1. Double Rainbow 1

2. 2 Bar at the Folies Bergères’ by Edouard Manet (1882)

3. 3 Chapter 35 The concept of optical interference is critical to understanding many natural phenomena, ranging from color shifting in butterfly wings (iridescence) to intensity patterns formed by small apertures. These phenomena cannot be explained using simple geometrical optics, and are based on the wave nature of light. In this chapter we explore the wave nature of light and examine several key optical interference phenomena. Interference 35-

4. 4 Huygen’s Principle: All points on a wavefront serve as point sources of spherical secondary wavelets. After time t, the new position of the wavefront will be that of a surface tangent to these secondary wavelets. Light as a Wave 35- Fig. 35-2

5. 5 Law of Refraction 35- Index of Refraction: Fig. 35-3 Law of Refraction:

6. 6 Wavelength and Index of Refraction 35- Fig. 35-4 The frequency of light in a medium is the same as it is in vacuum Since wavelengths in n1 and n2 are different, the two beams may no longer be in phase

7. 7 The geometrical explanation of rainbows given in Ch. 34 is incomplete. Interference, constructive for some colors at certain angles, destructive for other colors at the same angles is an important component of rainbows Rainbows and Optical Interference 35- Fig. 35-5

8. 8 Diffraction 35- Fig. 35-7 For plane waves entering a single slit, the waves emerging from the slit start spreading out, diffracting.

9. 9 Young’s Experiment 35- Fig. 35-8 For waves entering a two slit, the emerging waves interfere and form an interference (diffraction) pattern.

10. 10 The phase difference between two waves can change if the waves travel paths of different lengths. Locating Fringes 35- Fig. 35-10 What appears at each point on the screen is determined by the path length difference  L of the rays reaching that point. Path Length Difference:

11. 11 35- Fig. 35-10 Maxima-bright fringes: Minima-dark fringes: Locating Fringes

12. 12 Coherence 35- Two sources to produce an interference that is stable over time, if their light has a phase relationship that does not change with time: E(t)=E 0 cos(  t+  ) Coherent sources : Phase  must be well defined and constant. When waves from coherent sources meet, stable interference can occur. Sunlight is coherent over a short length and time range. Since laser light is produced by cooperative behavior of atoms, it is coherent of long length and time ranges Incoherent sources :  jitters randomly in time, no stable interference occurs

13. 13 Intensity in Double-Slit Interference 35- Fig. 35-12 E1E1 E2E2

14. 14 Fig. 35-13 Proof of Eqs. 35-22 and 35-23 35- Eq. 35-22 Eq. 35-23

15. 15 In general, we may want to combine more than two waves. For eaxample, there may be more than two slits. Prodedure: 1.Construct a series of phasors representing the waves to be combined. Draw them end to end, maintaining proper phase relationships between adjacent phasors. 2.Construct the sum of this array. The length of this vector sum gives the amplitude of the resulting phasor. The angle between the vector sum and the first phasor is the phase of the resultant with respect to the first. The projection of this vector sum phasor on the vertical axis gives the time variation of the resultant wave. Combining More Than Two Waves 35- E1E1 E2E2 E3E3 E4E4 E

16. 16 Interference from Thin Films 35- Fig. 35-15

17. hitt A 5:0-ft woman wishes to see a full length image of herself in a plane mirror. The minimum length mirror required is: A. 5 ft B. 10 ft C. 2.5 ft D. 3.54 ft E. variable: the farther away she stands the smaller the required mirror length 17

18. question Two thin lenses (focal lengths f 1 and f 2 ) are in contact. Their equivalent focal length is: A. f 1 + f 2 B. f 1 f 2 /(f 1 + f 2 ) C. 1=f 1 + 1=f 2 D. f 1 /f 2 E. f 1 (f 1 f 2 )=f 2 18

19. hitt The image of an erect candle, formed using a convex mirror, is always: A. virtual, inverted, and smaller than the candle B. virtual, inverted, and larger than the candle C. virtual, erect, and larger than the candle D. virtual, erect, and smaller than the candle E. real, erect, and smaller than the candle 19