## Abstract

A novel numerical algorithm is proposed to reconstruct a complex object from two Gabor in-line holograms. With this algorithm, both the real and imaginary parts of the complex amplitude of the wave front in the object field can be retrieved, and the “twin-image” noise is eliminated at the same time. Therefore, the complex refractive index of the object can be obtained without disturbance. Digital simulations are given to prove the effectiveness of this algorithm. Some practical experimental conditions are investigated by use of error estimation.

© 2003 Optical Society of America

## 1. Introduction

X-ray holography is a powerful technique for high-resolution imaging of biological, material, mineral, and fossil specimens. It has many advantages over other microscopy imaging techniques, such as optical microscopy and electron microscopy [1–3]. The resolution of x-ray holography is much higher than that of optical microscopy because the wavelength of x rays is much shorter than that of visible light. It could be several tens of nanometers at present. The penetrating length of x rays is much longer than that of an electron beam, which means that a thicker specimen can be observed by x rays. Soft-x-ray holography has been successfully used to reconstruct many kinds of small biological samples with high resolution [4]. Hard x rays can be used to penetrate a bulk sample such as a piece of fossil or ore several millimeters in size. However, sometimes it is difficult to obtain a high-contrast x-ray transmission image because the difference in the absorption coefficients of the different elements inside the sample is quite small. Thus phase-contrast imaging has attracted more attention in recent years [5–7]. Phase contrast can be much more sensitive than absorption contrast; therefore, by use of phase contrast a much higher signal-to-noise ratio and decreased radiation dose can be reached at the same time. Holography is highly suitable for phase-contrast imaging, because the complex amplitude of the original object wave front can be reconstructed with the digital holography technique. Then both the absorption and the phase shift caused by the object can be retrieved from the complex amplitude, and thus both phase-contrast and absorption-contrast images can be obtained.

Gabor in-line holography geometry has been adopted in most x-ray holographic experiments. This is because there is a lack of x-ray sources with adequate coherence and high-quality x-ray optical elements. The setup of in-line holography is quite simple: No optical elements are needed. However, there is a well-known disadvantage with Gabor in-line holography—twin-image noise. There are many kinds of digital techniques that can eliminate or diminish this noise, such as the iterative phase-retrieval algorithm [8,9] and the digital filter algorithm [10]. However, all these algorithms make the same assumption that the object should be purely absorptive, which means that the complex refractive index of the object must be real-valued. This requirement negates the advantage of x-ray holography for phase-contrast imaging. Therefore another type of reconstruction technique that can be used to deal with the complex object was developed. This technique is based on the use of two holograms that are recorded at different distances. In fact, one hologram means one known condition. The object that needs to be reconstructed is the unknown one. If the object is real-valued, then, theoretically, there is only one unknown that can be resolved by one hologram. For the complex-valued object, there are two unknowns: the real part and the imaginary part. So there are not enough known data from one hologram to resolve the two unknowns, and two holograms are necessary in this situation. The reconstruction from two holograms can be realized by different algorithms. Maleki and Devaney [11] have proposed a deconvolution algorithm that can recover a complex-valued object. However, this algorithm has to deal with the singularity problem, and the calculation is somewhat complex. Xiao *et al.* proposed another algorithm to eliminate the twin-image noise by subtracting a hologram recorded in the double distance from the reconstruction of the hologram recorded in the single distance [12]. This algorithm is relatively simple; however, only the real part of the complex object is restored. In the current paper we present a totally new algorithm also based on the two-hologram technique. Compared with the previous algorithms, our algorithm can reconstruct the complete complex amplitude of the original object wave front, both the real and imaginary parts, without the disturbance of twin-image noise, by a simple, noniterative technique. Digital simulations are given in this paper, which show the good reconstruction results and prove the effectiveness of our algorithm. Although our main focus is on high-resolution x-ray holography, this algorithm is also relevant for visible-light holography and opens up new possibilities for the reconstruction of in-line holograms recorded from phase objects. In this way, the lateral resolution of the reconstructed image could be enhanced in comparison with digital off-axis recording techniques, which are still restricted by the limited resolution of current commercially available CCD detectors [13].

## 2. Theory

Let us consider the formation of an in-line hologram. If the object in plane (*X,Y*) has a transmittance of *t*(*x,y*) and phase shift ϕ(*x,y*), the complex amplitude of the object wave front Ũ(*x,y*) becomes

$$={\mathit{\u0168}}_{0}(x,y)[1-a(x,y)]{e}^{i\varphi (x,y)},$$

where Ũ_{0}(*x,y*) is the incident wave front and *a*(*x,y*) is the pure absorption of the object. Let us assume that Ũ_{0}(*x,y*)=1, which indicates a plane-wave incident condition, and assume that *a*(*x,y*)≪1, | φ(*x,y*)|≪1. These assumptions can be easily satisfied because the complex refractive indices of all elements in high-energy x-ray regions are always near to 1. If the size of the object is not too large (less than 10 mm) and the object does not contain too many heavy atoms, the absorption and phase shift will be quite small. Then we have

$$=1-a(x,y)+i\varphi (x,y)-ia(x,y)\varphi (x,y)$$

$$\approx 1-\mathit{\xe3}(x,y).$$

We will neglect the high-order term of *a*(*x,y*)ϕ(*x,y*) in Eq. (2) and in the following equations. Here we define a complex absorption factor *ã*(*x,y*)=*a*(*x,y*)-*i*φ(*x,y*). Thus Re[*ã*(*x,y*)]=*a*(*x,y*) is the pure absorption and -Im[*ã*(*x,y*)]=φ(*x,y*) is the phase shift.

The complex amplitude distribution of the wave front in the hologram plane can be calculated by the Fresnel–Kichhoff diffraction integral,

where λ is the wavelength of the x ray and *z* is the distance between the object and the hologram. The exponential term exp[*i2*π*z*/λ] is a constant in a given (*x,y*) plane, and it will only add a constant phase shift to each pixel; thus we will not take this term into account in the following discussion. Now we define a convolution kernel function *h _{z}*(

*x,y*) as

then we have

where * represents the convolution operation. With Eqs. (2) and (5), the intensity distribution in the hologram plane can be written as

$$\approx 1-{\mathit{\xe3}}^{\ast}(x,y)\ast {h}_{z}^{\ast}(x,y)-\mathit{\xe3}(x,y)\ast {h}_{z}(x,y).$$

If we record two holograms in the distances of *z* and *2z*, we will have two intensity distributions *I _{1}*(

*x,y*) and

*I*(

_{2}*x,y*):

From these two equations and the relations of *h _{a}*(

*x,y*) *

*h*(

_{b}*x,y*)=

*h*(

_{a+b}*x,y*),

*h**

*(*

_{z}*x,y*)=-

*h*(

_{z}*x,y*),

*h*

_{0}(

*x,y*)=δ(

*x,y*) [10], we can obtain the following result,

Therefore we can obtain the real part of the complex absorption *ã*(*x,y*) as

Next, we let *ã*=*a _{r}*+

*ia*, and from Eq. (6) we have

_{i}Now we define *I*′(*x,y*) as

$$={a}_{i}\ast {h}_{z}^{\ast}-{a}_{i}\ast {h}_{z}.$$

Since *I _{1}*(

*x,y*) and

*a*(

_{r}*x,y*) are all known to us, then

*I*′(

*x,y*) becomes known as well. Then we can obtain

*a*(

_{i}*x,y*) from

*I*′(

*x,y*). In fact, from Eq. (11) we can obtain the following relations:

$$I\text{'}\ast {h}_{3z}={a}_{i}\ast {h}_{2z}-{a}_{i}\ast {h}_{4z},$$

$$I\text{'}\ast {h}_{5z}={a}_{i}\ast {h}_{4z}-{a}_{i}\ast {h}_{6z},$$

If we add up all the equations, we can derive

The final term in this equation is a residual error. It looks like a hologram with a recording distance of *2nz*. If this distance is large enough, the effect from errors is small and can be neglected. Therefore, we can obtain the distribution of *a _{i}* as

## 3. Simulations

We tested our new algorithm by computer simulation. First, we consider a simple one-dimensional sample to demonstrate our algorithm. The sample is an annular object with an internal diameter of 0.5 µm and an external diameter of 1.5 µm. The parameters in the simulation are as follows: λ=1 nm; *z*=1 mm; discrete sampling number *N*, 256; and total view field width, 16 µm. The absorption of the object is 10%, and the phase shift varies from 0 to 0.07, which is shown in Fig. 1 as curves (a) and (d). The reconstructions using our algorithm and an optical algorithm are shown in Fig. 1, from which we can see that the absorption and the phase shift are clearly retrieved by our algorithm.

Next, we study some factors that might affect the reconstruction’s quality. These factors include parameter of *M* in Eq. (14) and the position error of *z* in Eq. (3). In addition, because our algorithm is based on the weak absorption and weak phase-shift assumption according to Eq. (2), we study the reconstruction’s error as a function of the magnitude of the absorption and phase shift as well. Thus we define the rms absorption and phase errors as

$${E}_{p}={\sum}_{x}\frac{{\left\{\mathrm{Im}\left[\mathit{\xe3}\left(x\right)\right]-\varphi \left(x\right)\right\}}^{2}}{N}.$$

The value of *M* will affect the result of phase shift *E _{p}* only. The relations of

*M-E*with different

_{p}*N*are shown in Fig. 2(a). From these curves in Fig. 2(a) we can find that the minimum

*E*appear at different

_{p}*M*values for different

*N*; the optimized value of

*M*is approximately 17 or 45 for

*N*equal to 256 or 512. For large

*N*value, optimized

*M*value also becomes larger. Figures 2(b) and 2(c) are the

*E*and

_{a}*E*with different absorption and phase-shift magnitudes, from which we can find that the errors increase greatly when the magnitude of absorption or phase shift increases. Thus this algorithm is valid only for weak absorption and weak phase-shift situations, as we assumed before in Eq. (2).

_{p}In practical experiments, the recording distances of the two holograms must have some errors compared with the ideal value. These errors include distance error Δ*z* and lateral displacement error Δ*x*. The effect of these errors can be studied by digital simulations. First, we keep the recording distance *z* of the first hologram for 1 mm and vary the recording distance of the second hologram to see the effect caused by distance error. The distance of 2*z* is varied from 1.7 to 2.3 mm. The reconstructions are calculated as the function of 2*z*, shown in Fig. 3. The rms errors of the reconstruction are shown in Fig. 4(a). If we want both the rms absorption error and phase error to be less than 1×10^{-4}, the distance of the second hologram should be in the range of 1.92–2.08 mm, which means a position accuracy of ±4%. Figure 4(b) shows the rms errors as the function of lateral displacement Δ*x*, from which we can find that the lateral displacement has a magnificent effect on the reconstructions. The spacing interval of the plotted data in Fig. 4(b) is 1 pixel. Therefore, a lateral position accuracy of ±3 pixels or ±0.2 µm is need to ensure that the rms error is less than 1×10^{-3}.

Next, we test our algorithm with a relatively complicated two-dimensional object, shown in Fig. 5(a). It is an elliptical cell-like object with some inner structures. The absorption and phase shift are proportional to the density of the object. The maximum absorption and phase shift are 10% and 0.1 rad, respectively. There are two small squares in the upper-right and lower-left corners. These two squares act as pure phase objects and do not have any absorption in the formation of the two holograms. The phase shifts that they generate are 0.1 and 0.05 rad, respectively. Figures 5(b) and 5(c) are the holograms in the two different distances.

Figure 5(d) is the reconstruction of the real part of the complex amplitude in the object plane, corresponding to the absorption of the object. The object is clearly reconstructed, and the twin-image noise effect can hardly be seen in this image. The contrast of Fig. 5(d) is reversed compared with Fig. 5(a). This is because the gray level in Fig. 5(a) represents the transmissivity, not the absorption. The two squares can hardly be seen in Fig. 5(d), which is in agreement with our expectations, because the squares are pure phase objects and should not appear in the absorption image. Figure 5(e) shows the imaginary part of the object plane with the parameter *M*=45. Not only the cell-like object but also the two squares are clearly reconstructed in this figure.

## 4. Conclusion

In this paper we have proposed a new digital-holography reconstruction algorithm based on two holograms with which an object with complex refractive index can be reconstructed without the twin-image noise. This algorithm is noniterative and therefore does not have the problem of convergence that exists in the iterative phase-retrieval algorithm, and the time consumption is quite small as well. Theoretical analysis and simulation results show that it is an effective means to reconstruct the complex object and eliminate the twin-image noise as well. This algorithm is especially applicable for phase-type objects in x-ray holography, because better contrast and lower dosage can be achieved by this method compared with the absorption-contrast imaging technique.

## Acknowledgment

This research was supported under project 10105002 by the National Natural Science Foundation of China.

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