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The Hong Kong Polytechnic University 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: 2 E2 sin kx t 2 1 E1 sin kx t 1 Let 1 kx 1 2 kx 2 We have 1 E1 sin 1 t 2 E2 sin 2 t Resultant 1 2 E sin t E sin kx t E 2 E12 E22 2E1 E2 cos 2 1 E1 sin 1 E2 sin 2 tan E1 cos 1 E2 cos 2 tan I I1 I 2 2 I1 I 2 cos 2 1 E1 sin 1 E2 sin 2 E1 cos 1 E2 cos 2 interference term Two waves in phase result in total constructive interference: I max I1 I 2 2 I1 I 2 Two waves anti-phase result in total destructive interference: I min I1 I 2 2 I1 I 2 Optics II----by Dr.H.Huang, Department of Applied Physics 1 The Hong Kong Polytechnic University 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, kx2 t 2 kx1 t 1 k x2 x1 2 1 in a medium: 2 m x2 x1 2 1 2 nx2 x1 2 1 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 2 The Hong Kong Polytechnic University 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 . 1 E01 sin t 1 2 E02 sin t 2 Suppose: 1 2 E0 sin t E0 E01 cos 1 E02 cos 2 2 E01 sin 1 E02 sin 2 2 E01 sin 1 E02 sin 2 tan E01 cos 1 E02 cos 2 For multiple waves: N E0i sin t i E0 sin t i 1 E0 X 2 Y2 N X E 0i cos i i 1 and tan Y X N Y E0i sin i i 1 Optics II----by Dr.H.Huang, Department of Applied Physics 3 The Hong Kong Polytechnic University Superposition of Light Waves Example: Find the resultant of adding the sine waves: 2 10 sin t 4 1 20 sin t X 20 cos 0 10 cos Y 0 10 sin 3 10 sin t 12 4 15sin t 2 3 2 10 cos 15 cos 29.23 4 3 12 2 10 sin 15 sin 17.47 4 3 12 E X 2 Y 2 34 34 sin t 6 tan 1 Y X 30 Example: Find, using algebraic addition, the amplitude and phase resulting from the addition of the two superposed waves 1 E1 sin kx t 1 and 2 E2 sin kx t 2 , where 1=0, 2=/2, E1=8, E2=6, and x=0. 1 kx 1 0 arctan 2 kx 2 2 E E12 E22 2 E1 E2 cos 2 1 10 E1 sin 1 E2 sin 2 arctan 0.75 36.87 E1 cos 1 E2 cos 2 10 sin kx t 0.6435 Optics II----by Dr.H.Huang, Department of Applied Physics 4 The Hong Kong Polytechnic University Superposition of Light Waves Example: Two waves 1 E1 sin kx t and 2 E2 sin kx t are coplanar and overlap. Calculate the resultant’s amplitude if E1=3 and E2=2. E 2 E12 E22 2 E1 E2 cos 2 1 32 22 2 3 2 cos 1 E 1 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. time distance d d nd speed v cn c Optics II----by Dr.H.Huang, Department of Applied Physics 5 The Hong Kong Polytechnic University Superposition of Light Waves Standing Wave; Suppose two waves: I E0 I sin kx t I and R E0 R sin kx t R having the same amplitude E0I=E0R and zero initial phase angles. 1 2 2E0 I sin kx cos t Nodes at: x n , 2 n 0, 1, 2, 3,.... 1 x n Antinodes at: , 2 2 nodes or nodal points antinodes n 0, 1, 2, 3,.... Optics II----by Dr.H.Huang, Department of Applied Physics 6 The Hong Kong Polytechnic University Superposition of Light Waves Addition of Waves of Different Frequency: 2 E1 cosk2 x 2t 1 E1 cosk1 x 1t 1 2 2 E1 cosk p x p t cosk g x g t p 1 2 2 Group velocity: g vg 1 2 2 g kg dispersion relation =(k) 1 2 k1 k 2 vg k d dk Optics II----by Dr.H.Huang, Department of Applied Physics 7 The Hong Kong Polytechnic University Superposition of Light Waves Coherence: Frequency bandwidth: 1 2 Coherent time: t 1 Coherent length: x ct 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). 1 6 2 10 500 10 9 (b) : OPL n1d1 n2d2 1 0.90 1.5 0.10 1.05 m OPL 1.05 6 number of wavelengt hs 2 . 1 10 500 10 9 (c) : OPD 1.05 1 0.05 m OPD 0.05 5 6 6 (d ) : 10 2 . 1 10 2 . 0 10 500 10 9 (a) : number of wavelengt hs Optics II----by Dr.H.Huang, Department of Applied Physics 8 The Hong Kong Polytechnic University 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. OPL1 nd 1 m OPL2 na d1 d 5 ng d 2 d 4 nw d 3 1 0.78 1.5 0.02 1.33 0.20 1.076 m OPD OPL2 OPL1 0.076 m number of wavelengt hs 2 OPD OPD 0.076 5 1 . 52 10 500 10 9 9.55 105 radian Optics II----by Dr.H.Huang, Department of Applied Physics 9 The Hong Kong Polytechnic University 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+). s x, t 2E sin kx cos t s x, t 2 E sin kx cos t 2 E sin kx cost 2 E sin kx cost 2 2 E sin kx cos 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 10