Box-like spectral response of 2-D microring resonator arrays

Optics Communications 273 (2007) 105–113 www.elsevier.com/locate/optcom

Box-like spectral response of 2-D microring resonator arrays Chun-Sheng Ma a, Xin Yan a, Xian-Yin Wang a

a,b,*

, De-Lu Li

a

State Key Laboratory on Integrated Optoelectronics, College of Electronic Science and Engineering, Jilin University, Changchun 130012, China b Changchun University of Science and Technology, Changchun, 130012, China Received 7 June 2006; accepted 19 December 2006

Abstract In terms of the coupled mode theory and the transfer matrix technique, novel formulas of the transfer functions are presented for a 2-D microring resonator array. By using these formulas, transmission characteristics are analyzed for this kind of Si based polymer device. Simulation shows that by means of parallel-cascading multiple microrings in every filter element, the passband can be flattened, and the box-like spectral response can be formed; in terms of series-cascading such several filter elements, the passband becomes precipitous, and the non-resonant light and the crosstalk become weak.  2006 Elsevier B.V. All rights reserved. Keywords: Microring resonator array; Transfer function; Amplitude coupling ratio; Transmittance

1. Introduction The spectral response of the microring resonator (MRR) device plays an important role in optical communication networks. The single MRR is the simplest filter structure, which has two basic drawbacks [1–3]. One is that the Lorentzian spectral response of the single MRR is convex, which will require very accurate wavelength control and very precise fabrication technologies. Another is that the intensity of the non-resonant light in the single MRR transmission spectrum is strong, which will lead to large crosstalk of the device. In practical applications, MRR devices should have the so-called box-like spectral response to reduce the need for accurate wavelength control. Usually one can cascade multiple microrings in parallel or in series to form a MRR array to flatten the spectral response and increase the out-of-band signal rejection ratio. A parallel-cascaded MRR array can form a very flat box-like spectral response, but the sidelobes in the transmission spectrum are hardly dropped by varying

*

Corresponding author. Tel./fax: +86 431 85168270. E-mail address: [email protected] (X.-Y. Wang).

0030-4018/$ - see front matter  2006 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2006.12.031

the ring number [4–7]. A series-cascaded MRR array can greatly drop the intensity of the non-resonant light in the transmission spectrum by increasing the ring number, but its box-like spectral response is usually undulant more or less [8–11]. In order to further improve the filtering of MRR device, in this paper, we have designed a 2-D MRR array, which consists of several series-cascaded filter elements, each of which contains parallel-cascaded multiple microrings. By selecting proper values of parameters, such as the center distance of adjacent rings, the amplitude coupling ratio, the number of the filter elements, and that of the microrings in every filter element, very flat box-like spectral response can be obtained, the intensity of the non-resonant light can be dropped greatly, and hence the crosstalk can be reduced efficiently. This paper is organized as follows. First in Section 2, by using the coupled mode theory (CMT) and the transfer matrix technique (TMT), novel formulas of the transfer functions are derived for such a MRR array. Then in Section 3, by using these formulas, transmission characteristics are analyzed for this kind of Si based polymer device. Finally, based on the analysis and discussion, some conclusions are reached in Section 4.

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C.-S. Ma et al. / Optics Communications 273 (2007) 105–113

the coupling layer between the rings and the channels, respectively. In order to achieve the maximum output intensity of the drop port, all the rings should have the same radius R, and should have the same amplitude coupling ratio j and the same amplitude transmitting ratios t at all the coupling points [4–7]. The expressions of j and t are given in Ref. [12], which are real quantities and equal to jj and t defined in Refs. [13,14], and satisfy the relation j2 + t2 = 1. According to the universal relations for coupling between the ring and the channel [13], the relations of the amplitudes for the vth ring in the uth filter element are as follows:

2. Theory In terms of the CMT and TMT, in this section we derive the expressions of the transfer functions of the presented 2-D MRR array. Fig. 1a shows the diagram of a 2-D MRR array, which consists of M series-cascaded filter elements, each of which contains N parallel-cascaded rings and two straight waveguides, which are called channels. The channels are within the below plane, which are buried in the same medium, and the rings are within the above plane, which are clad by air, and a coupling layer exists between the rings and the channels. In every filter element, the N rings are placed on the top of two parallel channels. Assume that the total length of every channel is 2L1 + 2(N  1)L2 + L3, where L1, L2 and L3 are labeled in this figure. This figure is plotted for the case that the number of the filter elements M is even, in this case, the drop port is on the right of the bottom output channel; if M is odd, the drop port is on the left of the bottom output channel. Fig. 1b shows a basic structure of the vth ring in the uth ðiÞ filer element, where aðiÞ uv , buv (i = 1, 2, 3, 4; u = 1, 2, . . .,M; v = 1, 2, . . .,N) are the input and output amplitudes at the two coupling points, respectively. Fig. 1c shows the cross sections and the refractive index profiles of the rings and channels, where w1 and s1 are the core width and thickness of the rings, and w2 and s2 are those of the channels, respectively. Assume that the rings and channels have an identical core refractive index n1, but have different cladding refractive indices n3 and n2. Let d and n2 be the thickness and the refractive index of

a

ð1Þ ð3Þ ð1Þ ð2Þ ð1Þ buv ¼ tauv  jjauv ¼ tauv  jjbuv expðj/Þ;

ð1Þ

ð2Þ buv ð3Þ buv ð4Þ buv

ð2Þ

ð1Þ þ ¼ jjauv þ tbð3Þ uv expðj/Þ; ð2Þ ð3Þ ð4Þ ð4Þ tauv  jjauv ¼ tbuv expðj/Þ  jjauv ; ð2Þ ð3Þ ð4Þ jjauv þ tauv ¼ jjbuv expðj/Þ þ tað4Þ uv ; ð1Þ jjauv

¼ ¼ ¼

tað2Þ uv

ð3Þ ð4Þ

where / = pR(b  jaR), aR is the loss coefficient of the rings. Solving Eqs. (1)–(4), we can obtain ! ! ! ð4Þ ð4Þ auv buv 1 U2  V 2 V ¼ ; ð5Þ ð1Þ ð1Þ U V 1 auv buv where U¼

t½1  expðj2/Þ ; 1  t2 expðj2/Þ

V ¼

j2 expðj/Þ : 1  t2 expðj2/Þ

ð6Þ

The relations that follow are:

(1)

a0

a11

Input

Through 1

(1)

(1)

(1)

(4)

8 (4)

(1)

(1)

(1) au,v+1 buv auv buN auN bu,v+1 10

Through u

... (4)

(1)

...

(1)

bu1 au1

(4)

... (4)

(4)

...

(4) auN buN au,v+1 bu,v+1 auv buv

Through M

bu-1,1 (4)

Through u-1

... ...

bu

...

...

Through 2

Through M-1

(4)

au1 bu1

6

4

L1

(4)

L2

2

b

bM+1

bM1

(1)

(1)

(2)

(2) uv 6

buv auv 8

L3 10 8 6 4

buv a

2

4

(3)

(3) 2 uv

(4)

(4)

auv b

auv buv 0

n 3 s1 0

8 10

R

L1

L2

c

Drop

n3 w1 n1

Ring n3

R

n2 d w2 n2 s2 n1

Channel n2

n2

Fig. 1. (a) Diagram of a 2-D MRR array, (b) basic structure of the vth ring in the uth filer element, and (c) cross sections and refractive index profiles of the rings and channels.

C.-S. Ma et al. / Optics Communications 273 (2007) 105–113 ð4Þ auv

!

 ¼

ð1Þ buv



expðjw2 Þ

0

0

expðjw2 Þ

ð4Þ

bu;vþ1

! ð7Þ

;

ð1Þ

au;vþ1

where w2 = L2(b  jaL), aL is the loss coefficient of the channels. Substituting Eq. (7) into Eq. (5), we get ! ! ð4Þ ð4Þ bu;vþ1 buv ¼ Pv ; ð8Þ ð1Þ ð1Þ auv au;vþ1 where Pv is the amplitude transfer matrix, given by ! 1 ðU 2  V 2 Þ expðjw2 Þ V expðjw2 Þ Pv ¼ ; U V expðjw2 Þ expðjw2 Þ ðv ¼ 1; 2; . . . ; N  1Þ:

ð9Þ

Recurring Eq. (8) from v = 1 to N (when v = N, take ð4Þ ð4Þ ð1Þ ð1Þ L2 = 0, then w2 = 0, bu;N þ1 ¼ auN ¼ 0, and au;N þ1 ¼ buN Þ, we obtain !      ð4Þ P 11 P 12 0 0 bu1 P . . . P P  ¼ P 1 2 N 1 N ð1Þ ð1Þ ; ð1Þ buN buN P 21 P 22 au1 ð10Þ where PN is given by 1 PN ¼ U

U2  V 2 V

V 1

! ð11Þ

:

Thus from Eq. (10), we have ð4Þ

ð1Þ

bu1 ¼ P 12 buN ;

ð1Þ

buN ¼

¼

ð1Þ P 12 bu1;N

1 ð1Þ a : P 22 u1

ð12Þ ð1Þ au1

¼

P 12 ð1Þ expðjw3 Þ ¼ a expðjw3 Þ: P 22 u1;1

ð4Þ bu1;1

exp

ð1Þ

ð1Þ

ð4Þ

bM1 ¼

P u1 ð1Þ 12 a11 exp½jðu  1Þw3 : P u1 22

ð17Þ

3. Results and discussion In order to design such a Si based polymer MRR array with excellent features, in this section, we analyze the transmission characteristics of this device. We take the central resonant wavelength to be k0 = 1550 nm, the refractive index of polymer core of rings and channels to be n1 = 1.6278, that of polymer cladding surrounding the channels to be n2 = 1.465, and that of air cladding above and beside the rings to be n3 = 1, and other values of some parameters used in the calculation are given in relative figure captions. In our simulation, the bending characteristics of rings are investigated by using an efficient method proposed by Melloni et al. [15]. Both Melloni’s calculation and ours show that the accuracy of this method is in very good agreement with those of the beam propagation method (BPM) and the mode-matching method. This method is concluded to solve the propagation constant g of the bent modes from the following eigenvalue equations: n X ðRdpq þ cpq Þcq eq ¼ gep ; ðp ¼ 1; 2; . . . ; nÞ; ð18Þ

PM 12 ð1Þ a11 exp½jðM  1Þw3 : PM 22 ð1Þ

with cpq ¼

Z Z

xUp Uq dx dy;

ð19Þ

ð15Þ

nc ¼

P u1 ð1Þ 12 a exp½jðu  1Þw3 ; P u22 11

k ReðgÞ: 2pR

ð20Þ

ð1Þ

Using the relations bu ¼ buN expðjw1 Þ, a11 ¼ a0 exp ð4Þ fj½w1 þ ðN  1Þw2 þ w3 g, bMþ1 ¼ bM1 expfj½w1 þ ðN  1Þw2 þ w3 g, where w1 = L1 (b  jaL), from Eq. (15) we finally obtain the transfer function from the input port to the drop port D and that to the uth through port Bu as follows:  2  u1 2 bu  P  Bu ¼   ¼  12u expfj½2w1 þ ðN  1Þw2 þ uw3 g ; a0 P 22 ðu ¼ 1; 2; 3; . . . ; MÞ;

22

ð14Þ

ð13Þ

Substituting Eq. (14) into Eq. (12), we have buN ¼

0

where R is the radius of bent waveguide, eq is the eigenvector, Uq and cq = bq  jaq are the complex field profile and the complex propagation constant of the qth straight mode when the waveguide is incurved, and bq and aq are its real propagation constant and loss coefficient, respectively. These modes include the higher order modes that are below the cutoff, which are usually called quasimodes or leaky modes, and they have complex propagation constants because of the radiation [15]. The effective refractive index nc of the bent mode is simply determined by

Recurring Eq. (13) from u = u to 2, we obtain au1 ¼

   2 bMþ1 2 P M  12    D¼ ¼  M expfj½2w1 þ 2ðN  1Þw2 þ ðM þ 1Þw3 g : a  P

q¼1

Substituting Eq. (12) into the relation ðjw3 Þ, where w3 = L3 (b  jaL), we have ð1Þ au1

107

ð16Þ

3.1. Core size of rings and channels In order to realize the single mode propagation and reduce the polarization dependence of the device, we have optimized the core size of the rings and channels. Because the cores of the channels are buried in the same cladding n2, we can take their core width and thickness w2 = s2 = 1.50 lm, in this case, the birefringence between the Ex00 and Ey00 modes of the channels is zero, and only the Ex00 or Ey00 mode can propagate in the channels. Because the

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C.-S. Ma et al. / Optics Communications 273 (2007) 105–113

cores of the rings are clad different claddings n2 and n3, we should select the proper values of their core width and thickness w1 and s1 to greatly reduce the birefringence of the rings, which can be seen in the discussion of Fig. 2. Here, we choose w1 = 2.09 lm, s1 = 1.21 lm, in this case, only the Ex00 or Ey00 mode can propagate in the rings, and the Ex00 and Ey00 modes of all the rings and the channels nearly have an identical mode propagation constant b, as shown in Fig. 2. 3.2. Birefringence and central wavelength shift From the ring resonant equation 2pRnc = mk0, we can obtain the shift of the central wavelength between the Ex00 and Ey00 bent modes as dk0 ¼ k0 ðEx00 Þ  k0 ðEy00 Þ ¼

2pR 2pR ½nc ðEx00 Þ  nc ðEy00 Þ ¼ dnc ; m m ð21Þ

where dnc ¼ nc ðEx00 Þ  nc ðEy00 Þ is the birefringence between the Ex00 and Ey00 bent modes of the rings. Fig. 2 shows the effect of the ring radius Ron the effective refractive index nc, the birefringence dnc and the central wavelength shift dk0. We find that as R increases, nc decreases, while the birefringence dnc and dk0 increase. When R is 12.76 lm, then nc is about 1.5463 which is nearly equal to that of the channels, dnc is about 2.8 · 105, and dk0 is about 0.025 nm, both of which are very small. Therefore, only the Eypq mode is analyzed in the following simulation. 3.3. Ring radius and free spectral range (FSR) From the ring resonant equation 2pRnc = mk0, we can derive the expressions of the ring radius R and FSR as [12] mk0 ; 2pnc

FSR ¼

k0 nc k20 ¼ ; mng 2pRng

ð22Þ

The insertion loss contains the leakage loss of the higher refractive index substrate, the bending loss of the rings, and the propagation loss of the rings and channels. Calculation shows that when the thickness of the confined layer between the channels and the substrate is larger than 5 lm, the leakage loss caused by the Si substrate is dropped below 2.0 · 107 dB/cm, which is so small that it can be neglected. The bending loss of a ring can be determined by the following formula: Lb ðdBÞ ¼ 10log10 ½expð2ab  2pRÞ;

ð23Þ

where ab is the bending loss coefficient of the rings, of which the expression is given in Ref. [16]. Fig. 4 shows the effect of the ring radius R on the bending loss coefficient ab and the bending loss of a ring Lb. We find that as R increases, both ab and Lb decrease rapidly. When R is 12.76 lm, ab is about 2.0 · 102 dB/cm, and Lb is about 3.3 · 104 dB. Therefore, we only take account of the bending loss of the rings ab and the propagation loss ap, which includes the absorption loss of the materials and scattering loss mainly caused by the roughness of the sidewalls of the rings and channels, that is, aR = ab + ap for the rings and aL = ap for the channels, respectively. In the following calculations, we choose w1 = 2.09 lm, s1 = 1.21 lm w2 = s2 = 1.5 lm, m = 80, R = 12.76 lm, L1 = 2000 lm, L3 = 500 lm, and 2ap = 0.5 dB/cm.

(R=12.76μm, δ nc=-2.8x10 , -5

1.60 nc

δλ 0

δλ0=-0.03nm)

15 10

1.58

δ nc

5 0

1.56 (R=12.76μm, nc=1.5463)

1.54

-5 -10

0

10

20

30

40

-15 50

Ring Radius R (μm) Fig. 2. Effect of the ring radius R on the effective refractive index nc, the birefringence dnc and the central wavelength shift dk0 between the Ex00 and Ey00 bent modes of the ring, where w1 = 2.09 lm and s1 = 1.21 lm.

Free spectral Range FSR (nm)

20

x E00 E00

1.62

1.52

3.4. Insertion loss

25 y

-3

Effective Refractive Index nc

1.64

δ nc (x10 ) and δλ0 (nm)

R¼

where m is the resonant order of the rings, and ng = nc  kdnc/dk is the group refractive index. Fig. 3 shows the effect of the ring radius R on the FSR. We can see that as R increases, FSR decreases. If the FSR is selected too large, the ring radius R is too small, which will lead to large bending loss of the rings. On the contrary, if R is chosen too large, FSR will be too small, which will be difficult to realize the normal filtering of the device. When R is 12.76 lm, FSR is about 17.8 nm, and the relative resonant order m is 80.

50 40 30

(R=12.76 μm, FSR=17.8nm)

20 10 0

0

10

20

30

40

50

Ring Radiue R (μ m) Fig. 3. Effect of the ring radius R on the FSR, where w1 = 2.09 lm and s1 = 1.21 lm.

C.-S. Ma et al. / Optics Communications 273 (2007) 105–113 3

αb (dB/cm) and Lb (dB)

10

(R=12.76 μm, αb=2.0X10 dB/cm) -2

0

10

-4

(R=12.76 μm, Lb=3.3X10 dB) -3

10

2α b -6

10

Lb

-9

10

-12

10

8

10

12

14

16

18

20

Ring Radius R ( μm) Fig. 4. Effect of the ring radius R on the bending loss coefficient ab and the bending loss of a ring Lb, where w1 = 2.09 lm and s1 = 1.21 lm.

3.5. Transmission spectrum The transmission spectrum output from the uth through port and that from the drop port are defined as, respectively T u ðkÞ ðdBÞ ¼ 10log10 ½Bu ðkÞ; T ðkÞ ðdBÞ ¼ 10log10 ½DðkÞ:

ðu ¼ 1; 2; . . . ; MÞ;

ð24Þ ð25Þ

Output Spectrum T (dB)

Fig. 5 shows the effect of the center distance L2 of adjacent rings on the output spectrum T of the drop port. We can see that L2 should not be chosen arbitrary values, for

Output Spectrum T (dB)

example, L2 = 200, 400 lm, otherwise we cannot obtain the spectral response with fine shape. Calculation shows that when L2 equals the integral multiple of pR, for instance, L2 = 5pR, 10pR, the spectral response possesses better shape. When we select the value of L2 P 10pR, the box-like spectral response is formed. Because the amplitude coupling ratio j and the amplitude transmitting ratio t satisfy the relation j2 + t2 = 1, we can regard t as a function of j, that is, t = (1  j2)1/2, so we only investigate the effect of j on the transmittance of the device, which is equivalent to that of t. Fig. 6 shows the effect of the amplitude coupling ratio j on the output spectrum T of the drop port. We find that as j increases, the spectral response becomes flat and wide, and the sidelobes rise up. When j P 0.1, the box-like spectral response is formed. Fig. 7 shows the effect of the number of rings in every filter element N on the output spectrum T of the drop port. We can see that as N increases, the number of the sidelobes becomes large, but the height of the sidelobes is nearly unchanged. This means that it is hard to reduce the intensity of non-resonant light and the crosstalk of the device by varying the number of rings in every filter element. When N P 10, the box-like spectral response is formed. Fig. 8 shows the effect of the number of the filter elements M on the output spectrum T of the drop port. We find that as M increases, the box-like spectral response becomes narrow slightly, and the sidelobes drop down

0

0

-50

-50

-100

-100

-150

-150

L2=400 μm

L2=200 μm -200 1549.0

1549.5

1550.0

1550.5

-200 1551 .0 1549.0

0

0

-50

-50

-100

-100

-150

-150

1549.5

L2=5 π R -200 1549.0

1549.5

1550.0

109

1550.0

1550.5

1551.0

1550.5

1551.0

L2=10π R 1550.5

Wavelength λ (nm)

-200 1551.0 1549.0

1549.5

1550.0

Wavelength λ (nm)

Fig. 5. Effect of the center distance L2 on the output spectrum T of the drop port, where L1 = 2000 lm, L3 = 500 lm, R = 12.76 lm, 2ap = 0.5 dB/cm: j = 0.1, M = 4, N = 10, and L2 = 200, 400 lm, 5pR, 10pR.

C.-S. Ma et al. / Optics Communications 273 (2007) 105–113

Output Spectrum T (dB)

110 0

0

-50

-50

-100

-100

-150

-150

κ =0.05

Output Spectrum T (dB)

-200 1549.0

1549.5

1550.0

κ =0.10 1550.5

-200 1551.0 1549.0

0

0

-50

-50

-100

-100

-150

-150

1549.5

κ =0.15 -200 1549.0

1549.5

1550.0

1550.0

1550.5

1551.0

1550.5

1551.0

κ =0.20 1550.5

-200 1551.0 1549.0

Wavelength λ (nm)

1549.5

1550.0

Wavelength λ (nm)

Output Spectrum T (dB)

Fig. 6. Effect of the amplitude coupling ratio j on the output spectrum T of the drop port, where L1 = 2000 lm, L3 = 500 lm, R = 12.76 lm, 2ap = 0.5 dB/cm: L2 = 10pR, M = 4, N = 10, and j = 0.05, 0.1, 0.15, 0.20.

0

0

-50

-50

-100

-100

-150

-150

N=5

Output Spectrum T (dB)

-200 1549.0

1549.5

1550.0

N=10 1550.5

-200 1551.0 1549.0

0

0

-50

-50

-100

-100

-150

-150

1549.5

N=20 -200 1549.0

1549.5

1550.0

1550.0

1550.5

1551.0

1550.5

1551.0

N=40 1550.5

Wavelength λ (nm)

-200 1551.0 1549.0

1549.5

1550.0

Wavelength λ (nm)

Fig. 7. Effect of the number of rings in every filter element N on the output spectrum T of the drop port, where L1 = 2000 lm, L3 = 500 lm, R = 12.76 lm, 2ap = 0.5 dB/cm: L2 = 10pR, j = 0.1, M = 4, and N = 5, 10, 20, 40.

Output Spectrum T (dB)

C.-S. Ma et al. / Optics Communications 273 (2007) 105–113 0

0

-50

-50

-100

-100

-150

-150

M=2

Output Spectrum T (dB)

-200 1549.0

1549.5

M=3

1550.0

1550.5

-200 1551.0 1549.0

0

0

-50

-50

-100

-100

-150

-150

1549.5

M=4 -200 1549.0

1549.5

111

1550.0

1550.5

1551.0

1550.5

1551.0

M=5

1550.0

1550.5

-200 1551.0 1549.0

1549.5

1550.0

Wavelength λ (nm)

Wavelength λ (nm)

Fig. 8. Effect of the filter element number M on the output spectrum T of the drop port, where L1 = 2000 lm, L3 = 500 lm, R = 12.76 lm, 2ap = 0.5 dB/ cm: L2 = 10pR, j = 0.1, N = 10, and M = 2, 3, 4, 5.

greatly. This indicates that the intensity of non-resonant light and the crosstalk of the device can be reduced efficiently by increasing the number of the filter elements. From the discussion of Figs. 5–8, we can conclude that when we select L2 = 10pR, j = 0.1, M = 4, and N = 10, the spectral response is box-like, of which the 3-dB bandwidth is about 0.3 nm, the first pair of sidelobes are about 60 dB, and the insertion loss for the central resonant wavelength is about 2.3 dB. Fig. 9 shows the output spectrum T of the drop port and those T1, T2, T3 and T4 of the four through ports of the

-50 -100 -150 -200 -250 1546

We have optimized the core size to greatly reduce the birefringence and the polarization dependence between the differently polarized Ex00 and Ey00 modes, however, they

b

T

0

3.6. Spectrum shift caused by birefringence

Output Spectrum Tu (dB)

Output Spectrum T (dB)

a

designed device. We can see that the non-resonant light is mainly output from the first through port, while all the resonant light with the wavelength of k0 = 1550 nm is output from the drop port, and the maximum intensity of the subresonant peaks is about 65 dB.

1548

1550

1552

Wavelength λ (nm)

1554

T1

0

T2

-50 T3

-100 -150 T4

-200 -250 1546

1548

1550

1552

1554

Wavelength λ (nm)

Fig. 9. (a) Output spectrum T of the drop port and (b) those T1, T2, T3 and T4 of the four through ports, where L1 = 2000 lm, L3 = 500 lm, R = 12.76 lm, 2ap = 0.5 dB/cm: L2 = 10pR, j = 0.1, M = 4 and N = 10.

112

C.-S. Ma et al. / Optics Communications 273 (2007) 105–113 0

x

E00

-50

-100

-150

-200 1549.0

1549.5

1550.0

1550.5

1551.0

Wavelength λ (nm) Fig. 10. Shift of the output spectrum of the drop port between the Ex00 and Ey00 modes, where L1 = 2000 lm, L3 = 500 lm, R = 12.76 lm, 2ap = 0.5 dB/cm: L2 = 10pR, j = 0.1, M = 4 and N = 10.

still exist in the device more or less. The above device is designed for transmitting the Ey00 mode. If we let the Ex00 mode propagate in this device, we can examine the shift of the output spectrum of the drop port between the Ex00 and Ey00 modes, which is shown in Fig. 10. We find that the wavelength shift dk is about 0.024 nm, which is in good agreement with the value calculated from Eq. (21). 3.7. Spectrum shift caused by manufacturing tolerance Excellent MRR devices are dependent on accurate structural design and fine technology processing, however, manufacturing tolerances are hard to avoid in the fabrication of the MRR devices. Therefore, manufacturing tolerance analysis is important in the design and fabrication of MRR devices. Usually two kinds of main manufacturing tolerances exist in the fabrication of Si based polymer MRR devices, one is dn1 which is caused by the tuning of the core refrac-

Output Spectrum T (dB)

a

-4x -2x

0

0

2x

-50

-100

-150

1549.5

1550.0

1550.5

Wavelength λ (nm)

1551.0

n3 = 1 w1 = 2.09 lm s1 = 1.21 lm w2 = s2 = 1.5 lm M=4 N = 10 m = 80 R = 12.76 lm FSR = 17.8 nm j = 0.1 L1 = 2000 lm L3 = 500 lm L2 = 10pR 2ap = 0.5 dB/cm

0

-4

n1=4x10

k0 = 1550 nm n1 = 1.6278 n2 = 1.465

tive index n1, another is ds which results from the rotatingcoating of the core thickness s. Fig. 11 shows the shift of the output spectrum of the drop port caused by the manufacturing tolerances dn1 (when ds = 0) and ds (when dn1 = 0). We can see that when dn1, ds < 0, the output spectrum shifts to the left, in contrast, when dn1, ds > 0, the output spectrum shifts to the right compared to the case of dn1, ds = 0. When we take ds = 0 and dn1 = 4·, 2·, 2·, 4·104, the shift of the output spectrum is about 0.33, 0.16, 0.16, 0.33 nm, respectively, and when we take dn1 = 0 and ds = 4, 2, 2, 4 nm, the shift of the output spectrum is about 0.32, 0.16, 0.16, 0.32 nm, respectively, compared to the case of dn1,ds = 0. This means that precise manufacturing technologies are required, and allowed manufacturing tolerances should be restricted very strictly. It needs to point out that another output spectrum shift called the coupling-induced resonance frequency shift

b

0

-200 1549.0

Central resonant wavelength Refractive index of polymer core of rings and channels Refractive index of polymer cladding surrounding the channels Refractive index of air cladding above and beside the rings Core width of rings Core thickness of rings Core width and thickness of channels Number of filter elements Number of rings in every filter element Resonant order Ring radius Free spectral range Amplitude coupling ratio between ring and channel Distance Distance Center distance of adjacent rings Propagation loss coefficient

Output Spectrum T (dB)

Output Spectrum T (dB)

Table 1 Values of parameters of the designed 2-D Si based polymer MRR array

y

E00

-4

-2

0

0

2

s=4nm

-50

-100

-150

-200 1549.0

1549.5

1550.0

1550.5

1551.0

Wavelength λ (nm)

Fig. 11. Shift of the output spectrum of the drop port caused by the manufacturing tolerances dn1 and ds, where L1 = 2000 lm, L3 = 500 lm, R = 12.76 lm, 2ap = 0.5 dB/cm, L2 = 10pR, j = 0.1, M = 4 and N = 10: (a) ds = 0, dn1 = 4·, 2·, 0·, 2·, 4·104, and (b) dn1 = 0, ds = 4, 2, 0, 2, 4 nm.

C.-S. Ma et al. / Optics Communications 273 (2007) 105–113

(CIFS) also exists in this present MRR array, which is about 0.14 nm. The CIFS was analyzed in detail by Popovic´ et al. for a series-coupled 3-ring resonator in Ref. [17]. Considering the limitation of the length of the text, we no longer carry on the repeated analysis for this CIFS in this paper. In summary, the values of parameters of the designed Si based polymer MRR array are listed in Table 1. When M = 4 and N = 10, we can estimate from Table 1 and Fig. 1 that the size of this MRR device is [2L1 + 2(N  1)L2 + L3] · [w1 + M(2R)]  11.7 · 0.1 mm2. 4. Conclusions On the basis of the preceding analysis and discussion of the designed 2-D Si based polymer MRR array, conclusions are reached as follows. In terms of the general techniques of the CMT and TMT, we have derived novel formulas of the transfer functions of this device. Using the presented formulas, we can easily and conveniently perform the formulized analysis of the transmission characteristics, and obtain satisfactory simulated results, of which the accuracy can meet the needs of the engineering design of this kind of device. This present technique, we think, can also be applied to some other periodic coupling structure devices, such as photonic crystal defect cavity resonators. In order to get MRR devices with excellent features, accurate structural design and fine technology processing are required. In the design of the presented MRR array, we have optimized the core size of the rings and channels to greatly reduce the birefringence and the polarization dependence between the differently polarized Ex00 and Ey00 modes. To be precise, a very small birefringence of 2.8 · 105 exists in the device, resulting in a very small shift of the output spectrum between the Ex00 and Ey00 modes, which is about 0.024 nm. By parallel-cascading multiple rings in the filter elements, the box-like spectral response can be formed,

113

and by series-cascading several such filter elements, the intensity of the non-resonant light and the crosstalk can be reduced greatly. Properly selecting the central distance L2, the amplitude coupling ratio j, the number of the filter elements M, and that of the rings in every filter element N, we can obtain the box-like spectral response with appropriate shape. To be precise, when we choose L2 = 10pR, j = 0.1, M = 4 and N = 10, the 3-dB bandwidth is about 0.3 nm, the maximum intensity of the sub-resonant peaks is about 65 dB, and the insertion loss for the central resonant wavelength is about 2.3 dB. The size of this proposed 2-D microring resonator array is about 11.7 · 0.1 mm2, which can be realized in the fabrication of the device. References [1] L. Caruso, I. Montrosset, J. Lightwave Technol. 21 (2003) 206. [2] H. Haeiwa, T. Naganawa, Y. Kokubun, IEEE Photon. Technol. Lett. 16 (2004) 135. [3] M. Ogata, Y. Yoda, S. Suzuki, Y. Kokubun, IEEE Photon. Technol. Lett. 17 (2005) 103. [4] B.E. Little, S.T. Chu, J.V. Hryniewicz, P.P. Absil, Opt. Lett. 25 (2000) 344. [5] G. Griffel, IEEE Photon. Technol. Lett. 12 (2000) 810. [6] A. Melloni, Opt. Lett. 26 (2001) 917. [7] O. Schwelb, I. Frigyes, Microwave Opt. Technol. Lett. 44 (2005) 536. [8] G.T. Paloczi, Y. Huang, A. Yariv, Opt. Express 11 (2003) 2666. [9] O. Schwelb, I. Frigyes, Microwave Opt. Technol. Lett. 42 (2004) 427. [10] T. Hanai, S. Suzuki, Y. Hatakeyama, Y. Kokubun, Jap. J. Appl. Phys. 43 (2004) 5785. [11] J.K.S. Poon, J. Scheuer, S. Mookherjea, G.T. Paloczi, Y. Huang, A. Yariv, Opt. Express 12 (2004) 90. [12] C.S. Ma, X. Yan, Y.Z. Xu, X.Y. Wang, H. Li, D.M. Zhang, J. Opt.A: Pure Appl. Opt. 7 (2005) 135. [13] A. Yariv, Electron. Lett. 36 (2000) 321. [14] H.S. Lee, C.H. Choi, B.H.O.D.G. Park, B.G. Kang, S.H. Kim, S.G. Lee, E.H. Lee, IEEE Photon. Technol. Lett. 16 (2004) 1086. [15] A. Melloni, F. Carniel, R. Costa, M. Martinelli, J. Lightwave Technol. 19 (2001) 571. [16] E.A.J. Marcatili, Bell System Tech. J. 48 (1969) 2103. [17] M.A. Popovic´, C. Manolatou, M.R. Watts, Opt. Express 14 (2006) 1208.