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Geometrical Optics

Field: Optics

The study of light propagation in terms of rays.

Sequence of Expressions

Let Σ⊂R3\Sigma \subset \mathbb{R}^3 be the reflecting surface, and let P∈ΣP \in \Sigma be the point of incidence. Define n^∈R3\mathbf{\hat{n}} \in \mathbb{R}^3 as the unit normal vector to Σ\Sigma at PP. Let k^i∈R3\mathbf{\hat{k}}_i \in \mathbb{R}^3 be the unit vector representing the direction of the incident wave propagation (the incident ray). The angle of incidence, θi\theta_i, is defined by the relationship:\n\ncos⁡(θi)=∣k^i⋅n^∣\cos(\theta_i) = |\mathbf{\hat{k}}_i \cdot \mathbf{\hat{n}}| \n\nEquivalently, θi\theta_i is the unique angle in the interval [0,π/2][0, \pi/2] such that:\n\nθi=arccos⁡(∣k^i⋅n^∣)\theta_i = \arccos\left( |\mathbf{\hat{k}}_i \cdot \mathbf{\hat{n}}| \right)
Let r1\mathbf{r}_1 and r2\mathbf{r}_2 be the position vectors of the incident and refracted rays, respectively, intersecting the interface Σ\Sigma at point P\mathbf{P}. Let n\mathbf{n} be the unit normal vector to Σ\Sigma at P\mathbf{P}. Define the incident direction vector k1=r1′−r1\mathbf{k}_1 = \mathbf{r}_1' - \mathbf{r}_1 and the refracted direction vector k2=r2′−r2\mathbf{k}_2 = \mathbf{r}_2' - \mathbf{r}_2, where r1′\mathbf{r}_1' and r2′\mathbf{r}_2' are points infinitesimally further along the rays. The angle of incidence θ1\theta_1 and the angle of refraction θ2\theta_2 are defined by the directional cosines: cos⁡(θ1)=∣k1⋅n∣∣k1∣\cos(\theta_1) = \frac{|\mathbf{k}_1 \cdot \mathbf{n}|}{|\mathbf{k}_1|} and cos⁡(θ2)=∣k2⋅n∣∣k2∣\cos(\theta_2) = \frac{|\mathbf{k}_2 \cdot \mathbf{n}|}{|\mathbf{k}_2|}. The relationship between the angles and the refractive indices n1n_1 and n2n_2 of the media is given by Snell's Law, which mandates the equality of the tangential components of the wave vector: n1sin⁡(θ1)=n2sin⁡(θ2)n_1 \sin(\theta_1) = n_2 \sin(\theta_2).
Let r:R→R3\mathbf{r}: \mathbb{R} \to \mathbb{R}^3 be the position vector describing a ray path, parameterized by t∈Rt \in \mathbb{R}. Define the direction vector k^∈S2\hat{\mathbf{k}} \in \mathbb{S}^2 as a fixed unit vector. The condition for r(t)\mathbf{r}(t) to represent a set of parallel rays is that the tangent vector r′(t)\mathbf{r}'(t) must be constant and equal to k^\hat{\mathbf{k}} for all tt. Formally, this requires the ray path to satisfy the differential equation:\ndrdt=k^\frac{d\mathbf{r}}{dt} = \hat{\mathbf{k}} \nIntegrating this yields the parametric form of the ray: r(t)=r0+tk^\mathbf{r}(t) = \mathbf{r}_0 + t \hat{\mathbf{k}}, where r0=r(0)\mathbf{r}_0 = \mathbf{r}(0) is the initial position vector. The set of all such rays R\mathcal{R} is thus defined by the family of curves rr0,k^(t)=r0+tk^\mathbf{r}_{\mathbf{r}_0, \hat{\mathbf{k}}}(t) = \mathbf{r}_0 + t \hat{\mathbf{k}}, parameterized by the initial point r0∈R3\mathbf{r}_0 \in \mathbb{R}^3 and the fixed direction k^\hat{\mathbf{k}}.
Theorem

Reflection

Let I∈R3\mathbf{I} \in \mathbb{R}^3 be the unit vector representing the incident ray, and let N∈R3\mathbf{N} \in \mathbb{R}^3 be the unit normal vector to the reflecting surface at the point of incidence. The reflected ray vector R\mathbf{R} is defined by the vector reflection formula:\n\nR=I−2(I⋅N)N\mathbf{R} = \mathbf{I} - 2 \left( \mathbf{I} \cdot \mathbf{N} \right) \mathbf{N}\n\nFurthermore, the condition for specular reflection requires that the angle θi\theta_i between I\mathbf{I} and −N-\mathbf{N} (the direction into the surface) equals the angle θr\theta_r between R\mathbf{R} and N\mathbf{N} (the direction away from the surface). This is equivalent to the requirement that the component of I\mathbf{I} parallel to the surface is preserved, leading to the geometric constraint: R⋅N=−I⋅N\mathbf{R} \cdot \mathbf{N} = -\mathbf{I} \cdot \mathbf{N}.
Theorem

Refraction

Let r(s)\mathbf{r}(s) be a path parameterized by arc length ss connecting points r1\mathbf{r}_1 and r2\mathbf{r}_2 in two media, M1M_1 and M2M_2, separated by an interface Σ\Sigma. Let n(r)n(\mathbf{r}) be the spatially varying refractive index. The optical path length LL is defined by the action integral SS: \n\nS=∫r1r2n(r) dsS = \int_{\mathbf{r}_1}^{\mathbf{r}_2} n(\mathbf{r}) \, ds\n\nBy Fermat's Principle, the path r(s)\mathbf{r}(s) must satisfy the Euler-Lagrange equations derived from minimizing SS. If the interface Σ\Sigma is defined by r⋅n=0\mathbf{r} \cdot \mathbf{n} = 0, where n\mathbf{n} is the normal vector, and k1\mathbf{k}_1 and k2\mathbf{k}_2 are the wave vectors in M1M_1 and M2M_2 respectively, the continuity of the tangential component of the wave vector across the boundary Σ\Sigma requires that the generalized momentum component perpendicular to Σ\Sigma must satisfy the boundary condition derived from the Hamiltonian formulation of the Eikonal equation. Specifically, if θ1\theta_1 and θ2\theta_2 are the angles between k1\mathbf{k}_1 and k2\mathbf{k}_2 and the normal n\mathbf{n}, then the conservation of the component of the wave vector parallel to Σ\Sigma yields:\n\nn1sin⁡(θ1)=n2sin⁡(θ2)n_1 \sin(\theta_1) = n_2 \sin(\theta_2)
Let the optical medium be defined by a spatially varying refractive index n(r)n(\mathbf{r}). The propagation of the wavefront Φ(r,t)\Phi(\mathbf{r}, t) is governed by the eikonal equation, which is the scalar form of the wave equation in the high-frequency limit:\n(∂Φ∂r)2=n2(r)\left(\frac{\partial \Phi}{\partial \mathbf{r}}\right)^2 = n^2(\mathbf{r})\nwhere r=(x,y,z)\mathbf{r} = (x, y, z) and Φ\Phi is the eikonal function. The ray path r(s)\mathbf{r}(s) is parameterized by the arc length ss and follows the direction of the wave vector k=∇Φ\mathbf{k} = \nabla \Phi. The trajectory r(s)\mathbf{r}(s) must satisfy the differential equation:\ndds(n(r)drds)=∇n(r)\frac{d}{ds}\left(n(\mathbf{r}) \frac{d\mathbf{r}}{ds}\right) = \nabla n(\mathbf{r})\nThis equation, derived from the Hamiltonian formulation of the optical path length minimization, describes the continuous path of the ray through the inhomogeneous medium.
Let r(s)\mathbf{r}(s) be a parameterized path in R3\mathbb{R}^3 connecting points AA and BB. Define the refractive index n(r)n(\mathbf{r}) as a scalar field representing the local speed of light variation. The optical path length LL is given by the integral: L=∫ABn(r) dsL = \int_{A}^{B} n(\mathbf{r}) \, ds where dsds is the infinitesimal arc length element. By Fermat's Principle, the path r(s)\mathbf{r}(s) must satisfy the variational condition δL=0\delta L = 0. In a planar interface separating two media, M1M_1 and M2M_2, with refractive indices n1n_1 and n2n_2, and assuming the interface lies in the xyxy-plane, the path minimization leads to the generalized form of Snell's Law. If θ1\theta_1 and θ2\theta_2 are the angles of incidence and refraction, respectively, measured relative to the normal vector k^\mathbf{\hat{k}}, the condition is: n1sin⁡(θ1)=n2sin⁡(θ2)n_1 \sin(\theta_1) = n_2 \sin(\theta_2) This relationship is derived from the conservation of the component of the wave vector k\mathbf{k} tangential to the interface, specifically k∣∣=nk0csin⁡(θ)k_{||} = n \frac{k_0}{c} \sin(\theta), where k0k_0 is the vacuum wave number.
Let ki∈R3\mathbf{k}_i \in \mathbb{R}^3 and kr∈R3\mathbf{k}_r \in \mathbb{R}^3 be the unit direction vectors of the incident and reflected rays, respectively, and let n∈UnitSphere\mathbf{n} \in \text{UnitSphere} be the unit vector normal to the reflecting surface at the point of incidence. The Law of Reflection is mathematically expressed by the vector relationship:\nkr=ki−2(ki⋅n)n\mathbf{k}_r = \mathbf{k}_i - 2(\mathbf{k}_i \cdot \mathbf{n}) \mathbf{n}
Let the medium be defined by a static refractive index n(x):R3→R+n(\mathbf{x}): \mathbb{R}^3 \to \mathbb{R}^+. The path r(s)\mathbf{r}(s) of a light ray, parameterized by arc length ss, is a geodesic curve in the Riemannian manifold (R3,g)(\mathbb{R}^3, g), where the metric tensor gg is given by gij=n2(x)δijg_{ij} = n^2(\mathbf{x}) \delta_{ij}. The path r(s)\mathbf{r}(s) must satisfy the geodesic equation: \begin{equation*} \frac{d^2 x^k}{d s^2} + \Gamma^k_{ij} \frac{d x^i}{d s} \frac{d x^j}{d s} = 0 \end{equation*} where Γijk\Gamma^k_{ij} are the Christoffel symbols associated with gijg_{ij}. The Principle of Reversibility asserts that if r(s)\mathbf{r}(s) is a solution to this equation for s∈[0,L]s \in [0, L], then the curve r′(s)=r(L−s)\mathbf{r}'(s) = \mathbf{r}(L-s) is also a solution, provided the metric gijg_{ij} is independent of time tt. Specifically, the tangent vector k(s)=drds\mathbf{k}(s) = \frac{d\mathbf{r}}{ds} satisfies k(L−s)=−k(s)\mathbf{k}(L-s) = -\mathbf{k}(s) and the path remains a geodesic.
Let x∈R3\mathbf{x} \in \mathbb{R}^3 be the spatial coordinates and t∈Rt \in \mathbb{R} be time. Assume the wave propagation satisfies the homogeneous wave equation: (∂2∂t2−c2∇2)Φ(x,t)=0 \left( \frac{\partial^2}{\partial t^2} - c^2 \nabla^2 \right) \Phi(\mathbf{x}, t) = 0 where Φ\Phi is the phase function and cc is the wave speed. Define the initial wavefront S(t0)S(t_0) as the level set Φ(x,t0)=C0\Phi(\mathbf{x}, t_0) = C_0. The Principle of Huygens' Wavefronts asserts that the subsequent wavefront S(t)S(t) is the envelope of the secondary spherical wavelets emanating from every point x0∈S(t0)\mathbf{x}_0 \in S(t_0). Formally, the phase Φ(x,t)\Phi(\mathbf{x}, t) must satisfy the Eikonal equation in the limit of high frequency ω\omega: ∣∇Φ∣2=(ωc)2 \left| \nabla \Phi \right|^2 = \left( \frac{\omega}{c} \right)^2 The new wavefront S(t)S(t) is defined by the locus of points x\mathbf{x} such that the phase Φ(x,t)\Phi(\mathbf{x}, t) is constant, and this phase must be determined by the integral representation: Φ(x,t)=min⁡x0∈S(t0)(Φ0(x0)+1v∫t0tc2+∣∇Φ∣2dt′) \Phi(\mathbf{x}, t) = \min_{\mathbf{x}_0 \in S(t_0)} \left( \Phi_0(\mathbf{x}_0) + \frac{1}{v} \int_{t_0}^{t} \sqrt{c^2 + \left| \nabla \Phi \right|^2} dt' \right) where Φ0(x0)\Phi_0(\mathbf{x}_0) is the initial phase, and the minimum is taken over all paths connecting x0\mathbf{x}_0 to x\mathbf{x} on S(t0)S(t_0). The wavefront S(t)S(t) is thus the level set Φ(x,t)=C(t)\Phi(\mathbf{x}, t) = C(t).