## Abstract

We report the theoretical description and the experimental demonstration of an optical resonator formed by inserting a Fiber Bragg Grating (FBG) in a closed fiber loop. The spectral characteristics of such a resonator strongly depend on the reflectivity of the FBG. In the wavelength region where the FBG reflectivity *R* is negligible, the system behaves like a conventional ring resonator. On the other hand, when *R* is not vanishing, a split-mode structure can be observed, associated to the degeneracy removal of two counterpropagating resonant modes. The magnitude of the mode splitting can be used to sense small variations of the FBG physical parameters, such as length, temperature or group index. An example of strain sensing with this setup is reported, showing that the mode splitting is sensitive to a mechanical strain applied to the FBG, while it is almost insensitive to a strain applied to any other point of the resonator. This peculiar feature allows to perform cavity-enhanced, local strain measurements with a reduced sensitivity to environmental perturbations, which represents an important improvement in the framework of the fiber-optic sensors.

© 2013 Optical Society of America

## 1. Introduction

Fiber Bragg Gratings (FBGs) and Fiber Ring resonators are well-established optical devices widely utilized for their simple manufacturing, small dimensions, low cost and immunity to the EM radiation. FBGs and ring resonators are used in innumerable applications, ranging from telecommunications to physical, chemical and inertial sensing [1–6].

An experimental investigation of the spectral characteristics of fiber ring resonators has been reported in [7], focussing the attention on the power transfer characteristics in the undercoupled, critically coupled and overcoupled regimes. The advantage shown by these resonators consists in the possibility of obtaining tunable optical delay lines without requiring a high finesse by cascading different rings. In [8] a fiber ring optical resonator, formed by adiabatic fused-FBG couplers, has been proposed to control the coupling efficiency (via the grating strength) to switch the device into the three above mentioned operative regimes. In [9] the possibility of carrying out strain-free displacement measurements has been proposed by adopting a fiber ring containing a gap.

In this paper, we theoretically describe and experimentally demonstrate a novel fiber-optic strain sensor obtained by inserting a FBG in a closed fiber loop. The proposed sensing mechanism shows a cavity-enhanced sensitivity to a strain localized in the FBG region while being substantially insensitive to perturbations that affect the rest of the fiber loop.

#### FBG loop configuration

The optical fiber resonator configuration (see Fig. 1) is made by closing a conventional single-mode silica fiber which includes a FBG in a ring geometry using two fused fiber couplers.

As shown in Fig. 1, a linearly polarized optical field from a laser source (*E _{in}*) is launched through the optical fiber in order to excite the resonator modes via the evanescent coupler 1. A photodiode is placed at one of the ends of coupler 2 to monitor the cavity spectral response ${\left|{E}_{out1,2}\right|}^{2}$. Each coupler is modelled as a lumped element by the following transfer matrix, having a unitary determinant if the insertion loss is neglected [10]:

*E*

_{i}_{1}

*,*

_{c}*E*

_{i}_{2}

*are the input amplitudes of the coupler and*

_{c}*E*

_{o}_{1}

*,*

_{c}*E*

_{o}_{2}

*are the coupler outputs.*

_{c}*τ*and

*k*are the fractions of the optical amplitudes transmitted and coupled by the evanescent coupler, respectively. The FBG is also modelled via a transfer matrix [8]:

*E*

_{i}_{1},

*E*

_{i}_{2}are the amplitudes of the electric fields entering the FBG and

*E*

_{o}_{1},

*E*

_{o}_{2}are those of the electric fields exiting the FBG (see Fig. 1).

*t*and

*r*are the fractions of the optical field amplitudes transmitted and reflected via the FBG, respectively, and they are linked through the unitary determinant of Eq. (2), analogously to

*τ*and

*k*in Eq. (1). Being the FBG a dispersive element,

*t*and

*r*can be expressed as [8]:

*Θ*,

*Κ*and

*Δβ*can be expressed as functions of the effective fiber refractive index

*n*, the FBG index modulation depth Δ

*n*and its central wavelength

*λ*in the following way [8]:

_{b}*L*is given by the sum of ${\ell}_{b}$ and the remaining fiber length $\ell $, respectively. The spectral response,

*T*of the proposed device can be derived by utilizing the matrix formalism of [11]:

*L/2*due to the loss

*α*per unit length. The right hand side of Eq. (5) is a combination of two solutions with opposite phases, which are commonly referred to as “symmetric” and “antisymmetric” modes.

To describe the physical behaviour of this system, a simulation of the transmission spectrum was performed. It is shown in Fig. 2 along with the reflectivity curve of the FBG for wavelengths close to *λ _{b}*. In the spectral regions where

*r ~0*(left side of Fig. 2), the intracavity power circulates only in the direction of excitation (transmission is only detectable at the output port 1), and the symmetric and antisymmetric solutions of Eq. (5) coincide. In this case, referred to as “degenerate mode condition” [12], the device behaves as a conventional ring resonator. In the spectral region where

*r ≠ 0*instead, part of the intracavity power is scattered by the FBG and excites a counterpropagating mode. The subsequent redistribution of the intracavity power leads to a reduction of the signal detected at the output port 1. The non-vanishing

*R*also removes the degeneracy of the symmetric/antisymmetric solutions in Eq. (5), giving rise to a transmission spectrum with split resonant modes [12]. The magnitude of the mode splitting in the non-degenerate mode condition is directly related to the reflectivity of the FBG, and can be expressed, in analogy to the case of a waveguide microresonator [11], as:

#### The FBG loop as a strain sensor

It is easy to see from Eq. (5) and Fig. 3 that the system can transition from the degenerate to the non-degenerate mode by varying the FBG reflectivity. Since the latter is a function of $\lambda -{\lambda}_{b}$, this can be done by either tuning the laser across the FBG reflectivity curve, or by shifting the central Bragg wavelength. The amount of splitting of a resonance can thus be used to sense a variation of the FBG physical parameters.

We now consider in particular a perturbation of the cavity length Δ*L*. Like in any resonator, the variation of the FSR causes any resonance at wavelength λ to shift by an amount $\Delta \lambda =0.79\lambda \frac{\Delta L}{L}$, where 0.79 is the photoelastic coefficient of the fiber [12]. According to Eq. (6), a variation of the resonance splitting $\Delta {S}_{\lambda}$ is also associated to Δ*L*. In this respect, it is interesting to note that *S _{λ}* scales inversely with the overall cavity length $L={\ell}_{b}+\ell $, but also contains an implicit dependence on ${\ell}_{b}$ alone, through the reflectivity

*R*. The same perturbation can then result in different values of $\Delta {S}_{\lambda}$, depending on whether it affects $\ell $ or ${\ell}_{b}$. The sensitivity of

_{λ}*S*in these two cases can be obtained by differentiating Eq. (6) separately with respect to $\delta \epsilon =\frac{\delta \ell}{L}$ and $\delta {\epsilon}_{b}=\frac{\delta {\ell}_{b}}{L}$.

_{λ}*S*. However, if the strain applied involves the FBG length ${\ell}_{b}$, an additional term appears, associated to the change of the reflectivity

_{λ}*R*caused by the shift of the Bragg wavelength, as shown by Eq. (7b). It is important to notice that this term is independent of

_{λ}*L*, whereas

*S*scales inversely with it. Therefore, for a sufficiently long fiber loop, the splitting is only sensitive to local perturbations of the FBG length. This is a crucial advantage with respect to a traditional cavity with Bragg mirrors [1], where none of the observables available (transmitted power or resonance shift) can distinguish the strain applied to a specific point along the resonator.

_{λ}It is also interesting to notice that, unlike conventional cavity-based strain sensors, the sensitivity to variation of the FBG length depends on the reflectivity curve of the grating itself. To optimize the sensor, the terms on the right hand side of Eq. (7b) must be considered. In particular, $\frac{\partial {R}_{\lambda}}{\partial {\ell}_{b}}=\frac{\partial {R}_{\lambda}}{\partial {\lambda}_{b}}\frac{\partial {\lambda}_{b}}{\partial {\ell}_{b}}=\frac{\partial {R}_{\lambda}}{\partial {\lambda}_{b}}0.79{\lambda}_{b}\frac{\Delta {\ell}_{b}}{{\ell}_{b}}$ describes the overall response of the FBG reflectivity to a variation of its length. Its value increases for smaller ${\ell}_{b}$ and/or a narrower reflectivity curve. The factor $\frac{1}{2\sqrt{{R}_{\lambda}{\left(1-{R}_{\lambda}\right)}^{3}}}$ instead points out that the sensitivity is higher for either an extremely small or high FBG reflectivity. The optimal working point of the sensor is then a tradeoff between the maximum slope of the reflectivity curve and the maximum/minimum value of *R _{λ}*.

## 2. Experimental setup and results

The theoretical predictions on the described strain sensing mechanism were experimentally demonstrated using the setup shown in Fig. 3. The coherent radiation source is a narrow linewidth fiber laser (Koheras Adjustik) with center wavelength 1560.10 nm and a power of 42 mW, modulated by a ramp generator to visualize the cavity modes. The polarization state of the laser is controlled by a Lefevre-ring controller in order to minimize cavity resonances associated with the excitation of orthogonal polarization states [13]. The ring cavity is formed by using an FBG (measured reflectivity of 15%, Reflection Bandwidth 800 pm and insertion loss of 2.7 dB) integrated in a Single Mode Fiber (SMF) loop having $\ell $ = 5.98 m (FSR = 34.4 MHz). Radiation is injected in the ring resonator by means of two fiber couplers with 5% efficiency and an insertion loss of 0.2 dB. The signal exiting the FBG ring cavity is detected by a photodiode and visualized on a digital oscilloscope.

In a first instance, the laser source was tuned at the edge of the non-degenerate region of the spectrum, which represent the starting point of our measurements. Then, a strain *ε _{b}* was applied to the intracavity FBG by a piezoelectric transducer, as sketched in Fig. 3, in the direction of increasing splitting. A minimum strain of 25 n

*ε*was necessary to achieve a measurable mode splitting (1.4 MHz). Smaller splitting magnitudes are in fact hard to resolve because of the finite linewidth of the cavity resonances. A few examples of transmission spectra obtained for discrete values of the piezo voltage are reported in Fig. 4. The amount of strain corresponding to the different voltages was calculated from the resonance shifts $\Delta \lambda =0.79\lambda {\epsilon}_{b}$, also visible in Fig. 4.

In a second step, we applied an equivalent strain *ε* to a point of the fiber loop far from the FBG. To do so, voltages producing the same shifts of the interrogated cavity resonance were sent to another piezoelectric transducer (see Fig. 3). The splitting magnitudes associated to equivalent values of strain applied to the FBG and to the fiber loop are plotted in Fig. 5 along with the corresponding theoretical curves.

To calculate the latter, the grating was previously interrogated with a broadband spontaneous emission source (ASE). Its reflectivity curve was recorded with an optical spectrum analyzer and fitted to a theoretical model to retrieve the FBG parameters ${R}_{0}=0.15$, ${\ell}_{b}=6$ mm and $\Delta n=3.4\cdot {10}^{-5}$. These values were then substituted in Eqs. (3) and (6) to plot the theoretical curves of Fig. 5, that perfectly matches the experimental data.

The results above reported clearly show that the resonance splitting is only sensitive to variations of the FBG parameters and in fact immune from perturbations affecting all the rest of the resonator. This demonstrates the possibility to use the described scheme to measure small mechanical deformations, temperature or refractive index variations of the FBG with a cavity enhanced sensitivity and a reduced noise level. The resolution (i.e. the minimum detectable strain) of the demonstrated sensor depends on the ability to resolve the resonant wavelengths of the split modes, and therefore is ultimately limited by the finite linewidth of the cavity resonances (spectral resolution). However, the sensitivity of the splitting measurement can be improved by acting on the FBG parameters, as shown by Eq. (7b). With an optimized FBG design an overall resolution in the order of a few nε can be readily achieved, even with the current spectral resolution.

In conclusion, we have presented a theoretical description of a fiber Bragg grating enclosed in a fiber loop resonator, verified experimentally the model predictions and demonstrated that the device can be used to sense a localized strain applied to the FBG with a sensing mechanism immune to environmental perturbations of the fiber length. We believe that this simple scheme has strong potential for spectroscopic-based strain sensing with FBGs and may find several applications in the fields of opto-mechanics, quantum optics, physics of light-matter interactions and many others.

## Acknowledgements

This work was funded by Italian Ministry of Education and Research (MIUR) in the framework of PON01_01209 “BACKOP” project. We thank G. Notariale for technical assistance.

## References and links

**1. **G. Gagliardi, M. Salza, S. Avino, P. Ferraro, and P. De Natale, “Probing the ultimate limit of fiber-optic strain sensing,” Science **330**(6007), 1081–1084 (2010). [CrossRef] [PubMed]

**2. **S. Avino, A. Giorgini, M. Salza, M. Fabian, G. Gagliardi, and P. De Natale, “Evanescent-wave comb spectroscopy of liquids with strongly dispersive optical fiber cavities,” Appl. Phys. Lett. **102**(20), 201116 (2013). [CrossRef]

**3. **C. Ciminelli, C. M. Campanella, F. Dell’Olio, C. E. Campanella, and M. N. Armenise, “Label-free optical resonant sensors for biochemical applications,” Prog. Quantum Electron. **37**(2), 51–107 (2013). [CrossRef]

**4. **C. Ciminelli, C. E. Campanella, and M. N. Armenise, “Optimized design of integrated optical angular velocity sensors based on a passive ring resonator,” J. Lightwave Technol. **27**(14), 2658–2666 (2009). [CrossRef]

**5. **G. Gagliardi, M. Salza, P. Ferraro, P. De Natale, A. Di Maio, S. Carlino, G. De Natale, and E. Boschi, “Design and test of a laser-based optical-fiber Bragg-grating accelerometer for seismic applications,” Meas. Sci. Technol. **19**(8), 085306 (2008). [CrossRef]

**6. **T. T.-Y. Lam, G. Gagliardi, M. Salza, J. H. Chow, and P. De Natale, “Optical fiber three-axis accelerometer based on lasers locked to π phase-shifted Bragg gratings,” Meas. Sci. Technol. **21**, 094010 (2010). [CrossRef]

**7. **J. E. Heebner, V. Wong, A. Schweinsberg, R. W. Boyd, and D. J. Jackson, “Optical transmission characteristics of fiber ring resonators,” IEEE J. Quantum Electron. **40**(6), 726–730 (2004). [CrossRef]

**8. **J. M. Choi, R. K. Lee, and A. Yariv, “Ring fiber resonators based on fused-fiber grating add-drop filters:application to resonator coupling,” Opt. Lett. **27**(18), 1598–1600 (2002). [CrossRef] [PubMed]

**9. **F. Vollmer and P. Fischer, “Frequency-domain displacement sensing with a fiber ring-resonator containing a variable gap,” Sens. Actuators A Phys. **134**(2), 410–413 (2007). [CrossRef]

**10. **A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. **36**(4), 321 (2000). [CrossRef]

**11. **J. Čtyroký, I. Richter, and M. Sinor, “Dual resonance in a waveguide coupled ring micro-resonator,” Opt. Quantum Electron. **38**(9–11), 781–797 (2006). [CrossRef]

**12. **C. Ciminelli, C. E. Campanella, F. Dell’Olio, and M. N. Armenise, “Fast light generation through velocity manipulation in two vertically-stacked ring resonators,” Opt. Express **18**(3), 2973–2986 (2010). [CrossRef] [PubMed]

**13. **C. Ciminelli, C. E. Campanella, and M. N. Armenise, “Structural polarization conversion in integrated optical vertically stacked ring resonators,” Opt. Laser Technol. **48**, 294–301 (2013). [CrossRef]