## Purcell effect in nonlinear photonic structures: A coupled mode theory analysis

Optics Express, Vol. 16, Issue 17, pp. 12523-12537 (2008)

http://dx.doi.org/10.1364/OE.16.012523

Acrobat PDF (411 KB)

### Abstract

We develop a coupled mode theory (CMT) model of the behavior of a polarization source in a general photonic structure, and obtain an analytical expression for the resulting generated electric field; loss, gain and/or nonlinearities can also be modeled. Based on this treatment, we investigate the criteria needed to achieve an enhancement in various nonlinear effects, and to produce efficient sources of terahertz radiation, in particular. Our results agree well with exact finite-difference time-domain (FDTD) results. Therefore, this approach can also in certain circumstances be used as a potential substitute for the more numerically intensive FDTD method.

© 2008 Optical Society of America

## 1. Introduction

1. P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of Optical Harmonics,” Phys. Rev. Lett. **6**, 118–119 (1961). [CrossRef]

4. P. D. Maker, R. W. Terhune, M. Nisenhoff, and C. M. Savage, “Effects of Dispersion and Focusing on the Production of Optical Harmonics,” Phys. Rev. Lett. **8**, 21–22 (1962). [CrossRef]

3. J. A. Giordmaine, “Mixing of Light Beams in Crystals,” Phys. Rev. Lett. **8**, 19–20 (1962). [CrossRef]

2. M. Bass, P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Optical Mixing,” Phys. Rev. Lett. **8**, 18–18 (1962). [CrossRef]

5. 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]

6. N. Bloembergen and Y. R. Shen, “Quantum-Theoretical Comparison of Nonlinear Susceptibilities in Parametric Media, Lasers, and Raman Lasers,” Phys. Rev. **133**, A37–A49 (1964). [CrossRef]

7. G. Eckardt, R. W. Hellwarth, F. J. McClung, S. E. Schwartz, D. Weiner, and E. J Woodbury, “Stimulated Raman Scattering From Organic Liquids,” Phys. Rev. Lett. **9**, 455 (1962). [CrossRef]

9. J. Squier and M. Müller, “High resolution nonlinear microscopy: a review of sources and methods for achieving optimal imaging,” Rev. Sci. Instrum. **72**, 2855–2867 (2001). [CrossRef]

12. J. M. Dudley, C. Finot, D. J. Richardson, and G. Millot, “Self-similarity in ultrafast nonlinear optics,” Nature Phys. **3**, 597–603 (2007). [CrossRef]

10. Yuri Kivshar, “Spatial solitons: Bending light at will,” Nature Physics **2**, 729–730 (2006). [CrossRef]

11. Günter Steinmeyer, “Laser physics: Terahertz meets attoscience,” Nature Physics **2**, 305–306 (2006). [CrossRef]

8. G. P. Agrawal, A. Hasegawa, Y. Kivshar, and S. Wabnitz, “Introduction to the Special Issue on Nonlinear Optics,” IEEE J. Sel. Top. In Quantum Electron. **8**, 405–407 (2002). [CrossRef]

13. B. Ferguson and X.-Cheng Zhang, “Materials for terahertz science and technology,” Nature Materials **1**, 26–33 (2002). [CrossRef]

16. M. Soljačić and J. D. Joannopoulos, “Enhancement of non-linear effects using photonic crystals,” Nature Materials **3**, 211–219 (2004). [CrossRef] [PubMed]

17. G. DAguanno, 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]

19. J. Bravo-Abad, A. Rodriguez, P. Bermel, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Enhanced nonlinear optics in photonic-crystal microcavities,” Opt. Express **15**, 16161–16176 (2007). [CrossRef] [PubMed]

16. M. Soljačić and J. D. Joannopoulos, “Enhancement of non-linear effects using photonic crystals,” Nature Materials **3**, 211–219 (2004). [CrossRef] [PubMed]

20. T. Laroche, F. I. Baida, and D. Van Labeke, “Three-dimensional finite-difference time-domain study of enhanced second-harmonic generation at the end of a apertureless scanning near-field optical microscope metal tip,” J. Opt. Soc. Am. B **22**, 1045–1051 (2005). [CrossRef]

21. K. Sakoda and K. Ohtaka, “Optical response of three-dimensional photonic lattices: Solutions of inhomogeneous Maxwells equations and their applications,” Phys. Rev. B **54**, 5732–5741 (1996). [CrossRef]

22. K. Sakoda and K. Ohtaka, “Sum-frequency generation in a two-dimensional photonic lattice,” Phys. Rev. B **54**, 5742–5749 (1996). [CrossRef]

## 2. Coupled-mode-theory model

*P⃗*(

*r⃗*,

*t*) inside a photonic structure. Let (

*E⃗*

_{ν},

*H⃗*

_{ν}) label a mode of the source-free solutions to Maxwell’s equations obtained by using linear real indices of refraction of the photonic structure; while calculating the modes (

*E⃗*

_{ν},

*H⃗*

_{ν}), we assume that the photonic structure does not involve any loss, gain or nonlinearities. The effects of loss/gain will be addressed perturbatively below through the use of CMT. Nonlinear effects that are of interest for THz generation are included in the sense that the polarization source itself is generated through nonlinear effects, starting from some external electric fields of different frequencies; this polarization source, in turn excites the modes of the structure.

*D⃗*

_{ν}(

*r⃗*)=

*ε*(

*r⃗*)

*E⃗*

_{ν}(

*r⃗*), form a complete set[24], since they are eigenmodes of the hermitian operator

*P⃗*(

*r⃗*,

*t*) induces in the structure electromagnetic fields

*E⃗*(

*r⃗*,

*t*) and

*H⃗*(

*r⃗*,

*t*). If we denote the electric displacement vector by

*D⃗*(

*r⃗*,

*t*), then

*a*

_{ν}is the amplitude of the mode labeled by ν, normalized such that |

*a*

_{ν}|

^{2}is the total energy in that particular mode.

*k⃗*and polarization σ, hence ν is to be identified with (

*k⃗*,σ) in this case. For a photonic crystal, it is most convenient to label the modes by a band index

*n*, a polarization σ, and a Bloch wavevector

*k⃗*that lies in the first Brillouin zone. Therefore, we identify ν≡(

*n*,

*k⃗*,σ) for photonic crystals. For a finite photonic crystal structure, the allowed values of

*k⃗*consistent with boundary conditions, are discrete. However, for infinite photonic crystal structures, there is a continuum of allowed values of

*k⃗*, and the sum in Eq. (1) transforms into a discrete sum over n and an integral over

*k⃗*.

*ω*

_{ν}is the frequency of the mode labeled by ν, Γ

^{ν}

*and Γ*

_{rad}^{ν}

*are the rates of radiative (out of the structure) and absorptive decay, respectively, and Γ*

_{abs}^{ν}

*is the rate of gain. κ*

_{g}_{ν}

*S*

^{ν}

_{+}is the square root per unit time of the portion of the polarization source’s energy, that couples to the photonic structure; i.e. this term models the excitation of the mode of the structure by the polarization source. From Poynting’s theorem, κ

_{ν}

*S*

^{ν}

_{+}is given by

*t*is any reference time preceding the turn-on of the polarization source, and

_{o}*t*>

*t*.

_{o}*A*at time t is

*E⃗*(

*r⃗*,

*t*) is given by Eq. (6), and

*H⃗*(

*r⃗*,

*t*) can be obtained from Faraday’s law: ∇⃗×

*E⃗*(

*r⃗*,

*t*)=-∂[

*µ*

_{0}

*H⃗*(

*r⃗*,

*t*)]/∂

*t*,

*µ*

_{0}being the magnetic permeability.

## 3. Connection with Purcell effect

*p⃗*, embedded at position

*r⃗*in the structure. If we denote the dipole’s angular frequency by

_{o}*ω*, then the polarization is

_{s}^{ν}

*=0∀ν (no gain). In this case, the resulting mode amplitude*

_{g}*a*

_{ν}becomes

^{ν}=Γ

^{ν}

*+Γ*

_{rad}^{ν}

*, the total radiated power*

_{abs}^{ν}→0), we have Γ

^{ν}/{(

*ω*

_{ν}-

*ω*)

_{s}^{2}+(Γ

^{ν})

^{2}}→

*πδ*(

*ω*

_{ν}-

*ω*). Hence

_{s}*p̂*·

*E⃗*

^{*}

_{ν}(

*r⃗*)|

_{o}^{2}=|

*E⃗*

_{ν}(

*r⃗*)|

_{o}^{2}∀ν , and

*r⃗*. Therefore, the total radiated power (even from a nonlinear source) is proportional to the local density of states, as expected from the Purcell effect. It also increases quadratically with the dipole moment of the polarization source.

_{o}## 4. Connection with Doppler radiation in a PhC crystal

*ν⃗*in a PhC[25

25. C. Luo, M. Ibanescu, E. J. Reed, S. G. Johnson, and J. D. Joannopoulos, “Doppler Radiation Emitted by an Oscillating Dipole Moving inside a Photonic Band-Gap Crystal,” Phys. Rev. Lett. **96**, 043903 (2006). [CrossRef] [PubMed]

^{ν}=0 ∀ ν, we obtain

*n*,

*k⃗*,σ), and from Bloch theorem

*E⃗*(

_{nk⃗}*r⃗*)=

*e*(

^{ik⃗.r⃗}u⃗_{nk⃗}*r⃗*), where

*u⃗*(

_{nk⃗}*r⃗*) has the periodicity of the PhC, and hence only reciprocal lattice vectors

*G⃗*appear in its Fourier expansion

*u⃗*(

_{nk⃗}*r⃗*)=∑

*. Hence Eq. (14) becomes*

_{G⃗}e⃗_{nk⃗,G⃗}e^{iG⃗.r⃗}25. C. Luo, M. Ibanescu, E. J. Reed, S. G. Johnson, and J. D. Joannopoulos, “Doppler Radiation Emitted by an Oscillating Dipole Moving inside a Photonic Band-Gap Crystal,” Phys. Rev. Lett. **96**, 043903 (2006). [CrossRef] [PubMed]

*ω*≃

_{nk⃗σ}*ω*+(

_{S}*k⃗*+

*G⃗*).v⃗.

*E⃗*(

*r⃗*,

*t*) induced in the structure by a given external polarization source. Clearly, we need to have the largest possible density of modes that yield a significant value of the summand on the right hand side of Eq. (6). To achieve such large values of the summand, the phase-matched modes ought to have their resonance frequency

*ω*

_{ν}close enough to the frequency

*ω*of the source. Their polarization should also be matched to the source’s polarization, and their spatial overlap with the source’s extent should be considerable. This latter requirement suggests that the modes should be highly concentrated at the position of the source. And of course, we should attempt to reduce losses to the lowest possible level.

_{S}## 5. Numerical validation and terahertz generation

### 5.1. Optimizing the structure

^{(2)}nonlinearity, and that satisfies as many as possible of the criteria stated in section III. Before getting into the details of the structure, let’s outline briefly the fundamental principles. We propose to excite our PhC structure with a beam centered around an optical frequency; we denote the optical electric field by

*E⃗*. By optical rectification, this optical beam generates a polarization 𝓟

_{opt}*~χ*

^{THz}^{(2)}

*E*

_{opt}E^{*}

*in the terahertz frequency range[26*

_{opt}26. D. H. Auston, K. P. Cheung, J. A. Valdmanis, and D. A. Kleinman, “Cherenkov Radiation from Femtosecond Optical Pulses in Electro-Optic Media,” Phys. Rev. Lett. **53**, 1555–1558 (1984). [CrossRef]

*excites terahertz waves depends on several considerations pertaining to the PhC structure. First, one of course strives to use a material with χ*

^{THz}^{(2)}as large as possible. Next, another option is to increase the magnitude of

*E⃗*as much as possible; ultimately there is a limit to which one can pursue this approach, imposed by the optical breakdown threshold. Finally one can adjust the properties of the PhC structure at terahertz frequencies, to achieve high density of states and good spatial overlap between the terahertz modes and the source 𝓟

_{opt}*.*

^{THz}*r*=0.13

*a*,

*t*=0.04

*a*and

*n*=3.5 (close to the refractive index of GaP[28

_{r}28. G. Chang, C. J. Divin, J. Yang, M. A. Musheinish, S. L. Williamson, A. Galvanauskas, and T. B. Norris, “GaP waveguide emitters for high power broadband THz generation pumped by Yb-doped fiber lasers,” Opt. Express **15**, 16308–16315 (2007). [CrossRef] [PubMed]

29. T. Tanabe, K. Suto, J.-ichi Nishizawa, K. Saito, and T. Kimura, “Frequency-tunable terahertz wave generation via excitation of phonon-polaritons in GaP,” J. Phys. D: Applied Physics **36**, 953–957 (2003). [CrossRef]

*n*=2) of the transverse magnetic (σ=TM) modes is characterized by a saddle point where the band is ultraflat, and consequently the density of states is enhanced. The band structure was computed by preconditioned conjugate-gradient minimization of the block Rayleigh quotient in a planewave basis, using a freely available software package[30

30. S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express **8**, 173–190 (2001). [CrossRef] [PubMed]

*k*=0.1559(2

_{x}*π*/

*a*), where it has a width of only 1.27% around the central frequency

*f*=0.509(

*c*/

*a*); thereby if the targeted frequency is e.g.

*f*=1 THz, one needs to choose

*a*=152.7

*µ*m.

*z*(parallel to the rods), with

*k*=0.1559(2

_{x}*π*/

*a*), and by choosing

*a*appropriately, we can tune the structure to be optimized for any frequeny in the THz regime. Since we are interested in the very small frequency range around the saddle point of the second band, we consider a narrow-bandwidth excitation. To this end, we assume that the optical beam is sent through the waveguide centered at the origin, and is particularly chosen such that the current density

*J⃗*associated with the polarization source 𝓟

^{THz}*, has the form:*

^{THz}*k*=0.1559(2

^{s}_{x}*π*/

*a*) to ensure phase matching with the modes at the ultraflat portion of the second band. The angular frequency of the terahertz polarization source is

*ω*=0.509(2

_{S}*πc*/

*a*), and the values of the remaining parameters are:

*ζ*=0.02

*a*and

*τ*=100(

*a*/

*c*). The time

*t*is expressed in units of

*a*/

*c*. Since

*J⃗*points along

^{THz}*ẑ*, all the TM modes have their polarization perfectly matched to that of the source. For convenience in subsequent calculations, and because

*J⃗*(

^{THz}*x*,

*y*;

*t*) is separable in space and time, we write it as

*J⃗*(

^{THz}*x*,

*y*;

*t*)=0 for

*t*<0 and

*t*≥1000

*a*/

*c*.

*n*=2;

*k*=0.1559(2

_{x}*π*/

*a*),

*k*; σ=TM) and the terahertz polarization source. The degree of localization of the modes

_{y}*k*values. We computed the fields

_{y}30. S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express **8**, 173–190 (2001). [CrossRef] [PubMed]

*k*values. A representative plot is shown in Fig. 3 for

_{y}*k*=0. Despite the good localization of the modes of interest close to the source’s extent, this does not guarantee an optimum for the overlap integral

_{y}*n*=2. Indeed, for most of the different possible

*k*in the first brillouin zone, the integrand of the overlap integral

_{y}*x̂*, and hence integrates to a negligible value. That is, for

*n*=2 and for many

*k*, the integrand is observed to flip the sign at

_{y}*x*≃±0.25

*a*. One way to circumvent this problem is to periodically pole the structure every half period. More specifically, we propose to flip the sign of χ

^{(2)}at

*x*=±(0.25

*a*+ℓ0.5

*a*) where ℓ is a zero or positive integer, as illustrated in Fig. 4. Mathematically, this corresponds to multiplying the integrand in Eq. (22) by a factor

*q*(

*x*′), which is +1 in regions where χ

^{(2)}is positive, and -1 where χ

^{(2)}is negative. In this way, the integrand of the overlap integral 𝒪

_{n2;k⃗}preserves the same sign for most of

*x*, and therefore the integral evaluates to a substantially higher value than without any poling. Although this looks similar to quasi-phase matching[31], one should keep in mind that our motivation behind poling was to prevent the overlap integral of the ‘second’ band from vanishing. For the first band, the integrand of the overlap integral 𝒪

_{n=1;k⃗}does not change sign along

*x̂*, and so we wouldn’t have to pole the structure if we were interested in modes of the first band. Similarly, if we were interested in modes of the third band, we would have to use a poling configuration different from that used in the second band. Also note that although periodic poling leads to much more efficient coupling between the nonlinear polarization source and the second-band modes of the PhC structure, all the linear properties on which the numerical calculations are based, remain intact.

### 5.2. Calculation of generated energy

*E⃗*(

*r⃗*,

*t*) induced by the THz current source (Eq. (17)) in the periodically poled PhC structure, using Eq. (6). We consider a time of

*t*=1010(

*a*/

*c*) (which is 10

*a*/

*c*later than source turn-off), and we assume that

*a*=0 ∀(

_{nk⃗}*n*,

*k⃗*) prior to source turn-on at

*t*=0; this corresponds to the case where none of the modes of the PhC were excited before

*t*=0. We use only modes with

*k*=0.1559(2

_{x}*π*/

*a*)=

*k*, and we further assume Γ

^{s}_{x}*=Γ*

^{nk⃗}_{g}*=0. From Eq. (6), it is evident that we cannot proceed before computing the following two functions of*

^{nk⃗}_{abs}*k*

_{y}*u⃗*(

_{nk⃗}*r⃗*) has the same periodicity as the PhC. Consequently, 𝒪

*(*

^{poled}_{all space}*k*) becomes

_{y}*x*′. Hence the integral over all

*x*′ simplifies to the number 𝓝

*of periods in the x-direction times the integral over a single period, with*

_{x}*x*′ ranging from -

*a*/2 to

*a*/2. The integration over

*y*′ can also be taken to range between -

*a*/2 and

*a*/2, because the optical beam is sent through the central waveguide only, and thus

*J⃗*is zero for

^{THz}_{space}*y*′ outside the interval [-

*a*/2,

*a*/2]. Therefore

*a*, and calculate the TM fields

30. S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express **8**, 173–190 (2001). [CrossRef] [PubMed]

*TM*. While computing the modes, we make sure that all of them have their phases fixed relative to each other, and that they are all normalized in the same way, e.g. such that

*(*

^{poled}_{one period}*k*), as a function of

_{y}*k*, is shown in Fig. 5 (after poling is performed). Clearly, the integral takes values close to maximum for most of the

_{y}*k*’s. The reason for which it vanishes as

_{y}*k*→±0.5(2

_{y}*π*/

*a*) is that those modes have an extended node over the whole source, and consequently the integrand in Eq. (28) vanishes almost everywhere, either because the source is zero or because the electric field of the modes is zero. So whether the structure is poled or not, the overlap integral vanishes for modes with

*k*=±0.5(2

_{y}*π*/

*a*).

*(*

_{t}*k*) at

_{y}*t*=1010(

*a*/

*c*), using the previously calculated band structure. Eq. (6) becomes

*x*and

*y*directions, the

*k*values consistent with periodic boundary conditions, become dense enough that to a good approximation, the discrete sum over

_{y}*k*can be converted into an integral

_{y}*E⃗*(

*r⃗*,

*t*) at time

*t*=1010(

*a*/

*c*), over a region of space large enough in the

*y*direction, hoping that no energy would have left it by

*t*=1010(

*a*/

*c*). So, we take our computational domain to be a 2D box of size 1

*a*along

*x*and 80

*a*along

*y*. To save on computational memory, we use a spatial resolution of 64 gridpoints/

*a*only, and calculate the TM fields

*k*ranging between-

_{y}*π*/

*a*and

*π*/

*a*, again using the software package[30

**8**, 173–190 (2001). [CrossRef] [PubMed]

*k*, by 𝒪

_{y}*(*

^{poled}_{one period}*k*)·𝒯

_{y}_{1010(a/c)}(

*k*), and sum the resulting fields over all values of

_{y}*k*. Finally, we multiply the result by

_{y}*a*/

*π*and obtain the THz electric field induced by the optical beam in the PhC at time

*t*=1010(

*a*/

*c*). Note that we attempted to use a denser grid of

*k*values (334 equidistant

_{y}*k*points), without change in the final result. The electric energy density profile

_{y}*ε*(

*x*,

*y*)|

*E*(

*x*,

*y*,

*t*)|

^{2}, at

*t*=1010(

*a*/

*c*), can now be simply calculated over the computational domain, and is presented in Fig. 6.

*y*∈±[8

*a*,19

*a*]. Thus, to calculate the total emitted energy, it is sufficient to integrate the energy density at

*t*=1010(

*a*/

*c*) over a 2D box of size 1

*a*along

*x*, and large enough in the

*y*direction to enclose the regions

*y*∈±[8

*a*,19

*a*]. Setting

*a*=1, we obtain a value of 0.7108 for the total THz energy emitted in the PBG. To enable comparison with the FDTD results, we normalize the total emitted energy and express it in dimensionless units; that is, we divide the above-mentioned integral of the energy density in the poled PhC case, by the same quantity (0.1839) similarly calculated in an unpoled bulk of the same nonlinear material as that used in the PhC structure. We obtain a dimensionless value of 3.86.

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

*a*along

*x*, and 90

*a*along

*y*, and we discretize the structure with a resolution of 128 pixels/

*a*. We send the same source described before through the waveguide centered at

*y*=0, and we impose Bloch-periodic boundary conditions along

*x*, with a wavevector

*k*=0.1559(2

_{x}*π*/

*a*). At the boundaries in the y direction, we set up perfectly matched layers (PML), each of thickness 3

*a*. We simulate the effect of periodic poling by explicitely reversing the sign of the polarization source at

*x*=±0.25

*a*. To compute the total emitted energy at terahertz, we record the time evolution of the energy 𝓔 in a box of size 1

*a*(along

*x*)×80

*a*(along

*y*), centered at the origin. We also place flux calculation planes at

*y*=±40

*a*, and compute the flux that leaves through each of the flux planes as a function of time. The total emitted energy at a particular time

*t*is then given by the sum of the integral up to time

*t*of the net flux through the flux planes, and the energy remaining in the box surrounded by the flux planes, at time

*t*.

*t*=1010(

*a*/

*c*), and we show the result as a color contour plot in Fig. 8. The agreement between Fig. 6 and Fig. 8 is indeed remarkable; not only do we get a coincidence of the intervals

*y*∈±[8

*a*,19

*a*] in which the energy density is nonvanishing, but also the waveguides at which the energy density is maximum occur at exactly the same position, namely

*y*=±13

*a*, according to both methods (FDTD and CMT-based analytics). In addition to validating our analytical CMT-based formalism, the agreement of the results obtained from our analytical model with the exact FDTD results, suggests that our approach would work as a simpler alternative to the numerically intensive FDTD method. Our procedure has the advantage of being far less demanding than the brute-force FDTD technique, in terms of the computational time and resources, especially in problems involving frequencies that range over many orders of magnitude, such as terahertz generation by optical rectification. To get a concrete estimate, we mention that, although we assumed the terahertz polarization source to be given from the beginning, and dealt with terahertz frequencies only, the calculation of energy density profile shown in Fig. 6 took around 15 minutes on a single processor, while the FDTD calculations resulting in Fig. 7 took around 10 hours using 360 processors on a supercomputer. Finally, the FDTD calculations of Fig. 8 took ≃10 minutes using 32 processors on a supercomputer when a spatial resolution of 64 gridpoints/

*a*was used, and ≃2 hours using 360 processors when the spatial resolution was 128 gridpoints/

*a*. Note that the only assumption on which our analytical formalism is based is that CMT be valid, meaning that the rates Γ

^{ν}be much smaller than the frequency

*ω*

_{ν}for each mode ν of the photonic structure[23].

## 6. Conclusion

## Acknowledgments

## References and links

1. | P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of Optical Harmonics,” Phys. Rev. Lett. |

2. | M. Bass, P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Optical Mixing,” Phys. Rev. Lett. |

3. | J. A. Giordmaine, “Mixing of Light Beams in Crystals,” Phys. Rev. Lett. |

4. | P. D. Maker, R. W. Terhune, M. Nisenhoff, and C. M. Savage, “Effects of Dispersion and Focusing on the Production of Optical Harmonics,” Phys. Rev. Lett. |

5. | J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between Light Waves in a Nonlinear Dielectric,” Phys. Rev. |

6. | N. Bloembergen and Y. R. Shen, “Quantum-Theoretical Comparison of Nonlinear Susceptibilities in Parametric Media, Lasers, and Raman Lasers,” Phys. Rev. |

7. | G. Eckardt, R. W. Hellwarth, F. J. McClung, S. E. Schwartz, D. Weiner, and E. J Woodbury, “Stimulated Raman Scattering From Organic Liquids,” Phys. Rev. Lett. |

8. | G. P. Agrawal, A. Hasegawa, Y. Kivshar, and S. Wabnitz, “Introduction to the Special Issue on Nonlinear Optics,” IEEE J. Sel. Top. In Quantum Electron. |

9. | J. Squier and M. Müller, “High resolution nonlinear microscopy: a review of sources and methods for achieving optimal imaging,” Rev. Sci. Instrum. |

10. | Yuri Kivshar, “Spatial solitons: Bending light at will,” Nature Physics |

11. | Günter Steinmeyer, “Laser physics: Terahertz meets attoscience,” Nature Physics |

12. | J. M. Dudley, C. Finot, D. J. Richardson, and G. Millot, “Self-similarity in ultrafast nonlinear optics,” Nature Phys. |

13. | B. Ferguson and X.-Cheng Zhang, “Materials for terahertz science and technology,” Nature Materials |

14. | E. Mueller, “Terahertz radiation sources for imaging and sensing applications,” Photonics Spectra |

15. | E. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. |

16. | M. Soljačić and J. D. Joannopoulos, “Enhancement of non-linear effects using photonic crystals,” Nature Materials |

17. | G. DAguanno, 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 |

18. | A. Taflove, |

19. | J. Bravo-Abad, A. Rodriguez, P. Bermel, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Enhanced nonlinear optics in photonic-crystal microcavities,” Opt. Express |

20. | T. Laroche, F. I. Baida, and D. Van Labeke, “Three-dimensional finite-difference time-domain study of enhanced second-harmonic generation at the end of a apertureless scanning near-field optical microscope metal tip,” J. Opt. Soc. Am. B |

21. | K. Sakoda and K. Ohtaka, “Optical response of three-dimensional photonic lattices: Solutions of inhomogeneous Maxwells equations and their applications,” Phys. Rev. B |

22. | K. Sakoda and K. Ohtaka, “Sum-frequency generation in a two-dimensional photonic lattice,” Phys. Rev. B |

23. | H. A. Haus, |

24. | J. D. Joannopoulos, R. D. Meade, and J. N Winn, |

25. | C. Luo, M. Ibanescu, E. J. Reed, S. G. Johnson, and J. D. Joannopoulos, “Doppler Radiation Emitted by an Oscillating Dipole Moving inside a Photonic Band-Gap Crystal,” Phys. Rev. Lett. |

26. | D. H. Auston, K. P. Cheung, J. A. Valdmanis, and D. A. Kleinman, “Cherenkov Radiation from Femtosecond Optical Pulses in Electro-Optic Media,” Phys. Rev. Lett. |

27. | M. Ibanescu, E. J. Reed, and J. D. Joannopoulos, “Enhanced Photonic Band-Gap Confinement via Van Hove Saddle Point Singularities,” Phys. Rev. Lett. |

28. | G. Chang, C. J. Divin, J. Yang, M. A. Musheinish, S. L. Williamson, A. Galvanauskas, and T. B. Norris, “GaP waveguide emitters for high power broadband THz generation pumped by Yb-doped fiber lasers,” Opt. Express |

29. | T. Tanabe, K. Suto, J.-ichi Nishizawa, K. Saito, and T. Kimura, “Frequency-tunable terahertz wave generation via excitation of phonon-polaritons in GaP,” J. Phys. D: Applied Physics |

30. | S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express |

31. | R. W. Boyd, |

32. | A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, and G. Burr, “Improving accuracy by subpixel smoothing in the finite-difference time domain,” Opt. Lett. |

**OCIS Codes**

(190.4720) Nonlinear optics : Optical nonlinearities of condensed matter

(260.2110) Physical optics : Electromagnetic optics

(350.4238) Other areas of optics : Nanophotonics and photonic crystals

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: June 24, 2008

Revised Manuscript: July 17, 2008

Manuscript Accepted: July 18, 2008

Published: August 4, 2008

**Citation**

Rafif E. Hamam, Mihai Ibanescu, Evan J. Reed, Peter Bermel, Steven G. Johnson, Erich Ippen, J. D. Joannopoulos, and Marin Soljacic, "Purcell effect in nonlinear photonic structures: a coupled mode theory
analysis," Opt. Express **16**, 12523-12537 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-17-12523

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