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A side coupled adjacent resonators CROW (SC-CROW) structure is presented. The dispersion relation, group velocity and GVD are found numerically and analytically using a transfer matrix method. The structure shows fast and slow transmission of light. Zero group velocity is present at the ends of the Brillouin zone, but also in other parts of it. Infinite and finite SC-CROW structures show band gaps created by Bragg and ring resonances. Pulse propagation at slow group velocities is highly dispersive, but easier to manage at higher group velocities. The differences between SC-CROW, SCISSOR and CROW waveguides are studied in detail.
©2009 Optical Society of America
1. Introduction
During the past decade, microring resonators based slow-light structures received substantial attention due to their interesting properties and potential applications [1–4]. Much focus was devoted to two different types of slow-light structures: the CROW (Coupled Ring Optical Waveguide) [5–7], and the SCISSOR (Side Coupled Integrated Spaced Sequence of Resonators) [8, 9].
Both structures possess degeneracy of the CW and CCW propagating waves (in each cavity). This degeneracy is expressed in their dispersion relations, which allow a single group velocity (which could be either positive or negative) per frequency. This inherent symmetry limits the attainable flexibility, in particular, the ability to dynamically control the propagation velocity of a pulse.
In this paper, we introduce direct coupling between the cavities of a SCISSOR-like structure to create a new type of slow-light structure - the SC-CROW (Side Coupled CROW). The inherent degeneracy is lifted by the inter-cavity coupling, resulting in bending and splitting of the dispersion bands. At each frequency, there are four possible group velocities, which facilitate manipulation of light in a new manner: CROWs, SCISSOR, and other microring based photonic structures exhibit zero group velocities at the middle and edges of the Brillouin zone. In these regions the GVD (group velocity dispersion) is very large (in both structures), a property which prevents the exploitation of the low group velocity. In the SC-CROW structure on the other hand, zero group velocity regions are formed in the middle of the band gaps (see e.g. Fig. (2)). This is an attractive feature which enhances the flexibility of the SC-CROW structure for slow-light applications, such as delay-lines, low threshold lasers, etc. although strong dispersion issues must still be resolved.
The existence of an essentially tunable slow group velocity region inside the Brillouin zone is highly attractive for gain enhancement in lasers [10–14]. Slow group velocity can enhance light-matter interaction or, equivalently the gain per unit length. In Bragg or CROW based lasers [10–14], lasing is attained only at the band edges while in the SC-CROW structure the position of the zero group velocity inside the Brillouin zone can be modified.
2. Structure basics and transfer matrix analysis
A photograph of a SC-CROW structure is shown in Fig. 1. The device was fabricated by direct E-Beam lithography of SU-8 on Si/SiO2 wafer. For this specific structure, the microrings radius is 50µm, the waveguide width and thickness are 2µm and 1.6µm respectively and the gap between adjacent microrings and between the microrings and the I/O waveguides is 0.4µm. The resonance wavelengths of the rings are λ R=2πRn/m, where m is an integer, c is the velocity of light in vacuum; R is the radius and n is the effective index. The Bragg wavelengths are λL=Ln/m, where L is the length of the waveguide connecting adjacent rings and κ1, κ2, are the ring to waveguide coupling coefficients (here κ1=κ2), κ3 is the inter-ring coupling,
A single unit cell of the SC-CROW is marked in Fig. 1. The field in the cell can be represented by eight amplitude x 1..x 8 as defined in Fig. 1. The electric field amplitudes at adjacent unit cells can be connected by a transfer matrix m:
Invoking the Bloch Theorem, (1) can be used to derive the spectral properties of the structure for any set of the parameters n, ω, R, L, κi. In addition, it is also possible to find the electric field profile. The derivation of m (which is an 8×8 matrix) is straightforward but laborious and is given in Appendix A for clarity.
3. Dispersion relations, Group velocity, GVD
According to Bloch theorem the field at each unit cell is connected to the field at its neighbors by a phase shift x̲n+1=exp(ikeffL)x̲n. Substitution of the Bloch condition into (1) yields an eigenvalue problem which eiganvalues and eigenvectors are the Bloch wavenumber and field solutions:
where I is an 8×8 unit matrix. For ideal, lossless, coupling section (κ 2 i+t 2 i=1) the following dispersion relation can be derived:
where:
c 0(ω)={-κ 2 3(1+2t 2 1)+t 2 1[4 cos(2φ)+t 2 1cos(2φ-2θ)-2t 2 3cos(2θ)]+cos(2φ+2θ)}
c 1(ω)=-4t 1{[(-1+2κ 2 3)t 2 1-t 2 3+2t 2 1cos(2φ)]cos θ+κ 2 1cos(2φ+θ)}
c 2(ω)=2t 2 1[2κ 2 3-t 2 3+cos(2φ)+κ 2 3cos(2θ)]
c 3(ω)=-4κ 2 3 t 1cosθ
c 4(ω)=κ 2 3
φ=πRnω/c and θ=Lnω/c are the phases accumulated by propagation along half of a ring and along the connecting waveguides respectively. The group velocity is given by Vg=d ω/dkeff:
The group velocity dispersion, is then calculated according to GVD=d 2 k eff/dω 2. Figure 2 depicts typical dispersion, group velocity and GVD of an SC-CROW, for weak and strong coupling coefficients. The specific parameters are given in the figure caption.
For small coupling levels (Fig. 2A) the dispersion is similar to that of a CROW, but the degeneracy is removed showing the splitting of the bands. For large coupling (Fig. 2B) the dispersion resembles that of a SCISSOR structure, although band splitting occurs close to the ring resonances (l=1.523µm, l=1.571µm). For each frequency there are eight different eigenvectors representing eight different Bloch modes (four modes in each propagation direction) of the structure. If all eight modes are decaying, a bandgap is formed.
The field distribution in the infinite structure is determined by the eigenvectors of the transfer matrix m. Depending on the structure parameters and k eff, the energy may be localized in the microrings or in the waveguides. Figure 3 shows several examples of the Bloch waves in a SC-CROW structure. The frequencies of the Bloch modes shown in Figs. 3(a) and 3(b) are far from the resonance frequency of the individual microrings and, therefore, their power is concentrated primarily in the side wavguides. The frequencies of the Bloch modes shown in Figs. 3(c) and 3(d) are closer to the resonance and, hence, more power is focused in the microrings.
Fig. 2. Dispersion relations, group velocity and GVD for R=2.5µm, L=0.7πR, n=3.1. A) Weak coupling κ 1=κ 2=κ 3=0.2, B) Strong coupling κ 1=κ 2=κ 3=0.8.
4. Comparison to SCISSOR and CROW
Evidently, the SC-CROW unit cell is similar to that of the SCISSOR where the difference is the proximity of the micro-rings which yields the inter-cavity coupling. In the case κ 3=0 (SCISSOR), the coefficients c 3, c 4 of the dispersion relation (3, 4) vanish and the dispersion is reduced to the form c 0+2c 1·cos(k eff L)+2c 2·cos(2k eff L)=0, which is the SCISSOR dispersion relation [4]. This equation has four roots, of which two are degenerate. On the other hand, setting the waveguide-microring coupling coefficients to zero (κ 1=κ 2=0), a structure consisting of uncoupled CROW and two parallel waveguides is formed. In this case none of the coefficients c 0..c 4 vanish, although four of the roots are degenerate. It is therefore possible to view the realization of a SC-CROW structure either as CROW perturbed by waveguide to ring coupling, or as a SCISSOR perturbed by the inter-ring coupling. The later approach is advantageous for understanding the nature of the properties of the structure. Starting with a SCISSOR and gradually increasing κ3, we observe symmetrical splitting of the SCISSOR bands (Fig 2). The splitting is substantial at frequencies in the vicinity of the microring resonances, but almost non existent further away. Further increasing κ 3 gradually increases the splitting, creating two separate bands. The separation of the bands is therefore controlled by κ 3 while the profile (frequency dependence) of their mean value is determined by κ 1. The combination of these effects is the mechanism which removes the degeneracy of the CROW/SCISSOR, and generates the unique properties of the SC-CROW.
5. Finite structure analysis
The transfer matrix method can also be used for exploring the properties of a finite structure. Figure 4 depicts a finite structure SC-CROW incorporating four unit cells (as defined in appendix A) and two I/O sections (one at each side). The relation between the incoming and outgoing field vectors can be expressed by the following product of transfer matrices:
Note, that the input and output matrices, m out and m in, are different than the unit cell matrix. In order to realize the two I/O channel structure, different parameters for the first and last cells are used: setting κ 1=κ 2=κ 3=0 in the first and last cells, decouples the leftmost and rightmost half-rings (see the unit cell definition in appendix A) in these cells and a two channel SC-CROW is formed.
Assuming light is injected into channel x 1 on the left hand port (see Fig. 4), it is reflected into the channels on the left hand side of the structure, and transmitted to the channels on the right side. The dotted half-circles indicate the parts of the unit cells which were eliminated in order to realize the I/O sections. No light is injected into the channels on the right hand side, thus the field vectors and the left hand and right hand sides are:
The variables ri, bi which represent the amplitudes of the field are found (5) for each frequency to yield the spectral response of the finite structure (in the case of a two channel SC-CROW, r2=r3=b2=b2=0, and the finite structure can be represented by an equivalent 4X4 matrix).
In order to calculate the propagation of light in the finite structure, the initial mode must be represented as a linear combination of the eigenmodes .
Each of the eigenmodes has a different group velocity, which determines its propagation through the finite structure. The existence of several group velocities induces broadening and splitting of the injected signal, even if it is narrow banded, causing the complex propagation characteristics of the injected pulse. Figure 5 depicts the transmitted and reflected spectrum of a two channel finite SC-CROW. The ripples in the amplitude response stem from the resonances of the finite structure, where large variation of the phase is present. Another factor which might cause ripples is the mid-band zero group velocity regions. These occur for the structure shown in Fig. 5 at λ=1.512µm, λ=1.528µm and at λ=1.567µm. The wavelength λ=1.512µm is marked in Fig. 5, where the transmission and reflection functions show significant changes.
6. Pulse propagation
The results of the finite structure analysis yield the necessary information for understanding the propagation of a pulse through the finite SC-CROW. Understanding the dynamics of pulse propagation is important for non linear optics applications (the intensity of the light inside the resonator can be substantially larger than that of the injected signal), and for optical communication systems, where the information is being transmitted in pulses.
Fig. 5. Transmission (A,B) and reflection (C,D) of the SC-CROW shown in Fig. 4. Light is injected in port 1. R=2.5µm, L=0.7πR, κ 1=κ 2=0.8, κ 3=0.2. A, B: Transmission of light in ports 1 and 4. C,D: Reflection in ports 1 and 4.
In order to model the propagation of the injected pulse, it is separated into its harmonic components. The output spectrum is obtained by multiplying the injected spectrum by the SC-CROW frequency response (as in Fig. 5). Moreover, Since the transfer matrix describes the propagation of light along the whole structure, the evolution of the field in each of the unit cells can be found by taking the ingoing field vector (after finding the reflection coefficients), and multiplying it by a single transfer matrix for the propagation through each unit cell, yielding the pulse spectrum at all points of the structure. The time domain dynamics is then calculated by inverse Fourier transforming the spectrum at each point in the structure. The propagation of a pulse, injected into the upper left port, along the SC-CROW is shown in Fig. 6. The linked mpeg file (media 1) shows the temporal evolution of the field in the structure. The central wavelength of the pulse is 1.556µm which is in the linear dispersion regime of the dispersion curve in Fig. 2B, slightly left to the zero group velocity marked in the figure. As can be seen in the figure the pulse maintains its shape as it propagates through the structure although some broadening and the evolution of a long “tail” are also observed. The distortion is formed by the higher order dispersion of the finite structure at the wavelength of the pulse. Note that the pulse is extremely short (1.5ps, corresponding to bandwidth of ~300GHz or ~2.5nm) and, thus, the impact of the dispersion of the structure is substantial.
Fig. 6. Pulse propagation in a 300 unit-cells SC-CROW with high coupling (κi=0.8). The pulse width is 1.5 ps λ=1.556µm (Media 1 - pulse propagation, same parameters).
7. Conclusions
Micro-ring resonator based CROW and SCISSOR structures exhibit a doubly degenerate dispersion relations stemming from their symmetry for CW and CCW rotating waves. In the SC-CROW structure this degeneracy is removed by the existence of both inter-cavity and cavity-waveguide coupling, generating four different channels of propagation and reflection. As a result, four different group velocities may be present at a single frequency, thus allowing greater flexibility in manipulating the light properties. For small coupling coefficients, a SCISSOR-like band diagram is exhibited, but at stronger coupling the diagram is substantially modified. Slow light regions are formed at the edges of the Brillouin zone, as in CROWs and SCISSOR, but also inside it. This unique feature is advantageous because it enables precise engineering and tuning of the dispersion relation (and correspondingly the group velocity) to achieve linear propagation at any desired wavelength and bandwidth [15, 16].
Linear pulse propagation in the structure is shown at a relatively linear part of the dispersion relation with group velocity of Vg=0.13 c. Although some distortion is visible, most of the power is concentrated in the main (leading) lobe and can be used for data transmission. The SC-CROW facilitates a continuous and smooth transition between an effective SCISSOR structure (no inter-resonator coupling) and an effective CROW structure (no side coupling), thus allowing for much flexibility and engineering of the dispersion relation and light propagation properties.
Appendix A - SC-CROW transfer matrix
The transfer matrix analysis is done by taking one unit cell, and dividing it to 4 different sections: propagation (I), propagation and first coupling (II), propagation (III), second coupling (IV). Each section is described by a particular transfer matrix [17]. The final matrix m is achieved by successively multiplying these matrices.
Acknowledgment
The authors thank the Israel Ministry of Science and Technology for partially supporting this research.
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