We study the performance of amplitude computer-generated volume holograms (CGVH) in terms of efficiency and angular/frequency selectivity. We compare CGVHs to interferometrically-recorded amplitude volume holograms. Theoretical results show that amplitude CGVHs can increase the efficiency as well as the angular and wavelength selectivity relative to optically recorded amplitude volume holograms. We fabricate the CGVHs using femtosecond laser pulse micromachining in the bulk of glass and demonstrate results consistent with the theory. These results show that aperiodic three-dimensional structures provide the degrees of freedom necessary to improve the performance of volume diffractive optics. They suggest that, under certain circumstances, a departure from the Bragg paradigm provides enhanced volume diffraction properties.
©2007 Optical Society of America
Within the past decade the scientific community has been focused on exploring three dimensional (3D) periodic structures known as photonic crystals. However, aperiodic structures present additional flexibility to investigate novel phenomena and create new functionalities. In fact, one can envision volumetric structures implemented as systems of micro/nano-components (some of which may be periodic elements) designed for a particular set of optical functions. There are new micro and nano-fabrication techniques that can be adapted to the design requirements of such structures. Modified lithographic processes , holographic lithography [2, 3], and direct laser writing  are among the most promising techniques. In particular, femtosecond pulse laser micromachining of glass has shown promise for the fabrication of waveguides [5, 6], micro-lenses , polarization-sensitive elements [8, 9], arbitrary wavefront formation devices , and optical data storage systems .
In this paper, the particular focus is in the analysis and design of structures for arbitrary wavefront formation within a volume as they are capable of conducting a variety of optical functions. Diffractive structures are typically grouped into two general categories: thick/volume or thin diffractive optics. Furthermore, these two categories can each be subdivided into optically recorded or computer-generated. Of these four divisions, thin and thick optically recorded holograms have experienced extensive study as have thin computer-generated holograms. Thin holograms, both computer-generated and optically recorded, have shown much utility in the fields of imaging, optical filters, beam shaping, microscopy, and optical data processing, among others [12–14]. Volume holograms, on the other hand, provide potential for higher efficiency and greater capability for multiplexing information due to their angular and frequency selectivity properties [15–17].
The characteristics of periodic stratified gratings have been studied in several reports [18–21] while Ref.  demonstrated two-layer computer-generated holograms (CGHs). It is in aperiodic 3D volume computer-generated holography where insufficient research has been conducted. Ref.  presented the first computer generated holograms that extend in three dimensions forming a modulated 3D aperiodic structure.
Accordingly, in this work we investigate numerically and experimentally the performance of 3D structures termed computer-generated volume holograms (CGVHs). The potential benefits of CGVHs are in combining and improving useful characteristics of thin CGHs and optically recorded volume holograms, namely increased efficiency, arbitrary 3D wavefront encoding, and angular and frequency selectivity. Indeed, we show that these qualities can be controlled and enhanced by proper design.
2. Theory and design
2.1. Problem statement and modeling
The system under consideration is an arbitrary volumetric modulation of the index of refraction (real or imaginary) which is to be designed to perform a particular optical function. In this paper the modulation is imaginary so as to create absorption (or amplitude) modulation as the optical beam propagates through the otherwise transparent volume.
The volume is sampled in depth and modeled as discrete layers separated by a distance, as seen in Fig. 1. Wave propagation between layers is modeled by a scalar angular spectrum propagation technique  described by
where 𝓕xy and 𝓕xy -1 denote 2D Fourier and inverse Fourier transforms respectively, E is the complex amplitude of the field, ko is the free-space wavevector of the illumination, kt is the transverse wavevector, and Δz is the propagation distance. If the layers can be considered to be thin relative to the wavelength, the wave passing through a layer can be simply modeled by a transmission function. Otherwise, a more elaborate beam-propagation method should be implemented .
The CGVHs were designed as 1 to 10-layer, binary-amplitude structures with further design parameters of 1×128 pixels per layer, 3μm pixel size, 100μm total thickness (from first to last layer), and λ = 633nm was the illumination wavelength. For the purpose of investigating diffraction efficiency and selectivity, the target image (desired far-field reconstruction) was a single off-axis spot. The location of this spot corresponded to a plane wave diffracted to an angle of 3.37° from the input propagation direction in air. This angle was approximately halfway from the center to the edge of the far-field grid used.
The CGVH design was performed by optimization under the given constraints. We used an iterative projection onto convex sets algorithm [22,24–28] followed by a sequential binary search optimization to maximize the diffraction efficiency of the structures. The hybrid algorithm required acceptable computation time (< 1 minute on a 1.6 GHz PC). The design of one particular 4-layer CGVH is depicted in Fig. 2. The result resembles a binary grating although it is not periodic.
3. Diffraction efficiency
The theoretical far-field intensity pattern resulting from propagation through the structure depicted in Fig. 2 is shown in Fig. 3. The far-field intensity pattern is obviously asymmetric (not possible with thin, single-layer amplitude CGHs) and the designed diffracted spot is located by design at kt λ = 0.244, where kt is the transverse component of the wavevector. The simulated intensity of the diffracted spot is approximately the same as that of the DC (central) spot, which suggests a high diffraction efficiency relative to thin binary gratings (see Fig. 4).
The simulated diffraction efficiencies of the layered structures (computed as the energy contained in the target far-field pixel divided by the input energy) were as large as 14.1%, which is an improvement over the best optically recorded amplitude volume holograms for which the maximum efficiency is 3.7% and over thin single-layer amplitude holograms which have a maximum efficiency of 10.2% (see Fig. 4).
It is important to note that these results are for binary amplitude structures with allowable transmission values of 0 and 1. Therefore it is not unreasonable to think that using continuous amplitude or at least other binary transmission values may allow for even higher diffraction efficiencies. Indeed, for a single instance in designing an 8-layer structure using 0.2 and 1 as the allowable binary transmission values, the diffraction efficiency increased to 13.85% compared to 12.86% obtained with 0 and 1 as the binary transmission values. Therefore, it is apparent that the values shown in Fig. 4 are not necessarily global maxima, whereas the dotted lines included for comparison are for absolutely optimal periodic structures.
For comparison, a layered periodic structure was also designed under the same constraints by creating a binarized amplitude grating of the appropriate period, and shifting and repeating the structure to create a multi-layer volumetric grating. Each consecutive layer was shifted appropriately so that the first orders of each layer added constructively. Grating duty cycles of 25, 50, and 75% were investigated with the optimal efficiency occurring with a duty cycle of 50%. The fully periodic, 4-layer grating structure with 50% duty cycle provided a theoretical diffraction efficiency of only 8.4% (compared to 12.8% for the aperiodic structure).
To fabricate the CGVHs we used a technique recently developed known as femtosecond laser micromachining [5–11]. We used amplified femtosecond laser pulses at 800nm center wavelength focused with a microscope objective (NA=0.65) into the bulk of a glass substrate (Corning 2948). The laser system was a KM Labs 90MHz, 16fs Titanium Sapphire oscillator amplified by a Coherent RegA9050 amplifier at 100kHz repetition rate to the μJ level. The energy of the pulses exiting the focusing objective lens during the writing process was 0.85μJ and their duration was 60fs. The sample substrate was mounted on a computer-controlled 3-axis translation stage. The deepest layer was written first, 140μm beneath the surface, followed by each successive layer in reverse order to avoid distorting the focus of the writing beam. The written elements were 3μm × 384μm rectangles with 128 possible locations for each layer, which resulted in a 2D layer size of 384μm × 384μm. These layers were spaced appropriately (100μm/[N - 1], where N is the number of layers) such that the total depth of the written 3D structure was 100 μm and the first layer was 40 μm below the substrate surface.
5. Angular and frequency selectivity
The fabricated elements were mounted and centered on a rotation stage and the angular selectivity was experimentally investigated using a HeNe laser (632.8 nm) as the source and a power meter to detect the power in the diffracted spot. In order to compute efficiency the power incident on the structure was calculated from a measurement of the beam profile and the total incident power. Diffraction efficiency was then computed as the intensity in the diffracted spot divided by the power incident on the structure.
Simulations were conducted by varying the simulated illumination direction or wavelength. A good agreement between the experiment and the simulation was obtained and is shown in Fig. 5a. However, it is noticeable that the agreement decreases at large angles. This is due to the finite thickness of each fabricated layer (on the order of 7μm computed as the Rayleigh range of the focused writing beam) while the simulations assumed infinitely thin layers. Also seen in Fig 5a, is that the central peak for the 2-layer structure is slightly narrower [2.76° full width half maximum (FWHM)] than that of either 4- or 8-layer structures (4.02° and 4.20°). For comparison, the angular FWHM of the central peak of an optically recorded volume hologram of the same thickness and design parameters is 5.09°.
It is apparent in Fig. 5a that the angular free spectral range (FSR) (angular spacing between subsequent peaks) increases with the number of layers. This trend is reasonable since a larger number of layers more closely approximates a continuous volume hologram which has only a single peak. This is also consistent with the behavior of periodic multilayer holographic gratings .
Simulations were also done to investigate the frequency selectivity of the structures. As shown in Fig. 5b, the performance of the structures is on par with, but slightly sharper than, what would be expected from an optically recorded amplitude volume hologram of identical thickness. Perfect overlay is not expected since the simulated elements were layered, aperiodic, and binary as compared to periodic and continuous optically recorded holograms.
The efficiency of the four-layer structure was measured using an Argon Ion laser source at 457.9 and 514nm and a HeNe laser (λ = 633nm). At these three wavelengths the measured diffraction efficiency was 11.3%, 9.0% and 8.8% respectively. The experimental efficiency is lower than simulated, likely because the fabricated elements are not perfectly transmissive or absorptive, nor are they perfectly thin. However, the overall trend for the diffraction efficiency at these three wavelengths agrees with the simulation which shows that the efficiency does not change significantly over the visible spectrum.
An interesting comparison is to a layered periodic (non-continuous) structure as discussed in section 3. For a 4-layer periodic structure, the angular FWHM of the central peak was found by simulation to be 3.6° and the wavelength half-maximum cutoff was found by simulation to be 1600 nm. Both numbers are similar to those measured from the aperiodic results shown in Fig. 5. Therefore, it appears that the angular selectivity can be increased using a layered binary-amplitude structure rather than a continuous-amplitude volume hologram.
Selectivity is an adjustable parameter that can be controlled by the thickness (thicker elements are more selective, as seen from Fig. 5b), cross-section size, and by the designed diffraction angle. For example, one such structure of the same 100μm thickness but designed for a larger diffraction angle of 9.7° yielded a narrower angular selectivity of 1° and wavelength selectivity of 240nm (compared to equivalent optically recorded amplitude volume hologram selectivities of 1.8° and 363 nm).
We have probed the characteristics of aperiodic amplitude CGVHs. The efficiency of these structures is considerably higher (14%) than the efficiency of optimal optically recorded (periodic) amplitude volume holograms (3.7%) and thin, single-layer amplitude gratings (10.13%). It is also possible to increase the efficiency further through use of different binary transmission levels or continuous amplitude elements. The structures were also observed in both simulations and experiments to have angular and wavelength selectivity comparable to but higher than optically recorded amplitude volume holograms of the same 3D size and diffraction angle. These dependencies are adjustable by design.
While this study focused on volume amplitude modulation, it should be noted that aperiodic volume structures could be designed as phase elements in which case the diffraction efficiency should approach 100%. This will be the subject of future studies.
A general and intriguing consequence of this study is the suggestion that a departure from the periodic Bragg phenomenon could provide enhanced volume diffraction properties.
The authors would like to thank Wenjian Cai and Ted Reber for their previous efforts in this direction. This work was supported by the NSF under the NIRT award, DMI-0304650.
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