Lecture 27-Lecture 27-11Thin-Film Interference-Cont’d

Path length difference:

2l t

(Assume near-normal incidence.)

( 1/ 2)

m

m

destructive

constructive

0

n

where

• ray-one got a phase change of 180o due to reflection from air to glass.

• the phase difference due to path length is:

•then total phase difference:= ’+180.

22' '

n

ll

Lecture 27-Lecture 27-22 Two (narrow) slit Interference

• Upon reaching the screen C, thetwo wave interact to produce aninterference pattern consisting ofalternating bright and dark bands(or fringes), depending on theirphase difference.

Constructive vs. destructiveinterference

• According to Huygens’s principle,each slit acts like a wavelet. The the secondary wave fronts arecylindrical surfaces.

Young’s double-slit experiment

Lecture 27-Lecture 27-33 Interference Fringes

For D >> d, the difference in path lengths between the two waves is sindL • A bright fringe is produced if the path lengths differ by an integer number of wavelengths,

sin , 0, 1,d m m

• A dark fringe is produced if the path lengths differ by an odd multiple of half a wavelength,

sin ( 1/ 2) , 0, 1,d m m

y ~ D*tan(θ)~ D*(m+1/2)λ/d

y ~ D*tan(θ)~ D*mλ/d

Lecture 27-Lecture 27-44Intensity of Interference Fringes

Let the electric field components of the two coherent electromagnetic waves be

1 0

2 0

sin

sin( )

E E t

E E t

The resulting electric field component point P is then

1 2

0

0

sin sin( )

2 cos sin2 2

E E E

E t t

E t

2

0 0

202

4 cos2

m II E

I EI

Intensity is proportional to E2

I=0 when = (2m+1) , i.e. half cycle + any number of cycle.

Lecture 27-Lecture 27-55Dark and Bright Fringes of Single-Slit Diffraction

Lecture 27-Lecture 27-66Phasor Diagram

1

2

Lecture 27-Lecture 27-77Phasor Diagram for Single-Slit Diffraction

2 sinaN

total phase difference:

maxA

r

max22 sin sin

2 2

AA r

2

2ma

max m

xx

2

a

s)

2

(in

2( )I

I IA

I A

2I A

The superposition of wavelets can be illustrated by a phasor diagram. If the slit is divided into N zones, the phase difference between adjacent wavelets is

sin sin2

( / 2)

aaN

N

Lecture 27-Lecture 27-88Intensity Distribution 1

2 sinwhere

a

2

max

sin2( )

2

I I

maxima:

0 central maximum becausesin

1 0x

as xx

minima:

1, 2, 3,... 0m m

sin 02

sinor a m

1sinn )

2i 1

2(s aor m

or

Lecture 27-Lecture 27-99Intensity Distribution 2

• Fringe widths are proportional to /a.

• Width of central maximum is twice any other maximum.•Width = D*λ/a – D*(-1)λ/a = 2D*λ/a

• Intensity at first side maxima is (2/3)2 that of the central maximum.

for small

• y ~ D*θ •Bright fringe: D*(m+1/2)λ/a•Dark fringe: D*mλ/a•Width: D*λ/a except central maximum

y

Lecture 27-Lecture 27-1010Young’s Double-Slit Experiment Revisited

• If each slit has a finite width a (not much smaller than ), single-slit diffraction effects must be taken into account!

• Intensity pattern for an ideal double-slit experiment with narrow slits (a<<)

d

slit separation

Light leaving each slit has a unique phase. So there is no superimposed single-slit diffraction pattern but only the phase difference between rays leaving the two slits matter.

20

sin4 cos

dI I

where I0 is the intensity if one slit were blocked

D d

a

Lecture 27-Lecture 27-1111

Intensity Distribution from Realistic Double-Slit Diffraction

double-slit intensity sind

204 cosI I

replace by

2sin

mI

single-slit intensity envelope

sina

2

2 sin( ) (cos )mI I

Lecture 27-Lecture 27-1212Diffraction by a Circular Aperture

• The diffraction pattern consists of a bright circular region and concentric rings of bright and dark fringes.

• The first minimum for the diffraction pattern of a circular aperture of diameter d is located by

sin 1.22d

geometric factor

• Resolution of images from a lens is limited by diffraction.

• Resolvability requires an angular separation of two point sources to be no less than R where central maximum of one falls on top of the first minimum of the other:

1 1.22 1.22sinR d d

R R R

Rayleigh’s criterion

Lecture 27-Lecture 27-1313Diffraction Gratings

• Devices that have a great number of slits or rulings to produce an interference pattern with narrow fringes.

Types of gratings:

• transmission gratings• reflection gratings

• One of the most useful optical tools. Used to analyze wavelengths.

up to thousands per mm of rulings

D

D d

Maxima are produced when every pair of adjacent wavelets interfere constructively, i.e.,

sin , 0, 1,d m m

mth order maximum

Lecture 27-Lecture 27-1414

Spectral Lines and Spectrometer

• Due to the large number of rulings,the bright fringes can be very narrow and are thus called lines.

• For a given order, the location of aline depends on wavelengths, so light waves of different colors arespread out, forming a spectrum.

Spectrometers are devices that canbe used to obtain a spectrum, e.g.,prisms, gratings, …

Lecture 27-Lecture 27-1515X Ray Diffraction

• X rays are EM radiation of the wavelength on the order of 1 Å, comparable to atomic separations in crystals.

• X rays are produced, e.g., when core electrons in atoms are inelastically excited. They are also produced when electrons are decelerated or accelerated.

• Vacuum tubes, synchrotrons, …

Standard gratings cannot be used as X ray spectrometers.(Slit separation must be comparable to the wavelength!)

Von Laue discovered the use of crystals as 3-dimensional diffraction gratings.

Nobel 1914