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

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

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

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

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

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

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

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

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

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