# Littrow resonators and the critical coupling concept

Henri Benisty, Omer Khayam

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Henri Benisty, Omer Khayam. Littrow resonators and the critical coupling concept. 12th International Conference on Transparent Optical Networks (ICTON), Jun 2010, Munich, Germany. 10.1109/ICTON.2010.5549060. hal-00573190

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# Littrow Resonators and the Critical Coupling Concept

Henri Benisty, Member, IEEE, Omer Khayam

Laboratoire Charles Fabry de l'Institut d'Optique, CNRS, Univ. Paris-Sud

Campus Polytechnique, RD 128, 91127 Palaiseau Cedex, France

Tel: (33) 1 6543 32 86, Fax: (33) 1 6543 3318, e-mail: henri.benisty@institutoptique.fr

## ABSTRACT

We describe a class of open resonators formed by highly corrugated broad waveguides. The contra-directional coupling induced by the corrugation can be seen as a Littrow diffraction event on a reflection grating. The presence of both boundaries still discretizes the modes. The modes most affected by this Littrow diffractions are the higher order modes of the broad waveguide. The new modes that emerge from their coupling may have collectively a low group velocity in a sizable part of the Brillouin zone. For this, a critical coupling value has to be found, minimizing the minibands and maximizing the “stripes of minigaps” between them.

We give various implementations of this geometry, including an analogy with graphene nanoribbons. Furthermore, the analysis of the emerging field pattern of a toy-model shows phenomena previously identified in light localization studies, with large frequency intervals of “smooth” behaviour at “stripes of minigaps”, separated by narrow frequency intervals of quickly evolving patterns.

**Keywords:** periodic waveguide, multimode coupling, slow light, open cavity laser, critical coupling.

## 1. INTRODUCTION

Waveguides are usually considered under their single-mode realization, in order to safely transport the largest possible information bandwidth. Multimode systems, however, are a marvellous playground. One noticeable success is the so-called Multi-Mode interferometer. As for periodic multimode systems, photonic crystal waveguides investigations have revealed a number of possibilities [1,2], notably those based on the mini-stopband : based on this fundamental to high-order mode coupling phenomenon, various devices such as demultiplexers could be made [3-6].

This topic has a long history, although earlier, its potential was not recognized: for example, [7] reported about the mini-stopband in acoustics in 1986, and in [8] simulations on acoustic waveguide of a phononic crystal had similar feature (coarsely resolved).

Here, we discuss how such corrugated waveguides can become broad resonators with almost stationary modes. The situation of Littrow diffraction of a collection of modes onto the corrugated sides is found to be the simplest basis for our description. We believe that this essential simplicity is inspiring. Formally, we admit that there is no specific difference between a waveguide with highly corrugated edge and a fully periodic waveguide such as a guide made of a photonic crystal PhC1 in its transmission band embedded into a crystal PhC2 in its bandgap. However, the open aspect of the broad waveguide provides a large functional potential and also the possibility to easily couple the “inside” of the system with the outside, without the need of a semi-transparent mirror or other specific cavity boundary used in tunnelling mode.

In the Sec. 2, we start by reminding the essential physical ingredients of a broad waveguide and explaining why photonic crystal waveguides can be simply described from coupled modes of a “slab” waveguide. In Sec. 3, we introduce the critical coupling. We show how it gives rise to localization in open “Littrow resonators”. Such resonators were actually the initial incentive of this work, due to the observation of Littrow lasing [9-11]. In Sec. 4 we visit the topic of light localization with reference to field patterns (vortices) that emerge from disordered waveguide structures as studied in [12]. We show that the alternance of “steady” patterns and “fast changing patterns” upon scanning the frequency of light launched in the waveguide is also found in our resonator/waveguide systems. This hints at the fact that localization is not specific of disorder, and therefore bridges can be established between both aspects (for example as in [13]).

## 2. FROM A BROAD CORRUGATED WAVEGUIDE TO A LITTROW RESONATOR

The basic idea is to start from the modes {1...m...} of a simple broad slab waveguide of index n and width W, whose electric field is $E_m^\pm(x, z) = \exp(\pm jkz)\sin(m\pi x / W)$ with k the wavevector, x the transverse axis (Fig. 1a). Their angular frequency is $\omega_m = (c/n)[k^2 + (m\pi/W)^2]^{1/2}$. We write the role of corrugation of period a as a coupling between only contridirectional modes (forward $\leftrightarrow$backward), with traditional Coupled Mode Theory (CMT) notations [coupling constant $\kappa_{mm'}$, detuning $\delta_m$ vs. Bragg condition, envelope field $F_m^\pm(z)$ factorizing $\exp(\pm jKz)\sin(m\pi x / W)$ with $K = \pi / a$]:

$$ \frac{\partial F_m^\pm}{\partial z} = j\delta_m^\pm F_m^\pm + \sum_{m'=1...N} \kappa_{mm'} F_{m'}^\mp \quad (1) $$
The success of this strategy can be judged from the similarity with the dispersion of an actual photonic crystal waveguide (Fig. 1b).

<figure><img src="image_2.png" /><figcaption>Figure 1: (a) A broad photonic crystal waveguide and its slab equivalent; (b) CMT bandstructure calculation based on uncoupled modes and exact calculation (plane-wave expansion) for a “W15” PhC waveguide (15 missing rows in a triangular lattice).</figcaption></figure>

The initial modes (dashed lines) have the gentle hyperbolic shape of $\omega_m(k)$, but upon coupling with their folded counterparts, they form “minigap stripes” and “necklace dispersion branches”. The striking feature is that a single parameter was added here to the slab’s dispersion, with only a parity selection rule ($\kappa_{mm'} = \kappa_{smm'}$ with $s_{mm'}$ is 1 if $m, m'$ have same parity, zero otherwise). The single $\kappa$ value is thus found to be sufficient to describe all the intricate net of anticrossing, the smallest as well as the zone-center large ones. One of the reasons is the special fate of interaction of a given mode with several bands: anticrossing may not add up in this case, on the contrary, they may be reduced. Our particular interest, however, was not so much the region near the fundamental modes (which serves for the mini-stopband devices) but the zone-center flat region, which turns out to be flat. The necklace branches adopt “re-entrant” hyperbolic shapes, nearly the same as those of the original modes except for the centering at $k = K$.

It can be intuited that $\kappa$ is related to a plane-wave diffraction efficiency, by decomposing modes $m$ and $m'$ into their constituents. Hence it is natural to think of modulating the value of $\kappa$ as one would for diffraction efficiency. There is no hard limit, however, to the value of $\kappa$, whereas a diffraction efficiency is less than unity. The next section thus introduces the “critical coupling” as a threshold point when one increases $\kappa$ so as to force the local bandgaps at $k \sim K = \pi/a$ to interact strongly between them.

### 3. CRITICAL COUPLING

To understand the basic physics of “critical coupling” we look at the interaction of a reduced subset of forward and backward modes in Fig. 2, with a constant “FSR” (Free spectral range, separation).

<figure><img src="image_3.png" /><figcaption>Figure 2: Modification of the dispersion of coupled contradirectional modes upon increasing the coupling constant from 0 (a) to a value below critical coupling (b); at critical coupling $\kappa_c = \text{FSR}/\pi$ (c) and above it (d); (e) application of CMT as in Fig. 1b, but crossing the critical coupling condition.</figcaption></figure>

The main finding reported in [14] is that $\kappa_c = \text{FSR}/\pi$: Coupling should be a prescribed fraction of mode separation to produce flat bands. This formal “cartoon” of critical coupling is not a proof that the phenomenon
can occur in a real world system. However, we showed on the basis of PWE calculations of waveguide with
strong triangular corrugations that such a critical coupling can occur. The typical picture that emerges is well
accounted by looking at the application of CMT, as in Fig. 1b, but at a larger coupling value (Fig. 2e) Here as the
FSR increases with frequency, the critical coupling point is met only around a particular spectral domain.
Concerning PhC waveguides, we gave in [15] clues that their “native” form lies above critical coupling.

As for the emergence of two separate bands at the end of the “cartoon” (Fig. 2d), they have been related to “bright eigenstates” of the system, the more subtly flatter bands being the many “dark eigenstates” (dark modes) of the system [16]. The latter give rise to a large imaginary part between them, which endows them with different properties with losses: if small losses are added to all modes initially (representing a radiative coupling for instance), the losses of the bright modes will be much larger than those of the dark modes, following the trends in bandgap size.

<figure><img src="image_4.png" /><figcaption>Figure 3: (a) Ray-tracing design of a 45° Littrow resonator ; (b) layout for basic properties assessment, here angular tolerance ; (c) FDTD angular tolerance result ; (d) field of a prism shaped resonator ; (e) same for a purely planar system with grating couplers.</figcaption></figure>

We are now ready to consider Littrow resonators. In Fig. 3, we present a typical Littrow resonator corrugating
a single side for simplicity: the resonator material can be silicon (n = 3.45) and it can rest on silica (n = 1.5) with
total-internal reflection widely granted. In Fig. 3a, we picture a ray-tracing design, for a θ = 45° typical angle,

<figure><img src="image_5.png" /><figcaption>Figure 4: Localization study on a 18+18 modes model ; abscissa : frequency (FSR of initial modes = 10, FSR of coupled modes minibands = 5) ; (a) field in real space (log scale of intensity)in a fraction of the waveguide ; (b) zoom on the shaky region where speckle evolves rapidly ; (c) band structure for this critical coupling case.</figcaption></figure>
which amounts to tile triangles. The resonator photonic behaviour was probed by FDTD (finite difference time domain) using the PhotonDesign software. We varied the incident angle around at the critical coupling condition and “measured” transmission spectra of a basic system (Figs. 3b, 3c). More realistic realizations are shown with their field in Figs. 3d, 3e: a prism with AR coating and a planar system where the 45° coupling is replaced by a grating of period 2a. Quality factors up to $Q = 4000$ were assessed numerically in these systems [17]

## 4. LOCALIZATION

The field pattern at the output of a *coherently illuminated* multimode system generally exhibits some speckle. The statistics of the speckle pattern as a function of frequency was recently unfolded with a view to assess the features of various localization regimes [12]. In the strong localization regime using a tube filled of disordered spheres having a microwave Mie-type resonance, it was found that the pattern is slowly varying between two transmission resonance and rapidly varying at each resonance (see the movie associated with [12]). The specific statistics of these spatial patterns had universal features that we shall not address here.

We devised a similar computer experiment in a {18+18} modes model, explained in Fig.4 caption (details will be given later): The field patterns vs. frequency show that our “stripes of minigaps” produce similar effects.

## 5. CONCLUSION

In conclusion, there is a rich physics of broad periodic multimode waveguides, because the modal interactions confer them resonator aspects. They are therefore useful for lasers, localization or formation of “dark states”.

## REFERENCES

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