## 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 ${\text{\psi}}_{{z}_{1}}\left({x}_{1},{y}_{1}\right)$ at the recording plane a distance *z*_{1} away is given by

*k*= 2π/λ is the wave number of the plane wave. The diffracting aperture

*p*(

*x*

_{0},

*y*

_{0}) for a particle, is defined as the amplitude transmittance function,

*d*is the diameter of the particle. This aperture has circular symmetry, and under the far-field condition,

_{p}*d*

_{p}^{2}/λ

*z*

_{1}. Performing the integration yields the intensity distribution [15]

*J*

_{1}denotes the first-order Bessel function. Figure 1 shows the intensity distribution of a particle with a diameter of 70 µm.

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

*n*= 1, 2,...,

*n*$\in $

**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

*r*

_{0}of the end of the first lobe corresponds to the first zero-crossing position of the Bessel function:Thus, the recording distance of a particle is limited to the rangewhere

*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

*r*

_{0}=

*L*/2.

From Eqs. (4) and (5), the fringe spacing is Δ*r* = *d _{p}*/1.22. Therefore, the spatial resolution Δ

*x*of the CCD is expressed as

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

In reconstruction, the pattern is illuminated for reconstructing particles with ${h}_{-{z}_{1}}$, and the complex amplitude ${\text{\psi}}_{{z}_{0}}$ is reconstructed as

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 ${I}_{{z}_{0}}={\text{\psi}}_{{z}_{0}}*{\text{\psi}}_{{z}_{0}}^{*}$ of reconstructed particles. If we know the phase information of the hologram, the in-focus particles are reconstructed asTherefore 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 ${\text{\psi}}_{{z}_{1}k}$ of the first hologram *I*_{1} in the second hologram *I*_{2} with the distance Δ*z* between two holograms; (Step 2) replace the absolute value of the resulting complex amplitude with the square root of the second hologram *I*_{2}; (Step 3) back-reconstruct the new complex amplitude ${\text{\psi}}_{{z}_{2}k}^{\text{'}}$ to the first hologram; (Step 4) replace the absolute value of the reconstructed complex amplitude ${\text{\psi}}_{{z}_{1}k}$ with the square root of the first hologram *I*_{1}. These steps are formulated for the *k* th iteration as follows:

*z*

_{0}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.

In this retrieval process, the two recorded holograms must satisfy the range of Eq. (6) and the resolution of Eq. (7). However the distance Δ*z* has not been given the attention it deserves in the recording procedure. From Eq. (9), the complex amplitude at *z*_{1} is composed of two phases, namely the particle of ${\varphi}_{1}$ and the reference wave of *kz*_{1};

*m*. If the retrieved phase has the relationship ${\varphi}_{1}\approx {\varphi}_{1}^{\text{'}\text{'}}$, the distance Δ

*z*is derived from Eqs. (17) and (18) as

We have begun to verify this relationship using the holograms of Fig. 2(b). The position of hologram *I*_{1} is fixed at *z*_{1} that of the hologram *I*_{2} 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 $m\ge 6$, the peak shape becomes unstable in this condition. The SNR is reduced for $m\ge 10$. This is why Δ*z* is smaller than the far-field condition, 7.74 mm = ${d}_{p}^{2}/\lambda \ll \text{\Delta}z$. The number of iterations *N _{it}* 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 ${\u03f5}_{x}$ 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 ${\u03f5}_{\text{\Delta}z}$ in the *z* direction is lower than that of ${\u03f5}_{x}$ 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 Δ*z*. It is found that calibration of ${\u03f5}_{x}$ is sensitive, but that of ${\u03f5}_{\text{\Delta}z}$ 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 mm^{2} was observed by the two high-speed cameras (FASTCAM Mini UX100, Photron, frame rate: 4,000 fps, image size: 1024 × 1024 pixel^{2}, cell size: 10 µm × 10 µm) located at two positions *z*_{1} = 150 mm and *z*_{2} = *z*_{1} + Δ*z* = 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 [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).

In order to calibrate the misalignment for the target distance Δ*z* = 50 mm in the *z* direction, the installation distance between the two cameras was detected by changing the numerical condition of Δ*z* at each iterative loop of the phase retrieval method, as shown in Fig. 8. The distance was obtained as Δ*z* = 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.

## 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).

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