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THz Pyro-Optical Detector Based on LiNbO_{3} Whispering Gallery Mode Microdisc Resonator

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## Abstract

**:**

_{3}whispering gallery mode microdisc resonator as a potential bolometer detector in the THz range. The resonator is theoretically characterized in the stationary regime by its thermo-optic and thermal coefficients. Considering a Q-factor of 10

^{7}, a minimum detectable power of 20 μW was evaluated, three orders of magnitude above its noise equivalent power. This value opens up the feasibility of exploiting LiNbO

_{3}disc resonators as sensitive room-temperature detectors in the THz range.

## 1. Introduction

_{3}and other birefringent crystals are widely studied, and several detectors have already been developed [21,22]. One of the main disadvantages of LiNbO

_{3}is its high absorption coefficient in the THz regime, limiting the detection sensitivity. Furthermore, to obtain strong electro-optic effects, applications mainly involve the detection of high peak THz pulses. In this study we propose a new technique for THz sensing by means of a LiNbO

_{3}microdisc resonator. Besides its good sensitivity to high peak THz pulses, its small dimensions are extremely sensitive to temperature shifts due to CW (continuous wave) THz absorption as well. Whispering gallery mode (WGM) microresonators are optical resonators characterized by a high quality factor, up to 10

^{11}[23], and, therefore, they are suitable for high-precision sensing applications [24]. LiNbO

_{3}microdiscs were already widely used for thermal and electro-optic applications [25,26]. Like MgF

_{2}[27,28], its birefringent behavior permits extremely fine temperature measurements and stabilization by exploiting the different responses of the two polarization modes, transverse electric (TE) and transverse magnetic (TM) [25,29,30]. The choice of the LiNbO

_{3}crystal relies on its higher thermal dependence frequency shift when compared to other crystals such CaF

_{2}and MgF

_{2}[25,28,31]. The possibility of using WGM resonators such as bolometers was already proposed for the infrared (IR) region by Zhu [32] and Ioppolo [33]. The former study was concentrated on the thermal drift in a toroid silica microresonator promoted by the modulation of a CO

_{2}laser. The latter mainly theoretically examined the effect of IR absorption by a liquid-filled hollow WGM sphere resonator due to its thermal expansion. In this work we evaluate the extension of these concepts in the THz domain, proposing an additional application, as pyro-optical detector, to an existent structure [29,30]. In the first part, the thermal response of a LiNbO

_{3}microdisc illuminated by THz radiation is theoretically analyzed. The theory takes into account both the thermal expansion and the thermo-optic effect on the resonance shift. Furthermore, the thermal behavior of the device is studied, as well as its characteristic time, when in close contact with a heat sink. The final sensitivity and noise equivalent power are evaluated, pointing out the system limits.

## 2. Theory

_{3}having a radius r = 2 mm and height h = 1 mm, placed on an aluminum thermostat. LiNbO

_{3}has a density of ρ = 4.648 g/cm

^{3}[34], yielding to a mass m of about 0.058 g. Its thermodynamic properties are the specific heat C = 0.63 J/(g·K) and the thermal conductivity k = 3.92 W/(m·K) [35]. If THz radiation is directed onto the top surface of the microdisc, it will first be largely absorbed by the crystal, and then reflected by the Al thermostat and, finally, further absorbed by the crystal. The radiation will thus experience an effective optical thickness of 2h = 2 mm. In the THz frequencies above 0.6 THz, the absorption coefficient Α for LiNbO

_{3}is above 10 cm

^{−1}[36], we can expect that almost 90% of the radiation will be absorbed and converted into heat. In a more specific approach, given an incident power ${P}_{in}$ we can define the absorbed power:

_{3}. Considering the specific heat of the LiNbO

_{3}, it is possible to calculate the temperature increase due to the radiation absorption. In order to maximize the thermo-optical response of the device, we consider a z-cut microdisc and a TE mode propagating inside the resonator. Assuming an excitation wavelength at 1500 nm and a TE polarized beam, the thermo-optic coefficient of LiNbO

_{3}, α

_{to}, is about 3.02 GHz/K, while its drift due to thermal expansion is lower, α

_{te}≃ 2.8 GHz/K [25,37], yielding to a total coefficient of α

_{tot}= α

_{to}+ α

_{te}= 5.82 GHz/K. Therefore, even a small temperature variation can be detected as a frequency shift. The microresonator limit of detection (LOD) is strictly dependent on the quality factor, and often it is estimated at 1/20 of the resonance width [38]. In the case of a LiNbO

_{3}microdisc, it is possible to achieve quality factors (Q-factors) on the order of 10

^{7}[29,30], or even of 10

^{8}[39], corresponding to a sensitivity of δν = 0.1 MHz. Now, under these considerations, it is possible to convert this into a limit of detectable temperature $\Delta {T}_{lim}$ in which the signal to noise (S/N) ratio is equal to one.

_{2}discs by [27,28], where temperature was stabilized on the order of hundreds of nanokelvin [27,28]. As a first approximation, we consider the system as perfectly insulated, and it is possible to calculate the energy delivered by the THz radiation promoting the minimum detectable shift δν of the resonance peak due to the temperature increase $\Delta {T}_{lim}=\partial v/\alpha $.

_{res}is the minimum temperature detectable and C is the thermal capacity. Now, considering that ∆E = P∙t, from Equation (3) it is possible to calculate the THz radiation with power ${P}_{min}$ versus the irradiating time in order to have a δν shift response of the LiNbO

_{3}microdisc.

_{3}and the thermostat, as depicted in Figure 1.

_{3}transmission as a function of the frequency $\theta \left(\upsilon \right)$ is plotted in Figure 4.

_{2}WGM resonator with the same dimensions is considered, and simply inserting the C and ρ values for LiNbO

_{3}in the following formula: $\langle \partial {T}_{noise}^{2}\rangle ={k}_{b}{T}^{2}/\rho CV$, where V stands for the IR probe beam mode volume. In the case of the disc proposed here, the value obtained for $\langle \partial {T}_{noise}^{2}\rangle $ is ${\left(456nK\right)}^{2}$. This temperature value can then be converted to optical THz power by means of Equation (7). Still, considering χ(υ) = 1, finally the NEP [40] can be calculated by using Equation (5)

## 3. Discussion

^{9}[23,27,28]. Regarding the bandwidth, this study mainly concentrates on the feasibility of WGM-based THz bolometers, considering a stationary solution and the minimum power detectable in these conditions. Nevertheless, it is reasonable to expect a working frequency on the order of KHz, as was demonstrated in a similar study involving CO

_{2}lasers and a microtoroid resonator [32]. From Equations (5), (6) and (8) it is possible to obtain the complete information about the design of the resonator and the optimal choice of materials. Decreasing the thickness of the microdisc down to 100 μm [42] would increase the acquisition rate two orders of magnitude. However, it is worth noting that decreasing h increases the value of ${P}_{min}$ following Equation (8). Furthermore, a minor optical thickness will yield to lower radiation absorption. Adding these two effects together, in the regime where $h\ll {{\rm A}}^{-1}$, the minimum power detectable, ${P}_{min}$, will be proportional to $1/{h}^{2}$. Therefore, h cannot be decreased arbitrarily without any impact on ${P}_{min}$. A plot of Equations (5) and (8) of the dependence of the device thickness of $\tau $ and ${P}_{min}\left(\upsilon \right)$ is shown in Figure 5.

_{2}or CaF

_{2}microdiscs [43]. Moreover, it is possible to observe from Equation (8) that increasing the k value will also increase the NEP. It could be necessary, however, to find a compromise, since higher values of k, even if they help reduce the time constant, also decrease the temperature variation ΔT and, therefore, the system sensitivity. In the system proposed here, LiNbO

_{3}was chosen since it provides the highest sensitivity. Furthermore, since LiNbO

_{3}is a polar material, pyroelectric charges can be generated on its surface while the temperature changes due to THz radiation absorption [44]. The corresponding electric field can induce a further shift of the resonance and, therefore, enhance the system sensitivity. Nevertheless, a possible drawback can arise from the long lifetime of those charges, which can induce a baseline shift. A comparison of the values of $\tau $ and ${P}_{min}\left(\upsilon \right)$ obtained from microdiscs made of different bulk material, but having the same geometry and dimensions as the one proposed in this work is shown in Figure 6.

_{2}, LiNbO

_{3}is more suitable due to its faster response time. Furthermore, the values reported in Figure 6 were obtained considering χ(υ) = 1, and a proper study involving the efficiency dependence of the wavelength and the crystal type should be done.

## 4. Conclusions

_{3}microdisc resonators as room-temperature THz detectors. Since the crystal shows a high absorption in the THz range, its thermo-optical features were investigated in order to obtain a THz bolometer. The chosen dimensions and Q-factor (10

^{8}) were in agreement with standard values obtained in our lab [29,30]. The resonator was considered in ideal experimental conditions in which one side faced an ideal insulator and the other faced a perfect heat sink. A characteristic time of τ = 0.74 s for the stationary regime was found, and an equivalent NEP value of 9.66 nW·Hz

^{-1/2}was evaluated by considering the thermo-optical fluctuations.

^{9}[6] could push the detectable power down to 200 nW.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**One-dimensional model for theoretical parameter evaluation (

**a**). The 1-mm-thick lithium niobate disc has, on one side, a perfect insulator and, on the other one, a perfect heat sink. (

**b**) A 3D representation of the LiNbO

_{3}microdisc is placed on a metal heat sink and coupled by a tapered fiber.

**Figure 2.**Microdisc resonator thickness needed in order to absorb 10% (red), 50% (blue) and 95% (green) of the radiation versus the THz radiation frequency.

**Figure 3.**Absorption efficiency $\chi \left(\nu \right)$ versus the optical thickness x defined as $A\left(\nu \right)\xb7h$, and the values corresponding to an efficiency of 10% (red), 50% (blue) and 95% (green). On the abscissa on the top the dependence of $\chi \left(\nu \right)$ the frequency $\nu $ is shown, assuming the value of h = 1 mm, as in the case of this article.

**Figure 4.**Dependence of THz radiation transmission $\theta \left(\upsilon \right)$ from the THz radiation frequency $\upsilon $.

**Figure 5.**Dependence of the device thickness of $\tau $ (blue) and ${P}_{min}\left(\upsilon \right)$ (green), considering an absorption coefficient Α = 10 cm

^{−1}(corresponding to 0.6 THz).

**Figure 6.**Dependence from the device bulk material of $\tau $ (red) and ${P}_{min}\left(\upsilon \right)$ (blue).

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**MDPI and ACS Style**

Cosci, A.; Cerminara, M.; Conti, G.N.; Soria, S.; Righini, G.C.; Pelli, S. THz Pyro-Optical Detector Based on LiNbO_{3} Whispering Gallery Mode Microdisc Resonator. *Sensors* **2017**, *17*, 258.
https://doi.org/10.3390/s17020258

**AMA Style**

Cosci A, Cerminara M, Conti GN, Soria S, Righini GC, Pelli S. THz Pyro-Optical Detector Based on LiNbO_{3} Whispering Gallery Mode Microdisc Resonator. *Sensors*. 2017; 17(2):258.
https://doi.org/10.3390/s17020258

**Chicago/Turabian Style**

Cosci, Alessandro, Matteo Cerminara, Gualtiero Nunzi Conti, Silvia Soria, Giancarlo C. Righini, and Stefano Pelli. 2017. "THz Pyro-Optical Detector Based on LiNbO_{3} Whispering Gallery Mode Microdisc Resonator" *Sensors* 17, no. 2: 258.
https://doi.org/10.3390/s17020258