Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. Theory

Water wave transmission by an array of floating discs

An experimental validation of theoretical models of regular-water-wave transmission through arrays of floating discs is presented. The experiments were conducted in a wave basin. The models are based on combined potential-flow and thin-plate theories, and the assumption of linear motions. A low-concentration array, in which discs are separated by approximately a disc diameter in equilibrium, and a high-concentration array, in which adjacent discs are almost touching in equilibrium, were used for the experiments. The proportion of incident-wave energy transmitted by the discs is presented as a function of wave period, and for different wave amplitudes. Results indicate the models predict wave-energy transmission accurately for small-amplitude waves and low-concentration arrays. Discrepancies for large-amplitude waves and high-concentration arrays are attributed to wave overwash of the discs and collisions between discs. Validation of model predictions of a solitary disc's rigid-body motions are also presented.

Semi-analytic impedance modeling of three-dimensional photonic and metamaterial structures

We define the concept of an impedance matrix for three-dimensional (3D) photonic and metamaterial structures relative to a reference medium and show that it satisfies a matrix generalization of the basic algebraic properties of the wave impedance between homogeneous media. This definition of the impedance matrix is motivated by the structure of the Fresnel reflection and transmission matrices at the interface between the media. In the derivation of the Fresnel scattering matrices, the field in each medium is expressed by a Bloch mode expansion, with field matching at the interface being undertaken in a least-squares manner by exploiting a biorthogonality relation between primal and adjoint Bloch modes. A semi-analytic technique, based on the impedance matrix, is developed for modeling the scattering of light by 3D periodic photonic and metamaterial structures. The advantages (in design and intuition) of the formalism are demonstrated through two applications.

Long-wavelength mid-infrared reflectors using guided-mode resonance

We have proposed and fabricated a new mid-infrared reflector using the guided-mode resonance (GMR). The GMR reflector consists of subwavelength Ge grating on GaAs substrate with a low-refractive-index SiOx layer in between. With a total thickness of about 2 μm, a near-100% reflectivity at 8 μm has been obtained both theoretically and experimentally.

Narrow band, large angular width resonant reflection from a periodic high index grid at terahertz frequency

The property of a thin silicon membrane with periodic air slits of definite depth and width to exhibit under normal incidence a close to 100% ultra-narrow band reflection peak is demonstrated experimentally in the terahertz frequency range on a single-crystal silicon grid fabricated by submillimeter microsystem technology. An analysis based on the true modes supported by the grid reveals the nature of such resonances and permits to sort out those exhibiting ultra-narrow band.

Modal analysis of enhanced absorption in silicon nanowire arrays

We analyze the absorption of solar radiation by silicon nanowire arrays, which are being considered for photovoltaic applications. These structures have been shown to have enhanced absorption compared with thin films, however the mechanism responsible for this is not understood. Using a new, semi-analytic model, we show that the enhanced absorption can be attributed to a few modes of the array, which couple well to incident light, overlap well with the nanowires, and exhibit strong Fabry-Pérot resonances. For some wavelengths the absorption is further enhanced by slow light effects. We study the evolution of these modes with wavelength to explain the various features of the absorption spectra, focusing first on a dilute array at normal incidence, before generalizing to a dense array and off-normal angles of incidence. The understanding developed will allow for optimization of simple SiNW arrays, as well as the development of more advanced designs.

Experimental observation of evanescent modes at the interface to slow-light photonic crystal waveguides

We experimentally study the fields close to an interface between two photonic crystal waveguides that have different dispersion properties. After the transition from a waveguide in which the group velocity of light is v(g) ~ c/10 to a waveguide in which it is v(g) ~ c/100, we observe a gradual increase in the field intensity and the lateral spreading of the mode. We attribute this evolution to the existence of a weakly evanescent mode that exponentially decays away from the interface. We compare this to the situation where the transition between the waveguides only leads to a minor change in group velocity and show that, in that case, the evolution is absent. Furthermore, we apply novel numerical mode extraction techniques to confirm experimental results.

Coupled waveguide modes in hexagonal photonic crystals

We investigate the modes of coupled waveguides in a hexagonal photonic crystal. We find that for a substantial parameter range the coupled waveguide modes have dispersion relations exhibiting multiple intersections, which we explain both intuitively and using a rigorous tight-binding argument.

Bloch-mode extraction from near-field data in periodic waveguides

We demonstrate that the spatial profiles of both propagating and evanescent Bloch modes in a periodic structure can be extracted from a single measurement of an electric field at the specified optical wavelength. We develop a systematic extraction procedure by extending the concepts of high-resolution spectral methods previously developed for temporal data series to take into account the symmetry properties of Bloch modes simultaneously at all spatial locations. We illustrate the application of our method to a photonic crystal waveguide interface and confirm its robustness in the presence of noise.

Efficient coupling into slow light photonic crystal waveguide without transition region: role of evanescent modes

We show that efficient coupling between fast and slow photonic crystal waveguide modes is possible, provided that there exist strong evanescent modes to match the waveguide fields across the interface. Evanescent modes are required when the propagating modes have substantially different modal fields, which occurs, for example, when coupling an index-guided mode and a gap-guided mode.

Efficient slow-light coupling in a photonic crystal waveguide without transition region

We consider the coupling into a slow mode that appears near an inflection point in the band structure of a photonic crystal waveguide. Remarkably, the coupling into this slow mode, which has a group index ng>1000, can be essentially perfect without any transition region. We show that this efficient coupling occurs thanks to an evanescent mode in the slow medium, which has appreciable amplitude and helps satisfy the boundary conditions but does not transport any energy.

Modal method for classical diffraction by slanted lamellar gratings

We consider lamellar gratings made of dielectric or lossy materials used in classical diffraction mounts. We show how the modal diffraction formulation may be generalized to deal with slanted lamellar gratings and illustrate the accuracy and versatility of the new method through study of highly slanted gratings in a homogenization limit. We also comment on the completeness of the eigenmode basis and present tests enabling this completeness to be verified numerically.

Bloch waves in periodic multi-layered acoustic waveguides

The band spectrum and associated Floquet–Bloch eigensolutions, arising in acoustic and electromagnetic waveguides, which have periodic structure along the guide while remaining of finite width, are found. Homogeneous Dirichlet or Neumann conditions along the guide walls, or an alternation of them, are taken. Importantly, in some cases, a total stop band at zero frequency is identified providing space for low-frequency localized modes; geometric defects in the structured waveguide also create these modes. Numerical and asymptotic techniques identify dispersion curves and trapped modes. Some cases demonstrate maxima and minima of the spectral edges within the Brillouin zone and also allow for ultraslow light or sound. Imaging applications using anomalous dispersion to generate subwavelength resolution are possible and are demonstrated.

Dispersionless tunneling of slow light in antisymmetric photonic crystal couplers

We suggest a novel and general approach to the design of photonic-crystal directional couplers operating in the slow-light regime. We predict, based on a general symmetry analysis, that robust tunneling of slow-light pulses is possible between antisymmetrically coupled photonic crystal waveguides. We demonstrate, through Bloch mode frequency-domain and finite-difference time-domain (FDTD) simulations that, for all pulses with strongly reduced group velocities at the photonic band-gap edge, complete switching occurs at a fixed coupling length of just a few unit cells of the photonic crystal.

Application of the multiple-scattering method to analysis of systems with semi-infinite photonic waveguides

We propose a technique of compensating the spurious reflections implied by the multiple-scattering (MS) method, commonly used for analyzing finite photonic crystal (PC) systems, to obtain exact values of characteristic parameters, such as reflection and transmission coefficients, of PC functional elements. Rather than a modification of the MS computational algorithm, our approach involves postprocessing of results obtained by the MS method. We derive analytical formulas for the fields excited in a finite system, taking explicitly into account the spurious reflections occurring at the artificial system boundaries. The intrinsic parameters of the investigated functional element are found by fitting the results of MS simulations to those obtained from the formulas derived. Devices linked with one and two semi-infinite waveguides are analyzed explicitly; possible extensions of the formalism to more complex circuits are discussed as well. The accuracy of the proposed method is tested in a number of systems. The results of our calculations prove to be in good agreement with those obtained independently by other authors.

Efficient slow light coupling into photonic crystals

We study light coupling between two photonic crystal wave-guides, one of which supports slow light. We show theoretically that a short photonic crystal waveguide between the two that need to be coupled, can lead to a vanishingly small reflectivity. The design relies on the analogy with a lambda/4 anti-reflection layer in thin-film optics.We find that some of the usual relationships between the Fresnel coefficients at an interface no longer hold.

Evidence of a mobility edge for photons in two dimensions

A scaling analysis of conductance for photons in two dimensions is carried out and, contrary to widely held belief, we find strong evidence of a mobility edge. Such behavior is compatible with the existence of an Anderson transition for electronic systems under symplectic symmetry, and indeed we show that the transfer matrix in the photonic system we have modelled has such a symmetry. We verify single parameter scaling of the conductance and demonstrate the transition from the metallic phase to localization. Key parameters, including the critical disorder, the conductance, and the critical exponent of the localization length are calculated, and it is shown that the value of the critical exponent is similar to that for electronic systems with symplectic symmetry.

Wide-angle coupling into rod-type photonic crystals with ultralow reflectance

We describe the surprising phenomenon of near-perfect coupling from free space into uniform two-dimensional rod-type photonic crystals over a wide range of incident angles. This behavior is shown to be a generic feature of many rod-type photonic crystal structures that is related to strong forward scattering resonances of the individual cylinders. We explain these results using both semianalytic analysis and two-dimensional numerical calculations and identify the conditions under which efficient, wide-angle coupling can occur. The results may lead to more efficient designs for in-band photonic crystal devices such as superprisms and self-collimation based photonic circuits.

Linear embedding via Green's operators: a modeling technique for finite electromagnetic band-gap structures

We propose a modular electromagnetic modeling procedure for large finite electromagnetic band-gap (EBG) structures, called linear embedding via Green's operators. It is a diakoptic method based on the Huygens-Schelkunoff principle involving equivalent boundary current sources that electromagnetically characterize the enclosed domain of arbitrary shapes, as if it were a multiport system. In a cascade of embedding steps, separate reusable domains are combined to form larger domains. Device design often involves tuning local medium properties in a compact designated domain with a large environment. Through an additional embedding step the equivalent sources describing the environment can be transferred to the boundary of the designated domain, rendering subsequent design steps very fast. This two-stage optimization process is applied in the design of an EBG power splitter.

Conductance of photons in disordered photonic crystals

The conductance of photons in two-dimensional disordered photonic crystals is calculated using an exact multipole-plane wave method that includes all multiple scattering processes. Conductance fluctuations, the universal nature of which has been established for electrons in the diffusive regime, are studied for photons, in both principal polarizations and for varying disorder. Our simulations show that universal conductance fluctuations can be observed in H(||) (TE) polarization for weak and intermediate disorder while, for E(||) (TM) polarization, we show that the conductance variance is essentially independent of sample size but strongly dependent on disorder. The probability distribution of the conductance is also calculated in the diffusive and localized regimes, and also at their transition, for which the distributions for both polarizations are seen to be very similar.

Bloch mode scattering matrix methods for modeling extended photonic crystal structures. II. Applications

The Bloch mode scattering matrix method is applied to several photonic crystal waveguide structures and devices, including waveguide dislocations, a Fabry-Pérot resonator, a folded directional coupler, and a Y-junction design. The method is an efficient tool for calculating the properties of extended photonic crystal (PC) devices, in particular when the device consists of a small number of distinct photonic crystal structures, or for long propagation lengths through uniform PC waveguides. The physical insight provided by the method is used to derive simple, semianalytic models that allow fast and efficient calculations of complex photonic crystal structures. We discuss the situations in which such simplifications can be made and provide examples.