4.1 Introduction

In previous chapters, different methods for generating a phase-only hologram for a given 3-D object, such as the iterative Fresnel transform algorithm (IFTA), downsampling, and the patterned phase method, have been presented. While all these methods are effective in generating phase-only holograms, they all share the same prerequisite that a source image has to be available in the first place. This is not a problem in computer-generated holography (CGH), in which case the source images are provided in the form of numerical data, such as point cloud files or computer graphic models. However, in some applications such as remote sensing and microscopic holography, a hologram is captured directly from a real object based on acquisition methods like optical scanning holography and phase-shifting holography. For these cases, the source image is not available, and the methods that have been gone through previously are not applicable for deriving the phase-only hologram.

Intuitively, it could be possible to reconstruct the 3-D source image from a complex-valued hologram, with which a phase-only hologram can be generated with IFTA or other methods. However, there is a lack of an effective method that can reconstruct an arbitrary 3-D image from its hologram. In most of the methods developed to date, a hologram is reconstructed onto a series of regularly spaced depth planes. Subsequently, some sort of sharpness or focus index is measured for each region on the stack of reconstructed images, and the depth plane with the sharpest measurement among all its peers will be taken to be the focused plane of that particular region. Although such an approach is logical and quite sound, existing hologram reconstruction methods such as those discussed in [1–4] are computation intensive, and so only of use for reconstructing simple geometries. For more complicated object shapes, a region is likely to contain contents with a wider range of depths. When one area within a region is in focus, the remaining part may become blurred and the out-of-focus content may blend into the neighboring focused region(s). It is therefore not possible to use the sharpness index to locate the focused plane of each region unless the object space is thin or comprises of widely separated items.

Alternatively, it is possible to convert a complex-valued hologram into a phase-only hologram. There are two major difficulties with this approach. First, the process is lossy and irreversible, as the magnitude component of the hologram is removed. The amount of distortion incurred in the conversion is unpredictable. Second, as there is no information on the source image that is represented by the hologram, iterative techniques (for example, the Gerchberg–Saxton algorithm) cannot be applied to minimize the conversion error. This chapter describes six methods for converting a complex-valued hologram into a phase-only hologram: complex amplitude modulation (CAM); double-phase decomposition; uni-directional error diffusion (UERD); bi-directional error diffusion (BERD); localized error diffusion (LERD); and direct binary search (DBS). This last method is used for obtaining a binary phase-only hologram from the complex-valued hologram.

4.2 Complex Amplitude Modulation

Complex amplitude modulation is a method of encoding both the amplitude and phase components of a complex-valued function into a phase-only function. The idea of applying this concept in digital holography could be dated back to 1999, when J.A. Davis and his team succeeded in implementing phase-only optical filters for optical recognition and image processing [5]. Recently, the method has been applied to convert a complex-valued hologram into a phase-only hologram [6,7]. Upon illumination with a reference beam and filtering of the diffracted waves, the source image recorded in the phase-only hologram will be reconstructed as a visible image.

Here, a complex-valued hologram H(m,n)=|H(m,n)|expiθ(m,n) is to be converted into a phase-only hologram Hp(m,n) . After the hologram is generated, it will be illuminated by an inclined plane wave R(m,n)=|R(m,n)|expiθR(m,n) , making an angle of incidence θ_R along the vertical or horizontal direction for reconstructing the image recorded in the hologram.

Converting the hologram into a phase-only hologram with CAM is straightforward. The phase-only hologram is obtained simply by integrating the hologram H(m,n) and the off-axis plane wave into a phase function:

HP(m,n)=c exp{iβ|H(m,n)|cos[θ(m,n)−θR(m,n)]},(4.1)

where c and β are constant terms.

To view the reconstructed image, the phase-only hologram HP(m,n) is illuminated by the inclined reference plane wave R(m,n) that is making an angle of incidence θ_R along the vertical or horizontal direction. The diffracted wave on the reconstructed image plane is

DP(m,n)=c expiθR(m,n)exp{iβ|H(m,n)|cos[θ(m,n)−θR(m,n)]}.(4.2)

From Eq. (4.2) it can be seen that although the expression should contain the signal corresponding to the hologram H(m,n) , the diffracted wave DP(m,n) also includes components that are different from the hologram. The question is: How can the hologram signal H(m,n) be separated from the rest of the terms in DP(m,n) . It is shown in [6] that this is achieved by decomposing the expression in Eq. (4.2) into an infinite series based on the following property of the Bessel function:

expiscosψ=∑k=−∞∞ikJk(s)expikψ,(4.3)

where Jk(s) is the kth order of the Bessel function of the first kind.

Applying Eq. (4.3) to Eq. (4.2), the diffraction wave DP(m,n) is the sum of an infinite series of functions:

DP(m,n)=c∑−∞∞Jk[β|H(m,n)|]ikexp{−i[kθ(m,n)−(k+1)θR(m,n)]}.(4.4)

Eq. (4.4) is equivalent to the superposition of an infinite number of off-axis holograms; the reconstructed image of each one projects at a unique angle, which is an integer multiple of the off-axis illumination angle θ_R of the reference plane wave. For simplicity, the constant term c can be taken as unity.

In Eq. (4.4), the component corresponding to k=−1 ,

HCAM(m,n)=DP(m,n)|k=−1=−J−1[β|H(m,n)|]expiθR(m,n).(4.5)

The reconstructed images of higher-order diffraction with |k|>1 should be non-overlapping with the one corresponding to k=−1 if the angle of incidence θ_R of the reference wave is sufficiently large. Based on this assumption, an optical mask can be applied to extract HCAM(m,n) from the rest of the unwanted components. From Eq. (4.5), the −1 order of diffraction (denoted by HCAM(m,n) ) is related to the complex amplitude distribution of the hologram H(m,n) . If J−1[|H(m,n)|] is similar to H(m,n) , the source image can be reconstructed to a good approximation from this component. The reconstructed image of HCAM(m,n) will carry a certain amount of distortion, as the hologram has been changed to the function J−1[|H(m,n)|] , and the reconstructed image is overlaid with the zero-order ( k=0 ).

The first type of distortion, resulting from the modification of the hologram by J−1[|H(m,n)|] , can be alleviated by selecting a value for the constant term β that restricts the dynamic range of |H(m,n)| to the linear region of the Bessel function. To illustrate this point, a plot of the function J−1[x] for x=[0,5] is shown in Figure 4.1. From the diagram, the Bessel function is close to a linear function for x within the range [0,1.5].

Figure 4.1 Plot of Bessel function J₋₁[x] for x = [0,5].

The distortion caused by the overlaying of the zero-order component on the reconstructed image can be removed with an optical filter formed by a pair of lens arranged in a 4f filtering configuration.

Numerical simulation of the CAM method is conducted using the MATLAB code provided in Section 4.8.1. A complex-valued Fresnel hologram of the source image “Apple” in Figure 4.2(a) is first generated. Next, the phase-only hologram corresponding to the −1 order of diffraction signal HCAM(m,n) is extracted with Eq. (4.5). A reconstructed image of the phase-only hologram on the focused plane is thus obtained. The reconstructed images corresponding to β = 3.0, 2.0, and 0.50 are shown in Figures 4.2(b–d). In all these images, the dynamic range of brightness has been normalized to the range {0,255} , with 0 and 255 representing the lowest and the highest intensities, respectively. The attenuation of the brightness due to β , therefore, is not shown in the figures. The fidelities of the three reconstructed images, as compared with the original one in Figure 4.2(a), are 14.45 dB, 20.89 dB, and 30 dB for β=3 , β=2 , and β=0.5 . In general, the larger the peak-signal-to-noise-ratio (PSNR), the higher is the fidelity of the image. By observation, the quality of the reconstructed image is increased with decreasing values of β , which is also reflected by the increase in PSNR. However, from Eq. (4.5) and Figure 4.1, the smaller the value of β , the weaker the magnitude of the hologram HCAM(m,n) , and the dimmer will be its reconstructed image. Hence it is necessary to find a value of β that results in a reconstructed image with higher fidelity and sufficient brightness to create a favorable visual quality.

Figure 4.2 (a) Source image “Apple.” (b–d) Reconstructed image from H_CAM(m, n) with β = 3, 2, 0.5, respectively.

Another similar simulation is applied to convert the complex-valued hologram of the double-depth image “HK1” in Figure 1.12 into a phase-only hologram with the CAM method. The source image is divided into left and right sections located at axial distances of z1=0.07 m and z2=0.08 m from the hologram. A phase-only hologram is generated with β=0.5 , and the reconstructed images at the two focused planes are shown in Figure 4.3. From these two images, each side of the source image is reconstructed as a sharp image on its focused plane, with the other side being a blurred, defocused image. This simulation result demonstrates that the CAM method is capable of generating a phase-only hologram that can represent both intensity and depth dimension.

Figure 4.3 Reconstructed images of the phase-only hologram, obtained with the CAM method with β= 0.5. (a) On focused plane at 0.07 m; (b) on focused plane at 0.08 m.

4.3 Double-Phase Macro-Pixel Hologram

The double-phase macro-pixel hologram is based on the principle that if a complex number is bounded within the unit circle, it can be represented as the sum of two phase-only quantities. As a result, any complex quantity can be converted into a pure phase representation. Mathematically, given a complex number A expiθ with A within the range [0,1], the sum of two phase terms is expressed as

A expiθ=0.5[expiθ1+expiθ2],(4.6)

where A=cos(α) , θ1=θ+α , and θ2=θ−α . The simple mathematical relation in Eq. (4.6) suggests that this principle can be applied to convert a complex-valued hologram into a pair of phase-only images, which could be integrated in certain ways to form a double-phase-only hologram (DPH). From Eq. (4.6), it can be seen that as A is positive, the angle α=cos−1(A) is confined to the right half of the Euclidean plane.

The generation of a double-phase hologram was achieved by the pioneering attempt of C. Hsueh and A. Sawchuk [8] in the mid-1970s. In their investigation, a Fourier hologram was decomposed into a pair of phase components based on Eq. (4.6). Next, each Fourier cell (which can be interpreted as a hologram pixel) was split into two sub-cells, each encoding one of the phase terms. In the simplest case, a Fourier cell is evenly separated into two partitions, as shown in Figure 4.4(a). In each partition, a window of half the length of the cell is shifted vertically by an amount ( τ1 or τ2 ), which is proportionally to the phase quantity it represents.

Figure 4.4 (a) A Fourier cell with two vertical shifted sub-cells for representing two phase quantities. (b) Interleaving a pair of phase components with the 1 × 2 macro-pixel topology. (c) Interleaving a pair of phase components with the 2 × 2 macro-pixel topology.

With the availability of electronic accessible phase-only spatial light modulators (SLMs), the display of DPH has been simplified. In the work of J.M. Florence and R.D. Juday [9], a complex-valued Fresnel hologram H(m,n) was decomposed into a pair of phase-only images, θ1(m,n) and θ2(m,n) :

H(m,n)=|H(m,n)|expiθ(m,n)=0.5[expiθ1(m,n)+expiθ2(m,n)].(4.7)

Although both exponential terms on the right-hand side of Eq. (4.7) are phase-only quantities with constant magnitude of unity, they have to be combined to the complex-valued hologram. To address this issue, the two phase images are combined into a double-phase hologram θM(m,n) through spatial interleaving. A straightforward way to interleave the pair of phase images is to copying the odd and even rows of θ1(m,n) and θ2(m,n) to the odd and even rows of θM(m,n) , respectively. The adjacent pair of pixels in θM(m,n) forms a 1×2 macro-pixel, with each of its element (referred to as a sub-pixel) contributing from either θ1(m,n) or θ2(m,n) , as shown in Figure 4.4(b). In another words, the pair of phase images are each downsampled by two times and placed at non-overlapping areas in θM(m,n) .

When the double-phase hologram is illuminated by a coherent beam, the diffracted waves corresponding to θ1(m,n) and θ2(m,n) will be generated and merged on the reconstruction plane. However, due to the downsampling of these two signals, their frequency spectra will be added with aliasing images. In order to attenuate the aliasing images, and to merge θ1(m,n) and θ2(m,n), certain optical filtering or processing has to be applied so that the reconstructed image will be similar to that of the original hologram H(m,n) . As pointed out by V. Arrizón and D. Sanchez-de-la-Llave [10], the region in the reconstruction plane having high fidelity (measured in terms of signal-to-noise ratio) is rather small along the direction of the macro-pixel. To enhance the fidelity, the symmetrical 2×2 distribution of sub-pixels in Figure 4.4(c) is often adopted in forming a macro-pixel.

In the 2×2 macro-pixel arrangement, the pair of phase components θ1(m,n) and θ2(m,n) are downsampled by the orthogonal uniform sampling lattices L1(m,n) and L2(m,n) , as shown in Figure 4.5(a) and 4.5(b). Each lattice is in the form of a checkerboard pattern, with the sampled and non-sampled points assigned values of 1 (white) and 0 (black), respectively. The result of multiplexing θ1(m,n) and θ2(m,n) with the two sampling lattices is shown in Figure 4.5(c):

Figure 4.5 (a) Sampling lattice L₁(m, n). (b) Sampling lattice L₂(m, n). (c) Multiplexing θ₁(m, n) and θ₂(m, n) with L₁(m, n) and L₂(m, n).

θM(m,n)=L1(m,n)θ1(m,n)+L2(m,n)θ2(m,n).(4.8)

The Fourier transform of θM(m,n) is given by

θ˜M(ωm,ωn)=L˜1(ωm,ωn)*θ˜1(ωm,ωn)+L˜2(ωm,ωn)*θ˜2(ωm,ωn),(4.9)

where B˜(ωm,ωn) denotes the Fourier transform of an arbitrary function B(m,n) , and ∗ is the convolution operator. Eq. (4.9) is therefore rewritten as

θ˜M(ωm,ωn)=[θ˜1(ωm,ωn)+θ˜2(ωm,ωn)]+A˜S(ωm,ωn)=H˜(ωm,ωn)+A˜S(ωm,ωn).(4.10)

The first term on the right-hand side of Eq. (4.10) is the Fourier transform of the original hologram H(m,n) , while the rest of the terms are aliasing components comprising of shifted replicas of the baseband signal. If the sampling interval is small enough, the aliasing components are widely spaced so that they could be suppressed to a large extent with a low-pass filter. For example, an optical filter realized with a standard 4f configuration and an iris around the center of the Fourier plane can be employed to extract a low-pass version HL(m,n) of the original complex hologram signal.

Numerical simulation on the double-phase macroblock method in converting a complex-valued hologram into a phase-only hologram is conducted with the MATLAB code in Section 4.8.2. First, a complex-valued hologram is generated for the double-depth image “HK1” in Figure 1.12. Next, the complex-valued hologram is converted into a pair of phase-only holograms with the double-phase macroblock method, based on the 1×2 and the 2×2 macroblock topologies. Subsequently, each phase-only hologram is low-pass filtered with the bandwidth limited to one-quarter of its original value, so that aliasing errors at the high frequency range are suppressed. The reconstructed images at the two focused planes of the phase holograms generated with the 1×2 topology and the 2×2 macroblock topologies are shown in Figures 4.6(a, b) and 4.6(c, d), respectively.

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