1. Introduction

As a result of their rich band structures and unique dispersion relations, photonic crystals (PhCs) have been shown to display a variety of behaviors such as the superprism [1], self-collimation [2], anomalous and negative refraction [3–6]. In particular, the negative refraction of dielectric PhCs can be used to form images using a flat slab leading to subwavelength transversal resolution in the partial bandgap of the first band [6] and the second band [5].

For the left-handed metamaterials [7–9], although the physics of negative refraction were initially disputed, it is now widely accepted that the electric field, the magnetic field, and the wavevector form a left-handed triplet, the wavevector is antiparallel to the Poynting vector, and thus the phase velocity is opposed to the energy velocity. The result is a backward-traveling wave, and the refractive index related to phase is negative. When a wave travels in materials with alternating positive and negative refractive index, the accumulated phase can be compensated or even nulled or cancelled. Based on such phase cancellation, a new phase shifter has been demonstrated [10] using a left-handed metamaterial in which the geometrical size of the composite cells or atoms is much smaller than the working wavelength.

Although the dielectric photonic crystals can display negative refraction [5, 6], the mechanisms differ from those of left-handed metamaterials. For negative refraction at the second band and higher, an increase in the normalized frequency causes shrinkage of the equifrequency contours [5]. The resulting effective refractive index n _eff calculated from band structure is negative due to the inwardly directed group velocity [5]. Using plane wave decomposition from sampled field values [11], it has been shown that the fundamental wavevector of the Bloch modes in the first Brillouin zone is opposed to the energy flow. Because of this, it was expected that such a photonic crystal would exhibit left-hand behaviors. By applying a theoretical analysis of the Bloch modes using Fourier decomposition [12], it has been shown that the unfolded band structure cannot give a negative effective index of refraction. This contradiction raises a question about the fundamental wavevector of the Bloch modes.

For a plane wave in an isotropic medium, its phase velocity is associated with its wavevector, and the plane wave is a solution of Maxwell Equations [13]. In photonic crystals, the solution of Maxwell Equations is expressed as a sum of Bloch modes. Each of the modes may be assigned a phase velocity associated with its wavevector [14], which in general means that one may not define a global phase velocity [12]. However, each of these individual modes (spatial harmonics) is not itself a solution of Maxwell Equations [12]. When excited, the Bloch modes travel collectively in a definite direction governed by the group velocity [15]. We cannot distinguish the individual phases of each mode, but can only identify the overall phase. We define this to be the measurable phase.

While investigating the non-restricted imaging properties of finite-sized PhC slabs at the second band [16], we observed that when a plane wave is normally incident on to the slab, the waveform inside the slab almost maintains its plane wave nature in the direction of propagation, with some modulation or disturbance. Furthermore, when the plane wave passes through the slab, the exit beam is also a plane wave (excluding higher-order diffractions at the exit interface). Since both the input and the output are plane waves, we can say that the PhC slab introduces a macroscopic, measurable phase delay. Therefore, we can assign a phase velocity to the wavefronts inside the PhC slab. Consequently, a refractive index related to the phase delay which we term the phase-related refractive index can also be assigned. This raises several questions: is this phase-related refractive index the same theoretically as the effective refractive index calculated from the band structure? If different, is there any relation between them? Is it associated with any specific wavevector of the Bloch modes? Can the negative fundamental wavevector be predicted from the band structure?

Using a pseudo-interference method we showed previously that the phase-related refractive index is positive [17]. Around the same time Ref. [18] also found that the phase evolution inside PhC is positive. We have also carried out simulations of open cavities [19] which show that the phase-related refractive index deviates significantly from the effective index calculated from band structure. Therefore, the answer to the first question is that the two refractive indices are not the same. This suggests that the effective refractive index and the phase-related refractive index are still not satisfactorily understood, and there is ambiguity in determining the sign and the value of the indices. In this paper we aim to resolve this ambiguity, and it is arranged as follows: Section II hypothesizes the existence of a dominant wave and discusses the refractive indices; Section III analyzes the Bloch Modes numerically and validates the hypothesis. Section IV provides some discussions and gives conclusion.

2. Bloch modes, dominant wave and its refractive index

For conciseness, we directly use some known results. The discussion is limited to the two-dimensional (2D) photonic crystals. The wave amplitude u can be expressed as a sum of Bloch modes as [5]:

where k⃗′ is the fundamental wavevector [21], m and n are integers, u_m,n is the amplitude of the (m, n)_th mode, G⃗₁ and G⃗₂ are the reciprocal lattice vectors, and r⃗ is the position vector.

Our hypothesis is that, although the Bloch modes have an infinite number of spatial harmonics, the measurable phase is dominated by the wavevector that has the largest amplitude, which we define to be the dominant wave. The dominant wave can be determined from the solution of Maxwell Equations with approximate homogenous permittivity and the band structure. It can be shown (by adapting the 1-D model presented in [15]) that the amplitude of each Bloch mode u_m,n ∝ 1/q , where the quantity q can be written as

where e̅ is the averaged relative dielectric constant , given by ε̅ = fε_hole +(1-f)ε_host for TE waves and 1/ε̅ = f/ε_hole + (1- f)/ ε_host for TM waves (the subscripts denote hole and host materials), f is the filling factor, μ is the permeability, and ω is the time harmonic angular frequency. In the limit of zero hole-size, the only mode that is supported is the plane wave k⃗′ with $k\text{'}=n{k}_0=\omega \sqrt{\mu {\epsilon }_0\bar{\epsilon }}$ where k ₀ is the wavenumber in vacuum and n̅ = √ε̅. The inverse amplitude quantity q can be further simplified as (k⃗'/k ₀+(m G⃗₁+n G⃗₂)/k ₀)²-n̅². Defining the dominant wavevector as k⃗_M = k⃗'+m_M G⃗₁+n_M G⃗₂ where the subscript M denotes the dominant wave, its phase-related refractive index can be written as

for normal incidence. The dominant wave should therefore have its phase-related refractive index closest to the averaged refractive index. When the periodicity disappears, the material becomes homogenous. Thus the Bloch mode becomes a regular plane wave [20], and equation (3) leads to the material’s refractive index.

We take as an example the hexagonal photonic crystal in [5]. This has a refractive index of 3.6 for the host material with holes in air, and hole radius 0.4a resulting in an averaged value ε̅=9.5 for TE waves. Fig. 1 shows the reciprocal lattice and the reciprocal lattice vectors as G⃗₁ = x⃗2π/√3a + y⃗2π/a, G⃗₂ = x⃗2π/√3a-y⃗2π/a where x⃗ and y⃗ are the unit vectors along the axes, and a is the spatial lattice constant. For the first band, band structure calculation shows that the effective index ranges from 2.46 to 2.9. Thus the dominant wave is the fundamental wavevector itself, that is, m_M =n_M = 0, and its phase-related refractive index can be calculated from the band structure, and is the same as the effective refractive index n _eff. This is no surprise since from equation (3) both indices have the same definition. For the 2^nd band calculation from (2) shows that the mode m_M = n_M = 1 is the dominant wave with wavevector k⃗'+G⃗₁ + G⃗₂, which is similar to the result in [18]. Note that the vector G⃗₁+G⃗₂ is also a reciprocal space vector, which is ΓA⃗ shown in Fig. 1. The dominant wavevector is k⃗'+G⃗₁+G⃗₂ equal to k⃗'+x⃗4π/√3a which is always positive. This explains why the phase evolution is positive [18] and the phase-related refractive index is positive [17]. The contribution of other Bloch modes to the phase is simply to modulate its evolution, as illustrated from the pseudo-interference patterns [17].

 

Fig. 1. The reciprocal lattice of the hexagonal photonic crystals. First Brillouin zone is inside the blue hexagon.

Download Full Size | PDF

 

Fig. 2. Fourier decomposition of Bloch modes at the first band of the PhC with r/a=0.4 and host material n=3.6 for TE polarization

Download Full Size | PDF

3. Numerical Fourier decomposition of Bloch modes

To test validity of the above hypothesis, we numerically decompose the Bloch modes from the simulated complex filed (real and imaginary) inside the hexagonal photonic crystal by applying fast Fourier transform. The simulations were performed with the finite-difference time-domain (FDTD) method. To remove the possible effect of the second interface, we use a semi-infinite photonic crystal. The plane wave is incident on it normally from free space. Periodic boundary conditions are applied for the vertical ends perpendicular to the direction of light propagation, and perfectly-matched layers (PML) along the direction of light. The normal to the interface is along ΓM. The simulation stopped before the light reached the second surface.

We first investigate the first band, where the effective refractive index is positive as calculated from the band structure. The incident light is a TE-polarized plane wave (with E-field parallel to the holes). Fig. 2 shows the Fourier decomposition of the Bloch modes calculated from the complex field at the normalized frequency 0.156, where k_x (ΓM) and k_y (ΓK) are the wavenumbers along axes. The wavevector with the largest amplitude has a refractive index of 2.564, as defined in (3). The pseudo-interference technique gives a value of 2.59. From the band structure, the effective refractive index is calculated to be n _eff=2.562. Therefore, it is recognized that the dominant wavevector is the fundamental wavevector since no other wavevectors have a refractive index close to 2.56. To verify this result, some other simulations at different wavelengths were also done and similar results are obtained. Therefore we conclude that the effective index of refraction calculated from the band structure is the same as that of the fundamental wavevector of the Bloch modes at the 1^st band. This confirms our previous analysis. The wavevector sampling interval is 0.04 k ₀ in this paper, thus the precision of the calculated refractive index (as extracted from Fourier decomposition) is ± 0.04.

At the second band for the same photonic crystal, band structure calculation leads to n _eff=-1 at a normalized frequency of 0.30, while using pseudo-interference the phase-related refractive index is 2.87, which is positive. Obviously they are quite different. From simulation, the Fourier decomposition shown Fig. 3 gives a refractive index of 2.89 for the dominant wavevector, which is close to 2.87. This indicates that the dominant wave indeed determines the phase-related refractive index. From theory Equation (3) gives a refractive index of 2.84 for the dominant wavevector k⃗'+G⃗₁ + G⃗₂. For the fundamental wavevector Fig. 3 shows its refractive index to be -0.98, which is close to the value of n_eff as calculated from the band strucure. The relative amplitude of the fundamental wavevector to the dominant wave is 8.8% from the numerical results while using equation (2) gives 11.5%. The second negative vector from Fourier decomposition is -2.87k ₀ (corresponding to k⃗'-G⃗₁ -G⃗₂ which leads to -2.84k ₀). The second positive wavevector gives 6.74 k ₀, which is close to the theoretical value of 6.68 k ₀ for k⃗'+2G⃗₁ +2G⃗₂.

For the 2^nd band, we have also simulated oblique incidence cases using a Gaussian beam with PML conditions and obtained similar results. The value of ∣k⃗'/k ₀∣ is almost constant at the normalized frequency 0.30, confirming that the equifrequency contour is almost circular.

For a further demonstration, we performed an analysis of the third band of a photonic crystal using the same host material but with a ratio r/a of 0.35 for TM-polarization, the band structure (not given) shows that it has negative refraction. The dominant wave from equation (2) is again k⃗'+G⃗₁ +G⃗₂ and its refractive index is 2.19 at a normalized frequency of 0.428 (note that the equifrequency contour is almost hexagonal). The effective refractive index is -0.51 from the band structure. Fig. 4 shows the Fourier decomposition of the Bloch modes, which results in a fundamental wavevector refractive index of -0.53 and the dominant wavevector of 2.16. The pseudo-interference method gives a phase-related refractive index of 2.13, close to that of the dominant wave from our hypothesis and the Fourier decomposition. Some other wavevectors can also be identified from Fig. 4 and they agree with our hypothesis very well. For example, the second largest peaks have wavevectors (0.795 x⃗ ±2.35y⃗)k ₀, corresponding to wavevectors k⃗'+G⃗₁ and k⃗'+G⃗₂ which have theoretical values of (0.840 x⃗ ± 234 y⃗)k ₀. The good agreement between simulated results and theory can only be obtained with a negative fundamental wavevector, for which the refractive index has been calculated from band structure. This once again indicates that the folded band structure can correctly predict the behavior of the fundamental wavevector.

However, if the photonic crystal is highly anisotropic, the method can fail to correctly predict the dominant wave. For example, at a normalized frequency 0.4112 of the same PhC (r=0.35a) for TM polarization, the dominant wavevectors as determined by Fourier decomposition are (0.72 x⃗ ± 2.43 y⃗) k ₀, which correspond to k⃗'+G⃗₁ and k⃗'+G⃗₂, both of which differ from the predicted vector k⃗'+G⃗₁ +G⃗₂. Even though the hypothesis fails to predict the correct dominant wave in this situation, the numerical Fourier decomposition correctly give the wavevectors. For example, Fourier decomposition gives the fundamental wavevector of -0.76k ₀ which is close to -0.74 k ₀ calculated from band structure. Most importantly, it shows that the fundamental wavevector is indeed negative, and the phase velocity is positive.

 

Fig. 3. Fourier decomposition of Bloch modes of the same photonic crystal as in Fig. 2 but at the second band at normalized frequency 0.30.

Download Full Size | PDF

 

Fig. 4. Fourier decomposition of Bloch modes of the PhC (r/a=0.35) at the 3^rd band for TM-polarization at normalized frequency 0.428.

Download Full Size | PDF

4. Discussion and conclusion

We performed simulations at other normalized frequencies and obtained similar results (i.e., more isotropic equifrequency contours display good agreement between our theory and simulation). Therefore our hypothesis is more accurate for operating regimes where the band structure is less anisotropic. It also works well at long wavelength regime (ie, the 1^st band, even when it is highly anisotropic). At much higher bands with large value of a/λ the prediction can fail, probably because of strong scattering, highly anisotropic equifrequency contours (not shown) and/or the almost equal amplitudes of several wavevectors. Nonetheless, the results that we presented clearly show that the fundamental wavevector is negative and the phase-related refractive index is positive. A more accurate method to predict the dominant wave would be to use the plane wave expansion method. However we do not explore that in this paper.

It should be pointed out that the amplitudes of the simulated Bloch modes can be influenced by coupling condition, which is a function of the interface. However it is neither the effective refractive index, nor the phase-related refractive index that determine the coupling at the interface. The coupling of each Bloch mode is still a subject of investigation.

Our definition of the phase-related refractive index (equation (3)) means that the associated wavevector is directed out of the 1^st Brillouin zone at higher bands with negative refraction. It is always positive and determines only the phase evolution inside the photonic crystals but not the momentum conservation nor the refraction at the interface. This differs from the left-handed materials where the negative refractive index is due to material’s electric and magnetic resonances [8] and it determines the refraction at the interface. The fundamental wave can only dominate when a <<λ so that ∣G ₁ +G ₂∣ >> k ₀ , which is unlikely to occur at higher bands with negative refraction. The negative phase velocity obtained in [21] is due to the assumption of a phase term exp(jkx) in the field, which neglects the standing waves or Bloch resonances of photonic crystals [18, 22].

In summary, by taking polarization into account, we have hypothesized the existence of a dominant wave which governs the propagation of Bloch modes through photonic crystals with a positive phase velocity. This hypothesis is supported by the results of numerical studies in which we applied a Fourier transformation to the complex field inside semi-infinite photonic crystals. The results showed unambiguously that the fundamental wavevector has a refractive index which is the same as that calculated from the folded band structure in both sign and magnitude; the phase-related refractive index is associated with the dominant wave, and it is positive, thus there is no possibility of phase cancellation even through there is negative refraction due to a negative value of the fundamental wavevector.