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At the exit surface of a photonic crystal, the intensity of the diffracted wave can be periodically modulated, showing a maximum in the "positive" (forward diffracted) or in the "negative" (diffracted) direction, depending on the

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a r X i v : 0 8 0 4 . 1 7 0 1 v 1 [ p h y s i c s . o p t i c s ] 1 0 A p r 2 0 0 8
Pendell ¨osung effect in photonic crystals
S. Savo
1
, E. Di Gennaro
1
, C. Miletto
1
, A. Andreone
1
, P. Dardano
2
, L.Moretti
2
,
3
, V. Mocella
2
1
CNISM and Department of Physics, Universit
`
a di Napoli “Federico II”, Piazzale Tecchio 80, I-80125 Naples,ITALY
2
IMM - CNR, Sezione di Napoli, via P. Castellino 111,I-80131 Naples, ITALY
3
DIMET - University ”Mediterranea” of Reggio Calabria Localit Feo di Vito, I-89100 Reggio Calabria (Italy)emiliano.digennaro@na.infn.it
Abstract:
At the exit surface of a photonic crystal, the intensity of the diffracted wave can be periodically modulated, showing a maximumin the ”positive” (forward diffracted) or in the ”negative” (diffracted)direction, depending on the slab thickness. This thickness dependence isa direct result of the so-called Pendell¨osung phenomenon, consisting of the periodic exchange inside the crystal of the energy between direct anddiffracted beams. We report the experimental observation of this effect inthe microwave region at about 14
GHz
by irradiating 2D photonic crystalslabs of different thickness and detecting the intensity distribution of theelectromagnetic ﬁeld at the exit surface and inside the crystal itself.
© 2008 Optical Society of America
OCIS codes:
(050.1960) Diffraction theory; (260.2110) Electromagnetic theory; (290.4210)Multiple scattering; (999.9999) Photonic crystal
References and links
1. P. P. Ewald, “Zur Theorie der Interferenzen der R¨ontgenstrahlen,” Physik Z.
14
, 465 (1913).2. P. P. Ewald, “Crystal optics for visible light and X rays,” Rev. Mod. Phys.
37
, 46 (1965).3. A. Authier,
Dynamical Theory of X-ray Diffraction
(Oxford University Press,Oxford, 2001).4. N. Kato and A. R. Lang, “The projection topograph: a new method in X-ray diffraction microradiography,” ActaCryst.
12
, 249 (1959).5. D. Sippel, K. Kleinstu, and G. E. R. Schulze, “Pendell¨osungs-interferenzen mit thermischen neutronen an Si-einkristallen,” Phys. Lett.
14
, 174 (1965).6. C. G. Shull, “Observation of Pendell¨osung Fringe Structure in Neutron Diffraction,” Phys. Rev. Lett.
21
, 1585(1968).7. V. Mocella, J. Hrtwig, J. Baruchel, and A. Mazuelas, “Inﬂuence of the transverse and longitudinal coherence indynamical theory of X-ray diffraction,” J. Phys. D: Appl.Phys.
32
, A88 (1999).8. M. Born,
Athomtheorie des festen Zustandes
(Springer, Berlin, 1923).9. V. Mocella, “Negative refraction in Photonic Crystals: thickness dependence and Pendell¨osung phenomenon,”Opt. Express
13
, 1361 (2005).10. A. Balestreri, L. C. Andreani, and M. Agio, “Optical properties and diffraction effects in opal photonic crystals,”Phys. Rev. E
74
, 036,603 (2006).11. O. Francescangeli, S. Melone, and R. D. Leo, “Dynamical diffraction of microwaves by periodic dielectric me-dia,” Phys. Rev. A
40
, 4988 (1989).12. M. L. Calvo, P. Cheben, O. Martnez-Matos, F. del Monte, and J. A. Rodrigo, “Experimental Detection of theOptical Pendell¨osung Effect,” Phys. Rev. Lett.
97
, 084,801 (2006).13. V. Mocella, P. Dardano, L. Moretti, and I. Rendina, “A polarizing beam splitter using negative refraction of photonic crystals,” Opt. Express
13
, 7699 (2005).14. N. Kato, “The determination of structure factors by means of Pendell¨osung fringes,” Acta Cryst.
A25
, 119 (1969)
1. Introduction
Since the srcinal formulationof the diffractiontheory from Ewald [1], the Pendell¨osung effect
was predicted as a periodic exchange of energy between interfering wave-ﬁelds. The Germanterm comes from the formal analogy between the mechanical system composed by coupledpendula and the optical problem, where many waves contribute to the optical ﬁeld. In thisformal analogy pendulum is the counterpart of wave whereas the temporal dependence of themechanical problem corresponds to the spatial dependence in the considered optical problem[2]. Pendell¨osung is a relatively well known effect of Dynamical Diffraction Theory (DDT), a
rigorousformalismaccountingfor multiplescatteringeffectsthat are especiallyimportantin X-ray, electron and neutron diffraction from perfect crystals [3]. The requirement of high qualitycrystals explains why the ﬁrst experimental observation of the Pendell¨osung effect has beenobtainedin1959onlyinX-raymeasurements[4],andsomeyearslaterinneutrondiffraction[5,
6]. Recently, using the coherence of third generation synchrotron beams, Pendell¨osung fringesproduced by a plane wave exiting a Si crystal have been recorded [7].Photonic crystals (PhCs) are artiﬁcial periodic structures reproducing natural crystals at dif-ferent length scale. PhCs can control and manipulate the ﬂow of light in many different ways,since they exhibit a variety of properties, spanning from full photonic band gap to anomalousdispersion phenomena,includingsuperprism and negativerefraction effects. The wide range of characteristics shown by PhCs gave rise in the last decade to a multitude of new ideas for opto-electronic integrated devices and systems. Novel concepts of mirrors, waveguides, resonators,and frequency converters based on photonic crystals, to mention a few examples, have beenproposed. Indeed, the band theory of the electrons in solids, that is usually the main referenceof the PhC theory, is strongly inspired by DDT (as extensively discussed in the pioneeringBorn’s solid state textbook [8]), that represents one of the ﬁrst example of two-state theory -later became very popular in modern physics - such as up and down spins, electron and holepairs, etc.It is not surprising therefore that the Pendell¨osung effect has been predicted for photoniccrystals too. In 2D case, it has been thoroughly studied using both analytical and numericalmethods as a function of the PhC contrast index, beam incident angle, and light polarization[9]. Moreover,this studyhas beenextendedto opal3D photoniccrystals, wherethe dependenceof diffraction intensity as a function of the layers number has been investigated using a scatte-ring matrix approach [10]. On the experimental side the properties of microwave diffraction inperiodic structures have been reported in literature by measuring the pattern of backscatteredwaves in two dimensional artiﬁcial dielectric media [11]. Recently, the Pendell¨osung effect has
been detected also in the optical regime in volume holographic gratings, observing the oscilla-tory behavior of the angular selectivity of the diffracted light [12].In this work, we present an accurate theoretical study and precise measurements of the Pen-dell¨osungeffect in photoniccrystals by illuminatingwith a plane waves beam in the microwaveregion 2D square lattice PhCs having different number of rows (i.e. slabs with different thick-nesses) anddetectingthe electromagneticﬁeld insideandoutsideeachslab.We showthatunderparticular conditions the intensity of the diffracted wave at the exit surface can be periodicallymodulatedwith the slab thickness, presentinga maximumin the ”positive”(forwarddiffracted)or in the ”negative” (diffracted) direction. Moreover, we observe that inside the crystal the en-ergy is periodically exchanged between direct and diffracted beams.
2. Theoretical analysis
The Pendell¨osung effect in PhCs can be understood as a beating phenomenon due to the phasemodulationbetweencoexistingplanewave components,propagatingin the same direction.Thecoexistence is possible because such wavevectors are associated to two adjacent bands that are
overlapped, for a given frequency, in correspondence of suitably chosen PhC parameters.
λ
Μ
Γ
ω
π
λ )
Γ
Fig. 1. The band structure of the square-lattice PhC for the
TE
polarization. The red linerepresents the normalized frequency
ω
n
=
0
.
722 at which the Pendell¨osung effect takesplace (colour online).
In our case the 2D PhC consists of dielectric cylinders in air (dielectric permittivity
ε
r
=
8
.
6)arranged in a square geometry and having
r
/
a
=
0
.
255, where
r
is the cylinder radius and
a
isthe lattice constant. If
TE
polarization (electric ﬁeld parallel to the rods axis) is considered, anoverlap occurs between the forth and the ﬁfth mode for a normalized frequency
ω
n
=
ν
a
/
c
=
a
/
λ
=
0
.
722, as shown in Fig.1. Moreover, the crystal orientation is ﬁxed such that the normalat its surface is along the
XM
direction. Hence all possible wavevectors excited into the PhCwill have the same tangential component lying on
XM
.The Pendell¨osung phenomenon is analyzed in this context for an incident wavevector thatsatisﬁes the Bragg law [9, 13]. In Fig.2 we show in the reciprocal space the ﬁrst Brillouin zone
and the corresponding symmetry points for the square lattice PhC under study. Consideringthe reciprocal lattice vector that enforces the momentum conservation oriented along
Γ
X
in theﬁrst Brillouin zone, the Bragg law is fulﬁlled when the projection of the incident wavevectorcoincides with
Γ
X
, so that
k
h
is the diffracted wavevector whereas
k
i
is the incident one. Usingthe dispersion surfaces (or Equi-FrequencySurfaces, EFSs), that represent the loci of propagat-ing wavevectors for a ﬁxed frequency, we are then able to evaluate the relevant parameters of the beating effect. The wavevectors inside the PhC are determined by the intersection betweeneach EFS and the
XM
direction. Amongst the different intersections, only wavevectors havinggroup velocity oriented inside the crystal - in Fig. 2 in opposite direction respect to the externalnormal to the incident surface - will be effectively excited.Consider for instance the contribution of the incident wave: there is an interference betweentwo excited components, with the respective wavevectors pointing in two different directions.This produces a spatial periodic modulation along the wavevectors difference vector
∆
k
. Themodulation distance in the real space along the PhC normal direction is therefore
Λ
0
=
2
π
/
∆
k
.The same effect occurs also for the diffracted wave, giving rise to a spatial modulation with thesame length but 180
◦
out-of-phase in respect to the previous case.
Fig. 2. The reciprocal space with the ﬁrst Brillouin zone (dotted line) and symmetry pointsfor the square-lattice PhC. The contours for the normalized frequency
ω
n
=
0
.
722 are plot-ted. Arrows indicate the directions of group velocity
v
g
, whereas
ˆn
shows the normal to theincident surface (colour online).
As a consequence of the Pendell¨osung effect, the intensity
I
at the exit surface is harmon-ically modulated as a function of the thickness
t
[9]. When
t
is an even multiple of half thePendell¨osung distance, the beam at the exit surface is parallel to the incident beam, forming apositive angle respect to the PhC normal. On the other hand, when
t
is an odd multiple of
Λ
0
the beamat the exit surface is completelydirectedalongthe Bragg diffracteddirection,forminga negative angle respect to the PhC normal. Denoting by + and - the two possible directions atthe exit surface, this is summarized by:
t
=
2
m
Λ
2
⇒
max
(
I
+
)
t
= (
2
m
−
1
)
Λ
2
⇒
max
(
I
−
)
(1)where
m
=
1
,
2
,...
.Forcing the Pendell¨osung distance
Λ
0
be an even number of the lattice constant
a
, eq. (1) holdsfor any number
n
of PhC rows. In particular,
Λ
0
=
4
a
ensures that the intensity maxima of theexit waves changes periodically if
n
is even, and that the energy beam equally splits betweenpositive and negative direction if
n
is odd.From the EFSs analysis, assuming a TE polarization, we found that an angle
θ
i
=
43
.
8
◦
anda normalized frequency
ω
n
=
0
.
722 for the incident wave satisfy both the Bragg law and thepeculiar condition
Λ
0
=
4
a
.
3. Experimental setup
The experimentalresults are obtainedon2D PhCs havinga differentnumberofrows insertedina waveguide. First, the electromagnetic wave transmitted by the periodic structure is measuredat the exit of the PhC for different crystal thickness and its spatial distribution is shown. Then,the periodic modulation of the intensity of the diffracted waves with respect to
n
is reported.
Finally, a comparison along selected directions inside the photonic crystal between the elec-tric ﬁeld distribution measured and simulated using a Finite Difference Time Domain (FDTD)method is presented.Measurements are carried out by placing alumina rods with nominal permittivity
ε
r
=
8
.
6,radius
r
=
0
.
4
cm
and height
h
=
1
cm
in a square geometry with
r
/
a
=
0
.
255 (
a
=
1
.
57
cm
)sandwiched in an aluminum parallel-plate waveguide terminated with microwave absorbers.Since the loss tangent of alumina is extremely small at the frequency relevant for this work (tan
δ
<
10
−
4
), dielectric losses can be neglected. Due to the presence of metallic plates actingas mirrors, current lines that are perpendicularto the plates can be consideredas inﬁnitely long,as stated by the well-known mirror theorem. For the same reason the electric ﬁelds producedby these currents are constant along the same direction and thus the whole system acts as a 2Dstructure.The microwavephotoniccrystal is built in the shape of a 38
.
5
cm
wide slab (25 rod columns),with a thicknessthatcan bevariedaddingor removingrows.A dipoleantennais usedas source,oriented to produce an electric ﬁeld parallel to the rods axis and operating at the frequency of 13
.
784
GHz
,in orderto reproducethe same normalizedfrequency
a
/
λ
ofthe theoreticalmodel.Due to the waveguide characteristics, the TEM mode only can propagate up to 15
GHz
. Themaps of the real part of the electric ﬁeld are collected by using a HP8720C Vector Network Analyzer and another dipole antenna as a detector, that moves along the waveguide plane usingan x-y step motor. The thickness dependence has been investigated based on the observation of the beams at the exit of the crystal-air interface. We focused our analysis on structures with anumber of rows
n
ranging from 1 to 10.
Fig. 3. Schematic layout of the experiment carried out on the square-lattice PhC slab hav-ing 25 rods columns and a number of rows n varying from 10 to 1. The dashed line boxrepresents the scanned area during the measurements (colour online).
Fig. 3 shows the scheme of the measurement.Particular attention has been paid to the sourcecharacteristics. The incident beam has to be as collimated and directive as possible, ideallyconsisting of a single wavevector only. To realize the experiment, we inserted in the parallelplate waveguide two parallel microwave absorber stripes, having the role to ”guide” the elec-tromagnetic wave. The channel is 50
cm
long and 10
cm
wide, with tapered sidewalls in order

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