Nonlinear Optics

18.1 Nonlinear Polarisation

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

$P = \varepsilon_0(\chi^{(1)}E + \chi^{(2)}E^2 + \chi^{(3)}E^3 + \cdots)$

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

18.2 Second Harmonic Generation (SHG)

A beam of frequency $\omega$ generates light at $2\omega$ . 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 $d_{\text{eff} = \chi^{(2)}/2}$ is the effective nonlinear coefficient and $\Delta k = k_{2\omega} - 2k_\omega$ is the phase mismatch.

Phase matching: Maximum conversion occurs when $\Delta k = 0$ (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 $\chi^{(2)}$ every coherence length $\pi/\Delta k$ .

18.3 Other Nonlinear Processes

Process	Order	Description
SHG	$\chi^{(2)}$	$\omega + \omega \to 2\omega$
SFG	$\chi^{(2)}$	$\omega_1 + \omega_2 \to \omega_3$
Pockels effect	$\chi^{(2)}$	Linear electro-optic effect ( $\Delta n \propto E$ )
Optical Kerr effect	$\chi^{(3)}$	$n = n_0 + n_2 I$ (intensity-dependent refractive index)
Self-focusing	$\chi^{(3)}$	Beam collapses when $P > P_{\text{cr}}$
Two-photon absorption	$\chi^{(3)}$	Simultaneous absorption of two photons
Stimulated Raman/Brillouin	$\chi^{(3)}$	Inelastic scattering amplification

Self-phase modulation: The Kerr effect causes $\Delta n = n_2 I$ 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 $\lambda = 800$ nm ( $\omega$ ) and $\lambda = 400$ nm ( $2\omega$ ):

$n_o(800\,\text{nm}) = 1.6549$ , $n_e(800\,\text{nm}) = 1.5425$ (at $\theta = 29.2°$ )

$n_o(400\,\text{nm}) = 1.7030$ , $n_e(400\,\text{nm}) = 1.5665$ (at $\theta = 29.2°$ )

For Type I phase matching ( $o + o \to e$ ): $n_e(2\omega, \theta) = n_o(\omega)$ .

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

The coherence length without phase matching:

$\ell_c = \frac{\pi}{\Delta k} = \frac{\lambda}{4(n_e^{2\omega} - n_o^{\omega})}$

For typical values: $\ell_c \sim 5$ $\mu$ M. A 1 mm crystal is $\sim 200$ coherence lengths long, so phase matching is essential.

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

$\eta \approx \frac{8\pi^2 \times (2.0 \times 10^{-12})^2 \times 10^{-4} \times 10^{10}}{(1.6)^3 \times (400 \times 10^{-9})^2 \times 3 \times 10^8 \times 8.85 \times 10^{-12}} \approx 15\%$

Key Relationships

Effect	Susceptibility	Key Formula	Condition
Linear optics	$\chi^{(1)}$	$P = \varepsilon_0 \chi^{(1)} E$	Weak fields
SHG	$\chi^{(2)}$	$I_{2\omega} \propto d_{\text{eff}}^2 I_\omega^2 L^2 \,\text{sinc}^2(\Delta k L/2)$	Phase matching
Pockels effect	$\chi^{(2)}$	$\Delta n = n_0^3 r E / 2$	Non-centrosymmetric
Kerr effect	$\chi^{(3)}$	$n = n_0 + n_2 I$	All media
Self-focusing	$\chi^{(3)}$	$P_{\text{cr}} = \pi(0.61)^2 \lambda^2/(8n_0 n_2)$	$P > P_{\text{cr}}$

Common Pitfalls

Phase matching is essential: Without phase matching, the second-harmonic signal oscillates with crystal length, with the maximum efficiency at the coherence length $\ell_c = \pi/\Delta k$ . Beyond $\ell_c$ , back-conversion reduces the output.
$\chi^{(2)}$ requires non-centrosymmetry: In centrosymmetric media, all even-order nonlinearities vanish. Do not attempt SHG in glasses or cubic crystals like silicon without symmetry-breaking interfaces.
Kerr effect saturates at high intensity: The simple relation $n = n_0 + n_2 I$ holds only for $I \ll I_{\text{sat}}$ . At very high intensities, saturation, multiphoton absorption, and plasma generation modify the response.
Group velocity mismatch: For ultrashort pulses, the difference in group velocities between $\omega$ and $2\omega$ limits the interaction length. The walk-off length $L_{\text{walk-off}} = \tau_p / |v_g^{-1}(\omega) - v_g^{-1}(2\omega)|$ must exceed the crystal length.

Applications

Laser frequency conversion: SHG converts near-infrared Ti:sapphire laser output (800 nm) to blue/UV (400 nm). Sum-frequency generation produces tunable UV sources.
Electro-optic modulators: The Pockels effect enables high-speed optical modulators ( $>40$ GHz) for fibre-optic communications, using crystals like LiNbO $_3$ .
Ultrashort pulse generation: Kerr lens mode-locking (KLM) in Ti:sapphire lasers produces femtosecond pulses via self-focusing combined with an aperture.
Supercontinuum generation: Extreme spectral broadening in photonic crystal fibres, driven by self-phase modulation and soliton dynamics, produces octave-spanning spectra for frequency metrology.
Quantum optics: Spontaneous parametric down-conversion (SPDC) generates entangled photon pairs for quantum cryptography and quantum computing.

Connections to Other Topics

Quantum optics: SPDC is the workhorse for generating entangled photon pairs. The $\chi^{(2)}$ nonlinearity couples the vacuum field to signal and idler photons.
Femtosecond laser physics: The Kerr effect enables mode-locking, while self-phase modulation broadens the spectrum to support ultrashort pulses.
Solid-state physics: The nonlinear susceptibility tensor reflects crystal symmetry. Group theory determines which tensor components are nonzero for each crystal class.
Condensed matter: The electro-optic effect is used to characterise ferroelectric materials and domain structures.

Summary Table: Nonlinear Processes by Order

Order	Process	Application	Crystal Requirement
$\chi^{(1)}$	Linear refraction/absorption	Ordinary optics	Any
$\chi^{(2)}$	SHG, SFG, DFG, Pockels	Frequency conversion, modulators	Non-centrosymmetric
$\chi^{(2)}$	SPDC	Entangled photon pairs	Non-centrosymmetric
$\chi^{(3)}$	Kerr effect, SPM, XPM	Mode-locking, supercontinuum	All media
$\chi^{(3)}$	SRS, SBS	Amplifiers, lasers	All media
$\chi^{(3)}$	Two-photon absorption	Microscopy, lithography	All media
$\chi^{(3)}$	Self-focusing	Filamentation, damage	All media ( $n_2 > 0$ )

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