We present a study a new type of optical slow-light structure comprising a serpentine shaped waveguide were the loops are coupled. The dispersion relation, group velocity and GVD are studied analytically using a transfer matrix method and numerically using finite difference time domain simulations. The structure exhibits zero group velocity points at the ends of the Brillouin zone, but also within the zone. The position of mid-zone zero group velocity point can be tuned by modifying the coupling coefficient between adjacent loops. Closed-form analytic expressions for the dispersion relations, group velocity and the mid-zone zero vg points are found and presented.
© 2011 OSA
During the past decade, substantial research efforts were focused on slow light photonic structures and their applications. One of the most attractive approaches to realize such structures is to let the light propagate through a chain of coupled cavities. Significant reduction of the group velocity in such coupled cavity waveguides (CCWs) have been predicted and demonstrated experimentally. Because CCWs can be realized in chip-scales, the slowing effect opens up new avenues for numerous integrated optics applications e.g. delay lines, rotation sensors, optical memories, optical filters, wavelength converters and more [1–10]. In particular, micro-resonators based slow-light structures received substantial attention due to their interesting properties and potential applications [11–14]. Much focus was devoted to two different types of slow-light structures: the CROW (Coupled Resonator Optical Waveguide) [2,5,15], and the SCISSOR (Side Coupled Integrated Spaced Sequence of Resonators) [6,16]. Additional structures, exhibiting a combination of the properties of the SCISSOR and the CROW have been proposed and demonstrated as well [17–20].
One of the most important characteristics of periodic slow-light photonic structures is the formation of forbidden and allowed frequency bands in the dispersion relations of the structure. The edges of the forbidden gaps are characterized by very slow group velocity regimes and stopped-light (vg = 0) at the band-edge. In most of the more extensively studied slow-light structures (SLSs) such as the CROW and the SCISSOR, the zero group-velocity points are formed at the edges of the Brillouin zone (i.e. at K = 0 and K = π where K is the Bloch wavenumber) due to the Bragg resonances and the cavity resonances of the structure. Consequently, the dispersion relations of these structure exhibit cosine-like shapes and the ability to engineer and modify these relations is quite limited. Formation of zero group velocity points inside the Brillouin zone has also been shown to exist in structures allowing for two distinct, counter-propagating, light-paths such as the side-coupled adjacent resonators CROW (SC-CROW) [19,20] and coupled photonic crystal waveguides . In these cases the mid-zone zero group velocity points were found to emerge in the vicinity of the cavity [19,20] and Bragg  resonances. Mid Brillouin zone zero vg points were also found theoretically in the dispersion relations of optical microcoil resonator structures  for inter-winds coupling levels which exceed a certain value.
In this paper, we introduce a new type of slow-light structure consisting of a snake-like structure where optical coupling is introduced between adjacent curved sections (see Fig. 1 ). We find that unlike the CROW and the SCISSOR structures but similar to the SC-CROW, the serpentine optical waveguide (SOW) exhibits stopped-light points within the Brillouin zone. This is despite the fact that, unlike the SC-CROW, the SOW structure does not explicitly include resonators (although resonating optical paths do exist) and that the bands in the structure are formed by the Bragg condition (as in a conventional Bragg stack) stemming from the optical length of each unit cell. As discussed in Section 3 below (see also Fig. 3 ), the spectral response of the SOW is periodical with a free spectral range (FSR) which is determined by the phase accumulated by propagation along the optical length of a unit cell – k 0⋅n eff⋅(2πR + 2LB). The couplers in the structure serve as partial reflectors due to the geometry of the SOW, similar to the role of the interfaces between adjacent layers in a Bragg stack.
The position of the mid-zone stopped light point in the structure is determined by the coupling coefficients between the loops of the SOW (see Fig. 1), and therefore can be relatively easily modified and tuned, thus allowing much flexibility in the engineering and tailoring of the dispersion relations to specific applications. Similar to the optical microcoil resonator structure , the SOW provides two alternative paths for the light propagating through it – conventional propagation through the waveguide, and “tunneled” propagation through the couplers connecting adjacent unit cells. These two alternative routes are the key for the formations of the mid Brillouin zone zero group velocity regimes.
The existence of a tunable slow group velocity region inside the Brillouin zone is highly attractive for enhancing gain in lasers [22–26] and light-matter interactions. In Bragg or CROW based lasers [22–26], lasing is attained only at the band edges while in the structures such as the SC-CROW and SOW, the lasing point in the dispersion relation can be modified.
2. Structure parameters and transfer matrix
Figure 1 depicts a schematic of the SOW structure as well as the unit-cell and fields definitions. The structure is defined by three parameters: 1) The phase accumulation in a quarter-circle of the loop (section A); 2) The phase accumulation in the curved waveguide connecting the upper and lower loops (section B); 3) the coupling coefficient, κ, between the loops. Regarding the third parameter, we note that the structure can be generalized by allowing the coupling coefficients between the upper and lower loops to be different. However, such generalization does not modify substantially the properties of the structure and in order to simplify the analysis we strict ourselves to identical coupling.
A single unit cell of the SOW is marked in Fig. 1. The field in the cell can be represented by six amplitude E 1..E 6 as defined in Fig. 1. The electric field amplitudes at adjacent unit cells can be connected by a transfer matrix M:Eq. (1) can be used to derive the spectral properties of the structure as well as the field distribution for any set of the parameters.
3. Band structure, dispersion relations and group velocity
According to Bloch theorem the field at each unit cell is related to the field at its neighbors by a phase shift, i.e. , where K is the Bloch wavenumber and Λ is the length of a unit-cell. Substitution of the Bloch condition into Eq. (1) yields an eigenvalue problem which eiganvalues and eigenvectors are the Bloch wavenumber and field solutions:Eq. (2) into Eq. (3) yields the characteristic equation which constitute the dispersion relation of the structure, i.e. the relation between the frequency and the Bloch wavenumber. Extracting the dispersion relation from Eq. (3) is laborious but straight-forward:
Note, that the phase term in the left-hand side of Eq. (4) is the overall phase accumulated by a wave which propagates along the whole waveguide length of a unit cell – 4ϕA + 2ϕB = k 0⋅n eff⋅(2πR + 2LB), where k 0 is the wavenumber, n eff is the effective index of the waveguide mode, R is the radius of section A and LB is the overall length of section B (see Fig. 1) given by LB = 2αR.
As a concrete example we consider an SOW structure consisting of Silicon over insulator structure where the Si layer is 230nm thick and the waveguide width is 430nm. The radius of section A is R = 5μm and section B consists of two circular sections with radius R = 5μm and an angle of α = 58.5°. Note, that the choice of α not only affects the overall length of section B but also the width of the gap between the waveguide loops and, hence, the coupling coefficient.
Figure 2 illustrates the mode profile of the waveguide super-imposed on the index structure. At λ = 1.55mm the refractive indices of the Si and the silica are respectively n Si = 3.477 and n silica = 1.4455. The resulting effective index is n eff = 2.362 for the lowest TE mode. Figure 3(a) depicts the dispersion relation of an SOW structure with the above-mentioned cross-section and coupling coefficient of κ = 0.24. Figure 3(b) depicts the corresponding normalized group velocity (vg/Λ) for the band which is marked in Fig. 3(a). For simplicity, the dispersion of the waveguide cross-section and the material was neglected. Nevertheless, this dispersion can be readily introduced by modifying n eff according to the wavelength.
As can be expected, the dispersion relations consist of a series of bandgaps and passbands. The FSR, given by Δf FSR = c/n eff R(2π + 4α), is 2.45THz. Each passband consists of a pair of mirror image dispersion curves where the symmetry lines are located at frequencies which are “anti-resonant” with the optical length of the waveguide structure in each unit cell i.e. frequencies satisfying k 0⋅n eff⋅(2πR + 2LB) = 2π⋅(m + 1/2) where m is an integer. This can be easily obtained from Eq. (4) by substituting KΛ = ± π which yields cos(4ϕA + 2ϕB) = −1. The term “anti-resonant” is in quotation marks because there is no actual resonance or anti-resonance associated with this optical length.
Another interesting feature of the dispersion relations shown in Fig. 3 is the emergence of a zero group velocity point located within the Brillouin zone (at KΛ≈ ± 0.491π for the parameters chosen in Fig. 3). It should be noted that in the vicinity of the mid-zone zero group velocity point there are two possible (positive) group velocities for every frequency. For example, at f = 190.4THz there are two possible group velocities – v g/π = 1.06ps−1 and v g/π = 2.38ps−1. The reason for the two possible group velocities stem for the two paths through which the light can propagate along the structure – along the waveguide or by tunneling through the directional couplers. As a result, for a given frequency there are two possible Bloch modes (with different direct/tunneled propagation combination) which possesses different Bloch wavenumbers and, correspondingly, group velocities.
Figure 4 shows the dispersion relations of the SOW structure for various representative coupling coefficients. There are several interesting features which should be noted: 1) Once the coupling coefficient, κ, exceeds a certain value, a zero group velocity point is formed within the Brillouin zone; 2) As the coupling coefficient is increased, the zero group velocity point shifts to larger Bloch wavenumbers, K, in the dispersion diagram, approaching KΛ = 2π/3 for κ→1; 3) At κ = 0.5, the two curves forming each transmission band collide at K = 0. Increasing the coupling coefficient further results in a formation of a crossing of the two curves.
The features of the dispersion relations can be easily obtained from Eq. (4). Calculating the derivative of Eq. (4) with respect to ω and substituting vg = (dK/dω)−1 yields the group velocity at each point:
Substituting vg = 0 to Eq. (5) yields a quadratic equation for cos(KΛ) which solutions provide the zero vg point:
For meaningful solution, the absolute value of the right-hand side of Eq. (6) must be smaller than 1 thus posing a limit on the coupling coefficients for which a zero group velocity point is formed within the Brillouin zone is formed:
Figure 5 illustrates the position of the mid Brillouin-zone zero group-velocity point as a function of the coupling coefficient. Substituting κ = 1 in Eq. (6) yields the maximal Bloch number for which such point is formed – KΛ = cos−1(1/2).
It is interesting explore the group velocity vg = dω/dK for κ = 0.1, i.e. at the coupling level where the mid-zone zero group velocity is formed. Figure 6 depicts the group velocity of the structure defined in Section 3 for κ = 0.1. As shown in the figure, an inflection point accompanied by a flat region is formed around K = 0.
4. Finite difference time domain simulations
To verify the properties of the analytic model we studied the structure using finite difference time domain (FDTD) simulation tool. In order to calculate the dispersion relation of the structure we simulated a single unit cell with Bloch boundary conditions at x (corresponding to the horizontal coordinate in Fig. 1) and perfectly matched layers in all other boundaries. The dimensions and parameters of the structure were as detailed in Section 3.
Figure 7 depicts the dispersion relation of the structure calculated by the FDTD tool. The stars indicates the results of the simulations while the solid line indicate the corresponding analytic result based on Eq. (4), where the coupling coefficient and the effective index where calculated by an FDTD simulation and a TE semi-vectorial finite difference mode solver and were found to be n eff = 2.362 and κ = 0.245 respectively. The excellent agreement between the simulations and the analytic model is clear, thus validating the theoretical analysis of Section 3.
5. Practical considerations – finite SOW structure and propagation losses
The transfer matrix method can also be used for exploring the properties of a realistic SOW structure, incorporating propagation losses and finite length. Given a finite SOW structure consisting of N unit cells, one can relate the fields at the output to the fields at the input by a product of three matrices: 1) an input matrix, connecting the fields at the input to the fields at the first unit cell; 2) N-unit-cell transfer matrix which is given by MN = M N where M is given by Eq. (2); and 3) an output matrix, connecting the fields at the last unit cell to the fields at the output. Note that assuming the SOW is excited from one of its sides, the outputs of the structures are the transmitted field and the reflected field (see Fig. 1).
When analyzing the properties of a finite structure, the important parameters are the number of unit cells and the coupling coefficient. Figure 8 depicts a color representation of the transmission function (|t|2 in Fig. 1) of a lossless SOW structure consisting of five unit cells for coupling coefficients ranging from 0.01 to 0.99. The dependence of the transfer function on the coupling coefficient is quite complex. It seems that the SOW response can be divided into several sub-sections depending on the coupling level. At low coupling levels (κ<0.3) the transmission function consists of several transmission peaks (see inset 1). Although similar property is exhibited by several, finite, slow-light structures (e.g. CROW) it should be noted that the number of the transmission peaks decreases as the coupling coefficient is increased. This is in contrast to CROWs, for example, in which the number of transmission peaks is determined solely by the number of unit cells.
At larger coupling coefficients, asymmetric and narrow transmission peaks appear (see inset 2). These peaks grow wider as the coupling is further increased until they merge with the central peak and form a relatively flat, “box”-shaped spectral transmission (see inset 3). At larger coupling coefficients, a rather interesting transmission function emerges, consisting of three narrow peaks at the center and edges of the transmission band where the central peaks exhibits an EIT-like transmission profile (see Fig. 9 ).
In addition to the interesting details of the finite SOW transmission function, the global dependence of the transmission function on the coupling differs substantially from that of a CROW. Referring to Fig. 8, note that for very low and very high coupling levels the transmission function of the SOW approaches almost unity. In addition, the over-all transmission band becomes narrow as the coupling is increased, reaching a minimum at κ~0.35 and then broadening again as the coupling is further increased. The reason for this behavior is that for low coupling levels the SOW approaches a simple curved waveguide (which does not exhibit structure dependent spectral properties) while for κ→1 the strong coupling provide a shorter route for the light, with basically no back reflections, thus yielding an almost unity and wavelength independent spectral transmission. This is in contrast to CROW where the transmission band monotonically increases when the coupling in increased.
SOW consisting of more unit cells exhibit similar properties although the fine details vary. Figure 10(a) depicts the transmission function of a lossless SOW structure consisting of seven unit cells for coupling coefficients ranging from 0.01 to 0.99. Figure 10(b) illustrates a similar plot for a similar structure with waveguide propagation losses of −1dB/cm.
Referring to Fig. 10(a), the longer SOW structure exhibits spectral properties which are very similar to those of the five unit-cells SOW. There are some changes in the fine details though, e.g. in the number of transmission peaks at weak coupling, the coupling levels a which the “box”-shaped transmission function emerges, etc. but the global properties are similar. In addition, referring to Fig. 10(b), it should be noted that the introduction of practical loss levels does not substantially modifies the spectral properties of the SOW. In particular, the impact of loss can be observed only close to the edges of the transmission band (see also Section 6 below).
6. Pulse propagation
The results of the band structure analysis provide the necessary information for understanding the propagation of a pulse through the structure. 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.
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 SOW frequency response which is directly extracted from the dispersion diagram (Fig. 4). The time domain dynamics is then calculated by inverse Fourier transforming the spectrum at each point in the structure. Figure 11 illustrate the propagation of a 30ps wide pulse through an SOW with coupling coefficient of κ = 0.15. The central frequency is set to be 191.72THz (1564.8nm) which is approximately at the center of the slow-light band (the shallow and linear slope regime of the red curve in Fig. 4) which is formed around KΛ = 0.17π. The linked avi file (Media 1) shows the temporal evolution of the field in the structure. As seen in the figure the pulse maintains its shape as it propagates through the structure although some broadening is also observed. The distortion is formed by the GVD of the structure at the wavelength of the pulse.
The group velocity of the pulse propagating along the SOW can be extracted directly from the dispersion relations or from the temporal evolution of the pulse. For the case shown in Fig. 11 (κ = 0.15) the group velocity is found to be 0.35 Unit-cells/ps. Taking the length of the unit-cell to be the over-all length of the waveguide comprising it (i.e. its length for κ = 0 – a simple curved waveguide), the corresponding group velocity is found to be 0.14 × c. Reducing the coupling coefficient towards 0.1 would yield slower group velocities.
When the pulse is propagating through a finite structure, the spectral features of the finite device must be also considered. Figure 12 depicts the amplitude and the phase response of a SOW consisting of 7 unit-cells, identical in parameters to the infinite structure used in Fig. 11, for a lossless case (blue) and for −1dB/cm propagation loss (red). The figure is zoomed on the slow-light band spectral range as discussed above. The important points to note are that even for non-negligible losses, the structure is quite transparent (only −1.5dB transmission loss) and that the phase responses are almost identical.
The phase response allows for the extraction of the group delay which is in average over the pulse bandwidth approximately 40ps. This corresponds to a group velocity of ~0.2 unit-cells per ps which differs from that of an infinite SOW but within the same order of magnitude. The differences in the group velocities between the finite and the infinite structures is not surprising, in particular because the propagation takes place close to the edge of the pass band and the finite structure is rather short.
We proposed and analyzed a new slow-light structure consisting of coupled loops that form a serpentine-like optical structure. Unlike the most commonly studied slow-light structure, the band structure of the SOW exhibits a zero group velocity point located within the Brillouin zone. The frequency and, in particular, the Bloch wavenumber at which this point is formed is determined by the coupling coefficient between adjacent loops and is insensitive to the waveguide cross-section and length of the optical loops.
Transfer matrix formalism (TTF) was developed and was employed for the calculation of the dispersion relations of the structure. Very good agreement was found between the TTF and a FDTD analysis of the structure. An almost distortion-less propagation of a 30ps pulse in the structure is shown at the relatively linear part of the dispersion relation with group velocity in the order of 0.1 the speed of light in vacuum. Although some distortion is visible, most of the power is concentrated in the main (leading) lobe and can be used for data transmission.
An important point which must yet be addressed is the interfacing of the SOW with external I/O ports. In order to avoid back reflections at the interfaces between the I/O waveguides and the periodic structure, impedance matching sections must be placed at these interfaces (like in any finite periodic structure). The design of such sections deserves a comprehensive study and is, therefore, beyond the scope of this paper.
From the performances aspect, when compared with the CROW and the SCISSOR, all three devices are linear periodic photonic structures exploiting resonating optical paths to achieve low group velocity, there are both subjected to similar limitations (see e.g. [27,28]). In all structures the reduction of the group velocity is attained by forcing the light to circulate in closed loop, thus effectively extending the optical path. In practice the limitations on the achievable delays, bandwidth GVD, etc. stem primarily from the technology used for the realization of the structure (i.e. index contrast, intrinsic losses, etc.) and less from the specifically employed geometry of the device. Thus one can expect that CROWs, SCISSORs and SOWs employing similar waveguide structure exhibit similar performances from these aspects. The uniqueness and advantage of the SOW is that it can exhibit a low GVD zero group velocity region which can be tuned by the coupling coefficient, thus allowing for much flexibility and engineering of the dispersion relation and light propagation properties.
The authors thank the Israel Ministry of Science and Technology for partially supporting this research.
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