Formula Sheet
PHYC/ECE 464 (Laser Physics I)- University of New Mexico
Instructor: Mansoor Sheik-Bahae
OPTICAL SCIENCE & ENGINEERING
University of New Mexico
Hermite-Gaussian Beams
⎛ 2x ⎞
⎛ 2 y ⎞ w0
⎛
E ( x, y , z )
kr 2 ⎞
−1
H
i
= H m ⎜⎜
−
exp
⎜
⎟ × exp −i ⎡⎣ kz − (1 + m + p ) tan ( z / z 0) ⎤⎦
⎟⎟ p ⎜⎜ w( z ) ⎟⎟ w( z )
E0
w
z
q
z
(
)
2
(
)
⎝
⎠
⎝
⎠
⎝
⎠
2
1
1
λ0
π nw02
z ⎞
⎛ z02 ⎞ ,
ω 2π n
2
2⎛
,
,
=
−i
=
z
w
(
z
)
=
w
1
+
=
+
R
(
z
)
z
1
k=n =
0
0 ⎜
⎜
2 ⎟
2 ⎟
q ( z ) R( z ) π nw2 ( z )
λ
z
λ0
c
0
0 ⎠
⎝ z ⎠
⎝
(
)
Irradiance: I =< S >= ncε 0 E0 2
2
Fresnel’s Reflectivities:
Snell’s Law ni sin(θi ) = nt sin(θt )
n cos(θi ) − ni cos(θt ) tan(θi − θ t )
r|| = t
=
nt cos(θ i ) + ni cos(θt ) tan(θi + θt )
r⊥ = −
Intensity (Power) Reflectivity: R=|r|2
Brewster angle (from 1 to 2): θ B = tan −1 (n2 / n1 )
ni cos(θ i ) − nt cos(θ t )
sin(θ i − θ t )
=−
ni cos(θi ) + nt cos(θt )
sin(θi + θt )
Critical angle (from 1 to 2): θ c = sin −1 (n1 / n2 )
If n is complex then n → n = n + iκ in above expressions
Lens Transformation of a Gaussian beam:
1
1 1
=
−
Rout Rin f
⎛ 1 1 ⎞
1
= (n − 1) ⎜ − ⎟
f
⎝ R1 R2 ⎠
Lens-makers’ formula:
Fabry-Perot Transmission and Reflection (general; with gain or absorption):
T (θ , G0 ) =
G0 (1 − R1 )(1 − R2 )
(1 − G
0
R1 R2
)
2
+ 4G0 R1 R2 sin 2 (θ )
( R − R ) + 4G
(1 − G R R ) + 4G
2
R (θ , G0 ) =
1
2
0
2Δθ1/ 2 =
θ = kd =
0
2
1
2
0
4
Finesse F = π R1 R 2 = Δν FSR
1 − R1 R2
R1 R2 sin (θ )
2
R1 R2 sin 2 (θ )
Δν 1/ 2
Free Spectral Range: Δν = c = 1
FSR
τ RT
1
τp =
1 − R1 R2 2πΔν 1/ 2
2nd
Photon Lifetime:
τ RT
1 − G0 R1 R2
1/ 2 4
G0
R1 R 2
ω nd
c
for plane waves
Lorentzian line shape:
e.g. in natural or pressure broadened
Δν h / 2π
g (ν ) =
2
2
(ν −ν 0 ) + ( Δν h / 2 )
General Resonance Condition:
roundtrip phase change= q2π
Doppler broadened line shape
1/ 2
⎛ 4 ln 2 ⎞
g (ν ) = ⎜
⎟
⎝ π ⎠
2
⎡
⎛ ν −ν 0 ⎞ ⎤
exp ⎢( −4 ln 2 ) ⎜
⎟ ⎥
Δν D
⎢⎣
⎝ Δν D ⎠ ⎥⎦
1
1/ 2
with Δν D = ⎛⎜ 8kT ln2 2 ⎞⎟ ν 0
⎝ Mc
⎠
Formula Sheet (page 2)
PHYC/ECE 464 (Laser Physics I)- University of New Mexico
Instructor: Mansoor Sheik-Bahae
OPTICAL SCIENCE & ENGINEERING
University of New Mexico
Laser amplifier gain: ln G + G −1 = 0 where G0=exp(γ0Lg) is the small-signal gain, G=Iout/Iin
G0
Is Iin
ABCD Matrices ⎛ A B ⎞
⎜
⎝C
⎟
D⎠
Free space of length d
⎛1 d ⎞
⎜
⎟
⎝0 1 ⎠
⎛ r2 ⎞ ⎛ A B ⎞ ⎛ r1 ⎞
⎜ ⎟=⎜
⎜ ⎟
⎜ r ′ ⎟ ⎝ C D ⎟⎠ ⎜ r ′ ⎟
⎝ 2 ⎠
⎝1⎠
AD-BC=1
0 ⎞
⎛1
⎜
⎟
⎝ 0 n1 / n2 ⎠
Propagation in a medium of
length d and index n2=n
immersed in vacuum (n1=1).
⎛1 d / n⎞
⎜
⎟
1 ⎠
⎝0
Thin lens of focal length f
⎛ 1
⎜
⎝ −1/ f
0⎞
⎟
1⎠
Mirror with radius of curvature R
Spherical dielectric interface
0⎞
⎛ 1
⎜
⎟
⎝ −2 / R 1 ⎠
1
0 ⎞
⎛
⎜
⎟
⎝ (1 − n1 / n2 ) / R n1 / n2 ⎠
Photon Density (Photon Number per Volume)
dN p
dt
ABCD rule for
Gaussian Beams
Aq + B where
q2 = 1
Cq1 + D
Dielectric interface
(from n1 to n2)
=
G2S −1
τ RT
Np
V
=
I
hν c / ng
q ( z ) = z + iz 0
or
1
1
λ0
=
−i
q( z) R( z)
π nw ( z ) 2
Gaussian pulse propagation (broadening) in
dispersive media
⎛ z2 ⎞
τ p2 ( z ) = τ p2 0 ⎜1 + 2 ⎟ where
⎝ A0 ⎠
τ p2 0
and
2 β2
group velocity dispersion (GVD)
λ 3 d 2n λ 2
D
β2 =
=
2π c 2 d λ 2 2π c
dispersion length A 0 =
N p + N 2 cσ (Photon number growth dynamics due to stimulated and spontaneous emission processes)
S (survival factor)=R1R2 for a simple two mirror cavity,
Threshold condition: SG2=1 (linear cavity),
G2=roundtrip gain (exp(2γLg))
SG=1 (ring cavity)
Schawlow-Townes limit for laser linewidth: Δν osc ≈ 2π
At steady-state: γ = γ th =
γ0
hν
2
( Δν 1/ 2 )
Pout
where
1+ I / Is
I≈I++I-≈2I+ for a high-Q linear (standing-wave) or bidirectional ring cavity, I≈I+ for a unidirectional ring cavity.
Iout=Ta..T2I+ (T2 is the output coupling transmission and Ta.. are the transmission of other optical surfaces in
the path)
Q-Switching and Gain-Switching: Δtp≈τp (cavity photon lifetime)
Modelocking: Repetition Rate =1/Trt=2Lng/c (linear cavity), Pulsewidth: Δtp≈>1/Δν
Kerr-lens focal length: f nl π w4
8dn2 P
where n2 is the Kerr index defined as n=n0+n2I, d=thickness, P=peak
power, w= beam radius
Threshold current density in a diode laser: Jth= eNehthd/τr
Fundamental Physical Constants
Quantity
Symbol Value
Speed of light
c
Planck constant
h
Planck constant
h
Planck hbar
Planck hbar
Gravitation constant
G
Boltzmann constant
k
Molar gas constant
R
Charge of electron
e
Permeability of vacuum
Permittivity of vacuum
Mass of electron
Mass of proton
Mass of neutron
Avogadro's number
Stefan-Boltzmann constant
Rydberg constant
Bohr magneton
Bohr radius
Standard atmosphere
atm
1G = 10-4 T , 1 eV = 1.602 × 10-19 J, 1 dyne = 10-5 N, 1 erg = 10-7 J
OPTICAL SCIENCE & ENGINEERING
University of New Mexico