Laser cavity modes

The cavity of a laser may resonate in various ways during the process of generation of radiation. The cavity, which we can regard as a rectangular box with a square cross-section, has modes of oscillation, referred to as cavity modes, which are of two types, transverse and axial (or longitudinal). These are, respectively, normal to and along the direction of propagation of the laser radiation. 

The transverse modes are labelled TEM , where TEM stands for transverse electric and magnetic (field) m and I are integers that refer to the number of vertical and horizontal 

The variety of possible axial modes is generally of greater consequence. The various frequencies possible are given by Equation (9.2) so that the separation Av of the axial modes is given by 

For a cavity length of, for example, 50 cm the axial mode separation is 300 MFIz 

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The frequency of a single-mode laser inside the spectral gain profile of its active medium is mainly determined by the eigenfrequency of the active laser cavity mode. Therefore any instability of resonator parameters, such as variation of cavity length, mirror vibrations or thermal drifts of the refractive index will show up as frequency fluctuations and drifts of the laser line. 

Boyd and Gordon (1961) showed that explicit analytical expressions for the electric field distribution in the transverse modes may be obtained by allowing the limits of integration in equation (12.4) to tend to infinity. This is a valid approximation for stable resonators having Fresnel numbers which satisfy tlue condition F 1, i.e. the size of the mirror aperture is large compared with the size of the laser cavity mode. The condition that stable solutions exist for radiation which is propagated back and forth within the resonator is then equivalent to requiring that the field... 

Although 0-switching produces shortened pulses, typically 10-200 ns long, if we require pulses in the picosecond (10 s) or femtosecond (10 s) range the technique of mode locking may be used. This technique is applicable only to multimode operation of a laser and involves exciting many axial cavity modes but with the correct amplitude and phase relationship. The amplitudes and phases of the various modes are normally quite random. 

Spatial Profiles. The cross sections of laser beams have certain weU-defined spatial profiles called transverse modes. The word mode in this sense should not be confused with the same word as used to discuss the spectral Hnewidth of lasers. Transverse modes represent configurations of the electromagnetic field determined by the boundary conditions in the laser cavity. A fiiU description of the transverse modes requires the use of orthogonal polynomials. 

The linewidth Afl of such a single mode laser is determined by the bandwidth A of the laser cavity (which is inversely proportional to its g-factor), the laser frequency v and the output power P at this frequency. 

The laser interferometer consists of two coupled resonators, one containing the laser, the other the plasma under investigation (Fig. 10). The laser radiation, reflected back from mirror A/s, which contains phase information about the refractive index of the plasma, interferes with the laser wave in cavity A, resulting in an amplitude modulation of the laser output 267). This modulation can be related to the refractive index and therefore to the plasma frequency and electron density. With a curved rather than a planar mirror, the sensitivity can be increased by utilizing transverse cavity modes 268). 

In the case of a common lower level, the second absorption transition would show this narrowing effect when probed with a tunable monochromatic laser line. This example can be realized if atoms or molecules in a magnetic field are pumped by a laser, oscillating simultaneously on two cavity modes 324). if the Zeeman splitting of the probe equals the mode spacing of the laser, both transitions are pumped simultaneously and each laser mode selectively eats... 

These structures were recorded by a vectorial focal spot scanning in a spiral-by-spiral method rather in a raster layer-by-layer mode using a PZT stage. Such spiral structures fabricated in SU-8 have optical spot bands in near-lR [24], telecommunication [25], and 2-5 pm-IR region [26] or can be used as templates for Si infiltration [11]. It is obvious, that direct laser scanning is well suited for defect introduction into 3D PhC, as demonstrated in resin where a missing rod of a logpUe structure resulted in the appearance of a cavity mode in an optical transmission spectriun [27]. 

Since the discovery of lasers it has been known that a derivation of time-dependent equations governing interaction of molecules with electromagnetic cavity modes leads to the so-called spontaneous instabilities. These laser instabilities were also observed experimentally—even for the first laser built by Maiman in 1960. A random, periodic, or quasiperiodic train of spikes in a laser generation is a fundamental instability due to nonlinearity of laser equations. A comprehensive review of this specific laser-related topics was published in 1983 [14]. 

The case of a frequency mismatch between laser pumps and cavity modes was investigated [83], and for the first time, chaos in SHG was found. When the pump intensity is increased, we observe a period doubling route to chaos for Ai = 2 = 1. Now, for/i = 5.5, Eq. (3) give aperiodic solutions and we have a chaotic evolution in intensities (Fig. 5a) and a chaotic attractor in phase plane (Imaj, Reai) (Fig. 5b). 

The limit cycle is an attractor. A slightly different kind occurs in the theory of the laser Consider the electric field in the laser cavity interacting with the atoms, and select a single mode near resonance, having a complex amplitude E. One then derives from a macroscopic description laced with approximations the evolution equation... 

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