## Realization of asymmetric optical filters using asynchronous coupled-microring resonators

Optics Express, Vol. 15, Issue 15, pp. 9645-9658 (2007)

http://dx.doi.org/10.1364/OE.15.009645

Acrobat PDF (197 KB)

### Abstract

General filter architectures based on asynchronously-tuned, coupled-microring resonators are proposed for realizing optical filters with asymmetric spectral responses. Asymmetric filters enable more complex spectral shapes to be realized which can better meet the demands of more advanced applications than symmetric filters. By adjusting individual transmission zeros of the filter transfer function, the transition bands on the low and high-frequency sides of the passband can be separately optimized to achieve an optimum filter response. A method for synthesizing asymmetric spectral responses based on the energy coupling matrix will also be presented along with numerical examples of high-order asymmetric optical filters. These devices represent new microring-based architectures that can be explored for advanced applications in optical spectral shaping and dispersion engineering.

© 2007 Optical Society of America

## 1. Introduction

1. J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson, and P.-T. Ho, “Higher order filter response in coupled microring resonators,” IEEE Photon. Technol. Lett. **12**, 320–322 (2000). [CrossRef]

3. T. Barwicz, M. Popovic, P. Rakich, M. Watts, H. Haus, E. Ippen, and H. Smith, “Microring-resonator-based add-drop filters in SiN: fabrication and analysis,” Opt. Express **12**, 1437–1442 (2004). [CrossRef] [PubMed]

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

4. B. E. Little, S. T. Chu, J. V. Hryniewicz, and P. P. Absil, “Filter synthesis for periodically coupled microring resonators,” Opt. Lett. **25**, 344–346 (2000). [CrossRef]

5. R. Grover, V. Van, T. A. Ibrahim, P. P. Absil, L. C. Calhoun, F. G. Johnson, J. V. Hryniewicz, and P.-T. Ho, “Parallel-cascaded semiconductor microring resonators for high-order and wide-FSR filters,” J. Lightw. Technol. **20**, 900–905 (2002). [CrossRef]

6. K. Jinguji, “Synthesis of coherent two-port optical delay-line circuit with ring waveguides,” J. Lightwave Technol. **14**, 1882–1898 (1996). [CrossRef]

7. C. K. Madsen, “Efficient architectures for exactly realizing optical filters with optimum bandpass design,” IEEE Photon. Technol. Lett.10, 1136–1138 (1998). [CrossRef]

6. K. Jinguji, “Synthesis of coherent two-port optical delay-line circuit with ring waveguides,” J. Lightwave Technol. **14**, 1882–1898 (1996). [CrossRef]

7. C. K. Madsen, “Efficient architectures for exactly realizing optical filters with optimum bandpass design,” IEEE Photon. Technol. Lett.10, 1136–1138 (1998). [CrossRef]

8. V. Van, “Synthesis of elliptic optical filters using mutually-coupled microring resonators,” J. Lightw. Technol. **25**, 584–590 (2007). [CrossRef]

11. H. C. Bell, “Canonical asymmetric coupled-resonator filters,” IEEE Trans. Microwave Theory Technol. **30**, 1335–1340 (1982). [CrossRef]

## 2. General asynchronously-tuned coupled-microring filter topology

*N*microring resonators in which every resonator is assumed to be coupled to every other resonator. In the narrowband approximation where the filter bandwidth is much smaller than the free spectral range (FSR) of the microrings, the behavior of the system of coupled cavities in Fig. 1 may be conveniently described in terms of energy coupling in the time domain [13

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

_{i,j}as the energy coupling coefficient between microrings

*i*and

*j*. The input and output couplings µ

*and µ*

_{i}*represent the energy coupling between microring 1 and the input bus waveguide and between microring*

_{o}*N*and the output bus waveguide, respectively. The input, transmitted and reflected signals are denoted by

*s*,

_{i}*s*and

_{r}*s*, respectively. The transmitted signal gives rise to the drop-port response of the filter while the reflected signal yields the through-port response. For generality we also assume that all the resonators are detuned from each other, with the amount of detuning of microring

_{t}*i*measured from the center frequency

*ω*

_{0}of the filter passband given by Δ

*ω*=

_{i}*ω*-

_{i}*ω*

_{0}, where

*ω*is the resonant frequency of microring

_{i}*i*. Denoting

*a*as the energy amplitude in microring

_{i}*i*, the system of

*N*mutually-coupled microring resonators in Fig. 1 can be described by the matrix equation [8

8. V. Van, “Synthesis of elliptic optical filters using mutually-coupled microring resonators,” J. Lightw. Technol. **25**, 584–590 (2007). [CrossRef]

*s*=

*j*(

*ω*-

*ω*

_{0}),

**I**is the

*N*×

*N*identity matrix, and

**a**and

**b**are

*N*×1 vectors given by

**a**=[

*a*

_{1},

*a*

_{2},

*a*

_{3}, …

*a*]

_{N}*,*

^{T}*b*=[-

*jµ*, 0, 0, … 0]

_{i}s_{i}*.*

^{T}**M**in Eq. (1) represents the coupling topology and has the general symmetric form

**L**is a diagonal matrix which represents the rates of energy lost or extracted from the system. If all the microrings are assumed to be lossless, then

**L**is simply an

*N*×

*N*zero matrix except for the elements

*L*

_{11}=µ

^{2}

*/2 and*

_{i}*L*=µ

_{NN}^{2}

*/2, which represent the rates of energy coupling to the input and output bus waveguides, respectively. Solution of the matrix equation (1) gives the amplitudes*

_{o}*a*of the energies stored in the microrings, from which the transmitted and reflected spectral responses of the filter can be obtained via

_{i}8. V. Van, “Synthesis of elliptic optical filters using mutually-coupled microring resonators,” J. Lightw. Technol. **25**, 584–590 (2007). [CrossRef]

*z*and

_{k}*p*denote the zeros and poles, respectively, of the filter which are in general unpaired complex numbers, and

_{k}*a*and

_{k}*b*are the complex coefficients of polynomials

_{k}*Q*(

*s*) and

*P*(

*s*), which are of degrees

*N*and

*N*-2, respectively. The transfer function

*H*(

*s*) in (5) represents the response of the prototype filter with cut-off frequency

*ω*=1 rad/s. The filter synthesis procedure developed in [8

_{c}**25**, 584–590 (2007). [CrossRef]

14. A. E. Atia and A. E. Williams, “Narrow-bandpass waveguide filters,” IEEE Trans. Microwave Theory Technol. **20**, 258–265 (1972). [CrossRef]

18. R. J. Cameron, “Advanced coupling matrix synthesis techniques for microwave filters,” IEEE Trans. Microwave Theory Technol. **51**, 1–10 (2003). [CrossRef]

**25**, 584–590 (2007). [CrossRef]

*LC*resonators which can be described by the same coupling matrix

**M**in (2). Since

**M**is real and symmetric, it can be factored in the form [8

**25**, 584–590 (2007). [CrossRef]

15. A. E. Atia, A. E. Williams, and R. W. Newcomb, “Narrow-band multiple-coupled cavity synthesis,” IEEE Trans. Circuits Syst. **21**, 649–655 (1974). [CrossRef]

**T**is a real orthonormal matrix and Λ is a diagonal matrix containing the eigenvalues λ

*of*

_{k}**M**. The filter synthesis procedure thus involves determining the matrices Λ and

**T**for a given filter transfer function

*H*(

*s*) in (5). In the well-known coupling-matrix technique for synthesizing microwave filters, the elements of Λ and

**T**can be extracted from the short-circuit admittance parameters

**Y**

*of the equivalent circuit network, which can be expressed in the form [15*

_{sc}15. A. E. Atia, A. E. Williams, and R. W. Newcomb, “Narrow-band multiple-coupled cavity synthesis,” IEEE Trans. Circuits Syst. **21**, 649–655 (1974). [CrossRef]

*ξ*

^{(k)}

_{i,j}is the residue of admittance

*y*

_{i,j}at the pole λ

*, which is also an eigenvalue of the coupling matrix*

_{k}**M**. Thus the matrix

**Λ**, which is real, can be constructed from the poles λ

*of*

_{k}**Y**

*. Due to the reciprocal nature of the network, we have*

_{sc}*y*

_{12}=

*y*

_{21}and also

*y*

_{11}=

*y*

_{22}, or equivalently,

*ξ*

^{(k)}

_{12}=

*ξ*

^{(k)}

_{21}and

*ξ*

^{(k)}

_{11}=

*ξ*

^{(k)}

_{22}. The admittance residues

*ξ*

^{(k)}

_{11}are real while

*ξ*

^{(k)}

_{12}are imaginary, but unlike the case of symmetric filters,

*ξ*

^{(k)}

_{12}does not appear in conjugate pairs. From

*ξ*

^{(k)}

_{11}we can compute the input and output coupling coefficients of the microring filter via

**T**are next determined from

**T**can be obtained by Gram-Schmidt orthogonalization since the matrix

**T**is orthonormal. The coupling matrix

**M**is next determined from

**T**and Λ using (6). Finally, frequency scaling is applied to the prototype design to obtain the coupling coefficients for the filter with the specified bandwidth

*B*according to

_{i, j}→µ

_{i, j}(

*B*/2).

**Y**

_{sc}of the equivalent circuit network can be determined if the through-port response of the microring filter is known. Given a specified drop-port filter transfer function

*H*(

*s*) of the form in (5), the through-port response

*F*(

*s*) can be obtained from the factorization,

*F*(

*s*) is also the reflection coefficient Γ(

*s*) of the equivalent electrical circuit, we can obtain an expression for the input impedance

*Z*(

_{in}*s*) of the equivalent circuit from

*R*(

*s*) is an Nth-degree polynomial such that

*F*(

*s*)=

*R*(

*s*)/

*Q*(

*s*). From the above expression for

*Z*

*, the short-circuit admittance parameters*

_{in}**Y**

*sc*can be determined using a similar procedure as in [17

17. R. J. Cameron, “General coupling matrix synthesis methods for Chebyshev filtering functions,” IEEE Trans. Microwave Theory Technol. **47**, 433–442 (1999). [CrossRef]

*Z*(

_{in}*s*) is expressed as

*m*(

*s*) and

*n*(

*s*) are polynomials constructed from the complex coefficients

*a*and

_{k}*c*of

_{k}*Q*(

*s*) and

*R*(

*s*), respectively, according to

**Y**

_{sc}are given by [17

17. R. J. Cameron, “General coupling matrix synthesis methods for Chebyshev filtering functions,” IEEE Trans. Microwave Theory Technol. **47**, 433–442 (1999). [CrossRef]

*y*

_{11}(

*s*) and

*y*

_{12}(

*s*) in the form of (7), the poles λ

*and residues*

_{k}*ξ*

^{(k)}

_{11}and

*ξ*

^{(k)}

_{12}can be obtained, from which the coupling matrix

**M**can be determined.

**M**obtained using the above procedure may represent a coupling topology that is not realizable due to physical layout constraints, such as when a microring is required to couple to too many other microrings. Another non-realizable coupling topology, shown in Fig. 2, is the formation of a triplet or more generally, a circular arrangement consisting of an odd number of resonators. These structures lead to coupling between the forward and backward propagating modes in the microrings and results in a reflected wave at the input port. A non-realizable coupling topology may be converted into a realizable configuration by performing similarity transformations such as Jacobi rotation and reflection on the original matrix

**M**to obtain a new coupling matrix

**M**′ given by [15

15. A. E. Atia, A. E. Williams, and R. W. Newcomb, “Narrow-band multiple-coupled cavity synthesis,” IEEE Trans. Circuits Syst. **21**, 649–655 (1974). [CrossRef]

17. R. J. Cameron, “General coupling matrix synthesis methods for Chebyshev filtering functions,” IEEE Trans. Microwave Theory Technol. **47**, 433–442 (1999). [CrossRef]

**R**(

*θ*) is an

_{r}*N*×

*N*rotation matrix and

*θ*is the rotation angle chosen to annihilate unrealizable coupling elements in the original coupling matrix

_{r}**M**. In addition, we can also use similarity transformations to generate alternative coupling configurations with different sets of coupling coefficients and microring detunings. Out of these possible filter designs, an optimum device topology may then be chosen which has the minimum number of coupling elements as well as coupling values that are more easily realized. For example, it is generally desirable to minimize the number of negative coupling elements in the device since they require long optical coupling lengths (between 3

*π*/2 and 2

*π*) or precise layout geometry [10] and are thus more difficult to realize than positive couplings.

**M**are in general non-zero, indicating that the microrings will have some detunings with respect to the center filter frequency. These detunings are usually small and do not give rise to the Vernier effect, so that the transmission bands at adjacent FSRs are not suppressed. On the other hand, symmetric filters may be realized using either synchronous or asynchronous microrings. By allowing the resonators to take on detuning values, alternative asynchronous designs can be obtained which may be easier to implement than their synchronous counterparts. In this respect, asynchronicity provides an added degree of flexibility in the design of coupled-microring filters. Also, different filter topologies in general have different sensitivity performance with respect to variations in the coupling parameters. A simple tolerance analysis of coupled-microring filters with symmetric spectral responses has been given in [8

**25**, 584–590 (2007). [CrossRef]

## 3. Numerical examples of asymmetric optical filters

### 3.1. Seventh-order microring filter with 3 transmission zeros

^{th}-order microring filter with a 25GHz bandwidth. On the right-hand side of the passband, two transmission zeros are placed close to the band edge to achieve a very steep roll-off and an out-of-band rejection level of -55dB. On the left-hand side, the roll-off requirement is more relaxed so only one zero is needed, but the out-of-band rejection should be -65dB. The locations of the poles and zeros required to realize the filter transfer function are listed in Table 1. Also shown in the table are the poles of the short-circuit admittances,

*y*

_{11}and

*y*

_{12}, as well as their residues, ξ

_{11}and ξ

_{12}, which were obtained from the transfer function. In the filter synthesis procedure, the input and output energy coupling coefficients of the microring filter were first computed using Eq. (8) to give µ

*=µ*

_{i}*=1.2843, or 11.3816 after bandwidth scaling. The coupling matrix of the filter was obtained next using the procedure described in the previous section, yielding*

_{o}**M**is symmetric. The coupling topology described by the matrix

**M**above is not physically realizable with microring resonators since the matrix is almost full, requiring every microring to couple to every other microring except between resonators 1 and 7. To convert the above coupling topology into one which is physically realizable and much more simplified, we performed a series of rotations on the original matrix

**M**following a procedure described in [18

18. R. J. Cameron, “Advanced coupling matrix synthesis techniques for microwave filters,” IEEE Trans. Microwave Theory Technol. **51**, 1–10 (2003). [CrossRef]

*ω*,×10

^{9}rad/s) from the center frequency

*ω*

_{0}of each microring in the filter. In practice these small detunings may be realized by thermal-optic or electro-optic control of the index of the microring waveguides.

*N*-2 [12,18

18. R. J. Cameron, “Advanced coupling matrix synthesis techniques for microwave filters,” IEEE Trans. Microwave Theory Technol. **51**, 1–10 (2003). [CrossRef]

*L*satisfying 3

*π*/2<

*k*<2

_{c}L*π*, where

*k*is the per-unit-length coupling strength. Alternatively, it was recently suggested that negative coupling may also be achieved by a λ/8 shearing rotation of the coupling axis of the quadruplet [10]. It should also be noted that for microring resonators arranged in a loop-coupling configuration, such as the quadruplet in Fig. 3, the positions of the coupling points between adjacent microrings do affect the coupling signs so that care must be exercised in laying out the device geometry. By contrast, there is no restriction on the positions of the coupling points for microrings 3, 4 and 5.

_{c}*H*(

*s*) and

*F*(

*s*). It is seen that the synthesized responses agree very well with the specified filter characteristics. The placement of two transmission zeros on the right-hand side of the passband results in a much steeper roll-off on that side compared to the left-hand side. By varying the positions of the zeros, a trade-off between the roll-off rate and out-of-band rejection level can be obtained to suit a specific application. For comparison we also plotted the transmission response of a 7

^{th}-order Butterworth filter having the same bandwidth of 25GHz. The superior performance of the asymmetric filter is apparent from the much sharper band transitions and higher isolation levels compared to the Butterworth filter. In Fig. 5 we plotted the group delay response of the filter, which also exhibits an asymmetric spectral shape. Specifically, the group delay is seen to be higher near the right band edge, which corresponds to the steeper amplitude transition compared to the left-hand side.

### 3.2. Sixth-order microring filter with 2 transmission zeros

^{th}-order, 10GHz-bandwidth filter with two transmission zeros. The requirement here is to maximize the roll-off rate on the high-frequency side of the passband while maintaining a -45dB out-of-band rejection level. If there is no additional constraint specified for the low-frequency side of the passband, we can move both transmission zeros of the filter to the right-hand side. The poles and zeros of the filter transfer function which satisfies the above requirements are given in columns 1 and 2 of Table 2. The corresponding poles and residues of the short-circuit admittance parameters of the equivalent circuit are also shown in the table. Application of the filter synthesis procedure in Section 2 yielded the input and output coupling coefficients µ

*=µ*

_{i}*=6.8971 and optimized coupling matrix*

_{o}**M**′ given by

**M**in Eq. (21) was obtained using the approximate energy coupling formalism. To verify the validity of the energy model used, we also performed analysis of the synthesized microring filter using the power coupling method, which allows the exact device spectral responses to be determined. In the power analysis we assumed the microring resonators to have a nominal free spectral range of 2.5THz (20nm), which may be realized using a high-index material system such as Silicon-on-Insulator or III–V semiconductor compounds. Also, the filter passband was assumed to be centered at the 1.557µm wavelength. The exact amplitude responses at the drop port and through port, and the group delay of the filter obtained from the power analysis are shown by the blue dashed lines in Fig. 7(a) and Fig. 8. The results are seen to agree very well with the responses of the synthesized filter obtained using the energy-coupling approximation. For the group delay in Fig. 8, the spikes in the response obtained from the power analysis occur at the transmission zeros where the phase is not well defined. These frequencies also lie outside the passband and are thus usually of little interest. The power coupling method also allows us to investigate the broadband response of the filter over more than one free spectral range. In Fig. 9 we plotted the transmission characteristic of the filter at the drop port as a function of the wavelength across one FSR. It is seen that even though all the microrings in the filter design are detuned from each other, virtually no Vernier effect is observed since adjacent passbands are not suppressed or distorted, as shown by the insets of the figure, and the FSR of the filter is the same as that of the individual microrings.

## 4. Conclusion

## References

1. | J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson, and P.-T. Ho, “Higher order filter response in coupled microring resonators,” IEEE Photon. Technol. Lett. |

2. | B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, “Very high order microring resonator filters for WDM applications,” IEEE Photon. Technol. Lett. |

3. | T. Barwicz, M. Popovic, P. Rakich, M. Watts, H. Haus, E. Ippen, and H. Smith, “Microring-resonator-based add-drop filters in SiN: fabrication and analysis,” Opt. Express |

4. | B. E. Little, S. T. Chu, J. V. Hryniewicz, and P. P. Absil, “Filter synthesis for periodically coupled microring resonators,” Opt. Lett. |

5. | R. Grover, V. Van, T. A. Ibrahim, P. P. Absil, L. C. Calhoun, F. G. Johnson, J. V. Hryniewicz, and P.-T. Ho, “Parallel-cascaded semiconductor microring resonators for high-order and wide-FSR filters,” J. Lightw. Technol. |

6. | K. Jinguji, “Synthesis of coherent two-port optical delay-line circuit with ring waveguides,” J. Lightwave Technol. |

7. | C. K. Madsen, “Efficient architectures for exactly realizing optical filters with optimum bandpass design,” IEEE Photon. Technol. Lett.10, 1136–1138 (1998). [CrossRef] |

8. | V. Van, “Synthesis of elliptic optical filters using mutually-coupled microring resonators,” J. Lightw. Technol. |

9. | M. A. Prabhu and V. Van, “General two-dimensional coupled-cavity microring filter architectures,” in |

10. | M. A. Popovic, “Sharply-defined optical filters and dispersionless delay lines based on loop-coupled resonators and ‘negative’ coupling,” in |

11. | H. C. Bell, “Canonical asymmetric coupled-resonator filters,” IEEE Trans. Microwave Theory Technol. |

12. | A. E. Williams, J. I. Upshur, and M. M. Rahman, “Asymmetric response bandpass filter having resonators with minimum couplings,” U.S. patent 6337610 (2002). |

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

14. | A. E. Atia and A. E. Williams, “Narrow-bandpass waveguide filters,” IEEE Trans. Microwave Theory Technol. |

15. | A. E. Atia, A. E. Williams, and R. W. Newcomb, “Narrow-band multiple-coupled cavity synthesis,” IEEE Trans. Circuits Syst. |

16. | M. H. Chen, “Singly terminated pseudo-elliptic function filter,” COMSAT Technol. Rev. |

17. | R. J. Cameron, “General coupling matrix synthesis methods for Chebyshev filtering functions,” IEEE Trans. Microwave Theory Technol. |

18. | R. J. Cameron, “Advanced coupling matrix synthesis techniques for microwave filters,” IEEE Trans. Microwave Theory Technol. |

**OCIS Codes**

(130.3120) Integrated optics : Integrated optics devices

(250.5300) Optoelectronics : Photonic integrated circuits

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: May 7, 2007

Revised Manuscript: July 9, 2007

Manuscript Accepted: July 16, 2007

Published: July 19, 2007

**Citation**

Ashok M. Prabhu and Vien Van, "Realization of asymmetric optical filters using asynchronous coupled-microring resonators," Opt. Express **15**, 9645-9658 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-15-9645

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### References

- J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson and P.-T. Ho, "Higher order filter response in coupled microring resonators," IEEE Photon. Technol. Lett. 12, 320-322 (2000). [CrossRef]
- B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King and M. Trakalo, "Very high order microring resonator filters for WDM applications," IEEE Photon. Technol. Lett. 16, 2263-2265 (2004). [CrossRef]
- T. Barwicz, M. Popovic, P. Rakich, M. Watts, H. Haus, E. Ippen and H. Smith, "Microring-resonator-based add-drop filters in SiN: fabrication and analysis," Opt. Express 12, 1437-1442 (2004). [CrossRef] [PubMed]
- B. E. Little, S. T. Chu, J. V. Hryniewicz and P. P. Absil, "Filter synthesis for periodically coupled microring resonators," Opt. Lett. 25, 344-346 (2000). [CrossRef]
- R. Grover, V. Van, T. A. Ibrahim, P. P. Absil, L. C. Calhoun, F. G. Johnson, J. V. Hryniewicz and P.-T. Ho, "Parallel-cascaded semiconductor microring resonators for high-order and wide-FSR filters," J. Lightw. Technol. 20, 900-905 (2002). [CrossRef]
- K. Jinguji, "Synthesis of coherent two-port optical delay-line circuit with ring waveguides," J. Lightwave Technol. 14, 1882-1898 (1996). [CrossRef]
- C. K. Madsen, "Efficient architectures for exactly realizing optical filters with optimum bandpass design," IEEE Photon. Technol. Lett. 10, 1136-1138 (1998). [CrossRef]
- V. Van, "Synthesis of elliptic optical filters using mutually-coupled microring resonators," J. Lightw. Technol. 25, 584-590 (2007). [CrossRef]
- M. A. Prabhu and V. Van, "General two-dimensional coupled-cavity microring filter architectures," in IEEE Conference on Lasers and Electro-Optics, 2007, paper JTuA31.
- M. A. Popovic, "Sharply-defined optical filters and dispersionless delay lines based on loop-coupled resonators and ‘negative’ coupling," in IEEE Conference on Lasers and Electro-Optics, 2007, paper CThP6.
- H. C. Bell, "Canonical asymmetric coupled-resonator filters," IEEE Trans. Microwave Theory Technol. 30, 1335-1340 (1982). [CrossRef]
- A. E. Williams, J. I. Upshur and M. M. Rahman, "Asymmetric response bandpass filter having resonators with minimum couplings," U.S. patent 6337610 (2002).
- B. E. Little, S. T. Chu, H. A. Haus, J. Foresi and J.-P. Laine, "Microring resonator channel dropping filters," J. Lightw. Technol. 15, 998-1005 (1997). [CrossRef]
- A. E. Atia and A. E. Williams, "Narrow-bandpass waveguide filters," IEEE Trans. Microwave Theory Technol. 20, 258-265 (1972). [CrossRef]
- A. E. Atia, A. E. Williams and R. W. Newcomb, "Narrow-band multiple-coupled cavity synthesis," IEEE Trans. Circuits Syst. 21, 649-655 (1974). [CrossRef]
- M. H. Chen, "Singly terminated pseudo-elliptic function filter," COMSAT Technol. Rev. 7, 527-541 (1977).
- R. J. Cameron, "General coupling matrix synthesis methods for Chebyshev filtering functions," IEEE Trans. Microwave Theory Technol. 47, 433-442 (1999). [CrossRef]
- R. J. Cameron, "Advanced coupling matrix synthesis techniques for microwave filters," IEEE Trans. Microwave Theory Technol. 51, 1-10 (2003). [CrossRef]

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