Response to Comment on “Probing the Ultimate Limit of Fiber-Optic Strain Sensing”

Cranch and Foster argue that the strain resolution of our fiber resonator sensor is not limited by thermal noise. However, extension of common theoretical models to high-finesse passive resonators at low frequency needs further attention. Our findings are consistent with noise of thermodynamic nature for which no experimental evidence in the infrasonic range was available until recently. Thermally induced phase fluctuations in optical fibers have long been studied because they set a fundamental limit to strain resolution of sensors as well as frequency stability of fiber lasers (1, 2). In our original paper (3), we reported an unprecedented resolution level in strain sensing using a high-finesse fiber Bragg-grating (FBG) resonator interrogated by an optical-frequency-comb (OFC) stabilized diode laser. The OFC minimizes noise contribution from the laser, which is usually dominant in similar systems, particularly in the infrasonic frequency range (4, 5). Taking into account the laser-frequency noise when locked to the OFC, we estimated the ultimate limit of our technique to be 120 fε/√Hz at 2 Hz (3). Despite that, the strain resolution was found to be above the expected value. Cranch and Foster (6) compare our strain noise with that predicted by their model with an effective fiber length equal to twice the FBGs distance (analogous to fiber lasers) (7). We remark that the definition of effective length given in (7) is difficult to match with the popular Wanser’s model (8). Wanser’s theory provides the only analytical expression of thermal phase fluctuations for finite-cladding fibers. Over the past two decades, few experiments have confirmed this model for long-fiber interferometers—and those have been only in the acoustic range—whereas strong deviations were found at low frequency (9, 10). A similar theory was developed for fiber lasers, but a discrepancy with Wanser’s expression was pointed out, and the measured low-frequency noise was far from the predicted level as well (7). To explain this behavior, a different theoretical model was proposed for fiber lasers, based on the introduction of source terms in the stochastic heat diffusion equation (11). However, the extrapolation of the predictions of these models at low frequencies is still questionable. To our knowledge, no theory or experiment has been reported to date for high-finesse passive fiber resonators, which leaves open the question of which theoretical approach better describes the observed noise. In the absence of other models, we used Wanser’s formula with a finesse-dependent length to account for the power storage inside the cavity due to multiple reflections. Although the use of this scaling factor is arguable, it leads to a factor 8 in the noise calculation compared with the original Wanser’s formulation, and it describes well the observed noise. Finally, we note that the OFC noise-limited strain resolution is about a factor 6 below the Leave a comment (0) Prev | Table of