Advanced Vitreous State – The Physical Properties of...

Advanced Vitreous State – The Physical Properties of Glass

Active Optical Properties of Glass Lecture 20: Nonlinear Optics in Glass-Fundamentals

Denise KrolDepartment of Applied ScienceUniversity of California, DavisDavis, CA [email protected]

1

[email protected] Advanced Vitreous State - The Properties of Glass: Active Optical Properties of Glass 2

Active Optical Properties of Glass

2

1. Light emission(fluorescence, luminescence)

Optical amplification and lasing

2. Nonlinear Optical Properties

Optical transitions, spontaneous emission, lifetime, linebroadening, stimulated emission, population inversion, gain,amplification and lasing, laser materials, role of glass

Fundamentals: nonlinear polarization, 2nd-ordernonlinearities, 3rd-order nonlinearitiesApplications: thermal poling, nonlinear index, pulsebroadening, stimulated Raman effect, multiphotonionization


!

P(t) = "pi = #Nex(t) =Ne

2E0

m

1

$0

2#$ 2 # i$%

e#i$t

Oscillating dipole:

!

p = "ex(t)

Polarization:

Linear optics-classical electron oscillator

!

x(t) = x0e" i#t

=-eE

0

m

1

#0

2"# 2 " i#$

e"i#t

Solution: electron oscillates at driving frequency

Equation of motion for bound electron:

!

me

d2x

dt2

= "me#dx

dt"Kx " eE

0e"i$t

damping force binding force driving force

electron

ion core!

K = me"0

2

resonancefrequency


Frequency dependent dielectric constantelectron

ion core!

K = me"0

2

resonancefrequency

ω0

!

"D(#) =1+ $(#) =

P(t)

"oE(t)

=Ne

2

"om

1

#0

2%# 2 % i#&

!

P(t) = "0#E(t)

!

" #( ) =P(t)

$oE(t)

=Ne

2

$om

1

#0

2%# 2 % i#&

linear susceptibility


Nonlinear optics-anharmonic binding force

binding force

linear optics

!

F = "Kx

electron

ion core

Equation of motion for bound electron:

!

me

d2x

dt2

= "me#dx

dt"Kx " ax 2 + (.....) " eE

0e" i$t


F

x

!

F = "Kx " ax2

(+higher order terms)

nonlinear optics

x


Nonlinear optics-classical electron oscillator

Equation of motion for bound electron:electron

ion core

This equation has no general solution, we will solve it by using a perturbation expansion.We replace E(t) by λE(t) and seek a solution in the form:

!

x = "x(1) + "2x(2) + "3x(3) + ...The parameter λ is a parameter that characterizes the strength of the perturbation.It ranges continuously between 0 and 1 and we will set it to 1 at the end of the calculation.

Substitute expression for x in above equation of motion and group terms withequal powers of λ

!

me

d2x

dt2

= "me#dx

dt"Kx + ax 2 " e E

0e" i$t + cc[ ]


[email protected] Advanced Vitreous State - The Properties of Glass: Active Optical Properties of Glass 7!

x(2)(t) =

"ae2E0

2

m2

1

D(2#)D2(#)

e"i2#t

+1

D(0)D(#)D("#)

$

% &

'

( )

!

x(2)(t)" E(t)[ ]

2

!

D(" j ) = ("0

2 #" j

2) # i" j$

oscillates at 2ω “oscillates” at 0

Nonlinear optics-classical electron oscillator

!

x(1)(t) =

-eE0

m

1

"0

2#" 2 # i"$

e#i"t

!

x(1)(t)" E(t)

:

€

˙ ̇ x (1) + 2γ˙ x (1) +ω02x(1) = −eE(t) /m ( 4 )

2:

€

˙ ̇ x (2) + 2γ˙ x (2) +ω02x(2) + a x(1)[ ]

2= 0 ( 5 )

3:

€

˙ ̇ x (3) + 2γ˙ x (3) +ω02x(3) + 2ax(1)x(2) = 0


χ(2) resonances

!

x(2)(t) =

"ae2E0

2

m2

1

D(2#)D2(#)

e"i2#t

+1

D(0)D(#)D("#)

$

% &

'

( )

!

x(2)(t)" E(t)[ ]

2

oscillates at 2ω “oscillates” at 0

!

P(2)(t) = "Nex

(2)(t)

2nd-order nonlinear polarization

!

D(" j ) = ("0

2 #" j

2) # i" j$

!

" (2)(2#) =P(2)(2#)

E(#)2=aNe

3E0

2

m2

1

D(2#)D2(#)

2nd-order susceptibility (for SHG)

2nd-order susceptibility is enhanced when either ω or 2ω are equal to ω0

ω02ω

ω


χ(2) frequency terms

When the input field has 2 frequency components:

!

E( t) = E1e"i#1t + E

2e"i#2t

+ C .C

!

P(2)( t) = " (2)[E

1

2e#i2$1t + E

2

2e#i 2$2t + 2E

1E2e#i ($1+$2 )t + 2E

1E

2

*e#i ($1 #$2 ) t + c.c]

+ 2" (2)[E1E1

*+ E

2E2

*]

= P($n)e

#i$nt

n

% .

the 2nd-order polarization has many frequency components (mixing terms):

!

P(2"1) = # (2)E

1

2(SHG),

P(2"2) = # (2)E

2

2(SHG),

P(0) = 2# (2)(E1E1

*+ E

2E2

*) (OR),

P("1+"

2) = 2# (2)E

1E2

(SFG ),

P("1$"

2) = 2# (2)E

1E

2

*(DFG ).

Second harmonic generation

Optical rectification

Sum-frequency generationDifference-frequency generation


Nonlinear optics: general formalism

!

E(t) = E1e" i#1t + E

2e" i#2t + c.c.

The em field is expressed as sum of frequency components, for example

!

P( t) = P("n)e

#i"nt

n

$The induced polarization can be written as:

The frequency components ωn are determined by the order of theinteraction and the input frequencies

2nd-order interaction:input frequencies, E induced polarization frequencies, P

One ω1 2ω1, 0Two ω1, ω2 2ω1, 2ω2, 0, ω1+ω2, ω1-ω2Three ω1, ω2, ω3 2ω1, 2ω2, 2ω3 ,0, ω1+ω2, ω1+ω3, ω2+ω3, ω1-ω2, ω1-ω3, ω2-ω3

linear nonlinear

!

r P ( t) = " (1) #

r E (t)+ " (2) #

r E ( t)

2+ " (3) #

r E (t)

3+ # # #


3rd-order polarization and nonlinear index

!

P(3") = # (3)E0

3

P(") = 3# (3)E0

2E0

*

has frequency components:

!

P(3)( t) = " (3)E(t)3

Applied field, one input frequency

3rd order nonlinear polarization

!

E = E0e"i#t

!

P(") = # (1)E0

+ 3# (3)E0

2E0

*

!

P(") = (# (1) + 3# (3)E0E0

*)E

0= # (eff )E

0

!

" (eff ) = " (1) + 3" (3) E0

2

The total polarization at ω has linear and nonlinear contributions

!

n = n0

+ n2I

!

n2"# (3)

n2 = nonlinear index coefficient!

E0

2

" I

intensity


χ(3) - frequency terms

With three input fields: ω1, ω2, ω3

P(3) has the following frequency components:

3 1, 3 2, 3 3

1, 2, 3 2 1± 2, 2 1± 3, 2 2± 1, 2 2± 3, 2 3± 1, 2 3± 2

1+ 2+ 3, 1+ 2- 3, 1+ 3 2, 2+ 3- 1


2nd-order polarization

!

P(t) = P(1)(t) + P

(2)(t) = " (1)E(t) + " (2)E(t)2

!

P(t) = sin "t( ) + 0.5sin "t( )2

!

sin "t( )

!

"0.25cos 2#t( )

!

0.25

linear polarization

nonlinear polarization


Glass is isotropic

!

" (2) = 0

χ(2) requires lack of inversion symmetry

!

P(t) = P(1)(t) + P

(2)(t) = " (1)E(t) + " (2)E(t)2

!

P(2)(t) = " (2)E(t)2

!

"P(2)(t) = # (2) "E(t)[ ]2

!

P(2)(t) = "P(2)(t)#$ (2) = 0

In material with inversion symmetry


χ(2) tensor

P and E are vectors!So χ(n)’s are tensors

!

Px("

n+"

m) = #

xxx

(2)("

n+"

m,"

n,"

m)E

x("

n)E

x("

m)

!

Py ("n +"m ) = #yxx

(2)("n +"m,"n ,"m )Ex ("n )Ex ("m )


Nonlinear wave equation

!

"#2r E +

$ (1)

c2

% 2r E (1)

%t2

= "4&

c2

% 2r P

NL

%t2

Propogation of waves is described by nonlinear wave equation

Growth of SH wave depends on propagation length and is optimized when Δk=o(phase matching)

!

"A2

"z=4#ideff$2

2

k2c2

A1

2ei%kz

!

"A1

"z=8#ideff$1

2

k1c2

A2A1

*e% i&kz

which leads to a set of coupled differential equations (example SHG)

!

"k = k2# 2k

1


Nonlinear optics in glass

2nd-order nonlinearities In normal glasses χ(2)=0

3nd-order nonlinearities All materials, including glasses, have a χ(3)

In glass there are only three independent χ(3) tensor elements

χ(3) processes involve the interaction of 3 input waves to generate apolarization (4th wave) at a mixing frequency

-with 3 different input frequencies there are many possible output frequencies

Strength of generated signal depends on propagation length-optical fibers!

Phase matching: Δk=k4-k3-k2-k1=0

Advanced Vitreous State – The Physical Properties of...

Documents

Transcript of Advanced Vitreous State – The Physical Properties of...