Physics Theory

This section describes the physical principles and equations implemented in L-SURF.

Electromagnetic Wave Propagation

Light as Rays

L-SURF uses geometric optics (ray tracing) which is valid when:

\[\lambda \ll L\]

where \(\lambda\) is the wavelength and \(L\) is the characteristic size of obstacles. For visible light (400-700 nm) interacting with surfaces at meter scales, this is well satisfied.

Refractive Index

The refractive index \(n\) relates the speed of light in a medium to the vacuum speed:

\[n(\lambda, \vec{r}) = \frac{c}{v(\lambda, \vec{r})}\]

where:

  • \(c\) = speed of light in vacuum (2.998 × 10⁸ m/s)

  • \(v\) = phase velocity in the medium

  • \(\lambda\) = wavelength

  • \(\vec{r}\) = position (for spatially-varying media)

Dispersion Models

Sellmeier Formula (for optical glasses):

\[n^2(\lambda) = 1 + \sum_{i=1}^{3} \frac{B_i \lambda^2}{\lambda^2 - C_i}\]

where \(B_i\) and \(C_i\) are material-specific coefficients.

Cauchy Formula (empirical approximation):

\[n(\lambda) = A + \frac{B}{\lambda^2} + \frac{C}{\lambda^4}\]

Fresnel Equations

At an interface between two media, light splits into reflected and transmitted components. The amplitudes are governed by the Fresnel equations.

Geometry

       ↓ incident (θᵢ)        ↑ reflected (θᵣ)
          \                   /
           \                 /
            \               /
             \   n₁        /
═════════════════════════════════  interface
                  n₂
                   |
                   | transmitted (θₜ)
                   ↓

  • \(\theta_i\) = incident angle (from normal)

  • \(\theta_r\) = reflected angle (from normal)

  • \(\theta_t\) = transmitted (refracted) angle

  • \(n_1\) = refractive index of first medium

  • \(n_2\) = refractive index of second medium

Snell’s Law

The refraction angle is determined by Snell’s law:

\[n_1 \sin\theta_i = n_2 \sin\theta_t\]

Total Internal Reflection

When \(n_1 > n_2\) and \(\sin\theta_i > n_2/n_1\), total internal reflection occurs and no light is transmitted.

Fresnel Reflection Coefficients

The Fresnel equations give the amplitude reflection coefficients for s and p polarizations:

S-polarization (perpendicular to plane of incidence):

\[r_s = \frac{n_1 \cos\theta_i - n_2 \cos\theta_t}{n_1 \cos\theta_i + n_2 \cos\theta_t}\]

P-polarization (parallel to plane of incidence):

\[r_p = \frac{n_2 \cos\theta_i - n_1 \cos\theta_t}{n_2 \cos\theta_i + n_1 \cos\theta_t}\]

Reflectance (intensity ratios):

\[R_s = |r_s|^2, \quad R_p = |r_p|^2\]

Fresnel Transmission Coefficients

\[ \begin{align}\begin{aligned}t_s = \frac{2 n_1 \cos\theta_i}{n_1 \cos\theta_i + n_2 \cos\theta_t}\\t_p = \frac{2 n_1 \cos\theta_i}{n_2 \cos\theta_i + n_1 \cos\theta_t}\end{aligned}\end{align} \]

Transmittance (accounting for energy flux):

\[ \begin{align}\begin{aligned}T_s = \frac{n_2 \cos\theta_t}{n_1 \cos\theta_i} |t_s|^2\\T_p = \frac{n_2 \cos\theta_t}{n_1 \cos\theta_i} |t_p|^2\end{aligned}\end{align} \]

Energy Conservation

\[R_s + T_s = 1, \quad R_p + T_p = 1\]

Unpolarized Light

For unpolarized or randomly polarized light:

\[R = \frac{R_s + R_p}{2}, \quad T = \frac{T_s + T_p}{2}\]

Brewster’s Angle

At Brewster’s angle, p-polarized light has zero reflection:

\[\theta_B = \arctan\left(\frac{n_2}{n_1}\right)\]

For air-water interface: \(\theta_B \approx 53.1°\)

Polarization Tracking

L-SURF supports polarization tracking for more accurate simulations.

Polarization State

A ray’s polarization is described by:

  1. S-component intensity: \(I_s\)

  2. P-component intensity: \(I_p\)

  3. Total intensity: \(I = I_s + I_p\)

At each interface, components are multiplied by Fresnel coefficients:

\[ \begin{align}\begin{aligned}I_s^{\text{refl}} = R_s \cdot I_s^{\text{inc}}\\I_p^{\text{refl}} = R_p \cdot I_p^{\text{inc}}\\I_s^{\text{trans}} = T_s \cdot I_s^{\text{inc}}\\I_p^{\text{trans}} = T_p \cdot I_p^{\text{inc}}\end{aligned}\end{align} \]

Multiple Scattering

Ray Splitting

At each surface interaction, L-SURF can:

  1. Single mode: Generate only reflected OR refracted ray

  2. Splitting mode: Generate both reflected AND refracted rays

Splitting mode accurately tracks energy flow through multiple bounces.

Intensity Tracking

Ray intensity is tracked through propagation:

\[I_{\text{final}} = I_0 \prod_{i=1}^{N} R_i \cdot e^{-\alpha_i d_i}\]

where:

  • \(I_0\) = initial intensity

  • \(R_i\) = reflectance at bounce \(i\)

  • \(\alpha_i\) = absorption coefficient in medium \(i\)

  • \(d_i\) = distance traveled in medium \(i\)

  • \(N\) = number of bounces

Termination Criteria

Rays are terminated when:

  1. Intensity drops below threshold (typically 10⁻⁶ of initial)

  2. Maximum bounce count reached

  3. Ray exits simulation domain

  4. Ray trapped in total internal reflection cycle

Time-of-Flight

Ray Timing

Each ray tracks its accumulated time-of-flight:

\[t = t_0 + \sum_{i=1}^{N} \frac{d_i}{v_i} = t_0 + \sum_{i=1}^{N} \frac{n_i d_i}{c}\]

where:

  • \(t_0\) = initial time (emission time)

  • \(d_i\) = path length in segment \(i\)

  • \(v_i = c/n_i\) = phase velocity in segment \(i\)

  • \(c\) = speed of light in vacuum

Temporal Coherence

For pulsed sources, the initial time \(t_0\) can vary across the beam to model:

  • Pulse duration

  • Wavefront curvature

  • Beam divergence effects

Detection Time

At a detector, the time-of-arrival is:

\[t_{\text{detect}} = t_{\text{emit}} + \sum \frac{n_i d_i}{c}\]

This enables:

  • Time-resolved detection

  • Pulse spreading analysis

  • Group velocity dispersion studies

Wave Surface Interaction

Normal Vector Calculation

For curved surfaces (e.g., Gerstner waves), the local normal is computed from the surface gradient:

\[\vec{n} = \frac{\nabla f}{\|\nabla f\|}\]

where \(f(\vec{r}) = 0\) defines the surface.

For Gerstner waves:

\[z = \sum_{i=1}^{N} A_i \sin(k_i x - \omega_i t + \phi_i)\]

The normal vector includes contributions from all wave components:

\[ \begin{align}\begin{aligned}\frac{\partial z}{\partial x} = \sum_{i} A_i k_i \cos(k_i x - \omega_i t + \phi_i)\\\vec{n} = \frac{(-\partial z/\partial x, 0, 1)}{\sqrt{1 + (\partial z/\partial x)^2}}\end{aligned}\end{align} \]

Shadowing and Blocking

For rough surfaces, geometric shadowing/blocking occurs when:

  1. Shadowing: Incident ray blocked by surface feature

  2. Blocking: Reflected/refracted ray blocked by surface

L-SURF handles this through accurate ray-surface intersection testing.

Implementation References

The implementation follows standard texts:

  1. Born & Wolf - Principles of Optics (7th ed.) - Fresnel equations

  2. Hecht - Optics (5th ed.) - Electromagnetic wave theory

  3. Ishimaru - Wave Propagation and Scattering in Random Media - Rough surface scattering

  4. Tessendorf - Simulating Ocean Water (2001) - Gerstner waves