Double-layer broadband antireflection coatings forgrazing incidence angles

Jagdish C. Monga

This paper presents the methodology for the design of double-layer antireflection (AR) coatings forgrazing incidence angles for a given state of polarization. The designs are based on commonly usedthin-film materials with the optical constants that can be realized by using standard evaporationtechniques. The performance of some AR stacks has been computed, and the effect on spectral reflectancewith variation in the thickness of the high-index layer, angle of incidence, and the refractive indices of thematerials used for the inner and outer layers has been studied with a view to selecting a suitable designthat gives the lowest reflectance over the widest wavelength range. The AR stacks show a reflectance of< 0.5% over most parts of the visible and near-infrared regions of the spectrum.

Introduction

Dye lasers pumped by high peak power pulsed lasersare increasingly being used as a source of tunable andcoherent radiation for several applications. A prismbeam expander-grating combination is generally usedin these lasers to expand the incident beam in order toget narrow linewidth and to reduce the possibility ofdamage to the grating surface. 5 Since the magnifica-tion provided by a prism increases rapidly as theincident angle approaches 90',6 these are used atgrazing incidence angles.7' 8 Several authors have dis-cussed various principles that are involved in thedesign of prism beam expanders."' 2 The use of manyprisms in tandem to get the required magnificationand a large number of passes through these prismsbefore the buildup of radiation in the cavity results ina considerable reflection loss. Moreover, to utilizethese prism beam expanders in widely tunable lasers,it is important that the antireflection (AR) coating beeffective over a wide wavelength range. This makesthe use of broadband AR coatings essential to mini-mize reflection loss.

An AR coating at normal incidence, in its mostsimple form, consists of a quarter-wave thick layer ofa material with a suitable refractive index. At obliqueangles of incidence, the effective refractive index of amaterial is not only different from that at normal

The author is with the Laser & Plasma Technology Division,Bhabha Atomic Research Centre, Bombay 400 085, India.

Received 27 September 1990.0003-6935/92/040546-08$05.00/0.© 1992 Optical Society of America.

incidence but is also different for the two orthognalpolarizations. The required effective refractive indexof the material, whose single quarter-wave thick layercan produce zero reflectance, increases with angle ofincidence for p-polarized radiation and can be as highas 3.4 for an incident angle of 80°. These refractiveindices cannot be obtained with a single coatingmaterial. Even at lesser incident angles for which therequired effective refractive index is small, it may notalways be possible to obtain a material whose refrac-tive index exactly matches the required one. How-ever, it is possible to obtain these unusual refractiveindices by using two materials in an asymmetricdouble-layer period. The design of broadband ARcoatings for grazing incidence angles has not beenreported to the best of my knowledge. In an earlierpublication,'3 the design and performance of multi-layer AR coatings for grazing incidence angles effec-tive over a narrow wavelength range (- 30 nm) wasreported.

In this paper we present the methodology for thedesign of broadband AR coatings for grazing inci-dence angles and for a given state of polarization. Theperformance of some AR stacks has been computedand the effect on spectral reflectance with variation inthe thickness of the high-index layer, angle of inci-dence, and the refractive indices of the materials usedfor the inner and outer layers has been studied with aview to selecting a suitable design that gives thelowest reflectance over the widest wavelength range.The computed spectral characteristics of AR stacksbased on commonly used thin-film materials show areflectance of < 0.5% in the visible and near-infraredregions of the spectrum.

546 APPLIED OPTICS / Vol. 31, No. 4 / 1 February 1992

Design of Double-Layer Antireflection Coatings

Basic Design Procedure

Let us consider a double-layer coating with materialshaving refractive indices nj and n2 and phase thick-nesses Al and , on a substrate of refractive index nmand bounded by an incident medium of refractiveindex no as shown in Fig. 1. The characteristic matrixfor the film-substrate assembly is given by4

B cos 4) -sin 1Al cos 42 T sin 4)2

iTh sin Al cos 4 l i'rj2 sin 4)2 COs 41)2

(1)

where the phase thickness hi is related to the physicalthickness di of the material by the expression hi =(2rrIX)nidi cos Oi, i = 1 and 2, Oi being the angle atwhich radiation travels inside that material. Theeffective refractive indices 'r, l, 2, and ',m arecalculated from the refractive indices no, n n2, nm andthe angles 00, ol 02, Om in the corresponding media.The effective refractive indices for p and s polariza-tions are given by 'ri = n/cos Oi and i = n cos Oi,respectively, where the subscript i denotes the values0, 1, 2, and m. The angles O are related to the angleof incidence 00 by Snell's law, i.e., n sin 00 = n sinOi, i= 12,andm.

The admittance of the film-substrate assembly isgiven by

(em COs ()l COs 1)2- jq sin 4)2 sin,

(COs )1 COs 4)2 - sin Alj sin 4)2)

If this asymmetric period of two layers at a givenangle of incidence has to be equivalent to a quarter-wave thick layer of admittance A, the admittance ofthe film-substrate assembly must equal 2 /_Q . Substi-tuting this into Eq. (2) and equating the real andimaginary parts, we obtain

(5I2

Im2

) COs ()i COs )2 + ( 2 2 2 sin l sin 12 = , (3)~ 2 fI s1n snpmu

(2 2c - f)cos + sin 4)2 + (L - T)sin 4)j COs 4)2 = 0- (4)

Solving Eqs. (3) and (4), the expressions for the phasethicknesses of the two layers can be obtained asfollows:

tan2 Alj = ( _q 2

12)(I2- 2 12) (5)

'no

ni

2A n 2

9mit;\ nm

SUBSTRATE

Fig. 1. Schematic of a double-layer AR coating showing thevarious parameters used in the text.

polarization and at a given angle of incidence in Eqs.(5) and (6), the phase thicknesses of the two layerswith AR characteristics can be determined.

For moderate magnifications (~ 6), the prism beamexpanders are generally used at incident angles around700.9 To design an AR coating for an air-glass(nm = 1.52) interface at an incident angle of 700 forppolarization, the effective refractive indices 'r, m ofair and substrate, respectively, are calculated to be2.924 and 1.934, respectively. The effective refractiveindex of the material whose single quarter-wave thick

(1 2) + i(q sin A cos ,), + IQ2 COs ¢b1 sin 2 )

+ i cos Alsin 4)2 + I- sin Al cos 4)2)

(2)

layer would antireflect an air-glass interface can beobtained as q = [(2.924 x 1.934)]112 = 2.378. It maybe noted that for p polarization at 70° incidence,(qm - aq) is negative. Therefore, for solutions of Eqs.(5) and (6) to be real, either all the quantities("12 - I), (a - q), and (_1q2,I m2 - 22_ri2) must be nega-tive, or any two must be positive and the thirdnegative. The values of effective refractive indices q,and -'2 of the two materials whose suitable layerthicknesses would produce zero reflectance are shownby the shaded regions in Fig. 2. Possible layer indicesare assumed to be limited to the 1.38-2.50 range thatcorrespond to magnesium fluoride and titanium diox-ide as the materials with the lowest and the highestrefractive indices, respectively. There are three re-gions for which suitable layer thicknesses have ARcharacteristics. These regions correspond to the fol-lowing values of q and -12:

tan2 4 92 2()2.2 _ - 2)(N2 _ '1i2)( 1

2,1m

2- " 2 2I2)(i2 _ 92)

Region I(6) Region II

Region III

1.884 < 1 < 2.378,2.378 < X < 12291r2,1 .2 29 X2 < X < 2.378,

112 > 2.378,1.935 < 2 < 2.378,1.884 < 2 < 1.935.

Using the effective refractive indices for a given Based on these values of the effective refractive

1 February 1992 / Vol. 31, No. 4 / APPLIED OPTICS 547

INCIDENT MEDIUM

1 = 2.378

81 = 1.884

12 -

Fig. 2. Diagram showing the values of the effective refractiveindices m, and 2 for double-layer AR coatings on glass (nm = 1.52)forp-polarized radiation incident at 700. Suitable layer thicknessesof materials (shaded regions) can produce zero reflectance.

indices, suitable thin-film materials can be selectedfor double-layer AR coatings for grazing incidenceangles.

Selection of Thin-Film Materials

To select thin-film materials suitable for double-layergrazing incidence AR coatings, the conditions for theeffective refractive indices ('s) are converted intonormal incidence refractive indices (Ws) by the follow-ing expression:

2 1 + (11 - 42 sin2 O)12 (7)

Using Eq. (7) we find that, for region I, therefractive index of the inner layer, next to the sub-strate, should be greater than 2.135 and that of theouter layer, next to the incident medium, should liebetween 1.38 and 2.135. We can therefore selecttitanium dioxide (n = 2.3-2.5) as the material for theinner layer and any of a variety of materials, such asmagnesium fluoride (n = 1.38), silicon dioxide(n = 1.45), aluminum dioxide (n = 1.65), magnesiumoxide (n = 1.74), hafnium dioxide (n = 1.92), andzirconium dioxide (n = 2.12), for the outer layer. Forthe solutions that correspond to region II, the refrac-tive index of the outer layer should be more than2.135 and that of the inner layer should lie between1.52 and 2.135. In this region, the same materials asthose in region I can be used for designing double-layer AR stacks, but a high-index material must beused for the outer layer and a low-index material forthe inner layer. There is a limited choice of materialsfor the solutions that correspond to region III. Forthis region, the refractive index of the inner layershould lie between 1.38 and 1.52 and that of the outerlayer should lie between 2.05 and 2.135. We can selectzirconium dioxide (it = 2.12) for the outer layer, and

magnesium fluoride (n = 1.38) or silicon dioxide(n = 1.45) for the inner layer.

It may be noted that to design double-layer ARcoatings for an incident angle of 700, most of therefractory materials can be used that are known tohave good abrasion resistance, substrate adhesion,and high damage resistance properties. The asymmet-ric double-layer combination can be either of the form(pH, qL) or (pL, qH), where p, q denote the fractionsof quarter-wave thicknesses and H, L represent quar-ter-wave thick layers of high- and low-index materi-als, respectively. The best combination of materialsand the order in which these materials appear in thedouble-layer combination to obtain the lowest reflec-tance over the widest wavelength range can be se-lected after their spectral reflectance characteristicsare computed and analyzed.

General Design of Antireflection Stacks

To design AR coatings for an incident angle to 700, weselected the materials for the inner and outer layersfrom Fig. 2 so that suitable layer thicknesses of thesematerials can produce zero reflectance at the designwavelength and at the desired angle of incidence.Using titanium dioxide (n = 2.3) for the inner layerand silicon dioxide (n = 1.45) for the outer layer,which correspond to the solutions in region I of Fig. 2,and substituting the values of effective refractiveindices in Eqs. (5) and (6), we obtain the phasethicknesses Al of the outer layer as 18.1° and 161.90,and '2 of the inner layer as 51.6° and 128.4°. Thesevalues of the phase thicknesses have to be pairedcorrectly, which can be done by sketching a roughadmittance diagram. The phase thicknesses of thecorrectly paired AR stacks are air(18.10 L, 51.60 H)suband air(161.90L, 128.40H)sub, where sub refers to thesubstrate material. The AR stacks in terms of frac-tions of quarter-wave thicknesses at normal inci-dence would therefore become air(0.264L, 0.628H)airand air(2.363L, 1.564H)air, respectively.

Using titanium dioxide (n = 2.3) as the material forthe outer layer and zirconium dioxide (n = 2.12) asthe material for the inner layer, which correspond toregion II, we obtain the optical thicknesses of the ARstacks as air(0.246H, 0.939L)sub and air(1.946H,1.292L)sub. Similarly, we can choose zirconium diox-ide as the material for the outer layer and silicondioxide (n = 1.45) as the material for the inner layer.Using in Eqs. (5) and (6) the values of the effectiverefractive indices of these materials that correspondto region III, we obtain the phase thicknesses of theAR stacks as air(1.091H, 2.114L)sub and air(1.14L,0.512H)sub.

Results and Discussion

Effect of Thickness of the High-Index Layer

The spectral reflectance characteristics of the ARstacks, whose designs are given in the previoussection, are shown in Figs. 3-5. The reflectance at thecenter wavelength (600 nm) and at the desired angle

548 APPLIED OPTICS / Vol. 31, No. 4 / 1 February 1992

F

of incidence (70°) is zero for all the AR stacks.However, the characteristics show a variation inwavelength range over which a low reflectance can beachieved. The AR stack, air(0.264L, 0.628H)sub withn = 1.45 and n2 = 2.3, shows a reflectance of < 1%over a wavelength range from 460 to 850 nm (Fig.3A), while the AR stack, air(2.363L, 1.564H)sub, withthe same materials shows much narrower reflectancecharacteristics for which the reflectance remainsbelow 1% only from 565 to 630 nm (Fig. 3B). Simi-larly the AR stacks, air(0.246H, 0.939L)sub with nl =2.3 and n2 = 2.12 (Fig. 4A), and air(1.091H,2.114L)sub with n = 2.12 and n2 = 1.45 (Fig. 5A),show broader reflectance characteristics where thereflectance remains below 1% from 450 to 850 nm.

It may be noted that of the two possible combina-tions of layer thicknesses for a given set of materials,the broader reflectance characteristics are associatedwith the AR stack having a thinner high-index layer.In such a stack, the terminal point of the inner layerand the admittance locus of the outer layer lie on thesame side of the real axis of an admittance diagram.Any change in the position of the terminal point ofthe inner layer due to a change in wavelength tends tobe compensated by a suitable change in the admit-tance locus of the outer layer. In this case the endpoint of the complete assembly is not displaced too farfrom the ideal terminal point, i.e., the admittance ofthe incident medium. Therefore, the reflectance re-mains low for fairly large variations in wavelength. Itmay be seen (Fig. 5) that, when the thicknesses of thehigh-index layers for the two combinations are nearlyequal (1.091H and 1.14H), the AR stacks show nearlythe same reflectance characteristics.

Effect of Variation in the Refractive Index of the Outer andInner Layers

The spectral reflectance characteristics of some ARstacks having different materials for the outer layerare shown in Fig. 6. The inner layer for all these ARstacks has a refractive index of 1.92. The layerthicknesses for each set of materials have been ad-justed to produce zero reflectance for the centerwavelength at an incident angle of 700. Only those

0.04F

0.03

U

Li.ad(Z

, B

,I

0.02[

0.01

450 550 650 750 850

WAVELENGTH (nm)

Fig. 3. Spectral reflectance of double-layer AR coatings on glass(n = 1.52) forp-polarized radiation incident at 70: A, air(O.264L,0.628H)sub; B, air(2.362L, 1.564H)sub with n, = 1.45, n 2 = 2.30.

0.

U z

U 002 \

0.01-A

0.00-450 550 650 750 850

WAVELENGTH (nm)

Fig. 4. Spectral reflectance of double-layer AR coatings on glass(nm = 1.52) forp-polarized radiation incident at 70: A, air(0.246H,0.939L)sub; B, air(l.946H, 1.292L)sub with n, = 2.30, n 2 = 2.12.

layer combinations with thinner high-index layershave been selected. It may be seen that the AR stackwith an outer layer having lower refractive index(n, = 2.2) shows low reflectance over a narrowerwavelength range. The reflectance characteristicsbecome progressively broader as the refractive indexof the outer layer increases from 2.2 to 2.5. For an ARstack, air(0.966H, 1.073L)sub, having an outer layerof refractive index 2.5, the reflectance remains below0.1% from 450 to 750 nm.

The variation in spectral reflectance with refractiveindex of the inner layer is shown in Fig. 7. Therefractive index of the outer layer for all the stacks is2.3. The reflectance at the center wavelength is zerofor all the stacks since the layer thicknesses havebeen adjusted to give zero reflectance at this wave-length. The AR stack with an inner layer having ahigher refractive index (n2 = 2.12) shows narrowerreflectance characteristics that become broader as therefractive index of the inner layer decreases from 2.12to 1.74. The reflectance remains below 0.2% for an AR

0.02

ad_Uiz< A

U0.01

0.00450 550 650 750 850

WAVELENGTH (nm)

Fig. 5. Spectral reflectance of double-layer AR coatings on glass(nm = 1.52) forp-polarized radiation incident at 70: A, air(l.091H,2.114L)sub; B, air(l.14H, 0.521L)sub with n, = 2.12, n2 = 1.45.

1 February 1992 / Vol. 31, No. 4 / APPLIED OPTICS 549

n ,\nus uu 5

IIIIIIIIIIIIIII

IIIIII

0.010

0.008

U 0.006 Li. . . C 2.40

0.004 \ , ., /,- D 2.50

0.002 .,-

0.000=450 550 650 750 850

WAVELENGTH (nm)

Fig. 6. Spectral reflectance of double-layer AR coatings on glass(nm = 1.52) for p-polarized radiation incident at 70° for differentmaterials as the outer layer. The layer thicknesses for eachcombination of materials were adjusted to produce zero reflec-tance. n2 = 1.92 for all the stacks. A, air(0.843H, 0.475L)sub withn = 2.20; B, air(0.767H, 0.713L)sub with n1 = 2.30; C, air(0.786H,0.878L)sub with n = 2.40; D, air(O.966H, 1.073L)sub with n, =2.50.

stack, air(1.004H, 0.955L)sub with n1 = 2.3 and n2 =1.74, over a wavelength range from 450 to 750 nm.

It may be noted that broader reflectance character-istics are obtained when the refractive index of theouter layer is high and that of the inner layer is low.This is due to the fact that the thickness and spectralstability of the AR stacks depend on the refractiveindices of the layers. The most stable coatings arethose that have an inner layer whose effective refrac-tive index is close to that of the substrate and an outerlayer whose effective refractive index is close to thatof the incident medium. For the case of p-polarizedradiation at an incident angle of 70°, the effectiverefractive index of air is 2.924, which can be obtained

from a material of refractive index 2.75 [Eq. (7)].Therefore, the broadest reflectance characteristicscan be obtained for an AR stack whose inner layer hasa refractive index close to 1.52, the index of thesubstrate, and an outer layer close to 2.75. Thevariation in spectral reflectance with the refractiveindex of the inner layer for AR stacks having ahigh-index (n = 2.5) outer layer is shown in Fig. 8. Acomparative study of Figs. 7 and 8 shows that, for ARstacks with an inner layer having a refractive index of1.92 and an outer layer of index 2.3, the reflectanceremains below 0.2% from 490 to 730 nm, whereas thesame reflectance can be achieved for a wavelengthrange from 450 to 790 nm if the outer layer has arefractive index of 2.5. However, with increasingrefractive index of the inner layer, the improvementin wavelength range, for which low reflectance isobtained by increasing the refractive index of theouter layer, is only marginal.

For a certain combination of materials, it is possi-ble to have layer thicknesses that give zero reflectancewhether a material is used for the inner layer or forthe outer layer. For example, by using magnesiumoxide (n = 1.74) for the inner layer and titaniumdioxide (n = 2.3) for the outer layer, the AR stack,air(1.004H, 0.955L)sub, gives zero reflectance at thedesign wavelength. Similarly, an interchange of mate-rials, i.e., magnesium oxide for the outer layer andtitanium dioxide for the inner layer, gives an ARstack, air(0.323L, 0.581H)sub. The spectral reflec-tances of both these stacks are shown in Fig. 9 ascurves A and B, respectively. It may be seen that thebroader reflectance characteristics are associated withan AR stack, air(1.004H, 0.955L)sub, that has ahigh-index outer layer and a low-index inner layer.The order in which the materials appear in a double-layer AR coating is therefore important so as to havebroadband reflectance characteristics.

0.010,

,D

/

///B1 / /

/ // /

/'V

I...1

3

0.008_

adUzI.-Uad

Li.ad

0.006

0.004

0.002

0.00c4750 850

Fig. 7. Spectral reflectance of double-layer AR coatings on glass(nm = 1.52) for p-polarized radiation incident at 70° for differentmaterials as the inner layer. The layer thicknesses for eachcombination of materials were adjusted to produce zero reflec-tance. n, = 2.30 for all the stacks. A, air(1.004H, 0.955L)sub withnt = 1.74; B, air(0.793H, 0.718L)sub with n2 = 1.92; C, air(0.642H,0.725L)sub with 1t 2 2.0; D, aii'(O.246H, 0.94L)sub with t2 = 2.12.

1

\I s '\

II .450

/

/,/ A,

/* ,' ,/ /

A

'S.,/ . I

I I I . . 1 550 650 750 850

WAVELENGTH (nm)

Fig. 8. Spectral reflectance of double-layer AR coatings on glass(nm = 1.52) for p-polarized radiation incident at 70° for differentmaterials as the inner layer. The layer thicknesses for eachcombination of materials were adjusted to produce zero reflec-tance. n = 2.50 for all the stacks. A, air(0.966H, 1.073L)sub withn2 = 1.92; B, air(0.69H, 0.961L)sub with n 2 = 2.00; C, air(0.228H,1.028L)sub with L2 = 2.12.

550 APPLIED OPTICS / Vol. 31, No. 4 / 1 February 1992

0.010

0.008 l.1

U

L.adax

0.006

0. 004

X I

\.

450

0.002

550 650WAVELENGTH (nm)

l l | -.-~ u . . . . . . I . . . .n nnI

0 Ol Or

0.008

adUz

Uad

Liadax

0.006

0.004

0.002

-

- I

- I

- a- I

- a

450

#8B

I

550 650 750 850

WAVELENGTH (nm)

Fig. 9 Spectral reflectance of double-layer AR coatings on glass(n = 1.52) for p-polarized radiation incident at 700 by using thesame set of materials but interchangeably as the inner and outerlayers. The layer thicknesses were adjusted to produce zeroreflectance. A, air(1.004H, 0.955L)sub with n = 2.30 and n2 =

1.74; B, air(O.323L, 0.581H)sub with n, = 1.74 and n2 = 2.30.

Effect of Variations in Angle of Incidence

The variations in spectral reflectance of the ARstacks, air(0.264L, 0.628H)sub and air(2.363L,1.564H)sub, n, = 1.45 and n2 = 2.3, with angle ofincidence are shown in Figs. 10 and 11, respectively,in the form of equireflectance contours plotted over agrid of wavelength against angle of incidence. For thecase of an AR stack having a thinner high-index layer(Fig. 10), the reflectance remains below a specifiedvalue over a fairly large variation both in wavelengthas well as angle of incidence. For example, thereflectance remains below 2% over a wavelengthrange from 480 to 640 nm for a variation in the angleof.incidence from 660 to 740. For the AR stack havinga thicker high-index layer (Fig. 11), the reflectancechanges rapidly with wavelength but shows greaterstability with respect to changes in the angle ofincidence. Therefore, in contrast to the AR stack witha thinner high-index layer, the AR stack with a

Lii

0

Uz00

0adZ

REFLECTANCE

1 0.001

2 0.0023 0.005

4 0.008

5 0.0106 0.015

450 550 650 750 850

WAVELENGTH (nm)

Fig. 10. Equireflectance contours of a double-layer AR coating onglass (nm = 1.52) for p-polarized radiation incident at 700. The ARstack is air(0.264L, 0.628H)sub with n, = 1.45 and n2 = 2.30.

_safl

ax

(:2

zR

ad

0

0JzD

66 1. I I

REFLECTANCE

1 0-01

2 0.023 0.05

4 0.105 0.15

6 0.20

450 550 650 750 850

WAVELENGTH (nm)

Fig. 11. Equireflectance contours of a double-layer AR coating onglass (nm = 1.52) for p-polarized radiation incident at 700. The ARstack is air(2.362L, 1.564H)sub with n1 = 1.45 and n2 = 2.30.

thicker high-index layer shows a reflectance of < 2%over a narrow wavelength range from 565 to 620 nmonly, although over a fairly large variation in angle ofincidence from 660 to 740.

It may be noted that a change in phase thickness ofthe layers with variation in angle of incidence overthe range of interest (66° to 740) is very small incomparison with the change brought about by varia-tion in the wavelength (450-850 nm). For example,with a variation in the angle of incidence from 660 to740, the phase thickness of the high-index layer(n = 2.3) changes only by 1% whereas a similarchange in phase thickness results when the wave-length changes from 597 to 603 nm only. The ARstacks thus show greater stability in performancewith changes in the angle of incidence. The AR-coatedprism beam expanders can therefore be used at largerincident angles to obtain higher magnification with-out any increase in reflection loss.

The equireflectance contours of two AR stacks,air(1.004H, 0.955L)sub and air(1.091H, 2.114L)sub,having different materials for the inner and outerlayers are shown in Figs. 12 and 13, respectively. TheAR stack, air(1.004H, 0.955L)sub, with titaniumdioxide (n, = 2.3) as the high-index outer layer andmagnesium oxide (n2 = 1.74) as the low-index innerlayer shows much better spectral stability (Fig. 12).The reflectance remains below 0.5% over a widewavelength range from 450 to 850 nm. The equireflec-tance contours of the AR stack, air(1.091H,2.114L)sub, with zirconium dioxide (n, = 2.12) as theouter layer and silicon dioxide (n2 = 1.45) as the innerlayer are nearly symmetrical (Fig. 13). This meansthat, at a given wavelength or angle of incidence, thereflectance remains below a specified value for asimultaneous variation in these parameters.

Antireflection Coatings for s Polarization

There are situations where optical components arerequired to be used at grazing incidence angles forradiation polarized in a plane perpendicular to theplane of incidence.'5 The reflection loss for such a case

1 February 1992 / Vol. 31, No. 4 / APPLIED OPTICS 551

II

a -1n

REFLECTANCE

1 0.002

2 0.004

3 0.0064 0.0075 0.0096 0.011

7 0.0138 0.015

550 650 750 850

WAVELENGTH (nm)

Fig. 12. Equireflectance contours of a double-layer AR coating onglass (m = 1.52) for p-polarized radiation incident at 70°. The ARstack is sub(0.955L, 1.004H)air with n, = 2.3 and n2 = 1.74.

(s polarization) is much higher than that forp-polar-ized radiation. At an incident angle of 700, the reflec-tion loss from an air-glass interface for s-polarizedradiation is as high as 31%. AR coatings for s-polar-ized radiation can be designed based on the methodol-ogy presented above by using the correspondingeffective refractive indices in Eqs. (5) and (6).

For an incident angle of 70°, the effective refractiveindices of air and glass for s-polarized radiation werecalculated to be 0.342 and 1.195, respectively. Theeffective refractive index of the material whose singlequarter-wave thick layer would antireflect an air-glass interface is therefore 0.639. The possible valuesof the effective refractive indices of the materialswhose suitable layer thicknesses would give AR char-acteristics are shown by the shaded region in Fig. 14.The lower bound of this region is limited by magne-sium fluoride with an index of 1.38. There is only onepossible region and it corresponds to 2 > 1.89 and1.89 < - < 0.53512. One can select magnesiumfluoride or silicon dioxide for the low-index materialas the outer layer and titanium dioxide for the

REFLECTANCE

1 0.001

2 0.002

3 0.005

4 0.0085 0.010

6 0.015

1.0 _____ ___j-------M\XuLBY\L n 1.38

> =0.639

0.0 1.0 2.0 3 0

-712 -

Fig. 14. Diagram showing the values of the effective refractiveindices mq and 112 for a double-layer AR coating on glass (nm = 1.52)for s-polarized radiation incident at 70. Suitable layer thicknessesof materials (shaded region) can produce zero reflectance.

high-index material as the inner layer. Substitutingthe values of the effective refractive indices in Eqs. (5)and (6), we get the AR stacks with thinner high-indexlayers as air(1.558L, 0.661H)sub with n2 = 2.3 andn = 1.38, and air(1.396L, 0.92H)sub with n2 = 2.3and nl = 1.45. The computed spectral reflectances ofthese AR stacks centered at 600 nm are shown in Fig.15 as curves A and B, respectively. The reflectancecharacteristics are broader for the AR stack with anouter layer of lower refractive index. It may be notedthat this behavior is contrary to that observed in ARcoatings forp-polarized radiation. This is because theeffective refractive index of the incident medium islower than the substrate for s-polarized radiation.The reflectance remains below 2% over a wavelength

0.10

0.08

adz

U

LIvad

0.06

0.04

0.02

0.n

450 550 650 750 850

WAVELENGTH (nm)

Fig. 13. Equireflectance contours of a double-layer AR coating onglass (n = 1.52) for p-polarized radiation incident at 70°. The ARstack is sub(2.114L, 1.091H)air with n = 2.3 and 2 = 1.45.

500 550 600 650 700

WAVELENGTH (nm)

Fig. 15. Spectral reflectance of double-layer AR coatings on glass(nm = 1.52) on s-polarized radiation incident at 700 for differentmaterials as the outer layer. The layer thicknesses for eachcombination of materials were adjusted to produce zero reflec-tance. n = 2.3 for both stacks. A, air(1.558L, 0.661H)sub with n =1.38; B, air(1.396L, 0.92H1)sub with it = 1.45.

552 APPLIED OPTICS / Vol. 31, No. 4 / 1 February 1992

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Fig. 16. Variation of spectral reflectance with incident angle of adouble-layer AR coating on a fused silica substrate that wasdesigned to give zero reflectance at 770 for s-polarized radiation.The AR stack is air(1.49L, 0.925H)sub with nm = 1.463, nl = 1.38,and n2 = 2.40: curve A, 770; curve B, 790; curve C, 81°.

range from 570 to 635 nm for the AR stack withmagnesium fluoride as the outer layer (curve A).

Using fused silica (nm = 1.463) as the substratematerial, titanium dioxide (n2 = 2.4) and magnesiumfluoride (n, = 1.38) as the high- and low-index materi-als for the inner and the outer layers, respectively, anAR stack of air(1.49L, 0.925H)sub has been designedfor an incident angle of 770. The spectral reflectanceof this AR stack is shown in Fig. 16 for differentangles of incidence. At an incident angle of 770 (curveA), the reflectance remains below 2% from 580 to 620nm. As the incident angle increases, the reflectance ata given wavelength increases and the wavelength ofpeak transmission shifts toward the lower wave-length side. At an incident angle of 810 (curve C), thereflectance at the design wavelength is <3% andremains below 5% from 575 to 615 nm. This AR stackgreatly reduces the reflection loss that is as high as55% from an uncoated air-glass interface.

Conclusion

We have presented a methodology for the design ofbroadband double-layer AR coatings for grazing inci-dence angles. Almost all these AR stacks are based onthe refractory materials that are known to have goodabrasion resistance, substrate adhesion, and damageresistant properties."6 7 The refractive indices chosenwere those that can be obtained in the thin-film formin standard processing conditions by using electron-beam evaporation.>" The dispersion of the coatingmaterials was not taken into account while thespectral reflectances of AR stacks were calculated.Since the dispersion of these materials is fairly small,particularly in the 500-900-nm wavelength range,the spectral characteristics of the experimentallyproduced prototypes of the AR stacks do not differsubstantially from those reported here.

The author is grateful to D. D. Bhawalkar and ShriU. K. Chatterjee for their keen interest and constant

encouragement. The author thanks P. D. Gupta forthe many helpful discussions.

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