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Abstract
In terahertz (THz) photonics, there is an ongoing effort to develop thin, compact devices such as dielectric photonic crystal (PhC) slabs with desirable light–matter interactions. However, previous works in THz PhC slabs have been limited to rigid substrates with thicknesses ${\sim}100\,\,{\rm s}$ of micrometers. Dielectric PhC slabs have been shown to possess in-plane modes that are excited by external radiation to produce sharp guided-mode resonances with minimal absorption for applications in sensors, optics, and lasers. Here we confirm the existence of guided resonances in a membrane-type THz PhC slab with subwavelength (${\lambda _0}/6 {-} {\lambda _0}/12$) thicknesses of flexible dielectric polyimide films. The transmittance of the guided resonances was measured for different structural parameters of the unit cell. Furthermore, we exploited the flexibility of the samples to modulate the guided modes for a bend angle of $\theta \ge {5^ \circ }$, confirmed experimentally by the suppression of these modes. The mechanical flexibility of the device allows for an additional degree of freedom in system design for high-speed communications, soft wearable photonics, and implantable medical devices.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. INTRODUCTION
Communications-specific applications such as Wi-Fi enabled video calls, large data transfers, and entertainment streaming have pushed the bandwidth limits of non-optical components like copper wire, cable connections, etc. To achieve the higher bandwidths (${\ge} 100\;{\rm Gbit/s}$) required by the increased and future demand, researchers aim to develop new technologies designed to operate at terahertz (THz) frequencies. However, the majority of current devices and components are large and bulky with slow improvement for practical, compact THz platforms. The most popular candidates for planar, compact THz light manipulation are plasmonic metamaterials, subwavelength arrays of metallic resonators periodically distributed on a dielectric substrate. Applications of planar metamaterials include nonlinear enhancements [1–3], near-field focusing [4,5], and beam steering devices [6,7]. Additionally, due to the use of polyimide [8–10], polydimethylsiloxane (PDMS) [11] and carbon nanotube [12] substrates, THz metasurfaces are desirable for flexible photonics applications. But they are limited by intrinsic ohmic loss of metals at THz frequencies and requirements for fabrication of multilayer or composite structures.
To avoid the above limitations, photonic crystal (PhC) slabs have been employed to manipulate light–matter interaction with THz waves. In general, PhC slabs are two-dimensional (2D) planar devices composed of a dielectric material with an array of holes and periodicity in the range of the wavelength of interest. These planar 2D PhC slabs exhibit strong field confinement and interaction with THz waves with minimal absorption loss. In addition, the dielectric platforms of PhC slabs offer low-loss alternatives to metallic resonators.
As such, PhC slabs have been used to guide [13,14], filter [15,16], and enhance electric field distributions in small volumes [17,18]. Although light is usually confined within PhC slabs, certain guided modes can strongly couple with external radiation when they are normally incident to an in-plane resonance on the PhC surface. The in-plane mode, also described as the photonic band edge effect or the distributed feedback effect, can be utilized for lasing in PhCs [19,20]. Experimental demonstrations of the properties of 2D THz PhC slabs have been reported in silicon [21–32] and GaAs [33] media. The majority of reported THz PhC slabs have thicknesses in the range of THz wavelength [21–30,33] with a few demonstrations around 50 µm thickness [31–33]. For thin samples originating from rigid crystals, the PhC slabs are fragile and difficult to handle.
In this Letter, we report the existence of guided-mode resonances supported by a flexible membrane-type PhC (both 50 and 25 µm thick) operating at THz frequencies. Experimental measurements of the fabricated device confirm the existence of guided resonances that couple the in-plane resonance of the PhC slab with external radiation. Systematic studies of these resonances with respect to different thicknesses and hole geometries are presented, and curvature-dependent spectral measurements show active tunability of the guided modes as a function of bend angle. As such, our results pave the way for a new, flexible medium for dielectric PhC slabs with subwavelength thickness. Furthermore, the mechanical robustness of the devices allows for applications in flexible photonics, including integration with future biomedical and communications platforms.
2. MATERIALS AND METHODS
A schematic of the unit cell is given in Fig. 1(a) for the flexible PhC slab with air holes of a specific diameter $d$, radius $r$, and period $a$ in a square lattice. Next, we calculated the first 10 bands of the photonic band structure using the MIT Photonic Bands (MPB) package [34] for a 2D PhC slab [Fig. 1(b)] with dielectric constant $\epsilon = 3.4$ and $r/a = 0.3$ and plotted for both TE and TM polarizations [Fig. 1(c)]. Unlike silicon (${n_{\rm Si}} \sim 3.418$ at 1 THz) or GaAs (${n_{\rm GaAs}} \sim 3.59$ at 1 THz) PhC slabs, no clear photonic bandgap is present in our samples made of Kapton polyimide (${n_K} \sim 1.843$). However, we observed and verified through spectrographic analysis guided resonances originating from the in-plane resonant mode of the PhC slab coupled to the external THz radiation in the direction normal to the PhC plane [Fig. 1(d)]. The in-plane resonance leaks out from the surface as a guided resonance as it is coupled to the radiation.
Fig. 1. Flexible 2D THz PhC slab. (a) Unit cell of the 2D PhC slab with air hole radius $r$, diameter $d$, and cell period a. (b) Band dispersion calculated for the PhC slab in (a) with $r/a = 0.3$ and $\epsilon = 3.4$. The TM mode is dashed, and the TE mode is solid. (c) Representation of the mechanism for supporting guided modes where the incident THz wave couples with the in-plane resonant mode of the PhC slab. (d) Optical images of the fabricated PhC slab and highly flexible Kapton film (top-right insert). Noticeable cracking on the surface of the slab is attributed to possible thermal expansion and contraction during the ICP etching process.
Standard polyimide etch recipes reported in the literature are not applicable for Kapton films, which are uniquely synthesized for high temperature resistivity. To fabricate the air holes in our samples, we first patterned a titanium hard mask on the Kapton film, then dry etched with an inductively coupled plasma (ICP) etcher (Plasma-Pro 100 Cobra). The etching conditions are as follows: oxygen pressure of 20 mTorr, flow rate of 60 sccm, RF1 power at 75 W, and RF2 at power 1500 W with an approximate etch rate of ${\sim}1.05\;\unicode{x00B5} {\rm m/min}$. Then the titanium hard mask was removed with an etchant solution. A photo of a flexible Kapton polyimide film with an optical image of the PhC pattern on the sample is also shown [Fig. 1(d)].
Transmission measurements were performed with standard THz time-domain spectroscopy (THz-TDS) techniques. Two different experimental setups, at Rice and at Nanyang Technological University, were utilized to perform the measurements. The setups used output from a Ti:sapphire oscillator producing nearly (100–150 fs) pulses at 800 nm split into pump and probe beams. The pump and probe optical pulses served as generation and detection of the THz field in ZnTe crystals, respectively. The probe beam was delayed with respect to the pump beam using a delay stage, which in turn dictates the resolution of the system. Electric field amplitude and phase of the THz waveform are obtained by scanning the delay stage for a maximum resolution of 40 GHz. All measurements were done inside a box purged with dry air or nitrogen to remove excess water vapor. Numerical simulations were performed using a finite element method with periodic boundary conditions to emulate a 2D infinite array of unit cells. To ensure the accuracy of the simulations, the length scale of the mesh was set to ${\le} {\lambda _0}/10$ throughout the simulation domain, where ${\lambda _0}$ (600 µm) is the central wavelength of the incident radiation. The input and output ports were placed at $3{\lambda _0}$ from the PhC slab with open boundary conditions. In all numerical simulations, the permittivity value of $\epsilon = 3.4$ with the dielectric loss of 5% (tan $\delta = 0.05$) was applied for the dielectric Kapton layer.
3. RESULTS AND DISCUSSION
To study the behavior of the guided-mode resonances, we simulated (solid line) and measured (dotted line) the transmission of a 50 µm thick Kapton PhC slab with circular air holes of varying diameters [Fig. 2(a)]. Due to the periodicity of the unit cell (300 µm), such structures do not diffract normally incident THz radiation for frequencies less than 1 THz. The single guided resonance below 1 THz is observed between 0.9 and 1 THz in both experiment and simulation for all hole diameters. However, the experimental resonance has a larger linewidth compared to the simulations. The difference between the experiment and simulations is mainly due to finite sampling time, scattering losses, and possible disorder from slight imperfections or defects in the sample, which cause the experimental spectra to be dampened and inhomogeneously broadened. Recent measurements of comparable sharp resonances in the THz regime show a similar broadening in experiments for hole arrays in silicon slabs [35]. Unlike silicon PhC slabs with thicknesses ranging in the hundreds of micrometers [21–26], Fabry–Perot oscillations are not observed in our sample.
Fig. 2. Transmission of the guided modes as a function of hole diameter for 50 µm thick sample. (a) Simulated (solid line) and measured (dashed line) transmission spectra for circular air holes of different diameters in a square lattice. The spectra are offset vertically for clarity. (b) The hole fill fraction (black) and effective refractive index (red) are also calculated for different ${ r}/{ a}$ values. (c) Shifts in frequency of the guided resonances are extracted from experiment and simulation for increasing ${ r}/{a}$ values. (d) The measured modulation depth of the ${d} = 160 \;{\unicode{x00B5}{\rm m}}$ sample (red) and ${d} = 180 \;{\unicode{x00B5}{\rm m}}$ (blue) are plotted in parallel with a measured ${Q}$-factor insert for ${d} = 160 \;{\unicode{x00B5}{\rm m}}$.
To describe the dependence on hole geometry, Figs. 2(b) and 2(c) plot the fill fraction (FF) of the holes, the effective refractive index of the PhC slab (${n_{\rm eff}}$), and the frequency shift observed in the simulated and measured spectra as a function of $r/a$. The FF of the holes (${{\rm FF}_h}$) is calculated as ${(r/a)^2}$, while effective refractive index is determined from [21]
where ${ \epsilon _K} = 3.4$ for Kapton film and ${ \epsilon _h} = 1$ for air holes, while the fill fraction of Kapton (${{\rm FF}_k})$ is given as $1 - {{\rm FF}_h}$. The calculated and measured resonances shift towards higher frequencies for about 50 GHz from ${r}/{a} = 0 . 23$ (${ d} = 140 \;{\unicode{x00B5}{\rm m}}$) to ${r}/{ a} = 0 . 3$ (${ d} = 180 \;{\unicode{x00B5}{\rm m}}$). These shifts are accompanied by increased FF of the holes (${\sim}12\%$) and lowered effective refractive index (${\sim}0.08$) of the PhC slabs. The variation of the refractive index that we obtained here is very promising for the development of future flexible THz photonic components based on polymide films, such as a flexible gradient index (GRIN) metalenses.In order to quantify the resonances, modulation depth (MD) and quality factor (${Q}$ factor) from the guided modes are extracted in Fig. 2(d). ${Q}$ factor is defined as $Q = {f_i}/\Delta {f_i}$, where ${f_i}$ is the center frequency and $\Delta {f_i}$ is the full width at half-maximum (FWHM). The modulation depth is determined as ${\rm MD} = |({t_ {140} } - {t_i})/{t_ {140} }|$, where ${t_ {140} }$ is the transmission for ${d} = 140 \;{\unicode{x00B5}{\rm m}}$, and ${t_i}$ is the transmission for ${d} = 160$ and 180 µm, respectively. As shown in Fig. 2(d), the modulation depth of the PhC is gradually increased as the hole diameter is increased. Meanwhile, one can observe the ${ Q}$ factor linearly decreases as the hole diameter is increased. The measured modulation depth reaches a maximum value of about 85% around 0.95 THz, while the maximum measured ${Q}$ factor of the guided mode is about 11 when ${d} = 140 \;{\unicode{x00B5}{\rm m}}$.
The total ${Q}$ factor for our PhC device can be calculated from the following equation: $1/{Q_t} = 1/{Q_d} + 1/{Q_{\rm sw}}$, where ${Q_d}$ is the ${Q}$ factor due to dielectric losses and ${Q_{\rm sw}}$ is the ${Q}$ factor due to surface waves. For very thin substrates ($t \ll {\lambda _0}$), the loss due to surface waves, $1/{Q_{\rm sw}}$, is very small and can be neglected in calculation of the total ${Q}$ factor. This indicates that by reducing the thickness of the dielectric substrate, the ${Q}$ factor due to the surface waves can be improved, and as a result of that the total ${Q}$ factor of the device can be improved. Furthermore, the ${Q}$ factor due to dielectric losses can be improved by using low-loss materials. Hence, the total ${Q}$ factor is proportional to the substrate dielectric constant and inversely proportional to the substrate thickness.
To further analyze the guided modes, we reduced the thickness of the Kapton films to 25 µm (${\lambda _0}/12$) and fabricated PhC slabs with square holes in a square lattice [Fig. 3(a)]. Simulated and measured transmission amplitudes of the sample are plotted for varying hole diameters in Fig. 3(b). From the obtained results, the guided resonances that originate from periodic square holes and circular holes have very similar shape and linewidth for the same thickness. However, when the thickness is halved to 25 µm, the resonances exhibited extremely sharp dips compared to 50 µm. Indeed, a maximum measured ${Q}$ factor of about 31 is obtained when ${d} = 185 \;{\unicode{x00B5}{\rm m}}$ and linearly decreases as the hole’s diameter is increased. A maximum modulation depth of about 11.6% is observed for the guided resonance in measurements when ${d} = 185 \;{\unicode{x00B5}{\rm m}}$ with little modulation effect at the strong guided resonance frequency when ${d} = 190 \;{\unicode{x00B5}{\rm m}}$ [Fig. 3(d), dashed blue line]. Similar to the 50 µm sample, a minor frequency shift towards lower frequency is observed for decreasing hole diameters.
Fig. 3. Transmission of guided mode as a function of square hole diameter for 25 µm sample. (a) Unit cell of the PhC sample with parameters ${t} = 25 \;{\unicode{x00B5}{\rm m}}$, ${a} = 300 \;{\unicode{x00B5}{\rm m}}$, and variable diameter ${d}$. (b) Simulated (solid line) and measured (dotted line) transmission spectra for different diameters in a square lattice. The spectra are offset vertically for clarity. (c) Evolution of the ${Q}$ factor versus the hole diameter. Inset: optical microscope image of the fabricated PhC with square shaped air holes. (d) The frequency dependence of experimentally measured modulation depth for $d = 185\;\unicode{x00B5}{\rm m}$ and 190 µm.
The use of flexible substrates provides an unprecedented route to achieve frequency agility and/or amplitude modulation of metadevices due to modifications in the profiles and the periodicities of the structures when the substrates are bended or stretched. Next, we take advantage of the flexibility of our samples through the demonstration of active modulation of guided-mode resonance (Fig. 4). To mimic the experiment for active tunability, curvature-dependent simulations were performed by using the appropriate bending angles with the entire curved device designed in full (not only the elementary cell). The responses of the PhC slabs are simulated for different bending angles with their corresponding experimental results in Fig. 4(b), where we taped the 25 µm sample with ${d} = 160 \;{\unicode{x00B5}{\rm m}}$ on curved surfaces of angles 5° and 10° as illustrated and shown in Fig. 4(a). The result of Fig. 4 shows a red-shift corresponding to an increase in propagation constant $\beta$ at angles further from normal incidence. Previous work on rigid SiNx PhC slabs in the visible and NIR attributed angular-dependent shifts to the empty lattice approximation [36]. This trend is supported by experimental measurements, where the resonance dip becomes very small with a red-shift from $\theta { = 0^ \circ }$ to $\theta { = 5^ \circ }$ with the complete disappearance at $\theta { = 10^ \circ }$. Another possible explanation of the red-shift of the guided mode is an increase of the periodicity along the $y$ axis (i.e., $a^\prime \gt a$), which is caused by the modification of the profile of the PhC upon bending as illustrated in Fig. 4(a). The observed strong sensitivity of the guided modes to the surface curvature of PhC slabs confirms an additional degree of freedom to tune the resonances previously not reported for rigid THz PhC slabs.
Fig. 4. Curvature-dependent transmission of the guided mode. (a) Top panel: the figure illustrates the setup of bent PhC sample. Bottom panel: image of the curved PhC sample showing an approximately determined bent angle of $\theta { = 10^ \circ }$. (b) Simulated (solid line) and measured (dotted line) transmission are plotted for the 25 µm thick sample with bending angles $\theta { = 0^ \circ }$, 5°, and 10°.
4. CONCLUSION
To summarize, we have demonstrated the existence of the guided resonances in a membrane-type terahertz PhC slab from a flexible thin dielectric medium of Kapton polyimide films. The resonance originates from the coupling of the in-plane resonant mode on PhC plane to external radiation. The frequency positions of the resonances undergo a red-shift with increasing diameters of the air holes. For ultra-thin samples with subwavelength thickness (${\sim}{\lambda _0}/12$), the resonance dip is less pronounced and undergoes a red-shift from $\theta { = 0^ \circ }$ to $\theta { = 5^ \circ }$ with the complete disappearance at $\theta { = 10^ \circ }$. This allows for active tuning of the guided modes with respect to the curvature of the PhC slabs. The temperature resistance and the mechanical robustness of the Kapton films allow for easy integration of these PhC slabs into current microfabrication technology.
Funding
Air Force Office of Scientific Research (FA9550-16-1-0346); U.S. National Science Foundation (ECCS-1541959), (ECCS-1708315); Singapore Ministry of Education (MOE2016-T3-1-006); W. K. Keck Foundation.
Acknowledgment
T. A. S. acknowledges support from the CNS Scholars Program and C. K. acknowledges support in the form of the Just-Julian Graduate Research Assistantship. M. M. and R. S. acknowledge support from the Singapore Ministry of Education. J. K. acknowledges support by the U.S. National Science Foundation.
Disclosures
The authors declare no conflicts of interest.
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