Rigorous diffraction theory of optical reflection and transmission gratings

R I G O R O U S D I F F R A C T I O N T H E O R Y OF O P T I C A L R E F L E C T I O N A N D T R A N S M I S S I O N G R A T I N G S

1

RIGOROUS DIFFRACTION THEORY

OF OPTICAL REFLECTION

AND TRANSMISSION GRATINGS

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN

AAN DE TECHNISCHE HOGESCHOOL DELFT, OP GEZAG VAN DE RECTOR MAGNIFICUS IR. H.R. VAN NAUTA LEMKE, HOOGLERAAR

IN DE AFDELING DER ELEKTROTECHNIEK VOOR EEN COMMISSIE UIT DE SENAAT

TE VERDEDIGEN OP WOENSDAG 17 NOVEMBER 1971

TE 14 UUR

DOOR

PETRUS MARIA VAN DEN BERG

ELEKTROTECHNISCH INGENIEUR GEBOREN TE ROTTERDAM

BIBLIOTHELK

DER

TECHNISCHE HOGESCHOOL

DELFT

1971 BRONDER-OFFSET N.V. ROTTERDAM

IQS3 /ZO/

V

DIT PROEFSCHRIFT IS GOEDGEKEURD DOOR DE PROMOTOR

Aan mtjn Ouders Aan mijn Vrouw

COiJTENTS

SUMMARY 8

LIST OF MAJOR SYMBOLS 12

1 DIFFRACTION THEORY OF A REFLECTION GRATING

1. Introduction

2. Description of the configuration and statement of the problem

r r 3. Integral representations for 4> and ¥

'^ n n

A. Derivation of the integral equations

5. Numerical solution of the integral equations 6. Numerical and some experimental results Appendix A. Determination of the Green function Appendix B. Asymptotic approximation of K(.s \s) Appendix C. Derivation of the power relation References

DIFFRACTION THEORY OF A REFRACTION GRATING 1. Introduction

2. Description of the configuration and statement of the problem

r r t t 3. Integral representations for * , 'i , $ and ï

'^ n n n n 4. Derivation of the integral equations

5. Numerical solution of the coupled integral equations 6. i'-polarization and fl-polarization

7. Numerical results

Appendix A. Acceleration of the convergence of the series G and G

Appendix B. Derivation of the power relation References (261) (261) (263) (270) (273) (276) (280) (289) (290) (291) (292) 17 17 19 26 29 32 36 45 46 47 48 51 52 54 68 74 76 81 87

too

101 107

DIFFRACTION THEORY OF A COATED REFLECTION GRATING 109

1. Introduction 110 2. Description of the configuration and statement of

the problem 111

3. Integral representations for $ and m 119

" ^ n n

4. Derivation of the integral equations and their

numerical solution 122 5. Numerical results 125 Appendix A. Derivation of the power relation 131

References 133 DIFFRACTION THEORY OF A TRANSMISSION GRATING 135

1. Introduction 136 2. Description of the configuration and statement of

the problem 137 + + t t

3. Integral representations for $ , t , $ and t 146

"^ n n n n

4. Derivation of the integral equations and their

numerical solution 151 5. Numerical results 157 Appendix A. Derivation of the power relation 162

References 165

SAMENVATTING 167

LEVENSBERICHT 171

1

This part has been reprinted without change in pagination from Applied Scientific Research 2^ (1971), in which it has been published. The relevant page-numbers are placed in parentheses.

SUMMARY

In the present thesis we develop a diffraction theory of optical reflection and transmission gratings. All configurations under consid-eration are cylindrical in one direction and consist of an perfectly reflecting periodic boundary and/or a dielectric medium of which one boundary is periodic. When an electromagnetic wave is incident on a periodic structure of this kind the diffracted field consists of a dis-crete set of waves (the so-called spectral orders) propagating only in certain directions. The angles of diffraction of these spectral orders follow from the Rayleigh grating formula (to be modified in case dif-ferent media are present) and are determined by the angle of incidence, the wavelength of the diffracted waves in the relevant medium and the spatial period of the boundary. The amplitudes and phases of the dif-fracted waves depend on the particular shape and the period of the boundary, the boundary conditions to be imposed at the surface of the grating, as well as by the angle of incidence and the wavelength of the incident wave and the properties of the dielectric media if present. The recent availability of optical diffraction gratings in large quan-tities and of high quality has resulted into a new interest in the so-lution of the two theoretical problems connected with the diffraction of light by gratings, viz. the problem of spectral image formation (Cf. W. Werner, Imaging properties of diffraction gratings, Thesis, Delft University of Technology, 1970) and the calculation of the power dis-tribution amongst the different spectral orders. In this thesis atten-tion is paid to the latter problem. In all cases we consider the inci-dent wave is a monochromatic plane electromagnetic wave. To improve a grating's capability of blazing (i.e. to obtain specific preferred directions in which the diffracted power is concentrated) a complete description of the influence of the shape of the boundary on the power distribution will have to be developed. In order to solve the resulting boundary value problem we present a theory based on a rigorous solution of Maxwell's electromagnetic field equations.

In two cases the three-dimensional vectorial electromagnetic prob-lem can be reduced to two separate two-dimensional scalar ones, viz. one

SUMMARY

₉

corresponding to E- and the other to ff-polarization. This situation occurs if the incident wave has no spatial dependence in the direction of cylindricity of the grating and in the case where the boundary of the grating is a perfectly reflecting surface. If neither of these two conditions holds a coupling between the two mentioned sub-problems arises. In each of the two sub-problems the resulting scalar wave func-tion is a solufunc-tion of the two-dimensional Helmholtz equafunc-tions in the appropriate medium. All components of both the electric and magnetic field vector can be expressed in terms of the two scalar wave functions.

To obtain an expression for the diffracted scalar wave function, in each case a Green's function formulation of the problem is employed. The relevant Green's functions are chosen in such a way that integral representations are obtained, in which only the contributions from the remaining unknown functions on a single period of the grating occur. This is achieved by requiring that the Green's function consists of waves, which travel away from a plane "parallel" to the grating, while

in the direction parallel to the grating it possesses a phase variation exactly opposite to the one of the incident wave. Subsequently, the remaining unknown wave functions on a single period of the grating are determined with the aid of integral equations.

The numerical solution of these integral equations is achieved by applying the method of moments. The unknown functions are expanded in terms of a sequence of functions defined on the interval of integration. Through this method the integral equations are replaced by a system of

linear equations. Two versions of the method are employed:- the (dis-crete) Fourier transform and the cubic spline approximation. The method with Fourier coefficients is particularly useful in case the remaining integrals can be determined analytically. This case occurs when the grating is of the echellette type (the shape of the periodic boundary is then a triangle). The method of cubic spline approximation is partic-ularly useful, when the unknown functions in the integral equations can be expected to vary smoothly. Hence, the method of cubic spline approx-imation is preferable e.g. when the grating is sinusoidal. The remaining integrals are in this case determined numerically.

10 SUMMARY

integral equations are to be considered the truncation of the slowly converging series representation of the Green's function and the dis-cretisation of the singular integral equations. In order to decrease the truncation error in the series representation of the Green's func-tion and its derivatives, techniques for accelerating the convergence are employed. Further, special precautions are taken around the singu-lar points of the kernel of the integral equations in order to decrease the discretisation error of the integral equations. As an indication of the accuracy of the results obtained the power relation is used.

On practical grounds, only gratings with echellette and sinusoidal boundary are considered in some detail. This is, because echellette gratings can easily be manufactured by a flat bed ruling engine while the sinusoidal grating can be produced with the aid of a new technique employing the interference of two collimated beams of monochromatic coherent light. The sinusoidal variation of intensity thus produced is converted into corrugations on an optically flat surface by recording the fringes in a layer of photoresist. Numerical results for the two types of grating are presented and, wherever possible, compared with either results obtained in a different way by other authors or with some experimental results. Numerical and experimental results show to be in excellent agreement.

The thesis consists of four parts. In each part another configu-1

ration is considered. In the first part a diffraction theory of a re-fVeotion grating is given. This grating consists of an electrically perfectly conducting medium (i.e. a reflecting medium) with periodic boundary. In the second part a diffraction theory of a re fraction

grat-ing is given. This grating occupies a semi-infinite domain with period-ic boundary and consists of a non-conducting, lossless medium (i.e. a transparant medium). In the third part a diffraction theory of a coated reflection grating is given. This grating is a reflection grating cov-ered with a lossless dielectric coating, such that the top surface of the coating is a plane surface "parallel" to the reflecting surface of the grating. Employing the reflection and transmission properties of the plane top surface of the coating, the problem is reduced to a "mod-ified reflection grating problem". In the fourth part a diffraction

SUMMARY 11

tneory of a transmission grating is given. This grating consists of a lossless dielectric slab, of which one surface (the top surface) is plane and the other (the bottom surface) is periodic. Employing the reflection and transmission properties of the plane top surface of the grating, the problem is reduced to a "modified refraction grating prob-lem" .

All computations have been performed on the IBM 360/65 computer of the Computing Centre of the Delft University of Technology; the pro-grammes have been written in PL/1.

1

This part has been published in Applied Scientific Research 24^ (1971) pp. 261 - 293.

LIST OF MAJOR SYMBOLS

Latin symbols

Symbol Name

A coefficient in the boundary condition at a refracting

interface

b blaze angle of the echellette grating

C contour of integration

D period of the grating

E electric field vector

G Green's function

G„ modified Green's function in the case of ff-polarization

G. modified Green's function in the case of fl-polarization

h height of the sinusoidal grating

H_ magnetic field vector

k wave number

k wave vector

K truncation parameter in the system of linear equations K(.8 le) kernel function in integral equation

LIST OF MAJOR SYMBOLS 13

L line of integration

L arc length of period of the grating

n_ unit vector in the direction of the normal to a surface

N truncation parameter in series representation of Green's function

p parameter of the profile of the echellette grating

P point of observation

PT,{S) periodic cardinal spline function

Q point of observation

R reflection factor of the grating (of n-th spectral order)

R refraction vector of the grating (of n-th spectral order)

E

R reflection factor of a plane refracting surface in the n case of E-polarization (of n-th spectral order)

u

R reflection factor of a plane refracting surface in the _{n f e,}

case of fl-polarization (of w-th spectral order)

s arc length along period of the grating

S domain to which Green's theorem is applied

S(s) spline function

LIST OF MAJOR SYMBOLS

transmission factor of the grating (of n-th spectral order)

transmission factor of a plane refracting surface in the case of ^-polarization (of n-th spectral order)

transmission factor of a plane refracting surface in the case of ^-polarization (of n-th spectral order)

wave function (either E or H ) y y

Cartesian coordinate

grating efficiency in the case of £'-polarization

grating efficiency in the case of 5-polarization

unknown function in integral equation

unknown vector in vector integral equation

Cartesian coordinate

known function in integral equation

known vector in integral equation

Cartesian coordinate

X-component of the wave vector

LIST OF MAJOR SYMBOLS 15

Y,r 3-component of the wave vector

e scalar permittivity

6 angle indicating direction of propagation

X wavelength in free space

A surface of the grating

U scalar permeability

i angle indicating direction of propagation

wave function (= E )

y

wave function (= H )

y

unit vector in the direction of the tangent to a curve

A p p l . Sci. R e s . 24 J u l y 1971

DIFFRACTION THEORY

OF A REFLECTION GRATING

P. M. VAN DEN B E R G

Dept. of Electrical Engineering, Delft University of Technology, Delft, THE NETHERLANDS

Abstract

T h e reflection of a m o n o c h r o m a t i c p l a n e e l e c t r o m a g n e t i c w a v e b y a n electrically perfectly c o n d u c t i n g g r a t i n g is i n v e s t i g a t e d . T h e v e c t o r i a l e l e c t r o m a g -n e t i c p r o b l e m is r e d u c e d t o t w o s e p a r a t e scalar p r o b l e m s : t h o s e c o r r e s p o -n d i -n g t o E- a n d H-polarization respectively. A G r e e n ' s function f o r m u l a t i o n of t h e p r o b l e m is e m p l o y e d . F o r b o t h cases a n i n t e g r a l e q u a t i o n of t h e second k i n d for t h e r e m a i n i n g u n k n o w n function o n t h e surface of t h e g r a t i n g is derived. A n u m e r i c a l solution of t h i s i n t e g r a l e q u a t i o n is o b t a i n e d w i t h t h e aid of e i t h e r a (discrete) F o u r i e r t r a n s f o r m or a cubic spline a p p r o x i m a t i o n . S o m e n u m e r i c a l results of b o t h t h e echellette g r a t i n g a n d t h e sinusoidal g r a t i n g a r e p r e s e n t e d .

§ 1. Introduction

Reflection gratings are frequently employed in infrared

spectro-scospy. The relevant theoretical problem can be formulated as

follows. An incident monochromatic plane electromagnetic wave

gives rise to a discrete number of reflected waves, which propagate

in directions, following from the angle of incidence, the period of the

grating and the wavelength of the incident wave. The main

empha-sis of the present paper is lying upon the problem of the

determi-nation of the reflection factors of the different spectral orders. The

reflection grating is assumed to be electrically perfectly conducting

and cylindrical in one direction; hence the vectorial electromagnetic

problem can be reduced to two separate scalar problems,

corre-sponding to E- and H-polarization respectively. The resulting scalar

functions are solutions of the first and the second boundary value

problem respectively, for the two-dimensional Helmholtz equation.

-2 6 -2 p . M. VAN DEN BERG

Lord Rayleigh [1] proposed a method of solution, where he

as-sumed the discrete set of reflected, propagating and evanescent,

waves (together with the incident wave) to be a sufficient

de-scription of the total field to satisfy the boundary condition on the

surface of the grating. In this way, he reduced the problem to the

solution of an infinite system of linear, algebraic equations in the

case of E-polarization for a perfectly conducting sinusoidal grating.

Rayleigh's solution has been extended to the case of H-polarization

for the sinusoidal grating by Stroke [2]. The Rayleigh method of

solution has further been extended to gratings of a general periodic

profile by Petit [3-5] and others. With the availability of electronic

computers the system of linear equations could be solved

numeri-cally and comparison with experimental results showed that the

Rayleigh method is in general incorrect. The reason for this is that

the discrete set of reflected waves (together with the incident wave)

is not a sufficient description of the total field to satisfy the

bounda-ry condition at the grating (Lippmann [6], Wirgin [7], Petit [8]).

Other methods have been developed since.

Employing a Helmholtz representation of the reflected field,

Petit [9-11] obtained a rigorous expression for the reflected field

above the grating. Using the boundary conditions, he obtained, for

both cases of polarization, an integral equation of the first kind for

the remaining unknown function on the surface of the grating.

Em-ploying a (discrete) Fourier transform, he replaced the integral

equation by a system of linear equations, which were solved

nu-merically. In this way however, the case of H-polarization was not

directly solvable, because the resulting singular kernel of the

inte-gral equation is not integrable without special measures.

In the present paper we use a Green's function formulation of the

problem. Then, we obtain the same type of expression for the

re-flected field as presented by Petit. Using this expression we derive

an integral equation of the second kind, in which the singular point

of the kernel has been avoided. Both cases of polarization can now

be attacked, using the same method. A numerical solution is

ob-tained with the aid of either the (discrete) Fourier transform or with

the aid of the cubic spline approximation. In both methods the

integral equation is replaced by a system of linear equations.

In particular, two types of reflection gratings receive attention:

the echellette grating and the sinusoidal grating. Numerical results

DIFFRACTION THEORY OF A REFLECTION GRATING 2 6 3

for the echellette grating are compared with those obtained by

Petit [9-11] as well as some other experimental results. Numerical

results for the sinusoidal grating are compared with those obtained

by Pavageau [12-14]. By applying Poisson's summation formula,

Pavageau found that an integral equation of the second kind for

the current density along the whole grating can be reduced to one

for the current density along a single period of it. The latter integral

equation has been solved numerically by a method of successive

approximations. As far as comparable, the results are in very good

agreement.

§ 2. Description of the configuration and statement of the problem

The grating under consideration consists of an electrically perfectly

conducting cylindrical surface which is periodic in one direction.

The position of a point in space is given by its Cartesian coordinates

X, y, z. The y-axis is chosen parallel to the generators of the

cy-lindrical configuration of the grating, the ;t:-axis is chosen in the

direction of the periodicity (Fig. 1). The medium above the grating

is assumed to be electromagnetically linear, homogeneous, isotropic

and lossless, with (scalar) permittivity e and (scalar) permeability [i.

The complex representation of field quantities is used; the

com-plex time factor exp(—ico^) (i = imaginary unit, m = angular

fre-quency, t = time) is omitted throughout. All quantities are expressed

in Sl-units. A monochromatic, uniform plane wave is incident upon

the grating at angles <^o and 6o (see Fig. 2). </>o is the angle between

the y-axis and the direction of propagation of the incident wave and

00 is the angle between the negative 2-axis and the projection on

the {x, 2)-plane of the direction of propagation of the incident wave.

264

p . M. VAN DEN BERG

(2.1) Fig. 2. Orientation of the incident field.

The incident field E^ = E^{x, y, 2), H ' = H^{x, y, z} is then given by

£• = Eo exp(iao^ + i/3oy — iyo^),

W = Ho exp(iao^ + i^oy — iyoz),

where

ao^Asin(<^o)sin(0o),

yo^ksm{ct>o) cos(0o) = {k^ -fil- al)i with (k^- -pl-al)i;^ 0,

k ^(o{en)i = 27t/A with (sfi) ^ > 0,

(2.2)

with —7t/2 < 60 ^ •'^/2 and 0 < ^0 ^ ^t. In (2.2), A denotes the

wave-length in free space. Let the total field above the surface of the

grating be denoted by E = E{x, y, z), H = H{x, y, z), then the

re-flected field above the grating is introduced as

£ r def £ _ £;i

(2.3)

HT^H

- w.

The incident, the reflected and the total field satisfy the

source-free Maxwell equations

rot H + icaeE = 0,

DIFFRACTION THEORY OF A REFLECTION GRATING

265

Further the total field satisfies the boundary conditions on the

surface of the grating. At this perfectly conducting surface the

tangential component of the total electric field vector E should

vanish, hence

n X E = 0 on A, (2.5)

in which n = unit vector in the direction of the normal to the

surface A of the grating, pointing into the perfectly conducting

material of which the grating consists (see Fig. 3).

As the geometrical configuration is cylindrical in the y-direction,

we expect the y-dependence of all quantities to be the same as that

of the incident wave i.e.

E{x, y, z) = ê{x, z) exp(i/3oy),

H[x, y, z) = 3^{x, z)

exp(ij8oy)-This structure of the field is not in contradiction with either the

boundary conditions or the Maxwell equations. From Maxwell's

equations (2.4) it then follows that the x- and the 2-components of

ê and 3^ can be expressed in terms of êy and M'y through the

relations

(/j2 - ;8^) S^ = i^o{8éyldx) - UOIl{dj^yldz),

(/e2 - pl) ê^ = \Po[dêyjdz) + icoij.{8jeyldx),

(^2 - ,3^) ,r^ = i^oiëjeyjëx) + uos(8^yidz), • • '

(^2 _ Pi) .^^ = ipo{8jfyl8z) ~ ï(üE[8êyj8x).

Further, Sy and M'y satisfy the two-dimensional Helmholtz

equa-tions

8Zé'yl8x^ + 8^Syi8z^ + (/^2 _ ^2) Sy = 0,

8\^yl8x^ + 8^,?tyl8z^ + (^2 _ ^2^ .^,^ = 0.

From the boundary condition (2.5) it then follows that S'y and Jfy

satisfy the boundary conditions

éy = 0 on A,

(2.9)

n-V^u = 0 on A,

in which F = (BjSx) h + (Ö/Ö2) i^.

Since neither the wave equations (2.8) nor the boundary

con-ditions (2.9) lead to a coupling between êy and J^y, the vectorial

266

p . M. VAN DEN BERG

problem can be separated into the following two-dimensional scalar

problems:

(i) êy ^ 0 and J^y = 0 {E-polarization). In this case we

intro-duce as fundamental unknown quantity ^{x,z)^êy(x,z). Then

the incident wave function 0 ' = ^^[x, z) is given by

(pi = 00 exp(iaoA; — iyo-^)- (2.10)

The reflected wave function ^^ = ^^[x, z) satisfies the equation

8-^0^18X^ + 82$r/g^2 _|_ (^k2 _ pl) 0T = 0. (2.1 1)

The total wave function 0 = 0^ -\- 0'' satisfies the boundary

con-dition

<? = 0 on A (first boundary value problem). (2.12)

(ii) S'y = 0 and .^^y ^ 0 (H-polarization). In this case we

intro-duce as fundamental unknown quantity W(x, z) — J'fy(x, z). Then

the incident wave function ¥ " = W^(x, z) is given by

f i = Wo exp(iao^ - 1^02). (2.13)

The reflected wave function f"" = W''(x, z) satisfies the equation

8W^I8x^ + 8^Wrl8z^ + (/fe2 _ pl) y/r = 0. (2.14)

The total wave function ' ? = ¥ " - ) - ' ? " • satisfies the boundary

con-dition

rt-yw = 0 on A (second boundary value problem). (2.15)

Let the grating surface A be represented by f(x, z) = 0, then,

owing to the periodicity in the :*r-direction, f(x, z) has to satisfy the

condition

DIFFRACTION THEORY OF A REFLECTION GRATING 2 6 7

in which D denotes the spatial period (see Fig. 3). The periodicity

of the grating surface entails a quasi-periodicity in the reflected

field. Since exp(—icxo^) S^x,z) and exp(—iao%) J^^(x, z) are periodic

in x and the boundary conditions are periodic in x, it is expected

that exp(—iao^ï) S'^(x, z) and exp(—iao^ï) Ji^'^(x, z) are periodic in x.

Expansion of the relevant periodic functions in a Fourier series with

period D in x yields

S^(x,z)= S ^^(2) exp(ia„;*;),

r t = —oo

jltf'^(x, 2) = 2 -^K^) exp(ia„%)

(2.17)

, _ , „ V- 0 , ± 1 , ± 2 , . . . ) (2.19)

where

«» = cto + 27TO/Z). (2.18)

As in the domain 2max < 2 < oo (z^ax denotes the maximum value

of 2 on the grating) the reflected field is twice continuously

differ-entiable, we obtain, using the orthogonality of the functions

exp(ia„;i;) for all n on the interval xi< x < xi-^r D, from Maxwell's

equations

^2^^022 -f YISI = 0,

a2jr;;/322 + yls^i = o,

when 2max < 2 < oo,

where

Yl^.k^~Pl~<xl. (2.20)

In the domain 2max < 2 < oo, we require solutions of (2.19) in the

form of waves travelling away from the grating, hence

Sl(z) = Eiexp(iynz), n _ L , ^ 9 ^ ^9 9 n

(n = 0, ± 1 , ±2, ... 2.21)

J>iri(z) = lP„exp(iynz),

when 2max < 2 < oo,

where •,

yn^(k'^-p't-ocl)i with Re(yn)>0 a n d l m ( y „ ) > 0 . (2.22)

From (2.17) it follows that the reflected field can be written as an

infinite sum of plane waves either propagating or decaying

ex-268

p. M. VAN DEN BERG

ponentially in the positive 2-direction,

oo

S':(x,z)= Y, E;;exp(ia„A; + i7„2),

w h e n 2max < 2 < oo. (2.23)

3^'^(x,z)= 2 Hj;exp(ia„A; + iyre2)

ï i = —oo

We are dealing with propagating waves when y» is real, which is the

case if aj| < k"^ — p'^. In this case we can define an angle of reflection

dri with relation to the quantities a„, y„ and (k'^ — pl)^ = k sin(<^o)

such t h a t

oin = ksm(4>o) sin(öra),

(2.24)

y„ = y%sin(<^o) cos(0n),

where —7i/2 < 9» ^ 7t/2. We remark that 0^ is the angle between

the 2-axis and the projection on the (x, 2)-plane of the direction of

propagation of the reflected wave of spectral order n. In view of the

relation «K = «o + 2uw/ö we obtain the "grating formula"

sin(ö„) = sin(0o) + nX'ID

with X' = A/sin(</>o),

(M = 0, ± 1 , ± 2 , ...), (2.25)

5!;si=Ars; = x/o

Fig. 4. The geometrical construction of the angles of reflection of the re-flected waves of the different spectral orders from the grating formula

DIFFRACTION THEORY OF A REFLECTION GR.ATING 2 6 9

from which the angles of reflection 6^ of the propagating reflected

waves can be constructed (Fig. 4). If (;io = 7t/2 (JSQ = 0), we have

A' = A and we obtain the well-known grating formula already given

by Lord Rayleigh [1].

From (2.23) the fundamental unknown wave functions 0^ and

W^ are obtained as

CO

0^x, 2) = S K exp(i(x„x -)- iy„2),

"'~^°° when 2 m a x < 2 < o o . (2.26)

*P^(x, 2) = S Wl exp(ia.„A; + iy„2),

H= —OO

We note that 0nl^o and f ^ / f o, in both scalar cases, can be defined

as the reflection factor of the reflected wave of spectral order n.

Using the orthogonality of the functions exp(iam^) for all n on the

interval xi < x < xi -\- D in the domain 2inax < 2 < oo, we

ob-tain from (2.7), (2.23) and (2.26) the relations

(k-^ - pl) £^,„ = -Po<xn0l + w/xy^n.

£"• = 0'' V<n n' ^ r i ^ r V,n n>

(k^ - Pl) HI,, = - ^ o y ™ n + oye<xn0l,

from which it follows that all components of E\ and H], can be

ex-pressed in terms of 0\, and "P^. Hence, the problem has been solved

as soon as 0\ and f'^ have been calculated.

From (2.27) it follows t h a t the complex Poynting vector EJj X

X IF, (IP^ denotes the complex conjugate of HJ'J, which refers to

the power reflected in the spectral order n, can be written as

ElxW^ = w(E0l0i + liWlW'C)knl(k-^-pl) if fe„isreal, (2.28)

in which fe„ = (a„, j3o, y») is the wave vector of the reflected wave

of spectral order n. In the same manner the complex Poynting

vector Eo X HJ, which refers to the incident power, can be written as

Eo X HI = m(e0o0l + i^WoWl) fe/(^2 _ ^2)^ (2.29)

in which k = («Q, PQ, —yo) is the wave vector of the incident wave.

270 p. M. VAN DEN BERG

We observe that in the expressions for the complex Poynting

vector, and hence in the time-averaged power flow of the propagating

reflected waves as well as the incident wave, there is no coupling

be-tween the 0 and the W. The power flow in the case of E-polarization

and the power flow in the case of H-polarization are therefore

ad-ditive. From (2.28) and (2.29) we obtain

a>e0J'0;'*fe„/(A2 — Pl) for E-polarization,

" ~ wnWlW^lkni(k'^ - Pl) for H-polarization, (2.30)

if kn is real,

and

(üe0Q0*^kl(k'^ — Pl) for E-polarization,

EQ X HJ = ,^ (2.31)

(jo/iiWoWlkl(k'^ — P^) for H-polarization.

Then, the total power flow is the sum of the power flow for the

E-polarized waves and the power flow for the H-E-polarized waves. In

the next section we shall discuss how 0\, and f'' can be calculated.

Further we shall derive integral representations for these

quanti-ties.

§ 3. Integral representations for ^ ^ and W^

In order to derive an integral representation for 0^(x, z) and W^(x, z),

we apply the two-dimensional form of Green's theorem to a domain

S inside a simply closed contour C. For an interior point P, with

coordinates x^, 2p, of the domain S we obtain the integral

repre-sentations

0^(xv, 2p) = I {(n • F^r) Q _ $ r ( „ . yc)] ds,

when P inside C, (3.1)

^^(xv, 2p) = ! { ( « • VW^) G - W^(n • VG)} ds,

c

where G = G(xp, zp\x, z) denotes a suitably chosen two-dimensional

Green's function (n = unit vector in the direction of the outward

normal to C). For C we choose the closed contour consisting of the

straight lines L i and L2 parallel to the 2-axis, a period D apart,

to-gether with the curve L corresponding with a single period of the

grating profile, and the straight line L3 parallel to the %-axis at

2 = 23 > 2niax (see Fig. 5). The Green's function G = G(xp, zp\x, 2)

DIFFRACTION THEORY OF A REFLECTION GRATING 2 7 1

Fig. 5. Domain to which Green's theorem is applied.

has to satisfy the inhomogeneous Helmholtz equation

d^Gldx^ + 8'^Gj8z^ + (k'^ - pl) G = -d(xp - x,z-p- z), (3.2)

where d(x, z) is the two-dimensional delta function. The Green's

function is further chosen in such a way that in the integral

repre-sentations (3.1) only the contribution from L remains. This is

achieved by requiring that G consists of waves which travel away

from the plane 2 = zp. Then as zp < 23, it can be shown that the

contribution from L3 vanishes. Further, by requiring that G

pos-sesses at a fixed value of 2 a phase variation exactly opposite to

the one of 0^ and *P^, the contributions from Li and L2 cancel each

other. These requirements lead to the expression (see Appendix A)

00

G(xp,zp\x,z) = S (il2ynD)exp{i(Xn(xp — x)-iriyn\zp — z\}- (3-3)

? ( . = — 0 0

With this choice of the Green's function in (3.1) only a contribution

from L remains, i.e. (3.1) is replaced by

0r(xp, Zp) = \ {(n- F^"-) G - 0^n • VG)} ds,

when P above L. (3.4)

W^(xp, Zp) = J {(n-FÏ"") G - W^(n-VG)} ds,

L

On account of the boundary conditions 0 = 0 and n • yW = 0 on

L we prefer integral representations with 0, n • V0, W and n • VW,

in stead of 0^, n • V0^, ?f"" and n • VW^ in the integrands. These are

obtained by applying Green's theorem to a domain S' inside the

272

p . M. VAN DEN BERG

Fig. 6. Domain to which Green's theorem is applied.

simply closed contour C', which is defined as follows: C' is the closed

contour consisting of the straight lines Li and L2 parallel to the

2-axis, a period D apart, together with the curve L corresponding

with a single period of the grating profile, and the straight line L3

parallel to the ;t:-axis at 2 = 23 < 2^,;,, (2„ji„ denotes the minimum

value of 2 on the grating) (see Fig. 6). Keeping P above L as before

(Fig. 5), we obtain, as G is regular inside C',

0 = | { ( n • V0^) G - 0i(n • VG)} ds,

when P outside C', (3.5)

0 = § {(n-VWi)G-Wi(n-VG)}ds,

C'

(n = unit vector in the direction of the inward normal to C ) .

Choosing G(xp, zp\x, z) as given by (3.3), it can be shown t h a t

the contributions from Li and L2 cancel each other, while the

contribution from L3 also vanishes, since 0 ' , ¥ " and G all consist

of waves travelling away from the grating as 2 = 23 < z^•^J^. Hence

it follows that,

0 = J {(n • F01) G - 0i(n • VG)} ds,

when P above L. (3.6)

0 = I {(n-VWi)G ~Wi(n-VG)}ds,

L

Addition of (3.4) and (3.6), and using 0 = 0i + 0^ and W =

= ¥ " + ï^r leads to the desired expressions for the reflected fields

DIFFRACTION THEORY OF A REFLECTION GRATING 2 7 3

in a point P above the grating surface

0^xp, Zp) = J {(n • V0) G -0(n- VG)} ds,

when P above L. (3.7)

W^(xp, Zp) = j {(n • VW) G - W(n • VG)} ds,

L

Using the boundary conditions 0 = 0 and n • VW = 0 on L, we

obtain

0^(xp, Zp) = \ (nV0)G ds,

when P above L. (3.8)

W^(xp,zp) = - J 'F(n-VG)ds,

L

From the expansions (2.26) of 0^(x, z) and W^(x, z) in terms of 0','^

and f'• respectively, and from the integral representation (3.8) for

0'^(x, z) and W^(x, z) with the chosen Green's function (3.3), we

ob-tain as the integral representations for 0^^ and W^

01 = (i/2y„D) J (n-V0) exp(-ia„A; - iy„2) ds,

L

n = -(i/2y„D) j W(n-V) exp(-ia„.T - iy„2) ds,

(« = 0, ± 1 , ± 2 , ...). (3.9)

From these representations 0Jj and ï'Jj can be calculated as soon as

n • V0 and !f on L are known. Now, one way of calculating n • F 0

and V' on L is to derive integral equations for these unknown

functions. This procedure is discussed in the next section.

§ 4. Derivation of the integral equations

In this section we discuss the determination of n • F 0 and "/^ on L

with the aid of integral equations. These are obtained by applying

again Green's theorem to the domain in Fig. 5. However, we now

take the point P of observation on L. In stead of the representations

(3.4) for the reflected wave functions 0^ and ¥"", we now obtain

\0^(xp, zp) = f {(n • V0') G ~ 0^n • VG)} ds,

when P on L, (4.1)

^W^xp, Zp) =i{(n- FÏ"") G - W^(n • VG)}ds,

where | L denotes the Cauchy principal value of the relevant

inte-gral (see Flügge [15]) (this means that the singular point P on L

274

p . M. VAN DEN BERG

has been excluded symmetrically, after which the limiting value of

the integral has been taken). Analogously, application of Green's

theorem to the domain S' in Fig. 6 with the point P of observation

on L (compare (3.6)) yields the result

- i 0 i ( % P , 2p) = f {(n • F0») G ~ 0i(n • VG)}ds,

when P on L. (4.2)

-i,Wi(xp,zp) = i{(n-VWi)G-Wi(n-VG)}ds,

Addition of (4.1) and (4.2), and using 0 = 01 + 0 r and W =

= ¥ " -|- f"', leads to the equations

i 0 ( ^ P , 2p) - f {(n • F 0 ) G~0(n- VG)} ds = 0i(xp. zp),

iW(xp, Zp) ~ j {(n-VW)G - nn-VG)}ds = Wi(xp, Zp),

L

when P on L. (4.3)

Using the boundary conditions 0 = 0 and nVW = 0 on L, we

obtain the integral equation of the first kind for n • F 0 on L

- ƒ (n • F 0 ) G ds = 0^(xp, Zp) when P on L, (4.4)

and the integral equation of the second kind for "i?' on L

^^(xp, zp) + iW(n- VG) ds = ^^(xp, zp) when P on L. (4.5)

L

Now, we can derive other integral equations for n • V0 and W on L.

These are obtained by performing the operator F P = (c>l()xp) ix +

+ (8j8zp) iz on (3.4) and (3.6) and thereupon letting P approach L.

This yields

hVp0'(xp, zp) = ƒ { ( " • V0') VPG - 0^Vp(n • VG)} ds,

L WP1"{XP, Zp) = f {(n • VW^) VPG - f'Tp(n • FG)}ds, L

when P on L, (4.6)

and

- èFp0'(^p, 2p) = ƒ {(" • V0') VPG - 0»Fp(n • FG)} ds,

L

~WrWi(xp. Zp) = ƒ { ( « • VW^) VPG - '^iFp(n • FG)} ds,

L

when P on L. (4.7)

DIFFRACTION THEORY OF A REFLECTION GRATING 2 7 5

Addition of (4.6) and (4.7), and using 0 = 0 i + 0 r and W =

^\fii j ^ \f/T leads to the equations (with 0 = 0 and n • F V = 0

on L)

| F p 0 ( ^ p , zp) - ƒ (n • F 0 ) FpG ds = F P 0 ' ( ^ P . ^P). I,

WvW(xv, zp) + ƒ Wp(n • VG) ds = VvW^xp, zp),

L

when P on L. (4.8)

Multiplying through in (4.8) scalarly by tip (rip = unit vector in the

direction of the normal to L in the point P on L), we obtain the

integral equation of the second kind for n • F 0 on L,

h(np • F P ) 0(^p, ^p) - ƒ (n • F 0 ) ("p • FpG) ds = (np • VP) 0^(XP, zp) L

when P on L, (4.9)

and the integral equation of the first kind for ï ' on L (with

npVp0(xp, Zp) = 0 on L),

f ï ' ( n p - F p ) ( n - F G ) d s =

( M P - F P ) ' ? " ( A ; P ,

2p) when P on L. (4.10)

We remark that in the second equation of (4.6), (4.7) and (4.8) and

in (4.10) the limiting value of the integral in the point P on L cannot

be taken. Numerical evaluation of (4.10) is not possible without

special measures. Further, we remark that the integral equations

of the second kind for n • V0 on L (4.9) and for ï ' on L (4.5) have

the same structure. Since the unknown function in an integral

equation of the second kind also occurs outside the integral, in

general numerical solution of the integral equation of the second

kind gives less difficulties than the integral equation of the first

kind. For these reasons we shall choose as point of departure the

integral equations of the second kind

| n p - F p 0 ( ^ p . Zp) + I ( n - F 0 ) ( —np-FpG)ds = « p - F p 0 ' ( ^ p .

^P)-L

inxp, zp) +iW(n- VG) ds = Wi(xp, Zp),

L

when P o n L . (4.11)

from which n • F 0 and W on L can be calculated. In the next

section the procedure for solving these integral equations

numeri-cally will be discussed.

276

P . M. V A N DEN BERG § 5. Numerical solution of the integral equations

In this section we describe two methods of solving the integral

equations (4.11) numerically. Further we discuss how 0^,, and W^

can next be calculated with the aid of (3.9). To keep the discussion

in general terms, we introduce the quantities

( n - F 0 ) / 0 o for E-polarization,

X(x,z)'^^exp(—\a.ox)

WJWQ for H-polarization,

( n - F 0 ' ) / 0 o for E-polarization,

Ylx,z)'^^cxp(-i<xox) (5.1)

¥ " / f o for H-polarization,

— rip VpG for E-polarization,

K(xp,Zp\x,z)'^CXp{-iao(Xp-x)} TT . , •

a-n • VG tor H-polarizatioa-n.

li, further the grating profile is given by x = f(s) and z = g(s)

(0 < s < L), then, both integral equations of (4.11) have the same

form, i.e.

L

i-X(sp) + f K(sp I s) X(s) ds = y(sp) when 0 < sp ^ L. (5.2)

(I

In (5.2) X(s) is the unknown function, Y(s) is the known function

and ^ ( s p l s ) is the kernel of the integral equation of the second

kind. Let, further,

,(i/2y«D)exp(iaoA:—ia»A-—iy„2) ior E-polarization,

- " • ' ' ' ' ^ - ( i / 2 y „ D ) e x p ( i . o ^ ) x JorH-polarization,

X ( n - F ) e x p ( —ia„A;—iy„2)

0^/00 for E-polarization,

WJWQ for H-polarization,

(5.3)

then the expressions (3.9) for the calculation of 0'' and V'j have

the form

Rr, = iBr,{s) X(s) ds. (5.4)

II

(i) Solution with the aid of the (discrete) Fourier transform. Since

X(s), Y(sp) and K(sp|s) are periodic in sp and/or s, with period L,

R 1"*

_^n

DIFFRACTION THEORY OF A REFLECTION GRATING 2 7 7

we make use of the Fourier expansions

X(s) = -Z Xk exp{i(27zklL) s},

JC^ — o o o o

Y(sp) = S Y; exp{i(27t//L) sp}, (5.5)

ƒ = — oo oo oo

K(sp\s) = 2 S K^,kexp{i(2nJIL) sp - \(2-KklL) s}.

ƒ = —oo k= —oo

The Fourier coefficients Yj and Kj^j^ are defined as

Yi ^ Z.-1 j V(sp) exp{-i(27r//L) sp} dsp (5.6)

(I

and

L L

Kj, k = L-^ ƒ j K(spIs) exp{-i(27ty7Z.) sp + i(27iA/L) s} dsp ds. (5.7)

0 0

With (5.5) and an interchange of summation and integration, the

integral equation (5.2) can be replaced by the system of linear

equations

oo

2 (LKj,k + hSj,k)X,c=Yi (/ = 0 , ± l , d = 2 , ...), (5.8)

k= —oo

where dj^jc is Kronecker symbol: öj^jc = 0 if j ^ k, dk,k = 1- In

the same way (5.4) can be replaced by

oo

Rn = 2 LBn.liXjc, (5.9)

k= —oo

with

Bn, k ^ ' L-^ j Bn(s) exp{x(2TzklL) s} ds. (5.10)

0

In the numerical calculations we truncate the various series and

consequently the corresponding system of linear equations. The

number of equations (2K -\- 1) taken into account is chosen such,

that the prescribed accuracy is attained. Now, for each term Kj^ jc,

in which as a consequence of the truncation — K ^ ƒ ^ -j-K and

—K < A < -\-K, we can write

oo

K},k= S I<}.k,n. (5.11)

n= —oo

278

p . M. VAN DEN BERG

A numerical investigation of the right-hand side of (5.11) reveals

that the best approximation of Kj^k is obtained, when the series

in the right-hand side of (5.11) is truncated as follows

(A:/2) +

A-= S Kj,k,n, for E-polarization, when j ^ k;

(kl2)-N k + N

Kj,ic = 2 A'y, k, n, for both E- and H-polarization, when j = k;

k~N

= S K]^k,n, for H-polarization, when j ^ k;

W/2)-A'

(5.12)

in which N has to be chosen such, that the prescribed accuracy is

obtained.

(ii) Solution with the aid of cubic spline approximation. We

sub-divide t h e integration interval 0 < s < L by a mesh of points

0 = So < si < ... < SK = i . The value of the function X(s) in the

point Sfc is denoted by Xj; (^ = 0, 1 K). We approximate X(s)

by a function S(s), which is continuous together with its first and

second order derivatives on [0, L]. Further it is to coincide with a

cubic polynomial in each subinterval s^-i < s < s^ (/e = \,2, ...,K),

and it has to satisfy the condition S(sic) = Xk (k = 0, \, ..., K).

The function S(s), a continuous piecewise third-order polynomial,

satisfying

S(sk) = Xk (k = 0,\,...,K). (5.13)

and

S(J')(s+) =. S(ï')(Sfc) (/) = 0, 1,2; /e = 1,2, ...,/v - 1), (5.14)

is called a cubic spline (see Ahlberg [16]). Since X(s) is periodic in s

with period L, we have

5<ï')(0+) = S(J')(L-) (p = 0, 1,2). (5.15)

In this case the spline is called periodic. Hence the spline of

inter-polation of this type is (see Ahlberg [16])

X(s)^S(s) = S Pk(s)Xk, (5.16)

fc=i

DIFFRACTION THEORY OF A REFLECTION GRATING 2 7 9

said to be a periodic cardinal spline, satisfying

Pk(si)=dk,j (j = 0,l,...,K), (5.17)

n " ' ( s ; ) = ^ " ' ( V ) (/. = 0, 1, 2; ? = 1, 2, ..., A' - 1), (5.18)

and

P^^\0+) = P^^\L-) (P = 0,\,2)

when k = \,2, ..., K.

(5.19)

We can determine the splines (see Ahlberg [16])

Pk(s) = èk,] + Pl(Si)(s - Si) + iPl.(S;)(s - S;)2 +

Jr\Pl(Sj)(s-Si)^ (5.20)

(Sj-i < s s$ sy; k= 1,2, ..., K; j = 1, 2, ..., K).

by making use of the continuity (5.18) of the function Pk(s)

to-gether with its first and second order derivatives, and by making

use of the end condition (5.19).

Again, the integral equation (5.2) can be replaced by a bounded

system of linear algebraic equations. The solution of this system of

linear equations will approach the true solution of the integral

equation as the number of equations increases. With (5.2) and (5.16),

and an interchange of summation and integration, we obtain

iX(sp) + S [^PicKs) K(sp\s) ds} Xk = y(sp). (5.21)

k=l 0

If we substitute, in turn, sp = {s;} with [sj = s^} (j = \,2, ..., K)

into (5.21) and define Y(sj) = Yj, we obtain the system of linear

equations

S (A;,ic + l^uk)Xk = Yi (ƒ = 1, 2, ..„ K), (5,22)

in which

Kj,k^]Pk(s) K(sj\s) ds. (5.23)

0

In the same way, (5.4) can be replaced by

K

Rn = Yi Bn,kXk, (5.24)

280

p . M. VAN DEN BERG

with

Bn,k^iPk(s)Bn(s)ds. (5.25)

0

The method with the Fourier coefficients is particularly useful,

when the integrals (5.6), (5.7) and (5.10) can be determined

ana-lytically. When this is not the cast, it is advantageous to use the

method of spline approximation. In this case, the integrals (5.23)

and (5.25) are computed numerically. The latter method is

particu-larly useful, when we are dealing with a smooth function X(s). I n

connection with the numerical evaluation of (5.23) we observe t h a t

K(sp I s) is given by a slowly converging series. In order to make the

numerical solution of our problem feasible it is necessary to develop

techniques for evaluating K(sp\s) rapidly. This is achieved by

em-ploying an asymptotic approximation of /<'(sp|s) (see Appendix B).

§ 6. Numerical and some experimental results

In this section a number of numerical and some experimental

re-sults are presented. Two types of gratings receive attention: the

echellette grating and the sinusoidal grating. As an indication of

the accuracy of the numerical solution, we use the power relation

(derived in Appendix C)

S RnR; cos(en) =cos(do). (6.1)

propagatinpr wavds

First we present a number of results pertaining to the echellette

grating with a triangular profile (Fig. 7).

\ L angle of incidence 9»

period of the grating

DIFFRACTION THEORY OF A REFLECTION GRATING 2 8 1

In this case we have employed the numerical method with Fourier

coefficients. Then the integrals (5.6), (5.7) and (5.10) can be

de-termined analyticall}'. In Table I the accuracy of our method, with

integral equations of the second kind, is compared with the method

of Petit [9-11], who employed integral equations of the first kind.

TABLE I 2K + 1 2N + 1 ^0«S R^K\ = R<^K\ =•

P„<

70. R_ -1«*-1 -2^-2 cos{0J

Our results, with integral e q u a t i o n s of t h e second kind E-polarization 9 19 0.180 0.396 0.115 1.004 9 25 0.180 0.397 0.112 1.003 H-polarization 9 19 0.134 0.332 0.277 1.002 9 25 0.134 0.331 0.278 1.001 P e t i t ' s r e s u l t s , witli i n t e g r a l e q u a t i o n s of tlie first kind

E-polarization 9 19 0.179 0.397 0.107 0,998 13 27 0.180 0.397 0.107 0,999 H-polarization 9 19 0.142 0,321 0.279 0.992 Results for a symmetric echellette grating (/> = iD) with D = 1.25 (jtm and tan(6) = 0.3. The incident wave is normally incident (no = j8o = 0, cos{öo) = 1) with wavelength A = 0.546 (i,m.

In this table 2K + 1 = the total number of terms taken for the

approximate solution of the integral equation and 2A^ + 1 = the

total number of terms taken for approximating the kernel of the

integral equation. From the results we observe that (as expected)

an increase of K and N leads to an increased accuracy of the various

reflection factors (as far as can be concluded from the power

re-lation). In the case of E-polarization, the method based upon the

integral equation of the second kind converges slower than the

method with the integral equation of the first kind. The crucial

point in this is that, in order to obtain the same accuracy, we have

to take into account more terms in the kernel of the integral

equa-tion of the second kind. In t h e former case this kernel consists of a

normal derivative of the Green's function, whereas in the latter

case it is a nondifferentiated Green's function, the differentiation is

leading to a decrease in the rate of convergence. In t h e case of

H-polarization, on the contrary, t h e method based upon the integral

equation of the first kind - with special measures to take care of

the resulting singularity in the kernel (Petit [11]) - converges slower

282

p . M. VAN DEN BERG

than the method based upon the integral equation of the second

kind. Here, in order to obtain the same accuracy, we have to take

into account more terms in the kernel of the integral equation of the

first kind, because now this kernel consists of a double normal

de-rivative of the Green's function, in stead of a single normal

deriva-tive, the once more differentiating is again leading to a decrease in

the rate of the convergence.

In Fig. 8, the efficiency and the ellipticity in the (—l)st order

reflected wave as a function of the wavelength A are presented for a

150 [xm echellette grating with a blaze angle of 20 degrees and an

included groove angle of 90 degrees (Cf. Fig. 7). The efficiency in

the (—l)st order is defined as the fraction of the incident power

which is reflected in the (— l)st order reflected wave (this is R-iR*_j^)

by choosing the wavelength A and the angle of incidence oo such

that the angle between the incident wave and the (—l)st order

reflected wave is constant (here taken to be 32.72 degrees, because

of the available experimental facilities). If we let XE be the

efficien-cy in the case of E-polarization and XH the efficienefficien-cy in the case of

H-polarization, then, the ellipticity is defined as: ellipticity =

= (XE — XH)I(XE + XH). The numerical results obtained for the

ellipticity are compared with experimental data (Westerdij k [17]).

The numerical results are in excellent agreement with the

experi-mental ones. In Fig. 9, the efficiency and the elUpticity in the (— l)st

order reflected wave are presented for a 150 [xm echellette grating

with a blaze angle of 20 degrees and an included groove angle of

120 degrees. Near a wavelength of A = 75 \j.vn the reflected waves

of the (—3)rd and the (l)st order are very close to grazing emergence

(y_3 ^ 0 and yi ^ 0 ) . For the grating with an included groove

angle of 120 degrees, the limiting values of R-3R*_^ are nonvanishing

when y-3 -^ 0 and yi -^ 0, so t h a t we can expect an "anomalous"

behaviour of this grating in the reflected waves of other orders, for

example in the (—l)st order reflected wave (Cf. Fig. 9). For the

grating with an included groove angle of 90 degrees, however,

R-3R*_^ and RiRl vanish when y_3 - ^ 0 and yi ^ 0 , so that an

anomalous behaviour of the grating near the wavelength of A =

= 75 [im is not expected. On comparing the gratings with an

cluded groove angle of 120 degrees with the gratings with an

in-cluded groove angle of 90 degrees. Stroke [18] has predicted a

flatter ellipticity curve and as a consequence a higher efficiency

DIFFRACTION THEORY OF A REFLECTION GRATING 2 8 3 1.0 . 0.8 -0.6 0 . 4 0 . 2

-1 \—\ \ — I \ \ r

0.6 ••H0.4 0.2 0 . 2 0 . 4 0 . 6 -0

m—\—I—I—r

numerical results I I I I

1—\—\—rrm—\—1

e„ip,ici,y=|Ll|ti

X i n pm Fig. 8. Efficiency and ellipticity in the (—l)st order reflected wave as a function of the wavelength A(jSo = 0). The experimental data are reproduced

from Westerdij k [17].

XE = efficiency in the case of E-polarization. XH = efficiency in the case of H-polarization.

284

p . M. VAN DEN BERG

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ell

1

1 1 •pticity = 1 1 1 • ^

i

XE. XE

1

1 1 1 1 1 1 1 1 1 1 — o - numerical results

-J

\

\

-XH ^•XH

\

^ " " " ^ ^

^ ^ " ^ = ^

1

1

-1

20 40 60 80 100 120 UO • X inijm

Fig. 9. Efficiency and ellipticity in the (—l)st order reflected wave as a function of the wavelength A(/So = 0).

XE = the efficiency in the case of E-polarization. XH = the efficiency in the case of H-polarization.

DIFFR.\CTION THEORY OF A REFLECTION GRATING

285

1.0 0.8 0.6 0.4 0.2 1 -1 1 1 1 1 1 Vij!"

v J ^

_r^

_r

1J 0 lim '

1 1 1 J[y\

1 1 1 1 1 1 1

—o— numerical

>-^

"^

'^'^

/ ^

/

/

\

/ /

^

^

1 1 1 1 1 1 1 1 1 results < ^ ^ 1 1 1 " a~^ -20 40 60 80 100 1-20 140 . X in |jm 0.6 0.4 0.2 0 0.2 0.4 0.5 1 -1 -1 -1 ellipticity = -( 1 1 1 . \ f-XH E^XH 1 1 1 1 1 1 1 1 1 —0— numerical \ \

V.

1 1 1 1 1 1 1 1 1 results 1 1 1 — -t3 20 40 60 80 100 120 140 • A in p m

P^ig. 10, Efficiency a n d ellipticity in t h e ( — l ) s t o r d e r reflected w a v e a s a function of t h e w a v e l e n g t h A(/?o = 0).

XE = efficiency in t h e case of E-polarization. XH = efficiency in t h e case of H-polarization.

285

p . M. VAN DEN BERG

curve for the former ones. A comparison of Fig. 8 with Fig. 9 learns

that, apart from the anomaly in Fig. 9 near the wavelength A =

= 75 [xm, the ellipticity curve in Fig. 9 is slightly flatter than in

Fig. 8, the efficiency curve in Fig. 9 however being not higher.

Hence, Stroke's prediction does not hold. In Fig. 10, the efficiency

and the ellipticity in the (—l)st order reflected wave are presented

for a 150 [xm echellette grating with a blaze angle of 20 degrees and

an included groove angle of 90 degrees, but now the angle between

the incident wave and the (— l)st order reflected wave is 17 degrees.

Now, a comparison of Fig. 8 with Fig. 10 learns that the efficiency

curve is a higher one, though the ellipticity curves are the same

ones. So, a prediction as to the efficiency based upon the ellipticity

cannot be made.

In Fig. 11, the efficiency in the zero-order reflected wave is

pre-sented. When A > Ac = Ö{1 + .sin(öo)}, this efficiency should be

unity, because then only the zero-order reflected wave propagates.

The grating then can be considered as a high-pass filter in terms of

wavelengths. Experimentally, this effect has first been observed by

White [19].

• numerical results

1 I I I I I I L ^

200 240

. X in pm

Fig, 11. Efficiency in t h e zeroorder reflected w a v e as function of t h e w a v e -l e n g t h X(f)o = 0).

XE = efficiency in t h e case of E-polarization. XH = efficiency in t h e case of H-polarization.

DIFFRACTION THEORY OF A REFLECTION GRATING 2 8 7

angle of incidence do

i--[hl2)%\nUn>ID)

Fig. 12. G r a t i n g with a sinusoidal profile.

To obtain a single result for XE and XH in Figs. 8, 9, 10 and 11,

we have in the numerical work chosen 2K -f 1 = 15 and 2N -\- 1 =

= 31. For these values, the computing time on the Telefunken T R - 4

automatic computer amounts to about 300 seconds, for the two

cases of polarization together. The computing time for the same

programme on the IBM 360/65 computer amounts to about 60

seconds. The programme on the T R - 4 was written in the Algol 60

language; the programme on the IBM 360/65 was written in the

PL/1 language.

Finally, we present some results pertaining to the sinusoidal

grating whose reflecting surface is 2 = (A/2) s\n(2izxjD) (see Fig. 12),

with D = 1.25 (xm and h = 0.375, 0.500 and 0.700 fxm,

respective-ly. The integral equation of the second kind are solved with the aid

of a spline approximation. We subdivide the interval of integration

by a mesh of 20 equidistant points. The integrals (5.23) and (5.25)

are calculated numerically by choosing a 4-point Gauss-Legendre

quadrature (see Stroud and Secrest [20]) on each subinterval. We

use the asymptotic approximation of /^(spls) (see Appendix B)

with A^i = —N2, = 11. The relevant results are presented in Table I I ,

in which they are compared with those obtained by Pavageau [

12-14]. It is noticed that our results are in good agreement with those

obtained by Pavageau. The accuracy of our results seems to be

better, even in the last example, where h is greater than the

wave-length A. The computing time on the Telefunken T R - 4 amounts to

about 200 seconds, for the two cases of polarization together. The

288

p. M. VAN DEN BERG TABLE II h = 0.375 (xm R^lRtl = RiRl

s;?„R*cos(e„)

E-polarization our r e s u l t s 0,42229 0.01282 0.56988 1.00002 P a v a g e a u ' s results 0.41804 0.01587 0,56848 0.99989 H-polarization our results 0.00038 0,07862 0,88168 0,99996 P a v a g e a u ' s r e s u l t s 0.00038 0.08405 0,87002 0,99839 h = 0.500 fxm

R-iRU=RiR\

S / ? „ < cos(fl„) E-polarization our results 0.3352 0.1888 0.3311 0.9971 P a v a g e a u ' s results 0.3259 0.1882 0,3478 1.0029 H-polarization our r e s u l t s 0,7078 0.0924 0,1257 0.9964 P a v a g e a u ' s results 0,7150 0,0927 0.1253 1,0037 /I = 0,700 [xm R-2R—2 ^ ^ 2 ^ 2 S « „ i e * c o s { f l „ ) E-polarization our results 0.3443 0.1049 0,4772 0,9975 P a v a g e a u ' s results 0.3481 0.1078 0.4919 1.0209 H-polarization o u r results 0,1568 0,4082 0,1064 0.9949 P a v a g e a u ' s results 0.1576 0.4179 0,1081 1.0147 Results for a sinusoidal grating with D = 1.25 [xm and h = 0.375 (xm, 0.500 [xin and 0.700 ^ni, respectively. The incident field is normally incident (ao = ^0 = 0, cos(0o) = 1) with wavelength A = 0,546 (xm,

computing time for the same programme on the IBM 360/65 amounts

to about 40 seconds.

Acknowledgement

The author is indebted to Professor A. T. de Hoop of the

De-partment of Electrical Engineering, Delft University of

Technolo-gy, The Netherlands, for his suggestions and remarks.

DIFFRACTION THEORY OF A REFLECTION GRATING 2 8 9

A P P E N D I X A

Determination of the Green function. The Green function G =

= G[xp, Zp\x, z) introduced in Section 3 has to satisfy the

follow-ing requirements

8^Gl8x-^ + d^GI8z'^ + (k'^ - pl) G = -ö(xp ~ x,Zp- z), (A.l)

exp(iao^) G(xp, Zp\x, z) = periodic in x with period D, (A.2)

and

G consists of waves which travel away from the plane z = Zp. (A.3)

Expansion in a Fourier series with period D in x yields

exp(iao.T) G(xp, zp\x, z) =

oo

= exp(iao^p) S gn(zp\z) exp{i2-Kn(xp ~ x)lD} (A.4)

n = — oo

or, with «71 — cxQ + 2nnlD,

CO

G(xv, z-p\x, z) = S gn{^p\^) exp{\(Xn{xp — x)}, (A.5)

? È = — o o

Substitution of (A.5) in the Helmholtz equation (A.l) yields

oo

2 (c'2g„/&2 + ylgn) exp{iare(a:p — x)} = -d(Xp — X, Zp — z), (A.6)

Jt= —oo

where y,^ ^ k'^ ~ Pl — oil. Multiplying through in (A.6) by exp(ia.nx),

integrating from x = Xi to x = xi -\- D and using the orthogonality

of exp(ianx) for all n on the interval xi < x < xi + D,we obtain

S^gnldz^ + ylgn = -Ó(2P - z)iD. (A.7)

The solution of this one-dimensional Helmholtz equation, consisting

of waves travelling away from 2 = zp, is given by

gn(zp\z) = (i/2y„ö) exp{iy„ \zp — z\}, (A.8)

provided that y» is chosen such that both Re(yK) > 0 and Iin(y„) > 0

(this choice is in agreement with the one in Section 2). From (A.8)

and (A.5) it then follows that the Green function is obtained as

0 0

G(xp, zp\x,z)= S (iRynD) exp{iQ!„(;»;p — x) + iy„ |2p — z\}. (A.9)

n= - - 0 0

290

'P. M. VAN DEN BERG

A P P E N D I X B

Asymptotic approximation of K(sp\s). The kernel of the integral

equation (5.2) has the form

K(sp\s)= S { ( a „ / y „ ) j 3 / + ^ } e x p { i « „ ' r + i y „ | ^ | } , (B.l)

n— — oo

in which s^, 0S, '€ and S are functions of sp and/or .s. In order to

evaluate this sum, we write

( A ^ i - l )

K(sp\s)= 2 {(c.„/y„).is' + ^}exp{ia„'r + iy„|S'|} + K+ + A%

';i.= ( A ' 2 + l )

(B.2)

with

oo

A+ = S {(a»/7«) =^ + ^ } exp{ia„<g' + iy„ \^\}, (B.3)

and

K-= S {(<x„/y„) J!^ + ^ } explian'g'+ iy„ |i2*i}. (B.4)

n= —oo

The series (B.l) does not converge very rapidly, expecially as

\'3i\ ^ 0 . Consequently, we cannot neglect K+ and K-, even for

relatively large values oi Ni and —N2. However for relatively

large values of Ni and —N2,we remark that y« c^ iocn when n'^ Ni,

and yn c:^ —^ocn, when n < A'^2- This results into

0 0

/^+ ~ S (-iséf + ^ ) exp{ct«(i^ - li^l)}, (B.5)

and

iVa

X - ~ S ( + i j ^ + ^ ) e x p { a „ ( i ^ - f 1^1)}. fB.6)

n= —00

In view of the relations an = OLJ^^ -\- 2n(n — Ni)ID and a» =

= ajVj +

27T(W

— Nz)lD, we observe that K+ and K - are

con-verging geometric series. Application of the sum formula

0 0

2 ar» = a/(l - r) when |y| < 1, (B.7)

to (B.5) and (B.6) yields

i ^ + ~ ( - i j ^ + ^ ) e x p { a ^ / i ' ? - | ^ | ) } / [ l - e x p { + (27i/£))(i<^-|i^|)}],

(B.8)