1. Interference in One Dimension
We will assume that the waves have the same frequency, and travel to the right along the x-axis.
source

3. Path-length difference
Inherent phase difference
If the waves are initially in-phase
The optical path difference (OPD)
If 0 is constant waves are said to be coherent.

5. If x << λ constructive interference
If x = λ/2 destructive interference

6.  = 0, 2, 4, or any integer multiple of 2.
The condition of being in phase, where crests are aligned with crests and troughs with troughs, is that
 = 0, 2, 4, or any integer multiple of 2.
For identical sources, 0 = 0 rad , maximum constructive interference occurs when x = m ,
Two identical sources produce maximum constructive interference when the path-length difference is an integer number of wavelengths.

7.  =, 3, 5 or any odd multiple of .
The condition of being out of phase, where crests are aligned with troughs of other, that is,
 =, 3, 5 or any odd multiple of .
For identical sources, 0 = 0 rad , maximum constructive interference occurs when x = (m+ ½ ) ,
Two identical sources produce perfect destructive interference when the path-length difference is half-integer number of wavelengths.

9. The Superposition of Many Waves
The superposition of any number of coherent harmonic waves having a given frequency and traveling in the same direction leads to a harmonic wave of that same frequency.

10. The Mathematics of Standing Waves
A sinusoidal wave traveling to the right along the x-axis
An equivalent wave traveling to the left is
Where the amplitude function A(x) is defined as

11. Notes
The amplitude reaches a maximum value Amax = 2a at points where sin kx =1.
The displacement is neither a function of (x-vt) or (x+vt) , hence it is not a traveling wave.
The cos t , describes a medium in which each point oscillates in simple harmonic motion with frequency f= /2.
The function A(x) =2a sin kx determines the amplitude of the oscillation for a particle at position x.

12. The amplitude of oscillation, given by A(x), varies from point to point in the medium.
The nodes of the standing wave are the points at which the amplitude is zero. They are located at positions x for which
That is true if
Thus the position xm of the mth node is
Where m is an integer.

13. Example: Cold Spots in a Microwave Oven
“Cold spots”, i.e. locations where objects are not adequately heated in a microwave oven are found to be 1.25 cm apart.
What is the frequency of the microwaves?

14. If you listen to two sounds with very different frequencies, you hear two distinct tones.
But if the frequency difference is very small, just one or two Hz, then you hear a single tone whose intensity is modulated once or twice every second. That is, the sound goes up and down in volume, loud, soft, loud, soft, ……, making a distinctive sound pattern called beats.

15. When two light waves of different frequency interfere, they also produce beats.
Take E0 to be real.
For a nice demo of beats, check out:

16. The group velocity is the velocity of the envelope or irradiance: the math.
The carrier wave propagates at the phase velocity.
And the envelope propagates at the group velocity:
Or, equivalently, the irradiance propagates at the group velocity:

17. Calculating the Group velocity
vg º dw /dk
Now, w is the same in or out of the medium, but k = k0 n, where k0 is
the k-vector in vacuum, and n is what depends on the medium.
So it's easier to think of w as the independent variable:
Using k = w n(w) / c0, calculate: dk /dw = ( n + w dn/dw ) / c0
vg = c0 / ( n + w dn/dw) = (c0 /n) / (1 + w /n dn/dw )
Finally:
So the group velocity equals the phase velocity when dn/dw = 0,
such as in vacuum. Otherwise, since n increases with w, dn/dw > 0,
and:
vg < vf

18. Calculating Group Velocity vs. Wavelength
We more often think of the refractive index in terms of wavelength ,so let's write the group velocity in terms of the vacuum wavelength l0.

19. In regions of normal dispersion, dn/dw is positive
In regions of normal dispersion, dn/dw is positive. So vg < c for these frequencies.

20. The group velocity is less than the phase velocity in non-absorbing regions.
vg = c0 / (n + w dn/dw)
In regions of normal dispersion, dn/dw is positive. So vg < c for these frequencies.

21. The group velocity can exceed c0 when dispersion is anomalous.
vg = c0 / (n + w dn/dw)
dn/dw is negative in regions of anomalous dispersion, that is, near a
resonance. So vg can exceed c0 for these frequencies!
One problem is that absorption is strong in these regions. Also, dn/dw is only steep when the resonance is narrow, so only a narrow range of
frequencies has vg > c0. Frequencies outside this range have vg < c0.
Pulses of light (which are broadband) therefore break up into a mess.

22. Home Work