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WavePro 2011, June 8-11, Rethymno, Crete
Happy Birthday Costas !
WavePro 2011, June 8-11, Rethymno, Crete
Happy Birthday Costas !
NATO 1995 Elounda, WECS I Laguna Beach,
COST KTH Stockholm, Brussels
Hersonissos ,Agia Pelagia,….Rethymno 2011
Transport Properties of One-way EM
waveguide formed at the interface between
metal and 2D photonic crystal
Vladimir Kuzmiak1 Sergey Eyderman1, Mathias Vanwolleghem2
1 Institute of Photonics and Electronics, Czech Academy of
Sciences, v.v.i., Prague, Czech Republic
2 Faculty Département MMS, Institut d’Electronique Fondamentale
(CNRS UMR8622), Université Paris-Sud, 91405 Orsay, France
COST MP0702, GA CR No. 0004632
WavePro 2011, June 8-11, Rethymno, Crete
WavePro 2011, June 8-11, Rethymno, Crete
Outline
• Motivation
• Nonreciprocity – necessary conditions, role of the time-reverse symmetry breaking in nanodevices
• Nonreciprocal and one-way devices – concepts
• Spectral and transport properties of one-way waveguide
• Dependence of unidirectional behaviour on boundary conditions and when mg. field is modulated
• Discussion and conclusions
WavePro 2011, June 8-11, Rethymno, Crete
Motivation
• To explore physical mechanisms that may lead to nonreciprocal devices
• Provide hints to design compact integrated optical analog to one-way electronic device such is diode or transistor to contribute to the effort to make photonics more realistic alternative to electronic solutions
WavePro 2011, June 8-11, Rethymno, Crete
Nonreciprocity in the
context of the nanodevices
Effects of the disorder are
significantly influenced time-
reverse symmetry properties
the backreflection in the
presence of disorder may
constitute serious drawback
a new regime of photon transport
that supports only forward
propagating wave that is free of
disorder-induced backscattering
Reciprocal systems:
Non-reciprocal systems
with broken TR symmetry
WavePro 2011, June 8-11, Rethymno, Crete
[1] I. Vitebsky, J. Edelkind, E. Bogachek, and Uzi Landman:, Phys. Rev. B 55, 12566, 1997.
[2] A. Figotin and I. Vitebsky, Phys. Rev. E 63, 066609, 2001.
Nonreciprocity ω(k) ≠ ω(-k)
To satisfy a sufficient condition to ensure spectral non-reciprocity
(-k) (k) requires breaking both space- and time-reversal
symmetry [1]
A generic 1D periodic arrangement of magnetic and dielectric
components that give rise to strong spectral asymmetry [2] has
been suggested
WavePro 2011, June 8-11, Rethymno, Crete
Nonreciprocal devices - concepts
[3] F. D. M. Haldane and S. Raghu, arXiv:cond-mat/0503588v1.
[4] Z. Wang, et al.:, Phys. Rev. Lett. 100, 013905 (2008).
[5] Z. Wang, et al.:, Nature. 461, 772 (2009).
[6] H. Takeda and S. John, Phys. Rev. A 78, 023804 (2008).
[7] Z. Yu, et al.:, Phys. Rev. Lett. 100, 023902 (2008).
Design proposed in Ref. 3 employs photonic analogues
of quantum chiral edge states and relies on the
presence of a degenerate Dirac point in reciprocal
photonic crystal
Various designs of one-way waveguides operating at
microwave range proposed in Ref. [4]-[7].
WavePro 2011, June 11-14, Rehytmo, Crete
One-way EM waveguide
(a) (b)
[8] Z. Yu, G. Veronis, Z. Wang, and S. Fan, Phys. Rev. Lett. 100,
023902(2008).
[9] S. Eyderman, V. Kuzmiak, and M. Vanwolleghem, Proc. SPIE
Photonics Europe, Brussels, Belgium, April 2010, paper 7713-24.
WavePro 2011, June 11-14, Rehytmo, Crete
Waveguide A
Dielectric permittivity of the metal:
)(
)(00
01
01
)()(
22
22
2
1
i
i
ii
ii
i
b
b
b
b
p
11 9.8diel
srad /10.1 14 sradp /10.1 16
nma 100 ad 2.0 nma 50
WavePro 2011, June 8-11, Rethymno, Crete
Waveguide A
2B
SPSPL
2B
SPSPR
m
eBB
2
p
SP
WavePro 2011, June 8-11, Rethymno, Crete
One-way waveguide concepts vs. Bloch theorem
In order to achieve true one-way property it is crucial that
within a certain frequency range there appears a photonic
band that has a single sign for its group velocity over the
whole Brillouin zone.
(/a) = (-/a)
BAND STRUCTURE CALCULATION USING FURIER MODAL METHOD
BB B
… …
x
z
BB B
… …
BBBB BB
… …
x
z
x
z
Artificial periodization of the waveguide structure to be modeled (with
propagation along z). The dashed regions are the PML absorbers isolating the
“grating unit cells”. By a proper Fourier factorization in the x-direction the local
eigenmode basis can be described by discrete plane wave expansion.
out
in
S
inoutoutin
inoutoutin
in
out
B
F
TR
RT
B
F
,,
,,
in
ajk
in
inoutoutin
ajk
in
ajk
ininoutoutin
Be
F
TRe
Be
FRT
B
B
B
,,
,, 01
10 Af Bfl=
eigenmode scattering matrix S
S
Generalized eigenvalue problem with
applying Bloch boundary conditions which
allows to calculate band structures in the
frequency domain with frequency as the
independent parameter
WavePro 2011, June 8-11, Rethymno, Crete
Waveguide A
The field plots show the z-component of the Poynting vectors at the
reduced frequency 0.7ωp inside one unit cell of the structure shown on
the left. Lighter shades indicate positive Sz. In the obtained dispersion
diagram a forward and a backward plasmonic mode are found at this
frequency. The field plots show that +z and –z propagating plasmonic
modes are guided at opposite metallic interfaces.
ICTON 20010,June 27-July 1, Munich, Germany
Waveguide B
qualitative demonstration of the possibility to obtain similar
unidirectional regime by using a magnetic PhC and a nonmagnetic
metal with one way frequency range 0.366 < ω/ωp < 0.376
~ 363 nm < < 373 nm
Drude metal
fp = 2200 THz
= 1 THz
PML
garnet
13 p
erio
ds +
1 «
extr
a»
laye
r
x
z
d/a 0.23 central gap c 0.396 2c/a
p/7.25 = 0.396 2c/aa = 0.396 7.25 2c/p
= 1.066 p
B
ad
xx = 6.25 xy = -yx = -i*g (g=0.1)
Drude metal
fp = 2200 THz
= 1 THz
PML
garnet
13 p
erio
ds +
1 «
extr
a»
laye
r
x
z
d/a 0.23 central gap c 0.396 2c/a
p/7.25 = 0.396 2c/aa = 0.396 7.25 2c/p
= 1.066 p
B
ad
xx = 6.25 xy = -yx = -i*g (g=0.1)
WavePro 2011, June 8-11, Rethymno, Crete
Theoretical model Surface plasmon polaritons propagating along the interface between
two semi-infinite homogenous slabs fabricated from BIG and Ag
00
0
0
ig
ig
2
2
1)(
p
1.0,25.6 gzzyyxx
sradp /10381.1 16
srad /1028.6 12
[10] T. Tepper and C. Ross, J. Cryst. Growth., 255, 324 (2003)
WavePro 2011, June 8-11, Rehytmo, Crete
Dispersion relation Surface plasmon polaritons propagating along the interface between
two semi-infinite homogenous slabs fabricated from BIG and Ag
2222212
02
4)( xymxxm
x
KKkk
2321 4))(( xymxxmmmxxK
))(2(2 222 xxmxxmmxyiK
xxxyxx /2
2
2
1)(
p
[11] A. Boardman et al., New Journal of Physics 7 (2005) 191
WavePro 2011, June 8-11, Rethymno, Crete
Dispersion relation Surface plasmon polaritons propagating along the interface between
two semi-infinite homogenous slabs fabricated from BIG and Ag
[10] A. Boardman et al., New Journal of Physics 7 (2005) 191
WavePro 2011, June 8-11, Rethymno, Crete
Effect of Boundary Case conditions Case I
One-way propagation in the waveguide structure formed the interface
between metal and a MOPhC @ frequency ωc = ωp/√7.25 inside the band
gap: PML boundary conditions
WavePro 2011, June 8-11, Rethymno, Crete
Effect of Boundary Case conditions Case II
ωc = ωp/√7.25 inside the band gap: Perfect conductor(PC) boundary
conditions
WavePro 2011, June 8-11, Rethymno, Crete
Effect of Boundary Case conditions Case III
ωc = ωp/√7.25 inside the band gap: PC wall placed into middle of PhC
unit cell C
WavePro 2011, June 8-11, Rethymno, Crete
Conclusions
• Improved one-way waveguide that assumes significantly reduced mg. field ~ 0.01 T
• Spectral and transport characterics have been studied
• Dependence on boundary conditions and temporal modulation of the mg. field
• Some new and interesting phenomena observed
WavePro 2011, June 8-11, Rethymno, Crete
Conclusions
Future directions:
two major challenges to be addressed
1. reduce or completely avoid using mg. field,
2. scale operating wavelength to optical range
Costas !