1. Introduction

The ability to measure the near-field of microwave devices provides insight that is critical to study and design devices that are strongly impacted by near-field coupling techniques, such as antennas, feed mechanisms, detectors, characterization platforms, and microwave-based sensors. Several microwave field-mapping techniques exist, but optical-based methods enable high-sensitivity, high-resolution field mapping. One such method uses the electro-optic (EO) effect, in which an EO crystal is placed in the near field of the microwave device under test (DUT). Traditionally, an ultrafast pulsed laser is focused on the crystal and the laser beam experiences a polarization shift related to the local microwave field due to the EO effect. The polarization shift can be measured to map the electric field as a function of space by rastering either the DUT or the crystal and the laser beam [1–6]. Although this technique and its variants are effective and have shown significant improvements in sensitivity, minimizing invasiveness, and required components over the last two decades [7–9], the methods are based on the EO effect and thus involve monitoring a reflected laser beam. This increases complexity, time spent in optical alignment for free space setup, and can increase invasiveness and requirement of circulators for fiber-based setups.

An alternative, simpler optical field mapping technique based on optically induced conductance (OIC) can be used when only the relative electric field magnitudes are required. In the OIC method, a focused laser beam excites electron-hole-pairs (EHPs) in a semiconductor crystal that is located in the near-field of a driven microwave device. The generated EHPs increase the local conductivity, which increases the local loss and modulates the incident microwave signal proportional to the local electric field of the circuit element. Spatial resolution is limited by the motor step size, beam spot size, and eventually, the diffusion length of the EHPs [10]. Unlike the EO method which uses RF fields to modulate optical signals, the OIC method uses an optical signal to modulate the RF field, eliminating the need to monitor the laser beam and thus reducing the number of optical components. The initial proof of principle of the OIC method was demonstrated by mapping micron-scale fields of a single mode in a planar split-ring resonator (SRR) circuit, where the substrate of the circuit was the source of excited EHPs [10]. However, this limits the type of devices that can be mapped because it requires that the microwave circuit is fabricated on a photoactive, highly resistive substrate, precluding the use of common RF/microwave substrates such as printed circuit board (PCB) material or alumina. Additionally, microwave fields along the z-direction (optical axis) could not be effectively mapped using the fixed substrate. In this paper, we demonstrate an improved OIC method using an external, nanometer-scale thin film semiconductor and show that the relative electric field magnitude of several modes in insulating dielectric microwave resonators, placed on a non-photoconductive alumina substrate, can be mapped as a function of space in all three-dimensions (3D).

2. Experimental setup

A detailed schematic of the measurement technique is shown in Fig. 1. The DUT is a cylindrical dielectric resonator of diameter = 6 mm, height = 1 mm, and a large relative permittivity (ɛ ∼ 80) that is side-coupled to a microstrip transmission line. The resonator is glued to the substrate with a coupling distance of ∼0.16 mm between the edge of the resonator at the narrowest point and edge of the microstrip, allowing for a magnetic coupling feeding mechanism to excite several transverse electric (TE) modes. The microstrip of width = 0.635 mm is fabricated on a 0.635 mm thick alumina substrate and is designed for a characteristic impedance of ∼50 ohms. The backside ground plane and microstrip consist of a titanium tungsten adhesion layer and ∼3 µm of gold. Coaxial-to-microstrip connections are made using PCB to microstrip end-launch connectors on both ends of the microstrip. Port 1 of the DUT is driven by 6 dBm of continuous wave (CW) microwave signal, operating at the frequency of the mode that will be mapped. A laser diode beam of λ = 638 nm and peak power ∼30 mW, electrically modulated at 1 kHz, is focused on the external semiconductor crystal placed over the resonator. The commercially available crystal consists of a ∼600 nm thick silicon layer on a ∼475 µm thick sapphire substrate. The laser generates EHPs in the silicon layer which modulate the transmitted microwave signal. Port 2 of the DUT is connected to a Schottky diode RF detector, which passes the optically modulated signal to a lock-in amplifier (LIA) synchronized to the reference of the laser modulation. The laser is rastered in 100 µm steps by mounting it on an xy computer-controlled motor stage. The optically induced modulation of the transmitted signal is recorded from the LIA as a function of space.

 

Fig. 1. Schematic of the device under test (DUT) and the external crystal optically induced conductance (OIC) technique for 3D mapping of relative electric field magnitude. A commercial silicon-on-sapphire (SoS) is used as the external crystal; optically generated electron-hole-pairs (EHPs) modulate the transmitted microwave signal that is read out via a Schottky diode RF detector on a lock-in amplifier (LIA) as a function of spatial position.

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3. Results

3.1 Measured and simulated S-parameters

Several modes (fundamental and higher order) of cylindrical dielectric resonators are mapped to demonstrate the efficacy of the new OIC method and highlight its additional capabilities over the previous OIC method. Dielectric resonators have garnered significant interest in the filter design [11–15], antenna [16–20], and sensing [21–24] research areas because of their low loss, high quality factor, tunability, small footprint, and strong field confinement characteristics. Furthermore, recently, strong field confinement from resonant microwave structures has been exploited in optoelectronics material characterization [25], and infrared detection architectures [26,27]. In these applications, the measurement of fields from different resonant modes would provide significant insight to design and improve physical overlap between the microwave field and the loaded optoelectronic material. In Fig. 2, the unloaded measured and simulated S₂₁ parameters are plotted to determine the resonant frequencies of the DUT. The experimental S-parameters of the DUT can be obtained using a vector network analyzer. Short-open-load-thru (SOLT) calibrations are performed to move the reference plane to the coaxial side of the end launch connectors.

 

Fig. 2. (a) Measured (solid) and finite element method (FEM) simulated (dashed) S₂₁ parameter of the device; simulated vector electric (red) and magnetic (blue) field plots for the (b) TE_01δ, (c) TE_11δ, and (d) TE_21δ modes.

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Full-wave driven simulations are carried out in a commercial finite element method (FEM) solver. The simulation includes the microstrip and the dielectric resonator using the dimensions shown in Fig. 1. The relative permittivity of the substrate and resonator are set to be 10 and 80, respectively. Transition boundary conditions are used to model the finite thickness and conductivity of the metal microstrip and ground plane, and a scattering boundary condition encloses the DUT. Lumped ports are used on both sides of the microstrip for excitation and simulation of S-parameters.

3.2 Field mapping

We follow the standard nomenclature for cylindrical microwave DRs [17,28] for the resonant modes. First, the mode is determined to be transverse electric (TE), transverse magnetic (TM), or hybrid electromagnetic (HEM) by observing the electric field direction. As seen in Fig. 2, all the modes are TE modes with electric fields that are transverse to the normal of the cylinder. The indices of each mode correspond to field variations along the azimuthal, radial, and z-direction, respectively. In Fig. 3, the differences between the TE_01δ, TE_11δ, and TE_21δ resonant modes are clearly visible, demonstrating the utility of the OIC mapping to distinguish between different DR modes. As an example, the TE_21δ mode resembles a four-leaf clover and contains twice the azimuthal variation compared to the TE_11δ mode. This is expected as the first index in the nomenclature (corresponding to azimuthal variation) for the modes differs by a factor of two. The experimentally mapped modes are compared to the FEM simulation to validate measurements. The fields computed by the simulation agree with both the measured and expected difference in azimuthal variation between the two modes. Despite the reasonable agreement, slight deviations between the measurement and simulation are observed in both Figs. 2 and 3. These could be attributed to physical characteristics not captured by simulation, such as differences in the coupling gap and possible air gaps underneath the resonator resulting from a non-uniform layer of adhesive. Note that in Fig. 3, modes are oriented similarly and values are normalized to their maximum in both measured and simulated cases to allow facile comparison between fields on a single scale.

 

Fig. 3. Measured (a, b, c) and simulated (d, e, f) normalized electric field magnitudes for the TE_01δ (a, d), TE_11δ (b, e) and TE_21δ (c, f) modes of the dielectric resonator. The measurements are taken at 9.12 GHz, 11.33 GHz, and 12.64 GHz, and the simulated fields are plotted at 9.37 GHz, 11.63 GHz and 12.65 GHz for the TE_01δ, TE_11δ, and TE_21δ modes, respectively. Measurements are made with the silicon side placed directly on the surface of the resonator, and simulations show xy slices taken at a height of z = 600 nm away from the surface of the resonator to consider the thickness of the silicon layer in the experiment. All plots are oriented similarly and normalized to their respective maximum to enable clear comparisons on a single scale.

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Even though the OIC technique measures relative field magnitude, quantitative comparisons can still be made to determine the decay of a mode away from the surface. In Fig. 4, we plot the raw LIA output (without normalization) of the experimentally mapped fields for the fundamental TE_01δ mode (9.12 GHz) and the higher order TE_22δ mode (17.98 GHz). Comparison of the fields of the two modes near the top surface of the DR shows that the TE_01δ mode (Fig. 4(a)) has fields distributed between the center and extreme edges of the DR, whereas the TE_22δ mode (Fig. 4(b)) has field localized closer to the center of the DR and along the extreme edges. These observations are useful to design sensors and material characterization platforms, where the objective is to maximize overlap between local field hot-spots and the material under test (MUT). Thus, depending on lateral MUT size, the mode with the most field overlap can be chosen to enhance measurement sensitivity. Furthermore, the decay of the field for each mode can be determined by adjusting the height position of the absorber to help determine the suitable thickness of a MUT. As a proof of principle, the crystal is flipped such that the silicon absorber layer is above the sapphire handle ∼475 µm away from the top surface of the DR. The measured fields for the TE_01δ and TE_22δ at z = 475 µm are shown in Fig. 4(c) and 4(d), respectively. For the TE_01δ mode, there is a reduction in the field strength as the height increases, but the overall shape of the mode has very little change, as seen in Fig. 4(a) and 4(c). On the other hand, for the TE_22δ mode, the fields decay in strength as the height changes from near the surface to z = 475 µm and the overall shape of the mode also changes as the strong, rapidly changing fields along the radial direction near the edges of the resonator (changing from dark red to cyan) observed in Fig. 4(b) possess a weaker gradient (the edge retains a light cyan color) as seen in Fig. 4(d).

 

Fig. 4. Measured fields near the surface of the dielectric resonator (a, b), and at z = 475 µm away from the surface (c, d) for both the fundamental TE_01δ mode at 9.12 GHz (a, c) and the higher order TE_22δ mode at 17.98 GHz (b, d). The colorbar shows the unnormalized lock-in amplifier (LIA) output voltage.

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3.3 Loading effects

In Fig. 3, it is clear that there is minimal distortion caused by the SoS crystal laid on top of the resonator as the measured fields agree reasonably well with the simulated ones. However, in higher order modes with stronger field overlap with the load, there are deviations observed. To illustrate this, consider the higher order TE_12δ mode described in Fig. 5. In Fig. 5(a), the simulated field shows the strongest field confinement in the center of the DR, with additional weaker fields symmetrically away from the center, resulting in more radial variation compared to the TE_11δ mode. The measured fields in Fig. 5(b) show the strong center hot-spot and significant fields on the edges of the DR, but the additional weaker radial variation seen in the simulated fields (Fig. 5(a)) is not present. A second simulation was performed to determine if this effect is due to loading effects from the crystal. In this simulation, we added an additional loading component (10 mm × 10 mm × 475 µm, ɛ = 10 to represent sapphire) and placed it on top of the DR to emulate the experimental conditions. The TE_12δ fields from this simulation are plotted in Fig. 5(c), and it is observed that the loaded simulation is similar to the measurement as the additional radial variation observed in the Fig. 5(a) unloaded simulation is also missing. This shows that the experimentally observed field distortion is indeed a consequence of the crystal loaded on top of the DR for this particular mode; however, no significant differences were observed between unloaded and loaded simulations for the lower order modes mapped in Fig. 3. Thus, the current external crystal OIC system can accurately measure lower order modes with minimal field distortion, and future work will address loading effects of the crystal to more accurately measure higher order modes which are more impacted by the load.

 

Fig. 5. (a) Unloaded simulated, (b) measured, and (c) loaded simulated normalized electric field magnitude of the higher order TE_12δ mode; (d) Measured S₂₁ parameter of the TE_12δ mode for the unloaded and several different load types, with percentages beneath the resonant dip denoting the percent difference in resonant frequency with respect to the unloaded frequency.

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In Fig. 5(d), we plot the S₂₁ parameter with several different materials loaded on top of the DR to quantitatively describe and contrast distortions from loading effects. Loads with different permittivities and thicknesses are chosen for purpose of comparison—specifically, a thin glass cover slip (t = 170 µm), thick glass slide (t = 1 mm), the SoS used in the experiments (600 nm of Si on a 475 µm sapphire handle), and a thick silicon piece (t = 600 µm). In all cases, the unloaded resonant dip red-shifts toward lower frequencies as the introduced loads have permittivities greater than air. The values beneath each resonant dip in Fig. 5(d) correspond to the percent difference in resonant frequency between the loaded resonance and unloaded resonance. The thickness of the load has a significant effect on the measurement, as the thick glass slide has noticeably more weakening and red-shift in the resonance (4.17%) when compared to the thin glass cover slip (0.55%). The permittivity of the load also impacts the resonance, demonstrated by the silicon load shifting the resonance the most (5.48%) despite being only 60% of the thickness of the glass slide. These results elucidate the need to minimize the thickness and permittivity of the handle wafer to accurately map higher order modes. There is minimal loading from the glass cover slip in Fig. 5(d), though glass is an insulator and cannot serve as the source for EHPs. In the future, solution-processed photoconductive materials, such as colloidal quantum dots [29,30] or organic optoelectronic and photoconductive materials [31,32], could be spin-coated on the thin glass handle as the source of optically induced conductance. Depending on the type of device mapped, the absorber’s properties such as carrier lifetime, diffusion lengths, and thickness, could be chosen and designed to balance loading effects, sensitivity to weak RF fields, and spatial resolution along the optical axis.

4. Conclusion

In conclusion, we demonstrated a novel and improved method to map relative electric field magnitudes as a function of space using a method based on RF signal modulation via external crystal optically induced conductance. As a proof of principle, TE modes of a high permittivity DR side-coupled to a microstrip line were mapped using a nanoscale thin film, although the technique is applicable to a variety of microwave devices. Lower order modes such as the TE_01δ, TE_11δ, and TE_21δ were compared to FEM simulations with reasonable agreement. We also showed that by adjusting the height of the absorber, the field in the z-direction could also be mapped, which is useful to calculate field decay or constructing near-field approach curves. Finally, effects of loading i.e. weakening and shifting of the resonance were discussed in context of higher order modes with strong field overlap with the load. In the future, the use of a solution-processed absorber layer spin-coated on a thin, low-permittivity handle such as a glass cover slip, could result in effective field mapping with minimal distortion caused by loading effects.