# Christian Huygen Light is a wave, not merely a ray As waves propagate each point on the wavefront produces new wavelets. Wave Nature of Light

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1 Wave Nature of Light Christian Huygen Light is a wave, not merely a ray As waves propagate each point on the wavefront produces new wavelets Chapter 24 Wavelength Changes Wavelength of light changes in materials FREQUENCY DOES NOT 1.Calculate the wavelength of 400 nm light in water (n=1.33) [ans: 301 nm] 2.Calculate the wavelength of 680 nm red light in crown glass (n = 1.52) [ans: 447 nm] nm light passes from air into a certain liquid. In the liquid, the wavelength is 357 nm. Calculate n. [ans: 1.40] Mirages Hot air over puddles lowers refractive index of light Curves wavefronts so that it appears below the ground Diffraction Waves bend into shadow region when passing an obstacle Diffraction increases as size of holes approaches wavelength Ray model cannot explain this 1

2 Failure of the Ray Theory Particle/Ray theory predicts light would just pass through two slits Young s Experiments showed numerous lines Young s Experiment Interference Constructive Interference Waves are in phase Overlap and add together Destructive Interference Waves are out of phase by ½ l Waves cancel each other out Double Slit Diffraction Spectra Bending of light is due to diffraction Extra distance can produce either constructive or destrcuctive interference Constructive dsinq = ml (m= 0, 1, 2 ) Destructive dsinq = (m + ½)l (m= 0, 1, 2 ) Shows bands of light and dark Light = Maxima (constructive interference) Dark = Minima (destructive interference) 2

3 Double Slit: Ex 1 A screen has two slits, mm apart. It is set 1.20 m from a viewing screen. Calculate the angles for the first and second maxima. The light source is 500 nm. q 1 = 5 X 10-3 rad (remember 2p rad = 360 o ) q 1 = o q 2 = 1 X 10-2 rad q 2 = o Double Slit: Ex 1a Double Slit: Ex 2 How far apart will the first and second maxima be? y 1 = 6 mm y 2 = 12 mm The maxima are about 6 mm apart. White light passes through two slits 0.50 mm apart and an interference pattern is observed on a screen 2.5 m away. The violet fringe falls 2.0 mm away from the central white fringe. Estimate the wavelength of violet light. l v = 4 X 10-7 m l v = 400 nm similarly l r = 700 nm Small Angle Approximation x q L At small angles sinq q and tanq q (q is in radians) tan q = x so q = x L L Useful for finding location of minima and maxima Set your calculator to radians! Cannot use the assumption in degrees (must work through the sine and tangent function) 3

4 Degrees Constructive dsinq = ml Radians q = ml d y = mll (bright fringes) If a yellow light with a wavelength of 540 nm shines on a double slit with the slits cut mm apart, determine what angle you should look away from the central fringe to see the second order fringe? Where will the second order fringe appear if the screen is 75.0 cm from the source? d Destructive dsinq = (m + ½ )ll Destructive y = (m + ½ )ll (dark fringes) d (6.2 o, 8.14 cm) A screen is placed 3.0 m from a two-slit setup with the slits separated by 15 mm. If the wavelength of the light is nm, how far apart are the principal and m = 1 fringes? Diffraction Gratings (0.8 mm) Material with a large number of parallel slits Often 10,000 lines per centimeter More slits yield sharper peaks Transmission grating clear and passes the light Reflection grating fine lines on a metallic or mirror-like surface Two slits vs. 6 slits Diffraction formula CD as Diffraction Grating dsinq = ml (m = 0, 1, 2, 3, ) tanq = y L d is the size of the slit cannot use small angle approximation 4

5 Diffraction Grating: Ex 1 What is the d for a grating that is 10,000 lines/cm? 10,000 lines 100 cm = 1X10 6 lines 1 cm 1 m 1m d = 1 m/1 X 10 6 lines = 1 X 10-6 m/line Diffraction Grating: Ex 1a Calculate the angles for the 1 st and 2 nd order maxima of 400 nm light incident on this grating dsinq = ml sinq 1 = (1)(400 X 10-9 m) = X 10-6 m q 1 = 23.6 o sinq = ml d sinq 2 = (2)(400 X 10-9 m) = X 10-6 m q 2 = 53.1 o Calculate the angles for the 1 st and 2 nd order maxima of 700 nm light incident on this grating (ANS: q 1 = 44.4 o and q 2 = does not exist(sinq 2 >1) 5

6 Calculate the angles for the first through the 4 th maxima for 600 nm light incident on a grating of 5000 lines/cm. ANS: q 1 = 17.5 o q 2 = 36.9 o q 3 = 64.2 o q 4 = Does not exist White light (400 to 750 nm) strikes a grating containing 4000 lines/cm. Show that the blue (l=450 nm) third order spectrum overlaps the red (l=700 nm) second order spectrum. From the angles. (32.7 o, 34.1 o ) sinq = ml d sinq b = (3)(400 X 10-9 nm) = X 10-6 m q b = 32.7 o A diffraction grating with 12 thousand lines per cm separates a bright line at 24.5 degrees. What is the wavelength of the light? sinq = ml d sinq r = (2)(400 X 10-9 nm) = X 10-6 m q r = 34.1 o Single Slit Diffraction Single slits and objects with sharp edges refract light Diffraction pattern can produce bright spot in the middle of a coin (Fresnel) Further proof of the wave nature of light Bright spot 6

7 Diffraction Pattern through a small, square hole Electron Diffraction Single Slit Formula See similar pattern to double slit Minima occur at: Destructive Interference dsinq = ml (m = 0, 1, 2, 3, ) Constructive Interference No simple formula dsinq = (m+ ½)l [only an estimate] For a single slit 0.2 centimeters wide and a screen 1 meter distant, the second minimum occurs at a position 4.58 centimeter along the screen. Calculate the wavelength of light incident on the screen. (m=2) (800.0 nm (approx.)) 7

8 Light of 580 nm shines on a slit 0.30 mm wide. The observing screen is placed 2.0 m away. Find the position of the first dark fringe and second. dsinq = ml sinq = ml/d sinq 1 = (1)(580 X 10-9 m) = 1.9 X 10-3 (0.30 X 10-3 m) q 1 = o y 1 = 0.38 cm sinq 2 = (2)(580 X 10-9 m) = 3.9 X 10-3 (0.30 X 10-3 m) q 2 = o y 2 = 0.77 cm For a single slit experiment, determine how far from the central fringe the first order violet (λ = 350nm) and red (λ = 700nm) colors will begin to appear if the screen is 10 m away and the slit is cm wide. (m=1) Helium neon laser light (632.8 nm) is sent through a mm-wide single slit. What is the width of the central maximum on a screen 1.00 m from the slit? (4.22 mm) (7 mm, 14 mm) Single Slit: Ex 3 Light shines through a rectangular hole as shown. Will the diffraction be greater in the horizontal or vertical direction? Pattern will be greater in the vertical direction sinq = ml d As d decreases, q increases 8

9 Visible Spectrum 400 to 700 nm Can separate using a prism Glass violet light gets refracted more than red Rainbows m Red light is not refracted (bent) much Appears at the top of the rainbow Violet refracted much more Appears at the bottom 9

10 Wave Reflection n 2 > n 1 n 2 <n 1 Phase Change No Phase Change Odd # of Phase Changes Constructive Interference 2t = (m+ ½)l n Thin Film Equations Destructive interference 2t = ml n t = thickness of material n = refractive index of wedge/thin film(n=1 for air) Thin Film Equations Oil Film on Water Even # of Phase Changes Constructive Interference 2t = ml n Destructive interference 2t = (m+ ½)l n Extra distance is ABC ABC = 2t (t is the thickness of the film) When white light shines on the oil/water thin film (n oil =1.50, n water =1.33), red light is strongly reflected at 636 nm. Calculate the minimum thickness of the film. 1. Find #phase changes 2. Use appropriate formula Newton s Rings Curved glass placed on flat glass Produces light and dark bands Can be used to detect flaws in lenses (106 nm) 10

11 At point B, no phase change At point C, l phase change Thin Films: Ex 1 A fine wire (7.35 X 10-3 mm in diameter) is placed between two glass plates. How many light and dark bands appear for 600 nm light? No phase change at D Phase change at E Dark Bands 2t = ml n Destructive interference 2t = ml n=1 for air m =2t/l m = (2)(7.35 X 10-6 m)/(600 X 10-9 m) m = 24.5 (25 dark lines) Light Bands 2t = (m+ ½)l Constructive interference n 2t = (m+ ½)l n = 1 for air m + ½ =2t/l m = [(2)(7.35 X 10-6 m)/(600 X 10-9 m)] ½ m = 24 (24 bright lines) A soap bubble appears green (l=540 nm) as the viewer looks perpendicularly at the bubble (m=0). What is the minimum thickness of the bubble if n=1.35 for the soapy water? 2t = (m+ ½)l Constructive interference n 2t = l m = 0, perpendicular 2n t =(540 X 10-9 m) (4)(1.35) t = 1.00 X 10-7 m or 100 nm 11

12 Lens Coatings Glass surfaces reflect ~4% of light Can be a problem with multiple lens (cameras, microscopes) Thin coating(s) causes destructive interference (at least for one l) Prevents ghosting of the images Thin Films: Ex 3 What is the thickness of an optical coating MgF 2 (n=1.38) and is designed to eliminate light at 550 nm incident on glass (n=1.50) Both rays undergo a phase inversion 2t = (m+ ½)l n Destructive interference 2t = l m = 0, perpendicular 2n t =(550 X 10-9 m) (4)(1.38) t = 99.6 X 10-9 m or 99.6 nm Polarization Waves can be planepolarized Can use a slit to eliminate other directions Polarization of Light Light is unpolarized when produced by atoms Vibrations are in all planes at once Polaroid Sheets Long, parallel molecules Act as slits Polaroid Filters Light vectors can be resolved to vertical and horizontal components Polaroid filters will block ½ of the light Crossed polarizers will eliminate all light 12

13 Polaroid Formula I = I o cos 2 q I - Intensity of light that passes I o - Intensity of incident light q - Angle it hits the polarizer Polaroid: Ex 1 Unpolarized light passes first through a vertical filter, then through one that is 60 o to the vertical. First Filter I = ½ I o Second Filter I = I o cos 2 q I = (½ I o )(cos60 o ) 2 I = (½ I o )(0.5) 2 = 1/8 I o (60 o to the vertical) Polaroid: Ex 2 Light is passed through three filters. The first is vertical, the second 45 o to the vertical, and the third 45 o to the second (90 o from the vertical) What is the intensity and direction of the resultant light? First Filter I = ½ I o Second Filter I = I o cos 2 q I = (½ I o )(cos45 o ) 2 I = (½ I o )(0.5) = 1/4 I o (45 o to the vertical) Third Filter I = I o cos 2 q I = (1/4 I o )(cos45 o ) 2 I = (1/4 I o )(0.5) = 1/8 I o (90 o to the vertical) Note that if the middle filter is removed, no light will pass. 13

14 Polarization by Reflection Metallic surfaces do not polarize light Non-metallic object will preferentially absorb light perpendicular to the vertical A polaroid filter eliminates the reflected light and shows the light that passes vertically into the pond (look for the fish!) tanq p = n 2 n 1 q p = Brewster s Angle At Polarizing Angle, get 100% polarization Brewster s Angle Brewster s Angle: Ex 1 At what angle is sunlight fully plane-polarized from a lake. What is the refraction angle into the water? tanq p = n 2 n 1 tanq p = q p = 53.1 o Using Snell s Law, q r = 36.9 o Why is the Sky Blue? Scattering of light by N 2 molecules Short l (blue) scattered more than long l (red) At sunset, all blue is scattered out (more air to pass through). Only red remains Why are clouds white? Scattering is related to wavelength only if particles are smaller than the wavelength Light ~400 to 700 nm Molecules ~ 200 nm Larger particles - water scatter all wavelengths equally and appear white. 14

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