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Formulas & Tools

Refractive Index Dispersion Equations & Calculator
Useful Formulas & Constants
Formula
ne2 = 2.3849 – 1.259*10-22 + 1.079*10-22 + 1.6518*10-44 - 1.94741*10-66 + 9.36476*10-88
no2 = 2.35728 – 1.17*10-22 + 1.054*10-22 + 1.34143*10-44 – 4.45368*10-76 + 5.92362*10-88
n2 = 1 + 0.6961663*λ2/(λ2 – 0.00467914826) + 0.4079426*λ2/(λ2 – 0.0135120631) +0.8974794*λ2/(λ2 – 97.9340025)

n2 = 1 + 1.03961212*λ2/(λ2 – 0.00600069867) + 0.231792344*λ2/(λ2 – 0.0200179144) +1.01046945*λ2/(λ2 – 103.560653)

n2 = 1 + 0.5675888*λ2/(λ2 – 0.00252642999) + 0.4710914*λ2/(λ2 – 0.0100783328) +3.8484723*λ2/(λ2 –1200.55597)

n2 = 1 + 4.2980149*λ2/(λ2-0.19206302) + 0.62776557*λ2/( λ2-0.378782602) + 2.8955633*λ2/(λ2-46.9945952)

ne2 = 1 + 0.41344023*λ2/(λ2 – 0.00135737865) + 0.50497499*λ2/(λ2 – 0.00823767167) + 2.4904862*λ2/(λ2 – 565.107755)
no2 = 1 + 0.48755108*λ2/(λ2 – 0.001882178) + 0.39875031*λ2/(λ2 – 0.00895188847) +2.3120353*λ2/(λ2 – 566.135591)
ne2 = 1 + 1.5039759*λ2/(λ2 – 0.00548041129) + 0.55069141*λ2/(λ2 – 0.0147994281) + 6.5927379*λ2/(λ2 – 402.89514)
no2 = 1 + 1.4313493*λ2/(λ2 – 0.0052799261) + 0.65054713*λ2/(λ2 – 0.0142382647) + 5.3414021*λ2/(λ2 – 325.017834)
Etalon Formulas

 Two parameters completely specify an etalon: the free spectral range (FSR) and the finesse (ℑ). The FSR is the spacing (usually given in frequency) between transmission peaks.
The finesse is the ratio of the free spectral range to the full width at half maximum (FWHM) of the transmission peak and is directly related to the reflectivity of the surface R.

Free Spectral Range

c is the speed of light, n is the index of refraction of the etalon, and L is the thickness of the etalon.

At high finesse values (where R is very close to 100% or 1), R≈1-π/ℑ
 
 Finesse Reflectivity 
2 24%
4 47% 
6 60 % 
8 68 %
Finesse 
Reflectivity
10 73 % 
15 81 % 
20 85 % 




 
 
 
Wave Vector, Frequency, Wavelength & Wavenumbers

 Wave Vector, Frequency, Wavelength, Wavenumbers
An easy number to remember is a 1-pm linewidth is approximately 125 MHz ar 1550 nm. 

where
k = wave vector; v = frequency; w = 2πv = angular frequency; λ = wavelength; λ0 = wavelength in vacuum; n = refractive index

Wavelength (in vacuum), nm Frequency, THz Electron Volts, eV Wavenumber, cm-1
1561.42 192.00 0.80 6404.43
1550 193.41 0.80 6451.61
1320 227.12 0.94 7575.76
1064 281.76  1.17 9398.50 
980 350.91 1.27 10204.08
780 384.35 1.59 12820.51
632.8 473.76 1.96 15802.78
350 856.55 3.55 28571.43
 
International System of Units (SI) Prefixes

 

Factor Name Symbol
1021 zetta Z
1018 exa  E
1015 peta 
1012 tera 
109 giga 
106 mega
103 kilo
102 hecto
Factor Name Symbol
 10-2  centi  c
 10-3  mili  m
 10-6  micro  µ
 10-9  nano
 10-12  pico
 10-15  femto f
 10-18  atto a
 10-21  zepto z
 10-24  yocto
Common Material Properties

 

Material Refractive index, n ∆FSR*, MHz Thermal Expansion Coefficient α, ppm/°C Thermo-Optic Coefficient β or ∂n/∂T, ppm/°C  °
Air 1.0000 0.0 0.0 1.0
Fused Silica 1.444 13.1 0.55 6.57
Silicon 3.477 198.1 3.24 160
LASFN9 1.813 9.4 7.4 1.3

Change in FSR due to dispersive effects as measured from 1510 to 1570 nm for a 50 Ghz etalon

Nonlinear Crystal Thickness Limited by Group Velocity Mismatch (GVM)

 nonlinear crystal thickness limited by group velocity mismatch

Where t - pulse duration, c - speed of light, n - refractive index, λ - wavelength
Nonlinear Crystal Acceptances

 Nonlinear Crystal acceptances – Angular Δθ, Temperature ΔT, Spectral Δv – corresponding bandwidths at Full Width of Half Maximum (FWHM) of conversion efficiency.

Nonlinear Crystal Acceptances
Uniaxial Crystals Refractivity

 
Polar coordinate system for description of refractive properties of uniaxial crystal.


Uniaxial_Crystals.jpg

Whereas K – light propagation vector at phase matching conditions, Z – optical axis of crystal, θ – phase matching angle (or cut angle), φ – azimuthal angle.
Birefrigency Angle or Walk-off

 birefrigency angle or walk-off
Upper signs refer to negative crystal (no>ne) and the lower signs refer to positive one (ne>no).
Beam displacement because of walk–off:
Δ = L tan (ρ)

Whereas L – crystal length, ρ – walk-off angle.

BRF_W2.jpg
Reflection Air / Material

 Reflection Air/Material

Where n-refractive index, AOI - angle of incidence. 

Numerical Aperture

 numerical_aperture.jpg

numerical_apertures

Brewster's Angle

 The angle where only s-polarized light is reflected
BrewsterAngle.jpg

Gausian Beam

 A Gaussian beam spreads as follows,

GaussianBeam.jpg
where w(x) is the 1/e2 radius, λ is the wavelength, and x is the distance from the beam waist w0 where x=0.

A Rule of Thumb for Choosing a Lens

 ROT_.jpg
Where f is the lens focal length, d is the beam diameter at the focus, D is the 1/e2 diameter of the collimated beam.

Total Internal Reflection Angle

 total internal reflection angle

Where ntransmitted medium<nincident medium is required for total internal reflection.

Scaling Law for Laser Radiation Damage

 SLLRD_.jpg
Where E [J/cm2] is the damage threshold, t is pulse duration, E1 and t1 are the reference damage threshold and pulse duration.

Snell's Law

 Snellslaw.jpg

Phase Matching Types of Nonlinear Crystals

Negative crystals (no>ne)
Type 1 ko1+ko2=ke3(θ)  or “ooe interaction”
Type 2 ke1(θ)+ko2=ke3(θ)  or “eoe interaction”
Type 2 ko1+ke2(θ)=ke3(θ)   or “oee interaction”

Positive crystals (ne>no)
Type 1 ke1(θ)+ke2(θ)=ko3   or “eeo interaction”
Type 2 ko1+ke2(θ)=ko3   or “oeo interaction”
Type 2 ke1(θ)+ko2=ko3   or “eoo interaction”

Whereas k-wave propagation vector (k=2πn/λ); θ – phase matching angle in the crystal; o – ordinary polarization, e – extraordinary polarization; 1, 2, 3 indices – corresponds to wave vectors with longest (1), mid (2) and shortest (3) wavelengths.

Non Critical Phase Matching

 NCPM – when crystal phase matching angle equals 90º (θ = 90º). NCPM is achieved at special temperatures and/or wavelengths.

Physical Constants

Planck’s constant h = 6.6260755×10-34 J⋅s = 4.5×10-15 eV⋅s = 6.626×10-27erg⋅s
Dirac’s constant ħ = h/2π = 1.054×10-34 J⋅s = 1.054×10-27erg⋅s
Boltzmann’s constant kB = 1.380×10-16 erg/K = 8.62×10-5 eV/K = 1.380×10-23 J/K
kT = 25.9 meV at room temperature
     = 0.36 meV at liquid-helium temperature (4.2 K)
     = 6.7 meV at liquid-nitrogen temperature (4.2 K)
Velocity of light in vacuum c = 2.99792458×108 m/s
Electron charge e = 1.602×10-19 coulombs
Avogadro number Na = 6.0221367×1023 particles/mol
Permeability of vacuum μ= 4×10-7 T2⋅m3/J
                                                 = 12.566370614×10-7 T2⋅m3/J
Permittivity of vacuum ε0 = 1 / (μ0⋅c2)
                                             = 8.854187817×10-12 C2/J⋅m
Electron rest mass me = 9.1093897×10-31 kg
Proton rest mass mp = 1.6726231×10-27 kg
Neutron rest mass mn = 1.6749286×10-27 k