## Abstract

The linewidth of regenerative oscillators is enhanced by
amplitude–phase coupling of the oscillator field [Phys. Rev. **160**, 290
(1967)]. In laser oscillators, this
effect is well known for its impact on semiconductor laser
performance. Here, this coupling is studied in Brillouin lasers.
Because their gain is parametric, the coupling and linewidth
enhancement are shown to originate from phase mismatch. The theory is
confirmed by measurement of linewidth in a microcavity Brillouin
laser, and enhancements as large as $50 \times$ are measured. The results show that
pump wavelength and device temperature should be carefully selected
and controlled to minimize linewidth. More generally, this work
provides a new perspective on the linewidth enhancement effect.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

The linewidth of stimulated Brillouin lasers (SBLs) has received considerable attention for some time. SBLs based on optical fiber [1], for example, feature narrow linewidths that are useful for generation of highly stable microwave sources [2,3]. More recently, broad interest in microscale and nanoscale Brillouin devices [4] has focused attention on tiny, often chip-scale, SBLs in several systems [5–12]. These devices have high power efficiency [13] and provide flexible operating wavelengths [14], and their fundamental linewidth can be reduced to less than 1 Hz [12,13]. For these reasons, they are being applied in a range of applications including radio-frequency (RF) synthesizers [15,16], ring laser gyroscopes [12,17,18] and high-coherence reference sources [9].

SBLs derive gain through a process that is parametric in nature and for which scattering of an optical pump into a Stokes wave from an acoustic phonon must be phase matched [19,20]. When the phonon field is strongly damped, the process mimics stimulated emission. Nonetheless, phonon participation introduces dramatic differences into SBL linewidth behavior compared with conventional lasers. For example, while the conventional Schawlow–Townes laser linewidth [21] is insensitive to temperature, the fundamental SBL linewidth is proportional to the number of thermomechanical quanta in the phonon mode and therefore to the Boltzmann energy ${k_{\rm B}}\!T$ [13]. This dependence has been verified from cryogenic to room temperature [22]. Brillouin lasers can also oscillate on multiple lines through the process of cascade [13], in which an initial Stokes wave can serve to pump a second Stokes wave and so forth. Cascading introduces additional contributions to the SBL linewidth [23]. Finally, the parametric nature of the process means that pump phase noise couples through to the laser linewidth, although it is strongly suppressed by the phonon damping [24].

The fundamental linewidth of lasers is increased by the well-known linewidth enhancement factor $\alpha$ that characterizes amplitude–phase coupling of the field [25,26]. This quantity is best known for its impact on the linewidth of semiconductor lasers [27], and understanding, controlling, and measuring it have long been subjects of interest [28–30]. Here, the linewidth enhancement factor is studied in SBLs. The parametric nature of Brillouin gain is shown to strongly influence this parameter. Phase mismatch causes a nonzero $\alpha$ factor. Measurements of SBL frequency noise are used to determine $\alpha$ versus controlled amounts of phase mismatch, and the results are in good agreement with theory. Significant enhancements to the linewidth are predicted and measured, even when the SBL is operated only modestly away from perfect phase matching.

Amplitude–phase coupling occurs at a specified optical frequency when the real and imaginary parts of the optical susceptibility (equivalently, refractive index and gain) experience correlated variations subject to a third parameter. The ratio of the real to imaginary variation is the $\alpha$ parameter [27]. With a nonzero $\alpha$ parameter, noise that normally couples only into the laser field amplitude can also couple into the phase. And because phase fluctuations are responsible for the finite laser linewidth [25], the nonzero $\alpha$ factor thereby causes linewidth enhancement. For a physical understanding of how a nonzero $\alpha$ parameter arises within the SBL system, consider Fig. 1(a) (a detailed analysis is provided in Section 1 of Supplement 1). Optical pumping at frequency ${\omega _{\rm P}}$ on a cavity mode causes a Lorentzian-shaped gain spectrum through the Brillouin process. The Brillouin gain spectrum is frequency downshifted by the phonon frequency $\Omega$ (Brillouin shift frequency) relative to the pumping frequency. Laser action at frequency ${\omega _{\rm L}}$ is possible when a second cavity mode lies within the gain spectrum, which requires that $\Delta \omega \equiv {\omega _{\rm P}} - {\omega _{\rm L}}$ is close in value to $\Omega$. Perfect phase matching corresponds to laser oscillation at the peak of the gain (i.e., $\Delta \omega = \Omega$). Also shown in Fig. 1(a) is the refractive index spectrum associated with the gain spectrum according to the Kramers–Kronig relations. It is apparent that $\alpha$ (the ratio of the variation of real to imaginary susceptibility) will be zero for phase-matched operation, while it increases with increased frequency detuning relative to perfect phase matching.

Analysis (Sections 1 and 3 of Supplement 1) shows that the $\alpha$ factor enhancement of the fundamental SBL linewidth $\Delta {\nu _{{\rm SBL}}}$ is

where $\Delta {\nu _0}$ is the non-enhanced ($\alpha = 0$) SBL linewidth [given below in Eq. (3)] and the linewidth enhancement factor can be expressed using two equivalent frequency-detuning quantities relative to perfect phase matching:In the first quantity, phonon mode detuning $\delta \Omega \equiv \Omega - \Delta \omega$ is normalized by $\Gamma$, the Brillouin gain bandwidth (i.e., phonon decay rate constant). In the second quantity, optical mode detuning $\delta \omega \equiv \Delta \omega - {\rm FSR}$ (where FSR is the unpumped cavity free spectral range) is normalized by $\gamma$, the photon decay rate constant. Note that the sign of $\alpha$ changes to either side of perfect phase matching. Also, as an aside, $\delta \omega$ is the mode pulling induced by the Brillouin gain spectrum [13]. $\Delta {\nu _0}$ is given by

As a first step toward verification of Eqs. (1) and (2), it is necessary to measure the phase mismatch detuning at each point where the linewidth will be measured. The experimental setup and information on the high-$Q$ silica whispering gallery microcavity used to generate Brillouin laser action are provided in Fig. 1(b) and its caption. To vary the phase mismatch detuning, the pump laser wavelength ${\lambda _{\rm P}}$ is tuned, which is achieved by selecting different longitudinal modes within the same transverse mode family as pump and Stokes modes. This has the effect of varying $\Omega$ through the relationship $\Omega = 4\pi n{c_{\rm s}}/{\lambda _{\rm P}}$ (where $n$ is the refractive index and ${c_{\rm s}}$ is the speed of sound in the microcavity). Since $\Omega$ is not directly measurable in the experiment, we instead obtained information on the phase mismatch using $\delta \omega$, which requires measurement of $\Delta \omega$ and FSR.

The frequency $\Delta \omega$ is determined by first measuring the beating frequency of the pump and the SBL using a fast photodetector, followed by measurement of the detected current on an electrical spectrum analyzer. Beyond being influenced by mode pulling as noted above, this beating frequency is also slightly shifted via backaction of the amplitude–phase coupling (see Section 3 of Supplement 1) and the optical Kerr effect [33], both of which are proportional to the SBL powers. Therefore, to account for these effects, the beat note frequencies were measured at five different SBL power levels. Representative measurements performed at three pump wavelengths are shown in Fig. 2(a). The $y$-intercept of these plots provides the required beating frequency in the absence of the above effects, and a summary plot of a series of such measurements is provided as the blue square data points in Fig. 2(b). As an aside, the data point near 1559 nm is missing because of strong mode crossings at this wavelength in the SBL microcavity (i.e., higher-order mode families become degenerate with the SBL mode family).

To determine the FSR at each pumping wavelength, the mode spectrum of the resonator is measured by scanning a tunable laser whose frequency is measured using an RF calibrated interferometer [34]. The measured FSR is plotted versus wavelength as the dotted line in Fig. 2(b). Measurement of the FSR this way also ensured that pumping was performed on the same transverse mode family with which the pumping wavelength was tuned. This is important because the mode volume would change strongly if the mode family were to change. In Fig. 2(b), the phase-matching condition (gain center) occurs when FSR equals $\Delta \omega$ ($\delta \omega = 0$) at a pump wavelength around 1548 nm. We can also use the Brillouin shift at the gain center to infer that ${c_s} = 5845\;{\rm m/s}$, which is consistent with the material properties of silica [35].

Finally, $\gamma$ is determined via measurement of the cavity linewidth at each wavelength (equivalently, the total $Q$-factor ${Q_{\rm T}}$ of the resonator). By measuring both the linewidth and transmission on cavity resonance it is possible to extract both the intrinsic $Q$-factor ${Q_0}$ and external coupling $Q$ factor ${Q_{{\rm ex}}}$ at each wavelength ($1/{Q_{\rm T}} = 1/{Q_0} + 1/{Q_{{\rm ex}}}$). A plot of the results is provided in Fig. 2(c). The ${Q_0}$ values inferred this way are relatively constant across the measured modes, while ${Q_{{\rm ex}}}$ exhibits variation that reflects the wavelength dependency of the coupling condition. The $Q$ factors are significantly lower than those of state-of-the-art resonators of the same kind [8], which is intentional and increases the sensitivity of the noise measurement that follows. Using Eq. (2), the theoretical $\alpha$ factor as a function of wavelength from 1532 nm to 1563 nm is plotted in Fig. 2(b) (red circles). Deviations of beating frequency and the $\alpha$ factor from a linear trend are the result of variations in the total $Q$ factor across the measured wavelengths. The largest $\alpha$ factor is greater than 7, so a fundamental linewidth enhancement of more than $1 + {7^2} = 50$ is expected at the largest detuning values.

A frequency discriminator method [36,37] is used to measure the noise spectrum of the two-sided white frequency noise spectral density ${S_{\rm w}}$ of the SBL, as described in the Fig. 1(b) caption. The fundamental noise component in ${S_{\rm w}}$, defined as ${S_{\rm F}}$, is related to the fundamental SBL linewidth through $2\pi\! {S_{\rm F}} = \Delta {\nu _{{\rm SBL}}}$ [33] [$\Delta {\nu _{{\rm SBL}}}$ is given in Eq. (1)]. And the inverse power dependence contained in $\Delta {\nu _{{\rm SBL}}}$ is used to extract ${S_{\rm F}}$ from the measurement of ${S_{\rm w}}$. Data plots of ${S_{\rm w}}$ versus inverse power at three pumping wavelengths are given in the inset of Fig. 3 and reveal this power dependence. Of importance to this measurement is that optical pumping power was controlled by attenuation of the pump so that its phase noise was constant throughout the measurement. Therefore, only the intrinsic contribution to linewidth could cause the observed power dependence. The slope is equal to ${S_{\rm F}}$ normalized to an output power of 1 mW. Linear fitting provides the slopes that are plotted versus wavelength in the main panel of Fig. 3. The corresponding minimum measured fundamental noise is about ${S_{\rm F}} = 0.2\;{{\rm Hz}^2}/{\rm Hz}$ ($\Delta {\nu _{{\rm SBL}}} = 1.25\;{\rm Hz}$) near the phase-matching condition (gain center), and the maximum fundamental noise is more than ${S_{\rm F}} = 10\;{{\rm Hz}^2}/{\rm Hz}$ ($\Delta {\nu _{{\rm SBL}}} = 63\;{\rm Hz}$), corresponding to $50 \times$ noise enhancement, at the largest mismatch detunings. Comparison to Eq. (1) is provided via the green curve in Fig. 3. In this plot, ${Q_{\rm T}}$, ${Q_{{\rm ex}}}$, and $\alpha$ [Figs. 2(b) and 2(c)] measurements at each wavelength are used with no free parameters. $\gamma$ can be obtained from ${Q_{\rm T}}$, and we can infer $\Gamma /2\pi$ to be 34.7 MHz, assuming it is constant over the wavelength. Also, ${n_{{\rm th}}} = 572$ is used (corresponding to the operating temperature of 26.5°C). There is overall good agreement with the measured linewidth values. The conventional ${S_0} = \Delta {\nu _0}/(2\pi)$ (with $\alpha = 0$) is also plotted for comparison.

The nonzero intercept on the $y$ axis of the inset in Fig. 3 is believed to be related to transferred pump phase noise associated with imperfect Pound–Drever–Hall locking. This contribution will increase with increasing $\alpha$. Both it and the linewidth-enhancement-factor contribution to the pump phase noise are discussed in Section 3 of Supplement 1. We have also verified the $\alpha$ measurement results in another SBL resonator. Details can be found in Section 4 of Supplement 1.

As an aside, the relative intensity noise of the SBLs is another important characteristic of laser operation, and a typical measured spectrum is shown in Section 5 of Supplement 1. Also, as noted in the introduction, Brillouin lasers can provide multiline oscillation via cascade [13], and under these conditions additional terms appear in the linewidth expression [23]. In the context of the present discussion, it is therefore of interest to consider the impact of the $\alpha$ factor on linewidth under conditions of cascaded operation. This is done in Section 6 of Supplement 1.

We have studied the linewidth enhancement factor $\alpha$ in a Brillouin laser. A modification to the fundamental linewidth formula that incorporates the $\alpha$ factor was theoretically derived and then tested experimentally in a high-$Q$ silica whispering gallery resonator. Phase matching of the Brillouin process determines the sign and magnitude of $\alpha$. Under perfect phase-matching conditions, corresponding to laser oscillation at the Brillouin gain maximum, $\alpha = 0$. However, measurement and theory show that the mismatch (induced here by tuning of the pumping wavelength) leads to $\alpha$ factors greater than 7, yielding frequency noise and fundamental linewidth enhancement as large as $50 \times$. The sign of $\alpha$ can also be controlled through the sign of the frequency mismatch detuning. Although the phase-matching condition was controlled here via tuning of the pumping wavelength, it should also be possible to vary phase matching and therefore $\alpha$ through control of the temperature. This would vary the Brillouin shift frequency by way of the temperature dependence of the sound velocity. The results presented here stress the importance of proper pumping wavelength selection and observance of temperature control for narrow-linewidth operation of SBLs. These considerations will be important in all applications of these devices that are sensitive to frequency noise and linewidth.

## Funding

Air Force Office of Scientific Research (FA9550-18-1-0353); Caltech Kavli Nanoscience Institute.

## Acknowledgment

The authors thank Y. Lai, Q. Yang, and C. Bao for helpful discussions.

## Disclosures

The authors declare no conflicts of interest.

See Supplement 1 for supporting content.

## REFERENCES

**1. **L. Stokes, M. Chodorow, and H. Shaw, Opt. Lett. **7**, 509 (1982). [CrossRef]

**2. **X. S. Yao, Opt. Lett. **22**, 1329 (1997). [CrossRef]

**3. **P. T. Callahan, M. C. Gross, and M. L. Dennis, IEEE J. Quantum Electron. **47**, 1142 (2011). [CrossRef]

**4. **B. J. Eggleton, C. G. Poulton, and R. Pant, Adv. Opt. Photon. **5**, 536 (2013). [CrossRef]

**5. **M. Tomes and T. Carmon, Phys. Rev. Lett. **102**, 113601
(2009). [CrossRef]

**6. **I. S. Grudinin, A. B. Matsko, and L. Maleki, Phys. Rev. Lett. **102**, 043902
(2009). [CrossRef]

**7. **R. Pant, C. G. Poulton, D.-Y. Choi, H. Mcfarlane, S. Hile, E. Li, L. Thevenaz, B. Luther-Davies, S. J. Madden, and B. J. Eggleton, Opt. Express **19**, 8285 (2011). [CrossRef]

**8. **H. Lee, T. Chen, J. Li, K. Y. Yang, S. Jeon, O. Painter, and K. J. Vahala, Nat. Photonics **6**, 369 (2012). [CrossRef]

**9. **W. Loh, A. A. Green, F. N. Baynes, D. C. Cole, F. J. Quinlan, H. Lee, K. J. Vahala, S. B. Papp, and S. A. Diddams, Optica **2**,
225 (2015). [CrossRef]

**10. **N. T. Otterstrom, R. O. Behunin, E. A. Kittlaus, Z. Wang, and P. T. Rakich, Science **360**,
1113 (2018). [CrossRef]

**11. **K. Y. Yang, D. Y. Oh, S. H. Lee, Q. F. Yang, X. Yi, B. Shen, H. Wang, and K. J. Vahala, Nat. Photonics **12**, 297 (2018). [CrossRef]

**12. **S. Gundavarapu, G. M. Brodnik, M. Puckett, T. Huffman, D. Bose, R. Behunin, J. Wu, T. Qiu, C. Pinho, and N. Chauhan, Nat. Photonics **13**, 60 (2019). [CrossRef]

**13. **J. Li, H. Lee, T. Chen, and K. J. Vahala, Opt. Express **20**, 20170 (2012). [CrossRef]

**14. **J. Li, H. Lee, and K. J. Vahala, Opt. Lett. **39**, 287 (2014). [CrossRef]

**15. **J. Li, H. Lee, and K. J. Vahala, Nat. Commun. **4**, 2097 (2013). [CrossRef]

**16. **J. Li, X. Yi, H. Lee, S. A. Diddams, and K. J. Vahala, Science **345**,
309 (2014). [CrossRef]

**17. **J. Li, M.-G. Suh, and K. Vahala, Optica **4**,
346 (2017). [CrossRef]

**18. **Y.-H. Lai, M.-G. Suh, Y.-K. Lu, B. Shen, Q.-F. Yang, H. Wang, J. Li, S. H. Lee, K. Y. Yang, and K. Vahala, Nat. Photonics **14**, 345 (2020). [CrossRef]

**19. **Y. R. Shen and N. Bloembergen, Phys. Rev. **137**, A1787 (1965). [CrossRef]

**20. **R. W. Boyd, *Nonlinear Optics*
(Academic,
2003).

**21. **A. L. Schawlow and C. H. Townes, Phys. Rev. **112**, 1940 (1958). [CrossRef]

**22. **M.-G. Suh, Q.-F. Yang, and K. J. Vahala, Phys. Rev. Lett. **119**, 143901
(2017). [CrossRef]

**23. **R. O. Behunin, N. T. Otterstrom, P. T. Rakich, S. Gundavarapu, and D. J. Blumenthal, Phys. Rev. A **98**, 023832 (2018). [CrossRef]

**24. **A. Debut, S. Randoux, and J. Zemmouri, Phys. Rev. A **62**, 023803 (2000). [CrossRef]

**25. **M. Lax, Phys. Rev. **157**, 213 (1967). [CrossRef]

**26. **M. Lax, Phys. Rev. **160**, 290 (1967). [CrossRef]

**27. **C. Henry, IEEE J. Quantum Electron. **18**, 259 (1982). [CrossRef]

**28. **K. Vahala, L. C. Chiu, S. Margalit, and A. Yariv, Appl. Phys. Lett. **42**, 631 (1983). [CrossRef]

**29. **C. Harder, K. Vahala, and A. Yariv, Appl. Phys. Lett. **42**, 328 (1983). [CrossRef]

**30. **I. Henning and J. Collins, Electron. Lett. **19**, 927 (1983). [CrossRef]

**31. **M. Cai, O. Painter, and K. J. Vahala, Phys. Rev. Lett. **85**, 74 (2000). [CrossRef]

**32. **S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, Phys. Rev. Lett. **91**, 043902 (2003). [CrossRef]

**33. **H. Wang, Y.-H. Lai, Z. Yuan, M.-G. Suh, and K. Vahala, Nat. Commun. **11**, 1610 (2020). [CrossRef]

**34. **X. Yi, Q.-F. Yang, K. Y. Yang, M.-G. Suh, and K. Vahala, Optica **2**,
1078 (2015). [CrossRef]

**35. **J. R. Rumble, ed., *CRC Handbook of Chemistry and
Physics*, 100th ed. (CRC
Press, 2019).

**36. **M. Van Exter, S. Kuppens, and J. Woerdman, IEEE J. Quantum Electron. **28**, 580 (1992). [CrossRef]

**37. **H. Ludvigsen, M. Tossavainen, and M. Kaivola, Opt. Commun. **155**, 180 (1998). [CrossRef]