PS210 - Optical Techniques Section VI Section I Light as Waves, Rays and Photons Section II Geometrical Optics & Optical Instrumentation Section III Periodic and Non-Periodic (Aperiodic) Waves Section IV Electromagnetic Waves Section V Elements of Fibre Optics Section VI Interference Section VII Diffraction Section VIII Polarisation Section IX Light Sources Section X Light Detectors
1. The interaction of light waves with one another forms the basis of two fields of optics called interferometry and diffraction. It is important to appreciate the difference between the two effects: Interference occurs when two or more wavefronts are superimposed to give a resultant wave whose amplitude depends on their relative phases. Diffraction is the spreading of waves from a wavefront which is limited in extent occurring when all but a part of the wavefront is removed by an aperture or stop (such as in a grating). 2. NB: Two light waves with orthogonal polarisation cannot interfere! 3. In Interference the constituent waves are added together as phasors, usually as a geometrical series: a + ar + ar 2 + ar 3 +.. = a(1 - r n )/(1-r), while in Diffraction the waves are added by the mathematical summation process called integration.
4. We begin by looking at the following examples of INTERFERENCE: Young s Double Slit and Michelson s Two Beam Interferometer. Others include Newton s Rings,and the Fabry-Perot Multiple Beam Interferometer. 5. Recap: In all cases it is useful to recall that: (i) light of wavelength λ in air or vacuum has an effective wavelength of λ/n in a medium of a refractive index n (ii) the phase difference per unit length is equal to 2π / λ in air/vacuum and equal to 2 π n / λ in a medium. In other words when two waves, originally in phase, travel different distances z they are out of phase by an amount of 2πnz/λ (notice how these phase differences are fractions of 2π radians as z/λ is unitless since it is the ratio of two length quantities). We will see that the interference fringe patterns observed are contours - like isobars of equal pressure - which represent equal quantities such as height (Newton s Rings experiment) or inclination angle (Young s Double Slits experiment).
Young s Double Slit Experiment. 6. We can generate two beams by division of wavefront (Young s Double Slits) or division of amplitude (e.g., Michelson Interferometer) 7. The diagram below shows monochromatic light from a distant object (or laser) represented by plane wavefronts, striking a double slit with a centre to centre spacing of d (metres). P Plane Wavefronts d θ Viewing Screen Source: http://www.physicsoftheuniverse.com/images/quantum_double_slit.jpg
Young s Double Slit Experiment. 8. The light coming from each slit is shown as spherical wavefronts that overlap in space 9. Referring to the main diagram, At the observation point P the light coming from A travels a shorter distance l than the light from B. This path difference is: l = d Sinθ where θ is the angle the line OP makes with the normal to the screen at its centre of symmetry O 10 Then the phase difference (φ) between the waves AP and BP (shown on the diagram as RAYS) is 2π/λ times the path difference (d Sin θ), i.e., φ = (2π/λ) d Sinθ
Young s Double Slit Experiment. 11. When this phase difference is 2 π N, where N is an integer (±1, 2, 3..) then the two waves combine constructively and a bright band of light is seen at the observation point P. The condition for constructive interference is: d Sinθ = ± N λ ; N = 1, 2, 3, 12. Alternately when the phase difference is ± π, ± 3π etc the two waves interfere destructively and a dark band is observed. The condition for destructive interference is: d.sinθ = ± (N + ½) λ ; N = 1, 2, 3,. http://www.nowwhat.hk/images/longman/04.jpg http://www.matter.org.uk/schools/content/interference/formula.html
Michelson (Two Beam) Interferometer (Division of Amplitude) 11. An example of interference by division of amplitude occurs in the interferometer developed by Michelson in 1880. It consists of a partially silvered reflector D (beamsplitter) and two front silvered mirrors M 1 and M 2 placed at right angles to one another. A block of glass of the same composition and thickness as D, called the compensator C, is located parallel to D. Mirror M 1 S D C Source Mirror M 2
Michelson (Two Beam) Interferometer (Division of Amplitude) 12. Light from a monochromatic source S is amplitude divided at the beamsplitter D. One wave travels to, and back from, mirror M 1 ; the other wave travels through the compensator C to mirror M 2 and back to the point at which the two waves were divided (at D). At this location the waves recombine and travel to the observer at the point P. The waves arriving at the observation point P are out of phase because of the different path lengths they have travelled to and from the two mirrors. Let this distance as measured in air/vacuum be given by d. The phase difference φ associated with this path difference is: φ = (2π/λ)d 13. At the observation point P the superposition of the two monochromatic plane waves is an electric field E given by the vector sum of E 1 and E 2. Remember Phasors! E E 1 E 2
Michelson (Two Beam) Interferometer (Division of Amplitude) Note: Since the phase difference φ associated with this path difference is: φ = (2π/λ)d For constructive interference we require that this phase difference be an integer multiple of 2π, i.e., φ = (2π/λ)d = M(2π), where M = 1, 2, 3, or Mλ = 2d For destructive interference; (2π/λ)d = (2M+1)π or 2d/λ = 2M+1 => (M+1/2) λ = 2d E E 1 E 2
Michelson (Two Beam) Interferometer (Division of Amplitude) 14. The Intensity I of the combination of the two waves is then: I FINAL = E 1 2 + E 2 2 + 2.E 1 E 2 Cos φ or I FINAL = I 1 + I 2 + 2. [ (I 1 I 2 )]Cos φ ; φ = (2πd/λ) 15. If the beam splitter D is a 50:50 splitter then I 1 = I 2 and I FINAL = 2I + 2I Cos φ = 2I(1 + Cos φ) = 4I Cos 2 (φ/2) 16. The fringe pattern seen at the observation point P is a set of bright and dark bands. If the two mirrors M 1 and M 2 are accurately parallel then the fringe pattern seen at P is a set of circular rings. E E 1 E 2
Michelson (Two Beam) Interferometer (Division of Amplitude) 17. The Michelson Interferometer can be used to measure distances in terms of the wavelength of light, with a path length d related to a number M of fringes (counted at the observation point P) by the expression: 2d = Mλ 18. Another use of this interferometer is in determining the refractive index of gases. An evacuated optical cell of length l is placed in one of the optical paths of the interferometer. The gas, of refractive index n, is then released into the cell and causes M fringes to move across the field of view; this number is measured.the optical path distance between the two light waves changes from by 2l(n - 1). d = 2l(n-1) in this case, so we can write: Mλ = 4l(n 1) from which the refractive index n can be calculated.
Michelson (Two Beam) Interferometer (Division of Amplitude) 17. The Michelson Interferometer can be used to measure distances in terms of the wavelength of light, with a path length d related to a number M of fringes (counted at the observation point P) by the expression: 2d = Mλ 18. Another use of this interferometer is in determining the refractive index of gases. An evacuated optical cell of length l is placed in one of the optical paths of the interferometer. The gas, of refractive index n, is then released into the cell and causes M fringes to move across the field of view; this number is measured.the optical path distance between the two light waves changes from by 2l(n - 1). d = 2l(n-1) in this case, so we can write: Mλ = 4l(n 1) from which the refractive index n can be calculated.