Physics · Optics · Year 2

Nonlinear optics: the green laser

How does infrared light pass through a crystal and come out green?

The situation

In the evening, in an astronomy club, the teacher points out stars with a green laser pointer — the beam appears in the air as a thin green line. Surprisingly, there is no source of green light inside the pointer: the diode emits invisible 808 nm infrared light, and an Nd:YVO₄ crystal turns it into 1064 nm light. The green colour (532 nm) appears only in the KTP crystal — the frequency of the light is doubled. How can a material change the colour of light?

Topic

Nonlinear optics. Inside a green laser pointer: the frequency of infrared light is doubled

Key ideas

Second harmonic
Phase matching
Kerr effect

Formulas

P = ε_{0} (χ^{(1)} E + χ^{(2)} E^{2} + χ^{(3)} E^{3} + \dots)

Nonlinear polarisation: in a strong field the permittivity becomes dependent on the field.

η = \frac{8 π^{2} d_{e f f}^{2} L^{2} I}{ε_{0} c n_{ω}^{2} n_{2 ω} λ^{2}} {sinc}^{2} \frac{Δ k L}{2}, I_{2 ω} = η I \propto I^{2}

Second-harmonic efficiency (at small conversion): the harmonic intensity is proportional to the square of the fundamental intensity.

Δ k = \frac{4 π}{λ} (n_{2 ω} - n_{ω}), l_{k} = \frac{π}{Δ k}

Phase mismatch and coherence length. In BBO $Δ k = 0$ at $θ_{p m} \approx 22.8 °$ .

n = n_{0} + n_{2} I, P_{k r} = \frac{3.77 λ^{2}}{8 π n_{0} n_{2}}

The Kerr effect and the critical power for self-focusing.

i \frac{\partial u}{\partial ξ} + \frac{1}{2} \frac{\partial^{2} u}{\partial τ^{2}} + | u |^{2} u = 0, N^{2} = \frac{γ P_{0} T_{0}^{2}}{| β_{2} |}

The nonlinear Schrödinger equation: $ξ = z / L_{D}$ , $L_{D} = T_{0}^{2} / | β_{2} |$ . $N = 1$ is the fundamental soliton.

Theory

Nonlinear polarisation

The electric field of light displaces the electrons in a material and produces a polarisation $P$ . In weak light $P$ is proportional to the field: $P = ε_{0} χ^{(1)} E$ , and the permittivity $ε = 1 + χ^{(1)}$ does not depend on the field. When the laser field ( $10^{7}$ – $10^{9}$ V/m) approaches the field inside the atom (≈ $5 \cdot 10^{11}$ V/m), the electron's “spring” stops being linear: $χ^{(2)} E^{2}$ and $χ^{(3)} E^{3}$ are added to $P$ and the permittivity depends on the field. Because of $\cos^{2} ω t = \frac{1}{2} (1 + \cos 2 ω t)$ the $E^{2}$ term radiates a wave of double frequency — watch the “Polarisation” graph. $χ^{(2)}$ is non-zero only in crystals with no centre of symmetry (KTP, BBO, quartz crystal); in glass, water and air $χ^{(2)} = 0$ .

The second harmonic and the green pointer

In terms of photons: two 1064 nm photons combine to give one 532 nm photon ( $2 ℏ ω = ℏ \cdot 2 ω$ ). The probability of such a “meeting” is proportional to the density of the light, so the harmonic power is proportional to the square of the intensity: if the IR power rises 2 times, the green light rises 4 times. In the pointer an 808 nm diode pumps an Nd:YVO₄ crystal, which lases at 1064 nm. The KTP crystal sits inside the resonator — there the IR power is many times greater than at the output, and since $η \propto I$ the conversion is efficient. At the end an IR filter stops the remaining 1064 and 808 nm light; in cheap pointers it may be missing — invisible IR light is dangerous to the eye.

Phase matching

Each layer of the crystal radiates a harmonic wave. Because of dispersion $n_{2 ω} \neq n_{ω}$ : the harmonic and the fundamental wave travel at different speeds, $Δ k = (4 π / λ) (n_{2 ω} - n_{ω})$ . Waves born at different places begin to cancel one another after the coherence length $l_{c} = π / Δ k$ : in BBO along the optic axis $l_{c} \approx 13.5$ µm — the harmonic rises and falls. In a birefringent crystal the index of the extraordinary ray $n_{e} (θ)$ depends on the angle: at $θ_{p m} \approx 22.8 °$ , $n_{e} (2 ω, θ) = n_{o} (ω)$ , $Δ k = 0$ and the harmonic grows along the whole crystal as $L^{2}$ . Deviation from the angle suppresses the signal as ${sinc}^{2} (Δ k L / 2)$ : a longer crystal needs more exact tuning.

The Kerr effect and self-focusing

The third-order term $χ^{(3)}$ makes the refractive index depend on intensity: $n = n_{0} + n_{2} I$ . The centre of a Gaussian beam is brighter than its edge — $n$ is greater at the centre, and the beam makes a converging lens for itself. If the power is below $P_{c r}$ diffraction wins; if it is higher the beam is squeezed at a distance $z_{c} = z_{R} / \sqrt{P / P_{c r} - 1}$ and forms a filament, emitting white light (a supercontinuum). For fused silica (1064 nm) $P_{c r} \approx 4$ MW, in air (800 nm) ≈ 3 GW — powerful femtosecond lasers form filaments metres long in air. The same $n_{2} I$ gives self-phase modulation in time, and $χ^{(3)}$ also produces the third harmonic.

The nonlinear Schrödinger equation and solitons

In an optical fibre two effects act on a pulse: dispersion broadens it, while the Kerr effect redistributes the frequencies through phase modulation. Both are described by the NLSE. In anomalous dispersion ( $β_{2} < 0$ ) they balance, and a bright soliton $u = sech τ$ travels without changing shape ( $N = 1$ ); an $N = 2$ pulse is compressed and broadened with period $\frac{π}{2} L_{D}$ . In normal dispersion a dark soliton — a “hole” in the light — lives inside a continuous beam. In anomalous dispersion a continuous beam is unstable: small noise turns into a chain of pulses — modulation instability. In space the same equation describes a beam in a planar waveguide (with transverse $x$ in place of $τ$ ): the balance of diffraction and self-focusing gives bright and dark spatial solitons.

About the model

Model: SHG — plane wave, pump depletion $η = \tanh^{2} \sqrt{η_{0}}$ ; the walk-off of the beam in the crystal and the pulse shape are neglected; the Sellmeier formula for BBO and $d_{e f f} = 2$ pm/V. Kerr — the Gaussian beam is assumed to keep its shape (an aberration-free approximation). NLSE — lossless, without higher-order dispersion, solved by the split-step Fourier method; fibre: $β_{2} = - 20$ ps²/km, $γ = 1.3$ (W·km)⁻¹, $T_{0} = 1$ ps. The powers in the pointer are approximate model values.

Did you know?

The experiment regarded as the beginning of nonlinear optics was carried out in 1961 — a year after the laser was invented: Peter Franken's group passed the 694 nm light of a ruby laser through a quartz crystal and recorded an ultraviolet harmonic at 347 nm.
In 1834 the Scottish engineer John Scott Russell followed on horseback a solitary wave that ran a long way down a canal without changing shape — the first description of a soliton. Solitons in optical fibre were observed in 1980 by Mollenauer, Stolen and Gordon.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.