1. Two-beam interference:
Consider two waves represented by their electric field vectors E1 and E2:
(10-1)
(10-2)
- position vector r defines point P where the waves intersect to produce disturbance EP given by principle of superposition as:
- E1 & E2 are rapidly varying functions (VIS at 1014  1015 Hz)
- E1 & E2 average to zero over very short time intervals
- but measurement of these waves depends on detected energy of light beam
- irradiance Ee (W/m2) or radiant power density = time average of square of wave amplitude (to avoid mixing up with electric field, we use I to represent irradiance), that is,
(10-3)

2. Resulting irradiance at P is:
Two-beam interference:
Resulting irradiance at P is:
(10-4)
Irradiances of individual waves, I1 and I2
Interference term, I12
(10-5)
Note:
- if E1 & E2 are orthogonal, thus, their dot product = 0 and no interference occurs
- if E1 & E2 are parallel, interference term gives maximum contribution

3. Expanding and multiplying cosine factors results in
Two-beam interference:
Dot product is:
Defining:
we have
Expanding and multiplying cosine factors results in
Note: time averages indicated for each time-dependent factor

4. Averaging over a complete cycle of T:
Two-beam interference:
Averaging over a complete cycle of T:
. . . (10-6)
Where the phase difference between E1 & E2 is:
. . . (10-7)
Thus, interference term becomes:
. . . (10-8)
The irradiances terms I1 and I2 are then:
. . . (11-9a)
. . . (10-9b)

5. Thus eqn (10-5) is rewritten as:
Two-beam interference:
When E01  E02:
and:
Thus eqn (10-5) is rewritten as:
Whether interference term yields contructive or destructive interference depends on (cos ) in eqn (10-11)
. . . (10-10)
. . . (10-11)
- cos  > 0 results in constructive interference;
total constructive interference yields max irradiance
when
at (where m = 0, 1, 2, …) ; disturbances are in-phase
- cos  < 0 results in destructive interference;
total destructive interference yields min irradiance
at (where m = 0, 1, 2, …)
. . . (10-12)
. . . (10-13)

6. Special case when I1 = I2 = I0 : eqn (11-11) becomes
Two-beam interference:
Irradiance of interference fringes as a function of phase.
Special case when I1 = I2 = I0 :
eqn (11-11) becomes
When (equal amplitudes)
and  = 0 (in-phase); Imax = 4I0 and Imin = 0
 exhibits better contrast than the one above
. . . (10-14)

7. Fringe contrast  visibility is :
Two-beam interference:
Fringe contrast  visibility is :
. . . (10-15)
Consider other cases :
If the initial phase difference (1  2) varies randomly, waves are said to be mutually incoherent, and cos  becomes a time-dependent factor whose average = 0
- Interference is actually always taking place, but no pattern can be held long enough to be detected.
Therefore, in order to observe interference, some degree of coherence, that is, cos   0, is necessary
- e.g. two waves from independent sources such as incandescent bulbs or gas-discharge lamps, waves are mutually incoherent - you could not detect the interference pattern
- laser sources are independent but posses sufficient mutual coherence for interference to be observed over short periods of time

8. Inteference of Mutually Incoherent Fields
In practice, for E1 and E2 originating from different sources, this term =0
this is because, no source is perfectly monochromatic
To account for departures from monochromaticity, must allow for the phases to be functions of time i.e. the interference term takes the form:
For real detectors and for all but those laser sources with state-of-the art frequency stability, the time average term will be zero.
In such a case, the sources are term mutually incoherent and I=I1+I2
It is often said that light beams from independent sources, even if both sources are from the same kind of laser, do not interfere with each other. In fact, these fields do interfere but the interference term averages to zero over the averaging times of most real detectors.

9. Inteference of Mutually Coherent Beams
If light from the same laser source is split and then recombined at a detector, the time average term need not be zero
this is because the departures from monochromaticity of each beam, while still present, will be correlated since both beams are from the same source.
In this case,
if both beams travel path of equal duration
In such a case,  is a constant and the interference term takes the form:
Even if the fields travel paths that differ in duration by a time t, the phase difference resulting from the departure from monochromaticity, will still be nearly zero so long as t is less than the so-called coherence time, o , of the source.
Qualitatively, the coherence time of the source is the time over which departures from monochromaticity are small.

10. Young’s double-slit experiment: (wavefront division)
If S2P  S1P = m, waves arrive in phase at P, we have maximum irradiance - bright fringe
If S2P  S1P = (m + ½) , waves arrive 180° out-of-phase at P, we have total cancellation of irradiances of two waves - dark fringe

11. Assuming a << s, OP is almost exactly perpendicular to S1Q
Young’s double-slit experiment:
Assuming a << s, OP is almost exactly perpendicular to S1Q
Condition for constructive interference at point P is:
S2P  S1P =  = m  a sin 
Condition for destructive interference at point P is
 = (m + ½)  = a sin 
where m = 0, 1, 2, 3, …
Irradiance on screen at point determined by angle  is
[path difference related to phase difference through ]
. . . (10-16)
. . . (10-17)

12. m  Order of interference pattern
Young’s double-slit experiment:
Another approximation: y << s for points P near optical axis; thus sin   tan   y/s
Then,
At position when cos function is  1, constructive interference obtained
At position when cos function is 0, destructive interference obtained
Thus positions for bright fringes:
Separation between successive maxima is:
and minima located midway between them
Making hole spacing a smaller, fringe pattern expands
. . . (10-18)
. . . (10-19)
. . . (10-20)
m  Order of interference pattern

13. Double-slit interference with virtual sources :
Lloyd’s mirror
Coherent sources are:
S - real source
S’ - virtual (reflection)
When screen contacts mirror at M’, dark fringe observed at M’. This is because of phase shift of  at air-glass reflection (though optical-path difference between the two beams is zero)

14. Fresnel’s double mirrors
Double-slit interference with virtual sources :
Fresnel’s double mirrors
Coherent sources are the two virtual images of point source S, formed in the two plane mirrors M1 and M2. Direct light from S is not allowed to reach the screen.

15. Fresnel’s biprism Double-slit interference with virtual sources :
n = index of prism
Refracted light seems to come from two coherent, virtual sources S1 and S2.
Prism angle  is small(~1o); deviation angle m = (n  1); virtual source separation a = 2dm