Phase matching bandwidth - Northern Illinois University

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P. Piot, PHYS 630 – Fall 2008 Phase matching bandwidth I Δk Phase-matching only works exactly for one wavelength, say λ 0 . Since ultrashort pulses have lots of bandwidth, achieving approximate phase-matching for all frequencies is a big issue. The range of wavelengths (or frequencies) that achieve approximate phase-matching is the phase-matching bandwidth. [ ] 4 () () ( / 2) k n n ! " " " " # = $ 0 ! 0 2 ! Wavelength Refractive index n e n o 2 2 () ( / ) sinc ( / 2) sig I L L kL ! " # Recall that the intensity out of an SHG crystal of length L is: where: ( ) ) /2 ( n n ! ! " 2 ! !

Transcript of Phase matching bandwidth - Northern Illinois University

Page 1: Phase matching bandwidth - Northern Illinois University

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

!!

Page 2: Phase matching bandwidth - Northern Illinois University

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

Page 3: Phase matching bandwidth - Northern Illinois University

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)

Page 4: Phase matching bandwidth - Northern Illinois University

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.

Page 5: Phase matching bandwidth - Northern Illinois University

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

Page 6: Phase matching bandwidth - Northern Illinois University

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)

Page 7: Phase matching bandwidth - Northern Illinois University

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:

Page 8: Phase matching bandwidth - Northern Illinois University

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.

Page 9: Phase matching bandwidth - Northern Illinois University

P. Piot, PHYS 630 – Fall 2008

Effect of group velocity mismatch

Page 10: Phase matching bandwidth - Northern Illinois University

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

!! !

" #$ $= %& '( )

Page 11: Phase matching bandwidth - Northern Illinois University

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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