in frequency conversion of light in confined media
Emre Y¨uce,1 Georgios Ctistis,1 Julien Claudon,2 Jean-Michel G´erard,2 Willem L. Vos1
1
Complex Photonic Systems (COPS), MESA+ Institute for Nanotechnology, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands
2
CEA-CNRS-UJF “Nanophysics and Semiconductors” joint laboratory, CEA/INAC/SP2M, 17 Rue des Martyrs,
38054 Grenoble Cedex 9, France (Dated: August 13, 2015)
The generation of light with a controllable fre-quency is a long standing challenge in physics that draws continued attention, driven by novel emerging applications, and by an ongoing minia-turization of devices. In nonlinear optics quency conversion of light at a particular fre-quency is widely pursued. A modern approach to frequency conversion employs the confinement of light in resonances of nanophotonic structures, such as microcavities or waveguides. Supposedly, the physics of nanophotonic frequency conversion differs from traditional non-linear optics. Here, we unify these seemingly disparate views. We have reversibly switched the resonances of GaAs-AlAs microcavities in the near-infrared within 300 fs by the electronic Kerr effect. Light con-fined inside the cavity reveals a remarkable red shift or blue shift, depending on the timing of pump and probe pulses. From observations on cavities with a range of quality factors, we iden-tify the role of the local density of optical states (LDOS) available to the newly generated light fre-quencies. This concept from cavity quantum elec-trodynamics allows us to present a unified physics framework for frequency-conversion in both tra-ditional homogeneous and in nanophotonic me-dia.
The generation of optical frequencies starting from in-cident light with a particular frequency is widely pursued in nonlinear optics1–3. In well-known self-phase modula-tion1, an optical beam causes a nonlinear polarization in a spatially homogeneous nonlinear medium, leading to a temporal modulation of the phase φ(t) of the inci-dent wave, and an instantaneous frequency shift of the output light that is equal to the rate of change of the phase δω(t) = −∂δφ(t)/∂t. Since the frequency shift in-creases with incident light intensity and with interaction length, long fibers have become popular media to ob-tain frequency shifts by means of the electronic Kerr ef-fect1. Large super-continuum frequency shifts have been
obtained with Raman or soliton mechanisms, which are being applied in commercial white-light sources for
ultra-stable clocks or advanced microscopy2.
In nanophotonics the frequency of the confined light is converted by quickly changing the refractive index of the nanostructure in time, leading to a time-dependent change of light confined in one (or many) resonances4–13.
Since the nanoscale confinement leads to strong field-enhancements compared to homogeneous media, efficient conversion is feasible with low power. As the devices have a very small footprint they posses a great potential for on-chip integration4,13. As reviewed in Refs.4–6, the physics of nanophotonic frequency conversion apparently differs from the traditional case in at least three key features: (i) In nanophotonics, the output spectrum reveals dis-crete peaks at the system resonances (ω = ωc) instead of
a continuum as in traditional nonlinear optics.
(ii) In nanophotonics, the frequency shift δω(t) is inde-pendent of the tuning rate of the phase (∂δφ(t)/∂t) in contrast to the traditional case; instead the shift depends on the tuning rate of the resonance ωc(t).
(iii) In nanophotonics, the process should be faster than the resonance’s storage time (∂t < τc), which is not
rele-vant in traditional nonlinear optics.
The central question addressed in this work is how to reconcile the apparently different nonlinear physics be-tween traditional homogeneous media, and confining me-dia with resonances. Therefore, we consider in this paper a nanophotonic system - a planar microcavity - that sus-tains both a cavity resonance and a continuous density of optical states. We study by a pump-probe experi-ment the dynamics of frequency conversion of confined light when the cavity is switched in an ultrafast way via the electronic Kerr effect. Our planar GaAs-AlAs mi-crocavities consist of two Bragg mirrors that surround a λ-layer (See Supplementary for Sample and Setup). We study cavities with quality factors between Q = 390 ± 60 and Q = 890 ± 60 to vary the local density of optical states available to the generated light.
Figure 1(a) presents reflectivity spectra measured at several pump-probe delays ∆t on the Q = 390 cavity. The reference spectrum reveals the unswitched cavity res-onance at (ωc/2πc) = 7807.5 cm−1(λc = 1280.8 nm) and
100% maximum transient reflectivity. During the
80 90 100 Trans . R ef l. ( %) ∆t = -500 fs ∆t = -100 fs unswitched -1 Frequency (cm ) a 7750 7800 7850 7760 7810 7860 84 92 100 ωres b Measured 7760 7810 7860 84 92 100 T ransient Reflectivity (%) -1 Frequency (cm ) Time Delay ∆t (ps) −2 -1 0 1 c ωres Calculated
Figure 1: (a) Measured transient reflectivity spectra Rt(ω)
at time delays ∆t = −500 fs, and −100 fs, as well as an unswitched reference taken at ∆t = 2.8 ps where pump and probe have no temporal overlap. The high minimum
reflectiv-ity on resonance (Rmint = 85 %) is caused by the asymmetric
cavity design. Filled regions highlight how new frequencies exit the switched cavity in reflection geometry. Panels (b) and (c) are maps of measured and calculated transient reflectivity ver-sus time delay. The arrows indicate the direction of frequency conversion. White symbols (b) and the solid curve (c) mark
the resonance frequency of the cavity ωc(t).
ing event, the cavity resonance quickly shifts by 5.7 cm−1 to a lower frequency at maximum pump-probe overlap, and returns to the starting frequency immediately after the pump pulse has gone. The overlap is maximal at ∆t = −100 fs delay, since the incident probe light takes about 100 fs to charge the cavity before circulating inside. The shift to a lower frequency is caused by an increased refractive index, due to the positive non-degenerate Kerr coefficient of GaAs14.
While the microcavity is switched, escaping frequency-converted light is collected in reflection geometry. Here an interference occurs with probe light that has di-rectly reflected from the top mirror to yield a frequency-resolved transient reflectivity Rt(ω)14. The interference
appears as fringes at ∆t < 0 in Fig. 1(b), and is effec-tively a phase-sensitive heterodyne detection of light from the cavity that allows us to sensitively detect frequency-converted light outside the instantaneous resonance. The sensitivity allows us to observe interference over a much longer time (up to 2 ps) than the cavity decay time (τc = 300 fs).
Figure 1(a) reveals at ∆t = −100 fs delay a remark-able transient reflectivity in excess of 100% at frequen-cies above the cavity resonance (between 7813 cm−1 and 7851 cm−1). Simultaneously, the signal is depleted in comparison to the reference at frequencies below the
cavity resonance (between 7753 cm−1 and 7808 cm−1). Red-shifted light appears in Fig. 1(a) at a different de-lay ∆t = −500 fs, where excess transient reflectivity occurs below the cavity resonance (between 7756 cm−1
and 7789 cm−1) with a simultaneous depletion above the
resonance (between 7820 cm−1 and 7850 cm−1). These two features demonstrate that cavity-stored probe light is blue-shifted (or red-shifted) by as much as 50 cm−1, from below to above (or above to below) the unswitched cav-ity resonance. The δω = 50 cm−1 frequency shift of the stored light is much greater than the δωc= 5.7 cm−1shift
of the cavity resonance, which contradicts the nanopho-tonic view of peaks in the output centred on resonances (key feature (i)).
Figure 1(b) shows a map of measured transient re-flectivity spectra versus pump-probe delay ∆t. Blue-shifted light occurs at time delays between −100 fs and +100 fs, where the cavity resonance frequency ωc
in-creases, thereafter the confined light experiences a de-creasing refractive index, and a concomitant phase mod-ulation (∂δφ(t)/∂t) < 0. Red-shifted light appears at time delays between −900 fs and −100 fs, where ωc
decreases, hence the confined light experiences an in-creasing refractive index, and thus a phase modulation (∂δφ(t)/∂t) > 0. At delays ∆t < −500 fs, both blue- and red-shifted light appear, since both an increase and de-crease of the refractive index occur while the probe light circulates in the cavity. Here the magnitude of the red-shifted light is larger than that of the blue-red-shifted light, since the pump pulse first induces an increasing and then a decreasing refractive index while the stored light inten-sity decreases. To confirm these observations, we have calculated the nonlinear frequency conversion in reflec-tion with a time-resolved model (See Supplementary for Model), using only the pump fluence as an adjustable pa-rameter. Fig. 1(c) shows a map of frequency-converted spectra that agrees very well with the measured tran-sient reflectivity (Fig. 1(b)) regarding frequency shift, bandwidth, and timing. Our calculations yield blue- and red-shifted probe light at the same frequencies as in the measurements. As the timing of the blue- or red-shifts corresponds to minus the rate of change of the phase −(∂δφ(t)/∂t), one might naively conclude that the cause of the frequency shift (key feature (ii)) is settled in favor of traditional phase modulation. We will now see that this is not the case.
Figure 2 shows the blue-converted light signal as a function of the inverse quality factor 1/Q. The reflected light in excess of 100% is averaged over time delay from ∆t = −100 to 0 fs, and integrated over frequency, thereby mostly containing frequencies outside the cavity reso-nance (cf. Fig. 1). It is remarkable experimentally that the amount of frequency-converted light is inversely pro-portional with Q. In cavity quantum electrodynamics (QED), it is well-known that the quality factor Q gauges the local density of optical states (LDOS) on cavity
res-0 1x10-3 2x10-3 3x10-3 0 2 4 6 13 pJ/µm2 168 pJ/µm2 220 pJ/µm2 Int egra ted S ignal ( cm -1)
Inverse Quality Factor (1/Q)
LDOS, rv ( ,z)ω
Figure 2: Measured integrated blue-converted light, averaged over time delays between ∆t = −100 and 0 fs as a function of inverse quality factor 1/Q. Data are shown for different pump
pulse fluence 13, 168, 220 pJ/µm2(circles, triangles, squares),
and lines are linear fits to the data.
onance ρc(ωc, zc) that plays a central role in the Purcell
effect15,16. However, this notion would imply that the
signal would decrease with increasing LDOS, which is at odds with cavity QED intuition. On the other hand, it appears that the LDOS at frequencies outside the resonance can be identified with 1/Q (See Supplemen-tary for Theory). This contribution includes the LDOS-continuum from the surrounding vacuum ρv(ω, zc) that
tunnels into the cavity through the Bragg mirrors17. This
notion combined with our observations in Fig. 2 suggest that the color-converted intensity increases with the vac-uum LDOS. The increasing intensity of frequency verted light with 1/Q cannot be explained by only con-sidering the rate of change of phase, since the rate is the same for all cavities. Our conclusion that the LDOS plays a central role in color-conversion in confined media, in parallel with cavity QED, introduces novel physics in nonlinear optics3.
Fig. 2 also shows that the amount of frequency-converted light increases with pump pulse fluence. This is understood from the third-order electronic-Kerr non-linearity. The induced nonlinear polarization is expanded in the nonlinear index n = n0+ n2.Iputhat increases
lin-early with pump fluence Ipu3. At constant pump pulse
duration, an increased refractive index corresponds to an increased rate of change of the phase (∂δφ(t)/∂t), and thus an increased frequency shift. Regarding key feature (ii), we conclude that the frequency shift and intensity are related to both the LDOS and the rate of change of the phase.
To hone our new insights, we have performed numerical calculations. We have tuned the quality factor Q to vary the bandwidth of the resonant LDOS ρc(ω, zc) and the
amplitude of the continuum-LDOS ρv(ω, zc), and tuned
the switch pulse duration to vary the bandwidth of the nonlinear polarization PN L(ω
pr). Results for a low-Q
cavity switched by a long pulse are shown in Fig. 3(a-c). The nonlinear polarization PN L is non-zero over a
rv(ω,z )c 10-2 100 102 low-Q, broad NL P high-Q, narrow NL P c b g d LDO S ( n s /m ) a -250 0 250 e In t. C o n v e rs io n (% ) Model NL low-Q, narrow P -750 0 750 f -15 0 15 I 7750 7800 7850 -20 0 20 E x t. C o n v e rs io n ( % ) t t 1-T -R 7750 7800 7850 -20 0 20 h Frequency (cm-1) 7600 7800 8000 -4 0 4 -1 0 1 NL P -0.1 0.0 0.1 -0.5 0.0 0.5 N L P (n o rm .) -1 0 Analytic -1 0 -0.1 0.0 0.1 N L r ( ω ,z ) × P (n o rm .) c c rc(ω,z )c
Figure 3: Effects of LDOS and nonlinear polarization on
optical frequency conversion. Panels (a,d,g) show LDOS
and polarization spectra. The LDOS at the cavity center
ρc(ω, zc) is shown as black long-dashed curves, the vacuum
LDOS ρv(ω, zc) as gray dots, the normalized nonlinear
po-larization (PN L) as red curves. Panels (b,e,h) show internal
color conversion and analytic spectra at cavity center. Blue short-dash curves are the results of our numerical model, the
green curve is the analytic model (PN L· ρc(ω, zc)) (PN Land
PN L· ρ
c(ω, zc) are normalized to the maximum in each row).
Panels (c,f,i) show color-conversion spectra outside the cavity
calculated as (1 − Tt− Rt
) (purple curves), and red filled re-gions highlight the color-converted light. Panels (a-c) pertain
to a low-Q cavity (Q = 400) with a long pump (τpu= 900 fs),
(d-f ) to a high-Q cavity (Q = 7000) with the same pump
(τpu= 900 fs), and (g-i) to the same low-Q cavity (Q = 400)
with a short pump (τpu= 50 fs).
frequency range from 7750 cm−1 to 7870 cm−1, and the LDOS in the central λ−layer has a broad maximum of 20 ns/m on resonance, decreasing to 4 × 10−1 ns/m at
frequencies where PN L vanishes. Fig. 3(b) shows the
calculated conversion at the cavity center that extends from 7750 cm−1 to 7870 cm−1. We also plot an an-alytic model wherein the frequency conversion is pro-portional to the product (PN L· ρ
c(ω, zc)) of nonlinear
polarization and LDOS, which agrees well with the nu-merical results. Fig. 3(c) shows the externally detectable frequency conversion, plotted as the transient quantity (1 − Tt+ Rt), which discriminates simple elastic
scat-tering of a time-varying cavity resonance without con-version where 1 − Tt+ Rt = 0. The frequency shifted light in the cavity differs from the transient response out-side the cavity (See Supplementary for Theory). The re-sults show that the LDOS and the polarization conspire to yield a broad red-shifted frequency conversion down to 7740 cm−1, with a concomitant blue depletion as far as 7860 cm−1. The bandwidth of frequency conversion strongly exceeds the cavity linewidth (∆ωc = 18 cm−1),
confirming our observation that light is generated in a continuum outside the cavity resonance. While one natu-rally expects a cavity to be active over one linewidth ∆ωc,
that a much broader extent of the cavity LDOS spectrum is significant to frequency conversion, which notably in-cludes the vacuum LDOS that tunnels into the cavity ρv(ω, zc) (Fig. 3(g)). These results conclude the
discus-sion on the frequency-converdiscus-sion spectrum (key feature (i)): the spectral output is much broader than the reso-nance and given by PN L· ρ
c(ω, zc).
To understand the role of the quality factor, we show in Fig. 3(d-f) data for a high-Q cavity with the same nonlinear polarization and mode volume as in Fig. 3(a-c). The LDOS is narrow and strongly pronounced: on resonance the LDOS is an order of magnitude larger (4 × 102ns/m), and off-resonance an order of magnitude
lower (4 × 10−3ns/m) than in Fig. 3(a), in agreement with the Q-related LDOS scalings above. The LDOS is thus dominated by the cavity resonance, while the role of the vacuum LDOS ρv(ω, zc) is limited. As a result, the
frequency-converted spectrum is also narrow (Fig. 3(e)), as in Ref.6. The frequency-integrated conversion Fig. 3(f)
is much smaller than for the low-Q cavity (Fig. 3(c)), in agreement with our 1/Q observations in Fig. 2. The cavities’ LDOS spectrum is so narrow that it has little overlap with the nonlinear polarization, which explains the small frequency-integrated conversion. In the time-domain, this means that the resonance storage time is not matched to the duration of the pump or probe pulses. Hence, ultrafast frequency conversion of light is best done with a pulse duration matched to the cavity storage time (τpu' τc), as opposed to prior statements on key feature
(iii) above.
To understand the role of the nonlinear polarization, Fig. 3(g-i) shows results for a low-Q microcavity as in Fig. 3(a), with a much broader nonlinear polarization ex-tending over more than 400 cm−1bandwidth (Fig. 3(g)).
In agreement with our analytic model, we find a broader internally generated spectrum (Fig. 3(h)), and external spectrum (Fig. 3(i)) compared to (b) and (c), with red-shifted conversion extending beyond 7600 cm−1, and blue depletion below 8000 cm−1. These results indicate that in the limiting case of a broadband LDOS typical of a ho-mogeneous material, the output spectrum is determined by the nonlinear polarization, a situation typical for tra-ditional nonlinear optics.
We see in Figs. 3(b,e,h) a very good agreement be-tween the numerical results of the internal frequency-converted and an analytic model proportional to the product (PN L·ρc(ω, zc)). Considering by analogy Fermi’s
golden rule in cavity QED, one might wonder why the converted intensity is not proportional to the nonlinear polarization squared, assuming the polarization to play the role of transition dipole moment. For Fermi’s golden rule to hold, however, requires the bandwidth of the LDOS resonance to be much broader than the dipole’s bandwidth, known as the weak-coupling or Markov ap-proximation18. The inequality is strongly violated in the
present study, where the polarization bandwidth is
com-parable to the LDOS bandwidth. When the Markov ap-proximation is violated, the intensity can strongly differ from Fermi’s golden rule. For a homogeneous medium with a smooth LDOS, however, we expect an analogy to Fermi’s golden rule to hold; indeed such a relation can be identified in Ref.19. Measured Model a b blue red -2 0 2 4 -3 0 97 100 Transi ent R ef lectivity ( %) Excess Reflectivity (%) Time Delay ∆t (ps) -3 0 3 97 100 103
Figure 4: THz repeated (a) blue- and (b) red-converted light pulses versus time delay ∆t. The cavity is repeatedly switched
by two pump pulses (δtpu= 1.0 ps), and probed by two pulses
(δtpr = 2.0 ps). Solid curves are Rt(ω) calculated with our
dynamic model, filled regions highlight the frequency-converted pulses, horizontal dashed lines represent unswitched reflectiv-ity at 100%.
Recently, we reported that the electronic Kerr effect al-lows for repeated switching of a cavity resonance at THz rates20. We have extended this technique to generate
short trains of very fast - terahertz - repeated frequency shifted pulses, as shown in Figure 4, which is a first ap-plication of the new physics uncovered above. Moreover, since probe light stored in the cavity experiences solely a real refractive index, prospects are favorable to extend resonance-aided frequency conversion to single photons and other delicate quantum states of light, thus opening novel opportunities in quantum optics and cavity QED.
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Supplementary Information is not linked to the online version of this paper
Acknowledgements We thank Ad Lagendijk, Allard Mosk, Emanuel Peinke, Henri Thyrrestrup, and Elahe Yeganegi for useful discussions. This research was sup-ported by FOM, NWO, STW, and NWO-Nano.
Author Contributions All authors take full respon-sibility for the content of the paper.
Competing financial interests The authors declare no competing financial interests.
