Lei Li
email: lilei@ies.ac.cn

Pierre Boué

Lise Retailleau

Michel Campillo

Spatiotemporal correlation analysis of noise-derived seismic body waves with ocean wave climate and microseism sources

 Time variations of a noise-derived P-type phase are compared with those of the ocean wave heights and microseism sources.

 Do not equate a positive correlation with a causal relation when studying the links between noise sources and noise-derived signals.

 The derivation of seismic signals from ambient noise relies on the competition between the effective and ineffective sources.

Introduction

The incessant background vibrations of Earth had been observed as early as the birth of seismometers in the later 19th century [START_REF] Bernard | DigitSeis: A New Digitization Software for Analog Seismograms[END_REF][START_REF] Dewey | The early history of Seismometry (to 1900)[END_REF][START_REF] Ebeling | Inferring Ocean Storm Characteristics from Ambient Seismic Noise[END_REF]. They were termed "microseisms" due to their tiny amplitudes. With more apparatus deployed worldwide, it was soon recognized that microseisms are ubiquitous and irrelevant to seismicity.

The observation of microseisms aroused interests from various disciplines. Researchers linked the generation of microseisms to atmosphere processes and ocean wave activity. Meteorologists tried to employ land observations of microseisms to track remote oceanic storms (e.g., [START_REF] Harrison | Microseisms and Storm Forecasts[END_REF]. Since the mid-twentieth century, it has been well known that microseisms are excited by storm-driven ocean waves. The most energetic microseisms that dominate the seismic noise spectra, namely, the so-called secondary microseisms at seismic periods around 7 s [START_REF] Peterson | Observations and Modeling of Seismic Background Noise[END_REF], are excited by the nonlinear interactions between nearly equal-frequency ocean waves propagating in nearly opposite directions [START_REF] Longuet-Higgins | A Theory of the Origin of Microseisms[END_REF][START_REF] Hasselmann | A statistical analysis of the generation of microseisms[END_REF]. The periods of the excited secondary microseisms are half those of the colliding ocean waves. The excitation source is equivalent to a vertical pressure applied to the water surface, which is proportional to the product of the heights of the opposing equal-frequency waves. Due to this second order relation, moderate sea states can sometimes generate loud microseism noise [START_REF] Obrebski | How moderate sea states can generate loud seismic noise in the deep ocean[END_REF]. Thus, the presence of a strong microseism event does not necessarily imply a locally intense sea state.

By coupling the excitation theory of secondary microseisms proposed by [START_REF] Longuet-Higgins | A Theory of the Origin of Microseisms[END_REF] with the ocean wave action model, [START_REF] Kedar | The origin of deep ocean microseisms in the North Atlantic Ocean[END_REF] modeled the secondary microseism excitations in the north Atlantic, and validated the numerical modeling by comparing with inland seismological observations. Afterwards, more authors simulated the oceanic microseism sources and some reported the consistency between predictions and observations (e.g., [START_REF] Ardhuin | Ocean wave sources of seismic noise[END_REF][START_REF] Ardhuin | How ocean waves rock the Earth: Two mechanisms explain microseisms with periods 3 to 300s[END_REF][START_REF] Hillers | Global oceanic microseism sources as seen by seismic arrays and predicted by wave action models[END_REF][START_REF] Stutzmann | Modelling longterm seismic noise in various environments[END_REF][START_REF] Nishida | Teleseismic S wave microseisms[END_REF]. [START_REF] Stopa | Sea state trends and variability: consistency between models, altimeters, buoys, and seismic data (1979-2016[END_REF] compared the microseism simulations with real observations to validate their corrections to the global reanalysis wind fields, which systematically reduced the residuals in the wave hindcast over the past decades.

The seismic excitation by an oceanic microseism source is essentially akin to that by an earthquake, in that the seismic wavefield recorded at any point is a convolution of the source time function with the Green function of the propagating medium between source and receiver.

Their main difference lies in the source process. For earthquakes, the sudden rupture of faults leads to short-duration, impulsive source time functions. Isolated seismic phases are generally distinguishable from the seismograms. In contrast, the excitation of microseisms, approximated as Gaussian random process by some authors [START_REF] Peterson | Observations and Modeling of Seismic Background Noise[END_REF][START_REF] Steim | Theory and Observations -Instrumentation for Global and Regional Seismology[END_REF], is incessant, leading to long, random-like source time functions. The convolution mixture signals are not directly discernible from the seismograms. With array beamforming [START_REF] Rost | Array seismology: Methods and applications[END_REF] or correlation technique [START_REF] Campillo | Long-Range Correlations in the Diffuse Seismic Coda[END_REF][START_REF] Shapiro | Emergence of broadband Rayleigh waves from correlations of the ambient seismic noise[END_REF], specific phases from distant microseism sources have been identified from microseism noise records (e.g., [START_REF] Gerstoft | Global P, PP, and PKP wave microseisms observed from distant storms[END_REF][START_REF] Landès | Origin of deep ocean microseisms by using teleseismic body waves[END_REF][START_REF] Zhang | Resolving P-wave travel-time anomalies using seismic array observations of oceanic storms[END_REF][START_REF] Euler | Evidence for bathymetric control on the distribution of body wave microseism sources from temporary seismic arrays in Africa[END_REF][START_REF] Reading | Dominant seismic noise sources in the Southern Ocean and West Pacific, 2000-2012, recorded at the Warramunga Seismic Array, Australia[END_REF][START_REF] Gal | The frequency dependence and locations of short-period microseisms generated in the Southern Ocean and West Pacific[END_REF][START_REF] Liu | Source locations of teleseismic P, SV, and SH waves observed in microseisms recorded by a large aperture seismic array in China[END_REF][START_REF] Nishida | Teleseismic S wave microseisms[END_REF][START_REF] Meschede | The Effect of Water Column Resonance on the Spectra of Secondary Microseism P Waves[END_REF][START_REF] Meschede | Blind source separation of temporally independent microseisms[END_REF][START_REF] Retailleau | Toward high-resolution period-dependent seismic monitoring of tropical cyclones[END_REF]. The correlation technique is advantageous in that, by correlating the noise records at two receivers, explicit seismic signals can be derived. Noise-derived surface waves have been used to infer the azimuthal and seasonal changes of noise sources (e.g., [START_REF] Stehly | A study of the seismic noise from its longrange correlation properties[END_REF]. Noise-derived body waves can provide better constrains in imaging the noise sources [START_REF] Landès | Origin of deep ocean microseisms by using teleseismic body waves[END_REF]. Recently, deep body waves that propagate through the mantle and core have been extracted from ambient noise (e.g., [START_REF] Boué | Teleseismic correlations of ambient seismic noise for deep global imaging of the Earth[END_REF]Lin et al., 2013;[START_REF] Nishida | Global propagation of body waves revealed by cross-correlation analysis of seismic hum[END_REF][START_REF] Poli | Imaging the D″ reflector with noise correlations[END_REF][START_REF] Xia | Extraction of triplicated PKP phases from noise correlations[END_REF][START_REF] Spica | Lateral heterogeneity imaged by small-aperture ScS retrieval from the ambient seismic field[END_REF][START_REF] Retailleau | Ambient seismic noise imaging of the lowermost mantle beneath the North Atlantic Ocean[END_REF]. The noise-derived body waves are valuable for surveying the deep structure and for understanding the links between seismological observations and atmospheric/oceanographic phenomena. [START_REF] Hillers | Global oceanic microseism sources as seen by seismic arrays and predicted by wave action models[END_REF] made the first global-scale comparison between the oceanic microseism sources derived from seismological observations and oceanographic modeling. The seismologically derived data (time resolution: 13 days; spatial resolution: 2.5° latitude  5° longitude) are the global back-projections of near-zero-lag P signals generated from the cross correlations of microseism P waves at seismic array [START_REF] Landès | Origin of deep ocean microseisms by using teleseismic body waves[END_REF]. The modeled data (time resolution: 3 hours; spatial resolution: 1° latitude  1.25° longitude) are a global extension of the numerical simulation by [START_REF] Kedar | The origin of deep ocean microseisms in the North Atlantic Ocean[END_REF]. The two datasets are resampled to common resolutions for comparison. For the seismologically derived data, the back-projection is based on the relationship between the source-receiver distance and the horizontal slowness of teleseismic P wave. However, seismic phases that have common slownesses (e.g., P and PP waves) cannot be discriminated in this method [START_REF] Gerstoft | Global P, PP, and PKP wave microseisms observed from distant storms[END_REF][START_REF] Landès | Origin of deep ocean microseisms by using teleseismic body waves[END_REF]. Thus, the imaged sources are somewhat ambiguous. For the modeled data, coastal reflections of ocean waves, that can play a role in the ocean wave-wave interactions at near-coast regions [START_REF] Longuet-Higgins | A Theory of the Origin of Microseisms[END_REF][START_REF] Ardhuin | Ocean wave sources of seismic noise[END_REF], are neglected. Due to the resonance of seismic waves in the water columns, bathymetry can have significant effect on the excitation of microseisms [START_REF] Longuet-Higgins | A Theory of the Origin of Microseisms[END_REF][START_REF] Kedar | The origin of deep ocean microseisms in the North Atlantic Ocean[END_REF][START_REF] Hillers | Global oceanic microseism sources as seen by seismic arrays and predicted by wave action models[END_REF]. The importance to account for the bathymetric effect on the microseism P-wave excitations has been addressed in several studies (e.g., [START_REF] Euler | Evidence for bathymetric control on the distribution of body wave microseism sources from temporary seismic arrays in Africa[END_REF][START_REF] Gal | The frequency dependence and locations of short-period microseisms generated in the Southern Ocean and West Pacific[END_REF][START_REF] Meschede | The Effect of Water Column Resonance on the Spectra of Secondary Microseism P Waves[END_REF]. [START_REF] Hillers | Global oceanic microseism sources as seen by seismic arrays and predicted by wave action models[END_REF] considered the bathymetric effect, but using the amplification factors derived by [START_REF] Longuet-Higgins | A Theory of the Origin of Microseisms[END_REF] for surface waves. [START_REF] Rascle | A global wave parameter database for geophysical applications. Part 2: Model validation with improved source term parameterization[END_REF] established an oceanographic hindcast database that includes global oceanic secondary microseism sources of a 3-hour time resolution and a 0.5° spatial resolution. Coastal reflections were accounted for in the modeling [START_REF] Ardhuin | Ocean wave sources of seismic noise[END_REF].

Regarding the bathymetric effect on microseism excitations, [START_REF] Gualtieri | Modelling the ocean site effect on seismic noise body waves[END_REF] proposed the formulae for body waves based on ray theory. Concerning the localization of noise sources, Li et al. (2020) developed a double-array method that can estimate the respective slownesses of the interfering waves, and thereby, provide better constrains for the determination of the correlated seismic phases. The microseism sources that are effective for the derivation of seismic signals from noise records, can be mapped by back-projecting the noise-derived signals along the ray paths of the correlated phases. The double-array configuration eliminates the ambiguity in determining the effective source region (Fresnel zone). In this study, we integrate these new progresses to survey the associations of noise-derived body waves to ocean wave activity and microseism excitations. This paper is organized as follows. In section 2, we review the main results of [START_REF] Li | Observation and explanation of spurious seismic signals emerging in teleseismic noise correlations[END_REF] who reported the observation of a prominent P-type phase from the noise correlations between two regional seismic networks at teleseismic distance. The noise-derived phase has its spectral content concentrated in the period band of the secondary microseisms that are excited by the nonlinear ocean wave-wave interactions. In this paper, we denote the phase as Pdmc, corresponding to the fact that the phase originates from the correlation between the Direct P waves that transmit through the deep Mantle and the outer Core (microseism P and PKPab waves). In section 3, we estimate the temporal variations in the Pdmc amplitude and refute the associations to seismicity. In section 4, correlation analysis is used to unveil the spatiotemporal links of the Pdmc signal with the global oceanic wave climate and microseism sources. Last, we discuss the significance of this study in seismology, oceanography and climate science.

2 Noise-derived Pdmc phase [START_REF] Li | Observation and explanation of spurious seismic signals emerging in teleseismic noise correlations[END_REF] correlated the seismic noise records from two regional seismic networks at teleseismic distance: the FNET array in Japan and the LAPNET array in Finland (Fig. 1a). The continuous seismograms were divided into 4 h segments and whitened in the frequency domain.

Segments with large spikes (like earthquakes) were discarded. The available segments of each FNET-LPANET station pairs were correlated. For more technical details, see section 2 of Li et al. (2020). From the vertical-vertical components of the FNET-LAPNET noise correlations, they observed coherent spurious arrivals (the Pdmc phase named in the previous section) that emerged ~200 s earlier than the direct P waves (Fig. 1b). By estimating the respective slownesses of the interfering waves and their time delay, it is unveiled that a quasi-stationary phase interference between the teleseismic P waves at FNET and the PKPab waves at LAPNET, emanating from noise sources in the ocean south of New Zealand (NZ), lead to the noise-derived Pdmc phase (Fig. 1c). The quasi-stationary phase condition refers to that the interfering waves have no common path or common slowness, but the stack of correlation functions over a range of sources can still be constructive as an effect of finite frequency. This observation contrasts with the strict stationary phase condition that has been employed by [START_REF] Phạm | Earth's Correlation Wavefield: Late Coda Correlation[END_REF] to explain the spurious body phases in the earthquake coda correlations. The strict condition implies the existence of sources in the stationary-phase region, or say, the correlated waves have common ray paths or common slownesses. [START_REF] Li | Observation and explanation of spurious seismic signals emerging in teleseismic noise correlations[END_REF] substantiates the explanation of quasi-stationary phase for the observed Pdmc signals with numerical experiments based on ray theory and based on spectralelement modeling, and highlighted the discrepancies between (microseism) noise correlations and coda correlations.

The Pdmc phase has an apparent slowness of 4.6 s/deg, while the slownesses of the interfering P and PKPab waves are 4.7 s/deg and 4.2 s/deg, respectively. The dominant period of the Pdmc phase is 6.2 s, typical for secondary microseisms. The observation of the Pdmc phase is time-asymmetric (Fig. S1a). Its absence from the mirror side is ascribed to the faintness of the corresponding source in the low-latitude Atlantic (Fig. S1b).

There are several advantages to investigating the links between noise-derived signals and microseism sources with the Pdmc phase. First, the correlated P and PKPab waves are both prominent phases in the ballistic microseism wavefields. The Pdmc phase is easily observable from noise correlations, even between some single station pairs and on some single days (Fig. S2). Second, the isolation of Pdmc signals avoids potential bias caused by other prominent signals. Third, the effective sources are confined in a limited, unique region (Fresnel zone). In contrast, noise-derived surface waves have a broad Fresnel zone around the line across the correlated stations, and noise-derived P waves can have multiple Fresnel zones (see fig. 5 of Boué et al., 2014 for instance). The uniqueness of the effective source region can facilitate the study on the correlation between the noise-derived signals and the effective sources. Fourth, the correlated FNET and LAPNET networks are next to the northern Pacific and Atlantic, respectively, while the effective source region locates in the southern Pacific. The northern oceans have consistent seasonal variation pattern distinct from (reverse to) that of the southern oceans [START_REF] Stehly | A study of the seismic noise from its longrange correlation properties[END_REF][START_REF] Stutzmann | Global climate imprint on seismic noise[END_REF][START_REF] Landès | Origin of deep ocean microseisms by using teleseismic body waves[END_REF][START_REF] Hillers | Global oceanic microseism sources as seen by seismic arrays and predicted by wave action models[END_REF][START_REF] Reading | Dominant seismic noise sources in the Southern Ocean and West Pacific, 2000-2012, recorded at the Warramunga Seismic Array, Australia[END_REF]Turners et al., 2020). These geographical configurations make the observations easier to interpret. Last, there happens to be a seismic array (GEONET) in NZ next to the effective source region for the Pdmc phase. The seismic data from GEONET provide extra support to our study.

Correlation analysis

The sea state is composed of ocean waves at various frequencies and propagation directions. The nonlinear interaction between nearly equal-frequency ocean waves traveling in nearly opposite directions is equivalent to a vertical random pressure applied to the ocean surface [START_REF] Longuet-Higgins | A Theory of the Origin of Microseisms[END_REF][START_REF] Hasselmann | A statistical analysis of the generation of microseisms[END_REF], so that microseisms are generated. Figure 3 the microseism excitations at varying locations are independent [START_REF] Hasselmann | A statistical analysis of the generation of microseisms[END_REF], we note that the independence refers only to the phase information. The time variations of microseism source power are spatially associated. That is not surprising since the interacting ocean waves that excite microseisms could be driven by the same storms and swells can propagate freely over thousands of kilometers away [START_REF] Ardhuin | Observation of swell dissipation across oceans[END_REF]. We also notice there are high-r regions that may not be fully explained by the spatial association. These regions are characterized by low intensity of microseism excitations in Figs 3(a-b). A striking example is around [12°N, 88°E] in the Bay of Bengal (F in Fig. 3c). From Fig. 4, it can be seen that the source PSD series for [12°N, 88°E] is dominated by a single peak around May 1 st , coincident with the largest Pdmc peak. This coincidence leads to a high value of correlation coefficient. However, the Bay of Bengal is far away from the FNET-LAPNET great circle, which is inconsistent with the source imaging shown later in Fig. 5. Thus, the high correlation is spurious and does not imply a causality relationship between the microseism sources in the Bay of Bengal and the Pdmc signals.

Figures 3(g-h) show the correlation maps for hs, which will be discussed later.

As shown in Fig. 4, prominent peaks in the Pdmc series have correspondence in the source PSD series for the effective source at [47°S, 177°E]. However, there are some peaks in the latter without correspondence in the former (see the labeled dates in Fig. 4b for examples). Note that here the Pdmc strength is compared to the microseism source PSD at single point in Fig. 4, whereas the effective sources spread over a region. One needs to verify if the peak disparities observed from Figs 4(a-b) can be ascribed to the neglect of the spreading of the effective source region. To evaluate an overall microseism excitation in the effective source region, the bathymetric effect on P-wave excitation should be considered (in the previous analysis for single point locations, the consideration of bathymetric effect is unnecessary because a scaling over the source PSD series does not change the value of the correlation coefficient between Pdmc and source PSD). Using the equations proposed by [START_REF] Gualtieri | Modelling the ocean site effect on seismic noise body waves[END_REF] and the bathymetry around NZ (Fig. 5a), we compute the bathymetric amplification factors for P waves at a period of 6.2 s (Fig. 5b; see Fig. S3 for comparisons between the factors calculated following [START_REF] Longuet-Higgins | A Theory of the Origin of Microseisms[END_REF][START_REF] Gualtieri | Modelling the ocean site effect on seismic noise body waves[END_REF]. The factors vary largely with locations. Also, note that the Pdmc phase has different sensitivity to the sources in the effective region, or say, the sources make varying contributions to the Pdmc signal. The power of sources should be weighted in the averaging. We obtain the weights by back-projecting the beam power of noise correlations onto a global grid (Fig. 5c; see Supplementary for technical details). (1963), which have no constrains from seismological observations. One should consider the accuracy of the simulation: can we ascribe the peak disparities in Fig. 4 to the simulation error or not? The seismic noise records from the GEONET array adjacent to the effective source region provide the opportunity to validate the simulation. To obtain the daily microseism noise levels at GEONET, we apply the Hampel filter, a variant of the classic median filter, to the continuous seismograms to discard earthquakes and anomalous impulses. The filter replaces outliers with the medians of the outliers' neighbors and retains the normal samples. Technical details are provided in section S4 of the Supplementary. The resultant GEONET noise level exhibits a good correlation with the effective source intensity (r = 0.7). We thus deem that the numerical simulations are statistically reliable. When the effective source intensity is high, the GEONET noise level should also be high (see the peaks marked by dots in Fig. 6 for examples). However, due to the great spatiotemporal variability of noise sources in the effective region and the complexity of seismic waves propagating from ocean to land [START_REF] Ying | Propagation of microseisms from the deep ocean to land[END_REF][START_REF] Gualtieri | On the shaping factors of the secondary microseismic wavefield[END_REF], a larger peak in the source intensity series does not necessarily imply a larger peak in the noise level time series (e.g., see diamonds in Fig. 6 for examples). We also emphasize that a high GEONET noise level does not need to always have a correspondence in the source intensity (see squares in Fig. 6 for example), because the GEONET stations record microseisms emanating from noise sources all around, not only from the effective source region.

The above analysis explains the observed disparities between the Pdmc strength and the effective source intensity. From Fig. 6, one can see that the disparities primarily emerge in the shaded period when dominant microseism sources shift to the north hemisphere. The shading roughly separates the northern winter from the austral winter. The correlation between Pdmc strength and effective source intensity is low in the shaded period (r = 0.16), in contrast to the high correlation during the unshaded period (r = 0.74). Large Pdmc peaks always emerge on dates during the austral winter when the effective source intensity is much higher than its median, and meanwhile, noise levels at FNET and LAPNET are below their respective medians (see dots in Fig. 6 for examples). The seasonal variations of oceanic sources in the southern hemisphere are less strong than in the northern hemisphere (Fig. 3). On some dates (see triangles in Fig. 6 for examples), the effective source intensity can be considerable, but relevant Pdmc peaks are still missing. We notice that the corresponding microseism levels at FNET and LAPNET are obviously above their medians. Intensive ocean activity and microseism excitations in the north Pacific and Atlantic, lead to increased microseism noise levels at FNET and LAPNET. The Pdmc strength is anti-correlated with microseism noise levels at FNET (r = -0.12) and LAPNET (r = -0.18). We hereby conjecture that the microseism energy from the distant effective source region is dwarfed by the energetic microseisms excited by oceanic sources closer to the correlated FNET and LAPNET arrays, and consequently, Pdmc signals are overwhelmed by the background noise in the FNET-LAPNET cross-correlations. Last, we mention that the median threshold in Fig. 6 separates the major features of the time series described above, but there is no guarantee that it is a perfect threshold due to the nonlinear relationships between the Pdmc strength and the noise levels at the arrays. When computing the average wind speeds and wave heights, weights in Fig. 5(c) are used.

Discussions and conclusions

In this study, we explore the relations between the noise-derived Pdmc signals and global oceanic microseism sources using spatiotemporal correlation analysis. The effective source region E for the Pdmc phase is successfully identified from the correlation map in Fig. 3(c), which is consistent with that determined from the seismological back-projection in Fig. 5(c). The correlation map provides a convenient way to identify the effective sources of noise-derived seismic signals.

In our case, the seismic networks used for noise correlation are located in the northern hemisphere, while the effective source region is in the southern hemisphere. Ideally, we expect a correlation map with the following features: positive correlation with sources in the effective region, and negative or insignificant correlations with other inefficient sources. Positive correlation indicates a contribution to the construction of Pdmc signal from noise correlations, negative correlation implies an adverse impact, and insignificant correlation (decorrelation) means a negligible effect on the signal construction. However, we obtained a correlation map roughly showing that, the Pdmc signal is correlated with the southern sources and anti-correlated with the northern sources. The correlation with southern sources outside the effective region can be interpreted with the spatiotemporal correlation of the power of the microseism sources in the southern oceans, due to the large span of ocean storms and the long-range propagation of swells.

The anti-correlation with the northern sources, can partly be explained by the well-known reverse seasonal patterns of oceanic microseism excitations in the south and north hemispheres [START_REF] Stutzmann | Global climate imprint on seismic noise[END_REF][START_REF] Landès | Origin of deep ocean microseisms by using teleseismic body waves[END_REF][START_REF] Hillers | Global oceanic microseism sources as seen by seismic arrays and predicted by wave action models[END_REF][START_REF] Reading | Dominant seismic noise sources in the Southern Ocean and West Pacific, 2000-2012, recorded at the Warramunga Seismic Array, Australia[END_REF]. Another (relative to ineffective sources) imply more effective waves in the total wavefield, and thereby, a better quality for the noise-derived target signal. Note that not all waves emanating from the effective sources, but only those following specific ray paths, are effective. There might be multiple pairs of seismic phases that could contribute to the construction of the target signal.

However, their relative strength matters. As for the case of the Pdmc phase, the effective waves are P and PKPab, which are both prominent phases in the ballistic wavefield. [START_REF] Li | Observation and explanation of spurious seismic signals emerging in teleseismic noise correlations[END_REF] showed that the PcP-PKPab correlation and the PcS-PcPPcP correlation, could also lead to a signal at around the Pdmc emerging time. However, the PcP, PcS, and PcPPcP waves are weak phases in the ballistic wavefield, and thereby have minor contributions to the Pdmc signals. We emphasize that the sketch in Fig. 7(b) is only suitable for the ambient noise wavefields that are dominated by ballistic waves.

From Fig. 6, one can observe a high correlation between wind speed and wave height in region E (r = 0.74). It indicates that the ocean waves in region E are likely dominated by the waves forced by local winds. The correlation between wave height and microseism excitation is low (r = 0.25), implying a dominant role of the freely propagating swells in exciting the microseisms. Extreme sea state does not guarantee strong microseism excitation. That is not surprising according to the microseism excitation theory [START_REF] Hasselmann | A statistical analysis of the generation of microseisms[END_REF][START_REF] Longuet-Higgins | A Theory of the Origin of Microseisms[END_REF]: the excitation is proportional to the product of the heights of the colliding equal-frequency ocean waves. In lack of equal-frequency waves coming from opposite directions, even extreme wave climate cannot incite strong secondary microseisms. In contrast, for large peaks in the microseism excitation, the corresponding wave heights are generally moderate (e.g., on May 1st and 23rd). On these two dates, the low wind speeds but moderate wave heights in region E suggest that the ocean waves are dominantly the freely travelling swells from elsewhere, as also illustrated in the supplementary movie S1. Oppositely propagating equal-frequency swells collide with each other and incite strong microseisms. Our analysis and observations agree with those of [START_REF] Obrebski | How moderate sea states can generate loud seismic noise in the deep ocean[END_REF], who investigated a specific case that small swells from two We have described above the implications of this study in seismology and in understanding the process of microseism excitation. Now, we discuss the significance in oceanography and climate science. Well-documented historical ocean storms and wave climate are valuable for improving our understanding of climate change and global warming [START_REF] Ebeling | Inferring Ocean Storm Characteristics from Ambient Seismic Noise[END_REF]). However, modern satellite observations of ocean waves and storms have a history of merely decades. Microseisms are induced by storm-driven ocean waves [START_REF] Ardhuin | How ocean waves rock the Earth: Two mechanisms explain microseisms with periods 3 to 300s[END_REF][START_REF] Hasselmann | A statistical analysis of the generation of microseisms[END_REF][START_REF] Longuet-Higgins | A Theory of the Origin of Microseisms[END_REF]. The records of microseisms contain the imprint of climate [START_REF] Aster | Global trends in extremal microseism intensity[END_REF][START_REF] Stutzmann | Global climate imprint on seismic noise[END_REF]. Instrumental observation of microseisms has an over-century history, and started much earlier than the modern observations of ocean waves and storms. It has been a long-lasting effort for the seismological community to digitalize the historical analog seismograms [START_REF] Bernard | DigitSeis: A New Digitization Software for Analog Seismograms[END_REF][START_REF] Lecocq | On the Extraction of Microseismic Ground Motion from Analog Seismograms for the Validation of Ocean-Climate Models[END_REF]. Researchers expect that past seismic records can be used to recover undocumented historical ocean storms and wave climate [START_REF] Ebeling | Inferring Ocean Storm Characteristics from Ambient Seismic Noise[END_REF][START_REF] Lecocq | On the Extraction of Microseismic Ground Motion from Analog Seismograms for the Validation of Ocean-Climate Models[END_REF].

This study confirms that it is possible to detect remote microseism events (burst of microseism energy) with land observation of microseisms. We demonstrate that the noisederived Pdmc signals can be employed to monitor microseism events in a specific ocean region (Fig. 5). The remote monitoring of microseisms is promising as an aid to improving wave hindcast, in similar manners as demonstrated by [START_REF] Stopa | Sea state trends and variability: consistency between models, altimeters, buoys, and seismic data (1979-2016[END_REF]. The comparative analysis in Energetic microseism excitation does not always need extreme in situ wave heights, and extreme wave heights do not necessarily produce powerful microseisms [START_REF] Obrebski | How moderate sea states can generate loud seismic noise in the deep ocean[END_REF]and this study). It imply that secondary microseism events are not a perfect proxy for the extremal in situ wave climate. However, it does not mean the long-lasting attempt to monitor remote sea state and ocean storms with land observation of secondary microseisms is futile. In the Pdmc-hs correlation map (Fig. 3g), the largest r values do not fall in the effective region E as in the Pdmc-source correlation map (Fig. 3c), but in surrounding regions with moderate to high ocean wave activity (the bounded areas in Fig. 3g). We speculate that these regions could be the birthplaces of the colliding swells that generate the secondary microseisms in region E, or the ocean waves in these regions are driven by the same storms as the colliding waves in region E (see the spatial links of hs from Fig. 3h and supplementary movie S1). The detection of a microseism event could affirm the existence of the causative storms that generated the ocean waves propagating to the location of the microseism event, although the storms could be distant from the events.

Figure 1 .

 1 Figure 1. (a) Three regional broadband seismic networks used in this study: left, the LAPNET array in Finland (38 stations); center, the FNET array in Japan (41 stations); right, the GEONET array in New Zealand (46 stations). The histogram inset shows the distribution of the separation distances between the 1558 FNET-LAPNET station pairs. The center-to-center distance is 63° between LAPNET and FNET, and 85° between FNET and GEONET. The global inset shows the geographical locations of the three networks that are aligned on a great circle (dark line). (b) Annual FNET-LAPNET noise correlations that are filtered between 5 s and 10 s and stacked over time and in 0.1° inter-station distance bins. The spectrum inset indicates that the Pdmc phase has a 6.2 s peak period. (c) Ray paths of the interfering waves that generate the Pdmc phase. The effective source region is close to GEONET. 3 Temporal variations We extract the temporal variations of the Pdmc signals by beamforming the FNET-LAPNET noise correlations on a daily basis. The daily noise correlations are shifted and stacked by 𝐵(𝑡) = 〈𝐶 𝑡 + 𝑑 -𝑑 • 𝑝 〉,(1)with 〈•〉 the mean operator, 𝐶 and 𝑑 the correlation function and the distance between the ith FNET station and the jth LAPNET station, 𝑑 the reference distance (63°), 𝑝 the apparent

Figure 2 .

 2 Figure 2. Temporal variations in the strength of daily Pdmc signals, in comparisons with the daily cumulative seismic moments in NZ (pink line at bottom; for earthquake magnitudes above 2.0 in GEONET catalogue) and global large earthquakes (stars; magnitudes above 7.0 in USGS catalogue; see Table S1for a full list of earthquakes in 2008 above magnitude 5.5). The background image is composed of columns of daily envelopes of beamed FNET-LAPNET noise

  (a) shows a global map of average Power Spectral Density (PSD) of the equivalent surface pressure for a seismic period of 6.2 s, during the northern winter months of 2008. The hindcast PSD data are simulated by[START_REF] Ardhuin | Ocean wave sources of seismic noise[END_REF] and[START_REF] Rascle | A global wave parameter database for geophysical applications. Part 2: Model validation with improved source term parameterization[END_REF], based on the microseism excitation theory of[START_REF] Longuet-Higgins | A Theory of the Origin of Microseisms[END_REF] and[START_REF] Hasselmann | A statistical analysis of the generation of microseisms[END_REF]. The most energetic microseism excitations occur in the northern Atlantic south of Greenland and Iceland (near LAPNET), and in the northern Pacific between Japan and Alaska (near FNET). Figure3(b)shows the map for the austral winter months, with the strongest excitations occurring between NZ and Antarctic (near GEONET). The seasonal pattern of oceanic microseism excitations results from the same pattern of global wave climate (Figs3e-f). The seasonal pattern of the Pdmc strength agrees with that of the microseism excitation and wave climate in the effective source region south of NZ.We compute the correlation coefficient (denoted as r) between the Pdmc strength and the source PSDs at each grid point, and thereby obtain a global correlation map (Fig.3c). The largest r value for Pdmc and source PSD arises at [47°S, 177°E] in the effective source region (E in Fig.3c). The corresponding time series of daily source PSDs is plotted in Fig.4, in parallel with the Pdmc strength. Large peaks in the Pdmc series have good correspondence with large peaks in the source PSD series. From Fig.3(c), one can observe a broad region of positive r values (red colors; roughly, south Atlantic, south Pacific, and Indian ocean). However, the positive correlation does not imply a causality between the Pdmc phase and the sources outside the effective region E. We ascribe the apparent positive correlation to the spatial correlation of the time-varying microseism excitation. As shown inFig. 3(d), the source at[47°S, 177°E] in region E exhibits a similar pattern of apparent correlations with global sources as in Fig.3(c). Despite

  Figure 5(d) shows the map of annually averaged source PSDs surrounding NZ and Fig. 5(e) shows the map after the modulation of the bathymetric amplification factors in Fig. 5(b). The spatial patterns are altered significantly, indicating the importance to account for the bathymetric effect. The final source imaging that has been weighted by Fig. 5(c), is plotted in Fig. 5(f). It agrees well with the effective source region E determined from the correlation map in Fig. 3(c). Replacing the annual PSD map in Fig. 5(d) with daily PSD maps, we obtain maps like Fig. 5(f) for each date.Averaging over the map leads to the time series of daily intensity in the effective source region (labeled as effective source intensity in Fig.6). Averaging over a wide region has the advantage that the effects of potential source location errors due to the simplification of Earth model for fast travel time calculation, which have been addressed in some single array back-projection studies (e.g.,[START_REF] Gal | The frequency dependence and locations of short-period microseisms generated in the Southern Ocean and West Pacific[END_REF][START_REF] Nishida | Teleseismic S wave microseisms[END_REF], can be largely reduced. From Fig.6, one can see that the new effective source intensity series has almost the same peaks as the source PSD series for [47°S, 177°E] in Fig.4(b), suggesting that the observed peak disparities are caused by other reasons. Next, we investigate if the disparities are caused by errors in the simulation of hindcast data or if there are other physical explanations.The microseism source PSD data are simulated from the hindcast data of ocean wave directional spectra base on the excitation theory of[START_REF] Longuet-Higgins | A Theory of the Origin of Microseisms[END_REF] and Hasselmann

Figure 3 .

 3 Figure 3. (a) Global map of average PSD of oceanic microseism sources in 2008 northern winter months (Jan. to Mar. and Oct. to Dec.), for a seismic perod of 6.2 s. (b) Similar to (a) but for 2008 austral winter months (Apr. to Sep.). (c) Correlation map (corrmap) for the Pdmc strength and global microseism noise sources. Circles mark two regions with highest correlation coefficients: E, effective source region surrounding [47°S, 177°E] south of NZ; F, fake highlycorrelated region surrouding [12°N, 88°E] in the Bay of Bengal. (d) Correlation map for the source at [47°S, 177°E] and global sources. (e) Mean significant wave height (hs; four times the square root of the zeroth-order moment of ocean-wave frequency spectrum) in northern winter months. (f) Similar to (e) but for austral winter months. (g) Correlation map for the Pdmc strength and global wave heights. (h) Correlation map for wave heights at [47°S, 177°E] and global wave heights. The oceanographical hindcast data are provided by the IOWAGA products (Rascle & Ardhuin, 2013).

Figure 4 .

 4 Figure 4. True correlation (r = 0.73) between (a) the Pdmc strength from Fig. 2 and (b) the power of source at [47°S, 177°E] in the effective source region (E in Fig. 3c), and spurious correlation (r = 0.71) between Pdmc and (c) the power of source at [12°N, 88°E] in the Bay of Bengal (F in Fig. 3c).

Figure 5 .

 5 Figure 5. (a) Bathymetry around NZ. (b) Bathymetric amplification factors for P-type waves. (c) Imaging of effective sources obtained from the back-projection of the FNET-LAPNET noise correlations. (d) Annual average of source PSDs in 2008. (e) Source PSDs in (d) modulated by the factors in (b). (f) Source PSDs in (e) further modulated by the weights in (c).

Figure 6 .

 6 Figure 6. Temporal variations of daily Pdmc strength, microseism noise levels at three networks, and average wind speeds, wave heights and microseism excitations in the effective source region. The curves are normalized by their own maximums. Dashed horizontal lines denote their respective medians. Symbols mark some dates cited in the main text. When computing the effective source intensity, the bathymetric factors in Fig. 5(b) and weights in Fig. 5(c) are used.

  important reason is that compared to the remote effective sources in the south hemisphere, the northern sources closer to the correlated stations have larger impacts on the microseism noise levels at stations. Strong energy flux from the northern sources outshines the microseism energy coming from the distant effective sources. That deteriorates the construction of the Pdmc phase.The noise-derived Pdmc signals are primarily observable in the austral winter. That can be, on one hand, attributed to the stronger effective source intensity during that period, and on the other hand, to the relative tranquility in the northern oceans.In Fig.7, we summarize the classification of noise sources, the decomposition of wavefields, and the associations to the constituents of the inter-receiver noise correlation function. The diagram of Fig.7(a) explains the relationships using the case study of the Pdmc phase discussed above. We generalize Fig.7(a) to the derivation of an arbitrary signal (referred to as the target signal for convenience) from ambient noise wavefields (Fig.7b). The noise correlation function is composed of the target signal, any other signals and background noise. A source or a wave is called effective if it contributes to the construction of the target signal from noise correlations. Otherwise, it is called ineffective. The construction of the target signal is exclusively ascribed to the interference between the effective waves. Stronger effective sources

  storms meeting in the eastern Pacific generate loud microseism noise. There are also examples showing that wind waves can play a role in the excitation of microseisms, for instance, around July 31st when the local winds, wave height, and microseism excitations are all strong. Such examples are few. The good consistency between the temporal variations in the Pdmc strength, the effective source intensity and the NZ microseism noise level (Fig.6), provides extra supports to the analysis of the Pdmc observations and the quasi-stationary phase arguments proposed byLi et al. (2020). It also gives credits to the validity of the numerical modeling of oceanic microseism sources by[START_REF] Ardhuin | Ocean wave sources of seismic noise[END_REF] and[START_REF] Rascle | A global wave parameter database for geophysical applications. Part 2: Model validation with improved source term parameterization[END_REF].

Fig. 6

 6 Fig. 6 indicates that the remote event detection could be effective in the absence of strong sources near the stations, otherwise the detection could fail. Stations at low latitudes where wave climate and microseism excitation are relatively mild, or inland stations far from oceans, should have better performance in remote monitoring.

Figure 7 .

 7 Figure 7. (a) Sketch explanation for the relationships between microseism noise sources and the noise-derived Pdmc signal. (b) Generalization of diagram (a) for an arbitrary signal derived from ambient noise wavefields that are dominated by ballistic waves.

Figure S2 .

 S2 Figure S2. Inter-receiver noise correlations for all FNET-LAPNET station pairs: (a) stacked over the year of 2008; (b) on single day of 2008-05-01. The waveforms are windowed around the Pdmc phase. Dashed lines indicate inter-station distances. Back-projection imaging of noise sources: (c) using data from (a); (d) using data from (b).

Acknowledgments and data

The seismic data of FNET and LAPNET were provided by the National Research Institute for Earth Science and Disaster Resilience (http://www.fnet.bosai.go.jp/; last access: June 2018) and the Réseau Sismologique & Géodésique Français (http://www.resif.fr/; last access: June 2018), respectively. The seismic data of GEONET and the earthquake catalogue of New Zealand were provided by the GEONET Data Center (https://www.geonet.org.nz/; last access: June 2018). The global earthquake catalogue was provided by the U.S. Geological Survey (https://earthquake.usgs.gov/; last access: June 2018). The wind hindcast data were provided by the European Centre for Medium-Range Weather Forecasts (https://www.ecmwf.int/; last access: June 2018). The hindcast data of wave heights and microseism source PSDs were provided by the IOWAGA products (Rascle & Ardhuin, 2013).

The bathymetry data were extracted from ETOPO1 Global Relief Model (Amante & Eakins, 2009). The computations were performed mainly on the ISTerre cluster. This work was supported by Labex OSUG@2020 (Investissements d'avenir-ANR10LABX56) and the Simone and Cino del Duca Foundation, Insitut de France. The authors acknowledge the support from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement No 742335, F-IMAGE). We also acknowledge Anya M.

Reading and an anonymous reviewer for their valuable reviews that helped to improve the clarity of our paper, and special thanks to the editor for allowing the deadline extension during the COVID-19 pandemic.

Caption for Table S1 Caption for Movie S1

Text S1. FNET-LAPNET noise correlations

The correlation function CAB between two seismograms (SA and SB) is given by

The resultant CAB consists of an acausal part and a causal part, that correspond to the negative lags (𝜏 < 0) and the positive lags (𝜏 > 0), respectively. For efficiency, it is routine to compute the correlation function with the Fast Fourier Transform: Global maps of 6.2 s period secondary microseism sources and significant wave heights in 2008.

Text S2. Noise source imaging by back-projection

Assuming the interferometry between P waves at FNET and PKPab waves at LAPNET, we image the effective noise sources through the back-projection of the FNET-LAPNET noise correlations. We beam the FNET-LAPNET noise correlations and assign the beam power

) onto a 0.5°  0.5° grid as the probabilities of noise sources on the global surface. In the above equation, 〈•〉 means the average over 𝑥, 𝐶 is the correlation function between the i-th FNET station and the j-th LAPNET station, 𝑡 is the traveltime of the P wave from the s-th grid point to the i-th FNET station, and 𝑡 is the traveltime of the PKPab waves from the s-th source to the j-th LAPNET station. The inter-station noise correlations are windowed before the beamforming (Fig. S2a). The noise source imaging for the annually stacked noise correlations is plotted in Fig. S2(c). Only the region surrounding NZ is shown. Outside the region, hardly can the P wave reach FNET or the PKPab waves reach LAPNET. Besides a well-focused imaging of the expected source region in the ocean south of NZ, we notice a secondary spot to the west. In comparisons with the power map of oceanic microseism noise sources in Fig. 5(e), we ascribe it to the strong microseism excitation in the ocean south of Tasmania. We also back-project the daily noise correlations on 2008-05-01 (Fig. S2b), when the Pdmc phase reaches the largest strength through the year (Fig. 2). As shown in Fig. S2(d), an exclusive source region is imaged, which agrees with the dominant spot in Fig. S2(c). We filter the vertical components of the continuous seismograms around 6.2 s period. The seismograms are then divided into 15-min segments and the power of segments is computed. We apply the Hampel filter to the time series of noise power recursively. For each sample, we compute the local median and MAD of its eight neighbors (four before and four after). A sample is replaced by the median if it deviates from the median over three times of the MAD. The despiked time series is resampled from a 15-min interval to a 1-hour interval, by averaging over every four samples. Then, we apply the Hampel filter again and resample the time series to a 24hour interval. The averaging of noise levels over all stations of a seismic network leads to the time series of array noise level. Before the averaging, the Hampel filter is applied again, to discard possible anomalous values at some stations (see Fig. S3 for the example of GEONET).

The final time series of microseism noise levels for networks FNET, LAPNET and GEONET are shown in Fig. 6.


Francois Renaud

Gamal Mohamed

Anne-Laure Fehrembach

Evgueni Popov

Antoine Monmayrant

Olivier Gauthier-Lafaye

Quasi-total backward reflection with a CRIGF under oblique incidence

We report on a numerical study of Cavity Resonator Integrated Grating Filters (a grating coupler flanked with two distributed Bragg reflectors) under oblique incidence. At resonance, the diffracted energy is divided equally in four outcoupling directions, versus two directions only for the CRIGF under normal incidence and for the infinite grating coupler. We provide a simple model to demonstrate this fact. Then, we propose a configuration for which all the diffracted energy is reflected backward with respect to the direction of the incident beam, thus realizing a reflection filter free from specular reflectivity.

Introduction

Multilayer interference filters are the most common way to achieve spectral filtering. But, to be efficient, they require a large number of optical layers, which make them difficult and expensive to build, especially for infrared applications. An alternative is nanostructured components, like Guided-Mode Resonant Filters (GMRF), which allow an ultranarrow spectral filtering with a few number of optical layers [1]. GMRF rely on the excitation of a guided mode by the incident light thanks to a subwavelength grating coupler (GC). The resonance wavelength of the structure is directly linked to the period of the GC. The main issue of this kind of structure is their lack of angular acceptance [2,3]. Indeed, to have an efficient excitation of the guided mode, the incident beam must be perfectly collimated and the interaction area with the device must be large, otherwise the resonance peak is broaden and falls down.

A few years ago, a new type of structure has been proposed to generate resonance peaks spectrally as narrow as obtained with GMRF, but with a much greater angular acceptance: the Cavity Resonator Integrated Grating Filter 1 (CRIGF) [START_REF] Ura | 13th International Conference on Transparent Optical Networks[END_REF][START_REF] Kintaka | [END_REF]6]. A CRIGF consists of a grating coupler, like in the GMRF case, integrated on a waveguide resonator formed by two Distributed Bragg Reflectors (DBR), as shown in figure 1. Under normal incidence, there is a peak in the reflectivity spectrum at the resonance wavelength. This peak can go to a maximum of 90% with a spectral width less than 1 nm. CRIGF can be used as an end mirror in a laser cavity to stabilize the emission wavelength of a laser diode [7,[START_REF] Buet | 15th International Conference on Transparent Optical Networks (ICTON)[END_REF].

The behavior of CRIGF under normal incidence is now quite well understood. Using CRIGF under oblique incidence could be of interest for some applications. During the first semester of 2019, two conference papers reported on CRIGF for oblique incidence, the first one from our team, and the second one from S. Ura's team [START_REF] Kusuura | [END_REF][START_REF] Renaud | [END_REF]. In the present paper, our goal is to bring a further theoretical analysis of the behavior of CRIGF under oblique incidence. In the first part, using a rigorous numerical method, we show that there are four outcoupling directions of the guided mode when the CRIGF is illuminated under oblique incidence. We then explain theoretically this behavior, using a phenomenological approach involving the pole of the excited mode. Last, we study a configuration to control the emission directions of this CRIGF. We show an example where the incident light is quasi-totally backward reflected with respect to the direction of the incident beam.

CRIGF under oblique incidence

Figure 1 shows the CRIGF structure studied in this paper. The waveguide layer is made of Si 3 N 4 (refractive index 1.97) on a substrate of Su-8 polymer (refractive index 1.6). The different gratings are etched in a SiO 2 layer (refractive index 1.46).

The GC is designed to couple the incident wave into the TE 0 mode of the structure through the -1 diffracted order. The period Λ GC of the GC is calculated thanks to the coupling equation under oblique incidence for a grating periodic along one direction only:

±n ef f = sin θ -m λ Λ GC , (1)
where m is the diffraction order used for the coupling, n ef f is the effective index of the mode TE 0 , θ the incidence angle and λ the wavelength. The sign ± means that the mode can propagate forward (+) or backward (-) along x. The incident angle is chosen to be 30 o . To obtain a resonance peak around 850 nm from the excitation of the mode propagating backward along x (-n ef f ), with the first diffraction order, the period of the grating coupler (GC) is calculated to be Λ GC =390 nm. The DBR period Λ DBR is chosen to couple the mode propagating in the positive direction (with a normalized propagation constant +n ef f ) with the mode propagating in the negative direction (with a normalized propagation constant -n ef f ), thus:

Λ DBR = λ 2n ef f , (2)
from which we deduce Λ DBR =260 nm.

Figure 1: Scheme of the CRIGF structure under oblique incidence.

The GC is made of 21 periods with a filling factor of 0.45, and the DBR are made of 300 periods with a filling factor of 0.5. The phase section δ is used to change the length of the cavity between the two DBR, to adjust the spectral position of the resonance. The structure is simulated with the Fourier Modal Method (FMM) with 1801 Fourier coefficients (from -900 to 900) [START_REF] Rassem | [END_REF]. The incident beam is 4µm waist, and covers completely the GC (the length of the GC is 8.19µm). When the structure is illuminated with an oblique incident Gaussian beam (illumination angle of 30 o ), the light is outcoupled in four different directions: R + , R -, T + , T -. R + and T + are the specular reflection and transmission directions. They can be seen as the sum of a non-resonant term (which corresponds to the energy reflected and transmitted by the multilayer stack and that is not coupled into the guided mode) and a resonant term, which is the signal outcoupled from the waveguide layer at the resonance wavelength. The directions R -and T -correspond to the symmetric of R + and T + with respect to the z axis. They are free from non-resonant terms (not in the specular directions of reflection and transmission by the multilayer stack).

Figure 2 shows the reflectivity and transmittivity spectra of this CRIGF in each outcoupling direction calculated rigorously with the FMM. In dotted line, the total reflectivity (black) and total transmittivity (red) of the structure are plotted. One can see that the total reflectivity at the resonance is less than 50%, which is different from the normal incidence case (around 90% of reflectivity at the resonance). In solid line, the signals in the 4 outcoupling directions are plotted. At resonance, the reflectivity and transmittivity is almost equal to 25% in each direc- tion, the device behaves as a beam splitter. The outcoupling in the four directions can be understood by considering the different modes propagating inside the CRIGF (see figure 3). Indeed, when an incident beam is coming on the top of the structure (black arrow with the Gaussian symbol), a part of its energy is reflected and transmitted by the multilayer stack (non-resonant reflection R n.res and non-resonant transmission T n.res in black arrows) and the rest is coupled into one of the guided mode of the structure (solid blue arrow 1 in the gray layer) thanks to the -1 diffraction order of the grating coupler. The mode then propagates inside the waveguide layer until it reaches the left-hand side DBR. When going through the DBR, the mode is reflected in the opposite direction (dashed blue arrow 2 in the gray layer) and will be reflected by the right-hand side DBR. This configuration acts as a planar Fabry-Perot cavity for the guided mode, and the structure behaves as if there are two modes propagating in two opposites directions (solid and dashed arrows in the waveguide layer in gray), each of them carrying 50% of the initial energy. Following eq. 1, the mode represented with the arrow 1 (with normalized propagative constant -n ef f ) is outcoupled in the superstrate in the specular direction R + , while the mode represented with the arrow 2 (with normalized propagative constant +n ef f ) is outcoupled in the symmetric direction with respect to z, the direction R -. And each of these two modes is also outcoupled in transmission (T + for mode 1 and T -for mode 2).

In the next section, we demonstrate, using a phenomenological theory involving the pole associated with the resonance, that the energy of the mode is split equivalently in these four directions.

Phenomenological theory for the CRIGF under oblique incidence

In the case of a GMRF, the transmission of the structure represents a Fanotype anomaly [12] that can be written as the sum of a non-resonant transmission (transmission by the multilayer stack) and a Lorentzian resonant term [13,14,15]:

T 0 = T n.res + c t out λ -λ p 0 g - 0 , (3)
where T n.res corresponds to the transmission coefficient out of resonance, c t out is the outcoupling coefficient in transmission, g - 0 is the amplitude of the mode excited, λ p 0 is the eigenwavelength of the mode, solution of:

det[S(λ)] -1 = 0, (4)
with S the diffraction matrix of the GMRF.

A similar expression can be written for the reflectivity:

R 0 = R n.res + c r out λ -λ p 0 g - 0 , (5)
where R n.res corresponds to the reflection coefficient out of the resonance and c r out is the outcoupling coefficient in reflection.

For the following, we consider that 1. the non-resonant reflection is vanishing (multilayer designed to be an antireflection coating), and thus the non-resonant transmission modulus is unity, 2. the structure is symmetrical with respect to the z axis.

In the case were the non-resonant reflection is vanishing, eq. 5 writes

R 0 = c r out λ -λ p 0 g - 0 . (6)
From eq. 6, we can expect that the maximum of reflectivity is obtained for a wavelength close to the real part of the pole, λ Re(λ p 0 ). Moreover, for a structure symmetrical with respect to the z axis, it can be proved that the reflectivity maximum is 100% (provided that the materials are lossless [16]. Thus,

R 0 (λ = Re(λ p 0 )) = i c r out Im(λ p 0 ) g - 0 = exp (iψ r ), (7)
where ψ r is a real number. Thanks to the energy conservation, the transmittivity for λ = Re(λ p 0 ) is null, which entails, from eq. 3, that

i c t out Im(λ p 0 ) g - 0 = -T n.res , (8)
where T n.res modulus is unity under the first hypothesis mentioned above.

Here the subscript 0 stands for the values for a GMRF. Now we consider a CRIGF. The transmittivity (resp. reflectivity) in the specular direction is, as in the case of the GMRF, the sum of the non-resonant transmittivity (resp. reflectivity) and a resonant term coming from the outcoupling of the mode. As the incident beam illuminates only the area covered by the GC, we can assume that the non-resonant part does not change significantly when shifting to the CRIGF geometry (thus R n.res is negligible while T n.res has a modulus equal to one). We also assume that the length of the GC is sufficient to efficiently couple in the mode, hence the coupling conditions are the same for the GMRF and the CRIGF, which means that neither the outcoupling coefficients nor the imaginary part of the pole λ p of the CRIGF are significantly different from those of the GMRF. Thus, following eq. 6, we can write, for λ = Re(λ p )

R + (λ = Re(λ p )) = i c r out Im(λ p 0 ) g -, (9)
and

T + (λ = Re(λ p )) = T n.res + i c t out Im(λ p 0 ) g -, (10)
where g -is the amplitude of the mode propagating backward along x inside the CRIGF.

In the negative direction, the reflection and transmission are simply due to the outcoupling of the mode propagating in the positive direction, and due to the symmetry, the outcoupling coefficients are the same in positive and negative direction. Hence, the R -and T -coefficients have the same form as that given by eqs. 9 and 10, except that there is no non-resonant terms and that g -is replaced with g + , the amplitude of the mode propagating forward along x:

R -(λ = Re(λ p )) = i c r out Im(λ p 0 ) g + , (11)
and

T -(λ = Re(λ p )) = i c t out Im(λ p 0 ) g + . (12)
Now, for a system symmetrical with respect to the origin, the two modes propagating backward and forward along x in the CRIGF have the same amplitude, and each of them carries only half of the energy of the mode of GMRF considered above:

g + = g -= 1 2 g - 0 . (13)
From eqs. 7, 9, 11 and 13, we obtain

R + (λ = Re(λ p )) = R -(λ = Re(λ p )) = 1 2 exp (iφ r ), (14)
and from eqs. 8, 10, 12 and 13, we obtain

T + (λ = Re(λ p )) = -T + (λ = Re(λ p )) = 1 2 T n.res , (15)
and their modulus is equal to 0.5. In term of energy, it means that 25% of the energy goes in each of theses directions, as shown by the previous numerical results.

In the next part, we show how to obtain almost 100% of the energy in one of the four directions.

Control of the emission direction of the CRIGF

The control of four channels of emission is quite difficult. To return to a simpler case, where there are only two emission directions, a gold substrate is added at the bottom of the structure to remove the transmitted beams, as shown in figure 4. The signal R + T is the sum of the signal R + and the signal T + , which is reflected on the gold substrate. R - T corresponds to the sum of the signal R - and the signal T -(also reflected on the gold substrate).

With this structure, our goal is to obtain a total reflection in the R - T direction (no signal in R + T ). The control of the energy distribution between R + T and R - In figure 5, the reflected signals R + T and R - T are plotted versus the thickness of the Su-8 buffer layer. The first thing to be noticed is that the R + T signal (blue) is in phase opposition with the R - T signal (green). This can be explained by the composition of these two signals. Indeed, the R + T signal contains the nonresonant specular reflection, in addition to the outcoupling of the mode. The R - T signal contains only the resonant part that comes from the mode outcoupling.

The thickness of the Su-8 buffer layer is chosen to be approximately 1250 nm, which corresponds to a maximum of the R - T signal and a minimum of the R + T signal. The reflectivity spectrum of this structure is plotted on figure 6. Far from the resonance, the reflectivity in R + T is maximum, and corresponds to the non-resonant reflection due to the multilayer stack and the gold mirror. The signal in R - T is very weak, because the modes are not excited inside the structure, but not null due to the scattering of the incident beam at the GC edges.

At the resonance wavelength (around 849 nm), the signal coming from the outcoupling of the mode interferes destructively with the non-resonant part in the direction R + T , while the signal becomes maximum in the R - T direction: the incident light is quasi-totally backward reflected by the CRIGF.

We think that this configuration can be interesting when compared to the CRIGF used in normal incidence, because in the R - T direction, there is no Fabry-Perot interferences due to the substrate thickness, which are parasitic signals for filtering applications (for instance, the oscillations on the fig. 2 of [17] are due to the interferences coming from the thick substrate).

Conclusion

In this paper, we studied the behavior of the CRIGF under oblique incidence. Numerically, we show that the modes in the structure are outcoupled in four different directions, and that the energy in each of these directions is approximately the same. We gave a demonstration of this result thanks to a phenomenological approach. In order to reduce the number of outcoupling paths, we added a metallic mirror at the bottom of the structure, and show that a particular choice of the distance between the mirror and the waveguide lead to an outcoupling of 80% of the energy in the backward direction with respect to the incident beam. The main advantage of this configuration is to produce at resonance a reflected signal free from interference due to the substrate.

Figure 2 :

 2 Figure 2: Reflectivity and transmitivity spectra of the CRIGF in each outcoupling direction.

Figure 3 :

 3 Figure 3: Scheme of the different contributions in each outcoupling direction.

Figure 4 :

 4 Figure 4: Scheme of the CRIGF with a gold substrate to control the emission direction

Figure 5 :

 5 Figure 5: Evolution of the reflected signals R +T and R - T versus the thickness h 0 of the Su-8 buffer layer.

Figure 6 :

 6 Figure 6: Reflectivity spectrum of the CRIGF in the quasi-total backward reflection configuration.


François Renaud

Stéphane Calvez

Antoine Monmayrant

Evgeni Popov

Anne-Laure Fehrembach

Olivier Gauthier-Lafaye
email: olivier.gauthier-lafaye@laas.fr

Using Cavity Resonator Integrated Grating Filters for second harmonic generation

Keywords: non-linear optics, micro-resonator, subwavelength gratings

Cavity Resonator Integrated Grating Filters are micro-structured integrated resonators that exhibit resonances with Q factors in order of a thousand and that can be excited with focussed beams. These two characteristics make them particularly attractive as base components for non-linear optical generation. In this contribution, we experimentally demonstrate second harmonic generation in such devices under continuous wave excitation and show that the measured performance are in good agreement with theoretical predictions.

INTRODUCTION

Cavity Resonant Integrated Grating Filters (CRIGF) were introduced by Prof. Ura's team in 2012 [START_REF] Kintaka | Cavity-resonator-integrated guidedmode resonance filter for aperture miniaturization[END_REF] and have thus been thoroughly investigated by different teams. They allow narrow-band and efficient spectral filtering under tightly focused excitation [START_REF] Buet | High angular tolerance and reflectivity with narrow bandwidth cavity-resonator-integrated guided-mode resonance filter[END_REF], and can achieve polarization independence [START_REF] Kintaka | Polarization-independent guided-mode resonance filter with cross-integrated waveguide resonators[END_REF], and can be finely tuned by variational design [START_REF] Augé | Tunable graded cavity resonator integrated grating filters[END_REF], [START_REF] Kintaka | Grating-Position-Shifted Cavity-Resonator-Integrated Guided-Mode Resonance Filter[END_REF]. They have been demonstrated in different wavelength ranges from the near-infrared [START_REF] Buet | High angular tolerance and reflectivity with narrow bandwidth cavity-resonator-integrated guided-mode resonance filter[END_REF] to the mid-infrared [START_REF] Gauthier-Lafaye | Mid-infrared ($\sim$4.65μm) cavity resonator integrated grating filters[END_REF]. It has now been shown that their intrinsic properties stem from the excitation of a localized mode of high Q factor localized under the main grating coupler [START_REF] Rassem | Waveguide mode in the box with an extraordinary flat dispersion curve[END_REF] (see fig. 1).

Figure 1. schematic of a CRIGF highlighting its different parts : grating coupler (GC, blue), phase sections (PS, green) and distributed Bragg reflectors (DBR, red).

This localized mode with a high Q factor can be interesting for non-linear optical operations such as second harmonic generation (SHG) as it provides strong optical field enhancement within a small volume. We have recently demonstrated such SHG in these devices when made on a thin film of Lithium Niobate, a well-known material for its non-linear optical properties [START_REF] Renaud | Secondharmonic-generation enhancement in cavity resonator integrated grating filters[END_REF]. In this paper, we will detail recent experimental results obtained on such devices.

RESULTS

Samples are fabricated on a NanoLN-provided wafer made of a LiNbO3 substrate, a 2.3-µm-thick SiO2 spacer and a thin, 300-nm-thick LiNbO3 film oriented along its Z-cut direction. The grating layer was made of a 70-nm-thick Si3N4 layer deposited using low-temperature ICP-PECVD deposition. Electron-beam lithography was used to define the grating structures in a 130-nm-thick PMMA resist. After development, RIE dry etching was performed using a CHF3/O2 chemistry. It should be noted that the LiNbO3 layer acts as a stop etch during this fabrication step, resulting in smooth, homogeneous etching base. As a last step, a 460-nm-thick SiO2 capping layer was deposited using once again ICP-PECVD. This capping layer results in a low-reflectivity stack around the target pump wavelenght of 1.55 µm. For this wavelength of operation and the stack described above, the GC period is around 865 nm, with a DBR period half that of the GC.

The fabricated CRIGF exhibited sharp resonance peaks around 1.55 µm, and the SHG generation was found to follow exactly the resonances, as shown on figure 2.

Figure 2. Evolution of the SHG intensity with the pump wavelength for 4 different CRIGF periods indicated in

the inset. We will present results achieved on these filters and compare them to theoretical results. Routes towards higher SHG efficiency will also be presented.

ACKNOWLEDGEMENTS

Authors gratefully acknowledge support from the ANR under grant ANR-ASTRID-RESON. The authors also thankfully acknowledge the LAAS cleanroom team for technical support and technological expertise provided within the French RENATECH network of cleanroom facilities. F. Renaud acknowledges PhD grant from DGA under supervision from P. Adam.