Articles Optoelectronics
June 2010 Vol.55 No.17: 18341839 doi: 10.1007/s11434-009-3352-8
SPECIAL TOPICS:
Optimization design for polymeric S-shaped ridge waveguide LU RongGuo*, LIAO JinKun, TANG XiongGui, LI HePing & LIU YongZhi School of Optoelectronic Information, University of Electronic Science and Technology of China, Chengdu 610054, China Received January 7, 2009; accepted March 13, 2009
The dispersion characteristics and transverse field distribution of the fundamental and higher order modes are analyzed for polymeric ridge multimode waveguide by a new technique which is based on the combination of the effective index method and the variational method. An algorithm is implemented to study the effect of the structure parameters and dimensions on the dispersion curves. The optical field distribution of the fundamental and higher order modes for TM modes are computed. The single mode conditions of polymeric ridge waveguide are obtained. The relationship between the curvature radius and the bending loss of S-shaped ridge waveguide are studied with wide-angle finite-difference beam propagation method and effective index method. The conclusion is: when the curvature radius is larger than 5000 μm, the bending loss will not decrease distinctly even if the curvature radius increases, and the light can propagate stably in the S-shaped ridge waveguide. electro-optic polymer, S-shaped ridge waveguide, effective index method, variational method, wide-angle finite-difference beam propagation method
Citation:
Lu R G, Liao J K, Tang X G, et al. Optimization design for polymeric S-shaped ridge waveguide. Chinese Sci Bull, 2010, 55: 18341839, doi: 10.1007/s11434-009-3352-8
Electro-optic (EO) polymers have attracted much attention in recent years due to their great advantages such as low microwave dielectric constant, high electro-optic coefficient, and great processing flexibility over inorganic materials such as LiNbO3 or compound semiconductors [1–4]. Since polymeric S-shaped ridge waveguide is the key component of polymeric waveguide delay line, its optimization design is important [5]. In this paper, the dispersion characteristics and transverse field distribution of the fundamental and higher order modes are analyzed for polymeric S-shaped ridge waveguide by a new technique which is based on the combination of the Effective Index Method (EIM) [6] and the Variational Method (VM) [7,8]. The new technique combines the advantages of each method and avoids their disadvantages. EIM has the property that can be used to determine the dispersion curves very accurately with little calculations but it has difficulty in finding the field *Corresponding author (email:
[email protected])
© Science China Press and Springer-Verlag Berlin Heidelberg 2010
distribution, while VM can be used with very good accuracy to find the field distribution. Then the relationship between the curvature radius and the bending loss of S-shaped ridge waveguide are studied with Wide-Angle Finite-Difference Beam Propagation Method and Effective Index Method, and the optimum waveguide structure parameters are obtained.
1 Waveguide structure Figure 1 shows the structure of polymeric S-shaped ridge waveguide. It consists of two straight waveguide, two 180° bends and two 90° bends. The curvature radius of the 180° bends is smaller than that of the 90° bends so that the coupling between the straight waveguide and the bends can be avoided. Figure 2 shows the triple stack structure of the ridge waveguide. An E-O polymer layer (the core layer) is sandwiched between two cladding layers. The core layer is 25wt% ICP-E/polysulfone (n2 = 1.67), the cladding layers are NOA61 (n1 = 1.55) and UV15(n3 = 1.50).
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LU RongGuo, et al.
Chinese Sci Bull
June (2010) Vol.55 No.17
1835
§ J3 · J1 · ¸ arctg ¨ c2 ¸ © J2 ¹ © J2 ¹ §
J 2 w arctg ¨ c1 pʌ( p
0,1, 2,"),
(4)
where J ° 1 ® °¯J 3
Figure 1
2 2 neff k0 neff2 ,
(5)
2 2 k0 neff , neff3
2 2 2 2 /neff3 , for TM mode, /neff1 , c2 = neff2 for TE mode, c1 = neff2 c1=c2=1. Solving eq. (4), we get the effective indexes neff of TEpq or TMpq mode of the ridge waveguide, p and q are mode labels. The ridge waveguide can be equivalent to three dielectric slabs by using the EIM. The functions Y1(y), Y2(y) and X(x) are defined as the fields through these slabs as shown in Figure 2. In regions I and III:
Structure of S-shaped ridge waveguide.
Y1 ( y ) Figure 2
2 2 neff1 k0 neff ,J 2
Cross-section view and equivalent slab waveguide.
cos(J 2 d D ) exp[J 1 ( y d )], ° ®cos(J 2 y D ), °cos D exp(J y ), 3 ¯
y!d 0 y d , (6) y0
where J ° 1 ® °¯J 3
2 Dispersion characteristics and transverse field distribution of ridge waveguide 2.1
2 n12 , J 2 k0 neff1
2 k0 n22 neff1 ,
(7)
2 k0 neff1 n32 ,
Formula derivation
As shown in Figure 2, the ridge width is w, the upper and lower cladding’s refractive index is n1 and n3, respectively. The thickness of the core is d with a refractive index n2, the thickness of the ridge is D. In the equivalent slab waveguide, by applying Effective Index Method in regions I, II, III, respectively, we obtain the characteristic equation as follows:
J2H
§ J · § J · arctg ¨ c1 1 ¸ arctg ¨ c2 3 ¸ © J2 ¹ © J2 ¹ qʌ(q 0,1, 2,"),
D
2
2
2
k0 neffi n1 , J 2
2
k0 n2 neffi , J 3
k0
2ʌ / O0 .
· ¸ qʌ(q ¹
0,1, 2,......).
(8)
In region II: Y2 ( y )
cos(J 2c D E ) exp[J 1c( y D)], ° ®cos(J 2c y E ), °cos E exp(J c y ), 3 ¯
y!D 0 y D , (9) y0
where °J 1c ® °¯J 3c
(1)
where J1
§ J arctg ¨ c2 3 © J2
2
2
k0 neffi n3 ,
(2) (3)
In regions I, II, III, I = 1, 2, 3 respectively. In regions I and III, H = d, in region II, H = D. For fundamental mode q = 0, O0 is the wavelength in vacuum. For TE mode, c1 = c2 = 1, for TM mode, c1 = n22 /n12 , c2 = n22 /n32 . Solving eq. (1), we get the effective indexes neff of the three regions. According to the effective index profile, we consider the following characteristic equation:
E
2 n12 , J 2c k0 neff2
2 k0 n22 neff2 ,
(10)
2 k0 neff2 n32 ,
§ Jc · arctg ¨ c2 3 ¸ qʌ(q © J 2c ¹
0,1, 2,......).
(11)
In eqs. (8) and (11), for TE mode, c2 = 1; for TM mode, c2 = n22 /n32 , X(x) is expressed as: X ( x)
x w / 2, cos I exp[J 1cc( x w / 2)], ° cc w / 2 x w / 2, ®cos(J 2 ( x w / 2) I ), °cos(J ccw I ) exp[J cc( x w / 2)], x ! w / 2, 2 3 ¯
(12) where
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J cc k n 2 n 2 , J cc 0 eff eff1 2 ° 1 ® 2 2 °¯J 3cc k0 neff neff 3,
2 2 k0 neff2 , neff
distribution can be obtained. (13)
0,1, 2,......),
(14)
2 2 /neff1 , for TM mode, c1 in eq. (14), for TE mode, c1 = neff2 = 1. For more accurate optical field distribution, we consider the scalar wave equation
w2\ w2\ 2 ]\ 2 k02 [ n 2 ( x, y ) neff 2 wx wy
0.
(15)
By applying the result obtained above, the field distribution \(x, y) inside and outside the core can be assumed as
\ ( x, y )
X ( x)[Y1 ( y ) RY2 ( y )].
(16)
The functions X(x)Y1(y), and X(x)Y2(y) are approximate optical field distribution in regions I, III and region II, respectively. R is the ratio between X(x)Y1(y) and X(x)Y2(y). We multiply both sides of eq.(15) by \, then integrations are performed in the entire cross-section of the ridge waveguide, and we get
ª w2\
³³ «¬ wx
2
w2\ wy 2
June (2010) Vol.55 No.17
2.2 The dispersion characteristics of TE and TM modes
§ J cc · arctg ¨ c1 3 ¸ pʌ( p © J 2cc ¹
I
Chinese Sci Bull
º »\ dxdy ¼
2 2 k02 ³³ n 2\ 2 dxdy k02 neff ³³ \ dxdy
0.
As shown in Figure 2, w is the waveguide ridge width, the ridge height is 0.3 Pm. We analyze the TE and TM modes by varying waveguide ridge width w from 39 Pm, with a step 'w of 1 Pm, while other parameters are fixed. The curves are shown in Figure 3. Varying the wavelength varies from 1.01.7 Pm, with a step 'O of 0.1 Pm, when and other parameters are fixed, we can get Figure 4. It is obvious that the effective index monotonously decreases as the wavelength increases. when the ridge height varies from 0.11.0 Pm, with a step 'h of 0.1 Pm, and other parameters are fixed, we can get Figure 5. It shows that the fundamental mode’s effective index monotonously increases as the ridge height increases. There are three important conclusions: (1) The effective index changes slowly with the variation of ridge width. The curve is quite flat in Figure 3. But the effective index decreases distinctly when the orders of modes increase. (2) The effective index monotonously increases as the wavelength increases, and monotonously increases as the ridge height increases. (3) The effective index of TM mode is smaller than that of TE mode with the same order while other parameters are fixed. The single mode condition of polymeric ridge waveguide are obtained: the core thickness d 1.5 Pm, the ridge width w 5 Pm.
(17)
Substituting eq. (16) into eq. (17), we get aR 2 + bR + c
0,
(18)
where
a
2 2
³³ ( XX ccY 2 k02 neff
b
X 2Y2Y2¢¢ )dxdy k02 ³³ n 2 X 2Y22 dxdy
³³ X
2
Y22 dxdy,
1 cc 2 )dxdy ( XX ccY1Y2 X 2Y1Y ³³ 2 1 ³³ ( XX ccY1Y2 X 2Y1Y2cc)dxdy 2 2 2 k02 ³³ n 2 X 2Y1Y2 dxdy k02 neff ³³ X Y1Y2dxdy,
c
(19)
2 1
³³ ( XX ccY
(20)
X 2Y1ccY1 )dxdy k02 ³³ n 2 X 2Y12 dxdy
2 2 2 k02 neff ³³ X Y1 dxdy.
(21)
Substituting a, b, c into eq. (18) and solving it, we can get R. There are two solutions, We get the one with smaller absolute value. Substituting R into eq. (16), the transverse field
Figure 3 The curves of effective index vs. ridge width. (a) TM mode; (b) TE mode.
LU RongGuo, et al.
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June (2010) Vol.55 No.17
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3 The relationship between the curvature radius and the bending loss 3.1 Wide-angle finite-difference beam propagation method and padé approximants
Figure 4
Fundamental mode’s effective index vs. operating wavelength.
Figure 5
Fundamental mode’s effective index vs. ridge height.
2.3
Transverse field distribution of TM mode
The transverse field distribution of the fundamental and higher order modes for TM modes are computed by the technique which is based on the combination of the Effective Index Method and the Variational Method. The results are shown in Figure 6 and Figure 7. The wavelength is set as O0 = 1.55 Pm, structure parameters are set as D= 1.8 Pm, d = 1.5 Pm, w = 5 Pm and D = 1.9 Pm, d = 1.6 Pm, w = 6 Pm, respectively.
Finite-difference beam propagation methods (BPM) using the paraxial approximation have been applied successfully to the treatment of photonic integrated circuits containing shallow angles with regard to the direction of propagation. However, the method contains several important drawbacks, among which the most important is its limited angular range of principal propagation direction. This limitation has generated several techniques to correct the drawbacks. One approach for extending the accuracy of the BPM at wide angles has been the use of Padl approximants. In this paper, we analyze the relationship between the curvature radius and the bending loss with Wide-Angle Finite-Difference Beam Propagation Method and Padé approximants [9]. The scalar Helmholtz equation is expressed as
2 E n 2 k0 E
0,
(22)
where E(x, y, z) is the electric field, n(x, y, z) is the refractive index profile of waveguide, K0 is the wave number in a vacuum, kr=k0×nsubstrate. Let u ( x, y, z ) exp(iZt ) exp(ikr z ),
E ( x, y , z )
(23)
where Z is the frequency of light, kr=k0×nsubstrate, we write the scalar Helmholtz equation as
uzz 2ikr uz uxx (k 2 kr2 )u
0.
(24)
Eq. (24) is formally written as ikr ( 1 P 1)u ,
uz
where P is an operator defined by P
(25) 2
2
( w / wx ( k 2
kr2 )) / kr2 , the square root operator can be expanded by
Figure 6
Figure 7
The transverse field distribution of TM00 mode.
The transverse field distribution of TM10 mode.
Padé approximants, The expansion is given by the expression [9]: 1 P 1 | N m ( P ) Dn ( P ), where Nm(P) and Dn(p) are polynomials of order m and n, respectively. In the operator p, for the accuracy of simulation, Padé (2,2) approximants are used, then eq. (25) are given by recurrence eqs. (26), (27). uz
ikr ( N m ( P ) Dn ( P))u ,
(26)
uz
ik r
P 2 P2 4 u. 1 3P 4 P 2 16
(27)
The next step is the discretization of eq.(27) at point ( s, r 1/ 2), (s corresponds to the x coordinate, r corresponds to the z coordinate). First, we discretize it in the z coordinate, uz and u are expressed as uz=(ur+1–ur)/'z, u (u r 1 u r ) 2, then we get
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1 [1 P [ 2 P 2 r u , 1 K1 P K2 P 2
u r 1 where [1
(28)
1 16 ikr 'z 8, K1
3 4 ikr 'z 4, [ 2
ikr 'z 4, K 2
Chinese Sci Bull
3 4 1 16 ikr 'z 8, rewriting eq. (28), we get (1 a1 P)(1 a2 P) r u , (1 b1 P)(1 b2 P)
u r 1
(29)
where ai ,bi (i 1, 2) can be obtained by solving Polynomial Equation. We solve eq. (29) by two steps, the form of equation is expressed as (1 ai P ) r (i 1) 2 u . (1 bi P )
u r i 2
(30)
Then, we discretize it in the x coordinate, by using w2 u wx 2 (us 1 2us us 1 ) 'x 2 , we get Ausr1i 2 Busr i 2 Cusr1i 2
D,
(31)
where
bi ° A C 'x 2 k 2 r ° ° bi ª 2 2 º 2 ° B 1 2 « k kr 2 » kr ¬ 'x ¼ ° . ® a r ( i 1) 2 r ( i 1) 2 i °D u u s 1 s 1 ° 'x 2 kr2 ° ai ª 2 2 º ½ r (i 1) 2 ° 2 ° ®1 k 2 « k kr 'x 2 » ¾ us ¼¿ ¯ r ¬ ¯
(32)
By using the Thomas method, we can obtain the unknown field ur+1at z+'z by using the known field ur, where r and r + 1 respectively correspond to z and z+'z.
June (2010) Vol.55 No.17
the directional change within short distances, with large angles, and with low loss in order to increase packing density and stabilize optical circuits. It is well known that any bend in a dielectric waveguide causes losses by radiation. Two kinds of radiation loss must be distinguished. First, at a bend with constant radius of curvature power is lost by tangential radiation because of the finite speed of light in the cladding medium. Second, at any junction between two waveguides with different curvature radius there is a transition loss because of a mismatch of the field distributions [10]. The common method for reducing both the so-called pure bend and the transition losses is to increase the curvature radius until the total attenuation of the bend is smaller than an acceptable limit. Then, we study the relationship between the curvature radius and the bending loss of S-shaped rib waveguide. The wavelength is set as O0 = 1.55 Pm, the step length 'x = 'z = 0.5 Pm, and Cosh Beams are used as the exciting light. The analysis window is set as 900 Pm × 2000 Pm, and we simulate the optical field of the S-shaped waveguide with different curvature radius. Figures 810 show the bending loss with 400 Pm, 1000 Pm, 2000 Pm, and 5000 Pm curvature radius: the bending loss with 400 Pm curvature radius is high, and almost all the optical power emits from the S-shaped waveguide. When the curvature radius is 1000 Pm, most of the optical power can propagate a long distance (about 700 Pm) stably. There is almost no bending loss when the curvature radius is 2000 Pm and 5000 Pm. It is obvious that the bending loss is getting smaller and the light can propagate stably in the S-shaped ridge waveguide as the curvature radius increases. Figure 11 shows the relationship between the normalized average radiant power and the curvature radius of the waveguide. When the curvature radius is larger than 5000 Pm, the curve is flatter and thus there is a better stability for
3.2 Calculation of effective index As shown in Figure 2, the thickness of the core is 1.5 Pm, the ridge height is 0.3 Pm, the waveguide ridge width is 5 Pm, and the thickness of the lower and upper claddings is 3 Pm and 3.5 Pm, respectively. Solving eq. (1), we get the effective indexes neff of the three regions. By numerical calculation we obtain the transverse effective index distribution of TM fundamental mode in the ridge waveguide: n( x )
3.3
°neff 1 , x İ w / 2 , ® °¯neff 2 , x ! w / 2
neff1=1.6373, neff2 =1.6313, 'n
0.006.
(33)
Simulation analysis
It is desirable to change the direction of light propagation in optical delay line. So it is important to find the best curvature radius of the S-shaped ridge waveguide that can make
Figure 8 Bending loss with 400 Pm curvature radius. (a) The shape of waveguide; (b) optical field.
LU RongGuo, et al.
Chinese Sci Bull
June (2010) Vol.55 No.17
1839
Figure 11 Relationship between the normalized average radiant power and the curvature radius of waveguide.
Figure 9 Bending loss with 1000 Pm curvature radius. (a) The shape of waveguide; (b) optical field.
ters and dimensions on the dispersion curves. The optical field distribution of the fundamental and higher order modes for TM modes are computed. The single mode condition of polymeric ridge waveguide are obtained: the core thickness d1.5 Pm, the ridge width w5 Pm. The relationship between the curvature radius and the bending loss of the S-shaped ridge waveguide are studied with wide-angle finite-difference beam propagation method and effective index method. The conclusion is: when the curvature radius is larger than 5000 Pm, the bending loss will not decrease distinctly even if the curvature radius increases, and the light can propagate stably in the S-shaped ridge waveguide. This work was supported by Advance Research Program of Weapon Equipment, National Natural Science foundation of China (60736038), and National Hi-Tech Research and Development Program of China (2007A A01Z269). 1
2
Figure 10 Bending loss with 2000 Pm and 5000 Pm curvature radius. (a) 2000 Pm; (b) 5000 Pm.
3
optical power transmission, the bending loss will not decrease distinctly even if the curvature radius increases, and the light can propagate stably in the S-shaped ridge waveguide. This is of great value for reference.
4 5
6
4
Conclusions 7
The dispersion characteristics and transverse field distribution of the fundamental and higher order modes are ana lyzed for polymeric ridge multimode waveguide by a new technique which is based on the combination of the effecttive index method and the variational method. An algorithm is implemented to study the effect of the structure parame-
8 9 10
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