Frequency and spatial properties of laser radiation

Many other configurations and combinations of mirrors are possible but will not be described here. However, it should be noted that long-radius-hemispherical resonators, consisting of a curved mirror with ri 2L and a plane mirror (f2 = oo), are commonly used in high-power CW lasers. They constitute a compromise between mode volume, ease of alignment (resonator stabdity) and relatively low diffraction losses. 

A propagating electromagnetic wave, of which light is one type, must satisfy the complex wave equation 

The function U is the complex amplitude of the wave, and takes the general form 

If confined to a resonator, the wave has to obey the resonance boundary conditions the standing waves in the resonator are called modes. Particular functions U describe the different modes. Because of the nature of laser resonators, which normally are much longer than wide, one may conveniently split the description into longitudinal (often by convention the z-axis) and transverse (the x- and y-axes) components, or modes, with respect to the optical axis of the resonator. As it turns out, to a good approximation, one can use the longitudinal mode components to derive the resonance frequencies in the resonator these then are loosely addressed as longitudinal (frequency ) modes. The transverse mode components can be used to calculate the lateral intensity distribution of a laser beam in general one then speaks of transverse (intensity) modes. 

Let us first address the frequency modes and, for simplicity, restrict the discussion to those waves in the direction of the z-axis only. Thereafter, a brief description of the observed beam intensity patterns will be given. 

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