## Abstract

We start from a 2D photonic crystal nanocavity with moderate Q-factor and dynamically increase it by two order of magnitude by the joint action of coherent population oscillations and nonlinear refractive index.

© 2012 OSA

## 1. Introduction

Reducing cavity mode volumes and increasing the cavity lifetimes is a crucial trend in photonics [1, 2]. Micro or nanocavities with high-quality (*Q*) factors allow to strongly confine for long times the electromagnetic field in a very tiny spatial region. Ultimately, such properties may result in suitable conditions to enhance light-matter interaction [3] with applications in nonlinear optics [4–8], biosensing [9], microwave filtering [10], pulse storage [11–13], quantum information and quantum electrodynamics [14, 15], potentially in integrated platforms. The strategies to reduce the cavity volume and to increase the cavity lifetimes are multiple but they always consist of the implementation of clever geometrical designs and state of the art technologies in order to boost the quality factor *Q*. Those approches end up in cavities with extremely high *Q*-factors with relative small volumes [16, 17], with modest to high *Q*-factors and wavelength-limited volumes [18, 19], and in few particular cases to a combination of the two requirements [11, 20–22]. A rigorous theoretical analysis by C. Sauvan *et al.* [23] have shown that among the physical mechanisms involved in the geometrical *Q*-factor enhancement is the increase of the the group index *n _{g}* of the nanocavity mode. However ultra-high

*Q*-factor micro and nanocavity realizations are extremely sensitive to technological imperfections and residual absorption. Therefore, a main challenge is to build-up alternative avenues to improve and reconfigure the performance of a given nanocavity. This can be achieved by actively enhancing the lifetime starting from cavities with moderate, robust and reproducible performance.

Introducing a strongly dispersive material in a cavity has shown to be a way to change significantly the cavity lifetimes and resonance linewidths [24, 25]. This has been demonstrated in atomic physics using as dispersive media Cesium [24] or Rubidium [26] atoms in macroscopic cavities. The main mechanisms to achieve the strong material dispersion are coherent nonlinear interactions such as population trapping [24] and electromagnetically induced transparency (EIT) [25, 26]. In both cases, the interaction of the atomic systems with a driving (pump) field and a signal (probe) field frequency tuned to the cavity resonance leeds to a steep dispersion at the origin of a strong decrease of the group velocity and linewith narrowing. Coherent Population Oscillations (CPO) effect is also used to produce a strong refractive index dispersion. CPO can be implemented in any two-level system material [27–30] and two-level like system such as semiconductor quantum wells and dots [31–33]. Despite its simplicity, room temperature operation and wavelength tunability, it is only recently [34] that CPO has been proposed and implemented to achieve lifetime enhancement and linewidth narrowing.

Taken benefit form the strong nonlinear index dispersion of active media, cavity lifetimes enhancement can also be achieved. The nonlinear index dispersion has shown to be particularly strong in semiconductors nanotructures such us quantum wells [35]. Two main avenues are thus offered to improve the temporal and spectral response of a cavity: the atomic-like induced index dispersion and the nonlinear refractive index dispersion.

Here, we extend our recent results [36] and successfully combine these two approaches in solid state semiconductor nanocavities and demonstrate strong linewidth narrowing and lifetime enhancement in a 2D Photonic Crystal (2D PhC) nanocavity. Both CPO-induced dispersion and nonlinear dispersion are obtained in the quantum wells embedded in the nanocavity and contribute to the lifetime enhancement.

## 2. Experiment

The cavity, illustrated in Fig. 1, is a L3 2D PhC nanocavity formed by missing three holes on an otherwise periodical hole lattice. Such cavity is a good combination of small cavity mode volume, *V* ∼ (*λ*/*n*)^{3}, and reasonable high *Q*-factors of few 10^{3}, reaching about 10^{5} by engineering the size and the position of the holes surrounding the cavity [20]. However, we start here from a L3 nanocavity with moderate *Q*-factor, ∼ 4000, easily attainable and reproducible. Shown in Fig. 1, the sample is a 260-nm-thick InP semiconductor membrane having 4 InGaAsP/InGaAs quantum wells grown by metal-organic vapor phase epitaxy. The 2D PhC is a hexagonal air-hole lattice in the InP membrane with a lattice constant *a* = 450 nm, a hole radius of 120 nm and the two end holes are displaced along the cavity axis by 0.15*a* in order to increase the *Q*-factor. Preliminarily frequency domain measurements have shown that the cavity has an intrinsic *Q*-factor *Q*_{0} = 4030. This intrinsic *Q*-factor is a combination of a radiation loss limited *Q*-factor *Q _{rad}* = 6300 and of an absorption loss limited

*Q*-factor

*Q*

_{a}_{0}= 11200. The absorption coefficient of the QWs is

*α*

_{0}= 54 cm

^{−1}at the working wavelengths around 1570 nm and the overlap of the confined mode and the QWs is Γ = 20%.

The excitation of the nanocavity is made using a fibered cw laser having a spectral linewidth of 150 kHz [36]. It is tunable between 1490 nm and 1610 nm and delivers up to 15 mW output power. The coupling of the laser field into the nanocavity is done using a tapered fiber with a diameter of ∼ 1.5*μ*m which is located on the top of the InP membrane above the L3 nanocavity. The coupling efficiency dependents on the fiber diameter and on the gap between the fiber and the membrane, which optimally should be of the order of 100 nm. In our setup, the coupling efficiency into the cavity has been measured to about 7 %, mostly limited by the non-optimal fiber/membrane gap. Indeed, due to electrical charges accumulating on the surface of the fiber, the tapered fiber sticks to the membrane. The power that is launched into the tapered fiber, about 0.5 % of the nominal laser power, is of the order of few *μ*W, comparable to the saturation power of the quantum wells. The loaded *Q*-factor of the nanocavity with the tapered fiber is measured to be *Q _{l}* = 3752, corresponding to a cavity lifetime

*τ*= 3.1 ps. In order to avoid any thermal effects, the cw laser is modulated with a fibered acousto-optic modulator producing 100 ns duration square pulses repeated every 20

_{l}*μ*s, longer than the thermal dissipation time [37]. The CPO effect is induced by using a sine-wave intensity modulation with a time period longer than electron-hole recombination time

*τ*= 200 ps [27,29]. By driving a fibered Mach-Zehnder interferometer with a rf signal at frequency

_{r}*δ*, a 10% intensity modulation depth is applied to the square pulses. The rf signal is supplied by a pulse function arbitrary generator. This temporal sine-wave modulation, 1 +

*m*cos(2

*πδt*), with

*m*< 1 the modulation depth, corresponds in the Fourier space to three spectral components at

*ω*and

*ω*±

*δ*, associated respectively to the average power of the square pulse and the amplitude of the modulation. The three spectral components play the role of the pump and the probe exciting the QWs in the CPO regime as long as

*δ*< 1/

*τ*. The optical signal are then detected using an Avalanche Photo-Detector (APD) having a 1 GHz electrical bandwidth. The measurement of the delay is achieved with a home made lock-in detection whose outputs are sensitive to the in- and out-of-phase quadratures of the modulation. Basically, the lock-in detection consists first in amplifying the AC component of the electrical signal delivered by the APD and then mixing it with a reference sine-wave signal at the same frequency

_{r}*δ*. By changing the phase of the reference from 0° to 90°, the in and out quadrature, called

*X*and

*Y*are obtained at the output of the mixer and after removing the harmonic component at 2

*δ*using a low pass filter. The group delay experienced by the probe field (amplitude modulation) is given by

*τ*≃

_{g}*ϕ*/(2

*πδ*), where

*ϕ*= arctan(

*Y/X*) is the phase of the probe relative to the phase reference measured directly by the lock-in system. We stress here on the fact that it is the group delay which measured experimentally and not the cavity lifetime

*τ*. However, it is easy to show that

_{c}*τ*=

_{c}*τ*2 for the nanocavity we are using. A full discussion on this topic can be found in [38].

_{g}/Figures 2(a) and 2(b) show respectively the measured pump and modulation amplitude reflections from the nanocavity for different laser wavelengths tuned around the nanocavity resonance as a function of the laser power. The modulation amplitude reflection corresponds to the reflection of the probe in the Fourier space.

Figure 2(c) presents the corresponding group delay for *δ* =240 MHz. For a given wavelength, the group delay faithfully follows the evolution of the modulation amplitude reflection, reaching a maximum value for a particular laser power. On the other hand, the maximum group delay is also achieved on the steep slope of the pump reflection where a strong nonlinear behavior is clearly apparent for shorter wavelengths as we are getting closer to the nanocavity resonance. The maximum achieved delay is 685 ps for a laser power of about 11 mW, corresponding to 55 *μ*W pump power in the tapered fiber incident on the nanocavity and an optimal pump wavelength *λ _{M}* ≈ 1571.5 nm. This constitutes a strong enhancement of the cavity lifetime

*τ*= 342.5 ps (×110) when compared to the initial value of 3.1 ps.

_{c}The cavity lifetime enhancement is closely related to an equivalent narrowing of the resonance linewidth as shown in Fig. 3 representing the measured modulation amplitude transfer function of the nanocavity. The measurement is achieved for a laser power of 10 mW, corresponding to 50 *μ*W pump power. The pump wavelength is set near *λ _{M}* and the modulation frequency

*δ*is tuned from 50 to 400 MHz. From a spectral point of view, this is similar to measure the probe reflection as the probe frequency is tuned relatively to the pump frequency

*ν*=

_{p}*c/λ*. The resonance linewidth has a Half Width at Half Maximum (HWHM) of about 220 MHz. When considering the initial 25-GHz HWHM bandwidth of the nanocavity (

_{p}*τ*= 3.1 ps), this corresponds to a linewidth narrowing by a factor of 113, which is close to the enhancement observed for the nanocavity lifetime. Table 1 summarizes the initial properties of the nanocavity and the final ones.

_{c}## 3. Theory

The introduction of a strong dispersion in a cavity predicts an enhancement of the cavity *Q*-factor [39] given by:

*Q*is the initial quality factor of the cavity,

_{i}*n*is the refractive index of the cavity material, Γ the overlap integral between the cavity mode, and

_{b}*n*the group index associated to the strong dispersive medium embedded in the cavity. The bare CPO-induced dispersion in our L3 PhC nanocavity is characterized by the group index given by: [27, 40] where

_{g}*I*

_{0}=

*I*/

*I*is the normalized intensity with

_{sat}*I*and

*I*the pump and saturation intensities and

_{sat}*α*

_{0}the unsaturated absorption of the QWs. We have neglected here the dependance on the frequency

*δ*as in our case we are in the limit of

*δτ*≪ 1. Inserting the experimental parameters into Eq. (2) and using Eq. (1), a CPO group index

_{r}*n*= 26 and an enhancement of the

_{g}*Q*-factor by a factor 2.5 are obtained. However, the expression given by Eq. (1) does not include the nonlinear behavior of the QWs in the nanocavity which leads to the resonance shift and to the non-Lorentzian resonance shapes observed experimentally when the pump power is increased [36, 41].

To explain the experimental results, a full theoretical analysis based on the coupled mode theory is carried out. It is based on the equations describing the time evolution of the mode amplitude *a* in the cavity and of the carrier density *N* in the quantum wells. They are given by [34, 42, 43]:

*ω*

_{0}.

*τ*

_{a}_{0}=

*Q*

_{a}_{0}/

*ω*

_{0}is the intrinsic photon lifetime in the nanocavity limited by the absorption of the QWs.

*τ*=

_{l}*Q*/

_{l}*ω*

_{0}is the overall photon lifetime without pumping and defined by 1/

*τ*= 2/

_{l}*τ*+ 1/

_{e}*τ*+ 1/

_{rad}*τ*

_{a}_{0}, where

*τ*is the fiber coupling photon lifetime and

_{e}*τ*=

_{rad}*Q*/

_{rad}*ω*

_{0}the radiation photon lifetime. Δ =

*ω*

_{0}−

*α*/(2

_{H}*τ*

_{a}_{0}) −

*ω*, where

_{p}*ω*is the circular frequency of the pump laser.

_{p}*N*is the carrier density at transparency,

_{t}*α*is the Henry factor, proportional to

_{H}*n*

_{2}, connecting the real to the imaginary parts of the nonlinear susceptibility of the QWs, and |

*a*|

_{sat}^{2}the saturation energy.

*s*(

_{in}*t*) is the input signal whereas the reflected signal is deduced from: ${r}_{\mathit{out}}\left(t\right)=1/\sqrt{{\tau}_{e}}a(t)$. The solutions are derived by considering an intensity modulated excitation written as

*s*(

_{in}*t*) =

*s*

_{0}+

*s*

_{1}exp (

*j*2

*πδt*) +

*s*

_{−1}exp (−

*j*2

*πδt*), with

*s*

_{1}=

*s*

_{−1}≪

*s*

_{0}. Neglecting the high order terms, the output amplitude can also be written as

*r*(

_{out}*t*) =

*r*

_{0}+

*r*

_{1}exp (

*j*2

*πδt*) +

*r*

_{−1}exp (−

*j*2

*πδt*). The output power

*P*can thus be written:

_{out}*P*(

_{out}*t*) = |

*r*(

_{out}*t*)|

^{2}= |

*r*

_{0}|

^{2}+

*R*(

_{s}*δ*)cos[2

*πδt*−

*ϕ*(

*δ*)]. From

*P*(

_{out}*t*) we can deduce the reflected pump power

*R*= |

_{p}*r*

_{0}|

^{2}, the reflected modulation amplitude

*R*(

_{s}*δ*) and the group delay given by

*τ*≃

_{g}*ϕ*(

*δ*)/(2

*πδ*).

The theoretical evolutions of *R _{p}*,

*R*(

_{s}*δ*) and

*τ*(

_{g}*δ*) are then obtained using the following parameters:

*λ*

_{0}= 2

*πc*/

*ω*

_{0}= 1570.21 nm,

*α*= 25,

_{H}*Q*= 6300 and

_{rad}*Q*

_{a}_{0}= 11200. As already noticed, these values are either independently measured or inferred from experimental measurements. Note that we had to use

*Q*= 3000 in order to reproduce the width of the pump resonance [36]. This value is slightly different from that experimentally measured. This can be explained by the modification in the coupling conditions between the different experiments. In the numerical calculation, we consider

_{l}*s*

_{1}=

*s*

_{0}/80 which gives a modulation depth of 10%. The only free parameter is the value of the intracavity saturation energy |

*a*|

_{sat}^{2}. The theoretical results are presented in Figs. 2(d), 2(e), 2(f), and also in Fig. 3, in regard to the experimental ones. The experimental behaviors are well reproduced. However, for the highest laser powers and short wavelengths there is a discrepancy on the reflected modulation amplitude and delay that is quite reasonable considering the limited number of adjustable parameters. This is due to limitations both on the measurement of the steep resonances and on the determination of the theoretical parameters close to the highly nonlinear points. For the long wavelengths and high laser powers, both experimental and theoretical curves show negative delays. Whereas theory predicts a negative delay of −80 ps, experimentally we measure negative delays up to −300 ps, this result is not yet understood.

## 4. Conclusion

Producing highly confining small optical resonators is a difficult task with demanding design and fabrication performance. We demonstrated that by inducing a coherent non linear interaction such as coherent population oscillation and a strong nonlinear index it is possible to considerably boost the nanocavity Q-factor. Starting from a semiconuctor 2D PhC nanocavity with Q-factor of 3752 we achieved a Q-factor of 4×10^{5}, with converging spectral and temporal measurement results. For a given wavelength, close to the initial resonance, the Q enhancement is resonant with the pump power which allows to tune its value.

## Acknowledgments

We acknowledge support by the French Agence Nationale de la Recherche through the project CALIN (ANR 2010 BLAN-1002). These results are within the scope of C’Nano IdF and RTRA Triangle de la Physique; C’Nano IdF is a CNRS, CEA, MESR and Région Ile-de-France Nanosciences Competence Center. We thank Philippe Lalanne for helpful discussions.

## References and links

**1. **K. J. Vahala, “Optical microcavities,” Nature (London) **424**, 839–846 (2003). [CrossRef]

**2. **V. Ilchenko and A. Matsko, “Optical resonators with whispering-gallery modes-part ii: applications,” IEEE J. Sel. Top. Quantum Electron. **12**, 15–32 (2006). [CrossRef]

**3. **A. M. Yacomotti, F. Raineri, C. Cojocaru, P. Monnier, J. A. Levenson, and R. Raj, “Nonadiabatic dynamics of the electromagnetic field and charge carriers in high-*q* photonic crystal resonators,” Phys. Rev. Lett. **96**, 093901 (2006). [CrossRef] [PubMed]

**4. **M. F. Yanik, S. Fan, and M. Soljačić, “High-contrast all-optical bistable switching in photonic crystal microcavities,” Appl. Phys. Lett. **83**, 2739–2741 (2003). [CrossRef]

**5. **A. Shinya, S. Matsuo, Yosia, T. Tanabe, E. Kuramochi, T. Sato, T. Kakitsuka, and M. Notomi, “All-optical on-chip bit memory based on ultra high q ingaasp photonic crystal,” Opt. Express **16**, 19382–19387 (2008). [CrossRef]

**6. **Y. K. Chembo, D. V. Strekalov, and N. Yu, “Spectrum and dynamics of optical frequency combs generated with monolithic whispering gallery mode resonators,” Phys. Rev. Lett. **104**, 103902 (2010). [CrossRef] [PubMed]

**7. **M. Brunstein, A. M. Yacomotti, I. Sagnes, F. Raineri, L. Bigot, and A. Levenson, “Excitability and self-pulsing in a photonic crystal nanocavity,” Phys. Rev. A **85**, 031803 (2012). [CrossRef]

**8. **A. M. Yacomotti, P. Monnier, F. Raineri, B. B. Bakir, C. Seassal, R. Raj, and J. A. Levenson, “Fast thermo-optical excitability in a two-dimensional photonic crystal,” Phys. Rev. Lett. **97**, 143904 (2006). [CrossRef] [PubMed]

**9. **S. Arnold, D. Keng, S. I. Shopova, S. Holler, W. Zurawsky, and F. Vollmer, “Whispering gallery mode carousel –a photonic mechanism for enhanced nanoparticle detection in biosensing,” Opt. Express **17**, 6230–6238 (2009). [CrossRef] [PubMed]

**10. **D. Strekalov, D. Aveline, N. Yu, R. Thompson, A. Matsko, and L. Maleki, “Stabilizing an optoelectronic microwave oscillator with photonic filters,” J. Lightwave Technol. **21**, 3052–3061 (2003). [CrossRef]

**11. **T. Tanabe, M. Notomi, E. Kuramochi, A. Shinya, and H. Taniyama, “Trapping and delaying photons for one nanosecond in an ultra-small high-q photonic-crystal nanocavity,” Nat. Photonics **1**, 49–52 (2007). [CrossRef]

**12. **Q. Xu, P. Dong, and M. Lipson, “Breaking the delay-bandwidth limit in a photonic structure,” Nature Phys. **3**, 406–410 (2007). [CrossRef]

**13. **Y. Dumeige, “Stopping and manipulating light using a short array of active microresonators,” Europhys. Lett. **86**, 14003 (2009). [CrossRef]

**14. **T. Aoki, A. S. Parkins, D. J. Alton, C. A. Regal, B. Dayan, E. Ostby, K. J. Vahala, and H. J. Kimble, “Efficient routing of single photons by one atom and a microtoroidal cavity,” Phys. Rev. Lett. **102**, 083601 (2009). [CrossRef] [PubMed]

**15. **K. Rivoire, S. Buckley, A. Majumdar, H. Kim, P. Petroff, and J. Vuckovic, “Fast quantum dot single photon source triggered at telecommunications wavelength,” Appl. Phys. Lett. **98**, 083105 (2011). [CrossRef]

**16. **D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-q toroid microcavity on a chip,” Nature (London) **421**, 925–928 (2003). [CrossRef]

**17. **A. A. Savchenkov, V. S. Ilchenko, A. B. Matsko, and L. Maleki, “Kilohertz optical resonances in dielectric crystal cavities,” Phys. Rev. A **70**, 051804 (2004). [CrossRef]

**18. **E. Peter, I. Sagnes, G. Guirleo, S. Varoutsis, J. Bloch, A. Lemaitre, and P. Senellart, “High-q whispering-gallery modes in gaas/alox microdisks,” Appl. Phys. Lett. **86**, 021103 (2005). [CrossRef]

**19. **C. P. Michael, K. Srinivasan, T. J. Johnson, O. Painter, K. H. Lee, K. Hennessy, H. Kim, and E. Hu, “Wavelength-and material-dependent absorption in gaas and algaas microcavities,” Appl. Phys. Lett. **90**, 051108 (2007). [CrossRef]

**20. **Y. Akahane, T. Asano, B. Song, and S. Noda, “High-q photonic nanocavity in a two-dimensional photonic crystal,” Nature (London) **425**, 944–947 (2003). [CrossRef]

**21. **Y. Takahashi, H. Hagino, Y. Tanaka, B.-S. Song, T. Asano, and S. Noda, “High-q nanocavity with a 2-ns photon lifetime,” Opt. Express **15**, 17206–17213 (2007). [CrossRef] [PubMed]

**22. **J. Lu and J. Vuckovic, “Inverse design of nanophotonic structures using complementary convex optimization,” Opt. Express **18**, 3793–3804 (2010). [CrossRef] [PubMed]

**23. **C. Sauvan, P. Lalanne, and J. P. Hugonin, “Slow-wave effect and mode-profile matching in photonic crystal microcavities,” Phys. Rev. B **71**, 165118 (2005). [CrossRef]

**24. **G. Müller, M. Müller, A. Wicht, R.-H. Rinkleff, and K. Danzmann, “Optical resonator with steep internal dispersion,” Phys. Rev. A **56**, 2385–2389 (1997). [CrossRef]

**25. **M. D. Lukin, M. Fleischhauer, M. O. Scully, and V. L. Velichansky, “Intracavity electromagnetically induced transparency,” Opt. Lett. **23**, 295–297 (1998). [CrossRef]

**26. **H. Wang, D. J. Goorskey, W. H. Burkett, and M. Xiao, “Cavity-linewidth narrowing by means of electromagnetically induced transparency,” Opt. Lett. **25**, 1732–1734 (2000). [CrossRef]

**27. **M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Observation of ultraslow light propagation in a ruby crystal at room temperature,” Phys. Rev. Lett. **90**, 113903 (2003). [CrossRef] [PubMed]

**28. **M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science **301**, 200–202 (2003). [CrossRef] [PubMed]

**29. **E. Baldit, K. Bencheikh, P. Monnier, J. A. Levenson, and V. Rouget, “Ultraslow light propagation in an inhomogeneously broadened rare-earth ion-doped crystal,” Phys. Rev. Lett. **95**, 143601 (2005). [CrossRef] [PubMed]

**30. **P. Wu and D. V. G. L. N. Rao, “Controllable snail-paced light in biological bacteriorhodopsin thin film,” Phys. Rev. Lett. **95**, 253601 (2005). [CrossRef] [PubMed]

**31. **X. Zhao, P. Palinginis, B. Pesala, C. Chang-Hasnain, and P. Hemmer, “Tunable ultraslow light in vertical-cavity surface-emitting laser amplifier,” Opt. Express **13**, 7899–7904 (2005). [CrossRef] [PubMed]

**32. **N. Laurand, S. Calvez, M. D. Dawson, and A. E. Kelly, “Slow-light in a vertical-cavity semiconductor optical amplifier,” Opt. Express **14**, 6858–6863 (2006). [CrossRef] [PubMed]

**33. **A. El Amili, B.-X. Miranda, F. Goldfarb, G. Baili, G. Beaudoin, I. Sagnes, F. Bretenaker, and M. Alouini, “Observation of slow light in the noise spectrum of a vertical external cavity surface-emitting laser,” Phys. Rev. Lett. **105**, 223902 (2010). [CrossRef]

**34. **Y. Dumeige, A. M. Yacomotti, P. Grinberg, K. Bencheikh, E. Le Cren, and J. A. Levenson, “Microcavity-quality-factor enhancement using nonlinear effects close to the bistability threshold and coherent population oscillations,” Phys. Rev. A **85**, 063824 (2012). [CrossRef]

**35. **Y. H. Lee, A. Chavez-Pirson, S. W. Koch, H. M. Gibbs, S. H. Park, J. Morhange, A. Jeffery, N. Peyghambarian, L. Banyai, A. C. Gossard, and W. Wiegmann, “Room-temperature optical nonlinearities in gaas,” Phys. Rev. Lett. **57**, 2446–2449 (1986). [CrossRef] [PubMed]

**36. **P. Grinberg, K. Bencheikh, M. Brunstein, A. M. Yacomotti, Y. Dumeige, I. Sagnes, F. Raineri, L. Bigot, and J. A. Levenson, “Nanocavity linewidth narrowing and group delay enhancement by slow light propagation and nonlinear effects,” Phys. Rev. Lett. **109**, 113903 (2012). [CrossRef] [PubMed]

**37. **V. Moreau, G. Tessier, F. Raineri, M. Brunstein, A. Yacomotti, R. Raj, I. Sagnes, A. Levenson, and Y. D. Wilde, “Transient thermoreflectance imaging of active photonic crystals,” Appl. Phys. Lett. **96**, 091103 (2010). [CrossRef]

**38. **Q. Li, T. Wang, Y. Su, M. Yan, and M. Qiu, “Coupled mode theory analysis of mode-splitting in coupled cavity system,” Opt. Express **18**, 8367–8382 (2010). [CrossRef] [PubMed]

**39. **M. Soljačić, E. Lidorikis, L. V. Hau, and J. D. Joannopoulos, “Enhancement of microcavity lifetimes using highly dispersive materials,” Phys. Rev. E **71**, 026602 (2005). [CrossRef]

**40. **R. W. Boyd, M. G. Raymer, P. Narum, and D. J. Harter, “Four-wave parametric interactions in a strongly driven two-level system,” Phys. Rev. A **24**, 411–423 (1981). [CrossRef]

**41. **A. M. Yacomotti, F. Raineri, G. Vecchi, P. Monnier, R. Raj, A. Levenson, B. Ben Bakir, C. Seassal, X. Letartre, P. Viktorovitch, L. Di Cioccio, and J. M. Fedeli, “All-optical bistable band-edge Bloch modes in a two-dimensional photonic crystal,” App. Phys. Lett. **88**, 231107 (2006). [CrossRef]

**42. **H. A. Haus, *Waves and Fields in Optoelectronics* (Prentice-Hall, 1984).

**43. **E. Kapon, *Semiconductor Laser I: Fundamentals* (Academic Press, 1999).