Chapter 10. Interference of Light

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1 Chapter 10. Interference of Light Last Lecture Wave equations Maxwell equations and EM waves Superposition of waves This Lecture Two-Beam Interference Young s Double Slit Experiment Virtual Sources Newton s Rings Film Thickness Measurement by Interference

2 Two-Beam Interference r r Consider two waves E1 and E2 that have the same frequency ω : r r r E1 r t E01 k1 r t 1 r E r t E k r t r r (, ) = cos( ω + ε ) r r r r (, ) = cos( ω + ε ) Hecht, Optics, Chapter 9. (polarization direction)

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 x

5 The time averages are given by Two-Beam Interference The interference term is given by : phase difference

6 The total irradiance is given by Two-Beam Interference 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 When

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

8 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

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

10 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 : a θ y Δ s

11 Young s Double Slit Interference On the screen the irradiance pattern is given by Assuming that y << s : Bright fringes: Dark fringes:

12 Interference Fringes From 2 Point Sources

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

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

15 Interference With Virtual Sources: Lloyd s Mirror Light source mirror Rotation stage

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

17 Interference in in Dielectric Films

18 Analysis of Interference in in Dielectric Films

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

20 Analysis of Interference in 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 Constructive interference Destructive interference

21 Fringes of Equal Inclination Fringes arise as Δ varies due to changes in the incident angle: Constructive interference Destructive interference

22 Fringes of Equal Thickness When the direction of the incoming light is fixed, fringes arise as Δ varies due to changes in the dielectric film thickness : Constructive interference Destructive interference

23 Fringes of Equal Thickness: Newton s Rings

24 Fringes of Equal Thickness: Newton s Rings

Chapter 7. Interference of Light

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