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*
1
MOE Key Laboratory for Laser Plasmas and Collaborative Innovation Center of IFSA,
Department of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China
2
The 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
3
Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109, USA
*Correspondence and requests for materials should be addressed to:
Wenjie Wan ([email protected]) or Xianfeng Chen ([email protected])
This document contains supplementary information to the manuscript, where we demonstrated the
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.
1. 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 FWM in microresonators
(propagating in the same direction) are simplified as [2]:
d𝐴𝑝
d𝑡
2
𝑛2 𝜔𝑝
= 𝑖 𝑐𝐴
𝑒𝑓𝑓
(|𝐴𝑝 | 𝐴𝑝 + 4𝐴𝑝 𝐴𝑝 𝐴∗𝑠 𝑒 iΔ𝜔𝑡 𝑒 iΔl𝜙 + 4𝐴𝑝 𝐴𝑝 𝐴∗𝑖 𝑒 iΔ𝜔𝑡 𝑒 iΔl𝜙 ) = 0,
(1a)
d𝐴𝑠
d𝑡
𝑛 𝜔𝑠
= 𝑖 𝑐𝐴2
𝑒𝑓𝑓
2
(4|𝐴𝑝 | 𝐴𝑠 + 2𝐴𝑝 𝐴𝑝 𝐴∗𝑖 𝑒 −iΔ𝜔𝑡 𝑒 −iΔl𝜙 ),
(1b)
d𝐴𝑖
d𝑡
𝑛 𝜔𝑖
= 𝑖 𝑐𝐴2
𝑒𝑓𝑓
2
(4|𝐴𝑝 | 𝐴𝑖 + 2𝐴𝑝 𝐴𝑝 𝐴∗𝑠 𝑒 −iΔ𝜔𝑡 𝑒 −iΔl𝜙 ),
(1c)
Here, the subscripts p, s, i represent pump, signal and idler waves, respectively. A′ s are the
field amplitudes of each wave, with 𝐴𝑒𝑓𝑓 the modal effective area. n2 is the nonlinear refractive
index. Δω = 2ω𝑝 − 𝜔𝑠 − 𝜔𝑖 and Δl = 2𝑙𝑝 − 𝑙𝑠 − 𝑙𝑖 stand for frequency and angular
momentum detuning, respectively. And ϕ 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 causes a phase shift and does not contribute to energy
coupling of the waves.
When focusing on the specific scheme in our experiment, we only consider the phase matched
condition in the microcavity. In strong pump circumstance (𝑑𝐴𝑝 /𝑑𝑡 ≈ 0), the coupled equations
for phase-matched FWM can be further reduced as:
dAs
dt
dAi
dt
= 2iγP(2As + A∗i ),
(2a)
= 2iγP(2Ai + 𝐴∗𝑠 ),
(2b)
2
where P = |Ap | is the pump power. γi =
n2 𝜔𝑖
𝑐𝐴𝑒𝑓𝑓
≈ 𝛾 is the nonlinear coefficient, with
n2 be
the nonlinear refractive index and Aeff the modal effective area. Here, the phase of pump is set to
be zero, thus Ap 𝐴𝑝 term is also replaced by pump power P. The equations show that the signal
and idler light is coupled when in 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 G = 4iγP, and the FWM coupling coefficient g = 2iγP. The coefficients
differ by a factor of 2.
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, 2βm −
(𝛽𝑚−𝑁 + 𝛽𝑚+𝑁 ) = 0.
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 Δωm = (𝜔𝑚+1 − 𝜔𝑚 ) − (𝜔𝑚 − 𝜔𝑚−1 ) needs to be positive. The
condition restricts the size of silica microsphere. For signal wave of N factors of FSR away from
pump resonance, it requires NΔωm to be smaller than resonance width for phase-matched
process to occur.
3. 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 field of signal and idler in a microresonator can be re-written in the follow form:
dAs (𝜔𝑠 )
d𝑡
∗
= (−𝜅𝑜𝑠 − 𝜅𝑒𝑠 + 𝑖Δ𝜔𝑠 )𝐴𝑠 (𝜔𝑠 ) + 𝑖√2𝜅𝑒𝑠 𝐴𝑖𝑛
𝑠 (𝜔𝑠 ) + 𝑖𝑔𝐴𝑖 (𝜔𝑖 ),
(3a)
dAi (𝜔𝑖 )
d𝑡
= (−𝜅𝑜𝑖 − 𝜅𝑒𝑖 + 𝑖Δ𝜔𝑖 )𝐴𝑖 (𝜔𝑖 ) + 𝑖𝑔𝐴∗𝑠 (𝜔𝑠 ).
(3b)
Here, the subscripts p, s, i represent pump, signal and idler waves, respectively. A is the
normalized amplitude of a resonator mode, and |A|2 is equal to the total energy stored in the
cavity of a WGM. Ain
s is the input signal wave. κ0 , 𝜅𝑒 denote the intracavity and external cavity
decay rates, respectively. Δω = ω0 − ω + G is the frequency detuning of the wave from its
cavity resonance (ω0 ). The detuning Δωs and Δωi is dependent, which is restricted by energy
conservation. G is related to FWM XPM, which induced a nonlinear phase shift during FWM and
is included in the frequency detuning parameter. g is the nonlinear coupling strength between
signal and idler modes, which formulates the interference of the cavities’ decays. The value of g
is half of the FWM parametric gain G (See Eqn. (2)). Thus, it is possible where one can research
both weak and strong coupling 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 (g = 0). 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. While situations in coupled resonators or the like where 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 (
dAs
dt
= 0,
dAi
dt
= 0), the internal cavity mode of
signal light can be solved as
i√2κes 𝑋𝑖∗ (Δ𝜔𝑖 )
As = − 𝑋 (Δ𝜔
𝑠
∗
2
𝑠 )𝑋𝑖 (Δ𝜔𝑖 )−𝑔
𝐴𝑖𝑛
𝑠 ,
(4)
where X(Δω) = −κ0 − 𝜅𝑒 + 𝑖Δ𝜔. The output of the signal is then calculated as
𝑖𝑛
Aout
s = 𝐴𝑠 + 𝑖√2𝜅𝑒𝑠 𝐴𝑠 = (1 +
2𝜅𝑒𝑠 𝑋𝑖∗ (Δ𝜔𝑖 )
) 𝐴𝑖𝑛
𝑠 .
𝑋𝑠 (Δ𝜔𝑠 )𝑋𝑖∗ (Δ𝜔𝑖 )−𝑔2
(5)
𝐴𝑜𝑢𝑡
2
The transmission spectrum of signal is given by Ts = | 𝐴𝑠𝑖𝑛 | .
𝑠
We can also find the relation of idler output with respect to input signal to be
Aout
= 𝑖√2𝜅𝑒𝑠 𝐴𝑖 =
i
2𝑔√𝜅𝑒𝑠 𝜅𝑒𝑖
𝑋𝑖 (Δ𝜔𝑖 )𝑋𝑠∗ (Δ𝜔𝑠 )−𝑔2
𝐴𝑖𝑛
𝑠 .
(6)
Aout
2
And, the transmission of idler normalized with input signal is given by Ti = | 𝐴i𝑖𝑛 | .
𝑠
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 χ(3) is linked to the nonlinear coupling term g in
Equation (1). However it takes some efforts to convert χ(3) to g due to cavity enhancement
through a set of equations: g = 2γP, P = 2π ⋅ F ⋅ Pin and γ ≈ n2 𝜔⁄𝑐𝐴eff , where the pump
power in the cavity (𝑃), the external pump power (𝑃𝑖𝑛 ), the finesse of the cavity (𝐹), nonlinear
refractive index coefficient (n2 ), and modal effective area of the WGMs (Aeff ) 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 Pin =
0.6 mW. F can be measured experimentally as 𝐹 = FSR⁄Δω = 262 GHz⁄20 MHz = 1.31 × 104.
3
n2 of fused silica material can be calculated through n2 = 2𝑛2 𝜖
0 0𝑐
𝜒 (3) = 3.2 × 10−16 cm2 /W
given the value of third-order nonlinear susceptibility χ(3) is 2.5 × 10−22 m2 /V 2 [2]The
effective area can be numerically calculated through a finite element method as shown in Fig. S1
to obtain Aeff = 27.6 μm2. 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: Δωs = ωs − ωso , Δωi =
𝜔𝑖 − 𝜔𝑖𝑜 , where ω0 is the center of each WGM resonance. The energy conservation imposes the
constrain on ωs and ωi by ωs + ωi = 2ωp. Thus, the difference of frequency detuning of Δωs
and Δωi may occur in two cases as depicted in Fig. S2: (1) non-equidistantly distributed WGM
resonances, where the centers of resonances are not symmetrically located (𝜔𝑠𝑜 + 𝜔𝑖𝑜 ≠ 2𝜔𝑝𝑜 ).
The microcavity may be called a non-phase matched resonator. (2) shift of the pump wavelength
off its resonance center ωpo .
Fig. S2: Schematic of the definition and shift relation of each frequency detuning.
4. OIT Linewidth
The transmission of the signal reads: Ts = |1 +
2κes X∗i
Xs
2
| , where X = −κ0 − κe + iΔω =
X∗ −g2
i
−κ + iΔω. κ is total decay rate. Consider the case of zero detuning when Δωs = −Δωi = Δ
and assume the two decay rates of the signal and the idler are the same: 𝜅s = κi, the transmission
can be rewritten as
2
Ts = |1 +
2κes
|
g2 κ
g2
−κ)+iΔ(1+ 2 2 )
κ2 +Δ2
κ +Δ
(
(7)
In the linear case where 𝑔 → 0, the transmission can be reduced to a Lorentzian form Ts =
|1 +
2𝜅𝑒𝑠 2
| , the corresponding linewidth of the resonance dip is given by Γ = 𝜅 .
−𝜅+𝑖Δ
In the limit of Δ2 → 0 where OIT occurs when the frequency detuning is relative small,
the transmission can be read as:
2
Ts = |1 +
2κes
g2
κ
( −κ)+iΔ(1+
|
g2
)
κ2
(8)
which is still a Lorentzian form, its linewidth reads:
κ2 −g2
Γ = κ κ2 +g2
(9)
Equation(9) here gives a good estimation of the experimental linewidth. For example, (1)
When 𝑔 → 0, it reduces to Γ = 𝜅 as in the linear case without any gain. (2) When 𝑔 is small, the
linewidth is narrowing with increasing 𝑔, perfectly explaining the experimental observation in
Fig. 2c. (3) When the gain 𝑔 becomes larger than the decay (loss) κ, the linewidth function
g2 −κ2
becomes : Γ = κ g2 +κ2 , where OIT appears. Furthermore, when 𝑔~κ , the linewidth is
approaching zero, this explains the linewidth narrowing in the Fig.2d. (4) At last, when 𝑔~∞, Γ
becomes 𝜅 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.
5. Simulation results
To plot the transmission spectra of signal, Equ. 5 is used in the calculation. The corresponding
simulation parameters are set as follows:
1. κ0 = 𝜅𝑒 = 1 for both resonances at signal and idler waves. The FWHM of the resonance dip
at critical coupling γ = 2.
2. G = 2g is determined by FWM, see FWM coupling equations. The value of g, 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 g = 2γP.
3. Δωi = −Δωs − 𝛿 denotes the frequency detuning with respect to their own resonances. The
difference of frequency detuning of Δωs and Δωi may 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 g = 0, δ = 0, which corresponds to critical coupling of signal
without pump. Singularity at zero occurs due to discrete numerical calculation.
Figure S3.2 | The condition when g = 0.415, δ = 0
6. 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
c2 𝜆2
𝑐𝜆3
d2 𝑛
ΔFSR ≈ 4𝜋2 𝑛3𝑅2 ⋅ 𝐺𝑉𝐷 = − 4𝜋2 𝑛3 𝑅2 ⋅ d𝜆2 .
(7)
0.41c
Geometric dispersion of a WGM microsphere is given by ΔFSR ≈ − 2πnR 𝑚−5/3. Here, n is the
refractive index, R is the radius of the microsphere. GVD is the group velocity dispersion
parameter and is positive at λ > 1.3 μm for fused silica. The microspheres show rather weak
dispersion at around 1550 nm for sizes used in our experiments, as shown in Fig. S4. ΔFSR is
less than 1 MHz at 1550 nm for radius larger than 100 μm.
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.
7. Experiment details
Fig. S5. Experimental setup.
The experimental setup is shown in Fig. 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 μm, 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 μm [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 1X2 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 amplified pump 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 in Fig. 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 decreased under 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 in Fig. 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.