Computer-generated volume holograms (CGVHs) are gradient refractive index (GRIN) devices that consist of a superposition of multiple periodic diffraction gratings. Fabrication of these components for the visible range is difficult due to the small length-scale requirements but is more tenable in the terahertz (THz), as the length scales become more practical (≥ 10−5 m). We successfully utilized polymer-based 3D additive rapid-prototyping technology to fabricate, to our knowledge, the world’s first 3D THz CGVH in approximately 50 minutes, using $12 of consumables. This demonstration suggests that this technique could be extended to fabricate THz volumetric optics with arbitrary electromagnetic profiles.
© 2014 Optical Society of America
In recent decades, research in the terahertz (THz) region has expanded dramatically, driven by its nondestructive, noninvasive, and penetrative properties which have applications in imaging [1–4], security [5–9], communication [5,10], radar , spectroscopy [4,11], astronomy [5,12], and chemical detection [13,14]. Recent advancements have provided improved THz sources and detectors, expanding the range of applications. However, practical fabrication of THz volumetric components is challenging via traditional subtractive processes . We have found that additive rapid prototyping (RP) technology has made it possible for the development and fabrication of THz volumetric components. This fabrication technique yields the required sub-millimeter feature sizes of THz components, as verified by our earlier success fabricating high performance THz photonic crystals  and low-loss waveguides . We have applied this technique to the development and fabrication of THz computer-generated volume hologram (CGVH).
Unlike traditional holograms, computer-generated holograms (CGH) do not require a medium to record the interference pattern of the object and reference waves, instead, the object wavefront is digitally computed then encoded into a synthetic hologram [18–20]. A CGVH is a 3D version of the CGH; it exhibits similar properties but has stronger angular and wavelength selectivity [21,22]. CGVHs in any spectral band can be considered as a superposition of multiple diffraction gratings. Each grating is a periodic, continuous variation of refractive index, making CGVHs a type of gradient refractive index (GRIN) component. CGHs and CGVHs have many applications in the visible range such as high density optical data storage and processing [20,23–27], pulse shaping , imaging [29,30] and display [20,31] which have yet to be explored in the THz regime. Direct fabrication of these GRIN components in the visible range is difficult due to the small length-scale requirements (≤ 10−7 m). However, the problem is more feasible in the THz, as the length scales become more practical (≥ 10−5 m), and there exist inexpensive fabrication techniques such as the additive RP technology used in this effort.
2. CGVH design and RP fabrication
In our work, we used an Objet EDEN 350 Polyjet machine for fast and inexpensive fabrication of 3D THz components. The RP machine has a native print resolution of 42 μm (X) × 84 μm (Y) × 15 μm (Z). Printing on a 3D RP machine is analogous to an inkjet printer. A set of print heads is used to jet a 15 μm layer of photopolymer onto a build tray. This build tray is then lowered by 15 μm after each jetted layer to allow subsequent layers to be printed sequentially, ultimately forming the 3D structure. The EDEN 350 utilizes two types of ultraviolet curable photopolymers as its print media: model and support. The model photopolymer acts as the structure; the easy-to-remove support photopolymer allows for arbitrary complex 3D models to be fabricated. Normally, after the print process, the support photopolymer is simply pressure washed away, leaving the 3D structure behind. However, in this work, we used the support polymer to provide a second baseline refractive index in the final structure.
THz GRIN components can be fabricated by using mixtures of these photopolymers; however, we must have an understanding of the electromagnetic properties of the available EDEN 350 print media. The stock photopolymers were characterized by measuring the transmission response of a 3 mm thick fabricated sample slab with a Picometrix T-Ray 2000 THz time-domain spectrometer (THz-TDS), in which a short, time-domain pulse was propagated through a sample. A reference measurement without the sample was also recorded. The measured reference and sample data were Fourier transformed and compared to extract the frequency-dependent phase and amplitude of the sample. This comparison allows for the real permittivity (ε’) and the loss tangent (tan δ) to be numerically extracted  as shown in Fig. 1 (results for the model photopolymer).
The EDEN 350 is limited to two photopolymers for fabrication, thus we apply a simple effective-medium theory (EMT) formulation (Eq. (1)) to approximate the desired spatially-continuous refractive index variation in the THz CGVHs. In EMT, a sub-wavelength mixture of two materials appears to have an index of refraction that is the weighted volumetric average of the two constituent materials. The effective index of refraction can then be described as
Implementing the effective-medium approach required a modification to the interface with the RP machine as the standard interface was intended for large mechanical prototypes and not the many rapid variations in photopolymer layout required by the EMT method. We developed our own software interface to provide direct control of the photopolymer placement on the EDEN 350 in order to efficiently print these sub-wavelength features. The EMT approach was experimentally validated by designing several slabs with different effective indices, fabricating them on the RP system, and then characterizing their indices with the THz-TDS. The underlying index voxels (a volumetric unit cell of constant effective refractive index) were chosen to be 336 μm × 336 μm × 420 μm. Each index voxel was then further divided into 64 (43) sub-wavelength homogenous photopolymer voxels to approximate the desired index via EMT. These choices satisfy the EMT approximations at frequencies up to ~0.5 THz. We verified the efficacy of this EMT formulation by printing several 3 mm slabs with varying model/support ratios. Measurements on the THz-TDS verified that the indices of refraction of the slabs were within 1% of the intended indices.
Initially, we have designed and fabricated EMT-based CGVHs that contained a small number of simple, sinusoidal gratings. These diffraction gratings are spatially-continuous variations of refractive index which can be represented by grating vectors. The grating vector space is a mathematical representation of the hologram. A number of grating vectors were designed to satisfy Bragg closure between the illumination vector and the desired diffracted light vector  on the wave-normal circle. The required continuous index profile of the CGVH was then computed by taking the inverse Fourier transform of the k-space (spatial frequency domain) representation describing the grating vectors.
The general grating design process was to define the desired far-field angular pattern to be produced by the CGVH. The required outgoing wavevectors, combined with the known input wavevector then uniquely determine the grating vector (k-space) spectrum of the CGVH under the assumption of Bragg diffraction. This k-space description may then be converted into a description of the required spatial index variation via the inverse Fourier transform. It is important, however, to incorporate the physical constraints on the index profile into the design process for the k-space spectrum. From the standard Fourier relationships, the available bandwidth and discretization in k-space are determined based on the spatial resolution and spatial extent of the physical fabrication process. The discretization in k-space is given by , with the physical extent of the CGVH and the k-space bandwidth is with the physical fabrication resolution/discretization. The lossy nature of the support material at higher frequencies limited the longitudinal-extent of the hologram to approximately 10 mm. In the transverse dimensions, we selected a CVGH size of 50 mm for physical convenience and to match to the size of the source illumination. Any prospective CGVH design was then evaluated to determine if the required k-space spectrum fell within the available bandwidth and was well matched to the available resolution/discretization. A suitable grating vector would exhibit an on-Bragg frequency of 0.1 to 0.5 THz and a diffraction angle less than 1.195 radians (68.5°) (physical limit).
The first CGVH was designed with a grating period of 2.33 mm to optimally diffract 0.145 THz light at a diffraction angle of 1.04 radians (60°). The second CGVH was fabricated with a design grating period of 1.84 mm to produce Bragg diffraction at 0.76 radians (43°) and 0.234 THz. The third CGVH was multiplexed to contain both of the aforementioned gratings. Figure 2(a) is a simulated cross-sectional view of the multiplexed CGVH, showing the refractive index variation; Fig. 2(b) shows a picture of the fabricated multiplexed CGVH. Each CGVH has a physical size of 50.7 × 50.7 × 10.5 mm (X × Y × Z) which included a thin cladding of model material to protect the grating within.
3. Experimental characterization setup
A Picometrix T-Ray 2000 THz-TDS was similarly used for characterization of the fabricated CGVH components. The transmitter and CGVH were mounted together on an aluminum arm to a Newport PR50CC rotation stage such that a normal incidence is maintained as shown in Fig. 3(a).The rotation stage was centered underneath the CGVH to control the diffraction angle measured by the stationary receiver. This allowed for the horizontal far-field response of these CGVHs to be measured using the THz-TDS from 0° to 90° to capture their full performance. The rotation stage was controlled by a Newport ESP300 Universal Motion Controller. 15 measurements were taken at each angle and averaged; a spectral response (intensity vs. frequency) was generated for each angle. A second verification measurement was made with a Virginia Diodes continuous-wave (CW) system, consisting of a synthesizer and diode chain to produce THz signals, which were received by a broadband Golay cell as shown in Fig. 3(b). An HP 5350B frequency counter was used to verify the radio frequency (RF) input (10.8 - 20 GHz), which was subsequently multiplied by the diode chain (VDI WR5.2 × 2 and VDI WR3.4 × 3) to produce the appropriate THz output.
4. Experimental results
Figures 4(a)-4(c) are spectrograms that show the measured diffraction efficiencies (DE) of the CGVHs as a function of diffraction angle and frequency; the lighter areas correspond to regions of high DE. Here, DE is represented as the fraction of the total power at a given frequency that is diffracted into a given angle. By normalizing the efficiency to the total power diffracted into all angles, we are implicitly ignoring the impact of loss and scattering in the material. This allows us to fairly judge the efficiency of the grating itself. Figures 4(a) and 4(b) depict the performance of the 2.33 mm and 1.84 mm CGVHs respectively, both of which have the highest DE at their respective designed diffraction angle and on-Bragg frequency. The wide regions of high DE in the spectrograms are exaggerated by the logarithmic scale; however, there remains relatively good Bragg selectivity. The performance of the CGVH with a 1.84 mm grating period was also verified over a range of angles and frequencies by an experiment using a Virginia Diodes CW transmitter and Golay cell receiver. The results from that experiment are shown in the overlay in Fig. 4(b) and demonstrate qualitative agreement with the THz-TDS results. Figure 4(c) shows the DE of the multiplexed CGVH where two light regions of high DE can be observed, demonstrating that the hologram acts as a combination of the individual gratings.
Figures 4(d)–4(g) show slices of the DE data for the three holograms at specific angles and frequencies. In Figs. 4(d) and 4(f), the DE is plotted for the designed frequencies of 0.145 THz and 0.234 THz, respectively, as a function of measurement angle. In Figs. 4(e) and 4(g), the DE is plotted for the designed diffraction angles of 60° and 43°, respectively, as a function of frequency. In these figures, the DEs are plotted on a linear scale to demonstrate the Bragg selectivity at the designed frequencies and angles. As expected, the multiplexed CGVH demonstrates features of both the 1.84 mm and 2.33 mm gratings. Figure 4(e) shows the highest DE peak of 7.6% at ~0.144 THz which corresponds to the on-Bragg frequency of the 2.33 mm grating (solid, red). Also seen is a second order diffraction peak from the 2.33 mm grating at ~0.281 THz. Figure 4(g) shows the highest DE peak of 10.9% at ~0.230 THz which corresponds to the on-Bragg frequency of the 1.84 mm grating (dotted, black).
The DE of the multiplexed CGVH is noticeably lower than the single-grating DEs as the efficiency is directly impacted by the number of multiplexed gratings present in the CGVH. We would expect the efficiency of the multiplexed gratings to be half that of the measured efficiencies of the individual grating [33,34]. Indeed, an effect of approximately this magnitude is observed in Fig. 4(f) where, at 0.234 THz, the DE of the multiplexed CGVH (dashed, blue) is ~40% of the 2.33 mm single grating CGVH DE (solid, red). Additionally, the DE of the multiplexed grating (dashed, blue) is ~35% of the 1.84 mm single grating CGVH DE (dotted, black). Similar scaling can also be observed in Fig. 4(d).
This work has demonstrated the design, fabrication, and experimental testing, to our knowledge, of the world’s first THz computer-generated volume hologram. Furthermore, we have shown that additive RP technology can be a practical and inexpensive fabrication solution for creating complicated volumetric THz components—the THz CGVHs in this study required only 50 minutes and $12 of consumables. We are now investigating methods for incorporating a broader range of materials into the additive RP framework to provide increased control over the permittivity and permeability profiles of the resulting components. This will enable the fabrication of devices that have thus far been confined to theoretical study and 2D fabrication.
This research was supported by the National Science Foundation through grant 0925220. The authors thank C.K. Walker for providing the use of the THz characterization equipment, A. Young for expertise in the operation of the characterization equipment, and Z. Wu for the electromagnetic characterization routine. M.G. was previously part of the University of Arizona during this research and is currently at Duke University.
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