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Abstract
This work discusses the design and fabrication of a dual-plane terahertz (THz) hologram and an extended-depth-of-focus THz diffractive lens. The dual-plane THz hologram consists of 50 × 50 diffractive optical elements with identical element pixel size 1×1 mm, and the extended-depth-of-focus THz diffractive lens is designed with 25 concentric rings with identical ring width of 1 mm, resulting in same device dimension 50 mm × 50 mm. The height of the hologram pixels and concentric rings of the diffractive lens are optimized by nonlinear optimization algorithms with scalar diffraction theory based on Ray-Sommerfeld diffraction equation. Finite-Difference Time-Domain (FDTD) simulation results agree with optimization results obtained from the scalar diffraction theory for both the THz hologram and the THz diffractive lens. The demonstrated experimental results show that the proposed THz hologram and THz diffractive lens can generate the desired diffraction patterns. These diffractive structures have the potential to be applied in areas such as THz imaging, data storage, and displays.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Terahertz (THz) technology has broad applications, such as in biomedical sensing [1], modulation [2], wireless communications [3], imaging [4], and so on. THz diffractive optics are gaining wide attention in imaging and focusing THz systems [5–9]. Metasurface structures are good candidates for generating THz holograms due to its ability to project the desired target imaging by controlling the amplitude and phase of wavefronts [10–16]. Wang et al. [14] proposed wavelength-multiplexed holograms by constructing a specific metasurface structure based on subwavelength C-shaped metallic antennas. Letters ‘C’ and ‘N’ are generated at 0.5 and 0.63 THz, respectively, under linearly-polarized THz illumination. He et al. [15] demonstrated THz multiple-plane holograms by using C-shaped slot metallic antennas metasurface and Fresnel diffraction integral formula. Letters ‘C’, ‘N’, and ‘U’ are formed at the imaging distances 3, 4, and 5 mm, respectively, at 0.8 THz. Metasurfaces are also widely applied for THz metalens designs [17–19]. Wang et al. [17] demonstrated a THz metasurface-based metalens consisting of metallic C-shaped split-ring resonators. The experimental results agree with simulation results over the 0.5∼0.9 THz broad bandwidth. Jiang et al. [20] demonstrated a THz lens with axial focal depth of 10 mm by depositing a 100 nm gold film on a silicon substrate and etching V-shaped air holes, which requires micro photolithography technology. However, THz holograms and metalenses based on metasurface structures can be polarization-sensitive, limiting their further applications. Moreover, such designs require subwavelength metallic structures, complicating the fabrication process. As an alternative, diffractive optical elements are also suitable for THz hologram and lens design with the advantage of polarization-insensitivity. Moreover, the minimum feature size of the elements can be greater than the operating wavelengths.
Various optimization methods [21–24] for diffractive optics designs have been employed. However, optimization methods, such as genetic local search algorithm [21], iterative angular spectrum algorithm [22], and iterative Fourier-transform algorithms [23,24] can be time-consuming and require large computational resources. Here, the optimization variables are optimized by the modified direct binary search algorithm proposed in [25,26]. A 3D printer is utilized to fabricate the designed diffractive THz optics. The diffraction patterns of the device illuminated by a 0.3 THz Gaussian beam are captured through a Tera-1024 THz camera.
2. Design and simulation
The schematic of the dual-plane THz hologram is shown in Fig. 1(a). Two hologram masks are designed, and both consist of $50 \times 50$ square pixels with each pixel size $1{\rm{\;mm}} \times 1{\rm{\;mm}}$. The height of each pixel varies from 0 to 1.4 ${\rm{mm}}$ with 25 total height levels, leading to 58.3 ${\rm{\mu m}}$ step height size. Mask 2 functions as a dual-plane hologram with target images letter ‘U’ and ‘T’ at imaging distances ${D_1} = 50\;{\rm{mm}},\;$ and ${D_2} = 80\;{\rm{mm}}$, respectively, as shown in Fig. 1(a). For comparison, Mask 1 is designed to have only one imaging plane with target image letter ‘U’ at imaging distance 50 mm. The material used in the optimization and 3D printing fabrication is IFUN 3124 resin, which is commercially available. The refractive index of the resin at 0.3 THz is 1.7 + i0.03 obtained by transmission spectroscopy.
Fig. 1. (a) Schematic of a dual-plane THz hologram. (b) Pixel height distribution of hologram Mask 1 and Mask 2. Hologram normalized intensity distribution of (c) Mask 1 at 50 mm, and (d) Mask 2 at 50 mm and 80 mm, from the scalar diffraction theory optimization. Corresponding 3D FDTD simulation results of (e) Mask 1 at 50 mm, and (f) Mask 2 at 50 mm and 80 mm.
The optical mode of the i-th element of the hologram mask on the imaging plane is calculated by using Rayleigh-Sommerfeld diffraction equation [27], defined as follows:
The height of each pixel on the hologram masks are determined by a nonlinear optimization method [25] through minimizing a predefined loss function, which is defined as follows:
Lumerical full-wave 3D FDTD is applied to further verify the hologram masks design. The hologram mask is illuminated by an identical Gaussian beam used in the scalar diffraction optimization. The mesh accuracy is set to be 3, which is a good tradeoff between simulation accuracy and computational cost. Perfectly matched layer (PML) boundary conditions are applied in the x, y, and z directions. Figure 1(e) shows the 3D FDTD simulation result of the normalized electric field intensity at a distance of 50 mm for Mask 1, which matches the result obtained from the scalar diffraction theory, as shown in Fig. 1(c). For the dual-plane THz hologram Mask 2, the obtained FDTD simulation results at a distance 50 mm and 80 mm are shown in Fig. 1(f), consistent with the results shown in Fig. 1(d), which validates the optimization based on the scalar diffraction theory. Depicted in Supplement 1, Fig. S2, are details of non-normalized intensity levels and calculated efficiencies for each case.
Figure 2(a) shows the schematic of the THz depth-of-focus diffractive lens. The lens is designed to have depth-of-focus ${f_2} - {f_1}$. Two diffractive THz lenses are designed. Lens 1 has a focal length 50 mm, and Lens 2 has focal length from ${f_1}$= 50 mm to ${f_2} = \;$ 90 mm with depth-of-focus 40 mm. Both lenses have diameter 50 mm, and consist of concentric rings with identical width 1 mm. The height of each concentric ring in each design varies from 0 to 1.4 mm with total height levels 25, which is identical to the THz hologram masks design. Rayleigh-Sommerfeld diffraction equation and identical Gaussian incident beam are applied in the optimization. The heights of each concentric ring of the lens are optimized by applying the algorithm proposed in [26]. A modified figure-of-merit (FOM) from [28] defined in the optimization is as follows:
Fig. 2. (a) Schematic of THz depth-of-focus diffractive lens. (b) Pixel height distribution of THz diffractive Lens 1 and Lens 2. Normalized light intensity distribution on the x-z plane of (c) Lens 1, and (d) Lens 2, from the scalar diffraction theory optimization. Corresponding 3D FDTD simulation results of (e) Lens 1, and (f) Lens 2.
The optimized concentric ring height distributions for Lens 1 and Lens 2 are shown in Fig. 2(b). The normalized light intensity distribution after transmitting through the THz lenses on the x-z plane obtained from the scalar diffraction theory are shown in Fig. 2(c) and 2(d). From Fig. 2(c), light is focused at the focal plane of 50 mm for Lens 1. And for Lens 2, depth-of-focus from 50 mm to 90 mm is obtained, which is shown in Fig. 2(d). For further verification, the optimized height of the concentric rings of both lenses are imported into Lumerical 3D FDTD full wave simulations. The lenses are illuminated by a Gaussian beam. The boundary conditions and mesh accuracy are set identically to the hologram 3D FDTD simulations. Figures 2(e) and 2(f) show the normalized electric field intensity distribution from the 3D FDTD simulations for both lenses, which further confirms the optimization validation. Depicted in Supplement 1, Fig. S3, are details of non-normalized intensity levels and calculated efficiencies for each case.
3. Fabrication and measurement
The THz hologram masks and THz diffractive lenses are fabricated by utilizing a 3D printer (Elegoo Saturn) with Z-axis moving accuracy of 1.25 ${\rm{\mu m}}$ and XY resolution of 50 µm. IFUN 3124 resin is used as the 3D printing material. The x-y planes of the hologram masks and diffractive lens are aligned parallel to the 3D printer z-axis moving direction to achieve better printing accuracy for the pixel height. 0.4 mm substrate is added to the mask so ensure the strength of the mask. The printing time for each of them is about 3 hours. 3D printed hologram masks are shown in Figs. 3(a) and 3(b). Two diffractive THz lenses fabricated by the 3D printing are shown in Figs. 3(c) and 3(d).
Fig. 3. Tilted view of 3D printed THz hologram of (a) Mask 1, and (b) Mask 2. Tilted view of 3D printed THz diffractive lens of (c) Lens 1, and (d) Lens 2. (e) The experimental setup for the hologram and lens.
Figure 3(e) shows the THz experimental measurement setup. A terahertz source (Virginia Diodes, Inc) is utilized to generate the 0.3 THz Gaussian beam as the incident light. The diffraction patterns on the imaging and focusing planes of the devices are captured by a Tera-1024 THz camera. The camera consists of $32 \times 32$ pixel arrays with each pixel size $1.5 \times 1.5$ mm, resulting in an imaging profile size of $48 \times 48{\rm{\;mm}}$, which is only 2 mm smaller than the designed diffraction pattern size $50 \times 50{\rm{\;mm}}$ in the xy plane.
4. Results and discussion
The THz hologram imaging performance is determined by capturing the field intensity profile at the desired distance through the THz camera. Figure 4 shows the measured normalized field intensity profiles for both hologram masks. From Fig. 4(a), the letter ‘U’ is observed at an imaging distance of 50 mm for hologram Mask 1, matching the optimization results. For hologram mask 2, the letter ‘U’ is captured at an imaging distance of 50 mm, and the letter ‘T’ is observed at 80 mm, which functions as a dual-plane THz hologram. The discrepancy between the measurement results and simulated results will be discussed later with the lens design.
Fig. 4. Normalized intensity measurement of THz hologram (a) at distance of 50 mm for Mask 1, and (b) at distances of 50 mm, and 80 mm for Mask 2.
The focusing performance of the designed THz lenses is characterized by the point spread function (PSF) through the intensity measurement of at the desired focal planes. The full-width-at-half-maximum (FWHM) is evaluated at the focal planes. Diffraction-limited Gaussian with FWHM = ${\rm{\lambda }}$/(2NA) is chosen as the target PSF in the optimization. NA = ${\rm{sin}}({ta{n^{ - 1}}({D/2f} )} )$ is the numerical aperture. The diffraction-limited FWHM for Lens 1 is 1.12 mm at a focal length of 50 mm, which is close to the optimized FWHM of 1.22 mm obtained from scalar diffraction theory, shown in Fig. 5(a). And for Lens 2, the diffraction-limited FWHMs are 1.21, 1.39, 1.58, and 1.77 mm at focal lengths 55, 65, 75, and 85 mm, respectively. Compared to the optimized FWHMs 1.74, 3.51, 2.58, and 1.93 mm, the difference is larger than Lens 1. This is due to the limited optimization space with 25 concentric rings. Its performance can be further improved by increasing its optimization space. The FDTD simulations results of FWHMs and PSFs for both Lens 1 and Lens 2 are shown in Fig. 5(c) and 5(d). The FWHM of Lens 1 at focal distance 50 mm is 1.26 mm from FDTD simulation, as shown in Fig. 5(c). The FWHM difference between the scalar diffraction and FDTD is only 0.04 mm, which further verifies the validation of the scalar diffraction optimization accuracy. For Lens 2, as shown in Fig. 5(d), the obtained FWHMs from FDTD are 1.79, 3.78, 2.60, and 2.02 mm at focal lengths 55, 65, 75, and 85 mm, respectively, which are close to the FWHMs obtained from the scalar diffraction optimization. The THz measurement results of normalized intensity distribution for both lenses are shown in Fig. 5(e) and 5(f). Light is focused at the focal length of 50 mm for Lens 1, as shown in Fig. 5(e). As shown in Fig. 5(f), light is focused at focal lengths 55, 65, 75, and 85 mm for Lens 2. But its focusing performance is slightly worse than the Lens 1, which is expected from the results shown in Fig. 5(a) and 5(b).
Fig. 5. (a) 1D and 2D PSF at focal plane 50 mm for Lens 1, and (b) 2D PSF at focal planes 55, 65, 75, and 85 mm for Lens 2 obtained from the scalar diffraction theory. (c) 1D and 2D PSF at focal plane 50 mm for Lens 1, and (d) 2D PSF at focal planes 55, 65, 75, and 85 mm for Lens 2 obtained from full-wave 3D FDTD. (e) THz measurement of normalized intensity distribution for Lens 1 at focal plane distance of 50 mm. (f) THz measurement of normalized intensity distributions for Lens 2 at focal plane distances of 55, 65, 75, and 85 mm.
The difference between the measurement results and optimization results can stem from the measurement errors, material optical property variations, imperfect 3D printing and multimode gaussian profiles. Due to the low hardness of the cured resin, the THz lens with maximum thickness of 1.8 mm has some noticeable wrinkle. During the printing process, some resin residue is trapped in the concave location and will be cured by subsequent layer curing. Thus, we notice some unwanted accumulated cured resin at the lens valleys. In the measurement, the readout of each pixel corresponds to the averaged intensity of THz radiation on the pixel with dimension of 1.5mm x 1.5mm. As a result, THz image features smaller than 1.5 mm cannot be captured by the camera and a PSF with FWHM comparable to the pixel dimension will look bigger. Another measurement error came from the THz source consisting of multiple modes at the same frequency. Each mode has different focusing points, which results in larger beam size than expected. Above mentioned errors also explain why the experimental hologram images are blurrier than the simulated results.
5. Conclusion
In this work, a terahertz dual-plane hologram and an extended-depth-of-focus THz diffractive lens are demonstrated. The dual-plane hologram consists of $50 \times 50$ diffractive optical elements with each pixel size of $1\;mm \times 1$ mm, and the imaging distances are 50 mm and 80 mm. The extended-depth-of-focus THz diffractive lens consists of 25 concentric rings with identical ring width of 1 mm and achieved focus-of-depth from 50 mm to 90 mm. All the devices are designed at 0.3 THz. The height of the pixels and concentric rings are optimized by nonlinear optimization algorithms with scalar diffraction theory based on Ray-Sommerfeld diffraction equation. The implemented algorithms have a reduced number of iterations compared to other Fourier-based methods. FDTD simulation results match well with the optimization results obtained from the scalar diffraction theory for both THz hologram and THz lens. Experimental results and simulation results show that the proposed THz hologram and THz diffractive lens can generate desired diffraction patterns. Differences between experimental and simulation results are attributed to the quality of the incident THz beam profile and manufacturing errors.
Funding
National Science Foundation (ECCS #1936729).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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