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Analysis of response characteristics for polymer directional coupler electro-optic switches
Optics Communications 281 (2008) 5998–6005
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Analysis of response characteristics for polymer directional coupler electro-optic switches Chuan-Tao Zheng, Chun-Sheng Ma *, Xin Yan, Xian-Yin Wang, Da-Ming Zhang State Key Laboratory on Integrated Optoelectronics, College of Electronic Science and Engineering, Jilin University, Changchun 130012, PR China
a r t i c l e
i n f o
Article history: Received 21 July 2008 Received in revised form 4 September 2008 Accepted 4 September 2008
Keywords: Directional coupler Electro-optic switch Transmission power Switching voltage Switching time Cutoff switching frequency
a b s t r a c t By using the coupled mode theory, electro-optic modulation theory, conformal transforming method, image method and the proposed transfer matrix technique, novel expressions for the both cases of the low switching frequency and the ultra-high switching frequency are presented for analyzing the transmission powers, rise time, fall time, switching time and switching frequency of the polymer directional coupler electro-optic switches. Simulation results of an application based on the technique show that, the switching voltage and coupling length are about 1.457 V and 4.374 mm, respectively, and the switching time and cutoff switching frequency are about 32.8 ps and 114.7 GHz, respectively, for the designed switch. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction As the fast development of optical communication systems and technologies, optical switches and arrays have been playing an important role in information transferring, information exchange, optical cross-connect (OXC), optical add-drop multiplexer (OADM), and optical line protection (OLP), due to their great applications in optical signal processing, optical computer, optical instrument, equipment, and sensors [1–4]. Nowadays, in order to increase the bandwidth, capacity, and speed of the local and trunk optical networks, optical switches should be designed and fabricated with short switching time and high switching frequency, which should be up to the orders of ps and GHz, respectively [5–6]. Switching time and switching frequency depend on many factors, which involve the response time of the electro-optic polymer materials, the light-wave propagation velocity along the waveguide, the microwave propagation velocity along the travelingwave electrodes, the waveguide length, and the electrode structure of the switch. First, since the response time of almost all polymers, e.g. the polymer core AJ309 [7] used in this paper, can be as fast as subpicosecond or femtosecond, which is so fast compared with the period of the electric signal that it can be neglected, so we can reasonably think that, there exists no delay between the change of the electric signal and that of the refractive index of the electro-optic polymer. Moreover, the microwave propagation velocity depends * Corresponding author. Tel.: +86 431 85103909. E-mail address: [email protected] (C.-S. Ma). 0030-4018/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2008.09.021
on its effective refractive index, which can be calculated by using the characteristic parameters of the traveling-wave electrode. These parameters are obtained by solving the traveling-wave line equation through equaling the traveling-wave electrode to distributed parameter circuit [8]. In order to analyze the response characteristics including the switching time and switching frequency, a simulation technique is proposed in this paper. First in Section 2, the structural schematic of the switch and its traveling-wave electrode are presented. The push–pull electrodes are analyzed by utilizing the conformal transforming method and image method. The power transfer matrixes due to the change of the operation voltage applied on the traveling-wave electrodes are obtained by using the electro-optic modulation theory and coupled mode theory. Novel expressions are presented for analyzing the transmission powers, rise time, fall time, switching time, and switching frequency for the both cases of the low switching frequency and the ultra-high switching frequency. Then in Section 3, the simulation for the response characteristics is performed, which include the characteristic impedance, microwave effective refractive index, output powers, switching time, and cutoff switching frequency. Furthermore in Section 4, in order to check the accuracy of this technique, comparisons are carried out between the results calculated from this technique and those from the experimental method and extended point-matching method reported in Refs. [9,10]. Finally, a conclusion is reached in Section 5 based on the analysis and discussion.
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coefficient. Let b2 be the thickness of the upper/under buffer layers, n2 be the refractive index of the upper/under buffer layers and the cladding beside the rib, and a2 be their bulk amplitude attenuation coefficient. Let b3 be the thickness of the upper/under electrodes, n3 be its refractive index, and j3 be its bulk extinction coefficient. Let n4 be the refractive index of the cladding above the upper electrode, and a4 be its bulk amplitude attenuation coefficient. Denote Us as the switching voltage.
2. Model and theory 2.1. Model and structure Fig. 1 shows the structural diagram of the polymer directional coupler electro-optic switch, which consists of two identical parallel rib waveguides and a push–pull electrode structure. The coupling region is shown in Fig. 1a, where d is the coupling gap between the waveguides and L is the length of the coupling region. The cross-section of the coupling region is shown in Fig. 1b, where the structure of the rib waveguide is given as: air/upper electrode/ upper buffer layer/core/under buffer layer/under electrode/substrate, and only the core is electro-optic poled polymer material. The push–pull electrode structure consists of a pair of upper electrodes and a pair of under electrodes. During the poling, the applied poling voltage Upol is shown in Fig. 1b; during the operation, the applied operation voltage U is shown in Fig. 1c. Denote W as the electrode width, and G as the electrode gap. Let a be the core width, b1 be the core thickness, h be the rib height, n1 be the core refractive index, and a1 be its bulk amplitude attenuation
2.2. Electric field and refractive index change The electric field along the y-direction in the rib core is the sum of three parts as follows:
Eð1Þ y ðx; yÞ ¼ E1y ðx; yÞ þ E2y ðx; yÞ þ E3y ðx; yÞ;
ð1Þ
where E1y(x, y) is the uniform electric field caused by a pair of upper/under electrodes, which is determined by
E1y ðx; yÞ ¼
n22 U : 2n21 b2 þ n22 b1
ð2Þ
a P12
P10=P0 z =0 P11(0)=P0' P21(0)=0
z=L
A B
C D
d
P20=0
P11(L) P21(L)
P22
θ
Input passive region
Electro-optic active region
Output passive region
Upol
b n4,α 4
n2,α 2
y G upper electrode 0 d n2,α 2
upper buffer layer core electro-optic material
h
W n3,κ3
b3
n2,α 2
b2
a
n1,α 1
n4,α 4 x n2,α 2
b1
under buffer layer
n2,α 2
b2
under electrode
n3,κ3
b3
GND
c
U=Us
GND
U
U=0
~ n4
n4
y
W
G
n33,κ33 n2
upper electrode upper buffer layer core electro-optic material
0
n2
d h
b3 b2
a
n1
x
n2
b1
under buffer layer
n2
b2
under electrode
n3,κ3
b3
Fig. 1. (a) Structural diagram, (b) cross-section with poling voltage, and (c) cross-section with operation voltage in the electro-optic active region of polymer directional coupler electro-optic switches with push-pull traveling-wave electrodes.
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E2y(x, y) is the non-uniform electric field caused by the two upper coplanar electrodes, by using the conformal transforming method [11] and image method [12], which can be expressed as
E2y ðx; yÞ ¼ ð1 rÞ
1 X
ri E20;y ðx; y þ 2ib2 Þ;
ð3Þ
i¼0 n2 n2
g U dw dw where r ¼ n12 þn22 ; E20;y ðx; yÞ ¼ 2K ; ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; z ¼ x þ jy; 0 Im dz dz 2 2 2 2 2 1
ðg k z Þðg z Þ
2
0
G , and K = F(p/2, k) is the first kind ellipse integral. g ¼ G2 ; k ¼ Gþ2W E3y(x, y) is the non-uniform electric field caused by the two under coplanar electrodes, which can be expressed by E2y(x, y) as
E3y ðx; yÞ ¼ E2y ðx; b1 2b2 yÞ:
ð4Þ
Us to 0. Using the coupled mode theory, we can obtain the following results. (1) Assume that the light arrives at point z in the electro-optic active region at some moment. At this moment, the operation voltage U is changed from 0 to Us. In this case, the light propagates first for a length of z under U = 0 with the speed v = v0, and then for a length of L0 z under U = Us with the speed v = vi (i = 1, 2), where i represents the two different waveguides. So the output amplitudes at z = L0 can be determined as
The non-uniform electric field in the cladding layer is given by
Eyð2Þ ðx; yÞ ¼
1 X fr i E20;y ðx; y þ 2ib2 Þ þ r iþ1 E20;y ½x; y þ 2ði þ 1Þb2 g;
which will be used in the calculation of the distributed capacitance of the electrode. The overlap integral along the y-direction is introduced as
Cy ¼ G
ð6Þ
jE ðx; yÞj dxdy
0
where E (x, y) is the mode optical field distribution, and the integral region is over the electro-optic coupling region. Then according to the electro-optic modulation theory, the change of the refractive index of one waveguide core resulting from the applied voltage U can be determined as
DnðUÞ ¼
n31 U c Cy ; 2 33 G
ð7Þ
where c33 is the electro-optic coefficient of the poled polymer material. Then the refractive index of one waveguide core is decreased to n1 Dn, and that of another is increased to n1 + Dn. In this case, the mode effective propagation constants of the two waveguides are different from each other, which are denoted by b1 and b2, respectively, and let d = (b2 b1)/2.
Assume that the light power is only input from waveguide 1, that is, P1 ð0Þ ¼ R20 ¼ P 0 ; P 2 ð0Þ ¼ S20 ¼ 0, where R(0) = R0 and S(0) = S0 are the amplitudes of the input light. First we suppose that all the optical loss is neglected. For the convenience in the following analysis, we define two amplitude transfer matrixes A(z) and B(z) as
ð10Þ
P12;0 ðzÞ
¼ P0
P22;0 ðzÞ
jC 1 ðzÞj2
! ð11Þ
;
jD1 ðzÞj2
where
C 1 ðzÞ ¼ f1 ðL0 zÞf2 ðzÞ g 1 ðL0 zÞg 2 ðzÞ;
ð12aÞ
D1 ðzÞ ¼ jg 1 ðL0 zÞf2 ðzÞ jf1 ðL0 zÞg 2 ðzÞ:
ð12bÞ
(2) Assume that the light arrives at point z in the electro-optic active region at some moment. At this moment, the operation voltage U is changed from Us to 0. In this case, the light propagates first for a length of z under U = Us with the speed v = vi (i = 1, 2), and then for a length of L0 z under U = 0 with the speed v = v0. So the output amplitudes at z = L0 can be determined as
RðzÞ
SðzÞ
R0 : ¼ BðL0 zÞAðzÞ 0
ð13Þ
The relative output powers are given by
P12;0 ðzÞ P22;0 ðzÞ
¼ P0
jC 2 ðzÞj2
!
jD2 ðzÞj2
ð14Þ
;
where
2.3. Transmission power
AðzÞ ¼
R0 : ¼ AðL0 zÞBðzÞ 0
The relative output powers are given by
ð5Þ
1 ð1Þ E ðx; yÞjE0 ðx; yÞj2 dxdy U y ; RR 0 2
SðzÞ
i¼0
RR
RðzÞ
f1 ðzÞ jg 1 ðzÞ f2 ðzÞ jg 2 ðzÞ ; ðU–0Þ; BðzÞ ¼ ; ðU ¼ 0Þ; jg 1 ðzÞ f1 ðzÞ jg 2 ðzÞ f2 ðzÞ
C 2 ðzÞ ¼ f2 ðL0 zÞf1 ðzÞ g 2 ðL0 zÞg 1 ðzÞ; D2 ðzÞ ¼
jg 2 ðL0
zÞf1 ðzÞ
jf2 ðL0
zÞg 1 ðzÞ:
ð15aÞ ð15bÞ
We can understand from Eqs. (11) and (14) that for the different moment t and relative point z in the coupling region, the output powers at point z = L0 are different from each other, therefore, we can use Eqs. (11) and (14) to investigate the time response characteristics of the device. 2.4. Response under low switching frequency
ð8Þ
with
h i f1 ðzÞ ¼ cos ðd2 þ K 2 Þ1=2 z þ j g 1 ðzÞ ¼
K ðd2 þ K 2 Þ1=2
d
ðd2 þ K 2 Þ1=2 h i sin ðd2 þ K 2 Þ1=2 z ;
h i sin ðd2 þ K 2 Þ1=2 z ;
ð9aÞ ð9bÞ
f2 ðzÞ ¼ cosðKzÞ;
ð9cÞ
g 2 ðzÞ ¼ sinðKzÞ;
ð9dÞ
where K is the coupling coefficient. During the process of switching, the light propagating in the coupling region with a length of L0 = p/(2K) will be experienced two states of the operation voltage, that is, U is from 0 to Us or from
When the square wave switching voltage changes at low switching frequency, of which the microwave wavelength is assumed to be much larger than the electrode length, the entire electrode can be regarded to be lossless, and also it can be dealt as a lumped parameter circuit. In this case, when the operation voltage U changes suddenly from 0 to Us or from Us to 0, we can regard that the voltages at all points of the electrode are uniform, and they are changed at the same time. Let b be the mode propagation constant, and c is the velocity of the light in free space, then the propagation velocity can be deter0 mined as v = (k0c)/b. Denote L as the waveguide length of the input/output passive region. Assume that v0 is the light-wave propagation velocity when U = 0, and v1 and v2 are those of waveguide 1 and waveguide 2 when U = Us, then from Eqs. (11) and (14)
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we can obtain the expressions of the output powers P12,0 and P22,0 as follows:
a Travelling-wave line electrodes
(1) For the case of the operation voltage changing from 0 to Us
8 > P0 jC 1 ðL0 Þj2 ðt < t 0 þ L0 =v0 Þ > > > < P0 jC 1 ðL0 v1 ðt t0 L0 =v0 ÞÞj2 P12;0 ðtÞ ¼ > ðt 0 þ L0 =v0 6 t 6 t0 þ L0 =v0 þ L0 =v1 Þ > > > : P0 jC 1 ð0Þj2 ðt > t0 þ L0 =v0 þ L0 =v1 Þ 8 > P jD ðL Þj2 ðt < t 0 þ L0 =v0 Þ > > 0 1 0 > < P0 jD1 ðL0 v2 ðt t 0 L0 =v0 ÞÞj2 P22;0 ðtÞ ¼ > ðt 0 þ L0 =v0 6 t 6 t0 þ L0 =v0 þ L0 =v2 Þ > > > : P0 jD1 ð0Þj2 ðt > t 0 þ L0 =v0 þ L0 =v2 Þ:
+ U -
ð16Þ Travelling-wave line electrodes
ð17Þ
ð19Þ
i
ei
U z+dz,t C0dz -
0 2
4
6
8
10
Fig. 2. (a) Micro-element dz, and (b) equivalent lumped parameter circuit of dz of the traveling-wave electrodes.
where e0 is the free space permittivity, the integral region is along the surface of one electrode, and i represents four different regions surrounding the electrode. C 00 is the capacitance when the waveguide polymer material is replaced by air, which can also be calculated by taking the corresponding electric field in Eq. (23) after the replacement.The solution to Eq. (20) can be expressed as
ð24Þ
where U1 and I1 are the amplitudes offfi the voltage and current at qffiffiffiffiffiffiffiffiffiffi z = 0, respectively, and Z 0 ¼ 1=ðc C 0 C 00 Þ is named the characteristic resistance. When ZL = Z0, the electrode will work under the traveling-wave state, the microwave reflection (U1I1Z0/2) exp(Amz) will be zero. Let h1 be the original phase of the electric signal, then, Eq. (24) can be ultimately written as
UðzÞ ¼ U 1 exp½jðh1 bm zÞ;
ð25Þ
of which the expression in time-domain can be written as
uðz; tÞ ¼ U 1 cosðxm t bm z þ h1 Þ:
ð26Þ
Let vm be the microwave propagation velocity, then from Eqs. (11) and (14), for the case of U changing from Us to 0, we can obtain the expressions of the output powers P12,0 and P22,0 as follows:
ð20Þ (1) For the case of bm < b0, that is, vm > v0
ð21Þ
ð22Þ
Z ~ Ei ~ dSi ; si U
G0dz
UðzÞ ¼ ðU 1 þ I1 Z 0 =2Þ expðAm zÞ þ ðU 1 I1 Z 0 =2Þ expðAm zÞ;
qffiffiffiffiffiffiffiffiffiffiffiffiffi where nm ¼ C 0 =C 00 is defined as the microwave effective refractive index. C0 can be calculated by the following equation
X
U z,t
dz
ð18Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where Am ¼ ðR0 þ jxm L0 ÞðG0 þ jxm C 0 Þ ¼ am þ jbm ; xm is the angular frequency of the microwave, am and bm are its loss coefficient and propagation constant, respectively. Since the loss of the pffiffiffiffiffiffiffiffiffiffi electrode is am 0, that is, R0 0, G0 0, then bm ¼ xm L0 C 0 , which can be further written as [8]
C 0 ¼ 2e0
L0dz
R0dz
+
0
2
2pfm nm ; c
Iz
-
When the square wave switching voltage changes at ultra-high switching frequency, of which the microwave wavelength is comparative with or shorter than the length of the electrode. In this case, the distributed parameter effect will exist in the travelingwave electrodes, and the entire electrodes cannot be dealt as lumped parameter circuit. So we take a microelement dz of the electrode for analysis, which can be treated as a lumped parameter circuit as shown in Fig. 2a and b, where R0, L0, G0, and C0 are the characteristic resistance, inductance, conductance, and capacitance for the unit electrode length. Using the Kirchhoff’s Voltage Law (KVL) [13], we can obtain the following traveling-wave line equations as
bm ¼
b +
2.5. Response under ultra-high switching frequency
d UðzÞ ¼ A2m UðzÞ; dz2 2 d IðzÞ ¼ A2m IðzÞ; dz2
dz
z
(2) For the case of the operation voltage changing from Us to 0
8 > P jC ðL Þj2 ðt < t 0 þ L0 =v0 Þ > > 0 2 0 > < P0 jC 2 ðL0 v0 ðt t0 L0 =v0 ÞÞj2 P12;0 ðtÞ ¼ > ðt 0 þ L0 =v0 6 t 6 t0 þ L0 =v0 þ L0 =v0 Þ > > > : P0 jC 2 ð0Þj2 ðt > t0 þ L0 =v0 þ L0 =v0 Þ 8 > P0 jD2 ðL0 Þj2 ðt < t 0 þ L0 =v0 Þ > > > < P0 jD2 ðL0 v0 ðt t 0 L0 =v0 ÞÞj2 P22;0 ðtÞ ¼ > ðt 0 þ L0 =v0 6 t 6 t0 þ L0 =v0 þ L0 =v0 Þ > > > : P0 jD2 ð0Þj2 ðt > t 0 þ L0 =v0 þ L0 =v0 Þ:
zL
8 > P0 jC 1 ðL0 Þj2 ðt < t 0 þ L0 =v0 þ L0 =V m Þ > > > < P0 jC 1 ðL0 Lð1Þ Þj2 P12;0 ðtÞ ¼ > ðt 0 þ L0 =v0 þ L0 =V m t t 0 þ L0 =v0 þ L0 =v1 Þ > > > : P0 jC 1 ð0Þj2 ðt > t 0 þ L0 =v0 þ L0 =v1 Þ 8 > P0 jD1 ðL0 Þj2 ðt < t 0 þ L0 =v0 þ L0 =vm Þ > > > < P0 jD1 ðL0 Lð2Þ j2 P22;0 ðtÞ ¼ > ðt 0 L0 =v0 þ L0 =vm t t 0 þ L0 =v0 þ L0 =v2 Þ > > > : P0 jD1 ð0Þj2 ðt > t 0 þ L0 =v0 þ L0 =v2 Þ 0
ð23Þ
where L(1) = [vm(t t0 L /v0) L0][v1/(vm v1)] 0 [vm(t t0 L /v0) L0][v2/(vm v2)].
and
ð27Þ
ð28Þ
L(2) =
C.-T. Zheng et al. / Optics Communications 281 (2008) 5998–6005
(2) For the case of bm b0, that is, vm v0
ð29Þ
ð30Þ
(3) For the case of bm > b0, that is, vm < v0
ð31Þ
ð32Þ
3.2. Microwave effective index and characteristic impedance
t s ¼ maxðtd þ trise ; t d þ t fall Þ;
ð33Þ
fmcut
ð34Þ
When the loss of the device is considered, the mode propagation constant b will be changed to b ja, where a is the mode loss coefficient, and the output powers of the two waveguides should be modified to, respectively
P12 ðzÞ ¼ P 12;0 ðzÞ expð2azÞ;
P22 ðzÞ ¼ P 22;0 ðzÞ expð2azÞ:
ð35Þ
3. Simulation results In the following simulation, we select the values of the parameters as: the operation wavelength in free space is k0 = 1550 nm, the refractive index of the polymer core AJ309 is n10 = 1.643, its bulk amplitude attenuation coefficient is a1 = 2.0 dB/cm (the corresponding power loss coefficient is 2a1 = 4.0 dB/cm), and its electrooptic coefficient is c33 = 138 pm/V [7,14]; the refractive index of the polymer upper/under cladding layers and the cladding beside the core is n20 = 1.461, and its bulk amplitude attenuation coefficient is a2 = 0.25 dB/cm (the corresponding power loss coefficient is 2a2 = 0.5 dB/cm) [15]; the electrode is made of aurum, its refrac-
To get a low switching voltage and high electro-optic modulation efficiency, the electrode width and electrode gap should be optimized. According to our previous report of Ref. [17], we know that, the electrode gap should be taken as G = d, and when W increases from a, the switching voltage Us decreases slightly. Furthermore, from the analysis in Section 2.5, G and W will affect the
a
1.24
80
nm
1.22
60 1.20 50Ω
1.18 40 1.16 1.14
Z0 2
4
6
20 10
8
Characteristic Impedence Z0 (Ω )
0
where L(3) = [L0 vm(t t0 L /v0)][v0/(v0 vm)]. For the case of U changing from Us to 0, we can also get the expressions of P12,0(t) and P22,0(t) by replacing C1 and D1 in Eqs. (27)–(32) with C2 and D2, respectively, replacing C2 and D2 in Eqs. (31) and (32) with C1 and D1, respectively, and replacing v1 and v2 of L(i)(i = 1, 2) in Eqs. (27) and (28) with v0, respectively. For the above both cases of the low switching frequency and ultra-high switching frequency, we define the rise time trise and the fall time tfall, of which the meanings can be described as: trise is the time when the output power of one port changes from the minimum to 90% of the total power, and tfall is the time when that of another port changes from the maximum to 10% of the total power. The time from the moment when U starts to change to the moment when the output powers start to change is defined as the delay time td. So the switching time ts and cutoff switching frequency fmcut can be determined as
¼ 1=ðtrise þ t fall Þ:
First we perform the structural optimization of the device. The optimized results are as: the core width a = 3.0 lm, the core thickness b1 = 1.5 lm, the rib height h = 0.5 lm, the upper/under cladding layer thickness b2 = 1.5 lm, the minimum electrode thickness b3 = 0.15 lm, the coupling gap d = 3.7 lm, the slanting angle h = 1.75°, and the waveguide length of the input/output region L’ = 3.97 mm. This optimum structure guarantees only the Ey00 fundamental mode propagating in the waveguide, makes the propagation constants of the parts of waveguides with/without electrode be identical, and enables the mode propagation loss and the slanting loss to decrease as small as possible, which are about 2.286 dB/cm and 0.16 dB, respectively. In this case, the coupling length is L0 = 4.374 mm. The similar optimization methods and results can be seen in our previous report [17]. Under above optimized parameters, the light-wave effective index is neff = 1.5910, which will be considered in the optimization of the electrode parameters and the estimation of the response characteristics.
Electrode Width W (μ m)
b
55
1.195
nm 54 53
1.190
Z0
51
1.185 50Ω
1.180 3.0
52
3.5
4.0
4.5
50 49 5.0
Characteristic Impedence Z0 (Ω )
8 > P0 jC 2 ðL0 Þj2 ðt < t 0 þ L0 =v0 þ L0 =v0 Þ > > > < P0 jC 2 ðL0 Lð3Þ Þj2 P12;0 ðtÞ ¼ 0 0 > > ðt 0 þ L =v0 þ L0 =v0 6 t 6 t0 þ L =v0 þ L0 =vm Þ > > : 2 0 P0 jC 2 ð0Þj ðt > t0 þ L =v0 þ L0 =vm Þ 8 P0 jD2 ðL0 Þj2 ðt < t 0 þ L0 =v0 þ L0 =v0 Þ > > > > ð3Þ 2 > > < P0 jD2 ðL0 L Þj 0 P22;0 ðtÞ ¼ ðt 0 þ L =v0 þ L0 =v0 6 t 6 t0 þ L0 =v0 þ L0 =vm Þ > > > > P0 jD2 ð0Þj2 > > : ðt > t 0 þ L0 =v0 þ L0 =vm Þ
3.1. Structural optimization
Micro-wave Effective Index nm
8 2 0 > < P0 jC 1 ðL0 Þj ðt < t 0 þ L =v0 þ L0 =v0 Þ P12;0 ðtÞ ¼ P0 jC 1 ð0Þj2 > : ðt P t0 þ L0 =v0 þ L0 =v0 Þ 8 2 0 > < P0 jD1 ðL0 Þj ðt < t 0 þ L =v0 þ L0 =v0 Þ P22;0 ðtÞ ¼ P0 jD1 ð0Þj2 > : ðt P t0 þ L0 =v0 þ L0 =v0 Þ
tive index is n30 = 0.19, and its bulk extinction coefficient is j3 = 6.1 [16]; the layer upon the electrodes is air, its refractive index is n40 = 1.0, and its bulk amplitude attenuation coefficient is a4 = 0.
Micro-wave Effective Index nm
6002
Electrode Gap G (μ m) Fig. 3. Curves of the microwave effective index nm and characteristic impedance Z0 versus (a) the electrode width W, and (b) electrode gap G, where b3 = 0.15 lm, (a) G = 3.7 lm, and (b) W = 3.6 lm.
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min fmmax < c=kmin m ¼ 0:69 GHz, where we take km ¼ 100L0 43:74cm. Using Eqs. (16)–(19), Fig. 4 plots the curves of the output powers P12,0 and P22,0 versus the response time t, where the change of U is assumed to occur at the moment t0 = 0, the mode propagation loss is neglected, and P0 = 1. From the simulation results, we can understand that, the delay time is 10.5 ps, the rise time and fall time are about 15 ps and 19.6 ps, respectively, and the switching time is about 30.1 ps.
3.4. Response under ultra-high switching frequency From Section 3.3, the maximum microwave frequency corresponding to the lumped-circuit-treated electrodes is 0.69 GHz. When fm > 0.69 GHz, the response characteristics should be simulated using Eqs. (27)–(32). Fig. 5 shows the relations among the propagation powers P12,0, P22,0, the operation voltage U1 at z = 0 and the response time t, where t0 = 0, (a) U1 changes from 0 to
a Output Powers P12,0, P22,0
microwave effective index nm and characteristic impedance Z0, both of which will cause influences to the response characteristics. So G and W should be optimized for a compromise of high electrooptic modulation efficiency, match of impedance, and match between nm and neff. For the electrode structure described in Fig. 1, using Eqs. (22) and (23), Fig. 3 plots the curves of the microwave effective index nm and characteristic impedance Z0 versus (a) the electrode width W, and (b) the electrode gap G. From Fig. 3a, when W increases, the match between nm and neff can be realized, but this will result in the mismatch of the impedance. From Fig. 3b, the increase of G is an efficient way for the match between nm and neff, but this will decrease the electro-optic modulation efficiency and increase the switching voltage [17]. So we should select a compromise for the values of G and W. In order to match the characteristic impedance between the cable and the electrode for the great decrease of the reflection of the switching voltage signal, we choose the electrode gap and electrode width as G = 3.7 lm and W = 3.6 lm. In this case, the characteristic impedance is obtained, and a relative low switching voltage and a relative high microwave effective index are achieved, which are Z0 = 50.9 X, Us = 1.457 V, and nm = 1.19. Through above optimization on W and G, the device not only can be operated normally under a low switching voltage with no reflection of the switching signal, but also can have a relative high cutoff switching frequency. Notice that, during the optimization, first of all we guarantee that the switch can be operated normally, then, we make improvement for a larger operation bandwidth of the device. 3.3. Response under low switching frequency
trise
1.0
P12,0
0.8 0.6 0.4
ts td
0.2
P22,0
0.0 0
1.0
trise
P12,0
0.8 0.6 0.4
ts td
0.2
10
20
40
trise
0.6 0.4
ts td
0.2
P12,0 10
ts td
tfall
P12,0
0.0 10
20
30
30
40
1 0 0
1.0
40
30
60
90
120
P 22,0
0.8 0.6 0.4 0.2 0.0 0
0
20
2
P22,0
0.6
0.2
tfall
0.0
-1
0.8
0.4
c
Output Power P12,0,P22,0
Output Powers P12,0, P22,0
P22,0
Response Time t (ps)
Response Time t (ps)
1.0
40
0.8
0
30
30
trise
P22,0 0
20
1.0
tfall
0.0
b
b
U1 (V)
Output Powers P12,0, P22,0
a
10
Response Time t (ps)
Output Powers P12,0, P22,0
The wavelength of the low frequency switching voltage should be much larger than the length of the electrode, that is, km L0 , so the frequency of the operation voltage U should be
tfall
P 12,0
30
60
90
120
Response time t (ps)
Response Time t (ps) Fig. 4. Curves of output powers P12,0 and P22,0 versus the response time t under low microwave frequency, where (a) U changes from 0 to Us, (b) U changes from Us to 0, and t0 = 0.
Fig. 5. Relations among the propagation powers P12,0, P22,0, operation voltage U1 at z = 0, and the response time t under ultra-high microwave frequency, where (a) U1 changes from 0 to Us at the moment t0 = 0, (b) U1 changes from Us to 0 at the moment t0 = 0, and (c) U1 is a square wave switching signal.
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Rise/Fall Time trise, tfall (ps)
ts
tfall
6
38
trise 4
36
2
34
0 1.0
1.2
1.4
1.6
1.8
Switching Time ts (ps)
40
8
32 2.0
4. Discussion and error analysis
Microwave Effective Index nm Fig. 6. Effects of the microwave effective refractive index nm on the rise time trise, fall time tfall and the switching time ts when the operation voltage U changes from 0 to Us.
Us, (b) U1 changes from Us to 0, and (c) U1 changes periodically. We can observe from the results that, the rise time trise and the fall time tfall are less than those shown in Fig. 4, which are about 3.78 ps and 4.94 ps, respectively, and the delay time is 27.86 ps, so the switching time is about 32.8 ps. However, from Fig. 5c, we find that, the delay time causes no effect to the exchange of output powers. Besides, when the switching frequency is large enough, and when the output power of one port changes from the minimum to maximum, the output power will exchange again before increasing to the maximum due to the small period cycle of the microwave signal. In this case, the exchange of the output powers cannot be totally realized. So there exists the maximum switching frequency, which is named the cutoff switching frequency denoted by fmcut and will be discussed later. 3.5. Rise time, fall time, switching time, and switching frequency
4
10
3
10
2
cut
Switching Frequency fm (GHz)
Through the analysis in Section 2.5, the rise time trise and fall time tfall under ultra-high switching frequency will not be constants when the propagation velocity vm changes. Fig. 6 shows the effects of the microwave effective refractive index nm on the
10
1.2
1.4
1.6
1.8
rise time trise, fall time tfall and the switching time ts when the operation voltage U changes from 0 to Us. We can see that, when nm = neff, both the rise time and fall time become zero. This is because the propagation velocity of the square wave switching voltage matches that of the light-wave well enough. Fig. 7 shows the relation between the microwave effective refractive index nm and the cutoff switching frequency fmcut . We find that, when nm = neff, the cutoff switching frequency can be up to infinity. For the switch designed in this paper, it is about 114.7 GHz. The above excellent features given in Figs. 6 and 7 should be taken into consideration in the design of electro-optic switches and modulators.
2.0
Microwave Effective Index n m Fig. 7. Relation between the microwave effective refractive index nm and the cutoff switching frequency fmcut .
When an electro-optic modulator or switch with Mach–Zehnder Interferometer (MZI) structure or directional coupler (DC) structure possesses a similar traveling-wave electrode, its response characteristics can be similarly analyzed using this technique. The differences between them are: for the modulator, we need to calculate the 3 dB modulating bandwidth using the distributed parameters, whereas for the switch, we need to calculate the cutoff switching frequency; for the modulator, the signal propagating along the electrode is a modulation microwave signal, whereas for the switch, that is a switching voltage signal. In our presented switch, there is a difference between the effective index of the microwave nm = 1.19 and that of the light-wave neff = 1.5910. This is because the match between nm and neff, the match of impedance, and high electro-optic modulation efficiency cannot be achieved at the same time for a certain electrode dimension. So we optimize the electrode width and electrode gap firstly for the match of the impedance and high electro-optic modulation efficiency, and then for the match of effective indices between nm and neff, that is, firstly we guarantee that the switch can be operated normally under a low switching voltage, and then we make improvement for a larger operation bandwidth of the device. Therefore, the electrode width optimized in this paper is not equal to the core width a, which is different from that of the designed standard switch reported in our previous paper [17], which is equal to the core width a. This is an improvement for the match of the impedance and the match of effective index besides high electrooptic modulation efficiency. Since the parameters including the characteristic impedance Z0 and the microwave effective index nm are essential to estimate the bandwidth or the response performance of modulators or switches, in order to check the accuracy of this technique, a comparison is carried out between the simulation results of this technique and the experimental results reported in Ref. [9] and those calculated from the extended point-matching method reported in Ref. [10], which is shown in Table 1. Thus, we can conclude from the comparisons that the technique presented in this paper has satisfactory accuracy. 5. Conclusion By using the presented technique, the response characteristics are simulated for the polymer directional coupler electro-optic switches, which include the transmission powers, rise time, fall
Table 1 Comparisons between the calculated results from this technique and those from references Ref.
[9] [10]
Electrode parameters
Z0 (X)
nm
Width
Gap
Thickness
This
Ref.
Error (%)
This
Refs.
Error (%)
8 10
50 15
3 6
36.4 45.6
35 42
+4 +8
– 2.08
– 2.31
– 9.9
C.-T. Zheng et al. / Optics Communications 281 (2008) 5998–6005
time, switching time and cutoff switching frequency. Simulation results for the designed switch are as: the switching voltage is about 1.457 V, the coupling length is about 4.374 mm, the switching time is about 32.8 ps, and the cutoff switching frequency is about 114.7 GHz. Comparisons shows that the simulation results of this technique are in good agreement with the experimental and other simulated results reported in some other references. This indicates this technique has satisfactory accuracy. Acknowledgment The authors wish to express their gratitude to the National Science Foundation Council of China (the project number is 60706011), the ministry of education of China (the project number is 20070183087), and the Science and Technology Council of China (the project number is 2006CB302803) for their generous support to this work. References [1] X.J. Lu, M. Li, Opt. Components Mater. IV 6469 (2007) A4691.
6005
[2] T. Shibata, M. Okuno, T. Goh, T. Watanabe, M. Yasu, M. Itoh, M. Ishii, Y. Hibino, A. Sugita, A. Himeno, IEEE Photon. Technol. Lett. 15 (2003) 1300. [3] G.H. Hu, Y.P. Cui, B.F. Yun, C.G. Lu, Z.Y. Wang, Opt. Commun. 279 (2007) 79. [4] D.M. Yeo, S.Y. Shin, Opt. Commun. 267 (2006) 388. [5] Q. Wang, J.P. Yao, Opt. Express 15 (2007) 16500. [6] Y.H. Kim, U.C. Paek, W.T. Han, Appl. Opt. 44 (2005) 3051. [7] Y. Enami, D. Mathine, C.T. Derose, R.A. Norwood, J. Luo, A.K.Y. Jen, N. Peyghambarian, Appl. Phys. Lett. 91 (2007) 093505. [8] N.H. Zhu, Q. Wei, E.Y.B. Pun, P.S. Chung, Opt. Quantum Electron. 28 (1996) 137. [9] H. Zhang, M.C. Oh, A. Szep, W.H. Steier, C. Zhang, L.R. Dalton, H. Erlig, Y. Chang, D.H. Chang, H.R. Fetterman, Appl. Phys. Lett. 78 (2001) 3136. [10] D.J. Ren, M.S. Chen, X.M. Huang, Chin. Study Opt. Commun. 143 (2007) 47. [11] O.G. Ramer, IEEE J. Quantum Electron. 18 (1982) 386. [12] C. Sabatier, E. Caquot, IEEE J. Quantum Electron. 18 (1986) 32. [13] Y. Ken, Power Electron. Technol. 30 (2004) 30. [14] Y. Enami, C.T. Derose, D. Mathine, C. Loychik, C. Greenlee, R.A. Norwood, T.D. Kim, J. Luo, Y. Tian, A.K.Y. Jen, N. Peyghambarian, Nature Photon. 1 (2007) 180. [15] C. Pitois, C. Vukmirovic, A. Hult, Macromolecules 32 (1999) 2903. [16] W.G. Driscoll, W. Vaughan, Handbook of Optics, McGraw-Hill, New York, 1978. p. 7. [17] C.T. Zheng, C.S. Ma, X. Yan, X.Y. Wang, D.M. Zhang, Opt. Commun. 281 (2008) 3695.
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Analysis of response characteristics for polymer directional coupler electro-optic switches Chuan-Tao Zheng, Chun-Sheng Ma *, Xin Yan, Xian-Yin Wang, Da-Ming Zhang State Key Laboratory on Integrated Optoelectronics, College of Electronic Science and Engineering, Jilin University, Changchun 130012, PR China
a r t i c l e
i n f o
Article history: Received 21 July 2008 Received in revised form 4 September 2008 Accepted 4 September 2008
Keywords: Directional coupler Electro-optic switch Transmission power Switching voltage Switching time Cutoff switching frequency
a b s t r a c t By using the coupled mode theory, electro-optic modulation theory, conformal transforming method, image method and the proposed transfer matrix technique, novel expressions for the both cases of the low switching frequency and the ultra-high switching frequency are presented for analyzing the transmission powers, rise time, fall time, switching time and switching frequency of the polymer directional coupler electro-optic switches. Simulation results of an application based on the technique show that, the switching voltage and coupling length are about 1.457 V and 4.374 mm, respectively, and the switching time and cutoff switching frequency are about 32.8 ps and 114.7 GHz, respectively, for the designed switch. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction As the fast development of optical communication systems and technologies, optical switches and arrays have been playing an important role in information transferring, information exchange, optical cross-connect (OXC), optical add-drop multiplexer (OADM), and optical line protection (OLP), due to their great applications in optical signal processing, optical computer, optical instrument, equipment, and sensors [1–4]. Nowadays, in order to increase the bandwidth, capacity, and speed of the local and trunk optical networks, optical switches should be designed and fabricated with short switching time and high switching frequency, which should be up to the orders of ps and GHz, respectively [5–6]. Switching time and switching frequency depend on many factors, which involve the response time of the electro-optic polymer materials, the light-wave propagation velocity along the waveguide, the microwave propagation velocity along the travelingwave electrodes, the waveguide length, and the electrode structure of the switch. First, since the response time of almost all polymers, e.g. the polymer core AJ309 [7] used in this paper, can be as fast as subpicosecond or femtosecond, which is so fast compared with the period of the electric signal that it can be neglected, so we can reasonably think that, there exists no delay between the change of the electric signal and that of the refractive index of the electro-optic polymer. Moreover, the microwave propagation velocity depends * Corresponding author. Tel.: +86 431 85103909. E-mail address: [email protected] (C.-S. Ma). 0030-4018/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2008.09.021
on its effective refractive index, which can be calculated by using the characteristic parameters of the traveling-wave electrode. These parameters are obtained by solving the traveling-wave line equation through equaling the traveling-wave electrode to distributed parameter circuit [8]. In order to analyze the response characteristics including the switching time and switching frequency, a simulation technique is proposed in this paper. First in Section 2, the structural schematic of the switch and its traveling-wave electrode are presented. The push–pull electrodes are analyzed by utilizing the conformal transforming method and image method. The power transfer matrixes due to the change of the operation voltage applied on the traveling-wave electrodes are obtained by using the electro-optic modulation theory and coupled mode theory. Novel expressions are presented for analyzing the transmission powers, rise time, fall time, switching time, and switching frequency for the both cases of the low switching frequency and the ultra-high switching frequency. Then in Section 3, the simulation for the response characteristics is performed, which include the characteristic impedance, microwave effective refractive index, output powers, switching time, and cutoff switching frequency. Furthermore in Section 4, in order to check the accuracy of this technique, comparisons are carried out between the results calculated from this technique and those from the experimental method and extended point-matching method reported in Refs. [9,10]. Finally, a conclusion is reached in Section 5 based on the analysis and discussion.
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coefficient. Let b2 be the thickness of the upper/under buffer layers, n2 be the refractive index of the upper/under buffer layers and the cladding beside the rib, and a2 be their bulk amplitude attenuation coefficient. Let b3 be the thickness of the upper/under electrodes, n3 be its refractive index, and j3 be its bulk extinction coefficient. Let n4 be the refractive index of the cladding above the upper electrode, and a4 be its bulk amplitude attenuation coefficient. Denote Us as the switching voltage.
2. Model and theory 2.1. Model and structure Fig. 1 shows the structural diagram of the polymer directional coupler electro-optic switch, which consists of two identical parallel rib waveguides and a push–pull electrode structure. The coupling region is shown in Fig. 1a, where d is the coupling gap between the waveguides and L is the length of the coupling region. The cross-section of the coupling region is shown in Fig. 1b, where the structure of the rib waveguide is given as: air/upper electrode/ upper buffer layer/core/under buffer layer/under electrode/substrate, and only the core is electro-optic poled polymer material. The push–pull electrode structure consists of a pair of upper electrodes and a pair of under electrodes. During the poling, the applied poling voltage Upol is shown in Fig. 1b; during the operation, the applied operation voltage U is shown in Fig. 1c. Denote W as the electrode width, and G as the electrode gap. Let a be the core width, b1 be the core thickness, h be the rib height, n1 be the core refractive index, and a1 be its bulk amplitude attenuation
2.2. Electric field and refractive index change The electric field along the y-direction in the rib core is the sum of three parts as follows:
Eð1Þ y ðx; yÞ ¼ E1y ðx; yÞ þ E2y ðx; yÞ þ E3y ðx; yÞ;
ð1Þ
where E1y(x, y) is the uniform electric field caused by a pair of upper/under electrodes, which is determined by
E1y ðx; yÞ ¼
n22 U : 2n21 b2 þ n22 b1
ð2Þ
a P12
P10=P0 z =0 P11(0)=P0' P21(0)=0
z=L
A B
C D
d
P20=0
P11(L) P21(L)
P22
θ
Input passive region
Electro-optic active region
Output passive region
Upol
b n4,α 4
n2,α 2
y G upper electrode 0 d n2,α 2
upper buffer layer core electro-optic material
h
W n3,κ3
b3
n2,α 2
b2
a
n1,α 1
n4,α 4 x n2,α 2
b1
under buffer layer
n2,α 2
b2
under electrode
n3,κ3
b3
GND
c
U=Us
GND
U
U=0
~ n4
n4
y
W
G
n33,κ33 n2
upper electrode upper buffer layer core electro-optic material
0
n2
d h
b3 b2
a
n1
x
n2
b1
under buffer layer
n2
b2
under electrode
n3,κ3
b3
Fig. 1. (a) Structural diagram, (b) cross-section with poling voltage, and (c) cross-section with operation voltage in the electro-optic active region of polymer directional coupler electro-optic switches with push-pull traveling-wave electrodes.
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E2y(x, y) is the non-uniform electric field caused by the two upper coplanar electrodes, by using the conformal transforming method [11] and image method [12], which can be expressed as
E2y ðx; yÞ ¼ ð1 rÞ
1 X
ri E20;y ðx; y þ 2ib2 Þ;
ð3Þ
i¼0 n2 n2
g U dw dw where r ¼ n12 þn22 ; E20;y ðx; yÞ ¼ 2K ; ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; z ¼ x þ jy; 0 Im dz dz 2 2 2 2 2 1
ðg k z Þðg z Þ
2
0
G , and K = F(p/2, k) is the first kind ellipse integral. g ¼ G2 ; k ¼ Gþ2W E3y(x, y) is the non-uniform electric field caused by the two under coplanar electrodes, which can be expressed by E2y(x, y) as
E3y ðx; yÞ ¼ E2y ðx; b1 2b2 yÞ:
ð4Þ
Us to 0. Using the coupled mode theory, we can obtain the following results. (1) Assume that the light arrives at point z in the electro-optic active region at some moment. At this moment, the operation voltage U is changed from 0 to Us. In this case, the light propagates first for a length of z under U = 0 with the speed v = v0, and then for a length of L0 z under U = Us with the speed v = vi (i = 1, 2), where i represents the two different waveguides. So the output amplitudes at z = L0 can be determined as
The non-uniform electric field in the cladding layer is given by
Eyð2Þ ðx; yÞ ¼
1 X fr i E20;y ðx; y þ 2ib2 Þ þ r iþ1 E20;y ½x; y þ 2ði þ 1Þb2 g;
which will be used in the calculation of the distributed capacitance of the electrode. The overlap integral along the y-direction is introduced as
Cy ¼ G
ð6Þ
jE ðx; yÞj dxdy
0
where E (x, y) is the mode optical field distribution, and the integral region is over the electro-optic coupling region. Then according to the electro-optic modulation theory, the change of the refractive index of one waveguide core resulting from the applied voltage U can be determined as
DnðUÞ ¼
n31 U c Cy ; 2 33 G
ð7Þ
where c33 is the electro-optic coefficient of the poled polymer material. Then the refractive index of one waveguide core is decreased to n1 Dn, and that of another is increased to n1 + Dn. In this case, the mode effective propagation constants of the two waveguides are different from each other, which are denoted by b1 and b2, respectively, and let d = (b2 b1)/2.
Assume that the light power is only input from waveguide 1, that is, P1 ð0Þ ¼ R20 ¼ P 0 ; P 2 ð0Þ ¼ S20 ¼ 0, where R(0) = R0 and S(0) = S0 are the amplitudes of the input light. First we suppose that all the optical loss is neglected. For the convenience in the following analysis, we define two amplitude transfer matrixes A(z) and B(z) as
ð10Þ
P12;0 ðzÞ
¼ P0
P22;0 ðzÞ
jC 1 ðzÞj2
! ð11Þ
;
jD1 ðzÞj2
where
C 1 ðzÞ ¼ f1 ðL0 zÞf2 ðzÞ g 1 ðL0 zÞg 2 ðzÞ;
ð12aÞ
D1 ðzÞ ¼ jg 1 ðL0 zÞf2 ðzÞ jf1 ðL0 zÞg 2 ðzÞ:
ð12bÞ
(2) Assume that the light arrives at point z in the electro-optic active region at some moment. At this moment, the operation voltage U is changed from Us to 0. In this case, the light propagates first for a length of z under U = Us with the speed v = vi (i = 1, 2), and then for a length of L0 z under U = 0 with the speed v = v0. So the output amplitudes at z = L0 can be determined as
RðzÞ
SðzÞ
R0 : ¼ BðL0 zÞAðzÞ 0
ð13Þ
The relative output powers are given by
P12;0 ðzÞ P22;0 ðzÞ
¼ P0
jC 2 ðzÞj2
!
jD2 ðzÞj2
ð14Þ
;
where
2.3. Transmission power
AðzÞ ¼
R0 : ¼ AðL0 zÞBðzÞ 0
The relative output powers are given by
ð5Þ
1 ð1Þ E ðx; yÞjE0 ðx; yÞj2 dxdy U y ; RR 0 2
SðzÞ
i¼0
RR
RðzÞ
f1 ðzÞ jg 1 ðzÞ f2 ðzÞ jg 2 ðzÞ ; ðU–0Þ; BðzÞ ¼ ; ðU ¼ 0Þ; jg 1 ðzÞ f1 ðzÞ jg 2 ðzÞ f2 ðzÞ
C 2 ðzÞ ¼ f2 ðL0 zÞf1 ðzÞ g 2 ðL0 zÞg 1 ðzÞ; D2 ðzÞ ¼
jg 2 ðL0
zÞf1 ðzÞ
jf2 ðL0
zÞg 1 ðzÞ:
ð15aÞ ð15bÞ
We can understand from Eqs. (11) and (14) that for the different moment t and relative point z in the coupling region, the output powers at point z = L0 are different from each other, therefore, we can use Eqs. (11) and (14) to investigate the time response characteristics of the device. 2.4. Response under low switching frequency
ð8Þ
with
h i f1 ðzÞ ¼ cos ðd2 þ K 2 Þ1=2 z þ j g 1 ðzÞ ¼
K ðd2 þ K 2 Þ1=2
d
ðd2 þ K 2 Þ1=2 h i sin ðd2 þ K 2 Þ1=2 z ;
h i sin ðd2 þ K 2 Þ1=2 z ;
ð9aÞ ð9bÞ
f2 ðzÞ ¼ cosðKzÞ;
ð9cÞ
g 2 ðzÞ ¼ sinðKzÞ;
ð9dÞ
where K is the coupling coefficient. During the process of switching, the light propagating in the coupling region with a length of L0 = p/(2K) will be experienced two states of the operation voltage, that is, U is from 0 to Us or from
When the square wave switching voltage changes at low switching frequency, of which the microwave wavelength is assumed to be much larger than the electrode length, the entire electrode can be regarded to be lossless, and also it can be dealt as a lumped parameter circuit. In this case, when the operation voltage U changes suddenly from 0 to Us or from Us to 0, we can regard that the voltages at all points of the electrode are uniform, and they are changed at the same time. Let b be the mode propagation constant, and c is the velocity of the light in free space, then the propagation velocity can be deter0 mined as v = (k0c)/b. Denote L as the waveguide length of the input/output passive region. Assume that v0 is the light-wave propagation velocity when U = 0, and v1 and v2 are those of waveguide 1 and waveguide 2 when U = Us, then from Eqs. (11) and (14)
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we can obtain the expressions of the output powers P12,0 and P22,0 as follows:
a Travelling-wave line electrodes
(1) For the case of the operation voltage changing from 0 to Us
8 > P0 jC 1 ðL0 Þj2 ðt < t 0 þ L0 =v0 Þ > > > < P0 jC 1 ðL0 v1 ðt t0 L0 =v0 ÞÞj2 P12;0 ðtÞ ¼ > ðt 0 þ L0 =v0 6 t 6 t0 þ L0 =v0 þ L0 =v1 Þ > > > : P0 jC 1 ð0Þj2 ðt > t0 þ L0 =v0 þ L0 =v1 Þ 8 > P jD ðL Þj2 ðt < t 0 þ L0 =v0 Þ > > 0 1 0 > < P0 jD1 ðL0 v2 ðt t 0 L0 =v0 ÞÞj2 P22;0 ðtÞ ¼ > ðt 0 þ L0 =v0 6 t 6 t0 þ L0 =v0 þ L0 =v2 Þ > > > : P0 jD1 ð0Þj2 ðt > t 0 þ L0 =v0 þ L0 =v2 Þ:
+ U -
ð16Þ Travelling-wave line electrodes
ð17Þ
ð19Þ
i
ei
U z+dz,t C0dz -
0 2
4
6
8
10
Fig. 2. (a) Micro-element dz, and (b) equivalent lumped parameter circuit of dz of the traveling-wave electrodes.
where e0 is the free space permittivity, the integral region is along the surface of one electrode, and i represents four different regions surrounding the electrode. C 00 is the capacitance when the waveguide polymer material is replaced by air, which can also be calculated by taking the corresponding electric field in Eq. (23) after the replacement.The solution to Eq. (20) can be expressed as
ð24Þ
where U1 and I1 are the amplitudes offfi the voltage and current at qffiffiffiffiffiffiffiffiffiffi z = 0, respectively, and Z 0 ¼ 1=ðc C 0 C 00 Þ is named the characteristic resistance. When ZL = Z0, the electrode will work under the traveling-wave state, the microwave reflection (U1I1Z0/2) exp(Amz) will be zero. Let h1 be the original phase of the electric signal, then, Eq. (24) can be ultimately written as
UðzÞ ¼ U 1 exp½jðh1 bm zÞ;
ð25Þ
of which the expression in time-domain can be written as
uðz; tÞ ¼ U 1 cosðxm t bm z þ h1 Þ:
ð26Þ
Let vm be the microwave propagation velocity, then from Eqs. (11) and (14), for the case of U changing from Us to 0, we can obtain the expressions of the output powers P12,0 and P22,0 as follows:
ð20Þ (1) For the case of bm < b0, that is, vm > v0
ð21Þ
ð22Þ
Z ~ Ei ~ dSi ; si U
G0dz
UðzÞ ¼ ðU 1 þ I1 Z 0 =2Þ expðAm zÞ þ ðU 1 I1 Z 0 =2Þ expðAm zÞ;
qffiffiffiffiffiffiffiffiffiffiffiffiffi where nm ¼ C 0 =C 00 is defined as the microwave effective refractive index. C0 can be calculated by the following equation
X
U z,t
dz
ð18Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where Am ¼ ðR0 þ jxm L0 ÞðG0 þ jxm C 0 Þ ¼ am þ jbm ; xm is the angular frequency of the microwave, am and bm are its loss coefficient and propagation constant, respectively. Since the loss of the pffiffiffiffiffiffiffiffiffiffi electrode is am 0, that is, R0 0, G0 0, then bm ¼ xm L0 C 0 , which can be further written as [8]
C 0 ¼ 2e0
L0dz
R0dz
+
0
2
2pfm nm ; c
Iz
-
When the square wave switching voltage changes at ultra-high switching frequency, of which the microwave wavelength is comparative with or shorter than the length of the electrode. In this case, the distributed parameter effect will exist in the travelingwave electrodes, and the entire electrodes cannot be dealt as lumped parameter circuit. So we take a microelement dz of the electrode for analysis, which can be treated as a lumped parameter circuit as shown in Fig. 2a and b, where R0, L0, G0, and C0 are the characteristic resistance, inductance, conductance, and capacitance for the unit electrode length. Using the Kirchhoff’s Voltage Law (KVL) [13], we can obtain the following traveling-wave line equations as
bm ¼
b +
2.5. Response under ultra-high switching frequency
d UðzÞ ¼ A2m UðzÞ; dz2 2 d IðzÞ ¼ A2m IðzÞ; dz2
dz
z
(2) For the case of the operation voltage changing from Us to 0
8 > P jC ðL Þj2 ðt < t 0 þ L0 =v0 Þ > > 0 2 0 > < P0 jC 2 ðL0 v0 ðt t0 L0 =v0 ÞÞj2 P12;0 ðtÞ ¼ > ðt 0 þ L0 =v0 6 t 6 t0 þ L0 =v0 þ L0 =v0 Þ > > > : P0 jC 2 ð0Þj2 ðt > t0 þ L0 =v0 þ L0 =v0 Þ 8 > P0 jD2 ðL0 Þj2 ðt < t 0 þ L0 =v0 Þ > > > < P0 jD2 ðL0 v0 ðt t 0 L0 =v0 ÞÞj2 P22;0 ðtÞ ¼ > ðt 0 þ L0 =v0 6 t 6 t0 þ L0 =v0 þ L0 =v0 Þ > > > : P0 jD2 ð0Þj2 ðt > t 0 þ L0 =v0 þ L0 =v0 Þ:
zL
8 > P0 jC 1 ðL0 Þj2 ðt < t 0 þ L0 =v0 þ L0 =V m Þ > > > < P0 jC 1 ðL0 Lð1Þ Þj2 P12;0 ðtÞ ¼ > ðt 0 þ L0 =v0 þ L0 =V m t t 0 þ L0 =v0 þ L0 =v1 Þ > > > : P0 jC 1 ð0Þj2 ðt > t 0 þ L0 =v0 þ L0 =v1 Þ 8 > P0 jD1 ðL0 Þj2 ðt < t 0 þ L0 =v0 þ L0 =vm Þ > > > < P0 jD1 ðL0 Lð2Þ j2 P22;0 ðtÞ ¼ > ðt 0 L0 =v0 þ L0 =vm t t 0 þ L0 =v0 þ L0 =v2 Þ > > > : P0 jD1 ð0Þj2 ðt > t 0 þ L0 =v0 þ L0 =v2 Þ 0
ð23Þ
where L(1) = [vm(t t0 L /v0) L0][v1/(vm v1)] 0 [vm(t t0 L /v0) L0][v2/(vm v2)].
and
ð27Þ
ð28Þ
L(2) =
C.-T. Zheng et al. / Optics Communications 281 (2008) 5998–6005
(2) For the case of bm b0, that is, vm v0
ð29Þ
ð30Þ
(3) For the case of bm > b0, that is, vm < v0
ð31Þ
ð32Þ
3.2. Microwave effective index and characteristic impedance
t s ¼ maxðtd þ trise ; t d þ t fall Þ;
ð33Þ
fmcut
ð34Þ
When the loss of the device is considered, the mode propagation constant b will be changed to b ja, where a is the mode loss coefficient, and the output powers of the two waveguides should be modified to, respectively
P12 ðzÞ ¼ P 12;0 ðzÞ expð2azÞ;
P22 ðzÞ ¼ P 22;0 ðzÞ expð2azÞ:
ð35Þ
3. Simulation results In the following simulation, we select the values of the parameters as: the operation wavelength in free space is k0 = 1550 nm, the refractive index of the polymer core AJ309 is n10 = 1.643, its bulk amplitude attenuation coefficient is a1 = 2.0 dB/cm (the corresponding power loss coefficient is 2a1 = 4.0 dB/cm), and its electrooptic coefficient is c33 = 138 pm/V [7,14]; the refractive index of the polymer upper/under cladding layers and the cladding beside the core is n20 = 1.461, and its bulk amplitude attenuation coefficient is a2 = 0.25 dB/cm (the corresponding power loss coefficient is 2a2 = 0.5 dB/cm) [15]; the electrode is made of aurum, its refrac-
To get a low switching voltage and high electro-optic modulation efficiency, the electrode width and electrode gap should be optimized. According to our previous report of Ref. [17], we know that, the electrode gap should be taken as G = d, and when W increases from a, the switching voltage Us decreases slightly. Furthermore, from the analysis in Section 2.5, G and W will affect the
a
1.24
80
nm
1.22
60 1.20 50Ω
1.18 40 1.16 1.14
Z0 2
4
6
20 10
8
Characteristic Impedence Z0 (Ω )
0
where L(3) = [L0 vm(t t0 L /v0)][v0/(v0 vm)]. For the case of U changing from Us to 0, we can also get the expressions of P12,0(t) and P22,0(t) by replacing C1 and D1 in Eqs. (27)–(32) with C2 and D2, respectively, replacing C2 and D2 in Eqs. (31) and (32) with C1 and D1, respectively, and replacing v1 and v2 of L(i)(i = 1, 2) in Eqs. (27) and (28) with v0, respectively. For the above both cases of the low switching frequency and ultra-high switching frequency, we define the rise time trise and the fall time tfall, of which the meanings can be described as: trise is the time when the output power of one port changes from the minimum to 90% of the total power, and tfall is the time when that of another port changes from the maximum to 10% of the total power. The time from the moment when U starts to change to the moment when the output powers start to change is defined as the delay time td. So the switching time ts and cutoff switching frequency fmcut can be determined as
¼ 1=ðtrise þ t fall Þ:
First we perform the structural optimization of the device. The optimized results are as: the core width a = 3.0 lm, the core thickness b1 = 1.5 lm, the rib height h = 0.5 lm, the upper/under cladding layer thickness b2 = 1.5 lm, the minimum electrode thickness b3 = 0.15 lm, the coupling gap d = 3.7 lm, the slanting angle h = 1.75°, and the waveguide length of the input/output region L’ = 3.97 mm. This optimum structure guarantees only the Ey00 fundamental mode propagating in the waveguide, makes the propagation constants of the parts of waveguides with/without electrode be identical, and enables the mode propagation loss and the slanting loss to decrease as small as possible, which are about 2.286 dB/cm and 0.16 dB, respectively. In this case, the coupling length is L0 = 4.374 mm. The similar optimization methods and results can be seen in our previous report [17]. Under above optimized parameters, the light-wave effective index is neff = 1.5910, which will be considered in the optimization of the electrode parameters and the estimation of the response characteristics.
Electrode Width W (μ m)
b
55
1.195
nm 54 53
1.190
Z0
51
1.185 50Ω
1.180 3.0
52
3.5
4.0
4.5
50 49 5.0
Characteristic Impedence Z0 (Ω )
8 > P0 jC 2 ðL0 Þj2 ðt < t 0 þ L0 =v0 þ L0 =v0 Þ > > > < P0 jC 2 ðL0 Lð3Þ Þj2 P12;0 ðtÞ ¼ 0 0 > > ðt 0 þ L =v0 þ L0 =v0 6 t 6 t0 þ L =v0 þ L0 =vm Þ > > : 2 0 P0 jC 2 ð0Þj ðt > t0 þ L =v0 þ L0 =vm Þ 8 P0 jD2 ðL0 Þj2 ðt < t 0 þ L0 =v0 þ L0 =v0 Þ > > > > ð3Þ 2 > > < P0 jD2 ðL0 L Þj 0 P22;0 ðtÞ ¼ ðt 0 þ L =v0 þ L0 =v0 6 t 6 t0 þ L0 =v0 þ L0 =vm Þ > > > > P0 jD2 ð0Þj2 > > : ðt > t 0 þ L0 =v0 þ L0 =vm Þ
3.1. Structural optimization
Micro-wave Effective Index nm
8 2 0 > < P0 jC 1 ðL0 Þj ðt < t 0 þ L =v0 þ L0 =v0 Þ P12;0 ðtÞ ¼ P0 jC 1 ð0Þj2 > : ðt P t0 þ L0 =v0 þ L0 =v0 Þ 8 2 0 > < P0 jD1 ðL0 Þj ðt < t 0 þ L =v0 þ L0 =v0 Þ P22;0 ðtÞ ¼ P0 jD1 ð0Þj2 > : ðt P t0 þ L0 =v0 þ L0 =v0 Þ
tive index is n30 = 0.19, and its bulk extinction coefficient is j3 = 6.1 [16]; the layer upon the electrodes is air, its refractive index is n40 = 1.0, and its bulk amplitude attenuation coefficient is a4 = 0.
Micro-wave Effective Index nm
6002
Electrode Gap G (μ m) Fig. 3. Curves of the microwave effective index nm and characteristic impedance Z0 versus (a) the electrode width W, and (b) electrode gap G, where b3 = 0.15 lm, (a) G = 3.7 lm, and (b) W = 3.6 lm.
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min fmmax < c=kmin m ¼ 0:69 GHz, where we take km ¼ 100L0 43:74cm. Using Eqs. (16)–(19), Fig. 4 plots the curves of the output powers P12,0 and P22,0 versus the response time t, where the change of U is assumed to occur at the moment t0 = 0, the mode propagation loss is neglected, and P0 = 1. From the simulation results, we can understand that, the delay time is 10.5 ps, the rise time and fall time are about 15 ps and 19.6 ps, respectively, and the switching time is about 30.1 ps.
3.4. Response under ultra-high switching frequency From Section 3.3, the maximum microwave frequency corresponding to the lumped-circuit-treated electrodes is 0.69 GHz. When fm > 0.69 GHz, the response characteristics should be simulated using Eqs. (27)–(32). Fig. 5 shows the relations among the propagation powers P12,0, P22,0, the operation voltage U1 at z = 0 and the response time t, where t0 = 0, (a) U1 changes from 0 to
a Output Powers P12,0, P22,0
microwave effective index nm and characteristic impedance Z0, both of which will cause influences to the response characteristics. So G and W should be optimized for a compromise of high electrooptic modulation efficiency, match of impedance, and match between nm and neff. For the electrode structure described in Fig. 1, using Eqs. (22) and (23), Fig. 3 plots the curves of the microwave effective index nm and characteristic impedance Z0 versus (a) the electrode width W, and (b) the electrode gap G. From Fig. 3a, when W increases, the match between nm and neff can be realized, but this will result in the mismatch of the impedance. From Fig. 3b, the increase of G is an efficient way for the match between nm and neff, but this will decrease the electro-optic modulation efficiency and increase the switching voltage [17]. So we should select a compromise for the values of G and W. In order to match the characteristic impedance between the cable and the electrode for the great decrease of the reflection of the switching voltage signal, we choose the electrode gap and electrode width as G = 3.7 lm and W = 3.6 lm. In this case, the characteristic impedance is obtained, and a relative low switching voltage and a relative high microwave effective index are achieved, which are Z0 = 50.9 X, Us = 1.457 V, and nm = 1.19. Through above optimization on W and G, the device not only can be operated normally under a low switching voltage with no reflection of the switching signal, but also can have a relative high cutoff switching frequency. Notice that, during the optimization, first of all we guarantee that the switch can be operated normally, then, we make improvement for a larger operation bandwidth of the device. 3.3. Response under low switching frequency
trise
1.0
P12,0
0.8 0.6 0.4
ts td
0.2
P22,0
0.0 0
1.0
trise
P12,0
0.8 0.6 0.4
ts td
0.2
10
20
40
trise
0.6 0.4
ts td
0.2
P12,0 10
ts td
tfall
P12,0
0.0 10
20
30
30
40
1 0 0
1.0
40
30
60
90
120
P 22,0
0.8 0.6 0.4 0.2 0.0 0
0
20
2
P22,0
0.6
0.2
tfall
0.0
-1
0.8
0.4
c
Output Power P12,0,P22,0
Output Powers P12,0, P22,0
P22,0
Response Time t (ps)
Response Time t (ps)
1.0
40
0.8
0
30
30
trise
P22,0 0
20
1.0
tfall
0.0
b
b
U1 (V)
Output Powers P12,0, P22,0
a
10
Response Time t (ps)
Output Powers P12,0, P22,0
The wavelength of the low frequency switching voltage should be much larger than the length of the electrode, that is, km L0 , so the frequency of the operation voltage U should be
tfall
P 12,0
30
60
90
120
Response time t (ps)
Response Time t (ps) Fig. 4. Curves of output powers P12,0 and P22,0 versus the response time t under low microwave frequency, where (a) U changes from 0 to Us, (b) U changes from Us to 0, and t0 = 0.
Fig. 5. Relations among the propagation powers P12,0, P22,0, operation voltage U1 at z = 0, and the response time t under ultra-high microwave frequency, where (a) U1 changes from 0 to Us at the moment t0 = 0, (b) U1 changes from Us to 0 at the moment t0 = 0, and (c) U1 is a square wave switching signal.
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C.-T. Zheng et al. / Optics Communications 281 (2008) 5998–6005
Rise/Fall Time trise, tfall (ps)
ts
tfall
6
38
trise 4
36
2
34
0 1.0
1.2
1.4
1.6
1.8
Switching Time ts (ps)
40
8
32 2.0
4. Discussion and error analysis
Microwave Effective Index nm Fig. 6. Effects of the microwave effective refractive index nm on the rise time trise, fall time tfall and the switching time ts when the operation voltage U changes from 0 to Us.
Us, (b) U1 changes from Us to 0, and (c) U1 changes periodically. We can observe from the results that, the rise time trise and the fall time tfall are less than those shown in Fig. 4, which are about 3.78 ps and 4.94 ps, respectively, and the delay time is 27.86 ps, so the switching time is about 32.8 ps. However, from Fig. 5c, we find that, the delay time causes no effect to the exchange of output powers. Besides, when the switching frequency is large enough, and when the output power of one port changes from the minimum to maximum, the output power will exchange again before increasing to the maximum due to the small period cycle of the microwave signal. In this case, the exchange of the output powers cannot be totally realized. So there exists the maximum switching frequency, which is named the cutoff switching frequency denoted by fmcut and will be discussed later. 3.5. Rise time, fall time, switching time, and switching frequency
4
10
3
10
2
cut
Switching Frequency fm (GHz)
Through the analysis in Section 2.5, the rise time trise and fall time tfall under ultra-high switching frequency will not be constants when the propagation velocity vm changes. Fig. 6 shows the effects of the microwave effective refractive index nm on the
10
1.2
1.4
1.6
1.8
rise time trise, fall time tfall and the switching time ts when the operation voltage U changes from 0 to Us. We can see that, when nm = neff, both the rise time and fall time become zero. This is because the propagation velocity of the square wave switching voltage matches that of the light-wave well enough. Fig. 7 shows the relation between the microwave effective refractive index nm and the cutoff switching frequency fmcut . We find that, when nm = neff, the cutoff switching frequency can be up to infinity. For the switch designed in this paper, it is about 114.7 GHz. The above excellent features given in Figs. 6 and 7 should be taken into consideration in the design of electro-optic switches and modulators.
2.0
Microwave Effective Index n m Fig. 7. Relation between the microwave effective refractive index nm and the cutoff switching frequency fmcut .
When an electro-optic modulator or switch with Mach–Zehnder Interferometer (MZI) structure or directional coupler (DC) structure possesses a similar traveling-wave electrode, its response characteristics can be similarly analyzed using this technique. The differences between them are: for the modulator, we need to calculate the 3 dB modulating bandwidth using the distributed parameters, whereas for the switch, we need to calculate the cutoff switching frequency; for the modulator, the signal propagating along the electrode is a modulation microwave signal, whereas for the switch, that is a switching voltage signal. In our presented switch, there is a difference between the effective index of the microwave nm = 1.19 and that of the light-wave neff = 1.5910. This is because the match between nm and neff, the match of impedance, and high electro-optic modulation efficiency cannot be achieved at the same time for a certain electrode dimension. So we optimize the electrode width and electrode gap firstly for the match of the impedance and high electro-optic modulation efficiency, and then for the match of effective indices between nm and neff, that is, firstly we guarantee that the switch can be operated normally under a low switching voltage, and then we make improvement for a larger operation bandwidth of the device. Therefore, the electrode width optimized in this paper is not equal to the core width a, which is different from that of the designed standard switch reported in our previous paper [17], which is equal to the core width a. This is an improvement for the match of the impedance and the match of effective index besides high electrooptic modulation efficiency. Since the parameters including the characteristic impedance Z0 and the microwave effective index nm are essential to estimate the bandwidth or the response performance of modulators or switches, in order to check the accuracy of this technique, a comparison is carried out between the simulation results of this technique and the experimental results reported in Ref. [9] and those calculated from the extended point-matching method reported in Ref. [10], which is shown in Table 1. Thus, we can conclude from the comparisons that the technique presented in this paper has satisfactory accuracy. 5. Conclusion By using the presented technique, the response characteristics are simulated for the polymer directional coupler electro-optic switches, which include the transmission powers, rise time, fall
Table 1 Comparisons between the calculated results from this technique and those from references Ref.
[9] [10]
Electrode parameters
Z0 (X)
nm
Width
Gap
Thickness
This
Ref.
Error (%)
This
Refs.
Error (%)
8 10
50 15
3 6
36.4 45.6
35 42
+4 +8
– 2.08
– 2.31
– 9.9
C.-T. Zheng et al. / Optics Communications 281 (2008) 5998–6005
time, switching time and cutoff switching frequency. Simulation results for the designed switch are as: the switching voltage is about 1.457 V, the coupling length is about 4.374 mm, the switching time is about 32.8 ps, and the cutoff switching frequency is about 114.7 GHz. Comparisons shows that the simulation results of this technique are in good agreement with the experimental and other simulated results reported in some other references. This indicates this technique has satisfactory accuracy. Acknowledgment The authors wish to express their gratitude to the National Science Foundation Council of China (the project number is 60706011), the ministry of education of China (the project number is 20070183087), and the Science and Technology Council of China (the project number is 2006CB302803) for their generous support to this work. References [1] X.J. Lu, M. Li, Opt. Components Mater. IV 6469 (2007) A4691.
6005
[2] T. Shibata, M. Okuno, T. Goh, T. Watanabe, M. Yasu, M. Itoh, M. Ishii, Y. Hibino, A. Sugita, A. Himeno, IEEE Photon. Technol. Lett. 15 (2003) 1300. [3] G.H. Hu, Y.P. Cui, B.F. Yun, C.G. Lu, Z.Y. Wang, Opt. Commun. 279 (2007) 79. [4] D.M. Yeo, S.Y. Shin, Opt. Commun. 267 (2006) 388. [5] Q. Wang, J.P. Yao, Opt. Express 15 (2007) 16500. [6] Y.H. Kim, U.C. Paek, W.T. Han, Appl. Opt. 44 (2005) 3051. [7] Y. Enami, D. Mathine, C.T. Derose, R.A. Norwood, J. Luo, A.K.Y. Jen, N. Peyghambarian, Appl. Phys. Lett. 91 (2007) 093505. [8] N.H. Zhu, Q. Wei, E.Y.B. Pun, P.S. Chung, Opt. Quantum Electron. 28 (1996) 137. [9] H. Zhang, M.C. Oh, A. Szep, W.H. Steier, C. Zhang, L.R. Dalton, H. Erlig, Y. Chang, D.H. Chang, H.R. Fetterman, Appl. Phys. Lett. 78 (2001) 3136. [10] D.J. Ren, M.S. Chen, X.M. Huang, Chin. Study Opt. Commun. 143 (2007) 47. [11] O.G. Ramer, IEEE J. Quantum Electron. 18 (1982) 386. [12] C. Sabatier, E. Caquot, IEEE J. Quantum Electron. 18 (1986) 32. [13] Y. Ken, Power Electron. Technol. 30 (2004) 30. [14] Y. Enami, C.T. Derose, D. Mathine, C. Loychik, C. Greenlee, R.A. Norwood, T.D. Kim, J. Luo, Y. Tian, A.K.Y. Jen, N. Peyghambarian, Nature Photon. 1 (2007) 180. [15] C. Pitois, C. Vukmirovic, A. Hult, Macromolecules 32 (1999) 2903. [16] W.G. Driscoll, W. Vaughan, Handbook of Optics, McGraw-Hill, New York, 1978. p. 7. [17] C.T. Zheng, C.S. Ma, X. Yan, X.Y. Wang, D.M. Zhang, Opt. Commun. 281 (2008) 3695.