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P. Piot, PHYS 630 – Fall 2008
Phase matching bandwidth
I
Δk
Phase-matching only works exactly for one wavelength, say λ0.Since ultrashort pulses have lots of bandwidth, achievingapproximate phase-matching for all frequencies is a big issue.
The range of wavelengths (or frequencies) that achieve approximatephase-matching is the phase-matching bandwidth.
[ ]4
( ) ( ) ( / 2)k n n!
" " ""
# = $
0!
0
2
!Wavelength
Refra
ctive
inde
x
!
ne
!
no
2 2( ) ( / ) sinc ( / 2)sigI L L k L!" #
Recall that the intensity out of anSHG crystal of length L is:
where:
( ) )/ 2 (n n! !"
2
!!
P. Piot, PHYS 630 – Fall 2008
Phase matching bandwidth
!k (") =4#
"n(" )$ n(" / 2)[ ]
0 0 0 0
0 0
4( ) 1 ( ) ( ) ( / 2) ( / 2)
2k n n n n
! "# "## # "# # # #
# #
$ % $ %& &' = ( + ( () * ) *+ ,+ ,
because, when the input wavelength changes by δλ, the second-harmonic wavelength changes by only δλ/2.
The phase-mismatch is:
Assuming the process is phase-matched at λ0, let’s see what thephase-mismatch will be at λ = λ0 + δλ
x xBut the process is phase-matched at λ0
0 0
0
4 1( ) ( ) ( / 2)
2k n n
! "## # #
#
$ %& &' = () *+ ,
to first orderin δλ
P. Piot, PHYS 630 – Fall 2008
Phase matching bandwidth
The sinc2 curve will decrease by afactor of 2 when Δk L/2 = ± 1.39.
So solving for the wavelengthrange that yields |Δk | < 2.78/L
yields the phase-matchingbandwidth.
0
10 02
0.44 /
( ) ( / 2)FWHM
L
n n
!"!
! !=
# #$
0 0
0
4 12.78 / ( ) ( / 2) 2.78 /
2L n n L
! "## #
#
$ %& &' < ' <( )* +
I
Δk
FWHM
2.78/L-2.78/L
sinc2(ΔkL/2)
P. Piot, PHYS 630 – Fall 2008
Phase matching bandwidth examples
BBO KDP
The phase-matching bandwidth is usually too small, but it increases asthe crystal gets thinner or the dispersion decreases (i.e., thewavelength approaches ~1.5 microns for typical media).
The theory breaks down, however, when the bandwidthapproaches the wavelength.
P. Piot, PHYS 630 – Fall 2008
Group velocity mismatchInside the crystal the two different wavelengths have different groupvelocities.
Define the Group-VelocityMismatch (GVM):
0 0
1 1
v ( / 2) v ( )g g
GVM! !
" #
Crystal
As the pulse enters the crystal:
As the pulseleaves the crystal:
Second harmonic createdjust as pulse enters crystal(overlaps the input pulse)
Second harmonic pulse lagsbehind input pulse due to GVM
P. Piot, PHYS 630 – Fall 2008
Group velocity mismatch
0 / ( )v ( )
1 ( )( )
g
c n
nn
!!
!!
!
=
"#
0 0 0 00 0
0 0 0 0
( / 2) / 2 ( )1 ( / 2) 1 ( )
( / 2) ( )
n nn n
c n c n
! ! ! !! !
! !
" # " #$ $= % % %& ' & '
( ) ( )
00 0
0
1( ) ( / 2)
2GVM n n
c
!! !
" #$ $= %& '( )
Calculating GVM:
0
1 ( )1 ( )
v ( ) ( )g
nn
c n
! !!
! !
" #$= %& '
( )So:
0 0
1 1
v ( / 2) v ( )g g
GVM! !
" #
But we only care about GVM when n(λ0/2) = n(λ0)
P. Piot, PHYS 630 – Fall 2008
Effect of group velocity mismatch
Assuming that a very short pulseenters the crystal, the length of the ,SH pulse, δt, will be determined bythe difference in light-travel timesthrough the crystal:
! t =L
v g("0 / 2)#
L
v g("0 )= L GVM
Crystal
L GVM << ! pWe always try to satisfy:
P. Piot, PHYS 630 – Fall 2008
Effect of group velocity mismatch
L /LD
Second-harmonic pulse shape for different crystal lengths:
It’s best to use a very thin crystal. Sub-100-micron crystals are common.
!
LD "# p
GVM
Inputpulseshape
LD is the crystallength thatdoubles thepulse length.
P. Piot, PHYS 630 – Fall 2008
Effect of group velocity mismatch
P. Piot, PHYS 630 – Fall 2008
Effect of group velocity mismatchLet’s compute the second-harmonic bandwidth due to GVM.
Take the SH pulse to have a Gaussian intensity, for which δt δν = 0.44.Rewriting in terms of the wavelength,
δt δλ = δt δν [dν/dλ]–1 = 0.44 [dν/dλ]–1 = 0.44 λ2/c0
So the bandwidth is:
0
10 02
0.44 /
( ) ( / 2)FWHM
L
n n
!"!
! !#
$ $%
Calculating the bandwidth by considering the GVM yields the sameresult as the phase-matching bandwidth!
2 2
0 0 0 00.44 / 0.44 /
FWHM
c c
t L GVM
! !"!
"# =
00 0
0
1( ) ( / 2)
2GVM n n
c
!! !
" #$ $= %& '( )
P. Piot, PHYS 630 – Fall 2008
Difference frequency generation
ω1
ω1
ω3
ω2 = ω3 − ω1
Parametric Down-Conversion(Difference-frequency generation)
Optical ParametricOscillation (OPO)
ω3
ω2
"signal"
"idler"
By convention:ωsignal > ωidler
ω1
ω3 ω2
Optical ParametricAmplification (OPA)
ω1
ω1
ω3ω2
Optical ParametricGeneration (OPG)
Difference-frequency generation takes many useful forms.
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