Quasi-monochromatic

Nonmonochromatic Waves (1.16) Diffraction theory is readily expandable to non-monochromatic light. A formulation of the Kirchhoff-Fresnel integral which applies to quasi-monochromatic conditions involves the superposition of retarded field amplitudes. 

In the hrst case, the degree of self coherence depends on the spectral characteristics of the source. The coherence time Tc represents the time scale over which a held remains correlated this hme is inversely proportional to the spectral bandwidth Au) of the detected light. A more quantitative dehnition of quasi-monochromatic conditions is based on the coherence time all relevant delays within the interferometer should be much shorter than the coherence length CTc. A practical way to measure temporal coherence is to use a Michel-son interferometer. As we shall see, in the second case the spatial coherence depends on the apparent extent of a source. 

Although a strictly monochromatic wave, one for which the time dependence is exp( — itot), has a well-defined vibration ellipse, not all waves do. Let us consider a nearly monochromatic, or quasi-monochromatic beam ... 

The Stokes parameters of a quasi-monochromatic beam are given by... 

If two or more quasi-monochromatic beams propagating in the same direction are superposed incoherently, that is to say, there is no fixed relationship among the phases of the separate beams, the total irradiance is merely the sum of the individual beam irradiances. Because the definition of the Stokes parameters involves only irradiances, it follows that the Stokes parameters of a collection of incoherent sources are additive. 

In the derivations above of the Stokes parameters we began with monochromatic light and then extended our results to the more general case of quasi-monochromatic light. However, the operational definition of the Stokes parameters in terms of a set of elementary experiments involving a detector and various polarizers, as opposed to the formal mathematical definitions (2.80) and (2.84), is independent of any assumed properties of the beam. Unless otherwise stated, we shall assume that all beams of interest are quasi-monochromatic, which includes as a special case monochromatic light. 

Now let us assume that a monochromatic source of flux is placed in the plane of the entrance slit so that there is no constant phase relationship between the fields at any two given points in the slit. This, in itself, is a contradiction, because a perfect source monochromaticity implies both spatial and temporal coherence. By definition of coherence, a constant phase relationship would result. To eliminate the possibility of such a relationship, we must require the source spectrum to have finite breadth. Let us modify the assumption accordingly but specify the source spectrum breadth narrow enough so that its spatial extent when dispersed is negligible compared with the breadth of the slits, diffraction pattern, and so on. Whenever time integrals are required to obtain observable signals from superimposed fields, we evaluate them over time periods that are long compared with the reciprocal of the frequency difference between the fields. We shall call the assumed source a quasi-monochromatic source. 

The irradiance U is seen to be the sum of the irradiances due to each component plus a cross term 2SBut the assumption of complete incoherence and quasi-monochromaticity guarantees that no two components (f) and i2(t) could have a constant phase difference and still yield a... 

The wavelength of the emitted light depends on the choice of atoms A and B. Besides the principle of excimer formation, the technical parameters for the lamp and discharge are responsible for the quasi-monochromatic character of the emitted spectrum. The most important commercial excimers are formed by electronic excitation of molecules of rare gases (He2, Ne2, Ar2, Kr2, Xe2),... 

In most cases, low-pressure excimer UV sources show a dominant narrow-band (1-3 nm) transition. They can be considered to be quasi-monochromatic. 

Excimer lamps are quasi-monochromatic light sources available in UV wavelengths. The light is produced by silent electrical discharge through gas in the gap between two concentric quartz tubes. Electronically activated molecules are produced in the gas phase and decompose within nanoseconds to produce photons of high selectivity. This process is similar to the process in excimer lasers. 

Fig. 5.2. Schematic r.f. systems, (a) Simple heterodyne circuit, SI determines the pulse length, S2 switches the lens from transmit to receive, and A1 amplifies the reflected signal (b) quasi-monochromatic circuit the two oscillators and the pulse repetition frequency are phase-locked, and the final signal is lock-in detected (courtesy of John...

For quantitative applications the quasi-monochromatic circuit of Fig. 5.2(b) is better. The basic principles of this circuit are similar to those of Fig. 5.2(a), but there are some important differences. The r.f. oscillator and the local oscillator are two frequency synthesizers that are phase-locked to one another to give a difference frequency of precisely defined phase. This difference frequency is very much lower than in the simple heterodyne circuit,... 

For the determination of the spectral sensitivity complicated and expensive laboratory equipment are required in conjunction with experienced personnel. Such a facility should be capable in providing calibrated monochromatic radiation of sufficient power to be sensed by the broadband detector in question. A typical system should be comprised by a double monochromator coupled with a stable, high-power source (e.g. a 1000-Watt Xenon lamp). A spectrally calibrated detector should also be available to measure the quasi-monochromatic radiation emitted by this system. Such facilities are rarely available and only a few laboratories worldwide are able to cany out such characterizations. 

Neutrons with a wavelength of 5 to 10 A are often used. A narrow range of wavelengths is usually selected to give a quasi-monochromatic beam. This can be achieved by a crystal monochromator or, more usually, by a velocity-chopper system of rotating discs which select neutrons of a given velocity. 

The above derivation shows explicitly what approximations need to be adopted to obtain NLS Approximating K to second order in frequency and transverse wavenumber amounts to the paraxial, and quasi-monochromatic approximations for the linear wave propagation. The approximation in the nonlinear coupling Q also requires a narrow spectrum in order to be able to represent Q by a constant. 

---