Thermal Phase Noise Measurements in Optical Fiber ... - IEEE Xplore

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IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 48, NO. 5, MAY 2012

Thermal Phase Noise Measurements in Optical Fiber Interferometers Robert E. Bartolo, Member, IEEE, Alan B. Tveten, and Anthony Dandridge

Abstract— We present measurement data of fundamental thermal noise in a 40-m fiber optic Mach–Zehnder interferometer (MZI) using 80-µm-diameter optical fiber. To extend the measurements to low frequencies (below 500 Hz), the experimental setup is carefully designed to minimize ambient noise, thermal drift, and the phase and amplitude noise of the lasers. These experimental results are compared with theoretical predictions for the magnitude of the fundamental thermal noise in fiber, due to both thermodynamic temperature fluctuations and spontaneous length fluctuations. The experimental data, using two different solid-state lasers with two different emission wavelengths (1319 and 1550 nm), is in reasonable agreement with both theories over frequencies ranging from 20 Hz to 100 kHz. In terms of strain resolution, this paper demonstrates a fundamental thermal noise limit of approximately one femtostrain/rt(Hz) for a 40-m fiber optic MZI. Index Terms— 1/f noise, interferometers, optical fiber sensors, phase noise, thermal noise.

I. I NTRODUCTION

T

HERMAL noise has manifested itself in a variety of different contexts ranging from interferometric fiber optic sensors (for excellent review articles see [1, 2]), Er-doped fiber lasers [3], and in the dielectric optical coatings used in interferometric gravitational wave detectors [4]. Therefore, a detailed experimental and theoretical understanding of thermal noise is important, especially at frequencies below 1 kHz, since such fluctuations can be the limiting noise source in fiber optic sensing systems (for a recent example see [5]). In addition, thermal noise can also be significant in fiber optic sensing arrays where it reduces the utility of signal averaging over many sensing elements since the thermal noise in one arm of a compensating fiber optic Mach–Zehnder interferometer (MZI) is common to all sensing elements (see for example Fig. 11 in [1]). Furthermore, a recent theory of thermal noise fluctuations in fiber due to mechanical dissipation predicts a 1/f dependence that could be significant at low frequencies [6]. A widely-cited theoretical publication by Wanser, estimating the magnitude of thermal noise in fiber [7], was shown to be

Manuscript received July 14, 2011; revised January 20, 2012; accepted February 27, 2012. Date of publication March 19, 2012; date of current version April 20, 2012. This work was supported in part by the Office of Naval Research NRL 6.2 Base Funding. R. E. Bartolo is with the Department of Electrical and Computer Engineering, University of Maryland, College Park, MD 20742 USA (e-mail: [email protected]). A. B Tveten and A. Dandridge are with Optical Sciences Division, Naval Research Laboratory, Washington, DC 20375 USA (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JQE.2012.2190717

in good agreement with subsequent experimental results for 125 micron SMF-28 fiber [8, 9] (see also Figs. 4 and 5 in the review article by Kersey [2]). However, it was critical to measure the phase noise of 80 micron diameter fiber since it is the most commonly used fiber in interferometric acoustic sensors due to its ability to survive the typical bend radii seen with mandrel based sensors, as well as providing an improved packing density compared to 125 micron fiber. Despite its widespread use in fiber optic acoustic sensor systems (both production and pre-production) the thermal noise properties of this fiber had not been previously measured. Furthermore, for measurements in 125 micron fiber, the thermal noise at frequencies below 20 kHz were obscured by background environmental noise. In addition, the presentation of the theory by Wanser [7] was quite brief, lacking any details of the calculation. In contrast, a detailed theory of thermal noise in passive fiber was presented by Foster et al., as it applied to the case of DFB fiber lasers [10]. In this work, we present for the first time phase noise measurements for a 40 m MZI constructed using 80 micron diameter fiber where a unique fiber mandrel design and experimental approach was used to minimize the effects of environmental noise at low frequencies, down to 1 Hz. The experimental results are compared with both the Wanser theory and the theory by Foster et al. Furthermore, the unique ability to measure thermal noise down to relatively low frequencies (20 Hz) allowed for these experimental results to be compared with a recent theory for thermal noise fluctuations in fiber due to mechanical dissipation [6] that are only thought to be significant at these low frequencies. II. T HERMAL N OISE T HEORY In the Wanser theory [7], the rms amplitude of the phase noise fluctuations in fiber as a function of frequency, denoted as φrms (L,ω), was given by φrms (L, ω) √ (1) = 4π Sφφ (L, ω), Hz where the following analytic expression for the magnitude of the spectral function Sφφ (L,ω) was stated without derivation: 2 k B T 2 L dn Sφφ (L, ω) = + nα L F(ω). (2) 2κλ2 dT In the expression for Sφφ (L, ω), kB is the Boltzman constant, T the temperature, L = L1 + L2 the total length of fiber (the sum of the fiber lengths in each arm of the MZI), κ is thermal conductivity, λ the wavelength, dn/dT the refractive index temperature coefficient, neff the effective index, αL the

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BARTOLO et al.: THERMAL PHASE NOISE MEASUREMENTS IN OPTICAL FIBER INTERFEROMETERS

coefficient of linear expansion, where the frequency dependent term F(ω) is defined below. The theory depends critically on the precise estimates of these constants, where the most upto-date estimates are provided in Appendix A (see Table II) for the Fibercore SM 1500 [5.3/80] 80 μm fiber used in this work. In the limit that ω/v is negligible, where v is the velocity of light in the fiber, the frequency dependent term F(ω) is given by [7]: ⎞ ⎛ F(ω) = ln ⎝

4 + kmax

ω2 D2

4 + kmin

ω2 D2

⎠.

(3)

In this expression, D is the thermal diffusivity, kmax = 2/wo , where wo is the fiber mode-field radius, and kmin = 2.405/af , where 2af is the fiber outer diameter. It will be shown below that F(ω) decreases for frequencies above approximately 5 kHz, and is essentially constant at lower frequencies. Therefore a calculation of F(ω) in the zero frequency limit provides a useful estimate of the thermal noise magnitude, and is given by the following simple expression: 2a f kmax . (4) = 4 ln lim F(ω) = F(0) = 4 ln ω→0 kmin 2.405wo It can be seen that any precise comparison with the theory relies critically on the different fiber parameters, especially dn/dT, wo , and af , where the exact values are the subject of detailed measurements that must be referenced for the particular fiber used. As an additional test of the Wanser theory, the measurements to be discussed below utilize a different set of fiber parameters for the Fibercore 80 micron diameter fiber (see Appendix A) relative to previous measurements using Corning’s SMF-28 with a 125 micron outer diameter [8]. Furthermore, the two different lasers operating at different wavelengths (1319 nm and 1550 nm respectively) also provide an additional test of the Wanser theory. In the zero frequency limit, using the values for the constants in Appendix A for the 1319 nm laser, the magnitude of the thermal phase noise is given by 20log[φrms (L, 0)/rt(Hz)] = −125.5 dB re-rad/rt(Hz), approximately half a micro-radian per root Hz. Such levels of thermal noise are readily observable in interferometric sensor systems due to their highly optimized sensitivity combined with the use of low phase noise lasers. Note that the spread in experimental values for dn/dT published in the literature for various fibers is enough to easily raise or lower the absolute value for φrms (L,ω)/rt(Hz) by a dB or two. The theory by Foster [10] uses a slightly different notation and calculates the spectrum of temperature fluctuations (ST ) per unit length, where Sφφ (L,ω) = (kq)2 ST (ω). Here k is the wavenumber and nq = d(nL)/dT is the change in the optical path length with temperature, where this expression relates temperature fluctuations to phase fluctuations (see also footnote at the bottom of page 380 in [10]). Another possible source of intrinsic thermal noise in optical fibers has been proposed recently in terms of spontaneous length fluctuations caused by mechanical dissipation at low

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Fig. 1. Schematic of the experimental apparatus used to measure the thermal noise, including the solid-state laser source (left), the fiber-optic MZI cowound on an aluminum cylinder with a fiber-wrapped PZT (middle), where the two fiber outputs are inputs to a difference amplifier (right).

frequency [6]. According to the theory, the rms amplitude of the thermomechanical phase noise is given by 2πn 2πn rms L = = SL ( f ) φmech λ λ √ 2πn 2k B T Lϕ0 1 (rad/ H z), (5) = λ 3π E 0 A f where L is the length fluctuation, Eo is the bulk modulus, A is the cross-sectional area of the fiber, and ϕo is the loss angle that characterizes the mechanical dissipation. Numerical estimates for these parameters are as follows: The total fiber diameter = 160 μm for “80 micron” fiber (silica fiber plus acrylite coating), T = 295 K, E0 = 1.9×1010 Pa, ϕ0 = 1×10−2 as discussed here [6], λ = 1319 nm, and L = 80 m is the length of the fiber [see Appendix A]. At 100 Hz, 20log[φrms(L,ω)] = −124.4 dB re-rad/rt(Hz), a number that compares closely with the thermal noise value from the Wanser theory in the zero frequency limit. There is a condition for (5) to be valid, which is √ E 0 /ρ , f < 2L √ where√ E 0 /ρ is the sound speed in your medium. For optical fiber, E 0 /ρ ∼ 3645 m/s. Therefore, for a 40 m MZI, 80 m total fiber length, the above model should only be valid up to about 40 Hz. In addition, there has been no experimental verification for the loss angle ϕ0 for frequencies below 75 kHz [11]. III. E XPERIMENTAL S ETUP There are several technical challenges associated with measuring the thermal noise in an optical fiber MZI that become especially acute at low frequencies. One is ambient environmental pickup in the two arms of the MZI, which unless balanced, shows up as phase noise at the output. In order to minimize differential environmental pickup in both arms of the MZI used in this work, the optical fiber was co-wound on a solid aluminum cylinder with a diameter of 6 cm and a length of 10 cm [with minimal tension, less than 0.1 N] as illustrated in Fig. 1. This geometry allowed both arms to be co-wrapped in one layer on the surface of the aluminum cylinder thus minimizing thermal drift between the two arms of the MZI. Minimizing thermal drift also allowed a quadrature measurement to be made more effectively without active feedback. For each measurement, the interferometer was enclosed in a rigid aluminum box that was lined with an acoustically

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absorbing material consisting of several alternating layers of lead and open cell foam (denoted by the dashed lines in Fig. 1). This enclosure was mounted on an air bladder to reduce the effect of any building vibrations. The building vibrations were reduced by locating the interferometer on an air-floated optical table and performing measurements late in the evening and on weekends. Air currents and vibration were reduced by turning off the air-conditioning. In addition, the electronic spectrum analyzer was taken out of the room to minimize noise contributions from cooling fans and possible EMF pickup. More specifically, the measurements of the intrinsic thermally induced phase noise reported below utilized Fibercore SM 1500 [5.3/80] 80 μm diameter fiber. The MZIs were constructed using dual wavelength 3 dB couplers to allow measurements with a Lightwave Electronics Model 125 diodepumped 1319 nm Nd:YAG laser and an ATX Telecom Systems 1550 nm laser. A MZI was constructed from the Fibercore 80 μm fiber with a length of 40 meters in each arm. The path mismatch, L, for the 40 m MZI was less than 1 cm, thus greatly minimizing the effect of laser phase noise. For path-matched MZIs, the magnitude of the laser intensity noise can potentially be the dominant noise source, not the phase noise. A differential amplifier was employed to subtract off the laser intensity noise at quadrature, for the two MZI outputs, as depicted in Fig. 1. The phase noise for the two MZI outputs is anticorrelated in quadrature, therefore the differential amplifier effectively sums the phase noise from the two outputs together. The performance of the difference amplifier, as a function of the interferometer phase difference, has been modeled and the details presented in Appendix B. To obtain the optimum subtraction of the laser intensity noise using a differential amplifier, it was critical to obtain the measurement results while the interferometer was extremely close to quadrature. This was particularly important below 1 kHz since the magnitude of the relative intensity noise for the Nd:YAG has an approximate 1/fα functional dependence [12], increasing sharply at low frequencies with α greater than 1. In contrast, the thermal noise is expected to essentially be flat at low frequency according to (4). However the magnitude of the spontaneous length fluctuations are expected to also have a 1/fα noise dependence according to (5), where α = 0.5 for the noise amplitude (in contrast to the noise power where α = 1.0), thus making the details of the intensity noise subtraction more important. The power spectrum data was acquired from the output of the differential amplifier and measured with an HP 3562A signal analyzer. The measurements were acquired when the output of the differential amplifier was zero, which would correspond to the interferometer being in quadrature (See (B2) in Appendix B). Details of the experimental method used to obtain the calibration constant, to convert from voltage fluctuations to phase fluctuation in a MZI, can be found elsewhere (see Appendix A in [13]). In brief, two different calibration approaches can be used to give independent verification that the correct calibration constant was determined. The method used consistently in this work relies on evaluating the derivative, about quadrature (φ = π/2), of the equation describing the voltage signal of the MZI [V = A + Bcos(φ)]


Fig. 2. Phase noise data for the 80-μm fiber at 1319 nm using the Nd:YAG laser as the optical source (green curve). Theoretical curves for the different thermal noise theories (2) and the shot noise limit (gold curve at the bottom).

as a function of phase difference φ between the two arms of the interferometer. Note that B = B1 + B2 in the case of measuring the voltage output of the differential amplifier in quadrature [see Appendix B], as compared to measuring a single output of the MZI. Taking the inverse of this result yields the calibration constant, 1/B = 2rad/(Vmax − Vmin ), where Vmax (φ = 0) and Vmin (φ = π) are the maximum and minimum voltages observed on an oscilloscope as the phase of the interferometer drifts through multiples of π, or is driven using a voltage applied to the PZT on one arm of the MZI. For better low frequency resolution, the data was taken over three frequency ranges and combined in a plotting program to produce the final power spectrums to be discussed below. One additional challenge was maintaining the signal in quadrature while the spectrum analyzer accumulates a minimum of 50 averages, which was more problematic at lower frequencies where longer data acquisition times are needed. IV. E XPERIMENTAL DATA The phase noise measurement results using the 1319 nm Nd:YAG laser with the MZI near quadrature are shown in green in Fig. 2. The phase noise data displayed a 1/fα dependence at low frequencies, where an explanation for the observed inverse frequency dependence of the phase noise will be provided below. It can also be seen that, relative to previous measurements in shorter lengths of fiber (see Fig. 1b in Wanser [9] et al.), the ambient environmental pickup was clearly minimized at low frequencies. The phase noise also exhibits a shoulder at mid frequencies, with a gradual downturn at around 100 kHz. A comparison with the Wanser theory (2) is shown by the red curve using the fiber parameters listed in Appendix A. The agreement between experimental data and the functional dependence describing the roll off of the Wanser theory for frequencies above 1 kHz was excellent. However, to obtain precise quantitative agreement with the experimental data above 1 kHz, a small factor of 1 dB was subtracted from the red curve for the Wanser theory (again based on the fiber parameters in Appendix A). The subtraction of the 1 dB factor was reasonable and well within the experimental


Fig. 3. (a) Comparison of the phase noise at two different wavelengths, 1319 nm emitted from a Nd:YAG laser, and 1550 nm from a ATX solid-state laser. (b) Difference (or ratio in linear units) of the phase noise at the two different wavelengths [1319–1550 nm].

uncertainty (on the order of 10% on a linear scale, or 1 dB on a logarithmic scale) of determining both dn/dT and wo as discussed in Appendix A. In addition, an improved zero frequency expression by Wanser [14] lowers the noise by 0.5 dB for the fiber parameters used here. Plotted as the dashed line is a numerical evaluation of Foster’s theory for infinite cladding (see (28) and (29) in Foster et al. [10]), again for the fiber parameters listed in Appendix A. It can be seen, that without any offset adjustments, the Foster theory for infinite cladding gives excellent agreement above 1 kHz with the data and the Wanser theory offset by 1 dB. For both the Wanser and Foster theories assuming infinite fiber cladding, there is a slight departure from a constant value for frequencies below 500 Hz. It is also important to note that the observed level of thermal noise is 20 dB larger than the detector shot noise level as shown by the gold curve. For 0.5 mW on detector, the calculated shot noise, when converted from the equivalent voltage fluctuations to equivalent phase fluctuations, is −148.9 dB re rad/rt(Hz), more than 20 dB below the observed thermal noise. Therefore, thermal noise fluctuations are significant, and can often limit sensor performance well before other sources of noise. As predicted by the Wanser theory (2), the magnitude of the thermal noise is inversely proportional to wavelength. A comparison of the phase noise at two different wavelengths,

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1319 nm emitted from a Nd:YAG laser and 1550 nm from a ATX solid state laser respectively, are shown in Fig. 3(a). One would expect the 1319 nm laser to have a slightly larger thermal noise relative to the 1550 nm laser since the thermal noise amplitude scales inversely with wavelength. On a logarithmic scale the thermal noise would be larger by a factor of 1.4 dB [= 20log(1550/1319)], where the difference in mode field radius (wo ) between the two wavelengths contributes a much smaller correction (approximately 0.17 dB in the zero frequency limit shown in (4) since wo is in the denominator of a natural logarithm). The close agreement with the theory is illustrated in Fig. 3(b), where the difference on a logarithmic scale of the two data sets is shown superimposed on a line representing the theoretical ratio (1.40 dB discussed above). Clearly, there is good agreement with the scaling of the thermal noise for frequencies above 20 Hz for two completely different lasers, with different levels of intensity noise. Further analysis is necessary to explain the presence of the 1/f dependent phase noise at low frequencies, as shown in Fig. 2, where a clear deviation from the Wanser theory (black dotted curve) below 800 Hz can be seen. This analysis was also necessary in an effort to look for possible evidence of 1/f noise due to spontaneous length fluctuations in the fiber as described by (5). In Fig. 4 we examine possible sources or mechanisms for the 1/f dependence of the phase noise at low frequencies, below 1 kHz. The green curve is the measured phase noise of the MZI using the 1319 nm Nd:YAG laser, discussed previously in connection with Fig. 2. The black dotted curve is the Wanser thermal noise (2) calculated for 1319 nm (including the 1 dB correction factor), where a clear 1/f dependent deviation between the theory and experimental data can be seen starting at approximately 1 kHz. The red curve is data from a phase noise measurement from the differential amplifier taken approximately 90 degrees away from quadrature [(φ ∼0 degrees], which is not strictly speaking the same as an intensity noise measurement, but should nonetheless have a strong contribution from the intensity noise. A fit to this curve for 1/fα yields α ∼ 1.63, a clear departure from the Duan theory where α = 0.5. The blue curve is just this intensity noise replotted assuming 23 dB of intensity noise cancellation or subtraction by the differential amplifier. The exact amount of actual subtraction in the experiment was not known precisely, but 23 dB is reasonable from Fig. 6(a), assuming the phase of the MZI was φ ∼ 87 degrees, thus only 3 degrees from quadrature. In addition, subtracting 23 dB from the measured intensity noise data gives the best match to the observed phase noise data shown in green below approximately 20 Hz. The key question is whether the 1/f noise phase noise below 800 Hz is residual intensity noise not cancelled by the differential amplifier, or some other source of 1/f noise, for example the thermomechanical fluctuations predicted by Duan [6]. In Fig. 4(a) we compare the measured phase noise data with the combination (square root of the sum of the squares) of the Wanser theory (2) and the Duan theory (5). This curve yields an excellent fit with the data for frequencies above approximately 20 Hz. It should be clearly noted that the same phenomena was observed using a totally different laser, namely the Er-doped ATX solid state laser emitting at

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Fig. 4. (a) Experimental evidence for residual 1/f phase noise at low frequency due to spontaneous length fluctuations using the 1319-nm Nd:YAG laser. (b) Comparison of the measured phase noise with different combinations of the Wanser theory, plus the theory for spontaneous length fluctuations, and residual intensity noise fluctuations assuming 23 dB of cancellation from the differential amplifier. (c) Comparison of the laser phase noise floor for 1 cm of fiber for a Nd:YAG and a semiconductor DFB laser [15], relative to the measured fundamental thermal noise and detector shot noise.

1550 nm, with different levels of intensity noise. This can be seen by looking back at Fig. 3, where the phase noise for the Er-doped ATX laser was off by just 1.4 dB relative to the data for the Nd:YAG lasers above approximately 20 Hz. Below 20 Hz the phase noise data for the 1550 nm laser was likely residual intensity noise not cancelled by the differential amplifier, which is the likely explanation of the data for the Nd:YAG laser as well in the same frequency range. Along these lines we show in Fig. 4(b) a plot of the measured phase noise shown previously. Also shown in black squares, is the square root of the sum of the squares of the Wanser intrinsic thermal noise term (1), plus the thermomechanical noise term by Duan (5), and the residual out of quadrature intensity noise below 20 Hz (i.e., the blue curve where a


factor of 23 dB was subtracted off the measured data). It can be seen that the systematic combination of all three noise terms leads to an excellent agreement with the observed phase noise over the entire frequency range with the exception of all but the lowest frequencies, below 3 Hz. At these low frequencies there was insufficient data sampling for the FFT performed by the HP 3562 electronic spectrum analyzer which is limited to 801 data points per scan. The aqua colored curve shows an estimate of the phase noise where only the Wanser theory and the out of quadrature phase noise (−23 dB) were combined leaving an obvious gap from the measured phase noise spectrum. Nonetheless, the phase noise data from 20 Hz out to approximately 1 kHz, appears consistent with the Duan theory as predicted by (5), and also fills the gap between the observed phase noise and the Wanser theory. However, the excess noise at frequencies below 20 Hz, combined with the uncertainty in the parameters used in (5), prevents a more rigorous comparison of the present data with Duan’s theory. Further measurements with greatly reduced levels of low frequency intensity noise for example are necessary to document this effect more conclusively. Along these lines, in Fig. 4(c), to prove laser phase noise was not an issue in this work, phase noise data for the Nd:YAG for the 1 cm path imbalanced MZI is shown relative to the thermal noise curves and the shot noise. From Fig. 4(c), it can be seen that it would not be possible to use a semiconductor DFB laser to observe the level of fundamental thermal noise discussed here. Finally, it is useful to discuss the observed thermal noise floors in terms of strain resolution in light of recent work claiming to have probed the ultimate limits of fiber optic sensing in fiber Fabry-Perot (FP) cavities [5]. Using (1), the thermal noise limited strain resolution for our MZI in the zero frequency limit is given approximately by: √ √ 0.5μ r ad/ H z φrms (0, L)/ H z = (0.78)φrad (L = 80 m) 0.78 × 5.56 × 108 r ad 1.2 femto strain = √ Hz where φrms (L,0,) is given by (1) in the zero frequency limit, φrad (L = 80 m) is the total number of radians in 80 m of fiber, and the factor of 0.78 accounts for the stress-optic effect [16]. It can be seen that the thermal noise-limited strain sensitivity scales inversely with the square root of the length: √ √ 1 L φt herm (ω, L)/ H z ∼ ∼ √ . strain sensitivity = φrad (L) L L Therefore, in principle, the ultimate thermal noise-limited strain resolution of interferometric fiber optic sensors can be improved by increasing the path lengths in both arms of the MZI, however this becomes problematic in actual sensing systems in terms of wrapping fiber on a mandrel for lengths longer that about 40 m. Even in a 40 m MZI (with 80 m of total fiber length), femtostrain Hz−1/2 resolutions are readily achievable. This level of strain sensitivity is three orders of magnitude lower, relative to other approaches utilizing a fiber FP resonator and the Pound-Drever-Hall technique for example, where claims of thermal noise-limited strain resolutions at the sub-picostrain-Hz−1/2 level have been made [5].


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TABLE I T HERMO -O PTIC C ONSTANTS U SED IN (A1) Dispersion equation constants C0 C1 C2 C3

Numerical values 9.390590 0.235290 −1.318560 × 10−3 3.028870 × 10−4

However, upon close examination of the work by Gagliardi [5] et al., their claims of thermal noise-limited measurements in a fiber FP interferometer, for frequencies below 20 Hz, were questionable for reasons to be given below and as a Technical Comment in Science Magazine [17]. First of all, it is very difficult to observe the fundamental thermal noise floor in a fiber-optic FP cavity, compared to a MZI. In brief, one can greatly minimize L, the path imbalance of a MZI, in order to minimize the laser phase noise that scales with L, yet still have long interferometer arms that contribute to L (=L1 + L2 ) for the fundamental thermal noise term (2). This is critical for observing thermal noise above the residual laser phase noise, and other noise floors, such as intensity noise, shot noise, and electronic noise. However, with a FP cavity there is by definition a cavity spacing, LFP , that cannot be balanced to 0 in order to minimize laser phase noise as in the case of a MZI. For a FP cavity, L = LFP , where in contrast, for some MZIs used in fiber optic sensing, L approaches 0 (or at least less than 1 cm), and L can be on the order of one hundred meters to several km [8]. Furthermore, for the case of short fiber lengths (∼10 cm) used in [5], thermodynamically induced phase shifts will be very small and therefore difficult to measure as will be discussed below. It can be shown that in the infrasonic frequency range (1 mHz to 20 Hz) explored by Gagliardi, the fundamental thermal noise would be very difficult to observe in a FP cavity due to high levels of 1/f laser phase noise that are well documented at these low frequencies (see for example Figure 3B in [15]). These measurements [15] have shown that the laser phase noise at 1 Hz varies by two orders of magnitude, from 80 μrad-re-1 m/rt(Hz) for a highly coherent Nd:YAG, to about 8000 μrad-re-1 m/rt(Hz) for a semiconductor DFB laser. [Note that the laser phase noise scales with the path imbalance L, since φ = 2πneff L/λ, and these results are normalized to a 1 m path imbalance, i.e., in units of μrad-re-1 m/rt(Hz).] For a semiconductor DFB laser, like the one mentioned by Gagliardi, the magnitude of the phase noise for a FP cavity path length of 2 LFP = 0.26 m is 2000 μrad/rt(Hz) [−54 dB rerad/rt(Hz)], a factor of 4 less than the phase noise normalized to a 1 m path imbalance. However, the level of fundamental thermal noise scales with the square root of the fiber length based on (1) and is given by 0.02 μrad/rt(Hz) [or −154 dB re-rad/rt(Hz)] for a 0.26 m fiber length. This is a factor of 105 [100 dB] below the laser phase noise floor for a typical semiconductor DFB laser at 1 Hz, and below the shot noise for 0.5 mW of power on detector [see Fig. 4(c)]. So, even with the 40 to 60 dB noise reduction (a factor of 100 to 1000) from locking to an optical frequency comb (OFC), the laser phase noise for a semiconductor DFB laser at 1 Hz, not the thermal

Fig. 5. Plot of the thermo-optic coefficient dn/dT, shown in (A1), as a function of wavelength. See data sheet titled, “HPFS Fused Silica ArF Grade” at www.corning.com.

noise, will still be the dominant noise floor in determining the fundamental strain resolution for the fiber FP cavity cited in the work by Gagliardi [5]. Scaling the effective length of the FP cavity by F, the finesse, as Gagliardi has done, does not change the situation since the laser phase noise would also increase by F, while the thermal noise would only increase by the square root of F according to (1). Furthermore, as shown above, the thermal noise limited strain resolution would scale inversely with the square root of the effective length, leading to a strain resolution of approximately 3 fstrain/rt(Hz) for F = 110, not the 1 pstrain/rt(Hz) calculated by Gagliardi. V. C ONCLUSION We have performed detailed measurements of thermal noise in a 40 m MZI using 80 micron diameter fiber for the first time for two different laser sources operating at 1319 nm and 1550 nm. In addition, environmental noise at low frequencies was greatly minimized allowing for a systematic comparison of different thermal noise theories. These results show excellent agreement with the Wanser theory and the more recent theoretical treatment by Foster, to within the experimental uncertainties of the different fiber parameters listed in Appendix A. The phase noise data for both wavelengths is also consistent with the presence of a 1/f noise contribution whose origin at least seems consistent with spontaneous thermal length fluctuations predicted by Duan [6]. However, a larger degree of cancellation of laser intensity noise at low frequency would be necessary for a more definitive observation of this effect in future experiments. A PPENDIX A F UNDAMENTAL F IBER PARAMETERS A precise comparison with the theory for thermal noise depends critically on the exact values of the fundamental fiber parameters for dn/dT, the thermo-optic coefficient for the Fibercore SM 1500 [5.3/80] used in these measurements. There are several different estimates of dn/dT in the literature for the different glasses available [18, 19]. A product data sheet

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TABLE II S UMMARY OF THE C RITICAL F IBER PARAMETERS U SED IN THE T HERMAL N OISE T HEORIES BY WANSER , F OSTER , AND D UAN kB T ρ λ wo

1.38 × 10−23 JK−1 295 K 2.2 × 103 kg/m3 1319 nm 1550 nm 2.350 μm (at 1319 nm) 2.605 μm (at 1550 nm)

Boltzman constant Lab temperature Density of silica Optical wavelength mode field radius (wo )

d = 2af

80 μm

Cladding diameter (d), and radius (af )

kmax = 2/wo

8.51 × 105 /m (at 1319 nm) 7.68 × 105 /m (at 1550 nm)

Inverse mode field radius (wo )

kmin = 4.81/d κ Cv D = κ/(ρCv ) n

6.01 × 104 /m 1.37 W/(mK) 741 J/kg 0.82 × 10−6 m2 /s 1.457 (at 1319 nm) 1.457 (at 1550 nm)

αL

5.0 × 10−7 /K

dneff /dT v = c/n

9.520 × 10−6 /K (1319 nm) 9.488 × 10−6 /K (1550 nm) 2.06 × 108 m/s

HM−Z (L + W)

LS + LR = 40 m + 40 m = 80 m

Inverse cladding diameter (d) Thermal conductivity Specific heat Thermal diffusivity Index of refraction Coefficient of linear expansion Refractive index temp coefficient Effective light velociy Mach–Zehnder transfer function [total interferometer length]

from Corning Inc. [“HPFS Fused Silica ArF Grade”] provides the following formula describing dn/dT for silica glass, dn = Co + C1 λ−2 + C2 λ−4 + C3 λ−6 [with λ in μm] (A1) dT where the “C” prefactors are provided in Table I. A plot of (A1) is shown in Fig. 5. It can be seen that for two significant digits the thermo-optic coefficient is relatively constant at 9.5 × 10−6 over the wavelength range used in this work (1319 nm to 1550 nm).

Fig. 6. (a) Theoretical plot of the intensity noise cancellation term as a function of φ, 20log[(δB1 + δB2 )cos(φ)] on a logarithmic scale. This simulated data shows that the degree of intensity noise cancellation is an extremely narrow function of φ about the quadrature point. Also shown is 20log[(B1 + B2 )sin(φ)], the phase noise term that is relatively constant about quadrature, where the constants B1 and B2 are also used to calibrate the phase noise measurement. (b) Close-up of the degree of intensity noise cancellation as a function of φ.

the resultant output Y: A PPENDIX B M ATHEMATICS OF THE MZI O UTPUT C ONNECTED TO A D IFFERENTIAL A MPLIFIER As discussed in the main text of the paper, in order to cancel the laser intensity noise, the two detector outputs of the MZI were connected to a differential amplifier that subtracted the two signals [see Fig. 1]. The exact degree to which the laser intensity noise can be canceled at low frequencies, where 1/f laser intensity noise dominates, depends critically on the mathematical details to be presented below. [For an excellent review of the details of MZIs used for sensing see [20] and discussion surrounding Eq. 10.10]. The two outputs of the square law detectors can be written as A1 + B1 cos(φ) and A2 − B2 cos(φ). In general A1 is not equal to A2 , and B1 is not equal to B2 , however in practice these amplitudes can be closely matched by adjusting the gain (or input attenuation) on one of the optical detectors. The dc output of the differential amplifier is found by subtracting these two equations to obtain

Y = (A1 − A2 ) + (B1 + B2 ) cos(φ),

(B1)

where it can be seen that the differential amplifier output is a minimum when the MZI is in quadrature [i.e., when cos(φ)= 0 (for φ = π/2)]. The observed noise terms are found by taking a simple differential of this equation and are given by, (Y) = (δA1 − δA2 ) + (δB1 + δB2 ) cos(φ) +(B1 + B2 ) sin(φ)∗ δφ.

(B2)

Evaluating each of these terms individually we find that the difference δA1 − δA2 is small since they are correlated and adjusted to be nearly equal. The next term, (δB1 +δB2 )cos(φ), is the amplitude noise that goes to zero when the MZI is in quadrature [cos(φ) = 0 (for φ = π/2)]. The last term is the phase response and includes any phase noise and sensed phase signal that might be present. It is a maximum when the MZI is in quadrature (φ = π/2). The reduction in amplitude noise as a function of φ is plotted in Fig. 6, where it can be seen that the degree of


cancellation of the differential amplifier is quite sensitive to the magnitude of the cos(φ) term about quadrature (φ = π/2). Therefore it is critical to keep the interferometer as close to quadrature as possible. From Fig. 4 it was shown that at 1 Hz the intensity noise was dominated by 1/f noise that was about 60 dB larger than the thermal noise predicted by (1). To reduce the contribution of the amplitude noise to 10 dB below the expected thermal noise would require a noise reduction of about 70 dB at 1 Hz. From the plot shown in Fig. 6(b), the phase angle would need to be maintained about 0.02 degrees (3.5 × 10−4 radians) away from the exact quadrature point, something that would be quite difficult to achieve. In contrast, the phase noise calibration constant, 1/B, with the dimensions of volts/radian, where B = B1 + B2 , is relatively constant about quadrature. This is illustrated in Fig. 6(a), where calibration constant B is seen to be quite constant about quadrature. In fact one would have to go 60 degrees from the quadrature point to observe a 1 dB deviation in the accuracy of the calibration constant. ACKNOWLEDGMENT The authors would like to thank K. Wanser for several helpful discussions concerning his theory, and the latest values of dn/dT as shown in Appendix A. To G. Cranch for helpful discussions and the computer routines used to generate the values of the thermal noise theory by S. Foster, and thanks to S. Foster himself for helpful comments. To K. Nemata for first pointing out the Electronic Letters article by L. Duan, and finally to L. Duan himself for helpful discussion and calculations concerning his thermomechanical noise theory.

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[12] R. E. Bartolo, C. K. Kirkendall, V. Kupershmidt, and S. Siala, “Achieving narrow linewidth, low phase noise external cavity semiconductor lasers through the reduction of 1/ f noise,” Proc. SPIE: Novel In-Plane Semicond. Lasers V, vol. 6133, pp. 61330I-1–61330I-8, Jan. 2006. [13] R. E. Bartolo, A. Tveten, C. K. Kirkendall, P. W. Juodawlkis, W. Loh, and J. J. Plant, “Characterization of a low-phase-noise, high-power (370 mW), external-cavity semiconductor laser,” NRL Memorandum Rep. NRL/MR/567, Naval Research Laboratory, Washington DC, 2010. [14] K. H. Wanser, private communication, 2011. [15] R. Bartolo, A. Tveten, and C. K. Kirkendall, “The quest for inexpensive, compact, low phase noise laser sources for fiber optic sensing applications,” Proc. SPIE, vol. 7503, p. 750370, Oct. 2009. [16] C. D. Butter and G. B. Hocker, “Fiber optics strain-gauge,” Appl. Opt., vol. 17, no. 18, pp. 2867–2869, 1978. [17] G. A. Cranch and S. Foster, “Comment on ‘probing the ultimate limit of fiber-optic strain sensing,”’ Science, vol. 335, no. 6066, p. 286, Jan. 2012. [18] G. Ghosh, “Temperature dispersion of refractive indexes in some silicate fiber glasses,” IEEE Photon. Technol. Lett., vol. 6, no. 3, pp. 431–433, Mar. 1994. [19] J. M. Jewell, C. Askins, and I. D. Aggarwal, “Interferometric method for concurrent measurement of thermo-optic and thermal expansion coefficients,” Appl. Opt., vol. 30, no. 25, pp. 3656–3660, Sep. 1991. [20] A. Dandridge, “Fiber optic sensors based on the Mach–Zehnder and Michelson interferometers,” in Fiber Optic Sensors: An Introduction for Engineers and Scientists, E. Udd, Ed. New York: Wiley, 1991, pp. 271– 323.

Robert E. Bartolo (M’08) received the B.A. degree in economics and the B.S. degree in physics from the University of California, Irvine, in 1987, the M.S. degree in physics from the University of Maryland, College Park, in 1990, and the Ph.D. degree in physics from Purdue University, West Lafayette, IN, in 1995. He was an Assistant Research Scientist with the University of Maryland, before joining the Naval Research Laboratories, Washington DC, in 1999. He has worked on a variety of research topics, including phase coherent effects in nanostructures, semiconductor electroabsorption modulators, semiconductor expanded mode and photonics crystal lasers, fiber lasers, and noise properties of semiconductor lasers.

R EFERENCES [1] C. K. Kirkendall and A. Dandridge, “Overview of high performance fibre-optic sensing,” J. Phys. D, Appl. Phys., vol. 37, no. 18, pp. R197– R216, 2004. [2] A. D. Kersey, “A review of recent developments in fiber optic sensor technology,” Opt. Fiber Technol., vol. 2, no. 3, pp. 291–317, Jul. 1996. [3] S. Foster, G. A. Cranch, and A. Tikhomirov, “Experimental evidence for the thermal origin of 1/ f frequency noise in erbium-doped fiber lasers,” Phys. Rev. A, vol. 79, no. 5, pp. 053802-1–053802-7, May 2009. [4] G. M. Harry, A. M. Gretarsson, P. R. Saulson, S. E. Kittelberger, S. D. Penn, W. J. Startin, S. Rowan, M. M. Fejer, D. R. M. Crooks, G. Cagnoli, J. Hough, and N. Nakagawa, “Thermal noise in interferometric gravitational wave detectors due to dielectric optical coatings,” Class. Quantum Grav., vol. 19, no. 5, pp. 897–917, 2002. [5] G. Gagliardi, M. Salza, S. Avino, P. Ferraro, and P. De Natale, “Probing the ultimate limit of fiber-optic strain sensing,” Science, vol. 330, no. 6007, pp. 1081–1084, Nov. 2010. [6] L. Z. Duan, “Intrinsic thermal noise of optical fibres due to mechanical dissipation,” Electron. Lett., vol. 46, no. 22, pp. 1515–1516, Oct. 2010. [7] K. H. Wanser, “Fundamental phase noise limit in optical fibres due to temperature fluctuations,” Electron. Lett., vol. 28, no. 1, pp. 53–54, Jan. 1992. [8] K. H. Wanser, A. D. Kersey, and A. Dandridge, “Measurement of fundamental thermal phase fluctuations in optical fiber,” in Proc. 9th Int. Conf. Opt. Fiber Sensors, Florence, Italy, 1993, pp. 255–258. [9] K. H. Wanser, A. D. Kersey, and A. Dandridge, “Intrinsic thermal phase noise limit in optical fiber interferometers,” Opt. Photon. News, vol. 4, no. 12, pp. 37–38, 1993. [10] S. Foster, A. Tikhomirov, and M. Milnes, “Fundamental thermal noise in distributed feedback fiber lasers,” IEEE J. Quantum Electron., vol. 43, no. 5, pp. 378–384, May 2007. [11] B. M. Beadle and J. Jarzynski, “Measurement of speed and attenuation of longitudinal elastic waves in optical fibers,” Opt. Eng., vol. 40, no. 10, pp. 2115–2119, Oct. 2001.

Alan B. Tveten received the B.A. degree from Concordia College, Moorhead, MN, the M.A. degree from the University of Nebraska, Lincoln, and the Ph.D. degree from Colorado State University, Fort Collins. He was a Physics Instructor with Dana College, Blair, NE, Mankato State University, Mankato, MN, and Colorado State University, before joining as a research scientist with the Naval Research Laboratories, Washington DC, in 1979. Since 1979, he has been working with fiber optic sensors and their applications to the needs of the U.S. Navy.

Anthony Dandridge was born in Kent, England. He received the B.Sc. (hons.) and Ph.D. degrees in physics from the Sir John Cass School of Science and Technology, London University, London, U.K., and the City of London Polytechnic, London. He was a Lecturer of physics with the University of Kent, Canterbury, U.K., in 1979. Since 1980, he has been with Georgetown University, Washington DC, John Carrol University, Cleveland, OH, and the Naval Research Laboratories (NRL), Washington DC. In 1984, he became the Head of the Optical Sensor Section, NRL. In 1998, he was the Head of the Optical Techniques Branch, NRL. He has authored or co-authored over 400 technical reports, journal and conference publications. He holds a number of patents in the area of fiber optic sensing. His current research interests include fiber optic sensor systems, including acoustic, acceleration, and electromagnetic field sensing, as well as the properties of optical sources, multiplexing, and interrogation techniques for fiber sensors. Dr. Dandridge was a recipient of three Navy Group Achievement Award for his fiber optic work in 1989 (Towed Array), in 1993 (Magnetic Array), and in 2003 (Hull Array). In 2003, he received the Meritorious Civilian Service Award for his work on developing the fiber optic wide aperture array system for the Virginia class attack submarines.