## Abstract

This work discusses a novel approach for numerical wavefront reconstruction, which utilizes arbitrary phase step digital holography. The experimental results reveal that only two digital holograms and a simple estimation procedure are required for twin-image suppression, and for numerical reconstruction. One advantage of this approach is its simplicity. Only one estimate equation needs to be applied. Additionally, the optical system can be constructed from inexpensive, generally available elements. Another advantage is the effectiveness of the approach. The tolerance of the estimated value is less than 1% of the actual value, such that the quality of the reconstructed image is excellent. This novel approach should facilitate the application of digital holography and promote its use.

© 2007 Optical Society of America

## 1. Introduction

Holography, invented by Gabor in 1949 [1], is an important wavefront recording approach. In this approach, an unknown wavefront interferes with a reference wave which is mutually coherent, and has known amplitude and phase. The main problem encountered in recent developments of digital holography has been twin-image blurring during numerical reconstruction. Yamaguchi and Zhang proposed a solution, a method for suppressing such twin-images, which has led to substantial advances in phase shifting digital holography [2]. Although their approach is ideal for suppressing the twin-images that are formed during numerical reconstruction, it is not so useful for precise phase shifting in a serial, special, and stable shifting procedure. An interesting new approach that has often been much studied is the arbitrary phase steps approach [3–11]. The problem with this recording process is that differences exist between the actual and the theoretical phase errors. Even after triple recordings and complicated calculations, the results are still unsatisfactory. This work presents a novel approach that uses two arbitrary phase step digital holograms and a simple estimation equation to suppress the twin-images that are produced during numerical reconstruction. The experimental results reveal that this new approach is both simple and effective. The proposed approach has two advantages- the simplicity of the optical system and the procedure required and the ability to suppress the twin-images effectively by a simple estimation and numerical operation.

## 2. Principle

During holographic recording, a second wave, which is mutually coherent with the first, and of known amplitude and phase, is added to an unknown wave. The intensity of the sum of the two complex fields depends on both the amplitude and the phase of the unknown field.

$$={a}_{o}(x,y)\mathrm{exp}\left[j{\phi}_{o}(x,y)\right]$$

$$={a}_{o}^{2}+{a}_{r}^{2}+2{a}_{o}{a}_{r}\mathbf{cos}\left({\phi}_{o}-{\phi}_{r}\right),$$

Where *Ψ _{Obj}* denotes the object wave in the object plane;

*Ψ*denotes the object wave to be detected and recorded on the holographic plane; ℑ denotes the Fourier transform; ℑ

_{O}^{-1}denotes the inverse Fourier transform, and

*h*(

*x,y;z*) denotes the impulse response function of

_{1}*z*, which is the distance between the object and the holographic plane.

_{1}*Ψ*in Eq. (2) denotes the reference wave [12]. The intensity of the sum of the two waves is given by Eq. (3), which describes the first hologram.

_{R}In arbitrary phase step digital holography, an arbitrary retarding phase *Δφ* is added in the optical path. The intensity of the arbitrary phase step hologram, which is the second hologram, can now be written as,

$$\cong {a}_{o}^{2}+{a}_{r}^{2}+2{a}_{o}{a}_{r}\mathbf{cos}\left({\phi}_{o}-{\phi}_{r}\right)\mathbf{cos}\left(\Delta \varphi \right).$$

$$\left(\mathrm{For}\phantom{\rule{.2em}{0ex}}\mathrm{the}\phantom{\rule{.2em}{0ex}}\mathrm{det}\mathrm{ailed}\phantom{\rule{.2em}{0ex}}\mathrm{derivation}\phantom{\rule{.2em}{0ex}}\mathrm{see}\phantom{\rule{.2em}{0ex}}\mathrm{Appendix}\phantom{\rule{.2em}{0ex}}A.\right)$$

The value of *Δφ* must be estimated for twin-image suppression and numerical reconstruction. Accordingly, first assume that the operation is an off-axis holographic scheme, and then begin to analyze the acquired holograms. Equations (3) and (4) reveal that the difference between these two holograms is given only by the retarded phase *Δφ*. If this information can be separated from these two holograms, then the twin-images can be suppressed. Then, we considered about the Fourier spectrum of the hologram. Suppose in the off-axis holographic scheme, that the spectra of the terms in the hologram, given by Eq. (3), appear well-separated and can be considered independently, because the locations of distribution of the spectra in the holograms differ completely. Therefore, the linearity theorem can be applied to the Fourier transform of hologram *I _{H1}* and rewritten as Eq. (5). A similar idea can be applied to the second hologram

*I*. Now, the arbitrary phase information can be separated from the division of Fourier transform.

_{H2}Based on the above derivation, the Fourier spectra of the two acquired digital holograms appear to have the same separated distribution. Also, the Fourier spectra of the zero-order parts (*a*
^{2}
_{o}+*a*
^{2}
_{r}) of the two holograms have the same location of distribution, while the distribution of the Fourier spectra of the other parts of the holograms (including *ψ _{O}ψ**

_{R}and

*ψ**

_{O}*ψ*, given by 2

_{R}*a*cos(

_{o}a_{r}*φ*)) appear to be of the same character. Therefore, we suggest that the spectra of each parts of the hologram in the divisional operation can be regarded as independent of those of other parts. The following information about the arbitrary phase can be inferred;

_{o}-φ_{r}$$=1+\Im \left\{\mathrm{cos}\left(\Delta \phi \right)\right\}$$

Finally, based on Eq. (8) the value of the arbitrary phase *Δφ* can be easily estimated from two arbitrary phase step digital holograms. The constant on the right-hand side of Eq. (7) can be ignored. The inverse Fourier transform now makes a delta function *δ*.

## 3. Experimental setup and results

The experimental system shown in Fig. 1 is a modified Mach-Zehnder setup. An He-Ne laser with a power of 20*mW* and a wavelength of 632.8*nm* was used as the light source. Two λ/2-retarded wave-plates and a polarized beam-splitter were used to adjust the ratio of the intensity of the object wave to that of the reference wave. The two waves were collimated using spatial filters and lenses. A translation stage was adopted to add a retarded phase to the reference wave in the optical path. The hologram was recorded using a CCD sensor (Pixera-150SS CCD camera, 1040×1392 pixels, 0.484*cm*×0.65*cm*). A Newport resolution target (RES-1) was used as the object. It was placed at a distance of *z _{1}*=9.60

*cm*away from the CCD.

Figure 2(a) presents the first acquired hologram. The other was acquired after the distance of the translation stage was adjusted through 0.01*mm*, which is the minimum precision of the translation stage. Only the phase of the reference wave was assumed to change after the translation stage was applied. Restated, the wavefront of the reference wave, typically a plane wave, is quite similar to those of the first acquired hologram. An estimated value of *Δφ* was obtained by summing all elements in a 5×5 array in the center area according to Eq. (8). The retardation of the phase in the exact center of the calculated array was *Δφ*=1.4150*rad*. In fact, the estimated phase in the central area would be obtainable from a narrower area if the whole retarded phase in the recording procedure were to more uniform and precise. In the subsequent digital reconstruction, the zero-order image was suppressed using the numerical suppression approach, as described by formulated in Eqs. (9) and (10) [13], where |*ψ _{R}*|

^{2}is the intensity of the reference wave, and the distribution of the intensity is generally uniform. A uniform intensity can be subtracted by numerical operation for zero-order image suppression. Finally, after the operation of conjugate term suppressing and convolution with the impulse response through distance

*z*, only the object wave is determined; see Eq. (11) [6]. The derivation is given in detail in Appendix B.

_{1}Figure 2(a) shows the fringe pattern of the first acquired digital hologram in the off-axis recording geometry. Interference fringes that are characteristic of off-axis geometry are observed in this image. The spacing of fringes indicates an angle of about 0.09° between the object wave and the reference wave. (Calculated from the spacing *d* of the fringes with *d*=*λ*/sin(*θ*)) The fringe of the second hologram appears to be similar characteristic of off-axis geometry, as shown in Fig. 2(c). Also, the interference pattern in the second hologram appears to be similar, except for the addition of the retarded phase that caused the shift. Figures 2(b) and 2(d) present the Fourier spectra of Figs. 2(a) and 2(c), respectively. The difference between Figs. 2(b) and 2(d) seems not to be observed but the spectral images reveal that the well-separated areas of the Fourier spectra of the two acquired holograms are the same. Therefore the Fourier spectra of the various parts of the hologram can be regarded as independent in divisional operation. Additionally, the numerical reconstructed image in Fig. 2(a), which is presented in Fig. 2(e), indicates that blurring remains upon reconstruction of the traditional off-axis configuration. To suppress the blurring, the angle between the object wave and the reference wave exceed 2.89° in the traditional off-axis configuration when the reconstruction distance is 9.6*cm*. (The approach is described in “Numerical reconstruction and twin-image suppression using an off-axis Fresnel digital hologram,” currently being reviewed in Applied Physics B.) Therefore, the spacing between of fringe is less than 12*µm*, which limits the maximum spatial frequency which can be recorded on a CCD sensor, causing loss of complete hologram information in the recording procedure. However, the clear reconstructed image shown in Fig. 2(f) was obtained only after carrying out the numerical suppression and propagation through distance *z _{1}* according to Eqs (9), (10) and (11), respectively. (Equation (11) in particular would fails at

*Δφ*=±

*π*. [13]) The results reveal that the twin-image problem was mostly solved and the quality of the reconstructed image was enhanced. The information about the object wave, including the wavefront and the phase, was used effectively. A reconstructed image of a standard resolution target was obtained. The image indicates the spatial resolution of numerical reconstruction as high as 11.05

*µm*(Group 5 number 4; a detail image is shown in Fig. 2(h).) would be sustained to enhance the practicality in the digital holography applications.

## 4. Performance investigation and discussion

Figure 3 presents the performance of this approach for the estimation of an arbitrary step phase. During the simulation, arbitrary phase step holograms were first produced, then operated with the estimation and finally analyzed. A digital hologram was generated by object wave interference, as in Eq. (1). A plane wave acted as the reference wave, as described by Eq. (2), where the angle of interference between the object wave and the reference wave was 0.09o. The intensity ratio was 1:1. Another hologram, the arbitrary step hologram, was produced similarly, except for the addition of the phase *Δφ* to the reference wave. An estimated value for the two simulated holograms was then calculated, using Eq. (7), by summing the elements of a 3×3 array in the central area. The horizontal axis in Fig. (3) represents the added phase from 0 to 2*π rad*. The cosine of the added phase can be used to check the tolerance of the estimate obtained from Eq. (7). The results reveal that the estimates and the cosine of the added phase were very close, indicating that the estimation is satisfactory and effective, although cos(*Δφ*) equals both cos(2*π-Δφ*) and cos(-*Δφ*). The exact value of the step phase from the actual reconstruction procedure must be obtained.

However, the ignoring of [cos(*φ _{o}-φ_{r}-Δφ*)-cos(

*φ*)] in Appendix A can’t seem to satisfy general condition except for the small adding phase. In fact, the phase terms include the object and the reference fields as well as the arbitrary added phase. Also, the phase terms of the object and the reference fields vary in two-dimensional coordinates. Therefore, specific values of the phase term cannot be suited. Accordingly, we suggest that the ignored terms should be associated with two phase shifting holograms of zero-order term removed of unit amplitude. In fact, the difference between the two phase shifting holograms is very small, such that subtraction leaves information on the shifting phase only and can be neglected. Furthermore, Fig. 3 in this work that the approximation is feasible even though the added phase

_{o}-φ_{r}+Δφ*Δφ*is almost as large as 2

*π*, and the method of estimation is still effective.

## 5. Conclusions

This work presents a simple estimation approach for arbitrary phase step digital holography. This novel approach is both simple and effective. The feasibility of applying the approach to digital holography was demonstrated. Satisfactory results were obtained for the suppression of twin-images in numerical reconstructions. Filtering out these twin-images improved both the contrast in the reconstructed image and the image quality, such that information about the object wave was obtained and used effectively.

## Appendix

A. ${I}_{H2}={\mid {\psi}_{o}+{\psi}_{R}\mathrm{exp}\left(j\Delta \varphi \right)\mid}^{2}$ $={a}_{o}^{2}+{a}_{r}^{2}+2{a}_{o}{a}_{r}\mathrm{cos}\left({\phi}_{o}-{\phi}_{r}-\Delta \phi \right)$ $={a}_{o}^{2}+{a}_{r}^{2}+2{a}_{o}{a}_{r}\left[\mathrm{cos}\left({\phi}_{o}-{\phi}_{r}\right)\mathrm{cos}\left(\Delta \phi \right)+\mathrm{sin}\left({\phi}_{o}-{\phi}_{r}\right)\mathrm{sin}\left(\Delta \phi \right)\right]$ $={a}_{o}^{2}+{a}_{r}^{2}+2{a}_{o}{a}_{r}\left\{\mathrm{cos}\left({\phi}_{o}-{\phi}_{r}\right)\mathrm{cos}\left(\Delta \phi \right)+\frac{1}{2}\left[\mathrm{cos}\left({\phi}_{o}-{\phi}_{r}-\Delta \phi \right)-\mathrm{cos}\left({\phi}_{o}-{\phi}_{r}+\Delta \phi \right)\right]\right\}$ $\cong {a}_{o}^{2}+{a}_{r}^{2}+2{a}_{o}{a}_{r}\mathrm{cos}\left({\phi}_{o}-{\phi}_{r}\right)\mathrm{cos}\left(\Delta \phi \right)$

B. $\begin{array}{l}{I}_{H1}^{\text{'}}-\mathrm{exp}(-j\Delta \varphi ){I}_{H2}^{\text{'}}\hfill \\ =({\psi}_{O}{\psi}_{R}^{*}+{\psi}_{O}^{*}{\psi}_{R})-\mathrm{exp}(-j\Delta \varphi )\{{\psi}_{O}{[{\psi}_{R}\mathrm{exp}(j\Delta \varphi )]}^{*}+{\psi}_{O}^{*}{\psi}_{R}\mathrm{exp}(j\Delta \varphi )\}\hfill \\ ={\psi}_{O}{\psi}_{R}^{*}+{\psi}_{O}^{*}{\psi}_{R}-\mathrm{exp}(-j\Delta \varphi )[{\psi}_{O}{\psi}_{R}^{*}\mathrm{exp}(-j\Delta \varphi )+{\psi}_{O}^{*}{\psi}_{R}\mathrm{exp}(j\Delta \varphi )]\hfill \\ =[1-\mathrm{exp}(-j2\Delta \varphi )]{\psi}_{O}{\psi}_{R}^{*}\hfill \\ \hfill [1-\mathrm{exp}(-j2\Delta \varphi )=\text{const}.\\ \cong {\psi}_{O}{\psi}_{R}^{*}\hfill \end{array}$

## Acknowledgments

The authors would like to thank the National Science Council of the Republic of China, Taiwan, for financially supporting this research under Contract No. NSC 96-2221-E-451-009-MY3. Corresponding author: Chi Ching Chang’s e-mail address is chichang@mdu.edu.tw.

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