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Abstract
We present a design for optimal interfacing (I/O coupling) with slow-light structures consisting of coupled cavities. The I/O couplers are based on adding a small set of cavities with varying coupling coefficients at edges of the coupled cavities waveguide in order to match its impedance with that of the I/O waveguides. I/O efficiencies exceeding 99.9% are shown to be possible over a bandwidth which is larger than 50% of that of the coupled cavities waveguide. Consequently, the reflections at the edges of the slow-light structure are practically eliminated. We discuss the properties of the perfectly impedance matched slow-light structure as an effective (super-structure) cavity and study the impact of the number of cavities comprising the I/O coupler. We also consider in details the impact of errors and disorder in the I/O coupling sections.
© 2017 Optical Society of America
1. Introduction
The ability to control and modify the speed at which light propagates has received much attention and focus during the last few decades [1]. This is due to the fundamental importance of the speed of light as well as the numerous potential applications which such ability facilitates in nonlinear optics, telecom, sensing, and more [1]. Consequently, a wide variety of methods and techniques for controlling the speed of light (primarily the group velocity) have been developed and studied, demonstrating subluminal (vg<c) [2-6] and superluminal (vg>c) [7-10] group velocities.
Although the concept of fast light is somewhat counterintuitive and have been proposed for various important applications in sensing [10,11-13] and buffering[14-16], it seems that more efforts have been invested in slow light structures (SLSs) [1,4-7]. Slowing down the speed of light can be achieved by utilizing either atomic or photonic resonances [1]. Atomic-resonance-based structures have been shown to enable very slow group velocities. However, the photonic resonances based structures are more suitable for integration into practical systems. Consequently, numerous photonic SLSs, such as coupled cavities waveguides (also known as CROWs - coupled resonator optical waveguides), photonic crystal waveguides and many more, have been studied theoretically and experimentally[1,2,4-6].
Particularly, CROWs and coupled cavities structures have been studied as key ingredients for various applications [1,17] such as optical filters [18-20] and buffers[21,22], detectors[1], sensing [23-26] and nonlinear signal processing devices[27-29]. One of the main problems with finite CROW structure (as opposed to the infinite case) is that their spectral response consists of many narrow transmission peaks. This is because of the impedance mismatch between the slow-light Bloch waves of the CROW and the modes of the I/O waveguides[17]. As this problem is important to all SLSs, much effort has been invested in overcoming the impedance matching problem[18,19,30]. And indeed, a variety of methods for efficient light injection into the slow-light modes of photonic crystal waveguides has been proposed and studied[31-37]. However, most of the studies involving coupled cavities structures were focused on obtaining a desired spectral response (often flat-top for filtering applications) by optimizing the coupling coefficients between the cavities composing the structure[38-42]. The results of these optimizations yield non-periodic structures with dispersion properties and group index that are incompatible with that of an ideal (infinite) CROW. Moreover, the results of these studies cannot be used directly for constructing an efficient I/O coupler to finite CROW structure. Only few studies aiming directly at reducing the reflections at the ends of a CROW structure has been reported. Sumetsky et al. [43]presented a CROW design with flat transmission by trial and error optimization of first and last few coupling coefficients in the structure. This approach, however, was not optimized and utilized the Breit–Wigner formalism which becomes less accurate for cavities with low loaded Q-factor (i.e. large coupling coefficients). A related impedance matching approach was presented by Sanchis et al. [44]who used adiabatic tapering to reduce reflections in coupled PhC cavities structure. Chak and Sipe presented an analytic approach for optimizing the two coupling coefficients at each end of a finite CROW structure in order to eliminate the reflection at a single frequency in the passband[45]. Although this approach was utilized by many successive studies, it is limited by the maximal attainable flat bandwidth because of the two coupling coefficients modification restriction. Thus, a more general optimization approach for CROWs is still needed. We emphasize that as noted in[46], efficient I/O couplers allow for mimicking infinite CROW structures using small devices comprising only few cavities, thus facilitating the study of structural slow light.
More recently, Faggiani et al. [46]studied an efficient coupling scheme to a slow-light PhC waveguide mode from a faster-light mode of I/O PhC waveguides, to realize a unique light confinement approach which was designated as photonic speed bump. This structure was shown to exhibit spectral and spatial properties which are very similar to those of conventional cavities, although no roundtrip phase condition is required. The resonant property of this structure is manifested by a sharp increase in the local density of states at the frequency where the group velocities of the I/O waveguides and the SLS are matched by the taper sections.
In this letter we present a design approach for efficient I/O coupling to finite CROW structures by adding a small set of additional cavities with varying coupling coefficients at both ends of the CROW. Compared to the coupling approaches based on apodization techniques derived from filter theory, we find that our approach provides better coupling efficiency manifested by lower back reflection and ripple. We also find that a finite CROW which is coupled into using our approach exhibits dispersion properties which are almost identical to those of an infinite one. The impedance matched finite CROW structure studied here exhibits several properties which are very similar to those of the speed-bump structure presented in[46]. However, the excited slow light mode in the CROW is located at the center of the pass-band where the group velocity dispersion (GVD) and losses are minimal. This is in contrast to [46] where the excited mode in the SLS is located near the edge to the transmission band where GVD and losses are maximal.
The rest of the paper is organized as follows: in Section 2 we present the impact of impedance mismatch on the properties if a finite CROW structure. In Section 3 we introduce the I/O coupling design concept and study the impact of the coupler length on the quality and bandwidth of the impedance matching. We also compare the properties of the structure to those of similar CROW with impedance matching methods based on filter theory design. In Section 4 we investigate the spatial and spectral properties of the structure’s “mode”. In Section 5 we study the impact of disorder due to fabrication tolerances and in Section 6 we discuss the results and conclude.
2. I/O coupling to a CROW: the impedance matching problem
Figure 1(a) depicts a schematic of a finite CROW structure with an I/O coupling sections comprising a single ring and two coupling coefficient – κ1 and κ2. Due to symmetry consideration it is straightforward to show that the output section should be a mirror image of the input one. Figure 1(b) depicts the dispersion relations of an infinite CROW which is given by:
where K is the Bloch phase shift between adjacent unit-cells, κ is the coupling coefficient, Δω is the shift from the resonance frequency of the individual cavities and ΔνFSR is the free spectral range.
Fig. 1 (a) Schematic a CROW structure with a tapered I/O coupling scheme. (b) Dispersion relation of an infinite CROW structure. In this plot κ = 10−2.
As mentioned above, the properties of a finite CROW structure can differ substantially from those of the infinite structure. Figure 2 depicts the spectral transmission properties of an infinite CROW structure (solid black) and a finite structure consisting of 5 resonators (solid blue). The coupling coefficient between the cavities is 4x10−3 and for clarity reasons, the propagation losses are neglected at this stage. While the spectral response of the infinite CROW structure is a square shaped pass-band, the finite CROW exhibits 5 narrow transmission peaks. The striking difference between the properties of the finite and infinite structures can be explained completely by considering the finite CROW as a Fabry-Perot (FP) cavity with the dispersion properties of the infinite CROW [see Fig. 1(b)] and its characteristic impedance given by[30]:
where Z0 is the characteristic impedance of the conventional waveguides which couple light into and out of the CROW [see Fig. 1(a)] and the slow-down factor .Fig. 2 Transmission properties of infinite (solid black) and finite CROW structures calculated by transfer matrices (solid blue) and by FP model (green circles).
Due to the impedance mismatch between Z0 and ZCROW, optical Bloch wave in the CROW is reflected when it reaches the first and last couplers in the structure with the following (field) reflection coefficient:
Figure 2 also depicts the spectral transmission of the finite CROW structure as calculated by the FP model (green circles). As seen in the figure, the agreement between the FP model and the transfer matrices method is perfect. There are two important features to be noted from Fig. 2 – first, the sharpness of the transmission peaks which stems from the rather large mismatch between the impedances of the CROW and the conventional waveguides (in the case of Fig. 2 the ratio between them is ~32 at the middle of the passband and almost 1600 at the edges). Note that the peaks near the band edges become sharper due to the larger impedance mismatch (and higher reflectivity). The second feature is the non-uniform FSR of the equivalent FP cavity which stems from the nonlinear dispersion of the CROW [see Fig. 1(b)].
Clearly, the dramatic differences between the properties of finite and infinite CROW structures make it difficult to study CROW based SLSs as well as using them for practical applications. As these differences stem directly from the impedance mismatch problem, it is obvious that overcoming the mismatch is an important task.
3. Matching the impedances of the CROW and conventional waveguides
The most straightforward method to match the impedances on the I/O waveguides and that of the CROW Bloch modes without substantially modifying the dispersion relations (1) is to change the coupling coefficients of the first few cavities in the structure. Indeed, many studies have presented designs for such I/O couplers, primarily by using filter theory[18,38-42]. However, these studies were focused mainly on obtaining a flat-top, small ripple and high rejection ratio, bandpass filter consisting of a minimal number of resonators and no on efficient coupling into a CROW based SLS. Indeed, the structures designed in [18,38-42] are non-periodic and do not include a section which supports formal Bloch modes. Nevertheless, one may expect that the design of such filters can be used for efficient I/O coupling into a finite section of “periodic” (i.e. identical coupling coefficients) CROW. As shown below, such an approach provides a structure with dispersion relations and transmission properties which are quite similar to those of an infinite CROW, at least at the center of the bandpass. However, better mimicking of the infinite CROW structure can be obtained numerically optimizing the coupling coefficients using a rather straightforward cost function.
In order to match the impedances of the conventional waveguides and the CROW, the first few coupling coefficients of the structures are optimized numerically to obtain a broad and flat transmission function. The cost function to be minimized is the conventional least mean square summation of the difference between the CROW transmission and a rectangular shaped filter with bandwidth of 75% of that of the infinite CROW. The optimization approach which was employed here is the well-known the Nelder-Mead simplex search method described by Lagarias et al.[47]. This is a direct search method which proves to be efficient and fast for minimizing cost functions with few parameters (coupling coefficients in this case). Optimization of longer I/O sections would require more sophisticated stochastic optimization algorithms such as genetic optimization.
As discussed in Section 2, the infinite CROW structure exhibits a flat transmission profile because there is no backwards propagating (Bloch) wave in the structure. The spectral features of a finite CROW structure (see Fig. 2) are a direct consequence of the reflections (due to the impedance mismatch) at the ends of the CROW. The only way to obtain a flat, unity, transmission through a CROW over a wide bandwidth is by eliminating these reflections (or at least reducing them substantially). Thus, the outcome of the bandwidth optimization is a substantial reduction of the reflections. This is also discussed in details and demonstrated in Section 4 below which plots the intensities in the microrings comprising the CROW. The lack of intensity oscillations in an impedance matched structure as opposed to an unmatched one indicates that there is no leftwards propagating Bloch wave, i.e. there are no reflections. This is also shown clearly in Fig. 9 which presents the reflection coefficient Γ as a function of the coupling coefficients of the I/O sections. Thus, the result of the passband optimization is essentially the elimination (or at least reduction) of the reflection coefficient over the CROW passband.
Figure 3(a) compares the transmission functions of finite CROWs (10 resonators) with optimized I/O sections consisting of 2, 3 and 4 couplers to that of an infinite structure. The obtained coupling coefficients are listed in Table 1. Note that the transmission at the center of the passband is flat over at least 50% of the infinite CROW passband (even for the two couplers case) and that the bandwidth with the flat response increases as the I/O sections are extended. Figure 3(b) compares the slowdown-factor defined as of finite CROWs with varying I/O section lengths and that of an infinite CROW. As can be seen in the figure, as the length of the coupling sections is increased, the resulting group index converges to that of the infinite CROW. In fact for an I/O section comprising 4 couplers (i.e. additional 2 resonators at each side) the group index of the finite structure exhibits almost perfect match to that of the infinite one.
Fig. 3 Transmission function (a) and slow-down factor (b) of an infinite CROW (dashed black) and of finite structures (10 resonators) with tapered I/O sections consisting of 2 (solid blue), 3 (solid green) and 4 (solid red) couplers
Table 1. Coupling coefficients of I/O sections of different length.
As mentioned above, it is reasonable to assume that the approaches developed for the design of optical filters [18, 38-42] should be useful for obtaining efficient I/O coupling sections for CROWs. The straightforward method to utilize these approaches is to design an Mth order filter (where M is odd) which includes different coupling coefficients. The first coupling coefficients would serve as the I/O sections and the last one (which is the smallest) would be the coupling coefficient of the CROW. For example, designing the two-coupler I/O section as depicted in Fig. 1(a) requires the design of a filter comprising 5 resonators, yielding 3 coupling coefficients. The first 2 coefficients correspond to κ1 and κ2 while the third corresponds to κ.
Figure 4 compares the spectral properties and group indices of finite CROW structures with 2-couplers I/O sections designed according to filter theory (blue) and the numerical optimization described here (red). The finite CROW structures consist of 40 resonators (plus additional two for the I/O sections). The coupling coefficient between adjacent resonators (except the first and last two) is κ = 10−3. The filter theory based design is carried out according to the procedure described in [19,38] for a maximally flat filter. The resulting coupling coefficients are listed in Table 1. There are several striking differences between the spectral properties obtained by the two impedance matching approaches. The filter theory based I/O sections yields an overall broader transmission band than that of the numerically optimization approach. However, the ripple obtained by the first approach is larger, at least at the center of the passband. These properties are also manifested in the group velocities [represented by the slow-down factor in Fig. 4(b)] obtained by the two design approaches. The numerical optimization approach yields a smooth slow-down factor which is very close to that of the infinite CROW. The filter theory based approach, on the other hand, yields an oscillating slow-down factor where the maximal oscillations occur at the center of the CROW passband. Note, that the overall transmission band obtained by filter theory approach is broader (assuming the ripple is acceptable). However, the numerical optimization approach presented here yields finite CROW devices with spectral properties which mimic well those of an infinite device, at lease at a broad part of the passband in the vicinity of its center. Thus, the later design approach is more useful for practical experimental investigation of CROW based SLSs.
Fig. 4 Comparison between filter theory based and numerically optimized I/O sections design; (a) the spectral transmission of CROWs with filter theory based (blue) and numerically optimized (red) I/O sections. (b) The slow down factors of finite CROWs with filter theory based (blue), numerically optimized (red) and infinite CROW (dashed black).
It is also useful to compare the spectral properties and group indices of a finite CROW which was impedance matched by the approach presented here to that of Ref.[45]. As shown in Table 1, the coupling coefficients obtained by the numerical optimization approach described here for the two couplers case are very close to those obtained by the approach of Ref.[45]. It can, therefore, be expected that the spectral transmission properties of both structure would be similar. However, the numerical optimization approach allows for longer I/O coupling sections which can eliminate reflections over a wider bandwidth
Figure 5(a) depicts a comparison between the spectral transmission properties of a 10 micro-resonators CROW with numerically optimized three coupler long I/O sections (blue) to that of a structure designed according to the analytic approach of [45] (green). It is evident that the flat bandwidth of numerically optimized structure is substantially wider (by approximately 90% - see the inset of the figure) than that of the analytically optimized one. This is also manifested by the comparison between the intensity profiles in the CROW structures depicted in Fig. 5(b) (see Section 4 below for more detail on this plot). The flat profile of the numerically optimized structure (blue), which is plotted at a frequency detuned from the passband center by Δν = 2.2x10−3ΔνFSR indicates that the reflections at the CROW ends are almost canceled. This is in contrast to the oscillating intensity profile of the analytically optimized structure at the same frequency (green), which indicates non-vanishing reflection coefficients.
Fig. 5 Comparison between the spectral transmission (a) and intensity profiles (b) of impedance matched CROWS with three coupler numerical optimization (blue) and the approach of [45] (blue). Inset: zoom-in on the central part of the transmission band
4. Properties of the CROW’s “mode”
As clearly noted in[46], obtaining efficient impedance matching to a very slow propagating mode over a narrow bandwidth yields electromagnetic fields which are very similar to those of optical cavities. The “mode” of the finite (impedance matched) CROW structure exhibits many properties that are similar to those of the photonic speed-bump presented in [46], though there are some striking differences as discussed below.
Figure 6 depicts the field distributions in finite CROW structures, with (a) and without (b) impedance matching sections, at the center of the passband (maximal transmission). Each point represents the field intensity in one of the halves of the individual ring resonators where the blue squares and green circles correspond respectively to the forward and backwards propagating fields in the resonators [see Fig. 6(c)]. The first 3 (blue – input, green – through) and the last 4 (blue – output, green – add) resonators intensities indicated in both panels of Fig. 6 correspond to points along the I/O waveguides [see Fig. 6(c)]. The green points in resonators #1-3 indicate the power in the Through port (which vanishes at the center of the passband) while the green points in resonators #18-22 correspond to the power in the Add port (which is assumed to be zero). The corresponding blue points for these resonators correspond to the Input waveguide (#1-3) and the Drop (#18-22) in which the power is unity (maximal transmission).
Fig. 6 Field distribution in a finite CROW with (a) and without (b) impedance matching sections. The forwards and backwards propagating fields in the microrings are marked by blue squares and green circles respectively. (c) Schematics of the impedance matched CROW with indication of positions corresponding to the resonator index in panel (a).
The uniform intensity profile depicted in Fig. 6(a) clearly indicates the effectivity of the impedance matching as it proves that there are no backward propagating Bloch waves in the finite CROW structures. Note that the field in each of the individual microrings consists of forwards and backwards propagating parts (which are separated spatially). However, the absence of intensity oscillations in the CROW [as seen in Fig. 6(b)] indicates the successful excitation of a Bloch wave propagating in a single direction. We also note the difference in the intensities of the forward and backward propagating fields in each resonator. This difference corresponds to the net forwards power flow in the CROW. It should be clear that such difference also exists in the non-impedance-matched CROW [Fig. 6(b)]. However, due to the scale of the figure it is difficult to be observed.
The properties of the intensity distribution in the non-impedance-matched finite CROW [Fig. 6(b)] are very different from those of the impedance-matched case. The intensity distribution exhibits clear FP-like oscillations due to the existence of both forward and backward propagating Bloch waves. The intensity distribution is essentially a discretized version of the standing wave pattern existing in conventional FP cavities excited on resonance. We also note that the peak intensity in the non-impedance-matched structure is much larger than that in the impedance-matched one. This is because of the very large group index associated with the sharp resonance of the effective FP cavity resonance (see Fig. 2). We note that, similar to the impedance-matched case, the intensities in the forward and backward propagating sections of each microring are different due to the net power flow forwards. This small difference does not show clearly in Fig. 6(b) sue to the axes scale of the figure.
It is constructive to compare the spectral dependence of the spatial intensity distribution in the matched and non-matched CROW structures. Figure 7 depicts the field intensity distributions of the matched and non-matched finite CROW structures at several frequencies. As mentioned above, the spectral response of the non-impedance-matched CROW consists of several sharp peaks. The choice of frequencies at Fig. 7 corresponds to the five peaks located at the center of the passband. Referring to Fig. 7(b), it is clear that the intensity distribution at each of these frequencies corresponds to a FP-like mode of finite CROW. The number of intensity peaks is determined by the Bloch wavenumber corresponding to the pertaining frequency.
Fig. 7 Intensity profiles of finite CROW structures at several frequencies for the matched (a) and non-matched (b) cases. The legends on the left correspond to both panels.
In contrast to the previous case, the field intensity distributions in the impedance-matched structure exhibit a flat profile (indicating small reflection) over a broader spectral band [note the red and green curves in Fig. 7(a)]. The imperfect impedance matching is observed only towards the edge of the passband as relatively small oscillations (~20%) in the intensity distribution [blue and purple curves in Fig. 7(a)]. For comparison, Fig. 8 depicts the field distribution at the center of the passband, in finite CROW structure with impedance matching sections designed according to filter theory (second line in Table 1). The intensity pattern exhibit partial oscillations (~50% of the mean value) indicating non-vanishing reflection at the edges of the CROW. Note that although the ripple observed in the transmission properties is less than 10% (see Fig. 4), the impact of the impedance mismatch at the edges of the finite CROW introduce substantial modification of the intensity profile in the structure.
Fig. 8 Field intensity distribution in a finite CROW with impedance matching sections design based on filter theory.
Finally, as noted in [46], the properties of the field in the impedance matched point have many similarities to those of the modal field in conventional cavities. Although the resonant response of the CROW structure [Fig. 3(a)] may seem broad, its width essentially depends on the FSR of the individual resonators. For example, if the FSR of each resonator is ~1.5nm then the linewidth of the “mode” of finite (impedance matched) structure corresponding to Fig. 3(a) is merely 20pm. Moreover, this linewidth can be reduced almost arbitrarily simply by reducing the coupling coefficient between the adjacent resonators [48]:
We note that, as in [48], the frequency of this “mode” is independent of the CROW length. This is due to the absence of reflection at the CROW edges which eliminates the conventional roundtrip phase requirement. Another important property of every cavity mode is the Q-factor. The definition of the Q-factor in this structure is not obvious and requires special care because, unlike conventional cavities, the power in the structure does not decay exponentially but rather linearly in time. Let us assume that the structure is excited at the resonance frequency of the individual resonators comprising the CROW (corresponding to the center of the CROW passband). The time it takes the energy stored in the structure to exit depends on the group velocity and the number of resonators in the CROW. The group velocity in the CROW is given by
where L is the effective “length” of each resonator. In principle L may vary according to the specific shape of the resonator but as shown below its specific value is not important for determining the photon life-time or Q-factor of the structure. The time it takes the Bloch mode to propagate across a single resonator in the structure is given by:Consequently, the photon lifetime in the structure is simply , which on resonance (i.e. the center of the passband) equals to:We note that the time constant (7) is quite intuitive. The time it takes the energy to exit the structure increases with the length of the structure (NCROW), the roundtrip time in the individual resonators (~) and their Finesse (~). It should also be emphasized that due to the (approximately) constant rate in which the stored energy leaks out it is difficult to define the lifetime in the commonly used approaches such as where P is the power in the cavity.
5. Sensitivity to fabrication tolerances
Periodic structures such as the CROW are notoriously known for their sensitivity to disorder and deviation from perfect periodicity due to e.g. fabrication tolerances. The existence of such errors in the structure impairs the performances SLWs[30,49], introduce back reflections [50] and localization effects[51]. However, it has been shown that many of the properties of the CROW are only marginally impaired if the level of errors/disorder is sufficiently small[52]. However, as can be seen in Table 1 and Fig. 4, deviations from the optimal coupling coefficient may result in ripple and spatial intensity oscillations in the CROW (see Fig. 8) due to the non-vanishing reflectance at the edges of the structure.
In order to quantify the impact of errors and imperfections in the realization of the coupling sections, we show in Fig. 9 the reflection coefficient due to the impedance mismatch in the two couplers case (line 1 in Table 1) when the coupling coefficients deviate from the target values. The reflection coefficient is determined according to the intensity profile in the CROW. The field in the CROW is given by a superposition of the forward and backward propagating Bloch waves:
where n is the resonator number and Γ is the reflection coefficient. Γ can be obtained directly from the standing wave ratio [53] which can be extracted from the field profile in the CROW.Fig. 9 Field reflection coefficient as a function of the coupling coefficients of the I/O sections. The red circle marks the minimal reflection point.
Figure 9 depicts the dependence of the field reflection at the edges of a finite CROW on the coupling coefficients of the I/O sections. As shown in the figure, there is an area around the minimal reflection point (marked by the red circle) in which the reflection is very low and the impedance of the CROW is well matched to that of the conventional waveguides. Therefore, this I/O coupling approach is sufficiently robust to constitute a practical method for efficient interfacing with CROW structure.
6. Discussion and conclusions
The perfectly impedance matched finite CROW structure studied here constitutes an interesting platform which can serves both for fundamental research as well as for various applications. Particularly, this structure and design approach facilitates the study of periodic SLWs using a compact and robust device which is simple to fabricate and less sensitive to fabrication errors. It also serves as a cavity-like structure which is capable of confining light and enhancing the local intensity in a similar manner to that of plasmonic devices[46]. The enhancement of the intensity is obtained by dramatically reducing the group velocity inside the structure while eliminating reflections at its edges. In addition, the resonance wavelength of this “cavity mode” is independent of the structure length (number of resonators) while its effective photon lifetime increases with its length.
The structure studied here exhibits some similarity to the photonic speed bump structure[46]. Both structures utilize impedance matching techniques to slow down light at its entrance and re-accelerate at the exit in order to obtain field enhancement and cavity-mode-like intensity distribution. There are, however, many differences. The photonic speed bump structure matches the impedance of the input waveguide to that of a slow-light waveguide mode located close to the edge of the waveguide stop-band. As a result, the device is highly dispersive and would introduce distortions when excited by a pulse. The finite CROW structure studied here matches the impedance of the I/O waveguide to that of a Bloch mode located at the center of the CROW’s passband, where dispersion is minimal. Consequently, our structure can accommodate the broader bandwidth of a pulse without introducing substantial distortions. Another clear difference between the finite CROW and the photonic speed bump is the spectral transmission. While the CROW serves as a bandpass filter, the photonic speed bump exhibits a high-pass filter response which stem directly from the cutoff frequency of the PhC waveguide.
It should be noted that the finite CROW structure discussed here is design to be impedance matched to the frequency with the smallest group index (fastest group velocity) within the passband. This is in contrast to the photonic speed bump structure where the impedance is matches at a frequency at the edge of the passband where the group index is largest. While this may seem as a drawback, it should be recalled that effective group index in the CROW in determined by the coupling coefficient, κ, between the adjacent resonators which can be further reduced in order to obtain larger group index. Moreover, as the main limiting factor of SLWs is essentially the propagation loss (which increases rapidly with the ng[50,54-57]), the largest attainable (practical) group index is essentially independent of the choice of frequency. We emphasize that the CROW can be impedance matched also to Bloch mode located near the edges of the Brillouin zone [Fig. 1(b)]. However, as noted above, such configuration would not necessarily yield a more useful device.
The tolerances analysis described in Section 5 indicates that the range of coupling coefficients at which good impedance matching is obtained is quite narrow (see Fig. 9). In particular, the impedance matching conditions are sensitive to errors in the coupling coefficients located towards the inner part of the I/O sections. In the example presented here, the inner coupling coefficient, κ2, should not deviate from the target value (κ2 = 2x10−3) by more than 10%. Although such requirement may seem strict, there are several technological platforms which can provide the necessary tolerances. More specially, the surface nanoscale axial photonics (SNAP) concept, which was recently introduced by Sumetsky and associates[58-62], can provides the necessary accuracy and fabrication tolerances for obtaining impedance matching in coupled-cavities structures.
In terms of computation complexity, extremely high impedance matching was obtained by utilizing I/O sections with only three optimized coupling coefficients (Γ<10−10 for the coupling coefficients listed in raw 3 in Table 1). The optimization procedure is simple and fast, taking approximately 10 seconds on a regular laptop.
The introduction of the I/O sections at the ends of the finite CROW affects its spectral transmission properties and the field distribution in the structure. Nevertheless, while the spectral transmission can be made flat by achieving reasonable impedance matching, the field distribution is more sensitive to the mismatch as can be seen in Fig. 8. This sensitivity must be considered when utilizing CROW structures for practical applications, particularly for enhancing nonlinear processes which are sensitive to the local intensity profile.
Funding
Israel Science Foundation 949/14.
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