Extension of Wolter's multilayer reflection and transmission formulae for orthorhombic absorbing media

Volume

10. number

OPTICS COMMJNICATIONS

2

EXTENSION AND TRANSMISSION

OF WOLTER’s

FORMULAE

I;ebruary

MULTILAYER

1974

REFLECTION

FOR ORTHORHOMBIC

ABSORBING

MEDIA .!

W. SOHLER

Revised

Received manuscript

27 September 1973 received 12 November

1973

Walter’s multilayer rcflcction and transmission formulae are extended for the case, that the rnultilayer rystern ia composed of orthorhombic media. These are oriented in such a way, that two of their principal dielectric axes lit in the surface of a layer. The third axis forms with a second one the plane of incidcncc. An analysis of the complex roots in the formulae, always present when dealing with absorbing media, is given to choose the correct solutions. The extended formulae are well suited for a computer progam. They allow e.g. the computation of Attenuated ‘Total Reflection (ATR)-spectra of anisotropic crystals.

Walter’s multilayer formulae for stratified isotropic media [ I] are widely used in the optics of thin films (e.g. [2]), to calculate the reflectivity or transmissivity of a multilayer system. For anisotropic media, however, even for the simplest case of uniaxial crystals, such general formulae do not exist. In ref. [3] an expression is derived f’or the reflectivity of an uniaxial crystal, its optical axis lying in the surface and in the plane of incidence osr normal to the surface. For the same symmetry the reflectivity and transmissivity of a thin anisotropic layer is given in ref. [4]. Surface wave dispersion in an uniaxial slab has been calculated in ref. 1.51. In ref. [6] the reflectivity of the threemedia-system of an “Attenuated Total Reflection” (ATR)-arrangement is given for an uniaxial sample with its optical axis agairl in the surface and in the plane of incidence or normal to the surface. The anisotropy, caused by the influence of a magnetic field on one medium, is taken into account in [7], where Wolter’s reflection formula for a three-media system (ATR) has been modified. As all this work only considers special cases, it seems to be useful to have generalized formulae for a multilayer sysrem, composed of anisotropic layers, not only for a detailed analysis of ATR-spectra [8-l 11, * Work supported

by the Deutsche

Forschungsgemeinschaft.

but as well for multilayer structures. containing one or more anisotropic layers. It is the aim of this paper, to extend Wolter’s formulae, which are well suited for a computer program, for orthorhombic media. Their principal dielectric axes lie in the surface respectively normal to it and in, respectively normal. to the plane of incidence. We shall show, that a detailed analysis of the complex roots in the formulae. always present when dealing with absorbing media, is necessary to choose the correct solutions. As a result we shall see, that there are wave-solutions of Maxwell’s equations, which are damped in the -z-direction, though their planes of equal phase run in the tz-direction. The complex dielectric tensor E of crystals with symmetry as high or higher than that of the orthorhombic system can be diagonalized [ 121. Taking the x,y, z axes along the principal dielectric axes, we have I ExO 0 E = I 0 E?/0 \

and

D(r) = &E(T).

(1)

0 0 E,

1 Note, that D(r) includes the dielectric displacement and the current density of Maxwell’s equations as dis203

volume

IO. number

OPTIC‘S (‘0M~l1JNI<‘ATIONS

2

I:ebruar);

I974

cussed in [3, I? 1. We shall consider nonmagnetic matcrials. so that p = 1 md H(r) = B(v). The I‘ields E(v). D(V) and H(r)

k is ;I normalized

have the sane

cornplcx

of which is directed

phase factor

wave vector.

to the nomal

equal phase. Two of Maxwell’s

the real part

of the planes of’

equations

give

k x H(r) =

D(r).

(3)

k x E(r) =

H(r).

(4)

Eliminatmg

H(r)

to determine

between

the two eqs. (3) and (4).

the relationship

(k - E(r)) - k

kLE(r) =

between

k and E, WC pet

LET(Y)

(5)

We specialize k to he in the .Y -z plane, so that k = (li,, 0, IL-;).Then (5) yields ;I set of equations. which dccornposes ization

for two independent

off?(r),

namely

directions

of polar-

E,(Y) = (b:, 10. Ez) and E,, (Y)

= (0. ‘::),. 0). E,,(Y) is the ordinary wave. which propagates in the anisotropiz crystal as in 3n isotropic medium with ki = cJ.. For this polariration

with

k, =

11~,, *

sin II = C! 1

II,n is the index of refraction Then eq. (6) yields

boundaries parallel to the x ~9 plane Woltcr’s t‘ormulae are applicable without any modiiicatiorl. Therefore

k: =c

we restrict ourselves to the extraordinary wave. that ~VXI~SE(r) = E,(Y) snd k = k,. For this case (5) yields

corresponding

k.; fx + k:! EZ

the complex direction

(0)

Fresnel-equation

of propagation.

for our special case of

For given t‘,Yand cZ there

al-c four directions.

in which wave propagation sible. It’k, = (k ,,x, 0. Jc,~) is :I solution of(h),

is pos-

k2 =

0. --x-,,). k, = k, and k, = k3 are solutions too. Introducing the multilayer system 3s in Walter’s article (fig. I ). two of the principal dielectric axes 01‘

(kl,,

every anisotropic

_

layer lie in the x-1’ plane. whereas

the third axis coincides with the z axis. Then the boundary conditions for the tangential fields require. that ks is equal I‘or all possible ticld solutions in every medium (Snell’s law of refraction). This reduce, the number ot‘ possible directions of propagation to two.

(7) of the mth medium.

(8)

.Y

3 result,

already

which describe

EX t.. = 0,

lJ real

given in (31. X-i has two solutions. to the two possible ;m “incident”

roots of ccl. (8).

and ;I “reflected”

wave

in each medium. Commonly an “incident” wave should be associated in our coordinate system with l 0. :I “rellected” wave with Ke (k,) > 0 and Irn (k,) < 0. Then the waves al-e exponentially damped in that direction. the planes ol constant phase propagate with time. In the case ot’m isotropic Remind.

medium eq. (8) yields these two solutions. that Ini (CC,)< 0 because of the time deperr-

dcnce exp(+iw/) arid that ciliz = rli jKi3 with )I;? Ki > 0. i = s. _I’,2 [ 1I. In the case of 3n anisotropic niedium the conditions Ke (k,) < 0 and Im (k,) > 0 need not coirlcidc for ;iiI “incident” wave. ThereCole

one in every medium travclling upwards and ant‘ downwards. We assume. that the rnth medium is

that “incident” meant exponential z-dit-cctlon. i.e. Ini (k,) > 0. 2nd that damped in the the time-averaged Poynting vector has a negative : component. We shall show. that these postulations al-e

transparent and the angle 0 of incidence is real. so that

equivalent. The time-averaged

704

and retlection

we postulate.

Poynting

vector P(V) 1s given

Volume 10, number 2

OPTICS

[ 131 by the expression: P(r) =

&

February

COMMUNICATIONS

from the positive constant tive factor

lHO12 * c/h

1974

and the posi-

Re (E(r) X H*(r)) exp

sexp (

2~Im(k,)~z C

Ho is the amplitude

(9)

)

of H(r) (Y = 0 and t = 0). With

we can write k = E;/’ -z II2 and define a function P := ii; qcx ‘I< the real part of which is the z component of the time-averaged Poynting vector P(r), apart

An illustration of these complex functions is given in fig. 2. Dotted respectively dashed lines and areas are transformed to one another from picture to picture. We recognize, that the root of gwith Im (;‘I’) > 0 and Re (c1/2) < 0 yields the solution for an “incident” wave with Im (k,) > 0 and P< 0, whereas the other root yields the solution for a “reflected” wave with Im (k,) < 0 and P > 0. Note. that Re (k,) may be positive or negative for an “incident” as well as for a “reflected” wave, as stated above, depending upon the values of eX and eZ. A similar situation was found even in relation with isotropic absorbing media in an

Im( kz)

RA kz)

p=&-“Z~l” x

(a) Fig. 2. In (a)-(c)

(e)

lb) an illustration

of the two possible

roots

of the complex

IdI

function

is shown. Dotted and (dashed areas and the various lines of (a) in the complex cZ plane are transformed to (b) and (c). The d:,tted area in (a) is bounded by the positive axis of the complex plane and the half circle given by (Re (E,)-:(/* )* + (Im (~a))* = ~(1~. wave, the root in the lower half plane with a “reflected” The root z ‘1’ in the (lpper half plane is associated with an “incident” - ‘I2 with wave. The values of k, respectively P are obtained by multiplication respectively division of the appropriate root o ey’ = nXpjKX with n>,:, K~ > 0 (d). An “incident” wave is characterized by Im (k,) > 0, Re (P) < 0; a “reflected” wave by Im (k,) < 0 and Re (P) > 0.

205

IO. nurr1bcr2

Volume

investigation fortnulate

OPTICS

of surface

(‘OM~IUNKATIONS

l,ebruary

1974

WBVCS 1 131. Now we can

311 unambiguous

relationship

bctwcen

‘:-,Yand

/<,.. Eq. (3) yields:

The upper sign is valid for an “incident” wave. the lower t‘or 3 “reflected” wave with the cottvention, that both roots yield values with negative

imaginary

pat-ts.

Eq. ( IO) is valid in evety medium ot‘ fig. 1 by adding an indcs 1 to 1:,, I{,,. c,Y and L-_with 1 = 0. I, 2. .... 111. Comparing (I 0) with Woltcr‘\eqltations (4.0). we get

reduces

which

to Welter’s

cquation

(4.X) and

(4.10)

in the case

of an isotropic medium. The abbreviation pI l‘or the 2 dependence of the fields gives the analogous expression for Walter’s

equation

(5.4):

With these generalized expressions t‘or gI and pI all further formulae tot- TH-waves in Wolter’s article r-emain valid. The extended

formulae

reflectivity and transmissivity layer system. As an example.

permit

to calculate

of an anisotropic

we give the reflectivity

rnulti-

ot‘a single

boundary with vacuum and of a three-media system (e.g. 3 thin film or an ATK-arrangement) as the most important cases: a): Sirlglr

R

T,,

=

bourdary:

jr, I2 =

s

I

206

’

/

References II. U’oltcr. in: IlandbuL~h der Phyhik. LOI. 24 cd. S. I,luggc tsprtngcr, Herlin 1956). I21 A. Otto and u’. Sohlcr. Opt. <‘omm 3 (1971) 254. 131L..l’. Mostcllt~ Jr. and 1,. Woot~n, J. Opt. Sot. Am. 58 (1968)511. 141II. Schopper. %. Phy\. I32 (1951) 146. 28a (1973) 1055. ljl G. Bor\tcl. %. Nnturfurschun: 161V.V. Bryksm, D.N. Cfirlin and 1.1. Reshina, 1 IL. rvcrti. ‘I’?13 15 (1973) 1118. 171 l.L. Tyler, H. Fischer and R.J. Bell. Opt. (‘umm., in prcu. I81 V.V. Brpktin, D.N. Mirlin ‘rnd 1.1. Reshincl, JI:TP ILcttcr\ 16 (1972) 315. 191 I1.J. I alge and A. Otto. Phy\. Stat. Sol. (b) 56 (1973) 523. Il(jl <‘.ll. Perry, B. I ischcr and W. Buckel. S(>l. State (‘omm.. tn prva. ll.J. T:alge. A. Otto and W. Sohler. submitted to Phys. [ill stat. Sol. Hnndbuch drr Physik, vol. 20 (Springer. 1121 (G. S&essy. Berltn. 1928). I131 M. Born and 1.. Wolf. Principles ot‘ Optics (Pergamon Press, London, New York, Paris, Los Angeles. 1959 1. I141 A. Otto. Optik, 38 (1973) 566.

Ill