Formulas & Tools

Refractive Index Dispersion Equations & Calculator Useful Formulas & Constants

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

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