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Abstract
We demonstrate a passive, all-optical fiber frequency reference using a digitally enhanced homodyne interferometric phase readout. We model the noise contributions from fiber thermal noise and the coupling of double Rayleigh scattering in a digitally enhanced homodyne interferometer. A system frequency stability of 0.1 Hz/$\sqrt{\textrm {Hz}}$ is achieved above 100 Hz, which coincides with the double Rayleigh scattering estimate and is approximately a factor of five above the thermo-dynamic noise limit.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The performance of ultra-high sensitivity interferometry necessitates the use of stable, narrow linewidth laser sources. In applications where the free-running stability of these laser sources is insufficient, further laser frequency control is required. This is done by stabilizing the laser frequency to an optical reference of higher stability. At short integration timescales of below 1 second, the standard optical reference of choice is a high finesse ultra-low-expansion (ULE) cavity.
Depending on the amount of thermal noise and environmental isolation, ULE cavity systems achieve a typical frequency stability ranging from 0.1 Hz/$\sqrt {\textrm {Hz}}$ to 1 Hz/$\sqrt {\textrm {Hz}}$ at $\sim$1 Hz Fourier frequency [1–4]. With systematic efforts in isolation, the state-of-the-art ULE cavity is able to reach 0.04 Hz/$\sqrt {\textrm {Hz}}$ frequency stability at 1 Hz [5].
All-fiber optical frequency references present an interesting, but comparatively under-represented alternative to these Fabry-Perot type interferometers at a similar timescale. Potential advantages of a fiber system include frequency agility, lowered cost and better suitability for field-deployment due to their intrinsic alignment. However, fiber interferometers suffer from several limitations unique to guided wave optics. These include non-linear effects such as Brillioun scattering [6], Raman scattering [7] and Kerr self-phase modulation [8] which limit the optical power levels and coherence lengths that can be used. Linear limitations include Rayleigh back scattering (RBS) and photo-thermal effects [9–11], both of which must be contended with in order to optimize the phase fidelity of the interferometer. While photo-thermal effects can be effectively reduced by restricting the sensing optical power to the minimum required for shot noise limited sensitivity, or by implementing modest laser intensity stabilization, first order Rayleigh scattering remains an issue for interferometers operating in reflection.
A number of experiments have investigated the performance of fiber-based frequency references, with a typical frequency stability between 1 to 10 Hz/$\sqrt {\textrm {Hz}}$ at Fourier frequencies above 100 Hz being commonly achieved [12–18]. Surpassing the 1 Hz/$\sqrt {\textrm {Hz}}$ sensitivity limit in a similar Fourier frequency range has been a more significant undertaking and only demonstrated experimentally in a few instances [19–21]. The landmark sensitivity in this field however belongs to the work carried out by Kefelian et al, where using a 1 km fiber delay line in an unbalanced Michelson interferometer configuration, an output laser stability better than 0.1 Hz/$\sqrt {\textrm {Hz}}$ was achieved between 1-4 kHz [22]. This demonstrates the feasibility of fiber frequency references (FFRs) as an alternative to Fabry-Perot cavities, and to date still represents the state-of-the-art FFR performance.
To extend the 0.1 Hz/$\sqrt {\textrm {Hz}}$ stability beyond the Fourier frequency range achieved by Kefelian et al, two specific low-frequency challenges must be addressed. Firstly, reflection based fiber interferometers, such as a Michelson interferometer, are susceptible to first order RBS. This results in undesired interference between the main metrology field and the scatter field. As RBS is largely driven by laser frequency noise, it is exacerbated by the long optical coherence lengths typically used by FFRs and leads to an increased phase noise at infrasonic frequencies [23,24]. To mitigate RBS induced noise, a transmissive system is desirable as it avoids first order RBS. The second challenge involves the management of thermal related noise. This includes both noise induced by environmental temperature fluctuations and the fundamental thermal noise in optical fibers. Amongst these two noise sources, long lengths of fiber are typically dominated by temperature drift. Reduction of temperature effects therefore requires thorough and thoughtful thermal isolation.
Our primary goal in this work is to develop a fiber frequency reference that reaches the state-of-the-art frequency stability over a wider Fourier frequency range than that demonstrated in previous implementations. To address the aforementioned low frequency challenges, we adopt an arm-length mismatched Mach-Zehnder interferometer (MZI). The transmissive design removes sensitivity to first order RBS noise whilst the residual second order effects from double Rayleigh scattering (DRS) are significantly lower. In addition, the optical path length difference is extended from previous implementations to 15 km to allow higher sensitivity to laser frequency fluctuations [25,26]. The stability of the fiber reference is characterized by simultaneously interrogating a second, near-identical interferometer. Subtracting the individual readout signals then removes the common laser frequency noise, providing an estimate of their relative stability. These interferometers are placed in individual, passive, thermal-management chambers with a time constant of approximately 13 hours, allowing us to lower the onset frequency of temperature drift.
Although most existing FFR work utilize a feedback architecture to stabilize the laser source, here we use a fundamentally different paradigm, which plays to the strengths of fiber interferometry. By using the frequency agility available in fiber frequency references to measure the free-running laser frequency noise with high dynamic range, we operate using a feed-forward architecture to remove laser frequency noise from subsequent metrology tasks [27]. This approach is exemplified in the Laser Interferometer Space Antenna (LISA) mission where time delay interferometry (TDI) is used to synthesize a white light interferometer in post processing in order to remove laser frequency noise from the required metrology task - the measurement of gravitational waves [28,29]. Doing so enables the system to focus on phase fidelity without requiring the additional complexity and bandwidth limitations of active feedback or associated control noise.
The interferometric phase readout in our system is handled by digitally enhanced homodyne interferometry (DEHoI) [30]. DEHoI uses a four-level pseudo-random modulation sequence to step and sample between all four quadatures of the optical phase domain. The differential phase between the two MZI arms is recovered through a time-of-flight based correlation with the original sequence, while spurious interference outside of the selected range-gate are suppressed by the symbol length of the modulation sequence [31]. This allows for the reduction of RBS noise in reflection based systems and analogously second order DRS in transmissive interferometers.
Additional advantages of using DEHoI in the fiber frequency reference work involve the experimental simplicity and stability afforded by the technique. By removing active modulators from within the fiber interferometer and isolating the key optical paths, we are able to maximize the long-term stability of the interferometers [26]. The four-level phase stepping also provides a streamlined readout that is linear and robust whilst maintaining a large dynamic range that is able to track over many fringes.
In this paper, we demonstrate a fiber frequency reference architecture interrogated using a digitally enhanced homodyne phase readout. We provide a characterization of the main noise contributions including the effects of DRS noise and the limitations of the current DEHoI readout at high Fourier frequencies. In addition, we model fundamental fiber thermal noise, which our experimental results approach within a factor of 2 at 1 Hz Fourier frequency and a factor of 10 at high frequencies. We achieve an experimentally demonstrated relative frequency stability of 0.1 Hz/$\sqrt {\textrm {Hz}}$ above 100 Hz Fourier frequencies and 0.3 Hz/$\sqrt {\textrm {Hz}}$ at 1 Hz.
2. Experimental demonstration
Two nearly identical fiber Mach-Zehnder interferometers (IF1 and IF2), composed of single mode, polarization insensitive fiber components, are shown in Fig. 1. Both interferometers were used to measure the frequency dynamics of a free-running Koheras Adjustik E15 fiber laser centered at 1572 nm. The field used to interrogate both Mach-Zehnder fiber interferometers was encoded with a four-level pseudo-random phase modulation using a Gooch and Housego Fiber-Q acousto-optic modulator (AOM), with a centre RF frequency of 200 MHz and a DEHoI phase modulation frequency of 41.7 MHz. This modulated field was split between the two fiber interferometers, with each interferometer consisting of a short arm of approximately 1.5 m and a long arm of 15 km. Each interferometer was housed within individual passive thermal isolation chambers consisting of two 10 mm thick aluminum boxes, one inside the other. Thermal fluctuations inside the chambers were minimized by hermetically sealing the aluminum enclosures and housing each aluminum box inside a 5 cm thick Styrofoam box – creating a second order, low-pass thermal filter with a time constant of approximately 13 hours. Inside the inner aluminum enclosure, the fiber interferometer was constructed on a 10 mm thick metal plate mounted on shock-absorbing Sorbothane pads to provide mechanical isolation.
Fig. 1. The free running output of a 1572 nm fiber laser was encoded with the DEHoI modulation, and split to interrogate two identical Mach-Zehnder interferometers. The interferometers were housed in two passively-stabilized thermal chambers, and individually tracked the fluctuations in the laser frequency by reading out the interference between two length-mismatched arms. Following balanced photodetection and digitization, the dynamic phase evolution of each interferometer was recovered using DEHoI demodulation. Output phase data was then down-sampled and recorded to a networked host computer for further analysis.
We use the two output ports of each interferometer for balanced detection, obtaining higher phase signal amplitude and suppressing laser intensity noise. For each interferometer this was done using Insight BPD-1 balanced photodetectors followed by digitization through 250 MS/s analog-to-digital converters (ADC) on a National Instruments 5782 transceiver card. Real-time signal processing was carried out on a National Instruments 7975R Kintex 7 field-programmable gate array (FPGA). Once digitized, the FPGA based signal processing carried out two parallel DEHoI demodulation processes to recover the differential phase between the two arms for each interferometer. The decoded output was decimated to a sample rate of 8 kHz and transferred to a networked host computer for recording.
During the commissioning process, the DEHoI modulation frequency and code length was optimized to ensure reliable tracking of the laser frequency noise dynamics of the Koheras E15 laser source with the goal of optimizing readout bandwidth and code suppression. As code suppression is determined by the code length [31], we set the modulation frequency to the highest experimentally attainable frequency of 41.7 MHz to maximize available bandwidth. The second parameter, code length, then solely determines the absolute readout bandwidth, with longer codes reducing bandwidth in favor of code suppression. For longer DEHoI codes without sufficient bandwidth, the inability to resolve the laser frequency dynamics manifests itself as 2$\pi$ phase jumps in the interferometer readout (cycle slips), leading to a broadband increase in the interferometer phase noise. To maximize code suppression, we test multiple, increasing code lengths for their ability to continuously track the laser frequency dynamics. The optimal code length was determined as the maximum code length which maintains sufficient readout bandwidth to track laser frequency noise for >6 hours timescales. From this process, we arrive at an optimal 9 bit code length of 511 symbols (chips) where we maximize code suppression while retaining the ability to accurately track the laser frequency.
The main metric of interest is the relative stability of two interferometers which we compute by taking the difference of the two interferometer measurements of laser frequency noise. While the two interferometers differ in length by $\approx$ 30 m, we apply a Time-Delay Interferometry approach [25,27] to calibrate and synthesize a difference measurement that is independent of laser frequency noise. In Fig. 2 we plot 10 minutes of time domain data that has been converted into a frequency-domain amplitude spectral density, in units of both frequency and fractional stability. Trace (a) plots the free running laser frequency noise for the Koheras E15 as measured using a single fiber interferometer. From approximately 500 mHz to 1 kHz, this laser exhibits a 1/$\sqrt {f}$ frequency noise roll-off. A small feature, centered at 2 kHz, protrudes beyond this 1/$\sqrt {f}$ trend, likely due to noise peaking of a control loop within the laser, while at 400 mHz there is an abrupt transition to a 1/$f^3$ dependence that dominates low frequency stability.
Fig. 2. Two near-identical interferometers were used in this experiment to individually track the frequency fluctuations of the same source laser. Trace (a) plots the laser frequency spectral density as measured by a single interferometer, in both frequency stability (left) and fractional stability (right). As the two interferometers share the same frequency domain features, only one trace is plotted here. The subtraction of the two interferometer readouts (trace (b)) removes common-mode laser frequency noise, and is used to characterize their relative stability. A sensitivity of 0.1 Hz/$\sqrt {\textrm {Hz}}$ is achieved at frequencies above 100 Hz (blue). Differential coupling of the optical table resonances can be seen across the frequency range of 5 - 70 Hz (green), with distinct modes at 8 Hz, 23 Hz and 43 Hz. Between 400 mHz and 2 Hz, the interferometers are within a factor of 2 from fundamental fiber thermal noise (yellow). At infrasonic frequencies (< 400 mHz), we see the onset of temperature drift induced noise (pink).
The frequency spectral density of the relative interferometer stability, trace (b) of Fig. 2, can be broadly categorized into four regimes. The flat noise floor above 70 Hz (blue) reveals a broadband noise limitation. The slight roll-off above 2 kHz is due to an anti-aliasing filter prior to data decimation. Between 5 Hz and 70 Hz (green), we see mechanical and acoustic modes at 8 Hz, 23 Hz,43 Hz and 430 Hz. A further seismic measurement of the laboratory and the optical table was carried out using a separate commercial accelerometer. This revealed the 8 and 430 Hz resonances to be coupling from horizontal optical table movement, while the 43 Hz feature was identified to be a vertical table resonance. The 23 Hz mode did not directly correlate with table movement, and was instead hypothesized to be a mechanical mode of the fiber coil or isolation chamber. These modes rise up to two orders of magnitude above background frequency noise features.
At infrasonic frequencies, the system stability lands within a factor of 2 of the fundamental fiber thermo-mechanical noise between 400 mHz and 2 Hz (yellow). This is then succeeded by the onset of fiber temperature drift at lower Fourier frequencies (< 400 mHz, pink).
In Fig. 3, we compare the experimentally achieved sensitivity (trace (a)) with estimated levels of relevant noise sources (trace (b) - (d)). These noise mechanisms are discussed in detail in the following sections. Additive detection and quantization noises are more than two orders of magnitude below our experimental noise floor and excluded from Fig. 3.
Fig. 3. Fiber interferometer differential frequency noise spectral density - trace (a), fiber thermo-mechanical noise estimate - trace (b), fiber thermo-dynamic noise estimate - trace (c), double Rayleigh scatter noise estimate - trace (d). The slight roll-off in trace (a) beyond 2 kHz is due to our digital down-sampling filter.
3. Thermal noise
Thermal noise is a fundamental physical phenomenon and represents the ultimate achievable sensitivity for fiber optical sensors. Two potential thermal mechanisms contribute to phase error in the measurand, categorized by Duan as thermo-mechanical and thermo-dynamic [32–34]. At frequencies below $\approx 150$ Hz, thermo-mechanical is the dominant noise source amongst the two. Its contribution to phase error can be estimated for SMF-28 optical fiber at 1550 nm with the following expression [32]:
where the factor of $\sqrt {2}$ accounts for the incoherent addition between the two interferometers. Trace (b) of Fig. 3 plots this thermo-mechanical noise source. A previous direct analysis by Duan [35] assumed a Fourier frequency range well below the first longitudinal mechanical resonance of the fiber, at 400 mHz for our experimental parameters. This is extended to a wider bandwidth with the mode expansion approach proposed in a more recent publication [36], and validated in several experimental investigations [32,37,38]. As can be seen, our interferometer differential stability plotted in trace (a) of Fig. 3 lies within a factor of 2 of the thermo-mechanical noise limit between 400 mHz and 2 Hz.The thermo-dynamic noise contribution to our experiment can be estimated based on Eq. (19) from [36], with an extra factor of $\sqrt {4\pi }$ from conversion into a single-sided spectrum. This estimate is plotted in trace (c) of Fig. 3, intersecting with the thermo-mechanical noise estimate (trace (b)) at approximately 150 Hz. As shown in Fig. 3, although not a current limitation of our experiment, thermo-dynamic noise lies within a factor of 10 of our experimental noise floor above 150 Hz, and may impose a stability challenge in the future once we address dominant experimental noise sources.
4. Double Rayleigh scattering
The dynamics of Rayleigh scattering in optical fiber is often modeled as a randomly distributed, static array of scatter points driven by laser frequency noise, and typically associated with Rayleigh backscattering [39]. Our transmission architecture is immune to phase noise induced by first order RBS but still susceptible to DRS, which occurs when the back-reflected field undergoes a second scattering event. This reverses the scattering field direction of travel, realigning it with the metrology field [40].
As with first order RBS induced phase noise [24], the phase noise induced by DRS is dominated by coherent effects. In the case of DEHoI modulation with a highly coherent laser source, the optical coherence length $L_c$ is equivalent to the physical length of a DEHoI symbol (chip length) $L_{\textrm {chip}}$ [24]. We can determine the chip length using the speed of light in optical fiber and the DEHoI modulation frequency, $f_{\textrm {chip}}$, yielding the following expression:
Using Eq. (2) we can subsequently compute the total physical length of the modulation code within the interferometer fiber coil: $L_{\textrm {code}}$, as shown in Eq. (3):
The gating capability of DEHoI ensures that the majority of DRS induced phase noise is incoherent and strongly suppressed in the readout. However, coherence still remains within the single chip length ($L_{\textrm {chip}}$) being demodulated at the signal delay. For fiber lengths greater than the code length, the DEHoI modulation will repeat. This repetition results in the signal delay chip reoccurring at integer multiples of the code length $L_{\textrm {code}}$, all of which will be coherently demodulated. This means that DRS fields that scatter at these demodulation points, an integer multiple of $L_{\textrm {code}}$ extra distance relative to the metrology field, still interfere coherently and contribute to noise in the readout.
Figure 4 illustrates this scenario where the DRS time-of-flight delay re-synchronizes with the signal delay over multiple code lengths. In this illustration, the first scattering event reverses the metrology field’s direction of travel. Multiple second scattering events, represented by three dashed lines, occur at distances integer half codes ($L_{\textrm {code}}/2$) upstream, allowing the resultant DRS fields to gain a total round-trip delay of integer codes. These DRS fields carry the same code delay as the signal field and therefore appear indistinguishable to DEHoI, resulting in coherent demodulation. An analogous second scenario, also illustrated in Fig. 4, allows each DRS field to realign with the reference delay and couple into the readout. As we use a substantial length of fiber delay line ($L = 15$ km) and a relatively short modulation code (511 elements), these delayed, coherent DRS events become the dominant source of DRS noise within our system. To determine the noise contribution we must first start by determining the number of DEHoI codes present in our fiber coil using Eq. (4):
where $L$ is the length of the delay line. For our experimental parameters of $f_{\textrm {chip}} = 41.7$ MHz, 511 element code and $L = 15$ km, the code repeats approximately 6 times in the fiber. As this is a double-pass effect, the maximum total path length the DRS field can travel is therefore 12 complete codes prior to re-synchronizing with the demodulation delays and being coherently decoded.Fig. 4. An illustration of DRS fields re-synchronizing with the metrology field. When the additional round trip delay of the DRS field coincides with multiple digital codes (3 dashed lines of 2nd scattering event), the scatter field is realigned with the metrology field and its noise becomes coherently demodulated in the readout. As there are two metrology fields in our setup, the DRS field is able to interfere with both fields by obtaining the same code delay as either field.
To derive the expected noise contribution from this coherent DRS effect, we first consider the two metrology fields in our Mach-Zehnder interferometer interacting with the total DRS field, which can be written as:
We can further approximate the scattering coefficient $\alpha _S$ as equal to the attenuation coefficient $\alpha$, as scattering is the dominant source of loss in SMF-28e optical fiber at $1550$ nm. When considering DRS, $\gamma$ is squared due to each DRS event consisting of two scattering events. To determine the DRS phasor amplitude $\sqrt {P_{\textrm {sig}}\langle \gamma ^2\rangle }$, we modify Eq. (6) to give the second-order expression for $\langle \gamma ^2 \rangle$:
As we can see from Fig. 4, while the second scattering event must occur at specific locations with respect to the first scattering event location, the first scattering event can occur anywhere along the fiber. This distinction is manifested in the first and second scatter terms of Eq. (7), where the first backscatter is represented by the integrated loss over the entire fiber coil, $L$, and the second over a length of $L_{c}/2$, with the division of 2 due to double pass. The additional attenuation accrued by the scatter field is accounted for by the term $e^{-\alpha \eta L_{\textrm {code}}}$ within Eq. (7). The summation is further weighted by the factor $(2N-\eta )/2N$. This represents the fraction of second RBS scattering events that fit into the physical length of the fiber as a function of $\eta$.
Following the phasor analysis in [24], we derive the phase errors arising from DRS interfering with both the reference and signal fields as given in Eq. (5). The two errors are written as $\Delta \phi _{\textrm {err1}}$ and $\Delta \phi _{\textrm {err2}}$ respectively and can be calculated by:
We can see that the spectral characteristics of $\Delta \phi _{\textrm {err1}}$ and $\Delta \phi _{\textrm {err2}}$ are dependent on the dynamics of the DRS phasor, $\theta (t)$. As these DRS fields have traversed between 1 to 12 extra code lengths with respect to the desired metrology field, corresponding to time delays of between 12 and 144 microseconds, large amounts of laser frequency noise is coupled in through this process. These extensive delays also efficiently spread the DRS noise across the entire signal spectrum, allowing us to approximate the resulting noise spectral density by the RMS value of Eq. (9):
Substituting in Eq. (7) and adding the above two scenarios incoherently, we arrive at the broadband, single interferometer DRS noise floor of
This is then converted to frequency noise and accounted for our two interferometers, yielding a final broadband DRS relative noise floor of $0.12 \; \textrm {Hz}/\sqrt {\textrm {Hz}}$, shown in Eq. (12) and plotted in trace (d) of Fig. 3.
It can be seen from the above analysis that the DRS noise floor is dependent on the fiber coil length $L$, coherence length and therefore modulation frequency $f_{\textrm {chip}}$, and DEHoI code length $L_{\textrm {code}}$. We note that the modulation frequency $f_{\textrm {chip}}$ plays a relatively minor role in determining the DRS noise floor. This is because the extra physical length occupied by the code afforded by slower modulation is offset by the increased coherence length and therefore coupling of DRS at each demodulation code delay. Alternatively, lengthening the DEHoI code can effectively suppress DRS noise by reducing the number of code wraps within the fiber coil. However, a longer code requires longer integration time which reduces the code frequency, the signal update rate and the measurement bandwidth. In the current configuration, longer codes incur significant tracking challenges due to their lower code frequency, which filters the signal more aggressively, removing high frequency phase dynamics before the signal phase can be reconstructed. The inability of the low-pass-filtered and reconstructed phase to reliably follow the actual interferometer phase causes frequent cycle slipping in the time domain and substantially increased white noise in the signal spectrum. In optimizing interferometer sensitivity, we are therefore forced to maximize $f_{\textrm {chip}}$ and reduce code length until tracking errors and cycle slipping are eliminated. For our $15$ km fiber interferometer, this occurs with $f_{\textrm {chip}} = 41.7$ MHz and a code length of 511 symbols (chips).
5. Optimum fiber length
We now consider what the optimum fiber length would be in order to optimize broadband frequency noise sensitivity of our fiber frequency reference experiment. From Fig. 3, we identify two noise sources that may potentially dominate the signal spectrum above $150$ Hz: DRS induced phase noise, and thermo-dynamic noise. Both these noise sources are functions of interferometer fiber length. Figure 5 plots both DRS and thermo-dynamic equivalent frequency noise as a function of fiber interferometer arm length mismatch for three different code lengths: 511 (9 bit), 1023 (10 bit) and 2047 (11 bit) symbol codes. For the shortest 9 bit code (trace (b)), DRS noise dominates over all fiber lengths greater than $\approx$ 1.3 km. A DRS-limited sensitivity of $\approx$ 0.12 $\textrm {Hz}/\sqrt {\textrm {Hz}}$ is expected at 15 km (black cross) in close agreement with our experimental results plotted in Fig. 3. Below 1.3 km, there is less than one half of a full code resident in the fiber interferometer at any time, and all DRS noise is suppressed by DEHoI code orthogonality. At these short fiber lengths, the broadband sensitivity limit is expected to be thermo-dynamic phase noise at a level of 0.053 $\textrm {Hz}/\sqrt {\textrm {Hz}}$.
Fig. 5. Estimation of thermo-dynamic (a) and DRS noises (b - d) at 300 Hz Fourier frequency as a function of fiber coil length. At our current operating parameter of 41.7 MHz $f_{\textrm {chip}}$ and 511 element code (b), DRS remains a dominant noise source unless code wrapping is eliminated. This occurs at a substantially shorter coil length than our current setup of 15 km (black cross), approximately $L = 1.3$ km (red dot point), where an optimal high frequency sensitivity of 0.053 $\textrm {Hz}/\sqrt {\textrm {Hz}}$ can be expected. Additionally, we expect a more relaxed tracking requirement at shorter fiber lengths, potentially allowing the use a longer DEHoI code without incurring cycle slips. The corresponding DRS estimates at code lengths 1023 (c) and 2047 (d) are also plotted. Assuming tracking requirements are met, we can expect a thermo-dynamic noise limited frequency stability of 0.037 $\textrm {Hz}/\sqrt {\textrm {Hz}}$ at $L = 2.6$ km and 0.026 $\textrm {Hz}/\sqrt {\textrm {Hz}}$ at $L = 5.4$ km, respectively.
Figure 5 also demonstrates the advantages of using longer codes, provided no tracking errors are present, with a 10 bit code (trace (c)) reaching a predicted relative stability of 0.037 $\textrm {Hz}/\sqrt {\textrm {Hz}}$ at a fiber length of 2.6 km. An 11 bit code (trace (d)), meanwhile, reaches a predicted stability of 0.026 $\textrm {Hz}/\sqrt {\textrm {Hz}}$ at a fiber length of 5.4 km. These two parameters indicate shorter fiber coils with DEHoI code lengths at least twice the entire physical coil length are preferable.
However, at frequencies below a few Hz, thermo-mechanical noise will typically dominate over all broadband noise sources. From Eqn. 1, it is seen that thermo-mechanical frequency noise spectral density scales as $1/\sqrt {L}$, therefore improving with longer fiber coils. This means there is an implicit trade-off between broadband and low frequency stability when determining the optimum interferometer fiber length. For our current experiment, we have prioritized low frequency stability and have come close to reaching the thermo-mechanical noise limit at Fourier frequencies of around 1 Hz.
6. Conclusion
We have presented a fiber interferometer frequency reference architecture that achieves sub-Hz relative stability from 4 kHz to 300 mHz Fourier frequencies with the exception of mechanical modes at $\approx$ 8 Hz, 23 Hz, 43 Hz and 430 Hz. In addition, our interferometer reaches a broadband relative stability of 0.1 Hz/$\sqrt {\textrm {Hz}}$ above 100 Hz. We have identified the primary source of this broadband noise floor to be code coherent double Rayleigh scattering, confirmed through numerical modeling of DRS and its interaction with the digitally enhanced interferometry readout. We further show that the relative interferometer stability lies within a factor of two of the fiber thermo-mechanical noise floor at Fourier frequencies between 400 mHz and 2 Hz. In addition, at higher frequencies relative stability is within a factor of $\approx$ five from the thermo-dynamic noise floor of the fiber interferometer.
From the work presented here, we proceed to clearly define optimum fiber lengths and DEHoI modulation parameters to further improve the broadband noise and laser frequency tracking performance of the system. This analysis identifies a fundamental trade-off between low frequency thermo-mechanical noise stability and DRS or thermo-dynamic noise limited broadband stability which inform the design and implementation of future generation fiber frequency references.
Acknowledgments
The authors would like to acknowledge useful discussions on fiber thermal noise with Professor Lingze Duan.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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