Single-Slit Diffraction: Interference Caused by a Single “Slit” or “Hole in the Wall”

L L W W m = +1, -1, +2, -2, … but not: 0 For wide linear slit of width W>> λ: θ Min = θ m=1 ≈ λ / W For wide-diameter (D) circular opening, D>> λ: θ Min = θ m=1 ≈ 1.22 λ / D

A single slit in the wall, with a width of 0.020mm, is illuminated by a red laser beam with a wavelength of 680nm, incident from the left of the slit. (a)Find the angle θ 1 subtended at the center of the slit by the 1 st intensity minimum, closest to the central maximum (at θ = 0 o ), when observed at a very large distance L (meaning: L>>W) to the right of the slit. (b) Find the distance, in cm, between the central bright fringe (intensity maximum) and the 1 st dark fringes (intensity minima), closest to the central one, observed on a screen, 3.5m away to the right of the slit. (c) What is the largest possible angle, θ M, at which a dark fringe could (theoretically) still be observed on the screen in (b) ? (d) Find the total number of a dark fringes that could (theoretically) be observed on the screen in (b)! How wide, at least, would the screen have to be, to observe them all? (e) Suppose the slit is replaced by a circular aperture, i.e., a “round hole” in plain English, with a diameter of 0.020mm. Find the radius of the 1 st dark dark circle surrounding the central maximum on the screen. Also, find the angle θ Min = θ 1, subtended at the center of the hole by the radius of that 1 st dark circle (measured from the central maximum).

Diffraction Limits on Image Resolution in Optical Instruments

The headlights of a truck are mounted 1.5m apart and are seen by eye from a distance of 2km. Assume they send out yellow light with a wavelength of 600nm. The aperture of your eye (=opening through which light enters) has a diameter of 1cm. (a)Can you tell the truck apart from a motor cycle, a that viewing distance? I.e., can you resolve the two headlights? (b) At what (minimum) distance (truck--eye) would you not be able anymore to tell the truck apart from a motorcycle ? Hints: (1) Study the drawing on the next page: The headlights are the two object points (point light sources !), Q and R. (2) The two image points, Q’ and R’, are blurred on the retina: each is being “smeared out” into a diffraction pattern, as shown in the drawing. (3) If the Q’-diffraction intensity maximum (red) overlaps with the R’-diffraction maximum (blue) you cannot tell Q’ and R’ apart anymore: they are not resolved. The two headlights will then look like a single point light source to you, i.e., like a motorcycle. This happens when the spacing between Q’ and R’ becomes smaller than the width of either diffraction intensity maximum, i.e., stated in terms of angles, when: θ’ < θ Min

Images Q’ and R’ are resolved If θ’ > θ Min R R’ Q Q’