3D Printed Fabry–Pérot Filters for Terahertz Spectral Range

3D printing is evolving into a standard tool for prototyping various optical components suitable for application in the terahertz range of the electromagnetic spectrum. This work takes the next step in this evolution by demonstrating the fabrication and subsequent evaluation of Fabry–Pérot interferometers (FPIs). Large optical area (centimetre scale) Fabry–Pérot transmission filters have been 3D printed with polylactic acid (PLA) using a commonly used low-cost 3D printer. The advantages of the proposed approach include low cost, rapid prototyping and repeatability. Terahertz transmission measurements for two demonstrated filter designs realised to target optimisation of either signal transmission or spectral filter performance have been performed using terahertz time-domain spectroscopy (THz-TDS) and demonstrate good agreement with the simulated response in the operating spectral band of 0.30–0.75 THz (wavelengths from 1000 down to 400 μm). The critical spectral characteristics assessed were the filter peak transmission magnitude, central wavelength and full width at half-maximum (FWHM) of the transmission peak, as well as the free spectral range (FSR). The signal transmission levels were observed to reach beyond 90% for the first series of filters that targeted optimisation of this aspect; however, this was accompanied by diminished out-of-band rejection and broader transmission peaks in comparison to the second series of filters which targeted the overall performance. For the latter filter series, the resolution in terms of FWHM values of the transmission peaks was reduced to 40–50 GHz, with the out-of-band rejection approaching a ratio of 10:1. This level of spectral performance, along with the achieved signal peak transmission characteristics of 65–75%, provides adequate performance for many applications harnessing the terahertz spectral range.


Introduction
The progressive development of new sources, detectors and components for the terahertz frequency range has given rise to extensive interest in terahertz radiation and its applications in a wide range of fields [1]. The terahertz frequency/ wavelength range spans low-energy excitations that allow the probing of material interactions for practical applications [2]. For example, THz waves can pass through materials such as plastics, paint, paper and wood; thus, imaging can be performed through these materials [3,4], allowing non-destructive testing in the food and pharmaceutical industries to be demonstrated [5][6][7].
The evolution of 3D printing with various polymers has an immense potential to significantly benefit THz technology since it is one of the most cost-effective and efficient ways to produce reflectors and splitters for THz systems [8], which usually operate within the broad spectral range from 0.3 to 6 THz. Depending on the application, such systems require a source and a detector, THz waveguides, reflectors, splitters and other optical elements that are typically spectrally selective [9][10][11][12][13][14].
It has also been shown that compared to conventional methods, 3D polymerbased printed gratings, splitters and couplers can have low optical losses at THz wavelengths [15]. Furthermore, the rapid prototyping facilitated by 3D printing technology is highly attractive for fast evaluation and turnaround in the design of optical components. In contrast, the realisation of THz components based on conventional materials can require considerable time, resources and effort [8,14].
Widely adopted wavelength-selective optical filtering can be performed by Fabry-Pérot interferometers (FPIs), which utilise optical resonant cavities. FPIs function as narrow band-pass filters and consist of two mirrors separated by an air cavity [16]. The central wavelength of the FPI filter can be controlled by adjusting the extent of the gap (or cavity) between the two mirrors. As schematically depicted in Fig. 1, the FPI is a resonant optical device within which light transmission is sustained only for wavelengths that resonate within the cavity while non-resonant wavelengths are reflected. The transmittance through the filter is maximised at resonant frequencies and falls off rapidly away from these peaks. The order of each transmission peak refers to the number of half-wavelengths that fit into the cavity. The FPI resonance condition dictates that maximum transmittance for a given wavelength occurs when the extent of the gap between the mirrors, d, is an integer multiple of half the incident wavelength.
In the THz spectral range, band-pass Fabry-Pérot filters have previously been realised using multilayer dielectric mirrors based on z-cut quartz [17]. Various other dielectric materials have also been used to build multilayer THz mirrors, each having specific advantages and disadvantages [25][26][27]. For example, siliconbased mirrors can be fabricated using standard silicon-based processes. They

3
have been shown to have relatively low mechanical stability and require complex handling due to the very thin silicon layers of 10 μm thick [28]. Although previous approaches have provided filtering with adequate signal transmission and peak full-width half-maximum (FWHM) for a particular wavelength range [17,[25][26][27][28], they have often been limited because of complex fabrication technologies and low flexibility, warranting the investigation of 3D printing as a technology for THz applications.
This work demonstrates a functional and predictable Fabry-Pérot THz filter based on 3D-printed polylactic acid (PLA) mirrors and a printed stack assembly consisting of multiple PLA layers and air gaps, allowing for rapid prototyping and customisation of filter properties. The utilised mirrors consist of single and multiple 3D-printed PLA layers separated by predetermined air gaps. We demonstrate that the proposed designs offer adequate performance for many applications and can be Fig. 1 Idealised schematic representation of a Fabry-Pérot interferometer, consisting of two mirrors with a refractive index of n (n = 1.64 for PLA) separated by distance, d, within the medium refractive index, n 0 . Broadband incident light entering at a particular angle is for representation purposes only achieved using a commonly used low-cost extrusion printer and a readily available 3D printing plastic, such as the PLA polymer.

Material and Methods
We have chosen to use PLA, which is a biodegradable thermoplastic polymer and widely adopted 3D printing material. It is derived from natural products such as corn, sugarcane and potatoes. Its use is attractive because its preparation process consumes 25-55% fewer energy resources than petroleum-based polymers [29]. In the frequency range of 0.30-1.0 THz, PLA has a higher refractive index (n = 1.64) compared to other polymers such as high-density polyethylene (HDPE), nylon and acrylonitrile butadiene styrene (ABS), which are characterised by refractive index values around 1.55 [8].
3D printing using PLA relies on a low glass transition temperature (60 °C) and a melting point of 153 °C, with printing typically occurring at 190 °C. Due to the low thermal expansion coefficient value of 70 × 10 −6 °C −1 , the need for a heated print bed can be removed. The absorption coefficient and refractive index of PLA were obtained using terahertz time-domain spectroscopy measurements made on a 2-mmthick 3D-printed PLA layer exclusively printed for this purpose. The measured values for the absorption coefficient and refractive index are presented in Fig. 2a and b, respectively. They were used to predict the optical response of the filters via the analytical filter model across the investigated range of 0.3 to 0.75 THz and found to be in agreement with previous reports in the literature [8].

3
The 3D printing was performed using an Ender FDM 3D printer (3D Technology Co., Ltd., Shenzhen, China), considered one of the most common printers at an affordable cost. The stack build plate was replaced with a glass build plate to print the thinnest and flattest mirrors, and one surface of each mirror is likely to be very flat in comparison to the opposing surface. "Elmers disappearing purple" glue was used for adhesion to the glass since it is easy to remove from the glass plate after printing.
The PLA layers used to form the filters were printed with individual thicknesses of 116 ± 5 μm. This thickness corresponds to the minimum PLA layer thickness that was consistently achievable with the apparatus used in this study. The individual 3D-printed layers supported by thicker outer rings were used to construct two distinct series of devices, referred to in this paper as α-series and β-series. The α-series FPIs were formed between two 116-μm-thick PLA reflectors separated by a suitable air gap (PLA-air-PLA). Figure 3b schematically shows the three-dimensional expanded and cross-sectional views depicting the construction of α-series devices. These devices consisted of three different designs in which the air gap between the PLA reflectors was 330 μm (referred to as α-330), 430 μm (α-430) and 530 μm (α-530). PLA spacing rings of various thicknesses (330 μm, 430 μm and 530 μm) were designed and printed to establish the air cavity between the PLA reflectors. Clamping and holding rings helped to retain the air cavity between the two individual PLA reflectors and the formed filter stack. The PLA rings defined the optical aperture with inner/outer diameter of 15 mm/18 mm. The thicknesses of the PLA reflectors forming the FP filters and the PLA spacers were confirmed via digital micrometre measurements to have an accuracy of ± 5 μm. Schematic illustrating a 3D view as well as a cross-sectional view of stackable un-assembled α-series 3D-printed Fabry-Pérot filter where the FP cavity is formed by two single 116-μm-thick PLA layers, and (c) schematic illustrating a 3D view as well as a cross-sectional view of stackable unassembled β-series devices where the FP cavity is formed between two distributed Bragg reflectors (DBRs) each formed by a pair of 116-μm-thick PLA layers separated by a 116-μm air gap which was pre-defined by a step noticeable in the larger (red) ring. The thickness of the spacer ring defines the central air gap forming the FP cavity in both α-series and β-series devices The concept adapted for the realisation of β-series FPIs is presented in Fig. 3c. In contrast to α-series devices, which used single-PLA layers as reflectors, β-series devices used a 3-layer structure in which each consisted of two PLA layers separated by a 116-μm air gap. This is the concept of a 3-layer distributed Bragg reflector (DBR) formed by the refractive index contrast between the PLA-air-PLA layers. The intra-DBR air gap was facilitated by printing a 116-μm-high step in the outer supporting ring, which serves as a hard stop and separator between the two PLA layers to form the 3-layer DBR.
This intra-mirror air gap within the DBRs is kept constant for all β-series devices. The FP main cavity for the β-series devices was created by an air gap between two identical DBRs. The β-series consisted of three devices where the air gap between the DBRs was designed to be 330 μm (referred to as β-330), 430 μm (β-430) and 530 μm (β-530). Like the α-series, PLA spacer rings of a pre-defined thickness (330 μm, 430 μm and 530 μm) were printed to establish the main air cavity. The holding ring design ensured the individual layers were not compressed and that all the air gaps between the PLA layers maintained their integrity. Table 1 summarises the notation adopted to describe the fabricated filters listing the geometrical details and thicknesses of the various layers used in their construction.
The constrain associated with the used apparatus in terms of 116 ± 5 μm being the minimum reliably achievable PLA thickness had a notable impact on the fabricated DBRs. In an ideal design, a DBR is composed of alternating high-refractive-index and low-refractive-index layers of quarter-wavelength optical thickness. The ability to print layers and spacers with a minimum thickness of 116 ± 5 μm or multiples thereof placed a limitation for the realisation of the quarter-wavelength ideal design criteria. Therefore, targeting the spectral wavelength range of 400 to 1000 μm (0.75 down to 0.3 THz), the adopted layer and spacer thicknesses were chosen to be as close as practically possible to the quarter-wavelength optical thickness criteria.

Model
The measured optical transmission spectra of the filters have been compared against an analytical model constructed using optical transfer matrix methods to estimate the frequency-dependent transmission of a thin-film stack [30]. The simulation model relates the electric and magnetic fields of propagating waves at all thin-film interfaces via a characteristic matrix, with the electric field component of the incident light being treated as a scalar quantity. Along with interface transmission and reflection Fresnel coefficients, the model gives the complex transfer functions of the optical system and generates amplitude and phase response of the multilayer thinfilm stack. The input parameters to the model were the measured physical parameters of individual filter components, including PLA layer thickness, refractive index, absorption coefficient and the main FP cavity and intra-DBR air gaps. A detailed description of this theory can be found in the literature [30-32].

Measurements
A terahertz time-domain spectroscopy system (TeraPulse4000, Cambridge, UK) was used to characterise the PLA layers, which included reflectance measurements of the individual layers used to construct the α-series and β-series devices, as well as the transmission characteristics of the filter stacks. The specimen chamber was purged with dry nitrogen before measurements since atmosphere water vapour is a strong absorption of THz radiation. The THz spectrometer produces a useable spectrum over the range of 0.1-4 THz. Spectra from the system were acquired at a rate of 30 scans per second with a spectral resolution of 0.032 THz, with 1800 single point scans being averaged over a minute for each sample.

Results and Discussion
The measured THz reflectance spectra of the individual reflectors used in the construction of α-series and β-series devices are shown in Fig. 4 with a comparison to the transfer matrix model. It is evident that the measured and calculated data are in very good agreement for the single PLA layer reflectors used to construct α-series devices and PLA-air-PLA DBRs used to build β-series devices. The reflectance for the single-layer PLA mirror was below 20% across the investigated frequency range of 0.30-0.75 THz. However, the PLA-air-PLA DBR reached a peak value of 50% at a frequency of 0.46 THz. The measured and simulated transmittance spectra of α-series FP filters formed by pairs of single PLA layer reflectors separated by a main air cavity of 330 μm, 430 μm and 530 μm are shown in Fig. 5. It can be observed that the measured results are in good agreement with the calculated response irrespective of the main filter cavity. First-order transmission peaks can be observed for each of the three filters in the frequency range of 0.33-0.43 THz, whereas the second-order transmission peaks were observed in the range of 0.56-0.71 THz. As expected, the wavelength positions of the transmission peaks within each optical order are positively correlated with the extent of the gap between top and bottom reflectors. Consistently for all filters, the magnitudes of the first-order transmission peaks are near 90% or greater, and the magnitudes of the second-order transmission peaks are near 80% or greater.
For the α-430 filter, the central frequencies of the first and second-order transmission peaks are observed to be located at frequencies of 0.36 THz and 0.61 THz, respectively. FWHMs for the first-and second-order transmission peaks were estimated as 0.1 THz and 0.11 THz, respectively. The free spectral range (FSR), which corresponds to the wavelength separation between the 1st and 2nd order peaks, was 0.25 THz. Similar observations can be made in Fig. 5 for α-330 and α-530 filters, and the critical characteristics are summarised in Table 2.
The measured and simulated transmittance spectra of β-series FP filters are shown in Fig. 6. This series consisted of devices formed by DBR pairs separated by main air cavities of 330 μm, 430 μm and 530 μm. It can be ascertained that Consistently, for all β-series devices, the magnitudes of the first-order maximum transmission peaks were observed to be 64-76% and for the second-order transmission peaks to be around 58% or greater. For the β-430 filter, the FWHMs of the 1st and 2nd order peaks were estimated to be 0.039 THz and 0.043THz, respectively, and the free spectral range was found to be 0.18 THz. Similar observations can be made for other filters in the β-series. Spectral characteristics of β-series filters have been summarised in Table 2.
The reflectivity of mirrors used to construct the devices is the most relevant parameter that results in the differences in the observed performance between the α-and β-series filters. Nevertheless, the relatively poor reflectivity of below 20% measured and presented in Fig. 4 for single-PLA layer mirrors that were used to construct the α-series devices were adequate to achieve FP filter peak transmission of 90-94% (see Fig. 5). In this aspect, α-series devices outperformed the β-series. However, the α-series filters, in comparison to the β series, were observed to be characterised by reduced spectral performance, with broader transmission peaks and poor out-of-band rejection (transmission maximum-to-minimum ratio). For the β-series filters presented in Fig. 6 and summarised in Table 2, the measured transmission peaks were observed to be characterised by FWHM values of 40-50 GHz, with out-of-band rejection ratios approaching 10:1. This level of spectral performance for the β-series filters has been achieved with a signal peak transmission well above 50%, which may be the necessary lower bound for sufficient optical throughput to attain a satisfactory system signal-to-noise ratio [21].
It needs to be noted that during our experiments, a value of 190 °C was found to be the optimal temperature for printing our PLA layers at a print speed of 90 mm/s. At this printing temperature and speed, the boundaries between any structural grain-like order tends to be minimised; otherwise, significant light scattering would have been observed, resulting in degraded performance, which has been investigated recently in reference [33]. At higher printing temperatures and higher print speeds, the thickness of the printed layers was significantly greater than the targeted thickness of ~ 116 μm, and some imperfections such as a level of residual curvature, tilt and roughness were present in printed layers. Such imperfections partly contribute to the observed deviations between measured and simulated filter performance, which are more noticeable in Fig. 6 for β-series devices in comparison to transmission spectra presented for α-series devices in Fig. 5. It needs to be noted that any misalignment of layers during stacking can result in further deviations from the idealised scenario. However, we have observed that successive stepwise assembly and disassembly of each of the α-series and β-series filters by hand was adequately consistent for repeatedly achieving filter transmission spectra with less than ± 2% deviation. The constrain associated with the used apparatus in terms of 116 ± 5 μm being the minimum reliably achievable PLA thickness restricted us to print layers and spacers with a minimum thickness of 116 ± 5 μm or multiples thereof and placed a limitation for the realisation of the quarter-wavelength ideal design criteria. However, we were able to analytically compare using optical transfer matrix methods the performance of filters according to the realised geometry with the performance predicted for the ideal quarter-wavelength design criteria. This comparison demonstrated that no significantly notable performance improvement in terms of transmission peak FWHM or out-of-band rejection is to be expected even if we were able to print with thickness resolution allowing us exactly meet the quarter-wavelength ideal design criteria. We attribute this to the fortunate fact that with the minimum printable thickness of 116 ± 5 μm the quarter-wavelength criteria for various layer thicknesses were achieved to within a tolerance in the range of 10-20%.
We see the simplicity of the methods demonstrated in this paper to be most notable. While using the most common extrusion 3D printer (costing under $300) and one of the most common 3D printing plastics characterised by relatively poor optical properties, we were able to achieve performance levels that can be useful for variety of applications harnessing terahertz radiation. This perspective contributes further to the rapidly evolving additive manufacture of terahertz photonic components (see, for example ref. [34] for a recent review), which is relevant to the possibilities and challenges of this technique for realisation of components that in principle can extend beyond the terahertz range into the infrared and visible bands as technology matures.

Summary and Conclusion
This work has demonstrated the successful construction of PLA-based FP filters for the THz range. The constructed filters having a large optical area with centimetre-scale lateral dimensions have been 3D printed using one of the most common and low-cost 3D printers. The measured filter transmittance spectra were observed to agree with the simulated response, and repeated assembly and disassembly of the filters were shown to provide adequate reproducibility of filter transmission spectra with less than ± 2% deviation. Two filter types (α-& β-series) have been demonstrated in the frequency range from 0.30 to 0.75 THz. The peak transmission for α-series filters reached levels beyond 90%; however, this was accompanied by diminished out-of-band rejection and broader transmission peaks in comparison to β-series filters. Filter resolution in terms of FWHM values of the transmission peaks for the β-series were in the range of 40-50 GHz, with the out-of-band rejection that is approaching 10:1. This level of spectral performance, along with the achieved signal peak transmission characteristics of 65-75%, provides adequate performance for many applications harnessing the terahertz spectral range. Further enhancement of filter performance could be achieved by increasing the refractive index contrast between the printed layers and air, which can be realised by using different printing materials or impregnating the PLA filament with microparticles of, for example silicon or silicon carbide (SiC).
Funding This work was supported in part by the Australian Research Council, the Australian National Fabrication Facility and the Western Australian Government's Department of Jobs, Tourism, Science and Innovation.

Data Availability
The data that support the findings of this study are available from the authors on reasonable request.

Declarations
Ethical Approval Not applicable.

Competing Interests
The authors declare no competing interests.