Synthesis of a parallel-coupled ring-resonator filter - Dipartimento di ...

June 15, 2001 / Vol. 26, No. 12 / OPTICS LETTERS 917
Synthesis of a parallel-coupled ring-resonator filter
Andrea Melloni
Dipartimento di Elettronica e Informazione, Politecnico di Milano, Via Ponzio 34/5, 20133 Milan, Italy
Received January 3, 2001
An effective and exact synthesis technique for the design of parallel-coupled ring-resonator filters with a
maximally f lat stop-band characteristic of any order is presented. Simple closed-form formulas determine
the Q factor of each resonator and the coupling coefficients. The performances of these filters are discussed
for their applications as interleavers and channel-dropping filters in wavelength-division multiplexing systems.
© 2001 Optical Society of America
OCIS codes: 130.1750, 230.3120, 230.5750, 060.4230.
Selective bandpass filters are a key component in
modern dense wavelength-division multiplexing systems
that are characterized by small channel spacing
and high bit rates. The required characteristics are
high selectivity, that is, the ability to separate two
adjacent channels; high rejection out of band, which
guarantees low cross talk between channels; f lat
in-band response, a requirement that is often related
to maximum group-delay distortion; and, if possible,
low insertion losses. 1
In this Letter a simple and exact technique for the
synthesis of bandpass filters is presented, for the first
time to the author’s knowledge. Such a filter is realized
by the use of cascading ring resonators, as shown
in Fig. 1. The proposed technique is physically well
based, and it permits the design of filters with maximally
f lat (Butterworth) characteristics of any order
by use of closed-form formulas only. This type of filter
was recently discussed in Refs. 2 and 3. In Ref. 2
it was shown that a suitable apodization of the coupling
coefficients should improve the frequency response in
the rejected band. In spite of its title, the paper by
Griffel 3 does not come up to expectations, as it presents
an analysis of the filter structure only.
The filter is called a parallel-coupled or quarterwave-coupled
resonator filter. 4 Each ring resonator is
coupled to the adjacent resonator by means of two identical
sections of waveguide whose length Lc is an odd
multiple of a quarter of a wavelength. The incoming
wavelength-division multiplexing signal is applied to
input port P1. The channels whose wavelengths resonate
are dropped at port P2, while the others continue
toward the through port, P3. Port P4 is kept isolated
and can be used to add channels to the system. The
filter can operate on a single channel or, because of its
periodicity, on a number of interleaved channels.
The proposed technique is derived from modification
of a classic method used for the synthesis of microwave
stop-band filters. 4 The structure of Fig. 1, in fact, is
that of a stop-band filter, as the resonating frequencies
are dropped from the incoming signal without passing
through all the resonators. N is the order of the filter,
B is the bandwidth, f0 is the center frequency, and
FSR f0M, where M is an integer, is the free spectral
range of the filter. The technique consists in calculating
the Q factor of each resonator, which is simply
related to the N elements gq (with q 1 ... N) of the
prototype filter 4 by
where
Qq FSRgqB , (1)
gq p µ
2q 2 1
2 sin
2N p
∂
. (2)
The field coupling coefficients cq of the two couplers
in each ring are readily derived from the expression of
the finesse of a Fabry–Perot cavity 5 and result in
cq p2
2Qq 2 1 1 4Qq 2 p 2 1/2
2 1 . (3)
Note that the coupling coefficients cq are not affected
by the distance Lc between the rings. The shape and
the bandwidth of the filter response are determined
only by the Q factors of the cascaded resonators,
whereas the distance between the resonators inf luences
and even determines the out-of-band frequency
response of the filter.
For perfectly circular rings, the geometrical length
Lr of the rings is simply related to the FSR or to f0 by
c cM
Lr , (4)
neff FSR neff f0
where the requirement that M be an integer means
that all the unloaded rings resonate exactly at f0.
Length Lr includes the length of the two couplers of
each ring. In Eq. (4), neff is the effective refractive
index of the bent waveguide at center frequency
f0 and c is the speed of light. If the shape of the
rings is not circular, both the bending radius and the
refractive-index change along the ring itself and
Eq. (4) must be considered in an integral form.
Note that only the total length of the rings enters
into Eq. (4); the positions of the two couplers inside
Fig. 1. Filter with parallel-coupled ring resonators.
0146-9592/01/120917-03$15.00/0 © 2001 Optical Society of America

918 OPTICS LETTERS / Vol. 26, No. 12 / June 15, 2001
each ring are arbitrary. This means that the rings
should not necessarily be placed in line and, if necessary,
the filter can be folded as for the cross-grid
node consisting of a pair of microring resonators as discussed
in Ref. 6.
Length Lc of the coupling waveguides should be as
nearly equal as possible to an odd multiple of a quarter-wavelength
over the whole operating frequency
band of the filter. In other words, Lc must be the
smallest possible. Even more generally, 2Lc should be
a small odd multiple of a half-wavelength. However,
because of the large dimensions of the rings compared
with the wavelength, to fulfill this requirement the
rings cannot be placed too close to one another, to
avoid superposition, and this restriction affects the
operating frequency band of the filter. For circular
rings, the minimum distance is twice the ring radius.
When 2Lc is an odd multiple of half a wavelength,
the contributions of all the rings add in phase at the
resonating frequencies, and the signal is dropped at the
output port. However, when out of band, the contributions
do not add in phase, and at some frequencies
they even cancel completely and create transmission
nulls in the frequency response. This is a nice feature
of this type of filter, as the out-of-band rejection
increases.
A number of examples demonstrate the capability of
the proposed technique to design parallel-coupled filters
and also show both the advantages and the limitations
of this type of filter. The inf luence of distance
Lc on the frequency response of the filters is also investigated,
up to Lc Lr2. Larger values of Lc are
of small interest and are not discussed here. We carry
out the spectral analysis by cascading the transmission
matrices of the various building blocks of the structure.
As a first example, a bandpass filter with FSRB
10 is considered. Distance Lc has been chosen very
small, Lc Lr100l/4, where the subscript means the
odd multiple of a quarter-wavelength closer to Lr100.
The rings have a strongly elliptical shape, with a minimum
bending radius that can be supported only by
highly confined waveguides such as those described in
Refs. 2 and 6. As an example, a FSR of 100 GHz requires
a minimum bending radius equal to 15neff mm.
Higher FSRs require even smaller bending radii, which
depend on FSR 21 .
Figure 2 shows the frequency responses of the filters
for orders from 2 to 5, over two FSRs. The bandpass
characteristic is f lat (maximally f lat), and the bandwidth
at 23 dB is independent of order N. Note the
transmission zeros that appear between the dropped
bands: Such a feature permits a very low level of
cross talk to be attained, even with a low order of the
filter. Increasing the order makes the frequency response
more selective; the number of transmission zeros
increases, too.
If the distance between rings is much smaller then
the ring half-length, it is possible to take advantage
of the periodicity of the frequency response. The
smaller Lc, the higher the number of usable FSRs.
Figure 3 shows the frequency response of a fourthorder
filter with FSR 100 GHz, B 20 GHz, and
Lc Lr100l/4. As above, long and narrow rings
with a minimum bending radius of 15neff mm haveto
be packed very close to one another.
Note that both the shape of the dropped bands
and the out-of-band rejection remain acceptable for
15 FSRs. The deterioration in frequency response
is due to the variation of the phase shift induced in
the frequency by Lc, but the nulls between subsequent
dropped bands are maintained. The filter considered
in this example can be used as an interleaver to
separate odd from even channels in a 32-channel,
50-GHz spaced, wavelength-division multiplexing
system. As a rule of thumb, the number of channels
that can be managed by an interleaver produced by
parallel-coupled ring resonators is Lr3Lc.
To investigate how the distance between the rings
inf luences the frequency response, let us consider a
third-order filter with FSRB 10 and three values
of Lc. Figure 4 shows each frequency response of the
three filters with distances Lc rounded to l4. When
distance Lc is increased, the transmission nulls move
toward the dropped band, and the rejection decreases.
With Lc Lrpl/4, which corresponds to two circular
rings placed as close to each other as possible,
although the bandpass remains undistorted the rejection
out of band is unacceptable. Moreover, the filter
Fig. 2. Frequency responses for N 2...5and FSRB
10. Lc Lr100l/4.
Fig. 3. Frequency response for a fourth-order filter with
FSR 100 GHz, B 20 GHz, and Lc Lr100l/4.

Fig. 4. Frequency response of a third-order filter with
FSRB 10 and three values of Lc (rounded to l4).
Fig. 5. Frequency response of a fourth-order filter with
FSRB 30 and two values of Lc.
ceases to be periodic and cannot be used for interleaving.
A possible application in the case of large Lc (up
to Lc Lrp) consists of single-channel selection. As
an example, for N 3, FSR 1200 GHz, B 10 GHz,
June 15, 2001 / Vol. 26, No. 12 / OPTICS LETTERS 919
and Lc Lrpl/4, the rejected band is 0.9 FSR and
the ring radius is 40neff mm.
As a last example, a filter with Lc Lr2 is observed.
In this case, the two sections Lc considered
together have the same length as the rings themselves.
If distance Lc is equal to the odd multiple of a quarterwavelength
closer to Lr2, the filter has poor rejection,
as shown in Fig. 5 [labeled Lr2l/4]. In such a figure,
a fourth-order filter with FSRB 30 is considered.
If Lc Lr2, however, the frequency response is substantially
modified: The dropped bands widen considerably,
they are not maximally f lat, and the rejection
out-of-band values are excellent as the transmission
nulls move in. In this case the minimum bending radius
depends on the FSR only.
The filter characteristic is periodic, as in the first example.
Possible applications of this design are either
in interleaving or in single-channel selection. Without
entering into details, however, it can be said that a
perfect compensation between a transmission null with
ring resonance occurs at the center frequencies. This
is a critical situation, as any discrepancy with respect
to the ideal dimensions produces a deep notch in the
dropped bands.
In conclusion, the proposed synthesis technique
is precise and f lexible and uses only closed-form
formulas. Both channel-selective filters and interleavers
can be designed according to specifications of
bandwidth, FSR, out-of-band rejection, and selectivity.
The distance between the rings plays an important
role in the overall frequency response.
The author’s e-mail address is melloni@elet.polimi.it.
References
1. C. K. Madsen and J. H. Zhao, Optical Filter Design and
Analysis (Wiley, New York, 1999).
2. B. E. Little, S. T. Chu, J. V. Hryniewicz, and P. Absil,
Opt. Lett. 25, 344 (2000).
3. G. Griffel, IEEE Photon. Technol. Lett. 12, 810 (2000).
4. G. L. Matthaei, L. Young, and E. M. T. Jones, in
Microwave Filters, Impedance Matching Networks and
Coupling Structures (McGraw-Hill, New York, 1964).
5. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall,
Englewood Cliffs, N. J., 1984).
6. S. T. Chu, B. E. Little, W. Pan, T. Kaneko, and Y.
Kokubun, IEEE Photon. Technol. Lett. 11, 1426 (1999).