Chapter 7. Interference of Light

Transcription

1 Chapter 7. Interference of Light Last Lecture Superposition of waves Laser This Lecture Two-Beam Interference Young s Double Slit Experiment Virtual Sources Newton s Rings Film Thickness Measurement by Interference Stokes Relations Multiple-Beam Interference in a Parallel Plate

2 Two-Beam Interference r r Consider two waves E1 and E2 that have the same frequency ω : r r r r r r E1(, t) = E01cos( k1 ωt + φ1) = E01cos( ks1 ωt + φ1) r r r r r r E t E k t E ks t (, ) = cos( ω + φ ) = cos( ω + φ ) S 2 k r 1 k r 2 S 1

3 Two-Beam Interference Interference term

4 Two-Beam Interference The irradiances for beams 1 and 2 are given by : The interference term is given by r r r r E1 E2 = E01 E02cos( ks1 ωt+ φ1)cos( ks2 ωt+ φ2) r r r r 2 E1 E2 = 2E01 E02 cos( α ωt)cos( β ωt) r r = E01 E02 cos( α + β 2 ωt) + cos( β α) r r = E01 E02 cos( β α) r r = E E cosδ ( s ) ( ) { α ks + φ, β ks + φ } { 2cos( A)cos( B) = cos( A+ B) + cos( A B) } δ k s + φ φ : phase difference I = I + I + 2 I I cosδ E01 E r r when (same polarization) I = I + I + 2 I I cosδ for purely monochromatic light

5 Interference of mutually incoherent beams I = I + I + 2 I I cosδ { k( s1 s2) 2 t 1 t } cosδ = cos + φ () φ () mutual incoherence means φ 1 (t) and φ 2 (t) are random in time cosδ = 0 I = I1+ I2 Mutually incoherent beams do not interference with each other

6 Interference of mutually coherent beams I = I + I + 2 I I cosδ { k( s1 s2) 2 t 1 t } cosδ = cos + φ () φ () mutual coherence means [ φ 1 (t) - φ 2 (t) ] is constant in time I = I + I + 2 I I cosδ { ( 1 2) } cosδ = cosδ = cos k s s Mutually coherent beams do interference with each other

7 Interference of mutually coherent beams The total irradiance is given by There is a maximum in the interference pattern when This is referred to as constructive interference. There is a minimum in the interference pattern when This is referred to as destructive interference

8 Visibility Visibility = fringe contrast V I I max max + I I min min { 0 V 1 } When I = 4I I min = 0 max 0 Therefore, V = 1

9 Sources must be: Conditions for good visibility same in phase evolution in terms of time (source frequency) temporal coherence space (source size) spatial coherence Same in amplitude Same in polarization Normal Very good Very bad

10 Young s Double Slit Experiment Hecht, Optics, Chapter 9.

11 Young s Double Slit Interference Assume that y << s and a << s. The condition for an interference maximum is The condition for an interference minimum is Relation between geometric path difference and phase difference : 2π δ = kδ= Δ λ a θ y Δ s

12 Young s Double Slit Interference On the screen the irradiance pattern is given by Assuming that y << s : Bright fringes: Dark fringes: λs a Δ ymax =Δ ymin =

13 Interference Fringes From 2 Point Sources Figure 7-5

14 Interference Fringes From 2 Point Sources Two coherent point sources : P 1 and P 2

15 Interference With Virtual Sources: Fresnel s Double Mirror Hecht, Optics, Chapter 9.

16 Interference With Virtual Sources: Lloyd s Mirror Light source Figure 7-7 mirror Rotation stage

17 Interference With Virtual Sources: Fresnel s Biprism Hecht, Optics, Chapter 9.

18 7-4. Interference in Dielectric Films

19 Analysis of Interference in Dielectric Films Figure 7-12

20 Analysis of Interference in Dielectric Films The phase difference due to optical path length differences for the front and back reflections is given by

21 Analysis of Interference in Dielectric Films Also need to account for phase differences Δ r due to differences in the reflection process at the front and back surfaces Δ r Δ = Δ p + Δ r Δ r Constructive interference Destructive interference

22 Fringes of Equal Inclination Fringes arise as varies due to changes in the incident angle: θ i θ t Constructive interference Destructive interference Figure 7-13

23 7-5. Fringes of Equal Thickness When the direction of the incoming light is fixed, fringes arise as varies due to changes in the dielectric film thickness : Constructive interference Bright fringe Destructive interference Dark fringe

24 7-6. Fringes of Equal Thickness: Newton s Rings Figure 7-17

25 Newton s rings R-t m θ R R = r + ( R t ) R= m m r 2 m 2Rt m r + t 2t 2 2 m m m n r m t m << R Maxima (bright rings) when, t m λ λ Δ= 2 tm + = mλ ( nf = 1, Δ r = ) 2 2 r = m λr m Minima (dark rings) when, reflection transmission λ 1 Δ= 2tm + = m+ λ 2 2 r = 2 m mλr

26 7-7. Film-thickness measurement by interference

27 7-8. Stokes Relations in reflection and transmission E i is the amplitude of the incident light. The amplitudes of the reflected and transmitted beams are given by From the principle of reversibility Stokes relations r = e iπ r

28 7-9. Multiple-Beam Interference in a Parallel Plate I 0 I R I T Transmittance I T T I 0 Reflectance I R R I 0

29 Multiple-Beam Interference in a Parallel Plate I = E I R = E R 2 Find the superposition of the reflected beams from the top of the plate. The phase difference between neighboring beams is δ = kδ Δ = 2ndcosθ f t r t Given that the incident wave is t d r I T = E T 2

30 Multiple-Beam Interference in a Parallel Plate The reflected amplitude resulting from the superposition of the reflected beams from the top of the plate is given by Define Therefore

31 Multiple-Beam Interference in a Parallel Plate The Stokes relations can now be used to simplify the expression

32 Multiple-Beam Interference in a Parallel Plate The reflection irradiance is given by

33 Multiple-Beam Interference in a Parallel Plate The transmitted irradiance is given by

34 Multiple-Beam Interference in a Parallel Plate Minima in reflected irradiance and maxima in transmitted irradiance occur when Δ = 2ndcosθ = f t mλ Minima in transmitted irradiance and maxima in reflected irradiance occur when Δ= 2nd f cosθt = m+ 1 λ 2