This article was downloaded by: [Tufts University]On: 10 November 2014, At: 08:46Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

Optica Acta: International Journal ofOpticsPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/tmop19

Modal Theory for Thick HolographicGratings with Sharp Boundaries: I.General treatmentU. LANGBEIN & F. LEDERERPublished online: 14 Nov 2010.

To cite this article: U. LANGBEIN & F. LEDERER (1980) Modal Theory for Thick HolographicGratings with Sharp Boundaries: I. General treatment, Optica Acta: International Journal ofOptics, 27:2, 171-182, DOI: 10.1080/716099498

To link to this article: http://dx.doi.org/10.1080/716099498

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the“Content”) contained in the publications on our platform. However, Taylor & Francis,our agents, and our licensors make no representations or warranties whatsoeveras to the accuracy, completeness, or suitability for any purpose of the Content. Anyopinions and views expressed in this publication are the opinions and views of theauthors, and are not the views of or endorsed by Taylor & Francis. The accuracy of theContent should not be relied upon and should be independently verified with primarysources of information. Taylor and Francis shall not be liable for any losses, actions,claims, proceedings, demands, costs, expenses, damages, and other liabilitieswhatsoever or howsoever caused arising directly or indirectly in connection with, inrelation to or arising out of the use of the Content.

This article may be used for research, teaching, and private study purposes. Anysubstantial or systematic reproduction, redistribution, reselling, loan, sub-licensing,systematic supply, or distribution in any form to anyone is expressly forbidden. Terms& Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

OPTICA ACTA, 1980, voi . 27, NO . 2, 171-182

Modal theory for thick holographic gratings withsharp boundaries -j-

I. General treatment

U . LANGBEIN and F . LEDERERSektion Physik, Friedrich-Schiller-Universitat Jena,Jena, German Democratic Republic

(Received 8 November 1978; revision received 22 January 1979)

Abstract . In contrast to usual theories the propagation of the electromagneticfield through a thick sinusoidally modulated grating with sharp boundaries istreated as an electromagnetic scattering problem . This approach leads to thesolution of one integral equation rather than several differential ones and avoidsthe explicit consideration of crossing or boundary conditions . In this part allfundamental relations are derived by transforming the integral equation into a setof algebraic equations for the mode amplitudes of the fields involved . For theirtranslation invariance with respect to the x-y-plane unslanted reflection gratingsappear to play a particular role . In a second part of this paper the general relationswill be applied to unslanted transmission and reflection gratings . In the vicinity ofa Bragg case, which has different meanings for both grating types, fieldamplitudes and dispersion relations as well will be derived analytically and theirdependence on several hologram parameters and parameters of the incident fieldwill be plotted in various diagrams .

1 . IntroductionLet us consider the situation when an arbitrary but known electromagnetic field

falls onto an inhomogeneous medium with sharp boundaries . The resulting wavefields of interest inside and also outside the medium will be a consequence of complexinteraction between this field and the medium .

There are several ways for the determination of these wave fields. None of themleads to a usual boundary value problem even for a homogeneous medium withsharp boundaries, for the field values along the boundaries of the medium are notknown in general .

The determination of the boundary values by solving a separate integral equationis one approach called the 'non-local boundary value problem' [2, 3] . A morecommon approach consists in solving the differential equations inside and outsidethe medium simultaneously with the help of a suitable modal expansion for the fieldsinvolved . A subsequent fitting of the corresponding solutions along the boundariesof the medium has to be carried out by specifying the undetermined constants viacrossing conditions . An alternative approach, due to Wolf [3], avoids the explicit useof boundary or crossing conditions . Here the propagation problem in question istreated as an electromagnetic scattering problem characterized by an integralequation valid in whole space .

t Parts of this work were presented at the 1 . Nationale Tagung des Fachverbandes Optik,held in June 1978 at Jena, German Democratic Republic .

0030-3909/8012702 0171 80200 ( 1980 Tavlor Z Francis 1, rd

Dow

nloa

ded

by [

Tuf

ts U

nive

rsity

] at

08:

46 1

0 N

ovem

ber

2014

172

U. Langbein and F. Lederer

In this paper we investigate the propagation of the wave field bf(r, t) =&(r)exp (-i(o t) inside and outside a laterally unlimited volume grating with a sinusoid-ally modulated complex dielectric coefficient

e(r) = eo + a o + 2a 1 cos gr

when the incident wave

(1)

ei (r, t) = g i (r) exp (- iwt)

(2)

is given. The dielectric coefficient outside the grating is assumed to be FO . a1designates the modulation strength, g the reciprocal lattice vector .

Such volume gratings can be produced holographically and are of increasinginterest in various fields of modern optics [4] . Corresponding theoretical modelsdeveloped in microwave theory, integrated optics and holography base on differentmethods of solving the differential field equations

valid in the grating region only ; c is the velocity of light in vacuo . (For non-magneticmedia one can restrict one's attention to the electric field vector only .)

The results obtained differ in exactness and flexibility, depending on theparameters under discussion; see for instance the excellent review of Elachi [4] aswell as [5] .

We treat the propagation problem in question as an electromagnetic scatteringproblem and propose a solution of the integral equation

where the operator `grad div' acts on the unprimed variables only . Equation (4) isvalid throughout the space . It is equivalent to the system (3) of differential fieldequations including the crossing conditions connecting the partial fields along themedium boundaries [3] .

In the first part of this paper we derive the fundamental relations of our model bytransforming the integral equation (4) into a set of algebraic equations with the helpof suitable modal expansions . By introducing a trial modal expansion for theresulting field of'm inside the grating one succeeds in deriving rigorously a set ofalgebraic equations for the mode amplitudes and the wave vectors of the inner field .

~°(r)=gi(r)+(grad div+k2 ) JJJexp

medium

(E(r')-co)g(r')Gk(r-r')d3r', (4)

(iklr-r'I)

PO

Gk(r-r')=4nir-r'I (4 a)

curl curl 6'(r)-k 241(r)=0

valid outside the grating throughout the source-free space,

curl curl S(r) -k2S'(r)=k2(e(r)-eo)&(r),e a

(3 a)

~2k2= c2 Eo (3 b)

Dow

nloa

ded

by [

Tuf

ts U

nive

rsity

] at

08:

46 1

0 N

ovem

ber

2014

Modal theory for volume gratings

173

If the inner field is determined, the determination of the fields outside the grating isstraightforward . This procedure presented in part I gives several rigorous sets ofalgebraic equations for all field amplitudes and wave vectors of interest .

In the second part of this paper this general relations will be applied to unslantedtransmission and reflection gratings . If the incident field mode nearly fulfils a Braggcondition to be determined in detail, only some field modes have to be taken intoaccount and the solution of the resulting equations is obtained analytically . Itcomprises dispersion relations, transmission and reflection coefficients as well, validfor arbitrary complex parameters a o, a l (see equation (1)) and arbitrary gratingthickness d . The dispersion relations display the different interaction types oftransmission and reflection gratings just in the vicinity of a Bragg case . For unslantedtransmission gratings Kong's results [6] are verified .. An easy test of our calculus for the special case of an homogeneous layer is givenin [7] . The modulated half-space as extension of our problem to infinite gratingthickness is also covered in Part I . In particular, the homogeneous half-spaceproblem arises additionally when differences of the refractive indices on both sides ofthe grating are introduced .

`Travelling' unslanted transmission gratings were considered by Bhatia andNoble [9] using also an integral equation where the `effective' inner field appears, seee.g . [1] § 2.4 ., instead of the macroscopic ones . The difference between their and ourintegral equation is, apart from their taking into account a travelling modulation, dueto their effective (inner) field-concept, see e .g . [1] § 2 .4. Their integral equation canbe transformed into equation (4) of this paper by elimination of this effective fieldwith the help of the Lorentz-Lorenz formula . Their sets of algebraic equations arequite similar to those which we derive in Part I but were solved by otherapproximations than we use in Part II . A solution of equation (4) by the method ofsuccessive approximations which is especially useful for attenuated gratings is givenin [10] .

2 . Physical background of the integral equation and general modalexpansion for the inner fieldAccording to equation (4) the field 9(r) to be determined reveals a different

structure in the three areas z < 0, 0 < z <d, z > d (see the figure) :

Equation (5 a) shows the field (ff < existing on the left-hand side of the grating tobe composed of the incident field 1f' i and the reflected wave field 8M arising from thegrating and its boundaries . Formally the unchanged incident wave 9 i appears also inthe expressions for the inner field 1 m (equation (5 b)) and the right-hand field .9,(equation (5 c)) . That is why a splitting up of the term SM into two subterms will beexpected in the whole region z > 0, one of them being responsible for the `extinction'of the `wrong' wave 9 i in that region . This point of view corresponds to that of the so-called `Extinction theorem' [3] . The total field X > on the right-hand side of thegrating is the transmitted field .

z<0 : (5 a)

0'z_d : =

+00m'(~"m) (5 b)

z>d : &>-(ffi+(ff >'(0m) (5c)

Dow

nloa

ded

by [

Tuf

ts U

nive

rsity

] at

08:

46 1

0 N

ovem

ber

2014

174

U. Langbein and F . Lederer

gx =

Grating

Scheme illustrating the situation of the grating and the participating fields ; t(r)-dielectriccoefficient characterizing the grating, f, incident field, c§ inner field,transmitted field, 7-reflected field .

Obviously our central problem reduces to the solution of the integral equation forthe inner field &,,, . Knowing Sm the fields 6',, S, can be calculated straightfor-wardly via equations (5 a) and (5 c) .

Let the incident field f be an arbitrary wave field, the sources of which aresituated in the region z < 0 only . Propagating into the half-space z i 0, the mode-expansion of this field is [8]

1

cc.6,(r) _ ( 211)2

a (a, /3) exp (iy'r) da d/3, z, 0

(6)

where

(ejY> )=0,

+y

y = /(k2-a2-/32) °

Ep

(7)

(8)

(9)

(The imaginary part of y shall always be positive .) For the inner field 0'm to bedetermined we use the trial solutions

1if

cc

cc ••&m(r) = (2x) 2 f f

T_

Y,da d/3

{e= o (a, fl) eXp (ip a r)

+ e~,Q(a, /3) exp (i„t ,Qr){,if 9x,9%, 0, . and

1

x

(21) 2 f fdx d$

{e T+ ( a, /3) exp (ij r) + e .- ( a, /3) exp (i„. r)}

ifx

=0, (g,.=IgI=g) ; r=

yz-d

0< z< d

(10)

(10 a)

Dow

nloa

ded

by [

Tuf

ts U

nive

rsity

] at

08:

46 1

0 N

ovem

ber

2014

(The imaginary part of the square roots in equations (12) is to be chosen positivealways .)

The trial solutions (10) are generalizations of the simple ones for the inner field ofan homogeneous slab [7] towards a Bloch-mode expansion which takes modecoupling by the inhomogeneities of the medium into account . Except for gx =gy =0each 'Bloch-mode' denoted by index T is expected to split up into a set of submodes(index a), each of them possessing its own wave number Ka and ka respectively .

3. Solution of the basic integral equation (5 b)Using the Weyl-representation of the Green's function G,(r, r') [8]

G,k(r, r')= 87E

2J J

Yexp {i[a(x-x')+/3(y-Y)+ylz-z'I]}dotd/3

with respect to the figure, equation (5 b) takes the form

div+k2) B~~EOEO J J

dot df J'J~ dx' dy'

xJ

z dz'(e(r')-EO) .6m(r ) 1exp [i7'(r-r')]o

Y

Jd dz'(e(r')-Eo ).Qr')-exp [iy`(r-r')]

Y

The integral expression is our first subject where the function E(r), equation (1),

E(r) =Eo + aO +al [exp (igr) + exp ( - igr)]

(15)

and the &,§-mode expansions (10) and (10 a) respectively will be inserted explicitly .After that all integrations except an a, /3 one can be carried out . One gets a variety ofterms which can be collected in groups with respect to their propagation behaviour ifsuitable T redefinitions are introduced . Finally, differentation will be carried out andthe remaining field terms 4'i and &m in equation (14) will be replaced by their

&m(r) = Ii( r) + (grad

Modal theory for volume gratings 175

where

°IX+Tgx

+Tgy

GC+Tgx

JJ+ Tgv (11)hz,a

°T+=

a a

hi,a

(11 a)° T PT

(12)ht,a = I(K'- OC2- /32)+Tgz , l2i,a=- _,/(ko - CC2- a2)+Tg2,

pz- /(K2- O12- f3 2)+Tg,

°r= _ (K2- CC2- /3 2)+Tg . (12 a)

Dow

nloa

ded

by [

Tuf

ts U

nive

rsity

] at

08:

46 1

0 N

ovem

ber

2014

1 7 6

U. Langbein and F . Lederer

corresponding mode expansions (6), (10) and (10a) respectively. Subsequentordering of all field terms with respect to their propagation behaviour yields forgX, g Y # 0

0 =4~ 2~I dot d# YtY C-$oe~a+k2-µa 2

- oo

r=-ao a=-co

/2 r,a - Yr

x a e+ +a e+ +a e + )Jexp(iµ+ r)+ Y

E e- +k2-Rr,a ' a

( 0 r,a

1 r- l,a

1 r+ l,a

r,a

0 r,a

p2_72a=-ac

µr,a- Yr

x(aoer;a+ a1 exp (ig=d)ez_ l,a +a1 exp (- igZd)er+ 1,a)]exp (ip it)

+ 2coe,8r, o -

k2-~Y= OxYi (aoeQ+ale+ i,a+ale +l,a)[

a- - 40 YJI-tra - Yr)

exp ( - iµ~, ad) (k2 -Y2x®Y')

1+ (aoer-a +a1 exp (ig .d)er- l,aYr(/~r,a - Yr)

+ a 1 exp (- ig.d)er+ 1,a)JJ exP (iYt 'r)+ - C exP(iy,ad)(k 2 -yi OxYi)a=-~

MY, +Yr)

+

+

+ -Yr`O

Yr<

x (a0er,a+a1er-1,a+a1er+1,a)+kz

Yr(µr,a+yr)

x (aoeT,a+a1 exp (igad)er-l,a+a1 exp (-ig.d)er+1,a) ]exi <P (Yrr)

(16)

where

a+igX 1 for i=0YT= (i+gy

3i, 8 r,o =

(17)0 for TOO± YT

Yr=N/(k2- (a+TgX)2- ($+rgY) 2 )

(18)and the imaginary part of yr to be positive always .

The symbol Ox in equation (16) designates the dyadic product, for instance

a2 ap ay70Y=

up fl , flyay fly Y Z

For their different propagation behaviour, which reflects their mutual linearindependence mathematically, the amplitude expressions of the different wave typesin equation (16) must vanish separately . This gives the following system of algebraicequations

eo(µsu - k2)eia = (k2- µiaOµza)[e+ ] ,

,0(µr,a -k2)e*,a=(k2-/~r,a©µ=a)[es,,],

(19)

(20)

Dow

nloa

ded

by [

Tuf

ts U

nive

rsity

] at

08:

46 1

0 N

ovem

ber

2014

e,(r)=&1(r)+(grad div+k2

Modal theory for volume gratings

177

2eoyte18T,o=(k2-Yi(Dy>

)E Pr, ±k2) [ea]+(Flt.a+()Q2pk2) ip='off

0=(k2- Yi O Y<)

(µt,a - Yt) exp (ip,,A e+ + ft.a-Yt

-(u~a -k2)

[e...]Q -k2)

[e t,a] ,

where

[e+ ]=a oea+ale+ i,a+aie+i,a,

(23)

[era]=aoet,a+a1exp(ig=d)eT_i,a+a1exp( - igzd)et+1,a .

(24)

The whole set of equations (19)-(22) consists of two different types of equations .Each partial system (19) and (20) respectively forms a homogeneous one containingthe amplitudes of the inner field only . The condition for the existence of nontrivialsolutions (i .e . vanishing of the determinant of each system) permits the determin-ation of the wave numbers xa and ica to be solutions of two dispersion relations .

An assumed restriction to N i-modes of the inner field tf ,, yields a maximum of Ndifferent solutions x a , Ka respectively for each dispersion relation . In this case 2N2amplitudes eza of the inner field are to be determined . For this purpose there are2N(N- 1) linear independent vector equations (19), (20) . The additional 2N vectorequations which are needed are just given by equations (21) and (22) . Equation (21)connects the incident field amplitude with the amplitudes of the inner field and statesthe cancellation of the incident wave with the `wrong' wave number k inside themedium mentioned already .

For unslanted reflection gratings (gx =gy = 0) an analogous procedure yields,instead of equation (19)-(22) :

An inner field consisting of NT-modes possesses now 2N unknown amplitudes eT .Their determination is ensured by the 2(N-1) vector equations (19 a), (20 a) and thetwo vector equations (21 a), (22 a) .

So far the determination of the inner field has been considered . The fields outsidethe medium are given by equations (5 a), (5 c) . These relations can be considerablysimplified in the following manner . Insert the Weyl-expansion (13) into equation (4) .This gives immediately

gn2eo ff_ . da dJi ff_ dx'dy'

(22)

x {o dz' (e(r' ) -eo)G°„,(r' )

Yexp (iy ) ( r - r' ) .

(25)

+z

2 )et+ -_ (k2-

+eo(Nt - k

Nt+Oµt+ )[et ], (19a)

go ( .U"- 2 -k2)et =(k2- It Op )[ez ], (20 a)

µt + Y

(-(µt + y) exp

iµrd)2eoyel=(k 2

-Y'OY')z +]+[et (21 a)+2 -k2

x- 2-k2

[e -])>

0=(k2-Y`~Y`)E[et(µr - y) exp (iµrd) +

Flt - Y (22 a)2

]+

2k2 [e., ]~ .11.,+2 -k

j= -

Dow

nloa

ded

by [

Tuf

ts U

nive

rsity

] at

08:

46 1

0 N

ovem

ber

2014

1 78

U. Langbein and F . Lederer

Consider first X, and replace also e(r) by expression (15), offj by expression (6), f,,, byexpression (10) or (10 a) respectively which are known in principle now . Carry out allintegrations until only an a,#-one is left, differentiate and collect all field terms ofequal propagation behaviour . This gives for g x , g,. 0 0

1(r)=

x

p

k2 _Y>Ox Yr

>

a=

Rµr+,a'+Yr

>

dadp

z -k

+2 exp(utA[e Q]

T-

2YTEo(27)2ff +

'PI T 'z + Yz [eT,a)

exp (iy r)rT,a -k

k2-7"07"x

pr,a+Yr ++ eiST,O+

+2

2 [eT,a]2YTeo

a = -

µr,a - k

+ µT

y` exp ( - iuT,ad)[eT,a]

exp (iy r) .

( 26)Rr,a _k2

Since equation (21) is valid the second part of expression (26) vanishes and c9,(r)takes the form

1

da dl Y, et (a, P) exp (iy' r) .

(27)~>(r)=(27)2 ffx

r

Its amplitudes are in direct relation to the amplitudes of the inner field

et =k2 `Y~ OYr

(L+Yr([+I eXp

iFuT,ed)ea+

Ps, 2 + Yz [et, a] .

(28)2YrEO

a Or, - k

11T,a - k

In the particular case g,;=g„=0 one gets

S>(r)= (2 1 ) 2J J xdadJ3e>(a,fl)eXp(iy>r),

(27 a)

where now

e > - k2 -~ ©Y'

- + 2

k2 eXp (iµrd)[ei ] + µµz+ k2 [eT ]

(28 a)}Y o

PT

Starting with relation (25) the determination of of, follows in an analogous way butwithout using the explicit A` .-expansion (6) . The procedure involves the use ofequations (22), (22 a) respectively instead of equations (21), (21 a) and yields forgx,gy #0

< (r) = ol i(r) +1

2 if L Y_e' exp (iyt r) dot d/ .

(29)(27r

~ T--~

is composed of the incident wave A'j

the reflected field (see the figure), theamplitudes of which are

k2- Ys ®y

/tr,a`Yr +

N -YreT = -

+2 z [eT,a] + 2 2 exp ( - iiT .ad) [er,a)

(30)2YTEO

a

OT,a - k

Or-,O, - k

Dow

nloa

ded

by [

Tuf

ts U

nive

rsity

] at

08:

46 1

0 N

ovem

ber

2014

Modal theory for volume gratings

179

The unslanted reflection grating (g x =gy= 0) has to be considered separately again .One gets

9<(r)=g;(r)+ (27t)2 f f da d$e < (a, /3) exp (iy < r),

(29 a)

k2-7<07<

1( YT- yµT- Y

e

2YEO

T µi 2-k2[e+1 +µ= 2- k2exp(-iµ td)[e~ ]

(30 a)

Evidently equations (28) and (30), (28 a) and (30 a) respectively ensure the uniquedetermination of the fields f > , if < outside the volume grating if the inner field iff m isknown. The inner field is related to the incident field e; by equations (19)-(22),(19 a)-(22 a) respectively. It is worth mentioning that explicit calculation of the innerfield is not necessary when the relation o > =f>(9j) is of interest only .

For its translation invariance with respect to the x-y-plane the unslantedreflection grating plays a particular role in our formalism .

4. Particular cases4.1 . TE-polarization

It is no serious restriction of generality if one assumes gratings to be invariantwith respect to the y-coordinate, for instance

g=(gx, 0, gZ) .

(31)

In this case the determination of TE-polarized fields becomes a two-dimensionalproblem and the corresponding amplitude relations are especially simple . Wesuppose

e,=(0, e„ 0), e,~, a =(0, e.' a , 0), a i=(0, e , 0) .

(32)

The dyadic products do not appear now ; further simplification is obtained byinserting the relations (19) and (20), (19 a) and (20 a) respectively into the remainingexpressions. This gives for g,, :A 0 (now all amplitudes depend from mode number aonly)

Eo(µ~6 - k2)eia=k2[ea], (33)

Eo(lUs,a - k2)es,a = k2Les,a], (34)

2ye i8 t , o =a[(p=,a+Yz)esa+(Pz,a+yt)exp( - Zµsad)esa], (35)

0=Ya

Ys) exp (iu,,ad)eta+Yr)ei,a], (36)

2y,e~ _a

[(hs,a+Yz) exp (iµ,,ad)eia+(µt,a+Yz)e~,a], (37)

and for gX =O

2y ,e' _ - ~ [(ys,a - y )eta+yt) exp (-ip,,ad)et,a] (38)

(33 a)

a

z

2)et-+-kz ++ kEo(µ~ -

Lez ],

eo(µ., 2- k2)et =k2[ef ], (34 a)

Dow

nloa

ded

by [

Tuf

ts U

nive

rsity

] at

08:

46 1

0 N

ovem

ber

2014

180

U. Langbein and F . Lederer

2ye1=y[(uz+Y)eT +(µz-Y)eXp(-iµrd)eT],

(35a)T

0=Y[(ur-y) exp(iµtd)eT +(µz - Y)eT ],r

2Ye' =Y[(t + y) exp (iutd)eT +(µr+Y)e-],

(37 a)T

2Ye` _ - [(ur - Y)eT +(µr - Y)exp (-iµrd)e r ] .

(38a)r

In contrast to equations (8), (11), (17) all µ,y-wave vectors have only two componentsnow

4.2 . The periodically modulated half-spaceFinally we mention the case of unlimited grating thickness, for some consider-

ations sufficient already [4] . It leads to very simple amplitude relations (33)-(38) . Asbefore TE-polarization is assumed .

Let the sinusoidally modulated medium be localized in the half-space z >, 0. Thenthe characteristic integral equation has the form

Em(r)=Ei(r)+ik2

da

dx'{J (' ('z

dz'[e(r') - c o]Em(r')exp [iy`(r-r')]4 7rco

f-

f -

o

Y

(36 a)

+J

dz'[e(r')_eo]Em(r') eXp[iyY(r-r')]

t, r=(x,z) .

(43)

)Z

In contrast to equation (10) a trial solution for the inner field is sufficient now thatdoes not contain any e.-(F -amplitudes .

+

-(

a+tigz)

- (a+Tgx)(39)

ur, a=J(xQ-a2)+rg y7 µr,a =- ,l(i - a 2)+zgz, (40)

IL =(µr), itT =

(µT),(39a)

µr =J(K2 -a 2)+ig, µr=-J(x2-a2)+,rg, (40 a)

y

a+zgx

Z =( ± Yr(41)

(42)y,=,/(k 2 -(a+,rgx)2),

Y >- _( +y),(41 a)

y= J(k2-a 2 ) . (42 a)

Dow

nloa

ded

by [

Tuf

ts U

nive

rsity

] at

08:

46 1

0 N

ovem

ber

2014

instead of equation (38)

instead of equation (35 a)

instead of equation (38a)

Modal theory for volume gratings

181

Convergence problems may arise when making the z-integration in the second termof the whole integral in equation (43). The relevant expressions are

lim exp [iz'(°z,6+yA

(44)

but their disappearance is guaranteed by adding an infinitely small positiveimaginary part to ° t , a . The following modified amplitude relations are now obtained :Instead of equation (35)

2yte,5,,,o =Y-(°z,a + yT)e'T,a

2y,et = - Y (°T,a -YL)e+

a

2Ye` _ - Y(°t - Y)ei

gx:A 0

(45)

(46)

2ye;=Y(u,+y)e.,

(45 a)

(46 a)

Equations (33), (33 a) remain unchanged, equations (36), (36 a), (37), (37 a) do notappear for evidently there is no field on the right-hand side of the medium .

The extreme simple case of an homogeneous half-space becomes additionallyrelevant when differences in the refraction numbers on both sides of a modulatedslab are considered . The corresponding Em trial solution, as well as the amplituderelations, can be derived by straightforward simplification of equation (10) or from[7] .

5. Final remarksThe propagation of an electromagnetic field through a periodically modulated

slab was considered to be an electromagnetic scattering problem . One feature of thisapproach, resulting in more flexibility, is that the inner field can be determined by anintegral equation without respect to the fields arising outside the inhomogeneousmedium. If the inner field is known the fields outside the medium can be obtained bydirect integration without explicit consideration of crossing or boundary conditions .

Assuming an arbitrary orientated dielectric and/or absorption grating we derivedrigorously several sets of algebraic equations for the mode amplitudes and the wavevectors of all resulting fields as well .

Although not explicitly considered, our treatment is not restricted to simplegratings of type (1) but can be extended to a set of gratings, each of type (1), placedone after another and not necessarily in contact . Generalization is also possible tofunctions e(r) characterized by several reciprocal lattice vectors . The former caseenlarges the number of sets of algebraic equations relating the field amplitudes in thedifferent slabs . The latter one needs a generalized trial solution for the inner fieldreflecting the new symmetry of the medium .

Dow

nloa

ded

by [

Tuf

ts U

nive

rsity

] at

08:

46 1

0 N

ovem

ber

2014

182

Modal theory for volume gratings

La propagation d'un champ electromagnetique par un reseau de volume fortement limit&et sinusoidalment module est trait&e, au contraire des theories usuelles, cornme un problemede dispersion electromagnetique . Cette methode conduit a une equation d'int&grale dontsolution ne necessite pas explicitement la consideration des conditions aux transitions ou auxlimites comme dans le cas de la solution des equations differentielles de champ . A l'aide d'unetransformation des equations algebriques pour les amplitudes de modes des champs participesles equations fondamentales du modele sont d&rivees . A cause de leur invariance de translationrelative au plan x -y on doit consid&rer particulierement de noninclines reseaux de reflexion .

Dans une deuxieme partie de cette publication le modele est specialise dans de noninclinesreseaux de transmission et de reflexion . A proximite d'un cas de Bragg dont signification estdifferente pour les deux types de reseaux on peut calculer analytiquement les amplitudes dechamp et le relations de dispersion . A l'aide des diagrammes diff&rents l'influence desparam&tres d'hologramme et des propriet&s des champ incident sur les champs resultants estdiscut&e .

Die Ausbreitung eines elektromagnetischen Feldes durch ein scharf begrenztes sinusfor-mig moduliertes Volumengitter wird im Gegensatz zu i1blichen Theorien als elektromagneti-sches Streuproblem behandelt. Dieser Zugang fuhrt auf eine Integralgleichung, bei derenLosung Ubergangs- oder Randbedingungen nicht explizit beachtet werden mussen, wie dasbei der Losung des entsprechenden Satzes differentieller Feldgleichungen der Fall ist . DutchUmformung der Integralgleichung in algebraische Gleichungssysteme fur die Moden-amplituden der beteiligten Felder werden in diesem Teil die Grundgleichungen des Modellsabgeleitet . Wegen ihrer Translationsinvarianz bezuglich der x-y-Ebene besitzen ungeneigteReflexionsgitter hierbei eine Sonderstellung. In einem zweiten Teil dieser Arbeit wird dasModell auf ungeneigte Transmissions- and Reflexionsgitter spezialisiert . In der Nahe einesBraggfalles, dessen Bedeutung sich fur beide Gittertypen unterscheidet, lassen sich dieFeldamplituden and Dispersionsrelationen analytisch ermitteln . Anhand verschiedenerDiagramme wird der Einflul3 der Hologrammparameter and der Eigenschaften des ein-fallenden Feldes auf die resultierenden Felder diskutiert .

References[1] BORN, M., and WOLF, E ., 1964, Principles of Optics (Oxford, New York : Pergamon

Press) .[2] SEIN, J . J ., 1970, Optics Commun ., 2, 170 .[3] WOLF, E., 1973, Coherence and Quantum Optics, edited by L. Mandel and E. Wolf (New

York : Plenum Press), p . 339 . WOLF, E ., 1974, Paper presented at the Symposium onthe Mathematical Theory of Electromagnetism, University of Rome, 1974 .

[4] ELACHI, CH ., 1976, Proc. Inst . elect . electron . Engrs, 64, 1666 .[5] MAGNUSSON, R., and GAYLORD, T . K ., 1977, J. opt . Soc. Am ., 67, 1165 .[6] KONG, J . A., 1977, J. opt . Soc. Am ., 67, 825 .[7] LANGBEIN, U ., and LEDERER, F ., 1978, Wiss. Z. Friedrich Schiller-Univ . Jena ., 27,173 .[8] LALOR, E ., and WOLF, E., 1972, J. opt . Soc. Am ., 62, 1165 .[9] BI-IATIA, A. B ., and NOBLE, W. J ., 1953, Proc . R . Soc. A, 220, 356 .

[10] LEDERER, F., and LANGBEIN, U ., 1977, Opt. quant . Electron ., 9, 473 .

Dow

nloa

ded

by [

Tuf

ts U

nive

rsity

] at

08:

46 1

0 N

ovem

ber

2014