Review Imagine light incident on a thin optically clear medium in air such as a soap bubble Physics for Scientists Engineers 2 Spring Semester 2005 Transmitted light has no phase change Lecture 45 For reflected light If n1 n2 the phase of the reflected wave will be changed by half a wavelength If n1 n2 then there will be no phase change April 13 2005 Physics for Scientists Engineers 2 1 April 13 2005 Review 2 An interferometer is a device designed to measure lengths or changes in length using interference of light An interferometer can measure lengths or changes in lengths to a fraction of the wavelength of light using interference fringes m 0 m 1 m 2 The minimum thickness tmin that will produce constructive interference corresponds to t min air 4n An interferometer can be used to measure the thickness of a material or the index of refraction of a material by placing in one of the paths of the interfering light and counting the change in the number of fringes We get the same answer for the destructive interference of light passing from air to two clear optical media such that nair n1 n2 such as the coating on a camera lens April 13 2005 Physics for Scientists Engineers 2 2 Review 3 The criterion for constructive interference of light incident on a thin optically clear medium in air such as a soap bubble is 1 air 2t m 2 n Physics for Scientists Engineers 2 N material N air 3 April 13 2005 2tn 2t 2t n 1 Physics for Scientists Engineers 2 4 Diffraction by a Circular Opening Diffraction by a Circular Opening 2 We have considered interference through two slits and diffraction through a single slit This result is similar to the result from a single slit except for the factor of 1 22 Now we consider diffraction of light through a circular opening If one is using a circular lens to observe two distant points objects such as two stars whose angular separation is small diffraction limits the ability of the lens to distinguish these two objects Diffraction through a circular opening relates to observing objects with telescopes with circular mirrors and cameras that have circular lens The criterion for being able to separate two point objects is based on the idea that if the first image is centered on the first diffraction minimum of the second object the objects are just resolved The resolution of a telescope or camera is limited by diffraction phenomena This criterion is called Rayleigh s Criterion and is expressed as 1 22 R sin 1 d The first diffraction minimum from light with wavelength passing through a circular opening with diameter d is sin 1 22 d April 13 2005 Physics for Scientists Engineers 2 Where R is the minimum observable angular separation is the wavelength and d is diameter of the lens 5 April 13 2005 Examples of Resolution Physics for Scientists Engineers 2 6 Hubble Space Telescope The diameter of the Hubble Space Telescope is 2 4 m What is the minimum angular resolution of the Hubble Space Telescope Using Rayleigh s Criterion with green light of wavelength 550 nm we get R 1 22 550 10 9 m 2 8 10 7 2 4 m which corresponds to the angle subtended by a dime located 64 km away When the Hubble Space Telescope was first launched flaws were discovered in the main mirror that limited its ability to resolve images Resolved April 13 2005 Barely Resolved Physics for Scientists Engineers 2 A repair mission fixed the mirror so that it now functions at the diffraction limit Not Resolved 7 April 13 2005 Physics for Scientists Engineers 2 8 Double Slit Diffraction Double Slit Diffraction 2 We have discussed the interference pattern produced by two slits With diffraction effects the intensity of the interference pattern from double slits is given by For that analysis we assumed that the slits themselves were very narrow compared with the wavelength of light a sin I I max cos 2 For many sets of real world double slits the condition a is not met and we observe that not all the interference fringes have the same intensity Physics for Scientists Engineers 2 a d sin sin If the screen is placed a sufficiently large distance from the slits then we can write For these narrow slits the diffraction maxima are very wide and we saw peaks in the intensity that were the same intensity at all angles April 13 2005 2 9 Double Slit Diffraction 3 ay dy and L L On the next slide we calculate the intensity pattern for a double slit including interference and diffraction assuming L 2 0 m a 5 0 10 6 m d 1 0 10 5 m and 550 nm April 13 2005 Physics for Scientists Engineers 2 10 Real Life Two Slit Diffraction Pattern Diffraction minima Pattern for single slit diffraction April 13 2005 Pattern for double slit interference Physics for Scientists Engineers 2 11 April 13 2005 Physics for Scientists Engineers 2 12 Diffraction Gratings Diffraction Gratings We have discussed diffraction and interference for a single slit and for two slits Now we will discuss the application of diffraction and interference to a system of many slits Putting many slits together forms a device called a diffraction grating A portion of a diffraction grating is shown below In this drawing we see coherent light with wavelength incident on a series of narrow slits each separated by a distance d A diffraction grating has a large number of slits or rulings placed very close together A diffraction grating can also be constructed using an opaque material with grooves rather than actual slits A diffraction grating produces an intensity pattern that consists of narrow bright fringes separated by wide dark areas This characteristic pattern results from the use of many slits that produce destructive interference away from the maxima April 13 2005 Physics for Scientists Engineers 2 A diffraction pattern is produced on a screen a long distance L away 13 April 13 2005 Diffraction Gratings 2 We can calculate the path length differences for the paths shown on the previous slide Using an adjacent pair of rays the path length difference is x d sin The distance d is called the grating spacing To produce bright lines or constructive interference this path length difference must be an integer multiple of the wavelength so d sin m m 0 1 2 If the grating is W wide the number N of slits or gratings will be W d The values of m correspond to different bright lines Diffraction gratings are often specified in terms the number of slits or rulings per unit length n For m 0 we have the central maximum at 0 For m 1 we have the first order maximum We can obtain d from the
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