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The Brewster angle at an equal-permeability interface and the surviving $s$ -reflection

Physical scope

Deterministic example of the oblique-incidence interface model for two declared lossless media with equal permeabilities and indices n_1=1, n_2=3/2: the unique Brewster angle tan(theta_B)=n_2/n_1, r_p=0, t_p=1, the surviving s-reflection r_s=-5/13 with R_s+T_s=1 exactly. Excluded: measured refractive indices, absorption, interfaces between unequal permeabilities, unpolarised-beam statistics and every empirical claim.

Example

Adopt the setting of Oblique-incidence TE and TM Fresnel amplitudes, the flux budget, Brewster's angle and total internal reflection with two lossless media at rest, equal permeabilities $μ_{1} = μ_{2} = : μ > 0$ and declared refractive indices $n_{1} = 1$ , $n_{2} = 3 / 2$ (declared model parameters, not optical measurements), so that $ε_{j} = n_{j}^{2} / (c^{2} μ)$ and $K_{j} = n_{j} ω / c$ in the notation of Homogeneous isotropic medium parameters: phase speed, impedance, refractive index, complex propagation constant and skin depth. Then: (i) [Brewster angle] by Oblique-incidence TE and TM Fresnel amplitudes, the flux budget, Brewster's angle and total internal reflection(c) the TM amplitude vanishes exactly at the unique angle $θ_{B} \in (0, π / 2)$ with $\tan θ_{B} = n_{2} / n_{1} = 3 / 2$ , i.e. $\sin θ_{B} = 3 / \sqrt{13}$ , $\cos θ_{B} = 2 / \sqrt{13}$ ; the transmitted angle satisfies $n_{1} \sin θ_{B} = n_{2} \sin θ_{t}$ , hence $\sin θ_{t} = 2 / \sqrt{13}$ , and $θ_{B} + θ_{t} = π / 2$ ; (ii) [Amplitudes] at $θ_{i} = θ_{B}$ one has $Y_{1}^{p} = Y_{2}^{p} = \sqrt{13} / (2 c μ)$ , hence $r_{p} = 0$ and $t_{p} = 1$ ; the remaining $s$ -amplitudes are $Y_{1}^{s} = 2 / (\sqrt{13} c μ)$ , $Y_{2}^{s} = 9 / (2 \sqrt{13} c μ)$ and $r_{s} = \frac{2 / \sqrt{13} - 9 / (2 \sqrt{13})}{2 / \sqrt{13} + 9 / (2 \sqrt{13})} = - \frac{5}{13}, t_{s} = \frac{8}{13};$ (iii) [Flux budget] with $R_{s} = ∣ r_{s} ∣^{2} = 25 / 169$ and $T_{s} = (Y_{2}^{s} / Y_{1}^{s}) ∣ t_{s} ∣^{2} = (9 / 4) (64 / 169) = 144 / 169$ one has $R_{s} + T_{s} = 169 / 169 = 1$ exactly, while the TM channel has $R_{p} = 0$ , $T_{p} = 1$ ; (iv) [Scope] the numbers are declared model parameters; the example is a deterministic computation of the classical interface model, reports no measurement, and a pure TM incident wave has zero reflected field at the Brewster angle. For incident data with a nonzero TE component, the surviving reflected field is TE-polarised because $r_{p} = 0$ and $r_{s} \neq 0$ ; no unpolarised-beam statistics are modelled here.

Verification

technique · direct substitution in the oblique-incidence theorem and the medium parameters, with all numerical values kept exact

Given: The lossless two-media setting of Oblique-incidence TE and TM Fresnel amplitudes, the flux budget, Brewster's angle and total internal reflection with $μ_{1} = μ_{2} = μ > 0$ , $n_{1} = 1$ , $n_{2} = 3 / 2$ , $ε_{j} = n_{j}^{2} / (c^{2} μ)$ , and the Brewster angle $θ_{B}$ of A1.

[A1] The oblique-incidence theorem Oblique-incidence TE and TM Fresnel amplitudes, the flux budget, Brewster's angle and total internal reflection: its clause (a) gives the two admittance systems with $Y_{1}^{s} = k_{13} / (μ_{1} ω)$ , $Y_{1}^{p} = ε_{1} ω / k_{13}$ , $Y_{2}^{s} = k_{t 3} / (μ_{2} ω)$ , $Y_{2}^{p} = ε_{2} ω / k_{t 3}$ and the amplitude formulas; clause (b) gives the angle law $n_{1} \sin θ_{i} = n_{2} \sin θ_{t}$ , the normal fluxes $Y_{1}^{χ} ∣ E_{i}^{χ} ∣^{2} / 2$ , $- Y_{1}^{χ} ∣ E_{r}^{χ} ∣^{2} / 2$ , $Y_{2}^{χ} ∣ E_{t}^{χ} ∣^{2} / 2$ and $R_{χ} + T_{χ} = 1$ with $R_{χ} = ∣ r_{χ} ∣^{2}$ , $T_{χ} = (Y_{2}^{χ} / Y_{1}^{χ}) ∣ t_{χ} ∣^{2}$ ; clause (c) gives the unique Brewster angle with $\tan θ_{B} = n_{2} / n_{1}$ for $n_{1} \neq n_{2}$ and the nonzero $s$ -reflection there.

[F1] Medium parameters (Homogeneous isotropic medium parameters: phase speed, impedance, refractive index, complex propagation constant and skin depth): $K_{j} = ω (ε_{j} μ_{j})^{1 / 2} = ω n_{j} / c$ and $ε_{j} = n_{j}^{2} / (c^{2} μ)$ for $μ_{j} = μ$ . In the propagating interface setting of Oblique-incidence TE and TM Fresnel amplitudes, the flux budget, Brewster's angle and total internal reflection (Statement and clause (b)), $k_{13} = \sqrt{K_{1}^{2} - ∣ κ ∣^{2}} = K_{1} \cos θ_{i}$ and $k_{t 3} = K_{2} \cos θ_{t}$ for $∣ κ ∣ < K_{2}$ ; the cosine formulas follow from the stated positive normal roots and the principal angle convention.

[F2] Trigonometry: $\sin (\arcsin y) = y$ with sine strictly increasing on $[- π / 2, π / 2]$ (Principal inverse sine and inverse cosine); $\tan : (- π / 2, π / 2) \to R$ is a continuous strictly increasing bijection whose principal inverse is $\arctan : R \to (- π / 2, π / 2)$ , with $\tan (\arctan y) = y$ for every real $y$ (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); $\sin^{2} x + \cos^{2} x = 1$ and $\cos (- x) = \cos x$ (Pythagorean and parity identities for all six trigonometric functions on their natural domains); sine strictly increasing on $[0, π / 2]$ with $\sin 0 = 0$ (Signs, monotonicity intervals, and ranges of sine and cosine); and $\sin (x + π / 2) = \cos x$ (Quarter-turn values and shifts by pi/2 and pi).

1.1A1F2algebra

[The Brewster angle and its sine and cosine, clause (i)] Since $n_{1} = 1 \neq 3 / 2 = n_{2}$ and $μ_{1} = μ_{2}$ , A1 gives a unique Brewster angle $θ_{B} \in (0, π / 2)$ with $\tan θ_{B} = n_{2} / n_{1} = 3 / 2$ . By [F2], $\sin^{2} θ_{B} = \tan^{2} θ_{B} / (1 + \tan^{2} θ_{B}) = (9 / 4) / (13 / 4) = 9 / 13$ ; since $θ_{B} \in (0, π / 2)$ and sine is strictly increasing on $[0, π / 2]$ with $\sin 0 = 0$ , $\sin θ_{B} > 0$ , hence $\sin θ_{B} = 3 / \sqrt{13}$ ; and $\cos θ_{B} = \sin θ_{B} / \tan θ_{B} = (3 / \sqrt{13}) / (3 / 2) = 2 / \sqrt{13}$ .

2.1step 1.1A1F2algebra

[Transmitted angle and the right-angle relation, clause (i)] Snell's law $n_{1} \sin θ_{B} = n_{2} \sin θ_{t}$ of A1 gives $\sin θ_{t} = (2 / 3) (3 / \sqrt{13}) = 2 / \sqrt{13}$ . Since also $\cos θ_{B} = 2 / \sqrt{13} = \sin θ_{t}$ and $\sin (π / 2 - θ_{B}) = \cos θ_{B}$ by [F2], the equality $\sin θ_{t} = \sin (π / 2 - θ_{B})$ holds with $θ_{t}, π / 2 - θ_{B} \in (0, π / 2)$ ; sine is strictly increasing on that interval, hence injective, so $θ_{t} = π / 2 - θ_{B}$ , i.e. $θ_{B} + θ_{t} = π / 2$ .

3.1step 1.1step 2.1A1F1algebra

[The admittances, clause (ii)] By [F1], $K_{1} = ω / c$ , $K_{2} = 3 ω / (2 c)$ , $k_{13} = K_{1} \cos θ_{B} = (ω / c) (2 / \sqrt{13})$ and $k_{t 3} = K_{2} \cos θ_{t} = K_{2} \sin θ_{B} = (3 ω / (2 c)) (3 / \sqrt{13}) = 9 ω / (2 \sqrt{13} c)$ . With $ε_{1} = 1 / (c^{2} μ)$ and $ε_{2} = 9 / (4 c^{2} μ)$ , the admittances of A1 are $Y_{1}^{s} = k_{13} / (μ ω) = 2 / (\sqrt{13} c μ)$ , $Y_{2}^{s} = k_{t 3} / (μ ω) = 9 / (2 \sqrt{13} c μ)$ , $Y_{1}^{p} = ε_{1} ω / k_{13} = \sqrt{13} / (2 c μ)$ and $Y_{2}^{p} = ε_{2} ω / k_{t 3} = (9 / (4 c^{2} μ)) ω \cdot 2 \sqrt{13} c / (9 ω) = \sqrt{13} / (2 c μ)$ , so $Y_{1}^{p} = Y_{2}^{p}$ exactly.

4.1step 3.1A1algebra

[The amplitudes, clause (ii)] By A1 and step 3.1, $r_{p} = (Y_{1}^{p} - Y_{2}^{p}) / (Y_{1}^{p} + Y_{2}^{p}) = 0$ and $t_{p} = 2 Y_{1}^{p} / (Y_{1}^{p} + Y_{2}^{p}) = 1$ . For the $s$ -channel, factoring the common positive scale $1 / (\sqrt{13} c μ)$ from numerator and denominator, $r_{s} = (2 - 9 / 2) / (2 + 9 / 2) = (- 5 / 2) / (13 / 2) = - 5 / 13$ and $t_{s} = 2 \cdot 2 / (2 + 9 / 2) = 4 / (13 / 2) = 8 / 13$ .

5.1step 4.1A1algebra

[Flux budget, clause (iii)] By A1, $R_{s} = ∣ r_{s} ∣^{2} = 25 / 169$ and $T_{s} = (Y_{2}^{s} / Y_{1}^{s}) ∣ t_{s} ∣^{2} = (9 / 4) (64 / 169) = 144 / 169$ , so $R_{s} + T_{s} = (25 + 144) / 169 = 169 / 169 = 1$ exactly; and in the TM channel $R_{p} = ∣ r_{p} ∣^{2} = 0$ , $T_{p} = (Y_{2}^{p} / Y_{1}^{p}) ∣ t_{p} ∣^{2} = 1$ , so $R_{p} + T_{p} = 1$ as well.

6.1givenstep 5.1A1∎

[Scope, clause (iv)] All quantities are declared model parameters and the computation is exact rational or quadratic-irrational algebra of the interface model; no measured refractive index, no absorption, no unequal-permeability interface, no unpolarised-beam statistics and no empirical claim is used. Since $r_{p} = 0$ and $r_{s} \neq 0$ at $θ_{B}$ , an incident wave with a nonzero TE component has a surviving reflected TE component; no statement about random or unpolarised beams is made. No choice principle is used.

The Brewster angle at an equal-permeability interface and the surviving $s$ -reflection

Example

Verification

Depends on

Dependency tree · 2 levels

Sources

The Brewster angle at an equal-permeability interface and the surviving ss-reflection

Example

Verification

Depends on

Dependency tree · 2 levels

Sources

The Brewster angle at an equal-permeability interface and the surviving $s$ -reflection