Group selection

From Wikipedia, the free encyclopedia

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

 

Early explanations of social behaviour, such as the lekking of blackcock, spoke of "the good of the species". Blackcocks at the Lek watercolour and bodycolour by Archibald Thorburn, 1901.

Group selection is a proposed mechanism of evolution in which natural selection acts at the level of the group, instead of at the more conventional level of the individual.

Early authors such as V. C. Wynne-Edwards and Konrad Lorenz argued that the behavior of animals could affect their survival and reproduction as groups, speaking for instance of actions for the good of the species. In the 1930s, R.A. Fisher and J.B.S. Haldane proposed the concept of kin selection, a form of altruism from the gene-centered view of evolution, arguing that animals should sacrifice for their relatives, and thereby implying that they should not sacrifice for non-relatives. From the mid 1960s, evolutionary biologists such as John Maynard Smith, W. D. Hamilton, George C. Williams, and Richard Dawkins argued that natural selection acted primarily at the level of the individual. They argued on the basis of mathematical models that individuals would not altruistically sacrifice fitness for the sake of a group. They persuaded the majority of biologists that group selection did not occur, other than in special situations such as the haplodiploid social insects like honeybees (in the Hymenoptera), where kin selection was possible.

In 1994 David Sloan Wilson and Elliott Sober argued for multi-level selection, including group selection, on the grounds that groups, like individuals, could compete. In 2010 three authors including E. O. Wilson, known for his work on social insects especially ants, again revisited the arguments for group selection. They argued that group selection can occur when competition between two or more groups, some containing altruistic individuals who act cooperatively together, is more important for survival than competition between individuals within each group, provoking a strong rebuttal from a large group of evolutionary biologists and behavior analysts.

Early developments

Further information: natural selection and kin selection

Charles Darwin developed the theory of evolution in his book, Origin of Species. Darwin also made the first suggestion of group selection in The Descent of Man that the evolution of groups could affect the survival of individuals. He wrote, "If one man in a tribe... invented a new snare or weapon, the tribe would increase in number, spread, and supplant other tribes. In a tribe thus rendered more numerous there would always be a rather better chance of the birth of other superior and inventive members."

Once Darwinism had been accepted in the modern synthesis of the mid-twentieth century, animal behavior was glibly explained with unsubstantiated hypotheses about survival value, which was largely taken for granted. The naturalist Konrad Lorenz had argued loosely in books like On Aggression (1966) that animal behavior patterns were "for the good of the species", without actually studying survival value in the field. Richard Dawkins noted that Lorenz was a "'good of the species' man" so accustomed to group selection thinking that he did not realize his views "contravened orthodox Darwinian theory". The ethologist Niko Tinbergen praised Lorenz for his interest in the survival value of behavior, and naturalists enjoyed Lorenz's writings for the same reason. In 1962, group selection was used as a popular explanation for adaptation by the zoologist V. C. Wynne-Edwards. In 1976, Richard Dawkins wrote a well-known book on the importance of evolution at the level of the gene or the individual, The Selfish Gene.

Social behavior in honeybees is explained by kin selection: their haplodiploid inheritance system makes workers very closely related to their queen (centre).

From the mid 1960s, evolutionary biologists argued that natural selection acted primarily at the level of the individual. In 1964, John Maynard Smith, C. M. Perrins (1964), and George C. Williams in his 1966 book Adaptation and Natural Selection cast serious doubt on group selection as a major mechanism of evolution; Williams's 1971 book Group Selection assembled writings from many authors on the same theme.

It was at that time generally agreed that the primary exception of social group selection was in the social insects, and the explanation was limited to the unique inheritance system (involving haplodiploidy) of the eusocial Hymenoptera such as honeybees, which encourages kin selection, since workers are closely related.

Kin selection and inclusive fitness theory

Further information: Kin selection and inclusive fitness

Experiments from the late 1970s suggested that selection involving groups was possible. Early group selection models assumed that genes acted independently, for example a gene that coded for cooperation or altruism. Genetically-based reproduction of individuals implies that, in group formation, the altruistic genes would need a way to act for the benefit of members in the group to enhance the fitness of many individuals with the same gene. But it is expected from this model that individuals of the same species would compete against each other for the same resources. This would put cooperating individuals at a disadvantage, making genes for cooperation likely to be eliminated. Group selection on the level of the species is flawed because it is difficult to see how selective pressures would be applied to competing/non-cooperating individuals.

Kin selection between related individuals is accepted as an explanation of altruistic behavior. R.A. Fisher in 1930 and J.B.S. Haldane in 1932 set out the mathematics of kin selection, with Haldane famously joking that he would willingly die for two brothers or eight cousins. In this model, genetically related individuals cooperate because survival advantages to one individual also benefit kin who share some fraction of the same genes, giving a mechanism for favoring genetic selection.

Inclusive fitness theory, first proposed by W. D. Hamilton in the early 1960s, gives a selection criterion for evolution of social traits when social behavior is costly to an individual organism's survival and reproduction. The criterion is that the reproductive benefit to relatives who carry the social trait, multiplied by their relatedness (the probability that they share the altruistic trait) exceeds the cost to the individual. Inclusive fitness theory is a general treatment of the statistical probabilities of social traits accruing to any other organisms likely to propagate a copy of the same social trait. Kin selection theory treats the narrower but simpler case of the benefits to close genetic relatives (or what biologists call 'kin') who may also carry and propagate the trait. A significant group of biologists support inclusive fitness as the explanation for social behavior in a wide range of species, as supported by experimental data. An article was published in Nature with over a hundred coauthors.

One of the questions about kin selection is the requirement that individuals must know if other individuals are related to them, or kin recognition. Any altruistic act has to preserve similar genes. One argument given by Hamilton is that many individuals operate in "viscous" conditions, so that they live in physical proximity to relatives. Under these conditions, they can act altruistically to any other individual, and it is likely that the other individual will be related. This population structure builds a continuum between individual selection, kin selection, kin group selection and group selection without a clear boundary for each level. However, early theoretical models by D.S. Wilson et al. and Taylor showed that pure population viscosity cannot lead to cooperation and altruism. This is because any benefit generated by kin cooperation is exactly cancelled out by kin competition; additional offspring from cooperation are eliminated by local competition. Mitteldorf and D. S. Wilson later showed that if the population is allowed to fluctuate, then local populations can temporarily store the benefit of local cooperation and promote the evolution of cooperation and altruism. By assuming individual differences in adaptations, Yang further showed that the benefit of local altruism can be stored in the form of offspring quality and thus promote the evolution of altruism even if the population does not fluctuate. This is because local competition among more individuals resulting from local altruism increases the average local fitness of the individuals that survive.

Another explanation for the recognition of genes for altruism is that a single trait, group reciprocal kindness, is capable of explaining the vast majority of altruism that is generally accepted as "good" by modern societies. The phenotype of altruism relies on recognition of the altruistic behavior by itself. The trait of kindness will be recognized by sufficiently intelligent and undeceived organisms in other individuals with the same trait. Moreover, the existence of such a trait predicts a tendency for kindness to unrelated organisms that are apparently kind, even if the organisms are of another species. The gene need not be exactly the same, so long as the effect or phenotype is similar. Multiple versions of the gene—or even meme—would have virtually the same effect. This explanation was given by Richard Dawkins as an analogy of a man with a green beard. Green-bearded men are imagined as tending to cooperate with each other simply by seeing a green beard, where the green beard trait is incidentally linked to the reciprocal kindness trait.

Multilevel selection theory

Further information: Unit of selection

Kin selection or inclusive fitness is accepted as an explanation for cooperative behavior in many species, but there are some species, including some human behavior, that are difficult to explain with only this approach. In particular, it does not seem to explain the rapid rise of human civilization. David Sloan Wilson has argued that other factors must also be considered in evolution. Wilson and others have continued to develop group selection models.

Early group selection models were flawed because they assumed that genes acted independently; but genetically-based interactions among individuals are ubiquitous in group formation because genes must cooperate for the benefit of association in groups to enhance the fitness of group members. Additionally, group selection on the level of the species is flawed because it is difficult to see how selective pressures would be applied; selection in social species of groups against other groups, rather than the species entire, seems to be the level at which selective pressures are plausible. On the other hand, kin selection is accepted as an explanation of altruistic behavior. Some biologists argue that kin selection and multilevel selection are both needed to "obtain a complete understanding of the evolution of a social behavior system".

In 1994 David Sloan Wilson and Elliott Sober argued that the case against group selection had been overstated. They considered whether groups can have functional organization in the same way as individuals, and consequently whether groups can be "vehicles" for selection. They do not posit evolution on the level of the species, but selective pressures that winnow out small groups within a species, e.g. groups of social insects or primates. Groups that cooperate better might survive and reproduce more than those that did not. Resurrected in this way, Wilson & Sober's new group selection is called multilevel selection theory.

In 2010, Martin Nowak, C. E. Tarnita and E. O. Wilson argued for multi-level selection, including group selection, to correct what they saw as deficits in the explanatory power of inclusive fitness. A response from 137 other evolutionary biologists argued "that their arguments are based upon a misunderstanding of evolutionary theory and a misrepresentation of the empirical literature".

David Sloan Wilson and Elliott Sober's 1994 Multilevel Selection Model, illustrated by a nested set of Russian matryoshka dolls. Wilson himself compared his model to such a set.

Wilson compared the layers of competition and evolution to nested sets of Russian matryoshka dolls. The lowest level is the genes, next come the cells, then the organism level and finally the groups. The different levels function cohesively to maximize fitness, or reproductive success. The theory asserts that selection for the group level, involving competition between groups, must outweigh the individual level, involving individuals competing within a group, for a group-benefiting trait to spread.

Multilevel selection theory focuses on the phenotype because it looks at the levels that selection directly acts upon. For humans, social norms can be argued to reduce individual level variation and competition, thus shifting selection to the group level. The assumption is that variation between different groups is larger than variation within groups. Competition and selection can operate at all levels regardless of scale. Wilson wrote, "At all scales, there must be mechanisms that coordinate the right kinds of action and prevent disruptive forms of self-serving behavior at lower levels of social organization." E. O. Wilson summarized, "In a group, selfish individuals beat altruistic individuals. But, groups of altruistic individuals beat groups of selfish individuals."

Wilson ties the multilevel selection theory regarding humans to another theory, gene-culture coevolution, by acknowledging that culture seems to characterize a group-level mechanism for human groups to adapt to environmental changes.

MLS theory can be used to evaluate the balance between group selection and individual selection in specific cases. An experiment by William Muir compared egg productivity in hens, showing that a hyper-aggressive strain had been produced through individual selection, leading to many fatal attacks after only six generations; by implication, it could be argued that group selection must have been acting to prevent this in real life. Group selection has most often been postulated in humans and, notably, eusocial Hymenoptera that make cooperation a driving force of their adaptations over time and have a unique system of inheritance involving haplodiploidy that allows the colony to function as an individual while only the queen reproduces.

Wilson and Sober's work revived interest in multilevel selection. In a 2005 article, E. O. Wilson argued that kin selection could no longer be thought of as underlying the evolution of extreme sociality, for two reasons. First, he suggested, the argument that haplodiploid inheritance (as in the Hymenoptera) creates a strong selection pressure towards nonreproductive castes is mathematically flawed. Second, eusociality no longer seems to be confined to the hymenopterans; increasing numbers of highly social taxa have been found in the years since Wilson's foundational text Sociobiology: A New Synthesis was published in 1975. These including a variety of insect species, as well as two rodent species (the naked mole-rat and the Damaraland mole rat). Wilson suggests the equation for Hamilton's rule:

rb > c

(where b represents the benefit to the recipient of altruism, c the cost to the altruist, and r their degree of relatedness) should be replaced by the more general equation

rb_k + b_e > c

in which b_k is the benefit to kin (b in the original equation) and b_e is the benefit accruing to the group as a whole. He then argues that, in the present state of the evidence in relation to social insects, it appears that b_e>rb_k, so that altruism needs to be explained in terms of selection at the colony level rather than at the kin level. However, kin selection and group selection are not distinct processes, and the effects of multi-level selection are already accounted for in Hamilton's rule, rb>c, provided that an expanded definition of r, not requiring Hamilton's original assumption of direct genealogical relatedness, is used, as proposed by E. O. Wilson himself.

Spatial populations of predators and prey show restraint of reproduction at equilibrium, both individually and through social communication, as originally proposed by Wynne-Edwards. While these spatial populations do not have well-defined groups for group selection, the local spatial interactions of organisms in transient groups are sufficient to lead to a kind of multi-level selection. There is however as yet no evidence that these processes operate in the situations where Wynne-Edwards posited them.

Rauch et al.'s analysis of host-parasite evolution is broadly hostile to group selection. Specifically, the parasites do not individually moderate their transmission; rather, more transmissible variants – which have a short-term but unsustainable advantage – arise, increase, and go extinct.

Applications

Differing evolutionarily stable strategies

Further information: Evolutionarily stable strategy

The problem with group selection is that for a whole group to get a single trait, it must spread through the whole group first by regular evolution. But, as J. L. Mackie suggested, when there are many different groups, each with a different evolutionarily stable strategy, there is selection between the different strategies, since some are worse than others. For example, a group where altruism was universal would indeed outcompete a group where every creature acted in its own interest, so group selection might seem feasible; but a mixed group of altruists and non-altruists would be vulnerable to cheating by non-altruists within the group, so group selection would collapse.

Implications in population biology

Social behaviors such as altruism and group relationships can impact many aspects of population dynamics, such as intraspecific competition and interspecific interactions. In 1871, Darwin argued that group selection occurs when the benefits of cooperation or altruism between subpopulations are greater than the individual benefits of egotism within a subpopulation. This supports the idea of multilevel selection, but kinship also plays an integral role because many subpopulations are composed of closely related individuals. An example of this can be found in lions, which are simultaneously cooperative and territorial. Within a pride, males protect the pride from outside males, and females, who are commonly sisters, communally raise cubs and hunt. However, this cooperation seems to be density dependent. When resources are limited, group selection favors prides that work together to hunt. When prey is abundant, cooperation is no longer beneficial enough to outweigh the disadvantages of altruism, and hunting is no longer cooperative.

Interactions between different species can also be affected by multilevel selection. Predator-prey relationships can also be affected. Individuals of certain monkey species howl to warn the group of the approach of a predator. The evolution of this trait benefits the group by providing protection, but could be disadvantageous to the individual if the howling draws the predator's attention to them. By affecting these interspecific interactions, multilevel and kinship selection can change the population dynamics of an ecosystem.

Multilevel selection attempts to explain the evolution of altruistic behavior in terms of quantitative genetics. Increased frequency or fixation of altruistic alleles can be accomplished through kin selection, in which individuals engage in altruistic behavior to promote the fitness of genetically similar individuals such as siblings. However, this can lead to inbreeding depression, which typically lowers the overall fitness of a population. However, if altruism were to be selected for through an emphasis on benefit to the group as opposed to relatedness and benefit to kin, both the altruistic trait and genetic diversity could be preserved. However, relatedness should still remain a key consideration in studies of multilevel selection. Experimentally imposed multilevel selection on Japanese quail was more effective by an order of magnitude on closely related kin groups than on randomized groups of individuals.

Gene-culture coevolution in humans

Main articles: Dual inheritance theory and cultural group selection

 

Humanity has developed extremely rapidly, arguably through gene-culture coevolution, leading to complex cultural artefacts like the gopuram of the Sri Mariammam temple, Singapore.

Gene-culture coevolution (also called dual inheritance theory) is a modern hypothesis (applicable mostly to humans) that combines evolutionary biology and modern sociobiology to indicate group selection. It treats culture as a separate evolutionary system that acts in parallel to the usual genetic evolution to transform human traits. It is believed that this approach of combining genetic influence with cultural influence over several generations is not present in the other hypotheses such as reciprocal altruism and kin selection, making gene-culture evolution one of the strongest realistic hypotheses for group selection. Fehr provides evidence of group selection taking place in humans presently with experimentation through logic games such as prisoner's dilemma, the type of thinking that humans have developed many generations ago.

Gene-culture coevolution allows humans to develop highly distinct adaptations to the local pressures and environments more quickly than with genetic evolution alone. Robert Boyd and Peter J. Richerson, two strong proponents of cultural evolution, postulate that the act of social learning, or learning in a group as done in group selection, allows human populations to accrue information over many generations. This leads to cultural evolution of behaviors and technology alongside genetic evolution. Boyd and Richerson believe that the ability to collaborate evolved during the Middle Pleistocene, a million years ago, in response to a rapidly changing climate.

In 2003, the behavioral scientist Herbert Gintis examined cultural evolution statistically, offering evidence that societies that promote pro-social norms have higher survival rates than societies that do not. Gintis wrote that genetic and cultural evolution can work together. Genes transfer information in DNA, and cultures transfer information encoded in brains, artifacts, or documents. Language, tools, lethal weapons, fire, cooking, etc., have a long-term effect on genetics. For example, cooking led to a reduction of size of the human gut, since less digestion is needed for cooked food. Language led to a change in the human larynx and an increase in brain size. Projectile weapons led to changes in human hands and shoulders, such that humans are much better at throwing objects than the closest human relative, the chimpanzee.

Behaviour

In 2019, Howard Rachlin and colleagues proposed group selection of behavioural patterns, such as learned altruism, during ontogeny parallel to group selection during phylogeny.

Criticism

The use of the Price equation to support group selection was challenged by van Veelen in 2012, arguing that it is based on invalid mathematical assumptions.

Richard Dawkins and other advocates of the gene-centered view of evolution remain unconvinced about group selection. In particular, Dawkins suggests that group selection fails to make an appropriate distinction between replicators and vehicles.

The psychologist Steven Pinker concluded that "group selection has no useful role to play in psychology or social science", since it "is not a precise implementation of the theory of natural selection, as it is, say, in genetic algorithms or artificial life simulations. Instead it is a loose metaphor, more like the struggle among kinds of tires or telephones."

The evolutionary biologist Jerry Coyne summarized the arguments in The New York Review of Books in non-technical terms as follows:

Group selection isn't widely accepted by evolutionists for several reasons. First, it's not an efficient way to select for traits, like altruistic behavior, that are supposed to be detrimental to the individual but good for the group. Groups divide to form other groups much less often than organisms reproduce to form other organisms, so group selection for altruism would be unlikely to override the tendency of each group to quickly lose its altruists through natural selection favoring cheaters. Further, little evidence exists that selection on groups has promoted the evolution of any trait. Finally, other, more plausible evolutionary forces, like direct selection on individuals for reciprocal support, could have made humans prosocial. These reasons explain why only a few biologists, like [David Sloan] Wilson and E. O. Wilson (no relation), advocate group selection as the evolutionary source of cooperation.

 

Eusociality

From Wikipedia, the free encyclopedia

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

 

Co-operative brood rearing, seen here in honeybees, is a condition of eusociality.

Eusociality (from Greek εὖ eu "good" and social), the highest level of organization of sociality, is defined by the following characteristics: cooperative brood care (including care of offspring from other individuals), overlapping generations within a colony of adults, and a division of labor into reproductive and non-reproductive groups. The division of labor creates specialized behavioral groups within an animal society which are sometimes referred to as 'castes'. Eusociality is distinguished from all other social systems because individuals of at least one caste usually lose the ability to perform at least one behavior characteristic of individuals in another caste.

Eusociality exists in certain insects, crustaceans and mammals. It is mostly observed and studied in the Hymenoptera (ants, bees, and wasps) and in Isoptera (termites). A colony has caste differences: queens and reproductive males take the roles of the sole reproducers, while soldiers and workers work together to create a living situation favorable for the brood. In addition to Hymenoptera and Isoptera, there are two known eusocial vertebrates among rodents: the naked mole-rat and the Damaraland mole-rat. Some shrimp, such as Synalpheus regalis, are also eusocial. E. O. Wilson and others have claimed that humans have evolved a weak form of eusociality, but these arguments have been disputed.

History

The term "eusocial" was introduced in 1966 by Suzanne Batra, who used it to describe nesting behavior in Halictine bees. Batra observed the cooperative behavior of the bees, males and females alike, as they took responsibility for at least one duty (i.e., burrowing, cell construction, oviposition) within the colony. The cooperativeness was essential as the activity of one labor division greatly influenced the activity of another.

For example, the size of pollen balls, a source of food, depended on when the egg-laying females oviposited. If the provisioning by pollen collectors was incomplete by the time the egg-laying female occupied a cell and oviposited, the size of the pollen balls would be small, leading to small offspring. Batra applied this term to species in which a colony is started by a single individual. Batra described other species, wherein the founder is accompanied by numerous helpers—as in a swarm of bees or ants—as "hypersocial".

In 1969, Charles D. Michener further expanded Batra's classification with his comparative study of social behavior in bees. He observed multiple species of bees (Apoidea) in order to investigate the different levels of animal sociality, all of which are different stages that a colony may pass through. Eusociality, which is the highest level of animal sociality a species can attain, specifically had three characteristics that distinguished it from the other levels:

1. "Egg-layers and worker-like individuals among adult females" (division of labor)
2. The overlap of generations (mother and adult offspring)
3. Cooperative work on the cells of the bees' honeycomb

Weaver ants, here collaborating to pull nest leaves together, can be considered eusocial, as they have a permanent division of labor.

E. O. Wilson then extended the terminology to include other social insects, such as ants, wasps, and termites. Originally, it was defined to include organisms (only invertebrates) that had the following three features:

1. Reproductive division of labor (with or without sterile castes)
2. Overlapping generations
3. Cooperative care of young

As eusociality became a recognized widespread phenomenon, however, it was also discovered in a group of chordates, the mole-rats. Further research also distinguished another possibly important criterion for eusociality known as "the point of no return". This is characterized by eusocial individuals that become fixed into one behavioral group, which usually occurs before reproductive maturity. This prevents them from transitioning between behavioral groups and creates an animal society that is truly dependent on each other for survival and reproductive success. For many insects, this irreversibility has changed the anatomy of the worker caste, which is sterile and provides support for the reproductive caste.

Taxonomic range

Most eusocial societies exist in arthropods, while a few are found in mammals. Ferns may exhibit eusocial behavior amongst clones.

In insects

A swarming meat-eater ant colony

 

See also: Sexual selection in social insects and Identity in social insects

The order Hymenoptera contains the largest group of eusocial insects, including ants, bees, and wasps—those with reproductive "queens" and more or less sterile "workers" and/or "soldiers" that perform specialized tasks. For example, in the well-studied social wasp Polistes versicolor, dominant females perform tasks such as building new cells and ovipositing, while subordinate females tend to perform tasks like feeding the larvae and foraging. The task differentiation between castes can be seen in the fact that subordinates complete 81.4% of the total foraging activity, while dominants only complete 18.6% of the total foraging. Eusocial species with a sterile caste are sometimes called hypersocial.

While only a moderate percentage of species in bees (families Apidae and Halictidae) and wasps (Crabronidae and Vespidae) are eusocial, nearly all species of ants (Formicidae) are eusocial. Some major lineages of wasps are mostly or entirely eusocial, including the subfamilies Polistinae and Vespinae. The corbiculate bees (subfamily Apinae of family Apidae) contain four tribes of varying degrees of sociality: the highly eusocial Apini (honey bees) and Meliponini (stingless bees), primitively eusocial Bombini (bumble bees), and the mostly solitary or weakly social Euglossini (orchid bees). Eusociality in these families is sometimes managed by a set of pheromones that alter the behavior of specific castes in the colony. These pheromones may act across different species, as observed in Apis andreniformis (black dwarf honey bee), where worker bees responded to queen pheromone from the related Apis florea (red dwarf honey bee). Pheromones are sometimes used in these castes to assist with foraging. Workers of the Australian stingless bee Tetragonula carbonaria, for instance, mark food sources with a pheromone, helping their nest mates to find the food.

Reproductive specialization generally involves the production of sterile members of the species, which carry out specialized tasks to care for the reproductive members. It can manifest in the appearance of individuals within a group whose behavior or morphology is modified for group defense, including self-sacrificing behavior ("altruism"). An example of a species whose sterile caste displays this altruistic behavior is Myrmecocystus mexicanus, one of the species of honey ant. Select sterile workers fill their abdomens with liquid food until they become immobile and hang from the ceilings of the underground nests, acting as food storage for the rest of the colony. Not all social species of insects have distinct morphological differences between castes. For example, in the Neotropical social wasp Synoeca surinama, social displays determine the caste ranks of individuals in the developing brood. These castes are sometimes further specialized in their behavior based on age. For example, Scaptotrigona postica workers assume different roles in the nest based on their age. Between approximately 0–40 days old, the workers perform tasks within the nest such as provisioning cell broods, colony cleaning, and nectar reception and dehydration. Once older than 40 days, Scaptotrigona postica workers move outside of the nest to practice colony defense and foraging.

In Lasioglossum aeneiventre, a halictid bee from Central America, nests may be headed by more than one female; such nests have more cells, and the number of active cells per female is correlated with the number of females in the nest, implying that having more females leads to more efficient building and provisioning of cells. In similar species with only one queen, such as Lasioglossum malachurum in Europe, the degree of eusociality depends on the clime in which the species is found.

Termites (order Blattodea, infraorder Isoptera) make up another large portion of highly advanced eusocial animals. The colony is differentiated into various castes: the queen and king are the sole reproducing individuals; workers forage and maintain food and resources; and soldiers defend the colony against ant attacks. The latter two castes, which are sterile and perform highly specialized, complex social behaviors, are derived from different stages of pluripotent larvae produced by the reproductive caste. Some soldiers have jaws so enlarged (specialized for defense and attack) that they are unable to feed themselves and must be fed by workers.

Austroplatypus incompertus is a species of ambrosia beetle native to Australia, and is the first beetle (order Coleoptera) to be recognized as eusocial. This species forms colonies in which a single female is fertilized, and is protected by many unfertilized females, which also serve as workers excavating tunnels in trees. This species also participates in cooperative brood care, in which individuals care for juveniles that are not their own.

Some species of gall-inducing insects, including the gall-forming aphid, Pemphigus spyrothecae (order Hemiptera), and thrips such as Kladothrips (order Thysanoptera), are also described as eusocial. These species have very high relatedness among individuals due to their partially asexual mode of reproduction (sterile soldier castes being clones of the reproducing female), but the gall-inhabiting behavior gives these species a defensible resource that sets them apart from related species with similar genetics. They produce soldier castes capable of fortress defense and protection of their colony against both predators and competitors. In these groups, therefore, high relatedness alone does not lead to the evolution of social behavior, but requires that groups occur in a restricted, shared area. These species have morphologically distinct soldier castes that defend against kleptoparasites (parasitism by theft) and are able to reproduce parthenogenetically (without fertilization).

In crustaceans

Eusociality has also arisen in three different lineages among some crustaceans that live in separate colonies. Synalpheus regalis, Synalpheus filidigitus, Synalpheus elizabethae, Synalpheus chacei, Synalpheus riosi, Synalpheus duffyi, Synalpheus microneptunus, and Synalpheus cayoneptunus are the eight recorded species of parasitic shrimp that rely on fortress defense and live in groups of closely related individuals in tropical reefs and sponges, living eusocially with a typically a single breeding female and a large number of male defenders, armed with enlarged snapping claws. As with other eusocial societies, there is a single shared living space for the colony members, and the non-breeding members act to defend it.

The fortress defense hypothesis additionally points out that because sponges provide both food and shelter, there is an aggregation of relatives (because the shrimp do not have to disperse to find food), and much competition for those nesting sites. Being the target of attack promotes a good defense system (soldier caste); soldiers therefore promote the fitness of the whole nest by ensuring safety and reproduction of the queen.

Eusociality offers a competitive advantage in shrimp populations. Eusocial species were found to be more abundant, occupy more of the habitat, and use more of the available resources than non-eusocial species. Other studies add to these findings by pointing out that cohabitation was more rare than expected by chance, and that most sponges were dominated by one species, which was frequently eusocial.

In nonhuman mammals

Naked mole-rat, one of two eusocial species in the Bathyergidae

Among mammals, eusociality is known in two species in the Bathyergidae, the naked mole-rat (Heterocephalus glaber) and the Damaraland mole-rat (Fukomys damarensis), both of which are highly inbred. Usually living in harsh or limiting environments, these mole-rats aid in raising siblings and relatives born to a single reproductive queen. However, this classification is controversial owing to disputed definitions of 'eusociality'. To avoid inbreeding, mole rats sometimes outbreed and establish new colonies when resources are sufficient. Most of the individuals cooperatively care for the brood of a single reproductive female (the queen) to which they are most likely related. Thus, it is uncertain whether mole rats classify as true eusocial organisms, since their social behavior depends largely on their resources and environment.

Some mammals in the Carnivora and Primates exhibit eusocial tendencies, especially meerkats (Suricata suricatta) and dwarf mongooses (Helogale parvula). These show cooperative breeding and marked reproductive skews. In the dwarf mongoose, the breeding pair receives food priority and protection from subordinates and rarely has to defend against predators.

In humans

Further information: Group selection

An early 21st century debate focused on whether humans are prosocial or eusocial. Edward O. Wilson called humans eusocial apes, arguing for similarities to ants, and observing that early hominins cooperated to rear their children while other members of the same group hunted and foraged. Wilson argued that through cooperation and teamwork, ants and humans form superorganisms. Wilson's claims were vigorously rejected, not only because they were based on group selection, but also because human reproductive labor is not divided between castes. However, it has been claimed that suicide, male homosexuality, and female menopause evolved through kin selection, which, if true, would by some definitions make humans eusocial.

Evolution

Main article: Evolution of eusociality

Phylogenetic distribution

Eusociality is a rare but widespread phenomenon in species in at least seven orders in the animal kingdom, as shown in the phylogenetic tree (non-eusocial groups not shown). All species of termites are eusocial, and it is believed that they were the first eusocial animals to evolve, sometime in the upper Jurassic period (~150 million years ago). The other orders shown also contain non-eusocial species, including many lineages where eusociality was inferred to be the ancestral state. Thus the number of independent evolutions of eusociality is still under investigation. The major eusocial groups are shown in boldface in the phylogenetic tree.

Paradox

Prior to the gene-centered view of evolution, eusociality was seen as an apparent evolutionary paradox: if adaptive evolution unfolds by differential reproduction of individual organisms, how can individuals incapable of passing on their genes evolve and persist? In On the Origin of Species, Darwin referred to the existence of sterile castes as the "one special difficulty, which at first appeared to me insuperable, and actually fatal to my theory". Darwin anticipated that a possible resolution to the paradox might lie in the close family relationship, which W.D. Hamilton quantified a century later with his 1964 inclusive fitness theory. After the gene-centered view of evolution was developed in the mid 1970s, non-reproductive individuals were seen as an extended phenotype of the genes, which are the primary beneficiaries of natural selection.

Inclusive fitness and haplodiploidy

Further information: haplodiploidy

According to inclusive fitness theory, organisms can gain fitness not just through increasing their own reproductive output, but also via increasing the reproductive output of other individuals that share their genes, especially their close relatives. Individuals are selected to help their relatives when the cost of helping is less than the benefit gained by their relative multiplied by the fraction of genes that they share, i.e. when Cost < relatedness * Benefit. Under inclusive fitness theory, the necessary conditions for eusociality to evolve are more easily fulfilled by haplodiploid species because of their unusual relatedness structure.

In haplodiploid species, females develop from fertilized eggs and males develop from unfertilized eggs. Because a male is haploid, his daughters share 100% of his genes and 50% of their mother's. Therefore, they share 75% of their genes with each other. This mechanism of sex determination gives rise to what W. D. Hamilton first termed "supersisters" which are more related to their sisters than they would be to their own offspring. Even though workers often do not reproduce, they can potentially pass on more of their genes by helping to raise their sisters than they would by having their own offspring (each of which would only have 50% of their genes). This unusual situation, where females may have greater fitness when they help rear siblings rather than producing offspring, is often invoked to explain the multiple independent evolutions of eusociality (arising at least nine separate times) within the haplodiploid group Hymenoptera. While females share 75% of genes with their sisters in haplodiploid populations, they only share 25% of their genes with their brothers. Accordingly, the average relatedness of an individual to their sibling is 50%. Therefore, helping behavior is only advantageous if it is biased to helping sisters, which would drive the population to a 1:3 sex ratio of males to females. At this ratio, males, as the rarer sex, increase in reproductive value, negating the benefit of female-biased investment. 

However, not all eusocial species are haplodiploid (termites, some snapping shrimps, and mole rats are not). Conversely, many bees are haplodiploid yet are not eusocial, and among eusocial species many queens mate with multiple males, resulting in a hive of half-sisters that share only 25% of their genes. The association between haplodiploidy and eusociality is below statistical significance. Haplodiploidy alone is thus neither necessary nor sufficient for eusociality to emerge. However relatedness does still play a part, as monogamy (queens mating singly) has been shown to be the ancestral state for all eusocial species so far investigated.  If kin selection is an important force driving the evolution of eusociality, monogamy should be the ancestral state, because it maximizes the relatedness of colony members. 

Ecology

Many scientists citing the close phylogenetic relationships between eusocial and non-eusocial species are making the case that environmental factors are especially important in the evolution of eusociality. The relevant factors primarily involve the distribution of food and predators.

Increased parasitism and predation rates are the primary ecological drivers of social organization. Group living affords colony members defense against enemies, specifically predators, parasites, and competitors, and allows them to gain advantage from superior foraging methods.

With the exception of some aphids and thrips, all eusocial species live in a communal nest which provides both shelter and access to food resources. Mole rats, many bees, most termites, and most ants live in burrows in the soil; wasps, some bees, some ants, and some termites build above-ground nests or inhabit above-ground cavities; thrips and aphids inhabit galls (neoplastic outgrowths) induced on plants; ambrosia beetles and some termites nest together in dead wood; and snapping shrimp inhabit crevices in marine sponges. For many species the habitat outside the nest is often extremely arid or barren, creating such a high cost to dispersal that the chance to take over the colony following parental death is greater than the chance of dispersing to form a new colony. Defense of such fortresses from both predators and competitors often favors the evolution of non-reproductive soldier castes, while the high costs of nest construction and expansion favor non-reproductive worker castes.

The importance of ecology is supported by evidence such as experimentally induced reproductive division of labor, for example when normally solitary queens are forced together. Conversely, female Damaraland mole-rats undergo hormonal changes that promote dispersal after periods of high rainfall, supporting the plasticity of eusocial traits in response to environmental cues.

Climate also appears to be a selective agent driving social complexity; across bee lineages and Hymenoptera in general, higher forms of sociality are more likely to occur in tropical than temperate environments. Similarly, social transitions within halictid bees, where eusociality has been gained and lost multiple times, are correlated with periods of climatic warming. Social behavior in facultative social bees is often reliably predicted by ecological conditions, and switches in behavioral type have been experimentally induced by translocating offspring of solitary or social populations to warm and cool climates. In H. rubicundus, females produce a single brood in cooler regions and two or more broods in warmer regions, so the former populations are solitary while the latter are social. In another species of sweat bees, L. calceatum, social phenotype has been predicted by altitude and micro-habitat composition, with social nests found in warmer, sunnier sites, and solitary nests found in adjacent, cooler, shaded locations. Facultatively social bee species, however, which comprise the majority of social bee diversity, have their lowest diversity in the tropics, being largely limited to temperate regions.

Multilevel selection

Further information: Group selection

Once pre-adaptations such as group formation, nest building, high cost of dispersal, and morphological variation are present, between-group competition has been cited as a quintessential force in the transition to advanced eusociality. Because the hallmarks of eusociality will produce an extremely altruistic society, such groups will out-reproduce their less cooperative competitors, eventually eliminating all non-eusocial groups from a species. Multilevel selection has however been heavily criticized by some for its conflict with the kin selection theory.

Reversal to solitarity

A reversal to solitarity is an evolutionary phenomenon in which descendants of a eusocial group evolve solitary behavior once again. Bees have been model organisms for the study of reversal to solitarity, because of the diversity of their social systems. Each of the four origins of eusociality in bees was followed by at least one reversal to solitarity, giving a total of at least nine reversals. In a few species, solitary and eusocial colonies appear simultaneously in the same population, and different populations of the same species may be fully solitary or eusocial. This suggests that eusociality is costly to maintain, and can only persist when ecological variables favor it. Disadvantages of eusociality include the cost of investing in non-reproductive offspring, and an increased risk of disease.

All reversals to solitarity have occurred among primitively eusocial groups; none have followed the emergence of advanced eusociality. The "point of no return" hypothesis posits that the morphological differentiation of reproductive and non-reproductive castes prevents highly eusocial species such as the honeybee from reverting to the solitary state.

Physiological and developmental mechanisms

An understanding of the physiological causes and consequences of the eusocial condition has been somewhat slow; nonetheless, major advancements have been made in learning more about the mechanistic and developmental processes that lead to eusociality.

Involvement of pheromones

Pheromones are thought to play an important role in the physiological mechanisms underlying the development and maintenance of eusociality. In fact the evolution of enzymes involved both in the production and perception of pheromones has been shown to be important for the emergence of eusociality both within termites and in Hymenoptera. The most well-studied queen pheromone system in social insects is that of the honey bee Apis mellifera. Queen mandibular glands were found to produce a mixture of five compounds, three aliphatic and two aromatic, which have been found to control workers. Mandibular gland extracts inhibit workers from constructing queen cells in which new queens are reared which can delay the hormonally based behavioral development of workers and can suppress ovarian development in workers. Both behavioral effects mediated by the nervous system often leading to recognition of queens (releaser) and physiological effects on the reproductive and endocrine system (primer) are attributed to the same pheromones. These pheromones volatilize or are deactivated within thirty minutes, allowing workers to respond rapidly to the loss of their queen.

The levels of two of the aliphatic compounds increase rapidly in virgin queens within the first week after eclosion (emergence from the pupal case), which is consistent with their roles as sex attractants during the mating flight. It is only after a queen is mated and begins laying eggs, however, that the full blend of compounds is made. The physiological factors regulating reproductive development and pheromone production are unknown.

In several ant species, reproductive activity has also been associated with pheromone production by queens. In general, mated egg laying queens are attractive to workers whereas young winged virgin queens, which are not yet mated, elicit little or no response. However, very little is known about when pheromone production begins during the initiation of reproductive activity or about the physiological factors regulating either reproductive development or queen pheromone production in ants.

Among ants, the queen pheromone system of the fire ant Solenopsis invicta is particularly well studied. Both releaser and primer pheromones have been demonstrated in this species. A queen recognition (releaser) hormone is stored in the poison sac along with three other compounds. These compounds were reported to elicit a behavioral response from workers. Several primer effects have also been demonstrated. Pheromones initiate reproductive development in new winged females, called female sexuals. These chemicals also inhibit workers from rearing male and female sexuals, suppress egg production in other queens of multiple queen colonies and cause workers to execute excess queens. The action of these pheromones together maintains the eusocial phenotype which includes one queen supported by sterile workers and sexually active males (drones). In queenless colonies that lack such pheromones, winged females will quickly shed their wings, develop ovaries and lay eggs. These virgin replacement queens assume the role of the queen and even start to produce queen pheromones. There is also evidence that queen weaver ants Oecophylla longinoda have a variety of exocrine glands that produce pheromones, which prevent workers from laying reproductive eggs.

Similar mechanisms are used for the eusocial wasp species Vespula vulgaris. In order for a Vespula vulgaris queen to dominate all the workers, usually numbering more than 3000 in a colony, she exerts pheromone to signal her dominance. The workers were discovered to regularly lick the queen while feeding her, and the air-borne pheromone from the queen's body alerts those workers of her dominance.

The mode of action of inhibitory pheromones which prevent the development of eggs in workers has been convincingly demonstrated in the bumble bee Bombus terrestris. In this species, pheromones suppress activity of the corpora allata and juvenile hormone (JH) secretion. The corpora allata is an endocrine gland that produces JH, a group of hormones that regulate many aspects of insect physiology. With low JH, eggs do not mature. Similar inhibitory effects of lowering JH were seen in halictine bees and polistine wasps, but not in honey bees.

Other strategies

A variety of strategies in addition to the use of pheromones have evolved that give the queens of different species of social insects a measure of reproductive control over their nest mates. In many Polistes wasp colonies, monogamy is established soon after colony formation by physical dominance interactions among foundresses of the colony including biting, chasing, and food soliciting. Such interactions created a dominance hierarchy headed by individuals with the greatest ovarian development. Larger, older individuals often have an advantage during the establishment of dominance hierarchies. The rank of subordinates is positively correlated with the degree of ovarian development and the highest ranking individual usually becomes queen if the established queen disappears. Workers do not oviposit when queens are present because of a variety of reasons: colonies tend to be small enough that queens can effectively dominate workers, queens practice selective oophagy or egg eating, or the flow of nutrients favors queen over workers and queens rapidly lay eggs in new or vacated cells. However, it is also possible that morphological differences favor the worker. In certain species of wasps, such as Apoica flavissima queens are smaller than their worker counterparts. This can lead to interesting worker-queen dynamics, often with the worker policing queen behaviors. Other wasps, like Polistes instabilis have workers with the potential to develop into reproductives, but only in cases where there are no queens to suppress them.

In primitively eusocial bees (where castes are morphologically similar and colonies usually small and short-lived), queens frequently nudge their nest mates and then burrow back down into the nest. This behavior draws workers into the lower part of the nest where they may respond to stimuli for cell construction and maintenance. Being nudged by the queen may play a role in inhibiting ovarian development and this form of queen control is supplemented by oophagy of worker laid eggs. Furthermore, temporally discrete production of workers and gynes (actual or potential queens) can cause size dimorphisms between different castes as size is strongly influenced by the season during which the individual is reared. In many wasp species worker caste determination is characterized by a temporal pattern in which workers precede non-workers of the same generation. In some cases, for example in the bumble bee, queen control weakens late in the season and the ovaries of workers develop to an increasing extent. The queen attempts to maintain her dominance by aggressive behavior and by eating worker laid eggs; her aggression is often directed towards the worker with the greatest ovarian development.

In highly eusocial wasps (where castes are morphologically dissimilar), both the quantity and quality of food seem to be important for caste differentiation. Recent studies in wasps suggest that differential larval nourishment may be the environmental trigger for larval divergence into one of two developmental classes destined to become either a worker or a gyne. All honey bee larvae are initially fed with royal jelly, which is secreted by workers, but normally they are switched over to a diet of pollen and honey as they mature; if their diet is exclusively royal jelly, however, they grow larger than normal and differentiate into queens. This jelly seems to contain a specific protein, designated as royalactin, which increases body size, promotes ovary development, and shortens the developmental time period. Furthermore, the differential expression in Polistes of larval genes and proteins (also differentially expressed during queen versus caste development in honey bees) indicate that regulatory mechanisms may occur very early in development.

 

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)