Optics · Advanced Topics

Nonlinear Optics

When light is intense enough, the optical response of a medium becomes nonlinear — photons interact with each other through the medium. Nonlinear optics is the physics of lasers interacting with matter: frequency doubling, optical parametric amplification, solitons, and quantum light generation.

Prerequisites

Wave optics (Ch. WO) \cdot Lasers & coherent light (Ch. LA) \cdot Maxwell's equations (Ch. EM) \cdot E

lectrostatics: boundary (Ch. ES)

Learning Goals

Write $the nonlinear polarization expansion and identify which symmetry classes permit non-zero \chi$ . $\chi^{(3)}$
Explain the phase-matching condition for SHG, derive the coherence length, and contrast birefringent vs. quasi-phase-matching.
Describe optical parametric amplification (OPA) and spontaneous parametric down-conversion (SPDC) as sources of entangled photon pairs.
Derive the nonlinear Schrödinger equation for pulse propagation in fibre and identify the conditions for soliton formation.
Explain how a mode-lo $cked laser generates an optical frequency comb and how f-2f self-referencing stabilises f_$

NL.1 Nonlinear Polarization

The polarization of a medium in an electric field E:

P = \varepsilon_{0}(\chi^{(1)}E + \chi^{(2)}E^{2} + \chi^{(3)}E^{3} + \cdots) \qquad (nonlinear polarization)

(NL.1)

χ⁽¹⁾: linear susceptibility (index n = √(1+χ⁽¹⁾)). χ⁽²⁾: second-order (non-zero only in non-centrosymmetric materials — no inversion symmetry). χ⁽³⁾: third-order (present in all materials, responsible for Kerr effect).

At what intensity is χ⁽²⁾ important? When χ⁽²⁾E ~ χ⁽¹⁾: E ~ χ⁽¹⁾/χ⁽²⁾ ~ 10¹⁰ V/m (atomic field scale). For a 1 W laser focused to 1 μm²: I = 10⁹ W/m², E = √(2I/cε₀) ≈ 10⁶ V/m. Need pulsed lasers (MW–TW) to reach nonlinear regime. Modern OPAs achieve 10¹⁸ W/m².

NL.2 Second-Harmonic Generation

If E = E₀ cos(ωt), then χ⁽²⁾E² = χ⁽²⁾E₀²(1 + cos(2ωt))/2 — contains a component at 2ω. This is second-harmonic generation (SHG). The nonlinear wave equation:

\partial^{2}E_{2\omega}/\partialz^{2} - (n_{2\omega}^{2}/c^{2})\partial^{2}E_{2\omega}/\partialt^{2} = (1/c^{2}\varepsilon_{0}) \partial^{2}P_{2\omega}/\partialt^{2} \qquad (NL wave equation)

(NL.2)

Phase matching: efficient SHG requires the fundamental and second-harmonic waves to stay in phase. Phase mismatch: Δk = k_(2ω) − 2k_ω = 2ω(n_(2ω) − n_ω)/c. Without phase matching: I_(2ω) ∝ sinc²(ΔkL/2) — oscillates, maximum at L = π/(2Δk) (coherence length L_c ~ 10 μm in typical crystals).

Birefringent phase matching: use the ordinary and extraordinary rays of a birefringent crystal. Type I: n_e(2ω) = n_o(ω) by choosing the crystal angle.Quasi-phase-matching(QPM): periodically poled crystal (PPLN: Periodically Poled Lithium Niobate) with period Λ = 2L_c reverses χ⁽²⁾ every coherence length. Allows phase matching at any wavelength by choosing Λ. Conversion efficiencies > 50%.

Example NL.1 — SHG in KTP Crystal

$A 1064 nm Nd:YAG laser (P = 1 W, beam area A = 1 mm^{2}) is frequency-doubled in a 5 mm KTP c$ $rystal (\chi^{(2)} = 10 pm/V, n = 1.74). Estimate the conversion efficiency$ .

SHG intensity:

Under undepleted, plane-wave, phase-matched conditions, I_{2\omega} scales as |\chi^{(2)}|^{2} I_\omega^{2} L^{2}

. This gives the right dependence, but real efficiency also depends stron

gly on focusing, walkoff, coatings, absorption, and the effective nonlinear coefficient d_

Calculate:

I_\omega = P/A = 1/(10^{-6}) = 10^{6} W/m^{2}. This is modest intensity for nonlinear optics because the

beam area is large. A reasonable plane-wave estimate gives conversion on the order of 10^{-4}

to 10^{-3} for a few-mm crystal at this intensity

Total conversion:

So the output is in the sub-milliwatt to milliwatt range for this loose 1 mm^{2} beam. High 4

0–60% SHG efficiency is possible, but it requires much tighter focusing, a longer optimized crystal, quasi-phase matching, or an enhancement cavity.

NL.3 Optical Parametric Amplification

A pump photon at ω_p splits into signal (ω_s) and idler (ω_i) with ω_p = ω_s + ω_i —optical parametric amplification (OPA). The signal is amplified while the idler is generated. Phase matching: k_p = k_s + k_i.

OPAs can be tuned over wide ranges by adjusting the crystal angle or temperature. Optical parametric oscillators (OPOs) add a cavity — threshold when gain exceeds losses. Coverage: UV to mid-IR from a single pump laser. Applications: terahertz generation, frequency combs, squeezed light for quantum optics.

Spontaneous parametric down-conversion (SPDC): even at zero signal input, vacuum fluctuations seed the conversion. Produces entangled photon pairs— the workhorse source for quantum optics experiments. The signal and idler photons are entangled in polarization, momentum, and energy.

NL.4 Self-Phase Modulation and Solitons

The intensity-dependent refractive index (Kerr effect): n = n₀ + n₂I. For silica fiber: n₂ ≈ 2.6×10⁻²⁰ m²/W. An intense pulse modulates its own phase:

\Delta\phi = n_{2} I(t) \omega_{0} L/c \qquad (self-phase modulation phase shift)

(NL.3)

SPM broadens the pulse spectrum (creates new frequencies: chirp). Combined with anomalous group velocity dispersion (β₂ < 0, where the shorter-wavelength part travels faster), SPM can balance dispersion exactly — creating optical solitons: pulses that propagate without changing shape.

\partialA/\partialz + i\beta_{2}/2 \times \partial^{2}A/\partialt^{2} - i\gamma|A|^{2}A = 0 \qquad (nonlinear Schrödinger equation, fiber)

(NL.4)

The NLS equation is exactly solvable (inverse scattering). Soliton solution: A(z,t) = √P_0 sech(t/T₀) e^(iγP₀z/2). Modern submarine fiber-optic cables use soliton-like pulses and dispersion-managed transmission for Tbit/s data rates.

NL.5 Frequency Combs

A mode-locked laser emits pulses with repetition rate f_rep. In the frequency domain: a comb of modes equally spaced by f_rep, offset by f_CEO (carrier-envelope offset). The optical frequency comb:

f_{n} = n \times f_{rep} + f_{CEO} \qquad (optical frequency comb, n = integer)

(NL.5)

Self-referencing: use f-2f interferometry to measure f_CEO → fully determined comb. Accuracy: 10⁻¹⁹ fractional (limited by optical clocks). Applications: GPS, optical clock comparison, search for dark matter (variation of constants), exoplanet spectrograph calibration (radial velocities to cm/s precision). Nobel Prize 2005 (Hänsch and Hall).

Definition NL.1 — Common Traps

Nonlinear effects need high field strengths: $ordinary weak light usually sees only \chi^{(1)}$
Energy conservation is not enough: efficient frequency conversion also needs phase matching.
Self-phase modulation changes spectrum: time-dependent intensity creates time-dependent phase and frequency chirp.
Comb lines need both spacing and offset: $f_{rep} alone does not determine absolute optical frequencies$

Exercises — NL.1–NL.5 Nonlinear Optics

1.Explain why perfect phase matching is impossible in an isotropic dispersive medium for SHG. How does birefringent phase matching work, and what is quasi-phase-matching?

Straightforward

2.Describe spontaneous parametric down-conversion (SPDC) for producing entangled photon pairs. What are the energy and momentum conservation conditions? How are the photons entangled?

Intermediate

Find the pulse duration T_{0} (ps) and power P_{0} for a fundamental optical soliton in standard

single-mode fiber (\beta_{2} = -20 ps^{2}/km, \gamma = 1.3 W^{-1}km^{-1}) with P_{0} = 1 mW. What is the soliton p

riod?

Intermediate

4.Describe how a mode-

locked laser produces an optical frequency comb. How does f-2f interferometry stabilize f_

accuracy is achievable, and how is it used in exoplanet detection?

Challenging

Key Takeaways

$Nonlinear polarization: P = \varepsilon_{0}(\chi^{1}E + \chi^{2}E^{2} + \cdots). \chi^{2} non-zero only in non-centrosymmetric$ crystals.
$SHG: \chi^{2}E^{2} contains 2\omega component. Efficient only with phase matching: n(2\omega) = n(\omega$ .
$OPA: \omega_p \to \omega_s + \omega_i. SPDC: vacuum fluctuations generate entangled photon pairs$ .
$Kerr effect: n = n_{0} + n_{2}I. SPM: frequency broadening. With anomalous GVD \to solitons$ .
$NLS equation: soliton N=1 balance. A = \sqrtP_{0} sech(t/T_{0}) propagates without distortion$ .
$Frequency comb: f_{n} = nf_rep + f_{CEO}. f-2f stabilization \to 10^{-19} accuracy. GPS, spectrosco$ py, exoplanets.