Cascaded forward Brillouin lasing in a chalcogenide whispering gallery mode microresonator

Chalcogenide glasses have been recognized in the past decade as a candidate for low-threshold nonlinear optics because of their transparency in the infrared region, low glass transition temperature, and large nonlinearities.^1–5 They also exhibit many desirable photoinduced phenomena, including photocrystallization,⁶ photopolymerization,⁷ photodecomposition,⁸ photocontraction or expansion,⁹ photovaporization,⁸ and the photodissolution of metals such as silver.^8,10 These changes allow us to modify the optical characteristics of chalcogenide glasses, such as their electronic bandgap, refractive index, and optical absorption coefficient, to fabricate a wide variety of optical devices. Additionally, the low glass transition temperature of chalcogenide glasses allows fabricated devices to be reflowed to produce ultrasmooth optical surfaces.^11–13 

Stimulated Brillouin scattering (SBS) is one of the strongest third-order nonlinearities in most optical media. SBS manifests through the transfer of energy from the optical pump beam to an optical Stokes beam, either in the same or opposite direction as the pump. SBS is used in narrow-linewidth lasers,^14,15 slow light,¹⁶ optical cooling,¹⁷ optical isolators,¹⁸ and distributed sensing.¹⁹ The conservation of energy and momentum requires that ω_a = ω_p − ω_s and q = k_p ± k_s, where ω_a, ω_p, and ω_s are the frequencies and q, k_p, and k_s are the wavenumbers of the acoustic, pump, and Stokes waves, respectively. The sign in the momentum matching condition is positive (i.e., wavenumbers add up) for backward stimulated Brillouin scattering (BSBS) and negative for forward stimulated Brillouin scattering (FSBS). Since the pump and Stokes beams typically differ by no more than a few GHz in frequency, they are approximately equal in wavenumber, i.e., k_p ≈ k_s. Therefore, FSBS is mediated by phonons of low wavenumber, q ≈ 0, while BSBS is mediated by phonons of higher wavenumber, q ≈ 2k_p. Even though FSBS is understood to have theoretically stronger nonlinearity than BSBS with all else being equal,²⁰ many common resonator geometries do not have resonant acoustic modes for phonons with low wavenumber. Notably, fiber ring resonators do not support strong FSBS, although they are widely used for BSBS lasers.^21,22 This is because step-index fibers do not have acoustic eigenmodes for low-momentum phonons.

A resonator that satisfies phase-matching conditions for the optical pump, optical Stokes, and acoustic phonon waves exhibits SBS at sub-mW thresholds²³ due to the tight confinement of the optical fields. To exploit the many advantages of nonlinear optics using SBS in a small form factor, whispering gallery mode resonators are often used.^24,25 The resonance condition for the Stokes beam can be met by matching the free spectral range (FSR) of the resonator to the SBS shift so that the pump and Stokes beams can be excited to adjacent resonances with nearby azimuthal mode orders. Such resonators with FSR matching may support cascaded SBS. In fact, cascaded BSBS with GHz-scale shifts has been demonstrated in resonators of mm-scale diameter.^24–26 The FSR, unfortunately, scales inversely with the length of the resonator, which makes it difficult to fabricate a high-quality resonator that matches typical FSBS MHz shifts. For this reason, FSBS in a resonator has only been demonstrated between optical modes with different radial and azimuthal mode orders (i.e., intermodal FSBS).^27,28 This technique has the downside that it would be difficult to consistently reproduce the same Brillouin shift across devices, as the frequency difference between modes without at least one common mode order (radial or azimuthal) is highly sensitive to device geometry, placing a strict limit on fabrication tolerance. It is also not possible to demonstrate cascaded SBS using this technique since the higher-order modes with differing radial and azimuthal mode orders are not equally spaced in frequency. For instance, in the first report of FSBS in a microresonator, Bahl et al.²⁸ report forward Brillouin scattering between a pump mode (640, 2, 1) and Stokes mode (630, 1, 3) with a Brillouin shift of 57.8 MHz, where the numbers in brackets are, respectively, azimuthal, radial, and polar mode orders. Since there is no optical mode 57.8 MHz below the first Stokes mode (630, 1, 3), a cascaded process is not observed.

In this work, we avoid the technical difficulty of matching the FSR to the Brillouin shift by using intramodal Brillouin scattering (i.e., the Stokes beam is scattered within the same optical resonance as the pump beam). This is enabled by a chalcogenide glass microresonator platform with a forward Brillouin shift that is smaller than the width of the optical resonance. We rely on finite element modeling to choose a microresonator geometry that supports low-wavenumber resonant acoustic modes required to meet the phase matching condition. Using this approach, we demonstrate up to 25 orders of cascaded intramodal FSBS at a sub-mW threshold. The measured threshold of 0.91 mW is comparable to a previous report of 5 orders of cascaded BSBS in a silica microsphere at 0.6 mW.²⁹ 

In addition to the applications of Brillouin lasers in narrow-linewidth optical sources or slow-light generation, cascaded SBS can be used to improve optical sensors and photonic frequency synthesizers. The Brillouin frequency shift is cumulative over cascaded Stokes beams. Therefore, a small change in Brillouin shift leads to higher-order Stokes beams experiencing a greater net change in frequency than lower Stokes orders. This effect can be leveraged to improve the sensitivity of Brillouin-based gas sensors that rely on a change in the Brillouin shift as a function of gas concentration.³⁰ For example, an increase in the Brillouin shift of δf leads to the frequency of the 25th Stokes beam decreasing by 25 × δf, improving the sensitivity of the detector 25-fold.

Cascaded SBS on chip has been adopted to build photonic frequency synthesizers to produce low-noise radio frequency signals.^31–33 The technique presented in this paper does not require the free spectral range of the resonators to be matched to the Brillouin shift; therefore, we expect this work to loosen the fabrication tolerances for microresonators used in photonic frequency synthesizers. Additionally, since the forward Brillouin frequency shift is strongly dependent on the geometry of the resonator,²² we expect this result to advance the state of the art in low-noise tunable photonic frequency synthesizers.

High-quality chalcogenide microspheres were fabricated using commercially available arsenic sulfide (As₂S₃) fiber (IRFlex – IRF-S-6.5). First, the fiber was tapered down to ∼20 µm by heating the fiber to the glass transition temperature using a resistive heater and pulling on both sides using a motorized stage. The taper was then broken, and the tapered end was melted by bringing it close to a red-hot metal plate. The surface tension caused the molten glass to form a sphere, which was then allowed to cool down. It is possible to produce microsphere resonators with a Q-factor over 10⁷ using this technique. This is close to the theoretical upper limit due to absorption.^27,34

The optical and acoustic modes of the resonator were modeled using the commercial COMSOL multiphysics package.³⁵ Simulations were performed for intramodal SBS, where the pump mode and all Stokes orders have the same azimuthal, polar, and radial mode numbers. To satisfy this condition, the width of the resonance needs to be at least 300 MHz when the resonator is coupled to a tapered fiber (i.e., the loaded Q factor has to be less than 6.6 × 10⁵). Therefore, measurements and simulations were performed in the overcoupled regime. This is in sharp contrast to the common practice of using the highest-quality factor resonator that is possible to fabricate and undercoupling it for backward Brillouin lasers.

Since the resonances were all at least 300 MHz wide, several Stokes orders that were separated by a few tens of MHz would fit within the same resonance. The phase-matching condition requires that the azimuthal mode of the acoustic wave is the difference in mode numbers of the pump and Stokes beams. For FSBS, we hence have a standing acoustic wave with an azimuthal mode number equal to zero. Using the electric and displacement fields from the resonant optical and acoustic modes, respectively, we calculated the SBS gain in the resonator. The SBS gain of a single acoustic mode has a Lorentzian shape and a peak value of²⁰ 

where ω = ω_p ≈ ω_s is the frequency of the optical beams, Ω is the frequency of the acoustic wave or the SBS shift, ν_gp is the group velocity of the pump, ν_gs is the group velocity of the Stokes beam, ɛ is the permittivity of the material, ρ is the density, E_p and E_s are the unnormalized electric fields from the pump and Stokes beams, respectively, u_m is the unnormalized displacement vector from the resonant acoustic wave, and f is the net force due to electrostriction and radiation pressure. The angled brackets indicate surface integrals performed on a radial plane perpendicular to the direction of propagation of the optical beams.

The phonon quality factor Q_m is defined as twice the ratio of phonon frequency over the damping rate. Following the experimental results of Bahl et al.,²⁸ we assume that the phonon quality factor for an arsenic sulfide microresonator is limited by the phonon damping rate of the material. We estimate the damping rate as πΔν_B,³⁶ where Δν_B is the gain bandwidth, and obtain Q_m = 467 from the experimental results of Pant et al.³⁷ We also performed finite element simulations to include the phonon loss rate through the stem of the microresonator using COMSOL. Our simulations yielded Q_m to be about 800. Since the Q_m is difficult to measure experimentally and the simulations are strongly dependent on errors in reported material values, we use an order of magnitude approximation and take Q_m = 500 for our calculations.

Electrostriction force is a deforming force experienced by all dielectrics in the presence of non-uniform electric fields. The highly localized whispering gallery mode resonances in optical microresonators hence induce electrostriction forces in the bulk of the resonator. The electrostriction force is derived from the gradient of the electrostrictive tensor given by²⁰ 

where n is the refractive index, p_ijkl is the photoelastic tensor, and E_k and E_l are the net electric fields in the direction denoted by indices k and l.

The radiation pressure, on the other hand, is the pressure exerted on a dielectric interface due to the exchange of momentum between the dielectric and an electromagnetic field incident on its surface. The radiation pressure is derived from the Maxwell stress tensor. As the radiation pressure acts perpendicularly to the dielectric interface²⁰ and the acoustic eigenmode corresponding to the Brillouin shift oscillates parallel to the interface, the radiation pressure has no contribution to the Brillouin gain. Hence, we excluded radiation pressure from our calculations to reduce computational complexity.

When the pump power is below the Brillouin lasing threshold, pump photons scatter off thermally populated phonons to generate a very small amount of Stokes power. In this regime, we may use the small signal approximation (SSA) that the Stokes power is much lower than the pump power. We also ignore pump depletion and nonlinear losses. When the SSA is valid, the Stokes power coupled out of the resonator $Psout$ is related to the input Stokes power $Psin$ (which arises from thermal phonons) as²³ 

where τ is the coupling constant [Fig. 1(b)] and G is the round-trip envelope gain given by²³ 

Here, $Ppin$ is the input pump power, α is the total loss, and L is the length of the cavity. To account for the various losses, including material, scattering, and radiation losses, α is estimated from the measured Q factor as α = 2πn/Qλ.³⁸ At the Brillouin lasing threshold, the SSA breaks down as G approaches $1/τ$ and Eq. (3) goes to infinity. The pump power at which the SSA breaks down is therefore taken to be the lasing threshold.²³ While fitting the SSA model in Eq. (3) to the experimental data, $Psin$ only introduces a scaling factor in the order of 10⁻¹¹ if we assume only a single photon is generated per second per unit frequency by quantum fluctuations and thermal phonons (⁠$Psin=vghfs≈10−11$⁠, v_g is the group velocity, h is the Planck’s constant, and f_s is the Stokes frequency²³). Since α is the loss and Γ is obtained from simulations of the acoustic and optical eigenmodes of the system [Eq. (1)], the only remaining free parameter in the model is τ, the coupling constant. We extract τ from the experimental data to estimate the lasing threshold using the SSA model in Sec. III.

The experimental setup is shown in Fig. 2(b). An SMF-28 silica fiber was tapered to ∼500 nm radius by heating the fiber to above the glass transition temperature using a butane flame while stretching the fiber from both ends using motorized stages. The laser source was a single frequency mode-hop-free tunable CW laser in the 1550 nm band (Toptica CTL 1550). The resonator was mounted on a 3-axis piezo-actuated stage (Thorlabs MAX312D) with 20 nm resolution and brought within 100 nm of the tapered fiber. The system was imaged using a microscope with a long-distance objective [Mitutoyo 10× ICO 0.28 numerical aperture (NA), 33.5 mm working distance] from the top. The system was enclosed in a chamber to minimize disturbances due to temperature fluctuations and air currents. The transmitted light was measured using an InGaAs photodetector of 1.2 GHz bandwidth (Thorlabs DET01CFC) for the spectrum in Fig. 3(a) and an InGaAs photodetector of 40 MHz bandwidth (Thorlabs DET10C) for the spectrum in Fig. 5. A 20 dB amplifier was used with the high-bandwidth detector to make up for the lower responsivity. The signal was recorded using an oscilloscope and an RF spectrum analyzer (Keysight N9030B). A function generator provided a triangle wave signal to the laser to sweep the frequency by a few tens of GHz and a trigger signal to the RF spectrum analyzer and the oscilloscope. The laser was tuned into resonance by scanning for the characteristic dip in transmission across the tapered fiber on the oscilloscope. This dip is measured while operating at an undercoupled condition to measure the intrinsic Q factor of our resonators. The Stokes beams in the resonator are coupled back into the tapered fiber. The electrical spectrum analyzer was used to pick up beat notes between the Stokes beams and the pump that are typically too close for an optical spectrum analyzer to resolve. Using this setup, we observed the cascaded Brillouin spectrum shown in Fig. 3(a) from a 125 µm resonator of unloaded quality factor 1 × 10⁶ and the spectrum in Fig. 5 from a 100 µm resonator [Fig. 2(a)] of unloaded quality factor 6.2 × 10⁵.

Figure 3(b) shows a plot of peak frequencies from Fig. 3(a) as a function of Stokes order. The peak frequencies were found to lie at integer multiples of 34.5 MHz, with the highest detuning observed at 25 times the Brillouin shift (i.e., 25 Stokes orders). This is consistent with our simulations [Fig. 1(a)], which show a resonant acoustic wave at 33.2 MHz. The peak frequencies in Fig. 5 were similarly found to lie at integer multiples of 19.4 MHz. This is also consistent with our simulations for a 100 µm resonator showing resonant acoustic waves at 17.4 MHz. The mismatch in frequencies (34.5 MHz in the experiment vs 33.2 MHz in the simulation, 19.4 MHz in the experiment vs 17.4 MHz in the simulation) is attributed to the deviation in Young’s modulus of the commercial As₂S₃ fiber from the reported value of bulk material³⁹ and the perturbation in mode shape due to the stem of the resonator, which was not included in the simulation.

To estimate the lasing threshold, we used a microsphere of diameter 110 µm with a quality factor of 2.2 × 10⁶, showing cascaded Stokes beams separated by 18.3 MHz. As the pump power was reduced, the first Stokes line was the last to drop below the noise floor of the spectrum analyzer. We then slowly ramped up the pump power and recorded the power of the first Stokes line until the resonator suffered degradation in the Q factor.

Our experimental setup performs heterodyne measurements between the pump and Stokes beams to achieve high frequency resolution. However, since the electrical spectrum analyzer was used to pick up beat notes between the pump and Stokes beams, we needed to convert the measured signal S in dB to optical power units for measuring the lasing threshold. The detector current from superposing optical beams I₁ of frequency f₁ and I₂ of frequency f₂, for detector sensitivity R, is⁴⁰ 

Since the spectrum analyzer records the AC component of the signal, taking I₁ to be the pump and I₂ to be the Stokes, and using the subscripts p for pump and s for Stokes,

where S is the measured signal and I_ref is the internal reference current of the analyzer. To obtain Stokes power as a function of varying pump power given the measured value of S vs I_p,

The power of the first Stokes line as a function of pump power is shown in Fig. 3(c). Finite element simulations of this sphere revealed a resonant acoustic mode at 16.1 MHz that satisfies the phase matching condition and is close to the observed FSBS shift at 18.3 MHz. Using Eq. (1), we estimated the Brillouin gain Γ of the resonator to be 2.57 × 10⁵ m⁻¹ W⁻¹. The calculated value of Γ was used to fit the theoretical model in Eq. (3) to the experimental data to obtain the coupling constant τ and the threshold power. The small signal approximation breaks down at the Brillouin lasing regime, and the theoretical model goes to infinity at the threshold.²³ From the fit, we found the Brillouin lasing threshold to be at 910 ± 20 µW, for a coupling constant τ = 0.92. When the first-order Stokes power is plotted against pump power on a linear scale in Fig. 4, a knee is observed at around 0.76 mW, confirming a sub-mW lasing threshold. The small disagreement in threshold power measurement between the different methods (0.91 vs 0.76 mW) is attributed to the ambiguity in the knee method.⁴¹ As the quality factor degrades beyond about 1 mW of pump power, we cannot take measurements in the milliwatt regime.

We fabricated high-quality chalcogenide microresonators to study nonlinear effects in transmitted light through a coupled tapered fiber. We have observed beat notes between downconverted Stokes beams and the pump beam in an RF spectrum analyzer when the narrow-linewidth tunable laser at 1550 nm was tuned into resonance in a 125 µm sphere with a quality factor of 1 × 10⁶. The beat notes were observed at integer multiples of 34.5 MHz, up to 25 orders, which suggests a cascaded nonlinear downconversion process at resonance [Fig. 3(a)]. The observation is consistent with the simulation, which suggested that this was Brillouin scattering mediated by a resonant acoustic mode. The slight mismatch in frequency (33.2 MHz in the simulation as compared to 34.5 MHz in the experiment) was attributed to the deviation in Young’s modulus of the commercial As₂S₃ fiber from the reported value of bulk material³⁹ and the perturbation in mode shape from the stem of the microsphere, which was not included in the simulation. We also observed a cascaded Brillouin spectrum in a smaller resonator of diameter 100 µm with a quality factor of 6.2 × 10⁵ (Fig. 5). The Brillouin shift was observed to be 19.4 MHz, close to its predicted value of 17.4 MHz from the simulation.

Since the amplifier raised the noise floor of the spectrum in Fig. 3(a), the spectrum was averaged over 41 sweeps to increase the visibility of the peaks. However, the spectral lines were broadened to between ∼3 and ∼9 MHz because of the variation in acoustic eigenfrequency across sweeps due to thermal fluctuations. The power of the spectral lines was uneven since the amplifier circuit had parasitic absorptions at 105(2n + 1) MHz (n = 0, 1, 2, …). When the gain of the amplifier (nominally 20 dB) fell below 16 dB around 105, 315, 525, and 735 MHz, the signal was buried in the noise floor of the spectrum analyzer.

In contrast, the spectrum in Fig. 5 was taken in one shot (i.e., no time averaging) and without using an amplifier. An amplifier was not required for the spectrum in Fig. 5 on account of not coupling the resonator to the tapered fiber as strongly as we did for the resonator in Fig. 3(a). This is because we attempted to measure a GHz bandwidth comb with Fig. 3(a), which required the width of the resonance to be at the very least several GHz wide. This required that we had to overcouple the resonator until the loaded Q factor fell below 5 × 10⁴. On the other hand, we only needed the resonance for the measurement in Fig. 5 to be greater than a few 100 MHz wide. This condition was already met since the unloaded Q factor for the sphere used in Fig. 5 was 6.2 × 10⁵. As the Brillouin gain scales inversely with approximately the square of the loaded Q factor, the resonator used in Fig. 5 had much stronger Stokes lines. We could thus obtain the high signal-to-noise ratio in Fig. 5 without using an amplifier or averaging across multiple measurements.

We performed threshold measurements in a 110 µm resonator with a Q factor of 2.2 × 10⁶ showing spectral lines shifted from the pump by 18.3 MHz [Fig. 3(c)]. Finite element simulations of this microsphere revealed a resonant acoustic mode at 16.1 MHz, close to the observed FSBS shift. We theoretically calculated the forward Brillouin gain Γ to be 2.57 × 10⁵ m⁻¹ W⁻¹ from Eq. (1). We note that this is much higher than typical BSBS gain, as the gain is inversely proportional to the square of the Brillouin shift [Eq. (1)] for comparable overlap between the electrostriction force and the acoustic wave. For instance, a BSBS gain of 2.1 × 10² m⁻¹ W⁻¹ with a Brillouin shift of 7.7 GHz has been reported in the same material using a waveguide geometry.³⁷ Even with a significantly lower overlap between the optical and acoustic modes in a microsphere as compared to a waveguide, the simulated FSBS gain was over three orders of magnitude higher than the reported BSBS gain.

The calculated gain was used in a theoretical model for the Brillouin threshold.²³ The model showed excellent agreement with the experimental data [Fig. 3(c)]. The threshold for Brillouin lasing was predicted from the model at 910 ± 20 µW, close to the “knee” at 0.76 mW observed in Fig. 4. Beyond the Brillouin lasing threshold, cascaded Stokes lines were observed. Previous theoretical work has established that anti-Stokes beams will be generated along with Stokes beams in a waveguide that supports cascaded forward Brillouin scattering.⁴² The analysis also applies to resonators that are simultaneously resonant for the pump, Stokes, anti-Stokes, and acoustic waves. We confirmed the existence of anti-Stokes beams using an optical heterodyne method (see the supplementary material). As the theory agrees well with the experimental results, we can reliably confirm that we are observing cascaded intramodal forward Brillouin scattering in our resonators.

Soliton states or other low-threshold nonlinear processes were not observed due to normal dispersion both from the material¹¹ and the geometry.⁴³ Due to the high nonlinearity of chalcogenide glass, symmetry breaking is known to occur at powers of the order of a mW.³⁴ The damage threshold for chalcogenide microresonators is also known to be in the same range.⁴⁴ Therefore, it is likely that the power of the comb from a microsphere resonator cannot be dramatically increased without using an external amplifier. However, we note that there is room for improvement in the integral overlap between the electrostrictive force and the acoustic field in Eq. (1). It is possible that a higher overlap integral could be realized using a different resonator geometry that confines the standing acoustic wave more tightly (possibly a wedge or disk resonator⁴⁵), leading to Brillouin combs of higher optical power. It has also been reported that the Brillouin gain can be enhanced in sub-wavelength waveguides where the radiation pressure at the air-waveguide interface dominates the electrostrictive forces.^46,47 Exploring the prospects of cascaded FSBS combs in novel resonator geometries could be a direction for future research.

In conclusion, we demonstrate cascaded forward intramodal Brillouin scattering within a microresonator platform for the first time. We used a theoretical model to estimate the threshold for cascaded FSBS on a resonator and verified that it agrees with the experimental results. The resonator was excited using a tunable laser at 1550 nm, and beat notes between the pump and Stokes beams were observed at multiples of 34.5 MHz, corresponding to 25 orders of Stokes beams. We also established the existence of anti-Stokes beams generated through FSBS via a heterodyne measurement. We note the applications of cascaded forward Brillouin scattering in gas sensors and photonic radio frequency sources. This work could lay the foundation for future work into the SBS phenomena in the near and mid-infrared using chalcogenide optics.

More details of the heterodyne measurement of anti-Stokes beams generated via forward Brillouin scattering in the resonator can be found in the supplementary material.

The authors gratefully thank Lange Simmons (University of Colorado, Boulder), Dr. Omkar Supekar (University of Colorado, Boulder), Dr. Kyuyoung Bae (Radanta Corporation, Boulder, CO, USA), and Dr. Jiangang Zhu (Deepsight Technology, San Francisco, CA, USA). T. Shanavas thanks Professor Ethan Neil (University of Colorado, Boulder) for offering the computational physics class (Grant No. PHYS 5070) at the University of Colorado, Boulder. This work was funded by the Air Force Office of Scientific Research (Grant No. FA9550-19-1-0364) and the Office of Naval Research (Grant Nos. N00014-19-1-2251 and N00014-19-1-2382).

The authors have no conflicts to disclose.

Thariq Shanavas: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Michael Grayson: Conceptualization (equal); Formal analysis (equal); Investigation (equal). Bo Xu: Formal analysis (equal); Investigation (equal). Mo Zohrabi: Formal analysis (equal); Investigation (equal); Methodology (equal). Wounjhang Park: Funding acquisition (equal); Supervision (equal); Writing – review & editing (equal). Juliet T. Gopinath: Funding acquisition (equal); Project administration (equal); Resources (equal); Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.