T. Ozawa, J. Opt. 19, 065705 (2017).
[Crossref]

T. Tahara, T. Ito, Y. Ichihashi, and R. Oi, J. Opt. 22, 105702 (2020).
[Crossref]

Nature (1)

D. Gabor, Nature 161, 777 (1948).
[Crossref]

Opt. Express (1)

Opt. Lett. (7)

Opto-Electron. Adv. (1)

A. Vijayakumar, T. Katkus, S. Lundgaard, D. P. Linklater, E. Ivanova, S. H. Ng, and S. Juodkazis, Opto-Electron. Adv. 3, 200004 (2020).
[Crossref]

Other (1)

T. Tahara, S. Kikunaga, and Y. Arai, “Digital holography apparatus and digital holography method,” Japanese patentJP6308594 (18September2013).

Supplementary Material (1)

NameDescription
» Visualization 1       3D sensing of multiple fluorescence nanoparticles with a diameter of 0.2 um was conducted simultaneously by using incoherent digital holography. The proposed two-step phase-shifting interferometry was used.

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Figures (5)

Fig. 1.
Fig. 1. Setup of self-interference incoherent DHM with FINCH.
Fig. 2.
Fig. 2. Relationship between the two ${({{ \cos }^2}\phi + {{ \sin }^2}\phi)^{1/2}}$ values and the phase $\phi$ in the case where $|A(x,y){|^2} = 1$ , $M = 1$ , and $\alpha = \pi /{2}$ .
Fig. 3.
Fig. 3. Experimental results. (a) One of the recorded incoherent holograms. Images reconstructed by (b) single in-line hologram without PSI, (c) four-step PSI, and (d) the proposed two-step PSI.
Fig. 4.
Fig. 4. Experimental results for fluorescence nanoparticles in a 3D space. Images reconstructed (a), (b) from a single hologram with DIO and (c), (d) by the proposed PSI. Images obtained after numerical refocusing on (a), (c)  ${-}{1.9}\;{\rm mm}$ and (b), (d) 1.9 mm depths from the image sensor plane. Arrows indicate focused particles. Plots along (e)  $x$ -, $y$ -, and (f)  $z$ -axes and (g)  $x {\text -} z$ image for the reconstructed particle marked by the violet arrow (Visualization 1).
Fig. 5.
Fig. 5. Experimental results obtained by the proposed technique and CCS for fluorescent color particles. (a) Photograph of metal complex molecules. (b) One of the recorded wavelength-multiplexed fluorescence holograms. Images numerically focused on (c) 0, (d) 120, and (e) 170 mm depths from the image sensor plane. Arrows indicate the area and the focused particles. Brightness of (b)–(e) is enhanced.

Tables (1)

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Table 1. Quantitative Evaluations of the Reconstructed Images

Equations (17)

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I 1 ( x , y ) = | A ( x , y ) | 2 ( 1 + 2 M 1 + M cos ϕ ( x , y ) ) ,
I 2 ( x , y ) = | A ( x , y ) | 2 ( 1 + 2 M 1 + M cos ( ϕ ( x , y ) α ) ) ,
p | A ( x , y ) | 4 + q | A ( x , y ) | 2 + r = 0 ,
p = 2 ( 1 + M ) 2 ( 1 cos α ) 4 M sin 2 α ,
q = 2 ( 1 + M ) 2 ( 1 cos α ) [ I 1 ( x , y ) + I 2 ( x , y ) ] ,
r = ( 1 + M ) 2 [ I 1 2 ( x , y ) + I 2 2 ( x , y ) 2 I 1 ( x , y ) I 2 ( x , y ) cos α ] .
| A ( x , y ) | 2 = q ± q 2 4 p r 2 p .
| A ( x , y ) | 2 = I 1 ( x , y ) + I 2 ( x , y ) ± 2 I 1 ( x , y ) I 2 ( x , y ) ( 1 + cos α ) 1 cos α .
[ cos ϕ ( x , y ) ] + = ( 1 + M ) [ I 1 ( x , y ) | A ( x , y ) | 2 + ] 2 M | A ( x , y ) | 2 + ,
[ sin ϕ ( x , y ) ] + = ( 1 + M ) [ I 2 ( x , y ) I 1 ( x , y ) cos α | A ( x , y ) | 2 + ( 1 cos α ) ] 2 M sin α | A ( x , y ) | 2 + ,
[ cos ϕ ( x , y ) ] = ( 1 + M ) [ I 1 ( x , y ) | A ( x , y ) | 2 ] 2 M | A ( x , y ) | 2 ,
[ sin ϕ ( x , y ) ] = ( 1 + M ) [ I 2 ( x , y ) I 1 ( x , y ) cos α | A ( x , y ) | 2 ( 1 cos α ) ] 2 M sin α | A ( x , y ) | 2 ,
| A ( x , y ) | 2 = { q q 2 4 p r 2 p , w h e n { [ cos ϕ ( x , y ) ] + 2 + [ sin ϕ ( x , y ) ] + 2 } { [ cos ϕ ( x , y ) ] 2 + [ sin ϕ ( x , y ) ] 2 } 1 , q + q 2 4 p r 2 p , w h e n { [ cos ϕ ( x , y ) ] + 2 + [ sin ϕ ( x , y ) ] + 2 } < { [ cos ϕ ( x , y ) ] 2 + [ sin ϕ ( x , y ) ] 2 } 1 .
| A ( x , y ) | 2 cos ϕ ( x , y ) = ( 1 + M ) [ I 1 ( x , y ) | A ( x , y ) | 2 ] 2 M ,
| A ( x , y ) | 2 sin ϕ ( x , y ) = ( 1 + M ) [ I 2 ( x , y ) I 1 ( x , y ) cos α | A ( x , y ) | 2 ( 1 cos α ) ] 2 M sin α ,
I W M ( x , y ; α 1 , , α N ) = i = 1 N [ | A i ( x , y ) | 2 ( 1 + 2 M i 1 + M i cos [ ϕ i ( x , y ) α i ] ) ] ,
| A 2 ( x , y ) | 2 e j ϕ 2 ( x , y ) = 2 I W M ( x , y ; 0 , 0 ) I W M ( x , y ; 2 π , α 21 ) I W M ( x , y ; 2 π , α 21 ) 4 ( 1 cos α 21 ) + j I W M ( x , y ; 2 π , α 21 ) I W M ( x , y ; 2 π , α 21 ) 4 sin α 21 .

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