wavevector (phase) matching
DESCRIPTION
k 2 > k 1. n ( ). I (2 ). . 2 . z. Wavevector (Phase) Matching. Need k small to get efficient conversion - Problem – strong dispersion in refractive index with frequency in visible and near IR. - PowerPoint PPT PresentationTRANSCRIPT
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