南开大学光学工程内部课件Oct 19th

E1 (r , t ) E01 exp i(k1 r t 1 ) E2 (r , t ) E02 exp i(k2 r t 2 ).
Two plane waves meet at P
Superposition of two beams

The transmitted light is incident onto a screen containing two narrow slits
Young’s Double Slit Experiment

The symmetric narrow slits, S1 and S2 act as the two light sources The waves from the two slits come from the same source S0 and therefore are always in phase.
m


= 0, ±1, ±2, …
Interference Equations

Y:measured vertically from the zeroth order maximum Assumptions


L >>d,
d >>λ

y =LtanθLsinθ
I I1 I 2 2 I1I 2 cos(kd sin ) I1 I 2 2 I1I 2 cos(kdy / L)

Other Coherent Sources

Currently, it is much more common to use a laser as a coherent source The laser produces an intense, coherent,

monochromatic parallel beam over a width of several millimeters

Light from a monochromatic source goes through a narrow slit. The narrow width of the silt make sure transmitted light comes from a tiny region of the source which is coherent.


Therefore, they arrive in phase
Interference Patterns

The upper wave has to travel farther than the lower wave The upper wave travels one wavelength farther
d 7 /(n 1) 6.6 m
Lloyd’s Mirror

Produce an interference pattern with a single light source Wave reach point P either by a direct path or by reflection
P
Obviously,
2 * * * I E E E ( E1 E2 ) ( E1 E2 ) * * * * E1 E1 E2 E2 E1 E2 E2 E1 2 2 * E1 E2 E1 E2 c.c. I1 I 2 I inter
Iinter * E1 Βιβλιοθήκη E2 c.c.I1 E
2 1
,
I2 E
2 2
Interference Pattern
I I1 I 2 2 I1I 2 cos cos
where presents the crossing angle between the linear polarization directions of the two beams， and
k1 r 1 k2 r 2 .
parallel
I I1 I 2 2 I1I 2 cos
Relation between Polarization Directions
perpendicular
I I1 I 2
Coherence Conditions for Interference
Interference Equations

When total destructive interference occurs, a dark fringe is observed
This needs a path difference of an odd half wavelength δ = d sin θdark = (m + ½ ) λ
Two beam interference
Superposition of two beams
P
Assume the observation point P is far enough away from the sources so that the wavefronts at P can be regarded as planes. And we limit our discussion to linearly polarized waves.
In order to get stable intensity-modulated interference pattern by using two beams, two conditions must be satisfied
The two waves must maintain a constant phase with respect to each other, i.e. have the same frequency and constant initial phase The two electric fields have the non-zero projective components onto each other, i.e. the polarization directions cannot be perpendicular to each other


This is destructive interference

A dark fringe occurs
Young’s Experiment (Double-Slit Interference)
Interference Equations

The path difference, δ, is found from the tan triangle Optical Path Differenceδ = L = r2 – r1 = d sin


The reflected ray can be treated as a ray from the image S’ below the mirror
Interference Pattern from Lloyd’s Mirror

Young’s Double Slit Experiment

In 1801, Thomas Young demonstrated for the first time interference of light The coherent sources was gotten through:

mica, thickness d
1
Phase with meca
1 k ' d 2 nd /
Phase without meca
d
2
2 kd 2 d /
air, thickness d
2 d (n 1) / 7 (2 )
The mica thickness is :

Young’s Double Slit Experiment provides a method for measuring wavelength of the light
This experiment gave the wave model of light a great deal of credibility
Laser pointer
Young’s double-slit interference
Side view of the interference field
Interference Patterns

Constructive interference occurs at the center point The two waves travel the same distance
Interference Equations
This assumes the paths are parallel Not exactly, but a very good approximation (L>>d)

Interference Equations
I I1 I 2 2 I1I 2 cos(k ) I1 I 2 2 I1I 2 cos(kd sin )

For a bright fringe interference happens
where
total
constructive
2m k m

m = 0, ±1, ±2, … m is called the order number
When When