Nonlinear optics

From Wikipedia, the free encyclopedia

 

Structure of KTP crystal, viewed down b axis, used in second harmonic generation.

Nonlinear optics (NLO) is the branch of optics that describes the behavior of light in nonlinear media, that is, media in which the dielectric polarization P responds nonlinearly to the electric field E of the light. The nonlinearity is typically observed only at very high light intensities (values of atomic electric fields, typically 10⁸ V/m) such as those provided by lasers. Above the Schwinger limit, the vacuum itself is expected to become nonlinear. In nonlinear optics, the superposition principle no longer holds.

Nonlinear optics remained unexplored until the discovery in 1961 of second-harmonic generation by Peter Franken et al. at University of Michigan, shortly after the construction of the first laser by Theodore Harold Maiman. However, some nonlinear effects were discovered before the development of the laser. The theoretical basis for many nonlinear processes were first described in Bloembergen's monograph "Nonlinear Optics".

Nonlinear optical processes

Nonlinear optics explains nonlinear response of properties such as frequency, polarization, phase or path of incident light. These nonlinear interactions give 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), two photons are destroyed, creating a single photon at two times the frequency.
• Third-harmonic generation (THG), generation of light with a tripled frequency (one-third the wavelength), three photons are destroyed, creating a single photon at three times the frequency.
• High-harmonic generation (HHG), generation of light with frequencies much greater than the original (typically 100 to 1000 times greater).
• 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).
• Difference-frequency generation (DFG), generation of light with a frequency that is the difference between two other frequencies.
• Optical parametric amplification (OPA), 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).
• Optical parametric oscillation (OPO), generation of a signal and idler wave using a parametric amplifier in a resonator (with no signal input).
• Optical parametric generation (OPG), 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 (OR), generation of quasi-static electric fields.
• Nonlinear light-matter interaction with free electrons and plasmas.

Other nonlinear processes

• Optical Kerr effect, intensity-dependent refractive index (a ${\displaystyle \chi ^{(3)}}$ effect).
  • Self-focusing, an effect due to the optical Kerr effect (and possibly higher-order nonlinearities) caused by the spatial variation in the intensity creating a spatial variation in the refractive index.
  • Kerr-lens modelocking (KLM), the use of self-focusing as a mechanism to mode-lock laser.
  • Self-phase modulation (SPM), an effect due to the optical Kerr effect (and possibly higher-order nonlinearities) caused by the temporal variation in the intensity creating a temporal variation in the refractive index.
  • Optical solitons, an equilibrium solution for either an optical pulse (temporal soliton) or spatial mode (spatial soliton) that does not change during propagation due to a balance between dispersion and the Kerr effect (e.g. self-phase modulation for temporal and self-focusing for spatial solitons).
• Cross-phase modulation (XPM), where one wavelength of light can affect the phase of another wavelength of light through the optical Kerr effect.
• Four-wave mixing (FWM), can also arise from other nonlinearities.
• Cross-polarized wave generation (XPW), a ${\displaystyle \chi ^{(3)}}$ effect in which a wave with polarization vector perpendicular to the input one is generated.
• Modulational instability.
• Raman amplification
• Optical phase conjugation.
• Stimulated Brillouin scattering, interaction of photons with acoustic phonons
• Multi-photon absorption, simultaneous absorption of two or more photons, transferring the energy to a single electron.
• Multiple photoionisation, near-simultaneous removal of many bound electrons by one photon.
• Chaos in optical systems.

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.
• Raman scattering, interaction of photons with optical phonons.

Parametric processes

Nonlinear effects fall into two qualitatively different categories, parametric and non-parametric effects. A parametric non-linearity is an interaction in which the quantum state of the nonlinear material is not changed by the interaction with the optical field. As a consequence of this, the process is "instantaneous". Energy and momentum are conserved in the optical field, making phase matching important and polarization-dependent.

Theory

Parametric and "instantaneous" (i.e. material must be lossless and dispersionless through the Kramers–Kronig relations) nonlinear optical phenomena, in which the optical fields are not too large, can be described by a Taylor series expansion of the dielectric polarization density (dipole moment per unit volume) P(t) at time t in terms of the electrical field E(t):

${\displaystyle \mathbf {P} (t)=\varepsilon _{0}(\chi ^{(1)}\mathbf {E} (t)+\chi ^{(2)}\mathbf {E} ^{2}(t)+\chi ^{(3)}\mathbf {E} ^{3}(t)+\ldots ),}$

where the coefficients χ⁽ⁿ⁾ are the n-th-order susceptibilities of the medium, and the presence of such a term is generally referred to as an n-th-order nonlinearity. Note that the polarization density P(t) and electrical field E(t) are considered as scalar for simplicity. In general, χ⁽ⁿ⁾ is an (n + 1)-th-rank tensor representing both the polarization-dependent nature of the parametric interaction and the symmetries (or lack of) of the nonlinear material.

Wave equation in a nonlinear material

Central to the study of electromagnetic waves is the wave equation. Starting with Maxwell's equations in an isotropic space, containing no free charge, it can be shown that

${\displaystyle \nabla \times \nabla \times \mathbf {E} +{\frac {n^{2}}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\mathbf {E} =-{\frac {1}{\varepsilon _{0}c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\mathbf {P} ^{\text{NL}},}$

where P^NL is the nonlinear part of the polarization density, and n is the refractive index, which comes from the linear term in P.
Note that one can normally use the vector identity

${\displaystyle \nabla \times \left(\nabla \times \mathbf {V} \right)=\nabla \left(\nabla \cdot \mathbf {V} \right)-\nabla ^{2}\mathbf {V} }$

and Gauss's law (assuming no free charges, ${\displaystyle \rho _{\text{free}}=0}$),

${\displaystyle \nabla \cdot \mathbf {D} =0,}$

to obtain the more familiar wave equation

${\displaystyle \nabla ^{2}\mathbf {E} -{\frac {n^{2}}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\mathbf {E} =0.}$

For nonlinear medium, Gauss's law does not imply that the identity

${\displaystyle \nabla \cdot \mathbf {E} =0}$

is true in general, even for an isotropic medium. However, even when this term is not identically 0, it is often negligibly small and thus in practice is usually ignored, giving us the standard nonlinear wave equation:

${\displaystyle \nabla ^{2}\mathbf {E} -{\frac {n^{2}}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\mathbf {E} ={\frac {1}{\varepsilon _{0}c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\mathbf {P} ^{\text{NL}}.}$

Nonlinearities as a wave-mixing process

The nonlinear wave equation is an inhomogeneous differential equation. The general solution comes from the study of ordinary differential equations and can be obtained by the use of a Green's function. Physically one gets the normal electromagnetic wave solutions to the homogeneous part of the wave equation:

${\displaystyle \nabla ^{2}\mathbf {E} -{\frac {n^{2}}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\mathbf {E} =0,}$

and the inhomogeneous term

${\displaystyle {\frac {1}{\varepsilon _{0}c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\mathbf {P} ^{\text{NL}}}$

acts as a driver/source of the electromagnetic waves. One of the consequences of this is a nonlinear interaction that results in energy being mixed or coupled between different frequencies, which is often called a "wave mixing".

In general, an n-th order nonlinearity will lead to (n + 1)-wave mixing. As an example, if we consider only a second-order nonlinearity (three-wave mixing), then the polarization P takes the form

${\displaystyle \mathbf {P} ^{\text{NL}}=\varepsilon _{0}\chi ^{(2)}\mathbf {E} ^{2}(t).}$

If we assume that E(t) is made up of two components at frequencies ω₁ and ω₂, we can write E(t) as

${\displaystyle \mathbf {E} (t)=E_{1}\cos(\omega _{1}t)+E_{2}\cos(\omega _{2}t),}$

and using Euler's formula to convert to exponentials,

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

where "c.c." stands for complex conjugate. Plugging this into the expression for P gives

${\displaystyle {\begin{aligned}\mathbf {P} ^{\text{NL}}=\varepsilon _{0}\chi ^{(2)}\mathbf {E} ^{2}(t)&={\frac {\varepsilon _{0}}{4}}\chi ^{(2)}{\Big [}{E_{1}}^{2}e^{-i2\omega _{1}t}+{E_{2}}^{2}e^{-i2\omega _{2}t}\\&\qquad +2E_{1}E_{2}e^{-i(\omega _{1}+\omega _{2})t}\\&\qquad +2E_{1}{E_{2}}^{*}e^{-i(\omega _{1}-\omega _{2})t}\\&\qquad +\left(|E_{1}|^{2}+|E_{2}|^{2}\right)e^{0}+{\text{c.c.}}{\Big ]},\end{aligned}}}$

which has frequency components at 2ω₁, 2ω₂, ω₁ + ω₂, ω₁ − ω₂, and 0. These three-wave mixing processes correspond to the nonlinear effects known as second-harmonic generation, sum-frequency generation, difference-frequency generation and optical rectification respectively.

Note: Parametric generation and amplification is a variation of difference-frequency generation, where the lower frequency of 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

Most transparent materials, like the BK7 glass shown here, have normal dispersion: the index of refraction decreases monotonically as a function of wavelength (or increases as a function of frequency). This makes phase matching impossible in most frequency-mixing processes. For example, in SHG, there is no simultaneous solution to ${\displaystyle \omega '=2\omega }$ and ${\displaystyle \mathbf {k} '=2\mathbf {k} }$ in these materials. Birefringent materials avoid this problem by having two indices of refraction at once.

 

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

${\displaystyle E_{j}(\mathbf {x} ,t)=E_{j,0}e^{i(\mathbf {k} _{j}\cdot \mathbf {x} -\omega _{j}t)}+{\text{c.c.}}}$

at position ${\displaystyle \mathbf {x} }$, with the wave vector ${\displaystyle \|\mathbf {k} _{j}\|=\mathbf {n} (\omega _{j})\omega _{j}/c}$, where ${\displaystyle c}$ is the velocity of light in vacuum, and ${\displaystyle \mathbf {n} (\omega _{j})}$ is the index of refraction of the medium at angular frequency ${\displaystyle \omega _{j}}$. Thus, the second-order polarization at angular frequency ${\displaystyle \omega _{3}=\omega _{1}+\omega _{2}}$ is

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

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

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

The above equation is known as the phase-matching condition. Typically, three-wave mixing is done in a birefringent crystalline material, where the refractive index depends on the polarization and direction of the light that passes through. 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 or two of which have a different refractive index than the other one(s). Uniaxial crystals, for example, have a single preferred axis, called the extraordinary (e) axis, while the other two are ordinary axes (o). There are several schemes of choosing the polarizations for this crystal type. 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 (or type 0, or "zero")
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 uniaxial, which means that the e axis has a smaller refractive index than the o axes. In those crystals, type-I and -II phase matching 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 to the propagation vector. This would lead to beam walk-off, which limits the nonlinear optical conversion efficiency. Two other methods of phase matching avoid beam walk-off by forcing all frequencies to propagate at a 90° with respect to the optical axis of the crystal. These methods are called temperature tuning and quasi-phase-matching.

Temperature tuning is used when 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 temperature is controlled to achieve phase-matching conditions.

The other method is 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; in particular resonantly enhanced sum or difference frequency mixing in gasses is frequently used for extreme or "vacuum" Ultra Violet light generation. In common scenarios, such as mixing in dilute gases, the non-linearity is weak and so the light beams are focused which, unlike the plane wave approximation used above, introduces a pi phase shift on each light beam, complicating the phase matching requirements. Conveniently, difference frequency mixing with ${\displaystyle \chi ^{(3)}}$ cancels this focal phase shift and often has a nearly self-canceling overall phase matching condition, which relatively simplifies broad wavelength tuning compared to sum frequency generation. In ${\displaystyle \chi ^{(3)}}$ all four frequencies are mixing simultaneously, as opposed to sequential mixing via two ${\displaystyle \chi ^{(2)}}$ processes.

The Kerr effect can be described as a ${\displaystyle \chi ^{(3)}}$ as well. At high peak powers the Kerr effect can cause filamentation of light in air, in which the light travels without dispersion or divergence in a self-generated waveguide. At even 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 a few KeV. 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.

Example uses

Frequency doubling

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 nonlinear medium in a laser beam. While there are many types of nonlinear media, the most common media are crystals. 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, being transparent for both the impinging laser light and the frequency-doubled wavelength, and having high damage thresholds, which makes them resistant against the high-intensity laser light.

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 is significantly different from 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 at 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.

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.

 

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:

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

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:

${\displaystyle P_{\text{NL}}=\varepsilon _{0}\chi ^{(3)}(\Xi _{1}+\Xi _{2}+\Xi _{3})^{3},}$

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:

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

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 that 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 is higher in frequency such that ω₃ = ω + Δω, then the conjugate wave is of frequency ω₄ = ω − Δω. This is known as frequency flipping.

Angular and linear momenta in optical phase conjugation

Optical phase conjugation in the near field performs the reversal of classical rays, or retroreflection.

Classical picture

In classical Maxwell electrodynamics a phase-conjugating mirror performs reversal of the Poynting vector:

${\displaystyle \mathbf {S} _{\text{out}}(\mathbf {r} ,t)=-\mathbf {S} _{\text{in}}(\mathbf {r} ,t),}$

("in" means incident field, "out" means reflected field) where

${\displaystyle \mathbf {S} (\mathbf {r} ,t)=\epsilon _{0}c^{2}\mathbf {E} (\mathbf {r} ,t)\times \mathbf {B} (\mathbf {r} ,t),}$

which is a linear momentum density of electromagnetic field. In the same way a phase-conjugated wave has an opposite angular momentum density vector ${\displaystyle \mathbf {L} (\mathbf {r} ,t)=\mathbf {r} \times \mathbf {S} (\mathbf {r} ,t)}$ with respect to incident field:

${\displaystyle \mathbf {L} _{\text{out}}(\mathbf {r} ,t)=-\mathbf {L} _{\text{in}}(\mathbf {r} ,t).}$

The above identities are valid locally, i.e. in each space point ${\displaystyle \mathbf {r} }$ in a given moment ${\displaystyle t}$ for an ideal phase-conjugating mirror.

Quantum picture

In quantum electrodynamics the photon with energy ${\displaystyle \hbar \omega }$ also possesses linear momentum ${\displaystyle \mathbf {P} =\hbar \mathbf {k} }$ and angular momentum, whose projection on propagation axis is ${\displaystyle L_{z}=\pm \hbar \ell }$, where ${\displaystyle \ell }$ is topological charge of photon, or winding number, ${\displaystyle \mathbf {z} }$ is propagation axis. The angular momentum projection on propagation axis has discrete values ${\displaystyle \pm \hbar \ell }$.

In quantum electrodynamics the interpretation of phase conjugation is much simpler compared to classical electrodynamics. The photon reflected from phase conjugating-mirror (out) has opposite directions of linear and angular momenta with respect to incident photon (in):

${\displaystyle \mathbf {P} _{\text{out}}=-\hbar \mathbf {k} =-\mathbf {P} _{\text{in}}=\hbar \mathbf {k} ,}$

${\displaystyle {L_{z}}_{\text{out}}=-\hbar \ell =-{L_{z}}_{\text{in}}=\hbar \ell .}$

Nonlinear optical pattern formation

Optical fields transmitted through nonlinear Kerr media can also display pattern formation owing to the nonlinear medium amplifying spatial and temporal noise. The effect is referred to as optical modulation instability. This has been observed both in photo-refractive, photonic lattices, as well as photo-reactive systems. In the latter case, optical nonlinearity is afforded by reaction-induced increases in refractive index.

Molecular nonlinear optics

The early studies of nonlinear optics and materials focused on the inorganic solids. With the development of nonlinear optics, molecular optical properties were investigated, forming molecular nonlinear optics. The traditional approaches used in the past to enhance nonlinearities include extending chromophore π-systems, adjusting bond length alternation, inducing intramolecular charge transfer, extending conjugation in 2D, and engineering multipolar charge distributions. Recently, many novel directions were proposed for enhanced nonlinearity and light manipulation, including twisted chromophores, combining rich density of states with bond alternation, microscopic cascading of second-order nonlinearity, etc. Due to the distinguished advantages, molecular nonlinear optics have been widely used in the biophotonics field, including bioimaging, phototherapy,biosensing, etc.

Atomic, molecular, and optical physics

From Wikipedia, the free encyclopedia

 

Atomic, molecular, and optical physics (AMO) is the study of matter-matter and light-matter interactions; at the scale of one or a few atoms and energy scales around several electron volts. The three areas are closely interrelated. AMO theory includes classical, semi-classical and quantum treatments. Typically, the theory and applications of emission, absorption, scattering of electromagnetic radiation (light) from excited atoms and molecules, analysis of spectroscopy, generation of lasers and masers, and the optical properties of matter in general, fall into these categories.

Atomic and molecular physics

Atomic physics is the subfield of AMO that studies atoms as an isolated system of electrons and an atomic nucleus, while molecular physics is the study of the physical properties of molecules. The term atomic physics is often associated with nuclear power and nuclear bombs, due to the synonymous use of atomic and nuclear in standard English. However, physicists distinguish between atomic physics — which deals with the atom as a system consisting of a nucleus and electrons — and nuclear physics, which considers atomic nuclei alone. The important experimental techniques are the various types of spectroscopy. Molecular physics, while closely related to atomic physics, also overlaps greatly with theoretical chemistry, physical chemistry and chemical physics.

Both subfields are primarily concerned with electronic structure and the dynamical processes by which these arrangements change. Generally this work involves using quantum mechanics. For molecular physics, this approach is known as quantum chemistry. One important aspect of molecular physics is that the essential atomic orbital theory in the field of atomic physics expands to the molecular orbital theory. Molecular physics is concerned with atomic processes in molecules, but it is additionally concerned with effects due to the molecular structure. Additionally to the electronic excitation states which are known from atoms, molecules are able to rotate and to vibrate. These rotations and vibrations are quantized; there are discrete energy levels. The smallest energy differences exist between different rotational states, therefore pure rotational spectra are in the far infrared region (about 30 - 150 µm wavelength) of the electromagnetic spectrum. Vibrational spectra are in the near infrared (about 1 - 5 µm) and spectra resulting from electronic transitions are mostly in the visible and ultraviolet regions. From measuring rotational and vibrational spectra properties of molecules like the distance between the nuclei can be calculated.

As with many scientific fields, strict delineation can be highly contrived and atomic physics is often considered in the wider context of atomic, molecular, and optical physics. Physics research groups are usually so classified.

Optical physics

Optical physics is the study of the generation of electromagnetic radiation, the properties of that radiation, and the interaction of that radiation with matter, especially its manipulation and control. It differs from general optics and optical engineering in that it is focused on the discovery and application of new phenomena. There is no strong distinction, however, between optical physics, applied optics, and optical engineering, since the devices of optical engineering and the applications of applied optics are necessary for basic research in optical physics, and that research leads to the development of new devices and applications. Often the same people are involved in both the basic research and the applied technology development, for example the experimental demonstration of electromagnetically induced transparency by S. E. Harris and of slow light by Harris and Lene Vestergaard Hau.

Researchers in optical physics use and develop light sources that span the electromagnetic spectrum from microwaves to X-rays. The field includes the generation and detection of light, linear and nonlinear optical processes, and spectroscopy. Lasers and laser spectroscopy have transformed optical science. Major study in optical physics is also devoted to quantum optics and coherence, and to femtosecond optics. In optical physics, support is also provided in areas such as the nonlinear response of isolated atoms to intense, ultra-short electromagnetic fields, the atom-cavity interaction at high fields, and quantum properties of the electromagnetic field.

Other important areas of research include the development of novel optical techniques for nano-optical measurements, diffractive optics, low-coherence interferometry, optical coherence tomography, and near-field microscopy. Research in optical physics places an emphasis on ultrafast optical science and technology. The applications of optical physics create advancements in communications, medicine, manufacturing, and even entertainment.

History

The Bohr model of the Hydrogen atom

One of the earliest steps towards atomic physics was the recognition that matter was composed of atoms, in modern terms the basic unit of a chemical element. This theory was developed by John Dalton in the 18th century. At this stage, it wasn't clear what atoms were - although they could be described and classified by their observable properties in bulk; summarized by the developing periodic table, by John Newlands and Dmitri Mendeleyev around the mid to late 19th century.

Later, the connection between atomic physics and optical physics became apparent, by the discovery of spectral lines and attempts to describe the phenomenon - notably by Joseph von Fraunhofer, Fresnel, and others in the 19th century.

From that time to the 1920s, physicists were seeking to explain atomic spectra and blackbody radiation. One attempt to explain hydrogen spectral lines was the Bohr atom model.

Experiments including electromagnetic radiation and matter - such as the photoelectric effect, Compton effect, and spectra of sunlight the due to the unknown element of Helium, the limitation of the Bohr model to Hydrogen, and numerous other reasons, lead to an entirely new mathematical model of matter and light: quantum mechanics.

Classical oscillator model of matter

Early models to explain the origin of the index of refraction treated an electron in an atomic system classically according to the model of Paul Drude and Hendrik Lorentz. The theory was developed to attempt to provide an origin for the wavelength-dependent refractive index n of a material. In this model, incident electromagnetic waves forced an electron bound to an atom to oscillate. The amplitude of the oscillation would then have a relationship to the frequency of the incident electromagnetic wave and the resonant frequencies of the oscillator. The superposition of these emitted waves from many oscillators would then lead to a wave which moved more slowly.

Early quantum model of matter and light

Max Planck derived a formula to describe the electromagnetic field inside a box when in thermal equilibrium in 1900. His model consisted of a superposition of standing waves. In one dimension, the box has length L, and only sinusodial waves of wavenumber

${\displaystyle k={\frac {n\pi }{L}}}$

can occur in the box, where n is a positive integer (mathematically denoted by ${\displaystyle \scriptstyle n\in \mathbb {N} _{1}}$). The equation describing these standing waves is given by:

${\displaystyle E=E_{0}\sin \left({\frac {n\pi }{L}}x\right)\,\!}$.

where E₀ is the magnitude of the electric field amplitude, and E is the magnitude of the electric field at position x. From this basic, Planck's law was derived.

In 1911, Ernest Rutherford concluded, based on alpha particle scattering, that an atom has a central pointlike proton. He also thought that an electron would be still attracted to the proton by Coulomb's law, which he had verified still held at small scales. As a result, he believed that electrons revolved around the proton. Niels Bohr, in 1913, combined the Rutherford model of the atom with the quantisation ideas of Planck. Only specific and well-defined orbits of the electron could exist, which also do not radiate light. In jumping orbit the electron would emit or absorb light corresponding to the difference in energy of the orbits. His prediction of the energy levels was then consistent with observation.

These results, based on a discrete set of specific standing waves, were inconsistent with the continuous classical oscillator model.

Work by Albert Einstein in 1905 on the photoelectric effect led to the association of a light wave of frequency ${\displaystyle \nu }$ with a photon of energy ${\displaystyle h\nu }$. In 1917 Einstein created an extension to Bohrs model by the introduction of the three processes of stimulated emission, spontaneous emission and absorption (electromagnetic radiation).

Modern treatments

The largest steps towards the modern treatment was the formulation of quantum mechanics with the matrix mechanics approach by Werner Heisenberg and the discovery of the Schrödinger equation by Erwin Schrödinger.

There are a variety of semi-classical treatments within AMO. Which aspects of the problem are treated quantum mechanically and which are treated classically is dependent on the specific problem at hand. The semi-classical approach is ubiquitous in computational work within AMO, largely due to the large decrease in computational cost and complexity associated with it.

For matter under the action of a laser, a fully quantum mechanical treatment of the atomic or molecular system is combined with the system being under the action of a classical electromagnetic field. Since the field is treated classically it can not deal with spontaneous emission. This semi-classical treatment is valid for most systems, particular those under the action of high intensity laser fields. The distinction between optical physics and quantum optics is the use of semi-classical and fully quantum treatments respectively.

Within collision dynamics and using the semi-classical treatment, the internal degrees of freedom may be treated quantum mechanically, whilst the relative motion of the quantum systems under consideration are treated classically. When considering medium to high speed collisions, the nuclei can be treated classically while the electron is treated quantum mechanically. In low speed collisions the approximation fails.

Classical Monte-Carlo methods for the dynamics of electrons can be described as semi-classical in that the initial conditions are calculated using a fully quantum treatment, but all further treatment is classical.

Isolated atoms and molecules

Atomic, Molecular and Optical physics frequently considers atoms and molecules in isolation. Atomic models will consist of a single nucleus that may be surrounded by one or more bound electrons, whilst molecular models are typically concerned with molecular hydrogen and its molecular hydrogen ion. It is concerned with processes such as ionization, above threshold ionization and excitation by photons or collisions with atomic particles.

While modelling atoms in isolation may not seem realistic, if one considers molecules in a gas or plasma then the time-scales for molecule-molecule interactions are huge in comparison to the atomic and molecular processes that we are concerned with. This means that the individual molecules can be treated as if each were in isolation for the vast majority of the time. By this consideration atomic and molecular physics provides the underlying theory in plasma physics and atmospheric physics even though both deal with huge numbers of molecules.

Electronic configuration

Electrons form notional shells around the nucleus. These are naturally in a ground state but can be excited by the absorption of energy from light (photons), magnetic fields, or interaction with a colliding particle (typically other electrons).

Electrons that populate a shell are said to be in a bound state. The energy necessary to remove an electron from its shell (taking it to infinity) is called the binding energy. Any quantity of energy absorbed by the electron in excess of this amount is converted to kinetic energy according to the conservation of energy. The atom is said to have undergone the process of ionization.

In the event that the electron absorbs a quantity of energy less than the binding energy, it may transition to an excited state or to a virtual state. After a statistically sufficient quantity of time, an electron in an excited state will undergo a transition to a lower state via spontaneous emission. The change in energy between the two energy levels must be accounted for (conservation of energy). In a neutral atom, the system will emit a photon of the difference in energy. However, if the lower state is in an inner shell, a phenomenon known as the Auger effect may take place where the energy is transferred to another bound electrons causing it to go into the continuum. This allows one to multiply ionize an atom with a single photon.

There are strict selection rules as to the electronic configurations that can be reached by excitation by light—however there are no such rules for excitation by collision processes.