Waves & Oscillations

Transcription

1 Physics Waves & Oscillations Lecture 37 Interference Spring 2016 Semester Matthew Jones

2 Multiple Beam Interference In many situations, a coherent beam can interfere with itself multiple times Consider a beam incident on a thin film Some component of the light will be reflected at each surface and some will be transmitted Incident light Each transmitted beam will have a different phase relative to the adjacent beams. What is the total intensity of the reflected light?

3 Multiple Beam Interference All transmitted and reflected rays will be parallel They can be focused onto points P and P by lenses: Incident light What we need to know: Transmission and reflection coefficients Path length of refracted rays in the film

4 Multiple Beam Interference Reflection coefficients:and Transmission coefficients: and

5 Multiple Beam Interference Reflection coefficients:and Transmission coefficients: and

6 Multiple Beam Interference The additional phase in the film is always the same: = 2 cos If the initial phase is zero, then = = () = () = () In general: = () =

7 Multiple Beam Interference The total electric field on one side of the film: = This is in infinite sum of the form: Total electric field: = (when <1) = + 1

8 Multiple Beam Interferometry Simplifications: = =1 Total electric field: = 1 (1 ) 1 = = 1 1

9 Multiple Beam Interferometry The intensity of the light is = = (1 )(1 ) (1 )(1 ) = 2 (1 cos) 1+ 2 cos The intensity of the transmitted light is = cos

10 Multiple Beam Interferometry One more identity will clean this up a bit: cos=1 2sin ( 2) Reflected intensity: = Transmitted intensity: sin ( 2) 1+sin ( 2) 1 = 1+sin ( 2) The parameter = is called the coefficient of finesse Notice that = + We assumed that no energy was lost in the film

11 Multiple Beam Interferometry The function 1 = 1sin 2 is called the Airy function. Transmitted fraction

12 Multiple Beam Interferometry In practice, some fraction of the light will be absorbed Absorptance,, is defined by: =1 This modifies the transmitted intensity: = 1 () 1 Example: silver film, 50 nm thick, deposited on glass =0.94, =0.01, = = =1044

13 Multiple Beam Interference How sharp are the peaks? Width of one line: =4/ Ratio of line spacing to the width: = 2 = 2 Finesse, not to be confused with the coefficient of finesse. Previous example: 50

14 Fabry-Perot Interferometer Phase difference: = 4 =2 =2 cos Differentiate: Δ Δ =0 Δ =Δ

15 Fabry-Perot Interferometer = 2 Δ Δ Smallest resolvable wavelength difference: Δ = Δ 2 Minimum resolvable phase shift: Δ ~ =4/ Chromatic resolving power: R=

16 Fabry-Perot Interferometer Typical values: =50 =1 = R= 500 =2 10 Diffraction grating Michelson interferometer Fabry-Perot interferometer

17 Fabry-Perot Interferometer The effective gap between the surfaces can be adjusted by changing the pressure of a gas, or by means of piezoelectric actuators

18 Diffraction Light transmitted or diffused, not only directly, refracted, and reflected, but also in some other way in the fourth, breaking.

19 Huygens-Fresnel Principle Huygens: Every point on a wave front acts as a point source of secondary spherical waves that have the same phase as the original wave at that point. Fresnel: The amplitude of the optical field at any point in the direction of propagation is the superposition of all wavelets, considering their amplitudes and relative phases.

20 Single Slit Diffraction Examples with water waves Wide slit: waves are unaffected Narrow slit: source of spherical waves In between: multiple interfering point sources

21 Single Slit Diffraction 1 2 Think of the slit as a number of point sources with equal amplitude. Divide the slit into two pieces and think of the interference between light in the upper half and light in the lower half. Destructive interference when Minima when 2 sin= 2 sin=

22 Single Slit Diffraction sin tan=/ Minima located at =,=1,2,3, In general, the width of the image on the screen is not even close to.

23 Single Slit Diffraction Minima located at sin=,=1,2,3, Minima only occur when>. Waves from all points in the slit travel the same distance to reach the center and are in phase: constructive interference.

24 Fresnel and FraunhoferDiffraction Assumptions about the wave front that impinges on the slit: When it s a plane, the phase varies linearly across the slit: Fraunhofer diffraction When the phase of the wave front has significant curvature: Fresnel diffraction Fraunhofer Fresnell

25 Fresnel and FraunhoferDiffraction Fraunhofer diffraction Far field: / Fresnel Diffraction: Near field: wave front is not a plane at the aperture b is the smaller of the distance to the source or to the screen

26 Single-Slit FraunhoferDiffraction Light with intensity impinges on a slit with width Source strength per unit length: ℇ Electric field at a distance due to the length element : = ℇ =sin

27 Single-Slit FraunhoferDiffraction = ℇ Let =0be at the center of the slit. Integrate from 2to + 2: Total electric field: = ℇ / = ℇ ( = ℇ / ) sin sin 1 2 sin 1 2 sin ( )

28 Single-Slit FraunhoferDiffraction where = ℇ sin = 1 2 sin The intensity of the light will be sin = 0 = 0 sinc

29 Single-Slit FraunhoferDiffraction sin = 0 = 0 sinc Minima occur when =, =1,2, 1 2 sin sin sin

30 Single slit: Fraunhofer diffraction Adding dimension: long narrow slit Diffraction most prominent in the narrow direction. Emerging light has cylindrical symmetry

31 Rectangular Aperture Fraunhofer Diffraction Source strength per unit area: ℇ ℇ / / = ℇ / / / / / /, 0 sin sin

32 Rectangular Aperture, 0 sin sin 1 2 / 1 2 /