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Interference
Adrian Down
April 05, 2006
1 Conditions for interference
1.1 Review
Recall the result for interference of fully polarized radiation beams A and B. The combined irradiance is,
μ
I = |
EA|^2 + |
EB |^2 + |
EA||
EB | 2 <
J A†JB eı(φB^ −φA)
This formula led to the conditions for no interference,
- ωA 6 = ωB
- Unpolarized beams
- Perpendicularly polarized beams (J A†JB = 0)
- At a point halfway between a light and dark fringe of an interfenence pattern, where φB − φA = (n odd) · π 2
1.2 Newton’s rings: example of interference
Newton’s rings are produced by a thin lens on top of a reflective surface. The spacing between the lens and the mirror is on the same order as the wavelength of light in the visual spectrum. Interference occurs between light reflected from the lens and light reflected from the reflective surface below.
This phenomenon can be observed from oil floating on the surface of a puddle of water. Due to surface tension, the depth of the sheen of oil is not uniform across the surface of the puddle. The example of Newton’s rings shows that it is possible to have interfer- ence between two beams of the same frequency.
2 Interference of incompletely polarized light
2.1 General case
Thus far, we have discussed the case of a fixed angle φxy between the two beams. In this case, the electric fields are,
E(z, t) = <
Eeı(˜^ kz−ωt)
In the case that the light is not fully polarized, the relative phase of the components of the fields varies in time. This variation is slow on the order of the frequency of the wave. This slow variation is encapsulated in the complex electric fields,
E(z, t) = <
E(t)eı(˜^ kz−ωt)
When the beams A and B are not fully polarized, it is no longer the case that φA − φB is fixed. Even if the two beams are taken from the same mother beam, the phase between the waves could be a function of the time when the samples of the beam are taken. It is difficult in practice to create the conditions under which coherence occurs. Even with a vary high quality laser, there is some time dependence in the phase of the outgoing beam. Thus, it is desirable to discuss the more practical case of interference of incompletely polarized light.
2.2 Michelson Interferometer
The device consists of two mirrors and a screen. A beam enters and is reflected by a mirror at 45◦^ relative to the direction of the incoming radiation. There is a slit in the center of the mirror that allows some of the incoming radiation to be transmitted. This transmitted portion of the beam is reflected back to the other side of the angled mirror by a plane mirror. The radiation
The four fields in this case are,
EA(P ) = <
EAP (t)e−ıωt
EB (P ) = <
EBP (t)e−ıωt
EA(Q) = <
EAQ(t)e−ıωt
EB (Q) = <
EBQ(t)e−ıωt
3.2.2 Calculation
The difficulty is in the notation. The rest is just algebra. The wave crest propagates according to eı(k·r−ωt). The flight time τA B
of the wave crest from
P to Q is given by an integral along the path of which the beam follows,
ωτA B
Q
P
kA B
· drA B
The field at Q can be related to the field at point P at an earlier time. The relation of the physical fields implies a relation between the complex fields.
EA B
(Q, t + τA B
) = EA
B
(P, t)
⇔ E
A B Q(t^ +^ τ BA^ )e
−ıωτA B (^) = E
A B P (t)
⇔ E
A B Q(t^ +^ τ BA^ ) =^ E
A B P (t)e
ıωτA B
This holds for all time, so choose another time t′^ = t − τ , where τ = τB − τA. Substituting and using some algebra,
EBQ(t + τA) = EBP (t − τ )eıωτB
3.3 Irradiance
The irradiance at point Q is (neglecting the constant factors),
( 2
μ
IQ = |EAQ|^2 + |EBQ|^2 + 2<
EA Q∗ EBQ
Evaluating the irradiance at t + τA and substituting the expressions for the fields above,
EAQ (t + τA) = EPA (t)eıωτA^ EQB (t + τA) = EPB (t − τ )eıωτB ⇒ IQ(t + τA) = IQA (t + τA) + IQB (t + τA) + 2<
EA P ∗(t) · EBP (t − τ )eıωτ^
Taking the time average gives the final form of the expression for the ir- radiance of the combined beam. The last term represents the interference between the two beams,
〈IQA+ B〉 = IA^ + IB^ + 〈 2 <
EA P ∗(t) · EBP (t − τ )eıωτ^
The τ in this expression represents the path length between the points P and Q. This expression indicates that interference occurs because of phase differences due to path length differences between the two beams.
