1. Introduction

The ability of a photonic crystal (PhC) [1,2] to control the light dispersion relation (i.e., photonic band structure) with a high degree of freedom is an issue of scientific and practical importance. Tailoring of the band dispersion in PhC systems can give rise to anomalous dispersion characteristics and extremely slow group velocities (known as “slow light”) that cannot achieved in conventional homogeneous waveguides. These characteristics have opened up unprecedented and exciting possibilities in a wide range of photonic information technologies, including all-optical photonic circuit devices [3–5], negative refraction [6,7], and nonlinear optics [8–11]. For example, in line-defect PhC waveguides, a waveguiding mode located in a photonic band gap can produce a wavelength-scale sharp bend in the light propagation [3,4]. Also, specific features of the band dispersion relation and conservation of the tangential component of the wave vector at the boundary in PhCs enable the realization of super-prism, focusing, and negative refraction effects [6,7]. Furthermore, nonlinear optical effects and light-matter interactions can be enhanced by the extremely slow group velocity of light in PhCs [10,11]. Gaining an understanding of the photonic band structure and how it affects these unusual phenomena (both linear and nonlinear) is critical to advancing both fundamental research and technological applications of PhC waveguides, since many photonic properties of PhC waveguides and their underlying physical mechanisms are closely related to the shape of the photonic band structure [10].

The photonic band structure has been investigated experimentally by polarized-angular-dependent reflectivity measurements [12,13]. However, the photonic bands that can be probed with this technique are limited to those above the light cone. The light line in a vacuum (ω=ck) separates the observable region (ω>ck), in which the modes are oscillatory in air, from the region in which modes are evanescent in air and cannot couple with external free photons (ω<ck). However, engineering applications of two-dimensional (2D) PhC waveguides require the use of modes below the light line due to their vertical confinement and long lifetimes. The dispersion characteristics of waveguiding modes below the light line have been studied indirectly by interference measurements of transmission in a 2D PhC slab [14]. This method can deduce the group index (n_g) dispersion, but it cannot accurately determine the shape of the band dispersion due to the uncertainty in the origin position of the k-axis. Although determining the photonic band dispersion relation of the waveguiding modes below the light line is indispensable for understanding the unique properties of 2D PhC waveguides, experimental information on the specific shape of these band dispersions has not been obtained directly.

In this paper, we demonstrate, for the first time, the direct determination of the experimental photonic band structure below the light line in a 2D PhC waveguide using angle-resolved attenuated total reflection spectroscopy of a prism coupling arrangement over a wide frequency range. We observe sharp dips in the reflectance spectra originating from resonance coupling between the external evanescent wave from total reflection off the prism and the waveguiding modes in the 2D PhC waveguide. This provides clear information on individual band components by resolving the angle (i.e., wave vector k) and photon energy. Moreover, we describe the accuracy and suitability of this method by analyzing it by performing three-dimensional finite-difference time-domain (3D-FDTD) calculations of the field distribution and eigen-photon-energy of the observed photonic band modes for a model structure identical to the experimental arrangement of the prism and sample structure.

2. Experiment

Because waveguiding modes in 2D PhCs with k>ω/c cannot be matched to propagating modes incident from free space, they cannot be detected by external plane-wave excitation. To examine the experimental photonic band structure of these modes, we performed angle-resolved attenuated total reflection spectroscopy measurements. The incident radiation was coupled to waveguiding modes using a coupling prism in the Otto configuration [15]. The experimental geometry is shown in Fig. 1(a). A beam of radiation is incident from air through one face of a high-refractive-index prism. The beam passes through the prism and is incident on a second interface at an angle. For angles greater than a critical angle, the beam undergoes total internal reflection at the second interface and exits the prism at the third interface. At the surface of the reflection interface, a decaying evanescent wave exists on the air side of the prism, for which the parallel component of the wave vector satisfies k _//>ω/c. When the surface of the 2D PhC waveguide is brought close to (but not touching) the prism, this evanescent wave can couple to the guiding mode. A sharp dip originating from resonance between the evanescent wave and the waveguiding mode can be detected in the reflectance spectrum.

 

Fig. 1. (a) Schematic diagram showing the experimental geometry of angle-resolved attenuated total reflection spectroscopy and the 2D PhC waveguide used in the experiments and theoretical calculations. The collimated incident white light is transverse electric (TE) polarized. The coupling prism in the Otto configuration is used to couple incident radiation to waveguiding modes in the 2D PhC. (b) Cross-sectional SEM micrograph of the 2D PhC waveguide. The crystal structure of the patterned layer has a square lattice of circular air holes of radius 100 nm with a lattice constant of 300 nm. The inset shows the corresponding 2D first Brillouin zone and the high-symmetry lattice points.

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Figure 1(b) shows a cross-sectional scanning electron microscopy (SEM) micrograph of the constructed 2D PhC slab waveguide structure exhibiting high precision features at a sub-optical wavelength scale. An Ag cladding layer of thickness 500 nm was vapor-deposited on a Si substrate. Poly(methyl methacrylate) (PMMA) of thickness 500 nm, used as the waveguide core layer, was deposited on the Ag cladding by a spin coating and curing technique. The Ag cladding layer provides strong photon confinement due to its extremely low refractive index over a wide wavelength range compared to the low-index polymer core [13]. The 2D PhC slab was patterned with a square lattice of circular air holes of radius 100 nm and lattice constant 300 nm by electron beam lithography and inductivity coupled plasma dry-etching based on O₂/Ar plasmas, optimized to produce straight sidewalls. The detailed techniques for forming the straight sidewalls have been described elsewhere [16,17].

In the angle-resolved attenuated total reflection spectroscopy measurements for the fabricated structure, a collimated tungsten–halogen white light in the range 450–950 nm was used as the light source. A 45° right triangular prism constructed out of LaSF glass was used; it had a refractive index of n_p=1.87. The incident beam was focused onto the sample (100×100 µm²) using an achromatic long-focal lens (f=250 mm). The reflectivity spectra were detected using a corrected Czerny–Turner spectrometer (320-mm focal length) and a liquid nitrogen-cooled open-electrode charge coupled device (CCD). The incident and detected angles could be rotated by a stepping motor stage. The external incident angle θ_ext of the beams was scanned in the range 20–50°. For simplicity, the in-plane propagation lattice direction was set along the Γ-X (ϕ=0°) direction. The two-dimensional Brillouin zone and the symmetry points for a square lattice are shown in Fig. 1(b). All the measurements were made with transverse electric (TE) polarization of the incident light.

We carried out photonic band structure calculations for the 2D PhC slab waveguide using the 3D-FDTD method for Maxwell’s equations. The time-development Maxwell’s equations can be written in the following form:

where ε(r) is the permittivity, µ(r) is the permeability, and σ(r) is the conductivity, and are functions of the spatial coordinates. These equations can be discretized in space and time by a Yee-cell algorithm [18,19]. We used a 10⁵ discretization grid in the unit cell for the FDTD time-stepping formulas. An initial TE-polarized electric field with a low-symmetry distribution in the core was used to excite the TE-like modes. The evolution of the initial TE-polarized electromagnetic field was calculated using a discrete FDTD method. The model structure used for the calculations consisted of four layers: an air layer, the PMMA patterned core layer, the Ag metallic cladding layer, and the Si substrate layer. The thicknesses and the patterned square lattice of circular air holes were set to be the same as for the specimen used in the experiments. As a first approximation we used the equivalent relative permittivity for a metal (Ag cladding) ε(ω)=ε ₀+σ/iω. This method has a simple and general form and is very stable in FDTD calculations [13]. The refractive indexes of the dielectric layers were set to 1.49 for PMMA and to 3.75 for the Si layer, and the conductivity of the Ag cladding [20] was set to 3.10×10⁷ S/m. To simulate the 2D PhC waveguide structure in the computational mesh, Mur’s absorbing boundary condition was applied to the top and bottom boundaries normal to the plane of the waveguides. The Bloch boundary conditions over a unit cell of a PhC give the k vector dependence of the eigenmode. All the fields are obtained in the time domain in this method. In order to obtain the spectral information, the calculated fields need to be converted from the time domain into the frequency domain by taking their Fourier transform. The peaks of the spectral intensity correspond to the locations of the eigen-photon-energies of the band modes. The position-dependent field distributions of the band modes are also obtained by taking the Fourier transform of the transient field pattern. By performing the analysis in three dimensions, we were able to evaluate the accuracy and suitability of the technique for analyzing an experimental photonic band structure below the light line.

3. Results and discussion

Figure 2 shows typical angle-dependent attenuated total reflection spectra of the 2D PhC slab waveguide along the Γ-X lattice direction, taken with TE polarization. Several sharp dips are clearly observable, demonstrating that the evanescent field above the prism resonantly couples to in-plane photonic band modes in the 2D PhC. The observed resonance wavelengths clearly depend on the incident angle, which determines the in-plane wave vectors, k _//=(2π/λ) n_p sinθ_int, where λ is the wavelength of the incident light and θ_int=0° is normal incidence. The resonance dip at each resonance angle in the reflectivity is thus directly connected to the experimental dispersion curves, i.e., the experimental photonic band structure. Moreover, in regions of good coupling, the magnitude of the dip, which indicates the extent of the resonance coupling, was very large and had a maximum value of ΔR/R>80%. Since these features are sensitive to the profile of the etched holes and the separation distance between the prism and the PhC surface, it is clear that we have achieved precise nano-lithographical fabrication of the PhC waveguide structure and good coupling between the evanescent field and the photonic band mode for a short separation.

 

Fig. 2. Typical measured reflectance spectra for various angles of incidence with TE polarization along the Γ-X direction. For clarity, the spectra are shifted along the vertical axis by 0.3 with respect to each other.

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Figure 3 shows the experimental photonic band structure of the PhC along the Γ-X line obtained from the angle-dependent attenuated total reflection spectra, compared with the theoretical band structure calculated by the 3D-FDTD method for a model structure identical to the experimental sample structure. The straight solid line corresponds to the light line in a vacuum (ω=ck) and separates the region in which the modes are oscillatory in air (ω>ck) from the region in which modes are evanescent in air (ω<ck). In a standard angular-dependent reflectivity measurement without a coupling prism, the experimental bands that can be probed are limited to those above the light cone [12,13]. In angle-resolved attenuated total reflection spectroscopy, the in-plane wave vectors k _// of the incident plane-wave are increased in proportion to the prism index n_p, and hence many of the photonic band dispersions under the straight dashed line correspond to the boundary of the enlarged light cone in the prism (ω>ck/n_p). The features of the experimental band structure are in reasonably good agreement with the theoretical curves. This agreement demonstrates that it is realistically possible to design advanced fundamental research and technological applications of PhC slabs using this technology, since many optical properties of the waveguiding modes are closely related to the shape of the band structure under the light line in air.

 

Fig. 3. Experimental photonic band structure under the light line of the 2D PhC waveguide along the Γ-X line obtained by polarized angular-dependent reflection spectroscopy (closed circles) and the theoretical band structure calculated by the 3D finite-difference time-domain method (open squares). No empirical adjustments were made in the calculations. The field distributions and eigen-photon-energies of the bands indexed A, B, and C at the high-symmetry lattice point X are calculated as a function of the separation distance between the PhC and prism and shown in Figs. 4 and 5.

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Our results demonstrate the experimental specific shape of band dispersions under the light line, but we note that the short distance between the PhC and the prism to obtain good coupling may modulate the eigen-photon-energy of the band mode in the PhC. To examine the accuracy of our method, we should consider fluctuations in the eigen-photon-energy in the arrangement of the sample structure and the prism. We now examine the fluctuation of the field pattern and the eigen-photon-energy for several separation distances by 3D-FDTD calculations for a model structure identical to the experimental arrangement of the prism and sample structure. Figure 4 shows a cross-sectional view of the field distributions of the band modes under the light line in the z-x plane through the center of an air hole as a function of the separation distance (i.e., air-gap) d between the PhC and prism of the lowest photonic energy band indexed as A, the second lowest band indexed as B, and the third lowest band indexed as C, at the X point (see Fig. 3). In this figure, the maximum magnitude of the field is normalized to unity. The figure shows that the fields of the band modes under the light line are strongly confined in the waveguide core for large separation distances. The strength of the evanescent field is stronger at the PhC surface for shorter separation distances, resulting in increased coupling between the incident radiation and the band modes for a small air gap. This can be seen in the data as an oscillating mode in the prism at a separation distance less than 500 nm. Good coupling was observed at about 300 nm in each band mode.

 

Fig. 4. Electric field distributions in the z-x plane through the center of an air hole for several prism/2D PhC separation distances d at the X point of the three different photon energy bands indexed as A, B, and C in Fig. 3. The maximum of the electric field is normalized to unity. Perspective outlines of the prism/air interface in cross-section are superimposed on the diagram (red lines).

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Figure 5 shows the eigen-photon-energy of the band modes under the light line as a function of the separation distance d between the PhC and prism for three different energy bands indexed as A, B, and C at the point X (see Fig. 3). The eigen-photon-energy information was obtained by taking a Fourier transform of the fields in the time domain with an energy resolution of 13 meV at a photon energy of 2.0 eV. The photon energy domain for the shorter separation distances shifts towards the high photon energy side for each band. For the high-energy band C, the modulation of the eigen-photon-energy is relativity large. The quantity of the shift in the eigen-photon-energy with decreasing air-gap distance depends mainly on the vertical field distribution of the band mode. The lower energy bands A and B are the lowestorder waveguide mode and its electric field is almost perfectly confined in the waveguide core. On the other hand, the high-energy band C, near the light cone, has a double peak in the z direction, which indicates that the band corresponds to the second-order waveguide mode. This implies that this band mode has a relativity large amplitude in the air side region compared with the bands A and B, and is well coupled with the evanescent wave from the prism surface. This result concerning the differences in the quantity of the eigen-photon-energy shift agrees well with the behavior of the field distributions in Fig. 4. Judging from the field distribution changes in Fig. 4, the separation distance for which good coupling can be obtained is about 300 nm. The eigen-photon-energy shifts between separation distances of 300 nm (good coupling) and 3 µm (isolated) were +13 meV (Δhν/hν=0.6%) for band C and less than the energy resolution for bands A and B. In addition, the energy shifts were comparable with the computational error for our 3D-FDTD calculations, which was roughly estimated to be less than 1% from the convergence behavior of the eigenfrequencies. We found that the fluctuation in the photon energy domain between the band modes of the PhC with the prism and the isolated PhC were barely significant even at the well-coupled separation distance of 300 nm. These results clearly demonstrate that, except in the case of over coupling (d < 100 nm), the angle-resolved attenuated total reflection spectroscopy technique is accurate and suitable for directly determining the experimental photonic band structure below the light line in 2D PhC waveguides.

 

Fig. 5. Eigen-photon-energy of band modes as a function of the prism/2D PhC separation distance for the three different energy bands indexed as A, B, and C at the point X in Fig. 3.

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4. Conclusions

We have experimentally demonstrated for the first time the direct determination of the photonic band structure of waveguiding modes below the light line in a 2D PhC waveguide by angle-resolved attenuated total reflection spectroscopy measurements using a prism coupling arrangement over a wide frequency range. The measured photonic band structure is in good agreement with theoretical calculations. The accurate shape of the experimental band structure provides direct information on light dispersions and the propagating (both linear and nonlinear) properties of 2D PhCs. This information is crucial for understanding and designing the anomalous optical behavior of many photonic applications using PhC systems with a many degrees of freedom.