Introduction

Due to their spectral selectivity and long photon lifetimes, very high-quality (Q) factor whispering-gallery-mode1 (WGM) resonators have attractive functionalities for integrated or miniaturized photonics. In particular, it has been demonstrated that they can find applications in very sensitive sensing2,3, microwave signal generation4,5 or optical processing6,7,8,9,10,11, narrow-band and high-contrast optical filtering12,13 or high-coherence optical sources14,15,16,17,18. Although huge technological progresses have been done in chip-compatible integrated optics19, crystalline millimeter size WGM resonators obtained by thorough cleaning and polishing steps provide the highest resonance quality20,21 with Q-factors up to 1011. Another approach to reach very high Q-factors consists in using WGM resonators made of materials with optical gain22,23,24,25. The internal amplification is used to compensate for optical losses coming from surface absorption due for example to chemosorbed OH− ions in silica WGM resonators26,27,28 or surface scattering29. Moreover the pumping rate of the active medium give an additional degree of freedom to control the properties of the resonator30,31. In this paper, we present a new experimental configuration with a three port WGM rare-earth-doped microsphere: a dual in and out signal port and a third port which is used to inject the pump inside the WGM microsphere. This convenient experimental setup permits us to explore all the coupling regimes from undercoupling to spectral selective amplification in fluoride glass erbium-doped WGM microspheres. The coupling regime is infered from a careful analysis of the ringdown signal recorded using a frequency swept input signal30. In particular we show that this experimental technique can be used to obtain the dispersive properties of a highly transparent microsphere equivalent to a Gires-Tournois interferometer. Furthermore the technique is applied to measure very high intrinsic finesse values up to 4.0 × 107 associated with very low single pass absorption or ultrahigh Q-factors (>5 × 109) without the use of a fine-stabilized Laser21.

Results

System description

Figure 1 is a sketch of the experimental configuration. The evanescent tail of the mode of a tapered fiber is side coupled to the active WGM microcavity with the amplitude u(t). The coupling photon lifetime is denoted τe. The intrinsic cavity photon lifetime τ0 can be tailored by tuning the pumping rate through a half tapered fiber used to inject the pump field within the WGM microcavity. We assume that all the pumping effects (including the losses due to the pump taper signal scattering) are taken into account by the value of τ030: τ0 ≥ 0 for a passive cavity, in the case of an amplifying cavity −τe < τ0 < 0. In the high-finesse cavity assumption (all the resonators experimentally studied in this paper have a finesse F > 2.5 × 106), the output signal amplitude sout(t) is deduced from the input signal amplitude sin(t) by solving the following differential equation32,34:

where ω0 is the cavity resonance angular frequency. The amplitude cavity lifetime τ is defined by 1/τ = 1/τ0 + 1/τe and is related to the quality factor Q = ω0/Δω of the cavity by Q = ω0τ/2, here Δω = 2/τ is the cavity linewidth (in angular frequency units). In the linear stationary regime, the input signal can be written: and the system is fully described by its amplitude transfer function h = sout/sin:

The power transmission T(ω) = |h(ω)|2 measured through the output port is given by:

For a detuned cavity (|ω − ω0| → ±∞) we have T(ω) → 1 which comes from the fact that for a high-finesse (passive) resonator the antiresonance transmission values are such as . At resonance (ω = ω0) we can observe a dip with a Lorentzian shape (T(ω0) ≤ 1) in the case of a passive cavity or a Lorentzian peak (T(ω0) > 1) for an amplifying microresonator. The dispersion properties can be characterized by the group delay experienced by the system, it can be extracted from the induced phase shift on the input signal obtained from the phase shift ϕ(ω) by . In particular the resonant group delay is given by33:

The absolute and relative values of τ0 and τe determine the coupling regime of the system:

  • 0 < τ0 < τeundercoupling: 0 < T(ω0) < 1 and τg(ω0) < 0

  • τ0 = τecritical coupling34: T(ω0) = 0 and τg(ω0) → ±∞

  • τe < τ0overcoupling: 0 < T(ω0) < 1 and τg(ω0) > 0

  • τ0 → +∞ transparent cavity: T(ω0) = 1 and τg(ω0) = 2τe

  • τ0 < −τe frequency selective amplification35: T(ω0) > 1 and τg(ω0) > 0.

Figure 1
figure 1

Schematic view of a three-port WGM erbium-doped microsphere.

The signal is coupled thanks to a tapered fiber and the pump beam is coupled through the half taper.

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τg > 0 corresponds to a slow-light regime whereas τg < 0 is associated with a fast-light regime36,37,38,39. Ultrahigh-Q factor microcavities can have various applications depending on the operating regime. In the critical coupling, due to their wide free spectral range, microresonators can be used as selective notch filters. Used in the overcoupling, transparent or selective amplification regimes a microresonator behaves like an integrated delay line. In particular, the system can be a pure phase-shifter when the cavity is at transparency. An efficient resonant net gain can be obtained in the case of the selective amplification regime. From a practical point of view a thorough optical characterization is required to determine the properties of high-Q microresonators. The method usually carried out consists in using a very narrow-linewidth Laser which carrier angular frequency is linearly swept in time as input signal34,40: ω(t) = ωi + VSt/2. The bandwidth of the power transmission function T(ω) is deduced from the trace of the transmission T(t) in the time domain. To insure a stationary regime, the frequency sweeping rate VS has to be very slow compared to the sweeping speed V0 = 4/τ2 corresponding to one resonance scanned during the cavity lifetime. In Figure 2(a), the transmission T(t) of a critically coupled cavity and a very slow sweeping speed (VS = 10−3V0) is displayed. The width of the resonance in the time domain is 500τ or 500τ × 10−3V0 = 2/τ in the angular frequency domain. If the frequency speed is of the same order of magnitude as V0, the dynamic of the cavity mode must be taken into account21,30,41,42,43. The amplitude of the field is given by:

where we define:

with erf(z) and denoting the complex error function. Using Eqs. (1), (5) and (6), we can obtain an analytical expression for T(t). The transmission of the critically coupled cavity for a high speed frequency sweeping is plotted in Figure 2(c). The interference between the intracavity field and the input field creates some ringing oscillations on the transmission curve41. By analyzing this beat we determine the value of VS which is difficult to calibrate precisely from an experimental point of view. In a similar manner, the ringing signal from the cavity, is used to infer the coupling regime for a passive resonator30. Note that in the stationary regime this can be done unambiguously solely by interferometric44,45,46 or polarimetric47,48,49 techniques. For a perfectly transparent cavity (lossless cavity τ0 → +∞), the power transmission is T(ω) = 1 as can be seen in Fig. 2(e). Consequently in the stationary regime, information about the quality of the resonance and the dispersive properties of the cavity can only be obtained by measuring the phase shift ϕ(ω) = 2 arctan [(ω0 − ω)τe] [see Fig. 2(f)] using phase-sensitive methods44. When the frequency sweeping is high compared to V0 [Fig. 2(g)], the properties of the resonator can be deduced from a thorough analysis of the induced ringing oscillations in the transmission profile. For , the resonator is not perfectly loaded and the 2π phase shift disappears as illustrated in Fig. 2(h).

Figure 2
figure 2

Theoretical calculation of the transmission T(t) and the phase shift ϕ(t) of the amplitude transfer function in the dynamic regime: (a)–(d) for a critically coupled resonator and (e)–(h) for a transparent (or lossless) cavity.

We have used two different sweeping rates: (a), (b), (e) and (f) VS = 10−3V0 almost corresponding to the stationary regime; (c), (d), (g) and (h) VS = 10V0.

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Experimental setup

We used the experimental setup described in Fig. 3 to characterize the linear properties of erbium-doped ZBLALiP glass microspheres50. The fabrication process of the microspheres is described at section Methods. A counter-propagative pump field from a Laser diode at λP = 1480 nm is coupled to the resonator via a half-tapered fiber. Before starting the frequency sweeping of the probe signal, we identify and choose a resonance of the microcavity by analyzing its intrinsic emission with an optical spectrum analyzer (OSA). The probe signal reflected from the coupler C to the OSA is used to tune the probe Laser frequency to the chosen microsphere resonance. A narrow band tunable Laser operating around λS = 1560 nm is used as the frequency linearly scanned probe signal. The sweeping speed is around VS ≈ 2π × 1 MHz/μs which approximately corresponds to 4V0 for a quality factor Q = 109. The input probe signal is coupled to the WGM resonator via a tapered fiber. The coupling conditions (τe) can be adjusted by varying the distance between the microsphere and the taper. The mode matching between the pump and the probe waves is optimized by selecting the pump wavelength in the vicinity of the probe, here 1480 nm and 1560 nm. The signal extracted from the microsphere is first optically amplified by an erbium-doped fiber amplifier and then filtered with a narrow bandpass tunable filter (see section Methods). An amplified fast photodiode collects the output optical signal.

Figure 3
figure 3

Experimental setup.

ISO: optical isolator, VA: variable attenuator, OSA: optical spectrum analyzer, C: 3 dB coupler, PC: polarization controller, EDFA: Erbium-doped fiber amplifier, F: optical filter, PD: Amplified photodiode. PP and PS are respectively the pump and signal power at the taper inputs.

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Measurements

In Fig. 4 and Fig. 5 the ringing profiles T(t) obtained for two erbium-doped microspheres with two different concentrations and diameters D are shown. In these experiments we applied a pump power PP (defined in Fig. 3) between 3 mW and 11.4 mW. The power of the probe signal PS was chosen between 115 nW and 550 nW in order to avoid gain saturation. The ringing profiles are fitted using the analytical expressions given previously and the procedure described at the section Methods. As explained before, from the fit of the curves, we deduce τ0, τe as well as a (relatively) precise calibration of the frequency sweeping rate VS. The values of the two cavity lifetimes τ0 and τe fully characterize the linear properties of the microresonator, they give the coupling regime, the intrinsic Q-factor Q0 = ω0τ0/2, the loaded Q-factor Q, the resonant transmission (or gain) and the resonant group delay. All the experimental data have been gathered in Tab. I and plotted in the graph at Fig. 6 which shows the resonant transmission and group delay as a function of the intrinsic photon lifetime.

Table 1 Inferred linear parameters obtained from fitting of ringing profiles given in Fig. 4 and Fig. 5
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Figure 4
figure 4

Ringing profile recorded for two ZBLALiP microspheres for a 0.08 mol.% erbium-doped, D = 90 μm diameter microsphere.

The resonant transmissions T(ω0) and resonant group delays τg(ω0) deduced from the fits of the experimental curves A, B, E, H and I are reported in Fig. 6.

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Figure 5
figure 5

Ringing profile recorded for two ZBLALiP microspheres for a 0.1 mol.% erbium-doped, D = 145 μm diameter microsphere.

The resonant transmissions T(ω0) and resonant group delays τg(ω0) deduced from the fits of the experimental curves C, D, F, G, J and K are reported in Fig. 6.

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Figure 6
figure 6

Resonant transmission and group delay obtained from the fitting of the ringing profiles A-K provided in Fig. 4 and Fig. 5.

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Discussion

With the various microspheres all the different coupling regimes have been reached. For example, plot D of Fig. 5 shows a nearly perfect case of a critically coupled resonator. In the stationary regime such a resonator used as a notch filter would give a rejection of about 52 dB. From the fit of plot H (Fig. 4) we deduce an intrinsic Q-factor Q0 = 1.1 × 1010, for this resonator this corresponds to a loss-limited finesse F0 = λ0Q0/(nπD) ≈ 4.0 × 107 or a single-pass intensity absorption 1 − a2 = 2π/F0 = 1.6 × 10−7 assuming a refractive index n = 1.5 and λ0 = 2πc/ω0 = 1560 nm. In the case of a nearly transparent cavity as shown in plot I (Fig. 4) where , the stationary transmission spectrum would be T(ω) = 1 as theoretically shown in Fig. 2(e), unfortunately the phase shift properties of the resonator could not be measured. Using the transient analysis we can extract the coupling rate (τe = 1.5 μs) and deduce the dispersive properties of the system. Plot K (Fig. 5) is an example of a strongly resonant amplifier with a maximal gain of 19 dB associated with a bandwidth of 180 kHz. Finally, for a 220 μm diameter 0.1 mol.% erbium-doped microsphere we give in Fig. 7 a ringing profile corresponding to a loaded quality factor Q = 5.3 × 109 next to the critical coupling.

Figure 7
figure 7

Ringing profile measured from a 0.1 mol.% erbium-doped microsphere D = 220 μm diameter microsphere.

τ0 = 17.3 μs, τe = 17.5 μs and thus Q = 5.3 × 109.

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As a conclusion, we show here that the internal gain provided by the active medium of the cavity can be a way to compensate for glass-material limitation and permits the achievement of very high-Q optical microresonators up to 5 × 109. Depending on the coupling regime, erbium-doped microspheres can carry out diverse functions. i) At critical coupling, they can be used as a very selective notch filters with high rejection ratio. The frequency bandwidth of such a filter is 2/(πτ0). For a passive microresonator, it is thus limited by optical losses and set for a given structure whereas in the case of an active system, the bandwidth can be adjusted by tuning the pumping rate of the resonator. ii) Loss-compensated (Q0 → +∞) cavities could be used as transparent delay lines with a maximal delay τg(ω0) = 2τe that can be changed by modifying the coupling rate between the resonator and the access line. We have also shown that the cavity-ring-down-spectroscopy applied to loss-compensated resonators permits to measure their linear properties which is not possible when simply measuring the stationary power transmission T(ω). Moreover, very small absorption or ultrahigh intrinsic Q-factor Q0 can be measured with this method which implies possible applications in the domain of very sensitive sensors. iii) Finally in the spectral selective amplification regime, erbium-doped WGM resonator could be used as narrow-band amplifying filter for microwave or high-coherence photonic applications8,10.

Methods

Microsphere fabrication

The process is based on a microwave plasma torch which enables us to produce microspheres by melting of ZBLALiP glass50 powders. The plasma is generated using a microwave supply with a nominal oscillator frequency of 2.4 GHz and a power of 160 W. Argon (3 L/min output) is used as plasma gas and oxygen (4 L/min output) as sheath gas. Powders are axially injected and melt when passing through the plasma flame. Surface tension generates their spherical shape. Free spheres are collected down the chamber. The diameter of the spheres depends essentially on the grain size and varies from 10 to 250 μm. The spheres are then glued at the tip of drawn optical fibers (20 μm) permitting an easy handling.

Description of the experimental setup

The two tapers have a diameter smaller than 2 μm. The position of the microsphere is set and the two tapers are mounted on 3-axis micro-positioning stages in order to control their relative distance (typically less than the wavelength). The probe Laser is an external cavity Laser diode in the C-band (λS ≈ 1560 nm) with a long-term linewidth of 150 kHz. The Laser frequency is linearly swept over 6 GHz using a 100 Hz triangle 18 V peak to peak signal. The input signal power PS (Fig. 3) is weak (a few hundred nW) in order to avoid gain saturation inside the microsphere. Moreover due to taper aging during the experiments, insertion losses are such as we have to use an EDFA as optical preamplifier. The EDFA is a low noise amplifier with a maximal output of 18 dBm. An optical filter (F) is used to reject the amplified spontaneous emission (both from the EDFA and the microsphere) and the scattered pump signal. It is tunable in the range 1450–1650 nm and it has a minimal bandwidth around 6.25 GHz. The photodiode has a bandwidth of 1.2 GHz with a transimpedance gain between 102 and 104 V/A and a rise time between 1.8 and 25 ns. The pump Laser is a multimode Laser diode emitting at λP ≈ 1480 nm with a maximal power of 150 mW.

Cavity lifetime measurement procedure

The experimental transmission signal Texp(t) is normalized using the off-resonance value. In all our experiments and thus the bandwidth of the resonator is lower than 200 kHz corresponding to 0.2 μs for a sweeping speed of 2π × 1 MHz/μs. From Fig. 4, Fig. 5 and Fig. 7 we can deduce that the off-resonance value is taken for a detuning corresponding to almost 10 times the resonator bandwidth. This leads to a relative error (compared to the maximal transmission) less than 1% even for strongly amplifying resonators. The experimental signal is then compared to the theoretical signal Ttheo(t) depending on τ0, τe and VS calculated from Eqs. (5) and (6). A nonlinear optimization procedure on τ0, τe and VS is used to minimize the value of σ2 which reads:

where N is the number of experimental sampling points. In the general case we have checked that the error on the measurements was less than 10%. For critical coupling (plot D), where the relative value of τ0 and τe is crucial for the determination of T(ω0), we carried out several runs of optimization with very different starting points. All the inferred data were different by less than 0.1%. The method has been validated using fiber resonators30 for which the coupling coefficients can be independently measured. For a long four-port fiber resonator51 (length of 20 m and two 99%/1% couplers), we have checked the consistency of the method for overall Q-factors up to 3.3 × 109.