Ultrashort laser applications

Transcription

1 Ultrashort laser applications Prof. Dr. Cleber R. Mendonça Instituto de Física de São Carlos Universidade de São Paulo

2 University of Sao Paulo - Brazil students ~ undergrad grad. Sao Carlos Sao Paulo Piracicaba Santos Lorena Bauru Pirassunga Ribeirao Preto São Carlos

3 University of Sao Paulo, Sao Carlos

4 Sao Carlos Institute of Physics Professors: 97 Employers: 185 (technical and administration) Students: ~ 600 (undergrad) ~ 70 (grade Several research areas in Physics and Material Sciences

5 Photonics Groups research areas nonlinear spectroscopy of organic materials and inorganic materials novel methods in nonlinear optics control of light matter interaction by pulse shaping fs-laser microfabrication and micromachining

6 Summary Class #1 Introduction nonlinear optics Overview on ultrashort pulses Overview on Nonlinear Optics and light-mater interaction Class # Nonlinear Spectroscopy and fs-laser microfabrication Applications in nonlinear spectroscopy Applications in microfabrication

7 Laser pulse U.S.S Entreprise NCC-1701-E Length: m Width: 50.6 m Height: 88. m Lenght of light pulse: l ~ 15 m T = l/c = 50 ns

8 Laser pulse Death star diameter 10 Km Lenght of light pulse: l ~ 3 Km T = l/c = 10 s

9 Pulsed pumping: pulses on the order ~ µs Exciting (pumping) the laser medium with a short-pulse flash lamp (~1 µs) yields a reasonably short pulse. I(t) Long and complex pulse - this yields a pulse as long as the excited-state lifetime of the medium, which can be considerably longer than the pump pulse. - solid-state laser media have lifetimes in the µs; thus pulses are in the microseconds to milliseconds range.

10 Laser intensity Cavity Gain Cavity Loss (%) Q-switching: pulses on the order ~ ns 100 Q-switching principle: 0 t Preventing lasing (by introducing high loss) until the flash lamp finishes excitation Abruptly allowing the laser to lase. t The pulse duration is limited by how fast one can switch the round-trip time of the laser usually allows pulses ns t

11 Mode-locking: short pulses Locking the phases of the laser modes (frequencies) Intensity vs. time no pulse Time Intensity vs. time Ultrashort pulse! Time

12 continuous wave ultrashort pulses A constant and a delta-function are a Fourier-Transform pair Irradiance vs. time Spectrum Continuous wave: time frequency Ultrashort pulse: time frequency

13 short and long pulses The Fourier transform (uncertainty) principle: Intensity vs. time Spectrum Long pulse time frequency Short pulse time frequency

14 Laser power Media for ultrashort lasers Solid-state laser media have broad bandwidths

15 Dispersion Given the broad spectral band, dispersion is critical in ultrafast optics. Dispersion: dependence of the refractive index on wavelength. Two effects on a pulse (one in space and the other in time) Angular dispersion disperses a beam in space (angle): Group-velocity dispersion (GVD) disperses a pulse in time: v g (blue) < v g (red)

16 Ti:sapphire laser cristal Ti:safira Duration from fs.

17 Time scale of ultrashort pulses 1 fs = s

18 introduction how short is a femtosecond pulse?

19 Microfabrication Ti:Sapphire lasers 100 fs 50 fs 0 fs Very intense light Laser intensities ~ 100 GW/cm 1 x W/cm Laser pointer: 1 mw/cm (1 x10-3 W/ cm )

20 fs-laser micromachining Ti:Sapphire lasers 100 fs 50 fs 0 fs Very intense light

21 Nonlinear Optics Study of optical phenomena that occur when very intense light is used Nonlinear constitutive relationship P 0 ( E) E laser Optical properties Intensity Low frequency range (193) B ( H) H Nonlinear relationship between the field and the magnetic induction (solenoids and transformers) Saturation of of popoluation in spin levels in Magnetic Resonance (1948) Saturation of luminesncen in dyes (1941)

22 Nonlinear Optics 1961 Second Harmonic Generation P.A. Franken, et al, Physical Review Letters 7, p. 118 (1961) Peter A. Franken Birth of nonlinear optics as a separated area in optics

23 Nonlinear Optics One of the pioneers in nonlinear optics General formulation of nonlinear optical processes Nicolaas Bloembergen Nonline effects can be described in termos of classic eletromagnetism, with nonlinear susceptibilities included in the constitutive relationships Semiclassical treatments for nonlinear susceptibilities

24 Light-matter interaction: linear optics Lorentz Model harmonic oscillator Hendrik A. Lorentz k E 0 E e it m E << E inter. m d x dt dx m dt m x 0 ee with 0 k m

25 Steady state solution: electron oscilates with excitation frequency t i e x t x 0 ) ( ) ( 1 E i m e x with Thus, the oscillating dipole is given by t i e E i m e t x e t p 0 0 ) ( 1 ) ( ) ( Light-matter interaction: linear optics

26 The polarization can be written as N : dipoles density ) ( ) ( 1 ) ( ) ( 0 0 t E E i m Ne t Np t P Linear response With the suceptibility given by complex quantity i m Ne ) ( 1 ~ 0 0 Light-matter interaction: linear optics

27 Light-matter interaction: linear optics therefore, the complex index of refraction is written as n~ 1 1 ~ 1 ~... n i n are the Real and Imaginary parts of the complex index of refraction refraction absorption c n Re[~ ] Im ~

28 0 0 0 ) ( ) ( ) ( 1 m Ne n 0 0 ) ( ) ( ) ( m Ne Dispersion of n and n and do not depend on the light intensity Light-matter interaction: linear optics

29 Light-matter interaction: nonlinear optics high light intensities E rad. ~ E inter. How high should the light intensity be?

30 Light-matter interaction: nonlinear optics Inter-atomic electric field cw laser e = C r ~ 4 Å P w P = 0 W I w o = 0 m I = W/m 0 E ~ V/cm E o = V/cm

31 Light-matter interaction: nonlinear optics Inter-atomic electric field pulsed laser I = 100 GW/cm = W/m e = C r ~ 4 Å E ~ V/cm E o = V/cm

32 Light-matter interaction: nonlinear optics Perturbative nonlinear optics E ~ V/cm I ~ 100 GW/cm Extreme nonlinear optics (non perturbative) E ~ V/cm I ~ TW/cm

33 Light-matter interaction: nonlinear optics high intensities k E rad. ~ E inter. m anharmonic oscillator m d x dt dx m dt m x 0 max ee anharmonic term

34 Light-matter interaction: nonlinear optics In the equation of motion one considers in which a characterizes the anharmonicity (nonlinearity) The corresponding potential is given by Non-centresymmetric media

35 Light-matter interaction: nonlinear optics To solve the anharmonic oscillator Under the action of a field Assuming a x << 0 x Perturbative method The solution can be written as x x (1) x () x (3)...

36 Light-matter interaction: nonlinear optics The first order soluion x (1) is obtained by making a = 0 Whose solution we already know with x (1) e E0 ( ) m ( ) 0 i Then x (1) ( t) e m E D( ) D( ) ( ) 0 i

37 Light-matter interaction: nonlinear optics The second order solution x () is obtained by Using the results for x (1) we have Whose solution is with Then

38 Light-matter interaction: nonlinear optics For the monochromatic case, the polarization is given by from where we find P Nex Ne( x (1) x ()...) P N e / m) D( ) ( E E 3 N( e / m D() D ) a ( )... (1) 0 () 0 First and second order susceptibilities Therefore, the indiuced polarization induced inthe material is given by P (1) () 0 E 0 E...

39 Light-matter interaction: nonlinear optics Considering a more generic applied field In the anharmonic oscillator It can be seen that will present frequence components as Generating therefore, responses on such frequencies

40 Nonlinear optical response anharmonic oscillator hihg intensities E rad. ~ E inter. m d x dt dx m dt m x 0 max ee k m Hendrik A. Lorentz P Nonlinear polarization (1) () (3) 3 0( E E E...)

41 Light matter interaction: nonlinear optics nonlinear wave equation left right Light propagation in vacuum Matter-light interaction

42 Nonlinear optics Different terms in the nonlinear expansion of the polarization will be responsible for different nonlinear optical effects linear processes SHG THG Kerr effect

43 Second-order nonlinearities First we will study the nonlinear optical effects related to () starting from a more general description described in greater detail

44 Second-harmonic generation Lets consider the process of second-harmonic generation (SHG) where a laser beam described by impinges a crystal with () 0 The nonlinear polarization that is created in the crystal is described by which can be explicitly written as according to the driven wave equation Optical rectification (static electric field) Generation of radiation at the second harmonic frequency

45 Second-harmonic generation First demonstration of second-harmonic generation P.A. Franken, et al, Physical Review Letters 7, p. 118 (1961)

46 Second-harmonic generation Second Harmonic Generation in nonlinear crystals = 1064nm = 53nm observe that the crystal is transparent at both, and

47 Second-harmonic generation One common use of SHG is to convert IR light to visible, such as for example for Nd:YAG lasers Under proper experimental conditions, SHG can be so efficient that nearly all the incident power () is converted to second harmonic ()

48 Serious second-harmonic generation Frequency-doubling KDP crystals at Lawrence Livermore National Laboratory These crystals convert as much as 80% of the input light to its second harmonic. Additional crystals produce the third harmonic with similar efficiency!

49 Second-harmonic generation SHG can be visualized considering the interaction in terms of the exchange of photons between the various frequencies components of the field. two photons are destroyed and one at is created. Energy conservation holds for SHG E ω = 1E ω

50 Sum- and difference-frequency generation Suppose there are two different-color beams present: Using the second order contribution to the nonlinear polarization we find the nonlinear polarization given by

51 Sum-frequency generation The nonlinear polarization describing sum-frequency generation is given by the expression SFG is analogous to SHG, except that in this case the input beams are at different frequencies can be used to produce tunable radiation in the ultraviolet spectral region

52 Difference-frequency generation The difference-frequency generation is described by the nonlinear polarization DFG can be used to produce tunable IR radiation, by mixing tunable visible laser with a fixed frequency visible one For every photon 3 created a higher energy photon 1 is destroyed (input) a lower input frequency is created

53 Nonlinear optical response For centresymmetric materials (U(x) = U(-x)) That can be written as (1) (3) 3 P 0( E E...) P P (1) P (3) (1) (3) 0( I) E 0 ef E defining the effective susceptibility as ef (1) (3) I In such case, the refractive indes is given by ~ (3 ) (1) n 1 ef 1 ( I)

54 Nonlinear optical response For low dense optical media 1 1 ( ~ (1) (3) n I) Taking Real and Imaginary parts n~ n i n 1 ( 1 1 Re ~ ~ 1) (3) Re I 1 Im ~ 1 1) (3) Im I ( ~ n n 0 ni 0 I

55 Nonlinear optical response Third order processes: (3) ~ ( 3) Re ~ ~ (3) i Im (3) Nonlinear refraction n n 0 ni Nonlinear absorption 0 I n Re ~ (3) Im (3) self phase modulation lens-like effect two-photon absorption

56 Absorption cross-section N I N N 0 h I N h N 0 Two-photon absorption crosssection [cm 4 s] I 0 Considering the total absorption We can find the absorption cross-section The excitation rate is given by h I R 0 h I N h h I N R Two-photon absorption

57 Two-photon absorption Theoretically described in 1931 Tese de Doutorado U. de Göttingen "Über Elementarakte mit zwei Quantensprüngen Annals of Physics 9 (3): Maria Goeppert-Mayer Semi-classic treatment Two-photons are simultaneously absorbed in the same quantum act, promoting the molecule to an excited state with energy equivalent to the two photons absorbed.

58 Two-photon absorption: perturbation theory From the Schrödinger eq. With the Hamiltonian given by and where o is the Hamiltonian of the free atom with and assuming a monochromatic field

59 Two-photon absorption: perturbation theory The Schrödinger eq. in the presence of a time dependent potential whose solution can be written as Free atom solution Ĥ 0 with which substituted in the Schrödinger eq. leads to with e

60 Two-photon absorption: perturbation theory In order to solve this equation we used the perturbation method such that we have the following set of equations

61 Two-photon absorption: perturbation theory Linear absorption N = 1 Wihtout the field the system is at the ground state g Representing V mg as Thus, the eq. for is that results in

62 Linear absorption The probability of the atom be in a state m in at time t Since we have a distributions of transition frequencies m mg g line shape in this case

63 Linear absorption The transition rate for linear absoption is which is usually defined in terms of the absorption cross-section with

64 Two-photons absorption Two-photon absorption N = 1 and N = to obtain First we find which is used in the right hand side of the eq. wiht N = already known from linear absorption using

65 Two-photons absorption In this case, the convention for the representation of the various levels is illustrated in the diagram Therefore, we write

66 Two-photons absorption which results in for the two-photon absorption process Then, the probability of the atom being in a state n at a given time t Considering a line width for the final state we have

67 Two-photons absorption The transition rate for two-photon absorption is which in terms of the two-photon absorption cross-section with

68 Multi-photon absorption We can generalize the results to higher order processes ~ I 1 photon absorption ~ I photons absorption ~ I 3 3 photons absorption ~ I 4 4 photons absorption

69 Two-photons absorption 1961 Kaiser e Garrett: Excitação por absorção de dois fótons

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