Waves Part III Electromagnetic waves
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1 Waves Part III Electromagnetic waves
2 Electromagnetic (light) waves Transverse waves Transport energy (and momentum) Can travel through vacuum (!) and certain solids, liquids and gases Do not transport matter or charge From the Maxwell s Equations: wave equation with c vacuum = 1/ e 0 µ 0 Electromagnetic (linearly polarized) wave
3 The electromagnetic spectrum c = 299,792,458 m/s = 186, miles/s c air ~ c vacuum In vacuum: f(or ν) = c λ
4 Optical properties Interference: the superposition of two light beams to produce alternating regions of bright light and dim light Diffraction: the spreading of waves around corners (light waves, water waves) One convincing proof of the wave nature of light / contradicts geometric optics Huygens principle: every point on a wave front of light may be a source of secondary wave fronts / wavelets Coherence: two sources of light with the same frequency and a constant phase difference are called coherent Monochromatic: one color, only one wavelength/frequency
5 Path length difference Sources S1 and S2: coherent sources of light Different path lengths: different phases for both waves arriving at one point Constructive interferences: r 2 -r 1 = ml (m=0, m=±1, m=±2, m=±3 ) Destructive interferences: r 2 -r 1 = (m+½)l (m=0, m=±1, m=±2, m=±3 )
6 Young s double-slit interference (1802) INTERFERENCES ON THE SCREEN
7 Analysis (I) L d Slit S 1 q q»q q y q (Assumption: q»q correct, if L>>d) Slit S 2 Path lengths difference: DL = d sin q Incident Wave Screen
8 Analysis (II) L S 1 q y d S 2 For the light coming from S 2 to be in phase with the light coming from S 1 : DL = nl y = L tan q ; q small, tan q» sin q à y = L sin q DL = nl = d sin q y n = nll / d Position of n th bright fringe Out of phase : DL = (n+½) l = d sin q à y n = (n+½)ll / d (position of the n th dark fringe)
9 Single slit diffraction Far field diffraction = Fraunhofer diffraction (D>>a) Analysis similar to Young s two slits experiment b/c of Huygens principle (every point on a wave front of light may be a source of secondary wave fronts / wavelets)
10 Single slit diffraction analysis (I) a a/2 q Huygens principle: Waves 1 & 5, 2 & 6, 3 & 7 and 4 & 8 are out of phase (i.e. cancel one-byone) if: (a/2) sin q 1 = l/2 l/2
11 Single slit diffraction analysis (II) a a/4 q l/2 Huygens principle: One can repeat the exercise by considering a difference set of points along the slits (and a different diffraction angle) such that: (a/4) sin q 2 = l/2 Generalization (2n points): (a/2n) sin q n = l/2 Dark fringes in single slit diffraction: sin θ n = nλ a with n=±1, ±2 ±3
12 Exercise Single slit diffraction: a = 0.2 mm (2x10-3 m), l = 500 nm (500x10-9 m). What is the angular separation between n=1 and n=2 minima? sin q n = nl/a n = 1: sin q 1 = l/a = 500 x 10-9 / 2 x 10-3 = 2.5 x 10-4 à q 1» 2.5 x 10-4 rad n = 2: sin q 2 = 2l/a = 2 x 500 x 10-9 / 2 x 10-3 = 5 x 10-4 à q 1» 5 x 10-4 rad Dq = q 2 q 1 = (5 2.5) x 10-4 = 2.5 x 10-4 rad à Dq = 0.014º Angular width of the central maximum? Maximum between n=-1 and n=1: n = -1: sin q -1 = -l/a = -500 x 10-9 / 2 x 10-3 = -2.5 x 10-4 à q -1» -2.5 x 10-4 rad n = 1: sin q 1 = l/a = 500 x 10-9 / 2 x 10-4 = 2.5 x 10-4 à q 1» 2.5 x 10-4 rad Dq = 5 x 10-4 rad à Dq = 0.028º
13 Diffraction limit: microscopes a = nl with n>>1 If the observed object (size a or d) is many times larger than the wavelength l of the radiation observing it. However, if a (or d) is of the order of l or smaller, then diffraction effect will create significant blurring, then will completely destroy the image. Width: a (left fig.) or d (right fig.)
14 Wave-nature of light causes diffraction when light enters telescope. Instead of a point of light on telescope film, one get pattern - interference due to the different path lengths. Diffraction limit: telescopes Makes it difficult to resolve two close stars (bottom pic) One can define the resolving power a (in [rad]) of a telescope. α =1.22 λ D where l is the wavelength of the light and D is the diameter of the telescope. Note: the 1.22 comes from the position of the first dark circular ring of a Airy disk. Airy disk: In optics, the Airy disk and Airy pattern are descriptions of the best focused spot of light that a perfect lens with a circular aperture can make, limited by the diffraction of light (wikipedia).
15 Diffraction grating Diffraction grating: hundreds or thousands of slits per mm Application: light decomposition in individual wavelengths
16 Grating spectrograph
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