Lugiato–Lefever equation

The numerical models of lasers and the most of nonlinear optical systems stem from Maxwell–Bloch equations (MBE).

results in "mean-field" Suchkov-Letokhov equation (SLE) describing the nonstationary evolution of the transverse mode pattern.

[3] The model usually designated as Lugiato–Lefever equation (LLE) was formulated in 1987 by Luigi Lugiato and René Lefever [4] as a paradigm for spontaneous pattern formation in nonlinear optical systems.

The case of longitudinal patterns is intrinsically linked to the phenomenon of “Kerr frequency combs” in microresonators, discovered in 2007 by Tobias Kippenberg and collaborators,[8] that has raised a very lively interest, especially because of the applicative avenue it has opened.

denotes time, is linearly polarized and therefore can be treated as a scalar, we can express it in terms of the slowly varying normalized complex envelope

Some years later than,[4] there was the formulation of the longitudinal LLE, in which diffraction is replaced by dispersion.

From a mathematical viewpoint, the LLE amounts to a driven, damped, detuned nonlinear Schroedinger equation.

In some papers dealing with the longitudinal case one considers dispersion beyond the second order, so that Eq.

If we refer to the modes of the empty cavity, in the case of the uniform stationary solutions described by Eq.

(1), in the case of these stationary solutions E corresponds to a singlemode plane wave

(1) and (2) gives rise to four-wave mixing (FWM), which can generate other modes, so that the envelope

displays a spatial pattern: in the transverse plane in the case of Eq.

It is possible to generate also isolated intensity peaks,[13] that are called cavity solitons (see Fig.

Since cavity solitons can be “written”and “erased” one by one in the transverse plane like in a blackboard, they are of great interest for applications to optical information processing and telecommunications.

The FWM can give rise, for example, to processes in which pairs of photons of the longitudinal mode quasi-resonant with

[14] It is important to note that the instability which originates longitudinal patterns and cavity solitons in the LLE is a special case of the multimode instability of optical bistability, predicted by Bonifacio and Lugiato in [15] and first observed experimentally in.

This technique, introduced by Theodor Haensch[17] and John Hall[18] using mode-locked lasers, has led to myriad applications.

The work [8] demonstrated the realization of broadband optical frequency combs exploiting the whispering gallery modes activated by a CW laser field injected into a high-Q microresonator filled with a Kerr medium, that gives rise to FWM.

Since that time Kerr frequency combs (KFC), whose bandwidth can exceed an octave with repetition rates in the microwave to THz frequencies, have been generated in a wide variety of microresonators; for reviews on this subject see e.g.[19][20] They offer substantial potential for miniaturization and chip-scale photonic integration, as well as for power reduction.

Today KFC generation is a mature field, and this technology has been applied to several areas, including coherent telecommunications, spectroscopy, atomic clocks as well as laser ranging and astrophysical spectrometer calibration.

A key impetus to these developments has been the realization of Kerr cavity solitons in microresonators,[21] opening the possibility of utilizing Kerr cavity solitons in photonic integrated microresonators.

The longitudinal LLE (2) provides a spatio-temporal picture of the involved phenomena, but from the spectral viewpoint its solutions correspond to KFC.

The link between the topic of optical KFC and the LLE was theoretically developed in.

[21][22][23][24][25] These authors showed that the LLE (or generalizations including higher order dispersion terms) is the model which describes the generation of KFC and is capable of predicting their properties when the system parameters are varied.

The spontaneous formation of spatial patterns and solitons travelling along the cavity described by the LLE is the spatiotemporal equivalent of the frequency combs and governs their features.

The rather idealized conditions assumed in the formulation of the LLE, especially the high-Q condition, have been perfectly materialized by the spectacular technological progress that has occurred in the meantime in the field of photonics and has led, in particular, to the discovery of KFC.

results in the "mean-field" SLE equation where longitudinal derivative is absent:

Rigorous procedure [26] demonstrated that this precursor of LLE is applicable to modeling of the nonstationary evolution of the transverse mode pattern in the Disk laser (1966) .

The two photons that, as shown in Fig.4, are emitted in symmetrically tilted directions in the FWM process, are in a state of quantum entanglement: they are precisely correlated, for example in energy and momentum.

For instance, the difference between the intensities of the two symmetrical beams is squeezed, i.e. exhibits fluctuations below the shot noise level;[27] the longitudinal analogue of this phenomenon has been observed experimentally in KFC.

Figure 1. A light beam propagates along the direction. and are the transverse directions
Figure 2. Top view of ring cavity
Figure 3. Stationary curve of the normalized output intensity as a function of the normalized input intensity for . The stationary states in the segment with negative slope are unstable. The arrows show the hysteresis cycle which is covered when is increased and then decreased.
Figure 4. A four-wave mixing process in which two photons with are absorbed and two photons with and are emitted. , and are the components of the wave-vectors.
Figure 5. A typical pattern configuration that arises in the transverse planes in the output is a hexagonal pattern.
Figure 6. A typical Kerr cavity soliton in the transverse plane showing a bright peak in the dark background with diffraction rings.
Figure 7. Example of longitudinal pattern that travels along the cavity with the velocity of light in the medium and gives rise to a periodic sequence of pulses in the output.
Figure 8. Longitudinal Kerr cavity solitons.