The Reflection of Light from Glass with Double and Multiple Films

Antonin Vašiček

doi:10.1364/JOSA.37.000623

Journal of the Optical Society of America
Vol. 37,
Issue 8,
pp. 623-634
(1947)
•https://doi.org/10.1364/JOSA.37.000623

The Reflection of Light from Glass with Double and Multiple Films

Antonin Vašiček

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Antonin Vašiček, "The Reflection of Light from Glass with Double and Multiple Films," J. Opt. Soc. Am. 37, 623-634 (1947)

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Abstract

For the reflection of light from a thin film on glass, that is, from two boundary surfaces, the generalized Fresnel equations are:

\begin{array}{l} r_{p} e^{i δ_{p}} = \frac{{r_{p}}^{'} + {r_{p}}^{″} e^{- i x}}{1 + {r_{p}}^{'} {r_{p}}^{″} e^{- i x}}, & r_{s} e^{i δ_{s}} = \frac{{r_{s}}^{'} + {r_{s}}^{″} e^{- i x}}{1 + {r_{s}}^{'} {r_{s}}^{″} e^{- i x}} \end{array} .

This also can be interpreted as a reflection from a single surface of a hypothetical medium, the reflected components having amplitudes r_p and r_s and phases δ_p and δ_s. Similarly, the reflection from two thin films on glass, that is, from three boundaries, can be reduced to the case of a single film which in turn can be reduced to the reflection from a single surface. This process can be carried for any number of thin films on glass. The normal incidence problem is considered for one, two, and several films and limiting cases of interest in optical practice are calculated. The reflection formulae are simplified if multiple reflections in the films are disregarded. In conclusion the results for two thin films on glass are given. These may be used in practice to decrease or completely extinguish reflection, as well as to increase it.

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The reflecting power $ρ = {(\frac{{n_{1}}^{2} n - {n_{2}}^{2}}{{n_{1}}^{2} n + {n_{2}}^{2}})}^{2}$ in percent
$\begin{matrix} n = 1.6000, & n_{1} d_{1} = n_{2} d_{2} = \frac{λ}{4} \end{matrix}$

	n₁
n₂	1.200	1.400	1.600	1.800	2.000	2.200	2.400	2.600
1.200	5.325	13.737	23.018	31.947	40.025	47.130	53.251	58.524
1.400	0.651	5.325	12.440	20.366	28.207	35.527	42.152	48.049
1.51789	0	—	—	—	—	—	—	—
1.600	0.277	1.023	5.325	11.482	18.367	25.312	31.947	38.097
1.77088	—	0	—	—	—	—	—	—
1.800	2.850	0.027	1.362	5.325	10.745	16.814	23.018	29.051
2.000	7.238	1.466	0.014	1.662	5.325	10.171	15.577	21.164
2.02386	—	—	0	—	—	—	—	—
2.200	12.601	4.564	0.693	0.118	1.926	5.325	9.692	14.570
2.27684	—	—	—	0	—	—	—	—
2.400	18.367	8.700	2.850	0.277	0.277	2.159	5.325	9.304
2.52982	—	—	—	—	0	—	—	—
2.600	24.169	13.411	3.623	1.741	0.075	0.460	2.363	5.325

n	The reflecting power ρ = 0
	n₂ = 1.80		n₂ = 2.00		n₂ = 2.20
	n₁ simple refl.	n₁ multiple refl.	n₁ simple refl.	n₁ multiple refl.	n₁ simple refl.	n₁ multiple refl.
1.50	1.47232	1.46969	1.63887	1.63299	1.80657	1.79629
1.55	1.44798	1.44579	1.61180	1.60644	1.77681	1.76708
1.60	1.42477	1.42303	1.58593	1.58114	1.74835	1.73925
1.65	1.40260	1.40130	1.56119	1.55700	1.72108	1.71270
1.70	1.38139	1.38054	1.53750	1.53393	1.69495	1.68732
1.75	1.36109	1.36067	1.51480	1.51186	1.66990	1.66304

n₁	The reflecting power ρ = 0
	n = 1.51824, n₂ = 2.0000
	x₁	x₂	x₁+x₂
1.400	133° 04′ 17″	297° 13′ 48″	430° 18′ 05″
1.500	136° 48′ 20″	271° 15′ 00″	408° 03′ 20″
1.600	156° 00′ 30″	223° 15′ 23″	379° 15′ 53″
1.6231	180° 00′ 00″	180° 00′ 00″	360° 00′ 00″

The reflecting power $ρ = {(\frac{{n_{1}}^{2} n - {n_{2}}^{2}}{{n_{1}}^{2} n + {n_{2}}^{2}})}^{2}$ in percent
$\begin{matrix} n = 1.6000, & n_{1} d_{1} = n_{2} d_{2} = \frac{λ}{4} \end{matrix}$

	n₁
n₂	1.200	1.400	1.600	1.800	2.000	2.200	2.400	2.600
1.200	5.325	13.737	23.018	31.947	40.025	47.130	53.251	58.524
1.400	0.651	5.325	12.440	20.366	28.207	35.527	42.152	48.049
1.51789	0	—	—	—	—	—	—	—
1.600	0.277	1.023	5.325	11.482	18.367	25.312	31.947	38.097
1.77088	—	0	—	—	—	—	—	—
1.800	2.850	0.027	1.362	5.325	10.745	16.814	23.018	29.051
2.000	7.238	1.466	0.014	1.662	5.325	10.171	15.577	21.164
2.02386	—	—	0	—	—	—	—	—
2.200	12.601	4.564	0.693	0.118	1.926	5.325	9.692	14.570
2.27684	—	—	—	0	—	—	—	—
2.400	18.367	8.700	2.850	0.277	0.277	2.159	5.325	9.304
2.52982	—	—	—	—	0	—	—	—
2.600	24.169	13.411	3.623	1.741	0.075	0.460	2.363	5.325

n	The reflecting power ρ = 0
	n₂ = 1.80		n₂ = 2.00		n₂ = 2.20
	n₁ simple refl.	n₁ multiple refl.	n₁ simple refl.	n₁ multiple refl.	n₁ simple refl.	n₁ multiple refl.
1.50	1.47232	1.46969	1.63887	1.63299	1.80657	1.79629
1.55	1.44798	1.44579	1.61180	1.60644	1.77681	1.76708
1.60	1.42477	1.42303	1.58593	1.58114	1.74835	1.73925
1.65	1.40260	1.40130	1.56119	1.55700	1.72108	1.71270
1.70	1.38139	1.38054	1.53750	1.53393	1.69495	1.68732
1.75	1.36109	1.36067	1.51480	1.51186	1.66990	1.66304

Abstract

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Figures (6)

Tables (3)

Equations (94)

Journal of the Optical Society of America