Aude Perrineau

Jerome Van Der Woerd

Yves Gaudemer

Jing Liu-Zeng

Raphaël Pik

Paul Tapponnier

Robert Thuizat

Ronzhang Zheng

) J Van Der Woerd

Zheng Rongzhang

à la diffusion de documents scientifiques de niveau

Keywords: 10Be, 26Al, cosmogenic nuclide dating, river incision, Yellow River, northeastern Tibet

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Incision rate of the Yellow River in Northeastern Tibet constrained by 10Be and 26Al cosmogenic isotope dating of fluvial terraces: implications for catchment evolution and plateau building

1) Introduction

How and when the Tibetan plateau reached its present altitude and shape are important to investigate as the topography influences atmospheric circulations, thus the climate, but also because it may be used to constrain models of Tibetan plateau evolution (e.g., [START_REF] Tapponnier | Propagating extrusion tectonics in Asia: new insights from simple experiments with plasticine[END_REF][START_REF] Peltzer | Formation and evolution of strike-slip faults, rifts, and basins during the India-Asia collision: an experimental approach[END_REF]England & Houseman, 1986;[START_REF] Houseman | Crustal thickening versus lateral expulsion in the Indian-Asian continental collision[END_REF][START_REF] Royden | Surface deformation and lower crustal flow in eastern Tibet[END_REF]Fluteau et al., 1999;[START_REF] Tapponnier | Oblique stepwise rise and growth of the Tibet Plateau[END_REF][START_REF] Liu-Zeng | Quantifying landscape differences across the Tibetan plateau: implications for topographic relief evolution[END_REF]. In the stepped growth model of Tibet of [START_REF] Tapponnier | Oblique stepwise rise and growth of the Tibet Plateau[END_REF], northeastern Tibet is the youngest part of the plateau that formed from Pliocene time to present [START_REF] Meyer | Crustal thickening in Gansu-Qinghai lithospheric mantle subduction and oblique, strike-slip controlled growth of the Tibet Plateau[END_REF][START_REF] Métivier | Northeastward growth of the Tibet Plateau deduced from balanced reconstruction of two depositional areas: the Qaidam and Hexi Corridor basins, China[END_REF]. The region is characterized by narrow actively growing ranges (e.g., Tapponnier et al., 1990;[START_REF] Meyer | Crustal thickening in Gansu-Qinghai lithospheric mantle subduction and oblique, strike-slip controlled growth of the Tibet Plateau[END_REF][START_REF] Van Der Woerd | Rapid active thrusting along the northwestern range front of the Tanghe Nan Shan (western Gansu, China)[END_REF] that separate flat rapidly filling closed sedimentary basins [START_REF] Meyer | Crustal thickening in Gansu-Qinghai lithospheric mantle subduction and oblique, strike-slip controlled growth of the Tibet Plateau[END_REF], Métivier et al., 1998;[START_REF] Van Der Woerd | Couplage cinématique entre décrochements et chevauchements actifs dans le nord du Tibet[END_REF]. Such process may have been active in the central part of the Tibetan plateau, leading to the formation of a high topography at about 5000 m a.s.l.. From this point of view, the northeastern Tibetan region is a key place to unravel processes of the formation of the Tibetan Plateau. Understanding the different roles of tectonics, climate and the evolving drainage is thus important to constrain the formation of the present topography of Tibet (e.g., [START_REF] Clark | Surface uplift, tectonics, and erosion of eastern Tibet from large-scale drainage patterns[END_REF][START_REF] Liu-Zeng | Quantifying landscape differences across the Tibetan plateau: implications for topographic relief evolution[END_REF].

The Yellow River, together with the Jinsha, Lancang (Mekong), and Nu (Salween) rivers, drains the Tibetan plateau on its eastern rim (Figure 1). While its source is located on the high summer monsoon influenced central plateau, like the other large rivers, it does not escape the plateau to the southeast but is captured to the north after a hairpin loop in the Zoige (Roergai) basin (Figure 1; e.g., [START_REF] Harkins | Transient fluvial incision in the headwaters of the Yellow River, northeastern Tibet[END_REF]. As it flows back to the northwest, north of the Anyemaqen Shan, it enters regions of lesser precipitation that barely feed the river. This particularity makes the river sensitive to any north-south shift of the northern monsoon limit, which depends on climate variations (e.g., [START_REF] Benn | The role of the Indian summer monsoon and the mid-latitude westerlies in Himalayan glaciation: review and speculative discussion[END_REF][START_REF] Joussaume | Climat d'hier à demain[END_REF]. Both the fact that the river crosses a relatively dry northeastern Tibet allowing long-term preservation of the terraces and its high sensitivity to climate change probably in part explains the long sequence of fluvial terraces found all along the river from the Anyemaqen to the Ordos, in northern China.

The terraces of the Yellow River, near Gonghe (Figure 2) or near Lanzhou (e.g., [START_REF] Wang | Optically stimulated luminescence dating of sediments from the Yellow River terraces in Lanzhou: tectonic and climatic implications[END_REF][START_REF] Li | The environmental effects of the uplift of the Qinghai-Xizang Plateau[END_REF], have been commonly interpreted both to testify the first occurrence of the Yellow River in northeast Tibet and as a record of regional plateau surface uplift (e.g., [START_REF] Li | Magnetostratigraphic dating of river terraces: rapid and intermittent incision by the Yellow River of the northeastern margin of the Tibetan Plateau during the Quaternary[END_REF][START_REF] Li | The environmental effects of the uplift of the Qinghai-Xizang Plateau[END_REF][START_REF] Harkins | Transient fluvial incision in the headwaters of the Yellow River, northeastern Tibet[END_REF]. In this study, we will show that the terraces instead more likely indicate rapid incision of the river after a phase of damming and deposition of the Yellow River. In fact, probably better than anywhere else in Tibet, the terraces of the Yellow River across the Gonghe basin illustrate a process of plateau building that involves damming of the drainage by active range growth and rapid filling of closed sedimentary basins (e.g., [START_REF] Métivier | Northeastward growth of the Tibet Plateau deduced from balanced reconstruction of two depositional areas: the Qaidam and Hexi Corridor basins, China[END_REF][START_REF] Meyer | Crustal thickening in Gansu-Qinghai lithospheric mantle subduction and oblique, strike-slip controlled growth of the Tibet Plateau[END_REF].

In this paper, we describe the terraces of the Yellow River in the Gonghe basin. We present the 10 Be and 26 Al exposure ages obtained on four different terrace levels that allow to constrain both the incision history of the Yellow river across the Gonghe basin and the uplift rate of the surrounding ranges. We then discuss the implications on the basin history, and on the local and regional tectonics.

2) Geological and geomorphic setting 2.1 The Yellow River, basins and ranges

From the Zoige basin, east of Anyemaqen and south of the Kunlun fault to the Gobi platform, over a length of 500 km (or river length of >1000 km), the elevation drop of the Yellow River reaches about 2500 m as it crosses several sedimentary basins (Tongde, Gonghe, Guide, Xunhua, Linxia), all separated by actively growing ranges (Heka Shan, Waligong Shan, Kongma Shan, Laji Shan) (Figures 123). Along this stretch, the river crosses no less than 3 active strike-slip fault systems and about 6 active thrusts implying interactions of river dynamics and active tectonics. Across both the Kunlun and Haiyuan left-lateral strike-slip faults (Figure 1), the Yellow River is deflected 80-90 km left-laterally (e.g., [START_REF] Gaudemer | River offsets across active strike-slip faults[END_REF][START_REF] Gaudemer | Partitioning of crustal slip between linked, active faults in the eastern Qilian Shan, and evidence for a major seismic gap, the "Tianzhu gap", on the Western Haiyuan Fault, Gansu (China)[END_REF][START_REF] Van Der Woerd | Uniform postglacial slip-rate along the central 600 km of the Kunlun Fault (Tibet), from 26 Al, 10 Be and 14 C dating of riser offsets, and climatic origin of the regional morphology[END_REF]. Right-lateral movement along the Haiyen fault that connects with the more easterly trending Riyue-Laji-Kongma Shan thrusts contribute to the range growth along the northeastern border of the Guide basin. The Tongde basin is bounded to the north by the southvergent Heka Shan (Figure 1; [START_REF] Harkins | Transient fluvial incision in the headwaters of the Yellow River, northeastern Tibet[END_REF]. The Gonghe basin is limited to the north and south by active thrusts. In the south, thrusts are northvergent in the west and southvergent in the east, and their activity is attested by the magnitude Mw 6.9 1990 seismic event [START_REF] Chen | Source process of the 1990 Gonghe, China, earthquake and tectonic stress field in the northeastern Qinghai-Xizang (Tibetan) plateau[END_REF]. In the north, the Qinghainan Shan thrusts are southvergent to the west and are attested by folded and uplifted alluvial fans and terraces [START_REF] Van Der Woerd | Couplage cinématique entre décrochements et chevauchements actifs dans le nord du Tibet[END_REF] and northvergent to the east (Waligong Shan). Intensely faulted and folded Neogene series in front of the Ryue-Laji Shan (Fang et al., 2005) together with detrital zircon ages and their provenance from the Laji Shan [START_REF] Lease | Signatures of moutain building: Detrital zircon U/Pb ages from northeastern Tibet[END_REF] imply major shortening phases during the Plio-Quaternary along the northern Guide basin. Additional but less important thrusting also occurs along the western Linxia basin [START_REF] Parés | Northeastward growth and uplift of the Tibetan Plateau: magnetostratigraphic insights from the Guide Basin[END_REF][START_REF] Zheng | Late Cenozoic deformation subsequence in northeastern margin of Tibet: Detrital AFT records from Linxia Basin[END_REF], across the Lanzhou thrusts [START_REF] Wang | Optically stimulated luminescence dating of sediments from the Yellow River terraces in Lanzhou: tectonic and climatic implications[END_REF] and across the Tianjing-Mibo Shan north of the Haiyuan fault [START_REF] Gaudemer | Partitioning of crustal slip between linked, active faults in the eastern Qilian Shan, and evidence for a major seismic gap, the "Tianzhu gap", on the Western Haiyuan Fault, Gansu (China)[END_REF]. Most of the elevation change of the Yellow River occurs between Zoige and Linxia and the steepest river gradients are found in the gorges across the Waligong and Kongma Shan (Figure 3). [START_REF] Pan | Research upon the geomorphologic evolution of the Guide Basin and the development of the Yellow River[END_REF]Qinghai Geological Bureau, 1988; Figure 4e), similarly to the thrust of the eastern Waligong at the Guide basin western margin.

The most striking evidence for ongoing uplift is the impressive Longyang gorges carved by the Yellow River as it crosses the Waligong range (Figures 4a and4b). The slope of the Yellow River increases dramatically at the entrance of the Longyang gorges to about 0.5° (Figure 3). The height of the bedrock cliff reaches about 300-500 m in the gorge (Figure 4).

Other evidences for active tectonic uplift are given by the abraded bedrock abandoned above the gorge. At present, the bedrock slopes towards the west as expected if it were back-tilted to the west due to uplift of the Waligong on top of a west dipping thrust (Figure 5).

The Yellow river terraces across the Gonghe Basin

The Gonghe Basin is a northwest-southeast trending 250 x 50 km-wide sedimentary basin (Figures 1 and2). It is bounded to the north by the actively growing Qinghainan Shan and to the south by the Ngola Shan (or Gonghenan Shan) and the Heka Shan. It is filled with up to 1200 meters of Quaternary sediments (Qinghai Geological Bureau, 1988 ;[START_REF] Métivier | Northeastward growth of the Tibet Plateau deduced from balanced reconstruction of two depositional areas: the Qaidam and Hexi Corridor basins, China[END_REF]. Semi-arid climatic conditions prevail over most of the basin (annual precipitation average 250-350 mm;Domroes & Peng, 1988) and sand dune fields occupy the eastern part of it (Figures 2 and4c). The basin is almost entirely internally drained except in its eastern part where it is crossed by the Yellow River. In contrast with the Qinghai Lake basin located just north of the Qinghainan Shan (Figure 1) and occupied by the largest fresh water lake of China (4500 sq.m and about 27 m deep; [START_REF] Qin | The hydrological characteristics and water budget of Qinghai lake drainage basin[END_REF], only small lakes are visible across the Gonghe basin. Evidence for higher lake levels can be seen in abandoned lake shore lines or lake deposits (e.g. around Dalian lake or Chaka salt lake; Figure 6; van der Woerd, 1998).

Despite their proximity, both basins have slightly different present climatic regimes characterized by dryer conditions in Gonghe (Domroes & Peng, 1988).

As it crosses the Gonghe basin and incises through the sediment pile (Figure 4d), the Yellow River has abandoned a set of terraces that form huge fan shaped stairs across the entire 50 km-width of the basin (Figure 2). The freshness of the terraced landscape (Figure 4c) together with the fact that regressive erosion from the Yellow River gorge itself has not yet captured the central drainage of the Gonghe basin indicate that the Yellow River incision occurred recently as suggested by several authors (e.g., [START_REF] Li | The environmental effects of the uplift of the Qinghai-Xizang Plateau[END_REF][START_REF] Métivier | Northeastward growth of the Tibet Plateau deduced from balanced reconstruction of two depositional areas: the Qaidam and Hexi Corridor basins, China[END_REF][START_REF] Zhang | Evidence of a geological event and environmental change in the catchment area of the Yellow River at 0.15 Ma[END_REF][START_REF] Van Der Woerd | Couplage cinématique entre décrochements et chevauchements actifs dans le nord du Tibet[END_REF]. The large, up to 10 km-wide and 40-50 km long, terraces across the basin are mostly flat with slopes toward the north comparable to the present Yellow River slope (0.15-0.3°) in the same area (Figures 3a-3c). From the present river bed to the top we numbered the main and largest terrace levels T0 to T7 (Figures 2 and3). We designate the highest and largest terrace T7 knowing that it also corresponds to the regional surface level of basins around 3200-3250 m a.s.l. (Figures 2 and3). This surface can be easily recognized on satellite images or on digital elevation models (SRTM DEM) as it forms flat areas that merge smoothly with the piedmont bajadas of the surrounding ranges (Figures 2 and7a). A similar surface level correlates with the top of the sediment fill in the Guide and Tongde basin (Fothergill & Ma, 1999;Fang et al., 2005;[START_REF] Harkins | Transient fluvial incision in the headwaters of the Yellow River, northeastern Tibet[END_REF]. Interestingly, this level is also the highest paleo-lake stand of the Qinghai lake basin to the north (~ 3250 m, e.g., [START_REF] Lister | Lake Qinghai, China : closedbasin lake levels and the oxygen isotope record for ostracods since the latest Pleistocene, Paleogeography, Paleoclimatology[END_REF][START_REF] Rhode | Epipaleolithic/early Neolithic settlements at Qinghai Lake, western China[END_REF][START_REF] Rhode | Palaeoenvironmental and archaeological investigations at Qinghai Lake, western China: geomorphic and chronometric evidence of lake level history[END_REF][START_REF] Liu | Timing for high lake levels of Qinghai Lake in the Qinghai-Tibetan Plateau since the last interglaciation based on quartz OSL dating[END_REF]Fang et al., 2005).

T7 is clearly recognized as a fluvial terrace deposit at the entrance of the Gonghe basin north of Heka Shan (Figures 2 and7b). T7 is now covered by a 2 m-thick layer of eolian loess overlying a 15 cm-thick reddish paleosol (B-horizon; Figures 8a and8b; [START_REF] Pan | Research upon the geomorphologic evolution of the Guide Basin and the development of the Yellow River[END_REF]. The abandonment of T7 marks the onset of incision of the Yellow river in the Gonghe basin. The other major terrace levels (T6 to T1) are also composed of fluvial pebbles and gravels and are devoid or only covered by a thin loess cover across the Gonghe basin. Clear steep risers up to several tens of meters high separate the main terrace levels (Figures 2d and4c). In several places, as can be clearly seen on satellite images (Figure 2), these risers disappear under wind-blown sand dune fields because of favorable accumulation at the north-south trending risers which are almost perpendicular to the main northwest blowing winds.

For a long time the Yellow River terraces have been recognized as geomorphic markers of the active tectonics of northeastern Tibet during the Quaternary (e.g., [START_REF] Li | The environmental effects of the uplift of the Qinghai-Xizang Plateau[END_REF][START_REF] Lu | Geomorphologic evidence of phased uplift of the northeastern Qinghai-Tibet Plateau since 14 million years ago[END_REF][START_REF] Sun | Long-term fluvial archives in the Fen Wei Graben, central China, and their bearing on the tectonic history of the India-Asia collision system during the Quaternary[END_REF][START_REF] Li | Magnetostratigraphic dating of river terraces: rapid and intermittent incision by the Yellow River of the northeastern margin of the Tibetan Plateau during the Quaternary[END_REF]. From Lanzhou to Gonghe, their Early Pleistocene to late Pleistocene ages (e.g., [START_REF] Li | Magnetostratigraphic dating of river terraces: rapid and intermittent incision by the Yellow River of the northeastern margin of the Tibetan Plateau during the Quaternary[END_REF] have been interpreted to represent phases of regional plateau uplift. In contrast, [START_REF] Harkins | Transient fluvial incision in the headwaters of the Yellow River, northeastern Tibet[END_REF] have recently proposed, from a geomorphological analysis of the upper Yellow River gorge and its tributaries between Tongde and Zoige basins that the Yellow river propagated upstream due to headward incision.

From OSL and C-14 ages of low terraces 10 to 140 m above the present river bed in Tongde and upstream that range from 9 to 140 ka, (Figure 3), they infer an onset of incision from the top of the basin fill in Tongde at about 400-500 ka (Craddock et al., 2010). In more details, their results imply the upward propagation of a wave of incision starting around 500 ka in Tongde with incision rates of about 2.5 mm/yr that slowed down as incision progressed to less than 1 mm/yr at present [START_REF] Harkins | Transient fluvial incision in the headwaters of the Yellow River, northeastern Tibet[END_REF]; Figure 3).

The terraces in Gonghe have not been directly dated so far. Thermoluminescence (TL) dating of loess covering 7 terraces in Linxia give ages that range from 10 to 160 ka [START_REF] Li | Magnetostratigraphic dating of river terraces: rapid and intermittent incision by the Yellow River of the northeastern margin of the Tibetan Plateau during the Quaternary[END_REF] and due to the similar setting of terraces in Gonghe, the oldest of these ages has been attributed to the highest terrace in Gonghe [START_REF] Li | The environmental effects of the uplift of the Qinghai-Xizang Plateau[END_REF][START_REF] Li | Magnetostratigraphic dating of river terraces: rapid and intermittent incision by the Yellow River of the northeastern margin of the Tibetan Plateau during the Quaternary[END_REF][START_REF] Zhang | Evidence of a geological event and environmental change in the catchment area of the Yellow River at 0.15 Ma[END_REF]Fothergill & Ma, 1999). However, from paleomagnetostratigraphic studies in Guide and Linxia, it has been argued that the highest terrace in Guide postdates closely the shallowest dated sediments at 1.8 Ma (Fang et al., 2005), and that the Yellow river started incising in Gonghe at about 1.1 Ma [START_REF] Yan | Pleistocene magnetostratigraphy of the Gonghe Basin, NE Tibetan Plateau: headward incision of the Yellow River after 1.8 Ma[END_REF], in agreement with the inference of a post 500 ka incision of the Yellow River in Tongde and the upward propagation of a wave of incision [START_REF] Harkins | Transient fluvial incision in the headwaters of the Yellow River, northeastern Tibet[END_REF].

It is thus necessary to directly date the terraces, to determine their age of deposition and abandonment to constrain the incision history of the Yellow River and the rates of tectonic uplift of the narrow ranges cross-cut by the river.

3) Terrace dating, results and interpretation

Cosmogenic nuclide dating method

Cosmogenic nuclides are produced in situ by the interactions between secondary cosmic ray particles and surface rocks. Cosmogenic nuclides accumulate in rocks as long as they stay at/near to the surface of the Earth. 10 Be and 26 Al are mainly produced in quartz from O and Si by spallation and muonic capture, and the ubiquity of this mineral makes it a favoured target.

1 0

The concentration of cosmogenic radioactive nuclides in a rock, ! N , is given by:

! dN dt = P " #N (1)
where ! " is the decay constant of the nuclide considered and P is the production rate, which depends on the geographic location and altitude. For sub-surface samples, the production rate decreases exponentially with depth following:

! P(z) = P(0).exp "z # $ % & ' ( ) * (2)
where P(0) is the production rate at the surface, ! z the depth of the sample, ρ the density of the overlying material and ! " the attenuation length of the interacting particle. Analysing deep (shielded) samples may give access to the amount of inheritance. Indeed, samples situated below several times the penetration depth are shielded from cosmic radiations and their cosmogenic nuclide concentrations thus correspond to nuclides accumulated before being deposited, i.e., the inherited component. When inheritance and erosion are assumed to be negligible, then the minimum exposure age of a sample is given by:

!
! t min = 1 " ln 1# "N P $ % & ' ( ) . (3)
Taking into account erosion, inheritance and production by muons, the number of 10 Be atoms grows with time according to:

! N(z,"t) = N 0 (z,"t)exp(#$"t) + P n $ + % & n ' exp # % & n z ( ) * + , -+ . / 0 0 0 0 1 2 3 3 3 3 4 1 # exp #($ + % & n ' ( ) * + , -)"t ( ) * + , - ( ) * * + , --+ ! P m " exp # $ % m + $ % m & z ' ( ) ) ) ) * + , , , , - . /
/ / / 0 1 2 2 2 2 3 1# exp #(" + $ % m & ' ( ) * + , )4t ' ( ) * + , ' ( ) ) * + , , (4)
where ! N 0 (z,"t) is the inherited component, (n,m) subscripts for spallation by neutron and muons, respectively.

Samples were processed in the Cosmogenic Nuclides Laboratory of the University of

1 1
Strasbourg following standard methods described in [START_REF] Kohl | Chemical isolation of quartz for measurements of in-situproduced cosmogenic nuclides[END_REF]. 10 Be/ 9 Be and 26 Al/ 27 Al ratios were determined by accelerator mass spectrometry (AMS) at the ASTER (CEREGE, Aix-en-Provence) facility except two samples (Qi4-29-1 and Qi4-29-19, Table 1), which were measured at the Center for Accelerator Mass Spectrometry at Lawrence Livermore National Laboratory (Table 1). Natural 9 Be has been assumed to be not present in our samples after a set of samples has been checked by ICP-MS. The total 9 Be concentration is thus determined by the amount of carrier added to the sample.

The 26 Al concentrations were measured in a subset of samples to better constrain the exposure history of the terraces. As the stable isotope 27 Al is naturally present in quartz minerals, sample content of natural 27 Al was measured by ICP-MS before addition of 27 Al carrier. 10 Be and 26 Al have different decay constants and production rates, but the same production profiles.

They can thus be analysed together and compared. In case of a simple exposure history, without a period of burial, the 26 Al/ 10 Be nuclide ratio approximates 6.75 (Balco, 2009).

Nuclide concentrations were determined from the AMS measurements and the amount of corresponding stable isotope (Table 1). Ages were modelled using the CRONUS online calculator [START_REF] Balco | A complete and easily accessible means of calculating surface exposure ages or erosion rates from 10 Be and 26 Al measurements[END_REF]) version 2.2 (April 2009). Sub-surface sample concentrations have been modelled by fitting the sample concentrations following equation (2) (see also, [START_REF] Granger | Dating buried sediments using radioactive decay and muogenic production of 26Al and 10Be[END_REF] and using the constant surface production rate determined by the CRONUS calculator (see details below). We used the revised 10 Be decay constant of 5.1± 0.26 × 10 -7 a -1 of [START_REF] Nishiizumi | Absolute calibration of 10 Be AMS standards[END_REF] and the 26 Al decay constant of 9.83 ×10 -7 a -1 , following [START_REF] Balco | A complete and easily accessible means of calculating surface exposure ages or erosion rates from 10 Be and 26 Al measurements[END_REF]. Penetration depths used for depth profiles ages calculation are 160 g.cm -2 for neutrons and 1500 g.cm -2 for muons (Gosse & Phillips, 2001;[START_REF] Granger | Dating buried sediments using radioactive decay and muogenic production of 26Al and 10Be[END_REF]. We used densities of 2.65 for quartz, 2.7 for granite, 2.5 for a mix 1 2 gravel/pebble/cobble deposit, 1.8 for soil, 1.4 for loess. No topographic shielding correction was needed given the extent and flatness of the terraces. All other data used in the calculator, as advised by [START_REF] Balco | A complete and easily accessible means of calculating surface exposure ages or erosion rates from 10 Be and 26 Al measurements[END_REF] and [START_REF] Frankel | Terrestrial cosmogenic nuclide geochronology data reporting standards needed[END_REF], are given in Table 1.

The largest analytical uncertainty comes from the production rate estimation (Gosse & Phillips, 2001). [START_REF] Balco | A complete and easily accessible means of calculating surface exposure ages or erosion rates from 10 Be and 26 Al measurements[END_REF] estimated this uncertainty to approximate 10%. It is however larger for depth samples because uncertainties on penetration depths (6%), sample depth (0,5%) and density of overlying material (2-3%) must be added. Final uncertainties on production rates are then comprised between 10 and 12%. Uncertainties from the chemical processing are low (1-2%). Uncertainties from the AMS measurement are comprised between 1.8 and 3.5%. Total analytical uncertainties on ages, not considering modeling errors, thus range from 11% to 13% (Table 1).

We sampled four different terrace levels for cosmogenic dating, T7 and T5 in the basin, and two strath terrace levels in the gorge, WRB (Waligong right bank) and WLB (Waligong left bank) (Figure 2, Table 1).

Cosmogenic nuclide measurements

Terrace T7 samples

Stratigraphy

On the southern side of the Gonghe basin, north of Heka Shan and on the right bank of the Yellow River, clean exposure of the top 10 m of the T7 level can be seen in a gravel quarry used for road constructions (Figures 2c,7 and 8). The terrace conglomerate is a thick fill deposit, composed of gravels, pebbles and cobbles with a few 5 to 50 cm-thick and several 1 3 meters long sand lenses (Figure 8c). It is covered by a 15 cm-thick reddish paleosol (Figure 8b), itself covered by about 2 m of eolian loess (in the following sections, layer and sample depths are given relative to present ground surface, i.e., the top of the loess deposit). The conglomerate pile is marked by an angular unconformity at around 3.5-4 m depth. In an EW section, the ~5° west dipping pebble layers and sand lenses are intersected by overlying horizontal layers. In the NS section, the angular unconformity disappears, but a clear layer transition can be seen in the section approximately at the same depth (Figure 8). The two conglomerate units will be called C1 and C2 from bottom to top in the following sections.

The top 20 cm of the C2 conglomerate, just below the paleosol, is marked by a visible increase in the amount of whitish quartz-vein centimetric pebbles (Figure 9; Table 1). Our field observations and inspection of the quarry cliffs have not allowed to distinguish any other sedimentation interruption or other unconformities. Note that the latter would be difficult to detect since there is no obvious change in the nature and origin of the material deposited.

Therefore, in the following discussion, we will consider the section analyzed as a sequence of two depositional events (assuming that other sedimentation interruptions, if present, lasted only for short periods compared to the whole life time of the terrace), followed by the formation of a paleosol, itself followed by the deposition of loess.

Sampling

From depths of 2 to 7.5 m we collected 16 quartz-rich samples for cosmogenic exposure dating along two nearby vertical profiles (Figure 8, Table 1). At a depth of 2 m, a cobble was found above the paleosol but below the loess cover (sample Qi4-29-1). Below the paleosol, in unit C2, we collected a large number of centimeter-sized quartz-vein pebbles at a depth of 2.10 ± 0.10 m. These pebbles were sorted by size and analyzed as 4 separated amalgamated samples (samples Qi4-29-27a, b, c and d; Table1; Figure 9). Four additional samples are 4 individual quartz-rich cobbles (quartz vein or granitoid) from 2.3 to 3.2 m. In unit C1, three sand lenses were sampled at 3.7,15 and 23,respectively; Table 1). The 4 remaining samples are individual quartz-rich cobbles at 4. 00, 4.80, 6.15 and6.75 m (samples Qi4-29-16, 24, 21 and22; Table 1). Among those 16 samples, 13 were analyzed for 26 Al (Table 1).

10 Be nuclide concentrations 10 Be concentrations range from 235000 to 1500000 at/g(qtz) for most of the samples, except 2 that have highest concentrations of about 2 millions at/g(qtz) (Figure 10a, Table 1). Lowest concentrations are those of cobble samples Qi4-29-21 and Qi4-29-11 with, 235000 and 405000 at/g(qtz), respectively. The sand lens samples have larger concentrations of 445000, 505000 and 785000 at/g(qtz). The large concentrations of 1500000 at/g(qtz) is reached for samples right below the soil-conglomerate interface, samples Qi4-27a, b,c and d (Table 1).The whole sequence of 10 Be concentrations measured in the different samples do not show a monotonous exponential decrease with depth but imply either sequential deposition (e.g., Shaller et al., 2002;[START_REF] Matmon | Desert pavement-coated surfaces in extreme deserts present the longestlived landforms on Earth[END_REF] or a variable inheritance between samples (e.g., [START_REF] Anderson | Explicit treatment of inheritance in dating depositional surfaces using in situ 10 Be and 26 Al[END_REF][START_REF] Mériaux | Rapid slip along the central Altyn Tagh Fault: Morphochronologic evidence from Cherchen He and Sulamu Tagh[END_REF][START_REF] Mériaux | The Aksay segment of the Northern Altyn Tagh fault : tectonic geomorphology, landscape evolution and Holocene slip-rate[END_REF]LeDortz et al., 2009). In particular the different type of samples collected, individual cobbles, sand lenses or amalgamated pebbles may have various pre-exposure components (e.g., [START_REF] Oskin | Elevated shear zone loading rate during an earthquake cluster in eastern California[END_REF]. However, within each unit, C1 and C2, the sample concentrations show an exponential decrease with depth linked to their respective exposure histories (Figure 10).

Al nuclide concentrations

The 26 Al concentrations were measured in 13 samples (Table 1). The concentrations range from 2 to 9 millions at/g(qtz) and imply 26 Al/ 10 Be ratios that vary between 4.12 to 6.06 1 5

(Figures 10 and11; Table 1). Low 26 Al/ 10 Be ratio values usually result from differential decay of 10 Be and 26 Al caused by shielding from cosmic rays after a period of nuclide accumulation [START_REF] Bierman | Using in-situ cosmogenic isotopes to estimate rates of landscape evolution: a review from the geomorphic perspective[END_REF]. None of the samples have typical simple exposure surface 26 Al/ 10 Be ratio of 6.75 implying either shielding of the section or a significant component of inherited nuclides.

Disregarding a few anomalous samples, such as, Qi4-29-1 that is probably a reworked cobble found above the paleosol, Qi4-29-9 and Qi4-29-22 that have high 10 Be concentrations (large inheritance) and the sand samples, the 26 Al/ 10 Be ratio decreases slightly from 6 to 5.7 between 2 m to 6.15 m depth (Figure 10b). The sand lenses have lower 26 Al/ 10 Be ratios of about 4.8 to 5.06, which may indicate incorporation of material with inherited low 26 Al/ 10 Be ratios. We note that this might also be the case of present river sands, which have a ratio of about 5.28 (sample Qi4-43A, Table 1). We will jointly model (see below) the 10 Be concentration profile and the 26 Al/ 10 Be ratio to constrain the age of deposition and abandonment of the T7 level.

Terrace T5 samples

Geomorphic setting

Terrace T5 is one of the largest terraces southwest of Gonghe situated about 150 m below T7 (Figures 2,[START_REF] Li | Magnetostratigraphic dating of river terraces: rapid and intermittent incision by the Yellow River of the northeastern margin of the Tibetan Plateau during the Quaternary[END_REF]12). It is flat in its northern part where the surface is characterized by a tight pebble and cobble pavement (Figure 12). At places, sand dunes (barkanes) travel southeastwards across the terrace with the probable consequence of incorporation of modern sand below the pavement and abandonment of sandy patches at the terrace surface. We selected the wide paved surface because such surfaces are usually characterized by low erosion rates and are thus well suited for surface exposure dating (e.g., [START_REF] Van Der Woerd | Couplage cinématique entre décrochements et chevauchements actifs dans le nord du Tibet[END_REF][START_REF] Van Der Woerd | Long-term slip rate of the southern San Andreas Fault from 10 Be-26 Al surface exposure dating of an offset alluvial fan[END_REF][START_REF] Mériaux | Rapid slip along the central Altyn Tagh Fault: Morphochronologic evidence from Cherchen He and Sulamu Tagh[END_REF][START_REF] Matmon | Desert pavement-coated surfaces in extreme deserts present the longestlived landforms on Earth[END_REF].

Sampling and measurements 1 6

We sampled 7 well embedded cobbles on terrace T5 (Figure 12). The 7 samples have close 10 Be concentrations on the order of 3×10 6 at/g quartz (Table 1). This high nuclide concentration, which is twice the concentration of samples Qi4-29-27 from 2 m depth in T7, suggest that 1) T7 must have been shielded quite rapidly after deposition and 2) the T5 terrace surface has never been shielded and 3) inheritance and/or erosion, if present, have similar effects on the samples of T5. We will use the weighted mean of the 7 sample concentrations (3.21 ± 0.30 x 10e6 at/g(qtz); Table 1) as the 10 Be nuclide concentration of terrace T5 surface.

Waligong terrace samples

Geomorphology

It is across the Waligong ranges that the Yellow River has cut its most impressive gorge over a length of 30 km and a total drop of water level of 200 m (Figures 2 and3). Across the western Waligong, the Yellow River has carved a deep gorge with walls reaching about 500 m (Figures 3a,3b). The flat top of the gorge mimics the overall shape of the western Waligong (Figures 4 and5), and slopes towards the west into Longyangxia reservoir (Figure 2). In fact, in more details, the top of the gorge is formed by the progressive down cutting of the river resulting in abandonment of a sequence of strath terraces carved in the bedrock , which are covered by a thin deposit of fluvial gravel and cobbles (Figures 5 and13). These terraces now slope towards the west indicating that the western Waligong is actively growing and has progressively back tilted the terraces. While the main terrace levels can be clearly depicted in the landscape (Figures 5c and5d), their shape narrows near the gorge and their riser join with a low angle the gorge rims so that they become unclear near the gorge and we were unable to number and map them precisely.

7

Sampling and measurements

We sampled two sites on each side of the gorge (Figures 2 and13), Waligong right bank (WRB) and Waligong left bank (WLB), located at 2680 and 2940 m a.s.l., respectively. Given the difficulties of terraces mapping near the gorge rim, these two sites are not located within the regional Gonghe basin terrace stratigraphy. Elevation only is not sufficient as we suspect uplift to have occurred since their abandonment (see below). At both sites we sampled the bedrock as well as the overlying cobbles, 4 samples at WLB and 8 samples at WRB (Figure 13, Table 1).

On terrace WLB, samples have 10 Be concentrations that range from 0,56 to 2.6x106 at/g(qtz)

(Table 1). The lowest concentration (0,561x106 at /g(qtz)) is found in a piece of a highly fractured quartz vein sampled near the gorge cliff edge (sample Qi4-59, Figure 13a). Qi4-60 is a bedrock sample near the cliff edge and has a concentration of 1.82x106 at/g(qtz). Both remaining samples have higher concentrations, Qi4-62 is a protruding bedrock high (Figure 13a) and Qi4-61 is a cobble from the top of the terrace conglomerate cover (Figure 13a). We think that the average concentration of these two latter samples of 2.57±0.11 at/g(qtz) (Table 1) corresponds to the nuclide concentration accumulated since terrace abandonment and that the lower concentrations in the other samples are due to removal of the conglomerate cover and erosion of the fragile quartz vein outcrop.

On WRB terrace, samples have 10 Be concentrations that range from 1.2 to 1.9×10 6 at/g(qtz)

(Table 1). The 35% difference in concentration among the WRB samples may be explained by the relative positions of the samples in the section sampled (Figures 13c and13d). It appears that samples with the lowest concentration, Qi4-48 and Qi4-47 (Figure 11c) are bedrock samples close to the gorge cliff, which are no more covered by the terrace conglomerate. Sample Qi4-49 belongs to a bedrock high (Figure 13c) standing at the same level as the pebbly conglomerate top and show a similar 10 Be concentration as the 5 surface

1 8
cobbles Qi4-50 to 54 (Table 1). The distribution of concentrations of the WRB samples is thus best explained by removal of the conglomerate cover near the gorge cliff. The 10 Be concentration average of 1.89±0.02e6 at/g(qtz) of the 6 samples with highest concentrations correspond to the nuclides accumulated since terrace abandonment.

Age Modeling

Model age of T7

We modeled jointly the 10 Be concentration and the 26 Al/ 10 Be ratio in the depth profiles (e.g., [START_REF] Anderson | Explicit treatment of inheritance in dating depositional surfaces using in situ 10 Be and 26 Al[END_REF][START_REF] Granger | Dating buried sediments using radioactive decay and muogenic production of 26Al and 10Be[END_REF] considering a deposition scenario in three steps corresponding to the stratigraphy observed in the field (Figures 8 and10). Modelling results are presented in Figure 10 and take into account progressive increase in muongenic production at depth following analytical solutions described in [START_REF] Granger | Dating buried sediments using radioactive decay and muogenic production of 26Al and 10Be[END_REF] and calibrated to our site (Table 2). Disregarding samples with large inheritance and the sand samples, model 1 is a best fit adjusting all the 26 Al/ 10 Be ratios, and models 2 and 3 are limiting models toward short and long exposure times, respectively. In general, both the 10 Be concentration and the values of the 26 Al/ 10 Be ratio imply shielding by the loess during more than 150 ka (model 1). Involving fast erosion rates of the loess (0.01 cm/yr, model 3) very long time of loess shielding may be implied (400 ka) with the necessity to have an original loess thickness reaching several tens of meters.

Because the 2 m-thick loess deposit on top of the terrace is a shielding factor for cosmic rays (factor of decrease in production rate amounts to about 6) (e.g., [START_REF] Hetzel | Late Pleistocene/Holocene slip rate of the Zhangye thrust (Qilian Shan, 2 China) and implications for the active growth of the northeastern Tibetan Plateau[END_REF]Matmon et al., 2008) we have to consider its mode and timing of deposition. The presence of a paleosol below the loess (Figure 8) implies that the terrace surface remained free of loess during a certain time, i.e., quasi without any shielding. Without independent chronological 1 9 constraints on the loess age, several loess deposition modes can be modelled. In the first mode, the 2 m-thick loess is deposited instantaneously (or rapidly compared to the age of the terrace), in the second mode it was deposited continuously during a certain time (Figure 10c). Some constraints can be added from additional observations in Gonghe and elsewhere. As we mentioned above, loess is absent on the lower Yellow River terraces across the Gonghe basin, particularly on T5, so that the age of T5 may be the time when loess deposition stopped in the area (see below). In addition, 1 to 2 m-high loess sections described in northern Tibet generally show progressively younger ages towards the top (e.g., [START_REF] Peltzer | Formation and evolution of strike-slip faults, rifts, and basins during the India-Asia collision: an experimental approach[END_REF][START_REF] Van Der Woerd | Uniform postglacial slip-rate along the central 600 km of the Kunlun Fault (Tibet), from 26 Al, 10 Be and 14 C dating of riser offsets, and climatic origin of the regional morphology[END_REF][START_REF] Hetzel | Late Pleistocene/Holocene slip rate of the Zhangye thrust (Qilian Shan, 2 China) and implications for the active growth of the northeastern Tibetan Plateau[END_REF][START_REF] Rhode | Epipaleolithic/early Neolithic settlements at Qinghai Lake, western China[END_REF] over a time period of a few thousands of years. Another possible factor of shielding variation is involved if the loess cover is reworked by wind ablation implying a period of loess deposition followed by a period of loess erosion (model 3).

Assuming that sub-surface samples cannot loose any 10 Be, except by radioactive decay, the lowest concentrated and deepest samples in a depth-profile (below about 1.5 m, [START_REF] Anderson | Explicit treatment of inheritance in dating depositional surfaces using in situ 10 Be and 26 Al[END_REF] determine the minimum inherited component of the samples. We have thus considered the inheritance given by sample Qi4-29-21 (i.e., 235000 at/g, Table 1) in most models, and show only one alternative model (model 2) with an inheritance of 445000 at/g fitting the profile to the two sand samples with the lowest concentration (Qi4-29-23 and 15;

Figure 10,Table 2). Note that the present Yellow river is carrying sand that has a 10 Be concentration of about 300000 at/g quartz (Table 1; see also [START_REF] Harkins | Transient fluvial incision in the headwaters of the Yellow River, northeastern Tibet[END_REF]) that may be considered as the average inherited 10 Be component of the alluvial material transported by the river, assuming it is constant over time and not variable with material size.

While below the paleosol, the fits to the data mostly depends on the choice of the inheritance value, above the paleosol, the fits mostly depends on the onset of loess shielding (Figure 10a, Table 2). These models also show that, in all cases, deposition of conglomerate C2 occurred 2 0 shortly after deposition of conglomerate C1 (10 to 15 ka). Similarly, deposition of the loess cover occurred shortly (0 to 25 ka) after deposition of conglomerate C2.

As shown in Figure 10c (Table 2), when a long-lasting loess deposition process is considered, shielding starts to be efficient when this process last less than 20 ka. This implies, that shielding by the loess must have started a few thousands of years after the terrace has been abandoned and that the loess must have been deposited in a short time. Note that this is fully in agreement with the observation, that the terraces below T7 are not loess covered, so that they must have been abandoned after the loess deposition process ceased in the region and on T7 in particular. Sample Qi4-29-1, sampled above the paleosol and below the loess may be used as an additional constraint. It is clearly a reworked cobble as it is not emplaced in a fluvial deposit, but it must have been reworked at the time of the formation of the paleosol and before being capped by the loess. As can be seen in Figure 10a, models that are adjusted to this sample imply a minimum exposure of C2 of 10-15 ka. The same duration would be needed to account for soil formation above C2 and some surface reworking. The 10 Be concentration of this sample implies exposition under the loess during 150 ka.

To summarize, T7, or more precisely the top C2 conglomerate may have been deposited about 200 ka, and definitively incised by the Yellow River after 150 ka, some time before the 2 m loess cap accumulated on top of C2.

Model age of T5

Without independent constraints on erosion and inheritance for terrace T5, these parameters have to be estimated. Inheritance may be taken similar as for terrace T7 (235000 to 445000 at/g quartz) or similar to the present Yellow River sand 10 Be concentration of about 300000 at/g quartz (Table 1). Regarding erosion, studies have been carried out in the Gonghe basin 1 following intensification of agriculture and its consequence on desertification. Using the 137 Cs technique to assess erosion in drylands and sand shifting areas, erosion rates of about 1-2 mm/yr have been estimated over the last ~40 years [START_REF] Yan | 137 Cs dating of lacustrine sediments and human impacts on Dalian Lake, Qinghai Province, China[END_REF][START_REF] Zhang | Estimates of soil movement in a study area in Gonghe Basin, north-east of Qinghai-Tibet Plateau[END_REF].

These values of erosion are rather high [START_REF] Lal | Erosion history of the Tibetan Plateau since the last interglacial: constraints from the first studies of cosmogenic 10 Be from Tibetan bedrock[END_REF] and may only reflect the consequence of human activities. Lower values of 0.003-0.01 mm/yr were estimated for bedrock outcrops throughout Tibet [START_REF] Lal | Erosion history of the Tibetan Plateau since the last interglacial: constraints from the first studies of cosmogenic 10 Be from Tibetan bedrock[END_REF]. Again these values should be taken as maximum erosion rates for a flat alluvial terrace.

The average age of T5 calculated from 3.04±0.17×10 6 atoms/g is 106 ± 6 ka. Both the inheritance of 300000 at/g quartz and a maximum erosion rate of 0.002 mm/yr (Figure 14) imply subtractive and additive corrections of 15-20%, respectively. Without evidence that erosion rates may be larger, we thus conclude that the minimum age of T5 is 106 ± 15 ka.

Age model of Waligong

To calculate the age of the terraces we need to take into account the change in elevation due to their tectonic uplift (e.g., [START_REF] Hetzel | Late Pleistocene/Holocene slip rate of the Zhangye thrust (Qilian Shan, 2 China) and implications for the active growth of the northeastern Tibetan Plateau[END_REF]Ruszkiczay-Rüdiger et al., 2005a, 2005b) (Figure 15). Waligong Shan is composed of two sub-parallel ranges at the junction between the eastern termination of the Qinghainan and the Laji Shan closing the northeastern corner of the Gonghe basin (Figure 2). The western Waligong is a 15 km-wide and 50 km-long arcuate range trending roughly northwest-southeast (Figures 1 and2). Its asymmetric topographic shape, with a steep northeast and eastern flank under the summital crest that reaches 3500 m a.s.l. above a west dipping thrust and a western flank that slopes about 3.5° westward, (Figure 4e) indicate that it is an east-vergent crustal anticline. The eastern Waligong is less elevated but larger (20-30 km), trends northwest -southeast and makes up the western limit of the Guide basin. Similarly to the western Waligong, it has a steep eastern rim marked by a thrust at its base [START_REF] Pan | Research upon the geomorphologic evolution of the Guide Basin and the development of the Yellow River[END_REF] and grows as an east-vergent anticline (Figure 2). As can be seen 2 in the field (Figure 13) the strath terraces are carved into the Waligong Mesozoic granodiorite (Geological map of Qinghai, 1988).

A simple model (Figure 15) accounting for the tilt of the Waligong Shan and uplift of the terraces implies possible changes in elevation of 80 m for WRB terrace and of about 300 m for WLB terrace (Figure 15). For WRB, the in-situ production rate changes from 25.73 at/g/yr at 2600 m to 27.04 at/g/yr at 2680 m. This change in production rate of about 10 % remains small with regard to other uncertainties and may be neglected. For WLB the production rate change is more significant, varying from 26.6 at/g/yr at 2640 m to 31.93 at/g/yr at 2940 m, i.e. a change of 20 %. The real mean production rate of this terrace lies certainly between these end-values and we will take the mean between these bounds with a large error of 15%. The terrace ages are thus 74±10 ka for WRB and 90±15 ka for WLB. The maximum changes in elevation mentioned above imply maximum uplift rates for Waligong Shan between 1 and 3 mm/year. However, this model does not take into account the low 26 Al/ 10 Be ratio (5.23±0.27) obtained for sample Qi4-62 (WRG), which is a bedrock sample near the gorge cliff (Figure 13). This ratio implies that the surface bedrock needs to be exposed a first time (first cosmonuclide accumulation), then shielded during several hundred thousands years (ratio decreases; Figure 11) and then re-exposed until present. The cobble cover, likely deposited by the river, covered the surface before being progressively eroded to the present state (Figure 13). The beryllium concentrations of the cobbles would thus reflect the exposure duration since the river left, but bedrock concentrations are not totally related to this last phase of aggradation and erosion.

The surface would thus be exposed since ∼80 ka. Initiation of uplift likely began shortly afterwards at a minimum rate of 0.2 mm/year. Despite the fact that the 10 Be concentrations of the cobbles are higher on WLB than on WRB because of their higher elevation and production rate, simple calculations show that this scenario would lead to 10 Be concentrations 2 3 close to 2.0×10 6 at both sites, with a difference of ∼2.0×10 5 atoms instead of the ∼6.0×10 5 measured (Table 1).

4) Discussion

Onset age of incision in the Gonghe Basin

Whether the beginning of river incision corresponds to the first appearance of the Yellow River in the basin is an open question. Terrace sequences along the Yellow River differ markedly from Gonghe to Lanzhou. As explained above, terraces in the different basins seem completely unrelated to each other (ages, loess thickness, elevations). We cannot simply rely on these terraces to compare our chronology in the Gonghe Basin, but the timing of river incision should be coherent between the different basins [START_REF] Harkins | Transient fluvial incision in the headwaters of the Yellow River, northeastern Tibet[END_REF].

Several authors have proposed that incision in the Gonghe Basin began at around 150 ka.

(e.g., [START_REF] Li | The environmental effects of the uplift of the Qinghai-Xizang Plateau[END_REF][START_REF] Pan | Research upon the geomorphologic evolution of the Guide Basin and the development of the Yellow River[END_REF]. Evidences for a major change along the Yellow River 150 ka ago have been summarized by [START_REF] Zhang | Evidence of a geological event and environmental change in the catchment area of the Yellow River at 0.15 Ma[END_REF]. Periods of high regressive erosion from the Longyang gorges and the Sanmen gorges (in the middle reaches of the Yellow River north of the Ordos plateau, inset, Figure 1) are reported around that time [START_REF] Zhang | Age and environmental implications of the Sanmen Basin piercing in the middle reaches of Yellow River[END_REF][START_REF] Wang | Sedimentary records of environmental evolution in the Sanmen Lake Basin and the Yellow River running through the Sanmenxia Gorges eastward into the sea[END_REF]. This period of erosion in the Longyang gorges could have followed the folding of the fluvial sediments of the Gonghe formation [START_REF] Li | The environmental effects of the uplift of the Qinghai-Xizang Plateau[END_REF]. An increased sedimentation in the marine delta of the river could be related to this period of high erosion and to the climate warming of MIS-5 [START_REF] Imbrie | The orbital theory of Pleistocene climate: support from a revised chronology of the δ 18 O record[END_REF].

However, it has been proposed that incision began around 500 (±200) ka ago in the Tongde Basin [START_REF] Harkins | Transient fluvial incision in the headwaters of the Yellow River, northeastern Tibet[END_REF]Craddock et al., 2010), following the last 1.8 Ma lacustrine deposit near the top of the section in Guide [START_REF] Li | Magnetostratigraphic dating of river terraces: rapid and intermittent incision by the Yellow River of the northeastern margin of the Tibetan Plateau during the Quaternary[END_REF]. A direct implication is that incision in the Gonghe Basin began at the same time or before, but not after, which means that T7 must be much older than results from our investigations suggest. This can only be 2 4 achieved if a significant amount of erosion or a long period of shielding occurred and may imply that all terraces are significantly older. A 140 ka (OSL) terrace located 140 m above the Yellow River is located at an elevation of ∼2830 m in the Tongde basin [START_REF] Harkins | Transient fluvial incision in the headwaters of the Yellow River, northeastern Tibet[END_REF]Craddock et al., 2010). For comparison, an equivalent terrace in the Gonghe Basin would be near by T1, which is expected to be much younger than 140 ka.

Incision rates of the Yellow River in the Gonghe Basin

Terrace ages are plotted against their elevation on figure 16. The minimum and maximum long-term incision rates are ∼2 and ∼6 mm/year, respectively, with a mean rate of ∼4 mm/year. This mean rate fits well with the ages of T5 and T7. However, for all data it is needed to account for changes in the incision rate. The incision rate between the formation of T7 and T5 depends strongly on the age of T7, if erosion on T5 remains small. A period of high incision is needed between T5/WLB and WRB. The incision rate may have reached 9 mm/yr (10 mm/year maximum, if no elevation changes are taken into account for the Waligong terraces). Our data suggest that incision was first slow, then accelerated between ∼100 and 50 ka, before slowing down again. Such fast incision would have favoured the steep Waligong gorges formation.

A mean incision rate of 4 mm/year is higher than those mentioned in other places along the Yellow River in the literature, 0.7 mm/yr since 70 ka near Lanzhou [START_REF] Wang | Optically stimulated luminescence dating of sediments from the Yellow River terraces in Lanzhou: tectonic and climatic implications[END_REF], between 0.75 and 1 mm/yr since 166 ka near Linxia [START_REF] Li | Magnetostratigraphic dating of river terraces: rapid and intermittent incision by the Yellow River of the northeastern margin of the Tibetan Plateau during the Quaternary[END_REF] for the downstream part, between 0.9 and 1.0 mm/yr for the last 140 ka in the Tongde Basin [START_REF] Harkins | Transient fluvial incision in the headwaters of the Yellow River, northeastern Tibet[END_REF]. But it is in agreement with the 6 mm/yr since 93 ka in the Guide Basin proposed by [START_REF] Pan | Research upon the geomorphologic evolution of the Guide Basin and the development of the Yellow River[END_REF].

Climatic correlations and formation of the set of terraces

It is now recognized that climatic fluctuations play a key role in shaping fluvial terraces.

5

Correlation of terrace formation periods with climatic events is however not straightforward, because uncertainties inherent to the cosmogenic nuclides dating method often prevent from determining a precise correlation. Transitional periods between glacial and interglacial periods seem nevertheless to play a major role in forming fluvial terraces. These periods would be favorable for incision, while aggradation would occur during stable climatic phases, possibly both glacial and interglacial (e.g., [START_REF] Vandenberghe | Climate forcing of fluvial system development: an evolution of ideas[END_REF].

Loess in northeastern China was mainly deposited during glacial periods, while paleosols developed during interglacial phases, alternating loess and paleosol layers being correlated to climatic fluctuations (e.g., Ding et al., 2002). The fact that T7 gravels are directly overlain by a paleosol and this paleosol by a loess cover suggests that this terrace was abandoned during a deglaciation period and covered by loess during an aposteriori glacial event [START_REF] Pan | Evaluating the role of climate and tectonics during non-steady incision of the Yellow River: evidence from a 1.24 terrace record near Lanzhou, China[END_REF].

No such climatic indicators are available for the other terraces, although it is reasonable to assume that their formation followed a similar process, i.e., a formation as a consequence of dry/cold to wet/warm climate transition.

The terraces in the Gonghe Basin could have been formed during glacial periods and abandoned at the transition with warmer periods, when the river high incision power enabled it to break dams formed by the Waligong Shan.

Evidence for a Gonghe-Guide-Qinghai paleo-lake

Small lakes can be found in several places in the present Gonghe basin (Figure 6); Chaka Lake, in the northwest, has a present elevation of 3060 m; Dalian Lake, at an elevation of 2850 m [START_REF] Yan | 137 Cs dating of lacustrine sediments and human impacts on Dalian Lake, Qinghai Province, China[END_REF]; Duolonggou lake, north of Gonghe city, at an elevation of 3194 m. Lacustrine sediments can be found at several places in the Gonghe basin and at Dalian Lake for instance. These small lakes and sediments may be remnants of a larger lake occupying the bottom of the Gonghe basin (Figure 17). Such a lake may have covered adjacent intra-6 mountainous basins. It has been reported that sedimentary sequences in the Guide Basin were deposited under lacustrine conditions [START_REF] Parés | Northeastward growth and uplift of the Tibetan Plateau: magnetostratigraphic insights from the Guide Basin[END_REF]Fang et al., 2005;[START_REF] Li | Magnetostratigraphic dating of river terraces: rapid and intermittent incision by the Yellow River of the northeastern margin of the Tibetan Plateau during the Quaternary[END_REF].

Moreover, it is striking that the present level of the Qinghai Lake is exactly at the same altitude as Duolonggou lake and as T7 sampling site (3200 m). Qinghai lake exists since ~0.5

Ma [START_REF] Yuan | The formation and evolution of Qinghai Lake[END_REF] and its present maximum depth approximates 20 meters [START_REF] Colman | Late Cenozoic climate changes in China's western interior: a review of research on Lake Qinghai and comparison with other records[END_REF]. Reconstructions of Qinghai lake water level paleo-elevations lead to various chronologies [START_REF] Colman | Late Cenozoic climate changes in China's western interior: a review of research on Lake Qinghai and comparison with other records[END_REF][START_REF] Madsen | Age constraints on the late Quaternary evolution of Qinghai Lake, Tibetan Plateau[END_REF][START_REF] Rhode | Palaeoenvironmental and archaeological investigations at Qinghai Lake, western China: geomorphic and chronometric evidence of lake level history[END_REF][START_REF] Liu | Timing for high lake levels of Qinghai Lake in the Qinghai-Tibetan Plateau since the last interglaciation based on quartz OSL dating[END_REF].

The lake reached a maximum elevation of 3260 m (palaeo-shorelines older than 30 ka) and its level was particularly high during the penultimate glacial maximum (~150 ka). Other lakes in the region seemingly experienced the same high-stand event, the nearest in distance being Gahai Lake in the Qaidam Basin (Fan et al., 2010).

The Chinese literature discusses the existence of an eastern Qinghai paleo-lake covering the Qinghai, Gonghe, Xinghai and Xining basin (e.g., [START_REF] Pan | Research upon the geomorphologic evolution of the Guide Basin and the development of the Yellow River[END_REF]. It is based on a possible stratigraphic continuity between sediments in the Guide and Qinghai Basin. Such a lake would have broken into several parts due to the recent tectonic activity. Tectonic activity along the Qinghainan Shan fault likely began at the end of the Miocene or at the beginning of the Pliocene [START_REF] Métivier | Northeastward growth of the Tibet Plateau deduced from balanced reconstruction of two depositional areas: the Qaidam and Hexi Corridor basins, China[END_REF][START_REF] Meyer | Crustal thickening in Gansu-Qinghai lithospheric mantle subduction and oblique, strike-slip controlled growth of the Tibet Plateau[END_REF][START_REF] Van Der Woerd | Couplage cinématique entre décrochements et chevauchements actifs dans le nord du Tibet[END_REF]Fang et al., 2005;[START_REF] Lease | Signatures of moutain building: Detrital zircon U/Pb ages from northeastern Tibet[END_REF] separating the Qinghai and Gonghe Basin. A connection between the two basins was possible at the southeastern extremity of Qinghai Lake when the pass may have been at a lower elevation (present elevation: 3350 m). The Qinghai Basin and the western Gonghe Basin are now internally drained. [START_REF] Madsen | Age constraints on the late Quaternary evolution of Qinghai Lake, Tibetan Plateau[END_REF] proposed that Qinghai Lake began to form during the Middle or Late Pleistocene when the Riyue Shan uplift separated the Qinghai Basin from the Yellow River drainage system. Uplifting mountain ranges crossing the river course would have acted as boundaries for the eastern Qinghai paleo-lake, near the Longyang gorges (Gonghe Basin exit), Shanba gorges (Guide 7

Basin exit) or Jishi Gorges (Xunhua Basin exit) (Figure 17). The change from a lacustrine to a fluviatile environment would in this case correspond to the breaching of the topographic barriers and emplacement of the modern course of the Yellow River across the basins (Pares et al., 2003;Craddock et al., 2010).

However, no large lacustrine deposit can be found at present to attest more clearly the existence of such a lake, in particular at our sampling site of T7. Processes that lead to the formation of the wide paleo-regional base-level around 3200 m and of lakes present at this elevation stills need to be investigated, as well as the possible past connections between the different basins.

5) Conclusion

The Gonghe Basin is one of the widest basins of northeastern Tibet. It contains noteworthy wide terraces that testify to an important and recent water level decrease. Our results indicate that terraces were formed at the transition between glacial and interglacial periods. The upper one probably dates from the penultimate glacial maximum (MIS-6; [START_REF] Imbrie | The orbital theory of Pleistocene climate: support from a revised chronology of the δ 18 O record[END_REF].

Obtained ages show that incision occurs at a mean rate of ∼4 mm/year. Higher incision rates during some periods may be related to Waligong Shan uplift, although the tectonic signal remains difficult to separate from the river incision. Our results highlight the consequences of mountain growth and the interplay between tectonic and climatic processes in building the Tibetan Plateau. We provide chronological constraints supporting that this part of the plateau evolved rapidly in recent time from an internally drained but interconnected set of closed basins to an externally river drained plateau margin. 0 1) together with weighted mean.

Figure 10. (a) Plot of 10 Be concentrations of sub-surface samples of terrace T7 (see Table 1). Schematic stratigraphy of the section is shown. (b) 26 Al/ 10 Be ratio as a function of depth for a subset of samples of figure 10a. Models 1, 2 and 3 are fitted to both the 10 Be and the 26 Al/ 10 Be ratio (see text and Table 2). (c) Models showing the impact of various loess accumulation scenarios, first number is duration of loess deposit, second number is duration of exposure at the end of loess deposit. Conglomerates C1 and C2 where exposed each during 15 ka (Table 2).

Figure 11. 26 Al/ 10 Be ratio versus 10 Be for a subset of samples from the depth profile of terrace T7 (Table 1). Coloured boxes are allowed age and elevation ranges. Yellow River elevation taken at Waligong gorges entrance (= WRB site). Several scenarios and associated incision rates are proposed. For Waligong terraces, rates are mean rates calculated from the middle of boxes.

Maximum and mean total incision rates 6.0 mm/year and 4.0 mm/year, in orange.

Figure 17. Topographic contours at 3150, 3200 and 3250 m a.s.l. showing possible extensions of a regional paleolake or closed basin in the area of Qinghai and Gonghe basins before their separation due to lake capture by Yellow River and growth of the Qinghainan Shan. Lake surface elevation may have reached 3200-3250 m a.sl. This mapping, drawn from the present topography, does not account for large tectonic and morphologic changes. N 10 Be (x 10e6 at/g quartz)

Depth below present surface (cm)

(

1) (2) (2) (3 ) (3 )

  Craddock, W.H., E. Kirby, N.W. Harkins, H. Zhang, X. Shi, J. Liu, 2010. Rapid fluvial incision along the Yellow River during headward integration, Nature Geoscience, doi:10.1038/NGEO777. Ding, Z. L., Derbyshire, E., Yang, S. L., Yu, Z. W., Xiong, S. F. & Liu, T. S. 2002. Stacked 2.6-Ma grain size record from the Chinese loess based on five sections and correlation with the deep-sea δ 18 O record. Paleoceanography, 17(3), doi:10.1029/2001PA000725. Domroes, M. and G. Peng, 1988. The Climate of China. Springer, Berlin. 361 pages. England, P. C. & Houseman, G. A. 1986. Finite strain calculations of continental deformation: 2. Comparison with the India-Asia collision zone. Journal of Geophysical Research, 91, 3664-3676, doi:10.1098/rsta.1988.0135 Fan, Q.S., Lai, Z. P., Long, H., Sun, Y. J. & Liu, X. J. 2010. OSL chronology for lacustrine sediments recording high stands of Gahai Lake in Qaidam Basin, northeastern Qinhai-Tibetan Plateau. Quaternary Geochronology, 5, 223-227. Fang, X., Garzione, C., Van der Voo, R., Li, J., Fan, M., 2003. Flexural subsidence by 29 Ma on the NE edge of Tibet from the magnetostratigraphy of Linxia Basin, China. Earth and Planet. Sci. Lett., 210, 545-560. Fang, X., Yan, M., van der Voo, R., Rea, D. K., Song, C., Parés, J. M., Gao, J., Nie, J. & Dai, S. 2005. Late Cenozoic deformation and uplift of the NE Tibetan Plateau: evidence from high-resolution magnetostratigraphy of the Guide Basin, Qinghai Province, China. GSA Bulletin, 117, 1208-1225. Fluteau, F., G. Ramstein, and J. Besse, 1999. Simulating the evolution of the Asian and African monsoons during the past 30 Myr using an atmospheric general circulation model. J. Geophys. Res., 104(D10), 11,995-12,018. Fothergill, P. A. & Ma, H. 1999. Preliminary observations on the geomorphic evolution of the

Figures captions Figure 1 .

 captions1 Figures captions

Figure 2 .

 2 Figure 2. (a) SPOT satellite image mosaic of eastern Gonghe basin. White rectangle is figure

Figure 3 .

 3 Figure 3. (a) Slope variations along the Yellow River. River slope reaches its maximum

Figure 4 .

 4 Figure 4. (a) Field photo of the Longyang gorges, view towards the east. (b) Steep gorge cliff

Figure 5 .

 5 Figure 5. (a) Panoramic view of Waligong gorge. (b) View to west, towards Longyangxia

Figure 6 .

 6 Figure 6. Google Earth view of the Gonghe and Qinghai basins with present lakes or lake

Figure 7 .

 7 Figure 7. (a) View towards south of terrace T7 surface, (see location in figure 2). Surface T7

Figure 8 .

 8 Figure 8. (a) The top 10 m of terrace T7 in a gravel quarry on the right bank of the Yellow

Figure 9 .

 9 Figure 9. Mass distribution plot (probability density function) of amalgamated pebbles of

Figure 12 .

 12 Figure 12. (a) View to east of terrace T5 surface, showing a loose pavement of gravels and

Figure 13 .

 13 Figure 13. (a) View of Waligong left bank terrace (WLB) just above Longyang gorges (see

Figure 14 .

 14 Figure 14. Minimum exposure age range of T5 and influence of erosion. Erosion of about

Figure 15 .

 15 Figure 15. (a) to (d) Schematic reconstruction of progressive incision of Yellow River across

Figure 16 .

 16 Figure 16. Incision rate of the Yellow River deduced from terrace ages and elevations.

  Figure 2ab

  Figure11

  Figure12

  At the northeastern exit of the Gonghe basin, the Yellow River crosses the Waligong mountains carving the 500 m-deep Longyang Gorge (Longyangxia in Chinese) (Figures2 and 3). Since 1986, an artificial dam has been built at the entrance of the Longyang gorges forming the about 100 m deep Longyangxia Reservoir (water level at 2568 m in October 2004). The NW-SE trending Waligong Shan is composed of two massifs, the western and eastern Waligong, each one bounded to the east by active west-to southwest-dipping thrusts (Figure2). The western Waligong thrust juxtaposes the Waligong granite on top of red sandstones of Late Miocene-Pliocene age

Table 1 .

 1 10 Be and 26 Al analytical results of surface and sub-surface samples of the Yellow river terraces in the Gonghe basin..

Table 2 .

 2 Depth profile model parameters (see Figure10).

Table 1 .Table 2 .

 12 : AMS measurements at CAMS at Lawrence Livermore National Laboratory : see details of sample characteristics in Figure9. § : duplicate of the same sample after quartz leaching. Depth profile model parameters (see Figure10).

	Figure 17

*

C1 Erosion rate Exposure of C2 Erosion rate loess deposit Exposure loess Erosion rate of

										exposure
		(at/g quartz)	(ka)	(cm/yr)	(ka)	(cm/yr)	(ka)	(ka)	(cm/yr)	(ka)
	1	235000	15	0	45	0.005	1	150	0.0001	210
	2	445000	15	0	20	0	1	80	0	115
	3	235000	15	0	50	0		400	0.01	465
	models of	235000	15	0	15	0	50	10	0	90
	figure 10c	235000	15	0	15	0	25	70	0	125
		235000	15	0	15	0	10	120	0	160
		235000	15	0	15	0	1	150	0	181

Table 2 .

 2

Acknowledgements

Field access and field trip support was provided by Institute of Tibetan Plateau Research, Chinese Academy of Sciences, Beijing, and the Chinese Earthquake Administration, Beijing, and its local bureau in Xining. We acknowledge support from INSU-CNRS (programmes Relief de la Terre), from CNES-SPOT Image (Tectoscope program, image acquired through ISIS program). The 10 Be measurements were performed at the ASTER AMS French national facility (CEREGE, Aix-en-Provences), which is supported by the INSU-CNRS, the French Ministry of Research and Higher Education, IRD, and CEA. Two measurements were also made at Center for Accelerator Mass Spectrometry at Lawrence Livermore National Laboratory. We thank Ecole et Observatoire des Sciences de la Terre (Strasbourg, France) and Université de Strasbourg for support and making possible the operation of the cosmogenic laboratory of Institut de Physique du Globe de Strasbourg (UMR 7516) and Laboratoire d'Hydrogéologie et de Géochimie de Strasbourg (UMR 7517).


Philippe Delaye
email: philippe.delaye@iota.u-psud.fr

Gérald Roosen

Measuring low amplitude periodical phase shift without set-up stabilization: application to Pockels effect characterization

Keywords: 42.65.Hz, 42.62.Eh, 77.84.Dy, 78.20.Jq

We present an interferometric set-up for the measurement of low amplitude sinusoidal phase shifts that releases most of the constraints of usual architectures. It allows measurement without any stabilization of the set-up and can tolerate low quality fringe pattern. Due to its ease of use and its good sensitivity, this set-up can be used to measure low amplitude electro-optic coefficients with low applied voltage (less than 1V). Test measurements on LiNbO 3 and LiTaO 3 samples show good accordance between measured and theoretical Pockels phase shifts.

position. The experiment will thus be particularly sensitive to vibrations of the set-up and to thermal fluctuations of the studied crystal. In this letter we propose a new solution to this problem that works without the need of a stabilized interferometer and that even takes benefit of its drift and fluctuations for the measurement. This set-up thus enables to measure the very small phase shifts induced in Pockels crystals (such as LiNbO 3 or LiTaO 3 ) by AC voltage as low as some tens of mV.

In the interferometric set-up, a laser beam is separated in two arms by a beam splitter and recombined on a detector (using the same or another beam splitter). In one of the arms (the signal beam), we place the sample to be measured, whereas in the second arm a piezo-mirror allows to slowly vary the phase shift between the beams of the interferometer. When the interferometer is aligned (i.e. fringes are observed), the intensity received by the detector equals:

! I = I S + I R + 2r I S I R cos" t ( ) (1)
where I S is the intensity of the signal beam, in which is placed the Pockels crystal to be characterized and I R is the intensity of the reference beam. In this expression, r is a coefficient that renders the quality of the fringes observed on the detector. For perfectly aligned plane waves, we have r = 1 meaning that the phase shift between the beams does not vary on the surface of the detector. In the worst case of a large number of fringes on the surface of the detectors, meaning that the wavefronts are distorted or misaligned, we have r ≈ 0, and no modulation signal is delivered by the detector. In the general case, we have a value of r that varies between these two extrema, with a value of r that will have to be the closest as possible to 1 as the detected modulated signal is proportional to this value. Nevertheless, as we will see in the following, the technique we developed works whatever the quality of the fringes, and thus whatever the value of r (remembering that the measurement will be all the more easy and accurate than r will be close to 1, as the measured signal will be high). Finally, ! " t ( ) is the phase shift between the two interfering beams. It can be separated in two terms

! "(t) = # t ( ) + $# t ( ), with ! "# t ( ) = 2 "# RMS sin 2$ft ( )
$I t ( ) 1 2 4 4 4 3 4 4 4 (2)
3

In an usual interferometric set-up (homodyne detection) [START_REF] Aillerie | Measurement of the electro-optic coefficients: description and comparison of the experimental techniques[END_REF], the mean phase shift This stabilization at half fringe level is the main problem of usual homodyne detection schemes. In fact, the problem is not necessarily the stabilization by itself, but to find the right level of stabilization that will vary slowly if I R , I S or above all r vary. This is generally obtained by a stabilization (against thermal change, vibrations, air fluctuations, …) of the whole set-up during the measurement.

In our case, we use the same optical set-up and the same signal but we change the data processing. Our method does not require the stabilization of the interferometer to a given level, it will even use the naturally or artificially produced fluctuations of ! "(t) , without the necessity to know the exact value of this phase shift. This method can be seen as a low frequency analogous to the heterodyne detection where the ratio of height of the sideband to the carrier frequency amplitude give directly the phase modulation amplitude.

Considering the intensity received by the detector given by equation ( 2 are not independent and are correlated.

We will now show that if we measure simultaneously the RMS (Root Mean Square) value of the modulated signal ! "I RMS (supposed in all this study to be sinusoidal) and the mean value of the intensity, we can deduce the RMS value of the Pockels phase shift we want to determine. From equation (2), we deduce:

! "I RMS = #2r I S I R sin$ t ( ) "$ RMS (3) and 4
!

I Mean = I S + I R + 2r I S I R cos"(t) (4)
In these relation, we immediately recognize the parametric equation of an ellipse ( The same parameter should be obtained with a better accuracy using an adjustment of the whole ellipse. As the adjustment with the ellipse equation in not really easy and common, we have choosen another treatment, that will give more easily the same result, i.e. the determination of ! "# RMS using all the measured data.

Eliminating the slow phase shift ! "(t) from equations ( 3) and ( 4) (calculating

! cos 2 "(t) and ! sin 2 " t (
)), we arrive to the relation:

! "I RMS 2 = #"$ RMS 2 a1 1 2 3 I Mean 2 + 2"$ RMS 2 I S + I R ( ) a 2 1 2 4 4 3 4 4 I Mean + "$ RMS 2 4 r 2 I S I R # I S + I R ( ) 2
( ) a 3 1 2 4 4 4 4 3 4 4 4 4 (5)
We easily see that the graphical representation of this expression

! "I RMS 2 = f I Mean ( ) is a
parabola, which a1 parameter directly gives the RMS value

! "# RMS of the phase shift ! "# t ( ) due to
the voltage applied to the Pockels crystal. This measure is completely independent of the intensity of the different beams (I S and I R ), but also of the quality of the fringes (characterized by parameter r). Moreover the exact nature of the slow phase shift ! "(t) is not important. It does not need to vary according to a peculiar law (it can even be due to the natural drift of the interferometer) as soon as at least one fringe is scrolled ( ! "(t) > #), and the whole parabola is described by the signal with typically some cycles (two or three are usually sufficient).

Experimentally, the interferometer is of the Mach-Zehnder type, using a collimated and vertically polarized He-Ne laser emitting at 633nm (Figure 1). The Pockels crystal is placed in the signal arm. A half wave plate allows to choose the desired eigen-polarization of the crystal, a second half wave plate gives back a vertical polarization, selected by a second polarizer. At the output of the interferometer a lens makes the image of the crystal on the detector plane. The slow phase shift is created by a piezo-mirror inserted in the reference beam. The typical modulation frequency of the piezo-mirror is 20mHz with a displacement amplitude around 1µm. The detected signal is sent simultaneously on one input of an oscilloscope (with an internal low pass filter) to measure the mean intensity and on the input of a lock-in amplifier to measure the RMS value of the detected signal at the excitation frequency (the results of the measurement being sent on the other input of the oscilloscope). The Lock-In amplifier is also used to deliver the voltage applied to the Pockels crystal. Its RMS amplitude varies typically between 10mV and 2V, with a frequency varying between 1kHz and 100kHz (high frequency limitation of the Lock-In amplifier).

Typical signals are shown in Figure 2a. We see the mean signal that varies as the fringe is scrolled in front of the detector. In the same time the RMS value of the signal varies (visually in quadrature) with a maximum value at half fringe height and zero signal for absolute maximum and minimum of the mean intensity as expected. We see that tracing

! "I RMS = f I Mean
( ) , we obtain an ellipse as expected (Fig. 2b). After the treatment (tracing

! "I RMS 2 = f I Mean ( ) )
we obtain the expected parabola (Fig. 3), which fit allows to measure a Pockels phase shift of ! "# RMS = 13.047mrad, with an accuracy given by a standard deviation of 3µrad (in the following the standard deviation will be used as a measure of the accuracy).

We used this set-up to characterize two reference Pockels crystals of LiNbO 3 and LiTaO 3 .

These crystals are of the same class of symmetry (the 3m class). All their coefficients (electro-optic, elasto-optic, piezo-electric) are perfectly known and will be used to calculate the expected phase shift that will be compared to the experimental data given by the set-up. The crystals are cut along the crystallographic axes and are used in a transverse configuration with an electric field applied along the c-axis. The propagation is along the a-axis and the beam is either ordinary (i.e. vertically polarized along the b-axis) or extraordinary polarized (i.e. along the c-axis). The phase shift ( ! "# o and ! "# e for ordinary and extraordinary polarization respectively) acquired by the beam is then [START_REF] Yariv | Optical waves in crystals[END_REF][START_REF] Nye | Physical properties of crystals[END_REF]: ) where V is the applied voltage, L the thickness of the crystal along the beam propagation direction, d is the inter-electrode distance, and λ the used wavelength in vacuum. In these expressions the first term is due to the change of the dimension of the crystal because of the piezo-electric effect (with the piezo-electric coefficient d 31 ). The second term is due to the Pockels effect (with the unclamped electro-optic coefficient ! r ij T ). In both cases, n o is the ordinary refractive index and n e is the extraordinary refractive index. These expressions suppose that the crystal is free, i.e. that the frequency used is low compared to the piezo-electric resonance. The values of these parameters in the crystals we used, are summarized in Table 1.

! "# o = $ % V L d & ' ( ) * + 2 n o , 1
( )d 31 + n o 3 r 13 T [ ] (6)
! "# e = $ % V L d & ' ( ) * + 2 n e , 1
( )d 31 + n e 3 r 33 T [ ] (7
The first measurement is performed on a LiNbO 3 crystal at a fixed frequency as a function of the applied electric field, for both polarizations (Figure 4). We observed a linear dependence of the measured phase shift with a slope equal to ! "# o V =(2.186±0.003) mrad.V -1 and ! "# e V =(6.305±0.003) mrad.V -1 . These experimental values are in good accordance with the theoretical ones given in Table 1. In the insert of Figure 4, the variation of the phase shift with the applied voltage is showed in a log-log scale. This presentation shows that the linearity of the variation extends on the whole measurement range (more than two decades), until applied voltage as low as 10mV, corresponding to measured phase shifts as low as some tens of µrad. We also measure the response as a function of the frequency of the applied sinusoidal voltage (Figure 5). The sensitivity (i.e. the measured RMS phase shift normalized to the RMS applied voltage) is constant on the whole frequency range with a small increase that appears above 30kHz. In this range (i.e. below 30kHz), the mean value of the sensitivity is 6.31±0.06 mrad.V -1 , corresponding to a measurement accuracy of 1%. The increase corresponds to the beginning of a piezoelectric resonance situated at higher frequency (theoretically around 600kHz according to the dimensions and the orientation of the crystal). This behaviour confirms that the crystal is free in the condition of the measurement what justifies the expressions (equations ( 6) and ( 7)) used for the phase shift.

Similar measurements were performed in three LiTaO 3 crystals, with identical cut and dimensions. The obtained curves were similar to the one observed in LiNbO 3 , with nevertheless a slight dispersion of the sensitivity of the different crystals (Table 2) for the measurement at 25.7 kHz. Despite this dispersion the obtained values are in accordance with the theoretical value (Table 1). This dispersion was attributed to some problem in the application of the electric field, but considered independent of the measurement set-up. Indeed, a similar dispersion exists for both extraordinary and ordinary polarization. In all the crystals the ratio of the extraordinary phase shift to the ordinary one, is the same at a value of 3.66±0.05 that corresponds to the theoretical ratio. This point is confirmed by the fact that a measurement at lower frequency (183Hz) shows a lower dispersion in the measured values with ordinary polarization (Table 2). For some samples and polarisations, measurement had been performed several times on different days, each time putting on and removing the sample from the set-up, with a new alignment and optimization for each measurement. For example for the sample LTO1 with ordinary polarization at the frequency of 51.2kHz, 5 independent measurements were made giving sensitivities of 1.003, 0.998, 1.008, 1.010 and 1.013mrad.V -1 , what corresponds to a mean value of 1.006±0.006 mrad.V -1 , i.e. a relative accuracy of 0.6% typical of the measurement performed.

The presented Pockels material characterization set-up has several advantages compared to conventional set-ups. It is simple to use and does not require any stabilisation of the set-up. It is not sensitive to external perturbations such as vibrations or thermal effects in the measured sample. It is an interferometric geometry, so contrarily to the other simple and stabilisation independent set-up, the Sénarmont set-up, it can by proper cut of the crystal allow the measurement of diverse electrooptic coefficients independently. The sensitivity of the measurement (some tens of µrad without any peculiar precautions) allows to measure low amplitude electro-optic coefficients (such as ! r 13 T in LiNbO 3 or ! r 41 T in InP) without the need to apply high voltage (some volts are sufficient). The measurement does not require very high fringe quality, even if a higher contrast will lead to higher signal and easier and more accurate measurement (see Appendix). The ultimate performances of the presented set-up should be the same than the one obtained with a classical stabilized interferometric set-up perfectly optimized and aligned. Nevertheless these performances are accessible more easily with the proposed set-up, without the need of a servo-controlled stabilization, a vibration isolated breadboard, an air flow protection or a thermal control. It can be a rather cheap and good solution for people having occasionnal needs for moderately accurate phase modulation measurements (caracterization of a phase modulator, determination of electro-optic or piezo-electric coefficients of a crystal, … ), just using classical optical elements that are usually found in an optic laboratory.

Appendix : Accuracy of the measurement technique

The accuracy of the measurement technique can be theoretically estimated, using the fact that the phase shift is given by the ratio of the length of the axes to the ellipse when we trace The relative accuracy on the measured phase shift is given by ! " "# RMS ( )

! "I RMS = f I Mean ( ) .
"# RMS = "a a + "b b (A-1)
We will now estimate these different terms. We have first : the order of magnitude experimentally measured in our set-up (Fig. 3).

At low amplitude of the phase modulation we can determine the smallest measurable phase shift with our set-up, i.e. the phase modulation RMS value that would lead to a relative accuracy ! " "# RMS ( ) "# RMS = "a a = 1, equal to :

! "# RMS = 1+ C ( )" "I RMS Max ( ) 4CI Mean Max (A-4)
giving a numerical value of 7.5µrad, for our set-up. The order of magnitude of this value corresponds to the smallest value we were able to measure experimentally (i.e. 22µrad in Fig. 4).

  ), it can be split into two terms that evolve on very different time scales : a signal ! "I(t) modulated at the excitation frequency of the Pockels material and a mean intensity ! I Mean that varies slowly in time but is constant at the evolution time scale of ! "I(t) . The detector sees a quasi-uniform tint of an interference pattern, and the mean intensity ! I Mean is directly linked to the interference state of the fringe seen by the detector. The phase shift due to the drift of the set-up induces a displacement of the fringe pattern in front of the detector that corresponds to a change of the mean intensity ! I Mean and a slow variation of the amplitude of the modulated signal ! "I(t) . The important point is here that the temporal variations of the mean intensity ! I Mean and of the amplitude of the modulated signal ! "I(t)

!"

  (t) being the parameter), centered in the point (0, ! I S + I R ) with a small axis of length ! 4 r I S I R "# RMS and a great axis of length ! 4 r I S I R . The ratio of the length of the two axes of this ellipse gives the information we want, i.e. ! "# RMS . Nevertheless, the determination of the phase shift ! "# RMS through the ratio of the axes of the ellipse, only use a small part of the measured data (points around the extrema of the ellipse).

  The axes are given by ! a = 4 r I S I R "# RMS and ! b = 4 r I S I R . These parameters can be rewritten as a function of some other practical parameters of the fringe pattern, i.e. the maximum ! I Mean Max or the minimum ! I Mean Min intensity value, or the contrast C of the fringe pattern. For example,

  Min # "I Mean Max = "I Mean , is the standard deviation of the noise on the mean intensity. The relative error on b is independent of the measured phase shift, as b does not depend on it, but depends on the intensity and on the contrast of the fringes. The lower the intensity!I MeanMax , the lower the signal and the worse the relative accuracy, where as for a low contrast the fringes disappear, the mean intensity variation becomes smaller and is determined with a less good accuracy.Then, we have! a = 2"I RMS Max , with ! "I RMS Max = 2 I Mean Max # I Mean Min ( ) "$ RMS = 4CI Mean Max "$ RMS 1+C the noise on the output of the Lock-In amplifier giving the RMS value of the modulated signal. The relative error on "a" depends on the contrast of the fringes as a lower contrast means a smaller signal. It is the same for the maximum intensity of the fringe pattern that governs the measured modulated signal. The relative error also depends on the phase modulation amplitude as smaller phase modulation means smaller signal.At small value of the measured phase modulation, the preponderant term for the relative accuracy is ! "a a, that decreases when phase modulation increases until saturation towards a value governed by the term ! "b b at high values of the phase modulation. In our experiment, typical values we measure, are ! I Mean Max = 100mV and a contrast of the fringes easily better than 0.5. The performances of the detection is mainly governed by the electronic noise with typical values of ! "I Mean # 10µV and ! " "I RMS Max ( ) # 1µV . The accuracy at large value of the phase modulation would reach ! " "# RMS ( ) "# RMS = "b b = 0.03%, what correspond to

Figure 1 :

 1 Figure 1 : Experimental set-up for phase shift measurement. Pol : Polarizer, λ/2 : half wave plate, PM : Piezo-mirror, Det : Detector, S : sample. The laser beam is expanded and an image of the output face of the sample is made on the detector.

Figure 2 :Figure 3 :

 23 Figure 2 :(a) : Temporal evolution of the mean signal

Figure 4 :

 4 Figure 4 :Experimental evolution of the phase shift as a function of the applied voltage, with its linear fit, for ordinary polarization (square and dotted line) and extraordinary polarization (circles and full line). In the insert, the data for ordinary polarization are shown in log-log scale. The frequency of the sinusoidal applied voltage is 12.67kHz.

Figure 5 :

 5 Figure 5 : Evolution of the sensitivity of the Pockels LiNbO 3 crystal as a function of the frequency of the applied voltage.

  This requires the knowledge of the fringes contrast (to determine the half level). This value is obtained through a measurement of the maximum of the mean intensity

		I Mean reaching the
	detector (corresponding to	"(t) = 0), and the minimum mean intensity (corresponding to
		!
	!

! "(t) is stabilized to a constant value equal to ±π/2 (half fringe level), where the detected signal ! "I(t) is maximal.

! "(t) = π).

Table 1 :

 1 Parameters of the crystals used in the study

	LiNbO 3	LiTaO 3

Table 2 :

 2 Measurement sensitivity of LiTaO 3 crystals