Abstract . Diffraction and absorption losses of plasmon surface waves (PSW) propagating along a metallic grating are investigated numerically as a function of groove depth . A periodicity of diffraction losses is found to exist . The energy flow distribution (EFD) above and inside the grooves is calculated and a similarity between the PSW on shallow and deep gratings is established above the grooves, while inside the grooves of deep gratings totally hidden curls in EFD are found to form . . Introduction Recently it has been discovered [1] that a close connection exists between different types of phenomena on metallic gratings : plasmon surface waves (PSW) excitation, non-Littrow perfect blazing  It is well known that a pole of the scattering matrix corresponds to a solution of the homogeneous problem  where nM is the complex refractive index of the substrate . For highly conducting metals Re (aP) &gt; 1 and Im (aP) &gt; 0, the latter corresponding to the energy absorbed in the metal as the PSW propagates along the interface . As the periodic modulation is introduced (h ;0), the PSW may be coupled to a propagating diffraction order(s) in the upper medium provided a suitable wavelength to period ratio 2/d is chosen . Radiation losses appear as a consequence of this and Im (aP) grows rather rapidly (for the results presented in figure 1 d=0. 5 µm and 2=0 . 6328 gm) 

D Maystre

E Popov

L Tsonev

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JOURNAL OF MODERN OPTICS, 1990, VOL . 37, NO . 3, 379-387
Losses of plasmon surface waves on metallic grating
E . POPOV, L. TSONEV
Institute of Solid State Physics, Bulgarian Academy of Sciences,
blvd. Lenin 72, Sofia 1784, Bulgaria
and D. MAYSTRE
Laboratoire d'Optique Electromagnetique, Faculte des Sciences et
Techniques de Saint-Jerome, Universite d'Aix-Marseille III, Avenue
Escadrille Normandie-Niemen, F13397 Marseille Cedex 13, France
(Received 18 May 1989 and accepted 29 July 1989)
Abstract . Diffraction and absorption losses of plasmon surface waves (PSW)
propagating along a metallic grating are investigated numerically as a function of
groove depth . A periodicity of diffraction losses is found to exist . The energy flow
distribution (EFD) above and inside the grooves is calculated and a similarity
between the PSW on shallow and deep gratings is established above the grooves,
while inside the grooves of deep gratings totally hidden curls in EFD are found to
form .
1 . Introduction
Recently it has been discovered [1] that a close connection exists between
different types of phenomena on metallic gratings : plasmon surface waves (PSW)
excitation, non-Littrow perfect blazing [2] and total absorption of light at grazing
incidence [3] can all be attributed to certain parts of the same curve ; the trajectory
describing pole/zero of the scattering matrix as a function of groove depth h in the
complex plane of ka (ka is the projection of the incident wavevector k on the grating
plane and k=1k1=21t/)., where A is the wavelength) . A detailed tracing of this
trajectory is shown in figure 1 .
It is well known that a pole of the scattering matrix corresponds to a solution of
the homogeneous problem [4] . For a plane metal-air boundary (h=0) such a
solution, referred to as PSW, exists for TM polarization and has a complex
propagation constant:
( 1 )aP	
nM
=
(1 +nM) 1 J2 '
where nM is the complex refractive index of the substrate . For highly conducting
metals Re (aP) > 1 and Im (aP) > 0, the latter corresponding to the energy absorbed in
the metal as the PSW propagates along the interface .
As the periodic modulation is introduced (h ;0), the PSW may be coupled to
a propagating diffraction order(s) in the upper medium provided a suitable
wavelength to period ratio 2/d is chosen . Radiation losses appear as a consequence of
this and Im (aP) grows rather rapidly (for the results presented in figure 1 d=0.5µm
and 2=0.6328 gm) .
0950-0340/90 $3.00 © 1990 Taylor & Francis Ltd .
380
	
E. Popov et al .
0-09
-0 . 02
E
-0 . 13
a
(b)
Figure 1 . (a) Complex a-plane trajectory of pole/zero of the scattering matrix for a sinusoidal
aluminium grating with period d=0 . 5 µm at light wavelength 1=0 .6328µm. (b) x15
magnification of (a) in the vicinity of the point a=1 (reference [6]) .
Further increase of h turns the trajectory towards the line Re (a) = 1 . When the
pole crosses this line, it is transformed into a zero of the zeroth reflection order
(ZZRO) [4, 5] . As the grating becomes deeper and deeper, the trajectory of ZZRO
crosses the real a-axis at two points . The first one, aN • corresponds to the non-
Littrow perfect blazing [2] and the second, a G, to the grazing total absorption of light
[3] . Further growth of h causes a closing of the first loop in the trajectory and a
formation of a second loop above the real axis . When Re (ap) becomes greater than
unity, ZZRO is transformed back into a pole and a new branch of the dispersion
relation solution appears .
0-75 0 . 90 1-05
Re( oC )
(a)
Losses of PSWs on metallic gratings
	
381
As a consequence of this tracing, not only is a connection between some already
known phenomena established, but it also becomes possible to predict a new
anomaly-total absorption of light [6] and field enhancement [7] in deep and very
deep gratings . What remained was to find a reason (or physical interpretation) for the
existence of these loops in the pole/zero trajectory . The aim of this paper is to show
that the loops are due to the quasiperiodicity of the radiation losses as a function of
the groove depth. As a first step it is necessary to distinguish between the two types of
losses that exist for metallic gratings, namely diffraction (radiation) and absorption
losses .
2 . Separation of losses
As the PSW propagates along the grating surface, part of the energy it carries can
be radiated into the upper lossless medium and the other part is absorbed in the
metallic substrate . To separate these two types of losses it is necessary to calculate
(a) the energy flow passing through one period of the grating surface (Ie in
figure 2), and
(b) the energy flow going away from the grating (Id in figure 2) .
To attain accuracy, it is worth noting that a solution of the homogeneous rather
than the inhomogeneous problem is required, i .e. diffraction orders amplitudes have
to be calculated as a function of one of the orders (let us say the 0th order) without any
incident wave . This situation implies the existence of a pole of the scattering matrix .
Consequently the parameters of the system must ensure that such a solutuion of the
homogeneous problem exists . Once this requirement is fulfilled, we may set the
amplitude of the zeroth diffracted order equal to unity and calculate the others . Next,
normalization to unit energy flow through an infinite cross-section is necessary .
E
Figure 2 . General energy flow scheme for a plasmon surface wave propagating along the
grating surface .
382
	
E. Popov et al .
Let us consider the energy balance in the region ACDBFE of figure 2, including
one groove. Lines AC and BD are taken to be parallel to the radiation direction . The
total incident energy flow Ii reaching the region CAE along the x-direction has two
parts: Ii., is the flow through line AC in the upper lossless medium and Is describes
the incident energy flow in the absorbing substrate .
Numerical results, however, show that Iis is 10 3-105 times smaller than Iii and in
further discussions we shall ignore the infleunce of Ii. The incident energy flow I,
penetrates the region examined through the line AC . One part of it
I ; v, = Ii. exp [ - 2k Im (aP) d ]
is then simply transmitted over the next groove through the line BD . Another part Id
propagates through the line CD away from the grating . The third part Ie crosses the
grating surface and is absorbed in the metallic substrate . The energy flow Ie is equal
to the energy flow IAB through a straight line connecting the tops of the grooves, since
the upper medium is lossless . As AC -+oo, then Id tends towards a constant value
corresponding to the energy radiated by one groove . As there are no losses inside
ABCD, the following relation exists :
Iii-re. =Id +Ia =lip{1-exp[-2kIm(aP)d]} .
One can define relative diffraction and absorption losses as two ratios :
Figure 3 .
0
0
3. Diffraction (radiation) losses
A numerical code based on the differential formalism of Chandezon et al . [8] was
used according to the scheme of section 2 . An aluminium grating (refractive index
nM =1 .378+i7 .616) with a sinusoidal profile was considered . Figure 3 presents the
0 . 06
C:) O 3
E
0 . 65
h /d
Groove depth dependence of the imaginary part of a" : solid line-total imaginary
part, dashed line-diffraction part, and dotted line-absorption part .
1 . 3
rd 'd
Id
(2)
ra
Ii.-A.
la
Id + Ie '
Ia
(3)
Ii.-r. Id +Ia
Losses of PSWs on metallic gratings
	
383
groove depth dependence of the imaginary part of the pole aP and the contributions of
the diffraction ad =rd Im (aP) and absorption as=ra Im (ccP) parts . There is a gap in
their groove depth dependences, as we are only dealing with the case of Im (a) > 0 .
It must be pointed out that Im (c") is proportional to the losses: 2k Im (aP) is equal
to the decay constant of the plasmon . Hence in the following we speak of `losses'
rather than `imaginary parts' .
Absorption losses are almost independent of h. To be more precise, their
contribution to the groove depth dependence of total losses is quite small. A
quasiperiodicity of Im [aP(h)] and a d (h) is obvious and their minima appear almost
for the same groove depth values as the zeros of the -1st order diffraction efficiency
in the Littrow mounting (figure 4). Moreover, when the -1st order efficiency is zero
fot the Littrow mounting it is negligible in the entire region of angles of incidence
(the so-called 'antiblazing' of gratings [9]) . As can be concluded from the results
presented in figure 4, this general property is also valid for radiation of the PSW
propagating along a grating .
As is shown in the preceding paper [10], the existence of such an equivalence
between the properties of the flat surface and of deep gratings is due to a successive
formation of one or more energy flow curls in each groove when the grating depth
increases . When these curls are completely hidden inside the grooves the energy flow
distribution above the grooves becomes similar to the distribution above the flat
surface. Figures 5 and 6 represent the pictures of energy flow lines (locally tangential
to the direction of the Poynting vector) for two different values of groove depth :
0.058 pm and 0.395 gm, respectively . In these cases the PSW propagation constants
aP have almost equal imaginary parts . The energy flow distribution above the
grooves is almost identical for the two gratings : near the surface the lines are almost
parallel to the grating corresponding to the physical fact that the PSW propagates
1 . 0
0
0 0 . 65 1 . 3
h/d
Figure 4 . Groove depth dependence of the -1st order diffraction characteristics : solid
line-Littrow mounting efficiency (a=0 . 6328), dashed line-grazing incidence effici-
ency (a=0 . 99), dotted line-relative diffraction losses (r d) .
384
	
E. Popov et al .
from left to right . A small number of lines finish on the surface, leading to absorption
losses. Some of the upper lines turn away from the surface, corresponding to
radiation losses. As a result of both types of loss, the guided-wave amplitude
decreases as it propagates along the grating ; in terms of energy flow this fact is
represented by a decrease in the density of the lines .
The main difference between figures 5 and 6 can be found inside the grooves since
curls exist in each groove of a deep grating . Although some of the flow lines reach the
surface, there are closed curves in the deep grooves that separate the bottoms of the
grooves from the energy flow above the tops. This fact can explain why absorption
losses for deep gratings may take lower values than for shallow gratings and even for
flat surfaces (figure 3) .
8d
4d
0
d/5
0
0 d/2
(b)
d
Figure 5 . Energy flow distribution above the surface of a shallow grating of period 0 .5 gm
and groove depth 0 .058 gm: (a) above the grating, (b) inside the groove .
d
0
0
	
d/2 d
(b)
Figure 6 . Energy flow distribution above the surface of a deep grating of period 0 . 5 µm and
groove depth 0 . 395 µm: (a) above the grating, (b) inside the groove .
4. Absorption losses
Let us first consider shallow gratings (figures 5 and 7). For a flat surface the
direction of energy flow is almost parallel to the surface, consequently there are only
small losses . By increasing h, the flow lines are pulled up by the groove tops and
above them the power density of the electromagnetic field increases . Inside the
grooves the lines are spaced out and the field density decreases . Nevertheless, the
8d
4d
0
Losses of PSWs on metallic gratings 385
0 4d
(a)
8d
386
	
E. Popov et al .
value of the normalized average field density over one period increases with h as does
the grating surface area. Comparing the behaviour of the field density and the grating
surface area with absorption losses, the following conclusion can be drawn : the
increase of absorption losses is similar to the increase of the average field density and
is almost an order of magnitude greater than the growth in the area of the absorbing
surface. This has been confirmed in the case of deep gratings (figure 8) : A strong
a)
w
0 . 1
h /d
Figure 7 . Increase of absorption losses as a function of groove depth compared with the
change of field density on the top, in the middle and at the bottom of the groove : shallow
grating . Increase of grating surface area is also shown .
5 . 0
0
0 .7
0
1 .3
0 . 016
0
0 . 2
0
0
1 . 0
h /d
Figure 8 . Absorption losses compared with field density values on the top, in the middle and
at the bottom of the groove as a function of modulation depth for a deep grating .
Losses of PSWs on metallic gratings
	
387
connection and good correspondence between absorption losses and field density
behaviour on the top, at the bottom and in the middle of a groove can be seen in figure
8 . Moreover, under these conditions the absence of a strong connection between
absorption losses and grating surface area becomes evident--deep gratings (e.g .
h/d=0.72) can have only one half the absorption of a flat surface (figure 3) .
Acknowledgement
This work is completed under the financial support of the Ministry of Culture,
Science and Education under contract 648 .
References
[1] Popov, E., MASHEV, L ., and LOEWEN, E ., 1989, Appl. Optics, 28, 970 .
[2] MAYSTRE, D., CADILHAC, M., and CHANDEZON, J., 1981, Optica Acta, 28, 457 .
[3] MASHEV, L ., Popov, E., and LOEWEN, E ., 1988, Appl. Optics, 27,152 .
[4] See for example NEVIERE, M ., 1980, Electromagnetic Theory of Gratings, edited by
R. Petit (Berlin : Springer-Verlag), chap . 5 .
[5] MAYSTRE,D ., 1982, Electromagnetic Surface Modes, edited by D. Boardman (New York :
Wiley), chap . 17 .
[6] MASHEV, L ., Popov, E., and LOEWEN, E ., 1989, Appl. Optics, 28, 2538 .
[7] Popov, E., and TSONEV, L., 1989, Optics Commun ., 69, 193 .
[8] CHANDEZON, J., Dupuis, M, T., CORNET, G., and MAYSTRE, D., 1982, J. opt . Soc . Am .,
72,839 .
[9] MASHEV, L ., Popov, E., and MAYSTRE, D., 1988, Opt. Commun ., 67, 321 .
[10] Popov, E., TSONEV, L ., and MAYSTRE, D., J. mod. Optics (to be published) .


Losses of plasmon surface waves on metallic grating

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Losses of plasmon surface waves on metallic grating

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