Review If light waves are traveling from some point then the phase difference x can be related to the path difference between the two waves Physics for Scientists Engineers 2 The criterion for constructive interference is given by a path difference x given by x m m 0 m 1 m 2 Destructive interference will take place if the path difference x is a half wavelength plus an integer times the wavelength Spring Semester 2005 Lecture 42 1 x m 2 April 8 2005 Physics for Scientists Engineers 2 1 April 8 2005 Double Slit Interference m 0 m 1 m 2 Physics for Scientists Engineers 2 2 Double Slit Interference 2 Our first example of the interference of light is Young s double slit experiment Below we see that spherical wavelets emitted from a point in the center of the slit For this situation we assume that we have coherent light light with the same wavelength and phase This light is incident on a pair of slits Each slit is smaller than the wavelength of light The slits are separated by a distance d For each slit we will use a Huygens construction and assume all the light observed passing through each slit is due to wavelets emitted from a single point at the center of that slit April 8 2005 Physics for Scientists Engineers 2 3 We assume that the slit is much smaller than the wavelength of light so that we can represent the source of the wavelets with one point April 8 2005 Physics for Scientists Engineers 2 4 Double Slit Interference 3 Double Slit Interference 4 Now let s look at two slits like the one on the previous page If we place a screen to the right of the slits we will observe an alternating pattern of bright lines and dark lines corresponding to constructive and destructive interference between the light waves emitted from the two slits To quantify these lines of constructive interference we expand and simplify the plot on the previous page We have coherent light incident from the left and a source of spherical wavelets at the center of each slit We see that the gray dashed lines represent lines along which there is constructive interference April 8 2005 Physics for Scientists Engineers 2 5 April 8 2005 Double Slit Interference 5 To further quantify the two slit geometry we expand and simplify the figure on the previous slide In this figure we assume that we have placed the screen a large distance L away from the slits such that the lines r1 and r2 are parallel to each other and to the line drawn from the center of the two slits to point P A line drawn from a point midway between the two slits to the same point on the screen makes an angle with respect to a line drawn from the slits perpendicular to the screen We draw a line from S1 perpendicular to r1 and r2 making a triangle with sides d b and x The point P on the screen is a distance y above the centerline Physics for Scientists Engineers 2 6 Double Slit Interference 6 The two lines r1 and r2 represent the distance from the center of slit S1 and slit S2 respectively to a point P on a screen that is placed a distance L away from the slits April 8 2005 Physics for Scientists Engineers 2 The quantity x represents the path length difference between r1 and r2 7 April 8 2005 Physics for Scientists Engineers 2 8 Double Slit Interference 7 Double Slit Interference 8 For destructive interference the path length difference must be an integer plus a half times the wavelength This path length difference will produce different phases for light originating from the two slits and illuminating the screen at point P x d sin m The path length difference can be expressed in terms of the distance between the slits and the angle at which the light is observed sin A dark fringe on the screen signals destructive interference Note that for constructive interference and m 0 we obtain 0 which means that x 0 and we have a bright fringe at zero degrees For constructive interference this path length difference must be a multiple of the wavelength of the incident light April 8 2005 This bright fringe is called the central maximum m 0 m 1 m 2 Physics for Scientists Engineers 2 9 April 8 2005 Order of Interference Fringes 10 If the screen is placed a sufficiently large distance from the slits the angle will be small and we can make the approximation The order has a different meaning for bright fringes and for dark fringes sin tan y L For constructive interference We can express constructive interference as m 1 would give us the angle of the first order bright fringe m 2 would give us the second order fringe etc d sin d For destructive interference m 0 would give us the angle of the first order dark fringe y m L m 0 m 1 m 2 Rearranging this equation gives us the distance of the bright fringes from the central maximum along the screen 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 Physics for Scientists Engineers 2 Physics for Scientists Engineers 2 Fringes on a Distance Screen The integer m is called the order of the fringe April 8 2005 m 0 m 1 m 2 A bright fringe on the screen signals constructive interference x or x d sin d x d sin m 1 2 y 11 April 8 2005 m L d m 0 m 1 m 2 Physics for Scientists Engineers 2 12 Fringes on a Distance Screen 2 Double Slit Intensity on a Distant Screen Similarly we can express the distance of the dark fringes from the central maximum along the screen as 1 m L 2 y d We start our calculation of the intensity of light from a double slit by assuming that the light emitted at each slit is in phase The electric field of the light waves can be described by m 0 m 1 m 2 E Emax sin t where Emax is the amplitude of the wave and is the angular frequency These formulas allow us to locate the positions of the bright and dark fringes However we can also calculate the intensity of the light at any point on the screen April 8 2005 Physics for Scientists Engineers 2 13 Double Slit Intensity on a Distance Screen 2 When the light waves arrive at the screen from the two slits they have traveled difference distances and so can have difference phases that depend on the angle of observation April 8 2005 Physics for Scientists Engineers 2 14 Double Slit Intensity on a Distant Screen 3 The two phasors E1 and E2 are shown below Let s express the electric field of the light arriving at a given point on the screen from S1 as E1 Emax sin t …
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