SUPPLEMENTARY INFORMATION

Transcription

1 Supplemental Material Property Tables Cited in Main Text Table SI. Measured parameters for the sapphire-derived optical fibers Fiber Maximum Alumina Content Δn 10-3 Core Size Mole Percent (%) Weight Percent (%) (Peak Value) ( m) A B C D * SMF- 28 TM ~4 mole % GeO * Extrapolated from best-fit curve using data from Fibers A-C. Table SII. Measured parameters for the four fibers evaluated in this work Fiber Average Alumina Content (mole %) Measured Brillouin Frequency (GHz) Measured Brillouin Linewidth (MHz) Optical Mode Index LP 01 mode (1534 nm) A B C NR* D SMF- 28 TM ~4 mole % GeO * The SNR for this measurement was low for reasons to be discussed later in this paper. Accordingly, the spectral width measurement was not reliable (NR). The Brillouin spectrum could not be measured for this sample. Acoustic values are extrapolated using the additive model. Table SIII. Bulk material parameters determined from the fits to measured data Material Acoustic Velocity (m/s) Brillouin Linewidth Refractive Index Density (kg/m 3 ) Photoelastic Constant, p 12 (MHz) * SiO Al 2 O * From [30,31]. Linewidth at a Brillouin frequency of 11 GHz The photoelastic constant is assumed to have the same value as that of bulk sapphire [28]. NATURE PHOTONICS 1

2 Acoustic anti-guidance in the sapphire-derived fibers In order to justify neglecting the anti-guide nature of the fiber refer to Fig. S1, where the Fiber C compositional profile (as an example) and a plot of its fundamental optical mode are provided. As is discussed in the main text, the value of the acoustic velocity near the core center of this fiber is approximately 7000 m/s. Utilizing the model found in [19], and making an eight-layer stepwise approximation to the profile, three example modes were calculated (although there are many more) corresponding to this acoustic profile (with mode solutions at i, ii, and iii) as shown in Fig. S1(i-iii). Fig. S1. (a) Compositional profile for Fiber C shown with a plot of the calculated fundamental optical mode. Three example acoustic modes are calculated for this structure with mode solutions located at i, ii, and iii. (b) Calculated acoustic modes with mode solutions located at i, ii, and iii. The acosutic mode corresponding to position ii has the greatest overlap with the fundamental optical mode. Since its calculated waveguide loss is negligible, and the calculated phase velocity is within a few m/s of the core material value, measurements on this mode well-match the material values. As also found in Fig. 4(b) in the main text, large amplitude oscillations are computed for the cladding. These are the result of the anti-guide structure solutions becoming Hankel functions (i.e., radiation modes) thus giving rise to complex-valued propagation constants [S1]. Modes i and iii have larger calculated acoustic waveguide attenuations than does mode ii. Additionally, modes with phase velocities lower than that of mode ii form a ring pattern around the fundamental optical mode resulting in lower spatial overlap and diminished Brillouin interaction, 2 NATURE PHOTONICS

3 SUPPLEMENTARY INFORMATION which can be seen clearly in the mode-i plot. Similarly, modes with phase velocities greater than that of mode ii become highly oscillatory in the core region, also with diminished overlap with the optical mode. Thus, we conclude that mode ii has the largest spatial overlap with the optical mode and will dominate the Brillouin scattering data. The calculated propagation constant for mode ii gives rise to a modal velocity within a few m/s of the core material value. Furthermore, the calculated waveguide loss term (owing to the anti-guide nature of the core) is found to be negligible in comparison with the measured material acoustic damping coefficient (provided in main text). Thus, measurements of the Brillouin spectrum for the various fiber segments (if dominated by the equivalent of mode ii) approximate Brillouin measurements on the bulk material. Discussion of the bulk material characteristics A best fit to the measured refractive index data (see Fig. 1(b)), from pure silica up to Fiber B (~ 30 mole%, i.e. low-concentration ), yields a slope of approximately per mole percent. This slope is somewhat higher than the per mole% that is reported in Nassau [21]. Based on this observation, it is concluded that the mass density of the sapphire-derived optical fibers is larger than for the bulk samples described in [21]. Further supporting this conclusion is the fact that the slope of the measured refractive index values match well the comparison data provided in [33,S2] and those mass densities are also reported to be larger than that measured by Nassau [21]. The bulk value utilized in this work, due to the similarities with the measured refractive index, is therefore found by fitting the additive model to the H.S. curve in Fig. 2 of [21]. This bulk value should not be interpreted as the density of sapphire, but rather 100% amorphous alumina that possesses the same thermal and draw history as the sapphire-derived fibers. Clearly, the conditions under which the glasses were produced and fibers were drawn will cause further variations in these parameters. Fig. S2. Plot of acoustic velocity versus alumina concentration for two different mass densities. NATURE PHOTONICS 3

4 The additivity model utilizes the molar volume as the additivity parameter [30]. Therefore, with knowledge of the mass density (given the molar mass), the additive model can be applied to measured data in order to obtain the remaining bulk parameters, such as the acoustic velocity. The results obtained herein suggest a dependence of acoustic velocity on Al 2 O 3 concentration (in the low-concentration limit) of about +27 m/s/mol%, which is somewhat lower than that reported in [21] and considerably lower than reported in [31]. Interestingly, if the mass density alone is changed in the additive model to match the measurement in [21] (~2845 kg/m 3 ), a slope of 31 m/s/mol% is obtained, which is now in excellent agreement with [21]. The results of the two simulations are provided in Fig. S2. This provides further confidence in the enumerated value of the bulk mass density in the fibers described in this work. It is worth noting that the additive model leads to a curve for density that is sub-linear as a function of (mole percent) alumina. Thus, the bulk density used in these calculations do not match the linearly extrapolated, 100%- alumina values suggested in [21], but do match the low concentration values (up to the 7 mole% in [21]) identically. Additionally, inspection of Fig. 1(b) shows that the additive model further suggests that relationships between the other bulk parameters and alumina concentration also are not necessarily linear. In fact, the refractive index and acoustic velocity are sub-linear and superlinear, respectively. The slopes of the curves at low concentrations are driven by the bulk mass density, which could explain why the extrapolated bulk values do not match the simple linear fits-to-data (in the limit of 100% alumina) usually found in the literature. For example, extrapolating low-concentration acoustic velocity slope from the sapphire-derived fiber (27 m/s/mole%) to that for pure (100%) alumina gives rise to a bulk value of 8670 m/s, which is somewhat lower than reported in this work. Again, this results from the additive model predicting a super-linear dependence of the acoustic velocity on alumina concentration (in mole%), i.e. lower slopes at lower alumina concentrations. The plot of the Brillouin spectral width is dominated by the change (increase) in acoustic frequency as a function of alumina content, causing broadening with increasing alumina [32], and is not considered in this argument. Table SIV. Measured parameters for the three fibers Material Acoustic Velocity (m/s) Refractive Index Density (kg/m 3 ) Photoelastic Constant, p 12 Quartz 6024 * [S3] * [S4] 2648 [S5] 0.27 [S6] SiO Al 2 O [34] Sapphire * [39] 1.75 * [S7] 3980 [S8] [34] *Averages of the various crystal directions provided At a wavelength of 980 nm Finally, it is worth showing a comparison of the bulk parameters determined in this work with those of crystalline SiO 2 and Al 2 O 3. This is provided in Table SIV. Interestingly, the mass densities, along with the associated refractive indices, for crystalline SiO 2 and Al 2 O 3 are both larger than their amorphous, fiber-based counterparts, in nearly identical proportion for both materials. The acoustic velocity, also lower for the glassy configurations, on the other hand experiences a much larger decrease for Al 2 O 3 than for SiO 2 in going from crystalline to (theorized for alumina) glassy form. The photoelastic constant also is decreased somewhat (by about 6%) suggesting that the assumption made herein for the p 12 of sapphire may not be entirely 4 NATURE PHOTONICS

5 SUPPLEMENTARY INFORMATION correct, but is believed to be very close, given the proximity of the photoelastic constants of glassy and crystalline SiO 2. Setting p 12 of glassy bulk alumina to zero increases the calculated Brillouin gain for Fiber D (as the limiting case in this work) by only about 1 db. Fitting of the Brillouin spectral width As described in the Results and Discussion section of the main text, the Brillouin gain spectrum (BGS) is asymmetric most likely due to the excitation of a number of higher-order optical modes (HOMs). Since the measured Brillouin frequency shift (BFS) is directly proportional to the optical mode index, and the mode index decreases as the mode number is increased from the fundamental, the BFS decreases as the mode number is increased relative to the fundamental optical mode. Thus, observations of Brillouin scattering from HOMs will appear to the lower frequency side of the fundamental interaction. In the present case, since the output of the Brillouin testing apparatus (the circulator comprised of a purely single mode fiber) serves as both the transmit and receive port for the fiber under test, a considerable amount of spatial filtering is encountered at the join with the sapphire-derived fiber, thus limiting the observed number of HOMs in the BGS of the sapphire-derived fiber to merely a few. Fig. S3. Plot of the measured BGS for Fiber A (black solid curve), the fitted single- Lorentzian model (orange dashed curve), and a model including the HOM interactions (purple solid curve with boxes). The spectra have been normalized for comparison purposes. It is possible that the presence of these HOM interactions leads to a broadening of the Brillouin spectrum, due to the overlap between these spectra. However, in the present case, broadening is found to be negligible on the high frequency side of the observed BGS. Accordingly, in order to determine the Brillouin spectral width, a Lorentzian function was applied only to the high frequency side of the measured BGS. In order to further reinforce this, Fig. S3 shows the BGS measured from Fiber A on a narrower frequency scale, along with two models. The first model is the fitted single-lorentzian to the low frequency side of the BGS. The second is a simulation calculating the spectrum including the presence of HOMs, summed into the aggregate BGS. NATURE PHOTONICS 5

6 Since the Brillouin frequency of the next-higher LP mode, LP 02, is approximately one half-width (~ 50 MHz) from that of the fundamental mode, and is much weaker, the influence of the HOM modes is negligible with respect to the fitting method, broadening the high frequency side of the BGS by less than 2 MHz. Supplemental References S1 Dragic, P. Brillouin Suppression By Fiber Design. IEEE Summer Top. Meet. Ser., , S2 Aramaki S. & Roy, R. Revised Phase Diagram for the System Al 2 O 3 -SiO 2. J. Am. Ceram. Soc. 45, (1962). S3 Kushibiki, J. Ohtagawa, M. & Takanaga, I. Comparison of acoustic properties between natural and synthetic α-quartz crystals. J. Appl. Phys. 94, (2003). S4 Shuang Z. & Fuquan, W. The study on dispersive equation and thermal refractive index coefficient of quartz crystal. Acta. Photon. Sin. 35, (2006). S5 Langenhorst F. & Deutsch, A. Shock experiments on pre-heated α- and β-quartz: I. Optical and density data. Earth Plan. Sci. Lett. 125, (1994). S6 Narasimhamurty, T. Photoelastic Constants of α-quartz. J. Opt. Soc. Am. 59, (1969). S7 DeFranzo, A. Index of refraction measurement on sapphire at low temperatures and visible wavelengths. Appl. Opt. 32, (1993). S8 Jukuhara M. & Yamauchi, I. Temperature dependence of the elastic moduli, dilational and shear internal frictions and acoustic wave velocity for alumina, (Y)TZP and β -sialon ceramics. J. Mat. Sci. 28, (1993). 6 NATURE PHOTONICS