Françoise Lazarini

Pierre-Marie Lledo
email: pmlledo@pasteur.fr

Is Adult Neurogenesis Essential for Olfaction?

Keywords: GABA, Interneurons, Learning, Discrimination, Olfactory bulb, Memory

In mammals, new neurons are recruited into restricted brain areas throughout life.

Adult-born neurons produced in the subventricular zone migrate towards the olfactory bulb. Although thousands neurons reach this central structure every day, the functional impact of their integration into mature circuits remains debated. Recent investigations have revealed no striking sensory deficit per se when adult bulbar neurogenesis is challenged. However, cognitive functions, such as perceptual learning or olfactory memory, are clearly impaired. In this review, we highlight the role of network activity in shaping ongoing neurogenesis and in turn, how the integration of adult-born neurons refines preexisting network function, and consequently olfactory behavior.

Introduction

New neurons are continuously generated in two discrete areas of the adult brain, the subventricular zone (SVZ) of the lateral ventricles, and the subgranular zone of the hippocampus [START_REF] Kriegstein | The glial nature of embryonic and adult neural stem cells[END_REF]. The latter give rise to new granule neurons, which mature locally in the dentate gyrus, while the former produces neuroblasts undergoing a long migration along the rostral migratory stream (RMS) en route to the olfactory bulb (OB), the first olfactory relay in the CNS. Neural progenitors finish rostral migration in the core of the OB where they begin radial migration and mature into interneurons [START_REF] Kelsch | Watching synaptogenesis in the adult brain[END_REF]. Despite extensive cellular characterization of individual adult-born neurons, the impact of adult neurogenesis on OB circuit function and olfactory behavior is still unclear. It has been argued that ongoing adult neurogenesis is essential for structural maintenance of the OB circuit since blocking adult neurogenesis depletes the bulb of interneurons [START_REF] Imayoshi | Roles of continuous neurogenesis in the structural and functional integrity of the adult forebrain[END_REF]. This review will focus on new insights that indicate the function of adult bulbar neurogenesis goes beyond the mere maintenance of neuronal circuits.

In particular, we detail the functional characteristics of new neurons and how they shape mitral cell function and olfactory behavior.

Functions for old and new neurons

In rodents, olfaction is a key chemosensory modality that enables diverse essential functions such as food selection, danger detection and conspecific interactions. To fulfill this repertoire of functions, the olfactory system has to detect and discriminate odorants from a rich and varied olfactory environment, and then preserve this information in the form of memories. Remarkably, these tasks are highly flexible and adjustable by behavioral states (i.e., the context of sensory exposure), particularly by attention and motivation. Separate microcircuits in the OB achieve these functions (Figures 1 and2). For the vast majority of odorants, sensory detection triggers a behavioral response that is dictated by experiential associations. However, for a subset of odorants, stereotyped behaviors are induced both between individuals and upon repeated exposure for a given subject [START_REF] Stowers | Olfactory mechanisms of stereotyped behavior: on the scent of specialized circuits[END_REF].

Sensory information processing

In mammals, chemical stimuli are transduced into spike trains by olfactory receptor neurons. These sensory neurons release glutamate onto the distal dendrites of mitral and tufted cells, the two relay neurons of the OB circuit. Because the flow of information goes from the external world to the brain, this sensory input represents the so-called bottom-up pathway (Figure 3). However, the OB circuit is more than a mere relay station acting as a feedforward filter of sensory inputs. This circuit actively processes and refines sensory information using two classes of local inhibitory interneurons: Periglomerular cells and granule cells [START_REF] Wilson | Neural and behavioral mechanisms of olfactory perception[END_REF] (Figure 2). Periglomerular cells (GABAergic and dopaminergic interneurons), project to one or a few glomeruli and interact with olfactory nerve terminals and/or the primary dendrites of relay neurons [START_REF] Kosaka | Synaptic organization of the glomerulus in the main olfactory bulb: compartments of the glomerulus and heterogeneity of the periglomerular cells[END_REF]. Granule cells contribute to the network activity of the OB by inhibiting mitral cell dendrites via reciprocal dendrodendritic synapses, and by facilitating both mitral cell synchronization and network oscillations [START_REF] Bathellier | Circuit properties generating gamma oscillations in a network model of the olfactory bulb[END_REF][START_REF] Lagier | GABAergic inhibition at dendrodendritic synapses tunes gamma oscillations in the olfactory bulb[END_REF][9].

Odor discrimination, learning and memory are thought to depend on the synaptic interplay between interneurons and mitral cells [10][11][12]. Local synaptic inhibition is essential for odor-discrimination performance [13] and for olfactory perceptual learning [14] (see glossary). Granule cells might achieve this function by enabling the discharge patterns of mitral cells to diverge from similar and confusing sensory input patterns imposed by sensory receptor neurons (a task called pattern segregation).

Bulbar output patterns

After being processed and refined, odor information in the OB is transmitted to higher-order brain structures in the primary and accessory olfactory cortex. These centers receive excitatory input from mitral cells and presumably from associations between OB output patterns and stimuli in the environment. The behavioral state modulates the OB activity via the centrifugal fibers entering the bulbar circuit (Figures 1 and2). These centrifugal inputs are comprised of cortical, subcortical and modulatory projections [15]: 1) glutamatergic fibers from frontal cortex and olfactory cortex [16,17], 2) modulatory projections from the locus coeruleus (norepinephrin), the horizontal limb of the diagonal band of Broca (acetylcholine), and the dorsal raphe nucleus (serotonin) [18,19]. Centrifugal inputs shape the function of the bulbar network on various timescales (from milliseconds to days) and various contexts (see below). Interestingly, most of the centrifugal connections target granule cells [20], and tune the dendrodendritic inhibition of mitral cells [21,22]. On the other hand, centrifugal serotoninergic fibers preferentially target periglomerular cells and modulate odor input at the beginning of odor information processing, acting indirectly on olfactory receptor neuron terminals through a modulation of periglomerular inhibitory cells [23].

By modulating the activity of local interneurons, centrifugal inputs shape the spatial and temporal firing patterns of mitral cell activity, and thus are important for producing dynamic odor representation in the context of different behavioral states [23][24][25][26][27]. Much of the sensory input that enters the brain does so because we actively seek for sensory information through sniffing and others means, and because sensory investigation can often be triggered by environmental cues. In this case, centrifugal inputs to the OB provide the contextual information required for sensory formatting of odor representations, learning and memory. In short, this top-down pathway represents a mechanism by which the brain makes sense of the olfactory content (Figure 3). Olfactory bulb activity, and thus odor perception, is shaped by the interaction between the bottom-up and the top-down pathways. Experiments that manipulate the top-down pathway have been shown to influence olfaction. For instance, changing the impact of centrifugal cholinergic inputs to the OB circuit affects olfactory perception by changing rodents' ability to differentiate between perceptually similar odorants [28,29]. Similarly, experimental manipulation of noradrenergic activity in the OB has resulted in changes in olfactory perception and memory formation [START_REF] Coderre | Late effects of radiation on the central nervous system: role of vascular endothelial damage and glial stem cell survival[END_REF] and olfactory habituation memory [31]. Decreasing glutamatergic feedback from the olfactory peduncle to the OB has revealed a key role for centrifugal inputs in changing the formation of odor-reward associations, but not primary bulbar odor representations [32]. In this context, it is noteworthy that the first connections of adult-born neurons are established with centrifugal fibers, which can provide excitation to these young neurons soon after they arrive and integrate within the OB circuit [START_REF] Kelsch | Watching synaptogenesis in the adult brain[END_REF]33,34]. Centrifugal afferents offer a pathway by which new neuron activity, and thus olfactory function, can be shaped by the behavioral state and olfactory experience, including learning. Further studies will aim at characterizing the detailed synaptic mechanisms that links adult-generated neurons to centrifugal projections.

As we shall discuss below, both bottom-up and top-down pathways may regulate adult neurogenesis in a tightly orchestrated manner following temporal constraints defined by the progression of neuroblast maturation. According to this hypothesis, new neurons act as coincident detectors by appearing at the nexus between two key events: the spatial dimension and developmental stages (Figure 4).

Wiring adult-generated neurons into preexisting circuits

Thousands adult-born neurons arrive to the bulb per day and integrate into the existing neural circuit within just a few days [35]. About 95% of the new cells differentiate into granule cells, less than 3% become periglomerular cells [36], and a very small proportion give rise to glutamatergic juxtaglomerular neurons [37]. Each subtype of new bulbar interneuron originates from specific neural stem cells located in distinct regions of the SVZ-RMS system [38][39][40]. Adult-born neurons express GABA and glutamate receptors as soon as they start migrating [41] and when entering the deeper layers of the OB [34,42,43] before gradually increasing their responsiveness to both GABA and glutamate. A recent study on the dynamic pattern of synaptogenesis onto newborn neurons entering the OB has revealed the unexpected contribution of centrifugal afferents to new neuron maturation [34]. By observing that proximal synaptic structures on granule cell soma and dendrites form much earlier than the dendrodendritic contacts with resident mitral cells, this work shed light on the early function of centrifugal fibers on adult neurogenesis, consistent with a previous report [44]. The source of the centrifugal inputs that form early connections remains to be determined. Revealing their identity will be critical toward further understanding on the contribution of the top-down process in controlling OB adult neurogenesis.

The development of input synapses in adult-formed OB neurons is mostly complete by four weeks after the cell birth, and is sensitive to neuronal activity during a restricted period [33,45]. One-half of new neurons die during their process of dendritic branching and synaptogenesis between 15 and 40 days after cell birth, [45][46][47][48][49][50][51]. Most adult-generated neurons that survive this period of cell death will survive for several months and become indistinguishable from pre-existing neurons [46,48,49,[52][53][54], but see [47].

Having established that not only thousands adult-born neurons integrate into the existing OB circuit per day, but also about half of them survive, it was important to determine if the integration of new neurons influences OB-related behaviors. To address this issue, a popular strategy consists in ablating adult neurogenesis using systemic or genetic techniques. Unfortunately, no available technique can specifically reduce the population of adult-generated OB neurons. All reported ablation techniques present both advantages and undesired off-target effects, which may explain why discrepancies have emerged between these efforts (see Box 1). Further studies using selective and reversible ablation of adult neurogenesis should help elucidate the respective roles of neurogenic systems to animal behavior.

Interactions between sensory experience and adult neurogenesis

To elucidate the role of sensory inputs on adult neurogenesis many studies have used passive stimulation paradigms [55][56][57]. Passive olfactory experiences include either transient odorant exposure or olfactory enrichment lasting several days. As described below, recent findings demonstrate that some passive olfactory experiences, but not all, impact bulbar neurogenesis (see Box 2).

Effects of passive odorant exposure

Activity-dependent factors regulate many aspects of bulbar neurogenesis including synapse formation and the survival of new neurons [45,47,50,55,56,58,59]. Whereas passive/short-term exposure to odorants does not change the survival of new neurons [50,58,60,61], passive/long-term exposure to odorant-enriched environment enhances both SVZ proliferation [40] and the survival of new granule cells [55,59].

Conversely, long-term sensory deprivation reduces the survival of adult-born granule cells [45,56,59,62]. Interestingly, new cells that survive after sensory deprivation display an increased density of proximal input synapses in the unbranched apical dendrite [33]. It remains to be established whether adult-born granule cells survive in the absence of sensory input because they are able to compensate for their excitatory drive by enhancing their synapse density. Similarly, adult-born periglomerular neurons display activity-dependent survival. Sensory deprivation decreases [56], whereas olfactory enrichment [55] increases the survival of distinct populations of adult-born periglomerular neurons, supporting the hypothesis that their integration rate is dependent on the magnitude of sensory activity [59]. Also, sensory enrichment accelerates the development of their glutamatergic input synapses as visualized by genetic synaptic markers [63].

Impact of olfactory learning

Odor sampling behavior reflects not only current perception but also expectation following learning rules [24,64,65], highlighting once again the importance of the topdown pathway in processing the sensory information in the OB circuit. Olfactory perception is highly dependent on experience, as repeated exposures to similar odorants can improve olfactory discrimination (a task called olfactory perceptual learning) in rodents and humans [66,67]. Not surprisingly, adult neurogenesis also constitutes a substrate to the context-dependent modulation of odor responses (see Box 2). For instance, bulbar neurogenesis increases following olfactory perceptual learning [68]. Moreover, adult-generated neurons are preferentially involved in processing of experienced odorants rather than naïve odorants [68]. Similarly, bulbar neurogenesis increases following an active sensory experience such as associative olfactory learning (conditioned mice), but not by the simple passive exposure to odorants [50,58,60,61]. Moreover, olfactory learning in mice induces clustering of newborn cells in specific regions of the granule cell layer, 5 days post-conditioning, in an odorant-specific way [61]. In the time window when learning increases survival (i.e., 18-30 days after neuron birth), new granule cells begin receiving glutamatergic synaptic contacts. This suggests that the survival of adult-generated granule cells might be sensitive to experience during the period of early synaptogenesis.

Interestingly, olfactory learning induces a spatial distribution of newborn neurons, which is conserved after 30 days but not after 90 days post-conditioning, suggesting that cell death might occur in 30 to 90 day-old newborn cells [61]. Since learning to discriminate between two odorants reduces the survival of 38/45-day-old granule cells [50,69], but not 65-days-old neurons [50], it is possible that olfactory learning promotes the survival of specific 18-30 day-old neurons and simultaneously induces cell death of other 38-45-day-old neurons. This would explain why the previous olfactory learning tasks observe no net increase in granule cell survival.

Recent experiments demonstrate that adult-born granule cell synaptogenesis is sensitive to changes in synaptic input and have suggested that both synaptogenesis and neuronal survival occur after the neuron receives a minimum threshold of excitation from local glutamatergic inputs or centrifugal fibers [START_REF] Kelsch | Watching synaptogenesis in the adult brain[END_REF]33,70].

Further studies will have to clarify the respective functions of sensory-driven inputs and centrifugal fibers on the regulation of newborn neuron survival and synaptogenesis. Particularly, it will be important to decipher the mechanism by which some adult-generated neurons are "primed" to survive following a learning task that involves the top-down pathway.

Functions of adult-born neurons in the olfactory bulb circuit

The potential functions of adult-born neurons in the OB can be summarized into four main tasks: (1) maintenance of the OB circuit; (2) shaping sensory information processing; (3) supporting learning processes, and/or (4) mediating odor memory.

Below we discuss these non-exclusive options.

The contribution of adult neurogenesis to circuit maintenance

In the developing OB, most newborn granule cells reside in the superficial granule cell layer and can survive for the lifetime of the animal [48]. This phenomenon is reversed during adulthood although estimates for the precise rate of neuronal turnover remain debated. Some studies estimate that only a small proportion (~ 15%) of the total population of interneurons turnover in the granule cell layer [71,72].

Others suggest that most granule cells located deeper in the granule cell layer and half of the superficial ones are subjected to continuous turnover [START_REF] Imayoshi | Roles of continuous neurogenesis in the structural and functional integrity of the adult forebrain[END_REF]73]. These inconsistencies may result from different methodologies and should be solved in the future using different models. Interestingly, cell survival seems to depend on the quantity of newly produced neurons that reaches the OB: When bulbar neurogenesis is reduced after SVZ irradiation, the survival rate of the newborn neurons reaching the bulb is significantly prolonged [51], but see [73]. This observation could reflect a competition of adult-generated neurons for survival factors released by the neural targets, just as they compete with existing neurons for their innervations with synaptic projections.

Impact of adult neurogenesis on shaping sensory information processing

Adult-born neurons target mitral and tufted cells in the OB providing an additional inhibitory function to the bulbar circuit. However, adult-born neurons also can target local inhibitory interneurons yielding disinhibitory function in the OB circuit (Bardy et al., the Society For Neuroscience, 2009). Thus, the contribution of adult-generated neurons to bulbar physiology remains unclear and conflicting results have been reported regarding the functional outcomes of adult neurogenesis for olfaction [73][74][75]. One study found no impact of neurogenesis ablation on bulbar activity following efficient suppression of neurogenesis by irradiation [73]. In contrast, another study based on intraventricular infusion of the antimitotic agent Arac-C reported strong changes in mitral and tufted cell activity induced by depleting adult-born neurons, leading to impairment of short-term olfactory memory [75]. The contrasting results obtained in the two aforementioned studies highlight the need for further investigations to decipher the role of adult-generated neurons in modulating mitral cell activity. Finally, it is worth noting that a correlation between the turnover rate of adult bulbar neurogenesis and olfactory discrimination time was recently established [74]. This study used a pan-caspase inhibitor to impair the natural elimination of adult-born granule cells, and found that the time required to discriminate between two similar odorants was subsequently increased. Therefore, the continuous recruitment of adult-born neurons may lead to changes in the strength of the temporal binding of signals originating from different odorant receptors. This might be critical not only for adjusting odor discrimination, but also for perceptual learning as described above [76].

Impact of adult neurogenesis on learning processes

Previous studies have demonstrated that active sensory processing results in perceptual learning, associative learning, and memory formation, through changes in OB network activity [60]. Adult-generated neurons dedicated to the OB circuit seem not to be involved in perception per se (see Box 3), but rather in cognitive function (see Box 2). For instance, manipulation of bulbar neurogenesis revealed that the supply of new neurons is not necessary for odorant detection, discrimination and acquisition of an associative olfactory learning [START_REF] Imayoshi | Roles of continuous neurogenesis in the structural and functional integrity of the adult forebrain[END_REF]51,61]. On the other hand, adult OB neurogenesis is required for olfactory fear conditioning [73], olfactory perceptual learning [68] and long-term olfactory memory of an associative olfactory learning [51,61].

Developing adult-generated interneurons are unique in that their connectivity adapts to the degree of neuronal activity in the host circuit [START_REF] Kelsch | Watching synaptogenesis in the adult brain[END_REF]33,52,77,78]. They also trigger unique responses during odorant familiarization [79], suggesting a key role in olfactory processing and plasticity. This idea has received recent support from an electrophysiological study [80]. It was found that soon after synaptic integration, adult-born neurons displayed robust long-term potentiation not found in preexisting neurons. This ability is progressively lost as the cells mature over time. These findings are similar to those reported for adult-generated dentate gyrus neurons, which exhibit synaptic plasticity fading with time [81]. Interestingly, the synaptic plasticity of young adult-born OB interneurons occurs during a time window when new neurons are initially added to the circuit indicating that they might play a role in circuit plasticity required for learning. Although synaptic development is mostly completed by four weeks after cell birth [43,44,47,49,52], spines may still undergo rearrangement after this period indicating that mature OB interneurons still maintain some capacity for synaptic modification [52]. Thus, the precise stages at which new OB neurons play important role for olfaction remains to be characterized.

Role of adult neurogenesis in olfactory memory

The recruitment of new neurons in the OB circuitry might allow the association of otherwise temporally distinct input signals originating from bottom-up and/or topdown connections (Figure 4). Interestingly, newly-formed neurons receive first synapses from centrifugal fibers before establishing contact with afferent inputs from mitral cells [34,44]. Through these early contacts with the top-down pathway, newborn neurons are positioned to provide a top-down context to olfactory information. In addition, these early contacts may support memory based on integrating recursive signaling from the olfactory cortex and the OB. In line with this hypothesis, a correlation has been found between the degree of bulbar neurogenesis and short-term olfactory memory [55,82,83]. Moreover, blocking bulbar neurogenesis before and during odor enrichment prevents olfactory perceptual learning, and thus memory formation of experienced odorants in mice [68]. Early contacts with top-down fibers offer a mechanism by which odor experience increases newborn granule cell responsiveness and improves olfactory discrimination learning [14,58,68]. Altogether, mounting evidence supports the hypothesis that adult-generated neurons are required in the formation and/or shaping of olfactory memory circuits.

Halting adult neurogenesis by radiation, or Ara-C treatment, also modifies long-term odor associative memory tasks [51,61]. In contrast, long-term memory has been shown to be unaffected in mice producing a neuron-specific enolase-diphtheria toxin [START_REF] Imayoshi | Roles of continuous neurogenesis in the structural and functional integrity of the adult forebrain[END_REF], and in mice treated with Ara-C [75] despite the significant neurogenic reduction in the two latter studies. Details of the experimental design might have contributed to these discrepancies, including the strains of animal models used, the off-target effects of ablation methods and the behavioral paradigm used. Considering the latter option, it is worth noting that some authors [START_REF] Imayoshi | Roles of continuous neurogenesis in the structural and functional integrity of the adult forebrain[END_REF]75] employed a non-operant task whereas we [51] and others [61] used operant tasks. Operant conditioning differs from non-operant conditioning in that the animal is motivated to acquire a particular behavior to complete its task. Non-operant and operant conditioning paradigms for investigating olfactory abilities often lead to different conclusions, even from the same animal model, indicating they might support different olfactory functions [84][85][86]. The discrepancy between the two learning paradigms might originate from the participation of top-down centrifugal fibers brought into play during operant conditioning only.

Together, these data indicate that adult-born neurons are required for longterm memory of associative olfactory tasks involving active learning [51,61], but not for non-operant associative training [START_REF] Imayoshi | Roles of continuous neurogenesis in the structural and functional integrity of the adult forebrain[END_REF]75], but see [73]. These findings also indicate that the need for adult-born neurons might depend on the nature of the task. A recent study shows that the procerebrum of invertebrates, which is analogous to the mammalian OB, represents the storage site for associative olfactory memories and is directly involved in the memory retrieval process [87]. In the OB, adult-generated neurons may be the main targets of learning-induced changes originating from sensory inputs (the sensory space) and centrifugal fibers (the internal state), acting as key elements in the recall of memory traces (Figure 4). Similarly, a role for adult hippocampal neurogenesis in memory storage, rather than in learning set, was proposed in rodents [88]. Interestingly, as in the OB, developing immature hippocampal neurons might serve as pattern separators [89]. This function is particularly important for the accuracy of memory encoding. Pattern separation is also essential for the formation and use of memories derived from stimuli that are close in space or time.

Concluding remarks

Olfactory sensory processing begins in sensory neurons that transform chemical signaling into electrical patterns. This stage is unlikely to change over time during animal's experience. Then, mitral cells distribute neural activity widely to downstream structures and are influenced by a combination of sensory input and the behavioral state. This second transformation step constitutes a target not only for adult neurogenesis but also for top-down connections. Because the ability to process sensory information depends on the functional architecture and synaptic connectivity of the OB, adult neurogenesis has a particular impact on information processing.

Understanding the precise functions of adult bulbar neurogenesis will require elucidating the mechanisms that control the development of both the synaptic inputs and outputs of the adult-born OB interneurons, and how these precise sequence impact olfactory behavior.

Activity-dependent regulation of neurogenesis and experience-dependent participation of new neurons in olfactory information processing both highlight the relationship between adult neurogenesis and olfactory abilities. In this review, we have highlighted the potential functions of new neurons as the following: 1)

Maintaining the OB circuit. Newborn neurons might replace dying interneurons in the OB, thus providing a structural role for the maintenance of the network. 2) Tuning information processing. Adult-born neurons target mitral cells in the OB thus providing sufficient inhibitory activity necessary for sharp pattern segregation. In some circumstances, it might be vital to decorrelate and modify the odor representation that is going to be ultimately sent to downstream brain structures. 3) Role in learning. Because adult-born neurons express unique synaptic plasticity, these neurons are more suited than mature neurons to encoding new information. 4) Role in memory. As we propose, newborn neurons might be able to provide significant context to the information content due to their role as coincidence detectors. Even in the absence of sensory inputs, newborn neurons could be driven by centrifugal fibers, a feature that could represent part of a memory process.

Accumulating evidence suggests that modifications to adult neurogenesis alter olfactory processing in diverse ways. It will be of great interest to explore precisely at what stage of development the adult-generated OB neurons contribute to olfactory perception, odor memory, or fear responses. Are new neurons functionally important when they are young and more excitable? Or are they most important later when they are mature and fully integrated into the existing network? Also, future studies will have to be designed to selectively target distinct populations of adult-born interneurons (periglomerular cells versus granule cells). So far, the role for new periglomerular cells in olfactory information processing remains unknown. Finally, further studies on the relationship/cross-talk between adult-generated neurons and centrifugal projections will be certainly helpful to unravel the cellular and molecular mechanisms of odor discrimination and those involved in forming and restituting olfactory traces after learning. Clarification of these issues will highlight not only how OB neurogenesis impacts olfaction, but will inform the future development of cellular therapies for neurological disorders (Box 4). Authors declare having no conflicts of interest with this review.

Acknowledgments

Box 1. Strategies employed for challenging adult neurogenesis.

Most of the approaches today aim at impairing neurogenesis using radiation, generic anti-mitotic [91] or anti-proliferative [92] drug treatment, genetic manipulation or aging mice models. Studies using traditional knockout of genes essential for neurogenesis are interesting, but often result in conflicting data on olfactory behavior [84,[93][94][95].

Their interpretations are often limited by abnormalities of the whole brain structure or compensatory effects elicited during development.

Another alternative model of decreased neurogenesis is aging [93]. However, it is difficult to interpret unambiguously whether the sensory deficit found in this model results from the specific decrease of bulbar neurogenesis or to other impairment in the aging brain.

Treatments with antimitotic drugs, such as SVZ infusion of the antimitotic drug cytosine arabinosine (Ara-C), are also useful to halt adult SVZ neurogenesis [61,68,75]. However, these treatments are unspecific since they target all dividing cells, including neurovascular cell types and oligodendrocyte precursors cells [96]. In addition, Ara-C is also toxic for non-dividing cells, such as oligodendrocytes [96] and post-mitotic neurons [97]. Although no side effects were observed at low doses [68,75], slightly higher doses cause weight loss, bone marrow suppression, cerebellar neurotoxicity and reduction in motor coordination. In addition, antimitotic treatment probably reduces cell division throughout a large portion of the brain, even when the drug is focally infused into the parenchyma of the SVZ or into the cerebrospinal fluid of lateral ventricle. For instance, the effects of Ara-C after intraventricular infusion are therefore not limited to the SVZ, as recently shown by the subsequent 30-50 % reduction in hippocampal neurogenesis [61,75]. Remarkably, strong cognitive deficits have been reported to result from a similar reduction in the hippocampal neurogenesis [98]. Finally, treatment with antimitotic agents might disrupt the continuous production of olfactory sensory neurons necessary for olfactory function perception [99].

An alternative way to disrupt adult neurogenesis uses focal irradiation. Regionrestricted irradiation is more specific than focal infusion of antimitotic drugs, as only the irradiated area, but not the neighboring ones, exhibits a decrease in neurogenesis [51,73,89,100]. However, irradiation affects locally all dividing cells, including oligodendrocyte cell precursors [101], and some subtypes of neurovascular cells [102]. It also induces strong microglial activation, a hallmark of neuroinflammation, lasting for two months [103]. Thus, it is crucial to perform behavioral tests several months after irradiation.

Finally, adult neurogenesis can be suppressed by inducible recombination [98] or conditional transgenic ablation of neural stem cells [START_REF] Imayoshi | Roles of continuous neurogenesis in the structural and functional integrity of the adult forebrain[END_REF]104,105] or neuronal precursor cells [106]. These genetic approaches are less specific than focal irradiation or antimitotic drug treatment, since bulbar and hippocampal neurogenesis are both ablated, limiting thus the interpretation of behavioral analyzes.

Box 2. Operant associative olfactory learning versus non-operant olfactory learning.

Learning can be divided into distinct categories including non-associative forms such as habituation and sensitization, in which an animal learns about the properties of a single stimulus, and associative forms, in which an animal learns about the relationship between two stimuli (classical conditioning) or between a stimulus and the animal's own behavior (operant conditioning).

If when an animal first responds to a stimulus, it is neither rewarding nor harmful, the animal reduces subsequent responses. For example, a short-term exposure of a adult rat to an odorant, in the absence of any paired reinforcement, leads to a reduction in the proportion of mitral cells responding to that odorant with increased firing rates. This process called habituation reflects the progressive diminution of behavioral response probability with repetition stimulus.

Sensitization is another example of non-associative learning in which the progressive amplification of a response follows repeated administrations of a stimulus. Sensitization is part of adaptive as well as maladaptive learning processes in the organism.

Operant conditioning is a situation that leads to modify the occurrence and expression of a given behavior. Operant conditioning is distinguished from classical conditioning in that operant conditioning concerns changes of voluntary behavior.

Classical conditioning was first demonstrated by Ivan Pavlov (1927) [107]. It involves repeatedly pairing an unconditioned stimulus (which unfailingly evokes a reflexive response) with another previously neutral stimulus (which does not normally evoke the response). Following conditioning, the response occurs both to the unconditioned stimulus and to the other, unrelated stimulus.

Today, a central question in experimental Psychology is whether these different forms of learning are related to adult neurogenesis. When applied to olfaction, do the associative and non-associative forms of learning require the continuous arrival of newborn olfactory neurons? It seems to be the case (Figure I).

For instance, increasing newborn neuron survival in the OB by non-associative olfactory experience, leads to stronger short-term olfactory memory [55]. Similarly, enrichment to odors improves olfactory discrimination in adult rats [108] through a non-associative process called perceptual learning that requires intact adult neurogenesis [68]. Also, an olfactory memory deficit was reported in a spontaneous short-term memory paradigm after suppression of bulbar neurogenesis [75].

Adult-born neurons are also at play in associative olfactory learning. When adult neurogenesis is suppressed in the adult cricket, both olfactory learning and memory tested in an operant olfactory conditioning are impaired [82]. Similarly, correlative studies performed in rodents indicate a function of adult neurogenesis in associative learning [50,58,69]. Also, blocking of neurogenesis by irradiation or antimitotic treatment in mice impairs long-term retention of an associative olfactory task but not its acquisition [51,61]. Once selected to survive by learning, newborn neurons might change the neural representation of the learned odorants through shaping the physiology of the OB network. Following this view, it can be postulated that newborn neurons might be able to retain a memory trace of previous olfactory experience during an associative learning task. Finally, no deficit in olfactory memory was reported when adult neurogenesis was reduced by genetic ablation [START_REF] Imayoshi | Roles of continuous neurogenesis in the structural and functional integrity of the adult forebrain[END_REF]. This discrepancy might result from technical issues (specific context) and/or from the degree of difficulty of the discrimination tasks that has been used. Further experiments are required to clarify the relationship between the degree of adult neurogenesis and the complexity of discrimination learning task.

Box 3. Role of adult neurogenesis in olfactory discrimination

Olfactory discrimination is a primary function of OB granule cells [13]. As adultgenerated granule cells are a significant component of bulbar interneuron population, it was not surprising to find a link between the turnover rate of adult bulbar neurogenesis and the olfactory reaction time [74]. But surprisingly, most experiments aimed at manipulating adult bulbar neurogenesis failed to demonstrate a direct role of adult neurogenesis in odorant discrimination [START_REF] Imayoshi | Roles of continuous neurogenesis in the structural and functional integrity of the adult forebrain[END_REF]51,73,75]. Olfactory discrimination is clearly impaired in animals in which bulbar neurogenesis has been altered during embryogenesis [84,93,95] but not during adulthood [START_REF] Imayoshi | Roles of continuous neurogenesis in the structural and functional integrity of the adult forebrain[END_REF]51,73,75], suggesting that this task is not mediated by adult-generated neurons. The function of newborn neurons in odorant discrimination might be difficult to unveil because of their relatively small proportion compared to the population size of mature interneurons (but see [START_REF] Imayoshi | Roles of continuous neurogenesis in the structural and functional integrity of the adult forebrain[END_REF]). It might also be due to the eventual development of compensatory effects such as synaptic changes in mature neurons, or to the current methods used to challenge adult neurogenesis (see Box 1). Designing experiments that selectively activate or inhibit new neurons within the adult OB during odorant discrimination tasks of diverse difficulty are required to solve this still unsolved question.

Box 4: Outstanding questions

• At what stage of development do adult-generated OB neurons contribute to olfactory perceptual learning, odor memory, and fear responses?

• How does the orchestrated development of the synaptic inputs onto adult-born bulbar neurons impact olfactory abilities?

• Are new neurons functionally important when they are young and more excitable? Or are they most important later when they are mature and fully integrated into the existing network?

• Is there any role for new periglomerular cells in forming odor memories?

• What is the relationship/cross-talk between adult-generated neurons and centrifugal projections?

• What are the cellular and molecular mechanisms underlying olfactory perceptual learning and odor discrimination?

• What are the mechanisms involved in forming and restituting olfactory traces after learning?

• What is the contribution of adult neurogenesis to pheromone-related behaviors such as mating, social recognition and maternal behavior?

• What is the contribution of the constitutive adult bulbar neurogenesis to brain repair occurring during brain trauma, ischemia, neurodegenerative or inflammatory diseases?

Olfactory associative memory: Memory of an association between a given odorant stimulus and a reward. 6) the rostral migratory stream at the core (RMS). The OB receives inputs from olfactory sensory neurons (OSN) and from centrifugal fibers. The OB output is provided by mitral/tufted cells, which project to several parts of the olfactory cortex.

OSN project axons to glomeruli (white disks), in which they activate mitral/tufted cells and periglomerular cells. Centrifugal inputs modulate periglomerular and granule cell activities, which in turn modulate mitral cell activity.

Figure 2.

A canonical olfactory bulb circuit. The olfactory system includes the olfactory sensory neurons (OSN) located in the sensory epithelium, the relay neurons (mitral and tufted cells) of the OB, and cortical neurons in downstream structures, comprising the primary and accessory olfactory cortex. These centers include the anterior olfactory nucleus, which connects the OB through a portion of the anterior commissure, the olfactory tubercle, the piriform cortex (considered to be the primary olfactory cortex), the cortical nucleus of the amygdala and the entorhinal area [16,21,77]. The OB is the first relay of olfactory processing and encoding by three main types of neurons: (1) OSN detects odorants and transmits action potentials to the OB where it forms glutamatergic synapses with the dendrites of ( 2

:

  Figure I, Matthew Valley and Gabriel Lepousez and for comments on the manuscript.

Olfactory enrichment :

 enrichment Repeated exposure to novel odorants for several days in unique housing. Olfactory fear conditioning: Learning of an association between a neutral stimulus (odorant) and an aversive stimulus. The animal cannot choose to escape the conditioned stimulus upon testing. Olfactory habituation/spontaneous discrimination task: Assessment of nonassociative short-term memory, of spontaneous discrimination among dissimilar or similar odorants, unaltered by previous reinforcement. Olfactory habituation: Progressive decrease in investigating repeated similar odorant stimuli. Operant associative olfactory learning: A form of learning consisting of the acquisition of a certain behavior (for instance go and lick in a go-no go procedure) by pairing the neutral odorant stimulus with a reward. Pattern segregation: The separation of neuronal cell discharge patterns allowing divergence from similar and confusing input patterns.

Figure 1 .

 1 Figure 1. Schematic illustration of the OB structure and its connections. The OB is

  ) relay neurons, called mitral cells, and (3) local interneurons in target area called glomeruli. Mitral/tufted cells are glutamatergic neurons which project to the olfactory cortex. The activity of mitral/tufted cells is modulated by glumatergic input from OSNs, GABAergic input from interneurons called granule cells (GC), GABAergic and dopaminergic input from interneurons called periglomerular cells (PG), as well as centrifugal innervations (centrifugal fibers: CF). Unlike most central nervous system regions, inhibitory interneurons in the OB far outnumber principal cells by at least 50:1 [90]. The granule cells account for about 90% of all bulbar interneurons [90]. Periglomerular and granule cells are the primary targets of the centrifugal inputs to the OB. Sensory experience modulates the activity of glomeruli, of mitral/tufted cells and of granule cells. Centrifugal inputs play a key role in odor detection, discrimination and learning/memory by regulating sensory information. Newly generated interneurons continuously integrate into the existing neuronal circuit of the adult OB where they provide a source of plasticity that may allow behavioral adaptation to an ever-changing environment. Adult-born neurons target mitral cells in the OB thus providing an inhibitory function. Learning modulates the survival of newborn granule cells probably via centrifugal inputs. In this case, newborn granule cells might support long-term memory after learning.

Figure 3 .

 3 Figure 3. Regulation of the olfactory bulb activity. The neuronal activity of the OB

Glossary

Bottom-up pathway: The classical view of cortical processing proposes that sensory information evolves through a feed-forward hierarchy. In the first step of the olfactory system, the bottom-up pathway includes sensory signals from the olfactory epithelium to the olfactory bulb.

Coincidence detection: A process by which a neuronal element can encode information by detecting the incidence of simultaneous yet spatially segregated events. Coincidence detection enables a combinatorial code to achieve outcomes that cannot be generated by isolated stimuli and to increase the number of results attainable with a limited stimulus set. Coincidence detection, long appreciated to play a role in strengthening synaptic connections in mature nervous systems, can be important for establishing synaptic connections during neural maturation.

Long-term memory: Memory lasting several days or months.

Long-term potentiation: An increase in size of a synaptic response that indicates plasticity in synaptic connections, lasting one hour or more.

Non-operant associative olfactory task: Passive pairing of a context with an odorant stimulation that does not necessitate the learning of a specific behavior to get a reward.

Perceptual learning: A form of implicit memory that can be defined as an increased sensitivity to stimulus parameters that improves perceptual acuity due to experience.

Reinforced olfactory discrimination learning: Acquisition of a specific behavior in response to an odorant stimulus, paired with a positive reinforcer (reward) in a specific context (called conditioning: water or food deprivation for water or food reward, respectively).

Short-term memory: Memory lasting several minutes or hours.

Top-down pathway: Propagates information from higher (olfactory cortex, modulatory centers) to lower (olfactory bulb) levels. from two different behavioral contexts: Passive odorant exposure or active sensing.

Challenging OB network activity influences the recruitment of new neurons (black arrow), which in return will affect the ongoing activity of the host circuit. Note that adult neurogenesis is also sensitive to olfactory experience (i.e., sensory deprivation or odor-enriched environment) and to the behavioral state (i.e., learning to discriminate between two odorants) (black arrows). Thus, newborn neurons are able to associate a context to the information content at a particular stage of their maturation (temporal dimension). Information processing and refining within the OB circuitry might depend on both spatial and temporal dimensions, owing to the detection of coincidence by adult-generated neurons. This process might participate to the encoding of the odor representation, which is then sent to the olfactory cortex.


Artur Aleksanyan

Etienne Brasselet
email: etienne.brasselet@ubordeaux.fr

Spin-orbit photonic interaction engineering of Bessel beams

Keywords:

Interaction between the polarization and spatial degrees of freedom of a light field has become a powerful tool to tailor the amplitude and phase of light beams. This usually implies the use of space-variant photonic elements involving sophisticated fabrication technologies. Here we report on the optical spin orbit engineering of the intensity, phase, and polarization structure of Bessel light beams using a homogeneous birefringent axicon. Various kinds of spatially modulated free-space light fields are predicted depending on the nature of the incident light field impinging on the birefringent axicon. In particular, we present the generation of bottle beam arrays, hollow beams with periodic modulation of the core size, and hollow needle beams with periodic modulation of the orbital angular momentum. An experimental attempt is also reported. The proposed structured light fields may find applications in long-distance optical manipulation endowed with self-healing features, periodic atomic waveguides, contactless handling of high aspect ratio micro-objects, and optical shearing of matter

INTRODUCTION

Engineering the amplitude and phase of light beams is currently routinely used across various disciplines, but particularly in the field of optical manipulation [START_REF] Dholakia | Shaping the future of manipulation[END_REF][START_REF] Woerdemann | Advanced op tical trapping by complex beam shaping[END_REF]. This is usually done using spa tial light modulator technology that allows almost arbitrary beam shaping in a reconfigurable manner. Technology free options also exist, for instance, for the preparation of light beams endowed with one or several optical phase singularities. For instance, topo logical defects that spontaneously occur in liquid crystals have been shown to behave as efficient and reconfigurable singular phase masks for the generation of single or an array of optical vortices [START_REF] Barboza | Optical vortex induction via light matter interaction in liquid crystal media[END_REF]. Tailored technology free spin orbit photonic inter action engineering (SOPHIE) can also be achieved by using homogeneous optically anisotropic solid crystals, as earlier shown in [4 6] for diffracting beams and in [START_REF] King | Formation of higher order Bessel light beams in biaxial crystals[END_REF][START_REF] Khilo | Transformation of the order of Bessel beams in uniaxial crystals[END_REF] for nondiffracting beams; see [START_REF] Fadeyeva | Spatially engineered polarization states and optical vor tices in uniaxial crystals[END_REF] for a review.

Here we propose to use SOPHIE to impart various kinds of periodic modulation of the spatial distribution of intensity, phase, and polarization to nondiffracting Bessel like beams propagating in free space. As such, these results go beyond the previously dis cussed SOPHIE of paraxial [START_REF] Khilo | Transformation of the order of Bessel beams in uniaxial crystals[END_REF] and nonparaxial [START_REF] Khonina | Comparative investigation of nonparaxial mode propagation along the axis of uniaxial crystal[END_REF] Bessel beams that lead to spatially modulated Bessel beams in uniaxial dielectric media. This is made possible by using an axicon carved in a uni axial crystal. Its SOPHIE capabilities are theoretically outlined within the paraxial approximation, hence assuming flat enough axicons. Various incident fields impinging on the tip of the bi refringent axicon are considered: circularly polarized Gaussian beams, incoherent superposition of contracircularly polarized Gaussian beams, and circularly polarized Laguerre Gaussian beams, for which SOPHIE leads to distinct classes of spatially modulated fields. Then we report on experimental observations and discuss possible application potential for the optical manipu lation of atoms, micro objects, and soft matter.

GENERAL FRAMEWORK A. Transverse-to-Longitudinal Structuring

An option to realize free space spatial modulation of Bessel beams along the propagation direction is to interfere two copropagating or contrapropagating coherent Bessel beams. In the former case, periodic intensity modulation is obtained when the individual propagation constants differ [START_REF] Cerda | Experimental observation of interfering Bessel beams[END_REF]. On the other hand, the two beams can also have identical propagation constant magnitude when they travel in opposite directions [START_REF] Cizmar | Optical conveyor belt for delivery of submicron objects[END_REF]. It is also possible to use a single beam focused by an (isotropic) axicon provided that the incident beam is endowed by transverse spatial modulation, say with period Λ. Indeed, transverse modulation is mapped into longitudinal modulation with period Λ∕ tan γ (γ is the external geometrical tilt angle at the output of the axicon; see Fig. 1) along the line of focus of the axicon. Such a transverse to longitudinal structuring strategy has been realized by using higher order radial Laguerre Gaussian incident beams [START_REF] Mcgloin | Three dimensional arrays of optical bottle beams[END_REF]. However, the longi tudinal modulation obtained is limited to intensity modula tion, is only approximately periodic, and has a limited number of oscillations directly governed by the radial index p of the Laguerre Gauss mode used.

In our case, we exploit the transverse to longitudinal structur ing strategy mentioned above to achieve SOPHIE of Bessel beams using Laguerre Gauss beams having p 0. This is made possible when using a birefringent axicon by exploiting the anisotropy between radially and azimuthally polarized fields that are polari zation eigenstates for light propagating along the optical axis of a uniaxial medium [START_REF] Muys | Polarization eigenstates of an axially symmetric laser beam in a c cut uniaxial crystal[END_REF]. Before analyzing specific capabilities offered by SOPHIE, it is useful to present a general description in the case of axisymmetric incident paraxial fields impinging at normal incidence on an axisymmetric complex transmittance mask that acts as an axicon lens (defined by external geometrical angle γ) is endowed with slowly varying amplitude along the radial coordinate (defined by characteristic length Λ).

B. Field Description

Since Laguerre Gauss beams form an appropriate exact complete basis of the paraxial Helmholtz equation in the cylindrical co ordinate system, we consider an arbitrarily polarized incident Laguerre Gaussian beam with radial index p 0 and arbitrary azimuthal index l that propagates toward z > 0 and impinges on the transmittance mask at z 0, which also defines the beam waist location. Just before the mask, the complex representa tion of the incident field is written (omitting the common time factor e -iωt )

E l r; φ; z 0 -E l u l r∕we ilφ ac bc -; (1)
where E l is a constant amplitude factor, u l r∕w r∕w jl j exp r 2 ∕w 2 with w the beam waist radius, r; φ; z are the cylin drical coordinates, and c x iσy∕ 2 p refers to the circular polarization basis expressed in the Cartesian unit basis x; y; z with jaj 2 jbj 2 1.

Then we describe the transmitted field in the Debye approxi mation, where the field nearby the focus of an imaging system is obtained by integrating the propagated partial plane waves that reach the focus [START_REF] Wolf | Electromagnetic diffraction in optical systems. I. An integral rep resentation of the image field[END_REF]. Importantly, within this approximation, the plane wave spectrum is given by the transmitted field at z 0 . In the present case, the imaging lens is an axicon and, therefore, the partial waves of interest that reach on axis point M are limited to those coming from the transmittance mask apodized by an effective annular aperture (A) with typical angular width δθ λ∕Λ around the angle of view θ γ; see the sketch in Fig. 2. Following this sketch, the field at z (not too close from the mask) near the axis has the general form, up to a common phase propa gation factor,

E l r; φ; z > 0 ∝ Z A E 0 l z tan θ; ϕ; 0 e iks•r dΩθ; ϕ: (2)
In Eq. ( 2) above, E l ρ; ϕ; 0 E 0 l ρ; ϕ; 0 e -ikρ tan γ , where the phase factor accounts for the focusing properties of the axicon, with ρ the distance from a point P of the mask plane to the z axis. In addition, k 2π∕λ is the wavenumber in free space with λ the wavelength, s is the unit vector pointing from P to the focus point M, r is the position vector in the observation plane, and d Ωθ; ϕ sin θd θd ϕ is the elementary solid angle when look ing at P from M; see Fig. 2.

The evaluation of the structured light field therefore requires the determination of E 0 l ρ; ϕ; 0 . This is done in the next sub section in the particular case of interest, when the transverse modulation results from spin orbit interaction of light as it passes through a c cut birefringent axicon.

C. SOPHIE from a Birefringent Axicon

Within the paraxial approximation, the complex transmittance of the birefringent axicon is approximated by a 2 × 2 Jones matrix Tρ; ϕ operating on the transverse component of the incident field. Indeed, quoting Wolf and Li in [START_REF] Wolf | Conditions for the validity of the Debye integral rep resentation of focused fields[END_REF], it is known that, when the maximum tilt angle of a partial wave is less than 30°(i.e., γ < 30°), polarization effects are not very significant in the vicin ity of the focus. Such a statement has been elaborated in a pre vious pioneering study [START_REF] Richards | Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system[END_REF] and we will quantitatively comment on this point in Section 6. That said, one has

E l ρ; ϕ; 0 Tρ; ϕ • E l ρ; ϕ; 0 -; (3)
where T accounts both for the conical tilt imparted to the in cident field by the axicon and the spin orbit interaction of light as a result of different propagation constants for radially and azimuthally polarized eigenstates that are, respectively, associated with extraordinary (n e ) and ordinary (n o ) refractive indices of the uniaxial crystal. More precisely, n e β n jj n ⊥ ∕n 2 jj cos 2 β n 2 ⊥ sin 2 β 1∕2 for a partial wave that propagates at an angle β from the optical axis inside the birefringent axicon (see Fig. 1), and n o n ⊥ , where n ⊥;jj are the refractive indices of the uniaxial crystal perpendicular and parallel to the optical axis. The deriva tion of T follows from a straightforward Jones description, namely, Tρ; ϕ R ϕ Pρ Rϕ, where R is the rotation ma trix and P refers to the propagation in the local extraordinary/ ordinary frame of the birefringent axicon. From the following ex pressions in the linear polarization basis x; y,

Rϕ cos ϕ sin ϕ sin ϕ cos ϕ ; (4)
and Pρ e -ikρn e β-1α 0 0 e -ikρn ⊥ -1α ;

(5) one obtains after calculations the expression of T in the circular polarization basis c ; c -:

Tρ; ϕ exp i Δρ 2 ikρn ⊥ 1α × Cρ iSρe -2iϕ iSρe 2iϕ Cρ ; (6)
with Cρ cosΔρ∕2 and Sρ sinΔρ∕2, where Δρ is evaluated to the lowest order in α, noting that β ≃ 1 1∕n ⊥ α. This gives

Δρ n ⊥ 1 2 n 2 ⊥ n 2 ∥ ∕2n ⊥ n 2 ∥ α 3 kρ: (7)
Remarkably, Eq. ( 6) is reminiscent of the description of the spin orbit effects in a space variant uniaxial plate endowed with a radial optical axis distribution and a birefringent phase retardation Δ that scales linearly with the distance to the propagation axis, which echoes an analogy previously discussed in [START_REF] Loussert | Efficient scalar and vectorial singular beam shaping using homogeneous anisotropic media[END_REF]. Since γ ≃ n ⊥ 1α, Eqs. ( 3) and ( 6) allow us to identify the term E 0 l ρ; ϕ; 0 in Eq. ( 2). Indeed, noting that jΔρ∕kρn ⊥ 1αj ≪ 1 is always satisfied within the present paraxial frame work, one gets

E 0 l ρ; ϕ; 0 Cρ iSρe -2iϕ iSρe 2iϕ Cρ E l ρ; ϕ; 0 -: (8)
As one could expect, the 2 × 2 matrix in Eq. ( 8) corresponds to the spin orbit transformation that describes the dynamics of optical spin orbit coupling in uniaxial crystals [START_REF] Brasselet | Dynamics of optical spin orbit coupling in uniaxial crystals[END_REF]. Moreover, Eq. ( 8) allows identifying the characteristic length Λ of transverse modulation as ΔΛ 2π:

Λ 2λn ⊥ n 2 ∥ n ⊥ 1 2 n 2 ⊥ n 2 ∥ α 3 : (9)

D. Field Evaluation

Integration of Eq. ( 2) is carried out by inserting Eq. ( 8) into Eq. ( 2). First, integration along ϕ is done by noting that s • r r sin θ cosϕ φ and exploiting the identities R 2π 0 exp ix cosϕ φ il ϕdϕ 2πi l expil φJ l x and J l x 1 l J l x, where J l denotes the Bessel function of the first kind of order l. Then, integration along θ is performed, recalling that slowly varying transverse modulation of the field at z 0 im plies a distribution peaked on θ γ provided that δθ ≪ γ, hence, λ∕Λ ≪ γ [condition 1]. One can thus keep only the value of the integrand at θ γ provided that krδθ ≪ 1, hence r ≪ Λ∕2π [condition 2], and that zδθ ≪ Λ, hence z ≪ Λ 2 ∕λ [condition 3]. After calculations, one gets, in the circular polari zation basis,

E l r; φ; z > 0 ∝ E l u l z∕z max e il φ × CγzJ l γkr iSγzJ l -2 γkre -2iφ iSγzJ l 2 γkre 2iφ CγzJ l γkr a b ; (10)
where z max w∕γ is the characteristic quasi diffraction free dis tance of a Bessel beam generated by a typical axicon [START_REF] Mcgloin | Bessel beams: diffraction in a new light[END_REF]. Equation [START_REF] Khonina | Comparative investigation of nonparaxial mode propagation along the axis of uniaxial crystal[END_REF] emphasizes that SOPHIE leads to the transfor mation of an incident arbitrarily polarized Laguerre Gaussian beam with radial order zero and azimuthal order l into a coherent superposition of self imaged beams. Namely, the output field con sists of two c polarized beams endowed with Bessel features of order l and l 2 and two c -polarized beams involving orders l and l 2, all of them being spatially modulated along the beam propagation.

PLANE WAVE EXCITATION

SOPHIE of Bessel beams is first illustrated in the ideal situation of a circularly polarized incident plane wave, which implies the use of Eq. ( 10) in the limit w → ∞. The simulations are per formed using wavelength λ 633 nm and angle α 20°, and choosing the corresponding parameters of LiNbO 3 , which is the uniaxial material used in experiments (see Section 5), namely, n ⊥ 2.286 and n ∥ 2.203, which gives Λ ∼ 850λ. This allows us to check the three conditions for the validity of Eq. [START_REF] Khonina | Comparative investigation of nonparaxial mode propagation along the axis of uniaxial crystal[END_REF] given in Section 2.D. Namely, condition 1 is validated since λ∕Λ ∼ 10 -3 ≪ γ ∼ 0.45. On the other hand, condition 2 implies observation at distance r ≪ 85 μm from the axis, whereas con dition 3 implies observation at distance z ≪ 50 cm from the birefringent axicon, both conditions being fulfilled in Fig. 3, which shows the basic situation of a c polarized incident plane wave (l 0).

The total intensity distribution in the vicinity of the z axis is shown in Fig. 3(a) over the typical range 1 < z∕Λ∕γ < 10 that corresponds to nine self imaging spatial periods. The normalized intensity patterns of the c polarized components are also shown in Figs. 3(b) and 3(c). As expected from Eq. ( 10), the copolarized component (here c ) corresponds to a fundamental Bessel beam with spatially modulated on axis intensity, whereas the contrapo larized component (here c -) corresponds to a second order Bessel beam with a spatially modulated doughnut intensity profile. This is illustrated in Figs. 3(d) and 3(e) and Figs. 3(f ) and 3(g), where the transverse intensity (I ) and phase (Φ ) profiles associated with local intensity maximum are shown. In particular, the char acteristic on axis optical phase singularity with topological charge 2 associated with a second order Bessel beam is shown in Fig. 2(g). Noticeably, the sign of the latter topological charge is reversed when the helicity of the incident field is flipped, whereas the intensity patterns remain unchanged, which is a basic feature of SOPHIE.

LAGUERRE-GAUSSIAN BEAM EXCITATION A. General Considerations

Recalling that the above plane wave considerations represent an idealized proof of principle of SOPHIE of Bessel beams, a real istic approach is carried out by considering Laguerre Gaussian incident beams [see Eq. ( 1)] whose finite transverse spatial extent is dictated by the beam waist radius w. Assuming a large enough beam waist, the characteristic length Λ remains defined by Eq. ( 9), which typically implies w > 1 mm. A consequence of finite beam size is that on axis periodic intensity modulation has a Laguerre Gaussian envelope proportional to u 2 l z∕z max , as one can see from Eq. [START_REF] Khonina | Comparative investigation of nonparaxial mode propagation along the axis of uniaxial crystal[END_REF]. Therefore, only a limited number N of longitudinal oscillations may be of practical interest. Taking w 1 cm, which is the value used in simulations, the validity of the above three conditions is thus fulfilled and one gets N jΔwj∕2π ∼ 20.

B. Self-Imaged Optical Bottles

Since the decomposition into circularly polarized components shown in Fig. 3 exhibits alternating bell shaped [Fig. 3(b)] and doughnut shaped [Fig. 3(c)] intensity profiles along the z axis, this gives a basis to build up self imaged bottle beams, i.e., an array of zeros of intensity surrounded by bright regions in three dimensions [START_REF] Arlt | Generation of a beam with a dark focus sur rounded by regions of higher intensity: the optical bottle beam[END_REF]. Our idea is to prepare an incident field in order that SOPHIE leads to modulated Bessel beams of order 0 and 1 that are shifted along z by half the longitudinal period Λ∕γ.

For this purpose, there are several options. For instance, one can use an incident field that results from the incoherent super position of two Laguerre Gaussian beams: a c σ (σ 1) polar ized beam of order l 0 and a c -σ polarized beam of order l σ. Then, by selecting the c σ polarized component of the output field, one obtains the desired alternating sequence of bell shaped and doughnut shaped intensity profiles along the z axis. However, the two slowly varying intensity envelopes u 2 0 and u 2 1 [see Eq. ( 10)], are shifted along the z axis and lead to an array of op tical bottles where the transverse and longitudinal "intensity caps" differ. This is summarized in Fig. 4, which shows the individual contributions of the two incident beams with identical beam waist [Figs. 4(a) and 4(b)] and their sum [Fig. 4(c)]. Still, rather iso tropic optical bottles can be obtained at any desired place by adjusting the ratio between amplitudes E σ and E 0 [see Eq. ( 1)], as shown in Fig. 4(c). In that figure, the targeted location is z∕z max ≃ 0.4, and the corresponding ratio is E σ ∕E 0 ≃ 4.4. An enlargement of the latter bottle is displayed in Fig. 4(d). Both the transverse and longitudinal intensity profiles of the bottle are shown in Figs. 4(e) and 4(f ) [see dark dashed lines in Fig. 4(d)]. In addition, the isotropic character of the obtained bottle is assessed by looking at the intensity profile along the "intensity valleys" [see white dashed line in Fig. 4(d)], as shown in Fig. 4(f ). A rather good bottle is obtained, noting that the relative modulation between peak and valley intensity values around the three dimensional dark core is μ I peak I valley ∕ I peak 0.44.

The above limitations for obtaining a single quasi isotropic bottle can be circumvented by using two contracircularly polar ized Gaussian beams. In that case, the output intensity pattern does not depend on the selected circular polarization state. Indeed, whatever is the handedness of the selected output circular polarization, one gets the superposition between modulated Bessel beams of order 0 and 2 that are shifted along z by half the longitudinal period Λ∕γ. This is illustrated in Fig. 5, which is presented in a manner similar to Fig. 4 for an intensity ratio ≃2.1 between the c -σ and c σ polarized incident beams. However, this is at the expense of optical bottle isotropy (modulation parameter increases to μ 0.76), since the central doughnut intensity pat tern of a second order Bessel beam has a larger radius than a first order one, as emphasized in Figs. 5(d) 5(f ).

By proposing a SOPHIE approach based on a uniaxial axicon, these results contribute to previous efforts to generate free space single or multiple optical bottles using slabs of uniaxial [START_REF] Shvedov | Generation of optical bottle beams by incoherent white light vortices[END_REF][START_REF] Shvedov | Generation of vector bottle beams with a uniaxial crystal[END_REF] and biaxial [START_REF] Turpin | Optical vault: a reconfigurable bottle beam based on coni cal refraction of light[END_REF][START_REF] Loiko | Generating a three dimensional dark focus from a single conically refracted light beam[END_REF] crystals. This may be useful for Bessel beam based optical trapping and manipulation of microparticles, as done from the beginning of the 2000 s in a liquid environment [START_REF] Cizmar | Optical conveyor belt for delivery of submicron objects[END_REF][START_REF] Arlt | Optical microma nipulation using a Bessel light beam[END_REF][START_REF] Chavez | Simultaneous micromanipulation in multiple planes using a self reconstructing light beam[END_REF]. In particular, noting that the aspect ratio of the optical bottles obtained is typically Λ∕γλ ∼ 10 3 with the present para meters, the proposed approach should be especially useful in the case of a single or a chain of high aspect ratio micro objects. In particular, the photophoretic manipulation [START_REF] Shvedov | Optical guiding of absorbing nanoclusters in air[END_REF][START_REF] Desyatnikov | Photophoretic manipulation of absorbing aerosol particles with vortex beams: theory versus experiment[END_REF] of carbon nanotubes in gaseous media can be considered.

C. Corrugated Optical Hollow Pipes

Exploiting the above strategy, we can also access another kind of optical field modulation along the propagation direction, namely, corrugated optical hollow pipes that correspond to self imaged hollow beams. Among the possible options, one can use two con tracircularly polarized Laguerre Gaussian beams of order l ≠ 0 and select the c -jl j∕l polarized component of the output field. Indeed, one obtains in that case a superposition between modu lated Bessel beams of order l and l 2 sgnl that are shifted along z by half the longitudinal period Λ∕γ. In turn, corrugated hollow pipes with increasing inner core diameter are obtained as jl j increases. The results are shown in Figs. 6(a) 6(c) for l 1; 2; 3 and a ratio of amplitudes 1.33, 1.22, and 1.17 between the c and c -polarized incident beams, respectively. This kind of hollow pipe equips Bessel beam based atomic waveguides [START_REF] Arlt | Optical dipole traps and atomic waveguides based on Bessel light beams[END_REF] with periodic features.

D. Self-Imaged Orbital Angular Momentum Hollow Beam

We also propose another kind of spatially modulated field that consists of quasi nondiffracting beams with self imaged orbital angular momentum. This is achieved by using a c σ polarized Laguerre Gaussian beam of order l σ, and the results in the case of σ 1 are presented in Fig. 7. The hollow intensity map obtained is reminiscent of first order Bessel beams; see Fig. 7(a). More precisely, the output field is the superposition of a contra circularly polarized modulated Bessel beam of order l 1 that is shifted along z by half the longitudinal period Λ∕γ. Therefore, the orbital angular momentum density is modulated along the z axis. This is emphasized in Fig. 7(b), where the z invariant in tensity pattern is encoded as luminance, and the orbital angular Fig. 5. Same as in Fig. 4 in the case of two contracircularly polarized fundamental Gaussian beams (l 0) for a ratio of amplitude ≃2.1 between the c σ and c σ polarized incident beams. Fig. 6. Total intensity distribution nearby the z axis of corrugated optical hollow pipes of various inner diameter obtained in the case of two con tracircularly polarized Laguerre Gaussian beams of order l ≠ 0, by selecting the c jl j∕l polarized component of the output field. Results are shown for jl j 1; 2; 3 for a ratio of amplitudes 1.33, 1.22, and 1.17 between the c σ and c σ polarized incident beams.

momentum density M z is encoded in color scale. The latter is evaluated recalling that M z is proportional to the difference between the intensity of the beam carrying positive orbital angular momentum and that of the one carrying negative orbital angular momentum [START_REF] Soskin | Topological charge and angular momentum of light beams carrying optical vortices[END_REF]. That is to say, M z r; z ∝ σI -σ I σ , where I refers to the intensity of the c polarized circular component of the output field. In addition, the optical vortex skeleton of such a field is assessed in Figs. 7(c) and 7(d), which show the transverse phase profiles at two orbital angular momentum hot spot locations.

Arguably, a practical drawback of the latter option could be its inhomogeneous polarization state. However, this issue can be circumvented by preparing an incoherent superposition of c σ and c -σ polarized Laguerre Gaussian beams of order l σ with the same amplitude as the incident field. In that case, by selecting the c σ polarized component of the output field, one gets a uniformly polarized field endowed with self imaged orbital angular momen tum. Consequently, the output polarization state can be con trolled at will without affecting the orbital angular momentum content.

Noticeably, such optical fields unveil a novel application potential owing to their ability to transfer orbital angular momen tum in an alternating manner along the beam propagation direc tion. These beams are thus capable of shearing matter, which might be useful in soft matter viscoelastic measurements. Such "optical shearers" are another tool in completing the existing optical manipulation toolbox that already includes optical tweez ers [START_REF] Grier | A revolution in optical manipulation[END_REF], optical spanners [START_REF] Simpson | Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner[END_REF], optical stretchers [START_REF] Guck | The optical stretcher: a novel laser tool to micromanipulate cells[END_REF], and optical twisters [START_REF] Xavier | Three dimensional optical twisters driven helically stacked multi layered micro rotors[END_REF].

EXPERIMENTAL ATTEMPT

Our experimental approach is realized by using a birefringent axicon made of LiNbO 3 with angle α 20°and a 1 in. basis diameter, with 633 nm operating wavelength, as mentioned in Section 3. Here we present results that illustrate SOPHIE in the case of an incident c polarized Gaussian beam (Fig. 3) with beam waist radius evaluated as w ≃ 6.5 mm. Practically, we tried to image the circularly polarized components of the field near the z axis as a function of z by using various microscope objectives. However, we were not able to clearly distinguish Bessel beam fea tures, probably due to nonideal axisymmetry of the axicon itself. In fact, we recall that the fabrication of an axicon carved in LiNbO 3 remains a challenge itself. Therefore, instead of scanning the field along the z axis, we choose to analyze the output field at the distance z max from the birefringent axicon [see Fig. 8(a)], which can be directly observed on a screen. By doing so, one re trieves information on SOPHIE since the x; y observed field can be considered to be the longitudinal to radial projection of the field.

Experimental data are shown in Figs. the phase characteristics of SOPHIE are assessed by measuring the spatial distribution of the azimuth angle ψ of the polarization ellipse from the reduced Stokes parameters s 1 and s 2 following ψ 1∕2 arctans 2 ∕s 1 . Indeed, this gives access to the optical phase difference between the c components, δΦ Φ Φ -, from the relationship δΦ 2ψ [START_REF] Brasselet | Optical vortices from liquid crystal droplets[END_REF].

Comparison between observations and predictions can be as sessed within a ray optics description. Indeed, the characteristic diffraction length associated with the highest spatial frequency of the output beam just after the birefringent axicon, which scales as Λ 2 ∕λ, is of the order of 1 m and is typically two decades larger that z max . Therefore, one can approximate the field at z z max as the 1:1 inverted image of the field at z 0 ; see Eq. [START_REF] Khilo | Transformation of the order of Bessel beams in uniaxial crystals[END_REF]. Results are shown in Figs. 8(d) and 8(e), which exhibit fair agreement.

DISCUSSION

As mentioned in Section 2.C, we have restricted our approach to paraxial analysis, which is consistent with the magnitude of the involved partial wave tilt angle θ with respect to the propagation axis. It remains, however, instructive to quantify nonparaxial spin orbit effects, since they are arguably expected to contribute to SOPHIE of light. First, we notice that nonparaxial spin orbit contributions refer not only to higher order corrections of spin orbit effects due to the material anisotropy, but, indeed, they also arise from mere reflection and refraction of light as reviewed in [START_REF] Bliokh | Spin to orbital angular momentum conversion in focusing, scattering, and imaging systems[END_REF]. More precisely, for moderate angles, the magnitude of the latter contributions scale as ζ ⊥ θ 2 ∕4 and ζ z θ 2 p for the transverse and longitudinal parts of the field, respectively [START_REF] Bliokh | Spin to orbital angular momentum conversion in focusing, scattering, and imaging systems[END_REF].

Regarding the transverse field transformation made by the bi refringent axicon [see Eq. ( 8)], the relevant angle is β and gives ζ ⊥ ≃ 0.01. This should be compared with the nondiagonal spin orbit term in Eq. [START_REF] Khilo | Transformation of the order of Bessel beams in uniaxial crystals[END_REF], which scales as sinΔ∕2, and where Δ is the birefringent phase retardation arising from the material anisotropy of the uniaxial crystal. Since the periodic field modulation dis cussed in the present work is related with values of Δ covering a range that is several times larger than 2π, it is relevant to neglect the nonparaxial spin orbit contributions overall. On the other hand, longitudinal correction magnitude ζ z 0.28 implies the appearance of a vortex in the longitudinal field component of topological charge l σ with intensity corrections (∝ ζ 2 z ) of the order of ≲10%. The ensuing consequences should be evaluated by using a 3 × 3 matrix field transformation [START_REF] Bliokh | Spin to orbital angular momentum conversion in focusing, scattering, and imaging systems[END_REF], which can be the purpose of a future work.

Summarizing, we demonstrated that homogeneous optically anisotropic crystals can be used as SOPHIE optical elements capable of producing light fields with various kinds of self imaged properties without a specific wavelength requirement. In particu lar, we unveiled the realization of periodic bottle beams, corru gated hollow beams, and hollow needle beams with spatially modulated optical angular momentum, from a single incident beam of the Laguerre Gauss family. Extension to other kinds of paraxial beams (e.g., Hermite Gaussian or Ince Gaussian) is likely to extend the application potential of the proposed self imaged beam shaping technique, which may find use in optical trapping and manipulation of small scale objects and soft matter.

Funding. French National Research Agency (ANR) "Investments for the future" Programme IdEx Bordeaux LAPHIA(ANR 10 IDEX 03 02).

Fig. 1 .

 1 Fig.1. SOPHIE geometry. An axisymmetric light beam with finite characteristic transverse size w impinges on a birefringent axicon, defined by the angle α, carved in a uniaxial crystal whose optical axis (see double arrow) lies along its axis of symmetry. Ray tracing for incident rays parallel to the z axis at a distance w from the z axis allows one to define the diffraction free characteristic distance z max . β refers to the geometrical internal tilt angle with respect to the optical axis, and γ is the external angle. Λ refers to the characteristic spatial scale associated with transverse modulation of the field at the output of the axicon that is mapped into characteristic longitudinal modulation Λ∕ tan γ.

Fig. 2 .

 2 Fig.2. Notations used for the evaluation of the light field at a distance z from the transmittance mask located at z 0, near the propagation axis (in practice, up to several wavelengths from the z axis). See text for details.

Fig. 4 .

 4 Fig. 4. Intensity distribution near the z axis of the c σ polarized component of the field in the case of incoherent superposition of two Laguerre Gaussian beams: (i) a c σ (σ 1) polarized beam of order l 0 and (ii) a c σ polarized beam of order l σ. (a) Beam (i) alone. (b) Beam (ii) alone. (c) Beams (i) and (ii) together for a ratio of amplitude E σ ∕E 0 ≃ 4.4. (d) Enlargement of the region near z∕z max ≃ 0.4 that cor responds to an optical bottle. (e) Normalized transverse intensity profile; see vertical black dashed line in (d). (f) Solid curve. normalized longi tudinal intensity profile [see horizontal black dashed line in (d)]; dashed curve, intensity profile along the diagonal white dashed line in (d).

Fig. 3 .

 3 Fig. 3. (a) Total intensity distribution near the z axis in the case of a c polarized incident plane wave. The intensity distribution of the c polarized components are shown in (b) and (c), whereas the transverse intensity (I ) and phase (Φ ) profiles associated with a local intensity maximum (see dashed vertical lines) are shown in (d), (e) and (f), (g).

  8(b) and 8(c), where Fig. 8(b) represents the intensities I of the c polarized compo nents and the phase difference δΦ between the c components, respectively. The complementary periodic intensity modulation of I along the radial direction is the signature of the periodic modulation shown in Figs. 3(b) and 3(c), which corresponds to the c polarized incident Gaussian beam. On the other hand,

Fig. 7 .

 7 Fig. 7. (a) Total intensity near the z axis of a hollow beam with self imaged orbital angular momentum generated from an incident c Laguerre Gaussian beam of order l 1. (b) Orbital angular momen tum density M z r; z color map, where luminance is proportional to the total intensity shown in panel (a). Transverse phase profiles of the c and c polarized components are shown in (c) and (d), respectively.

Fig. 8 .

 8 Fig. 8. (a) Illustration of the experimental approach: the output light field is observed at a distance z z max from the birefringent axicon for a c polarized incident Gaussian beam. (b) and (d) Juxtaposed transverse intensity distribution I of the c polarized components of the output field at a distance z obs from the axicon above which a central shadow appears. (c) and (e) Corresponding phase difference δΦ between the c polarized components. (b) and (c) Experiments. (d) and (e) Simulations. Incident beam waist is w ≃ 6.5 mm.

Acknowledgment. This study has been carried out with financial support from the French State, managed by the French National Research Agency (ANR) in the frame of the Investments for the future Programme IdEx Bordeaux LAPHIA (ANR 10 IDEX 03 02).