optimizing shg efficiency
DESCRIPTION
Voigt notation: has 27 elements, but this can be simplified because . in a 3x3x3 matrix reduce to 18 elements in a 3x6 matrix. Optimizing SHG Efficiency. Crystal symmetry Reduces number of independent non-zero elements - PowerPoint PPT PresentationTRANSCRIPT
Optimizing SHG Efficiency
),(),;2(ˆ optimize toNeed |),;2(ˆ| EfficiencySHG )2(2)2( fdd effeff
)(ˆ)(ˆ),;2()2(ˆ )2(*)2( bk
ajijkieff eeded
Voigt notation: has 27 elements, but this can be simplified because . in a 3x3x3 matrix reduce to 18 elements in a 3x6 matrix
)2(ijkd )2()2(
ikjijk dd )2(
iJd
d11 d21 d31
d12 d22 d32
d13 d23 d33
d14 d24 d34
d15 d25 d35
d16 d26 d36
jk J11 122 233 3
23,32 413,31 512,21 6
)2(iJd
Crystal symmetry(1) Reduces number of independent non-zero elements(2) Relationships between non-zero elements
Note: all cubic crystal classes are isotropic for linear optics, but some are non-centrosymmetric, have some and are not isotropic for nonlinear optics. 0)2( iJd
Effect of Crystal Symmetry
System International Schönflies)1(
ij)2(
ijk)3(
ijkl
Number of Independent Elements
Standard Notations
System International Schönflies)1(
ij)2(
ijk )3(ijkl
Small dot = zero coefficient
Square dot= Zerocoefficient for Kleinman symmetry only
Similar connected dots= Equal coefficients
Open connected dots=Opposite in sign to closed dot
Connections via dashed lineValid for Kleinman symmetry only
Matrix for Biaxial, Uniaxial and Optically Isotropic Classes)2(iJd
Define axes Fields
Define:
)sin,sin cos,cos cos(ˆ)0,cos,sin(ˆ
)cos,sin sin,cos sin(ˆ
e
oeek
0 0 0 :Note
kekeee
e
o
eo
Assume birefringence small neglect difference in direction of and D
E
ekejijkoi
okojijkei
eedeoee
eede eoo
ˆˆˆ uniaxial ve
ˆˆˆ uniaxial ve- 1 Type)2(
)2(
okejijkoi
ekojijkei
eedeoeo
eedeeoe
ˆˆˆ uniaxial ve
ˆˆˆ uniaxial ve- 2 Type)2(
)2(
Calculation of for plane wave inputs )2(effd
d110 0
-d110 0
0 0 0
d140 0
0 -d140
0 -d110
e.g. crystal class 3,2
)2(iJd }ˆˆˆ2ˆˆˆˆˆˆ{ 21222111111
)2(ooeooeooeeff eeeeeeeeedd
3coscos}sincos3cos{cos
}cossincos2coscossincos{cos
1123
11)2(
23211
)2(
ddd
dd
eff
eff
)sin,sin cos,cos cos(ˆ)0,cos,sin(ˆ
e
o
ee
okojijkei eedeeoo ˆˆˆ uniaxial ve- I Type )2(
is normally fixed by the phase-matching condition and only can be variedFor this example, need 3 = 0, m = 0, /3 or 2/3 for optimum conversion!
)2(iJd
}ˆˆˆˆˆˆ2ˆˆˆ{ 22121211111)2(
ooeooeooeeff eeeeeeeeedd
3coscos}sincos3cos{cos
}coscoscossincos2sincoscos{
1123
11)2(
32211
)2(
ddd
dd
eff
eff
This result is the same as for Type I eoo!!
General Result for Kleinman Symmetry!Identical result for eoo and oeo, and eoe and oee!!!That is, can interchange the order of the o and e!
)sin,sin cos,cos (cosˆ)0,cos,sin(ˆ
e
oee
okejijkoi eedeoeo ˆˆˆ uniaxial ve II Type )2(
d110 0
-d110 0
0 0 0
d140 0
0 -d140
0 -d110
SHG With Finite Beams- previous results were valid for cw plane waves- much of NLO is done with pulsed lasers and finite beams- assume gaussian beams which propagate as linear “eigenmodes” when vector effects can be neglected, i.e. for beams many wavelengths wide
y
x
z
0
10
20
220
2
2
20
20
0
tan)( fieldfor (diameter) size"spot " minimum2
fieldfor (diameter) size"spot " )(2 1)(
Curvature of Radius 1)( range"Rayleigh " )(
)()()(
zzzw
zwzzwzw
zzzzRnwz
vac
..)(2)(
)(exp)(
)(21),(
2
2
20
0 cczR
kizw
tzkzizw
wtrE
E
Case I L<<z0 L2
00 |)(|)(21)( intensity axis"-on" EncI
))(
2exp(),0(),(
..)(exp )(21),(
20
2
20
2
0
wII
ccw
tkzitrE
E
Calculate harmonic power in negligible depletion regime. Assume k=0 and a Type I interaction
)(2
20
)2(]2)2([ 2
0
2
exp )()2(
)2,,( .. )2,,(21)( :SH
wefftzki
cn
diz
dzdccez,trE
EEE
Lwcn
diL eff
)(2exp)(
)2()2,,( 2
0
220
)2(
EE
2
)()2( 2
)()2( Note 00
202
0 wwww
2 beam narrower than beam!! This is a general result in NLO. The mixing of optical beams always leads to a nonlinear polarization narrower in space (and time) than any of the mixing fields.
)()(
)()(
)2(2)2()(
)2()2()2()2(
0
20
)2()(
20
20
0
znw
nwnwz
vac
nnvacvac
Rayleigh range the same for fundamental and harmonic
L
Case II: Optimum Conversion L 2z0()
- There is a trade-off between minimum beam width and crystal length L- As beam spreads, conversion efficiency goes down- Consider simplest case of Type I phase match, and negligibly small birefringence
0w
d
dA=2d
ddz
zwzww
cn
dcnP
L
L
eff 22
2
2
2
2
20
0
220
)2(
0 } ),(
2exp),(
)({2|)()2(
|)2(21)2(
E
Beam Profile
For detailed treatment, see G.D. Boyd and D.A. Kleinman, “Parametric Interaction ofFocused Gaussian Light Beams”, J. of Applied Physics, 39, No. 8, pp 3597-3639 (1968)
)()()2()(
3.4)()2(
068.1)( where7.5for occurs optimum
032
22
opt
00
LPcnn
dP
P
hzL
vac
eff
For small , h0() =,
For = 1, h0() 0.8
For >> 1, h0() -1 (note L >> 2z0!)
)()()2()(
2.3)()2(
032
22
LP
cnn
dP
P
vac
eff
)!!(not )()2( VNB )()(
)()2()(4
)()2(
2 where)(function Define
20
032
220
0
LLP
PLPhcnn
dP
P
zLh
vac
eff
For walk-off, h0() is reduced, see Boyd and Kleinman paper.
)(2)()2()(
4)()2( 0
032
22
Pz
cnn
dP
P
vac
eff
)(
)()2()(
2
)2 2
032
0
2)2(
P
nncw
Ld
P(ωω)P( eff
Pulsed Fundamental Finite BeamBeam Intensity Profile
t
- Assume a gaussian shaped temporal pulse
field!for is )( )(
2exp )0,(),( 2
2
tPtP
2)()2(
)(4exp )2(),( ),2( SH 2
22
tPtPtP
\ SH pulse is compressed in time relative to the fundamental pulseRecall SH spatial profile was also compressedGeneral Property of NLO: outputs compressed in space and time!
Normally it is the pulse energy that is measured.
dttPE ii ),()(
)0,2()(4
)2( )0,()(2
)( /22
PEPEadteat
)()2()()(
2
)0,()0,2(
)(1
)()2( 2
032
02/3
2)2(
22
nncw
Ld
PP
EE eff
Fundamental or Harmonic Beam Walk-Off
dze
cn
diyxLeyxz
L wzyx
effwyx
0)(
)][(220
)2()(2
020
22
20
22
)()2(
),,2,( )(),,,(
EEEE
Assume walk-off along y-axis only, small walk-off angle , on phase-matchAssume that w(z) w0
Interaction distance limited conversion efficiency reduced
trade-off between walk-off angle , beam size and length L<2z0)(zw
zyy
)( /2
)(2
)(2
)(2 define 0
000
wLLLw
Lw
zw
Lyu wowo
),()(
2exp)()2(
)2,( 1),( 20
220
)2(
0)( 2
uFw
xLcn
diLdeuF effu
EE
)()(
)()2()(
2
)(2 ),(2)( 23
020
2)2(2
PG
nncw
Ld
Pω)P( duuFG eff
e.g. KDP is uniaxial with the point group symmetry , and a transmission range of 0.35 to4.5 microns. The refractive indices with in microns are given by the general formulas
Example of Second Harmonic Generation
0535.1277580.0
)0014.0(0097.01295.2 ;
8984.577623.1
)0142.0(0101.02576.2 2
2
22
2
2
22
eo nn
(a) What input intensity I() is needed for 58% conversion for Type 1 SHG with 1.06m plane wave input?(b) For a low conversion efficiency of 10%, what is the fundamental power required for for a crystal of length L=2z0? What is w0 at the center of the crystal? What is w at the input?
m24
5129.1)2( 4709.1)2( 4942.1)( 4603.1)( oeoe nnnn
0
22
22
26.41 65942.0)sin(
)2()2()()2(
)()2(
)sin( ),2()(
PMPM
eo
oo
o
ePMeo
nnnn
nn
nn
For a negative uniaxial, Type I phase-match is eoo
4 )2sin()sin(ˆˆ 24For )2(
14)2( optPMeff ddm
pm/Vdpm/Vd eff 28.0)26.41sin(43.0ˆ 43.0ˆ 0)2()2(14
For conversion efficiencies > 10% with plane waves, it is necessary to first calculate theparametric gain length and then use to calculate the efficiency.For 58% conversion efficiency,
)/(tanh)(/)2( 2pgLII
.pgL
)( )/in ,()in (
)in (|ˆ|172.0 )cin (
)/in ,()in ()()(
)/in (|ˆ|22)in (
22/3
vac
)2(11
2
vac0
)2(11
sultgeneral recmMWInm
pm/Vdm
mWImcnn
Vmdm
effpg
effpg
222/3
11 1.6 )/()49.1(06.128.0172.01)cin ( GW/cmcmMWI
xxmpg
mrxx
xnnL PMeo
vacPM 64.0
)52.82sin(042.04101.06 )2sin(|)}2()2({|
4)(
0
-41
milliradsn
nnPM
o
oe 28 )2sin()2(
)2()2( angle off-Walk
(b) For L=2z0
)in ,()in ()in ()2()(
)]in (|ˆ[|10475.0
)()()2()(
|ˆ|2.3
)()2(
3vac
2
2)2(3
vac032
2)2(2
WPcmLμmnn
pm/Vdx
LPcnn
d
PP
eff
eff
eff
eff
KW6.10)( )in ,(06.149.1
]28.0[10475.01.0)()2( 33
23
PWP
xx
PP
μmzzwLwμmwLnw
vac6.4711]1[)
2(facet input at 6.33
2 2
0
2
00
20
e.g. LiNbO3 is a 3m uniaxial crystal. Consider both non-critical birefringent phase match (Type 1) and QPM wavevector-match in a L=1cm crystal with 1.06m input;(a) For birefringent phase-match calculate the input intensity required for plane wave input for 58% conversion into the harmonic. Assuming a beam input with L=2z0, find the input power required for 10% conversion efficiency. Find the angular bandwidth.(b) Repeat for QPM
(a) Type 1 eoo wavevector-match occurs very close to 1.06m input. The relevant nonlinear coefficient is )2(
31d
pm/Vdnnn oeo 95.5 32.2)2( 24.2)2()( 31
222/3
vac
)2(11 1.12)( )/in ,(
)in (
)in (|ˆ|172.0 )in ( MW/cmIcmMWI
nm
pm/Vdcm eff
pg
WWPx
x
WPcmLμmnn
pm/Vdx
PP eff
80 )in ,(06.124.2
]95.5[10475.01.0
)in ,()in ()in ()2()(
)]in (|ˆ[|10475.0
)()2(
33
23
3vac
2
2)2(3
02/1
4
2/1
108.0104x
06.1 )]2()2([4
)(
xnnLλ
PMeo
vacPM
And of course there is no walk-off!
(b) VpmdVpmdddp effeff /16 /28 2 1,For )2()2(33
)2(33
)2(
222/3
vac
)2(11 3.1)( )/in ,(
)in (
)in (|ˆ|172.0 )in ( MW/cmIcmMWI
nm
pm/Vdcm eff
pg
WWPxx
x
WPcmLμmnn
pm/Vdx
PP eff
9.0 )in ,(06.132.224.2
]18[10475.01.0
)in ,()in ()in ()2()(
)]in (|ˆ[|10475.0
)()2(
32
23
3vac
2
2)2(3
01 2.1|}4
)()]2()([)]2()({[|4
)(
vac
eeoovac
PM nnnnL
042.0)]()2([21
4)( ;083.0)()2( ;107.0)2()2( ;090.0)()(
ee
vaceeeoeo nnnnnnnn
The large angular bandwidth, no walk-off of “non-critical” phase-match are clear.Also, QPM allows tuning of the frequencies, a desirable feature.
Applications of QPM Engineering
y
x
Tunable SHG
By varying the QPM period in a direction orthogonalthe wave-vector matching condition,
different frequencies can be doubled.k = 2ke() - ke(2) + 2p/(y)
Enhanced Bandwidth
gxiKxiKiKx
effeff KKcceeedxd gg ..}{21)( )2()2(
Primary periodicity is modulated by a secondary, longer period grating
Wavevector Match at k+KKg=0
)(k
)(k
)2( k
K
gK
Wavelength
P(2)
Single Period Grating
Wavelength
P(2)
efficiencySHG in reduction vsbandwidth Increased :offTrade][)2( :bandwidth gratings Aperiodic
][)2( :bandwidth field 2 :grating uniform1
21
22121
/
//
kI
kILk
Instead of just one modulation, consider multiple, random modulation periods.Therefore there is a quasi-continuum of .gK
)}()2({ )(v)2(v max
maxmaxmax
g
max
g
ggg
g ncτL
cnL
nnc
LLL
Lmax – maximum length for walk-off to maintain interaction efficiency
vg – group velocity = c/ng ng – group index
Interaction efficiency greatly reduced when walk-off time pulse width t vg()
t vg(2)
Frequency Doubling of Ultrafast Pulses
1. Pulse walk-off between fundamental and harmonic is issue with fs pulses2. Wide bandwidths of fs pulses require short lengths3. Usually, efficiency scales with pulse energy for fs pulses FOM (%/nJ)
pulse match-phase
pulse
2
minpulsemin
2
match phase5.0
5.0
LL
FOMFOM limitedbandwidth Define max
2)2(
maxcwuf Ln
dL eff
To efficiently convert allwavelengths in the pulse requires
min
2min
2min2 5.0
222
)2
(sinc)2,(L
LkLLP pmpm
Material Wavevector-Match
FOMufpm2/(V2)
Efficiency%/nJ
L(mm)
PPLN Non-critical 710 95 0.4
LBO Non-critical 42 6 5.0
KTP Type 2 critical 20 1.5 1.5
pulsematch-phase
zPulse Compressed SHG
x,t x,t
I(x) I(x)
Spatially Shaped SHG Beams
- Pattern grating response in 1D (only) )(I
02w
)2( I
02wRegions of reduced SHG,
contoured for flat top beams
uniform grating
truncatedat 05.0 w
truncatedat 00.75w
“Flat top” beams
y