## Abstract

Conventional digital in-line holography requires at least two phase-shifting holograms to reconstruct an original object without zero-order and conjugate image noise. We present a novel approach in which only one in-line hologram and two intensity values (namely the object wave intensity and the reference wave intensity) are required. First, by subtracting the two intensity values the zero-order diffraction can be completely eliminated. Then, an algorithm, called partition calculation, is proposed to numerically remove the conjugate image. A preliminary experimental result is given to confirm the proposed method. The method can simplify the procedure of phase-shifting digital holography and improve the practical feasibility for digital in-line holography.

©2012 Optical Society of America

## 1. Introduction

In digital holography, the presence of zero-order diffraction and conjugate image adversely affects the quality of reconstructed image. Off-axis holography can resolve the problem to some extent, because the reconstructed image and the undesired terms can separately appear at different locations. However, the technique prevents the effective application of the space-bandwidth product of CCD camera, which further reduces the reconstructed image detail [1]. To make best use of the detector resolution in digital holography, it would be optimal to use an in-line configuration. However, in in-line holography, the reconstructed image at the center is fully superposed with the zero-order diffraction and the conjugate image. Therefore, it is necessary to remove the zero-order diffraction and the conjugate image for in-line holography. So far, a number of techniques on removing zero-order diffraction and conjugate image have been reported. Several methods, such as subtracting the mean value [2], subtracting the numerical generated intensity of the object and reference waves [3] and numerical space-shifting reconstruction [4], can partially or completely suppress the zero-order diffraction but maintain the conjugate image intact. Digital filtering techniques in frequency domain [5] and in space domain [6], as well as wavelet transform [7], can simultaneously eliminate the zero-order diffraction and the conjugate image. However, they are effective only in off-axis circumstance. Although the methods based on phase-shifting techniques [8–10] are able to eliminate the zero-order diffraction and the conjugate image for both off-axis circumstance and on-axis circumstance, it is hard to use the techniques in practical environments due to the complex of multiple hologram acquisition as well as the sensitivity of phase variation to the environment.

In this paper, we describe what we believe to be a novel approach based on partition calculation, with which the zero-order diffraction and the conjugate image can be removed very effectively in in-line holography. The entire process needs only recording a digital hologram and two intensity values named the object wave intensity and the reference wave intensity. By subtracting the two intensity values from the hologram, the zero-order diffraction can be completely removed. Then the algorithm of partition calculation is applied to remove the conjugate image. Lastly the reconstructed image without zero-order and conjugate image can be acquired. The technique is convenient in manipulation due to the numerical processing without any additional requirements to the recording optics but two shutters to record the object intensity and the reference intensity.

## 2. Theoretical derivation

A typical digital in-line holographic setup is shown in Fig. 1 . Under the condition of Fresnel approximation, the spherical reference wave on the plane of the hologram can be written as

*△ x*and

*△y*are the corresponding pixel sizes,

*rect()*is the rectangle function, λ is the wavelength of the laser used to record the hologram, A is the amplitude of the reference wave on the plane of the hologram,

*z*is the distance of the point reference light source from the hologram plane.

Assume that O(*x*, *y*) denotes the object wave on the plane of hologram. An in-line hologram and two intensity distribution (object wave intensity and reference wave intensity) are recorded sequentially and expressed as

By subtracting the two intensity values from the hologram, that is, by subtracting Eq. (3) and Eq. (4) from Eq. (2), we obtain a zero-order-free complex hologram

Equation (5) shows that *I _{C}(x, y)* still contains the conjugate image. In order to eliminate the conjugate image, we propose an algorithm of partition calculation. The complex hologram is divided equally to four regions, as shown in Fig. 2
. Coordinate of the centers of the four regions can be given as

New sub-coordinate systems are built up for each region and the centers of the four regions are regarded as the origins respectively. In the new sub-coordinate systems the reference wave in four regions can be described as

Where. ${R}^{\prime}({x}^{\prime},{y}^{\prime})=A\mathrm{exp}[i2\pi \frac{{x}_{R}{}^{2}+{y}_{R}{}^{2}}{2\lambda z}]\mathrm{exp}[i2\pi \frac{{{x}^{\prime}}^{2}+{{y}^{\prime}}^{2}}{2\lambda z}]$The sub-complex-holograms can be expressed in new sub-coordinate systems

The spatial frequency spectrum distribution of each sub-complex-hologram will be discussed. Assume that${R}^{\prime}(\xi ,\eta )=F\{{R}^{\prime}({x}^{\prime},{y}^{\prime})\}$, where *F* ^{denotes} Fourier transform operation. By utilizing the property of the Fourier transform, the spatial frequency spectrums of ${{R}^{\prime}}_{k}$and ${{R}^{\prime}}^{*}{}_{k}$ can be calculated

Under satisfying the condition imposed by Eq. (11), the spectrums of the object wave in each region can be extracted by filtering method and the reconstructed image and the conjugate image can be separated.

According to Eq. (11), the shorter the distance *z* of the point reference light source from the hologram plane is, the higher the object wave frequency which can be recorded will be. However, to satisfy Nyquist sampling rate, the distance *z* has a minimum value

*sw*denotes the effective space-bandwidth product of CCD camera.

_{CCD}In off-axis holography, to separate the spectrums of the zero-order term and the twin images, the frequency of the carrier wave must be at least three times as high as one of the object wave. Assuming that the carrier wave is a plane wave parallel to the *x-z* plane, the highest frequency of the object wave in *x* dimension must not exceed a maximum value $\frac{1}{8\Delta x}$. And in many cases, the frequency of the object wave is restricted to be equal in all direction, e.g. in digital holographic microscopy. So the maximum space-bandwidth product is $\frac{1}{16}s{w}_{CCD}$. Even though the object wave intensity and the reference wave intensity are also be recorded in off-axis holography, that is, the zero-order term can be eliminated by subtracting the two intensity values, the maximum space-bandwidth product is only up to $\frac{1}{4}s{w}_{CCD}$.

Therefore, the technique based on partition calculation can make better use of the effective space-bandwidth product of CCD camera than off-axis holography, which means the technique possesses better imaging performance. Conventional multi-exposure in-line holography can make full use of the effective space-bandwidth product of CCD camera and the highest object wave frequency which can be recorded is $\frac{1}{2\Delta x}$. Therefore, the reconstructed image by conventional in-line holography can have higher resolution than the one by our method. But it is hard to use the conventional in-line holography techniques in practical environments due to the sensitivity of phase variation to the environment during phase-shifting operation.

## 3. Algorithm illustration

First, the four matrixes are defined to obtain the four sub-complex-holograms in the four regions described in Fig. 2. They are

The sub-complex-hologram in each region is obtained by multiplication of the complex hologram expressed in Eq. (5) by the corresponding matrix defined in Eq. (15). We have

The frequency spectrums of the four sub-complex-holograms can be calculated by Fourier transform for Eq. (16). As shown in Fig. 3, the spectrums of object wave and the conjugate object wave in each sub-complex-hologram can be separated. In order to extract the spectrum of the object wave in each region, we define four triangular matrixes named TRI1, TRI2, TRI3 and TRI4

The number of lows and columns in the matrixes defined in Eq. (15) and in Eq. (17) is M and N, respectively. The spectrums of the object wave in each region can be extracted by calculating as follows

In fact, the spectrums extracted by Eq. (18) is the convolution of the spectrums of the object wave and the spectrums of the conjugate reference wave, that is, ${O}_{Rk}\text{(}\xi \text{,}\eta \text{)}={O}_{k}\text{(}\xi \text{,}\eta \text{)}\ast {R}_{k}{}^{*}\text{(}\xi \text{,}\eta \text{)}$.

Last, the object wave distribution in each region on the hologram plane can be acquired by multiplying reverse Fourier transform of Eq. (18) by the reference wave in each region

According to the type of the hologram, the sub-complex-amplitude-distribution of reconstructed image from each region can be computed. The four sub-complex-amplitude-distributions are superposed and the reconstructed image can be obtained.

In addition, due to linear operation of Fourier transform, the spectrums in each region extracted by Eq. (18) can be directly superposed. Then the object wave complex amplitude on the plane of the hologram can be obtained by multiplying reverse Fourier transform of the whole spectrums by the reference wave

The reconstructed image can be computed according to the type of the hologram. We adopt the method because of its relatively small amount of calculation workload. Figure 4 presents a schematic diagram for the implementation of partition calculation.

## 4. Experimental results and analysis

The experiments are carried out by taking the image-plane in-line geometry as an example. The optical setup is shown in Fig. 5 , in which a HeNe laser with a power of 20mW and wavelength of 632.8nm is used as the light source. The recording device adopts a CCD camera, the size of each pixel is 8.6μm × 8.3μm, the total numbers of the pixels is 1024 × 768, and the exposure speed is 16 frames per second. The distance of the point reference light source from the CCD camera is 12cm. The two shutters are adopted to record the object wave intensity and the reference wave intensity and are synchronized with the CCD camera by computer control. In this setup, an in-line hologram and two intensity values (object wave intensity and reference wave intensity) are sequentially recorded. The total time for recording the three images is about 0.3 seconds.

A phase resolution test chart fabricated in our laboratory is used as the recorded object. Figure 6(a) and Fig. 6(b) present the original in-line hologram and its frequency spectrum, respectively. By subtracting the object wave intensity and the reference wave intensity from the hologram, the frequency spectrum of zero-order term can be removed as shown in Fig. 6(c). But in Fig. 6(c) the frequency spectrum of object wave and conjugate object wave together exist. The complex hologram with removing zero-order term is divided to four equal regions as shown in Fig. 7 . Figure 8 demonstrates the frequency spectrums of the four sub-complex-holograms, in which the frequency spectrums of object wave and conjugated object wave in each region appear at symmetric locations. The frequency spectrum of object wave in each region is extracted as shown in Fig. 9 . The frequency spectrums extracted from the four regions are superposed and Fig. 10 displays the complete frequency spectrum of object wave. The complete frequency spectrum in Fig. 10 is computed by reverse Fourier transform and the object wave can be reconstructed. Figure 11 presents the phase image of the object. The result indicates that the zero-order diffraction and the conjugate image can be successfully removed and the clear reconstructed image can be acquired by the proposed method. All these results are exactly consistent with our theoretical prediction. The manipulation is convenient and feasible because the approach does not need conventional phase-shifting operation.

For comparison, we also present the phase distributions with the presence of the zero-order term and/or the conjugate image as shown in Fig. 12 . Figure 12(a) is the phase distribution of the reconstructed wave obtained directly from the recorded in-line hologram. The reconstructed wave field can be expressed as ${O}_{ZC}(x,y)={\left|O\right|}^{2}R+{\left|R\right|}^{2}R+O{\left|R\right|}^{2}+{O}^{*}{R}^{2}$, that is, the zero-order distribution and the conjugate image exist together with the reconstructed image. The phase distribution of the reconstructed wave is shown in Fig. 12(b) when the zero-order term is removed by subtracting the object wave intensity and the reference wave intensity and the reconstructed wave field is ${O}_{C}(x,y)=O{\left|R\right|}^{2}+{O}^{*}{R}^{2}$. Figure 12(c) is the phase distribution of the reconstructed wave when the conjugate image is removed by partition calculation for recorded in-line hologram and the reconstructed wave field is ${O}_{Z}(x,y)={\left|O\right|}^{2}{R}^{*}+{\left|R\right|}^{2}{R}^{*}+O{\left|R\right|}^{2}$. Apparently, the presence of the zero-order term and/or conjugate image adversely affects the quality of reconstructed image and the phase information of the reconstructed image is completely submerged in the sphere factor of the reference wave. As can seen in Fig. 11 and Fig. 12, the technique proposed in this paper can completely remove the zero-order diffraction and the conjugate image for in-line holography and can acquire high quality reconstructed image.

## 5. Conclusion

In conclusion, we have proposed an approach of partition computation which can successfully remove the zero-order diffraction and the conjugate image in the reconstruction. Both the theory and the experimental results demonstrate the performance of the novel approach is satisfactory. In addition, the approach is a significant breakthrough in digital in-line holography because all the conventional techniques for removing the zero-order diffraction and the conjugate image in on-axis holography need phase-shifting manipulation in which the optical setup is intricate and the phase variation is sensitive to the environment. Since the approach uses purely numerical operation and so does not need any of additional holograms and specific optical requirements, we believe it be a really convenient, practical and widely way in digital holography.

## Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grant No. 60877002, Natural Science Foundation of Zhejiang Province of China under Grant No. Z1080030 and Educational commission Foundation of Zhejiang Province of China under Grant No. Y201016465.

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