Abstract

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

1. Introduction

In-line digital holography, known as “Gabor holography” [1], is widely used in particle measurement (e.g., dispersed micro particles [2], plankton [3], snow crystals [4], and fuel droplets [5]) because of its simple setup and insensitivity to vibration. However, the method has a twin image problem [6] 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 [14]. 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 ψz1(x1,y1) at the recording plane a distance z1 away is given by

ψz1(x1,y1)=exp(jkz1)jλz1[1p(x0,y0)]exp{jk2z1[(x1x0)2+(y1y0)2]}dx0dy0,
where k = 2π/λ is the wave number of the plane wave. The diffracting aperture p(x0, y0) for a particle, is defined as the amplitude transmittance function,
p(x0,y0)={1,  x02+y02(dp/2)20,  x02+y02>(dp/2)2
where dp is the diameter of the particle. This aperture has circular symmetry, and under the far-field condition, dp2z1. Performing the integration yields the intensity distribution [15]
I(r)=|ψz1(r)|2=1πdp22λz1sin(πr2λz1)2J1(πdpr/(λz1))πdpr/(λz1)+(πdp24λz1)2[2J1(πdpr/(λz1))πdpr/(λz1)]2,
where J1 denotes the first-order Bessel function. Figure 1 shows the intensity distribution of a particle with a diameter of 70 µm.

 

Fig. 1 Hologram intensity log10I(r) of a particle against r derived from Eq. (3) for z1 = 100mm, dp = 70 µm and λ = 632.8 nm.

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The sine term in the right-hand side of Eq. (3) determines the fringe pattern of the signal. We obtain the difference between the nth and (n + 1)th fringe being 2π [16],

rn+12rn22rΔr=2λz.
Here, we derive rn2=2λzn (n = 1, 2,..., n N). Because of the low spatial frequency of charge-coupled devices (CCDs), the available particle information is contained in the first lobe [17]. The position r0 of the end of the first lobe corresponds to the first zero-crossing position of the Bessel function:
r=1.22λzdp.
Thus, the recording distance of a particle is limited to the range
dp2λ<z1<dpL/21.22λ,
where L/2 is the half width of the CCD elements. In this equation, the left-hand side is based on the Fraunhofer condition and the right-hand side is derived from 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 as

Δx<dp2.44=Δr2.
We find that the resolution depends on only the particle diameter, not on depth or wavelength.

The twin-image problem [18] occurs in the recording process of a hologram without phase information. For discussing this problem, we introduce the convolution kernel [19] defined as

hz1(x1,y1)1λz1exp[jkz1(x12+y12)].
With this kernel, Eq. (1) can be expressed more compactly as the two-dimensional convolution
ψz1=exp(jkz1)[1p(x0,y0)]*hz1.
The hologram intensity, Iz1, is recorded on a hologram as
Iz1=ψz1*ψ*z1=|1p*hz1|2=1p**hz1*p*hz1+|p*hz1|21p**hz1*p*hz1
Equation (10) is based on the assumption that the amplitude of the nonlinear term |p*hz1|2 is much smaller than that of the diffracted particle wave.

In reconstruction, the pattern is illuminated for reconstructing particles with hz1, and the complex amplitude ψz0 is reconstructed as

ψz0=Iz1*hz1=1p**h2z1p*hz1.
The first term on the right-hand side of Eq. (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 Iz0=ψz0*ψz0* of reconstructed particles. If we know the phase information of the hologram, the in-focus particles are reconstructed as
ψz0=ψz1*hz1=1p.
Therefore the phase information resolves the twin image problem in the recording process of in-line holography.

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 [14] 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 ψz1k 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 ψz2k' to the first hologram; (Step 4) replace the absolute value of the reconstructed complex amplitude ψz1k with the square root of the first hologram I1. These steps are formulated for the k th iteration as follows:

ψ'z2k=exp(jΔz)ψz1k*hΔz=|ψ'z2k|exp(jϕ2')
ψz2k=I2exp(jϕ2')
ψ'z1k=exp(jΔz)ψz2k*hΔz=|ψ'z1k|exp(jϕ1')
ψz1k=I1exp(jϕ1')
where ϕ1' and ϕ2' are the retrieved phases in the first and second hologram, respectively. The program exits from its iterative loop when the mode of the image intensity reconstructed at z0 by the retrieved complex amplitude of 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.

 

Fig. 2 (a) Phase retrieval method for two holograms; comparison of a simulated and reconstructed hologram between (b) phase retrieval method and (c) Gabor method. The holograms in (b) and (c) were recorded at z1 = 50 mm and z2 = 100 mm, respectively. The wavelength is 632.8 nm. We randomly dispersed 500 particles with a diameter of 70 µm in the same plane with a 512 pixel × 512 pixel area (pixel size: 10 µm × 10 µm).

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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 ϕ1 and the reference wave of kz1;

ψz1=I1exp(jϕ1)exp(jkz1).
The retrieved complex amplitude of Eq. (16) is also composed of the two retrieved phases as
ψz1=I1exp(jϕ1'')exp(jkmΔz),
where ϕ1'' and km are the retrieved phase of a particle and reference wave obtained by the distance Δz multiplied the integer m. If the retrieved phase has the relationship ϕ1ϕ1'', the distance Δz is derived from Eqs. (17) and (18) as

Δz=z1m           (m=1,2,..,   mN).

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 m6, the peak shape becomes unstable in this condition. The SNR is reduced for m10. This is why Δz is smaller than the far-field condition, 7.74 mm = dp2/λΔz. 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.

 

Fig. 3 Influence of m in Eq. (19) on (a) the signal-to-noise ratio (SNR) and (b) the number of iterations Nit. The SNR is defined as SNR=10log10(σp2/σb2), where σp and σb are the standard deviations of the particles and background, respectively, of the intensity at z0. The distance Δz is varied from 0.001 mm to 150 mm in increments of 0.001 mm.

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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 ϵx 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 ϵΔz in the z direction is lower than that of ϵx 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 ϵx is sensitive, but that of ϵΔz is not sensitive.

 

Fig. 4 SNR versus relative misalignment ϵx in the x direction and reconstructed planes. Holograms of particles shown in Fig. 2 are recorded at z1 = 50 mm and Δz = 50 mm. Hologram I2 is shifted in the x direction with respect to hologram I1.

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Fig. 5 SNR versus relative misalignment ϵΔz in the z direction. Holograms of Fig. 2 are recorded at z1 = 50 mm and Δz = 50 mm. Hologram, I2 is shifted in the z direction with respect to hologram I1. This condition corresponds to m = 1 in Fig. 3(a).

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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.

 

Fig. 6 Optical arrangement for recording two in-line holograms with two high-speed cameras for phase retrieval method.

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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 [21]. The vector field of Fig. 7(c) for the correction was calculated using particle image velocimetry [22] with the two reconstructed holograms of Figs. 7(a) and 7(b).

 

Fig. 7 Optical image distortion correction: (a, b) reconstructed holograms of particles; (c) vector field for image distortion correction.

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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.

 

Fig. 8 Mode of image intensity reconstructed at z0 by the retrieved complex amplitude.

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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.

 

Fig. 9 Experimental holograms of settling particles reconstructed by the phase retrieval method with two high-speed cameras (left) and the Gabor method (right). Each hologram contains particles with an average diameter of 70 µm that were settling from the upper edge of the hologram at approximately terminal velocity.

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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.

4. Conclusion

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.

Funding

Ministry of Education, Culture, Sports, Science and Technology of Japan (16K06080).

References and links

1. D. Gabor, “A new microscopic principle,” Nature 161(4098), 777–778 (1948). [CrossRef]   [PubMed]  

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]  

7. I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. 22(16), 1268–1270 (1997). [CrossRef]   [PubMed]  

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]  

10. Y. Awatsuji, A. Fujii, T. Kubota, and O. Matoba, “Parallel three-step phase-shifting digital holography,” Appl. Opt. 45(13), 2995–3002 (2006). [CrossRef]   [PubMed]  

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).

17. Y. Wu, X. Wu, Z. Wang, L. Chen, and K. Cen, “Coal powder measurement by digital holography with expanded measurement area,” Appl. Opt. 50(34), H22–H29 (2011). [CrossRef]   [PubMed]  

18. T. Latychevskaia and H. W. Fink, “Solution to the twin image problem in holography,” Phys. Rev. Lett. 98(23), 233901 (2007). [CrossRef]   [PubMed]  

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]  

References

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  1. D. Gabor, “A new microscopic principle,” Nature 161(4098), 777–778 (1948).
    [Crossref] [PubMed]
  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]
  7. I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. 22(16), 1268–1270 (1997).
    [Crossref] [PubMed]
  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]
  10. Y. Awatsuji, A. Fujii, T. Kubota, and O. Matoba, “Parallel three-step phase-shifting digital holography,” Appl. Opt. 45(13), 2995–3002 (2006).
    [Crossref] [PubMed]
  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).
  17. Y. Wu, X. Wu, Z. Wang, L. Chen, and K. Cen, “Coal powder measurement by digital holography with expanded measurement area,” Appl. Opt. 50(34), H22–H29 (2011).
    [Crossref] [PubMed]
  18. T. Latychevskaia and H. W. Fink, “Solution to the twin image problem in holography,” Phys. Rev. Lett. 98(23), 233901 (2007).
    [Crossref] [PubMed]
  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]

2011 (2)

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]

Y. Wu, X. Wu, Z. Wang, L. Chen, and K. Cen, “Coal powder measurement by digital holography with expanded measurement area,” Appl. Opt. 50(34), H22–H29 (2011).
[Crossref] [PubMed]

2010 (1)

J. Katz and J. Sheng, “Applications of holography in fluid mechanics and particle dynamics,” Annu. Rev. Fluid Mech. 42(1), 531–555 (2010).
[Crossref]

2009 (2)

2008 (1)

2007 (2)

T. Latychevskaia and H. W. Fink, “Solution to the twin image problem in holography,” Phys. Rev. Lett. 98(23), 233901 (2007).
[Crossref] [PubMed]

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]

2006 (1)

2005 (1)

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]

2004 (1)

Y. Awatsuji, M. Sasada, and T. Kubota, “Parallel quasi-phase-shifting digital holography,” Appl. Phys. Lett. 85(6), 1069–1071 (2004).
[Crossref]

2003 (1)

2000 (1)

S. Murata and N. Yasuda, “Potential of digital holography in particle measurement,” Opt. Lasers Eng. 32(7–8), 567–574 (2000).
[Crossref]

1997 (1)

1991 (1)

C. E. Willert and M. Gharib, “Digital particle image velocimetry,” Exp. Fluids 10(4), 181–193 (1991).
[Crossref]

1987 (2)

1976 (1)

G. A. Tyler and B. J. Thompson, “Fraunhofer holography applied to particle size analysis a reassessment,” J. Mod. Opt. 23(9), 685–700 (1976).

1972 (1)

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).

1948 (1)

D. Gabor, “A new microscopic principle,” Nature 161(4098), 777–778 (1948).
[Crossref] [PubMed]

Adolf, J.

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]

Amsler, P.

Awatsuji, Y.

Belas, R.

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]

Benz, S.

Cen, K.

Chen, L.

Denis, L.

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]

Ducottet, C.

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]

Fink, H. W.

T. Latychevskaia and H. W. Fink, “Solution to the twin image problem in holography,” Phys. Rev. Lett. 98(23), 233901 (2007).
[Crossref] [PubMed]

Fournel, T.

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]

Fournier, C.

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]

Fujii, A.

Gabor, D.

D. Gabor, “A new microscopic principle,” Nature 161(4098), 777–778 (1948).
[Crossref] [PubMed]

Gerchberg, R. W.

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).

Gharib, M.

C. E. Willert and M. Gharib, “Digital particle image velocimetry,” Exp. Fluids 10(4), 181–193 (1991).
[Crossref]

Harada, D.

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]

Hesse, E.

Honnery, D.

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]

Kaneko, A.

Katz, J.

J. Katz and J. Sheng, “Applications of holography in fluid mechanics and particle dynamics,” Annu. Rev. Fluid Mech. 42(1), 531–555 (2010).
[Crossref]

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]

Koyama, T.

Kubota, T.

Latychevskaia, T.

T. Latychevskaia and H. W. Fink, “Solution to the twin image problem in holography,” Phys. Rev. Lett. 98(23), 233901 (2007).
[Crossref] [PubMed]

Liu, G.

Lohmann, U.

Malkiel, E.

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]

Matoba, O.

Moehler, O.

Murata, S.

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]

S. Murata and N. Yasuda, “Potential of digital holography in particle measurement,” Opt. Lasers Eng. 32(7–8), 567–574 (2000).
[Crossref]

Nguyen, D.

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]

Nishio, K.

Onural, L.

L. Onural and P. D. Scott, “Digital decoding of in-line holograms,” Opt. Eng. 26(11), 261124 (1987).
[Crossref]

Osten, W.

Pedrini, G.

Place, A. R.

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]

Sasada, M.

Y. Awatsuji, M. Sasada, and T. Kubota, “Parallel quasi-phase-shifting digital holography,” Appl. Phys. Lett. 85(6), 1069–1071 (2004).
[Crossref]

Saxton, W. O.

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).

Schnaiter, M.

Scott, P. D.

Sheng, J.

J. Katz and J. Sheng, “Applications of holography in fluid mechanics and particle dynamics,” Annu. Rev. Fluid Mech. 42(1), 531–555 (2010).
[Crossref]

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]

Soria, J.

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]

Stetzer, O.

Tahara, T.

Tanaka, Y.

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]

Thompson, B. J.

G. A. Tyler and B. J. Thompson, “Fraunhofer holography applied to particle size analysis a reassessment,” J. Mod. Opt. 23(9), 685–700 (1976).

Tiziani, H.

Tyler, G. A.

G. A. Tyler and B. J. Thompson, “Fraunhofer holography applied to particle size analysis a reassessment,” J. Mod. Opt. 23(9), 685–700 (1976).

Ura, S.

Wang, Z.

Willert, C. E.

C. E. Willert and M. Gharib, “Digital particle image velocimetry,” Exp. Fluids 10(4), 181–193 (1991).
[Crossref]

Wu, X.

Wu, Y.

Yamaguchi, I.

Yasuda, N.

S. Murata and N. Yasuda, “Potential of digital holography in particle measurement,” Opt. Lasers Eng. 32(7–8), 567–574 (2000).
[Crossref]

Zhang, T.

Zhang, Y.

Annu. Rev. Fluid Mech. (1)

J. Katz and J. Sheng, “Applications of holography in fluid mechanics and particle dynamics,” Annu. Rev. Fluid Mech. 42(1), 531–555 (2010).
[Crossref]

Appl. Opt. (4)

Appl. Phys. Lett. (1)

Y. Awatsuji, M. Sasada, and T. Kubota, “Parallel quasi-phase-shifting digital holography,” Appl. Phys. Lett. 85(6), 1069–1071 (2004).
[Crossref]

Exp. Fluids (2)

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]

C. E. Willert and M. Gharib, “Digital particle image velocimetry,” Exp. Fluids 10(4), 181–193 (1991).
[Crossref]

J. Mod. Opt. (1)

G. A. Tyler and B. J. Thompson, “Fraunhofer holography applied to particle size analysis a reassessment,” J. Mod. Opt. 23(9), 685–700 (1976).

J. Opt. Soc. Am. A (1)

Jpn. J. Appl. Phys. (1)

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]

Nature (1)

D. Gabor, “A new microscopic principle,” Nature 161(4098), 777–778 (1948).
[Crossref] [PubMed]

Opt. Eng. (1)

L. Onural and P. D. Scott, “Digital decoding of in-line holograms,” Opt. Eng. 26(11), 261124 (1987).
[Crossref]

Opt. Express (1)

Opt. Lasers Eng. (1)

S. Murata and N. Yasuda, “Potential of digital holography in particle measurement,” Opt. Lasers Eng. 32(7–8), 567–574 (2000).
[Crossref]

Opt. Lett. (1)

Optik (Stuttg.) (1)

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).

Phys. Rev. Lett. (1)

T. Latychevskaia and H. W. Fink, “Solution to the twin image problem in holography,” Phys. Rev. Lett. 98(23), 233901 (2007).
[Crossref] [PubMed]

Proc. Natl. Acad. Sci. U.S.A. (1)

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]

Proc. SPIE (1)

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]

Other (2)

J. C. Russ, The Image Processing Handbook (CRC Press, 2016).

C. S. Vikram, “Particle Field Holography,” in Cambridge Studies in Modern optics (Cambridge University Press, 1992).

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Figures (9)

Fig. 1
Fig. 1 Hologram intensity log10I(r) of a particle against r derived from Eq. (3) for z1 = 100mm, dp = 70 µm and λ = 632.8 nm.
Fig. 2
Fig. 2 (a) Phase retrieval method for two holograms; comparison of a simulated and reconstructed hologram between (b) phase retrieval method and (c) Gabor method. The holograms in (b) and (c) were recorded at z1 = 50 mm and z2 = 100 mm, respectively. The wavelength is 632.8 nm. We randomly dispersed 500 particles with a diameter of 70 µm in the same plane with a 512 pixel × 512 pixel area (pixel size: 10 µm × 10 µm).
Fig. 3
Fig. 3 Influence of m in Eq. (19) on (a) the signal-to-noise ratio (SNR) and (b) the number of iterations Nit. The SNR is defined as SNR=10 log 10 ( σ p 2 / σ b 2 ) , where σ p and σ b are the standard deviations of the particles and background, respectively, of the intensity at z0. The distance Δz is varied from 0.001 mm to 150 mm in increments of 0.001 mm.
Fig. 4
Fig. 4 SNR versus relative misalignment ϵ x in the x direction and reconstructed planes. Holograms of particles shown in Fig. 2 are recorded at z1 = 50 mm and Δz = 50 mm. Hologram I2 is shifted in the x direction with respect to hologram I1.
Fig. 5
Fig. 5 SNR versus relative misalignment ϵ Δz in the z direction. Holograms of Fig. 2 are recorded at z1 = 50 mm and Δz = 50 mm. Hologram, I2 is shifted in the z direction with respect to hologram I1. This condition corresponds to m = 1 in Fig. 3(a).
Fig. 6
Fig. 6 Optical arrangement for recording two in-line holograms with two high-speed cameras for phase retrieval method.
Fig. 7
Fig. 7 Optical image distortion correction: (a, b) reconstructed holograms of particles; (c) vector field for image distortion correction.
Fig. 8
Fig. 8 Mode of image intensity reconstructed at z0 by the retrieved complex amplitude.
Fig. 9
Fig. 9 Experimental holograms of settling particles reconstructed by the phase retrieval method with two high-speed cameras (left) and the Gabor method (right). Each hologram contains particles with an average diameter of 70 µm that were settling from the upper edge of the hologram at approximately terminal velocity.

Equations (19)

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ψ z 1 ( x 1 , y 1 )= exp( jk z 1 ) jλ z 1 [ 1p( x 0 , y 0 ) ]exp{ jk 2 z 1 [ ( x 1 x 0 ) 2 + ( y 1 y 0 ) 2 ] }d x 0 d y 0 ,
p( x 0 , y 0 )={ 1,   x 0 2 + y 0 2 ( d p /2 ) 2 0,   x 0 2 + y 0 2 > ( d p /2 ) 2
I( r )= | ψ z 1 ( r ) | 2 =1 π d p 2 2λ z 1 sin( π r 2 λ z 1 ) 2 J 1 ( π d p r/( λ z 1 ) ) π d p r/( λ z 1 ) + ( π d p 2 4λ z 1 ) 2 [ 2 J 1 ( π d p r/( λ z 1 ) ) π d p r/( λ z 1 ) ] 2 ,
r n+1 2 r n 2 2rΔr=2λz.
r = 1.22λz d p .
d p 2 λ < z 1 < d p L/2 1.22λ ,
Δx< d p 2.44 = Δr 2 .
h z1 ( x 1 , y 1 ) 1 λ z 1 exp[ j k z 1 ( x 1 2 + y 1 2 ) ].
ψ z 1 =exp( jk z 1 )[ 1p( x 0 , y 0 ) ]* h z 1 .
I z 1 = ψ z 1 * ψ * z 1 = | 1p* h z 1 | 2 =1 p * * h z 1 * p* h z 1 + | p* h z 1 | 2 1 p * * h z 1 * p* h z 1
ψ z0 = I z 1 * h z 1 =1 p * * h 2 z 1 p* h z 1 .
ψ z 0 = ψ z 1 * h z 1 =1p.
ψ' z 2 k =exp( jΔz ) ψ z 1 k * h Δz =| ψ' z 2 k |exp( j ϕ 2 ' )
ψ z 2 k = I 2 exp( j ϕ 2 ' )
ψ ' z 1 k =exp( jΔz ) ψ z 2k * h Δz =| ψ ' z 1 k |exp( j ϕ 1 ' )
ψ z 1 k = I 1 exp( j ϕ 1 ' )
ψ z 1 = I 1 exp( j ϕ 1 )exp( jk z 1 ).
ψ z 1 = I 1 exp( j ϕ 1 '' )exp( jkmΔz ),
Δz= z 1 m            ( m=1,2,..,   mN ).

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