Off-axis digital holography generally uses a 2D-FFT based spatial filtering method to extract the complex object wave from an off-axis hologram. In this paper, we describe a novel single exposure complex object wave extraction method which can provide a faster solution than the FFT based spatial filtering approach while maintaining the reconstructed phase image quality. And also, we show that the proposed direct filtering scheme can provide more robust filtering capability to the off-axis spatial carrier frequency variation than the spatial filtering method.
©2013 Optical Society of America
Digital holography, which captures a hologram by using a CCD camera and reconstructs an object image numerically, has been researched extensively and applied in various fields such as phase-contrast imaging, polarization imaging, object recognition and surface form measurement [1–10]. As in the classical in-line holography configuration originally proposed by Gabor , a digitally reconstructed image from in-line digital holography also contains undesired zero-diffraction order and conjugate object twin image. Eliminating those undesired terms has been the main issue in in-line digital holography. In order to eliminate these terms accurately from the digital hologram, a phase shifting approach has been proposed . Phase shifting technique usually requires multiple recordings of phase shifted holograms. Previous studies on the in-line scheme claim that at least two holograms have to be measured to subtract the conjugate image from the hologram even when the zero order diffraction term can be removed by using the averaging technique . Also, as the off-axis geometry invented by Leith and Upatnieck gave a solution that could overcome the shortcomings of Gabor’s in-line scheme in practicability, the novel concept on the digital reference wave proposed by Cuche et al  opened various practical application fields in digital holography. Since the Fourier-transform based phase measurement method was proposed by Takeda et al , it has been one of the most useful techniques for calculating phase functions from interferogram data in the spatial [5, 7–10] and spectral domain [12, 13]. In the phase contrast off-axis digital holography, the zero order of diffraction and the twin object images can be separated spatially in the reconstructed plane just by performing the Fresnel transform. However, such approach prohibits the use of full pixels of the CCD sensor. For enhancing the reconstructed image quality of the off-axis digital holography while maintaining the benefit of single exposure scheme, the spatial filtering method processed in the Fourier transformed spatial frequency domain has been suggested although it has some additional computational load . The 2-dimensional fast Fourier transform (2D-FFT) based spatial filtering method has been regarded as the most general approach for various applications employing the off-axis digital holographic scheme. Recently, there were some attempts on direct complex object wave extraction methods based on complex wave retrieval algorithm  and direct coarse sampling theory  to overcome the disadvantages of the FFT-based approach since the FFT has its inherent drawbacks in terms of reconstructed image quality degradation due to energy leakage and limited reconstruction frame rate. The complex wave retrieval algorithm  can provide a useful alternative approach comparable with the FFT based spatial filtering method. In accordance to be addressed in that paper, however, the filtering speed seems to be at the similar level to the spatial filtering approach. And also, the direct coarse sampling algorithm  has some disadvantages in the sense that it needs to measure the DC term separately and subtract them from the off-axis hologram to apply the coarse sampling theory based on the carrier frequency image signal, which means it is a multiple exposure scheme.
In this paper, we describe a novel single exposure complex object wave extraction method which can provide a faster solution than the FFT based spatial filtering approach while maintaining the reconstructed phase image quality. Such complex object wave filtering speed enhancement arises from the fact that the proposed method can filter out only the complex object wave in the hologram plane directly without using any 2D-FFT calculation. And also, we show additional benefit of the proposed direct filtering method in term of more precise filtering capability.
2. Complex object wave direct filtering theory
The optical schematic of the off-axis digital holographic system used in this study is based on Mach-Zehnder interferometic scheme as depicted in Fig. 1 . A CCD camera records the hologram that results from the interference between the object wave O and the reference wave R. To obtain a proper off-axis hologram, the orientation of the reference mirror that reflects the reference wave is set such that the reference wave reaches the CCD with an incidence angle while the object wave propagates perpendicularly to the hologram plane 0xy.
In the hologram plane, the interference between the object wave and the reference wave creates the following hologram.Figure 2 shows the general procedure that is needed for object reconstruction in phase contrast off-axis digital holography [9, 10]. In order to compare theoretically the proposed complex object wave direct filtering method with the conventional spatial filtering we start the numerical reconstruction process with a computer generated off-axis hologram shown in Fig. 2(a). For this, we generate a virtual 100nm depth phase object written as ‘Opt’. As depicted in Fig. 2(a) through 2(e), we have to apply two times 2D-FFTs to perform the spatial filtering.
As the first step, we apply the 2D-FFT to transform the off-axis hologram to the 2-dimensional spatial frequency domain. (Fig. 2(b)) Then, we can extract the complex object wave O represented in the hologram plane 0xy as depicted in Fig. 2(d) and 2(e) by doing the inverse 2D-FFT after filtering out the undesired two terms. After the spatial filtering step, we multiply the object wave O in the hologram plane by the digital reference wave RD which is defined as follows [9, 10]:Equation (3) and Eq. (4) represent Single Fourier Transform Formulation (SFTF) and Convolution Formulation (CF), respectively .
Figure 2(g) and 2(h) represent the reconstructed amplitude and phase image in the observation plane obtained by using Eq. (4). The reconstructed 2-dimensional wave front map is an array of complex numbers. The phase image can be obtained by using the argument tan−1[Im(Ψ)/ReΨ)].Figure 3 depicts the numerical reconstruction procedure based on the proposed direct filtering approach. In contrast to the spatial filtering method, note that the off-axis hologram should be modified to apply the direct filtering method for extracting the complex object wave in the hologram plane as represented in Eq. (5). Since we can assume the reference wave is not changed once the optical system is set, and it can be measured and saved in advance, the following modified off-axis hologram IHM(k,l) can be regarded as a single acquisition hologram.
The main idea of this paper is based on the fact that the off-axis digital hologram can be regarded as an interferogram with a spatial carrier frequency. In temporal phase shifting based interferometry, the phase difference between the object and the reference wave φ can be obtained by Eq. (7) which is based on the 5 consecutive phase shifted interferogram intensities described in Eq. (6) .17].Figure 3(a) demonstrates in detail how the complex object wave in the hologram plane can be extracted. Let us assume that the off-axis hologram IH has N by N hologram size. Then, we can define column vectors from I1 to IN that satisfy IH = [I1 I2 I3 …. IN-1 IN]. As mentioned, we can subtract the reference wave intensity term from the original hologram IH as depicted in Fig. 3(a) and we use the modified hologram for the proposed direct calculation scheme. The modified 5 consecutive hologram column vectors described in Eq. (9) can be derived by replacing α in Eq. (6) with since the spatial phase shift interval caused by one CCD pixel size Δy is defined as in the digital reference wave RD described in Eq. (2), and by replacing the 5 consecutive scalar interferogram intensities from I1 to I5 in Eq. (2) with the kth row hologram intensity column vectors from to . The object and reference wave scalar amplitudes and can be replaced by the column vector and representing the kth column vector of the object and the reference amplitudes and defined in Eq. (5).Fig. 2, the direct filtering needs no 2D-FFTs and we can extract the direct filtered complex object wave ODF in the hologram plane 0xy as shown in Fig. 3(b) and 3(c) by using the following direct calculation algorithm which can be derived by using the modified 5 consecutive hologram column vectors defined in Eq. (9).Eq. (10). Since by the definition of the kth row visibility column vector γk, Eq. (11) can be derived easily by referring to Eq. (8). Ak can be calculated for each integer k ranging from 3 to N-2 once the phase column vector Φk is obtained. Here, Bk is a known column vector since the reference amplitude image B can be measured in advance. We can obtain the direct filtered complex object wave in the hologram plane which has finally (N-4) by N image size by sweeping the integer k that can vary from 3 to N-2 since each direct filtered column vector means . However, the direct filtered phase Φk in Eq. (10) contains the reference wave phase term . Therefore, we can obtain the correct complex object wave in the hologram plane by multiplying the digital reference wave RD depicted in Fig. 3(d) with the direct filtered complex object wave . Finally, we obtain the reconstructed wave front Ψ(m, n) in the same way as depicted in Fig. 3(e) and 3(f) by using the convolution formulation described in Eq. (4). Throughout the entire simulation process described in Fig. 2 and Fig. 3, the CCD pixel number is 1024 by 1024, the CCD pixel size Δx = Δy = 6.4μm, the laser wavelength λ = 635nm, the distance between the object and the CCD camera d is set to be 120mm. And also, the spatial directional vector kx = 2.0 × 0.00780 and ky = 2.0 × 0.00915. It should be noted that as in the 2D FFT based spatial filtering approach we assume the object wave spatial variation needs to be changed slowly over the applied spatial carrier frequency to use the proposed direct filtering scheme correctly.
Figure 4 depicts the reconstructed object phase line profiles corresponding to the central vertical line of the computer generated object ‘Opt’. As can be seen in Fig. 4, the reconstructed phase image quality based the proposed direct filtering approach is almost same as that obtained by using the spatial filtering scheme. Both results indicate that the reconstructed object height is around 100nm as we designed and they have similar fluctuation patterns in the edge area. Such fluctuation pattern comes from the used Fresnel transform process which employs a Fourier transform algorithm. Figure 4 shows that the direct filtering works correctly and can provide the comparable filtering capability with the spatial filtering approach.
3. Analysis on filtering capability
In order to clarify the advantages of the proposed direct filtering approach, we have additionally conducted some analysis between the two approaches in terms of calculation speed and filtering robustness to the spatial carrier frequency variation.
Figure 5 shows comparison results on the calculation time between the two approaches, which are obtained by varying the number of hologram image size N by N used for each process. In this study, we use an Intel Core2 CPU 6600 processor (2.4GHz) and Matlab 7.3 for the entire signal processing. Figure 5(a) represents the computation time required to obtain the object wave O in the hologram plane. The solid line indicates the spatial filtering result and the dotted result represents when the direct filtering method is applied. Figure 5(b) shows the direct filtering can calculate the object wave O in the hologram plane around one and half to two times faster compared with the spatial filtering approach. For the 2D FFT calculation of a N by N image, we need O(N2log2N) calculations . Since the direct filtering method is based on 2-dimensional array matrix calculation, the calculation number of the proposed direct filtering method can be represented as O(αN2). Here, α is a constant value related to the complexity of the 2D array matrix calculation. Therefore, the calculation time ratio between the two approaches can be O(αlog2N), which means the calculation time ratio versus the one axis hologram size N becomes a function of αlog2N as obtained in Fig. 5(b).
Secondly, we conducted the reconstructed phase image spatial resolution analysis by varying the off-axis carrier frequency kx and ky. For this task, we use a new computer generated USAF target phase object as illustrated in Fig. 6(a) . Here, we set the amplitude of the USAF target object to be 1. Figure 6(b) is the computer generated off-axis hologram obtained by the USAF target patterned phase object. Here, the computer generated USAF target has 900 by 900 image size. We set Δx = Δy = 6.4μm, λ = 635nm, d = 120mm.
Figure 7(a) and 7(d) represent the reconstructed amplitude and phase image, respectively which are obtained by using the UASF target hologram in Fig. 6(b) when no filtering scheme is applied. We can see that the object wave is not reconstructed correctly when no filtering scheme is used. Figure 7(b) and 7(e) correspond to the reconstructed amplitude and phase images, respectively when the spatial filtering is applied. Lastly, Fig. 7(c) and 7(f) shows the reconstructed amplitude and phase images obtained by using the proposed direct filtering approach. Here, we use the Convolution Formulation (Eq. (4) for reconstructing the object amplitude and phase images. As can be shown in Fig. 7, the proposed direct filtering approach works well for the object with more complicated pattern structures. In addition to the calculation speed enhancement, we have found that the direct filtering method can have additional benefit in filtering process in the sense that it can provide more robust results to the variation of the spatial carrier frequency as depicted in Fig. 8 . Figure 8(a) depicts the spatial frequency spectrum when kx = 1.5 × 0.00780 mm−1 and ky = 1.5 × 0.00915 mm−1. Figure 8(b) shows the results when kx = 2.0 × 0.00780 mm−1 and ky = 2.0 × 0.00915 mm−1. Likewise, Fig. 8(c) is when kx = 3.0 × 0.00780 mm−1 and ky = 3.0 × 0.00915 mm−1. Figure 8(d)-8(f) represent the zoomed reconstructed phase images of the dotted inner rectangular box illustrated in Fig. 7(e), which are obtained as we vary the off-axis carrier frequency kx and ky as shown in Fig. 8(a)-8(c). In the same way, Fig. 8(g)-8(i) correspond to the zoomed reconstructed phase images of the dotted rectangular box in Fig. 7(f), which is based on the direct filtering approach. Notice that even when the off-axis carrier frequency is not high enough as shown in Fig. 8(a), the direct filtering result as illustrated in Fig. 8(g) does not experience such serious spatial resolution degradation which can be seen in the spatial filtering result in Fig. 8(d). We claim that the proposed direct filtering scheme allow us to extract the object wave information almost perfectly without any high frequency information loss once the object wave spatial frequency spectrum is separated well from the DC term as shown in Fig. 8(a). In contrast, when the conventional rectangular shaped windowing based spatial filtering method is used, such spatial resolution degradation is inevitable for the low carrier frequency case since the windowing technique used in the spatial filtering process cannot extract the entire object information properly due to the spatial interruption by the DC and conjugate term.
4. Experimental results
In order to figure out the feasibility of the proposed direct filtering method, we conducted experiments based upon the schematic depicted in Fig. 1. Partially polarized light from a diode laser with the wavelength of 635 nm is collimated by a collimating optical sub-set. The collimated beam is divided into object and reference beams. A CCD camera with the pixel size of 6.4 μm × 6.4 μm records the off-axis hologram that results from the interference between the object wave O and the reference wave R. Two neutral density filters are used to adjust the object and the reference intensities. The measured hologram size is 768 by 768.
Figure 9(a) and 9(b) depict the reconstructed phase image obtained by using the spatial filtering and the proposed direct filtering approach, respectively. For all experiments, the convolution formulation (Eq. (4)) is used for reconstructing the phase image in the observation plane. As can be seen in the results, the direct filtering method can provide comparable reconstructed phase quality with that of the spatial filtering approach. Some phase image degradation around the edge area can be occurred when the hologram signal is not continuous. Such signal discontinuity comes mainly from dirt and defects on optics. We expect that such phase image degradation can be reduced to some amount by improving the cleanness level of the experimental environment. It should be noted that we can select kx as the carrier frequency just by defining row vectors instead of the column vectors based on the carrier frequency ky.
We proposed a novel single exposure complex object wave extraction method which can provide one and half to two times calculation speed enhancement while maintaining the reconstructed phase image quality in off-axis digital holography. And also, it is claimed that the direct filtering method can have additional benefit in filtering process in the sense that it can provide more robust results to the variation of the spatial carrier frequency. Furthermore, we expect the proposed direct filtering algorithm can be applied extensively to various application areas such as interferometry and ellipsometry which requires high speed accurate phase measurement related to frequency domain signal processing.
This work was supported partially by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MEST) (2012R1A1B3003346). And also, this research was supported by the R&D program for Industrial Core Technology through the Korea Evaluation Institute of Industrial Technology supported by the Ministry of Knowledge Economy in Korea (Grant No. 10040225).
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