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Design considerations of power splitters based on optical T-waveguide junctions
Optics Communications 277 (2007) 93–102 www.elsevier.com/locate/optcom
Design considerations of power splitters based on optical T-waveguide junctions Gebriel A. Gannat, S.S.A. Obayya
*
Institute of Advanced Telecommunications, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, UK Received 1 November 2006; received in revised form 4 May 2007; accepted 6 May 2007
Abstract In this paper, the problem of power splitting around different T-junction structure is investigated using the finite time difference domain method (FDTD). The variation of the width of one arm of T-junction has a significant impact on the transmitted, radiated and reflected power on both sides. The variation of the cavity parameters on one side of the T-junction lead to a dramatic changes of the ratio of power distribution in the waveguides. Furthermore, the variations of the gradient of the reflectors for symmetric and asymmetric T-junction resonators have shown a great effect on the transmission properties. Ó 2007 Elsevier B.V. All rights reserved.
1. Introduction Power splitters are very important components for the design of many useful photonic integrated circuits. Recently, many research efforts have been directed to enhance existing splitting techniques or to propose new techniques to be used in complex waveguides interconnections in microchips. Splitters can be classified as TE and TM splitters based on polarisation [1] or directional coupling [2], single mode splitters by using Y and T-junction configurations [3] and multimode interference splitters [4]. However, it is quite often required to split the optical power between arms with arbitrary angled positions as in T-junction structure demonstrated in [3]. In the last few decades, various numerical modelling techniques have been proposed in the literature for the analysis of photonic devices. Beam propagation method (BPM) is a very popular method to simulate waveguide components [5,6,13]. However, as most of BPM algorithms rely only on forwardly-propagating waves, they are not suitable for dealing with strongly reflecting devices such as the problem in hand [7]. Alternatively, the finite difference time domain *
Corresponding author. Tel.: +44 1792602560; fax: +44 1792602449. E-mail address: [email protected] (S.S.A. Obayya).
0030-4018/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2007.05.004
(FDTD) method is now thought to be one of the most general methods for analysing wide-class of photonic devices. For example, successful analysis using FDTD have been carried out for the dispersion properties in square hollow fibre with new cross section in [8] and optical switching using different wavelength controlling pulse using nonlinear directional coupler in [9]. Although the FDTD is computationally expensive it is proven to be an accurate tool for simulating reflecting optical structures when Courant stability condition and numerical dispersion are considered [10]. In this paper, the FDTD has been used to analyse the power splitting of different T-junctions, where different investigations have been carried out to explore the effect of different parameters of waveguide on splitting ratio of power transmitted in both arms of the waveguide. Parameters such as arm width, cavity diameters and reflector angle (as shown in Fig. 3) have been varied and sets of different results have been presented. These results clearly demonstrate the possibility of having variable power splitting ratio with minimum reflected and radiated powers. Similar structures to that shown in Fig. 3 have been proposed previously in [3] to report results on power splitting around T-junctions. However, the purpose of this paper is to investigate in detail the effect of resonator parameters with the aim to optimise the performance of the T-junction
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power splitter. In the literature, very interesting results have also been reported in studying the use of single and double mirrors around 90° corners [14]. Also, in this paper this idea of using mirrors will be adopted in the design of T-junction power splitter and compared with the design based on resonators. On the other hand, photonic crystal based power splitters have attracted some research efforts in the last few years and some good results have been reported in [15–18]. In general, the values of the total transmitted power reported in this paper are similar to those reported in [3,14], however, the main purpose of this paper remains to numerically present extensive investigation of the various possibilities of designing power splitters with different split ratios using resonators or mirrors around the corners of the T-junction. This paper is organised as follows: following this Introduction 1, a brief mathematical treatment of FDTD is given in Section 2. The results are explained and justified in detail in Section 3. Then in Section 4 conclusions are drawn.
domain boundaries to simulate the extension of the structure to infinity and modified set of Maxwell equation is used to develop the FDTD equations for UPML domain to absorb the propagated field at the boundaries of the computational domain and minimise the reflection [10]. The FDTD equation in side UPML are given below:
2. FDTD analysis
where the coefficients Lx, Cx, Ly, Cy, Lz, Cz and P are given by the following expressions:
Starting with Maxwell’s equations for a two-dimensional Yee’s cell, as shown in Fig. 1, the following FDTD equations can be obtained for TE-mode propagation [10]: " n # n n n Dt H y i;j H y i1;j H xi;j H xi;j1 nþ1 n Ezi;j ¼ Ezi;j þ ; ð1Þ e Dx Dy n n Dt Ezi;jþ1 Ezi;j n ¼ H H nþ1 ; ð2Þ xi;J xi;J l0 Dy n n Dt Eziþ1;j Ezi;j n ¼ H þ H nþ1 ; ð3Þ y i;j y i;j l0 Dx where Hx, Hy and Ez are the field components at x, y and z directions, respectively, Dx and Dy are the cells size on x and y directions, i and j are the number of cells on x and y direction, Dt is the time step, n is the number of iteration of the time step, e is the electrical permitivity and l0 is the permeability for the free space. The above equations apply to all cells inside the computational domain. Uniaxial perfectly matched layer (UPML) of 1 lm (20 cells) thickness has been wrapped around the computational
Hy Hx
ð5Þ ð6Þ ð7Þ ð8Þ ð9Þ
Lx ¼ 2e0 K xi;j rxi;j Dt;
ð10Þ
C x ¼ 2e0 K xi;j þ rxi;j Dt;
ð11Þ
Ly ¼ 2e0 K y i;j ry i;j Dt;
ð12Þ
C y ¼ 2e0 K y i;j þ ry i;j Dt;
ð13Þ
Lz ¼ 2e0 ; C z ¼ 2e0 ;
ð14Þ ð15Þ
P ¼ 2e0 Dt;
ð16Þ
where Dz is the electric flux density and Bx, By are the magnetic flux density along x and y directions, e0 is the free space permittivity and rx, ry are electric conductivity of UPML along x and y directions. The analysed structure is a two-dimensional structure in xy plane and the z direction is assumed to be uniform, therefore rz is assumed to be 0 and Kz = 1, where rx and ry are calculated using iDx
rx ¼ ðg1=Dx Þ rx0
rx0 ¼ Hx
Ez
(i , j)
ð4Þ
jDy
and ry ¼ ðg1=Dy Þ ry 0 :
ð17Þ
Due to stability problems during the simulation the value of g has been chosen to be 2 6 g 6 3, Kx and Ky are assumed to be equal to 1, where rx0 is calculated as follows:
Δ
x
Ly n 1 E þ C z Dnzi;j ; C y zi;j C y ei;j " n # n n n Lx n P H y i;j H y i1;j H xi;j H xi;j1 nþ1 Dzi;j ¼ D þ ; C x zi;j C x Dx Dy i Lz n 1 h n H xi;j þ C x Bnþ1 H nþ1 xi;j ¼ xi;j Lx Bxi;j ; C z l0 Cz n n Ly n P Ezi;jþ1 Ezi:j nþ1 B Bxi;j ¼ ; C y xi;j C y Dy i Lx n 1 h n H y i;j þ C y Bnþ1 H nþ1 y i;j ¼ y i;j Ly By i;j ; C x l0 Cx n Eziþ1;j Enzi;j Lz n nþ1 By i;j ¼ By i;j þ Dt ; Cz Dx
Enþ1 zi;j ¼
Δx
Hy
y Fig. 1. Positions of the electric field vector components of two dimensions Yee space lattice.
ln½Rð0Þ lnðgÞ 2gDxðgi 1Þ
and
ry 0 ¼
ln½Rð0Þ lnðgÞ ; 2gDyðgj 1Þ
ð18Þ
where R(0) is experimentally chosen to be e16. 3. Results A simple T-junction structure shown in Fig. 2, is first simulated using the developed FDTD code to validate its accuracy. As shown in this figure, the refractive indices of
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n=1 WL= 0.2 µm
n=3.2
WR = 0.2 µm
D2
TR
TL
D3
n=3.2
W=0.2 µm D1
Input
x
y Fig. 2. T-junction waveguide.
2
Eznþ1 ¼ Enzi;j uj sinð2pfnDtÞeððnDt3T Þ=T Þ ; i;j
1 0.9 0.8 0.7
0.4
where /j is the TE fundamental electric field profile, n is the number of time steps, Dt is the time step size and T is the width of the Gaussian pulse and T is chosen to be 5-fs. Reference points are chosen at different locations in the structure at which the time variation of the field at each point is recorded. As shown in Fig. 2, the reference points are labelled as D1, D2 and D3 to record the incident field, transmitted power in the left arm (TL) and transmitted power (TR), respectively. Once the transmitted power reaches the output terminals of the T-junction waveguide, the ratio of FFT of the reflected to incident field is calculated to compute the spectrum variation of the reflection coefficient. The transmission coefficients on both arms are also obtained in a similar way. As may be observed from Fig. 3, the total power transmitted on both arms at wavelength 1.55 lm, is about 30%, and the overall losses are about 45%, while the rest is reflected back toward the input arm. The obtained results as shown in Fig. 3 are in excellent agreement with the counterparts published in [3]. The field profile inside the structure shown in Fig. 4 can be obtained by injecting the same source-field (/j), modulated at wavelength 1.55 lm as stated in the equation below
0.3
ð20Þ
[3] [3]
0.5
ð19Þ
¼ Enzi;j uj sinð2pfnDtÞ: Eznþ1 i;j
*
|T| |R| |T| |R|
0.6
Power
the core and cladding are 3.2 and 1, respectively. The structure is discretised into a uniform mesh with cell size 20 nm and is terminated by 1 lm UPML to absorb the reflected power. To ensure the single mode propagation at 1.55 lm wavelength, the width of the waveguide is chosen to be 0.2 lm. All the results for the transmitted, reflected and radiated power are obtained by injecting a source-field along the transverse x direction, modulated at wavelength 1.55 lm and 5-fs wide Gaussian pulse. The source-field used is given as
*
*
*
*
*
*
*
0.2 0.1 0
1.5
1.55 Wavelength, λ (μm)
1.6
Fig. 3. Power spectrum for reflection, radiation and transmission in both arms of the T-junction waveguide.
Once the steady state reached, an algorithm through [11], is used to obtain the field profile in the structure, based on the numerical integration of a time-dependent signal over one period. A cos U ¼
N X N ð1 cosðxDtÞÞ f ðti Þ cosðxti Þ; p2 i¼1
A sin U ¼ N¼
T ; Dt
N X N ð1 cosðxDtÞÞ f ðti Þ sinðxti Þ; p2 i¼1
ð21Þ ð22Þ ð23Þ
where f is the modulating frequency, ti is the time step after reaching the steady state and Dt is time step size.
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|T| |R| |Prad |
0.7
Power
0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1 1.5
1.55 Wavelength, λ (μm)
1.6
Fig. 4. Field profile for T-junction waveguide at wavelength 1.55 lm.
Fig. 6. Power ratio for reflection, radiation and transmission for both arms for the enhanced waveguide in Fig. 5 at wavelength 1.55 lm.
As may be observed from Fig. 4, the density of the field profile lines in both arms are similar, however, they are low, meanwhile the high density circular spots at the input arm represent the reflected power. Furthermore, the lines around the cavity and the waveguide may present a high power radiation. It is very evident from the results shown in Fig. 3 and the field profile in Fig. 4 that our FDTD results are in strong agreement with the results published in [3,12]. In Fig. 5, the structure is modified so as to improve the transmitted power and minimise the radiated and reflected power. This is assumed to be achieved by adding resonant
cavities and top corner reflectors to the T-junction structure. The enhanced structure is symmetrical on both sides of the central line along x direction. As it may be observed from Fig. 6, the transmitted power on both arms of the structure is a maximum, about 98.4% (very close to the value of 98.6% reported in [3]), and the reflection and radiation are minimum, about 1.6%. This fact can be observed from the field profile shown in Fig. 7, where the number of field line on both arms are nearly identical and higher than the previous field profile presented in Fig. 4, the reflected power is very low as it may observed from low numbers of field lines in the input arm.
WL= 2 µm WR = 0.2 µm
2x n=3.2
y
D2
ΘR B
n=3.2
ΘL
A
a
D3 b
W=0.2 µm D1
x
y
Input
n=1
Fig. 5. T-junction waveguide with resonant cavities and 45° angle top reflectors, y = x = 0.4 lm.
G.A. Gannat, S.S.A. Obayya / Optics Communications 277 (2007) 93–102
Fig. 7. Field profile for the enhanced T-junction waveguide Fig. 5 at wavelength 1.55 lm.
To investigate how the variations of the width of the arm WR of the structure in Fig. 5, will affect the transmission power TL and TR, the FDTD algorithm has been run at wavelength 1.55 lm for different width for WR. In order to keep the thickness of both cavities on x direction identical, the thickness of the right arm cavity are governed by the following relation ð24Þ
b ¼ 0:66 lm W R ;
where WR varies from 0.2 lm to 0.5 lm with an increment of 0.05 lm. From the results presented in Fig. 8, it can be observed that the transmitted power (TL) has been slightly affected by the variation of WR. The transmitted power TL against the variation of WR, varies between 50% and 40% and on the other hand, a dramatic reduction can be observed on
1 0.9
97
the transmitted power in the right arm (TR), as TR starts at 50% at WR = 0.2 lm and reduces to 20% at WR = 0.5 lm. The effect on the reflection is negligible, about 3% and the effect on the radiation is significant, as it reaches about 37%, when WR is 0.5 lm. The significant change in the radiation is due to the increase of the strongly guided area due to the increase of the WR. The variation of WR will affect the cavity parameters and upset the resonance of the cavity. However, it may be observed from Fig. 8, when WR = 0.35 lm, the transmitted power TL jumped back to maximum, about 50% and the radiation about 12%. This can be an interesting point for future investigation. To validate the increase of the transmitted power at the point WR = 0.35 lm, the field profile at this point as it can be seen at Fig. 9, is obtained by an algorithm based on Eqs. (21) and (22). It is very evident from the number of field lines on the left arm and on the right, that the transmitted power on the left side (TL) is higher, while a low transmitted power and high radiation obtained on the right arm of the structure. The obtained results consolidate the fact that the radiation is proportional to the width of WR and this limits the tolerance of the symmetry of the structure, and therefore, requires a high degree of accuracy during the manufacturing process. For further investigation of the transmitted power in the enhanced structure illustrated at Fig. 5, the same fundamental field profile is used for electric field excitation at the same wave length 1.55 lm. The width of WR and WL chosen to be 0.2 lm and L = 45°, while R varies from 25° to 75° with an increment of 10°. It may be observed from the result presented in Fig. 10 that when the enhanced structure is identical on both sides of the symmetry line, the transmitted power TL and TR are found to be around 50% each, while the reflection and radiation nearly zero. Also it can be observed that at R = 35°, the transmitted power TL is about 57% can be obtained, while TR declines to
|TR| |TL| |R| |Prad|
0.8 0.7
Power
0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1 0.1
0.2
0.3
0.4
0.5
0.6
WR (μm) Fig. 8. Ratio of power transmissions on both arms, TL and TR, reflection and radiation for the modified waveguide in Fig. 5, when WL is 0.2 lm and WR is varying between (0.2 and 0.5) lm.
Fig. 9. Field profile for the modified structure in Fig. 3, when WL is 0.2 lm and WR = 0.35 lm at wavelength k = 1.55 lm.
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1
0.55
0.9
0.5
0.8
0.45
|TL| |TR | |R| |Prad |
0.35
0.7 0.6
Power
0.4
Power
|TR| |TL| |P rad| |R|
0.3 0.25 0.2
0.5 0.4 0.3
0.15
0.2
0.1
0.1
0.05
0
0
-0.1 20
-0.05
30
40
50
-0.1 0
10
20
30
40
50
60
70
60
70
80
θR
the modified structure shown in Fig. 11. The FDTD algorithms run for different values of , starting from 30° to 90°. As it may be observed from the results presented in Fig. 12, the transmitted power in the left arm TL, reaches the maximum, about 85% at = 35° and declines gradually to reach 50%, when around 90°. Meanwhile, the power transmitted at the right arm TR will be at minimum, about 10% when the value of is about 35° and will increase gradually as increased till it reaches its maximum 50% at = 90°. For the values of between 30° and 60° the radiated and reflected powers slightly increase. For further investigation to the obtained results, field profiles at certain values of are produced. The field profile presented in Fig. 13, has been obtained when = 35° and as it may be observed from the field profile, the field lines on the right arm are substantially higher in density than the lines on the left arm. This basically represents
41%. However, when R = 55°, TR will be slightly increased to 52% and TL will be decreased to 46%. For all the values of R < 35 or R > 55, the reflected and radiated powers start to increase and affect the transmitted power TL and TR. Therefore, it can be concluded that the accuracy of the cutting angle of the reflectors at the top corner of the cavity needs to be at the tolerance of about ±5°. To investigate the of splitting transmitted power with different ratio between the two arms, the structure has been modified by adding a reflector at the lower neck of the cavity as illustrated in Fig. 11. The assumption was to control the ratio of transmitted power on the right arm by changing the gradient of the cut or the reflector. The same field profile used in previous simulations has been used to simulate
n=1 WR =0.2 µm
WL = 0.2 µm
2x Reflectors
ΘR
D2
B
y ΘL
A
n=3.2
a
D3
b
Θ Reflector
D1
W=0.2 µm
Input
x
y
90
Fig. 12. Power ratio of transmission TL, TR, reflection and radiation against the angle for the modified structure in Fig. 11.
Fig. 10. Power ratio of transmissions (TL and TR), reflection and radiation of the modified waveguide in Fig. 5, when WR = WL = 0.2 lm, L = 45° and R is varying from 20° to 75°.
n=3.2
80
θ0
Fig. 11. T-junction structure with resonant cavity enhanced by 45° top reflectors and a lower corner reflector.
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99
that the radiated and reflected power is slightly lower than the values presented in Fig. 13. The field profile presented in Fig. 15 obtained when the value of chosen to be 75°. It is clear from the number of field lines on the field profile that the power transmitted in the right arm TR decreased, while the power on the left arm TL increased and this in agreement with the results obtained from Figs. 12 and 16. For further study, the structure is modified so that the resonator is replaced by a single mirror and a reflector as shown in Fig. 17, in a promise to improve the transmitted power and reduce the radiated and reflected powers on both arms of the proposed structure. This idea has been already proposed in [14] in the context of improving the transmission Fig. 13. Field profile for the modified structure in Fig. 11, when = 35° and wavelength k = 1.55 lm. 1
the transmitted power on both arms and agrees with the fact obtained from Fig. 12. For field profile presented in Fig. 14, the value has been chosen to be 60° and it can be observed from the obtained graph that the density of field lines on the right arm decreases, while increases on the left arm. Also it can be observed from the field lines
|TR| [θ =35]
|TR| [θ=60]
0.9 0.8 0.7
|TR| [θ=75]
Power
0.6 0.5
|TL| [θ=75]
|TL| [θ=60]
0.4 0.3
|TL| [θ=35]
0.2 0.1 0 -0.1 1.5
1.55
1.6
Wavelength, λ (μm)
Fig. 16. Power ratio of transmission TL, TR for different value of for the modified structure in Fig. 11. Fig. 14. Field profile for the modified structure in Fig. 11, when = 60° and wavelength k = 1.55 lm.
Reflector TL
n =3.2
n=3.2 TR
a Θ
Θ
Mirror
Input
x
Width D Θ
Θ
y Fig. 15. Field profile for the modified structure in Fig. 11, when = 75° and wavelength k = 1.55 lm.
Fig. 17. T-junction waveguide enhanced with a single mirror and reflector, h = 45°, width D = 0.311.
100
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around 90° bends. The values of the width D and mirror angle have been optimised so that maximum possible power transmission around the T-junction is obtained. Fig. 18 shows the variation of the transmitted, reflected and radiated power with the width D for a mirror angle, , of 45°. As may be shown from this figure, the maximum transmitted power of about 90% is obtained when the width D is 3.11 lm. Similarly, Fig. 19 shows the variation of the transmitted, reflected and radiated power with the mirror angle when the width D is 3.11 lm. As may be noted from the figure, the maximum transmitted power of about 93% is
obtained when the angle is 45°. The spectral variation of the transmitted, reflected and radiated power when the width D is 3.11 lm and the mirror angle is 45°, is shown in Fig. 20. As clearly shown in this figure, the maximum transmitted power at the wavelength k = 1.55 lm is about 93% and that value is nearly flat over a range of about 100 nm around the 1.55 lm wavelength. As suggested in [14], a double mirror structure may enhance the power transmission around a 90° corner better than the single mirror. Also, this idea has been extended here by studying the effect of using the double mirrors and reflectors arrangement with T-junction, as shown in Fig. 21, so to improve the power transmission. The modified structure is symmetrical in both sides of
1 0.9 1
0.8
0.9
0.7
0.8
|T| | P Rad| |R|
0.5 0.4
θ = 45
|T| |Prad| |R|
0.7 0.6
Power
Power
0.6
o
0.3
D= 0.311 μm o θ = 45
0.5 0.4 0.3
0.2
0.2
0.1 0.1
0
0.25
0.3
0.35
0
Width D (μm)
-0.1 1.5
Fig. 18. Transmitted power versus the width D for a single mirror T-junction waveguide of Fig. 17.
1.55 Wavelength, λ (μm)
1.6
Fig. 20. Transmitted power for a single mirror T-junction, h = 45°, a = 0.71 lm and width D = 0.311 lm at a wavelength k = 1.55 lm. 1 0.9
Double reflectors
0.8 0.7
Power
0.6
TL n=3.2
0.5
Θ2
|T| | Ref |
0.4
Width D
0.3
Θ1 Θ1
TR
a
Double Mirrors
Width D = 0.311 μm
x
0.2 0.1
y 30
40
50
60
70
θ (Degree) Fig. 19. Transmitted power versus the angle h for a single mirror waveguide of Fig. 17.
Input
0 20
n=3.2
Θ2
Fig. 21. T-junction waveguide enhanced with double mirrors and double reflectors, where h1 = 60°, h2 = 30°, a = 0.787 lm and width D = 0.367 lm.
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to be at 0.367 lm corresponding to a = 0.787 lm. As may be noted from Fig. 23, the maximum transmitted power obtained is about 97.5% and that is relatively flat over a range of more than 100 nm around the wavelength k = 1.55 lm. In general, as may have been observed from these results, the use of mirrors give slightly less values of transmitted power than the resonators, however, it may be more flat over a wider range of wavelengths. Similar ideas of using resonators and mirrors to design power splitters based on photonic crystals [15–18] will be investigated in detail in a future publication.
1
Power
0.9
0.8 θ1 = 45 θ1 = 50 θ1 = 55 θ1 = 60 θ1 = 65
0.7
4. Conclusion
D = 0.367 μm
0.6 20
25
30
35
40
45
θ2 (Degree) Fig. 22. Transmitted power versus angle h1 and h2 for a T-junction with double mirrors, where width D = 0.367 lm at a wavelength k = 1.55 lm.
1 0.9 0.8 |T| |Prad| |R|
0.7
Power
0.6 0.5
D = 0.367μm o θ1 = 60 o θ2 = 30
0.4 0.3
101
A power splitter based on optical waveguide T-junction structure has been analysed using the finite difference time domain (FDTD) method. The proposed structure is modified by adding resonators and reflectors. The performance of the modified structure has been dramatically improved as a maximum transmitted power of 98.4% and minimum reflected power of 1.6% has been obtained. Furthermore, the power splitting ratio between the two arms of the structure can be controlled by varying parameters such as, cavity dimensions and the angle R. A reflector at the lower corner of the cavity added in order the power splitting ratio between the two arms can be controlled by changing the gradient of the reflector. Moreover, the use of single and double mirrors instead of resonators have been considered for designing power splitters around T-junctions, and their performance have been compared with those using resonators. Very high values of the transmitted power have been reported here, similar to those in [3,14], and the effects of various parameters of either the resonator or the mirror, placed at the corner of the T-junction, on the power splitting ratio have been rigorously studied.
0.2
References
0.1 0 -0.1
1.5
1.55 Wavelength, λ (μm)
1.6
Fig. 23. Transmitted power for a T-junction with double mirrors, where h1 = 60°, h2 = 30°, a = 0.787 lm and width D = 0.367 lm at a wavelength k = 1.55 lm.
the central line along x direction. To obtain the maximum transmitted power the effect of the mirrors angles 1 and 2 on the transmitted power is thoroughly investigated. Fig. 22 shows the variation of the transmitted power with the angle 2 where the angle 1 is a parameter. As may observed from the curves that the maximum transmitted power of about 96% is obtained when 1 is 60° and 2 is about 30°. For further optimisation of the results of this double mirror, simulations run for different values of width D from 0.25 lm to 0.40 lm and the optimum width D found
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[11] M. De Pourcq, C. Eng, IEE Proc. H Microw. Antenn. Propag. 132 (1985) 360. [12] S.S.A. Obayya, IEEE J. Quantum Electron. 40 (5) (2004) 591. [13] S.S.A. Obayya, B.M. Rahman, H.A. El-Mikati, J. Lightwave Technol. 18 (3) (2000) 409. [14] R.L. Espinola, R.U. Ahmed, F. Pizzuto, M.J. Steel, R.M. Osgood Jr., Opt. Exp. 8 (9) (2001) 517.
[15] Gregory P. Nordin, Seunghyun Kim, JingboCai Cai, Jianhua Jiang, Opt. Exp. 10 (23) (2002) 1334. [16] S. Fan, S.G. Johnson, J.D. Joannopoulos, J. Opt. Soc. Am. B 18 (2) (2001) 162. [17] S. Kim, Hybrid photonic crystal and conventional waveguides structures, Dissertation, Huntsvelle, Alabama, 2004. [18] Y. Tsuji, K. Hirayama, T. Nomura, K. Sato, S. Nishiwaki, IEEE Photon. Technol. Lett. 18 (7) (2006) 850.
Design considerations of power splitters based on optical T-waveguide junctions Gebriel A. Gannat, S.S.A. Obayya
*
Institute of Advanced Telecommunications, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, UK Received 1 November 2006; received in revised form 4 May 2007; accepted 6 May 2007
Abstract In this paper, the problem of power splitting around different T-junction structure is investigated using the finite time difference domain method (FDTD). The variation of the width of one arm of T-junction has a significant impact on the transmitted, radiated and reflected power on both sides. The variation of the cavity parameters on one side of the T-junction lead to a dramatic changes of the ratio of power distribution in the waveguides. Furthermore, the variations of the gradient of the reflectors for symmetric and asymmetric T-junction resonators have shown a great effect on the transmission properties. Ó 2007 Elsevier B.V. All rights reserved.
1. Introduction Power splitters are very important components for the design of many useful photonic integrated circuits. Recently, many research efforts have been directed to enhance existing splitting techniques or to propose new techniques to be used in complex waveguides interconnections in microchips. Splitters can be classified as TE and TM splitters based on polarisation [1] or directional coupling [2], single mode splitters by using Y and T-junction configurations [3] and multimode interference splitters [4]. However, it is quite often required to split the optical power between arms with arbitrary angled positions as in T-junction structure demonstrated in [3]. In the last few decades, various numerical modelling techniques have been proposed in the literature for the analysis of photonic devices. Beam propagation method (BPM) is a very popular method to simulate waveguide components [5,6,13]. However, as most of BPM algorithms rely only on forwardly-propagating waves, they are not suitable for dealing with strongly reflecting devices such as the problem in hand [7]. Alternatively, the finite difference time domain *
Corresponding author. Tel.: +44 1792602560; fax: +44 1792602449. E-mail address: [email protected] (S.S.A. Obayya).
0030-4018/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2007.05.004
(FDTD) method is now thought to be one of the most general methods for analysing wide-class of photonic devices. For example, successful analysis using FDTD have been carried out for the dispersion properties in square hollow fibre with new cross section in [8] and optical switching using different wavelength controlling pulse using nonlinear directional coupler in [9]. Although the FDTD is computationally expensive it is proven to be an accurate tool for simulating reflecting optical structures when Courant stability condition and numerical dispersion are considered [10]. In this paper, the FDTD has been used to analyse the power splitting of different T-junctions, where different investigations have been carried out to explore the effect of different parameters of waveguide on splitting ratio of power transmitted in both arms of the waveguide. Parameters such as arm width, cavity diameters and reflector angle (as shown in Fig. 3) have been varied and sets of different results have been presented. These results clearly demonstrate the possibility of having variable power splitting ratio with minimum reflected and radiated powers. Similar structures to that shown in Fig. 3 have been proposed previously in [3] to report results on power splitting around T-junctions. However, the purpose of this paper is to investigate in detail the effect of resonator parameters with the aim to optimise the performance of the T-junction
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power splitter. In the literature, very interesting results have also been reported in studying the use of single and double mirrors around 90° corners [14]. Also, in this paper this idea of using mirrors will be adopted in the design of T-junction power splitter and compared with the design based on resonators. On the other hand, photonic crystal based power splitters have attracted some research efforts in the last few years and some good results have been reported in [15–18]. In general, the values of the total transmitted power reported in this paper are similar to those reported in [3,14], however, the main purpose of this paper remains to numerically present extensive investigation of the various possibilities of designing power splitters with different split ratios using resonators or mirrors around the corners of the T-junction. This paper is organised as follows: following this Introduction 1, a brief mathematical treatment of FDTD is given in Section 2. The results are explained and justified in detail in Section 3. Then in Section 4 conclusions are drawn.
domain boundaries to simulate the extension of the structure to infinity and modified set of Maxwell equation is used to develop the FDTD equations for UPML domain to absorb the propagated field at the boundaries of the computational domain and minimise the reflection [10]. The FDTD equation in side UPML are given below:
2. FDTD analysis
where the coefficients Lx, Cx, Ly, Cy, Lz, Cz and P are given by the following expressions:
Starting with Maxwell’s equations for a two-dimensional Yee’s cell, as shown in Fig. 1, the following FDTD equations can be obtained for TE-mode propagation [10]: " n # n n n Dt H y i;j H y i1;j H xi;j H xi;j1 nþ1 n Ezi;j ¼ Ezi;j þ ; ð1Þ e Dx Dy n n Dt Ezi;jþ1 Ezi;j n ¼ H H nþ1 ; ð2Þ xi;J xi;J l0 Dy n n Dt Eziþ1;j Ezi;j n ¼ H þ H nþ1 ; ð3Þ y i;j y i;j l0 Dx where Hx, Hy and Ez are the field components at x, y and z directions, respectively, Dx and Dy are the cells size on x and y directions, i and j are the number of cells on x and y direction, Dt is the time step, n is the number of iteration of the time step, e is the electrical permitivity and l0 is the permeability for the free space. The above equations apply to all cells inside the computational domain. Uniaxial perfectly matched layer (UPML) of 1 lm (20 cells) thickness has been wrapped around the computational
Hy Hx
ð5Þ ð6Þ ð7Þ ð8Þ ð9Þ
Lx ¼ 2e0 K xi;j rxi;j Dt;
ð10Þ
C x ¼ 2e0 K xi;j þ rxi;j Dt;
ð11Þ
Ly ¼ 2e0 K y i;j ry i;j Dt;
ð12Þ
C y ¼ 2e0 K y i;j þ ry i;j Dt;
ð13Þ
Lz ¼ 2e0 ; C z ¼ 2e0 ;
ð14Þ ð15Þ
P ¼ 2e0 Dt;
ð16Þ
where Dz is the electric flux density and Bx, By are the magnetic flux density along x and y directions, e0 is the free space permittivity and rx, ry are electric conductivity of UPML along x and y directions. The analysed structure is a two-dimensional structure in xy plane and the z direction is assumed to be uniform, therefore rz is assumed to be 0 and Kz = 1, where rx and ry are calculated using iDx
rx ¼ ðg1=Dx Þ rx0
rx0 ¼ Hx
Ez
(i , j)
ð4Þ
jDy
and ry ¼ ðg1=Dy Þ ry 0 :
ð17Þ
Due to stability problems during the simulation the value of g has been chosen to be 2 6 g 6 3, Kx and Ky are assumed to be equal to 1, where rx0 is calculated as follows:
Δ
x
Ly n 1 E þ C z Dnzi;j ; C y zi;j C y ei;j " n # n n n Lx n P H y i;j H y i1;j H xi;j H xi;j1 nþ1 Dzi;j ¼ D þ ; C x zi;j C x Dx Dy i Lz n 1 h n H xi;j þ C x Bnþ1 H nþ1 xi;j ¼ xi;j Lx Bxi;j ; C z l0 Cz n n Ly n P Ezi;jþ1 Ezi:j nþ1 B Bxi;j ¼ ; C y xi;j C y Dy i Lx n 1 h n H y i;j þ C y Bnþ1 H nþ1 y i;j ¼ y i;j Ly By i;j ; C x l0 Cx n Eziþ1;j Enzi;j Lz n nþ1 By i;j ¼ By i;j þ Dt ; Cz Dx
Enþ1 zi;j ¼
Δx
Hy
y Fig. 1. Positions of the electric field vector components of two dimensions Yee space lattice.
ln½Rð0Þ lnðgÞ 2gDxðgi 1Þ
and
ry 0 ¼
ln½Rð0Þ lnðgÞ ; 2gDyðgj 1Þ
ð18Þ
where R(0) is experimentally chosen to be e16. 3. Results A simple T-junction structure shown in Fig. 2, is first simulated using the developed FDTD code to validate its accuracy. As shown in this figure, the refractive indices of
G.A. Gannat, S.S.A. Obayya / Optics Communications 277 (2007) 93–102
95
n=1 WL= 0.2 µm
n=3.2
WR = 0.2 µm
D2
TR
TL
D3
n=3.2
W=0.2 µm D1
Input
x
y Fig. 2. T-junction waveguide.
2
Eznþ1 ¼ Enzi;j uj sinð2pfnDtÞeððnDt3T Þ=T Þ ; i;j
1 0.9 0.8 0.7
0.4
where /j is the TE fundamental electric field profile, n is the number of time steps, Dt is the time step size and T is the width of the Gaussian pulse and T is chosen to be 5-fs. Reference points are chosen at different locations in the structure at which the time variation of the field at each point is recorded. As shown in Fig. 2, the reference points are labelled as D1, D2 and D3 to record the incident field, transmitted power in the left arm (TL) and transmitted power (TR), respectively. Once the transmitted power reaches the output terminals of the T-junction waveguide, the ratio of FFT of the reflected to incident field is calculated to compute the spectrum variation of the reflection coefficient. The transmission coefficients on both arms are also obtained in a similar way. As may be observed from Fig. 3, the total power transmitted on both arms at wavelength 1.55 lm, is about 30%, and the overall losses are about 45%, while the rest is reflected back toward the input arm. The obtained results as shown in Fig. 3 are in excellent agreement with the counterparts published in [3]. The field profile inside the structure shown in Fig. 4 can be obtained by injecting the same source-field (/j), modulated at wavelength 1.55 lm as stated in the equation below
0.3
ð20Þ
[3] [3]
0.5
ð19Þ
¼ Enzi;j uj sinð2pfnDtÞ: Eznþ1 i;j
*
|T| |R| |T| |R|
0.6
Power
the core and cladding are 3.2 and 1, respectively. The structure is discretised into a uniform mesh with cell size 20 nm and is terminated by 1 lm UPML to absorb the reflected power. To ensure the single mode propagation at 1.55 lm wavelength, the width of the waveguide is chosen to be 0.2 lm. All the results for the transmitted, reflected and radiated power are obtained by injecting a source-field along the transverse x direction, modulated at wavelength 1.55 lm and 5-fs wide Gaussian pulse. The source-field used is given as
*
*
*
*
*
*
*
0.2 0.1 0
1.5
1.55 Wavelength, λ (μm)
1.6
Fig. 3. Power spectrum for reflection, radiation and transmission in both arms of the T-junction waveguide.
Once the steady state reached, an algorithm through [11], is used to obtain the field profile in the structure, based on the numerical integration of a time-dependent signal over one period. A cos U ¼
N X N ð1 cosðxDtÞÞ f ðti Þ cosðxti Þ; p2 i¼1
A sin U ¼ N¼
T ; Dt
N X N ð1 cosðxDtÞÞ f ðti Þ sinðxti Þ; p2 i¼1
ð21Þ ð22Þ ð23Þ
where f is the modulating frequency, ti is the time step after reaching the steady state and Dt is time step size.
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|T| |R| |Prad |
0.7
Power
0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1 1.5
1.55 Wavelength, λ (μm)
1.6
Fig. 4. Field profile for T-junction waveguide at wavelength 1.55 lm.
Fig. 6. Power ratio for reflection, radiation and transmission for both arms for the enhanced waveguide in Fig. 5 at wavelength 1.55 lm.
As may be observed from Fig. 4, the density of the field profile lines in both arms are similar, however, they are low, meanwhile the high density circular spots at the input arm represent the reflected power. Furthermore, the lines around the cavity and the waveguide may present a high power radiation. It is very evident from the results shown in Fig. 3 and the field profile in Fig. 4 that our FDTD results are in strong agreement with the results published in [3,12]. In Fig. 5, the structure is modified so as to improve the transmitted power and minimise the radiated and reflected power. This is assumed to be achieved by adding resonant
cavities and top corner reflectors to the T-junction structure. The enhanced structure is symmetrical on both sides of the central line along x direction. As it may be observed from Fig. 6, the transmitted power on both arms of the structure is a maximum, about 98.4% (very close to the value of 98.6% reported in [3]), and the reflection and radiation are minimum, about 1.6%. This fact can be observed from the field profile shown in Fig. 7, where the number of field line on both arms are nearly identical and higher than the previous field profile presented in Fig. 4, the reflected power is very low as it may observed from low numbers of field lines in the input arm.
WL= 2 µm WR = 0.2 µm
2x n=3.2
y
D2
ΘR B
n=3.2
ΘL
A
a
D3 b
W=0.2 µm D1
x
y
Input
n=1
Fig. 5. T-junction waveguide with resonant cavities and 45° angle top reflectors, y = x = 0.4 lm.
G.A. Gannat, S.S.A. Obayya / Optics Communications 277 (2007) 93–102
Fig. 7. Field profile for the enhanced T-junction waveguide Fig. 5 at wavelength 1.55 lm.
To investigate how the variations of the width of the arm WR of the structure in Fig. 5, will affect the transmission power TL and TR, the FDTD algorithm has been run at wavelength 1.55 lm for different width for WR. In order to keep the thickness of both cavities on x direction identical, the thickness of the right arm cavity are governed by the following relation ð24Þ
b ¼ 0:66 lm W R ;
where WR varies from 0.2 lm to 0.5 lm with an increment of 0.05 lm. From the results presented in Fig. 8, it can be observed that the transmitted power (TL) has been slightly affected by the variation of WR. The transmitted power TL against the variation of WR, varies between 50% and 40% and on the other hand, a dramatic reduction can be observed on
1 0.9
97
the transmitted power in the right arm (TR), as TR starts at 50% at WR = 0.2 lm and reduces to 20% at WR = 0.5 lm. The effect on the reflection is negligible, about 3% and the effect on the radiation is significant, as it reaches about 37%, when WR is 0.5 lm. The significant change in the radiation is due to the increase of the strongly guided area due to the increase of the WR. The variation of WR will affect the cavity parameters and upset the resonance of the cavity. However, it may be observed from Fig. 8, when WR = 0.35 lm, the transmitted power TL jumped back to maximum, about 50% and the radiation about 12%. This can be an interesting point for future investigation. To validate the increase of the transmitted power at the point WR = 0.35 lm, the field profile at this point as it can be seen at Fig. 9, is obtained by an algorithm based on Eqs. (21) and (22). It is very evident from the number of field lines on the left arm and on the right, that the transmitted power on the left side (TL) is higher, while a low transmitted power and high radiation obtained on the right arm of the structure. The obtained results consolidate the fact that the radiation is proportional to the width of WR and this limits the tolerance of the symmetry of the structure, and therefore, requires a high degree of accuracy during the manufacturing process. For further investigation of the transmitted power in the enhanced structure illustrated at Fig. 5, the same fundamental field profile is used for electric field excitation at the same wave length 1.55 lm. The width of WR and WL chosen to be 0.2 lm and L = 45°, while R varies from 25° to 75° with an increment of 10°. It may be observed from the result presented in Fig. 10 that when the enhanced structure is identical on both sides of the symmetry line, the transmitted power TL and TR are found to be around 50% each, while the reflection and radiation nearly zero. Also it can be observed that at R = 35°, the transmitted power TL is about 57% can be obtained, while TR declines to
|TR| |TL| |R| |Prad|
0.8 0.7
Power
0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1 0.1
0.2
0.3
0.4
0.5
0.6
WR (μm) Fig. 8. Ratio of power transmissions on both arms, TL and TR, reflection and radiation for the modified waveguide in Fig. 5, when WL is 0.2 lm and WR is varying between (0.2 and 0.5) lm.
Fig. 9. Field profile for the modified structure in Fig. 3, when WL is 0.2 lm and WR = 0.35 lm at wavelength k = 1.55 lm.
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1
0.55
0.9
0.5
0.8
0.45
|TL| |TR | |R| |Prad |
0.35
0.7 0.6
Power
0.4
Power
|TR| |TL| |P rad| |R|
0.3 0.25 0.2
0.5 0.4 0.3
0.15
0.2
0.1
0.1
0.05
0
0
-0.1 20
-0.05
30
40
50
-0.1 0
10
20
30
40
50
60
70
60
70
80
θR
the modified structure shown in Fig. 11. The FDTD algorithms run for different values of , starting from 30° to 90°. As it may be observed from the results presented in Fig. 12, the transmitted power in the left arm TL, reaches the maximum, about 85% at = 35° and declines gradually to reach 50%, when around 90°. Meanwhile, the power transmitted at the right arm TR will be at minimum, about 10% when the value of is about 35° and will increase gradually as increased till it reaches its maximum 50% at = 90°. For the values of between 30° and 60° the radiated and reflected powers slightly increase. For further investigation to the obtained results, field profiles at certain values of are produced. The field profile presented in Fig. 13, has been obtained when = 35° and as it may be observed from the field profile, the field lines on the right arm are substantially higher in density than the lines on the left arm. This basically represents
41%. However, when R = 55°, TR will be slightly increased to 52% and TL will be decreased to 46%. For all the values of R < 35 or R > 55, the reflected and radiated powers start to increase and affect the transmitted power TL and TR. Therefore, it can be concluded that the accuracy of the cutting angle of the reflectors at the top corner of the cavity needs to be at the tolerance of about ±5°. To investigate the of splitting transmitted power with different ratio between the two arms, the structure has been modified by adding a reflector at the lower neck of the cavity as illustrated in Fig. 11. The assumption was to control the ratio of transmitted power on the right arm by changing the gradient of the cut or the reflector. The same field profile used in previous simulations has been used to simulate
n=1 WR =0.2 µm
WL = 0.2 µm
2x Reflectors
ΘR
D2
B
y ΘL
A
n=3.2
a
D3
b
Θ Reflector
D1
W=0.2 µm
Input
x
y
90
Fig. 12. Power ratio of transmission TL, TR, reflection and radiation against the angle for the modified structure in Fig. 11.
Fig. 10. Power ratio of transmissions (TL and TR), reflection and radiation of the modified waveguide in Fig. 5, when WR = WL = 0.2 lm, L = 45° and R is varying from 20° to 75°.
n=3.2
80
θ0
Fig. 11. T-junction structure with resonant cavity enhanced by 45° top reflectors and a lower corner reflector.
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99
that the radiated and reflected power is slightly lower than the values presented in Fig. 13. The field profile presented in Fig. 15 obtained when the value of chosen to be 75°. It is clear from the number of field lines on the field profile that the power transmitted in the right arm TR decreased, while the power on the left arm TL increased and this in agreement with the results obtained from Figs. 12 and 16. For further study, the structure is modified so that the resonator is replaced by a single mirror and a reflector as shown in Fig. 17, in a promise to improve the transmitted power and reduce the radiated and reflected powers on both arms of the proposed structure. This idea has been already proposed in [14] in the context of improving the transmission Fig. 13. Field profile for the modified structure in Fig. 11, when = 35° and wavelength k = 1.55 lm. 1
the transmitted power on both arms and agrees with the fact obtained from Fig. 12. For field profile presented in Fig. 14, the value has been chosen to be 60° and it can be observed from the obtained graph that the density of field lines on the right arm decreases, while increases on the left arm. Also it can be observed from the field lines
|TR| [θ =35]
|TR| [θ=60]
0.9 0.8 0.7
|TR| [θ=75]
Power
0.6 0.5
|TL| [θ=75]
|TL| [θ=60]
0.4 0.3
|TL| [θ=35]
0.2 0.1 0 -0.1 1.5
1.55
1.6
Wavelength, λ (μm)
Fig. 16. Power ratio of transmission TL, TR for different value of for the modified structure in Fig. 11. Fig. 14. Field profile for the modified structure in Fig. 11, when = 60° and wavelength k = 1.55 lm.
Reflector TL
n =3.2
n=3.2 TR
a Θ
Θ
Mirror
Input
x
Width D Θ
Θ
y Fig. 15. Field profile for the modified structure in Fig. 11, when = 75° and wavelength k = 1.55 lm.
Fig. 17. T-junction waveguide enhanced with a single mirror and reflector, h = 45°, width D = 0.311.
100
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around 90° bends. The values of the width D and mirror angle have been optimised so that maximum possible power transmission around the T-junction is obtained. Fig. 18 shows the variation of the transmitted, reflected and radiated power with the width D for a mirror angle, , of 45°. As may be shown from this figure, the maximum transmitted power of about 90% is obtained when the width D is 3.11 lm. Similarly, Fig. 19 shows the variation of the transmitted, reflected and radiated power with the mirror angle when the width D is 3.11 lm. As may be noted from the figure, the maximum transmitted power of about 93% is
obtained when the angle is 45°. The spectral variation of the transmitted, reflected and radiated power when the width D is 3.11 lm and the mirror angle is 45°, is shown in Fig. 20. As clearly shown in this figure, the maximum transmitted power at the wavelength k = 1.55 lm is about 93% and that value is nearly flat over a range of about 100 nm around the 1.55 lm wavelength. As suggested in [14], a double mirror structure may enhance the power transmission around a 90° corner better than the single mirror. Also, this idea has been extended here by studying the effect of using the double mirrors and reflectors arrangement with T-junction, as shown in Fig. 21, so to improve the power transmission. The modified structure is symmetrical in both sides of
1 0.9 1
0.8
0.9
0.7
0.8
|T| | P Rad| |R|
0.5 0.4
θ = 45
|T| |Prad| |R|
0.7 0.6
Power
Power
0.6
o
0.3
D= 0.311 μm o θ = 45
0.5 0.4 0.3
0.2
0.2
0.1 0.1
0
0.25
0.3
0.35
0
Width D (μm)
-0.1 1.5
Fig. 18. Transmitted power versus the width D for a single mirror T-junction waveguide of Fig. 17.
1.55 Wavelength, λ (μm)
1.6
Fig. 20. Transmitted power for a single mirror T-junction, h = 45°, a = 0.71 lm and width D = 0.311 lm at a wavelength k = 1.55 lm. 1 0.9
Double reflectors
0.8 0.7
Power
0.6
TL n=3.2
0.5
Θ2
|T| | Ref |
0.4
Width D
0.3
Θ1 Θ1
TR
a
Double Mirrors
Width D = 0.311 μm
x
0.2 0.1
y 30
40
50
60
70
θ (Degree) Fig. 19. Transmitted power versus the angle h for a single mirror waveguide of Fig. 17.
Input
0 20
n=3.2
Θ2
Fig. 21. T-junction waveguide enhanced with double mirrors and double reflectors, where h1 = 60°, h2 = 30°, a = 0.787 lm and width D = 0.367 lm.
G.A. Gannat, S.S.A. Obayya / Optics Communications 277 (2007) 93–102
to be at 0.367 lm corresponding to a = 0.787 lm. As may be noted from Fig. 23, the maximum transmitted power obtained is about 97.5% and that is relatively flat over a range of more than 100 nm around the wavelength k = 1.55 lm. In general, as may have been observed from these results, the use of mirrors give slightly less values of transmitted power than the resonators, however, it may be more flat over a wider range of wavelengths. Similar ideas of using resonators and mirrors to design power splitters based on photonic crystals [15–18] will be investigated in detail in a future publication.
1
Power
0.9
0.8 θ1 = 45 θ1 = 50 θ1 = 55 θ1 = 60 θ1 = 65
0.7
4. Conclusion
D = 0.367 μm
0.6 20
25
30
35
40
45
θ2 (Degree) Fig. 22. Transmitted power versus angle h1 and h2 for a T-junction with double mirrors, where width D = 0.367 lm at a wavelength k = 1.55 lm.
1 0.9 0.8 |T| |Prad| |R|
0.7
Power
0.6 0.5
D = 0.367μm o θ1 = 60 o θ2 = 30
0.4 0.3
101
A power splitter based on optical waveguide T-junction structure has been analysed using the finite difference time domain (FDTD) method. The proposed structure is modified by adding resonators and reflectors. The performance of the modified structure has been dramatically improved as a maximum transmitted power of 98.4% and minimum reflected power of 1.6% has been obtained. Furthermore, the power splitting ratio between the two arms of the structure can be controlled by varying parameters such as, cavity dimensions and the angle R. A reflector at the lower corner of the cavity added in order the power splitting ratio between the two arms can be controlled by changing the gradient of the reflector. Moreover, the use of single and double mirrors instead of resonators have been considered for designing power splitters around T-junctions, and their performance have been compared with those using resonators. Very high values of the transmitted power have been reported here, similar to those in [3,14], and the effects of various parameters of either the resonator or the mirror, placed at the corner of the T-junction, on the power splitting ratio have been rigorously studied.
0.2
References
0.1 0 -0.1
1.5
1.55 Wavelength, λ (μm)
1.6
Fig. 23. Transmitted power for a T-junction with double mirrors, where h1 = 60°, h2 = 30°, a = 0.787 lm and width D = 0.367 lm at a wavelength k = 1.55 lm.
the central line along x direction. To obtain the maximum transmitted power the effect of the mirrors angles 1 and 2 on the transmitted power is thoroughly investigated. Fig. 22 shows the variation of the transmitted power with the angle 2 where the angle 1 is a parameter. As may observed from the curves that the maximum transmitted power of about 96% is obtained when 1 is 60° and 2 is about 30°. For further optimisation of the results of this double mirror, simulations run for different values of width D from 0.25 lm to 0.40 lm and the optimum width D found
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