Generation In Photonic Crystals Research Articles

Slow-light enhanced collinear second-harmonic generation in two-dimensional photonic crystals

We predict an enhanced efficiency of collinear second-harmonic generation in photonic crystals. This can be achieved by taking advantage of the complexity of the dispersion relation in periodic lattices. Thus, two essential conditions for increasing the nonlinear interaction, namely, phase matching and small group velocity (slow light), can be met simultaneously. The analytical results based on a modal approach are verified with rigorous simulations by means of the finite-difference time-domain method. The influence of material dispersion on optimal phase match is discussed.

Towards observation of sub-diffractive pulse propagation in photonic crystals

It is shown that sub-diffractive (self-collimating) propagation of ultrashort pulses of narrow light beams is possible in two dimensional planar photonic crystal slabs (in particular made with air holes in Si3N4 slab) where the sub-diffractive propagation of monochromatic beam of visible light has been demonstrated previously both theoretically and numerically. We found that the sub-diffractive propagations of the pulses of duration of more than 40fs are indistinguishable from that of monochromatic beams. However, for the pulses with duration less than 40fs their propagation peculiarities associated with asymmetric spatiotemporal misshaping, spatial and temporal broadening come into play. The effects are pronounced for the pulses of duration less than 20fs. The phenomena are shown for both TM and TE polarized light by means of numerical integration of 2D Maxwell’s equations with finite-difference time-domain technique.

A hardware acceleration of the time domain boundary integral equation method for the wave equation in two dimensions

We aimed at reducing the runtime of the time domain boundary integral equation method (TDBIEM) for the two-dimensional scalar wave equation by the help of a special-purpose computer MDGRAPE-2, which is a high-performance vector-parallel computer dedicated to computing long-range forces in classical molecular dynamics simulations. To this end, we presented a formulation to make MDGRAPE-2 compute the layer potential that is the most time-consuming in a run of the TDBIEM, considering to minimise the amount of data-transfer between MDGRAPE-2 and the host computer. The newly implemented TDBIEM code using three MDGRAPE-2s boards was about 109 times faster than the conventional code in an analysis of 5400 boundary elements and 640 time steps. We also demonstrated that the present hardware acceleration helps to perform large-scale simulations such as light propagation in photonic crystals.

Quantitative analysis of subdiffractive light propagation in photonic crystals

We present a qualitative study of the subdiffractive light beam propagation in photonic crystals, based on the theoretical analysis of full Maxwell equation system beyond the slowly varying envelope and paraxial approximations. We calculate the zero diffraction curve in the parameter space, we calculate the weak asymptotical broadening (spreading in transverse direction), and we also analyze the dependence of subdiffractive propagation on the polarization orientation (both for TM and TE polarization).

Imaging Single Quantum Dots in Three-Dimensional Photonic Crystals

Performing fluorescence wide-field microscopy we have imaged single semiconductor quantum dots deep inside a 3-dimensional photonic crystal prepared from colloidal polymer beads. Exploring the emission diffraction patterns in defocused images of quantum dots we demonstrate that the direction-dependent photonic stop band imprints an anisotropy to the angular emission of a single quantum dot. Hence a single, quasi-point-like emitter is manipulated to radiate its photons only to certain well-defined directions by means of the anisotropic light propagation in photonic crystals. The experiments thus provide new routes to evaluate local, frequency selective optical properties in 3-dimensional photonic crystals employing single emitters.

Theoretical prediction of thermooptical and structurally disordered sensitivities in metallo-dielectric photonic crystals

In this letter, we introduce a novel computational technique for predicting thermal and structural sensitivities in metallo-dielectric photonic crystals (PCs). The computational scheme is based on a hybrid formalism of the scattering matrix technique combined with the adjoint network method. The proposed technique can predict with high accuracy the impact of temperature fluctuations or structural disorders, to the lightwave propagation in PCs, without additional computational effort. Numerical simulations show that PC circuits based on metallic-metamaterial platforms are significantly less sensitive to temperature variations in comparison to usual dielectric PCs.

Thermo-optical sensitivity analysis in photonic crystal circuits based on semiconducting or metallic metamaterial constituents

We introduce a novel analysis technique for predicting thermo-optical sensitivities in photonic crystal (PC) circuits composed of either dielectric-semiconducting or metallic constituents. The proposed numerical analysis is based on a hybrid formalism of the scattering matrix technique combined with the adjoint network method. The proposed computational scheme can, with modest computational resources, predict with high accuracy, the effect of the temperature fluctuations to the light-wave propagation in PCs. Numerical simulations show that PC circuits based on metallic metamaterial platforms are significantly less sensitive to temperature variations than the usual dielectricor semiconducting PCs.

Light propagation in finite and infinite photonic crystals: The recursive Green’s function technique

We report a computational method based on the recursive Green's function technique for calculation of light propagation in photonic crystal structures. The advantage of this method in comparison to the conventional finite-difference time domain (FDTD) technique is that it computes Green's function of the photonic structure recursively by adding slice by slice on the basis of Dyson's equation. This eliminates the need for storage of the wave function in the whole structure, which obviously strongly relaxes the memory requirements and enhances the computational speed. The second advantage of this method is that it can easily account for the infinite extension of the structure both into air and into the space occupied by the photonic crystal by making use of the so-called ``surface Green's functions.'' This eliminates the spurious solutions (often present in the conventional FDTD methods) related to, e.g., waves reflected from the boundaries defining the computational domain. The developed method has been applied to study scattering and propagation of the electromagnetic waves in the photonic band-gap structures including cavities and waveguides. Particular attention has been paid to surface modes residing on a termination of a semi-infinite photonic crystal. We demonstrate that coupling of the surface states with incoming radiation may result in enhanced intensity of an electromagnetic field on the surface and very high $Q$ factor of the surface state. This effect can be employed as an operational principle for surface-mode lasers and sensors.

Fourier analysis of Bloch wave propagation in photonic crystals

The standard representation of an optical field propagating in a photonic crystal (PhC) is an electromagnetic Bloch wave. We present a description of these waves based on their Fourier transform into a series of electromagnetic plane waves. The contribution of each plane wave to the global energy and group velocity is detailed, and the valid domain of this decomposition is discussed. This description brings new insight to the fundamental properties of light propagation in PhCs. Most notably, it permits a continuous description of light propagation from the homogenous medium to the strongly modulated PhC case. It also provides an original physical understanding of negative refraction in PhCs and resolves inconsistencies that result from band folding.

Dual lattice photonic-crystal beam splitters

Light propagation in photonic crystals (PhC) is both sensitive to incident angle and wavelength. By combining two different PhC lattices, we utilize this effect to demonstrate a wavelength-dependent beam splitter with enhanced angular separation. The first lattice acts as a superprism that separates the incoming light according to wavelength, whereas the second lattice acts as an angular amplifier. We obtain 90° angular separation for two wavelengths separated by 70 nm (1300 nm regime) in a structure that is less than 10 μm long.

Electrically switchable third-harmonic generation in photonic crystals

We demonstrate electrically switchable third-harmonic generation in low refractive index contrast one-dimensional photonic crystals pumped by a near-infrared laser beam. A dramatic enhancement of the third-harmonic signal is observed on the short wavelength side of the photonic crystal stop band. The enhancement of the third-harmonic generation disappears when an electric field is applied, revealing the electrical switchability of the third-harmonic generation. The observed phenomenon of enhancement is explained theoretically with a coupled-mode model. We also show that up to a modulation frequency of 10 kHz the switching properties are supported in the photonic crystal.

Performance of nonlinear photonic crystal devices at high bit rates

I investigate theoretically the performance limits for all-optical switching devices incorporating photonic crystals made of nonlinear materials operating at high bit rates. I compare two switching techniques--shifting of the photonic bandgap and the more traditional interferometric method enhanced by slow wave propagation in photonic crystals-and show that the latter always has an advantage. I also demonstrate that the benefits provided by the photonic crystal in the interferometer are still severely limited by the dispersion and become significant only in materials combining high nonlinearity, high index contrast, and high damage threshold.

Compression and Quasi-soliton Propagation of Femtosecond Pulses in the Regime of Second Harmonic Generation in Photonic Crystals

Analysis of second harmonic generation in a photonic crystal with two types of nonlinearity in different layers of the periodic structure is performed. The results obtained show the possibility of generating both solitary double-frequency pulses with a duration of several femtoseconds and a train of such pulses.

Wavelet-based analysis of transient electromagnetic wave propagation in photonic crystals.

Photonic crystals and optical bandgap structures, which facilitate high-precision control of electromagnetic-field propagation, are gaining ever-increasing attention in both scientific and commercial applications. One common photonic device is the distributed Bragg reflector (DBR), which exhibits high reflectivity at certain frequencies. Analysis of the transient interaction of an electromagnetic pulse with such a device can be formulated in terms of the time-domain volume integral equation and, in turn, solved numerically with the method of moments. Owing to the frequency-dependent reflectivity of such devices, the extent of field penetration into deep layers of the device will be different depending on the frequency content of the impinging pulse. We show how this phenomenon can be exploited to reduce the number of basis functions needed for the solution. To this end, we use spatiotemporal wavelet basis functions, which possess the multiresolution property in both spatial and temporal domains. To select the dominant functions in the solution, we use an iterative impedance matrix compression (IMC) procedure, which gradually constructs and solves a compressed version of the matrix equation until the desired degree of accuracy has been achieved. Results show that when the electromagnetic pulse is reflected, the transient IMC omits basis functions defined over the last layers of the DBR, as anticipated.

Effective optical properties associated with wave propagation in photonic crystals of finite length along the propagation direction

The effective properties associated with the propagation of electromagnetic waves in photonic crystals (PCs) are analyzed for one dimensional photonic crystals to illustrate the difference in those properties between infinitely extended and finite size photonic crystals (along the wave propagation direction). It is shown that the multiple reflections at the two end surfaces of a PC result in oscillations in transmittance, phase velocity, group velocity, as well as effective refractive indexes neff, which are often determined experimentally from the transmitted or reflected wave. At the frequencies far away from the bandgap, the oscillation in both the transmittance and neff can be modeled by treating the PC as an effective medium with the effective properties derived from the dispersion curves of the corresponding infinitely extended PC. As the frequencies approaching the band edges, however, the oscillations in both transmittance and neff are markedly enhanced due to the reflections from the internal interfaces in the PC. The results clearly demonstrate the importance of the two end faces on the effective properties of a finite size PC, which can be quite different from those deduced for an infinitely extended PC.

Coherence properties of supercontinuum spectra generated in photonic crystal and tapered optical fibers

Numerical simulations have been used in studies of the temporal and spectral features of supercontinuum generation in photonic crystal and tapered optical fibers. In particular, an ensemble average over multiple simulations performed with random quantum noise on the input pulse allows the coherence of the supercontinuum to be quantified in terms of the dependence of the degree of first-order coherence on the wavelength. The coherence is shown to depend strongly on the input pulse's duration and wavelength, and optimal conditions for the generation of coherent supercontinua are discussed.

Numerical simulations and coherence properties of supercontinuum generation in photonic crystal and tapered optical fibers

Numerical simulations have been used to study broad-band supercontinuum generation in optical fibers with dispersion and nonlinearity characteristics typical of photonic crystal or tapered fiber structures. The simulations include optical shock and Raman nonlinearity terms, with quantum noise taken into account phenomenologically by including in the input field a noise seed of one photon per mode with random phase. For input pulses of 150-fs duration injected in the anomalous dispersion regime, the effect of modulational instability is shown to lead to severe temporal jitter in the output, and associated fluctuations in the spectral amplitude and phase across the generated supercontinuum. The spectral phase fluctuations are quantified by performing multiple simulations and calculating both the standard deviation of the phase and, more rigorously, the degree of first-order coherence as a function of wavelength across the spectrum. By performing simulations over a range of input pulse durations and wavelengths, we can identify the conditions under which coherent supercontinua with a well-defined spectral phase are generated.

Bidirectional beam propagation method for multilayered dielectrics with quadratic nonlinearity

By employing an iterative procedure based on a bidirectional beam propagation method, we develop a numerical algorithm to model light propagation in multilayered structures with second-order nonlinearities. Examples of the applicability of this technique in describing second harmonic generation in photonic crystals are given.

High Accuracy Finite-Difference Time-Domain Simulation of Light Propagation in Photonic Crystals.

We have introduced a high accuracy version of the Yee algorithm based on nonstandard finite differences for which the error is e∼(h/λ)2 compared with e∼(h/λ)6 for the ordinary one. This high accuracy allows the use of coarse numerical grids, which greatly reduces the computational cost. Realistic two-dimensional calculations can run on a personal computer, while three-dimensional ones can run on a fast workstation. Using this high accuracy algorithm, we developed computational models and methods to simulate optical propagation in photonic crystals, and to compute their transmission spectra, and bandgap structures. In addition, methodologies were developed to accurately represent arbitrary lattice structures on coarse grids.

Conversion efficiency for second-harmonic generation in photonic crystals

We derive an analytical expression for the conversion efficiency of second-harmonic generation (SHG) in a photonic crystal. The results obtained for the undepleted-pump limit allow us to describe the role played by the feedback and dispersion introduced by the photonic crystal and hence to optimize the SHG process.