Diffraction losses

To obtain the absolute sound attenuation in the coal slurry, the diffraction loss, the acoustic mismatch loss, the attenuation due to the Teflon window, and the oil coupling must be calculated. Thus, it is difficult to accurately determine the absolute attenuation. In practice, one measures the relative attenuation with respect to a standard. The attenuation of ultrasonic waves in a solid suspension is attributed to three major factors, namely, scattering, viscosity, and thermal effects. Although the presence of particles affects the fluid viscosity and thermal conductivity, the primary source of attenuation may be due to particle scattering. Hence, one may define the relative attenuation of the HYGAS coal slurry by comparing the slurry attenuation with that of the carrier fluid, i.e., the toluene/benzene mixture. This can be expressed by the equation... 

The energy stored cannot increase ad infinitum, so ultimately the power input to the cavity must equal the sum of the power dissipated resistively and lost by other mechanisms diffraction losses, coupling into the circuit etc. The power lost in the steady state is thus the input power Pin. 

Fig. 7.20 the resolution is greater than 100 dpi, there are 224 x 96 pixels, passively addressed with a line addressing time of 1.5 ms and a driving voltage of 19 V. The optical response is about 50 ms. The refractive index of the grating material is matched to the ordinary refractive index of the nematic liquid crystal to avoid diffractive losses from the grating. ... 

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. 

The experimental setup is shown in Fig. 1.18. The laser pulses are coupled into the resonator by carefully designed mode-matching optics, which ensure that only the TEMoo modes of the cavity are excited. Diffraction losses are minimized by spherical mirrors, which also form the end windows of the absorption cell. If the absorbing species are in a molecular beam inside the cavity, the mirrors form the windows of the vacuum chamber. For a sufficiently short input pulse (Tp < 7r), the output consists of a sequence of pulses with a time separation Tr and with exponentially decreasing intensities, which are detected with a boxcar integrator. For longer pulses (Tp > 7r), these pulses overlap in time and one observes a quasi-continuous exponential decay of the transmitted intensity. Instead of input pulses, the resonator can also be illuminated with cw radiation, which is suddenly switched off at f = 0. 

The so-called concentric resonator (Figure 3.5d), with r - -r2= L, represents the functional opposite of the plane-plane resonator. It is easiest to align, has the lowest diffraction loss and exhibits the smallest mode volume. For example, CW dye lasers incorporate this type of cavity (see Chapter 4.3) because of the short length of the active medium (the dye jet), strong focusing of the pump and resonator beams is necessary to cause efficient stimulated emission and to generate sufficient gain for laser action. However, this type of spherical resonator is not commonly used with any other laser. 

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. 

The overall efficiency of this system is much better than the WDXRF because of less restrictive solid angle losses and the diffraction losses. Although there is only a little absorption of low energy X-rays (since is so low for Be), the performance of the Si (Li) detector is limited at the low energy end... 

The attainable path difference As can be considerably increased by an optical delay line, placed in one arm of the interferometer (Fig. 4.28). It consists of a pair of mirrors, M3, M4, which reflect the light back and forth many times. In order to keep diffraction losses small, spherical mirrors, which compensate by collimation the divergence of the beam caused by diffraction, are preferable. With a stable mounting of the whole interferometer, optical path differences up to 350 m could be realized [4.24], allowing a spectral resolution of v/Av 10 This was demonstrated by measuring the linewidth of a HeNe laser oscillating at v = 5 x 10 Hz as a function of discharge current. The accuracy obtained was better than 5 kHz. 

These open resonators are, in principle, the same as the Fabry-Perot interferometers discussed in Chap. 4 we shall see that several relations derived in Sect. 4.2 apply here. However, there is an essential difference with regard to the geometrical dimensions. While in a common FPI the distance between both mirrors is small compared with their diameter, the relation is generally reversed for laser resonators. The mirror diameter 2a is small compared with the mirror separation d. This implies that diffraction losses of the wave, which... 

For our first example the diffraction losses of the plane FPI are about 5 x 10 and therefore completely negligible, whereas for the second example they reach 25% and may already exceed the gain for many laser transitions. This means that a plane wave would not reach threshold in such a resonator. However, these high diffraction losses cause nonnegligible distortions of a plane wave and the amplitude A(x, y) is no longer constant across the mirror surface (Sect. 5.2.2), but decreases towards the mirror edges. This decreases the diffraction losses, which become, for example, /Diffr 0 01 for N > 20. 

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. 

Resonators with curved mirrors may exhibit much lower diffraction losses than the plane mirror resonator because they can refocus the divergent diffracted waves of Fig. 5.5 (Sect. 5.2.5). 

The mode configurations of open resonators can be obtained by an iterative procedure using the Kirchhoff-Fresnel diffraction theory [5.17]. Concerning the diffraction losses, the resonator with two plane square mirrors can be replaced by the equivalent arrangement of apertures with size 2a) and a distance d between successive apertures (Fig. 5.7). When an incident plane wave is traveling into the -direction, its amplitude distribution is successively altered by diffraction, from a constant amplitude to the final stationary distribution An(x,y). The spatial distribution An(x,y) in the plane of the nth aperture is determined by the distribution An- (x, y) across the previous aperture. 

The amplitude attenuation factor C does not depend on x and y. The quantity Kd represents the diffraction losses and 0 the corresponding phase shift caused by diffraction. 

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. 

Note that wq and w do not depend on the mirror size. Increasing the mirror width 2a reduces, however, the diffraction losses as long as no other limiting aperture exists inside the resonator. 

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