## Abstract

Laser projection based on phase modulation promises several advantages over amplitude modulation. We examine and compare the merits of two phase modulation techniques; phase-only computer generated holography and generalized phase contrast, for the application of dynamic laser image projection. We adopt information theory as a guiding framework and analyze information-relevant metrics such as space-bandwidth product, output display resolution, efficiency, speckle noise, computational load and device requirements. The analysis takes into account the perspective of potential end-users.

©2008 Optical Society of America

## 1. Introduction

Beyond display applications, dynamic grayscale laser projections can serve as a generic technology for spatially controlled light-matter interaction that can benefit various applications, such as in materials processing [1], microscopy [2], and non-contact optical manipulation at microscopic scales [3], to name a few. Adapting the conventional amplitude-based display technology is a natural candidate and its potential for dynamic laser projection has been demonstrated in substrate patterning [4]. Alternative techniques offer promising enhancements and are being explored and developed. These include computer-generated holography [5], which offers robustness to dead pixels, and generalized phase contrast [6, 7], which achieves high efficiency with an encoding simplicity akin to conventional displays.

Computer-generated holography (CGH) extends Gabor’s holography [8] to generate desired fields instead of simply reading out previously recorded information. Conventional holography exploits interferometry to store a field’s amplitude and phase information as generalized gratings onto an intensity-dependent storage medium. The stored field is retrieved by a readout beam that diffracts through the recorded gratings, usually after processing the storage medium. In computer holography, the gratings are mathematically synthesized using numerical models, eliminating the need for the recording and allowing for the construction of three-dimensional images of the mathematical object upon diffractive readout. With its capacity for constructing three-dimensional fields, it seems reasonable to expect that it can only be better in projecting two-dimensional images, with the reduced complexity. Indeed, computer generated holography has been proposed as a generic display technology, from projection of planar images, to stereoscopic projections, and to true-three-dimensional projections. While CGH is a versatile technology that can be exploited over a range of display applications, this versatility does not guarantee that it is the most advantageous technique for dynamic projection of two-dimensional images. It is thus worth examining how it compares to other techniques in this respect.

The generalized phase contrast (GPC) extends Zernike’s phase contrast [9] to optically project two-dimensional fields on an output plane instead of simply visualizing naturally occurring phase inputs. Conventional phase contrast uses common path interferometry to visualize thin and unknown phase variations as intensity variations at an output plane. Subsequent to the initial proposal of using generalized phase contrast in image projection [6] and its experimental demonstration for the efficient projection of binary images [10], GPC has been successfully developed and shown to be a viable dynamic projection technology, especially for real-time interactive micro-manipulation [11]. GPC can project grayscale lattices [12] and is suitable for efficient laser projection of grayscale nonperiodic patterns and images [13, 14]. It can also accommodate non-uniform profiles, such as Gaussian beams [15]. While previous papers contained brief remarks that compared the performance of GPC relative to CGH, no systematic comparison of the generalized phase contrast technique relative to computer generated holography has yet been presented.

This work attempts to benchmark the performance of GPC for dynamic planar projections with respect to CGH. We adopt ideas from information theory to guide our analysis. Section 2 lays down the framework for benchmarking projection systems based on its information capacity. Section 3 compares the two techniques in terms of the information capacity and considers factors such as spatial degrees of freedom, temporal degrees of freedom, noise and efficiency, and discusses the impact of practical device constraints. Interrelations between these issues are also considered, together with the resulting performance tradeoffs. Section 4 presents our concluding remarks.

## 2. Pattern projection and information theory

Laser projection may be regarded as a communication system that transmits two-dimensional information from a sender to a receiver through a series of computational and optical components as illustrated in Fig. 1. Therefore, a meaningful metric for assessing quality and performance is the information capacity of the system. Let’s consider a projection system that has *N _{F}* independently controllable elements available at the output, which we will refer to as the number of degrees of freedom of the system. If each degree of freedom can have

*M*distinguishable levels then there are altogether ${M}^{{N}_{F}}$ possible output states, which is a measure of the information capacity of the system. The information capacity,

*N*, is commonly expressed in terms of number of the corresponding binary digits (bits)

_{C}Let’s consider a projection confined within a rectangular area bounded by {-0.5*L _{x}*≤

*x*≤0.5

*L*and -0.5

_{x}*L*≤

_{y}*y*≤0.5

*L*} and an effective band-limiting aperture that transmits spatial frequencies within {-0.5

_{y}*Δ*

*f*≤

_{x}*f*≤0.5

_{x}*Δ*

*f*and -0.5Δ

_{x}*f*≤

_{y}*f*≤0.5Δ

_{y}*f*}. To count

_{y}*N*we follow Lukosz’s thought experiment [17] of using tiled frequency information prior to imposing the band limit. This corresponds to discrete output elements at

_{F}*x*=0, 1/

*Δ*

*f*, …,

_{x}*m*/

*Δ*

*f*, where the edge of the projection corresponds to

_{x,}*m*=0.5

_{max}*L*

_{x}*Δf*and similarly along the y-axis. Thus, the number of degrees of freedom is given by

_{x}When there are numerous resolution elements such that 1≪{*L _{x}Δf_{x}*,

*L*} then the number of degrees of freedom becomes

_{y}Δf_{y}*N*≈

_{F}*L*

_{x}*L*, which is the familiar space-bandwith product. The +1 term in Eq. (2) guarantees available degrees of freedom when information is encoded only along one single dimension. A conventional communication channel does not use spatially-encoded information but exploits degrees of freedom along the time dimension instead. The temporal degrees of freedom can also be utilized in dynamic pattern projection to expand the degrees of freedom to

_{y}Δf_{x}Δf_{y}where *Δf _{t}* is the temporal bandwidth and

*L*is the length of time (duration) of the transmission.

_{t}The number of distinguishable states is limited by the signal-to-noise ratio (SNR) at the output. Assuming that the output, *s*+*n*, consists of the signal, *s*, and an additive noise, *n*, that is bandlimited and uncorrelated, then the number of distinguishable levels is M=(*s*+*n*)/*n*. The information capacity of a communication channel that considers the additive noise is

Lukosz proposed that *N _{F}* is invariant for an optical system and one can trade off the degrees of freedom between the different dimensions to achieve certain goals, such as superresolution [17]. Cox and Sheppard proposed expanding the invariant quantity to the information capacity to add the degrees of freedom for the z-axis and to include noise effects [18], as earlier considered by Fellgett and Linfoot for the space-bandwidth product [19]. The information capacity is useful for analyzing the performance of GPC-based and CGH-based laser projection systems. As we will discuss in the next sections these systems are also subject to various tradeoffs between the different degrees of freedom and noise.

The basic optical geometries of both systems are illustrated in Fig. 1. The projection plane in a GPC-based system is located at an image plane relative to the spatial light modulator (SLM) plane. This is similar to a conventional projector geometry that directly images amplitude modulation impressed upon an incident beam onto a projection plane. On the other hand, the CGH-based projects patterns at a conjugate Fourier plane relative to the SLM. This means that the required phase for encoding is coupled to a global transform (the Fourier transform) of the desired projection in contrast to the local pixel-by-pixel transform (imaging-like) phase encoding in GPC. The difference in encoding scheme arising from these complementary optical transforms strongly affects the performance in each of the projection techniques.

A GPC system images the input phase modulation at the output plane and synthesizes a reference wave using a phase contrast filter to visualize the phase images as intensity patterns. The phase-only modulation enables efficient pattern projection. Similarly, we will consider phase-only computer-generated holograms. Instead of projecting to the Fourier plane, the CGH approach can coherently superpose a lens function to any original Fourier phase hologram to shift the projection to an arbitrary Fresnel plane. However, this will increase the demands on the modulating device that may lead to suboptimal results. Hence, we will only consider Fourier phase CGHs in our comparison with GPC. In the same manner, we will also not consider the mapping-type CGH whose phase profile are derived from geometric ray tracing similar to conventional refractive beam-shaper design but are subsequently phase-wrapped into a diffractive structure. These mapping type elements have a restricted set of achievable patterns and, hence, are not well-suited for image projection.

## 3. Performance benchmarks

#### 3.1 Spatial degrees of freedom

The number of spatial degrees of freedom, (1+*L _{x}*

*Δ*

*f*)(1+

_{y}*L*

_{x}*Δ*

*f*), describes the number of independently addressable output elements and is a measure of the output display resolution. As with traditional display technology, it is reasonable to demand that laser projection should have a good output display resolution in order to sufficiently render fine details in the projections. The imaging geometry of GPC suggests that it is able to address as many output elements as those available in the SLM. This is subject to the band limits imposed by the lens apertures and can exploit high-resolution input devices using high numerical aperture imaging. At first glance, the Fourier relation between the SLM and projection planes in CGH suggests that the CGH output also matches the SLM pixel count. However, fundamental constraints require reducing the output display resolution in order to avoid speckled outputs.

_{y}The presence of speckles can be understood by examining some consequences of the Fourier transform relationship between the input and the projection planes in CGH. In general, a Fourier hologram that accurately reproduces a desired target output is a fully complex function and, hence, is lossy due to the coupled amplitude modulation. Hologram design algorithms iteratively search through different possible phase configurations of the target intensity for one whose inverse Fourier transform yields an intensity profile that matches the input illumination. Except for some special output patterns, a perfect match is never obtained. The imperfect match implies that, aside from single point projections, the efficiency will be typically less than 100% [20]. Furthermore, encoding the hologram phase on a slightly mismatched intensity profile applies a multiplicative spatial noise relative to the correct profile. This multiplicative noise inherently leads to intensity distortions and, possibly, even speckles at the output.

Aside from the multiplicative noise, speckles may also arise due to diffractive effects from the finite-sized CGH aperture. To understand this diffractive speckle mechanism, let us consider a phase-only CGH that is encoded using an SLM consisting of square pixels separated by distance *d* and without any interpixel dead space. Its output projection in the Fourier plane is given by [21]

where *Q*(*f _{x}*,

*f*) represents the desired projection pattern that is discretized on a regular grid and infinitely replicated over the reconstruction plane as a consequence of the sampling at the SLM plane; the sinc function represents an intensity roll-off envelope arising from the diffraction through the finite pixel apertures; and the convolution with

_{y}*A*(

*f*,

_{x}*f*) represents the diffraction broadening of the projected points due to the finite extent of the illumination at the SLM plane. The diffraction-broadened reconstruction points can lead to crosstalk between adjacent reconstruction points. With the output phase treated as an optimization free parameter in a phase-only CGH iterative design, neighbouring points inevitably end up with an unpredictable phase relationship. The pixel crosstalk thus results in speckled reconstructions as neighbouring pixels can randomly interfere constructively or destructively. The phase variation between adjacent reconstruction points can be smoothened as part of the optimization procedure to reduce the speckles even when the reconstruction points overlap [25]. However, this approach requires larger computational arrays which, given the same computational power, reduces the time bandwidth for dynamic image projection, as will be discussed in the next section.

_{y}A common procedure to address speckle noise is to define a smaller signal window within the output plane for the reconstruction and using the surrounding area as a so-called noise window. A CGH designer can then divert the noise away from the designated signal window. Deviations from the ideal performance of the modulating devices commonly yield zero-order beams in experimental implementations that the signal window must avoid. This can limit the signal window to one quadrant of the reconstruction plane to maintain the original square aspect ratio, which corresponds to a 4-fold resolution loss. In addition, one has to observe a minimum separation distance between the addressed reconstruction pixels within the signal window in order to avoid interpixel crosstalk. This requires illuminating through a broader input aperture, which calls for repetitive tiling of the derived phase pattern at the CGH plane. Producing completely isolated spots, such as in array illuminators, can require as much as 10×10 repetitions [16]. While tiling is easily done in static micro-optics CGH implementations, it can be strongly problematic for spatial light modulators required in dynamic applications. Following the discussion above, a 512×512 image may require a 10240×10240 SLM device!

Minimizing the distance between spots can alleviate the resolution loss and reduce the dead space between the reconstruction points, which may not be tolerable in some applications. Assuming uniform illumination through a circular input aperture, the distance between adjacent output spots relative to the diffraction-limited radius is given by [22]

where *f* is the focal length of the Fourier lens used, *λ* is the illumination wavelength, *T* is the period of the repeated cell and *D* is the diameter of the illuminating aperture. A suitable compromise is obtained when the relative distance is *σ _{R}*=2. At this spacing, the distance between spots equals the diameter of their central lobes and their first minima coincide. This occurs for

*D*=2.44

*T*, which requires 2.44 repetitions along the aperture diameter. Since the basic cell for a 512×512 single quadrant target is 1024×1024, the encoding device must have at least 2498×2498 pixels. This can still contain some noise as the rings surrounding the central spot will certainly exert a residual disturbance on the neighbouring spots.

The noisy output obtained when a CGH-based laser projection attempts to address as many output elements as those available in the SLM shows that the CGH-based laser projection has lower information capacity than an imaging system. Reducing the output speckles does not improve the information capacity but simply trades off between the different factors in Eq. (4). Constraining the output to a quadrant reduces *L _{x}* and

*L*while hologram tiling reduces the effective

_{y}*Δ*

*f*and

_{x}*Δ*

*f*. The steps described above effectively reduce the information capacity by ~24 compared to an imaging system with a similar numerical aperture and band limit. In accordance with the invariance of

_{y}*N*, one can also minimize speckle effects by exploiting temporal degrees of freedom. Instead of reducing the spatial degrees of freedom through the speckle reduction schemes described above, one can repetitively project a pattern with different speckles to an integrating detector for a time interval

_{C}*L*to distinguish the information from an averaged noise background. The reduced information capacity is apparent when one considers that a GPC-based projection can already transmit several different information for the same time interval.

_{t}#### 3.2 Temporal degrees of freedom

The high temporal bandwidth afforded by optical frequencies is severely limited by the modulator refresh rate and may be degraded further by the computational bandwith. A definite advantage of the GPC approach over phase-only CGH is in the computational expense required to determine a particular phase distribution that transforms into a desired intensity pattern. As a local phase-to-intensity mapping system, GPC requires virtually no computational load at all. This lends itself to true real-time dynamic operation that can be easily scaled to higher output display resolutions when required by the application. A CGH-based system that transmits pre-computed information can achieve modulator-limited temporal bandwidth but can be hampered by computational bandwidth in interactive systems especially when scaling to higher output resolutions.

The time involved in designing a phase-only CGH using direct binary search, genetic algorithms or simulated annealing render these optimization algorithms impractical for dynamic applications. Designs based on iterative Fourier transform algorithms yield considerably faster solutions. As discussed earlier, projecting a 512×512 image requires Fourier transform iterations involving arrays larger than 512×512 to avoid speckle problems. An output constrained to one quadrant requires at least 1024×1024 resolution elements. Even after numerous iterations, the noise arising from the approximate nature of the phase-only solution generally still spreads into the signal window. This requires a second set of iterations that diverts the noise away from the signal window [23].

The computational time required by these iterative steps can adversely impact the refresh rates in dynamic CGH-based grey level projections. Moreover, the output at this point, assuming suitable hologram repetition, either consists of isolated grey spots or of adjoining grey bumps with residual crosstalk. Avoiding bumpy reconstructions can be done by including the crosstalk effects during the iterative search. This requires more zero-padding in the iteration arrays in order to replicate the extent of the diffraction-limited spots and allow the algorithm to optimize around the interference effects.

The zero-padded iterations aimed at exploiting interference effects still go through the double set of iterations mentioned above. Depending on the target image or the choice of the initial phase during the iterations, the second set of iterations may stagnate in a state where isolated dark spots (optical phase singularities or “optical vortices”) persist within the signal window [24, 25] (see Fig. 2). One proposed solution is to first locate these vortices, then modify the phase to introduce vortices of opposite charge, and finally to perform further iterations to produce a clean image as the vortices annihilate and disappear during the propagation represented by the Fourier transforms.

It is easy to appreciate from the discussion above that designing a phase-only CGH to produce a proper grey-level image is burdened by an expensive computational load. Unless the entire hologram sequence is pre-computed prior to transmission, the computational burden for deriving the phase inputs can potentially become a bottleneck for the temporal bandwidth and restrict the temporal degrees of freedom in CGH-based projection. In contrast, GPC phase inputs are determined using straightforward and local phase-to-intensity mapping employed by the GPC-based projection method, which ensures that the temporal bandwidth afforded by the modulator is fully utilized.

#### 3.3 Signal-to-noise ratio and light efficiency

The projection efficiency is an important parameter from a practical perspective. Furthermore, the efficiency affects the strength of the projected signal and thus affects the information capacity. To incorporate the effect of efficiency, we rewrite the information capacity in Eq. (4) into

where *η* is the efficiency and s0 is the noise-free output signal for 100% efficiency. Equation (7) expresses the logical result that a system with zero efficiency has no information capacity. To study the projection efficiency, we again consider a square phase-only SLM containing N×N square pixels, each with side length *d*, without any interpixel dead space. When this SLM is used to encode phase information into an incident wave the pixelation results in replicated projections having a characteristic intensity roll-off due to the diffraction from the pixel apertures. For single beam scanning, the roll-off pegs the highest possible efficiency at the corners of the projection region to just 16% of the on-axis efficiency. The grayscale/contour plot for the spatial variation of efficiency in CGH single beam deflection is illustrated in the Fig. 3.

The roll-off and replication set an upper limit on the highest achievable efficiency for projections that employ a phase-only CGH. For Fourier array illuminators, this efficiency limit was determined by Arrizon and Testorf to be [16]

where *η*′_{ql} describes the desired efficiency (grey level) at the reconstruction point defined by coordinates (*q*,*l*) on a discrete grid separated by *fλ*/*T* within the signal window Ω_{s} at the projection plane. This efficiency limit, which was originally obtained for array illuminators, applies to general CGH-based laser projections as well since the projection pixels require sufficient separation as discussed above. Accounting for the proper separation in Eq. (8) predicts projections with maximum efficiency of ~52% when the image occupies the entire addressable region at the projection plane. The same efficiency limit is obtained when the projection is constrained to a single quadrant signal window. Our tests show that this efficiency limit holds whether the projection is a uniform grey level, a random grey image, or a typical image with a reasonable spread of grey levels like the Lena standard image. Achieving efficiency beyond 52% can only be achieved by a further reduction of the signal window to exclude the outer regions that tend to decrease efficiency. This consequently comes at a price of a further decrease in output display resolution.

To describe the efficiency of a GPC-based projection system, we start by describing its output intensity,

which is formed by the interference of a synthetic reference wave (SRW), *r _{S}*(

*x*’,

*y*’), with the image of the phase-modulated input,

*p*(

*x*’,

*y*’). For a circular input aperture with radius

*Δr*and a PCF having a circular phase-shifting region with radius

*Δ*

*f*the optimized SRW is given by the Hankel transform [7]

_{r}The SRW depends on the PCF size, *Δ*
*f _{r}*, as shown in Fig. 4(a). The SRW can extend beyond the signal region occupied by the image of the modulated input, which results in a residual intensity halo around the projected image. The energy lost to the halo region sets a fundamental limit on the achievable GPC efficiency. The GPC efficiency limit,

*η*, is obtained by accounting for the energy lost in the SRW tail via numerical integration of the expression:

_{max}The SRW profile and corresponding energy losses depend on the size of the phase contrast filter as illustrated in Fig. 4.

#### 3.4 Practical SLM devices: performance constraints

So far, our discussion mostly assumed that the two techniques are implemented on ideal spatial light modulating devices. It is known that the fundamental properties and limitations of real-world SLMs significantly impact the encoded and projected light [26]. We will now consider the demands imposed by the techniques upon the modulating device with an eye on the realistic performance that can be expected from practical devices. We will examine how the limitations of these devices influence the performance of the GPC and the phase-only CGH techniques, respectively. We consider issues related to SLM dead space, Fourier plane zero-order beam, phase bleeding, and phase stroke requirements.

#### 3.5. Interpixel dead space

One possible artefact of interpixel phase dead space is the appearance of a spurious zero-order beam that commonly forces the signal window away from the optical axis and limits the output display resolution [27]. Furthermore, the smaller pixel aperture reduces the energy throughput and broadens the roll-off so that more energy now channels to the higher-order replica of CGH-based projections. This effect is summarized in the relation [28]

where the fill factor, *r*, is the ratio of area of the active region to the total pixel area. The efficiency declines with decreasing *r* as plotted in Fig. 5. The efficiency plot, which shows 52% for 100% fill factor, was determined assuming that the CGH projection is constrained to one quadrant of the addressable region in the reconstruction plane. As mentioned in section 3.3, it is possible to project CGH-based images with higher efficiencies by projecting them to a smaller signal window and sacrificing resolution. The figure also includes the efficiency of phase-only CGH-based image projection to one quadrant relative to GPC-based projection to the entire addressable region of the reconstruction plane. GPC-based projection is expected to linearly decrease with *r* due to the decreasing energy throughput and where the initial efficiency at *r*=1 is taken to be 0.82 (the efficiency along the dotted line in Fig. 4(b). A horizontal plot of the relative performance would indicate that the fill factor variation has a similar effect on both GPC and CGH efficiency. The actual plot of the relative efficiency in Fig. 5 indicates that GPC is increasingly advantageous over CGH when using SLM devices with decreasing fill factors.

#### 3.6 Device modulation transfer function

Most devices fail to reach the needed phase strokes at higher spatial frequencies due to factors such as electrical timing [29] and addressing crosstalk between adjacent pixels. The crosstalk can be due to fringing effects of the fields in electrical addressing [30], the nature of the photoconductive mechanism in optical addressing [31], and various other mechanisms that contribute to the non-local response of the electro-optic medium. The net effect is “phase bleeding” or “phase blurring”, where the phase addressed to a particular SLM pixel gets blurred and bleeds to neighbouring pixels. The diffraction efficiency becomes frequency-dependent and introduces a major intensity roll-off in the projection [33].

Phase bleeding is problematic for a phase-only CGH since the CGH phase is phase-wrapped to within a 2π range. The sudden 2π jumps in phase-wrapped areas demand high frequency operation from the device. This is illustrated by Fig. 6, which shows a phase-only CGH and its reconstruction.

Phase bleeding yields erroneous encoding that degrades the reconstruction with spurious zero-order, ghost reconstructions, and noise [32]. A design that performs excellently in numerical simulations can give frustrating reconstructions on an actual SLM-device.

Minimizing the phase-wrapped regions by operating at lower spatial frequencies, while simultaneously avoiding the zero-order noise, comes with a further loss in resolution. It is also rather unfortunate that a phase-only CGH avoids the optical axis since this region is less susceptible to aberrations.

In contrast, GPC utilizes the zero-order beam to generate the reference wave. Hence, potential phase encoding errors that can affect the zero-order beam may be compensated for by choosing a suitable filter size and filter phase. With its symmetric 4*f* imaging configuration, aberrations in the Fourier plane introduced by the first lens are minimized by using a matching second lens. The ability of a GPC-based projection system to work with the limited modulation transfer function of existing devices is highlighted by our recent work, where we experimentally demonstrated laser projection of the Lena image at 74% efficiency [14].

#### 3.7 Required phase stroke

Except for binary phase holograms that throw away half of the energy to a twin image, a phase-only CGH generally requires at least a 2π phase stroke. Ignoring for the moment the limits on switching rate imposed by the computational costs, the required 2π phase modulation dynamic range sets another limit on the attainable switching rates. The current technique for grey image projection via GPC only demands a π phase stroke from the modulator and can thus benefit from the higher refresh rates expected from lower stroke devices in general. In SLMs utilizing electro-optic materials, for instance, a thinner medium can be used which is also preferred for being less susceptible to interpixel crosstalk and aberrations introduced by the medium.

## 4. Concluding remarks

Laser projection can be an enabling technology for various contexts and projection techniques will ultimately be assessed based on application-specific metrics. However, the information capacity is an objective measure that can pervade across a wide spectrum of potential applications and can compliment the other metrics. Undoubtedly, computer-generated holography is a highly versatile technique that is encountered in a plurality of applications. However, versatility does not guarantee that a technique will be the best option for a specific application. Our analysis illustrates the need for careful scrutiny before adopting available techniques. From the perspective of information capacity, our discussion indicates that there are several areas where a CGH-based laser projection system can be problematic and where a GPC-based system is a more attractive option. Nevertheless, we acknowledge that information capacity is a general measure and there are surely projection applications that give high premium on other performance measures and where a CGH-based approach will be preferable.

## Acknowledgement

We thank the Danish Technical Scientific Research Council (FTP) for supporting this research.

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