Z = E / H

; for the stated nonmagnetic media it is

Z_{0} / n

. A reflection coefficient is a reflected-to-incident complex amplitude ratio with an explicitly chosen field orientation. With this reflection-amplitude sign convention, using the question's formula for incidence inside silicon gives

r = \frac{Z _{0} /3.5 - Z _{0}}{Z _{0} /3.5 + Z _{0}} = - \frac{5}{9}, R = ∣ r ∣^{2} = \frac{25}{81} .

(1)

With a common tangential electric-field direction, the usual electric-field Fresnel equations instead give

r_{E} = (Z_{2} - Z_{1}) / (Z_{2} + Z_{1}) = + 5/9

for this incidence. The printed formula corresponds to the opposite reflected-field sign convention, for example the magnetic amplitude convention. It gives the same reflectance and the same round-trip factor

r^{2}

Each extra internal round trip multiplies the transmitted amplitude by

r^{2} e^{i δ}

, with

δ = 4 πn d / λ

. A geometric series therefore gives

A_{out} = A_{first} \sum_{j = 0}^{\infty} (r^{2} e^{i δ})^{j} = \frac{A _{first}}{1 - r ^{2} e ^{i δ}}, I \propto \frac{1}{1 + R ^{2} - 2 R cos δ} .

(2)

This is Fabry–Pérot transmission from multiple reflections. Assuming lossless silicon with wavelength-independent

n

, the full intensity transmittance between air regions is

T = \frac{( 1 - R ) ^{2}}{( 1 - R ) ^{2} + 4 R s i n ^{2} ( δ /2 )} .

(3)

Maxima have

2 n d / λ = j

, minima

2 n d / λ = j + \frac{1}{2}

. Here

2 n d = 7000 nm

, so within the specified tuning range the only maximum is $1400 nm$ , and the two minima are $1555.6 nm$ and $1272.7 nm$ . The contrast is

\frac{I _{m a x}}{I _{m i n}} = (\frac{1 + R}{1 - R})^{2} = (\frac{53}{28})^{2} = 3.583.

(4)

Figure 1.
Transmission contrast and resonance narrowing with reflectivity
. Original ideal lossless symmetric-etalon curves. The coating example uses chosen R=0.90, unchanged optical thickness and zero additional round-trip phase. **Details**.

Identical highly reflecting coatings increase the finesse: transmission maxima become narrower and the intensity between them falls. Ideal symmetric lossless coatings retain

T_{m a x} = 1

; absorption or unequal mirrors reduce it. The illustrated maxima stay fixed only because additional reflection wave phases are held unchanged. Real coating wave phases can also shift the resonances.