Theory of diffraction efficiency and anomalies of shallowmetal gratings of finite conductivity

Kenji UtagawaResearch Laboratory, Nippon Kogaku Company, Ltd., Shinagawaku, Tokyo, Japan

(Received 3 January 1978)

The boundary-value problem of diffraction from shallow metal gratings with finite conductivityis solved numerically by using an integral equation method. The electromagnetic field in the materialis described by means of a series expansion of evanescent waves. As suggested by Hessel and Oliner,grating anomalies can be divided into two types: a Rayleigh type and a resonance type. The Rayleigh-type anomaly characterized by its appearance just at the Rayleigh wavelength XR occurs for bothpolarizations, but its appearance has some differences, depending on polarization. In the case ofE polarization, the grazing mode is suppressed, while in case of Hi polarization, it is not. Theresonance-type anomaly can be observed only for HI1 polarization as a strong peak or a dip at thelonger-wavelength side of XR . It can be explained as the result of the resonance excitation of onesurface mode which corresponds to a surface plasmon, and its marked features are the extra absorp-tion of light and the relative phase shift of the diffracted field, which are not observed in Rayleigh-type anomalies.

1. INTRODUCTION

Unexpected rapid variations of the diffracted light intensityfrom reflection gratings were first observed experimentallyby Wood' in 1902, and were termed Wood's anomaly. He alsonoted that these anomalies were present only in H1I polariza-tion whose electric vector is perpendicular to the grooves ofthe grating.

In 1907 Rayleigh 2 developed a series expansion method,where the diffracted field was represented as a sum of planewaves whose amplitudes are determined from the boundaryconditions. In the case of H1I polarization, this theory showeda singularity at the wavelength where one of the spectral or-ders becomes just grazing, and he suggested that anomaliesare the result of the redistribution of diffracted light at thiswavelength. This wavelength has come to be called theRayleigh wavelength XR.

Later on, Ell anomalies were also found for deeply ruledgratings,3 and other phenomena called double anomalies4 wereobserved experimentally. In 1965 Hessel and Oliner5 pre-sented a guided wave approach to these phenomena and foundthat two types of anomalies can be distinguished: the Ray-leigh-type anomaly occurs just at XR, while the resonance-typeanomaly occurs sometimes at the wavelength far removedfrom XR. The resonance anomalies are generally observed asstrong peaks or dips even on shallow gratings. Ritchie et al. 6

showed that these resonance anomalies can be explained interms of an interaction between the incoming photon and asurface plasmon. In 1969 Beaglehole7 measured the magni-tude and the phase change of this photon field and also ob-served spontaneous emission by surface plasmons, which wasalso observed by Hutley et al. 8 and McPhedran9 .

On the other hand, numerical calculation of the diffractionefficiency for arbitrary-shaped gratings of infinite conduc-tivity was done by Pavageau and Bousquet,10 who employed,an integral equation formulation. This infinite conductivitytheory gave a good agreement with experimental results forinfrared regions. However, in the visible region, where theconductivity of metals cannot be assumed to be infinite, dis-crepancy between theory and experiment was observed.

Concerning finite conductivity theories, integral equationmethods"l-' 3 and a differential method'4 were presented.Fairly good agreement between finite theory and experimentwas obtained in the visible region.15 The differential methodshowed its utility in the ultraviolet region where the influenceof coated materials cannot be neglected.'6 As for Maystre'stheory, he assumd a one-dimensional unknown function a,which is defined on the grating surface and is closely relatedto the surface current, and he calculated the diffraction effi-ciency for arbitary-shaped gratings with various dielectricconstants. This integral equation, however, is very compli-cated and (p does not correspond to a real physical quantity;it is difficult to get a physical explanation for anomalies fromthis theory.

The purpose of this paper is to calculate and predict dif-fraction efficiencies, and moreover to give an explanation tothe behavior of anomalies in terms of the strength of the ex-cited evanescent modes in the grating: whether one or twoof them are suppressed or excited resonantly. An integralequation taking into account the dielectric constant of thegrating material is solved under the periodic boundary con-dition using a series expansion of evanescent waves. Strictlyspeaking, the applicability of this theory is restricted toshallow metallic gratings, diffraction efficiencies are calculatednumerically for several triangular profile gratings. Themarked difference between H11 and Ell efficiency for a deeplyruled grating having groove spacing comparable to the inci-dent wavelength is shown.

By examining the behavior of the. matrices which determinethe amplitude of every evanescent mode excited by the inci-dent field, the two different types of singularities, the Rayleightype and the resonance type, which are the origin of eachanomaly, can be easily distinguished. The well-known dis-persion relation of surface plasmons is derived from the con-dition which produces a resonance anomaly. Rayleigh-typeanomalies occur for both polarizations, but their appearancesare different, and for many cases HII Rayleigh-type anomaliesare overlapped with resonance anomalies and cannot be dis-tinguished.

Detailed accounts for many anomalous behaviors such as

333 J. Opt. Soc. Am., Vol. 69, No. 2, February 1979 0030-3941/79/020333-11$00.50 C 1979 Optical Society of America 333

double anomalies, extra absorption of light, and phase changesaround anomalous regions are also presented.

II. INTEGRAL EQUATION FORMULATION OF THETWO-DIMENSIONAL BOUNDARY VALUE PROBLEM

A. Integral equationLet us consider the diffraction problem in which an elec-

tromagnetic wave of time dependence exp(-iwt) travelingthrough the vacuum is incident upon a two-dimensional objectof dielectric constant E. If the electromagnetic wave is planepolarized and its magnetic vector is parallel to the plane ofincidence (Ell case), the diffracted electric field E(P) is givenby' 7

E(P) = (k 2- k) f G(PQ)E(Q) dS + A (P),

Q e Bi (1)

where

he = W/c, k0 = Ek 2

c being the light velocity in vacuum, A(P) is the incidentelectric field in the absence of the object, and Bi denotes theobject area. Green's function G is related to the Hankel

function of the first kind and is given by

G(r) = -(1/4)iHS1'(ker), r = IPQI. (2)

It should be noted that Eq. (1) is valid even when P is locatedinside of the object.

When the electric vector is parallel to the plane of incidence(HII case), the diffracted magnetic field H(P) is given by17

- H(P) = k- I2 G(PQ)H(Q) dSk2 k? JB1

iIC(~G(PQ)dSB(P)+( 1 - 1 ) i; H(Q) d(Qds + k2 1kh i JI an

k2 = k, P E Bi,

k 2 =ke, Pa Bi, 1,

(3)

1 1 +1 Pe1

k 2 k2 k?

where I denotes the boundary of the object, B(P) is the inci-dent magnetic field in the absence of the object, a/an is thesurface normal derivative, and n is the surface normal unitvector and is positive towards the inside of Bi.

B. Periodic boundary conditionLet us consider the diffraction grating defined by the

coordinate system shown in Fig. 1 having the periodicity

f(x + d) = f(x),

where z = f(x) represents the profile of the grating surface, andd is the period of the grating.

The incident plane wave is polarized as follows:

E = (O,ep(x,z),O), Ell case

H = (O,q,(x,z),O), H11 case.

The parallel subscript refers to the y direction which lies alongthe grating grooves. Dropping the time factor exp(-iwt), the

334 J. Opt. Soc. Am., Vol. 69, No. 2, February 1979

incident field I n-th diffracted order

e On

Z\= fcx)

dFIG. 1. Profile of a periodically ruled grating.

incident plane wave with incidence angle 0 can be representedby

(x,z) = exp(ike sinO x - ike cosO z). (4)

It is difficult to find engenfunctions of the Helmholtzequation for an arbitary-shaped boundary. But if we restrictthe material of the grating to metals [Re(E) <<-1], the mag-nitude of the electromagnetic field decreases rapidly in thematerial and the field can be described using a combinationof evanescent waves whose amplitudes are exponentially de-creasing towards the -z direction.

If we denote the field in the material by 0(x,z), the periodicboundary condition imposes the following relation:

(5)

The mth evanescent mode Om(xz) propagating towards thex direction can be written in a Floquet form as

Om (X,Z) = exp(ike sinO x)um (x,z),

where um(x,z) is a periodic function along the x direction withthe periodicity of the grating spacing d. Let us assume thatthe periodic function um (x,z) can be expressed as

um(x,z) = explimKx - p, (f(x) - z]J,

where

K = 27r/d.

Then the field in the material is given by

Om(XZ) = expliamx - pm[V(X) - zJ,am = mK + he sinO. (6)

The field Om in the material must satisfy the Helmholtzequations

AX =-00.

Substituting Eq. (6) into the above equation and, ignoring the

first and second derivatives of f(x), namely,

{f,(X)}2 -2ief'(X) + f"(X) <<1,

Kenji Utagawa 334

O(x + nd,z) = 4(x,z) exp(ikend sin0).

the coefficients Pm which determine the penetration depthof field are given by

Pm = (a 2 -k2)1/2 (7)

Using this approximation Am (x,z) does not exactly satisfy theHelmholtz equation except for the case of a plane boundary,but this will give a fairly good approximation to describe themth evanescent mode really produced in a shallow metallicgrating. Numerical calculation showed that nearly satisfac-tory results were obtained for very shallow gratings [f'(x) <0.05] and shallow gratings [f'(x) < 0.3] with sufficiently largedielectric constant [-Re(E) > 101.

These evanescent modes are orthogonal with each other andsatisfy the relation

d dx X dz nm)n = 6mn )f f -2Re(pm)

The electromagnetic field in the material therefore canexpressed by a linear combination of these evanescent moand is given by

¢0(X,Z) = E Cm bm(XZ),m

where Cm is the complex amplitude of the mth evanescmode.

Using the relation (5), the integration interval with respto the x direction is reduced to one grating period

S dx' X dz'G(x - x'; z -z')(x',z')

= j dx' Jf dz'Go(x - x'; z -z (x .

where the periodic Green's function is

Go(x - x'; z - z')

= X G(x - x' - nd; z - z') exp(ikend sinO).n

The Fourier integral form of the Hankel function18 is

G(r) =I H)(ker)4i

- -explixlz - z'I + ia(x - x')}da,4,7r1 - X

x = (ke2- a2 )1 /2

Then using the relation

E expl-ind(a - ke sinO)} = 27rd- 1 L 63(a - aj,

n n

the periodic Green's function is

Go(x - x'; z - z')

= (2id)-1 Ei Xn- explixnIz - z'J + i an (X - x), (10)n

where

Xn = (k2- a2)1/2.

(8)

l bedes

(Go(x - x'; z - z')an / z'f=(x')

= 1 + [f'(x)] 2 h1-/ 2 7 as n - l2d n \ Xn/

X expliXnlz -f(x')I + ian (x -X')), (12)where

s = sgn[z - f(x')].

This equation is not defined when z = f(x').

C. Infinite set of equationsIn order to get the diffraction field, we must determine the

electromagnetic field inside of the grating for a given incidentfield. For both polarizations, we assume that the electric andthe magnetic fields are given by

,P (x'z) = E Ckm(X',z'), Ell case (13)m

OH(x',z') = E C'mO(x',z'), H11 case (14)m

(9) where Obm is defined by Eq. (6) and CE, CH are the unknowncoefficients. We will determine these evanescent mode am-

ent plitudes from the integral Eqs. (1) and (3) by assuming bothP and Q are in the material. After Eqs. (4), (6), (10), and

iect (12)-(14) are substituted in the integral equations, we multiplythe integral equations by /s and carry out the integration overthe whole grating area, and we obtain

V8 I, [sm - Tsm1C = A8, Ell case (15)m

-E [ - Tsm + Ssm]CH = A,, H11 case (16)em

where

v. = (-ixo + p,)/2Re(p8 ),d

As = d-1 X exp[-i (a. - ao)x - ixof(x)] dx,

Tsm = ke(1 e) L Tsm(f),n=-co

dTsmf(n) = Re(p.)(ixnd2)-l J| dx exp[-i(as - an)X]

dx f dx' exp[-i(an - am)x']

X ff 5 x) fxf(x) d i -z'I

-p m [f(x ) - z'] P.p[f(x) Z-}J

Ssm=(1e) E Ssm(ln),n=--

dSsm(n) = Re(p8 ) d- 2 f dx exp[-i(a, - an)X]

dx f dx' exp[-i(an - am)x']

X eXpfl ) dz S(x - -n p(x)(f- Xn

X expliXn IZ - AX')l Ps VS[(X) - z11. (17)

Kenji Utagawa 335

(11)

The surface normal derivative of this periodic Green's func-tion is

335 J. Opt. Soc. Am., Vol. 69, No. 2, February 1979

The last equation is not defined when z = f(x'), but we cantake a principal value of integration.

Since we have already obtained the analytical form of theelectric and magnetic field inside of the grating, we can cal-culate the diffraction field straightforwardly. For that pur-pose, we represent the diffraction field by the sum of the planewaves

Eout(x,z) = L; DE exp(iacnx + ixnz), Ell casen

(18)Hout(x,z) = E; D' exp(ianx + iXnz), HII case

n

where Dn corresponds to the amplitude of the nth diffractionorder. On applying the integral Eqs. (1) and (3), we assumethat P is outside and Q is inside of the grating. When Eqs.(10), (12), (13), and (14) are substituted in the integral equa-tions, we obtain.

D' = _ UnmCE, Ell casem

H = E e 1 [Unm - Vnm]CH, H11 casem

(19)

(20)

where

k2(1-) 1 1Unm = .

2id Xn iXn + Pm

X go expf-i(an - am)x' - iXnf(x')]dx',

Vn. = 2d d (1 _a f (x ))

X exp[-i(a - am)x'-ixnf(x')]dx'. (21)

Once the amplitudes C', and CH1 are gotten from Eqs. (15)and (16), we can calculate the nth order diffraction field usingEqs. (19) and (20). The absolute efficiency In of the nthdiffraction order is

In = IDn 1 2(COSOn/COSO), (22)

where On is the angle between the z axis and the direction ofpropagation of the nth diffraction order. The relative phasechange between H11 and Ell polarization caused by diffraction

On = arg(D'/D'I. (23)

Then Eqs. (15) and (19) become

EC sm m = Asm

osm = 7mVs (6sm - Tsm);

E = CDnm m,

'Unm = ?ImUnm.

For the HII case, the new basic functions are

Hm = (mm

where

a2 + I poI2 Re(pm) 1/2 2 iEXo

lm 2 + 1Pm 12 Re(po) j iExo-po'

and

= E Mm

Then Eqs. (16) and (20) become

EY fsmCmH = As,m

Yism = (mvs El(Osm -Tsm + Ssm),

Dn _cVnm Mlm

CVnm = (mE_ (Unm - Vnm),

where OE and CH are the normalized evanescentplitudes.

(26)

(27)

(28a)

(28b)

(29)

(30)

(31)

mode am-

Now we can calculate the absorbed energy upon reflectionwhich is dissipated in the material through the imaginary partof the dielectric constant e. If we take the incident waveamplitude to be unity, the energy which is poured upon thelength d of the grating per unit time is (1/2)Eocd cosO, and theamount of energy absorbed in the material is

(112 )eocke Im(E) J dx J dz IEI 2,

fOf -

(32)

where Im(E) is the imaginary part of e and E is the electric fieldin the material. Then for the Ell case, the ratio of the lossenergy to the incident energy is

ke Im(E) pd f(X)dwxI dlIOEI 2ls d cosO JoJCX-~

D. Normalization of the evanescent mode amplitudesWhen a plane wave is incident upon a plane metal surface,

only one mode Co will be excited, and we normalize the am-plitude so as to make the value of Co unity. To do this for theEll case, we adopt the new basic functions P' instead of 0km asfollows:

,pE ='m= ml,

(24)where

IRe(pm) 1/2 2ixo

\ Re(po) / ixo-Poand

2 Im(E) k2 xo0- ) P 2 IC 12.Re(po) iXo - pol

(33)

For the H1 I case, loss energy calculation is somewhat com-plicated, because the energy dissipated in the material can beobtained only through the electric field. It can be calculatedfrom the obtained magnetic field by the equation

I IEI2= O k 12 + I I a'H 12

I kefaz I kef Ox

In this case, the contribution from the terms 0HCO¶H' appearingin the integration (32) is not zero. If we neglect these terms,the ratio of the loss energy to the incident energy can beroughly estimated as

mE= _ (?EOm M

(25)2Im(E~x 2 a+ IpoI2~

IH IS2 Im(e)xo ao I IP ( 32los- Re(po) l if Xo-poI2Z mm

336 J. Opt. Soc. Am., Vol. 69, No. 2, February 1979

(34)

Kenji Utagawa 336

If we define IT as

n+IT = E In + Iloss,

n=n-

MAW

(35)

where no- and n+ describe the highest negative and positivediffraction orders. The energy conservation law requires IT= 1, and this actually provides a check on the theory.

111. NUMERICAL RESULTS

A. Triangular profile grating in a Czerny-Turner mountingCalculation was done for triangular profile gratings. Three

parameters a, b, and d, which characterize the grating shape,are shown in Fig. 2. Then the grating profile is

b-x 0 x < a

f (x) = (36)

b x -d a < x < d.a - d

If we assume Czerny-Turner mounting with the angulardeviation in order -1 constant and equal to 2,o, the incidentangle 0 is given by the relation

K = 2keosspsin(0 - p). (37)

The surface plasmon dispersion curves at Al-vacuum planesurface (dotted curves) together with the incident and dif-fracted photon fields (solid curves) are shown in Fig. 3. Thesolid curve signified by m = 0, which was drawn using Eq. (37),indicates the relation between the energy and the x compo-nent of the momentum of the incident field, 0th evanescentmode and 0th diffracted field on the grating in a Czerny-Turner mounting with 2p = 34°.

As the surface plasmon dispersion curves lie entirely in thenonradiative regions (Ik I > w/c, i.e., the field of surface plas-mon on a plane surface is evanescent on both sides of thesurface), and the incident photon field lies in the radiativeregion (Ik j < w/c), coupling between incident light and surfaceplasmon cannot be expected when the surface is plane. Butif the surface is modulated with the periodicity d, a numberof modes (solid and dashed curves signified by m id 0) are

Z

C d

FIG. 2. Geometry of a triangular profile grating.

337 J. Opt. Soc. Am., Vol. 69, No. 2, February 1979

-10 -5 0 5 10

k,(pr')

FIG. 3. The relation between the energy and the x component of themomentum of every electromagnetic mode on a grating with d = 1.67 ,umand 2o = 34°. The solid curves are the propagating modes, and dashedcurves indicate the modes in the nonradiative regions. The dotted linesare the dispersion curves of surface plasmons on a plane Al surface.Double circles indicate the occurrance of a resonance anomaly. Verticallines indicate, as it were, the zone boundaries and when two double circlescome on the two of these lines located symmetrically, a double anomalywill be expected.

excited simultaneously by the incident light, and in this casemodes in the nonradiative regions become radiative. The xcomponent (kx = am) of the momentum of these modes isdifferent by the integral multiples of K = 27r/d from that ofthe incident field. And when one of these modes just corre-sponds to a surface plasma mode, strong excitation of thismode occurs. Since the magnetic field of the surface plasmonis perpendicular to the direction of propagation, this occursonly in H11 polarization.

For example, when the wavelength of the incident light is0.87 Atm, we can see from Fig. 3 that there are three diffractionorders: zeroth, first, and second. As the curve of the evan-escent mode m = -3 just intersects with the surface plasmondispersion curve at this wavelength, this mode is resonantlyexcited and the anomalous behavior of the diffracted field isexpected. From this figure, resonance anomalies are expectedto occur also at 1.41 and 0.80 Am, and indeed the calculatedresults show anomalies at these wavelengths (see Fig. 4).

D. Efficiency calculationFigure 4 shows the calculated H11 (solid line) and Ell (dashed

line) absolute efficiencies for 600 line/mm, 0.5 Am blazed Algrating in a Czerny-Turner mounting with 2zp = 34°. Thebottom curves are the first-order absolute efficiencies I_1and I', defined by Eq. (22), the middle curves are

n and En n~nnE, and the top curves are ITH and IT definedby Eq. (35). The curves IT indicate the accuracy of the cal-culation.

From the middle curve, we know the absorption of light bythe material. The dip around 0.8 ,4m is caused by large ab-sorption of Al, which is usually observed in the reflectioncoefficient curve of a plane Al mirror. The extraordinal ab-sorption around 1.41 ,m, where a resonance anomaly occurs,is caused by the strong field in the material due to the excitedsurface plasmon. In the H11 case, the bottom curve showsseveral anomalies, and each resonance anomaly occurs at thewavelength slightly longer than XR-

Kenji Utagawa 337

200f (C)

1.0

0.9

0.8

c 0.7a).Y 0.6

0 0.4Q?

< 0.3

0.2

0.1

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6.OAM)

FIG. 4. Theoretical absolute E11 (dashed line) and HII (solid line) efficienciesof an Al grating with d = 1.67, a = 1.63 gam, b = 0.247 Am and 2'p = 34°.(a) The bottom curves are I E and / a, the middle curves indicate 2 / andBy n and the upper curves are T E and I J. (b) Absolute efficiencies of thesecond order near their passing off. (c) Relative phase shift -1. Verticalarrows indicate the positions of XR-

Figure 4(b) shows the second-order absolute efficienciesnear their passing off. Depending on polarization, the be-havior of the two curves is largely different, the reason beingthat the Ell grazing mode is suppressed, while the H11 grazingmode is not. A detailed account will be given in Sec. IV.

Figure 4(c) shows the relative phase shift between H11 andE1I polarization defined by Eq. (23). We can infer from thefigure that when the incident light is linearly polarized (theangle between the electric vector and the incident plane is450), the first-order diffraction is almost linearly polarized inthis case except for the wavelength region where resonanceanomaly occurs. In the regions of resonance anomaly, a rapidchange of relative phase difference occurs, and on both sidesof the anomaly peak, the first-order diffracted light is ellip-tically polarized. This fact has already been found experi-mentally by Beaglehole. 7

In the calculation, integrations (17) and (21) were carriedout rigorously and the summation was done for I n I < Nmax,where Nmax was 10-20. The matrix size was so chosen as tocontain two modes in the nonradiative regions in each side.Computation time required for the whole calculation shownin Fig. 4, where the calculation was done at 60 wavelengths,was about 10 min with a core usage 40k (words).

IV. ANOMALIES OF METAL GRATINGS

A. Types of anomaliesAnomalies were observed usually for Hii polarization, but

338 J. Opt. Soc. Am., Vol. 69, No. 2, February 1979

-- - -- - -

I-2

Hi

where

Re(pm) iXm - Pm9mm 77mVM r .. - P m

1XM 1Xm Pm

(9sm ~- O.

Kenji Utagawa 338

.,VI - -I . N . . . - . - .

- "Ia - - i

(a)

I? f, t . t I T.

\- 1

It I

for deeply ruled gratings, anomalies were also observed evenfor Ell polarization. In 1962 Stewart and Gallawayl obtainednew experimental results regarding double anomalies. Whenthe coincidence of the passing off of new orders occurs it wasexpected that the overlapping of two anomalies should occur,but sometimes they did in fact remain separate.

Hessel and Oliner5 introduced a guided-wave approach andpointed out that two types of anomalies can be distinguished:a Rayleigh-type anomaly due to the emergence of a newspectral order at XR, and a resonance type which is related tothe excitation of a guided wave on the grating surface. On theother hand, Ritchie et al. 6 presented an explanation for H1Ianomalies in terms of a surface plasma resonance. Experi-ments were made using very shallow Al and Au gratings (600line/mm, blaze angle of 2°35'), and they showed evidence fora zone gap in the dispersion curve of the surface plasmon.This is another expression of the same phenomenon as doubleanomalies.

Beaglehole7 measured the magnitude and phase of the re-flected light in H11 anomaly regions. He also observed smallnoncoherent radiation resulting from spontaneous emissionby the surface plasmon in addition to the coherent radia-tion.

B. Explanation of anomaliesWe examine the physical background of these anomalous

behaviors except for the spontaneous emission observed byBeaglehole, which cannot be explained within the frameworkof the classical electromagnetic theory.

Two types of anomalies, the resonance type and the Ray-leigh type, can be explained in terms of the behavior of thematrices T and S. Let us assume that the grating has veryshallow triangular profile. When we expand each element ofthese matrices as power series in b and take only predominantterms, the diagonal elements of these matrices are given by

Tm 1 + Re(pm) ixm - PmtXm iXm - Pm

(1 - E)Re(pm)Smm .

ixm -Pm

and the off diagonal elements are

Tsm - ° (s Fd m),

Ssm - b/x 8 (s /- m);

the last elements Sam play a very important role on the occur-rence of the H1I anomaly.

For the Ell case, substituting the above equations into Eq.(26), we obtain

OF A-,j ]

lessII N.

.1 .2 .3 .4 .5 .6 .7 .B .9 1.0 1.1 1.2 1.3 1.4

FIG. 5. Variations of the value &,,, J'Y, and 1 sm with respect to as/ke.

Thus the E1 I surface mode amplitudes EM are given by

Em =-Am/6mm.

Figure 5 shows the value I nmmI versus amm/ke, and from thefigure we can see that I 16, j is nearly unity almost every-where. But when j aml l/ke 1, the term I Im, becomes in-finitely large, and the evanescent mode amplitude E issuppressed.

For the H1 I case, Eq. (30) becomes

, " -mV.m(1 - E2)Re(pm) / 2 E k2\

mm x i(m Xm + P*)(iEXm + Pm) am + 1

(38)

Yf 0b/XS when IxI 0 (s t m)

m 0 otherwise.

Equation (38) is completely right when the surface is plane.The off-diagonal elements yfsm are proportional to b, and aregenerally small in a shallow grating. But when a certain order,signified by i, becomes almost grazing (i.e., I xi 0 - 0), not onlythe diagonal element I Kii I but also the off-diagonal elementsJ im I become large (see Fig. 5), and these coupling terms playan importnat role in determining the amplitude of the evan-escent mode situated in the vicinity of the resonance position.In this case, for example, Eq. (30) can be written as

/.. -2-2 -2-1 W-2 '(

Of ,-,

-o _W,,0

curve of light in vacuum w = ck, and the position of the reso-nance anomaly in the efficiency curve will be far shiftedtowards the longer wavelength side of XR-

The surface mode amplitude OH near its resonance is

@iH = (Ai -'Y E 1 Ar OH Wii

[when axi - ±k,, Re(k.p)],

where the amplitudes of the other modes are

m AmWmm,

When ai = Re(ksp), namely, the denominator IJiiI is verysmall and the coupling terms I Yim I are still large (see Fig. 5),the evanescent mode amplitude eH grows up very large. Thisis the resonance excitation of the surface plasmon.

We must refer now to the matrices II and CV. The ampli-tude of the nth diffraction order Dn is given by Eq. (27) or (31)as the result of the interference among the radiated fields( Unm @ E or cVnm @H). Generally, the main contribution tothe nth diffracted field comes from the nth evanescent mode,the other modes, including modes in the nonradiative regionssuch as the surface plasma mode, also contribute to it largely.The elements of the matrices ?1 and At are very slowly varyingfunctions, and the anomalous behaviors of the solution aredetermined mainly from the matrices 6 and W. And if theamplitude or the phase of one evanescent mode should changefor some reason, this affects the type of interference amongthe radiated fields and changes the diffracted intensity. Thisis likely to happen near resonance anomalies where ampli-tudes and phases of the evanescent modes are changeable, andit is expected that the anomaly peak shape is very sensitiveto a slight change of the experimental conditions.

C. Plasma resonance anomaliesLet us examine the dominant peak around 1.41 ,um shown

in Fig. 4; its detailed peak shape is shown again in Fig. 6(a)

1.0

.o2

a)a,

0:f ..i OH A

) [H- IA-,

.5

As is shown in Fig. 5, the diagonal element I Wii I becomesvery small at the wavelength slightly larger than XR. Thisoccurs when ai = Re(k,,), where

! '/ 0.15

-_J , b' ,\__0.08

1.33 1.40 1.41 1.42 1.43 1.44

X (M)

1.4025I I t I I 1

1.39 1.40 1.41 1.42 1.43 1.44

X (X)

k~, = ke[E/(1 + E]P/2. (39)

This is just the surface plasmon dispersion relation shown inFig. 3. When the dielectric constant I EI becomes small, thesurface plasmon dispersion curve departs from the dispersion

339 J. Opt. Soc. Am., Vol. 69, No. 2, February 1979

FIG. 6. Plasma resonance anomalies of an Al (e =-122.7 + 40.3i) gratingwith various groove depths, the other conditions are the same with the caseshown in Fig. 4. (a) The bottom three curves are / 1 and the correspondingupper curves indicate the amount of absorption. (b) Dependence on thedielectric constant.

Kenji Utagawa 339

iHm(a)

,0.5 I

E J ---

I11.0 - I%

"II -)I -

1.4 1.5

I0.5

1.0 .

011

E,,, I .... I I-h

0.5 1E L -, .-7--,1.4 1.5

(b) II

nIItAlj

1.0 II1

1.4 1.5

0E.

F. ,,

0.5

0 5 r

1.0

same groove depth grating with f = -122.7 + Oi shown in Fig.6(b). Both show similar appearance but the latter shows noabsorption. We also see from the two curves shown in Fig.6(b), that when the dielectric constant I c becomes larger, thepeak becomes steeper and its peak position move towardsXR-

When two resonance anomalies are expected to coincide,sometimes the repulsion of two anomalies is observed. This

_ _ phenomenon is called double anomaly. One such example1.4 5 is shown in Fig. 8, and double anomalies are seen around 1.15

and 0.68 Am.

Around the wavelength where double resonance occurs, theevanescent mode amplitudes except for the two resonance

' 'modes are

m -Am/Imm.

Using these amplitudes, we can calculate the resonance modeamplitudes OH and &1H from the following equations:

0.5 1

1.4 1.5

0.5 !

1.4 1.5KyP) k Y)

FIG. 7. (a) Evanescent mode amplitudes of the case shown in Fig. 6(a)with b = 0.24 jim. (b) The amplitude @H2 of the case shown in Fig.. 6(a)with b = 0.08 Am. The outstanding peak appearing in the curve @2 near1.4 Am corresponds to the forced excitation of a surface plasmon. Thesolid curves indicate the modes in the region IamI < w/c, and the dashedcurves are the modes in the region ItarI > w/c. Vertical arrows indicatethe positions of XR-

together with the shallower groove case of b = 0.15 and b =

0.08 gim. When the depth of the groove increases, the peakbecomes broader and its peak position moves toward longerwavelengths from the position that is expected from the dis-persion curve of surface plasmons on a plane surface.

The gratings with b = 0.15 and b = 0.08 jim show strongerabsorption. This can be explained from the evanescent modeamplitudes shown in Fig. 7. The resonance of the mode eH2is sharper and stronger in the shallower groove case. Theexcited strong electromagnetic field loses its energy in themedium through the imaginary part of the dielectric constantand gives rise to the strong absorption of light. This is oneof the marked differences between the resonance-type and theRayleigh-type anomaly.

Influences of the dielectric constant on the resonanceanomaly are shown in Fig. 6(b). Compare the Al (E = -122.7+ 40.3i) grating of b = 0.24 Am shown in Fig. 6(a) with the

340 J. Opt. Soc. Am., Vol. 69, No. 2, February 1979

Hii -i + Wij * @,= Ai E " H _im 6' [ai Re(kp)],m d ij

9yji @iO + Wfjj @,O = Aj - E jmv. OH [aj -Re(ksp)].

The existance of the coupling terms 9tij and fIji are the originof the double anomaly. When I Ylii I and I Yjj I are smallestat a certain wavelength Xs,,p two plasma modes are expectedto be simultaneously excited, but the existance of the couplingterms naturally suppresses the amplitudes OH and OH, andthis brings a dip rather than a peak at this wavelength XspsThe peak occurs when

det I 9ii ijII yji Sibj

takes the minimum value, and this generally occurs on bothsides of the wavelength XAp, and this is the main cause of theso-called double anomaly.

To simplify the problem, we have neglected many off-di-agonal elements whose contribution is not large and also ne-

1.0

0.9

0.8

c 0.70.'

.Q 0.6

ci 0.5

' 0.4

a 0.3

0.2

0.1

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5

x (u)

FIG. 8. The first-order absolute efficiencies of an Al grating with d = 1.67jum, a = 1.63 Arm, b = 0.247 ,um, and 2(p = 00. Double arrows indicatethe coincidence of two Rayleigh wavelengths.

Kenji Utagawa 340

-IN N.

t --

It . . .-t t t t

t i. t. . t

1- I � , I . . . a I �_ � �

"-Z ----------

0.4 0.5 0.6 0.7 0.8 0.9 1.0k' ( &)

1.0

0.9

0.8

C 0.7

.) 0.6

I; 0.5

2 0.40U)E 0.3

0.2

0.1

0.4 0.5 0.6 0.7 0.8 0.9 1.0x (yt)

FIG. 9. The first-order absolute efficiencies of a grating with d = 1.0 Atm, a = 0.5 Aim, b = 0.24 ,m, and 2qp = 00. (a) Nearly perfect conductive case withE=-100000 + Oi. (b)Aluminum dielectric constant is used, and extraordinary absorption is seen in thevicinityoftheresonanceanomaly. The Hl efficienciesare largely different depending on the dielectric constant. (c) Efficiencies of the new appearing orders in case (b).

glected many additional terms which must be included in Eq.(37). To derive the surface plasmon dispersion relation onthe ruled surface, we must study directly

-= det[Jf']/det[[],

where the matrix P' is defined by replacing the mth columnof [X with [AJ]. Numerically, we will be able to find a complexvalue am = k rled which leads to det[W] = 0, and krpled(X) will

provide the surface plasmon dispersion relation on the ruledsurface. Using the value kSPed, similar treatment to thepole-zero approximation proposed by Maystre and Neviere1 9

will be possible.

D. Rayleigh-type anomaliesEfficiencies of deep groove gratings having the same profile

but different dielectric constant are shown in Fig. 9. Comparethe outstanding difference between H11 and Ell efficiencies ontheir dependence on the dielectric constant.

The resulting accuracy is better than 2% for the perfectconductive case shown in Fig. 9(a) for both polarizations. Butfor the case shown in Fig. 9(b), the energy excess in H11 po-larization is serious. This can be reduced in some degree usinga larger matrix size and double precision, but it seems that thisis mainly due to the incorrect estimation of the loss energy ilo.and the inadequacy of the field representation of this theoryfor the deep groove grating.

The Ell polarization anomaly observed in deep groovegratings as a cusplike peak [Fig. 9(a),(b)] is the Rayleigh-typeanomaly. As mentioned in the previous section, when onemode signified by i becomes just grazing, the correspondingdiagonal matrix element gii becomes infinity, and the grazingmode OE is suppressed. When the groove is deep enough andthe off-diagonal elements Cim is not small, this singularity

341 J. Opt. Soc. Am., Vol. 69, No. 2, February 1979

affects other modes and makes a slight change on the evan-escent mode amplitude at XR (Fig. 10). Moreover, the jumpof phase of the grazing mode at XR (OE 2 and COE in Fig. 10)

cHm

I

E

E

200'

100°

.............. 0.1.0 -100'

E -2V_ '..... ,,,,,.-300'

°. ' 0.oE ................

-100,

1.0 4f-no

J " 100,

E ;........... : O'

>0.5r 1E -,- -- -=0.6 0.7

0~.5J.. ................. .... 200'.200'0.5E...,..,..,.. ,1100'

1.0 1'°; . . .... ..... ... ............ O .,

1.0

0

-100'

0.5 ........... 11:?.. .5r . ...........17_ |- . .

0.5 r l

0.6 0.7

FIG. 10. Evanescent mode amplitudes of the case shown in Fig. 9(b).The dotted lines indicate the phases of these modes. Vertical arrows in-dicate the positions of XR-

Kenji Utagawa 341

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.4 0.5 0.6 0.7

c Em

0.5E f

changes the condition of interference among the diffractedfields, and causes the rearrangement of every diffracted in-tensity. For a deeply ruled grating, the grazing mode am-plitudes grow up rapidly from suppression (0'2 and 0E in Fig.10). The appearance of new orders becomes evident [Fig.9(c)], and the rearrangement of the diffracted intensities be-comes obvious. This is the origin of the Ell Rayleigh-typeanomaly.

In general, it is difficult to distinguish the H11 Rayleigh-typeanomaly from the resonance anomaly because they overlapeach other. For the perfect conductive case shown in Fig. 9(a),two types of anomalies which are expected to occur around0.57 gim are not distinguished. In Fig. 9(b), Al was chosen asmaterial and we see a double anomaly near 0.7 jim. And forboth cases, the efficiency curves are decreased suddenly at XR

due to the redistribution of energy to the new appearing or-ders. In H11 polarization, grazing modes are not suppressedso that the new orders appear abruptly [Fig. 9(c)], and thisbrings about the sudden decrease of the efficiencies of thepreviously existing orders.

To separate the resonance anomaly from the Rayleigh-typeanomaly, we assumed small dielectric constant E = -7.0 + O.Oiso as to shift the resonance peak position from XR. Keepingthe same configuration with the case shown in Fig. 4 exceptfor the dielectric constant, the calculated result is shown inFig. 11. In the figure, the resonance-type and the Rayleigh-type anomalies appear at 1.52 and 1.4 Aim, respectively. Theamplitude and the phase of every evanescent mode is shownin Fig. 12. From the figure we see that the resonance anomaly

c Hm

En

I

0.5 1 loc>. 10

0.

1.0 /.\ 1200-

1.7

1.3 1.4 1.5 1.6 1.7

...... ..............

E

200°

100'

1.0

E 10, 100,

I ,%

0.5 '100°

0 - -5 * . . O-.. ,..........

E-j

6

0.5

300'

200'

100°

0A (P)

I&Em

0.5 l -100'

L- - -200°............. 1..

0.5 '-200

1.3 1.,4 1.5 1.6 1.7

0.5.. ''''''':'''':''': 150

1.0

-100'

0.5 100'

---

0.5 . 1007

,- ,- ,- ,- 0-1.3 1.4 1.51.

X (P)

220-

210'

200'

.1 190'

180

170

1.0

0.9

0.8

c 0.7

:2 0.6

a. 0.5

0 0.4

< 0.3

0.2

0.1

1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

X (M)

FIG. 11. Absolute efficiencies of a grating with = -7.0 + 0.0i, d = 1.67Aim, a = 1.58 Aim, b = 0.241 Am, and 2V = 34°. (a) Solid lines indicatethe Hl efficiencies and dashed lines are the Ell efficiencies. (b) Relativephase shift o0. Vertical arrows indicate the positions of XR.

342 J. Opt. Soc. Am., Vol. 69, No. 2, February 1979

FIG. 12. Evanescent mode amplitudes of the case shown in Fig. 11.Dotted lines are the phases of these modes. Vertical arrows indicate thepositions of XR.

is the result of the resonance excitation of the surface plasmamode (in this case 042 around 1.52 Aim), and the Rayleigh-typeanomaly is the result of the redistribution of diffracted in-tensities which is caused by the abrupt phase change at theRayleigh wavelength.

V. SUMMARY

An integral equation was employed to solve the bound-ary-value problem of diffraction from metal gratings. Theelectromagnetic field in the material was represented by thelinear combination of a series of evanescent waves which weredescribed as the exponentially decreasing functions towardsthe -z direction.

Numerical calculation was done for triangular-shapedmetallic gratings. It was shown that the diffraction ef-ficiencies vary depending on the dielectric constant. Whenthe grating spacing is comparable to the wavelength of theincident light, the difference between H11 and Ell efficienciesbecomes obvious.

Moreover, from this theory, we obtain knowledge about thephase relation between H11 and Ell polarization after reflection:whether diffracted light is linearly polarized or ellipticallypolarized. It was also shown that the extraordinary absorp-tion of light occurs around the wavelength region of resonance

Kenji Utagawa 342

t

anomaly. And we will be able to find the condition that nearlyall the energy of the HII polarized incident light is absorbedin the material.

The calculation of the evanescent mode gave us someknowledge of the occurrence of various anomalies. The stateof affairs of the evanescent mode in the material is largelydifferent depending on polarization. In H11 polarization, whenone mode just coincides with a surface plasma mode, the res-onance excitation of this mode occurs and strong anomaly inthe diffracted intensity is expected (resonance anomaly).

When one of the diffracted orders becomes just grazing, asingularity in phase of the evanescent mode changes thecondition of interference among the diffracted fields andcauses the rearrangement of the diffracted intensities. Butas the E1 I grazing mode is suppressed, E1 I anomalies are notobserved in shallow groove gratings (Rayleigh-type anom-aly).

Since resonance anomalies are very sensitive to the grooveshape, dielectric constant, and mounting conditions, becausethe amplitude and the phase are changeable, extreme caremust be taken when we examine these anomalies experi-mentally.

ACKNOWLEDGMENTS

The author would like to thank H. Ikeda, senior scientist,

for supporting this work and Dr. Y. Itho for his helpful sug-gestions and criticisms upon reading this manuscript.

'R. W. Wood, Proc. R. Soc. Lond. 18, 396 (1902).2 Lord Rayleigh, Proc. R. Soc. Lond. A79, 399 (1907).3 H. C. Palmer, J. Opt. Soc. Am. 46, 50 (1956).4J. E. Stewart and W. S. Gallaway, Appl. Opt. 1, 421 (1962).5A. Hessel and A. A. Oliner, Appl. Opt. 4,1275 (1965).6R. H. Ritchie, E. T. Arakawa, J. J. Cowan and R. N. Hamm, Phys.

Rev. Lett. 21, 1530 (1968).7 D. Beaglehole, Phy. Rev. Lett. 22, 708 (1969).8 M. C. Hutley and V. M. Bird, Opt. Acta 20, 771 (1973).9 R. C. McPhedran and D. Maystre, J. Spectrosc. Soc. Jpn. 23, 13

Suppl. (1974).°0J. Pavageau and M. J. Bousquet, Opt. Acta 17, 469 (1970)."D. Maystre, Opt. Comm. 6, 50 (1972).12 H. A. Kalhor and A. R. Neureuther, J. Opt. Soc. Am. 63, 1412

(1973).13p. M. van den Berg and J. C. M. Borburgh, Appl. Phys. 3, 55

(1974).14 M. Neviere, P. Vincent, and R. Petit, Nouv. Rev. Opt. 5, 65

(1974).15 E. G. Loewen, D. Maystre, R. C. McPhedran, and I. Wilson, Jpn.

J. Appl. Phys. 14, Suppl. 14-1. (1975).16 M. C. Hutley, J. P. Verrill, R. C. McPhedran, M. Neviere and P.

Vincent, Nouv. Rev. Opt. 6, 87 (1975).17 B. I. Smirnov, A Course of Higher Mathematics (Pergamon, Lon-

don, 1964) Vol. 4.18J. Pavageau, R. Eido, and H. A. Kobeissi, C. R. Acad. Sci. Ser. B:264,

424 (1967).19 D. Maystre and M. Neviere, J. Opt. (Paris) 8,165 (1977).

Coupled surface plasmons in structures with thin metalliclayers

A. B. Buckman and C. KuoElectronics Research Center and Department of Electrical Engineering University of Texas at Austin, Austin, Texas 78712

(Received 22 September 1978; revised 8 November 1978)

If the negative-permittivity region in a layered structure is very thin ( S 300 A for Ag at 6328A excitation), the two surface plasmons at its interfaces couple in its interior to yield a coupled-surface-plasmon mode whose propagation constant increases with decreasing layer thickness. Thesemodes were observed by measuring reflectance versus angle of incidence through a high-index prismfor TM-polarized light. For very thin Ag films, the reflectance minimum diverges from the phase-matching condition, as determined from independently measured refractive indices and thicknesses.This divergence apparently arises from separation of the pole and the zero in the reflectance versuspropagation vector as the Ag thickness is decreased.

INTRODUCTION

Nonradiative surface plasmons on structures containingregions of negative dielectric permittivity E have been in-tensely studied in the past few years. Theoretically, it hasbeen shown that the electromagnetic boundary conditions atsurfaces bounding the negative-permittivity regions must bemodified to include longitudinal as well as transverse excita-tions.1 The effects of the longitudinal modes are, however,not expected to significantly alter experimental results; e.g.reflectance or transmittance versus excitation frequency orangle of incidence, unless the negative-permittivity region isextremely thin' (-50 A for potassium). Experimentally, a

variety of techniques for excitation of these surface plasmonshave been analyzed. 2-5 In general, excitation of a surfaceplasmon is associated with a change in the magnitude and/orphase of the reflection as frequency or angle of incidence isvaried. Interpretation of these reflection measurements iscomplicated by the coexistence of Brewster-like reflectionminima at certain incident angles and frequencies.6 To ourknowledge, experimental data reported to date on structurescontaining Ag, Au, or Al, for which Re(E) < 0 in the visible, areconfined to metal thicknesses greater than -300 A.

In this paper, we examine the effects of reducing metal-filmthicknesses to a few hundred A on the propagation of non-

343 J. Opt. Soc. Am., Vol. 69, No. 2, February 1979 0030-3941/79/020343-05$00.50 (P 1979 Optical Society of America 343