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Incoherent digital holography (IDH) can be realized by optical scanning holography or self-interference incoherent holography. Although IDH can exhibit high quality reconstruction due to its inherently speckle-free property, direct display of an incoherent hologram is a challenge because of its amplitude nonlinearity and the demand of complex modulation. In this paper we propose to compensate the amplitude nonlinearity at the object plane, and use bidirectional error-diffusion method to convert the complex-type incoherent Fresnel hologram to a phase-only Fresnel hologram for display. A spatial light modulator is used to reconstruct the phase-only hologram optically to demonstrate the validity of our proposed method.
© 2016 Optical Society of America
1. Introduction
In digital holography (DH), the pattern of interference fringes is recorded by a digital device, such as a CCD array, to generate a digital hologram. In the last decade, DH has become a promising technique for metrological and medical applications [1–4]. Recently, researchers have investigated the display of digital holograms for its potential application of holographic TV. However, it is still a challenge to display high-quality digital hologram because serious speckle always arises in the recording process of conventional DH. Although some techniques can eliminate the speckle noise, they usually demand heavy computation loading or numerous recordings and thus are not suitable for real-time holographic recording and display [5–7]. On the other hand, incoherent digital holography (IDH) is a good candidate to record speckle-free digital holograms [8–11]. The image quality of IDH is always much better than that of conventional coherent DH and thus is a good candidate for holographic display. There are two approaches to IDH. The first approach, including Fresnel incoherent correlation holography (FINCH) and Fourier incoherent single channel holography (FISCH), is based on generating two duplicates of an object wave to form self-interference [12–19]. Thus it can record a hologram of any kind of light sources. The other approach is optical scanning holography (OSH) [20, 21]. In OSH, a dynamic interference fringe pattern is first generated by a heterodyne interferometer, and the object target is two-dimensional (2D) raster scanned by this dynamic fringe pattern. The light scattered from the object target is detected by a single-pixel detector (for example, a photodetector), and demodulated by a lock-in amplifier as rows of hologram lines, which can be re-arranged to form a 2D hologram. It has been demonstrated that OSH can be operated in the incoherent mode [8–11]. The incoherent-mode of OSH, just like the first approach to IDH, also makes the recording of speckle-free holograms possible.
The key component of holographic display is a spatial light modulator (SLM). A SLM can modulate either the amplitude or the phase of incident light, but not both [22–24]. Intuitively, an off-axis amplitude-type incoherent hologram can be directly displayed by an amplitude-mode SLM. Nevertheless, the recording conditions (pixel pitch, object distance, etc.) are usually different from the reconstruction conditions. In addition, the intensity nonlinearity problem must be dealt with. Alternatively, complex holograms, in which the twin image and the zeroth order have been removed, are recorded by IDH. The complex hologram can be manipulated and consequently converted to an amplitude-only hologram or a phase-only hologram for display. Recently, Leportier et al. have converted a complex Fresnel hologram obtained from an incoherent-mode of OSH to a binary Fourier hologram by iterative direct binary search algorithm [9]. However, this iteration-based algorithm is time-consuming. Besides, the Fourier hologram demands specific reconstruction optics and is not favored in holographic 3D display. To resolve the above problems, In this paper we have employed OSH-based IDH and convert a complex Fresnel hologram to a phase-only Fresnel hologram by a recently proposed error diffusion method [25, 26] because the phase-only hologram always exhibits high diffraction efficiency. Meanwhile, the nonlinearity of intensity of the IDH is compensated. Finally, we use a liquid-crystal on silicon (LCoS) SLM to display the converted phase hologram. Experimental results prove that the proposed method can convert a display hologram from an incoherent hologram with high fidelity and high diffraction efficiency.
This paper is organized as follows. In section 2, we will review the properties of OSH and the related problems in optical reconstruction. In section 3, the proposed procedure, including the concept of bidirectional error diffusion (BERD) conversion method, will be reviewed. Experimental results are provided in section 4. We will give concluding remarks in section 5.
2. Optical scanning holography (OSH)
The principle as well as practical optical setups of OSH can be found generally in the literature [10, 11, 20, 21, 27–30]. In short, the complex amplitude recorded by the incoherent-mode of OSH is expressed as
where stands for the convolution calculation, denotes the coordinates centered at the recording plane; is the 3D intensity reflectivity of the object. andis the impulse response of OSH [20]; k is the wavenumber of light employed. The complex amplitude recorded by the incoherent-mode of OSH [Eq. (1)] is a complex hologram. In self-interference IDH, the hologram formula can also be expressed by Eq. (1) provided phase-shifting is applied to remove the zeroth order and the twin image. It should be noted that Eq. (1) is a little different from conventional coherent holography, in which the complex amplitude reflectance of the object, , is recorded in the hologram. Because , the intrinsic phase of the object is lost in recording an incoherent hologram, while the 3D location of the object is retained. Accordingly, the reconstructed image of the incoherent complex hologram is speckle-free and thus its quality is much better than that of a coherent hologram [11]. On the other hand, the problems including amplitude nonlinearity, pixel mismatch and the difficulty of complex modulation prevent the incoherent complex hologram from being optically displayed. In section 3 we will explain these problems and will propose methods for dealing with them.3. Conversion from an incoherent complex hologram to a phase-only Fresnel hologram
3.1 Amplitude nonlinearity
The original complex amplitude recorded by IDH [Eq. (1)] can be digitally reconstructed by performing propagation simulation to retrieve the field at a specified reconstruction plane [31],
where , and is the distance from the recording plane to the reconstruction plane. For simplifying the explanation, here we assume that the original object is planar and is located from the recording plane. By assuming that, Eq. (1) is reduced to , and the intensity reflection of the object can be retrieved if is the same as , that isIn optical display of the same hologram, it is the intensity of light to be viewed. Therefore, we will observe a reconstructed image with intensity distributionIt should be noted that the reconstructed intensity is the square of the reflectivity of object. In other words, gray scale of the reconstructed image is different from that of the true object. This nonlinearity problem encountered in display is demonstrated in Fig. 1. Figure 1(a) shows the absolute value of the reconstructed field () of the hologram in Eq. (1), while Fig. 1(b) shows , the reconstructed intensity, to simulate the optical reconstruction. The gray scale of Fig. 1(b) looks much different from that of the original object [Fig. 1(a)]. By using as the ideal image, the fidelity of the reconstructed intensity in Fig. 1(b) in terms of peak-signal-to-noise ratio (PSNR) is only 24.1 dB.
Fig. 1 (a) Absolute amplitude , i.e., the true intensity distribution, and (b) intensityof the reconstructed image carrying nonlinearity error.
To alleviate the problem of amplitude nonlinearity, we pre-modify the amplitude but retain the phase of the complex amplitude to compensate the nonlinear effect at the object plane. Explicitly, the modified complex amplitude is expressed as
where takes the phase of a complex function. The modified field can be propagated again to the destination plane, at which the complex field is converted to a phase-only hologram.3.2 Pixel mismatch
The size of the modified field is pixels of pitch . is first propagated to the destination plane with sufficient zero-padding [31]. Thus the size of the complex field at the destination plane,, is pixels with pitch . The pixel pitch of the LCoS SLM is . Accordingly, must be resized to pixels () with pitch by spectrum zero padding. First, we put , the Fourier spectrum of , to a blank matrix of size , and inverse Fourier transform the whole matrix [4]. If is shifted by related to the center of the blank matrix, a tilted carrior of the object light is introduced. As a result, an off-axis object lightcan be generated simultaneously in the resize procedure.
3.3 Complex modulation of spatial light modulator
The LCoS SLM can modulate either the phase or the amplitude of light. We adopt phase modulation to achieve better diffraction efficiency [32]. Thus the complex Fresnel hologram obtained in the last step must be converted to a phase-only Fresnel hologram. Here we use bidirectional error diffusion (BERD) method [25] for this conversion because BERD is iteration-free in comparison with time-consuming iteration-based algorithms [9]. BERD method can be parallelized and thus the computation speed is significantly increased with the cost of slightly worse fidelity [26]. In addition, BERD directly processes the hologram without altering the object distribution or phase, which also simplifies the conversion. Details of BERD method can be found in [25], and only a brief outline is provided here. To begin with, we assume that the complex field at the destination plane has been digitized with pixel pitch to , where and index the column and row, respectively. The digitized hologram is processed from the top row to the bottom row. Along the odd and even rows, the pixels are scanned from the left-to-right and the right-to-left directions, respectively. For each pixel, only the phase quantity is retained and the error is distributed in weighted proportion to the four adjacent pixels that have not been visited before. After visiting all pixels, a phase-only hologram is generated to address the SLM. The flow chart of the generation of a phase-only Fresnel hologram from a hologram from IDH is shown in Fig. 2 as a summary.
Fig. 2 Flow chart of the generation of a phase-only Fresnel hologram from a complex hologram recorded by IDH.
4. Experimental results
The parameters of the complex amplitude recorded by the incoherent-mode of OSH and the converted phase-only Fresnel hologram are listed in Table 1. As the phase-only hologram is generated, we perform digital reconstruction to verify the processing procedure. Figure 3(a) is the reconstructed image of the on-axis phase-only hologram at the reconstruction plane. The noise patterns at four corners are due to the BERD processing. These noise patterns have large off-set angles and will not disturb the on-axis object light. Figure 3(b) shows the zoom-in view of Fig. 3(a). The PSNR and diffraction efficiency of the reconstructed image is 51.4 dB and 2.26%, respectively. We have also focused the hologram at different planes. The focus-and-defocus effect remains, as shown in Fig. 3(c). Accordingly, the manipulation of the complex field in the conversion will not destroy the 3D information of the object. We also generated a conventional off-axis amplitude hologram using the same object field for comparison. The diffraction efficiency of the reconstructed image is extremely low (0.15%) and thus is not satisfactory.
Table 1. Parameters of the original complex field and the converted phase-only hologram
Fig. 3 Digitally reconstructed image of the on-axis phase-only Fresnel hologram (a), and its zoom-in view (b). (c) Selected portions of reconstructed images at different planes.
We have also used a LCoS SLM to optically reconstruct the converted phase hologram. The size of the LCoS SLM is pixels. Therefore, each time only a portion of the phase-only hologram is cropped, called hologram tile, to address the SLM. We used a lensless COMS camera at the reconstruction plane to acquire the reconstructed images. Thus the four-corner noise is outside the region of interest. The optically reconstructed image of a hologram tile from the on-axis phase-only hologram is shown in Fig. 4(a). The image cannot be identified because the residual zeroth order light due to the imperfect modulation of the LCoS SLM overlays on the on-axis reconstructed image. To solve this problem, we have used the off-axis object lightto generate an off-axis phase-only hologram. Both the procedure and result are similar to the on-axis case, except that the PSNR and diffraction efficiency is slightly worse (45.9 dB and 2.24%) now. Figure 4(b) shows the reconstructed image of a hologram tile from the off-axis phase-only hologram. Now the reconstructed image can be clearly observed, disregarding the artifacts due to hologram cropping. Finally, we have separately reconstructed images of nine hologram tiles of the off-axis phase-only hologram, and stitched these reconstructed images to a single full-size reconstructed image, which is shown in Fig. 4(c).
Fig. 4 Optically reconstructed images of hologram titles of (a) on-axis phase-only hologram, and (b) off-axis phase-only hologram. (c) Whole image stitched from multiple reconstructed images (see Visualization 1).
5. Conclusion
In this paper we have generated a phase-only Fresnel hologram obtained from incoherent digital holography. Issues of amplitude nonlinearity, pixel mismatch and complex-to-phase conversion have been addressed and solved. We have performed optical reconstruction as well as digital simulations to validate this technique. The resulting phase-only hologram is able to reconstruct high-definition 3D image of a real object with high diffraction efficiency related to amplitude hologram. In our demonstration, the incoherent hologram was recorded by OSH. We want to point out that holograms obtained from other incoherent holographic techniques, such as FINCH and FISCH, can also be converted by the proposed method. Due to the natural speckle-free property of incoherent holographic techniques, this technique can find promising applications in holographic TV.
Acknowledgments
This work is sponsored by the Ministry of Science and Technology of Taiwan under contract number MOST-103-2221-E-035-037-MY3.
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