Introduction to Nonlinear Optics Prof. Cleber R. Mendonca http://www.fotonica.ifsc.usp.br
Outline Linear optics Introduction to nonlinear optics Second order nonlinearities Third order nonlinearities Two-photon absorption Example of applications Conclusions
Linear optics vs Nonlinear optics Optics is a branch of physics that describes the behavior and properties of light and the interaction of light with matter. Explains optical phenomena. Nonlinear Optics The branch of optics that describes optical phenomena that occur when very intense light is used
Linear optics Maxwell equations ρ: charge density J: current density P: electric polarization M: magnetization
Linear optics P e M: response of the media to the applied field Electric Polarization p
Linear optics Maxwell equations can be combined, leading to a equation describing the electromagnetism Constitutive relationships r r P = χe r r M = χ H m r r J = σe response of the media to the applied field
Linear optics wave equation ( ρ = 0 ; J r = 0 ) left right Light propagation in vacuum Matter-light interaction
Linear optics E rad. << E inter. harmonic oscillator electron on a spring oscillation frequency k m e ω 0 = k me
electron on a spring Linear optics equation of motion
Linear optics harmonic oscillator Steady state: electron oscillates at driving frequency
Linear optics Oscillating dipole Polarization oscillator P( t) Ne 2 / = 2 0 0 ( 2 ω ω ) m E iωγ P = χe linear response
Linear optics by comparison ~ χ Ne 2 / = 2 0 0 m ( 2 ω ω ) iωγ which is a complex number then where and are the real and imaginary parts of the complex index of refraction refraction absorption
Linear optics
Linear optical process absorption refraction α 0 does not depend on light intensity n 0 does not depend on light intensity absorption of 10 % index of refraction 1.3
Nonlinear optics high light intensity E rad.~ E inter. How high should be the light intensity?
Nonlinear optics Inter-atomic electric field cw laser 2P πw P = 20 W I = w o = 20 μm 2 0 I = 3 10 10 W/m 2 e = 1.6 10-19 C r ~ 4 Å E ~ 1 10 10 V/m E o = 4 10 6 V/m
Nonlinear optics Inter-atomic electric field pulsed laser I = 10 GW/cm 2 = 10 10 13 W/m 2 E ~ 1 10 10 V/m E o = 1 10 8 V/m
Nonlinear optics high light intensity E rad.~ E inter. anharmonic oscillator anharmonic term
Nonlinear optics anharmonic oscillator potential energy charge displacement P nonlinear polarization response P = χ ( 1 ) E + χ ( 2 ) E 2 + χ ( 3 ) E 3 +... E
Nonlinear optics high light intensity E rad.~ E inter. anharmonic oscillator nonlinear polarization response P = χ ( 1 ) E + χ ( 2 ) E 2 + χ ( 3 ) E 3 +...
wave equation ( ρ = 0 ; J r = 0 ) Nonlinear optics left right Light propagation in vacuum Matter-light interaction
Nonlinear optics nonlinear expansion of the polarization r P = χ ( 1) r.e + χ ( 2 ) : r r EE + χ ( 3 ) r r r MEEE +... linear processes SHG THG Kerr effect
Nonlinear optics nonlinear expansion of the polarization
χ (2) Second order processes Nonlinear Optics ω 1 ω χ (2) ω 1 +ω 2 2 ω 1 ω 2 If ω 1 = ω 2 second harmonic generation: 2 ω optical retificatio: 0
Nonlinear Optics Second order processes ( 2 ) χ Second Harmonic Generation ω 2ω λ = 1064nm λ = 532nm
Second Harmonic Generation ( 2 ) χ 1- higher energy light 2- transparent material
Second Harmonic Generation ( 2 ) χ Phase Matching v ( ω) = v( 2ω )
Nonlinear Optics in medium with inversion symmetry and consequently
Nonlinear Optics Third order processes (3) χ ω 1 ω 1 +ω 2 +ω 3 ω 2 χ (3) ω 1 ω 2 ω 3 ω 3 ω 1 ω 2 +ω 3 If ω 1 = ω 2 = ω 3 Third harmonic generation: 3 ω Self phase modulation: ω
Third Harmonic Generation χ (3) ω 3ω ω Nonlinear media
Nonlinear Optics χ ( 2 ) = 0 (3) χ Third order processes Nonlinear polarization Third order polarization
Nonlinear Optics consequently and Kerr media n = n + 0 n2i
Nonlinear Optics Third order processes ( 3 ) χ n χ 2 ( 3 ) Kerr media: n = n0 + n2i Index of refraction depends on the light intensity
Kerr media: n + 0 2 Self phase modulation centre symmetric: = ( ) 3 n n I P NL = χ 3 E χ () 2 = 0 n 2 >0 Material behaves as a convergent lens x d f y z Sample
Optical switching low intensity ( 3 ) χ
Optical switching high intensity ( 3 ) χ Self action process
Optical switching response time: 10-15 s response time 1 10-9 1 10-15 1GHz 1 THz 1 million times faster
Nonlinear optics 2ω ω Nonlinear material ω? 3ω Intense light induced nonlinear response in the material Material change the light in a nonlinear way Self action effect
χ (3) is a complex quantity Nonlinear Optics
Third order processes: χ (3) Refractive process: n + 0 2 = 0 Absorptive process: = n n I α α + β I self-phase modulation lens-like effect nonlinear absorption two-photon absorption
Two-photon absorption (2PA) process Phenomenon does not described for the Classical Physics and does not observed until the development of the Laser. 1-photon absorption (Linear) 2-photon absorption (Nonlinear) Theoretical model: Maria Göppert-Mayer, 1931 Two photons from an intense laser light beam are simultaneously absorbed in the same quantum act, leading the molecule to some excited state with energy equivalent to the absorbed two photons.
two-photon absorption α = α 0 + βi 2ω ω Abs ω λ Applications: optical limiting fluorescence microscopy microfabrication
two-photon fluorescence α = α 0 + βi ω ω light emission fluorescence
localization of the excitation with 2PA dilute solution of fluorescent dye 2PA 1PA spatial confinement of excitation
excitation profile along z Radius, area and intensity of focused beam A( z ω( z ) ω0 1 + z z ) I( z ) = = 0 + = 2 2 πω = πω0 1 z0 E At 1 = A 1 2 πω 1 0 + z z z Normalized excitation rates (ignoring beam attenuation) 0 2 2 one photon I( z ) I( 0 ) z = 1 + z 0 2 two photon I( z ) I( 0 ) z = 1 + z 0 4
Applications Nonlinear interaction provides spatial confinement of the excitation fs-microfabrication α = α 0 α = α 0 + βi
Two-photon Applications polymerization -Two-photon polymerization Monomer + Photoinitiator Polymer light Photoinitiator is excited by two-photon absorption R2 PA I The polymerization is confined to the focal volume. 2 High spatial resolution
Applications -Two-photon polymerization Nature 412, 697-698 (2001) oscillator Venus statue Two-photon polymerization Opt. Exp. 12, 5521-5528 (2004) Bull
Applications -Two-photon two-photon polymerization photonic crystal J. W. Perry 20 µm
Other Applications studies-two-photon polymerization 3D cell migration studies in micro-scaffolds
Applications -Two-photon fluorescence microscopy Microscopy by two-photon fluorescence two-photon fluorescence microscopy 3D image of a cell Human chromosome Laboratory for Optics and Biosciences Ecole polytechnique
Applications -Two-photon fluorescence microscopy Fluorescent marker fluorophores
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