HAL Id: jpa-00249112
https://hal.archives-ouvertes.fr/jpa-00249112
Submitted on 1 Jan 1994
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Perturbative methods in theory of phase gratings
J. Harthong, P. Meyrueis
To cite this version:
J. Harthong, P. Meyrueis. Perturbative methods in theory of phase gratings. Journal de Physique III, EDP Sciences, 1994, 4 (2), pp.407-421. �10.1051/jp3:1994137�. �jpa-00249112�
Classification Physics Abstracts
02.60 42.20 42.40
Perturbative methods in theory of phase gratings
J. Harthong and P. Meyrueis
Laboratoire des systkmes photoniques, Ecole Nationale Supdrieure de Physique, 7 rue de l'universitd, 67084 Strasbourg Cedex, France
(Received15 February 1993, revised17 September1993, accepted19 October 1993)
Abstract. Perturbative methods are generally invoked for problems in which there is a small parameter. In the theory of phase gratings, the small parameter is the modulation amplitude of the refractive index, and the classical perturbative method is then the Born approximation. But it is well-known that the Bom approximation fails at the Bragg resonance, however small the modulation amplitude is. In this paper a perturbative method is presented, which is working at
Bragg resonance as well. A sequence of numbers (called the eigeni>alues of the problem) are
introduced they depend on the geometrical configuration (incidence angle, grating parameters). It is shown that the Bragg resonance occurs if (and only if~ two eigenvalues become equal. These
eigenvalues and the corresponding solutions of the equations can be expanded in powers of the modulation amplitude. The expansions are different according to whether the corresponding eigenvalue is simple or double. Explicit formulae or algorithms are given. Computing programs
have been written from them. These programs are efficient.
1. Introduction.
In a recent work Ii we have presented an altemative form of modal theory (see [2, 3]). This
altemative theory has a fringe benefit for periodic gratings the possibility of making
perturbative expansions in powers of the modulation amplitude in such a way that they can
work not only outside of the Bragg resonance region, but also within it.
The classical perturbative method is the Born approximation : it consists in solving the wave
equation
Af + k~(I + eq(x)) f
=
0
(where eq(x) is the permittivity modulation) as an expansion in powers of e. But it is well- known that the result of such an expansion is false when the Bragg resonance occurs it is
correct only if the diffracted light is negligible with respect to the transmitted one. The aim of
the present contribution is to show that a perturbational approach is possible at Bragg
resonance. For this we need in fact two different perturbational methods : one (called regular)
which will work outside of the Bragg region, and another (called singular) which will work
within the Bragg region. Of course these two methods give the same results in the intermediate
region.
Perturbational methods have several advantages.
At first, their precision (and therefore the computation times which are in inverse ratio to the
precision) can be adjusted to the characteristics of any given problem. Indeed, both
perturbational methods (the regular one and the singular one) permit expansions in powers of e at any order you like. The higher the order, the higher the precision and the computing time. It must be pointed out, however, that if the order is too large, the precision decreases generally
there is an optimal order up to which the perturbational series should be summed up. For
example, in dichromated gelatin, the modulation amplitude e is generally about 0.02 ; in
photopolymers (which are then thicker) e can be much smaller. Hence, in the former case the expansion must be carried to fourth or fifth order in the second case the same precision can be
obtained at first or second order.
Secondly, the computing times are shorter than for usual methods, such as coupled-wave
theories.
Thirdly, it appears that there are less problems of ill conditioning (the method spares matrix calculations). From a mathematical point of view, the Bragg resonance corresponds to
singularities (a mathematical analysis of this phenomenon will be presented in Sect. 3). In numerical computations using any classical method, such mathematical singularities are likely
to cause small denominators, however the computations are carried out. In this paper a specific
method is introduced for the singular case, in which the main cause of small denominators is aboided by cancelling out the small quantities.
Finally, it can be pointed out that considering small values of
e has physical interest, since almost all thick holograms have a modulation of sufficiently small amplitude to permit an
efficient use of the perturbative expansion proposed hereafter. The perturbational approach had been given up in the past because of the failure of the Born approximation ; it was a bad reason.
In section 2 we give a short review of the mathematical analysis given in reference [I]. This mathematical analysis is exact and will be the basic framework for the further developments. It will be developed for the particular case of E-polarization (extension to the case of H-
polarization is then straightforward). The perturbational approach of the present paper cannot be performed within the usual coupled-wave theory.
In section 3 we study the regular case, in which there is no resonance. We give explicit
recursive formulae for the expansion in powers of
e of the electromagnetic solution.
In section 4 the same is done for the singular case.
+2 (roll) ~ +2 (trfins)
~
+i (refi) ~ *'(trans)
=_
=
- ~
lncldent~wfive
~
~
~
grfitlng vector
0 (roll) -
_
i (refi) °~'~'~9 ~' (irons)
2. Short review of electromagnetic theory.
We set
a the angle of incidence,
w the angle of slant (for unslanted gratings w
= ± ar/2), f
=
A/d the normalized spatial frequency (d is the period of the grating),
e the modulation amplitude,
q(o) the modulation profile (normalized).
The grating vector has components sin w, 0, cos w along the axes Ox, Oy, Oz respectively.
For E-polarization we suppose thet the electric field is polarized in the direction parallel to the
y-axis. We call ko the wave number in the empty space (or air) and k
= Nko where N is the
mean refractive index in the holographic medium. In these conditions we have
electric field E
= (0, u (x, y, z), 0)
~'~~~~~~~~ ~~~~' ~ ~o ~
'
~' ~o ~
We start with the equation
Au (x, y, z) + k~ f(x, z) u(x, y, z)
= 0 (1)
where f(x, z is the permittivity (times I/~V~). Owing to the periodicity of the modulation and to the fact that it is a small perturbation of the constant mean permittivity we can write
f(x, z)
= I + eq(kf(x sin w + = cos w )), or, if we set p(x, z) = f(x sin
w + z cos w ),
f(x, z
=
I + eq(kp) q (the modulation profile of the grating) is a periodic function of period
2 ar. Then equation (I) becomes
Au (x, y, z + k~jl + e(kp(x, z))j u(x, y,
z =
0. (2)
The solution will be sought in the form u(x, y, z
=
e'~~~~ ~ Y(kz, kp(x, z)) (3)
where Y((, o) is an unknown function, periodic with respect to o
= kp. (For the sake of
brevity, we say periodic without other specification for functions of period 2 ar ; so the fact that the period is always 2 ar is understood.)
If we put (3) into (2) and expand the derivatives, we obtain a partial differential equation for Y in the two variables (
=
kz and o
= kp(x, z)
f~ ~~ + 2 if sin
w sin
a
~~
+ 2 f cos w
~~~
+
~~
+ (l sin~
a + eq(o )) Y
=
0. (4)
do do d( do d(
As Fourier did for the heat equation, we seek at first elementary solutions for (4) in the form Y((, 61
=
e'"'y(61. (51
Then a general solution of (4) will be a sum or a series of elementary solutions of this form.
Y((, o) must be periodic with respect to o, hence y(o must be periodic. If we report (5) in (4)
we obtain an ordinary differential equation in y :
f~y" + 2 if (« cos
w + sin w sin a y' + (1 «~ sin~
a + eq(o )) y
=
0. (6)
The problem is now reduced to the research of the periodic solutions of this equation. It is a second order linear differential equation with periodic coefficients. Such equations are well-
known. For example, if the profile q(o ) is a sinusoidal function, (6) is a Matthieu equation ;
Burkhardt [3] has reduced the wave equation (I) in the same way and has then solved the
diffraction problem by using the mathematical theory of Matthieu equations. This method of resolution is known as modal theory. Rather than solving the equation (6) by one of the usual
ways of modal theory, we propose to expand the periodic solutions in powers of
e.
Now, the problem of the wave propagation through the grating is completely solved by the
knowledge of the complete family of periodic solutions of (6), and such periodic solution can
occur only for a discrete family of values of « (eigenvalues) (see [Iii.
If e = 0, the problem can be solved by explicit analytic formulae, but the diffracted beams
are then vanishing, In order to get a perturbative method of computation, we start from the
unperturbed equation : the eigenvalues and the corresponding periodic eigensolutions can then be expressed analytically ; after that we shall compute an asymptotic expansion in powers of
e
by iteration.
3. Perturbative expansion in the regular case.
A periodic solution of (6) can be sought in the form of a Fourier series : y(6
=
zR~ e'~° (7)
In terms of the Fourier coefficients, the equation (6) becomes :
[-n~f~-2nf(«cos
w +sin w sina)+ I(1-«~-sin~ a)]R~+
+w
+F I ~n-j~j"0. (8)
j=-w
From Floquet's theorem we know that two fundamental solutions are of the form :
Y(6)
=
e'P°R(o)
z(o
=
e'~° s(o ) (9)
where R(o) and S(o) are periodic functions and p, T two parameters (possibly complex-
valued) which depend on «, a, w, q, and e, A periodic solution exists if and only if one of the two parameters p and T is integral,
If e
=
0, the equation (6) has constant coefficients and the eigenvalues of
« associated with
periodic solutions are then
«(° ~ '
= fN cos w + N/I (fN sin
w + sin n )~
«(°
= fN cos w ~/I (fN sin
w + sin a )~
and the corresponding eigensolutions are
(0+ ,N0
YN ~ ~
If e is small but not vanishing, the periodic eigenfunctions and the corresponding eigenvalues
are perturbed. In, an expansion in powers of e, the term of order 0 will be one of the eigenvalues
or eigensolutions for
e = 0 given above.
If R (6 ) is an eigensolution associated with an eigenvalue «, it has a Fourier expansion with coefficients R~
+w
R(o
= I R~ e'~° (lo)
n=-w
We can obtain the eigensoiutions (7) by expansions of each R~ and win powers of e
~ £ ~k k
n ~ n ~
'"° (II)
W = I W~ F~
= W~ I W~ E~.
km0 k~'
+w
If R(o
= jj R~ e'~° is a periodic solution of (6) we have by identification of the Fourier
n ~
series :
jn~f~+2nf(«cosw+sinasinw)+«~+sin~a-I]R~=e jj+w q~R~_j
j=-~
where qj are the Fourier coefficients of the function q(o ). Combining this with (6) we obtain jn~ f~ + 2 nf(«~ cos w + sin a sin
w ) + ml + sin~
a I +
+ w
+ 2 («o + nf cos w z «~ eJ + z z «, «j e'~ ' z R$ e~ = z z qi R~Ii~ e~
j »' , ml j ml km o mm'
f
w
(12) To alleviate the notations we introduce
A(
=
n~ f~
+ 2 nf(«~ cos w + sin a sin w + ml + sin~
a
= («~ + nf cos w )~ + (sin a + nf sin w )~ l
A(
= («~ + nf cos w «i (we will see later that this is 0 ) (13)
and for f
m 2 :
I -,
Al
= 2(«o + nf cos w ) «I + z «~ «i
_~
j =1
so that (12) becomes now :
lm
I I ~i ~R~ ~~ +W
~ i i qjRilj~ ~~
mmU (=0 mmf j=-~
It is obvious that for
e =
0 we have R (o
= Cte e'~° for some integer N, because the equation
(6) has then constant coefficients, and therefore (if we normalize to Cte
= I)
~o_~ (0l ifif n~N;n=N.
In addition, it can be agreed that R(
=
0 for km I. because any other agreement about the
Rl's will have an effect only on the N-th-order term of the Fourier series and therefore it will be cancelled by the preceding normalization for R. By identification of the coefficients in the series of (6) :
~" ~
~ ° ' ~~~l
"
° ~i~~~
~
+ w
~°~ ~
~ ~~~$ ~ ~,' ~n
~
i ~j~~
j ~ ~n N ~~~~)
j w
k m i + w
f°T 'N
~ 2 Al R~ + A$ Rl + I Al ~R~
~
i qj Ri-
j~ (14~)
k =' j w
(14a) is satisfied since A$ = n~ f~ + 2 nf(«o cos w + sin
a sin w + ml + sin~
a I (which vanishes for n
=
N because «o is the eigenvalue associated to the eigenfunction e'~° for
F = 0) and R(
= 0 if
n ~ N.
In (14b) Al
= qo = 0 for n =N and if we replace Al by its expression we see that
«1 =
0 : this implies that A(
=
0 for all other values of n, and therefore A(R(
= q~ _~ for
n ~N. So we have
~
"1 " (15)
~i ~n ~
(for n ~ N
n A(
The general recursion formula for m m 2 is then obtained from (14c) :
. for n
= N (14c) gives :
k=m-i +w
~~
~ i ~~ ~~~ ~ i ~j ~~ -~
k=' 1=-~
+~
~j ~j ~m-1N-j
j=-~
(the second equality holds because R(
=
0 for km I) and then by replacing A( by its expression from (13)
+~ m-1
2 («o + Nf cos
w mm = I ~J ~i I "f "m ~~~~
j=-~ t=1
.'and for n ~ N
+w m-I
A$R7
" I qj R7Ij~ i Al ~~ R( (17)
J=-W k='
These recursion formulas permit the iterative computation of the sequences «~,
At and RQ. The asymptotic expansions obtained by this way can be used for numerical
computations if e is small (it must at least be appreciably less than I) and if the denominators
do not vanish. These denominators are «o + nf cos w for n
= N and A( for n ~N. The
denominator «o + Nf cos w vanishes if cos o~ = 0 and the denominator A( vanishes if the
Bragg condition is satisfied for any order n. Indeed we have from (6) : A$
= («o + nf cos w )~ + (sin a + nf sin w )~ l
,
and «o, the eigenvalue for
e = 0, is given by :
«o=-fNcosw± ~/I-(fNsinw +sina)~
=-fNcosw+coso~
(to avoid the problem of the sign of cos o~ we consider that o~ can take values between
ar/2 and w for the negative case). So we have
«o + nf cos w = cos o~ + (n N ) f cos w
sin a + nf sin w
= sin a + Nf sin
w + (n N ) f sin w
= sin o~ + (n N f sin w
and then
Al
= (n N t j(n N t + 2 cos (w 6~)1 (18)
As expected this vanishes for n =N for other values of n this van vanish only if 2 cos (w o~)/f is an integer, or in other words if the Bragg condition is satisfied.
The conditions of vanishing denominators («o + Nf cos w
=
0 as well as A
=
0) mean that two of the unperturbed eigenvalues become equal in forming one double eigenvalue. This
can be shown easily as follows. The unperturbed eigenvalues are
~(N ~ l
=
fN cos w + cos 6N
"/~ '
= fN cos w cos 6~
Obviously «(~ ~
=
«(~ if cos o~
=
0. For the condition A(
=
0 we need a short analysis
«(~ ~ «(~'~
= (N' N f cos w + cos o~ cos o~.
,
o~ o~ o~. + o~
~~ ~ ~ ~°~ ~° ~ ~ ~~~
2 ~°~ 2 '
but from sin 6,, = sin a + nf sin
w we have
(N'- N ) f sin
w =
sin o~ sin o~,
=
2 sin
~~'
~
~~
cos
~~' ~~
By combining the two preceding results we obtain
~ (N ~
~ (N'~1 2 6N' °N 6N' + 6~
° ° sin w '~~~ 2 ~°~ ~° 2
('9)
Now it appears that if «(~~ and «(~'~J
are equal for N~ N' then o~, = ± ar + 2 w o~. Since o~ and o~, both must be diffraction angles associated to the same incidence
angle, we must have in addition (N N') f sin w
=
sin (2 w o~ sin o~, = 2 sin w cos (w o~).
This implies that cos (w o~)
= integer. f/2. The same could be done for the eigenvalues
«(~~J Finally,
we have shown that two eigenvalues of the same family (the family
«(~ ~
or the family «(~
can become equal in forming one double eigenvalue only at Bragg
incidence. The condition cos (w o~ = integer f/2 indeed means that the Bragg resonance
occurs for the N-th diffracted beam. It could be shown that two eigenvalues belonging to
different families lone in the family «(~ ~ and the other in the family «(~ can become equal only if they are conjugate (I.e. if they have the same index and differ only by the sign of
cos o~) ; then cos o~
=
0.
The previous short analysis shows that the perturbative expansion will be divergent if the incidence angle is close to a Bragg angle. Then the denominators we pointed out above will be too small, and will be iterated at each step of the recursive procedure; therefore the corresponding coefficients will grow up rapidly, so that the whole perturbative series will be
JOUR~AL DE PH~SfOLE III T 4 N' 2 FEBRUARY >u94 j~
