## Synthesis of high-order bandpass filters based on coupled-resonator optical waveguides (CROWs) |

Optics Express, Vol. 19, Issue 18, pp. 17653-17668 (2011)

http://dx.doi.org/10.1364/OE.19.017653

Acrobat PDF (1242 KB)

### Abstract

We present a filter design formalism for the synthesis of coupled-resonator optical waveguide (CROW) filters. This formalism leads to expressions and a methodology for deriving the coupling coefficients of CROWs for the desired filter responses and is based on coupled-mode theory as well as the recursive properties of the coupling matrix. The coupling coefficients are universal and can be applied to various types of resonators. We describe a method for the conversion of the coupling coefficients to the parameters based on ring resonators and grating defect resonators. The designs of Butterworth and Bessel CROW filters are demonstrated as examples.

© 2011 OSA

## 1. Introduction

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. 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]

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

4. 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]

5. M. Notomi, E. Kuramochi, and T. Tanabe, “Large-scale arrays of ultrahigh-Q coupled nanocavities,” Nat. Photonics **2**(12), 741–747 (2008). [CrossRef]

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

6. R. S. Tucker, P. C. Ku, and C. J. Chang-Hasnain, “Slow-light optical buffers: capabilities and fundamental limitations,” J. Lightwave Technol. **23**(12), 4046–4066 (2005). [CrossRef]

7. 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]

*κ*[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]

8. P. Chak and J. E. Sipe, “Minimizing finite-size effects in artificial resonance tunneling structures,” Opt. Lett. **31**(17), 2568–2570 (2006). [CrossRef] [PubMed]

9. M. Sumetsky and B. J. Eggleton, “Modeling and optimization of complex photonic resonant cavity circuits,” Opt. Express **11**(4), 381–391 (2003). [CrossRef] [PubMed]

10. J. Capmany, P. Muñoz, J. D. Domenech, and M. A. Muriel, “Apodized coupled resonator waveguides,” Opt. Express **15**(16), 10196–10206 (2007). [CrossRef] [PubMed]

*N*-resonator CROW is an

*N*-pole optical filter. The coupling coefficients of CROWs can be chosen to achieve desired properties such as maximally flat transmission (Butterworth filters) or maximally flat group delay (Bessel filters) over a prescribed bandwidth. Optical bandpass filters are important elements in optical signal processing, especially for wavelength-division-multiplexed (WDM) systems [11]. High-order bandpass filters based on coupled ring resonators have been extensively studied and experimentally demonstrated [12

12. B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, E. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. **16**(10), 2263–2265 (2004). [CrossRef]

16. S. J. Xiao, M. H. Khan, H. Shen, and M. H. Qi, “A highly compact third-order silicon microring add-drop filter with a very large free spectral range, a flat passband and a low delay dispersion,” Opt. Express **15**(22), 14765–14771 (2007). [CrossRef] [PubMed]

4. 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]

17. 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]

5. M. Notomi, E. Kuramochi, and T. Tanabe, “Large-scale arrays of ultrahigh-Q coupled nanocavities,” Nat. Photonics **2**(12), 741–747 (2008). [CrossRef]

18. D. Park, S. Kim, I. Park, and H. Lim, “Higher order optical resonant filters based on coupled defect resonators in photonic crystals,” J. Lightwave Technol. **23**(5), 1923–1928 (2005). [CrossRef]

19. R. Orta, P. Savi, R. Tascone, and D. Trinchero, “Synthesis of multiple-ring-resonator filters for optical systems,” IEEE Photon. Technol. Lett. **7**(12), 1447–1449 (1995). [CrossRef]

20. B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. **15**(6), 998–1005 (1997). [CrossRef]

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

22. A. Melloni and M. Martinelli, “Synthesis of direct-coupled-resonators bandpass filters for WDM systems,” J. Lightwave Technol. **20**(2), 296–303 (2002). [CrossRef]

20. B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. **15**(6), 998–1005 (1997). [CrossRef]

20. B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. **15**(6), 998–1005 (1997). [CrossRef]

## 2. Finite-length CROWs

*ω*

_{0}, and the inter-resonator coupling coefficient is

*κ*. For an input frequency

*ω*, the mode amplitude of the

*k*-th resonator can be written as

*a*(

_{k}*t*)exp[

*iωt*], where

*a*(

_{k}*t*) is the slowly-varying amplitude.

**15**(6), 998–1005 (1997). [CrossRef]

23. J. K. S. Poon and A. Yariv, “Active coupled-resonator optical waveguides. I. Gain enhancement and noise,” J. Opt. Soc. Am. B **24**(9), 2378–2388 (2007). [CrossRef]

*ω*≡

*ω*−

*ω*

_{0}and

*κ*is a real number. At steady state,

*da*/

_{k}*dt*= 0, and Eq. (1) becomes a recursive formula for

*a*,

_{k}*γ*is the solution of

*K*is a wave number and Λ is the distance between adjacent resonators. When

*ω*and

*K*represents the dispersion curve of the CROW (shown in Fig. 1(b)), which defines the CROW band within which light can propagate. Frequencies outside the CROW band are forbidden since

*K*is complex and

*, of the end resonators. When a CROW mode propagates to the boundary, the discontinuity between the CROW and the waveguide causes reflection, leading to Fabry-Perot-type oscillations. The reflection can be minimized by choosing 1/τ*

_{e}*properly. Figure 1(d) illustrates the difference between a finite-length and an infinite-length CROW at the boundary. In the case of a finite CROW, the*

_{e}*N*-th resonator is coupled to the output waveguide, while in the case of an infinite CROW, it is coupled to the next resonator. The differential equations for these two cases are respectivelyand

*N*-th resonator cannot tell the termination of the CROW. Since

*τ*=

_{e}*κ*and

*γ*= −

*i*, which corresponds to the center of the CROW band (Δ

*ω*= 0). Figure 1(e) compares the transmission spectra of 10-resonator CROWs with 1/

*τ*=

_{e}*κ*and 1/

*τ*= 0.1

_{e}*κ*respectively. For 1/

*τ*=

_{e}*κ*, the spectrum is flat at the band center. The ripple amplitudes increase at frequencies close to the band edge since the boundary is only matched for Δ

*ω*= 0. For 1/

*τ*= 0.1

_{e}*κ*, the ripples are large over the whole bandwidth. The optimal boundary condition 1/

*τ*=

_{e}*κ*leads to maximally-flat transmission spectrum for finite-size CROWs with uniform coupling coefficients. To further reduce the Fabry-Perot oscillations over the whole CROW band, one can taper the coupling coefficients to adiabatically transform between the CROW mode and the waveguide modes [9

9. M. Sumetsky and B. J. Eggleton, “Modeling and optimization of complex photonic resonant cavity circuits,” Opt. Express **11**(4), 381–391 (2003). [CrossRef] [PubMed]

10. J. Capmany, P. Muñoz, J. D. Domenech, and M. A. Muriel, “Apodized coupled resonator waveguides,” Opt. Express **15**(16), 10196–10206 (2007). [CrossRef] [PubMed]

## 3. Synthesis of bandpass filters based on CROWs

*N*identical resonators and is coupled to input and output waveguides (Fig. 2 ). All the

*N*+ 1 coupling coefficients are allowed to take on different values. The coupled-mode equations obeyed by the complex amplitudes of the

*N*resonators are

*i*Δ

*ωa*for each

_{k}*k*) and two coupling terms to the neighboring resonators, except for the first and last resonators which have only one neighbor. 1/τ

_{e}_{1}and 1/τ

_{e}_{2}are external losses of the first and last resonators due to coupling into the waveguides. The input mode with power

*μ*

_{1}. It can be shown from conservation of energy and time reversal symmetry that

**15**(6), 998–1005 (1997). [CrossRef]

*i*Δ

*ω*with the Laplace variable

*s*, Eq. (5) can be rewritten as

**which contains all the mode amplitudes can be solved by inverting A. The transmitted and reflected amplitude,**

*a**s*and

_{out}*s*are given respectively byandwhere

_{r}*s*with a leading term

*s*. Therefore,

^{N}*N*poles.

### 3.1 N-th order all-pole bandpass filters

*N*poles can be written aswhere

*k*,

*b*

_{N}_{-1},…,

*b*

_{0}are constants. Typical examples of all-pole filters are Butterworth, Chebyshev, and Bessel filters. We substitute

*s*with

*i*(

*ω*−

*ω*

_{0})/

*B*, where

*B*is a bandwidth parameter,

*T*(

*s*) then describes a bandpass filter which is centered at

*ω*

_{0}and of bandwidth scaled by

*B*. Figure 3(a) shows the transmission and group delay spectra of a Butterworth filter and a Bessel filter which feature maximally flat transmission and maximally flat group delay, respectively.

*N*-resonator CROW (Eq. (8)) and the transfer function of an

*N*-th-order all-pole lowpass filter (Eq. (9)) are both all-pole functions with

*N*poles, we present in what follows a formalism for designing the coupling coefficients of CROWs so that the amplitude transmission of the CROW is equal to the desired

*T*(

*s*).

### 3.2 Extraction of coupling coefficients for a desired filter response

*p*

_{1}through

*p*:where

_{N}*p*is the determinant of the buttom-right

_{k}*k*×

*k*submatrix of A (a principal minor of A). For example,

*p*is the determinant of A, and

_{N}*p*

_{1}= A

*. Each*

_{N,N}*p*is a polynomial in

_{k}*s*with a leading term

*s*. Once we know both

^{k}*p*and

_{N}*p*

_{N}_{-1}, all the coupling coefficients

*p*by

_{N}*p*

_{N}_{-1}, the quotient is

*p*

_{N}_{-1}by

*p*

_{N}_{-2}.

### 3.3 Coupling coefficients of Butterworth and Bessel CROWs

*N*= 4 Butterworth filter to demonstrate the extraction of coupling coefficients. The transfer function

*R*(

*s*) is unique and simple. The numerator of

*R*(

*s*) is

*s*. Table 2 lists the extracted coupling coefficients for Butterworth and Bessel filters of

^{N}*N*= 6 and 10. Note that the extracted coefficients are normalized by the bandwidth parameter

*B*, which can be selected to control the bandwidth of the CROW filter.

*ω*= −1 to 1. The coupling coefficients gradually increase toward the two ends of the CROW. This adiabatic transition of the coupling coefficients reduces the reflection at the boundary, and Butterworth CROWs are one of the optimal designs which remove the oscillations in the transmission spectra. Bessel CROWs, which possess maximally flat delay, do not have symmetric coupling coefficients. With the proper choice of

*R*(

*s*) in the power spectral factorization (see Appendix), the coefficients are nearly symmetric.

*N*= 6, 10 and 20. As the order increases, the transmission spectra become flatter in the passband and the roll-off at the band edges is steeper. To see how tolerant the Butterworth CROWs are under random change of the coupling coefficients, Fig. 4(b) shows the transmission spectra of

*N*= 10 Butterworth CROWs whose coupling coefficients are multiplied by a random variable which is uniformly distributed between 0.9 and 1.1. In other words, the standard deviation of the coupling coefficient is 5.8% of its original value. From the transmission spectra of 10 different simulations, the transmission is above 94% over most of the bandwidth.

### 3.4 CROWs with the presence of loss or gain

*s*with s + 1/

*τ*. Therefore, for an all-pole filter response

_{i}*T*(

*s*) designed for lossless resonators, the transmission in the presence of loss is given by

*τ*.

_{i}*τ*in the design to pre-compensate for the left shift due to the loss. This technique is called predistortion of the filters [25

_{i}25. A. M. Prabhu and V. Van, “Predistortion techniques for synthesizing coupled microring filters with loss,” Opt. Commun. **281**(10), 2760–2767 (2008). [CrossRef]

*T*(

*s*) in Eq. (9) are shifted to the right by 1/

*τ*. The constant in the numerator has to be decreased so that the magnitude of the new transfer function

_{i}*T*

_{0}(

*s*) is always smaller than or equal to 1. As a result,

*α*is a constant and is smaller than 1. In the presence of loss 1/

*τ*, the transfer function is

_{i}*α*.

*τ*. The factor

_{i}*α*is greater than 1 and is an amplification factor. Table 2 lists the predistorted design for

*N*= 10 Butterworth CROWs with 1/

*τ*= 0.05

_{i}*B*(lossy) and −0.05

*B*(amplifying) respectively. Their transmission spectra with and without loss/gain are plotted in Figs. 5(a) and 5(b) respectively. Since the group delay is greater at the band edge, frequencies near the band edge experience larger loss and gain. Consequently, the amplitude responses are predistorted accordingly before the loss or gain, as can be seen in Fig. 5. For Bessel filters, since the group delay is almost constant over the bandwidth, the characteristics of the filters remain the same in the presence of small gain or loss.

## 4. CROW filters based on microring resonators

*η*and

*t*are respectively the dimensionless coupling and transmission coefficients over the coupling region. Assuming the coupling is lossless,

26. J. K. S. Poon, J. Scheuer, S. Mookherjea, G. T. Paloczi, Y. Y. Huang, and A. Yariv, “Matrix analysis of microring coupled-resonator optical waveguides,” Opt. Express **12**(1), 90–103 (2004). [CrossRef] [PubMed]

*Waveguide-resonator coupling:*In Fig. 7(c), the two resonators in Fig. 7(a) are both coupled to an output waveguide with an external loss, 1/

*τ*. By writing down the coupled-mode equations of the two resonators and solving for the steady-state solution at

_{e}*ω*=

*ω*

_{0}, the amplitude transmission

*τ*. Figure 7(d) illustrates the corresponding structure for ring resonators. The condition that the transmission is unity at

_{e}*ω*=

*ω*

_{0}can be derived as

*τ*, we first use Eq. (15) to find an inter-resonator coupling

_{e}*η*which corresponds to a coupling coefficient

*η*.

_{i}*κ*in Table 2 to the microring couplings

*η*. The only constraint is that

*κ*does not exceed (

*π*/2)

*f*, or

_{FSR}*ω*/4, the maximal coupling which ring resonators with a free spectral range

_{FSR}*f*can achieve (see Eq. (15)). We consider examples of silicon microring CROWs. The mode index and group index of the silicon waveguides are respectively 2.4 and 4. The ring radius is 30 μm so that one resonant wavelength is at 1570.8 nm, and the free spectral range

_{FSR}*f*is 398 GHz. The bandwidth of the filters can be chosen by setting the bandwidth parameter

_{FSR}*B*. For example, the bandwidth of Butterworth filters is 2

*B*(Fig. 3(a)). If we choose

*B*=

*ω*·0.005, where

_{FSR}*η*for Butterworth and Bessel filters with

*B*=

*ω*·0.005 and

_{FSR}*B*=

*ω*·0.05 are listed in Table 3 .

_{FSR}*κ*in Table 2, calculated using CMT. Figure 8(a) shows the transmission spectra of Butterworth filters with

*B*=

*ω*·0.005. Since

_{FSR}*η*are sufficiently weak (the largest

*η*is 0.338), the two spectra are nearly identical. Figure 8(b) shows the spectra for

*B*=

*ω*·0.05, where the coupling is stronger. Although there are small passband ripples whose amplitude is about 0.0002, the spectrum still closely agrees with the desired response. Therefore, the conversion is valid even when

_{FSR}*η*is as high as 0.852, whereas the same

*κ*would be converted to

*η*= 1.102 using the formula proposed in [20

**15**(6), 998–1005 (1997). [CrossRef]

*κ*at the boundary will increase and might exceed the upper limit of

*κ*, (

*π*/2)

*f*. Therefore, resonators with large

_{FSR}*f*are beneficial. However, for ring resonators with very small radii, say less than 5 μm, the assumption of the transfer matrix formalism that the coupling region is sufficiently long compared to the wavelength is no long valid, and the coupling of modes will be complicated since modes in the opposite direction will also be excited. Figure 8(c) shows the spectra of transmission and group delay for an

_{FSR}*N*= 6 Bessel CROW with

*B*=

*ω*·0.05. Figure 8(d) compares the transmission spectra of Butterworth CROWs with 6 and 20 rings.

_{FSR}## 5. CROW filters based on grating defect resonators

*L*

_{1},

*L*

_{2},…,

*L*

_{N}_{+1}in Fig. 9(b)). These defect resonators form a grating CROW.

4. 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]

27. 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]

17. 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]

*a*and

*b*are the amplitudes of the forward and backward waveguide modes,

*δ*is the detuning of the propagation constant from the Bragg condition of the grating, and

*κ*(

_{g}*z*) is the coupling coefficient of the grating. In a grating CROW, the phase of

*κ*(

_{g}*z*) is shifted by

*π*at each quarter-wave-shifted defect, as shown in Fig. 9(b), while the amplitude remains the same. For an input mode

*a*(−∞) from the left, the field distribution and the transmission of a grating CROW with a given

*κ*(

_{g}*z*) can be solved using Eq. (17) with the boundary condition

*b*(∞) = 0.

*κ*in Table 2 to the lengths

*L*

_{1},

*L*

_{2},…,

*L*

_{N}_{+1}in grating CROWs applies methods similar to those employed in Section 4.

*Inter-resonator coupling:*Fig. 9(c) shows the case of two defects which are separated by a distance

*L*in an infinitely long grating. It was shown in [17

17. 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]

*v*is the group velocity of the waveguide mode. Since the Δ

_{g}*ω*is equal to the coupling coefficient

*κ*of the two resonators,where

*ω*≡

_{g}*κ*.

_{g}v_{g}*Waveguide-resonator coupling:*Fig. 9(d) shows a finite grating with two defects. The external loss 1/

*τ*of the defect modes into the waveguides is controlled by the length

_{e}*L*. The amplitude transmission of this grating at Bragg frequency can be solved, using Eq. (17), as

_{i}*κ*= 1/

*τ*. Therefore,

_{e}*κ*is 0.1/μm, so

_{g}*ω*/(2

_{g}*π*) is 1.19 THz. The transmission and group delay spectra of N = 6 Butterworth and Bessel filters with

*B*=

*ω*·0.05 are plotted in Fig. 10 .

_{FSR}## 6. Conclusion

*B*. Furthermore, predistortion techniques can be applied for the design of lossy or amplifying CROW filters.

## Appendix

*p*

_{N}_{-1}for the purpose of using Eq. (10) to extract the coupling coefficients of CROWs. The formalism is similar to the z-domain digital filter design described in [11,19

19. R. Orta, P. Savi, R. Tascone, and D. Trinchero, “Synthesis of multiple-ring-resonator filters for optical systems,” IEEE Photon. Technol. Lett. **7**(12), 1447–1449 (1995). [CrossRef]

*T*(

*s*) and

*R*(

*s*) are identical, as can be seen in Eqs. (11)a) and (11b). We assume that the numerator of

*R*(

*s*) is

*z*

_{1}through

*z*are the zeros of

_{N}*R*(

*s*). The goal of power spectral factorization is to find

*z*

_{1}through

*z*so that

_{N}*z*is selected from a pair (

_{i}*z*, −

*z**), where

*z*is a complex number, so there are at most 2

*combinations of zeros. In general,*

^{N}*p*(

*s*), and thus

*p*

_{N}_{-1}, are not real. For a filter with a complex

*p*

_{N}_{-1}, the resonant frequencies of the resonators have to be detuned, and Eq. (6) is modified as

*ω*

_{0}. Equation (10) is also modified by replacing

*s*with

*s*−

*δ*, so

_{i}*δ*can be extracted during the extracting process.

_{i}*R*(

*s*) can be chosen so that

*p*

_{N}_{-1}is real. Figure 11 shows three different choices of zeros for an

*N*= 7 Bessel filter. They correspond to different coupling coefficients

*κ*and frequency detuning

*δ*, as listed in Table 5 . The first one is often referred to as “minimum phase”, where all zeros are located inside the left-half

*s*-plane (Fig. 11(a)). It corresponds to zero frequency detuning and monotonically increasing

*κ*. In the second one, the zeros are all located at the first and third quadrants (Fig. 11(b)). The resulting values of

*κ*are symmetric, but the frequency detuning is nonzero since

*p*

_{N}_{-1}is complex. In our CROW filter design, we prefer a nearly symmetric

*κ*without frequency detuning. Consequently, we choose the zeros that are the most uniformly distributed around the origin and are complex conjugate pairs, as shown in Fig. 11(c). Note that although the three CROW filters in Table 5 look very different, they have the same

*T*(

*s*) and

*R*(

*s*) is different.

## Acknowledgments

## References and links

1. | A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. |

2. | J. K. S. Poon, L. Zhu, G. A. DeRose, and A. Yariv, “Transmission and group delay of microring coupled-resonator optical waveguides,” Opt. Lett. |

3. | F. N. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics |

4. | 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. |

5. | M. Notomi, E. Kuramochi, and T. Tanabe, “Large-scale arrays of ultrahigh-Q coupled nanocavities,” Nat. Photonics |

6. | R. S. Tucker, P. C. Ku, and C. J. Chang-Hasnain, “Slow-light optical buffers: capabilities and fundamental limitations,” J. Lightwave Technol. |

7. | A. Melloni, F. Morichetti, and M. Martinelli, “Four-wave mixing and wavelength conversion in coupled-resonator optical waveguides,” J. Opt. Soc. Am. B |

8. | P. Chak and J. E. Sipe, “Minimizing finite-size effects in artificial resonance tunneling structures,” Opt. Lett. |

9. | M. Sumetsky and B. J. Eggleton, “Modeling and optimization of complex photonic resonant cavity circuits,” Opt. Express |

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16. | S. J. Xiao, M. H. Khan, H. Shen, and M. H. Qi, “A highly compact third-order silicon microring add-drop filter with a very large free spectral range, a flat passband and a low delay dispersion,” Opt. Express |

17. | H. A. Haus and Y. Lai, “Theory of cascaded quarter wave shifted distributed feedback resonators,” IEEE J. Quantum Electron. |

18. | D. Park, S. Kim, I. Park, and H. Lim, “Higher order optical resonant filters based on coupled defect resonators in photonic crystals,” J. Lightwave Technol. |

19. | R. Orta, P. Savi, R. Tascone, and D. Trinchero, “Synthesis of multiple-ring-resonator filters for optical systems,” IEEE Photon. Technol. Lett. |

20. | B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. |

21. | V. Van, “Circuit-based method for synthesizing serially coupled microring filters,” J. Lightwave Technol. |

22. | A. Melloni and M. Martinelli, “Synthesis of direct-coupled-resonators bandpass filters for WDM systems,” J. Lightwave Technol. |

23. | J. K. S. Poon and A. Yariv, “Active coupled-resonator optical waveguides. I. Gain enhancement and noise,” J. Opt. Soc. Am. B |

24. | H. A. Haus, |

25. | A. M. Prabhu and V. Van, “Predistortion techniques for synthesizing coupled microring filters with loss,” Opt. Commun. |

26. | J. K. S. Poon, J. Scheuer, S. Mookherjea, G. T. Paloczi, Y. Y. Huang, and A. Yariv, “Matrix analysis of microring coupled-resonator optical waveguides,” Opt. Express |

27. | 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. |

28. | A. Yariv and P. Yeh, |

**OCIS Codes**

(130.2790) Integrated optics : Guided waves

(130.3120) Integrated optics : Integrated optics devices

(230.4555) Optical devices : Coupled resonators

(130.7408) Integrated optics : Wavelength filtering devices

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: July 18, 2011

Revised Manuscript: August 9, 2011

Manuscript Accepted: August 10, 2011

Published: August 23, 2011

**Citation**

Hsi-Chun Liu and Amnon Yariv, "Synthesis of high-order bandpass filters based on coupled-resonator optical waveguides (CROWs)," Opt. Express **19**, 17653-17668 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-18-17653

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