4.4 Show that the radii of the interference fringes of a plane-parallel Fabry-Perot interferometer, as shown in Figure 4.5(a), are approximately proportional to VO, VI, √2,... VN', where N' is an integer, provided that there is a zero-radius fringe. [Hint: Use Equation (4.10) and assume that is small, so that cos 01-02/2.1 2NπT = 4TT λο nd cos 0+ dr (4.10) 92 MULTIPLE-BEAM INTERFERENCE (a) Figure 4.5. Fabry-Perot interference fringes. (a) Monochromatic source: (b) non-monochromatic source. 4.5 A collimated beam of white light falls at normal incidence on a plate of glass of index n and thickness d. Develop a formula for the transmittance as a function of wavelength. Show that maxima occur at those wavelengths such that AN = 2nd N where A is the vacuum wavelength and N is an integer. The transmission function is thus periodic (in wavenumber or fre- quency) and is called a "channeled spectrum." 4.6 Using the result of Problem 4.5, find the maximum and min- imum transmittance of the channels for a plate of index of refraction 2.5. If the plate is 1 mm thick, find the wavelength separation between adjacent channels at a vacuum wavelength of 600 nm. 4.9 Fill in the steps in the derivation of the expression of the transfer matrix of a single film cos kl sin kl M n₁ -in, sin kl cos kl