8 24.1 Light waves and the coherent condition of waves ifA1≠A2then p=2nz,A=A1+A2(n=0,1,2,…) 4=(2n+1)z,A=4-42|(n=0,12,…) 里(14c0-a+= >Case 2 Y,(r, t)=A cos(kr -at+o) y(r,t)=y1(r,t)+y2(r,) A, cos(kr -at+o)+A cos(hr -at+o) s24.1 Light waves and the coherent condition of waves (kr, -at+o)+(kr, -at+o) y(r, t)=2A cos[ 2 cos[ (kr-at+o)-(k 2-+ 2 A, coS( )cos(r2+r1)-at+小l P(r, t)=2A, cos (2-G)cosl(r+2)-at+pI 2 Path difference The total amplitude k depends on (F2-r1 △r 6
6 if A1 ≠ A2 then (2 1) , ( 0,1,2, ) 2 , ( 0,1,2, ) 1 2 1 2 L L = + = − = = = + = n A A A n n A A A n ∆φ π ∆φ π cos( ) cos( ) ( , ) ( , ) ( , ) 1 1 1 2 1 2 = −ω + φ + −ω + φ = + A kr t A kr t Ψ r t Ψ r t Ψ r t ¾Case 2 ( , ) cos( ) ( , ) cos( ) 2 1 2 1 1 1 ω φ ω φ Ψ = − + Ψ = − + r t A kr t r t A kr t o2 r2 r1 p o1 §24.1 Light waves and the coherent condition of waves ( ) ] 2 )cos[ 2 2 cos( ] 2 ( ) ( ) cos[ ] 2 ( ) ( ) ( , ) 2 cos[ 2 1 2 1 1 1 2 1 2 1 ω φ ω φ ω φ ω φ ω φ + − + − = − + − − + ⋅ − + + − + = r r t kr kr k A kr t kr t kr t kr t Ψ r t A ( ) ] 2 ( ) cos[ 2 ( , ) 2 cos = 1 2 − 1 r1 + r2 −ωt + φ k r r k Ψ r t A r r r k − = ∆ λ π ( ) 2 2 1 The total amplitude depends on §24.1 Light waves and the coherent condition of waves Path difference
824.1 Light waves and the coherent condition of waves Ar=n, A=2A(n=0, 1, 2, . constructive Ar=(2n+1), A=0(n=0, 1, 2,)destructive In another words The phase difference 8 pk(巧2一)÷2z Ar=nh, Spath =2nT(n=0, 1, 2, . in phase △r=(2n+1) 2 =(2n+1)(n=0,1,2, O t of of phase s24.1 Light waves and the coherent condition of waves Same frequency and same direction of motion A, cos( at +ou) x2=A2 cos(at+p2) xsr +x Acos(ot+φ) A=A,+A A=y4+A2+24142cos(-q) A1sing+A2sin吗 P=arct A, cos+ A, coso
7 , 0 ( 0,1,2, ) 2 (2 1) , 2 ( 0,1,2, ) 1 L L ∆ = + = = ∆ = = = r n A n r n A A n λ λ constructive destructive In another words: = k r − r = ∆r λ π δ 2 ( ) The phase difference path 2 1 , (2 1) ( 0,1,2, ) 2 (2 1) , 2 ( 0,1,2, ) path path L L ∆ = + = + = ∆ = = = r n n n r n n n δ π λ λ δ π out of phase in phase §24.1 Light waves and the coherent condition of waves 2 cos( ) 1 2 2 1 2 2 2 A = A1 + A + A A φ −φ 1 1 2 2 1 1 2 2 cos cos sin sin arctg φ φ φ φ φ A A A A + + = cos( ) 1 2 = ω + φ = + A t x x x A A1 A2 r r r = + A r A1 r A2 r ω ω ω x 1 x 2 x φ φ1 φ 2 x Same frequency and same direction of motion cos( ) 1 = 1 ω + φ 1 x A t cos( ) 2 = 2 ω + ϕ 2 x A t §24.1 Light waves and the coherent condition of waves
824.1 Light waves and the coherent condition of waves x=a, cos( at +pu) x2=A2 cos(at+2) Y(r, t)=A, cos(kr -at+o) Y,(r, t)=A, cos(kr, -at+o) A=VA+A2+2A, 42 coskk( -) 2丌 =12421+cos2(△r) V24(1+cos Sath]=4A cos2-path 2 卯=r1si(+)+$m+小 A cos(kr +o)+A, cos(kr= +p) s24.1 Light waves and the coherent condition of waves 6. Wave intensity The power of transmitted by the wave de dP dt Po242 n?[kx-ax)+Bdydz The intensity of a wave is the average power transmitted by the wave through one square meter oriented perpendicular to the direction the wave is propagating de dydz dydz Aav 8
8 cos( ) 1 = 1 ω + φ 1 x A t cos( ) 2 = 2 ω + ϕ 2 x A t 2 2 [1 cos ] 4 cos ( )] 2 2 [1 cos 2 cos[ ( )] 2 2 path path 1 2 1 2 1 1 2 2 1 2 2 2 1 1 2 δ δ λ π A A A r A A A A A k r r A A = + = = + ∆ = + + − = cos( ) cos( ) sin( ) sin( ) arctg 1 1 2 2 1 1 2 2 φ φ φ φ ϕ + + + + + + = A kr A kr A kr A kr ( , ) cos( ) ( , ) cos( ) 2 2 2 1 1 1 ω φ ω φ Ψ = − + Ψ = − + r t A kr t r t A kr t §24.1 Light waves and the coherent condition of waves 6. Wave intensity The power of transmitted by the wave v A [ ] kx t y z t E P sin ( ) d d d d d 2 2 2 = = ρ ω −ω + φ The intensity of a wave is the average power transmitted by the wave through one square meter oriented perpendicular to the direction the wave is propagating. A v vA kx t t y z y z y z T P I T 2 2 0 av 2 2 2 av 2 1 sin [( ) ]d d d 1 d d 1 d d d ρ ω ρ ω ω φ = = = − + ∫ §24.1 Light waves and the coherent condition of waves
824.1 Light waves and the coherent condition of waves I=NPAOVocA2 For light oscillation: E=E coS( @t +Pu) 122-2 PEI@ ac Er 6. The methods of obtaining the coherent waves ① wavefront division ② amplitude division 824.2 Youngs double slit experiment 1. Installation and the phenomena of experiment This is a typical method of wavefront division
9 2 2 2 2 1 I = A v ∝ A r r ρ ω 6. The methods of obtaining the coherent waves 1wavefront division 2amplitude division §24.1 Light waves and the coherent condition of waves For light oscillation: cos( ) = 1 ω + φ 1 E E t 2 1 2 2 1 2 1 I = E v ∝ E r r ρ ω §24.2 Young’s double slit experiment 1. Installation and the phenomena of experiment This is a typical method of wavefront division
824.2 Youngs double slit experiment 2. Theoretical analysis Two light oscillations at point P ciden E,r,t) wave =Eo cos(hr-at+o E,(r, t) =Eo cos(hr -at+o) Path length difference r -r=sine eparation of the slits If d>>d 824.2 Youngs double slit experiment Phase difference Ar=-dsinB Path length differenc 4 The condition for constructive interference (bright fringe or maximum) and destructive interference(dark fringe or minimum)on the distant screen is
10 2. Theoretical analysis cos( ) ( , ) cos( ) ( , ) 0 2 2 0 1 1 ω φ ω φ = − + = − + E kr t E r t E kr t E r t Path length difference ∆r = r1 −r2 = d sinθ Two light oscillations at point P §24.2 Young’s double slit experiment Separation of the slits If D>>d ∆r The condition for constructive interference (bright fringe or maximum) and destructive interference (dark fringe or minimum)on the distant screen is Phase difference θ λ π ∆ λ π δ sin 2 2 path = r = d §24.2 Young’s double slit experiment