Kurt Busch, Marcus Diem, Meikel Frank, Antonio Garcia-Martin, Frank Hagmann, Daniel Hermann, Sergei Mingaleev, Suresh Pereira, Matthias Schillinger, and Lasha Tkeshelashvili
Since the invention of the laser, progress in Photonics has been intimately related to the development of optical materials which allow one to control the flow of electromagnetic radiation or to modify light-matter interaction. Photonic Crystals (PhCs) represent a novel class of optical materials which elevates this principle to a new level of sophistication. These artificial structures are characterized by two-dimensional (2D) or three-dimensional (3D) periodic arrangements of dielectric material which lead to the formation of an energy band structure for electromagnetic waves propagating in them. Recent advances in micro-structuring technology provide an enormous flexibility in the choice of material composition, lattice periodicity and symmetry of these arrangements allowing one to fabricate PhCs with embedded defect structures. As a consequence, the dispersion relation and associated mode structure of PhCs may be tailored to almost any need. This results in a potential for controlling the optical properties of PhCs that may eventually rival the flexibility in tailoring the properties of their electronic counterparts, the semiconducting materials.
One of the most striking features of PhCs is associated with the fact that suitably engineered PhCs may exhibit frequency ranges over which ordinary linear propagation is forbidden, irrespective of direction. These photonic band gaps (PBGs) lend themselves to numerous applications in linear, nonlinear and quantum optics. For instance, in the linear regime novel optical guiding characteristics through the engineering of defects such as microcavities, waveguides and their combination into functional elements, such as wavelength add-drop filters may be realized. Similarly, the incorporation of nonlinear materials into PBG structures is the basis for novel solitary wave propagation for frequencies inside the PBG. In the case of lattice-periodic Kerr-nonlinearities, the threshold intensities and symmetries of these solitary waves depend on the direction of propagation, whereas in the case of nonlinear waveguiding structures embedded in a 2D PBG material, the propagation characteristics strongly depend on the nature of the waveguides. Finally, the existence of complete PBGs allows one to inhibit spontaneous emission for atomic transition frequencies, deep in the PBG and leads to strongly non-Markovian effects, such as fractional localization of the atomic population for atomic transition frequencies in close proximity to a complete PBG.
The discovery of superrefractive phenomena such as the superprism effect and the resulting potential applications in telecommunication technology have recently attracted a lot of attention to the highly anisotropic nature of iso-frequency surfaces in the photonic band structure. Similarly, the tailoring of photonic dispersion relations and associated mode structures, group velocities, group velocity dispersions (GVDs) and effective nonlinearities through judiciously designed PhCs, allows one to explore regimes of nonlinear wave propagation in PhCs that hitherto have been virtually inaccessible. For instance, the existence of flat bands that are characteristic for 2D and 3D PhCs and the associated low group velocities may greatly enhance frequency conversion effects and may lead to improved designs for distributed-feedback (DFB) laser systems. Photonic crystals with embedded defects, such as microcavities and waveguiding structures, hold tremendous potential for the creation of photonic integrated circuits.
As in virtually any nano-photonic system, a careful theoretical analysis is of paramount importance when interpreting experimental data, and when predicting and realizing novel physical phenomena in PhCs. To date, photonic band structure calculations are used to determine and predict the dispersion relations of perfect, infinitely extended PhCs, and PhCs with simple defects such as isolated cavities and waveguides. More complex situations such as transmission and reflection from finite slabs of PhC-material or through waveguide bends are usually analyzed through direct simulations of Maxwell's equations, based on Finite-Difference Time-Domain (FDTD) or Finite Element (FE) methods. While these are perfectly legitimate approaches, which rest on some 30 years of experience, these techniques do require substantial computational resources and, as a consequence, modeling has been restricted to selected small scale PhC circuits. Moreover, certain computationally intensive aspects related to small scale PhC circuits, such as studies of the effect of fabricational tolerances and the optimization of device designs, still present serious challenges when working with FDTD or FE methods.
In this manuscript, we want to illustrate how the natural affinity of electromagnetic wave propagation in PhCs to the case of electron (wave) transport in semiconducting materials, allows us to devise a comprehensive and highly efficient theoretical framework for the qualitative, as well as quantitative determination of the optical properties of PhCs: Photonic band structure computations allow us to obtain photonic band structures and associated Bloch functions. Related physical quantities such as densities of states (DOS) and group velocities can be calculated with little additional work. Nonlinear PhCs can be studied through an appropriate multi-scale analysis that utilizes Bloch functions as carrier waves and leads to a natural generalization of the well-known slowly varying envelope approximation. Combining band structure calculations with elements from diffractive optics, enables us to determine the reflection and transmission properties of finite PhC-slabs. Finally, we show how defect structures in PhCs can be efficiently treated with the help of photonic Wannier functions. Moreover, this Wannier function approach allows us to formulate a PhC circuit theory, where a defect structure is replaced by the optical analogue of an impedance matrix.
1.2 Photonic band structure computation
The goal of photonic band structure computation is the solution of the wave equation for the perfect PhC, i.e., for an infinitely extended, strictly periodic array of dielectric material. The resulting dispersion relation and associated mode structure may then be further processed to derive related physical quantities such as DOS and group velocities. For simplicity of presentation, we consider in the remainder of the manuscript only 2D PhCs in the TM-polarized case. However, we want to emphasize that analogous considerations apply to the case of TE-polarized radiation in 2D PhCs, as well as to electromagnetic wave propagation in 3D PhCs and will give references where appropriate.
For TM-polarized radiation in 2D PhCs, the wave equation reduces to a single scalar equation for the z-component E([??]) of the electric field:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.1)
Here, c denotes the vacuum speed of light and [??] = (x, y) denotes a 2D position vector. The dielectric constant [[epsilon].sub.p]([??]) [equivalent to] [epsilon].sub.p]([??] + [??]) contains all the structural information of the PhC and is periodic with respect to the set [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of lattice vectors [??], generated by the primitive translations [[??].sub.i], i = 1, 2 which serve as a basis for the underlying PhC lattice. Equation (1.1) represents a differential equation with periodic coefficients and, therefore, its solutions obey the Bloch-Floquet theorem: The discrete translational symmetry of the lattice allows us to label the solutions with a wave vector [??] that is restricted to the first Brillouin zone (BZ) of the reciprocal lattice. This back-folding of the dispersion relation [omega]([??]) to the first BZ introduces a discrete band index n. The eigenmodes (Bloch functions) corresponding to eigenfrequency [w.sub.n]([??]) exhibit the Bloch-Floquet form of modulated plane waves
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.2)
Here, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is periodic with the lattice. A straightforward way of solving Eq. (1.1) and (1.2) is to expand all the periodic functions into a Fourier series over the reciprocal lattice G
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.3)
The Fourier coefficients [[eta].sub.??] are given by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.4)
where we have designated the volume of the Wigner-Seitz cell (WSC) by [V.sub.WSC]. Inserting this expansion into Eq. (1.1) and defining the coefficients [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], transforms the differential equation into an infinite matrix eigenvalue problem
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.5)
which must be suitably truncated to become accessible to an approximate numerical solution. Further details of the plane wave method (PWM) for 2D TE and 3D isotropic systems can be found, for instance, in and for anisotropic 3D systems in.
In Fig. 1.1(b), we show the band structure for TM-polarized radiation in a 2D PhC consisting of a square lattice (lattice constant a) of cylindrical air pores (radius [R.sub.pore] = 0.475a) in a silicon matrix ([epsilon] = 12) (for details on the fabrication of this structure, we would like to refer the reader to Chapter 4 of this book). This structure exhibits two complete 2D bandgaps. The larger, fundamental bandgap (20% of the midgap frequency) extends between w = 0.238 × 2[pi]c/a to w = 0.291 × 2[pi]c/a and the smaller, higher order bandgap (8% of the midgap frequency) extends from w = 0.425 × 2[pi]c/a to w = 0.464 × 2[pi]c/a. In the remainder of this chapter, this particular PhC will serve as the model problem for which we illustrate our solid state theoretical approach to the optical properties of PhCs.
1.2.1 Density of states
The photonic dispersion relation w.sub.n]([??]) gives rise to a photonic density of states (DOS), which plays a fundamental role for the understanding of the quantum optical properties of active material embedded in PhCs. The photonic DOS N(w) is defined by "counting" all allowed states with a given frequency w
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.6)
In Fig. 1.1(a) we depict the DOS for our model system, where the photonic band gaps are manifest as regions of vanishing DOS. Characteristic for 2D systems is the linear behavior for small frequencies as well as the logarithmic singularities, the so-called van Hove singularities, associated with vanishing group velocities for certain frequencies inside the bands. However, for applications to quantum optical experiments in photonic crystals, it is necessary to investigate not only the (overall) availability of modes with frequency w, but also the local coupling strength of an emitter at a certain position [??] in the PhC to the electromagnetic environment provided by the PhC. Consequently, it is the overlap matrix element of the emitter's dipole moment to the eigenmodes (Bloch functions) that is determining quantum optical properties such as decay rates etc.. This may be combined into the local DOS (LDOS), N([??], w), defined as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.7)
For an actual calculation, the integrals in Eq. (1.6) and Eq. (1.7) must be suitably discretized and one may again revert to the methods of electronic band structure calculations (see Ref.).
1.2.2 Group velocity and group velocity dispersion
In order to understand pulse propagation in linear and nonlinear PhCs, it is necessary to obtain group velocities and the group velocity dispersion (GVD) from the photonic band structure. In principle, this can be done through a simple numerical differentiation of the band structure, but in particular for the GVD, this becomes computationally complicated and great care must be exercised in order to avoid numerical instabilities. Therefore, we want to demonstrate how to obtain group velocities and GVD through an adaptation of the so-called [??] x [??]-perturbation theory (kp-PT) of electronic band structure theory. This approach has been applied to systems of arbitrary dimensions and will be particularly useful for the investigation of nonlinear effects in PhCs.
With the help of the Bloch-Floquet theorem Eq. (1.2), we may rewrite the wave equation (1.1) as an equation of motion for the lattice-periodic functions [u.sub.[??]]([??])
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.8)
where, [DELTA] = [[partial derivative].sup.2.sub.x] + [[partial derivative].sup.2.sub.y]. An inspection of Eq. (1.8) for the lattice-periodic [u.sub.[??]+[??]]([??])
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.9)
at a nearby wave vector [??] + [??] (|[??]| [pi]/a) suggests that we treat the second term on the l.h.s. as a perturbation to Eq. (1.8). In writing Eq. (1.9), we have introduced [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Comparing the perturbation series with a Taylor-expansion of frequency [[omega].sub.[??]+[??]] around [??], connects group velocities [[??].sub.[??]] = [[partial derivative].sub.[??]][[omega].sub.[??]] and GVD tensor elements [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] to expressions familiar from second order perturbation theory. Explicitly, we obtain for the group velocity
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.10)
and for the GVD tensor
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.11)
Here, we have used the notation [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for matrix elements of the operator O between Bloch functions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Despite their complicated appearance, these expressions can be evaluated rather easily using standard PWM and obtain very accurate, efficient and numerically stable results. In Fig. 1.2, we display the variation of the group velocities associated with bands 1, 3, and 5 of our model system. Clearly visible, are the extreme variations ranging from 0.5c for band 1 in the long wavelength (effective medium) limit, all the way to the almost vanishing group velocity of band 5 along the entire [LAMBDA]-X direction. This illustrates the huge parameter space of effective group velocities that can simultaneously be realized in PhCs.
1.3 Nonlinear photonic crystals
For large intensities of the light propagating through the photonic crystal, we should also account for the nonlinear polarization [P.sub.NL]([??], z), representing the nonlinear response of the materials that comprise the PhC. In this case, Maxwell's equations for the TM-polarized light propagating in PhCs take the form
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.12)
In writing this equation we have neglected the linear dispersion of the constituent materials, which is usually negligible compared to the dispersion associated with the photonic band structure.
The existence of PBGs, the tailoring of photonic dispersion relations and mode structures through judiciously designed PhCs, represent a novel paradigm for nonlinear wave interactions. To date, only a few works have been carried out for Kerr-nonlinearities or for [[chi].sup.(2)]-nonlinearities in PhCs. Moreover, the approximations involved in some of these works seriously limit the applicability of these theories to real PhCs. For instance, the study of Kerr-nonlinearities in 2D PhCs has been limited to weak modulations in the linear index of refraction. Similarly, the recent investigation of second harmonic generation in 2D PhCs failed to reproduce the well-known results for the limiting case of homogeneous materials.
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