Beam Dynamics in

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Konstantinos Makris Electrical Engineering
Department, Princeton University, USA
Superoscillatory diffractionless beams
Beam Dynamics in PT-waveguides and cavities
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Collaborative groups
R. El-Ganainy, and D.N.Christodoulides
College of Optics /CREOL, University of Central
Florida, USA
M. Segev
Technion, Israel
P. Ambichl, and S. Rotter
Institute of Theoretical Physics, TU-Wien,
Vienna, Austria
Z. Musslimani
Mathematics department, Florida State University,
USA

• G. Aqiang and G. Salamo University of Arkansas,
  USA
• C. E. Rüter and D. Kip - Clausthal University,
  Germany

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Overview

• Introduction to PT-symmetric Optics
• Physical characteristics of PT-symmetric
  potentials
• Group velocity in PT-symmetric lattices
• PT-symmetry breaking in Fabry-Perot cavities
• Conclusions

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Introduction to PT-symmetric Optics
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PT-symmetry in Quantum Mechanics
Should a Hamiltonian be Hermitian in order to
have real eigenvalues?
Schrödinger Equation
Parity and Time operators
PT symmetric Hamiltonian can exhibit entirely
real eigenvalue spectrum!
C. M.Bender et al, Phys. Rev. Lett., 80, 5243
(1998) C. M.Bender et al, Phys. Rev. Lett., 89,
270401 (2002) C. M.Bender et al, Phys. Rev.
Lett., 98, 040403 (2007) C. M.Bender,
Contemporary Physics, 46, 277 (2005)
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Quantum mechanics and Wave Optics
Paraxial Optics
Quantum Mechanics
Schrödinger equation
Paraxial equation of diffraction
Energy eigenvalues
Propagation constants
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PT symmetry in Optics
Nonlinear Schrödinger Equation
PT-symmetric potential
Typical parameters
R. El-Ganainy, K.G.Makris, D. N.
Christodoulides, and Z. H. Musslimani, Opt. Lett.
32, 2632 (2007). K. G. Makris, R. El-Ganainy, D.
N. Christodoulides, and Z. H. Musslimani, Phys.
Rev. Lett. 100, 103904 (2008). Z.H.Musslimani, K.
G. Makris, R. El-Ganainy, D. N. Christodoulides,
Phys. Rev. Lett. 100, 030402 (2008). K. G.
Makris, R. El-Ganainy, D. N. Christodoulides, and
Z. H. Musslimani, Phys. Rev. A 81, 063807
(2010). R. El-Ganainy, K.G.Makris, and D. N.
Christodoulides, Phys. Rev. A 86, 033813 (2012).
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Observation of PT-breaking in a passive coupler
Observation of PT-breaking in an active coupler
Experimental realization of PT-lattice
Paritytime synthetic photonic lattices, A.
Regensburger, C. Bersch, A. Miri, G.
Onishchukov, D. N. Christodoulides , and U.
Peschel Nature, 488, 167171 (09 August 2012)
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Photonic crystals
Negative index materials
PT-symmetric Optics
z
PT-symmetric waveguides
PT-symmetric cavities
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Physical characteristics of PTpotentials
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PT Phase transition in a single waveguide
Scarff potential
Exceptional point
Abrupt phase transition
Biorthogonality condition
0
Z. Ahmed, Phys. Lett. A, 282, 343 (2001) W.D.
Heiss, Eur. Phys. J. D 7, 1 (1999)
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Floquet-Bloch modes in real lattices
Discrete Diffraction
Floquet Bloch mode
k Bloch wavenumber, n number of band
D period
Orthonormality relation
Bandstructure
Superposition principle
Projection coefficients
Parsevals identity
D. N. Christodoulides, F. Lederer, and Y.
Silberberg, Nature, 424, 817 (2003).
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Bandstucture of a PT optical lattice
Exceptional point
Before phase transition
After phase transition
K. G. Makris, R. El-Ganainy, D. N.
Christodoulides, and Z. H. Musslimani, Phys. Rev.
Lett. 100, 103904 (2008).
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PTsymmetric optical cavities
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Scattering from Fabry-Perot PT cavities
g gain/loss
Scattering matrix
Basic relations
Exceptional point
Motion of Scattering matrix eigenvalues in the
complex plane
L. Ge, Y. D. Chong, and A.D. Stone, Phys. Rev.
Lett. 106, 093902 (2011).
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Relation between finite and open PT-cavities
Helmholtz equation in finite domain
(Cavity length)/2
General Robin Boundary Conditions
Gain-loss amplitude
P. Amblich, K.G.Makris, L. Ge, Y. D. Chong, and
S. Rotter, to be submitted (2013).
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Finite and open PT-cavities
Open scattering PT-system
Finite PT-system
Each eigenstate of S is also an eigenstate of an
effective Hamiltonian Heff with the appropriate
Robin boundary conditions
The effective Hamiltonian Heff is PT-symmetric
when
The 2D union of all the eigenvalue curves of Heff
for
is identical to the unbroken phase of the open
scattering problem
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Practical considerations for observing
PT-scattering in cavities
Symmetric output power below EP
Eigenvector of S-matrix
Eigenvector of S-matrix
broken
Asymmetric output power above EP
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Physical value of gain at the exceptional point
broken
Typical physical values
We need long cavities to observe PT-phase
transition
unbroken
g-mismatch tolerance
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Effect of incidence angle in scattering in
PT-cavities
It is experimentally easier if the angle of
incidence is non-zero
unbroken
unbroken
TE polarization
TM polarization
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Scattering coefficients in 2 layer PT-cavities
Reflectance from left to right
Reflectance from left to right
Transmittance
For both transmission
resonance points are below the EP
Normal incidence
L. Ge, Y. Chong, D. Stone, PRA 85, 023802 (2012)
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Multilayer Fabry-Perot PT-cavities
12 layers, TE, normal incidence
broken
broken
zoom
Multiple phase transitions
K.G.Makris, P. Amblich, L. Ge, S. Rotter, and D.
N. Christodoulides to be submitted (2013).
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Multilayer Fabry-Perot PT-cavities
TM-polarization
TE-polarization
12 layers
broken
broken
EP1
Closed paths of scattering eigenvalues in
complex plane
Experimentally, we do not need to scan the
length of cavity, but the angle
EP2
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Superoscillatory diffractionless beams
K.G. Makris
Electrical Engineering Department, Princeton
University, USA
E. Greenfield, and M. Segev
Physics Department, Solid State Institute,
Technion, Israel
D. Papazoglou, and S. Tzortzakis
Materials Science and Technology Department,
University of Crete, Heraklion, Greece Institute
of Electronic Structures and Laser, Foundation
for Research and Technology Hellas, Heraklion,
Greece
D. Psaltis
School of Engineering, Swiss Federal Institute of
Technology Lausanne (EPFL), Switzerland
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Optical Superoscillations
Superoscillatory field A field that locally has
subwavelength features but no evanescent waves.
Theoretical suggestion Optical super-resolution
with no evanescent waves
M. V. Berry, and S. Popescu, J. Phys. A Math.
Gen. 39, 6965 (2006) M. V. Berry, and M. R.
Dennis, J. Phys. A Math. Theor. 42, 022003
(2009) P. J. S. G. Ferreira and A. Kempf, IEEE
Trans. Signal Process., 54, 3732, (2006)
M. R. Dennis, A. C. Hamilton, and J. Courtial,
Opt. Lett. 33, 2976 (2008)
Optical Experiment Subwavelength focus in the
far field with no evanescent waves
N. I. Zheludev, Nature 7, 420 (2008) F. M.
Huang, et al., J. Opt. A Pure Appl. Opt. 9,
S285 (2007) F. M. Huang, and N. I. Zheludev,
Nano Lett. 9, 1249 (2009)
Fabrication of Superoscillatory lens
E. T. F. Rogers, et al. Nature Materials 11, 432
(2012).
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Diffraction-free beams in Optics
Helmholtz equation
Bessel beam of mth order

• They are stationary solutions of Helmholtz
  equation
• They have no-evanescent wave components
  (band-limited)
• They carry infinite power, thus they do not
  diffract

where
The lobes of a Bessel beam are always of the
order of l
Question Can we have diffractionless beams with
sub-l features?
Intensity profiles of Bessel beams
Answer YES, by using the concept of
superoscillations
m3
J. Durnin, J. Opt. Soc. Am. A 4, 651 (1987), J.
Durnin, J. J. Miceli, and J. H. Eberly, Phys.
Rev. Lett. 58, 1499 (1987).
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Stationary Superoscillatory beams
Stationary solution of Helmholtz equation
We force the field to pass through
N predetermined points in the x-y plane
The field as superposition of solutions of
Helmholtz equation
Solution of the problem
If the distances between the Pm points are
subwavelength, the coefficients cm will give us a
superoscillatory superposition
K. G. Makris and D. Psaltis, Opt. Lett. 36, 4335
(2011).
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Analytical form of a superoscillatory beam
We choose to write our field as superposition of
Bessel beams Jn
Polar coordinates
Superposition of Bessel beams
Specific example
Superoscillatory diffractionless beam
K. G. Makris and D. Psaltis, Opt. Lett. 36, 4335
(2011).
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Different 1D and 2D patterns
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Example 1 Superposition of J0,J1,J2 beams
zoom
subwavelength
3-point pattern
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Example 2 Superposition of J2,J6,J10 beams
Phase singularities on sub-wavelength scale
subwavelength
subwavelength
12-point pattern
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Experimental set-up
Superposition of two spatially translated J2
Bessel beams
Superpostion and not an interference effect
Diffraction limit
Wavelength
E. Greenfield, R. Schley, H. Hurwitz, J.
Nemirovsky, K.G. Makris, and M. Segev, Optics
Express, accepted (2013)
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Observation of superoscillatory beams
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w
w2.5 mm4l
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Conclusions
PT-symmetry in optical periodic potentials Group
velocity in PT-lattices PT- symmetric scattering
in cavities Relation between PT open and finite
systems Diffractionless superoscillatory
beams Observation of stationary superoscillatory
beams