1. Optics II----by Dr.H.Huang, Department of Applied Physics
Superposition of Light Waves
Principle of Superposition:
When two waves meet at a particular point in space, the resultant disturbance is simply the algebraic sum of the constituent disturbance.
Addition of Waves of the Same Frequency:
Let
We have
Resultant
interference term
Two waves in phase result in total constructive interference:
Two waves anti-phase result in total destructive interference:
Optics II----by Dr.H.Huang, Department of Applied Physics

2. Optics II----by Dr.H.Huang, Department of Applied Physics
Superposition of Light Waves
Coherent: Initial phase difference 2-1 is constant.
Incoherent: Initial phase difference 2-1 varies randomly with time.
Phase difference for two waves at distance x1 and x2 from their sources,
in a medium:
Optical Path Difference (OPD): n(x2-x1)
Optical Thickness or Optical Path Length (OPL): nt
Optics II----by Dr.H.Huang, Department of Applied Physics

3. Optics II----by Dr.H.Huang, Department of Applied Physics
Superposition of Light Waves
Phasor Diagram:
Each wave can be represented by a vector with a magnitude equal to the amplitude of the wave. The vector forms between the positive x-axis an angle equal to the phase angle .
Suppose:
For multiple waves:
Optics II----by Dr.H.Huang, Department of Applied Physics

4. Optics II----by Dr.H.Huang, Department of Applied Physics
Superposition of Light Waves
Example:
Find the resultant of adding the sine waves:
Find, using algebraic addition, the amplitude and phase resulting from the addition of the two superposed waves and , where 1=0, 2=/2, E1=8, E2=6, and x=0.
Optics II----by Dr.H.Huang, Department of Applied Physics

5. Optics II----by Dr.H.Huang, Department of Applied Physics
Superposition of Light Waves
Example:
Two waves and are coplanar and overlap. Calculate the resultant’s amplitude if E1=3 and E2=2.
Example:
Show that the optical path length, or more simply the optical path, is equivalent to the length of the path in vacuum which a beam of light of wavelength  would traverse in the same time.
Optics II----by Dr.H.Huang, Department of Applied Physics

6. Optics II----by Dr.H.Huang, Department of Applied Physics
Superposition of Light Waves
Standing Wave;
Suppose two waves: and
having the same amplitude E0I=E0R and zero initial phase angles.
nodes or
nodal points
antinodes
Nodes at:
Antinodes at:
Optics II----by Dr.H.Huang, Department of Applied Physics

7. Optics II----by Dr.H.Huang, Department of Applied Physics
Superposition of Light Waves
Addition of Waves of Different Frequency:
Group velocity:
dispersion relation =(k)
Optics II----by Dr.H.Huang, Department of Applied Physics

8. Optics II----by Dr.H.Huang, Department of Applied Physics
Superposition of Light Waves
Coherence:
Frequency bandwidth:
Coherent time:
Coherent length:
Example:
(a) How many vacuum wavelengths of =500 nm will span space of 1 m in a vacuum? (b) How many wavelengths span the gap when the same gap has a 10 cm thick slab of glass (ng=1.5) inserted in it? (c) Determine the optical path difference between the two cases. (d) Verify that OPD/ is the difference between the answers to (a) and (b).
Optics II----by Dr.H.Huang, Department of Applied Physics

9. Optics II----by Dr.H.Huang, Department of Applied Physics
Superposition of Light Waves
Example:
In the figure, two waves 1 and 2 both have vacuum wavelengths of 500 nm. The waves arise from the same source and are in phase initially. Both waves travel an actual distance of 1 m but 2 passes through a glass tank with 1 cm thick walls and a 20 cm gap between the walls. The tank is filled with water (nw=1.33) and the glass has refractive index ng=1.5. Find the OPD and the phase difference when the waves have traveled the 1 m distance.
Optics II----by Dr.H.Huang, Department of Applied Physics

10. Optics II----by Dr.H.Huang, Department of Applied Physics
Superposition of Light Waves
Example:
Show that the standing wave s(x,t) is periodic with time. That is, show that s(x,t)= s(x,t+).
Homework: 11.1; 11.3; 11.4; 11.5; 11.6
Optics II----by Dr.H.Huang, Department of Applied Physics