104 J. Opt. Soc. Am. B/Vol. 23, No. 1 /January 2006 Z. Zhang and M. Qiu
Coupled-mode analysis of a resonant channeldrop filter using waveguides with
mirror boundaries
Ziyang Zhang and Min Qiu
Laboratory of Optics, Photonics and Quantum Electronics, Department of Microelectronics and InformationTechnology, Royal Institute of Technology, Electrum 229, 164 40 Kista, Sweden
Received May 6, 2005; revised August 8, 2005; accepted August 26, 2005
. INTRODUCTIONhannel drop filters (CDFs) are the essential componentsf photonic integrated circuits and dense-wavelength-ivision-multiplexing optical communication systems.mong various existing devices, such as fiber Bragg grat-
ngs, Fabry–Perot filters, and arrayed waveguide grat-ngs, resonant CDF have attracted much attention ashey can potentially be used to select a single channelith a narrow linewidth. Photonic crystals, which offer aew way of controlling light propagation, are adequate forltracompact, highly wavelength-selective optical devicesnd serve as a good candidate for future CDFs.1,2
There have been different designs of CDFs in two-imensional (2D) photonic crystals3,4 and 2D photonicrystal slabs.5–11 One design is the surface-emitting type,hich consists of a waveguide and a cavity system. The
nput signal from the waveguide tunnels into the cavitynd is emitted in the vertical direction. The in-plane de-ign usually involves two waveguides (bus and drop) andcavity system with two degenerate modes of opposite
ymmetry. The channel to be selected comes from the busaveguide, tunnels into the cavity, and is eventually
ransferred to the drop waveguide.The operation principle of in-plane CDFs based on two
egenerate cavity modes has matured over the years.12–14
owever, so far there have not been detailed studies onn in-plane CDF system in which the waveguides are ter-inated by mirrors. In this paper, we will demonstrate
he effect of waveguides’ mirror boundaries on the drop ef-ciency of the filter. The cavity only needs to support aingle mode of arbitrary symmetry.
. FOUR-PORT SYSTEM WITH TWOAVEGUIDES AND A SINGLE-MODE
AVITYhis part of the analysis is similar to Ref. 13. We treat the
n-plane CDF as a four-port system. Placed between thewo waveguides is a cavity that supports a singletanding-wave mode. In practice, the cavity can be consid-red a microcavity in photonic crystals or some generali-ation of such a device. A schematic is shown in Fig. 1.he propagation mode in both waveguides is supposed toe single, nonleaky, and not highly dispersive at the reso-ant frequency of the cavity mode.The cavity–waveguide interaction length is fully con-
ained between the input and output reference planes.he amplitudes of the incoming waves into the system areenoted by S+i, and S−i are the amplitudes for the outgo-ng waves (i=1, 2, 3, 4). Let a be the amplitude of the cav-ty mode. The time evolution of the cavity mode can be ex-ressed by
da
dt= �j�0 −
1
�o−
1
�e−
1
��e�a + �1 exp�− j�d/2�S+1
+ �2 exp�− j�d/2�S+2 + �3 exp�− j��d/2�S+3
+ �4 exp�− j��d/2�S+4, �1�
here �0 is the resonant frequency, 1/�o is the decay rateue to loss, and 1/�e and 1/��e are the decay rates intoaveguide 1 (W1) and waveguide 2 (W2), respectively. �1nd � are coupling coefficients associated with the
2
006 Optical Society of America
fs
iaptirsFwt=
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da
t
wWws�t
�ml=
Fwms
Fn
Z. Zhang and M. Qiu Vol. 23, No. 1 /January 2006 /J. Opt. Soc. Am. B 105
orward- and backward-propagating modes in W1, andimilarly defined are �3 and �4 for W2.
We assume exp�j�t� time dependence for both the cav-ty mode and the propagation mode in the waveguidesnd choose port 1 as input with s+1=1. The rest of theorts are left open and matched with s+2=s+3=s+4=0. Inhe ideal case in which the system is symmetric and thentrinsic loss is negligible �1/�o=0�, the power transfer atesonance is �s−1�2= �s−2�2= �s−3�2= �s−4�2=0.25. The linehape of the transmission spectra is simply Lorentzian.rom the definition of the system Q factor Qs=�0 /��,here �� is the full width at half-maximum (FWHM) of
he transmission spectrum, it is easy to derive Qs� � /4.
ig. 1. Four-port system with a resonator placed between twoaveguides. The resonator supports a single standing-waveode. The system is symmetric with respect to the planes �1,2,
hown as the gray dashed lines.
ig. 2. Four-port resonant system with one waveguide termi-ated by a mirror.
0 e
. FOUR-PORT SYSTEM WITH ANE-MIRROR WAVEGUIDEOUNDARYow we apply a mirror at port 3. The input is still at port. In fact, the mirror can also be placed at port 2 or 4. Theesults are similar. The initial conditions are altered to+1=1, s+3=�s−3, and s+2=s+4=0, where � is the complexeflection coefficient defined by
� = � exp�j��, 0 � 1. �2�
For the moment we assume the mirror is frequency in-ependent; i.e., all channels are reflected with the samemplitude � and phase �.By power conservation, the outgoing waves can be writ-
en as13
s−1 = exp�− j�d�s+2 − exp�− j�d/2��2*a, �3�
s−2 = exp�− j�d�s+1 − exp�− j�d/2��1*a, �4�
s−3 = exp�− j��d�s+4 − exp�− j��d/2��4*a, �5�
s−4 = exp�− j��d�s+3 − exp�− j��d/2��3*a, �6�
here � and �� are the propagation constants in W1 and2, respectively. In this paper, we assume the twoaveguides are identical and the cavity mode is either
ymmetric or antisymmetric with respect to the planes1,2, shown as gray dashed lines in Fig. 2. Thus the equa-
ions above can be much simplified by setting
� = ��, �7�
�e = ��e, �8�
�i = �1/�e exp�j�i�,i = 1,2,3,4. �9�
i is either 0 or depending on the symmetric or antisym-etric property of the cavity mode. The decay rate is re-
ated to the Q factors. The intrinsic Q factor is Qo�0�o /2, and the coupling Q factor is Qe=�0�e /2.Using Eqs. (1)–(9), we can derive the outgoing waves:
S−1 = −exp�j��1 − �2 − �d��
� exp�j��3 − �4 − �d� + 2 +�e
�o+ j�� − �0��e
,
�10�
S−2 = exp�− j�d�1 −1
� exp�j��3 − �4 − �d�� + 2 +�e
�o+ j�� − �0��e
, �11�
iscaato
pfartpapt
(t
nlt
�m
inp
4MFoctnppTf=i2gt
Fp
106 J. Opt. Soc. Am. B/Vol. 23, No. 1 /January 2006 Z. Zhang and M. Qiu
S−3 = −exp�j��1 − �4 − �d��
� exp�j��3 − �4 − �d�� + 2 +�e
�o+ j�� − �0��e
,
�12�
S−4 = − exp�j��1 − �3
− �d��1 + � exp�j��3 − �4 − �d��
� exp�j��3 − �4 − �d�� + 2 +�e
�o+ j�� − �0��e
.
�13�
We assume the wave is totally reflected back at port 3,.e., �=1, and let �=�−�d+�3−�4, �3−�4 is decided by theymmetry property of the cavity mode, �3−�4=0 if theavity mode is even with respect to the mirror plane �2,nd �3−�4= ± if odd. One can vary the phase term � bydjusting the propagation constant � or the phase � in-roduced by the reflection at port 3. The intensities of theutgoing waves become
�s−1�2 =1
�2 +�e
�o+ cos ��2
+ �e2�� − ��0 − sin �
�e��2 , �14�
�s−2�2 =�1 +
�e
�o+ cos ��2
+ �e2�� − ��0 − sin �
�e��2
�2 +�e
�o+ cos ��2
+ �e2�� − ��0 − sin �
�e��2 , �15�
�s−3�2 = �s−1�2, �16�
�s−4�2 =2�1 + cos ��
�2 +�e
�o+ cos ��2
+ �e2�� − ��0 − sin �
�e��2 . �17�
It is also worth writing down the cavity mode equation
a =��e
cos � +�e
�o+ 2 + j�� − ��0 − sin �
�e���e
. �18�
From Eqs. (14)–(18), we notice that the reflection atort 3 has introduced a shift ��=sin � /�e in the resonantrequency. When �=k , the resonance shift is eliminated,nd, meanwhile, the resonant transmission at each porteaches either maximum or minimum. For a lossless sys-em, i.e., �=1 and 1/�o=0, when �=2k one ninth of lightower at resonant frequency is reflected back to port 1,nd the remaining eight ninths are divided evenly intoorts 2 and 4. When �=2k + , the resonant channel isotally reflected back to port 1.
The line shape of the transmission spectra from Eqs.14)–(17) still remains Lorentzian, whereas system Q fac-or is now
Qs =�0�e + sin �
2
1
2 +�e
�o+ cos �
. �19�
Recall that Qe=�0�e /2, and if Qe�1 we have
Qs =Qe
2 +Qe
Qo+ cos �
. �20�
If Qo�Qe, we can ignore the Qe /Qo term in the denomi-ator. The system Q factor and thus the transmission
inewidth can be adjusted between Qe /3 and Qe by varia-ion of the phase term �.
From Eq. (20) we also see that Qs and �� are related by. We will discuss more about this property in the two-irror system.We can get similar results when the mirror is applied
n port 4 or port 2. The power of the reflected resonant sig-al at port 1 ranges from 1/9 to 1, and the remaining twoorts with no mirror share the amount left.
. FOUR-PORT SYSTEM WITH TWO-IRROR WAVEGUIDE BOUNDARIES
rom the analysis in Section 3, we know that applyingnly one mirror boundary is not enough to achieve 100%hannel transfer. As shown in Fig. 3, we now terminatewo ports by mirrors in an attempt to improve the chan-el drop effect. The mirrors can be placed at port 2 andort 3 to achieve a forward channel drop or at port 2 andort 4 for a backward drop. Port 1 still works as the input.he analysis is similar, and we choose to work on the
ormer case. The initial conditions are then s+1=1, s+2�1s−2, s+3=�2s−3, and s+4=0, where �i=�i exp� j�i�,=1,2. The reflection coefficient �i can be different at portand port 3, but they are still frequency independent. To
eneralize this situation, we denote the distance betweenhe mirror and the cavity center by d1 and d2 separately.
We rewrite Eqs. (1) and (3)–(6) as
da
dt= �j�0 −
1
�o−
1
�e−
1
��e�a + �1 exp�− j�d1�s+1
+ �2 exp�− j�d2�s+2 + �3 exp�− j�d1�s+3
+ �4 exp�− j�d2�s+4, �21�
ig. 3. Four-port system with mirror boundaries applied toorts 2 and 3.
w
mat=it
�
w
t
c
Z. Zhang and M. Qiu Vol. 23, No. 1 /January 2006 /J. Opt. Soc. Am. B 107
The intensity of the outgoing waves are found to be
�s−1�2 =�cos �1 − �1� �e
�o+ �2 cos �2��2
+ �e2�1
2�� − ��0 +sin �1+�1�2 sin �2
�e�1��2
�2 +�e
�o+ �1 cos �1 + �2 cos �2�2
+ �e2�� − ��0 +
�1 sin �1+�2 sin �2
�e��2 , �26�
�s−4�2 =�1 + 2�1 cos �1 + �1
2��1 + 2�2 cos �2 + �22�
�2 +�e
�o+ �1 cos �1 + �2 cos �2�2
+ �e2�� − ��0 +
�1 sin �1+�2 sin �2
�e��2 , �27�
here a1,2=2�d2,1−�1,2+�� and ��=�1−�2=�4−�3.From Eq. (27), it is obvious that, for achieving maxi-um channel transfer, the mirror loss has to be minimal
nd cos �1=cos �2. To simplify the analysis, we assumehe two mirrors have the same reflection coefficient (�1�2=� and �1=�2=�) and the distance between the cav-
ty and each mirror is also the same �d1=d2=d�. The in-ensities of outgoing waves at port 1 and port 4 become
s−1�2 =�cos � − �
�e
�o− �2 cos ��2
+ �e2�2�� − ��0 +
�1+�2�sin �
�e� ��2
�2 +�e
�o+ 2� cos ��2
+ �e2�� − ��0 + 2� sin �
�e��2 ,
�28�
�s−4�2 =�1 + 2� cos � + �2�2
�2 +�e
�o+ 2� cos ��2
+ �e2�� − ��0 + 2� sin �
�e��2 ,
�29�
here �=2�d−�+��.The cavity mode becomes
a = exp�− j�d1��1/�e
�1 + � exp�− j��
�2 +�e
�o+ 2� cos �� + j�e�� − ��0 + 2� sin �
�e��
. �30�
A shift in resonant frequency is observed:
�� =2� sin �
�e=
�0
Qe� sin �. �31�
The intensity spectra remain Lorentzian, and the sys-em Q factor is
Qs =�0�e + 2� sin �
2�2 +�e
�o+ 2� cos ��
=Qe + � sin �
2 +Qe
Qo+ 2� cos �
Qe
2 +Qe
Qo+ 2� cos �
. �32�
Usually we have Qe�1, and � sin � on the numeratoran be neglected.
From the equations above, we can get some importantonclusions.
(a) From Eq. (29) the system can work as a filter ifwo identical mirrors are placed symmetrically with re-pect to the cavity center (�1=�2 and d1=d2). In the idealase, in which the mirror reflection is lossless ��=1� andhe system loss is also negligible �1/�o=0�, the channelith central frequency at �0+�� is completely dropped toort 4. Figure 4 shows a 3D surface plotted using Eq. (29)nder the assumptions Qe=Q0 /10=2�103 and �=1. Wean clearly see the drop efficiencies of this two-mirror sys-em at different � and � values.
(b) By one’s modifying the phase term �, the filteran work at different resonant frequencies and with dif-erent linewidths. However, the frequency shift ���� andhe system Q factor �Qs� are related to each other. Fromxpressions (31) and (32) and using sin2 �+cos2 �=1, wend
���
�0Qe�2
+ � Qe
2Qs−
Qe
2Qo− 1�2
= �2. �33�
ccording to Eq. (33), the relation between �� and Qs is alosed curve. As we shall see later in Section 6, cases (a)nd (b) can be favorable properties as they make postfab-ication tuning possible. For example, if the index of theaveguide arms can be tuned effectively (by electric,agnetic, or thermal means), the phase term can be
ig. 4. (Color online) Normalized transmission at port 4 versusand � under the assumptions Q =Q /10=2�103 and �=1.
e 0
mts
dpcWswilt
c(
niatl
tp=awii
rltltt(Fcp
5MIb
iawfwlpwtsnia
trwteotpc
Fs�Wt
108 J. Opt. Soc. Am. B/Vol. 23, No. 1 /January 2006 Z. Zhang and M. Qiu
odified in terms of the propagation constant �, and, inurn, both �� and Qs can be adjusted according to expres-ions (31) and (32).
(c) When �=1 and �=2k + , the output at port 4rops to zero, and all the channels are reflected back toort 1. In this special case, the cavity mode amplitude be-omes zero for all frequencies. Light cannot tunnel from1 into the cavity and thus is totally reflected back. The
ystem Q factor reaches its intrinsic value Qo. However, ife can generate light inside the cavity, it will be confined
n the cavity and in between the two mirrors. In this case,ight does not leak not from port 1 or port 4. We will verifyhis property in Section 6.
(d) This device can also work as a modulator for thehannel with central frequency �0. We let �=�0, and Eqs.28) and (29) become
�s−1�2 =�cos � − �
�e
�o− �2 cos ��2
+ �1 + �2�2 sin2 �
�2 +�e
�o+ 2� cos ��2
+ 4�2 sin2 �, �34�
�s−4�2 =�1 + 2� cos � + �2�2
�2 +�e
�o+ 2� cos ��2
+ 4�2 sin2 �. �35�
Without any loss, equations above are simplified into
�s−1�2 =1 − cos �
2, �36�
�s−4�2 =1 + cos �
2. �37�
Both the transmitted and the reflected signals are si-usoidally modulated by �. Hence, by one’s varying the
ndex of the waveguides or changing the mirror bound-ries, the output signal can be modulated via the phaseerm �. The function is similar to a Mach–Zehnder modu-ator, but the device can be made more compact.
Figure 5(a) shows the normalized intensity of theransferred ��s−4�2� and reflected ��s−1�2� signals versus thehase term � for the symmetric system (�1=�2 and d1d2) and lossless mirrors ��=1�. The value for Qo is takens ten times Qe. The maximum output at port 4 is 95.2%hen �=2k . Figure 5(b) also verifies the case discussed
n (c) listed above. When �=2k + , light does not tunnelnto the cavity and is totally reflected back.
The system performance will deteriorate when the mir-or loss is present, i.e., �1. In most cases, the mirroross and the cavity’s intrinsic loss play a major role, sincehe cavity–waveguide interaction length �d1+d2� is notong enough to introduce significant propagation loss dueo fabrication imperfections. The loss properties of thewo-mirror system can be determined from Eqs. (28) and29) or, in the more general case, from Eqs. (26) and (27).or example, assume �=0.9, Qo=10Qe, and �=2k ; inase (b), when the system works as a CDF, the maximumower transfer at resonance is reduced to 86%.
. FOUR-PORT SYSTEM WITH A RESONANTIRROR
n the previous analysis we have assumed the mirror toe frequency independent. Though the two-mirror system
n Section 4 can achieve 100% channel transfer, it reflectsll the other channels back to port 1, and other devicesill have to be used to separate the reflected channels
rom the input ones. If we can terminate the waveguidesith wavelength-selective mirrors so that only the se-
ected channel is reflected, the other channels can justass through. Although there exist various designs ofavelength-selective mirrors, one simple way is to use
he same single-mode cavity as a resonant mirror. Thechematic is shown in Fig. 6. Note that the passing chan-els from W1 do not couple into W2, and so the frequency-
ndependent mirror remains at port 3 to simplify thenalysis. Port 2 (5) is connected to a resonant mirror.We assume resonators a1 and a2 to be identical. Both of
hem support a single standing-wave mode with the sameesonant frequency �0. Although a1 decays into bothaveguides with rate 1/�e, a2 decays into only W1 with
he same rate. If the two resonators are placed farnough, their direct coupling can be neglected, and theynly indirectly couple with each other via W1. The nota-ions of the variables in Fig. 6 are similarly defined in therevious sections. The time evolution of the cavity modesan be expressed by
da1
dt= �j�0 −
1
�o−
2
�e�a1 + �1 exp�− j�d1/2�S+1
+ �2 exp�− j�d1/2�S+2�3 exp�− j�d1/2�S+3
+ � exp�− j�d /2�S , �38�
ig. 5. (Color online) (a) Reflection (solid curve) and transmis-ion (dashed curve) spectra of the channel with central frequency0 under the assumptions �1=�2, d1=d2, �=1, and Qo /Qe=10. (b)hen �= , the channel is totally reflected back. Light does not
unnel into the cavity and therefore undergoes no cavity loss.
4 1 +4
w
oTq
Fm d usin
Z. Zhang and M. Qiu Vol. 23, No. 1 /January 2006 /J. Opt. Soc. Am. B 109
da2
dt= �j�0 −
1
�o−
1
�e�a2 + �5 exp�− j�d2/2�s+5
+ �6 exp�− j�d2/2�s+6. �39�
By power conservations, we have
s−1 = exp�− j�d1�s+2 − exp�− j�d1/2��2*a1, �40�
s−2 = exp�− j�d1�s+1 − exp�− j�d1/2��1*a1, �41�
s−3 = exp�− j�d1�s+4 − exp�− j�d1/2��4*a1, �42�
s−4 = exp�− j�d1�s+3 − exp�− j�d1/2��*a1, �43�
ig. 6. Combined system with a single-mode resonator placed beirror by using the same single-mode cavity. Port 3 is terminate
3 s
−1 w
s−5 = exp�− j�d2�s+6 − exp�− j�d2/2��6*a2, �44�
s−6 = exp�− j�d2�s+5 − exp�− j�d2/2��5*a2, �45�
s−7 = exp�− j�d2�s+8, �46�
s−8 = exp�− j�d2�s+7. �47�
At the interface,
s±2 = s�5, �48�
s�4 = s±7. �49�
With the initial conditions, s+1=1, s+6=0, s+8=0,
two waveguides on the left. Port 2 (5) is connected to a resonantg an ordinary mirror that reflects all frequencies.
+3=�s−3, and �=� exp�j��, the outgoing waves become
For the ideal case, the selected channel should comeut only at port 8 and the passing channels at port 6.here should be no reflection back at port 1 for all fre-uencies. Let s =0, and we have
� = 1, �54�
� − �d1 − �� = 2k , �55�
��d1 + d2� = 2k + , �56�
here ��=� −� =� −� =� −� .
tween
1 2 3 4 5 6
scp
is
sac
itsa
tIaqfitmFcca
w
�−a=ota
pict
6TTstlhctbbpam
lTsbnTa
F�t=
Fcwcvd
110 J. Opt. Soc. Am. B/Vol. 23, No. 1 /January 2006 Z. Zhang and M. Qiu
Recall that ��=0 if the cavity mode is even with re-pect to the mirror plane �2 and ��= if odd. With theonditions above, the intensities of the outgoing waves atort 6 and port 8 become
�s−6�2 =� �e
�o�2
+ �e2�� − �0�2
�2 +�e
�o�2
+ �e2�� − �0�2
, �57�
�s−8�2 =4
�2 +�e
�o�2
+ �e2�� − �0�2
. �58�
The line shape of the transmission spectra is simplifiednto Lorentzian. The central frequency is at �0, and theystem Q factor is now
Qs =�0�e
2�2 +�e
�o�
=Qe
2 +Qe
Qo
. �59�
When the conditions in Eqs. (54)–(56) are matched, theystem can work as an ideal in-plane CDF. The reflectiont port 1 is eliminated, and, without loss, the selectedhannel at �0 can be completely transferred to port 8.
It is worth noting that other existing CDF designs us-ng two single-mode cavities require direct coupling be-ween them to achieve frequency degeneracy13 and to en-ure high drop efficiency; however, this requirement isvoided in our system.It is also important to study the influence of cavity de-
uning on the performance of the resonant mirror system.n the original coupled-mode equations (38) and (39), wellow the two cavities to resonate at slightly different fre-uencies, �01 (for cavity A1) and �02 (for cavity A2). Werst assume the two resonance frequencies are close sohat the waveguide mode is not affected and both cavityodes still decay into the waveguide with the same rate.ollowing the same procedure and assuming the phaseonditions from Eqs. (55) and (56) are still satisfied, wean derive the frequency response of the outgoing wavest port 8:
ig. 7. (Color online) Power transfer at port 8 versus frequency�−�02� /�02 and relative detuning ��02−�01� /�02 between thewo cavities under the assumptions Qe=Q0 /10=2�103 and �0.9.
�s−8�2 = � �1 + ���S2 + 1�e�
1�e
+ S2�1 + � + �eS1��2
, �60�
here
S1,2 =1
�e+
1
�o+ j�� − �01,02�. �61�
In Fig. 7, �s−8�2 is plotted versus the relative frequency�−�02� /�02 and the relative resonance detuning ��02�01� /�02 between the two cavities. The values in Eq. (60)re taken as Qe=Q0 /10=2�103 and �=0.9. When �02�01, the maximum power transfer at resonance is 86.1%wing to mirror losses and intrinsic cavity losses. Whenhere is a relative detuning of 0.05%, the power transfert frequency �02 further drops to 68.3%.In the extreme case when the two cavities become com-
letely detuned, the system is split into two noninterfer-ng parts. Port 1-4 works as the one-mirror system for thehannel with central frequency �01 as discussed in Sec-ion 3; port 5 and 6 serves as a resonant mirror for �02.
. TWO-MIRROR SYSTEM REALIZED INWO-DIMENSIONAL PHOTONIC CRYSTALShough the coupled-mode analysis above is general, weeek to verify it in a specific embodiment. Photonic crys-als, which bring up a new territory of molding the flow ofight, serve as a good candidate for demonstration. Thereave been many proposals on in-plane CDFs in photonicrystals with two open waveguides and a resonator sys-em supporting multiple modes.4,9–11 However, the em-odiment of the system with mirror boundaries has noteen well discussed. We hereby present a system in a 2Dhotonic crystal shown in Fig. 8. The lattice constant is L,nd the regular airhole radius R=0.36L. The dielectricaterial is InP with dielectric constant 10.5.Both waveguides are terminated by a photonic crystal
attice, which works as a perfect mirror in the 2D case.he central large airhole (labeled A) with radius 0.55Lupports a monopole cavity mode that is symmetric inoth in-plane directions.3,15 Without mirrors, the reso-ant frequency of the cavity mode �0=0.2627066�2 c /L�.he guided mode in both waveguides has a wave vector ofbout �0.25�2 /L at this frequency.3 A combination of 2D
ig. 8. (Color online) Four-port system realized in a 2D photonicrystal. The mirror is applied by one’s terminating the waveguideith a crystal lattice. The cavity A is constructed by one’s in-
reasing the radius of the central airhole to 0.55L, and it pro-ides a monopole mode that is symmetric in both in-planeirections.
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Z. Zhang and M. Qiu Vol. 23, No. 1 /January 2006 /J. Opt. Soc. Am. B 111
nite-difference time-domain (FDTD) techniques16 andadé approximation with Baker’s algorithm17 is used for
he Q-factor calculations. When both waveguides arepen, we find the in-plane Q factor of the cavity mode�=4435. The intrinsic Q factor Qo is larger than 2�106.� is related to Qe by Q�=1/ �Qe
−1+Qe�−1�, where Qe
�0�e /2 and Q�e=�0��e /2. The two waveguides are iden-ical; Qe should be twice as large as Q�, and thus Qe8870. When the distance between the two mirrors 2d is4L, the system Q factor Qs is calculated to be 2530, andhe resonant frequency is shifted to �00.2627261�2 c /L�. The Qs and �1 obtained from FDTDimulations match Eq. (33).
Figure 9 shows the normalized transmission intensity�s−4�2� versus frequencies for the case 2d=14L. The ra-ius of airholes labeled B in Fig. 8 is the same as regularirholes. The curve is the theoretical curve plotted usingq. (29), and the circles represent the data obtained byD FDTD transmission simulations. The power transfert resonant frequency is close to 100% owing to a largeo /Qe ratio.The phase term � can be varied in a number of ways,
.e., by modulating the index of the waveguides or chang-ng the mirror boundaries. In our simulations, we chooseo modify the radius �RB� of the airholes labeled B in Fig.. We find that, by increasing RB, � also increases, andhus both the resonant frequency and the system Q factorhange. We run FDTD simulations for each RB setup andet a series of discrete (��, Qs) values. As shown in Fig.
ig. 9. (Color online) (a) Intensity spectrum of the transferredignal at port 4. (b) Detailed spectrum around the central fre-uency. The curve is plotted using Eq. (29), and the circles areata obtained from FDTD simulations.
0(a), the data points obtained numerically match theurve plotted theoretically using Eq. (33).
The curve in Fig. 10(a) can be divided into four regionsepending on the phase term �. When �=k + /2, theesonant frequency shift becomes maximum, and the sys-em Q factor approaches Qe /2. When �=2k , the reso-ant frequency stays at �0, and the system Q factoreaches its minimum �Qe /4. Thus by adjusting the phaseerm �, one can facilitate post fabrication tuning. Tochieve large frequency tuning with little Q variations,ne can work on the lower half of the curve in Fig. 10(a),here the system Q does not change significantly in a
arge range of resonant frequency variations. Qe can alsoe reduced to increase the frequency tuning range, ac-ording to Eq. (31). On the other hand, the upper half ofhe curve can be used to tune the system Q factor at thexpense of a small shift in the resonant frequency.
When �=2k + , the system Q factor goes to its maxi-um Qo; however, light cannot couple from the wave-
uide (W1) into the cavity and is totally reflected back ife send in signals from port 1. On the other hand, if weanage to generate light inside the cavity, it will oscillate
etween the two mirrors and does not leak out from port 1r port 4. We put a point Gaussian source inside the cav-
ig. 10. (Color online) (a) System Q factor versus resonant fre-uencies. The curve is plotted using Eq. (33), and the circles rep-esent the data from FDTD simulations. (b) Hz field distributionhen �=2k + . A point Gaussian pulse is placed inside the cav-
ty as light source.
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112 J. Opt. Soc. Am. B/Vol. 23, No. 1 /January 2006 Z. Zhang and M. Qiu
ty as a light source in FDTD simulations, and, after suf-cient time steps, the Hz field distribution of the cavityode is shown in Fig. 10(b). The field concentrates in the
avity region and in between the two mirrors. The fieldmplitude near the two open ports is almost zero.For the resonant mirror system, the structure is shown
n Fig. 11. We apply a frequency-independent mirror atort 3 with reflection coefficient �. Cavity A2, which sup-orts the same mode as A1, works as a resonant mirror atort 6.The crystal lattice is the same as in the previous two-irror system. Both cavities are formed by our increasing
he airhole (A1, A2) radius to 0.55L. Some extra care muste taken so that the two cavity modes work at the sameesonant frequency and decay rate into W1. For conve-ience in the FDTD simulation, we fill the enlargedirhold A2 with a certain gas. The dielectric constant ofhe gas is tuned to be 1.18245. In practice, the radius of2 can be modified properly or the structural parametersf the neighboring airholes can be finely adjusted to en-ure the same degeneracy. The radius of B holes is set to.45L. The distance d1 and d2 are adjusted as shown inig. 11.Figure 12 shows the transmission spectra of �s−6�2 and
s �2. The solid curves are theoretical curves plotted us-
ig. 11. (Color online) Port 4 is terminated by a resonant mirroaterial inside hole A2 is tuned so that the cavity mode has the
ig. 12. (Color online) Intensity spectra of the dropped signal atort 6 and the transferred signal at port 8. The solid curves arelotted using Eqs. (57) and (58). The circles are data obtainedrom FDTD simulations.
−8
ng Eqs. (57) and (58). The circles represent data obtainedy 2D FDTD transmission simulations. The line shape re-ains Lorentzian. At resonant frequency, the signal
ower is dropped close to zero, and the power transferreds almost 100%.
. CONCLUSIONSn summary, we have investigated the multiport systemith a single standing-wave mode cavity and twoaveguides terminated by mirror boundaries. Coupled-ode analysis shows that, with the presence of mirrors,
oth the resonant frequency and the system Q factor areependent on the phase term introduced by reflection athe mirror and wave propagation between the two ports.he relation between the system quality factor and theesonant frequency is a closed curve. The two-mirror sys-em can work as a channel drop filter if two identical mir-ors are placed symmetrically to the cavity center. Withhe absence of mirror loss and negligible intrinsic systemoss, 100% channel transfer can be achieved. We improvehe system by applying a resonant mirror so that theassing channels are not affected. The uniqueness of theystem is that direct coupling between the two cavities isot required. We have also verified the coupled-modenalysis by demonstration of the two-mirror system in 2Dhotonic crystals. We believe this new concept of introduc-ng mirrors into resonant systems will find applications inhe designs of future photonic devices.
CKNOWLEDGMENTShis research was supported by the Individual Grants for
he Advancement of Research Leaders from the Swedishoundation for Strategic Research (SSF), the SSF Strate-ic Research Center in Photonics, and the Swedish Re-earch Council under project 2003-5501. Correspondinguthor Z. Zhang can be reached by e-mail [email protected].
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