W9_parallel_resonance.eps

420 Week 13: Interference and Diffraction

asin(θ) = (m+^1

2

)λ

As was the case forN-slit intereference secondary maxima, however, the exact

angles of the secondary maxima requires the solution of a transcendental equation

and not a formula as simple as this.

• Combined Interference and Diffraction:If one takes (e.g.) two slits, each of
  widtha, separated by a distanced > aand illuminated by light with wavelengthλ,
  the intensity on a distant screen is given by:

I(θ) = 4I 0 cos^2 (δ/2)

(

sin(φ/2)

φ/ 2

) 2

The resulting intensity is the usual two slit interference pattern,modulatedby the

so-called “diffraction envelope” of each slit indepedently.

• Diffraction Through Circular Apertures and Optical Instruments:A cir-
  cular aperture produces a diffraction pattern that qualitatively resembles that of
  a single slit with anaxially symmetriccentral maximum surrounded by rings of
  minima and ever-fainter secondary maxima. In many cases it is this diffraction
  of light from small or distant source points as it passes through theobjective lens
  of a microscope or telescope (respectively) that limits the resolution of optical in-
  struments. One can, of course, magnify objects almost without bound as far as
  geometric optics is concerned, but at some point diffraction makes further magnifi-
  cation pointless because neighboring source points in the field of vieware no longer
  resolvable according to the Rayleigh criterion at any greater magnification.
  The angle of thefirst minimum(dark ring around the central maximum) produced
  by a given wavelength of light is determined by the formula:

Dsin(θmin) = 1. 22 λ

whereDis the diameter of the circular aperture of the optical instrument. It is

beyond the scope of this course to derive this, but it is “reasonable” as an approx-

imation of the single slit result above. In almost all cases, we are only interest in

using this when the angles involved are very small, in which case we can write:

θmin= 1. 22 λ

D

• The Rayleigh criterion for wave-optic resolution with an optical instrument is then
  simply that the angle between the two source points as they enter the first lens of
  the microscope or telescope must exceed the angle to the first minimum of either
  one, or:

αincidence> θmin= 1. 22 λ

D

• Thin Film Interference:Light that strikes a thin transparent partially reflective
  film on top of a second reflective medium can interfere withitself provided that
  the film is thin enough that the total path difference between light reflected from
  the first versus the second surface is inside the coherence lengthof the light. Thin
  film interference is what makes soap bubbles and a drop of oil on water on dark
  pavement swirl with odd pastel colors.