On-chip high sensitivity laser frequency sensing with Brillouin mutually-modulated cross-gain modulation

We report the first demonstration of a photonic-chip based laser frequency sensor using Brillouin mutually-modulated cross-gain modulation (MMXGM) in a 7cm long chalcogenide waveguide. A large sensitivity (9.45mrad/kHz) of the modulation phase shift was demonstrated.


Introduction
Devices with large spectral sensitivity are becoming increasingly desirable in applications such as metrology, optical sensing, quantum information processing, and biomedical engineering [1][2][3][4]. Although the spectral sensitivity can be enhanced using slow light [5,6], the central measurement is usually that of the optical phase, and therefore requires the use of an interferometer. Recently, Sternklar et al. exploited a novel slow-light configuration known as Brillouin mutually-modulated cross-gain modulation (MMXGM). In this configuration, both a pump and Stokes signal are modulated at the same frequency; the Brillouin gain then introduces higher harmonics to the Stokes signal, with the first harmonic possessing a phase delay that is extremely sensitive to the value of the gain. By measuring the phase delay of the first harmonic, an extremely sensitive measurement of the detuning of the Stokes shift may be obtained. This technique relies on the gain associated with the Brillouin resonance rather than the dispersion, and so differs from frequency measurements that directly exploit the presence of slow-and fast-light near an SBS resonance [7][8][9][10][11][12][13][14]. With such a configuration, an impressive spectral sensitivity ∼6.5 mrad/kHz in the modulation phase was achieved for a modulation frequency of 100 Hz in a 3km long optical fiber [15]. The striking feature of this configuration is that it does not require an interferometer: the spectral sensitivity is enhanced in the modulation phase rather than the optical phase, which makes it convenient for many optical applications.
Recently, on-chip stimulated Brillouin scattering was demonstrated using the large Brillouin scattering cross-section (g B =0.74×10 −9 m/W), small effective mode area (A e f f =2.3 μm 2 ) and strong light-sound confinement in a ∼7 cm chalcogenide (As 2 S 3 ) rib waveguide [16]. Here, we report the first demonstration of photonic-chip laser frequency sensing using Brillouin MMXGM. For our on-chip Brillouin MMXGM demonstration a large spectral sensitivity of ∼9.5 mrad/kHz was measured for a modulation frequency of 50 kHz -this absolute sensitivity is comparable to the previous MMXGM-based measurements in kilometers of fiber. It has been noted in previous work, that provided the phase shift between the pump and Stokes signal can be made exactly equal to π radians, the sensitivity is inversely proportional to the length of the gain medium [15]. This suggests that on-chip devices that use Brillouin MMXGM can potentially be used to significantly enhance the achievable sensitivity. On-chip implementation of Brillouin MMXGM is also desirable from the point of view of the creation of compact on-chip integrated photonics devices. We discuss here the dependence of the sensitivity on the device length as well as on deviation from the phase shift of π, and show that the maximum spectral sensitivity is also limited by the resolution achievable in controlling the phase of the modulated Stokes signal. Figure 1 shows the concept of laser frequency sensing using Brillouin MMXGM in a chalcogenide (As 2 S 3 ) photonic-chip [17,18]. A pump (ω p ) and a Stokes signal (ω s ) are both modulated at the same frequency f mod f Brillouin and are counter-propagated through the chip. If the frequency difference between pump and Stokes signals is close to the Brillouin resonant frequency f Brillouin then the Stokes will experience Brillouin gain, and will undergo selective amplification at the points where the peaks of the modulated pump coincide with the troughs of the modulated Stokes. This effect distorts the Stokes wave and introduces higher harmonics. The first harmonic wave possesses a phase delay which is extremely sensitive to the detuning from the Brillouin resonance. It is important to note that the phase modulation due to MMXGM is fundamentally different from the phase changes induced by the dispersion of SBS, and can be made extremely sensitive to frequency detuning. The theory and principle of laser frequency sensing using Brillouin MMXGM has been previously derived in [14], for the specific case in which the modulation phase difference ϕ 2 between pump and Stokes is equal to π. In the following we give an outline of this theory, both to clarify the underlying physics and also to discuss the effects of deviations of ϕ 2 from the exact value of π.

Principle and theory
Assuming a modulation depth of α for the pump signal, the pump intensity I 1 (z,t) can be written at the end of the waveguide (z = L) as: where Ω = 2π f mod is the angular modulation frequency and K = Ωn/c is the modulation wavenumber for a waveguide with effective index n. The Stokes signal is modulated with modulation depth β . At the front facet (z = 0) the intensity I 2 (z,t) of the modulated Stokes is: where ϕ 2 is the modulation phase shift between the pump and the Stokes signals at the front facet of the sample as shown in Fig. 1. As it propagates through the waveguide the Stokes signal experiences Brillouin gain. Under the approximation that the pump is undepleted, at the output facet (z = L) the Stokes signal can be written [14]: where the gain G depends on detuning δ ω s from the Brillouin maximum according to the Lorentzian function: where G 0 = g B I 0 1 L is the peak gain and Γ B is the Brillouin linewidth, which is typically on the order of 10-50 MHz in most optical materials.
The expression for the amplified Stokes can be expressed equivalently as a sum of harmonics: where the coefficients i m (m ≥ 1) and phases θ m are functions of K, L, α, β , G and ϕ 2 . With weak modulation and KL 1, Eq. (3) can be expanded as In Eq. (6), the terms on the right are the DC gain of the Stokes, the modulation from the Stokes and the modulation from the pump, respectively. The total phase shift of the first harmonic θ 1 is whereĜ=Gα/β . In Eq. (7), the KL term represents the linear phase delay across the propagation medium and the inverse tangent denotes the nonlinear phase delay that occurs as a result of Brillouin MMXGM. This effect depends on the detuning of the central Stokes peak from the center of the Brillouin resonance, as given by Eq. (4). This change in the gain will in turn advance or delay the phase of the first harmonic. The spectral sensitivity S depends directly on the change in the first-harmonic phase Δθ 1 when there is a change in δ ω s : From Eq. (8) it can be seen that when ϕ 2 = π, the maximum sensitivity atĜ ≈ 1 will be inversely proportional to both L and f mod , yielding the same result as previously derived in [14] and [15]. In principle, provided an exact phase modulation shift of ϕ 2 = π can be reached, extremely high sensitivities can be attained by using short waveguides and low modulation frequencies.
In practice, however, there is a limit to the resolution with which the modulation phase can be determined. We now consider the modulation phase shift to be close but not equal to π, such that ϕ 2 = π + δ ϕ 2 , where δ ϕ 2 is small. Continuing the assumption that KL << 1, we find from Eq. (8) that the maximum sensitivity is This result shows that the limitation on the maximum sensitivity is determined by the larger of the quantities KL and δ ϕ 2 . We note that in Eq. (9), the sensitivity changes sign when KL ∼ δ ϕ 2 . This corresponds to a transition between the slow light and fast light regions [13,19]. Because KL is always positive, the situation when ϕ 2 is exactly equal to π corresponds to the slow-light configuration. As noted previously [15], there exist two sensing windows, spaced symmetrically around the Stokes frequency. Within each sensing window, and provided that KL is small, the phase shift θ 1 varies monotonically with δ ω s between either zero and π, for the slow-light configuration, or between zero and −π for the fast-light configuration. At the position of maximum sensitivity, which occurs in the middle of the sensing window, θ 1 is equal to ±π/2. From the above analysis, we note that Eq. (7) for the modulation phase shift was derived assuming only the first-order harmonic. However, the actual output Stokes wave (see Eq. (3) and Eq. (5)) contains higher-order harmonics. Figure 2 shows the theoretically calculated amplitudes of first-and second-order harmonics around the sensing window (centered around 17 MHz) as calculated using Eq. (10) and Eq. (14) from [14], for a 7 cm chalcogenide ribwaveguide used in the experiments. For these calculations we used the following parameters: n =2.4, Γ B =34 MHz, f mod =50 kHz,Ĝ 0 =αG 0 /β =2 and ϕ 2 =π. From Fig. 2, we note that the amplitude of the second-order harmonic is larger than that of the fundamental in the sensing window where the spectral sensitivity S is large. Therefore, in order to measure the phase change of first harmonic wave, higher-order harmonics must be removed in the experiment. In our experiments, we used an electrical filter to remove the higher harmonics.
The calculations shown in Fig. 2 were performed for a sensing window centered around 17 MHz. However, the position of the sensing window, defined by the detuning of its center frequency from the peak of the Brillouin resonance, varies with the value ofĜ 0 [15]. In Fig.  3 we show the calculated position of the sensing window and the corresponding maximum sensitivity S for different values ofĜ 0 . Typically the sensing window is a narrow frequency region around the center frequency. However, it is possible to apply the sensing technique in a large frequency range around the the peak of the Brillouin resonance by changing the value ofĜ 0 . From Fig. 3, we note that whenĜ 0 increased from 1.5 to 5.5, the maximum sensitivity varies slowly, changing less than 25%.  is the fiber polarization controller; SG is the signal generator to create Stokes wave; C1, C2 and C3 are circulators; EOML1, EOML2 and EOMH are intensity modulators; EDFA is the Erbrium doped fiber amplifier; OSA is the optical spectrum analyzer; D is optical detector; BPF is electronic band pass filter and OS is the oscilloscope. Blue wires indicate the electronic wire. Figure 4 shows the experimental set-up for the on-chip Brillouin MMXGM sensor. A tunable laser around 1550 nm was split with a 50/50 coupler to generate the pump and Stokes wave. After the Stokes signal was generated in one arm using the intensity modulator EOMH (electrooptic modulator of high frequency) driven by a signal generator SG, the light in both the arms was sinusoidally modulated by intensity modulators EOML1 and EOML2 (electro-optic modulator of low frequency) driven by function generators FG1 and FG2 at 50 kHz. The modulation phase could be controlled by adjusting the phase delay from the function generator FG2 triggered by FG1. Both the pump and signal were amplified by erbium doped fiber amplifiers (EDFAs) and coupled to a 7 cm long waveguide with a cross section 4μm×850nm through circulators (C1, C2) using lensed fibers. The total insertion loss was around 13 dB during the experiment, measured through port 3 of C1. The amplified Stokes signal was sent into a circulator C3 with a Bragg grating connected to port 2 (see Fig. 4). The 3dB-bandwidth of the Bragg grating was about 0.13 nm and one of its steep edges of the reflection peak was used to suppress the parasitic pump originated from the reflection at the chip input facet. The filtered signal was split using a 99/1 coupler where the 1% port was sent to the OSA and the 99% port was sent to a photo detector D. An electronic band pass filter BPF with a 2 kHz bandwidth was used before the detector to filter out the higher harmonics and DC component. The electronically filtered signal was observed on the oscilloscope OS.

Experiment
From Fig. 3, we note that the frequency at which the maximum sensitivity occurs depends on the value ofĜ 0 , which enables the application of this technique over a large frequency range around the peak of the Brillouin resonance (∼7.63 GHz). In our experiment, we first adjusted the pump and Stokes polarization to achieve a maximum gain (∼5-6 dB) at the peak of the Brillouin gain while keeping α≈β ≈0. In order to show that we can achieve large sensitivity for any frequency around the peak of the Brillouin resonance by adjusting the value ofĜ 0 , we arbitrarily picked a frequency detuning of Δ(δ ω s )∼25 MHz up-shifted from the peak of the Brillouin resonance. By observing the pump and Stokes waveform on OS, at port 2 of C1, we set ϕ 2 =π. The value ofĜ 0 was then varied by adjusting the modulation depths α and β (whilst keeping them small) so that θ 1 = π/2, which corresponds to the maximum sensitivity, occurs at Δ(δ ω s )∼25 MHz. Finally, we measured the sensitivity for a range of different values of ϕ 2 around a detuning δ ω s ∼25 MHz, and determined the value of ϕ 2 for which the sensitivity was maximized. With this optimized value of ϕ 2 , the frequency of the Stokes was detuned from δ ω s ∼25 MHz in steps of 100 kHz to observe the phase shift evolution of the first-harmonic on OS. In our experiments, the maximum sensitivity was obtained for a phase evolution from -π to 0, therefore occurring in the fast-light regime. We also observed phase evolution from the slow-light regime of 0 to π, however the sensitivity at this value of ϕ 2 was smaller. In the following we present here the results for the overall maximum sensitivity from the fast-light regime. Figure 5 shows the output waveform recorded on OS with the largest sensitivity we could obtain at the modulation frequency of 50 kHz, together with the normalized phase shift of the first harmonic wave. The black dashed curve in the center of Fig. 5(A) shows the evolution of the phase when the frequency detuning was changed. A phase shift evolution from -π to 0 was observed when the frequency was detuned from 24.876 MHz to 25.876 MHz in steps of 100 kHz. In our experiment, we observed a small distortion arising from the residual higher-order harmonics remaining after the band pass filter. The normalized phase shift plotted in Fig. 5 (B) shows a phase shift from -π to 0 when the frequency was detuned away from the peak of the Brillouin resonance. A maximum phase shift of Δθ 1 ∼0.95 rad was observed around detuning of 25.476 MHz (see Fig. 5(B)), corresponding to a maximum sensitivity of ∼9.5 mrad/kHz. The maximum attainable sensitivity is limited by the ability to measure the phase change of