The impedance boundary condition for a curved surface

PHYSICS REPORTS (Review Section of Physics Letters) 194, Nos. 5 & 6 (1990) 351-359. North-Holland

THE IMPEDANCE BOUNDARY CONDITION FOR A CURVED SURFACE

R. GARCIA-MOLINA* and A.A. MARADUDIN Department of Physics, University of California, Irvine, CA 92717, USA

and T.A. LESKOVA Institute for Spectroscopy, Academy of Sciences of the USSR, Troitsk, Moscow region 142092, USSR Abstract: We describe a systematic method for obtaining the impedance boundary condition for a p- or s-polarized electromagnetic field at the curved boundary between vacuum and a metal as an expansion in powers of the skin depth of the field in the metal. The surface is defined by the equation x3 = ~(x~), and the plane of incidence is the x~x3-plane. The first three terms in this expansion are obtained, and the conditions under which the result is valid are discussed. The boundary condition obtained is used in a determination of the dispersion curves for surface polaritons propagating across a one-dimensional metallic grating.

In the theoretical study of the diffraction of light from a metallic grating, or of the propagation of surface electromagnetic waves (surface polaritons) across a grating surface, the problem is greatly simplified if the electromagnetic field in the metal can be eliminated from consideration, so that only the field in the vacuum above it need be taken into account explicitly. In recent years the extinction theorem form of Green's theorem, together with Rayleigh's method, has been used successfully for this purpose, in studies of optical interactions at periodically corrugated [1] and randomly rough [2] surfaces. In this note we describe how the impedance boundary condition, due originally to Leontovich [3], can also be used for this purpose. It is necessary, first of all, to obtain the impedance boundary condition at a curved vacuum-dielectric interface. We therefore begin by describing a systematic method for obtaining this boundary condition for a p-polarized electromagnetic field as an expansion in powers of the skin depth of the field in the metal, and obtain the first three terms in the expansion. The first term in this expansion had been written down earlier by Rosich and Wait [4], while the second term has been obtained recently by Depine [5-7] by a rather different method. We then indicate how the corresponding result for an s-polarized electromagnetic field is obtained. We illustrate the results obtained by applying them to the study of the propagation of surface electromagnetic waves across periodically corrugated surfaces. For applications of the impedance boundary condition to the diffraction of light from a metallic grating the reader is referred to the papers by Rosich and Wait [4], Depine and Simon [8, 9], and Depine [5-7]. We consider the situation in which a metal is separated from vacuum by a surface defined by X3 = ~(Xl) , where ~(Xl) is a single-valued function of x 1. The region x 3 > ~(Xl) is vacuum, while the region x 3 < ~'(x1) is occupied by the metal, which is characterized by an isotropic, frequency-dependent dielectric constant e(to). * On leave from: Instituto de Ciencia de Materiales (CSIC), Facultad de Ciencias, C-XII, Universidad Autrnoma de Madrid, E-28049 Madrid, Spain. Permanent address: Departamento de Ffsica Aplicada, Universidad de Murcia, E-30071 Murcia, Spain. 0 370-1573/90/$3.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)

Electromagnetic interactions in condensed matter

352

We assume first a p-polarized wave whose plane of incidence is the x~x3-plane. We therefore work with the single, nonzero component of the magnetic field in the system, H2(x;t)= n2(x,x3160 ) exp(-i60t), whose amplitude satisfies a scalar wave equation in each region,

69XI

+ --

69X~

+

0)

>

H 2 (XlX3 60) = 0, 60

x 3 > st(x1)

(la)

x 3 < ((x~).

(lb)

<

+ OxZ3+ e(60) --~ H e (x~x3[60) = 0 ,

The nonzero components of the electric field amplitude in this polarization are given by ic

3

H:(xlx3 It°)

(2a)

E3(xix3160)- 60e ic Ox 3 1 H2(xlx31t°) '

(2b)

E, (X1X3 160) =

608 C~X3

where e is the dielectric constant of the medium in which the electric field is being calculated. The boundary conditions at the interface x 3 = ((x 1) are (3a)

H 2 (XlX3 [ 60)lx3=¢
o--;

>

1

8

<

H2(x,x3160)lx3= (xo - (60) en H2(x,x3160)lx,=~(x,) ,

(3b)

where d/dn is the derivative along the normal to the surface at each point, directed from the metal into the vacuum, 3 °909n~-~ n . V = {1 -{--[~'(x1)]2} -'/2 - ~ " ( X l ) -09Xl +

3 ) "

(4a)

In writing eq. (4a) we have used the result that the unit vector normal to the surface at each point is ti = {1 + [('(x1)12}-l/z(-~'(x1), O, 1).

(4b)

In eqs. (4) the prime denotes differentiation with respect to x,. We will define the surface impedance of the curved surface x 3 = ~'(x1) by the local relation

E~(x,x3160)lx3=,
=

Zp(X 1 160)H~(XlX3160)lx3=c(x,),

(B)

where E~ and H~ are the componentsof the electricand magneticfieldsin the vacuumthat are tangent to the curved surface at each point. In the present case E>(XIX3 160) "~- (/~)< E > ( x I x 3 I60))2 -

ic a H~(x,x 3 160) 60 o~n

(6a)

R. Garcia-Molinaet al., The impedanceboundary conditionfor a curved surface

353 (6b)

H> (XlX3 l to) = H~ (XlX3 l to)" It follows that

Z~(x~lto)=

E~(XlX3lto)lx,=axo n~(xlX3lto)lx,=ax,) ic toe(to)

ic (e/on)n~(xlX3lto)lx,=c(Xl) to n~(xlx3lto)lx3=ax,)

(e/en)H;(XlX~lto)l~3=ax~) H2 (xlx3 l to)lx~=axi)

(7)

'

where in the last step we have used the boundary conditions (3). We rewrite eq. (7) in the form

Zp(Xllto) =

ic to~(to) (1+

[~t(Xl)12}-l/2g(x 1 l to),

(8a)

[-~"(xl) 810Xl + °~lax3]H<2 (XlX3l to)lx3=aXl) g(x1

l to) =

H2 (XlX3 I)11 toXlx3=~'(Xl)

(8b)

The preceding results have the consequence that the impedance boundary condition (5) can be written alternatively as

--~'(Xl) ~X1 +

n2(x,x3lto)lx3=#(x 0 -

e(to)

g(x, lto)H;(XlX3lto)lx3=ax,).

(9)

Thus, if we have a way of obtaining Zp(xllto), eq. (5) provides a single boundary condition at the interface x 3 = ~'(Xx) that involves only components of the field in the vacuum. Before proceeding to outline a method for obtaining Zp(Xl[W ) we have to consider the following point. It is by no means obvious that a local relation of the type of eq. (5) exists between E~(Xl, ~'(xl)[ to) and H~(x~, ~'(Xx)I to). Indeed, a discussion [10] based on the extinction theorem [11] form of Green's theorem (see, e.g., ref. [12]) shows that in general the relation between these two quantities is nonlocal and has the form

Et(Xl, ~+(Xl)[to) = z~(o)(.~_ Ito)H(x~ I to) + T1 L~ ,7(1),,~,xI I to) ~xi d H(X1l to) 02

_ + ~1 ?1 Zp(2)(*llto) ~

H(xllto) + . . .

(lO)

where we have defined H(x I I to) =- H~(xl, ~'(xl) I to). In order that a relation of the type of eq. (5) hold, all terms past the first on the right-hand side of eq. (10) should be negligible compared to the first. This will be the case when the inequality kd ~ 1 is satisfied, where k is the component of the wave vector of the electromagnetic field parallel to the nominal surface x 3 = 0, while d(to) = (c/to)[-e(to)] -1/2 (Re d > 0, I m d < 0) is the (complex) skin depth of the metal [10]. This condition is usually well satisfied in the infrared portion of the optical spectrum, where Re e(to) is large and negative. The fact that the skin depth can play the role of a small parameter in the theory of the impedance

354

Electromagneticinteractionsin condensedmatter

boundary condition suggests that we seek the surface impedance in the case of a curved surface as an expansion in powers of d, in the frequency range where d is a small quantity, kd ~ 1. To accomplish this we replace the variables x~ and x 3 in eq. (lb) by X = Xl,

U = x 3 / d - ~(Xl)/d ,

(11)

and define

H~ (xlx3 l to) = H(xu [ to) . The equation for -~2

H(xulto)

(12)

is

-

r9

e

¢(x) = {1 + [~"(x)]2} -~/=

(13) (14)

Equation (8) now takes the form g(xxlto)= g ( x l t o ) -

1 (1

n(x0lto)

To solve eq. (13) we seek

d¢2(x) cgu

H(xu[ to) as

H(xulto) = H(°)(xu) + dn(1)(xu)

¢'(x)

n(xulto)l.=o.

(15)

an expansion in powers of d,

+ d2n(2)(xu)

+"',

(16)

substitute the latter into eq. (13), and equate to zero the coefficient of each power of d. In this way we obtain the recurrence relation satisfied by the {H("~(xu)},

(5

,x, a

+2~,(x) a_~du)H(n_l)(xu)

O~2

-¢2(x)--H("-2)(xu), cgx2

n=O, 1,2,....

(17)

The most general solution of eq. (17) for n = 0 that vanishes at u = - ~ is

H(°)(xulto) = A(x)

e ~(x~" ,

(18)

where A(x) is an arbitrary function of x. However, if [ - - ~ ' ( X 1 ) t ~ / t ~ X 1 + d/cTx3]n;(x1x3lto ) is to be < independent of the derivatives of H 2 (XlX3 [ to) with respect to x I on the surface x 3 = ~'(xl) , so that a local relation between those functions of the form given by eq. (8b) exist, we must choose A(x) to be a constant A, independent of x. In addition, it is convenient to solve eqs. (17) subject to the condition that H(")(xu)1,=0 = 0 for n -> 1, as the denominators on the right-hand side of eq. (15) then reduce to the constant A, while dH(xu Ito)/3x[u=0 = 0. The result of this procedure is the following expansion for K(xI[ to):

355

R. Garcia-Molina et al., The impedance boundary condition .for a curved surface

K(xll to) = {1 + [~'t(x1)]2}1/2 ( d

d

~"(Xl)

1 + ~ {1 + [sr'(x,)]2} 3/2

d2

[~'tt(x1)]2

)

8 {1 + [~"(x,)12}3 + O(d3) "

(19)

The first term on the right-hand side of this equation, together with eqs. (5) and (8a), yields the impedance boundary condition in the form used by Rosich and Wait [4], apart from some differences in notation, while the second term has been obtained recently by Depine [5-7]. We see from eq. (19) that the validity of the present results requires that the inequality ld[ ~'"(x,) I ,~ 1 be satisfied, in addition to the condition kd ~ 1, which permits the neglect of the derivatives of H(x, I to) with respect to x, on the right-hand side of eq. (10). Note that in the boundary condition obtained when eqs. (8a) and (19) are substituted into eq. (5) we evaluate the field components Et(XlX3[to ) and nt(x,x3lto ) on the curved surface x 3 = ~(Xl). We do not expand them in powers of ~'(Xl) about their values at x 3 =0. The result is therefore not a small-roughness approximation. In the case of s-polarization, we work with the single non-zero component of the electric field in the system, E2(x; t) = E2(XlX3lto) exp(-itot), whose amplitude also satisfies a scalar wave equation in each region,

~2

0)2)

ax---~3+ -~ E2 (x'x3 I °) t = O, to

<

-s--~ + c~x~ + e(to) ~ E2(xlx3lto)=O, ~X 1

The nonzero components of the HI(XIX3

Ito)-

H3(XIX 3 [to)=

X3>~(Xl) ,

(20a)

X3<~(X1).

(20b)

magnetic field amplitude in this polarization are given by

ic a

- - EE(XlXa[to) , tO t~X3 ic a - - Ez(XlX3lto). to ax 1

(21a) (21b)

The boundary conditions at the interface x 3 • ~(Xl) in this case are

E~ (x,x31to)lx~=axO = E~ (XlX31to)l.~=a~,) , a

>

a

<

e~ E= (xlx, I o,)l~,=.Xl) = ~ e2 (XlX31to)l~,=a~,) •

(22a) (22b)

The surface impedance is now defined by

E?(x,x31to)Ix,=axO= Z,(x, lto)n? (xlx3 l to)lx,=axo ,

(23)

where

> e~ (x,x, Ito) = E2(x~x~Ito),

(24a)

Electromagnetic interactions in condensed matter

356

H~(XlX3ito)=(h× H>(x,x3ito))2_ -~ ic ,~n <7 E2 (x'x3 >

I to).

(24b)

From eqs. (22), (23), and (24) we find that

E; (x,x3 l to)lx:<( l (iclto)( OlOn) E; (x,x l to)lx:<(.l

ZAx, lto)=

(25)

Thus, we obtain the result that

Zs(Xl

I to) =

e:(x,x31

to)l.~:~(~
3 × [-~ {1 + [ ~r'(x,)]2}-l/2(- ~r'(xl) ~-~xl

-1

0 ) E~(x,x31to)lx~::(.l)

•

(26)

Now, because g~(xlx3lto ) satisfies the same equation, eq. (20b), in the region x 3 < ~(Xl) as H2(xlx3[ to) does, viz. eq. (lb), we can use the result obtained in the case of p-polarization, eq. (8), to assert that

[--~'(X1)

alax3lE (x,x3ito)i.:<(xi)

OI3x I +

E~(XlX31to)lx~:~(.l)

(27)

= K(x11to) ,

where the function K(xl [ to) appearing here is the same function that is defined in eq. (8b). It follows, therefore, that 1

1

Zs(Xl [to) = (ic/to){1 + tr'"xl"'2"-'/2K:Xl t a t )1~ Ito) = - eztox'-'p:Xlt)LtIto) "

(28)

To illustrate the results just obtained, we apply them to the determination of the dispersion curves of (p-polarized) surface polaritons propagating across a large-amplitude metallic grating, whose dielectric constant we assume to be real and negative. Our starting point is eq. (9). The surface profile function in this case is a periodic function of x 1 with a period a. It follows from eq. (19) that K(xll to) is also a periodic function of x I with the same period. We therefore expand it in a Fourier series,

gp(to)e i(2~rp/a)xl.

K(Xllto)= E p=

(29)

--oo

The solution of eq. (la) in the region x 3 > ~(X1)ma x that vanishes at infinity and satisfies the Bloch condition imposed by the periodicity of the grating is oo

Hz (x,x31to)= ~ >

Ap(kto)eikpXl-"p (k'°)x3 ,

p=--c~

where

kp = k + (2~rp/a), with k the wave vector of the surface polariton, and where

(30)

R. Garcia-Molina et al., The impedance boundary condition for a curved surface

357

2 to2/c2 , ap(kto)= [k2 -(to2/c2)] 1/2 , kp>

(31a)

ap(kto) = -i[(to2/c 2) - k2l 1/2 , kp2 < to2/c2.

(31b)

We now invoke the Rayleigh hypothesis [13] and use the solution (30), which, strictly speaking, is valid only outside the selvedge region, in satisfying the boundary condition (9). In this way we obtain the equation satisfied by the coefficients {Ap(kto)}, (to2/c2)-

k,,k n

1 p~oo

Km_p(to)Ie_,(-a,,(kto)))A,(kto ) =0,

=

m=0,-+1,-+2,...,

(32a)

a

rim(Y) = _1 a f dx 1 e-i(2~rm/a)x1 eV~(xl)

(32b)

o

The dispersion relation for surface polaritons on the grating is obtained by equating to zero the determinant of the coefficients in this equation. The solution, to(k), of this dispersion relation is an even function of k, that is periodic in k with a period 2~r/a. We can therefore restrict k to the interval O
Electromagnetic interactions in condensed matter

358

I I I I SINUSOIOAL GRATING

o:5ooo~

0.5

I

~1/1 ,.~o= 5 0 0 A°

/ / .,,'I~So:O~, I

0.4 =0.3 3 3 0.2 0.1

/

-

/ 0.0

f

/

0.0

r

i

i

t

i

I

0.1

0.2

0.3

0.4

0.5

0.6

"" ~/o

k (t05cm -I ) Fig. 1. Dispersion curves in the nonradiative region for surface polaritons on a grating defined by the sinusoidal profile function ~(xl) = ~ocos(21rxJa) for a period a = 5000/k and two values of [0. The curve labeled ~'0= 0 A is that portion of the flat-surface dispersion curve which lies inside the first Brillouin zone. For ~'0= 500 A, we show the gap that opens up at k = ~r/a, and two branches of the dispersion curve. The solid curves are the exact results of ref. [14]; the open circles are the results obtained by the use of the impedance boundary condition. The dashed line is the light line to = ck.

I

I

I

I

I

~/I

SINUSOIDA.L GRATING 0.5

/ I / / ~ ~ . : 0 ~ .

5000A

_ o:

0.4 = I000

A

=0.3 3

0.2 /

0.1

-

~/o

/ 0.0

0.(

I

t

I

I

I

r

0.1

0.2

0.3

0.4

0.5

0.6

k (105crn -I) Fig. 2. The same as fig. 1 but for ~o = 1000,~. Note that in this case the upper branch of the'dispersion curve is pushed well into the radiative region.

upper branch of the dispersion curve has been driven into the radiative region, and we see only the lower branch. The open circles are the results obtained from eq. (32) with N = 6. The maximum error in the determination of the surface polariton dispersion curve caused by the use of eq. (32) is 5%. These results show that the use of the impedance boundary condition for the calculation of the dispersion curves of surface polaritons on even rather strongly corrugated metallic gratings yields results

R. Garcia-Molinaet al., The impedance boundary condition for a curved surface

359

that differ by only a few (4-5) per cent from the exact results, when the conditions for the validity of this approximation are satisfied. It is hoped that the results presented here will prove to be useful in the study of other optical interactions at periodically corrugated, or randomly rough, metal surfaces in the infrared region of the optical spectrum, as well as in the study of electromagnetic surface shape resonances (see, e.g., ref. [15]) in this frequency range.

Acknowledgements We are grateful to Dr. R.A. Depine for bringing refs. [7-9] to our attention. The work of A.A.M. was supported in part by Army Research Office Grant No. DAAL-88-K-0067. The work of R. G.-M. was supported by a NATO fellowship.

References [1] N.E. Glass, A.A. Maradudin and V. CeUi, Phys. Rev. B 27 (1983) 5150. [2] G. Brown, V. CeUi, M. Hailer, A.A. Maradudin and A. Marvin, Phys. Rev. B 31 (1985) 4993. [3] M.A. Leontovieh, Investigationsof Propagation of Radio Waves, Part II (Moscow, USSR, 1948); Appendix of Refraction and Reflection of Radio Waves, 13 papers by V.A, Fock, eds N. Logan and P. Blacksmith,REP. AD. 117276(US Government Printing Office, Washington,DC, 1957). [4] R.K. Rosich and J.R. Wait, Radio Sci. 12 (1977) 719. [5] R.A. Depine, J. Modern Opt. 34 (1987) 1135. [6] R.A. Depine, J. Opt. Soc. Am. A 4 (1988) 507. [7] R.A. Depine, Optik 79 (1988) 75. [8] R.A. Depine and J.M. Simon, Opt. Aeta 29 (1982) 1459. [9] R.A. Depine and J.M. Simon, Opt, Acta 30 (1983) 313. [10] R. Gareia-Molina, A.A. Maradudin and T.A. Leskova, unpublished work. [11] E. Wolf, in: Coherence and Quantum Optics, eds L. Mandi and E. Wolf (Plenum, New York, 1973) p. 339. [12] J.D. Jackson, Classical Electrodynamics(Wiley, New York, 1962) pp. 14--15. [13] Lord Rayleigh, Proc. R. Soc. (London) A 79 (1907) 399; Theory of Sound, Vol. II (Dover Publ., New York, 1945) p. 89. [14] B. Laks, D.L. Mills and A.A. Maradudin, Phys. Rev. B 23 (1981) 4965. [15] A.A. Maradudin and W.M. Visscher, Z. Phys. B 60 (1985) 215, and references therein.