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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 17 — Aug. 18, 2008
  • pp: 12523–12537
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Purcell effect in nonlinear photonic structures: A coupled mode theory analysis

Rafif E. Hamam, Mihai Ibanescu, Evan J. Reed, Peter Bermel, Steven G. Johnson, Erich Ippen, J. D. Joannopoulos, and Marin Soljačić  »View Author Affiliations


Optics Express, Vol. 16, Issue 17, pp. 12523-12537 (2008)
http://dx.doi.org/10.1364/OE.16.012523


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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

Since the emergence of nonlinear optics[1

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]

] in 1961, and the major breakthroughs[4

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

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

, 2

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

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

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

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]

] that subsequently marked its growth, the field of nonlinear optics has been producing continuous scientific excitement: Numerous nonlinear optical phenomena have been discovered, and have significantly impacted scientific progress in many respects[9

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

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

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

, 11

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

, 8

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]

]. In particular, the generation of terahertz radiation via nonlinear optical techniques, such as optical rectification, difference frequency generation and parametric generation, presents unique features[13

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

, 14

14. E. Mueller, “Terahertz radiation sources for imaging and sensing applications,” Photonics Spectra 40, 60 (2006).

] that are also proving to be very useful tools for biomedical imaging, sensing, spectroscopy, sample characterization, etc. This wealth of applications, made possible by nonlinear phenomena, has necessitated the perpetual quest for improving the efficiency of nonlinear optical processes. In addition to phase matching considerations, the Purcell effect[15

15. E. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).

], well known for its ability to control spontaneous emission rates, provides a hint on how to boost nonlinear conversion efficiencies, by modifying (increasing) the local density of photonic states (LDOS) at the location of the nonlinear polarization source. A light source embedded in a photonic structure is known to emit faster when the LDOS at the source frequency is larger, and this applies for light sources originating from nonlinear polarization as well. Photonic crystals constitute a versatile tool for tailoring LDOS, and hence they offer extraordinary opportunities to enhance nonlinear effects[16

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

], also by exploiting the Purcell effect.

So far, a few proposals have been made to exploit the Purcell effect in order to improve the nonlinear optical response of photonic crystals. For instance, the effective medium approach has been used to investigate how the density of states affects the conversion efficiency for second harmonic generation in a one dimensional photonic band gap (PBG) structure[17

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]

]. The finite-difference time-domain (FDTD) method[18

18. A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Norwood, 1995).

] could also be used to solve the problem numerically[19

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

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

, 20

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]

]. However, a better comprehension of the problem requires a more elaborate treatment, that elucidates the implications of the Purcell effect on efficient nonlinear conversion. A method using Green’s function has been developed[21

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

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

] to calculate the nonlinear optical response of a photonic crystal to an external polarization source. In this respect, temporal coupled mode theory[23

23. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, New Jersey, 1984).

] (CMT) seems to be of great promise, since it provides a physically intuitive framework that easily addresses the problem, where the photonic structure can also exhibit any sort of loss or gain.

In this paper, we use CMT to study the response of a general photonic structure to a traveling polarization source of arbitrary extent. As long as CMT is still valid (i.e. as long as the quality factors are large enough), absorptive and/or radiative losses, as well as gain, can be modeled. Given the current challenge of making high-power terahertz sources, we illustrate our general approach, by proposing, for the first time, a method for efficient generation of terahertz waves by optical rectification in a two-dimensional (2D) photonic crystal. For this purpose, we theoretically investigate sending an optical beam into an appropriately designed 2D nonlinear photonic crystal, and we use our CMT-based result to calculate the total energy radiated at terahertz frequencies. Next, we repeat the same calculation for the terahertz generated energy, but now using the exact FDTD method instead. The obtained close agreement between the CMT and FDTD results, not only validates our CMT approach, but it also suggests that our CMT treatment could serve as a substantially less numerically intensive alternative to the FDTD method, for certain systems of interest. For instance, in problems involving frequencies and wavelengths that range over many orders of magnitude, FDTD calculations are very difficult to perform. However, it is far less intricate to solve such problems by using our CMT-based formalism. A specific example of such cases is the generation of terahertz radiation by optical rectification, where a proper FDTD calculation should involve simultaneously wavelengths in both the optical and terahertz regimes, resulting in a computational burden that could go beyond the capabilities of currently available computers. Such complications, however, are not present if one uses our CMT formalism, instead.

The structure of the paper is as follows: In Sec. II, we develop the CMT model and calculate the generated electric field in the most general case. Then, in Sec. III, we discuss how the Purcell effect follows from our general result, by calculating the power emitted by a point dipole source. Finally, in Sec. IV, we apply our work to the specific problem of THz generation and compare our results with FDTD results.

2. Coupled-mode-theory model

To start with, we suppose that there exists a polarization source 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. {Dv(r)ε(r)} , where D⃗ ν(r⃗)=ε(r⃗)E⃗ ν(r⃗), form a complete set[24

24. J. D. Joannopoulos, R. D. Meade, and J. N Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 1995).

], since they are eigenmodes of the hermitian operator 1ε(r)×(×1ε(r)) . The polarization source 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 D(r,t)ε(r)=ε(r)E(r,t) can be expressed as a linear superposition[24

24. J. D. Joannopoulos, R. D. Meade, and J. N Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 1995).

] of Dv(r)ε(r)=ε(r)Ev(r) . Hence, we can write

E(r,t)=vav(t)Ev(r)d3rε(r)2Ev(r)2
(1)

where 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.

A note about the mode labeling is in place. For a uniform medium, the modes are labeled by the wavevector 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.

Treating each mode as a CMT port, the CMT equation for the mode amplitude a ν becomes[23

23. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, New Jersey, 1984).

]

davdt=iωvav(Γradv+ΓabsvΓgv)av+κvs+v
(2)

where ω ν is the frequency of the mode labeled by ν, Γν rad and Γν abs are the rates of radiative (out of the structure) and absorptive decay, respectively, and Γν g is the rate of gain. κν 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, κν S ν + is given by

κvs+v=d3rJ(r,t)·Ev*(r)d3rε(r)2Ev(r)2
(3)

ddt(av(t)Ξ(t))=Ξ(t)κvs+v
(4)

Hence

av(t)=av(to)eiωv(tto)e(Γradv+ΓabsvΓgv)(tto)
+totdtallspaced3rJ(r,t)·Ev*(r)d3rε(r)2Ev(r)2eiωv(tt)e(Γradv+ΓabsvΓgv)(tt)
(5)

where to is any reference time preceding the turn-on of the polarization source, and t>to.

Plugging this into Eq. (1), the electric field induced in the photonic structure becomes

E(r,t)=vEv(r)d3rε(r)2Ev(r)2av(to)eiωv(tto)e(Γrasv+ΓabsvΓgv)(tto)+
vEv(r)d3rε(r)2Ev(r)2totdtallspaced3rJ(r,t)·Ev*(r)eiωv(tt)e(Γradv+ΓabsvΓgv)(tt)
(6)

The total power radiated out of the photonic structure, at time t, is v2Γradvav(t)2 , and the flux through a surface A at time t is

AS·da=A{Real[E(r,t)]×Real[H(r,t)]}·da
(7)

where 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

To get more intuition on the above result, let us consider an oscillating point dipole of moment p⃗, embedded at position r⃗o in the structure. If we denote the dipole’s angular frequency by ωs, then the polarization is P(r,t)=pδ(3)(rro)eiωst . Let us further assume that Γν g=0∀ν (no gain). In this case, the resulting mode amplitude a ν becomes

av=iωsp·Ev*(ro)i(ωvωs)+(Γradv+Γabsv)eiωstd3rε(r)2Ev(r)2
(8)

Denoting Γνν radν abs, the total radiated power 𝒫=v2Γradvav2 is

𝒫=vΓradv(ωvωs)2+(Γv)22(ωs)2p·Ev*(ro)2d3rε(r)2Ev(r)2
(9)

In the limit of zero loss (Γν→0), we have Γν/{(ω ν-ωs)2+(Γν)2}→πδ(ω ν-ωs). Hence

𝒫=4π(ωs)2p2ε(ro)vε(ro)p̂·Ev*(ro)2δ(ωvωs)d3rε(r)Ev(r)2
(10)

If we further assume that there is perfect polarization match between the source and all the modes ν, then |·E⃗ * ν(r⃗o)|2=|E⃗ ν(r⃗o)|2∀ν , and

𝒫=4π(ωs)2p2ε(ro)g(ωs,ro)
(11)

where

g(ωs,ro)=vε(ro)Ev(ro)2δ(ωvωs)d3rε(r)Ev(r)2
(12)

is the local density of photonic states at r⃗o. 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.

4. Connection with Doppler radiation in a PhC crystal

To verify further the validity if our approach, we apply our general result (Eq. (6)) to another well-known special case: the radiation emitted by an oscillating dipole moving with a fixed velocity 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]

]. In this case, the current density can be simply expressed as

J(r,t)=joeiωstδ(rvt)
(13)

Plugging this into Eq. (6), and assuming that Γν=0 ∀ ν, we obtain

E(r,t)=vEv(r)d3rε(r)2Ev(r)2tdteiωstjo·Ev*(vt)eiωv(tt)
(14)

As we mentioned before, the modes of a PhC are labeled by ν≡(n,k⃗,σ), and from Bloch theorem E⃗nk⃗(r⃗)=eik⃗.r⃗u⃗nk⃗(r⃗), where u⃗nk⃗(r⃗) has the periodicity of the PhC, and hence only reciprocal lattice vectors G⃗ appear in its Fourier expansion u⃗nk⃗(r⃗)=∑G⃗e⃗nk⃗,G⃗eiG⃗.r⃗. Hence Eq. (14) becomes

E(r,t)=nkσEnkσ(r)d3rε(r)2Enkσ(r)2Gjo·enk,G*tdteiωstei(k+G)·vteiωnkσ(tt)
(15)

which simply evaluates to

E(r,t)=nkσEnkσ(r)d3rε(r)2Enkσ(r)2Gi[jo·(enk,G)*]ωnkσ[ωs+(k+G)·v]ei[ωs+(k+G)·v]t
(16)

which is consistent with Eq.(1) in ref(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]

) in the limit ωnk⃗σωS+(k⃗+G⃗).v⃗.

Thus, while our formalism properly reproduces the well-known predictions of the Purcell effect (Eq. (11)) and of Doppler radiation in PhCs (Eq. (16)), its scope extends further, because it analyzes in a systematic and quantitative way the behavior of an arbitrary polarization in any photonic structure. Note that the polarization source can have complicated spatiotemporal variations, and is not restricted to static collections of point dipoles.

A glance at Eq. (6) enables us to recognize the various possibilities for boosting the magnitude of the electric electric field 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 ωS 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.

5. Numerical validation and terahertz generation

5.1. Optimizing the structure

Having identified the essential features required to enhance the response of a photonic structure to an external polarization source, and motivated by the existing need for efficient high-power terahertz sources, we apply our formalism to the specific problem of terahertz generation by optical rectification. To this end, we select a nonlinear photonic crystal (PhC) structure that features a χ(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 E⃗opt. By optical rectification, this optical beam generates a polarization 𝓟THz(2) EoptE * opt in the terahertz frequency range[26

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]

]. According to our formalism, the rate at which this polarization 𝓟THz excites terahertz waves depends on several considerations pertaining to the PhC structure. First, one of course strives to use a material with χ(2) as large as possible. Next, another option is to increase the magnitude of E⃗opt 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 𝓟THz.

A promising PhC structure, in this respect, has been proposed by some of the current authors[27

27. M. Ibanescu, E. J. Reed, and J. D. Joannopoulos, “Enhanced Photonic Band-Gap Confinement via Van Hove Saddle Point Singularities,” Phys. Rev. Lett. 96, 033904 (2006). [CrossRef] [PubMed]

]. Here we consider a very similar structure consisting of a two-dimensional (2D) square lattice of rods in air. The spacing between the rods is denoted by a, the nonlinear susceptibility by χ(2), and the linear refractive index at terahertz frequencies by nr. The rods of radius r are connected along the x-direction by thin waveguides, each of thickness t and consisting of the same material as the rods. These waveguides are needed, in order to provide guiding for the optical light. A unit cell of the 2D periodic PhC structure is depicted in Fig. 1(a) as a color contour plot of the linear dielectric function ε(x,y). A similar contour plot is presented in Fig. 1(b), showing 5 units of the structure in each direction, together with the optical beam.

Fig. 1. (Color online) Color contour plot of the dielectric function ε(x,y) of the 2D PhC srtucture: (a) 1a×1a cell, and (b) 5a×5a cell, showing the optical beam through the central waveguide.

If we choose r=0.13a, t=0.04a and nr=3.5 (close to the refractive index of GaP[28

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

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]

], commonly used for terahertz generation by optical rectification), the second band (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]

]. The projected band diagram for σ=TM, shown in Fig. 2 along with a color contour plot of the second band, indicates that the second TM band is narrowest at kx=0.1559(2π/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=152.7µm.

Fig. 2. (Color online) (a) Projected band diagram of the first three bands. (b) Color contour plot of the second band, showing the saddle point where the band is narrowest.

So, the PhC structure we have chosen indeed has a considerable number of modes polarized along z (parallel to the rods), with kx=0.1559(2π/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 J⃗THz associated with the polarization source 𝓟THz, has the form:

JTHz(x,y;t)=ẑeikxsxy22ζ2eiωst(t500ac)22τ2
(17)

where ksx=0.1559(2π/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 ωS=0.509(2πc/a), and the values of the remaining parameters are: ζ=0.02a and τ=100(a/c). The time t is expressed in units of a/c. Since J⃗THz points along , all the TM modes have their polarization perfectly matched to that of the source. For convenience in subsequent calculations, and because J⃗THz(x,y;t) is separable in space and time, we write it as

JTHz(x,y;t)=JspaceTHz(x,y)·𝓕(t)
(18)

where

JspaceTHz(x,y)=ẑeikxsxy22ζ2
(19)

and

𝓕(t)=eiωst(t500ac)22τ2
(20)

We also set J⃗THz(x,y;t)=0 for t<0 and t≥1000a/c.

Up to this point most of the criteria for efficient THz generation have been met, and we are left with the issue of the spatial overlap between the modes (n=2; kx=0.1559(2π/a),ky; σ=TM) and the terahertz polarization source. The degree of localization of the modes En=2;(kx=0.1559(2πa),ky;σ=TM)(r) at the source’s extent can be checked by looking at the color contour plot of

wky(x,y)ε(x,y)En=2;(kx=0.1559(2πa),ky;σ=TM)(x,y)2
(21)

for several ky values. We computed the fields En=2;(kx=0.1559(2πa),ky;σ=TM)(r) by using the same 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]

], and calculated wky(x,y) for different ky values. A representative plot is shown in Fig. 3 for ky=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

Fig. 3. (Color online) Color contour plot of wky=0(x,y) .
𝒪nkallspaced3rJspaceTHz(r)·Enk*(r)
(22)

for n=2. Indeed, for most of the different possible ky in the first brillouin zone, the integrand of the overlap integral 𝒪n=2;(kx=kxs,ky) oscillates along , and hence integrates to a negligible value. That is, for n=2 and for many ky, the integrand is observed to flip the sign at x≃±0.25a. 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=±(0.25a+ℓ0.5a) 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 𝒪n2;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

31. R. W. Boyd, Nonlinear Optics (Academic Press, 2002).

], 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 𝒪n=1;k⃗ does not change sign along , 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.

Fig. 4. (Color online) Periodically poled PhC structure. The red portions correspond to the original orientation of the nonlinear crystal when χ(2) is positive, while in the blue portions, the crystal’s orientation is flipped resulting in a negative χ(2).

5.2. Calculation of generated energy

Let us now calculate the total emitted energy at THz frequencies. First, we need to calculate the electric field 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 10a/c later than source turn-off), and we assume that ank⃗=0 ∀(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 kx=0.1559(2π/a)=ksx, and we further assume Γnk⃗gnk⃗abs=0. From Eq. (6), it is evident that we cannot proceed before computing the following two functions of ky

𝒪allspacepoled(ky)allspaced3rq(x)JspaceTHz(r)·En=2;(kxs,ky)*(r)
=allspaced3rq(x)eikxsxy22ζ2ẑ.En=2;(kxs,ky)*(r)
(23)

and

𝒥t(ky)eiω(n=2;kxs,ky)t0tdt𝓕(t)eiω(n=2,kxs,ky)t
=eiω(n=2;kxs,ky)t0tdtei(ωsω(n=2;kxs,ky))t(t500ac)22τ2
(24)
Fig. 5. (Color online) Overlap integral 𝒪poledone period as a function of ky for modes at the narrowest portion (kx=0.1559(2π/a)) of the second band.

The integral in Eq. (23) can be simplified by noting that in a PhC, the electric fields can be written as

Enk(r)=eik·runk(r)
(25)

where u⃗nk⃗(r⃗) has the same periodicity as the PhC. Consequently, 𝒪poledall space(ky) becomes

𝒪allspacepoled(ky)=allspaced3rq(x)eikyyy22ζ2ẑ·un=2;(kxs,ky)*(r)
(26)

The integrand of this equation is periodic function of x′. Hence the integral over all x′ simplifies to the number 𝓝x of periods in the x-direction times the integral over a single period, with 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 J⃗THzspace is zero for y′ outside the interval [-a/2,a/2]. Therefore

𝒪allspacepoled(ky)=𝒩x·𝒪oneperiodpoled(ky)
(27)
𝒪oneperiodpoled(ky)=a2a2dxa2a2dyq(x)eikyyy22ζ2ẑ·un=2;(kxs,ky)*(r)
(28)

To evaluate this integral, we discretize space with a resolution of 256 gridpoints/a, and calculate the TM fields un=2;(kxs,ky)(r) based on the TM modes En=2;(kxs,ky)(r) computed by using the 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]

]. Note that the TE modes have their electric field polarized in the xy plane, and hence they don’t get excited by the polarization source. So, in all what follows, when we refer to PhC modes, we consider only the TM modes, and we omit the notation σ=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 onecelld3rε(r)En,k(r)2=1 . A plot of the overlap integral 𝒪poledone period(ky), as a function of ky, is shown in Fig. 5 (after poling is performed). Clearly, the integral takes values close to maximum for most of the ky’s. The reason for which it vanishes as ky→±0.5(2π/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 ky=±0.5(2π/a).

Next, we calculate 𝓣t(ky) at t=1010(a/c), using the previously calculated band structure. Eq. (6) becomes

E(r,t)=kyEn=2;(kxs,ky)(r)𝒩x𝒩y2𝒩x·𝒪oneperiodpoled(ky)𝓘t(ky)
(29)

Since the PhC is of infinite extent in the x and y directions, the ky values consistent with periodic boundary conditions, become dense enough that to a good approximation, the discrete sum over ky can be converted into an integral

E(r,t)=a𝒩y2π0.5(2πa)0.5(2πa)dkyEn=2;(kxs,ky)(r)𝒩x𝒩y2𝒩x·𝒪oneperiodpoled(ky)𝓘t(ky)
(30)
=aπ·πaπadkyEn=2;(kxs,ky)(r)𝒪oneperiodpoled(ky)𝓘t(ky)
(31)
Fig. 6. (Color online) CMT calculations: (a) Color contour plot of the unnormalized terahertz energy density ε(x,y)|E(x,y,t)|2 at t=1010(a/c), in a 2D box of size 1a×80a. (b) A zoom-in version of the plot in Fig. a, showing more details of the interval y∈±[8a,19a]. Note that the optical beam was originally sent through the waveguide at y=0

As we intended, all the emitted energy is still confined inside our large computational domain; more specifically, the energy density is non-vanishing only in the regions y∈±[8a,19a]. 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 1a along x, and large enough in the y direction to enclose the regions y∈±[8a,19a]. 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.

To confirm the validity of our analytical model in general, and of the result for the total emitted energy in the specific example of terahertz generation, we repeat exactly the same calculation for the terahertz emitted energy, but now instead of using our analytical formalism, we employ the finite-difference time-domain (FDTD) method[18

18. A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Norwood, 1995).

], using a freely available software package with subpixel smoothing for increased accuracy[32

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]

]. We consider a computational cell (centered at the origin) of size 1a along x, and 90a 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 kx=0.1559(2π/a). At the boundaries in the y direction, we set up perfectly matched layers (PML), each of thickness 3a. We simulate the effect of periodic poling by explicitely reversing the sign of the polarization source at x=±0.25a. To compute the total emitted energy at terahertz, we record the time evolution of the energy 𝓔 in a box of size 1a (along x)×80a (along y), centered at the origin. We also place flux calculation planes at y=±40a, 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.

Fig. 7. (Color online) FDTD calculation for the terahertz emitted energy in the PhC structure (not normalized to the bulk).

The time development of the energy 𝓔, and of the time integral of the net flux, is shown in Fig. 7, together with the total emitted energy, which is given by their sum. Again, we normalize the total energy emitted in the poled PhC to that emitted in the corresponding unpoled bulk, and we get a value of 3.94 in dimensionless units. Thus, our result for the normalized emitted energy at terahertz, differs only by ≃2% from the exact result computed by FDTD. To gain even more confidence in the validity of our analytical result, we perform FDTD calculations of the energy density profile in the poled PhC, at 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∈±[8a,19a] 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=±13a, 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 Γν be much smaller than the frequency ω ν for each mode ν of the photonic structure[23

23. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, New Jersey, 1984).

].

Fig. 8. (Color online) FDTD calculations: (a) Color contour plot of the terahertz energy density ε(x,y)|E(x,y,t)|2 at t=1010(a/c), in a 2D box of size 1a×80a. (b) A zoom-in version of the plot in Fig. a, showing more details of the interval y∈±[8a,19a]. Note that the optical beam was originally sent through the waveguide at y=0.

6. Conclusion

In conclusion, we developed an analytical model for calculating the electric field produced by an external polarization source in a photonic structure. Then, we investigated how the implications of this formalism can provide an insight on enhancing the efficiency of nonlinear effects. We illustrated the procedure, by applying it to the specific case of terahertz generation by optical rectification in a 2D nonlinear photonic crystal, and finally we checked our results against exact calculations based on the FDTD method. The results are in a good agreement with each other, thus validating our approach, and motivating us to propose our technique as a potentially simpler alternative to FDTD, as long as CMT is applicable.

Acknowledgments

Finally, we would like to acknowledge helpful discussions with Dr. Peter Rakich, Dr. Chiyan Luo, Ardavan Farjadpour Oskooi, Alejandro Rodriguez, Dr. Jorge Bravo-Abad, and Y. D. Chong. This work was supported in part by the Materials Research Science and Engineering Center Program of the National Science Foundation under award DMR 02-13282, the Army Research Office through the Institute for Soldier Nanotechnologies contract W911NF-07-D-0004 and the U.S. Department of Energy under award number DE-FG02-99ER45778. Furthermore, this material is based upon work supported by the National Science Foundation under the following NSF programs: Partnerships for Advanced Computational Infrastructure, Distributed Terascale Facility (DFT), and Terascale Extensions: Enhancements to the Extensible Terascale Facility. We also acknowledge support of the Buchsbaum award.

References and links

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]

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]

3.

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

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]

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]

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]

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]

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]

13.

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

14.

E. Mueller, “Terahertz radiation sources for imaging and sensing applications,” Photonics Spectra 40, 60 (2006).

15.

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

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]

18.

A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Norwood, 1995).

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]

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]

23.

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

24.

J. D. Joannopoulos, R. D. Meade, and J. N Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 1995).

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]

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]

27.

M. Ibanescu, E. J. Reed, and J. D. Joannopoulos, “Enhanced Photonic Band-Gap Confinement via Van Hove Saddle Point Singularities,” Phys. Rev. Lett. 96, 033904 (2006). [CrossRef] [PubMed]

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]

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]

31.

R. W. Boyd, Nonlinear Optics (Academic Press, 2002).

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]

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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References

  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]
  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]
  3. J. A. Giordmaine, "Mixing of Light Beams in Crystals," Phys. Rev. Lett. 8, 19-20 (1962). [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]
  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]
  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]
  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]
  10. Yuri Kivshar, "Spatial solitons: Bending light at will," Nature Phys. 2, 729-730 (2006). [CrossRef]
  11. Günter Steinmeyer, "Laser physics: Terahertz meets attoscience," Nature Phys. 2, 305-306 (2006). [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]
  13. B. Ferguson and X. -Cheng Zhang, "Materials for terahertz science and technology," Nature Mater. 1, 26-33 (2002). [CrossRef]
  14. E. Mueller, "Terahertz radiation sources for imaging and sensing applications," Photon. Spectra. 40, 60 (2006).
  15. E. Purcell, "Spontaneous emission probabilities at radio frequencies," Phys. Rev. 69, 681 (1946).
  16. M. Solja�?i�?, and J. D. Joannopoulos, "Enhancement of non-linear effects using photonic crystals," Nature Mater. 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 |(2) interactions," Phys. Rev. E. 64, 016609 (2001). [CrossRef]
  18. A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Norwood, 1995).
  19. J. Bravo-Abad, A. Rodriguez, P. Bermel, S. G. Johnson, J. D. Joannopoulos, and M. Solja�?i�?, "Enhanced nonlinear optics in photonic-crystal microcavities," Opt. Express 15, 16161-16176 (2007). [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]
  23. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, New Jersey, 1984).
  24. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 1995).
  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]
  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]
  27. M. Ibanescu, E. J. Reed, and J. D. Joannopoulos, "Enhanced Photonic Band-Gap Confinement via Van Hove Saddle Point Singularities," Phys. Rev. Lett. 96, 033904 (2006). [CrossRef] [PubMed]
  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]
  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]
  31. R. W. Boyd, Nonlinear Optics (Academic Press, 2002).
  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]

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