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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 Krol Department of Applied Science University of California, Davis Davis, CA 95616 dmkrol@ucdavis.edu

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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, line broadening, stimulated emission, population inversion, gain, amplification and lasing, laser materials, role of glass Fundamentals: nonlinear polarization, 2nd-order nonlinearities, 3rd-order nonlinearities Applications: thermal poling, nonlinear index, pulse broadening, stimulated Raman effect, multiphoton ionization

dmkrol@ucdavis.edu Advanced Vitreous State - The Properties of Glass: Active Optical Properties of Glass 3

P(t) = pi = Nex(t) = Ne2E0 m 1

2 2 i eit

Oscillating dipole: p = ex(t) Polarization:

Linear optics-classical electron oscillator

x(t) = x0eit = -eE0 m 1

2 2 i eit

Solution: electron oscillates at driving frequency Equation of motion for bound electron:

me d 2x dt 2 = me dx dt Kx eE 0eit

damping force binding force driving force electron ion core

K = me0

2

resonance frequency

dmkrol@ucdavis.edu Advanced Vitreous State - The Properties of Glass: Active Optical Properties of Glass 4

Frequency dependent dielectric constant

electron ion core

K = me0

2

resonance frequency

ω0

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

• E(t) = Ne2
• m

1

2 2 i

P(t) = 0E(t)

( ) = P(t)

• E(t) = Ne2
• m

1

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 dt 2 = me dx dt Kx ax 2 + (.....) eE0eit

damping force binding force driving force F x

F = Kx ax 2

(+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 with equal powers of λ

me d2x dt 2 = me dx dt Kx + ax 2 e E0eit + 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() ei2t + 1 D(0)D()D()

• x(2)(t) E(t)

[ ]

2

D( j) = (0

2 j 2) i j

• scillates at 2ω

“oscillates” at 0

Nonlinear optics-classical electron oscillator

x(1)(t) = -eE0 m 1

2 2 i eit

x(1)(t) E(t)

: ˙ ˙ x

(1) + 2γ˙

x

(1) + ω0 2x(1) = −eE(t)/m

2:

˙ ˙ x

(2) + 2γ˙

x

(2) + ω0 2x(2) + a x(1)

[ ]

2 = 0

3:

˙ ˙ x

(3) + 2γ˙

x

(3) + ω0 2x(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() ei2t + 1 D(0)D()D()

• x(2)(t) E(t)

[ ]

2

• scillates 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 = aNe3E0

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

ω0 2ω 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) = E

1ei1t + E2ei2t + C .C

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

1 2ei21t + E2 2ei 22t + 2E 1E2ei(1+2 )t + 2E 1E

2

*ei (1 2 )t + c.c]

+ 2 (2)[E

1E 1 * + E2E2 *]

= P(n)eint

n

• .

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

P(2 1) = (2)E

1 2

(SHG), P(2 2) = (2)E2

2

(SHG), P(0) = 2 (2)(E

1E 1 * + E2E2 *)

(OR), P(1+2) = 2 (2)E

1 E2

(SFG), P(12) = 2 (2)E

1 E

2

*

(DFG).

Second harmonic generation Optical rectification Sum-frequency generation Difference-frequency generation

dmkrol@ucdavis.edu Advanced Vitreous State - The Properties of Glass: Active Optical Properties of Glass 10

Nonlinear optics: general formalism

E(t) = E1ei1t + E2ei2t + c.c.

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

P(t) = P(n)eint

n

• The induced polarization can be written as:

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

2nd-order interaction: input frequencies, E induced polarization frequencies, P One ω1 2ω1, 0 Two ω1, ω2

2ω1, 2ω2, 0, ω1+ω2, ω1-ω2

Three 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 = E0eit

P() = (1)E0 + 3(3)E0

2E0 *

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

*)E0 = (eff )E0

(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 2t

( )

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

P

x(n + m) = xxx (2)(n + m,n,m)Ex(n)Ex(m)

P

y(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

2 r E + (1) c2 2 r E

(1)

t2 = 4 c2 2 r 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 = 4ideff2

2

k2c 2 A

1 2eikz

A

1

z = 8ideff1

2

k1c 2 A2A

1 *eikz

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

k = k2 2k1

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 a

polarization (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