- Email: [email protected]
Volume relaxation of quenched silica at room temperature monitored by whispering gallery mode resonance wavelength
Journal of Non-Crystalline Solids 476 (2017) 52–59
Contents lists available at ScienceDirect
Journal of Non-Crystalline Solids journal homepage: www.elsevier.com/locate/jnoncrysol
Volume relaxation of quenched silica at room temperature monitored by whispering gallery mode resonance wavelength
MARK
Lu Caib,1, Iwao Teraokaa,⁎, Yong Zhaob a b
Department of Chemical and Biomolecular Engineering, New York University, 6 MetroTech Center, Brooklyn, NY 11201, United States College of Information Science and Engineering, Northeastern University, Shenyang 110819, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Silica Quenching Volume relaxation Whispering gallery mode Frozen-in strain
We used a resonance wavelength shift of a whispering gallery mode to monitor volume relaxation at room temperature in cylindrical and spherical resonators fabricated by quenching the molten tip of a pure silica optical fiber. The high sensitivity of the method allowed measurement of the linear dimension at sub-ppm resolution. All the resonators registered a volume increase in the first few hours, in agreement with the volumetemperature equilibrium curve of type IV silica below the glass transition temperature. When heating of the fiber was insufficient, the volume increase was greater, ascribed to a larger volume of strained silica. We also formulated thermodynamics to estimate the temperature coefficients of polarizability at constant pressure and at constant density from resonance shift data obtained in experiments performed on relaxed silica resonators.
1. Introduction In a transparent dielectric medium of a circular cross section, light can travel near the surface via total internal reflection to complete a cycle [1]. This mode of wave propagation is called a whispering gallery mode (WGM). If the cycle contains an integral number of waves, the light superimposes constructively onto itself, and repeating this every cycle leads to a large amplitude. Since the intensity is proportional to the square of the amplitude, a strong wave can develop. The latter requires a strict condition for the wavelength, resulting in an extremely narrow resonance. In air, the Q factor can exceed 108. Different applications of WGM resonance have been proposed [2]. Among others, applications to sensing have been most intensively pursued, [3–7] as the resonance wavelength is sensitive to a small, localized refractive index (RI) change near the resonator surface in the surrounding medium. Since most of the wave travels within the resonator, the resonance wavelength is sensitive to a change of the resonator itself. Suppose the radius of a spherical resonator increases without changing its RI. Then, detecting a radius change of 0.1 ppm is not difficult with WGM. In practice, however, the radius change and the RI change occur at the same time, and it is often the case that they cancel each other to some extent. Nevertheless, the resultant sensitivity is among the highest, and WGM has been applied to detecting pressure, temperature, and stress changes [8–11].
⁎
1
The present report focuses on monitoring the changes in a silica resonator after it is fabricated by heating the tip of a pure silica optical fiber and then quenched. Silica has been extensively studied, [12–16] and its thermo-mechanical properties are well understood. The properties depend on the type of silica, distinguished by the starting material and processing [13]. There are four types of silica. Their fabrication methods and concentrations of OH are summarized in Table 1. The silica core of the fiber used in our experiment belongs to type IV, containing OH at about 0.2 ppm according to the supplier. Since silica is a preferred material for WGM resonators, their applications require that we have a good understanding of silica's properties and how the small variations in the starting material and processing affect the properties. Fortunately, monitoring the resonance wavelength shift can be a right tool to address this concern. Typically, a silica resonator is fabricated by exposing the tip of an optical fiber to a localized heat source to melt it. The surface tension results in a spherical resonator. When the heat source is turned off, the cooling takes just a second [17]. It means that the silica resonator is quenched. Therefore, the resonator is not at thermal equilibrium at least immediately after it is fabricated. Since the mobility of supercooled silica is low, it may take time to approach the equilibrium. This study finds how long it will take to get close to the equilibrium to the extent to stabilize the resonance wavelength. In the present report, we first develop a thermodynamic theory to express the resonator's RI and the resonance wavelength as a function of
Corresponding author. E-mail address: [email protected] (I. Teraoka). On leave.
http://dx.doi.org/10.1016/j.jnoncrysol.2017.09.022 Received 29 May 2017; Received in revised form 11 August 2017; Accepted 5 September 2017 Available online 12 September 2017 0022-3093/ © 2017 Elsevier B.V. All rights reserved.
Journal of Non-Crystalline Solids 476 (2017) 52–59
L. Cai et al.
With Eq. (4), Eq. (6) is rewritten to
Table 1 Fabrication methods and concentrations of OH for four types of silica. Silica type
Method of preparation
OH concentration/ppm
I
Electrical fusion of natural quartz in vacuum or inert gas Melting quartz crystal with oxy‑hydrogen flame Hydrolyzing SiCl4 when sprayed into oxy‑hydrogen flame Hydrolyzing SiCl4 with water-free plasma flame
Several
II III IV
6n∆n 1 ⎛ ∂α ⎞ ⎛ ∂ln a ⎞ ρ ⎛ ∂α ⎞ ⎤ =⎡ ⎢ α ⎝ ∂T ⎠ + 3 ⎝ ∂Τ ⎠ α ⎜ ∂ρ ⎟ ⎥ ΔT (n2 − 1)(n2 + 2) p p ⎝ ⎠T ⎦ ⎣ ρ ⎛ ∂α ⎞ ⎤ Δρ +⎡ ⎢1 + α ⎜ ∂ρ ⎟ ⎥ ρ ⎝ ⎠T ⎦ ⎣
150–400
In Eq. (7), the RI change by pressure at a constant temperature is expressed as
~ 1000 <1
6n ⎛ ∂n ⎞⎟ = 1 ⎡1 + ρ ⎛⎜ ∂α ⎞⎟ ⎤ ⎜ (n2 − 1)(n2 + 2) ⎝ ∂p ⎠T K⎢ α ⎝ ∂ρ ⎠T ⎥ ⎦ ⎣
temperature and density. In the theory, we will find a molecular-level expression of the resonance shift that will occur in volume relaxation post quenching and in a temperature change at around room temperature. We conducted these two lines of experiments on each of resonators of different diameters prepared by two different heating methods. After describing the experimental details, we will show the results and interpret them in the light of the theory.
∂p ∂p ρ ⎛⎜ ⎞⎟ = −V ⎛ ⎞ = K ⎝ ∂V ⎠T ⎝ ∂ρ ⎠T
Δλ Δn Δa Δn 1 ΔV Δn 1 Δρ = + = + = − λ n a n 3 V n 3 ρ
(10)
With Eq. (7),
Δλ (n2 − 1)(n2 + 2) ⎡ 1 ⎛ ∂α ⎞ ⎛ ∂ln a ⎞ ρ ⎛ ∂α ⎞ ⎤ = ⎢ α ⎝ ∂T ⎠ + 3 ⎝ ∂Τ ⎠ α ⎜ ∂ρ ⎟ ⎥ ΔT 6n2 λ p p ⎝ ⎠T ⎦ ⎣ +
ρ ⎛ ∂α ⎞ ⎤ 1 ⎫ Δρ ⎧ (n2 − 1)(n2 + 2) ⎡ ⎢1 + α ⎜ ∂ρ ⎟ ⎥ − 3 ⎬ ρ ⎨ 6n2 ⎝ ⎠ T ⎣ ⎦ ⎩ ⎭
(11)
For a density change at constant temperature,
∂ln λ ⎞ Δρ Δλ = ⎜⎛ ⎟ λ ⎝ ∂ln ρ ⎠T ρ
(12)
where (1) 2 2 ⎛ ∂ln λ ⎟⎞ = (n − 1)(n + 2) ⎡1 + ρ ⎜⎛ ∂α ⎟⎞ ⎤ − 1 ⎜ 2 ⎢ 6n α ⎝ ∂ρ ⎠T ⎥ 3 ⎝ ∂ln ρ ⎠T ⎣ ⎦
At constant pressure,
⎛ ∂α ⎞ = ⎛ ∂α ⎞ + ⎜⎛ ∂α ⎟⎞ ⎛ ∂ρ ⎞ ⎝ ∂Τ ⎠ p ⎝ ∂T ⎠ ρ ⎝ ∂ρ ⎠T ⎝ ∂Τ ⎠ p
1 ⎛ ∂ρ ⎞ 1 ∂V ∂ln a ⎞ = − ⎛ ⎞ = −3 ⎛ ρ ⎝ ∂Τ ⎠ p V ⎝ ∂Τ ⎠ p ⎝ ∂Τ ⎠ p
(3)
where (∂ ln a/∂ T)p represents the linear thermal expansion coefficient. For types I, II and IV silica, the linear thermal expansion coefficients are similar [18] and they are typically 0.52 ppm/K [19]. Therefore, Eq. (1) is now
⎛ ∂α ⎞ ⎛ ∂α ⎞ ⎛ ∂ln a ⎞ ⎛ ∂α ⎞ ⎤ Δα = ⎡ ⎢ ⎝ ∂T ⎠ + 3ρ ⎝ ∂Τ ⎠ ⎜ ∂ρ ⎟ ⎥ ΔT + ⎜ ∂ρ ⎟ Δρ p p ⎝ ⎠T ⎦ ⎝ ⎠T ⎣
(13)
With n = 1.447 and 1+ (ρ/α)(∂ α/∂ ρ)T = 0.64, we obtain (∂ ln λ/ ∂ln ρ)T = − 0.105. This coefficient indicates that a density decrease of 1 ppm causes a 0.105 ppm red shift of the resonance wavelength. Alternatively, 1 ppm red shift of resonance wavelength amounts to a 9.51 ppm decrease in density, and a 9.51 ppm increase in volume. Different experiments can measure Δ n due to a volume change at a constant temperature. In Eq. (7), setting ΔT to zero leads to
(2)
Since ρ ~ 1/V,
ρ ⎛ ∂α ⎞ ⎤ ΔV ∆n (n2 − 1)(n2 + 2) ⎡ =− ⎢1 + α ⎜ ∂ρ ⎟ ⎥ V n 6n2 ⎝ ⎠T ⎦ ⎣
(14)
With (ρ/α)(∂ α/∂ ρ)T = − 0.36, Δ n/n = − 0.23 × Δ V/V. A 1 ppm red shift of resonance wavelength amounts to a 2.17 ppm decrease in RI. This coefficient compares favorably with the one in Δ n/ n = −(n2p1/2)(Δ V/V), [16] where p1 ≊ 0.22 is the orientationally averaged elasto-optic coefficient. For an experiment in which the temperature changes at a constant pressure, an expression of the WGM resonance shift can be derived from Eq. (10) as
(4)
Lorentz-Lorenz equation relates the relative permittivity n to ρα as 2
(5)
For small changes,
Δρ 6n∆n Δα = + (n2 − 1)(n2 + 2) ρ α
(9)
For typical silica, n = 1.447 (at wavelength 1.3 μm), KT = 35.9 GPa, [20] and (∂ n/∂ p)T = 9.2 × 10− 12 Pa− 1 [21]. With these numbers, we can estimate, from Eq. (8), (ρ/α)(∂ α/∂ ρ)T as −0.36. The negative coefficient indicates that silica decreases its polarizability (per unit SiO2) with an increasing density. Now, we consider the WGM resonance shift:
We employ here thermodynamic analysis to express the resonance wavelength shift in terms of a change in the state of the glass that constitutes the resonator. To describe the state of glass including nonequilibrium states, we choose the temperature T and the density ρ as two independent variables, rather than T and pressure p. This provision allows describing the state when the glass relaxes toward the equilibrium to change its volume V at constant temperature and pressure. Since the resonance wavelength in air is proportional to the product of the refractive index (RI) n and radius a of the spherical resonator, we will first look at how n and a change with T and ρ, assuming that the changes are isotropic. Since most thermodynamic quantities are measured at constant T and p, we need to find formulas to convert these quantities into those at a given temperature and density. We model the glass as consisting of independent molecules with polarizability α at density ρ. When the temperature changes by ΔT and the density by Δρ, the polarizability changes according to
∂α ∂α Δα = ⎛ ⎞ ΔT + ⎜⎛ ⎟⎞ Δρ ⎝ ∂Τ ⎠ ρ ⎝ ∂ρ ⎠T
(8)
where K is the bulk modulus (reciprocal of the isothermal compressibility):
2. Theory of WGM resonance shift in air
n2 − 1 4π = ρα n2 + 2 3
(7)
Δλ ∂ln λ ⎞ =⎛ ΔT λ ⎝ ∂Τ ⎠ p
(6) 53
(15)
Journal of Non-Crystalline Solids 476 (2017) 52–59
L. Cai et al.
⎛ ∂ln λ ⎞ = 1 ⎛ ∂n ⎞ + ∂ln a n ⎝ ∂T ⎠ p ∂Τ ⎝ ∂Τ ⎠ p
The diameter of the resonator was estimated by analyzing the image. In each image analysis, the stem diameter of 300 μm was used as a reference. We also made resonators using FTx00UMT (x = 2, 3, 4; high OH; ThorLabs), but the peaks were about three times as broad as those made from FT300EMT. Therefore, these high-OH fibers were not actively employed in the study reported here.
(16)
The thermo-optic coefficient (∂n/∂T)p varies, depending on the type of silica. For type IV silica, it is 9.49 × 10− 6 K− 1 at 1.314 μm [20]. From Eqs. (3) and (6), we find that (∂n/∂T)p is related to (∂ α/∂T)p by
1 ⎛ ∂n ⎞ (n2 − 1)(n2 + 2) ⎡ 1 ⎛ ∂α ⎞ ∂ln a ⎤ = ⎢ α ⎝ ∂T ⎠ − 3 ∂Τ ⎥ n ⎝ ∂T ⎠ p 6n2 p ⎣ ⎦
3.2. Fabrication of tapers
(17)
We prepared a pair of pencil-shaped fiber tapers by etching coatingstripped single-mode fibers with a hydrofluoric acid solution. The two tapers were then aligned along a straight line with their two tips facing each other at a gap distance of ~210 μm. Subsequently, the coated portions of the fibers were glued onto a microscope slide. To monitor the temperature, a 0.8 mm glass-encapsulated 10 K thermistor (Analog Technologies, ATH10KR8T63S) was glued onto the slide near the gap of the two tapers.
where
∂ln a ρ ⎛ ∂α ⎞ 1 ⎛ ∂α ⎞ 1 ∂α = ⎛ ⎞ +3 ⎜ ⎟ ∂Τ α ⎝ ∂ρ ⎠T α ⎝ ∂T ⎠ ρ α ⎝ ∂T ⎠ p
(18)
from Eq. (2). With Eqs. (17) and (18), we estimate (1/α)(∂ α/∂ T)p and (1/α)(∂ α/∂ T)ρ as 19.96 and 19.40 ppm/K, respectively. 3. Experimental 3.1. Fabrication of a resonator
3.3. Coupling of the resonator to the tapers
We used a multimode fiber FT300EMT (low OH of 0.2 ppm, ThorLabs) to fabricate resonators. This fiber has a pure silica core and a thin TECS cladding, and is coated with Tefzel. The fiber was cut to an 8cm long section. The Tefzel coating was removed with a stripper for 3 cm from one of the ends. To remove the hard fluoropolymer cladding, the tip of the fiber was soaked in acetone, then wiped with a sheet of KimWipe. The tip of the bare silica fiber was then exposed to a local heating source that is either electric arc across a pair of tungsten electrodes or an oxy‑hydrogen torch. When the hydrogen torch was used to make a resonator of 423 μm, for example, the tip was exposed to the flame for as long as ~1 min. We continued to rotate the fiber around its axis to minimize bending. When the arc was used for the same size of resonator, the tip was exposed to two shots, each 2 s long. Thus, a resonator was formed at the tip of the optical fiber. The Tefzelcoated part was a stem to hold and position the resonator. The stem was mounted on a 3D translation stage. A total seven resonators of different diameters, ranging from 322 to 490 μm, were prepared using the electric arc by changing the exposure time. The longer the exposure time, the larger the diameter. With the hydrogen torch, six resonators of diameters from 350 to 527 μm were prepared. Fig. 1 shows micrographs taken with a stereo zoom microscope for some of these resonators. Those in part (a) were made by the electric arc; those in part (b) by the hydrogen torch. It is obvious that the electric arc formed resonators closer to a sphere than the hydrogen torch did.
The position of the resonator relative to the two tapers was adjusted by manipulating the 3D positioner. Rough positioning was done while observing the tapers and the resonator using a camera equipped with a microscope objective placed under the slide. Fine tuning for optimal positioning was accomplished while observing the resonance spectrum using the measurement system described below. 3.4. WGM measurement system A schematic diagram of our measurement system is shown in Fig. 2. The light from a distributed feedback (DFB) laser (FRL13DDR8-A31, FITEL) passes an isolator and leads to one of the tapered fibers that touch the resonator. The other tapered fiber picks up the WGM and leads to the photodetector. The current to the DFB laser is changed in a triangular wave at 24.40 Hz from 35 to 37.5 mA to scan the wavelength from 1314.5664 to 1314.5860 nm, unless otherwise specified. Each of the up and down current scans of the triangular wave consists of 1025 points of data with one being shared. The photodetector signal is digitized synchronously. We use the laser wavelength calibration of our earlier work that gives the wavelength as a function of the laser current [22]. A part of the light from the feed fiber transfers to the resonator via evanescent coupling. When the wavelength matches one of the resonance wavelengths, a strong mode is generated and circulates within the resonator. The 1025 data of the photodetector signal in each scan is a WGM spectrum. Fig. 1. Micrographs of resonators made by (a) electric arc and (b) hydrogen torch. The diameter at the equator is shown for each of the six resonators displayed here. The stem diameter is 300 μm.
54
Journal of Non-Crystalline Solids 476 (2017) 52–59
L. Cai et al.
temperature change in the isothermal volume relaxation experiment will be explained. Analysis of all the scans of resonance spectrum collected allows us to prepare a plot of a raw resonance shift (Δ λ/λ)raw with reference to the mean resonance wavelength in a reference period that is typically the first 1 s of the experiment. A temperature change Δ T is calculated with respect to the same reference period. Then, we calculate a temperature-compensated shift (Δ λ/λ)T-comp according to
Δλ ∂ln λ ⎛ Δλ ⎞ =⎛ ⎞ − ∆T ∂T ⎝ λ ⎠T − comp ⎝ λ ⎠raw
(19)
where ∂ lnλ/∂ T is the temperature coefficient of the resonance wavelength. The latter was evaluated in the cooling experiment, and we used that coefficient in converting the raw resonance shift into the temperature-compensated shift in the isothermal volume relaxation experiment.
Fig. 2. Schematic diagram of the measurement system. A zoomed view is shown for the resonator on a stem coupled to a pair of tapered fibers.
4. Results and discussion 3.5. Temperature monitoring 4.1. Isothermal volume relaxation experiment The thermistor mounted on the microscope slide was connected to a bridge circuit to convert the resistance to a voltage. The latter was digitized by a data logger at 4 Hz.
We use an example of isothermal experiment conducted for a resonator formed into a spherical shape with 450 μm diameter using arc to explain our method of spectrum data analysis. Fig. 3 shows some of the down-scan spectra collected in the first 2800 s continuous spectrum collection for the wavelength scan range of 19.53 pm. For this period, a total 68,095 spectra were collected. Only the down scan spectra were analyzed because of a better linearity between the laser current and the wavelength compared with the up scan. The resonance shifted to right, but the profile of the spectrum remained unchanged. We tracked the peak at around 1314.5748 nm in the spectrum data at 0.02 s. Fig. 4 shows a plot of the raw wavelength shift as a function of time since data collection started, together with a plot of the temperature. The temperature-compensated shift calculated from the raw shift and the temperature change is also shown. The temperature rose slowly, which is due to the greenhouse effect. We minimized the temperature rise by attaching aluminum foil onto the outer cover. Otherwise, the temperature would have risen a lot more. The temperature-compensated shift shows that a red shift continued for over 90 min, but the shift slowed down with time. The red shift indicates that the product of the RI and radius of the resonator continued
3.6. Passive temperature control Two covers were used to protect the precarious resonator–taper coupling from air flow disturbances and thermally insulate them. The smaller one made of polycarbonate sat above the microscope slide that had the two tapers glued on it. The other one was larger and contained the smaller cover, the slide stage, and the 3D translation stage. To minimize greenhouse effect, the larger cover was coated with aluminum foil. 3.7. Isothermal volume relaxation experiment One of the two experiments performed on each of the resonators monitored volume relaxation at room temperature. The resonator was rushed to coupling with the tapers immediately after taken out of the heating source. Collection of the spectrum data and temperature measurement started within 30 s. The two covers were put into place. The data were collected continuously for the first ~45 min. After a break of ~ 20 min, the data collection resumed and continued for ~2 min. The break and data collection were repeated once more. Throughout the experiment, the experimental setup, including the resonator–taper coupling and the covers, was held unchanged. Overall, this experiment monitored the resonance wavelengths and the temperature for ~ 90 min. The temperature rise during this period was < 0.5 K, although it was not actively regulated. We call this experiment an “isothermal volume relaxation experiment”. 3.8. Cooling experiment The other experiment was performed the next day or later to ensure the rapid volume change was over. In the second experiment, the resonator–taper pair, with the smaller cover, was heated with a 45 W halogen lamp for ~2 min. The temperature rose by ~2 K. Subsequently, the lamp was turned off, the resonance spectrum and temperature data were collected for ~ 20 min. We call this experiment a “cooling experiment”. 3.9. Resonance shift analysis The method of spectrum data analysis will be explained at the beginning of the Results and Discussion section for one of the experiments. Here, a method to subtract the resonance shift due to a
Fig. 3. Overlay of WGM resonance spectra at different moments in an isothermal volume relaxation experiment. The diameter of the resonator was 450 μm. The peak in the shade was tracked for estimating the resonance wavelength shift.
55
Journal of Non-Crystalline Solids 476 (2017) 52–59
L. Cai et al.
the size of the markers. We are reminded that (∂ ln λ/∂ln ρ)T = − 0.105 in silica, and therefore (∂ ln λ/∂ln V)T = 0.105. The red shift of the resonance indicates a volume increase (Fig. S1(a) and (b) in Supplemental materials show the plots). This result can be explained by using a diagram that describes the relationship between the temperature and volume in types I, II, and IV silica [18]. Fig. 7 was prepared mostly from Brückner's review article [13]. The solid curve indicates the equilibrium and has a minimum at around 1400–1500 °C, and the presence of the minimum is characteristic of silica. In contrast, other silicate glasses show a curve of simply increasing volume with temperature. When molten silica is quenched, it will follow the track indicated as “Quenching” to decrease its volume rapidly. When left at room temperature, it will relax to increase its volume, but the increase is limited by the mobility of the constituents at such a low temperature. Our spherical resonators were heated to well above 1500 °C, especially with the arc. It is known that optical fibers harbor residual strain as a result of quenching while being drawn. Heating silica to a sufficiently high temperature releases the strain, and when cooled or quenched, the volume-temperature relationship will follow the solid line in Fig. 7. We observed a volume increase in the isothermal experiment, which is explained as the volume relaxation of quenched silica at room temperature. In Fig. 7, the dashed line represents the volume–temperature relationship for silica under residual strain. The latter leads to a larger volume that is frozen in below the glass transition temperature [16]. The result in Fig. 6(a), showing a larger volume increase for smaller resonators, can be explained as a volume relaxation toward a greater volume, caused by the unmolten part of the fiber. The latter would leave the part near the surface under strain, although not as much as the fiber that we started with. The difference between the arc-formed resonators and those made by the torch can be also ascribed to the residual stress: The torch does not heat the fiber as much as the arc does. In Fig. 4, for example, (Δ λ/λ)total is ~2.3 ppm. This value leads us to estimate the volume increase as 21.9 ppm, that is an increase of radius by ~7.3 ppm (~ 1.64 nm). It is extremely small, and no other methods would detect this level of radius change. The RI decrease is ~5 ppm; again, it would be difficult to detect this small change with other methods. The noise in Fig. 4 is ~0.1 ppm. It means the resolution of radius change is ~0.071 nm or ~0.32 ppm.
Fig. 4. Isothermal experiment with a resonator of diameter 450 μm. Temperature monitored by a thermistor near the resonator, raw resonance shift (Δλ/λ)raw and resonance shift compensated for the temperature change, (Δλ/λ)T-comp, are plotted as a function of time since the measurement started. The plot of (Δλ/λ)raw was moved upward by 1 ppm for clarity.
to increase. These characteristics were observed in all of the resonators we tested. To estimate the total shift and the time constant of the shift from the data collected for a finite period of time, we did a curve fit with an empirical equation, (Δ λ/λ)T-comp = b0/(1 + b1 / t), where b0 and b1 are fitting parameters. The b0 ≡ (Δ λ/λ)total stands for the total shift at t = +∞, and the shift reaches a half of the total shift at t = b1 ≡ t1/2. Fig. 5(a) shows (Δ λ/λ)T-comp and the curve fit for three resonators of different diameters D fabricated with arc; Fig. 5(b) for three resonators fabricated with the hydrogen torch. The greater the diameter, the smaller the total shift for either series of resonators. The curve fit by the simple algebraic equation is satisfactory with R2 typically > 0.98. At first, we used a different equation, b0 / [1 + (b1 / t)γ], for the curve fit, but the value of γ was close to one for all the data. Therefore, we decided to set γ to one. The curve fit by (Δ λ/λ)total / (1 + t1/2 / t) was enforced on all of the results of (Δ λ/λ)T-comp. Fig. 6(a) shows a plot of the total shift as a function of the resonator diameter for the arc-formed resonators and hydrogen torch-formed resonators. Fig. 6(b) shows the half time t1/2. For both methods of resonator fabrication, the total shift decreases with an increasing resonator diameter. The longer the exposure time to the heat source, the smaller the shift. In Fig. 6(b), the half time is between 8 and 27 min, but the trend is not conclusive. Standard errors are within
4.2. Cooling experiment Fig. 8(a) shows a plot of the temperature and the resonance wavelength as a function of time since data collection started in one of the cooling experiments. The resonator of diameter 322 μm, made by the electric arc, was used here. In the cooling, the resonance shifted to a shorter wavelength. Fig. 8(b) shows a plot of the resonance wavelength as a function of temperature. The data lie along a straight line. Optimal fit by a linear relationship gives a slope of 0.008858 nm/K with R2 = 0.99984, which translates into Δ λ/λ = 6.74 ppm/K with a Fig. 5. Temperature-compensated resonance shift is plotted as a function of time for resonators of different diameters D, prepared by exposing a silica fiber to (a) electric arc and (b) hydrogen torch. Optimal curve fits by (Δλ/λ)T-comp = b0 / (1 + b1 / t), where b0 and b1 are fitting parameters, are also shown.
56
Journal of Non-Crystalline Solids 476 (2017) 52–59
L. Cai et al.
Fig. 6. (a) Total shift (Δλ/λ)total and (b) half time t1/2 estimated by curve fitting the plot of (Δλ/λ)T-comp vs time t by (Δλ/λ)total/(1 + t1/2/t) for resonators of different diameters D made by electric arc and hydrogen torch.
Table 2 Temperature coefficients for the resonators made by different heating method with different diameters. D/μm
Heating method
Temperature coefficient/(ppm/K)
322 490 435 423 376 391
Electric arc Electric arc Electric arc Electric arc Hydrogen torch Hydrogen torch
6.74 6.70 6.64 6.79 6.77 6.81
Fig. 7. Volume-temperature diagram of silica (types I, II, and IV).
standard error of 0.00061. Results for all the resonators studied in the cooling experiment are listed in Table 2. All of them give similar values. The mean value and the standard deviation of these six results are 6.74 ppm/K and 0.063 ppm/K, respectively. It appears that, in one day or longer, the resonators finished their volume relaxation, which gave essentially identical temperature coefficients for all the resonators tested. We did not conduct cooling experiments for the other resonators. From the T coefficient of 6.74 ± 0.06 ppm/K, we can use Eqs. (15) and (16) to estimate (∂ n/∂ T)p as 9.0 × 10− 6 ± 0.1 K− 1. This value is similar to 8.75 × 10− 6 K− 1, evaluated for another type of silica made by vapor-phase axial deposition (VAD) method [23], and 8.66 × 10− 6 K− 1, obtained for standard reference material (SRM) glass #739 [24], but is less than the values of type III silica that are > 10 × 10− 6 K− 1 [25,26]. We note here that (∂n/∂T)p depends on the wavelength and temperature, and varies for different types of silica. Using our value of (∂n/∂T)p, we can use Eq. (17) to estimate (1/α)(∂ α/ ∂ T)p as 19.0 ± 0.2 ppm/K, and with Eq. (18), (1/α)(∂ α/∂ T)ρ as 18.4 ± 0.2 ppm/K. It is obvious that the noise in cooling experiment in Fig. 5(a) is much
Fig. 9. WGM spectrum for a resonator of diameter 563 μm formed using electric arc. Eight scans of wavelength, demarcated by dashed lines, are combined into one with a wavelength scan range of 78.12 pm.
lower compared with isothermal experiment in Fig. 8(a). We roughly estimated the magnitudes of noise in the two figures as 0.1 ppm and 0.04 ppm, respectively. We saw a similar difference in all the experiments we did. We may ascribe the extra noises we saw in the isothermal experiment to movement of SiO2 during the volume relaxation: For the volume to change, the silica network must undergo internal rearrangement. The latter caused density fluctuations and lead to noises Fig. 8. (a) Temperature change and resonance shift vs. time and (b) temperature coefficient fitting in the cooling experiment.
57
Journal of Non-Crystalline Solids 476 (2017) 52–59
L. Cai et al.
Fig. 10. Q factors of isolated peaks for resonators of different diameters made by (a) electric arc and (b) hydrogen torch.
small the volume change. When the volume relaxation settled, the temperature coefficient of the silica resonator was reproducible and nearly identical for all the resonators tested. For spheres, the Q values of the resonators made by the hydrogen torch were generally lower than those made by the electric arc. For a cylinder with a minimal exposure to the heat source, it was the other way around.
in the time trace of resonance wavelength. When the relaxation had finished, the fluctuations disappeared, and the noise in the time trace was mostly due to system noise and ambient noise. 4.3. Q factor To investigate the effect of the resonator diameter and heating method on the Q factor of resonance, we collected eight spectra in different scan ranges, each 1.25 mA wide, and then combined them to one spectrum. Fig. 9 shows an example for the resonator of diameter 563 μm. Another example of combined spectrum is shown given in Fig. S2 of the Supplemental materials. We obtained a combined spectrum for each of five resonators made by the arc and five resonators made by the torch. A peak is either an isolated peak or a compound peak consisting of two or more nearby peaks. For each of the ten spectra, some isolated and symmetric peaks were selected to fit the spectrum with a Lorentzian [27] and evaluate the Q value. We plot the Q values for all the peaks analyzed in Fig. 10. For each resonator, there is a distribution in the Q value, as the coupling with the two tapers is different for each mode, and also the intrinsic Q factor is different. Still, we can see a difference in the distribution. In part (a) that shows the Q values for arcformed resonators, Q is small when the resonator is close to cylindrical, and increases as the resonator becomes more spherical. In part (b) that shows the Q values for torch-formed resonators, the distribution is more or less identical for different diameters. These results are reasonable, since heating smooths the resonator's surface and homogenizes the silica. The smoothing is more effective with a higher temperature provided by the arc than it is by the torch. Another possible cause for the lower Q in the torch-prepared resonators is that the hydrogen flame may have introduced OH into silica, leading to loss by absorption. It is interesting to see that the 300-μm, arc-formed resonator has smaller Q values compared with the 300-μm, torch-formed resonator. We ascribe this difference to the extremely short time of exposure to the heat source for the arc, resulting in a non-uniform heating.
Acknowledgement The study presented here was conducted while L. Cai stayed with I. Teraoka at NYU, made possible through China Scholarship Council. L. Cai thanks N. Luo for helping her with experiments. Appendix A. Supplementary data Supplementary data to this article can be found online at http://dx. doi.org/10.1016/j.jnoncrysol.2017.09.022. References [1] F. Vollmer, S. Arnold, Whispering-gallery-mode biosensing: label-free detection down to single molecules, Nat. Methods 5 (2008) 591–596. [2] Vladimir S. Ilchenko, Andrey B. Matsko, Optical resonators with whispering-gallery modes-part II: applications, IEEE J. Sel. Top. Quantum Electron. 12 (2006) 15–32. [3] M.R. Foreman, J.D. Swaim, F. Vollmer, Whispering gallery mode sensors, Adv. Opt. Photon. 7 (2015) 168–240. [4] Martin D. Baaske, Matthew R. Foreman, Frank Vollmer, Single-molecule nucleic acid interactions monitored on a label-free microcavity biosensor platform, Nat. Nanotechnol. 9 (2014) 933–939. [5] S.I. Shopova, R. Rajmangal, S. Holler, S. Arnold, Plasmonic enhancement of a whispering-gallery-mode biosensor for single nanoparticle detection, Appl. Phys. Lett. 98 (2011) 243104. [6] M.A. Santiago-Cordoba, S.V. Boriskina, F. Vollmer, M.C. Demirel, Nanoparticlebased protein detection by optical shift of a resonant microcavity, Appl. Phys. Lett. 99 (2011) 073701. [7] M.A. Santiago-Cordoba, M. Cetinkaya, S.V. Boriskina, F. Vollmer, M.C. Demirel, Ultrasensitive detection of a protein by optical trapping in a photonic-plasmonic microcavity, J. Biophotonics 5 (2012) 629–638. [8] A. Bianchetti, A. Federico, S. Vincent, S. Subramanian, F. Vollmer, Refractometrybased air pressure sensing using glass microspheres as high-Q whispering-gallery mode microresonators, Opt. Commun. 394 (2017) 152–156. [9] C.H. Dong, L. He, Y.F. Xiao, V.R. Gaddam, S.K. Ozdemir, Z.F. Han, G.C. Guo, L. Yang, Fabrication of high-Q polydimethylsiloxane optical microspheres for thermal sensing, Appl. Phys. Lett. 94 (2009) 231119. [10] U.K. Ayaz, T. Ioppolo, M.V. Ötügen, Wall shear stress sensor based on the optical resonances of dielectric microspheres, measurement science and technology, Meas. Sci. Technol. 22 (2011) 075203. [11] Qiulin Ma, Tobias Rossmann, Zhixiong Guo, Temperature sensitivity of silica microresonators, J. Phys. D. Appl. Phys. 41 (2008) 245111. [12] A.D. Yablon, Optical and mechanical effects of frozen-in stresses and strains in optical fibers, IEEE J. Sel. Top. Quantum Electron. 10 (2004) 300–311. [13] R. Brückner, Properties and structure of vitreous silica. I, J. Non-Cryst. Solids 5 (1970) 123–175. [14] B.H. Kim, Y. Park, T.J. Ahn, D.Y. Kim, B.H. Lee, Y. Chung, U.C. Paek, W.T. Han, Residual stress relaxation in the core of optical fiber by CO2 laser irradiation, Opt.
5. Conclusion We observed the volume relaxation process in quenched type IV silica via isothermal experiment of a WGM resonator. We utilized the high sensitivity of the resonance wavelength shift to the material changes in the silica resonator to observe ppm-level volume changes. A 1 ppm red shift of the resonance wavelength amounts to an 9.52 ppm increase in volume, and a 2.17 ppm decrease in RI. For the resonators with the same diameter, the one made by hydrogen torch exhibited a larger expansion than did the resonator made by the electric arc. For both methods, the longer the exposure time to the heating source, the 58
Journal of Non-Crystalline Solids 476 (2017) 52–59
L. Cai et al. Lett. 26 (2001) 1657–1659. [15] D.R. Perchak, J.M. O'Reilly, Observation of a volume minimum in computer simulations of supercooled, amorphous silica, J. Non-Cryst. Solids 167 (1994) 211–228. [16] A.D. Yablon, M.F. Yan, D.J. DiGiovanni, M.E. Lines, S.L. Jones, D.N. Ridgway, G.A. Sandels, I.A. White, P. Wisk, F.V. DiMarcello, E.M. Monberg, J. Jasapara, Frozen-in viscoelasticity for novel beam expanders and high-power connectors, J. Lightwave Technol. 22 (2004) 16–23. [17] H.N. Luo, H.S. Kim, M. Agarwal, I. Teraoka, Light turn-on transient of a whispering gallery mode resonance spectrum in different gas atmospheres, Appl. Opt. 52 (2013) 2834–2840. [18] R. Brückner, Metastable equilibrium density of hydroxyl-free synthetic vitreous silica, J. Non-Cryst. Solids 5 (1971) 281–285. [19] Corning® HPFS® 7979, 7980, 8655, Fused Silica. Corning, http://www.corning. com/media/worldwide/csm/documents/HPFS%20Product%20Brochure%20All %20Grades%202015_07_21.pdf. [20] HPFS® 8655, Fused Silica, Corning, http://www.corning.com/media/worldwide/
[21] [22] [23] [24] [25] [26] [27]
59
csm/documents/HPFS%208655%20Fused%20Silica%20Brochure%202013_08_15. pdf. K. Vedam, E.D.D. Schmidt, R. Roy, Nonlinear variation of refractive index of vitreous silica with pressure to 7 kbars, J. Am. Ceram. Soc. 49 (1966) 531–535. M. Agarwal, I. Teraoka, Whispering gallery mode dip sensor for aqueous sensing, Anal. Chem. 87 (2015) 10600–10604. A. Koike, N. Sugimoto, Temperature Dependences of Optical Path Length in Inorganic Glasses, vol. 56, Reports Res. Lab. Asahi Glass Co., Ltd, 2006, pp. 1–6. G. Ghosh, Model for the thermo-optic coefficients of some standard optical glasses, J. Non-Cryst. Solids 189 (1995) 191–196. T. Toyoda, M. Yabe, The temperature dependence of the refractive indices of fused silica and crystal quartz, J. Phys. D. Appl. Phys. 16 (1983) L97–L100. I.H. Malitson, Interspecimen comparison of the refractive index of fused silica, J. Opt. Soc. Am. 55 (1965) 1205–1209. I. Teraoka, H. Yao, N.H. Luo, Relationship between height and width of resonance peaks in a whispering gallery mode resonator immersed in water and sucrose solutions, Opt. Commun. 393 (2017) 244–251.
Contents lists available at ScienceDirect
Journal of Non-Crystalline Solids journal homepage: www.elsevier.com/locate/jnoncrysol
Volume relaxation of quenched silica at room temperature monitored by whispering gallery mode resonance wavelength
MARK
Lu Caib,1, Iwao Teraokaa,⁎, Yong Zhaob a b
Department of Chemical and Biomolecular Engineering, New York University, 6 MetroTech Center, Brooklyn, NY 11201, United States College of Information Science and Engineering, Northeastern University, Shenyang 110819, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Silica Quenching Volume relaxation Whispering gallery mode Frozen-in strain
We used a resonance wavelength shift of a whispering gallery mode to monitor volume relaxation at room temperature in cylindrical and spherical resonators fabricated by quenching the molten tip of a pure silica optical fiber. The high sensitivity of the method allowed measurement of the linear dimension at sub-ppm resolution. All the resonators registered a volume increase in the first few hours, in agreement with the volumetemperature equilibrium curve of type IV silica below the glass transition temperature. When heating of the fiber was insufficient, the volume increase was greater, ascribed to a larger volume of strained silica. We also formulated thermodynamics to estimate the temperature coefficients of polarizability at constant pressure and at constant density from resonance shift data obtained in experiments performed on relaxed silica resonators.
1. Introduction In a transparent dielectric medium of a circular cross section, light can travel near the surface via total internal reflection to complete a cycle [1]. This mode of wave propagation is called a whispering gallery mode (WGM). If the cycle contains an integral number of waves, the light superimposes constructively onto itself, and repeating this every cycle leads to a large amplitude. Since the intensity is proportional to the square of the amplitude, a strong wave can develop. The latter requires a strict condition for the wavelength, resulting in an extremely narrow resonance. In air, the Q factor can exceed 108. Different applications of WGM resonance have been proposed [2]. Among others, applications to sensing have been most intensively pursued, [3–7] as the resonance wavelength is sensitive to a small, localized refractive index (RI) change near the resonator surface in the surrounding medium. Since most of the wave travels within the resonator, the resonance wavelength is sensitive to a change of the resonator itself. Suppose the radius of a spherical resonator increases without changing its RI. Then, detecting a radius change of 0.1 ppm is not difficult with WGM. In practice, however, the radius change and the RI change occur at the same time, and it is often the case that they cancel each other to some extent. Nevertheless, the resultant sensitivity is among the highest, and WGM has been applied to detecting pressure, temperature, and stress changes [8–11].
⁎
1
The present report focuses on monitoring the changes in a silica resonator after it is fabricated by heating the tip of a pure silica optical fiber and then quenched. Silica has been extensively studied, [12–16] and its thermo-mechanical properties are well understood. The properties depend on the type of silica, distinguished by the starting material and processing [13]. There are four types of silica. Their fabrication methods and concentrations of OH are summarized in Table 1. The silica core of the fiber used in our experiment belongs to type IV, containing OH at about 0.2 ppm according to the supplier. Since silica is a preferred material for WGM resonators, their applications require that we have a good understanding of silica's properties and how the small variations in the starting material and processing affect the properties. Fortunately, monitoring the resonance wavelength shift can be a right tool to address this concern. Typically, a silica resonator is fabricated by exposing the tip of an optical fiber to a localized heat source to melt it. The surface tension results in a spherical resonator. When the heat source is turned off, the cooling takes just a second [17]. It means that the silica resonator is quenched. Therefore, the resonator is not at thermal equilibrium at least immediately after it is fabricated. Since the mobility of supercooled silica is low, it may take time to approach the equilibrium. This study finds how long it will take to get close to the equilibrium to the extent to stabilize the resonance wavelength. In the present report, we first develop a thermodynamic theory to express the resonator's RI and the resonance wavelength as a function of
Corresponding author. E-mail address: [email protected] (I. Teraoka). On leave.
http://dx.doi.org/10.1016/j.jnoncrysol.2017.09.022 Received 29 May 2017; Received in revised form 11 August 2017; Accepted 5 September 2017 Available online 12 September 2017 0022-3093/ © 2017 Elsevier B.V. All rights reserved.
Journal of Non-Crystalline Solids 476 (2017) 52–59
L. Cai et al.
With Eq. (4), Eq. (6) is rewritten to
Table 1 Fabrication methods and concentrations of OH for four types of silica. Silica type
Method of preparation
OH concentration/ppm
I
Electrical fusion of natural quartz in vacuum or inert gas Melting quartz crystal with oxy‑hydrogen flame Hydrolyzing SiCl4 when sprayed into oxy‑hydrogen flame Hydrolyzing SiCl4 with water-free plasma flame
Several
II III IV
6n∆n 1 ⎛ ∂α ⎞ ⎛ ∂ln a ⎞ ρ ⎛ ∂α ⎞ ⎤ =⎡ ⎢ α ⎝ ∂T ⎠ + 3 ⎝ ∂Τ ⎠ α ⎜ ∂ρ ⎟ ⎥ ΔT (n2 − 1)(n2 + 2) p p ⎝ ⎠T ⎦ ⎣ ρ ⎛ ∂α ⎞ ⎤ Δρ +⎡ ⎢1 + α ⎜ ∂ρ ⎟ ⎥ ρ ⎝ ⎠T ⎦ ⎣
150–400
In Eq. (7), the RI change by pressure at a constant temperature is expressed as
~ 1000 <1
6n ⎛ ∂n ⎞⎟ = 1 ⎡1 + ρ ⎛⎜ ∂α ⎞⎟ ⎤ ⎜ (n2 − 1)(n2 + 2) ⎝ ∂p ⎠T K⎢ α ⎝ ∂ρ ⎠T ⎥ ⎦ ⎣
temperature and density. In the theory, we will find a molecular-level expression of the resonance shift that will occur in volume relaxation post quenching and in a temperature change at around room temperature. We conducted these two lines of experiments on each of resonators of different diameters prepared by two different heating methods. After describing the experimental details, we will show the results and interpret them in the light of the theory.
∂p ∂p ρ ⎛⎜ ⎞⎟ = −V ⎛ ⎞ = K ⎝ ∂V ⎠T ⎝ ∂ρ ⎠T
Δλ Δn Δa Δn 1 ΔV Δn 1 Δρ = + = + = − λ n a n 3 V n 3 ρ
(10)
With Eq. (7),
Δλ (n2 − 1)(n2 + 2) ⎡ 1 ⎛ ∂α ⎞ ⎛ ∂ln a ⎞ ρ ⎛ ∂α ⎞ ⎤ = ⎢ α ⎝ ∂T ⎠ + 3 ⎝ ∂Τ ⎠ α ⎜ ∂ρ ⎟ ⎥ ΔT 6n2 λ p p ⎝ ⎠T ⎦ ⎣ +
ρ ⎛ ∂α ⎞ ⎤ 1 ⎫ Δρ ⎧ (n2 − 1)(n2 + 2) ⎡ ⎢1 + α ⎜ ∂ρ ⎟ ⎥ − 3 ⎬ ρ ⎨ 6n2 ⎝ ⎠ T ⎣ ⎦ ⎩ ⎭
(11)
For a density change at constant temperature,
∂ln λ ⎞ Δρ Δλ = ⎜⎛ ⎟ λ ⎝ ∂ln ρ ⎠T ρ
(12)
where (1) 2 2 ⎛ ∂ln λ ⎟⎞ = (n − 1)(n + 2) ⎡1 + ρ ⎜⎛ ∂α ⎟⎞ ⎤ − 1 ⎜ 2 ⎢ 6n α ⎝ ∂ρ ⎠T ⎥ 3 ⎝ ∂ln ρ ⎠T ⎣ ⎦
At constant pressure,
⎛ ∂α ⎞ = ⎛ ∂α ⎞ + ⎜⎛ ∂α ⎟⎞ ⎛ ∂ρ ⎞ ⎝ ∂Τ ⎠ p ⎝ ∂T ⎠ ρ ⎝ ∂ρ ⎠T ⎝ ∂Τ ⎠ p
1 ⎛ ∂ρ ⎞ 1 ∂V ∂ln a ⎞ = − ⎛ ⎞ = −3 ⎛ ρ ⎝ ∂Τ ⎠ p V ⎝ ∂Τ ⎠ p ⎝ ∂Τ ⎠ p
(3)
where (∂ ln a/∂ T)p represents the linear thermal expansion coefficient. For types I, II and IV silica, the linear thermal expansion coefficients are similar [18] and they are typically 0.52 ppm/K [19]. Therefore, Eq. (1) is now
⎛ ∂α ⎞ ⎛ ∂α ⎞ ⎛ ∂ln a ⎞ ⎛ ∂α ⎞ ⎤ Δα = ⎡ ⎢ ⎝ ∂T ⎠ + 3ρ ⎝ ∂Τ ⎠ ⎜ ∂ρ ⎟ ⎥ ΔT + ⎜ ∂ρ ⎟ Δρ p p ⎝ ⎠T ⎦ ⎝ ⎠T ⎣
(13)
With n = 1.447 and 1+ (ρ/α)(∂ α/∂ ρ)T = 0.64, we obtain (∂ ln λ/ ∂ln ρ)T = − 0.105. This coefficient indicates that a density decrease of 1 ppm causes a 0.105 ppm red shift of the resonance wavelength. Alternatively, 1 ppm red shift of resonance wavelength amounts to a 9.51 ppm decrease in density, and a 9.51 ppm increase in volume. Different experiments can measure Δ n due to a volume change at a constant temperature. In Eq. (7), setting ΔT to zero leads to
(2)
Since ρ ~ 1/V,
ρ ⎛ ∂α ⎞ ⎤ ΔV ∆n (n2 − 1)(n2 + 2) ⎡ =− ⎢1 + α ⎜ ∂ρ ⎟ ⎥ V n 6n2 ⎝ ⎠T ⎦ ⎣
(14)
With (ρ/α)(∂ α/∂ ρ)T = − 0.36, Δ n/n = − 0.23 × Δ V/V. A 1 ppm red shift of resonance wavelength amounts to a 2.17 ppm decrease in RI. This coefficient compares favorably with the one in Δ n/ n = −(n2p1/2)(Δ V/V), [16] where p1 ≊ 0.22 is the orientationally averaged elasto-optic coefficient. For an experiment in which the temperature changes at a constant pressure, an expression of the WGM resonance shift can be derived from Eq. (10) as
(4)
Lorentz-Lorenz equation relates the relative permittivity n to ρα as 2
(5)
For small changes,
Δρ 6n∆n Δα = + (n2 − 1)(n2 + 2) ρ α
(9)
For typical silica, n = 1.447 (at wavelength 1.3 μm), KT = 35.9 GPa, [20] and (∂ n/∂ p)T = 9.2 × 10− 12 Pa− 1 [21]. With these numbers, we can estimate, from Eq. (8), (ρ/α)(∂ α/∂ ρ)T as −0.36. The negative coefficient indicates that silica decreases its polarizability (per unit SiO2) with an increasing density. Now, we consider the WGM resonance shift:
We employ here thermodynamic analysis to express the resonance wavelength shift in terms of a change in the state of the glass that constitutes the resonator. To describe the state of glass including nonequilibrium states, we choose the temperature T and the density ρ as two independent variables, rather than T and pressure p. This provision allows describing the state when the glass relaxes toward the equilibrium to change its volume V at constant temperature and pressure. Since the resonance wavelength in air is proportional to the product of the refractive index (RI) n and radius a of the spherical resonator, we will first look at how n and a change with T and ρ, assuming that the changes are isotropic. Since most thermodynamic quantities are measured at constant T and p, we need to find formulas to convert these quantities into those at a given temperature and density. We model the glass as consisting of independent molecules with polarizability α at density ρ. When the temperature changes by ΔT and the density by Δρ, the polarizability changes according to
∂α ∂α Δα = ⎛ ⎞ ΔT + ⎜⎛ ⎟⎞ Δρ ⎝ ∂Τ ⎠ ρ ⎝ ∂ρ ⎠T
(8)
where K is the bulk modulus (reciprocal of the isothermal compressibility):
2. Theory of WGM resonance shift in air
n2 − 1 4π = ρα n2 + 2 3
(7)
Δλ ∂ln λ ⎞ =⎛ ΔT λ ⎝ ∂Τ ⎠ p
(6) 53
(15)
Journal of Non-Crystalline Solids 476 (2017) 52–59
L. Cai et al.
⎛ ∂ln λ ⎞ = 1 ⎛ ∂n ⎞ + ∂ln a n ⎝ ∂T ⎠ p ∂Τ ⎝ ∂Τ ⎠ p
The diameter of the resonator was estimated by analyzing the image. In each image analysis, the stem diameter of 300 μm was used as a reference. We also made resonators using FTx00UMT (x = 2, 3, 4; high OH; ThorLabs), but the peaks were about three times as broad as those made from FT300EMT. Therefore, these high-OH fibers were not actively employed in the study reported here.
(16)
The thermo-optic coefficient (∂n/∂T)p varies, depending on the type of silica. For type IV silica, it is 9.49 × 10− 6 K− 1 at 1.314 μm [20]. From Eqs. (3) and (6), we find that (∂n/∂T)p is related to (∂ α/∂T)p by
1 ⎛ ∂n ⎞ (n2 − 1)(n2 + 2) ⎡ 1 ⎛ ∂α ⎞ ∂ln a ⎤ = ⎢ α ⎝ ∂T ⎠ − 3 ∂Τ ⎥ n ⎝ ∂T ⎠ p 6n2 p ⎣ ⎦
3.2. Fabrication of tapers
(17)
We prepared a pair of pencil-shaped fiber tapers by etching coatingstripped single-mode fibers with a hydrofluoric acid solution. The two tapers were then aligned along a straight line with their two tips facing each other at a gap distance of ~210 μm. Subsequently, the coated portions of the fibers were glued onto a microscope slide. To monitor the temperature, a 0.8 mm glass-encapsulated 10 K thermistor (Analog Technologies, ATH10KR8T63S) was glued onto the slide near the gap of the two tapers.
where
∂ln a ρ ⎛ ∂α ⎞ 1 ⎛ ∂α ⎞ 1 ∂α = ⎛ ⎞ +3 ⎜ ⎟ ∂Τ α ⎝ ∂ρ ⎠T α ⎝ ∂T ⎠ ρ α ⎝ ∂T ⎠ p
(18)
from Eq. (2). With Eqs. (17) and (18), we estimate (1/α)(∂ α/∂ T)p and (1/α)(∂ α/∂ T)ρ as 19.96 and 19.40 ppm/K, respectively. 3. Experimental 3.1. Fabrication of a resonator
3.3. Coupling of the resonator to the tapers
We used a multimode fiber FT300EMT (low OH of 0.2 ppm, ThorLabs) to fabricate resonators. This fiber has a pure silica core and a thin TECS cladding, and is coated with Tefzel. The fiber was cut to an 8cm long section. The Tefzel coating was removed with a stripper for 3 cm from one of the ends. To remove the hard fluoropolymer cladding, the tip of the fiber was soaked in acetone, then wiped with a sheet of KimWipe. The tip of the bare silica fiber was then exposed to a local heating source that is either electric arc across a pair of tungsten electrodes or an oxy‑hydrogen torch. When the hydrogen torch was used to make a resonator of 423 μm, for example, the tip was exposed to the flame for as long as ~1 min. We continued to rotate the fiber around its axis to minimize bending. When the arc was used for the same size of resonator, the tip was exposed to two shots, each 2 s long. Thus, a resonator was formed at the tip of the optical fiber. The Tefzelcoated part was a stem to hold and position the resonator. The stem was mounted on a 3D translation stage. A total seven resonators of different diameters, ranging from 322 to 490 μm, were prepared using the electric arc by changing the exposure time. The longer the exposure time, the larger the diameter. With the hydrogen torch, six resonators of diameters from 350 to 527 μm were prepared. Fig. 1 shows micrographs taken with a stereo zoom microscope for some of these resonators. Those in part (a) were made by the electric arc; those in part (b) by the hydrogen torch. It is obvious that the electric arc formed resonators closer to a sphere than the hydrogen torch did.
The position of the resonator relative to the two tapers was adjusted by manipulating the 3D positioner. Rough positioning was done while observing the tapers and the resonator using a camera equipped with a microscope objective placed under the slide. Fine tuning for optimal positioning was accomplished while observing the resonance spectrum using the measurement system described below. 3.4. WGM measurement system A schematic diagram of our measurement system is shown in Fig. 2. The light from a distributed feedback (DFB) laser (FRL13DDR8-A31, FITEL) passes an isolator and leads to one of the tapered fibers that touch the resonator. The other tapered fiber picks up the WGM and leads to the photodetector. The current to the DFB laser is changed in a triangular wave at 24.40 Hz from 35 to 37.5 mA to scan the wavelength from 1314.5664 to 1314.5860 nm, unless otherwise specified. Each of the up and down current scans of the triangular wave consists of 1025 points of data with one being shared. The photodetector signal is digitized synchronously. We use the laser wavelength calibration of our earlier work that gives the wavelength as a function of the laser current [22]. A part of the light from the feed fiber transfers to the resonator via evanescent coupling. When the wavelength matches one of the resonance wavelengths, a strong mode is generated and circulates within the resonator. The 1025 data of the photodetector signal in each scan is a WGM spectrum. Fig. 1. Micrographs of resonators made by (a) electric arc and (b) hydrogen torch. The diameter at the equator is shown for each of the six resonators displayed here. The stem diameter is 300 μm.
54
Journal of Non-Crystalline Solids 476 (2017) 52–59
L. Cai et al.
temperature change in the isothermal volume relaxation experiment will be explained. Analysis of all the scans of resonance spectrum collected allows us to prepare a plot of a raw resonance shift (Δ λ/λ)raw with reference to the mean resonance wavelength in a reference period that is typically the first 1 s of the experiment. A temperature change Δ T is calculated with respect to the same reference period. Then, we calculate a temperature-compensated shift (Δ λ/λ)T-comp according to
Δλ ∂ln λ ⎛ Δλ ⎞ =⎛ ⎞ − ∆T ∂T ⎝ λ ⎠T − comp ⎝ λ ⎠raw
(19)
where ∂ lnλ/∂ T is the temperature coefficient of the resonance wavelength. The latter was evaluated in the cooling experiment, and we used that coefficient in converting the raw resonance shift into the temperature-compensated shift in the isothermal volume relaxation experiment.
Fig. 2. Schematic diagram of the measurement system. A zoomed view is shown for the resonator on a stem coupled to a pair of tapered fibers.
4. Results and discussion 3.5. Temperature monitoring 4.1. Isothermal volume relaxation experiment The thermistor mounted on the microscope slide was connected to a bridge circuit to convert the resistance to a voltage. The latter was digitized by a data logger at 4 Hz.
We use an example of isothermal experiment conducted for a resonator formed into a spherical shape with 450 μm diameter using arc to explain our method of spectrum data analysis. Fig. 3 shows some of the down-scan spectra collected in the first 2800 s continuous spectrum collection for the wavelength scan range of 19.53 pm. For this period, a total 68,095 spectra were collected. Only the down scan spectra were analyzed because of a better linearity between the laser current and the wavelength compared with the up scan. The resonance shifted to right, but the profile of the spectrum remained unchanged. We tracked the peak at around 1314.5748 nm in the spectrum data at 0.02 s. Fig. 4 shows a plot of the raw wavelength shift as a function of time since data collection started, together with a plot of the temperature. The temperature-compensated shift calculated from the raw shift and the temperature change is also shown. The temperature rose slowly, which is due to the greenhouse effect. We minimized the temperature rise by attaching aluminum foil onto the outer cover. Otherwise, the temperature would have risen a lot more. The temperature-compensated shift shows that a red shift continued for over 90 min, but the shift slowed down with time. The red shift indicates that the product of the RI and radius of the resonator continued
3.6. Passive temperature control Two covers were used to protect the precarious resonator–taper coupling from air flow disturbances and thermally insulate them. The smaller one made of polycarbonate sat above the microscope slide that had the two tapers glued on it. The other one was larger and contained the smaller cover, the slide stage, and the 3D translation stage. To minimize greenhouse effect, the larger cover was coated with aluminum foil. 3.7. Isothermal volume relaxation experiment One of the two experiments performed on each of the resonators monitored volume relaxation at room temperature. The resonator was rushed to coupling with the tapers immediately after taken out of the heating source. Collection of the spectrum data and temperature measurement started within 30 s. The two covers were put into place. The data were collected continuously for the first ~45 min. After a break of ~ 20 min, the data collection resumed and continued for ~2 min. The break and data collection were repeated once more. Throughout the experiment, the experimental setup, including the resonator–taper coupling and the covers, was held unchanged. Overall, this experiment monitored the resonance wavelengths and the temperature for ~ 90 min. The temperature rise during this period was < 0.5 K, although it was not actively regulated. We call this experiment an “isothermal volume relaxation experiment”. 3.8. Cooling experiment The other experiment was performed the next day or later to ensure the rapid volume change was over. In the second experiment, the resonator–taper pair, with the smaller cover, was heated with a 45 W halogen lamp for ~2 min. The temperature rose by ~2 K. Subsequently, the lamp was turned off, the resonance spectrum and temperature data were collected for ~ 20 min. We call this experiment a “cooling experiment”. 3.9. Resonance shift analysis The method of spectrum data analysis will be explained at the beginning of the Results and Discussion section for one of the experiments. Here, a method to subtract the resonance shift due to a
Fig. 3. Overlay of WGM resonance spectra at different moments in an isothermal volume relaxation experiment. The diameter of the resonator was 450 μm. The peak in the shade was tracked for estimating the resonance wavelength shift.
55
Journal of Non-Crystalline Solids 476 (2017) 52–59
L. Cai et al.
the size of the markers. We are reminded that (∂ ln λ/∂ln ρ)T = − 0.105 in silica, and therefore (∂ ln λ/∂ln V)T = 0.105. The red shift of the resonance indicates a volume increase (Fig. S1(a) and (b) in Supplemental materials show the plots). This result can be explained by using a diagram that describes the relationship between the temperature and volume in types I, II, and IV silica [18]. Fig. 7 was prepared mostly from Brückner's review article [13]. The solid curve indicates the equilibrium and has a minimum at around 1400–1500 °C, and the presence of the minimum is characteristic of silica. In contrast, other silicate glasses show a curve of simply increasing volume with temperature. When molten silica is quenched, it will follow the track indicated as “Quenching” to decrease its volume rapidly. When left at room temperature, it will relax to increase its volume, but the increase is limited by the mobility of the constituents at such a low temperature. Our spherical resonators were heated to well above 1500 °C, especially with the arc. It is known that optical fibers harbor residual strain as a result of quenching while being drawn. Heating silica to a sufficiently high temperature releases the strain, and when cooled or quenched, the volume-temperature relationship will follow the solid line in Fig. 7. We observed a volume increase in the isothermal experiment, which is explained as the volume relaxation of quenched silica at room temperature. In Fig. 7, the dashed line represents the volume–temperature relationship for silica under residual strain. The latter leads to a larger volume that is frozen in below the glass transition temperature [16]. The result in Fig. 6(a), showing a larger volume increase for smaller resonators, can be explained as a volume relaxation toward a greater volume, caused by the unmolten part of the fiber. The latter would leave the part near the surface under strain, although not as much as the fiber that we started with. The difference between the arc-formed resonators and those made by the torch can be also ascribed to the residual stress: The torch does not heat the fiber as much as the arc does. In Fig. 4, for example, (Δ λ/λ)total is ~2.3 ppm. This value leads us to estimate the volume increase as 21.9 ppm, that is an increase of radius by ~7.3 ppm (~ 1.64 nm). It is extremely small, and no other methods would detect this level of radius change. The RI decrease is ~5 ppm; again, it would be difficult to detect this small change with other methods. The noise in Fig. 4 is ~0.1 ppm. It means the resolution of radius change is ~0.071 nm or ~0.32 ppm.
Fig. 4. Isothermal experiment with a resonator of diameter 450 μm. Temperature monitored by a thermistor near the resonator, raw resonance shift (Δλ/λ)raw and resonance shift compensated for the temperature change, (Δλ/λ)T-comp, are plotted as a function of time since the measurement started. The plot of (Δλ/λ)raw was moved upward by 1 ppm for clarity.
to increase. These characteristics were observed in all of the resonators we tested. To estimate the total shift and the time constant of the shift from the data collected for a finite period of time, we did a curve fit with an empirical equation, (Δ λ/λ)T-comp = b0/(1 + b1 / t), where b0 and b1 are fitting parameters. The b0 ≡ (Δ λ/λ)total stands for the total shift at t = +∞, and the shift reaches a half of the total shift at t = b1 ≡ t1/2. Fig. 5(a) shows (Δ λ/λ)T-comp and the curve fit for three resonators of different diameters D fabricated with arc; Fig. 5(b) for three resonators fabricated with the hydrogen torch. The greater the diameter, the smaller the total shift for either series of resonators. The curve fit by the simple algebraic equation is satisfactory with R2 typically > 0.98. At first, we used a different equation, b0 / [1 + (b1 / t)γ], for the curve fit, but the value of γ was close to one for all the data. Therefore, we decided to set γ to one. The curve fit by (Δ λ/λ)total / (1 + t1/2 / t) was enforced on all of the results of (Δ λ/λ)T-comp. Fig. 6(a) shows a plot of the total shift as a function of the resonator diameter for the arc-formed resonators and hydrogen torch-formed resonators. Fig. 6(b) shows the half time t1/2. For both methods of resonator fabrication, the total shift decreases with an increasing resonator diameter. The longer the exposure time to the heat source, the smaller the shift. In Fig. 6(b), the half time is between 8 and 27 min, but the trend is not conclusive. Standard errors are within
4.2. Cooling experiment Fig. 8(a) shows a plot of the temperature and the resonance wavelength as a function of time since data collection started in one of the cooling experiments. The resonator of diameter 322 μm, made by the electric arc, was used here. In the cooling, the resonance shifted to a shorter wavelength. Fig. 8(b) shows a plot of the resonance wavelength as a function of temperature. The data lie along a straight line. Optimal fit by a linear relationship gives a slope of 0.008858 nm/K with R2 = 0.99984, which translates into Δ λ/λ = 6.74 ppm/K with a Fig. 5. Temperature-compensated resonance shift is plotted as a function of time for resonators of different diameters D, prepared by exposing a silica fiber to (a) electric arc and (b) hydrogen torch. Optimal curve fits by (Δλ/λ)T-comp = b0 / (1 + b1 / t), where b0 and b1 are fitting parameters, are also shown.
56
Journal of Non-Crystalline Solids 476 (2017) 52–59
L. Cai et al.
Fig. 6. (a) Total shift (Δλ/λ)total and (b) half time t1/2 estimated by curve fitting the plot of (Δλ/λ)T-comp vs time t by (Δλ/λ)total/(1 + t1/2/t) for resonators of different diameters D made by electric arc and hydrogen torch.
Table 2 Temperature coefficients for the resonators made by different heating method with different diameters. D/μm
Heating method
Temperature coefficient/(ppm/K)
322 490 435 423 376 391
Electric arc Electric arc Electric arc Electric arc Hydrogen torch Hydrogen torch
6.74 6.70 6.64 6.79 6.77 6.81
Fig. 7. Volume-temperature diagram of silica (types I, II, and IV).
standard error of 0.00061. Results for all the resonators studied in the cooling experiment are listed in Table 2. All of them give similar values. The mean value and the standard deviation of these six results are 6.74 ppm/K and 0.063 ppm/K, respectively. It appears that, in one day or longer, the resonators finished their volume relaxation, which gave essentially identical temperature coefficients for all the resonators tested. We did not conduct cooling experiments for the other resonators. From the T coefficient of 6.74 ± 0.06 ppm/K, we can use Eqs. (15) and (16) to estimate (∂ n/∂ T)p as 9.0 × 10− 6 ± 0.1 K− 1. This value is similar to 8.75 × 10− 6 K− 1, evaluated for another type of silica made by vapor-phase axial deposition (VAD) method [23], and 8.66 × 10− 6 K− 1, obtained for standard reference material (SRM) glass #739 [24], but is less than the values of type III silica that are > 10 × 10− 6 K− 1 [25,26]. We note here that (∂n/∂T)p depends on the wavelength and temperature, and varies for different types of silica. Using our value of (∂n/∂T)p, we can use Eq. (17) to estimate (1/α)(∂ α/ ∂ T)p as 19.0 ± 0.2 ppm/K, and with Eq. (18), (1/α)(∂ α/∂ T)ρ as 18.4 ± 0.2 ppm/K. It is obvious that the noise in cooling experiment in Fig. 5(a) is much
Fig. 9. WGM spectrum for a resonator of diameter 563 μm formed using electric arc. Eight scans of wavelength, demarcated by dashed lines, are combined into one with a wavelength scan range of 78.12 pm.
lower compared with isothermal experiment in Fig. 8(a). We roughly estimated the magnitudes of noise in the two figures as 0.1 ppm and 0.04 ppm, respectively. We saw a similar difference in all the experiments we did. We may ascribe the extra noises we saw in the isothermal experiment to movement of SiO2 during the volume relaxation: For the volume to change, the silica network must undergo internal rearrangement. The latter caused density fluctuations and lead to noises Fig. 8. (a) Temperature change and resonance shift vs. time and (b) temperature coefficient fitting in the cooling experiment.
57
Journal of Non-Crystalline Solids 476 (2017) 52–59
L. Cai et al.
Fig. 10. Q factors of isolated peaks for resonators of different diameters made by (a) electric arc and (b) hydrogen torch.
small the volume change. When the volume relaxation settled, the temperature coefficient of the silica resonator was reproducible and nearly identical for all the resonators tested. For spheres, the Q values of the resonators made by the hydrogen torch were generally lower than those made by the electric arc. For a cylinder with a minimal exposure to the heat source, it was the other way around.
in the time trace of resonance wavelength. When the relaxation had finished, the fluctuations disappeared, and the noise in the time trace was mostly due to system noise and ambient noise. 4.3. Q factor To investigate the effect of the resonator diameter and heating method on the Q factor of resonance, we collected eight spectra in different scan ranges, each 1.25 mA wide, and then combined them to one spectrum. Fig. 9 shows an example for the resonator of diameter 563 μm. Another example of combined spectrum is shown given in Fig. S2 of the Supplemental materials. We obtained a combined spectrum for each of five resonators made by the arc and five resonators made by the torch. A peak is either an isolated peak or a compound peak consisting of two or more nearby peaks. For each of the ten spectra, some isolated and symmetric peaks were selected to fit the spectrum with a Lorentzian [27] and evaluate the Q value. We plot the Q values for all the peaks analyzed in Fig. 10. For each resonator, there is a distribution in the Q value, as the coupling with the two tapers is different for each mode, and also the intrinsic Q factor is different. Still, we can see a difference in the distribution. In part (a) that shows the Q values for arcformed resonators, Q is small when the resonator is close to cylindrical, and increases as the resonator becomes more spherical. In part (b) that shows the Q values for torch-formed resonators, the distribution is more or less identical for different diameters. These results are reasonable, since heating smooths the resonator's surface and homogenizes the silica. The smoothing is more effective with a higher temperature provided by the arc than it is by the torch. Another possible cause for the lower Q in the torch-prepared resonators is that the hydrogen flame may have introduced OH into silica, leading to loss by absorption. It is interesting to see that the 300-μm, arc-formed resonator has smaller Q values compared with the 300-μm, torch-formed resonator. We ascribe this difference to the extremely short time of exposure to the heat source for the arc, resulting in a non-uniform heating.
Acknowledgement The study presented here was conducted while L. Cai stayed with I. Teraoka at NYU, made possible through China Scholarship Council. L. Cai thanks N. Luo for helping her with experiments. Appendix A. Supplementary data Supplementary data to this article can be found online at http://dx. doi.org/10.1016/j.jnoncrysol.2017.09.022. References [1] F. Vollmer, S. Arnold, Whispering-gallery-mode biosensing: label-free detection down to single molecules, Nat. Methods 5 (2008) 591–596. [2] Vladimir S. Ilchenko, Andrey B. Matsko, Optical resonators with whispering-gallery modes-part II: applications, IEEE J. Sel. Top. Quantum Electron. 12 (2006) 15–32. [3] M.R. Foreman, J.D. Swaim, F. Vollmer, Whispering gallery mode sensors, Adv. Opt. Photon. 7 (2015) 168–240. [4] Martin D. Baaske, Matthew R. Foreman, Frank Vollmer, Single-molecule nucleic acid interactions monitored on a label-free microcavity biosensor platform, Nat. Nanotechnol. 9 (2014) 933–939. [5] S.I. Shopova, R. Rajmangal, S. Holler, S. Arnold, Plasmonic enhancement of a whispering-gallery-mode biosensor for single nanoparticle detection, Appl. Phys. Lett. 98 (2011) 243104. [6] M.A. Santiago-Cordoba, S.V. Boriskina, F. Vollmer, M.C. Demirel, Nanoparticlebased protein detection by optical shift of a resonant microcavity, Appl. Phys. Lett. 99 (2011) 073701. [7] M.A. Santiago-Cordoba, M. Cetinkaya, S.V. Boriskina, F. Vollmer, M.C. Demirel, Ultrasensitive detection of a protein by optical trapping in a photonic-plasmonic microcavity, J. Biophotonics 5 (2012) 629–638. [8] A. Bianchetti, A. Federico, S. Vincent, S. Subramanian, F. Vollmer, Refractometrybased air pressure sensing using glass microspheres as high-Q whispering-gallery mode microresonators, Opt. Commun. 394 (2017) 152–156. [9] C.H. Dong, L. He, Y.F. Xiao, V.R. Gaddam, S.K. Ozdemir, Z.F. Han, G.C. Guo, L. Yang, Fabrication of high-Q polydimethylsiloxane optical microspheres for thermal sensing, Appl. Phys. Lett. 94 (2009) 231119. [10] U.K. Ayaz, T. Ioppolo, M.V. Ötügen, Wall shear stress sensor based on the optical resonances of dielectric microspheres, measurement science and technology, Meas. Sci. Technol. 22 (2011) 075203. [11] Qiulin Ma, Tobias Rossmann, Zhixiong Guo, Temperature sensitivity of silica microresonators, J. Phys. D. Appl. Phys. 41 (2008) 245111. [12] A.D. Yablon, Optical and mechanical effects of frozen-in stresses and strains in optical fibers, IEEE J. Sel. Top. Quantum Electron. 10 (2004) 300–311. [13] R. Brückner, Properties and structure of vitreous silica. I, J. Non-Cryst. Solids 5 (1970) 123–175. [14] B.H. Kim, Y. Park, T.J. Ahn, D.Y. Kim, B.H. Lee, Y. Chung, U.C. Paek, W.T. Han, Residual stress relaxation in the core of optical fiber by CO2 laser irradiation, Opt.
5. Conclusion We observed the volume relaxation process in quenched type IV silica via isothermal experiment of a WGM resonator. We utilized the high sensitivity of the resonance wavelength shift to the material changes in the silica resonator to observe ppm-level volume changes. A 1 ppm red shift of the resonance wavelength amounts to an 9.52 ppm increase in volume, and a 2.17 ppm decrease in RI. For the resonators with the same diameter, the one made by hydrogen torch exhibited a larger expansion than did the resonator made by the electric arc. For both methods, the longer the exposure time to the heating source, the 58
Journal of Non-Crystalline Solids 476 (2017) 52–59
L. Cai et al. Lett. 26 (2001) 1657–1659. [15] D.R. Perchak, J.M. O'Reilly, Observation of a volume minimum in computer simulations of supercooled, amorphous silica, J. Non-Cryst. Solids 167 (1994) 211–228. [16] A.D. Yablon, M.F. Yan, D.J. DiGiovanni, M.E. Lines, S.L. Jones, D.N. Ridgway, G.A. Sandels, I.A. White, P. Wisk, F.V. DiMarcello, E.M. Monberg, J. Jasapara, Frozen-in viscoelasticity for novel beam expanders and high-power connectors, J. Lightwave Technol. 22 (2004) 16–23. [17] H.N. Luo, H.S. Kim, M. Agarwal, I. Teraoka, Light turn-on transient of a whispering gallery mode resonance spectrum in different gas atmospheres, Appl. Opt. 52 (2013) 2834–2840. [18] R. Brückner, Metastable equilibrium density of hydroxyl-free synthetic vitreous silica, J. Non-Cryst. Solids 5 (1971) 281–285. [19] Corning® HPFS® 7979, 7980, 8655, Fused Silica. Corning, http://www.corning. com/media/worldwide/csm/documents/HPFS%20Product%20Brochure%20All %20Grades%202015_07_21.pdf. [20] HPFS® 8655, Fused Silica, Corning, http://www.corning.com/media/worldwide/
[21] [22] [23] [24] [25] [26] [27]
59
csm/documents/HPFS%208655%20Fused%20Silica%20Brochure%202013_08_15. pdf. K. Vedam, E.D.D. Schmidt, R. Roy, Nonlinear variation of refractive index of vitreous silica with pressure to 7 kbars, J. Am. Ceram. Soc. 49 (1966) 531–535. M. Agarwal, I. Teraoka, Whispering gallery mode dip sensor for aqueous sensing, Anal. Chem. 87 (2015) 10600–10604. A. Koike, N. Sugimoto, Temperature Dependences of Optical Path Length in Inorganic Glasses, vol. 56, Reports Res. Lab. Asahi Glass Co., Ltd, 2006, pp. 1–6. G. Ghosh, Model for the thermo-optic coefficients of some standard optical glasses, J. Non-Cryst. Solids 189 (1995) 191–196. T. Toyoda, M. Yabe, The temperature dependence of the refractive indices of fused silica and crystal quartz, J. Phys. D. Appl. Phys. 16 (1983) L97–L100. I.H. Malitson, Interspecimen comparison of the refractive index of fused silica, J. Opt. Soc. Am. 55 (1965) 1205–1209. I. Teraoka, H. Yao, N.H. Luo, Relationship between height and width of resonance peaks in a whispering gallery mode resonator immersed in water and sucrose solutions, Opt. Commun. 393 (2017) 244–251.