Theory of a Fabry-Perot type interferometer for surface polaritons

0 0 3 8 - 1 0 9 8 / 8 4 $3.00 + .00 Pergamon Press Ltd.

Solid State Communications, Vol. 50, No. 9, pp. 8 6 9 - 8 7 3 , 1984. Printed in Great Britain.

THEORY OF A F A B R Y - P E R O T TYPE INTERFEROMETER FOR SURFACE POLARITONS T.A. Leskova Institute of Spectroscopy of the USSR Academy of Sciences, Troitsk, Moscow r-n, 142092, USSR

(Received 12January 1984 by V.M. Agranovich) A theory of an interferometer for surface polaritons has been developed. Transmission and reflection coefficients have been calculated for surface polaritons crossing an interferometer on a metal surface. Two kinds of interference phenomena have been described. The experimental results by Z. Schlesginer and A.J. Sievers are discussed. SURFACE' electromagnetic waves (surface polaritons, SPs) are finding increasing application for studying optical properties of surfaces and thin films (see, e.g. [1 ] ). Therefore, one of the basic properties of SPs, their refraction at an interface, must be studied [2]. The important feature of the SP refraction is their possible transformation into bulk radiation. This effect may serve as a basis for the detection of surface waves just as the reverse effect (transformation of bulk radiation at an interface into a surface wave) may serve as a basis for the excitation of SPs. The SP refraction at the interface of two different metals and at the edge of a thin film depositec~ on metal surfaces has been investigated both analytically and numerically [ 3 - 8 ] . As it was shown in [4, 5] the SP diffraction at the edge of thin films may be used for studying the vibrational spectrum of thin f'rims (or transition layers). The local one-dimensional modes at the edge of the films can be studied in the same way [61 . A thin and limited (in the direction of propagation of SPs) film is an interferometer for SPs. In this paper we present the theory of the SP diffraction by such a surface interferometer. The experimental study o f such a case has been reported in the literature [9].

O~i" Oz

--+~(x)

~=

where

= (K 1 = k/(-g(X)

( ~: k

e2(~). The method of solution that will be employed is that used in the previous works [ 4 - 6 ] , and is based on the consideration of the electromagnetic field only in vacuum (at z ~> 0). In this case the presence of a metal surface is taken into account by the boundary condition which in our case has the form

xL,

e l ) 1/2 ,

0
k / ( - e2) lj2 , -~ ¢ o / c ,

We consider below only H-waves for which H = (0, 0). Let the SP be launched toward the interface x = 0 and its magnetic field be given by ~Uo = exp {iklx--tclz}, where kl = x/K] + k 2, Im kl > 0. The solution of Maxwell's equation in vacuum may always be represented as

9"d'= ~ o + f -

oo

d__ff_we_iWx e_~ F(w) 2rr v--K1 '

(2)

where v = ~ -- k 2 , Re v > 0. The substitution o f equation (2) into equation (1) at z = 0 and subsequent Fourier transformation of the obtained relationship yield the following integral equation for F(w):

1. GENERAL SOLUTION The physical system we consider here is depicted in Fig. 1. Each of the regions x < 0 (z < 0) and x > L (z < O) contains a metal characterized by a dielectric constant ex (w). The region 0 < x < L (z < 0) is also filled with a metal characterized by a dielectric constant

(1)

o,

dw'

K 2 -- K 1

ei(W-W')L

F(w) --_ ~ ~ i F(w') v' -- K1 e i(w+kOL

= -- i(K 2 -- K1) v'

=

~/(w'):

-

w+kx

--

--

1

w -- w' 1 (3)

k 2.

Equation (3) differs from the equations obtained in [ 4 - 6 ] , but it may also be solved analytically. From the analysis of equation (3) it is clear that the sought function F(w) as a function of w is analytic in the upper half-plane o f the complex variable w. We denote it by F,(w). On the other hand, the function eqWLF(w) 869

870

FABRY-PEROT TYPE INTERFEROMETER FOR SURFACE POLARITONS

Vol. 50. No. 9

in the upper and lower half-planes of w, respectively. The result is

••Vocuu•

SP

"////////////////////////4 MetaL: e I (w)

M e t a L : ~zCw)

F+(w)qJ+(w)- (e iwL c)+(w_~)+ ~b(w)]

q,_(w)

Fig. 1. The structure studied in this paper. is analytic in the lower half-plane of w, and we denote it by F_(w). The integral equation (3) may be identically transformed into the functional one

F+(w)eg(w) = eiwL ¢+(w) + (2(w),

(4)

K 2 --K 1 V--K

,

(5)

1

~b÷(w) = --i(K2 --K1)

- -e+ik~L W+kl

,

w--w +i6

oh(w) -

w+kl

i(K2 --K,)

w+k~

(6) dw' t%--~:1

~_(w)

q,_(w)J÷

(11)

(12)

Equation (8) can also be reformed in a similar fashion, thus -

1

-

q,_(w)

( e -iwL

(~(w) I

(13)

,~÷(w)]_

and + f_(w).

(7)

¢+(w) = -- ~+(w) (e -iwL ~(w) ]

~ + ~ I +.

The function ~+(w) defined by equation (6) is analytic in the upper half-plane of the variable w while the function f_(w) defined by equation (7) is analytic in the lower half-plane ofw. The second equation for F(w) can be obtained if we multiply equation (4) by e -iwL [10]: F_(w)~(w)

( eiu'L - (p.(w) O(w) ] +

and

F_(w) =

F(w') _~ 2rri v'--K1 w - - w ' - - i 6 f

1

q,+(w)

L

,

i(Kz--K1)

(10)

I ~(w) ~ = _ (eiWL ~+(w)]

~ dw' K2--K1 _ 2rri v'--~l

F ( w p) e - iw' L

x

¢+(w____)+ ~(w)] . • _(w) q,_(w)}_

The left-hand side of equation (10) is analytic in the upper half-plane, whereas the right-hand side is analytic in the lower half-plane of w. Both sides coincide at the real axis and, hence, must equal an integral function. It follows from Meixner's condition [11] that as Iwl ~ oo in the upper half-plane F÷(w) ~ [w[-r (r > 0) and as IwJ ~ oo in the lower half-plane F(w)~ Jwl -r (r > 0). Hence, by the extension of Liouville's theorem, the integral function must be the constant zero. Thus, we have

F+(w) -

where we have simplified the notation by defining • (w) = 1

=(e,~L

SP V//////////////////////? x i MetaL: ~1 (w)

,I,_(w)j÷

= (b+(w) + e - i w L q~(W).

~_(w---'--~ ~_(w----)"

It can readily be verified that F_(w) e-lWLF+(w). It follows from equation (4) that the expression (2) for the sought magnetic field may be written in the form =

91~= 3U° + f 2-rdW e-vZ v--~2-1 ( ~÷(w)

(8)

Consider first the solution of equation (4). The function q,(w) may be decomposed into the product of the two functions ~+(w) (see, e.g. [ 3 - 5 ] ) which are analytic and non-zero in the upper and lower halfplanes of the complex variable w, respectively, and satisfy a functional relationship q%(w) = q~_(-- w). Then equation (4) may be written as

F+(w)~+(w) = e+iwL d~+(w)+ (~(w)

(14)

\

+ ()(w) e-iWX).

(15)

/

Calculation of the magnetic field would, therefore, require determination of the functions q~+(w) and ~b(w) which satisfy; the pair of the integral equations (12) and (14). 2. AN APPROXIMATE SOLUTION FOR LARGE L

(9)

Now we decompose the right-hand side of equation (9) into the sum of the two functions which are analytic

We consider first the solution of equation (12) for the function ~(w). Taking into account the definition (7) we rewrite the left-hand side of equation (12) in the form

F A B R Y - P E R O T TYPE IN INTERFEROMETER FOR SURFACE POLARITONS

Vol. 50, No. 9

871

at the branch point w' = k and removing them outside the integral sign we find that the form of the function ~b(w) is given by

[m w K

~5 < < (

~(w) -

< ( ( ((

x k~

X

i(K2 - ~1) 'I'_(w) xP+(kl) w + kl

¢÷(kz)xP+(k:)

K2(K: - - ~ 1 ) gJ_(W) e ik~L k2 ( w - - k 2 )

~k Re w

-- g(k)¢p+(k)~I'_(w)c(k -- w, k -- k2),

(17)

where

g(k) =

(K2 - K : ) 2 k ~ I , + ( k ) ei(kL-n/4) ~r(k2 + k) x/2kL '

and ~0(ql,q2) = iL f dt

o

Fig. 2. The contour for the integral 12 .

[ 49(W)]

f-(W) -

i(1¢2 -+

t~l)

w +

i(K~--K,)

l )

w + kl

qJ](w +"

K2(K2 -- ~1) kO÷(w) eik~L ~b+(w) = q~(-- k2)V÷(k2) k2 (k2 ~ ~ - )

¢(w) ~ dw' 1 i(K2--K,) ~ _ - ~ ) = _J~ 27ri w' - w - i6 w' + kl

--g(k)q,+(w)[(f_(--k) 1 ~_(w')

f dw' 1 eiW~ ¢+(w') + 27ri w' - w + i6 qC(w') - I1 + I2. (16) The first integral can readily be calculated as -

i(t% - - g l )

1

• +(kl)

w+kl"

(18)

In a similar fashion an expression for the function ~b+(w) can also be obtained. Thus

1

Equation (12) can be written as

I1

4; e - t (t --iLql)(t -- iLq:)"

To calculate the second integral it is convenient to rewrite it using equation (5)

dw' W ''L 12 =_~ 2rd w ' - - w + i 6

~÷(w')~÷(w') ~(w')

Now we see that the singularities of the integrand in the upper half-plane are a pole at w' = k2 (k2 = x/g~ + k 2, Im k2 > 0, the equation ~ ( w ) = 0 is the dispersion relation for the SP in the region 0 < x < L) and a branch point at w' = k. The contour of the integral 12 may be deformed as shown in Fig. 2. The cut required for the definition of the single valued function ~ ( w ' ) runs along the vertical straight line from the point w' = k, so that the integrand decreases monotonously along the cut. IfkL >> 1 the main contribution into the integral along the cut is given by the integral along the small interval near the point w' = k. Evaluating the smooth and analytical functions ¢÷(w') and ~P÷(w')~/w' + k/(w' + k2)

i(~2---~)'1--k, k2 ]

(19)

i(K2 --tel) )] x ¢(k + w, k - - k 2 ) 4 k 2 - k l ~p(k+w,k--k, • The constants ~b+(k2), ¢+(k), ~b(-- k2), and f_(-- k) can be determined from a set of equations which is obtained by letting w = -- k2, w = -- k in equation (17) and letting w = k2, w = k in equation (19). Now we can determine the sought magnetic field. 3. SURFACE POLARITON REFLECTION AND TRANSMISSION The interaction of a SP with an interface of surfaces leads to the conversion of a portion of its energy into bulk radiation. As it was shown in [ 3 - 5 ] the predominant energy radiation direction is determined by the values ~1 and K2. When K1 and K2 are not very small as compared to k, so that the SP wave vectors k, and k2 differ considerably from k, very little energy is radiated into the directions almost parallel to the surface. Then, if the size of the interferometer L is large enough, the bulk wave radiated at the interface x = 0 does not interact with the interface x = L. Indeed, if inequalities x/(kl - - k ) L >> 1

(20a)

x/(k 2 -- k)L ~ 1

(205)

are satisfied, the function ¢(ql, q2) given by equation (18) is proportional to (kL) -1 . Therefore, it is

FABRY-PEROT TYPE INTERFEROMETER FOR SURFACE POLARITONS

872

negligibly small and may be omitted in equations (17) and (19). In this case the expressions for the functions ~(w) and ~÷(w) can be written in terms of the amplitude coefficients of the reflection and the transmission of the SPs by a single interface• From equation (15) it follows that the total magnetic field is a superposition of surface and bulk waves: contributions of the poles to the integral (15) describe surface waves, whereas the contribution of the vicinity of the saddle point determines the asymptotic form of the radiated field in the vacuum. After the integration we obtain for the SP amplitude reflection coefficient the following expression t l t 2 r 2 e2ik2 L e 2ik:L '

R = rl + 1--r~

(21)

where KI(K 2 - - K I )

1

2k]

q,2 (k,)

F1 =

and

the form of the well-known expressions in bulk crystaloptics.* The field at large distances may be calculated by means of the saddle-point method exactly as in [4-6]. The transformation coefficients of the energy of the incident SP into the energy of the transformed SPs and bulk radiation [4-6] in this case were calculated and the energy was found to be conserved identically. All the results for the SP transformation coefficients presented up to this point have been obtained upon the assumption of the validity of inequalities (20). As was shown in previous works [4, 5] the maximum flux of the energy of bulk waves is radiated at the angle a to the surface which is determined by the value a ~ K1/k at small K~. If the size L is not very large, the bulk wave radiated at the interface x = 0 can interact with the far edge of the interferometer. If aL < Ki1 the bulk wave can excite the SPs at the interface x = L. Now let us consider a physical system for which the inequality (20b) is satisifed, while the value K~ is so small that instead of equation (20a) we have

x / ( k l - k)L are, respectively, the amplitude reflection coefficients of the SPs with the wave vectors kl and k2 ; tl -

~:(K: -K1) ae.(k:)

k2(k2--kl) ~+(kl)

t 2 -~

and

-

kl(k2--kl) ~+(kl)

k2 --

1

7

x/Te -t

k2 - - k ~

J

dt t - i L ( k - - g l )

T =

~÷(k)l +k2 + k]

x r2 eik2L¢l(k -- kl) (22)

2.k, qr÷(ka)q~+(k) } +g I, ) (k2 + k - ~ 2 ~-k) ~p~(k-kl)

(23)

g(k) + t2 ~

e2ik~L

Finally, the amplitude of the surface wave in the region x > L, i.e. the SP amplitude transmission coefficient is

t2 e ik~L T - 1--r~ e 2ik~L"

t x t 2 e l k 2 L { 1--g(k)(2+(+~l ) 1 - - r g e 2ik2L

t~

T2 = 1 --r~ e : i ~ L "

(251

kl

Consequently, in this case the SP transformation coefficients can be obtained using equations (15), (17) and (19). The result for the amplitude transmission coefficient for the SP is

e-~z.

l - - r ~ e~ i ~ L ' tlr 2

In this case we obtain the expression for the function ~(k - - k l , k -- k2)

0

Here the amplitudes T1 and T~ are T~ -

(20c)

~I (k -- kl)

K~(K2 -- K~) q'.(k2)

= T1 eik2x e-K2z + :[2 e - i k 2 ( x - L )

< 1.

~(k - kl, k - k2) -

are the amplitude transmission coefficients of the SPs with the wave vectors kl and k2 [4, 5]. In the region 0 <: x < L the SP magnetic field is given by ~s

Vol. 50, No. 9

[

~o(k--kl,k--kl)

vl(k-kl)

~l(k-kl)

[

**(k)

k2--k,

kll --/~

[I

~ - ~ 1 ) }]"

~]

(26)

tl

Thus, the general form of the expressions for the SP amplitude transformation coefficients is the same as

(24)

*The seeming discrepancy in expression (21) for the SP reflection coefficient from the expression for the bulk wave reflection coefficient arises from the absence of the SP reflection coefficient symmetry.

Vol. 50, No. 9

F A B R Y - P E R O T TYPE INTERFEROMETER FOR SURFACE POLARITONS

873

It can easily be seen from equation (26) that the main contributions to the transmission coefficient are given by the first and the last terms. The first one is associated with the SPs propagating inside the interferometer region, the latter is due to the excitation of the SP in the region x > L as a result of the diffraction at the interface x = L o f the bulk waves radiated at the first interface x = 0. When x/(kl -- k)L ~ 1. equation (26) takes the simple form

0.8

O,6

E 0.4

T~

4K--!lelk~L+( l - 2 ~ - ! l - 2 K j ~ / 2 ( kK2 2~2

~2

-- k)L eikL

and the spatial period of the interference is

0.2

~,L t

0.1

I

0.2

I

2~r

(27) (28)

k2 - - k

0.3

L (cm)

Fig. 3. The energy transmission coefficient for surface polaritons as a function of L in the case when the Ag surface (tv.op = 3.8 eV, hP = 0.08 eV) is partly coated with a Ge film (e6e = 16, d = 0.2/a), v = 975 c m - ' . Solid line, 1W = 0. Dashed line, hi" = 0.08 eV. The magnetic field at large distances was obtained by the saddle-point method. The cumbersome but straightforward calculations of the transformation coefficients of the energy of the incident SP into the energy of the transformed SPs and bulk waves were carried out and the energy was found to be conserved identically in this case too.

As k2 differs only slightly from k, the interference period may be large. It should be noted that a bulk wave as a cylindrical wave is formed at such large distances from the interface asx, z >> max {X, Kiq}, where Kr 1 is the attenuation length for the field amplitude of an electromagnetic wave of the wavelength k. If the inequality (20c) is satisfied we cannot distinguish a cylindrical wave at the interface x = L. We should like to point out that a small change of the value k2, i.e. a small change in the dielectric constant o f the media filling the interferometer region, may drastically change the interference period. This interesting case will be discussed later.

4. DISCUSSION As it can be seen from equations (21) and (24), the intensities of the reflected and transmitted SPs oscillate as a function of the interferometer size L. The period of these oscillations is zSZ = 7r/k2. This interference pattern is associated with the multiple reflection of the SPs by the interferometer interfaces just as in bulk crystaloptics. A similar physical system was studied in [8] for cases of material discontinuity and o f the barrier whose region is filled with a dielectric characterized by a positive dielectric constant. However, as it was shown in [ 3 - 5 ] for infrared frequencies the SP reflection by a single interface is small. Therefore the oscillation amplitude is also small. If the dielectric constant of one of the media is very large (e.g. lell >> 1), the SP transmission by a single interface also becomes small. The energy of the incident SP in this case is mostly radiated in vacuum. As mentioned above, this bulk radiation may excite the SPs at the far edge of the interferometer. The total SP transmission coefficient IT 12 [equation (26)] is plotted as a function o f L in Fig. 3. The results presented in Fig. 3 are calculated using the experimental data by Z. Schlesinger and J.S. Sievers [9].

Acknowledgements - The author is grateful to Prof. V.M. Agranovich, Dr V.E. Kravtsov, and Prof. A.A. Maradudin for very helpful discussions. REFERENCES

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

Surface Polaritons (Edited by V.M. Agranovich & D.L. Mills). North-Holland, Amsterdam (1982). V.M. Agranovich, Usp. Fiz. Nauk 115, 199 (1975). L.A. Vainshtein, Factorization Method in Diffraction. Soy. Radio (1966). V.M. Agranovich, V.E. Kravtsov & T.A. Leskova, Zh. Exp. Teor. Fiz. 81, 11 (1981). V.M. Agranovich, V.E. Kravtsov & T.A. Leskova, SolidState Commun. 40, 687 (1981). V.M. Agranovich, V.E. Kravtsov & T.A. Leskova, Zh. Exp. Teor. Fiz. 84, 103 (1983). G.I. Stegeman, A.A. Maradudin & T.S. Rahman, Phys. Rev. B23, 2576 (1981). A.A. Maradudin, R.F. Wallis & G.I. Stegeman, Solid State Comrnun. 46,481 (1983). Z. Schlesinger & A.J. Sievers, Appl. Phys. Lett. 36, 409 (1980). D.S. Jones, The Theory of Electromagnetism. Macmillan, New York (1964). J. Meixner, Tech. Rept. EM-72, New York, Inst. Math. Sci., N.Y. University (1954).