Wavevector (Phase) Matching

z

I(2)

k2 > k1

- Need k small to get efficient conversion- Problem – strong dispersion in refractive index with frequency in visible and near IR

)]2()()[(2

)2()2()()(2

)2()(2

vac

vacvac

nnk

nknk

kkk

n = [n()-n(2)] 0 because of dispersion linear optics problem

n()

2

Solutions:1. Birefringent media2. Quasi-phase-matching (QPM)3. Waveguide solutions

Birefringent Phase-Matching: Uniaxial Crystals

Uniaxial Crystals

0 0

0 0

0 0

2on

2en

2on

= 0

“ordinary” refractive index “extraordinary” refractive indexon en

- optically isotropic in the x-y plane

- z-axis is the “optic axis” - for , any two orthogonal directions

are equivalent eigenmode axes

cannot phase-match for

zk ||

zk ||

x

y z

k

x

y z

k

in x-z planeeEin x-z plane, ne()

oE

along y-axis, noConvention:

- Note: all orthogonal axes in x-y plane are equivalent for linear optics- always in x-y plane- always has z component - angle from x-axis important for

oE

eE

)2(effd

2/12222 )](sin)()(cos)([)()(),(

PMoPMe

eoe

nnnnn

Type I Phase-Match1 fundamental eigenmode1 harmonic eigenmode+ve uniaxial ne>no

no(2) = ne(,)+ve uniaxial oee

Fundamental (need 2 identical photons)Harmonic (1 photon)

k = 2ke() – ko(2) = 2kvac()[ne(,)-no(2]

E

ne no

Non-critical phase-match =/2no(2) = ne()

x

y zeE

oE

x

y zeE

oCritical phase-match 0<</2no(2) = ne(,)

ne(,) no

ne

n()n()

k

Because of optical isotropy in x-y plane

for phase-matching lies on a cone

at an angle from the z-axis

lies in x-y plane

k

)2( oE

)2( and to is )( oe EkE

Note: does depend on angle from x-axis in x-y plane!!)2(effd

2/12222 )](sin)()(cos)([)()(),()2(

PMoPMe

eoeo

nnnnnn

)()()2()(

)2()()sin( )(sin1)(cosinsert 22

2222

eo

oo

o

ePMPMPM

nnnn

nn

Range of phase-match frequencies limited by condition ne() no(2)

Type I -ve uniaxial no>ne-ve uniaxial eoo no() = ne(,2)

Harmonic Fundamental

k = 2ko() – ke(2) = 2kvac()[no()-ne(,2]

)2()2()()2(

)()2()sin(

)(sin)2()(cos)2(

)2()2()2,()( 22

22

2222

eo

oo

o

ePM

PMoPMe

eoPMeo

nnnn

nn

nn

nnnn

Critical phase-match 0<</2

ne()no

ne

n()

Non-critical phase-match =/2

no

ne

n()

Type II Phase-Match2 fundamental eigenmodes1 harmonic eigenmode

+ve uniaxial oeo

Fundamentals, need 2 (orthogonally polarized) photons

Harmonic (1 photon)

)],()([21)2( eoo nnωn

k = ke() + ko() – ko(2) = kvac(){[ne(,)+no()] - 2no(2)}

)]()([21 )(

)],()([21 of Range

eoo

eo

nnn

nn

)()()}2()(){2(

)()2(2)(2)sin( 22

eo

ooo

oo

ePM

nnnnn

nnn

ne

no

n()

+ve uniaxial ne > no

-ve uniaxial eoeHarmonic Fundamental

)],()([21)2,( eoe nnn

)]()([21 )(

)],()([21 of Range

eoo

eo

nnn

nn

PMoPMe

eo

oPMoPMe

eo

nn

)nn

nnn

nn

2222

2222

sin)2(cos)2(

2()2(

)(sin)(cos)(

)()(21

k = ke() + ko() – ke(2)= kvac()[ne(,) + no()] - kvac(2)ne(,2)

= kvac(){[ne(,) + no()] - 2ne(,2)})]( )(),( of Range

eo

enn

n

Unique

n()

ne

no

Type II -ve uniaxial no > ne

ne(2,)

no(2)

ne(,)

no()

k

Z (optic) axis

PM

Poynting vectors

“Critical” Phase Match “Non-Critical” Phase Match

k

no() = ne(2)

Curves are tangent

)2sin()2(

)2()2( ncebirefringe smallfor

)2sin()2(

1)2(

12

),2( tan opticslinear 22

2

PMo

oe

PMeo

e

nnn

nn

n

Difference between the normals tothe curves represent spatial walk-offbetween fundamental and harmonic Reduces conversion efficiency

Type I eoo

“Critical” Versus “Non-Critical” Phase MatchHow precise must PM be? I(2) sinc2[kL/2= /2] 4/2 0.5

]),2(),2()([2

)],2()([22 PMPM

ePMeo

eo nnnLc

LcnnLk

0

22

2 ),2(21),2(),2(),2(

SHG) maximum (frommatch -phase from detuningangular

PMPMe

PMPMe

PMePMe

PMPMPM

nnnn

e.g. Type I eoo (-ve uniaxial)

)()( fno )],2()([2)],2()()[(2 vac eoeo nnc

nnkk

)2sin()}2()2({1

4)( ),2( Evaluating 22

PMeoPM

ΔkLe

nnLn

PM

Usually quote the “full” acceptance angle = 2PM

PM (Half width at half maximum)

I(2,)Note key role of birefringence

Non-collinear Phase-Matching

We have discussed only collinear wavevector matching. However, clearly it is possible to extend the wavelength range of birefringent phase-matching by tilting the beams.

Biggest disadvantage: Walk-off

Interaction limited to this region

expansionin |),2(21

next term need approached is )2/( matching"-phase critical-non" as diverges

22

2

PMe

PM

PM

n

2/1

)]2()2([4)(

eoPM nnL

λ

Small birefringence is an advantage in maintaining a useful angular bandwidth

Quasi-Phase-Matching

k = 2ke() – ke(2) + pK

- direction of is periodically reversed along a ferroelectric crystal)2(ijkd

Periodically poled LiNbO3(PPLN):

x

z

]2exp[)( )(333

)2(333 xpidxd

p

p

p’th Fourier component

}]2)2()(2[exp{),(

),;2()2(2)2(),2(

23

)(333

vac3

xpkkix

dn

kidx

xd

ee

p

p

E

E

Change phase-matching condition by manufacturing different

1a>0

a is the “mark-space ratio”

/2K

PPLN

A – perfect phase match with )(2)2( kk

B – QPM with p=1C - 0k

Quasi-Phase-Matching: Properties (1)

.|}4

)()]2()2([)]()2({[|4

)( 1

vacooee

vacPM nnnn

L

c/

n(2)n()

x

A modified form of“non-critical” phase-match

zE

k

1|| )sin(2 0 )12( 333)(

333333)(

333 pdppadpdad pp

The relative strengths of the Fourier components depend on a.)( peffd

k = 2ke() – ke(2) + pK0p Not useful since )()2( ee kk

Kkkp ee )2()(2 1 Not useful because )()2( ee kk

Kkkp ee )2()(2 1 Phase matching is possible

: ])/[sin(2 optimizing 333)1(

333 ppadd

333)2(

3331

43 ,

41for optimum 2 ddap

333)3(

333 32

43 ,

21 ,

61for optimum 3 ddap

\ Higher order gratings can be used to extend phase-matching to shorter wavelengths, although the nonlinearity does drop off, pdd p /2 333

)(333

Quasi-Phase-Matching: Properties (2)

2 21for optimum 1 333

)1(333 ddap

-fundamental and harmonic co-polarized- d(2)eff 16 pm/V (p=1)- samples up to 8 cms long- conversion efficiency 1000%/W (waveguides)- commercially available from many sources- still some damage issues

Right-hand side picture shows blue,green-yellow and red beams obtained by doubling 0.82, 1.06 and 1.3 mcompact lasers in QPM LiNbO3

State-of-the-art QPM LiNbO3

Solutions to Type 1 SHG Coupled Wave Equations-first assume negligible fundamental depletion valid to 10% conversion

),0(2

sinc)2(

)2,( 22)2(

EE kLeLdcn

iLkLi

eff

),0()2

(sinc)2()(

||2|)2,(|)2(

21)2,( 222

032

2)2(22

0

IkLL

cnn

dLcnLI eff

E

E(2) and E() are /2 out of phase at L=0!!!

e.g. Type I

2

E()

E()

E(2)

Large Conversion Efficiency (assume energy is conserved Kleinman limit)

.),()2,(~)(

),(

),(~)2(

)2,(

*13

)2(1

21

)2(3

kzieff

kzieff

ezzdcn

idz

zd

ezdcn

idzzd

EEE

EE

)(3

1

033)(

1

1

01131 )()0()2(

21)2,( )()0()(

21),( zi

tzi

t ezIcnzezIcnz

EE

0)0( ,1)0( with )0()]()([),(),( )( :onconservatiEnergy 3121

2131total tIzzzIzIzI

Field Normalization

)()(2 ~ ~ |),0(|),;(-2~~ :Defining 31)2(

skszωωωd

cn eff E

Normalized Coupling Constant NormalizedPropagation

Distance

NormalizedWavevector

Detuning

“Global Phase”

sin)()( sin)()()( 213311

dd

dd

cos

)()()( cos)()( )()(2

3

21

33131 dd

dd

dd

dds

dd

Inserting into coupled wave equations and separating into real and imaginary equations

]ln[sincos on manipulati someAfter

])()()(2[cos into )( and )(for ngSubstituti

321

3

21

331

dds

dd

sdd

dd

dd

dd

Integrated by the method of the variation of the parameters

) oft independen(constant )(21)()](1[cos 2

3323 zCs

)](1[2)()cos( 0 0)0( 2

3

33

sC

2/123

222

32/122

3213 )](

2)}(1[{]cos1)][(1[sin)()(

sSgnSgndd

Sgn is determined by the sign of boundary (initial) condition sine( ))0()0(2 31

The general solution is given in terms of Jacobi elliptic function )|( 41 uusn

21412223

412221 )4/(14/ )|()( )|(1)( ssuuusnuuusnu

|),0(~

)(sech)(

)tanh()(

)2(pg

1

3

E|eff

pg

pg

dcn

zz

zz

Solutions simplify for s=0,i.e. on phase-match

The conversion efficiency saturates at unity (as expected)

Δs=0.2

Δs0

(solid black line); (dashed black line); (red dashed line); (solid blue line, curve multiplied by factor of 4).

3~ 5.1~ 75.0~ 25.0~

The main (Δk=0) peak with increasinginput which means that the tuning bandwidthbecomes progressively narrower.The side-lobes become progressively narrowerand their peaks shift to smaller ΔkL.

Δs=0.2

z

I(2)

k2 > k1

Note the different shape of the harmonicresponse compared to low depletion case

Solutions to Type 2 SHG Coupled Wave Equations

2E3(2)

E1()

E2() . ),()2,(~)(

),(

),()2,(~)(

),(

),(),(~

)(22),2(

*13

)2(

22

*23

)2(

11

21)2(

33

kzieff

kzieff

kzieff

ezzdcn

izdzd

ezzdcn

izdzd

ezzdcn

izdzd

EEE

EEE

EEE

).,(),(),()0( )()0()(21),( 321

)(1

0

zIzIzIIezIcnz tzi

itiii

ii

E

Normalizations

)()()( ~ ~ )2()()(

4~~321

3213

0

3)2(

skszI

nnncd teff

31

2 1)( :ionsnormalizat for these that Note i i z

/)()0()( 2 iti IN

Physically useful solutions are given in terms of the photon fluxes N(), i.e. photons/unit area

Simple analytical solutions can only be given for the case Δs=0

)|)0(((0))0()(

)|)0(((0))0()( (0)(0) and 0for

)|)0(((0))( SHG II Typefor Solutions

12

211

12

22221

12

23

msnNNN

msnNNNNNs

msnNN

)0()0()0()0( define 2

221

22

21

1. No asymptotic final state2. All intensities are periodic with distance3. Oscillation period depends on input intensities

period

period

LK

KL

,0 asfunction elliptic

1)]1/()1[(2

)0()0(

1

2

m

33.0

)(3 N

)(1 N

)(2 N

Type 2 SHG: Phase-Matched