1980 : The Ikeda map

Christophe LETELLIER
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Kensuke Ikeda

The transmitted light by a Fabry-Pérot cavity containing a two level absorbermay produce some complex fluctuations. Kensuke Ikeda (Ristumeikan University, Kusutsu, Japan) found that when the detuning of the incident light with the absorber is taken into account, the transmitted field presents a chaotic behavior [1]. The device of the ring cavity is shown in Fig. 1. EI is the incident field, ET the transmitted field and, ER the reflected one. L is the length of the sample cell containing the two level absorber. Mirrors 1 and 2 have a partial reflectivity R while the mirrors 3 and 4 have total reflectivity. The difference equations proposed by Ikeda were then normalized  [2].

When the saturable absorption is ignored, the field  g= E / \Delta is governed by the difference equation [3]

g_{n+1} = a + R \, g_n \, e^{i \left[ \displaystyle \phi - \frac{p}{1 + |g_n|^2 \right]} \, .

Parameters of this equation are

  • the dimensionless input amplitude  a = \sqrt{T} \frac{E_{\rm I}}{\Delta} \, where T=1-R is the transmission function ;
  • the laser empty-cavity detuning  \phi = k L ;
  • p = \frac{\alpha_0 L}{2 \Delta} where \alpha_0 L is the linear absorption per pass.
  •  \Delta = \frac{\omega - \omega_{\rm ab}}{\gamma_\perp} is the laser two-level atom dimensionless detuning.

The difference equation on the dimensionless intracavity complex field amplitude g can be rewritten as a discrete map according to

    x_{n+1} = R + C_2 (x_n \cos \tau_n - y_n \sin \tau_n) \\[0.1cm] 
    y_{n+1} = C_2 (x_n \sin \tau_n + y_n \cos \tau_n) \, 


\tau_n = C_1 - \frac{C_3}{1+x_n^2+y_n^2} \, .

With parameter values R=1, C1=0.4, C2=0.9 and C3=6, a chaotic behavior is obtained as shown in Fig. 2. With these parameter values, the so-called Ikeda map can be viewed as a nice example illustrating the "Horseshoe map" introduced by Stephen Smale [4].

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Fig. 2. Chaotic behavior solution to the Ikeda map.

[1] K.Ikeda, Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system, Optical Communications, 30, 257-261, 1979.

[2] K. Ikeda, H. Daido & O. Akimoto, Optical turbulence : chaotic behavior of transmitted light from a ring cavity, Physical Review Letters, 45, 709-712, 1980

[3] S. M. Hammel, C. K. R. T. Jones & J. V. Moloney, Global dynamical behavior of the optical field in a ring cavity, Journal of the Optical Society of America B, 2 (4), 552-564, 1985.

[4] S. Smale, Differentiable dynamical systems. I. Diffeormorphisms, Bulletin of American Mathematical Society, 73, 747-817, 1967.

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