Perfectly blazed reflection gratings with rectangular grooves

Perfectly blazed reflection gratings with rectangular groovesJames W. Heath and Edward V. Jull

Department of Electrical Engineering, The University of British Columbia, Vancouver, British Columbia, Canada V6T 1 W5(Received 22 March 1978)

For plane electromagnetic wave incidence on a perfectly conducting periodic surface with a rectan-gular groove profile, the coefficients of the matrices governing the scattered mode amplitudes aregiven. These matrices decompose when the Bragg condition is satisfied, facilitating computation.Numerical results converge to those of the comb grating as the groove width approaches the period.Data for perfect blazing to the n = -1 spectral order (i.e., all the power is confined to this order)for TE, TM, and arbitrary polarization are given. Perfect blazing with arbitrary polarization fornear-grazing incidence is shown to be possible in principle with deep wide grooves. Numerical andexperimental results are shown also for narrow groove widths, effective for TM polarization only.

1. INTRODUCTION

Reflection gratings with perfectly conducting rectangulargrooves have attracted considerable recent interest.1 -5

Scattering by this surface can be analyzed rigorously, and canbe completely diffracted in one spectral order for both TE andTM polarizations simultaneously.3 A surface with thisdual-blazing feature is also relatively easily made, at least atmillimeter and longer wavelengths, and this effect has beendemonstrated experimentally.5 Numerical results, however,have been confined to groove-width-period ratios of from 0.5to 0.95 and the range of parameters for perfect dual blazinghas not been determined fully. Here it is shown that whenthe period satisfies the Bragg condition, as required for perfectblazing, the matrices that govern the scattered mode ampli-tudes decompose. This greatly reduces the computationaleffort required to find the surface parameters which provideperfect blazing. Numerical results for grooves wider, deeper,and narrower than considered earlier show several new fea-tures of this blazing phenomenon.

When the width of the rectangular groove equals the gratingperiod, the grating is called a comb. Scattering from a combcan be analyzed rigorously by techniques based on the calculusof residues6 and integral transforms,7 yielding exact resultsfor combs of depth sufficient that evanescent modes in thecomb are unaffected by it. The numerical results for widegrooves converge to these exact results, verifying the accuracyof this method in this limit and providing numerical data forscattering by a shallow comb and by a fin-corrugated surface.7

The set of TE and TM blazing depth curves given earlier3' 4

are completed with the presentation of curves for large andsmall groove-width-period ratios. Perfect blazing of therectangular groove profile for arbitrary polarization, alreadydemonstrated for 19.50 < Oi < 59.40, where Oi is the angle ofincidence from the normal, 5 is shown to be possible in prin-ciple for substantially larger values of Oi using finned surfaceswith deep grooves.

Experimental results here demonstrate blazing for arbi-trarily polarized incidence at Oi = 54.5° and the validity ofnumerical results for a surface with narrow grooves, effectivefor TM polarization only, and blazed for Oi = 64.25°.

11. SCATTERING MATRICES

As the analysis of scattering by the grating of Fig. 1 has al-ready been described 1 -3 ; here it is outlined only in detail suf-

ficient for definition of the resulting matrices. A plane wavenormally incident on the perfectly conducting rectangulargrooves is resolved into TM and TE components. For a TMwave incident at an angle Oi from the surface normal, themagnetic field in the free-space region (z > 0) can be writ-ten

Hy(xz) = Aie-jk(xsinOi-zcosOi)

+ A Ane -ik(xsinOn+zcosOn), n = 0,I1,+2,... (1)n=--

and similarly for the TE case with the electric field E, rep-lacing H, and the incident and scattered mode amplitudes,Bi and Bn, replacing Ai and An, respectively. k = 2ir/X is thepropagation constant,

sin6n = sin6i + nX/d, (2)

and

cosOn = (1 - sin2 On)1/2

- -j(sin2o, - 1)1/2

I sinO, I • 1

IsinOn I > 1. (3)

Equation (2) ensures that fields from adjacent grooves in-terfere constructively, while the negative imaginary root in(3) ensures that the evanescent modes decay away from thesurface. A time dependence eiwt is assumed.

The total fields in the grooves are

H= Cm cos [-(x + cos [km(z + h)],

Ey = E Dm sin [ a (x + sin[km(z + h)],

m = 0,1,2, ....

INCIDENTPLANE

WA VE

SCATTEREDHOMOGENEOUS

PLANEI WA VES

FIG. 1. A plane wave incident on a periodic surface with rectangulargroove profile.

1211 J. Opt. Soc. Am., Vol. 68, No. 9, September 1978 0030-3941/78/6809-1211$00.50 © 1978 Optical Society of America 1211

1.0

0 0 02 0.3 0.4 0.5h>^,

0.21

0.11

0.9s

0.6 0.7 08 09 1.O

FIG. 2. TM reflected power for large aspect ratios: 0, = 81.250, d = X/2 sinO,; - - -, aid = 1.0 (from Ref. 7); -, aid = 0.9999.

where Cm and Dm are the waveguide mode amplitudes and

km = [k2 - M 212m [ m (a )]

= j [ (-a ) I]

m irr < ka

mwr > k.a

Again the negative imaginary root is chosen so that evanescentmodes decay away from the surface. The other transversefields, E. and Hx, are obtained from Maxwell's equations.

Matching free-space and waveguide transverse field at z =0, and using the orthogonality of the modes, yields the fol-lowing matrix equations relating free-space to waveguidemode amplitudes:

C. km tan(kmh) bm = An

m=0 ik COSOn(4)

1/2 (,) 1/2 () j)1/2

k sinOn = ±:-a

k sinOn =-0,a

E = -() 2 1/2 cos[k sin0fn(a/2)]2mir/abrm ad) (k sinOn)2

- (m7r/a)2

m = 1,3,5, ...

. / 2 ) 1/2 sin[k sin0o(a/2)]2m7r/aad (k sinOn)2 - (mlr/a)2 '

m = 2,4,6,...

a)1/2 (j)m,k2d!

and

_ Dmbnm = Bn,m=1

where

= ( E e 1/2 sir[k sinOn(a/2)]2k sinOnnm - ad! (k sinOn)2

- (m7r/a) 2

2 1/2 cos[k sinO,,(a/2)]2k sinO,

i ad) (k sin0n ) 2- (m7r/a) 2

and

(5) 1,

e(2,

m = 0,2,4, ....

m = 1,3,5,...

m = 0m #d 0.

Aref and Bref are the specularly reflected mode amplitudes,formerly Ao and Bo in (1). The matrix equations to be solvedfor the waveguide mode amplitudes are

kmZ [Cm' tan(km'h)] ( i [Cmnb"m} k

rm'=o jn-- m k COSOn

-bm cot(km'h)) =-2Aicmo, (6)

1212 J. Opt. Soc. Am., Vol. 68, No. 9, September 1978

'1 I I I

-I

I-

I-

- I

I

IIIIIII -

0.8

§0.7

Ct0.-0 5

=0.5~:0

0.3

k sin0n = I-ra

Ao = Aref - Ai, Bo = Bref -Bi,

I I I I

l l l7

I

I

J. W. Heath and E. V. Jull 1212

FIG. 3. TM reflected power for small aspect ratios: 0, = 550, d = A/(2 sinOj).

and

E Dm, ( E mCnbnmi 5O m' cot(m=0 n'=- km

where

cmn = (-l)mb~m

Cmn = (-1)m+lbnm,

and rm is the Kronecker delta.

When the period d satisfies the Bragg con

kd sin0i = p-r, p = 1,2,3,..

the bracketed I I terms in (6) and (7) are rep'

amm = 2 E; cmnbnm', p = 1,n=-(p-1)

= cm(7p/2)b(-p/2)n' + 2 E cmnbnm', p = 2,4,6,....n=-(p/2-1)

where

M Ecmn = cmn or cmn,

bnm = bm or bE

for TM and TE polarization, respectively, for m and m' bothodd or even. ammo = 0 for all other m and i'. Thus the ma-trix equations decompose into separate equations governingodd and even waveguide mode amplitudes. Each matrix is1/4 the size of the original equation and each coefficient re-quires terms computed for only half the free-space modes,greatly increasing computational efficiency.


The waveguide mode amplitudes are found by choosing nand m large enough (n = 50, m = 10) and then solving (6) and

km(h)) (7) numerically. Under the Bragg condition (8), equationsfor the even and odd waveguide mode amplitudes can be

2BicEo, (7) solved separately. The free-space mode amplitudes are thenfound from (4) and (5). Larger values of n and m are used toensure that truncation errors are insignificant.

For perfect blazing of a surface the period must satisfy (8),with only two propagating free-space modes. If p = 1 andsinOi > 1/3 (Oi > 19.50), the n , 0 (specular reflection, 00 = Oi)and n = -1 (principal backscatter, 0-1 = -0i) are the only

dition, propagating modes. To find a "perfect" blazing depth for agiven angle the period is set by the Bragg condition, the aspect

(8) ratio aid chosen, and the groove depth varied until the spec-laced by ularly reflected power is less than 10-5 the incident power.

Varying h only is numerically efficient since [amn'] requires35.. computation only once.

Ill. NUMERICAL RESULTS

As the aspect ratio aid approaches unity the rectangulargroove profile resembles the comb profile. However, settingaid = 1.0 in (4)-(7) yields Aref = Ai, Bref = -Bi and all othermode amplitudes zero, i.e., reflection from a flat plate. Figure2 shows relative relected power vs h/X for a rectangular grooveprofile with aid = 0.9999, and for the corresponding combprofile, presented earlier.7 The discrepancy for small h/A isdue to the integral transform technique failing for small combdepths. The discrepancy for large h/X is because the TM1waveguide mode propagation constants for the two profilesare slightly different, resulting in a noticeable phase differencefor deep grooves.


20 30 40 50 60Oi (degrees)

70 80 90

FIG. 4. Blazing depth curves for large aspect ratios aid, d = X/2 singj;-- -, ad = 0.99,----, a/d = 0.995;-, a/d O.9999;---, ad=1.0 from [7].

0.3 I I

0.2 -

h,..

1.

1.

FIG. 5. Blazing depth curves for small aspect ratios aid, d = X/2 singj.


1.

h/A

0.

18 , I, l l l1 l, l l l

.6 -

.4 i

1TM TE,/

0[

8) / I

,4 ---

.2-

0.

0.

0

(n

As the aspect ratio goes to zero the rectangular grooveprofile becomes a flat plate with periodic slots, and setting aid= 0.0 yields reflection from a flat plate. The effect of verylittle incident power entering the grooves for small aid, is seenin Fig. 3; the range of groove depths over which there is a re-duction in reflected power decreases to a single value as aidapproaches 0.0, and at this depth there is no reflected power.This blazing depth is X/4 for all Oi, corresponding to an infinitereactance across the slot at the surface. The current excitedon the grating surface is entirely x directed but cannot enterthe slots.

The set of blazing depth curves, or curves of groove depthversus angle of incidence for perfect blazing, published ear-lier 3' 4 are completed by Fig. 4 for the TE and TM curves foraid = 0.99 to 1.0, and by Fig. 5 for the TM curves for a/d =1/100 to 1/3. The TE curves show the characteristics men-tioned earlier3 : an increase without bound as the TE1waveguide mode approaches cutoff, Oi = sin- 1 (a/d), and aminimum near cutoff of the TE2 mode, Oi = sin-1 (a/2d).Because the TEM waveguide mode is not cut off, TM polar-ization curves extend up to grazing incidence. The rapid risein the curves for the large aspect ratios for Di near 19.50 is dueto the existence of first the TM1 and then the TM 2 modes inthe grooves. In Fig. 4 the TM curves for the rectangulargroove profile converge to that for the comb profile given byEbbeson. 7 The curves in Fig. 5 converge to hA = 1/4 as aid

0.0.

The rectangular groove profile is blazed for both polariza-

I I I - .

1/100- -

1 -

64 \ 9

\\\"I

\ \\ \ I

\'\I

II I I

600 900


FIG. 6. Blazing depth curvesfor aid = 0.99, d = X/2 sinOi; -,lowest order; -- -, higher order.Simultaneous blazing occurs at:0; = 40.50 (h/X = 1.088), 0, =58.50 (h/X = 0.556), and 0, =73.1 WX/ = 1.049).

tions simultaneously wherever TE and TM curves for thesame aspect ratio intersect. The range of angles over whichthis can be done using the curves already presented is 19.50< Oi < 59.40.5 Table I is a list of these dual-blazing dimen-sions, from which it is evident that the angular limit of Oi =59.4° is where curves for the comb profile intersect. Morethan one blazing depth curve exists for any aspect ratio,4' 8 andsince all TM curves extend up to grazing incidence, it is pos-sible to use these higher-order curves, as shown in Fig. 6, tofind simultaneously blazed surfaces for angles of incidenceabove Oi = 59.4°. The intersections between the TE curvesof Fig. 4, and the deeper TM curves allow, in principle, thedesign of dual-blazed surfaces for these larger angles, but thedeep grooves required (e.g., at Di = 850 and aid = 0.9999, the

TABLE I. Dimensions of dual-blazed surfaces for shallow rectan-gular grooves.

0i (degrees) d/X h/X a/d hid

19.5 1.500 0.596 0.386 0.39820.0 1.462 0.652 0.391 0.44626.2 1.132 0.716 0.500 0.63130.0 1.000 0.715 0.567 0.71535.9 0.853 0.687 0.667 0.80740.0 0.778 0.650 0.736 0.83645.0 0.707 0.597 0.821 0.84450.0 0.653 0.566 0.896 0.86755.0 0.610 0.556 0.957 0.91059.4 0.581 0.557 1.000 0.959


groove depth is h/X = 3.05) have small tolerances. Thebandwidth of a blazed surface becomes narrow for deepgrooves.8 This is the major difficulty with dual-blazed sur-faces for large angles of incidence.

IV. EXPERIMENTAL RESULTS

Measurements were made of power reflected by corrugatedbrass surfaces over a range of incident angles using an exper-imental arrangement described elsewhere.9 With this ar-rangement it was more convenient to measure reflected ratherthan diffracted power. The surfaces were designed to scatteronly two modes and were blazed to the n = -1 mode for oneangle and at least one polarization. The surface dimensionswere 31X X 13X at 35 GHz, large enough for the effects of finitesize to be essentially negligible.10 At this frequency thegrooves could be milled to an accuracy of 40.003X, more thanadequate for these requirements. Difficulties due to finiteconductivity, surface roughness, and oxide coatings, whichmay occur at optical frequencies,4 are not encountered.

Figure 7 shows the measured and predicted specular re-flection from a plate designed for Oi = 64.25° with aid = 1/4and 56 grooves. This grating was designed to test the nu-merical results for the smaller aspect ratios and this aspectratio is the smallest that could be used conveniently.Agreement is very close for predicted and measured values;minima occur at the same angle, with a maximum measuredreduction in specular reflection of 31.5 dB. The angular rangeover which specular reflection is reduced by the corrugationsdepends primarily on the groove depth; shallow grooves areeffective over a larger angular range than deep grooves.


h

jl

TO 0

0 I Data for perfect dual blazing with shallow grooves is nowcomplete. Blazing is possible for grooves which are deeper

by integer multiples of a half-wavelength when there is onlysingle-mode propagation and by approximately this amountwhen other modes are present. It is possible in principle to

-10 obtain perfect dual blazing for angles of incidence much larger

than 59.4° with deep grooves and approximately comb grat-Z ings, but blazing with deeper grooves requires correspondingly

closer tolerances on the surface dimensions, and for this reason

will be often impractical. A more practical alternative might

t-20- be to accept less than perfect blazing by applying the Bragg

0cX condition (8) with p = 2 or 3. In this way near-perfect dualblazing may be achieved with wide but shallow grooves for the

Lu larger angles of incidence. This is presently being investi-

Lu-30

LuI

05 (a)6.o 64.25

-40

LL-10

-5055 60 65 70 75 Lu

E1 (degrees)

FIG. 7. Measured and calculated TM reflected power for a grating with 56grooves, X = 8.57 mm, d = 4.75 mm, a = 1.20 mm, and h = 1.78 mm. LA

ei~ 55-o lp

Figure 8 shows the measured and predicted behavior of a -30 -grating blazed for arbitrary polarization at Oi = 54.5°. Theexperimental results were presented earlier along with apredicted curve for a comb profile. 5 These predicted curves 4 0 50 60 70

for the rectangular groove profile correspond much more Ok (degrees)

closely to the measured values for both polarizations. Themaximum measured reduction is 99.5% and 99.9% for the TMand TE cases, respectively, although the surface dimensions 0 l l lare not exactly optimum for dual blazing at Oi = 54.5°. These (b)

results demonstrate the accuracy with which the behavior ofsuch a surface may be predicted.

V. CONCLUSIONS o 10

When the Bragg condition (8) is satisfied, the matrix \s

equation determining the waveguide mode amplitudes sepa- c]/rates into equations for even and odd numbered modes. This

happens for both TE and TM polarizations and provides a L -20 //major reduction in the effort of computing the groove depthfor perfect blazing. For groove widths nearly equal to the Wlperiod, numerical values for the scattered field are corre- ei=55\spondingly near those for the comb grating obtained by anentirely different method. This, as well as the close agree- -30-ment with measurements for a fin-corrugated surface, verifies

the accuracy of values for this limiting case. l/2iwitisiefciefr40 50 6 0 70

Blazing with grooves less than A/2 in width is ineffective for Oe (degrees)

TE polarization. In the limit, as the groove becomes a narrow FIG. 8. Measured and calculated reflected power for a grating with: 51

slot supporting the TEM mode only, the surface is perfectly grooves, X = 8.57 mm, d = 5.23 mm, a = 4.97 mm, and h = 4.80 mm. (a)

reflecting for TM polarization also unless the slot is almost TM polarization, (b) TE polarization. Experimental results (-O-) from

exactly X/4 in depth. Ref. 5, Fig. 5.

1216 J. Opt. Soc. Am., Vol. 68, No. 9, September 1978 J. W. Heath and E. V. Jull 1216

gated in connection with multipath interference reductionfrom metal building surfaces.

ACKNOWLEDGMENT

Work supported by the National Research Council ofCanada and the Ministry of Transport, Ottawa.

'A. Wirgin, "On the theory of scattering from rough lamellar surfaces,"Alta. Freq. 38, 327-331 (1969).

2A. Wirgin and R. Deleuil, "Theoretical and experimental investi-gation of a new type of blazed grating," J. Opt. Soc. Am. 59,1348-1357 (1969).

3A. Hessel, J. Shmoys, and D. Y. Tseng, "Bragg-angle blazing of dif-fraction gratings," J. Opt. Soc. Am. 65, 380-384 (1975).

4 J. L. Roumiguieres, D. Maystre, and R. Petit, "On the efficienciesof rectangular-groove gratings," J. Opt. Soc. Am. 66, 772-775(1976).

5E. V. Jull, J. W. Heath, and G. R. Ebbeson, "Gratings that diffractall incident energy," J. Opt. Soc. Am. 67, 557-560 (1977).

6 J. A. DeSanto, "Scattering from a periodic corrugated structure:Thin comb with soft boundaries," J. Math. Phys. 12, 1913-1923(1971); "Scattering from a periodic corrugated structure: Thincomb with hard boundaries," J. Math. Phys. 13, 336-341 (1972).

7G. R. Ebbeson, "TM polarized electromagnetic scattering from fin-corrugated periodic surfaces," J. Opt. Soc. Am. 66, 1363-1367(1976).

8 J. W. Heath, "Scattering by a conducting periodic surface with arectangular groove profile," M.A.Sc. Thesis, University of BritishColumbia (1977).

9E. V. Jull and G. R. Ebbeson, "The reduction of interference fromlarge reflecting surfaces," IEEE Trans. Antennas Prop. AP-25,565-570 (1977).

10 P. Facq, "Application des matrices de Toeplitz a la thborie de ladiffraction par les structures cylindriques periodiques limit6es,"D.Sc. Thesis, University of Limoges (1977).

Kinematic and dynamic light scattering from the periodic structureof a chiral smectic C liquid crystalStephen Garoff,* Robert B. Meyer, and Richard Barakatt

Division of Applied Sciences, Harvard University, Cambridge, Massachusetts 02138(Received 3 April 1978)

As in chiral nematic liquid crystals, one of the most striking optical properties of chiral smectic Cliquid crystalline phases is the selective reflection of certain wavelengths of light. This Bragg-likescattering is a direct consequence of the periodic nature of the helical structure of these phases. Inchiral nematics, the strong coupling of the light to the periodic structure leads to scattering in thedynamic (multiple-scattering) regime. In chiral smectic C phases, the strength of the coupling de-pends on the magnitude of the tilt angle between the long molecular axis and the layer normal. Forsmall tilt angles, the kinematic (single-scattering) regime of light scattering occurs. Further, due tothe nature of the modes of optical propagation in the chiral smectic C phase, some Bragg events occuras depolarized light scattering in the forward direction. A computer program has been written tocalculate the light scattering from a chiral smectic C sample. The results of these calculations demon-strate the behavior of the light scattering in both the kinematic and dynamic regimes.

1. INTRODUCTION

Some of the most dramatic optical properties of liquidcrystals are associated with the helical structure of chiral ne-matic and chiral smectic C phases. Chiral nematic and chiralsmectic C liquid crystalline phases exhibit selective reflectionof certain wavelengths of light. This Bragg-like scatteringis a direct consequence of the periodic nature of the helicalstructures of these phases. Papers have been published usingdynamic scattering theory (which takes into account strongcoupling and multiple scattering of the light by the structure)to explain the reflection spectra from chiral nematic sam-ples.1-5 The kinematic scattering theory (which holds forweak coupling and single scattering of the light by the periodicstructure) cannot be applied to most chiral nematics since thecoupling to the light is too strong. However, in the chiralsmectic case each scattering regime can be separately stud-ied.

In chiral nematic liquid crystals orientational ordering isimposed on the somewhat elongated, rigid molecules com-posing the phase. Basically, the long axes of the molecules

in any small volume lie parallel and the local symmetry isuniaxial. However, the molecules have a helical arrangementin which the long axis (the director d) precesses slowly abouta direction perpendicular to the direction of the long axes.

In a chiral smectic C phase, the long axes of the moleculesare again parallel; however, in this case the molecules are ar-ranged in layers. The long molecular axes tilt at an angle 0relative to the layer normal and this tilt direction slowly pre-cesses about a direction parallel to the layer normal (see Fig.1). The layer spacing is on the order of 0.002 ,um and the pitch(the repeat distance of the helix) is on the order of 3 pum. Thetilt angle and the pitch are temperature dependent; the tiltangle decreases as temperature is increased and goes contin-uously to zero at the second-order transition to the smecticA phase.

The purpose of this paper is to examine both the kinematicand dynamical theories of light scattering as applied to a chiralsmectic C. Further, the results of a numerical algorithm areused to demonstrate the predictions of these theories. Theoutline of the paper is as follows. In Sec. II we review some

1217 J. Opt. Soc. Am., Vol. 68, No. 9, September 1978 0030-3941/78/6809-1217$00.50 ( 1978 Optical Society of America 1217

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Perfectly blazed reflection gratings with rectangular grooves