Imaging with a Thick Lens. Consider a glass lens of refractive index n , the z axis and thickness d , and two spherical surfaces of equal radii R as shown in Figure 3.1. Determine the ray-transfer matrix of the system between the two planes at distances d 1 and d 2 from the vertices of the lens. The lens is placed in air ( n = 1 ) . Show that the system is an imaging system (i.e., the input and output planes are conjugate) if 1 z 1 + 1 z 2 = 1 f or s 1 s 2 = f 2 , where z 1 = d 1 + h s 1 = z 1 − f z 2 = d 2 + h s 2 = z 2 − f and h = ( n − 1 ) f d n R 1 f = n − 1 R [ 2 − n − 1 n d R ] The points F 1 and F 2 are known as the front and back focal points, respectively. The points P 1 and P 2 are known as the first and second principal points, respectively. Show the importance of these points by tracing the trajectories of rays that are incident parallel to the optical axis. Figure 3.1: Imaging with a thick lens. P 1 and P 2 are the principal points and F 1 and F 2 are the focal points. Imaging with a Thick Lens. Consider a glass lens of refractive index n , the z axis and thickness d , and two spherical surfaces of equal radii R as shown in Figure 3.1. Determine the ray-transfer matrix of the system between the two planes at distances d 1 and d 2 from the vertices of the lens. The lens is placed in air ( n = 1 ) . Show that the system is an imaging system (i.e., the input and output planes are conjugate) if 1 z 1 + 1 z 2 = 1 f or s 1 s 2 = f 2 , where z 1 = d 1 + h s 1 = z 1 − f z 2 = d 2 + h s 2 = z 2 − f and h = ( n − 1 ) f d n R 1 f = n − 1 R [ 2 − n − 1 n d R ] The points F 1 and F 2 are known as the front and back focal points, respectively. The points P 1 and P 2 are known as the first and second principal points, respectively. Show the importance of these points by tracing the trajectories of rays that are incident parallel to the optical axis. Figure 3.1: Imaging with a thick lens. P 1 and P 2 are the principal points and F 1 and F 2 are the focal points.
Transmission Through a Diffraction Grating. (a) The thickness of a thin transparent plate varies sinusoidally in the x direction, d(x,y) = 1/2 d0 [1+cos(2πx/Λ)] as illustrated in Figure 1.1. Show that the complex amplitude transmittance is T(x,y) = h0 exp[-j 1/2(n-1)k0 d0 cos(2πx/Λ)] where h0 = exp[-j 1/2(n+1)k0 d0]. (b) Show that an incident plane wave travelling at a small angle θi with the z direction is transmitted in the form of a sum of plane waves travelling at angles θq = θi+q λ/Λ. Hint: Expand the periodic function T(x,y) in a Fourier series. Figure 1.1: A thin transparent plate with periodically varying thickness serves as a diffraction grating. It splits an incident plane wave into multiple plane waves travelling in different directions.