Konstantinos Makris
Electrical Engineering Department, Princeton University, USA
Beam Dynamics in
PT-waveguides and
cavities
Superoscillatory
diffractionless
beams
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
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
Introduction to PT-symmetric Optics
PT-symmetry in Quantum Mechanics
Should a Hamiltonian be Hermitian in order to have real eigenvalues?
Parity and Time operators
 pˆ   pˆ
Pˆ  
 xˆ   xˆ
 pˆ   pˆ

Tˆ   xˆ  xˆ
 i  i

PT-potential
Schrödinger Equation

 2  2
i

 V ( x)
2
t
2m x
V *  x  V x
PT symmetric Hamiltonian can exhibit entirely real eigenvalue spectrum!
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)
*C.
Quantum mechanics and Wave Optics
Paraxial Optics
Quantum Mechanics
z
x
Paraxial equation of diffraction
E 1  2 E
i

 k0 n ( x ) E  0
2
z 2k x
Propagation constants
Schrödinger equation
2

 2
i

 V ( x)
2
t
2m x
Energy eigenvalues
PT symmetry in Optics*
iU z  U xx  VR  x   iVI  x   U  U U  0
Nonlinear
Schrödinger
Equation
2
V x   V  x
*
PT-symmetric potential
Typical parameters
X
VR  x 
VI  x 
0.5m  0  1.6m
nRmax  103
g  a  30cm1
G
nImax  5 104
L
*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).
Observation of
PT-breaking
in a passive coupler
Observation of
PT-breaking
in an active coupler
Experimental
realization of PT-lattice
Parity–time synthetic photonic lattices,
A. Regensburger, C. Bersch,
A. Miri, G. Onishchukov,
D. N. Christodoulides , and U. Peschel
Nature, 488, 167–171 (09 August 2012)
Photonic crystals
Negative index materials
PT-symmetric
Optics
z
PT-symmetric
waveguides
PT-symmetric
cavities
Physical characteristics
of PT–potentials
PT Phase transition in a single waveguide*
V  x   sec h2  x   iV0 sec h  x  tanh  x 
Scarff
potential
Exceptional point



V0  1.25, real eigenvalues
V0  1.25, complex eigenvalues
Abrupt phase transition

*
u
m
   x  un  x  dx  dnnm

0
*Z.
Ahmed, Phys. Lett. A, 282, 343 (2001)
W.D. Heiss, Eur. Phys. J. D 7, 1 (1999)
Biorthogonality
condition
Floquet-Bloch modes in real lattices*
z
n  k , x  exp i  n  k  z  Floquet Bloch mode
Discrete Diffraction
k : Bloch wavenumber, n : number of band
D period

*

 m  k ', x  n  k , x  dx n,m  k  k ' Orthonormality relation
x

H  x 
Bandstructure
  D

An  k  n  k , x  dk Superposition
principle
n 1  D
An  k  


n*  k , x  H  x  dx
Projection
coefficients




H  x  dx 
2
  D

n 1  D
An  k  dk
2
Parseval’s
identity
D  10 m a  4.4 m   2.1103
*D.
N. Christodoulides, F. Lederer, and Y. Silberberg, Nature, 424, 817 (2003).
Bandstucture of a PT optical lattice*
V  x   4 cos
2
 x   iV0 sin  2 x  
V0  0.5 real eigenvalues
 If

V0  0.5 complex eigenvalues
 If
V0  0.5
Exceptional
point
Before phase transition
n k   n  k 
n  k, x   n*  k, x 
After phase transition
*K.
G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Phys. Rev. Lett. 100, 103904 (2008).
PT–symmetric
optical cavities
Scattering from Fabry-Perot PT cavities*
g: gain/loss
G
L
broken
r t '
S g  

t
r
'


Scattering
matrix
S  g  S  g   I
unbroken
*
P  S  g   P  S  g 
Basic
relations
0 1
P

1 0
P  S †  g   P  S *  g 

    1, below PT threshold

*



   1, above PT threshold
*L. Ge, Y. D. Chong, and A.D. Stone, Phys. Rev. Lett. 106, 093902 (2011).
Exceptional
point
Motion of Scattering matrix
eigenvalues in the complex plane
Relation between finite and open PT-cavities*
Helmholtz equation
in finite domain
(Cavity length)/2
zz    z  k02  0
  z   * z

    L      L 

Gain-loss amplitude
*P. Amblich, K.G.Makris, L. Ge, Y. D. Chong, and S. Rotter, to be submitted (2013).
General Robin
Boundary Conditions
Finite and open PT-cavities
Finite PT-system
zz    z  k02  0
    L      L 

Open scattering PT-system
 aout 
 ain 
 ain 

  S         
 bout 
 bin 
 bin 
Each eigenstate of S is also an eigenstate of an effective
Hamiltonian Heff with the appropriate Robin boundary conditions
     1
  i   1   1
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
Practical considerations for observing PT-scattering in cavities
a
G
L
Eigenvector
of S-matrix
aei
Symmetric output power
below EP
Eigenvector
of S-matrix
 n1  1.5  ig , n2  1.5  ig ,

 L  5 m, 0  1 m
broken
Asymmetric output power above EP
Physical value of gain at the exceptional point
g  0.1  g phys  12000cm1
g  0.001  g phys  120cm1
g  0.0001  g phys  12cm1
broken
Typical physical values
n  1.5  ig , 0  1 m, L ~ 3mm
g ~ 4 104  g phys ~ 48cm 1
We need long
cavities to observe
PT-phase transition
g-mismatch
10% tolerance
unbroken
Effect of incidence angle in scattering in PT-cavities
 n1  1.5  ig , n2  1.5  ig ,

 L  5 m, 0  1 m

It is experimentally easier if the angle of incidence is non-zero
unbroken
TE polarization
unbroken
g PT  0.11
TM polarization
Scattering coefficients in 2 layer PT-cavities*
R r
Reflectance
from
left to right
2
R'  r'
T  t  t'
2
2
Reflectance
from
left to right
2
Transmittance
For  ~ 19
both
transmission resonance
points are below the EP
o
Normal
incidence
R  0, T  1
L. Ge, Y. Chong, D. Stone, PRA 85, 023802 (2012)
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).
Multilayer Fabry-Perot PT-cavities
TM-polarization
TE-polarization
12 layers
broken
broken
EP1
Experimentally,
we do not need to
scan the length of
cavity, but the angle
EP2
Closed paths
of scattering
eigenvalues in
complex plane
TE,  0
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
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).
Diffraction-free beams in Optics*
 2U  2U  2U
2



k
0U  0
2
2
2
z
x
y
U  x, y, z   J m  ar  eim ei z
r  x 2  y 2 , tan   y / x
Helmholtz
equation
Bessel beam
of mth order
k0  2 
m=0
•They are stationary solutions of Helmholtz equation
•They have no-evanescent wave components (band-limited)
•They carry infinite power, thus they do not diffract
 2  kx2  k y2  k02  0  a  k0
where
a2  k2  kx2  k y2
  1 m  0  a  2   m 1   no evanescent waves
The lobes of a Bessel beam are always of the order of 
Question: Can we have diffractionless beams with sub- features?
Answer: YES, by using the concept of superoscillations
m=3
*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).
m=3
Intensity profiles
of Bessel beams
Stationary Superoscillatory beams*
U  x, y, z   f  x, y  exp i z 
Pm  xm , ym  , m  1,..., N
We force the field to pass through N
predetermined points in the x-y plane
N
f  x, y    cm f m1  x, y 
The field as superposition of
solutions of Helmholtz equation
m 1
Solution of the problem
c M
1
f
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).
Stationary solution of Helmholtz equation






 f 0  r1  f1  r1  ... f N 1  r1  


 f 0  r2  f1  r2  ... f N 1  r2  
M 

...
...
...
...


 f 0  rN  f1  rN  ... f N 1  rN  
N N
c  [c1 c2 ...cN ]T
rn   xn , yn 
f  [ f  r1  f  r2  ... f  rN ]T
Analytical form of a superoscillatory beam*
We choose to write our field as superposition of Bessel beams Jn
 r1,1    ,  ,  r2 ,2   0,0 ,  r3 ,3    ,0
3
f  x, y    cm J m1  ar  e 
i m 1
m 1
Polar
coordinates
Superposition of
Bessel beams
n
1 g1 1 g 2 1 


M  1 g1  2  g 2  2   where
1 g1  3 g 2  3 

f


c


 [0 1 0]T
 [1 0  8  a  ]
2 T

 gn  m   rm eim  1  n  1 , n  1, 2, m  1, 2,3
2


 c1  f  r2 

1
c2   f  r3   f  r1    a 

 c3  4  f  r1   f  r3   2 f  r2   a 2
Specific example

2
U  r,  , z   J 0  ar   8  a   J 2  ar  ei 2 ei z


K. G. Makris and D. Psaltis, Opt. Lett. 36, 4335 (2011).
Superoscillatory
diffractionless beam
Different 1D and 2D patterns
Example 1: Superposition of J0,J1,J2 beams
  1 m, a  2m1
zoom
 
rI   max U

w  FWHM
2
so _ region
w  FWHM ~ 400nm
3-point pattern
I max
rI ~ 1 70
subwavelength
Example 2: Superposition of J2,J6,J10 beams
Phase singularities on sub-wavelength scale
  1 m
rI ~ 1 300
w ~ 300nm subwavelength
12-point pattern
a  2 m1
Experimental set-up*
 E1  A21ei m kz  J 2 (ar )

i  m  kz 
E

A
e
J 2 (a  r  r ')
 2
22
Diffraction limit:
 /(2  NA)  19 m
Wavelength:   633nm
Superposition of two spatially
translated J2 Bessel beams
Superpostion and not
an interference effect
*E. Greenfield, R. Schley, H. Hurwitz, J. Nemirovsky, K.G. Makris, and M. Segev, Optics Express, accepted (2013)
Observation of superoscillatory beams
w=2.5 m~4
rI  1 100
w
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