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Abstract
Fano resonance theoretically is an effective approach for sensitivity enhancement in photonic sensing applications, but the reported methods suffer from complicated structure and fabrication, narrow dynamic range, etc. In this article, we propose a photonic thermometer with sub-millikelvin resolution and broad temperature measurement range implemented by a simple waveguide-microring Fano structure. An air hole is introduced at the center of the coupling region of the waveguide of an all-pass microring resonator. The effective refractive index theory is used to design its equivalent phase shift and therefore the lineshape of the Fano resonance. Experimental results showed that the quality factor and the Fano parameter of the structure were invariant in a broad temperature range. The wavelength-temperature sensitivity was 75.3 pm/℃, the intensity-temperature sensitivity at the Fano asymmetric edge was 7.49 dB/℃, and the temperature resolution was 0.25 mK within 10℃ to 90℃.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Corrections
14 September 2023: A typographical correction was made to the author affiliations.
1. Introduction
Fano resonance is a unique physical phenomenon of interference between a narrow discrete state and a broad continuum state which could be implemented with numerous materials and physical properties. One of the main features of Fano resonance is its asymmetric line profile, which leads to a higher quality-factor and a steeper slope-rate compared with the corresponding Lorentzian resonance. Fano resonance has been widely adopted in many applications, such as photonic sensing [1], photonic communication [2,3] and imaging [4]. For the purpose of practice and mass production, Fano resonance has been implemented with integrated photonic devices with various structures including photonic crystals [5], plasmonics [6], metamaterials [7], etc. Among all of these structures, high-quality optical microresonators [8–13] have been intensively researched for implementing ultra-high sensitivity photonic sensors for biochemical [14], refractive index [15], and temperature [16] sensing.
Photonic thermometer precisely measures thermal-induced spectrum shift which is an overall effect of the thermo-refractive and the thermo-elastic effects of device material [17]. It traces the measurement temperature quantity to the optical frequency which could be obtained and kept at a very low uncertainty level [18]. In addition to the determination of temperature by measuring spectrum shift, the other essential method is obtaining temperature fluctuations by measuring intensity variations of a slightly off-resonance signal at the asymmetric edge of the line profile, by which the sensitivity and hence the resolution is further improved. In accordance with those advanced properties, metrologists expect to elevate the performance to micro-resonator typed photonic thermometer, which intrinsically has high stability, high resolution, and low uncertainty, to challenge the performance of metrological purposed standard platinum resistor thermometer and the corresponding traditional electric thermometry systems [19–21].
By utilizing Fano resonance, Li et al. theoretically predicted a photonic approach based on gain-loss coupled microresonators could enhance the temperature measurement sensitivity up to 8 orders of magnitude greater than that of two indirectly coupled lossy cavities, and he deduced its resolution could reach down to 10−13 K [22]. Kong et al. reported a theoretical metal-insulator-metal (MIM) waveguide-based temperature sensor with a sensitivity of 0.36 nm/℃, but the gap between cavities is only 20 nm which required an extremely stringent constraint to fabrication tolerance [23]. Qiu et al. proposed a Fano-thermometer based on a silicon eye-like double-rings structure which has a sensitivity of 48.8 pm/℃ and 1.596 dB/℃ [24]. A high-sensitivity Fano resonance temperature sensing method with integrated silicon Bragg reflectors was demonstrated by Chang et al. [25], while this device is constrained by complicated post-processing to control the phase plate thickness for tuning Fano lineshape. Thermometer based on cone-shaped in-wall capillary-based microsphere resonators had a sensitivity of 10.9 pm/°C within 0 to 100 ℃ [26], but temperature-induced variations of coupling coefficient resulted in a large line profile changing that hurt measurement resolution. Fano thermometer based on thin-film lithium niobate (TFLN) Suzuki phase lattice (SPL) photonic crystal achieved temperature measurement sensitivity up to 770 pm/℃ [27], but the lineshape changed with temperature because of thermal coefficient mismatch of silica and TFLN. The above researches predicted and preliminary proved that the microcavity typed Fano photonic structures could be expected to be a practical thermometer. However, the actual performance of the fabricated devices and corresponding measurements are limited by their low sensitivity, or complicated fabrication, or narrow dynamic range etc.
In this article, we propose a sub-millikelvin temperature measurement approach based on Fano resonance which is implemented with a simple waveguide-microring structure. It’s featured in keeping its high sensitivity, resolution over a wide temperature range. An air-hole was introduced in the waveguide coupling region, and the initial phase of the continuum state was controlled by the effective refractive index of the coupling region to obtain a specific Fano parameter and a high-quality-factor Fano resonance. The thermo-refractive effect and the thermo-elastic effects brought wavelength shifts of the spectrum of the Fano resonance, but the waveguide-microring structure ensured that the difference between the eigenfrequencies of discrete and that of the continuum states was a constant in a broad temperature range. Therefore, Fano parameters and quality factors of the resonance have a high stability. With the determination of resonance frequency and edge intensity, the proposed method maintains its high sensitivity and sub-millikelvin resolution.
2. Principle
Figure 1(a) shows the physical model of Fano resonance symbolized as a system of two coupled driven oscillators. A strongly-damped oscillator 1 with its eigenfrequency ${\omega _\textrm{1}}$ and damping constant ${\gamma _1}$ and a weakly-damped oscillator 2 with the eigenfrequency ${\omega _\textrm{2}}$ and ${\gamma _2}$ interacts under a weak coupling regime where $|g |\ll |{{\gamma_1}} |$. When the oscillator 1 is driven by an external field $f$, such as electromagnetic waves, mechanical vibration, etc., with angular frequency $\omega $. The amplitude of oscillator 1 around the frequency ${\omega _\textrm{2}}$ is [28]:
Fig. 1. Fano resonance in waveguide-cavity structure. (a) Model of Fano resonance by a system of two coupled driven oscillators; (b) air-hole waveguide coupled microring resonator.
Figure 1(b) shows the structure of the proposed device. $R$ is the radius of the microring, and w are the width of the bus waveguide and the microring. An air-hole with a diameter d is drilled in the waveguide at the center of the coupling region. The transfer matrix of its coupling region can be written as:
With Eq. (2), it is easy to write the transfer function of the device:
The phase shift of the self-coupling transfer function can be calculated by
where ${n_{eff}}$ is the effective refractive index of the section containing an air-hole. The ${n_{eff}}$ can be obtained by where ${n_{hole}}$ is the refractive index of the material in the air-hole.As shown in Fig. 1(a), the temperature increment works on waveguide and resonator which will result in a change in the two eigenfrequencies of the states. As long as the drifts of the eigenfrequencies of the two states are identical along with the temperature variation, the air-hole-induced phase shift $\Delta {\phi _1}$ always represents the equivalent phase of the continuum states since the temperature variation will not bring extra phase difference between the two states.
We obtained the relation between Fano parameter q and $\Delta {\phi _1}$:
Therefore, we can design the q related Fano features in accordance with the refractive index of the materials, hole diameter, and geometrical features of the all-pass filter.With the proposed device, we can conduct temperature measurements that actually acquire spectrum shift and voltage variation of the asymmetric edge of the Fano resonance lineshape, and it is an overall result of the thermo-optic effect and the thermal expansion effect. The temperature-induced resonant wavelength shift $\Delta \lambda $ can be written as:
The high slope rate of the edge of Fano lineshape can be used to improve the temperature sensitivity to overcome the limitation of spectrometer resolution. The temperature sensitivity can be increased by
where $k\textrm{ }[{dB\textrm{/}nm} ]$ is the slope rate of steep Fano edge, and ${S_V}\ [dB/ \ ^{\circ}{\rm{C}}]$ is the slope-temperature sensitivity. ${S_\lambda }\ [pm/ \ ^{\circ}{\rm{C}}]$ is the wavelength-temperature sensitivity.3. Experiments
3.1 Design of Fano parameters
To obtain a designated Fano lineshape with a high quality-factor and an enhanced edge slop rate, we firstly performed numerical simulations by sweeping the diameter, $d$, of the air-hole. The width of the bus waveguide was set to 540 nm and the diameter of the microring was 20 µm. With an evaluated typical fabrication capability, the round-trip amplitude attenuation ${a_{rt}}$ was set to 0.99. Figure 2(a) shows the normalized transmission spectrum calculated by Eq. (3) with the diameter of air-hole from 0 to 500 nm. As shown in Fig. 2(a), when $d = 0$, the transmission spectrum is a typical Lorentz lineshape which is a special case of Fano resonance that q equals infinity. With the increment of the diameter of air-hole, the effective refractive index of bus waveguide in the coupling region decreases. It results in a set of gradually turned Fano resonance. The Fano parameter q decreases from infinity to 0.34, and the slope rate of Fano resonance varies as a quadric.
Fig. 2. Numerical results of Fano lineshapes and slope rate determination. (a) Fano resonances with air-hole diameter from 0 to 500 nm; (b) The slope rate of different Fano resonances.
As shown in Fig. 2(b), the slope rate of the asymmetric Fano edge reaches their maximum when the diameter d equals 360 nm where the Fano parameter q equals 0.9. The slope of the Fano-like lineshape is 108.3 dB/nm, which is about 1.6 times to that of conventional Lorentz resonance.
We used finite-difference time-domain (FDTD) method to simulate the field distribution of the device at the resonance peak and dip with the parameter $d = 360\textrm{ }nm$. Figure 3(a) shows that constructive interference between resonant states of waveguide and microring in the coupling region results in a light field transmission at air-hole. On the contrary, there is destructive interference between two states as shown in Fig. 3(b), which appears as a strong reflection at the air-hole.
Fig. 3. The FDTD simulation results. (a) and (b) are the optical field distributions in the coupling region at the Fano resonance peak and dip, respectively.
3.2 Fabrication and experiment setup
The air-hole waveguide-microring device was fabricated on a silicon-on-insulator wafer with a 220 nm silicon layer and a 2 µm buried silicon dioxide layer. The whole device was air cladded. Figure 4(a) is the image of scanning electric microscopy of the device. The actual diameter of the drilled circular hole was 368 nm. The width of the waveguide was 537 nm. The diameter of the microring was 20 µm. The gap between the waveguide and the microring was 83 nm. In addition, we fabricated a conventional all-pass microring resonator with the same geometric parameters but without the hole for the comparison of Fano and Lorentz lineshape.
Fig. 4. (a) The vertical SEM view of fabricated air-hole waveguide-microring device; (b) Schematic of temperature sensing system setup with PC (polarization controller), WM (wavelength meter), WH (waveguide holder), TC (temperature controller) and PD (photodetector).
Figure 4(b) shows the experimental setup. An external cavity diode laser (Toptica, CTL1550) was used as a laser source. The probe beam was split into 90/10 by a fiber splitter. One path was fed to a wavelength meter (Toptica, WSU10-IR2) to monitor the scanning wavelength. The other was connected to a three-pads polarization controller. The probe laser was coupled in and out of the sensor through a pair of photonic crystal grating couplers [29]. The transmitted intensity signal was measured by an InGaAs photodetector (Thorlabs, PDA10CS2). The system used a proportion integration differentiation temperature controller (Thorlabs, TED200C) and a waveguide holder (Suruga Seiki, F274-18) with a pre-calibrated negative temperature coefficient (NTC) thermistor to provide a series of temperatures. Laser scan and data acquisition were implemented through a data acquisition card (National Instrument, USB-6361).
4. Results
Figure 5 shows the normalized transmission spectrum of the sensor at room temperature. It is evident that the spectrum is a combination of the designed Fano resonances with an envelope of the spectrum determined by the grating coupler. The Fano resonance reaches its maximum extinction ratio of 9.57 dB near the wavelength of 1552 nm. A comparison between the measured Fano and corresponding Lorentzian transition spectrum at room temperature around 1552 nm is shown in Fig. 5(b). The full width at half maximum (FWHM) of the Fano resonance which equals wavelength difference at the transmission peak and dip is 78 pm. The corresponding quality factor of Fano resonance is 19,895, while the quality factor of the Lorentz line is 15,836. The above result shows that the proposed device will give a high wavelength-temperature resolution than the traditional all-pass microring resonator. Fano parameter is fitted as that q equals 1. The slope rate of Fano resonance is 140.2 dB/nm. Comparingly, the slope rate of the Lorentz line is 99.9 dB/nm. That is, the designed Fano resonance slope is 1.4 times greater than that of the Lorentz resonance. These results show that the designed Fano resonance has a strong capability to enhance measurement sensitivity.
Fig. 5. (a) Measured normalized transmission spectrum for the Fano resonance of air-hole waveguide-microring device; (b) Transmission spectra for Fano resonance (red solid line) and Lorentz-like lineshape (black solid line) by fine wavelength scanning near 1550 nm.
We conducted a series of temperature sensing experiments. The temperature points were chosen according to the NTC calibration chart to keep the best accuracy, and the temperature range covered 10.1 ℃ to 50.7 ℃. As shown in Fig. 6(a), as the temperature increases, the center wavelength of the Fano resonance offers a clear redshift. The wavelength-temperature relationship of the device is shown in Fig. 6(b). The wavelength-temperature sensitivity is 75.3 pm/℃. Unlike the results of other Fano structures, the lineshape and above performance are very stable over the experimental temperature scale.
Fig. 6. (a) Experimental normalized transmission of Fano resonance at different temperatures; (b) Fitting curve (red solid line) of experiment data (blue squares) showing the wavelength shift as a linear function of temperature.
To prove the capability of sensitivity enhancement, we evaluated temperature-induced voltage variation at several temperature points which were 19.9 ℃, 24.8 ℃, 30.4 ℃, respectively. Because of the nonlinearity of the slope-temperature sensing method in dB/℃, we measured the slope-temperature sensitivity as shown in Fig. 7. The peak value of the measured Fano transmission spectrum is 3.3 V, so that the slope of Fano resonance we measured is equal to 39 V/nm, from Eq. (8) we can get the theoretical slope-temperature sensitivity of 2.94 V/℃ which is equal to 5.39 dB/℃. By fitting the experiment data, the temperature-voltage sensitivity at three temperature points is 3.86 V/℃, 4.04 V/℃ and 4.09 V/℃ (7.07 dB/℃, 7.40 dB/℃, 7.49 dB/℃), respectively. With a laser intensity stabilization which could reach signal to noise ratio up to 104:1, supposing electric circuit has a 1 mV measurement resolution, the results show that, at the above temperature points, the proposed approach could reach resolutions of 259 µK, 248 µK and 244 µK, respectively.
Fig. 7. Experimental temperature-voltage sensing at fix-wavelength points in 19.9 ℃ (green square), 24.8 ℃ (red circle) and 30.4 ℃ (blue triangle) of the device.
5. Discussions
The proposed temperature sensor has a very stable lineshape over a wide range of temperature, which is indicated by the temperature-voltage sensing curves at different temperature points that are almost parallel shown in Fig. 7. However, the theoretical data includes temperature points outside the linear region, so the measured sensitivity of each temperature point is higher than the theoretical value. The reason for that is that the second derivative near the peak of Fano lineshape is nonzero, the slope rate declines when the temperature variation is beyond the linear region. Thus, the average temperature measurement resolution in the linear region is highest. Therefore, in the actual precision sensing based on Fano resonance, the linear region should be considered.
Because the equivalent phase of the continuum state of the proposed device is almost constant over a wide temperature range, the measurements keep its high sensitivity. The phase of the device at 10 ℃ and 90 ℃ was calculated numerically, as shown in Fig. 8(a). At different temperature points, the Fano resonance appears at 1552 nm and 1558 nm, respectively. At a temperature difference of 80 ℃, we calculated the total phase difference near the resonant dip of two Fano resonances. The numerical result shows that the change in the total phase of the two Fano resonances at different temperature points is only 0.6 degree, which proves that the Fano parameter of our device is almost constant over a large temperature range. So, our device has high stability in wide-range temperature sensing.
Fig. 8. (a) Total phases of the device in different temperatures (red and green solid line) and the difference between them (blue solid line); (b) Temperature-induced Fano parameter variation (red square) and slope rate variation (blue star).
To obtain the temperature-Fano parameter relationship plotted in Fig. 8(b), we fit Fano resonances at each temperature point in Fig. 6(a). In the range of 10.1 ℃ to 50.7 ℃, the change rate of the Fano parameter q is only $\textrm{5} \times \textrm{1}{\textrm{0}^{\textrm{ - 5}}}\textrm{/}^{\circ}{\rm{C}}$. At a temperature point of 24.8 ℃, the Fano parameter is lowest, which probably comes from the perturbation of the base-line. The Fano parameter only increases by 0.005 at the temperature range from 10 ℃ to 90 ℃, which shows that the variation of the Fano resonance slope in the wide temperature range is extremely small, so that the Fano lineshape is almost unchanged. This is because the phase change of the continuum state and the discrete state is consistent under a wide range of temperature, which makes the total phase change of the system basically unchanged (as shown in Fig. 8(a)), so the interference of the two resonant states is unchanged. Therefore, the Fano resonance temperature sensing system based on the air-hole waveguide-microring device has high temperature-lineshape stability, which maintains sub-millikelvin temperature sensing in a wide temperature range.
6. Summary
We proposed a practical and reliable temperature sensor with sub-millikelvin resolution and broad temperature sensing range by implementing a Fano resonance on a simple waveguide-microring structure. We added an air-hole at the center of the waveguide of the coupling region and designed the phase of the continuum state of Fano resonance by means of the effective refractive index theory to obtain a steep intensity slope rate. The continuum state interfered with the discrete states formed in microring, resulting in a designated Fano resonance with Fano parameter of 0.9. We fabricated the device and preliminarily proved its performance. The measured wavelength-temperature sensitivity was 75.3 pm/℃, and the intensity-temperature sensitivity was 7.49 dB/℃ within temperature from 10 ℃ to 90 ℃. The corresponding fine resolution was 0.25mK. This performance can be improved by taking advanced fabrication and measurement techniques.
Funding
National Key Research and Development Program of China (2017YFF0206102); National Natural Science Foundation of China (61675020, 61775183, 91950119).
Disclosures
The authors declare no conflicts of interest.
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