Ikeda Map: Chaos in a Nonlinear Optical Cavity

Light bouncing between two mirrors

The Ikeda map models a real physical device: a ring cavity, formed by mirrors, that is fed by a continuous laser beam and filled with a nonlinear medium whose refractive index depends on the local light intensity. Kensuke Ikeda derived a discrete map in 1979 describing how the complex electric field inside the cavity evolves from one round trip to the next. In its common two-dimensional real form the map is:

t = 0.4 - 6 / (1 + x^2 + y^2)
x_next = 1 + u * (x*cos(t) - y*sin(t))
y_next = u * (x*sin(t) + y*cos(t))

Here u is a pump-strength parameter set by the incoming laser intensity and cavity losses, and the angle t is not constant -- it depends on the current field intensity x^2 + y^2, which is exactly the intensity-dependent phase shift that makes the medium nonlinear. A field that is strong at one point in the cavity gets rotated by a different angle than a field that is weak, and that self-referential twisting, repeated every round trip, is the source of the chaos.

From a single fixed point to a spiral of chaos

At low pump strength u, the map settles to a single stable fixed point: the cavity field reaches a steady intensity and stays there, round trip after round trip. As u increases, that fixed point loses stability through a sequence of bifurcations -- period doubling, much like the logistic map -- and eventually the trajectory never settles down at all, instead wandering forever over a spiral-shaped strange attractor. The spiral shape is a direct fingerprint of the map's own construction: the intensity-dependent rotation angle t means points at different distances from the origin get rotated by different amounts, winding the attractor into the characteristic multi-armed spiral you see traced out on screen.

Why optical physicists care about a bounded rotation

Ikeda's original motivation was to explain optical bistability and instability observed in real ring-cavity laser experiments: under strong enough pumping, the transmitted light intensity does not settle to a steady value but instead flickers, oscillates, or becomes fully chaotic, and this map reproduces that transition parameter-for-parameter with the real optics. It became one of the first clean experimental confirmations that deterministic chaos -- not equipment noise -- can be the actual cause of an irregular output beam, a result that mattered for the design of optical switches and bistable devices built around exactly this kind of nonlinear cavity.

A cousin of the Henon map, wearing polar clothes

Structurally, the Ikeda map shares the Henon map's recipe of one nonlinear stretch (here, the intensity-dependent rotation) followed by a linear contraction (the pump term u scaling the rotated field back down), and both produce attractors with fractal, non-integer dimension born from repeated stretch-and-fold cycles. The visible difference is entirely the geometry of the nonlinearity: Henon's quadratic term folds the plane along a parabola, while Ikeda's rotation angle folds it along a spiral, which is why the two attractors look so different even though the underlying stretch-fold mechanism generating the chaos is the same idea.

Frequently asked questions

Try it live

Everything above runs in your browser — open Ikeda Map and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.