Gabriel Mouahid

Mintsa Rodrigue

Khalid Nguema

Salem Al Mashikhi

Mohamed Al Yafae

Hélène Idris

Moné

Host-parasite life-histories of the diurnal vs . nocturnal chronotypes of Schistosoma mansoni : adaptive significance

objectives To optimise host-to-host transmission, digenean trematodes (parasites) synchronise their cercarial emission patterns with the aquatic activities of their vertebrate hosts. Schistosoma mansoni has two different circadian chronotypes: a diurnal shedding pattern with a mean peak occurring at 11:00 h, and a nocturnal shedding pattern with a mean peak occurring at 20:00 h. We analysed the life-history variations between these two chronotypes at the levels of the parasite and its hosts.

methods For each chronotype, we quantified three life-history traits related to the parasite (prepatent period, infection rate and cercarial production) and analysed the morphometry and the morphology of the parasite eggs; we also quantified three life-history traits related to the snail intermediate host (shell diameter, fecundity and survival rate) and one life-history trait related to the experimental definitive host (survival rate). A phylogeny based on the mitochondrial cytochromeoxidase gene was made on samples of both chronotypes.

results Life-history analysis revealed significant variations between the two chronotypes. Lifehistory traits were optimal for both the parasite and the snail host for the diurnal chronotype compared to the nocturnal one. The new chronotype behaved like an allopatric population towards its snail host. Phylogenetic analysis supports the hypothesis of a lateral transfer of S. mansoni from humans to Rattus rattus. These results were interpreted in terms of an ongoing sympatric speciation.

conclusion The nocturnal chronotype of S. mansoni showed non-adapted life-history traits in its relation with the snail intermediate host Biomphalaria pfeifferi. The emergence of this new phenotype is probably linked to divergent natural selection.

keywords Schistosoma mansoni, nocturnal chronotype, cercarial emergence pattern, circadian rhythm, sympatric speciation specific time of the day, tens or even hundreds of swimming larvae (cercariae) that infect humans or reservoir hosts, such as rodents, by active penetration when the hosts come into contact with water.

. The recent discovery of an exclusively nocturnal chronotype in Schistosoma mansoni [2], linked to its adaptation to a rodent host, showed that this species may have undergone significant genetic variation to switch from a photophase emission to exclusively emission during the scotophase. This switch in circadian behaviour raises questions since it should have been accompanied by some other traits, especially those responsible for the host-parasite life-histories. The present work aims to compare the life-history

Introduction

In human parasitic diseases, the species Schistosoma mansoni (digenean trematode) is an important agent responsible for bilharziasis or intestinal schistosomiasis. It is a gonochoric parasite that nests in the venous circulatory system of the intestine, where it performs its sexual reproduction producing large quantities of embryonated eggs, many of which leave the circulatory system to reach the intestine and are expelled with stools. Contact of the eggs with freshwater causes hatching liberating the swimming larvae (miracidia). Each miracidium infects, by active penetration, the snail intermediate host of the genus Biomphalaria where asexual multiplication occurs. After the prepatent period, the infected snails shed, at a 1 Host-parasite life-histories of the diurnal vs. nocturnal chronotypes of Schistosoma mansoni: adaptive significance Gabriel Mouahid 1 , Rodrigue Mintsa Nguema2 , Khalid M. Al Mashikhi3 , Salem A. Al Yafae 3 , Mohamed A. Idris 4and H el ene Mon e 1 traits of human-adapted diurnal and rodent-adapted nocturnal chronotypes of S. mansoni in order to understand how this behavioural diversity could be interpreted in terms of some biological concepts such as sympatry, allopatry and parasite speciation.

Material and methods

Biological material

We used two populations of Schistosoma mansoni originating from Dhofar region (Oman) characterised by a different cercarial shedding pattern. The diurnal chronotype was originally collected from human definitive hosts, whereas the nocturnal chronotype was originally collected from naturally infected Rattus rattus and Biomphalaria pfeifferi [START_REF] Mouahid | A new chronotype of Schistosoma mansoni: adaptive significance[END_REF]. The biological material for the present study has been collected for both chronotypes according to each aim, phylogeny and life-history traits.

Regarding phylogeny, the adult worms were from the G0 to the G5 generations. Twenty-four adult male specimens of S. mansoni from Dhofar (Oman) were used: 10 worms of the diurnal chronotype and 14 worms of the nocturnal chronotype (Table 1). We also used two adult male worms of Schistosoma rodhaini from Burundi known for having nocturnal chronotype.

Regarding the life-history traits we studied the survival of the vertebrate experimental host (the mouse) towards the two chronotypes using data collected from the G2 to the G27 generations. The two chronotypes were then maintained in the laboratory using laboratory bred pigmented Biomphalaria pfeifferi as intermediate hosts, and Swiss OF1 mouse as definitive host. Life cycles were maintained as follows: eggs of S. mansoni were collected from the liver and small intestine of Swiss OF1 mice previously infected with 120 or 80 cercariae. The liver was ground and mixed in a saline solution (8.5& NaCl) and filtered through a series of four sieves with pore sizes of 425, 180, 106 and 45 lm. The eggs on top of the 45 lm sieve were placed in a glass beaker that contained well water (at 25 °C) to allow the egg hatching. After egg hatching, miracidia were collected under a binocular microscope using a Pasteur pipette and placed one by one in 24 well plates. Each well received a healthy pigmented B. pfeifferi in the presence of well water at 25 °C. The 24 well plates were then placed in a room at a constant temperature (25 °C), balanced photoperiod (light/dark: 12 h/12 h) and balanced photophase (06:00 am to 06:00 pm) for 24 h before grouping them in an aquarium with 2 liters of well water and fed ad libitum with fresh lettuce.

The life-history traits of each member of the host-parasite combination S. mansoni/B. pfeifferi were studied using the G14 generation. For each chronotype, the study was conducted twice, using two populations of B. pfeifferi originating from two waterbodies, Siginitti 54°28 0 08.99″ (E); 17°08 0 36.44″ (N) (Experiment 1) and Tibraq 54°19 0 35.09″ (E); 17°05 0 58.11″ (N) (Experiment 2). We considered these experiments as equivalent since the snails were originating from the same drainage basin (Wustah mountains, Dhofar, Oman) and since they were shown to segregate in PCA analysis based on microsatellites genetic comparison [START_REF] Nguema | Genetic diversity, fixation and differentiation of the freshwater snail Biomphalaria pfeifferi (Gastropoda, Planorbidae) in arid lands[END_REF]. The snail mean diameters at the beginning of the experiments are presented in Table 2; they were not significantly different whatever the chronotype and whatever the experiment.

Life-history traits parameters

The interaction between the rodent experimental definitive host and S. mansoni was studied by analysing the survival data of female Swiss OF1 mice, commonly used in the laboratory for the maintenance of life cycles. The study was conducted with two cercarial doses per mouse: 120 and 80 cercariae. For the 120 cercarial dose, we collected 144 data from G2 to G6 generation (2003-2010) for the diurnal chronotype and 96 data from G4 to G13 (2004-2010) for the nocturnal chronotype. For the 80 cercarial dose, we collected 66 data from G11 to G21 (2013-2017) for the diurnal chronotype and 90 data from G15 to G27 (2012-2017) for the nocturnal chronotype.

The interaction between B. pfeifferi and S. mansoni was studied by analysing six life-history traits, three were related to the parasite and three to the snail host.

The parameters measured for S. mansoni were prepatent period (number of days from miracidial penetration to the first cercarial emission), infection rate (number of snails successfully infected divided by the number of surviving exposed snails), and cercarial production (number of cercariae produced per infected snail per day). Cercarial production was measured from the first day of cercarial emission until the 91st day post-infection. The cercariae produced daily by each snail were filtered through a Nitrel filter (/ = 25 lm), stained with 5% Lugol solution, and counted under a binocular microscope. Six B. pfeifferi among the positive snails were used to measure cercarial production. When one of them died, it was immediately replaced by another positive snail of the same combination of similar size, until stock exhaustion. For each chronotype, we also studied the morphometry of 50 eggs recovered from mice faeces. We retained only the eggs containing a living miracidium. Morphometry was done under a microscope.

Three parameters were measured weekly for the snail intermediate host: growth (shell diameter), fecundity and survival, from the time of infection (week 0) until the end of the experiment which lasted 13 weeks. Shell diameter was measured with a digital caliper. Fecundity was determined by measuring the mean weekly number of eggs per snail.

DNA extraction, PCR amplification and sequencing

The adult male worms were preserved in 95% ethanol and stored for DNA extraction. DNA from each worm was extracted using QIAmp â DNA Mini Kit (Qiagen) as described in the manufacturer's protocol. The polymerase chain reaction (PCR) was employed to amplify the cytochrome C oxidase gene (Cox1) using: two primers Cox1schist F5 0 -TCTTTRGATCATAAGCG-3 0 and Cox1-schist-R 5 0 -TAATGCATMGGAAAAAAACA-3 0 [4] and two internal primers Cox-man-F1 5 0 -AATTTTAATAGGTG-GATTTGG-3 0 and Cox-man-R1 5 0 -AATCCATAATAAC-CAAACGAC-3 0 [START_REF] Mouahid | A new chronotype of Schistosoma mansoni: adaptive significance[END_REF]. Each PCR reaction was carried out in a total volume of 25 ll: DNA 2 ll, 59 Green GoTaq TM Buffer (Promega) 5 ll; MgCl 2 25 mM, 1 ll; dNTPs 2 mM, 2.5 ll; primers 10 lM, each 1 ll; Go Taq TM DNA Polymerase (Promega) 0.3 ll and H 2 O milliQ, 12.2 ll. This mix was placed in an Eppendorf Mastercycler. PCR cycling conditions were as follows: initial denaturation step at 94 °C for 120 s; denaturation at 94 °C for 30 s, annealing at 50 °C for 30 s and elongation at 72 °C for 120 s, for 40 cycles; extension step at 72 °C for 7 min. In order to assess the validity of results obtained from each PCR assay, PCR negative assay (H 2 O milliQ added as template) was done. PCR products of the expected size (1100 bp) were obtained. DNA bands were isolated from a 1% agarose gel and purified using Wizard â SV Gel and PCR Clean-Up System (Promega). PCR products were sequenced using a dideoxy-dye-terminator method (CEQ TM DTCS-Quick Start kit, Beckman Coulter) and a CEQ TM 8000 apparatus (Beckman Coulter). Sequences were obtained using the CEQ TM 8000 sequence analysis software.

Phylogenetic analysis

The different steps of the phylogenetic analysis were conducted on MEGA 7.0.9 beta software [START_REF] Kumar | MEGA7: molecular evolutionary genetics analysis version 7.0 for bigger datasets[END_REF]. DNA multiple sequence alignments were performed according to codons using MUSCLE program [START_REF] Edgar | MUSCLE: multiple sequence alignment with high accuracy and high throughput[END_REF]. Poorly aligned and highly variable region alignments were automatically removed using GBlocks 0.91b [START_REF] Castresana | Selection of conserved blocks from multiple alignments for their use in phylogenetic analysis[END_REF][START_REF] Dereeper | Fr: robust phylogenetic analysis for the non-specialist[END_REF][START_REF] Dereeper | BLAST-EXPLORER helps you building datasets for phylogenetic analysis[END_REF]; http://molevol.cmima.csic.e s/castresana/Gblocks_server.html) with the following three options: smaller final blocks, gap positions within the final blocks and less strict flanking positions. Phylogenetic tree was obtained via maximum-likelihood (ML) method on the MEGA software. Different models of DNA evolution for coding sequences and associated parameters were estimated using MEGA software [START_REF] Nei | Molecular Evolution and Phylogenetics[END_REF] and the best parameters were identified based on the Bayesian Information Criterion (BIC). An ML tree was then computed with MEGA software using the best nucleotide substitution model Hasegawa-Kishino-Yano (HKY+G) and validated via bootstrap procedure with 1000 replicates. The tree was built using only haplotypes according to each host from which each chronotype was originated (Humans, Rattus rattus and Biomphalaria pfeifferi). We also used S. mansoni Cox1 sequence from Puerto Rico (accession number AF216698), S. mekongi (accession number AF217449) and S. japonicum (accession number NC_002544). The accession numbers of the sequences used in this study are MK172830, MK172831 and MK172832 for the diurnal chronotype and MK172833 and MK172834 for the nocturnal chronotype (Table 1).

Statistical analyses

Fisher's exact test (infection rate), the Mann-Whitney test (U) and autocorrelation were used as statistical analysis tools for the various comparisons. Survival rates were examined by Kaplan-Meier survival analysis and survival curves were compared using the log rank test. All data were analysed using StatView v.

Results

Parasite life-history traits

In either experiment (1 or 2, see Table 3), mean prepatent period was significantly shorter in the diurnal than the nocturnal chronotype (P < 0.05); infection rates did not significantly differ between the two chronotypes; cercarial production of the diurnal chronotype was significantly higher than that of the nocturnal chronotype (P < 0.05). Cercarial production followed a rhythmic pattern (Figure 1a, b andd) with alternating periods of high and low production, except for the diurnal chronotype in experiment 2 (Figure 1c). Between two periods of high or low cercarial production, the autocorrelation test indicated a periodicity of 14 days for the diurnal chronotype and of 15-20 days for the nocturnal chronotype.

There was no significant difference in the measurements of the eggs of diurnal and nocturnal chronotypes. However, whatever the chronotype, the morphological study of the eggs collected in the stools revealed the presence of a polymorphism based on the shape of the egg shell with the presence of two morphotypes A and B (Figure 2): eggs with A morphotype had the classical form of S. mansoni, whereas eggs with B morphotype showed a beginning of curvature at the opposite end of the lateral spine but not as pronounced as in S. rodhaini eggs, and the proportion of the B morphotype was significantly higher in the nocturnal chronotype (Chi-square = 9.18; P = 0.0034).

Host life-history traits

In either experiment (1 or 2) and for either chronotype used, B. pfeifferi's growth curves were close, with rapid growth during the first 3 weeks followed by a slowdown until week 13 (Table 4; Figure 3). In experiment 1, mean weekly shell diameter showed no significant difference between the two chronotypes, whereas in experiment 2, mean weekly shell diameters were significantly smaller with nocturnal chronotype since week 2 post-infection. Snails infected with diurnal S. mansoni chronotype continued to spawn even though the values (Table 4) were far below the values of healthy snails (145.00 AE 23.24 to 159.80 AE 10.43 for a period of 13 weeks). Snails infected with the nocturnal S. mansoni chronotype produced a oriented at the opposite side of the lateral spine. The diurnal chronotype contained 92% of the A morphotype whereas the nocturnal chronotype only contained 64%. The rest of the eggs presented the B morphotype; thus tiny amount of eggs or did not lay eggs at all. Egg-laying began in the 3rd week post-infection and was not continuous. In the 4 host-parasite combinations studied, the proportion of egg-laying snails was very small and laying could take place until the 13th week. Survival of B. pfeifferi differed between experiment 1 and 2 (Table 4). In experiment 1 no significant difference was found between the chronotypes while in experiment 2, the survival of the snails infected with the nocturnal chronotype was significantly shorter.

Mice survival (in days) according to the cercarial dose and to the chronotype was as follows: (i) For the cercarial dose of 120 it was 127 days AE 5 (N = 144) for the diurnal chronotype and 149 days AE 10 (N = 96) for the nocturnal chronotype. (ii) For the cercarial dose of 80 it was 145 days AE 8 (N = 66) for the diurnal chronotype and 172 days AE 13 (N = 90) for the nocturnal chronotype. For both cercarial doses, the Mann-Whitney U-test showed no significant difference between the two chronotypes (P = 0.5 and P = 0.8 for each cercarial dose, respectively). However, mice survival increased with decreasing cercarial dose with a significant difference only for the diurnal chronotype (P = 0.01 and U = 3748).

Phylogeny

The phylogenetic results based on the 1054 bp Cox1 analysis revealed the presence of 3 haplotypes in the diurnal chronotype whose accession numbers are MK172830 for 7 individuals represented by Sm-DC1/Hs, MK172831 represented by Sm-DC2/Hs and MK172832 for 2 individuals, represented by Sm-DC3/Hs. Regarding the nocturnal chronotype, two accession numbers were given, MK172833 for the 10 adult worms (Sm-NC1/Rr to Sm-NC10/Rr) collected from naturally infected host Rattus rattus and MK172834 for the 4 adult worms (Sm-NC1/ Bp to Sm-NC4/Bp) collected from naturally infected Biomphalaria pfeifferi and after mouse experimental infection. Whatever the host, only one haplotype, identical to the diurnal haplotype Sm-DC1/Hs, was observed in the nocturnal chronotype (Table 1 and Figure 4).

Discussion

We explored the host-parasite life-history traits thanks to the dual and remarkable presence of two S. mansoni chronotypes (diurnal vs. nocturnal) in the same region and found that the diurnal and the nocturnal chronotypes had different life-history traits. The diurnal chronotype had a significantly shorter prepatent period, the same infection rates in the snails and a significantly higher production, inducing higher growth, fecundity and survival in its snail host are factors that will enhance its transmission to the vertebrate host. Thus diurnal chronotype transmission is higher that nocturnal chronotype transmission. This allows us to say that the diurnal chronotype of S. mansoni seems to be better adapted to its sympatric snail intermediate host than the nocturnal chronotype S. mansoni. In fact, the diurnal parasite cercarial production than the nocturnal chronotype.

Regarding the host life-history traits, shell diameter, fecundity and survival of the snails infected with the diurnal chronotype were either the same or significantly higher than those of the snails infected with the nocturnal chronotype and mouse survival did not differ between the two chronotypes. From the parasite's point of view, reducing its pre-patent period, enhancing cercarial chronotype behaves normally, with its snail host coming from a sympatric population whereas the nocturnal chronotype behaves as if the snail host is from an allopatric population. For example, Ibikounl e et al. showed that the prepatent period was significantly shorter for S. mansoni and for S. haematobium from Toho-Todougba (Benin) on their sympatric snail hosts B. pfeifferi and Bulinus globosus, respectively, compared to the pre-patent periods obtained for different allopatric populations of the snail hosts [START_REF] Ibikounl E | Life-history traits indicate local adaptation of the schistosome parasite, Schistosoma mansoni, to its snail host, Biomphalaria pfeifferi[END_REF][START_REF] Ibikounl E | Snail intermediate host/Schistosoma haematobium relationships from three transmission sites in Benin (West Africa)[END_REF]. The "allopatric" behaviour of the nocturnal chronotype supports the hypothesis of its recent appearance. The morphology of the parasite eggs also differed between the chronotypes: the percentage of morphotype B (presence of a curvature) was significantly higher in the eggs of the nocturnal chronotype. This trait has been observed in a Guadeloupean transmission site, where R. rattus was involved in the life cycle of a diurnal S. mansoni population [START_REF] Th Eron | Early and late shedding patterns of Schistosoma mansoni cercariae: ecological significance in transmission to human and murine hosts[END_REF]. However, no explanation can be given at present on the possible (or not) implication these differences may have on schistosome transmission.

Whatever the chronotype, the cercarial production showed generally high and low cercarial production periods with a rhythmic pattern where periods of high production alternated with periods of low production. Such rhythmic variation in the cercarial production was shown in S. mansoni from Brazil on Biomphalaria glabrata from Brazil by [START_REF] Th Eron | Dynamics of larval populations of Schistosoma mansoni in Biomphalaria glabrata: I Rhythmic production of cercariae in monomiracidial infections[END_REF] and in S. bovis from Spain on Planorbarius metidjensis from Morocco [START_REF] Mouahid | Schistosoma bovis: variability of cercarial production as related to the snail hosts: Bulinus truncatus, B. wrighti and Planorbarius metidjensis[END_REF]. These variations were closely related to the pattern of cercariogenesis and sporocystogenesis processes [START_REF] Th Eron | Dynamics of larval populations of Schistosoma mansoni in Biomphalaria glabrata: II Chronobiology of the intra-molluscal larval development during the shedding period[END_REF].

Molecular analysis revealed a greater diversity within the diurnal than the nocturnal chronotype and argues in favour of an emergence of the nocturnal phenotype from the diurnal phenotype. This is supported by a previous molecular study of various strains of S. mansoni, which showed that the lateral transfer from humans to rats is likely to have been recent and localised [START_REF] Morgan | Origin and diversification of the human parasite Schistosoma mansoni[END_REF]. The scenario of the nocturnal chronotype emergence is probably linked to a diversifying natural selection that leads to its selection. The definitive host Rattus rattus, which constitutes a contrasting environment with a nocturnal behaviour compared to humans, allows this new trait to be selected. The diurnal chronotype widened its host spectrum by passing over the rat; it is not a host shift. We do not know which mechanism was at the origin of the chronobiological divergence of S. mansoni from Dhofar. Mutation and/or genetic drift are two main mechanisms. Many authors agree that divergent natural selection is a fundamental step in the process of speciation among sexually reproducing eukaryotes. The ethological divergence in the profiles of the cercarial emission (diurnal or nocturnal) constitutes good conditions for a phenomenon of sympatric speciation (in the sense of Coyne and Orr [START_REF] Coyne | Speciation[END_REF]) to be triggered. Mating might not occur because of differences in cercarial shedding time. This behavioural divergence could act as a prezygotic reproductive barrier, reducing the possibility of males and females to meet in the same definitive host (Human or rodent) and consequently to contribute to a barrier to gene flow between the two chronotypes. We hypothesise that this prezygotic isolation is likely to create, in the long term, post-zygotic isolation as it currently occurs between species of schistosomes of the same group (terminal or lateral spine egg group).

Indeed, we are aware that the concept of species in the genus Schistosoma is to be taken with care. Since the 1970s, species of the same group are known to hybridise with a postzygotic isolation from the 1st to 3rd generation (depending on the crossed species and on the reciprocal crossing) due to high decrease in the miracidial viability and infectivity [START_REF] Taylor | Hybridisation experiments on five species of African Schistosomes[END_REF]. In schistosomes, no complete cessation of sexually mediated gene exchange exists and natural hybridisation between different species of schistosome in Africa explains the growing number of cases of introgressed species, such as S. haematobium introgressed by S. bovis [START_REF] Mon E | Introgressive hybridizations of Schistosoma haematobium by Schistosoma bovis at the origin of the first case report of Schistosomiasis in Corsica (France, Europe)[END_REF]. If the gene flow is not interrupted between the two chronotypes, which is perfectly conceivable since we are within the same species, we could imagine the emergence of natural viable hybrid progeny with an ultradian chronotype (diurnal and nocturnal) adapted to both humans and rats.

Figure 1

 1 Figure 1 Mean daily cercarial productions of diurnal (a and c) and nocturnal (b and d) Schistosoma mansoni chronotype from Biomphalaria pfeifferi from Siginitti (a and b (Experiment 1)) and from Tibraq (c and d (Experiment 2)). Bars represent standard errors.

Figure 2

 2 Figure 2 Whatever the chronotype, diurnal vs. nocturnal, Schistosoma mansoni eggs showed two morphotypes: (a) eggs with a typical morphology of S. mansoni (L = 143 lm) and (b) eggs with, at the opposite end of the lateral spine, a beginning of curvature but not as pronounced as in Schistosoma rodhaini (L = 130 lm). (c) S. rodhaini typical egg (L = 139 lm). L: egg length.

Figure 3

 3 Figure 3 Mean weekly diameter (AE SE) of Biomphalaria pfeifferi (a: Experiment 1 and b: Experiment 2) infected with the diurnal (triangle) and the nocturnal (square) Schistosoma mansoni chronotypes during 13 weeks post infection. *Significant difference (Mann-Whitney U-test; P < 0.05).

Figure 4

 4 Figure 4 ML tree based on mitochondrial cytochrome oxidase I sequences of Schistosoma mansoni individual adults according to each chronotype, diurnal vs. nocturnal. This analysis was computed using a 1054 bp-long multiple alignment of 24 S. mansoni Omanese samples, one sample of S. mansoni from Puerto Rico (NMRI), two samples of S. rodhaini from Burundi, one sample of S. japonicum and one sample of S. mekongi for the outgroups. Node supports are indicated by the ML bootstrap values.

Table 1

 1 Schistosoma mansoni adult specimens used for the molecular analysis. Three haplotypes (*) were found for the diurnal chronotype. Only one haplotype, identical to the diurnal haplotype Sm-DC1/Hs, was observed for the nocturnal chronotype

	Specimen	COI accession number	Generation	Country, region, locality	Latitude (East)	Longitude (North)
	Sm-DC1/Hs*	MK172830	G 1	Oman, Dhofar region, Sheer locality	54°11 0 10.41″	17°11 0 41.21″
	Sm-DC2/Hs*	MK172831
	Sm-DC3/Hs*	MK172832
	Sm-DC4/Hs	MK172830	G 2
	Sm-DC5/Hs	MK172830	G 3
	Sm-DC6/Hs	MK172832
	Sm-DC7/Hs	MK172830
	Sm-DC8/Hs
	Sm-DC9/Hs
	Sm-DC10/Hs
	Sm-NC1/Rr	MK172833	G 0	Oman, Dhofar region, Tibraq locality	54°19 0 35.09″	17°05 0 58.11″
	Sm-NC2/Rr
	Sm-NC3/Rr
	Sm-NC4/Rr
	Sm-NC5/Rr		G 1
	Sm-NC6/Rr
	Sm-NC7/Rr
	Sm-NC8/Rr
	Sm-NC9/Rr		G 3
	Sm-NC10/Rr		G 5
	Sm-NC1/Bp	MK172834	G 0
	Sm-NC2/Bp
	Sm-NC3/Bp
	Sm-NC4/Bp

Sm, Schistosoma mansoni; DC, diurnal chronotype; NC, nocturnal chronotype; Hs, Homo sapiens; Rr, Rattus rattus; Bp, Biomphalaria pfeifferi; G, generation.

Table 2

 2 Diameter of Biomphalaria pfeifferi (mean in mm AE SE) at their exposure to diurnal and nocturnal chronotypes for experiments 1 and 2

	Experiment	Diurnal	Nocturnal
	1	4.26 AE 0.13 (N = 32)	4.53 AE 0.17 (N = 19)
			NS
	2	4.53 AE 0.17 (N = 45)	4.30 AE 0.15 (N = 31)
		NS	NS

N, Number of snails used; NS, non significant (Mann-Whitney U-test).

  [START_REF] Kumar | MEGA7: molecular evolutionary genetics analysis version 7.0 for bigger datasets[END_REF] (SAS Institute Inc), SPSS (Version 10.05, Chicago, IL, USA), and GraphPad InStat (Version 3.05, GraphPad Software, La Jolla, CA, USA). Means and standard-errors (SEs) for all values were given. A P-value less than 0.05 was considered significant.The French "Direction D epartementale de la Protection des Populations" provided (i) authorisation to our laboratory (UMR 5244) for experiments on animals (living vertebrates) N°: C 66-136-01 with prefectoral order N°: 2012-201-0008 and (ii) certificate for animal experimentation for Dr H el ene Mon e, authorisation N°: C 66.11.01; articles N°: R 214-87, R 214-122 and R 215-10. Housing, breeding and animal care adhered to national ethical requirements.

Table 3

 3 Diurnal vs. nocturnal Schistosoma mansoni life-history traits on Biomphalaria pfeifferi from Siginitti (experiment 1) and from Tibraq (Experiment 2)

		Experiment 1		Experiment 2
		Diurnal (N = 32)	Nocturnal (N = 19)	Diurnal (N = 45)	Nocturnal (N = 31)
	Pre-patent period (mean day AE SE)	26.56 AE 1.33	32.38 AE 0.73*	28.30 AE 0.26	30.29 AE 0.35*
	Infection rate (%)	64	42 (NS)	44	68 (NS)
	Cercarial production/day/snail	372 AE 68 (N = 6)	221 AE 35* (N = 6)	464 AE 74 (N = 6)	183 AE 16* (N = 6)
	(mean number of cercariae AE SE)

NS, non-significant; SE, Standard-Error; *P < 0.05 (Mann-Whitney U-test for pre-patent period and cercarial production; Fisher exact test for infection rate).

Table 4

 4 Biomphalaria pfeifferi from Siginitti (Experiment 1) and from Tibraq (Experiment 2) life-history traits, infected either with the diurnal or the nocturnal Schistosoma mansoni chronotype

		Experiment 1		Experiment 2
		Diurnal (N = 18)	Nocturnal (N = 8)	Diurnal (N = 20)	Nocturnal (N = 17)
	Shell diameter (mm)	8.48 AE 0.12	8.49 AE 0.16 NS	8.52 AE 0.12	7.92 AE 0.11*
	Fecundity	7.44 AE 5.64	0.75 AE 0.80 NS	9.88 AE 4.06	0.00
		13/18: 0 egg	7/8: 0 egg	10/20: 0 egg	17/17: 0 egg
	Survival (%)	90	100 NS	85	47*

Snail growth is expressed by the shell diameter and Fecundity is expressed by the mean number of eggs/week/snail. NS, non significant; *P < 0.05 (Mann-Whitney U-test). Survival is determined at the 13th week post-infection.

IHPE Laboratory, UMR5244, UM, CNRS, IFREMER, University of Perpignan Via Domitia, Perpignan, France

Institut de Recherche en Ecologie Tropicale Libreville, Libreville, Gabon

Directorate General of Health Services, Salalah, Sultanate of Oman

Department of Microbiology and Immunology, College of Medicine and Health Sciences, Sultan Qaboos University, Muscat, Sultanate of Oman

G. Mouahid et al. Diurnal vs. nocturnal chronotypes of Schistosoma mansoni

Acknowledgement

This research was financially supported by UPVD, CNRS, Ministry of Health of Oman, and by the French National Agency for Research (ANR) [grant ANR-17-CE12-0005-01] CHRONOGET to HM.


Emilie Sakat
email: emilie.sakat@c2n.upsaclay.fr

Léo Wojszvzyk

Jean-Jacques Greet

Jean-Paul Hugonin

Christophe Sauvan
email: christophe.sauvan@institutoptique.fr

Enhancing light absorption in a nanovolume with a nanoantenna: theory and gure of merit

Keywords: Nanophotonics, Plasmonics, Nanoantenna, Absorption, Local Density of Optical States Light absorption and emission are two fundamental processes of light-matter interac-

We study light absorption by a dipolar absorber in a given environment, which can be a nanoantenna or any complex inhomogeneous medium. From rst-principle calculations, we derive an upper bound for the absorption, which decouples the impact of the environment from the one of the absorber. Since it is an intrinsic characteristic of the environment regardless of the absorber, it provides a good gure of merit to compare the ability of dierent systems to enhance absorption. We show that, in the scalar approximation, the relevant parameter is not the eld enhancement but the ratio between the eld enhancement and the local density of states. Consequently, a plasmonic structure supporting hot spots is not necessarily the best choice to enhance absorption. We also show that our theoretical results can be applied beyond the scalar approximation and the plane-wave illumination.

Absorption inside subwavelength objects is weak but dierent strategies can be used to circumvent this limitation. The absorber properties, the illumination, or the environment can be engineered. Many works have been dedicated to the optimization of the absorption by a single nanoparticle 1521 . For an absorber in a homogeneous medium, the physical bounds of the problem are known; they depend on the multipolar character of the particle. The maximum absorption cross-section of a molecule or a nanoparticle that behaves like a pure electric dipole (dipolar approximation), is 3λ 2 /(8πn 2 ), with λ the wavelength and n the surrounding refractive index [START_REF] Tretyakov | Maximizing Absorption and Scattering by Dipole Particles[END_REF][START_REF] Grigoriev | Optimizing Nanoparticle Designs for Ideal Absorption of Light[END_REF] . This upper bound can only be reached if the absorber polarizability α matches a precise value, α opt = i3λ 3 /(8π 2 n). Subwavelength particles that go beyond the dipolar approximation oer additional degrees of freedom and are governed by dierent physical bound 1621 . Absorption by an ensemble of particles in a homogeneous medium is yet another related problem with a dierent upper limit [START_REF] Hugonin | Fundamental limits for light absorption and scattering induced by cooperative electromagnetic interactions[END_REF] .

Plunging the absorber in a complex medium or modifying the illumination oers additional possibilities to tailor the absorption 10,23,24 . In this work, we consider a subwavelength absorber in a complex environment (inhomogeneous or not) as depicted in Fig. 1. We focus on the absorption inside the absorber and do not discuss dissipation in the surroundings, if any. To fully exploit the control possibilities oered by the environment, it is crucial to know what is the relevant gure of merit. The absorption density is proportional to the local electric-eld intensity |E(r)| 2 25 , which results from the environment and the absorber. It is highly desirable to decouple both contributions to provide a gure of merit that is intrinsic to the environment. Only then can we properly compare the ability of dierent structures to modify absorption.

Let us draw a parallel with spontaneous emission. In a complex medium, it is modied by a factor proportional to the photonic local density of states (LDOS) and the situation is more complex: the eld enhancement is not the only electromagnetic property of the environment that drives the absorption. For instance, it has been recently

shown that, in a scalar quasi-static case, the absorption cross-section is inversely proportional to the square of the LDOS 23 . The work in Ref.

23 considers absorption in a given system absorber + environment but does not decouple the contribution of the absorber from the impact of the environment. It does not study the physical bounds of the problem. Until now, no upper bound kind of Purcell factor analogue has been derived for the problem of an absorber in a complex environment.

Antenna theory provides a solution in one specic case.

For an antenna receiving a signal from the direction (θ, φ), the maximum absorption cross-section of the load is G a (θ, φ)λ 2 /(4πn 2 ), with G a the antenna gain 3234 . This upper bound is reached if the load is impedance-matched with the antenna. The gain is dened for an emitting antenna as the fraction of the wall-plug power radiated in the direction (θ, φ). Unfortunately, the G a λ 2 /(4πn 2 ) limit only applies to plane-wave illumination and to antennas working in the scalar approximation when one component of the electromagnetic eld is dominant. In cases where the vectorial nature of the electromagnetic eld cannot be neglected, the problem remains open. Moreover, the antenna point of view highlights the gain, whereas other derivations underline the eld enhancement and the LDOS 23 . It is thus important to generalize existing results, while enlightening the link between them.

In this work, we derive a general upper bound for the power dissipated in a subwavelength absorber surrounded by any complex inhomogeneous environment. Owing to its subwavelength dimensions, the absorber is treated in the electric-dipole approximation. On the other hand, the electromagnetic properties of the environment are treated rigorously without any approximation. The upper bound is independent of the absorber; it entirely depends on the environment and the illumination. Thus, it provides a relevant gure of merit for comparing the ability of dierent systems to enhance absorption. This gure of merit results from the interplay between two electromagnetic properties of the environment, the eld enhancement and the Green tensor. We show under which assumptions the gain is a relevant parameter. We nally discuss under which conditions the system can reach the upper bound.

We apply the theory to a few emblematic examples of nanophotonics: a plasmonic dimer nanoresonator, dielectric nanoantennas, and a silicon-on-insulator (SOI) ridge waveguide.

We evidence that a plasmonic system providing extremely large eld enhancements is not necessarily an optimal choice to increase absorption.

We consider the problem illustrated in Fig. 1(a). An absorbing subwavelength particle (in red) is placed in a complex absorbing or non-absorbing environment (in gray) and illuminated by an incident eld E inc . We focus on the absorption inside the particle. We consider passive materials and use the exp(-iωt) convention for time-harmonic elds, with ω = 2πc/λ. The following derivations apply to any incident eld and any environment geometry. We refer to the environment as the antenna, but no particular assumption is made on its geometry.

The total electric eld can be written as

E(r) ≡ E b (r) + E s (r)
, where E b is the eld in the absence of the absorber, see Fig. 1(b), and E s is the eld scattered by the absorber. It is the eld radiated by the current source induced inside the absorber by the exciting eld.

We now make the sole assumption of our derivation. We assume that the subwavelength particle scatters light like an electric dipole p located at r = r 0 . The scattered eld is then E s (r) = µ 0 ω 2 G(r, r 0 )p, with G the Green tensor of the antenna alone.

The power dissipated in the particle is the dierence between extinction and scattering, P a = P e -P s 25 . Within the dipole approximation, P e = -1 2 ωIm(p † E b ) and P s = ωp † gp, with g ≡ 1 2 µ 0 ω 2 Im[G(r 0 , r 0 )] (see Supporting Information). It follows

P a = - 1 2 ωIm(p † E b ) -ωp † gp , (1)
where E b ≡ E b (r 0 ) and p † is the conjugate transpose of p. The induced dipole is p =

ε 0 α(ω)[E b (r 0 ) + µ 0 ω 2 S(r 0 , r 0 )p],
with α(ω) the polarizability tensor of the particle and S = G -G 0 , G 0 being the Green tensor of the homogeneous medium of refractive index n that surrounds the particle.

In order to derive an upper bound that is independent of the absorber, we calculate the maximum of P a = f (p). In a complex medium composed of passive materials, the quantity ωu † gu gives the total electromagnetic power emitted by the electric-dipole source u (see Supporting Information). Therefore, u † gu 0 whatever the vector u and g is a semi-denite positive matrix. We now use this important property to derive the maximum of P a = f (p). For that purpose, we have to rewrite the dissipated power under the form P a = -ωu † gu + A, where A is independent of p. A few algebraic manipulations of Eq. ( 1) lead to (details can be found in (see Supporting Information))

P a = -ω p † + i 4 E † b g -1 g p - i 4 g -1 E b + ω 16 E † b g -1 E b . (2)
Now the dissipated power takes the form P a = -ωu

† gu + A, with u = p -i 4 g -1 E b and A = ω 16 E † b g -1 E b . Since g is a semi-denite positive matrix, we know that -ω(p † + i 4 E † b g -1 )g(p -i 4 g -1 E b ) 0.
This readily leads to an upper bound for the absorption, which is independent of the absorber and depends only on the environment and the illumination,

P a P max a , with P max a = ω 16 E † b g -1 E b . (3)
According to Eq. ( 2), the upper bound is reached for an optimal dipole p opt = i 4 g -1 E b .

The optimal polarizability that yields the maximum absorption is then (see Supporting

Information) α opt (ω) = c 2 ω 2 S * (r 0 , r 0 ) -i ωn 3πc I -1 . ( 4
)
with S * the conjugate of S. Any other polarizability in the same environment necessarily absorbs less light. Equation ( 4) can be seen as a vectorial generalization of the usual scalar impedance-matching concept [START_REF] Balanis | Antenna theory: Analysis and design[END_REF][START_REF] Greet | Impedance of a Nanoantenna and a Single Quantum Emitter[END_REF] .

We dene the absorption eciency η a = P a /P inc as the fraction of the incident power that is absorbed inside the particle. The maximum absorption eciency is

η max a = ω 16
E † b g -1 E b P inc . (5)
If the incident eld is a plane wave, we rather dene a maximum absorption cross-section by normalizing P max a with the incident Poynting vector 1

2 ε 0 cn|E inc | 2 , σ max a = 3λ 2 8πn 2 g 0 E † b g -1 E b |E inc | 2 , (6)
where g 0 I ≡ 1 2 µ 0 ω 2 Im[G 0 (r 0 , r 0 )] = ω 3 n/(12πε 0 c 3 )I. Note that, in a homogeneous environment, g = g 0 I, E b = E inc , and we recover the usual result σ max a = 3λ 2 /(8πn 2 ).

Equations ( 3)-( 6) form the central result of this article. They deserve a few comments before we illustrate their consequences on a couple of examples. The upper bound is independent of the absorber; it solely depends on the antenna and the incident eld. Thus, Eqs. ( 5) and ( 6) provide meaningful gures of merit for comparing the ability of dierent systems to enhance absorption. These novel gures of merit result from an interplay between the local eld E b provided by the sole antenna and the imaginary part of its Green tensor g.

In the case where the vectorial nature of the electromagnetic eld can be neglected, the general upper bound can be replaced by an approximate form. Let us assume that the eld E b and the Green tensor g are dominated by a single component, the z component. Within this scalar approximation, Eq. ( 6) reduces to

σ max a ≈ 3λ 2 8πn 2 |E bz | 2 |E inc | 2 g 0 g zz = 3λ 2 8πn 2
Intensity enh. LDOS enh. .

The upper bound appears to be the ratio between the intensity enhancement

|E bz | 2 /|E inc | 2
and the LDOS enhancement g zz /g 0 . The fact that a LDOS enhancement is detrimental can be intuitively understood as follows: a larger LDOS increases the radiation of the induced dipole, i.e., it increases the scattering at the expense of absorption.

It can be shown using reciprocity that Eq. ( 7) is equivalent to

σ max a = G a (θ, φ)λ 2 /(4πn 2 ),
with G a the antenna gain (see Supporting Information). The upper bound derived with antenna theory is thus a particular (scalar) case of the general result in Eq. ( 6). Two equivalent points of view can be adopted. They provide dierent conclusions and it is interesting to consider both of them. Equation ( 7) evidences that antennas supporting hot spots are a good choice if, and only if, the formation of hot spots is not concomitant with a large LDOS. On the other hand, the antenna gain is the product of the radiative eciency and the directivity, G a (θ, φ) = η r D(θ, φ) 32 . This second point of view underlines that (i) directional structures perform better and (ii) dielectric objects (η r = 1) are better choices than plasmonic structures with η r < 1.

Let us illustrate the theory with a few examples. First, we study a plasmonic dimer nanoresonator and a Yagi-Uda antenna made of silicon nanospheres. Both can be treated in the scalar approximation. Secondly, we consider structures that can only be studied with a fully vectorial formalism: silicon nanospheres in a L-shape conguration and a SOI ridge waveguide.

In the case of a single-mode antenna, LDOS and eld enhancement are not independent since both are driven by the excitation of an eigenmode 36 . Both are resonant with the same spectral prole and increasing one necessarily enhances the other. Thus, quite counterintuitively, the value of the upper bound in Eq. ( 7) is not correlated to the presence of hot spots. To illustrate this conclusion, let us consider a dimer made of two gold nanorods illuminated by a plane wave. Only the dipole mode at λ = 3 µm is excited. Figure 2(a) displays the spectra of the maximum absorption cross-section and the intensity enhancement at the gap center: The antenna is embedded in a medium of refractive index n = 1.5.

|E b | 2 /|E inc | 2 is resonant whereas σ max a is not.
Since the upper bound in the scalar case is proportional to the radiative eciency η r and the directivity, we switch to a Yagi-Uda antenna made of silicon nanospheres, which is known to be directional 38 . We keep a plane-wave illumination and a wavelength of λ = 3 µm. Thus, silicon is transparent and η r = 1. The maximum absorption cross-section σ max a is represented on Fig. 3(a) as a function of the incident angle θ for r 0 located in between the reector and the rst director. The incident plane wave is either TM (black curve) or TE (red curve) polarized.

For θ = 0 • , σ max a is one order of magnitude larger than 3λ 2 /(8πn 2 ). This evidences that a dielectric directional antenna can provide larger absorption enhancements than a plasmonic antenna supporting hot spots. Note that, for a xed geometry, the upper bound changes with the incident angle, whereas the optimal polarizability given by Eq. ( 4) is constant.

Once the polarizability has been chosen, one is sure to reach the upper bound whatever the incident eld.

Before considering fully vectorial examples, we discuss the possibility to reach the upper bound on the Yagi-Uda example. The optimal polarizability corresponds for instance to a sphere of radius 176 nm lled with a material of relative permittivity ε opt = -6.46+1.22i; the relation between the electric-dipole polarizability, the sphere radius, and the permittivity has been rigorously calculated with Mie theory. Such a permittivity at λ = 3 µm can be obtained with highly-doped semiconductor nanocrystals [START_REF] Mendelsberg | Extracting reliable electronic properties from transmission spectra of indium tin oxide thin lms and nanocrystal lms by careful application of the Drude theory[END_REF][START_REF] Mendelsberg | Understanding the Plasmon Resonance in Ensembles of Degenerately Doped Semiconductor Nanocrystals[END_REF] . We have checked with a rigorous numer- ical method that inserting this nanosphere in the Yagi-Uda antenna yields an absorption that is indeed equal to the upper bound (see Supporting Information). We have also tested the robustness of the optimal polarizability. Figure 3 Such a structure cannot be described in the scalar approximation, especially for positions r 0 outside the symmetry planes. In that case, Eq. ( 7) is not valid. The system can only be Calculations are performed with a rigorous multipole method 38,39 . characterized by the vectorial upper bound of Eq. ( 6). The latter depends on the absorber position and it is important, for a given nanoantenna, to evaluate where the particle could reach the best absorption. Let us study a last example where the incident eld is not a plane wave. It allows us to evidence the generality of the absorption upper bound that can be used in a variety of photonic structures. We consider an absorber located above a SOI ridge waveguide. In this integrated conguration, the incident eld is a guided mode. By applying Eq. ( 5), we have calculated the maximum absorption eciency above the waveguide, see Fig. 5. The absorber is moved along the horizontal white line located 20 nm above the waveguide; the incident eld is either the fundamental quasi-TE mode (solid black curve) or the quasi-TM mode (dashed red curve). The latter provides an absorption eciency of almost 50%, even for an absorber located outside the waveguide core.

In conclusion, we derived an upper bound for the problem of absorption by a dipolar absorber in a complex environment, homogeneous or not. The derivation relies on a vectorial formalism and is valid for any environment and any illumination. Since it decouples the environment from the absorber, the upper bound provides a meaningful gure of merit for comparing the intrinsic ability of dierent structures to enhance absorption in a nanovolume. Moreover, it allows seeking, for one given structure, the optimal position for the absorber.

In the scalar approximation, the relevant parameter is not the eld enhancement but the ratio between eld enhancement and LDOS. Thus, although placing an absorber in a hotspot of a plasmonic structure can increase the absorption 10 , plasmonic antennas supporting hot spots are not necessarily the best candidates to reach high absorption enhancements. Larger enhancements can be achieved with dielectric directional antennas. In cases where the scalar approximation is not valid or the illumination is not a plane wave, the situation is more complex and our general upper bound is the good gure of merit to characterize the problem.

Since it relies on electromagnetic calculations (eld enhancement and Green tensor) that are nowadays performed routinely, we think that this work opens new avenues for understanding and optimizing absorption in subwavelength volumes inserted in complex media. Thanks to

tion 1 , 2 .

 12 Absorption converts electromagnetic energy carried by photons to internal energy of matter carried by electrons or phonons. Controlling the absorption in small volumes is a major issue for numerous applications, such as photovoltaics 35 , sensing by surface-enhanced infrared absorption (SEIRA) 68 or enhanced infrared photoexpansion nanospectroscopy 9 , thermal emission 10,11 , photothermal eects at the nanoscale for enhanced photocatalysis and nano-chemistry 1214 .

29 .FIG. 1 :

 291 FIG. 1: (a) Subwavelength absorber (in red) in an arbitrary environment (in gray, referred to as "nanoantenna") illuminated by an incident eld E inc . We focus on the absorption inside the particle. (b) Same problem without the absorber. The eld E b is the eld scattered by the environment alone.

Figure 2 ( 4 FIG. 2 :

 242 FIG. 2: Nanoantennas in the scalar approximation. Dimer composed of two gold cylinders illuminated by a normal-incident plane wave polarized along the cylinder axis. The point M (r 0 ) is the gap center. (a) Spectra of the maximum absorption cross-section σ max a (left axis, black curve) and the intensity enhancement (right axis, dashed red curve) for L = 414 nm, R = 15 nm, and w = 20 nm. (b) Maximum absorption cross-section and intensity enhancement as a function of the gap width w (the cylinder length is tuned to keep the resonance xed at λ = 3 µm). Calculations are performed with an aperiodic Fourier modal method dedicated to body-of-revolution objects 37 . The gold permittivity is given in (see Supporting Information).

FIG. 3 :

 3 FIG. 3: Nanoantennas in the scalar approximation. Yagi-Uda antenna composed of silicon spheres (refractive index 3.436). The reector (radius 498 nm) is separated from the rst director by 700 nm. The four directors (radius 382 nm) are equally spaced by 250 nm. The point M (r 0 ) is the center between the reector and the rst director. (a) Maximum absorption cross-section σ max a

  (b) shows the absorption cross-section at normal incidence in TM polarization obtained by varying the absorber permittivity around the optimum. Since the cross-section varies smoothly around the maximum, we have a good exibility on the choice of the permittivity.Let us now consider a L-shape antenna made of three silicon spheres, see Figs. 4(a)-(b).

2 FIG. 4 :

 24 FIG. 4: Beyond scalar approximation. L-shape antenna made of three silicon spheres (radius 430 nm) equally spaced by 1.54 µm and embedded in a medium of index n = 1.5 at λ = 3 µm.

( a )

 a Intensity enhancement and (b) maximum absorption cross-section as a function of the position r 0 in the (x, y) plane for z = 232 nm. The map of the intensity enhancement is clearly dierent from the spatial distribution of the maximum absorption cross-section.

  Figures 4(a)-(b) show respectively the spatial distributions of the intensity enhancement and the maximum absorption cross-section in the (x, y) plane for z = 232 nm. The antenna is illuminated from the bottom by a plane wave propagating along the z axis and polarized linearly along the white arrow. The blue cross in Fig. 4(b) marks the position where the upper bound is maximum. It does not correspond to the maximum of |E b | 2 /|E inc | 2 , which evidences, once again, that seeking hot spots is not sucient to maximize the absorption.

WaveguideFIG. 5 :

 5 FIG. 5: Beyond scalar approximation and plane-wave illumination. SOI ridge waveguide at λ = 1.5 µm. The silicon ridge (n = 3.5, width 500 nm, thickness 250 nm) lies over a silica substrate (n = 1.44). Maximum absorption eciency as a function of the position along the dashed white line located 20 nm above the waveguide for the fundamental quasi-TE mode (solid black curve) and the quasi-TM mode (dashed red curve). The insets give the electric-eld intensity distributions of both modes.

  . For an emitter coupled to a resonant cavity, the emission enhancement has a well-known upper bound, the Purcell factor, which is intrinsic to the resonator and independent of the emitter

2729 . The

upper bound is reached if emitter and resonator fulll a few matching conditions spectral, spatial, and in polarization

Supporting Information Available

The following le is available free of charge: Supporting Information for Publication.pdf: we provide in this document some additional elements concerning the numerical calculations and the derivations of the upper bound and of the optimal polarizability presented in the main text.