Foteini Karampoula

Efstathios Giaouris
email: stagiaouris@aegean.gr

Julien Deschamps

Agapi I Doulgeraki

George-John E Nychas

Florence Dubois-Brissonnet
email: florence.dubois-brissonnet@agroparistech.fr

Hydrosol of Thymbra capitata is a highly efficient biocide against biofilms of Salmonella Typhimurium: real-time visualization of bacterial inactivation by CLSM

Keywords: Disinfection, hydrosol, natural antimicrobial agent, benzalkonium chloride, anti-biofilm, confocal laser scanning microscopy

Salmonella is recognized as one of the most significant enteric foodborne bacterial pathogens. In recent years, the resistance of pathogens to biocides and other environmental stresses, especially when these are embedded in biofilm structures, has led to the search and development of novel antimicrobial strategies capable of displaying both high efficiency and safety. To this direction, the aim of the present work was to evaluate the antimicrobial activity of hydrosol of the Mediterranean spice Thymbra capitata against both planktonic and biofilm cells of Salmonella Typhimurium, and to compare its action with the one of benzalkonium chloride (BC), a commonly used industrial biocide. In order to achieve this, the disinfectant activity following 6 min treatments was comparatively evaluated for both disinfectants by calculating their concentrations needed to achieve the same log-reductions against both types of cells. Their bactericidal effect against biofilm cells was also comparatively determined by in situ and real-time visualization of cell inactivation through the use of time-lapse confocal laser scanning microscopy (CLSM). Interestingly, results revealed that hydrosol was almost equally effective against both biofilms and planktonic cells, whereas 200 times higher concentration of BC was needed to achieve the same effect against biofilm compared to planktonic cells. Similarly, time-lapse CLSM revealed the significant advantage of the hydrosol to easily penetrate within the biofilm structure and quickly kill the cells, despite the 3D-structuration of Salmonella biofilm. food-contact surfaces. It is an aqueous solution which can easily be rinsed out from surfaces, it does not have the strong smell of the EO and it is a by-product of EOs distillation procedure without any industrial application. Consequently, hydrosol obviously could be of great value to combat biofilms and thus to improve product safety not only for the food industries, but probably also for many other industries which experience problems related to biofilms.

Importance

The results of this paper highlight the significant antimicrobial action of a natural compound that is hydrosol of T. capitata, against both planktonic and biofilm cells of a common foodborne pathogen. Hydrosol has numerous advantages as disinfectant of

Introduction

Salmonella is an important foodborne pathogenic bacterium that remains the most frequent causative agent of food-borne outbreaks occurring in the European Union (EU). In 2013, this accounted for 22.5 % of the total outbreaks reported [START_REF] Efsa | Trends and sources of zoonoses, zoonotic agents and food-borne outbreaks in 2013[END_REF]. That year, a total of 82,694 confirmed cases of salmonellosis were reported by 27 EU member states, with an average notification rate of 20.4 cases per 100,000 people. The serovars Enteritidis and Typhimurium were the most commonly involved representing respectively 39.5 % and 20.2 %, of all reported serovars in confirmed human cases. It is also noted that while the most important sources of food-borne Salmonella outbreaks are eggs and egg products, Salmonella was also detected in many other types of foods as well [START_REF] Efsa | Trends and sources of zoonoses, zoonotic agents and food-borne outbreaks in 2013[END_REF].

Although the native habitat of Salmonella is considered to be the intestinal tract of diverse vertebrates, this genus has been shown to be also able to survive for extended periods of time in non-enteric habitats. It is strongly believed that its survival and persistence in these non-host environments may be attributed to its great ability to form biofilms on various inanimate surfaces [START_REF] Dubois-Brissonnet | Adaptation of Salmonella to antimicrobials in food-processing environments[END_REF][START_REF] Giaouris | Attachment and biofilm formation by Salmonella in food processing environments[END_REF]. Biofilms are consortia of microorganisms adherent to each other and/or to a surface and embedded in a scaffold of self-produced extracellular polymeric substances (EPS) [START_REF] Costerton | Microbial biofilms[END_REF][START_REF] Giaouris | Infra-and inter-species interactions within biofilms of important foodborne bacterial pathogens[END_REF]. Thus, there is extended documentation on the ability of Salmonella to attach to various food-contact surfaces (such as stainless steel, plastic and cement) and form biofilms under in vitro conditions [START_REF] Chia | Attachment of different Salmonella serovars to materials commonly used in a poultry processing plant[END_REF][START_REF] Giaouris | Effect of temperature, pH, and water activity on biofilm formation by Salmonella enterica Enteritidis PT4 on stainless steel surfaces as indicated by the bead vortexing method and conductance measurements[END_REF][START_REF] Giaouris | The adherence of Salmonella Enteritidis PT4 to stainless steel: The importance of the air-liquid interface and nutrient availability[END_REF][START_REF] Marin | Biofilm development capacity of Salmonella strains isolated in poultry risk factors and their resistance against disinfectants[END_REF][START_REF] Oliveira | Comparison of the adhesion ability of different Salmonella enteritidis serotypes to materials used in kitchens[END_REF], while Salmonella cells have also been recovered from surfaces and products of various food industries [START_REF] Er | Antimicrobial and antibiofilm effects of selected food preservatives against Salmonella spp. isolated from chicken samples[END_REF][START_REF] Finn | Phenotypic characterization of Salmonella isolated from food production environments associated with low-water activity foods[END_REF][START_REF] Jakociune | Demonstration of persistent contamination of a cooked egg product production facility with Salmonella enterica serovar Tennessee and characterization of the persistent strain[END_REF][START_REF] Piras | Influence of temperature, source, and serotype on biofilm formation of Salmonella enterica isolates from pig slaughterhouses[END_REF]. During the last decades, it has become increasingly clear that biofilms are the predominant mode of bacterial life in most environments [START_REF] Bridier | Resistance of bacterial biofilms to disinfectants: a review[END_REF][START_REF] Giaouris | Attachment and biofilm formation by foodborne bacteria in meat processing environments: Causes, implications, role of bacterial interactions and control by alternative novel methods[END_REF][START_REF] Lindsay | What food safety professionals should know about bacterial biofilms[END_REF].

Biofilms formed in food processing environments are of special importance since they may act as a persistent source of product contamination which may lead to food spoilage and / or transmission of diseases [START_REF] Dubois-Brissonnet | Adaptation of Salmonella to antimicrobials in food-processing environments[END_REF][START_REF] Giaouris | Attachment and biofilm formation by Salmonella in food processing environments[END_REF][START_REF] Stocki | Persistence of Salmonella on egg conveyor belts is dependent on the belt type but not on the rdar morphotype[END_REF][START_REF] Srey | Biofilm formation in food industries: a food safety concern[END_REF].

Poor sanitation of food-contact surfaces is believed to be an essential contributing factor in foodborne disease outbreaks. Disinfection procedures using either physical or chemical methods have been extensively used over the years to reduce or eliminate bacteria found on food-contact surfaces. However, current methods of sanitation of food facilities are not always sufficiently efficient because of the great resistance displayed by many foodborne bacteria, especially when these are grown into biofilms [START_REF] Bridier | Resistance of bacterial biofilms to disinfectants: a review[END_REF][START_REF] Langsrud | Cross-resistance to antibiotics of Escherichia coli adapted to benzalkonium chloride or exposed to stressinducers[END_REF][START_REF] Moen | Subminimal inhibitory concentrations of the disinfectant benzalkonium chloride select for a tolerant subpopulation of Escherichia coli with inheritable characteristics[END_REF]. Several mechanisms have been proposed to be involved in biofilm resistance to disinfectants, including: (i) limitations to the free diffusion of antimicrobial agents through the biofilm matrix, (ii) variability in the physical and chemical microenvironments within the biofilm (e.g., varied conditions of pH, osmotic strength, or nutrients) leading to varied levels of metabolic activity and cell adaptive responses, (iii) mutations and horizontal transfer of genes coding for resistance mechanisms (e.g. detoxifying membrane transporters), (iv) differentiation of bacterial cells into physiological states less susceptible to treatments (e.g. dormant, viable but not culturable, VBNC), together with the presence of extremely resistant "persister" cells and (v) bacterial protection among multi-species consortium [START_REF] Bridier | Resistance of bacterial biofilms to disinfectants: a review[END_REF][START_REF] Giaouris | Attachment of Salmonella spp. to food contact and product surfaces and biofilm formation on thm as a stress adaptation and survival strategies[END_REF]. This persistence of pathogens in food environments despite continuous disinfection led to the search and development of novel antimicrobial strategies capable of displaying both high efficiency and safety. The growing negative consumer perception against artificial synthetic chemicals has shifted this research effort toward the development and application of environmental friendly disinfectants. The latter should display high lethal activity against biofilms in small concentrations, be safe and easily degraded in the environment [START_REF] Bridier | Resistance of bacterial biofilms to disinfectants: a review[END_REF][START_REF] Giaouris | Attachment and biofilm formation by foodborne bacteria in meat processing environments: Causes, implications, role of bacterial interactions and control by alternative novel methods[END_REF][START_REF] Simoes | A review of current and emergent biofilm control strategies[END_REF].

Plant extracts can provide unlimited opportunities for microbial control, owing to their great chemical diversity [START_REF] Borges | Insights on antimicrobial resistance, biofilms and the use of phytochemicals as new antimicrobial agents[END_REF][START_REF] Burt | Essential oils: their antibacterial properties and potential applications in foods -a review[END_REF][START_REF] Negi | Plant extracts for the control of bacterial growth: efficacy, stability and safety issues for food application[END_REF]. In recent years, several reports demonstrating the antibacterial effect of crude essential oils (EOs) and / or their active components against bacteria embedded in biofilms have been published [START_REF] Knowles | Antimicrobial action of carvacrol at different stages of dual-species biofilm development by Staphylococcus aureus and Salmonella enterica serovar typhimurium[END_REF][START_REF] Lambert | A study of the minimum inhibitory concentration and mode of action of oregano essential oil, thymol and carvacrol[END_REF][START_REF] Neyret | Plant-derived compounds as natural antimicrobials to control paper mill biofilms[END_REF][START_REF] Perez-Conesa | Inactivation of Listeria monocytogenes and Escherichia coli O157:H7 biofilms by micelle-encapsulated eugenol and carvacrol[END_REF][START_REF] Sandasi | The effect of five common essential oil components on Listeria monocytogenes biofilms[END_REF][START_REF] Mith | Antimicrobial activities of commercial essential oils and their components against food-borne pathogens and food spoilage bacteria[END_REF][START_REF] Faleiro | Antibacterial and antioxidant activities of essential oils isolated from Thymbra capitata L. (Cav.) and Origanum vulgare L[END_REF]. However, there are still strict limitations on the practical application of these compounds for the disinfection of industrial surfaces, arising from their strong hydrophobic nature, which hampers their efficient rinsing out from surfaces after a disinfection program, and their intense smell. Hydrosols, which are by-products of the EOs distillation procedure (these are situated just under the organic phase at the end of steam distillation), are aqueous solutions and may thus be easily rinsed out from surfaces. In addition, they do not present the strong smell of the EOs and they do not have any industrial application until now [START_REF] Garneau | Chemical composition of the hydrosol and the essential oil of three different species of the Pinaceae Family : Picea glauca (Moench) Voss., Picea mariana (Mill.) BSP, and Abies balsamea (L.) Mill[END_REF]. Interestingly, the antimicrobial action of the hydrosol fraction of Satureja thymbra EO has been demonstrated against single-and mixed-species biofilms composed of technological, spoilage and pathogenic bacteria [START_REF] Chorianopoulos | Disinfectant test against monoculture and mixed-culture biofilms composed of technological, spoilage and pathogenic bacteria: bactericidal effect of essential oil and hydrosol of Satureja thymbra and comparison with standard acid-base sanitizers[END_REF], but to the best of our knowledge, this is the only report available on the anti-biofilm action of hydrosols. In addition, antibacterial activity of hydrosols of several aromatic plants against planktonic pathogens has been shown [START_REF] Chorianopoulos | Characterization of the essential oil volatiles of Satureja thymbra and Satureja parnassica: Influence of harvesting time and antimicrobial activity[END_REF][START_REF] Sagdic | Sensitivity of four pathogenic bacteria to Turkish thyme and oregano hydrosols[END_REF][START_REF] Sagdic | Antibacterial activity of Turkish spice hydrosols[END_REF][START_REF] Tornuk | Efficacy of various plant hydrosols as natural food sanitizers in reducing Escherichia coli O157:H7 and Salmonella Typhimurium on fresh cut carrots and apples[END_REF][START_REF] Vermaa | Chemical composition and antimicrobial potential of aqueous distillate volatiles of Indian peppermint (Mentha piperita) and spearmint (Mentha spicata)[END_REF]. Using such compounds for the disinfection of surfaces in food industrial environments could thus be an interesting way to valorize them [START_REF] Giaouris | Attachment and biofilm formation by foodborne bacteria in meat processing environments: Causes, implications, role of bacterial interactions and control by alternative novel methods[END_REF].

For the better characterization and understanding of biofilm resistance mechanisms, development of innovative microscopy techniques, such as confocal laser scanning microscopy (CLSM), together with improvements in fluorescent labeling have emerged. Direct investigation of biocide reactivity within the native structure of biofilms can now provide in situ important information on the dynamics of biocide action and spatial heterogeneities of bacterial susceptibility within these structurally and physiologically heterogeneous sessile consortia [START_REF] Bridier | Methods for biofilms constituents and turnover: destructive and nondestructive methods[END_REF][START_REF] Bridier | Dynamics of biocide action in Pseudomonas aeruginosa biofilms[END_REF][START_REF] Davison | Spatial and temporal patterns of biocide action against Staphylococcus epidermidis biofilms[END_REF][START_REF] Takenaka | Direct visualization of spatial and temporal patterns of antimicrobial action within model oral biofilms[END_REF].

Taking into account all the previous, the aim of the present work was to evaluate the antimicrobial activity of hydrosol of the Mediterranean spice Thymbra capitata against both planktonic and biofilm cells of Salmonella Typhimurium and to compare its action with the one of benzalkonium chloride (BC), a commonly used industrial biocide. In order to achieve this, the antimicrobial activities of both disinfectants were first comparatively evaluated against 24 h planktonic and biofilm cells (on stainless steel coupons) by plate counting. Concentrations needed to achieve after 6-min treatments the same log-reductions of both type of cells were thus quantitatively determined. Afterwards, the bactericidal effect of the two tested disinfectants, when these were used at some selected concentrations (based on the previously obtained results), was comparatively evaluated by monitoring in real-time and in-situ inactivation of biofilm cells on polystyrene microplates through time-lapse confocal laser scanning microscopy (CLSM) analysis.

Materials and methods

Bacterial strain, growth conditions, and preparation of bacterial suspension

The bacterium used in this study was Salmonella enterica subsp. enterica serovar Typhimurim strain CDC 6516-60 (ATCC 14028) isolated from animal tissue of 4-week-old chickens [START_REF] Jarvik | Short-term signatures of evolutionary change in the Salmonella enterica serovar Typhimurium 14028 genome[END_REF]. Before each experiment, the microorganism, stored in cryovials at -80 o C, was resuscitated in two successive subcultures (7 h and 16 h) in Tryptone Soy Broth (TSB, Biomérieux, France) at 37°C. The working culture was a third subculture, incubated at 20°C for 24h and subsequently harvested by centrifugation (5000 g, 10 min, at 20 °C) for disinfectant testing. For the biofilm formation, the pellets were re-suspended and diluted in 10 ml of 150 mM NaCl in order to obtain a bacterial suspension adjusted at 10 6 cells/mL.

Antibacterial agents

The hydrosol of Thymbra capitata (pharmacy Provata, Athens, Greece) and the quaternary ammonium compound, benzalkonium chloride C14 (BC: MW, 368.04; puriss., anhydrous, 99.0%, Fluka, France) were used to carry out disinfectant tests.

Different concentrations of hydrosol (25-75% for planktonic cells; 50-100% for biofilms) and BC (0.0006-0.002 % for planktonic cells; 0.125-0.5 % for biofilms) were prepared by diluting appropriate quantities of each antimicrobial compound in sterilized deionized water the day of the experiment.

Chemical analysis of Thymbra capitata hydrosol and its related essential oil

T. capitata hydrosol was chemically analyzed by Solid Phase Micro-Extraction followed by Gas Chromatography -Mass Spectrometry (SPME GC/MS) [START_REF] Kopsahelis | Molecular characterization and molasses fermentation performance of a wild yeast strain operating in an extremely wide temperature range[END_REF]. Hydrosol samples (10 mL) together with 3 g NaCl were placed into a 20 mL headspace vial fitted with a Teflon-lined septum sealed with an aluminum crimp seal, through which the SPME fiber 50/30 mm divinylbenzene/carbozen in poly-dimethyl-siloxane 2 cm (Supelco, Bellefonte, PA, USA) was introduced. 4-methyl-2-pentanol was used as an internal standard at a final concentration 0.8 mg/L. The mix was equilibrated at 60 °C for 45 min. After head-space extraction, the fiber was transferred to the GC injection port where the absorbed compounds were thermally desorbed for 3 min at 240 °C.

The essential oil was diluted in pure hexane (1:9), 3 g NaCl were added to 10 mL of diluted oil and the mixture was then chemically analyzed by either SPME GC/MS analysis as described above or by GC/MS analysis following filtration using 0.22 μm filters and direct injection (1 μL). Gas chromatography-mass spectrometry (GC/MS) analysis was carried out with a 6890N system (Agilent Technologies, USA) equipped with an HP-5MS column (30 m, 0.25 mm inner diameter, film thickness 0.25 µm, Agilent technologies) and coupled with an Agilent Technologies 5973Networked mass detector. Column temperature was set at 35 °C for 6 min, increased at 60 °C at the rate of 2 °C/min, kept constant for 5 min, increased again to 200°C at the rate of 5 °C/min and then to 250°C at the rate of 25 °C/min and finished constant for 6 min. Helium was used as the carrier gas (linear velocity of 1.8 or 1.5 mL/min for SPME or direct analysis, respectively). The injector was operated in splitless mode for SPME analysis and in split mode (1:50 split ratio) for direct injection analysis. Both injector and detector temperatures were 240 °C. The mass spectrometer was operated in the electron impact mode with the electron energy set at 70 eV and 45-400m/z scan range. Results acquired were processed by ChemStation integrated software (Agilent Technologies) and constituents were identified by comparing mass spectra with reference spectra from NB575, stack and Wiley275 libraries, by spectra from standard compounds (in house libraries) and by determining kovats' retention indexes and comparing them with those reported in the literature. In SPME GC/MS analysis, the volatile compounds were semiquantified by dividing the peak areas of the compounds of interest by the peak area of the IS and multiplying this ratio by the initial concentration of the IS (expressed as mg/L), while in direct injection GC/MS analysis the % area of volatile compounds was estimated. The peak areas were measured from the full scan chromatograph using total ion current (TIC). Each experiment was carried out in duplicate and the mean data are presented.

Biofilm formation

Biofilms of S. Typhimurium were grown on 1-cm² stainless steel (SS) AISI 204 coupons (Goodfellow, Cambridge Science Park, UK) and also on the wells of 96-well polystyrene microtiter plates (Greiner Bio-One 655090, France) with a Clear base (polystyrene; 190±5 µm thick) [START_REF] Bridier | The biofilm architecture of sixty opportunistic pathogens deciphered using a high throughput CLSM method[END_REF]. Before use, the SS coupons were cleaned with surfactant RBS 35 (Société des traitements chimiques de surface, Lambersart, France), rinsed with deionized water and settled in the wells of a 24-well polystyrene microtiter plate (Techno Plastic products, Switzerland) [START_REF] Neyret | Plant-derived compounds as natural antimicrobials to control paper mill biofilms[END_REF][START_REF] Meylheuc | Adsorption of biosurfactant on solid surfaces and consequences regarding the bioadhesion of Listeria monocytogenes LO28[END_REF]. 1 mL or 250 µL of bacterial subculture, as prepared above (~ 10 6 cells/mL), were respectively poured into the wells of the 24-well (containing SS coupons) and 96-well microplates. Adhesion on SS coupons or on polystyrene wells was done by sedimentation for 2 h at 20 °C.

Subsequently, the planktonic bacterial suspension was removed and 1 mL (or 250 µL) of TSB was added in each well. Microtiter plates were incubated at 20 °C for 24 h without shaking to allow biofilm development (on either SS coupons or the polystyrene wells).

Antimicrobial testing against planktonic cells

Planktonic cells were challenged with the disinfectants using the EN 1040 standard protocol [START_REF]Chemical disinfectants and antiseptics -Basic bactericidal activity -Test method and requirements (phase 1)[END_REF]. Briefly, 1 ml of a 100-fold diluted solution of the working culture (prepared as previously described) was centrifuged at 5000 g for 10 min.

Subsequently, the pellet (~10 7 cells) was resuspended in 1 mL of each antimicrobial solution (at different concentrations) and left in contact for 6 min at 20°C. The antimicrobial action was halted by transfering a volume (1:9) to a quenching solution (3 g/L L-α-phosphatidyl cholin, 30 g/L Tween 80, 5 g/L sodium thiosulfate, 1 g/L Lhistidine, 30 g/L saponine) for 10 min. Serial dilutions were then prepared and survivors were enumerated on Tryptic Soy Agar (TSA) using the 6 × 6 drop count method [START_REF] Chen | A 6 x 6 drop plate method for simultaneous colony counting and MPN enumeration of Campylobacter jejuni, Listeria monocytogenes, and Escherichia coli[END_REF].

The control was performed in the same way with sterile deionized water instead of the disinfectant. The logarithm reduction achieved was the difference between the log10 of the survivors after the test with deionized water (control) and the log10 of the survivors after the test with the antimicrobial agent. For each condition, three to seven replicates were performed.

Antimicrobial testing against biofilms formed on stainless steel coupons

Following biofilm formation on SS coupons, the planktonic suspension was removed from each well and each coupon was once rinsed with 1 ml of 150 mM NaCl.

Afterwards, each coupon was immediately challenged with 1 mL of each disinfectant solution (at different concentrations) for 6 min at 20°C (the same time as for planktonic cells). Antimicrobial action was halted by placing coupons for 10 min at 20°C in the quenching solution, as previously described. The survivors were removed from the surface by scratching with a plastic rake (folded pipette cone) in a standardized way (horizontal/vertical/oblique) and enumerated by plate counting as previously described.

Control was performed in the same way with sterile deionized water instead of the disinfectant and log reductions were calculated as previously described. For each condition, three to height replicates were performed.

Calculation of resistance coefficients of the two biocides

The resistance coefficient (Rc) was determined for each biocide. Rc is equal to Cbiofilm/Cplanktonic, where Cbiofilm corresponds to the biocide concentration required to kill a given level of biofilm cells and Cplanktonic to the concentration needed to kill the same level of planktonic cells (15).

Antimicrobial testing against biofilms formed on polystyrene evaluated by time-lapse microscopy

This method allows the direct investigation of biocide reactivity within the native structure of biofilms [START_REF] Bridier | Dynamics of biocide action in Pseudomonas aeruginosa biofilms[END_REF]. To achieve this, cells were initially labelled with a viability fluorescent marker and subsequently submitted to disinfection. The antimicrobial action induces cell membrane's permeabilization and subsequently loss of fluorescence that represents cell's death.

Biofilms formed on polystyrene wells were initially stained with the esterase viability marker Chemchrome V6 (AES Chemunex, Ivry-sur-Seine, France) which can penetrate passively into the cell where it is cleaved by cytoplasmic esterases, leading to the intracellular release of fluorescent residues (green fluorescence). After the 24 h biofilm growth period, 100 µL of the medium were gently removed from each well and replaced with 100 μl of V6 solution (diluted 1:100 in B16 buffer) [START_REF] Bridier | Dynamics of biocide action in Pseudomonas aeruginosa biofilms[END_REF]. Microplate was incubated in the dark for 20 min at 37 °C in order to reach fluorescence equilibrium.

Afterwards, the whole liquid part above the biofilm was gently removed from each well and biofilms were once rinsed with 100 μl of 150 mM NaCl to eliminate any excess of fluorescent dye. All the wells were refilled with 100 μl of 150 mM NaCl, except those wells where the action of 100% hydrosol would be checked (no dilution).

Image analysis of fluorescence loss and estimation of inactivation kinetic parameters

The GinaFiT, a freeware add-in for Microsoft Excel was used to model inactivation kinetics [START_REF] Geeraerd | GInaFiT, a freeware tool to assess non-log-linear microbial survivor curves[END_REF]. This tool enables testing of nine different types of microbial survival models, and the choice of the best fit depends on five statistical measures (i.e., sum of squared errors, mean sum of squared errors and its root, R 2 , and adjusted R 2 ). During the present study, the "shoulder log-linear tail", "log-linear tail", or "log-linear" inactivation models were fitted to the fluorescence intensity curves obtained from the CLSM image series during biocide treatment. Two inactivation kinetic parameters were then extracted from this fitting: Sl, the shoulder length (min) that corresponded to the length of the lag phase (time period where fluorescence remains unreduced), and kmax, the inactivation rate (min -1 ).

Statistical analysis

All statistical analyses (oneway ANOVA, linear regression) were performed using JMP v8.0 software (SAS, Cary, USA). Significance was defined as a P value associated with a Fisher test value lower than 0.05.

Results

Chemical analysis of T. capitata hydrosol and essential oil

The chemical composition of T. capitata hydrosol was analyzed in order to identify the various antimicrobial compounds that it might contain. As hydrosol is a byproduct of EO distillation procedure, it was interesting to also comparatively analyze T. capitata EO. Chemical composition of T. capitata hydrosol and EO are presented in Table 1.

This analysis revealed that T. capitata hydrosol is a complex mixture containing 24 constituents. Its major compound is carvacrol (946,3 mg/L) followed by 1-octen-3ol (10,8 mg/L), terpinen-4-ol (9,3 mg/L), borneol (6,6 mg/l), α-terpineol (2,7 mg/l), pcymene (1,2 mg/l), γ-terpinene (1,2 mg/l), eugenol (0,9 mg/l), linalool (0,8 mg/l) and thymol (0,8 mg/l). On the other hand, the main constituents of T. capitata EO are carvacrol (28.3%), γterpinene (31.2%) and p-cymene (20%). Other components such as α-terpinene, α-thujene and α-pinene, E-caryophyllene, thymol were also found in this analysis. Interestingly, among the 24 hydrosol compounds, 14 were also detected in the EO, while ten oxygenated compounds were detected in hydrosol but not in the EO.

Antimicrobial activity of hydrosol and BC against planktonic and biofilm cells on stainless steel coupons

Disinfectant efficacies of T. capitata hydrosol and BC were tested against 24 h S. Typhimurium planktonic and biofilm cells on SS coupons. Cell density in biofilms after 24 h of development reached 7.47 ± 0.46 log cfu/cm 2 . To be able to make comparisons, the cell density of planktonic suspension was also adjusted to a similar level (7.96 ± 0.23 log cfu/ml).

Log reductions of planktonic (log cfu/ml) and biofilm cells (log cfu/cm 2 ) achieved after 6 min exposure to different concentrations of BC (0.0006 -0.5%) and T.

capitata hydrosol (37.5 -100%) are presented in Table 2.

First of all, biofilm cells have been found to demonstrate significant greater resistance to both biocides compared to the planktonic cells. Thus, higher concentrations of each biocide were required to kill the same number of biofilm cells compared to planktonic ones. More particularly, a 6 log reduction was achieved after treatment of planktonic cells with 0.0012 % BC, whereas 0.25 % BC was required to achieve the same log reduction of biofilm cells. An approximate 5 log reduction of planktonic and biofilm cells was obtained by applying respectively 42% and 75 % hydrosol solutions.

Rc for two different and representative log-reductions and mean Rc are presented in Table 2 for the two biocides. Based on these results, BC has a significantly higher Rc (208.3) than hydrosol (1.6). This actually means that BC needs to be 208 fold more concentrated to exert the same effect against biofilm cells than against planktonic ones, whereas hydrosol is almost equally efficient against both types of cells.

Real time visualization of the biocide action against S. Typhimurium biofilm cells on polystyrene by confocal laser scanning microscopy

Antimicrobial efficiency of BC and T. capitata hydrosol against 24 h S. Typhimurium biofilms was evaluated using real-time visualization of fluorescence loss by CLSM. During control experiments (treatment with distilled water), a fluorescence loss less than 10% ± 3% of initial fluorescence was observed, after 25 min of treatment.

Before adding the disinfectant, the biovolume and thickness of Salmonella biofilm were respectively 575155±212356 µm 3 and 55.4±10.4 µm. Three illustrative experiments for treatments with BC 0.5%, hydrosol 100% and hydrosol 75% are presented in Figures 123. All sections into biofilms treated with the same biocide were fitted by using the same inactivation model (BC 0.5% "shoulder log-linear tail", hydrosol 100% 'log linear tail', hydrosol 75% 'log linear') demonstrating that each biocide acts with the same type of kinetics into the biofilm structure. A few images corresponding to the maximum projection of the whole biofilm as a function of time are represented under each Figure . From the Figures 123, it can firstly be observed that fluorescence loss is much quicker following treatment of biofilms with hydrosol 100%

(and even 75 %) compared to BC 0.5%.

Inactivation parameters obtained following all repetitions conducted in this study (including the illustrative experiments described above) are shown in Table 3.

Quantitative comparison of these inactivation parameters revealed some very interesting findings for the two biocides. SI for BC (10.7 ± 7, R 2 =0.98 ±0.08) was significantly different and markedly higher than the close to zero SI for hydrosol 100% (0.2 ± 0.4 R 2 =0.99 ±0.04). Not only SI value was high, but there was also high standard deviation. In order to explain this big variability, SI values were correlated with biofilm biovolume, and it was found a significant important linear correlation (Prob>F 0.0432* p=0.05) but Rsquare of the model was extremely low (R 2 =0.26). No significant difference was recorded in SI at the different sections inside the biofilm during the application of BC 0.5 %. Nevertheless, in the 1 st section, mean SI (11.34 ± 1.7) was a little higher than 2 nd (10.93 ± 1.7) which was a little higher than 3 rd (10.58 ± 1.7).

With regard to the inactivation rates, kmax for BC (0.51 ± 0.40, R 2 =0.98 ±0.08) was statistically significant lower than kmax presented for 100% hydrosol (0.82 ± 0.36, R 2 =0.99 ±0.04). Consequently, BC 0.5 % has lower bactericidal effect on S. Typhimurium biofilms than hydrosol 100%, which was very efficient from the first seconds of its application leading to total biofilm eradication within 2-3 minutes. No significant difference was observed in kmax values at different sections with the biofilms whatever the disinfectant.

Comparing the two concentrations of hydrosol, hydrosol 75% resulted in a much lower kmax (0.24 ± 0.17) than hydrosol 100% and a higher SI (3.44 ± 2.79). For hydrosol 75%, no significant differences were observed in kmax and SI within the different sections inside the biofilm, just like for hydrosol 100%. It should be noted that hydrosol 50% was also tested in this study but it didn't exhibit sufficient bactericidal activity presenting an almost zero inactivation rate (0.03 ± 0.02) (data not shown).

Discussion

Thymbra capitata is a species of aromatic plants native to the Mediterranean region of southern Europe, North Africa, and the Middle East. It belongs to the family of Lamiaceae, which includes many widely used culinary herbs (such as basil, mint, rosemary, sage, savory, marjoram, oregano, and thyme). The anti-biofilm activity of its EO has already been described [START_REF] Benbelaid | Antimicrobial activity of some essential oils against oral multidrugresistant Enterococcus faecalis in both planktonic and biofilm state[END_REF][START_REF] Palmeira-De-Oliveira | The anti-Candida activity of Thymbra capitata essential oil: effect upon pre-formed biofilm[END_REF], but not the one of its hydrosol. It was first important to characterize hydrosol composition by SPME GC-MS in comparison with the one of the original essential oil (EO). The main constituents of T. capitata EO were oxygenated phenolic monoterpene carvacrol, monocyclic monoterpene γ-terpinene and the alkylbenzene p-cymene, as found in previous studies [START_REF] Bakhy | Wild Thymbra capitata from Western Rif (Morocco): essential oil composition, chemical homogeneity and yield variability[END_REF][START_REF] Salas | Chemical composition and antioxidant activity of the essential oil of Thymbra capitata (L.) Cav. in Spain[END_REF][START_REF] Hortigon-Vinagre | Thymbra capitata essential oil prevents cell death induced by 4-hydroxy-2-nonenal in neonatal rat cardiac myocytes[END_REF]. However, carvacrol content in this study was much lower (20%) than previous reported concentrations (60-70%). In addition, other components detected here, such as monocyclic monoterpene α-terpinene, bicyclic monoterpenes, as α-thujene and α-pinene, acyclic monoterpenes myrcene or linalool and the sesquiterpene E-caryophyllene, were also described as important compounds [START_REF] Salas | Chemical composition and antioxidant activity of the essential oil of Thymbra capitata (L.) Cav. in Spain[END_REF][START_REF] Galego | Antioxidant activity of the essential oils of Thymbra capitata, Origanum vulgare, Thymus mastichina and Calamintha baetica[END_REF][START_REF] Salgueiro | Chemical composition and antifungal activity of the essential oil of Thymbra capitata[END_REF]. The chemical composition of hydrosols of aromatic herbs produced in the same extraction process of EOs is not commonly studied. Their chemical analysis is the subject of only a limited number of publications [START_REF] Garneau | Chemical composition of the hydrosol and the essential oil of three different species of the Pinaceae Family : Picea glauca (Moench) Voss., Picea mariana (Mill.) BSP, and Abies balsamea (L.) Mill[END_REF][START_REF] Garneau | Chemical composition and stability of the hydrosols obtained during essential oil production. I. The case of Melissa officinalis L. and Asarum canadense L[END_REF][START_REF] Inouye | A comparative study on the composition of forty four hydrosols and their essentiel oils[END_REF][START_REF] Lei | Chemical composition of essential oils and hydrosols from fresh flowers of Cerasus subhirtella and Cerasus serrulata from East China[END_REF][START_REF] Paolini | Comparison of liquid-liquid extraction with headspace methods for the characterization of volatile fractions of commercial hydrolats from typically Mediterranean species[END_REF].

However, to the best of our knowledge, the chemical composition of T. capitata hydrosol has not been reported until now. In this study, the chemical analysis revealed that this is a complex mixture containing part of the EO components. Thus, among 24 constituents that were detected, 14 were in common with the EO. However, some components, such as α-terpinene, myrcene, camphene and limonene, were missing in the hydrosol. Most of these missing compounds are completely insoluble in water, while some others probably disappear quickly from the acidic hydrosol with diverse degradation mechanisms [START_REF] Lei | Chemical composition of essential oils and hydrosols from fresh flowers of Cerasus subhirtella and Cerasus serrulata from East China[END_REF]. Some of the components that were found only in hydrosol and not in the EO could derive from chemical reactions occurring during hydro-distillation or extraction procedure before the GC analysis [START_REF] Garneau | Chemical composition and stability of the hydrosols obtained during essential oil production. I. The case of Melissa officinalis L. and Asarum canadense L[END_REF].

In this study, T. capita hydrosol was found to have a high bactericidal activity against planktonic cells, as 1.2±0.2, 2.8±0.5 and 4.7±0.7 log reductions were achieved by applying 37.5, 40 and 42 % (v/v) respectively. Information on the evaluation of different hydrosols as disinfectants is available [START_REF] Chorianopoulos | Characterization of the essential oil volatiles of Satureja thymbra and Satureja parnassica: Influence of harvesting time and antimicrobial activity[END_REF][START_REF] Sagdic | Sensitivity of four pathogenic bacteria to Turkish thyme and oregano hydrosols[END_REF][START_REF] Sagdic | Antibacterial activity of Turkish spice hydrosols[END_REF][START_REF] Tornuk | Efficacy of various plant hydrosols as natural food sanitizers in reducing Escherichia coli O157:H7 and Salmonella Typhimurium on fresh cut carrots and apples[END_REF][START_REF] Vermaa | Chemical composition and antimicrobial potential of aqueous distillate volatiles of Indian peppermint (Mentha piperita) and spearmint (Mentha spicata)[END_REF]. However, to the best of our knowledge, the antimicrobial action of T. capita hydrosol fraction has not been recorded yet.

Based on our results, T. capitata hydrosol presented a significant anti-biofilm action (complete eradication with 100% (v/v), approximate 5 log reduction with 75%) (Table 2). Antibacterial action of Satureja thymbra hydrosol against 5-days mono-and multi-species biofilms of Pseudomonas putida, Salmonella enterica and Listeria monocytogenes has been previously shown [START_REF] Chorianopoulos | Disinfectant test against monoculture and mixed-culture biofilms composed of technological, spoilage and pathogenic bacteria: bactericidal effect of essential oil and hydrosol of Satureja thymbra and comparison with standard acid-base sanitizers[END_REF], but it is the only one reporting anti-biofilm action of hydrosols. T. capitata and S. thymbra are similar species which belong to the same family of Lamiaceae (also called Labiatae). Although it is difficult and often misleading to compare results obtained in different studies, lower exposure time was actually required in the current study for the complete eradication of biofilm (6 min instead of 60 min) and a lower dose (75% instead of 100%) was also sufficient [START_REF] Chorianopoulos | Disinfectant test against monoculture and mixed-culture biofilms composed of technological, spoilage and pathogenic bacteria: bactericidal effect of essential oil and hydrosol of Satureja thymbra and comparison with standard acid-base sanitizers[END_REF].

The Rc observed for hydrosol was very low (1.6) showing that this is almost equally active against both planktonic and biofilm cells (Table 2). In the literature, the Rc values of various biocides could range from 1 to 1000 depending on the considered species and antimicrobial agent [START_REF] Bridier | Resistance of bacterial biofilms to disinfectants: a review[END_REF]. Among all biocides reported, only hydrogen peroxide and phenol exhibited the minimum Rc=1. Biocides with the second lowest Rc (=4) were of plant origin (oregano, carvacrol, thymol and eucalyptus oil) [START_REF] Bridier | Resistance of bacterial biofilms to disinfectants: a review[END_REF]. It is noteworthy that hydrosol had a lower Rc than the ones of its main components carvacrol or thymol alone (however tested against biofilms of different bacterial species), revealing a very promising anti-biofilm agent.

BC (Rc 208.3) was shown much less efficient compared to hydrosol against Salmonella biofilms. This thus needed to be more than 200 times more concentrated to exert the same effect on biofilm cells than on planktonic ones. In the literature, the Rc values for BC range from 10 to 1000, but in most cases these are above 50 [START_REF] Bridier | Resistance of bacterial biofilms to disinfectants: a review[END_REF]. Typical sanitizing concentrations of quaternary ammonium compounds, such as BC, range from 0.02 to 0.1% (v/v), but these are more generally applied at 0.02% (v/v) [START_REF] Gaulin | Disinfectants and sanitizers for use on food contact surfaces[END_REF]. According to some authors, the recommended user concentration of BC can be a little bit higher at 0.07% (v/v) [START_REF] Wong | Comparative susceptibility of planktonic and 3-day-old Salmonella Typhimurium biofilms to disinfectants[END_REF]. However, in the present study it was demonstrated that BC at 0.125% has an inadequate bactericidal effect against S. Typhimurium biofilm cells, resulting only in a 2 log reduction, whereas a sufficient 6 log reduction was only achieved with BC 0.25% (Table 2). Similarly, 0.02% BC failed to eradicate 48 h S. Typhimurium patterns of inactivation indicate the existence of transport limitations which is in accordance with previous references presuming that the restricted penetration of BC into biofilms might be one of the key processes explaining the resistance of biofilms to this biocide [START_REF] Bridier | Dynamics of biocide action in Pseudomonas aeruginosa biofilms[END_REF]. However, contrarily to P. aeruginosa ATCC 15442 biofilm [START_REF] Bridier | Dynamics of biocide action in Pseudomonas aeruginosa biofilms[END_REF], no significant difference in inactivation kinetics at different depths of the biofilm were obtained, even if there was approximately 40 sec more delay in the biocide action of BC 0.5% at each section situated deeper into the biofilm. This is probably due to differences in biofilm thickness and in composition of matrix for the two bacterial species.

On the contrary, hydrosol 100% caused a uniform and direct linear loss of fluorescence in biofilm cell clusters of S. Typhimurium, suggesting that the slightly greater resistance of the biofilm compared to planktonic cells observed in this study for this biocide could not be due to limitations affecting its penetration into the biofilm. In addition, hydrosol 100% displayed a relatively high inactivation rate resulted in a rapid and total cells permeabilization throughout the biofilm within a few minutes (Fig. 2, Table 3).

The composition of hydrosol can explain its tremendous efficiency. T. capitata hydrosol's main constituent is carvacrol with its concentration to be 943.6 mg/L (0.0943%). Several studies have demonstrated that carvacrol has both bacteriostatic and bactericidal activity against foodborne microorganisms including Salmonella enterica [START_REF] Knowles | Antimicrobial action of carvacrol at different stages of dual-species biofilm development by Staphylococcus aureus and Salmonella enterica serovar typhimurium[END_REF][START_REF] Nazer | Combinations of food antimicrobials at low levels to inhibit the growth of Salmonella sv.Typhimurium: a synergistic effect ?[END_REF][START_REF] Ravishankar | Carvacrol and cinnamaldehyde inactivate antibiotic-resistant Salmonella enterica in buffer and on celery and oysters[END_REF][START_REF] Zhou | Synergistic effect of thymol and carvacrol combined with chelators and organic acids against Salmonella Typhimurium[END_REF]. The effectiveness of carvacrol as a natural antimicrobial compound is well established and its mechanism of action is believed to be associated with structural and functional damage to cellular membranes [START_REF] Burt | Essential oils: their antibacterial properties and potential applications in foods -a review[END_REF][START_REF] Lambert | A study of the minimum inhibitory concentration and mode of action of oregano essential oil, thymol and carvacrol[END_REF][START_REF] Zhou | Synergistic effect of thymol and carvacrol combined with chelators and organic acids against Salmonella Typhimurium[END_REF][START_REF] White | Antibacterial efficacy of phosvitin, carvacrol, or nisin alone or combined against foodborne human enteric pathogens[END_REF][START_REF] Xu | The antibacterial mechanism of carvacrol and thymol against Escherichia coli[END_REF]. It has also been previously demonstrated as an anti-biofilm compound. Biofilm eradication concentration (BEC, 5 log reduction) of carvacrol against L. monocytogenes biofilms on SS coupons and microplates was obtained at 5 mM (0,0750 mg/L) [START_REF] Upadhyay | Antibiofilm effect of plant derived antimicrobials on Listeria monocytogenes[END_REF]. Similarly, it was reported that carvacrol BEC against S. aureus and S. epidermidis biofilms on microplates ranged from 0.125 to 0.5 % (v/v) [START_REF] Nostro | Effects of oregano, carvacrol and thymol on Staphylococcus aureus and Staphylococcus epidermidis biofilms[END_REF]. Low levels (2 and 3 log CFU/g) of S. Enteritidis and E. coli O157:H7 were inactivated when radish seeds were treated with 4000 ppm and 8000 ppm (0.4-0.5%) of carvacrol for 60 min [START_REF] Landry | Effectiveness of a spontaneous carvacrol nanoemulsion against Salmonella enterica Enteritidis and Escherichia coli O157:H7 on contaminated broccoli and radish seeds[END_REF]. The markedly effective antimicrobial and anti-biofilm action of the T. capitata hydrosol against S. Typhimurium could thus be partly attributed to its high carvacrol content. However, other constituents of this hydrosol, such as terpinen-4-ol, thymol and eugenol, have also previously shown antibacterial properties [START_REF] Burt | Essential oils: their antibacterial properties and potential applications in foods -a review[END_REF][START_REF] Nazer | Combinations of food antimicrobials at low levels to inhibit the growth of Salmonella sv.Typhimurium: a synergistic effect ?[END_REF][START_REF] Nostro | Effects of oregano, carvacrol and thymol on Staphylococcus aureus and Staphylococcus epidermidis biofilms[END_REF][START_REF] Devi | Eugenol (an essential oil of clove) acts as an antibacterial agent against Salmonella typhi by disrupting the cellular membrane[END_REF][START_REF] Mondello | In vivo activity of terpinen-4-ol, the main bioactive component of Melaleuca alternifolia Cheel (tea tree) oil against azole-susceptible and -resistant human pathogenic Candida species[END_REF]. Their concentrations in the hydrosol were much lower compared to carvacrol's concentration, but synergistic action can occur between all these compounds and can explain the overall high efficiency of the hydrosol. In previous studies, combinations of different compounds from EOs, in particular carvacrol, thymol and eugenol, showed synergistic activity [START_REF] Neyret | Plant-derived compounds as natural antimicrobials to control paper mill biofilms[END_REF][START_REF] Nazer | Combinations of food antimicrobials at low levels to inhibit the growth of Salmonella sv.Typhimurium: a synergistic effect ?[END_REF][START_REF] Garcia-Garcia | Bactericidal action of binary and ternary mixtures of carvacrol, thymol, and eugenol against Listeria innocua[END_REF][START_REF] Hyldgaard | Essential oils in food preservation: mode of action, synergies, and interactions with food matrix components[END_REF].

The high efficiency of hydrosol is also probably due to its high water solubility. We can thus assume that active molecules of the hydrosol can better diffuse in aqueous medium around planktonic bacteria or biofilm cells and their efficiency get improved compared to the EOs that need initial solubilization in an organic solvent (ethanol, DMSO) before introduction into an aqueous medium.

Conclusions

It is of great importance to take into consideration that bacteria can manage to survive in many harsh environments, like those encountered in food processing, mainly being enclosed into biofilms, sessile consortia which present higher resistance against many environmental stresses (e.g. application of antimicrobial agents) compared to their planktonic counterparts. Although the study of biofilms has come into sharp focus in recent years, more research is required to further understand the intricate mechanisms accounting for biofilm recalcitrance and to develop efficient methods to control them.

Both methods used in this study led to the same conclusion for both biocides. BC, which is a common industrial disinfectant, proved to have insufficient anti-biofilm activity at user's recommended concentration. On the contrary, T. capita hydrosol was clearly demonstrated as a highly efficient anti-biofilm agent. The findings of this study i.e. the efficiency to kill biofilm cells through a hydrosol fraction, an aqueous solution, which is easily rinsed out from surfaces, and does not have the strong smell of the EO, may be of great value not only for the food industries, but probably also for many other industries which experience problems related to biofilms. This obviously could be an interesting alternative and / or supplementary way to combat biofilms, being one innovative, natural and environmentally safe antimicrobial agent. Indeed as hydrosols are by-products of EOs distillation procedure, without any industrial application until now, it could be useful and smart to apply such natural compounds for the disinfection of surfaces in these environments. However, before its extensive use as disinfectant in the food industry and other environments facing hygiene problems, more safety studies based on toxicology data should be carried out. Rc= Cbiofilm/Cplanktonic, where Cbiofilm corresponds to the biocide concentration required to kill a given level of biofilm cells and Cplanktonic to the concentration needed to kill the same level of planktonic cells.

  intensity of green fluorescence was quantified by the LCS Lite confocal software (Leica microsystems). Intensity curves showing fluorescence loss were extracted separately for four different sections within the biofilms (from the attachment surface to the top of biofilm) (see results for distances), as well as for their maximum projection, which is the 2D projection of the 3D biofilm structure and represents the fluorescence in the whole biofilm. Intensity values were normalized by dividing the fluorescence intensity recorded at the different time points by the initial fluorescence intensity values obtained at the same location. Three-dimensional projections of biofilm structure were reconstructed using the Easy 3D function of the IMARIS 7.0 software (Bitplane, Switzerland). Quantitative structural parameters of the biofilms (i.e. biovolume, density and thickness), were calculated using ICY, an open community platform for bioimage informatics, created by Quantitative Image Analysis Unit at Pasteur Institute (http://www.bioimageanalysis.org/).

Figure 2 .

 2 Figure 2. Quantification of fluorescence intensity during T. capitata hydrosol 75%

Figure 3 .

 3 Figure 3. Quantification of fluorescence intensity during T. capitata hydrosol 100%

Table 1 .

 1 Constituents of T. capitata hydrosol and EO, as identified by SPME GC/MS,

	862
	863	and their (w/v) concentration or percentage composition
		Compounds	KΙ	Hydrosol	Essential oil
				mg/L	%area SPME
		Toluene	<800	0.2
		2-methyl-Butanoic acid, methyl ester	<800	0.3
		N-ethyl-1,3-dithioisoindoline	821	0.6
		(Z)-3-Hexen-1-ol	847	0.7
		α-Thujene	912	0.1	2.5
		.alpha.-Pinene	917	0.1	2.7
		Camphene	933		0.8
		Mesitylene	957	0.2
		.beta.-Pinene	964		0.7
		1-Octen-3-ol	980	10.8	0.2
		.beta.-Myrcene	990		2.0
		3-octanol	997	0.4
		.alpha.-Phellandrene	999		0.3
		3-Carene	1005		0.1
		α-Terpinene	1013		2.4
		p-Cymene	1020	1.3	2.3
		Eucalyptol	1026	0.1	nd
		Limonene	1028		0.8
		trans-β-Ocimene	1061		0.1
		γ-Terpinene	1063	1.2	35.3
		1-methyl-4-(1-methylethenyl)-Benzene	1105	0.1	0.1
		α-Terpinolene	1109		0.1
		Linalool	1123	0.9	0.8
		Borneol	1172	6.6	0.3
		Terpinen-4-ol	1181	9.3	0.3
		α-Terpineol	1224	2.7	<0.1
		1-Isopropyl-2-methoxy-4-methylbenze	1250		0.7
		D-Carvone	1257		<0.1
		Thymol	1299	0.8	1.1
		Carvacrol	1332	946.3	20.4
		p-Cymen-7-ol	1323		1.2
		4-Hydroxy-3-methylacetophenone	1329		0.2
		Eugénol	1368	1.2
		Phenol, 5-methyl-2-(1-methylethyl)-, acetate	1377		0.1
		Anethole	1381	0.2
		Caryophyllene	1421		0.4
		Ethanone	1428	0.1
		1-Acetyl-4-methyldibenzofuran	1451	0.2
		Phenol, 2,4-bis(1,1-dimethylethyl)	1517	0.1
		(-)-Spathulenol	1581		nd
		Caryophyllene oxide	1587		0.1
		(α-Caryophylladienol)	1640		nd
		2,3,4,6-Tetramethylphenol	2079		nd
		Hexanedioic acid, bis(2-ethylhexyl) ester	>2200		nd
	864
	865

Table 2 .

 2 Log reductions of planktonic (log cfu/ml) and biofilm cells (log cfu/cm 2 ) after 6 min disinfection treatments with BC and T. capitata hydrosol and resistance

	coefficients Rc
	Biocide	Log reduction	Concentration	Rc*	Mean Rc
				(%)
		Planktonic	Biofilm	Cplanktonic Cbiofilm
		(cfu/ml)	(cfu/cm 2 )
	BC	>7.5	>7.5	0.0022	0.5
		6.2 ±0.6	6.1 ±0.6	0.0012	0.25	208.3
		2.7 ±0.3	2.2 ±0.1	0.0006	0.125	208.3
						208.3
	Hydrosol >7.5	>7.5	45	100
		4.7 ±0.7	4.7 ±0.1	42	75	1.8
		1.2 ±0.2	1.2 ±0.2	37.5	50	1.3
						1.6

Table 3 :

 3 Inactivation parameters [kmax, inactivation rate (min -1 ); Sl, shoulder length (min)] determined for each biocide and its respective concentrations for four different sections within the biofilms, as well as for their maximum projection, which represents the fluorescence in the whole biofilm. Mean values are presented ±

	standard errors.
	Biocide		kmax	SI
	(n replicates)
	Hyd 100%	Max projection	0.82±0.09	0.2±0.13
	(n=18)	Section 1	0.82±0.09	0.56±0.13
		Section 2	0.85±0.09	0.28±0.13
		Section 3	0.84±0.09	0.26±0.13
		Section 4	0.84±0.09	0.27±0.13
	Hyd 75%	Max projection	0.24± 0.06	3.44±2.79
	(n=7)	Section 1	0,25± 0.06	7.04±2.79
		Section 2	0.25± 0.06	3.69±2.79
		Section 3	0.24± 0.06	3.59±2.79
		Section 4	0.21± 0.06	5.05±2.79
	BC 0.5%	Max projection	0.51±0.09	10.70±1.15
	(n=18)	Section 1	0.64±0.09	11.34± 1.67
		Section 2	0.57±0.09	10.93±1.67
		Section 3	0.48±0.09	10.5±1.71
		Section 4	0.37±0.09	11.53±1.89

Acknowledgements

Authors acknowledge the European Cooperation in Science and Technology (COST) Action FA1202 BacFoodNet: "A European network for mitigating bacterial colonization and persistence on foods and food processing environments" (http://www.bacfoodnet.org/) for facilitating collaboration and providing FK a short term scientific mission in UMR Micalis, Inra, AgroParisTech, France.

The authors also thank Dr. Yiannis Kourkoutas for carrying out the SPME GC/MS analysis at the Laboratory of Applied Microbiology and Biotechnology, Department of Molecular Biology and Genetics, Democritus University of Thrace, Greece.

biofilm on concrete (only 0.22 log-reduction) [START_REF] Corcoran | Commonly used disinfectants fail to eradicate Salmonella enterica biofilms from food contact surface materials[END_REF]. It was also reported that BC 0.75 % was required to eradicate 3 days biofilms of S. Typhimurium [START_REF] Wong | Comparative susceptibility of planktonic and 3-day-old Salmonella Typhimurium biofilms to disinfectants[END_REF]. All these findings point out that BC, a nevertheless very common industrial disinfectant, is not effective against S. Typhimurium biofilms when this is applied at the recommended or generally used concentrations. This constitutes a common phenomenon for many industrial sanitizers, since the standards for testing disinfectant efficiency, such as European EN 1040 method [START_REF]Chemical disinfectants and antiseptics -Basic bactericidal activity -Test method and requirements (phase 1)[END_REF], widely utilize planktonic cultures and results do not reflect the efficacy against bacteria in a biofilm state. However, it should be noted that the use of BC at high concentrations required for biofilm eradication may be not easily applied in food industry because of the risk of leaving residues in food. BC maximum residue level (MRL) for food and feed, laid down in Regulation (EC) No 396/2005, is 0.5 mg/kg (June 2014). Moreover, 0.1% is the maximum concentration of BC that does not produce primary irritation on intact skin or act as a sensitizer [START_REF] Block | Disinfection, sterilization and preservation[END_REF].

In the present study, spatial and temporal dynamics of the biocide actions were evaluated towards biofilms in real time and in situ by confocal laser scanning microscopy (CLSM). The spatial information obtained from CLSM analysis is of great importance because biofilms are known to present structural and physiological heterogeneity [START_REF] Joux | Use of fluorescent probes to assess physiological functions of bacteria at single-cell level[END_REF][START_REF] Rani | Spatial patterns of DNA replication, protein synthesis, and oxygen concentration within bacterial biofilms reveal diverse physiological states[END_REF][START_REF] Stewart | Diffusion in biofilms[END_REF]. The obtained temporal information can provide clues about the protective mechanisms of cells into the biofilm. For example, when a tolerant subpopulation is present, the shape of the inactivation curve should be concave up, and when a reaction-diffusion interaction limits the rate of access of the antimicrobial agent into the biofilm the shape of the inactivation curve should be concave down [START_REF] Cogan | Effects of persister formation on bacterial response to dosing[END_REF][START_REF] Dodds | Modeling biofilm antimicrobial resistance[END_REF].

We have examined here the spatial action of both biocides in S. Typhimurium biofilms (Fig. 123, Table 3). BC 0.5% exhibited high SI values which means that there is an important initial time period at which it is totally ineffective against biofilm cells. Such List of tables: List of figures:


domain: phys spi

text:
Hui Peng
email: davidpeng1749@gmail.com

Dividing Diffraction Pattern of Rotating Grating into Three Portions and Studying Each Portion Respectively ---Revisiting Rotating Grating Experiments

Keywords: grating, diffraction pattern, diffraction order, rotating grating, twisting grating, inclined incident light beam, orientation-dependence of diffraction pattern, angle-dependence of diffraction pattern

The phenomena of the rotating grating have been studied and have the varieties application. The diffraction normal order 𝑚 ! that is at the intersection of the grating's normal vector and the screen is introduced and is the function of the incident angle of light, the grating spacing and wavelength.

We suggest the following: (1) the diffraction normal order 𝑚 ! and the zeroth order (𝑚 = 0) divide a diffraction pattern into three portions; each portion corresponds to a range of diffraction orders, 𝑚 " , terms of 𝑚 and 𝑚 # ; (4) the diffraction pattern of a rotating grating is non-uniform.

𝑚 "" , 𝑚 """ , and is described by specific equation, respectively; (2) the number of the diffraction orders "m" in each portion is determined by the incident angle of light, the grating spacing, and wavelength;

(3) the diffraction angles and positions of the diffraction orders of each portion can be presented in

! ! = 3 ! ! = 2 ! !! = 1 ! = 0 ! !!! = -1

Introduction

The diffraction pattern of a grating is described by the grating equations that states that a grating of spacing "d", when a light beam incident at an angle to the grating, creates a diffraction pattern with diffraction orders, "m", at discrete angles 𝜃 $ . The diffraction angle, 𝜃 $ , is measured from the grating normal. The phenomena are referred as "conical pattern" [START_REF] Harvey | Description of Diffraction Grating Behavior in Direction Cosine Space[END_REF][START_REF] Braig | An EUV beam-splitter based on conical grazing incidence diffraction[END_REF] or as "tilted grating" [START_REF] Ninad | Novel cases of diffraction of light from a grating: Theory and experiment[END_REF] or as "oblique incident light beam" [START_REF] Heuberger | Light diffraction from a phase grating at oblique incidence in the intermediate diffraction regime[END_REF] or as "twisted grating" [START_REF]Interference and Diffraction[END_REF] or as "rotating grating" rotating CW and CCW around one, two and three axes sequentially [START_REF] Peng | Double Slit Experiment Still Has Much to Offer (6): Curved, Expanded and Inclined Interference Patterns -Dynamic Double Slit/Cross-Double Slit Experiments[END_REF][START_REF] Peng | Double Slit Experiment Still Has Much to Offer (7): Particle Nature of Photons, Normal Diffraction Pattern and Curved Diffraction Pattern Emerging in Same Grating Experiment[END_REF][START_REF] Peng | Double Slit Experient Still Has Much to Offer (9): ---Continuously Curved Diffraction Patterns of 2D-Cross-Grating Experiments[END_REF][START_REF] Peng | Non-Uniform, Curved, Expanded and Inclined Diffraction Pattern of 1D-Grating Experiments ---Rotating Grating Around Three Axes[END_REF][START_REF] Peng | Non-Uniform Diffraction Pattern of 2D-Cross-Grating -Rotating Grating Around Three Axes[END_REF][START_REF] Peng | Math Model of Rotating Grating -CW/CCW Rotation Angle-Dependence[END_REF].

The standard grating equation for the rotating grating has the form

𝑑𝑠𝑖𝑛(𝜃 % ) ± 𝑑𝑠𝑖𝑛(𝜃 $ ) = 𝑚𝜆
where the "+" and "-" are used when 𝜃 % and 𝜃 $ are on the same side and opposite side of the grating normal. And treating all angles as positive.

However, it has been pointed out that for the single diffraction pattern of a given grating, to study systematically the chrematistics of the grating pattern and to understand the optical meaning, the pattern may be divided into different portions based on the different locations of the different diffraction orders [START_REF] Peng | Math Model of Rotating Grating -CW/CCW Rotation Angle-Dependence[END_REF]. To describe a diffraction pattern clearly, different grating equations may be used for different portions.

First, introducing the normal diffraction order 𝑚 ! , hereafter referred it as the "normal order", that is the function of the grating spacing, wavelength and the incident angle of light. Then we show the followings in detail: [START_REF] Harvey | Description of Diffraction Grating Behavior in Direction Cosine Space[END_REF] the normal order 𝑚 ! and the zeroth order 𝑚 = 0 divide a diffraction pattern into three portions; each portion corresponds to a range of diffraction orders "m", and is described by specific equations, respectively; (2) the number of the diffraction orders "m" in each portion is determined by the grating spacing, wavelength and the incident angle of light; [START_REF] Ninad | Novel cases of diffraction of light from a grating: Theory and experiment[END_REF] characterizing the diffraction angle and positions of the diffraction orders of each portion; (4) the diffraction pattern of a rotating grating is non-uniform.

Experimental Setup and Right-hand Rule

Experimental Setup Figure 1 shows the setup. The laser light propagates along negative X direction.

The slits of the grating are along the Y axis. Next section shows how we divide the diffraction pattern of a rotated grating into three portions, and each portion is described by corresponding equation.

We have shown [START_REF] Peng | Double Slit Experiment Still Has Much to Offer (6): Curved, Expanded and Inclined Interference Patterns -Dynamic Double Slit/Cross-Double Slit Experiments[END_REF][START_REF] Peng | Double Slit Experiment Still Has Much to Offer (7): Particle Nature of Photons, Normal Diffraction Pattern and Curved Diffraction Pattern Emerging in Same Grating Experiment[END_REF] that when the grating is rotated around two or three axes, the CW rotations and CCW rotations make different. Let us define the normal diffraction order 𝑚 ! (hereafter denoted as "normal order") and its position. Figure 3 is used to define the normal order 𝑚 ! , which shows that the grating rotates CCW around the Y axis by 𝜃 % (according to the right-hand rule, Figure 2), which is equivalent to the light beam shining on the grating at the oblique angle 𝜃 % . The line OO' is the grating normal, the point O' is on the screen. The "𝐵𝐷 = 𝑑" is the spacing between two slits of the grating. The beam-1 and beam-2 travelling along path-1 and path-2, respectively, and meet at the point 𝑂'. After passing the grating, there is no path deference between two beams, i.e., 𝐷𝑂 ' = 𝐵𝑂′. The O" corresponds to the diffraction zeroth-order 𝑚 = 0. The O' corresponds to the normal order 𝑚 ! . Denote 𝑂𝑂" = 𝐿.

Grating Equations for

We define 𝑚 ! > 0 and 𝜃 % > 0 for the CCW rotation of the grating. The standard grating equation, 𝑑𝑠𝑖𝑛(𝜃 % ) + 𝑑𝑠𝑖𝑛(𝜃 $ ) = 𝑚𝜆, becomes,

𝑑𝑠𝑖𝑛(𝜃 % ) = 𝑚𝜆. (1a)
Definition of normal order 𝒎 𝑵 :

𝑚 ! = ( ) 𝑠𝑖𝑛(𝜃 % ) (1b)
Eq. 1b defines the normal order 𝒎 𝑵 . The 𝑚 ! is not necessary to be an integer.

Example-3-1: Figure 4 shows examples of Eq. 1b, where different spacings are in terms of # slits/mm, different incident angle is 𝜃 % , and the red laser of wavelength 𝜆 = 0.65 𝜇𝑚 is used. Therefore, the first diffraction order positioned above 𝑚 ! is 𝑚 = 2. There is only one diffraction order between the zeroth order 𝑚 = 0 and the normal order 𝑚 ! , which is 𝑚 = 1. Normal order m N vs. spacing, wavelength (650nm) and incident angle (Eq. 1b)

m_N: at θi=30 degree m_N: at θi=45 degree m_N: at θi=60 degree m_N: at θi=75 degree Example-3-3 (Figure 4): for 𝑑 = 100 *%+,-$$ = 10𝜇𝑚, 𝜆 = 0.65𝜇𝑚, 𝜃 % = 60 # , 𝑠𝑖𝑛60 # = 0.866, 𝑚 ! = 13.3. Therefore, the first diffraction order positioned above 𝑚 ! is 𝑚 = 14. There is 13 of diffraction orders between the zeroth order 𝑚 = 0 and the normal order 𝑚 ! , which are 𝑚 = 1 -13.

Experiment-3-1 to -3-3 show that the larger the spacing d, the larger the value of the normal order 𝑚 ! and thus, the larger the number of the diffraction orders between 𝑚 and 𝑚 ! .

Position of normal order 𝒎 𝑵 on the screen

Let's define the position of the normal order 𝑚 ! (Figure 3):

ℎ $! = 𝑂′𝑂". (2)
The point O' corresponds to the position of 𝑚 ! . The position ℎ ! of 𝑚 ! is at the intersection of the grating normal and the screen. Calculating the position ℎ ! , we have,

ℎ $! = 𝐿 𝑡𝑎𝑛(𝜃 % ) (3a) ℎ $! = 𝐿 𝑚 ! ) (./-0 " . ( 3b
)
Figure 5. Position ℎ $! of normal order vs. incident angle 𝜃 % (Eq. 3a)

Example-3-4 (Figure 5): for a relatively small incident angle, for example, 𝜃 % ≤ 30 # , we approximately have a linear relation between ℎ $! and 𝜃 % , ℎ $! ≈ 𝐿𝜃 % .

To express ℎ $! in terms of the normal order 𝑚 ! , Eq. (1b) gives

𝑠𝑖𝑛(𝜃 % ) = 12+(0 " ) 5[12+(0 " )] # 89 = 𝑚 ! ) ( . (4)
Solving Eq. ( 4), we obtain

𝑡𝑎𝑛(𝜃 % ) = 𝑚 ! $ % ; 9<=$ ! $ % > # (5)
Substituting Eq. ( 5), Eq. (3a) becomes

ℎ $! = 𝑚 ! ? $ % ; 9<=$ ! $ % > # (3c)
The ℎ $! can be expressed by 𝑚 ! (Eq. 3c) or by 𝜃 % (Eq. 3a) or by both 𝑚 ! and 𝜃 % (Eq. 3b). Positin h mN of normal order m N (Eq. 3a)

Dividing Diffraction Pattern into Three Portions

Now let us divide the diffraction pattern of a CCW rotating grating into three portions.

First portion: the first portion 𝑚 " is above 𝑚 ! , i.e., 𝑚 " > 𝑚 ! > 𝑚, where 𝑚 = 0

Second portion: the second portion 𝑚 "" is between 𝑚 ! and 𝑚, i.e., 𝑚 ! > 𝑚 "" > 𝑚.

Third portion: the third portion is below 𝑚 = 0, i.e., 𝑚 """ < 𝑚 and 𝑚 """ < 0.

The subscripts "I, II and II" indicate First-, Second-and Third portion respectively.

The first portion and second portion are divided by the normal order 𝑚 ! . The second portion and third portion are divided by the zeroth-order 𝑚 = 0.

Figure 6. Schematic of three portions

The plan of the article: start from (1) the first portion 𝑚 " with angle 𝜃 $" ; then (2) reduce 𝜃 $" to zero, i.e., go to the normal order 𝑚 ! ; then (3) go to second portion 𝑚 "" with angle 𝜃 $"" ; then (4) enlarging 𝜃 $"" to 𝜃 % , i.e., 𝜃 $"" goes to the zero order 𝑚 = 0; then (5) go to third portion by enlarging 𝜃 $"" to 𝜃 $""" .

Grating Equations for First Portion:

𝒎 𝑰 > 𝒎 𝑵 > 𝟎

Grating Equations

When 𝑚 " > 𝑚 ! > 0, 𝑌 $" is above 𝑂', i.e., the incident angle 𝜃 % and diffraction angle 𝜃 $" are on the same side of grating normal (Figure 7). Where 𝑌 $" is the position of the diffraction order 𝑚 " . The path-difference (phase difference) is,

𝐴𝐵 + 𝐵𝐶.
For far field, path-1, 𝑂𝑌 $" and path-2 can be treated as approximately parallel to each other. Where 𝐴𝐵 = 𝑑𝑠𝑖𝑛(𝜃 % ) and 𝐵𝐶 = 𝑑𝑠𝑖𝑛(𝜃 $" ).

A B O D ! ! ! ! Path-1 Path-2 " " " "" " """ " # " = 0 ! $" ! $"" ! $"""
𝑑𝑠𝑖𝑛(𝜃 % ) + 𝑑𝑠𝑖𝑛(𝜃 $" ) = 𝑚 " 𝜆. ( 6
)
𝜃 $" = 𝑎𝑟𝑐𝑠𝑖𝑛 P𝑚 " ) ( -𝑠𝑖𝑛 (𝜃 % )Q. ( 7a
)
Substituting Eq. (1b) into Eq. ( 6), we obtain the grating equations in terms of 𝑚 " and 𝑚 ! (Figure 8).

𝑑𝑠𝑖𝑛(𝜃

$" ) = 𝜆[𝑚 " -𝑚 ! ] (8)
𝜃 $" = 𝑎𝑟𝑐𝑠𝑖𝑛 P ) ( (𝑚 " -𝑚 ! )Q (7b)
Example-3-5 (Figure 8): let us consider an example: 𝜆 = 0.65 𝜇𝑚, the incident angle

𝜃 % = 45 # , For 𝑑 9### , 𝑚 ! = 1.09, then 𝑚 " = 2 For 𝑑 A## , 𝑚 ! = 2.

Limitation of Number of Diffraction Orders 𝒎 𝑰

Eq. (7a) and Eq. (7b) limit the value and numbers of the integer diffraction orders 𝑚 " .

Since 𝑠𝑖𝑛(𝜃 $" ) ≤ 1, thus Eq. 8 gives

) ( [𝑚 " -𝑚 ! ] < 1, namely 𝑚 " < ( ) + 𝑚 ! (9a)
Substituting Eq. (1b) into Eq. (9a), the number of diffraction orders "𝑚 " " is proportional to spacing d and satisfy

𝑚 " < ( ) [1 + 𝑠𝑖𝑛𝜃 % ] (9b)
Example-3-6 (Figure 9): let us consider an example: 𝜆 = 0.65 𝜇𝑚, the incident angles Eq. 9b (Figure 9) shows the maximum values of the diffraction orders 𝑚 " of the grating with different densities of slits and different incident angles. The grating with low density of slits has larger value of diffraction orders 𝑚 " . On the contrary, the grating with high density of slits has a smaller value of diffraction orders.

𝜃 % = 30 # ,

Position of Diffraction Orders 𝒎 𝑰 on Screen

Next, let us calculate the positions of the diffraction orders 𝑚 " on the screen (Figure 10).

For 𝐿 ≫ 𝑑, the three rays r, 𝑟 9 and 𝑟 G are essentially treated as being parallel, thus, 𝜃 ≈ 𝜃 $" .

Denoting h $" = 𝑌 $ 𝑂", we have

h $" = 𝐿 tan(θ H + θ IJ ) (10)
The spacing between two adjacent diffraction orders is

∆ℎ $"B<G ≡ h $"B -h $"G = 𝐿 [tan(θ H + θ IJB ) -tan(θ H + θ IJG )] = 0.83
Distance between diffraction orders 3 and 4 is

∆ℎ $"K<B ≡ h $"K -h $"B = 𝐿 [tan(θ H + θ IJK ) -tan(θ H + θ IJB )] = 6.63
Therefore, the distances between two adjacent diffraction orders are non-uniform.

Grating Equations for Second Portion:

𝒎 𝑵 > 𝒎 𝑰𝑰 > 𝟎

Grating Equations

When 0 < 𝑚 "" < 𝑚 ! , 𝑌 $ is in between the point 𝑂' and the point O", i.e., the incident angle 𝜃 % and diffraction angle 𝜃 $"" are on the opposite sides of the grating normal, and 𝜃 % > 𝜃 $"" (Figure 11). Where 𝐴𝐵 > 𝐷𝐶, 𝐴𝐵 = 𝑑𝑠𝑖𝑛(𝜃 % ), 𝐷𝐶 = 𝑑𝑠𝑖𝑛(∠𝐶𝐵𝐷) = 𝑑𝑠𝑖𝑛(𝜃 $"" ), 𝜃 % > 𝜃 $"" and 𝑚 "" > 0.

The grating equation for this situation in terms of 𝜃 % and 𝜃 $"" is

𝑑𝑠𝑖𝑛(𝜃 % ) -𝑑𝑠𝑖𝑛(𝜃 $"" ) = 𝑚 "" 𝜆. (12)
Substituting Eq. (1b) into Eq. ( 12), we have the grating equation in terms of 𝑚 "" and 𝑚 # ,

𝑠𝑖𝑛(𝜃 $"" ) = ) ( [𝑚 ! -𝑚 "" ] (13a)
𝜃 $"" = arcsin P

) ( (𝑚 ! -𝑚 "" )Q (13b)
The larger the value of the integer "𝑚 "" ", i.e., 𝑌 $"" moves to O', the smaller the diffraction angle 𝜃 $"" .

Example-3-8 (Figure 12): let us consider an example: 𝜆 = 0.65 𝜇𝑚, the incident angle 𝜃 % = 45 # .

Addition to Figure 8, Figure 12 shows the diffraction angles vs. the diffraction orders 𝑚 " and 𝑚 "" .

Figure 12. Diffraction angles vs. the combination of diffraction orders 𝑚 " and 𝑚 "" When a diffraction order is closer to the normal order 𝑚 ! , its diffraction angle is smaller.

For 𝑑 9### = 9$$ 9### *%+,-= 1 𝜇𝑚, 𝑚 ! = 1.09, 𝑚 "" = 1 For 𝑑 A## , 𝑚 ! = 2.18, 𝑚 "" = 1,2 A B O O' D ! ! ! "## ! "## d ! ! " "## C Path-

Limitation of Number of Diffraction Orders 𝒎 𝑰𝑰

Note that Eq. (13a) limits the value of "𝑚 "" ". Since 0 < 𝑠𝑖𝑛(𝜃 $"" ) < 1, thus Eq. 13a gives 0 <

) (

[𝑚 ! -𝑚 "" ] < 1, namely, the integer diffraction orders "𝑚 "" " satisfy

𝑚 "" > 𝑚 ! - ( ) (14a)
𝑚 ! > 𝑚 "" (14b)
Substituting Eq. (1b) into Eq. (14a), the integer diffraction orders "𝑚 "" " has to satisfy

𝑚 "" > ( ) [𝑠𝑖𝑛(𝜃 % ) -1] (14c)
By definition, the integer 𝑚 "" > 0, Eq. 14c is satisfied, which is what we discussed in Section 3.3, i.e., 0 < 𝑚 "" < 𝑚 ! . (14d)

Position of Diffraction Orders 𝒎 𝑰𝑰 on Screen

Next let us determine the position of the diffraction order 𝑚 "" .

For 𝐿 ≫ 𝑑, the three rays 𝑌 $"" 𝑂, 𝑌 $"" 𝐷 and 𝑌 $"" 𝐵 are essentially treated as being parallel, thus,

∠𝑌 $"" 𝑂𝑂′ ≈ 𝜃 $"" . (15)
Denoting 𝑂"𝑌 $"" = ℎ $"" , we have

ℎ $"" = 𝐿 tan (𝜃 % -𝜃 $"" ) (16)
The spacing between two adjacent diffraction orders is

∆ℎ $"" = 𝐿 [tan(θ H -θ IJJ89 ) -tan(θ H -θ IJJ )]. (17)
Therefore, the diffraction pattern of the rotating grating is nonuniform. (1) Using Eq. 14c to determine 𝑚 "" . We have 𝑚 "" = 1.

Example

(2) Using Eq. 13b, 𝜃 $"" = arcsin P (3) ℎ $"" = 𝐿 tan(𝜃 % -𝜃 $"" ) = 𝐿 tan 19. 9 # = 0.36𝐿

Grating Equations for 𝒎 = 𝟎

The path difference before passing through the grating is 𝐴𝐵 = 𝑑𝑠𝑖𝑛𝜃 % . The path difference after passing through the grating is 𝐶𝐷 = 𝑑𝑠𝑖𝑛𝜃. Figure 13 shows that 𝜃 $# = 𝜃 = 𝜃 % , so 𝐴𝐵 = 𝐶𝐷, i.e., 𝐴𝐵 -𝐶𝐷 = 0, the net path difference is zero. Thus, the grating equation is

𝑑𝑠𝑖𝑛𝜃 % -𝑑𝑠𝑖𝑛𝜃 $# = 0 = 𝑚𝜆, (18)
𝑚 = 0.
The diffraction order 𝑚 = 0 separates the Second portion (described in Section 3.4) and third portion.

Grating Equations for Third

Portion: 𝒎 𝑰𝑰𝑰 < 𝟎

Grating Equations

When 𝑌 $ is below the point O" (𝑚 = 0), we define 𝑚 """ < 0. The incident angle 𝜃 % and diffraction angle 𝜃 $""" are on the opposite sides of the grating normal, and 𝜃 $""" ≥ 𝜃 % (Figure 14). The diffraction angle for the third portion starts from 𝜃 $""" = 𝜃 % , which leads to the zeroth order 𝑚 = 0, and then gradually increase, such that 𝑚 """ = -1, -2 ….

We have the path difference,

𝐴𝐵 -𝐷𝐶.

Where 𝐴𝐵 < 𝐷𝐶, 𝐴𝐵 = 𝑑𝑠𝑖𝑛(𝜃 % ) and 𝐷𝐶 = 𝑑𝑠𝑖𝑛(∠𝐶𝐵𝐷) = 𝑑𝑠𝑖𝑛(𝜃 $""" ).

Figure 14. Schematic of Third Portion of CCW rotating grating

The grating equation for this situation in terms of 𝜃 % and 𝜃 $""" is

A B O O' D ! ! ! ! Path-1 Path-2 Path-1 Path-2 m = 0 O" C ! "# ! Grating A B O O' D ! ! ! "### d ! ! " "### C Path-1 Path-2 Path-1 Path-2 O" ! "###
Grating 𝑑𝑠𝑖𝑛(𝜃 % ) -𝑑𝑠𝑖𝑛(𝜃 $""" ) = 𝑚 """ 𝜆.

(19) Substituting Eq. (1b) into Eq. ( 19), we have the grating equation in terms of 𝑚 """ and 𝑚 ! ,

𝑠𝑖𝑛(𝜃 $""" ) = ) ( [𝑚 ! -𝑚 """ ] (20a) 𝜃 $""" = arcsin P ) ( (𝑚 ! -𝑚 """ )Q (20b)
Note that the third portion starts from O" and contain the diffraction orders below O".

Limitation of Number of Diffraction Orders 𝒎 𝑰𝑰𝑰

Note that Eq. (20a) limits the value of "𝑚 """ ". Since 0 ≤ 𝑠𝑖𝑛(𝜃 $""" ) ≤ 1, thus Eq. 20a gives

0 < ) ( [𝑚 ! -𝑚 """ ] < 1. (21a) 𝑚 """ > P𝑚 ! - ( ) Q, (21b)
𝑚 """ > ( ) [𝑠𝑖𝑛𝜃 % -1]. (21c)
Example-3-10: At the zeroth order 𝑚 = 0, 𝜃 $""" = 𝜃 % = 45 # .

For 𝜃 % = 45 # , 𝑑 9### , 𝑚 ! = 1.09, 𝑚 """ ≥ 𝑚 ! -( ) = -0.45. Thus 𝑚 """ = 0.

For 𝜃 % = 45 # , 𝑑 A## , 𝑚 ! = 2.18, 𝑚 """ ≥ 𝑚 ! -( ) = -0.9. Thus 𝑚 """ = 0.

For 𝜃 % = 45 # , 𝑑 B## , 𝑚 ! = 3.62, 𝑚 """ ≥ 𝑚 ! - ( ) = -1.5.
Thus 𝑚 """ = -1.

For 𝜃 % = 45 # , 𝑑 G## , 𝑚 ! = 5.44, 𝑚 """ ≥ 𝑚 ! - ( ) = -2.25. Thus 𝑚 """ = -1, -2.
For 𝜃 % = 45 # , 𝑑 9## , 𝑚 ! = 10.88, 𝑚 """ ≥ 𝑚 ! -( ) = -4.5. Thus 𝑚 """ = -1, -2, -3, -4.

Figure15. Diffraction angles vs. diffraction orders 𝑚 " , 𝑚 "" and 𝑚 """

Addition to Figure 12, Figure15 shows the diffraction angles vs. diffraction orders 𝑚 " , 𝑚 "" and 𝑚 """ .

Position of Diffraction Orders 𝒎 𝑰𝑰𝑰 on Screen

We denote the position of diffraction orders 𝑚 """ on screen 𝑂"𝑌 $""" = ℎ $""" . Figure 16 shows, for 𝐿 ≫ 𝑑, path-1, 𝑂𝑌 $""" and path-2 can be treated as parallel, and thus ∠𝑌 $""" 𝑂𝑂" = 𝜃 $""" -𝜃 % Thus, the position of diffraction orders 𝑚 """ is ℎ $""" = -𝐿 𝑡𝑎𝑛(𝜃 $""" -𝜃 % ).

(

) 22
Note that the minus sign indicates that the point 𝑌 $""" is below the point O".

Example-3-11:

For 𝑑 A## , 𝜃 % = 30 # . 𝑚 ! = ( ) 𝑠𝑖𝑛(𝜃 % ) = 1.53, 𝑚 """ ≥ 𝑚 ! - ( )
= -1.53. Using Eq. 21c to determine 𝑚 """ . We have 𝑚 """ = -1. Using Eq. 20b, 𝜃 $""" = arcsin P

) ( (𝑚 ! -𝑚 """ )Q = 55. 6 # .
Using Eq. 22, ℎ $""" = -𝐿 tan(𝜃 $""" -𝜃 % ) = -0.48𝐿

Experiment-3-1:

For the grating of 𝑑 A## , the incident angle is 𝜃 % = 30 # , then 𝑚 ! = 1.53. The experimental result is shown in Figure 15. O"

! ! Grating ! ! = 3 ! ! = 2 ! !! = 1 ! = 0 ! !!! = -1
From Eq. 7a, 𝜃 $" = 𝑎𝑟𝑐𝑠𝑖𝑛 P𝑚 " ) ( -𝑠𝑖𝑛 (𝜃 % )Q, for 𝑚 " = 4, we obtain 𝜃 $"<K = 53 # . Eq. 10, h $" = 𝐿 tan(θ H + θ IJ ), gives the position of h $" from the position of the zeroth order,

h $"<K = 8𝐿
Let us compare with the diffraction orders 𝑚 " = 3 and 𝑚 " = 2.

For 𝑚 " = 3, we obtain 𝜃 $"<B = 28 # and h $"<B = 1.6𝐿.

For 𝑚 " = 2, we obtain 𝜃 $"<G = 8. 68 # and h $"<G = 0.8𝐿.

For 𝑚 "" = 1, we obtain 𝜃 $"" = 10. 1 # and h $"" = 0.36𝐿.

For 𝑚 """ = -1, we obtain 𝜃 $""" = 55. 6 # and h $""" = -0.48𝐿.

The diffraction order 𝑚 " = 4 is much far from the zeroth order and much dimmer to be detected.

Grating Equations for Rotating CW Grating Around Y axis

When the grating is rotating around one axis, say the Y axis, there is no difference between CW and CCW rotations. However, we have shown that when the grating is rotated around two or three axes sequentially, the CW and CCW rotations make different.

Normal Diffraction Order "𝒎 𝑵 "

Definition of Normal Diffraction Order "𝒎 𝑵 " for CW rotating Grating

Figure 18 shows that the grating rotates CW around the Y axis by 𝜃 % , which is equivalent to the light beam shining on the grating at the oblique angle 𝜃 % . The line OO' is the grating normal. The spacing between two slits is "d". The beam-1 and beam-2 along path-1 and path-2, respectively, meet at the point 𝑂' . After passing the grating, there is no path deference between two beams, i.e., 𝐷𝑂 ' = 𝐵𝑂′.

The O" corresponds to the zeroth order 𝑚 = 0. The O' corresponds to 𝑚 ! .

We define 𝑚 ! < 0 for the CW rotating grating. For the CCW rotating grating, 𝑚 ! > 0 in Eq. 1b.

For the case of 𝐿 ≫ 𝑑, path-1, OO' and path-2 are approximately parallel to each other. For the light beams arriving at the point O' on the grating normal, the diffraction angle 𝜃 $ ! = 0. The "AB" is the path-deference of light propagating along path-1 and path-2, ∠𝑂′𝑂𝑂" = 𝜃 % , we have

𝑑𝑠𝑖𝑛(𝜃 % ) = -𝑚 ! 𝜆. (23a)
Definition of the normal order 𝒎 𝑵 : 𝜃 % > 0.

𝑚 ! = - ( ) 𝑠𝑖𝑛(𝜃 % ). (23b)

Position of normal order 𝒎 𝑵 on screen

Defining 𝑂′𝑂" = ℎ $! , then we define

ℎ $! = -𝐿 𝑡𝑎𝑛(𝜃 % ). (24a)
The ℎ $! is determined by the 𝜃 % and ℎ $! < 0.

Eq. 24a has the same form as Eq. 3a. However, ℎ $! > 0 in Eq. 3a for the CCW rotating grating.

To find 𝑡𝑎𝑛(𝜃 % ) in terms of the diffraction order m and 𝑚 ! , Eq. (23a) gives

𝑡𝑎𝑛(𝜃 % ) = -𝑚 ! ) ( ; 9<=<$ ! $ % > # (25)
Using Eq. ( 25), Eq. (24a) becomes

ℎ $! = 𝑚 ! ? ( ) ; 9<=$ ! $ % > # < 0 (24b)
Note that for CCW rotating grating, the normal order 𝑚 ! is on the positive side of the zeroth order 𝑚 = 0, while for CW rotating grating, the normal order 𝑚 ! is on the negative side of the zeroth order.

Dividing Diffraction Pattern into Three Portions

Now let us divide the diffraction pattern of a CW rotating grating into three portions (Figure 19).

! ! Path-1 Path-2 O" ! ! Grating " = 0 " " " ## " # " ### ! !" ! !"" ! !""" ! !
First portion: the first portion is above O", i.e., 𝑚 " > 𝑚, where 𝑚 = 0, thus 𝑚 " > 0.

Second portion: the second portion is between 𝑚 ! and 𝑚 = 0, i.e., 𝑚 > 𝑚 "" > 𝑚 ! , 𝑚 "" < 0.

Third portion: the third portion is below 𝑚 ! , i.e., 𝑚 """ < 𝑚 ! < 0.

The subscripts "I, II and II" indicate First-, Second-and Third portion respectively.

The first portion and second portion are divided by the zeroth order 𝑚 = 0. The second portion and third portion are divided by the normal-order 𝑚 ! .

First Portion:

𝒎 𝑰 > 𝟎

Grating Equation

When 𝑚 " > 0, 𝑌 $" is above 𝑂", i.e., 𝜃 $" > 𝜃 % (Figure 20). The incident angle 𝜃 % and diffraction angle 𝜃 $" are on the opposite sides of the grating normal, and 𝜃 $" > 𝜃 % . The diffraction angle for the first portion starts from 𝜃 $" = 𝜃 % , which leads to the zeroth order 𝑚 = 0, and then gradually increase 𝜃 $" , such that 𝑚 " = 1, 2 ….

We have the path difference,

𝐷𝐶 -𝐴𝐵,

where 𝐷𝐶 > 𝐴𝐵, 𝐴𝐵 = 𝑑𝑠𝑖𝑛(𝜃 % ) and 𝐷𝐶 = 𝑑𝑠𝑖𝑛(∠𝐶𝐵𝐷) = 𝑑𝑠𝑖𝑛(𝜃 $" ). ℎ $" = 𝑂"𝑌 $" .

(𝑚 " -𝑚 ! )Q (27b) ℎ $" = 𝐿 tan (𝜃 $" -𝜃 % ) (28)
The larger the value of the integer "𝑚 " ", the larger the diffraction angle 𝜃 $" and the larger the position "ℎ $" ". O" ! #

Limitation on number of diffraction order 𝒎 𝑰

Eq. 26 and Eq. 27a limit the value of "𝑚 " ", since 𝑠𝑖𝑛(𝜃 $" ) ≤ 1, thus, Lowering 𝑌 $" again, 𝑌 $" becomes 𝑌 $"" that is between O' and 𝑂". So 0 > 𝑚 "" > 𝑚 ! , i.e., 𝜃 $"" < 𝜃 % . The incident angle 𝜃 % and diffraction angle 𝜃 $"" are on the opposite sides of the grating normal. The path difference is,

𝑚 " < ( ) [1 -𝑠𝑖𝑛(𝜃 % )] (29a

𝐷𝐶 -𝐴𝐵,

where 𝐷𝐶 < 𝐴𝐵, 𝐴𝐵 = 𝑑𝑠𝑖𝑛(𝜃 % ) and 𝐷𝐶 = 𝑑𝑠𝑖𝑛(∠𝐶𝐵𝐷) = 𝑑𝑠𝑖𝑛(𝜃 $"" ), ℎ $"" = 𝑂"𝑌 $"" (Figure 22). (30)

𝑠𝑖𝑛(𝜃 $"" ) = ) ( [𝑚 "" -𝑚 ! ] (31a) 𝜃 $"" = arcsin P ) ( (𝑚 "" -𝑚 !) Q (31b) ℎ $"" = -𝐿 tan (𝜃 % -𝜃 $"" ) (32)
Eq. (31a) limits the value of "𝑚 "" ". Since 𝑠𝑖𝑛(𝜃 $"" ) ≤ 1, thus, When 𝑌 $""" is below 𝑂', i.e., 𝑚 ! > 𝑚 """ , the incident angle 𝜃 % and diffraction angle 𝜃 $""" are on the same side of the grating normal. Defining 𝑂"𝑦 $""" = ℎ $""" . We have the path difference (Figure 23),

𝐴𝐵 + 𝐵𝐶,

where 𝐴𝐵 = 𝑑𝑠𝑖𝑛(𝜃 % ) and 𝐵𝐶 = 𝑑𝑠𝑖𝑛(𝜃 $""" ).

Figure 23. Schematic of Third portion of CW rotating grating

The grating equation is,

𝑑𝑠𝑖𝑛(𝜃 % ) + 𝑑𝑠𝑖𝑛(𝜃 $""" ) = -𝑚 """ 𝜆. ( 34
)
Substituting Eq. (23b) into Eq. (34), we obtain

𝑠𝑖𝑛(𝜃 $""" ) = ) ( [𝑚 ! -𝑚 """ ] (35a)
𝜃 $""" = arcsin P

) ( (𝑚 ! -𝑚 """ )Q (35b) ℎ $""" = -𝐿 𝑡𝑎𝑛(𝜃 % + 𝜃 $""" ) (36)
Eq. (35a) limits the value of "𝑚 """ ", since 𝑠𝑖𝑛(𝜃 $""" ) ≤ 1, thus

𝑚 """ > 𝑚 ! - ( ) . (37)

Summary and Discussion

A B O O' ! ! ! ! " "### C ! "### Path-1 Path-2 Path-1 Path-2 ! "### ! ! O"

Grating

For the diffraction pattern of a grating rotating CW and CCW around one axis that parallels the slits of the grating, we break the pattern into 3 portions. The three portions are divided by the normal order 𝑚 ! and the zeroth-order, 𝑚 = 0, respectively, then describe each portion respectively.

We show that (1) Each portion is described by specific equations; " ""

" = 0 = 0 A B O O' D ! ! ! ! Path-1 Path-2 Path-1 Path-2 ! ! m = 0 ! ! O" Grating A B O O' ! ! d Path-1 Path-1 Path-2 Path-2 O" ! ! Grating D 𝑑𝑠𝑖𝑛(𝜃 $" ) + 𝑑𝑠𝑖𝑛(𝜃 % ) = 𝑚 " 𝜆, 𝑑𝑠𝑖𝑛(𝜃 $" ) -𝑑𝑠𝑖𝑛(𝜃 % ) = 𝑚 " 𝜆. 𝜃 $" = 𝑎𝑟𝑐𝑠𝑖𝑛 P ) ( (𝑚 " -𝑚 ! )Q, 𝜃 $" = arcsin P ) ( (𝑚 " + 𝑚 ! )Q h $" = 𝐿 tan(θ H + θ IJ ), ℎ $" = 𝐿 tan (𝜃 $" -𝜃 % ) 𝑚 " < ( ) + 𝑚 ! , 𝑚 " < ( ) -𝑚 ! 4
) Second portion:

CCW rotation: 𝑚 ! > 𝑚 "" > 0, CW rotation: 0 > 𝑚 "" > 𝑚 ! 𝑑𝑠𝑖𝑛(𝜃 % ) -𝑑𝑠𝑖𝑛(𝜃 $"" ) = 𝑚 "" 𝜆 𝑑𝑠𝑖𝑛(𝜃 $"" ) -𝑑𝑠𝑖𝑛(𝜃 % ) = 𝑚 "" 𝜆 𝜃 $"" = arcsin P

) ( (𝑚 ! -𝑚 "" )Q 𝜃 $"" = arcsin P ) ( (𝑚 ! -𝑚 "" )Q ℎ $"" = 𝐿 tan (𝜃 % -𝜃 $"" ) ℎ $"" = 𝐿 tan (𝜃 % -𝜃 $"" )
𝑚 "" > 𝑚 ! - = arcsin < 𝜆 𝑑 (𝑚 ! -𝑚 """ )?

Order: 𝑚 𝑚 ! = -𝑑 𝜆 𝑠𝑖𝑛𝜃 # 𝑚 " < 𝑑 𝜆 + 𝑚 ! 𝑚 "" > 𝑚 ! -𝑑 𝜆 𝑚 """ > 𝑚 ! -𝑑 𝜆

Position: h

ℎ $! = -𝐿 𝑡𝑎𝑛(𝜃 # ) h $" = 𝐿 𝑡𝑎𝑛(𝜃 $" -𝜃 # ) ℎ $"" = -𝐿 tan (𝜃 # -𝜃 $"" ) ℎ $""" = -𝐿 𝑡𝑎𝑛(𝜃 $""" + 𝜃 # )

Acknowledgement

The author would like to thanks Dr. Sandro Faetti for helpful discussions.

  Diffraction angles (degree) Diffraction orders: m_III, m=0, m_II and m_I Diffraction angles vs. diffraction orders m I , m II , m III, m=0 and grating spacing, wavelength (650nm), incident angle (45 degree) 1000 lines/mm 500 lines/mm 300 lines/mm 200 lines/mm 100 lines/mm

Figure 1 .

 1 Figure 1. Schematic Setup Right-Hand Rule: Figure 2 shows the right-hand rule for determine CW and CCW rotation of the grating along an axis.

Figure 2 .

 2 Figure 2. Right-Hand Rule for determining Rotation Direction

Rotating Grating CCW Around Y axis 3 . 1 . 3 . 1 . 1 .

 31311 Normal Diffraction Order "𝒎 𝑵 " Definition of Normal Diffraction Order "𝒎 𝑵 "

Figure 3 .

 3 Figure 3. Schematic of normal order "𝑚 ! " In the case of 𝐿 ≫ 𝑑, path-1, OO' and path-2 are approximately parallel to each other. For the light beams arriving at the point O' on the grating normal, the diffraction angle 𝜃 $ ! = 0. The "AB" is the path-deference between path-1 and path-2.

Figure 4 .Example- 3 - 2 (

 432 Figure 4. Correlation of 𝑚 ! (m_N), d, 𝜆 and incident angle 𝜃 % (Eq. 1b) Example-3-2 (Figure 4): for 𝑑 = 1000 *%+,-$$ = 1𝜇𝑚, 𝜆 = 0.65𝜇𝑚, 𝜃 % = 60 # , we have 𝑚 ! = 1.3.

  mm 500 lines/mm 300 lines/mm 200 lines/mm 100 lines/mm 50 lines/mm Normal order m N Spacing between slits of grating

Figure 7 .

 7 Figure 7. Schematic of First Portion of CCW rotating grating The standard grating equation for this situation is

2 , 5 For 8 ForFigure 8

 2588 Figure 8 Correlation of diffraction angle 𝜃 $" , 𝑚 " , 𝑚 ! , d, 𝜆 (650nm), at 𝜃 % = 45 # .

11 )Figure 10 .

 1110 Figure 10. Schematic of Position of diffraction order 𝑚 " Example-3-7: For a given grating of 𝑑 = 500 *%+,-$$ = 2 𝜇𝑚, 𝜆 = .65 𝜇𝑚, the incident angle is 𝜃 % =

Figure 11 .

 11 Figure 11. Schematic of Second Portion of CCW rotating grating We have the path difference, 𝐴𝐵 -𝐷𝐶,

3 For 5 For

 35 orders: m I and m II Diffraction angles of First-portion and Second-portion vs. wavelength (650nm), spacing, incident angle and diffraction orders (Eq. 7b and Eq. 13b) 1000 lines/mm 500 lines/mm 300 lines/mm 200 lines/mm 100 lines/mm For 𝑑 B## , 𝑚 ! = B.BB×#.E#E .FA = 3.62, 𝑚 "" = 1,2,𝑑 G## , 𝑚 ! = A×#.E#E .FA = 5.44, 𝑚 "" = 1,2,3,4,𝑑 9## , 𝑚 ! = 9#×#.E#E .FA = 10.88, 𝑚 "" = 1,2,3,4,5,6,7,8,9,10

- 3 - 9 :

 39 For a given grating of 𝑑 = 500 *%+,-$$ = 2 𝜇𝑚, 𝜆 = 0.65 𝜇𝑚, the incident angle is 𝜃 % = 30 # , then we have 𝑚 ! =

Figure 13 .

 13 Figure 13. Schematic of zeroth order of CCW rotating grating

Figure 16 .

 16 Figure 16. Schematic of deriving position of diffraction orders 𝑚 """

Figure 17 .[ 1 +

 171 Figure 17. Experiment-3-1: Picture of diffraction ordersThe total number of the diffraction orders is 𝑚 """ + 𝑚 "" + 𝑚 " + 𝑚(= 0) =6:𝑚 " < ( ) [1 + 𝑠𝑖𝑛𝜃 % ] = 4.6thus, 𝑚 " = 2, 𝑚 " = 3, 𝑚 " = 4,

Figure 18 .

 18 Figure 18. Schematic of normal diffraction order "𝑚 ! " of CW rotating grating

Figure 19 .

 19 Figure 19. Schematic of First-, Second and Third Portions of CW rotating grating

Figure 20 .

 20 Figure 20. Schematic of First Portion of CW rotating grating The diffraction equations are, 𝑑𝑠𝑖𝑛(𝜃 $" ) -𝑑𝑠𝑖𝑛(𝜃 % ) = 𝑚 " 𝜆.(26)𝑠𝑖𝑛(𝜃 $" ) = ) ( [𝑚 " -𝑚 ! ](27a)𝜃 $" = arcsin P ) (

4 . 4 .

 44 Grating Equation for: 𝒎 = 𝟎Lowering 𝑌 $" until reaching O". The line OO" is the zeroth order that dividing the first portion and the second portion (Figure21).

Figure 21 . 4 . 5 .

 2145 Figure 21. Schematic of zeroth order dividing First and Second PortionsThe diffraction angle is equal to the incident angle, 𝜃 $" = 𝜃 % .

Figure 22 .

 22 Figure 22. Schematic of Second portion of CW rotating grating The diffraction equations are,

4 . 4 .

 44 Grating Equation for Third Portion: 0 > 𝑚 ! > 𝑚 """

( 2 ) 1 ) 2 ) 3 )

 2123 The distances between two adjacent diffraction orders are non-uniform; (3) The number of the diffraction orders is limited by the spacing of the grating, wavelength and the incident angle. Let us summarize the contents of this article below. Zero order 𝒎 = 𝟎: CCW and CW rotations CCW rotation CW rotation 𝑑𝑠𝑖𝑛𝜃 % -𝑑𝑠𝑖𝑛𝜃 $# = 0 = 𝑚𝜆 The equations describing the diffraction orders of both the CW and CCW rotation of the grating are the same. 𝒎 𝑵 : normal diffraction order (normal order) CCW rotation: 𝑚 ! > 0, CW rotation: 𝑚 ! First portion:CCW rotation: 𝑚 " > 𝑚 ! > 0: CW rotation: 𝑚 " > 0 > 𝑚 !

  Third portionCCW rotation: 𝑚 ! > 0 > 𝑚 """ , CW rotation: 0 > 𝑚 ! > 𝑚 """ % ) -𝑑𝑠𝑖𝑛(𝜃 $""" ) = 𝑚 """ 𝜆, 𝑑𝑠𝑖𝑛(𝜃 % ) + 𝑑𝑠𝑖𝑛(𝜃 $""" ) = 𝑚 """ 𝜆 𝜃 $""" = arcsin P) ( (𝑚 ! -𝑚 """ )Q 𝜃 $""" = arcsin P ) ( (𝑚 """ -𝑚 ! )Q ℎ $""" = -𝐿 𝑡𝑎𝑛(𝜃 $""" -𝜃 % ) ℎ $""" = -𝐿 𝑡𝑎𝑛(𝜃 % + 𝜃 $""" ) 𝑚 """ > 𝑚 ! -( ) 𝑚 """ > 𝑚 ! -( ) CCW Rotating Grating: 𝑚 ! > 0 Grating Definition of 𝑚 ! 𝑚 ! > 0 First Portion: 𝑚 " 𝑚 " > 𝑚 ! Second Portion: 𝑚 "" 𝑚 "" < 𝑚 ! , 𝑚 "" > 0Third Portion: 𝑚 """ 𝑚 """ < 0Equation 𝑚 ! = 𝑑 𝜆 𝑠𝑖𝑛𝜃 # 𝑠𝑖𝑛𝜃 # + 𝑠𝑖𝑛𝜃 $" = 𝑚 " 𝜆 𝑑 𝑠𝑖𝑛𝜃 $" = 𝜆 𝑑 [𝑚 " -𝑚 ! ] 𝑠𝑖𝑛𝜃 # -𝑠𝑖𝑛𝜃 $"" = 𝑚 "" 𝜆 𝑑 𝑠𝑖𝑛𝜃 $"" = 𝜆 𝑑 [𝑚 ! -𝑚 "" ] 𝑠𝑖𝑛𝜃 # -𝑠𝑖𝑛𝜃 $""" = 𝑚 """ = 𝐿 𝑡𝑎𝑛(𝜃 # ) h $" = 𝐿 𝑡𝑎𝑛(𝜃 # + 𝜃 $ ) ℎ $"" = 𝐿 𝑡𝑎𝑛(𝜃 # -𝜃 $"" ) ℎ $""" = -𝐿 𝑡𝑎𝑛(𝜃 $""" -𝜃 # )CW Rotating Grating:𝑚 ! < 0 Grating Definition of 𝑚 ! 𝑚 ! < 0First Portion: 𝑚 " 𝑚 " > 0 Second Portion: 𝑚 "" 𝑚 "" > 𝑚 ! , 𝑚 "" < 0Third Portion: 𝑚 """ 𝑚 """ < 𝑚 ! Equation 𝑚 ! = -𝑑 𝜆 𝑠𝑖𝑛(𝜃 # ) 𝑠𝑖𝑛𝜃 $" -𝑠𝑖𝑛𝜃 # = 𝑚 " 𝜆 𝑑 𝑠𝑖𝑛𝜃 $" = 𝜆 𝑑 [𝑚 " -𝑚 ! ]𝑠𝑖𝑛𝜃 $"" -𝑠𝑖𝑛𝜃 # = 𝑚 "" 𝜆 𝑑 𝑠𝑖𝑛𝜃 $"" = 𝜆 𝑑 [𝑚 "" -𝑚 ! ] 𝑠𝑖𝑛𝜃 # + 𝑠𝑖𝑛𝜃 $"""

  45 # , 60 # , 75 # , and the spacing of the grating 𝑑 is 1000 lines/mm, 300 lines/mm, 200 lines/mm, 100 lines/mm and 50 lines/mm respectively.

Figure 9. Ranges of diffraction order 𝑚 " are determined by 𝜃 % , d and 𝜆 (Eq. 9b)