Nonlinear Optics | Physics - Wyatt's Notes

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 ( $n_2 > 0$ )

flowchart TD
    A[22_Nonlinear Optics] --> B[Key Concepts]
    A --> C[Core Principles]
    A --> D[Practical Applications]
    B --> E[Fundamental definitions]
    C --> F[Design patterns]
    D --> G[Real-world usage]

Intuition

Linear optics assumes the medium’s response is proportional to the applied field, but at high intensities, nonlinear effects emerge. The second-order nonlinearity generates harmonics at twice the frequency, used in green laser pointers. Third-order effects include self-focusing, where a beam modifies the refractive index and collapses under its own intensity. Phase matching ensures that nonlinear contributions add constructively over the interaction length. Four-wave mixing and parametric amplification enable optical frequency conversion and amplification. These effects are weak at ordinary light levels but become dominant in focused laser beams, opening applications from frequency doubling to optical computing.

Cross-References

Lasers — High-intensity laser light is the primary driver of nonlinear optical effects; mode-locked lasers produce the peak powers needed for SHG and Kerr lensing.
Coherence Theory — Phase matching in nonlinear crystals requires coherence between the fundamental and harmonic fields.
Fourier Optics — Spatial filtering and beam propagation in nonlinear media use the Fourier transform relationship between near and far fields.
Calculus
Linear Algebra
Vector Calculus
Quantum Computing

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