*R*, sensor linearity and sensitivity are analyzed. A calibration using an optical attenuator is reported to validate the model.

_{M}## 1.

## Introduction

Fiber-optic intensity sensors (FOS) based on multimode^{1} (MM) and single mode^{2} (SM) fibers need a self-referencing method to minimize the influences of long-term aging of source characteristics, as well as short-term fluctuations in optical power loss in the leads to and from the transducer. Time division, wavelength normalization,^{1}
^{3} and frequency-based self-referencing methods^{4}
^{5} based on MM fibers, and a Michelson topology with SM fibers^{6} have been reported.

In this letter, we propose a novel frequency-based approach using a ring resonator (RR), with an improved sensitivity. Its principle and properties are discussed and tested.

## 2.

## Theoretical Analysis

The new sensing scheme is a RR and a FOS (see Fig. 1). The RR operates under an incoherent regime, so τ≫Tc, where Tc is the source coherence time and τ is the loop transit time. The RR relative output power, P_{3}/P_{1}
, is given by:

## (1)

$$g\sqrt{\frac{{K}^{2}+{[(1-2\cdot K)\cdot H]}^{2}+2\cdot K\cdot (1-2\cdot K)\cdot H\cdot \mathrm{cos}(\omega \cdot \tau )}{1+{(K\cdot H)}^{2}-2\cdot K\cdot H\cdot \mathrm{cos}(\omega \cdot \tau )},}$$*m*is the measurand, F(m) is the FOS calibration curve, γ and

*K*are the coupler excess loss and coupling coefficient ω is the modulating signal, pulsation α is the fiber attenuation coefficient in dB/km,

*A*is an attenuation, and

*L*is the loop length. The FOS modulates the RR loss,

*H*, and the output power frequency P

_{3}/P

_{1}, see inset of Fig. 1 for K∈(0−0.5); there is a constant maximum if cos(ωτ)=+l , and a dependent on

*H*minimum if cos(ωτ)=−l . The

*frequency normalization method*is based on the sinusoidal modulation of the optical power source at two frequencies f

_{1}and f

_{2}, as seen in Figs. 1 and 2. In this method, the measurement parameter is R

_{M1}:

## (3)

$${R}_{M1}=\frac{\left|\frac{{P}_{3}}{{P}_{1}}\right|(\omega ,\tau )}{|\frac{{P}_{3}}{{P}_{1}}{|}_{|\mathrm{cos}(\omega \tau )=1}}=\frac{\left|{P}_{3}\right|(\omega ,\tau )}{|{P}_{3}{|}_{|\mathrm{cos}(\omega \tau )=1}}.$$*two ports normalization method*uses a single frequency (f

_{1}), a coupler inside the RR for measuring P

_{4}, and two down leads under identical external conditions; and the measurement parameter is R

_{M2}:

## (4)

$${R}_{M2}=\frac{|{P}_{3}|(\omega \text{,}\tau )}{|{P}_{4}{|}_{|\mathrm{cos}(\omega \tau )=-1}}.$$## (5)

$$\frac{1}{{R}_{Mi}}\left(\frac{\partial {R}_{Mi}}{\partial m}\right)=\frac{\partial {R}_{Mi}}{{R}_{Mi}\partial H}\left(\frac{\partial H}{\partial m}\right)={S}_{Mi}{k}_{1}{S}_{F}$$_{F}=δF/δm the FOS sensitivity, k

_{1}is a constant and i=1, 2 for the frequency and two ports normalization method, respectively. This system sensibility is enhanced by S

_{Mi}. If f

_{1}is the resonance frequency, S

_{M1}is given by: So S

_{M1}tends to ∞ if H→H

_{0}=K/(1–2K), The presence of noise limits the real value of the sensitivity. S

_{M1}is plotted at Fig. 2, for a f

_{1}frequency of 1,302 MHz, in a RR with a loop length of 1067 m. There is an inflection point for every

*K*at the H

_{0}value. For every quiescent point, a certain

*K*can be selected for achieving high sensitivities. S

_{M2}behaves quite similar to S

_{M1}.

## 3.

## Measurements

The experimental setup is made of a LD of 1.5 μm, with 5 MHz linewidth, internally modulated with a signal coming from the tracking generator of a RF spectrum analyzer. The sensing scheme (see Fig. 1) is made of a polarization maintaining 2×2 variable ratio fiber coupler with pigtails of 1 m, 1067 m of standard SM fiber, and an attenuator simulating the FOS. f_{1}
is 1.302 MHz, f_{2}
is 1.207 MHz, and K=0.22. The calibration curves, for both self-referencing methods, are reported in Fig. 3. There is a great agreement between theory and measurements, and the system reveals good sensitivity compared to other topologies;^{5} even though f_{1}
is not in the resonance frequency. Measurements variations, around 4, could be improved using a low coherence source in order to decrease the source induced noise.

## 4.

## Conclusions

Two different self-referencing methods for intensity fiber-optic sensors are described and their sensitivities are theoretically analyzed. The proposed scheme, using RR operating under incoherent regime, is flexible because the operation point and sensitivity is controlled by a coupling coefficient. Experimental calibration curves are reported validating the utility of the model developed. This configuration has a better sensitivity to other topologies.

## Acknowledgments

We wish to thank J. M. Sa´nchez-Pena and S. Vargas. This work was supported by Comision Interministerial de Ciencia y Tecnologı´a (TIC2003-03783).