Cavity modes transverse mode

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. 

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. 

In order to realize the extremely small linewidths, attainable because of the temporal coherence of induced emission, care must be taken to ensure that the threshold condition for induced oscillation is fulfilled for only one mode. The transverse modes can be eliminated by an appropriate choice of the cavity dimensions, introducing... 

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). 

The CO2 lasers were also investigated in connection with chaotic behavior, and here we mention the most important papers in the field. The chaotic behavior associated with a transverse mode structure in a cw CO2 laser was observed in 1985 [40]. In the CO2 laser with elastooptically modulated cavity length, a period doubling route to chaos was also found [41]. 

The phenomenon of unstable combustion results from a self-amplifying interaction between combustion processes and the. acoustic oscillations of the gas within the rocket motor. The unexpected appearance of combustion instability in any rocket generally terminates its mission thru motor case rupture from overpressure, disruption of guidance systems by severe vibration, or thrust malalignment. Both axial mode and transverse mode instabilities are observed (Ref 45). In the case of the transverse mode the characteristic wave time is usually that required to travel radially around the proplnt cavity whereas the characteristic time for the axial mode is the time for the wave to travel from end to end in the combustion chamber. Double-base proplnts predominantly are prone to transverse wave instabilities and infrequently to those in the axial mode, while composite proplnts appear to go unstable mostly in the axial mode. In the case of transverse instability chamber pressures have been known o double whereas in axial mode instabilities artificially induced by pulsing the chamber pressure at lOOOpsi, the pressure excursion may reach 300—400psi. A review of recent theoretical combustion modeling for combustion instability has been made by Price (Ref 47)... 

In these equations and hereafter we use simplified symbols for quantities referring to the P-branch line v, 7—l- c—I, /. Namely, Xvy instead of Xu,y-i etc. The second term in (2) represents the rate of spontaneous emission into the oscillating cavity modes. 1 is approximately the effective solid angle subtended by the mirrors after several reflections. (Alternatively, e is the fraction of stable transverse modes.) After threshold eS j is negligible, eS j- Xvj- i i i " i important only before threshold —as a source of noise photons to trigger-on the lasing process. The spontaneous emission terms in (1) are given by =A j+iN y, A j is the Einstein coefficient. In infrared lasers where typically A ) 10s. S, ... 

Diode lasers have an extremely small cavity. Most lasers in the power range below 200 mW (CW) are single-mode lasers, i.e. the height and width are so small (a few pm) that only one transversal mode is excited. This implies that the radiation ean, in prineiple, be focused into a diffraction-limited spot. However, beeause the eavity is only a few pm long, the light is emitted over a wide angle. The general beam profile of a laser diode is shown in Fig. 7.1. 

If the reflectivity is very high, diffraction losses may become dominant, in particular for cavities with a large separation d of the mirrors. Since the TEMqo mode has the lowest diffraction losses, the incoming laser beam has to be mode-matched by a lens system to excite the fundamental mode of the resonator but not the higher transverse modes. Similar to intracavity absorption, this technique takes advantage of the increased effective absorption length Leff = LUX — R), because the laser pulse traverses the absorbing sample 1/(1 - R) times. 

In a rectangular cavity, electromagnetic waves are classified as transverse electric (TE) or transverse magnetic (TM) modes. AU the field combinations can be obtained by the superposition of TE and TM modes. TE modes are defined as the waves that have no electric field component in a defined propagation direction. In this discussion, the propagation direction is assumed to be the f-direction. Similarly, TM modes have no magnetic field component in the f - direction. By assuming a cavity with dimensions, a X b X d in the x-,y-, and z-directions, respectively, the frequencies at which nontrivial solutions of the Helmholtz Equation occur are... 

FIGURE 4 Two distinct transverse modes oscillating over different spatial regions of the laser optical cavity. 

This discussion barely touches on the topics of cavity modes and high-resolution laser technology. Laser light also propagates in transverse cavity modes [5], which can introduce fine structure superimposed on the axial mode frequencies if lasing is not confined to the lowest-order TEMqo transverse mode. Ring dye lasers are currently the most widely used frequency-stabilized high-resolution lasers. 

The spatial profile of a laser beam at the exit of the laser aperture is determined by the geometry of the laser cavity and the propagation of light within the cavity. The transverse electromagnetic modes are represented by TEM ... 

In such a cavity, the transverse (horizontal) size of the atomic mode can be expressed in terms of the distance L from the surface of the mirror to the classical turning point, which determines the length of the cavity. For estimation purposes, the gradient force potential can be taken to be stepped near the surface of the mirror. The shape of the surface of the mirror in the simplest approximation can he treated as a paraboloid of revolution,... 

Thus we arrive at the important conclusion that a finite chain having N +1 particles has N normal modes of longitudinal vibration. The same arguments used above can be generalized to three dimensions and it can be shown that two transverse modes of vibration are also possible. Therefore, a chain of N +1 atoms can have 3N discrete modes of vibration. Thus the fc-vector, which was continuous for an infinite chain, is now quantized into 3N discrete states. The energy of each of these states is assigned to be hoj. Because of the close resemblance of these quantized vibrational states to photons in a cavity, these quantized waves are called phonons. 

The fractional power loss per transit, 1 - Yq, obtained by Fox and Li (1961) for the lowest-order transverse modes of a plane mirror cavity having circular apertures, is shown in Fig.12.3 as a function of the Fresnel number... 

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... 

Fig.12.5. Definitions of the parameters, R2, and L for an optical cavity and the radii characterizing the Gaussian field distribution of the laser transverse modes.

This problem demonstrates the link between the widths of the cavity response and the transmission peaks of this well-known interferometer. However, the fringe pattern observed with a Fabry-Perot etalon should not be confused with the mode pattern of a laser having a plane-parallel resonator. The Fabry-Perot etalon is normally used with the plates so close together that all the transverse modes of the corresponding optical cavity are virtually degenerate in frequency. The plane wavefronts assumed in the discussion of the theory of the etalon are composed of an infinite sum over the transverse modes of the cavity. 

Calculate the frequency separation between the transverse modes of an almost plane-parallel cavity in which two concave mirrors of 50 m radius of curvature are separated by a distance of 1 m. (Ans ... 

There are differences between photons and phonons while the total number of photons in a cavity is infinite, the number of elastic modes m a finite solid is finite and equals 3N if there are N atoms in a three-dimensional solid. Furthennore, an elastic wave has tliree possible polarizations, two transverse and one longimdinal, in contrast to only... 

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