3.1.1 Dual SLM Holographic Display System

To overcome this problem, a holographic display formed by the cascade of a pair of SLMs is proposed by C. Stolz et al. [1] and R. Tudela et al. [2]. The optical setup is shown in Figure 3.1. The first SLM modulates the amplitude of the illumination beam according to the amplitude component of the hologram, while the second SLM modulates the phase of the optical wave according to the phase component of the hologram. The light wave that emerges from the two SLMs will be equivalent to the holographic signal emitted from the hologram, from which a reconstructed image of the 3-D scene can be observed. Although this is a straightforward implementation, the setup is bulky and expensive as two SLMs are involved. Due to the fine resolution of the display devices, precise optical alignment is required to ensure the correct amplitude and phase modulation of every light beam that passes through every pair of corresponding pixels of the two SLMs. Nevertheless, the method is effective and further research works have been conducted along this direction. An alternative two-SLM approach has been reported by L. Zhu and J. Wang [3] and A. Siemion et al. [4], who decomposed a complex-valued hologram into a pair of phase components, and displayed them with two cascaded phase-only SLMs. The concept has been extended by M. Makowski and his co-workers for displaying color Fourier holograms [5].

Figure 3.1 Concept of a dual, magnitude–phase coupled SLM holographic display system.

3.1.2 Split SLM Holographic Display System

The optical setup in Figure 3.1 can be much simplified if only a single SLM is used in the display system. Such an attempt has been made by Liu et al. [6] with the setup shown in Figure 3.2. Instead of separating the real and the imaginary components into two different SLMs, they are multiplexed on a single SLM. Referring to Figure 3.2, an amplitude-only SLM is uniformly split into two areas. The real component HR(m,n) and the imaginary component HI(m,n) of a complex-valued hologram are each displayed on one of these two areas. When the SLM is illuminated with a coherent wave, the optical beam that passes through each section is amplitude-modulated by the hologram pixels on it. The modulated light waves from the two sections are then merged to form the holographic signal of the hologram through some optical devices. From the holographic signal, a visible 3-D image of the object scene that is represented by the original complex hologram will be reconstructed.

Figure 3.2 A partitioned, single SLM holographic display system.

A different solution, which is also based on the split SLM configuration, has been proposed by H. Song et al. [7]. A hologram is decomposed into a pair of phase components, with each of them being displayed in separate sections on a phase-only SLM. Similar to [6], the SLM is illuminated with a coherent beam, and the optical waves emitted from the two sections are merged optically to give the complex-valued holographic signal.

There are two major problems associated with the split SLM display system. First, as a single SLM is used to display the two components of a complex-valued hologram, the display area is reduced by two times. The size of the hologram, and hence the space–bandwidth product, are decreased by the same factor. As a result, the field of vision of a hologram displayed on a split SLM is smaller than that of a single SLM. Second, optical accessories such as lenses and high-resolution gratings are required to combine the optical wave from the two sections of the SLM. The alignment of the SLM and the optical accessories has to be very precise. Otherwise, the two components of the hologram cannot be integrated to recover the original holographic signal and are heavily distorted on the reconstructed image. In view of these issues, it is desirable if a hologram can be encapsulated into a single component (either the amplitude or the phase component), so that it can be displayed with a single, non-partitioned SLM.

3.1.3 Amplitude-Only SLM Holographic Display System

In the study of optical holography in Section 1.4, a hologram can be represented solely by its magnitude component if it is mixed with an inclined reference wave. This is the classic way of recording hologram onto a photographic film. Following the same principle, an amplitude-only SLM can be used to display an off-axis hologram as the values of all the pixels are real quantities. As explained in Section 1.4, there are two major drawbacks. First, the reconstructed image of an off-axis hologram is accompanied by the zero-order beam and the twin image, both of which may contaminate the virtual image if the pixel size of the SLM is not fine enough. Second, the optical efficiency of an amplitude-only hologram is rather low, as hologram pixels are translucent (i.e., having transmissivity between opaque and transparent), which blocks or fails to reflect some of the light energy.

A second solution is to display the phase component of a hologram alone by modulating the phase angle of the optical beam that impinges on it. Being different from an amplitude-only hologram, the optical efficiency of a phase-only hologram is high, as the transmissivity of hologram pixels is close to 100%, and only the phase angle of the optical beam is changed. However, direct removal of the magnitude component of a hologram will lead to heavy distortion in the reconstructed image. To overcome this problem, the hologram has to be generated in a way that the magnitude component is more or less homogeneous. Such kind of holograms are referred to as phase-only holograms. In the past few decades there have been numerous research efforts to explore methods for generating phase-only holograms. In the subsequent part of this chapter, different methods that have been developed in recent years on the generation of phase-only Fresnel hologram will be reviewed.

3.2.1 Generating Phase-Only Hologram for a Single-Depth Image

The Gerchberg–Saxton algorithm, often referred to as GSA [8], is by far one of the most popular approaches for generating a digital phase-only hologram. The algorithm, named after its two inventors, Gerchberg and Saxton, was originally used to retrieve the phase of a pair of related light distributions. The principles of GSA have been extended to the generation of digital phase-only Fourier holograms. A Fourier hologram is a complex-valued image that is obtained from the Fourier transform of an image. Similar to what was mentioned in Section 3.1, it will be desirable to convert it into a phase-only component to increase the optical efficiency.

Figure 3.3 depicts a variation of the GSA known as the “iterative Fourier transform algorithm” (IFTA), which is applied to generate a phase-only Fourier hologram. The pair of light distributions are the intensity I(m,n) of the source object on the image plane and its digital Fourier hologram H(m,n) on the hologram plane. The objective is to derive a hologram so that the reconstructed image of its phase component HP(m,n) is similar to I(m,n) . Intuitively, this seems to contradict the theory of holography as the phase component of a hologram alone is insufficient to recover the original image. However, this dilemma can be alleviated if HP(m,n) is being solely required to reconstruct a modified version of I(m,n) given by J(m,n)≈I(m,n)eiϕ(m,n) , where eiϕ(m,n) is a pure phase term. The latter provides an extra dimension of freedom for partially absorbing the distortion caused by dropping the magnitude component of the hologram. As human perception only responds to the intensity of light, introduction of the extra phase term does not affect the appearance of the image. In IFTA, computation of a phase-only hologram HP(m,n) is obtained through repetitive rounds of iterations, explained below.

Figure 3.3 The iterative Fourier transform algorithm (IFTA) for generating a phase-only Fourier hologram.

To start with, let t denote the current round of iteration, and Gt(m,n)=eiϕt(m,n) is the phase-only hologram obtained in the current epoch. Initially, at t=0 , each pixel in the phase-only hologram ϕt=0(m,n) is assigned a random value within the range [−π,π] .

Next, the phase-only hologram is inverse Fourier transformed to a complex image Jt(m,n)=|Jt(m,n)|eiθt(m,n) on the image plane, as given by

Jt(m,n)=|Jt(m,n)|eiθt(m,n)=F−1[Gt(m,n)].(3.1)

Subsequently, an amplitude constraint is imposed with which the magnitude of Jt(m,n) is replaced by I(m,n) , resulting in a phase-added image IPt(m,n)=I(m,n)eiϕt(m,n) . The phase-added image IPt(m,n) is then Fourier transformed into a Fourier hologram Ht(m,n)=|Ht(m,n)|eiϕH;t(m,n) :

Ht(m,n)=|Ht(m,n)|eiϕH;t(m,n)=F[IPt(m,n)],(3.2)

and the phase component ϕH;t(m,n) of the hologram is retained as the phase-only hologram Gt+1(m,n) , completing the current round of iteration. The conversion of the signals from the image plane to the hologram plane, and vice versa, is commonly referred to as forward and backward propagation, respectively.

The above process is repeated for a certain number of epochs, or until the difference between I(m,n) and |Jt(m,n)| is below a preset threshold. Note that in Eq. (3.2), the amplitude constraint is imposed so that the magnitude of the image to be converted to the hologram is always enforced to be identical to the original image I(m,n) . The phase term eiϕt(m,n) is allowed to vary freely according to the recent round of iteration. This has the effect of absorbing part of the errors caused by retaining only the phase component of the hologram. Upon completion of the process at the Tth epoch, the phase component GT(m,n)=eiϕT(m,n) is taken to be the phase-only hologram HP(m,n) . With more iterations, the reconstructed image of HP(m,n) will become increasingly similar to the source image I(m,n) .

The IFTA for generating Fourier phase-only holograms can be easily extended to the generation of Fresnel phase-only holograms, simply by replacing the forward/inverse Fourier transform with the forward/inverse Fresnel transform in Figure 3.3. This IFTA approach reported in [9].

There are several major disadvantages of the IFTA. First, for an arbitrary source image it is almost impossible to estimate how many rounds of iterations are required to generate the desired phase-only hologram (i.e., one with a reconstructed image that is close to the original one). Second, the reconstructed image is quite noisy as a result of the distortions caused by removing the magnitude component of the hologram. Third, the errors that can be absorbed into the phase term eiϕT(m,n) are quite limited. As a result, it is unlikely that the quality of the reconstructed image of HP(m,n) can be improved significantly simply by increasing the number of iterations. Fourth, as shown in Figure 3.3, the IFTA is only applicable for generating the phase-only hologram of a planar image that is parallel to the hologram plane. The subsequent parts of this section will demonstrate a number of methods for improving the performance of IFTA.

Simulation of the IFTA for generating the Fresnel hologram of a single-depth image is illustrated with the MATLAB code shown in Section 3.4.1. The object space adopted in the simulation is the single-depth image “Apple,” shown in Figure 3.4(a), which is located at an axial distance of 0.08 m from the hologram. The image is converted into a phase-only Fresnel hologram with the IFTA method. The reconstructed images on the focused plane of the phase-only holograms that are generated with one, three, and five iterations of the IFTA are shown in Figure 3.4(b–d). In the reconstructed images, only the region that is occupied by the image of the apple is shown. The reconstructed images are similar to the source image, and contaminated with a certain amount of random noise. With more iterations, the noise level is lowered and the reconstructed images become clearer.

Figure 3.4 (a) Source image “Apple.” (b–d) Reconstructed image from a phase-only Fresnel hologram that is generated with one, three, and five round(s) of IFTA, respectively.

3.2.2 Enhanced IFTA: Mixed-Region Amplitude Freedom Method

In Section 3.2.1, the reconstructed image of a phase-only hologram generated with the IFTA suffers noise contamination and slight distortion. The noise signal is caused by the removal of the magnitude component in the hologram, and although part of it can be absorbed into the phase term eiϕT(m,n) through the IFTA, the remaining ones will be distributed rather uniformly as random noise in the reconstructed image. In other words, every part of the reconstructed image will be added with a certain amount of random noise. After a certain number of iterations, there will not be further improvement of the noisy appearance of the reconstructed image. To overcome this problem, an enhanced IFTA method known as “mixed-region amplitude freedom” (MRAF) was proposed in [10]. The method was intended for designing holographic atom traps, but it can also be applied in generating phase-only Fourier or Fresnel holograms. In essence, the source image of the object is housed inside a new image of a larger size, so that the majority of noise resulting from the IFTA can be dispatched to the region that is not occupied by the source image. Details of the MRAF method are shown in Figure 3.5.

Figure 3.5 The “mixed-region amplitude freedom” (MRAF) for generating a phase-only Fresnel hologram, with the incorporation of the signal and the noise regions.

The source image I(m,n) is first moved to the center of a larger canvas, resulting in a bigger image IE(m,n) . The part that is occupied by I(m,n) (i.e., the dotted region) and the remaining areas on IE(m,n) are labeled as the “signal region” S and the “noise region,” respectively. Initially, at time t=0 , the pixels of the phase-only hologram GE;t=0(m,n) are assigned random values within the range [−π,π].

Next, the phase-only hologram is inverse transformed to the reconstructed image JE;t(m,n)=|JE;t(m,n)|eiϕt(m,n) . On the image plane, the image IPE;t(m,n) is obtained by applying an amplitude constraint on JE;t(m,n) :

IPE;t(m,n)={I(m,n)eiϕt(m,n)if (m,n)∈SJt(m,n)eiϕt(m,n)otherwise.(3.3)

Equation (3.3) shows that the amplitude constraint enforces the signal region of the reconstructed image to duplicate the source image, whereas in the area covered by the noise region the reconstructed image is not altered. Grossly speaking, the noise region is not subject to amplitude constraint and can be altered freely in the iterative process. As the noise region is separated from the signal region containing the source image, it can be used to accommodate part of the distortions that are caused by removing the magnitude component of the hologram.

The amplitude constraint image is then forward transform to its hologram, and the phase component is retained as the phase-only hologram GE;t(m,n) , completing the current round of iteration. The above process is repeated until certain criteria, such as reaching the maximum number of iterations, have been met. In each round of iteration, the image within the signal region S is preserved, while the signal in the noise region will be altered, progressively shifting the noise signal from the signal region to the noise region.

Simulation of the MRAF with the incorporation of the signal and noise regions is conducted with the program given in Section 3.4.2. The reconstructed images of the phase-only holograms that are generated with one, three, and five iterations of the MRAF method are shown in Figure 3.6(a–c). The reconstructed image is composed of the image of the apple in the signal region, together with a noisy background in the noise region. Magnified views of the images in the signal region are shown in Figure 3.6(e–g), and for comparison the original image “Apple” is shown again in Figure 3.6(d). Compared with the results in Figure 3.4(b–d), it is evident that the noise contamination on the reconstructed images has been reduced.

Figure 3.6 (a–c) Reconstructed image from a phase-only Fresnel hologram that is generated with one, three, and five rounds of MRAF. (d) Source image “Apple.” (e–g) Reconstructed image from a phase-only Fresnel hologram that is generated with one, three, and five round(s) of MRAF.

3.2.3 Noise Reduction with IFTA Multiple Frame Averaging

The quality of a phase-only hologram HP(m,n) generated with IFTA is dependent on the initial randomization of the phase values of the hologram pixels. Suppose the object space is a single planar image I(m,n) that is parallel to the hologram plane, the reconstructed image I(m,n) of the phase-only hologram can be expressed as

J(m,n)=I(m,n)+ℵ,(3.4)

where ℵ represents the noise contamination of the reconstructed image, which is dependent on the initial condition.

To reduce the noise contamination, IFTA is applied to generate a set of phase-only holograms of the same input image, each initialized with different phase randomization. Each of the holograms is referred to as a sub-hologram. Due to the deviation in initial conditions, the reconstructed images of each sub-hologram will be added with a different noise signal. The expectation of the reconstructed image J(m,n) obtained from averaging the reconstructed images of all the sub-holograms is given by

(3.5)E[J(m,n)]=E[I(m,n)]+E[ℵ]=I(m,n)+E[ℵ].

If the initial randomization is based on white noise, it is likely that the noise in the result will also be similar to white noise. With a sufficient number of samples, E[ℵ] will be close to a homogeneous background and E[J(m,n)] is practically equivalent to the source image.

The above principle can be applied to reduce the noise signal in an IFTA-generated phase-only hologram, and is outlined as follows. To start with, a set of phase-only sub-holograms representing the same source image is generated with the IFTA based on different initial randomization. Subsequently, the sub-holograms are displayed in rapid succession on an SLM. Due to the persistence of vision of human perception, the sequence of reconstructed images will appear to be merged into a single image. If the number of sub-holograms is sufficiently large, the reconstructed image will be close to the source image.

Simulation of this IFTA multiple frame averaging (IFTA-MFA) noise reduction method is conducted here. Similar to previous simulations, the object space is the single-depth image “Apple” in Figure 3.4(a). In the simulation, two or more phase-only sub-holograms are generated with the IFTA under different initial randomizations of the holograms. The magnitude of the reconstructed images of these sub-holograms on the focused plane are averaged to simulate the persistence-of-vision effect.

The reconstructed images obtained with the averaging of 2–5 frames of phase-only sub-holograms that are generated with one iteration of IFTA are shown in Figures 3.7(a–d). Compared with the results in Figure 3.4, it is evident that the noise contamination of the reconstructed images has been reduced, even though only one iteration has been applied in the generation of each phase-only sub-hologram.

Figure 3.7 Averaging multiple frames of reconstructed images of phase-only Fresnel holograms that are generated with one iteration of IFTA, but under different initial random noise (a)–(d) averaging 2–5 frames, respectively.

The IFTA-MFA has one major disadvantage. In order to achieve better noise reduction, more sub-hologram frames have to be generated and displayed at high speed on the SLM. As a result, the SLM has to operate at a substantially higher refresh rate. With the assumption that the frame rate of a holographic video clip is at the video rate of 25 hologram frames per second, the video refresh rate for five sub-holograms will be 5×25=125 frames per second.

3.2.4 Generating Phase-Only Hologram of a Multi-Depth Object with IFTA

The IFTA described previously is employed in generating a phase-only hologram of a single-depth image that is parallel to the hologram plane. In practice, object points in a 3-D object space could be residing in different depth planes. A 3-D object space can be represented as a sequence of uniformly spaced layers Λ=[Ik(m,n)]|1≤k≤K . Each layer is the intensity of a source image that is parallel to the hologram plane, and contains the pixels that are within the layer. In [11], IFTA was extended to the generation of an object scene constituted by multiple layers of planar images. The process is depicted in Figure 3.8.

Figure 3.8 The IFTA for generating a phase-only Fresnel hologram of multi-depth image planes.

Initially, at time t=0 , the phase-only hologram Ht(m,n) is randomly generated, with each hologram pixel assigned a random phase angle in the range [0,2π) . The magnitude of all the hologram pixels is set to a constant value of unity, resulting in an initial phase-only hologram. The phase-only hologram is then back propagated to each of the image planes at layer k , where 1≤k≤K , resulting in a reconstructed image Jk;t(m,n) :

Jk;t(m,n)=|Jk;t(m,n)|eiϕk;t(m,n),(3.6)

where ϕk;t(m,n) is the phase component of the reconstructed image Ht(m,n) on the kth focused plane. An amplitude constraint image IPk;t(m,n) is imposed on the image at each reconstruction plane by replacing the magnitude of Jk;t(m,n) with the intensity of the corresponding source image Ik(m,n) :

IPk;t(m,n)=Ik(m,n)eiϕk;t(m,n).(3.7)