Waves Part III Electromagnetic waves

Waves Part III Electromagnetic waves

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

The electromagnetic spectrum c = 299,792,458 m/s = 186,282.397 miles/s c air ~ c vacuum In vacuum: f(or ν) = c λ

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

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 )

Young s double-slit interference (1802) INTERFERENCES ON THE SCREEN

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

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)

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)

Single slit diffraction analysis (I) a a/2 q 1 1 2 3 4 5 6 7 8 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

Single slit diffraction analysis (II) a a/4 q 2 1 2 3 4 5 6 7 8 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

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

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.)

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).

Diffraction grating Diffraction grating: hundreds or thousands of slits per mm Application: light decomposition in individual wavelengths

Grating spectrograph