Nonlinear optics

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

Nonlinear optics (NLO) is the branch of optics that describes the behaviour of light in nonlinear media, that is, media in which the polarization density P responds non-linearly to the electric field E of the light. The non-linearity 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.

History

The first nonlinear optical effect to be predicted was two-photon absorption, by Maria Goeppert Mayer for her PhD in 1931, but it remained an unexplored theoretical curiosity until 1961 and the almost simultaneous observation of two-photon absorption at Bell Labs  and the discovery of second-harmonic generation by Peter Franken et al. at University of Michigan, both shortly after the construction of the first laser by Theodore 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.
• Half-harmonic generation, the special case of OPO or OPG when the signal and idler degenerate in one single frequency,
• 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).
  • Self-diffraction, splitting of beams in a multi-wave mixing process with potential energy transfer.
• 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 (electric dipole moment per unit volume) P(t) at time t in terms of the electric 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 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) (see crystal optics). 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).

A device producing the phase-conjugation effect is known as a phase-conjugate mirror (PCM).

Principles

Vortex photon (blue) with linear momentum ${\displaystyle \mathbf {P} =\hbar \mathbf {k} }$ and angular momentum ${\displaystyle L=\pm \hbar \ell }$ are reflected from perfect phase-conjugating mirror. Normal to mirror is ${\displaystyle {\vec {n}}}$ , propagation axis is ${\displaystyle {\vec {z}}}$. Reflected photon (magenta) has opposite linear momentum ${\displaystyle \mathbf {P} =-\hbar \mathbf {k} }$ and angular momentum ${\displaystyle L=\mp \hbar \ell }$. Because of conservation laws PC mirror experiences recoil: the vortex phonon (orange) with doubled linear momentum ${\displaystyle \mathbf {P} =2\hbar \mathbf {k} }$ and angular momentum ${\displaystyle L=\pm 2\hbar \ell }$ is excited within mirror.

One can interpret optical phase conjugation 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.

Reversal of wavefront means a perfect reversal of photons linear momentum and angular momentum. The reversal of angular momentum means reversal of both polarization state and orbital angular momentum. Reversal of orbital angular momentum of optical vortex is due to the perfect match of helical phase profiles of the incident and reflected beams. Optical phase conjugation is implemented via stimulated Brillouin scattering, four-wave mixing, three-wave mixing, static linear holograms and some other tools.

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.

Four-wave mixing technique

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

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.

Common SHG materials

Dark-red gallium selenide in its bulk form

Ordered by pump wavelength:

• 800 nm: BBO
• 806 nm: lithium iodate (LiIO₃)
• 860 nm: potassium niobate (KNbO₃)
• 980 nm: KNbO₃
• 1064 nm: monopotassium phosphate (KH₂PO₄, KDP), lithium triborate (LBO) and β-barium borate (BBO)
• 1300 nm: gallium selenide (GaSe)
• 1319 nm: KNbO₃, BBO, KDP, potassium titanyl phosphate (KTP), lithium niobate (LiNbO₃), LiIO₃, and ammonium dihydrogen phosphate (ADP)
• 1550 nm: potassium titanyl phosphate (KTP), lithium niobate (LiNbO₃)

Raman scattering

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Raman_scattering

Raman scattering or the Raman effect /ˈrɑːmən/ is the inelastic scattering of photons by matter, meaning that there is an exchange of energy and a change in the light's direction. Typically this involves vibrational energy being gained by a molecule as incident photons from a visible laser are shifted to lower energy. This is called normal Stokes Raman scattering. The effect is exploited by chemists and physicists to gain information about materials for a variety of purposes by performing various forms of Raman spectroscopy. Many other variants of Raman spectroscopy allow rotational energy to be examined (if gas samples are used) and electronic energy levels may be examined if an X-ray source is used in addition to other possibilities. More complex techniques involving pulsed lasers, multiple laser beams and so on are known.

Light has a certain probability of being scattered by a material. When photons are scattered, most of them are elastically scattered (Rayleigh scattering), such that the scattered photons have the same energy (frequency, wavelength and color) as the incident photons but different direction. Rayleigh scattering usually has an intensity in the range 0.1% to 0.01% relative to that of a radiation source. An even smaller fraction of the scattered photons (approximately 1 in 10 million) can be scattered inelastically, with the scattered photons having an energy different (usually lower) from those of the incident photons—these are Raman scattered photons. Because of conservation of energy, the material either gains or loses energy in the process.

Rayleigh scattering was discovered and explained in the 19th century. The Raman effect is named after Indian scientist C. V. Raman, who discovered it in 1928 with assistance from his student K. S. Krishnan. Raman was awarded the Nobel prize in Physics in 1930 for his discovery. The effect had been predicted theoretically by Adolf Smekal in 1923.

History

The elastic light scattering phenomena called Rayleigh scattering, in which light retains its energy, was described in the 19th century. The intensity of Rayleigh scattering is about 10⁻³ to 10⁻⁴ compared to the intensity of the exciting source. In 1908, another form of elastic scattering, called Mie scattering was discovered.

The inelastic scattering of light was predicted by Adolf Smekal in 1923 and in older German-language literature it has been referred to as the Smekal-Raman-Effekt. In 1922, Indian physicist C. V. Raman published his work on the "Molecular Diffraction of Light", the first of a series of investigations with his collaborators that ultimately led to his discovery (on 28 February 1928) of the radiation effect that bears his name. The Raman effect was first reported by Raman and his coworker K. S. Krishnan, and independently by Grigory Landsberg and Leonid Mandelstam, in Moscow on 21 February 1928 (one week earlier than Raman and Krishnan). In the former Soviet Union, Raman's contribution was always disputed; thus in Russian scientific literature the effect is usually referred to as "combination scattering" or "combinatory scattering". Raman received the Nobel Prize in 1930 for his work on the scattering of light.

In 1998 the Raman effect was designated a National Historic Chemical Landmark by the American Chemical Society in recognition of its significance as a tool for analyzing the composition of liquids, gases, and solids.

Instrumentation

An early Raman spectrum of benzene published by Raman and Krishnan.

 

Schematic of a dispersive Raman spectroscopy setup in a 180° backscattering arrangement.

Modern Raman spectroscopy nearly always involves the use of lasers as an exciting light source. Because lasers were not available until more than three decades after the discovery of the effect, Raman and Krishnan used a mercury lamp and photographic plates to record spectra. Early spectra took hours or even days to acquire due to weak light sources, poor sensitivity of the detectors and the weak Raman scattering cross-sections of most materials. The most common modern detectors are charge-coupled devices (CCDs). Photodiode arrays and photomultiplier tubes were common prior to the adoption of CCDs.

Theory

The following focuses on the theory of normal (non-resonant, spontaneous, vibrational) Raman scattering of light by discrete molecules. X-ray Raman spectroscopy is conceptually similar but involves excitation of electronic, rather than vibrational, energy levels.

Molecular vibrations

Raman scattering generally gives information about vibrations within a molecule. In the case of gases, information about rotational energy can also be gleaned. For solids, phonon modes may also be observed. The basics of infrared absorption regarding molecular vibrations apply to Raman scattering although the selection rules are different.

Degrees of freedom

For any given molecule, there are a total of 3N degrees of freedom, where N is the number of atoms. This number arises from the ability of each atom in a molecule to move in three dimensions. When dealing with molecules, it is more common to consider the movement of the molecule as a whole. Consequently, the 3N degrees of freedom are partitioned into molecular translational, rotational, and vibrational motion. Three of the degrees of freedom correspond to translational motion of the molecule as a whole (along each of the three spatial dimensions). Similarly, three degrees of freedom correspond to rotations of the molecule about the ${\displaystyle x}$, ${\displaystyle y}$, and ${\displaystyle z}$-axes. Linear molecules only have two rotations because rotations along the bond axis do not change the positions of the atoms in the molecule. The remaining degrees of freedom correspond to molecular vibrational modes. These modes include stretching and bending motions of the chemical bonds of the molecule. For a linear molecule, the number of vibrational modes is 3N-5, whereas for a non-linear molecule the number of vibrational modes is 3N-6.

Vibrational energy

Molecular vibrational energy is known to be quantized and can be modeled using the quantum harmonic oscillator (QHO) approximation or a Dunham expansion when anharmonicity is important. The vibrational energy levels according to the QHO are

${\displaystyle E_{n}=h\left(n+{1 \over 2}\right)\nu =h\left(n+{1 \over 2}\right){1 \over {2\pi }}{\sqrt {k \over m}}\!}$,

where n is a quantum number. Since the selection rules for Raman and infrared absorption generally dictate that only fundamental vibrations are observed, infrared excitation or Stokes Raman excitation results in an energy change of ${\displaystyle E=h\nu ={h \over {2\pi }}{\sqrt {k \over m}}}$

The energy range for vibrations is in the range of approximately 5 to 3500 cm⁻¹. The fraction of molecules occupying a given vibrational mode at a given temperature follows a Boltzmann distribution. A molecule can be excited to a higher vibrational mode through the direct absorption of a photon of the appropriate energy, which falls in the terahertz or infrared range. This forms the basis of infrared spectroscopy. Alternatively, the same vibrational excitation can be produced by an inelastic scattering process. This is called Stokes Raman scattering, by analogy with the Stokes shift in fluorescence discovered by George Stokes in 1852, with light emission at longer wavelength (now known to correspond to lower energy) than the absorbed incident light. Conceptually similar effects can be caused by neutrons or electrons rather than light. An increase in photon energy which leaves the molecule in a lower vibrational energy state is called anti-Stokes scattering.

Raman scattering

Raman scattering is conceptualized as involving a virtual electronic energy level which corresponds to the energy of the exciting laser photons. Absorption of a photon excites the molecule to the imaginary state and re-emission leads to Raman or Rayleigh scattering. In all three cases the final state has the same electronic energy as the starting state but is higher in vibrational energy in the case of Stokes Raman scattering, lower in the case of anti-Stokes Raman scattering or the same in the case of Rayleigh scattering. Normally this is thought of in terms of wavenumbers, where ${\displaystyle {\tilde {\nu }}_{0}}$ is the wavenumber of the laser and ${\displaystyle {\tilde {\nu }}_{M}}$ is the wavenumber of the vibrational transition. Thus Stokes scattering gives a wavenumber of ${\displaystyle {\tilde {\nu }}_{0}-{\tilde {\nu }}_{M}}$ while ${\textstyle {\tilde {\nu }}_{0}+{\tilde {\nu }}_{M}}$ is given for anti-Stokes. When the exciting laser energy corresponds to an actual electronic excitation of the molecule then the resonance Raman effect occurs, but that is beyond the scope of this article.

A classical physics based model is able to account for Raman scattering and predicts an increase in the intensity which scales with the fourth-power of the light frequency. Light scattering by a molecule is associated with oscillations of an induced electric dipole. The oscillating electric field component of electromagnetic radiation may bring about an induced dipole in a molecule which follows the alternating electric field which is modulated by the molecular vibrations. Oscillations at the external field frequency are therefore observed along with beat frequencies resulting from the external field and normal mode vibrations.

 

The different possibilities of light scattering: Rayleigh scattering (no exchange of energy: incident and scattered photons have the same energy), Stokes Raman scattering (atom or molecule absorbs energy: scattered photon has less energy than the incident photon) and anti-Stokes Raman scattering (atom or molecule loses energy: scattered photon has more energy than the incident photon)

The spectrum of the scattered photons is termed the Raman spectrum. It shows the intensity of the scattered light as a function of its frequency difference Δν to the incident photons, more commonly called a Raman shift. The locations of corresponding Stokes and anti-Stokes peaks form a symmetric pattern around the RayleighΔν=0 line. The frequency shifts are symmetric because they correspond to the energy difference between the same upper and lower resonant states. The intensities of the pairs of features will typically differ, though. They depend on the populations of the initial states of the material, which in turn depend on the temperature. In thermodynamic equilibrium, the lower state will be more populated than the upper state. Therefore, the rate of transitions from the more populated lower state to the upper state (Stokes transitions) will be higher than in the opposite direction (anti-Stokes transitions). Correspondingly, Stokes scattering peaks are stronger than anti-Stokes scattering peaks. Their ratio depends on the temperature, and can therefore be exploited to measure it:

${\displaystyle {\frac {I_{\text{Stokes}}}{I_{\text{anti-Stokes}}}}={\frac {({\tilde {\nu }}_{0}-{\tilde {\nu }}_{M})^{4}}{({\tilde {\nu }}_{0}+{\tilde {\nu }}_{M})^{4}}}\exp \left({\frac {hc\,{\tilde {\nu }}_{M}}{k_{B}T}}\right)}$

Selection rules

In contrast to IR spectroscopy, where there is a requirement for a change in dipole moment for vibrational excitation to take place, Raman scattering requires a change in polarizability. A Raman transition from one state to another is allowed only if the molecular polarizability of those states is different. For a vibration, this means that the derivative of the polarizability with respect to the normal coordinate associated to the vibration is non-zero: ${\displaystyle {\frac {\partial \alpha }{\partial Q}}\neq 0}$. In general, a normal mode is Raman active if it transforms with the same symmetry of the quadratic forms ${\displaystyle (x^{2},y^{2},z^{2},xy,xz,yz)}$, which can be verified from the character table of the molecule's point group. As with IR spectroscopy, only fundamental excitations (${\displaystyle \Delta \nu =\pm 1}$) are allowed according to the QHO. There are however many cases where overtones are observed. The rule of mutual exclusion, which states that vibrational modes cannot be both IR and Raman active, applies to certain molecules.

The specific selection rules state that the allowed rotational transitions are ${\displaystyle \Delta J=\pm 2}$, where ${\displaystyle J}$ is the rotational state. This generally is only relevant to molecules in the gas phase where the Raman linewidths are small enough for rotational transitions to be resolved.

A selection rule relevant only to ordered solid materials states that only phonons with zero phase angle can be observed by IR and Raman, except when phonon confinement is manifest.

Symmetry and polarization

Monitoring the polarization of the scattered photons is useful for understanding the connections between molecular symmetry and Raman activity which may assist in assigning peaks in Raman spectra. Light polarized in a single direction only gives access to some Raman–active modes, but rotating the polarization gives access to other modes. Each mode is separated according to its symmetry.

The symmetry of a vibrational mode is deduced from the depolarization ratio ρ, which is the ratio of the Raman scattering with polarization orthogonal to the incident laser and the Raman scattering with the same polarization as the incident laser: ${\displaystyle \rho ={\frac {I_{r}}{I_{u}}}}$ Here ${\displaystyle I_{r}}$ is the intensity of Raman scattering when the analyzer is rotated 90 degrees with respect to the incident light's polarization axis, and ${\displaystyle I_{u}}$ the intensity of Raman scattering when the analyzer is aligned with the polarization of the incident laser. When polarized light interacts with a molecule, it distorts the molecule which induces an equal and opposite effect in the plane-wave, causing it to be rotated by the difference between the orientation of the molecule and the angle of polarization of the light wave. If ${\displaystyle \rho \geq {\frac {3}{4}}}$, then the vibrations at that frequency are depolarized; meaning they are not totally symmetric.

Stimulated Raman scattering and Raman amplification

The Raman-scattering process as described above takes place spontaneously; i.e., in random time intervals, one of the many incoming photons is scattered by the material. This process is thus called spontaneous Raman scattering.

On the other hand, stimulated Raman scattering can take place when some Stokes photons have previously been generated by spontaneous Raman scattering (and somehow forced to remain in the material), or when deliberately injecting Stokes photons ("signal light") together with the original light ("pump light"). In that case, the total Raman-scattering rate is increased beyond that of spontaneous Raman scattering: pump photons are converted more rapidly into additional Stokes photons. The more Stokes photons that are already present, the faster more of them are added. Effectively, this amplifies the Stokes light in the presence of the pump light, which is exploited in Raman amplifiers and Raman lasers.

Stimulated Raman scattering is a nonlinear optical effect. It can be described using a third-order nonlinear susceptibility ${\displaystyle \chi ^{(3)}}$.

Requirement for space-coherence

Suppose that the distance between two points A and B of an exciting beam is x. Generally, as the exciting frequency is not equal to the scattered Raman frequency, the corresponding relative wavelengths λ and λ' are not equal. Thus, a phase-shift Θ = 2πx(1/λ − 1/λ') appears. For Θ = π, the scattered amplitudes are opposite, so that the Raman scattered beam remains weak.

• A crossing of the beams may limit the path x.

Several tricks may be used to get a larger amplitude:

• In an optically anisotropic crystal, a light ray may have two modes of propagation with different polarizations and different indices of refraction. If energy may be transferred between these modes by a quadrupolar (Raman) resonance, phases remain coherent along the whole path, transfer of energy may be large. It is an Optical parametric generation.
• Light may be pulsed, so that beats do not appear. In Impulsive Stimulated Raman Scattering (ISRS), the length of the pulses must be shorter than all relevant time constants. Interference of Raman and incident lights is too short to allow beats, so that it produces a frequency shift roughly, in best conditions, inversely proportional to the cube of the pulse length.

In labs, femtosecond laser pulses must be used because the ISRS becomes very weak if the pulses are too long. Thus ISRS cannot be observed using nanosecond pulses making ordinary time-incoherent light.

Inverse Raman effect

The inverse Raman effect is a form of Raman scattering first noted by W. J. Jones and B.P. Stoicheff. In some circumstances, Stokes scattering can exceed anti-Stokes scattering; in these cases the continuum (on leaving the material) is observed to have an absorption line (a dip in intensity) at ν_L+ν_M. This phenomenon is referred to as the inverse Raman effect; the application of the phenomenon is referred to as inverse Raman spectroscopy, and a record of the continuum is referred to as an inverse Raman spectrum.

In the original description of the inverse Raman effect, the authors discuss both absorption from a continuum of higher frequencies and absorption from a continuum of lower frequencies. They note that absorption from a continuum of lower frequencies will not be observed if the Raman frequency of the material is vibrational in origin and if the material is in thermal equilibrium.

Supercontinuum generation

For high-intensity continuous wave (CW) lasers, stimulated Raman scattering can be used to produce a broad bandwidth supercontinuum. This process can also be seen as a special case of four-wave mixing, wherein the frequencies of the two incident photons are equal and the emitted spectra are found in two bands separated from the incident light by the phonon energies. The initial Raman spectrum is built up with spontaneous emission and is amplified later on. At high pumping levels in long fibers, higher-order Raman spectra can be generated by using the Raman spectrum as a new starting point, thereby building a chain of new spectra with decreasing amplitude. The disadvantage of intrinsic noise due to the initial spontaneous process can be overcome by seeding a spectrum at the beginning, or even using a feedback loop as in a resonator to stabilize the process. Since this technology easily fits into the fast evolving fiber laser field and there is demand for transversal coherent high-intensity light sources (i.e., broadband telecommunication, imaging applications), Raman amplification and spectrum generation might be widely used in the near-future.

Applications

Raman spectroscopy employs the Raman effect for substances analysis. The spectrum of the Raman-scattered light depends on the molecular constituents present and their state, allowing the spectrum to be used for material identification and analysis. Raman spectroscopy is used to analyze a wide range of materials, including gases, liquids, and solids. Highly complex materials such as biological organisms and human tissue can also be analyzed by Raman spectroscopy.

For solid materials, Raman scattering is used as a tool to detect high-frequency phonon and magnon excitations.

Raman lidar is used in atmospheric physics to measure the atmospheric extinction coefficient and the water vapour vertical distribution.

Stimulated Raman transitions are also widely used for manipulating a trapped ion's energy levels, and thus basis qubit states.

Raman spectroscopy can be used to determine the force constant and bond length for molecules that do not have an infrared absorption spectrum.

Raman amplification is used in optical amplifiers.

The Raman effect is also involved in producing the appearance of the blue sky (see Rayleigh Scattering: 'Rayleigh scattering of molecular nitrogen and oxygen in the atmosphere includes elastic scattering as well as the inelastic contribution from rotational Raman scattering in air').

Raman spectroscopy has been used to chemically image small molecules, such as nucleic acids, in biological systems by a vibrational tag.