Open Access
Add to BibSonomy
Abstract
This paper introduces a simple method for the measurement of the relative permittivity and the Pockels coefficient of electro-optic (EO) materials in a waveguide up to sub-THz frequencies. By miniaturizing the device and making use of plasmonics, the complexities of traditional methods are mitigated. This work elaborates the fabrication tolerance and simplicity of the method, and highlights its applicability to various materials, substrates and configurations. The method is showcased using drop-casted perovskite barium titanate (BaTiO3, BTO) nano-particle thin-films and it has previously been used to measure epitaxial thin film BTO. In this work we show the effective relative permittivity of drop casted BTO to be εeff ∼ 30 at 200 MHz, dropping to ∼ 18 at 67 GHz and similarly, the effective Pockels coefficient was found to be reff ∼ 16 at 350 MHz and ∼ 8 at 70 GHz. These values are a factor > 50 below the values found for thin film BTO. Yet, the fact that the method can be applied to such different samples and Pockels strengths gives testimony to its versatility and sensitivity.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Measuring relative permittivity and electro-optic characteristics of both existing and novel electro-optically active materials at highest speeds becomes vital if they are to be deployed for on-chip photonic EO devices, such as modulators. The lack of a simple characterization method up to 100 GHz and above hinders the evaluation of known and novel materials and slows the progress to surpass current device performance [1]. The key property of EO materials is their non-linear susceptibility, specifically the 2nd and 3rd order susceptibility coefficients $\chi _{ijk}^{(2 )}\; ({{\omega_i}\; ;{\omega_j},{\omega_k}} )$ and $\chi _{ijkl}^{(3 )}\; ({{\omega_i}\; ;{\omega_j},{\omega_k},{\omega_l}} )$ respectively, which are related to the relative permittivity [2]. In optical communications, the 2nd order susceptibility is of particular interest because of its high speed and is often represented by the Pockels coefficient (${r_{ijk}}$). Materials with large Pockels coefficient are of paramount importance to decrease size and power consumption of EO devices [3,4]. Large values at radio frequencies (RF) can occur e.g. due to phononic polarization mechanisms [5]. But exactly these can be strongly frequency dependent, especially within the technologically relevant RF range. Therefore, accurate determination of these parameters is essential for developing advanced photonic devices.
Current state-of-the-art characterization methods for relative permittivity and electro-optic (EO) effects at RF can be divided into bulk and thin-film methods. Bulk methods for measuring relative permittivity include impedance spectroscopy [6] and resonant cavity perturbation [7] methods. For EO characterization in bulk materials, methods such as Electro-Optic Sampling [8–11] are used. However, these bulk methods have limitations: they require expensive development of homogeneous bulk materials, are difficult to probe at RF frequencies and may necessitate high voltages. Even more, it is electro-optic active thin films that get increasingly more relevant. Yet, the material properties of such thin films can be quite different from the bulk counterparts. Various factors, such as interface effects, stress and strain caused by the lattice structure can significantly influence the material properties, rendering bulk methods inadequate. This makes thin-film methods and characteristics essential for integrated device designs. Techniques like waveguide-based [12–15] and transmission line methods [16,17] are employed for relative permittivity measurement at RF, while methods like integrated photonic devices [18,19] and free space polarization shifts [20] are used for EO thin-film characterization. Yet, while the former lacks the simplicity of characterization at high speeds due to travelling wave electrodes, the latter is error-prone because of the small interaction length of a few 10s to 100s of nanometer. Further, the measurement of the relative permittivity and the Pockels coefficient for the same material deposition within the same device is rarely performed, although both are important information for designing high-speed devices. The general lack of high-speed characterizations of nonlinear materials in literature indicates the necessity of a simple and versatile approach to characterize the relative permittivity and EO strength up to sub-THz speeds.
In this work, we present a characterization method providing access to the simultaneous measurement of both the EO effect and the relative permittivity at highest RF frequencies. The method is demonstrated using perovskite barium titanate (BaTiO3, BTO) nano particles and is designed to be mostly substrate-independent, accommodating various materials and configurations. The method can be applied to thin-films that are deposited in a top-down or bottom-up approach. By offering a simple and fabrication-tolerant routine to characterize these parameters up to highest frequencies, we hope to propel the research on electro-optic active materials, which are becoming more and more relevant for the photonics community.
2. Method
The simultaneous characterization of relative permittivity and electro-optic strength in materials at high speeds presents various challenges. While 100s of micrometer large electric devices can be considered static (having an equipotential throughout the device) at low RF frequencies, they become complex travelling wave structures as the electric wavelength shrinks to similar length scales [21]. Miniaturizing such devices to a few micrometers allows to push the bandwidth to hundreds of GHz as shown by Burla et al. [22] and thereby working with a lumped-element approach. Advantages and properties of this miniaturization are discussed in the following sections.
2.1 Principle
To measure the relative permittivity and the electro-optic coefficient, a device is required that facilitates a large overlap between electric RF signals and optical fields with the material being characterized. We strive for a lumped-element characterization to be able to measure across a large frequency range, therefore necessitating a plasmonic approach. The resulting small scale of the device produces strong electric fields with a substantial electric and optical field overlap that provide large nonlinear interactions despite of the small length. This is in contrast to photonic approaches, which provide insufficient modulation strength for devices of small length due to the wide electrode spacing and small overlap. Finally, of particular interest for measuring at highest frequencies is the favorable RF field enhancement in plasmonic devices [23,24] as the potential output power of most high-speed sources drastically decreases with increasing frequency.
A device geometry that combines all these key elements while remaining versatile in its applicability is depicted in Fig. 1. The core elements of such a device comprises three components:
Fig. 1. Illustration of a plasmonic device for characterization of the relative permittivity and the nonlinear electroptic strength. In blue the silicon waveguide and two grating couplers. In yellow the electric gold pads which simultaneously serve as a plasmonic waveguiding section. The sideview (a) shows the silicon waveguide on any arbitrary substrate (grey). The sideviews (b,c) illustrate the difference of a top down and bottom up structure for the plasmonic section, where green is the deposited nonlinear material after (respectively before) the gold deposition.
Optical waveguides (blue): Achieved through a high-refractive-index material like silicon, by which the gratings and waveguides are formed, that guide the continuous-wave (CW) laser and convert the photonic mode into the plasmonic and vice versa [25].
Electrical contacts (yellow): They are the electrical contact pads and form the plasmonic slot. Reducing the distance between probe contacts and slot plays a crucial role in minimizing parasitic capacitances and inductances, affecting the device’s bandwidth.
Plasmonic slot waveguide (green): The plasmonic slot serves simultaneously as the waveguide and the electrical contact. Since the empty slot exhibits no nonlinearity, filling the slot with any electro-optic active material allows to measure its relative permittivity and electro-optic Pockels coefficient.
The fabrication of the device is detailed in Section 2.2. Once the device is fabricated it first needs to be characterized electronically before accurate conclusions about the electro-optic strength can be made. For guidance, Fig. 2 summarizes the process flow and the dependencies of the measurement method described in this work.
Fig. 2. The process flow to determine the relative permittivity $\varepsilon (\omega )$ and the Pockels parameter $r(\omega )$ is illustrated and further explained in the text. The cross dependencies of the various measurements are illustrated with arrows. The acronyms are VNA: Vector Network Analyzer, OSA: Optical Spectrum Analyzer. CSR: Carrier Sideband Ratio.
The electrical characterization and with it the extraction of the relative permittivity ε(ω) at high frequencies is performed by using a vector network analyzer (VNA). For this, a microwave probe is used to contact the plasmonic device directly on top of the slot and thus essentially acts as a lumped-element termination in the microwave domain. We then measure the reflection parameter ${S_{11}}$ of the device, from which we can infer the complex impedance and thereby to the device capacitance ${C_{\textrm{Slot}}}(L )+ {C_{\textrm{Pads}}}$, with typical values in the order of tens of Femtofarad. Through a device length ($L$) sweep, the pad capacitance (${C_{\textrm{Slot}}}$) and the stray capacitances (${C_{Pads}})$ can be calibrated.
This approach ensures high sensitivity and detailed insights into the device's impedance up to highest frequencies. The analytic simplicity of the method largely depends on its lumped element character. Therefore, to achieve highest bandwidths it largely depends on the possibility to electrically contact the slot in its proximity. The exact process and the subsequent permittivity extraction is detailed in Section 2.4.
In a second step follows the electro optic characterization. The determination of the nonlinear electro-optic (Pockels) coefficient relies on the sum and difference frequency generation (SFG/DFG) of an optical carrier with the electrical signal within the plasmonic slot. Due to the plasmonic slot, only the generated transverse electric (TE) modes will be guided in the plasmonic section, which enables TE probing of the material. The carrier and the generated SFG and DFG are then coupled out and measured with an optical spectrum analyzer (OSA). To assess the frequency response, various RF frequencies are mixed with the carrier and subsequently measured, as is exemplary shown in Fig. 3. The strength of the measured sideband powers then directly relates to the Pockels coefficient, as further elaborated in Section 2.5.
Fig. 3. The figure shows multiple exemplary OSA measurements of mixing an optical carrier with various electric RF frequencies (here ∼ 15–70 GHz, bright to dark). This measurement was done with a 10 dBm laser and 10 dBm RF power input. Resolving frequencies below ∼ 15 GHz requires a higher resolution OSA, not shown here. The measured sideband strength is the result from the electro-optic strength of our material interacting with the carrier and RF signal. The resulting carrier sideband ratio (CSR) then is used to determine the Pockels coefficient. Note, the plot is not normalized in order to indicate expectable losses - here a 20 $\mu $m plasmonic slot length with ${\sim} $0.7 dB/ $\mu $m. The huge measurable CSR is the reason for the methods EO sensitivity.
2.2 Wide platform compatibility and large tolerances
Bottom-Up & Top-Down: Because of the little requirements on the waveguides and electrical contacts, the device is suitable for characterizing materials using the top-down and bottom-up approach.
In the top-down approach, materials such as organic molecules or nano particles are deposited on already fabricated devices [26]. This enables the characterization of most synthesized electro optic materials. On the other hand, in the bottom-up approach, thin films such as lithium niobate or BTO are deposited before device fabrication. This is of outmost interest for the heterogeneous integration of electro optic devices on more advanced platforms [27–29]. The result of such a bottom-up approach can be seen in our previous work [30].
Substrate Independence: A key advantage of this method is its independence from the substrate with the only constraints that the substrate must be insulating and that it must have a lower refractive index than the optical waveguides. Apart from that, the plasmonic character strongly confines the optical mode and makes it largely unaffected by surrounding materials. The short device lengths further provide high tolerances towards fabrication related propagation losses. This means in most cases, waveguides can be made from high-index amorphous silicon (instead relying on the silicon-on-insulator platform), which is compatible with most substrates. It is important to note, that the plasmonic devices can be fabricated using common fabrication materials like gold and silicon. Thus, novel technological process development and optimizations aren't necessary due to the many deposition and patterning processes already known.
Growth Quality: The plasmonic approach is suitable for all kind of growth qualities. There's no need to have a very smooth surface or to be free from defects. Only a small area and few 10s of nanometer thickness are required. As the plasmonic devices are only a few micrometers long, any extra losses from roughness or other imperfections of the materials are typically insignificant.
Overall, these characteristics make for a quick and simple way to characterize materials, both in top-down and bottom-up designs. The method therefore is also suitable for unconventional material substrates, which are often used for the development of novel thin film materials. For example, the device size would also allow to be built on small, synthesized flakes of exotic compositions. This can be very useful for testing a wide range of nonlinear materials quickly, long before a high quality thin can be produced.
2.3 Fabrication
In the top-down approach undertaken in this study, the device was first fabricated, followed by the deposition of BTO cubes. This process utilized a commercially available silicon-on-insulator (SOI) wafer, consisting of a 220 nm silicon layer atop a 3 µm thick oxidized silicon wafer. The oxide serves as the lower index surrounding necessary for the silicon waveguides. Initially, the 220 nm silicon was etched using an inductively coupled plasma (ICP) process, with approximately 150 nm of hydrogen silsesquioxane (HSQ) employed as a protective mask. Any remaining HSQ was subsequently removed with a brief 1:7 buffered HF etch. The metal layer was deposited through a high-contrast e-beam process using Polymethylmethacrylate (PMMA) as a lift-off medium. A thin titanium adhesion layer, followed by a gold layer, was then deposited using an electron beam evaporator with a vertical deposition angle.
The nonlinear BTO colloidal nanoparticles (${\sim} $ 10 nm) were synthesized as described in detail in [31]. To avoid coating the entire chip with BTO particles, a PMMA lift-off mask was applied precisely on top of the plasmonic slots. Considering that the oleic acid-passivated nanoparticles dissolve in non-polar solvents, a 9:1 Hexane:Octane mixture was used to disperse 1 mg of colloids in 2 ml of solvent, partially countering coffee-stain effects. It was crucial that these solvents did not dissolve the PMMA mask, allowing for the subsequent lift-off process. The suspension was sonicated, evenly drop-casted onto the PMMA mask, and left to dry at room temperature. This process, resulting in approximately 30 nm thick films, was repeated several times until the slot height was surpassed. To prevent the already deposited colloids from dissolving, a ligand exchange was performed, replacing the oleic acid with thiocyanate (SCN) without affecting the PMMA mask. The SCN molecules’ shorter length allowed for denser packing of the nanoparticles. Afterward, the PMMA mask was removed with acetone, revealing the lift-off result with a scanning electron microscope (SEM) as shown in Fig. 4(a), designed with a trapezoidal shape to minimize optical reflections. Figure 4(b) illustrates a transmission electron microscope (TEM) cross-section of the plasmonic slot, showcasing the dense filling. The color differences between the cubes are due to varying electron scattering strengths resulting from the random lattice orientation.
Fig. 4. (a) A top-view SEM image of the plasmonic slot section with Si access waveguides (blue), gold contacts (yellow) and the patterned BTO nano-particles (green). In (b) a TEM cross section of a plasmonic slot is displayed, showing the dense but random filling of the BTO particles within the slot. The color differences of the cubes are a result of different scattering of electrons due to the random orientation of the lattice. The white hole at the bottom is likely to be a defect from lamella preparation.
This fabrication process is adaptable to other substrates, as well as the bottom-up approach. For instance, in prior work we deposited the nonlinear material prior to the device fabrication [30]. Adapting the process can generally be done with an HSQ mask and a pure argon ICP etch, which allows to pattern most nonlinear materials into a ridge that can act as a slot in a plasmonic waveguide. The access waveguides in silicon can also be fabricated by depositing a 220 nm amorphous silicon layer, followed again by etching with an HSQ mask. In both cases, removing the HSQ is optional, as the short propagation length minimizes losses from entrapped hydrogen within the residual resist. The gold deposition and lift off process remains the same across all fabrication procedures.
2.4 Effective permittivity
Determining effective permittivity requires accurate modeling of the measurement setup and the device under test (DUT). As illustrated in Fig. 5, the electrical measurement setup is divided into two sections: the measurement setup and the DUT.
Fig. 5. Equivalent circuit of the measurement setup as explained in detail in the text. The black lines represent the electric circuit, while the red line corresponds to the optical paths. The accompanying TEM image offers a detailed insight into the device components that correlate with the depicted equivalent circuit.
Measurement Setup: A RF power source is used to generate the electrical signal, which is guided through a RF cable to the RF probe. As indicated, usually a bias-T is added to pole and characterize a material’s dependency on static electric fields. The signal then propagates through an RF cable to the RF probe. Thus, the electrical calibration needs to be conducted at the probe plane (${V_{\textrm{Probe}}}$) to isolate the device's response from the setup.
Permittivity Measurement: To measure the relative permittivity, a VNA is used that simultaneously acts as the RF power source and measures the reflected signal (scattering parameter ${S_{11}}$). To deduce the DUT’s electric properties from the VNA measurement, the device needs to be modeled by a simple equivalent circuit, as illustrated within the dashed box in Fig. 5. The plasmonic slot and the electrical pads act as two parallel capacitors. The potential leakage through the plasmonic slot is captured with a parallel resistance ${R_1}$, while contact and pad resistance are modeled by ${R_2}$. The accompanying TEM image serves to visualize these components further. Given the compactness of the pad, it can be treated as a lumped element. The presence of any inductive impedance is negligible compared to the capacitive part. As a result, the device’s impedance can be expressed as
The imaginary part of the measured inverse impedance allows us to isolate the slot capacitance ${C_{\textrm{Slot}}}({L,\omega } )$ contribution. To do so, the plasmonic device length L is varied and the resulting total capacitance $C({L,\; \omega } )$ is fitted with a length independent pad capacitance ${C_{\textrm{Pads}}}(\omega )$ and a linear length dependent ${C_{\textrm{Slot}}}({L,\omega } )$ via the imaginary part of the equation.
Permittivity Simulation: Once the capacitance contribution of the slot is measured, the resulting capacitance from the 2D cross section of the plasmonic slot needs to be simulated. We replicated the geometry from Fig. 4, which is typical for lift-off processes. If no TEM image is available, the most crucial factor in the geometry is not the exact angle of the sidewalls but the narrowest distance present, which can also be imaged with a SEM. Given that the relative permittivities of all materials involved are known except for the thin film, we can deduce the permittivity of the nonlinear material by correlating the simulated capacitance with the measurement. This would also allow to resolve the granularity of the individual BTO cubes; however, we chose to treat the thin film as an effective medium to focus on the characterization method. In example, to determine the effective isotropic material response, the simulation can be as simple as shown in Fig. 6.
Fig. 6. A 2D simulation showing the electric field strength distribution within the plasmonic slot. The cross section is modeled according to the imaged TEM in Fig. 4. The cubes are modeled as one isotropic material in order to treat it as an effective dielectric layer.
The result of the effective layer analysis of our thin film is shown in Fig. 7. We can observe a drop in permittivity with frequency, as it is expected for BTO compounds [11,30,32]. We measured the effective relative permittivity of the BTO nano particle thin film to be ${\varepsilon _{\textrm{eff}}}\; \sim \; 30$ at 200 MHz and dropping to ${\varepsilon _{\textrm{eff}}}\; \sim \; 18$ at 67 GHz. It is noteworthy that the relative drop is at similar frequencies and of comparable magnitude for nanoparticles of ${\sim} $10 nm in size as it is for bulk crystals [33]. It also should be mentioned, that VNA measurements become inherently noisier at low frequencies. This is because the large device capacitance leads to ever larger impedances with lower frequencies ($Z\; \sim \; 1/\omega C$), which will reach the resolution limits of the VNA to accurately measure the reflection coefficient of e.g., 99.999% for a 50 $\mathrm{\Omega }$ to 10 M$\mathrm{\Omega }$ interface.
Fig. 7. The resulting relative permittivity from the ${S_{11}}$ VNA reflection measurement. The solid line is a moving average of the data points.
2.5 Pockels coefficient
In order to determine the Pockels coefficient, we need to establish the connection between the geometrical cross section of the device and link it to the electro-optic sideband measurements from Fig. 3. The link between the device’s behaviour and the applied voltage is established through the half-wave voltage ${V_\pi }$, which is the required applied voltage at the device to induce a π phase shift. The relationship with the Pockels coefficient becomes clear when we assess the change in the propagation mode index $\mathrm{\Delta }n(V )$, essential for attaining a π phase shift ($\mathrm{\Delta }\varphi = \pi \,)$
where V is the applied voltage, ${k_0}$ is the vacuum wavenumber of the carrier and L the device length. With the mode index’s dependency on the applied voltage $\mathrm{\Delta }n(V )$, we can use the differential $\mathrm{\Delta }n(V )= \frac{{\partial n(V )}}{{\partial V}}\; V$ to deduce the voltage ${V_\mathrm{\pi }}$ needed to achieve a full π phase shiftThe crucial component $\frac{{\partial n(V )}}{{\partial V}}$ in this equation enables us to determine ${V_\mathrm{\pi }}$. Using the Pockels equation $\mathrm{\Delta }n = r{n^3}E\Gamma /2$ with $E = V/d$, $k = 2\pi /\lambda $ and taking its derivate in Eq. (4) gives
where $\lambda $ is the wavelength, d is the electrode gap, n the refractive index, r the Pockels coefficient, L the device length and $\Gamma = \mathrm{\Delta }{n_{\textrm{effective}}}/\mathrm{\Delta }{n_{\textrm{material}}}$ the field interaction factor. In many instances this factor $\Gamma $ is similar to the overlap between the electric RF and optic fields [27]. This result can serve to give a first analytical approximation for ${V_{\pi \; }}$.However, in the case of plasmonics $\varGamma $ is not merely the field overlap, as this factor can also exceed unity ($\varGamma > 1$). For instance, narrow plasmonic slots can have $\varGamma > 1$ because of a slow-down of the group velocity, which is not captured with a simple overlap picture [27,34]. Yet, an accurate determination of $\varGamma $ is crucial. Therefore, in most cases the determination of the factor $\frac{{\partial n(V )}}{{\partial V}}$ requires a 2D simulation.
${\boldsymbol{V}_{\boldsymbol{\pi}}}$ Simulation: In order to obtain the factor $\frac{{\partial n(V )}}{{\partial V}}$, we use an optic mode simulation of the plasmonic slot cross section. To fully characterize this relationship, three simulations are necessary:
- First, the optical mode is simulated, see Fig. 8(a), which provides the plasmonic mode’s effective index with no electric field applied $n({V = 0} )$.
- Then the RF simulation from the previous section determines the RF electric field distribution across the slot (Fig. 6) for an applied voltage ${V_{\textrm{Device}}}$. The resulting fields then are used to update the refractive index locally within the electro-optic active material according to the Pockels effect $\mathrm{\Delta }{n_{\textrm{ij},\,\textrm{EO material}}} ={-} {r_{\textrm{ij}}}{n^3}{E_\textrm{j}}/2$ with a first guessing value for the Pockels coefficient ${r_{\textrm{ij}}}$, e.g ${r_{\textrm{ij}}} = $100 pm/V. The resulting spatial distribution of the index alteration, due to the RF field and the Pockels effect, is shown in Fig. 8(b), which is the origin of the change in the plasmonic mode’s propagation index.
- Finally, a second optical mode simulation, with the updated the nonlinear material’s refractive index, provides the new propagation index $n({V = {\textrm{V}_{\textrm{Device}}}} )$
Fig. 8. (a) A 2D simulation showing the normalized optic field strength within the plasmonic slot using the same geometry as in Fig. 6. (b) The distribution of the normalized index change within the EO layer resulting from the RF simulation and the Pockels effect. The simulation shows that the strong field enhancement and EO interaction foremost takes place between the electrodes and therefore defines the probing area.
Using the resulting index difference $\partial n = n({{\textrm{V}_{\textrm{Device}}}} )\, - \,n(0 )$ and the assumed voltage drop $\partial V = \; {\textrm{V}_{\textrm{Device}}} - 0$, we can calculate the simulation’s ${V_\pi }$ for the assumed geometric cross section using Eq. (4) for a specific device length L. Note that in an effective medium approach, the tensor elements are replaced by the isotropic coefficient ${r_{\textrm{eff}}} = {r_{ij}}$.
${\boldsymbol{V}_{\boldsymbol{\pi \; }}}$ Measurement: To measure the half wave voltage ${V_\pi }$, the setup illustrated in Fig. 5 is used with a RF power source instead of a VNA in order to achieve higher output powers. The RF signal is mixed with the laser via the plasmonic device. The resulting SFG & DFG sideband strength is then measured with an OSA, as shown in Fig. 3. The sidebands’ and the carrier’s magnitude can now be used to determine the measured ${V_{\pi ,\, \textrm{measured}}}$, through the following expression [35]:
${\boldsymbol{V}_{\mathbf{Device}}}$ Measurement: Determining the exact voltage drop ${V_{\textrm{Device}}}$ across the electrodes is critical for Eq. (6). Therefore, we have to consider both the setup losses of the signal forwarded to the probe and the signal at the probe. Further, we must consider that the voltage across a capacitive device (such as our plasmonic device) depends on both the incident as well as the reflected wave.
To compensate for the electrical losses (${\alpha _{\textrm{Setup Loss}}}$) of the setup (foremost cable losses), the power of the electrical signal at the probe is determined as ${P_{\textrm{Fwd},\textrm{Probe}}} = {P_{\textrm{Fwd}}} \cdot {\alpha _{\textrm{Setup Loss}}}$. Consequently, we can find the amplitude of the voltage wave at the probe by
The voltage dropping off across the plasmonic slot comprises of both the incident as well as the reflected signal. The reflected field can be seen from the ${S_{11}}$ measurement as per our previous section
Here, we used again that the pad resistance ${R_2}$ is negligible compared to the capacitive impedance and thus conclude that the voltage across the modulator slot (${V_{\textrm{Slot}}})$ is the same as the device voltage (${V_{\textrm{Device}}}$) in the calibration plane. The ${S_{11}}$ scattering value’s frequency dependency can be understood from the present implementation where we feed the signal from a 50 $\mathrm{\Omega }$ source to the device and thus have ${S_{11}} = ({{Z_{\textrm{Device}}} - {Z_{50}}} )/({{Z_{\textrm{Device}}} + {Z_{50}}} )$. In the case of the plasmonic device, the impedance ${Z_{\textrm{Device}}}\; $ is mainly capacitive and thus increases with low frequency. This effectively leads to an increase of the effective voltage across the device ${V_{\textrm{Device}}}$ at lower frequencies. More precisely, for lower frequencies ($Z\; \sim \; 1/\omega C \to \infty $ and thus ${S_{11}}$ = 1) causing the device voltage ${V_{\textrm{Device}}}$ to be twice the incoming amplitude ${V_{\textrm{Device}}} = 2\; {V_{\textrm{Fwd},\,\textrm{Probe}}}$. As frequency increases, the capacitive impedance decreases, leading to a corresponding reduction in ${V_{\textrm{Slot}}}$. This behaviour aligns with the RC roll-off determined by the total capacitance and the 50 $\mathrm{\Omega }$ source impedance. Figure 9 shows that this roll-off becomes already significant, because the device capacitance is rather large due to the high effective dielectric permittivity of BTO and the narrow plasmonic gap. If the measured RC roll-off impairs the analysis, the device’s capacitance can easily be reduced by e.g. increasing the plasmonic gap width, reducing the device length or adding another capacitance in series.
Fig. 9. The measured frequency dependent factor ${V_{\textrm{Device}}}/\; {V_{\textrm{Fwd},\,\textrm{Probe}}} = \; |{1 + {S_{11}}({L,\omega } )} |$ for a device with $L$ = 10 $\mathrm{\mu}\textrm{m}$. The strong roll-off is due to the large capacitance of the device. As this defines the voltage drop across the EO material, this becomes increasingly relevant at higher frequencies.
Pockels Coefficient: Finally, the Pockels coefficient ${r_{\textrm{measured}}}$ can now be determined by using its inverse linear relationship with ${V_{\pi \; }}\sim 1/r$, see Eq. (5). By comparing the simulation ${V_{\pi ,\, \textrm{sim}}}$ with the measured ${V_{\pi ,\, \textrm{measured}}}$, we find
Fig. 10. The measured frequency dependent Pockels Coefficient ${r_{\textrm{eff}}} = {r_{\textrm{measured}}}$ of the BTO nano particles, deduced from the measured SFG sidebands.
While this method effectively measures the in-device associated relative permittivity and Pockels coefficients, it cannot determine the complete tensor. This limitation arises because the measurements cannot access arbitrary spatial directions, unlike more versatile bulk crystal methods. Nonetheless, with further refined analysis of the material, like its crystal orientation or its composition, the simulation model can be arbitrarily detailed [30]. This extends the method’s capability to infer on specific electro-optic tensor elements for specific regions (e.g. the cubes alone) and tensor orientations that align in-plane.
3. Conclusion
This work has introduced a method for accurately measuring both the relative permittivity and Pockels coefficient of electro-optic (EO) active materials, extending the range of characterization to the sub-THz range, only limited by the available equipment and the RC cutoff of the device. The device miniaturization and the use of plasmonics in this method has addressed and mitigated the challenges and complexities found in traditional approaches. This method's fabrication tolerance, versatility, and simplicity have been thoroughly elaborated and verified using perovskite barium titanate nano particles and epitaxial BTO thin films, as shown in our previous work [30]. The results, revealing the effective relative permittivity of drop-cast BTO nano particle thin films to be ${\varepsilon _{\textrm{eff}}}\; \sim \; $30 at 200 MHz and dropping to ∼ 18 at 67 GHz, along with the Pockels coefficient of ${r_{\textrm{eff}}}\; \sim $ 16 at 350 MHz and ∼ 8 at 70 GHz, underline the sensitivity and potential of this approach. The analysis also shows the detrimental consequences that interstitial low permittivity materials can have on the achieved effective Pockels strength.
The simplicity of this method opens new avenues for the precise characterization of various materials on various platforms, advancing the field of thin film materials for electric and EO devices at highest frequencies. Its application extends beyond drop-casted nano particles, offering a robust tool for the broader scientific community. The method's adaptability and sensitivity emphasize its significance and the exciting prospects it holds for future research and applications in the field of optically active materials.
Funding
Horizon 2020 Framework Programme (101092766, 101096909, 871391, 871658); Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung (200021_208094).
Acknowledgments
We gratefully thank the operations team of BRNC and ScopeM for their support and assistance put into this work. We further thank Prof. Wang for enabling the S11 VNA measurements. Electro-optic measurements have been recorded with the free and open-source tool “LabExT”.
Disclosures
The authors declare no conflicts of interest.
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.
References
1. A. Rahim, A. Hermans, B. Wohlfeil, et al., “Taking silicon photonics modulators to a higher performance level: state-of-the-art and a review of new technologies,” Adv. Photonics 3(02), 024003 (2021). [CrossRef]
2. R. C. Miller, “Optical second harmonic generation in piezoelectric crystals,” Appl. Phys. Lett. 5(1), 17–19 (1964). [CrossRef]
3. J. E. Ortmann, F. Eltes, D. Caimi, et al., “Ultra-low-power tuning in hybrid barium titanate–silicon nitride electro-optic devices on silicon,” ACS Photonics 6(11), 2677–2684 (2019). [CrossRef]
4. K. Liu, C. R. Ye, S. Khan, et al., “Review and perspective on ultrafast wavelength-size electro-optic modulators,” Laser Photonics Rev. 9(2), 172–194 (2015). [CrossRef]
5. A. K. Hamze, M. Reynaud, J. Geler-Kremer, et al., “Design rules for strong electro-optic materials,” npj Comput. Mater. 6(1), 130 (2020). [CrossRef]
6. J. T. S. Irvine, D. C. Sinclair, and A. R. West, “Electroceramics: Characterization by Impedance Spectroscopy,” Adv. Mater. 2(3), 132–138 (1990). [CrossRef]
7. O. Klein, S. Donovan, M. Dressel, et al., “Microwave cavity perturbation technique: Part I: Principles,” Int. J. Infrared Millimeter Waves 14(12), 2423–2457 (1993). [CrossRef]
8. J. M. Wiesenfeld, “Electro-optic sampling of high-speed devices and integrated circuits,” IBM J. Res. Dev. 34(2.3), 141–161 (1990). [CrossRef]
9. M. S. T. Nagatsuma, M. Yaita, and K. Takeya, “Electro-optic probing technology for ultrahigh-speed IC diagnosis,” IEEE Instrumentation and Measurement Technolgy Conference, 1476-1483 vol.1473 (1994). [CrossRef]
10. J. A. Valdmanis, G. Mourou, and C. W. Gabel, “Picosecond electro-optic sampling system,” Appl. Phys. Lett. 41(3), 211–212 (1982). [CrossRef]
11. M. Zgonik, P. Bernasconi, M. Duelli, et al., “Dielectric, elastic, piezoelectric, electro-optic, and elasto-optic tensors of BaTiO3 crystals,” Phys. Rev. B 50(9), 5941–5949 (1994). [CrossRef]
12. Y. Kok Yeow, “Materials characterization using microwave waveguide system,” in Microwave Systems and Applications, K. G. Sotirios, ed. (IntechOpen, Rijeka, 2017), p. Ch. 14.
13. H. Jiqing, A. Sligar, C. Chih-Hung, et al., “A grounded coplanar waveguide technique for microwave measurement of complex permittivity and permeability,” IEEE Trans. Magn. 42(7), 1929–1931 (2006). [CrossRef]
14. J. Ge, T. Xia, and G. Wang, “Design and optimization methodology of coplanar waveguide test structures for dielectric characterization of thin films,” J. Electron Test 36(2), 183–188 (2020). [CrossRef]
15. K. Byoungjoong, C. Jeiwon, C. Changyul, et al., “Nondestructive measurement of complex permittivity and permeability using multilayered coplanar waveguide structures,” IEEE Microw. Wireless Compon. Lett. 15(5), 381–383 (2005). [CrossRef]
16. M. D. Janezic, D. F. Williams, V. Blaschke, et al., “Permittivity characterization of low-k thin films from transmission-line measurements,” IEEE Trans. Microwave Theory Tech. 51(1), 132–136 (2003). [CrossRef]
17. Y. Liu, L. Chen, C. Y. Tan, et al., “Broadband complex permeability characterization of magnetic thin films using shorted microstrip transmission-line perturbation,” Rev. Sci. Instrum. 76(6), 063911 (2005). [CrossRef]
18. K. Alexander, J. P. George, J. Verbist, et al., “Nanophotonic Pockels modulators on a silicon nitride platform,” Nat. Commun. 9(1), 3444 (2018). [CrossRef]
19. G. F. Feutmba, L. da Silva, N. Singh, et al., “High frequency characterization of PZT thin-films deposited by chemical solution deposition on SOI for integrated high speed electro-optic modulators,” Opt. Mater. Express 13(7), 2120 (2023). [CrossRef]
20. K. J. Kormondy, Y. Popoff, M. Sousa, et al., “Microstructure and ferroelectricity of BaTiO(3) thin films on Si for integrated photonics,” Nanotechnology 28(7), 075706 (2017). [CrossRef]
21. P. Kharel, C. Reimer, K. Luke, et al., “Breaking voltage–bandwidth limits in integrated lithium niobate modulators using micro-structured electrodes,” Optica 8(3), 357 (2021). [CrossRef]
22. M. Burla, C. Hoessbacher, W. Heni, et al., “500 GHz plasmonic Mach-Zehnder modulator enabling sub-THz microwave photonics,” APL Photonics 4(5), 056106 (2019). [CrossRef]
23. C. Haffner, W. Heni, Y. Fedoryshyn, et al., “All-plasmonic Mach–Zehnder modulator enabling optical high-speed communication at the microscale,” Nat. Photonics 9(8), 525–528 (2015). [CrossRef]
24. W. Heni, C. Haffner, D. L. Elder, et al., “Nonlinearities of organic electro-optic materials in nanoscale slots and implications for the optimum modulator design,” Opt. Express 25(3), 2627–2653 (2017). [CrossRef]
25. M. Ono, H. Taniyama, H. Xu, et al., “Deep-subwavelength plasmonic mode converter with large size reduction for Si-wire waveguide,” Optica 3(9), 999 (2016). [CrossRef]
26. L. R. Dalton, J. Leuthold, B. H. Robinson, et al., “Perspective: Nanophotonic electro-optics enabling THz bandwidths, exceptional modulation and energy efficiencies, and compact device footprints,” APL Mater. 11(5), 21 (2023). [CrossRef]
27. A. Messner, D. Moor, D. Chelladurai, et al., “Plasmonic, photonic, or hybrid? Reviewing waveguide geometries for electro-optic modulators,” APL Photonics 8(10), 100901 (2023). [CrossRef]
28. M. Kohli, D. Chelladurai, A. Messner, et al., “Plasmonic ferroelectric modulator monolithically integrated on SiN for 216 GBd data transmission,” J. Lightwave Technol. 41(12), 3825–3831 (2023). [CrossRef]
29. M. Xu, Y. Zhu, F. Pittalà, et al., “Dual-polarization thin-film lithium niobate in-phase quadrature modulators for terabit-per-second transmission,” Optica 9(1), 61 (2022). [CrossRef]
30. J. Winiger, K. Keller, D. Moor, et al., “PLD Epitaxial Thin-Film BaTiO3 on MgO − Dielectric and electro-optic properties,” Adv. Mater. Interfaces (2023).
31. D. Caruntu, T. Rostamzadeh, T. Costanzo, et al., “Solvothermal synthesis and controlled self-assembly of monodisperse titanium-based perovskite colloidal nanocrystals,” Nanoscale 7(30), 12955–12969 (2015). [CrossRef]
32. A. von Hippel, “Ferroelectricity, Domain Structure, and Phase Transitions of Barium Titanate,” Rev. Mod. Phys. 22(3), 221–237 (1950). [CrossRef]
33. K. Laabidi, M. D. Fontana, M. Maglione, et al., “Indications of two separate relaxators in the subphonon region of tetragonal BaTiO 3,” Europhys. Lett. 26(4), 309–314 (1994). [CrossRef]
34. C. Haffner, W. Heni, Y. Fedoryshyn, et al., “Plasmonic organic hybrid modulators—scaling highest speed photonics to the microscale,” Proc. IEEE 104(12), 2362–2379 (2016). [CrossRef]
35. S. Yongqiang, Y. Lianshan, and A. E. Willner, “High-speed electrooptic modulator characterization using optical spectrum analysis,” J. Lightwave Technol. 21(10), 2358–2367 (2003). [CrossRef]
