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Total internal reflection: decaying transmitted field, vanishing normal flux and the reflection phase

Physical scope

Deterministic example of total internal reflection in the lossless interface model for declared indices n_1=3/2, n_2=1 at sin(theta_i)=4/5: the unit-modulus reflection factor with its phase, the evanescent transmitted field with decay factor e^{-beta x_3}, the vanishing normal and positive tangential averaged fluxes. Excluded: measured indices, absorption, information-speed or signal-propagation claims, and every empirical statement.

Example

Adopt the setting of Oblique-incidence TE and TM Fresnel amplitudes, the flux budget, Brewster's angle and total internal reflection with equal permeabilities $μ_{1} = μ_{2} = : μ > 0$ and declared indices $n_{1} = 3 / 2$ , $n_{2} = 1$ (declared model parameters, not measurements), choose the tangential wavevector $κ = (6 ω / (5 c)) e_{1}$ , so $\hat{κ} = e_{1}$ and $\hat{y} = n \times \hat{κ} = e_{2}$ , and take the transverse-electric polarisation $e_{j} = E_{j} \hat{y}$ . Let the incidence angle be given by $\sin θ_{i} = 4 / 5$ , which exceeds the critical angle $θ_{c}$ with $\sin θ_{c} = n_{2} / n_{1} = 2 / 3$ , so $∣ κ ∣ = K_{1} \sin θ_{i} = (6 / 5) (ω / c) > (ω / c) = K_{2}$ . Then with $β = \sqrt{∣ κ ∣^{2} - K_{2}^{2}} = (\sqrt{11} / 5) (ω / c)$ and $k_{13} = (ω / c) (3 / 2) (3 / 5) = (9 / 10) (ω / c)$ : (i) [Reflection] $Y_{1}^{s} = 9 / (10 μ c)$ , $Y_{2}^{s} = i \sqrt{11} / (5 μ c)$ and $r_{s} = \frac{9 / 10 - i \sqrt{11} / 5}{9 / 10 + i \sqrt{11} / 5}, ∣ r_{s} ∣ = 1, t_{s} = \frac{9 / 5}{9 / 10 + i \sqrt{11} / 5}, ∣ t_{s} ∣^{2} = \frac{324}{125};$ the reflection factor is the phase $r_{s} = e^{- 2 i φ}$ with $φ = \arctan ⁣ (2 \sqrt{11} / 9) \approx 0.63514703$ (about ${36.39124}^{\circ}$ ), i.e. the reflected wave acquires the phase $- 2 φ \approx - 1.27029406$ (about $- {72.78249}^{\circ}$ ); (ii) [Transmitted field] the transmitted field is $E_{t} (x, t) = e^{- β x_{3}} Re (E_{t} \hat{y} e^{i (∣ κ ∣ x_{1} - ω t)})$ with $β = (\sqrt{11} / 5) (ω / c)$ : it decays by the factor $e^{- \sqrt{11} / 5} \approx 0.51514$ when $x_{3}$ increases by $c / ω$ , its period-averaged normal flux vanishes identically, $⟨ S_{t} \cdot n ⟩ \equiv 0$ , while its tangential averaged flux is positive, $⟨ S_{t} \cdot \hat{κ} ⟩ = e^{- 2 β x_{3}} \frac{(6 / 5) (ω / c) ∣ E_{t} ∣^{2}}{2 μ ω} = e^{- 2 β x_{3}} \frac{3 ∣ E_{t} ∣^{2}}{5 μ c}$ ; (iii) [Flux budget] $∣ r_{s} ∣^{2} + (Re Y_{2}^{s} / Y_{1}^{s}) ∣ t_{s} ∣^{2} = 1 + 0 = 1$ , so the whole incident normal averaged flux is reflected, while the evanescent field carries a tangential averaged flux only; (iv) [Scope] the example is an exact deterministic computation of the lossless interface model for declared parameters; it reports no measurement, and the presence of a tangential evanescent flux is not a claim about information speed or signal propagation.

Verification

technique · direct substitution in the oblique-incidence and medium plane-wave theorems with exact rational and quadratic-irrational arithmetic, and an explicit phase computation via Euler's form

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 $μ_{1} = μ_{2} = μ > 0$ , $n_{1} = 3 / 2$ , $n_{2} = 1$ , the tangential wavevector $κ = (6 ω / (5 c)) e_{1}$ , the transverse-electric data $e_{j} = E_{j} \hat{y}$ with $\hat{y} = e_{2}$ , and the incidence data $\sin θ_{i} = 4 / 5$ .

[A1] The oblique-incidence theorem Oblique-incidence TE and TM Fresnel amplitudes, the flux budget, Brewster's angle and total internal reflection: clause (a) and (d) give $Y_{1}^{s} = k_{13} / (μ_{1} ω)$ , $Y_{2}^{s} = k_{t 3} / (μ_{2} ω)$ , the amplitude formulas, and in the evanescent branch $∣ r_{s} ∣ = 1$ , $∣ t_{s} ∣^{2} = 4 (Y_{1}^{s})^{2} / ((Y_{1}^{s})^{2} + ∣ Y_{2}^{s} ∣^{2}) > 0$ together with the flux budget; clause (d) also records that the transmitted field decays and its normal averaged flux vanishes.

[A2] The medium plane-wave structure Plane-wave fields in a homogeneous Ohmic medium: amplitude conditions, dispersion, averaged flux and the evanescent branch: the transmitted evanescent field of clause (e) with the substitution $n^{'} = n$ , $κ_{0} = κ$ : the exact factorization, the vanishing normal component $⟨ S \cdot n^{'} ⟩ \equiv 0$ and the positive tangential averaged flux $⟨ S \cdot {\hat{κ}}_{0} ⟩ = ∣ κ_{0} ∣ ∣ e ∣^{2} e^{- 2 β n^{'} \cdot x} / (2 μ ω)$ for the transverse-electric data $e = E_{0} (n^{'} \times {\hat{κ}}_{0})$ .

[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 the common permeability. The selected evanescent wavevectors in Oblique-incidence TE and TM Fresnel amplitudes, the flux budget, Brewster's angle and total internal reflection give $k_{13}^{2} = K_{1}^{2} - ∣ κ ∣^{2}$ , $β^{2} = ∣ κ ∣^{2} - K_{2}^{2}$ and $k_{t 3} = i β$ .

[F2] Complex arithmetic and modulus (Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive): $\overline{z + w} = \overline{z} + \overline{w}$ , $\overline{z w} = \overline{z} \overline{w}$ , $∣ z w ∣ = ∣ z ∣ ∣ w ∣$ , $∣ z ∣^{2} = z \overline{z}$ and $∣ z ∣ \geq 0$ with $∣ z ∣ = 0$ only for $z = 0$ ; in particular a complex number of modulus one is nonzero and the sum of a complex number and its conjugate is twice its real part.

[F3] Trigonometry and phases: $\arctan$ is the principal inverse tangent with $\tan (\arctan y) = y$ (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 $1 + \tan^{2} x = \sec^{2} x$ (Pythagorean and parity identities for all six trigonometric functions on their natural domains); cosine is strictly decreasing on $[0, π]$ with $\cos (π / 2) = 0$ (Signs, monotonicity intervals, and ranges of sine and cosine, The zero sets of sine and cosine and the least positive common period 2 pi); and for real $y$ , $\exp (i y) = \cos y + i \sin y$ with $∣ \exp (i y) ∣ = 1$ ( $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ ).

1.1givenA1F1algebra

[The geometry and the evanescent data] With $κ = (6 ω / (5 c)) e_{1}$ one has $\hat{κ} = e_{1}$ , $∣ κ ∣ = (6 / 5) (ω / c)$ and $\hat{y} = n \times \hat{κ} = e_{3} \times e_{1} = e_{2}$ ; $K_{1} = (3 / 2) (ω / c)$ , $K_{2} = ω / c$ by [F1], so $∣ κ ∣ = K_{1} \sin θ_{i}$ with $\sin θ_{i} = 4 / 5$ , and $∣ κ ∣ > K_{2}$ because $6 / 5 > 1$ . Hence the evanescent branch applies: $β = \sqrt{∣ κ ∣^{2} - K_{2}^{2}} = \sqrt{(36 / 25 - 1)} (ω / c) = \sqrt{11 / 25} (ω / c) = (\sqrt{11} / 5) (ω / c)$ and $k_{13} = \sqrt{K_{1}^{2} - ∣ κ ∣^{2}} = \sqrt{(9 / 4 - 36 / 25)} (ω / c) = \sqrt{81 / 100} (ω / c) = (9 / 10) (ω / c)$ ; the critical angle satisfies $\sin θ_{c} = n_{2} / n_{1} = 2 / 3 < 4 / 5 = \sin θ_{i}$ , both angles being in $(0, π / 2)$ .

2.1step 1.1A1F2algebra

[Admittances and reflection data, clause (i)] By A1 and [F1], $Y_{1}^{s} = k_{13} / (μ_{1} ω) = 9 / (10 μ c)$ , $Y_{2}^{s} = k_{t 3} / (μ_{2} ω) = i β / (μ ω) = i \sqrt{11} / (5 μ c)$ and $t_{s} = 2 Y_{1}^{s} / (Y_{1}^{s} + Y_{2}^{s}) = (9 / (5 μ c)) / ((9 / 10 + i \sqrt{11} / 5) / (μ c)) = (9 / 5) / (9 / 10 + i \sqrt{11} / 5)$ ; the numerator $9 / 10 - i \sqrt{11} / 5$ of $r_{s} = (Y_{1}^{s} - Y_{2}^{s}) / (Y_{1}^{s} + Y_{2}^{s})$ is the complex conjugate of its denominator, so $∣ r_{s} ∣ = 1$ by [F2] (and A1). For the transmitted modulus, $∣ t_{s} ∣^{2} = 4 (Y_{1}^{s})^{2} / ((Y_{1}^{s})^{2} + ∣ Y_{2}^{s} ∣^{2})$ by A1; computing $(Y_{1}^{s})^{2} = 81 / (100 μ^{2} c^{2})$ and $∣ Y_{2}^{s} ∣^{2} = 11 / (25 μ^{2} c^{2}) = 44 / (100 μ^{2} c^{2})$ gives $∣ t_{s} ∣^{2} = (324 / 100) / (125 / 100) = 324 / 125$ .

2.2step 1.1A1A2algebra

[The transmitted field and its fluxes, clause (ii)] By $∣ t_{s} ∣^{2}$ -normalised data we take the transmitted electric amplitude $E_{t} \hat{y}$ with $E_{t} = t_{s} E_{i}$ ; the evanescent structure of [A2] applied with $n^{'} = n = e_{3}$ , $κ_{0} = κ$ , $β$ as in step 1.1 and $e = E_{t} \hat{y} = E_{t} (n^{'} \times {\hat{κ}}_{0})$ gives exactly $E_{t} (x, t) = e^{- β x_{3}} Re (E_{t} \hat{y} e^{i (∣ κ ∣ x_{1} - ω t)})$ , the identity $⟨ S_{t} \cdot n ⟩ \equiv 0$ , and the tangential averaged flux $⟨ S_{t} \cdot \hat{κ} ⟩ = e^{- 2 β x_{3}} ∣ κ ∣ ∣ E_{t} ∣^{2} / (2 μ ω)$ ; substituting $∣ κ ∣ = (6 / 5) (ω / c)$ gives $⟨ S_{t} \cdot \hat{κ} ⟩ = e^{- 2 β x_{3}} (6 / 5) (ω / c) ∣ E_{t} ∣^{2} / (2 μ ω) = e^{- 2 β x_{3}} 3 ∣ E_{t} ∣^{2} / (5 μ c)$ . The decay over a distance $c / ω$ is the factor $e^{- β c / ω} = e^{- \sqrt{11} / 5}$ , whose decimal display in clause (ii) is the stated rounding.

3.1step 2.1F2F3algebra

[The reflection phase, clause (i)] Put $Y_{1} : = 9 / 10$ and $β^{'} : = \sqrt{11} / 5$ in the units $1 / (μ c)$ and $α : = β^{'} / Y_{1} = 2 \sqrt{11} / 9$ , so that $r_{s} = (1 - i α) / (1 + i α)$ ; let $φ : = \arctan α \in (0, π / 2)$ and $w : = 1 + i α$ , $∣ w ∣ = \sqrt{1 + α^{2}}$ . By [F3], $\tan φ = α$ and $1 + \tan^{2} φ = \sec^{2} φ$ , so $\cos^{2} φ = 1 / (1 + α^{2}) = 1 / ∣ w ∣^{2}$ ; since $\cos φ > 0$ (cosine is strictly decreasing on $[0, π]$ with $\cos (π / 2) = 0$ and $φ \in (0, π / 2)$ ), $\cos φ = 1 / ∣ w ∣$ and $\sin φ = α \cos φ = α / ∣ w ∣$ . Euler's form of [F3] then gives $w = ∣ w ∣ (\cos φ + i \sin φ) = ∣ w ∣ e^{i φ}$ and $\overline{w} = ∣ w ∣ e^{- i φ}$ by [F2], so $r_{s} = \overline{w} / w = e^{- 2 i φ}$ . The decimal values displayed in clause (i) are the stated roundings of $φ = \arctan (2 \sqrt{11} / 9)$ and of the reflected phase $- 2 φ$ .

4.1givenstep 2.1step 2.2A1F2∎

[Flux budget and scope, clauses (iii)-(iv)] By A1, $∣ r_{s} ∣^{2} + (Re Y_{2}^{s} / Y_{1}^{s}) ∣ t_{s} ∣^{2} = 1$ ; here $Re Y_{2}^{s} = 0$ by step 2.1, so the identity reads $1 + 0 = 1$ : the whole incident normal averaged flux is reflected, while the tangential averaged flux of step 2.2 is positive and transports no normal flux. All quantities are declared model parameters; the computation is deterministic classical interface algebra, reports no measurement, and neither the tangential evanescent flux nor the reflection phase is a claim about information speed, signal propagation or empirical data. No choice principle is used.

Total internal reflection: decaying transmitted field, vanishing normal flux and the reflection phase

Example

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