UNIVERSITY PHYSICS II CHAPTER 24 Physical optics s24.1 Light waves and the coherent condition of waves 1. Light waves Light wave is a small part of the whole electromagnetic wave spectrum. Wavelength(m) 700mm650600550500450400mm
1 1. Light waves §24.1 Light waves and the coherent condition of waves Light wave is a small part of the whole electromagnetic wave spectrum
824.1 Light waves and the coherent condition of waves 2. The interference phenomena of waves 824.1 Light waves and the coherent condition of waves 3. Coherence and the conditions of coherence OA superposition of waves may give rise to variations in the resulting amplitude of the total wave disturbance, known as interference @the conditions of coherence >The same physical type of waves, and same direction of the oscillation The same frequency; )A phase difference that is independent of time The phase difference is the difference between the individual phases of the two waves
2 §24.1 Light waves and the coherent condition of waves 2. The interference phenomena of waves 3. Coherence and the conditions of coherence 1A superposition of waves may give rise to variations in the resulting amplitude of the total wave disturbance, known as interference. 2the conditions of coherence ¾The same physical type of waves, and same direction of the oscillation; ¾ The same frequency; ¾A phase difference that is independent of time. The phase difference is the difference between the individual phases of the two waves. §24.1 Light waves and the coherent condition of waves
8 24.1 Light waves and the coherent condition of waves 4. The phase difference Simple harmonic oscillation x(t=Acos(at+o) Time( A---the amplitude 2兀 a---the angular frequency a 小- the initial phase at+o---phase 824.1 Light waves and the coherent condition of waves Sinusoidal(harmonic) waves Y(x, t)=Acos(lr-@t+o) △x Wave at t=△t Wave at t=0 k 2r ---angular wave number
3 4. The phase difference Simple harmonic oscillation x(t) = Acos(ωt +φ ) A---the amplitude ω---the angular frequency φ---the initial phase ω t + φ ---phase T π ω 2 = §24.1 Light waves and the coherent condition of waves Sinusoidal(harmonic) waves Ψ (x,t) = Acos(kx −ωt +φ ) λ 2π k = ---angular wave number §24.1 Light waves and the coherent condition of waves
8 24.1 Light waves and the coherent condition of waves If two waves Y(r, t)=Acos(kr-@t+Ou) ￥2(x,D)=Acos(kx-ort+中2) The phase difference of the two waves 6=(k n2)-(kx-or+)=中2- If two waves Y(x, t)=Acos(hx, -at+o) Y2(x, t)=Acos(Kx-at+ The phase difference of the two waves 2丌 ath at+o)-(l-at+o) )==44x 824.1 Light waves and the coherent condition of waves 5. The f principle of superpose ition and interference of waves If the amplitudes are not too large, the total wave disturbance at any point x and time t is the sum of the individual wave disturbance (x,D)=1(x,D)+￥2(x,D)+3(x,D)+ Case 1 Y(x, t)=A, cos( lx -@t +u) p2(x, t) t+p2) y(x,D)=y1(x,t)+y2(x,) A, cos(kr-at+o1)+A, cos(kx-at +o2)
4 ( , ) cos( ) ( , ) cos( ) 2 2 1 1 Ψ ω φ Ψ ω φ = − + = − + x t A kx t x t A kx t If two waves The phase difference of the two waves 2 1 2 1 δ = (kx −ωt +φ ) − (kx −ωt +φ ) = φ −φ ( , ) cos( ) ( , ) cos( ) 2 2 1 1 Ψ ω φ Ψ ω φ = − + = − + x t A kx t x t A kx t If two waves The phase difference of the two waves kx t kx t ∆x λ π δ ω φ ω φ 2 ( ) ( ) path = 2 − + − 1 − + = §24.1 Light waves and the coherent condition of waves 5. The principle of superposition and interference of waves If the amplitudes are not too large, the total wave disturbance at any point x and time t is the sum of the individual wave disturbance. Ψ (x,t) =Ψ1 (x,t) +Ψ2 (x,t) +Ψ3 (x,t) +L cos( ) cos( ) ( , ) ( , ) ( , ) 1 1 1 2 1 2 = −ω +φ + −ω +φ = + A kx t A kx t Ψ x t Ψ x t Ψ x t ( , ) cos( ) 1 = 1 − ω + φ 1 Ψ x t A kx t ( , ) cos( ) 2 = 1 − ω + φ 2 Ψ x t A kx t ¾Case 1 §24.1 Light waves and the coherent condition of waves
8 24.1 Light waves and the coherent condition of waves Y(x, t =2A, cos kx-o+中)+(kx-ot+中 (hr-at+ou-ckr-at+e) cos 2 2A, coS( )cos(hr-at n+中2 2 2 Y(x, t)=2A, cos( n2- )cos(kx-at+ +吗2 2 4p=2n丌,A=2A1(n=0,2,…) constructive 4小p=(2n+1)z,A=0(n=0,1,2,…) destructive 824.1 Light waves and the coherent condition of waves YI(x, t) 平(x0)xn) Y(r t y(r,) 2(x,n) △φ=0 △p=丌 p=2x/3 Hxt)
5 ) 2 )cos( 2 2 cos( ] 2 ( ) ( ) cos[ ] 2 ( ) ( ) ( , ) 2 cos[ 2 1 1 2 1 1 2 1 2 1 φ φ ω φ φ ω φ ω φ ω φ ω φ + − + − = − + − − + ⋅ − + + − + = A kx t kx t kx t kx t kx t Ψ x t A ) 2 ) cos( 2 ( , ) 2 cos( 2 1 1 2 1 φ φ ω φ φ + − + − Ψ x t = A kx t (2 1) , 0 ( 0,1,2, ) 2 , 2 ( 0,1,2, ) 1 L L = + = = = = = n A n n A A n ∆φ π ∆φ π constructive destructive §24.1 Light waves and the coherent condition of waves ∆φ = 0 ∆φ = π ∆φ = 2π 3 ( , ) and ( , ) 2 1 Ψ x t Ψ x t ( , ) 1 Ψ x t ( , ) 2 Ψ x t ( , ) 1 Ψ x t ( , ) 2 Ψ x t Ψ(x,t) Ψ(x,t) Ψ(x,t) Ψ Ψ Ψ Ψ Ψ Ψ §24.1 Light waves and the coherent condition of waves
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
