Inspecting and plotting the refractive index

NonlinearCrystals.jl provides access to refractive index data for a wide range of birefringent nonlinear crystals. Built-in crystals like LBO, BBO, or KTP_F are readily available as objects. Each crystal is defined in its own file under the crystal_data directory.

Depending on the nature of the crystal—unidirectional or bidirectional—its birefringence is described via two or three principal refractive indices that depend on both wavelength and temperature. These are provided as SellmeierFunction objects and can be accessed as fields of the crystal:

For unidirectional crystals (e.g., BBO):

BBO.n_o_principal   # ordinary axis
BBO.n_e_principal   # extraordinary axis

For bidirectional crystals (e.g., KTP_F):

KTP_F.n_X_principal
KTP_F.n_Y_principal
KTP_F.n_Z_principal

Here's how to evaluate basic refractive and dispersion properties for a principal axis:

julia> using NonlinearCrystals, Unitful

julia> n = refractive_index(LBO.n_X_principal, 1064u"nm", 300u"K")   # Refractive index for polarization along the principal X axis at 1064 nm and 300 K
1.564762196788999

julia> ng = group_index(LBO.n_X_principal, 1064u"nm", 300u"K")   # Group index
1.58131698872782

julia> β₂ = β2(LBO.n_X_principal, 1064u"nm", 300u"K")   # Group velocity dispersion (GVD)
1.7872825157935612e-26 s^2 m^-1

All inputs and outputs of NonlinearCrystals.jl have physical units based on Unitful.jl. It is easy to convert those if needed:

julia> β₂ |> u"fs^2/mm"
17.872825157935612 fs^2 mm^-1

To analyze thermal effects, you can also compute the thermo-optic coefficient:

julia> dn_dT = dn_dtemp(LBO.n_X_principal, 1064u"nm", 300u"K")  # ∂n/∂T at 1064 nm and 300 K
-2.3793331118400023e-6 K^-1

Querying refractive indices

These principal indices apply when the electric field is aligned with the corresponding dielectric axis. For arbitrary propagation directions or polarizations, you can compute direction-dependent refractive indices, specified for polar angles θ and ϕ:

julia> RefractionDataHiLo(45u"°", 10u"°", LNB_C, 532u"nm"; temp= 349u"K")

Crystal:                  LNB (Lithium Niobate, congruent melt with mole ratio Li/Nb 0.946)
Wavelength (nm):          532.0
k angles:                 θ: 45.00°, ϕ: 10.00°
k direction:              [0.696, 0.123, 0.707]    
Temperature:              349.00 K (75.85 °C)
───────────────────────────────────────────────────────────────────────────
Refractive index type:    hi (o)                    lo (e)                   
Refractive index:         2.324                     2.279                    
Group index:              2.586                     2.523                    
Walkoff angle (mrad):     0.000                     37.905                   
S direction:              [0.696, 0.123, 0.707]     [0.669, 0.118, 0.733]    
E direction:             ±[-0.174, 0.985, -0.0]    ±[-0.722, -0.127, 0.68]  
D direction:             ±[-0.174, 0.985, -0.0]    ±[-0.696, -0.123, 0.707] 
β₂ (fs²/mm):              896.823                   825.030                  
β₃ (fs³/mm):              544.009                   489.841

The output has the following entries:

Field	Description
Crystal	Description of the nonlinear crystal.
Wavelength	Wavelength of light for which refractive indices are computed.
k angles	Polar (θ) and azimuthal (ϕ) angles in degrees, specifying the propagation direction of the wavevector k.
k direction	Cartesian unit vector of the wavevector direction in the dielectric crystal frame.
Temperature	Temperature at which refractive properties are computed.
Refractive index type	hi and lo refer to the higher and lower refractive indices at the given angle. In certain cases (unidirectional crystals like LNB or when k lies within one of the principal planes), hi and lo can be assigned to the ordinary and extraordinary polarizations often referred to in textbooks.
Refractive index	Refractive index of the two polarizations, relevant for assigning hi and lo.
Group index	Group index of the two polarizations describing the velocity of the pulse envelope relative to the speed of light in vacuum.
Walkoff angle (mrad)	Angle between the Poynting vector S and the wavevector k. Characterizes spatial walkoff between the polarizations.
S direction	Direction of the Poynting vector (energy flow) for each polarization.
E direction	Electric field direction (polarization vector) for each mode. ± indicates that the handedness/phase is not defined here. E is perpendicular to the corresponding S direction
D direction	Electric displacement direction, which lies in the index ellipsoid surface and is perpendicular to k.
β₂ (fs²/mm)	Group velocity dispersion (or second-order dispersion) calculated via d²k/dω².
β₃ (fs³/mm)	Third-order dispersion per unit length calculated via d³k/dω³.

Plotting refractive index vs. wavelength

You can visualize the dispersion behavior using:

plot_refractiveindex(LBO)

To see the temperature dependence:

plot_refractiveindex(LBO; temp=[293.15u"K", 413.15u"K", 533.15u"K"])

This will generate interactive plots of refractive index as a function of wavelength at different temperatures.