1. Introduction

Negative refraction has attracted much attention recently [1–13, 15–23]. In contrast to double-negative metamaterials (also known as left-handed materials) [1–5], single negative materials [2, 5] and indefinite materials [6], photonic crystals (PhCs) made of synthetic periodic dielectric materials can exhibit negative refraction behaviors that are solely determined by the characteristics of their band structures and equifrequency contours [7–13].

In general, there are two categories of negative refraction in PhCs. The first is in the transmission band (the second band and higher), where an increase of normalized frequency leads to the shrinkage of the equifrequency contours surrounding the Γ point. The effective refractive index n _eff is negative [9] due to the inwardly directed group velocity according to definition [14]. Design calculations are based on the band structures [9]. It has been demonstrated that the fundamental wavevector of the Bloch modes in the first Brillouin zone is indeed opposed to the energy flow [15]. However, such opposition does not mean a negative phase velocity [16–18]. In fact, the phase velocity and the phase related refractive index is positive and is governed by the dominant wave [18].

The second category of negative refraction occurs inside the partial bandgap (most previous studies have considered the first band only). It was originally suggested that in this case negative refraction is due to a “negative-definite effective photonic mass” and strong anisotropy [10]. However, this concept of photonic mass does not allow analytic design for negative imaging. Although there have been some theoretical and experimental investigations of imaging using this type of photonic crystal, for example [10–12, 19–23], few papers made further investigation into the design of such photonic crystals.

Here we seek to establish conditions for negative refraction in the partial bandgap. We introduce a simple explanatory model and show that strong anisotropy is not a necessary condition, and imaging is not necessarily restricted and adjacent to the interface as was shown in [10] and [12].

2. Conditions for negative refraction

Previous papers investigated the negative refraction in the first band and have not explicitly and systematically summarized the conditions. By analyzing previous results and based on our own investigation which will be described in more detail below, we hypothesize the conditions for negative refraction in the partical bandgap: 1) the equifrequency contours are convex around another symmetrical point other than the Γ point (for example, in[10] this is the M point for a square lattice), and 2) the equifrequency contour shrinks to this point as the normalized frequency increases. Note that these conditions do not require any anisotropy and do not restrict this negative refraction behavior to the 1^st band.

The complex amplitude of a wave inside a photonic crystal can be written in the form of Bloch modes [14]

where $\overrightarrow{k}\prime$ is the fundamental wavevector, u_m,n is the amplitude of individual Bloch mode (m,n), and is a periodic function of space, ${\overrightarrow{G}}_1$ and ${\overrightarrow{G}}_2$ are the reciprocal space vectors in the first Brillouin zone, and r⃗ is the space vector. For a photonic crystal with a square lattice constant a (Fig. 1), the magnitude of the reciprocal space vector is given by

By examining the 1^st band of the photonic crystal investigated in [10], we can find that at lower frequencies, the equifrequency contours surround the Γ point (the Brillouin center). Since the fundamental wavevector k′ is zero at Γ, this Γ point is the reference center for waveneumbers. However, at higher frequencies above the X point, that is, beginning at some point inside the partial bandgap, the equifrequency contours surround the M point (condition (1)). These are shown in Fig. 1 and are calculated from the band structure using BandSolve® (RSoft Inc.). Further, as the normalized frequency increases, the equifrequency contours shrink (condition (2)). Therefore, negative refraction occurs. Under these conditions, the wavenumber reference point must be shifted to the M point (as demonstrated by the shift of the airline in [10]). Since the absolute value of the relative wavevector at the M point is k _ΓM=G ₁, Eq. (1) should be modified as

where

is the local wavevector relative to the M point. Referring to Fig. 1, the projected wavevector on MΓ of this local wavevector $(\overrightarrow{\mathrm{MA}}\prime )$ is always opposed to the projection of the incident wavevector $\left(\overrightarrow{\Gamma A}\right)$ , and also opposed to that of the group velocity (the norm of the equifrequency contour, not shown in Fig. 1) which points away from Γ on the contours.

 

Fig. 1. The equifrequency contours calculated from band structure and the conservation of the momentum. The interface is along NN.

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Fig. 2 The effective refractive index (solid red line) and the local effective refractive index (black line) calculated from band structure. The circles and diamonds are numerical results extracted from simulations in Sec.III.

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We define an effective refractive index as the ratio of the (local) fundamental wavenumber and the wavenumber in free space (if the equifrequency contours are square-like, such an effective index may be considered as the principal value). From the band structure calculated using BandSolve® (RSoft Inc.) and verified with PWE Band Solver® (OptiWave Inc.), the solid (red) line shown in Fig. 2 is the effective index calculated for the photonic crystal in [10] where Ω is the normalized frequency. With equation (4) we also calculated the refractive index of the local wavevector at different frequencies with negative refraction, shown in Fig. 2 as solid (black) line in absolute values.

Based on the concept of negative-definite mass, originally [10] negative imaging was found to occur in the range of Ω=0.186 to 0.198, which is denoted in Fig.2 by the two vertical dashed lines labeled as “NI Region [10]”. However, based on the conditions we present here, and employing the concept of the local wavevector (4), all of the frequencies within the partial gap which have equifrequency contours surrounding M can experience negative refraction. This range of normalized frequency extends from 0.17 to 0.236 (labeled as “NI Region Eq. (4)” in Fig. 2) whereas the partial bandgap begins at 0.164. According to (4), the local effective refractive index lies between -1.62 to 0.0 in this frequency range whereas the effective refractive index calculated from the band structure is 2.54 to 3.00 for the fundamental wavevector.

Around Ω=0.20, the local refractive index is close to -1, indicating that negative imaging should be possible. This differs somewhat from the result published in [10] where the central normalized frequency for negative refraction was given as 0.192. Equation (4) yields a local effective refractive index is -1.09 at this frequency. This discrepancy is probably due to differences in the implementation of the different software tools used to perform the analysis.

3. Numerical validation and discussion

To validate the above hypotheses, we performed some numerical simulations for two-dimensional photonic crystals. The first simulations were designed to validate the effective refractive index calculations from the band structures. The second simulations were to validate the two conditions mentioned above. Although our simulations were based on specific photonic crystals, the results can be applied to any other similar photonic crystals. All the simulations were performed using the finite-difference time-domain (FDTD) method (OptiFDTD 7) from OptiWave®, with a spatial sampling of 10nm and the Courant limit time step size.

For the calculation of effective refractive index, we used a semi-infinite PhC with an air interface along ΓM with 60 by 30 holes (the PhC parameters are the same as in [10]). To excite the Bloch modes, a plane wave is used with H-polarization where the magnetic field is in parallel to the holes. For normal incidence, the periodic boundary conditions and the perfect-matched-layers are applied accordingly. The resulting complex fields (real and imaginary) in the entire PhC simulation domain are then Fourier-transformed to decompose the Bloch modes. As shown previously [18], the band structure predicts the behavior of the fundamental wavevector, which can be used to calculate the effective index. The extracted effective refractive index from such simulations is shown in Fig. 2 as small circles. Using Eq. (4), the local refractive index is also calculated and is shown in Fig. 2 with small diamonds. It can be seen from Fig. 2 that the simulation results agree well with the calculation from the band structure and Eq. (4). For example, at Ω=0.192 the extracted effective index has a value of 2.535. From the band structure, it is 2.54; k _ΓM (G ₁) is obtained from simulation as 3.67 k ₀ which agrees well with the theoretical value 3.68 k ₀ from Eq. (2).

Furthermore, to show the location of wavevectors in the reciprocal space, the reciprocal lattice is also numerically obtained through Fourier transformation of the field where a cylindrical source is excited at the center of the infinite photonic crystal and four periodic boundaries are used. Figure 3 shows the 2D reciprocal lattice (corresponding to the equifrequency contour shown in Fig. 1) of this photonic crystal at Ω=0.192. This should be compared to Fig. 1 (calculated from band structure). From the numerical values of various wavevectors given in the above paragraph, the two results from different methods are the same. The red spots correspond to the location of the wavevectors k⃗′+(mG⃗ ₁+nG⃗ ₂) which are obtained using another simulation with light incident normally onto the semi-infinite PhC, and their brightness of the red spots shows the relative amplitude of the Bloch modes. The crossing point of the two solid lines is the Γ point with an absolute zero wavenumber and the crossing point between the horizontal solid line and the vertical dashed line is the M point. It can be seen that the fundamental wavevector is located on the left-hand side of the equifrequency contour surrounding the M point where the light is incident from left to right. Fig. 3 also clearly shows that the fundamental wavevector is the dominant wavevector, and is located within the first Brillouin zone. This differs from the case of negative refraction at higher transmission bands where the dominant wave is not in the first Brillouin zone [18].

We then analyze imaging properties using a 7-row PhC slab with a fixed lattice constant 0.349 µm. The object is a Gaussian source (in practice, a narrow waveguide can be used) with a half-width of 0.1µm located at 0.6 µm in front of the slab. Figure 4 shows the modulus of the magnetic field at different normalized frequencies Ω. At Ω=0.168 in Fig. 4(a) and 0.192 in Fig. 4(b), there is no image (intensity peak) found. Analysis shows that image formation begins around the normalized frequency of 0.196. At Ω=0.200 in Fig. 4(c) an image can be found with an intensity peak on the left side of the slab, and at Ω=0.225 in Fig. 4(d) the image is clearly not adjacent to the PhC. Furthermore, it can be seen that, as the normalized frequency increases, the image distance increases. This is to be expected, given the decreasing value of the local refractive index shown in Fig. 2.

 

Fig. 3. The reciprocal lattice space and the first few Bloch modes (red spots) from simulations.

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Fig. 4(a) and (b) (a) Ω=0.168 (no image) (b) Ω=0.192 (no image) Fig. 4. (a) (b) Imaging of a 7- row PhC slab at four different normalized frequencies.

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Fig. 4(c) and (d) (c) Ω=0.200(image formed) (d) Ω=0.225 (non-adjacent image) Fig. 4. (c) (d) Imaging of a 7-row PhC slab at four different normalized frequencies.

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Fig. 5. 2^nd band of a photonic crystal with rod in air (rod-radius-period ratio 0.38 and dielectric constant 8.9)

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From Fig. 4 it can be seen that the origin of the adjacent and restricted imaging properties reported in [12] is due to the square-like contours at lower frequencies, which tends to collimate light. In other words; since the incident waves are almost collimated by the photonic crystal, imaging is independent of the object locations (restricted imaging [12]). As the normalized frequency increases, the equifrequency contours become rounded. Therefore image locations depend on the object. To confirm the above analysis, we numerically calculated the image position from simulation as the object was shifted by 0.1 µm towards or away from the slab. At Ω=0.200, the image is shifted 0.10µm and 0.09 µm respectively. At Ω=0.225 (local effective refractive index -0.443) the image is shifted 0.11 µm and 0.10 µm. Thus imaging is indeed not restricted. Note that a claim of imaging may not be justified if only the outgoing convex wavefronts (the instantaneous field) is observed since it is possible that the intensity decreases monotonically. If the evanescent waves do not dominate image formation, an intensity peak should be found to identify an image [11]. Images given in Fig. 4 passed this test.

It is possible to have negative refraction in higher bands within the partial band gap. As an example, Fig. 5 shows the band structure of the 2^nd band of a photonic crystal with rod in air (dielectric constant is 8.9, rod radius is 0.38a) of TM polarization. The highest normalized frequency is 0.4 at Γ. Below this frequency the normal negative refraction in the transmission band is similar to [9]. The equifrequency contours from Ω=0.38 to 0.44 surround the M point and the two conditions given in Section II for negative refraction are satisfied. Figure 6 shows imaging at Ω=0.41, and Fig. 7 shows the equifrequency contours. It can be seen that, at this normalized frequency, the equifrequency contour is almost rounded, and the image is not adjacent to the interface. Further simulations show that imaging is also not restricted. Therefore, negative refraction can be extended to higher bands.

However, if the two conditions are not satisfied as occurs for example the K point in Fig. 5, negative refraction may not occur. In this case, although condition (1) is satisfied, that is, the equifrequency contours surround and are convex to the K point, the equifrequency contours expand with increasing normalized frequency, which contravenes condition (2).

Since refraction is governed by the band structures, one may not use the local effective index to predict the exact location of the images using the Veselago relation [1, 13]. In other words, although the local refractive index is helpful in intuitively understanding negative refraction in the partial band gap, its value does not guarantee the Veselago relation, in particular if the equifrequency contours are not circular, and the normalized frequency deviates from the ideal value for imaging.

Note that the two conditions given in this paper are sufficient: whether all of them are necessary needs further investigation. Note also that to compare the theory and experiments is a challenge, since fabrication errors will significantly modify the theoretical band structure. Nonetheless, quantitatively the imaging phenomenon can be observed experimentally [20–22].

 

Fig. 6. Negative imaging at Ω=0.41 in the partial bandgap at the 2^nd band (rod in air with permittivity 8.9 and radius-period ratio 0.38).

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Fig.7. Equifrequency contours of the PhC in Fig. 6 (Notice the contours surrounding M satisfy the two conditions of negative refraction).

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4. Conclusion

We have proposed the two sufficient conditions for negative refraction in the partial bandgap. Under these conditions, since the operational reference point is shifted from Γ to M, we introduced the concept of the local wavevector Eq. (4) which is always negative. Therefore the negative refraction can be explained intuitively by the antiparallelism between the local wavevector and the group velocity. The local negative refractive index can be obtained from the photonic crystal (the periodicity) and its band structure using Eq. (4), which provides a guideline for the design of such photonic crystals. Neither anisotropy nor the 1^st band is a prerequisite for negative refraction in such a photonic crystal. The restricted and surface-adjacent imaging properties are due to the collimation effect of the equifrequency contour and the lower operational normalized frequency than the ideal value. When the contour becomes circular, imaging is non-restricted. At higher normalized frequencies, imaging is not adjacent to the slab surface. Such negative refraction and imaging have been extended to higher bands with the partial bandgap when the two conditions are satisfied. Note that in a practical quasi-2D PhC such as those fabricated on SOI wafers, waveguiding in the third dimension will reduce the effective refractive index and the band structure will differ quantitatively from the true 2D results calculated here. However, although the specific numerical results will be modified, we expect that the same phenomenon can be observed.