In this paper, we derive the exact dispersion relation of one dimensional periodic coupled-resonator optical waveguides of finite length, from which the reduced group velocity of light is obtained. We show that the group index strongly depends on the number of cavities in the system, especially for operation at the center frequency. The nonlinear phase sensitivity shows an enhancement proportional to the square of the group index (or light slowing ratio). Aperiodic coupled ring-resonator optical waveguides with optimized linear properties are then synthesized to give an almost ideal nonlinear phase shift response. For a given application and bandwidth requirement, the nonlinear sensitivity can be increased by either decreasing resonator length or by using higher-order structures. The impact of optical loss, including linear and two-photon absorption is discussed in post-analysis.
©2004 Optical Society of America
The ability to reduce the group velocity of light and enhance the light-matter interaction has been demonstrated with the use of artificial optical resonances [1, 2, 3]. Slow-light refers to the situation that the group velocity vg is much less than the free-space speed of light c. The group velocity is determined by the derivative of the radial frequency ω with respect to the propagation constant k, given the optical dispersion relation k(ω). A dispersion relation has been established for the coupled resonator optical waveguide (CROW) in the tight-binding approximation and simplified with the assumption of nearest neighbor coupling [4, 5]. In one-dimensional periodic optical slow-light structures, such as the thin-film micro Fabry-Perot-cavity arrays and the coupled ring-resonator arrays shown in Fig. 1, the solution for Bloch-wave propagation gives more explicit description of the dispersion relation .
Slow group velocity benefits the efficiency of nonlinear processes. The nonlinear phase sensitivity enhancement is predicted to be proportional to group index (or slowing ratio) squared as (c/vg )2 [6, 8], which can only be evaluated when the dispersion relation has been solved. While the theoretically-derived dispersion relations [4, 6] are strictly valid only for infinitely long structures, they have been applied to experimental measurements on finite two-  and three-dimensional  photonic bandgap structures, where a group velocity as low as 10-2c was estimated . In this paper, we derive the exact dispersion relation and the group velocity for finite one-dimensional periodic coupled resonator waveguides. The theoretical derivations are verified with numerical simulations. The dispersion relation shows that at resonances within the transmission miniband , light propagation can be slower in finite structures than in the infinite structure of the same architecture. The study of the nonlinear sensitivity at the center frequency verifies that the nonlinear phase sensitivity enhancement is proportional to the square of the slowing ratio in finite structures. We also demonstrate the application of a filter design technique  to optimize aperiodic coupled resonator structures to produce a nearly ideal nonlinear phase shift response. For given application and bandwidth requirement, the nonlinear sensitivity can be increased by either decreasing resonator length or by using higher-order structures.
2. Coupled ring-resonator waveguide
The transfer matrix method has been widely used in the analysis of one-dimensional periodic optical structures, such as coupled-resonator waveguides [6, 11] and photonic band-gap structures , where a unit cell is replicated by periodicity. The amplitudes of the input and output waves of each unit cell are related by the transfer matrix M̂ as
where the superscripts + and - indicate the forward and backward going waves respectively. For unit cells having real refractive index, M̂ can be expressed in a general form as 
where t 1 and r 1 are the volume transmission and reflection coefficients related by |r 1|2+|t 1|2=1. Usually t 1 and r 1 are complex quantities and are frequency dependent. M̂ is a unimodular matrix, i.e. det(M̂)=1, as the system is assumed lossless.
The architecture of the unit cell specifically determines the volume coupling coefficients. Figure 1 emphasizes two representative unit cells in coupled-resonator waveguides (the shaded portions). Figure 1(a) is the integrated Fabry-Perot-cavity type unit cell that can be realized using thin-film technology, where the coupling element is a partially reflective mirror formed by a stack of thin-film layers of quarter-wavelength thickness. (An alternative implementation is a one dimensional photonic crystal defect waveguide .) Given specific materials and the design wavelength, there is no extra control over the coupling strength except the formation of the mirror stack; hence the transmission and reflection coefficients are frequency dependent (usually of complex quantities at non-resonant frequencies) and cannot be arbitrarily chosen. The wave propagating through the coupled Fabry-Perot-cavity structure develops standing-waves in the cavities. Figure 1(b) shows the corresponding travelling-wave type unit cell formed by a ring-resonator. The coupler is usually assumed to be non-dispersive and has real and constant energy coupling ratio. The transfer matrices of these two unit cells are identical if the mirror stack is replaced with a complex coupler; however, the thickness of the mirror has to be taken into consideration to determine the effective cavity length . For simplification purposes, we take the coupled ring-resonator waveguide as an example in this paper to analyze the dispersion relation of finite coupled-resonator optical waveguides; nonetheless, the methodology is applicable to thin-film coupled-resonator structures.
To study the coupled ring-resonator waveguide shown in Fig. 1 (b), we assume that all couplers are identical and ideal, i.e. no coupling loss and no backward reflection, so that the operation is solely determined by the coupling (r) and transmission (t) coefficients also related by r 2+t 2=1. The average circumference of each ring is L. Considering a lossless system, the transfer matrix of the ring-resonator unit cell  is
where , d=L/2 is the sum of two L/4 arcs for the ring-resonator, and k=nω/c is the material wave number where n is the effective refractive index, ω=2πc=λ is the radial frequency, and λ is the wavelength in free space.
Comparing the elements in matrices (3) and (2), the volume transmission of the ring-resonator unit cell is given as
and the volume transmittance is T 1=|t 1|2=t 2, which is of the same value as the coupler transmittance.
2.1. Infinite waveguide
where β is the propagation constant of the Bloch wave. This expression is consistent with Yariv’s derivation [4, 5] after some straightforward simplifications. Differentiating the frequency ω with respect to the propagation constant returns the group velocity 
which indicates the group index (or group velocity slowing ratio) is
For resonance frequencies (i.e. kd=2mπ with integer m), ng =n/t, meaning that the group velocity is 1/t times slower than the phase velocity ν=c/n in the underlying material. The weaker the coupling between cavities, the more strongly the light is slowed; however, there is a tradeoff between the light slowing ratio and the range of frequencies that can propagate through the waveguide. The bandwidth Δω depends on the coupling coefficient as 
where FSR=c/(nL) is the free spectral range. The bandwidth increases with increasing coupling efficiency, which in turn weakens the light slowing ability.
2.2. Finite waveguide
The assumption of Bloch-wave propagation is only valid for infinite periodic structures; therefore the resulting conclusions may not be applied to finite structures. Any fabricated waveguide is of finite length. To study such a finite structure, viewing the whole waveguide as a composite unit cell with complex waveguide transmission coefficient tN and reflection coefficient rN with relation |tN |2+|rN |2=1, the transfer matrix  analog to Eq. (2) is
where the complex tN can be described by its amplitude and phase as
in which TN is the waveguide transmittance and ϕN is the total phase shift determined by
Here kN is the propagation constant in the finite waveguide, being described in the dispersion relation of Eq. (15), and LN is the total waveguide length. For an N-cavity waveguide (i.e. N+1 unit cells), LN =(N+1)d. Assuming that all the unit cells are identical, the overall transfer matrix of the N-cavity waveguide can also be computed from the unit cell transfer matrix M̂ using 
Then substituting Eq. (4) and solving for tN , we get
Rewriting tN in the form of Eq. (10), with sin(kd)=t cos(βd) from Eq. (5) and sin(Nβd)=sin(βLN -βd)=sin(βLN ) cos(βd)-cos(βLN ) sin(βd), the exact dispersion relation of the finite N-cavity coupled ring-resonator waveguide is
where β is determined from Eq. (5).
The group delay per unit length is calculated by differentiating the propagation wave vector kN with respect to radial frequency as
where RHS reprents the right-hand-side of Eq. (15), and ngN=c/vgN is the group index and vgN is the group velocity in the finite structure. With some mathematical efforts, the group index is derived from Eqs. (15) and (16) as
where ng is the slowing ratio of the infinitely long waveguide given in Eq. (7). The total group delay is then calculated as
Determination of the miniband width is from Eq. (8), which is valid for both infinite and finite structures.
2.2.1. Linear response
Figure 2 shows the linear transmission properties of an 11-cavity (solid lines) and a 10-cavity (dashed lines) coupled ring-resonator waveguide with average ring length of L=50 µm (FSR=4 THz) and miniband Δν=FSR/3; the coupling coefficient is t=0.5 (the lower the t, the sharper and deeper each resonance). In the top figure, observing the transmittance Iout =Iin at the central frequency, the 11-cavity waveguide produces full transmission, but the 10-cavity waveguide is at a local minimum; they are out-of-phase. Moving towards the band edge, the two transmittance responses gradually become in-phase. The total linear phase in the miniband is determined by the number of cavities, where each ring contributes one π phase shift. Much of the phase, however, is accumulated at the edge of the band, manifested by the band-edge spikes in the group delay responses shown in Fig. 2 (c). This is characteristic of auto-regressive (AR) optical filters. (In the transmission transfer function tN given by Eq. (14), tN possesses N poles and (N+1)=2 zeros at the origin, and is thus characterized as an AR filter .) The dispersion relations ω(k) are plotted in Fig. 3 (a). Also drawn for comparison are the dispersion relation of an infinite waveguide (thick solid line), which is a smooth curve, and the dispersion relation of the underlying material (thick dashed line), which is a straight line. The plots of normalized group velocity vg/c (Fig. 3(b)) and group index (Fig. 3(c)) show clearly the difference resulting from the ripples in the dispersion relation of finite waveguides. While approaching the band edge, the group index at some frequencies increases significantly, theoretically reaching infinity in an infinite system, resulting in zero group velocity [4, 7].
Observing Fig. 3(b) at the central frequency, the 11-cavity waveguide shows stronger ability to slow light propagation even than the infinite structure, but there’s no light slowing effect in the 10-cavity waveguide; it has the same group velocity as the material. This can be explained by studying the group index ngN at νm , where sin(kd)=0, and hence cos(βd)=0 from Eq. (5). The expression for ngN is reduced to
showing the dependence of slowing ratio on the length of the waveguide. For a waveguide having an odd number of cavities, we have sin(βLN )=0 and cos(βLN )=1, so that
which is 1/t times larger than the value for that of an infinite structure, meaning that the propagation of light on resonance is slower in the finite waveguide even though the propagation distance (waveguide length) is shorter. This phenomenon results from the miniband resonances at specific wavelengths. The resonance bandwidth δν at νm , i.e. the full-width at half-magnitude (FWHM), is much narrower than the width of the entire miniband Δν, which is also the resonance bandwidth for infinite structures. Otherwise, if N is an even number, we have cos(βLN )=0 and sin(βLN )=1 instead, thus
revealing that the structured waveguide is just like the unstructured material; no light slowing occurs. However, if a working frequency other than νm is chosen, such as at neighboring resonance peaks, which forces cos(βLN )=1 and sin(βLN )=0, light slowing does occur for even-N cavity waveguide, as shown in Fig. 3(b)(c). To create effective slow-light periodic structures operating at the center of the miniband, the waveguide needs to contain odd number of cavities.
2.2.2. Nonlinear response
In Fig. 4, the intensity transmittance and nonlinear phase shift response of the 11-cavity waveguide (11th order AR filter) are shown as functions of normalized input intensity n 2 I in, where n 2 is the nonlinear coefficient of the underlying material and I in is the incident intensity. All the couplers are considered to be linear and intensity independent. With increasing n 2 I in, the transmittance goes down and the nonlinear phase change ΔΦ accumulates gradually. When the incident intensity is high enough, both the transmittance and phase shift responses experience rapid transitions (emphasized by arrows), and switch to a higher stable branch. Beyond this point, if the intensity is reduced, the transmittance and phase shift do not switch back immediately to the lower branch but stay on the higher branch until the input intensity is lower than the point of low-to-high transition, as depicted in the figure. This phenomenon, known as optical bistability (or multistability in this case), can occur when an optical system possesses both nonlinearity and feedback, as here. The region within the hysteresis loop, illustrated by the dotted lines, represents the unstable region. Optical bistability is a useful phenomenon for making optical switches and optical logic gates .
The nonlinear phase sensitivity defined as dΔΦ=dIin is an important metric of the performance of nonlinear devices. Prediction of the nonlinear phase sensitivity at low intensity input can be obtained from the linear filter response, which is fully controlled by two parameters: the coupling coefficient t and ring length L. Figure 5 shows the normalized nonlinear phase sensitivity dΔΦ=d(n 2 Iin ) (proportional to /n 2 where is the effective nonlinear refractive index of the waveguide) of the 11-cavity waveguide as a function of coupling coefficient and cavity length, respectively. Both figures are computed at the central resonant wavelength λm=500 nm. Figure 5(a) indicates that the nonlinear phase sensitivity is proportional to 1=t 4, hence proportional to the group index squared according to Eq. (9) or inversely proportional to group delay squared with Eq. (18) as
if varying the coupling strength. Figure 5(b) shows that if the cavity length is varied, the group index stays constant and the nonlinear phase sensitivity is proportional to cavity length or group delay, as
given constant coupling efficiency. A similar relationship also holds for increasing the number of cavities (i.e. filter order) instead of increasing cavity length. In both of these situations, the total group delay is increased, but the group index is constant (at νm ). According to Eq. (8), the finesse F=FSR=Δν∝1=sin-1(t); which is therefore constant with L and N; however, δν scales inversely with L and N. In situations where t is constant, then the nonlinear sensitivity is proportional to 1/δν. Obviously, the most efficient way to enhance the nonlinear phase sensitivity is to control the coupling coefficient, but at the loss of overall bandwidth (i.e. Δν). Vertical couplers with precise coupling efficiency have been demonstrated using a thermal wafer bonding technique .
3. Optimizing coupled ring-resonator waveguide as a nonlinear phase shifter
Optical bistability is undesirable for a nonlinear phase shifter. An ideal nonlinear phase shifting element should have constant transmission fraction with increasing intensity until at least a π phase shift is achieved. The lower the intensity needed to obtain the π phase shift, the better the nonlinear sensitivity. To avoid optical bistability and get better nonlinear phase response, an optical structure should be optimized to produce a flat-top magnitude response with a steep linear phase response within the passband, as shown in Fig. 6; a purely periodic structure produces strong ripples unless the finesse is extremely low (i.e. t≈1 in Eq. (8)). Light incident on the optimized waveguide will be transmitted with efficiency given by the magnitude response, but will also experience a phase change due to the phase response. As the light intensity increases, the overall filter response will redshift due to intensity-induced changes in the structure components, which are themselves constructed from (weakly) nonlinear materials. Ideally, under weak detuning, the transmitted intensity fraction will not change given a flat-topped magnitude response, but the phase at the output will change due to the steep linear phase response within the passband. The slope of the phase determines the group delay. In effect, what this approach does is to amplify the intrinsic nonlinearity of a material, where the efficiency of the process improves with increasing filter group delay. Strong detuning in multi-resonator systems can result in distortions of the filter response, however.
3.1. Optimized waveguide response
Finite coupled ring-resonator waveguides with finesse of 40 are designed using the methodology described by Melloni , where a prototype filter is synthesized using a classic microwave filter design method and then substituted with optical components, such as coupled ring-resonators. The synthesis procedure has coupling coefficients in each stage optimized to produce the ideal response as Fig. 6 shows. Because of the variation in coupling coefficients, this finite coupled ring structure is not periodic. Waveguides having 4 THz FSR (i.e. average ring length is 50 µm) and ~100 GHz bandwidth with 3 to 11 rings are designed, and the optimized coupling coefficients are listed in Table 1, showing that these aperiodic systems are symmetric. The linear spectral responses are shown in Fig. 7. With increasing filter order, the passband gradually approaches the ideal square shape. In the passband, the transmission ripples have been minimized and the phase is approximately linear. The total phase change as well as the inband phase increase with filter order. Some fraction of the total phase change locates outside the band to reduce the inband transmission ripples. Since there are no independent zeros in AR filters, much of the inband phase is built up at the edge of the passband and hence results in group delay spikes. By adding independent zeros into the transfer function to create auto-regressive moving-average (ARMA) optical filters, better phase and group delay responses can be produced . The group delay at the central frequency is linearly proportional to the filter order as shown in Fig. 8 for N≥3.
Figure 8(b) compares the physical size of the waveguides with the underlying material that produces the same group delay. The waveguide design is much more compact. Taking the 9th order filter as an example, the physical length is 250 µm producing about 21.6 ps group delay; to produce the same group delay, a material length of 4.3 mm is required. There’s about 17 times size reduction. For structures of much higher orders, an asymptotic size reduction of 25 is estimated by dividing the slopes of the lines in Fig. 8(b).
3.2. Optimized nonlinear phase shift
The nonlinear phase shift responses of the optimized coupled ring-resonator waveguides are shown in Fig. 9 as a function of normalized input intensity n 2 I in. The incident frequency is at the center of the passband. Because of the flattened linear response, optical bistability has been reduced. The nonlinear phase sensitivity is almost constant at low incident intensity, as shown in Fig. 9(c), and scales proportionally with filter order (N≥3) as shown in Fig. 10, where plotted for comparison is the nonlinear phase sensitivity of the underlying material of equal group delay. In the case of the 9th order filter, there’s about 10 times improvement. For much higher order filters, the asymptotic enhancement is about 12 obtained from the line slopes. If we define a figure-of-merit as
which takes into account increase in nonlinearity and reduction in physical length, where the superscripts “w” and “m” are for waveguide and material, respectively, we obtain FOM about 170 for the 9-cavity waveguide.
Examining the nonlinear phase shift, higher order waveguides require less input intensity to produce a π phase shift, i.e. lower n 2 Iπ, and also produce higher transmittance. For example, for the 9th order waveguide, a π phase shift is obtained at n 2 Iπ=5×10-6 with almost 100% transmittance, while n 2 Iπ=1.3×10-5 with transmittance less than 70% is obtained for a 5th order waveguide.
For a single artificial resonance, the nonlinear sensitivity scales linearly with the product of finesse and quality factor . The same is true for multi-resonator optical waveguides [10, 19]. In a specific optical application, the bandwidth is usually given; hence, the nonlinear sensitivity is simply proportional to FSR or finesse, as Fig. 10 shows, where (n 2 I π/4)-1 and (n 2 Iπ )-1 produced by a 7-cavity waveguide are studied as a function of FSR. Note that this case is different to that described by Eq. (23) where the finesse was constant.
When both bandwidth and FSR are chosen, the nonlinear sensitivity can also be improved by increasing the filter order. The nonlinear sensitivity versus filter order at normalized intensity n 2 I π/4 and n 2 Iπ is plotted in Fig. 11 (a) and (b), respectively. Figure 11 (a) shows that the nonlinear sensitivity (roughly proportional to (n 2 I π/4)-1 since π/4 is a relatively small phase change) scales linearly with filter order. In Fig. 11 (b), although there’s large nonlinear change with increasing filter order from 3 to 7, (n 2 Iπ )-1 also approximates a linear relation to filter order if N≥7. In both figures, nonlinear sensitivity of a bulk material is plotted for comparison. For the 9-cavity waveguide, the nonlinear sensitivity enhancement factor is about 10.3 and 11.7 at n 2 I π/4 and n 2 Iπ , respectively; considering the size reduction, the FOM is about 177 and 202, respectively. FOMs of higher order waveguides can be approximated from the slope of the lines in Fig. 11, which is around 300. Figure 12 (a) shows the nonlinear response at frequencies of ν=νm ±Δν/4 for the 9-cavity structure. Because of the flattened linear filter response, transmittance above 90% can be obtained at the normalized incident intensity n 2 Iπ where a π phase change is produced, demonstrating a broadband nonlinearity. The value of n 2 Iπ decreases with increasing frequency across the passband. Compared with the unstructured material at νm (thick-solid line in Fig. 12) (a), the coupled resonator structure requires much less intensity to produce a π phase shift, and at a given moderate incident intensity, the nonlinear phase shift produced by the coupled resonator structure is much greater, although they allow the same group delay.
4. Discussion and conclusions
In the previous sections, the exact dispersion relation and the group velocity for finite coupled ring-resonator waveguides are derived, from which the nonlinear phase sensitivity at low intensity input can be predicted. In purely periodic waveguides, at the resonance frequency, the nonlinear phase sensitivity enhancement is inversely proportional to the coupling strength to the fourth power or proportional to slowing ratio squared if varying the coupling coefficient, and proportional to single resonator size if varying cavity length with constant coupling. Because of resonance peaks in the transmission miniband, optical multistability is observed.
Aperiodic coupled ring-resonator optical waveguides with optimized linear response were synthesized to give almost ideal nonlinear phase shift response. For a given application and bandwidth requirement, the nonlinear sensitivity can be increased with either decreasing single resonator length or using higher-order structures. In an aperiodic 9-cavity ring-resonator waveguide having 4 THz FSR and ~100 GHz bandwidth, if considering both nonlinear enhancement and size reduction, a FOM ~200 is expected for a π nonlinear phase shift, compared to the underlying material producing the same group delay.
In the post-analysis, we can take the linear and nonlinear optical loss of the ring resonators into consideration that were neglected in the design procedure. In Fig. 12 (b), the optimized aperiodic 9-cavity ring-resonator waveguide is studied with the presence of linear absorption and two photon absorption (TPA). The intensity transmission is reduced to 0.6 by linear absorption (α=1cm-1) at low incident intensity, while the amplitude response maintains flattened transmission characteristics up to a π phase change under moderate TPA (K=0.03  or T~0.75). In the presence of loss, a slightly greater value of n 2 Iπ is obtained, resulting in slightly reduced nonlinear sensitivity. For large TPA (K=0.1 or T~2:5), the transmission at incident intensity n 2 Iπ is further decreased, and the nonlinear sensitivity is reduced as well.
This research was sponsored by the United States Army Research Office grant 41374-PH-YIP.
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