Formulae and Constants:
e = 1.60 x 10-19 Coulombs
E = h f = hc / λ
& momentum, p = h / λ
1 eV = 1.60 x 10-19 J
h = Planck’s const = 6.63 x 10-34 J•s
= 4.14 x 10-15 eV•s
____
hc = 1240 eV•nm
8
c = “speed of light” = 1 / √ (ε0 µ0) = 3 x 10 m/s
c=λf
Traveling wave equation:
y(x,t) = A sin [k(x - vt)] = A sin (kx - ωt) = A sin [ (2π / λ) x - ( 2π / T) t ]
where
k = 2π / λ
ω = 2π f = 2π / T
v=ω/k
2-slit interference
θm locates maxima (centers of bright spots) for 2 slits, N slits, gratings
a sin θm = m λ (m = 0, 1, 2, ...)
where a = slit separation
Polarization
• intensity of polarized light transmitted proportional to the amplitude squared:
Malus’ Law I = I0 cos2θ
• by reflection -- Brewster’s angle = polarizing angle -- tan θp = n2/n1
• by scattering
For a crystal with thickness d, the phase difference between the 2 rays (ordinary and
extraordinary) is Φ = [2π / λ0] * |n⊥ -n|| | * d
Photoelectric Effect
hf = φ + Ek or hc / λ = φ + eV0
or hc / λ = hc / λc + eV0
Blackbody Radiation:
Wien displacement law
λmax T = 2.880 x 106 nm•K
Stefan-Boltzmann law
energy / (time•area) = σ T 4
σ = 5.67 x 10 -8 W / (m2 K4)
Bohr Model of Atom
Allowed energy levels: En = -13.6 eV / n2
Energies of photons that can be emitted or absorbed = [exactly] | En1 -En2 |
Reflection/refraction
n=c/v
λn = λ / n
Snell’s Law: n1 sin θ1 = n2 sin θ2
For step-index fibers
___________
Numerical Aperture = N.A. = no sin θm = √ n12 -n22
___________
Skip distance = Ls = n2 d / √ n12 -n22
# of modes = (1/2) [ π d (N.A.) / λ ] 2
in order to be single-mode, a fiber diameter d must be:
d < 2.4 λ / π (N.A.)
For fibers in general:
Loss of power in dB = α L = 10 dB log10 (P1 / P2)
where α is the attenuation coefficient
Lenses/Mirrors
Textbook sign conventions: [these assume light is coming from the left !]
• if the object/image is to the left of the vertex (where lens/mirror crosses axis), its distance
u/v is negative
• radii of curvature follow the same rule as above, so diverging mirrors have positive f =
focal lengths, but diverging lenses have negative focal lengths
Object/image location
Magnification
Mirror Equations
1/u + 1/v = 2/r = 1/f
m = hi / ho = - v / u
Lens Equations
- 1/u + 1/v = 1/f
m = hi / ho = v / u
For Mirrors: f = radius of curvature / 2
Lens Maker’s Equation: 1/f = [(n2 - n1 ) / n1] [ 1/r1 -1/r2 ]
3 standard rays for ray diagrams:
• in thru focus, out parallel to axis
• in parallel, out thru focus
• in thru center, undeviated