Reflectionless multichannel wavelength demultiplexer in a transmission resonator configuration
ABSTRACT A wave demultiplexer system with N channels in a twodimensional photonic crystal is proposed. The demultiplexer is realized by the coupling among an ultralowquality factor microcavity and N resonators with highquality factor. The coupling mode theory is employed to analyze the behavior of this system. The analytic results reveal that the reflection is fully absent at the peak frequencies for all channels. The simulations obtained by multiplescattering method and experimental results in the microwave region show that the analysis is valid. This method might also be valuable for the design of other alloptical functional circuits.
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160IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 39, NO. 1, JANUARY 2003
Reflectionless Multichannel Wavelength
Demultiplexer in a Transmission
Resonator Configuration
Chongjun Jin, Shanhui Fan, Shouzhen Han, and Daozhong Zhang
Abstract—A wave demultiplexer system with
twodimensional photonic crystal is proposed. The demultiplexer
is realized by the coupling among an ultralowquality factor
microcavity and
resonators with highquality factor. The
coupling mode theory is employed to analyze the behavior of
this system. The analytic results reveal that the reflection is fully
absent at the peak frequencies for all channels. The simulations
obtained by multiplescattering method and experimental results
in the microwave region show that the analysis is valid. This
method might also be valuable for the design of other alloptical
functional circuits.
channels in a
Index Terms—Microcavity, photonic crystal, resonant couple,
wavelengthdemultiplexer.
I. INTRODUCTION
W
Commercially available wavelength demultiplexer systems,
such as planar lightwavecircuitbased arraywaveguide gratings
[1]–[3] and fiber gratings [4], tend to occupy a footprint of
at least one centimeter square. There is therefore tremendous
interest in developing devices that are more compact.
Recently, photonic crystal based wavelength demultiplexers
attracted much attention [5]–[11] as potentially viable schemes
for device miniaturization. Photonic crystals are artificial struc
tures composed of periodically arrayed dielectric media [12],
[13]. These structures have been found to be very useful in
the design of alloptical devices [14]. For wavelength demul
tiplexer applications, photoniccrystalbased mechanisms that
have been proposed include the channeldrop tunneling scheme
[5]–[7], the superprism effect [11], and the frequencyselective
droppingofphotonsfromwaveguidechanneltothesurrounding
media [8]–[10]. Among all of these mechanisms, the most com
pact configurations involve the interactions of waveguides with
microcavities[5]–[10].Forpracticalimplementations,however,
it is crucial that the reflection be eliminated. Previously, the
elimination of reflection is achieved by side coupling the cavity
to the waveguides, and by adjusting the symmetry properties of
AVELENGTH demultiplexers play an important role
in enhancing the capacity of optical communications.
Manuscript received January 31, 2002; revised August 13, 2002. This work
was supported by the NNSFC, by the Chinese National Key Basic Research
Special Fund (CNKBRSF), and by the National Research Center for Intelligent
Computing Systems under Grant 99104.
C.Jin,S.Han,andD.ZhangarewiththeOpticalPhysicsLaboratory,Institute
of Physics and Center for Condensed Physics, Chinese Academy of Sciences,
Beijing, 100080, China.
S. Fan is with the Department of Electrical Engineering, Stanford University,
Stanford, CA 943054085 USA.
Digital Object Identifier 10.1109/JQE.2002.806188
(a)
(b)
Fig. 1.
multichannel. (b) Schematic electronic circuit of the multichannel filter.
(a) Schematic optical circuit for realizing a wave demultiplexer with
the cavity modes [5]. Doing so, however, tends to require de
tailed tunings of the cavity properties, which represent a chal
lenge in device designs and fabrications.
In this paper, we introduce a photonic crystal demultiplexer
structure based upon the coupling among a microcavity with an
ultralowquality factor and transmission resonators with high
quality factors, as shown schematically in Fig. 1(a). The config
uration consists of
transmission resonators,
a microcavity with lowquality factor. Each resonator possesses
a resonant frequency at a particular signal frequency channel,
to allow for a frequencyselective transmission of the frequency
channel into the corresponding waveguide. In general, in trans
mission resonator configurations, signals at frequency channels
away from the resonant frequency are reflected. Here, however,
we introduce a theoretical criterion to show that by appropriate
design of the waveguide branches, the overall reflection from
the demultiplexer device can, in fact, be completely eliminated
for all frequency channels. We substantiate our theoretical anal
ysis with numerical and experimental work. In such, our design
represents an optical analog of a classic electronic filter circuit,
as shown in Fig. 1(b), where
rated by
inductance–capacitance (LC) resonant circuits with
near 100% efficiency and no reflections [15].
branches, and
channels can be perfectly sepa
00189197/03$17.00 © 2003 IEEE
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JIN et al.: REFLECTIONLESS MULTICHANNEL WAVELENGTH DEMULTIPLEXER IN A TRANSMISSION RESONATOR CONFIGURATION161
(a)
(b)
Fig. 2.
composed of twodimensional photonic crystal. (b) Corresponding straight
waveguide.
(a) Cross section of the wave demultiplexer with two channels
Let us start by considering the case with only two channels
before we generalize the analysis to
a structure consisting of an input waveguide split into two out
going waveguides at a Tbranch, as shown in Fig. 2(a). Within
each outgoing waveguide, we place a resonator, labeled
,respectively.Eachoftheresonatorsissymmetricalwithtwo
cylinders at either side of the defect cylinder. Furthermore, the
Tbranchwaveguideandthebentwaveguidescanbeconsidered
as an ultralowquality factor microcavity, labeled as
Therefore, we can describe the whole optical system in terms
of the coupling among the three microcavities, and deduce its
properties using coupled mode theory. As the losses of the cavi
ties can be designed to be very small, and the coupling between
the cavities
and is negligible, the coupling equations can
be written as [17]
channels. We consider
and
[16].
(1)
(2)
(3)
(4)
where
of the resonant modes for
the decay rate of the microcavity
through the port
the input and output waves at the port
represents the coupling coefficient between resonant cavities.
Using (1)–(4), the transmittivity
, respectively, and the reflectivity , can be determined as in
(5)–(7), shown at the bottom of the page.
For a full demultiplexing operation in a wavelengthdivi
sionmultiplexing (WDM) system, we would like to completely
transmit each frequency channel to an appropriate corre
sponding output waveguide without any reflection or crosstalk.
In this twochannel WDM device, therefore, we will be pri
marily interested in the response function of the system in
the vicinity of the resonant frequencies
we would like the reflectivity to vanish, i.e.,
frequency channels
and. Also, at each frequency channel
or, we would like the corresponding transmittivity,
, respectively, to reach 100%, while the other transmittivity
becomes zero, in order to eliminate crosstalk. From (5)–(7),
shown at the bottom of the page, one could easily check that
this ideal behavior can be achieved provided that the following
three criteria are satisfied:
and aretheresonant frequenciesand theamplitudes
(
due to the power escape
,and denote the field amplitudes of
), respectively, is
, respectively, and
andat the portsand
and. Ideally,
, for both
or
(8)
(9)
(10)
Here, in (8) and (10),
cavities in the outgoing waveguides.
Let us give a brief physical discussion of each of these three
criteria above. Equation (8) indicates that the frequency chan
nels of the signals should fall within the resonant line shape
represents the label for the
(5)
(6)
(7)
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162IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 39, NO. 1, JANUARY 2003
Fig. 3.Schematic picture of the coupling between two resonators.
of the branch; this should hold true for any broadband splitter
structures [14]. Equation (9) requires that the microcavities
andshould possess sufficiently highquality factors in order
to provide the necessary distinction between nearest neighbor
channels.SucharequirementistypicallysatisfiedforanyWDM
microcavity filter structure. The key designing consideration is
then encapsulated in (10). Simply put, this equation relates the
coupling constants between the cavities to the decaying rate of
the cavities to each individual port.
Weshallnowseehow(10)canbesatisfied,byconsideringthe
coupling constant between two cavities that are connected with
a waveguide, as shown in Fig. 3. Our treatment closely follows
HausandLai’stheoryoncoupledcavitystructures[18].Briefly,
thecoupledmodetheoryequationsthatdescribethestructurein
Fig. 3 are
(11)
(12)
(13)
(14)
(15)
(16)
Here,
time of the two cavities, respectively,
coming and outgoing waves for the th cavity at either the input
or the output waveguide, while
and outgoing waves for the th cavity in the connecting wave
guide. Furthermore, due to the wave propagation within the
waveguide, we have
and,are the frequencies and the decaying
and are the in
and are the incoming
(17)
and
(18)
Combining (14), (16)–(18), we have
(19)
and
(20)
whichcanbepluggedbackinto(11)and(12)todetermineanef
fective coupling constant between the two cavities. In this way,
one could write the coupling constant between the two cavities
as
(21a)
(21b)
Comparing (21) with (10), we see that the criterion outlined
above can indeed be achieved with
, and. Therefore, it can be deduced that
. Thus, ideal performance for our
demultiplexer can be realized by using a symmetric waveguide
and a symmetric branch, and by choosing the appropriate phase
shift between the branch and the highQ cavities.
We note that, in general, the phase
However, since the entire signal bandwidth is typically only on
the order of a few percent of the carrier frequency, the variation
of
over the entire signal bandwidth should be small. Also, in
general the phase shift can only determined by detailed electro
magnetic simulations. The analytic calculations above however
establish the basic criteria for the design procedure. Finally, for
our purposes here, the choice of phase factor at
avoids the diverging point of the coupling constant
foreleadstoasimplifiedformalism.Inthecaseswhere
the coupling constant between the cavities diverge, and it is no
longer appropriate to describe the system in terms of the cou
pling coefficients . However, even in this case, there is still a
valid physical solution to (11)–(16). Such a physical solution
corresponds to the situation where the resonant amplitudes in
the two resonators oscillates exactly in phase.
We shall emphasize that a symmetric branch, as character
ized by equal decay of the resonance to each of the branches,
generates a finite reflection [16] by itself. In this design, it is
precisely this finite reflection from the branch that cancels the
reflected amplitude from the highQ resonances, resulting in an
overall reflectionless operation at every frequency channel for
the demultiplexer.
To visualize the behavior of our analytic theory, we plot the
resultsofequations(5)–(7)inFig.4,intheregimewherethecri
teria(8)–(10)aresatisfied.Forconcreteness,wehavearbitrarily
chosenthenormalizedresonantfrequenciesanddecayrates(
), (,), and (,
and (0.96, 0.0005), respectively. It can be found that
indeed approach 100% at the resonant frequencies, while the
reflection vanishes at these frequencies. Thus, our theory does
indeed predict the presence of an ideal demultiplexer.
We can generalize the results, as described above for a two
channel demultiplexer, to a system with
structure is shown in Fig. 5, where each port outputs a spe
cific frequency channel. Using similar theoretical derivations,
,
is frequency dependent.
and there
,
,
) as (1.0, 0.5), (1.04, 0.0005),
and
output ports. The
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JIN et al.: REFLECTIONLESS MULTICHANNEL WAVELENGTH DEMULTIPLEXER IN A TRANSMISSION RESONATOR CONFIGURATION 163
Fig. 4.
in Fig. 2(a). assuming that (? , ??? ), (? , ??? ) and (? , ??? ) are (1.0, 0.5),
(1.04, 0.0005 ), (0.96, 0.0005) respectively.
Transmission spectra for a twochannel wave demultiplexer, as shown
the transmittivity at a particular port
(22), shown at the bottom of the page.
Equation (22) describes a system that is represented in
Fig. 1(a). To apply (22) to a system as shown in Fig. 5, we note
that the essence of the derivation is the presence of a symmetric
branch with the reflection amplitude equal to the transmission
amplitude into the each of the branch. It is, therefore, conceiv
able that such a branching structure can be built with a treelike
structure. We will investigate this further in later works.
From (22), it can be clearly seen that, at a frequency channel
, when the criteria as described by (8)–(10) are satisfied,
thetransmittivitiesofallportsare zeroexceptfor port
it reaches 100%. No reflection can be found at the input port
for all frequency channels. In order to visualize the prop
agation properties described by (22), the transmission spectra
of each port in a 16channel wave demultiplexer are plotted in
Fig. 6, where assuming that
is 0.0001. It is clear that the transmittivity at each port does ap
proach100%atresonantfrequency.Thatisan
demultiplexer.
To test the validity of the above approach, the multiple
scattering method [19], [20] is employed to simulate the
propagation of waves in these structures. An experiment in
themicrowaveregionisalsocarriedout,thedetailedexperiment
setup can be found elsewhere [21]. Here, we adopt the structure
as shown in Fig. 2(a), the parameters of this photonic crystal
wave demultiplexer are chosen as follows. The photonic crystal
with a lattice constant
is composed of 0.182 radius dielectric
cylinders imbedded in a background dielectric medium. The
dielectric constants of the cylinders and background are 8.0
can be expressed as in
,where
is 0.5, ()
channelswave
Fig. 5.
in twodimensional photonic crystal.
Schematic optical circuit for realizing ? channels wave demultiplexer
Fig. 6.
that ??? is 0.5 and ??? (? ? ? ? ??) is 0.0001.
Transmission spectra for a 16channel wave demultiplexer assuming
and 1.03. respectively. The first band gap of this crystal for
Spolarized waves is in the range of 0.36 0.46
is the light velocity in vacuum. Point defect is created by
replacing a selected cylinder with a dielectric cylinder with
different radius and dielectric constant.
special cylinders and serve as two point defects, and their
radius and dielectric constants are 0.182 , 1.0 and 0.273 ,
8.0 respectively. In the simulation, we integrate the energy
fluxes at two outlets respectively. Meanwhile, the energy flux
at the outlet of a corresponding straight waveguide shown in
Fig.2(b)isalsocalculated.Forclarity,wedefinearenormalized
transmission spectrum, which is a ratio of the flux at the outlet
of the demultiplexer to that of the straight waveguide from
each outlet. From the spectrum. one can confirm whether the
reflection from the input port exists at the peak frequencies.
The measured and simulated transmission spectra are plotted
, where
andare two
(22)
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164IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 39, NO. 1, JANUARY 2003
Fig. 7.
outlets of the demultiplexer.
Measured and simulated renormalized transmission spectra at the
in Fig. 7, where diamond and solid circles are the experimental
values obtained from
and
solid and dashed lines are the corresponding simulations. It
is found that the transmission to the
Lorentzian peak at a resonant frequency of 0.4420 (
the transmission to
port also shows a single Lorentzian
peak at a resonant frequency of 0.4068 (
both peaks have their maximal transmissivity approaching
100%. Because both simulations and experiments presented
in this paper incorporate the reflection from the interfaces
at the edges of the crystal and therefore does possess some
error that over 100% transmission appears at the peak of the
port–3 spectrum. However, the presence of large transmitted
amplitudes at both frequencies is a strong and clear evidence
that the cancellation of reflection amplitude is at work here.
From energyconservation considerations, the existence of a
nearcomplete transmission at these two frequencies indicates
that the reflection is small for both frequencies for the structure.
Furthermore, both the experimental and simulated results agree
well with each other, except that there exists a little peak in
both of
andports which might be caused by very small
interaction between microcavities
reveal that directly coupling resonant tunneling effect can be
used to design wavelength demultiplexer with no reflection
at the peak frequencies.
In conclusion, we introduce a wavelength demultiplexer
operatingbythecouplingamongalowqualityfactorcavityand
microcavities with highquality factors. This optical circuit
is similar to the lowfrequency multichannel filter electronic
circuit. The merit of this optical circuit is that the design
of such a circuit is very simple because one might build
a defect to extract a peak frequency and the technological
complexityisreducedcomparedwiththechanneldroptunneling
wave demultiplexer. Furthermore, the reflection at the peak
frequencies can be fully eliminated compared with the other
method where considerable reflection appears. Eventhough our
simulation and experiment are working in a twodimensional
photonic crystal, our theory based on the coupling theory
can be valid for both two and threedimensional photonic
ports, respectively, while the
port exhibits a single
), while
). Moreover,
and. These facts
crystals. This method might also be valuable for the design
of other alloptical functional circuits.
ACKNOWLEDGMENT
The authors would like to thank Prof. Z. Zhang from HKUST
for his program of multiscattering method. They also thank the
National Research Center for Intelligent Computing System for
use of the Dawning2000A parallel computer.
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JIN et al.: REFLECTIONLESS MULTICHANNEL WAVELENGTH DEMULTIPLEXER IN A TRANSMISSION RESONATOR CONFIGURATION 165
Chongjun Jin was born in Zhejiang Province, China, in 1969. He received the
Ph.D. degree in physics from Harbin Institute of Technology, Harbin, China, in
1997.
HehasbeenwiththeInstituteofPhysics,ChineseAcademyofSciences,Bei
jing, China, since 1997, where he is currently an Associate Research Scientist.
He is also a Research Assistant at the Department of Electrical and Electronics
Engineering, University of Glasgow, Glasgow, U.K. His main research interests
are as follows: applications of photonic crystals in the fields of optoelectronics,
preparation of threedimensional photonic crystals through the selfassembly
method, nonlinear effects of photonic crystals, simulation of the propagation of
electromagneticwavesinphotoniccrystals,andpreparationoftwodimensional
photonic crystals using wet and dry etching.
Shanhui Fan received the Ph.D. degree in physics from the Massachusetts In
stitute of Technology (MIT), Cambridge, in 1997.
Hehas beenanAssistant ProfessorofElectrical EngineeringatStanfordUni
versity since 2001. Previously, he was a Research Scientist at the Research Lab
oratory of Electronics, MIT. His research interests are in computational and the
oretical studies of the basic properties and applications of micro and nanopho
tonic structures. He has published 47 journal articles, has given more than 20
invited talks at U.S. and international conferences, and holds 11 patents.
Dr.FanisamemberoftheAmericanPhysicalSocietyandtheOpticalSociety
of America.
Shouzhen Han was born in Hebei Province, China, in 1976. He received the
Master’s degree from Beijing Polytechnic University, Beijing, China, in 2002.
He is currently working toward the Ph.D. degree at the Institute of Physics,
Chinese Academy of Sciences, Beijing, China.
Daozhong Zhang was born in Sichuan Province, China, in 1943. He graduated
from the Chinese University of Science and Technology in 1965.
Since then, he has been with the Institute of Physics, Chinese Academy of
Sciences, Beijing, China, where he is currently a Research Scientist and the
Director of the Optical Physics Laboratory. His research interests include laser
spectroscopy of atoms and molecules, nonlinear optics, photonic crystals, and
light localization. He has published more than 100 papers.