Temperature Estimation Procedure

Determining the temperature of the photonic crystal membrane is more difficult than it appears at first sight. Conventional spectral methods cannot be used as these assume a blackbody emission spectrum, which the photonic crystal clearly doesn’t emit. Alternatively, the fragile membrane cannot be touched by a ther- mocouple as this would potentially cause a short-circuit, also provide a cooling effect and certainly damage the crystal. Even the substrate temperature cannot be used, as it is not directly heated and it has very different thermal emission properties compared to the patterned photonic crystal membrane.

In order to obtain an accurate estimation of the temperature of the photonic crystal membrane, we exploited the thermo-optic effect of silicon. The thermo- optic effect is the change in a material’s refractive index due to a change in temperature. The wavelength of the resonant modes of the photonic crystal slab depend on the effective index of the mode which is related to the refractive index of the doped silicon. Therefore, any change in the index of the silicon changes the effective index of the mode and thus the wavelength of resonant mode. This effect provides a very nice dependency between the temperature of the silicon and the wavelength of the resonance. However, this relationship does need to be calibrated and has a different dependency for each of the different resonant modes as the percentage overlap of the mode and the silicon is specific for each mode.

Due to the nature of the material (doped silicon) and the high temperatures

that the membrane is expected to reach (≈1000 K) data on the temperature

dependent refractive index for doped silicon is not available. This prevented the use of simulations to calibrate the temperature dependent wavelength position of each of the resonances. An alternative procedure is to take the device and heat it up on an external hot plate and measure the reflection spectrum at each increment of temperature change. This way the temperature of the entire device will be known and can be related to the spectral position of the resonance as the temperature changes.

To execute this procedure the sample was placed onto an external heater and the surface temperature of the chip was measured using a thermocouple attached to the surface placed away from the actual photonic crystal membrane. By placing the sample on a heater of high thermal mass, we can assume that the entire sample, including its surface, is in thermal equilibrium. The maximum surface temperature we were able to reach using the external heater was 740 K. We measured the reflection spectrum at each step of the heating process and tracked the spectral position of the reflection resonance at ≈1.2 µm. The reflection spectra were measured using the setup-A configuration.

Figure 3.25 shows the measured reflection spectrum of the photonic crystal membrane (period 605 nm and hole radius 120 nm) for the resistively heated membrane with an applied bias range from 0 V to 78 V. The reflection measure- ment for the chip on the external heater is very similar and so is not included. As the photonic crystal membrane heats up, the refractive index of the silicon in- creases and so does the effective index of the resonance mode, so the mode shifts to longer wavelengths. Each of the reflection curves shows a strong reflection peak at≈1.2µm. The peak is a superposition of multiple Lorentzian resonances i.e. there are two resonant peaks visible at the room temperature spectrum, 0

1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 0 10 20 30 40 50 60 70 80 0 0.2 0.4 0.6 0.8 1.0 Volt (V) Wavelength (µm) Reflection

Figure 3.25: Measured reflection spectra of the phonic crystal membrane for a range of applied voltages: 0 V, 10 V, 20 V, 30 V, 40 V, 42 V, 44 V, 46 V, 48 V, 50 V, 52 V, 54 V, 56 V, 58 V, 60 V, 62 V, 66 V, 68 V, 70 V, 72 V, 74 V, 76 V, 78 V. The measurement were done using setup-A measurement configuration.

V, because the measurement spectra are taken using setup-A and so collect light from a cone of angles. Also these reflection spectra are a little different to the ones shown in Fig. 3.23 (completely uniform crystal slab on oxide) and Fig. 3.24e (crystal slab on oxide with four missing rows of holes every 25µm) because these measurements are made on the fabricated device, which is suspended with oxide ridge supports, so the reflection properties have changed somewhat.

Two Lorentzian curves are fitted to accurately track the position of the two resonant wavelengths at 1.2µm. The fit is shown in Fig. 3.26a, where a narrower wavelength range and a select number of reflection resonances are replotted from the 3D plot in Fig. 3.25. The two fits for the room temperature (0 V) and the high bias (76 V) spectra are shown. The same fitting procedure was applied to the reflection data measured with the device on the external heater, not included here. This method allows us to determine the relationship between the temper- ature of the photonic crystal membrane and the resonant reflection wavelength. The black solid line in Fig. 3.26b shows this relationship with a measured ther- mal coefficient of 0.07 nm/K for the short wavelength resonance (the magenta Lorentzian fit) for the sample measured on the external heater. Since we were unable to achieve surface temperatures higher than 740 K with the available heater, we extrapolated the line to higher temperatures as a very good linear fit to the data was achieved, the dotted segment of the plot in Fig. 3.26b.

Figure 3.26 shows the measured reflection spectrum for the photonic crystal slab resistively heated, to make the graph readable only seven of the applied

1.16 1.18 1.20 1.22 1.24 0 0.2 0.4 0.6 0.8 1.0 Wavelength (µm) R efl ec tan ce 1.17 1.19 1.21 1.23 300 500 700 900 1100 Reflectance Peak (µm) Te m pe ra tur e (K ) 0 V 30 V 50 V 56 V 60 V 66 V 70 V 76 V Lorentzian (a) Lorentzian Thermal coeff. 0.07 nm/K (b) Linear extrapolation Short wavelength Lorentzian peak fit fit 30 V 50 V 56 V 60 V 66 V 70 V 76 V

Figure 3.26: (a) Measured reflection spectrum for the resistively heated photonic crystal membrane, at room temperature (0 V) and for seven different applied voltages: 30 V, 50 V, 56 V, 60 V, 66 V, 70 V and 76 V. Two Lorentzians are fitted to the reflec- tion resonances (shown here for the case of room temperature and one of the higher obtained temperature with a voltage of 76 V) in order to track the peak wavelength of the resonances accurately. (b) The black solid line represents the thermal coefficient (0.07 nm/K) for the photonic crystal membrane which was calculated using an external heater that reached a maximum surface temperature of 740 K. The linear extrapola- tion of this coefficient to higher temperatures is also shown. The solid magenta dots represent the peak position of the left hand (short wavelength) reflection resonance (indicated with the magenta Lorentzian fit in panel (a)) as the voltage increases, with the seven reflection resonances from panel (a) highlighted. Using the calculated (and extrapolated) thermal coefficient value, the applied voltage can be accurately mapped onto the corresponding temperature value [57].

voltages are plotted: 30 V, 50 V, 56 V, 60 V, 66 V, 70 V and 76 V. For the high voltage case of 76 V, the two Lorentzians almost completely overlap and hence have very similar resonant wavelengths. Figure 3.26b shows how the temperature for each reflection measurement is determined by using the peak of the reflection resonance for the short wavelength Lorentzian fit mapped onto the thermal coef- ficient line, with each of the reflection resonances shown in panel (a) individually marked in (b).

I would like to comment on the linear extrapolation in Fig. 3.26b, on its accuracy and is any deviation form the linear behaviour expected at the high temperatures. The linear fit to 740 K represents how the peak wavelength of the optical resonance in the slab shifts as the temperature increases. Up to 740 K the relation is very linear with an almost perfect linear fit achieved with the measured data, however, what happens beyond this?

The resonant wavelength of the mode depends on a number of things; the refractive index of the material (and also the surrounding medium), the thick-

ness of the slab, the period of the structure and the hole radius. If any of these parameters change the resonant wavelength will shift. By heating the structure all of these parameters will change but with varying magnitudes. The thermo- optic effect relates how a material’s refractive index changes as the temperature changes. For intrinsic bulk silicon the refractive index shift from room tempera- ture to 750 K is approximately linear [71]. No data is available for highly doped silicon material. The thermal expansion of the silicon material will increase the size of the structural parameters; slab thickness and period. The crystal lat- tice constant for intrinsic bulk silicon also increases approximately linearly with temperature to over a 1000 K [72].

Although these trends point to a linear response at high temperatures, it is difficult to isolate how each parameter change contributes to the resonant wavelength shift as the temperature increases and would require investigation via simulation. However, what I can say is that I expect the behaviour to be closely linear and that there will be, atleast, no major deviations from a linear- like response. For there to be a significant deviation from linear behaviour there would need to be a significant change with one of the parameters where the mode profile and/or mode confinement changes suddenly, e.g. when the material softens or starts to melt. At this point the linear relationship will most certainly be gone. For the devices used here no such behaviour was observed, the material and the crystal structure did not degrade at the high temperatures. The optical response of the heated structure was re-examined at room temperature after heating and the resonance wavelength had not changed. Therefore, it is safe to say that the material and photonic crystal structure are stable at the high temperatures. Finally, I would like to point out that this technique is used as an estimation procedure and to give a ball park figure of the operating temperature.