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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 8 — Apr. 9, 2012
  • pp: 9249–9263
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Designing coupled-resonator optical waveguides based on high-Q tapered grating-defect resonators

Hsi-Chun Liu and Amnon Yariv  »View Author Affiliations


Optics Express, Vol. 20, Issue 8, pp. 9249-9263 (2012)
http://dx.doi.org/10.1364/OE.20.009249


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Abstract

We present a systematic design of coupled-resonator optical waveguides (CROWs) based on high-Q tapered grating-defect resonators. The formalism is based on coupled-mode theory where forward and backward waveguide modes are coupled by the grating. Although applied to strong gratings (periodic air holes in single-mode silicon-on-insulator waveguides), coupled-mode theory is shown to be valid, since the spatial Fourier transform of the resonant mode is engineered to minimize the coupling to radiation modes and thus the propagation loss. We demonstrate the numerical characterization of strong gratings, the design of high-Q tapered grating-defect resonators (Q>2 × 106, modal volume = 0.38·(λ/n)3), and the control of inter-resonator coupling for CROWs. Furthermore, we design Butterworth and Bessel filters by tailoring the numbers of holes between adjacent defects. We show with numerical simulation that Butterworth CROWs are more tolerant against fabrication disorder than CROWs with uniform coupling coefficient.

© 2012 OSA

1. Introduction

A coupled-resonator optical waveguide (CROW) consists of a sequence of weakly coupled resonators in which light propagates through the coupling between adjacent resonators [1

1. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24(11), 711–713 (1999). [CrossRef] [PubMed]

]. Both the bandwidth and the group velocity are dictated solely by the inter-resonator coupling strength. Such a unique waveguiding mechanism has found applications such as optical delay lines, optical buffers, optical bandpass filters, interferometers, and nonlinear optics [2

2. F. N. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1(1), 65–71 (2007). [CrossRef]

5

5. A. Melloni, F. Morichetti, and M. Martinelli, “Four-wave mixing and wavelength conversion in coupled-resonator optical waveguides,” J. Opt. Soc. Am. B 25(12), C87–C97 (2008). [CrossRef]

].

When the grating is strong, such as periodic air holes in a silicon waveguide, the length of each grating-defect resonator can be as short as a few microns. High density of resonators is important for optical buffers since the delay-bandwidth product is proportional to the number of resonators. CROWs based on such small resonators have been experimentally demonstrated in silicon waveguides [10

10. S. Nishikawa, S. Lan, N. Ikeda, Y. Sugimoto, H. Ishikawa, and K. Asakawa, “Optical characterization of photonic crystal delay lines based on one-dimensional coupled defects,” Opt. Lett. 27(23), 2079–2081 (2002). [CrossRef] [PubMed]

]. However, the major limitation was the intrinsic propagation loss due to radiation [11

11. A. Martínez, J. García, P. Sanchis, F. Cuesta-Soto, J. Blasco, and J. Martí, “Intrinsic losses of coupled-cavity waveguides in planar-photonic crystals,” Opt. Lett. 32(6), 635–637 (2007). [CrossRef] [PubMed]

]. Highly confined modes lead to large spatial Fourier components which are phase-matched with the lossy radiation modes. The resulting low quality factor of the resonators (Q<1000) leads to power decay time constant of approximately 1 picosecond, limiting applications such as optical delay lines. Because of the coupling to the higher-order (radiation) modes, coupled-mode equations which consider only forward and backward guided modes are no longer valid. Consequently, the design of grating resonators based on strong gratings usually relies on three-dimensional simulation of the entire structures.

When CROWs are based on high-Q resonators, the coupling to radiation modes is negligible, so the coupled-mode equations are valid. The coupled-mode formalism which we will present in Section 2 is useful for the analysis and design of grating-defect resonators and CROWs. After showing the systematic design of high-Q tapered resonators in Section 3, we will demonstrate the control of inter-resonator coupling for CROWs in Section 4 and filter design based on tailoring the coupling coefficients along a CROW in Section 5. Filter design not only optimizes the transmission and dispersive properties of CROWs but also improves the tolerance of CROWs against fabrication disorder, as will be discussed in Section 6. Finally, it is worth mentioning that we design the resonators and CROWs to resonate at the Bragg wavelength of the grating in order to ensure that the resonant wavelength will not change with the number of holes. If the resonant wavelength is not at the Bragg wavelength, especially near the band edge, an extra phase section will be required when cascading resonators for CROWs, as will be shown in the appendix.

2. Coupled-mode formalism for grating CROWs

A Bragg grating is a periodic perturbation to a waveguide. A grating with a period Λ couples counter-propagating modes with a propagation constant β if the phase-matching condition is satisfied, i.e. 2π/Λ = 2β. The coupled-mode equations relating the amplitudes of the forward mode a and the backward mode b are given by [17

17. A. Yariv and P. Yeh, Photonics, 6th ed. (Oxford University Press, 2007).

]
dadz=iδa+iκg*(z)bdbdz=iδbiκg(z)a,
(1)
where δββB is the detuning from the Bragg condition, βB≡π/Λ, and κg(z) is the coupling coefficient of the grating. The absolute value and phase of κg(z) represent the strength and phase of the grating respectively. κg(z) is a constant for a uniform grating. If the grating strength is tapered, |κg(z)| varies along the grating. For a grating structure distributed between z = 0 and z = Lg and an input a(0) = 1 from z = −∞, the general approach of solving the transmission and the field distribution is as follows: (i) Set the boundary condition at the output as a(Lg) = 1 and b(Lg) = 0 (no input from z = ∞). (ii) Propagate a and b from z = Lg back to z = 0 analytically or numerically, using Eq. (1). (iii) Divide the resulting a(z) and b(z) by a(0) to recover the input amplitude a(0) = 1.

A grating-defect CROW consists of a sequence of defects, where adjacent defect modes interact with each other via their evanescent fields, as shown in Fig. 1(b). κg(z) alternates between κg and −κg. The inter-resonator coupling is determined by the spacing between defects, denoted as L. For a grating structure consisting of only QWS defects (i.e. a real κg(z)), the field distribution at δ = 0 for an input a(0) = 1 can be derived as
a(z)=a(Lg)cosh(zLgκg(z')dz')
(2a)
and
b(z)=ia(Lg)sinh(zLgκg(z')dz'),
(2b)
where a(Lg)=1/cosh(0Lgκg(z')dz') is the transmitted amplitude. We consider an inter-resonator spacing L and L1 = LN + 1 = L/2 at the boundary, which guarantees cosh(0Lgκg(z')dz')=1 and thus unity transmission a(Lg) = 1. The energy stored in the grating, Estored=0Lg(|a|2+|b|2)dz/vg, can be derived as Lg[sinh(κgL)/κgL]/vg, where vg is the group velocity of the waveguide mode. Since the input power |a(0)|2 is 1, the group velocity in the grating-defect CROW is given by
vg,CROW=LgEstored=vgκgLsinh(κgL).
(3)
The slowing factor is a function of κgL.

The time-domain inter-resonator coupling coefficient, denoted as κ, can be obtained by solving the frequency splitting of two coupled defect resonators separated by L, which results in
κ=κgvgexp(κgL).
(4)
with the assumption of exp(−κgL)1 [9

9. H. A. Haus and Y. Lai, “Theory of cascaded quarter wave shifted distributed feedback resonators,” IEEE J. Quantum Electron. 28(1), 205–213 (1992). [CrossRef]

, 18

18. H. C. Liu and A. Yariv, “Synthesis of high-order bandpass filters based on coupled-resonator optical waveguides (CROWs),” Opt. Express 19(18), 17653–17668 (2011). [CrossRef] [PubMed]

]. The group velocity at resonant frequency is thus vg,CROW=2κL=vg[2κgL/exp(κgL)], which, in the limit of exp(−κgL)1, is the same as Eq. (3). Finally, we consider a single grating-defect resonator, as shown in Fig. 1(a). The resonator is coupled to input and output waveguides via gratings of length L. The loss rate of the mode amplitude due to coupling to each waveguide is given by 1/τe=κgvgexp(2κgL) [18

18. H. C. Liu and A. Yariv, “Synthesis of high-order bandpass filters based on coupled-resonator optical waveguides (CROWs),” Opt. Express 19(18), 17653–17668 (2011). [CrossRef] [PubMed]

]. The total loss rate is 1/τ=2/τe, and the quality factor of the resonator is
Q=12ωτ=ωexp(2κgL)4κgvg.
(5)
Q is an exponential function of L.

Figures 2(a)
Fig. 2 Spectra of (a) transmission and (b) group delay of N = 10 grating-defect CROWs with inter-defect spacing L = 200 μm (blue) and L = 300 μm (green). κg = 0.01/μm.
and 2(b) show the spectra of transmission and group delay of 10-resonator grating-defect CROWs with L = 200 μm and L = 300 μm respectively. We choose a group index of 4 and a weak grating with κg = 0.01/μm. The lengths of the first and last grating sections, L1 and LN + 1, are chosen to be L/2 to match the CROW section to the waveguides [18

18. H. C. Liu and A. Yariv, “Synthesis of high-order bandpass filters based on coupled-resonator optical waveguides (CROWs),” Opt. Express 19(18), 17653–17668 (2011). [CrossRef] [PubMed]

]. According to Eq. (4), the coupling coefficients of the two CROWs differ by a factor of exp(κgΔL) = e, which agrees with the bandwidths and the group delay shown in Fig. 2.

3. High-Q tapered grating-defect resonators in silicon waveguides

3.1 Numerical characterization of Bragg gratings

Given a grating with a hole radius r and a period Λ, we can determine its Bragg wavelength and κg by simulating the transmission and reflection of the grating. For a grating with a constant κg between z = 0 and z = L and an input a(0) = 1, the phase of the reflected mode b(0) can be derived from Eq. (1) as θr = −π/2−sin−1(δ/κg) if L is sufficiently long. The Bragg wavelength (δ = 0) can be obtained at the condition θr = −π/2. κg can be determined from the transmitted power at Bragg wavelength, |a(L)|2 = 1/cosh2(κgL).

Figure 3(b) shows the calculated κg(r) and Λ(r) for a Bragg wavelength of 1570 nm. Since the area of holes is proportional to r2, κg(r) is quadratic at small radii. At larger radii, κg(r) becomes linear and the slope starts decreasing, since κg corresponds to the first-order Fourier component of the grating. On the other hand, the perturbation of the propagation constant corresponds to the constant term of the Fourier components, so Λ(r) is nearly a quadratic curve. Note that for a hole radius of 100 nm, κg is 1.49/μm, which is 16% of the propagation constant and thus corresponds to a very strong grating.

3.2. Design of high-Q tapered grating-defect resonators

We consider grating-defect resonators with 4 and 6 tapered holes respectively. The objective is to find an α which minimizes the radiation loss. Figure 5(b) shows the field distribution on one side of the defect for α = 0, 0.55, and 1 based on their κg(z). To estimate the radiation loss, we integrate the spatial Fourier spectrum over the radiation continuum. Figure 5(c) shows the portion of energy in the radiation continuum, ηrad, as a function of α for 4 and 6 tapered holes respectively. Tapering the grating reduces ηrad by more than 3 orders of magnitude. The effect of 6 tapered holes is better than that of 4 tapered holes. The minimum of ηrad occurs at α = 0.48 and 0.55 for 4 and 6 tapered holes respectively. This result shows that tapered gratings with α~0.5, corresponding to a field distribution of approximately exp[−(Δz)3/2], are better than linear tapers with Gaussian distribution. Their field distributions are shown in Fig. 5(b), and the spatial Fourier spectra are shown in Fig. 5(d). Compared to α = 1, while the spectrum of α = 0.55 is larger at higher frequencies, it is an order of magnitude smaller within the radiation continuum. As a result, the taper profile with α = 0.55 constitutes an optimal design.

4. CROWs based on tapered grating-defect resonators

Grating CROWs are formed by cascading the high-Q grating-defect resonators designed in Section 3. Figure 6
Fig. 6 Schematic drawing of the first two resonators of a grating-defect CROW.
shows the first two resonators of a grating CROW. The inter-resonator coupling is controlled via the number of holes between neighboring defects, denoted as m. m includes the number of tapered holes (2nt) and the number of regular holes (nreg). In the appendix, we will show that when cascading two symmetric grating resonators with external quality factors Q1 and Q2 respectively, the coupling coefficient is given by
κ=ω4Q1Q2.
(7)
If the resonant frequency is not at the Bragg wavelength, an additional waveguide section between two resonators is required for appropriate coupling in CROWs (see appendix). We have shown in Eq. (5) that Q is proportional to exp(2κgL). For a tapered resonator, the relation is modified as
Qexp[20Lκg(z)dz]=exp[2i=1nκg,iΛi]=exp[2i=1ntκg,iΛi]exp[2nregκgΛ],
(8)
which breaks down into the contribution of each hole in the tapered region and the regular grating, respectively. Therefore, Q can be written as
Q=Q0a2nreg,
(9)
where Q0 is the quality factor of a resonator with only the tapered region (no regular hole) and a≡exp(κgΛ) = 1.849. We fit the curve in Fig. 5(e) (α = 0.55, nt = 6) with Eq. (9) and obtain Q0 = 548 and a = 1.848. If we cascade two resonators with nreg,1 and nreg,2 respectively, we obtain the inter-resonator coupling coefficient given by
κ=ω4Q1Q2=ω4Q0anreg,
(10)
where nreg = nreg,1 + nreg,2 is the total number of regular holes between two defects. For the first or last resonators, the external loss rate to the waveguides is given by (also shown in the appendix)
1τe=ω4Q=ω4Q0a2nreg,
(11)
where nreg is the number of regular holes in the first or last grating section. To match between the CROW and the waveguides, we require κ = 1/τe [18

18. H. C. Liu and A. Yariv, “Synthesis of high-order bandpass filters based on coupled-resonator optical waveguides (CROWs),” Opt. Express 19(18), 17653–17668 (2011). [CrossRef] [PubMed]

]. Therefore, the number of holes in the first and last sections is one half of the number of holes in the middle sections, i.e. m1 = mN+1 = m/2.

Figures 7(a)
Fig. 7 Spectra of (a) transmission and (b) group delay of 10-resonator grating-defect CROWs with m = 12, 14, and 16 respectively.
and 7(b) show the spectra of transmission and group delay of N = 10 CROWs for m = 12, 14, and 16 respectively (nreg = 0, 2, and 4). The band center is at 1569.2 nm. Both the bandwidth and the group delay are dictated by m. The bandwidth is equal to 4κ and the group delay at the band center is given by N/(2κ). Adding two holes between defects results in a factor of a2 = 3.415 in κ, which agrees with simulation. Note that the maximal transmission for m = 12 is only 0.955. This is due to the strong index contrast between the waveguide and the grating section, which scatters light to the radiation modes. The strong index contrast can be reduced by tapering the grating at the input [13

13. A. R. M. Zain, N. P. Johnson, M. Sorel, and R. M. De La Rue, “Ultra high quality factor one dimensional photonic crystal/photonic wire micro-cavities in silicon-on-insulator (SOI),” Opt. Express 16(16), 12084–12089 (2008). [CrossRef] [PubMed]

]. For larger m, the radiation loss increases due to the longer delay. The transmitted power can be written as exp(−ωτ/Qi), where τ is the group delay. The delay for m = 16 is 106.3 ps, which leads to a transmitted power of 0.943. Including the scattering loss at the input, the transmission drops to about 0.9, which agrees with the simulation.

5. Filter design based on grating CROWs

High-order bandpass filters with optimized transmission and dispersive properties can be realized in CROWs if the coupling coefficients are allowed to take on different values. For example, Butterworth filters exhibit maximally flat transmission, while Bessel filters possess maximally flat group delay. For a desired filter response, the coupling coefficients which determine the transfer function of CROWs can be derived [18

18. H. C. Liu and A. Yariv, “Synthesis of high-order bandpass filters based on coupled-resonator optical waveguides (CROWs),” Opt. Express 19(18), 17653–17668 (2011). [CrossRef] [PubMed]

, 20

20. V. Van, “Circuit-based method for synthesizing serially coupled microring filters,” J. Lightwave Technol. 24(7), 2912–2919 (2006). [CrossRef]

]. Table 1

Table 1. Coupling Coefficients of N = 10 Butterworth and Bessel CROWs

table-icon
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lists the couplingcoefficients of N = 10 Butterworth and Bessel filters respectively. These coupling coefficients are normalized to a chosen bandwidth parameter B.

In grating-defect CROWs, the coupling coefficients are translated to the numbers of holes based on Eqs. (10) and (11). However, these numbers of holes are in general non-integers. Table 2

Table 2. Numbers of Regular Holes of N = 10 Butterworth and Bessel CROWs

table-icon
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lists the numbers of regular holes corresponding to the two filters in Table 1. The bandwidth parameter B is chosen as 2·κ(nreg = 4) so that its bandwidth is equal to those of CROWs with uniform coupling and nreg = 4 (m = 16 in Fig. 7). Since κ is an exponential function of nreg, we can add an arbitrary Δnreg in order to change the bandwidth without having to rederive all the nreg. A non-integer nreg can be realized by an integer number nint = nreg of identical holes which are equivalent to a fraction γ = nreg/nreg of a regular hole. For example, 3.6 regular holes are equivalent to 4 × 0.9 holes. This can be seen in
anreg=exp[nregκgΛ]=exp[nintκg(r)Λ(r)].
(12)
Therefore, we need to determine the hole radius r whose κg(r)Λ(r) is equal to γκgΛ. This can be done by interpolating the curve of κgΛ versus r. For example, the radius of a γ = 0.9 hole is 90.1 nm. If nreg is negative, we can reduce the hole sizes starting from the outermost tapered holes. An alternative way is to choose a resonator with fewer tapered holes, such as the resonators with nt = 4 designed in Section 3.

Figures 8(a)
Fig. 8 Spectra of transmission and group delay of (a) an N = 10 Butterworth grating CROW and (b) an N = 10 Bessel grating CROW.
and 8(b) show the spectra of transmission and group delay of an N = 10 Butterworth CROW and an N = 10 Bessel CROW, respectively. The values of transmission at the band center are both 0.916, indicating a small intrinsic loss due to the large group delay in addition to the scattering loss at the input. Substracting the scattering loss which corresponds to a transmission of 0.955, the intrinsic Q can be obtained from the intrinsic loss and the groupdelay and is determined to be 2.34 × 106 and 1.97 × 106 for Butterworth and Bessel CROWs, respectively. Therefore, the tailoring of coupling coefficients does not degrade the Q of the resonators. In practice, the Q of the resonators may degrade due to imperfection of fabrication. With larger intrinsic loss, the transmission spectrum of Butterworth CROWs may be distorted since the loss is proportional to the group delay. A predistortion technique can be applied to pre-compensate for the distortion provided that the Q can be estimated and is uniform over the CROW [18

18. H. C. Liu and A. Yariv, “Synthesis of high-order bandpass filters based on coupled-resonator optical waveguides (CROWs),” Opt. Express 19(18), 17653–17668 (2011). [CrossRef] [PubMed]

]. Since the group delay of Bessel CROWs is flat within the bandwidth, the transmission spectrum is not distorted by uniform resonator loss.

6. Effect of fabrication disorder on grating-defect CROWs

The major limitation in the experiment of CROWs has been the unavoidable fabrication imperfection which leads to disorder in the resonant frequency of each resonator and the coupling coefficients. The disorder distorts the CROW response and limits the minimum bandwidth which CROWs can be designed with. The yield of CROWs drops as the number of resonators is increased or as the CROW bandwidth is decreased. The effect of disorder on CROWs has been investigated in the literature [21

21. C. Ferrari, F. Morichetti, and A. Melloni, “Disorder in coupled-resonator optical waveguides,” J. Opt. Soc. Am. B 26(4), 858–866 (2009). [CrossRef]

, 22

22. S. Mookherjea and A. Oh, “Effect of disorder on slow light velocity in optical slow-wave structures,” Opt. Lett. 32(3), 289–291 (2007). [CrossRef] [PubMed]

]. In this section we analyze the disorder effect on grating-defect CROWs.

The oscillations in the passband can be reduced by applying Butterworth filter design. Figures 9(d) and 9(e) show the simulated spectra of Butterworth CROWs with the same bandwidth and the same δω as in Figs. 9(b) and 9(c). The average oscillation amplitudes are reduced to 4.4 dB and 1.1 dB, respectively. The disorder is a perturbation to an ideal CROW and can be taken as scatterers in the CROW. For CROWs with uniform coupling coefficient, the boundaries between the CROW and the waveguides cause reflection and form a cavity. Disorder in uniform CROWs can be thought of as scatterers in a cavity which can cause large oscillations. In a Butterworth CROW, the coupling coefficients are tailored to adiabatically transform between the CROW and the waveguides, thereby removing the cavity and reducing the amplitude of oscillations.

7. Conclusion

We have demonstrated a systematic approach to design high-Q tapered grating-defect resonators, control the inter-resonator coupling, and design high-order grating CROW filters. The formalism based on coupled-mode theory is valid in strong gratings, with the help of 3D simulations for the characterization of gratings. The optimized Q of 2.16 × 106 is an order of magnitude higher than the theoretical Q of grating-defect resonators designed in the literature. Based on these high-Q resonators, CROWs which are shorter than 60 μm exhibit a group delay of more than 100 ps while maintaining a transmission of 0.9. The control of inter-resonator coupling via the number of holes provides a convenient way of designing coupled-resonator structures. Furthermore, we demonstrated the design of tenth-order Butterworth and Bessel filters which possess maximally flat transmission and group delay, respectively. Besides flat transmission, Butterworth CROWs are more robust against fabrication disorder compared to CROWs with uniform coupling coefficient.

The grating-defect CROWs designed in this paper are attractive for their small footprints, high quality factors, and their natural coupling to input and output waveguides. The coupled-mode formalism developed in this work can be further applied to other types of strong grating structures to minimize the coupling to radiation modes and reliably calculate the transfer function of grating structures when the coupling to radiation modes is negligible.

Appendix. Inline coupling of resonators

Inline resonators are, by definition, fabricated, cascaded, and coupled in a single waveguide. The objective of this appendix is to derive the inter-resonator coupling coefficient as a function of individual quality factors and the length of the coupling waveguide.

Figure 10(a)
Fig. 10 Schematic drawings and the corresponding grating structures of (a) a symmetric resonator and (b) inline coupling of two resonators.
shows a symmetric inline resonator, i.e. the coupling to the waveguides is equally strong on both sides. In a symmetric grating-defect resonator, the number of holes on both sides of the defect is equal. The time-domain coupled-mode equations of the structure in Fig. 10(a) are
dadt=(iω1τi2τe)aiμsinsout=iμasr=siniμa,
(13)
where a is the resonator mode amplitude, sin, sr, and sout are the input, reflected, and transmitted mode amplitudes respectively, ω is the resonant frequency, 1/τi and 1/τe are the intrinsic loss and the external loss to each waveguide respectively, and μ is the waveguide-resonator coupling. It can be shown that μ=2/τe using conservation of energy. The quality factor of the resonator is given by Q = ωτ/2, where 1/τ = 1/τi + 2/τe is the total loss rate. In the regime where intrinsic loss is negligible (the linear region in Fig. 5(d)), Q = ωτe/4. Therefore, we obtain 1/τe = ω/(4Q) if Q is given.

In Fig. 10(b), we consider two inline resonators cascaded in a waveguide. The inter-resonator coupling is via the coupling waveguide of length d. The coupled-mode equations of the two resonators and the coupling waveguide are given by
da1dt=(iω11τe1)a1iμ1seiθda2dt=(iω21τe2)a2iμ2s+eiθs+=seiθiμ1a1s=s+eiθiμ2a2.
(14)
The notations are shown in Fig. 10(b). θ is the phase accumulated in the propagation. The resonant frequencies and the external losses of the two resonators can be different in general. The intrinsic losses and the coupling to the other resonators or waveguides have been ignored and can be added to the equations. Combining the last two equations of Eq. (14), s+ and s- can be expressed as linear combinations of a1 and a2, and Eq. (14) can be rewritten as coupled-mode equations of two directly coupled resonators (also shown in Fig. 10(b)):
da1dt=i(ω1cotθτe1)a1icscθτe1τe2a2da2dt=i(ω2cotθτe2)a2icscθτe1τe2a1.
(15)
The coupling leads to detuning of resonant frequencies and a coupling coefficient which depend on the round-trip phase of the coupling waveguide cavity, 2θ:
Δω1,2=cotθτe1,2κ=cscθτe1τe2.
(16)
When designing CROWs, we require identical resonant frequencies. If the frequency detuning Δω is nonzero, the resonators in a CROW may experience different frequency detuning. For example, the frequencies of the first and last resonators are less detuned since they only couple to one resonator, while the other resonators have two neighbors. Therefore, we require Δω = 0, which corresponds to 2θ = π, a totally destructive interference in the coupling cavity. 2θ = π also leads to a minimal coupling coefficient. The cavity round-trip phase includes the phase of reflection from the grating and the propagation phase in the cavity. At the Bragg wavelength, the reflection phase θr = −π/2−sin−1(δ/κg) is −π/2. Therefore, the round-trip phase with d = 0 is already π. This is an important reason why we choose to work at the Bragg wavelength. If the resonance is near the band edge, an additional coupling waveguide of length d is required to satisfy a round-trip phase of π [24

24. H. C. Liu, C. Santis, and A. Yariv, “Coupled-resonator optical waveguides (CROWs) based on grating resonators with modulated bandgap,” in Slow and Fast Light (Toronto, Canada, 2011), p. SLTuB2.

]. The coupling coefficient for the grating-defect resonators in this paper is thus given by
κ=1τe1τe2=ω4Q1Q2,
(17)
which proves Eq. (7).

Acknowledgments

The authors thank Christos Santis for helpful discussions. This work was supported by National Science Foundation and The Army Research Office.

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H. A. Haus and Y. Lai, “Theory of cascaded quarter wave shifted distributed feedback resonators,” IEEE J. Quantum Electron. 28(1), 205–213 (1992). [CrossRef]

10.

S. Nishikawa, S. Lan, N. Ikeda, Y. Sugimoto, H. Ishikawa, and K. Asakawa, “Optical characterization of photonic crystal delay lines based on one-dimensional coupled defects,” Opt. Lett. 27(23), 2079–2081 (2002). [CrossRef] [PubMed]

11.

A. Martínez, J. García, P. Sanchis, F. Cuesta-Soto, J. Blasco, and J. Martí, “Intrinsic losses of coupled-cavity waveguides in planar-photonic crystals,” Opt. Lett. 32(6), 635–637 (2007). [CrossRef] [PubMed]

12.

P. Velha, E. Picard, T. Charvolin, E. Hadji, J. C. Rodier, P. Lalanne, and D. Peyrade, “Ultra-high Q/V Fabry-Perot microcavity on SOI substrate,” Opt. Express 15(24), 16090–16096 (2007). [CrossRef] [PubMed]

13.

A. R. M. Zain, N. P. Johnson, M. Sorel, and R. M. De La Rue, “Ultra high quality factor one dimensional photonic crystal/photonic wire micro-cavities in silicon-on-insulator (SOI),” Opt. Express 16(16), 12084–12089 (2008). [CrossRef] [PubMed]

14.

E. Kuramochi, H. Taniyama, T. Tanabe, K. Kawasaki, Y. G. Roh, and M. Notomi, “Ultrahigh-Q one-dimensional photonic crystal nanocavities with modulated mode-gap barriers on SiO2 claddings and on air claddings,” Opt. Express 18(15), 15859–15869 (2010). [CrossRef] [PubMed]

15.

Q. M. Quan, P. B. Deotare, and M. Loncar, “Photonic crystal nanobeam cavity strongly coupled to the feeding waveguide,” Appl. Phys. Lett. 96(20), 203102 (2010). [CrossRef]

16.

Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425(6961), 944–947 (2003). [CrossRef] [PubMed]

17.

A. Yariv and P. Yeh, Photonics, 6th ed. (Oxford University Press, 2007).

18.

H. C. Liu and A. Yariv, “Synthesis of high-order bandpass filters based on coupled-resonator optical waveguides (CROWs),” Opt. Express 19(18), 17653–17668 (2011). [CrossRef] [PubMed]

19.

J. W. Mu and W. P. Huang, “Simulation of three-dimensional waveguide discontinuities by a full-vector mode-matching method based on finite-difference schemes,” Opt. Express 16(22), 18152–18163 (2008). [CrossRef] [PubMed]

20.

V. Van, “Circuit-based method for synthesizing serially coupled microring filters,” J. Lightwave Technol. 24(7), 2912–2919 (2006). [CrossRef]

21.

C. Ferrari, F. Morichetti, and A. Melloni, “Disorder in coupled-resonator optical waveguides,” J. Opt. Soc. Am. B 26(4), 858–866 (2009). [CrossRef]

22.

S. Mookherjea and A. Oh, “Effect of disorder on slow light velocity in optical slow-wave structures,” Opt. Lett. 32(3), 289–291 (2007). [CrossRef] [PubMed]

23.

H. C. Liu, C. Santos, and A. Yariv, “Coupled-resonator optical waveguides (CROWs) based on tapered grating-defect resonators,” in CLEO (San Jose, USA, 2012).

24.

H. C. Liu, C. Santis, and A. Yariv, “Coupled-resonator optical waveguides (CROWs) based on grating resonators with modulated bandgap,” in Slow and Fast Light (Toronto, Canada, 2011), p. SLTuB2.

OCIS Codes
(140.4780) Lasers and laser optics : Optical resonators
(230.4555) Optical devices : Coupled resonators
(230.5298) Optical devices : Photonic crystals
(130.7408) Integrated optics : Wavelength filtering devices

ToC Category:
Integrated Optics

History
Original Manuscript: February 7, 2012
Revised Manuscript: March 30, 2012
Manuscript Accepted: April 2, 2012
Published: April 6, 2012

Citation
Hsi-Chun Liu and Amnon Yariv, "Designing coupled-resonator optical waveguides based on high-Q tapered grating-defect resonators," Opt. Express 20, 9249-9263 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-8-9249


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References

  1. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett.24(11), 711–713 (1999). [CrossRef] [PubMed]
  2. F. N. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics1(1), 65–71 (2007). [CrossRef]
  3. A. Melloni, F. Morichetti, C. Ferrari, and M. Martinelli, “Continuously tunable 1 byte delay in coupled-resonator optical waveguides,” Opt. Lett.33(20), 2389–2391 (2008). [CrossRef] [PubMed]
  4. F. N. Xia, M. Rooks, L. Sekaric, and Y. Vlasov, “Ultra-compact high order ring resonator filters using submicron silicon photonic wires for on-chip optical interconnects,” Opt. Express15(19), 11934–11941 (2007). [CrossRef] [PubMed]
  5. A. Melloni, F. Morichetti, and M. Martinelli, “Four-wave mixing and wavelength conversion in coupled-resonator optical waveguides,” J. Opt. Soc. Am. B25(12), C87–C97 (2008). [CrossRef]
  6. J. K. S. Poon, L. Zhu, G. A. DeRose, and A. Yariv, “Transmission and group delay of microring coupled-resonator optical waveguides,” Opt. Lett.31(4), 456–458 (2006). [CrossRef] [PubMed]
  7. M. Notomi, E. Kuramochi, and T. Tanabe, “Large-scale arrays of ultrahigh-Q coupled nanocavities,” Nat. Photonics2(12), 741–747 (2008). [CrossRef]
  8. T. J. Karle, D. H. Brown, R. Wilson, M. Steer, and T. F. Krauss, “Planar photonic crystal coupled cavity waveguides,” IEEE J. Sel. Top. Quantum Electron.8(4), 909–918 (2002). [CrossRef]
  9. H. A. Haus and Y. Lai, “Theory of cascaded quarter wave shifted distributed feedback resonators,” IEEE J. Quantum Electron.28(1), 205–213 (1992). [CrossRef]
  10. S. Nishikawa, S. Lan, N. Ikeda, Y. Sugimoto, H. Ishikawa, and K. Asakawa, “Optical characterization of photonic crystal delay lines based on one-dimensional coupled defects,” Opt. Lett.27(23), 2079–2081 (2002). [CrossRef] [PubMed]
  11. A. Martínez, J. García, P. Sanchis, F. Cuesta-Soto, J. Blasco, and J. Martí, “Intrinsic losses of coupled-cavity waveguides in planar-photonic crystals,” Opt. Lett.32(6), 635–637 (2007). [CrossRef] [PubMed]
  12. P. Velha, E. Picard, T. Charvolin, E. Hadji, J. C. Rodier, P. Lalanne, and D. Peyrade, “Ultra-high Q/V Fabry-Perot microcavity on SOI substrate,” Opt. Express15(24), 16090–16096 (2007). [CrossRef] [PubMed]
  13. A. R. M. Zain, N. P. Johnson, M. Sorel, and R. M. De La Rue, “Ultra high quality factor one dimensional photonic crystal/photonic wire micro-cavities in silicon-on-insulator (SOI),” Opt. Express16(16), 12084–12089 (2008). [CrossRef] [PubMed]
  14. E. Kuramochi, H. Taniyama, T. Tanabe, K. Kawasaki, Y. G. Roh, and M. Notomi, “Ultrahigh-Q one-dimensional photonic crystal nanocavities with modulated mode-gap barriers on SiO2 claddings and on air claddings,” Opt. Express18(15), 15859–15869 (2010). [CrossRef] [PubMed]
  15. Q. M. Quan, P. B. Deotare, and M. Loncar, “Photonic crystal nanobeam cavity strongly coupled to the feeding waveguide,” Appl. Phys. Lett.96(20), 203102 (2010). [CrossRef]
  16. Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature425(6961), 944–947 (2003). [CrossRef] [PubMed]
  17. A. Yariv and P. Yeh, Photonics, 6th ed. (Oxford University Press, 2007).
  18. H. C. Liu and A. Yariv, “Synthesis of high-order bandpass filters based on coupled-resonator optical waveguides (CROWs),” Opt. Express19(18), 17653–17668 (2011). [CrossRef] [PubMed]
  19. J. W. Mu and W. P. Huang, “Simulation of three-dimensional waveguide discontinuities by a full-vector mode-matching method based on finite-difference schemes,” Opt. Express16(22), 18152–18163 (2008). [CrossRef] [PubMed]
  20. V. Van, “Circuit-based method for synthesizing serially coupled microring filters,” J. Lightwave Technol.24(7), 2912–2919 (2006). [CrossRef]
  21. C. Ferrari, F. Morichetti, and A. Melloni, “Disorder in coupled-resonator optical waveguides,” J. Opt. Soc. Am. B26(4), 858–866 (2009). [CrossRef]
  22. S. Mookherjea and A. Oh, “Effect of disorder on slow light velocity in optical slow-wave structures,” Opt. Lett.32(3), 289–291 (2007). [CrossRef] [PubMed]
  23. H. C. Liu, C. Santos, and A. Yariv, “Coupled-resonator optical waveguides (CROWs) based on tapered grating-defect resonators,” in CLEO (San Jose, USA, 2012).
  24. H. C. Liu, C. Santis, and A. Yariv, “Coupled-resonator optical waveguides (CROWs) based on grating resonators with modulated bandgap,” in Slow and Fast Light (Toronto, Canada, 2011), p. SLTuB2.

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