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A concise design of 16 × 16 polymer AWG with low insertion loss and crosstalk
Optik 125 (2014) 920–923
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
A concise design of 16 × 16 polymer AWG with low insertion loss and crosstalk Yingchao Xu a,b,∗ , Hongyi Lin b a b
College of Chemistry and Chemical Engineering, Xiamen University, Xiamen 361005, China Department of Mathematics and Physics, Xiamen University of Technology, Xiamen, China
a r t i c l e
i n f o
Article history: Received 26 December 2012 Accepted 18 May 2013
Keywords: DWDM Arrayed waveguide grating (AWG) Insertion loss Crosstalk
a b s t r a c t In this paper, a 16-channel arrayed waveguide grating multiplexer (AWG) has been designed using polymer materials with 1.5% refractive index difference. Certain important parameters are optimized using the coupling mode theory and Beam Propagation Method. The factors that affect the insertion loss and the crosstalk are analyzed in this paper. In our design we introduced the parabolic taper structure and evaluated the suitable number of the arrayed waveguide, obtaining a total insertion loss of 2.19 dB. For obtaining a low crosstalk we evaluate the pitches of adjacent input/output waveguides X and arrayed waveguides d as different values. We chose the value of X about 2.5 times of d by enlarging the pitches of adjacent input/output waveguides. The crosstalk of the designed AWG is lower than −40 dB. © 2013 Elsevier GmbH. All rights reserved.
1. Introduction
1.1. Analysis of insertion loss and crosstalk
Dense Wavelength Division Multiplexing (DWDM) enormously satisfies the demand of modern optical communication: high speed and large capacity. From the end of the last century, the development speed of DWDM is exceptionally swift and violent. The DWDM technology is mature day by day and stimulates the development of the correlation component. Arrayed waveguide grating (AWG) [1] is one of the most promising devices for multiplexer/demultiplexer in Dense Wavelength Division Multiplexing (DWDM) system because of its large numbers of channels, low insertion loss [2], little device dimension, high stability and a potential of integration with other polymer devices. Heretofore, the fabrication of AWG is still in research at home and abroad. We use thermally cross-linkable FPEEK [3] as the optical waveguide material, the refractive index difference of the core and the cladding is = n1 − n2 /n1 = 1.5%. The core size of the AWG is only 1 cm × 2 cm. This is very favorable for improving the integrated level of the integrated planar light wave circuit. The factors that affect the insertion loss and the crosstalk are analyzed in this paper. The methods we studied in the paper to reduce the insertion loss and the crosstalk are very concise, practical and efficient. The total loss (containing the material loss) of the designed AWG multiplexer is 2.19 dB. And the crosstalk is under −40 dB.
Fig. 1 shows the schematic diagram of half a polymer AWG multiplexer studied in this work. This device is composed of 2N + 1 input rectangular waveguides, 2N + 1 output rectangular waveguides, two focusing slabs, and an AWG which contains 2M + 1 arrayed rectangular waveguides, all of which are integrated on the same substrate of silicon. In this paper we denote b as the core thickness, a as the core width, d as the pitch of adjacent arrayed waveguides, X as the pitch of adjacent I/O waveguides, L as the path length difference of adjacent arrayed waveguides. m as the diffraction order, and 0 as the wavelength and central wavelength in free-space, respectively, as the wavelength spacing, as the angular spacing, in as the input angle, and out as the input angle, 0 as the orientation angle of the central waveguide. When the light is launched into the input slab and diffracted in the output slab, in order to match the light phase, it must satisfy the following grating equation:
∗ Corresponding author at: College of Chemistry and Chemical Engineering, Xiamen University, Xiamen 361005, China. E-mail addresses: [email protected], [email protected] (Y. Xu). 0030-4026/$ – see front matter © 2013 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2013.05.194
ns d sin in + nc L + ns d sin out = m.
(1)
According to Eq. (1), the expressions of , L, f, FSR can be derived as follows: L =
m0 nc
(2)
=
m ng ns d nc
(3)
Y. Xu, H. Lin / Optik 125 (2014) 920–923
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Fig. 2. The relation of AWG crosstalk and adjacent I/O waveguides separation X. Fig. 1. Schematic diagram of half an AWG multiplexer.
FSR =
f =
0 nc mng
(4)
ns d2 nc m ng
(5)
With ng = nc −
dnc d
(6) Fig. 3. The relation of AWG crosstalk and adjacent I/O waveguides separation X.
1.1.1. The insertion loss The insertion loss of the AWG is consist of diffraction loss of the input/output slab, bent loss caused by arrayed waveguides and I/O channels, leakage loss resulted from high refractive index substrate, and propagation loss. Generally, the bent loss is just about 5 × 10−5 dB, and the leakage loss is about 0.01 dB. They are relatively quite small compared with other factors. The propagation loss derives the absorption and dispersion of the waveguide material when light transmitting in the waveguides. Its typical value is 2 dB. So the propagation loss has little relation with the structure of the AWG multiplexer. So in this work we mainly analyze the diffraction loss.The diffraction loss in the input slab is defined as [4]:
⎡
Lossin = −10 log10 ⎣ dif
M k=−M
k+ı/2 k−ı/2
/2
−/2
⎤
P0 ()d
⎦
(7)
P0 ()d
where ı = 2 arcsin (a/2f) ≈ a/f. As the waveguide width a is much smaller than the pitch of adjacent arrayed waveguides d. So the diffraction loss in the input slab is very large. Introducing the linear tapered waveguides at the entrance end of the arrayed waveguide is the common method to reduce the diffraction loss of the input slab waveguide. In this paper, we introduced the parabolic tapered waveguide in the same place. And it is more effective reducing loss. The diffraction loss in the output slab is defined as [4]: dif
Loss0 (t )(dB) = −10 log10 0 (l )
= −10 log10
vmax
P(l)
v=−vmax P(l + vm
(8)
where 0 is the diffraction efficiency of the output slab waveguide, max = int(/2 m ), m = 0 /ns d, P(l m ) and P(l m + m ) is defined by: P(0 ) = E 2 (0 )
(9)
Here E( 0 ) is the normalized distant field of all the arrayed waveguides in the output slab. dif As Fig. 2 shows Loss0 (l ) reduces as the pitch of adjacent arrayed waveguides d decreases. So we can minimize d as possible.
1.1.2. The crosstalk According to the overlap integral method [5], the relation between the crosstalk and adjacent I/O waveguides separation X is obtained as shown in Fig. 3. We can see if X is enlarged properly, a desired crosstalk can be obtained. 2. Parameter optimization 2.1. Central wavelength and wavelength spacing For our design, we select the central wavelength 0 = 1550 nm, and the wavelength spacing = 0.8 nm according to the criterion of the International Telecommunication UnionTelecommunication (ITU-T). 2.2. Central orientation angle 0 and bent radius of arrayed waveguides We set the central orientation angle 0 as 45◦ and the bent radius of arrayed waveguides as 1200 m. 2.3. Size and pitch of adjacent input/output waveguides and arrayed waveguides We choose that the ratio between the waveguide width and thickness is a/b = 1. According to the light waveguide theory, the maximum width of the rectangular waveguide is relation to the following formula: amax = 2−3/2 ()
−1/2
/n1
(10)
is the relative refractive index difference between n1 and n2 , n1 is the refractive index of the waveguide core. Introduce the known data to formula (Eq. (10)), we can get: amax = 3 m, b = 3 m. In order to reduce the crosstalk of the AWG while not making the insertion loss increase, we evaluate the pitches of adjacent input/output waveguides X and the pitches of adjacent arrayed waveguides d as different values. In our design, we choose the value of X about 2.5 times of d by enlarging the pitches of adjacent input/output waveguides and decreasing the pitches of adjacent arrayed waveguides. We choose X = 12 m, d = 5.6 m for the balance of low insertion loss and low crosstalk.
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Y. Xu, H. Lin / Optik 125 (2014) 920–923
2.4. The number of the arrayed waveguides The number of the arrayed waveguides should be many enough so as to the diverging light field of the slab waveguide can be completely collected by the arrayed waveguides. Generally it should be more than 5 times of the number of the input waveguides. But it cannot be too large; otherwise the structure of the AWG is difficult to design. So the number of the arrayed waveguide has a minimum. Considering the relation between the diffraction efficiency of the input slab waveguide and the number of the arrayed waveguide 2M + 1 (see Fig. 4), we choose 2M + 1 = 81, and at this value is close to 100%, the diffraction loss is minimized.
Table 1 Design parameters.
2.5. The parabolic tapered waveguides In our design, we introduced the parabolic tapered waveguides at the connection place of the output waveguides and the slab waveguides, and between the slab waveguides and the arrayed waveguides. The parabolic taper structure is smoother than the linear one. For a linear taper, the width of the taper at any place is according to:
W (z) = W1 + (W0 − W1 )
1−Z Ltap
Fig. 6. The relation between the diffraction efficiency of the input slab waveguide and the taper width.
(11)
Central wavelength (nm) Wavelength spacing (nm) Refractive index of waveguide core Refractive index of waveguide cladding The core size (m2 ) Diffraction order Pitch of adjacent input/output waveguide (m) Pitch of adjacent arrayed waveguide (m) Length difference of adjacent arrayed waveguide (m) Focal length of slab waveguide (m) Free spectral range (nm) Number of arrayed waveguides
0 n1 n2 a×b m X d L R FSR N
1550 0.8 1.517 1.495 3×3 85 12 5.6 86.8 1500.3 18.17 81
where Z is along the propagation direction, Ltap is the length of the taper, ı is the tapering angle, W1 , W0 are the entrance width and the exit width of the taper, respectively. Because ı < 1◦ , W Ltap , so a parabolic taper with the same ı is only a little shorter than the linear one (see Fig. 5). W0 , W1 , Ltap and ı have the following geometrical relation: W0 = W1 − 2Ltap tan ı
(12) ı = 0.7◦ ,
In our design, we chose W1 = 5.6 m, W0 = 3 m (the width of the core). According to Eq. (12), Ltap ≈ 106.4 m, but this value is for the linear taper. For the parabolic taper, we chose a smaller value 100 m. Fig. 6 shows the relation between the diffraction efficiency of the input slab waveguide and the taper width. We can see when W1 = d (here is 5.6 m), is as high as 95%.
Fig. 7. The transmission spectrum for all 16 channels.
3. Simulation results The designed parameters are shown in Table 1 and the simulated transmission spectrum for all 16 channels is shown in Fig. 7. We can see the crosstalk is under −40 dB and the insertion loss is under 1.1 dB. Considering the propagation loss of FPEEK material is just 0.5 dB/cm (obtained by cutback method), and the size of the AWG is only 1 cm × 2 cm. The central orientation angle 0 is 45◦ and the bent radius of arrayed waveguides is 1200 m. So the waveguide length is: (2 − 1.2 × 21/2 ) + 1/4 × 2 × 1.2 = 2.188 cm. The total loss of the designed AWG should be 2.19 dB. Fig. 4. The relation between the diffraction efficiency of the input slab waveguide and the number of the arrayed waveguide 2M + 1.
Fig. 5. The geometrical configuration for a taper.
4. Summary We have designed a 16 × 16 AWG based on polymer material FPEEK. The parabolic taper structure has been introduced and optimized to reduce the diffraction loss. The total insertion loss of the device is 2.19 dB. We evaluate the pitches of adjacent input/output waveguides X and arrayed waveguides d as different values. We choose the value of X about 2.5 times of d by enlarging the pitches of adjacent input/output waveguides and have obtained a low crosstalk under −40 dB. The method is concise and helpful to the design of AWG.
Y. Xu, H. Lin / Optik 125 (2014) 920–923
Acknowledgments This work is supported by the S&T plan projects of Fujian Provincial Education Department (JA12254) and Fujian Provincial University Outstanding Young Scientific Talents cultivation project (JA11228). References [1] M.K. Smit, New focusing and dispersive planar component based on optical phased array, Electron. Lett. 24 (7) (1988) 385–386.
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[2] A. Sugita, A. Kaneko, K. Okamoto, et al., Very low insertion loss arrayedwaveguide grating with vertically tapered waveguides, Photon. Technol. Lett. 12 (9) (2000) 1180–1182. [3] F. Wang, C.H. Ma, W. Sun, et al., Arrayed waveguide grating multiplexer with high thermal stability on silicon, Opt. Laser Technol. 37 (7) (2005) 527–531. [4] C.H. Ma, W.B. Guo, D.M. Zhang, et al., Analytical modeling of loss characteristics of a polymer arrayed waveguide multiplexer, Opt. Laser Technol. 34 (8) (2002) 621–630. [5] M. Pascual, P. Daniel, C. Jose, Modeling and design of arrayed waveguide gratings, J. Lightwave Technol. 20 (4) (2002) 661–674.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
A concise design of 16 × 16 polymer AWG with low insertion loss and crosstalk Yingchao Xu a,b,∗ , Hongyi Lin b a b
College of Chemistry and Chemical Engineering, Xiamen University, Xiamen 361005, China Department of Mathematics and Physics, Xiamen University of Technology, Xiamen, China
a r t i c l e
i n f o
Article history: Received 26 December 2012 Accepted 18 May 2013
Keywords: DWDM Arrayed waveguide grating (AWG) Insertion loss Crosstalk
a b s t r a c t In this paper, a 16-channel arrayed waveguide grating multiplexer (AWG) has been designed using polymer materials with 1.5% refractive index difference. Certain important parameters are optimized using the coupling mode theory and Beam Propagation Method. The factors that affect the insertion loss and the crosstalk are analyzed in this paper. In our design we introduced the parabolic taper structure and evaluated the suitable number of the arrayed waveguide, obtaining a total insertion loss of 2.19 dB. For obtaining a low crosstalk we evaluate the pitches of adjacent input/output waveguides X and arrayed waveguides d as different values. We chose the value of X about 2.5 times of d by enlarging the pitches of adjacent input/output waveguides. The crosstalk of the designed AWG is lower than −40 dB. © 2013 Elsevier GmbH. All rights reserved.
1. Introduction
1.1. Analysis of insertion loss and crosstalk
Dense Wavelength Division Multiplexing (DWDM) enormously satisfies the demand of modern optical communication: high speed and large capacity. From the end of the last century, the development speed of DWDM is exceptionally swift and violent. The DWDM technology is mature day by day and stimulates the development of the correlation component. Arrayed waveguide grating (AWG) [1] is one of the most promising devices for multiplexer/demultiplexer in Dense Wavelength Division Multiplexing (DWDM) system because of its large numbers of channels, low insertion loss [2], little device dimension, high stability and a potential of integration with other polymer devices. Heretofore, the fabrication of AWG is still in research at home and abroad. We use thermally cross-linkable FPEEK [3] as the optical waveguide material, the refractive index difference of the core and the cladding is = n1 − n2 /n1 = 1.5%. The core size of the AWG is only 1 cm × 2 cm. This is very favorable for improving the integrated level of the integrated planar light wave circuit. The factors that affect the insertion loss and the crosstalk are analyzed in this paper. The methods we studied in the paper to reduce the insertion loss and the crosstalk are very concise, practical and efficient. The total loss (containing the material loss) of the designed AWG multiplexer is 2.19 dB. And the crosstalk is under −40 dB.
Fig. 1 shows the schematic diagram of half a polymer AWG multiplexer studied in this work. This device is composed of 2N + 1 input rectangular waveguides, 2N + 1 output rectangular waveguides, two focusing slabs, and an AWG which contains 2M + 1 arrayed rectangular waveguides, all of which are integrated on the same substrate of silicon. In this paper we denote b as the core thickness, a as the core width, d as the pitch of adjacent arrayed waveguides, X as the pitch of adjacent I/O waveguides, L as the path length difference of adjacent arrayed waveguides. m as the diffraction order, and 0 as the wavelength and central wavelength in free-space, respectively, as the wavelength spacing, as the angular spacing, in as the input angle, and out as the input angle, 0 as the orientation angle of the central waveguide. When the light is launched into the input slab and diffracted in the output slab, in order to match the light phase, it must satisfy the following grating equation:
∗ Corresponding author at: College of Chemistry and Chemical Engineering, Xiamen University, Xiamen 361005, China. E-mail addresses: [email protected], [email protected] (Y. Xu). 0030-4026/$ – see front matter © 2013 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2013.05.194
ns d sin in + nc L + ns d sin out = m.
(1)
According to Eq. (1), the expressions of , L, f, FSR can be derived as follows: L =
m0 nc
(2)
=
m ng ns d nc
(3)
Y. Xu, H. Lin / Optik 125 (2014) 920–923
921
Fig. 2. The relation of AWG crosstalk and adjacent I/O waveguides separation X. Fig. 1. Schematic diagram of half an AWG multiplexer.
FSR =
f =
0 nc mng
(4)
ns d2 nc m ng
(5)
With ng = nc −
dnc d
(6) Fig. 3. The relation of AWG crosstalk and adjacent I/O waveguides separation X.
1.1.1. The insertion loss The insertion loss of the AWG is consist of diffraction loss of the input/output slab, bent loss caused by arrayed waveguides and I/O channels, leakage loss resulted from high refractive index substrate, and propagation loss. Generally, the bent loss is just about 5 × 10−5 dB, and the leakage loss is about 0.01 dB. They are relatively quite small compared with other factors. The propagation loss derives the absorption and dispersion of the waveguide material when light transmitting in the waveguides. Its typical value is 2 dB. So the propagation loss has little relation with the structure of the AWG multiplexer. So in this work we mainly analyze the diffraction loss.The diffraction loss in the input slab is defined as [4]:
⎡
Lossin = −10 log10 ⎣ dif
M k=−M
k+ı/2 k−ı/2
/2
−/2
⎤
P0 ()d
⎦
(7)
P0 ()d
where ı = 2 arcsin (a/2f) ≈ a/f. As the waveguide width a is much smaller than the pitch of adjacent arrayed waveguides d. So the diffraction loss in the input slab is very large. Introducing the linear tapered waveguides at the entrance end of the arrayed waveguide is the common method to reduce the diffraction loss of the input slab waveguide. In this paper, we introduced the parabolic tapered waveguide in the same place. And it is more effective reducing loss. The diffraction loss in the output slab is defined as [4]: dif
Loss0 (t )(dB) = −10 log10 0 (l )
= −10 log10
vmax
P(l)
v=−vmax P(l + vm
(8)
where 0 is the diffraction efficiency of the output slab waveguide, max = int(/2 m ), m = 0 /ns d, P(l m ) and P(l m + m ) is defined by: P(0 ) = E 2 (0 )
(9)
Here E( 0 ) is the normalized distant field of all the arrayed waveguides in the output slab. dif As Fig. 2 shows Loss0 (l ) reduces as the pitch of adjacent arrayed waveguides d decreases. So we can minimize d as possible.
1.1.2. The crosstalk According to the overlap integral method [5], the relation between the crosstalk and adjacent I/O waveguides separation X is obtained as shown in Fig. 3. We can see if X is enlarged properly, a desired crosstalk can be obtained. 2. Parameter optimization 2.1. Central wavelength and wavelength spacing For our design, we select the central wavelength 0 = 1550 nm, and the wavelength spacing = 0.8 nm according to the criterion of the International Telecommunication UnionTelecommunication (ITU-T). 2.2. Central orientation angle 0 and bent radius of arrayed waveguides We set the central orientation angle 0 as 45◦ and the bent radius of arrayed waveguides as 1200 m. 2.3. Size and pitch of adjacent input/output waveguides and arrayed waveguides We choose that the ratio between the waveguide width and thickness is a/b = 1. According to the light waveguide theory, the maximum width of the rectangular waveguide is relation to the following formula: amax = 2−3/2 ()
−1/2
/n1
(10)
is the relative refractive index difference between n1 and n2 , n1 is the refractive index of the waveguide core. Introduce the known data to formula (Eq. (10)), we can get: amax = 3 m, b = 3 m. In order to reduce the crosstalk of the AWG while not making the insertion loss increase, we evaluate the pitches of adjacent input/output waveguides X and the pitches of adjacent arrayed waveguides d as different values. In our design, we choose the value of X about 2.5 times of d by enlarging the pitches of adjacent input/output waveguides and decreasing the pitches of adjacent arrayed waveguides. We choose X = 12 m, d = 5.6 m for the balance of low insertion loss and low crosstalk.
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Y. Xu, H. Lin / Optik 125 (2014) 920–923
2.4. The number of the arrayed waveguides The number of the arrayed waveguides should be many enough so as to the diverging light field of the slab waveguide can be completely collected by the arrayed waveguides. Generally it should be more than 5 times of the number of the input waveguides. But it cannot be too large; otherwise the structure of the AWG is difficult to design. So the number of the arrayed waveguide has a minimum. Considering the relation between the diffraction efficiency of the input slab waveguide and the number of the arrayed waveguide 2M + 1 (see Fig. 4), we choose 2M + 1 = 81, and at this value is close to 100%, the diffraction loss is minimized.
Table 1 Design parameters.
2.5. The parabolic tapered waveguides In our design, we introduced the parabolic tapered waveguides at the connection place of the output waveguides and the slab waveguides, and between the slab waveguides and the arrayed waveguides. The parabolic taper structure is smoother than the linear one. For a linear taper, the width of the taper at any place is according to:
W (z) = W1 + (W0 − W1 )
1−Z Ltap
Fig. 6. The relation between the diffraction efficiency of the input slab waveguide and the taper width.
(11)
Central wavelength (nm) Wavelength spacing (nm) Refractive index of waveguide core Refractive index of waveguide cladding The core size (m2 ) Diffraction order Pitch of adjacent input/output waveguide (m) Pitch of adjacent arrayed waveguide (m) Length difference of adjacent arrayed waveguide (m) Focal length of slab waveguide (m) Free spectral range (nm) Number of arrayed waveguides
0 n1 n2 a×b m X d L R FSR N
1550 0.8 1.517 1.495 3×3 85 12 5.6 86.8 1500.3 18.17 81
where Z is along the propagation direction, Ltap is the length of the taper, ı is the tapering angle, W1 , W0 are the entrance width and the exit width of the taper, respectively. Because ı < 1◦ , W Ltap , so a parabolic taper with the same ı is only a little shorter than the linear one (see Fig. 5). W0 , W1 , Ltap and ı have the following geometrical relation: W0 = W1 − 2Ltap tan ı
(12) ı = 0.7◦ ,
In our design, we chose W1 = 5.6 m, W0 = 3 m (the width of the core). According to Eq. (12), Ltap ≈ 106.4 m, but this value is for the linear taper. For the parabolic taper, we chose a smaller value 100 m. Fig. 6 shows the relation between the diffraction efficiency of the input slab waveguide and the taper width. We can see when W1 = d (here is 5.6 m), is as high as 95%.
Fig. 7. The transmission spectrum for all 16 channels.
3. Simulation results The designed parameters are shown in Table 1 and the simulated transmission spectrum for all 16 channels is shown in Fig. 7. We can see the crosstalk is under −40 dB and the insertion loss is under 1.1 dB. Considering the propagation loss of FPEEK material is just 0.5 dB/cm (obtained by cutback method), and the size of the AWG is only 1 cm × 2 cm. The central orientation angle 0 is 45◦ and the bent radius of arrayed waveguides is 1200 m. So the waveguide length is: (2 − 1.2 × 21/2 ) + 1/4 × 2 × 1.2 = 2.188 cm. The total loss of the designed AWG should be 2.19 dB. Fig. 4. The relation between the diffraction efficiency of the input slab waveguide and the number of the arrayed waveguide 2M + 1.
Fig. 5. The geometrical configuration for a taper.
4. Summary We have designed a 16 × 16 AWG based on polymer material FPEEK. The parabolic taper structure has been introduced and optimized to reduce the diffraction loss. The total insertion loss of the device is 2.19 dB. We evaluate the pitches of adjacent input/output waveguides X and arrayed waveguides d as different values. We choose the value of X about 2.5 times of d by enlarging the pitches of adjacent input/output waveguides and have obtained a low crosstalk under −40 dB. The method is concise and helpful to the design of AWG.
Y. Xu, H. Lin / Optik 125 (2014) 920–923
Acknowledgments This work is supported by the S&T plan projects of Fujian Provincial Education Department (JA12254) and Fujian Provincial University Outstanding Young Scientific Talents cultivation project (JA11228). References [1] M.K. Smit, New focusing and dispersive planar component based on optical phased array, Electron. Lett. 24 (7) (1988) 385–386.
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[2] A. Sugita, A. Kaneko, K. Okamoto, et al., Very low insertion loss arrayedwaveguide grating with vertically tapered waveguides, Photon. Technol. Lett. 12 (9) (2000) 1180–1182. [3] F. Wang, C.H. Ma, W. Sun, et al., Arrayed waveguide grating multiplexer with high thermal stability on silicon, Opt. Laser Technol. 37 (7) (2005) 527–531. [4] C.H. Ma, W.B. Guo, D.M. Zhang, et al., Analytical modeling of loss characteristics of a polymer arrayed waveguide multiplexer, Opt. Laser Technol. 34 (8) (2002) 621–630. [5] M. Pascual, P. Daniel, C. Jose, Modeling and design of arrayed waveguide gratings, J. Lightwave Technol. 20 (4) (2002) 661–674.