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:

.. math::

   \lambda \ll L

where :math:`\lambda` is the wavelength and :math:`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 :math:`n` relates the speed of light in a medium to the vacuum speed:

.. math::

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

where:

- :math:`c` = speed of light in vacuum (2.998 × 10⁸ m/s)
- :math:`v` = phase velocity in the medium
- :math:`\lambda` = wavelength
- :math:`\vec{r}` = position (for spatially-varying media)

Dispersion Models
^^^^^^^^^^^^^^^^^

**Sellmeier Formula** (for optical glasses):

.. math::

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

where :math:`B_i` and :math:`C_i` are material-specific coefficients.

**Cauchy Formula** (empirical approximation):

.. math::

   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
~~~~~~~~

.. code-block:: text

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

- :math:`\theta_i` = incident angle (from normal)
- :math:`\theta_r` = reflected angle (from normal)
- :math:`\theta_t` = transmitted (refracted) angle
- :math:`n_1` = refractive index of first medium
- :math:`n_2` = refractive index of second medium

Snell's Law
~~~~~~~~~~~

The refraction angle is determined by Snell's law:

.. math::

   n_1 \sin\theta_i = n_2 \sin\theta_t

Total Internal Reflection
^^^^^^^^^^^^^^^^^^^^^^^^^

When :math:`n_1 > n_2` and :math:`\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):

.. math::

   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):

.. math::

   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):

.. math::

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

Fresnel Transmission Coefficients
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. math::

   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}

Transmittance (accounting for energy flux):

.. math::

   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

Energy Conservation
^^^^^^^^^^^^^^^^^^^

.. math::

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

Unpolarized Light
~~~~~~~~~~~~~~~~~

For unpolarized or randomly polarized light:

.. math::

   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:

.. math::

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

For air-water interface: :math:`\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**: :math:`I_s`
2. **P-component intensity**: :math:`I_p`
3. **Total intensity**: :math:`I = I_s + I_p`

At each interface, components are multiplied by Fresnel coefficients:

.. math::

   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}}

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:

.. math::

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

where:

- :math:`I_0` = initial intensity
- :math:`R_i` = reflectance at bounce :math:`i`
- :math:`\alpha_i` = absorption coefficient in medium :math:`i`
- :math:`d_i` = distance traveled in medium :math:`i`
- :math:`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:

.. math::

   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:

- :math:`t_0` = initial time (emission time)
- :math:`d_i` = path length in segment :math:`i`
- :math:`v_i = c/n_i` = phase velocity in segment :math:`i`
- :math:`c` = speed of light in vacuum

Temporal Coherence
~~~~~~~~~~~~~~~~~~

For pulsed sources, the initial time :math:`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:

.. math::

   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:

.. math::

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

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

For Gerstner waves:

.. math::

   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:

.. math::

   \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}}

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