Nonlinear interactions in periodic and quasi-periodic structures

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Two-dimensional nonlinear frequency
converters A. Arie, A. Bahabad, Y. Glickman, E.
Winebrand, D. Kasimov and G. Rosenman Dept. of
Physical Electronics, School of Electrical
Engineering Tel-Aviv University, Tel-Aviv,
Israel FRISNO 8, Ein Bokek 2005
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Nonlinear optical frequency mixers
Single wave and direction
Multiple wavelengths
Multiple directions
Multiple wave and directions
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• Questions about Nonlinear Mixers
• How to design them?
• Single wave and direction 1D periodic modulation
  of c(2).
• Multiple wave, single direction 1D
  quasi-periodic modulation.
• Multiple waves and directions 2D periodic and
  quasi-periodic modulation of c(2). Design using
  Dual Grid Method
• 2. How to produce them?
• Domain reversal in ferroelectrics (LiNBO3, KTP,
  RTP) using electric field
• Through planar electrodes
• High voltage atomic force microscope
  (sub-micron resolution)
• Electron beam poling
• 3. What can we do with them?

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Electric field poling of ferroelectric crystals
Technological mature, Commercially
available Limited resolution (gt4 mm), long
processing time.
M. Yamada et al, Applied Phys. Lett. 62, 435
(1993) G. Rosenman et al, Phys. Rev. Lett. 73,
3650 (1998)
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High Voltage Atomic Force Microscope Poling
Order of magnitude improvement in poling
resolution. Main technological problem writing
time (1 week (!) for 1mm ? 150mm sample) Y.
Rosenwaks, G. Rosenman, TAU
Applied Phys. Lett. 82, 103 (2003) Phys. Rev.
Lett. 90, 107601 (2003)
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Forward vs. Backward nonlinear frequency
conversion
Example SHG of NdYAG laser in PPKTP. Forward
SHG Fundamental SH propagate in the same
direction. Phase matching requirement For m1,
QPM period 9 mm Backward SHG Fundamental, SH
propagate in opposite directions. Phase matching
requirement For m1, QPM period 0.14 mm
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HVAFM resolution (period 1 mm) not short enough
for Backward SHG. Solution Characterize by
non-collinear SHG.
Collinear SHG
Backward SHG
Non-collinear SHG
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Characterization by non-collinear QPM SHG
S. Moscovich et al, Opt. Express 12, 2336 (2004)
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Multiple-wavelengthQPM nonlinear interactions

• Examples of multiple interactions using the
  2nd-order nonlinear coefficient, d(z)
• Dual wavelength SHG, w1 ? 2w1 , w2 ? 2w2
• Frequency tripling,
• w1 ? 3w1 SHG w1 ? 2w1 SFG w1 2w1 ? 3w1
• All optical splitting and all-optical deflection
• Aperiodic modulation of the nonlinear coefficient
  is required in order to obtain high efficiency
  simultaneously for two different processes.
• Suggested solution quasi-periodic modulation of
  the nonlinear coefficient?.
• ?K. Fradkin-Kashi and A. Arie, IEEE J. Quantum
  Electron. 35, 1649 (1999).

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Quasi-periodic structures (QPS)

• Quasi-periodic patterns found in nature, e.g.
  quasicrystals? and studied by mathematicians
  (Fibonacci) and crystalographers.
• A quasi-periodic structure can support more than
  one spatial frequency. The Fourier transform of
  one-dimensional QPS has peaks at spatial
  frequencies
• - m,n integers.
• - Note Two characteristic frequencies.
• ?D. Schetman et al., Phys. Rev. Lett. 53, 195
  (1984).

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Quasi-crystals in nature
Scanning electron micrographs of single grains of
quasicrystals
Typical diffraction diagram of a quasicrystal,
exhibiting 5-fold or 10-fold rotational symmetry
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Quasi-periodic structures (QPS) cont.

• Nonlinear optics building blocks ? ferroelectric
  domains with different widths and reversed
  polarization.
• Order of blocks ? quasi-periodic series zn
• - E.g. Fibonacci series
• Fourier transform has peaks at spatial
  frequencies
• - m,n integers ? irrational.
• - Note Two characteristic frequencies.

(average lattice parameter)
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Fourier transform relations between structure and
efficiency
E.g. for SHG
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Direct frequency triplingusing GQPS in KTP

• Theory

LSLLSLLSLSLLS
?k

• Experiment

SHG
THG
hexp0.00003 /W2
SFG
K. Fradkin-Kashi et al, Physical Review Letters
88, 023903 (2002).
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Two-dimensional Nonlinear Periodic Structure
Proposed by V. Berger, Phys. Rev. Lett., 81, 4136
(1998) Modulation methods for nonlinear
coefficient are planar methods - both available
dimensions can be used for nonlinear processes.
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2D Periodic Lattices
2D Real Lattice A set of points at
locations Where m,n are integers, and a1, a2 are
(primitive) translation vectors. There are 5
Bravais lattices in 2D Examples Square lattice
Hexagonal lattice 2D Lattice basis (atoms)
gt Crystal 2D Lattice basis (nonlinear
domain) gt Nonlinear superlattice
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The Five 2D Bravais Lattices

• C. Kittel, Introduction to solid state physics

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The Reciprocal Lattice
Define primitive b1, b2, vectors of the
Reciprocal Lattice such that The Reciprocal
Lattice points are given by In crystals, the
Reciprocal Lattice is identical to a scaled
version of the diffraction pattern of the
crystal. The nonlinear susceptibility can be
written as a Fourier series in the Reciprocal
Lattice E.g., for hexagonal lattice of
cylinders (with circle filling factor f),
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QPM in 2D Nonlinear Structure
Consider SHG example The QPM condition is a
vector condition
Where G is a vector in the Reciprocal
Lattice. May phase matching possibilities exist
in 2D lattice.
?V. Berger, Phys. Rev. Lett. 81, 4136 (1998).
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2D rectangular pattern in LiNbO3
Optical microscope
AFM topographic scan
Fresnel diffraction (60 cm)
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Experimental setup
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Experimental determination of Reciprocal Lattice
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Angular input-output relations
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2D QPM structures with annular symmetry
An annular structure with period 25 microns
800 micron
1mm
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Geometrical considerations
Phase matching can occur at all
directions. Different processes can be phase
matched at different angles.
We can calculate the angle of the second harmonic
using the law of cosines.
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QPM in 2D Quasi-Periodic Nonlinear Structures
2D quasi-periodic structures offer further
extension to the possible phase matching
processes. One can have several phase matching
directions and along each direction to phase
match several different processes. Main problem
How to design a nonlinear mixer that
phase-matches several interactions in a multitude
of directions
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Design using the Dual Grid Method
Well known algorithm for the design of
quasi-crystals 1. Ensures minimum separation
between lattice points. Step 1 defining the
required spectral content
Ron Lifshitz, TAU Socolar, Steinhardt and Levine,
Quasicrystals with arbitrary orientational
symmetry Phys. Rev. B. vol.32, 5547-5550 (1985).
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Step 2 Creating a grid based on the vectors
defined in Step 1

• The grid is dual (transformable) to a lattice
  containing the spectral content defined by the
  vectors in step 1

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Step 3 Generating a quasi-periodic lattice

• The lattice is based on a topological
  transformation of the grid from step 2

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Step 4 Creating a nonlinear superlattice

• The vertices of the lattice mark the location of
  a repeated cell characterized by a uniform
  nonlinear permittivity value
• Design of the repeated cell shapes the spectrum
  energy distribution

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How to construct the quasi-crystal?

1. Start with D two-dimensional mismatch vectors
2. Add D-2 components to each of the mismatch
   vectors
3. One obtains D D-dimensional vectors
4. Find the dual basis
   , where a is two-dimensional and b is D-2
   dimensional, such that
5. Put a cell at a subset of the positions given

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DGM design example in 1D Quasi-periodic
superlattice

• Choosing required spectral content e.g. two
  parallel phase-mismatch vectors.

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Step 2 Creating a 1D grid

• Creating the grid each vector defines a family
  of lines in the direction of the vector and with
  separation inversely proportional to its
  magnitude

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Step 3 Creating a 1D Quasi-periodic super lattice

• Transforming the grid to a lattice The order of
  appearance of the lines from each family (its
  topology) determines the order of the lattice
  building blocks

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Design of a nonlinear color fan
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Color fan II
Color-fan designed for fundamental of 3500nm
Microscope image
Image processed for FFT
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Color fan III
Arrows indicating positions of required
mismatch wave-vectors
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Color fan IV
Diffraction Image
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Phase-matching methods and corresponding
conditions on wavevector difference
1. J. A. Giordmaine, Phys. Rev. Lett. 8, 19
(1962) P.D. Maker et al., 8, 21 (1962).2. J. A.
Armstrong et al., Phys. Rev. 127, 1918 (1962).3.
S.-N. Zhu et al., Science 278, 843 (1997), K.
Fradkin-Kashi and A. Arie, IEEE J. Quantum
Electron. 35, 1649 (1999). 4. V. Berger, Phys.
Rev. Lett. 81 4136 (1998).
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Nonlinear devices utilizing multiple phase
matching possibilities

1. Ring cavity mixers
2. Multiple harmonic generators
3. Nonlinear prisms and color fans
4. Omni-directional mixers
5. All-optical deflectors and splitters

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Nonlinear deflection and nonlinear splitting
All-optical deflection of wy as a function of
pump wz. Step 1 collinear SHG of pump wzwz
gt(2w)z Step 2 noncollinear DFG of SH and cross
polarized input signal (2w) z-wy gtnoncollinear
wy at angle q with respect to input beam.
All-optical splitting of wy into two directions
Step 1 collinear SHG of pump wzwz gt(2w)z
(same as above) Step 2 Simultaneous noncollinear
DFG of SH and cross polarized input signal into
two different directions
S. M. Saltiel and Y. S. Kivshar, Opt. Lett. 27,
921 (2002).
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All-optical deflection splitting
Saltiel and Kivshar, Opt. Lett. 27, 921 (2002)
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Summary

• New methods for poling ferroelectrics offer
  improved resolution and larger design
  flexibility
• Sub-micron resolution using HVAFM poling.
  Characterized by non-collinear SHG.
• Modified E-beam poling. Characterized by 2D NLO
• Design fabrication of a 1D quasi-periodic
  structure for multiple-wavelength nonlinear
  interactions.
• Phase-match any two arbitrarily chosen
  interactions.
• Dual wavelength SHG and frequency tripling
  demonstrated in KTP.
• Multiple-wavelength interactions by 2D periodic
  nonlinear structures.
• 15 different phase matching options measured in
  2D rectangular pattern.
• Annular symmetry device recently realized
• Further extension by 2D quasi-periodic structures
• Dual grid method offer a possibility to phase
  match several interactions in a multitude of
  directions.
• Next steps experimental realizations,
  demonstration of devices, finding useful
  applications.