P O L I T D i p a r t i Motivation h E 2 E - 1 - PowerPoint PPT Presentation

Nonlinear optics in the short pulse regime: basics and practice M. Marangoni Physics Department, Politecnico di Milano (Italy) Institute of Photonics and Nanotechnology of CNR (Italy) e-mail: marco.marangoni@.polimi.it P O L I T D i p a r t i

Motivation h � � E 2 E - 1 GOLDEN RULE � � � ATOMS � MOLECULES

Optical frequency comb synthesizers Yb: fibre � � = 1.05 � m Er: fibre Ti: Sapphire � SC = 0.6-1.6 � m � � = 1.55 � m � � = 0.8 � m � SC = 0.8-2.2 � m f rep = 0.1-1 GHz � SC = 0.4-1.2 � m P < 80 W f rep = 0.1-0.25 GHz f rep = 0.07-10 GHz P < 1 W P < 3 W

How to change spectral range ? SECOND ORDER NONLINEAR OPTICS !! 2 � 1 � 1 2 � 2 � (2) � 2 � 1 + � 2 � 1 - � 2 � � � � � � � ( ) exp( ) exp( ) E t A i t A i t cc 1 1 2 2 Optical rectification Second harmonic generation (SHG) � � * * 2 � � � � � � � � � � ( 2 ) ( 2 ) 2 ( 2 ) ( 2 ) ( ) ( ) 2 [ exp( 2 ) P t E t A A A A A i t 1 1 2 2 1 1 � � � � 2 * � � � � � � � � � � � � � � exp( 2 ) 2 exp[ ] 2 exp[ ] ] A i t A A i t A A i t cc 2 2 1 2 1 2 1 2 1 2 Difference frequency generation (DFG) Sum frequency generation (SFG)

OUTLINE � Equations governing a cw second order parametric process � The problem of phase matching � The equations of linear pulse propagation � Parametric processes in the femtosecond pulse regime � Examples: analytical and numerical discussion

The photons picture � � 2 � 2 � � (2) SHG � � � � 1 + � � 2 = � � 3 � 1 � 3 = � 1 + � 2 � 2 � (2) � 3 SFG � 2 � 1 k 1 + k 2 = k 3 � 3 � 2 = � 3 - � 1 � 1 � 3 � (2) DFG � 1 � 2

Optical parametric amplification (OPA) & optical parametric generation (OPG): what are they ? They are the same process as DFG, but differ in the initial conditions � pump � 1 � 3 � 3 � (2) � 2 = � 3 - � 1 � 1 � 2 idler signal In DFG, � 3 and � 1 have comparable energies and you look for an intense � 2 � In OPA, � 1 has an energy 100-10000 times lower than � 3 and you look for a � strong amplification of � 1 ( � 1 acts as a seed) In OPG, � 1 photons come from vacuum noise and you are looking for � extreme parametric gains (10 nJ � > 10 11 photons !!)

�������������������������� ������������������������������ � 1 � 3 � (2) HR@ � 1 T@ � 1 You may enclose your crystal in an optical cavity � ����������������������������������������� � ����������������������������������������������������������������������������������� � OPTICAL PARAMETRIC OSCILLATOR

Femtosecond OPOs vs. OPAs: Femtosecond OPOs � are pumped by simple laser oscillators � provide high repetition rates (100 MHz) � have low output energy (nJ level) � require matching of the OPO cavity length to pump laser � large yet not huge oscillation bandwidth Femtosecond OPAs � require pumping by amplified laser systems � provide low repetition rates (1-100 kHz) � have high output energy ( � J-mJ level) � are easy to operate (no length stabilization) � ultrabroad bandwidth, up to the few-cycles regime

The wave equations for second order parametric processes

The wave equation for nonlinear optical media � ������������������������������������������������������������������������ charges and currents, we get the wave equation 2 2 � � 1 1 E P 2 � � - E 2 2 2 2 � � � c t c t 0 0 0 � The polarization of the medium is made of a linear and a nonlinear contribution P = P L + P NL � For a continuous wave, the linear polarization is P L = � 0 ( � r - 1 ) E � Making the scalar approximation and considering a plane wave, the propagation equation becomes 2 2 � � 1 1 E P 2 � � NL - E 2 2 2 2 � � � t t c c 0 0

The slowly varying envelope approximation � Starting from the scalar propagation equation 2 2 � � 1 P E 2 NL � � � - E 0 2 2 2 � � c t t � � � � � � � ( , ) ( ) exp we look for a solution E z t A z i kz t � � � � � � � ( , ) ( ) exp P z t P z i k z t with NL P � By substitution, we get the equation � � � 2 2 � � d A dA � � � � � � � p � 2 2 2 exp ik k A A P i k k z 0 2 2 dz dz c 2 d A dA �� 2 � Assuming ik (slowly varying envelope approximation, 2 dz dz 2 � � � � � dA SVEA) we get the equation � � p � exp i P i k k z 0 2 dz k

The nonlinear polarization in second-order parametric interactions � Consider the superposition of three waves at frequencies � 1 , � 2 and � 3 with � 1 + � 2 = � 3 � � � � � � � � � � � � � � � ( , ) ( ) exp ( ) exp E z t A z i k z t A z i k z t 1 1 1 2 2 2 � � � � � � ( ) exp A z i k z t 3 3 3 � By second order nonlinear effect, the following polarizations are generated at the three frequencies � � � � � � � � � � � 2 * P ( z , t ) d A ( z ) A ( z ) e xp i k k z t 1 0 2 3 3 2 1 NL e ff � � � � � � * � � � � � ( , ) 2 ( ) ( ) exp P z t d A z A z i k k z t 2 0 1 3 3 1 2 NL eff � � � � � � � � � � � ( , ) 2 ( ) ( ) exp P z t d A z A z i k k z t 3 0 1 2 1 2 3 NL eff where d eff is an effective second order nonlinear coefficient

Three-frequency interaction in a second order nonlinear medium � Consider three waves at � 3 (pump) , � 1 (signal) and � 2 (idler) , with � 1 + � 2 = � 3 . We obtain the following equations � � d A � � 1 eff * 1 � � exp i A A i kz 2 3 � z n c 1 � � d A � � 2 eff * 2 � � exp i A A i kz 1 3 � z n c 2 � � d A � � 3 eff 3 � � � exp i A A i kz 1 2 � z n c 3 where � k = k 3 - k 2 - k 1 is the wave vector mismatch between the three waves Setting � k = 0 is crucial to get highly efficient energy transfer between the interacting waves

OPA/DFG solution for small pump depletion � By neglecting pump depletion (A 3 = cost.) and assuming an input beam at the signal frequency � 1 and no input at the idler frequency � 2 (A 2 (0) = 0) the coupled differential equations admit the solution: � � � 2 � � � � � � � � 2 0 1 I L I sinh gL � � 1 1 2 � g � � � 2 � � � � � � � 2 0 2 I L I sinh gL 2 1 � 2 g 1 with g and � given by: � � 2 � � d eff � k � � 1 2 � � � 2 � 2 � g I 3 � 3 � � 2 n n n c 1 2 3 0 the latter representing a figure of merit for the parametric gain. The presence of a phase-mismatch clearly affects such gain.

Parametric gain � In the high gain approximation ( � L >>1) and under phase-matching ( � k = 0): one has: � ( 0 ) ( 0 ) I I � � � � 1 1 2 � � � � ( ) exp 2 ( ) exp 2 I L L I L L 1 2 � 4 4 1 � This allows us to define a parametric gain: � � � � � � 1 1 d I L � � � � � � � 1 2 eff 1 2 2 � � G exp L exp I L � � 3 � 3 0 4 4 2 I � n n n c � � � 1 1 2 3 0 For high gain we need high pump intensity (ultrashort pulses are good!), large nonlinear coefficient d ef f and high signal and idler frequencies The gain is exponential since the presence of a seed photon at the signal wavelength stimulates the generation of an additional signal photon and of a photon at the idler wavelength. Due to the symmetry of signal and idler, the amplification of an idler photon stimulates in turn the generation of a signal photon. Therefore, the generation of the signal field reinforces the generation of the idler field and viceversa, giving rise to a positive feedback

Parametric gain: examples with BBO Red-pumped BBO crystal Blue-pumped BBO crystal: 7 10 5 mm higher gain because BBO 4 mm 6 10 � p = 0.8 � m � � � 1 � 3 mm � s = 1.2 � m 2 5 10 Parametric Gain 9 10 2 mm 4 mm 4 10 8 10 3 mm 3 10 BBO 7 10 2 mm � p = 0.4 � m Parametric Gain L = 1 mm 2 10 6 10 � s = 0.6 � m 1 10 5 10 L = 1 mm 4 0 10 10 20 40 60 80 100 2 ) Pump Intensity (GW/cm 3 10 2 10 G. Cerullo and S. De Silvestri, 1 10 20 40 60 80 Rev. Sci. Instrum. 74 , 1 (2003). 2 ) Pump Intensity (GW/cm