antireflection surface structure: dielectric layer(s) over a high spatial-frequency surface-relief...

Antireflection surface structure: dielectric layer(s) overa high spatial-frequency surface-relief grating ona lossy substrate

Elias N. Glytsis and Thomas K. Gaylord

With proper design, a dielectric (or lossy) overcoated high spatial-frequency rectangular-groove grating on alossy substrate can exhibit zero reflectivity. A procedure for designing these structures based on impedancematching and the effective index of the grating in the long wavelength limit is presented. The needed fillingfactor and groove depth of the grating to produce antireflection behavior are calculated for a given complexindex of refraction of the substrate and index and thickness of the coating layer. The analysis is applicable toany wavelength and angle of incidence and for either TE or TM polarization. It is shown that multiple zero-reflectivity solutions occur. Necessary and sufficient conditions for zero-reflectivity solutions are derived inthe appropriate parameter space. The analysis is also extended to multiple dielectric (or lossy) overlayers.Since the treatment is based on the equivalence of the grating to a lossy layer in the long wavelength limit, thetreatment also includes as an intermediate step the design of a dielectric (or lossy) overcoated homogeneouslossy layer on a lossy substrate. The designs of zero-reflectivity gold gratings overcoated by a single dielectriclayer are presented for wavelengths in the 0.44-12.0-ium range. The sensitivity of these resulting structures tochanges in the angle of incidence, the coating thickness, the coating index, the filling factor, and the groovedepth are presented. Furthermore, the design of dielectric overcoated gold gratings that are simultaneouslyantireflecting for both TE and TM polarizations is presented.

1. Introduction

Low reflectivity surfaces, or antireflection surfaces,on dielectric or lossy materials have been the subject ofa great deal of interest in recent years for a wide varietyof applications. Thin film coatings are often used toproduce low reflectivity. Alternatively, a high spatial-frequency periodic surface (a diffraction grating) on adielectric, semiconductor, or metal can also behave likean antireflection surface.1-7 Areas of application in-clude high power lasers (windows, polarization-selec-tive mirrors, absorbers), solar cells and photodetectors(for improving their efficiency), and elimination ofunwanted reflections from metals.

Absorption phenomena in surface-relief metallicgratings have been known for many years. Wood8

suggested that some of the power incident on a metallicgrating can be absorbed rather than reradiated. Lat-

The authors are with Georgia Institute of Technology, School ofElectric Engineering, Atlanta, Georgia 30332-0250.

Received 4 April 1988.0003-6935/88/204288-17$02.00/0.© 1988 Optical Society of America.

er, this reduced-reflection phenomenon was experi-mentally observed9 10 and theoretically analyzed10-18

for both transverse electric (TE) polarization andtransverse magnetic (TM) polarization. A similarphenomenon has been described by Shah and Ta-mir1920 and Azzam et al.

21 in multilayered structures

that are lossy.Total absorption of the incident radiation has been

observed by Hutley and Maystre22 for TM polarizationusing high spatial-frequency (all except the zero orderare cutoff) surface-relief gratings on metallic surfaces.The total absorption of light by surface-relief gratingshas recently been studied for TM incident polarizationby Depine et al.

2 3 and Mashev et al.2 4 These studies

are applicable for cycloidal or sinusoidal gratings only,and the the resulting light absorption is narrow bandwith respect to the incident wavelength and angle ofincidence. Gaylord et al.

7 have calculated the re-quired single homogeneous layers or high spatial-fre-quency rectangular-groove surface-relief gratings onlossy substrates to produce zero reflectivity.

Dielectric overlayers can be used to minimize thereflectivity of lossy substrates.25 26 It has also beenobserved1316 27-3' that dielectric overlayers on the topof surface-relief gratings affect the reflection proper-ties of the gratings. The antireflection effect of astructure consisting of a dielectric coating over a sur-

4288 APPLIED OPTICS / Vol. 27, No. 20 / 15 October 1988

Region 1 n

dc Region c ' § nc

2 Region 2 \ n2 -jK2

Region 3I n

Fig. 1. Geometry of an electromagnetic wave in a lossless medium(region 1) incident on a lossless overlayer (region c) on top of a lossylayer (region 2), which is on top of a lossy substrate (region 3). Thecontinuous lines correspond to propagating waves, while the dashed

lines correspond to evanescent waves.

face-relief grating has been characterized in the litera-ture as "a curious property."' 6

Two-dimensional smoothly graded crossed dielec-tric gratings or moth eye surfaces also exhibit antire-flection behavior, as has been experimentally observedand described by Wilson and Hutley32 and Derrick etal.33 These structures somewhat resemble the corru-gated walls used in microwave anechoic chambers.

The present work is a generalization of the analysisof Ref. 5 (for dielectric gratings) and of Ref. 7 (for lossygratings) to the case of dielectric (or lossy) overcoateddielectric or lossy gratings. In this work the requiredzero-reflectivity thickness and complex refractive in-dex of the equivalent homogeneous layer on a lossysubstrate, overcoated by an arbitrary number of di-electric (or lossy) layers, are calculated. These over-layers are sometimes needed to provide a smooth ex-ternal surface for grating protection or foraerodynamic or hydrodynamic applications. In addi-tion, as shown in this paper, the inclusion of the over-layers can make the grating easier to fabricate.

A lossy rectangular-groove grating is specified by itsfilling factor F representing the fraction of the gratingperiod occupied by the grating material, by its groovedepth d2 and by its equivalent complex index of refrac-tion n2 - K2, where the imaginary part is the extinctionratio for the material. The filling factor and groovedepth of a high spatial-frequency rectangular-groovegrating that will produce zero reflectivity can be calcu-lated.5 7 The calculated grating parameters F and d2make the grating equivalent to a single homogeneouslossy layer in the long wavelength limit (wavelength islarge compared to grating period). Thus the homoge-neous lossy layer calculation may be considered as anintermediate step in calculating the parameters of theantireflection grating. In the limiting case, withoutdielectric overlayers, these results reduce to those pre-viously reported in Ref. 7. The analysis used for thecalculation of the equivalent homogeneous lossy coat-ing is rigorous impedance matching approach. Thestack of dielectric (or lossy) overlayers along with thesuperstrate are replaced by an equivalent superstrate,

and the problem can be reduced to the three-layerproblem of Ref. 7. The analysis is applicable for bothTE and TM polarizations and for any angle of inci-dence and wavelength. It is shown that multiple zero-reflectivity solutions exist for both TE and TM polar-izations. In fact, for an arbitrary complex refractiveindex of the substrate, any dielectric superstrate, andany number of dielectric (or lossy) overlayers of arbi-trary thickness and refractive index, there are multiplegroove depths together with the corresponding fillingfactors that produce antireflection behavior for eitherTE or TM polarization and for an arbitrary angle ofincidence. The inclusion of the overlayers increasesthe free parameters of the problem and gives addition-al design flexibility. Consequently, gratings with fill-ing factors closer to 50% and small (compared to theincident wavelength) groove depths can be designed.Necessary and sufficient conditions to have a solutionclose to a desired filling factor are derived in the appro-priate parameter space.

The analysis initially includes a single dielectricoverlayer and then is generalized to an arbitrary num-ber of layers. Some limiting cases are discussed. Ex-ample results for gold gratings with an incident freespace wavelength in the range from X0 = 0.44 to 12.0 mand a dielectric overlayer (n = 1.5) of arbitrary butconstant thickness are presented to demonstrate thedescribed analysis. The dependence of the zero-re-flectivity solution on the thickness of the dielectricoverlayer and on its refractive index is also demon-strated. Comparison of the zero-reflectivity solutionsfor the uncoated7 and overcoated gratings (this work)is presented, and general comments are made aboutthe advantages and disadvantages of the insertion ofdielectric overlayers. Finally, the sensitivity of thereflectivity of an antireflection grating (designed usingthe procedure described in this work) is examined as afunction of the filling factor and the groove depth ofthe grating. Zero reflectivity solutions are presentedfor both TE and TM polarizations over the entirerange of wavelengths and for angles of incidence be-tween 0 (normal incidence) and 900 (grazing inci-dence). Finally, the addition of one or more overlayerscan tune the zero-reflectivity solution due to the in-crease of the free parameters of the problem. Thusantireflection gratings insensitive to the incident po-larization can be designed.

II. Homogeneous Overcoated Layer and GratingGeometry

The geometry of a single homogeneous lossy layer(region 2) of thickness d2, overcoated by a dielectriclayer (region c) of thickness d, on a lossy substrate(region 3) is depicted in Fig. 1. Region 1 is assumed tobe lossless and is characterized by a refractive index ni.Region c is the coating region and for a lossless dielec-tric is characterized by its refractive index n. Regions2 and 3 are lossy and characterized by their complexrefractive indices n 2 - K2 and n3 - jK3, respectively,where the imaginary parts K2 and K3 are the extinctioncoefficients of those regions. An electromagnetic

15 October 1988 / Vol. 27, No. 20 / APPLIED OPTICS 4289

ni

I'l

n \ -K2

| ' niK3

(b)

II N1 Z Za

I t NZ I , - _K ' z 1Cj- _4 N2 -j 2 Z

N3- 3I: N.3-iK3 ~

(c) (d)

Fig. 2. (a) Diffraction geometry of a lossy rectangular-groove surface-relief high spatial-frequency grating overcoated by a dielectric layer.(b) The equivalent layer model for the grating in the long wavelength limit. (c) The normal-incidence equivalent layer model.(d) Transmission line model of the layer model of (c). (e) Transmission line model equivalent to model (d). (a)-(c) the continuous lines corre-

spond to propagating waves, while the dashed lines correspond to evanescent waves.

plane wave in region 1 (superstrate) is incident at anangle k1 on the dielectric coating (region c). On refrac-tion, homogeneous plane waves propagate in region cand inhomogeneous plane waves (plane waves withnonconstant amplitudes along their phase fronts)propagate in the lossy regions 2 and 3. The quantitiesAc 02, and 53 are the angles of refraction in regions c, 2,and 3, respectively. The angles 2 and 03 are complexdue to the complex character of the correspondingrefractive indices of regions 2 and 3. All angles ofrefraction are determined by the phase matching re-quirements (Snell's law) across the boundaries, that is,

nj sino, = n sine, = (n2 - jK2 ) sinO2

= (n3 - jK3 ) sing3l (1)

The incident wave may be either TE or TM polarized.The lossless superstrate medium is typically air (n, =1.0). The diffraction geometry of the surface-reliefrectangular-groove lossy grating which is equivalent tothe homogeneous lossy layer in the high spatial-fre-quency limit is shown in Fig. 2(a). Again the dielectricoverlayer is included. The grooves of the grating areassumed to be filled by the dielectric material of theoverlayer and are periodically spaced in the x directionwith period A. The grating vector K is given by K =(27r/A)I, where x is the unit vector in the x direction,and the boldface variable indicates a vector quantity.The depth of the grooves is d2. In a similar manner tothe homogeneous lossy layer geometry (Fig. 1), an elec-tromagnetic plane wave in the superstrate (region 1) isincident at an angle 'l on the dielectric overlayer (re-gion c) and then at an angle q0 on the lossy grating(region 2) on the lossy substrate (region 3) producingboth forward-diffracted and backward-diffractedwaves. Any incident polarization may be decomposedinto components with either the electric field or themagnetic field parallel to the grating grooves. Thefirst case corresponds to TE polarization, and the elec-tric field is perpendicular to the grating vector (E l K).The second case corresponds to TM polarization, andthe magnetic field is perpendicular to the grating vec-tor (1.1K). In this analysis the plane of incidence isthe x-z plane, and the grating vector also lies in that

plane. Thus there is no coupling between TE and TMpolarized waves. The angles of diffraction of the back-ward-diffracted (reflected) waves are given by thegrating equation (phase matching condition) iXo/n, =A(sinfo + sinkov), where i is the diffracted order, X isthe free space wavelength, p, is the angle of incidencein region 1 (superstrate) of refractive index nj, and 01iis the angle of diffraction in region 1 of the ith-orderbackward-diffracted wave. The angles of diffractionof the forward-diffracted (transmitted) waves are giv-en by the grating equation io = A(n, sinq5 + n3 sinO3 i),where 93i is the angle of diffraction in region 3 of theith-order forward-diffracted wave.

In the present work the primary case of interest isthat of high spatial-frequency gratings in which allforward- and backward-diffracted orders, other thanthe i = 0 orders, are cut off. In this situation, all the i >+1 and i < -1 forward and backward orders are eva-nescent. The i = 0 forward-diffracted order is not cutoff but decays as it enters the lossy substrate. Conse-quently, only the i = 0 backward (reflected) orderexists as a homogeneous propagating wave.

Ill. Single Homogeneous Layer Model

A. Layer Analysis

The amplitudes of the normalized electric field Erreflected back to region 1 and the normalized field Ettransmitted into region 3 are related by

(2)(EL) =- R. exp(d)f?, 2 exp(X2d2)A23 (Et)

where r = r1crc2r23,

Rij = + +pjj) for ij = 1,c,2,3, (3)

and i = diag(-yj,--yj), where i is a diagonal 2 X 2matrix with elements y and -y,, with i = c,2. Theamplitude transmissivities Tij and the amplitude re-flectivities Pij of the interfaces between media i and jare defined as functions of the parameters of the analy-sis in the following section. The complex variable yi isthe complex (in general) propagation coefficient ofregion i and is given by yi = (i + ji) cos,0 for i = c,2.


n3-jK 3

(a)

zin- 1

2

(e)

FA �- ""': N

The quantity ai is the attenuation coefficient of regioni (a, = 0 since the overlayer was assumed lossless anda2 = koK 2 , where ko = 2r/Xo), and fli is the phase con-stant (, = kon, and 2 = kon 2). The + sign in Eq. (3)applies for ElK (TE) polarization, and the - signapplies for H K (TM) polarization. The amplitudeof the normalized electric field reflected is [from Eq.(2)]

a grating period, Ey is approximately the same inthe grooves and in the ridges of the grating. Thus theaverage value of DY in the grating region [Fig. 2(a),region 2] is DY = EONEIKEY = eO[(1 - F)n2 COS20 + F(n 3- jK3)2 cos2 03]Ey, where a bar over a variable indicatesaverage value, NEIK is the effective index of the grat-

Er = ± pl, exp(rc + r 2 ) + Pc2 exp(-rc + r 2 ) + P23 exp(-rF - r 2 ) + P1cPc2P23 exp(rc - r 2 )

exp(rc + r2) + plcpc2 exp(-rc + r 2) + P2P23 exp(r - r2) + PlcP23 exp(-rc - r2)

where the + and - signs apply for ElK and HIKpolarizations, respectively, and ri = -yidi, for i = c,2. Ina following section it will be shown that the electricfield reflected Er may also be expressed in terms ofcharacteristic impedances.

B. Characteristic Impedance (Transmission Line)Representation

The amplitude transmissivities and reflectivitiesmay be expressed in terms of the characteristic imped-ances Z1, Zc, Z2, and Z 3 of the four regions (Fig. 1),respectively. The characteristic impedance of a loss-less region is real while the characteristic impedance ofa lossy region is complex. For El K (TE) incidentpolarization, the transmissivities rij and the reflectivi-ties pij(ij = 1,c,2,3) are

Tij = 2Zj/(Z 1 + Zi),

(5)

Pij = (Zj - Zi)/(Zj + Zi),

where ij = 1,c,2,3, and the characteristics impedancesfor this polarization are given by

Zi = Z0I(ni - jKi) coski = Z0 (Nj - jKj), (6)

where i = 1,c,2,3,Zo is the characteristic impedance offree space, and Ni and Ki are the normal-incidenceeffective refractive index and extinction coefficient,respectively. If region i is lossless, then Ki = 0. ForH K (TM) incident polarization, the transmissivitiesand reflectivities are

Tij = 2Z3 costkj/(Zj + Z1) cos4;,(7)

Pij = (Zj - Zi)/(Zj + Zi),

where ij = 1,c,2,3, and the characteristic impedancesfor this polarization are

Zi = Zo cosoi/(ni -jKi) = Z0 /(Nj - jKj), (8)

for i = 1,c,2,3.

C. Effective Index of Grating at Long Wavelengths

The effective refractive index of the grating may beapproximated using the electromagnetic boundaryconditions on the electric field E and the displacementD in the long wavelength limit and the average fields inthe grating region.5'7'3437 For El K (TE) polariza-tion, the electric field is parallel to the grating grooves.Since the tangential electric field must be continuousand the free space wavelengths is long compared to

ing for E K polarization, and EO is the permittivity offree space. The effective refractive index of the grat-ing NEIK is a complex effective index and is given by

NE±K = (n2 - K2) COS02 = [(1 - F)n2 cos2 p0

+ F(n3 - j3)2 COS203]1/2

= [(1 - F)N2 + F(N3 -jK3 )2 ] 112 . (9)

The normal-incidence effective refractive indices Niand extinction coefficients Ki(i = 1,c,2,3) can be de-fined using the expressions for the characteristic im-pedances given by Eqs. (6) and (8). For incident E l K(TE) polarization, the real-valued effective quantitiesare

Ni = Re[(ni - jKj) coski],

(10)

Ki =-Im[(ni - jKi) COSOJ],

where Re and Im denote real and imaginary parts,respectively, and i = 1,c,2,3.

For HI K (TM) polarization, the magnetic field ofthe incident plane wave is parallel to the gratinggrooves. Since the normal component of the electricdisplacement must be continuous and the wavelengthis long compared to a grating period, D. is approxi-mately the same in the grooves and ridges. Therefore,the average value of E isE = [1/EoNHK]DX = [(1 -F)cos2Oc/n 2 + F cos2 03/(n3 - K3)2 ] (D.eo), and thus thecomplex effective index NH1K polarization is

N JKZ = [(1- F) COS2 0,/n2

+ F cos203 /(n3 - jK3)2]-1/2

= [(1 - F)/N2 + F/(N3- jK3)2]-1/2. (11)

For incident H I K (TM) polarization, the real-valuedeffective quantities for normal-incidence are

N = Re[(ni - jKi)/cotSi,

(12)

Ki = -Im[(ni - jK)/cos],

where again i = 1,c,2,3.In summary, in the long wavelength limit, the grat-

ing has the same reflection and transmission charac-teristics as the equivalent homogeneous lossy layer.This is shown in Figs. 2(a) and (b). The normal inci-dence equivalent model is shown in Fig. 2(c).


(4)

: r

IV. Conditions for Zero Reflectivity at Long Wavelengths

A. Impedance Matching

The optical (or microwave) characteristics of therectangular-groove metallic surface-relief grating canbe very accurately modeled in the long wavelengthlimit by a single homogeneous layer with complex re-fractive index NEIK [Eq. (9)] for E lK (TE) polariza-tion and NHIK [Eq. (11)] for H K (TM) polarization.In addition, single homogeneous layers may be conve-niently modeled as transmission lines with parametersdetermined by the characteristics of the correspondingregions (for example, layers that correspond to regionsc and 2 in Fig. 1). The termination load of that trans-mission line can be determined by the characteristicsof region 3 (Fig. 1). The transmission line model of thelayered model of Fig. 1 is shown in Fig. 2(d).

The complex input impedance Zin for the combina-tion of transmission lines of characteristic impedancesZC and Z2 and lengths d, and d2, respectively, terminat-ed with a load of impedance Z3 , is given by

alent zero-reflectivity equation to Eq. (14) can bewritten as

Zf[tanh(kK2d2) + j tan(k0N2d2)] +

Z2 - , - jZ, tan(koNV'd,)

[ 1 -(ZI/Z,) tan(koKd)J

[1 + j tanh(koK 2d2) tan(kN 2d2 )] -

Z - jZ tan(k0 AV'd,)Z3 C~~~~, [tanh(kJCK2 d2 +1-j(Zl/Zc) tan(koK'd,)

i tan(k0 N2 d2)] = 0. (15)

Equation (15) is valid for both E l K and H K polar-izations provided the corresponding characteristic im-pedances and effective and primed indices are used.From Eq. (15) it is observed that if an equivalentcharacteristic impedance Zjc is defined so that

Z - jZc tan(koNcdc)

1-i(Z 1IZ,) tan(koKdc)(16)

Z2 [Z3 + Z2 tanh(r 2 )] + Z, tanh(rF)[Z 2 + Z3 tanh(r2 ) 1Z[Z 2 + Z3 tanh(r 2 )] + Z2 tanh(r,) [z3 + Z2 tanh(r 2)i (

where ri = yidi, and yi is the complex propagationcoefficient of the transmission line given by yi = (ai +jfli) cosoi = a + jif (i = c,2; for dielectric overlayer ac =0). The real part a is the attenuation coefficient ofthe transmission line, and the imaginary part fl3 is thephase constant. Using the identity tanh(a + jb)[tanh(a) + j tan(b)]/[1 + j tanh(a) tan(b)], the hyper-bolic tangents of Eq. (13) can be written as tanh(rc) = jtan(fl'dc) and tanh(rc) = [tanh(a'2 d2) + tan(3 2d2)]/[1+ j tanh(a'd 2) tan(3'2d2)]. The phase attenuation co-efficients can be written as = koN' and = kOK'respectively. For TE polarization the primed vari-ablesN andK areN = N andK' = Kifor i = c,2. ForTM polarization the corresponding primed variablesareN = Re[(Ni -jKi) cos25j] andK = -Im[(N -jKj)cos2

0i]. Substituting the characteristic impedances,Eq. (6) or (8), the primed quantities, and either Eq.(10) or (12) into Eq. (13), the input impedance Zin maybe expressed as a function of d2 , d, Xo, n, n2, n3, r, 02,

and 03.

The normalized electric field Er may be expressed interms of the input impedance Zin and the characteris-tic impedance Z1 of the first region as Er = 1(Zin -Z1(Zin + Z) (where the + sign corresponds to TE inci-dent polarization and the - sign corresponds to TMincident polarization). This may be straightforwardlyshown using Eqs. (4), (5) or (7), and (13). To have zeroreflected electric field, the input impedance Zin of Eq.(13) must be matched to the characteristic impedanceZ1, of the first region, as shown in Fig. 2(d). That is,

zero reflectivity Zin = Z1. (14)

Substituting the hyperbolic tangents (as has beenmentioned previously) into Eq. (14) produces a com-plex equation that specifies the condition for zero re-flectivity for E 1K or H11K polarization. The equiv-

Eq. (15) has the same form as the corresponding equa-tion of the three-layer problem of Ref. 7 (where nocoating is included). From Fig. 2(e), the zero-reflec-tivity condition can be stated as

zero reflectivity Zin = zic1 (17)

where Zin is shown in Fig. 2(d). Using the equivalentcharacteristic impedance Z a three-layer problemcan be identified, and this situation is depicted in Fig.2(e). Thus, the four-layer problem can be trans-formed into a three-layer problem by combining theeffects of the input region 1 and region c (coating).This effective input region has a characteristic imped-ance Z (instead of Z1), a corresponding effective re-fractive index Nl, and an extinction coefficient Kjc.Those quantities can be defined from the equation Z1,= Zol(Nc - jK1,). Using the previous equation andEq. (16), the quantities N1 c and K1, are

NN N 2[1 + tan2 (koNcd,)]

N 2 + N tan2(kN'dc)

NC(N 2-N 2) tan(k6N'dc)

Kl N + Ni tan (k N'cdc)

(18)

(19)

Equations (18) and (19) are valid for both TE and TMpolarization if the corresponding equations for N1, Nc,and Nc are used. Substituting Eqs. (6) or (8), (10) or(12), (16), (18), (19), and the definitions of N' and K'(i = c,2) into Eq. (15), an equivalent condition for zeroreflectivity is obtained. This equation is applicablefor both TE and TM polarization, and it can be sepa-rated into a real-part equation and an imaginary-partequation. The real-part equation is


(NN3- K1 K3) tanh(koK2d2) +(N1 ,K3 + KCN3) tan(koN2 d2) +

(NCN2 - N2N3 - KjK2 + K2K3) +

(NK 2 - K2N 3 + KjN 2 - N2 K 3) X

tanh(koK2 d2) tan(koN2d 2) -

(N2 - K2) tanh(kK2 d2 ) - 2N2K2 tan(kN2d2) = O.

The imaginary-part equation is

-(N1 ,K3 + K1CN3) tanh(koK2d2) +

(N,N3 - K1,K3) tan(koN2d2 ) -(N1K2-K2N3 + K1,N2-N2K3) +(N 1 CN2 - N2 N3 + K1jK 2 + K2K3 ) x

tanh(kK 2 d2) tan(koN2d2) -

(N2 - K2) tan(koN2 d2) + 2N2K2 tanh(kJK2 d2) = O.

complex equation and may be separated into a real-part equation and an imaginary-part equation. Theseare

(23)

(20)

(21)

Equations (20) and (21) are valid for both ElK andHlK polarizations provided that the correspondingeffective quantities are used [Eqs. (10) for El K andEqs. (12) for HlK]. At this point some special casescan be discussed. If there is no coating, n, = n1 and d,can be arbitrary. In this case N1 c = N, K1i = 0, andEqs. (20) and (21) reduce to the corresponding equa-tions of the no coating problem.7 The same conditionis satisfied (N, = N1 and K1i = 0) when d, = 0 or whenkoN'dc = m-r (m = 1,2,.. .). In the latter case thecoating has no effect on the reflectivity of the struc-ture.

The given quantities (parameters) in the problemare the incident index ni (region 1), the incident angle01, the coating index nc and thickness dc (region c), thesubstrate index n3, the substrate extinction coefficientK3, and the free space wave vector magnitude ko (orequivalently the incident free space wavelength X0).The extinction coefficient Kc is also a parameter in thecase of a lossy coating, which is discussed in AppendixB. Using the phase-matching condition, Eq. (1), theangles bc and 03 can be specified. Therefore, theknown quantities in Eqs. (20) and (21) are N1 c, Kjc, N 3 ,K3, and ko. The unknown quantities associated withregion 2 (lossy layer or equivalent grating region) areN2 , K2, and d2 (since N 2 and K'2 can be expressed asfunctions of N2 and K 2 as shown in Appendix A).Thus there are two equations [Eqs. (20) and (21)] andthree unknowns (N2 , K2 , and d2). As a result theproblem remains underspecified. Taking into ac-count the equivalence between the complex refractiveindex of the lossy homogeneous layer and the complexeffective index of the grating, a third equation is devel-oped in the following sections.

B. E K (TE) Polarization

Since the homogeneous layer model is a representa-tion of the grating in the long wavelength limit, thecomplex effective refractive index of the layer, N2 -

jK 2 , must be equal to the effective index of the grating.Thus for El K (TE) polarization,

N 2 -jK 2 = NEIK, (22)

where NEIK is given by Eq. (9). Equation (22) is a

N 2K2 = FN3K3, (24)

respectively. Equations (20), (21), (23), and (24) rep-resent four independent equations with four un-knowns (N2, K2, F, and d2). Due to the transcendentalnature of Eqs. (20) and (21) analytic solutions are notpossible. However, these equations can be solved nu-merically with the procedure described in Appendix A.The resulting values of the filling factor F and thegroove depth d2 completely specify the rectangular-groove surface-relief grating that will exhibit zero re-flectivity for E K (TE) polarization. A special caseof the overcoated grating is when d, = 0. In this casethe material (of refractive index n, different from air)fills the grating grooves.

C. H 1 K Polarization

For H K (TM) polarization the complex refractiveindex of the layer, N 2 - jK2, must be equal to theeffective index of the grating for this polarization, thatis,

N 2 - jK2 = NHIK, (25)

where NHIK is given by Eq. (10). The complex Eq.(25) may be separated into a real-part equation and animaginary part equation, which are

-2[(1-F)(N2 + K2)2 + FN2(N2 - K2)]

[(1- F)(N 2- K2) + FN2]2 + [2(1 - F)N3K3 ]2

N2K 2FN4N2 K3 ,(27)

[(1 - F)(N3-K2) + FN] 2 + [2(1- K]2

respectively. Equations (20), (21), (26), and (27) like-wise represent four equations with four unknowns (N2,K 2 , F, and d2) and can be solved numerically as de-scribed in Appendix A. Similar to the TE polarizationcase, the resulting values of the filling factor F and thegroove depth d2 completely specify the surface-reliefrectangular-groove grating that will exhibit zero re-flectivity for H K (TM) polarization.

V. Generalization for Multiple Overlayers

The analysis that has been presented for a singleoverlayer can be easily generalized to include multipledielectric overlayers. The refractive indices of themultiple dielectric coatings are denoted by nCm andtheir thicknesses by dcm, where m = 1,2,.. . ,N, and N isthe total number of dielectric overlayers. The geome-try of the lossy homogeneous layer (region 2) on a lossysubstrate (region 3), overcoated by multiple overlayers(regions c1 ,c2 ,. .. ,cn) is shown in Fig. 3. The corre-sponding geometry for an overcoated surface-reliefrectangular-groove grating is shown in Fig. 4. Usingthe homogeneous layer analysis described in Sec. II.A,the normalized electric field reflected can be calculat-ed by Eq. (2) where, instead of R1 C exp(AXd,)R 2, the


N 2-K 2 = (1 - 17)N 2 + F(N 2- K 2),2 2 C 3 3

factor R1,1 exp(Ac1 dc1)flC1 2 exp(Ac 2 d0 2 ) .. RCN-lcN

exp(cNdN)RCN2 should be used. Furthermore, =T1cl'crc2 . .. TCN2r23. As a result, a similar but morecomplicated version of Eq. (4) can be found for Er.This expression is equivalent to the transmission linemodel expression. The transmission line model of themulticoated homogeneous lossy layer is equivalent tothe one shown in Fig. 2(d), but Zc is replaced by thecascaded series of ZC1 ,ZC2 , ... ZCN. Using a proceduresimilar to the one described previously, the zero-reflec-tivity condition can be written as Zin = Zjc [equivalentto Eq. (17)], with Z1, including all the overlayers andthe superstrate. The characteristic impedance Zin re-mains the same, since it depends only on the character-istics of regions 2 and 3. The quantity Z 0 as given byEq. (16) differs due to the multiple overlayers. Thecharacteristic impedances and the effective quantitiesof the multilayer problem are defined similarly to Eqs.(5)-(8), (10), and (12). Since Zjc = Z0 /(N 0 -jKlc)

only the values of N1 c and K1 c will be changed toinclude the effect of all the overlayers. Those valuescan be calculated by the following recursive equations:

X.-.Ncl[l + tan2(DI)]

XIl N [N0 , - Y..1 tan(D1 )]2 + [XI-, tan(D1 )]2

[Y-1 + N0, tan(DI)] [N, - Y-1. tan(D] - Xi' tan(D^)

=cN~, [1N, - Y tan(DI)] 2 + [Xl. 1 tan(DI)] 2

where DI = k0 Nc1 d,1, Xo = N 1 , Yo = 0, and = 1,2, ... ,N.Equations (28) and (29) are valid for both EL K (TE)and HK (TM) polarizations provided the corre-sponding values for N,, and N', are used. The quanti-ties N,, and K 0 are

N10 = XN,

Kj0 = YN.

(30)

(31)

The zero-reflectivity condition is again a complexequation that can be decomposed into a real-partequation and into an imaginary-part equation that arethe same as Eqs. (20) and (21), respectively, but withN1 and K1 given by Eqs. (30) and (31). To form againa system of four equations with four unknowns (N2, K2,F, and d 2), Eqs. (23) and (24) or Eqs. (26) and (27) canbe used for ElK (TE) or HlK (TM) polarization.The only difference is that N, should be replaced byNON, since the nth dielectric coating fills the gratinggrooves.

In summary, the inclusion of multiple dielectriccoatings does not change the general form of the solu-tion of the problem. The effect of the multiple coat-ings is concentrated in the quantities N1 and K1l. Asa result the procedure that is described in Appendix Adoes not change except that instead of N, the quantityNON must be used. The case of lossy coatings is dis-cussed in Appendix B.

n,Region 1

d -Region c c nc

dC Region c nc2C2

dc., Region c 1 CXcN_1

dc Region CN CNn

d, Region 2 02 'j 2- J'K2

Region 3 n3-jK3

3

Fig. 3. Geometry of multiple overlayers (regions c,c2, .cn) ontop of a lossy layer (region 2) on top of a lossy substrate (region 3).

(28)

(29)

Region

01/

d0 Region , nc,

dc Region c, nx

'dc. Region c n., iN-Z ce9Tn C.,

d'r-, Region C., nCN,

dC2 Region C,

d, Rion 2

Region 3, n, -JK.

0

Fig. 4. Diffraction geometry of the multiple overcoated lossy rect-angular-groove surface-relief high spatial-frequency grating.

lent surface-relief grating may be formulated by ex-pressing the impedance matching conditions [Eqs.(20) and (21)] as

tan(koN 2 d2 ) = E + G tanh(kJK 2 d2 ), (32)

tanh2 (kKC2d2 ) + P tanh(kK 2 d2) + 1 = 0 (33)

VI. Calculational Procedure

To aid in numerical calculation, the problem of de-termining the lossy homogeneous layer and the euiva-

where E, G, and P are defined in Appendix A. Thegiven parameters of the problem are X0, n1 , 01 n, d, n3,and K3. (In the case of multiple coatings nc and d arereplaced by ncln c2 ... ,nCn and ddc 2 , . ,dn, respec-


tively.) The filling factor F is in the range between 0and 1. The calculational procedure starts with thecomputation of the effective quantities N 1 , N0 , N3 , andK3 for a given incident polarization. Then the fillingfactor F is increased from 0 to 1. For every F value thecoefficients E, G, P, N 2 , and K'2 can be computed asfunctions of F as described in Appendix A. Conse-quently, the only unknown of Eqs. (32) and (33) is d2.Solving Eqs. (32) and (33) independently, the root lociof d2 as functions ofF can be computed. Any intersec-tion of the two root loci represents a solution d2 thatsatisfies both Eqs. (32) and (33) simultaneously for thesame value of F. Then for that specific value of F, thequantities N2 , K2 and n 2 , K2 can be computed as de-scribed in Appendix A.

Inspection of Eq. (32) reveals that multiple solutionsare possible, each corresponding to a different branchof the tan(koK'2d2) function. These branches are la-beled with an index m. The range corresponding tothe mth branch is (2m - 1) zr/2 < kON'2 d2 < (2m + 1)ir/2for m = 0,1,2,..., since N' > 0 (for a component ofpropagation along the incident wave vector), and onlypositive values of the thickness d2 are allowed. Anexample root loci diagram is shown in Fig. 5, where thea branch corresponds to the root locus of Eq. (33),while the branches labeled 1-5 correspond to the firstfive branches of Eq. (32). The intersection points A1,to A5 give the first five solution sets (n 2 , K2, F, and d2).Note that discontinuities in the branches of Eq. (32)occur due to the change in sign of the coefficients E andG as functions of F.

VIl. Necessary and Sufficient Conditions for Solution

An acceptable solution set [N2,K2,F,d2] or[n2,K2,F,d2] of Eqs. (32) and (33) must satisfy the in-equalities: 0< F< 1, d2 > 0,N 2 >0,n 2 >0,and K2 >0,K2 > 0. Equation (33) is a second-order algebraicequation with respect to p = tahn(koK'2 d2 ) and can bewritten as

p2+pp + 1 = . (34)

Since K'2,d2 > 0, and p = tanh(koK'2 d2 ) < 1, one of theroots of Eq. (34) should lie in the interval (0,1). If -2< P < + -, then Eq. (34) has only negative or complexsolutions. If P < -2, the only solution of Eq. (34) inthe interval (0,1) is

0 < p1 = tanh(kOKd 2) = < 1, (35)

where P is defined in Appendix A. Consequently, thenecessary condition for existence of an acceptable so-lution of Eqs. (32) and (33) is

P = P(F;X0,Xo,,n,n,,d,,n3,K3) = P(F) <-2, (36)

where the filling factor F is a parameter in the aboveinequality. Applying Eq. (36) it is possible to computethe range of filling factor values, for which zero-reflec-tivity solutions may be found, given the incident freespace wavelength X0, the angle of incidence 01, thematerial parameters nlncn3,K3, the thickness of thecoating layer d0, and the incident polarization, E l K or

ROOT LOCI

a0

0

0.0 0.2 0.4 0.6 0.8 1.0

FILLING FACTOR, F

Fig. 5. Rootloci of Eqs. (32) and (33). The a branch correspondstosolutions of Eq. (33), while the m = 1-5 branches correspond tosolutions of Eq. (32). Due to the change in sign of coefficients E andG of Eq. (32), branch discontinuities are possible. Intersectionpoints A1,A2, . A5, . . ., correspond to zero-reflectivity solutions.

HK [since all parameters in Eq. (Al) depend on thepolarization]. Inequality (36) guarantees an accept-able solution of Eq. (33) but not of Eq. (32). To obtaina sufficient condition for an acceptable solution ofboth Eqs. (32) and (33), Eq. (32) should also have anacceptable solution common with the solution of Eq.(33). Substituting d from Eq. (34) into Eq. (32) gives

tan[N Ntanh1 p 2 -4)E-G p_ P -

(37)

or

Q(F;XO,0k,n1,nc,dc,n3,x3 ) = Q(F) = O. (38)

If Eq. (38) has a solution F* so that 0 < F* < 1 for thegiven Xo, q1, n1 , nc, n 3, K3, and d,, then it is guaranteedthat the system of Eqs. (32) and (34) has an acceptablesolution, i.e., a zero-reflectivity high spatial-frequencysurface-relief rectangular-groove grating or an equiva-lent lossy layer can be designed. If 0 < F,, F2 < 1, andQ(F) and P(F) are both continuous in the interval[F,F 2], the necessary and sufficient conditions for zeroreflectivity solution are

Q(F1)Q(F2) < 0,

P(F) <-2, forF FF< F2. (39)

If inequalities (39) are satisfied, there is a value of thefilling factor F*, a groove depth d*, and n*,K2 (of theequivalent lossy layer) that satisfy both Eqs. (32) and(34) and consequently Eq. (14). The valuesF*,d2,n*,K* comprise a zero-reflectivity solution set andcan be found using the calculational procedure de-scribed in the previous section. In case of multiplegratings instead of n and d the parametersnclnc 2 . . . nCN and d 1,d 2 ... ,dcN should be used.

Vil. Limiting Dielectric Case

The case of multiple dielectric overlayers over adielectric substrate or the equivalent case of the multi-


coated dielectric surface-relief rectangular-groovegrating on a dielectric substrate can be derived as aspecial case of the general lossy problem that has beenpresented. Regions 2 and 3 of Figs. 1 and 2 (or of Figs.3 and 4) are dielectric; therefore, K2 = K3 = . As aconsequence K 3 = K' = K2 = K2 = 0. In addition N2 =N2 as it can be shown using Eqs. (A16)-(A21). Usingthe above relations, Eqs. (20) and (21) are simplified to

KirN 3 tan(kGN2d2) + N 0 N2 - N2N 3 = 0,

(N 0N3 - N2) sin(k0 N2d2) = 0-

The solution of the system of Eqs. (40) and (41) isN 2 = (N,,N3) ,

d2 rN2tan' [(NIC) N 3--N + m 2N27rN2 N3 KjC N

El K Polarization

dc/Xo= 1, n0 = 1.5, m,= O0

0.3

U. 0.2Le

(40) E

(41) 3U. 0.1

(42)

(43)

where m = 0,1,2,.... In the uncoated case Eqs. (42)and (43) reduce to N2 = (NlN 3 )1/2 and d2 = (2m + 1)X0/4(NN 3)1/2 , that have been calculated for the specialcase of normal incidence ( = 0) in Ref. 5. From thevalue of N2 of Eq. (42) the value of n2 is found to be n2 =

(N2 + n 2 sin 201)1/2 for ElK (TE) polarization and n2

= N2f [1 + (1 + 4ni sin 2 1I/N2)1/2]/211/2 for HK (TM)polarization. Thus n2 and d2 specify the homogeneouslayer of region 2. This layer is equivalent to the sur-face-relief grating in the long-wavelength limit. Thefilling factor for both polarizations can be computedusing Eqs. (9) and (10) and are

F = (N0N3 - N2)/(N2 - N) (44)

for TE polarization and

F = N3(NN 3 - N)/N,(N - N) (45)

for TM polarization. If no coating is included, thenNi = N = N1 and F = N1l(N + N3) for TE polariza-tion and F = N3/(N1 + N3) for TM polarization. Thelatter values of F have been found in Ref. 5 for normalincidence.

IX. Required Gratings for Gold Surfaces

A. Wavelength Dependence

Some example designs of antireflection gratings ongold substrates are presented. As an intermediateresult, antireflection lossy coatings on lossy substratesare designed. The gratings (or coatings) are overcoat-ed by one dielectric overlayer. The calculational pro-cedure described previously and the normalized equa-tions of Appendix A were solved for free spacewavelengths from X = 0.44 to 12.0,gm (violet to infra-red). The tabulated refractive indices of bulk gold7,38were used in the computations. The refractive indexof the dielectric overlayer is n = 1.5 (constant for therange of free space wavelengths). As mentioned previ-ously, multiple solutions are possible, each corre-sponding to a different branch of Eq. (32).

For El K (TE) polarization at normal incidence onovercoated gold gratings, the calculated values of thefilling factor F and the normalized groove depth d2/XOfor the corresponding overcoated grating are presented

0.0

with coating 1.5

10

1.0 lw0w

00EC0

0.5 IiiN

0z

0.5 1 2 5 10

WAVELENGTH, X, (pum)

Fig. 6. Filling factor F and the normalized groove depth d2/XO toproduce zero reflectivity for ELK polarization in the range of freespace wavelengths from X = 0.44 to 12.0 Am at normal incidence

with a coating (of n, = 1.5 and d0/Xo = 1) and without a coating. 7

in Fig. 6, compared with the corresponding antireflec-tion values without the coating.7 The normalizedthickness of the coating is d = 1 over the entirerange of incident wavelengths. Only the m = 2 branchsolutions are included. When an overlayer is used, thebranch solution appears to have discontinuities, sincethe solution presented belongs to the m = 1 branchsolution for X < 0.5 ,um and belongs to the m = 2branch solution for X > 0.50 ,um. The m = 1 branchsolution does not exist in the no coating case. Evenwhen the coating is included, the m = 1 branch solutiondoes not exist for 0 > 0.50,gm. As an example inter-mediate step, the calculated real n2 and imaginary K2parts of the complex refractive index of the lossy ho-mogeneous layer (required for antireflection behaviorat Xo = 0.5,Mm) are n 2 = 1.2123 and K2 = 0.20018 (for m= 2), respectively. (The corresponding antireflectionvalues without the coating are n2 = 0.78590 and K2 =0.21655 for m = 2.) Both the refractive index n2 andthe extinction coefficient K2 increase with the inclusionof the overlayer, and they are both <1.5 (= n) for theentire range of wavelengths used. The filling factor ofthe antireflection grating increases with the inclusionof the overlayer while the groove depth decreases.Both of these effects are desirable in the design ofantireflection gratings for E K (TE) polarization,since this polarization is characterized by small fillingfactors and large groove depths when there is no coat-ing. The m = 1 or 2 (depending on the wavelengthrange) branch solutions give the largest filling factorand the shallowest groove depth for El K incidentpolarization. Similarly to the no coating case,7 evenwhen a dielectric overlayer of n = 1.5 is used, thefilling factor tends toward small values (but with aslower rate than the no coating filling factor case) asthe free space wavelength increases.

ForH K (TM) polarization at normal incidence onovercoated gold, the calculated values of filling factor


F and the normalized groove depth d2/Xo for the corre-sponding overcoated grating are presented in Fig. 7,compared with the corresponding values without thecoating.7 For this incident polarization, the m = 1branch solution exists, and it gives the largest fillingfactor and the shallowest groove depth. Similarly forEl K polarization the optical thickness of the dielec-tric overlayer was held at constant d,/Xo = 1. Again, asan example intermediate step, the calculated real n2and imaginary K2 parts of the complex refractive indexof the lossy homogeneous layer (required for antire-flection behavior at X0 = 0.5 gm) are n2 = 2.15621 and K2= 0.40947 (for m = 1), respectively. (The correspond-ing antireflection values without the coating are n2 =1.8474 and K2 = 0.41372 for m = 1.) Both the refractiveindex n2 and the extinction coefficient K2 do not changeappreciably from their corresponding values withoutcoating. However, both the filling factor and the nor-malized groove depth of the grating decrease with theinclusion of the overlayer. This effect is desirable indesigning gratings closer to 50% filling factor. In con-trast to the El K case, the refractive index is always>1.5 (= nj), while the extinction coefficient K2 remainsalways <1.5. Similarly to the HK no coating case,the filling factor of the grating tends toward unity(with a slower rate) as the wavelength increases.

B. Angle of Incidence Dependence

Antireflection surface-relief gratings (and their cor-responding dielectric overcoated lossy homogeneouslayers) can also be designed for oblique incidence forboth ElK and HIK polarizations. Example calcu-lations for gold substrates are presented for a freespace wavelength of 0.5 Aim. The real and imaginaryparts of the complex refractive index of gold at 0.5 gmare n3 = 0.80 and K3 = 1.82. The refractive index of thedielectric overlayer is again n, = 1.5. For zero reflec-tivity with E l K polarization, the filling factor and thenormalized groove thickness of the surface-reliefequivalent in the long-wavelength limit grating areshown in Fig. 8 compared with the values withoutcoating.7 The normalized thickness of the coating isd/X = 0.15. No discontinuities appear for the chosencoating thickness. However, only the m 2 2 branchsolutions exist. In Fig. 8 only the m = 2 branch solu-tion is shown. Again, as an example intermediatestep, the calculated real n2 and imaginary K2 parts ofthe complex refractive index of the lossy homogeneouslayer (required for antireflection behavior at 300 angleof incidence) are n2 = 1.27692 and K2 = 0.14862 (for m =2), respectively. (The corresponding antireflectionvalues without the coating are n2 = 0.83570 and K2 =0.15547 for m = 2.) The refractive index increasesmonotonically, while the extinction coefficient de-creases monotonically, both in a similar fashion withthe no coating case. The filling factor of the gratingdecreases monotonically as the angle of incidence in-creases. However, the groove depth remains almostinsensitive to the angle of incidence even when theoverlayer is used. An important difference when acoating is included is that there is no zero-reflectivity

H.l K Polarization

dc/X= 1 nc = 5 =

UL.

0I-

'0a:

zJ:U1

I-

uJ0.

W0

LU00(30LU

N

0Z

WAVELENGTH, X (pm)

Fig. 7. Filling factor F and the normalized groove depth d2/XO toproduce zero reflectivity for H 1K polarization in the range of freespace wavelengths from X0 = 0.44 to 12.0 ,um at normal incidencewith a coating (of n, = 1.5 and d,/Xo = 1) and without a coating. 7

0.2

0-'0

(32

E I K Polarization

d0/Xo= 0.15, n0 = 1.5, X.= 0.5pm7.5

I-

M

0.EL

W

0

LU

00

0

c

L)

o~~~~~~~~~~~~~~

z

0 15 30 45 60 75 90

ANGLE OF INCIDENCE, 4, (degrees)

Fig. 8. Filling factor F and the normalized groove depth d2/X toproduce zero reflectivity for ELK polarization for a free spacewavelength X0 = 0.5 um as a function of the angle of incidence with a

coating (of n, = 1.5 and d,/Xo = 0.15) and without a coating. 7

solution for 01 > -80°. [The necessary condition ofEq. (36) is not satisfied for 01 > 80°]

For zero reflectivity with HK polarization, thefilling factor and the normalized groove thickness ofthe surface-relief equivalent in the long-wavelengthlimit grating are shown in Fig. 9 compared with thevalues without coating.7 The normalized coatingthickness is d,/Xo = 0.10. No branch discontinuitiesappear for the chosen parameter values. Again, as anexample intermediate step, the calculated real n2 andimaginary K2 parts of the complex refractive index ofthe lossy homogeneous layer (required for antireflec-tion behavior at 30° angle of incidence) are n2 = 2.3630and K2 = 1.0493 (for m = 1), respectively. (The corre-sponding antireflection values without the coating aren2 = 1.8519 and K2 = 0.46931 for m = 1.) Both the


0I-

U

'0

U.z

IL

0.3-

1<

I-

0.LU

0_CC3o

(L

00

0

N

'0

0Z

0 15 30 45 60 75 90

ANGLE OF INCIDENCE, 0, (degrees)

Fig. 9. Filling factor F and the normalized groove depth d2 /X toproduce zero reflectivity for HI1K polarization for a free spacewavelength from X0 = 0.5 ,um as a function of the angle of incidencewith a coating (of n, = 1.5 and d/Xco = 0.10) and without a coating. 7

UL

o6 0.2

'0

zI-

0.1

E I K Polarization2 n0 = .5, X= 0.5pm, b= 0°

2 \2/

1 / \ ; ;/y3

0.0 0.1 0.2 0.3

NORMALIZED COATING THICKNESS, dC/X.

2.0-

n 1.5-

refractive index and the extinction coefficient decreasemonotonically as functions of the angle of incidenceinstead of increase as in the no coating case. Thefilling factor decreases and the groove depth increasesmonotonically as functions of the angle of incidence.No zero reflectivity solution can be found for 01 >82° [since again Eq. (36) does not hold]. In all the

presented plots (Figs. 6-9) the computed solutions aregiven with solid dots. A continuous line, calculated bycubic spline interpolation, has been provided for easeof interpretation.

C. Dielectric Coating Thickness Dependence

Inspection of Eqs. (18) and (19) reveals that N1, andK1, are periodic functions of the coating thickness d,with a period Xo/2N'. Thus the solution set of theproblem [n2,K2,F,d2/Xo] is a periodic function of thecoating thickness. This is an additional degree offreedom that can be exploited for designing antireflec-tion coatings.

For E K polarization, the calculated filling factor Fand normalized groove depth d2/X0 are presented inFigs. 10(a) and (b), respectively, as functions of thenormalized coating thickness for normal incidence. Inthis case nc = 1.5, and the normalized period is 1/(2N')= 1/(2nc) = 1/3. As an example intermediate step, thecalculated real n2 and imaginary K2 parts of the com-plex refractive index of the lossy homogeneous layer(required for antireflection behavior at dc/Xo = 0.10coating optical thickness) are n2 = 1.18407 and K2 =0.22441 (for m = 2), respectively. The branch solu-tions m = 2-5 are included. The m = 1 branch solutiondoes not exist for the used Xo = 0.5-gim free spacewavelength. Discontinuities in the branch solutionscan be observed in Figs. 10(a) and (b). These are dueto changes in the sign of the coefficients E and G of Eq.(32) as was mentioned before. Wherever there is adiscontinuity in the branch solution the actual solu-tion may be continuous. For example, in the gapbetween the m = 4 and m = 5 branch solutions [Fig.

(3

'0

0cc

1.0-

0.5-

0.0

(a)

E K Polarizationn0 = 1.5, No= 0.5pm, 01 = °°

5_ 5

4

3~~~~~~~

22

0.1 0.2 0.3

NORMALIZED COATING THICKNESS, d/X.

(b)

Fig. 10. Filling factor F and the normalized groove depth d2/X toproduce zero reflectivity for EK polarization for a free spacewavelength X = 0.5 jum as a function of the coating thickness dX atnormal incidence with a coating (of n = 1.5). Branch solutions for

m = 2-5 are presented.

10(a)], the solution for the filling factor may be contin-uous and may belong either to the m = 4 or the m = 5branch solution [but in Fig. 10(a) it was not computed].This result is because there may be more than one mbranch solution. (The m branch intersects the abranch in Fig. 5 more than once.) In Fig. 10(a) it isshown that for the specific wavelength, angle of inci-dence, and coating material the largest filling factorthat can be achieved is -0.30 (or 30%) for d/X 0o 0.04+ (1/3), 1 = 0,1,2 ....

For H1K polarization, the calculated filling factorF and normalized grgove depth d2/Xo are presented inFigs. 11(a) and (b), respectively, as functions of thenormalized coating thickness for the same coating (n,= 1.5) and at normal incidence. As an example inter-mediate step, the calculated real n2 and imaginary K2parts of the complex refractive index of the lossy ho-mogeneous layer (required for antireflection behaviorat d/Xo = 0.10 coating optical thickness) are n2 =2.4216 and K2 1.1185 (for m = 1), respectively. Thebranch solutions for m = 1-5 are shown. Similar to theEl K case, discontinuities appear in the solutions forthe H1K case. The minimum value (for m = 1) of the


_X.u l , , ,u.

H l K Polarization

n =1.5, X 0=0.5pm,0,=0°

0IU.

0zfl

E I K Polarization

dr/X.= 0.1, X.= 1 pm, = 0°

NORMALIZED COATING THICKNESS, d / Ao

(a)

H I K Polarization

nr= 1.5, X= 0.5pm, 0 = 0°

COATING INDEX, nc

Fig. 12. Filling factor F and the normalized groove depth d2/Xo toproduce zero reflectivity for ELK polarization for a free spacewavelength X0 = 1.0 um as a function of the coating refractive indexn, at normal incidence with a coating (of d,/Xo = 0.10). Branch

solutions for m = 2 and 3 are presented.

_ 55

_ A ~~~~~~~~~4

4

31~

For E K polarization at normal incidence, the cal-culated filling factor F and normalized groove depthd2/X0 are presented in Fig. 12 as functions of the refrac-tive index of the coating n,. The incident wavelengthis Xo = 1.0 gim, and the normalized coating thickness isd,/Xo = 0.10. As an example intermediate step, thecalculated real n2 and imaginary K2 parts of the com-plex refractive index of the lossy homogeneous layer(required for antireflection behavior at n, = 2.0) and n2= 1.0137 and K2 = 0.08861 for m = 2, respectively. Thebranch solutions for m = 2 and 3 are shown. Again, them = 1 solution does not exist. It is observed that for acoating of n0 5.0, the filling factor is F 0.30, whichrepresents a big improvement compared to the 0.0179and 0.0415 values. Further improvement is possibleby tuning the coating thickness d,/Xo.

For H 1 K polarization at normal incidence, the cal-culated filling factor F and normalized groove depthd2/X0 are presented in Fig. 13 as functions of the refrac-tive index of the coating n,. As an example intermedi-ate step, the calculated real n2 and imaginary K2 partsof the complex refractive index of the lossy homoge-neous layer (required for antireflection behavior at n,= 2.0) are n2 = 18.976 and K2 = 5.2354 (for m = 1),respectively. The incident wavelength is Xo = 1.0 Aim,and the normalized coating thickness is d,/Xo = 0.10.The branch solutions for m = 1 and 2 are shown. It isobserved that for a coating of n, n 5.0, the filling factoris F 0.55, which represents a large improvementcompared to the 0.9696 and 0.9339 values. Furtherimprovement is possible by tuning the coating thick-ness d,/Xo.

In Secs. IX.C and D, the tunability of the zero-reflectivity solution due to the new parameters d, andn, was demonstrated. These additional degrees offreedom can lead to much better antireflection gratingdesigns.

4 3

3 2

0.0 0.1 0.2 0.3

NORMALIZED COATING THICKNESS, d. /X 0

(b)

Fig. 11. Filling factor F and the normalized groove depth d2/XO toproduce zero reflectivity for H IK polarization for a free spacewavelength Xc = 0.5 ,um as a function of the coating thickness dIXo atnormal incidence with a coating (of n, = 1.5). Branch solutions for

m = 1-5 are presented.

filling factor is F = 0.386 for d,/Xo 0.275 + 1(1/3), 1 =0,1,2, ....

D. Dielectric Coating Refractive-index Dependence

It has already been mentioned that the insertion of adielectric overlayer increases the filling factor forEl K polarization and decreases the filling factor for,H 1 K polarization. From Fig. 6 (E l K polarization)the computed zero reflectivity filling factors for anincident free space wavelength of 1.0 gim (the real andimaginary parts of the complex refractive index of goldat 0.5 Aim are n3 = 0.22 and K3 = 6.71) are 0.0179 and0.0415 without coating and with a coating of n, = 1.5,respectively. Similarly, from Fig. 7 (H11K polariza-tion) the corresponding values of the filling factor are0.9696 and 0.9339 without a coating and with a coatingof n, = 1.5, respectively. These values of the fillingfactor are not very practical since they are too small forE l K polarization or too large for H 1 K polarization.However, if coatings of different materials are used,more practical filling factors can be calculated leadingto better designs.


0U

'0(321

2.0'

3iC-

0

Q

LU

00

(3

0

0z

1,

3~

0LU0

0(3

0c

'a

N

Z0

1.5-

1.0'

0.5'

0.nI

4

H I K Polarizationd,/Xo= 0.1 Xo= 1 pm, 0 = 0°

4 5

COATING INDEX, n.

1.0-0.10

'0

3o

a.LUIaLUI

0.05 °(3'0aLU

0Z

0LU 0.8-0UiLU

ir 0.6-LUj00

IL- 0.4-

z0

U"'0.2-

nr= 1.5, a,=0.dr,/X0 = 1, No= 0.5,m

FEIK= 0.167FH1K= 0.408

E1K

HIK

0.0 0.2 0.4 0.6NORMALIZED GROOVE DEPTH, d / X,

Fig. 13. Filling factor F and the normalized groove depth d2/Xc toproduce zero reflectivity for H LK polarization for a free spacewavelength X = 1.0 Am as a function of the coating refractive indexn, at normal incidence with a coating (of d/Xc = 0.10). Branch

solutions for m = 1 and 2 are presented.

X. Reflectivity Characteristics

Using the procedure described in Sec. VI and theequations of Appendix A, the zero reflectivity equa-tions (20) and (21) can be solved. Those equationsrepresent the impedance matching condition Zin = Zior equivalently Zin = Z. For the given examples, thelossy overcoated homogeneous layer on a lossy sub-strate and the corresponding overcoated surface-reliefgrating on a lossy substrate appear to have the imped-ance of free space to an incident plane wave. Thus noreflected wave exists. If the substrate is sufficientlythick, all the incident power will be absorbed.

To examine the optical characteristics of both theovercoated lossy homogeneous layers on lossy sub-strates and the corresponding overcoated surface-re-lief high spatial-frequency gratings on lossy sub-strates, the ratio of the power reflected to the incidentpower was calculated with parameters obtained previ-ously but with a wide range of thicknesses (d2/X0) orfilling factors F (or equivalently n2 - jK2). Using thehomogeneous lossy layer analysis described (due to theequivalence of layers and gratings in the long wave-length limit), a null in the reflected power is calculatedexactly at the value of groove depth predicted for bothpolarizations when the predicted filling factor (orequivalently n2 - jK2) is kept constant. The reflectedpower oscillates with increasing thickness d2/XO asshown in Fig. 14 for E K and H 1 K polarizations, ford,/Xo = 1.0 and n = 1.5 at 0.5 Am free space wave-length. For both polarizations and for a general mthbranch solution, the mth minimum of the reflectivitybecomes the null, and all other minima are nonzero.For E K polarization, sometimes instead of the mthminimum, the (m - 1)th minimum becomes a null.This occurs when there is no m = 1 branch solution orwhen the other branches do not have multiple solu-tions. In the case presented in Fig. 14, the m = 2branch solution has two solutions, as can be observedfrom Fig. 10. Thus the m = 2 minimum will be the null

Fig. 14. Fraction of power reflected as a function of groove depthfor a ELK polarized wave with X = 0.5 ,m normally incident on anovercoated (with n = 1.5 and d/Xo = 1) gold rectangular-groovesurface-relief high spatial-frequency grating of filling factor F =0.166676. The zero reflectivity occurs at a normalized groove depthd2/I\ = 0.594587 (for m = 2). The parameters of the lossy layerequivalent to the grating are n2 = 1.21228 and K2 = 0.200184 (for m =2). In the case of H11K polarization the filling factor is F =0.408266, and the zero-reflectivity normalized groove depth is d2/XO= 0.061358 (form = 1). Theparametersofthelossylayerequivalent

to the grating are 2 = 2.15621 and K2 = 0.409474 (for m = 1).

as shown in Fig. 14. For E K polarization the result-ing minima tend to be more rounded than the minimafor H I1K polarization. As the normalized groovedepth (or equivalently the normalized thickness of thelossy layer) increases to infinity, the reflected poweroscillations damp out, and the reflectivity asymptoti-cally approaches a value of R = [(Nl - N2 ) 2 + (K1 -K2)2]/[(N1 + N2 )2 + (K1 + K2 )2 ], where N2 - K2 is theeffective index of region 2, and NKl are given byEqs. (18) and (19), respectively. When the normalizedgroove depth, d2/Xo, predicted by the analysis is keptconstant and the filling factor of the grating (or equiva-lently n2 - jK2) varies from 0 to 1, the resulting reflec-tivity is as shown in Fig. 15 for ElK and HI1K polar-izations. It is observed that the reflectivity tends to berounded around the predicted zero-reflectivity valueof the filling factor for both polarizations.

Xl. Discussion and Summary

For the dielectric overcoated rectangular-groovesurface-relief gratings of high spatial frequency, ascalculated by the impedance matching procedure (Sec.IV), the effective index of the structure is matched tothat of the surrounding medium. Thus there is noreflected wave. If the substrate is sufficiently thick,the structure absorbs all incident power.

The inclusion of one or more dielectric overlayersoffers additional degrees of freedom for design flexibil-ity. For El K (TE) polarization the addition of adielectric overlayer increases the filling factor for an-tireflection. For HK (TM) polarization the fillingfactor decreases. The change in the filling factor re-sults in an improved design for both polarizations,since TE polarizations, since TE polarization is gener-


1.0-

C 0.6-0

-

0I-LU

Lu

0

F

U:'0

0.6

0.4

0.2

n = 1.5, )= 0°dr,/X.= 1 o= 05JM

(d, /XO)EK 0.594

(d2/X 0 )K= 0.061

K

0.0 1 x :0.0 0.2 0.4 0.6 0.8 1.0

FILLING FACTOR, F

Fig. 15. Fraction of power reflected as a function of filling factor fora ELK polarized wave with X = 0.5 um normally incident on anovercoated (with n, = 1.5 and d,/Xo = 1) gold rectangular-groovesurface-relief high spatial-frequency grating of groove depth d2/XO =

0.594587. The zero reflectivity occurs atafillingfactorF = 0.166676(for m = 2). The parameters of the lossy layer equivalent to thegrating are n2 = 1.21228 and K2 = 0.200184 (for m = 2). In the case ofHI1K polarization the normalized groove depth is d2/XO = 0.061358,and the zero-reflectivity filling factor is F = 0.408266 (for m = 1).The parameters of the lossy layer equivalent to the grating are n2 =

2.15621 and K2 = 0.409474 (for m = 1).

ally characterized by small values of filling factor whileTM polarization is characterized by large values offilling factors. The dielectric overlayer causes theantireflection groove depth to decrease for both polar-izations. Again this effect is an improvement sincesmaller groove depth gratings are easier to fabricate.The overlayer brings the second solution for TE polar-ization (m = 2, since usually the m = 1 solution does notexist) close to the second solution for TM polarization.For example, this is observed in Figs. 10(a), (b) and11(a), (b). The thickness of the dielectric overlayergives the additional flexibility to match the solutionsof the two polarizations. From those figures it is ob-served that for dc/Xo 0.035 + (1/3) (1 = 0,1,2,...),both polarizations have the same filling factor, F 0.290, and the same groove depth, d2/Xo 0.3207.The m = 2 solutions are matched for TE and TMpolarization. Thus the resulting antireflection grat-ing is insensitive to the incident polarization! The frac-tion of the power reflected is shown in Fig. 16 as afunction of the filling factor at the designed value d,/Xo= 0.3207 for both polarizations. It is shown that bothpolarizations have a zero (0.00027 for TE and 0.00045for TM) at the same filling factor. Using the addition-al degrees of freedom of the overlayer, surface-reliefrectangular-groove antireflection gratings can thus bedesigned for random polarization. Being able to varythe refractive index of the material also has a positiveeffect in the grating design. From Fig. 13 it is observedthat the filling factor increases for TE polarization anddecreases for TM polarization as the layer refractiveindex increases. This can be particularly useful atlonger wavelengths. If N overlayers are used, the freeparameters increase to 2N (nCl,nC2c...,ncN, and

dr /O= 0.32, Xo= 0.5 pm

LU

0. 0-U

LI 0.6

Z0EI

0.0U. 0. . . . . .

FILLING FACTOR, F

Fig. 16. Fractions of power reflected as a function of filling factor ofan overcoated (with n = 1.5 and dXo = 0.035) gold rectangular-groove surface-relief high spatial-frequency grating of groove depthd2/X = 0.32065, which has very low reflectivity for both E K andH K polarizations at Xo = 0.50 ,um for filling factor F = 0.289897.

d,14d2 . . . dN) . Thus the design can be improved fur-ther. In addition, if the incident wave is of very highintensity, the free parameters may be adjusted to re-duce the peak electric field within the structure so as toavoid dielectric breakdown.

Other observations, similar to those of Ref. 7, aboutthe zero-reflectivity cases presented are the following:(1) The real part of the complex refractive index n2does not necessarily lie in the range n, < n2 < n (2)The real part of the complex refractive index n2 is notnecessarily less than unity if n3 < 1. (3) Zero reflectiv-ity is possible in principle even for large extinctioncoefficients (e.g., K3 = 67.5 for gold at 0 = 10.0 gm).(4) Zero reflectivity is possible even when the real partof the substrate index n3 is less than unity. (5) Therequired zero-reflectivity thicknesses are not simplyodd multiples of X0/4 but are given by the branchsolutions that have been described, and the first solu-tion can be either larger or smaller than X0/4.

The analysis presented is applicable for any wave-length. Even though example results were presentedfor the violet to the infrared range of wavelengths, thetreatment is also valid for microwave wavelengths.Fabrication for these wavelengths might be considera-bly easier than in the optical range since the gratingperiods could be much larger. However, with highresolution electron-beam lithography, feature sizes assmall as 30 nm have been produced ,39 thus makingfabrication of optical antireflection gratings also possi-ble.

For antireflection behavior, it is not an inherentrequirement that the surface-relief grating have a highspatial frequency. The present analysis, however,treats the high spatial frequency grating case. Thislong-wavelength limit is achieved when o/A >> 1 andrepresents the limit in which the homogeneous lossylayer and the grating are completely equivalent (thesame characteristics with wavelength, angle of inci-dence, thickness, etc.). Nevertheless, surface-relief


gratings with relatively large periods can exhibit zeroreflectivity, as previously observed.2 In these cases,propagating backward-diffracted orders other thanthe zero order are also present. Minimizing the reflec-tivity in the zero order does not necessarily minimizethe diffraction efficiencies of the other orders. Thepresent analysis can in fact be used to obtain estimatesof the values needed for antireflection behavior bylarge period gratings.

The equivalence of gratings and homogeneous layersin the limit of small periods indicates that a stairstepsurface-relief profile consisting of N steps could beused to simulate a coating of N individual homoge-neous layers that can also be overcoated. Thus the useof a stairstep grating would allow further broadband-ing both in wavelength and angle of incidence.

In summary, the required thickness and complexrefractive index of single homogeneous layers on lossysubstrates, dielectric overcoated, to produce zero re-flectivity have been calculated by a rigorous imped-ance matching approach. The analysis is applicablefor both ELK and HlK polarizations, any angle ofincidence, any free space wavelength, and an arbitrarynumber of dielectric (or lossy) overlayers. The fillingfactor and the groove depth of a rectangular-groovesurface-relief overcoated grating equivalent to the sin-gle homogeneous lossy layer in the long-wavelengthlimit are also calculated. The method reduces to thatpreviously found for uncoated lossy or dielectric sur-face-relief gratings. It is an extension of previouswork57 for application to lossy gratings with dielectric(or lossy) overlayers. It has been shown that multiplezero-reflectivity solutions exist for both TE and TMpolarizations and for any angle of incidence, wave-length, and arbitrary number of overcoatings. Exam-ple zero-reflectivity gold gratings covered by a singledielectric overlayer (glass of n = 1.5) for incident freespace wavelengths in the range from 0.44 to 12.0 ,mhave been presented. The additional flexibility of theoverlayers in the antireflection design has been high-lighted. Antireflection gratings for random polariza-tion incidence can be fabricated using overlayers ofsuitable refractive index and thickness.

This research was sponsored in part by a grant fromthe Joint Services Electronics Program under contractDAAG29-84-K-0024.

B = 2FN3K3/Nl,, (A5)

C = N2 - (N2N3 - K 2K3 + K2KIN/f,

D = K 2 - (N2K3 + N3K 2 -N2K,)INIc,

E = _(C2 + D2)/(CI3-DR,-BC + AD),

G = -(CN 3 + DR3 - AC - BD)/(CR3 - DS3 - BC + AD),

where

N3 = N3 -(KK31Nlc),

R3 = K3 + (KcN3/N).

(A6)

(A7)

(A8)

(A9)

(A10)

(All)

The unknown quantities, N2, K2, F, and d2, may bedetermined using the iterative numerical proceduredescribed in Sec. VI. The rectangular surface-reliefgrating is completely specified by the filling factor Fand the groove depth d2. The refractive index n2 andthe extinction coefficient K2 of the lossy layer may beobtained using

(A12)(n+/2)'/2B ,

[(A 2 +B 2)1/2 - A] /2

K2 = (n,/2)[(A2 + B2) 1 2 - A(A123

where

Ao = [(1 - F)n 2 + F(n3 - jK3)l1n.,

Bo = 2Fn3K3/nc-

(A14)

(A15)

For HK (TM) polarization, the quantities N'2,K'2and the quantities introduced in Eq. (Al) are

N2 = '/2[N2(l ro) SOK2 ],

K' = /2[K2(l + r) soN 2],

(A16)

(A17)

where N2 and K2 are given by Eqs. (A2) and (A3) asbefore but with A and B as given below and where

rO = +C/1 cos(0/2),

s = -C4i sin(0/2),

= [(N 2 -2nj sinq5)2 + K2]/ 2 [(N2 + 2n, sinok)2 + K2] 2

N 2+ K 2

(A18)

(A19)

Appendix A: Equations Used in the CalculationalProcedure

The value of P in Eq. (33) isN,+RG+DE-A-BG (Al)

DG

For ElK (TE) polarization, the quantities N',K',and the new quantities introduced in Eq. (Al) are

N2 = N2 = [(Nc/2)"/2B]I[(A2 + B2 ]1/2 - A 2, (A2)

K2 = K2 = (Nc/2)[(A2 + B2)1/2 - A]I}/2, (A3)

and the parameters A, B, C, D, E, and G areA = [(1 - )N + F(N3 - K)]/Nl,, (A4)

(A20)

6 a-' K2 ' -)(N 2 + 2n, sin0)

+ a-' (N-K2 ) - 2 tn-N2 -2n, sinl 1

,K 2\

N2)(A21)

Either the upper or the lower signs are chosen in bothEqs. (A16) and (A17) so that K'2 0 in order to havedecreasing exponential fields as required for lossy ma-terials. Only one set of these signs will simultaneouslymake N'2 and K'2 0 as required. The other quanti-ties for HK polarization are

N2[( - F)(N2 + K2)2 + FN2(N2-K2)]A--- 3 [-( K)+ FNC]+[(1-)K3 (A22)Nj [(1 - F)(N - K) + FN2]2 + [2( - NK]


n2 =

(A13)

B 1 2FN4N3K 3 (A23)Nlc [(1 - F)(N2-K 2) + FN2]2 + [2(1 - )N3K3]2

and C, D, E, and G are given by Eqs. (A6)-(A9), respec-tively. The unknown quantities N2, K2, F, and d2 maybe obtained using the iterative numerical proceduredescribed. The refractive index n2 and the extinctioncoefficient K2 of the lossy layer for HI1K polarizationmay be obtained using Eqs. (A12) and (A13), respec-tively, with

A 2n, sin 2k(1 r)(1 r,)2 + S2

Bo = 4- 2n, sin2'kjS c(1 4-ro )2 + S2

(A24)

(A25)

where the signs are chosen as before so that N'2 0 andK' > 0.

In case of multiple coatings, N1 and K1,, are comput-ed from Eqs. (30) and (31), and N, is replaced by NCN-

Appendix B: Generalization for Lossy Overlayers

It is possible for the coating materials to be absorp-tive in a given wavelength band. Then the designprocedure will be slightly changed due to the complexrefractive index, n, - jK, of the lossy overlayer. Inthat case the expression for Z1, [Eq. (16)] will change,since tanh(r,) is now more complicated due to thecomplex instead of the purely imaginary character ofr,. Thus the expressions for N1,, and K1,, will be trans-formed to

N,(ala3 + a2a4) - K,(a2a3 - aja4)a2+a2a3 4 a

K , = ,(a 2a3 - aja 4 ) + K,(ala3 + a2a4) (B2)

where a1 = N 1 - N, tanh(koK',d 0) - K, tan(koNd)a2 = N, tan(koNd) - N1 tanh(koKd,) tan(koNcd,)-K 0 tanh(koKd), a3 = N0 +Ktanh(koK d) Xtan(koN'dc) - N1 tanh(koK'd c), and a4 = Kc + N1 Xtan(koN d c) - Nc tanh(koK'dc) tan(koKcdc).

In addition to the change in N1 c and K1c, the effec-tive grating indices given by Eqs. (9) and (10) willchange since nc is replaced by nc - jKc. Thus thecoefficients A and B defined in Appendix A will changefor both polarizations. For E 1K polarization Eqs.(A4) and (A5) are replaced by

A = [(1 - F)(N - K 2) + F(N3 - K2)]/N,, (B3)

B = 2[(1 - F)NK, + FN3K3]/Nc, (B4)

The case of multiple lossy coatings can also be treat-ed straightforwardly. Thus the recursive Eqs. (28)and (29) are replaced by

Nc(aja3, + a2 a4 ) - Kc(a2 a3 - alia 41 )

a21 + a 1

Nc1(a2 1a3 - alia 41 ) + Kc(alla3l + a21a41)

' ~~~a3l + al

(B7)

(B8)

with

a,, = XI-, + Y1., tanh(kK',dc,) tan(kN,d,,)

-NcI tanh(koK',dc,) - Kc, tan(koNc1 d,,),

a21 = Y-, + NCI tan(koN,,d,) - Kc1 tanh(koK'cdc,)

- XI-, tanh(koc,dc,) tan(koN'cd,),

a31 = N,1 + Kc, tanh(kOcK',d,,) tan(ko.Ncdc,)

- 1X1- tanh(kKc,d,) - Y1.. tan(koNcdc,),

a41 = Kc, + XI-, tan(kO.Nc,dc1) - Nc, tanh(kocI(d,,,)

X tan(kON1,,d,,) - Y,, tanh(kcKc,dc,),

and Xo = Nl, Y = 0, Nic = XN, Klc = YN, = 1,2, ... ,N,and N is the total number of overlayers. In addition,instead of Nc and Kc, NcN and KCN should be used inEqs. (B3)-(B6).

References1. M. G. Moharam and T. K. Gaylord, "Diffraction Analysis of

Dielectric Surface-Relief Gratings," J. Opt. Soc. Am. 72, 1385(1982).

2. M. G. Moharam and T. K. Gaylord, "Rigorous Coupled-WaveAnalysis of Metallic Surface-Relief Gratings," J. Opt. Soc. Am.A 3,1780 (1986).

3. R. C. Enger and S. K. Case, "High-Frequency HolographicTransmission Gratings in Photoresist," J. Opt. Soc. Am. 73,1113(1983).

4. R. C. Enger and S. K. Case, "Optical Elements with UltrahighSpatial-Frequency Surface Corrugations," Appl. Opt. 22, 3220(1983).

5. T. K. Gaylord, W. E. Baird, and M. G. Moharam, "Zero-Reflec-tivity High Spatial-Frequency Rectangular-Groove DielectricSurface-Relief Gratings," Appl. Opt. 25, 4562 (1986).

6. Y. Ono, Y. Kimura, Y. Ohta, and N. Nishida, "AntireflectionEffect in Ultrahigh Spatial-Frequency Holographic Relief Grat-ings," Appl. Opt. 26,1142 (1987).

7. T. K. Gaylord, E. N. Glytsis, and M. G. Moharam, "Zero Reflec-tivity Homogeneous Layers and High Spatial-Frequency Sur-face-Relief Gratings on Lossy Materials," Appl. Opt. 26, 3124(1987).

8. R. W. Wood, "Anomalous Diffraction Gratings," Phys. Rev. 48,928 (1935).

9. M. C. Hutley and V. M. Bird, "A Detailed Experimental Study ofthe Anomalies of a Sinusoidal Diffraction Grating," Opt. Acta20, 771 (1973).

and for HK polarization Eqs. (B22) and (B23) arereplaced by

A = 1 (1 - F)(NN2 - K)(N2 + K2)2 + F(N2 + K 2)(N2- K2)N1, [(1 - F)(N3 -K3) + F(N2 - K2)]2 + [2(1 - F)N3K3 + FNK,]2

1 2(1-F)NKc(N2 + K3) + 2FN3 K3 (N2 + K 2)

(B5)

(B6)


Nic [(1 - F)(N3 - K3) + F(Nc - KC)]2 + [2(1-F)N3 K3 + FNK,]2

Nj =

::

10. J. Hdgglund and F. Selberg, "Reflection, Absorption, and Emis-sion of Light of Opaque Optical Gratings," J. Opt. Soc. Am. 56,1031 (1966).

11. R. C. McPhedran and D. Maystre, "A Detailed TheoreticalStudy of the Anomalies of a Sinusoidal Diffraction Grating,"Opt. Acta 21, 413 (1974).

12. D. Maystre and R. Petit, "Brewster Incidence for Metallic Grat-ings," Opt. Commun. 17, 196 (1976).

13. G. W. Ford and W. H. Weber, "Electromagnetic Interactions ofMolecules with Metal Surfaces," Phys. Rep. 113, 195 (1984).

14. D. Maystre and M. Nevibre, "Sur une M6thode d'Rtude Theori-tique Quantitative des Anomalies de Wood des Reseaux deDiffraction: Application aux Anomalies de Plasmons," J. Opt.Paris 8, 168 (1977).

15. M. Neviere, D. Maystre, and P. Vincent, "Application du Calculdes Modes de Propagation a l'Etude Theoretique des R6seauxRecouverts de Di6lectrique," J. Opt. Paris 8, 231 (1977).

16. E. G. Loewen and M. Nevibre, "Dielectric Coating Gratings: ACurious Property," Appl. Opt. 16, 3009 (1977).

17. J. R. Andrewartha, J. R. Fox, and I. J. Wilson, "ResonanceAnomalies in the Lamellar Gratings," Opt. Acta 26, 69 (1979).

18. J. R. Andrewartha, J. R. Fox, and I. J. Wilson, "Further Proper-ties of Lamellar Grating Resonance Anomalies," Opt. Acta 26,197 (1979).

19. V. Shah and T. Tamir, "Brewster Phenomena in Lossy Structur-es," Opt. Commun. 23, 113 (1977).

20. V. Shah and T. Tamir, "Anomalous Absorption by Multi-Lay-ered Media," Opt. Commun. 27, 383 (1981).

21. R. M. A. Azzam, E. Bu-Habib, J. Casset, G. Chassaing, and P.Gravier, "Antireflection of an Absorbing Substrate by an Ab-sorbing Thin Film at Normal Incidence," Appl. Opt. 26, 719(1987).

22. M. C. Hutley and D. Maystre, "The Total Absorption of Light bya Diffraction Grating," Opt. Commun. 19, 431 (1976).

23. R. A. Depine, V. L. Brudny, and J. M. Simon, "Phase Behaviornear Total Absorption by a Metallic Grating," Opt. Lett. 12,143(1987).

24. L. B. Mashev, E. K. Popov, and E. G. Lowen, "Total Absorptionof Light by a Sinusoidal Grating near Grazing Incidence," Appl.Opt. 27, 152 (1988).

25. K. Javily and R. M. A. Azzam, "Coating a Transparent or Ab-sorbing Substrate by a Transparent Thin Film for MinimumReflectance of Unpolarized Light at Oblique Incidence," Optik72, 75 (1986).

26. M. Lesser, "Antireflection Coatings for Silicon Charge-CoupledDevices," Opt. Eng. 26, 911 (1987).

27. M. C. Hutley, J. F. Verrill, and R. C. McPhedran, "The Effect ofa Dielectric Layer on the Diffraction Anomalies of an OpticalGrating," Opt. Commun. 11, 207 (1974).

28. J. M. Elson, L. F. DeSandre, and J. L. Stanford, "Analysis ofAnomalous Resonance Effects in Multilayer-Overcoated, LowEfficiency Gratings," J. Opt. Soc. Am. A 5, 74 (1988).

29. L. Mashev and E. Popov, "Diffraction Efficiency Anomalies ofMulticoated Dielectric Gratings," Opt. Commun. 51,131 (1984).

30. L. Mashev and E. Popov, "Zero-Order Anomaly of DielectricCoated Gratings," Opt. Commun. 55, 377 (1985).

31. L. B. Mashev and E. G. Loewen, "Anomalies of All-DielectricMultilayer Coated Reflection Gratings as a Function of GrooveProfile: An Experimental Study," Appl. Opt. 27, 31(1988).

32. S. J. Wilson and M. C. Hutley, "The Optical Properties of 'MothEye' Antireflection Surfaces," Opt. Acta 29, 993 (1982).

33. G. H. Derrick, R. C. McPhedran, D. Maystre, and M. Nevibre,"Crossed Gratings: A Theory and its Applications," Appl.Phys. 18, 39 (1979).

34. 0. Wiener, "Die Theorie des Mischkorpers fur das Feld derStationdren Stromung," Abh. Math. Phys. K. Sachs. Akad.Wiss. Leipzig 32, 509 (1912).

35. Z. Hashin and S. Shtrikman, "A Variational Approach to theTheory of the Effective Magnetic Permeability of MultiphaseMaterials," J. Appl. Phys. 33, 3125 (1962).

36. R. B. Stephens and P. Sheng, "Acoustic Reflections from Com-plex Strata," Geophysics 50, 1100 (1985).

37. M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford,1980).

38. G. Hass and L. Hadley, "Optical Properties of Metals," in Amer-ican Institute of Physics Handbook, D. E. Gray, Ed. (McGraw-Hill, New York, 1972), p. 6-119.

39. R. E. Burge, "Zone Plates for X-Ray Microscopy at 300-A Reso-lution," J. Opt. Soc. Am A 3(13), P21 (1986).

Robert S. Kidwell of Bell Aerospace at the 1987 OSA Annual Meet-ing. Photo: F. S. Harris, Jr.


antireflection surface structure: dielectric layer(s) over a high spatial-frequency surface-relief...

Documents