Nonlinear optics

From Wikipedia, the free encyclopedia

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

 

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):

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

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 a 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

${\displaystyle {\begin{aligned}\mathbf {P} ^{\text{NL}}&=\varepsilon _{0}\chi ^{(2)}\mathbf {E} ^{2}(t)\\[3pt]&={\frac {\varepsilon _{0}}{4}}\chi ^{(2)}\left[{E_{1}}^{2}e^{-i2\omega _{1}t}+{E_{2}}^{2}e^{-i2\omega _{2}t}+2E_{1}E_{2}e^{-i(\omega _{1}+\omega _{2})t}+2E_{1}{E_{2}}^{*}e^{-i(\omega _{1}-\omega _{2})t}+\left(\left|E_{1}\right|^{2}+\left|E_{2}\right|^{2}\right)e^{0}+{\text{c.c.}}\right],\end{aligned}}}$

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 ${\displaystyle L=\pm \hbar \ell }$ is 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. Examples of pattern formation are spatial solitons and vortex lattices in framework of Nonlinear Schrödinger equation.

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

See also: Second-harmonic generation

 

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)

Adaptive optics

From Wikipedia, the free encyclopedia

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

 

A deformable mirror can be used to correct wavefront errors in an astronomical telescope.

 

Illustration of a (simplified) adaptive optics system. The light first hits a tip–tilt (TT) mirror and then a deformable mirror (DM) which corrects the wavefront. Part of the light is tapped off by a beamsplitter (BS) to the wavefront sensor and the control hardware which sends updated signals to the DM and TT mirrors.

 

Play media

An artist's impression of adaptive optics.

The wavefront of an aberrated image (left) can be measured using a wavefront sensor (center) and then corrected for using a deformable mirror (right)

Adaptive optics (AO) is a technology used to improve the performance of optical systems by reducing the effect of incoming wavefront distortions by deforming a mirror in order to compensate for the distortion. It is used in astronomical telescopes and laser communication systems to remove the effects of atmospheric distortion, in microscopy, optical fabrication and in retinal imaging systems to reduce optical aberrations. Adaptive optics works by measuring the distortions in a wavefront and compensating for them with a device that corrects those errors such as a deformable mirror or a liquid crystal array.

Adaptive optics should not be confused with active optics, which works on a longer timescale to correct the primary mirror geometry.

Other methods can achieve resolving power exceeding the limit imposed by atmospheric distortion, such as speckle imaging, aperture synthesis, and lucky imaging, or by moving outside the atmosphere with space telescopes, such as the Hubble Space Telescope.

History

Adaptive thin shell mirror.

Adaptive optics was first envisioned by Horace W. Babcock in 1953, and was also considered in science fiction, as in Poul Anderson's novel Tau Zero (1970), but it did not come into common usage until advances in computer technology during the 1990s made the technique practical.

Some of the initial development work on adaptive optics was done by the US military during the Cold War and was intended for use in tracking Soviet satellites.

Microelectromechanical systems (MEMS) deformable mirrors and magnetics concept deformable mirrors are currently the most widely used technology in wavefront shaping applications for adaptive optics given their versatility, stroke, maturity of technology and the high resolution wavefront correction that they afford.

Tip–tilt correction

The simplest form of adaptive optics is tip–tilt correction, which corresponds to correction of the tilts of the wavefront in two dimensions (equivalent to correction of the position offsets for the image). This is performed using a rapidly moving tip–tilt mirror that makes small rotations around two of its axes. A significant fraction of the aberration introduced by the atmosphere can be removed in this way.

Tip–tilt mirrors are effectively segmented mirrors having only one segment which can tip and tilt, rather than having an array of multiple segments that can tip and tilt independently. Due to the relative simplicity of such mirrors and having a large stroke, meaning they have large correcting power, most AO systems use these, first, to correct low order aberrations. Higher order aberrations may then be corrected with deformable mirrors.

In astronomy

Astronomers at the Very Large Telescope site in Chile use adaptive optics.

 

Play media

Laser being launched into the night sky from the VLT Adaptive Optics Facility.

Atmospheric seeing

Further information: Astronomical seeing

When light from a star passes through the Earth's atmosphere, the wavefront is perturbed.

 

The Shack–Hartmann sensor is one type of wavefront sensor used for adaptive optics.

Negative images of a star through a telescope. The left-hand panel shows the slow-motion movie of a star when the adaptive optics system is switched off. The right-hand panel shows the slow motion movie of the same star when the AO system is switched on.

When light from a star or another astronomical object enters the Earth's atmosphere, atmospheric turbulence (introduced, for example, by different temperature layers and different wind speeds interacting) can distort and move the image in various ways. Visual images produced by any telescope larger than approximately 20 centimeters are blurred by these distortions.

Wavefront sensing and correction

An adaptive optics system tries to correct these distortions, using a wavefront sensor which takes some of the astronomical light, a deformable mirror that lies in the optical path, and a computer that receives input from the detector. The wavefront sensor measures the distortions the atmosphere has introduced on the timescale of a few milliseconds; the computer calculates the optimal mirror shape to correct the distortions and the surface of the deformable mirror is reshaped accordingly. For example, an 8–10 m telescope (like the VLT or Keck) can produce AO-corrected images with an angular resolution of 30–60 milliarcsecond (mas) resolution at infrared wavelengths, while the resolution without correction is of the order of 1 arcsecond.

In order to perform adaptive optics correction, the shape of the incoming wavefronts must be measured as a function of position in the telescope aperture plane. Typically the circular telescope aperture is split up into an array of pixels in a wavefront sensor, either using an array of small lenslets (a Shack–Hartmann wavefront sensor), or using a curvature or pyramid sensor which operates on images of the telescope aperture. The mean wavefront perturbation in each pixel is calculated. This pixelated map of the wavefronts is fed into the deformable mirror and used to correct the wavefront errors introduced by the atmosphere. It is not necessary for the shape or size of the astronomical object to be known – even Solar System objects which are not point-like can be used in a Shack–Hartmann wavefront sensor, and time-varying structure on the surface of the Sun is commonly used for adaptive optics at solar telescopes. The deformable mirror corrects incoming light so that the images appear sharp.

Using guide stars

Natural guide stars

Because a science target is often too faint to be used as a reference star for measuring the shape of the optical wavefronts, a nearby brighter guide star can be used instead. The light from the science target has passed through approximately the same atmospheric turbulence as the reference star's light and so its image is also corrected, although generally to a lower accuracy.

A laser beam directed toward the centre of the Milky Way. This laser beam can then be used as a guide star for the AO.

The necessity of a reference star means that an adaptive optics system cannot work everywhere on the sky, but only where a guide star of sufficient luminosity (for current systems, about magnitude 12–15) can be found very near to the object of the observation. This severely limits the application of the technique for astronomical observations. Another major limitation is the small field of view over which the adaptive optics correction is good. As the angular distance from the guide star increases, the image quality degrades. A technique known as "multiconjugate adaptive optics" uses several deformable mirrors to achieve a greater field of view.

Artificial guide stars

An alternative is the use of a laser beam to generate a reference light source (a laser guide star, LGS) in the atmosphere. There are two kinds of LGSs: Rayleigh guide stars and sodium guide stars. Rayleigh guide stars work by propagating a laser, usually at near ultraviolet wavelengths, and detecting the backscatter from air at altitudes between 15–25 km (49,000–82,000 ft). Sodium guide stars use laser light at 589 nm to resonantly excite sodium atoms higher in the mesosphere and thermosphere, which then appear to "glow". The LGS can then be used as a wavefront reference in the same way as a natural guide star – except that (much fainter) natural reference stars are still required for image position (tip/tilt) information. The lasers are often pulsed, with measurement of the atmosphere being limited to a window occurring a few microseconds after the pulse has been launched. This allows the system to ignore most scattered light at ground level; only light which has travelled for several microseconds high up into the atmosphere and back is actually detected.

In retinal imaging

Artist's impression of the European Extremely Large Telescope deploying lasers for adaptive optics

Ocular aberrations are distortions in the wavefront passing through the pupil of the eye. These optical aberrations diminish the quality of the image formed on the retina, sometimes necessitating the wearing of spectacles or contact lenses. In the case of retinal imaging, light passing out of the eye carries similar wavefront distortions, leading to an inability to resolve the microscopic structure (cells and capillaries) of the retina. Spectacles and contact lenses correct "low-order aberrations", such as defocus and astigmatism, which tend to be stable in humans for long periods of time (months or years). While correction of these is sufficient for normal visual functioning, it is generally insufficient to achieve microscopic resolution. Additionally, "high-order aberrations", such as coma, spherical aberration, and trefoil, must also be corrected in order to achieve microscopic resolution. High-order aberrations, unlike low-order, are not stable over time, and may change over time scales of 0.1s to 0.01s. The correction of these aberrations requires continuous, high-frequency measurement and compensation.

Measurement of ocular aberrations

Ocular aberrations are generally measured using a wavefront sensor, and the most commonly used type of wavefront sensor is the Shack–Hartmann. Ocular aberrations are caused by spatial phase nonuniformities in the wavefront exiting the eye. In a Shack-Hartmann wavefront sensor, these are measured by placing a two-dimensional array of small lenses (lenslets) in a pupil plane conjugate to the eye's pupil, and a CCD chip at the back focal plane of the lenslets. The lenslets cause spots to be focused onto the CCD chip, and the positions of these spots are calculated using a centroiding algorithm. The positions of these spots are compared with the positions of reference spots, and the displacements between the two are used to determine the local curvature of the wavefront allowing one to numerically reconstruct the wavefront information—an estimate of the phase nonuniformities causing aberration.

Correction of ocular aberrations

Once the local phase errors in the wavefront are known, they can be corrected by placing a phase modulator such as a deformable mirror at yet another plane in the system conjugate to the eye's pupil. The phase errors can be used to reconstruct the wavefront, which can then be used to control the deformable mirror. Alternatively, the local phase errors can be used directly to calculate the deformable mirror instructions.

Open loop vs. closed loop operation

If the wavefront error is measured before it has been corrected by the wavefront corrector, then operation is said to be "open loop". If the wavefront error is measured after it has been corrected by the wavefront corrector, then operation is said to be "closed loop". In the latter case then the wavefront errors measured will be small, and errors in the measurement and correction are more likely to be removed. Closed loop correction is the norm.

Applications

Adaptive optics was first applied to flood-illumination retinal imaging to produce images of single cones in the living human eye. It has also been used in conjunction with scanning laser ophthalmoscopy to produce (also in living human eyes) the first images of retinal microvasculature and associated blood flow and retinal pigment epithelium cells in addition to single cones. Combined with optical coherence tomography, adaptive optics has allowed the first three-dimensional images of living cone photoreceptors to be collected.

In microscopy

In microscopy, adaptive optics is used to correct for sample-induced aberrations. The required wavefront correction is either measured directly using wavefront sensor or estimated by using sensorless AO techniques.

Other uses

GRAAL is a ground layer adaptive optics instrument assisted by lasers.

Besides its use for improving nighttime astronomical imaging and retinal imaging, adaptive optics technology has also been used in other settings. Adaptive optics is used for solar astronomy at observatories such as the Swedish 1-m Solar Telescope and Big Bear Solar Observatory. It is also expected to play a military role by allowing ground-based and airborne laser weapons to reach and destroy targets at a distance including satellites in orbit. The Missile Defense Agency Airborne Laser program is the principal example of this.

Adaptive optics has been used to enhance the performance of classical and quantum free-space optical communication systems, and to control the spatial output of optical fibers.

Medical applications include imaging of the retina, where it has been combined with optical coherence tomography. Also the development of Adaptive Optics Scanning Laser Ophthalmoscope (AOSLO) has enabled correcting for the aberrations of the wavefront that is reflected from the human retina and to take diffraction limited images of the human rods and cones. Development of an Adaptive Scanning Optical Microscope (ASOM) was announced by Thorlabs in April 2007. Adaptive and active optics are also being developed for use in glasses to achieve better than 20/20 vision, initially for military applications.

After propagation of a wavefront, parts of it may overlap leading to interference and preventing adaptive optics from correcting it. Propagation of a curved wavefront always leads to amplitude variation. This needs to be considered if a good beam profile is to be achieved in laser applications. In material processing using lasers, adjustments can be made on the fly to allow for variation of focus-depth during piercing for changes in focal length across the working surface. Beam width can also be adjusted to switch between piercing and cutting mode. This eliminates the need for optic of the laser head to be switched, cutting down on overall processing time for more dynamic modifications.

Adaptive optics, especially wavefront-coding spatial light modulators, are frequently used in optical trapping applications to multiplex and dynamically reconfigure laser foci that are used to micro-manipulate biological specimens.

Beam stabilization

A rather simple example is the stabilization of the position and direction of laser beam between modules in a large free space optical communication system. Fourier optics is used to control both direction and position. The actual beam is measured by photo diodes. This signal is fed into analog-to-digital converters and then a microcontroller which runs a PID controller algorithm. The controller then drives digital-to-analog converters which drive stepper motors attached to mirror mounts.

If the beam is to be centered onto 4-quadrant diodes, no analog-to-digital converter is needed. Operational amplifiers are sufficient.