Open Access
Add to BibSonomy
Abstract
Self-heterodyne beat note measurements are widely used for the experimental characterization of the frequency noise power spectral density (FN–PSD) and the spectral linewidth of lasers. The measured data, however, must be corrected for the transfer function of the experimental setup in a post-processing routine. The standard approach disregards the detector noise and thereby induces reconstruction artifacts in the reconstructed FN–PSD. We introduce an improved post-processing routine based on a parametric Wiener filter that is free from reconstruction artifacts, provided a good estimate of the signal-to-noise ratio is supplied. Building on this potentially exact reconstruction, we develop a new method for intrinsic laser linewidth estimation that is aimed at deliberate suppression of unphysical reconstruction artifacts. Our method yields excellent results even in the presence of strong detector noise, where the intrinsic linewidth plateau is not even visible using the standard method. The approach is demonstrated for simulated time series from a stochastic laser model including 1/f-type noise.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Narrow-linewidth lasers exhibiting low phase noise are core elements of coherent optical communication systems [1–3], gravitational wave interferometers [4–6] and emerging quantum technologies, including optical atomic clocks [7,8], matter-wave interferometers [9,10] and ion-trap quantum computers [11,12]. For many of these applications, the performance depends critically on the laser’s intrinsic (Lorentzian) linewidth [13,14], which is typically obscured by additional $1/f$-like noise [15–20]. Because of this so-called flicker noise, the linewidth alone is not a well-defined quantity and needs to be specified for a given measurement time. A detailed characterization of the phase noise requires the measurement of the corresponding power spectral density (PSD).
The measurement of the frequency noise power spectral density (FN–PSD) is challenging as the rapid oscillations of the laser’s optical field cannot be directly resolved by conventional photodetectors. A standard method that is widely used for the characterization of the FN–PSD is the delayed self-heterodyne (DSH) beat note technique [15,21–25], which allows to extract the phase fluctuation dynamics from a slow beat note signal in the radio frequency (RF) regime. The DSH method requires some post-processing of the measured data in order to reconstruct the FN–PSD of the laser by removing the footprint of the interferometer. In this paper we describe an improved post-processing routine based on a parametric Wiener filter that avoids typical reconstruction artifacts which occur in the standard approach.
This paper is organized as follows: In Sec. 2, we describe the experimental setup and provide a model of the measurement that takes detector noise into account. In Sec. 3, we review the Wiener deconvolution method with particular emphasis on its application to the DSH measurement. We discuss a family of frequency-domain filter functions and their capabilities in restoring the FN–PSD of the laser. In Sec. 4, we present a novel method, that allows for a precise estimate of the intrinsic linewidth even at low signal-to-noise ratio (SNR), when the onset of the intrinsic linewidth plateau is overshadowed by measurement noise. The approach is demonstrated for simulated time series in Sec. 5. We close with a discussion of the method in Sec. 6.
2. Delayed self-heterodyne beat note measurement
In the DSH measurement method, see Fig. 1, the light of a laser is superimposed with the frequency-shifted (heterodyne) and time-delayed light from the same source. The frequency shift $\Delta \omega _{\mathrm {AOM}}$ (typically several tens of MHz) is realized with an acousto-optic modulator (AOM) and the delay $\tau _{d}$ is implemented via long fibers. If the delay is larger than the coherence time of the laser, the delayed light can be regarded as a statistically independent second laser with the same frequency and noise characteristics. The DSH method allows to down-convert the optical signal to a beat note signal in the RF domain, that can be resolved by electronic spectrum analyzers. Unlike other methods, the DSH method does not require stabilization of the laser to an optical reference (e.g., a frequency-stabilized second laser). Moreover, the frequency noise characteristics can be measured over a broad frequency bandwidth. A detailed description of the experimental setup and the post-processing procedure can be found in Ref. [26].
Fig. 1. Experimental setup for the DSH beat note measurement. The laser beam is separated by an AOM, where one arm of the signal is frequency shifted and delayed by a long fiber. Both beams are superimposed at a photodetector, which captures only the slow beat note signal.
After down-conversion and $I$–$Q$ demodulation (Hilbert transform) is carried out by the spectrum analyzer, the detected in-phase and quadrature signals read [26]
From the measured time series $I\left (t\right )$, $Q\left (t\right )$ one easily obtains the phase fluctuation difference
In the frequency domain, the relation between the phase fluctuations $\delta \phi \left (t\right )$ and $\Delta \phi \left (t\right )$ reads
In the standard post-processing routine [15,26], Eq. (5) is solved for $S_{\delta \phi,\delta \phi }\left (\omega \right )$ by division through $\left |H\left (\omega \right )\right |^{2}=2\left (1-\cos {\left (\omega \tau _{d}\right )}\right )$. This approach has two notable shortcomings: First, it does not take into account the detector noise and thereby fails at increased measurement noise levels. Second, the transfer function has roots at $\omega _{n}=2\pi n/\tau _{d}$, $n\in \mathbb {Z}$, which turn to poles in its inverse. Hence, the reconstructed PSD exhibits a series of reconstruction artifacts [27–29], resulting from an uncontrolled amplification of the detector noise.
3. Parametric Wiener filters
In this section, we present the Wiener deconvolution method for reconstructing hidden signals from noisy time series data. Besides the well-known Wiener filter, we introduce power spectrum equalization (PSE) as an important representative of the group of parametric Wiener filters [30].
Let $x\left (t\right )$ denote the time series of a hidden signal of interest that is measured by an experimental setup characterized by a convolution kernel $h\left (t\right )$. In the case of the DSH measurement described above, this is $h\left (t\right )=\delta \left (t\right )-\delta \left (t-\tau _{d}\right )$. Furthermore, let $\xi \left (t\right )$ denote additive Gaussian white measurement noise. Then the experiment yields an observed time series
The process noise and measurement noise are assumed to be uncorrelated $\langle x\left (t\right )\xi \left (t'\right )\rangle =0$. One seeks for an optimal reconstruction $\hat {x}\left (t\right )$ of the hidden signal
where the (de-)convolution kernel $g\left (t\right )$ minimizes the reconstruction error.In this paper, our main interest is the reconstruction of PSDs of hidden signals in the frequency domain, for which we introduce the Fourier space representation of Eq. (6)
From Eq. (7), we obtain the relations between the PSD of the measured time series $S_{z,z}\left (\omega \right )$, the hidden signal $S_{x,x}\left (\omega \right )$ and noise $S_{\xi,\xi }\left (\omega \right )$ and the reconstructed signal PSD $S_{\hat {x},\hat {x}}\left (\omega \right )$ as
In the following, we discuss different candidates for the filter function $G\left (\omega \right )$. Their performance is assessed with regard to the reconstruction of the FN–PSD of a semiconductor laser [16,18,19] from DSH measurements. The transfer function of the self-heterodyne interferometer is
and we assume the hidden signal and noise PSDs asFig. 2. Comparison of different filters $G\left (\omega \right )$ for FN–PSD reconstruction. (a) Analytical signal Eq. (10) and detector noise PSD Eq. (11) along with the observed spectrum and the exact SNR. Parameters in the plot are $\nu =1.4$, $C=10^{11}\,\mathrm {Hz}^{\nu +1}$, $S_{\infty }=10^{3}\,\mathrm {Hz}$, $\sigma =10^{-5}\,\mathrm {s}^{1/2}$ and $\tau _{d}=10$ µ$\mathrm {s}$. (b) The inverse filter Eq. (13) yields a reconstructed PSD with spurious peaks at the pole frequencies $f_{n}=n/\tau _{d}$, $n\in \mathbb {Z}$. The intrinsic linewidth plateau is obscured by detector noise and can not be recovered using the inverse filter. (c) The Wiener filter (Eq. 14) induces sharp dropouts in the reconstructed PSD at the pole frequencies and does not follow the hidden signal at low $\mathrm {SNR}<1$. (d) Power spectrum equalization Eq. (15) yields an exact reconstruction of the hidden signal’s PSD if the exact SNR is provided. The singularities at the pole frequencies are bounded from above by the SNR, which enables exact compensation of both the detector noise and the interferometer effects.
In Eq. (10), $S_{\infty }$ determines the intrinsic laser linewidth, which is obscured by additional colored noise of power-law type with $0.8\lesssim \nu \lesssim 1.6$ (flicker noise). The functional form of Eq. (10) is consistent with theoretical models and experimental observations for frequencies well below the relaxation oscillation (RO) peak (typically at several GHz). The level of phase measurement noise, cf. Eq. (3), is specified by $\sigma$ and the corresponding frequency measurement noise PSD is a quadratic function of the frequency. The model PSDs (Eqs. (10)–(11)) imply the signal-to-noise ratio
Figure 2 shows that different filters $G\left (\omega \right )$ can lead to vastly different results for $S_{\hat {x},\hat {x}}\left (\omega \right )$. In the following section, we discuss their behavior in more detail.
3.1 Inverse filter
The inverse filter is given by the inverse of the transfer function
Using Eq. (8), the corresponding reconstruction of the PSD of the hidden signal is found to be
3.2 Wiener filter
Wiener filtering achieves an optimal trade-off between inverse filtering and noise removal. It subtracts the additive noise and reverses the effects of the interferometer simultaneously. The Wiener filter is obtained from minimizing the mean square error of the time-domain signal at an arbitrary instance of time, see Appendix B.1. In the frequency domain, the Wiener filter reads
We note that in the absence of detector noise ($\mathrm {SNR}\left (\omega \right )\to \infty$), the Wiener filter Eq. (14) reduces to the inverse filter Eq. (13). Although the Wiener filter provides an optimal reconstruction of the time-domain signal, it fails in reconstruction of the corresponding PSD. Using Eq. (8), we find
Thus, the Wiener filter systematically underestimates the hidden signal. Moreover, we find that the reconstructed signal is exactly zero at the frequencies $\omega _n$ (roots of the transfer function) $S_{\hat {x},\hat {x}}^{\mathrm {(Wiener)}}\left (\omega _n\right ) =0$, see Fig. 2(c). At low SNR, the Wiener filter overemphasizes noise removal such that the reconstructed signal is too low at high frequencies, cf. Figure 2(c).
3.3 Power spectrum equalization
Besides the standard Wiener filter, there exist several variants of the method which are collectively referred to as parametric Wiener filters [30]. An important one is power spectrum equalization (PSE), which is tailored to minimize the quadratic error of the reconstructed PSD, see Appendix B.2. The corresponding filter function reads
Fig. 3. Reconstruction artifacts in the PSE filter method with incorrectly estimated SNR. (a) Overestimation ($S_{\infty }^{\mathrm {est}}=5S_{\infty }$) and (b) underestimation ($S_{\infty }^{\mathrm {est}}=0.2S_{\infty }$) of the intrinsic linewidth $S_\infty$ leads to spurious oscillations and spikes in the reconstructed spectrum $S_{\hat {x},\hat {x}}^{\mathrm {(PSE)}}$. In the case of misspecification of the SNR, the maxima of $\left |G_{\mathrm {PSE}}\right |^{2}$ are no longer bounded by the exact SNR. The assumed $\overline {S}_{x,x}$, which has the functional form Eq. (10) and enters the SNR estimate, is shown as a red dashed line. The method described in Sec. 4. aims at minimizing the deviation between $\overline {S}_{x,x}$ and $S_{\hat {x},\hat {x}}^{\mathrm {(PSE)}}$ in order to estimate the true value of the intrinsic linewidth parameter $S_{\infty }$. (c) The corresponding objective function Eq. (17) features a sharp minimum at the exact value.
Finally, we note that all the filter candidates discussed in this section can be written in a unified way as parametric Wiener filters of the following form:
4. Intrinsic linewidth estimation at low signal-to-noise ratio
In the previous section, it was shown that the PSE filter can provide an excellent reconstruction of the hidden signal’s PSD if the exact SNR is supplied. At first glance, this approach appears to be rather impractical, since the specification of the exact SNR already anticipates the actual measurement result to a certain degree. One might therefore worry that arbitrary reconstructions could be generated. It turns out, however, that the PSE filter method introduces characteristic reconstruction artifacts when the specified SNR is incorrect, see Fig. 3. These reconstruction artifacts are easily recognized to be unphysical, such that the incorrect SNR estimate can be rejected. Based on this observation, we develop a method that simultaneously reconstructs both the PSD of the hidden signal as well as the correct SNR, by minimizing these artifacts.
In the following, we employ again the analytic model PSDs Eqs. (10)–(11). For the sake of simplicity, we assume that the parameters $C$ and $\nu$ can be accurately estimated from the data, since the low-frequency part of the signal is only negligibly affected by measurement noise. Furthermore, the detector noise level $\sigma$ can be accurately estimated from the detected signal $S_{z,z}$ at the frequencies $\omega _n=2\pi n/\tau _d$, $n\in \mathbb {Z}$ (where the transfer function is zero, see Eq. (8a) and Eq. (9)), cf. Fig. 2. The only free parameter to be estimated then is $S_{\infty }$.
Figure 3(a)–(b) shows the effects of over- and underestimation of $S_{\infty }$ in the analytical model. Due to the mismatch between the filter function $\left |G_{\mathrm {PSE}}\left (\omega \right )\right |^{2}$ and the observed spectrum $S_{z,z}\left (\omega \right )$, spurious spikes (reconstruction artifacts) show up at frequencies $\omega \approx \omega _{n}$ in the reconstructed spectrum $S_{\hat {x},\hat {x}}^{\mathrm {(PSE)}}\left (\omega \right )$. At large frequencies these spikes are damped out in both $\left |G_{\mathrm {PSE}}\left (\omega \right )\right |^{2}$ and $S_{z,z}\left (\omega \right )$, but their product yields a wrong value of the intrinsic linewidth plateau. We introduce an objective function $D\left (S_{\infty }\right )$ that penalizes this deviation (i.e., the “inconsistency”) between the reconstructed signal $S_{\hat {x},\hat {x}}^{\mathrm {(PSE)}}\left (\omega ;S_{\infty }\right )$ (depending on the assumed SNR as a function of estimated $S_{\infty }$) and the implicitly assumed signal $\overline {S}_{x,x}\left (\omega ;S_{\infty }\right )$ obeying the functional form Eq. (10) as
5. Application to stochastic laser dynamics
In this section, we demonstrate the method described in Sec. 4. for simulated time series. In Sec. 5.1, we introduce a stochastic laser model including non-Markovian colored noise, that generates realistic time series with frequency drifts as commonly observed for diode lasers. In Sec. 5.2, we apply the linewidth estimation method to simulated DSH measurement data.
5.1 Stochastic laser rate equations
We consider a Langevin equation model for a generic single-mode semiconductor laser
Table 1. List of parameter values used in stochastic time series simulation.
The Langevin forces describe zero-mean Gaussian colored noise with the following non-vanishing frequency-domain correlation functions:
The white noise part of the model includes a quantum mechanically consistent description of light-matter interaction fluctuations [31]. Moreover, we have included three independent $1/f$-type noise sources with power-law exponents $\nu _{P}$ and $\nu _{N}$, respectively. The colored noise amplitudes are taken as $\sigma _{P}\left (P\right )=2P\sigma _{P,0}$ and $\sigma _{N}\left (N\right )=\sqrt {N}\sigma _{N,0}$ (modeling Hooge’s law [32,33]). The noise correlation functions Eq. (22) are formulated at the unique noise-free steady state $\big (\overline {P},\overline {N}\big )$. The full nonlinear system of Itô-type stochastic differential equations used for simulation is given in Appendix C. The numerically simulated FN–PSD is shown in Fig. 4 along with (semi-)analytical approximations. All parameter values used in the simulations are listed in Table 1.
Fig. 4. (a) Numerically computed FN–PSD from simulated time series using the stochastic laser rate Eq. (18). Analytical approximations are derived from linearization of the model at the noise-free steady state. White noise (WN) and different colored noise (CN) contributions are indicated separately. (b) Zoom on relaxation oscillation (RO) peak.
5.2 Intrinsic linewidth estimation
We apply the method described in Sec. 4. to simulated DSH measurements. The simulation is carried out in two steps: First, the stochastic laser rate Eq. (18) are simulated using the Euler–Maruyama method (time step $\Delta t=50\,\mathrm {ps}$). In the second step, the DSH measurement is simulated by evaluation of Eq. (1), which includes addition of Gaussian white measurement noise. The simulated $I$–$Q$ data are used to generate the time series $\Delta \phi$ according to Eq. (2). The observed spectrum is computed from $S_{z,z}\left (\omega \right )=\omega ^{2}S_{\Delta \phi,\Delta \phi }\left (\omega \right )$ and shown in Fig. 5 (a). For recovery of the original FN–PSD, the PSE filter method is applied to the simulated FN–PSD $S_{z,z}\left (\omega \right )$. In the estimation procedure, see Algorithm 1, the frequency range is restricted to frequencies below the RO peak to ensure validity of the analytical model (10).
Fig. 5. Application of the linewidth estimation method to simulated time series data. (a) PSDs of the hidden signal, the detector noise and the measured data along with their analytic values. (b) Reconstructed PSD using the inverse filter. The inset zooms in on a region with low SNR, where the reconstructed PSD clearly deviates from the true signal. One observes reconstruction artifacts at the pole frequencies. (c) The PSE filter method yields an accurate reconstruction of the hidden signal even at low SNR that is free of reconstruction artifacts. The analytic model has been fitted to the PSD to give the SNR according to the method described in Sec. 4. (d) Minimization of the objective function Eq. (17) yields a sharp estimate of the intrinsic linewidth parameter. Inset: Monte Carlo sampling of the simulated DSH experiment yields an estimated intrinsic linewidth $S_{\infty }= 498.8 \pm 13.4\,\mathrm {Hz}$. The distribution has been fitted to an exponentially modified Gaussian.
The optimal reconstruction of the hidden FN–PSD is shown in Fig. 5(c) along with corresponding SNR estimate and the measurement noise PSD. The PSE filter yields a significantly better reconstruction than the inverse filter method, which contains the characteristic reconstruction artifacts and deviates clearly from the hidden signal at increased measurement noise, see Fig. 5(b). The objective function Eq. (17) evaluated for the simulated stochastic data is shown in Fig. 5(d). Just like in Sec. 4, the objective function features a sharp minimum near at the exact value.
In order to estimate the accuracy of the method, we have sampled the simulation of the DSH measurement $10^4$ times and evaluated the statistics of the obtained linewidth estimates, see inset of Fig. 5(d). For the parameters given in Table 1, we found a linewidth of $498.8 \pm 13.4\,\text {Hz}$, i.e., a relative error of about 2.7 %. This Monte Carlo approach to error estimation inherently takes into account both the sampling error as well as the corresponding estimation uncertainties of the parameters $C$, $\nu$, and $\sigma$, and the propagation of these errors to the resulting accuracy of $S_{\infty }$.
Algorithm 1. Estimation of intrinsic linewidth parameter S∞
6. Discussion
The method presented in Sec. 4. not only provides an artifact-free reconstruction of the hidden FN–PSD, but also allows to extract the intrinsic linewidth when it is obscured by measurement noise. The procedure, however, relies on the specification of the frequency-dependent SNR in the form of the analytical model Eqs. (10)–(11). As we have demonstrated in Fig. 3, incorrect SNR estimates lead to reconstruction errors, which are identified as such via inconsistencies with the assumed functional form Eq. (10) of the hidden PSD. This a priori assumption of the functional form, however, is well validated [16,18,19], so that no false bias is imposed here. Instead, our method exploits this additional prior knowledge about the physics of the problem to extract additional information from the measured data that is not used in the inverse filter method.
Even though we restricted the parameter estimation problem in Secs. 4. and 5.2 to a single unknown variable, it should be straightforward to extend the method to a multivariate (nonlinear) minimization problem where all parameters characterizing the SNR are estimated simultaneously. Furthermore, it would be interesting to apply the estimation method to the reconstruction of the relative intensity noise (RIN) PSD, which is typically more obscured by detector noise.
In principle other estimation methods can also be employed for reconstruction of the FN–PSD from noisy time series data. For example, Zibar et al. [34] have used an extended Kalman filter to estimate the effect of amplifier noise on the phase noise PSD of a laser. The disadvantage of this method, however, is that it requires a (comprehensive) mathematical model of the dynamical system under measurement, which imposes a significant overhead. Moreover, the application of Kalman filters to problems with large delay (like the DSH-measurement), is notoriously difficult and computationally heavy [35]. In contrast, the strength of parametric Wiener filters is that they are independent of assumptions on the underlying state space model and simple to implement. Moreover, since the method is formulated in the frequency-domain, it does not suffer from computational burden due to the large delay.
Lastly, we comment on the applicability of the DSH method with subcoherent delay, as assumed in the simulations presented above. It is well known that the DSH method with subcoherent delay yields an optical power spectrum with complicated structure, which depends sensitively on the delay and therefore must be interpreted carefully [17,22]. The same holds true for the very similar self-homodyne method at nearly coherent and subcoherent delay [36,37]. We thus avoid determining the (intrinsic) linewidth directly from the optical power spectrum in this work. Instead, we aim at the reconstruction of the FN–PSD of the single laser, which contains all relevant information, but is difficult to obtain because of the roots of the transfer function as elaborated above. We remark that this approach does not make any simplifying assumptions on the statistical dependency of the phase noise at different times and is not restricted to the use of delays exceeding the coherence time.
7. Summary
We have presented an improved post-processing routine based on a parametric Wiener filter, that yields a potentially exact reconstruction of the FN–PSD (without any reconstruction artifacts) from DSH beat note measurements. The method, however, requires an accurate estimate of the frequency-dependent SNR, which can be consistently obtained by deliberate suppression of the characteristic reconstruction artifacts. In this way, both the footprint of the interferometer as well as the detector noise can be removed with high accuracy. Remarkably, the method thus allows for the reconstruction of the intrinsic linewidth (white noise) plateau even when it is entirely obscured by measurement noise. The approach has been demonstrated for simulated time series based on a stochastic laser rate equation model including non-Markovian $1/f$-type noise.
Appendix
A. Effective phase measurement noise
We seek for an approximation of the effective phase measurement noise and its two-time correlation function. Starting from Eq. (1), we expand for small noise
Expansion at the CW state with $P\left (t\right )=\overline {P}+\delta P\left (t\right )$ and $\phi \left (t\right )=\overline {\Omega }t+\delta \phi \left (t\right )$ yields
We approximate the two-time correlation function
By neglecting the rapidly oscillating cross-correlation term, we arrive at Eq. (3).
B. Derivation of the frequency domain filter functions
B.1. Wiener filter
We consider the mean square error between the hidden signal $x\left (t\right )$ and its reconstruction Eq. (6b)
Fourier transform and substitution of 7 yields
B.2. Power spectrum equalization
We seek for an optimal reconstruction $S_{\hat {x},\hat {x}}\left (\omega \right )$ of the PSD that minimizes the quadratic error
Starting from $\langle \hat {X}\left (\omega \right )\hat {X}^{*}\left (\omega '\right ) \rangle = 2\pi S_{\hat {x},\hat {x}}\left (\omega \right )\delta \left (\omega -\omega '\right )$, we substitute Eq. (7). Assuming $\left \langle X\left (\omega \right )\Xi ^{*}\left (\omega '\right )\right \rangle =0$, we arrive at
The last line allows to rewrite the expression for the reconstruction error as
Minimization of the error by variation of the filter $G\left (\omega \right )\to G\left (\omega \right )+\varepsilon \delta G\left (\omega \right )$ yields
C. Itô-type stochastic differential equations
The Langevin Eqs. (18) can be written as a system of Itô-type stochastic differential equations
Here, $\mathrm {d}W\sim \mathrm {Normal}\left (0,\mathrm {d}t\right )$ denotes the increment of the standard Wiener processes (Gaussian white noise). Wiener processes with different sub- and superscripts are statistically independent. Construction of the colored noise sources $\mathcal {F}_{P,\phi,N}$ is described in Appendix D.
D. Colored noise
Colored noise sources $\mathcal {F}_{P,\phi,N}$ (subscripts are omitted in the following) are modeled as a superposition of independent Ornstein–Uhlenbeck (OU) fluctuators (Markovian embedding) [38]
where $A$ is a normalization constant (see below), $n$ is the number of OU fluctuators andThe fluctuators are statistically independent, i.e., $\mathrm {d}W_{i}\left (t\right )\mathrm {d}W_{j}\left (t\right )=\delta _{i,j}\,\mathrm {d}t$. From the stationary covariance $C_{X_{i},X_{j}}\left (\tau \right )=\left \langle X_{i}\left (t+\tau \right )X_{j}\left (t\right )\right \rangle =\delta _{i,j}\,\exp {\left (-\gamma _{i}\left |\tau \right |\right )}$, we obtain the auto-correlation function of the colored noise
The corresponding PSD is obtained according to the Wiener–Khinchin theorem [39] as
In the following, we consider a power-law distribution
The integral can formally be solved by a hypergeometric function. More insight, however, is gained by considering the asymptotic limit $\gamma _{0}\to 0$ and $\gamma _{\infty }\to \infty$, which leads to
Hence, the PSD exhibits a power-law type frequency-dependency
Funding
This work was funded by the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG) under Germany’s Excellence Strategy (EXC 2046: MATH+ (project AA2-13)); Berlin Mathematics Research Center MATH+ (project AA2-13).
Disclosures
The authors declare no conflicts of interest.
Data availability
MATLAB source codes to reproduce the simulation data are publicly available [40].
References
1. K. Kikuchi, “Fundamentals of coherent optical fiber communications,” J. Lightwave Technol. 34(1), 157–179 (2016). [CrossRef]
2. K. Zhou, Q. Zhao, X. Huang, C. Yang, C. Li, E. Zhou, X. Xu, K. K. Wong, H. Cheng, J. Gan, Z. Feng, M. Peng, Z. Yang, and S. Xu, “kHz-order linewidth controllable 1550 nm single-frequency fiber laser for coherent optical communication,” Opt. Express 25(17), 19752 (2017). [CrossRef]
3. H. Guan, A. Novack, T. Galfsky, Y. Ma, S. Fathololoumi, A. Horth, T. N. Huynh, J. Roman, R. Shi, M. Caverley, Y. Liu, T. Baehr-Jones, K. Bergman, and M. Hochberg, “Widely-tunable, narrow-linewidth III-V/silicon hybrid external-cavity laser for coherent communication,” Opt. Express 26(7), 7920 (2018). [CrossRef]
4. B. Willke, K. Danzmann, M. Frede, P. King, D. Kracht, P. Kwee, O. Puncken, R. L. Savage, B. Schulz, F. Seifert, C. Veltkamp, S. Wagner, P. Weßels, and L. Winkelmann, “Stabilized lasers for advanced gravitational wave detectors,” Class. Quantum Grav. 25(11), 114040 (2008). [CrossRef]
5. B. P. Abbott, R. Abbott, R. Adhikari, et al., “LIGO: the laser interferometer gravitational-wave observatory,” Rep. Prog. Phys. 72(7), 076901 (2009). [CrossRef]
6. K. Dahl, P. Cebeci, O. Fitzau, M. Giesberts, C. Greve, M. Krutzik, A. Peters, S. A. Pyka, J. Sanjuan, M. Schiemangk, T. Schuldt, K. Voss, and A. Wicht, “A new laser technology for LISA,” Proc. SPIE 11180, 111800C (2019). [CrossRef]
7. A. D. Ludlow, M. M. Boyd, J. Ye, E. Peik, and P. O. Schmidt, “Optical atomic clocks,” Rev. Modern Phys. 87(2), 637–701 (2015). [CrossRef]
8. Z. L. Newman, V. Maurice, C. Fredrick, T. Fortier, H. Leopardi, L. Hollberg, S. A. Diddams, J. Kitching, and M. T. Hummon, “High-performance, compact optical standard,” Opt. Lett. 46(18), 4702 (2021). [CrossRef]
9. A. Peters, K. Y. Chung, and S. Chu, “High-precision gravity measurements using atom interferometry,” Metrologia 38(1), 25–61 (2001). [CrossRef]
10. O. Carraz, F. Lienhart, R. Charrière, M. Cadoret, N. Zahzam, Y. Bidel, and A. Bresson, “Compact and robust laser system for onboard atom interferometry,” Appl. Phys. B 97(2), 405–411 (2009). [CrossRef]
11. N. Akerman, N. Navon, S. Kotler, Y. Glickman, and R. Ozeri, “Universal gate-set for trapped-ion qubits using a narrow linewidth diode laser,” New. J. Phys. 17(11), 113060 (2015). [CrossRef]
12. I. Pogorelov, T. Feldker, C. D. Marciniak, L. Postler, G. Jacob, O. Krieglsteiner, V. Podlesnic, M. Meth, V. Negnevitsky, M. Stadler, B. Höfer, C. Wächter, K. Lakhmanskiy, R. Blatt, P. Schindler, and T. Monz, “Compact ion-trap quantum computing demonstrator,” PRX Quantum 2(2), 020343 (2021). [CrossRef]
13. C. Henry, “Phase noise in semiconductor lasers,” J. Lightwave Technol. 4(3), 298–311 (1986). [CrossRef]
14. H. Wenzel, M. Kantner, M. Radziunas, and U. Bandelow, “Semiconductor laser linewidth theory revisited,” Appl. Sci. 11(13), 6004 (2021). [CrossRef]
15. K. Kikuchi and T. Okoshi, “Dependence of semiconductor laser linewidth on measurement time: evidence of predominance of 1/f noise,” Electron. Lett. 21(22), 1011 (1985). [CrossRef]
16. K. Kikuchi, “Effect of 1/f-type FM noise on semiconductor-laser linewidth residual in high-power limit,” IEEE J. Quantum Electron. 25(4), 684–688 (1989). [CrossRef]
17. L. B. Mercer, “1/f frequency noise effects on self-heterodyne linewidth measurements,” J. Lightwave Technol. 9(4), 485–493 (1991). [CrossRef]
18. Y. Salvadé and R. Dändliker, “Limitations of interferometry due to the flicker noise of laser diodes,” J. Opt. Soc. Am. A 17(5), 927–932 (2000). [CrossRef]
19. G. M. Stéphan, T. T. Tam, S. Blin, P. Besnard, and M. Têtu, “Laser line shape and spectral density of frequency noise,” Phys. Rev. A 71(4), 043809 (2005). [CrossRef]
20. S. Spießberger, M. Schiemangk, A. Wicht, H. Wenzel, G. Erbert, and G. Tränkle, “DBR laser diodes emitting near 1064 nm with a narrow intrinsic linewidth of 2 kHz,” Appl. Phys. B 104(4), 813–818 (2011). [CrossRef]
21. T. Okoshi, K. Kikuchi, and A. Nakayama, “Novel method for high resolution measurement of laser output spectrum,” Electron. Lett. 16(16), 630 (1980). [CrossRef]
22. L. Richter, H. Mandelberg, M. Kruger, and P. McGrath, “Linewidth determination from self-heterodyne measurements with subcoherence delay times,” IEEE J. Quantum Electron. 22(11), 2070–2074 (1986). [CrossRef]
23. H. Tsuchida, “Laser frequency modulation noise measurement by recirculating delayed self-heterodyne method,” Opt. Lett. 36(5), 681 (2011). [CrossRef]
24. M. Schiemangk, S. Spießberger, A. Wicht, G. Erbert, G. Tränkle, and A. Peters, “Accurate frequency noise measurement of free-running lasers,” Appl. Opt. 53(30), 7138 (2014). [CrossRef]
25. Z. Bai, Z. Zhao, Y. Qi, J. Ding, S. Li, X. Yan, Y. Wang, and Z. Lu, “Narrow-linewidth laser linewidth measurement technology,” Front. Phys. 9, 768165 (2021). [CrossRef]
26. M. Schiemangk, “Ein Lasersystem für Experimente mit Quantengasen unter Schwerelosigkeit,” Ph.D. thesis, Humboldt University Berlin (2019).
27. W. Lewoczko-Adamczyk, C. Pyrlik, J. Häger, S. Schwertfeger, A. Wicht, A. Peters, G. Erbert, and G. Tränkle, “Ultra-narrow linewidth DFB-laser with optical feedback from a monolithic confocal Fabry–Perot cavity,” Opt. Express 23(8), 9705–9709 (2015). [CrossRef]
28. S. Wenzel, O. Brox, P. D. Casa, H. Wenzel, B. Arar, S. Kreutzmann, M. Weyers, A. Knigge, A. Wicht, and G. Tränkle, “Monolithically integrated extended cavity diode laser with 32 kHz 3 dB linewidth emitting at 1064 nm,” Laser Photonics Rev. 16(12), 2200442 (2022). [CrossRef]
29. R. R. Kumar, A. Hänsel, M. F. Brusatori, L. Nielsen, L. M. Augustin, N. Volet, and M. J. R. Heck, “A 10-kHz intrinsic linewidth coupled extended-cavity DBR laser monolithically integrated on an InP platform,” Opt. Lett. 47(9), 2346 (2022). [CrossRef]
30. J. S. Lim, Two-Dimensional Signal and Image Processing (Prentice Hall, 1990).
31. L. A. Coldren, S. W. Corzine, and M. L. Mašanović, Diode Lasers and Photonic Integrated Circuits (Wiley, 2012).
32. F. N. Hooge, “1/f noise sources,” IEEE Trans. Electron Devices 41(11), 1926–1935 (1994). [CrossRef]
33. I. A. Garmash, M. V. Zverkov, N. B. Kornilova, V. N. Morozov, R. F. Nabiev, A. T. Semenov, M. A. Sumarokov, and V. R. Shidlovskii, “Analysis of low-frequency fluctuation of the radiation power of injection lasers,” J. Sov. Laser Res. 10(6), 459–476 (1989). [CrossRef]
34. D. Zibar, J. E. Pedersen, P. Varming, G. Brajato, and F. D. Ros, “Approaching optimum phase measurement in the presence of amplifier noise,” Optica 8(10), 1262 (2021). [CrossRef]
35. A. Gopalakrishnan, N. S. Kaisare, and S. Narasimhan, “Incorporating delayed and infrequent measurements in extended Kalman filter based nonlinear state estimation,” J. Process Control 21(1), 119–129 (2011). [CrossRef]
36. H. Ludvigsen, M. Tossavainen, and M. Kaivola, “Laser linewidth measurements using self-homodyne detection with short delay,” Opt. Commun. 155(1-3), 180–186 (1998). [CrossRef]
37. I. Attia, E. Wohlgemuth, O. Balciano, R. J. Cohen, Y. Yoffe, and D. Sadot, “Laser linewidth characterization via self-homodyne measurement under nearly-coherent conditions,” Opt. Express 30(9), 14492 (2022). [CrossRef]
38. S. M. Kogan, Electronic Noise and Fluctuations in Solids (Cambridge University Press, 1996).
39. R. Kubo, M. Toda, and N. Hashitsume, Statistical Physics II: Nonequilibrium Statistical Mechanics, vol. 31 of Springer Series in Solid-State Sciences (Springer, 1991).
40. M. Kantner and L. Mertenskötter, “Supplementary code to “Accurate evaluation of self-heterodyne laser linewidth measurements using Wiener filters,”” Zenodo (2023), https://doi.org/10.5281/zenodo.7728408.
