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Prove the relation for the Brewster angle [Eq. (2–43)]. 4. Show that Eqs. (2–45) and (2–46) are satisfied for a single boundary between two non-absorbing media. 5. What is the angle of refraction for a plane wave incident at the critical angle but from the opposite side of the boundary? 6. 5: (a) The wave is incident normal to the surface; (b) The wave is incident at 45◦ to the surface. C HAPTER 3 L INEAR SYSTEMS THEORY The previous chapter reviewed the basic principles underlying light propagation phenomena that are of interest for this book.

As a result, we see that in the medium the linear phase represents a different propagation angle αn which is related to the incidence angle by the relation sin αn = sin α . n (4–40) This relation is identical with Snell’s law [Eq. (2–37)] as derived from the boundary conditions. In a similar way we may obtain the transformation of the quadratic phase factor, Q 1 1 → Qn , R nR (4–41) which indicates that the radius of curvature, R, of a spherical wave incident on a dielectric medium is changed to nR within the medium.

3–3. What is the relation between the two FTs? 9. Prove the symmetry relation: If the object is real, g(x, y) = g ∗ (x, y), then its FT is symmetric: G(u, v) = G∗ (−u, −v). 10. Find the FT of the pattern shown in Fig. 3–4. Discuss the result. 30 LINEAR SYSTEMS THEORY Figure 3 2. The shaded area has a value 1 while the rest has the value 0. Figure 3 3. The shaded area has a value 1 while the rest has the value 0. Figure 3 4. The shaded area has a value 1 while the rest has the value 0. 11. Show that the FT of a 1D function, f (x), is given by F (x)δ(y), where F (x) is the FT of f (x).

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A system of optical design by Arthur Cox

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