Light turn-on transient of a whispering gallery mode
resonance spectrum in different gas atmospheres
Huiyi Natalie Luo, Heejoo Shua Kim, Monica Agarwal, and Iwao Teraoka*
Department of Chemical and Biomolecular Engineering
Polytechnic Institute of New York University
Six MetroTech Center, Brooklyn, New York 11201, USA
*Corresponding author: [email protected]
We examined the resonance spectrum change after turning on the light to feed the fiber
taper evanescently coupled to a silica whispering gallery mode (WGM) resonator
surrounded by different gases at different pressures. The resonance shifted to a longer
wavelength, indicating a temperature rise, before reaching a steady state. The increment
was proportional to the power of the light and approximately reciprocally proportional to
the thermal conductivity of the surrounding gas, whereas the rate of the shift was
approximately proportional to the thermal conductivity. The temperature rise, caused by
absorption of intense WGM in silica, was significant even when the wavelength scan range
contained only a few tall resonance peaks. We then estimated the power of heat generation
and the mean power of WGM during the wavelength scan.
OCIS codes: 120.6810, 260.5740, 350.3950.
1 Introduction
Efforts to utilize whispering gallery mode (WGM) dielectric resonators for sensing a change in the environment are continuing [1–3]. The temperature of the resonator [4], the refractive index
(RI) of the surrounding medium [5–7], and molecules adsorbed on the resonator surface [8–10] are examples of the environment. In a typical sensing scheme, the wavelength of the light source is scanned repeatedly to follow the shift of resonance wavelength in response to the change. The shift experiment is carried out, after the spectrum becomes steady. It is required that the spectrum not shift by other mechanisms during the measurement. When the transparent resonator is in air or another gas, the quality factor of WGM easily exceeds 1 107. A small change in the environment is sufficient to shift the resonance by as much as its linewdith, thus promising highly sensitive detection. The steadiness is crucial to successful applications of the resonator that take advantage of the narrow width.
Here we reveal the importance of the steady state by examining changes in the spectrum after we switch on the light leading to a silica resonator in air as we continue to scan the source wavelength. The spectrum shifted to a longer wavelength before reaching a steady pattern. We repeated this experiment by placing the spherical resonator in different gases at different pressures. We used a fast scan and a relatively low power to avoid thermal broadening [11–13].
The scan places the sphere in resonance only for a fraction of time, thus the sphere is heated only in those periods. Nevertheless, it is sufficient to shift the resonance wavelengths.
Recently, Ganta et al. studied the resonance shift transient after a high-power, visible laser illuminating a spherical silica resonator in free space was turned off [14,15]. The laser heats up the resonator by ~1 K to increase the RI of the resonator and thus red shift the resonance lines
2 for probe wavelengths. Turning off the laser brings back the resonance lines to the original wavelengths. The method we adopt here, in contrast, uses a probe laser in near infrared to cause a temperature increase. Ganta et al. repeated the experiment in air, helium, and nitrogen at different pressures ranging from 1 atm to ~0.1 mTorr. In each transient, they evaluated the effective thermal conductivity of the ambient and found that the conductivity remained unchanged until the pressure dropped below ~10 Torr and a further decrease in the pressure led to decreasing conductivity to approach zero. The conductivity drop in He occurred at higher pressures than it did in air and N2. They analyzed the pressure dependence in the light of thermal accommodation coefficients that account for the efficiency of heat transfer from the heated silica to the impinging gas molecules.
WGM in a sphere undergoing a temperature change
Each WGM in a spherical resonator is specified by the polarization of light, the number of waves around the circular orbit, the number of peaks of the mode intensity along the direction normal to the propagation direction on the sphere surface, and the number of peaks in the radial direction. The high RI contrast of silica in air allows scores of radial modes to be excited by evanescently coupling a taper to feed the sphere with light at the equator. Furthermore, variations in the number of peaks in the meridional direction can easily reach hundred. Therefore, the
WGM spectrum is extremely busy, and typically close to hundred modes can be observed in a wavelength scan over just 30 ppm of the center wavelength.
The high Q intensifies WGM. Absorption of light by silica turns a tiny fraction of the mode energy into heat without compromising the Q value. When the wavelength is scanned repeatedly, the sphere is heated as the wavelength of the feed light matches one of the WGM 3 resonance lines. The coupling efficiency varies from mode to mode, depending on the positions of the coupling taper, and so does the heating. The heating occurs where the WGM has a strong presence, mostly in a band around the equator where the taper touches. When the light starts to feed the resonator while the scan continues, the sphere’s temperature increases, resulting in shifts of the resonance peaks to longer wavelengths, mostly through an increase in RI [16].
Thermal conductivity of silica (s = 1.38 W/m∙K) is greater compared with gases (see
Table 1) [17]. The heat generated in the WGM orbit, Pheat, propagates first to the rest of the resonator (radius a) and then dissipates into the surroundings (thermal conductivity ). In the time scale to follow the change toward the steady state, we can consider that the sphere’s temperature is uniform all the time [15]. The resonator’s heat exchange with the surroundings occurs through conduction and radiation. The latter contribution was negligible in our experiments except in vacuum. With the heat conduction alone, the temperature will rise by
, (1) with a rate constant
(2) ,
where Csp is the sphere’s heat capacity. The temperature change can be monitored by the shift
of resonance wavelength , since our sphere has
(3) .
4 The latter was estimated from the resonance shifts in response to the resonator’s temperature increase in several steps.
The thermal conductivity of a gas at a given temperature T is proportional to the product of the mean free path (MPF) of gas molecules and the pressure p. Using the collisional cross section , MPF is expressed as
, (4)
where kB is Boltzmann constant [18]. Therefore, is independent of p, as long as MPF is shorter compared with the characteristic length of temperature gradient around the sphere. It can be shown that the latter length is the sphere radius a. As p decreases, MPF increases to become comparable to a. That pressure pc is given as
. (5)
Below this pressure, the gas is in the Knudsen regime, where collisions with the sphere surface dominate over collisions between molecules. Table 1 lists , MPF at 300 K, 1 atm, and pc for a =
161.0 µm for the gases used here.
Experimental
Our experimental system (Fig. 1) uses a head-on pair of tapers touching a microsphere. The tapers were made by etching single mode fibers (Corning, smf28e+) in a silicone-covered hydrofluoric acid solution [19] and glued onto a microscope slide with a tip-to-tip distance of
~50 µm. They are parallel but displaced by ~20 µm to minimize direct transmission of light
5 across the taper gap. A microsphere (see the inset, a = 161.0 µm), made by exposing a tip of another single mode fiber to methane–oxygen flame, sat on top of the taper pair. The position of the sphere was finely adjusted in three dimensions to optimize the coupling. The sphere–taper assembly, the positioner for the microsphere, and a wireless CMOS camera with a microscope objective (mounted on another positioner) were housed in a vacuum chamber connected to a vacuum pump and a gas cylinder. A vacuum gauge (Ashcroft) monitored the gauge pressure. The fibers entered the chamber via a house-made feedthrough.
The wavelength of a DFB laser (FITEL 13DDR8-A31) was controlled by the laser current. The light was fed into the microsphere through one of the tapers after passing an isolator
(OE Market), an attenuator (OZ Optics; 5dB unless otherwise specified), and a mechanical switch (OpLink OFMS 522212). The power after the switch is 0.92 mW at 35 mA. As the laser current was scanned in a triangular wave, the intensity of light picked up by the other taper was digitized. Each of the up and down scans consists of 1025 channels of data (one channel is shared) at 50 kHz sampling, and therefore the frequency of the triangular wave is 24.40 Hz. The current scan from 35 to 40 mA covers wavelengths from 1302.8989 to 1302.9320 nm. The light intensity signal as a function of the laser current is busy with scores of peaks, and is essentially a spectrum of resonance lines. We analyzed the down-scan spectra only, since the wavelength changes linearly with the laser current, as opposed to a curved relationship in the up scan. The light was fed to the microsphere by manually operating the switch, asynchronous to the wavelength scan. We used a switch to turn on the feed light already in a steady wavelength scan, rather than turning on the light source.
6 For a given environment (gas type, pressure), light turn-on experiments were conducted at ~300 K in triplicate. Data collection continued at least for 55 s, following each turn-on event.
The number of scanned spectra collected for each turn-on transient is more than 1,300.
Among readily available gases (O2, N2, CO2, He, Ar), the thermal conductivity variations are rather small. O2 and N2 have nearly identical values, and so do CO2 and Ar. To have different values, we mixed He and N2, He and O2, and He and CO2 at different molar ratios (1:2, 1:1, 2:1,
4:1, 6:1) and estimated the conductivity for each mixture according to the formulas given in literature [20].
We analyzed the turn-on transient using a “zone shift” method. A section of 100 channels in a spectrum of a given down scan was compared with different sections of the same width in the spectrum of the preceding scan. The comparison was run backwards, starting with the last scan in a given turn-on experiment, until the first scan was reached that is not entirely a background noise of the photodetector. The shift c in the channel number that optimizes the overlap between the two sections in the two spectra was evaluated. From the rise of the photodetector signal, we could easily tell in which scan (can be an up scan) and at which channel the light was turned on. Then, each identified section in each down scan was labeled with the time counted from the light turn on event. The shift rate dc/dt was calculated for each section as the ratio of c to t2 – t1, where t1 and t2 are the time at the centers of the two sections, and was given a time stamp of (t1 + t2)/2. The shift was evaluated for each of eight sections covering the
101th to 900th channels of the spectrum. Since we analyzed the down scan only, the data obtained from one experiment consist of groups interspaced by more than a half of the scan period of the triangular wave. Combining three independent experiments narrows the gaps. As 7 the turn on was asynchronous, the gaps remained in some experiments. Details of the “zone shift” method will be published elsewhere.
8 Results and discussion
The turn-on event shows as an abrupt appearance of resonance lines in one of the scans. Figure 2 compares the spectrum that captured the event (first spectrum, a), the next two spectra (b, c) and a spectrum (d) after a steady state was reached (~44 s). The optical switch does not have a chattering, at least in our sampling rate. The part of the spectrum in (a) after the turn-on event
(455th channel) is similar to a part of (b). However, the zoomed spectra in (e) and (f) indicate that each peak exhibits a different shift: the earlier peak (located toward right in (e)) shifts more than does the later peak. A fraction of the scan period is sufficient to change the peak-to-peak spacing, as the resonator undergoes rapid changes during the early scans immediately after the turn on. With time, the spectral pattern shifts slowly to a greater current (longer wavelength).
The spectrum in (c) is similar to the one in (d), except the shift. Thus, the channel shift between adjacent scans or the channel shift rate dc/dt decreases to zero with time. Note that this experiment does not follow the build up of WGM (opposite of cavity ring down) that will complete within 1 µs.
We repeated the experiment with different pump-in powers of the laser in different gases at –2˝ Hg (or 0 for air) using different attenuators from 0dB to 20dB. Figure 3 compares the plots of channel shift rate dc/dt as a function of time t since the light turn-on event for the resonator in ambient air. The shift rate decays to zero, following a curve approximated by an exponential decay. The slopes of the plots for different powers in a semi-logarithmic scale are nearly identical, and the main difference between the plots is vertical shift. Upon close inspection, we find that the early data deviate upward. The deviation may have captured the spread of the heat from WGM’s hot zone to the rest of the sphere before the heat starts to dissipate into the 9 surroundings, as the resonance monitors the temperature in the hot zone. Each curve was fit by a sum of two exponential decays. The main decay rate is almost similar, and the total shift, estimated by integrating dc/dt with time, is nearly proportional to the power of light, as seen in the inset of the figure. The proportionality indicates that the shift was caused by an intense
WGM absorbed by the microsphere and turned into heat. The results were similar with the other gases.
In Figure 3, the plots at high power, especially the one at 0dB, shows undulation in the decay. The latter is caused by broadening and red shift of some resonance lines partly due to heating by some of the high-Q modes [21]. The broadening appears more prominently in the up scan, causing slightly different patterns in the up and down scans. However, shifts of the peak positions by the temporarily uneven heating are not small in the down scan, causing undulation.
When the attenuation level was 5 dB or greater, we found that the laser’s pump-in power is sufficiently low to have nearly identical up and down scan spectra, indicating that the heating occurred rather uniformly during the scans, not concentrated in the periods of the broadened peaks. The plot of dc/dt was close to a straight line except the early part. All the results were obtained using a 5dB attenuator except those in Figure 3.
Figure 4 (a) shows a plot of the decay rate , estimated by curve fitting the dc/dt data excluding the early part with an exponential decay, at different pressures of air. To estimate the total shift, we curve-fitted the dc/dt data with a sum of two exponential decays and integrated the curve. The results are shown in Fig. 4 (b). Both and the total shift remain unchanged until the pressure drops to near zero (~ –760 Torr). This result agrees with the thermal conductivity’s independence of pressure unless the pressure is lowered to decrease the mean free path of gas molecules to the sphere radius. See Table 1; the values of pc are a lot lower than –711 Torr, the 10 lowest pressure that gives plateaus in the plots. A similar result was obtained by Rosenberger et al. in air and helium [14,15]. Our vacuum system cannot control or measure the pressure close to
–30˝ Hg, so we focus on the pressure range where dc/dt plot is independent of pressure.
Figure 5 compares the plot of the channel shift rate dc/dt as a function of time t for He,
N2, O2 and CO2 at –10˝ Hg. The faster decay at the early stage, followed by a slow exponential decay, is seen for all the gases. The decay rates are different among the three gases: The greater the thermal conductivity, the faster the decay.
We repeated the experiments and data analysis at –2˝, –10˝, and –20˝ Hg for each of N2,
O2, He, and CO2. The plots of dc/dt in each gas nearly overlap on top of each other (not shown).
Figure 6 shows the decay rate of the main decay component for different gases at different pressures (≥ –20˝ Hg) including mixtures. Most of the data fall along a straight line. The dashed line is a linear fit to the data for pure gases, and is nearly parallel to the dash-dotted line that
3 represents eq 2. We estimated Csp as 28.5 µJ/K from the density (2.20 g/cm ) and specific heat
(0.740 J/g·K) of silica, assuming that the resonator is a sphere of a = 161.0 µm. Thus, we find that the observed spectral shift was caused by heating of the resonator and release of the heat into the surroundings. The dashed line has a positive intercept, indicating an extra conduit of heat flow, most likely the stem made of silica. We evaluate the contribution of the stem in a simple model as follows. The stem of radius b and thermal conductivity s will change the effective
2 conductivity through the sphere surface to eff = + (b/2a) (s–). If the heat generation in the sphere were not limited to the hot zone of the WGM, but rather uniform on the sphere, the plot of
2 as a function of would have an intercept of (4a/Csp)(b/2a) s and a nearly identical slope of
11 2 (4a/Csp)[1–(b/2a) ] (solid line in Figure 6). The solid line lies above most of the data. We consider that the latter difference results, as the hot zone of WGM is away from the stem, thereby making the heat release through the stem less efficient compared with the uniform heating.
We used eq 3 to convert the total shift to the overall temperature change T in the turn-on event. The plot of T as a function of (Fig. 7) is on a straight line with a slope of –1 for pure gases, as anticipated from the mechanism. It is common to Figs. 6 and 7 that the data for the mixtures deviate from those for the pure gases. The deviation is most serious for a mixture of He with CO2, the heaviest among the molecules studied here. We speculate that non-uniform mixing near the sphere surface caused the deviation. We are currently studying the shift in response to a pressure change in different gas environments. These further studies will help explain the deviation observed.
The temperature rise in CO2 is ~0.8 K with a 5dB attenuator. In N2, the resonance shift by
T (0.65 K) at –5dB is 6.3 ppm, ~300 times as large as the peak width. To decrease the shift to the peak width, the pump-in power needs to be attenuated by 30dB, but that would make the photodetector signal too weak.
We can estimate Pheat from the line of slope –1 in Figure 7, since T = Pheat/(Csp), obtained from eqs 1 and 2. The result is Pheat = 63 µW. The mean power of WGM (averaged over the scan range), , is related to Pheat as
(6)
12 where abs is the attenuation coefficient of the 1.3 µm light by absorption. We note here that abs, contributed by the tail of UV absorption and the tail of the overtone mode of OH stretch, minimizes at around 1.3 µm. There is no definite literature value for abs, and we estimate it as follows: The overall attenuation coefficient of single-mode silica fiber at 1.3 µm is 7.74 10–5
–1 –5 –1 m [22]. One seventh comes from absorption [23], and therefore abs = 1.1 10 m [24].
Then, we have an estimate of = 5.7 kW. However, this is just the mean value. When the feed wavelength matches the resonance wavelength of a tall peak in the spectrum, the power is even larger. This estimate is a lot larger compared with the power of light that transfers from the taper to the sphere, which is expected to be less than 0.1 mW. The high power is made possible by extremely high Q, as the intensity of the mode is proportional to Q2.
Conclusions
We found that the microsphere is at an elevated temperature that results from a balance between heat generation by absorption of light and heat release into the surroundings. The temperature difference from the ambient is approximately proportional to the reciprocal of the thermal conductivity of the surrounding gas. The effect of heating was sufficiently large to shift the resonance by two orders of magnitude when the feed light was attenuated by 10 dB from 0.92 mW laser power in the feed fiber. The heat balance in the resonator is precarious. A disturbance in the air will change the temperature and fluctuate the resonance wavelengths. The problem may be more serious in a small-volume resonator such as a toroid that has a small area available for heat release.
13 Acknowledgement
We acknowledge a support from NSF through grant No. 1004015.
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23. From the chart in http://www.invocom.et.put.poznan.pl/~invocom/C/P1-
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24. The estimates of the scattering component and absorption component vary from report to
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17 Table 1. Thermal conductivity and collisional cross section at 300 K, mean free path (MPF) at 300K, 1 atm, and the pressure pc that makes MPF equal to the sphere radius 161.0 µm.
a 2 b Gas , W/m∙K , nm MPF, nm pc, Pa He 0.156 0.21 195 87
N2 0.0260 0.43 95 42
O2 0.0262 0.40 102 45
CO2 0.0168 0.52 79 35 a) From reference 17, b) From reference 18.
18 Fig. 1. (Color online) Measurement system for whispering gallery modes in different atmospheres. A microsphere resonator sits on top of the gap between a head-on pair of tapers.
Light from a laser is fed to one of the tapers, and the other taper collects the light from the resonator. The dashed line represents a vacuum chamber. A micrograph of the microsphere resonator is included.
19 Fig. 2. (Color online) Resonance spectra of a sphere (161.0 µm radius) in –2˝ Hg N2 in a 5 mA down scan (1302.8989 – 1302.9320 nm) of the wavelength of a laser (–5dB), shown as a function of the channel number (1–1025). The spectra were collected from right to left. The light was turned on in the 18th scan (a). Spectra of the 19th (b), 20th (c), and 1100th (d) scans are horizontally displaced to align the corresponding peaks along the dashed line. (e) and (f) zoom parts of (a) and (b). Each nearly vertical line connects the peaks that belong to the same mode.
20 Fig. 3. (Color online) Channel shift rate dc/dt, plotted as a function of time t since the light was brought into a sphere of radius 161.0 µm in ambient air. Plots are shown for different levels of attenuation in the feed light, indicated adjacent to the plots. For each attenuation, results compiled from three experiments are shown. (Inset) The total shift in channels vs relative power of the feed light (1 = 0dB).
21 Fig. 4. (Color online) Decay rate of dc/dt (a) and the total shift (b), plotted as a function of gauge pressure of air. The sphere radius was 161.0 µm. A shift of +100 translates into a temperature increase of 0.255 K.
22 Fig. 5. (Color online) Channel shift rate dc/dt, plotted as a function of time t since the light was brought into a sphere of radius 161.0 µm in He, N2, O2, and CO2, each at –10˝ Hg. For each gas, results compiled from three experiments are shown. The N2, O2, and CO2 data are multiplied by
3, 9, and 27, respectively, to avoid overlap.
23 Fig. 6. (Color online) Decay rate of dc/dt, plotted as a function of thermal conductivity of gas,
. The data are shown for pressures of He, N2, O2, and CO2 at –20˝, –10˝, and –2˝ Hg and for pressures of mixtures between –10˝ and 0˝ Hg. The dash-dotted and solid lines are theoretical estimates assuming an isolated sphere and a sphere on a 125–µm stem, respectively. The dashed line is a linear fit to the data of pure gases.
24 Fig. 7. (Color online) Temperature rise T, plotted as a function of the decay rate . The pressures are identical to those in Fig. 6. The straight line has a slope of –1.
25