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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 24 — Nov. 26, 2007
  • pp: 16161–16176
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Enhanced nonlinear optics in photonic-crystal microcavities

Jorge Bravo-Abad, Alejandro Rodriguez, Peter Bermel, Steven G. Johnson, John D. Joannopoulos, and Marin Soljačić  »View Author Affiliations


Optics Express, Vol. 15, Issue 24, pp. 16161-16176 (2007)
http://dx.doi.org/10.1364/OE.15.016161


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Abstract

Nonlinear photonic-crystal microresonators offer unique fundamental ways of enhancing a variety of nonlinear optical processes. This enhancement improves the performance of nonlinear optical devices to such an extent that their corresponding operation powers and switching times are suitable for their implementation in realistic ultrafast integrated optical devices. Here, we review three different nonlinear optical phenomena that can be strongly enhanced in photonic crystal microcavities. First, we discuss a system in which this enhancement has been successfully demonstrated both theoretically and experimentally, namely, a photonic crystal cavity showing optical bistability properties. In this part, we also present the physical basis for this dramatic improvement with respect to the case of traditional nonlinear devices based on nonlinear Fabry-Perot etalons. Secondly, we show how nonlinear photonic crystal cavities can be also used to obtain complete second-harmonic frequency conversion at very low input powers. Finally, we demonstrate that the nonlinear susceptibility of materials can be strongly modified via the so-called Purcell effect, present in the resonant cavities under study.

© 2007 Optical Society of America

1. Introduction

Since the early days of nonlinear optics, optical resonators have been seen as an attractive way to enhance nonlinear optical phenomena, such as a frequency conversion processes [1

1. R. W. Boyd, Nonlinear Optics (Academic Press, California, 1992).

] (in optical parametric oscillators) or optical bistability properties [2

2. H. M. Gibbs, Optical Bistability: Controlling Light with Light (Academic Press, Orlando, FL, 1985).

]. Traditionally, these nonlinear optical resonators have consisted of a nonlinear material located between two partially transmitting mirrors. Although interesting in their own right, the application of these nonlinear Fabry-Perot interferometers for designing all-optical logical devices is rather limited, as they can not fulfill the requirements in size, switching time and operating power of practical integrated optical systems.

As a consequence of these recent advances in nanophotonic fabrication, many of the nonlinear phenomena previously analyzed in conventional nonlinear etalons are being revisited within the context of PhC cavities [17

17. E. Centeno and D. Felbacq, “Optical bistability in finite-size nonlinear bidimensional photonic crystals doped by a microcavity,” Phys. Rev. B 62, R7683–R7686 (2000). [CrossRef]

40

40. A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, and G. W. Burr “Improving accuracy by subpixel smoothing in the finite-difference time domain,” Opt. Lett. 31, 2972–2974 (2006). [CrossRef] [PubMed]

]. Although it is true that the physical mechanisms producing nonlinear phenomena (such as optical bistability or nonlinear frequency conversion) in PhC cavities are similar to those observed in their conventional counterparts, it has been demonstrated that PhC microresonators enhance the performance of traditional nonlinear devices by several orders of magnitude. In addition, due to their design versatility, PhC cavities can be used as the basis of completely new configurations performing all-optical logic functions, such as all-optical transistor action [25

25. M. F. Yanik, S. Fan, M. Soljacic, and J. D. Joannopoulos, “All-optical transistor action with bistable switching in a photonic crystal cross-waveguide geometry,” Opt. Lett. 28, 2506–2508 (2003). [CrossRef] [PubMed]

]. Moreover, PhC resonators offer new fundamental ways of tailoring optical nonlinearities by using the so-called Purcell effect. Finally, in this context, note that not only PhC resonators can lead to a strong enhancement of nonlinear phenomena, but also nonlinear effects can be enhanced by using slow-light properties of PhCs, via the corresponding band egdes [41

41. G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidakovic, J. A. Levenson, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Photonic band edge effects in finite structures and applications to χ(2) interactions,” Phys. Rev. E 64, 016609 (2001). [CrossRef]

, 42

42. M. L. Povinelli, S. G. Johnson, and J. D. Joannopoulos, “Slow-light, band-edge waveguides for tunable time delays,” Opt. Express 13, 7145–7159 (2005). [CrossRef] [PubMed]

] or by means of coupled-cavity waveguides [18

18. M. Soljacic, S. G. Johnson, S. Fan, M. Ibanescu, E. Ippen, and J. D. Joannopoulos, “Photonic-crystal slow-light enhancement of non-linear phase sensitivity,” J. Opt. Soc. Am. B 19, 2052–2059 (2002). [CrossRef]

, 43

43. Y. Xu, R. K. Lee, and A. Yariv, “Propagation and second-harmonic generation of electromagnetic waves in a coupled-resonator optical waveguide,” J. Opt. Soc. Am. B 17, 387–400 (2000). [CrossRef]

, 44

44. J. E. Heebner, R. W. Boyd, and Q. H. Park, “Slow light, induced dispersion, enhanced nonlinearity, and optical solitons in a resonator-array waveguide,” Phys. Rev. E 65, 036619 (2002). [CrossRef]

]

It is important to mention that, together with the aforementioned rapid development of the experimental techniques, during the last decade there has also been an important growth in large-scale computing technologies. Thus, the combination of pure numerical methods as the non-linear finite-difference-time domain method (FDTD) [37

37. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech, Norwood, MA, 2000).

] (which simulates Maxwell’s equations with no approximation apart from discretization), with analytical approaches such as coupled-mode theory [38

38. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, NJ, 1984).

] or perturbation theory [19

19. M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E 66, 055601(R) (2002). [CrossRef]

] allow a complete characterization of the electromagnetic response of the nonlinear PhC cavities under study. In particular, all the theoretical calculations shown in this paper have been obtained by using one of the tools just mentioned. However, since this work is mainly focused in explaining the physical mechanisms responsible for the observed results, we refer to the reader to more specialized references for details on the numerical calculations and analysis [37

37. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech, Norwood, MA, 2000).

40

40. A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, and G. W. Burr “Improving accuracy by subpixel smoothing in the finite-difference time domain,” Opt. Lett. 31, 2972–2974 (2006). [CrossRef] [PubMed]

].

This manuscript reviews three examples of nonlinear optical processes that can be dramatically enhanced by PhC resonators. The first example in Section 2 reviews the mechanism of optical bistability, and shows how PhC cavities enhance it in a striking fashion. The second example in Section 3 shows how PhCs can dramatically lower the threshold for 100% efficient harmonic generation compared to conventional approaches. The third example in Section 4 models the origin of nonlinearity at an atomic level, and shows how it can be tailored by its PhC enviroment. In particular, it gives rise to PhC cavities that enhance not only the field but also the nonlinear coefficient. These results are summarized in Section 5.

Fig. 1. Sketch of a system composed by an optical resonator coupled symmetrically to both an input and output ports. ωc is the corresponding resonant frequency and Γ is the width of the resonance. Pin and Pout label the incoming and outgoing powers through the structure, respectively. Inset shows the typical linear transmission spectrum corresponding to this system.

2. Optical bistability in photonic crystal cavities

For pedagogical reasons, let us start by considering the general system sketched in Fig. 1. It consists of a resonant nonlinear cavity coupled to both an input and an output port. This optical resonator can be a conventional nonlinear Fabry-Perot resonator (i.e., a slab of nonlinear material situated between two partially transmitting mirrors) as well as a PhC microcavity (a defect in an otherwise perfectly periodic PhC). In the linear regime, light is transmitted between the input and the output ports by means of a resonant tunneling process. Thus, if we assume that the input and output ports are two single-mode waveguides, the ratio between the input and outgoing powers (Pin and Pout, respectively) is characterized by a Lorentzian shape (see inset of Fig. 1)

PoutPin=11+((ωωc)Γ)2
(1)

where Γ is the width of the resonance and ωc is the corresponding resonant frequency. Note that from Γ and ωc we can obtain the quality factor of the cavity Q=ωc/2Γ.

PoutPin=11+(PoutP0Δ)2
(2)

where P 0 is the so-called characteristic power of the cavity (this magnitude will be discussed in more detail below) and Δ is the frequency detuning normalized by the width of the resonance, Δ=(ωc-ωp)/Γ. Figure 2(b) displays Pout/Pin as a function of Pout for Δ=3.

To illustrate this point, consider the structure displayed in Fig. 3(a). It is formed by a two-dimensional (2D) PhC composed by high-ε dielectric rods embedded in a low-ε dielectric material. A defect has been introduced at the center of the structure by slightly increasing the size of one of the rods. This central defect is coupled symmetrically to two single mode PhC waveguides on the left and right [19

19. M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E 66, 055601(R) (2002). [CrossRef]

]. If we assume that the central rod is made by a Kerr-nonlinear material, this structure can be considered to be a PhC implementation of the system sketched in Fig. 1. Thus, this structure displays the same kind of bistability properties described previously: if we send light through a PhC waveguide, the input—output relation will exhibit bistability (two stable solutions for a given input power), provided that the frequency detuning of the external illumination and the resonant cavity frequency is large enough. This is confirmed by the numerical calculations shown in Fig. 3(b), where the results obtained from both perturbation theory and the nonlinear FDTD method have been plotted (see green line and blue dots, respectively). In this case it has been assumed Δ=3.8. The crucial point is that although the quality factor shown in Fig. 3(a) is just 500, similar cavities can be designed to show Q ~106 (for instance, just by increasing the number of layers surrounding the defect), while they are confining light in regions of subwavelength size, something that could not be reached in a straightforward manner by using conventional Fabry-Perot devices. This, taking into account the scale law for the characteristic power P 0 we have deduced above, can lead to a reduction of the typical values of P 0 by several orders of magnitude.

Fig. 2. (a) Evolution of the transmission spectra through the system sketched in Fig. 1 when the refractive index of the resonator is increased by δn. As can be seen in this panel, δn shifts the original resonant frequency of the cavity ωc (dashed line) towards the frequency of the external illumination ωp (blue dashed line). (b) Dependence of Pout/Pin as a function of the outgoing power for Δ=3 (see text for details on this magnitude). (c) Same function as (b) but this time Pout is plotted as a function of Pin for several values of Δ. Dotted lines display the unstable branches of the hysteresis loop for each case.
Fig. 3. (a) Photonic crystal implementation of the system sketched in Fig. 1. The PhC is made by a periodic two dimensional distribution of high dielectric rods (εH=12.25, yellow regions in the figure) in a low-ε background (εL=2.25). The rods have a radius of r=0.25a. A point defect, introduced by increasing the radius of the central rod to r=0.33a, is symmetrically coupled to two single mode PhC waveguides on the left and right. The electric field pointing into the page is depicted with positive (negative) values in red (blue). (b) Computed dependence of the output power (Pout) as a function of the input power (Pin) for the structure shown in panel (a) when the central rod is assumed to be made by a nonlinear Kerr-like material. Green line displays the results obtained from a perturbation theory analysis while the blue dots correspond to the result of a nonlinear FDTD simulation. Dashed lines represent the unstable branch of the bistable loop.

At this point, let us mention that, in contrast to conventional electronic logical gates, in which most of the operating power is dissipated, in the scheme analyzed in this Section only a small fraction of this power is absorbed by the structure (typically only ~10% of the operating power is absorbed [46

46. M. Notomi, Personal communication (2007).

], i.e., 7 fJ in the system analyzed in [32

32. T. Tanabe, M. Notomi, S. Mitsugi, A. Shinya, and E. Kuramochi, “Fast bistable all-optical switch and memory on a silicon photonic crystal on-chip,” Opt. Lett. 30, 2575–2577 (2005). [CrossRef] [PubMed]

]). If we take into account that the energy employed to operate this kind of photonic logical gates can be reutilized in some other parts of the system, it can be stated that the performance of these structures start being comparable to that corresponding to their electronic counterparts (in which the typical value of the power consumed in a single logical gate of modern day microprocessors is of the order of 1 fJ). Thus, these results challenge the traditional belief that all-optical logic processing based on nonlinear optical devices is not feasible due to the weak nonlinearities of naturally existing materials.

Summarizing, it has been shown that, due to their unique light confinement mechanism, PhC microcavities are particularly suitable for geometric enhancement of nonlinearities. In addition, optical bistability of PhC microcavities can be used as the basis of complex devices performing all-optical logical operations, such as integrated optical isolation [22

22. M. Soljacic, C. Luo, J. D. Joannopoulos, and S. Fan, “Nonlinear photonic crystal microdevices for optical integration,” Opt. Lett. 28, 637–639 (2003). [CrossRef] [PubMed]

], logical AND gates [25

25. M. F. Yanik, S. Fan, M. Soljacic, and J. D. Joannopoulos, “All-optical transistor action with bistable switching in a photonic crystal cross-waveguide geometry,” Opt. Lett. 28, 2506–2508 (2003). [CrossRef] [PubMed]

] and all-optical transistor action [25

25. M. F. Yanik, S. Fan, M. Soljacic, and J. D. Joannopoulos, “All-optical transistor action with bistable switching in a photonic crystal cross-waveguide geometry,” Opt. Lett. 28, 2506–2508 (2003). [CrossRef] [PubMed]

, 33

33. M. Notomi, A. Shinya, S. Mitsugi, G. Kira, E. Kuramochi, and T. Tanabe, “Optical bistable switching action of Si high-Q photonic-crystal nanocavities,” Opt. Express 13, 2678–2687 (2005). [CrossRef] [PubMed]

]. Importantly, due to their characteristic size, switching time and high integrability, this new class of optical processing devices (and similar approaches [47

47. Q. F. Xu and M. Lipson, “Carrier-induced optical bistability in Silicon ring resonators,” Opt. Lett. 31, 341–343 (2006). [CrossRef] [PubMed]

]) have many of the desired features for their on-chip implementation. Thus, we believe that the results reviewed in this section will pave a way to the future optical integrated devices based on enhanced nonlinearities inside PhC microcavities.

3. Harmonic generation in photonic crystal cavities

The most basic approach to harmonic generation involves two propagating modes interacting through a nonlinear medium, usually via a χ (2) (Pockels) or χ (3) (Kerr) nonlinearity [1

1. R. W. Boyd, Nonlinear Optics (Academic Press, California, 1992).

, 41

41. G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, Y. Dumeige, P. Vidakovic, J. A. Levenson, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, “Photonic band edge effects in finite structures and applications to χ(2) interactions,” Phys. Rev. E 64, 016609 (2001). [CrossRef]

, 48

48. S. Pearl, H. Lotem, Y. Shimony, and S. Rosenwaks, “Optimization of laser intracavity second-harmonic generation by a linear dispersion element,” J. Opt. Soc. Am. B 16, 1705–1711 (1999). [CrossRef]

54

54. P. P. Markowicz, H. Tiryaki, H. Pudavar, P. N. Prasad, N. N. Lepeshkin, and R. W. Boyd, “Dramatic enhancement of third-harmonic generation in three-dimensional photonic crystals,” Phys. Rev. Lett. 92, 083903 (2004). [CrossRef] [PubMed]

]. In this case, light at one frequency co-propagates with the generated light at the harmonic frequency. This scheme poses a series of challenges. First, because each field accumulates a different phase as it travels through the waveguide or medium, a phase-matching condition between the two wavelengths must be satisfied in order for the two modes to couple efficiently [55

55. V. Berger, “Second-harmonic generation in monolithic cavities,” J. Opt. Soc. Am. B 14, 1351–1360 (1997). [CrossRef]

, 56

56. Y. Dumeige and P. Feron, “Wispering-gallery-mode analysis of phase-matched doubly resonant second-harmonic generation,” Phys. Rev. A 74, 063804 (2006). [CrossRef]

]. Second, the pump power required to achieve maximum nonlinear conversion can be quite high. Instead one employs a cavity to trap the light at the input frequency and/or the output frequency, and it turns out that not only does this greatly reduce the power requirement, but it can also enable 100% conversion in principle. The most common scheme is a singly-resonant cavity, in which the input frequency is trapped and the harmonic frequency immediately escapes, so that all of the light is eventually converted if the lifetime is long enough, and negligible down-conversion occurs [27

27. T. V. Dolgova, A. I. Maidykovski, M. G. Martemyanov, A. A. Fedyanin, O. A. Aktsipetrov, G. Marowsky, V. A. Yakovlev, G. Mattei, N. Ohta, and S. Nakabayashi, “Giant optical second-harmonic generation in single and coupled microcavities formed from one-dimensional photonic crystals,” J. Opt. Soc. Am. B 19, 2129–2140 (2002). [CrossRef]

,57

57. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962). [CrossRef]

74

74. L. Scaccabarozzi, M. M. Fejer, Y. Huo, S. Fan, X. Yu, and J. S. Harris, “Enhanced second-harmonic generation in AlGaAs/AlxOy tightly confining waveguides and resonant cavities,” Opt. Lett. 31, 3626–3628 (2006). [CrossRef] [PubMed]

]. A tantalizing possibility, however, is to use a doubly-resonant cavity [35

35. A. Rodriguez, M. Soljacic, J. D. Joannopoulos, and S. G. Johnson, “χ(2) and χ(3) harmonic generation at a critical power in inhomogeneous doubly resonant cavities,” Opt. Express 15, 7303–7318 (2007). [CrossRef] [PubMed]

], in which the nonlinear interaction is enhanced by trapping both the input and the harmonic frequencies. In this case, both up- and down-conversion of the frequency must be included, but it turns out that there is a critical pump power (much lower than the power for a singly resonant cavity) at which 100% conversion can be achieved.

Fig. 4. Schematic diagram of waveguide-cavity system. Input light from a waveguide (left) at one frequency ω 1 is coupled to a doubly-resonant cavity (with resonances at ω 1 and ω 2, with respective lifetimes Q 1 and Q 2) and converted to a cavity mode at another frequency ω 2 by a χ (2) process. The converted light is radiated back into the waveguide at both frequencies.

We now turn to the details of enhanced nonlinear conversion in cavities by focusing first on the particular case of χ (2) nonlinearities. It can be shown that in a single-resonant cavity the critical power P in required for high conversion efficiency scales as VHG/Q [75

75. A. Di Falco, C. Conti, and G. Assanto, “Impedance matching in photonic crystal microcavities for second-harmonic generation,” Opt. Lett. 31, 250–252 (2006). [CrossRef] [PubMed]

] (note that the scaling of P in is similar to that corresponding to the critical power P 0 defined in the former Section). However, in a doubly-resonant nonlinear cavity, in addition to the coupling coefficients between the two fields (~1/VHG), the important figures of merit are the lifetimes Q 1 and Q 2 at the frequencies ω 1 and ω 2=2ω 1, respectively, as depicted schematically in Fig. 4. As shown in [35

35. A. Rodriguez, M. Soljacic, J. D. Joannopoulos, and S. G. Johnson, “χ(2) and χ(3) harmonic generation at a critical power in inhomogeneous doubly resonant cavities,” Opt. Express 15, 7303–7318 (2007). [CrossRef] [PubMed]

], in this case, P in scales as VHG/Q21 Q2 (actually, the singly-resonant cavity can be considered as a special case of small Q 2). The additional factor of Q 2 plays a crucial role in decreasing the critical power. For example, for a singly resonant macroscopic cavity displaying second-harmonic generation at P in=1W operating at a bandwidth Q ~1000, one can immediately reduce this operating power to milliwatt levels and by further reducing VHG to microscopic sizes, one can in principle obtain microwatt levels [35

35. A. Rodriguez, M. Soljacic, J. D. Joannopoulos, and S. G. Johnson, “χ(2) and χ(3) harmonic generation at a critical power in inhomogeneous doubly resonant cavities,” Opt. Express 15, 7303–7318 (2007). [CrossRef] [PubMed]

]. Full expressions for operating powers are obtained by solving explicit coupled-mode equations for the cavity modes [35

35. A. Rodriguez, M. Soljacic, J. D. Joannopoulos, and S. G. Johnson, “χ(2) and χ(3) harmonic generation at a critical power in inhomogeneous doubly resonant cavities,” Opt. Express 15, 7303–7318 (2007). [CrossRef] [PubMed]

].

Fig. 5. Plot of conversion efficiency Pω2out/Pin (black), and reflection Pω1out/Pin vs. P in for the schematic geometry in Fig. 4 (Here in/out denotes input/output power at frequency ω). The maximum conversion efficiency is achieved at the expected critical power P 0. To compute this figure, we have chosen conservative modal parameters ω 1=0.3 2πc/a, Q 1=104, Q 2=2Q1, 1/VHG≈10-5a-3 (where a is the characteristic length scale of the system, see Ref. [35] for further details on this calculation).

Similar phenomena occur if one considers third-harmonic generation in a doubly resonant χ (3) cavity, with cavity frequencies ω 1 and ω 3=3ω 1 and corresponding Q values Q 1 and Q 3. (It is also possible to perform third-harmonic generation via a χ (2) medium, using a combination of second-harmonic and sum-frequency generation [72

72. G. McConnell, A. I. Ferguson, and N. Langford, “Cavity-augmented frequency tripling of a continuous wave mode-locked laser,” J. Phys. D: Appl.Phys 34, 2408–2413 (2001). [CrossRef]

,76

76. K. Koch and G. T. Moore, “Singly resonant cavity-enhanced frequency tripling,” J. Opt. Soc. Am. B 16, 448–459 (1999). [CrossRef]

]). Again, there is a coupling coefficient ~1/VHG. Again, there is a critical power P 0 where 100% conversion is possible, by matching the conversion and reflection rates. Here, the critical power P0~VHG/(Q3/21Q1/23), which for Q 1=Q 3 gives a similar figure of merit as that found in the bistability case VHG/Q2.

All of these statements can be precisely derived from perturbation and coupled-mode theory [35

35. A. Rodriguez, M. Soljacic, J. D. Joannopoulos, and S. G. Johnson, “χ(2) and χ(3) harmonic generation at a critical power in inhomogeneous doubly resonant cavities,” Opt. Express 15, 7303–7318 (2007). [CrossRef] [PubMed]

]. However, χ (3) media introduce an additional wrinkle: self-phase modulation (SPM), where the nonlinearity shifts the cavity frequencies in addition to generating the harmonic. In the bistability phenomenon, SPM was the source of the entire effect, but here it poses a problem because, as the input power is increased, the cavity frequencies shift out of resonance. This shift can be countered by appropriately detuning the cavity frequencies beforehand [35

35. A. Rodriguez, M. Soljacic, J. D. Joannopoulos, and S. G. Johnson, “χ(2) and χ(3) harmonic generation at a critical power in inhomogeneous doubly resonant cavities,” Opt. Express 15, 7303–7318 (2007). [CrossRef] [PubMed]

].

4. Tailoring optical nonlinearities via the Purcell effect

In this section, the impact of the Purcell effect upon the strength of the Kerr nonlinear coefficient is discussed. The Purcell effect, first discovered in 1946 [77

77. E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681–686 (1946).

], is a phenomenon whereby a complex dielectric environment strongly enhances or suppresses spontaneous emission (SE) from a dipole source [78

78. D. Kleppner, “Inhibited spontaneous emission,” Phys. Rev. Lett. 47, 233–236 (1981). [CrossRef]

81

81. D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vuckovic, “Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal,” Phys. Rev. Lett. 95, 013904 (2005). [CrossRef] [PubMed]

]. Recently, it was shown that using the Purcell effect for frequencies close to an atomic resonance can substantially influence the resultant Kerr nonlinearity for light of all (even highly detuned) frequencies [36

36. P. Bermel, A. Rodriguez, J. D. Joannopoulos, and M. Soljacic, “Tailoring optical nonlinearities via the Purcell effect,” Phys. Rev. Lett. 99, 053601 (2007). [CrossRef] [PubMed]

].

Optical nonlinearities are caused by atomic or molecular resonances: the closer the frequency ω becomes to the resonant frequency ωba, the stronger the atom-photon coupling becomes and the larger the nonlinear effects. On the other hand, operating too close to resonance generally leads to large absorption loss, so instead one operates at a detuned frequency ω=ωba+Δ. Intuitively, modifying the SE rate of the resonance changes a property of the resonance which causes nonlinearity, so it should modify the nonlinear process in some way. The key point is that the atomic resonance and the optical mode lie at different frequencies: this makes it possible for a photonic crystal to suppress SE at ωba via a photonic band gap, which increases χ (3) as shown below, while simultaneously operating at a probe frequency ω outside the gap. In fact, one can additionally simultaneously engineer optical resonant cavities at ω, which further enhance nonlinear effects by concentrating the fields.

Moreover, as we show below, this enhancement effect displays some unexpected properties. For example, while increasing SE strengthens the resonance by enhancing the interaction with the optical field, it actually makes the optical nonlinearity weaker. Furthermore, phase damping (e.g., through elastic scattering of phonons), which is detrimental to most optical processes, plays an essential role in this scheme, because in its absence, these effects disappear for large detunings (i.e., the regime in which low loss switching can take place).

In order to approach this problem quantitatively, we start with a simple, generic model displaying Kerr nonlinearities: a collection of two-level systems. The corresponding complex Kerr susceptibility has been calculated in the steady state limit using the rotating wave approximation [1

1. R. W. Boyd, Nonlinear Optics (Academic Press, California, 1992).

, 36

36. P. Bermel, A. Rodriguez, J. D. Joannopoulos, and M. Soljacic, “Tailoring optical nonlinearities via the Purcell effect,” Phys. Rev. Lett. 99, 053601 (2007). [CrossRef] [PubMed]

, 82

82. S. John and T. Quang, “Resonant nonlinear dielectric response in a photonic band gap material,” Phys. Rev. Lett. 76, 2484–2487 (1996). [CrossRef] [PubMed]

], and is given by:

χ(3)=43Nμ4T1T22(ΔT2i)h¯3(1+Δ2T22)2,
(3)

where N is the number of two-level systems, µ is their dipole moment, T -1 1 is the rate of population decay, T -1 2=[(1/2)T -1 1+γ phase] is the rate of phase damping, and Δ≡ω-ωba is the detuning of the incoming wave of frequency ω from the electronic resonance frequency ωba. For large detunings ΔT 2≫1, one obtains the approximation that:

Reχ(3)43Nμ4(1h¯Δ)3T1T2.
(4)

Of course, there are many types of materials to which a simple model of noninteracting two-level systems does not apply. However, it has been shown that χ (3) nonlinearities of some semiconductors such as InSb (a III–V direct bandgap material) can be treated as a collection of independent two-level systems with energies given by the conduction and valence bands, and yield reasonable agreement with experiment [83

83. D. Miller, S. Smith, and B. Wherrett, “The microscopic mechanism of 3rd-order optical nonlinearity in InSb,” Opt. Commun. 35, 221–226 (1980). [CrossRef]

]. The predicted nonlinear coefficient displays the same scaling with lifetimes as Eq. (4), so the considerations that follow should also apply for such semiconductors.

Fig. 6. A 7×7 square lattice of dielectric rods (ε=12.25) in air, with a single defect rod in the middle. On top of the dielectric structure outlined in black, the Ez field is plotted, with positive (negative) values in red (blue). A small region of nonlinear material, e.g., a CdSe nanocrystal, with transition frequency ωelec, is placed in the defect rod.

Now, consider the effects of changing the SE properties for systems modeled by Eq. (4), in which χ (3) scales as T1/T2. Oftentimes, the phase coherence time T 2 will be much smaller than T1 [84

84. M. Nielsen and I. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, England, 2000).

], so that T 2≈γ-1 phase will remain nearly unchanged even if T1 is altered by the Purcell effect. In that case, the enhancement of the real part of χ (3), denoted by η, will be given by ηT 1,purcell/T 1,vac, which means that the suppression of SE will enhance nonlinearities. This comes about because a larger T 1 increases the virtual lifetime for nonlinear processes to occur [85

85. J. J. Sakurai, Modern Quantum Mechanics (Addison-Wesley, Reading, MA, 1994).

]. Conversely, Purcell enhancement suppresses nonlinearities by reducing this virtual lifetime.

The rate of phase damping is crucial to this scheme, because as in the limit that phase damping is controlled exclusively by the SE rate (i.e., T 2≈2T 1), the ratio T 1/T 2 in Eq. (4) will not be altered by changes in SE, and therefore, the nonlinearity will revert to its normal value for large detunings (i.e., the regime in which lossless switching can take place).

It is also interesting to note that this enhancement scheme will generally not increase nonlinear losses, which are a very important consideration in all-optical signal processing. If the nonlinear switching figure of merit ξ is defined by ξ=Reχ (3)/(λ Imχ (3)) [86

86. G. Lenz, J. Zimmermann, T. Katsufuji, M. E. Lines, H. Y. Hwang, S. Spalter, R. E. Slusher, S. W. Cheong, J. S. Sanghera, and I. D. Aggarwal, “Large Kerr effect in bulk Se-based chalcogenide glasses,” Opt. Lett. 25, 254–256 (2000). [CrossRef]

], then ξ purcell/ξ vacuum=T 2,purcell/T 2,vacuum≥1, for all cases of suppressed SE.

The general principle described thus far should apply for any medium where the local density of states (DOS) is substantially modified provided that one is in the weak-coupling regime [87

87. V. Savona, L. C. Andreani, P. Schwendimann, and A. Quattropani, “Quantum well excitons in semiconductor microcavities: unified treatment of weak and strong coupling regimes”, Solid State Comm. 93, 733–739 (1995). [CrossRef]

]. In the following, we show how this effect would manifest itself in a PhC system, thus illustrating how strong nonlinear suppression or enhancement effects could be achieved in practical physical systems. Our system consists of a seven by seven 2D square lattice of dielectric rods (ε=12.25 and radius 0.25a) in air, with a two-level system placed in the middle defect rod of radius r=0.35a, as illustrated in Fig. 6.

First, consider the magnitude of the enhancement or suppression of SE in this system. Clearly, since there are several periods of high contrast dielectric, one expects to observe two distinct effects. First, there will be a substantial but incomplete suppression of emission inside the bandgap. Second, there will be an enhancement of SE outside the bandgap (since the DOS is shifted to the frequencies surrounding the bandgap), and also close to the cavity resonance. For an atom polarized in the direction out of the 2-D plane, only the TM polarization need be considered. The enhancement of SE obtained in a time-domain simulation is plotted in Fig. 7(a).

Fig. 7. (a) Numerical calculation of the enhancement of SE for the set-up in Fig. 6, given by the ratio of the rate of emission in the PhC, T -1 1,purcell, divided by the emission rate in vacuum, T -1 1,vac. (b) Kerr enhancement η≡Reχ(3)purcell/Reχ (3) vac as a function of electronic transition frequency (ω elec) for a system of dielectric rods in air, with the parameter values listed in the text.

Some recent work has demonstrated that single nanocrystals can demonstrate predominantly radiative decay in vacuum even at room temperature, e.g., single CdSe/ZnS core-shell nanocrystals with a peak emission wavelength of 560 nm have Γrad≈39Γnr, where the radiative lifetime T 1,rad=25.5 ns [88

88. X. Brokmann, L. Coolen, M. Dahan, and J. P. Hermier, “Measurement of the radiative and nonradiative decay rates of single CdSe nanocrystals through a controlled modification of their spontaneous emission,” Phys. Rev. Lett. 93, 107403 (2004). [CrossRef] [PubMed]

]. Extrapolating from low temperature results [89

89. H. Shinojima, “Optical nonlinearity in CdSSe microcrystallites embedded in glasses,” IEICE Trans. Electron. E90-C, 127–134 (2007). [CrossRef]

], we estimate T 2≈0.13 fs at room temperature.

When these data are combined with the data from Fig. 7(a), we can calculate the enhancement of the real part of the Kerr coefficient, η, as a function of the atomic transition frequency ω elec, when probed at a frequency ω ph=0.508(2π c/a) (the cavity’s resonant frequency). The results are shown in Fig. 7(b). Enhancements of up to a factor of 21 are predicted in the regions in which SE is suppressed, for these exact parameter values. This set-up has the advantage of allowing simultaneous probe field enhancement (through the concentration of the field in the defect mode), along with nonlinear coefficient enhancement (through the suppression of spontaneous emission). The enhancement factor is less than the theoretical maximum value of 40 obtained from the expression for maximum nonlinear enhancement that takes the quotient of T1 values with and without complete SE suppresion for this material because the SE is suppressed only by a factor of about 43, as shown in Fig. 7(a). However, a much bigger PhC, which suppresses SE very strongly, should approach the theoretical maximum enhancement. If a single nanocrystal with the parameters above proves difficult to use in practical devices, note that even bulk samples of similar nanocrystals have been shown to yield a significant radiative decay component, corresponding to Γrad≈Γnr [90

90. D. V. Talapin, A. L. Rogach, A. Kornowski, M. Haase, and H. Weller, “Highly luminescent monodisperse CdSe and CdSe/ZnS nanocrystals synthesized in a hexadecylamine-trioctylphosphine oxide-trioctylphospine mixture,” Nano Lett. 1, 207–211 (2001). [CrossRef]

]. Thus, it has been predicted that with strong suppression of radiative decay, nonlinear enhancement of a factor of two or more could be observed at room temperature, for certain materials.

Summarizing, it has been shown that the Purcell effect can be used to tailor optical nonlinearities [36

36. P. Bermel, A. Rodriguez, J. D. Joannopoulos, and M. Soljacic, “Tailoring optical nonlinearities via the Purcell effect,” Phys. Rev. Lett. 99, 053601 (2007). [CrossRef] [PubMed]

]. We have illustrated in an exemplary system how enhancements of Kerr nonlinearities of at least one order of magnitude should be achievable. This phenomenon is caused by strong modification of the local DOS near the resonant frequency. Thus, our treatment can easily be applied to analyze the Kerr nonlinearities of two-level systems in almost any geometrical structure in which the Purcell effect is substantial (e.g., PhC fibers [91

91. N. M. Litchinitser, A. Abeeluck, C. Headley, and B. Eggleton, “Antiresonant reflecting photonic crystal optical waveguides,” Opt. Lett. 27, 1592–1594 (2002). [CrossRef]

] or optical cavities). It also presents a reliable model for a variety of materials, such as quantum dots, atoms, and certain semiconductors. Finally, note that the physical principle described in this Section (i.e., the strong modification of nonlinearities via the Purcell effect) should have a general character, and thus may apply to coefficients at other orders, such as χ (5), and to other materials (e.g. where two-level approximation does not apply).

5. Summary and conclusions

In summary, all these phenomena have already begun to enable the design of novel all-optical signal processing devices whose operating powers and switching times are orders of magnitude smaller than those corresponding to traditional nonlinear optical devices. These properties, combined some other key features of these devices, such as their micrometric size and their high integratibility, could make nonlinear photonic crystal cavities one of the most important actors in the development of future photonic integrated technology.

Acknowledgements

This workwas supported in part by the Materials Research Science and Engineering Center Program of the National Science Foundation under Grant No. DMR 02-13282, the Army Research Office through the Institute for Soldier Nanotechnologies Contract No. W911NF-07-D-0004, and the U.S. Department of Energy under Grant No. DE-FG02-99ER45778.

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OCIS Codes
(190.0190) Nonlinear optics : Nonlinear optics
(190.4360) Nonlinear optics : Nonlinear optics, devices

ToC Category:
Nonlinear Optics for Functional Devices and Applications

History
Original Manuscript: September 4, 2007
Revised Manuscript: November 12, 2007
Manuscript Accepted: November 12, 2007
Published: November 21, 2007

Virtual Issues
Focus Serial: Frontiers of Nonlinear Optics (2007) Optics Express

Citation
Jorge Bravo-Abad, Alejandro Rodriguez, Peter Bermel, Steven G. Johnson, John D. Joannopoulos, and Marin Soljacic, "Enhanced nonlinear optics in photonic-crystal microcavities," Opt. Express 15, 16161-16176 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-24-16161


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