32 Interference Diffraction 1 2 3 4 5

32. Interference & Diffraction 1. 2. 3. 4. 5. 6. Coherence & Interference Double-Slit Interference Multiple-Slit Interference & Diffraction Gratings Interferometry Huygen’s Principle & Diffraction The Diffraction Limit

How do optical principles ultimately govern the amount of information that can be stored on a DVD? Ans: Dim of bit >

32. 1. Coherence & Interference Coherence Waves are coherent they maintain a constant phase relation. ( also same freq & wavelength ) Steady interference patterns occur only when waves are coherent. Coherence length = typical length of wave train. Waves from different sources cannot be coherent. Interference occurs only for waves from same source but travelling on different paths.

Destructive & Constructive Interference crest meets trough Destructive interference: crest meets crest Constructive interference: Caveat:

Application: More CD Music Beams reflected from pits out of phase with those from flats. Reflected beams less intense wherever pit is present.

32. 2. Double-Slit Interference fringes Bright fringes: Dark fringes: L >> d << :

Example 32. 1. Laser Light Two slits 0. 075 mm apart are located 1. 5 m from a screen. Laser light shining through the slits produces an interference pattern whose 3 rd order bright fringe is 3. 8 cm from the screen center. Find the light’s wavelength. red light from He-Ne laser

Intensity in the Interference Pattern Neglect difference in amplitude: small max. at

32. 3. Multiple-Slit Interference & Diffraction Gratings For a N-slit system: Condition for constructive interference: Since Condition for destructive interference is: ( m = integer but not multiples of N )

Diffraction Gratings Diffraction gratings: several cm’s across with thousands of slits Made by photoreducing images of // lines, or by ruling with diamond stylus on Al-plated glass. Two types: Transmission gratings. Reflection gratings. Maxima condition position of max depends on for m 0. ( m = order of dispersion) Spectrometer: sharp lines if N large.

Example 32. 2. Grating Spectrometer Light from glowing hydrogen contains discrete spectral lines called H and H , with wavelengths of 656. 3 nm & 486. 1 nm, respectively. Find the 1 st order angular separation between these wavelengths in a spectrometer that uses grating with 6000 slits per cm. maxima condition

Resolving Power barely resolved Consider an N-slit grating. The mth order maximum of a spectral line with wavelength is at angle max given by The closest minimum is at angle min given by For a spectral line with wavelength , we have These two lines can be resolved only if or = resolving power of the grating

Example 32. 3. “Seeing” a Double Star A double star system appears as a single point to a telescope. The H line of its massive stationary star is at = 656. 272 nm; that of its orbiting companion is Doppler shifted to = 656. 329 nm while moving away from Earth. If a spectrometer has 5000 lines, what order spectrum will resolve these H lines? m=3

X-Ray Diffraction X-ray: ~ 0. 1 nm = 1 A Criterion for constructive interference : Bragg condition

Other Gratings Anything with regularly spaced structures can act as a diffraction grating. E. g. , l Rainbow seen on underside of CD or DVD. l Acoustic-optic modulators steers light beams using acoustic waves as gratings. l Laser printers & digital copiers also use AOMs for photo-printing.

GOT IT ? 32. 2. If you increase the number of slits in a grating while keeping the spacing the same, what happens to No change (a) the positions of the intensity maxima in the interference pattern for a given wavelength; increases (b) the intensity of the maxima; and narrower (c) the width of the maxima ?

32. 4. Interferometry : Split light into beams, send them on different paths, then recombine. Usage: Very sensitive technique for measuring displacements, time intervals, & other differences in quantities.

Thin Films Light is partially reflected at both surfaces of a thin film interference. Upon incident on larger n medium: reflected wave is inverted ( = ). Upon incident on smaller n medium: reflected wave suffers no phase-shift ( = 0 ). Normal incident; constructive interference:

Newton’s ring Anti-reflection coating: 3 layers with suitable thickness &

Conceptual Example 32. 1. A Soap Film Figure shows a soap film in a circular ring with different colored bands that runs horizontally across the film. Why do these bands occur? Gravity makes film thicker near bottom. for constructive interference varies with height. Multiple bands with colors repeating : multiple orders of interference

Making the Connection A 20 cm high soap film is 1. 0 m thickat the bottom, tapering to near zero thickness at the top. If it is illuminated with 650 -nm light, how many bright bands appear? Constructive interference : The film’s refractive index is essentially that of water, n = 1. 33. Number of bands = mmax + 1 = 3 i. e. , m = 0, 1, 2, 3 ( 4 bright bands )

The Michelson Interferometer Laser Interferometer Gravitational wave Observatory ( LIGO ) Multiple reflection in 4 -km arms effective length of 300 km.

Example 32. 4. Sandstorm A sandstorm has pitted the aluminum mirrors of a desert solar-energy installation, and engineers want to know the depth of the pits. They construct a Michelson interferometer with a sample from one of the pitted mirrors in place of one flat mirror. With 633 -nm laser light, the interference pattern in figure results. What is the approximate depth of the pit? Extra path length = 0. 2

32. 5. Huygen’s Principle & Diffraction Huygen’s Principle : All points on a wave front acts as point sources of spherically propagating wavelets. New wave front is the surface tangent to all these wavelets.

Diffraction Plane wave incident on barrier with hole. Large hole ( d >> ) : negligible diffraction Small hole ( d < ) : significant diffraction Geometrical optics are valid for systems with dimensions >>

Single-Slit Diffraction Light from y cancelled by that from y+a/2 if : or Light from y cancelled by that from y+a/4 if : or … Destructive interference: a> m = 0 give = 0 is the central maximum. c. f. maxima of multi-slit diffraction

Intensity in Single-Slit Diffraction Phase difference from points at ends of slit: is the intensity of central maximum. Minima at or

Multiple-Slits & Other Diffraction Systems Slit width not negligible Diffraction patterns from light passing sharp edges Straight edge Circular aperture with crosshairs

32. 6. The Diffraction Limit 2 far-away incoherent sources illuminating a slit 2 single-slit diffraction patterns ( but no interference pattern ). Large angular separation: diffraction patterns distinct Small angular separation: diffraction patterns overlap Note: angular separation of the sources = that of the patterns =

Fully resolved Barely resolved (Rayleigh criterion ) Unresolved Rayleigh criterion: Central max. of one pattern coincides with 1 st min. of the other.

1 st min. of single slit diffraction is given by For optical systems << a Rayleigh criterion for a slit becomes Rayleigh criterion for a circular aperture : Diffraction limit: minimum size of resolvable object set by diffraction. Interferometer: combine data from widely separated telescopes to produce a large effective aperture size. Commonly used in radio telescopes.

GOT IT ? 32. 3. You try to resolve details within a cell but they look fuzzy even at the highest power of your microscope. Which of the following might help: (a) substitute an eyepiece with shorter focal length, (b) put a red filter over the white light source used to illuminate the microscope slide; or (c) put a blue filter over the white light source?

Example 32. 5. Asteroid Alert An asteroid 20 106 km away appears on a collision course with Earth. What’s the minimum size for the asteroid that could be resolved with the 2. 4 -m-diameter diffraction-limited Hubble Space Telescope, using 550 -nm reflected sunlight? Minimum size for the asteroid that could be resolved :

Application: CD to DVD to Blu-ray laser: IR laser: red light laser: blue light Content: 74 min audio Content: 2 hr vidio Content: 4. 5 hr HD vidio