- Email: [email protected]
One solution to decrease the threshold of all-optical switches
15 May 1999
Optics Communications 163 Ž1999. 181–184
One solution to decrease the threshold of all-optical switches Zhuo Zhang ) , Yunan Sun Department of Optical Engineering, Beijing Institute of Technology, Beijing 100081, China Received 2 November 1998; received in revised form 17 February 1999; accepted 13 March 1999
Abstract Based on the opposite response of self-defocusing and self-focusing media to the light intensity changes, a novel design is investigated in detail. Numerical analyses show the device can be used at relatively low threshold. At the same time, the analyses also show some factors influence the threshold, such as the nonlinear index coefficient and the linear index difference. q 1999 Published by Elsevier Science B.V. All rights reserved. Keywords: Polymer–glass waveguide all-optical switches; Threshold; Nonlinearity
1. Introduction All-optical switches aroused a good deal of interest due to their attractive features, such as great speed and huge bandwidth. However, a major obstacle to the practical realization of a variety of nonlinear waveguide devices is the relatively high switching power required. Here, we describe the recent progress in decreasing the switching power. On one hand, improving the strategy for structure design is to modify the waveguide geometry and to induce the guided wave instability w1–6x. On the other hand, choosing materials with higher three-order nonlinearity, for example, semiconductors, glasses, organic polymer. Particularly, the organic conjugated polymers are attractive candidates for fabrication of all-optical switches due to their large three-order nonlinearities with subpicosecond responses w7,8x. In this paper, we propose a polymer–glass waveguide all-optical switch. The device is composed of a symmetric Y-branch ion-exchanged glass waveguide with a strip of self-defocusing polymer loaded on top of one branch, and a strip of self-focusing polymer loaded on top of the other
)
Corresponding author
branch Žsee Fig. 1.. Fig. 1 shows the schematic graph of the Y-branch polymer–glass waveguide all-optical switch. Due to the opposite response of self-defocusing medium and self-focusing medium to the light intensity changes, it reduces the switching power and improves the switching efficiency. The paper is organized as follows. Section 2 gives the theoretical basis for the device and the factors affecting the threshold of all-optical switch. In Section 3, we give the numerical results and analyses. Finally, the conclusions are drawn in Section 4.
2. Theoretical basis Fig. 2 shows the geometry of the Y-branch waveguide. Then, the nonlinear refractive indices in two channels are n2i s n 2pi q a i < E < 2
Ž1.
where n i , n pi , a i are nonlinear refractive index, linear refractive index, and nonlinear index coefficient, respectively. For self-defocusing medium, i s 1, a 1 - 0; for self-focusing medium, i s 2, a 2 ) 0. According to the Fermat Theorem, the light always bends into the dense medium. So, we assume that when
0030-4018r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 9 . 0 0 1 3 8 - 8
Z. Zhang, Y. Sun r Optics Communications 163 (1999) 181–184
182
where k a s < b 2 y k 2 n2a < 1r2 and E1q, E1y can be uniquely determined from the boundary conditions.
x 1 s w1 y Fig. 1. Schematic of a symmetric Y-branch polymer–glass waveguide all-optical switch.
1 q1
cny1
x 2 s yS y w 2 y
the intensity in one branch is equal to that in the other branch, the index of both branches always are the same. Thus, we obtain the following equality: n 2p1 q a 1 < E y
Ž x.<
2
s n 2p 2 q a 2 < E y
Ž x.<
2
(Ž
qi2 s
pi2 s
< Ey Ž x . < 2 s
n 2p1 y n2p 2
Ž3.
a2 y a1
mi s
For the low input intensity, n1 ) n 2 , the light is put out from arm 1 according to the Fermat Theorem. When the input intensity is increased somewhat, such that n1 f n 2 , there exist outputs in two arms. Finally, when the input intensity is strong enough, we have n1 - n 2 , output completely from arm 2. The power related to the switching is called threshold, which is discussed in more detail in Section 3. Therefore, the equi-power point is related to the threshold. From Eq. Ž3., we assume that threshold is proportional to the linear index difference, and is inversely proportional to the nonlinear index coefficient in the media. The majority of all-optical switches use self-focusing medium or self-defocusing medium separately. Here in our design, two different nonlinear media Žself-focusing and self-defocusing. are utilized, covered on top of two branches, respectively. This practice greatly increases the switching efficiency, and shall be discussed in Section 3. If only the guided TE modes are considered Žin the steady state., the electric field of guide wave in the waveguide can be written as E Ž x , z ,t . s E y Ž x . exp w j Ž v t y b z . x
Ž4.
By virtue of the effective index method w3x, the channel waveguides can be converted into a five-layer nonlinear slab waveguide, as shown in Fig. 2. Then, we can obtain the electric field of guide wave for the equivalent slab waveguide as follows:
°
E 1 exp w yk aŽ x y w 1 . x p 1 cn w q 1Ž x y x 1 . < m 1 x
~E
q 1
EyŽ x . s
¢
exp Ž k a x . q E 1y exp Ž yk a x .
1
q2
< m1
p1
cny1
E2 p2
< m2
Ž6.
where
Ž2.
then
E1
ž / ž / 2
b 2 y k 2 n2pi . q 2 k 2a i Ci ,
qi2 q b 2 y k 2 n2p i k 2a i
,
qi2 q b 2 y k 2 n2p i
Ž7.
2 qi2
Ci s k 2 Ž n 2pi y n 2s . q
1 2
k 2a i Ei2 Ei2
where k s vrC is the wave number in the free space. And C is integration constant. It is nonzero because the nonlinear medium is bounded. According to the boundary condition, we can obtain the dispersion relation:
As
q2 k a B Ž 1 q e 2 k ,S . q q1Ž 1 y e 2 k aS . k a k a B Ž 1 y e 2 k ,S . q q1Ž 1 q e 2 k aS .
A s Tn Tny1
q2
ž / ka
Ž8.
y q2 w 2
B s Tn q1w1 y Tny1
q1
ž / ka
where TnŽ w < m. is Jacobian–elliptic cotangent function w9x, w , m are its phase angle and modulus, respectively. Then the modal power in the waveguide can be evaluated by
w1 - x w1 ) x ) 0 0 ) x ) yS
p 2 cn w q 2 Ž x y x 2 . < m 2 x
yS ) x ) yS y w 2
E2 exp w k aŽ x q S q w 2 . x
x - yS y w 2
Ž5.
Fig. 2. Geometry of an equivalent five-layer nonlinear waveguide.
Z. Zhang, Y. Sun r Optics Communications 163 (1999) 181–184
183
integration of Poynting vector over the transverse crosssection in the following way:
b Ps
2 vm 0
q`
Hy` < E Ž x . < y
2
dx
Ž9.
3. Numerical results and analysis We select the parameters of nonlinear waveguide w3x as follows: n a s 1.511, n f1 s 1.60, n f 2 s 1.561, n g s 1.516, a 1 s 1.0 = 10y15 m2rV 2 , a 2 s 1.0 = 10y15 m2rV 2 , l s 1.3 mm, w1 s w2 s 4.0 mm, S s 8.2 mm. Where n s , n f1, n f 2 and n g are indices of substrate, self-defocusing medium, self-focusing medium and ion-exchanged glass channels, respectively. According to the dispersion relation, Eq. Ž8., and energy conservation, we obtain the numerical results. Fig. 3 shows the power–Neff diagram of two channels. From this figure, we get three pieces of important information. Ž1. There exists the switching phenomenon. In the diagram, the power peaks of the different channels lie at different points. Furthermore, when power in the self-defocusing branch reaches its peak, the power in the selffocusing branch is approximately zero. Ž2. There exists the threshold. We define the highest power of the self-defocusing branch as the threshold where the self-focusing branch just begins to open. At the same time, we assume to draw a straight line parallel to the horizontal axis across the curves, when the line lies below the threshold, there are three cross points. This means that there are three propagation states. Only when the power exceeds the threshold, is there a cross point and one propagation state can exist. It means that when power is greater than the threshold, there exists an output completely from the self-focusing branch. Ž3. The slopes of both curves are finite values. This means that the switching response is a process, but not an abrupt change. Fig. 4 shows the power–Neff diagram with different nonlinear index coefficient in the self-defocusing branch,
Fig. 3. The power–Neff diagram of different branches and the parameters listed below: n f 1 s1.600, n f 2 s1.561, a 1 sy1.0= 10y1 5 , a 2 s1.0=10y1 5 .
Fig. 4. The power–Neff diagram with different nonlinear index coefficient of self-defocusing medium and the parameters listed below: n f 1 s1.600, n f 2 s1.561, a 2 s1.0=10y1 5.
and illustrates how the nonlinear index coefficient affects the threshold. The threshold is inversely proportional to the nonlinear index coefficient according to Eq. Ž3.. From the diagram, with the nonlinear index coefficient of the selfdefocusing medium decreased, the threshold is increased. This result conforms to the above analyses. In addition, the threshold of using the self-defocusing and self-focusing media at the same time is actually lower than when only the self-focusing medium is used. On the other hand, the switching response changes with the nonlinear index coefficient of the self-defocusing medium. When the nonlinear index coefficient decreases, the slope of the curve increases. Finally, the curve tends to an infinite slope when the nonlinear index coefficient of the self-defocusing medium is equal to zero. This means the switching response is abrupt. Fig. 5 shows the power–Neff diagram of the self-defocusing branch with different values of n 2p1 y n2p 2 , and illustrates how the linear refractive index difference affects the threshold. The threshold is proportional to the linear refractive indices difference. With the difference reduced, the threshold of all-optical switch decreases obviously.
Fig. 5. The power–Neff diagram of the self-defocusing branch with different linear index difference and parameters listed below: n f 1 s1.600, a 1 sy1.0=10y1 5 , a 2 s1.0=10y1 5 .
184
Z. Zhang, Y. Sun r Optics Communications 163 (1999) 181–184
4. Conclusions In this paper, we discuss the way to decrease the threshold in the all-optical switch, and propose a novel low-threshold all-optical switch. It is composed of a symmetric Y-branch glass waveguide with a strip of self-defocusing polymer loaded on top of one branch, and a strip of self-focusing polymer loaded on top of the other branch. The numerical results show the structure actually decreases the threshold a great deal. At the same time, we get the factors, the linear index difference and the nonlinear index coefficient, affecting the threshold of all-optical switch. The threshold is proportional to the linear index difference, and is inversely proportional to the nonlinear index coefficient. In addition, we find that the nonlinear index coefficient of self-defocusing medium is a factor affecting the switching response. The lower the nonlinear index coefficient, the more acute the switching response to the effective index changes. This needs to be considered when designing an all-optical switch.
Acknowledgements The work was funded by the Natural Science Foundation of China.
References w1x J. Guoliang et al., Acta Optica Sinica 16 Ž1996. 1332, Chinese. w2x S. Radic et al., Opt. Lett. 19 Ž1994. 1789. w3x J.Y. Chen et al., Appl. Opt. 33 Ž1994. 3375. w4x J.Y. Chen et al., Opt. Commun. 98 Ž1993. 201. w5x R.W. Micallef et al., Opt. Commun. 147 Ž1998. 259. w6x R.W. Micallef et al., Electron. Lett. 33 Ž1997. 80. w7x L. Chunfei et al., Physics 23 Ž1994. 521, Chinese. w8x D. Neher et al., Synth. Met. 49–50 Ž1992. 21. w9x Z. Zhang, Y. Sun, Opt. Technol., submitted.
Optics Communications 163 Ž1999. 181–184
One solution to decrease the threshold of all-optical switches Zhuo Zhang ) , Yunan Sun Department of Optical Engineering, Beijing Institute of Technology, Beijing 100081, China Received 2 November 1998; received in revised form 17 February 1999; accepted 13 March 1999
Abstract Based on the opposite response of self-defocusing and self-focusing media to the light intensity changes, a novel design is investigated in detail. Numerical analyses show the device can be used at relatively low threshold. At the same time, the analyses also show some factors influence the threshold, such as the nonlinear index coefficient and the linear index difference. q 1999 Published by Elsevier Science B.V. All rights reserved. Keywords: Polymer–glass waveguide all-optical switches; Threshold; Nonlinearity
1. Introduction All-optical switches aroused a good deal of interest due to their attractive features, such as great speed and huge bandwidth. However, a major obstacle to the practical realization of a variety of nonlinear waveguide devices is the relatively high switching power required. Here, we describe the recent progress in decreasing the switching power. On one hand, improving the strategy for structure design is to modify the waveguide geometry and to induce the guided wave instability w1–6x. On the other hand, choosing materials with higher three-order nonlinearity, for example, semiconductors, glasses, organic polymer. Particularly, the organic conjugated polymers are attractive candidates for fabrication of all-optical switches due to their large three-order nonlinearities with subpicosecond responses w7,8x. In this paper, we propose a polymer–glass waveguide all-optical switch. The device is composed of a symmetric Y-branch ion-exchanged glass waveguide with a strip of self-defocusing polymer loaded on top of one branch, and a strip of self-focusing polymer loaded on top of the other
)
Corresponding author
branch Žsee Fig. 1.. Fig. 1 shows the schematic graph of the Y-branch polymer–glass waveguide all-optical switch. Due to the opposite response of self-defocusing medium and self-focusing medium to the light intensity changes, it reduces the switching power and improves the switching efficiency. The paper is organized as follows. Section 2 gives the theoretical basis for the device and the factors affecting the threshold of all-optical switch. In Section 3, we give the numerical results and analyses. Finally, the conclusions are drawn in Section 4.
2. Theoretical basis Fig. 2 shows the geometry of the Y-branch waveguide. Then, the nonlinear refractive indices in two channels are n2i s n 2pi q a i < E < 2
Ž1.
where n i , n pi , a i are nonlinear refractive index, linear refractive index, and nonlinear index coefficient, respectively. For self-defocusing medium, i s 1, a 1 - 0; for self-focusing medium, i s 2, a 2 ) 0. According to the Fermat Theorem, the light always bends into the dense medium. So, we assume that when
0030-4018r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 9 . 0 0 1 3 8 - 8
Z. Zhang, Y. Sun r Optics Communications 163 (1999) 181–184
182
where k a s < b 2 y k 2 n2a < 1r2 and E1q, E1y can be uniquely determined from the boundary conditions.
x 1 s w1 y Fig. 1. Schematic of a symmetric Y-branch polymer–glass waveguide all-optical switch.
1 q1
cny1
x 2 s yS y w 2 y
the intensity in one branch is equal to that in the other branch, the index of both branches always are the same. Thus, we obtain the following equality: n 2p1 q a 1 < E y
Ž x.<
2
s n 2p 2 q a 2 < E y
Ž x.<
2
(Ž
qi2 s
pi2 s
< Ey Ž x . < 2 s
n 2p1 y n2p 2
Ž3.
a2 y a1
mi s
For the low input intensity, n1 ) n 2 , the light is put out from arm 1 according to the Fermat Theorem. When the input intensity is increased somewhat, such that n1 f n 2 , there exist outputs in two arms. Finally, when the input intensity is strong enough, we have n1 - n 2 , output completely from arm 2. The power related to the switching is called threshold, which is discussed in more detail in Section 3. Therefore, the equi-power point is related to the threshold. From Eq. Ž3., we assume that threshold is proportional to the linear index difference, and is inversely proportional to the nonlinear index coefficient in the media. The majority of all-optical switches use self-focusing medium or self-defocusing medium separately. Here in our design, two different nonlinear media Žself-focusing and self-defocusing. are utilized, covered on top of two branches, respectively. This practice greatly increases the switching efficiency, and shall be discussed in Section 3. If only the guided TE modes are considered Žin the steady state., the electric field of guide wave in the waveguide can be written as E Ž x , z ,t . s E y Ž x . exp w j Ž v t y b z . x
Ž4.
By virtue of the effective index method w3x, the channel waveguides can be converted into a five-layer nonlinear slab waveguide, as shown in Fig. 2. Then, we can obtain the electric field of guide wave for the equivalent slab waveguide as follows:
°
E 1 exp w yk aŽ x y w 1 . x p 1 cn w q 1Ž x y x 1 . < m 1 x
~E
q 1
EyŽ x . s
¢
exp Ž k a x . q E 1y exp Ž yk a x .
1
q2
< m1
p1
cny1
E2 p2
< m2
Ž6.
where
Ž2.
then
E1
ž / ž / 2
b 2 y k 2 n2pi . q 2 k 2a i Ci ,
qi2 q b 2 y k 2 n2p i k 2a i
,
qi2 q b 2 y k 2 n2p i
Ž7.
2 qi2
Ci s k 2 Ž n 2pi y n 2s . q
1 2
k 2a i Ei2 Ei2
where k s vrC is the wave number in the free space. And C is integration constant. It is nonzero because the nonlinear medium is bounded. According to the boundary condition, we can obtain the dispersion relation:
As
q2 k a B Ž 1 q e 2 k ,S . q q1Ž 1 y e 2 k aS . k a k a B Ž 1 y e 2 k ,S . q q1Ž 1 q e 2 k aS .
A s Tn Tny1
q2
ž / ka
Ž8.
y q2 w 2
B s Tn q1w1 y Tny1
q1
ž / ka
where TnŽ w < m. is Jacobian–elliptic cotangent function w9x, w , m are its phase angle and modulus, respectively. Then the modal power in the waveguide can be evaluated by
w1 - x w1 ) x ) 0 0 ) x ) yS
p 2 cn w q 2 Ž x y x 2 . < m 2 x
yS ) x ) yS y w 2
E2 exp w k aŽ x q S q w 2 . x
x - yS y w 2
Ž5.
Fig. 2. Geometry of an equivalent five-layer nonlinear waveguide.
Z. Zhang, Y. Sun r Optics Communications 163 (1999) 181–184
183
integration of Poynting vector over the transverse crosssection in the following way:
b Ps
2 vm 0
q`
Hy` < E Ž x . < y
2
dx
Ž9.
3. Numerical results and analysis We select the parameters of nonlinear waveguide w3x as follows: n a s 1.511, n f1 s 1.60, n f 2 s 1.561, n g s 1.516, a 1 s 1.0 = 10y15 m2rV 2 , a 2 s 1.0 = 10y15 m2rV 2 , l s 1.3 mm, w1 s w2 s 4.0 mm, S s 8.2 mm. Where n s , n f1, n f 2 and n g are indices of substrate, self-defocusing medium, self-focusing medium and ion-exchanged glass channels, respectively. According to the dispersion relation, Eq. Ž8., and energy conservation, we obtain the numerical results. Fig. 3 shows the power–Neff diagram of two channels. From this figure, we get three pieces of important information. Ž1. There exists the switching phenomenon. In the diagram, the power peaks of the different channels lie at different points. Furthermore, when power in the self-defocusing branch reaches its peak, the power in the selffocusing branch is approximately zero. Ž2. There exists the threshold. We define the highest power of the self-defocusing branch as the threshold where the self-focusing branch just begins to open. At the same time, we assume to draw a straight line parallel to the horizontal axis across the curves, when the line lies below the threshold, there are three cross points. This means that there are three propagation states. Only when the power exceeds the threshold, is there a cross point and one propagation state can exist. It means that when power is greater than the threshold, there exists an output completely from the self-focusing branch. Ž3. The slopes of both curves are finite values. This means that the switching response is a process, but not an abrupt change. Fig. 4 shows the power–Neff diagram with different nonlinear index coefficient in the self-defocusing branch,
Fig. 3. The power–Neff diagram of different branches and the parameters listed below: n f 1 s1.600, n f 2 s1.561, a 1 sy1.0= 10y1 5 , a 2 s1.0=10y1 5 .
Fig. 4. The power–Neff diagram with different nonlinear index coefficient of self-defocusing medium and the parameters listed below: n f 1 s1.600, n f 2 s1.561, a 2 s1.0=10y1 5.
and illustrates how the nonlinear index coefficient affects the threshold. The threshold is inversely proportional to the nonlinear index coefficient according to Eq. Ž3.. From the diagram, with the nonlinear index coefficient of the selfdefocusing medium decreased, the threshold is increased. This result conforms to the above analyses. In addition, the threshold of using the self-defocusing and self-focusing media at the same time is actually lower than when only the self-focusing medium is used. On the other hand, the switching response changes with the nonlinear index coefficient of the self-defocusing medium. When the nonlinear index coefficient decreases, the slope of the curve increases. Finally, the curve tends to an infinite slope when the nonlinear index coefficient of the self-defocusing medium is equal to zero. This means the switching response is abrupt. Fig. 5 shows the power–Neff diagram of the self-defocusing branch with different values of n 2p1 y n2p 2 , and illustrates how the linear refractive index difference affects the threshold. The threshold is proportional to the linear refractive indices difference. With the difference reduced, the threshold of all-optical switch decreases obviously.
Fig. 5. The power–Neff diagram of the self-defocusing branch with different linear index difference and parameters listed below: n f 1 s1.600, a 1 sy1.0=10y1 5 , a 2 s1.0=10y1 5 .
184
Z. Zhang, Y. Sun r Optics Communications 163 (1999) 181–184
4. Conclusions In this paper, we discuss the way to decrease the threshold in the all-optical switch, and propose a novel low-threshold all-optical switch. It is composed of a symmetric Y-branch glass waveguide with a strip of self-defocusing polymer loaded on top of one branch, and a strip of self-focusing polymer loaded on top of the other branch. The numerical results show the structure actually decreases the threshold a great deal. At the same time, we get the factors, the linear index difference and the nonlinear index coefficient, affecting the threshold of all-optical switch. The threshold is proportional to the linear index difference, and is inversely proportional to the nonlinear index coefficient. In addition, we find that the nonlinear index coefficient of self-defocusing medium is a factor affecting the switching response. The lower the nonlinear index coefficient, the more acute the switching response to the effective index changes. This needs to be considered when designing an all-optical switch.
Acknowledgements The work was funded by the Natural Science Foundation of China.
References w1x J. Guoliang et al., Acta Optica Sinica 16 Ž1996. 1332, Chinese. w2x S. Radic et al., Opt. Lett. 19 Ž1994. 1789. w3x J.Y. Chen et al., Appl. Opt. 33 Ž1994. 3375. w4x J.Y. Chen et al., Opt. Commun. 98 Ž1993. 201. w5x R.W. Micallef et al., Opt. Commun. 147 Ž1998. 259. w6x R.W. Micallef et al., Electron. Lett. 33 Ž1997. 80. w7x L. Chunfei et al., Physics 23 Ž1994. 521, Chinese. w8x D. Neher et al., Synth. Met. 49–50 Ž1992. 21. w9x Z. Zhang, Y. Sun, Opt. Technol., submitted.