Chapter 8. Second-Harmonic Generation and Parametric Oscillation 8.0 Introduction Second-Harmonic generation : Parametric Oscillation : Reference : R.W. Boyd, Nonlinear Optics, Chapter 1. The nonlinear Optical Susceptibility

The Nonlinear Optical Susceptibility General form of induced polarization : where, : Linear susceptibility : 2nd order nonlinear susceptibility : 3rd order nonlinear susceptibility : 2nd order nonlinear polarization : 3rd order nonlinear polarization Maxwell’s wave equation : Source term : drives (new) wave

Second order nonlinear effect Let’s us consider the optical field consisted of two distinct frequency components ; : Second-harmonic generation : Sum frequency generation : Difference frequency generation : Optical rectification # Typically, no more than one of these frequency component will be generated  Phase matching !

Nonlinear Susceptibility and Polarization 1) Centrosymmetric media (inversion symmetric) : Potential energy for the electric dipole can be described as Restoring force : Equation of motion : Damping force Coulomb force Restoring force

Purtubation expansion method : Assume, Each term proportional to ln should satisfy the equation separately : Damped oscillator Second order nonlinear effect in centrosymmetric media can not occur !

2) Noncentrosymmetric media (inversion anti-symmetric) : Potential energy for the electric dipole can be described as Restoring force : Equation of motion : Damping force Coulomb force Restoring force

Similarly, : Report Assume, Each term proportional to ln should satisfy the equation separately Solution : : Report

Example) Solution for SHG Put general solution as Similarly, : Report

Susceptibility Polarization : : linear susceptibility : SHG : SFG : DFG : OR

<Miller’s rule> - empirical rule, 1964 is nearly constant for all noncentrosymmetric crystals. # N ~ 1023 cm-3 for all condensed matter # Linear and nonlinear contribution to the restoring force would be comparable when the displacement is approximately equal to the size of the atom (~order of lattice constant d) : mw02d=mDd  D=w02/d : roughly the same for all noncentrosymmetric solids. (non-resonant case) : used in rough estimation of nonlinear coefficient. : good agreement with the measured values

Qualitative understanding of Second order nonlinear effect in a noncentrosymmetric media

2w component

General expression of nonlinear polarization and Nonlinear susceptibility tensor General expression of 2nd order nonlinear polarization : where, 2nd order nonlinear susceptibility tensor # Full matrix form of : SHG : SFG : SFG : SHG

Example 1. SHG Example 2. SFG

Properties of the nonlinear susceptibility tensor 1) Reality of the fields are real measurable quantities : 2) Intrinsic permutation symmetry

3) Full permutation symmetry (lossless media : c is real) 4) Kleinman symmetry (nonresonant, c is frequency independent) intrinsic : Indices can be freely permuted !

Define, 2nd order nonlinear tensor, ## If the Kleinman’s symmetry condition is valid, the last two indices can be simplified to one index as follows ; and, can be represented as the 3x6 matrix ; : 18 elements

Again, by Kleinman symmetry (Indices can be freely permuted), dil has only 10 independent elements : : Report

Example 1. SHG : Report Example 2. SFG

8.2 Formalism of Wave Propagation in Nonlinear Media Maxwell equation Polarization : Assume, the nonlinear polarization is parallel to the electric field, then Total electric field propagating along the z-direction : where, and

w1 term (slow varying approximation) Text

Similarly,

8.3 Optical Second-Harmonic Generation Neglecting the absorption ; where, Assume, the depletion of the input wave power due to the conversion is negligible

Phase-matching in SHG Output intensity of 2nd harmonic wave : Conversion efficiency : Phase-matching in SHG Maximum output @ : phase-matching condition If : decreases with l Coherence length : measure of the maximum crystal length that is useful in producing the SHG (separation between the main peak and the first zero of sinc function)

Technique for phase-matching in anisotropic crystal Example) Phase matching in a negative uniaxial crystal

Experimental verification of phase-matching , there exists an angle qm at which , so, if the fundamental beam is launched along qm as an ordinary ray, the SH beam will be generated along the same direction as an extraordinary ray. Example (p. 289) Experimental verification of phase-matching Taylor series expansion near : Report

Second-Harmonic Generation with Focused Gaussian Beams If z0>>l, the intensity of the incident beam is nearly independent of z within the crystal Total power of fundamental beam with Gaussian beam profile :

So, Conversion efficiency : : identical to (8.3-5) for the plane wave case (*) P(2w) can be increased by decreasing w0 until z0 becomes comparable to l # It is reasonable to focus the beam until l=2z0 (confocal focusing) (**) Example (p. 292)

Second-Harmonic Generation with a Depleted Input Considering depletion of pump field, Define, (8.2-13)  where, SHG : Let’s consider a transparent medium : , and perfect phase-matching case :

: Total energy conservation Define, : Total energy conservation Initial condition : # : 100% conversion [2N(w photons)  N(2w photons)]

Conversion efficiency :

8.4 Second-Harmonic generation Inside the Laser Resonator # Second-harmonic power Pump beam power # Laser intracavity power :  Efficient SHG SH output power :

8.5 Photon Model of SHG Annihilation of two Photons at w and a simultanous creation of a photon at 2w - Energy : w+ w=2w - Momentum :

8.6 Parametric Amplification : # Special case : w1=w2 (degenerate parametric amplification) Analogous Systems : - Classical oscillators - Parasitic resonances in pipe organs(1883, L. Rayleigh) : - RLC circuits Example) RLC circuit

Steady-state solution : Assuming Put, where, Steady-state solution : (degenerate parametric oscillation) Phase matching Threshold condition

Optical parametric Amplification Polarization of 2nd order nonlinear crystal :

(phase-matching), and also depletion of field due to (8.2-13),  where, When (lossless), (phase-matching), and also depletion of field due to the conversion is negligible, where,

Qualitative understanding of parametric oscillation : Solution : Qualitative understanding of parametric oscillation : # Initially w1(or w2) is generated by two photon spontaneous fluorescence or by cavity resonance # w2(or w1) wave increases by difference frequency generation between w3 and w1(or w2) # w1(or w2) wave also increases by difference frequency generation between w3 and w1(or w2) # w2(or w1) wave : Signal [A(0)=0] # w2(or w1) wave : Idler [A(0)>0]

Initial condition : Photon flux :

8.7 Phase-Matching in Parametric Amplification Put,

General solution :

Phase-Matching : Report Example) Phase-matching by using a negative uniaxial crystal : Report

8.8 Parametric Oscillation where, (8.8-1)

Think of propagation inside a cavity as a folded optical path. Even though Eq. (8.8-1) describe traveling-wave parametric interaction, it is still valid if we Think of propagation inside a cavity as a folded optical path. If the parametric gain is equal to the cavity loss (threshold gain), So, absorption in crystal, reflections on the interfaces, cavity loss(mirrors, diffraction, scattering), … Condition for nontrivial solution : : Threshold condition for parametric oscillation

Threshold pump intensity : If we choose to express the mode losses at w1 amd w2 by the quality factors, respectively, Decay time (photon lifetime) of a cavity mode : (4.7-5) Temporal decay rate : and Threshold pump intensity : Pump intensity : Threshold pump intensity :

Example) Absorption loss = 0 (4.7-5), (4.7-3)  : given by only the cavity mirror’s reflectivity Example (p. 311)

8.9 Frequency Tuning in Parametric Oscillation Phase-Matching condition : If the phase matching condition is satisfied at the angle, q=q0 And, we have

Neglecting the second order terms, (w3 is a fixed frequency, and if we use an extraordinary ray for the pump) (If we use ordinary rays for the signal and idler) Parametric oscillation frequency with the angle :

Example) Frequency tuning by using a negative uniaxial crystal

8.11 Frequency Up-Conversion : Sum Frequency Generation Phase-matching condition : Solution : where,

# Oscillating function with z (cf : parametric oscillation) therefore Power : # Oscillating function with z (cf : parametric oscillation)

Conversion efficiency : Typically, conversion efficiency is small Example (p. 318)