ShaneLogic/complex-refractive-index-relations

Complex Refractive Index Relations

When to Use

• Relating optical constants to dielectric function in absorbing media
• Calculating ε₁ and ε₂ from measured nᵣ and κ (or vice versa)
• Analyzing optical properties of semiconductors with absorption
• Converting between optical and electrical response parameters

Core Relations

The complex dielectric function ε = ε₁ + iε₂ is related to optical constants:

ε₁ = nᵣ² - κ²
ε₂ = 2 × nᵣ × κ

Where:

• ε₁ = Real part of complex dielectric function
• ε₂ = Imaginary part of complex dielectric function
• nᵣ = Refractive index
• κ (kappa) = Extinction coefficient

Physical Meaning

ε₁ (Real Part):

• Represents the dispersive (energy storage) component
• Related to polarization and refractive index
• Describes how the medium responds to the electric field

ε₂ (Imaginary Part):

• Represents the absorptive (energy dissipation) component
• Directly related to optical absorption coefficient α
• Describes energy loss in the medium

nᵣ (Refractive Index):

• Determines phase velocity of light in the medium
• Related to ε₁ in non-absorbing materials

κ (Extinction Coefficient):

• Determines attenuation of light amplitude
• Related to absorption coefficient: α = 4πκ/λ

Constraints

• Assumes damping/absorption is present (κ > 0)
• For non-absorbing materials (κ = 0): ε₁ = nᵣ² and ε₂ = 0
• Valid for homogeneous, isotropic media

Inverse Relations

To solve for nᵣ and κ given ε₁ and ε₂:

nᵣ = √[(|ε| + ε₁) / 2]
κ = √[(|ε| - ε₁) / 2]

where |ε| = √(ε₁² + ε₂²)

Variables

| Symbol | Description | Type | |--------|-------------|------| | ε₁ | Real part of complex dielectric function | real | | ε₂ | Imaginary part of complex dielectric function | real | | nᵣ | Refractive index | real | | κ | Extinction coefficient | real |