Open Access
Add to BibSonomy
Abstract
We present and establish a versatile analytical model that allows overall analysis and optimization for the phase noise performance of the delay interferometer based optical phase-locked loop (OPLL). It allows considering any type of lasers with arbitrary frequency noise properties while taking into account the contributions from various practical noise sources, thus enabling comprehensive investigation for the complicated interaction among underlying limiting factors. The quantitative analysis for their evolution along with the change of the delay of the interferometer unveils the resulting impact on the fundamental limit and dynamics of the output phase noise, leading to a well-balanced loop bandwidth and sensitivity thus enabling the overall optimization in terms of closed-loop noise performance. The tendencies observed and the results predicted in terms of coherence metrics in numerical verification with different lasers have testified to the precision and effectiveness of the proposed model, which is quite capable of acting as a design tool for the insightful analysis and overall optimization with guiding significance for practical applications.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Highly coherent laser sources exhibiting high spectral purity, stability, and low phase noise in either frequency-fixed or swept operations are of vital significance in various applications such as spectroscopy [1], ranging (i.e. lidar) [2,3], quantum metrology [4], atomic clocks [5], fiber optic-sensing [6,7], optical spectrum analyzer [8], optical coherence tomography [9]. For instance, from the viewpoint of metrology, lidar, and sensing, a high degree of coherence is essential to achieving high resolution, precision, and accuracy. Nevertheless, the linewidth broadening due to the inevitable intrinsic phase noise as well as the instability stemming from the spontaneous emission, cavity vibration, pumping noise, external disturbance, and so on [10–12] are still quite detrimental, thus impairing the performance in applications.
In this context, different techniques have been proposed for phase noise suppression, i.e. linewidth narrowing, and frequency stabilization. It can be realized via the popular Pound-Drever-Hall [13,14] technique implemented by locking the laser onto an ultra-stable Fabry-Perot cavity. The required ultra-stable high-finesse cavity and the inconvenience in frequency tunability should be taken into consideration for different scenarios.
Alternative approach relying on actively locking to an unbalanced interferometer using an optical phase-locked loop (OPLL) has also addressed lots of attention thanks to its effective coherence enhancement for both frequency-fixed and swept lasers [15–19]. Although the trade-off between the loop bandwidth and discrimination gain, namely the sensitivity, for their dependence on the unbalance delay has brought about a strict limit on the performance, the compact, feasible, versatile, and practical features have made it appealing in fields from fundamental research to state-of-the-art applications. According to the operational principle, in order to maximize the efficiency of such systems in various scenarios with different lasers and varied noise conditions, an appropriate design should allow for systematic optimization of the performance taking into account these limiting factors and the potential noise sources.
Most of the well-established models, on one hand only take into account certain kinds of noises, on the other hand quite often only consider lasers with regular power-law frequency noise spectrum (FNS), thus losing precision and accuracy in predicting the noise performance. A recent solution on the optimal choice for the delay of the unbalanced interferometer was promoted and established mainly concerning lasers with specific noise properties via the fitting of their FNS [20,21], in which other important factors such as fiber thermal noise and reference phase noise have been hardly taken into consideration or being insufficient in precision. One popular classic model [22] has assumed an asymptotically frequency-independent noise at low-frequency region. While, nevertheless, recently a non-flat frequency response is observed in fiber phase noise, indicating the existence of a different type of thermal fluctuation associated with optical fibers [23–25]. Recently, a system model has been developed especially for phase noise in MHz frequency range, yet the employed fiber thermal noise model finds difficulties in describing noises at low-frequency regimes [26]. Therefore, it is indeed an urgent requirement to establish an analytical model that not only allows for analyzing lasers with practically any type of FNS but also be able to account for the contributions from different types of noise sources, thus enabling comprehensive analysis and overall optimization of the unbalanced interferometer based OPLL for lasers with arbitrary frequency noise properties.
In this paper, we present an elaborate analytical model for overall phase noise analysis and performance optimization for unbalanced delay interferometer based OPLL system. This model is not only capable of analyzing any type of lasers with arbitrary frequency noise properties, even for lasers with irregular FNS, but also permits taking into consideration the contributions from practical noise sources such as reference oscillator and fiber thermal noise. Therefore, it allows quantitatively describing the complicated interaction of different kinds of noises and thus the evolution with respect to the delay of the unbalanced interferometer, unveiling the sophisticated relation of these limiting factors to the eventual output phase noise performance. With the proposed model, the comprehensive discussion for the dominant factors at different conditions is enabled and precise prediction for the optimal delay required together with the associated coherence metrics can be realized, giving rise to a well-balanced loop bandwidth, discrimination gain, and sensitivity thus leading to the overall optimization for the performance. Numerical verifications concerning several different types of laser sources at various noise conditions have been carried out, in which the resulting optimized unbalanced interferometer delays that lead to the optimal output phase noise performance in terms of FNS and linewidth have confirmed the capability of the proposed model. This thus offers an evaluation tool for the fundamental limits and intuitive design guidance for practical systems.
2. OPLL model
2.1 Operation mechanism
A typical unbalanced delay interferometry based laser stabilization system that is comprised of a delay self-heterodyne (DSH) in the form of unbalanced Mach-Zehnder interferometer (UMZI) is illustrated in Fig. 1. The light from the laser source is split into two parts. One is sent to the UMZI for phase locking and the other is provided for out-of-loop applications. The former is further divided and travels through two arms, where one is inserted with an acousto-optic frequency shifter (AOFS) providing a frequency shift $\omega _A$ and the other includes a fiber delay $\tau _d$. They are recombined and then injected onto a balanced photodetector (BPD).
Fig. 1. Schematic for UMZI based OPLL system, PMC, polarization-maintained coupler; BPD, balanced photodetector; AOFS, acousto-optic frequency shifter; PFD, phase-frequency discriminator.
The output photocurrent can be then given by $i_{\mathrm {PD}}(t)=\cos [\varphi _{\mathrm {Laser}}^n(t)-\varphi _{\mathrm {Laser}}^n(t-\tau _\mathrm {d})+\omega _\mathrm {A}t+\omega _0\tau _\mathrm {d}]$, where $\omega _0$ represents the original laser frequency and $\varphi _{\mathrm {Laser}}^n(t)$ denotes the laser phase noise. The reference oscillator offers a stable reference with frequency $\omega _\mathrm {R}$ that is chosen to be equal to $\omega _\mathrm {A}$ for heterodyne detection. The output error signal from the phase-frequency discriminator (PFD) is then sent to a loop filter (LF) with proper loop processing. The resulting feedback control signal is used to correct the laser frequency fluctuations via various types of frequency actuators such as intra-cavity piezoelectric transducer (PZT) stretcher for the fiber laser, current driver for semiconductor laser (SCL) and diode laser, and external electro-optic modulators to realize closed-loop stabilization.
The small signal model is sketched in Fig. 2, where the transient response of the system is described in the Fourier domain, in which the noises are regarded as deviations from the steady state. The LF is designed relying on a proportional integrator with the transfer function $F_{\mathrm {LF}}(\omega )=K_\mathrm {L}(1+K_\mathrm {I}/j\omega )$, where $K_\mathrm {L}$ and $K_\mathrm {I}$ hold for the gain and the cut-off frequency, respectively, and $\omega$ represents Fourier angular frequency. The influence of the total loop delay $\tau _L$ is taken into consideration by the term $e^{-j\omega \tau _\mathrm {L}}$. The laser source acts as an optical voltage-controlled oscillator as represented by $F_{\mathrm {Laser}}(\omega )$, which in general consists of an integer in addition to the frequency modulation response of the laser. In this context, we define the forward gain of the phase locking system as:
In addition to the intrinsic laser phase noise, the overall output phase noise is composed of the contributions from the noise sources within the loop, including the intrinsic phase noise of the reference oscillator, fiber thermal noise of the UMZI, shot noise during photodetection, and electrical thermal noise, as denoted by $\varphi _{\mathrm {ref}}^{n}(\omega )$, $\varphi _{\mathrm {UMZI}}^{n}(\omega )$, $\varphi _{\mathrm {SN}}^{n}(\omega )$, and $\varphi _{\mathrm {Th}}^{n}(\omega )$, respectively. Accordingly, their phase noise spectrum (PNS) can be additionally described by $S_{\varphi _{\mathrm {ref}}^{n}(\omega )}$, $S_{\varphi _{\mathrm {UMZI}}^{n}(\omega )}$, $S_{\varphi _{\mathrm {SN}}^{n}(\omega )}$, and $S_{\varphi _{\mathrm {Th}}^{n}(\omega )}$, respectively. It is worth noting that care should be taken when dealing with the phase fluctuations induced by environmental instability such as temperature drift, mechanical vibrations, and acoustic disturbances can be efficiently mitigated with proper insulations of the loop. This can be efficiently mitigated with proper protection of the loop such as vacuum enclosures, seismic isolation provided by modern anti-vibration platforms, thermal insulation, and acoustic isolation box [27]. In this work, the isolation is implemented by housing the OPLL structure in an iron sound-proof box equipped with sound insulation and thermal isolating materials. This box is additionally placed on a passive vibration isolation platform in our experiments.
Note that the UMZI configuration is almost naturally immune to Rayleigh backscattering (RBS) induced phase noise degradation. In the case of other loop configurations such as Michelson interferometer, such effects due to RBS can still be mitigated by introducing frequency shifts [28]. Moreover, for lasers with high intrinsic coherence or locked with respect to long fiber delays, double Rayleigh scattering (DRS) related phase noise may come into play as modeled in [29]. Since such conditions can be hardly satisfied for the commercial laser considered in our case, it is feasible to neglect this noise source without loss of generality. Nevertheless, if necessary, these two noise sources can be readily accounted for with the proposed model by directly importing their respective PNS into the loop analysis.
The contributions of these noise sources are compared in Fig. 3(a) in terms of PNS in order to illustrate their respective characteristics. A reference operating at 40 MHz is yielded by a direct digital frequency synthesizer (DDS) that is synchronized to a highly stable atomic clock, whose phase noise is directly measured by a spectrum analyzer.
Fig. 3. (a) PNS for different noise sources; (b) different lasers with their measured FNS used for the numerical verification for the proposed model.
Concerning the UMZI, since the environment induced instability can be largely mitigated as mentioned above, the influences due to the intrinsic thermal fluctuations of the fiber become the leading factor. This can be described by the thermoconductive noise [30] induced from the spontaneous thermal expansion and thermo-optic effect. The PNS of the general thermoconductive noise is given by $S_{\mathrm {TC}} (\omega )$ that is mainly determined by the phase velocity, thermal diffusivity, mode field radius, and outer diameter of the fiber and is positively related to the delay $\tau _{\mathrm {d}}$ and the square of the temperature T. Using the typical parameters for Corning standard SMF-28 optical fibers [31,32], the resulting $S_{\mathrm {TC}}(\omega )$ for 200 m and 4 km fiber at 300 K at the optical wavelength of 1550 nm are drawn in tangerine and purple dotted line, respectively in Fig. 3(a), indicating an approximate white noise behavior at the low-frequency region. In addition, it is observed that there is a slow roll-off at about 1-10 kHz while it decreases when above approximately 10 kHz [33], testifying to the frequency dependency of $S_{\mathrm {TC}} (\omega )$.
Nevertheless, the thermal induced variation is usually regarded as a random process, any single type of power-law noise could hardly provide an accurate description, especially in low-frequency regions where slow drift dominates. A 1/f behavior for fiber thermal noise has been observed in infrasonic frequencies mainly attributed to the fiber mechanical dissipation [34]. This so-called thermomechanical noise was modeled as $S_{\mathrm {TM}}(\omega )=\alpha _{\mathrm {TM}} \mathrm {T} \tau _{\mathrm {d}} / \omega$ and has been experimentally verified , where $\alpha _{\mathrm {TM}}$ is a constant representing material properties [35]. It is mainly determined by the effective refractive index of the fiber, the optical wavelength, mechanical loss angle, Young’s modulus of the fiber material, and the cross-sectional area of the fiber, whose parameters can be found from the aforementioned Corning standard SMF-28 [31,32,34]. The experimental data for the loss angle is approximately 0.01 at frequencies between 75 and 200 kHz. However, so far, there is no direct experimental evidence showing that this value is available at frequencies below 1 kHz [36]. We nonetheless make the same assumption of 0.01 that was used in the previous verification experiments for loss angle [32,35]. Here, we introduce $S_{\mathrm {TM}}(\omega )$ into the proposed model to enhance the precision for the analysis of the noise contribution. To give an intuitive illustration, fiber thermal noise in different conditions for 200 m and 4 km standard single-mode fiber (SSMF) are calculated and presented, respectively in Fig. 3(a). According to these models, it can be observed that the PNS of both fiber thermal noises are proportional to $\tau _{\mathrm {d}}$ and thus the root mean square (RMS) phase noise is proportional to $\sqrt {\tau _{\mathrm {d}}}$. This is quite different from the other noises since they are not dependent on fiber delay. Apart from these significant low-frequency noises, the shot noise in addition to the electrical thermal noise is also taken into account. The former is calculated with the 1 mW optical power used in the analysis and the 0.9508 A/W responsivity for the BPD (Thorlabs PDB480C). While the latter is dominated by the electrical noise induced from the PFD, which is $\sim 1\times 10^{-14}$ $\mathrm {rad}^2/\mathrm {Hz}$ in this demonstration, constituting the system noise floor as presented by the red dotted curve.
A precise description of the intrinsic laser phase noise is necessary to accurately predict the output phase noise. This can be achieved by measuring the actual FNS using the short-delayed self-heterodyne (SDSH) method [37]. Thanks to the proposed analytical model that allows directly importing the actual noise of the laser, it is not restricted for lasers with noise models of only certain kinds of noises such as white frequency noise. Thus, this versatile model readily applies to any practical lasers with arbitrary frequency noise properties.
2.2 Noise transfer and analysis
Following the small signal analysis, the impact of the aforementioned noise sources on the output phase noise are all taken into account in the closed-loop case, where the closed-loop transfer function can be then written as
It could be observed that in general, $H_{\mathrm {cl}}(\omega )$ behaves in a low-pass fashion according to the transfer properties of all the comprised stages within the loop. This indicates the fact that in the low-frequency region, the stability and noise performance of the output is largely dominated by the reference oscillator and other in-loop noise sources. Meanwhile, the closed-loop output phase can be given by:
In order to study the statistic characteristics for the output noise, the output FNS, $S_{v_{\textrm {out}}^{n}}(\omega )$ containing the contributions of all the above noises can be further derived based on Eq. (2) as
According to the effect of the low-pass $H_{\mathrm {cl}}(\omega )$ and high-pass $H_{\mathrm {e}}(\omega )$ the eventual output phase noise of the laser strongly depends on the UMZI delay $\tau _{\mathrm {d}}$, which not only addresses the inevitable trade-off for the loop bandwidth and discrimination gain, namely sensitivity, but also in turn determines how, and to what extent other noise sources will impact on the output phase noise. It is thus of great importance to establish an analysis model for the optimization of UMZI delay concerning the interaction and evolution of different noise sources for laser with arbitrary frequency noise properties.
2.3 Evolution of output laser FNS
On the basis of Eq. (4), we take into consideration the FNS of different laser sources and other noise sources described above to explore the performance optimization with respect to $\tau _{\mathrm {d}}$. Three different laser sources with distinct frequency noise properties are discussed, respectively. The system is built as a type II loop with lead-lag filters to cope with the nonlinearity and hysteresis effect in the frequency modulation (FM) response for both fiber laser [38,39] and SCL [15]. In order to satisfy the loop stability condition, the 3 dB cut-off frequency and the proportional gain $K_{\mathrm {L}}$ of the LF are set to be one tenth of the UMZI FSR and inversely proportional to UMZI delay, respectively, which also helps to optimize the loop response. Meanwhile, the conversion gain $K_{\mathrm {P}}$ is unity and the loop delay $\tau _{\mathrm {L}}$ is zero, both of which can be adjusted according to actual situations if necessary. For the demonstrative purpose, we take three typical kinds of lasers: fiber laser ($\sim$ 5 kHz linewidth), external cavity laser (ECL) ($\sim$ 50 kHz linewidth) [40], and SCL ($\sim$ 800 kHz linewidth) [15] in the analysis whose FNS are sketched in Fig. 3(b). It should be noted that owing to the limited response for the intra-cavity piezo in most of the fiber lasers, which is usually several tens of kHz, the achievable loop bandwidth is strictly limited to ten or even several kHz range, according to loop stability requirement. Subjected to this practical issue, the loop bandwidth for fiber laser at short UMZI delays is adjusted to 10 kHz from a practical point of view.
As aforementioned, except for the influence on $H_{\mathrm {cl}}(\omega )$ and $H_{\mathrm {e}}(\omega )$, the variation of UMZI delay $\tau _{\mathrm {d}}$ introduces changes in fiber-delay-dependent noises, making the evolution of the output FNS more complicated. First, the performance of the UMZI based phase-locking is evaluated with the commercial fiber laser at different $\tau _{\mathrm {d}}$ from 200 m to 20 km as compared in Fig. 4(a), where the corresponding loop bandwidth can be inferred by the location of the bumps. For the sake of clarity, $\tau _{\mathrm {d}}$ has been normalized to the intrinsic coherence length $\mathrm {L_c}$ of the laser without loss of generality. Besides the significant frequency noise suppression, the changes in $\tau _{\mathrm {d}}$ also greatly alters the output FNS. Basically, a longer delay should allow for a higher gain but with a smaller loop bandwidth as clearly manifested in the numerical results, where the in-loop suppression enhances and the gain bump shifts towards the low-frequency region along with the increase of $\tau _{\mathrm {d}}$.
Fig. 4. Evolutions for the output FNS and for various noise components at different UMZI delay. (a) (b) (c) are the output FNS of the fiber laser, ECL, and SCL, respectively. (d) (e) (f) are in the case of fiber laser source, the comparison of contributions from several noise sources at the fiber delay of 200 m, 4 km, and 20 km, respectively.
At the same time, the contributions from different noise sources also vary along with $\tau _{\mathrm {d}}$. It can be directly derived from Eq. (2) and (4) that the closed-loop and the error transfer factors $|H_{\mathrm {cl}}(\omega )|^2$ and $|H_{\mathrm {e}}(\omega )|^2$ are in approximately inverse and direct proportion to the square of $\tau _{\mathrm {d}}$, respectively, especially in the low-frequency region. On one hand, for noises that are independent on $\tau _{\mathrm {d}}$, their impacts on the output FNS are largely governed by the factor $|H_{\mathrm {cl}}(\omega )|^2f^2$, corresponding to an inversely proportional contribution with respect to ${\tau _{\mathrm {d}}}^2$. Among these, the reference is usually the leading noise source that we can take as the dominance without loss of generality. On the other hand, for $\tau _{\mathrm {d}}$ dependent noises, in particular the fiber thermal noise, since it is intrinsically proportional to $\tau _{\mathrm {d}}$, it is interesting to compare their contributions at different $\tau _{\mathrm {d}}$. In principle, due to their unique dependence in regard of $\tau _{\mathrm {d}}$, the former exhibits a faster decline than the latter when $\tau _{\mathrm {d}}$ increases. Therefore, in general, a longer UMZI fiber delay makes the OPLL system less sensitive to the distributed fiber noise and reference phase noise. Besides, concerning $|H_{\mathrm {e}}(\omega )|^2$ induced high-pass filtering, the contribution of laser frequency noise is more serious as $\tau _{\mathrm {d}}$ increases. This leads to the fact that at longer delays, fiber thermal noise gradually becomes the chief component in the second term in Eq. (3) and (4). This way, the output FNS within the loop bandwidth is the result of the sophisticated interaction among these noises as verified by the tendency presented in Fig. 4(d), 4(e), and 4(f). It appears that the fiber thermal noise is not stronger than the reference noise until delay approaches $\sim$4 km and overtakes when extended to 20 km. While the overall in-band frequency noise is dominated by that of the laser when beyond 4 km delay because the rising rate of $|H_{\mathrm {e}}(\omega )|^2S_{v_{\textrm {Laser}}^{n}}(\omega )$ is higher than the decline rate of $|H_{\mathrm {cl}}(\omega )|^{2}S_{\varphi _{\text {UMZI}}^{n}}(\omega ) f^{2}$. For frequency higher than the LF’s cut-off frequency, $|H_{\mathrm {e}}(\omega )|^2$ remains almost unchanged regardless of $\tau _{\mathrm {d}}$ while other $|H_{\mathrm {cl}}(\omega )|^{2}$ related terms quickly degrade. Therefore, in this region, the output basically coincides with the dominating laser FNS and hardly changes with $\tau _{\mathrm {d}}$.
Similar trends can be expected for lasers with distinct frequency noise properties, as sketched in Fig. 4(b) and 4(c) for ECL and SCL, respectively. The in-band frequency noise firstly decreases and then rises along with the increase of $\tau _{\mathrm {d}}$ though the delays at which the transition occurs vary for different lasers. It is found that for a higher laser FNS, only a shorter delay is necessary to reach the transition, inferring the possibility to find an optimum delay for minimizing the output noise as it plays a decisive role among the interplay of different noise sources, thus the overall optimization for the system performance. Worth noting that this overall optimization can also be important and beneficial for the generation of frequency chirped laser [17–19] using UMZI based OPLL. This is, however, out of the scope of this work, and a more versatile model that is able to include arbitrary frequency agility will be pursued in our future research.
3. System optimization
For quantitative optimization of such sophisticated interplay among these noise sources, the coherence metric in terms of laser linewidth is taken to evaluate the performance for the output frequency noise. Concerning lasers with possibly irregular frequency noise properties, which in particular usually occurs for lasers under OPLL control, the equivalent linewidth calculated using the FNS with power area method (PAM) [41] is applied for ease of estimating the evolution at varying UMZI delays.
3.1 Optimal UMZI delay in the OPLL
The predicted equivalent linewidth for the same fiber laser as in the last section is illustrated in Fig. 5(a), where the evolution with increasing delay has been studied. It could be observed that under the identical condition for all the noises, the equivalent linewidth quickly improves and then degrades along with the increase of $\tau _{\mathrm {d}}$ and the optimal linewidth of $\sim$2.19 Hz is achieved at about 3.52 km, corresponding to 0.24 times of the intrinsic coherence length as shown in Fig. 5(b) with the normalization mentioned above.
Fig. 5. Equivalent linewidth evolution of OPLL output for a fiber laser, (a) with increased UMZI delay, (b) with UMZI delay normalized to intrinsic coherence length.
As discussed above, in cases when $\tau _{\mathrm {d}}$ is not very long, the reference is usually the leading noise source over the others, which seriously limited the in-band noise performance. Therefore, further improvement in the resulting coherence needs to consider applying with an ultra-low noise reference which can be verified as compared in Fig. 5. Though being referenced to a highly stable atomic clock, the noise of the DDS can still be substantially improved by a low noise analog signal generator. At the optimal delay of $\sim$3.52 km where originally the reference noise dominates, or even at a shorter delay, via further reducing the phase noise of the reference by 10 dB, a lower linewidth, corresponding to a higher degree of coherence can be realized as illustrated by the red dashed curve in Fig. 5. In particular, if with an ideally noiseless reference, the prediction in blue dashed curve constitutes a lower bound for the output FNS, namely the optimum linewidth and coherence that can be achievable.
Except for UMZI delay, another potential optimization lies in the choice of an appropriate reference. Based on the discussion above, such an optimal reference should stay well below the fiber thermal noise of the UMZI or at least smaller than the contribution of the laser frequency noise, thus being hardly dominant. Taken as a useful criterion, it acts as a design guideline that allows predicting the reference noise level which is needed to achieve the optimal output phase noise performance.
Besides, the modeling of the fiber thermal noise for the UMZI has an important influence on the overall analysis and the eventual optimization. As shown in Fig. 5, the performance only considering traditional thermoconductive noise, namely in the absence of the 1/f component in the fiber thermal noise, is drawn in green dotted line. The resulting linewidth is entirely lower than that in cases when different power-law components are involved, namely at least with both thermoconductive and thermomechanical noises, while the optimal delay is reduced to 3.37 km. This suggests the fact that the low-frequency noise makes a non-trivial influence on the phase noise performance. Thus, stabilization for the UMZI is becoming necessary, where proper temperature control in addition to acoustic and vibration isolation can be used to mitigate the environmental fluctuations that are sensitive for low-frequency regimes. During the modeling process, one could find that the inappropriate placing and winding of fiber might bring about extra noise and deterioration. Incorrect placement may add extra dissipation under external excitation and could potentially raise the loss angle over the entire, or a portion of the interested frequency span that finally gives rise to fiber thermal fluctuations.
3.2 Application for different kinds of laser sources
Similar tendencies can also be observed for lasers with distinct frequency noise properties, with practical or ideally noiseless references, respectively, as observed from Fig. 6. Discrepancies are found for different lasers that the optimal delays are achieved at different locations with respect to their normalized coherence length, which can be intuitively understood because though the same condition is preserved for other noise sources, the contribution from the laser frequency noise differs a lot for different lasers. Particularly, for all these different lasers, obvious overlaps occur between cases with practical or ideal references when the delay is extended to longer than the intrinsic coherence length. This is probably because in this region the laser frequency noise is dominant over all other noises, thus the output becomes nearly irrelevant to other system noises. When the delay is much shorter than optimal values, it is also interesting to find that the tendency for the evolution of different laser sources coincides due to the dominance of the system noise, i.e. second term in Eq. (4), since laser frequency noise is still weak under loop action. Furthermore, for lasers with a stronger frequency noise, generally a longer optimal delay relative to the intrinsic coherence length is expected. This is probably because a higher gain is necessitated when one needs to cope with a stronger noise, especially in low-frequency region.
Fig. 6. Equivalent linewidth evolution of OPLL output for three different kinds of lasers with practical or ideally noiseless references, (a) with increasing UMZI delay, (b) with UMZI delay normalized to intrinsic coherence length.
As observed from the comparison in Fig. 6, it allows finding an optimal configuration for the UMZI based OPLL in order to achieve a high coherence, namely narrow linewidth and low phase noise for lasers with arbitrary FNS. Their evolutions as sketched in Fig. 7 unveil the fact that though the ratio between the optimal delay and intrinsic coherence length is not fixed, for lasers with higher intrinsic coherence, in general longer optimal delay is expected. Worth mentioning that when we take a different look in terms of normalization, such tendency is more precisely explained that the optimal ratio slightly decreases along with the increase of coherence length. Thus, the optimal UMZI delay is closely related to the shape of the intrinsic laser FNS and those for other noise sources. Therefore, the proposed model that allows analyzing practical lasers with arbitrary FNS is of great importance for the appropriate OPLL design as well as comprehensive optimization.
Fig. 7. Optimal UMZI delay for lasers with different coherence lengths with and without normalization to intrinsic coherence length.
4. Conclusion
In this work, an analytic model that allows comprehensive analysis of the output for phase noise and optimization of the performance for unbalanced delay interferometer based OPLL system applicable to lasers with arbitrary frequency noise properties has been presented. Taking into account the contributions from practical noise sources including reference oscillator and fiber thermal noise, a comprehensive study of the closed-loop phase noise and quantitative analysis of output laser FNS is offered. With the increasing delay of the unbalanced interferometer, the contributed trends and change rates from different noise sources vary with each other, leading to the complicated interaction and the evolution of eventual output phase noise performance. Applying the model to a fiber laser, an ECL, and a SCL, similar tendencies are found: with increasing interferometer delay, the loop bandwidth decreases and the in-band output frequency noise firstly decreases and then rises, while the out-band frequency remains coinciding with the input laser FNS. Associating with equivalent linewidth evaluation, under the identical condition for all the noises, the change of closed-loop laser linewidth at varied interferometer delay is analyzed and the optimal interferometer delay could be solved for different laser sources. Numerical demonstrations at various noise conditions show that the choice of reference phase noise level, environment stability, and placement of unbalanced interferometer also have great impacts on the system performance. Comparing different types of laser sources, it appears that a longer optimal delay is expected for lasers with higher intrinsic coherence, though the ratio between delay and the coherence length slightly decreases with the increase of the latter. This model provides an insight into the dynamic impacts from practical noise sources on the system performance, offering useful guidance for system design including interferometer delay optimization.
Funding
National Natural Science Foundation of China (61827807, 61805014).
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.
References
1. P. Del’Haye, O. Arcizet, M. L. Gorodetsky, R. Holzwarth, and T. J. Kippenberg, “Frequency comb assisted diode laser spectroscopy for measurement of microcavity dispersion,” Nat. Photonics 3(9), 529–533 (2009). [CrossRef]
2. J. Riemensberger, A. Lukashchuk, M. Karpov, W. Weng, E. Lucas, J. Liu, and T. J. Kippenberg, “Massively parallel coherent laser ranging using a soliton microcomb,” Nature 581(7807), 164–170 (2020). [CrossRef]
3. N. Kuse and M. E. Fermann, “Frequency-modulated comb LIDAR,” APL Photonics 4(10), 106105 (2019). [CrossRef]
4. C. L. Degen, F. Reinhard, and P. Cappellaro, “Quantum sensing,” Rev. Mod. Phys. 89(3), 035002 (2017). [CrossRef]
5. A. D. Ludlow, M. M. Boyd, J. Ye, E. Peik, and P. O. Schmidt, “Optical atomic clocks,” Rev. Mod. Phys. 87(2), 637–701 (2015). [CrossRef]
6. J. C. Juarez, E. W. Maier, K. N. Choi, and H. F. Taylor, “Distributed fiber-optic intrusion sensor system,” J. Lightwave Technol. 23(6), 2081–2087 (2005). [CrossRef]
7. J. Von Der Weid, R. Passy, G. Mussi, and N. Gisin, “On the characterization of optical fiber network components with optical frequency domain reflectometry,” J. Lightwave Technol. 15(7), 1131–1141 (1997). [CrossRef]
8. M. Bishof, X. Zhang, M. J. Martin, and J. Ye, “Optical spectrum analyzer with quantum-limited noise floor,” Phys. Rev. Lett. 111(9), 093604 (2013). [CrossRef]
9. I. Grulkowski, J. J. Liu, B. Potsaid, V. Jayaraman, J. Jiang, J. G. Fujimoto, and A. E. Cable, “High-precision, high-accuracy ultralong-range swept-source optical coherence tomography using vertical cavity surface emitting laser light source,” Opt. Lett. 38(5), 673–675 (2013). [CrossRef]
10. M. Osinski and J. Buus, “Linewidth broadening factor in semiconductor lasers - An overview,” IEEE J. Quantum Electron. 23(1), 9–29 (1987). [CrossRef]
11. S. Gundavarapu, G. M. Brodnik, M. Puckett, T. Huffman, D. Bose, R. Behunin, J. Wu, T. Qiu, C. Pinho, N. Chauhan, J. Nohava, P. T. Rakich, K. D. Nelson, M. Salit, and D. J. Blumenthal, “Sub-hertz fundamental linewidth photonic integrated Brillouin laser,” Nat. Photonics 13(1), 60–67 (2019). [CrossRef]
12. B. Young, F. Cruz, W. Itano, and J. Bergquist, “Visible lasers with subhertz linewidths,” Phys. Rev. Lett. 82(19), 3799–3802 (1999). [CrossRef]
13. R. Drever, J. L. Hall, F. Kowalski, J. Hough, G. Ford, A. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31(2), 97–105 (1983). [CrossRef]
14. J. Millo, D. Magalhães, C. Mandache, Y. Le Coq, E. English, P. Westergaard, J. Lodewyck, S. Bize, P. Lemonde, and G. Santarelli, “Ultrastable lasers based on vibration insensitive cavities,” Phys. Rev. A 79(5), 053829 (2009). [CrossRef]
15. W. Xie, Q. Zhou, F. Bretenaker, Z. Xia, H. Shi, J. Qin, Y. Dong, and W. Hu, “Fourier transform-limited optical frequency-modulated continuous-wave interferometry over several tens of laser coherence lengths,” Opt. Lett. 41(13), 2962–2965 (2016). [CrossRef]
16. P. A. Roos, R. R. Reibel, T. Berg, B. Kaylor, Z. W. Barber, and W. R. Babbitt, “Ultrabroadband optical chirp linearization for precision metrology applications,” Opt. Lett. 34(23), 3692–3694 (2009). [CrossRef]
17. C. Greiner, B. Boggs, T. Wang, and T. Mossberg, “Laser frequency stabilization by means of optical self-heterodyne beat-frequency control,” Opt. Lett. 23(16), 1280–1282 (1998). [CrossRef]
18. V. Crozatier, G. Gorju, F. Bretenaker, J. Le Gouët, I. Lorgeré, C. Gagnol, and E. Ducloux, “Phase locking of a frequency agile laser,” Appl. Phys. Lett. 89(26), 261115 (2006). [CrossRef]
19. J. Qin, Q. Zhou, W. Xie, Y. Xu, S. Yu, Z. Liu, Y. Tong, Y. Dong, and W. Hu, “Coherence enhancement of a chirped DFB laser for frequency-modulated continuous-wave reflectometry using a composite feedback loop,” Opt. Lett. 40(19), 4500–4503 (2015). [CrossRef]
20. A. P. Hilton, P. S. Light, L. Talbot, and A. N. Luiten, “Optimal design for spectral narrowing and fast frequency sweep of an interferometer-stabilized laser,” Opt. Lett. 45(1), 45–48 (2020). [CrossRef]
21. T. DiLazaro and G. Nehmetallah, “Phase-noise model for actively linearized frequency-modulated continuous-wave ladar,” Appl. Opt. 57(21), 6260–6268 (2018). [CrossRef]
22. Y. T. Chen, “Use of single-mode optical fiber in the stabilization of laser frequency,” Appl. Opt. 28(11), 2017–2021 (1989). [CrossRef]
23. G. Gagliardi, M. Salza, S. Avino, P. Ferraro, and P. De Natale, “Probing the ultimate limit of fiber-optic strain sensing,” Science 330(6007), 1081–1084 (2010). [CrossRef]
24. G. A. Cranch and S. Foster, “Comment on ‘Probing the ultimate limit of fiber-optic strain sensing’,” Science 335(6066), 286 (2012). [CrossRef]
25. G. Gagliardi, M. Salza, S. Avino, P. Ferraro, and P. De Natale, “Response to comment on ‘Probing the ultimate limit of fiber-optic strain sensing’,” Science 335(6066), 286 (2012). [CrossRef]
26. M. Parniak, I. Galinskiy, T. Zwettler, and E. S. Polzik, “High-frequency broadband laser phase noise cancellation using a delay line,” Opt. Express 29(5), 6935–6946 (2021). [CrossRef]
27. F. Kéfélian, H. Jiang, P. Lemonde, and G. Santarelli, “Ultralow-frequency-noise stabilization of a laser by locking to an optical fiber-delay line,” Opt. Lett. 34(7), 914–916 (2009). [CrossRef]
28. H. Jiang, F. Kéfélian, P. Lemonde, A. Clairon, and G. Santarelli, “An agile laser with ultra-low frequency noise and high sweep linearity,” Opt. Express 18(4), 3284–3297 (2010). [CrossRef]
29. Y. Zhang, C. P. Bandutunga, T. G. McRae, M. B. Gray, and J. H. Chow, “Double Rayleigh scattering in a digitally enhanced, all-fiber optical frequency reference,” Opt. Express 29(17), 26319–26331 (2021). [CrossRef]
30. K. H. Wanser, “Fundamental phase noise limit in optical fibres due to temperature fluctuations,” Electron. Lett. 28(1), 53–54 (1992). [CrossRef]
31. Corning Inc., “Corning optical fiber product information,” http://www.corning.com.
32. J. Dong, J. Huang, T. Li, and L. Liu, “Observation of fundamental thermal noise in optical fibers down to infrasonic frequencies,” Appl. Phys. Lett. 108(2), 021108 (2016). [CrossRef]
33. K. H. Wanser, A. D. Kersey, and A. Dandridge, “Intrinsic thermal phase noise limit in optical fiber interferometers,” Opt. Photonics News 4(12), 37–38 (1993). [CrossRef]
34. L. Duan, “General treatment of the thermal noises in optical fibers,” Phys. Rev. A 86(2), 023817 (2012). [CrossRef]
35. R. E. Bartolo, A. B. Tveten, and A. Dandridge, “Thermal phase noise measurements in optical fiber interferometers,” IEEE J. Quantum Electron. 48(5), 720–727 (2012). [CrossRef]
36. B. M. Beadle and J. Jarzynski, “Measurement of speed and attenuation of longitudinal elastic waves in optical fibers,” Opt. Eng. 40(10), 2115–2119 (2001). [CrossRef]
37. Q. Zhou, J. Qin, W. Xie, Y. Tong, Z. Liu, Y. Dong, and W. Hu, “Dynamic frequency-noise spectrum measurement for a frequency-swept DFB laser with short-delayed self-heterodyne method,” Opt. Express 23(22), 29245–29257 (2015). [CrossRef]
38. Y. Feng, W. Xie, Y. Meng, L. Zhang, Z. Liu, W. Wei, and Y. Dong, “High-performance optical frequency-domain reflectometry based on high-order optical phase-locking-assisted chirp optimization,” J. Lightwave Technol. 38(22), 6227–6236 (2020). [CrossRef]
39. J. Qin, L. Zhang, W. Xie, R. Cheng, Z. Liu, W. Wei, and Y. Dong, “Ultra-long range optical frequency domain reflectometry using a coherence-enhanced highly linear frequency-swept fiber laser source,” Opt. Express 27(14), 19359–19368 (2019). [CrossRef]
40. K. Numata, J. Camp, M. A. Krainak, and L. Stolpner, “Performance of planar-waveguide external cavity laser for precision measurements,” Opt. Express 18(22), 22781–22788 (2010). [CrossRef]
41. Q. Zhou, J. Qin, W. Xie, Z. Liu, Y. Tong, Y. Dong, and W. Hu, “Power-area method to precisely estimate laser linewidth from its frequency-noise spectrum,” Appl. Opt. 54(28), 8282–8289 (2015). [CrossRef]
