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. Table of Opticks, 1728 Cyclopaedia Optics (appearance or look in ancient Greek) is a branch of physics that describes the behavior and properties of light and the interaction of light with matter. ... Prism splitting light Light is electromagnetic radiation with a wavelength that is visible to the eye (visible light) or, in a technical or scientific context, electromagnetic radiation of wavelengths that are studied in the field of optics. ... In electrostatics, the polarization is the vector field that results from permanent or induced electric dipole moments in a dielectric material. ... In physics, an electric field or E-field is an effect produced by an electric charge (or a time-varying magnetic field) that exerts a force on charged objects in the field. ... Lasers range in size from microscopic diode lasers (top) with numerous applications, to football field sized neodymium glass lasers (bottom) used for inertial confinement fusion, nuclear weapons research and other high energy density physics experiments. ...

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);
• Self-phase modulation (SPM), a Χ³ effect.
• 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.

Second Harmonic Generation (SHG) is a subcategory of nonlinear optics in physics. ... Self-phase modulation (SPM) is a nonlinear optical effect of light-matter interaction. ... Sum Frequency generation is the process in which light beams at two frequencies interact in some special conditions to produce light at a new frequency equal to the sum of the interacting frequencies. ... An optical parametric amplifier, abbreviated OPA, is a laser light source that emits the light of variable wavelengths by an optical parametric amplification process. ... An optical parametric oscillator (OPO) converts a input laser wave (called pump) into two output waves of lower frequency () by means of nonlinear optical interaction. ... Spontaneous parametric down-conversion is an important process in quantum optics. ... State Peace and Development Council is the official name of the government of Myanmar (formerly known as Burma). ...

Other nonlinear processes

• Optical Kerr effect, intensity dependent refractive index;
  • Self focusing;
    • Kerr-lens modelocking (KLM).
  • Self-phase modulation (SPM);
    • Optical solitons.
  • 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.
• Chaos in Optical Systems

The Kerr effect or the quadratic electro-optic effect is a change in the refractive index of a material in response to the intensity of an external electric field. ... Kerr-lens modelocking is a method of modelocking lasers via a nonlinear optical process known as the optical Kerr effect. ... Self-phase modulation (SPM) is a nonlinear optical effect of light-matter interaction. ... In mathematics and physics, a soliton is a self-reinforcing solitary wave caused by nonlinear effects in the medium. ... Four-Wave Mixing is an intermodulation distortion in optical systems, similar to the third order intercept in electrical systems. ... Raman scattering or the Raman effect is the inelastic scattering of a photon which creates or annihilates an optical phonon. ... In physics, a phonon is a quantized mode of vibration occurring in a rigid crystal lattice, such as the atomic lattice of a solid. ... Brillouin scattering occurs when light in a medium (such as water or a crystal) interacts with density variations and changes its path. ... Photoionisation is a physical process in which a photon strikes an atom, ion or molecule, resulting in the ejection of an electron. ... To meet Wikipedias quality standards, this article or section may require cleanup. ...

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.

The Pockels effect, or Pockels electro-optic effect, is the production of birefringence in an optical medium induced by a constant or varying electric field. ... Electro-optic modulator is an optical device in which a signal-controlled element is used to modulate a beam of light. ... An acousto-optic modulator (AOM) consists of a piezo-electric transducer which creates sound waves in a material like glass or quartz. ...

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. Nd:YAG is an acronym for Neodymium-doped Yttrium Aluminum Garnet, a compound that is used as the lasing medium for certain solid-state lasers. ... Part of a Ti-sapphire oscillator. ...

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. Beta barium borate (β-barium borate, BBO or β-BaB2O4) is a crystal frequently used for frequency mixing and other nonlinear optics applications. ... A huge KDP crystal grown at LLNL to be cut into slices and used on the National Ignition Facility. ... Potassium titanyl phosphate (KTiOPO4) or KTP is a nonlinear optical material which is commonly used for frequency doubling diode pumped solid-state lasers such as Nd:YAG and other neodymium-doped lasers. ... Lithium niobate (LiNbO3) is a compound of niobium and lithium. ... A calcite crystal laid upon a paper with some letters showing the double refraction Birefringence, or double refraction, is the decomposition of a ray of light into two rays (the ordinary ray and the extraordinary ray) when it passes through certain types of material, such as calcite crystals, depending on... Polymer is a generic term used to describe a very long molecule consisting of structural units and repeating units connected by covalent chemical bonds. ...

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) P(t) at time t in a medium can be written as a power series in the electrical field: In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ...

.

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 no inversion symmetry. If we write It has been suggested that this article or section be merged into permittivity. ... Inversion has different meanings in different fields of knowledge: Something that is inverted or the process by which an inverse is obtained. ... Square with symmetry group D4 Symmetry is a characteristic of geometrical shapes, equations, and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ...

,

where c.c. denotes the complex conjugate (E1 and E2 being the incident beams of interest), the second-order term in will read In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. ...

,

where the summation is over

.

The six combinations (n_x,m_x) 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. In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ...

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. For a non-technical introduction to the topic, please see Introduction to Quantum mechanics. ...

Phase matching

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

,

at position , with the wave vector , where c is the velocity of light and n(ω_j) the index of refraction of the medium at angular frequency ω_j. Thus, the second-order polarization angular frequency ω₃ is

.

At each position , the oscillating second-order polarization radiates at angular frequency ω₃ and a corresponding wave vector . Constructive interference, and therefore a high intensity ω₃ field, will occur only if

.

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. The refractive index (or index of refraction) of a material is the factor by which the phase velocity of electromagnetic radiation is slowed in that material, relative to its velocity in a vacuum. ...

One undesirable effect of angle tuning is that the optical frequencies involved are not collinear 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. Quasi Phase Matching is a technique in Nonlinear Optics which allows a positive net flow of energy from pump frequency to signal and idler frequencies. ...

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 a 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. Self-phase modulation (SPM) is a nonlinear optical effect of light-matter interaction. ... An optical parametric amplifier, abbreviated OPA, is a laser light source that emits the light of variable wavelengths by an optical parametric amplification process. ...

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. Image File history File links Download high resolution version (289x679, 9 KB) Summary Cooled high pressure noble gas is accelerated by a supersonic nozzle into a tube. ... Image File history File links Download high resolution version (289x679, 9 KB) Summary Cooled high pressure noble gas is accelerated by a supersonic nozzle into a tube. ...

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 two-cycle laser pulse, one electron is first extracted from the atom by the first half of the cycle and then smashed back by the second half. A part of the wave function hits the core and produces an attosecond light flash expending up to 100 eV. This is called high harmonics generation. The laser must be linearly polarized, so that the electron does not cycle around the ion and it must be of low enough intensity to not accelerate the electron to relativistic velocities, as the electron would then be pushed behind the ion. The laser beam is only loosely focused to get a 5 cm waist. As the degree of the Taylor series rises, it approaches the correct function. ... An attosecond is an SI unit of time equal to 10-18 of a second. ... eV may stand for: electronvolt eingetragener Verein, i. ...

The gas should have a pressure of about 10 mbar 5 cm after the waist and 1E-8 mbar everywhere else to avoid absorption. Cooling, pulsing, and using a capillary helps to achieve this pressure gradient. Dispersion limits the shortness of the pulses and the efficiency. Cooled high pressure noble gas is accelerated by a supersonic nozzle into a tube. A laser beam crosses this jet and high harmonics are generated. A bump makes the gas jet jump over the hole where the laser beam exits. This set-up even pumps gas out of the exit hole. Dispersion can mean any of several things: A phenomenon that causes the separation of a wave into components of varying frequency. ...

Optical phase conjugation

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). Download high resolution version (1280x1234, 839 KB) Wikipedia does not have an article with this exact name. ... Download high resolution version (1280x1234, 839 KB) Wikipedia does not have an article with this exact name. ... Retroreflectors are clearly visible in a pair of bicycle shoes. ...

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. Holography (from the Greek, Όλος-holos whole + γραφή-graphe writing) is the science of producing holograms, an advanced form of photography that allows an image to be recorded in three dimensions. ...

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:

where E_j 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:

P_NL = ε₀χ⁽³⁾(Ξ₁ + 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:

.

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

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

;

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.

References

1. Scientific American, December 1985, "Phase Conjugation," by Vladimir Shkunov and Boris Zel'dovich.
2. Scientific American, January 1986, "Applications of Optical Phase Conjugation," by David M. Pepper.
3. Scientific American, October 1990, "The Photorefractive Effect," by David M. Pepper, Jack Feinberg, and Nicolai V. Kukhtarev.

Common SHG materials

• 806 nm light : LiIO3.
• 860 nm light : KNbO3.
• 980 nm light : KNbO3.
• 1064 nm light : KDP, LBO and BBO.
• 1319 nm light : KNbO3, BBO, KDP, LiIO3, LiNbO3, and KTP.

See also: Category:Nonlinear optical materials

KDP Potassium DiHydrogen Phosphate, KH2PO4. ... Lithium triborate (LiB3O5, or LBO) is a non-linear optics crystal. ... Beta barium borate (β-barium borate, BBO or β-BaB2O4) is a crystal frequently used for frequency mixing and other nonlinear optics applications. ... Beta barium borate (β-barium borate, BBO or β-BaB2O4) is a crystal frequently used for frequency mixing and other nonlinear optics applications. ... KDP Potassium DiHydrogen Phosphate, KH2PO4. ... Hexagonal crystal structure of LiNbO3 Lithium niobate (LiNbO3) is a compound of niobium and lithium. ... Potassium titanyl phosphate (KTiOPO4) or KTP is a nonlinear optical material which is commonly used for frequency doubling diode pumped solid-state lasers such as Nd:YAG and other neodymium-doped lasers. ...

See also

• Born-Infeld action
• Category:Nonlinear optical materials

In physics, the Born-Infeld theory is a nonlinear generalization of electromagnetism (see nonlinear electrodynamics). ...

External links

• Encyclopedia of laser physics and technology, with content on nonlinear optics, by Rüdiger Paschotta