Wave Optics Review: Interference and Optical Elements

1. Chapter 24 Wave Optics

2. Review – optical elements

3. Optical equations • Sign conventions p (+) real on the side light goes in q (+) real on the side light comes out f (+) for focusing optics R check sign of f M (+) for upright images

4. Interference Sections 1 – 3

5. Wave Optics • The wave nature of light explains various phenomena • Interference • Diffraction • Polarization

6. Principle of Superposition • Two waves travelling in opposite directions will pass straight through each other • When the waves are on top of each other, the two amplitudes add up to get the total amplitude • This is the cause of interference effects

7. Huygen’s Principle • Christian Huygens (1629 – 1695) assumed that light is a form of wave motion rather than a stream of particles • Huygens’ Principle is a geometric construction for determining the position of a new wave at some point based on the knowledge of the wave front that preceded it • All points on a given wave front are taken as point sources for the production of spherical secondary waves, called wavelets, which propagate in the forward direction with speeds characteristic of waves in that medium • After some time has elapsed, the new position of the wave front is the surface tangent to the wavelets

8. Huygens’ Construction for a Plane Wave • At t = 0, the wave front is indicated by the plane AA’ • The points are representative sources for the wavelets • After the wavelets have moved a distance cΔt, a new plane BB’ can be drawn tangent to the wavefronts

9. Huygens’ Construction for a Spherical Wave • The inner arc represents part of the spherical wave • The points are representative points where wavelets are propagated • The new wavefront is tangent at each point to the wavelet

10. Diffraction • Huygens’ principle requires that the waves spread out after they pass through narrow slits • This spreading out of light from its initial line of travel is called diffraction • In general, diffraction occurs when waves pass through small openings, around obstacles or by sharp edges

11. Interference • Light waves interfere with each other much like water waves do • All interference associated with light waves arises when the electromagnetic fields that constitute the individual waves combine

12. Conditions for Interference • For sustained interference between two sources of light to be observed, there are two conditions which must be met • The sources must be coherent • They must maintain a constant phase with respect to each other • The waves must have identical wavelengths

13. Producing Coherent Sources • Light from a monochromatic (single wavelength) source is allowed to pass through a narrow slit, So • The light from the single slit is allowed to fall on a screen containing two narrow slits, S1 and S2 • The first slit is needed to insure the light comes from a tiny region of the source which is coherent • The waves emerging from the second screen originate from the same wave front and therefore are always in phase

14. Producing Coherent Sources • Currently, it is common to use a laser as a coherent source • The laser produces an intense, coherent, monochromatic beam over a width of several millimeters • The laser light can be used to illuminate multiple slits directly

15. Young’s Double Slit Experiment • Thomas Young first demonstrated interference in light waves from two sources in 1801 • The light from the two slits form a visible pattern on a screen • The pattern consists of a series of bright and dark parallel bands called fringes • Constructive interference occurs where a bright fringe appears • Destructive interference results in a dark fringe

16. Resulting Interference Pattern Active Figure: Young's Double-Slit Experiment

17. Interference Patterns • Constructive interference occurs at the center point • The two waves travel the same distance • Therefore, the waves arrive in phase • A bright fringe occurs • Path difference is δ = 0

18. Interference Patterns, 2 • The upper wave travels one-half of a wavelength farther than the lower wave • The trough of the bottom wave overlaps the crest of the upper wave • Therefore, the waves arrive out of phase • This is destructive interference • A dark fringe occurs • Path difference is δ = 1/2λ

19. Interference Patterns, 3 • The upper wave travels one wavelength farther than the lower wave • Therefore, the waves arrive in phase • Again, constructive interference results • A bright fringe occurs • Path difference is δ = λ

20. Interference Equations • The path difference is δ = r2 – r1 • If L is much greater than d (L » d) • The paths are approximately parallel • The path difference is found from the smaller triangle to be • δ = r2 – r1 = d sin θ

21. Interference Equations, 2 • For a bright fringe, produced by constructive interference, the path difference must be either zero or some integral multiple of the wavelength • δ = d sin θbright = m λ • m = 0, ±1, ±2, … • m is called the order number • When m = 0, it is the zeroth order maximum • When m = ±1, it is called the first order maximum • etc.

22. Interference Equations, 3 • For a dark fringe, produced by destructive interference, the path difference must an odd half wavelength • δ = d sin θdark = (m + ½) λ • m = 0, ±1, ±2, …

23. Interference Equations, 4 • The positions of the fringes can be measured vertically from the zeroth order maximum • y = L tan θ L sin θ • for bright fringes sin θ = m λ / d • for dark fringes sin θ = (m + ½) λ / d • Assumptions • L>>d • d>>λ • Approximation • θ is small and therefore the approximation tan θ sin θ can be used

24. Interference Equations, final • For bright fringes • For dark fringes

25. Uses for Young’s Double Slit Experiment • Young’s Double Slit Experiment provides a method for measuring the wavelength of the light • This experiment gave the wave model of light a great deal of credibility • It is inconceivable that particles of light could cancel each other, as in the case of destructive interference