Physics 1202: Lecture 28Today’s Agenda

• Announcements:

– Midterm 2: solutions

• HW 8 this FridayHW 8 this Friday

• Diffraction– Review

• Polarization – Reflection by surface

Diffraction

Experimental Observations:(pattern produced by a single slit ?)

So we can calculate where the minima will be !

sin = ± m /a m=±1, ±2, …

Why is the central maximum so much stronger than the others ?

So, when the slit becomes smaller the central maximum becomes ?

Diffraction patterns of two point sources for various angular separation of the sources

Resolution(circular aperture)

min = 1.22 ( / a)

Rayleigh’s criterionfor

circular aperture:

Two-Slit Interference Pattern with a Finite Slit Size

Idiff = Imax [ sin (/2) / (/2) ]2

Diffraction (“envelope” function):

= 2 a sin () /

Itot = Iinter . Idiff

Interference (interference fringes):Iinter = Imax [cos (d sin / ]2

smaller separation between slits => ?

smaller slit size => ?The combined effects of two-slit and single-slit interference. This is the pattern produced when 650-nm light waves pass through two 3.0- mm slits that are 18 mm apart. Animation

ApplicationX-ray Diffraction by crystals

Can we determine the atomic structure of the crystals, like proteins, by analyzing X-ray

diffraction patters like one shown ? A Laue pattern of the enzyme

Rubisco, produced with a wide-band x-ray spectrum. This

enzyme is present in plants and takes part in the process of

photosynthesis.

Yes in principle: this is like the problem of determining the slit separation (d)

and slit size (a) from the observed pattern, but much much more

complicated !

Determining the atomic structure of crystalsWith X-ray Diffraction (basic principle)

2 d sin = m m = 1, 2, ..Crystalline structure of sodium chloride (NaCl). length of the cube edge is a = 0.562 nm.

Crystals are made of regular arrays of atoms

that effectively scatter X-ray

Bragg’s Law

Scattering (or interference) of two X-rays from the

crystal planes made-up of atoms

Polarization of light• Recall E&M wave

• This is an example of linearly polarized light– Electric field along a fixed axis ( here y )

y

x

z

• Most light source are nonpolarized– Electric field along random axis

22_all_imgs_in_ppt

E

Polarizers• Made of long molecules (organic polymers)

– Block electric field along their length– Electric field perpendicular passes through

• So Eafter=E cos

• Recall that I ~ E2

E

E

I = I0 cos2 Malus’s law

Polarization of Electromagnetic Waves (light)

as view along directionof propagation

linearly polarized

unpolarized

I = Imax cos2

Polarization by Absorption

E = Emax cos

Two polarizers have polarization angles set at 1 =15o and 3 = 90o with respect to vertical axis, as show on the Figure below.

(a) By what factor is the intensity of unpolarized light attenuated going through both polarizers (If/Ii = ? ).

EXAMPLE:

If/Ii = cos2(90-15) = 0.07

If/Ii = cos2(45-15) = 0.75If/Ii = cos2(90-45) = 0.55

If/Ii = 0.5x0.75 x 0.55 = 0.21 !

I = Imax cos2

unpolarized

polarized polarized

(b) Does the attenuation factor increase or decrease if a third polarizer is inserted in-between the two polarizers, with polarization angle 2 =45o ?

Unpolarized light through linear polarizer: If/Ii = <cos 2()> = 0.50

If/Ii = 0.5

If/Ii = 0.50 x 0.07 = 0.04 !

Polarization by Reflection

Brewster’s angle:

n1 sin p = n2 sin 2

note : 2 = 90 - p

using : n1 = 1 and n2 = n

n = sin p / cos p

Or: n = tan p

for n =1.55 p = 57o When unpolarized light is

incident on a reflecting surface, the reflected and refracted beams are partially polarized.

The reflected beam is completely polarized when the angle of incidence equals the polarizing angle (Brewster’s angle)

EXAMPLE

a. 22.2°b. 7.7°c. 16.6°d. 36.9°e. 53.1°

How far above the horizon is the moon when its image, reflected in a calm lake, is completely polarized ? (nwater = 1.33)

n = tan p

p = tan-1(1.33) = 53.1o

Polarization by Double Refraction

Unpolarized light incident on a calcite crystal splits into an ordinary (O) ray and an extraordinary (E) ray

which are polarized in mutually perpendicular directions.

A calcite crystal produces a double image because it is a

birefringent (no=1.658, nE=1.486) material.

•There are materials where the speed of light is not the same in all directions. •Such (birefingent) materials thus have two indexes of refraction•And light beam splits into two beam, ordinary (O) and extraordinary (E) ray•E and O rays are also polarized in mutually perpendicular directions.

Application

A plastic model of an arch structure under load conditions observed between perpendicular polarizers. Such patterns are useful in the optimal design of architectural components.

Bright area ==> max stress

Dark area => minimum stress

• Some materials become birefringent when stressed (glass, plastic,..)• Since the birefringence effectively rotates the polarization of the light when such an object is placed between the “polarizer” and “analyzer” oriented at 90o, only the light passing thorough stressed portion of the material will be observed. This provides a way to “image” stress in model plastic structures.