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Oblique-incidence TE and TM Fresnel amplitudes, the flux budget, Brewster's angle and total internal reflection

Physical scope

Conditional oblique-incidence Fresnel theorem for two homogeneous isotropic lossless scalar media at rest at a plane interface, under the stated interface identities: the TE and TM admittance systems, amplitude formulas, the flux budget R_chi+T_chi=1 in the propagating branch, the Brewster condition and the total-internal-reflection identities |r_chi|=1 with decaying transmitted field. Excluded: absorbing, dispersive, anisotropic, moving or nonlinear media, the ideal conductor, existence or uniqueness of interface-consistent data, signal- and group-velocity claims, and all empirical claims.

Statement

Assume the following plane-interface setting with the same stationary interface identities as Phase matching at a plane interface: common tangential wavevector and frequency, the reflection law and Snell's law: two homogeneous isotropic lossless scalar media at rest (medium 1 in ${x_{3} < 0}$ , constants $ε_{1}, μ_{1} > 0$ ; medium 2 in ${x_{3} > 0}$ , constants $ε_{2}, μ_{2} > 0$ ; both $σ = 0$ ), the plane interface $S = {x_{3} = 0}$ with normal $n = e_{3}$ , and the plane-wave structure of Plane-wave fields in a homogeneous Ohmic medium: amplitude conditions, dispersion, averaged flux and the evanescent branch. Let $ω > 0$ , let $κ \in R^{2}$ (identified with the tangential plane ${x_{3} = 0}$ ) satisfy $0 < ∣ κ ∣ < K_{1} : = ω (ε_{1} μ_{1})^{1 / 2}$ , put $k_{13} : = \sqrt{K_{1}^{2} - ∣ κ ∣^{2}} > 0$ and define wavevectors $k_{i} : = κ + k_{13} n, k_{r} : = κ - k_{13} n, k_{t} : = κ + k_{t 3} n,$ where $k_{t 3} \in C$ satisfies $k_{t 3}^{2} = K_{2}^{2} - ∣ κ ∣^{2}$ with $K_{2} : = ω (ε_{2} μ_{2})^{1 / 2}$ and is selected by non-growth in the $x_{3} > 0$ half-space: $k_{t 3} = \sqrt{K_{2}^{2} - ∣ κ ∣^{2}} > 0 (∣ κ ∣ < K_{2}), k_{t 3} = 0 (∣ κ ∣ = K_{2}), k_{t 3} = i β, β : = \sqrt{∣ κ ∣^{2} - K_{2}^{2}} > 0 (∣ κ ∣ > K_{2}) .$ [This transmitted phase data is the interface-consistent, bounded choice: the propagating and evanescent branches of Plane-wave fields in a homogeneous Ohmic medium: amplitude conditions, dispersion, averaged flux and the evanescent branch(d),(e); the common $(κ, ω)$ is declared here for each actual wave, also when a reflected amplitude is zero or the critical transmitted root is zero; the nonzero phase-matching theorem explains this matching where its hypotheses apply, and an absent reflected field has no determined phase labels.] Fix the orthonormal tangential frame $\hat{κ} : = κ / ∣ κ ∣$ and $\hat{y} : = n \times \hat{κ}$ , and consider the two polarisations: (s) [transverse electric] $e_{i} = E_{i} \hat{y}$ , $e_{r} = E_{r} \hat{y}$ , $e_{t} = E_{t} \hat{y}$ with $E_{i}, E_{r}, E_{t} \in C$ , $E_{i} \neq 0$ , and $h_{j} : = k_{j} \times e_{j} / (μ_{j} ω)$ ; (p) [transverse magnetic] $h_{i} = H_{i} \hat{y}$ , $h_{r} = H_{r} \hat{y}$ , $h_{t} = H_{t} \hat{y}$ with $H_{i}, H_{r}, H_{t} \in C$ , $H_{i} \neq 0$ , and electric amplitudes $e_{j} : = - (k_{j} \times \hat{y}) H_{j} / (ω ε_{j})$ (the same computation shows that in both cases the pairs satisfy the amplitude conditions of Plane-wave fields in a homogeneous Ohmic medium: amplitude conditions, dispersion, averaged flux and the evanescent branch(b) in their media). For each polarisation define the amplitudes along the fixed tangential directions $E_{j}^{(s)} : = e_{j} \cdot \hat{y}$ and, in the case (p), $E_{j}^{(p)} : = e_{j} \cdot \hat{x}$ with $\hat{x} : = \hat{κ}$ ; assume the four stationary interface identities of Phase matching at a plane interface: common tangential wavevector and frequency, the reflection law and Snell's law hold on $R^{2} \times I$ for a nonempty open interval $I$ . Then:

(a) [Amplitude system] for $χ = s$ in every branch, and for $χ = p$ when $k_{t 3} \neq 0$ , the tangential interface identities reduce to $E_{i}^{χ} + E_{r}^{χ} = E_{t}^{χ}, Y_{1}^{χ} (E_{i}^{χ} - E_{r}^{χ}) = Y_{2}^{χ} E_{t}^{χ},$ with the admittances $Y_{1}^{s} = \frac{k_{13}}{μ_{1} ω}, Y_{1}^{p} = \frac{ε_{1} ω}{k_{13}}, Y_{2}^{s} = \frac{k_{t 3}}{μ_{2} ω}, Y_{2}^{p} = \frac{ε_{2} ω}{k_{t 3}} (k_{t 3} \neq 0) .$ Write $k_{j 3} : = k_{j} \cdot n$ for the signed components, so $k_{i 3} = k_{13}$ and $k_{r 3} = - k_{13}$ . Here $Y_{1}^{χ} > 0$ uses the positive incident normal magnitude $k_{13}$ , whereas the signed reflected component is $k_{r} \cdot n = - k_{13}$ ; $Y_{1}^{χ}$ is not defined from that negative component. No $Y_{2}^{p}$ is defined at $k_{t 3} = 0$ . Hence, whenever $Y_{1}^{χ} + Y_{2}^{χ} \neq 0$ , $r_{χ} : = \frac{E_{r}^{χ}}{E_{i}^{χ}} = \frac{Y_{1}^{χ} - Y_{2}^{χ}}{Y_{1}^{χ} + Y_{2}^{χ}}, t_{χ} : = \frac{E_{t}^{χ}}{E_{i}^{χ}} = \frac{2 Y_{1}^{χ}}{Y_{1}^{χ} + Y_{2}^{χ}}$ (The two-equation interface admittance system: solution formulas and the normal-flux identity(a)); the reflected and transmitted amplitude symbols are defined by these fixed-direction conventions, and sign conventions for full vector amplitudes may reverse a reflected amplitude relative to this convention.

(b) [Propagating branch and Snell's law] if $∣ κ ∣ < K_{2}$ then all three wavevectors are real, $Y_{1}^{s}, Y_{1}^{p}, Y_{2}^{s}, Y_{2}^{p} > 0$ are real, and with $θ_{j} \in (0, π / 2)$ defined by $\sin θ_{j} = ∣ κ ∣ / ∣ k_{j} ∣$ one has $θ_{r} = θ_{i}$ and $n_{1} \sin θ_{i} = n_{2} \sin θ_{t}$ , $n_{j} = c (ε_{j} μ_{j})^{1 / 2}$ (Phase matching at a plane interface: common tangential wavevector and frequency, the reflection law and Snell's law(c), Homogeneous isotropic medium parameters: phase speed, impedance, refractive index, complex propagation constant and skin depth); the period-averaged normal fluxes are $Y_{1}^{χ} ∣ E_{i}^{χ} ∣^{2} / 2$ , $- Y_{1}^{χ} ∣ E_{r}^{χ} ∣^{2} / 2$ and $Y_{2}^{χ} ∣ E_{t}^{χ} ∣^{2} / 2$ , so $R_{χ} : = ∣ r_{χ} ∣^{2}, T_{χ} : = \frac{Y_{2}^{χ}}{Y_{1}^{χ}} ∣ t_{χ} ∣^{2}, R_{χ} + T_{χ} = 1.$

(c) [Brewster condition] In the finite-admittance branches, $r_{χ} = 0$ if and only if $Y_{1}^{χ} = Y_{2}^{χ}$ ; in particular, if $μ_{1} = μ_{2}$ and $∣ κ ∣ < K_{2}$ , for unequal indices $n_{1} \neq n_{2}$ , $r_{p} = 0$ holds at exactly one admissible incidence angle $θ_{B} \in (0, π / 2)$ , characterised by $\tan θ_{B} = n_{2} / n_{1}$ , and $r_{s} \neq 0$ at that angle. For equal indices $n_{1} = n_{2}$ (and equal permeabilities), the media are identical, $Y_{1}^{χ} = Y_{2}^{χ}$ and $r_{s} = r_{p} = 0$ , $t_{s} = t_{p} = 1$ for every allowed incidence angle; there is no unique distinguished Brewster angle.

(d) [Total internal reflection] if $∣ κ ∣ > K_{2}$ (which requires $n_{1} > n_{2}$ , $θ_{i} : = \arcsin (∣ κ ∣ / K_{1}) \in (0, π / 2)$ , $∣ κ ∣ = K_{1} \sin θ_{i}$ and $θ_{i} > θ_{c} : = \arcsin (n_{2} / n_{1})$ ), then $Y_{2}^{s} = i β / (μ_{2} ω)$ and $Y_{2}^{p} = - i ε_{2} ω / β$ are purely imaginary, $∣ r_{s} ∣ = ∣ r_{p} ∣ = 1, t_{s}, t_{p} \neq 0, ∣ t_{χ} ∣^{2} = \frac{4 (Y_{1}^{χ})^{2}}{(Y_{1}^{χ})^{2} + ∣ Y_{2}^{χ} ∣^{2}} > 0,$ the transmitted field decays as $e^{- β x_{3}}$ and its period-averaged normal flux vanishes identically while its tangential averaged flux does not vanish (Plane-wave fields in a homogeneous Ohmic medium: amplitude conditions, dispersion, averaged flux and the evanescent branch(e)); the reflected wave carries the whole incident normal averaged flux.

(e) [Critical angle] if $∣ κ ∣ = K_{2}$ then $k_{t 3} = 0$ : the transmitted wave travels along the interface, its tangential electric amplitude vanishes in the (p) case and $Y_{2}^{s} = 0$ in the (s) case, and $r_{s} = 1$ , $t_{s} = 2$ , $r_{p} = - 1$ , $t_{p} = 0$ with vanishing transmitted normal averaged flux. The critical TM branch is solved from the nonsingular equations $E_{i}^{p} + E_{r}^{p} = k_{t 3} H_{t} / (ω ε_{2}) = 0$ and $H_{i} + H_{r} = H_{t}$ , not by dividing by $k_{t 3}$ ; they give $H_{r} = H_{i}$ and $H_{t} = 2 H_{i}$ . The transmitted TM electric vector can have a nonzero normal component even though its tangential amplitude is zero.

(f) [Scope] the statements are conditional on the interface identities and on the given wave data; no existence or uniqueness of interface-consistent data is claimed. Excluded: anisotropic media, complex $ε$ or $μ$ and absorption in either medium (only the lossless dielectric case with the evanescent transmitted branch is covered), dispersive media, moving media, the ideal conductor, nonlinear response, and all claims about signal speeds, group velocities, information transfer or empirical data. The statements for $Y_{1}^{χ} + Y_{2}^{χ} = 0$ (which cannot occur in the stated branches, where $Re Y_{1}^{χ} > 0$ and $Re Y_{2}^{χ} \geq 0$ ) are excluded; the notation 'angle of refraction' is used only where $k_{t}$ is real.

Facts & Assumptions

Given: The two-media lossless setting of the Statement, its common $(κ, ω)$ and wavevectors $k_{i}, k_{r}, k_{t}$ , the two polarisation data families, and the four stationary interface identities [A1] on $R^{2} \times I$ .

[A1]

The four stationary interface identities of Phase matching at a plane interface: common tangential wavevector and frequency, the reflection law and Snell's law hold for all $(y, t) \in R^{2} \times I$ .

[F1]

Plane-wave structure (Plane-wave fields in a homogeneous Ohmic medium: amplitude conditions, dispersion, averaged flux and the evanescent branch): the amplitude conditions $k_{j} \cdot e_{j} = 0$ , $k_{j} \times e_{j} = ω μ_{j} h_{j}$ , $k_{j} \cdot h_{j} = 0$ , $k_{j} \cdot k_{j} = ε_{j} μ_{j} ω^{2}$ and the solution form $h_{j} = k_{j} \times e_{j} / (μ_{j} ω)$ ; the lossless pointwise identities $B = \hat{k} \times E / v$ and period-averaged flux $⟨ S ⟩ = ∣ e ∣^{2} k / (2 μ ω)$ for real $k$ ; and, in the evanescent branch, the factorization with decay $e^{- β n^{'} \cdot x}$ , the vanishing normal averaged flux and the positive tangential averaged fluxes for the two transverse polarisations.

[F2]

The angle laws (Phase matching at a plane interface: common tangential wavevector and frequency, the reflection law and Snell's law(c)): for real wavevectors at a lossless interface, $θ_{r} = θ_{i}$ and $n_{1} \sin θ_{i} = n_{2} \sin θ_{t}$ with the same angle conventions, together with the parameters $v_{j}$ and $n_{j} = c / v_{j}$ of Homogeneous isotropic medium parameters: phase speed, impedance, refractive index, complex propagation constant and skin depth.

[F3]

Admittance algebra (The two-equation interface admittance system: solution formulas and the normal-flux identity): (a) the solution formulas for the two-equation system; (b) for real $Y_{1} > 0$ the identity $∣ r ∣^{2} + (Re Y_{2} / Y_{1}) ∣ t ∣^{2} = 1$ with $∣ t ∣^{2} = 4 Y_{1}^{2} / ∣ Y_{1} + Y_{2} ∣^{2} > 0$ ; (c) for real $Y_{1} > 0$ and purely imaginary $Y_{2}$ the identities $∣ r ∣ = 1$ and $∣ t ∣^{2} = 4 Y_{1}^{2} / (Y_{1}^{2} + ∣ Y_{2} ∣^{2}) > 0$ ; (d) the real-positive specialisation agreeing with The two-equation interface system at normal incidence.

[F4]

Period averages (Period averages of products of real parts of complex exponentials) and the stored-energy/flux definitions $u = \frac{1}{2} (E \cdot D + B \cdot H)$ , $S = E \times H$ of Energy balance and Ohmic dissipation for the constant nondispersive linear model.

[F5]

Geometry of the fixed frame: $\hat{x} = \hat{κ} = κ / ∣ κ ∣$ , $\hat{y} = n \times \hat{κ}$ , with the right-handed orthonormal triad relations $\hat{x} \times \hat{y} = n$ , $\hat{y} \times n = \hat{x}$ , $n \times \hat{x} = \hat{y}$ and the reversed signs; the cross product is bilinear with $⟨ u \times v, w ⟩ = \det [u v w]$ , and $u \times v$ is orthogonal to $u$ and $v$ (The cross product in $R^{3}$ , The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , The cross product is bilinear, alternating, and orthogonal to both factors).

[F6]

Medium parameters (Homogeneous isotropic medium parameters: phase speed, impedance, refractive index, complex propagation constant and skin depth): $K_{j} = ω (ε_{j} μ_{j})^{1 / 2}$ , $v_{j} = (ε_{j} μ_{j})^{- 1 / 2}$ , $n_{j} = c (ε_{j} μ_{j})^{1 / 2}$ . The wavevector definitions and selected roots in this item's Statement give $k_{13}^{2} = K_{1}^{2} - ∣ κ ∣^{2}$ and $k_{t 3}^{2} = K_{2}^{2} - ∣ κ ∣^{2}$ with its declared branch selection.

[F7]

Complex arithmetic, conjugation, modulus (The complex numbers as $R [x] / (x^{2} + 1)$ , with the real embedding and imaginary unit $i$ , $C = R [x] / (x^{2} + 1)$ is a field, every element is uniquely $a + b i$ , and every nonzero element has inverse $(a - b i) / (a^{2} + b^{2})$ , Real and imaginary parts, complex conjugation, and modulus, Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive) with nonnegative square roots (Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ ) and the existence of complex square roots (Every complex number has a square root, by an explicit Cartesian formula).

[F8]

Trigonometry: $\sin (\arcsin y) = y$ with sine strictly increasing on $[- π / 2, π / 2]$ (Principal inverse sine and inverse cosine); $\tan$ restricts to a strictly increasing bijection $(- π / 2, π / 2) \to R$ with the principal inverse (The principal inverse tangent $\arctan : R \to (- π / 2, π / 2)$ ); $\tan x = \sin x / \cos x$ (Tangent, cotangent, secant, and cosecant on their exact natural domains); $\sin^{2} x + \cos^{2} x = 1$ with $1 + \tan^{2} x = \sec^{2} x$ (Pythagorean and parity identities for all six trigonometric functions on their natural domains); sine is strictly increasing on $[0, π / 2]$ and cosine strictly decreasing on $[0, π]$ with $\sin 0 = 0$ , $\cos 0 = 1$ and $\cos (π / 2) = 0$ (Signs, monotonicity intervals, and ranges of sine and cosine, The zero sets of sine and cosine and the least positive common period 2 pi).

[F9]

The laboratory setting, fields and units (Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields, The electric and magnetic fields as fixed-time spatial vector fields, Framework-local SI units, the vacuum constants and the exact/measured status of their values) and the declared lossless branches (Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch).

Proof

technique · direct; verify the amplitude conditions for the two polarisation families, reduce the tangential interface identities to scalar admittance systems in the fixed frame, and read off the propagating, Brewster, evanescent and critical cases from the admittance algebra and the medium plane-wave structure

1.1givenF1F5F7algebra

[Both polarisation families satisfy the amplitude conditions] For (s), $k_{j} \cdot e_{j} = E_{j} k_{j} \cdot \hat{y} = 0$ because $k_{j} = κ + k_{j 3} n$ and $\hat{y} ⊥ κ, n$ ; and $k_{j} \times e_{j} = E_{j} k_{j} \times \hat{y} = E_{j} (κ \times \hat{y} + k_{j 3} n \times \hat{y}) = E_{j} (∣ κ ∣ n - k_{j 3} \hat{x})$ by [F5], which equals $ω μ_{j} h_{j}$ with $h_{j} = k_{j} \times e_{j} / (μ_{j} ω)$ by definition. For (p), $e_{j} = - (k_{j} \times \hat{y}) H_{j} / (ω ε_{j})$ and $k_{j} \cdot e_{j} = 0$ because $k_{j} \cdot (k_{j} \times \hat{y}) = 0$ by the orthogonality in [F5]; further $k_{j} \times e_{j} = - (k_{j} \times (k_{j} \times \hat{y})) H_{j} / (ω ε_{j}) = [(k_{j} \cdot k_{j}) \hat{y} - k_{j} (k_{j} \cdot \hat{y})] H_{j} / (ω ε_{j}) = (ε_{j} μ_{j} ω^{2}) \hat{y} H_{j} / (ω ε_{j}) = ω μ_{j} H_{j} \hat{y} = ω μ_{j} h_{j}$ , using $k_{j} \cdot \hat{y} = 0$ and $k_{j} \cdot k_{j} = ε_{j} μ_{j} ω^{2}$ of [F1]. In (s), $k_{j} \cdot h_{j} = 0$ by cross-product orthogonality, and $k_{j} \times h_{j} = - ω ε_{j} e_{j}$ follows by expanding $k_{j} \times (k_{j} \times e_{j}) = k_{j} (k_{j} \cdot e_{j}) - (k_{j} \cdot k_{j}) e_{j}$ and using dispersion. In (p), $k_{j} \cdot h_{j} = 0$ because $\hat{y}$ is perpendicular to $k_{j}$ , and $k_{j} \times h_{j} = - ω ε_{j} e_{j}$ is the definition of $e_{j}$ . Thus all four amplitude conditions hold in both families, including zero reflected or transmitted amplitudes.

2.1givenA1step 1.1F5F7algebra

[The tangential interface identities in the fixed frame] By the orthogonal decomposition of [F5], the tangential parts of the electric amplitudes are: for (s), $e_{j}^{τ} = e_{j} = E_{j} \hat{y}$ ; for (p), $e_{j}^{τ} = (k_{j 3} H_{j} / (ω ε_{j})) \hat{x}$ because $e_{j} = (k_{j 3} \hat{x} - ∣ κ ∣ n) H_{j} / (ω ε_{j})$ from $k_{j} \times \hat{y} = ∣ κ ∣ n - k_{j 3} \hat{x}$ by [F5]. The magnetic tangential traces are obtained as follows: $h_{j} = H_{j} \hat{y}$ in (p), and in (s) $h_{j} = E_{j} (κ \times \hat{y} + k_{j 3} n \times \hat{y}) / (μ_{j} ω) = E_{j} (∣ κ ∣ n - k_{j 3} \hat{x}) / (μ_{j} ω)$ with tangential part $- k_{j 3} E_{j} \hat{x} / (μ_{j} ω)$ and, for the reflected wave with $k_{r 3} = - k_{13}$ , tangential part $+ k_{13} E_{r} \hat{x} / (μ_{1} ω)$ . Since the interface identities hold for all $(y, t)$ with the common $(κ, ω)$ , each has the form $Re (W e^{i (κ \cdot y - ω t)}) = 0$ for a constant tangential vector $W$ . Since $n = e_{3}$ , $n \times (w_{1}, w_{2}, 0) = (- w_{2}, w_{1}, 0)$ , so the cross-product trace identities are equivalent to these real-part identities. Fixing any $y$ and allowing $t$ to vary in $I$ , the time-line proof of the amplitude-vanishing clause (a) of [F1] gives $W = 0$ . Concretely: the tangential electric identity gives $E_{i} + E_{r} = E_{t}$ in (s) and $E_{i}^{(p)} + E_{r}^{(p)} = E_{t}^{(p)}$ in (p); the tangential magnetic identity gives $k_{t 3} E_{t} / (μ_{2} ω) = k_{13} (E_{i} - E_{r}) / (μ_{1} ω)$ in (s) and $H_{i} + H_{r} = H_{t}$ in (p), where $E_{j}^{(p)} = k_{j 3} H_{j} / (ω ε_{j})$ by definition, and when $k_{t 3} \neq 0$ , converting $H_{j} = Y_{j}^{p} E_{j}^{(p)}$ for $j = i, t$ and $H_{r} = - Y_{1}^{p} E_{r}^{(p)}$ (the reflected sign entering through $k_{r 3} = - k_{13}$ ) turns the last identity into $Y_{1}^{p} (E_{i}^{(p)} - E_{r}^{(p)}) = Y_{2}^{p} E_{t}^{(p)}$ . Writing $Y_{1}^{s} = k_{13} / (μ_{1} ω)$ , $Y_{2}^{s} = k_{t 3} / (μ_{2} ω)$ , $Y_{1}^{p} = ε_{1} ω / k_{13}$ and $Y_{2}^{p} = ε_{2} ω / k_{t 3}$ when $k_{t 3} \neq 0$ gives the two systems of (a).

3.1step 2.1F3F7algebra

[Solution of the systems, part (a)] In the (s) branch, $Y_{1}^{s} > 0$ is real and positive, and $Y_{2}^{s}$ is real positive for $∣ κ ∣ < K_{2}$ , zero at the critical point, and $i β / (μ_{2} ω)$ for $∣ κ ∣ > K_{2}$ ; in every case $Re (Y_{1}^{s} + Y_{2}^{s}) = Y_{1}^{s} + Re Y_{2}^{s} \geq Y_{1}^{s} > 0$ , so $Y_{1}^{s} + Y_{2}^{s} \neq 0$ . In the (p) branch with $k_{t 3} \neq 0$ , $Re Y_{1}^{p} = Y_{1}^{p} > 0$ and $Re Y_{2}^{p} = 0$ or $> 0$ , so again $Y_{1}^{p} + Y_{2}^{p} \neq 0$ . The admittance lemma F3 therefore applies to each scalar system of step 2.1 and gives $r_{χ} = (Y_{1}^{χ} - Y_{2}^{χ}) / (Y_{1}^{χ} + Y_{2}^{χ})$ and $t_{χ} = 2 Y_{1}^{χ} / (Y_{1}^{χ} + Y_{2}^{χ})$ ; in particular $t_{χ} \neq 0$ because its numerator $2 Y_{1}^{χ} \neq 0$ .

3.2step 2.1F4F5F6F7algebra

[Critical angle, part (e)] Let $∣ κ ∣ = K_{2}$ ; then $k_{t 3} = 0$ by [F6], so $Y_{2}^{s} = 0$ and no $Y_{2}^{p}$ is defined (a division by $k_{t 3}$ is not performed). For (s), the system of step 2.1 reads $E_{i} + E_{r} = E_{t}$ and $Y_{1}^{s} (E_{i} - E_{r}) = 0$ with $Y_{1}^{s} > 0$ , so $E_{r} = E_{i}$ and $E_{t} = 2 E_{i}$ , i.e. $r_{s} = 1$ and $t_{s} = 2$ . For (p), $E_{t}^{(p)} = k_{t 3} H_{t} / (ω ε_{2}) = 0$ and $E_{i}^{(p)} = k_{13} H_{i} / (ω ε_{1})$ , $E_{r}^{(p)} = - k_{13} H_{r} / (ω ε_{1})$ by the fixed-direction definitions of step 2.1; the tangential electric identity $E_{i}^{(p)} + E_{r}^{(p)} = 0$ gives $H_{r} = H_{i}$ , and the tangential magnetic identity $H_{i} + H_{r} = H_{t}$ gives $H_{t} = 2 H_{i}$ ; hence $r_{p} = E_{r}^{(p)} / E_{i}^{(p)} = - H_{r} / H_{i} = - 1$ and $t_{p} = E_{t}^{(p)} / E_{i}^{(p)} = 0$ . The transmitted normal averaged flux vanishes: $⟨ S_{t} \cdot n ⟩ = \frac{1}{2} Re ((e_{t} \times {\overline{h}}_{t}) \cdot n)$ by [F4], and with $h_{t} = H_{t} \hat{y}$ and $e_{t} = - (k_{t} \times \hat{y}) H_{t} / (ω ε_{2}) = - ∣ κ ∣ H_{t} n / (ω ε_{2})$ (because $k_{t} = κ$ and $κ \times \hat{y} = ∣ κ ∣ n$ by [F5]), the vector $e_{t} \times {\overline{h}}_{t}$ is a multiple of $n \times \hat{y}$ , which is orthogonal to $n$ : $(e_{t} \times {\overline{h}}_{t}) \cdot n = 0$ . In the TE case $e_{t} = E_{t} \hat{y}$ and $h_{t} = ∣ κ ∣ E_{t} n / (μ_{2} ω)$ , so $e_{t} \times {\overline{h}}_{t}$ is tangential and its normal component is also zero. The transmitted TM electric vector $- ∣ κ ∣ H_{t} n / (ω ε_{2})$ has a nontrivial normal component whenever $H_{t} \neq 0$ .

4.1step 3.1F1F2F3F6F8algebra

[Propagating branch: angles and fluxes, part (b)] Let $∣ κ ∣ < K_{2}$ . Then $k_{t 3} = \sqrt{K_{2}^{2} - ∣ κ ∣^{2}} > 0$ and all three wavevectors are real by [F6]; the four admittances of step 2.1 are real and positive. With $∣ k_{i} ∣ = ∣ k_{r} ∣ = K_{1}$ and $∣ k_{t} ∣ = K_{2}$ from $∣ k_{j} ∣^{2} = ∣ κ ∣^{2} + k_{j 3}^{2}$ , the angles $θ_{j} : = \arcsin (∣ κ ∣ / ∣ k_{j} ∣) \in (0, π / 2)$ are well defined by [F8] since $0 < ∣ κ ∣ < \min {K_{1}, K_{2}}$ ; then $\sin θ_{r} = ∣ κ ∣ / K_{1} = \sin θ_{i}$ , so $θ_{r} = θ_{i}$ by the strict increase of sine and of $\arcsin$ in [F8], and $K_{1} \sin θ_{i} = ∣ κ ∣ = K_{2} \sin θ_{t}$ , which is $n_{1} \sin θ_{i} = n_{2} \sin θ_{t}$ after multiplying by $c / ω$ and using $n_{j} = c K_{j} / ω = c (ε_{j} μ_{j})^{1 / 2}$ by [F6]; this is the same angle law as [F2]. The period-averaged normal fluxes: for (s), the flux of [F1] is $⟨ S_{j} ⟩ = ∣ e_{j} ∣^{2} k_{j} / (2 μ_{j} ω)$ with $∣ e_{j} ∣^{2} = ∣ E_{j}^{(s)} ∣^{2}$ and $k_{j} \cdot n = k_{j 3}$ , so the normal fluxes are $k_{j 3} ∣ E_{j}^{(s)} ∣^{2} / (2 μ_{j} ω)$ , i.e. $Y_{1}^{s} ∣ E_{i}^{(s)} ∣^{2} / 2$ , $- Y_{1}^{s} ∣ E_{r}^{(s)} ∣^{2} / 2$ , $Y_{2}^{s} ∣ E_{t}^{(s)} ∣^{2} / 2$ ; for (p) the same $⟨ S_{j} \cdot n ⟩ = ∣ e_{j} ∣^{2} k_{j 3} / (2 μ_{j} ω)$ equals $Y_{j}^{p} ∣ E_{j}^{(p)} ∣^{2} / 2$ up to the reflected sign, because $∣ e_{j} ∣^{2} = μ_{j} ∣ H_{j} ∣^{2} / ε_{j}$ and $∣ E_{j}^{(p)} ∣ = ∣ k_{j 3} ∣ ∣ H_{j} ∣ / (ε_{j} ω)$ give $∣ e_{j} ∣^{2} k_{j 3} / (2 μ_{j} ω) = k_{j 3} ∣ H_{j} ∣^{2} / (2 ε_{j} ω)$ for the incident and transmitted signs, while the reflected signed component $k_{r 3} = - k_{13}$ produces the minus sign, as in the Statement. With $R_{χ} = ∣ r_{χ} ∣^{2} = ∣ E_{r}^{χ} ∣^{2} / ∣ E_{i}^{χ} ∣^{2}$ and $T_{χ} = (Y_{2}^{χ} / Y_{1}^{χ}) ∣ t_{χ} ∣^{2} = Y_{2}^{χ} ∣ E_{t}^{χ} ∣^{2} / (Y_{1}^{χ} ∣ E_{i}^{χ} ∣^{2})$ , the budget $R_{χ} + T_{χ} = 1$ is F3 applied with the real positive $Y_{1}^{χ}, Y_{2}^{χ}$ .

4.2step 3.1F1F3F6F7algebra

[Total internal reflection, part (d)] Let $∣ κ ∣ > K_{2}$ ; then $K_{2} < K_{1}$ , so $n_{2} < n_{1}$ , and $k_{t 3} = i β$ with $β = \sqrt{∣ κ ∣^{2} - K_{2}^{2}} > 0$ by [F6]. Hence $Y_{2}^{s} = k_{t 3} / (μ_{2} ω) = i β / (μ_{2} ω)$ and $Y_{2}^{p} = ε_{2} ω / (i β) = - i ε_{2} ω / β$ are purely imaginary by [F7], while $Y_{1}^{χ} > 0$ is real; the admittance lemma F3 gives $∣ r_{s} ∣ = ∣ r_{p} ∣ = 1$ and $∣ t_{χ} ∣^{2} = 4 (Y_{1}^{χ})^{2} / ((Y_{1}^{χ})^{2} + ∣ Y_{2}^{χ} ∣^{2}) > 0$ , so $t_{χ} \neq 0$ . The transmitted wave is the evanescent branch of [F1] with $n^{'} = n$ , $κ_{0} = κ$ and $k_{t} \cdot k_{t} = K_{2}^{2} = ε_{2} μ_{2} ω^{2}$ : the field factorizes with the decay factor $e^{- β x_{3}}$ , its period-averaged normal flux vanishes identically, and its tangential averaged flux is strictly positive; since $∣ r_{χ} ∣ = 1$ , the reflected normal averaged flux equals minus the incident normal averaged flux, i.e. the reflected wave carries the whole incident normal averaged flux.

5.1step 3.1step 4.1F3F6F8algebra

[Brewster condition, part (c)] In each finite-admittance branch the formula of step 3.1 gives $r_{χ} = 0$ exactly when the numerator $Y_{1}^{χ} - Y_{2}^{χ}$ vanishes, i.e. $Y_{1}^{χ} = Y_{2}^{χ}$ . Assume now $μ_{1} = μ_{2} = : μ$ and $∣ κ ∣ < K_{2}$ , so that the (p) admittances are real positive. The condition $Y_{1}^{p} = Y_{2}^{p}$ reads $ε_{1} / k_{13} = ε_{2} / k_{t 3}$ ; substituting $ε_{j} = n_{j}^{2} / (c^{2} μ)$ and $k_{j 3} = (ω n_{j} / c) \cos θ_{j}$ (from $k_{13} = K_{1} \cos θ_{i} = (ω n_{1} / c) \cos θ_{i}$ and $k_{t 3} = K_{2} \cos θ_{t} = (ω n_{2} / c) \cos θ_{t}$ ) gives $n_{1} \cos θ_{t} = n_{2} \cos θ_{i}$ . Together with Snell's law $n_{1} \sin θ_{i} = n_{2} \sin θ_{t}$ of step 4.1, squaring both and eliminating $\sin^{2} θ_{t}$ gives $(n_{1}^{2} - n_{2}^{2}) [n_{2}^{2} - (n_{1}^{2} + n_{2}^{2}) \sin^{2} θ_{i}] = 0$ : indeed $n_{1}^{2} (1 - \sin^{2} θ_{t}) = n_{2}^{2} \cos^{2} θ_{i}$ and $\sin θ_{t} = (n_{1} / n_{2}) \sin θ_{i}$ yield $n_{1}^{2} n_{2}^{2} - n_{1}^{4} \sin^{2} θ_{i} = n_{2}^{4} - n_{2}^{4} \sin^{2} θ_{i}$ , i.e. $n_{2}^{2} (n_{1}^{2} - n_{2}^{2}) = (n_{1}^{2} - n_{2}^{2}) (n_{1}^{2} + n_{2}^{2}) \sin^{2} θ_{i}$ . If $n_{1} \neq n_{2}$ , division by the nonzero first factor gives $\sin^{2} θ_{i} = n_{2}^{2} / (n_{1}^{2} + n_{2}^{2})$ ; since $0 < θ_{i} < π / 2$ and sine is strictly increasing on $[0, π / 2]$ with $\sin 0 = 0$ by [F8], this determines the unique admissible angle $θ_{B} : = \arcsin (n_{2} / \sqrt{n_{1}^{2} + n_{2}^{2}}) \in (0, π / 2)$ , This angle is admissible: with $D : = \sqrt{n_{1}^{2} + n_{2}^{2}}$ , its tangential wavenumber is $K_{1} n_{2} / D < K_{2}$ because $n_{1} / D < 1$ . Snell's law therefore gives $\sin θ_{t} = n_{1} / D$ and, by positivity of both cosines and the Pythagorean identity, $\cos θ_{i} = n_{1} / D$ , $\cos θ_{t} = n_{2} / D$ . Thus $n_{1} \cos θ_{t} = n_{2} \cos θ_{i}$ , establishing $Y_{1}^{p} = Y_{2}^{p}$ and $r_{p} = 0$ ; this proves sufficiency as well as uniqueness. At this angle $\cos θ_{B} > 0$ : cosine is strictly decreasing on $[0, π]$ with $\cos (π / 2) = 0$ by [F8], so $0 < θ_{B} < π / 2$ gives $\cos θ_{B} > \cos (π / 2) = 0$ . The Pythagorean identity of [F8] then gives $\cos θ_{B} = \sqrt{1 - \sin^{2} θ_{B}} = n_{1} / \sqrt{n_{1}^{2} + n_{2}^{2}}$ , so $\tan θ_{B} = \sin θ_{B} / \cos θ_{B} = n_{2} / n_{1}$ by the definition of tangent in [F8]; no other admissible angle has this tangent, because tangent is strictly increasing on $(0, π / 2)$ by [F8]. Also $r_{s} \neq 0$ at $θ_{B}$ , because $Y_{1}^{s} = Y_{2}^{s}$ would give $k_{13} = k_{t 3}$ (the $μ$ factors being equal), hence $K_{1}^{2} = ∣ κ ∣^{2} + k_{13}^{2} = ∣ κ ∣^{2} + k_{t 3}^{2} = K_{2}^{2}$ and $n_{1} = n_{2}$ , contrary to hypothesis. If instead $n_{1} = n_{2}$ and $μ_{1} = μ_{2}$ then $ε_{1} = ε_{2}$ and the two media are identical: $k_{13} = k_{t 3}$ and $ε_{1} = ε_{2}$ give $Y_{1}^{χ} = Y_{2}^{χ}$ , so $r_{s} = r_{p} = 0$ and $t_{s} = t_{p} = 1$ at every admissible angle (where the (p) formula applies), with no distinguished Brewster angle.

6.1givenstep 3.2F1F3F9∎

[Scope, part (f)] All clauses are conditional on the given wave data and the interface identities [A1]; the argument used the amplitude conditions and solution structure of [F1], the admittance algebra [F3], the angle conventions of [F2] and [F8], the period averages and stored-energy flux of [F4], and the complex arithmetic of [F7]; the ideal-conductor limit, absorption, dispersion, anisotropy, moving and nonlinear media, signal-speed and group-velocity statements and all empirical claims are excluded. The denominator $Y_{1}^{χ} + Y_{2}^{χ}$ is nonzero in every considered branch because $Re Y_{1}^{χ} > 0$ and $Re Y_{2}^{χ} \geq 0$ ; the interface identities are stated hypotheses on the noncompact plane (for compact interfaces, Stationary macroscopic interface jump conditions for D, B, E and H), and no choice principle is used, all constructions being finite-dimensional algebra in the given data.

Oblique-incidence TE and TM Fresnel amplitudes, the flux budget, Brewster's angle and total internal reflection

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources