Basics of Optics of multilayer systems

104 Pages·0.664 MB·English

by  Furman Sh.A., Tikhonravov A.V.

From the book
Basics of
OPTICS OF MULTILAYER SYSTEMS
Sh. A. Furman
A.V. Tikhonravov
Edition Frontieres, Gif-sur-Yvette,
1992
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1 SPECTRAL CHARACTERISTICS OF MULTILAYER
COATINGS: THEORY 4
1.1 Electromagneticfleldinlayeredmedia;spectralcharacteristicsof
layered media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.1 Layered medium electromagnetic fleld equations . . . . . 4
1.1.2 Amplitude transmittance and re(cid:176)ectance of the layered
medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.1.3 The Fresnel formulas . . . . . . . . . . . . . . . . . . . . . 15
1.1.4 Other ways of determining amplitude transmittance and
re(cid:176)ectance of the layered medium. . . . . . . . . . . . . . 15
1.1.5 Transmittance, re(cid:176)ectance and absorptance . . . . . . . . 17
1.2 Principal recurrent methods for calculating layered media spec-
tral coe–cients . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.2.1 Matrix calculating method . . . . . . . . . . . . . . . . . 18
1.2.2 Description of optical coating and presentation of calcu-
lation results . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.2.3 Recurrent formulas for the amplitude transmittance and
re(cid:176)ectance . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.2.4 Approximate formulas . . . . . . . . . . . . . . . . . . . . 26
1.2.5 Other recurrent methods . . . . . . . . . . . . . . . . . . 27
1.2.6 The admittance phase plane. . . . . . . . . . . . . . . . . 30
1.3 Some common properties of multilayer coatings . . . . . . . . . . 36
1.3.1 The relation between the characteristic matrix and spec-
tral coe–cients of a multilayer coating . . . . . . . . . . . 37
1.3.2 Relations between spectral characteristics for the direct
and reverse waves . . . . . . . . . . . . . . . . . . . . . . 41
1.3.3 Transmittance and re(cid:176)ectance of the combination of two
multilayer subsystems . . . . . . . . . . . . . . . . . . . . 43
1.4 The multilayer coatings optical properties: impact of variations
in the layer parameters. . . . . . . . . . . . . . . . . . . . . . . . 46
1.4.1 Spectral coe–cients derivatives with respect to layer pa-
rameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
1.4.2 Impact of the layer parameter errors on the multi-
layercoatingsspectralcoe–cients: probabilityassessment
method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
1.4.3 Multilayer coating spectral properties: in(cid:176)uence of sepa-
rate layer parameter variations . . . . . . . . . . . . . . . 53
1
1.5 Multilayer periodic systems. Quarter-wave dielectric mirrors . . . 55
1.5.1 Characteristic matrix of a periodic multilayer system . . . 55
1.5.2 Quarter-wave mirror properties at the central wavelength 58
1.5.3 Width of the quarter-wave mirror high-re(cid:176)ectance zones . 61
1.5.4 Principalpropertiesofaquarter-wave dielectric mirrorin
the proximity to the central wave length . . . . . . . . . . 65
1.5.5 Dielectric mirrors on a metal substrate . . . . . . . . . . . 68
1.6 Approximate formulas for dielectric narrow bandpass fllters . . . 70
1.6.1 Commonexpressionsforspectralcoe–cientsoffllter-type
systems near central wavelength . . . . . . . . . . . . . . 71
1.6.2 Research of narrow bandpass fllters with a half-wave cen-
tral layer . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
1.7 Synthesis methods based on merit function optimization . . . . . 78
1.7.1 Merit function selection . . . . . . . . . . . . . . . . . . . 79
1.7.2 Merit function optimization . . . . . . . . . . . . . . . . . 81
1.7.3 General pattern of the synthesis and the problem of the
starting design choice . . . . . . . . . . . . . . . . . . . . 86
1.8 Synthesis method based on the needle-like variations of the re-
fractive index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
1.8.1 Needle-like variations of the refractive index . . . . . . . . 92
1.8.2 The principal idea of the method . . . . . . . . . . . . . . 96
1.8.3 Description of the method . . . . . . . . . . . . . . . . . . 99
2
Preface
Multilayer optical coatings are successfully employed in various flelds, such as
optical and scientiflc instrumentation manufacturing, astronomy, spectroscopy,
medicine,etc. Agreatnumberoflaboratoriesandexpertsareinvolvedindesign-
ing and manufacturing modern optical coatings. At present, in order to receive
best results one needs to be aware of all aspects of design and production of
multilayer optical coatings. The goal of this book is to provide a comprehen-
sive outlook on modern problems in multilayer optics. A characteristic feature
of the book is an attempt to make it useful for a wide range of experts from
specialists of practical applications to specialists elaborating the theory of thin
fllm coatings.
3
Chapter 1
SPECTRAL
CHARACTERISTICS OF
MULTILAYER
COATINGS: THEORY
1.1 Electromagnetic fleld in layered media;
spectral characteristics of layered media
It is essential to begin the section with deriving layered medium fleld equations
from the Maxwell equations as sometimes errors occur in calculating due to an
inconsistent use of some flnal formulas. It is evident from the following that
a change of the sign in the time exponential factor results in the change of all
values for complex conjugates. Misunderstanding is sometimes caused by the
fact that difierent authors use formulas obtained with difierent choice of the
time exponential factor. Also, introducing the re(cid:176)ectance through a magnetic
or an electric fleld results in difierent signs in the flnal re(cid:176)ectance formulas.
Knowledgeoftheprincipalstepsinderivingformulashelpstoeliminatepossible
errors.
1.1.1 Layered medium electromagnetic fleld equations
Fig. 1.1 presents the basic physical model dealt with in studying multilayer
optical coatings. Two homogeneous semi-inflnite isotropic media are separated
byasetofplaneparallelisotropiclayers,non-limitedineitherdirections,whose
permittivityandconductivitydependononespatialcoordinateperpendicularto
the boundary between layers and media. A plane electromagnetic wave incites
from the flrst medium to the boundary. This causes a plane re(cid:176)ected wave in
4
Fig. 1.1. Multilayer optical coating model.
the flrst medium and a plane transmitted wave in the second one.
Thegivenmodeladequatelydescribesmultilayercoatingsasthewidthofthe
coatingalwaysexceedsmanytimesthewavelengthoftheincidentwaveandthe
total thickness of the coating. Further, we consider these both media and the
layers non-magnetic (magnetic permeability is equal to 1) and assume an ab-
sence of volume charges, both assumptions are quite standard in the multilayer
optics. These assumptions admitted, the Maxwell equations in the medium are
as follows:
1@~ "@~ 4…(cid:190)
¢ ~ = H; ¢ ~ = E + ~: (1.1.1)
£E ¡c @t £H c @t c E
Here ~ is an electric vector, ~ is a magnetic vector, " is permittivity, (cid:190) is
E H
electrical conductivity, c is the velocity of light.
The ~; ~ fleldsatisflestheequations(1.1.1)atthe"and(cid:190) continuitypoints.
E H
At the " and (cid:190) discretion points, i.e., at the boundaries between the layers ~
and H~ tangential components ought to be continuous. A study of the fleld inEa
layered medium, in efiect, comes down to the solution of equation (1.1.1) with
specifled boundary conditions when the fleld is excited by an incident plane
wave.
Letusdenotethespacevariable,perpendiculartotheboundaryoflayers,as
z and direct the z axis outwardly, to where the incident wave (Fig. 1.1) comes
from. Let the incident wave be a plane monochromatic !-frequency wave. Let
us assume a time dependence of the fleld be exp(i!t) 1 and hold
1This selection of the time factor gives the sign of the imaginary part of the complex
refractive index most frequently used in multilayer optics; it will feature the negative sign.
5
~ =E~ exp(i!t); ~ =H~ exp(i!t):
E H
Here vectors E~ and H~ depend only on the spatial variables.
Substituting these expressions into (1.1.1) and further reducing by exp(i!t)
exponential factor, we obtain
! ! 4…(cid:190)
¢ E~ = i H~; ¢ H~ =i "E~ + E~: (1.1.2)
£ ¡ c £ c c
Typically the wavelength of incident wave in vacuum is used as a spectral
parameter characterizing a monochromatic wave:
c
‚= :
2…!
Another convenient spectral parameter is the wavenumber of the incident
wave in vacuum k =!=c, related to the wavelength by the equation:
2…
k = :
‚
It is also convenient for further use to introduce a complex permittivity of
the medium deflned as
4…(cid:190)
"~=" i :
¡ c
In conformity with the previously arranged direction of the z-axis, the com-
plexpermittivityturnstobeafunctionofavariablez. Takingintoaccountthe
above assignations, equations (1.1.2) will take the following form:
¢ E~ = ikH~; ¢ H~ =ik~"(z)E~: (1.1.3)
£ ¡ £
Let us decompose the electromagnetic fleld of the wave into two compo-
nents, the flrst being the S-component, with the electric vector perpendicular
to the incidence plane, the other one, the P-component, featuring the plane-
parallel electric vector. Let us select a coordinate system so that the x-axis be
perpendicular to the plane of incidence and the y-axis be in this plane.
Let us denote the angle of incidence as (cid:176) , the permittivity of the outer
a
space where the incident wave comes from as " ; the permittivity of the second
a
outer space, traditionally referred to as substrate, as " . Let us then consider
s
equations for the S-component and the P-component of the fleld separately.
The S-polarization case. In this case, the electric vector E~ has only one
component E difierent from 0.
x
Fig. 1.2showstheyz-cross-sectionofthelayeredstructureinquestion. The
x-axis is directed backwards, so the E~ vector is perpendicular to the yz-plane,
away from the observer. The scalar form of the vector equations (1.1.3) gives:
H =0; @E =@z = ikH ; @E =@y =ikH ;
x x y x z
¡
As is seen from below, the change of the sign before i!t is equivalent to the substitute of
the complex values obtained below for the complex conjugates. Note, that in a number of
studies,includingthewidelyknownbook"PrinciplesofOptics"byM.BornandE.Wolf,the
exponentialfactortakestheformofexp(¡i!t). Sothecomplexrefractiveindexinthatbook
hasapositiveimaginarypart.
6
Fig. 1.2. Electric and magnetic vectors orientation in the S- and
P-polarization cases.
@H =@y @H =@z =ik"~(z)E ;
z y x
¡
@H =@x=0; @H =@x=0: (1.1.4)
z y
The flrst equation shows that the magnetic vector has only H and H
y z
components difierent from 0. Thus, the magnetic vector lies in the yz-plane,
i.e., in the incidence plane. It is an evidence of the crosswise fleld of the plane
electromagnetic wave, with the electric, the magnetic and the wave vectors
forming the right-hand triple. It follows from the two other equations that the
magnetic vector, hence, the electric vector, as well, depend only on y and z.
Substituting H and H from the second and third equations into the fourth
y z
one, we obtain:
@2E @2E
2 + x +k2"~(z)E =0: (1.1.5)
@y2 @z2 x
Letusapplythevariableseparationmethodtotheequation, inotherwords
let us seek its solution in the form:
E (y;z)=u(z)g(y):
x
By substituting this expression into (1.1.5), dividing it by u(z)g(y) and sep-
arating the variables, we obtain
7
1 dg2 1 d2u
= +k2"~(z)u(z) : (1.1.6)
¡g(y)dy2 u(z) dz2
• ‚
It follows from this equation that both its parts can depend neither on y
or on z and, by this very fact, are equal to a certain constant value. This
constant will be further determined from the condition of the fleld excitation
by an incident plane wave. For the time being, it is convenient to assume it
equal to k2fi2, where k is the wavenumber of the incident wave, and fi is a
value undetermined so far. Then we obtain the following equation for the g(y)
function from (1.1.6):
d2g
+k2fi2g(y)=0: (1.1.7)
dy2
Let us select its solution in the following form:
g(y)=exp(ikfiy):
Thispresentationdoesinnowayafiectitscommoncharacter,asthenumer-
ical constant placed in a general case before the exponential, can be included
into the u(z) function and the sign of fi has not yet been determined. So E is
x
written down in the following form
E (y;z)=u(z)exp(ikfiy):
x
Itfollows fromthesecondandthirdequations ofthesystem(1.1.4) that H
y
andH haveasimilartypeofdependenceony. Then,weobtainfromthethird
z
equation the following:
H (y;z)=fiu(z)exp(ikfiy):
z
Assume
H = (cid:192)(z)exp(ikfiy):
y
¡
Thus, the vector functions E~ and H~ are written down as
E~ = u(z);0;0 exp(ikfiy);
f g
H~ = 0; (cid:192)(z);fiu(z) exp(ikfiy): (1.1.8)
f ¡ g
Heretheflrst,thethird,theflfthandthesixthequationsofthesystem(1.1.4)
are satisfled. The remaining two equations provide a system for determining
the u(z) and (cid:192)(z) functions. By substituting the above expressions for the fleld
components into the equations and reducing them by exp(ikfiy), we obtain
du d(cid:192)
=ik(cid:192); =ik "~(z) fi2 u: (1.1.9)
dz dz ¡
The equations (1.1.9) are valid for the fl£eld at eve⁄ry continuity points of
complex permittivity "~(z), i.e., everywhere at the " and (cid:190) continuity points. At
the " and (cid:190) discretion points, i.e., at the boundaries of the layers tangential
components of the E~ and H~ ought to be continuous. The E~ and H~ tangential
8
components are equal to u(z)exp(ikfiy) and - (cid:192)(z)exp(ikfiy). Since the conti-
nuity conditions ought to be satisfled at the layer boundary at any y value, it
follows, then, that they-dependenceinany layer, intheouter space, andin the
substrate is the same, while the u(z) and (cid:192)(z) functions are continuous at the
boundaries of layers.
Let us now determine the constant fi. The fleld of the incident plane wave
can be presented as 2
~ =E~ exp( i~k~r+i!t):
A
E ¡
Here E~ is the fleld amplitude,~k is the wave vector of the incident wave, ~r
A
is the coordinate vector of the observation point.
Inahomogeneousisotropic mediumfeaturing" permittivity, thewave vec-
a
tor is equal to ~k = kp" ~l, where~l is a unit directing vector. Let the incident
a
wave spread in the direction shown in Fig. 1.2. Then the directing vector is
~l= 0; sin(cid:176) ; cos(cid:176) ;
a a
f ¡ ¡ g
and the incident wave fleld depends on the coordinates in the following way:
~ =E~ exp[ikp" (zcos(cid:176) +ysin(cid:176) )+i!t]:
a a a a
E
Hence, we obtain
fi=p" sin(cid:176) : (1.1.10)
a a
This equation represents the Snell law. Actually, a is the same in all layers,
the outer space, and the substrate. But in the latter fi=p" sin(cid:176) and so
s s
p" sin(cid:176) =p" sin(cid:176) : (1.1.11)
a a s s
The Snell law (1.1.11) is valid also in case of absorbing substrate. In this
case complex permittivity "~, should be substituted into (1.1.11) instead of " .
s s
The (cid:176) angle in this case will also be complex, which, in the physical sense
s
implies a non- coincidence of the plane of the constant wave phase with that of
the constant wave amplitude.
TheP-polarizationcase. NowtheH~ vectorhasonlyH componentdifierent
x
from zero, and the E~ vector has two E and E components. The vector equa-
y z
tions (1.1.3) written in the scalar form give the following system of equations:
@E =@y @E =@z = ikH ; @E =@x=0; @E =@x=0; (1.1.12)
z y x z y
¡ ¡
E =0; @H =@z =ik"~(z)E ; @H =@y = ik"~(z)E :
x x y x z
¡
Similar to the S-case, we establish that the fleld components depend only
on y and z and the y-dependence takes the form of exp(ikfiy). Let us represent
H as
x
2The selection of the sign preceding i~k~r is determined by the selection of the sign before
i!t. The plane of the constant phase i~k~r¡!t=const spreads with time in the direction of
thevector~k.
9