Nonlinear reflection from a dense saturable absorber: from stability to chaos

1 October 1997

OPTICS COMMUNICATIONS ELSEVIER

Optics Communications 142 (1997) 66-70

Nonlinear reflection from a dense saturable absorber: from stability to chaos Enrique Conejero Jarque a,1,Victor Malyshev

b*2

a Departamento de Fisica Aplicada, Uniuersidad de Salamanca, 37071 Salamanca, Spain b All-Russian Research Center ” Var:ilm State Optics Institute”, 199035 Saint Petersburg, Russia

Received 3 March 1997; revised 6 May 1997; accepted 15 May 1997

Abstract Non-linear reflection of an intense plane wave from a vacuum-dense absorber interface is considered under the conditions of resonance and normal incidence. In addition to the stationary and self-oscillation regimes reported previously, chaotic behaviour is found. 0 1997 Elsevier Science B.V.

1. Introduction Although widely discussed during the last decade [l131, the problem of nonlinear resonant optical response of dense two-level media still focuses the attention of many scientists. This occurs first because the present system is a good model for studying resonance phenomena and second because it reveals a number of interesting effects, including optical bistability [1,2,4,6,10,13-151 and ultrafast optical switching [7-9,111. Such peculiarities are the result of two kinds of coupling among the atoms of the medium: (i) via the near-zone dipole-dipole forces [ 1,2,5-9,11,13] and (ii) via the far-zone radiation field [3,4,10,12,13]. The former may produce an abrupt change in nonlinear dielectric susceptibility, separating the medium into two regions of different refraction indices [ 13,151, while the latter, under saturating intensities of the incident field, may yield a self-reflected wave inside the medium [3]. In this communication, we describe a time-domain analysis of the reflection from a “vacuum-dense two-level medium” interface under conditions of fixed external field amplitude and normal incidence, looking for non-stationary responses. We leave aside the problem of sliding incidence when bistability and related phenomena occur at

values of the incidence angle close to the angle of total internal reflection (see theoretical predictions [ 16- 181 and experimental observations [ 19,201 of these effects). Also, we do not address here the problem of a limited beam, reserving this for further studies. At high densities of two-level emitters, in addition to the self-oscillation regime reported recently [lo], reflectivity behaviour is chaotic. To our knowledge, in the model considered chaotic behaviour has never been mentioned. More detailed study of this new regime will be provided elsewhere.

2. General formalism In our description of the nonlinear boundary value problem, for a linearly polarized monochromatic plane wave (k,,w,) normally incident upon the plane of a semi-infinite (x 2 0) two-level medium, we follow the approach of Ref. [6] using the integral version of the wave equation, The whole set of equations then reads e(t,r)

= ea(r)exp(iO +#amexp(i&-E’I)R([‘,r)d[‘, /

aR/a7= -[i(6+ 6,W) ’E-mail: [email protected]. * E-mail: [email protected].

aW/&-=

0030.4018/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved. PII SOO30-4018(97)00275-7

(i/2)(eR*

+ l]R-ieW,

- e*R)

- (1 + W)/T,.

(la) (lb) (lc)

E. Conejero Jarque, V. Malysheu / Optics Communications 142 (1997166-70 Here, e = pET2/fi and e, = pE,,T2/fi are respectively the dimensionless slowly varying in time amplitudes of the average (Maxwell) and incident electric fields; CL is the transition dipole moment and T2 is the relaxation time of the electric polarization; R/2 is the amplitude of the off-diagonal density matrix element; W is the population difference; V= 2~-~~N~T~/fi, where N,, is the atom density; 6 = (02, - o,,)T2 is the detuning between the laser is ( wO> and the transition (w,,) frequencies: 7, = T,/T, the normalized relaxation time of the population; 5 = k, x, and T = t/T2. The term 6,W, where 6, = (4r/3>~.~~N~T~/h, comes from the local field correction [1,21] and describes the resonance frequency shift which depends on the population difference. The new resonance condition is therefore 6’ = 6 - 6, = 0, now used instead of 6 = 0. We have neglected retardation effects in Eq. (la) since, for the parameters used in our simulation, all the effects are determined by a boundary region of the size of a few vacuum wavelengths and hence, the actual time of the passage of light through it has the order of magnitude of the optical period while all the characteristic times of are much the problem discussed (T,,T,,h/l*.E,,A~‘,A;‘) larger. One can therefore consider the field as propagating instantaneously [ 10,131. The reflected wave amplitude that we are mainly interested in can be simply expressed as e,.(t) = e(O,t) - e,(t). The reflection coefficient is then given by the equation r(t) = le,(t)/e,(t)l.

3. Steady-state

analysis

Let us restore first the basic facts of the steady-state analysis before studying the temporal evolution of our system. Taking a constant amplitude of the incident field and assuming the existence of a stationary solution (aR/&= a W/dr = 0), one can exclude the electric polarization R from Eqs. (1) and obtain a closed equation for the electric field amplitude 3

X [ c( le12) - 11e( 5’)W.

61

as the resonance frequency shift with the population of the upper level. In the presence of the local field, the population difference obeys a cubic equation [l]

qr

6’+ 6,(W+

I)]‘+

1) = -le1’7,.

It is straightforward to check that Eq. (4) has three distinct real roots (and thus may show bistability) when 6, > 4 (see Ref. [14]), ]e]‘r, > 2 and 6’ E [&“,c?“~‘], where the values of 6”“, S’(‘) depend on 6, and tend to [ -6,,fi] when 6, B 1 [13]. Since Eq. (4) has local origin, it is reasonable to refer to this effect as local bistability. It results in the appearance of a population domain-wall inside the material, whose location may be stable or may oscillate depending on the incident field amplitude. This phenomenon does not require saturating values of the incident field amplitude. Nevertheless, we go no further into this aspect here since it was studied in detail in a previous publication 1131. Another kind of bistability and even multistability (of non-local origin) appears when the parameter !P is sufficiently high. It is originated by the existence of a gap in the polariton spectrum of the medium. The latter is given by the dispersion equation ( w/c)~E,(o,~) = k2 (see, for example, Refs. [22,23]), where &r( w,k) is the dielectric function in the absence of the field (in our case it does not depend on the wave-vector k). Thus the non-linear optical response (in particular, the reflection) may show multistable behaviour when the incident frequency is inside the gap, 6’ E [-2ly,-(2V)-‘1 [4,10,13]. At saturating values of the incident field, reflection occurs inside the material rather than on the boundary, giving rise to a distributed feedback responsible for the multi-valued optical response mentioned above. Fig. 1 shows an example of such behaviour. The spectral regions in which the local and non-local types of bistability appear are only partly overlapped and so one of them may be discriminated by choosing the appropriate detuning. If one selects the detuning inside the

(2)

Here, i+6+6,w 8( ]e]‘) = 1 + 2!P 1+(S+6,W)2+]e]2r,

(3)

is the nonlinear permittivity, which includes the saturation effect at high values of the electric field intensity as well 5

’ It is straightforward to show that Eq. (2) is exactly equivalent to the differential form d*e/d[* + .z(lel*)e = 0 with 8(le12) given by Eq. (3).

10

15

20

25

INCIDENT FIELD AMPLITUDE, e0 Fig. 1. A sample of the multi-valued coefficient at ly = 18 and 8’ = - 25.

behaviour

of the reflection

68

E. Conejero Jarque, V. Malyshec /Optics

Communications

142 (1997) 66-70

spectral region - !P < 6’ < - 6,) then local bistability will be suppressed while the non-local one will exist. This is the case we are interested in. The next section addresses a time-domain analysis of the non-local type of multistability based on non-stationary equations (1).

4. Time-domain

analysis

As previously reported [ 10,131, it is possible to obtain a stable hysteresis loop of the reflection (in the case of a multi-valued solution of Eq. (2) by slowly sweeping the incident field amplitude up and down. If the amplitude is raised, the stationary regime fails and the reflection shows self-oscillations. Eqs. (1) comprise a set of five real nonlinear equations with a nonlocal coupling via the wave equation (la). Hence, the dimensionality of our system is infinite. It is therefore quite reasonable to expect more complicated behaviours of the optical response than the auto-oscillations described above.

3 .Z 5 -z m 5 e Y % $

1

1 o-2

I

I

103 102 10' 100 1 0.' 1 o-2

lo3t

i

102 10' 100 1 o-1 102

'

I

I

0

25

50

I, I

II' 75

I 100

FREQUENCY, Tz I T

1

Fig. 3. Power spectra corresponding

to the temporal series of Fig.

0.5

0 1

05

0 1

0.5

0 0

25

50 TIME,

75

100

T I T,

Fig. 2. Temporal dynamics of the high density system (Y’ = 18) for three different values of the external field amplitude. The incident frequency is in the middle of the polariton band gap of the medium (6’ = - 25). Values of the incident field amplitude are: (a) e, = 16, (b) eO = 17 and (c) eO = 19. Auto-oscillations and an irregular response are appreciable.

We performed a series of numerical calculations, carry ing them out in the following way: we begin with a fixed value of the incident field amplitude and wait for a regular pattern (in the case of one existing). We then change the amplitude stepwise and wait again, and so on. In all calculations, we took r, = T,/T, = 1. The stability of the integration scheme was examined by decreasing the integration steps. The results are shown in Figs. 2-4. Fig. 2 shows a typical evolution of the reflection for three values of the external field. At e, = 16, one can observe a simple periodic pattern of the reflection coefficient oscillating between approximately r = 0.25 and r = 1, with a frequency of the order of T; ‘. However, when the dimensionless incident field is increased to e. = 17, the periodic pattern is destroyed and one sees a rather irregular response, although traces of periodicity are conserved. Raising the field amplitude further, up to e, = 19, we apparently recover a quasi-periodic regime of reflection, as can be seen in Fig. 2c. If the driving field amplitude exceeds e, = 19, entirely irregular oscillations of reflection (similar to those at e, = 17 and not shown in Fig. 2) are observed. Fourier analysis is a powerful tool when studying dynamical systems (see, for instance Ref. [24]). In Fig. 3, we show the results of calculations of the power spectra

E. Conejero Jarque, V. Malysheu / Optics Communications

142 (19971 66-70

69

5. Conclusion The nonlinear optical response of a saturable absorber (reflection, for instance) may show a route from a stable regime to a chaotic one. This requires both saturating values of the driving field and a frequency within the polariton band gap. The effects may be seen in materials which show polariton splitting. The self-oscillatory regime of the reflection may be used to obtain a pulsed reflection signal under a cw-excitation.

(b)

Acknowledgements

IO

-

0

-

0

We thank L. Roso for reading the manuscript and for helpful discussions on the results. Partial support from the Spanish Direcci6n General de Investigaci6n Cientifica y Tecnologica (under grant PB95-0955) and from the Junta de Castilla y Ledn are acknowledged. V. Malyshev thanks the Salamanca University, where this work was carried out, for its hospitality.

10 ELECTRIC

30

20 FIELD,

REAL

40

PART

Fig. 4. Maps of the imaginary part of the field at the boundary versus its real part corresponding to the temporal dynamics of Fig. 2.

corresponding to the temporal series shown in Fig. 2. When calculating these spectra, we neglect the first temporal cycles, thus eliminating transient effects. At the lowest value of the field, the spectrum is discrete and is composed of harmonics of a basic frequency, according to the oscillatory behaviour. By contrast, the spectrum related to the intermediate value of the field consists of some significant broadened frequency peaks over a broad-band continuum. This is a sign of chaotic dynamics. At the highest value of the field, one sees linear combinations of at least two basic frequencies, but the background is not negligible. Such behaviour may be related either to a case of quasi-periodicity [24] or chaos. Finally, in Fig. 4 we have represented a projection of the phase space dynamics: the imaginary part of the electric field at the boundary versus its real part. As seen in Fig. 4, this two-dimensional representation confirms the conclusions drawn above regarding the type of temporal dynamics of our system. Despite the infinite dimensionality of the system, the map corresponding to e, = 16 shows how the system evolves towards a limit cycle which will be eventually reached when the time tends to infinity. When e, = 17, the system cannot find an attractor, providing new evidence of chaotic behaviour. At e, = 19, a dense map is obtained. Again, it is impossible to distinguish between quasi-periodic or chaotic behaviour.

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