Sergio Estupiñán
email: sergio.estupinan@unige.ch

Ingdahl Kasper

Nicolas Andkjaer
email: nicolas.szilas@unige.ch

Szilas

Engagement in Interactive Digital Storytelling: Sampling without spoiling

Keywords: Interactive Digital Storytelling, Evaluation, Engagement, Continuation Desire, Disruptiveness, Replayability, User Research

Interactive Digital Storytelling (IDS) enables users to influence the unfolding of a story at the plot level using Artificial Intelligence techniques for generating and balancing narrative paths on the fly. Despite their promise, most IDS systems are still unable to consistently deliver engaging user experiences. To further understand the characteristics of this new media, particularly user engagement, we apply the concept of Continuation Desire to an existing Interactive Narrative to dynamically sample engagement during a play session. We use a comparative study to assess the effects of interrupting users during gameplay. This study found no evidence that, if done properly, interrupting the experience spoils it. We find no significant impact on the desire to play again as a result of introducing interruptions either.

Introduction

Interactive Digital Storytelling (IDS) is a relatively new media whose rationale is to empower users to influence the course of narrative events of a computer-based system [START_REF] Molnar | Interactive Digital Narrative[END_REF][START_REF] Murray | Hamlet on the Holodeck: the future of narrative in cyberspace[END_REF]. The unfolding narrative is dynamically generated using Artificial Intelligence (AI) for narrative generation and balancing. In IDS, users would 'feel' the unfolding story as being their own, since it is the outcome of their own actions that drives it. The high degree of freedom, increasing complexity, and range of possible narratives, makes it difficult to keep the structure of the narrative engaging and meaningful. Such variability is also problematic from an evaluation perspective. The focus of evaluation must be on the Interactive Narrative Experience (INE), not the final product.

Current evaluation of IDS systems has mainly focused on the global appreciation of the INE inquiring users about their experience after a play session using surveys and standardized questionnaires. There is, however, a lack of research in the IDS community into the factors that determine, during run time, if and why an IDS system is perceived as meaningful and engaging by the users at a given time.

In game studies, one such concept is engagement [START_REF] Schoenau-Fog | Hooked! -Evaluating engagement as continuation desire in interactive narratives[END_REF], which has been used to explain the user involvement with the system, albeit not specifically with a narrative focus. Engagement theories have previously been used to measure games as processes, and given their narrative focus, they could fit well with IDS. This article introduces a method inspired by engagement research and performs a comparative study to measure the impact of such a method on users.

Related Work

Early research on the evaluation of IDS has provided basic results beyond the mere display of output examples [START_REF] Cavazza | Characters in Search of an author: AI-based Virtual Storytelling[END_REF] and the evaluation of the product. Ad-hoc questionnaires have been used in an attempt to evaluate the general qualities of systems (e.g. eval. B in [START_REF] Barber | Generation of adaptive dilemma-based interactive narratives[END_REF]). Another line of research used objective metrics to measure the user experience (e.g. aggregating interaction logs). Qualitative studies have also been conducted based on interviews and content analyses [START_REF] El-Nasr | Experiencing interactive narrative: A qualitative analysis of Facade[END_REF]. In the video games domain, related research has focused on the evaluation of in-game metrics as well as fine-tuning parameters within games [START_REF] Sundstedt | Game Analytics[END_REF]. Log analyses [START_REF] Kim | Tracking real-time user experience (TRUE): a comprehensive instrumentation solution for complex systems[END_REF] and interrupting questionnaires have also been used, although not necessarily in relation to the narrative dimension. It is only with the IRIS project [START_REF] Klimmt | The UX-tool: Measurement device and documentation for empirical assessment of the UX in IS[END_REF] that a IDS-specific evaluation framework was proposed, focusing on the general assessment of users' Interactive Narrative Experience after using an IDS system [START_REF] Roth | Experiencing Interactive Storytelling[END_REF]. While all of the above self-report approaches are a posteriori, there are other methodologies which have proceeded differently. To tackle the problem of measuring engagement in a condensed way, Continuation Desire (CD) attempts to perform sampling during an experience: users are interrupted and asked to quantify to which extent they want to continue the experience, what they want to do next and why they want to continue [START_REF] Schoenau-Fog | Evaluation of Continuation Desire As an Iterative Game Development Method[END_REF][START_REF] Schoenau-Fog | Hooked! -Evaluating engagement as continuation desire in interactive narratives[END_REF]. This approach produces a continuous CD curve and accounts of the narrative, making it possible to trace the evolution of CD as the Interactive Narrative unfolds.

The multiple facets associated with the term engagement, including flow, immersion, and enjoyment, share this common trait: the willingness to continue taking part in the activity, which is what Continuation Desire measures. It has therefore been argued that CD acts as an indicator of engagement [START_REF] Schoenau-Fog | Hooked! -Evaluating engagement as continuation desire in interactive narratives[END_REF], although the relation is oneway.

This answer to the problem of a posteriori approaches does not come without problems however: Injecting questionnaires into an experience could disrupt and bias the experience, which should be avoided [START_REF] Schoenau-Fog | Evaluation of Continuation Desire As an Iterative Game Development Method[END_REF]. This study addresses this problem.

Engagement in Interactive Digital Storytelling

Given the unsuitability of relying only in non-disruptive or psychophysiological measures for engagement research, the challenge becomes to minimize the disturbance of more intrusive methods so as not to bias the results. The scope of this article is limited to designing and implementing an in-game Continuation Desire evaluation approach and to measure the disruption caused. We believe that such a method should meet these requirements: 1) Gather fine-grained data during the interaction with an IDS system, and 2) The measuring instrument should not bias the results.

Experiment Design

To test the hypothesis that introducing interruptions during the experience will not have a significant negative impact on the experience, we implemented a CD questionnaire into the IDS system "Nothing For Dinner" 1 and set up a comparative study using selfreporting metrics disruptiveness and replayability desire. Two independent groups were recruited using a scientific crowdsourcing platform2 : Interruption group (N=35), and Control group (N=42). There were 36 males and 41 females, 18-40 years old (M=27.3 y/o, SD=5.95 y/o), speaking English as a first language. There was no recruitment requirement for computer proficiency or gaming profile. Compensation was 1.50 £ for around 23 minutes, the study was taken online on their own computers.

Metrics

The content of the interruption questionnaire is not evaluated in this article, it is mentioned as a proposal for a way to measure user appreciation of the INE. The relevant metrics for this experiment are: 1) Disruptiveness; how much users felt the interruptions were in the way of the experience (measured in the Interruption group), and 2) Replayability desire; how willing the user was to try the experience again (sampled in both groups).

Interruption Design

Triggering. The moment of interruption must be chosen to avoid generating user discomfort (e.g. while the user is reading or selecting an action) and also since actions may occur in parallel in the system (e.g. agents may decide to interact with each other or with the player). We implemented the following interruption triggering algorithm:

1) Play session starts 2) Wait 'x' seconds 3) Wait until all User-Initiated Actions (UIA) are completed 4) Watch for new UIA 5) Wait until UIA is completed 6) Wait for system dialogs to end 7) Wait 'y' seconds 8) Is there any active UIA? YES) Go to 6 9) Trigger interruption, pausing the system 11) Has the interruption appeared 'z' times? NO) Go to 2 12) Wait for 't' seconds 13) End

In step 5, the algorithm waits for all User-Initiated Actions (UIA) to finish so the user can see the outcome of their actions prior to each interruption. User Interface. We consider that the UI design should meet the following characteristics: 1) Users can observe the state of their session while the questions are asked, so they feel they are in the system (non-masking interface), and 2) The elements giving questions, as well as users' means of answering, should mimic the existing system to ensure continuity between system and sampling.

4.3

Questionnaire Contents Continuation Desire. The Engagement Sampling Questionnaire (ESQ) [START_REF] Schoenau-Fog | Hooked! -Evaluating engagement as continuation desire in interactive narratives[END_REF] was presented in each interruption: 1) "Do you want to continue the experience?" (Likert scale), 2) "What makes you want/not want to continue the experience?", and 3) "What do you want to do next in the game?". The last question is relevant from a narrative perspective since it assesses the audience's anticipatory behavior, a core property of narratives [START_REF] Eco | Lector in fabula: Le rôle du lecteur ou la Coopération interprétative dans les textes narratifs[END_REF].

Affective state. Users reported their affective state during each interruption using the Self-Assessment Manikin (SAM) [START_REF] Bradley | Measuring Emotion: The Self-Assessment Semantic Differential Manikin and the[END_REF], a validated cross-cultural affective assessment tool in three dimensions (Valence, Arousal, Dominance). Chosen for its fast learning curve and rapid reporting, users were shown SAM in the preliminary phase of the study. The affective results are not in the scope of this article.

Procedure

Upon being informed about the financial compensation and accepting the research consent, participants started the experiment by being redirected from the scientific crowdsourcing website to the study website. The procedure was:

1. Training questions (Interruption group): SAM is introduced and explained. 2. Anticipation: 1) "I want to begin playing" (Likert scale). 2) "What makes you want or not want to begin playing?" (freeform text). 3. The experience is presented in full. When the interruption procedure has run its course or time is reached for the Control group, the post-questionnaire is presented. 4. Post-game question.: "I want to try the game again" (Likert scale). 5. Regarding the interruptions (Interruption group): "The in-game interruptions spoiled my gaming experience" (Likert scale), "Overall, filling out the interruption questions was:" (Very difficult -Very easy, Single Ease Question)

We used seven-point Likert scales since they provide a good balance between degrees of freedom and ease of use. Participants received the compensation at the end.

Results

Interruption distribution:

From the start of the experience to Interruption 1, two clusters were found: C1, size = 20, 5.63 minutes elapsed in average (SD=51.32 s), results interval [222, 400] s, and C2 2, size = 15, 8.50 minutes elapsed in average (SD=63.9 s), results interval [426, 650] s. Difficulty: The perceived difficulty filling out the interruption questionnaire was calculated using the Single Ease Question, scoring 5.79 (Moderately easy) in average (SD=1.12). The calculated 95% Confidence Interval ranged from 5.42 to 6.16, where 4 corresponds to Neutral, and 7 to Very easy.

Replayability desire: The Control group scored 3.07 whereas the Interruption group 2.82 (Disagree a little) in average. An independent two-tailed t-test was performed, finding no significant difference (p = 0.536). The calculated Cohen's d is 0.1362, so the effect size is very small.

Discussion and Conclusion

The main objective of this article is to assess the disruptiveness of the proposed approach for measuring Continuation Desire (an indicator of the presence of engagement) within the Interactive Digital Storytelling domain. A comparative experiment was made using the IDS system "Nothing For Dinner" with an Interruption group (N=35) and a Control group (N=42), each session lasted around 23 minutes.

The obtained results indicate that the interruption triggering system performed consistently, spreading the interruptions evenly over the entire session. More importantly, it was found no major perception of disruptiveness when using the proposed approach for the sampling of Continuation Desire. In terms of replayability, we found no statistically significant effect when introducing the interruptions. The content of the interruption questionnaire was not analyzed in this article, it remains the basis for future work assessing its usefulness compared to existing evaluation approaches. These findings support the notion that in-experience sampling could be performed without creating major impact on users. It remains our long-term goal to advance the evaluation of IDS, for which we consider this study a first successful step.

Figure 1 .

 1 Figure 1. Elapsed time between Question Rounds in seconds. Next, 97% of Interruption 2 triggered around 2.90 min in average (SD=55.52 s) interruptions being evenly distributed around 94 and 261 s after the first interruption. Finally, 94% of Interruption 3 triggered around in average 2.54 min after Interruption 2 (SD=51.73 s), uniformly spaced around 85 and 227 seconds. Disruptiveness: Participants in the Interruption group had a neutral opinion of interruptions having spoiled their experience, with an average of 4.11 (SD=1.61), 95% Confidence Interval ranging from 3.57 to 4.63 (4 = Neutral, 5 = Agree a little).Difficulty: The perceived difficulty filling out the interruption questionnaire was calculated using the Single Ease Question, scoring 5.79 (Moderately easy) in average (SD=1.12). The calculated 95% Confidence Interval ranged from 5.42 to 6.16, where 4 corresponds to Neutral, and 7 to Very easy.Replayability desire: The Control group scored 3.07 whereas the Interruption group 2.82 (Disagree a little) in average. An independent two-tailed t-test was performed, finding no significant difference (p = 0.536). The calculated Cohen's d is 0.1362, so the effect size is very small.

http://nothingfordinner.org/

https://prolific.ac/


Quentin Gyger
email: quentin.gyger@etu.unige.ch

Nicolas Szilas
email: nicolas.szilas@unige.ch

A Taxonomy of Synchronous Communication Modalities in Online Games

Keywords: synchronous communication, taxonomy, online video game, game design, communication modalities, chat, voice over internet protocol, emote, predefined message, indicative signal

This paper discusses the benefits, limits and specificities of communication modalities in online video games and proposes a corresponding taxonomy. This taxonomy could serve as a guideline for game designers who want to integrate communication into the game mechanics.

Synchronous Communication: Different Modalities

Online games have a social component by definition and since any social act requires communication, it's essential to understand how communication is addressed in online video games. This communication has different topics [START_REF] Peña | An analysis of socioemotional and task communication in online multiplayer video games[END_REF], dispatched to various modalities [START_REF] Jensen | The effect of communication modality on cooperation in online environments[END_REF] that have their own benefits and limitations. Communication naturally occurs when it comes to achieve the game's goals, however, there is a greater amount of socioemotional discussion [START_REF] Peña | An analysis of socioemotional and task communication in online multiplayer video games[END_REF]. Victory and defeat being inherent to game's components, knowing how to communicate with different modalities in cooperative and competitive team games is essential. Even though the trend for online games is Voice over Internet Protocol (VoIP), there is still massive written communication in chat [START_REF] Suznjevic | Hack, Slash, and Chat: A study of players' behavior and communication in MMORPGs[END_REF][START_REF] Chen | Communication, coordination, and camaraderie in World of Warcraft[END_REF] because of its own benefits such as persistence or the ability to easily follow who said what in a conversation [START_REF] Smith | Voice-to-text chat conversion for remote video game play[END_REF]. Based on the analysis of most popular games [START_REF]Most Popular Core PC Games | Global[END_REF] and additional independent games, we found that the communication modalities mainly present in online games consists of either a written chat, a VoIP (integrated or not in the game), emotes (messages often accompanied by an animation, performed via a command), or predefined message (PM) for basic exchanges (i.e. say hi, good game, have fun). In several games there are also indicative signals (IS), which are alerts (in the form of different signals) used to provide to the other players some pieces of information. Two common IS are pings which are "a combination of animation and sound indicating a point of interest" [START_REF] Wuertz | Why Players use Pings and Annotations in Dota 2[END_REF] and freely drawing line in a gamespace such as annotations in Dota 2 [START_REF] Wuertz | Why Players use Pings and Annotations in Dota 2[END_REF]. Pings could be more elaborate than just a general indication, such as the Smart ping 1 in League of legends [START_REF] Chipteck | [END_REF] which allows to make specific announcements. In addition, some of these modalities can be mixed, but they generally just coexist in games, according to the kind of game, the players' needs and the evolution of the games. Indeed, they can greatly influence the ability of players to communicate, to spend good time in games and reach their goal in an optimal way. Therefore, these modalities are important in game design.

Communication at Different Levels

In online games, the conversation may be diegetic or extradiegetic, depending on whether its elements transgress or not the fiction [START_REF] Bordwell | Narration in the fiction film[END_REF] established in the game. Diegetic conversations, also called in character (IC) or in role-play (IRP) conversations, place the players in their character's role. Extradiegetic conversations, also called out of character (OOC) conversations place the players as themselves playing a game. A modality of communication like in-game chat or VoIP (inbuild or independent) can be used as much for diegetic or extradiegetic discussions [START_REF] Nevelsteen | A Survey of Characteristic Engine Features for Technology-Sustained Pervasive Games[END_REF]. However, depending on the context of the game and the place given to the role-play, some modalities can take the player out of the game [START_REF] Smith | Voice-to-text chat conversion for remote video game play[END_REF]. For example, if a young woman plays the role of an old wizard, voice communication can disrupt the other players' immersion. But the same modality embedded in the gameplay can be very immersive too, for example in DayZ (a 3D survival FPS game), the voice comes directly from the character and the player can speak for his or her character. Integrating communication as an element of game design in itself can allow the player to stay in the flow [START_REF] Csikszentmihalyi | Flow. The Psychology of Optimal Experience[END_REF], but an extradiegetic discussion does not necessarily leave the player out of the flow. Indeed, in competitive games, players often communicate as players controlling a character and don't argue in role-play, but their discussions remain at the same diegetic level and a roleplay communication could appear discordant. Therefore, the communication must be thought of in such a way that players can remain at the desired diegetic level or can travel to the different levels at the same time.

A distinction is necessary at this level: such a correspondence regarding the diegetic level does not mean that one should not make use of metalepsis, which introduces extradiegetic into a diegetic narration [START_REF] Genette | Figures III[END_REF]. Indeed, the use of mastered metalepsis has interesting virtues, including in serious games [START_REF] Allain | Exploration de la métalepse dans les" serious games" narratifs[END_REF]. The assumption here is that back and forth switches between different diegetic levels must be followed by the different players to maintain a consistent communication and keep them immersed in the game. Moreover, role-players frequently navigate back and forth between their IC and OOC status, and sometimes they even do meta-gaming (introduce OOC knowledge in their role-play).

1

"Smart Ping is a radial menu that includes four alerts [(danger, on my way, assist me, enemy missing)]. These can be used to communicate with your team faster than taking the time to type and [break the] flow of gameplay" [START_REF] Chipteck | [END_REF]. These alerts are displayed with sound and their own visual effect and are visible to teammates on the terrain and the minimap.

Benefits and Limits of Communication Modalities

All communication modalities have inherent advantages and limitations. The benefits and limits of the different communication modalities discussed in this section are intrinsic or technical characteristics and do not constitute an exhaustive list, but they highlight important particularities that are used in game design to reinforce specific aspects of a game.

A chat is a common feature in online games, it allows players to freely converse and this modality of conversation is easy to follow, because the messages remain displayed in sequence along with the utterer's name. In some games, conversations can also be displayed on top of the characters to add role-play immersion. The VoIP feature is very widespread, because it allows players to express themselves instantly without writing on a keyboard, which can provide an effective communication (without interrupting the gameplay). The predefined message mostly have a socioemotional role, but can also be task-oriented. They allow to exchange basic messages, essentially courtesy, without the need to moderate the words of the players. They are usually limited in number but quick to use like in Hearthstone 2 [14]. They can also be displayed in a radial menu such as Communication wheel in Overwatch [START_REF]Communication Wheel[END_REF] which offers eights communication's options and also allow to communicate on specific goals of the game (i.e. say group up, need healing). The emote feature consists of prerecorded sentences (written in or spoken by the character) accessible to other players around the player character in the game (i.e. John thanks you for your help, John is pointing at you). This allows role-play players to take action on their characters. Some emotes have not only a written or spoken component, but also a visual animation (i.e laughing, dancing, drinking) performed by the character. They are usually enabled by clicking on shortcuts or typing keywords in chat (i.e. /dancing). Among IS, the ping feature has mainly a strategic communication function: it allows to quickly communicate information about some game's specific objectives. The information they give can be contextual if the player pings a specific object (i.e. tower, opponent or ally players) [START_REF] Wuertz | Why Players use Pings and Annotations in Dota 2[END_REF]. The drawing feature can also have the same purpose without the contextual part [START_REF] Wuertz | Why Players use Pings and Annotations in Dota 2[END_REF] and allow to give information about a direction or movement, they can also be used for socioemotional communication (i.e. drawing a smiley or congratulations words).

The benefits and limitations of these modalities will be assessed across five dimensions (see Fig. 1). The first dimension is called gameplay coexistence and represent the players' ability to communicate without interrupting their actions in the game. Multicommunication describes the possibility given to the player to follow multiple conversations through the same modality at the same time (without overload of the channel). Persistence represents the time interval during which the message will remain visible before disappearing. Expressive power represents the varieout restriction. Multilingual is the possibility to communicate with people who speak 2 "short quotes that heroes can [utter]during a game. Each emote comes as a sound bite, and in a written form, displayed in a speech bubble next to the character's portrait." [START_REF] Allain | Exploration de la métalepse dans les" serious games" narratifs[END_REF]. They're accessible by right-clicking on the portrait. another language without personal knowledge of this language (online games can be played in different countries). Figure 1 displays the characteristics of modalities according to these five dimensions.

Modalities of Communication: Towards a Taxonomy

The purpose of this section is not to define an exhaustive list of the modes of communication present in every kind of online games, but to give a current overview of communication in online video games [START_REF]Most Popular Core PC Games | Global[END_REF] to inspire the game designers. Table 1 below is a proposal for classifying these different modalities using their characteristics (see Fig. 1) and characteristics of communication useful to cooperate, have fun or achieve goal in game. For this reason, we introduce in the taxonomy the diegetic aspect described in the Section 2 and the socioemotional and task-oriented function. Socioemotional messages can be positive or negative and include message that show solidarities, tension relief, agreement, disagreement, tension or antagonisms. Task-oriented communication are asking for or giving an opinion, suggestion or task orientation [START_REF] Peña | An analysis of socioemotional and task communication in online multiplayer video games[END_REF]. Theses function are essentials for the communication of online games and can be reached with different modalities.

In online games such as Massively Multiplayer Online Games (MMORPG), Multiplayer Online Battle Arena (MOBA) and First-Person Shooters (FPS), to name a few, synchronous communication can be written in a chat or vocal when it occurs in inbuilt or independent VoIP (i.e. Discord, Ventrilo, Teamspeak). These competitive games may also contain IS (more or less elaborate) that could be visual and/or audio indications to communicate game's tasks through HUD (head-up display) alerts. Recent successful FPS like Fornite Battle Royal seems to prioritize vocal and PM through a radial menu, but a chat could also fulfill a valid communication function, as the modality and its restrictions would be the same for all players [START_REF] Herring | Fast talking, fast shooting: Text chat in an online first-person game[END_REF]. Other games without teamplay like Collectible Card Game (CCG), racing or fighting video games could prefer written PM to fulfill a basic socioemotional or task-oriented communication function. Games with Role-Playing (RPG) components could have emotes to give an immersive dimension and encourage players to have diegetic conversations. As we have seen above, the modalities have some intrinsic characteristics. Indeed, their persistence could be low as in VoIP and IS or high as in chat. Multicommunication could be described as the possibility for many people to communicate without saturating the discussion channel (i.e. many people can chat at the same time, while few can talk (VoIP) at the same time because channel's overload). The gameplay coexistence is an important component based on the ability of players to communicate without interrupting their action in the game. Especially in fast games, efficiency could be crucial in which case it's important that communication coexists with the gameplay. There are also some modalities' technical characteristics like the Expressive power, denoting the player's ability to communicate freely. It should be noted that a limited communication is not necessarily bad, because it could provide great moderation and the possibility to easily translate messages, which allows the game to be easily multilingual.

Conclusion

Communication is an important component of game design, as it may reinforce game's qualities such as immersion, coordination, efficiency or sociability of players. In this article, after having identified 6 main synchronous communication modalities, we proposed a taxonomy of such modalities according to a series of dimensions and the communication's characteristics. This taxonomy could serve as a guideline for game designers who want to optimally integrate communication into the gameplay. In addition, the taxonomy proposes a design space in which some regions are unexplored in current online games, which may suggest new communication techniques to be invented in the future.

Fig. 1 .

 1 Fig. 1. Communication modalities' characteristics. This graphic has been made in relation to the intrinsic and technical properties of the different modalities and could be different depending on the context or the modalities' design.

Table 1 .

 1 Taxonomy of synchronous communication in online video games. Modalities characteristics are classified by levels (L = low, M = medium, H = high).

				Synchronous communication in online games
			Communication Characteristics			Modalities Characteristics
		Sign-system Verbal Non-verbal Written Vocal Visual Static Visual gestures Audio	Function Task-oriented Socioemotional	Diegetic level Diegetic Extradiegetic	Gameplay Coexistence	Intrinsic Multicommunication	Persistence	Technical Expressive Power Multilingual
	Chat	x			x	x	x	x	L	H	H	H	L
	VoIP		x		x	x	x	x	H	L	L	H	L
	PM	x			x	x		x	H	M	L	L	H
	Emote	x	x	x		x	x		M	M	L	M	H
	Ping		x	x	x			x	H	M	L	L	H
	Drawing			x	x	x		x	M	M	L	M	H

Table 1

 1 doesn't constitute an absolute taxonomy of synchronous communication in online games, because some characterizations could be changed depending on the context or integration of the modality. Instead, Table1illustrates the common use of the communication modalities in online games.


B Busson

L Dalstein

Nonlinear Optical Response of a Gold Surface in the Visible Range: A Study by Two-Color Sum-Frequency Generation Spectroscopy. Part II: Model for Metal Nonlinear Susceptibility

We present a modeling of the nonlinear optical response of a metal surface in order to account for recent experimental results from two-color Sum-Frequency Generation experiments on gold. The model allows calculating the surface and bulk contributions, and explicitly separates free and bound electron terms. Contrary to the other contributions, the perpendicular surface component is strongly model-dependent through the surface electron density profiles. We consider three electron density schemes at the surface, with free and bound electrons overlapping or spilling out of the bulk, for its calculation. The calculated SFG signals from the metal rely only on bulk quantities and do not need an explicit definition of the density profiles. In the particular case of gold, when the free electrons overlap with the bound ones or spill out of the bulk, the free electron response completely dominates through the perpendicular surface terms. When the bound electrons spill out, the situation is more balanced, still in favor of the free electrons, with lower amplitudes and different dispersion lineshapes. As for silver, the free electron contributions dominate, and the calculated slow amplitude growth from blue to red follows the experimental trends.

I. Introduction

In a previous paper 1 , we have measured the experimental effective nonlinear susceptibility of a polycrystalline gold film in an infrared-visible Sum-Frequency Generation (SFG) experiment while tuning the visible wavelength over the visible range (435-705nm) (two-color Sum-Frequency Generation, 2C-SFG). Using the vibrationnally resonant response of an adsorbed thiol monolayer, we could extract the absolute phase and amplitude lineshape of gold in these conditions. In this paper, we propose to model the nonlinear response of metals, while explicitly accounting for free and bound electron contributions, as the base for a simulation of the experimental data.

Such models have been extensively develop ped for second-harmonic generation (SHG), and far less for SFG. They consist of two parts: the first one phenomenologically describes the generation process itself, determining the sources of nonlinear polarization and deriving the expressions of the generated electromagnetic waves, both at the interface and in the far field, they give birth to. The second part establishes the link between a solid state physics description of a metal bulk and surface, and the nonlinear coefficients introduced in the phenomenological part.

Phenomenological models describing the nonlinear optical response at the interface between two materials appeared soon after the discovery of nonlinear optics 2 . The formulations for the SHG nonlinear polarization produced in reflection by a surface and for the electric fields radiated in the far field were derived by several authors [3][4][5][6][7][8][9] , and the essentials are summarized below. The nonlinear sources decompose into bulk and surface terms, even if part of the bulk response also formally contributes to an equivalent surface dipolar source, making it difficult to experimentally separate intrinsic bulk and surface contributions 6,10,11 . Over time, several refinements have improved the description of the nonlinear processes at the surface of a centrosymmetric bulk: an unambiguous definition of the quantities involved (polarization sources, nonlinear susceptibilities) has been derived 4,11 ; a rigorous separation between surface and bulk contributions, each assigned to a different Green function formalism for the generation of the second harmonic electric field 5 has been established; symmetry rules have been applied to both surface and bulk contributions, reducing the number of independent susceptibility coefficients, in particular for isotropic and cubic bulks 9,12,13 .

As for SFG, emphasis has early been put on its specificity in the infrared-visible configuration, namely the molecular response 14 adsorbed on a substrate. The substrate contribution is often reduced to a vibrationnally constant term, either used as a support for the interference with the molecular contribution for phase analysis 15,16 , or on the contrary suppressed by pulse shaping and time-delay methods 17 . The phenomenological description of SFG of the surface and bulk from a material has therefore been given much less attention than in SHG, except in specific cases for which the bulk contribution is important like chiral liquids 18 , quadrupolar SFG response 19 or for a clear separation between the bulk and surface contribution 20 . It has anyway benefitted from all the knowledge gained for the SHG case. For an isotropic system, the basic equations may be found in Ref. 2 and 21. We note that the form of the bulk contribution is much more complicated than for SHG.

For the analysis of experiments performed with fixed incoming wavelengths, or far from resonant phenomena, the phenomenological model may be sufficient provided that constant values are assigned to the various χ (2) components, extracted from the data recorded as a function of an external parameter (for example the polarization states of light 10,22 , angles of incidence 23 , and azimuth angle 24 ). For a spectroscopic study [23][24][25] , one must take dispersion into account as the various nonlinear susceptibility tensor components will vary with the incoming wavelengths, whereas simple rules (e.g. Miller's rule) do not apply in resonant conditions. Such studies require modeling the sources of the SHG and SFG as a function of the wavelengths and material properties. For metals, the source of nonlinearity is the electrons, separated in conduction (free) and valence (bound) electrons as a consequence of the band structure. For SHG, nonlinear coefficients stem from electronic properties through the so-called Rudnick-Stern coefficients 10,[25][26][27] , which reduce the formalism to three dispersive numbers a, b and d (standing for surface perpendicular, surface parallel and bulk components, respectively) 4,26 . Most of such analyzes of the origin of χ (2) devote to the free electron response only, for which the R-S coefficients were originally estimated close to unity. This is well suited for some metals (e.g. aluminum and silver 25 ) but of limited use for gold in the visible range as bound electrons contribute to the optical properties of gold above 2eV through interband transitions. A simple free electron model has been shown inadequate to describe the nonlinear response of gold in the visible range 25 . Even for the free electron gas, some resonances as a function of the frequency are expected, mostly in the high energy range towards the plasma frequency of the metal 28,29 . Some authors have analyzed separately the interband contributions 9,27,30,31 , but there is no consensus on whether they participate to SHG through an additional contribution to χ (2) or only by modifying the linear dielectric function 27,30,32,[START_REF] Liebsch | Electron. Excit. Met. Surfaces[END_REF] . Other groups use density functional theory 28,[START_REF] Weber | [END_REF] or a generalized Thomas Fermi model account at the same time both free and bound electron contributions and their interaction in nonlinear optics still remains a challenge. Using a long history in modeling first the electromagnetic fields at surfaces 36,37 , then the nonlinear SHG and SFG responses of electrons in metals 7,38,39 , one group proposed a way to implement the complex electronic structure of transition metals in the interband transition regime into the Rudnick-Stern coefficients 40 .

In this paper, we evaluate the nonlinear responses of gold and silver in a 2C-SFG experiment in their theoretical frame 38,40 in order to compare them to the experimental data in a forthcoming step. The metal is described by its free and bound electron densities and contributions to the total dielectric function. We investigate three schemes for the nonlinear optical response, which differ by the relative positions of their electron density profiles as a function of depth and the associated coupling between free and bound contributions. We show that the free electrons dominate the total SFG response, but that the amplitude and dispersion of the effective nonlinear susceptibility of gold, and the weight of the bound electron terms, strongly depend on the profile chosen.

II. Origin and calculation of the surface and bulk contributions to SFG

The phenomenological description is reviewed in Appendix A. In order to apply it, we must evaluate the bulk polarization, and the surface parameter χS after an additional integration step over z. This requires to explicitly model the electron density of the metal (i.e. its material properties) and link it to its optical response. We consider the metal as a continuous assembly of dipoles described by the classical model for the displacement of an electron, bound to its rest position by a harmonic force and a submitted to both a resisting force and the time-dependent electromagnetic field of light 38,40 . This harmonic oscillator approximation has been shown consistent with more accurate descriptions of SHG 7 and allows a calculation of the contribution to the material nonlinear response. Such a description may constitute a good approximation for a bound electron, for which a full quantum mechanical calculation would however rigorously be more appropriate 30 , or a free electron when the restoring force vanishes. Solving the equations for the displacements in a Taylor expansion at second order in the electric fields and keeping only the ω3 term gives the expression of the nonlinear polarization PB(r,ω3). The free electron case may also be recovered by considering the hydrodynamic equations for the velocity field of the free electron gas, while neglecting the pressure term 9,21,38 . Refined theories exist which explicitly take this extra term into account 4,39,41 . Few examples exist of a theory comprising both free and bound electron contributions with no ad hoc or unknown parameter included 9,30,31 . The metal is described as the superimposition of two electron populations, a free electron gas (density n0 f ) and a bound electron density (n0 b ), with n0 f + n0 b = n0. The gold interband contribution from the bound electrons shall not be neglected when considering that, for a threshold around 2eV 42 , the SFG beam in our experiments is always able to excite them. For silver, the energies involved do not allow to directly excite the interband transitions, but the bound electrons still contribute to the total dielectric function. This implies that the off-resonance approximation sometimes made in the presence of interband transitions cannot apply here, at least for gold 27,30 . Starting from here, all the quantities defined in the phenomenological model have to be split between free and bound electron terms when appropriate, with the corresponding superscripts. The bulk dielectric function of the metal is therefore written as 40 :

  f f b b i i i 0 i 0 1 4 n 4 n          (1)
where αi f (resp. αi b ) represent the electronic polarizabilities of the free (resp. bound) electrons at frequency ωi, and separated into the two components by εi -1 = (εi f -1) + (εi b -1), where ε -1 represents the resonant part of the dielectric functions. The polarizabilities quantify the amplitude of the electronic response to the excitation by light, and the second order nonlinear response is therefore proportional to a product of two electronic polarizabilities. As the free and bound electronic polarizabilities differ, their contributions to the SFG signal must be evaluated separately, then summed up, immersed in a medium which dielectric function encompasses free and bound parts together.

The bulk polarization is obtained by plugging the electric fields, considered as plane waves with wavevectors k1 and k2 propagating inside medium 2, into the nonlinear polarization:

    i i . i i i , e    k r E r E r E , i=(1,2) (2)
The complex wavevectors are evaluated by continuity of the component parallel to the surface and with the help of the dispersion relation of the metal (ki)²=εi ωi²/c² for the perpendicular one. The field gradients in the bulk arise from the propagation of the plane waves through their phases. This leads to:

            1 2 i . f /b f /b f /b B 3 1 1 1 2 1 1 2 1 f /b f /b 2 2 1 2 2 2 1 2 , ie D . . D . .            k k r P r k E E k E E k E E k E E (3a) with       f b B 3 B 3 B 3 , , ,      P r P r P r (3b)
where coefficients Di f/b and Δi f/b are given by equations ( 45) to (47) of Ref. 38 for which the electron density and dielectric functions refer respectively to free and bound electrons. We recall their expressions in Appendix B. When working with a single crystal, additional terms, which depend on the crystal orientation at the surface, should be added to account for the bulk anisotropic contribution 5,13 . We note that D1 and D2 have a purely dipolar origin, as well as Δ2 when the approximation ω2<<ω3 is valid. The electric field generated by this bulk polarization follows a complex expression recalled in Ref. 38, equation (A9). This equation may be tracked back through Ref. 21 to the original work of Bloembergen and Pershan 2 .

The electric field gradients giving birth to the surface terms have a completely different origin. They arise in the z direction from the rapid variations of the electron densities when crossing the interface. The long wavelength approximation ensures continuity of the parallel components of the electric fields and the perpendicular ones of the displacement fields along the electron density gradients 43 , and validates the dropping of the terms related to field curls in the dipolar term of the nonlinear polarization in the selvedge region 38 . In order to evaluate the surface terms, the local nonlinear polarization is calculated from equation (A1) as a function of z, while both the electron densities and the dielectric functions

f f b b i i i (z) 1 4 n (z) 4 n (z)       (4)
become z-dependent in the interface region 43,44 , then integrated between external medium and bulk as usual 5,[START_REF] Heinz | Nonlinear Surf. Electromagn. Phenom[END_REF] to calculate the surface nonlinear susceptibility components. Equation (A1) becomes:

    f / b f / b f / b f / b f / b f / b f / b B 3 3 3 1 1 2 2 2 1 f / b f / b f / b 1 2 1 2 2 1 f / b f / b f / b 2 1 1 2 1 2 3 n (z) z n (z) ( ) e 1 n (z) 2e 1 n (z) 2e                      P E E E E E E E E E E E     (5)
where the first term is the depolarization part of the SFG field. Each line in equation ( 5) relates to the dipolar, quadrupolar and magnetic contributions, respectively, and the gradients now only relate to the z-direction. As a consequence, the dipolar (resp. magnetic) contribution does not appear in the parallel (resp. perpendicular) component of the nonlinear polarization.

The origin of the dipolar term is different from the quadrupolar and magnetic contributions. For the former, direct coupling of one incoming electric field and one electric field gradient generates a polarization at the SFG frequency proportional to the SFG polarizability α3. For the latter, the coupling of the incoming electric fields generates local quadrupoles and magnetic dipoles, which gradients take part of the total polarization. This accounts for their proportionalities to polarizabilities at the incoming frequencies only. This has major consequences in the IR-visible SFG case applied to metals like gold and silver, which may be qualitatively described by ω2<<(ω1,ω3) and |α2 b |<<|α2 f |. As will be checked below, we therefore expect that the magnetic and quadrupolar contributions to the bound electron polarization should be negligible compared to free electron terms. This implies that, for bound electrons, the total parallel surface contribution is also small, whereas the bulk and perpendicular terms are from dipolar origin only. As for the free electron contributions, it has already been noticed by Petukhov 21 that the Δ1 and Δ2 terms almost vanish for a free electron gas, leaving D1≈D2 as leading terms. As a consequence, it appears that the full bulk contribution (i.e. for free and bound electrons) is purely dipolar under reasonable approximations. This contributes to simplify the modelling of the IR-visible SFG bulk processes in metals.

The free (resp. bound) electron component of the polarization parallel to the surface f B,x P (z) (resp.

b B,x P (z) ) becomes:

f f f f f 1 2 2 1 2 1 B,x 2 1x 2z 1 1z 2 x 3 2 3 1 n (z) n (z) P (z) 1 E E 1 E E 2e z (z) z (z)                                           (6)
The difference between ω1 and ω2 induces a partial amplitude transfer from xzx to xxz terms, as a consequence of the interference between magnetic and quadrupolar contributions. Integration according to equation (A2) is straightforward, and the xxz and xzx components therefore do not depend on the electron density profiles across the interface 40 . Integral leads to the following expressions for the xxz and xzx components of χS :

f / b f / b f / b f / b 1 2 1 S,xxz 1 2 f / b 3 0 1 12 1 b ( , ) 4 4 2n e             (7a) f / b f / b f / b f / b 1 2 2 S,xzx 2 1 f / b 3 0 1 12 1 b ( , ) 4 4 2n e             (7b) f b 1 2 1 S,xxz S,xxz S,xxz 1 2 0 3 1 12 1 b( , ) 2n e 4 4                (7c) f b 1 2 2 S,xzx S,xzx S,xzx 2 1 0 3 1 12 1 b( , ) 2n e 4 4                (7d) with the simple values b f/b (ω1,ω2)= b f/b (ω2, ω1) = -1.
The last two equations define the overall χS components by adding the free and bound electron contributions. Care must be taken as global parameters b(ω1,ω2) and b(ω2,ω1) now differ from -1.

On the contrary, because of the dipolar contribution, integration of the perpendicular component (zzz) has to be explicitly performed. No term in the selvedge nonlinear polarization arises from two electric fields parallel to the surface, leading as usual to a vanishing zxx contribution to χS 21,38 . For the zzz term, the z-components of the selvedge nonlinear polarizations induced by the free (resp. bound) electrons P f B,z(z) (resp. P b B,z(z)), evaluated inside the metal, are

f f f f f 1 2 3 1 2 B,z 3 3 1 2 2 1 f f f 1 2 1z 2z 1 2 1 1 P (z) n (z) e (z) (z) z (z) (z) z (z) 1 1 n (z) E E z ( z ) ( z )                                           (8)
where the quantities not explicitly depending on z are the bulk ones, and one has to integrate them as in equation (A2). Following equation ( 4), the integrand becomes a function of n f (z) and n b (z) only, which are supposed continuous in the following. In order to actually calculate these zzz terms, we therefore must make some assumptions on these quantities.

We consider the three extreme cases for the electron distributions: two split electron distributions (Figure 1(b),(d)) and an overlapping electron model (Figure 1(c)). In the first scheme (s1, Figure 1(b)), the free electrons spill out from the bulk whereas bound electrons don't, so their respective electron density gradients do not overlap as a function of depth. It is known that, for a free metallic surface, the free electrons may spill out of the bulk (over a depth far below one nanometer) [START_REF] Liebsch | Electron. Excit. Met. Surfaces[END_REF]43 , thus enhancing their contribution to the surface terms. We thus consider that the free electron density as a function of depth varies from zero to its bulk value n0 f (reached for z=z0) while n b (z)=0, then remains constant for z<z0 while the interband electron density raises from zero to its bulk value n0 b (Figure 1(b)). For a complete analysis, we may consider two additional configurations: in scheme 2 (s2, Figure 1(c)), the free and bound electron distributions overlap as a function of depth, whereas in scheme 3 (s3, Figure 1(d)), the bound electrons spill out more than the free electrons symmetrically to scheme 1. As stated above, the model chosen for the electron density profiles has no influence on the bulk and parallel surface terms. Consequently, equations ( 3) and ( 7) remain unchanged for schemes 1, 2 and 3 and we concentrate in the evaluation of the perpendicular zzz term.

Integration of the z-component of the nonlinear polarization in scheme 1 leads to a sum of three contributions: a pure free electron (a f ) one arising from the surface above z0; a free electron contribution induced by the variation in the bound electron density across the interface (a bf ); a pure bound electron contribution (a b ) screened by the bulk free electron density:

f f s1,f f bf 1 2 S,zzz 1 2 3 s1 1 2 s1 1 2 f f f 0 1 2 1 1 1 a ( , ) a ( , ) 2n e 4 4                     (9a) b b s1,b b 1 2 S,zzz 1 2 3 s1 1 2 b b b 0 1 2 1 1 1 a ( , ) 2n e 4 4              (9b) and s1 s1,f s1,b 1 2 S,zzz S,zzz S,zzz 3 s1 1 2 0 1 1 1 a ( , ) 2n e 4 4               (9c)
defines the global parameter as1, with

      f f f f j 3 1 2 f f s1 1 2 i f f f f f f f {i, j,k} k 1 2 2 3 3 1 c.p.of {1,2,3} 1 a ( , ) 2 1 ln 
                                      (10a)
where c.p. denotes circular permutation of {1,2,3},

f f f f b b 3 bf 1 2 3 3 i i s1 1 2 0 f f f f i 1 3 1 2 3 i i i 1 1 1 2 a ( , ) 1 I 1                                      (10b) f f f b b b b 3 b 1 2 3 3 1 2 i s1 1 2 1 f f f f f i 1 1 2 3 1 2 3 i i 1 2 1 a ( , ) I                                     (10c)
The technical details of the integration steps, along with definitions and properties of I0 and I1 integrals can be found in the Appendix C. The result is identical to Ref. 40, while we correct here some original confusion in the attribution of each term. For scheme 3, the results are identical provided that superscripts b and f are exchanged in equations ( 9) and (10).

In scheme 2, the integration is performed in the same way as above, leading to the new values for the free electron and interband contributions to parameter as2.

f f s2,f f 1 2 S,zzz 1 2 3 s2 1 2 f f f 0 1 2 1 1 1 a ( , ) 2n e 4 4                  (11a)           f f f f f f f j 3 1 2 f f 1 2 3 s2 1 2 i f {i, j,k} 1 2 3 1 2 2 3 3 1 k c.p.of {1,2,3} b 3 f f f i 3 1 2 1 i f i 1 i 1 a ( , ) 2 ln 1 1 I 1 1                                                                (11b)
For the interband contribution, one may simply exchange εi f and εi b in equations (11). We also may check that formula (11b) turns down to (10a) when the bound electron contribution vanishes (εi b = 1 and εi = εi f ).

We immediately see that the zzz contribution (as well as the parallel one for obvious reasons) depends neither on the spatial evolution of the electron density distributions (i.e. n f/b (z) are arbitrary continuous functions) nor on any intrinsic length characteristic of the interface, related for example to the thickness of the interface layer or to the spatial extent of the electron density gradients. For that respect, the present model allows by its simplicity to avoid the delicate definition of the density profiles, which has been shown to raise severe issues in the hydrodynamic model 35 . On the contrary, it only depends on the scale-free relative positions of the electron densities, and of course on their bulk values. This means that, in this modeling, the magnitudes of the gradients (of the electric fields and electron densities) are compensated by their spatial extents after integration into the surface nonlinear polarization. However, as we will see below, integration leads to huge differences between the various schemes, in particular as a consequence of self-and cross-screening between free and bound contributions.

In order to illustrate the results of the modelling of the nonlinear response of metals, we briefly present the results for the simple case of silver, then we focus on the more complex case of gold, for which a detailed analysis will be performed in the third part of this article.

III. The nonlinear SFG response of silver

The dielectric function of silver is tabulated from Yang et al. [START_REF] Yang | [END_REF] (sample C, corrected for Kramers-Kronig consistency). In order to separate the free and bound electron parts, we use their fits of the free electron behavior by a Drude model in the near infrared range, with parameters ωP=8.90eV and Γ=36.57meV.

          2 f P i i i 1 i (12)
In the visible range, the bound electron dielectric function follows: εi b = 1+εi -εi f for i= (1,3). It encompasses a contribution from ε∞, which we suggest to correct to 4 instead of the tabulated value of 5, in order to comply with the original data of Ref. . For the infrared beam, the remaining bound electron contribution is supposed real but unknown, we therefore performed our simulations with ε2 b =2 40 .

The various contributions to the effective χ (2) for silver are displayed on Figure 2. As may be anticipated, the effect of the interband terms is rather weak, especially at low energy, except marginally for scheme 3. The free electron contribution dominates, for scheme 1 and 2 through the zzz term, and for scheme 3 through the surface parallel and bulk terms. All contributions share a common phase within a 30° range, except for a sign change for the bound and cross terms in scheme 3. Figure 3 shows the total effective nonlinear susceptibility of silver in the visible range, for two infrared wavenumbers: 2900cm -1 (representative of studies in the CH stretch region) and 1000cm -1 (for the phenyl ring collective modes), under the hypothesis n0 b = n0 f . Schemes 1 and 2 compare very well, showing the expected free electron behavior, with a regular increase in amplitude from blue to red. Scheme 3 differs as a consequence of the more pronounced interference between the three free electron terms, which compensate each other to exhibit a rather flat total contribution. The relative amplitude between schemes 1 and 2 on one side and 3 on the other side varies from 2 to 5 from blue to red. The phases for the three schemes are the same. These curves do not change for a ratio Rbf=n0 f /n0 b varying from 0.1 to 1 (dominant free electron character). For the low energy infrared case, the amplitudes increase (for schemes 1 and 2) by up to 20 percent as a consequence of a higher infrared free electron polarizability, whereas the phases decrease by a low 8 to 13°. Comparing these curves to experimental data remains difficult because recording the absolute SFG response of a silver bare surface requires an accurate absolute calibration of the SFG spectrometer 25 and does not provide with the phase parameter. To our knowledge, there is no such study available in the literature. After adsorption of a molecular monolayer, data analysis allows to extract a phase parameter of the silver response and an amplitude ratio at a fixed or tunable visible wavelength. As for the amplitudes of the silver SFG signals, only a few SFG experiments tune the visible wavelength on such systems [47][48][49][50][51] and even fewer provide with useful parameters 47,49 . Using the same data analysis as in Ref. 1, i.e. by comparing molecular and metal SFG responses, the published data which explicitly quantify the amplitude of the silver contribution are then compatible with the present model under schemes 1 and 2 (i.e. with a dominant free electron zzz response), with a silver SFG contribution slowly growing in amplitude from blue to red as compared to the molecular one.

However, comparison to the present results remains difficult in a straightforward manner because the behavior shown in Figure 3 is then modulated by the molecular SFG response 1 . The latter strongly depends on the nature of the molecule adsorbed at the silver surface: its symmetries dictates on the nonvanishing molecular hyperpolarizabilities, and its orientation (tilt and azimuth angles) impact on the relationships between molecular hyperpolarizability and nonlinear susceptibility contributions 52 .

In addition, these are modulated by their own Fresnel factors, which introduce dispersion in the visible range both in amplitude and in phase, and by wavelength-dependent Raman polarizabilities, which may become resonant on the high energy side 50,51 . Finally, care must be taken that i) the silver surfaces under study are often chosen as single crystals (for which additional terms for the nonlinear susceptibility are required); ii) the results may vary between co-propagating and counter-propagating SFG geometries 53,54 , and iii) several studies are performed under electrochemical control, in which case the electrochemical potential modulates the amplitude and phase of the silver signals, in particular the free electron part 25,55,56 . Experimentally, the silver SFG response remains weak (contrary to gold), although experimental evidence may vary on this point 48,57,58 . Consequently, the uncertainties on the amplitude, and even more on the phase, extracted from curve fitting of the silver response become rather big (considering that the interference pattern between molecular and weak metal nonlinear contributions is less pronounced). It has also been shown, using the example of a thiophenol monolayer on silver 57 , that curve fitting may not lead to unique set of parameters 59 . We can indeed check in the literature that the fitted silver phase varies over rather wide ranges 48,50,53,57,[60][61][62] but, without a full molecular analysis predicting the phase of the vibrational peaks 1 , this parameter alone is not a useful probe of the present theory. In addition, many experiments involve molecules which are doubly resonant, or either electronically 63 or vibrationally 64 modified by adsorption on metal, which dismisses them as reliable molecular probes. There definitely lacks a dedicated study of an adapted molecule (e.g. long chain alkanethiol) adsorbed on polycrystalline silver, for which SFG spectra would be recorded and fully analyzed at enough wavelengths covering the visible and near infrared ranges.

IV. The nonlinear SFG response of gold

We now focus on the more interesting case of gold since there is a strong interference between free and bound electron contributions in the visible range. The dielectric function is tabulated from Olmon et al. 65 (template-stripped sample). In order to separate the free and bound electron parts, we fit the free electron behavior by a Drude model in the near infrared range and find parameters ωP=8.80eV and Γ=49.4meV. The bound electron dielectric function is evaluated as for silver, with ε2 b =2.

For the three schemes, the bound electron contribution is considered as a whole, even if the valence electrons fall into two families: two electrons in the band closest to the Fermi level (upper d-band) experience interband transitions in the visible and near-UV, while the eight other electrons lie in bands with lower energies 66 (lower d-bands). We could therefore split the bound electron contributions in two parts by considering the upper and lower d-bands as two distinct and interacting dipolium contributions. A refined model explicitly separating both contributions is currently under investigation. At this stage, for the simulations below, we follow the choice of Ref. 40 and use the arbitrary values n0 b = n0 f in order to illustrate the balance between free and bound electron terms.

A. Fresnel factors

The evolution of the amplitudes and phases of the Fresnel factors for zzz, xxz, xzx and bulk terms is presented on Figure 4. The absolute value for zzz is around a hundred times smaller than for xzx, and ten times smaller than for xxz. This stems from the fact that the ratio Fx(ωi)/Fz(ωi) is roughly proportional to the index of refraction of gold (ni). A second consequence of this is seen on the phase curves: at the low energy end of the visible spectrum, the influence of interband transitions becomes smaller and the refractive index of gold is roughly imaginary for the three beams. All Fresnel factors share at this point a common phase, except for a sign difference between Fxxz and Fxzx on one side, and Fzzz and Fbulk on the other side. From the low value of Fzzz, one may thus be tempted to neglect the contribution from the a term in the following. However, it has been shown by SHG that the weakness of the Fresnel contribution to zzz may be counterbalanced by the high value of the a source term itself 10,11 , thus producing perpendicular and parallel contributions comparable in intensity 24 . The situation is analogous in SFG, although Fresnel factors for xxz and xzx additionally differ as a consequence of the intense screening of the infrared field inside gold through its high dielectric function. It is interesting to note that the amplitudes, especially for zzz, show two maxima in the middle of the visible range, related to the excitation of the interband transitions of gold by the visible and SFG beams, respectively. More precisely, considering the formulas (A6) for x and z Fresnel factors, and taking into account the fact that cos(θi 2 ) is a complex number close to 1 with a small imaginary part, x and z contributions are maximum when 1+ni cosθi 1 (resp. ni (1+ni cosθi 1 )) is roughly minimum, that is when the modulus of the index of refraction (or the dielectric function) is minimum. This happens around ωi=500nm, the effect being more peaked for z because of the additional ni factor. This has already been noticed by Ref. 28 for SHG, but in our case this leads to a double peak corresponding to the two beams involved in the visible range. The behavior of the zzz contribution alone may account for the existence of the experimental maximum seen in Ref. 1. At this stage, taking into account the Fresnel effects, which was not done in Ref. 67, could cast some doubt on the influence of surface electronic states as postulated therein, and attribute the experimental maxima to a simple linear optical effect. However, only the full SFG response in the far field (i.e. the effective nonlinear susceptibilities) is measured, and we need to evaluate both the intrinsic χ (2) amplitudes and their Fresnel screening factors. In addition, the phase parameter gives an additional track to estimate the accuracy of the modeling. The surface xxz and xzx, as well as the bulk contribution, do not depend on the scheme. Figure 5 displays the dispersion of the corresponding effective second order susceptibilities in amplitude and phase for ε2 b =2. Components xxz and xzx roughly share the same amplitudes and, as expected from the above analysis, they solely originate from the free electrons. As for the bulk, free and bound electrons equally contribute at high energy, whereas free electrons dominate at the other end of the spectrum. Nevertheless, the bulk free electron contribution (growth from blue to red) is strongly influenced by the interband part of the dielectric function, as shown by the presence of two broad maxima corresponding roughly to the excitation of the interband transitions by the visible and SFG photons, with a prominent peak around 600nm. The same behavior is visible to a lower extent for surface parallel terms. The bulk interband contribution closely follows the interband dielectric function, with a profile showing a decrease towards higher wavelengths and a maximum in the 500nm region. This confirms the analysis of Ref. 30, which have shown that the bound electron contribution to the bulk SHG response is negligible except when the incoming photon energy matches that of the interband transition, as it happens in Figure 5 (c). In addition, they remark that, apart from this resonance case, their effect appears only through their modification of the dielectric function (i.e. in the free electron contribution). All phases share a close to common value at 700nm (between 62 and 90°) and a similar raise from red to blue, with a slope increasing from the xxz case (phase span around 70°) to the bulk bound electron terms (phase span around 210°).

B. Surface and bulk contributions

The free electron terms do not depend on the value chosen for ε2 b , whereas the interband contribution to xxz and xzx is negligible. In addition, it can be checked that the interband contribution to the bulk term does not depend either on ε2 b . The leading terms for the bulk interband contribution (D2 and Δ2) arise from the strong gradients of the infrared electric field and do not depend on the infrared bound electron polarizability. As a whole, the full surface parallel and bulk terms are therefore independent of the choice of the infrared dielectric function for bound electrons. We now turn to the zzz contribution to the surface response, the only model-dependent one. Figure 6 shows the dispersion in amplitude and phase of the various contributions to effective nonlinear susceptibilities in the three schemes. The behaviors of the three models are dramatically different: schemes 1 and 2 show no substantial difference, contrary to scheme 3. In schemes 1 and 2, the inner contribution involving bound electrons (χ bf and χ b ) is negligible (by at least a factor 10 2 and 10 4 , respectively) as compared to χ f . This is due to the huge difference in field gradients in the infrared between free and bound electrons, in favor of the free electron response. The zzz part of the surface term therefore turns down to a nearly pure free electron response, with a characteristic increase towards the low energy region comparable to a Drude dielectric function. On the contrary, in scheme 3, the high infrared field gradients of the free electrons are screened by the bound electron background. The total response is more of the bound electron type, which accounts for the smaller overall amplitude due to its smaller field gradients. In addition, the bound electron contribution is rather flat, as is the modulus of the bound dielectric function. The phases of the dominant terms (thus also of the total contributions) stay rather constant as a function of the wavelength for the three schemes, leading to schemes 1,2 on one side and 3 on the other being almost opposite in sign. We again acknowledge the analogy with the dielectric functions, which also show a phase difference around 180° between free and bound electrons.

As for the phases of all terms in Figures 5 and6, their evolutions are either rather flat or experience a change in their values around 500nm and/or 600nm. This corresponds to the wavelengths for which the Fresnel contributions show a maximum (and the total dielectric function a dip) as explained above, and also to the turning point where the free electron dielectric function becomes greater than the interband one. Finally, as was noticed for the Fresnel factors, all surface and bulk terms share an almost universal phase at 700nm, except for a sign difference for the zzz contribution in scheme 3. We therefore expect the phase to remain constant for the total effective susceptibility of gold at this wavelength.

Figure 6 therefore evidences a dramatic evolution from scheme 1 to scheme 3, with an effective total zzz term evolving from a pure free electron to a nearly pure bound electron contribution (with a lower amplitude). In other words, the (unscreened) external electron density mainly drives the overall behavior, with a natural higher amplitude for the free electron part. In addition, both amplitudes lineshapes and relative phases of the total contributions follow the trends of the dielectric function of their leading terms.

Tuning parameter ε2 b does not influence much the zzz contribution under schemes 1 and 2. As only the terms involving the interband polarizabilities depend on it, their weaknesses compared to the free electron contributions makes the effect of this parameter negligible. With a closer look, it may be checked that χ bf (scheme 1) and χ b (scheme 2) do not vary with ε2 b . Only χ b in scheme 1 is influenced by this parameter, but its amplitude is hundred times less than that of χ bf . We may therefore expect that ε2 b has no influence at all on the total nonlinear susceptibilities in schemes 1 and 2. As for scheme 3, the χ b term increases whereas the χ bf term decreases with ε2 b , with very close phases. As a consequence, their sum slowly grows but in limited amounts. Comparing Figure 5 and6, in schemes 1 and 2 the zzz absolute amplitudes prevail over the surface parallel and bulk contributions, whereas in scheme 3 the latter will dominate the full response, except at high energy. The total effective susceptibility of gold for the three schemes appears on Figure 7 under the hypothesis of equal contributions from free and bound electrons. The amplitude lineshapes in schemes 1 and 2 look very similar indeed to the zzz contribution (i.e. of pure free electron type), but the overall amplitude is higher and the resonances around 500 and 600nm change from dips to humps, showing that the effect of surface parallel and bulk contributions is small but not negligible. This is confirmed by the phase curves in Figure 7(b), which show a raise in the average phase with distinguishable resonant structures. As a whole, for these schemes, the SFG response of gold is essentially of free electron origin, except a small contribution from the bulk term. As for scheme 3, as expected the total effective susceptibility follows the trends of the surface parallel and bulk terms, with a clear effect of the two resonances on the amplitude, whereas the phase exhibits the same raise from red to blue, with intermediate values between xxz/xzx and bulk ones. Again, the total effective susceptibility is mainly of the free electron type, but the influence of the bound electron contributions from zzz and bulk terms should this time not be neglected. In addition, it may be enhanced depending on the actual value of n0 f /n0 b , which will be done in a future step.

V. Discussion

These results show that, for schemes 1 and 2 with n0 f =n0 b , the nonlinear susceptibility of gold originates almost completely in the free electrons, as is often assumed 32 . The interband contribution may however not be omitted in the frame of scheme 3. For silver, the three schemes favor the free electron response, even if the dispersion curves differ between them. For both metals, the influence of bound electrons mostly shows up through the dielectric functions. The three schemes considered here represent extreme and schematic configurations for which the zzz component is analytically calculable without knowledge of the actual functions describing the electron density profiles. Of course, the actual profiles may be more accurately described by an intermediate configuration between two of these extreme schemes, leading to an intermediate behavior of the simulated zzz contribution (Figure 6) and total effective susceptibility (Figure 7). We plan to refine our simulations in this way in the future. However, it is not possible to consider a model for which the screening of the bound electron terms would be lower than for scheme 3.

It is interesting to compare the absolute SFG intensities radiated by a silver (Figure 3) and a gold (Figure 7) surface. The present simulations predict that their effective susceptibilities, for example for experiments with a 532nm visible laser, should compare, and, as they share common bulk free electron and valence d electron densities, so should the experimental intensities. We know that this is not the case, and this may be due to an underestimation of the actual contribution of the bound electrons of gold. In addition, as will be studied in details in the next part of this article, the excitation of surface states 67 may lead to a resonant enhancement of the surface gold response, accounting for the difference between gold and silver.

We use the bulk polarizabilities all over the calculations, including in the selvedge. It may be considered that, in this region, the polarizabilities differ from the bulk as a consequence of the change in the atomic coordination in the topmost layer. In particular, we could expect a raise in the bound electron polarizability due to lower constraints towards the air side than towards the bulk metal side. This could impact on the balance between free and bound electron terms in the total nonlinear surface susceptibility.

The bound electron spill out explored in scheme 3 is rather unusual. Matranga et al. 25 have shown that the SHG response of a metal surface with an adsorbed alkanethiol layer corresponds to an electron deficient interface as a consequence of the electronegativity of sulfur. As the on-top Au-S or Ag-S bond mostly involves the electrons from the 6s (resp. 5s) band 68 , one may consider that these s electrons do not contribute to the free electron dielectric function anymore, because their polarizabilities are much reduced 21 . This is equivalently described by a decrease in the free electron density close to the interface, whereas the bound electron density is unchanged. Scheme 3 therefore takes into account the fact that the metal-S bonding creates a vacancy in free electrons leading to the situation symmetric to scheme 1, with the free electron density spilling less out than the bound electron one. The present case of sulfur covalently bound to gold and silver should not be confused with the situation of alkali adatoms or overlayers on top of transition metals, which lead to an enhancement of SHG as a consequence of the extreme polarizability of alkali metals [START_REF] Liebsch | Electron. Excit. Met. Surfaces[END_REF] . We note that the situation should be different for an adsorbate-free metal surface, for which a true free electron spill out is expected. However, in this case, the method used in Ref. 1 would not be usable for the internal calibration of the evolution of the SFG intensities as a function of the visible wavelength.

We evaluate the free and bound electron polarizabilities from their respective dielectric functions, separated from the full tabulated dielectric function of silver and gold. Such a split is always somehow arbitrary, and alternate choices may alter the balance between free and bound contributions to the effective nonlinear susceptibility. The present modeling is self-consistent in the frame of classical mechanics and has proven its efficiency 7 , but may still remain too coarse, for example when modeling the bound electrons as a continuous dipolium. In addition, the local Drude model, although widely used for such simulations, leads to a rapid growth of the dielectric function in the infrared, having a strong impact on the value of a f (ω1,ω2) under the long wavelength approximation. A more realistic model, either nonlocal 12,37,69 , including a hydrodynamic pressure term 4,39,41 , or even more sophisticated 43 , and the limitation of the rise of a f 40 , could help refine the definition of silver and gold properties and improve the accuracy of the modeled free electron response.

We see that the weight of the zzz contribution, with respect to the others, varies substantially between schemes 1, 2 and 3. In the scheme 3 for example, it even becomes smaller than the bulk and surface parallel terms above 500nm, and of the same order of magnitude below. This is not what is usually reported in the literature, where the surface perpendicular contribution is found bigger that the others 10,11,24 as in our schemes 1 and 2, in spite of unfavorable Fresnel coefficients (taking care of the definition of the χ (2) tensor in terms of internal or external fields, and of the inclusion of some part of the bulk contribution into the surface terms 11 ). However, as recalled in the introduction, most of the analyses published on the metal surfaces use SHG as a technique, which mostly differs from the SFG case by the absence of an infrared beam, characterized by a very high free electron and a very low bound electron contribution to the dielectric function. This dramatically impacts on the balance between the various terms, including through the modulation of Fresnel reflectivity, and prevents from a direct comparison of dispersion data between SFG and SHG. In addition, SHG studies often investigate bare metal surfaces, either in vacuum, air 70 , water 24 or under electrochemical potential control 25,71 . As explained above, for such surfaces, the free electron term, and the zzz contribution, may indeed become dominant as a consequence of spill out.

VI. Conclusion

In this paper, we applied a model for the calculation of the surface (parallel and perpendicular to the interface) and bulk contributions to the effective nonlinear susceptibility of silver and gold in an infrared-visible SFG experiment. The model explicitly takes into account the free and bound electron densities, and the dispersive behavior in the visible range only requires their respective dielectric functions as inputs. Our model is one of the few which takes into account the free and bound electrons together 31 , and we aimed at doing so using a limited amount of unknown or free parameters. For that respect, we believe that it could be useful for a variety of applications. The perpendicular surface contribution requires additional hypotheses on the relative distributions of the free and bound electron density profiles across the interface. However, in the three cases considered, only their relative overlaps as a function of the depth across the interface and their bulk values matter, and not their actual shapes. The present study provides a general frame for the analysis of the intrinsic and effective nonlinear susceptibilities of gold and silver by second order nonlinear optics. An extension to the other metals is straightforward, and it should be interesting to consider the influence of the interband transitions on the nonlinear response in the cases of silver and platinum, for which they occur at higher and lower energy than the visible range, respectively. The three schemes describing the characteristic electron distributions make it adaptable to a variety of experimental cases. We believe that further improvements of this model will lead to a more sophisticated and accurate description of such systems.

As long as the bound electrons do not spill out of the bulk more than the free electrons, the latter, mainly through the surface perpendicular term, almost completely account for the total SFG response, even if the interband transitions modulate it through the SFG and visible dielectric functions. When the bound electrons are allowed to spill out of the free electron density, their contribution becomes more significant, but still minoritary except at high energy. Again the free electron dominates, this time through the surface parallel and bulk terms, and with a lower overall amplitude. This study proves that the actual shapes of the electron densities at the interface (or at least their overlaps), which are extremely difficult to experimentally measure on any given sample, should remain the key parameter in order to correctly analyze a gold surface by nonlinear optics.

In the next step, we plan to use this model in order to account for the experimental data detailed in Part I 1 , by studying the influence of the balance between free and bound contributions, and a putative resonance with a surface state 67 .

Appendix A: Quick review of the phenomenological description Two light beams, modeled by plane waves at frequencies ω1 (visible, angle of incidence θ1 1 ) and ω2 (infrared, θ2 1 ) propagate in air (medium 1, ε air =ε 1 =1) and coincide at the metal surface (medium 2, dielectric function ε metal  ε, with εi = ε(ωi) = (ni)² where n is the complex index of refraction of the metal), supposed isotropic. This hypothesis is reasonable when a polycrystalline surface is used in the experiments. Even if the (111) surface orientation is preferred in this case, macroscopic isotropy is recovered as a consequence of random orientation between the microscopic crystallites. The SFG beam (ω3 = ω1 + ω2) is detected in reflection at the phase matching angle θ3: ω3sinθ 1 3 = ω1sinθ 1 1 + ω2sinθ 1 2 (superscripts refer to the medium, subscript to the beam). All beams are p-polarized. Across the interface, quantities ni sinθi remain constant in media 1 and 2: sinθ 1 i = ni sinθ 2 i. The thiol intermediate layer, considered non-absorbing and very thin as compared to all wavelengths, is blank for the propagation of light plane waves 44 .

The incoming electric fields create in the bulk a nonlinear second-order dipolar polarization P(r,ω3), a quadrupolar polarization Q(r,ω3) and a magnetization M(r,ω3), each one quadratic in the local electric fields or their gradients through specific susceptibility tensors χ (2) 18,72,73 . The dipolar polarization may be split into a "pure" dipolar term, proportional to the electric fields, and two terms involving also the field gradients and the magnetic fields 73 . These contributions are equivalent to a unique bulk polarization with [START_REF] Raab | Multipole Theory in Electromagnetism[END_REF] :

        B 3 3 3 3 3 1 i c , , . , , 2          P r P r Q r r    (A1)
For a cubic crystal, the pure dipolar term vanishes for symmetry reasons, and the bulk polarization involves only terms proportional to electric field gradients. Once PB is known all over the bulk, the total bulk contribution to the emission of SFG radiation has been calculated 2,21 . Additional surface terms arise from the gradients experienced by the electric fields when crossing the interface 9 . They are well described by an equivalent surface dipolar source 5 , decomposed in terms parallel (called x/y below) and perpendicular (z) to the surface. This selvedge contribution is usually calculated in the longwavelength approximation as a consequence of the small thickness of this interface slab. If translational symmetry is assumed in the surface plane, the local effective nonlinear polarization arises at the interface from the integration along z of the nonlocal contributions [START_REF] Shen | [END_REF] and we may then define the surface polarization as 73 :

    S 3 B 3 I z, dz     P P (A2)
where integration runs over the interface slab. It is parametrized by a surface nonlinear susceptibility tensor χS (we drop superscript (2) from now on for clarity), for which the formalism usually devoted to the emission of SFG radiation by molecular monolayers at interfaces applies 76 :

        S 3 S 3 1 2 1 2
; , :

        P E E (A3)
and the surface contribution to the SFG intensity becomes:

  2 3 2 3 eff p 3 S,ppp p 1 p 2 3 2 1 3 8 I ( ) I ( )I ( ) c cos         with 2 p i p i c I ( ) ( ) 2     E (A4)
where the electric fields are evaluated in medium 1. In this equation, the effective surface nonlinear susceptibility tensor χS eff encompasses the transmittivity and reflectivity conditions for the electric fields at the boundary, wrapped into Fresnel coefficients. We locate the sources of nonlinearity, and the electric fields which generate it, inside the metal. This is not the most common habit for SHG: in general, the incoming fields are evaluated inside the metal, but the nonlinear polarization outside 6,24,38,77 . However, our convention allows the best separation between intrinsic bulk and surface contributions in SHG 11 . Even if this choice has no impact on the effective and far field quantities, the definition of χS depends on this conventional choice 6,11,[START_REF] Heinz | Nonlinear Surf. Electromagn. Phenom[END_REF] .

We have, for an isotropic interface:

F F F F          (A5a) 1 1 1 zzz z 3 z 1 z 2 3 1 2 F F ( )F ( )F ( )sin sin sin        (A5b) 1 1 1 xxz x 3 x 1 z 2 3 1 2 F F ( )F ( )F ( )cos cos sin        (A5c) 1 1 1 zxx z 3 x 1 x 2 3 1 2 F F ( )F ( )F ( )sin cos cos        (A5d) 1 1 1 xzx x 3 z 1 x 2 3 1 2 F F ( )F ( )F ( ) cos sin cos        (A5e)
With the definitions above, the Fresnel factors for the surface terms have the form:

2 i x i 2 1 i i i 2 cos F ( ) cos n cos       and 2 1 i i z i 2 1 i i i i 2n cos 1 F ( ) cos n cos n             (A6)
It can be shown 21,38 that the corresponding Fresnel factor for the bulk contribution is

bulk z 3 z 1 z 2 F F ( )F ( )F ( )     (A7)
In a final step, the Rudnick-Stern dimensionless coefficients 10,[25][26][27] allow to compare the models to a pure free electron gas response. Following the definitions of 38 , we write:

1 2 S,zzz 3 1 2 0 1 1 1 a( , ) 2n e 4 4            (A8a) 1 2 1 S,xxz 1 2 0 3 1 12 1 b( , ) 2n e 4 4             (A8b) 1 2 2 S,xzx 2 1 0 3 1 12 1 b( , ) 2n e 4 4             (A8c)
where n0 is the bulk electron density; e the absolute value of the electron charge; indices 1,2,3 stand for visible, infrared and SFG, respectively. Parameters a and b refer to the incoming electric fields evaluated inside the metal, and creating the second order polarization outside, as is, to our knowledge, their universal convention. As we are interested in the absolute phases, the sign convention is also important. It depends on the orientation of the z-axis (towards or away from metal surface) and consequently on the integration in equation (A2). We adopt the convention usually used in SFG 21 , with the z-axis pointing away from the interface, whereas Ref. 40, and several others 77 studying SHG, use the opposite. This choice accounts for the differences in the signs of a and b parameters and allows a coherent comparison of our results with the analysis of Ref. 1. Most available models show that the surface zxx term vanishes (although an effective zxx contribution originating in the bulk term is often included in the surface term 10,11 ) because no gradients are experienced by the parallel components of the incoming electric fields [START_REF] Heinz | Nonlinear Surf. Electromagn. Phenom[END_REF] . Originally in SHG, parameter b equals -1 and is therefore dispersionless, contrary to a 4,25 , whose amplitude may experiment resonant behaviors. However, all susceptibility components show dispersion in SFG (equation A8), and, as each term is modulated by the appropriate Fresnel coefficient, the overall effective terms all show a dependency on the incoming and generated wavelengths.

Appendix B: Coefficients for the bulk response

The D and Δ coefficients appearing in equation ( 3) have the following expressions:

f / b f / b f / b f / b 1 2 i i f / b 0 1 1 1 D d n e 4 4        with f / b f / b 3 1 2 i f / b 2 i i 1 d 1         (B1) f / b f / b f / b f / b 1 2 i i f / b 0 1 1 1 n e 4 4          with f / b f / b 3 3 i i f / b 3 i i 1 1            (B2)
Appendix C: Calculation of the free and bound electron contributions to the SFG response We follow the derivation of Ref. 36 and 38. The interface is defined as the region where the electron density varies from zero to its bulk value (and so does the dielectric function). Continuity is assumed for the perpendicular (z) and parallel (x) components of the displacement and electric fields, respectively. The local nonlinear polarization in this region, which sums up dipolar, quadrupolar and magnetic contributions, is therefore z-dependent. The full surface contribution to the nonlinear polarization at ω3=ωSFG is obtained by integrating this z-dependent local value across the interface. Integration of the x-component is straightforward and leads to the result presented in the main text. In schemes 1, 2 and 3, the contributions from free and bound electrons are evaluated separately, then added up. The z-components of the local nonlinear polarizations induced by the free (resp. bound) electrons P3z f (z) (resp. P3z b ) are given by equation ( 16):

f f f f f 1 2 3 1 2 B,z 3 3 1 2 2 1 f f f 1 2 1z 2z 1 2 1 1 P (z) n (z) e (z) (z) z (z) (z) z (z) 1 1 n (z) E E z ( z ) ( z )                                           (C1)
where the quantities not depending explicitly on z are the bulk ones. Integration over the interface layer gives the surface nonlinear polarization: where integration may be extended to interval (-∞;+∞) for convenience.

Using f / b f / b f / b i i (z) 1 4 n (z)     , the integrand is transformed into a sum of rational fractions of n f (z) and n b (z) only. The results of the main text may be deduced by use of the following procedures. The integrations are split between [z0;+∞) and (-∞;z0]. For scheme 1 (and conversely scheme 3 after swapping f and b superscripts), in the first half, n b (z) vanishes and n f (z) varies from n0 f to 0, thus:

  f f f f f i i i i f 0 n (z) (z) 1 4 n (z) 1 1 1 u 1 n                 . (C3)
The integration variable is changed from z to u=n f (z)/n0 f , which varies from 1 to 0. In the second half, n f (z) remains equal to n0 f while n b (z) varies from n0 b to 0. Integration in this case runs over variable u=n b (z)/n0 b , varying from 1 to 0, using:

    b b b i i f b b f f i i i i i b f f 0 i i 1 1 n (z) (z) 4 n (z) 1 1 u n                           . (C4)
In scheme 2, both electron densities vary together and One may then choose u=n f (z)/n0 f as the integration variable, as for scheme 1. The calculations involve integrals of the form:

1 0 1 2 3 0 0 1 2 3 dx J (q ,q ,q ) J (1 q x)(1 q x)(1 q x)       , with q1≠q2≠q3 (C6) 1 0 1 2 3 0 3 2 0 1 2 3 dx I (q ,q ;q ) I (q ) (1 q x)(1 q x)(1 q x)       (C7)

1 1 1 2 3 1 3 2 0 1 2 3
x dx I (q ,q ;q ) I (q ) (1 q x)(1 q x)(1 q x)     

 (C8)

A direct calculation is possible after partial fraction decomposition. As an example, 1 i 0 1 i {i, j,k} 1 2 1 3 i j i k c.p.of {1,2,3} q q J ln(1 q ) c.p. ln(1 q ) (q q )(q q ) (q q )(q q )

           (C9)
where c.p. denotes circular permutation of q1, q2, q3.

For the calculation of formula (C1) above, in order for example to recover symmetric expressions between 1, 2 and 3 indices, the reader may find the following relations useful:

i 1 i 0 i 0 q I (q ) I (q ) J   1 I (q ) I (q ) I (q ) 2J (1 q )(1 q )(1 q )        (C11) ln(1 q ) 1 I (q ) I (q ) I (q )

(1 q )(1 q )(1 q ) (q q )(q q )

            (C12)
1 0 1 2 0 2 3 0 3 1 2 3 1 q I (q ) q I (q ) q I (q ) 1 (1 q )(1 q )(1 q )        (C13)

1 1 1 2 1 2 3 1 3 0 1 2 3
1 q I (q ) q I (q ) q I (q ) J (1 q )(1 q )(1 q )

       (C14)
Knowing the expression of the surface nonlinear polarization (C2) leads to the surface χ (2) components (equation (A3)), which may be expressed as functions of a f/b (ω1,ω2) and b f/b (ω1,ω2) through equations ( 7), ( 9) and ( 11), and plugged into equation (A5) to calculate the effective susceptibilities.

Figure 1 :

 1 Figure 1: (a) Sketch of the experimental configuration. (b,c,d) Schemes of the distributions of the electron densities n f (z) (resp. n b (z)) of free (resp. bound) electrons as a function of depth inside the metal for schemes 1 (b), 2 (c), and 3 (d).

Figure 2 :

 2 Figure 2: Amplitudes (a,c) and phases (b,d) of the various components of surface parallel (xzx+xxz), bulk (a,b) and surface perpendicular zzz (c,d) contributions to the effective nonlinear susceptibilities of silver. On the left, red color stands for surface parallel term and blue for bulk. Squares: free electrons, dashed lines: bound electrons. On the right, red, black and green stand for schemes 1, 2 and 3, respectively. Squares: free electrons, dashed lines: bound electrons, open circles: cross term (bf). Infrared wavenumber is fixed at 2900cm -1 and ε2 b =2.

Figure 3 :

 3 Figure 3: Total effective susceptibility of silver in amplitude (a) and in phase (b) for schemes 1, 2 and 3. Red, black and green stand for schemes 1, 2 and 3, respectively. Infrared wavenumber is fixed at 2900cm -1 (plain lines) or 1000cm -1 (dashed lines). Constant numerical factors are dropped, ε2 b =2. and n0 f =n0 b =1.

Figure 4 :

 4 Figure 4: Amplitudes (a) and phases (b) of the xxz (dots), xzx (plain lines), zzz (squares) and bulk (dashed lines) Fresnel factors of gold versus the visible wavelength (λvis). Infrared wavenumber is fixed at 2900cm -1 . Phases of the bulk and zzz factors are equal. Several amplitudes are multiplied for clarity by a factor 10 (xxz) or 100 (zzz, bulk).

Figure 5 :

 5 Figure 5: Amplitudes (a,c) and phases (b,d) of the various components of xxz, xzx (a,b) and bulk (c,d) contributions to the effective nonlinear susceptibilities of gold. Red color stands for xxz term, black for xzx and blue for bulk. Squares: free electrons, dashed lines: bound electrons, plain lines: total contributions. Infrared wavenumber is fixed at 2900cm -1 and ε2 b =2. Amplitude for the interband part of xxz and xzx contributions is multiplied by 100 for clarity. Constant numerical factors are dropped and n0 f =n0 b =1 is used to calculate the full contribution.

Figure 6 :

 6 Figure 6: Lineshapes of the amplitudes (a,b) and phases (c) of the various contributions to the effective χzzz of gold according to schemes 1 (red) and 2 (black) in (a,c) and 3 (green) in (b,c). Squares: free electrons, dashed lines: bound electrons, open circles: cross-terms (bf), plain lines: total contributions. In the inset of panel (a), a logarithmic scale allows to visualize the smallest contributions. Infrared wavenumber is fixed at 2900cm -1 and ε2 b =2. Constant numerical factors are dropped and n0 f =n0 b =1 is used to calculate the full contribution.

Figure 7 :

 7 Figure 7: Total effective susceptibility of gold in amplitude (a) and in phase (b) for schemes 1, 2 and 3. Open circles: scheme 1, plain line: scheme 2, dashed line: scheme 3. Infrared wavenumber is fixed at 2900cm -1 and ε2 b =2. Constant numerical factors are dropped and n0 f =n0 b =1.

in order to circumvent the problem of a priori definition of the electron density profile. Taking into