Optical depth

From Wikipedia, the free encyclopedia

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

Aerosol Optical Depth (AOD) at 830 nm measured with the same LED sun photometer from 1990 to 2016 at Geronimo Creek Observatory, Texas. Measurements made at or near solar noon when the Sun is not obstructed by clouds. Peaks indicate smoke, dust and smog. Saharan dust events are measured each summer.

In physics, optical depth or optical thickness is the natural logarithm of the ratio of incident to transmitted radiant power through a material. Thus, the larger the optical depth, the smaller the amount of transmitted radiant power through the material. Spectral optical depth or spectral optical thickness is the natural logarithm of the ratio of incident to transmitted spectral radiant power through a material. Optical depth is dimensionless, and in particular is not a length, though it is a monotonically increasing function of optical path length, and approaches zero as the path length approaches zero. The use of the term "optical density" for optical depth is discouraged.

In chemistry, a closely related quantity called "absorbance" or "decadic absorbance" is used instead of optical depth: the common logarithm of the ratio of incident to transmitted radiant power through a material. It is the optical depth divided by log_e(10), because of the different logarithm bases used.

Mathematical definitions

Optical depth

Optical depth of a material, denoted $\tau$, is given by:$${\displaystyle \tau =\ln \left(\frac{{\mathrm{\Phi }}_{\mathrm{e}}^{\mathrm{i}}}{{\mathrm{\Phi }}_{\mathrm{e}}^{\mathrm{t}}}\right)=-\ln T}$$where

• ${\mathrm{\Phi }}_{\mathrm{e}}^{\mathrm{i}}$ is the radiant flux received by that material;
• ${\mathrm{\Phi }}_{\mathrm{e}}^{\mathrm{t}}$ is the radiant flux transmitted by that material;
• $T$ is the transmittance of that material.

The absorbance $A$ is related to optical depth by:$${\displaystyle \tau =A\ln 10}$$

Spectral optical depth

Spectral optical depth in frequency and spectral optical depth in wavelength of a material, denoted ${\displaystyle {\tau }_{\nu }}$ and ${\displaystyle {\tau }_{\lambda }}$ respectively, are given by: $${\displaystyle {\tau }_{\nu }=\ln \left(\frac{{\mathrm{\Phi }}_{\mathrm{e},\nu }^{\mathrm{i}}}{{\mathrm{\Phi }}_{\mathrm{e},\nu }^{\mathrm{t}}}\right)=-\ln T_{\nu }}$$$${\displaystyle {\tau }_{\lambda }=\ln \left(\frac{{\mathrm{\Phi }}_{\mathrm{e},\lambda }^{\mathrm{i}}}{{\mathrm{\Phi }}_{\mathrm{e},\lambda }^{\mathrm{t}}}\right)=-\ln T_{\lambda },}$$ where

• ${\displaystyle {\mathrm{\Phi }}_{\mathrm{e},\nu }^{\mathrm{t}}}$ is the spectral radiant flux in frequency transmitted by that material;
• ${\displaystyle {\mathrm{\Phi }}_{\mathrm{e},\nu }^{\mathrm{i}}}$ is the spectral radiant flux in frequency received by that material;
• ${\displaystyle T_{\nu }}$ is the spectral transmittance in frequency of that material;
• ${\displaystyle {\mathrm{\Phi }}_{\mathrm{e},\lambda }^{\mathrm{t}}}$ is the spectral radiant flux in wavelength transmitted by that material;
• ${\displaystyle {\mathrm{\Phi }}_{\mathrm{e},\lambda }^{\mathrm{i}}}$ is the spectral radiant flux in wavelength received by that material;
• ${\displaystyle T_{\lambda }}$ is the spectral transmittance in wavelength of that material.

Spectral absorbance is related to spectral optical depth by: $${\displaystyle {\tau }_{\nu }=A_{\nu }\ln 10,}$$$${\displaystyle {\tau }_{\lambda }=A_{\lambda }\ln 10,}$$ where

• ${\displaystyle A_{\nu }}$ is the spectral absorbance in frequency;
• ${\displaystyle A_{\lambda }}$ is the spectral absorbance in wavelength.

Relationship with attenuation

Attenuation

Main article: Attenuation

Optical depth measures the attenuation of the transmitted radiant power in a material. Attenuation can be caused by absorption, but also reflection, scattering, and other physical processes. Optical depth of a material is approximately equal to its attenuation when both the absorbance is much less than 1 and the emittance of that material (not to be confused with radiant exitance or emissivity) is much less than the optical depth: $${\displaystyle {\mathrm{\Phi }}_{\mathrm{e}}^{\mathrm{t}}+{\mathrm{\Phi }}_{\mathrm{e}}^{\mathrm{a}\mathrm{t}\mathrm{t}}={\mathrm{\Phi }}_{\mathrm{e}}^{\mathrm{i}}+{\mathrm{\Phi }}_{\mathrm{e}}^{\mathrm{e}},}$$$${\displaystyle T+ATT=1+E,}$$ where

• Φ_e^t is the radiant power transmitted by that material;
• Φ_e^att is the radiant power attenuated by that material;
• Φ_eⁱ is the radiant power received by that material;
• Φ_e^e is the radiant power emitted by that material;
• T = Φ_e^t/Φ_eⁱ is the transmittance of that material;
• ATT = Φ_e^att/Φ_eⁱ is the attenuation of that material;
• E = Φ_e^e/Φ_eⁱ is the emittance of that material,

and according to the Beer–Lambert law, $${\displaystyle T=e^{-\tau },}$$so:$${\displaystyle ATT=1-e^{-\tau }+E\approx \tau +E\approx \tau ,\text{if}\tau \ll 1\text{and}E\ll \tau .}$$

Attenuation coefficient

Optical depth of a material is also related to its attenuation coefficient by:$${\displaystyle \tau ={\int }_0^l\alpha (z)\mathrm{d}z,}$$where

• l is the thickness of that material through which the light travels;
• α(z) is the attenuation coefficient or Napierian attenuation coefficient of that material at z,

and if α(z) is uniform along the path, the attenuation is said to be a linear attenuation and the relation becomes: $${\displaystyle \tau =\alpha l}$$

Sometimes the relation is given using the attenuation cross section of the material, that is its attenuation coefficient divided by its number density:$${\displaystyle \tau ={\int }_0^l\sigma n(z)\mathrm{d}z,}$$ where

• σ is the attenuation cross section of that material;
• n(z) is the number density of that material at z,

and if ${\displaystyle n}$ is uniform along the path, i.e., ${\displaystyle n(z)\equiv N}$, the relation becomes:$${\displaystyle \tau =\sigma Nl}$$

Applications

Atomic physics

In atomic physics, the spectral optical depth of a cloud of atoms can be calculated from the quantum-mechanical properties of the atoms. It is given by$${\displaystyle {\tau }_{\nu }=\frac{d^{2}n\nu }{2\mathrm{c}\hslash {\epsilon }_0\sigma \gamma }}$$where

• d is the transition dipole moment;
• n is the number of atoms;
• ν is the frequency of the beam;
• c is the speed of light;
• ħ is the reduced Planck constant;
• ε₀ is the vacuum permittivity;
• σ is the cross section of the beam;
• γ is the natural linewidth of the transition.

Atmospheric sciences

See also: Beer–Lambert law

In atmospheric sciences, one often refers to the optical depth of the atmosphere as corresponding to the vertical path from Earth's surface to outer space; at other times the optical path is from the observer's altitude to outer space. The optical depth for a slant path is τ = mτ′, where τ′ refers to a vertical path, m is called the relative airmass, and for a plane-parallel atmosphere it is determined as m = sec θ where θ is the zenith angle corresponding to the given path. Therefore,$${\displaystyle T=e^{-\tau }=e^{-m{\tau }^{\prime }}}$$The optical depth of the atmosphere can be divided into several components, ascribed to Rayleigh scattering, aerosols, and gaseous absorption. The optical depth of the atmosphere can be measured with a Sun photometer.

The optical depth with respect to the height within the atmosphere is given by $${\displaystyle \tau (z)=k_{\text{a}}{w}_1{\rho }_{0}H{e}^{-z/H}}$$ and it follows that the total atmospheric optical depth is given by $${\displaystyle \tau (0)=k_{\text{a}}{w}_1{\rho }_{0}H}$$

In both equations:

• k_a is the absorption coefficient
• w₁ is the mixing ratio
• ρ₀ is the density of air at sea level
• H is the scale height of the atmosphere
• z is the height in question

The optical depth of a plane parallel cloud layer is given by $${\displaystyle \tau =Q_{\text{e}}{\left[\frac{9\pi L^{2}HN}{16{\rho }_l^2}\right]}^{1/3}}$$where:

• Q_e is the extinction efficiency
• L is the liquid water path
• H is the geometrical thickness
• N is the concentration of droplets
• ρ_l is the density of liquid water

So, with a fixed depth and total liquid water path, $\tau \propto N^{1/3}$.

Astronomy

Main article: Optical depth (astrophysics)

In astronomy, the photosphere of a star is defined as the surface where its optical depth is 2/3. This means that each photon emitted at the photosphere suffers an average of less than one scattering before it reaches the observer. At the temperature at optical depth 2/3, the energy emitted by the star (the original derivation is for the Sun) matches the observed total energy emitted.

Note that the optical depth of a given medium will be different for different colors (wavelengths) of light.

For planetary rings, the optical depth is the (negative logarithm of the) proportion of light blocked by the ring when it lies between the source and the observer. This is usually obtained by observation of stellar occultations.

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.