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Oscillation and chaos in a Fabry-Perot bistable cavity with gaussian input beam
PHYSICS LETTERS
Volume 92, number 5
OSCILLATION
AND CHAOS IN A FABRY-PEROT
15 November 1982
BISTABLE CAVITY
WITH GAUSSIAN INPUT BEAM
W.J. FIRTH and E.M. WRIGHT Department of Physics, Heriot-WattUniversity,Riccarton, Edinburgh, Scotland, UK Received 30 August 1982
We report first results of numerical calculations on Ikeda instabilities in a folded Fabry-Perot resonator pumped by a gaussian beam matched to the cavity. Oscillations of periods tR, 2f~ and 4tR, followed by quasi-periods 2.5tR and 5tR, are observed en route to chaotic responses. The tR oscillation arises from nonlinear nonreciprocity.
Introduction. Ikeda ] l] first reported branch instabilities in a nonlinear ring resonator (NLRR), since when a great deal of theoretical and numerical work has been concentrated on this area [2]. Among these papers, Moloney et al. [3] report calculations incorporating transverse effects in the NLRR. We have also followed this path [4] and have seen period-doubling and chaos in the NLRR in a number of models, in-
pout
eluding the method used below. Our results are in qualitative agreement with those of Moloney et al. The Fabry-Perot or folded resonator (NLFP) is more attractive from an experimental viewpoint than the NLRR, because of the extra nonlinearity due to the double traverse of the medium. Using a plane wave analysis Firth [5] showed that the NLFP also shows positive branch instabilities. In this letter we
la1
Fig. 1. (a) Pout versus Pin: I = 0.01,~ = 25,d/Rm = 0.952, R = 0.9, &, = 0.2; the transverse grid size is 4Wo. Region (1) shows tR oscillation: bifurcation and chaos appears in region (2). 64 Gauss-Laguerre abscissae (plus the central point) were used in the calculations. (b) Resonator configuration.
0 031-9163/82/0000-0000/$02.75
0 1982 North-Holland
211
Volume 92, number 5
PHYSICS LETTERS
15 November 1982
include transverse effects by considering a NLFP with curved mirrors pumped by a gaussian beam matched to the fundamental mode (cylindrical symmetry assumed). We compare the results with those obtained from the corresponding plane wave case.
Flu,
Basic theory. Our model system is a thin slice of medium with linear absorption but nonlinear refractive index (x(~) and adiabatic approximations), symmetrically enclosed by two external mirrors forming a stable cavity with fundamental mode radius W0 (fig. lb). This cavity is driven by a gaussian pum beam of the ! same radius. We scale axially (z) to ZcFV,and transversely (r) to W,, and scale fields so as to yield integrated powers in units ofP, (critical power: see refs. [6,7] for a discussion). We thus obtain
Bo(j, t + tm) = BIG, t) e-dexp{-(i/a)(l
=o.F+kiV;F-2i[]F12
+2)Bi2]F,
(la)
+ (no/c) a/at]B
= -cd3 +$iVfB
X ]IFoo’, t)12 + 21BI0‘,
- 2i[21F12 t IB12]B.
(lb)
The unequal coefficients in square brackets at the rhs demonstrate nonlinear nonreciprocity and arise from the induced phase grating [5,8]. The output or circulating powers are given by
X
2rdr
lE12.
(2)
0
With new variables z’ = z, tF = t - (n,/c)z, tg = t + (no/c)(z - I), the left hand sides of eqs. (la,b) become aF/az’ and -aB/az’ respectively. The medium length I is assumed small enough that diffraction effects can be ignored in the medium. There is a problem in that e.g. F will depend on tB as well as tF. This reflects the fact that F meets wavefronts that left z = I (propagating in the opposite direction) at different times. However, we are expecting temporal oscillations of the envelopes F, B of the order tR = 2(d + nl)Z)/c, ISI 3whereas the medium transit time is t, = noZ/c. We assume tR 3- t,, therefore we can, as a first approximation, integrate the resulting eq. (1 a) as if it were steady state. Describing the envelopes E(q, t) by vectors E(q, t) [or E(j, t)], where ri are discrete coordinate values, we obtain
212
t)121).
(3)
- e-zaZ)
PIFoo’, 012 + lBIo’, t)1211.
(4)
Propagation from the medium to the mirror and back again is governed by the following integral equation (we ignore diffraction losses at the mirrors) [9] :
-ikw~ * Ein(r)= 7
J r’dr’ Jo((kW~/B)rr’)
0
t r’2)]Eout(r’)
,
(5)
withA, B the elements of the appropriate ABCD matrix. We convert this integral equation to a matrix equation by using Gauss-Laguerre quadrature [9]. The above mentioned ri are chosen to be the GaussLaguerre abscissae. Thus if Be(t) is the field vector on exit from the left face of the NL medium, and Fo(t + i(tR - t,)) the returning field vector after propagating over the external cavity, then Fo(r + ;(tR - tm)) = fl
i==J
- e-zd)
Using a similar argument for the field propagated from z’ = Zto 0, we obtain
X exp [(ikWo/2B)A(r2
[a/a2 + &,/c)a/at]F
[-a/az
t + tm) = Fo(j, t) e-dexp{-(i/or)(I
)
exp(i$$O) % Be(t) + %a (6)
where R is?he input mirror reflectivity, Q. the cavity detuning, M the matrix propagator, Pi, the input power reaching the medium face and a the vector describing a cavity gaussian. A similar relation without the driving term, connects B,(t t i(tR - t,)) and Fl(t) at the other face. The resulting equations are iterated, and we obtain steady state, oscillatory and chaotic responses depending on the input power Pin. We have in effect, a system of (65 in the calculations) plane wave “Ikeda oscillators” coupled by diffraction in the external cavity strongly so in our case, since the cavity is near confocal. We choose a high-diffraction cavity in order to avoid part beam switching [3,10] since our primary concern here is investigating instabilities. PZane wave results. Before presenting and discussing our numerical results we briefly discuss the correspond-
Volume 92, number 5
15 November 1982
PHYSICS LETTERS
Xl0’
SCALED INPUT
INTENSITY
Fig. 2. Plane wave calculations: #o = -2.846 (other parameters as in 0. 1). Figure shows the scaled input-output characteristic for the case that the nonreciprocal terms are retained. Region (1) shows IR oscillation, region (2) shows oscillatory and chaotic behaviour.
ing plane wave problem
as a basis for comparison. Firth’s paper [5] gives the basic theory. For our particular case we find (fig. 2) (a) a stable lower branch which switches up to (b) an unstabk position of the upper branch showing round trip, i.e. tR, oscillations. In fact there is a sizeable portion of the upper branch which oscillates in this fashion. As the driving field is increased these oscillations die out and a steady response ensues, but further increase leads to (c) Ikeda oscillation with period -1.4 tR (estimate from Fourier spectrum which consists of one fairly sharp spike), leading very quickly into apparently chaotic behaviour. Stability was never recovered above this point. The stability edges for the tR oscillation are accurately predicted by Firth’s theory [5]. This theory also predicts that if we remove the nonreciprocal terms in eqs. (l), tR oscillations are forbidden to all orders. For a physical interpretation of this phenomenon it is useful to regard the cavity as containing two wave trams at any time, spaced tR/2 apart, and interacting with each other (via $3)) twice per round trip. In the steady state, these trains are iden-
tical in amplitude and phase. In tR oscillation this symmetry is broken - the two trains have unequal amplitudes and impose unequal phase shifts on each other. Only the nonreciprocal interaction (i.e. the phase grating) can force such a splitting. This oscillation is thus the first dynamic instability attributable to nonlinear nonreciprocity [8]. Gaussian beam results. Fig. la shows the output power for the system as a function of input power (grating terms retained). Considerable qualitative similarity to the plane wave results obtained above are noticed. In particular the region of tR oscillation appears here also. Here it has literally, an extra dimension, in that where the counterpropagating fields have the same spot size in the linear case, here the nonlinearity breaks that symmetry, as the fields dilate and contract in the course of the tR oscillation. However, in contrast to the NLRR where a rich variety of beam profiles appear (quasi-gaussian, doughnut shaped etc. [3]), the profiles we observe during oscillation remain quasi-gaussian. If the tR oscillation is just an extension of a plane 213
Volume
92, number
5
PHYSICS
UNITS
15 November
Fig. 3. Time series (integrated power) for various in fig. 1. (a) Pin = 75, @R), (b) Pin = 120, (2rR),
x10' TIE
LETTERS
input (c) Pin
(at&.
SW2
Ial
70
a 68
I
I
I
x10’ TIME
Fig. 4. Time series (integrated
214
UNITS
TRl2
power)
TIME
for (a) quasi
2.5tR oscillation
P.I,, = 160, (b) chaotic
UNITS
TR/2
response,
Pin = 220, in fig. 1.
1982
powers =
140,
15 November 1982
PHYSICS LETTERS
Volume 92. number 5
b) ___..__ ..__._.__ _...._ .-..--__---.---T-----
(4
-21 ++
~~~~~___~~~~~~
_
__
_ _^
_. i.--_-..~
.--
0
-3
__-=~--‘-.~__ -18
-II
__r--q.__i--
--_.
-
X12’ REAL(E)
REAL(E)
+ +
++
:
+
+
+ + ..-_-__a__.-_le --38 ..-_w ._._ _~~~~_~~~~_._.~~~~~__ L_~~-_..25_ 3031 23 +
-25)
+
+
+
4
--e-----G
6
REAL(E)
R!aLcE>
Fig. 5. Phase-space attractors: Real (Fi) versus Imag (Fl) [Real Q versus Imag (E)], for& r = 0.55, (c)r = 1.06, (d) r = 1.57. wave phenomenon,
the higher unstable region shows qualitative differences from the plane wave results. Here we observe a period doubling bifurcation sequence, with base period 2tR. Fig. 3 shows tR, 2tR and 4t, oscillations in the integrated output power (these oscillations occur in the intensity at each radius also): we have also seen period St, ln spectra and evidence of 16tR. At higher input powers we observe oscillations which apparently contain numerous commensurate and incommensurate frequencies. In particular we see a quasi-oscillation which is predominantly 2.5tR which “period doubles” to a predominantly 3, oscillation. The sequence then seems to proceed to chaos - fig. 4 shows the time series (integrated power) *r and more notably, fig. 5 shows the *i The Fourier spectrum for this case is structureless.
16
xw1
XIV
= 220 at various radii: (a) r = 0, (b)
phase space diagrams at different radii. The on-axis attractor is of the spiral type found by Ikeda for the NLRR [ 11, but at finite radii the nature of the attractors is less clear, though this may be due in part to the smaller amplitude at high radii. These period doubling sequences have also been observed with other cavity geometries. We now consider the case where the nonreciprocal terms are removed. As in the plane wave case the region of tR oscillation disappears. The lower branch is stable as is the upper branch just above switch up. As the input power is increased the system starts oscillating with base period 2tR. In conclusion, we have presented the first predictions of period-doubling and chaos in a finite FabryPerot resonator, together with the first dynamic in-
215
PHYSICS LETTERS
Volume 92, number 5 stability
arising from nonlinear
E.M. Wright acknowledges the Science
and Engineering
nonreciprocity.
financial support from Research
Council.
References [l] K. Ikeda, H. Daido and 0. Akimoto, Phys. Rev. Lett. 45 (1980) 709. [2] See e.g.: 12th IQEC digest, Appl. Phys. B28 (June/July 1982). [ 31 J.V. Moloney, F.A. Hopf and H.M. Gibbs, 12th IQEC digest, Appl. Phys. B28 (June/July 1982) p. 98; J.V. Moloney and H.M. Gibbs, 12th IQEC digest, Appl. Phys. B28 (June/July 1982) p. 100.
216
15 November 1982
[4] W.J. Firth, E. Abraham and E.M. Wright, 12th IQFC digest, Appl. Phys. B28 (June/July 1982) p. 170. [5] W.J. Firth, Opt. Commun. 39 (1981) 343. [6] J.H. Marburger and F.S. Felber, Phys. Rev. Al7 (1978) 335. [7] W.J. Firth and E.M. Wright, Opt. Commun. 40 (1982) 233. [8] A.E. Kaplan and P. Meystre, Opt. Commun. 40 (1982) 229. [9] R.L. Sanderson and W. Streifer, Appl. Opt. 8 (1969) 131 [lo] N.N. Rosanov and V.E. Semenov, Opt. Commun. 38 (1981) 435.
Volume 92, number 5
OSCILLATION
AND CHAOS IN A FABRY-PEROT
15 November 1982
BISTABLE CAVITY
WITH GAUSSIAN INPUT BEAM
W.J. FIRTH and E.M. WRIGHT Department of Physics, Heriot-WattUniversity,Riccarton, Edinburgh, Scotland, UK Received 30 August 1982
We report first results of numerical calculations on Ikeda instabilities in a folded Fabry-Perot resonator pumped by a gaussian beam matched to the cavity. Oscillations of periods tR, 2f~ and 4tR, followed by quasi-periods 2.5tR and 5tR, are observed en route to chaotic responses. The tR oscillation arises from nonlinear nonreciprocity.
Introduction. Ikeda ] l] first reported branch instabilities in a nonlinear ring resonator (NLRR), since when a great deal of theoretical and numerical work has been concentrated on this area [2]. Among these papers, Moloney et al. [3] report calculations incorporating transverse effects in the NLRR. We have also followed this path [4] and have seen period-doubling and chaos in the NLRR in a number of models, in-
pout
eluding the method used below. Our results are in qualitative agreement with those of Moloney et al. The Fabry-Perot or folded resonator (NLFP) is more attractive from an experimental viewpoint than the NLRR, because of the extra nonlinearity due to the double traverse of the medium. Using a plane wave analysis Firth [5] showed that the NLFP also shows positive branch instabilities. In this letter we
la1
Fig. 1. (a) Pout versus Pin: I = 0.01,~ = 25,d/Rm = 0.952, R = 0.9, &, = 0.2; the transverse grid size is 4Wo. Region (1) shows tR oscillation: bifurcation and chaos appears in region (2). 64 Gauss-Laguerre abscissae (plus the central point) were used in the calculations. (b) Resonator configuration.
0 031-9163/82/0000-0000/$02.75
0 1982 North-Holland
211
Volume 92, number 5
PHYSICS LETTERS
15 November 1982
include transverse effects by considering a NLFP with curved mirrors pumped by a gaussian beam matched to the fundamental mode (cylindrical symmetry assumed). We compare the results with those obtained from the corresponding plane wave case.
Flu,
Basic theory. Our model system is a thin slice of medium with linear absorption but nonlinear refractive index (x(~) and adiabatic approximations), symmetrically enclosed by two external mirrors forming a stable cavity with fundamental mode radius W0 (fig. lb). This cavity is driven by a gaussian pum beam of the ! same radius. We scale axially (z) to ZcFV,and transversely (r) to W,, and scale fields so as to yield integrated powers in units ofP, (critical power: see refs. [6,7] for a discussion). We thus obtain
Bo(j, t + tm) = BIG, t) e-dexp{-(i/a)(l
=o.F+kiV;F-2i[]F12
+2)Bi2]F,
(la)
+ (no/c) a/at]B
= -cd3 +$iVfB
X ]IFoo’, t)12 + 21BI0‘,
- 2i[21F12 t IB12]B.
(lb)
The unequal coefficients in square brackets at the rhs demonstrate nonlinear nonreciprocity and arise from the induced phase grating [5,8]. The output or circulating powers are given by
X
2rdr
lE12.
(2)
0
With new variables z’ = z, tF = t - (n,/c)z, tg = t + (no/c)(z - I), the left hand sides of eqs. (la,b) become aF/az’ and -aB/az’ respectively. The medium length I is assumed small enough that diffraction effects can be ignored in the medium. There is a problem in that e.g. F will depend on tB as well as tF. This reflects the fact that F meets wavefronts that left z = I (propagating in the opposite direction) at different times. However, we are expecting temporal oscillations of the envelopes F, B of the order tR = 2(d + nl)Z)/c, ISI 3whereas the medium transit time is t, = noZ/c. We assume tR 3- t,, therefore we can, as a first approximation, integrate the resulting eq. (1 a) as if it were steady state. Describing the envelopes E(q, t) by vectors E(q, t) [or E(j, t)], where ri are discrete coordinate values, we obtain
212
t)121).
(3)
- e-zaZ)
PIFoo’, 012 + lBIo’, t)1211.
(4)
Propagation from the medium to the mirror and back again is governed by the following integral equation (we ignore diffraction losses at the mirrors) [9] :
-ikw~ * Ein(r)= 7
J r’dr’ Jo((kW~/B)rr’)
0
t r’2)]Eout(r’)
,
(5)
withA, B the elements of the appropriate ABCD matrix. We convert this integral equation to a matrix equation by using Gauss-Laguerre quadrature [9]. The above mentioned ri are chosen to be the GaussLaguerre abscissae. Thus if Be(t) is the field vector on exit from the left face of the NL medium, and Fo(t + i(tR - t,)) the returning field vector after propagating over the external cavity, then Fo(r + ;(tR - tm)) = fl
i==J
- e-zd)
Using a similar argument for the field propagated from z’ = Zto 0, we obtain
X exp [(ikWo/2B)A(r2
[a/a2 + &,/c)a/at]F
[-a/az
t + tm) = Fo(j, t) e-dexp{-(i/or)(I
)
exp(i$$O) % Be(t) + %a (6)
where R is?he input mirror reflectivity, Q. the cavity detuning, M the matrix propagator, Pi, the input power reaching the medium face and a the vector describing a cavity gaussian. A similar relation without the driving term, connects B,(t t i(tR - t,)) and Fl(t) at the other face. The resulting equations are iterated, and we obtain steady state, oscillatory and chaotic responses depending on the input power Pin. We have in effect, a system of (65 in the calculations) plane wave “Ikeda oscillators” coupled by diffraction in the external cavity strongly so in our case, since the cavity is near confocal. We choose a high-diffraction cavity in order to avoid part beam switching [3,10] since our primary concern here is investigating instabilities. PZane wave results. Before presenting and discussing our numerical results we briefly discuss the correspond-
Volume 92, number 5
15 November 1982
PHYSICS LETTERS
Xl0’
SCALED INPUT
INTENSITY
Fig. 2. Plane wave calculations: #o = -2.846 (other parameters as in 0. 1). Figure shows the scaled input-output characteristic for the case that the nonreciprocal terms are retained. Region (1) shows IR oscillation, region (2) shows oscillatory and chaotic behaviour.
ing plane wave problem
as a basis for comparison. Firth’s paper [5] gives the basic theory. For our particular case we find (fig. 2) (a) a stable lower branch which switches up to (b) an unstabk position of the upper branch showing round trip, i.e. tR, oscillations. In fact there is a sizeable portion of the upper branch which oscillates in this fashion. As the driving field is increased these oscillations die out and a steady response ensues, but further increase leads to (c) Ikeda oscillation with period -1.4 tR (estimate from Fourier spectrum which consists of one fairly sharp spike), leading very quickly into apparently chaotic behaviour. Stability was never recovered above this point. The stability edges for the tR oscillation are accurately predicted by Firth’s theory [5]. This theory also predicts that if we remove the nonreciprocal terms in eqs. (l), tR oscillations are forbidden to all orders. For a physical interpretation of this phenomenon it is useful to regard the cavity as containing two wave trams at any time, spaced tR/2 apart, and interacting with each other (via $3)) twice per round trip. In the steady state, these trains are iden-
tical in amplitude and phase. In tR oscillation this symmetry is broken - the two trains have unequal amplitudes and impose unequal phase shifts on each other. Only the nonreciprocal interaction (i.e. the phase grating) can force such a splitting. This oscillation is thus the first dynamic instability attributable to nonlinear nonreciprocity [8]. Gaussian beam results. Fig. la shows the output power for the system as a function of input power (grating terms retained). Considerable qualitative similarity to the plane wave results obtained above are noticed. In particular the region of tR oscillation appears here also. Here it has literally, an extra dimension, in that where the counterpropagating fields have the same spot size in the linear case, here the nonlinearity breaks that symmetry, as the fields dilate and contract in the course of the tR oscillation. However, in contrast to the NLRR where a rich variety of beam profiles appear (quasi-gaussian, doughnut shaped etc. [3]), the profiles we observe during oscillation remain quasi-gaussian. If the tR oscillation is just an extension of a plane 213
Volume
92, number
5
PHYSICS
UNITS
15 November
Fig. 3. Time series (integrated power) for various in fig. 1. (a) Pin = 75, @R), (b) Pin = 120, (2rR),
x10' TIE
LETTERS
input (c) Pin
(at&.
SW2
Ial
70
a 68
I
I
I
x10’ TIME
Fig. 4. Time series (integrated
214
UNITS
TRl2
power)
TIME
for (a) quasi
2.5tR oscillation
P.I,, = 160, (b) chaotic
UNITS
TR/2
response,
Pin = 220, in fig. 1.
1982
powers =
140,
15 November 1982
PHYSICS LETTERS
Volume 92. number 5
b) ___..__ ..__._.__ _...._ .-..--__---.---T-----
(4
-21 ++
~~~~~___~~~~~~
_
__
_ _^
_. i.--_-..~
.--
0
-3
__-=~--‘-.~__ -18
-II
__r--q.__i--
--_.
-
X12’ REAL(E)
REAL(E)
+ +
++
:
+
+
+ + ..-_-__a__.-_le --38 ..-_w ._._ _~~~~_~~~~_._.~~~~~__ L_~~-_..25_ 3031 23 +
-25)
+
+
+
4
--e-----G
6
REAL(E)
R!aLcE>
Fig. 5. Phase-space attractors: Real (Fi) versus Imag (Fl) [Real Q versus Imag (E)], for& r = 0.55, (c)r = 1.06, (d) r = 1.57. wave phenomenon,
the higher unstable region shows qualitative differences from the plane wave results. Here we observe a period doubling bifurcation sequence, with base period 2tR. Fig. 3 shows tR, 2tR and 4t, oscillations in the integrated output power (these oscillations occur in the intensity at each radius also): we have also seen period St, ln spectra and evidence of 16tR. At higher input powers we observe oscillations which apparently contain numerous commensurate and incommensurate frequencies. In particular we see a quasi-oscillation which is predominantly 2.5tR which “period doubles” to a predominantly 3, oscillation. The sequence then seems to proceed to chaos - fig. 4 shows the time series (integrated power) *r and more notably, fig. 5 shows the *i The Fourier spectrum for this case is structureless.
16
xw1
XIV
= 220 at various radii: (a) r = 0, (b)
phase space diagrams at different radii. The on-axis attractor is of the spiral type found by Ikeda for the NLRR [ 11, but at finite radii the nature of the attractors is less clear, though this may be due in part to the smaller amplitude at high radii. These period doubling sequences have also been observed with other cavity geometries. We now consider the case where the nonreciprocal terms are removed. As in the plane wave case the region of tR oscillation disappears. The lower branch is stable as is the upper branch just above switch up. As the input power is increased the system starts oscillating with base period 2tR. In conclusion, we have presented the first predictions of period-doubling and chaos in a finite FabryPerot resonator, together with the first dynamic in-
215
PHYSICS LETTERS
Volume 92, number 5 stability
arising from nonlinear
E.M. Wright acknowledges the Science
and Engineering
nonreciprocity.
financial support from Research
Council.
References [l] K. Ikeda, H. Daido and 0. Akimoto, Phys. Rev. Lett. 45 (1980) 709. [2] See e.g.: 12th IQEC digest, Appl. Phys. B28 (June/July 1982). [ 31 J.V. Moloney, F.A. Hopf and H.M. Gibbs, 12th IQEC digest, Appl. Phys. B28 (June/July 1982) p. 98; J.V. Moloney and H.M. Gibbs, 12th IQEC digest, Appl. Phys. B28 (June/July 1982) p. 100.
216
15 November 1982
[4] W.J. Firth, E. Abraham and E.M. Wright, 12th IQFC digest, Appl. Phys. B28 (June/July 1982) p. 170. [5] W.J. Firth, Opt. Commun. 39 (1981) 343. [6] J.H. Marburger and F.S. Felber, Phys. Rev. Al7 (1978) 335. [7] W.J. Firth and E.M. Wright, Opt. Commun. 40 (1982) 233. [8] A.E. Kaplan and P. Meystre, Opt. Commun. 40 (1982) 229. [9] R.L. Sanderson and W. Streifer, Appl. Opt. 8 (1969) 131 [lo] N.N. Rosanov and V.E. Semenov, Opt. Commun. 38 (1981) 435.