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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 95616dmkrol@ucdavis.edu
1
dmkrol@ucdavis.edu 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
dmkrol@ucdavis.edu Advanced Vitreous State - The Properties of Glass: Active Optical Properties of Glass 3
!
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
dmkrol@ucdavis.edu Advanced Vitreous State - The Properties of Glass: Active Optical Properties of Glass 4
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
$om
1
#0
2%# 2 % i#&
linear susceptibility
dmkrol@ucdavis.edu Advanced Vitreous State - The Properties of Glass: Active Optical Properties of Glass 5
Nonlinear optics-anharmonic binding force
binding force
linear optics
!
F = "Kx
electron
ion core
Equation of motion for bound electron:
!
me
d2x
dt2
= "me#dx
dt"Kx " ax 2 + (.....) " eE
0e" i$t
damping force binding force driving force
F
x
!
F = "Kx " ax2
(+higher order terms)
nonlinear optics
x
dmkrol@ucdavis.edu Advanced Vitreous State - The Properties of Glass: Active Optical Properties of Glass 6
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[ ]
damping force binding force driving force
dmkrol@ucdavis.edu 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
dmkrol@ucdavis.edu Advanced Vitreous State - The Properties of Glass: Active Optical Properties of Glass 8
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
dmkrol@ucdavis.edu Advanced Vitreous State - The Properties of Glass: Active Optical Properties of Glass 9
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
dmkrol@ucdavis.edu Advanced Vitreous State - The Properties of Glass: Active Optical Properties of Glass 10
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
$The induced polarization can be written as:
The frequency components ωn are determined by the order of theinteraction and the input frequencies
2nd-order interaction:input frequencies, E induced polarization frequencies, P
One ω1 2ω1, 0Two ω1, ω2 2ω1, 2ω2, 0, ω1+ω2, ω1-ω2Three ω1, ω2, ω3 2ω1, 2ω2, 2ω3 ,0, ω1+ω2, ω1+ω3, ω2+ω3, ω1-ω2, ω1-ω3, ω2-ω3
linear nonlinear
!
r P ( t) = " (1) #
r E (t)+ " (2) #
r E ( t)
2+ " (3) #
r E (t)
3+ # # #
dmkrol@ucdavis.edu Advanced Vitreous State - The Properties of Glass: Active Optical Properties of Glass 11
3rd-order polarization and nonlinear index
!
P(3") = # (3)E0
3
P(") = 3# (3)E0
2E0
*
has frequency components:
!
P(3)( t) = " (3)E(t)3
Applied field, one input frequency
3rd order nonlinear polarization
!
E = E0e"i#t
!
P(") = # (1)E0
+ 3# (3)E0
2E0
*
!
P(") = (# (1) + 3# (3)E0E0
*)E
0= # (eff )E
0
!
" (eff ) = " (1) + 3" (3) E0
2
The total polarization at ω has linear and nonlinear contributions
!
n = n0
+ n2I
!
n2"# (3)
n2 = nonlinear index coefficient!
E0
2
" I
intensity
dmkrol@ucdavis.edu Advanced Vitreous State - The Properties of Glass: Active Optical Properties of Glass 12
χ(3) - frequency terms
With three input fields: ω1, ω2, ω3
P(3) has the following frequency components:
3 1, 3 2, 3 3
1, 2, 3 2 1± 2, 2 1± 3, 2 2± 1, 2 2± 3, 2 3± 1, 2 3± 2
1+ 2+ 3, 1+ 2- 3, 1+ 3 2, 2+ 3- 1
dmkrol@ucdavis.edu Advanced Vitreous State - The Properties of Glass: Active Optical Properties of Glass 13
2nd-order polarization
!
P(t) = P(1)(t) + P
(2)(t) = " (1)E(t) + " (2)E(t)2
!
P(t) = sin "t( ) + 0.5sin "t( )2
!
sin "t( )
!
"0.25cos 2#t( )
!
0.25
linear polarization
nonlinear polarization
dmkrol@ucdavis.edu Advanced Vitreous State - The Properties of Glass: Active Optical Properties of Glass 14
Glass is isotropic
!
" (2) = 0
χ(2) requires lack of inversion symmetry
!
P(t) = P(1)(t) + P
(2)(t) = " (1)E(t) + " (2)E(t)2
!
P(2)(t) = " (2)E(t)2
!
"P(2)(t) = # (2) "E(t)[ ]2
!
P(2)(t) = "P(2)(t)#$ (2) = 0
In material with inversion symmetry
dmkrol@ucdavis.edu Advanced Vitreous State - The Properties of Glass: Active Optical Properties of Glass 15
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 )
dmkrol@ucdavis.edu Advanced Vitreous State - The Properties of Glass: Active Optical Properties of Glass 16
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
dmkrol@ucdavis.edu Advanced Vitreous State - The Properties of Glass: Active Optical Properties of Glass 17
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
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