To investigate the frequency dispersion of the guided waves it is necessary to consider the frequency dependence of bothεi and µi in Eq. (11.14). Although
the specific form of the refractive index of a photonic crystal depends of the band structure [106], in this analysis the forms for the composite negative in- dex material, as given in Eqs. (11.1) and (11.2), will be used. Based on earlier analysis the defining parameters are chosen to be:ωp,2/2π= 10 GHz,ω0,2/2π =
4 GHz and F = 0.56[70, 80]. In this case, the region of simultaneously neg- ative ε2 and µ2 ranges from 4 GHz to 6 GHz. The values of ωp,1/2π = 2 GHz and ω0,1/2π = 1 GHz were then chosen so that ε1 and µ1 are always positive and ε2µ2 > ε1µ1 in this range. Examples of dispersion curves for the modes of Fig. 11.5 are plotted in the top curves of Fig. 11.6. These curves correspond to the solutions: (a) (a,A) with L = 0.1 cm, (b) (b,B) with L = 1 cm(i.e., the
0.8 1.0 1.2 −3.6 −3.5 −3.4 −3.3 5.2 5.3 5.4 0.018 0.022 0.026 Frequency (THz) 0 0.5 1.0 1.5 −2 0 2 4 4.8 4.9 5.0 −10 0 10 Frequency (THz) 2 2.5 3.0 3.5 β (1/cm) 1.5 2.0 2.5 v g (cm/ns) 4.8 5.0 5.2 −0.3 −0.2 −0.1 0 Frequency (THz) β 2 (ns 2 /cm) νc (a) (b) (c)
Figure 11.6: Top: propagation constantsβ, middle: group velocitiesvg and bottom: group velocity dispersion parametersβ2. These solutions correspond to the modes: (a)
(a,A), (b)(b,B), and (c)(c,Γ)and(d,Γ)of Fig. 11.5 where in (c) the solid and dashed lines correspond to the strongly(c,Γ)and weakly(d,Γ)localised modes, respectively.
Chapter 11 1D Negative Refractive Index Materials
twoH2y,1 solutions), and (c) (c,Γ) and (d,Γ) with L= 2 cm(i.e., the degenerate
H3y,1solutions) where the solid line corresponds to the strongly localised mode and the dashed line is the weakly localised mode. From this it is seen that of the two solutions offered by each designated mode type, one has a dispersion curve with a positive slope and the other a negative slope. Significantly, as
vg = 1/β1 = dω/dβ [Eq. (3.17)], this implies that the sign of the group veloc- ities will also be different and thus each mode can support both forward and backwards propagating waves [113]. This is confirmed in the middle curves of Fig. 11.6 wherevgis plotted explicitly for each of the modes. For the case of the H2y,1modes it is the non-surface wave which has a negativevg, and for theH3y,1 modes it is the tightly confined mode. In addition, it can also be seen that for the case of the degenerate modes, as the frequency is increased the two solu- tions forβconverge until they reach a cutoff frequencyνcassociated with their intersection.3 As a result, this convergence of the two solutions means that as
the frequency approachesνc,vgapproaches zero so that the propagating mode
will be slowed considerably. Thus this waveguide offers a convenient method for generating “fast” light (vg < 0), “slow” light (vg c) and perhaps to even
trap light (vg = 0). The possibility to slow or trap light has many potential
applications such as optical data storage, optical memories and quantum com- puting. Furthermore, as the light-matter interaction is enhanced for low vg,
slow light can be used to observe nonlinear processes such as harmonic gener- ation and four-wave mixing in even weakly nonlinear materials [114].
The group velocity dispersion (GVD) of the guided modes as calculated via:
β2 = d2β/dω2 [Eq. (3.16)] is then plotted for each of the modes in the bottom curves of Fig. 11.6. In all cases the GVD parameter is quite large, particu- larly for the degenerate modes where the frequency approachesνc(the region of low vg), and can be either anomalous [(a,A) and (c,Γ)] or normal [(b,B)
and (d,Γ)]. Such large dispersion is typical behavior of the GVD at the band edges of photonic crystals [115] and for these particular modes, the dispersion can be around 9 orders of magnitude larger than that of conventional mate- rials such as silica fibres (20 ps2km−1). This makes these waveguides idea for dispersion management and particularly for use in integrated circuits where short device lengths are favoured. In addition, by exploiting the reduced non-
3Although the strongly localised mode exists for frequencies below those plotted here, the
Chapter 11 1D Negative Refractive Index Materials
linear threshold/large GVD combination it should be possible to investigate nonlinear effects such as optical soliton formation (see Section 3.8.1).
The energy flux of the guided (degenerate) modes in Fig. 11.6(c), which is char- acterised by thezcomponent of the Poynting vector [Eq. (10.21)], has also been calculated. Since for backwards waves the Poynting vector and the wave- vector point in opposite directions, it is expected that the energy flux of the modes will also have opposite signs [73]. The total power flux through the core and cladding regions of the waveguide are calculated as,
Pcore=
core
Szdxdy, Pclad =
clad
Szdxdy. (11.15)
For both modes the power flux inside the core is opposite to that in the cladding (see middle row of Fig. 11.5). However, on calculating the total normalised en- ergy flux defined as:
P = Pcore+Pclad
|Pcore|+|Pclad|
, (11.16)
Fig. 11.7 shows that total energy flows in a positive direction for the weakly localised mode and a negative direction for the strongly localised mode, in agreement with the signs of vg. It is worth noting that by definition, |P| < 1
and P → 1 as the mode becomes poorly confined andP → −1 as the mode becomes tightly confined. The significant feature of this result is that as the solutions converge atνc, the energy fluxes inside and outside the guide exactly cancel so that the total energy flux vanishes. Importantly, in their analysis for a negative index planar waveguide, Shadrivov et al. showed that atP = 0the energy flowed in a double-vortex structure so that most of the energy remained localised inside the wave packet [80]. Thus as the energy flux goes to zero, the guided modes do not disintegrate and an analogous result for the modes of a channel waveguide can be expected.
480 485 490 495 500 −1 0 1 Frequency (THz) Energy Flux
Chapter 11 1D Negative Refractive Index Materials
In Figs. 11.6 and 11.7 the dispersion and energy flux of typical modes have been examined as functions of the frequency. Alternatively, it is also useful to consider the dependence of the mode properties on the waveguide width
L. Fig. 11.8 shows (a) the propagation constant and (b) the normalised energy flux for the H3y,1 modes at a fixed frequency, ω/2π = 5 GHz, where again the solid line corresponds to the strongly localised mode(c,Γ)and the dashed line is the weakly localised mode (d,Γ). As expected, these have similar forms to the previous curves for varying frequency except that this time the two solu- tions converge asLis decreased until they reach a cutoff lengthLc. Thus these results suggest that the propagating mode can be slowed simply by adiabati- cally decreasing the waveguide width. Furthermore, by decreasing the width to the critical lengthLcit should be possible to stop the light completely. Thus it is expected that a simple waveguide structure such as that shown in the inset of Fig. 11.8 should act as an optical trap, where the frequency of light that can be trapped is determined by the range of the waveguide width.
4 6 8 10 12 β (1/ µ m) 0.20 0.22 0.24 0.26 −1 0 1 L (µm) Energy Flux L c Lc L
Figure 11.8: (a) Propagation constant and (b) normalised energy flux of theH3y,1 solu-
tions from Fig. 11.6(c) as functions of the waveguide widthL. The solid and dashed lines correspond to the strongly(c,Γ)and weakly(d,Γ)localised modes, respectively. Inset: design for an optical trap.
Chapter 11 1D Negative Refractive Index Materials