Hybrid Multiphase Fresnel Lenses on Silicon Wafers for Terahertz Frequencies

The hybrid multiphase Fresnel lenses (H-MPFLs) were developed on silicon wafer at the selected frequency of 585 GHz. For this reason, the design of a standard MPFL was modified thoroughly in various outer zone areas in order to reduce the complexity and manufacturing time of the diffractive optical elements by employing the direct laser ablation processes. The phase offset was found by precise control over the phase shift of incoming radiation in steps of π/12, revealing its optimal value to be of +π/4 independently on the hybridization order of the lens, the focusing gain of which was found to be up to 10 % higher than that achieved with a standard design MPFL. The numerical modeling data were confirmed by experiments demonstrating the effective THz beam focusing with H-MPFL samples to the diffraction-limited spot size. The proposed development procedure can be further modified by scaling the hybrid lens design to other frequencies with different zone numbers and/or by employing other materials suitable for THz photonics integration on-chip with the semiconductor devices.


I. INTRODUCTION
A S THE terahertz (THz) science and technologies progress rapidly in recent years, there is a request for optical components of high performance and less fabrication difficulty, which integration on-chip with the semiconductor devices would be less demanding [1], [2], [3]. A beam shaping with the diffractive optical elements (DOEs) grabs attention of the THz community because of their miniature size (order of wavelength), light weight, and compatibility with on-chip technologies with respect to reflective and refractive elements [4], [5], [6]. Depending on the dedicated function for example focusing, redirecting, and splitting of incoming radiation, a typical DOE may possess different structural forms based on the design rules of wave optics [7]. The multi-phase Fresnel lens (MPFL) is a successful example of DOE which defines the beam shape by describing the phase and amplitude distribution of incident wave at the focal plane. In general, MPFL may have different zone profiles like binary or multilevel step patterns which make a phase shift of incoming radiation resulting in constructive wave interference at a predefined distance along an optical axis. In a particular case, the thickness of material is adjusted to introduce the phase offsets strictly related to the value of the effective refractive index. Different tools such as mechanical polishing, hot pressing, and 3D printing have been used for the development of the MPFLs as the discrete components of different low refractive index materials such as polymers, Teflon, polyamides, and paraffins which possess small optical losses in the sub-THz frequency range [8], [9], [10], [11]. All those materials and fabrication processes make bottleneck for the developments at higher THz frequencies and for the integration of MPFLs on-chip with other THz devices [2], [3]. Researchers moved to higher frequencies by making use of silicon (Si) with low absorption losses typical for many semiconductor crystals [12], [13]. Reactive ion etching [14] and photo-lithography [15] are the two most common methods used to manufacture Si-based MPFLs; however, they are tedious if several processing steps need to be modified [16]. A mask-less direct laser ablation (DLA) technology was proposed recently [17], [18] as a flexible tool for the development of MPFLs with different number of phase quantization levels (Q), which demonstrated good focusing performance in a wide frequency range from 0.6 up to 5 THz [19], [20]. Large Fresnel losses due to high refractive index of Si were proposed to compensate by fabrication of anti-reflection coatings within the same DLA process [21], [22], [23]. Notably that the diffraction efficiency of Si-based MPFLs with large Q value can reach almost 100% [19]. However, manufacturing of the MPFL with a large number of phase quantization levels is a challenging task because the complexity of step profile increases going toward the outer zones [20]. On the other hand, the design of lens with a high numerical aperture, defined as the ratio between the diameter and the focal distance, needs to be considered in the case of THz device manufacturing on-chip. Indeed, DLA processing of Si for THz technologies makes specific requirements for the surface smoothness and This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ dimensions of the phase-delay-material used especially in the area of most outer zones [13], [24]. It is worth noting that only with the appearance of the DLA technology, many designs of DOEs become possible to realize crystal materials in the THz range [25], [26], [27]. Thus, new designs and optimization procedures of phase step profiles require a detailed investigation. The first attempts of simplification of the outermost zones of MPFLs by phase correction technique were made theoretically and experimentally by several groups [28], [29], [30]. However, to the best of our knowledge, the development of hybrid MPELs (H-MPFLs) with a similar or even better focusing performance than the standard design MPFL has not been reported.
In this work, the hybrid MPFLs were developed on highresistivity float zone (HRFZ) Si wafer for radiation focusing at the selected frequency of 585 GHz. To realize the H-MPFL, we modified the design of standard MPFL with the phase quantization levels of eight (Q = 8), which has been previously demonstrated to be a good tradeoff between fabrication time, design complexity, and focusing performance [19], [20]. Simplification and successful optimization of outer zone parameters of standard MPFL lead to less demanding DLA fabrication process without deterioration of the focusing gain. The focal length, F, of 13 mm and the F-number of 1 were selected as initial parameters for the design. All MPFL samples were fabricated by the DLA technology, demonstrating very good THz beam focusing in agreement with numerical data. We believe that our findings are useful for further developments of laser-processed THz components and for THz photonics integration on-chip with semiconductor devices.

II. NUMERICAL MODELING
A loss-free Si of 0.5 mm thickness (h) was selected for the development of the MPFL samples of different shapes and complexites. Namely, both-sides polished, (100) crystal orientation HRFZ-Si wafers with a resistivity of 100 kΩ cm and a refractive index of 3.45 were employed. The properties of HRFZ-Si after laser ablation have been investigated in the THz range elsewhere [24].
The standard MPFL was designed using the following equations [6]: here r n is the outer radius of nth subzone (concentric ring), λ is the design wavelength, used value λ = 0.512 mm. The depth of single subzone, d, was described by equation here m is the integer number value which will be discussed later, ε is the refractive index of HRFZ-Si, the value of which was measured by THz time-domain spectroscopy to be approximately 3.45. To have a complete set of quantization levels, the total number of Fresnel zones (FZ) and concentric rings (n) in the design were fixed to integers of 3 and 3 * Q, respectively. Thus, the total diameter of the lenses was estimated to be about D = 2 * r 24 = 13.02 mm.
In the case of standard MPFL, the parameter d was kept the same for all FZ. However, for lenses of hybrid design, the depth of subzones was varied with respect to the used value of Q. A finite difference time domain method using CST software was used to analyze and optimize the focusing performance of the samples. The focusing gain was defined as the squared ratio between the field amplitude with (E MPFL ) and without (E 0 ) lens under research. The simulations were performed by choosing perfectly matched layer boundary conditions in order to absorb the transmitted and reflected waves from the structure. Symmetric conditions were enforced in X (perfect electric) and Y directions (perfect magnetic) to reduce simulation time and working memory requirements. The calculations were performed assuming the steady-state energy criterion using a multifrequency plane wave with a field amplitude of 1 V/m irradiation onto the sample. The intensity distribution along the optical axis and at the focal plane was calculated and analyzed. Fig. 2 shows simulated focusing gain characteristics obtained for the selected samples. Calculated beam profiles normalized to input radiation for the standard MPFL are shown in Fig. 2 by black solid lines. The gain value without taking into account a reflection loss was found to be about 20 dB (about 100 times) which is in agreement with data obtained from the analytical performance description of the MPFL used [6], [20].
The optimization of phase profile in the selected outer FZ areas of the H-MPFL samples was performed by using a discrete step size of one-eighth (m/8) of the depth initial value at the subzone area in (2) until the lens reached a maximum value of the focusing gain. For this reason, an integer m value was varied in the range from −4 to +4, which corresponds to a phase change of π/12 for the selected wavelength. The optimization results in terms of the focusing gain, beam size, focal distance, and #F-number of the MPFL samples with hybrid phase profile with respect to the standard MPFL as a function of optimization factor (m) are shown in Fig. 3.
The change of the depth in subzones area modifies the lens performance independently on the hybridization order of H-MPFL samples. Larger focusing gain was found by making the depth of subzones deeper for both hybrid design lenses than the actual subzone depth instead of making them shallower [see Fig. 3(a)]. The maximum gain was found when the optimization factor of +3 was added to the actual subzone depth of H-MPFL1 and H-MPFL2 samples, which corresponds to the phase increment of π/4. Moreover, we see the reduction in full-width half maximum (FWHM) of focused beam size at the same m = + 3 [see Fig. 3(b)]. In addition, Fig. 3(c) and (d) demonstrate the modification of focal length and #F-number with very similar values as for the standard MPFL sample with d = 26 μm, when the optimization factor for H-MPFL samples is of about three-eighths of the actual subzone depth, i.e., m = + 3. It is worth noting that the maximum gain for standard MPFL samples is achieved when m = 0, i.e., the performance of standard design diffractive lenses is optimal in terms of focusing gain, therefore, analytical (1) and (2) accurately describe geometry parameters of the MPFL.  Numerical optimization results are summarized in Table I. The H-MPFL1 and H-MPFL2 samples with factor m = +3 demonstrated maximum focusing gain and were treated as optimal design lenses, resulting in up to 10 % (+0.4 dB) higher gain in comparison to the standard MPFL [see also red and blue curves in Fig. 2(a) and (b)]. On other hand, the optimization of the H-MPFL3 samples was not carried out as their initial focusing gain was found to be small. Namely, the H-MPFL3 sample with m = 0 demonstrated the focusing gain to be up to 2 times smaller with respect to the reference sample and even smaller as compared to the MPFL4, which is a standard design MPFL with Q = 8 and m = 0 but with only two sets of FZ, namely FZ 1 and FZ 2. The increase in focusing gain for the optimized H-MPFL samples was due to constructive interference of the wavefront thanks to precise control over a phase shift of incoming radiation.
Further increase in the subzone depth, cases m > + 3, resulted in the destructive interference and reduction of the focusing gain and FWHM values. Moreover, from the results shown in Fig. 2(a), we clearly see a change in sidelobe level from 0.23 dB for standard MPFL to 4.05 dB for H-MPFL1 sample with an increase of absolute focusing gain at the central beam by up to 10% (+0.4 dB). Further modification of the hybrid lens design from H-MPFL1 to H-MPFL2 resulted in two times smaller sidelobe magnitude (2.48 dB) maintaining approximately the same focusing gain values at the central beam (see Table I).

III. RESULTS EXPERIMENTAL VALIDATION
Three samples in total, two of optimized design (H-MPFL1 and H-MPFL2) and one of standard MPFL design were fabricated for operation at the frequency of 585 GHz. The zone plates were patterned by an industrial-scale pulsed laser (Atlantic 60, EKSPLA UAB) with a pulse duration of 10 ps, operation wavelength of 1064 nm, scan speed of 856 mm/s (47 % spot overlap), 32 μm laser spot size diameter, 11.8 J/cm 2 laser irradiation fluence, hatch angle rotation of 45 deg after each scan, and etch depth of 0.2 μm per layer which allowed to maintain precise control over the profile shape of the MPFLs. Microscope pictures of the fabricated samples are shown in the bottom row of Fig. 1. The step profiles of each sample scanned with "Veeco Dektak 150" profilometer demonstrate a good fit to the design shape as shown from the center towards edge side in the top row of Fig. 1.
The outer subzones become increasingly narrower with growing distance from the center in a standard design MPFL. Therefore, a small spot size of the laser beam is needed to produce a large diameter lens and ensure sufficient fabrication resolution in the outer zone areas. Whereas the outer subzones in the hybrid lenses are wider but a bit deeper. Although the removed material volume in the standard design lens is smaller than that in the hybrid lenses, the feasibility to use a large diameter laser beam leads to a faster and less demanding fabrication process of those lenses. For example, if we consider the narrowest ring of the standard and hybrid design to be, respectively, 100 and 150 μm, then we increase proportionally the laser spot and 1.5 times faster fabricate hybrid lens of the optimized design.
Focusing performance of the samples was explored experimentally by using a THz continuous wave system operating at 585 GHz frequency. Setup is shown in Fig. 4. Briefly, it is composed of RF generator, amplifier, collimator (HDPE lens), reflecting off-axis parabolic mirror (Gold Mirror), microbolometric THz detector placed on scanning multiaxis translation stage, and lock-in amplifier for the signals readout and storage in the personal computer. The radiation generated by Schottky diode-based AMC (Amplification Multiplication Chain) was outcoupled via a horn antenna. The measurements were done by scanning with the THz detector placed on an automated 3D axis stage. This allowed us to evaluate the intensity distribution along the optical axis (Z-axis) and in the focal plane (X-and Y-directions) with respect to the sample.
The measurements revealed a Gaussian beam shape at the sample plane in experiments with FWHM = 14 mm (see inset  in Fig. 4). Notably, the plane wave excitation was used in the numerical simulation resulting in overestimated gain values in comparison to the experimental data. We would assume that the aperture efficiency with the reflection losses in our experiment was within the range of 38%.
Experimental data of the THz beam focusing with different samples are shown in Fig. 5. The focal distance was found to be around 12 mm (z-direction) similar for all samples. It is clear that the beam is perfectly focused as an Airy disc seen in the focal plane as the side lobes around the central focal spot. The  It is worth noting that the magnitude of the intensity change was found the same as predicted by numerical modeling. Fig. 6 demonstrates the focusing gain characteristics obtained experimentally and numerically for different lens samples. Both hybrid design lenses demonstrated the same #F-number and the focusing gain approx. Ten percent (0.4 dB) higher than values are achieved with the reference MPFL samples. Detailed comparison of experimental and modeling results for fabricated samples is done in Table II. The first columns of the simulation and experimental data show the focused beam FWHM measured along the Y-axis direction. From the beam width percentage ratio, B, of the beam size of hybrid, FWHM H , to the beam size of standard, FWHM S , design samples, i.e., B = FWHM H / FWHM S , we conclude that the focused beamwidth of optimized H-MPFL samples is narrowed by up to 8% compared to the standard MPFL in simulation, whereas from the experimental data, a similar change of FWHM values was found. The comparison of focusing gain was evaluated as the attenuated gain, A, indicating the ratio between the focusing gain of standard, G S , and hybrid, G H , design MPFL samples, i.e., A = G S , / G H . Here A < 1 indicates the focusing gain of the hybrid design lens to be higher in comparison to standard design MPFL. By comparing the A values, we clearly see that the optimized H-MPFL samples demonstrated slightly better focusing performance both in theory and in experiment. It is worth to mention that by measuring the ratio between intensities at the focused beam center and the side of first-order lobe, the diffraction efficiency was estimated to be about 93 % (98 %) in the experiment (in modeling), the value of which was in agreement with the analytical description of the MPFL [6], [20].
We can also see from Table II a noticeable change in the focusing gain of about 7 % and beam size of about 6 % in the experiment values for optimized H-MPFL 2 compared to optimized H-MPFL 1 samples. Such differences were in the range of experimental errors as the simulation data demonstrated a considerably smaller difference between respective parameters of the optimized hybrid samples being only about 0.9% and 0.3%, respectively (see also Figs. 2 and 3).
Thus, obtained results validate a successful optimization process of the hybrid lenses with reduced complexity of phase profile. The H-MPFL samples demonstrated in the experiment slightly better focusing performance as those of standard design MPFL samples. The increment in the focusing gain of hybrid lenses was attributed to smaller shadowing effects and more efficient constructive interference of the spherical wavefront because of precise control over the phase shift of incoming radiation at the outer FZ by making them deeper [31], [32], [33]. Achieved overall focusing gain, which was not a concern of this research work, from the optimized H-MPFL samples was up to 18.7 dB (20.4 dB) in the experiment (numerical modeling), can be further increased with a larger amount of the FZ [6], [20].

IV. CONCLUSION
To conclude, we have developed for selected frequency the H-MPFLs with two times smaller number of steps in the phase profile at most outer FZ areas. Optimal phase offset was found to be of +π/4 independently on the hybridization order of the H-MPFL samples, the focusing gain of which was found to be up to 10% higher in comparison to a standard design MPFL. Thus, we proposed a new approach for the development of much simpler and less complex DOEs which can be scaled to other frequencies and integrated on semiconductor chips with THz detectors and emitters.