This paper presents a phase retrieval method for digital holography with two high-speed cameras in particle measurement. The conditions for recording two holograms are derived theoretically. We focus in particular on the distance between the two holograms. The relative misalignment of the two holograms is also evaluated by numerical simulations. Finally, we experimentally compared settling particles reconstructed by the presented method and the Gabor method.
© 2016 Optical Society of America
In-line digital holography, known as “Gabor holography” , is widely used in particle measurement (e.g., dispersed micro particles , plankton , snow crystals , and fuel droplets ) because of its simple setup and insensitivity to vibration. However, the method has a twin image problem  that occurs when an obstacle between in-focus objects and out-of-focus objects at the in-focus plane in the reconstructed hologram. The problem is caused by a lack of recorded phase information in the hologram.
Several approaches have been developed for recording the information using the phase-shifting method [7, 8]. However, it is diﬃcult to observe objects moving at high speed because the method requires at least two holograms recorded at diﬀerent times. To overcome this diﬃculty, a phase-shifting array device was developed for recording multiple holograms simultaneously [9–11]. However, the phase-shifting method is limited by having to be implemented on a vibration-isolation table because of its sensitivity to vibration.
The phase retrieval method [12, 13] solves such problems by using in-line holography to record two or more holograms at diﬀerent positions. This iterative-propagation approach is based on the Gerchberg-Saxton algorithm . The intensity of the hologram corresponds to modulus constraint, and the propagation is carried out by using Fresnel-Kirchhoﬀ diﬀraction theory. However, this method has been used to observe only static objects because of the requirement of two or more holograms; it has not yet been applied to the measurement of moving particles at high speed.
In this paper, we use two high-speed camera to apply the phase retrieval method to particle measurement. The condition for recording two holograms is derived theoretically. We focus in particular on the distance between two holograms, and we use numerical simulations to evaluate their relative misalignment. Finally, the misalignment calibration is demonstrated experimentally. We also compare settling particles reconstructed by the present method and the Gabor method.
2. Phase retrieval method with two cameras in digital holography
2.1 Digital holography in particle measurement
We consider the case in which a particle is illuminated by a unit plane wave of wavelength λ. The complex amplitude at the recording plane a distance z1 away is given by15]Figure 1 shows the intensity distribution of a particle with a diameter of 70 µm.17]. The position r0 of the end of the first lobe corresponds to the first zero-crossing position of the Bessel function:Eq. (5) with r0 = L/2.
From Eqs. (4) and (5), the fringe spacing is Δr = dp/1.22. Therefore, the spatial resolution Δx of the CCD is expressed asEq. (1) can be expressed more compactly as the two-dimensional convolutionEquation (10) is based on the assumption that the amplitude of the nonlinear term is much smaller than that of the diﬀracted particle wave.
In reconstruction, the pattern is illuminated for reconstructing particles with , and the complex amplitude is reconstructed asEq. (11) leads to a uniform component, and the second and third terms lead to the real and virtual images, respectively, of the particles. The suppression of the in-focus and out-of-focus particles causes the out-of-focus image in the intensity of reconstructed particles. If we know the phase information of the hologram, the in-focus particles are reconstructed as
2.2 Phase retrieval method in digital holography
The information can be retrieved by using a phase retrieval (P.R.) method based on the Gerchberg–Saxton algorithm  which requires two holograms for use with holography [13,20]. The method consists of four steps between two holograms as shown in Fig. 2(a): (Step 1) reconstruct the complex amplitude of the first hologram I1 in the second hologram I2 with the distance Δz between two holograms; (Step 2) replace the absolute value of the resulting complex amplitude with the square root of the second hologram I2; (Step 3) back-reconstruct the new complex amplitude to the first hologram; (Step 4) replace the absolute value of the reconstructed complex amplitude with the square root of the first hologram I1. These steps are formulated for the k th iteration as follows:Eq. (16) becomes the maximum value. This mode corresponds to the background intensity because the area of the background is wider than the area of particles in the focused plane. The mode increases as the phase approaches the retrieved phase. Simulated and reconstructed holograms of Figs. 2(b) and 2(c) show that the eﬀect of the twin image has been removed by using the phase retrieval method.
In this retrieval process, the two recorded holograms must satisfy the range of Eq. (6) and the resolution of Eq. (7). However the distance Δz has not been given the attention it deserves in the recording procedure. From Eq. (9), the complex amplitude at z1 is composed of two phases, namely the particle of and the reference wave of kz1;Eq. (16) is also composed of the two retrieved phases asEqs. (17) and (18) as
We have begun to verify this relationship using the holograms of Fig. 2(b). The position of hologram I1 is fixed at z1 that of the hologram I2 is varied in the z direction. Figure 3(a) shows the signal-to-noise ratio (SNR) has peaks where m in Eq. (19) takes an integer value. When , the peak shape becomes unstable in this condition. The SNR is reduced for . This is why Δz is smaller than the far-field condition, 7.74 mm = . The number of iterations Nit can be approximated as a linear function of m as shown in Fig. 3(b). The phase of the reference wave is retrieved quickly in the case of small m. These results indicate that the phase retrieval method needs to retrieve not only the phase of the particles, but also the phase of the reference wave.
2.3 Evaluation of relative misalignment between two holograms
A relative misalignment between the two holograms is evaluated numerically as the SNR of the reconstructed amplitude in the xy plane and the z direction. The relative misalignment in the x direction is evaluated for a rotation and a shift in the xy plane. Figure 4 shows that the SNR of the phase retrieval method is lower than that of the Gabor method when the relative misalignment in the x direction is more than two pixels. Reconstructed particles are also blurred by misaligned particles. Phase information has to be retrieved from the pixel in the same xy position or the misalignment may be up to one-third of the particle diameter. Relative to the results in Fig. 4, the eﬀect of the relative misalignment in the z direction is lower than that of as shown in Fig. 5. If there is no misalignment in the x direction, as is also shown in Fig. 3(a), the SNR is greater than that of the Gabor method at all Δz. It is found that calibration of is sensitive, but that of is not sensitive.
3. Experimental results
Two holograms of particles were recorded for the phase retrieval method by using two high-speed cameras without lenses and a He-Ne laser (wave length: 632.8 nm), as illustrated schematically in Fig. 6. Particles with an average diameter of 70 µm were settling from a particle feeder at a height of H = 150 mm. An area of 5.12 × 5.12 mm2 was observed by the two high-speed cameras (FASTCAM Mini UX100, Photron, frame rate: 4,000 fps, image size: 1024 × 1024 pixel2, cell size: 10 µm × 10 µm) located at two positions z1 = 150 mm and z2 = z1 + Δz = 200 mm for the case of m = 3. Because the phase retrieval method is insensitive to vibration, none of the experiments were conducted on a vibration-isolation table.
As mentioned in Sec. 2.3, a relative misalignment between the two holograms must be corrected in the phase retrieval method with two cameras. First of all, the misalignment of a rotation and a shift in the xy plane was calibrated by the image distortion correction . The vector field of Fig. 7(c) for the correction was calculated using particle image velocimetry  with the two reconstructed holograms of Figs. 7(a) and 7(b).
In order to calibrate the misalignment for the target distance Δz = 50 mm in the z direction, the installation distance between the two cameras was detected by changing the numerical condition of Δz at each iterative loop of the phase retrieval method, as shown in Fig. 8. The distance was obtained as Δz = 49.34 mm as the maximum value. The misalignment between the target and the installation distance was calibrated using an xyz stage.
Figure 9 shows a comparison of reconstructed experimental holograms of the settling particles using the phase retrieval method with two cameras and the Gabor method. The characteristic parts of the reconstructed holograms are magnified in the central part of each panel.
In the example of a low concentration of particles in Fig. 9(a), in the phase retrieval method compared to the Gabor method, the particle shape is more clearly visible on the reconstructed plane, and a clearer fringe pattern is also observed because of twin-image elimination. However, in the example of a high concentration of particles in Fig. 9(b), clusters of particles are also more clearly observed using the presented method.
In this paper, we derived the conditions for recording a particle using the phase retrieval method with two holograms. The signal-to-noise ratio of the reconstructed hologram and the number of iterations are especially dependent on the distance between the two holograms. Moreover, the signal-to-noise ratio is strongly aﬀected by the relative misalignment between the two holograms in the xy plane as compared to the z direction. The calibration procedure for this misalignment was also experimentally demonstrated. We conducted the experiments to observe the settling particles without the need for a vibration-isolation table. Fringe patterns and reconstructed particles were clearly observed by two high-speed cameras at 4,000 fps under both high and low particle concentrations. Therefore, we conclude that the phase retrieval method with two high-speed cameras could be used more widely in particle measurement because of its twin-image elimination, insensitivity to vibration, and simplicity of optical setup.
Ministry of Education, Culture, Sports, Science and Technology of Japan (16K06080).
References and links
2. S. Murata and N. Yasuda, “Potential of digital holography in particle measurement,” Opt. Lasers Eng. 32(7–8), 567–574 (2000). [CrossRef]
3. J. Sheng, E. Malkiel, J. Katz, J. Adolf, R. Belas, and A. R. Place, “Digital holographic microscopy reveals prey-induced changes in swimming behavior of predatory dinoflagellates,” Proc. Natl. Acad. Sci. U.S.A. 104(44), 17512–17517 (2007). [CrossRef] [PubMed]
4. P. Amsler, O. Stetzer, M. Schnaiter, E. Hesse, S. Benz, O. Moehler, and U. Lohmann, “Ice crystal habits from cloud chamber studies obtained by in-line holographic microscopy related to depolarization measurements,” Appl. Opt. 48(30), 5811–5822 (2009). [CrossRef] [PubMed]
5. D. Nguyen, D. Honnery, and J. Soria, “Measuring evaporation of micro-fuel droplets using magnified DIH and DPIV,” Exp. Fluids 50(4), 949–959 (2011). [CrossRef]
6. J. Katz and J. Sheng, “Applications of holography in fluid mechanics and particle dynamics,” Annu. Rev. Fluid Mech. 42(1), 531–555 (2010). [CrossRef]
8. S. Murata, D. Harada, and Y. Tanaka, “Spatial phase-shifting digital holography for three-dimensional particle tracking velocimetry,” Jpn. J. Appl. Phys. 48(9S2), 09LB01 (2009). [CrossRef]
9. Y. Awatsuji, M. Sasada, and T. Kubota, “Parallel quasi-phase-shifting digital holography,” Appl. Phys. Lett. 85(6), 1069–1071 (2004). [CrossRef]
11. Y. Awatsuji, T. Tahara, A. Kaneko, T. Koyama, K. Nishio, S. Ura, T. Kubota, and O. Matoba, “Parallel two-step phase-shifting digital holography,” Appl. Opt. 47(19), D183–D189 (2008). [CrossRef] [PubMed]
12. G. Liu and P. D. Scott, “Phase retrieval and twin-image elimination for in-line Fresnel holograms,” J. Opt. Soc. Am. A 4(1), 159–165 (1987). [CrossRef]
13. Y. Zhang, G. Pedrini, W. Osten, and H. Tiziani, “Whole optical wave field reconstruction from double or multi in-line holograms by phase retrieval algorithm,” Opt. Express 11(24), 3234–3241 (2003). [CrossRef] [PubMed]
14. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diﬀraction plane pictures,” Optik (Stuttg.) 35, 237–246 (1972).
15. G. A. Tyler and B. J. Thompson, “Fraunhofer holography applied to particle size analysis a reassessment,” J. Mod. Opt. 23(9), 685–700 (1976).
16. C. S. Vikram, “Particle Field Holography,” in Cambridge Studies in Modern optics (Cambridge University Press, 1992).
19. L. Onural and P. D. Scott, “Digital decoding of in-line holograms,” Opt. Eng. 26(11), 261124 (1987). [CrossRef]
20. L. Denis, C. Fournier, T. Fournel, and C. Ducottet, “Twin-image noise reduction by phase retrieval in in-line digital holography,” Proc. SPIE 5914, 148–161 (2005). [CrossRef]
21. J. C. Russ, The Image Processing Handbook (CRC Press, 2016).
22. C. E. Willert and M. Gharib, “Digital particle image velocimetry,” Exp. Fluids 10(4), 181–193 (1991). [CrossRef]