# Lect03 - P Diffraction Spectroscopy Incident...

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P Incident Wave (wavelength λ ) y L a δ Diffraction & Spectroscopy

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Overview Overview z Multiple-slit Interference formula* z Diffraction Gratings z Optical Spectroscopy z Spectral Resolution z Single-Slit Diffraction* z Interference + Diffraction z X-ray Crystallography *Derivations in Appendix
Last Lecture: General properties of N Last Lecture: General properties of N - - Slit Interference Slit Interference The positions of the principal maxima of the intensity patterns always occur at φ = 0, ± 2 π , ± 4 π, . .. is the phase between adjacent slits] ( i.e., dsin θ = ± m λ , m = 0, 1, 2,… ) . The principal maxima become taller and narrower as N increases. The intensity of a principal maximum is equal to N 2 times the maximum intensity from one slit. The width of a principal maximum goes as 1/N. The # of zeroes between adjacent principal maxima is equal to N-1. The # of secondary maxima between adjacent principal maxima is N-2. 0 −2π I 0 16I 1 N=4 0 −2π I 0 25I 1 N=5 0 −2π I 0 9I 1 N=3 −λ/ d 0 λ/ d φ θ φ θ −λ/ d 0 λ/ d −λ/ d 0 λ/ d φ θ How do we calculate these interference patterns… How do we calculate these interference patterns…

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Act 1 Light interfering from 10 equally spaced slits initially illuminates a screen. Now we double the number of slits, keeping the spacing constant. What happens to the net power I on the screen? a. stays same (I) b. doubles (2I) c. increases x4 (4I)
Act 1 Light interfering from 10 equally spaced slits initially illuminates a screen. Now we double the number of slits, keeping the spacing constant. What happens to the net power on the screen? a. stays the same b. doubles c. increases by 4 If we double the number of slits, we expect the net power on the screen to double. How does it do this… z The location and number of the principle maxima (which have most of the power) does not change. z The principle maxima become 4x brighter. z But they also become only half as wide. (A/2) z Therefore, the net power (integrating over all the peaks) increases two-fold, as we would expect. (P goes as IA) We will soon see that we often use such an array of slits (also called a “diffraction grating”) to perform very precise metrology, e.g, spectroscopy, crystallography, etc.

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N N - - Slit Interference Slit Interference Summary Summary (see Appendix for derivation) (see Appendix for derivation) z The Intensity for N equally spaced slits is given by: L y and d sin d = = θ λ δ π φ 2 2 1 ) 2 / sin( ) 2 / sin( = N I I N * y L d θ z As usual, to determine the pattern at the screen (detector plane), we need to relate φ to θ or y = Ltan θ : ** φ is the phase difference between adjacent slits. * Your calculator can probably plot this up. Give it a try. ** Note: You will not be able to use the small angle approximations if d ~ λ.
Example Problem #1 In an N-slit interference pattern, at what angle θ min does the intensity first go to zero?

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Lect03 - P Diffraction Spectroscopy Incident...

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