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Formula Sheet
PHYC/ECE 463
Advanced Optics I
harmonic plane wave: E = Re{E0 exp(iωt − ik .r + ϕ )}
Poynting vector S =
1
μ0
E×B
ni sin(θi ) = nt sin(θt )
ρπ =
k=
Irradiance: I = S = ε 0 nc E0 2
UNM/Fall 2007
nω 2π n
=
c
λ0
2
Snell’s Law
nt cos(θi ) − ni cos(θt ) tan(θ i − θ t )
=
nt cos(θi ) + ni cos(θ t ) tan(θi + θt )
Fresnel’s Reflectivity
ρσ = −
ni cos(θi ) − nt cos(θt )
sin(θ i − θ t )
=−
ni cos(θi ) + nt cos(θt )
sin(θi + θt )
1 − ρ 2 = ττ '
R =| ρ |2
R +T =1
If n is complex then n → n = n − iκ ≡ ε / ε 0 ≡ 1 + χ in above expressions
Absorption coefficient (K or α ) and skin depth (δ): α ≡
2
δ
=
4πκ
λ0
Classical Electron Oscillator Model:
ω p2
χ= 2
ω0 − ω 2 + iω / τ
where ω p =
Nq 2
(plasma frequency)
m0ε 0
Drude model for metals: ω0 → 0
Group Velocity vg =
dω
dk
Light pressure (on perfecting absorbing surface) P =
I
c
Prism with apex angle α at minimum deviation angle θD:
⎛ α + θD ⎞
sin ⎜
⎟
⎝ 2 ⎠
n=
⎛α ⎞
sin ⎜ ⎟
⎝2⎠
Numerical Aperture (NA) of an optical fiber:
n0 sin(θ max ) = n 2f − nc2
f# (f-number)=f/D
Lens-makers’ formula:
⎛ 1 1 ⎞
1
= (n − 1) ⎜ − ⎟
f
⎝ R1 R2 ⎠
Gaussian imaging formula (thin lens)
1 1 1
= +
f S S'
(refractive sphere)
n '− n n n '
= +
R
S S'
Paraxial Ray Tracing Matrices:
Propagation
0⎞
⎛ 1
⎟ (length d and index n)
⎜
⎝d / n 1⎠
Refractive surface
⎛ 1 −P ⎞
⎟ where P=(n’-n)/R
⎜
⎝0 1 ⎠
Mirror with radius of curvature R
⎛ 1 −2 / R ⎞
⎜
⎟
1 ⎠
⎝0
Thick lens (Thickness Dl , index nl)
PP ' Dl ⎞
⎛ P ' Dl
− P − P '+
⎜1 − n
nl ⎟
l
⎜
⎟
⎜ Dl
⎟
PDl
1−
⎜ n
⎟
nl
l
⎝
⎠
P=(nl-n)/R, P’=(n’-nl)/R’
Thin lens of focal length f
⎛ 1 −1/ f ⎞
⎜
⎟
1 ⎠
⎝0
Separated Doublet
d
1 1
d ⎞
⎛
⎜1 − f n − f − f + f f n ⎟
2 b
1
2
2 1 b
⎜
⎟
⎜ d
⎟
d
1−
⎜ n
⎟
f1nb
b
⎝
⎠
between conjugate planes
n' ⎞
⎛
⎜ mα n M 12 ⎟
⎜⎜
⎟
mx ⎟⎠
⎝ 0
between principal planes
⎛ 1 M 12 ⎞
⎜
⎟
⎝0 1 ⎠
for telescopic Systems
n'
⎛
⎞
⎜ mα n 0 ⎟
⎜⎜
⎟⎟
⎝ M 21 mx ⎠
position of principal planes
⎛ n ⎞
D=⎜
⎟ (1 − M 11 )
⎝ M 12 ⎠
⎛ n' ⎞
D' = ⎜
⎟ (1 − M 22 )
⎝ M 12 ⎠
Contact Doublet Achromatization:
f1V1 + f 2V2 = 0
where V =
nd − 1
is the Abbe number
n f − nc
Physical constants:
speed of light
electronic charge
permittivity of vacuum
electronic mass
Planck constant
c~ 2.998×108
q~ 1.602×10-19
ε0~8.854×10-12
m0~9.1094×10-31
h~6.626×10-34
Near distance of a normal eye: d0=250 mm
m s-1
C
F/m
Kg
J.s