Transcript ppt 6.5MB - Weizmann Institute of Science

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
E1
E1
Single wave and
direction
Multiple
wavelengths
E1
E1
Multiple
directions
E1
Multiple wave
and directions
 2  21
 2  21
3  31
 4  41
 2  21
 2  21
 2  21
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Questions about Nonlinear Mixers
1. 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 (>4 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
RbTiOPO4
High Voltage AFM
Cantilever
Laser
DV~15 kV
Ferroelectric Sample
Scanner
590 nm
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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Electron Beam Induced Ferroelectric Domain Breakdown
G. Rosenman, E. Weinbrand, Patent Pending, 2004
a.
LiNbO3
LiNbO3
b.
c.
LiNbO3
LiNbO3
electron drop
LiNbO3
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Forward vs. Backward nonlinear frequency conversion
Example: SHG of Nd:YAG laser in PPKTP.
Forward SHG: Fundamental & SH propagate in the same direction.
Phase matching requirement:
k 2
2m
 2k 

k
For m=1, QPM period = 9 mm
k
kqpm
k2
Backward SHG: Fundamental, SH propagate in opposite directions.
Phase matching requirement:
k
k 2
k
2m
 2k 

k2
kqpm
For m=1, 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
RbTiOPO4
590
nm
S. Moscovich et al, Opt. Express 12,
2336 (2004)
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Multiple-wavelength
QPM nonlinear interactions
• Examples of multiple interactions using the 2nd-order nonlinear
coefficient, d(z):
– Dual wavelength SHG, 1  21 , 2  22
– Frequency tripling,
1  31 [ SHG: 1  21  SFG: 1  21  31 ]
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 onedimensional QPS has peaks at spatial frequencies:
2m 2n
k m, n 
a

b
- 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:
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LSL LSLSL
• Fourier transform has peaks at spatial frequencies:
2 (m  n )
k m, n 
D  L  S (average lattice
;
L  S
parameter)
- m,n integers;  irrational.
- Note: Two characteristic frequencies.
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Fourier transform relations between structure and efficiency
L
E.g. for SHG:
E2  E2   d ( z ) exp( iDk  z )dz
0
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Direct frequency tripling
using GQPS in KTP
• Theory
efficiency
SFG
LSLLSLLSLSLLS…
SHG
Dk
• Experiment
THG
SHG
hexp0.00003 %/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
r  ma1  na2
Where m,n are integers, and a1, a2 are (primitive) translation vectors.
There are 5 Bravais lattices in 2D
Examples:
Square lattice:
a1  a2 ,   900
Hexagonal lattice:
a1  a2 ,   1200
2D Lattice + basis (atoms)
=>
2D Lattice + basis (nonlinear domain) =>
Crystal
Nonlinear superlattice
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The Five 2D Bravais Lattices
a2
a1
Simple
cubic
a2
a2
a1
Hexagonal
a1
Rectangular
a2
a1
Centered cubic
•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
bi  a j  2 ij
The Reciprocal Lattice points are given by
G  kb1  lb2
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:
c ( 2 )( r )   kG  exp( iG  r )
GRL
E.g., for hexagonal lattice of cylinders (with circle filling factor f),
( 2 ) J1 ( G R )
K G  4 fc
GR
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QPM in 2D Nonlinear Structure
Consider SHG example:
The QPM condition is a
vector condition:
k2  2k  G  0
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
AFM topographic scan
Optical microscope
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
Es ( q ) 
iws E p Ei*
nsWc
 d (r ) J (qr )rdr
0
q  Dk x2  Dk y2  Dk , r  x 2  y 2
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Geometrical considerations
Phase matching can occur at all directions.
Different processes can be phase matched at different angles.
k2w
Gn  n  2 / Period  Dk
2k w
We can calculate the angle of the second harmonic using
the law of cosines.
(2k w ) 2  (k 2 w ) 2  (Gn ) 2
cos  
4k w k 2 w
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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 phasematches 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
Dk 5
Dk4
Dk1
Dk 3
Dk2
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
2
Dk1
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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
 c(2)
 c(2)
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How to construct the quasi-crystal?
1. Start with D two-dimensional mismatch vectors
(Dkm(1) ,..., Dkm( D) ), [ m  1,2]
2. Add D-2 components to each of the mismatch vectors
(qm(1) ,...qm( D ) ), [ m  3..D]
3. One obtains D D-dimensional vectors
K ( j )  (Dk ( j ) , q ( j ) )
( j)
( j) ( j)
4. Find the dual basis A  (a , b ) , where a is twodimensional and b is D-2 dimensional, such that
A(i )  K ( j )  2 ij
5. Put a cell at a subset of the positions given  n j a
j
( j)
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DGM design example in 1D:
Quasi-periodic superlattice
• Choosing required
spectral content –
e.g. two parallel
phase-mismatch
vectors.
Dk1 
2

Dk2  2
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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
2

Dk1
2
1
Dk2
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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
S 1
L 


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Design of a nonlinear color fan
Designed through the DGM method
ω
ω+ω
2ω
4ω
ω + 2ω
3ω
2ω
2ω + 2ω
4ω
3ω
+2
k
k
Dk12
+23
2+24
k
k2
k4
Dk24
k2
k3
Dk1 2 3
k2
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Color fan II
Color-fan designed for fundamental of 3500nm
20 mm
Microscope image
20mm
Image processed for FFT
37
Color fan III
Numerical Fourier transform
of the mathematical lattice
FFT of the device image
* 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 y as a function of pump z.
Step 1: collinear SHG of pump: z+z =>(2)z
Step 2: noncollinear DFG of SH and cross polarized input signal
(2) z-y =>noncollinear y at angle  with respect to input beam.
All-optical splitting of y into two directions
Step 1: collinear SHG of pump: z+z =>(2)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).
42
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, finding44useful
applications.