Dielectric coated gratings: a curious property

E. G. Loewen and M. Neviere

Theoretical calculations predict that under certain special conditions a shallow fine pitch diffraction grating,given a precise dielectric overcoating, acts as a complete absorber of incident light of a specific wavelengthand polarized in the P plane. Despite lack of a physical explanation, the phenomenon was confirmed by ex-periment.

The development of space and plasma spectroscopyhas, in the last decade, brought with it a great increasein interest in the VUV region. Aluminum is by far thebest reflector known for this work, but only if free ofoxide layers that absorb. The solution has been toprotect fast fired Al coatings with thin layers of MgF2which keep oxygen from attacking the aluminum.However, such layers can have significant effects on theefficiency behavior of a diffraction grating.

On the experimental side, Palmer' has shown thatdiffraction anomalies formed in P polarized light canbe greatly enhanced by overcoating a grating with adielectric. Cowan and Arakawa2 have shown that forS polarized light, anomalous features are markedly af-fected by quite thin dielectric layers. Hutley et al.

3

have obtained sets of experimental data which showthat the presence of a dielectric layer can give rise tostrong and sharp anomalies, even for P polarized light,of a type never found on bare gratings.

On the theoretical side, a first formalism using infiniteconductivity theory4 predicted that the presence of adielectric layer would give rise to a sharp P planeanomaly near the blaze wavelength. A more recenttheory,5 6 taking into account both the dielectric layerand the finite conductivity of the metal, confirmed theexistence of such anomalies and showed remarkablyclose conformance 6 with the experimental results ofHutley et al.

3 The effects of very thin layers (X/20 andX/40) on a grating used with S polarized light were alsostudied and showed an important shift and broadeningcompared to the anomalies of the bare grating. Butuntil now, reasons for the new anomalies in both S andP polarizations were not well understood.

E. G. Loewen is with Bausch & Lomb, Inc., Rochester, New York14625. M. Nevibre is with Laboratoire d'Optique Electromagnetique,Facult6 des Sciences St. Jerome, 13397 Marseille cedex 4, France.

Received 4 December 1976.

In a recent paper,7 we studied the leaky modes of aplane and a corrugated dielectric waveguide, boundedby a metal on one of its sides. As in the case of gratingcouplers,8 we showed that the propagation constants ofthese leaky modes are the complex zero of the deter-minant of a known complex matrix M and can be de-termined using the differential formalism.5 68 But fora corrugated waveguide, this determinant also hascomplex poles which can be determined in the samemanner. As we did for the plasmon anomalies of a baregrating, 9 we demonstrated that the location and theshape of the anomalies of a dielectric coated grating canbe deduced from the values of these complex zeros andpoles using a simple formula. The proof showed thatthese anomalies are due to the guiding phenomenon of'the corrugated waveguide. This new theory was usedto study the evolution of the anomalies when an arbi-trary parameter is varied. It was found that for a givengrating, with a sufficient groove depth, and for P po-larization, an incident plane wave can be completelyabsorbed, provided that the grating is covered with adielectric layer of a specific thickness. This remindsus of the phenomenon of total absorption found for abare grating in S polarization 0' 9 as recently verified'and related to the existence of surface plasmon waves.But for P polarization and a dielectric coated grating,one would not expect it, because in the P plane there areno surface plasmons on bare gratings.

To verify this unexpected behavior, the followingexperiment was conducted. We chose a 2400-g/mm,10022' blazed aluminum grating and coated it withMgF2. The wavelength X was chosen so as to produceonly one spectral order, the zero order, when incidenton the grating near normal. Figure 1 shows the tra-jectories of the complex poles a, and zeros ap of thedeterminant of the M matrix for this grating as a func-tion of the thickness e of the dielectric layer, as given bythe theory for X = 0.492 ,im. As we have shown, theangle of incidence 0 for which the dielectric coatedgrating presents an anomaly can be easily deduced fromthe real part of a, or ap by the simple relation

November 1977 / Vol. 16, No. 11 / APPLIED OPTICS 3009

sinO + (/d) = R(R) R(ap). (1)

Moreover, the value of the minimum of the efficiency(where efficiency &n in the nth order is defined as theratio of the energy diffracted in this order to the incidentenergy) of this anomaly is simply given by the squareof the ratio of the imaginary parts of a, and ap 79

26min | .(Z) (2)6'(ap)

We can see on Fig. 1 that for e < 0.12 Am we are belowthe cutoff region, and no guided wave can propagate inthe structure. But for e > 0.12 ,gm, we begin to get aguided mode, and we find a zero and a pole whoseimaginary parts are quite different. As a result, Eq. (2)enables us to predict a very strong anomaly, with aminimum close to zero for all the thicknesses between0.13 ,gm and 0.16 gim, the zero value being reached the-oretically for e = 0.1520 gim. If we choose e = 0.13 gim,for example, Eqs. (1) and (2) show, using the values ofa. and ap shown on Fig. 1, that an anomaly appears for0 = -9.8° and that its minimum is 4.4%.

All the results can be verified by direct computationsof the efficiency in the vicinity of the predicted anomaly,using the rigorous method to solve the boundary valueproblem described in Ref. 5. Figure 2 gives the ef-ficiencies for three different thicknesses of the dielectriclayer. For e = 0.11 gi (curve 1), since no guided mode

:f (9P)_f (C P

lo-2

0

-10-2

e 0.200

0.190

/0180

0170

600

0 '90

_ 140

/0.130

0o 125

T u t .l2O

1.0 1 0.160 1.10 1.1s

.170

.180

0. 90

e0.200

exists, the efficiency curve is smooth with values closeto the reflectance of the dielectric coated metal. Fore = 0.13 Am (curve 2) we calculate an anomaly whichagrees exactly with the aforementioned predictions.For e = 0.1520 Am (curve 3), since a, is on the real axis,we get an anomaly whose minimum is reached for 0 =7.50 and is equal to zero. This predicts that if a planewave falls on this grating with this critical angle of in-cidence, there is no diffracted energy. The incidentenergy is completely absorbed by the grating and servesonly to heat the metal.

The imaginary parts of a, and a1p arise from a com-bination of the complex index of the metal and thecorrugated geometry of the surface. The constants ofthe dielectric layer play only a minor role here, and thelosses in the MgF 2 were considered negligible.

Since the real part of a, and a1p varies when e is var-ied, as one can see in Fig. 1, the angle of incidence cor-responding to the minimum is very sensitive to thethickness of the dielectric layer as well as its refractiveindex. As shown in Fig. 3, it is also very sensitive withrespect to the wavelength. Thus, these anomaliesshould constitute an accurate testing for verifying thetheory.

Figure 4 gives measured efficiency curves as a func-tion of 0 for a thickness e = 0.13 ,m and four values forthe wavelength X. The efficiency of this grating, whichbehaves nominally like a mirror, is generally close to thereflectance of dielectric coated metal, i.e., between 0.8and 0.9. But for X = 0.492 ,im, when 0 becomes close to-70, this efficiency drops sharply to just 5.5%. This hasto be compared to the 4.4% given by the theory. The70 angle of incidence is slightly different from the 9.80predicted.

A similar small discrepancy is observed for the othervalues of X, but the experiment of Fig. 4 and the theoryof Fig. 3 agree in showing the sensitivity of the locationof this anomaly with respect to the wavelength and inpredicting the direction of the shift. Moreover, severalcomputations have shown that the location of thisanomaly is also very sensitive to other parameters, such

*(.P)

i V())

e=o. 152,um2

-12 -11 -10 -9

I I Il I

'I II. V,.

-8 -7

0.5

-o-6 -5 -4 e

Fig. 1. Trajectories of the complex pole a and zero p of the de-terminant of the M matrix when the thickness e (in um) of the di-electric is varied. The results hold for a 2400.g/mm, 10°22' blazed

aluminum grating, covered with MgF2, and X = 0.492 ,um.

Fig. 2. Theoretical efficiency curves for the grating in Fig. 1 coveredwith various dielectric layers of thickness e (in um) as a function ofthe angle of incidence 0 in degrees (A = 0.492 ,um). Direct computa-

tion using differential formalism.

3010 APPLIED OPTICS / Vol. 16, No. 11 / November 1977

-

-

0-12 -11 -10 -9 -8 -7 -6 -5 -4 e

Fig. 3. Theoretical efficiency curves for the grating of Fig. 1 coveredwith a 0.13-jum thick layer of MgF 2 and four different wavelengths.

, £.

-11 -10 -9 -8 -7 -6 -5 -4 -:

In view of the rather critical experimental confir-mation of the unexpected theoretical discovery of sharpP-plane absorption phenomena it would be very satis-fying if a good physical explanation could be provided.Unfortunately, this is not the case. The surface plas-mon explanation3"11 (i.e., collective resonances of elec-trons under the surface of the grating) that is appro-priate for describing similar phenomena observed inS-plane polarization and bare metal gratings is notapplicable. As far as we know, there are no plasmonsfor bare gratings in P polarization, or is there reason toexpect them from the simple addition of a dielectricovercoating. The understanding of these strong ab-sorptions of energy will require further develop-ments.

We can now speculate on the use of such dielectriccoated gratings for filtering. By appropriate choice ofgroove spacing, groove depth, and thickness of dielectriccoating, the location of an anomaly can be tuned. Itthen becomes possible to absorb polarized radiation atone wavelength completely. In a previous paper,9 it wasshown that such filtering is also possible in the S plane,if the required groove depth can be produced. How-ever, from a practical point of view, the approach de-scribed here is easier to realize.

0.5

0

Fig. 4. Experimental efficiency curves for e = 0.13 gm and fourdifferent wavelengths.

as groove depth, groove shape, the thickness, and therefractive index of the layer. Also we must rememberthat at 2400 g/mm, it is very difficult to obtain theperfectly triangular profile used in the theory. In ad-dition, we only know approximately the value of thethickness of MgF 2 used in the experiments. Consid-ering these limitations, we can consider the experi-mental verification as satisfactory.

The theory previously developed 7 to explain andstudy anomalies introduced by dielectric coatings ongratings has been successfully verified. This demon-strates that these anomalies are related to the complexzeros and poles of the M matrix introduced in Ref. 8. Itis easy to show that these zeros and poles are, respec-tively, the poles and zeros of the scattering matrix (S)first introduced in nuclear physics and used in theelectromagnetic theory by several authors.9 12-15 Asshown by the recent developments of the SingularityExpansion Method (SEM),'16 the determination of thesepoles and zeros may be useful for the study of diffractionproblems.

References1. C. H. Palmer, J Opt. Soc. Am. 42, 269 (1952).2. J. J. Cowan, and E. T. Arakawa, Z. Phys. 235, 97 (1970).3. M. C. Hutley, J. F. Verrill, and R. C. MacPhedran, Opt. Commun.

11,207 (1974).4. M. Neviere, M. Cadilhac, R. Petit, Opt. Commun. 6, 34 (1972).5. M. Nevibre, P. Vincent, and R. Petit, Nouv. Rev. Opt. 5, 65

(1974).6. M. C. P. Hutley, J. C. Verrill, R. MacPhedran, M. Neviere, and

P. Vincent, Nouv. Rev. Opt. 6, 84 (1975).7. M. Neviere, D. Maystre, and P. Vincent, J. Opt. in press

(1977).8. M. Neviere, P. Vincent, R. Petit, and M. Cadilhac, Opt. Commun.

9, 48 (1973).9. D. Maystre, and M. Neviere, J. Opt. in press (1977).

10. D. Maystre, and R. Petit, Opt. Commun. 14, 196 (1976).11. M. C. Hutley and D. Maystre, Opt. Commun. 14, 431 (1976).12. J. L. Uretsky, Ann. Phys. 33, 400 (1965).13. M. Neviere, M. Cadilhac, and R. Petit, IEEE Trans. Antennas

Propag. AP-21, 37 (1963).14. P. Vincent, D. Maystre, and R. Petit, Proceedings of the Fifth

Colloquium of Microwave Communication, G. Bognar, Ed.(Akademiai Kiado, Budapest, 1974), ET-405.

15. M. Neviere P. Vincent, Opt. Acta 23, 557 (1976).16. C. E. Baum, The Singularity Expansion Method, in Transient

Electromagnetic Fields, L. B. Felsen Ed. (Springer, Heidelberg,1976), pp. 129-179.

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