Review If coherent light with wavelength is incident on a single slit of width a a single slit diffraction pattern will be produced Physics for Scientists Engineers 2 The angles of the minima are given by a sin m m 1 2 3 m is the order of each minimum If we project this diffraction pattern on a screen a large distance L away from the slit the position on the screen measured from the center line is given by Spring Semester 2005 Lecture 44 y April 12 2005 Physics for Scientists Engineers 2 1 April 12 2005 Review 2 2 where m 1 2 3 Physics for Scientists Engineers 2 2 Thin Film Interference The intensity of light passing through a single slit I relative to Imax that we would get if there were no slit is sin I I max m L a a sin Another way of producing interference phenomena is partial reflection of light from the front and back layers of thin films A thin film is an optically clear material with thickness on the order few wavelengths of light Examples of thin films include the walls of soap bubbles and thin layers of oil floating on water When the light reflected off the front surface constructively interferes with the light reflected off the back surface of the thin film we see the color corresponding to the wavelength of light that is interfering constructively April 12 2005 Physics for Scientists Engineers 2 3 April 12 2005 Physics for Scientists Engineers 2 4 Thin Film Interference 2 Thin Film Interference 3 When light travels from an optical medium with an index of refraction n1 into a second optical medium with index of refraction n2 several things can happen Let s begin our analysis of thin films by studying a thin film with index of refraction n in air as shown below 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 be reflected In this case the phase of the light can be changed depending on the index of refraction of the two optical media If n1 n2 the phase of the reflected wave will be changed by half a wavelength An angle of incidence is shown for the light waves in the figure for clarity If n1 n2 then there will be no phase change April 12 2005 Physics for Scientists Engineers 2 We will assume that light is incident perpendicular to the surface of the thin film 5 April 12 2005 Thin Film Interference 4 The reflected wave undergoes a phase shift of half a wavelength when it is reflected because nair n At the front surface some of the light is transmitted and some is reflected The reflected light will not be considered The transmitted light has no phase shift and emerges from the film and interferes with the light that was reflected when the light originally entered the film The light that is transmitted has no phase shift and continues to the back surface of the film The transmitted light has traveled a longer distance than the originally reflected light and has a phase shift given by the path length difference that is twice the thickness t of the film At the back surface again part of the wave is transmitted and part of the wave is reflected The transmitted light passes through the film completely The reflected light has no phase shift because n nair and travels back to the front surface of the film Physics for Scientists Engineers 2 6 Thin Film Interference 5 When the light wave reaches the boundary between air and the film part of the wave is reflected and part of the wave is transmitted April 12 2005 Physics for Scientists Engineers 2 7 April 12 2005 Physics for Scientists Engineers 2 8 Thin Film Interference 4 Thin Film Interference 5 The fact that the originally reflected light has undergone a phase shift and the transmitted light has not means that the criterion for constructive interference is given by 1 x m 2t 2 m 0 m 1 m 2 t min The wavelength of the light traveling in air is related to the wavelength of the light traveling in the film by April 12 2005 1 air 2t m 2 n m 0 m 1 m 2 The minimum thickness tmin that will produce constructive interference corresponds to The wavelength refers to the wavelength the light traveling in the thin film which has index of refraction n We can then write air 4n Note that this result applies only to the case where of a material with index of refraction n and air on both sides like a soap bubble air n Physics for Scientists Engineers 2 9 April 12 2005 Example Lens Coating Physics for Scientists Engineers 2 10 Example Lens Coating 2 Many high quality lenses are coated to prevent reflections This coating is designed to set up destructive interference for light that is reflected from the surface of the lens We assume that the light is incident perpendicularly on the surface of the coated lens as shown below Assume that the coating is MgF2 which has ncoating 1 38 and the lens is glass with nlens 1 51 Question What is the minimum thickness of the coating that will produce destructive interference for light with a wavelength of 550 nm April 12 2005 Physics for Scientists Engineers 2 Light reflected at the surface of the coating will undergo a phase change of half a wavelength because nair ncoating The light transmitted through the coating has no phase change 11 April 12 2005 Physics for Scientists Engineers 2 12 Example Lens Coating 3 Example Lens Coating 4 Light reflected at the boundary between the coating and the lens will undergo a phase change of half a wavelength because ncoating nlens Thus the minimum thickness for the lens coating to provide destructive interference corresponds to m 0 This reflected light will travel back through the coating and exit with no phase change t min Thus both the light reflected from the coating and from the lens will have suffered a phase change of half a wavelength Note that this formula is the same as the one we found for constructive interference in a film with air on both sides To analyze thin film interference one must always take into account the phase changes at the boundary Thus the criterion for destructive interference is 1 air 2t m 2 ncoating April 12 2005 An even number of phase changes is the same as no phase changes m 0 m 1 m 2 Physics for Scientists Engineers 2 air 550 10 9 m 9 96 10 8 m 99 6 nm 4n 4 1 38 An odd number of phase changes is the same as one phase change 13 April 12 2005 Interferometer Physics for Scientists Engineers 2 14 Interferometer 2 An interferometer is a device designed to measure lengths or changes in length using interference of light A photograph and drawing
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