601
Appendix 8: Rigorous Coupled Wave Theory
of Simple and Multiplexed Gratings
A8.1 Introduction
Moharam and Gaylord [1] were the rst to show how coupled wave (CW) theory could be formulated
without approximation. This led to a simple computational algorithm that could be used to solve the
wave equation exactly. Although earlier approaches such as the Modal method [2] were also rigorous,
they involved the solution of a transcendental equation for which a general unique algorithm could not
be dened. This came in contrast to the simple Eigen formulation proposed by Moharam and Gaylord.
Here we provide a derivation of the rigorous coupled wave (RCW) theory for the more general spatially
multiplexed grating. We shall then show how the resulting equations simplify to Moharam and Gaylord’s
equations for the simple phase grating. Rather than solving these equations using an Eigen method, we
employ an alternative approach using Runge–Kutta integration.
*
A8.2 Derivation of RCW Equations
For brevity, we shall limit discussions to the lossless case with isotropic permittivity
†
and we shall
employ the σ-polarisation for which the Helmholtz equation may be written:
∂
∂
+
∂
∂
−=
2
2
2
2
2
0
u
x
u
y
uγ (A8.1)
where u is the transverse (z) electric eld and the parameter
γββκ
µ
µ
µµ
22
1
2=−
−+
{}
⋅−⋅
=
∑
ee
ii
N
Kr Kr
(A8.2)
denes the multiplexed grating.
‡
Following the notation established in Chapter 12, we now consider the
case of illumination of the grating by a wave of the form
uy e
ikxky
xy
()
()
<=
+
0
(A8.3)
where
k
k
xc c
yc c
=−=
=−=
βθψβ
βθψβ
µµ
µµ
sin( )sin()
cos( )cos(
Φ
Φ ))
∀µ
(A8.4)
*
This may be programmed with exceptional ease using Mathematica from Wolfram Research Inc using the NDSolve
function.
†
In 1989, Glytis and Gaylord extended the RCW theory to anisotropic media and spatially multiplexed gratings.
‡
This is just the polychromatic version of Equation 12.107.
602 Appendix 8
In both the front region (y < 0) and the rear region (y < d), the average index is assumed to be n
0
.
Now, the Helmholtz eld u(x, y) may be consistently expanded as follows:
uxyuye
lll
ik lK lK x
l
xx x
(,) ... ()
...
( ...)
=
++ +
123
11 22
3321
123
3
=−∞
∞
=−∞
∞
=−∞
∞
=−
∑∑∑
=
ll
lll
ik x
l
uye
x
... ()
...
∞∞
∞
=−∞
∞
=−∞
∞
=
∑∑∑
∏
ll
il Kx
N
e
x
21
1
σσ
σ
(A8.5)
This expression may be substituted into Equations A8.1 and A8.2. On taking the Fourier transform and
applying orthogonality, we then arrive at the following RCW equations:
klKu
xx
N
llll
N
+
−
=
∑
σσ
σ
β
1
2
2
123
...
(() ()
...
...( ).
y
u
y
y
u
llll
llll
N
−
∂
∂
=
−
2
2
1
123
123
2βκ
σ
σ
.. . ...( )...
() ()
l
iK y
llll l
iK y
N
y
N
y
ye
uy
e
σ
σ
σ
+
{
+
−
123
1
}}
=
∑
σ 1
N
(A8.6)
A8.3 Simplification in the Case of Simple Non-Multiplexed Grating
For the case of the simple sinusoidal grating, the transformation
uy uye
ll
ik lK y
yy
()
ˆ
()
()
=
+
(A8.7)
reduces Equation A8.6 to the more usual form
∂
∂
++
∂
∂
=+ ++
2
2
2
2
ˆ
()
()
ˆ
()
()(
uy
y
ik lK
uy
y
klKkl
l
yy
l
xx y
KKuyuyuy
yl
ll
)
ˆ
()
ˆ
()
ˆ
()
22
11
2−
{}
−+
{}
−+
ββκ
(A8.8)
used by Moharam and Gaylord [1].
A8.4 Derivation of Boundary Conditions
In the zones in front of and behind the grating where κ
σ
= 0, Equations A8.6 is reduced to the simpler
constant index equations:
() ()
...
klKuy
xx
N
llll
N
+−
−
∂
=
∑
σσ
σ
β
22
1
2
123
uu
y
y
llll
N123
2
0
...
()
∂
=
(A8.9)
These equations dene which l modes can propagate in the exterior regions. They have simple solutions
of the form
uAe
llll ll ll
iklK
NN
xx
N
123123
22
1
... ...
()
=
∑
−+
=
β
σσ
σ
−− +
+
∑
=
y
llll
iklK
Be
N
xx
N
123
22
1
...
()
β
σσ
σ
y
(A8.10)
where the square roots are real for undamped propagation.
*
Accordingly, we may deduce that the front
solution comprising the illumination wave and any reected modes must be of the form
*
Note that there are modes that propagate inside the grating but which show damped propagation outside.
603Appendix 8
uxyeeue
ik x
iky
llll
l
i
x
x
N
N
(,) ...
...
=+
−
=−∞
∞
−
∑
β
22
123
ββ
σσ
σ
σσ
σ
22
1
1
−+
+
=
=
∑
∑
()klKy
ik lK
xx
N
xx
N
e
=−∞
∞
=−∞
∞
=−∞
∞
∑∑∑
x
lll
321
(A8.11)
Likewise, the rear solution comprising all transmitted modes must be of the form
uxyue
llll
l
iklK
N
N
xx
(,) ...
...
()
=
=−∞
∞
−+
∑
=
123
22
β
σσ
σ
11
1
3
N
xx
N
y
ik lK x
ll
e
∑
∑
+
=−∞
∞
=
∑
σσ
σ
221
=−∞
∞
=−∞
∞
∑∑
l
(A8.12)
By demanding continuity of the tangential electric eld and the tangential magnetic eld at the boundar-
ies y = 0 and y = d, we may now use these expressions to dene the boundary conditions required for a
solution of Equation A8.1 within the multiplexed grating. At the front surface, these are
iku
du
dy
iklK
x
y
x
β
β
22
000
000
0
2
1
20−− =
−−+
=
(())
(
...
...
1122
2
0
123
123
0
xx ll l
lll
y
lK u
du
dy
++ =
=
...) ()
...
...
(A8.13)
And at the rear surface, they take the form
iklK lK ud
du
xx xlll
lll
β
2
11 22
2
123
123
−+ ++ =()()
...
...
ddy
yd
=
(A8.14)
The modes available for external (undamped) propagation are calculated using the condition
β
2
> (k
x
+ l
1
K
1x
+ l
2
K
2x
+ …)
2
(A8.15)
Note, however, that there are internal propagating modes that nevertheless do not propagate outside the
grating, and these must be retained.
A8.5 Numerical Solution of RCW Equations
Moharam and Gaylord [1] solved the single grating equations (Equation A8.8) using a state-space for-
mulation in which solutions are obtainable through the eigenvalues and eigenvectors of an easily dened
coefcient matrix. However, as mentioned above, we can also solve the more general equations (Equation
A8.6), subject to the boundary conditions in Equations A8.13 and A8.14, using simple Runge–Kutta inte-
gration. This is a practical method as long as the number of component gratings within the multiplexed
grating is relatively small.
Diffraction efciencies of the various modes are dened as
η
β
lll
xx xx
y
ll
klKlKlK
k
u
123 1
2
11 22 33
2
...
( ...)
=
−+ +++
223 123
llll
u
... ...
*
(A8.16)
where the elds in this equation are dened either at the front boundary in the case of reected modes or
at the rear boundary in the case of transmitted modes. Note that we are treating the lossless case here, so
the sum of all transmitted and reected efciencies totals to unity.
*
*
In the case of the front reected 000... mode, one uses
η
β
000
22
000 000
11
... ... ...
()
()
=
−
−−
k
k
uu
x
y
*
.
604 Appendix 8
A8.5.1 Comparison of Kogelnik’s Theory and PSM Theory with RCW Theory
Equation A8.6, subject to the boundary conditions in Equations A8.13 and A8.14, is solved by Runge–
Kutta integration. This permits the rigorous calculation of the diffraction efciencies of all modes, which
are produced by a general grating. Figure A8.1 shows an example for a simple reection grating and a
simple transmission grating at Bragg resonance.
*
In the case of the reection grating, a very high index
modulation has been assumed. Nevertheless, the Kogelnik/parallel stacked mirror (PSM) estimation is
still only 20% out, and it is clear that most of the “dynamics” of the grating are associated with the +1
reected mode as both PSM and Kogelnik’s CW theories assume. In the case of the transmission holo-
gram, a relatively high index modulation is assumed and also a large incidence angle with respect to the
grating planes. Here we see again only a small departure from the Kogelnik/PSM estimation but also the
presence of the +2 mode.
A8.5.2 Comparison of N-PSM Theory with RCW Theory
In order to compare the N-PSM model [4] developed in Chapter 12 with the RCW theory, we investigate
the typical spatially multiplexed grating, which is illustrated in Figure A8.2. This grating is composed
of two simple gratings that have been sequentially recorded in the same material using the same laser
wavelength of 532 nm and the same incidence angle of 30°. The component gratings have differing slants
and different grating constants and give rise to the multiplexed grating structure shown in Figure A8.2c.
In Figure A8.3, the diffraction efciency at Bragg resonance as determined by the N-PSM model
†
is compared for different grating thicknesses and different index modulations to an RCW calculation.
Typically, 14 modes are retained in the Runge–Kutta integration, the higher order modes being many
orders of magnitude smaller than the lower ones. Only the reected modes are plotted. Figure A8.3a
shows an extreme case where an index modulation of n
1
= 0.3 for each of the two component gratings of
Figure A8.2c is assumed. It is clear that in this case, there are quite important differences between the
N-PSM model and the RCW model. In addition, quite a few of the higher order modes allowed under
the RCW theory start to oscillate at non-negligible amplitudes, including modes that require both grat-
ings present to propagate. However, this is indeed an extreme case, and one might have anticipated this
from the results of Moharam and Gaylord [1]. As the index modulation of each grating drops in Figure
A8.3b through d, we see a better and better agreement between the two models. Figure A8.3d represents
*
Note that at Bragg resonance, the PSM and Kogelnik models give the same predictions.
†
Conventional N-CW theory [5] gives an identical prediction to N-PSM at Bragg resonance here.
0 1 2 3 4 5
0.2
0.4
0.6
0.8
1.0
0 1 2 3 4 5
0.2
0.4
0.6
0.8
1.0
PSM /Kogelnik
PSM /Kogelnik
+1 (Reflected)
+1 (Transmitted)
0 (Reflected)
0 (Transmitted)
-1 (Transmitted)
0 (Transmitted)
+1 (Transmitted)
+2 (Transmitted)
-1 (Transmitted)
0
+1
-1
0
-1
+1
0
-1
+1
+2
d/Λ
d/Λ
ηη
(a) (b)
FIGURE A8.1 Diffraction efciency (η
σ
) versus normalised grating thickness according to RCW theory and compared
to the PSM and Kogelnik theories at Bragg resonance for (a) the simple reection grating (n
0
= 1.5, n
1
/n
0
= 0.331/2, θ
c
= θ
r
=
50°, ψ = 30°, λ
c
= λ
r
= 532 nm) and (b) the simple transmission grating (n
0
= 1.5, n
1
/n
0
= 0.121/2, θ
c
= θ
r
= 80°, ψ = 60°,
λ
c
= λ
r
= 532 nm).
605Appendix 8
a typical multiplexed grating made using a modern material such as photopolymer or dichromated gela-
tine. The N-PSM and RCW models therefore produce extremely good agreement here, and higher order
modes only account for less than 1% of the total diffraction.
In Figure A8.4, a comparison of the N-PSM model and the RCW theory for the off-Bragg case is pre-
sented. In particular, the diffractive efciency of the multiplexed grating of Figure A8.2c is investigated
as the replay wavelength is changed whilst keeping the replay angle of incidence xed at 30°. Although
a complex analytic solution of the N-PSM equations is available for this problem, for pure convenience,
one can solve the N-PSM equations numerically using Runge–Kutta integration. In Figure A8.4a, an
index modulation for each of the component gratings of n
1
= 0.03 is used; this would be rather typical
for diffractive elements made from photopolymer or dichromated gelatin. Agreement between the two
theories is clearly excellent, particularly in the primary diffractive band. In Figure A8.4b and c, we plot
for comparison the N-PSM diffractive efciencies when one or another of the two component gratings is
removed from the multiplexed element.
It is interesting to note that the diffractive efciency does not peak in Figure A8.4a for both compo-
nents at the Bragg angle. This is because the dominant grating within the multiplexed element depletes
the reference wave disproportionately around resonance. As the wavelength is changed away from reso-
nance, there rapidly becomes more reference wave available within the grating for signal generation by
the second component grating. Even though the intrinsic efciency of this second grating drops away
from Bragg resonance, the increased reference wave left by the rst grating more than compensates for
this, leading to the characteristic hollow curve with symmetric peaks away from resonance.
Figure A8.4d repeats the case of Figure A8.4a but with individual index modulations of n
1
= 0.15.
Here the differences between the N-PSM and RCW solutions are somewhat larger as might be expected.
(a)
(b)
(c)
FIGURE A8.2 An example of a spatially multiplexed phase reection grating. The grating, whose (x,y) index distribution
is shown in (c) is formed by the sequential recording of the two simple gratings shown in (a) and (b). Each diagram shows a
section of size 0.5 μm by 2 μm. Each simple grating has been recorded with a reference beam angle of Φ
c
= 30° and with a
wavelength of 532 nm. One grating has a slope of ψ
1
= 20° and the other has a slope of ψ
2
= −20°. Note that the form of the
multiplexed grating in (c) is fundamentally different from the characteristic linear shape of its component simple gratings
of (a) and (b). Note also that identical index modulations for each of the two component gratings have been assumed in (c).
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