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Wyatt's Notes — University

Nonlinear Optics

18.1 Nonlinear Polarisation

When the electric field is strong (e.g., laser), the polarisation develops nonlinear terms:

P=ε0(χ(1)E+χ(2)E2+χ(3)E3+⋯ )P = \varepsilon_0(\chi^{(1)}E + \chi^{(2)}E^2 + \chi^{(3)}E^3 + \cdots)

The second-order susceptibility χ(2)\chi^{(2)} is nonzero only in non-centrosymmetric media. The third-order χ(3)\chi^{(3)} exists in all media.

18.2 Second Harmonic Generation (SHG)

A beam of frequency ω\omega generates light at 2ω2\omega. The intensity of the second harmonic:

I_{2\omega} = \frac{2\omega^2 d_{\text{eff}^2 I_\omega^2 L^2}{n_\omega^2 n_{2\omega} c^3 \varepsilon_0}\,\text{sinc}^2\!\left(\frac{\Delta k\,L}{2}\right)}

Where deff=χ(2)/2d_{\text{eff} = \chi^{(2)}/2} is the effective nonlinear coefficient and Δk=k2ω−2kω\Delta k = k_{2\omega} - 2k_\omega is the phase mismatch.

Phase matching: Maximum conversion occurs when Δk=0\Delta k = 0 (momentum conservation). Techniques:

  • Birefringent phase matching: Exploit the different refractive indices for ordinary and extraordinary polarisations.
  • Quasi-phase matching: Periodically pole the nonlinear crystal to reverse the sign of χ(2)\chi^{(2)} every coherence length π/Δk\pi/\Delta k.

18.3 Other Nonlinear Processes

ProcessOrderDescription
SHGχ(2)\chi^{(2)}ω+ω→2ω\omega + \omega \to 2\omega
SFGχ(2)\chi^{(2)}ω1+ω2→ω3\omega_1 + \omega_2 \to \omega_3
Pockels effectχ(2)\chi^{(2)}Linear electro-optic effect (Δn∝E\Delta n \propto E)
Optical Kerr effectχ(3)\chi^{(3)}n=n0+n2In = n_0 + n_2 I (intensity-dependent refractive index)
Self-focusingχ(3)\chi^{(3)}Beam collapses when P>PcrP > P_{\text{cr}}
Two-photon absorptionχ(3)\chi^{(3)}Simultaneous absorption of two photons
Stimulated Raman/Brillouinχ(3)\chi^{(3)}Inelastic scattering amplification

Self-phase modulation: The Kerr effect causes Δn=n2I\Delta n = n_2 I which broadens the spectrum of ultrashort pulses. Combined with dispersion, this leads to soliton formation in optical fibres (a balance between Kerr self-focusing and anomalous dispersion).

Worked Example 18.1: Phase Matching in BBO Crystal

Beta-barium borate (BBO) is a common nonlinear crystal for SHG of 800 nm Ti:sapphire laser light.

The relevant refractive indices at λ=800\lambda = 800 nm (ω\omega) and λ=400\lambda = 400 nm (2ω2\omega):

no(800 nm)=1.6549n_o(800\,\text{nm}) = 1.6549, ne(800 nm)=1.5425n_e(800\,\text{nm}) = 1.5425 (at θ=29.2°\theta = 29.2°)

no(400 nm)=1.7030n_o(400\,\text{nm}) = 1.7030, ne(400 nm)=1.5665n_e(400\,\text{nm}) = 1.5665 (at θ=29.2°\theta = 29.2°)

For Type I phase matching (o+o→eo + o \to e): ne(2ω,θ)=no(ω)n_e(2\omega, \theta) = n_o(\omega).

Using Sellmeier equations, the phase matching angle is found to be θPM≈29.2°\theta_{\text{PM} \approx 29.2°}.

The coherence length without phase matching:

ℓc=πΔk=λ4(ne2ω−noω)\ell_c = \frac{\pi}{\Delta k} = \frac{\lambda}{4(n_e^{2\omega} - n_o^{\omega})}

For typical values: ℓc∼5\ell_c \sim 5 μ\muM. A 1 mm crystal is ∼200\sim 200 coherence lengths long, so phase matching is essential.

The conversion efficiency for perfect phase matching with a 10 mm crystal at Iω=100I_\omega = 100 MW/cm2^2:

η≈8π2×(2.0×10−12)2×10−4×1010(1.6)3×(400×10−9)2×3×108×8.85×10−12≈15%\eta \approx \frac{8\pi^2 \times (2.0 \times 10^{-12})^2 \times 10^{-4} \times 10^{10}}{(1.6)^3 \times (400 \times 10^{-9})^2 \times 3 \times 10^8 \times 8.85 \times 10^{-12}} \approx 15\%