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Bistability and nonlinear standing waves in an experimental transmission line
PHYSICS LETTERS A
Physics LettersA 174 (1993) 250—254 North-Holland
Bistability and nonlinear standing waves in an experimental transmission line P. Marquie and J.M. Bilbault Physique Non Lindaire: Laboratoire Ondes et Structures Cohérentes, EP J 0005 CNRS, Faculté des Sciences, 6 Blvd. Gabriel, 21000 Dijon, France Received 28 July 1992; revised manuscript received 11 December 1992; accepted for publication 18 December 1992 Communicatedby A.R. Bishop
The bistability of the transfer function and the spatial dependence of the voltage envelope are investigated near the lower gap on a finite electrical transmission line. The nonlinear standing wave behaviour and the bistability in our experiments agree with our theoretical predictions, which take into account the damping ofthe line.
1. Introduction In recent years, many theoretical and numerical studies [1—41have been devoted to the nonlinear transmission properties of bilayers, superlattices and networks. Near a gap, the transmissivity of these systems exhibits bistability and becomes important once the amplitude ofthe incident sinusoidalwave is larger than a certain threshold. These remarkable properties of infinite systems are related to the existence of gap solitons [5,6]. In finite systems, a nonlinear standing wave (NLSW) with a slowly varying envelope appears [7—9],which is sometimes also called a gap soliton [10,11], as it is for infinite systems. Experimentally, although bistability was already investigated on electrical transmission lines [12] and in optics [131, there is no evidence of bistability related to the behaviour of NLSWs with the carrier wave frequency close to a gap edge. Therefore, it is important to investigate the behaviour of simple systems which are not superlattices but present a “natural gap” as defined in ref. [111. In this case, the theoretical approaches and experimental investigations are often more simple. In this Letter we study the nonlinear transmission properties close to a gap of a finite dispersive electrical line [14, 1 5 matched to an infinite impedance. The theoretical analysis takes into account the dissipation of the electrical components and leads to 250
a set of two coupled nonlinear Schrodinger (NLS) equations. The solutions of these equations allow us to calculate the voltage transfer function, the voltage envelope along the line and to predict bistable properties. These results are well fitted by our experiments.
2. Theoretical calculations As illustrated in fig. 1, we consider a nonlinear network with a finite number of cells. Each cell contains the linear inductance L1 in series, the parallel linear inductance L2 and the capacitance C of a biased varicap. This capacitance is voltage depen400nF
BB11~~
V~PU~
2V1
L2
- - -
~
________ BB11~
- - - _______________
_____________________
~=
.~
Fig. 1. Representation of the electrical transmission line. Each BB 112 diodewith a nonlinear capacitance is biased by 2 V through a resistance of 5 M~.Linear capacitances Cd~~ are used to block the dc biased current but have no effect in the considered frequency range.
0375-9601/93/S 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
Volume 174, number 3
PHYSICS LETTERS A
dent [16] and is biased by a constant voltage Vo, C( V0 + V~)=
=
C~(1
—
2a V,, + 3flV~), (2.1)
where the coefficients a and fi are positive. With L1 and L2 are respectively a resistance r and a conductance g. If we consider the nth cell (see fig. 2), by applying Kirchhoff’s laws, straightforward calculations lead to the wave equation of the nonlinear transmission line for the voltage V~(t), ~ + u~JV~ + u ~( rC0 + gL1)
~
dt+gVn)+rCou2o
In this expression, X=(x— Vgt),
2t,
t=
6=k~x—w
1,t, and e is a small parameter, k.. is the wavenumber of the carrier wave, v5 is the linear wave group velocity in the system, w1, is the angular frequency ofthe carrier wave. Then we apply the reductive perturbation method [18,19 J to separate the envelope scale from the carrier wave scale, 2r~andand g=we e2g assume dissipation terms such as r= 1. To order e we get the linear dispersion relation, (2.4) where w0 = u0\/~correspondsto the lower gap edge, due to the parallel linear inductance L2. In the fol-
~V~
+~~~($
8 March 1993
~ (—aV~+flV~)
lowing, we restrict our study to the behaviour of the system near this gap edge with k,~ =0, co~= w02 and the z.~=0.Next, under these conditions, to order f envelope evolution is given by the NLS equation,
~ (2.5) =
~2V2
a
—~—~
—
where f_-L
/?
~92~~3
(2.2)
~
-
1/L2 and u~=l/L1C0. Following Remoissenet [17], we now consider modulated waves withregard a slowly varying envelope in time and space with to the carrier wave frequency and we add the dc and second harmonic terms, respectively ~ij(X, r) and ~i~(X, r), to the fundamental one ~1~(X,‘r) in order to take the charge asymmetry into account. Thus, we focus on the long wavelength range, so the discrete index n becomes a continous variable x, and we suppose that the solution of eq. (2.2) is expressed as V(x, t)=~j(X, r)e’° 2[1~j(X,r)+~(X,r)e2’~}+c.c. +~
(2.3)
In —~
L
where ~ and ~A’X are respectively the first and second derivatives of ~ with respect to t and X. The dispersion, nonlinearity and dissipation coefficients are
2/2w 2), P_—u0 0, Q=w~(~fl—~a [‘ 0=g1/2C0.
(2.6)
We notice that the dissipation term I’o only depends on the conductance g1, that is, the dissipation due to the resistance r~is negligible. Instead ofsolving directly the NLS equation (2.5) which contains a dissipative term, we consider the envelope voltage amplitude ~j(X, r) as a superposition of forward and backward propagating waves, ~(X, r)=[~ exp(iK÷X)+~ exp(—iK...X)] Xexp(—iQ’r), (2.7) where Q represents the detuning between the incoming wave frequency and the carrier wave frequency
1
2.By sub-
‘7TT~1D-~__I—-:J-
and can be expressed by Q= (w—a0)/~
(O~ ~
(V)~
I
i,.
c
L
I 2
g ~
I
~
Fig. 2. Schematic representation ofthe nih unit cell. C( V,~)is the nonlinear capacitance of the BB 112 varicap, while r and g are introduced to take into account the respective damping of inductances L1 and L2.
lation in exp stituting expression (iK÷X)and (2.7)exp in ( eq.iK...X). (2.5), (Terms we get such a reas exp[i(2K÷+K)X}, are ne~ected because give no phase-matching.) The coefficientsthey of —
...
exp(iK+X) and exp( —iKX) must be zero, which leads first to a set of two coupled nonlinear Schrädinger equations, 251
Volume 174, number 3
PHYSICS LETTERS A
ii~+P’%~÷1x+iTo7~ 2)~c=0, +Q(I~ I2+2I~iiI
(2.8)
+Q(2~l2+l~ij2),~=0,
(2.9)
8 March 1993
I
I
I
I
C
which finally reduces to Q—PK~.+ir
0+Q(l ~
~1 12)=0, 2+2I~4iI2)=0.
~
(2.10) (2.11)
Q—PK~+iTo+Q((~l In order to solve this system in K. 1., K_, ~4 and ~i, we have to consider the boundary conditions. Indeed (see fig. 1) the finite electrical transmission line is matched at X=0 with a real impedance R1, while at X= —1, the voltage generator has no internal impedance. Let us consider the simplest case of an infinite impedance R,, which allows us to obtain analytical results. The boundary conditions at X= 0 give voltage and current continuity, which leads to V~—_V_=~V1,K+=K_=K=a+ib,
a
IllilillI
0
2
—
-
*
4)
.. *1
(2.14)
—
/,
B
..~., ~‘ 1 ,4.
‘
,.“
,.~ ‘
________________
T=
~Vi~
,‘
I
, /
~~
—
/
-.‘
\
By means of eq. (2.7), the envelope voltage amplitude is obtained as a function of the slow spatial coordinate X 2(bX) . (2.15) V(X)=2V~~Jcos2(aX) +sh The voltage amplitude given by the generator is = V( —1), consequently, the voltage transfer function (defined as the ratio between the envelope voltage amplitude, respectively at the end and at the beginning of the line) becomes
2
I
I
E (213)
1.5
imental results. Points A, B and C are characteristic experimental states chosen for the envelope study.
(2.12)
a=
1
Vinput (V)
Fig. 3. Voltage transfer function T through the whole line versus the input parameter ~ Analytical calculations (solid line) are for £2 = 1.13 x I0~rad s~,while crosses correspond to exper-
where the real and imaginary parts of the wave vector are, respectively,
r
111111
0.5
A
Ii._P,’II
I
0 10 20 30 Fig. 4. Voltage amplitudeCell envelopes Number versus cell number in the
The solid line is anal~’ticallycalculated for point A offig. 3, while diamonds represent experimental In the same way, the dashed line and the crosses representresults. respectively transmission line.
analytical and experimental envelopes for point B of fig. 3.
versus X. The values of the parameters are chosen as
(2.16)
whichasis analysed close possible toin those the next of section. the experimental line From eqs. (2.l2)-(2.14), we note for r=o (non-
In figs. 3 and 4, we have respectively represented T versus V~ 111,and the envelope voltage amplitudes
dissipative case) that we get b = 0 and K= a. This
252
=
_____________________
~cos~(al) +sh2(b1)~
means that K still depends on
V.~2~ For
this ideal
Volume 174, number 3
PHYSICS LETTERS A
case, the transfer function and the NLSW envelope are only slightly modified relative to that occurring in reality.
3. Experimental study The experimental transmission line, see fig. 1, is composed of 27 cells, and the individual components have been carefully selected to minimize the inhomogeneities. The linear inductances are L1=220±5 IIH and L2=470±lO ~.tH and their ohmic measured losses are respectively: r= 5 ~ and g= 0.9 x lO~~ The nonlinear capacitance, which consists of a varicap diode (BB 112), is biased by a constant voltage V0 2 V. Under these conditions one has: C0=330±lOpF, a=0.2l V’, and /1=0.0197 In the experiment, an incident sinusoidal wave generated by a voltage generator with internal resistance 50 L~,is launched at the input of the transmission line. The output of the line is open. The voltage waveform is observed on a numerical oscilloscope with fast Fourier transform (FFT) processing. Oscilloscope probes with high impedance (10 M~)and small capacitance (15 pF) are chosen in order to avoid parasitic reflections. The lower gap edge lies at w0=2.54X 106 rad s’, and 2werads—’ have: P=2.69x 106 rads’, V and J’~=l.36x l0~ Q=—7.76x pF~. Thel0~ incident wave frequency is about 450 kHz, and lies just above w 0. 2, as defined in (2.7), The accuracy Q= (w—w0)/e is poor and weinconsider Q as an adjustable parameter. Our experimental results show that harmonics are generated (about 10%); however, in order to cornpare to the theoretical calculations, in a first approxirnation we only consider the fundamental term of the voltage, given by the FFT processing. The experimental voltage transfer function is represented in fig. 3. It is well fitted by the theoretical curve if we choose Q= l.13x l0~rad s’. In the linear approximation, that is for weak input voltages, the voltage transfer function is almost constant: T~1.2 at point A. The spatial evolution of the voltage envelope amplitude corresponding to point A is represented in fig. 4: the electrical length of the line exceeds three half-wavelengths, as shown either by the experimental crosses or the theoretical curve ~—‘
8 March 1993
(solid line). Note that the output of the line corresponds to a maximum because the output is open. When the input voltage amplitude overcomes a certain threshold, the system switches to a higher transmitting state and the transfer function becomes twice as large as in the linear case. To point B of fig. 3 corresponds the spatial envelope evolution represented in fig. 4. In this case, the electrical length is less than three half-wavelengths. For the maximum of the transfer function (point C in fig. 3) the theoretical envelope corresponds exactly to three halfwavelengths with a maximum at the output and a minimum at the input. It is difficult to check experimentally this prediction because the voltage is close to the lower threshold and therefore very unstable. Wedding and Jager [12] have interpreted the appearance ofbistability by an asymmetrical frequency shift in a resonance curve. In fact they have considered their line as a “black box”. Here, we have related the bistability to the NLSW properties. A careful reader should have noticed that at first sight the bistability is not apparent in (2.16) as it is related to V1. However, the existence of bistability is related implicitly to V~,111 (experimental parameter). In conclusion, our experimental measurements of the bistability and the shape of envelope waves are in good agreement with our theoretical although we have neglected harmonics predictions and inhomogeneities. Our study that amplitude in a nonlinear sion line the confirms input voltage acts transmisas a free parameter which modifies the wavelength in the systern. When the line is open, as in our case, and contains an odd number of half-wavelengths, the system exhibits an important behaviour regarding transmission near a gap edge. Our experimental results suggest that a transmission line is an interesting model to study the properties of nonlinear systems with complex behaviour.
Acknowledgement The authors are grateful to A.C. Scott of Lyngby University (Denmark) and M. Remoissenet ofDijon University (France) for helpful discussions. 253
Volume 174, number 3
PHYSICS LETTERS A
References [I] Wei Chen and D.L. Mills, Phys. Rev. B 36 (1987) 6269. [21 D. Barday and M. Remoissenet, Phys. Rev. B 41 (1990) 10387. [31Wei Chen and D.L. Mills, Phys. Rev. B 35 (1987) 524. [4]K.L.StokesandA.Puri,Opt.Lett. 15(1990)986. [5]C. Martijn de Sterke andJ.E. Sipe, Phys. Rev. A 43 (1991) 2467. [6]C.MartijndeSterke,Phys.Rev.A45(1992)2012. [7] Wei Chen and D.L. Mills, Phys. Rev. Lett. 58 (1987)160. [8] C. Martijn de Sterke and i.E. Sipe, Phys. Rev. A 38 (1988) 5149. [9] J.M. Bilbault and M. Remoissenet, J. Appi. Phys. 70 (1991) 4544.
254
8 March 1993
[101 D.L. Mills and SE. Trullinger, Phys. Rev. B 36 (1987) 947. [IlID. Barday and M. Remoissenet, Phys. Rev. B 43 (1991) 7297. [12] B. Wedding and D. Jager, AppI. Phys. Lett. 41(1982)1028. [131 N.D. Sankey, D.F. Prelewitz and T.G. Brown, Appl. Phys. Lett.60 (1992) 1427. [141K. Fukushima, M. Wadati and Y. Narahara, J. Phys. Soc. Japan49 (1980) 1593. [l5]Y. Nejoh, Phys. Scr. 31(1985)415. [16] T. Kofane, B. Michaux and M. Remoissenet, J. Phys. C 21 (1988) 1395. 117] M. Remoissenet, Phys. Rev. B 33 (1986) 2386. [18] T. Taniuti and N. Yajima, J. Math. Phys. 10 (1969) 1369. [19]D.J.KaupandA.C.Newell,Phys.Rev.B18(1978)5162.
Physics LettersA 174 (1993) 250—254 North-Holland
Bistability and nonlinear standing waves in an experimental transmission line P. Marquie and J.M. Bilbault Physique Non Lindaire: Laboratoire Ondes et Structures Cohérentes, EP J 0005 CNRS, Faculté des Sciences, 6 Blvd. Gabriel, 21000 Dijon, France Received 28 July 1992; revised manuscript received 11 December 1992; accepted for publication 18 December 1992 Communicatedby A.R. Bishop
The bistability of the transfer function and the spatial dependence of the voltage envelope are investigated near the lower gap on a finite electrical transmission line. The nonlinear standing wave behaviour and the bistability in our experiments agree with our theoretical predictions, which take into account the damping ofthe line.
1. Introduction In recent years, many theoretical and numerical studies [1—41have been devoted to the nonlinear transmission properties of bilayers, superlattices and networks. Near a gap, the transmissivity of these systems exhibits bistability and becomes important once the amplitude ofthe incident sinusoidalwave is larger than a certain threshold. These remarkable properties of infinite systems are related to the existence of gap solitons [5,6]. In finite systems, a nonlinear standing wave (NLSW) with a slowly varying envelope appears [7—9],which is sometimes also called a gap soliton [10,11], as it is for infinite systems. Experimentally, although bistability was already investigated on electrical transmission lines [12] and in optics [131, there is no evidence of bistability related to the behaviour of NLSWs with the carrier wave frequency close to a gap edge. Therefore, it is important to investigate the behaviour of simple systems which are not superlattices but present a “natural gap” as defined in ref. [111. In this case, the theoretical approaches and experimental investigations are often more simple. In this Letter we study the nonlinear transmission properties close to a gap of a finite dispersive electrical line [14, 1 5 matched to an infinite impedance. The theoretical analysis takes into account the dissipation of the electrical components and leads to 250
a set of two coupled nonlinear Schrodinger (NLS) equations. The solutions of these equations allow us to calculate the voltage transfer function, the voltage envelope along the line and to predict bistable properties. These results are well fitted by our experiments.
2. Theoretical calculations As illustrated in fig. 1, we consider a nonlinear network with a finite number of cells. Each cell contains the linear inductance L1 in series, the parallel linear inductance L2 and the capacitance C of a biased varicap. This capacitance is voltage depen400nF
BB11~~
V~PU~
2V1
L2
- - -
~
________ BB11~
- - - _______________
_____________________
~=
.~
Fig. 1. Representation of the electrical transmission line. Each BB 112 diodewith a nonlinear capacitance is biased by 2 V through a resistance of 5 M~.Linear capacitances Cd~~ are used to block the dc biased current but have no effect in the considered frequency range.
0375-9601/93/S 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
Volume 174, number 3
PHYSICS LETTERS A
dent [16] and is biased by a constant voltage Vo, C( V0 + V~)=
=
C~(1
—
2a V,, + 3flV~), (2.1)
where the coefficients a and fi are positive. With L1 and L2 are respectively a resistance r and a conductance g. If we consider the nth cell (see fig. 2), by applying Kirchhoff’s laws, straightforward calculations lead to the wave equation of the nonlinear transmission line for the voltage V~(t), ~ + u~JV~ + u ~( rC0 + gL1)
~
dt+gVn)+rCou2o
In this expression, X=(x— Vgt),
2t,
t=
6=k~x—w
1,t, and e is a small parameter, k.. is the wavenumber of the carrier wave, v5 is the linear wave group velocity in the system, w1, is the angular frequency ofthe carrier wave. Then we apply the reductive perturbation method [18,19 J to separate the envelope scale from the carrier wave scale, 2r~andand g=we e2g assume dissipation terms such as r= 1. To order e we get the linear dispersion relation, (2.4) where w0 = u0\/~correspondsto the lower gap edge, due to the parallel linear inductance L2. In the fol-
~V~
+~~~($
8 March 1993
~ (—aV~+flV~)
lowing, we restrict our study to the behaviour of the system near this gap edge with k,~ =0, co~= w02 and the z.~=0.Next, under these conditions, to order f envelope evolution is given by the NLS equation,
~ (2.5) =
~2V2
a
—~—~
—
where f_-L
/?
~92~~3
(2.2)
~
-
1/L2 and u~=l/L1C0. Following Remoissenet [17], we now consider modulated waves withregard a slowly varying envelope in time and space with to the carrier wave frequency and we add the dc and second harmonic terms, respectively ~ij(X, r) and ~i~(X, r), to the fundamental one ~1~(X,‘r) in order to take the charge asymmetry into account. Thus, we focus on the long wavelength range, so the discrete index n becomes a continous variable x, and we suppose that the solution of eq. (2.2) is expressed as V(x, t)=~j(X, r)e’° 2[1~j(X,r)+~(X,r)e2’~}+c.c. +~
(2.3)
In —~
L
where ~ and ~A’X are respectively the first and second derivatives of ~ with respect to t and X. The dispersion, nonlinearity and dissipation coefficients are
2/2w 2), P_—u0 0, Q=w~(~fl—~a [‘ 0=g1/2C0.
(2.6)
We notice that the dissipation term I’o only depends on the conductance g1, that is, the dissipation due to the resistance r~is negligible. Instead ofsolving directly the NLS equation (2.5) which contains a dissipative term, we consider the envelope voltage amplitude ~j(X, r) as a superposition of forward and backward propagating waves, ~(X, r)=[~ exp(iK÷X)+~ exp(—iK...X)] Xexp(—iQ’r), (2.7) where Q represents the detuning between the incoming wave frequency and the carrier wave frequency
1
2.By sub-
‘7TT~1D-~__I—-:J-
and can be expressed by Q= (w—a0)/~
(O~ ~
(V)~
I
i,.
c
L
I 2
g ~
I
~
Fig. 2. Schematic representation ofthe nih unit cell. C( V,~)is the nonlinear capacitance of the BB 112 varicap, while r and g are introduced to take into account the respective damping of inductances L1 and L2.
lation in exp stituting expression (iK÷X)and (2.7)exp in ( eq.iK...X). (2.5), (Terms we get such a reas exp[i(2K÷+K)X}, are ne~ected because give no phase-matching.) The coefficientsthey of —
...
exp(iK+X) and exp( —iKX) must be zero, which leads first to a set of two coupled nonlinear Schrädinger equations, 251
Volume 174, number 3
PHYSICS LETTERS A
ii~+P’%~÷1x+iTo7~ 2)~c=0, +Q(I~ I2+2I~iiI
(2.8)
+Q(2~l2+l~ij2),~=0,
(2.9)
8 March 1993
I
I
I
I
C
which finally reduces to Q—PK~.+ir
0+Q(l ~
~1 12)=0, 2+2I~4iI2)=0.
~
(2.10) (2.11)
Q—PK~+iTo+Q((~l In order to solve this system in K. 1., K_, ~4 and ~i, we have to consider the boundary conditions. Indeed (see fig. 1) the finite electrical transmission line is matched at X=0 with a real impedance R1, while at X= —1, the voltage generator has no internal impedance. Let us consider the simplest case of an infinite impedance R,, which allows us to obtain analytical results. The boundary conditions at X= 0 give voltage and current continuity, which leads to V~—_V_=~V1,K+=K_=K=a+ib,
a
IllilillI
0
2
—
-
*
4)
.. *1
(2.14)
—
/,
B
..~., ~‘ 1 ,4.
‘
,.“
,.~ ‘
________________
T=
~Vi~
,‘
I
, /
~~
—
/
-.‘
\
By means of eq. (2.7), the envelope voltage amplitude is obtained as a function of the slow spatial coordinate X 2(bX) . (2.15) V(X)=2V~~Jcos2(aX) +sh The voltage amplitude given by the generator is = V( —1), consequently, the voltage transfer function (defined as the ratio between the envelope voltage amplitude, respectively at the end and at the beginning of the line) becomes
2
I
I
E (213)
1.5
imental results. Points A, B and C are characteristic experimental states chosen for the envelope study.
(2.12)
a=
1
Vinput (V)
Fig. 3. Voltage transfer function T through the whole line versus the input parameter ~ Analytical calculations (solid line) are for £2 = 1.13 x I0~rad s~,while crosses correspond to exper-
where the real and imaginary parts of the wave vector are, respectively,
r
111111
0.5
A
Ii._P,’II
I
0 10 20 30 Fig. 4. Voltage amplitudeCell envelopes Number versus cell number in the
The solid line is anal~’ticallycalculated for point A offig. 3, while diamonds represent experimental In the same way, the dashed line and the crosses representresults. respectively transmission line.
analytical and experimental envelopes for point B of fig. 3.
versus X. The values of the parameters are chosen as
(2.16)
whichasis analysed close possible toin those the next of section. the experimental line From eqs. (2.l2)-(2.14), we note for r=o (non-
In figs. 3 and 4, we have respectively represented T versus V~ 111,and the envelope voltage amplitudes
dissipative case) that we get b = 0 and K= a. This
252
=
_____________________
~cos~(al) +sh2(b1)~
means that K still depends on
V.~2~ For
this ideal
Volume 174, number 3
PHYSICS LETTERS A
case, the transfer function and the NLSW envelope are only slightly modified relative to that occurring in reality.
3. Experimental study The experimental transmission line, see fig. 1, is composed of 27 cells, and the individual components have been carefully selected to minimize the inhomogeneities. The linear inductances are L1=220±5 IIH and L2=470±lO ~.tH and their ohmic measured losses are respectively: r= 5 ~ and g= 0.9 x lO~~ The nonlinear capacitance, which consists of a varicap diode (BB 112), is biased by a constant voltage V0 2 V. Under these conditions one has: C0=330±lOpF, a=0.2l V’, and /1=0.0197 In the experiment, an incident sinusoidal wave generated by a voltage generator with internal resistance 50 L~,is launched at the input of the transmission line. The output of the line is open. The voltage waveform is observed on a numerical oscilloscope with fast Fourier transform (FFT) processing. Oscilloscope probes with high impedance (10 M~)and small capacitance (15 pF) are chosen in order to avoid parasitic reflections. The lower gap edge lies at w0=2.54X 106 rad s’, and 2werads—’ have: P=2.69x 106 rads’, V and J’~=l.36x l0~ Q=—7.76x pF~. Thel0~ incident wave frequency is about 450 kHz, and lies just above w 0. 2, as defined in (2.7), The accuracy Q= (w—w0)/e is poor and weinconsider Q as an adjustable parameter. Our experimental results show that harmonics are generated (about 10%); however, in order to cornpare to the theoretical calculations, in a first approxirnation we only consider the fundamental term of the voltage, given by the FFT processing. The experimental voltage transfer function is represented in fig. 3. It is well fitted by the theoretical curve if we choose Q= l.13x l0~rad s’. In the linear approximation, that is for weak input voltages, the voltage transfer function is almost constant: T~1.2 at point A. The spatial evolution of the voltage envelope amplitude corresponding to point A is represented in fig. 4: the electrical length of the line exceeds three half-wavelengths, as shown either by the experimental crosses or the theoretical curve ~—‘
8 March 1993
(solid line). Note that the output of the line corresponds to a maximum because the output is open. When the input voltage amplitude overcomes a certain threshold, the system switches to a higher transmitting state and the transfer function becomes twice as large as in the linear case. To point B of fig. 3 corresponds the spatial envelope evolution represented in fig. 4. In this case, the electrical length is less than three half-wavelengths. For the maximum of the transfer function (point C in fig. 3) the theoretical envelope corresponds exactly to three halfwavelengths with a maximum at the output and a minimum at the input. It is difficult to check experimentally this prediction because the voltage is close to the lower threshold and therefore very unstable. Wedding and Jager [12] have interpreted the appearance ofbistability by an asymmetrical frequency shift in a resonance curve. In fact they have considered their line as a “black box”. Here, we have related the bistability to the NLSW properties. A careful reader should have noticed that at first sight the bistability is not apparent in (2.16) as it is related to V1. However, the existence of bistability is related implicitly to V~,111 (experimental parameter). In conclusion, our experimental measurements of the bistability and the shape of envelope waves are in good agreement with our theoretical although we have neglected harmonics predictions and inhomogeneities. Our study that amplitude in a nonlinear sion line the confirms input voltage acts transmisas a free parameter which modifies the wavelength in the systern. When the line is open, as in our case, and contains an odd number of half-wavelengths, the system exhibits an important behaviour regarding transmission near a gap edge. Our experimental results suggest that a transmission line is an interesting model to study the properties of nonlinear systems with complex behaviour.
Acknowledgement The authors are grateful to A.C. Scott of Lyngby University (Denmark) and M. Remoissenet ofDijon University (France) for helpful discussions. 253
Volume 174, number 3
PHYSICS LETTERS A
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