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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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1E12 2

1E12 2

13 3 14 4

1E

1E

Multiple wavelengths

Multiple directions

Multiple wave and directions

Nonlinear optical frequency mixers

Single wave and direction

1E12 2 12 2

12 2

3

Questions about Nonlinear Mixers

1. How to design them?

Single wave and direction: 1D periodic modulation of (2).

Multiple wave, single direction: 1D quasi-periodic modulation.

Multiple waves and directions: 2D periodic and quasi-periodic modulation of (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

M. Yamada et al, Applied Phys. Lett. 62, 435 (1993)

G. Rosenman et al, Phys. Rev. Lett. 73, 3650 (1998)

Technological mature, Commercially available

Limited resolution (>4 m), long processing time.

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V~15 kV

Scanner

Cantilever

Laser

Ferroelectric Sample

High Voltage AFM

590 nm

RbTiOPO4

High Voltage Atomic Force Microscope Poling

Applied Phys. Lett. 82, 103 (2003)

Phys. Rev. Lett. 90, 107601 (2003)

Order of magnitude improvement in poling resolution.

Main technological problem: writing time (1 week (!) for 1mm 150m sample)

Y. Rosenwaks, G. Rosenman, TAU

6LiNbO3

Electron Beam Induced Ferroelectric Domain BreakdownG. Rosenman, E. Weinbrand, Patent Pending, 2004

LiNbO3

a.LiNbO3

c. LiNbO3

b.LiNbO3

electron drop

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Forward vs. Backward nonlinear frequency conversionExample: SHG of Nd:YAG laser in PPKTP.

Forward SHG: Fundamental & SH propagate in the same direction.

Phase matching requirement:

For m=1, QPM period = 9 m

Backward SHG: Fundamental, SH propagate in opposite directions.

Phase matching requirement:

For m=1, QPM period = 0.14 m

mkk

222

mkk

222

k k

k2

kqpm

kqpm

k k2k

8

k kk2

2

k k

k2

k2 2kkk2

2

Collinear SHG

Non-collinear SHG

Backward SHG

HVAFM resolution (period ~ 1 m) not short enough for Backward SHG.

Solution: Characterize by non-collinear SHG.

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Characterization by non-collinear QPM SHG

590 nm

RbTiOPO4

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,

– Frequency tripling, [ SHG: SFG:

All optical splitting and all-optical deflection

• AperiodicAperiodic modulation of the nonlinear coefficient is required in order to obtain high efficiency simultaneously for two different processes.

• Suggested solution: quasi-periodicquasi-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 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: TwoTwo characteristic frequencies.

D. Schetman et al., Phys. Rev. Lett. 53, 195 (1984).

bn

am

k nm 22

,

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Quasi-crystals in natureQuasi-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 blocksbuilding blocks ferroelectric domains with different widths and reversed polarization.

• Order of blocksOrder of blocks quasi-periodic series {zn}:

- E.g. Fibonacci series:

• Fourier transform has peaks at spatial frequencies:

;

- m,n integers; irrational.

- Note: TwoTwo characteristic frequencies.

SL

nmk nm

)(2, SLD (average lattice parameter)

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L L L L LS S S

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Fourier transform relations between structure and efficiency

dz)zkiexp()z(dEEL

0

22 E.g. for SHG:

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efficiency

SHG

SFG

exp %/W2

• Theory

• Experiment

LSLLSLLSLSLLS…

k

THG SHG

SFG

Direct frequency triplingDirect frequency triplingusing GQPS in KTPusing GQPS in KTP

K. Fradkin-Kashi et al, Physical Review Letters 88, 023903 (2002).

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Two-dimensional Nonlinear Periodic StructureTwo-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 Lattices2D 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) => Crystal

2D Lattice + basis (nonlinear domain) => Nonlinear superlattice

21 anamr

021 90, aa

021 120, aa

18•C. Kittel, Introduction to solid state physics

a1

a2

Simple cubic

a1

a2

Hexagonal

a1

a2

Rectangular

a1

a2

Centered cubic

The Five 2D Bravais LatticesThe Five 2D Bravais Lattices

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The Reciprocal LatticeThe Reciprocal LatticeDefine 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),

ijji ab 2

21 blbkG

RLG

G)2( )rGiexp(k)r(

RGRGJ

fKG)(

4 1)2(

20V. Berger, Phys. Rev. Lett. 81, 4136 (1998).

QPM in 2D Nonlinear StructureQPM in 2D Nonlinear Structure

0Gk2k2

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.

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2D rectangular pattern in LiNbO2D rectangular pattern in LiNbO33

AFM topographic scan

Fresnel diffraction (60 cm)

Optical microscope

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Experimental setupExperimental setup

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Experimental determination of Reciprocal LatticeExperimental determination of Reciprocal Lattice

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Angular input-output relationsAngular input-output relations

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2D QPM structures with annular symmetry2D QPM structures with annular symmetryAn annular structure with period ~ 25 microns

~1mm

~ 800 m

icron

2222

0

*

,

)()()(

yxrkkkq

rdrqrJrdWcn

EEwiqE

yx

s

ipss

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Phase matching can occur at all directions.Different processes can be phase matched at different angles .

wk2

wk2kPeriodnGn /2

We can calculate the angle of the second harmonic using the law of cosines.

wwn

ww

kk

Gkk2

2222

4

)()()2(cos

Geometrical considerationsGeometrical considerations

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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

QPM in 2D Quasi-Periodic Nonlinear StructuresQPM in 2D Quasi-Periodic Nonlinear Structures

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Design using the Dual Grid MethodDesign using the Dual Grid Method

1k

2k3k

4k

5k

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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1

2

k

Step 2: Creating a grid based on the vectors Step 2: Creating a grid based on the vectors defined in Step 1defined 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 latticeStep 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 superlatticeStep 4: Creating a nonlinear superlattice

)( 2)( 2

• 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?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

]2,1[),,...,( )()1( Dkk

]..3[),,...( )()1( Dqq D

),( )()()( jjj qkK

),( )()()( jjj baA

ijji KA 2)()(

j

jjan )(

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DGM design example in 1D: DGM design example in 1D: Quasi-periodic superlattice Quasi-periodic superlattice

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

2

1 k

22 k

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12

2

k

1

2

k• Creating the grid:

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

Step 2: Creating a 1D gridStep 2: Creating a 1D grid

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1S

L

• 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

Step 3: Creating a 1D Quasi-periodic super Step 3: Creating a 1D Quasi-periodic super latticelattice

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ω + ω 2ω

ω + 2ω 3ω

2ω + 2ω 4ω

2ω

3ω

4ω

ω

Designed through the DGM method

21 kk k

2k 2k

42 k4k

k

3k321 k

+2

2+24 +23

2k

Design of a nonlinear color fanDesign of a nonlinear color fan

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m20

Color-fan designed for fundamental of 3500nm

20m

Microscope image Image processed for FFT

Color fan IIColor fan II

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Numerical Fourier transform of the mathematical lattice

FFT of the device image

* Arrows indicating positions of required mismatch wave-vectors

Color fan IIIColor fan III

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Diffraction Image

Color fan IVColor fan IV

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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 Nonlinear devices utilizing multiple phase matching possibilitiesmatching 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 splittingAll-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.

S. M. Saltiel and Y. S. Kivshar, Opt. Lett. 27, 921 (2002).

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

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All-optical deflection & splittingAll-optical deflection & splitting

Saltiel and Kivshar, Opt. Lett. 27, 921 (2002)

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SummarySummary• 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.