Rayleigh distance

In optics and spectroscopy, resolution is often limited by diffraction. To a good approximation, the spread function may appear as a single-slit diffraction pattern (Section II). If equal-intensity objects (spectral lines) are placed close to one another so that the first zero of one sine-squared diffraction pattern is superimposed on the peak of the adjacent pattern, they are said to be separated by the Rayleigh distance (Strong, 1958). This separation gives rise to a 19% dip between the peaks of the superimposed patterns. 

Brief reflection on the sampling theorem (Chapter 1, Section IV.C) with the aid of the Fourier transform directory (Chapter 1, Fig. 2) leads to the conclusion that the Rayleigh distance is precisely two times the Nyquist interval. We may therefore easily specify the sample density required to recover all the information in a spectrum obtained from a band-limiting instrument with a sine-squared spread function evenly spaced samples must be selected so that four data points would cover the interval between the first zeros on either side of the spread function s central maximum. In practice, it is often advantageous to place samples somewhat closer together. 

Fig. 2 shows the CFRP-sandwich specimen and the transducer mounted on the scanner. Fig. 23 presents a C-scan of the specimen as first interesting result. Only the defects visible from the outside are indicated. The distance between transducer and specimen was smaller than the focal length, so that the angle of incidence at the edge of the sound beam converts the longitudinal waves to Rayleigh-waves in the specimen. These waves provide a very sharp image of the surface. This method opens the possibility for a non-contact acoustic microscope. 

Figure 24. The acquisition camera image of the Keck LGS with the segmented mirror "unstacked." The brightening on the left is the Rayleigh scatter of light from the laser. The 36 spots show an elongation increasing with distance from the laser source, which is left of the camera.

From the 2D instantaneous Rayleigh temperature fields, such as shown in Figure 7.1.8, isotemperature contours can be obtained and they clearly show that the distance between the isothermal contours strongly varies at different locations, being deeply perturbed by turbulence, especially on the fresh reactants side. 

The resolution or "resolving power" of a light microscope is usually specified as the minimum distance between two lines or points in the imaged object, at which they will be perceived as separated by the observer. The Rayleigh criterion [42] is extensively used in optical microscopy for determining the resolution of light microscopes. It imposes a resolution limit. The criterion is satisfied, when the centre of the Airy disc for the first object occurs at the first minimum of the Airy disc of the second. This minimum distance r can then be calculated by Equation (3). 

The important break through in the understanding of cavitation came in 1917 when Lord Rayleigh [12] published his paper On the pressure developed in a liquid during the collapse of a spherical cavity . By considering the total collapse of an empty void under the action of a constant ambient pressure Pq, Rayleigh deduced both the cavity collapse time t, and the pressure P in the liquid at some distance R from the cavity to be respectively (Eqs.2.25 and 2.26). 

The first term of the last equation represents Rayleigh scattering, which occurs at the excitation frequency Vex- The second and the third terms correspond to the Stokes and anti-Stokes frequencies of (vex - w) and (vex + r v). Here, the excitation frequency has been modulated by the vibration frequency of the bond. It is important to note that Raman scattering requires that the polarizability of a bond varies as a fimction of distance, that is (dajdr) must be greater than zero if a Raman line is to appear. 

---