- Email: [email protected]
Haping the switching characteristics of nonlinear directional couplers
1 September
1995
OPTICS COMMUNICATIONS ELSZVIER
Optics Communications
119 (1995) 347-351
Haping the switching characteristics of nonlinear directional couplers Hamid Hatami-Hanza,
P.L. Chu
Optical Communications Groups, School of Electrical Engineering, University of New South Wales, Sydney 2052, Australia Received 14 March 1995
Abstract We derive a Fourier transform relationship between the coupling coefficient and the fractional switched power of a nonlinear directional coupler. This relationship is then applied to choose the appropriate coupling coefficient to give the desired switching characteristic of the coupler. It is shown that nonlinear couplers with raised cosine or gaussian coupling functions yield a switching characteristic with very large extinction ratio. On the other hand, a suitably chosen sinusoidal coupling function renders the coupler to function as a &i-state switch.
1. Introduction Nonlinear directional coupler forms the backbone of optical switches and logic gates. A great deal of effort has been devoted to the analysis of the switching characteristics of parallel couplers (constant coupling coefficient) [ I-31. While these couplers are structurally simple, their switching characteristics invariably have large side lobes in the post-switching region which contribute to undesirable cross-talks between the channels. The switching characteristic of a nonlinear coupler resembles that of the frequency response of a linear filter. It has been shown [4-6] that, for linear waveguide couplers used as wavelength filters, the sidelobes can be reduced if the coupling between the channels is a variable function of the propagation distance. However, as far as we are aware, very little work has been published on shaping the coupling coefficient of the nonlinear directional coupler to yield the desired switching characteristics. The purpose of this paper is to show that, the power output of a nonlinear coupler is the Fourier transform of the coupling coefficient in 0030-4018/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDIOO30-4018(95)00331-2
the post-switching region. We can therefore use this relationship to select a suitable coupling coefficient to tailor the switching characteristics. We then show that if the coupling coefficient takes the form of either a raised cosine or gaussian function, the sidelobe or the crosstalk (extinction ratio) is vastly improved from the case when the coupling coefficient is a constant. Similar Fourier relationship has been obtained for linear waveguide couplers used as wavelength filters but not as a nonlinear optical switch [ 4,6]. Nonlinear couplers with variable coupling coefficients have been considered by several authors [7-91 but they address problems different from the present one.
2. Fourier relation We consider a nonlinear coupler as shown in Fig. 1. If U and V are the mode amplitudes of the individual waveguides, their evolution along the coupler can be described by
H. Hatatni-Hanza, P.L. Chu / Optics Communications I I9 (1995) 347-351
348
L input port
1 straight
through
pcni
n12
We now proceed to derive the Fourier relationship between the coupler output V(L) and the coupling coefficient c(z) which is valid in the post-switching region, i.e. in the region where the power-length product is larger than the critical switching point. In this region, the fractional coupled power p is small such that (p) ’ -=KI p I. In this case, we can write from Eqs. (4) and (5):
1
2 nzz cross
input port 2
Fig.
over port
I. Schematic view of a nonlinear coupler in general.
1O
IPl*(1+4
=:p
l+lPl*
(8)
and Eq. (3) becomes where 6= (p, - p2) I2 and P,and /3* are the respective linear propagation constants of the two waveguides, and A & are the respective change of the corresponding propagation constants due to nonlinear effect. They are therefore functions of the intensities of the modes U and V. c(z) is the linear coupling coefficient of the coupler. Following the approach by Kolgenik [ 41 applied to linear couplers, Eqs. ( 1) and (2) can be transformed into a single Ricatti equation:
dp
z
=j(2S+AP,-Ap2)p+jc(z)(l-p2),
(3)
where p= V/U. We note that for Kerr nonlinearity, A/3,,, are linear functions of the total power in the respective waveguide and if we neglect the cross phase modulation, we can write Ap,=
]Ul’=
1+
/VI*=
(9)
we can write it in the
p= ~ ei(zs+Po)Z
(10)
and Eq. (9) can be rewritten as du z =jc(z)[e
-j(2S+Pok+
&2
&*S+Po)z
(11)
I.
Since cr << 1, the second term at the right hand side of Eq. ( 11) can be neglected, and integrating the resultant equation over the whole length of the coupler gives
!p,fJl 0
c(~‘)
IPI*’
-p*).
Since p is a complex quantity, form
PO
e-j(2mT+po)h’ &‘+g(O)
.
(12)
-cc
where PO is the normalised total power and is related to the actual input power Pti, by P,= Iu(‘+
dp - =j(2S+P,)p+jc(z)(l dz
ko@‘tin
7’
(6)
effl
where Aeffl is the effective cross-sectional area of waveguide 1 and nr2 is its nonlinear Kerr coefficient. k, is the free space wave number; and
Eq. ( 12) gives the desired result in that the fractional coupler output power is equal to the Fourier transform of the coupling coefficient c( z’) . It is noted that z’ = z/L and c( z’) = 0 outside the region 0 to L. The transformed variable is (26 + PO)L. When 8 # 0, the coupler is asymmetric. Without loss of generality, we will concentrate our attention to symmetric coupler, i.e. S # 0. It is now clear that the transformed variable is the power-length product, Pd. The accuracy of the Fourier relationship is greatest near 1V(L) 1 = 0 but deteriorates as V(L) increases. Inspite of this, it gives
H. Hatami-Hanza.
P.L. Chu / Optics Communications
us an insight into choosing the appropriate coupling coefficient to yield a desired switching characteristic. After c(z) is chosen, the actual switching characteristic can be obtained by direct numerical solution of Eq. (3).
3. Shaping of switching characteristics Let us consider several examples of shaping the desired switching characteristics of the nonlinear coupler. To do so, we replace c(z) by its normalised quantity w(z) :
c(z) = mW(2
j w(z) di)-‘, 0
(13)
where 7rl2 accounts for the coupler equal to half beat length, i.e. complete switching. If w(z) is a constant, the switching characteristic exhibits sine-function-like behaviour with large sidelobes. This is readily seen from Eq. ( 12). Now, if we wish to get rid of the sidelobes, Eq. ( 12) indicates that coupling coefficient in the form of raised cosine or gaussian functions will give the desired characteristics. We therefore choose the raised cosine function as W(Z) = [ 1
+cos(2T(z~L’2))] (14)
and the gaussian function as ( 15) Substituting these functions to Eq. ( 13) and subsequently to Eq. (3) and performing the numerical integration of the resulting equation gives the actual switching characteristics. Fig. 2a shows the switching characteristics of the three types of couplers: conventional constant coupler, raised cosine coupler and gaussian coupler. As expected, the conventional coupler has large sidelobes whereas the raised cosine coupler has negligible sidelobes and the gaussian coupler has no sidelobes at all. However, the switching sensitivity, i.e. the switching slope, decreases as we move from conventional coupler to gaussian coupler. Fig. 2b shows the extinction ratio of the coupler as a function of the power-length product where the extinction ratio is defined as
119 (1995)
347-351
Ext.Ratio = 10 Log,& 1p( *).
349
(16)
The zero dB extinction ratio is the point at which the outputs from both waveguides are equal. This is also called the critical point. For conventional nonlinear coupler with constant coupling coefficient, the powerlength product at the critical point is always 27r. As we move from conventional coupler to gaussian coupler, the critical point increases as shown in Fig. 2b. However, the extinction ratios of raised cosine and gaussian couplers are generally much larger than that of the conventional coupler. This implies that the resulting crosstalk between the waveguides is reduced. When 1p[ < 1, the extinction ratio is negative indicating that more power remains in the original channel and very little is switched to the cross channel. In Fig. 2b, the absolute value of the extinction ratio has been plotted. It is thus necessary to interpret the curves such that the actual extinction ratio is positive between the origin and the first zero, and negative between the first zero and the second zero and then the sign alternates thereon. The Fourier relationship given by Eq. ( 12) also shows that if the coupling coefficient c(z) is a periodic function, sidebands of the switched power (+ will occur. The amplitudes of these sidebands increases with the amplitudes of the harmonics of c( z) and their locations depend on the frequencies of the harmonics. Of course, the fundamental component will not be affected. This is indeed the case as obtained from numerical integration of Eq. (3) when the normalised coupling coefficient w(z) is given by
w(z) = 1 +A,
cos
(17)
wheref, is the spatial modulation frequency and A, is the modulation amplitude. Fig. 3 shows the switching characteristics of this coupler for different values offm and A,,,. It can be seen that the first sideband can be made nearly as large as the fundamental by increasing the modulation index A,. A coupler with this characteristic can be used as a tri-state switch, i.e. the output state of the coupler depends on three levels of the input power. Furthermore, the position of the first sideband can be shifted to any desired power-length product since it is determined by the modulation frequency&.
350
H. Hatami-Hanza,
P.L. Chu /Optics
Communications
119 (1995) 347-351
(a)
raised
cosine
v) ::
0.6
Li
z
0 n
0.4
power-length product
80.00
conventioal _ ____---
_.
raised
cosine
gaussian
60.00 E s
\
0 ._ 3 s
‘*
‘\ ,
\
40.00
switching
critical
points
.o Ti .E z cu 20.00
0.00 0.00
10.00
15.00
20.00
25.00
30.00
power-length product Fig. 2(a). Normalised power at cross over port ( 1V(‘/PO) versus the power-length for different coupling coefficients. Here a= I, S= 0. (b) Extinction ratio of the nonlinear coupler switch versus the power-length for different coupling coefficient. Here CY = 1, 6= 0.
4. Conclusions In conclusion, we have derived a Fourier relation
between the coupling coefficient of a nonlinear coupler and the fractional switched output power. This rela-
H. Hatami-Hanza, P.L. Chu I Optics Communications 119 (1995) 347-351
351
fm=l.O,A,=l.O ,
/‘
\
fm~1.5,
I
‘\
‘.O/
power-length Fig. 3. Norma&d Here a= 1. S=O.
power at cross over port ( ( VI ‘/PO) versus the power-length
tionship is useful in choosing the appropriate coupling coefficient to yield a desired switching characteristic. In particular, we have shown that coupling coefficient in the form of raised cosine and gaussian functions give much larger extinction ratio than the conventional coupler with conbtant coefficient function. Furthermore, a coupler with sinusoidal coupling coefficient can be made to behave as a &i-state switch.
References LI ] M. Jensen, IEEE J. Quantum Electron. QE-18 (1982) 1580. 121 S. Trillo, S. Wabnitz, E.M. Wright and G.I. Stegeman, Optics Lett. 13 (1988) 672.
A, ~1 .O
,‘\
product for a nonlinear coupler with modulated
coupling coefficient.
[3] M.N. Islam, Optics L&t 15 (1989) 1257. [4] H. Kolgenik, Bell. Syst. Tech. J. 55 ( 1976) 109. [5] H.A. Haus and N.A. Whitaker,J. Appl. Phys. J&t. 46 1.
( 1985)
[6] R.C. Alfemess and P.S. Cross, IEEE J. Quantum Electron. 14 (1978)
843.
[ 71 V. Leutheuser, U. Langbein and F. Lederer, Optics Comm. 75 (1990) 251. and F. Lederer, Optics Comm. 102 ( 1993) 478. [9] H. Hatami-Hanza and P.L. Chu, Proc. 19th Australian Conference on Optical Fibre Technology ( 1993) p. 5 1.
181E. Weinert-Raczka
1995
OPTICS COMMUNICATIONS ELSZVIER
Optics Communications
119 (1995) 347-351
Haping the switching characteristics of nonlinear directional couplers Hamid Hatami-Hanza,
P.L. Chu
Optical Communications Groups, School of Electrical Engineering, University of New South Wales, Sydney 2052, Australia Received 14 March 1995
Abstract We derive a Fourier transform relationship between the coupling coefficient and the fractional switched power of a nonlinear directional coupler. This relationship is then applied to choose the appropriate coupling coefficient to give the desired switching characteristic of the coupler. It is shown that nonlinear couplers with raised cosine or gaussian coupling functions yield a switching characteristic with very large extinction ratio. On the other hand, a suitably chosen sinusoidal coupling function renders the coupler to function as a &i-state switch.
1. Introduction Nonlinear directional coupler forms the backbone of optical switches and logic gates. A great deal of effort has been devoted to the analysis of the switching characteristics of parallel couplers (constant coupling coefficient) [ I-31. While these couplers are structurally simple, their switching characteristics invariably have large side lobes in the post-switching region which contribute to undesirable cross-talks between the channels. The switching characteristic of a nonlinear coupler resembles that of the frequency response of a linear filter. It has been shown [4-6] that, for linear waveguide couplers used as wavelength filters, the sidelobes can be reduced if the coupling between the channels is a variable function of the propagation distance. However, as far as we are aware, very little work has been published on shaping the coupling coefficient of the nonlinear directional coupler to yield the desired switching characteristics. The purpose of this paper is to show that, the power output of a nonlinear coupler is the Fourier transform of the coupling coefficient in 0030-4018/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDIOO30-4018(95)00331-2
the post-switching region. We can therefore use this relationship to select a suitable coupling coefficient to tailor the switching characteristics. We then show that if the coupling coefficient takes the form of either a raised cosine or gaussian function, the sidelobe or the crosstalk (extinction ratio) is vastly improved from the case when the coupling coefficient is a constant. Similar Fourier relationship has been obtained for linear waveguide couplers used as wavelength filters but not as a nonlinear optical switch [ 4,6]. Nonlinear couplers with variable coupling coefficients have been considered by several authors [7-91 but they address problems different from the present one.
2. Fourier relation We consider a nonlinear coupler as shown in Fig. 1. If U and V are the mode amplitudes of the individual waveguides, their evolution along the coupler can be described by
H. Hatatni-Hanza, P.L. Chu / Optics Communications I I9 (1995) 347-351
348
L input port
1 straight
through
pcni
n12
We now proceed to derive the Fourier relationship between the coupler output V(L) and the coupling coefficient c(z) which is valid in the post-switching region, i.e. in the region where the power-length product is larger than the critical switching point. In this region, the fractional coupled power p is small such that (p) ’ -=KI p I. In this case, we can write from Eqs. (4) and (5):
1
2 nzz cross
input port 2
Fig.
over port
I. Schematic view of a nonlinear coupler in general.
1O
IPl*(1+4
=:p
l+lPl*
(8)
and Eq. (3) becomes where 6= (p, - p2) I2 and P,and /3* are the respective linear propagation constants of the two waveguides, and A & are the respective change of the corresponding propagation constants due to nonlinear effect. They are therefore functions of the intensities of the modes U and V. c(z) is the linear coupling coefficient of the coupler. Following the approach by Kolgenik [ 41 applied to linear couplers, Eqs. ( 1) and (2) can be transformed into a single Ricatti equation:
dp
z
=j(2S+AP,-Ap2)p+jc(z)(l-p2),
(3)
where p= V/U. We note that for Kerr nonlinearity, A/3,,, are linear functions of the total power in the respective waveguide and if we neglect the cross phase modulation, we can write Ap,=
]Ul’=
1+
/VI*=
(9)
we can write it in the
p= ~ ei(zs+Po)Z
(10)
and Eq. (9) can be rewritten as du z =jc(z)[e
-j(2S+Pok+
&2
&*S+Po)z
(11)
I.
Since cr << 1, the second term at the right hand side of Eq. ( 11) can be neglected, and integrating the resultant equation over the whole length of the coupler gives
!p,fJl 0
c(~‘)
IPI*’
-p*).
Since p is a complex quantity, form
PO
e-j(2mT+po)h’ &‘+g(O)
.
(12)
-cc
where PO is the normalised total power and is related to the actual input power Pti, by P,= Iu(‘+
dp - =j(2S+P,)p+jc(z)(l dz
ko@‘tin
7’
(6)
effl
where Aeffl is the effective cross-sectional area of waveguide 1 and nr2 is its nonlinear Kerr coefficient. k, is the free space wave number; and
Eq. ( 12) gives the desired result in that the fractional coupler output power is equal to the Fourier transform of the coupling coefficient c( z’) . It is noted that z’ = z/L and c( z’) = 0 outside the region 0 to L. The transformed variable is (26 + PO)L. When 8 # 0, the coupler is asymmetric. Without loss of generality, we will concentrate our attention to symmetric coupler, i.e. S # 0. It is now clear that the transformed variable is the power-length product, Pd. The accuracy of the Fourier relationship is greatest near 1V(L) 1 = 0 but deteriorates as V(L) increases. Inspite of this, it gives
H. Hatami-Hanza.
P.L. Chu / Optics Communications
us an insight into choosing the appropriate coupling coefficient to yield a desired switching characteristic. After c(z) is chosen, the actual switching characteristic can be obtained by direct numerical solution of Eq. (3).
3. Shaping of switching characteristics Let us consider several examples of shaping the desired switching characteristics of the nonlinear coupler. To do so, we replace c(z) by its normalised quantity w(z) :
c(z) = mW(2
j w(z) di)-‘, 0
(13)
where 7rl2 accounts for the coupler equal to half beat length, i.e. complete switching. If w(z) is a constant, the switching characteristic exhibits sine-function-like behaviour with large sidelobes. This is readily seen from Eq. ( 12). Now, if we wish to get rid of the sidelobes, Eq. ( 12) indicates that coupling coefficient in the form of raised cosine or gaussian functions will give the desired characteristics. We therefore choose the raised cosine function as W(Z) = [ 1
+cos(2T(z~L’2))] (14)
and the gaussian function as ( 15) Substituting these functions to Eq. ( 13) and subsequently to Eq. (3) and performing the numerical integration of the resulting equation gives the actual switching characteristics. Fig. 2a shows the switching characteristics of the three types of couplers: conventional constant coupler, raised cosine coupler and gaussian coupler. As expected, the conventional coupler has large sidelobes whereas the raised cosine coupler has negligible sidelobes and the gaussian coupler has no sidelobes at all. However, the switching sensitivity, i.e. the switching slope, decreases as we move from conventional coupler to gaussian coupler. Fig. 2b shows the extinction ratio of the coupler as a function of the power-length product where the extinction ratio is defined as
119 (1995)
347-351
Ext.Ratio = 10 Log,& 1p( *).
349
(16)
The zero dB extinction ratio is the point at which the outputs from both waveguides are equal. This is also called the critical point. For conventional nonlinear coupler with constant coupling coefficient, the powerlength product at the critical point is always 27r. As we move from conventional coupler to gaussian coupler, the critical point increases as shown in Fig. 2b. However, the extinction ratios of raised cosine and gaussian couplers are generally much larger than that of the conventional coupler. This implies that the resulting crosstalk between the waveguides is reduced. When 1p[ < 1, the extinction ratio is negative indicating that more power remains in the original channel and very little is switched to the cross channel. In Fig. 2b, the absolute value of the extinction ratio has been plotted. It is thus necessary to interpret the curves such that the actual extinction ratio is positive between the origin and the first zero, and negative between the first zero and the second zero and then the sign alternates thereon. The Fourier relationship given by Eq. ( 12) also shows that if the coupling coefficient c(z) is a periodic function, sidebands of the switched power (+ will occur. The amplitudes of these sidebands increases with the amplitudes of the harmonics of c( z) and their locations depend on the frequencies of the harmonics. Of course, the fundamental component will not be affected. This is indeed the case as obtained from numerical integration of Eq. (3) when the normalised coupling coefficient w(z) is given by
w(z) = 1 +A,
cos
(17)
wheref, is the spatial modulation frequency and A, is the modulation amplitude. Fig. 3 shows the switching characteristics of this coupler for different values offm and A,,,. It can be seen that the first sideband can be made nearly as large as the fundamental by increasing the modulation index A,. A coupler with this characteristic can be used as a tri-state switch, i.e. the output state of the coupler depends on three levels of the input power. Furthermore, the position of the first sideband can be shifted to any desired power-length product since it is determined by the modulation frequency&.
350
H. Hatami-Hanza,
P.L. Chu /Optics
Communications
119 (1995) 347-351
(a)
raised
cosine
v) ::
0.6
Li
z
0 n
0.4
power-length product
80.00
conventioal _ ____---
_.
raised
cosine
gaussian
60.00 E s
\
0 ._ 3 s
‘*
‘\ ,
\
40.00
switching
critical
points
.o Ti .E z cu 20.00
0.00 0.00
10.00
15.00
20.00
25.00
30.00
power-length product Fig. 2(a). Normalised power at cross over port ( 1V(‘/PO) versus the power-length for different coupling coefficients. Here a= I, S= 0. (b) Extinction ratio of the nonlinear coupler switch versus the power-length for different coupling coefficient. Here CY = 1, 6= 0.
4. Conclusions In conclusion, we have derived a Fourier relation
between the coupling coefficient of a nonlinear coupler and the fractional switched output power. This rela-
H. Hatami-Hanza, P.L. Chu I Optics Communications 119 (1995) 347-351
351
fm=l.O,A,=l.O ,
/‘
\
fm~1.5,
I
‘\
‘.O/
power-length Fig. 3. Norma&d Here a= 1. S=O.
power at cross over port ( ( VI ‘/PO) versus the power-length
tionship is useful in choosing the appropriate coupling coefficient to yield a desired switching characteristic. In particular, we have shown that coupling coefficient in the form of raised cosine and gaussian functions give much larger extinction ratio than the conventional coupler with conbtant coefficient function. Furthermore, a coupler with sinusoidal coupling coefficient can be made to behave as a &i-state switch.
References LI ] M. Jensen, IEEE J. Quantum Electron. QE-18 (1982) 1580. 121 S. Trillo, S. Wabnitz, E.M. Wright and G.I. Stegeman, Optics Lett. 13 (1988) 672.
A, ~1 .O
,‘\
product for a nonlinear coupler with modulated
coupling coefficient.
[3] M.N. Islam, Optics L&t 15 (1989) 1257. [4] H. Kolgenik, Bell. Syst. Tech. J. 55 ( 1976) 109. [5] H.A. Haus and N.A. Whitaker,J. Appl. Phys. J&t. 46 1.
( 1985)
[6] R.C. Alfemess and P.S. Cross, IEEE J. Quantum Electron. 14 (1978)
843.
[ 71 V. Leutheuser, U. Langbein and F. Lederer, Optics Comm. 75 (1990) 251. and F. Lederer, Optics Comm. 102 ( 1993) 478. [9] H. Hatami-Hanza and P.L. Chu, Proc. 19th Australian Conference on Optical Fibre Technology ( 1993) p. 5 1.
181E. Weinert-Raczka