Technical Comments

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

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Science  20 Jan 2012:
Vol. 335, Issue 6066, pp. 286
DOI: 10.1126/science.1205452

Abstract

Gagliardi et al. (Reports, 19 November 2010, p. 1081) described an ultrahigh-resolution fiber-optic strain sensor. This comment addresses an error in the calculation of the fundamental thermodynamic noise in optical fiber resonators and shows that the actual thermodynamic noise level is lower by a factor of at least 11 than that calculated.

Gagliardi et al. (1) reported a fiber-optic resonator capable of achieving a strain resolution approaching 10−13 ε Hz–1/2 at infrasonic frequencies below 10 Hz. The system demonstrated the highest-resolution strain sensitivity at frequencies less than ~10 Hz published to date, and we commend the author on an excellent achievement. However, the calculated level of fundamental thermal noise exhibited by such an optical cavity is incorrect and not consistent with previously published theoretical models and experimental measurements. The method used by Gagliardi et al. to calculate thermodynamically induced phase fluctuations, employing an effective cavity length for a high–quality factor (high-Q) optical cavity, is not valid because the thermal-noise-limited sensitivity is unaffected by the cavity storage time and is only dependent on fiber length and material properties. We believe that this error may lead to the misconception that fundamentally limited strain resolution is readily achievable in short-length, passive fiber-optic resonators. We show that, in view of the actual thermodynamic noise level, this limit is very difficult to attain in a passive resonator, at frequencies below ~10 Hz, with current laser technology.

Thermodynamically induced phase fluctuations can affect many applications, such as fiber-optic sensors (2, 3) and laser stabilization (4, 5). Glenn (6) presented the first rigorous theoretical study of thermal noise in single-mode fiber. An alternative formulation leading to a simplified expression for thermal noise in optical fiber was later presented by Wanser (7), who subsequently provided strong experimental evidence to support the validity of the proposed model (8) and which has been widely applied since. More recent theoretical work has provided a further formulation for equilibrium thermal noise in passive optical fibers and fiber lasers (9), which agrees within a few percent with Wanser’s formulation. Assuming insulating boundary conditions at the outer cladding interface and taking parameters typical for conventional single-mode fiber, a cavity length of 0.13 m yields thermal phase fluctuations in a fiber Fabry-Perot cavity of (2Sϕ)1/2 = 3.4 × 10−8 rad/Hz1/2 at 1 Hz. This corresponds to a strain resolution, δεth = 28 × 10−15 ε/Hz1/2, a value lower by a factor of ~11 than the lowest value calculated by Gagliardi et al. To resolve phase fluctuations of this order requires a laser frequency noise less than ~5 Hz/Hz1/2, which is lower by a factor of at least 4 than that achieved by the stabilized semiconductor laser used in (1). The thermal-noise-limited strain resolution is shown in Fig. 1 alongside the strain resolution achieved in (1). At low frequency, the theoretical thermal noise is quite sensitive to the boundary conditions at the interface of the fiber cladding, where we assume ideal insulating conditions (solid line in Fig. 1). However, even if one assumes an infinite cladding, the theoretical thermal noise limit at 1 Hz is still close to an order of magnitude lower than that calculated in (1) (dotted line in Fig. 1).

Fig. 1

Theoretical thermal-noise-limited strain sensitivity, δεth, for 0.13-m fiber Fabry-Perot cavity: (solid line) insulating boundary condition and (dotted line) infinite cladding. Also shown is the measured strain resolution according to (1) (red line), the measured strain resolution of a distributed feedback (DFB) fiber laser (9), and a possible 1/f1/2 noise contribution according to (15) (dashed line). (Inset) The dependence of the thermal-noise-limited strain resolution versus the effective cavity length.

Thermodynamically induced phase fluctuations in optical fiber are indistinguishable from mechanically induced phase fluctuations. Therefore, the assertion by Gagliardi et al. that the thermal-noise-limited strain resolution is dependent on cavity storage time (i.e., finesse) has no physical basis and leads to a large error in the calculated thermal-noise-limited strain sensitivity. It should be noted that variability in material parameters is not sufficient to account for the discrepancy between the thermal noise calculations presented here and those presented in (1). Experimentally measured thermal noise spectra in optical fiber interferometers by Wanser (8) are more than an order of magnitude lower than noise spectra reported in (1) at frequencies above 1 kHz, taking into account the differences in fiber length.

Because of a lack of sufficiently stable lasers, measurement of equilibrium thermal noise has been achieved only in balanced, long path length (>50 m) interferometers (8) and fiber laser strain sensors (911), albeit at frequencies typically extending beyond 100 Hz. No studies, to date, have characterized equilibrium thermal noise at frequencies below 100 Hz in passive optical cavities because of the exceptional stability requirements of the probing laser. However, recent work has demonstrated a single-frequency fiber laser locked to a high quality factor (Q) cavity (12) with frequency stability below 1 Hz/Hz1/2 above 1 Hz. It is interesting to note that thermal-noise-limited strain resolution is found to scale as Leff1/2, as illustrated in the inset of Fig. 1. In the current work, it follows that thermal noise in a fiber Fabry-Perot cavity is more readily resolved with a shorter cavity.

Measurements of thermal noise at low frequencies in passive optical-fiber cavities would establish the validity of current theories over these frequency ranges. In single-frequency fiber lasers, frequency fluctuations are found to be limited by equilibrium thermal noise above ~1 kHz with a 1/f1/2 noise contribution observed at lower frequencies (9). Strong evidence suggests this 1/f1/2 noise to be associated with radiative transitions (spontaneous emission) in the gain medium (13, 14) and, as such, is referred to as nonequilibrium thermal noise. Experimental evidence confirms the absence of this noise mechanism in passive optical fiber (11). More recent theoretical work has proposed a fundamental source of 1/f1/2 noise in passive optical fiber due to mechanical dissipation (15). A 1/f1/2 dependence is predicted with a magnitude exceeding equilibrium thermal noise at frequencies below ~50 Hz. An estimate of the magnitude of this noise source in a 0.13-m fiber Fabry-Perot cavity is also illustrated in Fig. 1 (dashed line). It is evident from the data obtained by Gagliardi that, if present, this noise source may not be resolvable without further environmental isolation to reduce the 1/fn (n ~ 1) noise observed below 1 Hz.

Reaching thermal-noise-limited strain resolution in passive optical fiber at very low frequencies is thus a formidable challenge. At present, thermal-noise-limited strain sensitivity can only be achieved in short, active optical cavities based on fiber lasers (3). Both the absence of probing lasers with sufficient frequency stability at frequencies below 1 Hz and attaining sufficient environmental isolation may present major obstacles to achieving the full aspiration of (1).

Supporting Online Material

www.sciencemag.org/cgi/content/full/335/6066/286-a/DC1

SOM Text

Table S1

References

References and Notes

  1. Acknowledgments: G.A.C. was funded by the Naval Research Laboratory 6.2 base program.
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