JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

Theoretical and Experimental Investigation of a New Type of Blazed Grating*

ARmAND WIRGIN

Groupe Fillrage et Diiffraction, Instilut d'Optique, 3, Bd. Pasteur, Paris 15e, France

AND

ROGER DELEUIL

Laboratoire d'Optique, FacultM des Sciences de Marseille, Saint Jerdine, Marseille 136, France(Received 27 January 1969)

A rigorous electromagnetic theory of the diffraction of light by blazed lamellar gratings has been de-veloped. It is applied to calculate the diffracted powver distribution of four gratings (R 1 -R 4 ) whose grooveshave the following depths and widths (bl) in units of grating period a: (0.433, 0.750), (0.333, 0.667), (0.250,0.500), and (0.200, 0.400). All four gratings are theoretically blazed in the minus-first order for 300 incidenceand for the wavelength equal to the grating period. The blaze is, however, broad band, and the minus-first-order theoretical power-conversion efficiency (P-1) for the most interesting grating, R3 , exceeds 0.8, 0.9, 0.95,and 0.98 in the wavelength bands 0.84 a-1.49 a, 0.85 a-1.47 a, 0.87 a-1.43 a, and 0.98 a-1.31 a, respectively,for linearly polarized light. These surprising performances, which far exceed those obtainable with echelettegratings insofar as the incident light is linearly polarized, have been experimentally verified with the help ofa microwave interference spectrometer, and point to the potential usefulness of blazed lamellar-reflectiongratings in far-infrared spectroscopy.INDEX HEADINGS: Grating; Reflectance diffraction; Scattering.

Let us consider the properties of plane metallic grat-ings, that is, structures obtained by ruling a series ofparallel equidistant grooves on a plane metallic surface.In all that follows, the gratings will be idealized byassuming them to be perfectly reflecting and of infinitespatial extent (i.e., infinite number of infinitely longgrooves). We shall qualify as theoretical any of theresults pertaining to idealized gratings, The influenceof the finite spatial extent of the gratings, as well asthat of the illuminating beam, is well known' 2 and notof immediate interest in this paper, whereas the finiteconductivity of the metallic support produces a lossof reflectance similar to that observed with planemirrors.3

Let an idealized grating, whose period is a, be placedin the vacuum, and be illuminated by a monochromaticplane wave (wavelength X) whose propagation vectork is perpendicular to the groove axis and strikes thegrating at the angle OA with respect to the normal tothe plane of the grating. The presence of the gratinggives rise to one or more reflected plane waves, Bo,B-, ... , Bn .*. . , which propagate in the directions0. (n= O. i 1, ***) relative to the grating normal. Theseangles are related to 6A, a, and X by the grating equation

riI < -,/2sinOn= sinOi+nX/a; i |sinO | •1 (1)

Lw/2<06<37r/2.

If we now suppose the incident wave to be poly-chromatic, then the previous relation indicates that

* Work supported by the Direction des Recherches et Moyensd'Essais, Grant No. 269/65 and its extensions.

IF. A. Jenkins and H. E. Wllite, Fundantentals of Optics(McGraw-Hill Book Co., New York, 1957).

2 G. W. Stroke, in HIandbuclz der Physik Vol. 29, S. Fhigge, Ed.(Springer-Verlag, Berlin, 1967).

3 R. Petit, Compt. Rend. 258, 1429 (1964).

there is angular dispersion (as a function of A) ineach order (i.e., each value of n) of diffraction, exceptin the specular zeroth order. Therefore, for N orderssatisfying the constraint I sinO, I • 1, the grating createsan N-fold spectral degeneracy.

However, the spectrum is observed in only one orderat a time, the energy going into the other orders beingcompletely wasted. It is usually desirable to have theobserved spectrum as luminous as possible, so that away must be found of concentrating the incident energyinto a sole diffraction order.

One method of, at least, simplifying the problem, isto reduce the order of degeneracy by reducing thegrating period a with respect to the mean wavelength ofthe spectrum. In the limit, only the minus-first andzeroth orders survive, but even if we neglect the factthat the angular dispersion and resolving power (fornon-idealized gratings) are at their lowest values in theminus-first order, the problem still remains of reducingthe power that is reflected into the useless zeroth order.

Another possibility, called blazing, involves shapingthe grooves in some convenient manner. It occurredto Rayleigh4 and Wood' that a grating with triangulargrooves (echelette grating, Fig. 1) could be made toconcentrate a substantial fraction of the incident energyinto only one grating order. Their idea is that situationsexist in which the rays reflected by one of the inclinedfacets (angle f relative to the grating plane) of thegrooves have the same direction as a given (non-zero)grating order m; this happens for that wavelength XEm(the blaze wavelength), which satisfies the relation

Am= (-2a/n) sin:3 cos(8-0j). (2)

4 Lord Rayleigh, Phil. Mag. 14, 60 (1907); Proc. Roy. Soc.(London) A79, 399 (1907).

5 R. W. Wood, Proc. Phys. Soc. (London) 18, 396 (1902);Phil. Mag. 4, 396 (1902).

1348

VOLUME 59, NUMBER 10 OCTOBER 1969

BEW TYPE OF BLAZED GRATING

NInh

N_E)

rn

11

N

OL

FIG. 1. Echelette grating. FIG. 2. Lamellar grating.

Equation (2) contains no real information about theactual power reflected into the blazed order at the blazewavelength, nor at other wavelengths. These powercalculations can be carried out with the help of formulasobtained from conventional scalar diffraction theory.6

The difficulty is that, in practice, the periods of diffrac-tion gratings are comparable to the mean wavelengthof the observed light, and in these circumstances thediffraction phenomena are poorly described by consid-erations based on scalar theory. In particular, the latteris incapable of taking into account the influence of thepolarization of the illumination, which is often apredominant factor in the determination of the magni-tude of the blaze effect.7' 8

For this reason, much work has been devoted in thepast ten years2

&A4 to improving the theoretical descrip-tion of the grating action. The obtained results (insome cases experimental7' 9"5-' 7) have been useful inexplaining some aspects of the influence of the specificprofile of echelettes on the scattered energy distribution,but other questions, such as whether it is preferable forthe grooves to be right triangles, and just how tooptimize the blaze effect for maximum reflectance andbandwidth, have not yet been answered.

THEORETICAL ANALYSIS OF A NEW TYPEOF BLAZED GRATING

Practically all the gratings used in spectroscopy areof the echelette type. The question can reasonably beasked whether other types of gratings can be blazed,

6 R. P. Madden and J. Strong, Concepts of Classical Optics(Freeman, San Francisco, 1958).

7 G. W. Stroke, Phys. Letters 5, 45 (1963).8 R. Petit, Rev. Opt. 45, 249, 353 (1966).9 G. W. Stroke, Rev. Opt. 39, 291 (1960).0 B. A. Lippmann, J. Opt. Soc. Am. 43, 408 (1953).

W. C. Meecham, J. Acoust. Soc. Am. 28, 370 (1955).2 P. Bousquet, Japan J. Appl. Phys. 4, suppl. 1, 549 (1965).

'3 A. Wirgin, Rev. Opt. 43, 449 (1964); 44, 20 (1965).'4 J. Pavageau, thesis, Ed. de la Rev. Opt., Paris (1966).'5 R. Deleuil, Opt. Acta 16, 23 (1969).16 J. M. Proud, P. Tamarkin and W. C. Meecham, J. Appl.

Phys. 28, 1298 (1957).1" A. Hadni, E. Decamps, D. Grandjean, and C. Janot, Compt.

Rend. 250, 2007 (1960).

and if so, whether they might be more efficient thanechelettes.

The lamellar grating (Fig. 2), with grooves having arectangular profile, if examined from a purely geo-metrical-optics point of view, is seen to produce abackscatter for wavelengths

XAg= (- 2a/m) sin6i. (3)

This blaze effect is similar to that obtained withechelettes except that it is produced, not by one reflec-tion, but by two or more (even-number) reflectionswithin the grooves. Similar effects can be observed forhigher-order polygonal groove shapes.

These simplified hints of the possibility of producingblaze behavior, which proved to be so useful in thedevelopment of echelette gratings, are the points ofdeparture for the present analysis of the lamellargrating. To get the full picture, however, we mustcalculate the powers radiated into the grating orders,and in doing this, take into account all of the effectswhich might have a more or less important influenceon the power distribution. The only way to do this isto solve the electromagnetic boundary-value problemwhich rigorously translates the interaction between theincident light and the grating.

The problem can be stated in the following terms.The twice-differentiable complex function U(x,z) issought in the half-infinite domain S(x,z) above thegrating contour C(x,z). U, together with its gradient,is continuous and square integrable in any finite sub-domain of S and satisfies the three relations""18-21

(A+k2) U(P) = O, PES, (4)

2 (1-e) U(P)+2 (1-0)a.U(P)=0, PEC, (5)

(0,-ik) [U(P)- Ui(P)-A U. (P*)]-O (r-1);r=(P2+Z2)E Xx pES. (6)

P= P(x,z) is an arbitrary point whose coordinates are

"8 A. Wirgin, Compt. Rend. B262, 385 (1966).19 A. Wirgin, Compt. Rend. B262, 579 (1966).20 A. Wirgin, Compt. Rend. B262, 870 (1966).21 A. Wirgin, Compt. Rend. B262, 1032 (1966).

October 1969 1349

ARMAND WIRGIN AND ROGER DELEUIL

(x,z), An is the normal derivative, A is an arbitraryconstant, P*=P*(x,-z) is the image point to P withrespect to the plane z=0, k is the free-space wavenumber, equal to 27r/X=co/c where co is the angularfrequency of the incident and total fields (the timefactor exp-iwt has been suppressed).

It is understood that the field is monochromatic andpolarized in the direction of the y coordinate (which isconsequently ignorable) parallel to the groove axis.Two cases are considered, corresponding to the twopolarization eigenstates (identified by the index ,3) ofthe field; in the first case (#=-1, El, polarization)the incident and total electric fields have only onenonzero component-the y components-which aredesignated by the functions Uj and U, respectively,and in the second case (8 = +1, E± polarization), it isthe magnetic-field vectors which are parallel to thegrooves, Uj and U representing the incident and totaltransverse magnetic-field components. The incidentfield is a plane wave, represented analytically by theform

isi= sinGiUj(P)=expik(sjx-wjz); jw= cosOi (7)

I64iI <7r/2.

Equation (4), which results from Maxwell's equationsin the vacuum, describes the propagation of the fieldU in the domain S. Equation (5), which is also a con-sequence of Maxwell's equations, is the boundarycondition at the surface of the grating (which, werecall, is perfectly conducting), and Eq. (6) is the modi-fied Sommerfeld radiation condition, which states thatthe field at infinity is decomposable into the incidentplane wave, a specularly reflected plane wave, and anoutgoing cylindrical wave.

The solution to the problem is sought with the helpof Green's theorem.

First, we divide S into a series of subdomains

Floquet condition

U(Pna) = U(P) expiksina, (11)

which is a consequence of Eqs. (4), (5) and (7). Inaddition, the following definitions apply to Eq. (10)

P' =P'(x',z') P*'=P*' = (x, -z'),

Po=p'(X,0), pna=Pna(x+na,z),

Fo+(P'; P)=Z Gp+(P'; Pna) expiksana,

(12)

g(P'; P)=4iIoo('){k(P'; P)]

=4iHo(l) (kR) =g(x'-x,z'-z),

R = I [(x'-x)2+ (z'- z)2 1.

Furthermore g is the two-dimensional free-space Green'sfunction, Ho' is the Hankel function, and Gg+ satisfiesthe Sommerfeld radiation condition, in addition to theboundary condition

'(l-f)Gg+(P'; PI)+2 (1+,/)d2G0+(P'; PI) = 0,

V., x. (13)

The use of Poisson's sum formula'3 results in theidentity

F#+(P'; P) =, (-2ikaw.)-1 expiksn (x'-x)n

X~ex-pikWn I Z'-zl +3 expikcon I z'+z I], (14)where

(15)

S(P) =S+(P)+ Sn(p) (Z =n n

nc

n1-co(8)

S+ is the domain z> 0,VY, S. is the groove domain-b<z<0, 0<x,<1, xn is the local x coordinate forthe nth groove, where

xn=x-na+1/2, (9)

I is the groove width and b is the groove depth. Theapplication of Green's theorem in St gives

1/2

U(P') = Ui(P,)+1Ui(P*2)+f [(1-1) U(P')J1/2

X0z,r# (P; PI) -2 (I +#)Fe (PI; PI) 0zU (PI) ]dx;

P'ES+, (10)

where use has been made of Eqs. (4)-(6), and of the

Consequently Eq. (10) can be shown to reduce to

U(P') = Ui(P') + A. expik (snX'+Conz');n

PEES+, (16)where

1/2 dx

A n=ln~o+ { -[ (1 -I)U((P)-1/2 a

+2(1+±1) (ikco.)-'&dU(P0)J exp-iksnx, (17)

an,m being the Kronecker delta.Green's theorem can be applied, in a similar manner,

to the domain Sn, so that

I

U (Pn)= [GB-(Pn'; Pno) d9 U (Pno)

-U(P 0)a.Gp-(Pn'; Pn0)]dXn; Pn 'Sn. (18)

1350 Vol. 59

G�+ (P'; P) = g (P'; P) +Og (P; P*),

Sn=sj+2n7r1ka, COn = + (I - Sn') 1-

NEW TYPE OF BLAZED GRATING

The following definitions apply to Eq. (18)

Pn =Pn (XIn,Z/), PnI = P (XnO), Pn=P.(X.,z),

Go7-(Pn/; P.) =E Eg(x.'-x,,+2m1,z'-z)

+i0g (x,'+x,+2m1,z'-z)

+±g3g (xn'-x,+2m1,z'+z+2b)

+g (xn'+xn+2m1,z'+z+2b)]; (19)

Go satisfies the Sommerfeld radiation condition andthe same boundary conditions as U on the walls of thegrooves [Eq. (5)]. Once again, the Poisson sumformula permits us to establish the identity

GF (P.'; P.) = E m(2k1-Ym)'1[coskam(xn'-Xn)m=(1-0) /2

+1 coskam (xn'+xn)][2 (1-13) sinky (z'+b)

+2 (l+13) coskym(z'+b)] expi[kbym+'?r(1+13)]; (20)

where

am=mr/kl, ym= (1am2)l, eo=1, emo,=2. (21)

Consequently Eq. (18) reduces to

00

U(P,,,) = E nRm[cosk(avmxn.'-ym(z'+b))m= (1-0) /2

+0 cosk(amxn'+'ym(z'+b))]; Pn'&Sn,

and

(22)

nRm=Rm expiksina (23)

in view of the fact that U satisfies Eq. (11).Equation (22) is a representation of the groove

fields in terms of waveguide modes, whereas Eq. (16)is the Rayleigh plane-wave representation of the fieldin the region exterior to the grooves. It will be notedthat Eq. (22) cannot be put in the same form as Eq.(16); this confirms Lippmann's discovery'" that Ray-leigh's original theoryi of diffraction gratings is inexact.

In Eq. (16), the set ao, for which the con are real,corresponds to the undamped plane waves (i.e., thepreviously mentioned grating orders) that are respons-ible for the spectral lines that are observed at infinity.The quantities of interest are the powers, P. (relativeto the incident power), radiated into the grating orders,since they give a measure of the intensities of thespectral lines. These powers are related to the plane-wave amplitudes by

Pn =A fl|2Wn/wi, (24)

and conservation of energy dictates that

Ad P. = 1. (25)nEan.

The next, and more difficult, task is to establish theequations that enable us to calculate the A n. We makeuse of the continuity conditions of U and VU thatappear in the statement of the boundary-value problem,and which imply

U(x,O+) = U(x,O-) =U(x,O) = U(PI)

aU(x,0+) = adU(x,0O)-- aU(x,o) = azU(P°); (26)

na-1/2<x<na+l/ 2 , yV(z±=lim (zd4! el )).e-O

By applying Eq. (26) to Eq. (10), we find","

[-2 (1 -1)a, +12 (1+0)][U(P"')-2Ui(P")]1/2

= (1-13) (k2+d., 2) 1/dxU(P2)F+(P+'; PI)

11/2

- (1+1A) J dxF#+(PO+'; PO)aU(PO);1/2

I x'jl<1/2, (27)

where P" = P" (x'0) and P°' --'= PE'(x',0+). We thenreplace U(P"') and aU(P") in the previous relationby the expansions obtained from Eq. (22), and applythe quadratures

1/2 dx'2 j E1[2(1-3) sinkayjxo'+ (1+fl) coskajxo'](.");

wr1/2 1

whereby we obtain

oo

FjmRm =T; jj=(1-13), 2(1-fl)+1,.-*, (28)m= (1-0) /2

Fjm and Tj being defined as

Tj= [2 (1-0)2wCy7'-1+' (1+1)ejj]Co + seckb'yj,

Fjm = -aj,m+ (/ai) sinkbym seckb-yj

, [21 _) +2 (1+13) 6 ]CnmCnj+

n 2-y. con

IdDCnm46 = 2 - expLiksn (-1/2)

X[Ej (1 -1)i sinkamP+± (1+1) coska<m]. (29)

On the other hand, by inserting into Eq. (17) theexpansions for U(P0 ) and adU(P') obtained from Eq.(22), we find the following linear transformationbetween A n and Rn

An=Pb3tnO+ E HnJR.m; n=0, 4f1, d2 2, * *(30)m= (1-) /2

1351October 1969

ARIMAND WIRGIN AND ROGER DELEUIL

where

H.rn = (1 -fl)+- (1 +,)-"'jCnJ- sinkbley. (31)

The problem, as we see from Eqs. (28) and (30),reduces to inverting the infinite-order matrix equation

FR= T (32)

and calculating the matrix product

A-,1= HR (33)

[1 is the vector (1,0,0, * * .)].Two very serious difficulties are involved in these

operations: the matrices are of infinite order and theelements of F are infinite series [see Eq. (29)]. Wemust therefore resort to an approximate numericaltechnique. We simultaneously truncate the matricesappearing in Eq. (32) and (33), as well as the series

contained in F, in such a way that we are left with3'

E Fjm, (f)RmC 2 II) =Tj;m=(I-fl) /2

j= (-)2 (1- + 1, * ,M, (34)

A.( ') =133no+ Z HnmRn(M);In -(I-#) /2

n=0, 4-l, **-, hM, (35)

0,5 1,5

0 0,5 1 1,5

L+ll , I ' - d/0 0,5 1 1,5

FIG. 3. Grating R,: b=0.433a, 1=0.750a, oi=3 0 '. Theoreticaldiffraction efficiency (In) vs a/A in the orders n=O, -2 (top)and blazed order n = -1 (bottom) for the two polarization statesE,, and E±.

0 0,5 1 1,5

FIG. 4. Grating R,: b=0.333a, 1=O.667a, Oi=30Q. Theoretical Po(top) and P- 1 (bottom) vs a/X.

R.C") and AX(M) being the Mth order approximationsto Rm and A-, and Fjm (M) being the Mth order approxi-mation to Fjm obtained by replacing the series rn=:in the latter by the series Fn=_m1`. Equations (34) and(35) are solved, and M is incremented, thereby leadingto a set of approximations for An. This process isrepeated until the A ,/M) (M= 1,2, **) differ from eachother by no more than some preassigned small quantity.

Experience shows that, in the case in which thegrating period is comparable to the wavelength, con-vergence is generally attained for M of the order of ten.

1352 Vol. 59

NEW TYPE OF BLAZED GRATING

For this order of approximation, the entire calculationis executed by a UNIVAC 1108 computer. The adoptedsolution is physically meaningful because the algorithmis strictly energy conserving for all M.

By choosing the grating parameters on the basis ofthe insight gained from Eq. (3) and with the help of amethodical trial- and error-optimization procedure, wehave been able to obtain the results22 23 depicted inFigs. 3-6, which represent the relative scattered powers

1,5

0 0,5 1 1,5

FIG. 5. Grating R3: b=0.250a, l=0.500a, Oi=30. Theoretical P0 ,P- 2 (top) and P-, (bottom) vs a/X.

in the different grating orders as functions of the wave-number (in units of grating period a).

EXPERIMENTAL CONSIDERATIONS

Inevitably associated with the measurement, in theoptical domain (visible and near-infrared), of thediffraction phenomena associated with gratings, are thedifficulties of accurately ruling and then determiningthe exact form of the grooves. Although progress has

22 A. Wirgin, Rev. Opt. 47, 333 (1968).23 A. Wirgin, French patent, BF No. prov. 159952, 19 July

1968.

0 1 1,5 2

FIG. 6. Grating R4 : b = 0.200a, I = 0.400a, Oi = 30°. Theoretical P0 ,P- 2 (top) and P-, (bottom) vs a/X.

recently been made in electron-microscope techniques2 4

that determine the characteristic parameters (such asthe inclination of the groove facets of an echelette)of certain profiles, the same techniques reveal that theseprofiles differ significantly from the theoretical profiles,because of blank roughness and roughness created bythe cutting tool. Whereas measurement of scattered-power distributions is quite easy in the visible, thesame is far from being true in the infrared, where, onthe other hand, the grating profile is easier to defineaccurately.

In the hertzian domain (e.g., millimeter waves),however, it is possible to work with gratings havingperiods that range from several millimeters to two orthree centimeters. Ruling such gratings presents no in-surmountable difficulty; the distance between groovescan be kept equal over many periods, to an accuracyof several parts in one thousand, the grooves have anexcellent surface finish (X/200, if desired), and oncethe ruling is terminated, it is possible to determineaccurately the form of each groove. It is then a simple

24 J. P. Chauvineau, L. Constanciel, A. Marraud, and R. Petit,Rev. Opt. 46, 417 (1967).

1353October 1969

1354ARMAND WIRGIN AND ROGER DELEUIL

matter to evaluate the influence of the groove profileon the scattered-power distribution of the grating.

The apparatus used in this experiment has alreadybeen described elsewhere, 5 -25 so that we shall here giveno more than its principal characteristics after brieflyrecalling the principles that entered into its conception.

The basic instrument is a microwave spectrometerthat operates in the K'a band (11.3> X> 7.5 mm). Ourinstrument, being an interferometer, permits us toeliminate the influence of power variations of the sourceand to suppress errors due to the nonlinearity of thedetector. We thus conserve all the advantages of a nullmethod, provided that we rigorously stabilize thefrequency.

The interferometer is depicted in Fig. 7. The emittedpower is divided into two (equal) parts by a directionalcoupler. A fraction of this power traverses the referencebranch constituted by a calibrated attenuator (CA)and a phase changer (PC). The second fraction isguided to the focus of the paraboloid that illuminatesthe grating to be studied. An arm, which rotates aroundan axis situated in the grating plane and lying parallelto the grooves, is terminated by a second paraboloidthat explores the diffracted field. The signal capturedby this mirror is superposed, in a second directionalcoupler, onto the reference signal.

The measurement consists of canceling the detectedsignal by acting on the reference-branch components.The power scattered into a given order is determinedby comparing the signals furnished by the grating andthen by a plane metallic mirror. The mirror has thesame height as the grating and its area is such thatit receives the same incident flux as the grating.

With the wavelengths used here, the gratings haveonly a limited number, N,, of grooves. However, assoon as N0 is greater than about twelve, gratingsbehave, for all practical purposes, as if they wereinfinite. Therefore, we have chosen 50>1VN>20, witha blank size of 300X260 mm. The diameter of theparabolic mirrors is 530 mm so that the ratio X/D staysclose, to 1/60. In addition, as the distance betweenthe emitting antenna and the grating is greater than3 m, the incident wave can be considered to be practi-cally plane.

The apparatus, as described, permits measurementswith less than three percent relative error and hasalready been used to obtain several interesting resultswith diffraction gratings.l5 '2 '

INTERPRETATION OF RESULTSAND CONCLUSIONS

Examination of the dispersion curves (P. vs X) in theblazed order of a typical echelette grating8 reveals that(a) the curves are bell-shaped, (b) the El, and E±

25 R. Deleuil and F. Varnier, Compt. Rend. B267, 1074 (1968).

curves are shifted with respect to one another, so thatthe blaze wavelength (that wavelength for which theefficiency is maximum), which is theoretically inde-pendent of polarization, is, in fact, not the same forthe two polarizations, (c) the efficiency maximum ismore pronounced for one polarization than for theother. The first observation, (a), means that the blazeeffect has a certain bandwidth and is not at its maximumthroughout this band. Typically, with an ideal echelette,it is possible to concentrate 80% of the incident powerinto one order in a band of one-half octave for a givenpolarization. In the best case, the bandwidth can attain0.7 octave for 80% efficiency or 0.3 octave for 90%efficiency. When, by a suitable choice of the grooveparameters, the El, and E± curves are united withinthe same band, the width of this band is usually nogreater than 0.25 octave for 80% or more theoreticalefficiency. Generally, the effect of uniting the two curvesis to reduce the maximum heights of both. A conse-quence of (b) is that the blaze formula is incorrect foreither or both polarizations; this is not surprising, inview of our previous comments.

As with echelettes, the dispersion curves of blazedlamellar gratings in the blazed order (Figs. 3-6) aremore or less bell shaped, and the blaze dominates inone or the other polarization for a given wavelength.The latter effect causes very annoying unnecessarypower losses if the radiation emitted by the source isunpolarized. This is less of a problem in ir spectroscopesbecause the light, before striking the grating, undergoesa series of reflections which tend to polarize it; itshould, in principle, be necessary only to align thegrating relative to the major axis of the polarizationellipse, in order to obtain the maximum efficiency.

Among the gratings we have considered, R3 and R4do not exhibit genuine blaze effects in ElI polarizationwhereas R1 and R2 produce very pronounced blazes inboth polarizations, albeit in two distinct domains.Within these domains, grating R, can be used to analyze

FIG. 7. CA=calibrated attenuator, D=detector, DC=direc-tional coupler, FI = ferrite isolator, G =galvanometer, K = klys-tron, PC=phase changer, PS=power supply, RC=referencecavity, SC=stabilizing circuit, VA=variable attenuator, gV=microvoltmeter.

1354 Vol. 59

NEW TYPE OF BLAZED GRATING

the spectrum of polarized light by working in the E,,orientation for the smaller wavelengths, and afterrotation of the grating with respect to the incidentbeam, in the El orientation for the larger wavelengths.In this way, more than 71%0 theoretical efficiency isobtained in the minus-first order in a band extendingfrom 0.89 a, to 1.49 a which is equivalent to 0.45-0.75 g

(i.e., practically the whole visible region, which extendsfrom 0.40-0.72 ,u) for a grating of period 0.5 g. In thesame manner, R1 can be made to concentrate more than80%7 of the power into the minus-first order within the

1

`f\\

f \

fo \\

I \

0,8

0,6'

0,4.

0,2'

E//

\ of/0\ ,,'/

-1

0 0,5

I

'0,8

0,6~

0,4'

0,2

1 1,5

Po

--------

III

I

l l

III'I \\ //

II

I,,, ..0 I O,

I'

0 1 0.5

FIG. 8. Gratingization (bottom).,experimental.

1 ' 1,5

R 2. Po in E,, polarization (top) and E± polar-Dashed curve is theoretical and points are

almost contiguous bands 0.91-1.20 a and 1.26-1.48 a;R2 can be used with greater than 90% efficiency in thesame order within the bands 0.79-0.95 a and 1.11-1.47 a.

Grating R3 is the most interesting of the four. In theEs orientation, it produces more than 90% efficiency

,/L

aix1,5

FIG. 9. Grating R3. Po in El (top) andE± (bottom) polarizations.

in a band extending over three quarters of an octave,and more than 95% efficiency for a bandwidth of 0.64octave. Given a period of 0.5 ,I, R3 could be used toexamine the whole spectrum of visible wavelengthswith less than 10% theoretical light loss. In practice, thisloss is greater, owing to the imperfect reflectivity ofthe metallic substrate. Or, if the period were 500 L,

it could be used for far-infrared spectroscopy with morethan 90% efficiency in the band 430-740 I. This is apractical as well as theoretical figure, for, in the farinfrared, most metals behave as near-perfect reflectors.Another interesting feature of R3 is the practically flatpeak of its dispersion curve in the blazed order; thiscould be very useful for absolute intensity measure-ments of spectra. In cases in which the bandwidth ofthe spectrum is larger and there is plenty of light, R4

would be the appropriate grating to use, because itconcentrates more than 80% of the power into a singleorder, over 0.85 octave.

I Po1

0,8

0.6

0,4

- -- - -

't-e+'*-

0,2

-4l*0 0,5 1

- heala

__4 j

1355October 1969

'k 41

1,5

Fe

- ___J� �ii

i

II

II

II

ARMAND WIRGIN AND ROGER DELEUIL

0.8

0,6

0.4

0.2

P

0 0.5 1,5

0,8

0,61

P

. 11

0 0,5

FiG. 10. Grating R 4. Poin E,, (top) and E± (bottor

The aim of the experiments was to veoretical properties of gratings R2, R 3 andFigs. 4-6, by measuring the diffraction-elvariations in the different orders as a fuifor a fixed (300) incident angle. It woulmost interesting to fix our attention o0order, n = -1; however, continuous explowas not feasible in the domain of inter<2.0), because of the geometry of th(diameters of the parabolic mirrors and tlfrom the grating) which made measuremenin a 100 sector, 40°> S-L=0-,-180'0.87<a/X<1.19), about the back-scatterthe other hand, examination of P-2interest, because P- 2 = 0 for a/X <1.33,finally limited the measurements to t]zeroth order, which could be continuouwithin the whole domain of variation oftion, whenever it was necessary andchecked the energy-balance relation [Emeans of several power measurementsorders.

The experiment, therefore, reduced to lment of Po in the two polarization eigenst~El, for each of the gratings R2, R3 and R4

are depicted in Figs. 7-10, wherein thecorrespond to the theoretical curves and I

6

Tf__

so that wehose of thesly exploredi/X. In addi-possible, werq. (25)] by.n the other

;he measure-;tes, E,, and. The resultsdashed linesthe symbol k

TABLE I.

Theory Experiment

P0 0.60 0.56P-1 0.35 0.38P-2 0.05 0.04

Total 1.00 0.98

20 C. H. Palmer, J. Opt. Soc. Am. 42, 268 (1952).27 C. H. Palmer, F. C. Evering, and F. C. Nelson, Appl. Opt.

4,1271 (1965).28 R. WV. Wood, Phil. Mag. 23, 315 (1912); Phys. Rev. 48, 928

(1935).

11

1356 Vol. 59

i\,

4i �#\

represents the experimental point and its margin oferror.

There is, on the whole, excellent agreement betweentheory and experiment. This is particularly true forgrating R3 (Fig. 9); in El, polarization, all of the experi-mental points fall on the theoretical curve, to withinthe margin of error and in EP polarization, the blazeeffects in intensity and bandwidth, predicted by thetheory, are verified. In addition, the Wood anomaly26 -2 8

Ey M occurs at its theoretical position a/X= 1.333, whichshows that the grating constant a was conserved with

ap, good precision from one groove to the next. However,2 while the theory predicted a bright anomaly-with Po

attaining 1.0-the experiment gave Po= 0.825. Thisresult is not surprising, because, as we have alreadypointed out,'" relative variations of the order of 3%

AAf \ of the groove depths and widths can sometimes cause9El variations of the diffraction efficiencies that attain

30%.The situation is less satisfactory in the cases of

gratings R2 and R 4 (Figs. 8 and 10). The slight shiftof P0 , in Fig. 8, towards the long wavelengths is mainlydue to small variations of the groove parameters,which arise because of play in the cutting tool. The

a/x results depicted in Fig. 10 cannot be as easily explained.2 The measurements confirm the theory in E,, polariza-

tion except near a/X= 1.333 (which marks the onset) polarizations, of scattering into the minus-second order). An energy-

balance check revealed that some of the energy missingfrom the zeroth order reappeared in the minus-first

rify the theo- order (see Table I). For the other polarization, the blazei R 4 given in effect is more or less verified and apparently superiorficiency (P.) (for 0.67 a/X - 1.0) to what is theoretically predicted,nction of a/X whereas the Wood anomaly, as before, is correctly.d have been positioned but less bright (0.61 vs 1.0) than predicted.n the blazed A measurement of P-L, for a/X-0.82 showed thatration of P-, energy was not conserved within the experimental,St (0.5 <a/X limit of error. At the same time, a background ofLe apparatus scattered energy was observed outside the allowedieir redsenergyts impossible scattering directions (Oo and 0-l), which can be attrib-°>200 (i.e., uted to unsatisfactory surface polish. This shows thatdirection. On the imposed tolerances of X/200 were not respected inwas of little the ruling of grating R4, and that even in the microwave

1

NEW TYPE OF BLAZED GRATING

domain, much attention has to be given to the problemof surface-finish quality.

In conclusion, the experiments have, on the whole,verified the unusual properties of lamellar gratingspredicted by the electromagnetic theory. To the bestof our knowledge, these performances far exceed thosethat have been obtained, until now, with echelettes(except for unpolarized light) and point to the potentialusefulness of blazed lamellar gratings in far-ir spectros-

JOURNAL OF THE OPTICAL SOCIETY OF AMERICA

copy23 and in frequency-modulated beam-scanningradar devices.29

ACKNOWLEDGMENT

The authors wish to thank the referee for his helpfulcriticism.

29 A. Wirgin, French patent, BF No. prov. 179261, 19 December1968.

VOLUME 59, NUMBER 10 OCTOBER 1969

October 1969 1357