# Nonlinear optics

Nonlinear optics (NLO) is the branch of optics that describes the behaviour of light in nonlinear media, that is, media in which the dielectric 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 those provided by pulsed lasers.

Nonlinear optics gives rise to a host of optical phenomena:

## 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;
• Spontaneous parametric down conversion (SPDC), the amplification of the vacuum fluctuations in the low gain regime;
• Optical rectification, generation of quasi-static electric fields.
• Four-wave mixing (FWM), can also arise from other nonlinearities.

## 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:

## 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 dihydrogen phosphate), KTP (potassium titanyl phosphate), 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. However, organic polymeric materials are set to take over from crystals as they are cheaper to make, have lower drive voltages and superior performance.

### Theory

A number of nonlinear optical phenomena can be described as frequency-mixing processes. If the induced dipole moments of the material respond instantaneously to an applied electric field, the dielectric polarization (dipole moment per unit volume) ${\displaystyle P(t)}$ at time ${\displaystyle t}$ in a medium can be written as a power series in the electrical field:

${\displaystyle P(t)\propto \chi ^{(1)}E(t)+\chi ^{(2)}E^{2}(t)+\chi ^{(3)}E^{3}(t)+\cdots }$.

Here, the coefficients ${\displaystyle \chi ^{(n)}}$ are the ${\displaystyle n}$-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 no inversion symmetry. If we write

${\displaystyle E(t)=E_{1}e^{i\omega _{1}t}+E_{2}e^{i\omega _{2}t}+{\rm {c.c.}}}$,

where c.c. denotes the complex conjugate (E1 and E2 being the incident beams of interest), the second-order term in will read

${\displaystyle P^{(2)}(t)\propto \sum \chi ^{(2)}n_{0}E_{1}^{n_{1}}E_{2}^{n_{2}}e^{i(m_{1}\omega _{1}+m_{2}\omega _{2})t}+{\rm {c.c.}}}$,

where the summation is over

${\displaystyle (n_{0},n_{1},n_{2},m_{1},m_{2})=(1,2,0,2,0),\;(1,0,2,0,2),\;(2,2,0,0,0),\;(2,0,2,0,0),\;(2,1,1,1,-1),\;(2,1,1,1,1)}$.

The six combinations ${\displaystyle (n_{x},m_{x})}$ correspond, respectively, to the second harmonic of ${\displaystyle E_{1}}$, the second harmonic of ${\displaystyle E_{2}}$, the optically rectified signals of ${\displaystyle E_{1}}$ and ${\displaystyle E_{2}}$, the difference frequency, and the sum frequency. A medium that is thus pumped by the fields ${\displaystyle E_{1}}$ and ${\displaystyle E_{2}}$ will radiate a field ${\displaystyle E_{3}}$ with an angular frequency ${\displaystyle \omega _{3}=m_{1}\omega _{1}+m_{2}\omega _{2}}$.

Note: in this description, ${\displaystyle \chi ^{(2)}}$ is a scalar. In reality, ${\displaystyle \chi ^{(2)}}$ 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

${\displaystyle E_{j}(\mathbf {x} ,t)=e^{i(\omega _{j}t-\mathbf {k} _{j}\cdot \mathbf {x} )}}$,

at position ${\displaystyle \mathbf {x} }$, with the wave vector ${\displaystyle \mathbf {k} _{j}=n(\omega _{j})\omega _{j}/c}$, where ${\displaystyle c}$ is the velocity of light and ${\displaystyle n(\omega _{j})}$ the index of refraction of the medium at angular frequency ${\displaystyle \omega _{j}}$. Thus, the second-order polarization angular frequency ${\displaystyle \omega _{3}}$ is

${\displaystyle P^{(2)}(\mathbf {x} ,t)\propto E_{1}^{n_{1}}E_{2}^{n_{2}}e^{i(\omega _{3}t-(m_{1}\mathbf {k} _{1}+m_{2}\mathbf {k} _{2})\cdot \mathbf {x} )}}$.

At each position ${\displaystyle \mathbf {x} }$, the oscillating second-order polarization radiates at angular frequency ${\displaystyle \omega _{3}}$ and a corresponding wave vector ${\displaystyle \mathbf {k} _{3}=n(\omega _{3})\omega _{3}/c}$. Constructive interference, and therefore a high intensity ${\displaystyle \omega _{3}}$ field, will occur only if

${\displaystyle \mathbf {k} _{3}=m_{1}\mathbf {k} _{1}+m_{2}\mathbf {k} _{2}}$.

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. This phase matching technique is called angle tuning. Typically a crystal has three axes, one of which has a different refractive index than the other ones. This axis is called the extraordinary (e) axis, while the other two are ordinary axes (o). There are several schemes of choosing the polarizations. If the signal and idler have the same polarization, it is called "Type-I phase-matching", and if their polarizations are perpendicular, it is called "Type-II phase-matching". However, other conventions exist that specify further which frequency has what polarization relative to the crystal axis. These types are listed below, with the convention that the signal wavelength is shorter than the idler wavelength.

Phase-matching types (${\displaystyle \lambda _{p}\leq \lambda _{s}\leq \lambda _{i}}$)
Polarizations Scheme
Pump Signal Idler
e o o Type I
e o e Type II (or IIA)
e e o Type III (or IIB)
e e e Type IV
o o o Type V
o o e Type VI (or IIB or IIIA)
o e o Type VII (or IIA or IIIB)
o e e Type VIII (or I)

Most common nonlinear crystals are negative unaxial, which means that the e axis has a smaller refractive index than the o axes. In those crystals, type I and II phasematching are usually the most suitable schemes. In positive uniaxial crystals, types VII and VIII are more suitable. Types II and III are essentially equivalent, except that the names of signal and idler are swapped when the signal has a longer wavelength than the idler. For this reason, they are sometimes called IIA and IIB. The type numbers V–VIII are less common than I and II and variants.

One undesirable effect of angle tuning is that the optical frequencies involved do not propagate collinearly with each other. This is due to the fact that the extraordinary wave propagating through a birefringent crystal possesses a Poynting vector that is not parallel with the propagation vector. This would lead to beam walkoff which limits the nonlinear optical conversion efficiency. Two other methods of phase matching avoids beam walkoff by forcing all frequencies to propagate at a 90 degree angle with respect to the optical axis of the crystal. These methods are called temperature tuning and quasi-phase-matching.

Temperature tuning is where the pump (laser) frequency polarization is orthogonal to the signal and idler frequency polarization. The birefringence in some crystals, in particular Lithium Niobate is highly temperature dependent. The crystal is controlled at a certain temperature to achieve phase matching conditions.

The other method quasi-phase matching. In this method the frequencies involved are not constantly locked in phase with each other, instead the crystal axis is flipped at a regular interval Λ, typically 15 micrometres in length. Hence, these crystals are called periodically-poled. This results in the polarization response of the crystal to be shifted back in phase with the pump beam by reversing the nonlinear susceptibility. This allows net positive energy flow from the pump into the signal and idler frequencies. In this case, the crystal itself provides the additional wavevector k=2π/λ (and hence momentum) to satisfy the phase matching condition. Quasi-phase matching can be expanded to chirped gratings to get more bandwidth and to shape an SHG pulse like it is done in a dazzler. SHG of a pump and Self-phase modulation (emulated by second order processes) of the signal and an optical parametric amplifier can be integrated monolithically.

### Higher-order frequency mixing

The above holds for ${\displaystyle \chi ^{(2)}}$ processes. It can be extended for processes where ${\displaystyle \chi ^{(3)}}$ is nonzero, something that is generally true in any medium without any symmetry restrictions. Third-harmonic generation is a ${\displaystyle \chi ^{(3)}}$ 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 ${\displaystyle \chi ^{(3)}}$ as well.

At high intensities the Taylor series, which led the domination of the lower orders, does not converge anymore and instead a time based model is used. When a noble gas atom is hit by an intense laser pulse, which has an electric field strength comparable to the Coulomb field of the atom, the outermost electron may be ionized from the atom. Once freed, the electron can be accelerated by the electric field of the light, first moving away from the ion, then back toward it as the field changes direction. The electron may then recombine with the ion, releasing its energy in the form of a photon. The light is emitted at every peak of the laser light field which is intense enough, producing a series of attosecond light flashes. The photon energies generated by this process can extend past the 800th harmonic order up to 1300 eV. This is called high-order harmonic generation. The laser must be linearly polarized, so that the electron returns to the vicinity of the parent ion. High-order harmonic generation has been observed in noble gas jets, cells, and gas-filled capillary waveguides.

### Optical phase conjugation

File:PhaseConjugationPrinciple.en.png
Comparison of a phase conjugate mirror with a conventional mirror. With the phase conjugate mirror the image is not deformed when passing through an aberrating element twice.

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).

One can interpret this nonlinear optical interaction as being analogous to a real-time holographic process. In this case, the interacting beams simultaneously interact in a nonlinear optical material to form a dynamic hologram (two of the three input beams), or real-time diffraction pattern, in the material. The third incident beam diffracts off this dynamic hologram, and, in the process, reads out the phase-conjugate wave. In effect, all three incident beams interact (essentially) simultaneously to form several real-time holograms, resulting in a set of diffracted output waves that phase up as the "time-reversed" beam. In the language of nonlinear optics, the interacting beams result in a nonlinear polarization within the material, which coherently radiates to form the phase-conjugate wave.

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:

${\displaystyle \Xi _{j}(\mathbf {x} ,t)={\frac {1}{2}}E_{j}(\mathbf {x} )e^{i(\omega _{j}t-\mathbf {k} \cdot \mathbf {x} )}+{\mbox{c.c.}}}$

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:

${\displaystyle P_{\mbox{NL}}=\epsilon _{0}\chi ^{(3)}(\Xi _{1}+\Xi _{2}+\Xi _{3})^{3}}$

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:

${\displaystyle P_{\omega }={\frac {1}{2}}\chi ^{(3)}\epsilon _{0}E_{1}E_{2}E_{3}^{*}e^{i(\omega t-\mathbf {k} \cdot \mathbf {x} )}+{\mbox{c.c.}}}$.

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

${\displaystyle E_{4}={\frac {i\omega l}{2nc}}\chi ^{(3)}E_{1}E_{2}E_{3}^{*}}$

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

${\displaystyle E_{4}(\mathbf {x} )\propto E_{3}^{*}(\mathbf {x} )}$;

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.

## Common SHG materials

File:GaSeBulk.jpg
Dark-Red Gallium Selenide in its bulk form.