Critical slowing down at a bifurcation Available to Purchase

N. B.

 

Tufillaro

and

A. M.

 

Albano

, “,”  , – ();

T. M.

 

Mello

and

N. B.

 

Tufillaro

, “,”  , – ();

H. T.

 

Savage

,

C.

 

Adler

,

S. S.

 

Antman

, and

M.

 

Melamud

, “,”  , ();

O.

 

Maldonado

,

M.

 

Markus

, and

B.

 

Hess

, “,”  , – ();

S. T.

Vohra

,

F.

Bucholtz

,

D. M.

Dagenais

, and

K. P.

Koo

, “,” , – ().

S. M.

 

Rezende

,

A.

 

Azevedo

,

A.

 

Cascon

, and

J.

 

Koiller

, “,”  , – ();

D. J.

Mar

,

L. M.

Pecora

,

F. J.

Rachford

, and

T. L.

Carrol

, “,” , – ().

G. Nicolis and I. Prigogine, Self-Organization in Non-Equilibrium Systems (Wiley, New York, 1977);

J.-C.

 

Roux

, “,”  , – ();

J. L. Hudson and O. E. Rössler, “Chaos and complex oscillations in stirred chemical reactions,” in Dynamics of Nonlinear Systems, edited by V. Hlavacek (Gordon & Breach, New York, 1985).

C.

 

Normand

,

Y.

 

Pomeau

, and

M. G.

 

Velarde

, “,”  , – ();

P. Bergé, Y. Pomeau, and Ch. Vidal, Order within Chaos (Wiley, New York, 1984);

R. P.

Behringer

, “,” , – ().

The literature is too extended to be cited here. A good overview of the problem is presented in

H.

Haken

, “,” , – (), which also contains other interesting examples.

H. G. Solari, M. A. Natiello, and G. B. Mindlin, Nonlinear Dynamics (IOP, Bristol, UK, 1996);

S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering (Perseus, Cambridge, MA, 2001).

F.

 

De Pasquale

,

Z.

 

Racz

,

M.

 

San Miguel

, and

P.

 

Tartaglia

, “,”  , – ();

R.

 

Kapral

and

P.

 

Mandel

, “,”  , – ();

G.

 

Broggi

,

A.

 

Colombo

,

L. A.

 

Lugiato

, and

P.

 

Mandel

, “,”  , – ();

C.

Van den Broeck

and

P.

Mandel

, “,” , – ().

H.

Zeghlache

,

P.

Mandel

, and

C.

Van den Broeck

, “,” , – ().

P.

 

Mandel

and

T.

 

Erneux

, “,”  , – ();

M.

 

Lefebvre

,

D.

 

Dangoisse

, and

P.

 

Glorieux

, “,”  , – ();

L.

 

Fronzoni

,

F.

 

Moss

, and

P. V. E.

 

McClintock

, “,”  , – ();

W.

 

Scharpf

,

M.

 

Squicciarini

,

D.

 

Bromley

,

C.

 

Green

,

J. R.

 

Tredicce

, and

L. M.

 

Narducci

, “,”  , – ();

M. C.

 

Torrent

and

M. San

 

Miguel

, “,”  , – ();

F. T.

 

Arecchi

,

W.

 

Gadomski

,

R.

 

Meucci

, and

J. A.

 

Roversi

, “,”  , – ();

S. T.

 

Vohra

,

F.

 

Bucholtz

,

K. P.

 

Koo

, and

D. M.

 

Dagenais

, “,”  , – ();

D.

 

Bromley

,

E. J.

 

D’Angelo

,

H.

 

Grassi

,

C.

 

Mathis

, and

J. R.

 

Tredicce

, “,”  , – ();

S. T.

Vohra

,

L.

Fabiny

, and

K.

Wiesenfeld

, “,” , – ().

V. N.

 

Chizhevsky

and

P.

 

Glorieux

, “,”  , – ();

V. N.

 

Chizhevsky

,

R.

 

Corbalán

, and

A. N.

 

Pisarchik

, “,”  , – ();

K.

Pyragas

, “,” , – ().

P. Mandel, Theoretical Problems in Cavity Nonlinear Optics (Cambridge U.P., New York, 1997).

J. Y.

Bigot

,

A.

Daunois

, and

P.

Mandel

, “,” , – ().

The laser is a modulated visible laser diode kit, Thorlabs, Model S1031, $≈$250.$ A stabilized power supply can be purchased for this system from the same manufacturer. The modulating signal is coupled, from a standard function generator, directly into the laser circuit with a jack connector.

High Speed Silicon Detector, 1 GHz bandwidth, Model DET 210 by Thorlabs, $≈$100.$

This system is no longer sold by Thorlabs and we do not know of another company selling this device. A possible replacement for this laser system could be the PMA Laser Module by Power Technology, 〈www.powertechnology.com〉: a laser module with built-in analogic modulation and the protections which are necessary to make it a useful student lab tool.

The spontaneous emission, always present in measurable amounts in semiconductor lasers, is not considered here. In practice, no special operation is required, because the spontaneous contribution remains hidden in the trace noise of the oscilloscope when the vertical amplification is correctly chosen for the visualization of the lasing output.

The logarithmic functions are natural logarithms throughout the paper.

A discussion for class B lasers (Ref. 19) can be found in Sec. V.A of Ref. 20 (read the commentary to Figs. 6 and 7 in that section).

J. R.

Tredicce

,

F. T.

Arecchi

,

G. L.

Lippi

, and

G. P.

Puccioni

, “,” , – ().

G. L.

Lippi

,

L. M.

Hoffer

, and

G. P.

Puccioni

, “,” , – ().

The horizontal scale used in Fig. 6(b), chosen to display a whole period of the triangular driving, does not possess enough resolution to show the detail of the transition: an overshoot of the intensity beyond the triangular shape, with damped oscillations. This can only be seen by triggering around $V*$ and expanding the scale. However, one can recognize an indication of a more complex evolution from the jagged trace at the sharp front.

We remark that some students, even very bright ones, will oscillate between considering the explanation obvious, when looking at the theory, and totally wrong, when sitting in front of the experiment. It may take some time and patient discussion to convince them of the fact that the experimental results truly agree with the theoretical interpretation and that their intuition and some of the experimental visualizations suggest the wrong conclusion.

With digital oscilloscopes that provide the derivative of the signal, it is possible to use this feature (when the detector signal is not too noisy), to determine the “end-time” for the measurement of the delay. The derivative changes sign at the inflection point of the switch-up, a point that depends directly on the signal shape rather than on a threshold level set by the user. The advantage of this measurement technique, relative to the mid-height reference that we have discussed so far, is that the oscilloscope provides the end-time information without further manipulation by the user; its disadvantage is that if the signal is too noisy, then the information coming from the derivative may be difficult to exploit. In the tests we performed, however, the zero crossing of the intensity derivative always proved to be a reliable indicator.

D.

Bromley

,

E. J.

D’Angelo

,

H.

Grassi

,

C.

Mathis

,

J. R.

Tredicce

, and

S.

Balle

, “,” , – ().

This upward shift in the crossing point to nonzero intensity values is related to the fact that due to the continuous sweep, the laser maintains a certain degree of memory of its state at the previous instant. As a consequence, the initial slope of the laser intensity versus time is smaller than it ought to be, and the extrapolated straight lines meet at a nonzero vertical coordinate. A discussion of the various consequences on the output intensity of a class-B laser of sweeping the pump can be found in Ref. 24. As an illustration, the flatter slope in the initial phases of lasing for growing pump values can be recognized clearly in Fig. 6(b), where the linear response regime appears to be doubled; the upsweep corresponds to the lower of the two curves. In Fig. 8 the same effect is less visible because of the scale on which the curves are plotted.

G. P. Agrawal and N. K. Dutta, Long-Wavelength Semiconductor Lasers (Van Nostrand-Reinhold, New York, 1986).

W. Scharpf, M. Squicciarini, D. Bromley, C. Green, J. R. Tredicce, and L. M. Narducci, in Ref. 9.

In a semiconductor laser, as in all Class B lasers (Ref. 19), the excess population inversion necessary to initiate the optical amplification process gives rise to an additional macroscopic time delay. A discussion of this contribution, which is nonlinear and is expected to deform the distribution of points measured in Fig. 7, is beyond the scope of this paper. An understanding of its physical origin can be gained from Sec. V.A of Ref. 20.

P.

 

Jung

,

G.

 

Gray

,

R.

 

Roy

, and

P.

 

Mandel

, “,”  , – ();

J.

Zemmouri

,

B.

Ségard

,

W.

Sergent

, and

B.

Macke

, “,” , – ().