P Incident Wave (wavelength l) y L a Lecture 3 Diffraction & Spectroscopy We will discuss N split interference (diffraction) and single split diffraction Multi slit Interference, I = 4I1cos 2(f/2) (Fo[.]
P Lecture 3: Diffraction & Spectroscopy Incident Wave (wavelength l) y a L We will discuss N-split interference (diffraction) and single-split diffraction (2 slits) Multi-slit Interference, I = 4I1cos2(f/2) (For point sources, I1 = constant.) and Single-slit Diffraction, I1(q) (For finite sources, I1 = I1(q).) to obtain Total Interference Pattern, I = 4I1(q)cos2(f/2) (Remember how f is related to q: f/2p = d/l = (dsinq)/l d q/l) Content Multiple-slit Interference formula Diffraction Gratings Optical Spectroscopy Spectral Resolution Single-Slit Diffraction Interference + Diffraction Applications: X-ray Crystallography General properties of N-Slit Interference 9I 16I1 16 N=3 g(Ix) h(I x) 25I 25 N=4 N=5 20 10 h5( I x) 10 00 10-2p 10-l/d 00 0x 2p l/d 10 10 f q 0 10-2p 10-l/d 00 0x 2p l/d 10 10 f q 00 10-2p 10-l/d 00 0x 2p 10 f l/d 10 q • The positions of the principal maxima of the intensity patterns always occur at f = 0, 2p, 4p, [f is the phase between adjacent slits] (i.e., dsinq = ml, m = 0, 1, 2,…) • The principal maxima become taller and narrower as N increases • The intensity of a principal maximum is equal to N2 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 exercise 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 exercise 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 If we double the number of slits, we expect the net power on the screen to double How does it this… The location and number of the principle maxima (which have most of the power) does not change The principle maxima become 4x brighter But they also become only half as wide Therefore, the net power (integrating over all the peaks) increases two-fold, as we would expect 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 N-Slit Interference – Summary The Intensity for N equally spaced slits is given by: sin( Nf / 2) I N = I1 sin(f / 2) * y q d As usual, to determine the pattern at the screen (detector plane), we need to relate f to q or y = Ltanq: f d d sinq d q y = = and q 2p l l l L ** L f is the phase difference between adjacent slits * Note: we can not be able to use the small angle approximations if d ~ l Example In an N-slit interference pattern, at what angle qmin does the intensity first go to zero? (In terms of l, d and N.) qmin ? l/d q Example In an N-slit interference pattern, at what angle qmin does the intensity first go to zero? (In terms of l, d and N.) qmin ? sin( Nf / 2) I N = I1 sin( f / 2) l/d q has a zero when Nfmin/2 = p, or fmin = 2p/N But f = 2p(d sinq)/l 2pd q/l = 2p/N Therefore, qmin l /Nd As the illuminated number of slits increases, the peak widths decrease! This is a general feature: Wider slit features narrower patterns in the “far field” Optical spectroscopy – how we know about the world • Quantum mechanics definite energy levels, e.g., of electrons in atoms or molecules • When an atom transitions between energy levels emits light of a very particular frequency • Every substance has it’s own “signature” of what colors it can emit • By measuring the colors, we can determine the substance, as well as things about it’s surroundings (e.g., temperature, magnetic fields), whether it’s moving (via the Doppler effect), etc Optical spectroscopy is invaluable in materials research, engineering, chemistry, biology, medicine… But how we precisely measure wavelengths??? ... N-split interference (diffraction) and single-split diffraction (2 slits) Multi-slit Interference, I = 4I1cos2(f/2) (For point sources, I1 = constant.) and Single-slit Diffraction, I1(q) (For... Content Multiple-slit Interference formula Diffraction Gratings Optical Spectroscopy Spectral Resolution Single-Slit Diffraction Interference + Diffraction Applications: X-ray Crystallography... 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 N-Slit Interference – Summary