April 12, 2005 Physics for Scientists&Engineers 2 1Physics for Scientists &Physics for Scientists &EngineersEngineers 22Spring Semester 2005Lecture 44April 12, 2005 Physics for Scientists&Engineers 2 2ReviewReview! If coherent light with wavelength ! is incident on a singleslit of width a, a single-slit diffraction pattern will beproduced! The angles of the minima are given by• m is the order of each minimum! If we project this diffraction pattern on a screen a largedistance L away from the slit, the position on the screen,measured from the center line, is given bya sin!= m" m = 1, 2, 3,...( )y =m!La m = 1,2, 3,...( )April 12, 2005 Physics for Scientists&Engineers 2 3Review (2)Review (2)! The intensity of light passing through a single slit I relativeto Imax that we would get if there were no slit isI = Imaxsin!!"#$%&'2 where !=(a)sin*April 12, 2005 Physics for Scientists&Engineers 2 4Thin Film InterferenceThin Film Interference! Another way of producing interference phenomena ispartial reflection of light from the front and back layers ofthin films! A thin film is an optically clear material with thickness onthe order few wavelengths of light! Examples of thin films include the walls of soap bubbles andthin layers of oil floating on water! When the light reflected off the front surfaceconstructively interferes with the light reflected off theback surface of the thin film, we see the colorcorresponding to the wavelength of light that is interferingconstructivelyApril 12, 2005 Physics for Scientists&Engineers 2 5Thin Film Interference (2)Thin Film Interference (2)! When light travels from an optical medium with an index ofrefraction n1 into a second optical medium with index ofrefraction n2, several things can happen! The light can be transmitted through the boundary• In this case the phase of the light is not changed! A second process that can occur is that the light can bereflected• In this case, the phase of the light can be changed depending on theindex of refraction of the two optical media• If n1 < n2, the phase of the reflected wave will be changed by half awavelength• If n1 > n2 then there will be no phase changeApril 12, 2005 Physics for Scientists&Engineers 2 6Thin Film Interference (3)Thin Film Interference (3)! Let’s begin our analysis of thin films by studying a thin filmwith index of refraction n in air as shown below! We will assume that light is incident perpendicular to thesurface of the thin film! An angle of incidence is shown for the light waves in thefigure for clarityApril 12, 2005 Physics for Scientists&Engineers 2 7Thin Film Interference (4)Thin Film Interference (4)! When the light wave reaches the boundary between air andthe film, part of the wave is reflected and part of the waveis transmitted! The reflected wave undergoes a phase shift of half awavelength when it is reflected because nair < n! The light that is transmitted has no phase shift andcontinues to the back surface of the film! At the back surface, again part of the wave is transmittedand part of the wave is reflected! The transmitted light passes through the film completely! The reflected light has no phase shift because n > nair andtravels back to the front surface of the filmApril 12, 2005 Physics for Scientists&Engineers 2 8Thin Film Interference (5)Thin Film Interference (5)! At the front surface, some of the light is transmitted andsome is reflected! The reflected light will not be considered! The transmitted light has no phase shift and emerges fromthe film and interferes with the light that was reflectedwhen the light originally entered the film! The transmitted light has traveled a longer distance thanthe originally reflected light and has a phase shift given bythe path length difference that is twice the thickness t ofthe filmApril 12, 2005 Physics for Scientists&Engineers 2 9Thin Film Interference (4)Thin Film Interference (4)! The fact that the originally reflected light has undergone aphase shift and the transmitted light has not means thatthe criterion for constructive interference is given by! The wavelength ! refers to the wavelength the lighttraveling in the thin film, which has index of refraction n! The wavelength of the light traveling in air is related tothe wavelength of the light traveling in the film by!x = m +12"#$%&'(= 2t m = 0, m = ±1, m = ±2,...( )!=!airnApril 12, 2005 Physics for Scientists&Engineers 2 10Thin Film Interference (5)Thin Film Interference (5)! We can then write! The minimum thickness tmin that will produce constructiveinterference corresponds to! Note that this result applies only to the case where of amaterial with index of refraction n and air on both sides,like a soap bubblem +12!"#$%&'airn= 2t m = 0, m = ±1, m = ±2,...( )tmin=!air4nApril 12, 2005 Physics for Scientists&Engineers 2 11Example: Lens CoatingExample: Lens Coating! Many high quality lenses are coated to prevent reflections! This coating is designed to set up destructive interferencefor light that is reflected from the surface of the lens! Assume that the coating is MgF2, which has ncoating = 1.38and the lens is glass with nlens = 1.51! Question:! What is the minimum thickness of the coating that willproduce destructive interference for light with awavelength of 550 nm?April 12, 2005 Physics for Scientists&Engineers 2 12Example: Lens Coating (2)Example: Lens Coating (2)! We assume that the light is incident perpendicularly on thesurface of the coated lens as shown below! Light reflected at the surface of the coating will undergo aphase change of half a wavelength because nair < ncoating! The light transmitted through the coating has no phasechangeApril 12, 2005 Physics for Scientists&Engineers 2 13Example: Lens Coating (3)Example: Lens Coating (3)! Light reflected at the boundary between the coating andthe lens will undergo a phase change of half a wavelengthbecause ncoating < nlens! This reflected light will travel back through the coatingand exit with no phase change! Thus both the light reflected from the coating and fromthe lens will have suffered a phase change of half awavelength! Thus the criterion for destructive interference ism +12!"#$%&'airncoating= 2t m = 0, m = ±1, m = ±2,...( )April 12, 2005 Physics for Scientists&Engineers 2 14Example: Lens Coating (4)Example: Lens Coating (4)! Thus the minimum thickness for the lens coating to providedestructive interference corresponds to m = 0! Note that this formula is the same as the one we found
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