Supplementary Information for Mid-Infrared Optical Frequency Combs at 2.5 µm based on Crystalline Microresonators

Transcription

1 1 Supplementary Information for Mid-Infrared Optical Frequency Combs at 2.5 µm based on Crystalline Microresonators C. Y. Wang 1,2,3,, T. Herr 1,2,, P. Del Haye 1,3,7, A. Schliesser 1,2, J. Hofer 1,6, R. Holzwarth 1,3, T. W. Hänsch 1,4, N. Picqué 1,4,5*, T. J. Kippenberg 1,2* 1 Max-Planck Institut für Quantenoptik, Hans-Kopfermann Strasse 1, D Garching, Germany 2 École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland 3 Menlo Systems GmbH, Am Klopferspitz 19a, D Martinsried, Germany 4 Ludwig-Maximilians-Universität München, Fakultät für Physik, Schellingstrasse 4/III, München, Germany 5 Institut des Sciences Moléculaires d Orsay, CNRS, Bâtiment 350, Université Paris-Sud, Orsay, France 6 Current affiliation: Paul Scherrer Institut, 5232 Villigen PSI, Switzerland 7 Current affiliation: National Institute of Standards and Technology, 325 Broadway, Boulder CO 80305, USA These authors contributed equally to this work * To whom correspondence should be addressed. nathalie.picque@mpq.mpg.de and tobias.kippenberg@epfl.ch

2 2 SUPPLEMENTARY FIGURES Supplementary Figure S1. Thermal Instability in CaF 2 resonators. (a) Experimentally observed oscillatory behaviour in cavity transmission in a CaF 2 resonator (Q~2 10 8, radius~2 mm) when the pump laser frequency is fixed and near a resonance. This means the cavity resonance oscillates with time. (b) Simulated cavity transmission based on the model of two different time scales for thermal refractive index effect and thermal expansion effect and a negative dn/dt of the material.

3 3 SUPPLEMENTARY TABLES Supplementary Table S1. Materials properties studied in this work. Here we use a more conservative estimation as high-q resonators are needed. Transparency ( a Bulk material GVD dn/dt Window in the mid-ir b (10-5 1/K) c Fused silica (SiO 2 ) 0.19 to 2.4 µm Silicon Nitride (Si 3 N 4 ) 0.5 to 6 µm normal >0 CaF to 7 µm anomalous -1.1 MgF to 6 µm anomalous 0.1 BaF to 10 µm anomalous LiF 0.2 to 5 µm anomalous -1.6 NaF 0.16 to 8 µm anomalous -1.3 Sapphire (Al 2 O 3 ) 0.25 to 4.5 µm anomalous 1 Quartz Crystal 0.18 to 3.5 µm anomalous -0.5 a We use the number from Ref. [42] for fused silica. Silicon nitride data is from Ref. [43]. CaF 2, MgF 2, BaF 2, LiF and NaF data are from Ref. [44] and [45]. Sapphire data is from [46]. Quartz crystal data is from Ref. [45]. b. A positive GVD is anomalous. can be obtained from c Fused silica, CaF 2, MgF 2, sapphire and quartz crystal data are from Corning datasheets. BaF 2, LiF and NaF data are from Crystran Ltd datasheets. No number for silicon nitride was found in literature, but from experimental experience it should be positive. All data were measured at room temperature.

4 4 Supplementary Table S2. Material properties of several semiconductor materials transparent in the mid-infrared. Bandgap E g ( Bulk material GVD in the mid- IR a dn/dt (10-5 1/K) b Silicon (Si) 1.1 ev [47, 48] normal 17 Germanium (Ge) 0.66 ev [49] normal 45 Aluminum Nitride 6.2 ev - Anomalous 2-3 (AlN) Gallium Arsenide (GaAs) 1.42 ev - Anomalous above 7 µm 14.7 Indium Phosphide (InP) 1.34 ev - Anomalous above 5.5 µm 20 a. A positive GVD is anomalous. can be obtained from b Si and Ge data are from arxiv:physics/ , which is consistent with the numbers in Crystran Ltd datasheets. AlN data is from Ref. [50]. GaAs data is from Crystran Ltd datasheet. InP data is from Ref. [51].

5 5 SUPPLEMENTARY NOTE Thermal instability When light is coupled into a whispering-gallery mode resonator, the absorption of photons causes an increase in temperature which induces a resonance frequency shift. This resonance frequency shift comes from a change in optical path length with temperature and has two contributions: The temperature-dependent refractive index ; Thermal expansion the cavity radius. It can be easily shown that the temperature-dependent resonance wavelength can be approximated by, where denotes the cold cavity resonance wavelength. For materials which have both positive dn/dt and α like fused silica, heating of the cavity would shift the resonances to higher frequencies. Therefore, scanning the pump laser from a lower to higher frequency would lead to an artificial broadening of the resonance (triangular shape), whereas scanning from a higher to lower frequency would lead to an artificial narrowing of the resonance [30]. The triangular-shaped broadening of the resonance provides a thermal feedback loop to lock the cavity resonance to the laser: As the pump frequency sits within the thermal triangle, frequency fluctuation in the pump induces power fluctuation in the cavity, which moves the cavity resonance in the same direction as the pump frequency. Therefore, the cavity resonance stays frequency locked to the pump. This thermal self-locking is the underlying mechanism that enables stabilization of Kerr combs [18]. Due to the thermal self-locking, one comb mode can be directly accessed via the frequency of the pump laser, whereas the mode spacing as a second degree of freedom is controlled by changing the optical path length of the microcavity via the pump power-dependent refractive index change of the resonator.

6 6 A very important aspect is that the thermo-refractive index change and thermal expansion can happen on very different time scales. Thermo-refractive index change depends on the relative temperature of the optical mode volume and is nearly instantaneous to heating by absorption; for thermal expansion the whole resonator radius has to be taken into account, which means the time scale is defined by the speed of heat conduction to the inner part of the resonator. Therefore, thermal expansion can be regarded as a slower effect compared to thermo-refractive effect. This is especially the case in crystalline resonators, since the radius of the cavity is in general much bigger than the radius of the optical mode. If dn/dt and α have opposite signs like in CaF 2 resonators, when the pump laser is set at a fixed frequency near the resonance, a thermal instability in the form of a reoccurring drop and rise in transmission is observed (Supplementary Figure S1 (a)). This blinking behaviour means the cavity mode is drifting back and forth with respect to the fixed pump laser frequency, so the pump laser oscillates between in- and out-ofresonance with time. Therefore, materials with opposite signs of dn/dt and α like CaF 2 do not allow for thermal self-locking and active locking scheme or injection locking is needed to lock the laser frequency to the cavity mode. However, active locking schemes such as Pound- Drever-Hall technique are usually limited to rather low pump power, and in this way one loses one degree of freedom to control and stabilize the comb. A similar oscillatory dynamics of a high-q PDMS-coated silica microtoroid is reported in Ref. [41] where the observed dynamics is explained by two competing thermal effects of different time constants. Here similar argument can also be used to model and simulate the oscillatory dynamics in resonators with opposite signs of dn/dt and α. The simulation is based on three coupled differential equations of two temperatures and (one of the mode volume and the other of the cavity on average) and the intra-cavity power. The two temperatures are functions of heat diffusion (with different time constants and ) and intracavity power, whereas the intracavity power is a function of the laser detuning and thermal resonance shift: (S1)

7 7 (S2) (S3) where Δ denotes the laser detuning. If 1 stands for slow thermal expansion and 2 for fast thermo-refractive effect, in the simulation we estimate the ratio of time constants to be, which corresponds to. We also assume that the ratio The coefficients b 1 and b 2 correspond to the resonance frequency shifts per Kelvin and the ratio can be approximated by. The linewidth κ is normalized to one. Supplementary Figure S1 (b) shows the simulated cavity transmission as a function of time, which shows good agreement with the observed dynamics.

8 8 SUPPLEMENTARY REFERENCES 41. He, L.N., et al., Oscillatory thermal dynamics in high-q PDMS-coated silica toroidal microresonators. Optics Express, (12): p Kim, K.S., et al., Measurement of the Nonlinear Index of Silica-Core and Dispersion-Shifted Fibers. Optics Letters, 19(4), (1994). 43. Ikeda, K., et al., Thermal and Kerr nonlinear properties of plasma-deposited silicon nitride/silicon dioxide waveguides. Optics Express, 16(17), (2008). 44. Milam, D., M.J. Weber, and A.J. Glass, Nonlinear Refractive-Index of Fluoride- Crystals. Applied Physics Letters, 31(12), (1977). 45. Adair, R., L.L. Chase, and S.A. Payne, Nonlinear Refractive-Index of Optical- Crystals. Physical Review B, 39(5), (1989). 46. Major, A., et al., Dispersion of the nonlinear refractive index in sapphire. Optics Letters, 29(6), (2004). 47. Zlatanovic, S., et al., Mid-infrared wavelength conversion in silicon waveguides using ultracompact telecom-band-derived pump source. Nature Photonics, 4(8), (2010). 48. Bristow, A.D., N. Rotenberg, and H.M. van Driel, Two-photon absorption and Kerr coefficients of silicon for nm. Applied Physics Letters, 90(19), (2007). 49. Sheikbahae, M., et al., Dispersion of Bound Electronic Nonlinear Refraction in Solids. Ieee Journal of Quantum Electronics, 27(6), (1991). 50. Watanabe, N., T. Kimoto, and J. Suda, The temperature dependence of the refractive indices of GaN and AlN from room temperature up to 515 degrees C. Journal of Applied Physics, 104(10), (2008). 51. Martin, P., et al., Accurate Refractive-Index Measurements of Doped and Undoped Inp by a Grating Coupling Technique. Applied Physics Letters, 67(7), (1995).