SLOW LIGHT AND FROZEN MODE REGIME IN PHOTONIC CRYSTALS

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April, 2007
SLOW LIGHT AND FROZEN MODE REGIME IN PHOTONIC
CRYSTALS
Alex Figotin and Ilya Vitebskiy University of
California at Irvine Supported by MURI grant
(AFOSR)
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What are photonic crystals? Simplest examples of
periodic dielectric arrays
1. Each constitutive component is perfectly
transparent, while their periodic array may not
transmit EM waves of certain frequencies. 2.
Strong and controllable spatial dispersion,
particularly at ? L. 3. Photonic crystals
should be treated as genuinely heterogeneous
media no effective homogeneous medium
can imitate a photonic crystal.
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Electromagnetic dispersion relation in photonic
crystals
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Each stationary point is associated with slow
light, but there are some fundamental differences
between these three cases.
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What is the frozen mode regime? Example of a
plane wave incident on a lossless semi-infinite
photonic crystal
- What happens if the incident wave frequency is
equal to that of slow mode with vg 0 ? - Will
the incident wave be converted into the slow mode
inside the photonic crystal, or will it be
reflected back to space? Assuming that the
incident wave amplitude is unity, let us see what
happens if the slow mode is related to (1) RBE,
(2) SIP, (3) DBE.
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Assuming that the incident wave amplitude is
unity, lets see what happens if the slow mode is
related to (1) a regular band edge, (2) a
stationary inflection point, or (3) a degenerate
band edge.
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Regular BE
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Frozen mode profile at different frequencies
close to SIP
In all cases, the incident wave has the same
polarization and unity amplitude.
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DBE case
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Frozen mode profile at different frequencies
close to DBE
In all cases, the incident wave has the same
polarization and unity amplitude.
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Summary of the case of a plane wave incident on
asemi-infinite photonic crystal supporting a
slow mode. - The case of a regular BE the
incident wave is reflected back to space without
producing slow mode in the periodic structure. -
The case of a stationary inflection point the
incident wave can be completely converted into
the slow mode with infinitesimal group velocity
and huge diverging amplitude. - The case of a
degenerate photonic BE the incident wave is
totally reflected back to space, but not before
creating a frozen mode with huge diverging
amplitude and vanishing energy flux.
Regular band edge
Stationary inflection point
Degenerate band edge
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Transfer matrix formalism
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What kind of periodic structures can support the
frozen mode regime?
SIP
DBE
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2. Frozen mode regime in bounded photonic crystals
So far we have discussed the frozen mode regime
in lossless semi-infinite periodic structures.
What happens to the frozen mode regime if the
photonic crystal has finite dimensions?
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EM wave incident on a finite photonic slab
different possibilities
a) The incident wave is partially transmitted
through the photonic slab. b) There is no
transmitted wave if a mirror or an absorber is
present. -----------------------------------------
--------------------------------------------------
- We start with the case (a), involving incident,
transmitted, and reflected waves. Then we turn to
the case (b), where there is no transmitted wave
at all.
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Transmission band edge resonances near a regular
photonic band edge (generic case)
Finite stack transmission vs. frequency.?g
regular photonic band edge
Smoothed field intensity distribution at the
frequency of first transmission resonance
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Giant transmission band edge resonances near a
degenerate photonic band edge
Finite stack transmission vs. frequency. ?g
degenerate photonic band edge
Smoothed Field intensity distribution at
frequency of first transmission resonance
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Regular BE vs. degenerate BE
A stack of 10 layers with degenerate photonic BE
performs as well as a stack of 100 layers with
regular photonic BE !
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Example Transmission band edge resonance in
periodic stacks of 8 and 16 double layers.
Smoothed electromagnetic energy density
distribution inside photonic cavity at frequency
of transmission band edge resonance
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Frozen mode profile at frequency of a giant
transmission band edge resonance a) with a
mirror at the right-hand boundary, b) without
the mirror.
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Publications 1 A. Figotin and I. Vitebsky.
Nonreciprocal magnetic photonic crystals.Phys.
Rev. E 63, 066609, (2001) 2 A. Figotin and I.
Vitebskiy. Electromagnetic unidirectionality in
magneticphotonic crystals. Phys. Rev. B 67,
165210 (2003). 3 A. Figotin and I. Vitebskiy.
Oblique frozen modes in layered media.Phys. Rev.
E 68, 036609 (2003). 4 J. Ballato, A. Ballato,
A. Figotin, and I. Vitebskiy. Frozen light in
periodicstacks of anisotropic layers. Phys. Rev.
E 71, 036612 (2005). 5 G. Mumcu, K. Sertel, J.
L. Volakis, I. Vitebskiy, A. Figotin. RF
Propagation in Finite Thickness Nonreciprocal
Magnetic Photonic Crystals. IEEE Transactions on
Antennas and Propagation, 53, 4026 (2005) 6 A.
Figotin and I. Vitebskiy. Gigantic transmission
band-edge resonance inperiodic stacks of
anisotropic layers. Phys. Rev. E72, 036619,
(2005). 7 A. Figotin and I. Vitebskiy.
Electromagnetic unidirectionality and frozen
modesin magnetic photonic crystals. Journal of
Magnetism and Magnetic Materials, 300, 117
(2006). 8 A. Figotin and I. Vitebskiy. "Slow
light in photonic crystals" (Topical
review),Waves in Random Media, Vol. 16, No. 3,
293 (2006). 9 A. Figotin and I. Vitebskiy.
"Frozen light in photonic crystals with
degenerate band edge". Phys. Rev. E74, 066613
(2006)
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Auxiliary slides
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Fig. 2. Absorption versus frequency of a periodic
stack with DBE at ? ?d (a) The vacuum PS
mirror arrangement shown in Fig. 1(a). (b) The
vacuum PS vacuum arrangement shown in Fig.
1(b). N 8 is the number of unit cells in the
periodic stack. Black and blue curves correspond
to two different values of absorption coefficient
? of the isotropic B layers. In either case (a)
or (b), larger absorption coefficient (the black
curve) gives higher absorption peaks at
frequencies of transmission band-edge resonances.
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Frozen mode regime in the presence of negative
absorption (one of the constitutive components is
a gain medium).
Transmission dispersion of a periodic stack with
different values of negative absorption (gain) ?.
Solid red curve corresponds to ? 0. Observe the
sharp difference between a regular TBER (just
below ?a ) and a giant TBER (just below ?d ).
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Transmission/reflection dispersion of a periodic
stack with different values of negative
absorption (gain) ?. Compared to the previous
slide, the magnitude of negative absorption here
is larger. The difference between the regular
TBER (just below ?a ) and the giant TBER (just
below ?d ) is now extreme.