Crystal optics

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Crystal_optics

Crystal optics is the branch of optics that describes the behaviour of light in anisotropic media, that is, media (such as crystals) in which light behaves differently depending on which direction the light is propagating. The index of refraction depends on both composition and crystal structure and can be calculated using the Gladstone–Dale relation. Crystals are often naturally anisotropic, and in some media (such as liquid crystals) it is possible to induce anisotropy by applying an external electric field.

Isotropic media

Typical transparent media such as glasses are isotropic, which means that light behaves the same way no matter which direction it is travelling in the medium. In terms of Maxwell's equations in a dielectric, this gives a relationship between the electric displacement field D and the electric field E:

${\displaystyle \mathbf{D}={\epsilon }_0\mathbf{E}+\mathbf{P}}$

where ε₀ is the permittivity of free space and P is the electric polarization (the vector field corresponding to electric dipole moments present in the medium). Physically, the polarization field can be regarded as the response of the medium to the electric field of the light.

Electric susceptibility

In an isotropic and linear medium, this polarization field P is proportional and parallel to the electric field E:

${\displaystyle \mathbf{P}=\chi {\epsilon }_0\mathbf{E}}$

where χ is the electric susceptibility of the medium. The relation between D and E is thus:

${\displaystyle \mathbf{D}={\epsilon }_0\mathbf{E}+\chi {\epsilon }_0\mathbf{E}={\epsilon }_0(1+\chi )\mathbf{E}=\epsilon \mathbf{E}}$

where

${\displaystyle \epsilon ={\epsilon }_0(1+\chi )}$

is the dielectric constant of the medium. The value 1+χ is called the relative permittivity of the medium, and is related to the refractive index n, for non-magnetic media, by

${\displaystyle n=\sqrt{1+\chi }}$

Anisotropic media

In an anisotropic medium, such as a crystal, the polarisation field P is not necessarily aligned with the electric field of the light E. In a physical picture, this can be thought of as the dipoles induced in the medium by the electric field having certain preferred directions, related to the physical structure of the crystal. This can be written as:

${\displaystyle \mathbf{P}={\epsilon }_0\mathit{\chi }\mathbf{E}.}$

Here χ is not a number as before but a tensor of rank 2, the electric susceptibility tensor. In terms of components in 3 dimensions:

${\displaystyle \left(\begin{array}{c}{P}_x\\ P_y\\ P_z\end{array}\right)={\epsilon }_0\left(\begin{array}{ccc}{\chi }_{xx}& {\chi }_{xy}& {\chi }_{xz}\\ {\chi }_{yx}& {\chi }_{yy}& {\chi }_{yz}\\ {\chi }_{zx}& {\chi }_{zy}& {\chi }_{zz}\end{array}\right)\left(\begin{array}{c}{E}_x\\ E_y\\ E_z\end{array}\right)}$

or using the summation convention:

${\displaystyle P_i={\epsilon }_0\sum _{j\in \{x,y,z\}}{\chi }_{ij}{E}_j.}$

Since χ is a tensor, P is not necessarily colinear with E.

In nonmagnetic and transparent materials, χ_ij = χ_ji, i.e. the χ tensor is real and symmetric. In accordance with the spectral theorem, it is thus possible to diagonalise the tensor by choosing the appropriate set of coordinate axes, zeroing all components of the tensor except χ_xx, χ_yy and χ_zz. This gives the set of relations:

${\displaystyle P_x={\epsilon }_0{\chi }_{xx}{E}_x}$

${\displaystyle P_y={\epsilon }_0{\chi }_{yy}{E}_y}$

${\displaystyle P_z={\epsilon }_0{\chi }_{zz}{E}_z}$

The directions x, y and z are in this case known as the principal axes of the medium. Note that these axes will be orthogonal if all entries in the χ tensor are real, corresponding to a case in which the refractive index is real in all directions.

It follows that D and E are also related by a tensor:

${\displaystyle \mathbf{D}={\epsilon }_0\mathbf{E}+\mathbf{P}={\epsilon }_0\mathbf{E}+{\epsilon }_0\mathit{\chi }\mathbf{E}={\epsilon }_0(I+\mathit{\chi })\mathbf{E}={\epsilon }_0\mathit{\epsilon }\mathbf{E}.}$

Here ε is known as the relative permittivity tensor or dielectric tensor. Consequently, the refractive index of the medium must also be a tensor. Consider a light wave propagating along the z principal axis polarised such the electric field of the wave is parallel to the x-axis. The wave experiences a susceptibility χ_xx and a permittivity ε_xx. The refractive index is thus:

${\displaystyle n_{xx}=(1+{\chi }_{xx}{)}^{1/2}=({\epsilon }_{xx}{)}^{1/2}.}$

For a wave polarised in the y direction:

${\displaystyle n_{yy}=(1+{\chi }_{yy}{)}^{1/2}=({\epsilon }_{yy}{)}^{1/2}.}$

Thus these waves will see two different refractive indices and travel at different speeds. This phenomenon is known as birefringence and occurs in some common crystals such as calcite and quartz.

If χ_xx = χ_yy ≠ χ_zz, the crystal is known as uniaxial. (See Optic axis of a crystal.) If χ_xx ≠ χ_yy and χ_yy ≠ χ_zz the crystal is called biaxial. A uniaxial crystal exhibits two refractive indices, an "ordinary" index (n_o) for light polarised in the x or y directions, and an "extraordinary" index (n_e) for polarisation in the z direction. A uniaxial crystal is "positive" if n_e > n_o and "negative" if n_e < n_o. Light polarised at some angle to the axes will experience a different phase velocity for different polarization components, and cannot be described by a single index of refraction. This is often depicted as an index ellipsoid.

Other effects

Certain nonlinear optical phenomena such as the electro-optic effect cause a variation of a medium's permittivity tensor when an external electric field is applied, proportional (to lowest order) to the strength of the field. This causes a rotation of the principal axes of the medium and alters the behaviour of light travelling through it; the effect can be used to produce light modulators.

In response to a magnetic field, some materials can have a dielectric tensor that is complex-Hermitian; this is called a gyro-magnetic or magneto-optic effect. In this case, the principal axes are complex-valued vectors, corresponding to elliptically polarized light, and time-reversal symmetry can be broken. This can be used to design optical isolators, for example.

A dielectric tensor that is not Hermitian gives rise to complex eigenvalues, which corresponds to a material with gain or absorption at a particular frequency.

African Australians

From Wikipedia, the free encyclopedia

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

 

African Australians

Total population
326,673 (2021 census) 1.3% of Australia's population
New South Wales 75,942 (1.02%)  Queensland 64,112 (1.32%)  Western Australia 66,744 (2.61%)  Northern Territory 2,660 (1.08%)  Victoria 90,640 (1.47%)  South Australia 17,607 (1.03%)  Tasmania 3,434 (0.66%)  Australian Capital Territory 5,504 (1.37%)
Languages
Australian English African languages
Religion
Christianity Islam traditional African religions

African Australians are Australians descended from any peoples of Sub-Saharan Africa, including naturalised Australians who are immigrants from various regions in Sub-Saharan Africa and descendants of such immigrants. At the 2021 census, the number of ancestry responses categorised within Sub-Saharan African ancestral groups as a proportion of the total population amounted to 1.3%. Note that Australian official statistics are based on country of origin not race, hence African immigrants of European descent (such as White South Africans) and their descendants are included as African Australians.

Large-scale immigration from Africa to Australia is only a recent phenomenon, with Europe and Asia traditionally being the largest sources of migration to Australia. African Australians come from diverse ethnic, cultural, linguistic, religious, educational and employment backgrounds.

History

An agricultural officer from Ghana visiting Queensland under the Special Commonwealth African Assistance Plan, 1960s

Large-scale immigration from Africa to Australia is only a recent phenomenon, with Europe and Asia traditionally being the largest sources of migration to Australia.

Coins minted by the Tanzanian medieval kingdom of Kilwa Sultanate have been found on the Wessel Islands. They are the oldest foreign artefacts ever discovered in Australia. Other people descended from African emigrants later arrived indirectly via the First Fleet and 19th century multicultural maritime industry. Notable examples are Billy Blue, John Caesar, and Black Jack Anderson.

Migrants from Mauritius have also been arriving in Australia since before federation in 1901. They came as convicts, prospectors who sought Victoria's goldfields, or skilled sugar workers who significantly helped to develop Queensland's sugar industry.

Following the 1823 Demerara Slave Rebellion in British Guiana, several hundreds of enslaved Africans who had participated in the rebellion were deported to Queensland, Australia.

The Special Commonwealth African Assistance Plan enabled students from British Commonwealth African countries, including from Ghana, to travel to Australia during the mid-1960s. More than 70 percent of those from West African countries remained in Australia following military coup d'états in their countries of birth.

However, immigration from Africa to Australia generally remained limited until the 1990s, thus compared to other established European and American countries, African Australian community remains new in the country itself.

In 2005–06, permanent settler arrivals to Australia included 4,000 South Africans and 3,800 Sudanese, constituting the sixth and seventh largest sources of migrants, respectively.

Demographics

African Australians are Australians of direct Sub-Saharan African ancestry. They are from diverse racial, cultural, linguistic, religious, educational and employment backgrounds. The majority (72.6%) of African emigrants to Australia are from southern and eastern Africa. The Australian Bureau of Statistics classifies all residents into cultural and ethnic groups according to geographical origin.

Migration streams

People of South African ancestry whose parents were both born in Australia as a fraction of total residents

Some of the most significant migration streams as of 2011-2012 were as follows:

• Other immigrants from Africa arrived via humanitarian programs, mostly from East Africa. In the 2011–2012 fiscal year, these individuals were mainly from Burundi (44/79), Congo (143/158), the Democratic Republic of the Congo (370/454), Eritrea (244/294), Malawi (57/71), Rwanda (44/62), and Tanzania (40/67).
• Additionally, other immigrants from Africa arrived through a family reunion migration stream. In the 2011–2012 fiscal year, these individuals were primarily from Ethiopia (412/802), Ghana (152/202), Guinea (33/62), Liberia (82/129), Sierra Leone (106/140), Somalia (164/420), and Uganda (37/67).
• A significant number of African migrants have come to Australia through a skilled migration stream. In the 2011–2012 fiscal year, these individuals were chiefly from Kenya (188/415), Mauritius (228/303), Nigeria (126/250), South Africa (4,239/6,307), Zambia (35/115), and Zimbabwe (467/848).
• Some African immigrants have also arrived via a secondary migration from New Zealand, where they are citizens.

Broadcasting services for African migrants

Multicultural broadcaster Special Broadcasting Service (SBS) broadcasts in five African languages on radio, including Nuer and Dinka of South Sudan, Swahili of Tanzania and the African Great Lakes region, Tigrinya of Eritrea and Amharic of Ethiopia. Arabic broadcasting began with a 6am service by SBS in 1975, and from 2016, SBS began a year-long trial of SBS Arabic 24, a 24/7 digital radio station and website. It continues today and includes an Arabic24 podcast. An English language program, simply called SBS African (nicknamed the African Hour) was broadcast until 2017, when it was cut from schedule. 2ME Radio Arabic also broadcasts in Arabic throughout Australia.

Social status

As Africans only began to migrate to Australia in larger numbers much later than Africans were brought to the United States as slaves, and those who settled in parts of Europe, African Australian status is largely a new challenge for Australian authorities, and it is acknowledged that widespread racism against Africans is not uncommon in Australia.

Relationship to Indigenous Australians

Further information: Black Australians

The concept of how the American notion of "blackness" was adopted and adapted by Aboriginal civil rights activists has been little known or understood in the US. In 2011, the Museum of Contemporary African Diasporan Arts in New York mounted an exhibition of Indigenous Australian art, concerned with making connections between the current civil rights and spiritual movements of Indigenous Australians and that of black people in America and elsewhere.

A 2012 study looked at attitudes towards African immigrants in Western Australia, based on a survey of 184 Australians, examining the quantitative data for use in developing strategies to combat prejudice, and the media's role in the development of negative attitudes. It compared the results of the study with those previously found in looking at attitudes towards Indigenous and Muslim Australians.

Natasha Guantai, in response to Roxane Gay's initial implication that the only "black people" in Australia would be of African descent, wrote "In the dominant Australian narrative, blacks are regarded as Aboriginal. This is a narrative with little space for non-Indigenous black Australians". Guantai goes on to highlight some differences in the experience of the various groups - Indigenous Australians, immigrants from Africa, the black descendants of settlers, and black people who arrive from other white-majority countries such as the UK or the US.

In 2018 Kaiya Aboagye, a PhD student of Ghanaian, Aboriginal, South Sea and Torres Strait Islander heritage, underlined the African connection to Aboriginal Australians, citing "long histories of African/Indigenous relationships both inside and outside Australia", despite the many and varied origins and experiences of blackness among peoples in the Global South.

Relationship with the criminal justice system

In 2021, it was reported that African Australians, predominantly of South Sudanese descent, comprised 19 percent of young people in custody in Victoria, despite making up less than 0.5 percent of the overall population. Previously, in 2013 Victoria Police settled a racial profiling complaint lodged by members of the African community by agreeing to review its procedures. A 2020 study in the Australian and New Zealand Journal of Criminology found that South Sudanese-born individuals were significantly overrepresented in as perpetrators of "crimes against the person", such as robbery and assault, but that "rates for less serious crimes, such as public order and drug offences, have remained stable and relatively low for South Sudanese-born youth".

Organised crime

In 2016, the Liberal Party began to campaign against what it identified as "South Sudanese gangs" in Melbourne, following riots at the Moomba Music Festival in the city. This campaign was criticised by local community leaders, and the Australian Greens MP Adam Bandt said it was using "race to win votes and whip up hatred". South Sudanese Australians commit around 1% of crime in Melbourne, which is higher than their share of the population (0.14%), but is not adjusted for the low average age of the South Sudanese-born population, which can account for their over-representation in the statistics.

In 2018, then-Prime Minister Malcolm Turnbull described the supposed presence of South Sudanese gangs in Melbourne as a "real concern", with then-Home Affairs Minister Peter Dutton claiming that Melburnians were afraid to leave their homes at night due to gang-related violence. Then-Victorian Premier Daniel Andrews rejected Turnbull's comments.

The debate on "African gangs" in Melbourne was a key part in the Victorian Liberal Party's campaign for the 2018 state election under then-Opposition Leader Matthew Guy.

Criminologists and the police commissioners of Melbourne say that episodes of youth criminality occurring in Melbourne do not amount to "gang activity" or organised crime, according to the definition used by law enforcement. The debate around so-called "African gangs" was highly racialised and resulted in many examples of racist discourse on social media, leading Anthony Kelly, executive officer of of Melbourne's Flemington and Kensington Community Legal Centre, to describe it as a "racialised moral panic". The aftermath of the panic caused black people in Melbourne to fear that they would be arrested simply for congregating in public spaces, with South Sudanese people reporting high levels of targeting by police.

African Australian identity

Main article: African Australian identity

 

African Australian identity is the objective or subjective state of perceiving oneself as an African Australian and as relating to being African Australian. As a group identity, "African Australian" can denote pan-African ethnic identity, as well as a diasporic identity in relation to the perception of Africa as a homeland.

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 (when the electric field of the light is >10⁸ V/m and thus comparable to the atomic electric field of ~10¹¹ 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 was 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):

${\displaystyle \mathbf{P}(t)={\epsilon }_0\left({\chi }^{(1)}\mathbf{E}(t)+{\chi }^{(2)}{\mathbf{E}}^2(t)+{\chi }^{(3)}{\mathbf{E}}^3(t)+\dots \right),}$

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

${\displaystyle \mathrm{\nabla }\times \mathrm{\nabla }\times \mathbf{E}+\frac{n^2}{c^2}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{t}^2}\mathbf{E}=-\frac{1}{{\epsilon }_0{c}^2}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{t}^2}{\mathbf{P}}^{\text{NL}},}$

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

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

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

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

to obtain the more familiar wave equation

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

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

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

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:

${\displaystyle {\mathrm{\nabla }}^2\mathbf{E}-\frac{n^2}{c^2}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{t}^2}\mathbf{E}=\frac{1}{{\epsilon }_0{c}^2}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{t}^2}{\mathbf{P}}^{\text{NL}}.}$

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:

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

and the inhomogeneous term

${\displaystyle \frac{1}{{\epsilon }_0{c}^2}\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{t}^2}{\mathbf{P}}^{\text{NL}}}$

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

${\displaystyle {\mathbf{P}}^{\text{NL}}={\epsilon }_0{\chi }^{(2)}{\mathbf{E}}^2(t).}$

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

${\displaystyle \begin{array}{rl}{\mathbf{P}}^{\text{NL}}& ={\epsilon }_0{\chi }^{(2)}{\mathbf{E}}^2(t)\\ & =\frac{{\epsilon }_0}{4}{\chi }^{(2)}\left[{E_1}^2{e}^{-i2{\omega }_{1}t}+{E_2}^2{e}^{-i2{\omega }_{2}t}+2E_1{E}_2{e}^{-i({\omega }_1+{\omega }_2)t}+2E_1{E_2}^{\ast }{e}^{-i({\omega }_1-{\omega }_2)t}+\left({\left|E_1\right|}^2+{\left|E_2\right|}^2\right)e^0+\text{c.c.}\right],\end{array}}$

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 ${\displaystyle {\omega }^{\prime }=2\omega }$ and ${\displaystyle {\mathbf{k}}^{\prime }=2\mathbf{k}}$ 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 ${\displaystyle \Vert {\mathbf{k}}_j\Vert =\mathbf{n}({\omega }_j){\omega }_j/c}$, 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 ${\displaystyle \Vert {\mathbf{k}}_3\Vert =\mathbf{n}({\omega }_3){\omega }_3/c}$. Constructive interference, and therefore a high-intensity ${\displaystyle {\omega }_3}$ field, will occur only if

${\displaystyle {\overrightarrow{\mathbf{k}}}_3={\overrightarrow{\mathbf{k}}}_1+{\overrightarrow{\mathbf{k}}}_2.}$

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

(${\displaystyle {\lambda }_p\le {\lambda }_s\le {\lambda }_i}$)

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

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

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:

${\displaystyle P_{\text{NL}}={\epsilon }_0{\chi }^{(3)}({\mathrm{\Xi }}_1+{\mathrm{\Xi }}_2+{\mathrm{\Xi }}_3{)}^3,}$

resulting in generation of waves with frequencies given by ω = ±ω₁ ± ω₂ ± ω₃ in addition to third-harmonic generation waves with ω = 3ω₁, 3ω₂, 3ω₃.

As above, the phase-matching condition determines which of these waves is the dominant. By choosing conditions such that ω = ω₁ + ω₂ − ω₃ and k = k₁ + k₂ − k₃, this gives a polarization field:

${\displaystyle P_{\omega }=\frac{1}{2}{\chi }^{(3)}{\epsilon }_0{E}_1{E}_2{E}_3^{\ast }{e}^{i(\omega t-\mathbf{k}\cdot \mathbf{x})}+\text{c.c.}}$

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

${\displaystyle E_4=\frac{i\omega l}{2nc}{\chi }^{(3)}{E}_1{E}_2{E}_3^{\ast },}$

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

${\displaystyle E_4(\mathbf{x})\propto E_3^{\ast }(\mathbf{x}),}$

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

${\displaystyle \mathbf{S}(\mathbf{r},t)={ϵ}_0{c}^2\mathbf{E}(\mathbf{r},t)\times \mathbf{B}(\mathbf{r},t),}$

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 ${\displaystyle \hslash \omega }$ also possesses linear momentum ${\displaystyle \mathbf{P}=\hslash \mathbf{k}}$ and angular momentum, whose projection on propagation axis is ${\displaystyle L_{\mathbf{z}}=\pm \hslash \ell }$, 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 ${\displaystyle \pm \hslash \ell }$.

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):

${\displaystyle \begin{array}{rl}{\mathbf{P}}_{\text{out}}& =-\hslash \mathbf{k}=-{\mathbf{P}}_{\text{in}}=\hslash \mathbf{k},\\ {L_{\mathbf{z}}}_{\text{out}}& =-\hslash \ell =-{L_{\mathbf{z}}}_{\text{in}}=\hslash \ell .\end{array}}$

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.

Connecting bulk properties to microscopic properties

Molecular nonlinear optics relate optical properties of bulk matter to their microscopic molecular properties. Just as the polarizability can be described as a Taylor series expansion, one can expand the induced dipole moment in powers of the electric field: ${\displaystyle \mathit{\mu }={\mathit{\mu }}_{\mathbf{0}}+\alpha \cdot \mathbf{E}+\frac{1}{2}\beta :\mathbf{E}\mathbf{E}}$, where μ is the polarizability, α is the first hyperpolarizability, β is the second hyperpolarizability, and so on.

Novel Nonlinear Media

Certain molecular materials have the ability to be optimized for their optical nonlinearity at the microscopic and bulk levels. Due to the delocalization of electrons in π bonds electrons are more easily responsive to applied optical fields and tend to produce larger linear and nonlinear optical responses than those in single (𝜎) bonds. In these systems linear response scales with the length of the conjugated pi system, while nonlinear response scales even more rapidly.

Green Fluorescent Protein (GFP) chromophore p-hydroxybenzylideneimidazolinone (HBDI) used in nonlinear bioimaging is an example of a pi-conjugated donor-acceptor (D-π-A) chromophore.

One of the many applications of molecular nonlinear optics is the use in nonlinear bioimaging. These nonlinear materials, like multi-photon chromophores, are used as biomarkers for two-photon spectroscopy, in which  the attenuation of incident light intensity as it passes through the sample is written as ${\displaystyle \frac{-dI}{dx}=\frac{N\delta I^2}{\hslash \omega }}$.

where N is the number of particles per unit volume, I is intensity of light, and δ is the two photon absorption cross section. The resulting signal adopts a Lorentzian lineshape with a cross-section proportional to the difference in dipole moments of ground and final states.

Similar highly conjugated chromophores with strong donor-acceptor characteristics are used due to their large difference in the dipole moments, and current efforts in extending their pi-conjugated systems to enhance their nonlinear optical properties are being made.

Common second-harmonic-generating (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)