Donglai Gao

Wen-Li Chen

Run-Tao Zhang

Ye-Wei Huang

Hui Li

Multi-modal vortex-and rain-wind-induced vibrations of an inclined flexible cable

Keywords: flexible cable, vortex induced vibrations, rain-wind induced vibrations, multi-modal behaviours

We reproduce multi-modal vortex-induced vibrations (VIVs) and multi-modal rainwind-induced vibrations (RWIVs) of an inclined and yawed cable in wind tunnel tests.

The flexible cable model has low mass and low damping. First, the cable model is kept dry and exposed to uniform airflow; it experiences first-, second-and third-mode VIVs with the increase of wind speed. The structural responses of VIVs are analysed and the frequency lock-in phenomenon is observed for different modes of VIVs of the flexible cable. In addition to VIVs of the dry cable, RWIVs are excited by guiding water rivulets on the cable surface from a water tube. The first-, second-and third-mode RWIVs of the flexible cable are observed and identified at a much higher range of incoming wind speeds than that of the VIVs. To further explore the origin of the multi-modal behaviours of RWIVs, the upper rivulet is guided and restricted to forming locally along the cable. Experimental results reveal that the higher-and multiple-mode RWIVs can be excited by a local rivulet, even under a lower wind speed. Finally, RWIVs are compared to VIVs to uncover their underlying similarities and relationships.

Introduction

Because of their inherent flexibility, low damping and relatively low mass, long stay cables of cable-stayed bridges are sensitive to wind-induced vibrations. Vortex-induced vibrations (VIVs) and rain-wind-induced vibrations (RWIVs) have been issues of great concern to the bridge engineering community, because the large-amplitude vibrations may lead to fatigue and durability problems. For example, cable-deck connections may suffer from failure, owing to the breakdown of protections against corrosion [START_REF] Pacheco | Keeping cables calm[END_REF][START_REF] Poston | Cable-stay conundrum[END_REF].

When subjected to wind, stay cables are characterised by flow separation and alternating vortex shedding into the wake. Periodic vortex shedding can result in large unsteady cross-flow forces acting on the cable [START_REF] Brika | Vortex-induced vibrations of a long flexible circular cylinder[END_REF]. If the shedding frequency approaches an order of the modal frequency of the flexible cable, a cable-wake resonance develops. This fluid-structure synchronization, often referred to as 'lock-in', may induce significant cross-flow vibrations of the cables. In bridge engineering, multimodal VIVs on stay cables are often associated with the non-uniformity of the incoming flow, because bridges are usually constructed in the atmospheric boundary layer [START_REF] Matsumoto | Field observation of the full-scale wind-induced cable vibration[END_REF].

Chen et al. [START_REF] Chen | Numerical Simulation of Vortex-Induced Vibrations of Inclined Cables under Different Wind Profiles[END_REF] conducted a numerical study, suggesting that a 100-m inclined cable would experience multi-modal VIVs under different velocity profiles. Chen et al. [START_REF] Chen | An experimental investigation on vortex induced vibration of a flexible inclined cable under a shear flow[END_REF] experimentally investigated a flexible cable subject to sheared incoming flow. Results revealed that the cable model experienced single-and multi-modal VIVs under different incoming velocity profiles, and the vortex shedding frequencies from the cable synchronised well with the cable's modal frequencies.

Hikami and Shiraishi [START_REF] Hikami | Rain-wind induced vibrations of cables stayed bridges[END_REF] found that the amplitude of wind-induced vibrations of stay cables could be greatly amplified if they are simultaneously subject to light-to-moderate rain conditions. Owing to great vibration amplitudes, RWIVs have become a major concern for bridge engineers over the past three decades. Vigorous efforts have been made to explore the excitation mechanism of severe cable vibrations under rain and wind conditions. Various research philosophies and approaches, including theoretical modelling [START_REF] Yamaguchi | Analytical study on growth mechanism of rain vibration of cables[END_REF]- [START_REF] Lemaitre | Rainwater rivulet running on a stay cable subject to wind[END_REF], field observation [START_REF] Ni | Field observations of rain-windinduced cable vibration in cable-stayed Dongting Lake Bridge[END_REF]- [START_REF] Zuo | Interpretation of field observations of wind-and rain-windinduced stay cable vibrations[END_REF], computational fluid dynamics [START_REF] Li | A numerical and experimental hybrid approach for the investigation of aerodynamic forces on stay cables suffering from rainwind induced vibration[END_REF][START_REF] Cheng | Multi-scale simulation of rainwater morphology evolution on a cylinder subjected to wind[END_REF] and wind tunnel tests [START_REF] Flamand | Rain-wind induced vibration of cables[END_REF]- [START_REF] Du | Aerodynamic characteristics of an inclined and yawed circular cylinder with artificial rivulet[END_REF] have been employed to obtain a better understanding of the behaviours and excitation mechanisms of RWIVs. This type of severe cable vibration, jointly induced by rain and wind, is characterised by much lower frequencies and much larger amplitudes than conventional VIVs. It is generally recognised that RWIVs often occur for wind-speed ranges of 8 to 15 m/s, in the frequency range of 1-3 Hz and with yaw angles 20-60°, as concluded by Flamand [START_REF] Flamand | Rain-wind induced vibration of cables[END_REF]. Additionally, the upper rivulet on the leeward side of the cable surface and its circumferential oscillation play a crucial role in the excitation of RWIVs. Several wind tunnel tests [START_REF] Flamand | Rain-wind induced vibration of cables[END_REF]- [START_REF] Du | Aerodynamic characteristics of an inclined and yawed circular cylinder with artificial rivulet[END_REF] have been conducted to gain insights into the excitation mechanism of RWIVs; they considerably improved our understanding of this phenomenon. However, during previous wind tunnel tests, rigid and segmental models were usually employed because of the limitation of the test sections. Thus far, the multimodal behaviours of RWIVs on a stay cable have never been reproduced or investigated in a wind tunnel test. An alternative approach to studying multi-modal RWIVs is field monitoring. Ni et al. [START_REF] Ni | Field observations of rain-windinduced cable vibration in cable-stayed Dongting Lake Bridge[END_REF] measured the response characteristics and excitation mechanisms of stay cables experiencing RWIVs on the Dongting Lake Bridge, P. R.

China. Field monitoring was realised by employing accelerometers, anemometers and rain gauges. The dominant frequency of RWIVs corresponded to the third mode frequency of the cable, and the first five modes were found to participate. Zuo et al. [START_REF] Zuo | Field observation of vortex-and rain-windinduced stay-cable vibrations in a three-dimensional environment[END_REF] developed a long-term full-scale monitoring system on the Fred Hartman Bridge, United States, to investigate the characteristics of stay-cable vibrations. They attempted to interpret both VIVs and RWIVs in the context of the three-dimensional nature of the environment. By investigating the similarities and differences between VIVs and RWIVs, they argued that RWIVs might be a vortex-induced type of excitation that is different from conventional vortex shedding. This exhibited similarities to the research findings of Matsumoto et al. [START_REF] Matsumoto | Rain-wind-induced vibration of inclined cables at limited high reduced wind velocity region[END_REF], in which RWIV was explained as a type of VIV at a high reduced velocity. It was shown by experiments that the upper rivulet on the cable surface and the incoming turbulence played crucial roles in the mechanism of the VIV at a high reduced velocity. Zuo and Jones [START_REF] Zuo | Interpretation of field observations of wind-and rain-windinduced stay cable vibrations[END_REF] reported a type of large-amplitude vibration of dry cables at a high reduced velocity. They proposed that the largeamplitude cable vibrations, with or without rainfall, might originate from the same type of vortex-shedding mechanism that is different from the von Kármán vortex shedding.

Moreover, such a mechanism would be promoted and enhanced by the upper rivulet.

Experimental setup 2.1 Introduction to the laboratory facilities

Our experimental study was performed in the larger test section of the Joint Laboratory of Wind Tunnel and Wave Flume (WTWF), Harbin Institute of Technology, P. R. China.

The closed-loop wind tunnel laboratory has two test sections, a larger one and a smaller one, as sketched in Fig. 1(a). The dimension of the larger test section is 6.0 m (width) × 3.6 m (height) with a length of 50 m, whereas that of the smaller one is 4.0 m (width) × 3.0 m (height) ×25 m (length). A wave flume, as deep as 26 m, lies beneath the larger test section. The wave flume is separated from the larger test section by movable and perforated floors, as noted in Figs. 1(a) and 1(c). The perforated floor allows an immediate water drainage into the wave flume. Therefore, the larger section of the WTWF laboratory was suitable for performing our experimental study of RWIVs.

Flexible cable model and experimental setup

To excite multi-modal vibrations, the flexible cable model needs to be sufficiently long, within the limitations of the larger test section of WTWF. A cable having a total length, L , of 8.31 m and an outer diameter, D , of 98.36 mm was manufactured to reproduce multi-modal VIVs and RWIVs for our experiment. The flexible cable model realises low mass and low damping ratios on a 12-mm diameter steel wire rope inside with multiple layers of foam taps wrapped around it, as illustrated in Fig. 1(b). These are the key factors influencing the occurrence of VIVs and, particularly, RWIVs. A polyolefin heat-shrink tube was adopted as the outside coating to create a smooth surface for the flexible cable model. The model was tensioned by two rigid support frames, both firmly fixed between the tunnel floor and the ceiling. Thus, the support frames were static, experiencing no detectable vibrations during the wind tunnel tests. The upper rivulet plays a crucial role in the occurrence of RWIVs. Spraying water in the wind tunnel has been a long-established method of forming an upper rivulet for the reproduction of RWIVs [START_REF] Hikami | Rain-wind induced vibrations of cables stayed bridges[END_REF][START_REF] Li | A numerical and experimental hybrid approach for the investigation of aerodynamic forces on stay cables suffering from rainwind induced vibration[END_REF]. However, some recent studies [START_REF] Jing | Study on the role of rivulet in rain-windinduced cable vibration through wind tunnel testing[END_REF][START_REF] Jing | Excitation mechanism of rain-wind induced cable vibration in a wind tunnel[END_REF] successfully excited RWIVs by guiding water lines on the cable surface. It was verified that this approach had the same effects as spraying water. In our study, a plastic water tube with an inner diameter of 5 mm was glued onto the upper end of the cable model to guide the upper water rivulet and to generate RWIVs, as shown in Fig. 2(b). It is like the configuration made by Jing et al. [START_REF] Jing | Study on the role of rivulet in rain-windinduced cable vibration through wind tunnel testing[END_REF][START_REF] Jing | Excitation mechanism of rain-wind induced cable vibration in a wind tunnel[END_REF], who placed a large water tank outside the wind tunnel, allowing the water flow to be controlled by a valve. In our experiments, the outlet flow rate from the water tube was constantly kept to 21.14 ml/s. This flow rate is relatively large but highly necessary for the reproduction of RWIVs. We found that the upper rivulet could hardly be formed with smaller tube diameters. Thus, the RWIVs could never be reproduced. However, during the RWIVs, only a small amount of water was witnessed to have been blown out by the wind. Even less water contributed to the formation of the upper rivulet. Therefore, most of the guided water are manipulated by gravity to form the lower rivulet. Therefore, the lower rivulet was found to be notably thick, resulting in added mass to the cable model.

Experiment results

Multi-modal VIVs of the flexible cable

The root-mean-square (RMS) values of the acceleration and vibration displacements (both in-plane and out-of-plane) of the flexible cable model are collected and plotted against incoming wind speeds in Fig. 3. The displacement time-histories were obtained by integrating the recorded accelerations. The amplitudes, (A), in the present study, were normalised by dividing by the cable diameter, (D). When the wind speed gradually increased from 1.33 m/s, the dry cable remained initially stable. As the wind speed exceeded 1.67 m/s, cable vibrations could be observed. Afterwards, the cable vibrations became larger with the increase of wind speed. When the wind speed reached 1.92 m/s, the vibration amplitudes obtained a local maximum. Then, the cable vibrations witnessed a downward trend, whereas the wind speed continuously increased. At an incoming wind speed of 2.71 m/s, the vibration amplitudes reached the local minimum.

Then, the first-mode frequency lock-in ended, and the second-mode lock-in developed.

U

Therefore, the first-mode VIVs occurred in the wind speed range of 1.67-2.71 m/s.

Additionally, the in-plane vibrations were much more profound than the out-of-plane vibrations for each test case, which is consistent with the field monitoring results of Zuo et al. [START_REF] Zuo | Field observation of vortex-and rain-windinduced stay-cable vibrations in a three-dimensional environment[END_REF] and Zuo and Jones [START_REF] Zuo | Interpretation of field observations of wind-and rain-windinduced stay cable vibrations[END_REF]. As the wind speed continued to increase, second and third mode VIVs occurred. The phenomena of higher modes VIVs were like that of the first mode, but the frequency lock-in range was found to expand gradually.

Moreover, the local maximum of the cable vibrations for the second-and third-mode

VIVs were found at an incoming wind speed of 3.42 and 5.42 m/s, respectively.

Multi-modal RWIVs of the flexible cable

f 3 =6.36 Hz
The wind tunnel tests for RWIVs started with a wind speed of 7.92 m/s and gradually increased in steps of 0.42 m/s. First, the upper water rivulet failed to form at low wind speed because it dropped or slipped down to the lower surface of the cable model.

Consequently, the wind-induced vibrations of the flexible cable were observed to be minimal, and no RWIVs signal was recorded by the accelerometers. At a wind speed of 8.76 m/s, some water rivulets intermittently climbed up the cable surface. However, no continuous upper rivulet was observed along the cable, and thus no RWIVs were generated. However, when the incoming wind speed shifted from 10.01 m/s to 10.43 m/s, an upper rivulet gradually formed along the entire cable, moving periodically around the cable surface. The lower rivulet was clear, but nearly static. Meanwhile, a violent cable vibration was easily detected. Therefore, multi-modal behaviours of RWIVs in a stay cable were successfully reproduced in our wind tunnel tests. RWIVs at this wind speed were dominated by its first vibration-shape mode. As the wind speed continued to increase, the amplitudes of RWIVs gradually enhanced. As the wind speed changed from 13.76 m/s to 14.18 m/s, the vibration amplitudes were found to decrease while remaining notable, as shown in Fig. 10(a). In this case, the mode shape indicates that the cable vibration was dominated by the second mode because the node was nearly fixed at L/2. When the wind speed increased to 14.60 m/s, the dominant vibration mode was changed to its third mode, as revealed by the frequency analysis in Fig. 10(b). Then, the cable vibrated at its third mode with its wind speed getting higher. When the incoming wind speed was in the range of 14.18-14.60 m/s, the amplitudes of cable vibrations diverged because of the mode-switch scenario ( [START_REF] De Langre | Frequency lock-in is caused by coupled-mode flutter[END_REF][START_REF] Violette | A linear stability approach to vortexinduced vibrations and waves[END_REF]). During RWIVs, the in-plane vibration amplitudes were much larger than the out-of-plane vibrations, as can be seen in Fig. 10(a). This agrees with the field monitoring results by Zuo et al. [START_REF] Zuo | Field observation of vortex-and rain-windinduced stay-cable vibrations in a three-dimensional environment[END_REF] and Zuo and Jones [START_REF] Zuo | Interpretation of field observations of wind-and rain-windinduced stay cable vibrations[END_REF]. In our study, three different test cases, at incoming wind speeds The frequency spectrum of the cable vibrations is plotted in Fig. 14 to demonstrate that the cable experienced second-mode RWIVs. 16 presents their frequency spectrum. The phenomenon is like the first and second RWIVs, but with a higher order frequency. The higher-mode RWIVs tend to occur at higher wind speed. These characteristics of RWIVs are similar to those of the multi-modal VIVs. It is believed that a higher wind speed contributes to a larger vortex shedding frequency from the cable model, and that the wake frequency can couple with the higher mode of the cable's frequency. This Unlike VIVs, the dominant frequencies of RWIVs were found to deviate from the modal frequencies of the dry cable in still air. During our experimental campaign, before and after wind tunnel tests, we conducted two free vibration tests for each case to ensure that no tension slack occurred in the steel wire ropes. Thus, the frequency deviation during experiments was caused by the joint action of changing tension forces and the added mass of water. According to Irvine [START_REF] Irvine | Cable Structures[END_REF], the vibration frequency of a cable can be described as

2 S n f L m , (1)
where n denotes the n th mode, is the axial tension and S m is the cable mass.

Owing to the existence of lower and upper rivulets, the added mass always leads to a decrease of the vibration frequency. With a larger vibration amplitude, the effects of the added mass are minimal because only a small amount of water is retained on the cable surface (i.e., the rivulets become thin). Conversely, when the vibration amplitude is small, the effects of added mass become more profound. However, a larger vibration amplitude contributes to an increase of axial tension force and, therefore, an increase f 3 =6.01 Hz of the vibration frequency, as can be derived from Eq. ( 1). This observation has also been confirmed and reported by previous studies (see Fig. 11 (a) in Ref. [START_REF] Violette | A linear stability approach to vortexinduced vibrations and waves[END_REF]).

In first-mode RWIVs, the vibration amplitude was large, thus the tension force was substantially larger and the added mass was small. Thus, the vibration frequency increased by 0.36 Hz. However, when the flexible cable experienced third-mode RWIVs, the vibration amplitudes became smaller and more water was retained on the cable surface (i.e., the rivulets became thick). Thus, the effects of added mass became dominant, whereas the effect of the increased axial tension force greatly shrunk, resulting in a slightly smaller frequency than that of the dry cable in still air. In the case of the second-mode RWIVs, the effects of the two factors roughly overcame each other.

Thus, the vibrations frequencies were close to that of the dry cable.

RWIVs excited by locally distributed rivulets

As shown in the field monitoring results reported by Zuo et al. [START_REF] Zuo | Field observation of vortex-and rain-windinduced stay-cable vibrations in a three-dimensional environment[END_REF], in-situ RWIVs often exhibit multiple modes. Besides the three-dimensional incoming airflow, the distribution of upper rivulets is also suspected to play a crucial role in the occurrence of multi-modal RWIVs. During bridge engineering, stay cables are usually as long as 100 m. Therefore, rivulets cannot possibly be formed along the entire cable length.

Inspired by this, we investigated the RWIVs excited by locally distributed upper rivulets.

The locally distributed upper rivulets were generated by changing the position of the water supply and using a pipe to prevent rivulets from forming along the entire cable length. We designed four types of locally distributed upper rivulets, as shown in Fig. RWIV responses of the cable with different guided rivulets at an incoming wind speed V= 11.26 m/s are measured and presented in Fig. 18. For the test case of LR1 (i.e., the upper rivulet is restricted to form in the middle L/2), the RWIV is identified as the first mode, as shown in Fig. 18. However, the vibration amplitudes were smaller than those in Fig. 11, when the upper rivulet was formed along the entire cable. Additionally, when U the upper rivulet was guided to form in the upper L/3 and was prevented from slipping down to the lower span of the cable, for test case LR2, the RWIVs were identified as second-mode at the same wind speed. When the rivulet formation length was limited to the upper L/2, for test case LR3, the RWIVs were also identified as second-mode, but the vibration amplitudes were notably larger than those of test case LR2. Furthermore, because the upper rivulet could form in the upper 2L/3, for test case LR4, RWIVs of the flexible cable exhibited multiple modes. By performing an FFT analysis to identify the frequency components, the cable vibrations were found to be jointly dominated by the first and second modes, as shown in Fig. 18.

In the previous study of Ni et al. [START_REF] Ni | Field observations of rain-windinduced cable vibration in cable-stayed Dongting Lake Bridge[END_REF], they reported that the overall dominant mode of the RWIVs was the third mode, but the dominant mode can differ for various response segments. In our experimental study, whereas the cable length was limited and much smaller than real cables, the RWIVs with different modes and amplitudes were successfully reproduced by guiding differently and locally distributed upper rivulets.

This implies that the local distributions of upper rivulets contributed to exciting the multi-modal RWIVs, providing another possible explanation for the field observation of the RWIVs of real cables often exhibiting multiple modes.

Discussion

According to the research findings of Matsumoto et al. [START_REF] Matsumoto | Rain-wind-induced vibration of inclined cables at limited high reduced wind velocity region[END_REF], RWIV was explained as a

VIV, but at a high reduced velocity region. In their wind tunnel tests, they found two dominant frequency components: one for Kármán vortex shedding and another with a much lower frequency. The latter was the driving force of RWIV. Moreover, as reported by Zuo et al. [START_REF] Zuo | Field observation of vortex-and rain-windinduced stay-cable vibrations in a three-dimensional environment[END_REF], substantial similarities exist between VIVs and RWIVs with in-situ field monitoring, perhaps owing to a vortex-induced excitation that is different from conventional Kármán vortex shedding. A low-frequency vortex, for instance, is a driving factor, as shown by Matsumoto et al. [START_REF] Matsumoto | Rain-wind-induced vibration of inclined cables at limited high reduced wind velocity region[END_REF]. In our study, both the higher-mode VIVs and RWIVs tended to develop at a higher wind speed. For VIVs, the frequency of the vortex shedding from the cable became larger as the wind speed increased and synchronised with higher-mode cable frequencies. A similar excitation scenario of RWIVs was found. Additionally, frequency overlap and mode-switching of VIVs have been reported in experiments and theoretical predictions [START_REF] De Langre | Frequency lock-in is caused by coupled-mode flutter[END_REF][START_REF] Violette | A linear stability approach to vortexinduced vibrations and waves[END_REF][START_REF] Facchinetti | Vortex-induced travelling waves along a cable[END_REF]. While they were not observed in the VIVs in our experiments, it is interesting to find the frequency overlap and mode switch phenomena in RWIVs, as shown in Fig. 10. Based on these experimental results, we conclude that RWIVs exhibited notable similarities to VIVs, as suggested by Zuo et al. [START_REF] Zuo | Field observation of vortex-and rain-windinduced stay-cable vibrations in a three-dimensional environment[END_REF] and Matsumoto et al. [START_REF] Matsumoto | Rain-wind-induced vibration of inclined cables at limited high reduced wind velocity region[END_REF]. This undiscovered connection between the two is the key to disclosing the true excitation mechanism of RWIVs. Apart from increasing wind speeds, locally distributed upper rivulets were also found to play a crucial role in exciting multi-modal RWIVs, even under lower wind speeds. Therefore, increased wind speeds and local upper rivulets are two main sources of multi-modal RWIVs in cables.

Conclusions

By employing one flexible cable model, we reproduced and investigated multi-modal VIVs and RWIVs in wind tunnel tests. We made progress with the study of multi-modal RWIVs of stay cables with our wind tunnel tests. RWIVs were measured on the same framework as VIVs, and the main conclusions are as follows:

1) When the dry cable model was subjected to uniform airflow, it was found to experience first-, second-and third-mode VIVs with increased wind speeds. The phenomena of frequency lock-ins, observed in different modes of VIVs of the flexible cable, were similar.

2) It is feasible to reproduce multi-modal RWIVs in a wind tunnel. The key factors are minimizing the damping ratio, increasing the length and reducing the mass of the cable model. The present study verified that guiding water lines on the cable surface can excite the RWIVs.

3) Large-amplitude RWIVs were excited under the joint action of wind and rain.

Higher modes of the RWIVs on a flexible cable tended to develop at higher wind speeds, and the mode-switch phenomenon was observed, whereas the incoming wind speed remained the same. Underlying links between the two seemingly different types of wind-induced cable vibrations should not be overlooked; they remain to be explored in future studies.

Figure 1 .

 1 Figure 1. Cable model and experimental setup: (a) 3D-sketch of the WTWF laboratory; (b) cross-section of the flexible cable model and (c) the inclined flexible cable model.

Figure 2 .

 2 Figure 2. Cable model configuration: (a) spatial position of the inclined cable and (b)guiding water on the upper end of the cable to generate upper rivulet, thus exciting RWIVs.

Figure 3 .Figure 4 .Figure 5 .Figure 6 .Figure 7 .Figure 8 .Figure 9 .

 3456789 Figure 3. RMS of accelerations (a) and the normalised vibration amplitudes (b) of the inclined cable with the increase of wind speed.

Figure 11 .

 11 Figure 11. Accelerations (a) and the normalised displacement responses (b) of the first mode RWIVs at 11.26 m/s.

Figure 12 .

 12 Figure 12. Frequency spectrum of the first mode RWIVs.

Figure 13 .

 13 Accelerations (a) and the normalised displacement responses (b) of the second mode RWIVs at 14.18 m/s.

Figure 14 .

 14 Figure 14. Frequency spectrum of the second-mode RWIVs.

Figure. 15

 15 Figure. 15 illustrates the vibration time-histories of the third-mode-dominated RWIVs

Figure 15 .

 15 Figure 15. Accelerations (a) and normalised displacement responses (b) of thirdmode RWIVs at 15.43 m/s.

Figure 16 .

 16 Figure 16. Frequency spectrum of third-mode RWIVs.

17

 17

Figure 17 .

 17 Figure 17. Sketch of the locally distributed upper rivulets.

LR1

f 1 =2. 30 HzFigure 18 .

 13018 Figure 18. The normalised displacement responses (left) and frequency spectrum (right) of the flexible cable under different rivulet distributions. In every subplot, the bluedashed line denotes in-plane signals and the red-solid line denotes the out-of-plane signals.

4 ) 5 )

 45 The RWIVs of the flexible cable were found to exhibit different vibration modes and amplitudes, owing to the different distributions of the guided upper rivulet. The experiments suggest that the local distributions of upper rivulet contributed to exciting the multi-modal RWIVs, even under lower wind speeds. Therefore, the effects of locally distributed upper rivulets provided another possible cause for the occurrence of multi-modal RWIVs, generally observed in real cables of cablestayed bridges. RWIVs on flexible cables exhibit similarities to VIVs in wind tunnel tests.

  In the present study, we reproduce multi-modal VIVs and multi-modal RWIVs by employing an inclined flexible cable model in a wind tunnel test. The VIVs are

	reproduced under a smooth incoming airflow, and the RWIVs are excited under the
	joint action of a smooth incoming flow and a guided upper rivulet.
	This paper is organised as follows. Details of the flexible cable model and the
	experimental setup are given in Section 2. Experiment results, including multi-modal
	VIVs and multi-modal RWIVs, excited by either increasing wind speed or local rivulets
	of the flexible cable, are presented and analysed in Section 3. Conclusions are drawn in
	Section 4.

Table 1

 1

	Support frame
	Support frame
	Cable model
	Accelerometers	L/6 Knot
	Wind speed

. Additionally, the mass of the cable model was measured to be 1.03 kg/m. As shown in Fig.

2

(a), the cable's yaw angle, , was set as 45°, and its inclination angle, , was decided by the size of the test section to be 23.39°. Both are appropriate conditions for the generation of RWIVs.

Table 1 .

 1 Modal frequencies and damping ratios of the flexible cable model.

		In-plane		Out-of-plane
	Mode	Frequency (Hz) Damping ratio (%) Frequency (Hz) Damping ratio (%)
	First	2.31	0.41	2.14	0.39
	second	4.17	0.43	4.03	0.42
	Third	6.36		6.21

Acknowledgements

This study was funded by the National Natural Science Foundation of China (NSFC) through Grants 51378153, 51578188 and 51722805. D. G also acknowledges the financial support from China Scholarship Council (CSC 201606120189).


Mariama Akodad

Guillaume Schurtz

Jérôme Adda

Florence Leclercq

François Roubille

Management of valvulopathies with acute severe heart failure and cardiogenic shock


L Moreaux

O Sandre

J Mertz

Membrane imaging by second-harmonic generation microscopy

We present a detailed analysis of the generation of second-harmonic radiation from biological membranes labeled with a styryl dye. In particular, we consider the high-numerical-aperture limit appropriate to highresolution microscopy in which an excitation beam is tightly focused from the side onto a membrane surface. In this limit the active surface area that contributes to second-harmonic generation (SHG) depends only on the tightness of the beam focus and the SHG radiation is confined by phase matching into two well-defined off-axis lobes. We derive expressions for the SHG radiation power, angular distribution, and polarization dependence in the cases of ideal or nonideal molecular alignment in the membrane and uniaxiality of the molecular hyperpolarizability. We define an SHG cross section similar to that used in two-photon-excited fluorescence (TPEF) to permit direct comparison of the two imaging modalities. Finally, we corroborate our results with experiments based on the excitation of a styryl dye in giant unilamellar vesicles with a mode-locked Ti:sapphire laser.

INTRODUCTION

Nonlinear optics is proving to be a powerful tool for biological imaging. Fluorescence microscopy by two-photon excitation [START_REF] Denk | Two-photon laser scanning fluorescence microscopy[END_REF][START_REF] Lakowicz | Multiphoton excitation of biochemical fluorophores[END_REF][START_REF] Xu | Multiphoton excitation of molecular fluorophores and nonlinear laser microscopy[END_REF] has become a laboratory standard, and three-photon excitation has demonstrated its feasibility. [START_REF] Maiti | Measuring serotonin distribution in live cells with three-photon excitation[END_REF][START_REF] Xu | Multiphoton fluorescence excitation: new spectral windows for biological nonlinear microscopy[END_REF] More recently, nonlinear microscopies based on radiative harmonic generation have also been applied to the imaging of surfaces [START_REF] Dadap | Measurement of the vector character of electric fields by optical second-harmonic generation[END_REF][START_REF] Gauderon | Threedimensional second-harmonic generation imaging with femtosecond laser pulses[END_REF] and biological samples. Second-harmonic generation (SHG) has been used to image the intrinsic nonlinear susceptibility of muscle tissues [START_REF] Guo | Secondharmonic tomography of tissues[END_REF] and to image living cells labeled with styryl dyes. [START_REF] Peleg | Non-linear optical measurement of membrane potential around single molecules at selected cellular sites[END_REF][START_REF] Campagnola | High-resolution nonlinear optical imaging of live cells by second harmonic generation[END_REF] Third-harmonic generation has also been used to provide microscopic images of cells and plant samples. [START_REF] Yelin | Laser scanning thirdharmonic-generation microscopy in biology[END_REF][START_REF] Muller | '3D-microscopy of transparent objects using thirdharmonic generation[END_REF] Harmonic generation is an optical phenomenon involving coherent radiative scattering, whereas fluorescence generation involves incoherent radiative absorption and reemission. As such, fluorescence and harmonic images are derived from fundamentally different contrast mechanisms. Moreover, because of its coherent nature, harmonic radiation is usually highly directional and depends critically on the spatial extent of the emission source, making a full description of harmonic generation more complicated than fluorescence generation. We limit our discussions here to the description of molecular SHG only, with the understanding that many of our techniques may be generalized to higher-order harmonics.

It is well known that SHG of dipolar origin cannot arise from a medium that possesses an inversion symmetry. Therefore molecular SHG is usually studied in geometries in which the radiating molecules are spatially ordered. A common geometry is that of a planar interface or a surface that imparts a preferential orientation to a thin layer of molecules. [START_REF] Heinz | Determination of molecular orientation of monolayer adsorbates by optical second-harmonic generation[END_REF] Typically, the layer is illuminated at an oblique angle by a laser beam, resulting in the emission of SHG beams in well-defined reflection and transmission directions. [START_REF] Bloembergen | Second harmonic reflected light[END_REF][START_REF] Shen | The Principles of Nonlinear Optics[END_REF] Because the laser beam is usually unfocused or weakly focused, both the driving laser field and the resultant SHG fields can be treated as simple collimated beams with well-defined wave vectors, and the layer of molecules is most appropriately described in this geometry by a macroscopic surface susceptibility that is independent of the illuminated surface area.

In contrast, if SHG is intended to provide microscopic image resolution, the driving laser beam must be focused to a small spot size. The driving field can no longer be considered a simple plane wave as above, and the structure of the resultant SHG radiation becomes critically dependent on the particular geometry of the driving field near the focal center. The use of a macroscopic surface susceptibility to quantify the SHG emission becomes inappropriate in this case, and one must resort to a more refined description of the surface nonlinearity for length scales much smaller than the radiation wavelength. In particular, it becomes necessary to redefine a surface nonlinearity starting from the level of individual molecular hyperpolarizabilities (or nonlinear molecular cross sections).

In this paper we provide a detailed theoretical description of SHG emission appropriate to a high-resolution scanning microscope configuration that uses a tightly focused excitation beam. The sensitivity of SHG emission to tight focusing has already been recognized for bulk nonlinear crystals. [START_REF] Boyd | Nonlinear Optics[END_REF] Because our attention is directed to biological imaging we consider here the different geometry of molecular SHG from membrane surfaces, as has been demonstrated experimentally in the imaging of cells and vesicles. [START_REF] Campagnola | High-resolution nonlinear optical imaging of live cells by second harmonic generation[END_REF][START_REF] Lewis | Second-harmonic generation of biological interfaces: probing the membrane protein bacteriorhodopsin and imaging membrane potential around GFP molecules at specific sites in neuronal cells of C. elegans[END_REF][START_REF] Moreaux | Membrane imaging by simultaneous second-harmonic generation and two photon microscopy[END_REF] We begin by characterizing the SHG emission from a single hyperpolarizable molecule in terms of a nonlinear scattering cross section. We then extend our discussion to the characterization of SHG emission from a planar distribution of molecules. In particular, we adopt a formalism that specifically permits a direct comparison between the radiated powers in the case of SHG and in the corresponding case of two-photonexcited fluorescence (TPEF); the latter is already well established in biological microscopy. Indeed, recent results show that simultaneous SHG and TPEF can be obtained from the same collections of molecules. Finally we corroborate our theoretical formalism with experiments based on a full characterization, in terms of power, angular distribution, and polarization, of the SHG radiation obtained from giant unilamellar vesicles labeled with styryl dye. Our research here serves as a follow-up to provide complete theoretical and experimental support for the results presented in Ref. 18.

THEORY OF SHG MICROSCOPY

A. Single Molecule

We begin by defining the SHG cross section of a single molecule. The most straightforward definition entails calculating the total radiated SHG power and dividing this by the square of the incident driving field intensity. In particular, a molecule is considered an elemental dipole radiator driven by the excitation light according to its first hyperpolarizability. For simplicity we assume that the excitation field is linearly polarized in the z ˆdirection and begin by examining only the ␤ zzz component of this hyperpolarizability, denoted ␤. If the excitation light has frequency , the induced dipole moment at frequency 2 will be given by

2 ϭ 1 /2␤E 2 z ˆ, (1)
where E is the excitation field amplitude and we have used the Taylor convention. [START_REF] Willets | Problems in the comparison of theoretical and experimental hyperpolarizabilities[END_REF] The radiated secondharmonic far field at an inclination from the z axis is

E 2 ͑ ͒ ϭ Ϫ 2 2 0 c 2 r sin͑ ͒exp͑ Ϫ2i͓t͔͒ ˆ, ( 2
)
where 0 is the free-space permittivity, c is the speed of light, r is the observation distance from the dipole, and [t] is the corresponding retarded time. The resultant power per differential solid angle at an inclination , in units of photons/second, may be expressed as

P 2 ͑ ͒ ϭ 3 16 SHG sin 2 ͑ ͒I 2 , ( 3
)
where I is the excitation intensity in units of (photons/ second)/area and

SHG ϭ 4n 2 ប 5 3n 2 0 3 c 5 ͉␤͉ 2 (4)
(n and n 2 are the indices of refraction at and 2; ប is Planck's constant). We have defined SHG such that the total SHG power, obtained by integration of Eq. ( 3) over all solid angles, reduces to the simple expression

P SHG ϭ 1 /2 SHG I 2 (5)
(the extra factor of 1/2 stems from our description of powers in units of photons/second rather than in watts).

We recall that the fluorescence power emitted by a dipole undergoing two-photon excitation can be expressed similarly as

P TPEF ϭ 1 /2 TPEF I 2 , ( 6
)
where TPEF is the two-photon fluorescence (or action) cross section, defined by the two-photon absorption cross section multiplied by the fluorescence quantum yield. [START_REF] Xu | Measurement of two-photon excitation cross section of molecular fluorophores with data from 690 nm to 1050 nm[END_REF] In view of the similarity between Eqs. ( 5) and ( 6), we may regard SHG as the cross section for SHG of an individual dipole. In particular, TPEF and SHG may be expressed in the same units for direct comparison. We note here that SHG depends on the square of the magnitude of the molecular first hyperpolarizability, whereas TPEF depends on the imaginary part of the second hyperpolarizability. 21 As such, SHG tends to be much smaller than TPEF in practice.

B. Coherent Summation

We now turn to the case of a collection of molecules driven by a field E and generalize the procedure described above to evaluate the resultant SHG radiation angular distribution and power. We restrict ourselves to the geometry shown in Fig. 1, in which the dipoles are assumed to be uniformly distributed in a two-dimensional xy plane (membrane plane) and illuminated by a highly focused excitation beam propagating side-on in the x ˆdirection and polarized in the yz plane (the fact that we use a side-on geometry rather than a head-on geometry will become evident). Our basic strategy consists in coarsegrain averaging the molecular dipole moments over membrane surface areas whose dimensions are small compared with the radiation wavelength but large enough to encompass large numbers of molecules. In this way we may define a local induced macroscopic dipole moment per unit molecular surface density:

2,i ͑ x, y ͒ ϭ 1 /2E 2 ͑ x, y ͒ ͚ j,k ͗␤ ijk ͘ ˆj ˆk , (7)
where E (x, y) is the complex amplitude of the driving field at the position (x, y) on the membrane, ⑀ ˆis its polarization direction, ͗␤͘ symbolizes a local ensemble average of the molecular hyperpolarizabilities (we shall describe this average in more detail below), and we have neglected local field effects. We emphasize that our term ''macroscopic'' here is still microscopic relative to wavelength scales. Thus all the molecules taken in the coarse-grain average at a given position (x, y) are driven in phase with one another, and the net local dipole moment per unit area generated at this position is simply N s 2 (x, y), where N s is the molecular surface density. The phase of 2 (x, y) at different positions is determined by the phase of the driving field E (x, y). To calculate the radiation generated by a global membrane surface, we then treat the local dipole moments at each position on the membrane plane as elemental radiators and coherently sum their radiated electric fields while taking their relative phases and amplitudes into account.

As we emphasized in Section 1, the excitation field cannot be regarded as a simple collimated beam here, and indeed a full calculation that takes diffraction and polarization into account shows that the field's amplitude and phase cannot even be expressed analytically in the case of tight focusing. [START_REF] Born | Principles of Optics[END_REF] It has been shown, however, that as far as nonlinear interactions are concerned one may closely approximate a tightly focused field by assuming that its amplitude is Gaussian in both the axial and the lateral directions about the focal center and that its phase near the focal center progresses linearly. That is, we can write [START_REF] Guo | Secondharmonic tomography of tissues[END_REF] where w x and w y are the axial and the lateral field waists, respectively (explicit expressions for these may be found in App. B of Ref. 23), k ϭ n /c is the local wave vector, and is a parameter that characterizes the phase shift experienced by a Gaussian beam in the vicinity of a focal center. This phase shift is commonly referred to as a Gouy shift or a phase anomaly. [START_REF] Born | Principles of Optics[END_REF] In the case of weak focusing, may be approximated by (1 Ϫ 2/k 2 w y 2 ), whereas for tight focusing this expression tends to be a slight overestimate (see Appendix A). We emphasize again that the field in Eq. ( 8) differs from a collimated beam in two respects: It experiences a local axial amplitude variation about the focal center as well as a phase shift. Both of these effects play a considerable role in the structure of the resultant SHG.

E ͑ x, y ͒ ϭ ϪiE exp ͩ Ϫ x 2 w x 2 Ϫ y 2 w y 2 ϩ ik x ͪ , (
By adopting the coordinate system illustrated in Fig. 1, we proceed to calculate the second-harmonic far field radiated in the direction (, ). We begin by writing the contribution to the radiated electric field from the local induced polarization per unit molecular surface density and at the focal center only [indicated by the superscript (0) ; 2 (0) ϭ 2 (0, 0)]. This is given by

E 2 ͑ 0 ͒ ͑ , ͒ ϭ ͩ E 2 ͑ 0 ͒p ͑ , ͒ E 2 ͑ 0 ͒s ͑ , ͒ ͪ ϭ r M • 2 ͑ 0 ͒ , ( 9
)
where ϭ 2 / 0 c 2 and M is the projection matrix, defined by M ͫ ˆͬ ϭ ͫ Ϫsin cos sin cos cos 0 cos Ϫsin ͬ . [START_REF] Campagnola | High-resolution nonlinear optical imaging of live cells by second harmonic generation[END_REF] That is, E 2 p is the amplitude of E 2 in the ˆdirection and E 2 s is the amplitude of E 2 in the ˆdirection. The total radiated field in the propagation direction (, ) is then given by the coherent summation of the contributions from the local induced polarization at all points x and y, with their associated spatially dependent phase shifts taken into account. We find then that

E 2 ͑ , ͒ ϭ N s r ͵ ͵ M • 2 ͑ x, y ͒ ϫ exp͓Ϫik 2 ͑
x cos ϩ y sin sin ͔͒dx dy, (11) where k 2 ϭ 2n 2 /c. On integration, Eq. ( 11) yields

E 2 ͑ , ͒ ϭ NA͑, ͒E 2 ͑ 0 ͒ ͑ , ͒, (12)
where we have introduced the parameters

N ϭ 2 w x w y N s , (13)
A͑, ͒ ϭ exp ͭ Ϫ k 2 2 8 ͓w x 2 ͑ cos Ϫ Ј͒ 2 ϩ w y 2 ͑ sin sin ͒ 2 ͔ ͮ (14)
and Ј ϭ n /n 2 . The physical meanings of N and A(, ) are as follows: N defines the effective total number of molecules that contribute to the generation of second-harmonic light. This definition is identical to that detailed in Ref. 23 for two-photon excitation, though here it applies to a two-dimensional geometry. Accordingly, the effective total surface area that produces second-harmonic light is simply N/N s . Because the molecules involved in SHG are spatially distributed on this surface area, the resultant angular profile of the SHG radiation is much more complicated in structure than that of a simple elemental radiator. In particular, we observe from Eq. ( 12) that the emission pattern of the SHG is the product of two terms: The first is the pattern that arises from an elemental dipole radiator E 2 (0) (, ) at the focal center, and the second is an angular modulation term defined by the scalar function A(, ). As we shall see below, the effect of the latter modulation term on the radiation structure is quite dramatic.

Finally, the radiated second-harmonic powers in the p and s polarization and in the solid angle defined by (, ) are given by

P 2 p,s ͑ , ͒ ϭ 1 /2n 2 0 cr 2 ͉E 2 p,s ͑ , ͉͒ 2 , ( 15
)
and the total radiated power obtained by integration over all angles becomes

P SHG ϭ P SHG p ϩ P SHG s ϭ 4 3 n 2 0 c 2 N 2 ϫ ͓⌰ x 2,x ͑ 0 ͒2 ϩ ⌰ y 2,y ͑ 0 ͒2 ϩ ⌰ z 2,z ͑ 0 ͒2 ͔, ( 16
)
where we have introduced the angular structure parameters

⌰ x ϭ 3 8 ͵ A 2 ͑ , ͒sin 3 dd, ⌰ y ϭ 3 8 ͵ A 2 ͑ , ͓͒1 Ϫ sin 2 sin 2 ͔sin dd, (17)
⌰ z ϭ 3 8 ͵ A 2 ͑ , ͓͒1 Ϫ sin 2 cos 2 ͔sin dd.
The contributions to the total power from the various directional components of the induced second-harmonic dipoles are therefore readily identified. We note that if A(, ) ϭ 1, as is the case for a single elemental dipole radiator, then ⌰ x ϭ ⌰ y ϭ ⌰ z ϭ 1.

C. Membrane Hyperpolarizability

In the derivation of Eqs. ( 15) and ( 16), no assumptions have been made about the dipole vector 2 (0) , which may be taken as arbitrary. For our considerations, however, we shall assume that 2 (0) is generated by molecules that are quasi-uniaxial. That is, if we define a coordinate system (X, Y, Z) that is proper to a molecule, then the molecular first hyperpolarizability is dominated largely by the single component ␤ zzz . Moreover, we assume that the molecules are preferentially aligned in the lipid membrane such that, on average, the molecular Z axis is in the same direction as the z axis defined by the membrane, though we do not otherwise specify the distribution that governs tilt angles ␣.

The molecular X and Y axes, in turn, are assumed to be uniformly randomly oriented. In particular, we discount any possible correlation between the X and Y orientations and tilt angle ␣. With the above assumptions in mind, we may derive a coarse-grained local ensemble average for the hyperpolarizability that is valid over regions whose dimensions are small compared with the optical wavelength but large enough to encompass a large number of molecules. Expressed in the membrane coordinate system (x, y, z), the nonzero components of this first hyperpolarizability are given by

␤ xxz ϭ ␤ xzx ϭ ␤ yyz ϭ ␤ yzy ϵ ␤ ϩ , ␤ xyz ϭ ␤ xzy ϭ Ϫ␤ yxz ϭ Ϫ␤ yzx ϵ ␤ Ϫ , ␤ zxx ϭ ␤ zyy ϵ ␤ t , ␤ zzz ϵ ␤ z , (18)
where we have used the definitions

␤ ϩ ϭ ͗cos 3 ␣͘B ϩ ϩ 1 /2͗sin 2 ␣ cos ␣͑͘B Z Ϫ B T ͒, ␤ Ϫ ϭ 1 /2͗3 cos 2 ␣ Ϫ 1͘B Ϫ , ␤ t ϭ ͗cos 3 ␣͘B T ϩ 1 /2͗sin 2 ␣ cos ␣͑͘B Z ϩ B T Ϫ 2B ϩ ͒, ␤ z ϭ ͗cos 3 ␣͘B Z ϩ ͗sin 2 ␣ cos ␣͑͘B T ϩ 2B ϩ ͒, (19)
where ͗ ͘ signifies an ensemble average and

B ϩ ϭ 1 /2͑␤ XXZ ϩ ␤ YYZ ͒, B Ϫ ϭ 1 /2͑␤ XYZ Ϫ ␤ YXZ ͒, B T ϭ 1 /2͑␤ ZXX ϩ ␤ ZYY ͒, B Z ϭ ␤ ZZZ . ( 20
)
Having derived the effective molecular hyperpolarizability in the membrane frame, we may now derive the induced second-harmonic dipole at the focal center 2 (0) from Eq. ( 7). We consider the case when the driving field is polarized in the yz plane and write ϭ (0, sin , cos ), where represents the inclination of the polarization direction from the z axis. A straightforward application of Eqs. ( 7) and ( 18) then yields

2 ͑ 0 ͒ ϭ 1 /2 E 2 ͑ ␤ Ϫ sin 2, ␤ ϩ sin 2, ␤ z cos 2 ϩ ␤ t sin 2 ͒. (21)
Finally, by inserting Eq. ( 21) into Eqs. ( 9), (12), and (15), we obtain the p and s polarization components of the radiated second-harmonic power.

D. Uniaxial Hyperpolarizability

To distill the basic features of our results, we restrict ourselves here to the simplifying case in which the molecules in the membrane are perfectly oriented in the z ˆdirection (i.e., ␣ ϭ 0) and strictly uniaxial, such that the only nonzero component to the molecular first hyperpolarizability is ␤ ZZZ (i.e., ␤ ϩ ϭ ␤ Ϫ ϭ ␤ t ϭ 0; ␤ z ϭ ␤ ZZZ ). We also begin by considering the case in which the driving field is polarized in the z ˆdirection (i.e., ϭ 0). This is the same case as examined in subsection 2.A for a single molecule, with the only difference being that here we consider the effects of a planar distribution of molecules. As before, we find at the focal center

P 2 p ͑ , ͒ ϭ 3 16 SHG N 2 A 2 ͑ , ͒cos 2 cos 2 I 2 , ( 23
)
P 2 s ͑ , ͒ ϭ 3 16 SHG N 2 A 2 ͑ , ͒sin 2 I 2 , ( 24
)
and the integrated total power is

P SHG ϭ 1 /2⌰ z N 2 SHG I 2 . ( 25
)
For a single radiating dipole, ⌰ z ϭ 1; however, for a distribution of many dipoles, ⌰ z Ͻ 1 in general. The angular pattern of the radiated SHG light is governed mainly by the function A(, ). The structure of A(, ) critically depends on the interaction area and on Ј and exhibits two symmetric peaks, at peak ϭ Ϯcos Ϫ1 (Ј) (Fig. 2). This bidirectional nature of the SHG stems from the fact that the excitation light is subject to an effective increase in wavelength owing to the phase anomaly near the focal center. As a result, the SHG radiation must be emitted at an angle Ϯ peak relative to the excitation propagation direction to be properly phase matched. We assume throughout this paper that is roughly constant within the SHG excitation area. This approximation is justified in Appendix A. We emphasize here that is intimately linked to the interaction area and may not be regarded as an independent parameter. When the interaction area is reduced, then so too is , which leads to a widening of the angular separation between the two lobes as well as a broadening of the lobes themselves. In particular, a reduction in the axial and lateral waists of the excitation beam leads to a broadening of the lobe profiles along the and directions, roughly respectively.

A direct comparison can be made between the total powers emitted in the cases of SHG and TPEF. Noting that the total TPEF power scales with N, we find that

P SHG P TPEF ϭ 2ͱ2⌰ z N SHG TPEF , ( 26
)
where the factor 2ͱ2 stems from the volume contrast of a three-dimensionally Gaussian fluorescence excitation volume. [START_REF] Mertz | Molecular photodynamics involved in multiphoton excitation fluorescence microscopy[END_REF] Because the SHG radiation is emitted into two well-defined lobes, small-angle approximations about Ϸ Ϯ peak and Ϸ 0 may be used to provide an estimate of the structure parameter ⌰ z . A straightforward evaluation of Eqs. (17) with these approximations yields

⌰ z Ϸ 3Ј 2 k 2 2 w x w y ͱ 1 Ϫ Ј 2 . ( 27
)
Bearing in mind the approximate dependence of the focal area on the numerical aperture (NA) of the excitation beam (w x ϳ NA Ϫ2 ; w y ϳ NA Ϫ1 ), we can infer the following relations: N ϳ NA Ϫ3 , I ϳ NA 2 , and for low NA we can approximate ⌰ z ϳ NA 2 (see Appendix A). We find then that the total emitted SHG power is roughly independent of NA for NA Ͻ 0.8. For higher NA's, the SHG power diminishes. In comparison, the total emitted TPEF power scales as NA in a two-dimensional geometry.

E. Polarization Anisotropy

In our discussions so far we have assumed that the membrane geometry was planar. In our experiments, however, the membranes are in fact spherical, with diameters typically in the 20-50-m range. The excitation beam waist at the focal center is typically of submicrometer size; hence the assumption that the membrane is planar over this dimension is entirely valid. We note, however, that the membrane coordinate frame as defined above is not the same as the laboratory coordinate frame. In particular, if the excitation beam polarization is linear and fixed in the laboratory frame, it will appear to tilt in the membrane frame, depending on which portion of the membrane surface is illuminated. If we scan only along an equatorial cross section of the membrane, the polarization direction will rotate relative to the membrane's z axis with an angle , which spans 0 to 2. In the case of perfectly aligned uniaxial molecules (␤ ϩ ϭ ␤ Ϫ ϭ ␤ t ϭ 0), the induced local dipole moment for an angle as obtained from Eq. ( 21) is simply

2 ͑ 0 ͒ ϭ 1 /2 E 2 ͑ 0, 0, ␤ z cos 2 ͒. ( 28
)
The resultant p and s polarizations of the radiated field in the local membrane coordinate frame are then obtained from Eq. ( 9), and the total radiation power generated locally along the membrane equator is found to vary as cos 4 .

Experimentally it is difficult to isolate the p and s components of the radiated field because these are not fixed in the laboratory frame. We can transform them, however, into orthogonal polarization components, which are fixed in the laboratory frame through the relation

ͩ E 2 ʈ E 2 Ќ ͪ ϭ ͫ cos͑ Ϫ ͒ Ϫsin͑ Ϫ ͒ sin͑ Ϫ ͒ cos͑ Ϫ ͒ ͬͩ E 2 p E 2 s ͪ . (29)
Here E 2 ʈ and E 2 Ќ represent the radiated electric field components parallel and perpendicular, respectively, to the excitation beam's polarization direction. These latter polarization components can be isolated quite simply, as is shown below.

Following steps similar to those used in deriving Eq. ( 16), we can derive the total radiated SHG powers along the ʈ and Ќ polarizations. These are given by Fig. 2. Left, an excitation beam propagating in the x direction and polarized along the z axis is focused (side-on) onto the membrane of a labeled lipid vesicle. Only a small surface area (thick segment; side view) of this much larger vesicle contributes to SHG. Phase matching between the SHG and excitation fields causes the SHG radiation to be double peaked in the forward direction. Right, far-field power distribution of the SHG radiation.

P SHG ʈ ϭ 4 3 n 2 0 c 2 N 2 2,z ͑ 0 ͒2 ͓͑⌰ z Ϫ 2⌰ z Ј͒cos 2 ϩ ⌰ z Ј͔, (30)
P SHG Ќ ϭ 4 3 n 2 0 c 2 N 2 2,z ͑ 0 ͒2 ͓͑⌰ z Ϫ 2⌰ z Ј͒sin 2 ϩ ⌰ z Ј͔, (31)
where we have introduced the auxiliary angular structure parameter

⌰ z Ј ϭ 3 8 ͵ A 2 ͑ , ͒ ϫ ͓͑1 Ϫ cos ͒ 2 sin 2 cos 2 ͔sin dd. ( 32
)
Although the above relations are exact, in most cases of interest ⌰ z Ј is considerably smaller than ⌰ z and can safely be neglected. We then find that the powers emitted into the orthogonal polarization components P SHG ʈ and P SHG Ќ vary, respectively, as cos [START_REF] Dadap | Measurement of the vector character of electric fields by optical second-harmonic generation[END_REF] and cos 4 sin 2 along the membrane equator. We emphasize that the above results apply to rigorously uniaxial and well-oriented molecules only. In practice, not all molecules have these attributes, as will be seen below.

SHG CROSS SECTION

The molecule used in our experimental investigation is the lipophilic styryl dye N-(4-sulfobutyl)-4-(4-(dihexylamino)styryl)pyridinium (Di-6-ASPBS). The donor-(bridge)-acceptor structure of this molecule permits significant charge transfer along its major Z axis, resulting in a large first hyperpolarizability component ␤ ZZZ . [START_REF] Marder | Approaches for optimizing the first hyperpolarizability of conjugated organic molecules[END_REF][START_REF] Kogej | Mechanisms for enhancement of twophoton absorption in donor-acceptor conjugated chromophores[END_REF][START_REF] Chemla | Nonlinear Optical Properties of Organic Molecules and Crystals[END_REF] Donor-(-bridge)-acceptor molecules have been found to be well described by a two-state model. [START_REF] Shen | The Principles of Nonlinear Optics[END_REF][START_REF] Oudar | Optical nonlinearities of conjugated molecules. Stilbene derivatives and highly polar aromatic compounds[END_REF] When driven near resonance, ␤ ZZZ may be approximated by

␤ ZZZ ͑ Ϫ2, , ͒ ϭ 2 ប 2 eg 2 ⌬ ͫ 1 ͑ eg Ϫ 2 ϩ i⌫ ͒͑ eg Ϫ ϩ i⌫ ͒ ϩ 1 ͑ eg Ϫ ϩ i⌫ ͒͑ eg ϩ Ϫ i⌫ ͒ 1 ͑ eg ϩ 2 Ϫ i⌫ ͒͑ eg ϩ Ϫ i⌫ ͒ ͬ , ( 33
)
where eg and eg are, respectively, the transition frequency and the dipole transition moment, ⌬ is the difference between excited-state and ground-state dipole moments and ⌫ is a phenomenological damping constant.

Although no direct experimental measurements of ␤ ZZZ in membrane are available, the absorption spectrum of Di-6-ASPBS in membrane allows us to infer a charge transfer energy ប eg ϭ 2.56 eV, a transition moment eg Ϸ 10 D, and a damping factor ⌫ Ϸ 0.19 eV, all specific to a membrane environment. In addition, previous electrochromism measurements of similar dyes in membrane [START_REF] Loew | Charge shift probes of membrane potential. A probable electrochromic mechanism for ASP probes on a hemispherical lipid bilayer[END_REF] allow us to estimate that ⌬ Ϸ 16 D for Di-6-ASPBS. We may therefore predict a large static first hyperpolarizability of roughly ␤(0) Ϸ 2 ϫ 10 Ϫ48 C m 3 V Ϫ2 . This value is enhanced when the molecule is excited near resonance. In particular, for an excitation wavelength of 880 nm, we find that ␤ Ϸ 1 ϫ 10 Ϫ47 C m 3 V Ϫ2 , or, equivalently, SHG Ϸ 1 ϫ 10 Ϫ3 GM (1 GM ϭ 10 Ϫ50 cm 4 /photon s Ϫ1 ). For comparison, the TPEF cross section of Di-6-ASPBS in membrane at the same excitation wavelength has been estimated to be TPEF Ϸ 30 GM, based on TPEF measurements in ethanol. We note that, despite the fact that SHG is ϳ4 orders of magnitude smaller than TPEF for a single molecule, we may benefit from the fact that SHG field amplitudes add coherently in the case of a large number of molecules, as shown in Eq. (26). In practice and for standard dye labeling densities, we can easily obtain SHG and TPEF powers that are comparable, as we show below.

EXPERIMENTAL ANALYSIS A. SHG and TPEF Imaging of Membranes

Our experimental apparatus is a home-built scanning SHG TPEF microscope, the basics of which are shown in Fig. 3. The excitation source is a mode-locked Ti:sapphire laser (Spectra-Physics), which delivers ϳ80-fs pulses at an 81-MHz repetition rate. The laser light is focused into the sample with a water-immersion microscope objective (Olympus, LUMPlanFL60ϫW/IR) and the resultant SHG is collected in the forward direction, while the TPEF is collected in the backward direction. Our sample consists of giant unilamellar vesicles (GUV's) made from a pure lipid in water. These serve as model systems for the study of SHG radiation from membrane surfaces. The lipid type is 1,2-dioleoyl-sn-glycero-3phosphocholine (Avanti, DOPC), which is in the fluid phase L ␣ at room temperature. The vesicles are labeled at 1 mol. % with Di-6-ASPBS; preparation and labeling are described elsewhere. [START_REF] Sandre | Dynamics of transient pores in stretched vesicles[END_REF] Calcium ions are added to promote adhesion between adjacent vesicles, and the glass slide is coated with poly-L-lysine (Sigma P8920) to prevent sticking and bursting.

Figure 4 shows SHG and TPEF images of three GUV's of nearly equal radii that have adhered to form a foam. The GUV's are labeled with Di-6-ASPBS and acquired under the conditions described above. SHG scales with the Fig. 3. Experimental layout: a Ti:sapphire laser beam is focused into a sample with a microscope objective (MO). The transmitted SHG is collected with a condenser (C), bandpass (BP) filtered, and detected with a photomultiplier tube (PMT). The transmitted laser light is blocked with a colored glass filter (F). The TPEF from the sample is epicollected, discriminated with a dichroic mirror (DM), bandpass (BP) filtered, and detected with a PMT. Three-dimensional images are formed by scanning the laser focal spot in the lateral directions with galvanometermounted mirrors, and in the axial direction by translating the MO.

square of the excitation intensity and hence leads to the same intrinsic three-dimensional resolution as TPEF, here 510 nm lateral and 1.9 m axial. An excitation power of less than 1 mW at the sample provides approximately equal measured powers for both SHG and TPEF, allowing the images to be acquired simultaneously with integration times of ϳ10 s/pixel. The measured radiated powers are in good agreement with those predicted by the model developed in Section 3. In particular, if we estimate our dye surface density to be N s ϭ 1.5 ϫ 10 16 m Ϫ2 , and take into account the fact that essentially all the SHG light is collected because it is highly directional whereas only ϳ20% of the radiated fluorescence is epicollected, we find that P SHG /P TPEF Ϸ 0.5. To confirm that the light collected in the forward direction is indeed SHG in origin, we routed both the forward and the backward signals into a spectrograph. The SHG spectrum revealed a sharp peak at 440 nm, whereas the TPEF spectrum was broadly distributed about 580 nm. [START_REF] Moreaux | Membrane imaging by simultaneous second-harmonic generation and two photon microscopy[END_REF] The leftmost two vesicles in Fig. 4(a) have nearly equal TPEF brightness; the third vesicle is less bright, presumably owing to reduced labeling. Because SHG scales with the square of the label surface density, the contrast between the two bright vesicles and the third, dimmer, vesicle is enhanced in Fig. 4(b). We also clearly illustrate here the coherent nature of SHG radiation, as opposed to the incoherent nature of TPEF. In regions where the dipole distributions are symmetric, such as in the adhesion zones between the adjacent vesicles, the SHG vanishes, whereas the TPEF does not. We note that the cancellation of SHG is almost perfect in the left zone. It is less perfect in the right zone owing to the disparity in the surface labeling density.

B. SHG Radiation Pattern

To characterize the angular pattern of the SHG radiation we removed the condenser from our collection optics and replaced it with an oil-immersion microscope objective (Olympus, UPlanFL60ϫ1.25 Oil Iris). We then imaged the back aperture of this objective onto a CCD camera (Cohu 4912/Scion Corporation LG-3 frame grabber). This allowed us to visualize directly the emission angle . Figure 5 illustrates the SHG radiation pattern obtained when the laser scanning was restricted to a small patch of membrane on the equatorial slice of a vesicle. The average power at the focus was maintained below 5 mW, permitting a CCD integration time of ϳ5 ms/pixel. The two lobes that occur at Ϯ peak are manifest. These are the emission directions that correspond to phase matching of the SHG and excitation fields, as predicted by theory. As shown, however, the two lobes are not perfectly cylindrically symmetric about ϭ 0. In particular, the left lobe curves outward rather than inward. This may be the result of the local membrane curvature, which is not taken into account in our theory. In particular, when the membrane curvature was reversed, the lobe symmetry also became reversed. We also point out that there was a slight index-of-refraction mismatch between the insides and the outsides of our vesicles. As a result, the lobe that is directed toward the inside of a vesicle experiences a slight distortion owing to the lensing effect when it exits the vesicle, which may further contribute to the observed asymmetry in the lobes.

When the laser was allowed to scan over an entire vesicle cross section, the angular radiation pattern became annular, as shown in Fig. 6. The radiation pattern was observed to be highly sensitive to the inclination of the local membrane plane relative to the propagation axis. For example, in images generated from cross sec-Fig. 4. Simultaneous (a) TPEF and (b) SHG images of three vesicles labeled with Di-6-ASPBS that have adhered to form a foam. The radiating Di-6-ASPBS molecules are more-or-less symmetrically distributed in the adherence regions between the vesicles. In the left region, a symmetric distribution results in a nearly perfect cancellation of SHG. In the right region, the cancellation is imperfect because of a disparity in labeling density. As opposed to SHG radiation, TPEF is independent of molecular distribution because it is incoherent. tions below the equatorial plane, two annular rings became apparent. The origin of these rings is made clear in Fig. 6. We note that the radius of the annular ring obtained from the equatorial plane provides a precise determination of peak . In our case, from Fig. 6(a) and correcting for the index mismatch that is caused by the condenser glass, we find experimentally peak Ϸ 24°, corresponding to Ј Ϸ 0.91. This value of Ј is somewhat larger than that predicted from numerical calculations (see Appendix A). A possible reason for this discrepancy is that, as mentioned above, one lobe experiences a slight deviation on transit through the vesicle while the other does not. Furthermore, we assumed in our calculations that the back aperture of the excitation objective was completely and uniformly filled. Such was not the case in practice, meaning that our focal spot was probably somewhat larger than specified.

C. SHG Polarization Analysis

To conclude our experimental analysis, we briefly examine some polarization anisotropy properties of the SHG radiation obtained from vesicle membranes. In particular, we consider the case in which the excitation light is linearly polarized and we can select the corresponding parallel and cross-polarized components of the detected SHG radiation. We perform this selection simply by inserting a linear polarizer between the condenser and the SHG detection photomultiplier tube in Fig. 2. Depending on the orientation of this polarizer relative to the excitation polarization detection, we can effectively isolate the P SHG ʈ and P SHG Ќ components of the SHG radiation, as described in Subsection 2.E. Our results, for example, for Di-6-ASPBS labeling of GUV's are presented in Fig. 7. We note that we do not fully recover the respective cos [START_REF] Dadap | Measurement of the vector character of electric fields by optical second-harmonic generation[END_REF] and cos 4 sin 2 angular dependencies for P SHG ʈ and P SHG Ќ , as predicted from our simplified model. Though there may be several explanations for this, the most probable is that the assumptions that Di-6-ASPBS is strictly uniaxial and that it is perfectly aligned in the z ˆdirection when it is inserted into a membrane are overly simplistic. For example, it has been suggested [START_REF] Dirk | The contribution of electrons to second harmonic generation in organic molecules[END_REF] , yielding a closer fit to experiment. Moreover, even in the case of a rigorously uniaxial polarizability, any deviations from perfect alignment would further complicate matters by introducing a nonzero 2,y (0) component. Finally, we assumed in Subsection 2.E that the active SHG surface area remained constant everywhere along the membrane equator; that is, we assumed that w z ϭ w y in the focal plane and that the two waist dimensions could be freely interchanged. This is not strictly true in the case of a linearly polarized excitation beam.

Suffice it to say that a full polarization analysis of the SHG radiation from membranes remains rather complex owing to the many possible hyperpolarizability components that may be involved, as evidenced by Eqs. ( 18)- (20). Such a comprehensive analysis is beyond the scope of this paper.

CONCLUSION

Our purpose in this paper was to lay some theoretical foundations for high-resolution SHG imaging of membranes. Inasmuch as SHG imaging may be performed simultaneously with TPEF imaging, we have taken particular care to establish a formalism that permits direct comparison between the two modalities. Similarities between SHG and TPEF emission powers are that both scale with the square of the excitation intensity, that they may both be defined in terms of commensurate molecular cross sections, and that their active focal areas (or volumes) are independent of the sample size and depend only on the focus tightness. Differences lie in the fact that SHG power scales with the square of the number of molecules, whereas TPEF power scales only linearly with number. Moreover, a structure parameter is necessary for proper quantification of SHG power, which depends on the extent of the active focal area. We note finally that our formalism, which is essentially derived from regarding a distribution of hyperpolarizable molecules as a phased-array antenna, is readily extendible to the generation of harmonics of higher order. In all these cases, effective molecular sections may be rigorously defined, and an active focal area (or volume, as in the case of third-harmonic generation in bulk) may be readily identified.

APPENDIX A

A general theoretical framework for describing the electromagnetic field near the focus of a tightly focused beam has been proposed by Richard and Wolf. [START_REF] Richards | Electromagnetic diffraction in optical systems. II. Structure of the image field in aplanetic system[END_REF] This framework, based on diffraction theory, may be applied to linear or nonlinear optical microscopy. In the case of TPEF microscopy, the signals are incoherent, and only the local field intensity, or correspondingly the field amplitude, need be calculated about the focal center. In the case of SHG microscopy, however, the signals are coherent, and the situation becomes more complicated. Both the local field amplitude and phase must be calculated about the focal center.

It is well known that any focused beam incurs a net phase shift after it passes through its focus. This phenomenon, commonly referred to as a phase anomaly, has a profound influence on the spatial structure of SHG radiation from a membrane. On closer inspection the phase shift, which is /2 exactly at the focal center, is found to vary approximately linearly with distance along the propagation axis near the focal center. As such, a focused beam behaves much as a plane wave near the focal center, though with an effective wave vector that has been modified relative to that of an unfocused beam. The effective wave vector of a focused beam near its focal center may be written as k , where k is the wave vector of a corresponding unfocused beam and in general is less than 1. Because the phase of the induced secondharmonic polarizability is governed by the phase of the excitation field, and because appreciable SHG power is produced only in directions that permit proper phase matching between this induced polarizability and the resultant SHG field, we find that, because Ͻ 1, the radi- ated SHG power in the forward ( ϭ 0) direction is weak. The radiated SHG power along ϭ Ϯ peak , however, is appreciable because the fields in these directions are in phase and add coherently. Both peak and the total radiated SHG power depend critically on ; hence a detailed evaluation of this parameter is warranted.

We use the integral formulas derived in Ref. 22 to examine how varies with the NA of the focusing objective of the excitation beam. To begin, we evaluate only at the focal center and consider the case n 2 Ϸ n . We may define a focusing angle NA ϭ sin Ϫ1 (NA/n ) to characterize the angular range of the excitation light. A comparison of peak ϭ cos Ϫ1 () and NA for several NA's is shown in Fig. 8. We observe that peak / NA is roughly constant and less than 1, from which we obtain the general rule of thumb that a collection NA equal to the excitation NA is sufficient for essentially all the SHG light to be collected. In particular, for low NA's we find that Fig. 8.

Variations of the ratio peak / NA [where NA ϭ sin Ϫ1 (NA/n)] and of the parameter (1 Ϫ 2 ) 1/2 / 2 as a function of the NA of a water-immersion microscope objective, assuming a uniformly backfilled aperture and linear polarization. The plots are obtained from full numerical evaluations valid for arbitrary NA. For low NA, peak → NA /ͱ2. Fig. 9. Spatial profile of the phase-anomaly parameter (shades) and of the normalized intensity-squared distribution (contours) about the focal center (0, 0) of an 880-nm-wavelength laser beam focused by water-immersion objective with a NA of 0.9. The axial propagation direction is x; the lateral direction is y. Profiles were calculated assuming a uniformly backfilled aperture and linear polarization. is relatively constant within the major portion of the intensity-squared profile. peak ϭ NA /ͱ2. We also provide a plot of the parameter (1 Ϫ 2 ) 1/2 / 2 as found in expression (27). This parameter is found to vary linearly with NA for low to modest NA's.

Finally, throughout this paper we have assumed that is roughly constant over the entire membrane surface area that contributes to SHG radiation. To justify this assumption we illustrate the numerical evaluation of the (x, y) near the focal center and compare this with the corresponding intensity distribution of the excitation beam (Fig. 9) for our experimental case of interest (NA of 0.9). We observe that varies only slightly within the major portion of the intensity-squared distribution, and hence the approximation that is constant over the active surface area in which SHG is produced is quite reasonable. In our case, we find numerically that Ϸ 0.88.

Fig. 1 .

 1 Fig. 1. Coordinate system defining the SHG emission direction.The membrane surface (shaded; z ϭ 0) is approximated to be planar at the length scales considered. The focused excitation beam propagates in the ϩx direction. The resultant SHG is mostly confined to an interaction area schematically depicted as a dotted ellipsoid and is radiated in the directions defined by and (thick arrow), with polarization components parallel to (E 2 p ) and perpendicular to (E 2 s ) the emission plane (shown dashed).

Fig. 5 .

 5 Fig. 5. CCD image of the back aperture of the SHG collection objective. The excitation beam is scanned only over a small portion of an equatorial slice of a GUV membrane. The doublepeaked angular distribution of the SHG radiation is apparent.

Fig. 6 .Fig. 7 .

 67 Fig.6. Stack of CCD images of the back aperture of the SHG collection objective. The excitation beam, incident from above in the insets, is scanned through full cross sections of a GUV at various latitudes. The equatorial cross section is scanned for image (a), leading to a single annular ring at peak , whereas latitudes below the equator are scanned for images (b) and (c). The CCD images were integrated over times long enough to permit multiple scans. From image (a) we deduce that peak Ϸ 24°.

  that a molecule closely related to Di-6-ASPBS may possess a significant transverse polarizability component B T in the molecule frame [and hence ␤ t in the membrane frame; see Eqs. (18)-(20)]. Such a transverse component would add a sin 2 component to 2,z (0) and hence would significantly modify the expected angular dependencies of P SHG

	ʈ	and
	P SHG Ќ

Vol. 17, No. 10 / October 2000 / J. Opt. Soc. Am. B

Author manuscript of article published on https://doi.org/10.1364/JOSAB.17.001685

͑ 0 ͒ ϭ 1 /2␤ z E 2 z ˆ, (22)which again is the induced dipole moment per unit molecular surface density. The radiated power from the distribution of induced dipole moments about the focal center, expressed in photons/second, is then given by

ACKNOWLEDGMENTS

The authors gratefully acknowledge financial support by the Insitut Curie and by the Centre National de Recherche Scientifique. We thank M. Blanchard-Desce for providing the dye molecules and S. Charpak and C. Boccara for helpful assistance. L. Moreaux was supported by a Bourse Docteur Inge ´nieur.

J. Mertz's e-mail address is jerome.mertz@espci.fr.