Wavelength dependence of the Brillouin spectral width of boron doped germanosilicate optical fibers

Pi-Cheng Law; Peter D. Dragic

doi:10.1364/OE.18.018852

1. Introduction

Brillouin scattering in optical fibers has been finding an increasing number of applications. More traditional of such applications are, for example, distributed sensing [1,2] and light amplification [3], while newer applications include slow-light systems [4] and ultra-long coherence length lasers [5], among others. Recently, novel fibers have been realized with the rationale that it is possible to tailor the acoustic properties, i.e. an acoustic index profile (AIP), of an optical fiber, sometimes largely independently of its optical properties [6–8]. Undoubtedly, these and newer innovations will lead to an even broader range of applications for Brillouin scattering, both spontaneous (SpBS) and stimulated (SBS).

Effectively tailoring the AIP of an optical fiber necessitates a fairly accurate understanding of how fiber dopants influence a number of acoustic properties of the optical fiber, namely the acoustic velocity [9] and viscosity [10,11] (or acoustic wave loss due to material damping). These parameters largely set the observed Brillouin frequency shift, ν_B, and Brillouin spectral width, Δν_B, respectively. The latter is important in designing fibers with Brillouin gain coefficients (BGC, units of m/W) of certain strength [12], including maximizing or minimizing this value for a specific system or application.

It is traditionally held [13] that there is a wavelength- (or acoustic frequency-) squared dependence of the Brillouin spectral width, increasing with decreasing optical wavelength. This links back to a dependence of the acoustic attenuation on the square of the acoustic frequency [10,14], which is then linked to the optical wavelength through the Bragg condition, $λ_{a} = λ_{o} / 2 n$ with n being the index of refraction. The subscripts ‘o’ and ‘a’ will denote the optical and acoustic values, respectively.

This first order analysis, however, neglects the fact that there is an additional frequency-dependent dynamic viscosity term in the expression for the Brillouin spectral width [10]. This term has a maximum viscosity value at some glass temperature (or fictive temperature [15]), that is dependent on the acoustic frequency. We will show that when the temperature of the peak of the viscosity curve at hypersonic frequencies lies far from a fiber’s fictive temperature [16], Δν_B acquires a simple frequency-squared dependence. However, we will also show that when the peak of viscosity lies in the vicinity of a fiber’s fictive temperature, then a departure from this rule can be experienced. From [11], one may conclude that this is potentially the case with boric oxide as its peak viscosity at hypersonic acoustic frequencies lies around 1000 ~2000 K, warranting some further investigation.

In this paper, we investigate the wavelength dependence of Δν_B for boric oxide-doped germanosilicate fibers. A total of eight fibers are investigated at both 1534 nm and 1064 nm. Four of these are B + Ge-doped silica that were drawn from the same preform, but at different draw temperatures (1900 °C, 1950 °C, 2050 °C, and 2150 °C), while the remaining three B + Ge fibers include two different samples from Nufern of East Granby, CT and one purchased from Newport Corp. The latter three fibers are highly-B + Ge-doped photosensitive fibers. A sample of Corning SMF-28^TM is included in the analysis as the eighth and a reference fiber. We find that each of the B-doped fibers follows a sub-wavelength-squared dependence of the Brillouin spectral width, while the Corning fiber appears to obey the wavelength-squared law. A possible explanation for this behavior is offered in some detail using the theory found in [10]. Understanding this phenomenon for other and less traditional dopants then becomes important for wavelength-specific applications.

2. Experimental setup

This experiment uses an optical heterodyne detection system to measure the Spontaneous Brillouin Scattering (SpBS) spectra [17]. Since we are measuring at two wavelengths (1534 nm and 1064 nm), the systems we employed were both Er and Yb fiber amplifier-based. More details of the Er-based system can be found in [18], while Fig. 1 illustrates the block diagram of the Yb-based experimental configuration which is described here. A Nd:YAG laser operating at 1064.263 nm and linewidth of less than 10 kHz works as a seed laser. The output light of the seed laser is isolated and then amplified by a first, in-house assembled, ytterbium doped fiber amplifier (YDFA#1) providing about 200mW of saturated output power. The amplified seed signal then passes through Port 2 of an optical circular. The optical circular is produced from HI-1060^TM fiber which utilizes Ge dopant ions and has about 1.5m of this fiber at each of the three ports. Therefore, in addition to the signal from the fiber under test (FUT), the Brillouin frequency shifts from the HI-1060^TM fiber are also observed. The amplified seed signal passes into the FUTs with an optical single mode launch generating the Brillouin scattered signals which pass back through port 3 of the circulator and into YDFA#2. The Brillouin scattered signals (reflected backward) are adequately amplified by YDFA#2 (small signal gain) before being heterodyned with a fast PiN detector (New Focus 1534). Finally, an Agilent PSA series electrical spectrum analyzer (ESA) measures the acoustic modes carried by the Brillouin scattered signals.

Fig. 1 Experimental apparatus for measurement of the Brillouin spectrum at 1064 nm. A narrow linewidth YAG laser operating at 1064 nm is boosted and passed through a circulator to the test fiber. The Stokes’ signal and local oscillator signal are then preamplified before being heterodyned with a fast detector.

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Due to adopting the heterodyne approach, the fundamental local oscillator (LO) signal is required for mixing. As such, we point out that the LO signal came from appropriately cleaving the output end of the FUT. The Fresnel reflection from the end of the FUT provides an adequate LO signal to achieve low-noise heterodyne detection, but was limited via a cleave angle so as to not saturate YDFA#2.

In the experiments, YDFA#1 works as a power amplifier to boost the seed laser output power. Meanwhile, YDFA#2 works as a detector preamplifier to increase the scattered and reflected signals, including both the SpBS and seed signals. In order to obtain trustworthy data, YDFA#1 should provide adequate power to generate SpBS signals, while not generating Stimulated Brillouin Scattering (SBS) with apparent gain. This may result in an observed linewidth that is narrower than the true spontaneous spectrum [19]. Therefore, amplifier characteristics, including pump power, etc. were adjusted to avoid any spectral narrowing effects due to SBS gain.

YDFA#2 plays the very important role of amplifying the SpBS and seed signals to an adequate level for the fast PiN detector. We used 120cm of heavily-doped YDF and a 200mW pump laser diode in designing YDFA#2. Due to the lower signal power in this amplifier stage, we expect that significant ASE noise may come up and therefore the fiber length was selected in order to minimize this ASE noise. We also note that the reflected signals have different power levels. Apparently, the seed signal power can be multiple orders of magnitude larger than the SpBS signal power. Therefore, the seed signal could saturate YDFA#2 resulting in less gain being available to the SpBS signal. Therefore, as described above, we adjusted the cleave angle of the FUT to reduce the seed signal power reflected into YDFA#2 to that of small signal levels. Figure 2(a) and 2(b) are the output power and gain of YDFA#2 versus the input signal power at 1064nm at 221mW pumping power. It is found that YDFA#2 has a small signal gain in the range from 0dBm to −20dBm (or less) (insertion and splicing losses to the input power in the system makes these values seem large) with around 17dB net gain. Its performance allows the fast PiN to produce an adequate heterodyne signal.

Fig. 2 a) Available YDFA#2 power for the Brillouin gain measurements and b) Gain vs. signal power for the heterodyne optical preamplifier showing the region of small signal gain.

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3. Experimental results

Eight fibers are investigated in this study. Four of these (Fibers 1 – 4) were drawn from the same B-doped germanosilicate preform, but at different draw temperatures, as outlined in Table 1 . These fibers have core diameters of 10 μm and the concentrations of B₂O₃ and GeO₂ were measured via Electron Probe Micro Analysis (EPMA) to be about 6.2 wt% and 10.8 wt%, respectively. The decreasing Stokes’ frequency with decreasing draw temperature (see Table 1) can largely be attributed to an increased required draw tension at lower temperatures resulting in a larger residual fiber strain, consistent with the results found in [20,21], however, this analysis is beyond the scope of this work. Complete details for the remaining three B + G fibers (Fibers 5-7) are not available, but since they each have similar Stokes’s shifts, they probably have very similar dopant compositions [9]. Additionally, due to their application as photosensitive fiber, it is believed that they each possess similar optical properties and therefore similar proportions of germania and boric oxide as compared with Fibers 1 – 4. The final fiber that is tested is Corning SMF-28^TM, containing about 4.0 mol% GeO₂ (also EPMA data).

Table 1. Fitted spectral widths and measured Stokes’ shifts for the various fibers at 1534 nm and 1064 nm.

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In Fig. 3 are provide the refractive index profiles (RIPs) of both SMF-28^TM (measured by Interfiber Analysis Inc.) and Fiber 4 (provided by Nufern). Fibers 1 – 3 have similar RIPs to Fiber 4, but with slightly greater index differences due to the differing draw conditions [22]. In particular, the NAs of Fibers 1 – 4 are 0.141, 0.127, 0.112, and 0.107, respectively. Furthermore, because our fibers are acoustically guiding (and acoustically multimode) structures, the Brillouin spectrum will be dominated by light scattering from acoustic modes of the acoustic waveguide [23]. Since we are focusing on the fundamental L₀₁ acoustic mode in this work, a plot of the fundamental mode (normalized field) is also provided in Fig. 3 (red curves) for both 1064 nm and 1534 nm, with the acoustic mode at 1534 nm being insignificantly wider. It is clear that the fundamental acoustic mode is well-confined to the uniform region of the fiber (about 1 to 4.5 microns in position). More specifically, it is calculated that the L₀₁ acoustic power confinement in this ring region is > 99% in all Fibers 1 – 4 at both wavelengths. That the modes at the two wavelengths have essentially the same spatial distribution within a uniform region of the fiber helps to remove uncertainty in the measurements caused by non-uniform acoustic profiles. In other words, we assume that measurements on the L₀₁ acoustic mode (acoustic velocity, viscosity) will be close to the material values in the uniform portion of the waveguide at both wavelengths. Finally, the similarity of the SMF-28^TM RIP to that of Fibers 1 – 4 provides confidence that SMF-28^TM can serve as a reasonable control or reference fiber.

Fig. 3 A plot of the measured RIPs of SMF-28^TM (dashed line) and Fiber 4 (solid line). Both fibers have a similar central dip. Also shown is a plot of the L₀₁ acoustic mode (Fiber 4) at 1534 nm (slightly wider in the central region) and 1064 nm. Their amplitudes were adjusted for visual clarity with respect to the RIPs.

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Figure 4 provides an example of Brillouin spectra obtained at 1534 nm and 1064 nm (Fiber 4). This data is provided as ‘typical’ of the spectra obtained for the fibers of the present study. In both cases, the contributions due to the circulator, in addition to several higher-order acoustic modes (HOAMs), are observed. The HOAMs are clearly visible in the spectra, but not directly pointed out. The dashed line shown in Fig. 4 represents the best fit to the L₀₁ acoustic mode.

Fig. 4 Typical spectrum obtained at 1534 nm and 1064 nm (Fiber 4). Several acoustic modes are observed including peaks due to the measurement apparatus (circulator). The best fit to the fundamental mode is also shown (dashed line) to make the HOAMs more obvious.

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Provided in the inset of each plot is the fitted spectral width (Δν_B) of the L₀₁ acoustic mode and the measured peak frequency (ν_B). It is worthy of note that the ratio of the peak frequencies is slightly different than the ratios of the wavelengths. In particular, $ν_{B}^{1064} / ν_{B}^{1534} = 1.446$ , while 1534/1064 = 1.442. Most of this difference can be attributed to chromatic dispersion in the glass since the material index of refraction is slightly higher at 1064 nm.

As a point of reference, a segment of SMF-28^TM was also tested for comparison. The results are shown in Fig. 5 , along with the best fit to the L₀₁ mode. HOAMs are visible at both wavelengths. To compare the spectra, we first point out that the ratio of the square of the Brillouin frequencies is 2.08, while that of the spectral widths is 2.06. Therefore, we can conclude that the lightly Ge-doped silica fiber appears to follow the frequency-squared law.

Fig. 5 Spectra of SMF-28^TM obtained at 1534 nm and 1064 nm. Some HOAMs are visible. The best fit to the fundamental mode is also shown (dashed line).

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On the other hand, inspection of Table 1 discloses that this is not the case for the B-Ge doped fibers. Each spectral width measured at 1064nm is smaller than expected (i.e. relative to that measured at 1534 nm and expected from a λ^- ²-dependence), such that the ratio of the spectral widths is smaller than 2.08, suggesting that B-Ge doped fibers do not follow the frequency-squared law. To explain this, we will consider (in the next section) the effect of an additional frequency-dependent dynamic viscosity in the expression for the Brillouin spectral width [24]. We will show that this term acts to reduce the expected ratio of the spectral widths at the two wavelengths in these fiber types. That is, the dynamic viscosity will contribute some factor (less than 1) to the frequency-squared law to reduce the spectral width at 1064 nm relative to that expected from the measured value at 1534 nm.

Fibers 1 – 4 are the same B-Ge doped fibers, only associated with different draw temperatures (1900°C to 2150°C, respectively). Therefore, the spectral widths in this case may be strongly related to the fibers’ acquired fictive temperatures [11]. However, we point out that the fiber draw temperature is not typically equal to, but instead usually greater than its acquired fictive temperature [15] due to some glass relaxation in the draw process. Within this context, we observe that from 1900°C to 2150°C the ratio of the Brillouin spectral widths gradually gets larger. Meanwhile, the spectral widths of 1060PS, HPS980, and F-SBG have similar ratios to that of Fiber 4. This suggests that each of these four fibers may have similar fictive temperatures, possibly due to similar draw temperatures. This will be discussed in greater detail in the next section. Figure 6 shows the ratio of the spectral widths plotted as a function of fiber draw temperature for Fibers 1 – 4.

Fig. 6 A plot of the ratio of Δν_B at 1064 nm to Δν_B at 1534 nm versus the draw temperatures of Fibers 1, 2, 3, and 4.

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It is worth pointing out that Fiber 1 shows an abnormal spectral width at 1534 nm. From Table 1, the spectral width at 1534 nm of this fiber appears to be anomalous in that it has broken a trend by being larger than that of the next-higher-temperature fiber. This observation suggests that there may be more effects in the relationship for the spectral width that may not be accounted for in this work. And as a result, the 1900 °C data point in Fig. 6 also appears visually anomalous. We are continuing to investigate the cause of this anomaly.

4. Discussion

4.1 Theory

Table 1 suggests that the B-Ge doped fibers have an anomalous behavior in the wavelength-dependence of the Brillouin spectral width that departs from the usual frequency-squared law. To explain this, we will first investigate the derivation of the Brillouin spectral width. There are some phenomena to explain the internal friction which is equal to the Brillouin spectral width divided by the Stokes’ frequency shift. Anderson and Bommel proposed that structural relaxation may be responsible for the internal friction in fused silica and speculated that the lateral motion of the oxygen atom in the Si-O₄ tetrahedron probably results in the internal friction effect [25]. Next, based on the glassy states associated with certain activation energies, the tunneling model was developed to explain the low-temperature anomalies of amorphous solids, but at the higher temperature the thermally activated relaxation process rather than the tunneling process seems to dominate the anomalous properties in vitreous silica [26] and in vitreous germania [27]. We follow the theory (thermally activated relaxation process) which can be found in greater detail in [24] but is presented here for the convenience of the reader. The wave number q of acoustic phonons defines the elastic wave with respect to the angle θ between the incident and scattered light. Since Brillouin scattering is a backward Stokes’ wave scattering, θ = 180°, giving rise to

q = \frac{2 n}{λ_{o}} \sin (θ / 2) = \frac{2 n}{λ_{o}} = \frac{1}{λ_{a}} = \frac{ω}{V_{a}} .

In Eq. (1) n is the refractive index of the scattering media,

λ_{o}

is the wavelength of the incident light,

λ_{a}

is the acoustic wavelength, ω is the acoustic frequency (ω from [24] is identical to the Stokes’ frequency shift ν_B of the present analysis), and

V_{a}

is the acoustic velocity.

The width of the Brillouin spectrum will be controlled by the viscosity η´ (in the case that the Landau-Placzek ratio γ is 1). Therefore, in addition to the wavelength λ, the complete equation for the Brillouin width $Δ ω_{B}$ (Δν_B) will include the dynamic viscosity η´(ω) which is dependent on acoustic frequency ω as follows:

Δ ω_{B} = q^{2} \frac{η^{'} (ω)}{ρ_{o}} = {(\frac{2 n}{λ_{o}})}^{2} \frac{η^{'} (ω)}{ρ_{o}} .

Meanwhile, the dynamic viscosity η´(ω) is related to the imaginary part,

M^{″}

, of the complex mechanical modulus,

M^{*} (ω)

(the energy dissipated via friction) by

η^{'} (ω) = \frac{M^{″}}{ω} = M_{2} \frac{τ}{1 + {(ω τ)}^{2}} .

Here

M_{2}

is the relaxational modulus and τ is a characteristic time constant for the relaxational process. We point out that the real part of the complex mechanical modulus is the longitudinal acoustic modulus M′ from which the longitudinal acoustic velocity can be determined.

First, we note that ω is fixed via the Bragg condition for a fixed laser wavelength. Second, a temperature variation is built into Eq. (3) from the fact that $M^{″} (ω)$ depends on both ω and τ. This time constant is dependent on the temperature T [24]. More specifically, the time constant is related to the activation energy $E_{A}$ , Boltzmann’s constant k, and temperature T by the Arrhenius function,

τ = τ_{o} e^{\frac{E_{A}}{k T}} .

In addition, the frequency-dependent kinematic viscosity ν´(ω) [10] is defined as

ν^{'} (ω) = \frac{η^{'} (ω)}{ρ_{o}} .

From Eqs. (3), (4), and (5) we can get the relationships between the kinematic viscosity, ν´(ω), and the loss modulus,

M^{″}

, giving rise to

ν^{'} (ω) = \frac{η^{'} (ω)}{ρ_{o}} = \frac{M^{″}}{ρ_{o} ω} = \frac{M_{2}}{ρ_{o}} \frac{τ}{1 + {(ω τ)}^{2}}

and

ξ = ν^{'} (ω) \cdot ω = \frac{η^{'} (ω) \cdot ω}{ρ_{o}} = \frac{M^{″}}{ρ_{o}} = \frac{M_{2}}{ρ_{o}} \frac{ω τ}{1 + {(ω τ)}^{2}} = \frac{M_{2}}{ρ_{o}} \frac{ω τ_{0} e^{\frac{E_{A}}{k T}}}{1 + {(ω τ_{o} e^{\frac{E_{A}}{k T}})}^{2}} .

Continuing, pure B2O3, is found to be best represented by three principal relaxation mechanisms [24], including non-planar BO₃ distortions, impurity diffusion, and final network disintegration, contributing to the specific loss modulus,

M^{″} / ρ_{o}

(viscous dissipation). This transforms Eq. (7) (a single mechanism relationship) into a linear combination of the three individual mechanisms as follows:

ξ = ν^{'} (ω) \cdot ω = \frac{η^{'} (ω) \cdot ω}{ρ_{o}} = \frac{M^{″}}{ρ_{o}} = \sum_{n = 1}^{3} g_{n} \frac{M_{2}}{ρ_{o}} \frac{ω τ_{0, n} e^{\frac{E_{A, n}}{k T}}}{1 + {(ω τ_{o, n} e^{\frac{E_{A, n}}{k T}})}^{2}} .

In Eq. (8)

g_{n}

is the weight factor for the relaxational mechanism n with the definition

g_{n} M_{2} = V_{0, n}

. Then, Eq. (8) is substituted into Eq. (2) to yield Eq. (9) below, which we will use to explain the apparent anomalous behavior of the spectral width ratio, Δω_B(1064nm)/Δω_B(1534nm),

Δ ω_{B} = {(\frac{2 n}{λ_{o}})}^{2} \cdot \frac{η^{'} (ω)}{ρ_{o}} = {(\frac{ω}{V_{a}})}^{2} \cdot \sum_{n = 1}^{3} \frac{V_{0, n}}{ρ_{o}} \frac{τ_{0, n} e^{\frac{E_{A, n}}{k T}}}{1 + {(ω τ_{o, n} e^{\frac{E_{A, n}}{k T}})}^{2}} .

The fact that the B-Ge doped fibers have a smaller measured spectral width ratio than expected suggests that we need to consider not only the wavelength (Bragg-matched acoustic frequency) but also the viscous relaxation mechanism terms in Eq. (9). It is the ratio of the summation terms (η´(1064nm)/η´(1534nm)) that contributes the factor of less than 1 to the frequency-squared law (Δω_B(1064nm)/Δω_B(1534nm) = (ω_1064nm/ω_1534nm)² ≈ (1534nm/1064nm)² = 2.08) thus reducing the spectral width ratio. This seems to be a plausible explanation for the observed anomalous behavior of the B-Ge doped fibers.

4.2 Analysis

To start, we use the pre-exponential time constant, $ω τ_{o, n}$ , and the activation energy, $E_{A, n}$ , of pure B₂O₃ [24] to reproduce Fig. 5(a) from [11] which is the loss modulus, $M^{″}$ , as a function of the glass temperature. In [11], the incident light source is a single mode Argon-ion laser operating at 514.5nm. In Fig. 7 the resulting curve (orange line) includes all three reported [11,22] mechanisms of non-planar BO₃ distortions, impurity diffusion, and final network disintegration. At 773K the dissipation mechanism with $E_{A, 1}$ = 36.5 kJ/mol and $ω τ_{o, 1} = 3 \times 10^{- 3}$ is the non-planar BO₃ distortions (blue line). At 1023K the dissipation mechanism with $E_{A, 2}$ = 44.8 kJ/mol and $ω τ_{o, 2} = 3 \times 10^{- 3}$ is the impurity diffusion (green line) which is very weak in [11] and may not be applicable in the present case, but it is retained only for consistency. Above 1023K the dissipation mechanism with $E_{A, 3}$ = 67.3 kJ/mol and $ω τ_{o, 3} = 2 \times 10^{- 2}$ is the final network disintegration (red line). The final network disintegration continuously increases the viscous dissipation with increasing temperature beyond 1500 K, and this mechanism clearly dominates the specific loss modulus at high temperature. In addition, we reproduced the fitting curves in order to extract the $V_{0, n}$ , which are then used to extrapolate $M^{″}$ of pure B₂O₃ to 1064nm and 1534nm. The extrapolation of the loss modulus to these two wavelengths is presented next and is shown in Fig. 8 .

Fig. 7 The reproduced curve from [11] of the loss modulus M” vs. Temperature at 514.5 nm in pure B₂O₃. There are three dissipation mechanisms affecting the resulting aggregate curve (orange): non-planar BO₃ distortions (blue), impurity diffusion (green), and network disintegration (red).

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Fig. 8 Extrapolated loss modulus vs. temperature at 1534 nm (solid line) and at 1064 nm (dashed line) including the three relaxation mechanisms.

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In Fig. 8 the curves show that the modulus not only depends on the temperature but also on the operating wavelength. In the comparison of the solid (1534 nm) and dashed (1064 nm) lines, the low temperature peaks may not immediately show that there are apparent shifts, but we point out that at the middle valley around 900K the loss modulus at 1064nm is very close to that at 1534nm due to the overlap of the tails of the two dominant dissipation mechanisms and broad curves. Then, at higher temperatures, the curves diverge in that the upper end of the loss modulus shifts from 1300K to 1400K (at 1 GPa), that is, the third peak has moved towards higher temperature for the smaller optical wavelength. This further implies that the third dissipation mechanism will strongly dominate at high temperature at all wavelengths, while it is possible that other unknown mechanisms with higher activation energy could also be present. In summary, the main result that is seen in Fig. 8 is that the peaks of the loss modulus curves shift to higher temperatures with decreasing optical wavelength. The loss modulus curves are also observed to be broader at a shorter optical wavelength.

Because the resulting curves shift to higher temperature with decreasing wavelength, we can see that the value of $M^{″}$ at 1064nm, for example at a temperatures higher than 1000K in Fig. 8, will be lower than that at 1534nm, while we point out to the reader in Eqs. (2) and (3) the relationships among $M^{″}$ , η´(ω), and $Δ ω_{B}$ . Therefore, taking each of these together, we may conclude that in addition to a wavenumber-dependence, $Δ ω_{B}$ will decrease when $M^{″}$ decreases due to a decrease in η´(ω). This effect is observed in the whole range of possible as-drawn fiber fictive temperatures (note the draw temperatures of Fibers 1 – 4). This suggests that the observed Brillouin spectral width of fibers derived from boric oxide-containing glasses will be smaller at 1064 nm than expected when compared with measurements at 1534 nm assuming the frequency-squared law.

We would like to point out that the spectral widths shown in Table 1 are consistent with an increasing spectral width (from Fibers 1 to 4 in order) due to an increasing fiber fictive temperature resulting from increasing fiber draw temperature. In particular, by inspecting Fig. 8, the $M^{″}$ value increases when the temperature increases (beyond ~900K) and thereby $Δ ω_{B}$ would be expected to increase with increasing fictive temperature. Therefore, we may conclude that Table 1 confirms this trend whereby $Δ ω_{B}$ increases from 99MHz to 120MHz at 1064 nm as the draw temperature increases from 1950°C to 2150°C.

Considering the additional term, η´(ω) in Eq. (2), the spectral width is related to the glass temperature. From the fitting curves of Fig. 7 (Eq. (8)), we extract the $V_{0, n}$ in order to calculate the expected spectral width of pure B₂O₃ through Eq. (9). This result is shown in Fig. 9 for the three wavelengths, 514 nm, 1064 nm, and 1534 nm. As before, we arrive at the conclusion that the fictive temperature of the fibers increases with increasing measured spectral width in Table 1. However, further complicated by the presence of germania (as well as silica) in the fiber which also influences the spectral width, it is difficult to determine a precise fictive temperature for the fibers. We are continuing to investigate how to determine these parameters from the available data. But we can speculate that since the spectral width of silica (the dominant species in the fiber) is narrower than that of boric oxide [16], and the measured spectral widths lie in the vicinity of 100 MHz at 1064nm, the fictive temperatures of the fiber samples are someplace > 1100 K.

Fig. 9 Calculated Brillouin spectral width vs. temperature for pure bulk boric oxide at three wavelengths.

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From Fig. 9, it may be expected that, in contrast to Ge-doped fibers [28], $Δ ω_{B}$ should increase as the temperature is increased for B-doped fibers. To confirm this, Fiber 4 was placed inside of a thermally-controlled chamber and $Δ ω_{B}$ was measured as a function of temperature at 1534nm. The result of this measurement is shown in Fig. 10 . It has an initially decreasing then increasing spectral width with increasing temperature. The decreasing spectral width can probably be partly attributed to Ge-related mechanisms in the fiber [28] due to Ge doping, while the increase in spectral width can be attributed to B.

Fig. 10 Brillouin spectral width vs. fiber temperature for Fiber 4 at 1534 nm. We see that the spectral width increases with increasing temperature at higher temperatures.

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Next, we consider the ratios of the measured spectral widths at 1064nm to those at 1534nm for the B₂O₃ fibers. From the data presented in Table 1 one may conclude that Eq. (2), with an additional term $η^{'} (ω) / ρ o$ , is most representative of the observed spectral widths. In order to confirm this, Fig. 11 provides the ratio of the Brillouin spectral width at 1064 nm to that at 1534 nm as a function of temperature (ratio of the green to blue curves in Fig. 9). We see that the ratio is lower than the ratio of the square of the acoustic frequencies at temperatures lower than around 3000K. This suggests that the additional term contributes a factor (η´(1064nm)/η´(1534nm)) of less than 1 to the frequency-squared law, while it decreases with decreasing temperature. In contrast, when the temperature is larger than around 3000K, this factor reaches unity and therefore the ratio approaches a constant value of about 2.08, corresponding to the ratio of the square of the acoustic frequencies. Thus, beyond this temperature the spectral width ratio follows the frequency-squared law. This suggests that at temperatures much higher than the peak of the viscosity curve the wavelength-dependence of the Brillouin spectral width reduces to the frequency-squared law. Silica (or predominantly silica) fibers, for example, would then be expected to follow the frequency-squared law since the peak of the viscosity curve would always lie well-below (at cryogenic temperatures) any expected fictive temperature of a derived fiber [16]. This is evident in the SMF-28^TM fiber (see data in Table 1), which contains about 4mol% GeO₂ in silica.

Fig. 11 The ratio of the spectral width at 1064nm to 1534nm for pure B₂O₃.

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More specifically along these lines, in contrast to the B + Ge fibers, the lightly Ge-doped silica fiber data shows that the spectral width roughly follows the frequency-squared law. For a simulation of silica glass, we refer to Fig. 12(a) , where based on the work found in [16,29] a rough example of an arbitrary glass with a single-mechanism and a peak viscosity at cryogenic temperatures is provided. (Fig. 12(b) shows the corresponding spectral width dependence on temperature at a wavelength of 1064nm.) It is seen that the curve reaches the ratio value greater than 2.0 at temperatures above 200K. Therefore, for all possible fiber fictive temperatures (T > room or ambient temperature), the silica glass has already reached this constant ratio value.

Fig. 12 (a) The ratio of the spectral width at 1064nm to 1534nm and (b) the spectral width of 1064nm for the pure silica simulation.

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With respect to the fibers in Table 1, we can see that there does indeed appear to be a dependence of measured spectral width on the fiber draw temperature, and that the ratios of observed spectral widths may in fact be a way to gauge the fictive temperature of a fiber. Along these lines, one may conclude that fibers 5, 6, and 7 are drawn at temperatures close to that of fiber 4 (2150 °C). Another observation is that in our data, the ratio from Fiber 2 to Fiber 4 changes from 1.57 to 1.76 with a change in draw temperature of 200 K. We will initially neglect the anomalous data point observed for Fiber 1. This is consistent with the slope of the curve in Fig. 11 in the region of 1400K to 2400K. Furthermore, since silica follows the frequency-squared law for any anticipated fiber fictive temperature, the addition of it to pure boric oxide is expected to increase the ratio of the spectral widths (to closer to the frequency-squared law) calculated in Fig. 11 for regions less than about 3000K. Thus, since our maximum observed ratio is 1.76, we can state based on Fig. 11 that the fibers’ fictive temperatures probably lie below 1900K. From this observation, it is clear that the fictive temperature is less than the draw temperature, consistent with some glass relaxation during the fiber draw process.

Finally, we point out that this is a coarse analysis, and not a complete one. While the study above assumes bulk B₂O₃, the present case is in reality complicated by the presence of germania (as well as silica) in the fiber which also influences the spectral width. Therefore, in order to comprehend the B + G fiber characteristics, we need to further understand the various relationships among dopants, temperature, precise oxide concentrations, index of refraction, acoustic modes, and so on. We are working to extend this analysis with a model for several mixed materials. This may prove to be useful in the design of optical fibers with tailored Brillouin properties for a number of different applications.

5. Conclusion

We have presented Brillouin spectral width measurements of B₂O₃ and GeO₂ co-doped silica fibers at two wavelengths, 1064 nm and 1534 nm. Our data indicates that fibers containing boric oxide do not follow the conventional frequency-squared law for the Brillouin spectral width. Instead the measured ratio appears to depart from the ratio of the square of the acoustic frequencies. We have presented an analysis for bulk B₂O₃ in order to explain this behavior. It was proposed that since the peak of the viscosity versus temperature curve lies within the vicinity (or greater than that) of the anticipated fiber fictive temperature, a usually-neglected dynamic viscosity term reduces this ratio. In contrast, at fiber fictive temperatures much higher than the peak of the viscosity curve, the wavelength-dependence of the Brillouin spectral width reduces to the frequency-squared law.

Acknowledgements

This work was supported in part by the Joint Technology Office (JTO) through their High Energy Laser Multidisciplinary Research Initiative (HEL-MRI) program entitled “Novel Large-Mode-Area (LMA) Fiber Technologies for High Power Fiber Laser Arrays” under ARO subcontract # F014252. The authors are grateful to Nufern of East Granby, CT for the custom fabrication of Fibers 1 – 4, the fiber samples, and for several fruitful discussions with their team. The authors would also like to acknowledge Professor Gary R. Swenson for his full support.

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Fiber	1534 nm		1064 nm		Ratio $Δ ν_{B}^{1064} / Δ ν_{B}^{1534}$
Fiber	Δν_B (MHz)	ν_B (GHz)	Δν_B (MHz)	ν_B (GHz)	Ratio $Δ ν_{B}^{1064} / Δ ν_{B}^{1534}$
1) 1900°C	72	10.057	98	14.535	1.36
2) 1950°C	63	10.118	99	14.640	1.57
3) 2050°C	64	10.179	105	14.717	1.64
4) 2150°C	68	10.195	120	14.746	1.76
5) 1060PS	98	8.525	170	12.333	1.73
6) HPS980	115	8.476	201	12.237	1.75
7) Newport F-SBG	122	8.478	210	12.240	1.72
8) Corning SMF-28^TM	30	11.008	61	15.922	2.06

Wavelength dependence of the Brillouin spectral width of boron doped germanosilicate optical fibers

Abstract

1. Introduction

2. Experimental setup

3. Experimental results

4. Discussion

4.1 Theory

4.2 Analysis

5. Conclusion

Acknowledgements

References and links

Cited By

Figures (12)

Tables (1)

Equations (9)

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