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Useful Formulas & Constants
Refractive Index Dispersion Equations & Calculator
Physical Constants
Etalon Formulas
Wave Vector, Frequency, Wavelength & Wavenumbers
International System of Units (SI) Prefixes
Non Critical Phase Matching
Phase Matching Types of Nonlinear Crystals
Nonlinear Crystal Thickness Limited by Group Velocity Mismatch (GVM)
Nonlinear Crystal Acceptances
Uniaxial Crystals Refractivity
Birefrigency Angle or Walk-off
Reflection Air / Material
Numerical Aperture
Brewster's Angle
Gausian Beam
A Rule of Thumb for Choosing a Lens
Total Internal Reflection Angle
Scaling Law for Laser Radiation Damage
Snell's Law
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 μ0 = 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
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.
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
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 |
| Factor | Name | Symbol |
| 1021 | zetta | Z |
| 1018 | exa |
E |
| 1015 | peta |
P |
| 1012 | tera |
T |
| 109 | giga |
G |
| 106 | mega | M |
| 103 | kilo | k |
| 102 | hecto | h |
| Factor | Name | Symbol |
| 10-2 | centi | c |
| 10-3 | mili | m |
| 10-6 | micro | µ |
| 10-9 | nano | n |
| 10-12 | pico | p |
| 10-15 | femto | f |
| 10-18 | atto | a |
| 10-21 | zepto | z |
| 10-24 | yocto | y |
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 |
Non Critical Phase Matching
NCPM – when crystal phase matching angle equals 90º (θ = 90º). NCPM is achieved at special temperatures and/or wavelengths.
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.
Nonlinear Crystal Thickness Limited by Group Velocity Mismatch (GVM)
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.
Birefrigency Angle or Walk-off
Beam displacement because of walk–off:
Δ = L tan (ρ)
Whereas L – crystal length, ρ – walk-off angle.
Where n-refractive index, AOI - angle of incidence.
Numerical Aperture
Brewster's Angle
The angle where only s-polarized light is reflected
Gausian Beam
A Gaussian beam spreads as follows,
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
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
Where ntransmitted medium<nincident medium is required for total internal reflection.
Scaling Law for Laser Radiation Damage
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
