Fresnel number

Because of the high gain and wide gain bandwidth of RGH lasers, high-Fresnel-number stable resonators, which are... 

Output resonance is reached (for a droplet with a fixed radius) when specific wavelengths within the inelastic emission profile correspond to MDR s with different n,t values. For those wavelengths, the droplet can be envisioned as an optical cavity with a large Fresnel number and Q-factors which are dependent on the specific n and I values. The portion of the inelastic radiation detected is that allowed to "leak" out of the droplet cavity. However, it is the internal field distributions of the electromagnetic waves at X, and X, which are best described by spherical harmonic functions and not by plane waves as in the case of an extended medium, that affect the nonlinear optical interactions. Such interactions in droplets can be illustrated by several well known examples in nonlinear spectroscopy of liquids in an optical cell. 

Fresnel number of an aperture Parameter related to the deviation of the field behind an aperture from the geometrical optics field. 

FIGURE 4 Diffraction pattern of a ring-shaped aperture with internal diameter 0.8 times the external one, illuminated by a plane wave at a distance corresponding to a Fresnel number of 15. The three-dimensional plot was obtained by using an improved fast Fourier transform algorithm. [From Luchini, P. (1984). Comp. Phys. Commun. 31,303.]... 

An important parameter in diffraction theory is the so-called Fresnel number of a diffracting system ... 

It should be noted that these two definitions are oversimplified in reality, the boundary between the Fraunhofer and Fresnel approximations is not so clear-cut, since the Fresnel-number criterion is a rather crude delimiter. Regardless, for conceptual reasons and simplicity, we only will discuss Fraunhofer diffraction here and restrict the principal treatment to the two most commonly encountered (useful) diffraction objects, namely an elongated straight slit (e.g. as encountered in spectrometers) and a circular aperture (pin holes, diaphragms, any circular lens). 

F Fresnel number min minimum sensitizer-activator distance... 

Equation (84) assumes that the diffiaction angle Aq/5 equals the geometric solid angle Sll which corresponds to a Fresnel number... 

It can be shown [5.18] that all resonators with plane mirrors that have the same Fresnel number also have the same diffraction losses, independent of the special choice of a, d, or X. 

The general integral equation (5.28) cannot be solved analytically, therefore one has to look for approximate methods. For two identical plane mirrors of quadratic shape (2a), (5.28) can be solved numerically by splitting it into two one-dimensional equations, one for each coordinate x and y, if the Fresnel number N = a l(dX) is small compared with dld), which means if a < id X) . Such numerical iterations for the infinite strip resonator have been performed by Fox and Li [5.19]. They showed that stationary field configurations do exist and computed the field distributions of these modes, their phase shifts, and their diffraction losses. 

It can be shown [5.1,5.24] that in nonfocal resonators with large Fresnel numbers N the field distribution of the fundamental mode can also be described by the Gaussian profile (5.32). The confocal resonator with d = R can be replaced by other mirror configurations without changing the field configurations if the radius Rf of each mirror at the position zo equals the radius R of the wavefront in (5.37) at this position. This means that any two surfaces of constant phase can be replaced by reflectors, which have the same radius of curvature as the wave front - in the approximation outlined above. 

The diffraction losses of a resonator depend on its Fresnel number N = a /dX (Sect 5.2.1) and also on the field distribution A(x, y,z = d/2) at the mirror. The fundamental mode, where the field energy is concentrated near the resonator axis, has the lowest diffraction losses, while the higher transverse modes, where the field amplitude has larger values toward the mirror edges, exhibit large diffraction losses. Using (5.31) with z = djl the Fresnel number j dX) can be expressed as... 

The confocal resonator with the smallest spot sizes at a given mirror separation d according to (5.39) also has the lowest diffraction losses per round-trip, which can be approximated for circular mirrors and Fresnel numbers N > hy 15.1]... 

Fig. 5.11. Diffraction losses of some modes in a confocal and in a plane-mirror resonator, plotted as a function of the Fresnel number N...

In Fig. 5.32, the ratio Kio/koo of the diffraction losses for the TEMio and the TEMqo modes in a symmetric resonator with = g2 = g is plotted for different values of g as a function of the Fresnel number N. From this diagram one can obtain, for any given resonator, the diameter 2a of an aperture... 

Figure 5.6 a Fresnel zones on mirror Mi, as seen from the center A of the other mirror M2 b the three regions of dja with the Fresnel number A > 1, A = 1, and A < 1... 

In order to estimate the magnitude of diffraction losses let us make use of a simple example. A plane wave incident onto a mirror with diameter la exhibits, after being reflected, a spatial intensity distribution that is determined by diffraction and that is completely equivalent to the intensity distribution of a plane wave passing through an aperture with diameter la (Fig. 5.5). The central diffraction maximum at 0 = 0 lies between the two first minima at d = Xlla (for circular apertures a factor 1.2 has to be included, see, e.g., [306]). About 16 % of the total intensity transmitted through the aperture is diffracted into higher orders with 0 > X/la. Because of diffraction the outer part of the reflected wave misses the second mirror M2 and is therefore lost. This example demonstrates that the diffraction losses depend on the values of a, d, X, and on the amplitude distribution A x,y) of the incident wave across the mirror surface. The influence of diffraction losses can be characterized by the dimensionless Fresnel number... 

This states that the diffraction losses of a plane mirror resonator can be neglected if the Fresnel number Np is larger than the number n of transits through the resonator. 

Let us first consider the selection of transverse modes. In Sect. 5.2.3 it was shown that the higher transverse TEM , modes have radial field distributions that are less and less concentrated along the resonator axis with increasing transverse order n or m. This means that their diffraction losses are much higher than those of the fundamental modes TEMoo (Fig. 5.12). The field distribution of the modes and therefore their diffraction losses depend on the resonator parameters snch as the radii of curvature of the mirrors Ri, the mirror separation d, and, of conrse, the Fresnel number (Sect. 5.2.1). Only those resonators that fiilfiU the stabiUty condition [291, 314]... 

In Fig. 5.34, the ratio yio/yoo of the diffraction losses for the TEMio and the TEMoo modes in a symmetric resonator with gi = gi = g is plotted for different values of g as a function of the Fresnel number Np. From this diagram one can obtain, for any given resonator, the diameter 2a of an aperture that suppresses the TEMio mode but still has sufficiently small losses for the fundamental TEMoo mode with beam radius w. In gas lasers, the diameter 2a of the discharge tube generally forms the limiting aperture. One has to choose the resonator parameters in such a way that a 3w/2 because this assures that the fundamental mode nearly fills the whole active medium, but still suffers less than 1 % diffraction losses (Sect. 5.2.6). 

Figure 5.34 Ratio yio/yoo of diffraction losses for the TEMio and TEMqo modes in symmetric resonators as a function of the Fresnel number Np for different resonator parameters g = -d/R...

According to Fig. 5.12 the Fresnel number Np should be smaller than 0.8, in order to increase the losses of the TEMio mode above 10 %. The Fresnel number is defined as Ap = where w. is the beam waist of the fundamental mode... 

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