February 15, 2005 / Vol. 30, No. 4 / OPTICS LETTERS 427

Observation of Kerr nonlinearity in microcavities at roomtemperature

Hossein Rokhsari and Kerry J. Vahala

Thomas J. Watson Laboratories of Applied Physics, California Institute of Technology, Pasadena, California 91125

Received August 19, 2004

We have devised and experimentally verified a method for observation of the optical Kerr effect in micro-cavities at room temperature. The technique discriminates against the much larger and typically dominantthermal component of nonlinearity by using its relatively slow frequency response. Measurement of the Kerrcoefficient or equivalently of the third-order nonlinear susceptibility of the cavity material is demonstrated fora silica microcavity. With this approach, useful information about the characteristic thermal response timein microresonators can also be acquired. © 2005 Optical Society of America

OCIS codes: 190.3270, 190.4360, 190.1900, 230.5750.

Ultrahigh-quality factor (high-Q) whispering gallerymode (WGM) microcavities (e.g., microspheres andmicrotoroids) are ideally suited for observing non-linear optical effects with extremely low (submicro-watt) threshold powers.1 – 5 The combination of highcirculating power made possible by the high qualityfactor and strong confinement of this power withinmode volumes of the order of hundreds of cubic mi-crometers leads to intercavity circulating intensitiesin excess of gigawatts per square centimeter withonly 1 mW of input power (assuming Q values ofapproximately 100 million in microtoroid resonatorswith principal diameters of �50 mm). This level ofintensity is sufficient to unveil nonlinear phenomenaeven in materials such as silica that are weaklynonlinear. The optical Kerr effect and a variety ofits applications have been the subject of numerousresearch activities around the world.6 The observa-tion of this phenomenon in microcavities, however, isproblematic, as it is accompanied by relatively largerthermal nonlinear effects.4,5,7 The refractive index ofsilica, which is also a function of temperature, variesas the circulating power is partially absorbed by themedium, and the cavity temperature subsequentlyincreases. This larger effect (thermo-optic effect)masks the Kerr-driven variations in the refractiveindex. Treussart et al.5 found evidence for Kerr bista-bility in microresonators by immersing microspheresin a superf luid helium bath at 2 K. This cryogenicsetup greatly reduces the sensitivity of WGM reso-nances to temperature and allows the Kerr effect tobe dominant and hence observable.

In this Letter we propose and experimentally verifya method for observing the optical Kerr effect in micro-cavities at room temperature. This technique alsoallows us to estimate the x �3� or equivalently the n2coeff icient in silica. The method relies on the rela-tively long response times of thermal effects comparedwith that of the almost instantaneous Kerr effect. Tomanipulate this distinction in favor of the Kerr effectwe use two beams (pump and probe) individuallyresonant with two WGM resonances of an opticalmicrocavity. Modulation of the pump power resultsin modulation of the cavity’s refractive index and

consequently in modulation of the resonant frequencyof the WGM with which the probe beam is in reso-nance. The probe beam’s power transmission beyondthe resonator waveguide junction therefore acquiresa strong Fourier component at the modulation fre-quency of the pump beam. This component can bemeasured with a lock-in amplif ier. When the pump’smodulation frequency becomes sufficiently faster thanthe thermal effects, the signal detected by the lock-inamplifier is due purely to the Kerr effect.

The modulated probe power (with frequency V) isgiven by the following equation:

DPVprobe

Pprobe� PV

pumpQtotalprobeQeff

pump2n2lpump

pn2VeffC�x� , (1)

where Pprobe is the probe power coupled into the reso-nator and PV

pump is the fraction of pump power coupledinto the cavity and carrying modulation frequency V.The modulated probe power (DPV

probe) depends on therefractive-index change in the cavity caused by themodulated pump power (PV

pump), which is proportionalto the n2 coefficient, the quality factor of the cavityat pump wavelength (Qeff

pump), and the cross sectionof the optical mode. The changes in refractive indexof the cavity result in variation of the probe power’stransmission, which depends greatly on the slope ofthe transmission profile versus the linewidth or, inother words, the quality factor of the WGM coupled tothe probe signal. The total quality factor of the op-tical mode at the probe frequency (nprobe) is given byQtotal

probe, which includes both intrinsic losses of the cav-ity and loading by the optical waveguide and can beobtained from linewidth measurements in the desiredcoupling regime. Transfer of modulation from pumpto probe beam also depends on detuning of the cav-ity’s resonant frequency from that of the probe beam,which, measured in units of cavity linewidth, is de-noted x. The functional form of this dependence isgiven by C�x� � 2x�1 1 x2, a number between 1 and 0depending on the value of x. The effective qualityfactor (Qeff

pump, defined below) of the resonator at thepump wavelength (lpump) determines the enhancement

0146-9592/05/040427-03$15.00/0 © 2005 Optical Society of America

428 OPTICS LETTERS / Vol. 30, No. 4 / February 15, 2005

of the pump power in the cavity8:

Pcavitypump

Ppump�

lpumpQeffpump

p2nR, (2)

where R and n denote the radius and the refractiveindex, respectively, of the cavity. Equation (3) belowshows how Qeff

pump can be obtained from coupling pa-rameter K, defined as K � Q0�Qext, where Q0 is theintrinsic quality factor of the optical mode and Qext isthe quality factor associated with the coupling to theoptical waveguide:

Qeffpump � Q0

K�1 1 K�2

. (3)

Finally, note the role of small mode volumes (Veff ) inEq. (1) in reducing the required optical power for ob-servation of nonlinear effects.

The experimental setup that we used to observe andmeasure the Kerr nonlinearity of silica microcavitiesis shown in Fig. 1. Two tunable external-cavity diodelasers were used to generate pump and probe beamsin the 1550- and 1480-nm bands, respectively. Thepump and probe laser frequencies are simultaneouslytuned to two optical WGM resonances of a toroidalmicrocavity9 and launched into a tapered opticalfiber by a WDM coupler. Tapered optical f iberswith waist diameters of 1 3 mm were fabricatedby the traditional technique of simultaneous f lameheating and pulling of standard single-mode (SMF-28)fibers.10 Efficient and ideal coupling to the opticalmodes of microtoroid cavities is possible with thesewaveguides.11 A top-view optical micrograph of a mi-crotoroid evanescently side coupled to a tapered opticalfiber is shown in Fig. 1. The output of the 1550-nmlaser is modulated with a Mach–Zehnder modulator(bandwidth, 20 GHz) driven by a rf function generator(Func. Gen.) that generates sinusoidal waves withfrequencies as high as 80 MHz. Because of residualtransmitted modulated pump power in the probechannel, the detected signal could potentially maskthe relatively small modulation in the probe power,and therefore two cascaded wavelength-selectivecouplers are used to ensure high extinction (above38 dB) of the pump power in the probe channel.The pump and probe laser frequencies are adjustedwhile the detected pump and probe transmissionsare monitored by an oscilloscope to ensure that bothfrequencies are tuned to the desired proximity of theresonant frequencies of the cavity. The probe laserwas slightly detuned from resonance to maintain 50%transmission through the junction [i.e., C�x� � 0.8].The microtoroid is coupled almost critically to bothpump and probe signals in these experiments. Ahigh-precision piezoelectric transducer stage (10-nmsteps) was used to position the microtoroid relative tothe tapered fiber and thereby to control the couplingto the resonator. With the laser frequencies tunedto proximity of the desired WGM resonances andwith the pump power modulated, the probe power

is detected and the photocurrent is processed bythe lock-in amplifier with the output of the functiongenerator as its reference signal. The amplif ier wasoperated in the �r, u� setting, where r (in volts) and u

(in degrees) are the amplitude and the relative phaseof the detected signal at the reference frequency (V).

The output of the lock-in amplifier (r, in microvolts)is plotted in Fig. 2 versus the modulation frequency.This plot is normalized to the frequency response of theentire system with the microtoroid decoupled from thefiber taper. Therefore the plot shown in Fig. 2 isthe pure response that results from cross modulationof the probe beam by the pump beam. Thermaleffects can follow the modulation of the pump powerup to �V � 10 kHz; hence a f lat response is observed

Fig. 1. Experimental setup used for characterizing theKerr nonlinearity of microcavities. The picture is a top-view optical micrograph of a toroid microresonator that hasbeen evanescently side coupled to a tapered optical f iber.Laser sources at 1550- and 1480-nm bands are used forpump and probe beams, respectively.

Fig. 2. Measured amplitude modulation of the probe beamas a function of the modulation frequency of the pumppower. The dotted lines show a 3-dB corner frequency of�25 kHz where the modulation of the pump power becomescomparable to or faster than the thermal response time ofthe resonator. The second roll-off is due to limited band-width of the cavity, which does not allow the pump powerin the resonator to build up instantaneously. The f lat re-sponse in the middle shows the fast Kerr effect.

February 15, 2005 / Vol. 30, No. 4 / OPTICS LETTERS 429

below this frequency. At higher frequencies, however,a decline in the amplitude response is observed, witha 3-dB corner frequency of �25 kHz, suggesting athermal response time of the order of 6 ms. As wasmentioned in Ref. 4, the fastest thermal response timeis associated with the conduction of heat generated inthe optical mode volume to the cavity bulk materialand can be put into the following compact form:

tthermal ��dR�2

D�

1D

µ2Rl2

p2n2

∂2�3, (4)

where dR is the effective thickness of the optical modelocalized near the surface of the cavity and D is thetemperature conductivity of glass (8 3 1023 cm2�s).For the microtoroid under study here with a major ra-dius of 36 mm and a minor diameter or thickness of�8 mm, the toroid geometry becomes similar to thatof a sphere with respect to thermal effects within themodal volume. Applying Eq. (4) to this sample pre-dicts a thermal time constant of 5 ms, in close agree-ment with the value derived from Fig. 2.

Returning to Fig. 2, we can see that the decline in theamplitude response continues until it reaches a secondplateau from �600 kHz to 10 MHz that is far beyondthe modulation speeds that thermal effects can follow.To ensure that this f lat response was not due to thenoise level that arose from residual pump power in the1480-nm channel or from rf cross talk, we switchedoff the probe power during data acquisition in the f latportion of the spectrum. This resulted in a decreasein the lock-in response by a factor of 40, thereby con-firming that the measured response is associated withmodulation of the probe wave. Also, when the probebeam was detuned far away from resonance [C�x� � 0],the signal dropped to the noise level, demonstratingthat cross modulation of pump and probe occursonly in the cavity. Furthermore, by calibrating thelock-in amplifier we could infer the modulated probepower (i.e., DPV

probe). Plugging this inferred power intoEq. (1), we obtain an n2 value of �6 6 4� 3 10216 cm2�W,which agrees well with measured n2 values for fusedsilica (n2 � 3 3 10216 cm2�W). We therefore at-tribute the f lattened response region in Fig. 2 to Kerrnonlinearity.

The optical modes coupled to the pump and probebeams had total quality factors of 107 and 2 3 106,respectively. The power levels coupled into the reso-nator to generate the curve in Fig. 2 were �15 and�10 mW for pump and probe beams, respectively.Equation (1) shows that the signal level at frequencyV can be preserved for lower-Q optical modes at a costof higher optical input powers. We confirmed this factby repeating the same measurement for lower-qualityfactor WGM resonances both within the same cavityand within other microtoroids. Not only has the f lat

Kerr response been observed and the inferred n2coefficient obtained in lower Q samples but also theprobe-to-pump power ratio has been reduced to lessthan 5%, which completely rules out the possibility offeedback from the probe to the pump.

The f lat Kerr response in the plot ultimately falls offbecause of the limited bandwidth of the optical mode.The optical field in the cavity is built up in a finiteperiod that is roughly equal to the inverse of the op-tical mode linewidth. At modulation frequencieshigher than the cavity linewidth the modulation of thepump power cannot fully build up within the cavityand be transferred to the probe beam. The WGMexcited by the pump in the sample shown in Fig. 2 hada loaded Q of �107, which translates to a linewidthof �20 MHz at 1550 nm. By using lower-Q opticalmodes, on the other hand, we could extend the Kerrplateau to higher frequencies.

In conclusion, in the present research we havedemonstrated a powerful approach to observing Kerrnonlinearities in microcavities with extremely lowoptical powers. The simple experimental setup androom-temperature operation of these measurementsare attractive for the study of nonlinear phenomenain microcavity structures. The method discussed iscapable of measuring the nonlinear susceptibility ofthe cavity material and can be used to characterizethe n2 coefficients of various materials packaged in amicrocavity type structure.

This research was supported by the Defense Ad-vanced Research Projects Agency, the Caltech LeeCenter, and the National Science Foundation. H.Rokhsari’s e-mail address is rokhsari@caltech.edu.

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