Etienne Sirot
email: sirot@univ-ubs.fr

Pierrick Blanchard

Anne Loison

Olivier Pays

How vigilance shapes the functional response of herbivores

Keywords: Antipredator strategies, foraging strategies, functional response, herbivores, predators, trophic cascades, vigilance

Abstract

1. Herbivores in quest of food have to attend areas where predators also concentrate their attacks. They have to deal with two conflicting demands, the need of feeding and that of avoiding predation. Using a theoretical approach, we study the influence of these antagonistic constraints on the shaping of an herbivore's foraging strategy and their consequences in terms of functional response.

2. In our model, the animal may curtail predation risk by being vigilant while foraging and by limiting its time of presence in the feeding area. Being vigilant means either stopping all feeding activities to scan for predators (exclusive vigilance) or using the time spent handling resources to do so, with reduced detection capacities (routine vigilance). Using an optimization criterium, we thus determine how long the animal will stay in the feeding area and how it will share its time and attention while present there, as a function of resource density and predation risk.

3. All behavioural variables show smooth but sometimes non-monotonic variations in response to changes in these environmental conditions. When these variations are taken into account, the instantaneous intake rate of the herbivore, as well as the total biomass of plant consumed, increase in a decelerating but continuous manner with the patch's richness, in qualitative agreement with a type II functional response. This smooth pattern is explained by the possibility for the herbivore, when food abundance rises, to progressively redirect its attention to routine antipredator vigilance.

4. The relative importance of routine vigilance with respect to exclusive vigilance declines when predation risk increases, as well as the duration of the patch's visit and the total quantity of food eaten. The exact shape of the functional response is thus mediated by subtle adjustments of behavioural and attentional priorities, in response to variations in both food availability and predation risk.

Introduction

Functional response, that is, the relationship between the quantity of resource available and the quantity actually eaten by each consumer, is a major driver of population dynamics [START_REF] Gotelli | A primer of Ecology[END_REF]. Identifying realistic functional responses remains challenging, however, because many activities interfere with feeding and influence individual intake rates in the wild (e.g., [START_REF] Skalski | Functional responses with predator interference: viable alternatives to the Holling type II model[END_REF]. Antipredator defense, in particular, can monopolize large portions of foraging time and influence both the time spent in the feeding area and the quantity of food collected by each individual (e.g., [START_REF] Hochman | Patch use, apprehension, and vigilance behavior of Nubian Ibex under perceived risk of predation[END_REF]. Its effects on an animal's functional response may thus be high (Fortin et al., 2004a;[START_REF] Baker | Vigilance and the functional response of granivorous foragers[END_REF].

The question of relative investment in feeding and in antipredator defense is particularly acute in wild herbivores, which must devote long periods to foraging every day (e.g., [START_REF] Mysterud | The relative roles of body size and feeding type on activity time of temperate ruminants[END_REF], during which they are often exposed to high levels of threat (e.g., [START_REF] Mónus | Is foraging time limited during winter ? -A feeding experiment with tree sparrows under different predation risk[END_REF]. The activities of a foraging herbivore indeed encompass a set of combined decisions simultaneously related to its metabolic needs, to the characteristics of plant patches, and to the modalities of predator activity (e.g., [START_REF] Pöysä | Feeding-vigilance trade-off in the teal (Anas crecca): effects of feeding method and predation risk[END_REF][START_REF] Berger-Tal | Complex state-dependent games between owls and gerbils[END_REF][START_REF] Blanchard | Coping with change of predation risk across space and time through complementary behavioral responses[END_REF]. Within this context, the way the animal gathers information about its environment is of major importance, for different and conflicting reasons. First, attention dedicated to the food patch allows a forager to locate resources or advantageous feeding places [START_REF] Underwood | Vigilance behavior in grazing African antelopes[END_REF].

Second, observation of the surroundings, through vigilance, gives a better chance of detecting and escaping predators [START_REF] Lind | Determining the fitness consequences of antipredation behavior[END_REF]. Although an animal may sometimes receive concomitant signals of different nature [START_REF] Lima | Back to the basics of antipredatory vigilance: can nonvigilant animals detect attacks?[END_REF], it cannot respond to each of them with maximal efficiency. All depends on the way its attention is shared, which reflects its priorities [START_REF] Dukas | The cost of limited attention in blue jays[END_REF][START_REF] Hochman | Patch use, apprehension, and vigilance behavior of Nubian Ibex under perceived risk of predation[END_REF].

The present study deals with this question, by investigating the topic of time-and attention sharing strategies for a foraging herbivore exposed to predation, and its functional consequences. To this end, we use a theoretical approach and take into account different features of herbivore behaviour which, to our knowledge, have both received empirical support and motivated theoretical inquiry, but have not been merged within a common theoretical framework.

First, we consider that an herbivore may respond to predation threat both by limiting the time it spends in the foraging area, where predators often concentrate their attacks [START_REF] Valeix | Does the risk of encountering lions influence African herbivore behavior at waterholes?[END_REF], and by being more vigilant while present in this area [START_REF] Lendrem | Predation risk and vigilance in the blue tit (Parus caeruleus)[END_REF].

Considering these two possibilities does indeed considerably enlarge the panel of possibilities for the animal's responses to predators' presence [START_REF] Brown | Vigilance, patch use and habitat selection: foraging under predation risk[END_REF][START_REF] Sirot | Adjustments in compound defensive strategies in response to variation in predation risk[END_REF], and field studies confirm simultaneous adjustments of patch time and vigilance in response to variations in predation risk (e.g., [START_REF] Lima | Maximizing feeding efficiency and minimizing time exposed to predators: a trade-off in the black-capped chickadee[END_REF][START_REF] Toïgo | Vigilance behavior in lactating female Alpine ibex[END_REF][START_REF] Embar | Risk management in optimal foragers: the effects of sightlines and predator type on patch use, time allocation and vigilance in gerbils[END_REF].

Second, we consider the different types of antipredator vigilance that exist in the wild.

An herbivore may indeed totally stop feeding to scan for predators, but it may also be vigilant while handling seeds, fruits or leaves [START_REF] Popp | Effects of food handling time on scanning rates among American goldfinches[END_REF][START_REF] Cowlishaw | A simple rule for the cost of vigilance: empirical evidence from a social forager[END_REF] or while chewing [START_REF] Illius | Costs of vigilance in foraging ungulates[END_REF]Fortin et al., 2004b). This situation is however expected to lower its ability to detect predators, because handling postures may partially prevent scanning (for animals handling discrete food items; [START_REF] Kaby | What limits predator detection in blue tits (Parus caeruleus): posture, task or orientation?[END_REF] and because chewing hinders sensorial abilities (for grazers; [START_REF] Blanchard | Induced or routine vigilance while foraging[END_REF]. Following a terminology that applies to both kinds of herbivores, we will thus consider two distinct types of vigilance [START_REF] Baker | Are the costs of routine vigilance avoided by granivorous foragers?[END_REF][START_REF] Périquet | Influence of immediate predation risk by lions on the vigilance of prey of different body size[END_REF]. First, exclusive vigilance implies an interruption of all feeding activities and provides the best detection capacities. Second, routine vigilance allows the individual to scan for predators while handling food, including chewing (see [START_REF] Spalinger | Mechanism of foraging in mammalian herbivores: new models of functional response[END_REF], with reduced detection capacities. Because they monopolize attention, both exclusive and routine vigilance are incompatible with food search.

In the first part of this work, and for the sake of comparison, we study the simplified situation where the herbivore is not subject to predation. It thus dedicates all its attention to the exploration of the food patch. The possibility to detect resources is however greater when the animal raises its head, be it handling food or not, than when it collects food, because it then dominates the food patch, can turn its head in any direction, and can move (e.g., [START_REF] Hopewell | Grazing and vigilance by Soay sheep on Lundy island: influence of group size, terrain and the distribution of vegetation[END_REF][START_REF] Baker | Vigilance and the functional response of granivorous foragers[END_REF]. Our aim is to measure the influence of these varying levels of exploration capacities on the forager's feeding rate, hence, on the shape of its functional response.

In the second part, we study the influence of predation threat on the herbivore's behavioural choices and functional response. The decisions the animal has to make thus concern the duration of its foraging bout, the sharing of its attention between food patch exploration and predator detection, and its relative investment in exclusive and routine vigilance. This model differs from the first one in the sense that it is not purely mechanistic, but involves the derivation of optimal strategies [START_REF] Maynardsmith | Optimization theory in evolution[END_REF]. Finally, we study the influence of these strategies on the herbivore's functional response.

The model

Influence of food patch observation on functional response.

Here we consider that the herbivore is not subject to predation. Its attention is thus entirely dedicated to food detection. We assume that food detection is only possible when the animal raises its head, even if it is handling food, because it can then both observe the patch and move. By contrast, movements and food detection are hindered when the animal's head clings to the substrate to crop food. In this case, and for the sake of simplicity, we assume that finding new food items is precluded.

The proportions of time respectively devoted to exclusively searching for food, s, to searching for food while handling food, h, and to cropping food, c, verify:

s + h + c = 1
(1).

When exclusively searching for food, the individual finds food with rate aR, where R denotes resource density and a the area explored per unit of searching time. Similarly, food is found with rate a'R when the animal is searching for food while handling, with a' ≤ a, because handling may interfere with searching. The overall rate of food discovery is thus given by:

f = saR + ha'R (2).
The amount of time necessary to crop one unit of food is denoted by C, and the amount of time necessary to handle it, once it has been cropped, by H. The functional response is then given by the intake rate of the herbivore, i, expressed as a function of resource density R. Note that we always have i ≤ f, because the animal can never eat more food than it finds.

Optimal time allocation for an herbivore subject to predation.

We now consider that the individual is foraging under the threat of predation. As a response to the perceived level of threat, it adjusts its levels of routine and exclusive vigilance while foraging, as well as the time it spends on the food patch [START_REF] Favreau | Predators, food and social context shape the types of vigilance exhibited by kangaroos[END_REF].

The animal can never search for food and scan for predators at the same time. The time spent handling food is thus now shared between 'searching food while handling'

(proportion of time h f ) and 'routine vigilance' (proportion of time h r ), with

h f + h r = h.
Similarly, the time spent observing without handling food is shared between 'exclusively searching for food' (proportion of time s) and 'exclusive vigilance' (proportion of time v).

The fifth activity is 'cropping food' (proportion of time c), which is incompatible with observation.

We finally have:

s + c + h f + h r + v = s + c + h + v = 1 (3).
In addition, the individual may at any moment decide to leave the food patch and join a refuge where it is sheltered from predation, but cannot feed. The sixth behavioural variable is thus the time T spent on the patch, with T ≤ T max , where T max denotes maximum foraging time, and T max -T the time spent in the refuge.

As in the preceding model, the quantity of food detected per unit of searching time is

given by equation ( 2) and the actual intake rate by i, with i ≤ f. The quantity of food consumed during the patch visit is given by: Q = iT. The final reward for the forager is an increasing but decelerating function of this quantity:

F(Q) = 1 -exp(-Q) (4)
, where is a constant (see [START_REF] Houston | General results concerning the trade-off between gaining energy and avoiding predation[END_REF]. This reward must however be weighted by the probability of surviving predatory attacks throughout the foraging period. This probability is called S and predatory attacks are assumed to occur randomly, following a Poisson process.

Survival probability S depends on the level of risk , which denotes the instantaneous probability of a predatory attack. Attacks may occur at any moment, but the probability of escaping them increases with the part of its attention the animal devotes to vigilance. More precisely, the herbivore is captured when under an attack with a probability P that decreases when its level of exclusive vigilance, v, and, to a lower extent, its level of routine vigilance, h r , increase, because exclusive vigilance is more efficient than routine vigilance. The link between the probability of capture and vigilance levels is given by:

P(v,h r ) = exp(-(Av + Bh r ))
, with A ≥ B, and the chance of surviving predatory attacks over the entire period spent on the patch by: S(v,h r ,T) = exp(-P(v,h r )T) (5).

The final reward for the forager, when the probability of surviving predatory attacks is taken into account, is thus given by: Fitness

= S(v,h r ,T)F(Q) (6).
This quantity is used to derive the optimal strategy. Because of the high number of variables, this strategy has to be derived numerically. This means that, for a given set of parameters' values, we systematically explore the space of behavioural variables s, c, h f, h r , v and T, and determine, using equation ( 6), the set of variables that maximizes fitness. This set corresponds to the optimal strategy.

Using this procedure, we allow parameters' values to vary in order to evaluate the influence of each of them, as well as their interactions, on the optimal strategy. The results presented below are representative of the different tendencies emerging during this exploration.

The significance of the different variables and parameters is summarized in Table 1.

Results

Influence of food patch observation on functional response.

Here the herbivore is not subject to predation. The aim of this first model is to explore how a mere mechanistic description of its foraging activity, which incorporates differences in food detection efficiency when head position changes, alters the shape of the functional response.

Foraging time is shared between exclusively searching for food (proportion of time s), cropping (proportion of time c) and handling food (proportion of time h). Food detection only occurs when the animal raises its head, be it handling food or not, and the rate of food discovery is given by equation (2).

The intake rate i corresponds to the quantity of food actually consumed per unit of time, with i ≤ f. Two situations are possible.

First, the quantity of food discovered per unit of foraging time, f, may be fully processed during this unit of foraging time. This means that the animal can eat all the food it finds (i.e., i = f) and happens when food density in the environment is not too high. In this situation, harvest rate is limited by encounter rate with resources, and we have:

c + h = Cf + Hf = Ci + Hi ≤ 1 (7).
Re-arrangement of equations ( 1), ( 2) and ( 7) leads to the following expression for the intake rate, as demonstrated in the Appendix: i (8).

Equation ( 8) is from a mathematical point of view equivalent to a type II functional response. Thus, when the density of resources initially increases, the intake rate rises at a decelerating rate (see figure 1), and the proportion of time spent exclusively searching for food s decreases, while the proportions of time spent cropping and handling increase, because more food is consumed (see figure 2).

We remark that the role played by the handling activity in the functional response's equation is variable. All depends on the ability of the herbivore to discover food while already handling. If this ability is high (i.e., a' close to a), then the handling activity does not slow the feeding process, because handling fully contributes to searching. Intake rate is only slowed by the cropping rate. If this ability is low (i.e., a' close to 0), the effect of handling time adds up to that of cropping.

Second, the quantity of food discovered per unit of foraging time may exceed the processing capacity of the animal. This happens when Cf + Hf > 1, which is achieved when : R > R' = 1/(a'H) (9), R' being the resource density above which the animal finds more food than it can eat.

The animal's intake rate is then lower than its discovery rate (i.e., i < f) and it does not search for food any more in an exclusive manner (i.e., s = 0), as the time spent handling food is sufficient to detect all the resources it will eat (see figure 2). We have then:

c + h = Ci + Hi = 1, which gives: c = C/(C + H), h = H/(C + H) and i = 1/(C + H) (10).
The intake rate thus does not increase any more with food density when this density exceeds threshold value R' (see equation ( 9)). Both cropping and handling thus contribute to limit intake rate and determine its maximum value. This maximum rate is nevertheless reached at a lower density of resource if handling is largely compatible with searching for food (see figure 1).

Optimal time-allocation for an herbivore subject to predation.

Influence of food density.

The situation investigated in the present part is more complex, first, because the herbivore now shares its foraging time between 5 different activities (see figure 3), second, because it also decides how long to stay on the patch (see figure 4). Additionally, the amount of food actually consumed may be measured both by the instantaneous intake rate i and by the total quantity Q of food consumed when the patch's visit ends (figure 4).

A general conclusion emerging from our results is that an increasing density of food allows the herbivore to consume more resource while investing more in antipredator defense (see figure 4). Both instantaneous intake rate i and total quantity of food consumed Q do indeed always increase with R (figure 4), while increased defense first takes the form of higher vigilance levels, then, of a reduction of the time spent on the patch (see figures 3 and 4).

What initially allows the herbivore to spend more time in vigilance is that, as food becomes easier to find, the time spent searching for it decreases (i.e., s decreases, see figure 3). The herbivore can thus feed more while spending less time searching for food, which allows it to be more vigilant. Note that, at low food density, i.e., when feeding is limited by encounter rate with resource, exclusive vigilance is the only type of vigilance present. Routine vigilance begins to be observed precisely when no more time is spent in pure observation of the food patch (i.e., h r ≠ 0 when s = 0, see figure 3). The reason is that s = 0 means that the time spent handling resources is sufficient for the animal to discover all the food it consumes.

As a consequence, and as was described in the first part of the study, any further increase in R leads the animal to detect more food while handling than it can actually eat. It then begins to devote part of its handling time to vigilance. From this moment on, the intake rate increases with R not only because food is found more easily, but also because the overall proportion of time dedicated to feeding rises, exclusive vigilance being progressively replaced by routine vigilance. In parallel, an increasing feeding rate allows the animal to leave the food patch earlier (figure 4).

We remark that, while the variations in certain variables are non-monotonic (i.e., h f and v) and if some others may sometimes be equal to 0 (i.e., s and h r ), the combined fluctuations in these variables and in the time dedicated to the patch's visit produce very smooth variations in the feeding rate of the animal while on the patch (i) and in the total quantity of food consumed (Q) (see figure 4). Both quantities regularly increase, with a decelerating rate, with food density.

To evaluate the relative roles of behavioural adjustments on the patch and of patch residence time in this process, we also explore the situation where the possibility to find shelter in a refuge is cancelled, the animal being forced to spend maximum time T max on the patch. Its response to an increasing density of food is then presented in figures 5 and 6.

Compared to the preceding situation, we see that all activities involved in feeding are reduced, for the benefit of exclusive vigilance (figure 5). Food intake rate thus decreases, as the animal invests more time in the most efficient type of vigilance. Nevertheless, by staying longer on the patch, it finally consumes a larger quantity of food, provided it survives predators' attacks.

The functional response curve is thus slightly lowered if we consider intake rate, and slightly raised if we consider the total quantity eaten. In both cases, its general shape remains unaltered (figure 6).

Influence of predation risk.

Here we investigate the influence of predation risk on the herbivore's behaviour, in conditions of low (figures 7 and 8) and high (figures 9 and 10) food density. As previously, we remark that, when food is scarce, the animal devotes part of its time to searching for food exclusively and none to routine vigilance (i.e., s ≠ 0 and h r = 0, see figure 7), while, when food is abundant, the opposite pattern is observed (see figure 9).

When food is scarce, the animal initially responds to an increase in predation risk by progressively reducing the proportions of time devoted to searching, cropping, and handling resources, and becomes more vigilant (see figure 7). Attention is thus progressively reorientated towards predator detection. Then, the animal begins to reduce the time it spends on the patch, without changing anymore the proportions of time dedicated to its different activities (figures 7 and8). An increasing level of risk thus always leads to a reduction of plant removal, first, because the animal feeds less while on the patch, second, because it spends less time there.

Similar patterns are observed at high food density (see figures 9 and 10), with three important differences. First, the overall level of vigilance is higher; second, routine vigilance is also present; third, the level of risk that leads the animal to curtail the patch's visit is lower.

An intriguing fact is that, if the animal globally increases its vigilance when the level of risk rises, this increase must be only attributed to exclusive vigilance, routine vigilance showing an opposite trend. The reason is that, here, all consumed food is found during the handling process, which means that the quantity of food consumed is proportional to h f . Since this quantity of food is also proportional to the total proportion of time spent handling it (i.e., to h = h f + h r ), h f and h r are proportional to one another. Thus, mechanistically, as the quantity eaten declines, the level of routine vigilance also declines. This, in turn, reinforces the need for the animal to increase its level of exclusive vigilance.

Discussion

Because alertness and efficient foraging are generally not fully compatible, defining attentional priorities is a permanent obligation for a foraging herbivore [START_REF] Hochman | Patch use, apprehension, and vigilance behavior of Nubian Ibex under perceived risk of predation[END_REF]. The main reason is that, while a feeding individual has to focus its attention towards the interior of its feeding area [START_REF] Kaby | What limits predator detection in blue tits (Parus caeruleus): posture, task or orientation?[END_REF][START_REF] Makowska | Vigilance during food handling in grey squirrels, Sciurus carolinensis[END_REF], a vigilant one gazes, listens or smells outwards [START_REF] Lima | Back to the basics of antipredatory vigilance: can nonvigilant animals detect attacks?[END_REF]. Vigilance is the most extreme form of apprehension, that is, the part of a forager's attention that is redirected from food search to antipredator awareness (Hochman & Kotler, 2004). In the wild, the level of apprehension is indeed variable and notably increases with the perceived level of threat, at the expense of feeding efficiency [START_REF] Dall | Attention, 'apprehension' and gerbils searching in patches[END_REF][START_REF] Clark | The behavioral ecology of a cognitive constraint: limited attention[END_REF][START_REF] Kotler | Apprehension and time allocation in gerbils: the effects of predatory risk and energetic state[END_REF]. In connection with these findings, we study the strategies determining attentional priorities in an herbivore subject to predation. The level of apprehension is here reflected by the sharing of time between food search and antipredator vigilance, which may differ between the periods when the animal is already handling food and the periods when it is not.

The first model ignores antipredator behaviours and considers that the animal focuses all its observation capacities on food detection. It is used as a reference to later explore the situation where a part of these capacities is devoted to vigilance. The main assumption of this model is that resources may only be detected when the animal is not busy cropping food, because it can then orientate its head freely and move within the patch. The resulting equation linking resource density to the forager's intake rate is equivalent to a type II functional response when resource density is not too high, then, reaches a fixed maximum value (see eqs. ( 7) and ( 9)). Food intake rate is initially slowed down by the cropping process, then, at high resource densities, fixed by the combined effects of cropping and handling times. These results coincide with the predictions of [START_REF] Spalinger | Mechanism of foraging in mammalian herbivores: new models of functional response[END_REF] model on herbivore functional responses (see transition between process 1 and process 3 in [START_REF] Spalinger | Mechanism of foraging in mammalian herbivores: new models of functional response[END_REF] and [START_REF] Smart | Measuring the functional responses of farmland birds: an example for a declining seed-feeding bunting[END_REF]. The positive but decelerating influence of food abundance on intake rate that characterizes the type II functional response is well documented in grazers [START_REF] Hudson | Foraging ecology of bison in aspen boreal habitats[END_REF][START_REF] Gross | Functional response of herbivores in food-concentrated patches: tests of a mechanistic model[END_REF] and in seed-eating animals [START_REF] Barnard | Flock feeding and time budgets in the house sparrow (Passer domesticus L.)[END_REF][START_REF] Baker | Vigilance and the functional response of granivorous foragers[END_REF], and so is the fact that, when food biomass is high, the intake rate becomes mainly governed by the forager's capacities to process this food (Wilmshurst et al., 1999). We

show here that the switching between these two situations occurs when the quantity of food discovered during the handling process matches the quantity of food eaten. At lower food densities, the animal must devote a part of its foraging time to exclusively searching for food (see Fig. 1). At higher food densities, this activity vanishes and the animal begins to find more food than it can eat, which means that part of the information it gathers about its environment is lost. Things change when the herbivore is allowed to re-orientate its attention towards predator detection.

In this respect, feeding and scanning have long been considered as mutually exclusive [START_REF] Barnard | Flock feeding and time budgets in the house sparrow (Passer domesticus L.)[END_REF][START_REF] Glück | An experimental study of feeding, vigilance and predator avoidance in a single bird[END_REF]. An increasing feeding rate should then mechanically lead to lower vigilance. Negative correlations between resource density and vigilance are indeed documented in situations where feeding hinders observation of the surroundings [START_REF] Underwood | Vigilance behavior in grazing African antelopes[END_REF][START_REF] Pöysä | Feeding-vigilance trade-off in the teal (Anas crecca): effects of feeding method and predation risk[END_REF]; see also [START_REF] Pays | Detecting predators and locating competitors while foraging: an experimental study of a medium-sized herbivore in an African savanna[END_REF]. In many cases however, higher levels of vigilance are observed when food is abundant (see [START_REF] Beauchamp | How does food density influence vigilance in birds and mammals?[END_REF]. A first explanation is that animals living in rich environments are able to sustain high levels of reserves. They can thus invest more in vigilance. This response is observed both on a short time scale after a meal [START_REF] Pravodusov | Body mass, ambient temperature, time of day, and vigilance in tufted titmice[END_REF] and after prolonged periods of intense feeding [START_REF] Bednekoff | Florida scrub-jays (Aphelocoma coerulescens) are sentinels more when well-fed (even with no kin nearby)[END_REF]. A second reason is that feeding enhances vigilance when these activities are compatible with one another, which routine vigilance makes possible [START_REF] Popp | Effects of food handling time on scanning rates among American goldfinches[END_REF]Fortin et al., 2004b). Intake rate and overall investment in vigilance can then vary concomitantly, and high biomass of plants be beneficial to predator detection (Fortin et al., 2004b). The present model accounts for these two possibilities. It shows that animals living in rich environments conciliate high intake rates and high levels of vigilance while on food patches, thanks to routine vigilance, and leave these patches with high levels of reserves. In fact, during our analyses we observed that making routine vigilance more effective does not only increase its use, but also allows the animal to leave the patch sooner, thanks to its higher feeding rate. The advantage of routine vigilance is thus twofold.

The level of routine vigilance is however also strongly influenced by predation risk.

More generally, the quantity of plants eaten by the herbivore, as well as its level of exposure to predation, are driven by an important behavioural plasticity, notably concerning the sharing of its attention between food and predator detection. The patterns highlighted here are indeed absent when searching for food cannot take place during handling [START_REF] Lima | Vigilance and diet selection: the classical diet model reconsidered[END_REF] and when routine vigilance is not allowed to vary in response to predation risk variation (Fortin et al., 2004a).

Identifying routine vigilance in the wild is possible, but requires a part of interpretation [START_REF] Benhaiem | Hunting increases vigilance levels in roe deer and modifies feeding site selection[END_REF][START_REF] Favreau | Predators, food and social context shape the types of vigilance exhibited by kangaroos[END_REF]. Animals which feed on the ground, then lift their head and gaze outwards while handling or chewing, can be considered as being performing routine vigilance [START_REF] Baker | Are the costs of routine vigilance avoided by granivorous foragers?[END_REF][START_REF] Favreau | Short-term behavioural responses of impalas in simulated antipredator and social contexts[END_REF], but the distinction is far from clear for animals that browse plant or tree foliage at head level i.e., when head is up. Moreover, the situation is complexified by a differential use of senses for predator detection and for food search, and by the way handling food interferes with their accuracy [START_REF] Blanchard | Induced or routine vigilance while foraging[END_REF].

Nevertheless, empirical studies confirm the combined influence of predation risk and resource abundance on herbivores' behaviour in the wild, and the complex interactions between routine vigilance, food search, and exclusive vigilance [START_REF] Kotler | Apprehension and time allocation in gerbils: the effects of predatory risk and energetic state[END_REF]Pays et al., 2012). Thus, Fortin et al. (2004b), [START_REF] Benhaiem | Hunting increases vigilance levels in roe deer and modifies feeding site selection[END_REF] and [START_REF] Robinson | Foraging-vigilance trade-offs in a partially migratory population: comparing migrants and residents on a sympatric range[END_REF] show that a high density of palatable plants allows grazing ungulates to increase their level of routine vigilance, while, at low densities of resources, searching for food takes too much time, and they are forced to mostly rely on exclusive vigilance (see [START_REF] Illius | Costs of vigilance in foraging ungulates[END_REF]). The influence of resource availability is, however, strongly mediated by variations in predation risk, which is the major driver of vigilance [START_REF] Favreau | Predators, food and social context shape the types of vigilance exhibited by kangaroos[END_REF]. An increasing perception of danger generally triggers higher levels of exclusive vigilance, which competes with feeding [START_REF] Blanchard | Induced or routine vigilance while foraging[END_REF][START_REF] Périquet | Influence of immediate predation risk by lions on the vigilance of prey of different body size[END_REF][START_REF] Van Der Meer | The effect of simulated African wild dog presence on anti-predator behavior of kudu and impala[END_REF][START_REF] Favreau | Short-term behavioural responses of impalas in simulated antipredator and social contexts[END_REF]Bergval et al., 2016), and possibly engenders a reduction of routine vigilance [START_REF] Hochman | Patch use, apprehension, and vigilance behavior of Nubian Ibex under perceived risk of predation[END_REF][START_REF] Favreau | Predators, food and social context shape the types of vigilance exhibited by kangaroos[END_REF].

Our model also incorporates the possibility for the herbivore to spend less time in the foraging area. The duration of the patch's visit is then conjointly determined by food availability and predation risk. When food is scarce, the animal uses all available foraging time, even if the risk is present. At higher food densities, it responds to an increasing level of risk by leaving the patch sooner and looking for shelter. Field studies confirm that, when the time available is short, or the density of food low with regards to energetic needs, the herbivores will forage as long as they can (Illius & Gordon, 1987;[START_REF] Van Der Veen | Daily routines and predator encounters in yellowhammers Emberiza citrinella in the field during winter[END_REF], even if this makes them more vulnerable to predation [START_REF] Daly | Behavioural modulation of predation risk: moonlight avoidance and crepuscular compensation in a nocturnal desert rodent, Dipodomys merriami[END_REF], and provided they do not regularly experience low risk periods during which they can concentrate their feeding activity [START_REF] Kotler | Moonlight avoidance in gerbils reveals a sophisticated interplay among time allocation, vigilance and state-dependent foraging[END_REF]. By contrast, under less critical conditions, animals subject to a high level of risk will leave the area for a refuge sooner [START_REF] Lendrem | Predation risk and vigilance in the blue tit (Parus caeruleus)[END_REF][START_REF] Kotler | Apprehension and time allocation in gerbils: the effects of predatory risk and energetic state[END_REF] and having eaten less [START_REF] Hochman | Patch use, apprehension, and vigilance behavior of Nubian Ibex under perceived risk of predation[END_REF]. Estimating the real impact of herbivores on plant populations, in particular, their equilibrium densities, thus requires to not only measure the instantaneous intake rate of these animals, but also how long, and how much, they feed every day.

Ultimately, we show that when all types of behavioural adjustments are taken into account, the curves linking resource density to the total quantity of food consumed show the same trends as the type II functional response (see figure 4). In other words, the two-steps mechanism highlighted in the first part of the study vanishes when adaptive antipredator responses are incorporated in the functional response. The situation remains complicated, however, since the quantity of resource actually eaten is also highly dependent on predation risk (see figures 6 and 8). The dynamics of the whole trophic chain is thus affected, the herbivore's behavioural responses to predation risk enhancing the cascading effects that predators create across the three trophic levels (see [START_REF] Schmitz | Trophic cascades: the primacy of traitmediated indirect interactions[END_REF][START_REF] Preisser | Scared to death? The effects of intimidation and consumption in predator-prey interactions[END_REF].

Antipredator defense is known to influence population dynamics in a deep and complex manner (e.g., [START_REF] Křivan | Effects of optimal antipredator behavior of prey on predator-prey dynamics : the role of refuges[END_REF][START_REF] Brown | Vigilance, patch use and habitat selection: foraging under predation risk[END_REF][START_REF] Rinaldi | Remarks on antipredator behavior and food chain dynamics[END_REF]. In addition, natural situations are complexified by numerous effects, such as resource depletion, competition among foragers, and temporal variations in predation risk (e.g., [START_REF] Mitchell | Density-dependent harvest rates by optimal foragers[END_REF][START_REF] Sirot | Adjustments in compound defensive strategies in response to variation in predation risk[END_REF]. Nevertheless, we show here that the derivation of an herbivore's functional response, a prerequisite for the exploration of interactions among populations, not only requires to include vigilance, but also the extent to which it varies across situations.

Moreover, the possibility for the consumer to simultaneously consume plants and scan for predators, through variable levels of routine vigilance, will affect vital rates in the three populations. The influence of this process on the dynamics of the community thus certainly deserves to be explored. Parameter values: a = 1, a' = 0.8 (for intake rate i) or 0.3 (for intake rate i'), C = 1.5, H = 2.

Figure 1 :

 1 Figure 1: Functional response: intake rate as a function of food density R.

Figure 2 :

 2 Figure 2: Proportions of time dedicated to exclusively searching for food, s, cropping, c, and handling food, h, as a function of resource density R. Parameter values: : a = 1, a' = 0.8, C = 1.5, H = 2.

Figure 3 :

 3 Figure 3: Proportions of time dedicated to exclusively searching for food, s, cropping, c, searching for food while handling, h f , routine vigilance, h r , and exclusive vigilance, v, as a function of resource density R. Parameter values: a = 1, a' = 0.8, C = 1.5, H = 2, A = 2, B = 1,  = 2,  = 0.15, T max = 8.

Figure 4 :

 4 Figure 4: Intake rate, i, total quantity of food consumed, Q, and duration of patch visit, T, as a function of resource density R. These quantities are linked by relationship: Q = iT. Parameter values: same as figure 3.

Figure 5 :

 5 Figure 5: Proportions of time dedicated to exclusively searching for food, s, cropping, c, searching for food while handling, h f , routine vigilance, h r , and exclusive vigilance, v, as a function of resource density R, when time spent on the patch is fixed. Parameter values: a = 1, a' = 0.8, C = 1.5, H = 2, A = 2, B = 1,  = 2,  = 0.15, T = T max = 8.

Figure 6 :

 6 Figure 6: Intake rate, i, total quantity of food consumed, Q, and duration of patch visit, T, as a function of resource density R, when time spent on the patch is fixed. These quantities are linked by relationship: Q = iT. Parameter values: same as figure 5.

Figure 7 :

 7 Figure 7: Proportions of time dedicated to exclusively searching for food, s, cropping, c, searching for food while handling, h f , routine vigilance, h r , and exclusive vigilance, v, as a function of predation risk , when food density is low. Parameter values: a = 1, a' = 0.8, C = 1.5, H = 2, A = 2, B = 1,  = 2, R = 0.2, T max = 8.

Figure 8 :

 8 Figure 8: Intake rate, i, total quantity of food consumed, Q, and duration of patch visit, T, as a function of predation risk when food density is low. These quantities are linked by relationship: Q = iT. Parameter values: same as figure 7.

Figure 9 :

 9 Figure 9: Proportions of time dedicated to exclusively searching for food, s, cropping, c, searching for food while handling, h f , routine vigilance, h r , and exclusive vigilance, v, as a function of predation risk , when food density is high. Parameter values: a = 1, a' = 0.8, C = 1.5, H = 2, A = 2, B = 1,  = 2, R = 1, T max = 8.

Figure 10 :

 10 Figure 10: Intake rate, i, total quantity of food consumed, Q, and duration of patch visit, T, as a function of predation risk when food density is high. These quantities are linked by relationship: Q = iT. Parameter values: same as figure 9.

Table 1 :

 1 Variable and parameters: Area of food discovery when the animal is exclusively searching for food a' (< a)Area of food discovery when the animal is searching while already handling food A Effect of exclusive vigilance on the probability of escaping predation B (< A)Effect of routine vigilance on the probability of escaping predation

	Behavioural
	variables
	s	Proportion of time spent exclusively searching for food
	c	Proportion of time spent cropping food
	h (=h f + h r ) Proportion of time spent handling food
	h f	Proportion of time spent searching food while handling food
	h r	Proportion of time spent in routine vigilance
	v	Proportion of time spent in exclusive vigilance
	T	Patch residence time
	Parameters
	a
		Rate of predatory attacks
	R	Resource density
	T

max Maximum residence time on the food patch  Speed with which terminal fitness saturates with the quantity of food eaten

Acknowledgements: We thank Burt P. Kotler and an anonymous reviewer for their insightful remarks.

Authors' contributions: ES and OP conceived the ideas. ES built the model and wrote the first draft. OP prepared the figures. PB, AL, and OP have improved the manuscript with significant comments. All authors gave final approval for submission.

Appendix: Derivation of the functional response.

The relationships between the herbivore's foraging behaviour and its feeding efficiency connect behavioural variables s, c and h to intake rate i. Here we consider the situation where the animal can eat all the food it finds. As demonstrated in the text (see equation ( 2)), we have: i = saR + ha'R.

(A1).

In addition, s + c + h = 1 (A2), because there are only three possible activities (see equation ( 1)).

Finally, the proportions of time respectively devoted to cropping and to handling food are given by: c = Ci (A3) and h = Hi (A4), because the quantity of food consumed per unit of time is i.

Since there are 4 variables, simultaneous resolution of equations ( A1), (A2), (A3) and (A4) allows to compute each of them as a function of resource density R. In particular, we get: i (A5).


Mengjia Wang

Zhijin Huang

Roland Salut

Miguel Angel Suarez

Huihui Lu
email: thuihuilu@jnu.edu.cn

Nicolas Martin

Thierry Grosjean
email: thierry.grosjean@univ-fcomte.fr

Plasmonic helical nanoantenna as a converter between longitudinal fields and circularly polarized waves

Introduction

Controlling the polarization vector of light provides a key approach to manipulating lightmatter interaction on the nanoscale. [START_REF] Novotny | Principles of nano-optics[END_REF] In turn, tailoring the local optical response of subwavelength structures brings new prospects to light polarization control. Recent advances in plasmonics and metamaterials led to numerous novel polarization systems combining the merits of high compactness, broadband operation and new functionalities. [START_REF] Gansel | Gold helix photonic metamaterial as broadband circular polarizer[END_REF][START_REF] Chen | Experimental confirmation of miniature spiral plasmonic lens as a circular polarization analyzer[END_REF][START_REF] Yu | A broadband, background-free quarter wave plate based on plasmonic metasurfaces[END_REF][START_REF] Wang | Subwavelength polarization optics via individual and coupled helical traveling-wave nanoantennas[END_REF] So far, polarization control mainly refers to the transverse vector components of a propagating light wave. At small scale however, the 3D nature of light cannot be avoided and a longitudinal optical component arises and can even become dominant for strongly confined fields. This is for instance observed in tightly focused radially-polarized beams [START_REF] Quabis | Focusing light to a tighter spot[END_REF][START_REF] Novotny | Longitudinal field modes probed by single molecules[END_REF] and surface plasmons tailored by plasmonic lenses, [START_REF] Lerman | Demonstration of nanofocusing by the use of plasmonic lens illuminated with radially polarized light[END_REF][START_REF] Chen | Direct acceleration of an electron in infinite vacuum by a pulsed radially-polarized laser beam[END_REF] tapered tips 10,11 and their variants. [START_REF] Chen | Experimental confirmation of miniature spiral plasmonic lens as a circular polarization analyzer[END_REF][12][13][14][15] In these cases, the perfect cylindrical symmetry of the near-fields leads to an electromagnetic picture similar to that of a longitudinal radiating dipole (i.e., parallel to the field axis of symmetry). Such longitudinal fields play an important role in super-resolution microscopy, [START_REF] Quabis | Focusing light to a tighter spot[END_REF]16 fluorescent sensing and imaging, [START_REF] Novotny | Longitudinal field modes probed by single molecules[END_REF]17,18 particle acceleration, 19 laser machining, [START_REF] Niziev | Influence of beam polarization on laser cutting efficiency[END_REF] material processing, [START_REF] Weber | Effects of radial and tangential polarization in laser material processing[END_REF][START_REF] Toyoda | Transfer of light helicity to nanostructures[END_REF] optical spin-orbit interaction [START_REF] Yao | Orbital angular momentum: origins, behavior and applications[END_REF][START_REF] Bliokh | Transverse and longitudinal angular momenta of light[END_REF][START_REF] Aiello | From transverse angular momentum to photonic wheels[END_REF][START_REF] Neugebauer | Magnetic and electric transverse spin density of spatially confined light[END_REF][START_REF] Rodríguez-Fortuño | Near-field interference for the unidirectional excitation of electromagnetic guided modes[END_REF] and advanced light shaping. [START_REF] Bauer | Observation of optical polarization Möbius strips[END_REF][START_REF] Maucher | Creating Complex Optical Longitudinal Polarization Structures[END_REF][START_REF] Wan | Generation of exotic optical polarization Möbius strips[END_REF] Longitudinal electric fields are intrinsic to small scale optics. Therefore, longitudinal-totransverse field conversion requires the concept of a broadband and directional nanoantenna to efficiently interconnect near-fields and far-fields [START_REF] Novotny | Antennas for light[END_REF][START_REF] Biagioni | Nanoantennas for visible and infrared radiation[END_REF] and fulfill the needs for interconversion between two orthogonal electromagnetic states. Common resonant nanoantennas have a modest ability to tune light polarization and show far-field patterns of low directionality.

Spiral zone plates [START_REF] Guo | Review of the functions of Archimedes' spiral metallic nanostructures[END_REF] have shown the ability to convert an incoming circularly polarized propagating light into confined longitudinally-oriented electric fields. [START_REF] Chen | Experimental confirmation of miniature spiral plasmonic lens as a circular polarization analyzer[END_REF]15,[START_REF] Chen | Efficient miniature circular polarization analyzer design using hybrid spiral plasmonic lens[END_REF][START_REF] Guo | A spiral plasmonic lens with directional excitation of surface plasmons[END_REF][START_REF] Mao | Far-field radially polarized focal spot from plasmonic spiral structure combined with central aperture antenna[END_REF][START_REF] Cheng | Adjustable 3D plasmonic Archimedes spiral lens for optical manipulation[END_REF] The reciprocal configuration has been demonstrated as well. [START_REF] Rui | Beaming circularly polarized photons from quantum dots coupled with plasmonic spiral antenna[END_REF] However, besides architectures much larger than the typical size of the light confinements showing dominant longitudinal electric field, which represents limits for small scale applications, their operation suffer from two limits.

First, their polarization properties are usually strongly wavelength-dependent. We however note that the chromatism of spiral zone plates can be relaxed using logarithmic-spiral nanoslits of varying width [START_REF] Liu | Twisted focusing of optical vortices with broadband flat spiral zone plates[END_REF][START_REF] Mehmood | Broadband spin-controlled focusing via logarithmic-spiral nanoslits of varying width[END_REF] or more complex architectures. [START_REF] Bachman | Spiral plasmonic nanoantennas as circular polarization transmission filters[END_REF] Second, the light released in free space by a spiral nanoslit needs substantial diffraction to acquire a Gaussian form and propagate self-similarly. This can be an advantage if ones uses these microdevices for light focusing, [START_REF] Mao | Far-field radially polarized focal spot from plasmonic spiral structure combined with central aperture antenna[END_REF][START_REF] Mehmood | Broadband spin-controlled focusing via logarithmic-spiral nanoslits of varying width[END_REF] but it may represent a limitation in the design of a polarization optics. Plasmonic helical nanoantennas have been recently proposed to create and tailor chiroptical effects with direct implication in polarization control. [START_REF] Gansel | Gold helix photonic metamaterial as broadband circular polarizer[END_REF][START_REF] Wang | Subwavelength polarization optics via individual and coupled helical traveling-wave nanoantennas[END_REF][START_REF] Schäferling | Helical plasmonic nanostructures as prototypical chiral near-field sources[END_REF][START_REF] Esposito | Programmable extreme chirality in the visible by helix-shaped metamaterial platform[END_REF][START_REF] Kosters | Core-shell plasmonic nanohelices[END_REF][START_REF] Woźniak | Chiroptical response of a single plasmonic nanohelix[END_REF][START_REF] Wang | Chiroptical transmission through a plasmonic helical traveling-wave nanoantenna, towards on-tip chiroptical probes[END_REF][START_REF] Höflich | Resonant behavior of a single plasmonic helix[END_REF] So far, helical nanoantennas were excited with transversely polarized propagating waves. Here, we study the polarization properties of a single plasmonic helix positioned in a nonradiative plasmonic focus showing dominant longitudinal electric field component (perpendicular to the focusing plane, [START_REF] Lerman | Demonstration of nanofocusing by the use of plasmonic lens illuminated with radially polarized light[END_REF][START_REF] Chen | Direct acceleration of an electron in infinite vacuum by a pulsed radially-polarized laser beam[END_REF] parallel to the helix axis). The nano-helix is designed to operate in the traveling-wave regime, also known as the axial-mode in the radiofrequency domain, [START_REF] Kraus | Antennas and wave propagation[END_REF] thereby gaining a directional far-field pattern and a broadband operation. [START_REF] Wang | Subwavelength polarization optics via individual and coupled helical traveling-wave nanoantennas[END_REF] Introducing tapered geometries to the helical structure is a common approach to further enhancing its spectral invariance. [START_REF] Kraus | Antennas and wave propagation[END_REF] As an alternative to nano-helix tapering, [START_REF] Gansel | Tapered gold-helix metamaterials as improved circular polarizers[END_REF] we show that varying the pitch length along the helix broadens the operation bandwidth by at least 2.25 times (measurements are limited by our laser bandwidth). The reciprocal process is verified as well, where a circularly polarized incoming optical wave is converted into a tiny light spot of dominant longitudinal electric field generating an axis-symmetrical diverging surface plasmons all around the nanoantenna.

Compared with microscale spiral devices, individual plasmonic helical nanoantennas offer a much smaller footprint together with the prospect of the directional release of a circularly polarized light directly in the form of a Gaussian-like beam. [START_REF] Wang | Subwavelength polarization optics via individual and coupled helical traveling-wave nanoantennas[END_REF] Helical nanoantennas thus provide a new degree of freedom in the control of light polarization at small scale.

Results and discussion

To demonstrate the concept of a converter between longitudinal electric fields and circularly polarized waves, we positioned a helical nanoantenna at the center of a circular slit which perforates a thin gold film on a glass substrate (Fig. 1). The nanoantenna consists of a lefthanded four-turn gold-coated carbon helix lying on a 100-nm cylindrical pedestal. First, the skeleton of the subwavelength structure is realized by growing a 105-nm broad carbon wire on a 100-nm thick gold layer with focused ion beam induced deposition (FIBID). [START_REF] Esposito | Programmable extreme chirality in the visible by helix-shaped metamaterial platform[END_REF][START_REF] Esposito | Three dimensional chiral metamaterial nanospirals in the visible range by vertically compensated focused ion beam induced-deposition[END_REF] Two helix configurations, with a varying or constant pitch, are obtained by adjusting the FIBID parameters (see Figs. 1(e) and (f), respectively). The graded pitch length is achieved from the vertical gradients of both precursor concentration and decomposition rate of the FIBID process above the substrate. The resulting helix exhibits average pitch angles of 22.3°, 20.7°, 16.0°and 12.1°from the bottom to the top turn, respectively. The constant-pitch helix (Fig. 1(f)) is obtained by increasing the dwell time by about 30% from the first turn to compensate those gradient effects. [START_REF] Esposito | Three dimensional chiral metamaterial nanospirals in the visible range by vertically compensated focused ion beam induced-deposition[END_REF] We find a homogeneous pitch angle of about 19.5 • along the entire structure. The fabricated helical nanostructures are then plasma-sputtered with a 25-nm thick gold layer via a glancing angle deposition technique: [START_REF] Wang | Subwavelength polarization optics via individual and coupled helical traveling-wave nanoantennas[END_REF][START_REF] Kosters | Core-shell plasmonic nanohelices[END_REF][START_REF] Woźniak | Interaction of light carrying orbital angular momentum with a chiral dipolar scatterer[END_REF][START_REF] El Beainou | Electrical resistivity and elastic wave propagation anisotropy in glancing angle deposited tungsten and gold thin films[END_REF] the metal target is tilted with an angle of 80 • from the helix axis and it is rotated during the deposition time at a constant speed of 2 rev•min -1 . Finally, the gold film is milled with a focused ion beam to form a 16-µm diameter and 200-nm wide circular slit surrounding the nanoantenna (see Fig. 1(d)). The helix pedestal coincides with the center of the circular aperture. Upon illumination with a radially polarized beam from the backside, the circular slit launches cylindrical surface plasmons which converge towards the center of the slit. [START_REF] Lerman | Demonstration of nanofocusing by the use of plasmonic lens illuminated with radially polarized light[END_REF] We numerically simulate the electric field distribution of the resulting near-field interference pattern within a circular slit of outer diameter and width equal to 7 µm and 200 nm, respectively. The plasmonic focus at center is defined by an out-of-plane electric field (Fig. The helical nanoantenna is designed to operate in the traveling-wave regime at telecommunication wavelengths. [START_REF] Wang | Subwavelength polarization optics via individual and coupled helical traveling-wave nanoantennas[END_REF] The geometrical requirements on a low-frequency helical antenna to operate in the travelling-wave regime are a circumference that approaches the targeted wavelength and a pitch angle value in between 12 • and 14 • . [START_REF] Kraus | Antennas and wave propagation[END_REF] With its 505-nm outer diameter, our plasmonic helical nanoantenna satisfies the condition of an outer circumference (of 1.59 µm) matching the targeted telecommunication wavelengths. Optimum polarization conversion has then been numerically found for pitch angles ranging from 10 • to 40 • . Used as a light-plasmon coupler, the circular slit plays no role in the travelling-wave properties of the plasmonic helix.

In travelling-wave regime, surface plasmons guided along the nanohelix and are released in free space in the form of directional optical waves (see Fig. S1 and Movie S1 in the Supporting Information). The helix can thus be interpreted as a subwavelength chiral waveguide which propagates vortex surface plasmons carrying orbital angular momentum (OAM). The OAM is imparted to the guided surface plasmons by the chiral nanostructure itself and is thus independent of the excitation conditions. When circular polarization is generated by the helical nanoantenna, its vortex guided mode (of charge ±1 depending on the helix handedness) is released as freely propagating waves carrying spin angular momentum (SAM) of ±1 per photon, owing to spin-orbit interaction [START_REF] Wang | Subwavelength polarization optics via individual and coupled helical traveling-wave nanoantennas[END_REF][START_REF] Bliokh | Spin-orbit interactions of light[END_REF] (see Fig. 1(a)). Reciprocally, an impinging circularly-polarized wave excites the twisted mode of the helix for only one polarization handedness, [START_REF] Wang | Chiroptical transmission through a plasmonic helical traveling-wave nanoantenna, towards on-tip chiroptical probes[END_REF] a phenomenon which refers to circular dichroism [START_REF] Gibbs | Plasmonic nanohelix metamaterials with tailorable giant circular dichroism[END_REF][START_REF] Esposito | Nanoscale 3D chiral plasmonic helices with circular dichroism at visible frequencies[END_REF] and chiroptical transmission. [START_REF] Gansel | Gold helix photonic metamaterial as broadband circular polarizer[END_REF][START_REF] Kosters | Core-shell plasmonic nanohelices[END_REF][START_REF] Wang | Chiroptical transmission through a plasmonic helical traveling-wave nanoantenna, towards on-tip chiroptical probes[END_REF][START_REF] Höflich | Resonant behavior of a single plasmonic helix[END_REF] In the following, we define that a left-handed helical nanoantenna generates a left-handed circular polarization, i.e., the electric field turns counter-clockwise when one looks along the propagation direction of the outcoming wave. Reciprocally, a left-handed plasmonic helical nanoantenna is selectively excited for an incoming left-handed circular polarization.

The concept of a nanohelix as a converter between longitudinal electric fields and circularly polarized waves originates from the ability of low-frequency helical antennas to interconnect circularly polarized freely propagating waves and the fundamental guided mode of a metal wire. [START_REF] Kraus | Antennas and wave propagation[END_REF] Such a subwavelength wire mode has been shown to exist at optical frequencies and to match the longitudinal electric field of a confined light 10,13,[START_REF] Chang | Quantum optics with surface plasmons[END_REF] To demonstrate the generation of circular polarization from a longitudinal electric light field, we project a focused radially polarized beam onto the back of our sample (Fig. 2(a)).

When crossing a rotating LP, such a doughnut beam (Fig. corresponds to the direct transmission of the impinging light through the circular slit. When the rotating LP is inserted, this original circular pattern breaks into two symmetric lobes aligned with the transmission axis of the analyzer (Fig. 2(c), the four last panels on the right), revealing that the light directly outcoupled from the circular slit is radially polarized.

The spot located at the center of the bright ring corresponds to the light released by the helical nanoantenna. Since no light is observed in-between the two nanostructures, the distal helix is solely excited with surface plasmons. The optical waves leaving the helix show a constant intensity regardless the LP direction. When the LP is replaced by the circular analyzer, the central spot is visible only when the analyzer selectively transmits left circular polarization (see Figs. 2(e) and (f)). In contrast, when the analyzer selectively transmits the other handedness, a faint doughnut-like pattern is observed (Fig. 2 (f) and inset). This doughnut-like beam is known to be a second-order vortex which originates from angular momentum conservation between the guided mode of a plasmonic structure and freely propagating waves. 15,[START_REF] Garoli | Helicity locking of chiral light emitted from a plasmonic nanotaper[END_REF] This residual vortex component slightly reduces the helicity of the off-axis outcoming waves. With our fabricated helix, the maximum intensity of this second-order vortex-like field is 0.11 times as high as the overall beam's whereas a ratio of 0.14 is numerically anticipated (cf. Methods and Fig. S2 in the Supporting Information).

Owing to its radial polarization, the surrounding light ring remains unchanged for the two handedness of the circular analyzer.

The spectral properties of the helix are investigated by measuring the on-axis degree of circular polarization (DOCP) of the leaving optical waves as a function of the wavelength.

The DOCP refers to the distribution of radiated photons prepared in the spin states +1 and of the two DOCP maxima would suggest that the spectral condition for a conversion from longitudinal electric field to circular polarization mainly scales with the helix radius, which appears to be consistent with the design rules for radiofrequency helical antennas. [START_REF] Kraus | Antennas and wave propagation[END_REF] For the helix of a constant pitch (Fig. 1(f)), the DOCP remains larger than 0.99 within a wavelength range of 80 nm. Although the plasmonic helix operates in the nonresonant traveling-wave regime, [START_REF] Wang | Subwavelength polarization optics via individual and coupled helical traveling-wave nanoantennas[END_REF][START_REF] Wang | Chiroptical transmission through a plasmonic helical traveling-wave nanoantenna, towards on-tip chiroptical probes[END_REF] its periodic nature limits its spectral bandwidth by providing a wavelength-dependent operation. By breaking the helix periodicity, i.e., by using the varyingpitch helix (Fig. 1(e)), we extend the spectral invariance of the nanoantenna to a broader wavelength range exceeding the 180-nm bandwidth of our laser.

By combining our experimental measurements to the theoretical prediction of the optical properties of a nanoslit (see Section S1 of the Supporting Information), we estimate that the efficiency with which the constant and varying-pitch helices convert incoming surface plasmons into freely propagating waves is of 26.5% and 23.7%, respectively. Such values represent a lower limit of the helix's efficiency as the fabricated nanoslit is expected to be less efficient in converting light into surface plasmons than its theoretical model. Current nanofabrication facilities do not ensure nanoscale slits with perfectly parallel edges and sharp corners.

In a reciprocal approach, a circularly polarized focused wave is projected onto the varyingpitch helix and the back of the sample is imaged with a high numerical aperture microscope objective coupled to an infrared camera (Fig. 3(a); cf. Methods). Figures 3(b) -(e diffraction by the slit, can be also found at the output of spiral-shaped structures. [START_REF] Mehmood | Broadband spin-controlled focusing via logarithmic-spiral nanoslits of varying width[END_REF] When a LP is inserted before the camera, the original circular pattern in the images is turned into a two-lobe distribution aligned with the LP axis, which reveals a radial polarization state of the outcoming waves. Such an axially-symmetric optical field at the structure output originates from an isotropic longitudinally-oriented dipole source of surface plasmons at the center of the circular slit (i.e., at the position of the helix). When the helicity of the incident light is flipped, the intensity of the output optical pattern decreases by 2.8 times and the outcoming light loses its radial polarization state (see Figs. 3(f)). The residual optical pattern originates from a direct transmission of the incident light through the slit. The chiral optical response shown in Figs. 3 (d) and (f) evidences the central role of the helix in the light transmission process since the rest part of the plasmonic platform is achiral. The conversion from circular polarization to longitudinal electric field occurs when the incoming polarization handedness is identical to the helix's, which agrees well with past experimental observations on helical plasmonic nanoantennas. [START_REF] Gansel | Gold helix photonic metamaterial as broadband circular polarizer[END_REF][START_REF] Wang | Chiroptical transmission through a plasmonic helical traveling-wave nanoantenna, towards on-tip chiroptical probes[END_REF] Fig. 3 thus shows the ability of helical nanoantennas to convert circular polarization in the far-field into a longitudinal electric field in the near-field over the 180-nm spectral bandwidth of our laser. This broadband operation is consistent with that shown in Fig. 2.

Upon illumination with left circular polarization, a left-handed helix excites diverging cylindrical surface plasmons at the metal-air interface (Fig. 4(a)). As they hit the circular slit, these surface waves are converted into radially polarized freely propagating waves together with focused surface plasmons on the opposite metal-substrate interface. We observe in Fig. 4 (c) the longitudinal electric component E z of these transmitted surface plasmons (in amplitude and phase) plotted 20 nm beneath the metal layer. The phase is selectively represented within the circular slit. Due to computation limitations, the circular slit is shrinked to a diameter of 7 µm. The transmitted longitudinal near-field within the circular slit is constituted by concentric fringes of constant phase, which evidence a near aberrationfree plasmonic focusing on the backside of the metal layer. Similar transmitted plasmonic field is obtained when the circular slit is excited with a longitudinal electric dipole (i.e., oriented along 0z) positioned just above the slit center (see Fig. 4(d)). The good agreement between the two electromagnetic pictures of Figs. 4(c) and (d) suggests that upon excitation with left circular polarization, a left-handed helix behaves as a longitudinallyoriented dipole source (i.e., parallel to 0z) of plasmons. We thus numerically confirm that the helical nanoantenna converts impinging circularly-polarized propagating waves into longitudinally-oriented optical near-fields generating axis-symmetrical cylindrical surface plasmons. In this reciprocal approach, the helical nanoantenna absorbs the angular momentum from the impinging light to generate twist-free axis-symmetrical near-fields. We predict that the above-described effect pertains in the 1.2 µm to 2.0 µm wavelength range (see Fig. 5). Such a broadband operation is attributed to the waveguide mode nature of the surface plasmons travelling along the helical nanoantenna. According to Fig. 2(d), a broader operation bandwidth could be expected with a varying-pitch helical nanoantenna.

Conclusion

We demonstrate an individual helical nanoantenna to convert between the circular polar-

Methods

To examine the optical response of the fabricated helices, our experimental setup is built onto a Nikon TE2000 inverted microscope. In the experimental characterization of Fig. 2, a near-infrared light (λ=1.47 -1.65 µm) emerging from a tunable laser (Yanista Tunics-T100S) is coupled to a single-mode polarization-maintaining fiber (P3-1500PM-FC-2, Thorlabs) and collimated with an achromatic fiber collimator (RC08APC-P01, Thorlabs). The output beam passes successively through an achromatic linear polarizer (LPNIR100-MP2, Thorlabs) and a space-variant half-wave plate (WPV10L-1550, Thorlabs) to produce radially polarized light.

The resulting doughnut beam is then focused onto the gold layer with a (4x, 0.1) microscope objective from the backside. A piezo stage (Mad city labs) is used to align the circular slit regarding the focused beam. The sample is imaged in transmission mode with a microscope objective (Leitz Wetzlar NPL FL 63x, 0.9) coupled to an infrared camera (GoldEye model G-033, Allied Vision Technologies GmbH). A proper field lens is inserted before the camera. To investigate the polarization state of the transmitted light, two different analyzers are positioned in between the objective and the field lens. These analyzers are either a rotating linear polarizer (LPNIR100-MP, Thorlabs) or the same fixed linear polarizer with a rotating quarter-wave plate (AHWP05M-1600, Thorlabs) inserted in front. In the reciprocal configuration shown in Fig. 3, the illumination and collection benches are inverted. The collimated laser beam passes through the circular polarizer and it is then focused onto the helix with the (63x, 0.9) microscope objective. The light transmitted back in the substrate is detected with the (25x, 0.4) microscope objective coupled to the infrared camera. The polarization state of the outcoming light is analyzed with the rotating linear polarizer. In order to compare the transmission intensities regarding two opposite incident handedness, the raw optical images (Figs. 3(d) and (f), left panels) are integrated over the circular area enclosed by the slit boundary.

All the numerical simulations are carried out using the 3D Finite Difference Time Domain (FDTD) method available on a commercial software from Synopsis. The plasmonic core-shell nanowire forming the helical nanoantenna is modeled as a 105-nm diameter carbon cylinder covered with a 25-nm thick gold layer. Such a wire is wound up in the form of a 4-turn helix and is also used to terminate the structure on one end with a 100-nm long pedestal lying on a 100-nm thick gold layer covering a glass substrate. To numerically verify the polarization conversion principle of our nanoantennas, the simpler constant-pitch configuration of the helix is consistently considered. The resulting helical nanoantenna has an outer diameter of 505 nm and a total height of 1.66 µm, which corresponds to a homogeneous pitch angle of approximately 20.7 • . The cylindrical pedestal is located at the the center of the 200 nm wide circular slit of a 7 µm diameter that surrounds the nanoantenna. Due to computation limits, the modeled circular slit has a shorter diameter than its experimental counterpart. For the illumination from the substrate, a radially polarized beam is projected onto the circular slit at normal incidence. The beam waist is 8 µm wide, align with the symmetry axis of the circular slit and located 350 nm away from the metal-glass interface. In the reciprocal case, the helix is illuminated from the top with a circularly polarized Gaussian beam whose 1.5-µm waist is aligned with the helix axis and set at a distance of 0 nm from the summit of the nanostructure. The non-uniform grid resolution varies from 30 nm for portions at the periphery of the simulation to 5 nm within and near the helix and the circular slit.

Figure 1 :

 1 Figure 1: Plasmonic helical nanoantenna as a subwavelength converter between longitudinal electric fields and circularly polarized light waves. (a) Schematics of the plasmonic platform used for demonstration: the helix pedestal is set at the center of a circular slit engraved in a thin gold layer. The operation principle of the platform is represented in a conversion from a longitudinal field to circular polarization. The radially polarized incoming beam projected on the back of the sample is converted by the circular slit in plasmonic focused waves. Positioned within the plasmonic focus showing a dominant out-of-plane (longitudinal, with respect to the focusing plane) electric field component (of zero longitudinal angular momentum), the left-handed helical nanoantenna develops vortex surface plasmons of charge l=-1. This twisted plasmon mode is released in free space as left-handed circularly polarized light (σ=-1 per photon) owing to spin-orbit interaction. 53 RP: radial polarization. LCP: left circular polarization. SP: surface plasmons. LoP: out-ofplane/longitudinal field. (b) and (c) Numerically predicted longitudinal (|E z |, out-of-plane) and transverse (|E t |, in-plane) amplitudes of the electric optical field, respectively, plotted 20 nm above a 100-nm thick gold layer engraved with a 7 µm diameter circular slit. The circular slit is illuminated with a radially polarized beam from the substrate. (b) and (c) are normalized to the maximum of the electric field amplitude |E t | 2 + |E z | 2 . (c) is saturated for a better view of the field pattern within the circular slit. (d) Scanning electron micrograph of a fabricated platform. (e) and (f) Scanning electron micrographs of two plasmonic helices of varying and constant pitch length, respectively.

1

  (b)) surrounded by an in-plane radially-polarized doughnut field of weaker intensity (Fig.1(c)). The out-of-plane (longitudinal, with respect to the focusing plane) field component (E z ) interferes constructively at the focus whereas the in-plane (transverse, with respect to the focusing plane) field component (E t ) undergoes a destructive interference phenomenon at the center of the circular slit.[START_REF] Lerman | Demonstration of nanofocusing by the use of plasmonic lens illuminated with radially polarized light[END_REF][START_REF] Chen | Direct acceleration of an electron in infinite vacuum by a pulsed radially-polarized laser beam[END_REF] Such a 3D electromagnetic picture is similar to that of an electric dipole oriented along 0z.[START_REF] Novotny | Principles of nano-optics[END_REF] Reciprocally, a longitudinally-oriented dipole-like source of light positioned at the center of the circular slit generates axis-symmetrical diverging surface plasmons that are converted back into a radially polarized freely-propagating optical waves by the slit. 11

Figure 2 :

 2 Fig.2(b)). We then image the overall structure in the far-field with a microscope objective coupled to an infrared camera (cf. Methods). The polarization state of the emerging light is analyzed either with a rotating linear polarizer (LP) or a quarter-wave plate (QWP) followed by a linear polarizer (i.e., a circular analyzer). These polarization analyzers are inserted before the camera. Without analyzer, the image shows an individual spot at the center of a 16-µm diameter narrow light ring (Fig.2(c), left panel). The ring pattern

- 1 .Figure 3 :

 13 Figure 3: Conversion from circular polarization to longitudinal electric field. (a) Schematics of the experimental setup: the left-handed helix is illuminated from the top. LP: linear polarizer. QWP: quarter-wave plate. OBJ: microscope objective. (b-f) Five series of optical images acquired with the varying-pitch helical nanoantenna at wavelengths ranging from 1.47 µm to 1.62 µm by steps of 50 nm. The incoming circular polarization is lefthanded in (b-e) and right-handed in (f). (d) and (f) are acquired at the same wavelength of 1.57 µm and with the same camera dynamics. Each series is constituted by a raw optical image (on the left) and four polarization-analyzed images obtained when a linear polarizer is positioned in front of the camera. Four orientations of the linear polarizer, represented with white arrows in the figures, are considered. Intensity attenuation in the polarizationanalyzed images (induced by the presence of a polarizer in front the camera) is numerically compensated. Image size: 25 × 25 µm 2 .

Figure 4 :

 4 Figure 4: The helical nanoantenna converts from circular polarization to longitudinal electric field. (a,b) Schematics of the diverging surface plasmons (blue arrows) excited (a) from a left-handed helical nanoantenna upon illumination with left circular polarization (LCP), and (b), from an (longitudinal) electric dipole oriented perpendicular to the top surface of the metal film (ED). (c,d) Simulated amplitude (upper panel) and phase (downer panel) of the longitudinal electric component (E z ) of the surface plasmons generated at the metal-glass interface by the circular slit, upon the excitation conditions of (a) and (b), respectively. The E z component is plotted 20 nm beneath the metal-glass interface. The modeled helix has a constant pitch length. The computation parameters are detailed in the Method section.

Figure 5 :

 5 Figure 5: Broadband conversion from circular polarization to longitudinal electric field. Simulated amplitude (upper panel) and phase (downer panel) of the longitudinal electric field E z of the transmitted surface plasmons at the gold-glass interface, with the setup shown in Fig. 4(a). These near-fields are plotted 20 nm beneath the substrate-metal interface and at various wavelengths ranging from 1.0 to 2.2 µm. The modelled helix has a constant pitch length.

  ization of a freely propagating wave and longitudinal electric fields in the near-field, thus providing a new degree of freedom in light polarization control. When positioned in the longitudinally-polarized focus of converging surface plasmons, an individual helical nanoantenna is shown to radiate circularly-polarized light that is directly shaped into a Gaussian beam. Reciprocally, upon illumination with circular polarization, the plasmonic helix generates localized longitudinal near-fields which excites axis-symmetrical diverging surface plasmons on a thin metal film. Being non-resonant, the helical nanoantenna operates over a broad spectral range, which is shown to be further widened by breaking the helix periodicity (as an alternative to the well-known tapered helices). Moreover, our fabricated nanoantennas show excellent stability in time during acquisitions as well as unchanged optical performances even three years after their fabrication. As a new broadband and subwavelength polarization optics, individual plasmonic helices hold promise for applications such as optical information processing, polarimetry, miniaturized displays, optomagnetic data storage, microscopy, sensing and communications. As an example, plasmonic helices may couple longitudinally oriented quantum emitters to right-and left-handed single photons, thus paving the way towards new architectures in polarization-encoded optics and spin-encoded photon qubits with applications in quantum information and opto-spintronics.

  All field simulations are conducted in the continuous wave regime at various wavelengths ranging from 1 µm to 2.2 µm. The total computation volume spans 10 µm in the x and y directions. It extends 1.5 µm below the gold layer in the glass substrate and terminates 4 µm beyond the top of the helix in air. The field distributions shown in Figs.1(c) and (d) are calculated without the helical nanoantenna and with an illumination from the substrate (incoming radial polarization). Being of shorter diameter than its experimental counterpart, the modelled circular slit over-adds a stronger background to the optical signal from the helix. In that context, the results shown in Figs.4(d) and 5 are obtained by subtracting the simulated fields with the helix from those anticipated without the nanoantenna to filter out the background signal from the slit. concentric metallic rings under radially polarized illumination. Nano Lett. 2009, 9, 4320-4325. (10) Stockman, M. I. Nanofocusing of optical energy in tapered plasmonic waveguides. Phys. Rev. Lett. 2004, 93, 1-4. (11) Tugchin, B. N. et al. Plasmonic tip based on excitation of radially polarized conical surface plasmon polariton for detecting longitudinal and transversal fields. ACS Photonics 2015, 2, 1468-1475. (12) Janunts, N. A.; Baghdasaryan, K. S.; Nerkararyan, K. V.; Hecht, B. Excitation and superfocusing of surface plasmon polaritons on a silver-coated optical fiber tip. Opt. Commun. 2005, 253, 118-124. (13) Ginzburg, P.; Nevet, A.; Berkovitch, N.; Normatov, A.; Lerman, G. M.; Yanai, A.; Levy, U.; Orenstein, M. Plasmonic resonance effects for tandem receiving-transmitting nanoantennas. Nano Lett. 2011, 11, 220-224. (14) Zhang, S.; Wei, H.; Bao, K.; Håkanson, U.; Halas, N. J.; Nordlander, P.; Xu, H. Chiral surface plasmon polaritons on metallic nanowires. Phys. Rev. Lett. 2011, 107, 1-5. (15) Garoli, D.; Zilio, P.; Gorodetski, Y.; Tantussi, F.; De Angelis, F. Beaming of helical light from plasmonic vortices via adiabatically tapered nanotip. Nano Lett. 2016, 16, 6636-6643. (16) Dorn, R.; Quabis, S.; Leuchs, G. Sharper focus for a radially polarized light beam. Phys. Rev. Lett. 2003, 91, 233901. (17) Macklin, J. J.; Trautman, J. K.; Harris, T. D.; Brus, L. E. Imaging and time-resolved spectroscopy of single molecules at an interface. Science 1996, 272, 255-258. (18) Sick, B.; Hecht, B.; Novotny, L. Orientational imaging of single molecules by annular illumination. Phys. Rev. Lett. 2000, 85, 4482-4485.

Acknowledgement

The authors are indebted to Katja Höflich for helpfull discussions. This work is funded by the EIPHI Graduate School (contract ANR-17-EURE-0002), the Region Bourgogne Franche-Comte, the French Agency of Research (contract ANR-18-CE42-0016), the I-SITE BFC (contract ANR-15-IDEX-0003). It is supported by the French RENATECH network and its FEMTO-ST technological facility.

Supporting Information Available

Numerical analysis of the helicity of the optical waves released by an individual helical nanoantenna positioned within a plasmonic focus. Numerical analysis of the operation principle of the plasmonic helix, evidencing a travelling wave regime. Estimation of the efficiencies with which the constant and varying-pitch helical nanoantennas convert longitudinal fields from confined surface plasmons into circularly polarized freely propagating waves.