Question

A digital camera has a lens that can be approximated as a thin lens of focal length 6 mm, located a variable distance x from an image CMOS pixel plane. The pixel array has dimensions of 1000x1000 and the pixel size is 3 microns. 1. If x- 8 mm, then the image on the pixel plane is in sharp focus for an object a distance So in front of the lens. Determine So If an object at nearly infinite distance from the lens consists of 2 point sources (think of two distant stars). What angular separation between these points will correspond to one pixel of separation on the camera if the distance to the plane is adjusted to 6 mm. a. b. c.A hair of width 50 microns is located 30 millimeters from the lens. The lens is adjusted to be a distance x from the image plane so the hair is in sharp focus on the image plane. Determine x. On the image, what is the apparent diameter of the hair in pixels? d. Review the extra problem 3 from problem set 1. Using this camera determine the number of pixels of separation between the two Balmer transitions: 4 to 2 (green) 3 to 2 (red) Assume the diffraction grating is near the lens and has line spacing of 0.1 mm.

Answer 1

Given that (a) thin less of focal length = 6mm we know that thin lens equation is given by 1/v - 1/mu = 1/f we know, f = 6mm v = x = 8mm 1/8 - 1/-s = 1/6 1/8 - 1/6 = -1/s_0 S_0 = infinity (b) To calculate the angular separation theta = pixel/x = = 3 times 10^-6/6 times 10^-3 = 0.005 radians theta = 0.005 radians \r\n (c) A hair of width 50 microns is located 30 millimeters from the lens. We have to calculate apparent diameter of the hair in pixels. 1/v - 1/mu = 1/f 1/v - 1/(-30) = 1/6 1/v = 1/6 - 1/30 = 5 - 1/30 = 4/30 v = 7.5 m from magnification m = h_i/h_0 = v/u h_i/50 = (7.5/30) h_i = 7.5 times 50/30 = 12.5 micron h_i = 0.0472 pixels