Coupled Heat Transfer and Ray Tracing Equations
Under static conditions, the heat equation can be written as
where
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ρ (SI unit: kg/m3) is the density,
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Cp (SI unit:  J/(kg·K)) is the heat capacity at constant pressure,
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T (SI unit: K) is the temperature,
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Q (SI unit: W/m3) is the heat source, and
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k (SI unit:  W/(m·K)) is the thermal conductivity.
The propagation of each ray can be modeled using a set of coupled first-order equations:
where k (SI unit: rad/m) is the wave vector, ω (SI unit: rad/s) is the angular frequency, t (SI unit: s) is time, and q (SI unit: m) is the ray position vector. In an isotropic medium, the frequency and wave vector are related by the expression
where c = 299,792,458 m/s is the speed of light in a vacuum and n (dimensionless) is the refractive index. If the medium is absorbing, then the intensity of the ray decreases and it deposits some power in the domain as it passes through. This is described in greater detail in the section Attenuation Within Domains in Theory for the Geometrical Optics Interface.
A bidirectional coupling between the Geometrical Optics and Heat Transfer interfaces may be required for any of the following reasons:
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The refractive index is temperature dependent.
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The medium undergoes thermal expansion and the refractive index is strain dependent.
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The medium undergoes thermal expansion, causing rays be reflected or refracted at different angles when entering or leaving the medium.
When none of these conditions is met, the Geometrical Optics interface affects the Heat Transfer interface, and a unidirectional coupling is sufficient.