Chapter 8 SecondHarmonic Generation and Parametric Oscillation 8

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 Nonlinear Optics Lab. Hanyang Univ.

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

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 Optics Lab. Hanyang Univ.

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 Nonlinear Optics Lab. Hanyang Univ.

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 ! Nonlinear Optics Lab. Hanyang Univ.

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 Nonlinear Optics Lab. Hanyang Univ.

Similarly, Assume, Each term proportional to ln should satisfy the equation separately Solution : : Report Nonlinear Optics Lab. Hanyang Univ.

Example) Solution for SHG Put general solution as Similarly, : Report Nonlinear Optics Lab. Hanyang Univ.

Susceptibility Polarization : : linear susceptibility : SHG : SFG : DFG : OR Nonlinear Optics Lab. Hanyang Univ.

<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) : mw 02 d=m. Dd D=w 02/d : roughly the same for all noncentrosymmetric solids. (non-resonant case) : used in rough estimation of nonlinear coefficient. : good agreement with the measured values Nonlinear Optics Lab. Hanyang Univ.

Qualitative understanding of Second order nonlinear effect in a noncentrosymmetric media Nonlinear Optics Lab. Hanyang Univ.

2 w component Nonlinear Optics Lab. Hanyang Univ.

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

Example 1. SHG Example 2. SFG Nonlinear Optics Lab. Hanyang Univ.

Properties of the nonlinear susceptibility tensor 1) Reality of the fields are real measurable quantities : 2) Intrinsic permutation symmetry Nonlinear Optics Lab. Hanyang Univ.

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

Define, 2 nd 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 3 x 6 matrix ; : 18 elements Nonlinear Optics Lab. Hanyang Univ.

Again, by Kleinman symmetry (Indices can be freely permuted), dil has only 10 independent elements : : Report Nonlinear Optics Lab. Hanyang Univ.

Example 1. SHG : Report Example 2. SFG Nonlinear Optics Lab. Hanyang Univ.

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 Nonlinear Optics Lab. Hanyang Univ.

w 1 term (slow varying approximation) Text Nonlinear Optics Lab. Hanyang Univ.

Similarly, Nonlinear Optics Lab. Hanyang Univ.

8. 3 Optical Second-Harmonic Generation Neglecting the absorption ; where, Assume, the depletion of the input wave power due to the conversion is negligible Nonlinear Optics Lab. Hanyang Univ.

Output intensity of 2 nd harmonic wave : Conversion efficiency : Phase-matching in SHG Maximum output @ If : phase-matching condition : 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) Nonlinear Optics Lab. Hanyang Univ.

Technique for phase-matching in anisotropic crystal So, Example) Phase matching in a negative uniaxial crystal Nonlinear Optics Lab. Hanyang Univ.

# If , 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 Nonlinear Optics Lab. Hanyang Univ.

Nonlinear Optics Lab. Hanyang Univ.

Second-Harmonic Generation with Focused Gaussian Beams If z 0>>l, the intensity of the incident beam is nearly independent of z within the crystal Total power of fundamental beam with Gaussian beam profile : Nonlinear Optics Lab. Hanyang Univ.

So, Conversion efficiency : : identical to (8. 3 -5) for the plane wave case (*) P(2 w) can be increased by decreasing w 0 until z 0 becomes comparable to l # It is reasonable to focus the beam until l=2 z 0 (confocal focusing) (**) Example (p. 292) Nonlinear Optics Lab. Hanyang Univ.

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 : Nonlinear Optics Lab. Hanyang Univ.

Define, : Total energy conservation Initial condition : # : 100% conversion [2 N(w photons) N(2 w photons)] Nonlinear Optics Lab. Hanyang Univ.

Conversion efficiency : Nonlinear Optics Lab. Hanyang Univ.

8. 4 Second-Harmonic generation Inside the Laser Resonator # Second-harmonic power # Laser intracavity power : Pump beam power Efficient SHG SH output power : Nonlinear Optics Lab. Hanyang Univ.

8. 5 Photon Model of SHG Annihilation of two Photons at w and a simultanous creation of a photon at 2 w - Energy : w+ w=2 w - Momentum : Nonlinear Optics Lab. Hanyang Univ.

8. 6 Parametric Amplification : # Special case : w 1=w 2 (degenerate parametric amplification) Analogous Systems : - Classical oscillators - Parasitic resonances in pipe organs(1883, L. Rayleigh) : - RLC circuits Example) RLC circuit Nonlinear Optics Lab. Hanyang Univ.

Assuming Put, where, Steady-state solution : (degenerate parametric oscillation) Phase matching Threshold condition Nonlinear Optics Lab. Hanyang Univ.

Optical parametric Amplification Polarization of 2 nd order nonlinear crystal : Nonlinear Optics Lab. Hanyang Univ.

(8. 2 -13), where, When (lossless), (phase-matching), and also depletion of field due to the conversion is negligible, where, Nonlinear Optics Lab. Hanyang Univ.

Solution : Qualitative understanding of parametric oscillation : # Initially w 1(or w 2) is generated by two photon spontaneous fluorescence or by cavity resonance # w 2(or w 1) wave increases by difference frequency generation between w 3 and w 1(or w 2) # w 1(or w 2) wave also increases by difference frequency generation between w 3 and w 1(or w 2) # w 2(or w 1) wave : Signal [A(0)=0] # w 2(or w 1) wave : Idler [A(0)>0] Nonlinear Optics Lab. Hanyang Univ.

Initial condition : Photon flux : Nonlinear Optics Lab. Hanyang Univ.

8. 7 Phase-Matching in Parametric Amplification Put, Nonlinear Optics Lab. Hanyang Univ.

General solution : Nonlinear Optics Lab. Hanyang Univ.

Phase-Matching Example) Phase-matching by using a negative uniaxial crystal : Report Nonlinear Optics Lab. Hanyang Univ.

8. 8 Parametric Oscillation where, (8. 8 -1) Nonlinear Optics Lab. Hanyang Univ.

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, Condition for nontrivial solution : absorption in crystal, reflections on the interfaces, cavity loss(mirrors, diffraction, scattering), … : Threshold condition for parametric oscillation Nonlinear Optics Lab. Hanyang Univ.

If we choose to express the mode losses at w 1 amd w 2 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 : Nonlinear Optics Lab. Hanyang Univ.

Example) Absorption loss = 0 (4. 7 -5), (4. 7 -3) : given by only the cavity mirror’s reflectivity Example (p. 311) Nonlinear Optics Lab. Hanyang Univ.

8. 9 Frequency Tuning in Parametric Oscillation Phase-Matching condition : If the phase matching condition is satisfied at the angle, q=q 0 And, we have Nonlinear Optics Lab. Hanyang Univ.

Neglecting the second order terms, (w 3 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 : Nonlinear Optics Lab. Hanyang Univ.

Example) Frequency tuning by using a negative uniaxial crystal Nonlinear Optics Lab. Hanyang Univ.

8. 11 Frequency Up-Conversion : Sum Frequency Generation Phase-matching condition : Solution : where, Nonlinear Optics Lab. Hanyang Univ.

therefore Power : # Oscillating function with z (cf : parametric oscillation) Nonlinear Optics Lab. Hanyang Univ.

Conversion efficiency : Typically, conversion efficiency is small Example (p. 318) Nonlinear Optics Lab. Hanyang Univ.