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Submitted on 1 Jan 1964
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Computation of optical characteristics of dielectric multilayers
Kazuo P. Miyake
To cite this version:
Kazuo P. Miyake. Computation of optical characteristics of dielectric multilayers. Journal de Physique, 1964, 25 (1-2), pp.255-257. �10.1051/jphys:01964002501-2025500�. �jpa-00205751�
255.
COMPUTATION OF OPTICAL CHARACTERISTICS OF DIELECTRIC MULTILAYERS Par KAZUO P. MIYAKE,
Institute for Optical Research, Kyôiku University, Tokyo.
Résumé. 2014 Un programme de calculateur électronique a été imagine, ce qui est commode pour calculer les différentes caractéristiques optiques de couches multiples diélectriques. Ce programme est utilisé pour trouver des séparatrices soit seules, soit comprises entre deux prismes, dans le visible, pour une incidence de 45°. On a trouvé que des couches multiples GHHLHLHLLG, GHHLHLHLG, GLLHLHLLG et GHHLHLHG sont utiles comme séparatrices placées entre
deux prismes et que des couches multiples ALHLHG et ALHLHLLG sont valables aussi comme
séparatrices. A et G représentent respectivement l’air (n = 1) et le verre (ng = 1,52) et L et H
des couches de MgF2 (n = 1,38) et ZnS (n = 2,30) respectivement, d’épaisseur optique 03BB/4
dans les conditions de travail.
Abstract. 2014 A program for an electronic computer has been worked out, which is convenient for calculating various optical characteristics of dielectric multilayers. This program is used to find achromatic half-prisms and half-mirrors in the visible wavelength region for 45° incidence. It has been found that multilayers GHHLHLHLLG, GHHLHLHLG, GLLHLHLLG and GHHLHLHG are useful as half-prisms, and multilayers ALHLHG and ALHLHLLG are also good as half-mirrors, where A and G represent air (n =1) and glass (ng =1. 52) respectively and L
and H layers of MgF2 (n = 1.38) and ZnS (n = 2.30) respectively, whose optical thickness is
equal to 03BB/4 in the working condition.
JOURNAL DE PHYSIQUE TOME 25, JANVIER-FÉVRIER 1964,
It is troublesome to compute the reflectivity of multilayers, except in special cases in which compo- nent layers have exclusively optical thicknesses of X/4 or X/2 for the central wavelength x.
Thanks to development of a high speed auto-
matic computer, this trouble has been overcome, and it is now possible to get useful informations on
the multilayer design.
The present paper describes a program for an electronic computer to compute optical charac-
teristics of multilayers and its applications to the multilayer design and the film thickness control.
It is to be noted in this connection that similar programs have been worked out by Laikin [1] and Berning [2].
Description of programs. - There are two essen- tial points of the present program which are as follows :
1) Computing optical characteristics such as
intensity reflectance, phase changes at reflection, amplitude ratio and relative phase difference bet-
ween p- and s-components, at a given angle of
incidence and given wavelength, and giving thereby
numerical data useful for the polarimetric analysis
of multilayers of given construction.
2) Plotting spectral reflection curves of multi-
layers while the layers are being deposited and supplying informations useful for the thickness control of each layer under deposition.
The computation is performed in the follwing
way: A transparent material is deposited on a
substrate such as glass or on the multilayer already deposited. During the evaporation, spectral cha-
racteristics can be obtained for each equal amount
of deposition of layers up to the specified thickness.
It is also possible to compute the optical charac-
teristics of the multilayers of which the uppermost-
layer thickness exceeds the prescribed value. If
the result of computation for a certain layer under deposition is not required, unnecessary compu- tation can be skipped by specifying the number of
variations of the film thickness as zero. Such
computations are repeated for each layer consti- tuting the whole multilayer, and the computation cycle will be completed after the characteristics of the complete multilayer with prescribed construc-
tion have been obtained.
The formula adopted for numerical computation
in the program is a well-known one in which the
multilayer is replaced by a virtual reflecting sur- face, and the computation is carried on by iterative
method. The number of layers must not exceed fifty. The refractive indices of the substrate, uppermost medium and ambient medium during
the evaporation may be specified at will. The computation of the spectral characteristic is effec- ted by changing the wavenumber instead of the
wavelength.
The line printer prints out in one line the number
of layers under computation, the thickness of the
uppermost layer, the wavenumber, the intensity
reflectance and than 8 for both p- and s-compo- nents respectively, the amplitude ratio and the
relative phase difference between two polarized components. Spectral reflection curves can be
plotted automatically by the line printer.
A great deal of computation has been performed by this program to obtain half-prisms and half-
mirrors having flat spectral characteristics in the visible wavelength region.
Design of achromatic half-prism and half-
mirrors. - As the number of alternating X/4-Iayers
of high and low refractive indices increases, the intensity reflection for the central wavelength
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01964002501-2025500
256
increases, but the wavelength region in which spectral reflection curve can be considered flat becomes narrower. To obtain an achromatic mul-
tilayer, this defect must be overcome.
It is well known that a X/2-layer behaves for the
wavelength À as if the layer did not exist. There- fore, if the h/2-layer is located about the middle of alternating h/4-layers, the reflectivity of this multilayer becomes very low for the wavelength X
and the multilayer has a narrow transmission band there. It will be undesirable that a À/2- layer should exist in alternating layers if a flat spectral chatacteristic is required.
A X/2-layer has the same effect for the wave-
length 2À and (2/3) X as a quarter-wavelength layer,
and contributes to increase reflection at these wave-
lengths. Therefore, the reflectivity decreases at
the central wavelength and increases on both sides
of it.
With these two facts taken into considerations, investigations have been made systematically of
the optical characteristics of multilayers which
have a X/2-layer on either one or both sides of
alternating À/4-layers. The present program is very convenient for dealing with problems of this
sort.
As the result, the following multilayers have been
found good as half-prisms. Spectral reflection
curves of these half-prisms are shown in figure 1.
It has also been found that the three-layers prism of GLLHLG has an achromatic characte- ristic and shows the reflectivities of nearly 50 %
and zero for s- and p-components respectively.
Its reflection curve is shown in figure 2. This prism is useful as a half-prism when the light is linearly polarized.
The allowable aperture angle of the above-men-
tioned prisms is :1: ca. 50 from the normal in the air.
A half-mirror has a different condition, because
it comes in contact with air at its surface. As the refractive index of air is unity and lower than those of ZnS and MgF2, the uppermost ZnS- and MgF 2- layers behave differently from each other. A À/2- layer of ZnS in contact with air has a similar but
more pronounced effect than that in half-prisms,
because of the difference in refractive indices between air and ZnS being much larger than that
between glass and ZnS, while a a/4-layer of MgF2 exposed to air has the same effects as x /2-layers in half-prisms. As its optical thickness is half of the
latter, the rate of change in phase retardation with wavenumber due to optical thickness is half that
FIG. 1. - Spectral reflection curves of half-prisms.
FIG. 2. -Spectral reflection curves of multilayerGLLHLG.
of Ã/2-layer. This property is very effective in
making the spectral characteristic flat in half- mirrors.
As a result of systematic computations, the
257
following multilayers have been found to be good
half-mirrors..
Reflection curves of these mirrors are plotted in figure 3.
FIG. 3. - Spectral reflection curves of half-mirror.
Data for thickness control. - Data for control-
ling the film thickness during deposition of these multilayers can be obtained easily. The three- layer GLLHLG is taken as an example. The reflectivity for the Hg e-line (5 461 A) incident
at 450 is plotted in figure 4 with the film thickness
as abscissa.
The reflectivity of the multilayer can also be
calculated easily in a series of computations when
the thickness of the layer under deposition exceeds
FIG. 4. - Reflectivity for Hg e-line as function
of film thickness.
FIG, 5. - Spectral reflection curves
of multilayer GLLHLG during deposition.
the prescribed value. The curves for this case are
represented by dotted curves.
Spectral reflection curves of multileyers during deposition can be plotted with the film thickness of the uppermost layer as a parameter. Those for the normal incidence are shown in figure 5.
This feature of the program is useful for control-
ling the film thickness by observation of the spec- tral reflection curve.
REFERENCES
[1] LAIKIN (M.), J. Opt. Soc. Am., 1960, 50, 721. [2] BERNING (J. A.) and BERNING (P. H.), J. Opt. Soc.
Am., 1960, 50, 813.
