Review Coherent light with wavelength incident on two narrow slits separated by a distance d produces an interference pattern on a screen located a large distance L from the slits Physics for Scientists Engineers 2 The position of the bright fringes from the center line is given by y Spring Semester 2005 Physics for Scientists Engineers 2 m 0 m 1 m 2 The position of the dark fringes from the center line is given by 1 m L 2 y m 0 m 1 m 2 d Lecture 43 April 10 2005 m L d 1 April 10 2005 Review 2 Physics for Scientists Engineers 2 2 Review 3 The integer m is called the order of the fringe Coherent light with wavelength incident on two narrow slits separated by a distance d produces an interference pattern on a screen located a large distance L from the slits with intensity given by The order has a different meaning for bright fringes and for dark fringes dy I 4I max cos 2 L For constructive interference m 1 would give us the angle of the first order bright fringe m 2 would give us the second order fringe etc For destructive interference m 0 would give us the angle of the first order dark fringe m 1 would give us the second order fringe etc For both bright and dark fringes the first order fringe is the one closest to the central maximum April 10 2005 Physics for Scientists Engineers 2 3 April 10 2005 Physics for Scientists Engineers 2 4 Diffraction Diffraction 2 Any wave passing through an opening that is about the same size as the wavelength of the wave will experience diffraction The same applies to light waves For example to the right we show coherent light incident on an opening with dimension comparable to the wavelength of the light Diffraction means that the wave will spread out on the other side of the opening rather than having the opening cast a sharp shadow In addition if light passes through a narrow slit it will produce an interference pattern called a diffraction pattern Rather than casting a sharp shadow light spreads out on the other side of the opening Light passing a sharp edge will also exhibit a diffraction pattern April 10 2005 Physics for Scientists Engineers 2 Huygens principle can describe this spreading out and we can use a Huygens construction to quantify the diffraction phenomenon 5 April 10 2005 Diffraction 3 In this case the light far from the edge of the barrier continues to travel in a straight line The light near the edge of the barrier seems to bend around the barrier is described by the sources of wavelets near the edge The resulting light waves on the right side of the opening can undergo interference and produce a characteristic diffraction pattern Diffraction phenomena cannot be described by geometric optics Often the resolution of an optical instrument can be limited by diffraction effects rather than geometric effects Light waves can also encounter barriers such as the one shown Physics for Scientists Engineers 2 6 Diffraction 4 We can describe this spreading out by using a Huygens construction and assuming that spherical wavelets are emitted at several points inside the opening April 10 2005 Physics for Scientists Engineers 2 7 April 10 2005 Physics for Scientists Engineers 2 8 Single Slit Diffraction Single Slit Diffraction 2 At a distant screen we will observe an intensity pattern characteristic of diffraction We start our quantitative description of diffraction by studying the diffraction of light through a single slit of width a that is comparable to the wavelength of light that is passing through the slit This intensity pattern will consist of bright and dark fringes We will approach the calculation using a Huygens construction For the case of two slit interference we were able to work out the equations for the bright fringes based on constructive interference We assume that the light passing through the single slit is described by spherical wavelets emitted from a distribution of points located in the slit For diffraction we will analyze the dark fringes as destructive interference because constructive interference is beyond the scope of this book The light emitted from these points will superimpose and interfere based on the path length difference for each wavelet at each position April 10 2005 Physics for Scientists Engineers 2 9 April 10 2005 Single Slit Diffraction 3 Physics for Scientists Engineers 2 Single Slit Diffraction 4 To study the interference let s expand and simplify our previous figure for single slit diffraction We start with light emitted from the top edge of the slit and from the center of the slit as shown We assume coherent light with wavelength incident on a slit with width a that produces an interference pattern on a screen a distance L away To analyze the path difference we show an expanded version of our figure to the right Here we assume that the point P on the screen is far enough away that the rays r1 and r2 are parallel and make an angle with the central axis We employ a simplifying method of analyzing pairs of light waves emitted from points in the slit April 10 2005 Physics for Scientists Engineers 2 10 11 April 10 2005 Physics for Scientists Engineers 2 12 Single Slit Diffraction 5 Single Slit Diffraction 6 Therefore the path length difference for these two rays is sin Now let s consider four rays instead of two x a sin or x a 2 2 The criterion for the first dark fringe is x a sin 2 2 Although we chose one ray originating from the top edge of the slit and one from the middle of the slit to locate the first dark fringe we could have used any two rays that originated a 2 apart inside the slit April 10 2005 Physics for Scientists Engineers 2 13 Here we choose a ray from the top edge of the slit and three more rays originating from points spaced a 4 apart April 10 2005 Single Slit Diffraction 7 This equation describes the second dark fringe At this point we could take six pairs and eight pairs and describe the third and fourth dark fringes etc The path length difference between the pairs of rays r1 r2 r2 r3 r3 r4 is given by The result is that the dark fringes from single slit diffraction can be described by a sin m x a sin or x a 4 4 sin tan y L a sin or a sin 2 4 2 Physics for Scientists Engineers 2 m 1 2 3 If the screen is placed a sufficiently large distance from the slits the angle will be small and we can make the approximation The dark fringe is given by April 10 2005 14 Single Slit Diffraction 8 We can expand the previous drawing to represent the case of the screen being far away as shown to the right
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