Control of dispersion in photonic crystal waveguides using group symmetry theory

Pierre Colman; Sylvain Combrié; Gaëlle Lehoucq; Alfredo De Rossi

doi:10.1364/OE.20.013108

The linear and nonlinear propagation of waves depends crucially on the dispersion. The ability to control the dispersion is therefore a highly desired property of the physical medium supporting the propagation. For instance, the minimization of chromatic distortion in optical fibers [1] is the main concern in optical communications. Moreover, some nonlinear interactions e.g. parametric conversion/generation, involving different frequencies, are completely determined by the dispersion, through the phase matching condition [2]. In optical fibers and waveguides, dispersion control is possible because the waveguide dispersion, which can be controlled through the geometry, can compensate for the material dispersion, which is material-dependent [3]. In an optical waveguide, e.g. the so-called photonic wires, the dispersion can be controlled by changing the height and the width of the high index silicon wire on top of the silica layer [3]. The ability to generate a prescribed dispersion function, also referred as dispersion engineering [4, 5], is a more demanding goal. By offering many degrees of freedom in the design, photonic crystals (PhC) fibers [6] and waveguides [4, 7–10] have proven to enable dispersion engineering.

The particular dispersion of line-defect photonic crystal waveguides is often described as the result of an anti-crossing between index-guided (e.g. guided by total internal reflection) and gap-guided modes (which only exist in PhCs) [11]. However the inherent complexity of photonic crystals, as opposed to simpler optical structures, limits the understanding of the mechanisms underlying the control of the dispersion. As a result, in order to tune a PhC waveguide, the design is optimized regardless of the nature of the underlying mechanism, e.g. the changes in the mode field distribution or in the coupling between modes. This said, dispersion engineering has been demonstrated in PhC waveguides by optimizing the coupling of two waveguides [7,8] or by coupling modes with different polarizations [12].

The main idea of this paper is to define a suitable perturbation of PhC waveguide such that the even mode, which is commonly considered for PhC waveguide devices, anti-crosses with the odd mode. The parity here is defined relative to the plane parallel to the propagation direction and perpendicular to the PhC slab. Those two modes are typically found in line-defect PhC waveguides and are uncoupled because they have different symmetries; however, the odd mode is rather regarded as a nuisance limiting the spectral extent of single mode operation. Here, we show that, owing to even-odd mode coupling, a variety of useful dispersion profiles can be controlled by a single parameter, which is a quite desirable feature. More generally, the idea of coupling different modes in a PhC waveguide, as a mean to to control the dispersion in PhCs has been considered [13–16]; here we show how group symmetry theory can provide the right guidance to the design. We have fabricated a waveguide according to these design rules and found out that, not only the achievement of the desired dispersion profile was straightforward, owing to the very simple design rule, but the experimental evidence also allowed us to conclude that the propagation loss of such a waveguide were very low, compared to our initial design.

To fix the ideas, let us consider first the case of two coupled PhC waveguides [7] within the formalism of coupled mode theory (CMT). The dispersion of the super-mode resulting from the coupling γ(k) of the modes of the individual waveguides, with dispersion ω_wg_{1,2} (k), results from the diagonalisation of:

| \begin{matrix} ω_{w g {1}} (k) & γ (k) \\ γ^{*} (k) & ω_{w g {2}} (k) \end{matrix} |

Introducing the coupling results into a modification of the dispersion of the original modes, which can be controlled within some extent by playing with some parameters, such as the radius of the holes or the separation between the two waveguides. We point out that γ(k) could be modified, in principle, almost independently from ω_wg_{i} i = {1, 2}.

Here, we explore the possibility of coupling the two modes of a single waveguide in order to control the dispersion. Operating with a single waveguide, rather than a set of coupled waveguides, has many practical advantages: the excitation (i.e. the coupling) of the mode is easier and the linear and nonlinear effective areas are smaller, resulting into stronger light-matter interaction, a desirable feature in nanophotonics. Specifically, we consider the odd mode, defined relative to the plane of symmetry x – z, where x is the direction of the propagation and z is perpendicular to the 2D PhC lattice.

Let us consider a line-defect waveguide with dispersion represented in Fig. 1(c), calculated using the 3D plane-wave expansion method implemented in the MPB code [17]. The corresponding parameters, normalized with the lattice period, are: radius r/a = 0.26, slab thickness h/a = 0.41, width of the line defect W/a = 0.95 √3, outward displacement of the first row of holes s/a = 0.14, with radius reduced to r₁/a = 0.23. The refractive index of the material is assumed to be 3.17. Here, the parameters have been chosen in order to lower the edge of the odd mode (at k = 0.5K, K being the reciprocal lattice vector); indeed, here the frequency spacing is only δfa/c = 610⁻³. If a suitable coupling between these two modes, which are now closer in frequency, is introduced, then the anti-crossing will result into a change in the dispersion. As a consequence, controlling the coupling strength would enable, for example, the flattening of the lower frequency branch.

Fig. 1 PhC waveguide without (a) and with (b) perturbation. (c) Corresponding band diagram of the initial (perturbed) structure in black (red) line. The cyan markers indicate the modes for which the field distribution is shown in Figs. 2(a) and 2(b). The thin blue line corresponds to the dispersion obtained for a different perturbation as discussed in Fig. 2(d).

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Fig. 2 Spatial distribution of the H_z-field, calculated at K=0.445, (a) Even mode and Odd mode. (b) even-like and odd-like mode for a dielectric perturbatation as presented in (c). (c) Calculated |γ_ee|, |γ_oo| and |γ_eo| coefficients for T=0.15a. Inset : corresponding Δε and its dominant symmetry (colors refers to opposite signs). (d) Idem as (c) but holes are moved in the same direction so the symmetry of Δε is now B₁ instead of A₂.

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Proceeding in the same way as for the coupled PhC waveguides, we now consider the coupling of even and the odd mode within the formalism of the coupled mode theory. The related eigenvalue problem is formulated by including the substrate mode which is mainly of the index-guided type. This is necessary because that mode might also interact with the other modes. The off-diagonal terms represent the coupling induced by a suitable modification of the PhC design. Namely the purpose here is to get a simple way to control the dispersion by tuning only the interaction between the odd and the even modes, without suffering from possible interactions with the other modes.

| \begin{matrix} ω_{odd} (k) & γ_{o e} (k) & γ_{o s} (k) \\ γ_{o e}^{*} (k) & ω_{even} (k) & γ_{e s} (k) \\ γ_{o s}^{*} (k) & γ_{e s}^{*} (k) & ω_{substrate} (k) \end{matrix} |

Within the CMT formalism (which is a first order perturbation theory), the coupling between γ_eo(k) is proportional to the spatial overlap between the odd and even modes, weighted by the perturbation Δε (r⃗) of the dielectric function ε(r⃗) associated to the PhC structure, i.e.:

γ_{o e} (k) = \iint {\vec{E}}_{odd}^{*} (\vec{r}) \cdot Δ ε (\vec{r}) \cdot {\vec{E}}_{even} (\vec{r}) d \vec{r}

In general, an arbitrary perturbation of the dielectric function ε(r⃗) would also introduce the other diagonal and off-diagonal terms into Eq. (2).

\begin{array}{r} γ_{o o} (k) = \iint {\vec{E}}_{odd}^{*} (\vec{r}) \cdot Δ ε (\vec{r}) \cdot {\vec{E}}_{odd} (\vec{r}) d \vec{r} \\ γ_{e e} (k) = \iint {\vec{E}}_{even}^{*} (\vec{r}) \cdot Δ ε (\vec{r}) \cdot {\vec{E}}_{even} (\vec{r}) d \vec{r} \\ γ_{s s} (k) = \iint {\vec{E}}_{substrate}^{*} (\vec{r}) \cdot Δ ε (\vec{r}) \cdot {\vec{E}}_{substrate} (\vec{r}) d \vec{r} \\ γ_{o s} (k) = \iint {\vec{E}}_{odd}^{*} (\vec{r}) \cdot Δ ε (\vec{r}) \cdot {\vec{E}}_{substrate} (\vec{r}) d \vec{r} \\ γ_{e s} (k) = \iint {\vec{E}}_{even}^{*} (\vec{r}) \cdot Δ ε (\vec{r}) \cdot {\vec{E}}_{substrate} (\vec{r}) d \vec{r} \end{array}

As a consequence, the perturbation would not only introduce even-odd mode coupling, but would also mix these modes with the substrate mode (γ_es, γ_os); as well with any other modes-which were not considered here-; and induce a frequency offset of the single dispersion profile through the diagonal terms. Hence the result of the perturbation is uncertain. If γ_es = 0 and γ_os = 0, the dispersion is then easier to control, at least from a qualitative point of view.

Hereafter we give a procedure to generate a perturbation Δε (r⃗) such that γ_eo (k) is non-zero, but also dominates also substantially over the other terms. Let us first look at the integrand ${\vec{E}}_{odd}^{*} (\vec{r}) \cdot Δ ε (\vec{r}) \cdot {\vec{E}}_{even} (\vec{r})$ . The integral can be non-zero only if this is an even function. Considering that the planar PhC waveguide belongs to the group of symmetry C_2v, that statement is equivalent to requiring the integrand to have at least the element of symmetry A₁ (namely, the fully symmetric function). Conversely, if that condition is not respected, then integrand will have both positive and negative parts which would mutually cancel out.

Proceeding similarly as in [18], the Bloch modes in a PhC with a 2D hexagonal lattice can be labelled according to Table 1. Based on the mode field distribution of the unperturbed structure in Fig. 2(a), we conclude that the even and the odd modes belong to the symmetry subgroups B₁ and B₂ respectively. The symmetry condition on the integrand in γ_oe (k) is then formulated as:

A_{1} = B_{2} \otimes X \otimes B_{1}

where X is the symmetry of Δε (r⃗), still to be determined. It is found that X must have the symmetry A₂. This implies that the dielectric perturbation is anti-symmetric with respect both to the x = 0 and the y = 0 symmetry planes. This also implies that γ_oo (k) = γ_ee (k) = γ_ss (k) = γ_se (k) = 0, i.e., the only non-zero elements are γ_eo (k) and γ_so (k). Indeed, since the even and the substrate modes have the same symmetry, they are not coupled together by a such perturbation. It can be observed that the coupling γ_so (k) between the odd and the substrate modes is minimized if the initial frequency spacing between the two modes is large and if the changes of the dielectric function Δε (r⃗) is localized where the field of the even and odd modes is much stronger than that of the substrate mode. This is in general the case in the region close to the line defect (see Fig. 2(a)).

Table 1. C2v point group symmetry table

View Table

Based on that analysis, a modified PhC geometry is generated by translating the first row of holes by T/a = 0.1 along x in opposite directions, as is shown in Fig. 1(b). It is straightforward to verify that the change introduced respects the A₂ symmetry (Fig. 2(d)). As a consequence, the original symmetry of the waveguide is now broken. The corresponding dispersion (Fig. 1(c)) now exhibits a kink near K = 0.445 and the lower (upper) band is translated to a lower (higher) frequency (spacing has now increased to δfa/c = 1.510⁻²). The origin of the kink (i.e. zero group velocity dispersion points) is related to the change from a crossing behaviour (even and odd uncoupled) to an anti-crossing one, which implies that the modes repel each other.

We point out that this analysis applies to other design, for instance that based on the B₁ perturbation, as used in ref. [10]. There, the even mode and the substrate modes of a simple line-defect waveguide are very close in frequency. Thereby the introduction of the B₁ readily produces a strong change in the dispersion.

Because the waveguide symmetry is broken, the strict classification of the modes as odd or even is not applicable. We therefore label the lower (upper) frequency mode as even-like (odd-like). The spatial distribution of the modes corresponding to these new bands is shown in Fig. 2(b) and clearly reveals the lack of a well defined parity (odd or even). The modification of the spatial distribution of the modes, i.e. their hybridisation through mode coupling, can be connected to the dispersion. In order to elucidate this point, we calculate the diagonal and off-diagonal terms introduced in Eq. (2), which are shown in Fig. 2(c). A closer look in the field distribution reveals that the perturbation still keeps some features with the A₁ symmetry and, as a result, small diagonal contributions γ_ee and γ_oo appear into the operator in Eq. (2). We repeated the same calculation with a perturbation with equal magnitude but a different symmetry (namely B₁), as shown in Fig. 2(d). Interestingly, while γ_ee and γ_oo are basically unchanged, the coupling between odd and even modes has disappeared. The dispersion (blue line on Fig. 1(c) has no zero GVD points or equivalently, no anti-crossing is observed. This comparison confirms the minor role played by the diagonal contributions which are only responsible for a moderate steepening of the dispersion around K = 0.37, but do not create any inflexion point. It is now apparent that the features introduced in the dispersion are mainly related to the contribution γ_eo above, which depends on T/a, a parameter defining the degree of asymmetry introduced in the structure. Therefore the change of the dispersion is controlled by an unique parameter.

Let us consider now the dispersion, namely the group index as a function of the wavelength, as the parameter T is changed (Fig. 3(c)). Indeed, while a flat-band over more then 10nm is obtained with T=0.05 (delay-bandwidth product of 0.15), the other values generate a dispersion where the group dispersion changes sign twice and a local maximum of the group index increasing sharply as T is further increased.

Fig. 3 (a) Group index dependence as a function of the wavelength calculated for T= 0.05a, 0.1a, 0.2a. (b) Corresponding measured dispersion. The lattice period is a = 465nm.

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We have fabricated a sample based on GaInP membrane [19] with the above specified design and we measured the dispersion using an interferometric technique [20]. The use of mode adapters [21] prevents any coupling issue, although the mode is slightly asymmetric. The result is shown is Fig. 3(b). The agreement with the predicted dispersion is very good. Residual differences (e.g. for T/a = 0.05) can be attributed to the fabrication tolerances (e.g. the radius of the holes). The important point is that the transition from one of these dispersion profiles to another follows continuously the change in the parameter T; which can be further adjusted in order to reach the desired dispersion profile.

An important practical aspect in dispersion-engineered waveguides is their robustness against disorder, particularly their ability in the regime of low group velocity. For instance, a group index of 40 is reached for T/a = 0.2 (Figs. 3(a)–3(b)). Thus a group delay of about 200 ps was obtained on a 1.5 mm long waveguide. This is quite a respectable target, comparable to the state of the art for delay lines based on nanophotonics [22–25]. Furthermore, in this slow light regime, the related increase of propagation losses is limited to about 5dB. For example, this allowed demonstrating efficient Four Wave Mixing in PhCs [26]. The excitation of a mode lacking of a well-defined symmetry might potentially be an issue, based on of mode-matching considerations (see Fig. 2(b) for instance). However, we experimentally found that coupling is very good except when the second zero of the group velocity dispersion is reached, namely when the wavelength is larger than 1545 nm (Figs. 3(c)–3(d)), which leaves enough margin for the use of this device. It must also be pointed out that in the calculation of the dispersion, the symmetry cannot be used to further decrease the computation time.

In conclusion, we have revisited the concept of dispersion tailoring based on mode coupling and we have implemented it in a single-mode line-defect PhC waveguide. The group theory analysis was used to determine a suitable perturbation of the PhC structure such that only the even and the odd modes are coupled. The change in the dispersion, and, particularly, the onset of two zero group dispersion points, is explained through the anti-crossing related to the even-odd mode coupling. A variety of dispersion profiles, including flat-band, is controlled through a single geometrical parameter, representing the degree of asymmetry which is introduced into the PhC structure. A variety of waveguides has been fabricated according to this design with relatively low propagation loss in the slow light regime, in particular a group velocity of c/40 has been achieved on a long waveguide, resulting in a measured delay of 200ps.

The same concept can also be extended to the coupling between two arbitrary modes (not only odd and even ones). This result is relevant to a broad range of applications, particularly parametric effects (e.g. Four-Wave Mixing) and it enables the optimization of the phase matching. We believe that it could be applied to other systems as well (e.g. PCF) and enable an even more accurate control of the dispersion.

Acknowledgments

This work has been supported by the 7th framework of the European commission through the GOSPEL project (www.gospel-project.eu) and COPERNICUS (www.copernicusproject.eu) projects. We thank M. Santagiustina, G. Eisenstein and S. Trillo for enlightening discussions.

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	E	C₂ (z)	σ_v (xz)	σ_v (yz)

A₁	1	1	1	1
A₂	1	1	−1	−1
B₁	1	−1	1	−1
B₂	1	−1	−1	1

A₂ ⊗ A₂	1	1	1	1	=A₁
B₁ ⊗ B₂	1	1	−1	−1	=A₂

Control of dispersion in photonic crystal waveguides using group symmetry theory

Abstract

Acknowledgments

References and links

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Figures (3)

Tables (1)

Equations (5)

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