6. Measurement of Pockels coefficients in optical fibers

The elasto-optic effect consists on the variation in the refractive index generated by any strain applied to the fiber. The correspondent elasto-optic coefficients are usually determined by measuring the optical activity induced by a mechanical twist and the phase change induced by longitudinal strain [32, 33]. This technique relies on the use of the conventional axial modes propagating through the fiber. Since these modes are essentially transverse to the axis of the fiber [34], the anisotropy of the elasto-optic effect does not show up. On the contrary, WGMs have a significant longitudinal component; hence, their optical fields experience the anisotropy of the elasto-optic effect intrinsically. In the last years, researchers have demonstrated a number of fiber devices in which the longitudinal components of the electromagnetic modes are significant, such as microfibers [35] and microstructured optical fibers with a high air-filling fraction [36]. For these cases, the measurement and characterization of the anisotropy of the elasto-optic effect and its Pockels coefficients are of high interest. Roselló-Mechó et al. reported a technique based on the different wavelength shifts of TE- and TM-WGM resonances in a fiber under axial strain, to measure these coefficients [37]. This technique has the additional advantage that, since it does not involve the conventional modes of the fiber, there is no need that the FUTs are single mode in order to carry out the measurements. Then, the coefficients can be measured at different wavelengths to determine their dispersion; this is a limitation of the usual technique based on the optical activity which is overcome by means of WGM technique [38].

According to Eq. (1), a variation in the refractive index will tune the WGM resonances in wavelength. In this case, an axial strain will be applied to the FUT in order to induce this variation in the index, due to the elasto-optic effect. This feature was applied in different works in order to tune the WGM resonances

resonances was measured as the laser power was increased, at two different points, one within the irradiated section and one outside it. The data does not show any sign of saturation of the heating, at this range of power. The temperature of the irradiated section increased linearly, at a rate of 26:48 � 0:15 ºC=W, and at

(a) Direct measurement of the loss as the PS980 fiber is irradiated. (b) Heating of the PS980 fiber as a function

PS980 36:<sup>9</sup> � <sup>0</sup>:7 26:<sup>48</sup> � <sup>0</sup>:15 0:<sup>718</sup> � <sup>0</sup>:014 52 � 3 6200 � 400 120 � 23 PS1250 40:<sup>1</sup> � <sup>0</sup>:8 30:<sup>80</sup> � <sup>0</sup>:17 0:<sup>768</sup> � <sup>0</sup>:014 50 � 3 6600 � 400 131:14 SM1500 >40<sup>1</sup> <sup>1</sup>:<sup>20</sup> � <sup>0</sup>:03 < 0:03<sup>2</sup> <sup>190</sup> � 50 370 � 90 1.95<sup>4</sup> H2-SMF28 28:<sup>8</sup> � <sup>0</sup>:5 23:<sup>48</sup> � <sup>0</sup>:13 0:<sup>815</sup> � <sup>0</sup>:012 n/a<sup>2</sup> <sup>5600</sup> � 400 n/a<sup>1</sup>

ΔT=P ºCð =WÞ

WGM technique Direct measurements

α2=α<sup>1</sup>

Irradiated Pristine Irradiated Pristine

α (dB/km)

0:718 � 0:014 ºC=W in the pristine region. The ratio between these values, that is,

This process was repeated for all the different fibers mentioned before: PS1250, SM1500, and hydrogenated SMF28, at 1550. Table 1 includes the results from the measurements and the corresponding analysis: α2=α<sup>1</sup> was obtained for each of them

The results, compiled in Table 1, allow establishing several conclusions of interest. First, as expected, the absorption coefficient is substantially increased due to the UV irradiation. As a consequence, even for signals of moderate powers, FBGs might experience shifts and chirps that should be taken into account [31]. Second,

et al. analyzed the measurements to demonstrate that these results lead to the conclusion that scattering loss increases at a higher rate than absorption loss [11]. Finally, Eq. (6) can be used to calculate the absolute value of the absorption and scattering coefficients by taking into account the values of h and a for a silica

<sup>2</sup> =αabs

<sup>1</sup> was calculated from the

<sup>1</sup> . Roselló-Mechó

<sup>2</sup> =αabs

the ratio αabs

148

Figure 9.

1

2

3

4

Table 1.

Nonavailable.

Nominal value.

Below detection limit.

Cutback measurement.

of the illumination power.

<sup>1</sup> � <sup>α</sup>abs <sup>2</sup> =αabs 1

Applications of Optical Fibers for Sensing

<sup>2</sup> =αabs

technique based in WGMs.

<sup>1</sup> , is 36:9 � 0:7 ºC=W.

Measurement of thermal heating and loss coefficient of different fibers.

from the direct measurement of the loss, while αabs

the results show that α2=α<sup>1</sup> is systematically higher than αabs

[39, 40]. However, there was not any mention to the different behaviors of TE- and TM-WGM.

An axial strain introduces a refractive index perturbation in an isotropic, cylindrical MR, due to the elasto-optic effect, which will be different for the axial (Δnz) and transversal directions (Δnt):

$$\frac{\Delta n\_t}{n\_0} = -p\_{et}\varepsilon; \qquad p\_{et} \equiv \frac{n\_0^2}{2} (p\_{12} - \nu (p\_{11} + p\_{12})), \tag{8}$$

$$\frac{\Delta n\_x}{n\_0} = -p\_{ex}\varepsilon; \qquad p\_{ex} \equiv \frac{n\_0^2}{2} \left(p\_{11} - 2\nu p\_{12}\right),\tag{9}$$

into account the Sellmeier coefficients for the value of the refractive index at

1531 nm 1064 nm

Whispering Gallery Modes for Accurate Characterization of Optical Fibers' Parameters

The measurements were repeated at 1064 nm, to study the dispersion of the elasto-optic effect. Results at both wavelengths are compiled in Table 3 and are compared with those reported in the literature. Both sets of measurements are in good agreement, and the small differences might be due to the fact that the technique based in WGM measures the pij of the cladding material (i.e., fused silica), while in the case of the other techniques, the coefficients are determined by the material of the fiber core, which is usually silica doped with other elements.

Present work Literature

0.113 @ 633 nm [32] 0.121 @ 633 nm [41]

0.252 @ 633 nm [32] 0.270 @ 633 nm [41]

In this chapter, we described a technique based on the excitation of WGMs around cylindrical MRs, to measure properties of the MR material. The resonant nature of the WGMs confers this technique with high sensitivity and low detection limits. Also, the technique allows measuring these parameters with axial resolution; hence, it is possible to detect changes of the parameters point to point along the MR. The technique has been applied to different experiments. Mainly, thermo-optic effect and elasto-optic effect have been investigated in silica fibers. The variation in the index, due to a change in the temperature or strain, rules the shift in wavelength of the WGM resonances. When the technique was applied to different types of fibers and components, different information were obtained from the experiments. In particular, we measure temperature profiles in pumped, rareearth doped fibers and in FBGs; the absorption coefficient in irradiated photosensitive fibers; and the Pockels coefficients in telecom fibers. Novel results were obtained: for example, it was possible to measure absorption and scattering loss coefficients separately, and, also, the anisotropy of the elasto-optic effect was observed experimentally. The information provided by the WGM-based technique might help to optimize the fabrication procedures of doped fibers and fiber

This work was funded by Ministerio de Economía y Competitividad of Spain and

FEDER funds (Ref: TEC2016-76664-C2-1-R) and Generalitat Valenciana (Ref: PROMETEOII/2014/072), Universitat de València (UV-INV-AE16-485280). X. Roselló-Mechó's contract is funded by the FPI program (MinECo, Spain, BES-2014-068607). E. Rivera-Pérez's contract is funded by the Postdoctoral Stays in

Foreigner Countries (291121, CONACYT, Mexico).

1531 nm and Poisson's ratio of ν ¼ 0:17 � 0:01.

p<sup>11</sup> 0.116 0.131

DOI: http://dx.doi.org/10.5772/intechopen.81259

p<sup>12</sup> 0.255 0.267

Comparison of experimental pij values with those reported in the literature.

7. Conclusions

Table 3.

components as FBGs or LPGs.

Acknowledgements

151

where n<sup>0</sup> is the unperturbed index of the MR, pij are the elasto-optic coefficients, ν is Poisson's ratio, and ε is the strain applied to the MR. The coefficients pet and pez are the effective elasto-optic coefficients, which are defined for simplicity. According to the reported values for the elasto-optic coefficients for fused silica (p<sup>11</sup> ¼ 0:121, p<sup>12</sup> ¼ 0:27 [41], ν ¼ 0:17 [42]), the ratio Δnt=Δnz≈6:97; hence, it is expected that the strain introduces a significant differential anisotropy. With this in mind, Maxwell's equations will be solved considering the uniaxial tensor given by Eq. (2). The solutions, as mentioned before, split in two families of WGM, the TE and TM modes, whose resonant frequencies will be obtained by solving Eqs. (3) and (4).

The refractive index perturbation is not the only factor to take into account when evaluating the wavelength shift of WGM resonances due to strain: the radius a of the MR also varies with it according to Poisson's ratio, Δa=a ¼ �νε.

With all these ideas in mind, the relative shift of the WGM resonances, ΔλR=λR, can be characterized as a function of the strain, for TE- and TM-WGMs. Figure 10a shows an example of the anisotropic behavior of TE- and TM-WGM. The strain applied to the MR was 330 με for both polarizations, and the measured wavelength shift was different for each of them: 0.18 nm for TE-WGM and 0.11 nm for TM-WGM. ΔλR=λ<sup>R</sup> was measured as a function of the strain in detail at 1531 nm, for both polarizations; the results are shown in Figure 10b. A linear trend in both cases can be observed: the slopes of the linear regressions that fit the experimental values are <sup>s</sup>TE ¼ �0:<sup>369</sup> � <sup>0</sup>:<sup>006</sup> με�<sup>1</sup> for the TE- and <sup>s</sup>TM ¼ �0:<sup>201</sup> � <sup>0</sup>:<sup>004</sup> με�<sup>1</sup> for the TM-WGM. The ratio sTE=sTM ¼ 1:84 shows the anisotropy of the elasto-optic effect. From these values, it is possible to calculate the elasto-optic coefficients pij with its uncertainties (see [37] for a more detailed description of the procedure), by taking

### Figure 10.

(a) Wavelength shift of TE-/TM-WGM resonances for ε ¼ 330 με. (b) Measurement of the wavelength shift as a function of the strain.

Whispering Gallery Modes for Accurate Characterization of Optical Fibers' Parameters DOI: http://dx.doi.org/10.5772/intechopen.81259


Table 3.

[39, 40]. However, there was not any mention to the different behaviors of TE- and

¼ �pet <sup>ε</sup>; pet � <sup>n</sup><sup>2</sup>

¼ �pez <sup>ε</sup>; pez � <sup>n</sup><sup>2</sup>

are the effective elasto-optic coefficients, which are defined for simplicity. According to the reported values for the elasto-optic coefficients for fused silica (p<sup>11</sup> ¼ 0:121, p<sup>12</sup> ¼ 0:27 [41], ν ¼ 0:17 [42]), the ratio Δnt=Δnz≈6:97; hence, it is expected that the strain introduces a significant differential anisotropy. With this in mind, Maxwell's equations will be solved considering the uniaxial tensor given by Eq. (2). The solutions, as mentioned before, split in two families of WGM, the TE and TM modes, whose resonant frequencies will be obtained by solving Eqs. (3)

An axial strain introduces a refractive index perturbation in an isotropic, cylindrical MR, due to the elasto-optic effect, which will be different for the axial (Δnz)

0

where n<sup>0</sup> is the unperturbed index of the MR, pij are the elasto-optic coefficients, ν is Poisson's ratio, and ε is the strain applied to the MR. The coefficients pet and pez

The refractive index perturbation is not the only factor to take into account when evaluating the wavelength shift of WGM resonances due to strain: the radius

With all these ideas in mind, the relative shift of the WGM resonances, ΔλR=λR, can be characterized as a function of the strain, for TE- and TM-WGMs. Figure 10a shows an example of the anisotropic behavior of TE- and TM-WGM. The strain applied to the MR was 330 με for both polarizations, and the measured wavelength shift was different for each of them: 0.18 nm for TE-WGM and 0.11 nm for TM-WGM. ΔλR=λ<sup>R</sup> was measured as a function of the strain in detail at 1531 nm, for both polarizations; the results are shown in Figure 10b. A linear trend in both cases can be observed: the slopes of the linear regressions that fit the experimental values are <sup>s</sup>TE ¼ �0:<sup>369</sup> � <sup>0</sup>:<sup>006</sup> με�<sup>1</sup> for the TE- and <sup>s</sup>TM ¼ �0:<sup>201</sup> � <sup>0</sup>:<sup>004</sup> με�<sup>1</sup> for the TM-WGM. The ratio sTE=sTM ¼ 1:84 shows the anisotropy of the elasto-optic effect. From these values, it is possible to calculate the elasto-optic coefficients pij with its uncertainties (see [37] for a more detailed description of the procedure), by taking

(a) Wavelength shift of TE-/TM-WGM resonances for ε ¼ 330 με. (b) Measurement of the wavelength shift as

a of the MR also varies with it according to Poisson's ratio, Δa=a ¼ �νε.

0

<sup>2</sup> <sup>p</sup><sup>12</sup> � <sup>ν</sup> <sup>p</sup><sup>11</sup> <sup>þ</sup> <sup>p</sup><sup>12</sup>

<sup>2</sup> <sup>p</sup><sup>11</sup> � <sup>2</sup>νp<sup>12</sup>

, (8)

, (9)

TM-WGM.

and (4).

Figure 10.

150

a function of the strain.

and transversal directions (Δnt):

Applications of Optical Fibers for Sensing

Δnt n0

> Δnz n0

Comparison of experimental pij values with those reported in the literature.

into account the Sellmeier coefficients for the value of the refractive index at 1531 nm and Poisson's ratio of ν ¼ 0:17 � 0:01.

The measurements were repeated at 1064 nm, to study the dispersion of the elasto-optic effect. Results at both wavelengths are compiled in Table 3 and are compared with those reported in the literature. Both sets of measurements are in good agreement, and the small differences might be due to the fact that the technique based in WGM measures the pij of the cladding material (i.e., fused silica), while in the case of the other techniques, the coefficients are determined by the material of the fiber core, which is usually silica doped with other elements.
