Kerr-lens mode-locking (KLM) is a method of

mode-locking Mode locking is a technique in optics by which a laser can be made to produce pulses of light of extremely short duration, on the order of picoseconds (10−12 s) or femtoseconds (10−15 s). A laser operated in this way is sometimes r ...

laser A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word "laser" is an acronym for "light amplification by stimulated emission of radiation". The fir ...

s via the

nonlinear optical Nonlinear optics (NLO) is the branch of optics that describes the behaviour of light in ''nonlinear media'', that is, media in which the polarization density P responds non-linearly to the electric field E of the light. The non-linearity is typic ...

Kerr effect The Kerr effect, also called the quadratic electro-optic (QEO) effect, is a change in the refractive index of a material in response to an applied electric field. The Kerr effect is distinct from the Pockels effect in that the induced index chang ...

. This method allows the generation of pulses of light with a duration as short as a few femtoseconds. The optical

is a process which results from the nonlinear response of an optical medium to the

electric field An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field fo ...

of an

electromagnetic wave In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visib ...

. The

refractive index In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, or ...

of the medium is dependent on the field strength. Because of the non-uniform power density distribution in a

Gaussian beam In optics, a Gaussian beam is a beam of electromagnetic radiation with high monochromaticity whose amplitude envelope in the transverse plane is given by a Gaussian function; this also implies a Gaussian intensity (irradiance) profile. This ...

(as found in laser resonators) the refractive index changes across the beam profile; the refractive index experienced by the beam is greater in the center of the beam than at the edge. Thus a rod of an active Kerr medium functions as a lens for high intensity light. This is called

self-focusing Self-focusing is a non-linear optical process induced by the change in refractive index of materials exposed to intense electromagnetic radiation. A medium whose refractive index increases with the electric field intensity acts as a focusing lens ...

and in extreme cases leads to material destruction. In the laser cavity short bursts of light will then be focused differently from continuous waves. To favor the pulsed mode over continuous-wave, the cavity could be made unstable for continuous-wave operation, but more often a low stability is a by-product of a cavity design putting emphasis on aperture effects. Older designs used a hard aperture, that simply cuts off, while modern designs use a soft aperture, that means the overlap between the pumped region of the gain medium and the pulse. While the effect of a lens on a free laser beam is quite obvious, inside a cavity the whole beam tries to adapt to this change. The standard cavity with flat mirrors and a thermal lens in the laser crystal has the smallest beam width on the end-mirrors. With the additional Kerr lens the width on the end-mirror gets even smaller. Therefore, small end-mirrors (hard aperture) favor pulses. In Ti:Sapphire oscillators telescopes are inserted around the crystal to increase the intensity. For a soft aperture consider an infinite laser crystal with a thermal lens. A laser beam is guided like in a glass fiber. With an additional Kerr lens the beam width gets smaller. In a real laser the crystal is finite. The cavity on both sides features a concave mirror and then a relative long path to a flat mirror. The continuous-wave light exits the crystal end face with a larger beam width and slight divergence. It illuminates a smaller area on the concave mirror, leading to a small beam-width on the way to the flat mirror. Thus

diffraction Diffraction is defined as the interference or bending of waves around the corners of an obstacle or through an aperture into the region of geometrical shadow of the obstacle/aperture. The diffracting object or aperture effectively becomes a s ...

is stronger. Because of the divergence the light is effectively coming from a point farther apart and leads to more convergence after the concave mirror. This convergence is balanced with diffraction. The pulsed light exits the end face with a smaller beam width and no divergence. Thus it illuminates a larger area on the concave mirror and is less convergent afterwards. So both continuous waves and pulsed light fronts are mirrored back onto themselves. A cavity close to a confocal one means to be close to instability, which means the beam diameter is sensitive to cavity changes. This emphasizes the modulation. With a slightly asymmetric cavity prolonging the cavity emphasizes diffraction and even makes it unstable for continuous-wave operation, while staying stable for pulsed operation. The length of the medium used for KLM is limited by

group velocity The group velocity of a wave is the velocity with which the overall envelope shape of the wave's amplitudes—known as the ''modulation'' or ''envelope'' of the wave—propagates through space. For example, if a stone is thrown into the middl ...

dispersion. KLM is used in Carrier envelope offset control.

Starting a Kerr-lens modelocked laser

Initiation of Kerr-lens modelocking depends on the strength of the nonlinear effect involved. If the laser field builds up in a cavity the laser has to overcome the region of continuous-wave operation, which often is favoured by the pumping mechanism. This can be achieved by a very strong Kerr-lensing that is strong enough to modelock due to small changes of the laser field strength (laser field build-up or stochastic fluctuations). Modelocking can also be started by shifting the optimum focus from the continuous-wave operation to pulsed operation while changing the power density by kicking the end mirror of the resonator cavity (though a piezo mounted, synchronous oscillating end-mirror would be more 'turn key'). Other principles involve different nonlinear effects like saturable absorbers and saturable Bragg reflectors, which induce pulses short enough to initiate the Kerr-lensing process.

Modelocking – evolution of the pulse

Intensity changes with lengths of nanoseconds are amplified by the Kerr-lensing process and the pulselength further shrinks to achieve higher field strengths in the center of the pulse. This sharpening process is only limited by the bandwidth achievable with the laser material and the cavity-mirrors as well as the dispersion of the cavity. The shortest pulse achievable with a given spectrum is called the

bandwidth-limited pulse A bandwidth-limited pulse (also known as Fourier-transform-limited pulse, or more commonly, transform-limited pulse) is a pulse of a wave that has the minimum possible duration for a given spectral bandwidth. Bandwidth-limited pulses have a con ...

Chirped mirror A chirped mirror is a dielectric mirror with chirped spaces—spaces of varying depth designed to reflect varying wavelengths of lights—between the dielectric layers (stack). Chirped mirrors are used in applications like lasers to reflect ...

technology allows to compensate for timing mismatch of different wavelengths inside the cavity due to material dispersion while keeping the stability high and the losses low. The Kerr effect leads to the Kerr-lens and

Self-phase modulation Self-phase modulation (SPM) is a nonlinear optical effect of light–matter interaction. An ultrashort pulse of light, when travelling in a medium, will induce a varying refractive index of the medium due to the optical Kerr effect. This variatio ...

at the same time. To a first approximation it is possible to consider them as independent effects.

Applications

Since Kerr-lens modelocking is an effect that directly reacts on the electric field, the response time is fast enough to produce light pulses in the visible and near infrared with lengths of less than 5 femtoseconds. Due to the high electrical field strength focused ultrashort laser beams can overcome the threshold of 10¹⁴ W cm⁻², which surpasses the field strength of the electron-ion bond in atoms. These short pulses open the new field of

ultrafast optics In optics, an ultrashort pulse, also known as an ultrafast event, is an electromagnetic pulse whose time duration is of the order of a picosecond (10−12 second) or less. Such pulses have a broadband optical spectrum, and can be created by m ...

, which is a field of

nonlinear optics Nonlinear optics (NLO) is the branch of optics that describes the behaviour of light in ''nonlinear media'', that is, media in which the polarization density P responds non-linearly to the electric field E of the light. The non-linearity is typica ...

that gives access to a completely new class of phenomena like measurement of electron movements in an atom (attosecond phenomena), coherent broadband light generation ( ultrabroad lasers) and thereby gives rise to many new applications in optical sensing (e.g. coherent laser radar, ultrahigh resolution

optical coherence tomography Optical coherence tomography (OCT) is an imaging technique that uses low-coherence light to capture micrometer-resolution, two- and three-dimensional images from within optical scattering media (e.g., biological tissue). It is used for medical ...

), material processing and other fields like

metrology Metrology is the scientific study of measurement. It establishes a common understanding of units, crucial in linking human activities. Modern metrology has its roots in the French Revolution's political motivation to standardise units in Fran ...

(extremely exact frequency and time measurements).

References and notes

{{reflist # D. E. Spence, P. N. Kean, and W. Sibbett, Opt. Lett. 16, 42(1991). # M. Piche, Opt. Commun. 86, 156(1991). # B. Proctor, E. Westwig, and F. Wise, Opt. Lett. 18, 1654(1993). # V. Magni, G. Cerullo, and S. De Silvestri, Opt. Commun. 101, 365(1993). Nonlinear optics