P. Piot, PHYS 630 – Fall 2008

Nonlinear optics• To date we considered dielectric media characterized by a linear

relation between polarization and E-fied

• Now we consider media characterized by a nonlinear relationbetween E and P

!

P = "0#E

!

P = a1E +1

2a2E

2 +1

6a3E

3 + ...

= "0 #E + # (2)E 2 + # (3)E 3 + ...( )= "0#E + 2dE 2 + 4# (3)E 3 + ...

2nd order 3rd order

P. Piot, PHYS 630 – Fall 2008

Nonlinear wave equation I• From Maxwell’s equation and now considering

• We obtain a nonlinear wave equation

where P is usually written as

!

"2E #

1

c0

2$t

2E = µ

0$t

2P

!

P = "0#E + P

NL

!

PNL

= 2dE2

+ 4" (3)E 3+ ...

!

D = "0E + P

P. Piot, PHYS 630 – Fall 2008

Nonlinear wave equation II• Using

• The nonlinear wave equation can be rewritten

with

• Born approximation consist in adopting an iterative solution for theNL wave equation

!

"2E #

1

c2$t

2E = S

S = µ0$t

2PNL

!

c = c0/n

n2

=1+ "

“source”

radiationS E

S(E)

P. Piot, PHYS 630 – Fall 2008

• Assume higher order than second order are negligible

• Take

• Corresponding polarization is

Second harmonic generation (SHG) I

!

PNL

= 2dE2

!

E =1

2E(")ei"t + c.c.( )

!

E = 2d1

4E(")ei"t + c.c.( ) E *

(")e# i"t + c.c.( )

= dEE * + d EEe2i"t + c.c.( )

dcoptical rectification

2ω2nd harm. generation

E

P

= +

P. Piot, PHYS 630 – Fall 2008

• Use to frequency-double lasers

Second harmonic generation (SHG) II

P.A. Franken, et al, Physical Review Letters 7, p. 118 (1961)

P. Piot, PHYS 630 – Fall 2008

Second harmonic generation (SHG) III

• The actually published results…

Input beamThe second harmonic

Note that the very weak spot due to the second harmonic is missing.It was removed by an overzealous Physical Review Letters editor,who thought it was a speck of dirt.

P. Piot, PHYS 630 – Fall 2008

• Consider a plane wave propagating along z, and write the paraxial Hemholtzequation

• Can be approximated as

where Δk=k(2ω)-k(ω) so finally

need Δk=0 to achieve maximumE(2ω) electric field (this was nottaken care in the 1961 experiment)

• This is referred to as “phase matching”

Phase matching I

!

"#

2E(2$) % 2ik&

zE(2$)[ ]e%k(2$ )z = µ

0&t

2dE

2($)( )e%k($ )z

!

"zE(2#) = AdE

2(#)e$kz

!

E(2")#sin$kz

$kz

Poor phase matching

Good phase matching

P. Piot, PHYS 630 – Fall 2008

Phase matching II

which will only be satisfied when:

Unfortunately, dispersion prevents this from ever happening!

So we’re creating light at ωsig = 2ω.

0

2 2 ( )polk k nc

!!= =

0 0

(2 )( ) (2 )

sig

sig sigk n nc c

! !! != =

sig polk k=

! 2!Frequency

Refra

ctive

in

dex

And the k-vector of the polarization is:

The phase-matching condition is:

The k-vector of the second-harmonic is:

!

n(2") = n(")

P. Piot, PHYS 630 – Fall 2008

Phase matching III

We can now satisfy the phase-matching condition.

Use the extraordinary polarizationfor ω and the ordinary for 2ω.

Birefringent materials have different refractive indices for different polarizations. Ordinary and extraordinary refractive indicescan be different by up to ~0.1 for SHG crystals.

! 2!FrequencyRe

fract

ive in

dex

!

ne

on

ne depends on propagation angle, so we can tune for a given ω.Some crystals have ne < no, so the opposite polarizations work.

!

no(2") = n

e(")

P. Piot, PHYS 630 – Fall 2008

Phase matching IV

0

ˆ2 cos

2 ( )cos

pol

pol

k k k k z

k nc

!

"" !

#= + =

$ =

r r r

2(2 )sig

o

k nc

!!=

ˆˆcos sink k z k x! != "r

ˆˆcos sink k z k x! !" = +r

z

x

But: So the phase-matchingcondition becomes:

θ

θ

!

n(2") = n(")cos#

•Phase matching via non-colinear overlap oftwo ω beams

P. Piot, PHYS 630 – Fall 2008

Application of noncollinear phase matching

•Noncollinear phasematching can be usedto measure the durationof ultra-short pulse