Beta Phase: Square45 is currently in beta testing. Expect some features or content to be incomplete or missing.
45

Physical Optics

Field: Optics

The study of the wave properties of light.

Sequence of Expressions

Definition

Wavefronts

Let r∈R3\mathbf{r} \in \mathbb{R}^3 be the spatial position vector and Φ(r,t)\Phi(\mathbf{r}, t) be the phase function of an electromagnetic wave, satisfying the Helmholtz equation ∇2Φ+k2Φ=0\nabla^2 \Phi + k^2 \Phi = 0 (for monochromatic waves, k=ω/ck = \omega/c). A wavefront Σ\Sigma at time t0t_0 is defined as the level set of the phase function: Σ={r∈R3∣Φ(r,t0)=C}\Sigma = \{\mathbf{r} \in \mathbb{R}^3 \mid \Phi(\mathbf{r}, t_0) = C \}. The geometry of this surface is governed by the eikonal equation, which states that the magnitude of the wave vector k=∇Φ\mathbf{k} = \nabla \Phi must satisfy: \n\n∣∇Φ(r,t0)∣=k|\nabla \Phi(\mathbf{r}, t_0)| = k \n\nThis implies that the wavefront Σ\Sigma is an isophase surface, and its normal vector n\mathbf{n} is proportional to the gradient of the phase: n∝∇Φ\mathbf{n} \propto \nabla \Phi. Furthermore, the propagation of the wavefront is described by the characteristic manifold of the wave equation, where the phase Φ\Phi is a solution to the Hamilton-Jacobi equation derived from the eikonal approximation.
Let E1(r,t)\mathbf{E}_1(\mathbf{r}, t) and E2(r,t)\mathbf{E}_2(\mathbf{r}, t) be two monochromatic electromagnetic fields propagating in a linear, isotropic medium, satisfying the homogeneous wave equation: ∇2Ej−1v2∂2∂t2Ej=0\nabla^2 \mathbf{E}_j - \frac{1}{v^2} \frac{\partial^2}{\partial t^2} \mathbf{E}_j = 0 for j=1,2j=1, 2. Define the resultant field Etotal(r,t)\mathbf{E}_{total}(\mathbf{r}, t) via the principle of superposition: Etotal(r,t)=E1(r,t)+E2(r,t)\mathbf{E}_{total}(\mathbf{r}, t) = \mathbf{E}_1(\mathbf{r}, t) + \mathbf{E}_2(\mathbf{r}, t). Assume the fields can be represented by their complex amplitudes: Ej(r,t)=E0j(r)ei(kj⋅r−ωt)\mathbf{E}_j(\mathbf{r}, t) = \mathbf{E}_{0j}(\mathbf{r}) e^{i (\mathbf{k}_j \cdot \mathbf{r} - \omega t)}. The intensity II (proportional to the time-averaged Poynting vector magnitude) of the resultant field is given by: I(r)=12μ0c∣Etotal(r,t)∣2=I1(r)+I2(r)+2I1(r)I2(r)cos⁡(Δϕ(r))I(\mathbf{r}) = \frac{1}{2\mu_0 c} \left| \mathbf{E}_{total}(\mathbf{r}, t) \right|^2 = I_1(\mathbf{r}) + I_2(\mathbf{r}) + 2\sqrt{I_1(\mathbf{r}) I_2(\mathbf{r})} \cos(\Delta \phi(\mathbf{r})) where Ij(r)=12Re(E0j⋅E0j∗)I_j(\mathbf{r}) = \frac{1}{2} \text{Re}(\mathbf{E}_{0j} \cdot \mathbf{E}_{0j}^*) is the intensity of the individual waves, and Δϕ(r)=k1⋅r−k2⋅r+ϕ01−ϕ02\Delta \phi(\mathbf{r}) = \mathbf{k}_{1} \cdot \mathbf{r} - \mathbf{k}_{2} \cdot \mathbf{r} + \phi_{01} - \phi_{02} is the phase difference, determining the constructive (Δϕ=2πn\Delta \phi = 2\pi n) or destructive (Δϕ=π(2n+1)\Delta \phi = \pi (2n+1)) interference conditions.
Let E(r,t)E(\textbf{r}, t) be the complex electric field amplitude at position r=(x,y,z)\textbf{r} = (x, y, z) and time tt. Assume the field propagates from an aperture plane z=0z=0 to a observation point r=(x,y,z)\textbf{r} = (x, y, z) where z>0z>0. Let E0(r⊥)E_0(\textbf{r}_\bot) be the field amplitude on the aperture plane r′=(x′,y′,0)\textbf{r}' = (x', y', 0), where r⊥=(x′,y′)\textbf{r}_\bot = (x', y'). The field E(r,t)E(\textbf{r}, t) at r\textbf{r} is given by the Kirchhoff-Huygens diffraction integral:\n\nE(x,y,z)=1iλ∫AE0(x′,y′)[eikRR] dx′dy′E(x, y, z) = \frac{1}{i\lambda} \int_{A} E_0(x', y') \left[ \frac{e^{ikR}}{R} \right] \, dx' dy' \n\nwhere k=ω/ck = \omega/c is the wave number, λ\lambda is the wavelength, AA is the aperture area, and R=(x−x′)2+(y−y′)2+z2R = \sqrt{(x-x')^2 + (y-y')^2 + z^2} is the distance from the source point (x′,y′,0)(x', y', 0) to the observation point (x,y,z)(x, y, z). For the Fraunhofer approximation (far-field), where zz is large and the phase term is approximated by the planar wave phase exp⁡(ik(x−x′)2+(y−y′)22z)\exp\left(i k \frac{(x-x')^2 + (y-y')^2}{2z} \right), the integral simplifies to:\n\nE(x,y,z)≈eikziλz∫AE0(x′,y′)eikz(xx′+yy′) dx′dy′E(x, y, z) \approx \frac{e^{ikz}}{i\lambda z} \int_{A} E_0(x', y') e^{i\frac{k}{z} (x x' + y y')} \, dx' dy'
Let k∈R3\mathbf{k} \in \mathbb{R}^3 be the unit wave vector defining the direction of propagation, and let ω\omega be the angular frequency. The electric field E(r,t)\mathbf{E}(\mathbf{r}, t) of a monochromatic, transverse electromagnetic wave propagating in a linear, isotropic, and homogeneous medium is given by:\nE(r,t)=E0cos⁡(k⋅r−ωt)+H0sin⁡(k⋅r−ωt)\mathbf{E}(\mathbf{r}, t) = \mathbf{E}_0 \cos(k \cdot \mathbf{r} - \omega t) + \mathbf{H}_0 \sin(k \cdot \mathbf{r} - \omega t) \nwhere E0\mathbf{E}_0 and H0\mathbf{H}_0 are the complex amplitude vectors of the electric and magnetic fields, respectively. Polarization is characterized by the amplitude vector E0\mathbf{E}_0, which must satisfy the transverse condition: E0⋅k=0\mathbf{E}_0 \cdot \mathbf{k} = 0. Furthermore, the polarization state is defined by the ratio of the components of E0\mathbf{E}_0 projected onto an orthonormal basis (e^1,e^2)(\hat{\mathbf{e}}_1, \hat{\mathbf{e}}_2) spanning the plane perpendicular to k\mathbf{k}: E0=E01e^1+E02e^2\mathbf{E}_0 = E_{01} \hat{\mathbf{e}}_1 + E_{02} \hat{\mathbf{e}}_2. The degree of polarization is quantified by the Stokes parameters S=(S0,S1,S2,S3)\mathbf{S} = (S_0, S_1, S_2, S_3), where S0=⟨E2⟩S_0 = \langle E^2 \rangle, S1=⟨Ex2−Ey2⟩S_1 = \langle E_x^2 - E_y^2 \rangle, S2=⟨2ExEy⟩S_2 = \langle 2 E_x E_y \rangle, and S3=⟨2ExEz⟩S_3 = \langle 2 E_x E_z \rangle (assuming a coordinate system where k\mathbf{k} is aligned with zz for simplicity, or generalized components otherwise).
Let Ei,Hi\mathbf{E}_i, \mathbf{H}_i be the incident fields and Er,Hr\mathbf{E}_r, \mathbf{H}_r be the reflected fields in Medium 1, and Et,Ht\mathbf{E}_t, \mathbf{H}_t be the transmitted fields in Medium 2. Assume the interface lies in the xyxy-plane. The boundary conditions require continuity of the tangential components of E\mathbf{E} and H\mathbf{H}. Define the relative parameters ϵr1,μr1\epsilon_{r1}, \mu_{r1} and ϵr2,μr2\epsilon_{r2}, \mu_{r2}. The reflection coefficients rsr_s (s-polarization, perpendicular to the plane of incidence) and rpr_p (p-polarization, parallel to the plane of incidence) are given by:\n\nFor ss-polarization:\nrs=η2cos⁡θi−η1cos⁡θtη2cos⁡θi+η1cos⁡θtr_s = \frac{\eta_2 \cos \theta_i - \eta_1 \cos \theta_t}{\eta_2 \cos \theta_i + \eta_1 \cos \theta_t}\n\nFor pp-polarization:\nrp=η2cos⁡θt−η1cos⁡θiη2cos⁡θt+η1cos⁡θir_p = \frac{\eta_2 \cos \theta_t - \eta_1 \cos \theta_i}{\eta_2 \cos \theta_t + \eta_1 \cos \theta_i}\n\nWhere ηk=μrk/ϵrk\eta_k = \sqrt{\mu_{rk} / \epsilon_{rk}} is the intrinsic impedance of Medium kk, and θi\theta_i and θt\theta_t are the angles of incidence and transmission, respectively, related by Snell's Law: sin⁡θi/sin⁡θt=ϵr2/ϵr1\sin \theta_i / \sin \theta_t = \sqrt{\epsilon_{r2} / \epsilon_{r1}}. The transmission coefficients tst_s and tpt_p are related to the reflection coefficients by tk=1+rkt_k = 1 + r_k (assuming μr1=μr2=1\mu_{r1} = \mu_{r2} = 1).
Let Ei\mathbf{E}_{i} and Er\mathbf{E}_{r} be the incident and reflected electric field amplitudes, respectively, at an interface separating two media with relative permittivities ϵ1\epsilon_{1} and ϵ2\epsilon_{2}. Let θi\theta_{i} be the angle of incidence and θt\theta_{t} be the angle of transmission, satisfying Snell's Law. The reflection coefficient for the electric field amplitude, rr, is defined by the ratio of the reflected field amplitude to the incident field amplitude. For the perpendicular polarization (TE mode, E⊥n^\mathbf{E} \perp \hat{\mathbf{n}}), the coefficient is:\nr⊥=η2cos⁡(θi)−η1cos⁡(θt)η2cos⁡(θi)+η1cos⁡(θt)\mathbf{r}_{\perp} = \frac{\eta_{2} \cos(\theta_{i}) - \eta_{1} \cos(\theta_{t})}{\eta_{2} \cos(\theta_{i}) + \eta_{1} \cos(\theta_{t})} \nwhere ηj=μj/ϵj\eta_{j} = \sqrt{\mu_{j}/\epsilon_{j}} is the intrinsic impedance of medium jj. For the parallel polarization (TM mode, E∥n^\mathbf{E} \parallel \hat{\mathbf{n}}), the coefficient is:\nr∥=η2cos⁡(θt)−η1cos⁡(θi)η2cos⁡(θt)+η1cos⁡(θi)\mathbf{r}_{\parallel} = \frac{\eta_{2} \cos(\theta_{t}) - \eta_{1} \cos(\theta_{i})}{\eta_{2} \cos(\theta_{t}) + \eta_{1} \cos(\theta_{i})} \nAlternatively, the reflectance RR (ratio of reflected intensity to incident intensity) is given by R=∣r∣2R = |r|^2. For the perpendicular case, R⊥=∣η2cos⁡(θi)−η1cos⁡(θt)η2cos⁡(θi)+η1cos⁡(θt)∣2R_{\perp} = \left| \frac{\eta_{2} \cos(\theta_{i}) - \eta_{1} \cos(\theta_{t})}{\eta_{2} \cos(\theta_{i}) + \eta_{1} \cos(\theta_{t})} \right|^2. For the parallel case, R∥=∣η2cos⁡(θt)−η1cos⁡(θi)η2cos⁡(θt)+η1cos⁡(θi)∣2R_{\parallel} = \left| \frac{\eta_{2} \cos(\theta_{t}) - \eta_{1} \cos(\theta_{i})}{\eta_{2} \cos(\theta_{t}) + \eta_{1} \cos(\theta_{i})} \right|^2.
Let E1(r,t)\mathbf{E}_1(\mathbf{r}, t) and E2(r,t)\mathbf{E}_2(\mathbf{r}, t) be two monochromatic electric fields propagating in vacuum, defined by the solutions to the wave equation ∇2E−1c2∂2E∂t2=0\nabla^2 \mathbf{E} - \frac{1}{c^2}\frac{\partial^2 \mathbf{E}}{\partial t^2} = 0. Assume the fields can be represented as: \begin{align*} \mathbf{E}_1(\mathbf{r}, t) &= \mathbf{A}_1 e^{i (\mathbf{k}_1 \cdot \mathbf{r} - \omega t)} \mathbf{\hat{e}}_1 \\ \mathbf{E}_2(\mathbf{r}, t) &= \mathbf{A}_2 e^{i (\mathbf{k}_2 \cdot \mathbf{r} - \omega t)} \mathbf{\hat{e}}_2 \end{align*} where Aj\mathbf{A}_j are the amplitudes, kj\mathbf{k}_j are the wave vectors, and ω\omega is the angular frequency. The resultant electric field is Eres(r,t)=E1(r,t)+E2(r,t)\mathbf{E}_{\text{res}}(\mathbf{r}, t) = \mathbf{E}_1(\mathbf{r}, t) + \mathbf{E}_2(\mathbf{r}, t). Constructive interference occurs at a point r\mathbf{r} and time tt if the phase difference Δϕ=k1⋅r−k2⋅r+(k2−k1)⋅r\Delta \phi = \mathbf{k}_1 \cdot \mathbf{r} - \mathbf{k}_2 \cdot \mathbf{r} + (\mathbf{k}_2 - \mathbf{k}_1) \cdot \mathbf{r} satisfies the condition Δϕ=2πn\Delta \phi = 2\pi n, where n∈Zn \in \mathbb{Z}, and the polarization vectors e^1\mathbf{\hat{e}}_1 and e^2\mathbf{\hat{e}}_2 are aligned, such that the resultant amplitude is maximized: \begin{equation*} \left| \mathbf{E}_{\text{res}}(\mathbf{r}, t) \right| = \left| \mathbf{A}_1 + \mathbf{A}_2 \right| \end{equation*}. This condition is equivalent to the path difference ΔL=∣r1−r2∣⋅k^\Delta L = |\mathbf{r}_1 - \mathbf{r}_2| \cdot \mathbf{\hat{k}} satisfying ΔL=nλ\Delta L = n\lambda, where λ=2π/∣k∣\lambda = 2\pi/|\mathbf{k}| is the wavelength.
Let Σ\Sigma be the interface separating two isotropic media, Medium 1 and Medium 2. Define the unit normal vector N\mathbf{N} to Σ\Sigma. Let k1\mathbf{k}_1 and k2\mathbf{k}_2 be the wave vectors of the incident and refracted electromagnetic waves, respectively. The refractive indices n1n_1 and n2n_2 are defined by the ratio of the speed of light in vacuum cc to the phase velocity viv_i in the medium: ni=c/vin_i = c/v_i. The angle of incidence θ1\theta_1 and angle of refraction θ2\theta_2 are defined by the projections of the wave vectors onto the plane perpendicular to N\mathbf{N}. The conservation of the tangential component of the wave vector k∥\mathbf{k}_{\parallel} across the boundary Σ\Sigma mandates that: \n\nk1⋅t=k2⋅t\mathbf{k}_1 \cdot \mathbf{t} = \mathbf{k}_2 \cdot \mathbf{t} \n\nwhere t\mathbf{t} is any unit vector tangent to Σ\Sigma. This leads to the scalar relationship:\n\nn1sin⁡(θ1)=n2sin⁡(θ2)n_1 \sin(\theta_1) = n_2 \sin(\theta_2)
Let E(x,t)\mathbf{E}(\mathbf{x}, t) be the electric field satisfying the homogeneous wave equation (∂2∂t2−c2∇2)E=0\left(\frac{\partial^2}{\partial t^2} - c^2 \nabla^2\right) \mathbf{E} = 0 in R3\mathbb{R}^3. Define the initial wavefront S(t0)S(t_0) as the surface where E(x,t0)=E0(x)\mathbf{E}(\mathbf{x}, t_0) = \mathbf{E}_0(\mathbf{x}). Huygens' Principle asserts that the field E(x,t)\mathbf{E}(\mathbf{x}, t) at a point x\mathbf{x} and time tt is given by the superposition integral over the initial surface S(t0)S(t_0): E(x,t)=14πρ∮S(t0)E0(x′)1ρ⋅n dS′\mathbf{E}(\mathbf{x}, t) = \frac{1}{4\pi\rho} \oint_{S(t_0)} \mathbf{E}_0(\mathbf{x}') \frac{1}{\rho} \cdot \mathbf{n} \, dS' where ρ=∣x−x′∣\rho = |\mathbf{x} - \mathbf{x}'| is the distance, x′\mathbf{x}' is a point on S(t0)S(t_0), and n\mathbf{n} is the unit normal vector to S(t0)S(t_0) pointing towards x\mathbf{x}. This formulation captures the constructive interference of secondary wavelets.
Let E(r,t)\mathbf{E}(\mathbf{r}, t) be the electric field satisfying the wave equation ∇2E−μϵ∂2E∂t2=0\nabla^2 \mathbf{E} - \mu \epsilon \frac{\partial^2 \mathbf{E}}{\partial t^2} = 0 in a medium characterized by ϵ(r)\epsilon(\mathbf{r}) and μ(r)\mu(\mathbf{r}). Assume a monochromatic wave solution of the form E(r,t)=A(r)eikS(r)/ℏ\mathbf{E}(\mathbf{r}, t) = \mathbf{A}(\mathbf{r}) e^{i k S(\mathbf{r}) / \hbar}, where k=ω/ck = \omega/c and S(r)S(\mathbf{r}) is the eikonal function. The ray optics approximation is defined by the Eikonal equation, which mandates that the phase function S(r)S(\mathbf{r}) satisfies:\n(∇S(r))2=n2(r)c2ℏ2k2\left(\nabla S(\mathbf{r})\right)^2 = \frac{n^2(\mathbf{r})}{c^2} \hbar^2 k^2 \nwhere n(r)=ϵ(r)μ(r)/ϵ0μ0n(\mathbf{r}) = \sqrt{\epsilon(\mathbf{r})\mu(\mathbf{r})}/\sqrt{\epsilon_0 \mu_0} is the refractive index. The ray trajectory r(s)\mathbf{r}(s) parameterized by arc length ss is then governed by the Hamiltonian system derived from the phase function S(r)S(\mathbf{r}):\ndrds=∇S(r)\frac{d \mathbf{r}}{d s} = \nabla S(\mathbf{r})\ndds(12∣∇S(r)∣2)=Fext(r,s)\frac{d}{d s} \left( \frac{1}{2} \left| \nabla S(\mathbf{r}) \right|^2 \right) = \mathbf{F}_{ext}(\mathbf{r}, s) \nwhere Fext\mathbf{F}_{ext} represents any external forces or material inhomogeneities not accounted for by the background medium parameters.