Wave Optics-1Wave Nature of LightLight is a wave, an electromagnetic wave. The wavelength of visible light is very small. Visible light: = 400 nm (violet) 700 nm (red)c = f , f = c / , = c / fWave-like effects are difficult to detect because of the small wavelength. In many situations, light behaves like a ray, exhibiting no obvious wave-like behavior.Newton (late 1600's) did not believe that light was a wave since he always observed ray-behavior. Wave-like behavior was not clearly observed until around 1800.Review of Constructive/Destructive interference of Waves:Consider 2 waves, with the same speed v, the same wavelength , (and therefore same frequencyf = c / ), traveling in the same (or nearly the same) direction, overlapping in the same region of space: If the waves are in phase, they add constructive interferenceIf the waves are out of phase, they subtract destructive interferenceLast update: 1/14/2019 Dubson Phys2020 Notes, University of Colorado big hole D >> : ray-behaviorLight passing through hole in wall: tiny hole D : wave-behaviorwavefronts=+Wave Optics-2If wave in nearly the same direction:Huygen's Principle: Each point ona wavefront (of given f, ) can beconsidered to be the source of aspherical wave.To see interference of light waves, you need a monochromatic (single ) light source, which is coherent (nice, clean plane wave). This is not easy to make. Most light sources are incoherent (jumble of waves with random phase relations) and polychromatic (many different wavelengths).Young's Double slit experiment (1801) :Last update: 1/14/2019 Dubson Phys2020 Notes, University of Colorado =+addsubtractplane wavespherical wave (same , f )speed cccwall with infinitesimal holeWave Optics-3What do you expect to see on the screen? Ifyou believe light is a ray, then you expect tosee 2 bright patches on the screen, one patchof light from each slit.But here is what you actually see: A series ofbright and dark fringes: wave interferenceHow do we explain this? Considerthe 2 slits as 2 coherent pointsources of monochromatic light.Two sources are coherent is they have the same wavelength (and therefore the same frequencyf ) and they emit peaks and troughs in sync, in phase. Each slit (source) emits light in all forward directions, but let us consider only the parts of the waves heading toward a particular point on the screen.Last update: 1/14/2019 Dubson Phys2020 Notes, University of Colorado monochromatic plane waved2 slitsscreencLdposition x on screenIntensity IdxIWave Optics-4If the screen is far away (L >> d), then the rays from the two slits to the same point on the screen are nearly parallel, both heading in the same direction, at the same angle .The ray from the lower slit has to travel further by an extra distance (d sin to reach the screen. This extra distance is called the path difference. When the path difference (p.d.) is one full wavelength, or 2 full wavelengths, or an integer number of wavelengths, then the waves will arrive in phase at the screen. There will be constructive interference and a bright spot on the screen.p.d. d sin m , m = 0, 1, 2, ...= q = l (constructive interference)But if the path difference is ½ wavelengths or 3/2 wavelengths, etc, then there will be destructiveinterference at the screen and the screen will be dark there.12p.d. d sin (m ) , m = 0, 1, 2, ...= q = + l (destructive interference)Notice that the formula p.d. = d sin is NOT a definition of path difference. It is a formula for path difference in a specific situation, namely when the screen is "at infinity". The definition of path difference is this: p.d. = (distance to one source) – (distance to the other source)A plot of brightness (intensity) vs. angle position on the screen:Last update: 1/14/2019 Dubson Phys2020 Notes, University of Colorado dd sinto screendd sinWave Optics-5Maxima at angles where sin = m / d (rads) [Recall sin (rads) if Young's experiment was the first real proof that light is a wave. If you believe that light is a ray, there is no way to explain the destructive interference seen on the screen. In the ray-view, when you hit a screen with two rays, the brightness of the 2 rays always adds and you see a bright spot there. It is impossible to explain destructive interference of two light sources, unless you admit that light is a wave.Single Slit Diffraction"Diffraction" = interference due to infinitely-manysources packed infinitely close via Huygen'sPrinciple. Huygen's Principle says that a slit that isilluminated by a plane wave can be consider to befilled with an array of coherent point sources.Last update: 1/14/2019 Dubson Phys2020 Notes, University of Colorado sin I (intensity)m = 0 m = 1m = 1m = 2m = 20dddDslit filled with imaginary Huygen's sourcesWave Optics-6Consider the light from just two of the infinitely-many sources: one at the top of the slit, and one exactly in the middle of the slit. When thepath difference between these two sourcesand the screen is ½ wavelength, that is,when Dsin2 2lq =, then the light fromthese two source interfere destructivelyand no light from those two sourcesilluminates the screen at that particularlyangle . But notice that all the sources can grouped in pairs, with each pair's membersD/2 apart. The light from all the sources (the entire slit) cancel in pairs, andthere is no light at the position on the screen at the angle such thatDsin , or Dsin2 2lq = q = l. The angle sin / Dq = l is the firstintensity minimum on the screen. The intensity pattern on the screen looks like this:Last update: 1/14/2019 Dubson Phys2020 Notes, University of Colorado sin I (intensity)DDD0DD(D/2) sinto screenD/2D/2D/2Wave Optics-7The angular width of the "central maximum" is 2(rads) DlDq =. Notice that in the limit, D (slit width becomes as small as the wavelength of light), the central max becomes so broad, that we get spherical wave behavior.Last update: 1/14/2019 Dubson Phys2020 Notes, University of Colorado D > D Wave Optics-8Diffraction GratingA diffraction grating is an array of many narrow slits with a uniform inter-slit spacing d. A grating with
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