1. Introduction

Gabor developed a novel two-step, lens-less imaging process, originally called wavefront reconstruction, commonly referred to as in-line holography [1]. Although holography is ideal for recording three-dimensional information about objects, its usefulness is limited by conjugate and zero-order images. In in-line holography, the reconstructed image inevitably suffers from the presence of the zero-order image and its conjugate, which can overlap with the reconstructed image and cause blurring. This problem was solved when Leith and Upatnieks [2] presented the off-axis reference beam technique. However, their technique prevents the effective application of CCD, because the object is reconstructed from only a fraction of the complete hologram information. Yamaguchi and Zhang [3] proposes a solution that adopts multiple recordings with phase shifts of 0, π/2, π and 3π/2 to remove undesired terms from a digital hologram without sacrificing any active areas of a CCD. However, a phase-shifting digital holographic system depends on the recording of digital holograms on a vibration-free optical table. This method is applicable only in a laboratory and cannot be applied on moving targets during the recording procedure. Takaki [4], Cuche [5] and Zhang [6] proposed and discussed several method for suppressing the zero-order image and twin-images but the practical application is not satisfactory, constrain considered the using of the phase-shifting digital holographic based scheme and off-axis scheme. Poon [7] presented the optical scanning holography, but the additional element and employed technique would raise the complication of application. However Kreis et al. [8, 9] recently developed single-exposure in-line digital holography, which suppresses the dc term, representing the contribution of the reference wave to the zero-order, by subtracting of the average intensity of the hologram. However, this simple operation maintains the twin images intact, and only partially suppresses the zero-order image if the object wave intensity is not uniform [9].

This work proposes a novel holographic reconstruction approach to implement the reconstruction that adopts a digital hologram and suppress the major blurring that is caused by zero-order images. From the acquired digital hologram and known recording conditions the intensities of the object and reference waves can be generated in a personal computer and operated in suppression. The major advantage is that the generated intensities exactly conform to the zero-order images of reconstruction even if the distribution of the object wave is not uniform. Additionally, the generation is performing in numerical operation only. Therefore this novel approach can be accomplished by using the generated intensities to suppress the major blurring in numerical reconstruction of the single digital hologram. Moreover this proposed approach can solve these constraints on the performance of single-exposure digital holography and simplify the procedure of phase-shifting digital holography.

2. Theoretical derivation

Holographic recording adds, to each unknown wave, a second wave, which is coherent with the known wave and has a known amplitude and phase. The intensity of the sum of the two complex fields then depends on both the amplitude and the phase of the unknown field. Accordingly, if ψ ₀ denotes the object wave to be detected and reconstructed, and ψ _R denotes the reference wave with which ψ ₀ interferes, then the intensity of the sum is denotes as [10],

The first two terms on the right-hand side of Eq. (1) must be eliminated to suppress the zero-order image, where |ψ ₀|² denotes the intensity of the object wave, and |ψ _R|² denotes the intensity of the reference wave. The reference wave may be a plane wave for convenient reconstruction, and is known as the amplitude and phase. Therefore subtracting |ψ _R|² from the hologram in Eq. (1) and squaring the result yields,

Since (ψ ₀ ψ _R ^*)² and (ψ ₀ ^* ψ _R)² are mutually conjugate, the imaginary part of their complex amplitudes are canceled out and should be hidden. Therefore, for convenient operation, both two terms, (ψ ₀ ψ _R ^*)² and (ψ ₀ ^* ψ _R)², can be substituted approximately for |ψ ₀ ψ _R ^*|². Equation (2) can then be rewritten as

In practical cases, I _H+|ψ _R|² exceed |ψ ₀|²/2, the influence of | y/0| /2 can be neglected. Finally, the intensity of the object wave can be rewritten as the follow relation.

The above expression, Eq. (5), can be adopted to suppress the zero-order image in numerical reconstruction, where |ψ _R|² can be numerically generated by a computer. In the right-hand side of Eq. (5), all terms are real number forms, and the other two terms can be subtracted from the digital hologram I _H to suppress the zero-order images. However, the left-hand side of Eq. (5) has two mutually conjugate terms. The imaginary part of the complex amplitude that describes two terms of left-hand side of Eq. (5) is canceled out, and only the real part appears. Therefore the operation that subtracts the intensities of object and reference wave from the digital hologram can be accomplishing for describing the object wave and its conjugation term in hologram plane. ψ ₀ ψ _R ^*+ψ ₀ ^* ψ _R exactly describe the object term and its conjugate, that is, perform the numerical reconstruction without the blurring of zero-order image. But, in recording procedure the intensity ratio of the object wave and reference wave thus is significant and can be exactly used to generate the intensity of reference wave. Moreover, the original object image in the object plane can be obtained by performing the numerical reconstruction. In later work we adopt an off-axis holographic scheme to demonstrate the results for avoiding the confusion of suppression in in-line holography.

3. Simulation

In the simulation procedure, the intensity ratio of object wave to reference wave is assumed to be 1:1. The input image which denotes as ψ _{0 Obj} and shown in Fig. 1(a) is set at a distance z=28.60cm away from the hologram plane, at a wavelength of 632.8nm. Figure 1(b) shows a typical example of a simulated digital hologram. This hologram is generated by interference of the reference wave ψ _{0 R} and object wave ψ _{0 O}. The object wave is obtained by performing the propagation of input image as Eq. (6) [11], where k denotes the wave number, (x, y) and (ζ, η) denote the coordinate of object plane and hologram plane, respectively. A plane wave is used as the reference wave, described in Eq. (7) [12]. Where a_r denotes the amplitude a_r=1 and kx, ky denotes the two components of the wave vector k_x=338cm^-1 and k_y=324cm^-1.

 

Fig. 1. Simulation results. (a) inputted image; (b) digital hologram; (c) reconstructed image; (d) Fourier power spectrum of (c); (e) reconstructed image obtained by novel suppression approach, (f) Fourier power spectrum of (e).

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Figure 1(c) presents the reconstructed image obtained by performing the numerical propagation of z=28.60cm with the digital hologram, as Eq. (8) [13]. This figure indicates that the different terms of reconstruction produce a well-separated distribution. However, the reconstructed image is mostly superposed on the zero-order image, and thus cannot be resolved. Hence, the useful information about the object cannot be obtained if the zero-order image is not suppressed.

Figure 1(d) shows the calculated two-dimensional Fourier spectrum of the reconstructed image Fig. 1(c). Meanwhile Fig. 1(d) is produced by operating the fast Fourier transform of reconstructed wave from Eq. (8) [5, 6]. The image of spectrum exactly indicated the similar well-separation and appeared the distribution of power. Figure 1(e) shows the reconstruct image that obtained by performing the same reconstruction operation as Eq. (8) after the numerical suppressing process by using Eq. (5), that is, accomplishing the numerical reconstruction which ψ ₀ ψ _R+ψ ₀ ^* ψ _R is substituted for I _H. The figure obviously indicates that the object image and its conjugate image can be resolved effectively. Additionally, the spectrum in Fig. 1(f) exhibits only two similar power distributions. Only the object image and its conjugate are therefore inferred to exist.

4. Experimental results and analysis

The experimental setup shown in Fig. 2 is a Mach-Zehnder setup. An He-Ne laser with a power of 20mW and a wavelength of 632.8nm used as a light source. Two λ/2 retarded wave-plates and a polarized beam splitter are adopted to adjust the intensity ratio of the object wave to reference wave. Both the two waves are collimated as plane waves by a spatial filter and a lens. The angle between the object wave and reference wave is 0.2705°. Transparent characters ‘中正’ of size 1mm×2mm, written on a photographic slide, are used as the object. The hologram is recorded using a CCD sensor (Pixera-150SS CCD camera, 1392×1040 pixels, 0.650cm×0.484cm). The distance z is 28.60cm between the CCD and object, and the intensity ratio of the object wave to reference wave is 1:1. This novel approach can be applied to the in-line and off-axis digital holography. The experimental results are referred to both in-line and off-axis holograms, that present the acquired digital holograms in Figs. 3(a) and 3(c).

 

Fig. 2. Schematic diagram of the optical setup. SF: spatial filter; PBS: polarized beam splitter; BS: beam splitter.

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Fig. 3. Experimental results. (a), (c) holograms acquired by off-axis and in-line digital holographic scheme respectively; (b), (d) reconstructed images obtained by the novel suppression approach and numerical reconstruction from (a), (c).

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Figures 3(b) and 3(d) show the reconstructed images obtained by performing the novel suppression approach and numerical reconstruction of the acquired digital holograms. Meanwhile, the intensity of reference wave is generated by multiplying the proportion of intensity ratio with an operation array. The intensity of object wave gets from the operation of Eq. (4). Therefore the zero-order image can be completely suppressed numerically by subtracting the generated intensities from the hologram. Also the experimental results are consistent with the simulation. It demonstrates the feasibility in practical applying. In particular, the noise in the optically generated reference wave must be controlled during the recording process. The intensity of reference wave in proportion to the hologram needs to be known during the suppressing process. Moreover, experimental results indicate that the proposed suppression approach is suitable for use in the general recording condition that the intensity of the reference wave is larger than that of the object wave. Because the neglected term of Eq. (3) would affect the result of suppression, since the intensity of the object wave is larger than the other.

Finally, the performance of the zero-order image suppression relates to vary with intensity ratio in recording procedure is simulated. Figure 4 plots the proportion of the Fourier power spectrum of reconstructed zero-order image to that of the object term. This figure demonstrates that the proportion of residual zero-order image never exceeds 1.2%. It reveals that the performance of this approach is satisfactory. The arrow in Fig. 4 indicates the reconstructed image which the zero-order image is suppressed successfully even under critical conditions. Accordingly, the reconstructed object images can be resolved clearly.

 

Fig. 4. The proportion of the Fourier power spectrum of reconstructed zero-order image to object term.

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5. Conclusion

In conclusion, this study presents an effective and simple approach to suppress the zero-order image from a digital hologram. The advantage of this approach is that the experimental procedure and suppression operation are simple. Additionally, the experimental results indicate that the performance of the novel approach is satisfactory, that is, the numerical suppression of zero-order image is effective. Since the zero-order image is filtered, both the contrast and the quality of the reconstructed image are improved. In the future, this approach will be adopted to reconstruct the profile of the phase object thus clearly showing the phase information.