Supplementary information for
Optically Induced Transparency In a Micro-cavity
Yuanlin Zheng1, Jianfan Yang2, Zhenhua Shen1, Jianjun Cao1,
Xianfeng Chen1*, Xiaogan Liang3, and Wenjie Wan1,2*
1MOE Key Laboratory for Laser Plasmas and Collaborative Innovation Center of IFSA,
Department of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China
2The University of Michigan-Shanghai Jiao Tong University Joint Institute,
The State Key Laboratory of Advanced Optical Communication Systems and Networks,
Shanghai Jiao Tong University, Shanghai 200240, China
3Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109, USA
*Correspondence and requests for materials should be addressed to:
Wenjie Wan () or Xianfeng Chen ()
This document contains supplementary information to the manuscript, where we demonstratedthe nonlinear coupling of different order WGMs in a Kerr microcavity.The effect is realized in a silica microsphere by scanning FSR spectrum of signal light with pump injection under degenerate four-wave mixing condition. This result is an altered FSR transmission of signal light to an EIT-like spectrum in certain conditions.
- Four-wave mixing in WGM microcavities
For a degenerate FWM as considered here, the governing equations are presented in the temporal domain[1]. This is helpful for the treatment of the coupled mode equations for FWM in microcavities, as will be discussed later.Assuming small-signal approximation(i.e., the pump field is not depleted), the coupled mode equations for degenerate FWMin microresonators(propagating in the same direction) are simplified as[2]:
, (1a)
, (1b)
, (1c)
Here, the subscripts represent pump, signal and idler waves, respectively. are the field amplitudes of each wave, with the modal effective area. is the nonlinear refractive index.and stand for frequency and angular momentum detuning, respectively. And is the azimuth angle.For weak signal and idler (the pump is considered non-depleted), we include the cross phase modulation (XPM) and four-wave mixing (FWM) terms. The XPM term only causesa phase shift and does not contribute to energy coupling of the waves.
When focusing on the specific scheme in our experiment, we onlyconsider the phase matched condition in the microcavity. In strong pump circumstance (),the coupled equations for phase-matched FWM can be further reduced as:
, (2a)
, (2b)
whereis the pump power. is the nonlinear coefficient, with be the nonlinear refractive index and the modal effective area. Here, the phase of pump is set to be zero, thus term is also replaced by pump power.The equations showthat the signal and idler light is coupled whenin the presence of pump.The XPM term only cause a frequency shift and is separated from FWM coupling term, thus they can be treated separately.We can define the XPM coefficient as , and the FWM coupling coefficient . The coefficients differ by a factor of 2.
- FWM phase matching condition in WGM microcavities
For FWM to be efficient in microcavity, the phase matching condition must be satisfied. In WGM microcavities, such as microsphere, angular momentum is conserved intrinsically when signal and idler are located at resonances symmetrical with respect to that of the pump, that is,.
Nonlinearity in FWM also alters mode structure.The index change of signal and idler induced by XPM is twice that of SPM on pump. The result is that two cavity lines can only be dispersion-compensated about a pump cavity mode in zero or anomalous dispersion region[3].The variation of FSR needs to be positive. The condition restricts the size of silica microsphere.For signal wave of factors of FSR away from pump resonance, it requires to be smaller than resonance width for phase-matched process to occur.
- Theoretical modal of WGM mode coupling via FWM in microresonator
The origin of EIT-like structure in the transmission spectrum of coupled resonators lies in the interference of the cavities’ decays[4].By referring to Eqn. (2) and coupled mode theory, the internal cavity fieldof signal and idler in a microresonatorcan be re-written in the follow form:
, (3a)
. (3b)
Here, the subscripts represent pump, signal and idler waves, respectively. isthe normalized amplitude of a resonator mode, andis equal to the total energy stored in the cavity of a WGM.is the input signal wave.denote the intracavity and external cavity decay rates, respectively.is the frequency detuning of the wave from its cavity resonance (). The detuning andis dependent, which is restricted by energy conservation.isrelated to FWM XPM, which induced a nonlinear phase shift during FWM and is included in the frequency detuning parameter. is the nonlinear coupling strength between signal and idler modes, which formulates the interference of the cavities’ decays. The value of is half of the FWM parametric gain (See Eqn. (2)). Thus,it is possiblewhere one can research both weak and strongcoupling conditions in one microcavity.The input signal is small compared with the pump so as the small-signal approximation holds. When pump light is not present, the nonlinear coupling vanishes (). The coupling equations degenerate into a single WGM coupling occasion. Here,the coupling between the two WGMs (of different frequencies) is realized via nonlinear wave mixing.Whilesituations in coupled resonators or the likewhere two WGMs are coupling in linear regime, the coupled resonances need to overlap in frequency, which poses difficulty in finding a co-resonant wavelength.
By considering the steady sate of the system (), the internal cavity mode of signal light can be solved as
,(4)
where.The output of the signal is then calculated as
.(5)
The transmission spectrum of signal is givenby.
We can also find the relation of idler output with respect to input signal to be
.(6)
And, the transmission of idler normalized with input signal is givenby.
Under zero dispersion situation, the frequency detuning for the two resonances is equal but with opposite sign. When we scan the signal wavelength in one direction, the energy conservation forces the wavelength of idler to sweep in the opposite direction.The transmission spectra are shown in Fig. 2 in the manuscript. The modal presented here is general, and is not limited to FWM.
Here the third-order nonlinear susceptibility is linked to the nonlinear coupling term g in Equation (1). However it takes some efforts to convert to g due to cavity enhancement through a set of equations:, and, where the pump power in the cavity (), the external pump power (), the finesse of the cavity (), nonlinear refractive index coefficient (), and modal effective area of the WGMs () can be estimated experimentally or numerically. For the current experimental setup, the external input pump power is around 3mW, with around 20% coupling efficiency through the taper fiber, resulting . F can be measured experimentally as . of fused silica material can be calculated through given the value of third-order nonlinear susceptibility is [2]The effective area can be numerically calculated through a finite element method as shown in Fig. S1 to obtain . As a result, the calculated value of g is 0.46, close to the value we use for theoretical simulation.
Figure S1.Electric field distribution of WGM of the microsphere (diameter 250 um, wavelength 1550 nm).
For simplicity and without loss of physics, the frequency shift due to XPM in FWM process can be ignored. The frequency detuning of signal and idler is reduced as: , where is the center of each WGM resonance. The energy conservation imposes the constrain on and by .Thus, the difference of frequency detuning of and may occur intwo cases as depicted in Fig. S2: (1) non-equidistantly distributed WGM resonances, where the centers of resonances are not symmetrically located (). The microcavity may be called a non-phase matched resonator. (2)shift of the pump wavelength off its resonance center .
Fig. S2: Schematic of the definition and shift relation of each frequency detuning.
- OIT Linewidth
The transmission of the signal reads: , where . is total decay rate. Consider the case of zero detuning when and assume the two decay rates of the signal and the idler are the same: , the transmission can be rewritten as
(7)
In the linear case where, the transmission can be reduced to a Lorentzian form , the corresponding linewidth of the resonance dip is given by .
In the limit of where OIT occurs when the frequency detuning is relative small, the transmission can be read as:
(8)
which is still a Lorentzian form, its linewidth reads:
(9)
Equation(9) here gives a good estimation of the experimental linewidth. For example, (1) When , it reduces to as in the linear case without any gain. (2) When s small, the linewidth is narrowing with increasing , perfectly explaining the experimental observation in Fig. 2c. (3) When the gain becomes larger than the decay (loss) , the linewidth function becomes : , where OIT appears. Furthermore, when , the linewidth is approaching zero, this explains the linewidth narrowing in the Fig.2d. (4) At last, when , becomes again, indicating the peak like in Fig.2e has a linewidth limit depending on its original linear decay rate. Also in this case, there will be significant nonlinear effect contributed into the system due to the increasing intensity, which may further narrow the linewidth, similar to the process of a laser.
- Simulation results
To plot the transmission spectra of signal, Equ. 5 is used in the calculation. The correspondingsimulation parameters are set as follows:
- for both resonances at signal and idler waves. The FWHM of the resonance dip at critical coupling.
- is determined by FWM, see FWM coupling equations. The value of , as shown in the coupling equation, indicates the coupling strength and is proportional to the intensity of pump intensity, and thus can be controlled by the later. The factor.
- denotes the frequency detuning with respect to their own resonances. The difference of frequency detuning ofandmay be introduced due to non-equidistantly distributed WGM resonances or the shift of pump wavelength.
Figure S3 gives the simulation results at typical conditions of critical coupling and OIT. As shown, the phase response across the resonance is not distorted during the OIT process. Hence, the slow light effect (group velocity lower than c) can only be multiplied around 2-3 times, much less the atomic counterpart in EIT. Moreover, the fast light effect (group velocity faster than c) is absent in the current setup.
Figure S3.1 | The condition when, which corresponds to critical coupling of signal without pump. Singularity at zero occurs due to discrete numerical calculation.
Figure S3.2| The condition when
- Dispersion in WGM microsphere
The dispersion of microsphere can be estimated by considering its material dispersion and geometric dispersion. The model for the dispersion of WGM microresonators has been well established in Ref. [5, 6]. The variation of FSR induced by material dispersion is
. (7)
Geometric dispersion of a WGM microsphere is given by . Here, is the refractive index, is the radius of the microsphere.is the group velocity dispersion parameter and is positive at for fused silica. The microspheres show rather weak dispersion at around 1550 nm for sizes used in our experiments, as shown inFig. S4. isless than 1 MHz at 1550 nm for radius larger than.
Fig. S4: Variation of FSR of silica microspheres.
The resonance linewidth of the microspheres in the experiment is about 20 MHz. Thus the resonance detuning between signal and idler is close to zero even in dozens FSRs range. In the experiment, the detuning of the two resonances is achieved by detuning the pump frequency.
- Experiment details
Fig. S5. Experimental setup.
The experimental setup is shown inFig. S5, a narrow-linewidth tunable laser (TLB-6700 Velocity) is used as the pump source. Another narrow-linewidth tunable laser (Agilent 81682A) is used as the input signal light source. Both are polarization controlled and evanescently coupled into a silica microsphere through a tapered fiber. The microsphere is fabricated by electric arc fusing the tip of a stripped standard single mode fiber (SMF). The sizes of microspheres are selected to avoid normal dispersion. The microsphere is mounted on a nano 3D transducer stage for precise position controlling. The diameter is measured to be 265 , corresponding to an FSR of 2.15 nm at 1550 nm. Anomalous dispersion condition for FWM at 1550 nm requires that the diameter of the microsphere to be larger than 136 [3]. The taper fiber is also self-made by heating-and-pulling method. Efficient coupling requires the effective index of tapered fiber equals that of the excited WGMs, and is experimentally achieved by scanning the microsphere along the fiber tapering region. The transmitted light is split by a coupler, with one arm detected by an optical spectrum analyzer and the other filtered by a commercial CWDM (channel span 20 nm). Firstly, we scan the pump in very low input intensity to measure Q factors and determine free spectral range (FSR) spectra of the cavities. The amplifiedpump light is locked to a resonant mode of the microsphere using thermal self-locking effect by scanning its frequency from high to low frequencies. Successful mode locking is confirmed by both monitoring the transmission power and the generation of stable frequency comb, which also indicates the fulfillment of phase matching condition for FWM. Another purpose of comb generation is that one can determine precisely the wavelengths of WGM resonances at other frequencies. This is utilized to make sure that the frequency swept signal light overlaps a desired resonance.One generated frequency comb spectrum is shown inFig. S6. The signal light is then swept around an adjacent comb line (multiple FSRs away from pump)to make sure wavelength overlapping and gradually decrease the pump power under comb generation threshold. Finally, each wavelength is separated by the CWDM and detected independently. During the experiment, the polarization of each wave is controlled for optimal performance.
Fig. S6. Experimentally measured frequency comb spectrum.FSR = 2.1 nm. The pump wavelength is 1548.52 nm. Pump power is 6.8 mW, slightly above threshold.OSA resolution is 0.02 nm.
In our experiment, the pump and another two frequency lines are chosen to be 1535.64, 1548.52 and 1561.68 nm. The wavelength of signal is repeatedly scanned around a frequency comb line at 1535.64 nm. This is to avoid florescence background of the EDFA for signal detection and also for CWDM filtering. The wavelength overlapping is confirmed by an optical spectrum analyzer (OSA) with a resolution of 0.02 nm. This is one method to make sure the scanned modified spectrum is due to FWM, other than mode splitting. Launching the signal at input power of 0.1 mW, the pump intensity is then gradually decreasedunder the comb generation threshold. The spectrum is shown in Fig. S7, which proves the occurrence of FWM in the microsphere and no other frequencies are generated.
Fig. S7: Experimentally measured spectrum of FWM waves. Signal wave is repeatedly sweeping across its resonance. The linewidth of idler wave seems narrower, because FWM only occurs when signal wave is resonant and the data is occasionally recorded. OSA resolution is 0.2 nm.
The signal and generated idler after filtering out by a CWDM is simultaneously detected and monitored by an oscilloscope. The measured transmission spectrum is shown in Fig. 3 and 4 in the manuscript. As the frequency of signal is scanned through the resonance, there is a peak in the resonance dip and at the same time the generation of idler. The offset of the idler intensity is the unfiltered fluorescence background. The result verifies that the modified transmission spectrum originates from the FWM process.
As shown inFig. S4, the actual dispersion in large silica microsphere is actually quite small for WGMs only several FSRs apart. In the observation of Fano-like effect experiment, the pump wavelength is tuned. However, the actual pump wavelength shift is much larger than the offset (MHz) as show in Fig. 3 in the manuscript. This is because the thermally locked pump would pull the resonances along with it when its wavelength is shifted. The peak of idler transmission is synchronized with the peak of signal as observed in the oscilloscope in the experiment. Thus, the frequency detuning of idler wave is flipped in accordance with theoretical prediction.
Besides, as can be seem from Eq. 2, the effect is not only limited to FWM demonstrated here. Other nonlinear effect in WGM resonators may also have similar effects. The exploration of FWM in microresonators here has its advantages for optical network compatibility. Moreover, the signal and control light is separable, which is more favorable for all-optical switching and processing.
Reference
1.Kippenberg, T.J., Nonlinear optics in ultra-high-Q whispering-gallery optical microcavities 2004, California Institute of Technology.
2.Boyd, R.W., Nonlinear Optics (Third Edition). 2008: Academic Press.
3.Agha, I.H., et al., Four-wave-mixing parametric oscillations in dispersion-compensated high-Q silica microspheres. Physical Review A, 2007. 76(4): p. 043837.
4.Maleki, L., et al., Tunable delay line with interacting whispering-gallery-moderesonators. Optics Letters, 2004. 29(6): p. 626-628.
5.Del'Haye, P., et al., Optical frequency comb generation from a monolithic microresonator. Nature, 2007. 450(7173): p. 1214-1217.
6.Del'Haye, P., Optical Frequency Comb Generation in Monolithic Microresonators. 2011.