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# Nonlinear optics

Nonlinear optics is the branch of optics that describes the behaviour of light in nonlinear media, that is, media in which the polarization P responds nonlinearly to the electric field E of the light. This nonlinearity is typically only observed at very high light intensities such as provided by pulsed lasers.

Nonlinear optics gives rise to a host of optical phenomena:

 Contents

## Frequency mixing processes

• Second harmonic generation (SHG), or frequency doubling, generation of light with a doubled frequency (half the wavelength);
• Sum-frequency generation (SFG), generation of light with a frequency that is the sum of two other frequencies (SHG is a special case of this);
• Third harmonic generation (THG), generation of light with a tripled frequency (one-third the wavelength) (usually done in two steps: SHG followed by SFG of original and frequency-doubled waves);
• Difference-frequency generation (DFG), generation of light with a frequency that is the difference between two other frequencies;
• Parametric amplification, amplification of a signal input in the presence of a higher-frequency pump wave, at the same time generating an idler wave (can be considered as DFG);
• Parametric oscillation, generation of a signal and idler wave using a parametric amplifier in a resonator (with no signal input);
• Parametric generation, like parametric oscillation but without a resonator, using a very high gain instead;
• Optical rectification, generation of quasi-static electric fields.

## Other nonlinear processes

• Optical Kerr effect, intensity dependent refractive index;
• Self focusing;
• Self-phase modulation (SPM);
• Cross-phase modulation (XPM);
• Four-wave mixing (FWM), can also arise from other nonlinearities.
• Raman scattering, interaction of photons with optical phonons;
• Raman amplification,
• Optical phase conjugation.
• Brillouin scattering, interaction of photons with acoustic phonons;
• Optical phase conjugation.
• Two-photon absorption, simultaneous absorption of two photons, transferring the energy to a single electron;
• Multiple photoionisation, near-simultaneous removal of many bound electrons by one photon.

## Related processes

In these processes, the medium has a linear response to the light, but the properties of the medium are affected by other causes:

• Pockels effect, the refractive index is affected by a static electric field; used in electro-optic modulators;
• Acousto-optics, the refractive index is affected by acoustic waves (ultrasound); used in acousto-optic modulators .

## Frequency-mixing processes

One of the most commonly-used frequency-mixing processes is frequency doubling or second-harmonic generation. With this technique, the 1064-nm output from Nd:YAG lasers or the 800-nm output from Ti:sapphire lasers can be converted to visible light, with wavelengths of 532 nm (green) or 400 nm (violet), respectively.

Practically, frequency-doubling is carried out by placing a special crystal in a laser beam under a well-chosen angle. Commonly-used crystals are BBO (β-barium borate), KDP (potassium dihydrophosphate), and lithium niobate. These crystals have the necessary properties of being strongly birefringent (necessary to obtain phase matching, see below), having a specific crystal symmetry and of course being transparent for and resistant against the high-intensity laser light.

### Theory

A number of nonlinear optical phenomena can be described as frequency-mixing processes. In general, the dielectric polarization P(t) at time t in a medium can be written as a power series in the electrical field:

Failed to parse (unknown function \propto): P(t) \propto \chi^{(1)} E(t) + \chi^{(2)} E^2(t) + \chi^{(3)} E^3(t) + \cdots

.

Here, the coefficients χ(n) are the n-th order susceptibilities of the medium. For any three-wave mixing process, the second-order term is crucial; it is only nonzero in media that have a broken inversion symmetry. If we write

$E(t) = E_1 e^{i \omega_1 t} + E_2 e^{i \omega_2 t} + {\rm c.c.}$,

where c.c. denotes the complex conjugate, the second-order term in will read

Failed to parse (unknown function \propto): P^{(2)}(t) \propto \sum\chi^{(2)} n_0 E_1^{n_1} E_2^{n_2} e^{i (m_1\omega_1 + m_2\omega_2) t } + {\rm c.c.}

,

where the summation is over

$(n_0,n_1,n_2,m_1,m_2) = (1,2,0,2,0),\; (1,0,2,0,2),\; (2,2,0,0,0),\; (2,0,2,0,0),\; (2,1,1,1,-1),\; (2,1,1,1,1)$.

The six combinations (nx,mx) correspond, respectively, to the second harmonic of E1, the second harmonic of E2, the optically rectified signals of E1 and E2, the difference frequency, and the sum frequency. A medium that is thus pumped by the fields E1 and E2 will radiate a field E3 with an angular frequency ω3 = m1ω1 + m2ω2.

Note: in this description, χ(2) is a scalar. In reality, χ(2) is a tensor whose components depend on the combination of frequencies.

Parametric generation and amplification is a variation of difference frequency generation, where the lower-frequency one of the two generating fields is much weaker (parametric amplification) or completely absent (parametric generation). In the latter case, the fundamental quantum-mechanical uncertainty in the electric field initiates the process.

### Phase matching

The above ignores the position dependence of the electrical fields. In a typical situation, the electrical fields are traveling waves described by

$E_j(\mathbf{x},t) = e^{i(\omega_j t - \mathbf{k}_j \cdot \mathbf{x})}$,

at position $\mathbf{x}$, with the wave vector $\mathbf{k}_j = n(\omega_j)\omega_j/c$, where c is the velocity of light and nj) the index of refraction of the medium at angular frequency ωj. Thus, the second-order polarization angular frequency ω3 is

Failed to parse (unknown function \propto): P^{(2)} (\mathbf{x}, t) \propto E_1^{n_1} E_2^{n_2} e^{i (\omega_3 t - (m_1 \mathbf{k}_1 + m_2 \mathbf{k}_2)\cdot\mathbf{x})}

.

At each position $\mathbf{x}$, the oscillating second-order polarization radiates at angular frequency ω3 and a corresponding wave vector $\mathbf{k}_3 = n(\omega_3)\omega_3/c$. Constructive interference, and therefore a high intensity ω3 field, will occur only if

$\mathbf{k}_3 = m_1 \mathbf{k}_1 + m_2 \mathbf{k}_2$.

The above equation is known as the phase matching condition. Typically, three-wave mixing is done in a birefringent crystalline material (I.e., the refractive index depends on the polarization and direction of the light that passes through.), where the polarizations of the fields and the orientation of the crystal are chosen such that the phase-matching condition is fulfilled.

### Higher-order frequency mixing

The above holds for χ(2) processes. It can be extended for processes where χ(3) is nonzero, something that is generally true in any medium without any symmetry restrictions. Third-harmonic generation is a χ(3) process, although in laser applications, it is usually implemented as a two-stage process: first the fundamental laser frequency is doubled and then the doubled and the fundamental frequencies are added in a sum-frequency process. The Kerr effect can be described as a χ(3) as well.

### Optical phase conjugation

It is possible, using nonlinear optical processes, to exactly reverse the propagation direction and phase variation of a beam of light. The reversed beam is called a conjugate beam, and thus the technique is known as optical phase conjugation (also called time reversal, wavefront reversal and retroreflection).

The most common way of producing optical phase conjugation is to use a four-wave mixing technique, though it is also possible to use processes such as stimulated Brillouin scattering. A device producing the phase conjugation effect is known as a phase conjugate mirror (PCM).

For the four-wave mixing technique, we can describe four beams (j = 1,2,3,4) with electric fields:

$\Xi_j(\mathbf{x},t) = \frac{1}{2} E_j(\mathbf{x}) e^{i (\omega_j t - \mathbf{k}\cdot\mathbf{x})} + \mbox{c.c.}$

where Ej are the electric field amplitudes. Ξ1 and Ξ2 are known as the two pump waves, with Ξ3 being the signal wave, and Ξ4 being the generated conjugate wave.

If the pump waves and the signal wave are superimposed in a medium with a non-zero χ(3), this produces a nonlinear polarization field:

PNL = ε0χ(3)1 + Ξ2 + Ξ3)3

resulting in generation of waves with frequencies given by ω = ±ω1 ±ω2 ±ω3 in addition to third harmonic generation waves with ω = 3ω1, 3ω2, 3ω3.

As above, the phase-matching condition determines which of these waves is the dominant. By choosing conditions such that ω = ω1 + ω2 - ω3 and k = k1 + k2 - k3, this gives a polarization field:

$P_\omega = \frac{1}{2} \chi^{(3)} \epsilon_0 E_1 E_2 E_3^* e^{i(\omega t - \mathbf{k} \cdot \mathbf{x} ) } + \mbox{c.c.}$.

This is the generating field for the phase conjugate beam, Ξ4. Its direction is given by k4 = k1 + k2 - k3, and so if the two pump beams are counterpropagating (k1 = -k2), then the conjugate and signal beams propagate in opposite directions (k4 = -k3). This results in the retroreflecting property of the effect.

Further, it can be shown for a medium with refractive index n and a beam interaction length l, the electric field amplitude of the conjugate beam is approximated by

$E_4 = \frac{i \omega l}{2 n c} \chi^{(3)} E_1 E_2 E_3^*$

(where c is the speed of light). If the pump beams E1 and E2 are plane (counterpropagating) waves, then:

Failed to parse (unknown function \propto): E_4(\mathbf{x}) \propto E_3^*(\mathbf{x})

that is, the generated beam amplitude is the complex conjugate of the signal beam amplitude. Since the imaginary part of the amplitude contains the phase of the beam, this results in the reversal of phase property of the effect.

Note that the constant of proportionality between the signal and conjugate beams can be greater than 1. This is effectively a mirror with a reflection coefficient greater than 100%, producing an amplified reflection. The power for this comes from the two pump beams, which are depleted by the process.

The frequency of the conjugate wave can be different from that of the signal wave. If the pump waves are of frequency ω1 = ω2 = ω, and the signal wave higher in frequency such that ω3 = ω + Δω, then the conjugate wave is of frequency ω4 = ω - Δω. This is known as frequency flipping.