New topic: Diffraction only one slit, but wide From Last time Two-source interference: Interference-like pattern from a single slit. For a slit: a θ central width ~ 2 Diffraction grating Week3HW on Mastering Physics due Fri. Sep. 18 Diffraction = interference from many sources Week2HW due Fri. Sep. 11 Huygen s principle Long wavelength: wide pattern Short wavelength narrow pattern Overlapping diffraction patterns Two sources ->two diffraction patterns. Huygen s principle: each portion of the slit acts as a source of waves These sources interfere according to path-length difference. Width central max determined by aperture. Angular separation θ Larger aperture gives better angular resolution For a circular aperture (e.g. lens) θ min = 1.22 D 3 36 Lick refractor at UC-Berkeley Diffraction from other objects D Light diffraction by pinhead Large aperture -> good angular resolution θ min = 1.22 General effect Clearest w/single wavelength D 5 6 1
Another source of phase difference Interference summary In some cases reflection gives phase shift Waves start in phase Travel different distances (extra path length = δ) No longer in phase when combined (Phase diff φ) n1 n2>n1 π phase shift Longer path Here, δ = /2 Phase diff π Crest aligns with trough Destructive interference Shorter path n1 n2<n1 no phase shift 7 180 (π radians) phase shift from reflection Thin film interference no phase shift from reflection air: n=1 Black t n>1 Extra path length~2t /n Thin film air: n=1 Contributions to the phase difference Phase difference from reflection Colors changing with thickness 9 Top reflection has π phase shift, bottom not Phase difference from path length difference Path length difference = 2t Gives phase difference 2π 2t /n 10 Phase difference = π + 2π Reflection phase shift 1 2t = m + 2 n 2t = m n 2mπ 2t ( /n ) constructive (2m + 1)π destructive Convert to phase (m = 0,1,2 ) (m = 0,1,2 ) # wavelengths in extra path length constructive interference destructive interference What happens when: t << all in light? Constructive int. condition for some? Biological iridescence = 11 Some organisms seem to reflect incredibly vivid colors. Not by pigment, but interference! 12 2
Waves and geometry Interference and diffraction demonstrate that light is a wave. Doesn t always appear as a straight of light but sometimes it almost does! constructive = 446nm Geometric optics: Tracing the path of light s Thursday, Sep. 4 Phy208 Lecture 2 13 What is a light? Light is a line in the direction along which light energy is flowing. What does a light do? Light s travel forever in straight line unless they interact with matter (reflection, refraction, absorption) Wavefronts (crests of waves) Ray enters eye -> you can see the light source What about diffraction? Light really behaves as a wave The concept of a light is an approximation i.e. a lie Light s from point source Light s are not always parallel. E.g. light bulb visible from all directions Rays must be traveling in all directions Light perpendicular to local wavefront (crest of wave). Wavelength << aperture size, s are good approximation 3
Interaction of light with matter Absorption Reflection Reflection and Refraction Direction of light can be changed by Reflection (lets you see an object) Refraction (transmits light thru object) at an interface between different materials Reflection/refraction occur at interfaces between different materials And all occur simultaneously Refraction Interface Air Plastic Ray is the incident Ray is the reflected Ray is refracted into the lucite Ray is reflected inside the lucite Ray is refracted as it enters the air from the lucite When are materials different? For reflection/refraction materials are different if they have different index of refraction Light propagates at different speed in different materials. Due to interaction of electromagnetic wave with atoms in material. v = c n Vacuum Material Index of refraction 1.00 exactly Air (actual) 1.0003 Air (accepted) 1.00 Ice 1.31 Water 1.33 Ethyl Alcohol 1.36 Oil 1.46 Pyrex glass 1.46 Crown glass 1.52 Polystyrene plastic 1.59 Flint glass 1.66 Diamond 2.41 c=speed of light in vacuum What do you think? Pyrex stirring rod (n=1.46) dipped into beaker of Wesson oil (n=1.46). What happens to the rod? A. Appears dark B. Appears bright C. Appears invisible D. Appears curved E. Appears inverted Beaker of Wesson oil Pyrex stirring rod No reflection/refraction if index of refraction is same. Angle of incidence = angle of reflection Reflection Incident θ i Reflected Why θ i =? Christian Huygens modeled this i690 Said that each point on wavefront acts as source of spherical wavelets Superposition of wavelets gives reflected plane wave such that θ i = θ i Multiple reflections Apply θ i = at each surface trace 4
What about refraction? Refraction occurs when light moves into medium with different index of refraction. Result: light direction bends according to Snell s law sinθ 1 = n 2 sin Why Snell? Can analyze in exactly the same way Light moves at different speed in different media θ i θ i,1 n 2 Angle of refraction n 2 > v 2 <v 1 Refraction angle Quick quiz Reflected Reflected Which of these fluids has the smallest index of refraction (highest light speed)? A B v 2 <v 1 n 2 > v 2 >v 1 n 2 < slower in medium 2 faster in medium 2 n 2 > n 2 < Refracted bent Refracted bent toward normal away from normal A. Fluid A B. Fluid B C. Fluid C D. All equal C Numerical Example A beam of light is traveling underwater, aimed up at the surface at 45 away from the surface normal. Part of it is reflected back into the water, and part is transmitted into the air. Air n 2 =1.00 Quick quiz A trout looks up through the surface at the setting sun, and at the moon directly overhead. He sees n 2 =1.0 =1.33 Water =1.33 θ 1 =45 sinθ 1 = n 2 sin sin = n 2 sinθ 1 = 0.94 = arcsin n 1 sinθ 1 = 70 n 2 A. Moon directly overhead, sun ~ parallel to water surface B. Moon directly overhead, sun ~ 40 above water surface C. Moon ~ 40 from vertical, sun ~ parallel to water surface D. Moon and sun aligned at 40 from vertical. 5
Total Internal Reflection Is possible when light is directed from > n 2 refracted s bend away from the normal Critical angle: angle of incidence that will result in an angle of refraction of 90 (sin = 1) Optical Fibers Plastic or glass light pipes Applications: Medicine: endoscope (light can be directed even if bent and the surgeon can view areas in the body using a camera.) Telecommunications The cladding has a lower n than the core For water: sinθ c = 1 1.333 = 0.75 θ c = 48.75 6