Nonlinear Optics - Fotonica.ifsc.usp.br - USP

Introduction to Nonlinear Optics<br />

fs-laser microfabrication<br />

Prof. Dr. Cleber R. Mendonca

Sao Carlos<br />

University of Sao Paulo - Brazil<br />

students 77.000<br />

52.000 undergrad.<br />

25.000 grad.<br />

employers 15.000<br />

professors 6.000<br />

• Sao Paulo<br />

• Sao Carlos (9.000)<br />

• Ribeirao Preto

University of Sao Paulo – in Sao Carlos

Picosecond laser<br />

Laser Nd:YAG<br />

Qswitched/modelocked<br />

532nm and 1064 nm<br />

100 ps<br />

150-fs laser system<br />

Amplifier Ti:safira<br />

775 nm<br />

150 fs<br />

800 μJ

30-fs amplified laser system

microfabrication lab

Part 1: Introduction to Nonlinear Optics<br />

Prof. Dr. Cleber R. Mendonca

Linear optics<br />

Outline<br />

Part 1: Introduction to nonlinear optics<br />

Introduction to nonlinear optics<br />

Second order nonlinearities<br />

Third order nonlinearities<br />

Two-photon absorption<br />

Part 2: fs-laser microfabrication<br />

Applications of nonlinear optics in<br />

laser microfabrication

Linear optics vs Nonlinear optics<br />

Optics is a branch of physics that describes the behavior and properties of light<br />

and the interaction of light with matter. Explains optical phenomena.<br />

Nonlinear Optics<br />

The branch of optics that describes optical phenomena that occur when very<br />

intense light is used


Maxwell equations<br />

ρ: charge density<br />

J: current density<br />

P: electric polarization<br />

M: magnetization


P e M: response of the media to the applied field<br />

Electric Polarization<br />

p


Maxwell equations can be combined, leading to an equation<br />

describing the electromagnetism<br />

Constitutive relationships<br />

r r<br />

P = χE<br />


M = χ H m<br />


J = σE<br />

response of the media<br />

to the applied field


wave equation ( = 0 ; J = 0 )<br />

r<br />

ρ<br />

left<br />

Light propagation in vacuum<br />

right<br />

Matter-light interaction

harmonic oscillator<br />


electron on a spring<br />

k<br />

E rad.

electron on a spring<br />

equation of motion<br />

Linear optics


harmonic oscillator<br />

Steady state: electron oscillates at driving frequency

Oscillating dipole<br />

Polarization oscillator<br />

P(<br />

t)<br />


Ne m<br />

= E<br />

2<br />

ω − iωγ<br />

( 2<br />

−ω<br />

)<br />


2 /<br />


P = χE<br />

linear response

y comparison<br />

then<br />


~ Ne<br />

χ = 2<br />

m<br />


ω −ω<br />

) − iωγ<br />


2 /<br />


which is a complex number<br />

where and are the real and imaginary parts of the complex index of refraction<br />

refraction absorption

Linear optics

absorption<br />

Linear optical process<br />

refraction<br />

α 0 does not depend on light intensity n 0 does not depend on light intensity<br />

absorption of 10 % index of refraction 1.3

Nonlinear optics<br />

high light intensity<br />

E rad. ~ E inter.<br />

How high should be the light intensity ?

Inter-atomic electric field<br />

e = 1.6 × 10 -19 C<br />

r ~ 4 Å<br />

E ~ 1 × 10 10 V/m<br />


cw laser<br />

P = 20 W<br />

w o = 20 μm<br />

I<br />

2P<br />

=<br />

πw<br />

I = 3 × 10 10 W/m 2<br />

E o = 4 × 10 6 V/m<br />


0

Inter-atomic electric field<br />

E ~ 1 × 10 10 V/m<br />


pulsed laser<br />

I = 10 GW/cm 2 =<br />

10 × 10 13 W/m 2<br />

E o = 1 × 10 8 V/m


anharmonic oscillator<br />

anharmonic term<br />


E rad. ~ E inter.

P<br />


potential energy<br />

χ<br />


E<br />

charge displacement<br />

nonlinear polarization response<br />

( 1 )<br />

+<br />












...<br />


E







nonlinear polarization response<br />









E rad. ~E inter.<br />





+ ...

wave equation ( = 0 ; J = 0 )<br />


ρ<br />

left<br />

Light propagation in vacuum<br />


right<br />

Matter-light interaction

P = χ<br />

nonlinear expansion of the polarization<br />

( 1)<br />


. E + χ<br />

linear<br />

processes<br />




(<br />

: EE<br />

+ χ<br />

3 )<br />

r r r<br />

MEEE<br />

+ ...<br />

SHG THG<br />

Kerr effect


nonlinear expansion of the polarization



Second order processes<br />

ω 1<br />



χ (2)<br />

–If ω 1 = ω 2<br />

– second harmonic generation: 2 ω<br />

– optical retification: 0<br />

ω 1 +ω 2<br />

ω 1 −ω 2



Second Harmonic Generation<br />

Phase Matching<br />

( ω) v(<br />

2ω)<br />

v =

Second order processes<br />


ω 2ω<br />




λ = 1064nm λ = 532nm<br />

)





1- higher energy light<br />

2- transparent material


in medium with inversion symmetry<br />

and consequently

Third order processes<br />









χ (3)<br />

–If ω 1 = ω 2 = ω 3<br />

– Third harmonic generation: 3 ω<br />

– Self phase modulation: ω<br />

ω 1 +ω 2 +ω 3<br />

ω 1 −ω 2 −ω 3<br />

ω 1 −ω 2 +ω 3



2 )<br />



Nonlinear polarization<br />

Third order polarization<br />




Third order processes

consequently<br />

and<br />


Kerr media<br />

n n0<br />

n2I<br />

+ =

χ (3) is a complex quantity<br />

Nonlinear Optics

Third order processes: χ (3)<br />

Refractive process:<br />

n n0<br />

n2I<br />

+ =<br />

• self-phase modulation<br />

• lens-like effect<br />

Absorptive process:<br />

α = α + β 0 I<br />

• nonlinear absorption<br />

• two-photon absorption

Two-photon absorption (2PA) process<br />

Phenomenon does not described for the Classical Physics and does not<br />

observed until the development of the Laser.<br />

1-photon absorption<br />

(Linear)<br />

2-photon absorption<br />

(Nonlinear)<br />

Theoretical model: Maria Göppert-Mayer, 1931<br />

Two photons from an intense laser light beam are simultaneously absorbed<br />

in the same “quantum act”, leading the molecule to some excited state with<br />

energy equivalent to the absorbed two photons.

2ω<br />

two-photon absorption<br />

ω<br />

ω<br />

Applications:<br />

Abs<br />

optical limiting<br />

fluorescence microscopy<br />

microfabrication<br />

α = α0<br />

+ βI<br />

λ

2PA<br />

localization of the excitation with 2PA<br />

dilute solution of fluorescent dye<br />

1PA<br />

spatial confinement of excitation

excitation profile along z<br />

Radius, area and intensity of focused beam<br />

I(<br />

one photon<br />

two photon<br />

⎛ z<br />

ω(<br />

z ) = ω0<br />

⎜<br />

⎜1+<br />

⎝ z<br />

2 2<br />

A( z ) = πω = πω0<br />

z ) =<br />


At<br />


∝ =<br />

A<br />


⎞<br />

⎟<br />

⎠<br />

1 ⎟<br />



⎛ z ⎞<br />

⎜ +<br />

⎝ z ⎠<br />



0 1<br />

⎛<br />

πω ⎜ +<br />

⎝<br />

z<br />


Normalized excitation rates<br />

(ignoring beam attenuation)<br />





z ) ⎛<br />

= 1<br />

0 ⎜ +<br />

) ⎝<br />

z ) ⎛<br />

= 1<br />

0 ⎜ +<br />

) ⎝<br />










−2<br />





−4<br />





2

This is the end of Part 1<br />

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