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
1. H. Z. Jin, H. Wang, Y. P. Zhang, Y. Li, and P. Z. Qiu, “The influence of structural parameters of CCD on the reconstruction image of digital holograms,” J. Mod. Opt. 55(18), 2989–3000 (2008). [CrossRef]
]. 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
2. T. M. Kreis and W. P. P. Jüptner, “Suppression of the dc term in digital holography,” Opt. Eng. 36(8), 2357–2360 (1997). [CrossRef]
], subtracting the numerical generated intensity of the object and reference waves [3
3. G. L. Chen, C. Y. Lin, M. K. Kuo, and C. C. Chang, “Numerical suppression of zero-order image in digital holography,” Opt. Express 15(14), 8851–8856 (2007). [CrossRef] [PubMed]
] and numerical space-shifting reconstruction [4
4. Y. C. Dong and J. Wu, “Space-shifting digital holography with dc term removal,” Opt. Lett. 35(8), 1287–1289 (2010). [CrossRef] [PubMed]
], can partially or completely suppress the zero-order diffraction but maintain the conjugate image intact. Digital filtering techniques in frequency domain [5
5. E. Cuche, P. Marquet, and C. Deperursinge, “Spatial filtering for zero-order and twin-image elimination in digital off-axis holography,” Appl. Opt. 39(23), 4070–4075 (2000). [CrossRef] [PubMed]
] and in space domain [6
6. L. H. Ma, H. Wang, Y. Li, and H. J. Zhang, “Elimination of zero-order diffraction and conjugate image in off-axis digital holography,” J. Mod. Opt. 56(21), 2377–2383 (2009). [CrossRef]
], as well as wavelet transform [7
7. J. W. Weng, J. G. Zhong, and C. Y. Hu, “Digital reconstruction based on angular spectrum diffraction with the ridge of wavelet transform in holographic phase-contrast microscopy,” Opt. Express 16(26), 21971–21981 (2008). [CrossRef] [PubMed]
], 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
8. I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. 22(16), 1268–1270 (1997). [CrossRef] [PubMed]
10. T. Nomura and M. Imbe, “Single-exposure phase-shifting digital holography using a random-phase reference wave,” Opt. Lett. 35(13), 2281–2283 (2010). [CrossRef] [PubMed]
] 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
Fig. 1 A typical digital in-line holographic setup.
. Under the condition of Fresnel approximation, the spherical reference wave on the plane of the hologram can be written as
Where M and N are the horizontal and vertical pixel numbers of the CCD camera, △ x
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
shows that IC(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
Fig. 2 The schematic diagram of the four regions and new coordinate system
. 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
The sub-complex-holograms can be expressed in new sub-coordinate systems
denote the object wave and the conjugate object wave in each region respectively.
The spatial frequency spectrum distribution of each sub-complex-hologram will be discussed. Assume that
, where F denotes
Fourier transform operation. By utilizing the property of the Fourier transform, the spatial frequency spectrums of
can be calculated
Then the Fourier transform of Eq. (8)
can be performed
Where * denotes convolution operation. According to Eq. (10)
, the frequency spectrums of object wave and conjugate object wave in each region are modulated to symmetrical high-frequency domain, as shown in Fig. 3
Fig. 3 (a)-(d) are the schematic maps for the frequency spectrum distribution in each region.
. In order to separate completely the twin images, the highest spatial frequency of the object wave
must not exceed a maximum value
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
Where max() is to take the larger value of these two values. By substituting Eq. (12)
in Eq. (11)
and assuming that
for simplicity of the notation, we can obtain
Therefore, the maximum space-bandwidth product of the object wave which can be recorded by the proposed technique can be calculated
denotes the effective space-bandwidth product of CCD camera.
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 . 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 . 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 .
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 . 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,
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.
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
Fig. 4 A schematic diagram of the implementation of partition calculation.
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
Fig. 5 Image-plane in-line holography geometry.
, 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)
Fig. 6 (a) digital in-line hologram, (b) the frequency spectrum of the hologram, (c) the frequency spectrum without zero-order term.
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
Fig. 7 Sub-complex-hologram in each region
. Figure 8
Frequency spectrum in each region corresponding with Fig. 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
Fig. 9 Extracted frequency spectrum of conjugate object wave in each region.
. The frequency spectrums extracted from the four regions are superposed and Fig. 10
Fig. 10 The whole frequency spectrum
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
Fig. 11 The reconstructed phase image
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
Fig. 12 (a) the reconstructed image with the presence of the zero-order term and the conjugate image, (b) the reconstructed image with removing the zero-order term, (c) the reconstructed image with removing the conjugate image.
. 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
, 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
. 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
. 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.