A tungsten-filament lamp having a straight filament that is in diameter is used as a source for an, interference experiment. At what distance from the lamp must the aperture be placed in order that the transverse coherence width is at least 0. If a double-slit aperture is used, why should the slip oriented parallel to the lamp filament? We now proceed to treat the more general case of multiple-era n interference.
The most common method of producing a large number of mutually coherent beams is by division of amplitude. The division occurs by multiple reflection between two parallel, partially reflecting surfaces. These surfaces might be cmnitransparent mirrors, or merely the two sides of a film or slab of transparent material, and so forth. The situation is illustrated in Figure 4. Here, for simplicity, the reflecting surfaces are considered to be two thin, identical s nsireflecting mirrors. The primary beam is partially reflected and partially transmitted at the first surface.
The transmitted part is subsequently reflected back and forth betwec the two surfaces as shown.
Eot2r4 Ept2r2 EOt2 Figure 4. Paths of light rays in multiple reflection between two parallel mirrors. Now a phase change may occur on reflection, hence r is, in general, a complex number.
Introduction to classical and modern optics
We can express it, accordingly, as e act,. As we have shown previously in Section 2. Equation 4. In the more general c,,,tsc in which tile are and is easy to show that th pry:. We then find the corresponding fracti i1al transmission to be 7r, T2 T max 4. Ac- 1. This me an cording to C, : Lion 4. Fabry and ea dn. If One plate spacing can be mechanif1a. The surfaces must be extren AA'y flat and parallel in order to obtain the maximum fringe sharpness.
An ordinary optical flat of 4 wavelength flatness is not got.. Rather, a flatnne. If a broad cat, interference fringes in the orm of concentric appear in the 1" cal plane of the focusing lens Figure mg x n,:a be ohser-t ed v sually or photographed. A given nstant value of. The in t.
The intensity distribution will be a superposition of two fringe systems as indicated in Figure 4. Here the two components are assumed to be of equal intensity. Graph of intensity distribution for two monochromatic lines in Fabry-F'erot interferometry. Now the two frequencies a and w' can be said to be resolved if there is a dip in the intensity curve. A useful convention for resolution in the case of multiple-beam interference is known as the Taylor criterion. According to this convention, two equal lines are considered to be resolved if the individual curves cross at the half-intensity point, so that the total intensity at the saddle point is equal to the maximum intensity of either line alone.
From Equations 4. By increasing the order of interference, the resolving power with a given reflectance can be made as large as desired. However, the free spectral range then diminishes, and so a compromise must be chosen in any given aapplication. For a given vale of the mirror separation the resolving power, in principle, can be increased indefinitely by making the reflectance closer and closer to unity.
However, a practical limit is imposed by absorption in the reflecting surface which reduces the intensity of the transmitted fringes, as discussed in Section 4. Silver and aluminum films, deposited by vacuum evaporation, have long been used for Fabry-Perot instruments. The useful reflectance with these metal films, as limited by absorption, is only about 80 to 90 percent. More recently, multilayer dielectric films have been used for Fabry-Perot work. With such films, discussed in the next section, useful reflectances approaching 99 percent can be achieved.
A good Fabry-Perot instrument can easily have a resolving power of 1 million, which is 10 to times that of a prism or small-grating spectroscope. For a more complete discussion of Fabry-Perot interferometry, and of other instruments used in high-resolution spectrosac- copy, see References  and  listed at the end of the book. Optical surfaces having virtually any desired reflectance and transmittance characteristics may be produced by means of thin film coatings.
These films are usually deposited on glass or metal substrates by high-vacuum evaporation.
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The well-known use of antireflecting coatings for camera lenses and other optical instruments is only one of the many practical applications of thin films. Other applications include such things as heat-reflecting and heat-transmitting mirrors "hot" and "cold" mirrors , one-way mirrors, optical filters, and so forth. First consider the case of a single layer of dielectric of index n1 and thickness l between two infinite media of indices no and nT Figure 4.
For simplicity we shall develop the theory for normally incident light. The modifications for the general case of oblique in- 4. Wave vectors and their dv ssociated electric fields for the case of normal incidence on a single dielectric layer. The amplitude of the electric vector of the incident beam is Ep.
That of the reflected beam is 'p', and that of the transmitted beam is ET. The electric-field amplitudes in the film are E1 and E1' for the forward and backward traveling waves, respectively, as indicated in the figure.
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The boundary conditions require that the electric and magnetic fields be continuous at each interface. The phase factors c and e-ikt result from the fact that the wave travels through a distance from one interface to the other. Now suppose that we have N layers numbered 1, 2, 3, nN and thicknesses 11, 12, having indices of refraction n1, n2, n3, 13,. IN, respectively.
In the same way that we derived Equation 4. Each transfer matrix is of the form given by Equation 4. The overall transfer matrix M is the product of the individual transfer matrices. I acl os. Fabry-Perot Interference Filter A Fabry-Perot type of filter consists of a layer of dielectric having a thickness of wavelength for some wavelength X0 and bounded on both sides by partially reflecting surfaces.
In effect, this is a Fabry-Perot etalon with a very small spacing. The result is a filter that has a transmission curve given by the Airy function Equation 4. Higher-order peaks also occur for wavelengths 2 X0,! The spectral width of the transmission band depends on the reflectance of the bounding surfaces. Fabry-Perot filters were first made with silver films to produce the necessary high reflectance, but now they are usually made entirely of multilayer dielectric films.
The latter are superior to the metal films because of their higher reflectance and lower absorption. Figure 4. Find the maximum and minimum transmittance of the inter- ferometer. What is the value of the reflecting finesse and of the coefficient of finesse? The mirrors of a Fabry-Perot resonator for a laser are coated to give a reflectance of 0. Find the value of the fringe width in wavelength and in frequency at a wavelength of n ai.
N' , where N' is an integer, provided that there is a zero-radius fringe. Develop a formula for the transmitt ance as a function of wavelength. The transmission function is thus periodic in wavenumber or frequency and is called a "channeled spectrum. If the plate is 1 mm thick, find the wavelength separation between adjacent channels at a vacuum wavelength 4.
Close examination of the shadow edge reveals that some light goes over into the dark zone of the geometrical shadow and that dark fringes appear in the illuminated zone. This "smearing" of the shadow edge is closely related to another phenomenon, namely, the spreading of light after passing through a very small aperture, such as a pinhole or a narrow slit, as in Young's experiment.
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The collective name given to these departures from geometrical optics is diffraction. The essential features of diffraction phenomena can be explained qualitatively by Huygens' principle. This principle in its original form states that the propagation of a light wave can be predicted by assuming that each point of the wave front acts as the source of a secondary wave that spreads out in all directions.
The envelope of all the secondary waves is the new wave front. We shall not attempt to treat diffraction by a direct application of Huygens' principle. We want a more quantitative approach. Our strat- egy will be to cast Huygens' principle into a precise mathematics form known as the Fresnel-Kirchhof formula. This formula will then be applied to various specific cases of diffraction of light by obstacles and apertures.
By "grad, " is meant the normal component of the gradient at the surface of integration. We let the volume enclosed by the surface of integra- tion include the point P.