Complex object wave direct extraction method in off-axis digital holography

Daesuk Kim; Robert Magnusson; Moonseob Jin; Jaejong Lee; Won Chegal

doi:10.1364/OE.21.003658

1. Introduction

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 [1], 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 [2]. 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 [3]. 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 [9] opened various practical application fields in digital holography. Since the Fourier-transform based phase measurement method was proposed by Takeda et al [11], 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 [10]. 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 [14] and direct coarse sampling theory [15] 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 [14] 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 [15] 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.

Fig. 1 Schematic diagram of the Mach-Zehnder type off-axis digital holography used for both theoretical simulations and experiments in this study.

Download Full Size | PDF

In the hologram plane, the interference between the object wave $O = A \exp (j φ_{O})$ and the reference wave $R = B \exp (j φ_{R})$ creates the following hologram.

I_{H} (k, l) = | O |^{2} + | R |^{2} + R^{*} O + R O^{*} = A^{2} + B^{2} + 2 A B \cos (φ_{O} - φ_{R})

Here, R* and O* denote the complex conjugate of the reference and object wave, respectively. 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.

Fig. 2 Sequential reconstruction steps of the conventional spatial filtering based phase contrast off-axis digital holography: (a)off-axis hologram, (b)Fourier transformed spatial frequency domain data, (c)spatially filtered frequency domain data(undesired terms are removed), (d)-(e) inversely Fourier transformed data (amplitude and phase data, respectively), (f)phase map of the digital reference wave, and (g)-(h)reconstructed object wave (amplitude and phase data, respectively)

Download Full Size | PDF

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 R_D which is defined as follows [9, 10]:

R_{D} (k, l) = A_{R} ​ \exp [i (2 π / λ) (k_{x} k Δ x + k_{y} l Δ y)]

Here, A_R is the amplitude of the digital reference wave which is set to be 1 in the simulation code, and k_x and k_y are the two components of the wave vector that must be adjusted such that the propagation direction of R_D matches as closely as possible with that of the experimental reference wave R. Then, we obtain the final reconstructed wave front Ψ(m, n) in the observation plane 0ξη by using either of the following Fresnel transformations. Equation (3) and Eq. (4) represent Single Fourier Transform Formulation (SFTF) and Convolution Formulation (CF), respectively [16].

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)].

\begin{array}{l} Ψ_{S F T F} (m, n) = \frac{\exp (i 2 π d / λ)}{i d λ} \exp [\frac{i π}{λ d} (m^{2} Δ ξ^{2} + n^{2} Δ η^{2})] \times F {R_{D} (k, l) O (k, l) \exp [\frac{i π}{λ d} (k^{2} Δ x^{2} + l^{2} Δ y^{2})]} \end{array}

\begin{array}{l} Ψ_{C F} (m, n) = \frac{\exp (i 2 π d / λ)}{i d λ} F^{- 1} {F [(R_{D} (k, l) O (k, l)] F (\exp [\frac{i π}{λ d} (k^{2} Δ x^{2} + l^{2} Δ y^{2})])} \end{array}

Here, k, l, m, n are integers. d and λ are the distance between the CCD and the observation plane and the wavelength of the laser source, respectively. F denotes the 2D fast Fourier transformation. Δx and Δy are the sampling intervals in the hologram plane and Δξ and Δη are the sampling interval in the observation plane. 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 I_H^M(k,l) can be regarded as a single acquisition hologram.

I_{H}^{M} = I_{H} - | R |^{2} = | O |^{2} + R^{*} O + R O^{*} = A^{2} + 2 A B \cos (φ_{O} - φ_{R})

Here, the reference wave R represents

B \exp (j φ_{R})

. What we need to extract from the modified hologram I_H^M is the complex object wave

O = A \exp (j φ_{O})

in the hologram plane so that the Fresnel transform can be applied for reconstructing the object wave in the observation plane.

Fig. 3 Sequential reconstruction steps of the proposed direct filtering based phase contrast off-axis digital holography: (a)modified off-axis hologram [Complex object wave direct calculation procedures: 5 consecutive hologram column vectors are used to calculate the corresponding direct filtered complex object wave column vector], (b)-(c) directly calculated complex object wave in the hologram plane (amplitude and phase data, respectively), (d)phase map of the digital reference wave, and (e)-(f)reconstructed object wave in the observation plane (amplitude and phase data, respectively).

Download Full Size | PDF

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].

\begin{array}{l} I_{1} = A + B + 2 \sqrt{A B} \cos (ϕ - 2 α) \\ I_{2} = A + B + 2 \sqrt{A B} \cos (ϕ - α) \\ I_{3} = A + B + 2 \sqrt{A B} \cos (ϕ) \\ I_{4} = A + B + 2 \sqrt{A B} \cos (ϕ + α) \\ I_{5} = A + B + 2 \sqrt{A B} \cos (ϕ + 2 α) \end{array}

Here, A and B are the intensities of the object and reference beams, respectively. α is an arbitrary temporal phase shift.

φ = \tan^{- 1} [\frac{1 - \cos 2 α}{\sin α} (\frac{I_{2} - I_{4}}{2 I_{3} - I_{5} - I_{1}})]

The visibility of the interferogram defined as

γ = \frac{2 \sqrt{A B}}{A + B}

can also be calculated by using the same 5 phase shifted interferogram intensities as follows [17].

γ = \frac{\sqrt{4 {(I_{2} - I_{4})}^{2} + {(2 I_{3} - I_{5} - I_{1})}^{2}}}{4 I_{0} \sin α \sqrt{(\sin^{2} ϕ + \sin^{2} α \cos^{2} ϕ)}}

Here, I₀ is the total intensity A + B. 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 I_H has N by N hologram size. Then, we can define column vectors from I₁ to I_N that satisfy I_H = [I₁ I₂ I₃ …. I_N-1 I_N]. As mentioned, we can subtract the reference wave intensity term

{| R |}^{2} = B^{2}

from the original hologram I_H as depicted in Fig. 3(a) and we use the modified hologram

I_{H}^{M}

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

2 π k_{y} Δ y / λ

since the spatial phase shift interval caused by one CCD pixel size Δy is defined as

2 π k_{y} Δ y / λ

in the digital reference wave R_D described in Eq. (2), and by replacing the 5 consecutive scalar interferogram intensities from I₁ to I₅ in Eq. (2) with the k_th row hologram intensity column vectors from

I_{k - 2}^{M}

to

I_{k + 2}^{M}

. The object and reference wave scalar amplitudes

\sqrt{A}

and

\sqrt{B}

can be replaced by the column vector

A_{k}

and

B_{k}

representing the k_th column vector of the object and the reference amplitudes

A

and

B

defined in Eq. (5).

\begin{array}{l} I_{k - 2}^{M} = A_{k}^{2} + 2 A_{k} B_{k} \cos [Φ_{k} - 2 (2 π k_{y} Δ y / λ)] \\ I_{k - 1}^{M} = A_{k}^{2} + 2 A_{k} B_{k} \cos [Φ_{k} - (2 π k_{y} Δ y / λ)] \\ I_{k}^{M} = A_{k}^{2} + 2 A_{k} B_{k} \cos Φ_{k} \\ I_{k + 1}^{M} = A_{k}^{2} + 2 A_{k} B_{k} \cos [Φ_{k} + (2 π k_{y} Δ y / λ)] \\ I_{k + 2}^{M} = A_{k}^{2} + 2 A_{k} B_{k} \cos [Φ_{k} + 2 (2 π k_{y} Δ y / λ)] \end{array}

Here,

Φ_{k}

represents the k_th row column vector of the phase difference between the object and the reference wave

φ_{O} - φ_{R}

. Compared with the procedures demonstrated in Fig. 2, the direct filtering needs no 2D-FFTs and we can extract the direct filtered complex object wave O^DF 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).

Φ_{k} = \tan^{- 1} [\frac{1 - \cos (4 π k_{y} Δ y / λ)}{\sin (2 π k_{y} Δ y / λ)} (\frac{I_{k - 1} - I_{k + 1}}{2 I_{k} - I_{k + 2} - I_{k - 2}})]

A_{k} = \frac{\sqrt{4 {(I_{k - 1} - I_{k + 1})}^{2} + {(2 I_{k} - I_{k + 2} - I_{k - 2})}^{2}}}{8 B_{k} \sin (2 π k_{y} Δ y / λ) \sqrt{[\sin^{2} Φ_{k} + \sin^{2} (2 π k_{y} Δ y / λ) \cos^{2} Φ_{k}]}}

We can calculate Φ_k by using Eq. (10). Since

A_{k} = \frac{γ_{k} (A_{k}^{2} + B_{k}^{2})}{2 B_{k}}

by the definition of the k_th row visibility column vector γ_k, Eq. (11) can be derived easily by referring to Eq. (8). A_k can be calculated for each integer k ranging from 3 to N-2 once the phase column vector Φ_k is obtained. Here, B_k 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

O^{D F} = A \exp (j Φ) = [O_{3}^{D F} ​ O_{4}^{D F} O_{5}^{D F} .... O_{N - 3}^{D F} O_{N - 2}^{D F}]

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

O_{k}^{D F}

means

A_{k} \exp (j Φ_{k})

. However, the direct filtered phase Φ_k in Eq. (10) contains the reference wave phase term

φ_{R}

. Therefore, we can obtain the correct complex object wave in the hologram plane

O = A \exp (j φ_{O})

by multiplying the digital reference wave R_D depicted in Fig. 3(d) with the direct filtered complex object wave

O^{D F} = A \exp (j Φ)

. 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 k_x = 2.0 × 0.00780 and k_y = 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.

Fig. 4 Comparison between the reconstructed object phase line profiles obtained from (a)the central vertical line of Fig. 2(f) which is based on the spatial filtering and (b) that of Fig. 3(f) obtained by using the direct filtering.

Download Full Size | PDF

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 [18]. 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).

Fig. 5 (a) Computation time required to obtain the complex object wave in the hologram plane versus the hologram size N, and (b) ratio between the computation time of the spatial filtering method and that of the direct filtering method versus the hologram size N.

Download Full Size | PDF

Secondly, we conducted the reconstructed phase image spatial resolution analysis by varying the off-axis carrier frequency k_x and k_y. 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.

Fig. 6 (a)Computer generated USAF target phase object and (b)off-axis hologram generated by using the USAF target object.

Download Full Size | PDF

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 k_x = 1.5 × 0.00780 mm⁻¹ and k_y = 1.5 × 0.00915 mm⁻¹. Figure 8(b) shows the results when k_x = 2.0 × 0.00780 mm⁻¹ and k_y = 2.0 × 0.00915 mm⁻¹. Likewise, Fig. 8(c) is when k_x = 3.0 × 0.00780 mm⁻¹ and k_y = 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 k_x and k_y 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.

Fig. 7 Reconstructed amplitude and phase images, respectively: (a) and (d) when no filtering is used, (b) and (e) when the spatial filtering applied, and (c) and (f) when the direct filtering is employed.

Download Full Size | PDF

Fig. 8 Filtering capability analysis on the variation of the off-axis spatial carrier frequency: (a)spatial frequency spectrum when k_x = 1.5 × 0.00780 mm⁻¹ and k_y = 1.5 × 0.00915 mm⁻¹, (b) when k_x = 2.0 × 0.00780 mm⁻¹ and k_y = 2.0 × 0.00915 mm⁻¹, and (c)when k_x = 3.0 × 0.00780 mm⁻¹ and k_y = 3.0 × 0.00915 mm⁻¹, (d)-(f) zoomed reconstructed phase images of the dotted inner rectangular box in Fig. 7(e), 7(g)-(i) zoomed reconstructed phase images of the dotted rectangular box in Fig. 7(f).

Download Full Size | PDF

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 k_x as the carrier frequency just by defining row vectors instead of the column vectors based on the carrier frequency k_y.

Fig. 9 Experimental results: (a) reconstructed object phase map obtained by using the spatial filtering approach and (b) that obtained by using the proposed direct filtering method.

Download Full Size | PDF

5. Conclusion

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.

Acknowledgments

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).

References and links

1. D. Gabor, “A new microscopic principle,” Nature 161(4098), 777–778 (1948). [CrossRef] [PubMed]

2. I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. 22(16), 1268–1270 (1997). [CrossRef] [PubMed]

3. Y. Takaki, H. Kawai, and H. Ohzu, “Hybrid holographic microscopy free of conjugate and zero-order images,” Appl. Opt. 38(23), 4990–4996 (1999). [CrossRef] [PubMed]

4. F. Dubois, M. L. Requena, C. Minetti, O. Monnom, and E. Istasse, “Partial spatial coherence effects in digital holographic microscopy with a laser source,” Appl. Opt. 43(5), 1131–1139 (2004). [CrossRef] [PubMed]

5. D. Beghuin, E. Cuche, P. Dahlgren, C. Depeursinge, G. Delacretaz, and R. P. Salathe, “Single acquisition polarization imaging with digital holography,” Electron. Lett. 35(23), 2053–2055 (1999). [CrossRef]

6. B. Javidi and D. Kim, “Three-dimensional-object recognition by use of single-exposure on-axis digital holography,” Opt. Lett. 30(3), 236–238 (2005). [CrossRef] [PubMed]

7. D. G. Abdelsalam, B. J. Baek, Y. J. Cho, and D. Kim, “Surface form measurement using single-shot off-axis Fizeau interferometer,” J. Opt. Soc. Korea 14(4), 409–414 (2010). [CrossRef]

8. D. G. Abdelsalam, R. Magnusson, and D. Kim, “Single-shot, dual-wavelength digital holography based on polarizing separation,” Appl. Opt. 50(19), 3360–3368 (2011). [CrossRef] [PubMed]

9. E. Cuche, F. Bevilacqua, and C. Depeursinge, “Digital holography for quantitative phase-contrast imaging,” Opt. Lett. 24(5), 291–293 (1999). [CrossRef] [PubMed]

10. E. Cuche, P. Marquet, and C. Depeursinge, “Spatial filtering for zero-order and twin-image elimination in digital off-axis holography,” Appl. Opt. 39(23), 4070–4075 (2000). [CrossRef] [PubMed]

11. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72(1), 156–160 (1982). [CrossRef]

12. D. Kim, S. Kim, H. J. Kong, and Y. Lee, “Measurement of the thickness profile of a transparent thin film deposited upon a pattern structure with an acousto-optic tunablefilter,” Opt. Lett. 27(21), 1893–1895 (2002). [CrossRef] [PubMed]

13. D. Kim, H. Kim, R. Magnusson, Y. J. Cho, W. Chegal, and H. M. Cho, “Snapshot phase sensitive scatterometry based on double-channel spectral carrier frequency concept,” Opt. Express 19(24), 23790–23799 (2011). [CrossRef] [PubMed]

14. M. Liebling, T. Blu, and M. Unser, “Complex-wave retrieval from a single off-axis hologram,” J. Opt. Soc. Am. A 21(3), 367–377 (2004). [CrossRef] [PubMed]

15. K. Khare and N. George, “Direct coarse sampling of electronic holograms,” Opt. Lett. 28(12), 1004–1006 (2003). [CrossRef] [PubMed]

16. T. Colomb, F. Montfort, J. Kühn, N. Aspert, E. Cuche, A. Marian, F. Charrière, S. Bourquin, P. Marquet, and C. Depeursinge, “Numerical parametric lens for shifting, magnification, and complete aberration compensation in digital holographic microscopy,” J. Opt. Soc. Am. A 23(12), 3177–3190 (2006). [CrossRef]

17. P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26(13), 2504–2506 (1987). [CrossRef] [PubMed]

18. Wikipidia, “Fast Fourier Transform,” http://en.wikipedia.org/wiki/Fast_Fourier_transform.

Complex object wave direct extraction method in off-axis digital holography

Abstract

1. Introduction

2. Complex object wave direct filtering theory

3. Analysis on filtering capability

4. Experimental results

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (9)

Equations (11)

Optics Express