**4. 2D Fresnel holography**

Previous sections have been concerned with far-field (or Fraunhofer) diffraction, in which the RPF *Fxy* and hologram *huv* are related by the Fourier transform:

$$F\_{\mathbf{x}y} = \mathcal{F}[h\_{\mathbf{u}\mathbf{v}}] \tag{41}$$

In the near-field (or Fresnel) propagation regime, RPF and hologram are related by the Fresnel transform which, using the same notation, can be written as

$$F\_{\mathbf{x}y} = \mathcal{FR}[h\_{\mathbf{u}\mathbf{v}}] \tag{42}$$

and the RPF *Fxy* at a distance *z* is related to the *P* × *Q*-pixel hologram *huv* of feature size Δ*<sup>x</sup>* × Δ*<sup>y</sup>* by Fresnel diffraction so that

$$F\_{xy} = \mathcal{F}\mathcal{R}[h\_{\mu\upsilon}] = F\_{xy}^{(1)} \cdot \mathcal{F}\left[f\_{\mu\upsilon}^{(2)}h\_{\mu\upsilon}\right] \tag{43}$$

where

18 Will-be-set-by-IN-TECH

highly saturated primaries associated with laser-based display system, but that the speckle artefacts traditionally associated with this method of projection are substantially suppressed.

Fig. 8. Projected image at WVGA resolution resulting from a phase-only holographic projection system, employing the techniques described in this chapter, manufactured by Light Blue Optics Ltd. In this instance, *λ<sup>r</sup>* = 642 nm, *λ<sup>g</sup>* = 532 nm and *λ<sup>b</sup>* = 445 nm.

**4. 2D Fresnel holography**

Δ*<sup>x</sup>* × Δ*<sup>y</sup>* by Fresnel diffraction so that

Several methods can be combined in a holographic projector in order to reduce speckle. In particular, the use of multiple holograms per video frame is beneficial to the speckle contrast; since *N* phase-independent subframes per video frame are shown within the eye's integration period, then the eye acts to add *N* independent speckle patterns on an intensity basis, and the contrast of the low-frequency components of the speckle in the field *Vxy* falls as *N*1/2. Due to computational and LC switching speed limitations, *N* cannot be increased indefinitely so additional methods can be combined to further reduce the speckle contrast. The presence of an intermediate image plane between the lens pair *L*<sup>3</sup> and *L*<sup>4</sup> makes it straightforward to employ optical speckle reduction techniques, as previously presented by Buckley (2008c).

Previous sections have been concerned with far-field (or Fraunhofer) diffraction, in which the

In the near-field (or Fresnel) propagation regime, RPF and hologram are related by the Fresnel

and the RPF *Fxy* at a distance *z* is related to the *P* × *Q*-pixel hologram *huv* of feature size

*xy* · F *f* (2) *uv huv* 

*Fxy* <sup>=</sup> FR[*huv*] = *<sup>F</sup>*(1)

*Fxy* = F[*huv*] (41)

*Fxy* = FR[*huv*] (42)

(43)

RPF *Fxy* and hologram *huv* are related by the Fourier transform:

transform which, using the same notation, can be written as

$$F\_{xy}^{(1)} = \frac{\Delta\_{\mathbf{x}} \Delta\_{\mathbf{y}}}{j\lambda z} \exp\frac{j2\pi z}{\lambda} \exp\frac{j\pi}{\lambda z} \left[ \left(\frac{\mathbf{x}}{P\Delta\_{\mathbf{x}}}\right)^2 + \left(\frac{y}{Q\Delta\_{\mathbf{y}}}\right)^2 \right] \tag{44}$$

and

$$f\_{\mu\upsilon}^{(2)} = \exp\frac{\dot{f}\pi}{\lambda z} \left(\mu^2 \Delta\_x^2 + v^2 \Delta\_y^2\right). \tag{45}$$

so that the dimensions of the RPF are *<sup>λ</sup><sup>z</sup>* <sup>Δ</sup>*<sup>x</sup>* <sup>×</sup> *<sup>λ</sup><sup>z</sup>* <sup>Δ</sup>*<sup>y</sup>* , consistent with the size of RPF in the Fraunhofer diffraction regime as per Schnars & Juptner (2002). The Fresnel diffraction geometry is illustrated in Figure 9.

Fig. 9. Fresnel diffraction geometry. When the hologram *huv* is illuminated by coherent light, the RPF *Fxy* at a distance *z* is determined by Fresnel (or near-field) diffraction.

As previously shown by Dorsch et al. (1994); Fetthauer et al. (1995), it is straightforward to generalize hologram generation algorithms to the case of calculating Fresnel holograms. Here, the OSPR algorithm 1 is employed, replacing the conventional Fourier transform step by the discrete Fresnel transform of equation 43. The samples of the discrete Fresnel transform are found to be distributed as

$$\begin{split} \mathfrak{R}[F\_{\mathbf{xy}}(\mathbf{0})] &\sim \mathcal{N}[\mu\_{I}P\_{\prime}\sigma\_{r}^{2}P^{\prime}Q^{\prime}] \\ \mathfrak{G}[F\_{\mathbf{xy}}(\mathbf{0})] &\sim \mathcal{N}[\mu\_{i}P\_{\prime}\sigma\_{i}^{2}P^{\prime}Q^{\prime}] \\ \mathfrak{R}[F\_{\mathbf{xy}}] &\sim \mathcal{N}[\mathbf{0}\_{\prime}(\sigma\_{r}^{2}+\sigma\_{i}^{2})P^{\prime}Q^{\prime}/2] \\ \mathfrak{G}[F\_{\mathbf{xy}}] &\sim \mathcal{N}[\mathbf{0}\_{\prime}(\sigma\_{r}^{2}+\sigma\_{i}^{2})P^{\prime}Q^{\prime}/2]. \end{split} \tag{46}$$

where *P*� = *P*Δ<sup>2</sup> *<sup>x</sup>* and *Q*� = *Q*Δ<sup>2</sup> *y*.

The use of Fresnel holography has in two beneficial effects. Firstly, the diffracted near-field at the propagation distance *z* does not contain the conjugate image evident in the Fraunhofer region, in which *z* is necessarily greater than the Goodman (1996) distance. Second, because Fresnel propagation is characterized by a distance *z*, it is evident that the hologram incorporates lens power determined by the properties of the computed hologram, rather than the optical system. It therefore follows that the lens count in a holographic projection system could be reduced simply by removing *L*<sup>3</sup> of Figure 7, employing instead a Fresnel hologram which encodes the equivalent lens power *z* = *f*3.

Fig. 11. Optical setup (a) and resultant RPF (b) of a lens-sharing projector design, utilizing a Fresnel hologram with *z* = 100mm displayed on the microdisplay. Polarizers are omitted for clarity. The demagnification caused by the combination of *L*<sup>4</sup> and the hologram causes

(a) (b)

SLM

Computer-Generated Phase-Only Holograms for Real-Time Image Display 297

of approximately three. Polarizers were used to remove the large zero order associated with Fresnel diffraction, but have been omitted from Figure 11(a) for clarity. The angle of reflection

An example image, projected on a screen and captured in low-light conditions with a digital camera, is shown in Figure 11(b). The RPF has been optically enlarged by factor of approximately three due to the demagnification of the hologram pixels and, as the architecture is functionally equivalent to the simple holographic projector of Figure 7, the image is in focus

A 3D hologram of an object is simply a recording of the complex electromagnetic field (produced by light scattered by the object) at a plane in front of the object. By Huygens' principle as detailed in Hecht (1998), if we know the EM field distribution on a plane *P*, we can propagate Huygens wavelets through space to evaluate the field at any point in 3D space. As such, the plane hologram encodes all the information necessary to view the object from any position and angle in front of the plane and hence is, in theory, optically indistinguishable from the object. In practice, limitations in the pixel resolution of the recording medium restricts the viewing angle which, as in the 2D case, varies inversely with the pixel size Δ,

Consider a plane, perpendicular to the z-axis, intersecting the origin, and one point source emitter of wavelength *λ* and amplitude *A* at position (*x*, *y*, *z*) behind it. The field *h*(*u*, *v*)

with *r* =

(*u* − *x*)

<sup>2</sup> <sup>+</sup> (*<sup>v</sup>* <sup>−</sup> *<sup>y</sup>*)

<sup>2</sup> + *z*<sup>2</sup> (47)

present at the plane (*u*, *v*, *z* = 0) - i.e. the hologram *h*(*u*, *v*) - is given by

*λ r* 

*<sup>j</sup>λr*<sup>2</sup> exp <sup>2</sup>*j<sup>π</sup>*

at all points and, due to the use of Fresnel holography, the conjugate image is absent.

optical enlargement of the RPF by a factor of approximately three.

200mm

*f* = 36mm

was also kept small to avoid defocus aberrations.

Fibre-coupled laser, = 532 nm

**5. 3D holography**

as given by equation 37.

*<sup>h</sup>*(*u*, *<sup>v</sup>*) = *ZA*

## **4.1 Holographic projector with variable demagnification**

In the Fourier projection system of Figure 7 the demagnification *D* of the hologram pixels, and the concomitant enlargement of the RPF, is determined optically and is given by the ratio *D* = *f*3/ *f*4. The use of a Fresnel hologram displayed on a dynamically addressable microdisplay, however, would allow for a novel variable demagnification effect since the effective focal length of the Fresnel hologram encoding *L*<sup>3</sup> could be varied simply by recomputing the hologram.

An experimental verification of this variable demagnification principle was performed by removing *L*<sup>3</sup> of Figure 7 and setting *f*<sup>4</sup> = 100 mm. Three Fresnel holograms were calculated using OSPR with *N* = 24 subframes, each of which were designed to form a target image in the planes *z* = 100 mm, *z* = 200 mm and *z* = 400 mm. A microdisplay with pixel pitch Δ*<sup>x</sup>* = Δ*<sup>y</sup>* = 13.62*μ*m was used to display the holograms, and the resulting RPFs - which were reconstructed at *λ* = 532 nm and imaged onto a non-diffusing screen - were captured with a digital camera. The results are shown in Figure 10, and clearly show the RPF scaling caused by the variable demagnification introduced by each of the Fresnel holograms.

Fig. 10. Experimental results of the variable demagnification principle described. The scale of the RPFs (a) to (c) is determined by the effective focal length *z* of *N* = 24 sets of Fresnel holograms displayed on a dynamically addressable microdisplay.

### **4.2 Lens sharing in a holographic projector**

In the previous section, it was shown that lens *L*<sup>3</sup> of the demagnification lens pair could be removed by encoding the equivalent lens power into the hologram. From inspection of Figure 7, it is clear that the same argument could also be applied to *L*<sup>2</sup> of the beam-expansion lens pair. It follows that, if *f*<sup>2</sup> = *f*3, the common lens can be shared between the beam-expansion and demagnification assemblies by encoding it into a Fresnel hologram displayed on a reflective microdisplay. The remaining lens *L*<sup>4</sup> is typically the smallest in the optical path in order to maximize the demagnification *D*.

An experimental projector was constructed to demonstrate the lens-sharing concept, and the optical configuration is shown in Figure 11(a). A fiber-coupled laser was used to illuminate the same reflective microdisplay, which displayed *N* = 24 sets of Fresnel holograms each with *z* = 100 mm. Since the light from the fiber end was highly divergent, this removed the need for lens *L*1. The output lens *L*<sup>4</sup> had a focal length of *f*<sup>4</sup> = 36 mm, giving a demagnification *D*

Fig. 11. Optical setup (a) and resultant RPF (b) of a lens-sharing projector design, utilizing a Fresnel hologram with *z* = 100mm displayed on the microdisplay. Polarizers are omitted for clarity. The demagnification caused by the combination of *L*<sup>4</sup> and the hologram causes optical enlargement of the RPF by a factor of approximately three.

of approximately three. Polarizers were used to remove the large zero order associated with Fresnel diffraction, but have been omitted from Figure 11(a) for clarity. The angle of reflection was also kept small to avoid defocus aberrations.

An example image, projected on a screen and captured in low-light conditions with a digital camera, is shown in Figure 11(b). The RPF has been optically enlarged by factor of approximately three due to the demagnification of the hologram pixels and, as the architecture is functionally equivalent to the simple holographic projector of Figure 7, the image is in focus at all points and, due to the use of Fresnel holography, the conjugate image is absent.

### **5. 3D holography**

20 Will-be-set-by-IN-TECH

296 Advanced Holography – Metrology and Imaging

In the Fourier projection system of Figure 7 the demagnification *D* of the hologram pixels, and the concomitant enlargement of the RPF, is determined optically and is given by the ratio *D* = *f*3/ *f*4. The use of a Fresnel hologram displayed on a dynamically addressable microdisplay, however, would allow for a novel variable demagnification effect since the effective focal length of the Fresnel hologram encoding *L*<sup>3</sup> could be varied simply by recomputing the

An experimental verification of this variable demagnification principle was performed by removing *L*<sup>3</sup> of Figure 7 and setting *f*<sup>4</sup> = 100 mm. Three Fresnel holograms were calculated using OSPR with *N* = 24 subframes, each of which were designed to form a target image in the planes *z* = 100 mm, *z* = 200 mm and *z* = 400 mm. A microdisplay with pixel pitch Δ*<sup>x</sup>* = Δ*<sup>y</sup>* = 13.62*μ*m was used to display the holograms, and the resulting RPFs - which were reconstructed at *λ* = 532 nm and imaged onto a non-diffusing screen - were captured with a digital camera. The results are shown in Figure 10, and clearly show the RPF scaling caused

(a) *z* = 100 mm (b) *z* = 200 mm (c) *z* = 400 mm

Fig. 10. Experimental results of the variable demagnification principle described. The scale of the RPFs (a) to (c) is determined by the effective focal length *z* of *N* = 24 sets of Fresnel

In the previous section, it was shown that lens *L*<sup>3</sup> of the demagnification lens pair could be removed by encoding the equivalent lens power into the hologram. From inspection of Figure 7, it is clear that the same argument could also be applied to *L*<sup>2</sup> of the beam-expansion lens pair. It follows that, if *f*<sup>2</sup> = *f*3, the common lens can be shared between the beam-expansion and demagnification assemblies by encoding it into a Fresnel hologram displayed on a reflective microdisplay. The remaining lens *L*<sup>4</sup> is typically the smallest in the optical path

An experimental projector was constructed to demonstrate the lens-sharing concept, and the optical configuration is shown in Figure 11(a). A fiber-coupled laser was used to illuminate the same reflective microdisplay, which displayed *N* = 24 sets of Fresnel holograms each with *z* = 100 mm. Since the light from the fiber end was highly divergent, this removed the need for lens *L*1. The output lens *L*<sup>4</sup> had a focal length of *f*<sup>4</sup> = 36 mm, giving a demagnification *D*

holograms displayed on a dynamically addressable microdisplay.

**4.2 Lens sharing in a holographic projector**

in order to maximize the demagnification *D*.

by the variable demagnification introduced by each of the Fresnel holograms.

**4.1 Holographic projector with variable demagnification**

hologram.

A 3D hologram of an object is simply a recording of the complex electromagnetic field (produced by light scattered by the object) at a plane in front of the object. By Huygens' principle as detailed in Hecht (1998), if we know the EM field distribution on a plane *P*, we can propagate Huygens wavelets through space to evaluate the field at any point in 3D space. As such, the plane hologram encodes all the information necessary to view the object from any position and angle in front of the plane and hence is, in theory, optically indistinguishable from the object. In practice, limitations in the pixel resolution of the recording medium restricts the viewing angle which, as in the 2D case, varies inversely with the pixel size Δ, as given by equation 37.

Consider a plane, perpendicular to the z-axis, intersecting the origin, and one point source emitter of wavelength *λ* and amplitude *A* at position (*x*, *y*, *z*) behind it. The field *h*(*u*, *v*) present at the plane (*u*, *v*, *z* = 0) - i.e. the hologram *h*(*u*, *v*) - is given by

$$h(\mu, v) = \frac{ZA}{j\lambda r^2} \exp\left(\frac{2j\pi}{\lambda}r\right) \text{ with } r = \sqrt{\left(\mu - x\right)^2 + \left(v - y\right)^2 + z^2} \tag{47}$$

(a) (b)

Computer-Generated Phase-Only Holograms for Real-Time Image Display 299

(c) (d) (e)

This chapter has described a number of technical innovations that have enabled the realization

By defining a new psychometrically determined optimization metric that is far more suited to human perception than the conventional MSE measure, a method for the generation of phase-only holograms which results in perceptually pleasing video-style images was demonstrated. This allows the realization of phase-only holographic video projection systems which, for the first time, overcome the twin barriers of the computational complexity of calculating diffraction patterns in real time and the poor quality of the resultant images. Using these techniques, the chapter has demonstrated algorithms and methods for the generation of 2D and 3D images in the Fraunhofer and Fresnel regimes. As shown in simulation and by preliminary experiment, the RPFs produced by the calculated holograms exhibit a substantial improvement in quality and a reduction in computation time on the scale

A number of commercially available products, notably from Light Blue Optics Inc. (2010), now employ variants of this technology. This chapter, and the information contained herein, contains a thorough description of state-of-the-art holographic projection technology and provides a complete reference to enable an interested reader to simulate, construct and

of orders of magnitude compared to the other techniques demonstrated thus far.

Fig. 12. Simulated RPFs produced at pinhole aperture locations *K*<sup>1</sup> (a), and *K*<sup>2</sup> (b), and

experimental results (c)-(e) from Cable (2006).

of a real-time, phase-only holographic projection technology.

characterize a 2D or 3D phase-only holographic projector.

**6. Conclusion**

If we regard a 3D scene as *M* sources of amplitude *Ai* at (*xi*, *yi*, *zi*), the linear nature of EM propagation results in the total field hologram *h*(*u*, *v*)

$$h(\boldsymbol{u}, \boldsymbol{v}) = \sum\_{i=1}^{M} \frac{z\_i A\_i}{j \lambda r\_i^2} \exp\left(\frac{2j\pi}{\lambda} r\_i\right) \text{ with } r = \sqrt{(\boldsymbol{u} - \boldsymbol{x}\_i)^2 + (\boldsymbol{v} - \boldsymbol{y}\_i)^2 + z\_i^2} \tag{48}$$

If we wish to sample *h*(*u*, *v*) over the region *umin* ≤ *u* ≤ *umax*, *vmin* ≤ *v* ≤ *vmax* to form a *P* × *P* hologram *huv*, then *ri* becomes:

$$r\_i = \sqrt{\left(u\_{\min} + u \frac{u\_{\max} - u \min}{P} - x\_i\right)^2 + \left(v\_{\min} + v \frac{v\_{\max} - v \min}{P} - y\_i\right)^2 + z\_i^2} \tag{49}$$

In Algorithm 2 we present a version of OSPR that generates *N* full-parallax 3D holograms *h* (*n*) *uv* , *<sup>n</sup>* <sup>=</sup> <sup>1</sup> ··· *<sup>N</sup>*, for a given set of *<sup>M</sup>* point sources *Ai*, *<sup>i</sup>* <sup>=</sup> <sup>1</sup> ··· *<sup>M</sup>*, at positions (*xi*, *yi*, *zi*) in the image plane (*x*, *y*, *z*).

**inputs** : *M* point sources of amplitude *Ai* at position (*xi*, *yi*, *zi*), *M*, *N* **output**: *N* binary phase holograms *h* (*n*) *uv* of size *<sup>P</sup>* <sup>×</sup> *<sup>P</sup>* pixels

$$\begin{array}{l|l} \text{for } n \leftarrow 1 \text{ to } N/2 \text{ do} \\ \qquad \text{Let } h\_{uv}^{(n)} = \sum\_{i=1}^{M} \frac{\sum\_{j} \alpha\_{i}}{j \omega\_{i}^{n}} \exp\left(j \Phi\_{i}^{(n)} + \frac{2j\pi}{\lambda} r\_{i}\right) \text{ with } r\_{i} \text{ as equation 49 and where } \Phi\_{i}^{(n)} \text{ is} \\ \qquad \text{unformly distributed in the interval } [0; 2\pi] \\ \qquad \text{Let } m\_{uv}^{(n)} = \mathfrak{K} \left\{ g\_{uv}^{(n)} \right\} \text{ where } \mathfrak{K} \{\cdot\} \text{ represents the real part} \\ \qquad \text{Let } m\_{uv}^{(n+N/2)} = \mathfrak{J} \left\{ g\_{uv}^{(n)} \right\} \text{ where } \mathfrak{J} \{\cdot\} \text{ represents the imaginary part} \\ \qquad \text{Let } h\_{uv}^{(n)} = \begin{cases} -1 \text{ if } m\_{uv}^{(n)} < 0 \\ 1 \text{ if } m\_{uv}^{(n)} \ge 0 \end{cases} \\ \end{cases}$$

**end**

**Algorithm 2:** The OSPR algorithm modified to calculate *N P* × *P* pixel full-parallax 3D holograms *huv* for a given set of *M* point sources *Ai*.

To test this algorithm, we consider the calculation of *N* = 8 holograms of resolution 512 × 512 and size 2 mm × 2 mm centered at the origin of our plane *P*, giving a pixel size of Δ = 4 *μ*m and hence a viewing angle of around 9 degrees under coherent red illumination *λ* = 632 nm. The 3D scene used was a set of *M* = 944 point sources that formed a wireframe cuboid of dimensions 12 cm × 12 cm × 18 cm, located at a distance of 1.91 m from the plane.

The simulated RPFs produced were calculated by propagating Huygens wavelets from the *N* holograms *h* (*i*) *uv* in turn through a pinhole aperture *K* onto a virtual screen (a plane perpendicular to the line from the center of the cube to the pinhole), and recording the intensity distribution on the screen <sup>|</sup>*F*(*i*) *xy* | 2; as before, the time-averaged percept is *Vxy* = 1 *<sup>N</sup>* <sup>∑</sup>*<sup>N</sup> i*=1 *F*(*i*) *xy* 2 . Simulated views of the hologram from two positions - *K*<sup>1</sup> = (0.20, −0.39, 1.95) and *K*<sup>2</sup> = (0.39, −0.39, 1.92) - are shown in Figures 12(a)-(b) together with experimental results from Cable (2006) in Figures 12(c)-(e).

(a) (b)

Fig. 12. Simulated RPFs produced at pinhole aperture locations *K*<sup>1</sup> (a), and *K*<sup>2</sup> (b), and experimental results (c)-(e) from Cable (2006).
