Christophe Finot
email: christophe.finot@u-bourgogne.fr

F Chaussard

Sonia Boscolo

Simple guidelines to predict self-phase modulation patterns

We present a simple approach to predict the main features of optical spectra affected by self-phase modulation (SPM), which is based on regarding the spectrum modification as an interference effect. A two-wave interference model is found sufficient to describe the SPM-broadened spectra of initially transform-limited or up-chirped pulses, whereas a third wave should be included in the model for initially down-chirped pulses. Simple analytical formulae are derived, which accurately predict the positions of the outermost peaks of the spectra.

INTRODUCTION

Self-phase modulation (SPM) is one of those very fascinating effects discovered in the early days of nonlinear optics [START_REF] Shimizu | Frequency Broadening in Liquids by a Short Light Pulse[END_REF] and among the first examples of nonlinear optical phenomena to which novices of the field are usually exposed [START_REF] Rottwitt | Nonlinear Optics: Principles and Applications[END_REF][START_REF] Boyd | Nonlinear optics[END_REF]. It refers to the phenomenon by which an intense optical beam propagating in a Kerr medium induces through the nonlinearity of the medium a modulation of its phase that is proportional to its own intensity profile. Fiber optics is a convenient testbed for the experimental study of nonlinear optical phenomena owing to the possibility for a propagating beam to undergo large amounts sof nonlinearity without suffering from spatial modifications or thermal effects [START_REF]Nonlinear Fiber Optics[END_REF]. The earliest observation of SPM in optical fibers was made in 1978 [START_REF] Stolen | Self-phase modulation in silica optical fibers[END_REF], and the phenomenon has been extensively studied since then. For an input pulsed beam, the time-dependent phase change induced by SPM is associated with a modification of the optical spectrum, which depends on the frequency modulation (chirp) of the pulse electric field. If the pulse is initially Fourier-transform-limited or up-chirped SPM leads to spectral broadening, whereas an initially down-chirped pulse is spectrally compressed by the effects of SPM [START_REF] Stolen | Self-phase modulation in silica optical fibers[END_REF][START_REF] Markaryan | Spectral compression of ultrashort laser pulses[END_REF]. For strong SPM, the optical spectrum can exhibit strong oscillations.

SPM is often regarded as being harmful for optical communication systems or high-power pulse generation [START_REF] Richardson | High power fiber lasers: current status and future perspectives[END_REF]. However, when conveniently managed, SPM can be a precious ingredient that has stimulated a tremendous amount of applications, including the generation of ultra-short pulses [START_REF] Gustafson | Subpicosecond pulse generation using the optical Kerr effect[END_REF][START_REF] Tomlinson | Compression of optical pulses chirped by self-phase modulation in fibers[END_REF], the compensation of dispersion through solitonic effects [START_REF]Nonlinear Fiber Optics[END_REF][START_REF] Dudley | The cancellation of nonlinear and dispersive phase components on the fundamental optical fiber soliton : a pedagogical note[END_REF], the generation of wavelengthmultiplexed sources [START_REF] Yusoff | A 36-channel x 10-GHz spectrally sliced pulse source based on supercontinuum generation in normally dispersive highly nonlinear holey fiber[END_REF], the frequency shifting of ultra-short pulse sources [START_REF] Chung | Megawatt peak power tunable femtosecond source based on selfphase modulation enabled spectral selection[END_REF], the contrast enhancement of ultra-short pulses [START_REF] Buldt | Temporal contrast enhancement of energetic laser pulses by filtered self-phase-modulation-broadened spectra[END_REF], several applications in ultrafast optical signal processing [START_REF] Finot | All-optical fiber-based ultrafast amplitude jitter magnifier[END_REF][START_REF] Finot | Experimental demonstration of an ultrafast all-optical bit-error indicating scheme[END_REF], and the characterization of optical pulses [START_REF] Lin | Optical pulsewidth measurement using selfphase modulation[END_REF][START_REF] Baudin | Fiber-based measurement of temporal intensity and phase profiles of an optical telecommunication pulse through self-phase modulation[END_REF] and waveguides [START_REF] Kim | Measurement of the nonlinear index of silica-core and dispersion-shifted fibers[END_REF].

In this paper, by extending an approach initially developed in the context of sinusoidally phase modulated continuous waves [START_REF] Finot | Impact of a temporal sinusoidal phase modulation on the optical spectrum[END_REF], we present a novel theoretical treatment of SPM based on a spectral interference model. The typical oscillatory character of optical spectra affected by SPM indeed originates from strong excursions of the instantaneous frequency, so that in general there are contributions from different times to the Fourier integral for a given frequency component. Depending on the exact frequency, these contributions may constructively add up or cancel each other [START_REF] Shimizu | Frequency Broadening in Liquids by a Short Light Pulse[END_REF]. After introducing the situation under investigation, we present the intuitive but accurate description of the SPM spectral patterns that is afforded by our proposed approach. Simple analytical formulae are derived, which accurately predict the positions of the outermost peaks of the spectra. In the last section, we discuss both qualitatively and quantitatively the different scenarios that are observed depending on the initial chirp of the pulses.

PRINCIPLE AND SITUATION UNDER INVESTIGATION

Pulse propagation in a single-mode optical fiber is modeled with the standard nonlinear Schrödinger equation [START_REF]Nonlinear Fiber Optics[END_REF]:

2 2 2 2 1 | | , 2 i z t          (1)
where (z,t) is the complex envelope of the pulse electric field, z is the propagation coordinate, t is the retarded time, 2 is the group-velocity dispersion parameter, and  is the Kerr-nonlinearity coefficient accounting for both the nonlinear refractive index n2 and the fiber's effective cross-sectional area. We consider an initial pulse having the waveform      

/ /

PT  is satisfied), the solution to Eq. ( 1) is simply

  ( , ) (0, ) exp ( , ) , NL z t t i z t     ( 2
)
where NL(z,t) = B I(t) is the SPM-induced nonlinear phase shift. Here, B = P0 z is the maximum phase shift that occurs at the pulse center located at t = 0 and is widely known as the B-integral. In the presence of loss or gain, the physical propagated length z is replaced with an effective length defined as Leff = [1 -exp(-z)] / , where α > 0 (< 0) accounts for loss (gain). The temporally varying nonlinear phase implies a time dependence of the instantaneous optical frequency NL(t) = -dNL/dt, which in turn translates into changes in the pulse spectrum. The shape of the spectrum S() is obtained by taking the Fourier transform of Eq. ( 2):

  2 ( ) (0, ) exp ( ) exp( ) . S t i B I t i t dt        (3)
In general, the spectrum depends on the pulse shape and the initial chirp imposed on the pulse. In the general case, S() cannot be calculated analytically. However, for large values of the B-integral, many important insights into S() can be obtained using the method of stationary phase [START_REF] Cubeddu | Self-Phase Modulation and "Rocking" of Molecules in Trapped Filaments of Light with Picosecond Pulses[END_REF].

For an input pulse with a parabolic shape, the SPM-induced frequency chirp NL is a strictly monotonic function of time, so the pulse has a different instantaneous frequency at each point. In this case, no spectral interference occurs and, thus, the spectrum does not feature any oscillatory structure [START_REF] Parmigiani | Ultra-flat SPM-broadened spectra in a highly nonlinear fiber using parabolic pulses formed in a fiber Bragg grating[END_REF]. Conversely, for common transformlimited bell-shaped pulses, such as Gaussian or hyperbolic secant pulses, NL has a non-monotonic temporal variation; it reaches maximum positive and minimum negative values, m and min, and approaches zero as t becomes infinitely large. It has therefore been proposed to estimate the magnitude of SPM-induced spectral broadening by simply calculating  = m -min. A more accurate measure of spectral broadening is provided by the root-mean-square (rms) spectral width rms [START_REF] Pinault | Frequency broadening by self-phase modulation in optical fibers[END_REF]. The typical time dependence of NL for these bell-like pulse waveforms also means that for min < NL < m the same chirp occurs at two values of t, showing that the pulse has the same instantaneous frequency at two distinct points. Qualitatively speaking, these two points represent two waves of the same frequency but different phases that can interfere constructively or destructively depending on their relative phase difference. The characteristic oscillatory structure in the pulse spectrum is a result of such interference [START_REF] Shimizu | Frequency Broadening in Liquids by a Short Light Pulse[END_REF]. Mathematically, the Fourier integral in Eq. (3) gets dominant contributions at the two values of t at which the chirp is the same. These contributions may add up in phase or out of phase. Indeed, it is possible to use the stationary phase method to obtain an approximate expression of S() that is valid for large values of B [START_REF] Cubeddu | Self-Phase Modulation and "Rocking" of Molecules in Trapped Filaments of Light with Picosecond Pulses[END_REF]. Note that for initially chirped pulses, in general the same chirp may occur at more than two values of t. The approach that we propose here is to evaluate the spectrum affected by SPM by calculating the intensity in the interference pattern associated with the different instants in the pulse that have the same instantaneous frequency. A measure of the spectrum extent is then provided by the position of the outermost peak of intensity.

SPM PATTERN PREDICTION OF INITIALLY TRANSFORM-LIMITED PULSE

A. Analysis of the spectrum of a Gaussian pulse

Analysis

In order to introduce the basis of our discussion, we first consider the case of the widely used Fourier-transform-limited Gaussian pulse with the temporal intensity profile:

I(t) = G(t) = exp(-(t/T0) 2 )
, where T0 is the half-width at 1/e intensity point. Given the evenness of the intensity function and the oddness of the chirp function, we can restrict our study to the positive times only. The chirp (t) = NL(t) for such a pulse [Fig.

1(b)] is   1 00 ( ) , Bt t H G t TT      ( 4
)
where Hn is the n th -order Hermite polynomial. The extrema of the chirp function can be easily found by solving the equation

  2 2 0 0 () 0, d t B t H G t dt T T        (5)
which yields the following maximum value of  on the positive time tm: 15) and the corresponding approximation to the pulse profile are plotted with dashed green lines.

0 2 . 1 exp 2 m m m T t B t              (6)
As can be seen in Fig. 1(b), a given instantaneous frequency 0 between 0 and m is observed at two different instants, t1 and t2 (with t2> t1), whose separation t = t2 -t1 is a decreasing function of 0. These instants are the two roots of the equation (t) = 0:

2 0 10 2 0 21 , m m m m t t W e t t W e                                   (7)
where Wn is the Lambert W function, with n = 0, -1 for the main and second branch, respectively. The total phase difference T between t1 and t2 is the result of two contributions. The first contribution D is due to the time delay t:

  00 . D t       (8)
This quantity can be represented graphically in Fig. 1(b) as the area of the cyan rectangle, and is a non-monotonic function of 0 (Fig. 2). The second contribution NL to the total phase offset arises from SPM:

        0 2 1 2 1 ( ) ( ) , NL NL NL t t B G t G t          ( 9
)
and can be represented as the opposite of the area under the curve (t) between t = t1 and t = t2 in Fig. 1(b). It is apparent that NL is a negative and monotonically increasing function of 0 that has the maximum negative value of -B at 0 = 0 and is zero when 0 = m. The total phase difference between t2 and t1 is therefore

  0 . T D NL          (10)
It can be geometrically interpreted as the opposite of the area of the region in Fig. 1(b) that is bounded by the graph of and the horizontal line 0 (yellow region), and it is a negative and monotonically increasing function of 0 with the same extrema as NL (Fig. 2). 8), ( 9) and ( 10) are represented by red, blue and purple solid curves, respectively. Their toy-model approximations 'D, 'NL and 'T based on Eq. ( 17) are represented by dash-dotted curves. The green and cyan horizontal lines represent the values of the total phase offset causing constructive and destructive interference, respectively [Eqs. ( 13) and ( 12)].

With the knowledge of the instants t1 and t2 and the total phase difference T between them, we can calculate the distribution of intensity I(0) in the pattern associated with the interference between these two times:

    0 1 2 1 2 0 ( ) ( ) 2 ( ) ( ) cos . 2 T I G t G t G t G t             (11)
Equation [START_REF] Yusoff | A 36-channel x 10-GHz spectrally sliced pulse source based on supercontinuum generation in normally dispersive highly nonlinear holey fiber[END_REF] indicates that there is destructive interference causing intensity minima at frequencies for which

(2 1) , 2 T m       (12)
with m being a negative integer number (Fig. 2, cyan lines). From Eq. ( 12), one may obtain the convenient relationship between the number of minima k on one side of the spectrum and the value of B derived by Cubeddu et al. in [20]: B = (2k-1) . Intensity maxima or constructive interference occur when

2 2 T m      ( 13
)
at negative integer m. In particular, the position ωM of the outermost peak of intensity corresponds to m = 0 in Eq. ( 13): It is worth noting that T being not a linear function of 0, the frequencies for which constructive and destructive interference occurs are not equally spaced, and the outermost intensity peak is the widest [START_REF] Wang | High-resolution spectra of self-phase modulation in optical fibers[END_REF]. Figure 3 compares the spectral intensity distribution I(0) given by Eq. ( 11) with the exact spectrum S() numerically calculated from Eq. ( 3) at a level of SPM corresponding to B = 20 rad, and apparently indicates excellent qualitative agreement between the two spectral patterns. The positions of the intensity extrema predicted by Eqs. ( 12) and ( 13) are in quantitative agreement with the actual positions. The evolutions of the intensities of the two interfering waves at the instantaneous frequency 0, I1 = G(t1) and I2 = G(t2), with 0 that are plotted in Fig. 2(b) highlight that the lower 0, the higher the intensity difference between these two waves. This brings about a decreasing contrast in the interference pattern I(0) with decreasing 0. It is also noteworthy that our model is intrinsically unable to predict the outer decreasing wings of the spectrum.

  2 TM      (14)
We have also compared the evolutions of I(0) and S() with the level of SPM accumulated in the fiber. The results shown in Fig. 4(a) confirm both the qualitative agreement between the spectral pattern predicted by Eq. ( 11) and the numerical evaluation of the spectrum from Eq. ( 3), and the accuracy of the analytical prediction for the positions of the spectrum's extrema. It is also apparent from Fig. 4(a) that the maximum instantaneous frequency excursion contained in the spectrum m or the rms spectral width rms = rms, 0 [1+ (0.877 B) 2 ] 1/2 derived by Pinault et al. in [START_REF] Pinault | Frequency broadening by self-phase modulation in optical fibers[END_REF] (rms, 0 is the initial rms spectral width of the pulse) may not be the most intuitive quantities to describe the spectrum's expansion generated by SPM and, especially, the position of the outermost spectral peak that is very relevant for many recent applications of SPM [START_REF] Chung | Megawatt peak power tunable femtosecond source based on selfphase modulation enabled spectral selection[END_REF][START_REF] Buldt | Temporal contrast enhancement of energetic laser pulses by filtered self-phase-modulation-broadened spectra[END_REF]. Our approach is also able to describe the evolution of the spectral intensity at the central frequency 0 = 0, for which T = -B, with the B-integral. A simple sinusoidal function can indeed qualitatively explain the resulting pattern: I(B, 0=0)  1+ sin(B)/2 [Fig. 4(b)]. This can be of interest for nonlinear signal processing applications based on optical band-pass filtering at the central frequency [START_REF] Finot | All-optical fiber-based ultrafast amplitude jitter magnifier[END_REF][START_REF] Finot | Experimental demonstration of an ultrafast all-optical bit-error indicating scheme[END_REF].

Toy model

One point of the analysis so far that could be slightly inconvenient is the use of the Lambert W function to determine the interfering instants t1 and t2. While such a function is nowadays widely implemented in professional scientific software, it prevents the development of a fully analytical approach to the problem. Simplified but fully tractable expressions giving the main features of the spectral interference process can be obtained by expanding the chirp function (t) in Taylor series about t = tm to second order:

  2 2 2 2 2 0 ( ) , 1 2. 2 m mm mm m t t A t t d A dt T t              ( 15
)
As we can see in Fig. 1(b) (dashed green curve), this expansion provides a rather good approximation to the actual function in the vicinity of m.

Within this approximation, the interfering instants t'1 and t'2 are simply the roots of a quadratic equation, (t') = 0, hence they are separated by

'' 0 21 ' 2 , m m m T t t t        ( 16
)
and the linear and nonlinear phase differences between them are 8), ( 9) and [START_REF] Dudley | The cancellation of nonlinear and dispersive phase components on the fundamental optical fiber soliton : a pedagogical note[END_REF] as long the instantaneous frequency remains close to its maximum value m (Fig. 2, dash-dotted curves). Accordingly, Eq. ( 14) giving the position of the outermost spectral peak simplifies to

  ' ' ' . 2 T M m T           (18)
This equation can be readily solved to yield

1/3 2/3 2/3 0 '. 2 m Mm T     (19)
For a Gaussian pulse Eq. ( 19) takes the form

2 2 3 3 1 ' 1 exp 1 1.84 . 43 M m m B B                            (20)
Therefore, our toy model explicitly predicts a B-dependent correction factor for the position of the outermost spectral peak relative to the maximum instantaneous frequency excursion. As can be seen in Figs. 3 and4 (purple dashed curves), the prediction based on Eq. ( 20) is entirely plausible, thereby being more convenient for practical applications than rms or m.

B. Analysis of other pulse shapes

We have also used our proposed method to describe the SPMbroadened spectra of initially transform-limited pulses with hyperbolic secant, Lorentzian and super Gaussian temporal intensity profiles. In the case of a hyperbolic secant pulse with I(t) = sech 2 (t/T0), the chirp is given by (t) = 2 B I(t) tanh(t/T0), and attains a maximum value of m=3 -3/2 / T0 at tm = T0 atanh(3 -1/2 ). The toy model supplies the same expression for the position of the outermost spectral peak as that of Eq. [START_REF] Cubeddu | Self-Phase Modulation and "Rocking" of Molecules in Trapped Filaments of Light with Picosecond Pulses[END_REF]. We note that for this pulse shape, an exact calculation of the optical spectrum affected by SPM is available [START_REF] Phan Huy | Characterization of self-phase modulation in liquid filled hollow core photonic bandgap fibers[END_REF]. However, this calculation remains quite technical and not fully intuitive.

The Lorentzian profile I(t) = (1+(t/T0) 2 ) -4 entails the chirp (t) = 4 B (1+(t/T0) 2 ) -3 /T0 with a maximum value of m = 25 B / (54 tm) at tm = T0 5 -1/2 . The toy-model analysis leads to the following prediction of the position of the outermost spectral peak: The results obtained for the hyperbolic secant and Lorentzian pulse shapes are summarized in Fig. 5, and demonstrate the excellent ability of our approach to reproduce the overall shape and the extreme points of the spectrum. However, similarly to the case of a Gaussian pulse, the amplitude of the outermost peak is underestimated. This stems from the fact that the stationary-phase approximation ceases to be valid in the vicinity of m. 13) and ( 12), respectively.

Figure 6 shows the results obtained for a super-Gaussian pulse given by I(t) = exp(-(t/T0) 2m ) with m = 3. In this case, the chirp m has a maximum point at tm = T0 (1-1/(2m)) 1/2m , which can be approximated to T0 for large m. With this approximation, we get m = 2 B m e -1 / T0. Thus, using the toy model, we obtain the following expression for the position of the outermost spectral peak:

2 2 3 3 12 ' 1 exp 1 1.32 . 2 2 3 M m m B B                            (22)
We can see in Fig. 6 that our method can accurately predict the lateral extreme points of the spectrum. However, a significant discrepancy between the spectral intensity distribution I(0) and the actual spectrum S() can be observed at the low frequencies. In particular, S() features a marked peak at 0 = 0, which cannot be reproduced by our approach. In fact, most of the pulse energy remains in the central peak because the SPM-induced chirp is nearly zero over the central region of the pulse as a consequence of the nearly uniform intensity of a super-Gaussian pulse for |t| < T0. This central peak may make the first minimum from the center slightly fuzzy, hence care should be taken when applying the formula for the number of minima in the SPMbroadened spectrum derived by Cubeddu et al. to experimental data. 13) and ( 12), respectively.

SPM PATTERN PREDICTION OF LINEARLY CHIRPED INPUT PULSE A. General discussion on the appearance of spectral interference fringes

In this section, we discuss the effects of an initial linear chirp on the SPMmodified pulse spectrum. Such a chirp can be easily generated through propagation in a linear dispersive medium such as a pair of diffraction gratings, fiber Bragg gratings or a hollow core optical fiber. Linearly chirped pulses are widely used in chirped pulse amplification to mitigate undesirable SPM [START_REF] Richardson | High power fiber lasers: current status and future perspectives[END_REF][START_REF] Strickland | Compression of amplified chirped optical pulses[END_REF]. However, the residual SPM can be sufficient to degrade the quality of the pulses [START_REF] Perry | Self-phase modulation in chirped-pulse amplification[END_REF]. We focus our discussion on an input Gaussian pulse, but the method can be extended to other bell-shaped waveforms [START_REF] Boscolo | Impact of initial pulse shape on the nonlinear spectral compression in optical fibre[END_REF]. Note that part of the discussion can also aid to qualitatively understand the initial stage of wavebreaking occurring in a nonlinear dispersive fiber, as described by Anderson et al. [START_REF] Anderson | Wavebreaking in nonlinear optical fibers[END_REF]. For a Gaussian pulse with an initial linear chirp (C ≠ 0), the total chirp after undergoing SPM is given by

  1 00 1 ( ) . 2 tC t H B G t TT         (23)
Hence, the extreme points of the chirp te verify the following equation:

  2 0
,

e e t H G t R T     ( 24
)
where R = C/B is the ratio of the initial chirp coefficient to the level of SPM experienced. The function H2(t/T0) G(t) is plotted in Fig. 7(a) and has values comprised between -2 (attained at t = 0) and 4 exp(-3/2) (attained at ts = (3/2) 1/2 ). In other words, for R below Rmin = -2 or above Rmax = 4 exp(-3/2), the chirp (t) is a strictly monotonic function as shown in panels b1 or b5 of Fig. 7 (blue curves). Hence, for this range of values of R, we should not expect spectral interference to occur and the spectrum should not show any oscillatory structure. When R = C = 0 (panel b3), we recover the case of an unchirped input pulse studied in Sec. 3. For -Rmin < R < 0 (see panel b4 as an example), the chirp has a local minimum and a local maximum, with the maximum Cm = (tCm) located on the trailing edge of the pulse at

0 0 1 2 4 exp . 2 4 2 Cm T R tW          (25)
For R = Rmin, the two extrema merge and the chirp has an inflexion point at t = 0 (panel b5, red curve). For 0 < R < Rmax, the chirp may feature minimum and a maximum on the trailing edge of the pulse as illustrated in panel b2. The maximum point is given by Eq. ( 25), while the minimum value Cmin = (tCmin) occurs at

0 1 1 2 4 exp . 2 4 2 Cmin T R tW           (26)
For R = Rmax, the chirp exhibits an inflexion point at ts = tCm = tCmin (panel b1, red curve). The different behaviors of the chirp function according to the ratio of its linear and nonlinear parts impose remarkably different features on the pulse spectrum. To illustrate this point, we have plotted in Fig. 8 the evolution of the spectrum with the ratio R at a level of SPM corresponding to B = 20 rad. The results highlight the distinctly different spectral regimes that are observed depending on R: for R > Rmax, the central part of the spectrum does not show any oscillations. For 0 < R < Rmax, two intense peaks are visible in the spectrum and some ripple develops on the spectrum's boundaries. For Rmin < R < 0, the spectrum features an oscillatory structure, which is different from the sinusoidal variation discussed in Sec. 3. Finally, for R < Rmin, no oscillatory structure is visible in the spectrum and most of the pulse energy is focused on the center of the spectrum. Indeed, the pulse experiences spectral compression, a phenomenon that is well documented in the literature and has found many practical applications [START_REF] Stolen | Self-phase modulation in silica optical fibers[END_REF][START_REF] Markaryan | Spectral compression of ultrashort laser pulses[END_REF][START_REF] Finot | Design rules for nonlinear spectral compression in optical fibers[END_REF][START_REF] Oberthaler | Spectral narrowing of ultrashort laser pulses by self-phase modulation in optical fibers[END_REF]. In the following subsection, we discuss some simplifications of the spectral interference model that can be advantageously used to describe the SPM-induced spectral pattern of an initially chirped Gaussian pulse. Then, we discuss the two regimes where the pulse spectrum exhibits an apparent oscillatory structure and explain the origin of the spectral features that are induced by the initial chirp of the pulse.

B. Simplified model

The maximum point of the chirp tCm given by Eq. ( 25) is very important for the description of the SPM pattern. However, once again the presence of the Lambert W function would prevent the development of a fully analytical approach. We can circumvent this difficulty by considering an approximate formulation of the problem that is valid in the highly nonlinear propagation regime for which R is close to zero. To this end, we approximate the function H2(t/T0)G(t) around the point t = tm by its first-degree Taylor polynomial:

    51 22 2 00 2 . m te H G t t t TT      ( 27
)
The green dashed line in Fig. 7(a) highlights the validity of this approximation over a rather wide range of possible values of R. Accordingly, Eq. ( 25) simplifies to and the peak value of the chirp is obtained as

Equation ( 29) indicates that for C > 0, the maximum instantaneous frequency excursion contained in the pulse spectrum is increased relative to the case of an unchirped input pulse. Conversely, for slightly negative values of R, the maximum frequency is smaller than that for an unchirped pulse. The black dash-dotted curve in Fig. 8 confirms the validity of the approximate formula of Eq. ( 29) for small values of R.

To estimate the position of the outermost peak of the spectrum, we can use the method described in Sec. 3. To be rigorous, we should include in the total phase difference between pulse parts having the same instantaneous frequency a phase term accounting for the initial chirp of the pulse. However, we have found that for small values of R, this term does not affect the prediction of the position of the outermost peak 'CM. Hence, for small R we can utilize Eq. ( 20) with m replaced by 'Cm. The expression for 'CM so obtained is plotted in Fig. 8 with a purple dash-dotted curve and confirms the pertinence of the simplification made.

C. Normal initial chirp

In this section, we focus on an input Gaussian pulse with a linear normal (positive) chirp such that the ratio R is below Rmax. The time-dependence of the total chirp of the pulse for C = 4 and B = 20 rad is shown in Fig. 9.

We can clearly see that the chirp has a local maximum Cm at tCm < ts and a local minimum Cmin at tCmin > ts. Accordingly, for a given instantaneous frequency 0 such that 0 < 0 < Cmin, only one point in the pulse corresponds to this frequency, hence no spectral interference occurs. By contrast, for Cmin < 0 < Cm, three instants: t1 < tCm, tCm < t2 < tCmin, and t3 > tCmin have instantaneous frequency equal to 0. However, one should note that as t3 is very far from the center of the pulse, the pulse intensity at this point I3 = G(t3) is very low and can be neglected. This reduces the problem to a two-wave interference process similar to the one discussed in Sec. 3 for the case of an unchirped input pulse. As a result, spectral interference only occurs at the boundaries of the spectrum as can be seen in Fig. 10. When R approaches Rmax, Cmin andCm become very close to each other. This results in the development of a single peak on each side of the spectrum, which thus acquires a 'batman-ear' shape. Accordingly, the use of a linearly chirped input pulse may be a convenient alternative to a triangular-shaped pulse for efficiently copying information onto two different frequency channels [START_REF] Latkin | Doubling of optical signals using triangular pulses[END_REF][START_REF] Verscheure | Pulse doubling and wavelength conversion through triangular nonlinear pulse reshaping[END_REF]. In the opposite limit of R approaching zero, which occurs for a low input chirp or for a high level of SPM accumulated in the fiber, the oscillations cover the major part of the spectrum. 13) and ( 12), respectively.

D. Anomalous initial chirp

Finally, we discuss the case of an input Gaussian pulse with an anomalous (negative) chirp. As previously mentioned, for R < Rmin the pulse is spectrally compressed by the effects of SPM [START_REF] Stolen | Self-phase modulation in silica optical fibers[END_REF][START_REF] Markaryan | Spectral compression of ultrashort laser pulses[END_REF][START_REF] Finot | Design rules for nonlinear spectral compression in optical fibers[END_REF][START_REF] Oberthaler | Spectral narrowing of ultrashort laser pulses by self-phase modulation in optical fibers[END_REF]. We are interested here in values of R above Rmin, which correspond to the propagation regime in which the spectrum expands again after the point of maximum spectral focusing and, thus, spectral interference can arise. The temporal chirp profile of the pulse for C = -4 and B = 20 rad is shown in Fig. 11. Contrary to the case of a normal initial chirp, spectral interference can now occur for any value of 0 such that -Cm < 0 < Cm. For a given 0 > 0, the pulse has this instantaneous frequency at three distinct points: 0 < t1 < tCm, t2 > tCm, and t3 < -tCm.

In Fig. 12(a) we compare the spectrum I(reconstructed from the interference between t1 and t2 with the actual spectrum S() obtained from Eq. (3). We can see that the two-wave interference model can conveniently reproduce the positions of the extrema of the oscillating pattern. However, as can also be seen in Fig. 12(b1), the details of the pattern are not qualitatively reflected. Indeed, S() has a more complex structure than a sinusoidal-like variation, exhibiting an overmodulation and sharper features. To reproduce these properties, one has to take into account the third wave located at t3. The resulting spectral intensity distribution is represented by a purple solid curve in Fig. 12(a) and is clearly in much better agreement with the actual spectrum than the two-wave interference pattern. The results shown in Fig. 12(b2) substantiate the necessity of including the third wave in the interference model for an accurate qualitative description of the spectral pattern engendered by SPM. 13) and ( 12), respectively.

CONCLUSIONS

We have described a simple theoretical approach to predict the main features of the spectra of optical pulses affected by SPM, which is based on regarding the optical spectrum modification as an interference effect. We have shown that a two-wave interference process is sufficient to describe the SPM-broadened spectra of initially Fourier-transformlimited pulses or pulses with an initial positive linear chirp, and to accurately predict the extreme values of the spectra. Simplified but fully tractable closed formulae have been derived for the positions of the outermost peaks of the spectra, which provide a more plausible measure of the spectrum extent than the approximate or rms expressions for the spectral bandwidth that are commonly used. We have also discussed the various spectral regimes that are observed depending on the ratio of the initial chirp of the pulses and the level of SPM accumulated in the fiber, and shown that in the case of negatively chirped input pulses, the description of the SPM spectral patterns requires the inclusion of a third wave in the interference model.

The present approach has been discussed in the context of dispersionless passive propagation, but its extension to include gain or loss is straightforward. Our approach can also help better understand qualitatively the genesis of peculiar spectrum shapes of laser pulses, such as the batman ear spectra observed in all-normal dispersion lasers [START_REF] Renninger | Pulse shaping and evolution in normal-dispersion mode-locked fiber lasers[END_REF] or Mamyshev oscillators [START_REF] Liu | Megawatt peak power from a Mamyshev oscillator[END_REF]. While the present discussion focuses on SPM, the concept can also be applied to other nonlinear modulations of the phase of a pulse, such as the modulation generated by cross-phase modulation [START_REF] Islam | Cross-phase modulation in optical fibers[END_REF] or an external modulator [START_REF] Plansinis | Spectral changes induced by a phase modulator acting as a time lens[END_REF][START_REF] Audo | Nonlinear spectrum broadening cancellation by sinusoidal phase modulation[END_REF][START_REF] Boscolo | Enhanced nonlinear spectral compression in fibre by external sinusoidal phase modulation[END_REF].

Funding Information. Institut Universitaire de France (IUF).

Fig. 1 .

 1 Fig. 1. Temporal intensity and chirp profiles for a transform-limited Gaussian pulse after undergoing SPM at a B-integral of 20 rad. The extreme points of the chirp function and the solutions of the equation = 0 are indicated with red and blue lines, respectively. The cyan and yellow regions represent the phase difference D and the opposite of the total phase offset T, respectively. The approximation to the chirp function based on Eq. (15) and the corresponding approximation to the pulse profile are plotted with dashed green lines.

Fig. 2 .

 2 Fig. 2. Variation of the different phase differences between the two interfering times at the frequency 0, t1 and t2, with 0 for a transformlimited Gaussian pulse after undergoing SPM at B = 20 rad. The phase differences D, NL and T as calculated from Eqs. (8), (9) and (10) are represented by red, blue and purple solid curves, respectively. Their

Fig. 3 .

 3 Fig. 3. (a) Spectral intensity distribution I(0) calculated from Eq. (11) (blue solid curve) compared to the spectrum S() obtained from Eq. (3) (red solid curve) for a transform-limited Gaussian pulse after undergoing SPM at B = 20 rad. The green and cyan dotted lines represent the frequencies at which constructive and destructive interference occurs, as predicted by Eqs. (13) and (12), respectively. The red and purple vertical dash-dotted lines represent the maximum frequency m from Eq. (6) and the toy-model prediction for the position of the outermost spectral peak 'M [Eq. (20)]. (b) Evolutions of the intensities of the two interfering waves at the frequency 0, I1 = G(t1) and I2 = G(t2), with 0 (purple and green curves, respectively).

Fig. 4 .

 4 Fig. 4. (a) Evolution of the spectrum of a transform-limited Gaussian pulse with the B-integral accumulated in the fiber. The spectral intensity distribution I(0) calculated from Eq. (11) (panel 1) is compared to the spectrum S() obtained from Eq. (3) (panel 2). The evolutions of the rms spectral width rms, the position of the outermost spectral peak M predicted by Eq. (14), and the value predicted by the toy 'M [Eq.

( 20 )

 20 ] are plotted with red, white and dark blue dash-dotted lines, respectively. The green and cyan dotted open circles represent the frequencies at which constructive and destructive interference occurs, as predicted by Eqs. (13) and (12), respectively. (b) Evolution of the spectral intensity at the central frequency with the accumulated Bintegral as obtained by numerical evaluation of Eq. (3) (red curve) and by the relationship: I(B, 0=0)  1+sin(B)/2 stemming from Eq. (11) (blue curve).

  Here, the nonlinear phase contribution 'NL has been obtained by using a first-order Taylor expansion of the pulse intensity profile I(t) about tm:I(t) = I(tm) -m (t -tm)/B [Fig. 1(a), green dashed curve]. The predictions from Eq. (17) show fairly good agreement with the results obtained from Eqs. (

Fig. 5 .

 5 Fig. 5. SPM patterns of transform-limited hyperbolic secant (panels 1) and Lorentzian (panels 2) pulses. (a) Spectral intensity distribution I(0) (blue solid curve) compared to the spectrum S() obtained from Eq. (3) (red solid curve) for a B-integral accumulated in the fiber of 20 rad. (b) Evolution of the spectrum S() with the B-integral. The red and purple vertical dash-dotted lines represent the maximum frequency m and the toy-model prediction for the position of the outermost spectral peak 'M [Eqs. (20) and (21)]. The green and cyan dotted lines or open circles represent the frequencies which constructive and destructive interference occurs, as predicted by Eqs. (13) and[START_REF] Chung | Megawatt peak power tunable femtosecond source based on selfphase modulation enabled spectral selection[END_REF], respectively.

Fig. 6 .

 6 Fig. 6. SPM pattern of a transform-limited super-Gaussian pulse with m = 3. (a) Spectral intensity distribution I(0) (blue solid curve) compared to the spectrum S() obtained from Eq. (3) (red solid curve) for a Bintegral accumulated in the fiber of 20 rad. (b) Evolutions of I(0) (panel 1) and S() (panel 2) with the B-integral. The red and purple vertical dash-dotted lines represent the maximum frequency m and the toymodel prediction for the position of the outermost spectral peak 'M [Eq. (22)]. The green and cyan dotted lines or open circles represent the frequencies at which constructive and destructive interference occurs, as predicted by Eqs. (13) and[START_REF] Chung | Megawatt peak power tunable femtosecond source based on selfphase modulation enabled spectral selection[END_REF], respectively.

Fig. 7 .

 7 Fig. 7. (a) Plot of H2(t/T0)G(t) for an initially chirped Gaussian pulse. The green dashed line is the linear approximation to the function in the vicinity of tm given by Eq. (26). The gray dash-dot horizontal lines represent the values of R used in the different subplots b. (b) Temporal chirp profiles of the pulse for different values of R: (b1), R = 1.5 (blue curve) and R = Rmax (red curve); (b2) R = 0.5; (b3) R = 0; (b4) R = -1; (b5) R = -2.5 (blue curve) and R = Rmin (red curve).

Fig. 8 .

 8 Fig. 8. Evolution of the spectrum of a chirped Gaussian pulse with the ratio R = C/B for B = 20 rad, as obtained from numerical integration of Eq. (3). The white dotted horizontal lines represent the notable values of R. The green dotted horizontal lines indicate the R-values used in Figs. 9 and 11. The red, blue, black and purple dash-dotted curves represent the frequencies Cm, Cmin, 'Cm and 'CM, respectively.

Fig. 9 .

 9 Fig. 9. Temporal intensity (panel a) and chirp (panel b) profiles for an initially chirped Gaussian pulse with C = 4 after undergoing SPM at a Bintegral of 20 rad. The extreme points of the chirp function and the solutions of the equation = 0 are indicated with red and blue lines, respectively.

Fig. 10 .

 10 Fig. 10. SPM pattern of an initially chirped Gaussian pulse with C = 4. (a) Spectral intensity distribution I(0) (blue solid curve) compared to the spectrum S() obtained from Eq. (3) (red solid curve) for a B-integral accumulated in the fiber of 20 rad. (b) Evolutions of I(0) (panel 1) and S() (panel 2) with the B-integral. The red and purple vertical dashdotted lines represent the extreme frequencies Cm and Cmin, and the toy-model prediction for the position of the outermost spectral peak 'CM, respectively. The green and cyan dotted lines or open circles represent the frequencies at which constructive and destructive interference occurs, as predicted by Eqs. (13) and[START_REF] Chung | Megawatt peak power tunable femtosecond source based on selfphase modulation enabled spectral selection[END_REF], respectively.

Fig. 11 .

 11 Fig. 11. Temporal intensity (panel a) and chirp (panel b) profiles for an initially chirped Gaussian pulse with C = -4 after undergoing SPM at a Bintegral of 20 rad. The extreme points of the chirp function and the solutions of the equation = 0 are indicated with red and blue lines, respectively.

Fig. 12 .

 12 Fig. 12. SPM pattern of an initially chirped Gaussian pulse with C = -4. (a) Spectral intensity distribution I(0) resulting from the interference of two and three waves (blue and purple solid curves respectively) compared to the spectrum S() obtained from Eq. (3) (red solid curve) for a B-integral accumulated in the fiber of 20 rad. (b) Evolutions of I(0) when two and three interfering waves are considered (panels 1 and 2, respectively) and S() (panel 3) with the B-integral. The red and purple vertical dash-dotted lines represent the maximum frequency Cm, and the toy-model prediction for the position of the outermost spectral peak 'CM, respectively. The green and cyan dotted lines or open circles represent the frequencies at which constructive and destructive interference occurs, as predicted by Eqs. (13) and[START_REF] Chung | Megawatt peak power tunable femtosecond source based on selfphase modulation enabled spectral selection[END_REF], respectively.

Acknowledgment. We thank J. Fatome, B. Kibler and K. Hammani for fruitful discussions.


G Jourdan

G Torricelli

J Chevrier

F Comin

Tuning the effective coupling of an AFM lever to a thermal bath

Keywords: numbers: 07, 10, Cm, 85, 85, +j

Fabrication of Nano-Electro-Mechanical-Systems (NEMS) of high quality is nowadays extremely efficient. These NEMS will be used as sensors and actuators in integrated systems. Their use however raises questions about their interface (actuation, detection, read out) with external detection and control systems. Their operation implies many fundamental questions related to single particle effects such as Coulomb blockade, light matter interactions such as radiation pressure, thermal effects, Casimir forces and the coupling of nanosystems to external world (thermal fluctuations, back action effect). Here we specifically present how the damping of an oscillating cantilever can be tuned in two radically different ways: i) through an electro-mechanical coupling in the presence of a strong Johnson noise, ii) through an external feedback control of thermal fluctuations which is the cold damping closely related to Maxwell's demon. This shows how the interplay between MEMS or NEMS external control and their coupling to a thermal bath can lead to a wealth of effects that are nowadays extensively studied in different areas.

Introduction

General context

NEMS with dimensions truly nanometric are fascinating objects. They are now described in numerous reviews [START_REF] Ekinci | Nanoelectromechanical systems[END_REF][START_REF] Roukes | Plenty of room, indeed[END_REF] and are fascinating in many respects. They are geometrically and chemically very well defined model objects, with numerous potential applications. Well known examples are oscillators such as silicon levers or carbon nanotubes. As NEMS contain a large number of atoms (usually much more than a million), they are most often described as continuous objects but their linear dimensions can make them geometrically close to complex molecules and to viruses. They are used for example as accelerometers, mass detectors with sensitivity close to the atomic level [START_REF] Ekinci | Ultrasensitive nanoelectromechanical mass detection[END_REF], and electro-mechanical filters at high frequencies. The description and analysis of their properties inherently touch some very fundamental issues related to electron or photon coupling with a mechanical oscillator. Driving effects in this field are certainly the Casimir force due to the zero point motion of the electro-magnetic field in vacuum, and observation of the quantum behavior of a NEMS oscillator [START_REF] Schwab | Putting mechanics into quantum mechanics[END_REF][START_REF] Lahaye | Approaching the Quantum Limit of a Nanomechanical Resonator[END_REF][START_REF] Arcizet | Radiation-pressure cooling and optomechanical instability of a micromirror[END_REF][START_REF] Kleckner | Sub-kelvin optical cooling of a micromechanical resonator[END_REF][START_REF] Gigan | Self-cooling of a micromirror by radiation pressure[END_REF]. It is interesting to notice that most groups involved in this research are readers of Braginsky's pioneering works related to measurements of ultra small displacements often in the context of gravitational wave detection [START_REF] Braginsky | Systems with Small Dissipation[END_REF][START_REF] Braginsky | Quantum Measurement[END_REF]. Small energies involved in NEMS excitations make them very sensitive to many external noises. Specifically one can mention measurement back action effects and Brownian motion as they are always operated in contact with a thermal bath. Due to thermal fluctuations, it is hardly possible to define true quasi-static transformations as these NEMS are driven from one stable state to another. In the last decade, in the context of molecular motors, this point has been addressed both theoretically and experimentally. It shows that far before touching any quantum limit, thermal fluctuations that are usually negligible in mechanics dealing with current objects (although not in electronic with strong Johnson noise often limiting performances), become a central ingredient in the description of NEMS behaviour. A decade ago, general theorems in statistical physics [START_REF] Jarzynski | Nonequilibrium equality for free energy differences[END_REF][START_REF] Crooks | Nonequilibrium measurements of free energy differences for microscopically reversible markovian systems[END_REF] such as the Jarzynski equality, an equation that connects the free energy difference between two equilibrium states and the work needed to travel between these two states through nonequilibrium processes, and recently experimentally studied [START_REF] Douarche | An experimental test of the jarzynski equality in a mechanical experiment[END_REF], have given a frame to analyse the non quasi-static processes induced by thermal fluctuations that are involved in NEMS operation at room temperature. This has already been applied to AFM experiments [START_REF] Hyuk | Fluctuation theorems of brownian particles controlled by a maxwell's demon[END_REF][START_REF] Noy | Direct determination of the equilibrium unbinding potential profile for a short dna duplex from force spectroscopy data[END_REF]. In this paper, we show how it is possible to alter the resonance curve width of an oscillator coupled to a thermal bath at temperature T. We then present the result of two independent experiments performed using a UHV Omicron AFM and a home made AFM. The two examples presented here have been chosen so that in one case it is the coupling to the thermal bath that has been directly changed tuning the electro-mechanical coupling, whereas in the second case, this coupling has remained the same but a very fast control of the thermal fluctuations has been set resulting in a very efficient thermal energy extraction from the oscillator.

Report on two AFM experiments

Experimental report that thermally induced fluctuations of a mechanical nano or micro-oscillator can be directly and efficiently influenced without changing the bath temperature is then the central point of this paper. To exemplify this point, we present two different AFM experiments. AFM levers are used as a model system for MEMS and NEMS. Although the oscillator has micrometer dimensions, displacements and distances of interaction are truly nanometric. Both experiments have in common the experimental control of the thermally induced fluctuations of the mechanical oscillator connected to a thermal bath. The signature of the dissipative coupling efficiency onto the resonance of a mechanical oscillator is as usual identified in the frequency resonance width. In the first case, a true increase of the dissipation is observed due to an added dissipative channel as the oscillator becomes part of an electrical circuit that includes a large resistance. In the second case in contrast, an apparent increase of dissipation is in fact due to detailed damping of thermal fluctuations by an external feedback loop. The resonance width clearly increases but a simultaneous and large decrease of oscillator thermal energy is observed and is analysed as a strong reduction in temperature.

AFM: increase damping due to Johnson noise

As classically modelled by the Langevin equation, coupling of AFM levers or of micro/nanooscillators to a thermal bath is essentially described by a viscous force -Γv in the dynamical equation. A characteristic time τ = m/Γ can be associated with the friction coefficient Γ, where m is the oscillator mass. The oscillator resonance frequency ω res is much larger than 1/τ . Short time scale thermal fluctuations with a known force spectral density are described by the temperature and the same friction coefficient Γ through a constant spectral density of noise:

S F = 2k B T Γ (1)
The time τ = m/Γ as discussed in reference [START_REF] Braginsky | Systems with Small Dissipation[END_REF] characterizes the long time oscillator behaviour and the kinetic of heat exchanges with the thermal bath. For time t << τ , the oscillator keeps the memory of its state at t = 0, whereas for time t >> τ , this memory is lost and the average oscillator displacement is fixed by the thermal bath and described by the fluctuation-dissipation theorem (energy equipartition). Measurement of τ from the oscillator resonance characteristics, essentially from the resonance width, enables us to investigate specific channels of energy dissipation in the thermal bath. We here present the study of a dissipation channel associated with Johnson noise present in the set up described in figure 1.

The set up consists of a gold coated sphere with a radius R s = 40 µm glued at the free end of an AFM cantilever creating a capacitance with the sample as the opposite electrode (figure 1). The microsphere polarization circuit is controlled by a DC generator and includes a large resistance due to a poor metallic coating of the probe (this resistance is shown in following section to be 30 MΩ). When applying a DC voltage V c , an attractive capacitive force between the sphere and the sample surface is then generated and described by:

F e = 1 2 C ′ 0 V 2 c (2)
In the sphere plane configuration, the derivative of the capacitance is given at small distances (d << R s ) by:

C ′ 0 = - 2πǫ 0 R s d (3)
This electromechanical coupling turns out to be a new damping channel for the main degree of freedom x o of the mechanical oscillator, since the latter is now connected to the thermal bath associated to the resistance.

The damping process

In this section we show that the mechanical energy can be dissipated by the joule effect in addition to other mechanical friction processes described by Γ 0 (phonons collisions, etc.).

The potential between the sample and the sphere amounts to

V c = V 0 -RI (figure 1(b)).
Therefore, to first order in I, a contribution to the force directly proportional to the current is generated. According to the equation (2) this contribution is given by:

F e = -C ′ 0 V 0 RI (4)
For small displacement in first order of x 0 and I:

Q = C(d 0 + x o )V c = (C 0 + C ′ 0 x o )V 0 -RC 0 I (5)
The charge Q located on the sphere depends on the lever motion x o because of the electromechanical coupling. The last term can be neglected when compared to Q, because the working frequency range determined by the resonance (ω res /2π = 5009.8 Hz) is well below the cut frequency 1/RC 0 (at 100 nm, a pessimistic estimation gives 12000 Hz if the sphere is modelled by a flat 50 µm radius disk) of the low pass circuit. It results a current proportionnal to the speed of the lever according to the relation I = dQ/dt = C ′ 0 ẋo V 0 , the force thus obtained turns out to be a damping process:

F e = -(C ′ 0 ) 2 V 2 0 R ẋo (6)
Therefore the resistance transforms the current generated by the lever displacement into heat with a friction coefficient dissipation rate is clearly observed when a bias voltage is applied to the system. As shown in figure 2, at a distance of 90 nm width increase from 30 rad/s up to 52 rad/s is observed in the resonance curve when the applied voltage is changed from 0 to 0.5 V. This is a clear evidence that the increased dissipation has an electrostatic origin. It implies therefore the existence of a dissipative channel that is made efficient as the sphere surface voltage is applied in agreement with equation [START_REF] Arcizet | Radiation-pressure cooling and optomechanical instability of a micromirror[END_REF]. It means that this added dissipative channel is externally tuned by the applied voltage V 0 . The effective friction coefficient Γ R = mγ R can be used to analyse the continuous increase of the resonance width as the sphere surface distance d is decreased at constant applied voltage of 0.5 V . The behaviour of the width parameter γ = γ 0 +γ R is consistent with a 1/d 2 power laws predicted by equations ( 3) and ( 6): The mass of the oscillator is dominated by the microsphere and estimated to m = 2.8 10 -10 kg (ρ sphere = 1040 kgm -3 ). In figure 3, a resistance of 30 MΩ is used to quantitatively describe the experimental result. This is the single adjustable parameter as all other quantities are experimentally measured in this model geometry.

Γ R = (C ′ 0 ) 2 V 2 0 R.
γ R = 4π 2 ǫ 2 0 R 2 s d 2 V 2 0 R m (7)

The fluctuation dissipation theorem: thermal equilibrium

The new dissipative channel can then be externally controlled as the electromechanical coupling is made efficient using a circuit that includes a dissipative element such as a resistance. However this increased damping does not mean that mechanical position fluctuations are more damped than without any coupling. Conversely the Johnson noise V J introduced by the resistance in the circuit above turns out to be a new source of thermal excitation of the lever (figure 1). It indeed produces an additional fluctuating force on the sphere through the crossing term 2V 0 V J :

F J = C ′ 0 V 0 V J (8)
whose power density spectrum is given by S

F J = (C ′ 0 ) 2 V 2 0 S V J with S V J = 2k B T R.
Although not original, it remains interesting for consistency of the analysis to show that the results comply with the fluctuation dissipation theorem which establishes that the damping rate Γ is related to the Langevin fluctuation force F L :

Γ = 1 2k B T < F L (t)F L (0) > dt (9)
For the noisy force F J , this comes from equation ( 8) and S V J = V J (t)V J (0)dt:

Γ J = (C ′ 0 V 0 ) 2 R (10)
This is the rate Γ R derived in equation [START_REF] Arcizet | Radiation-pressure cooling and optomechanical instability of a micromirror[END_REF]. Since the oscillator is in thermal equilibrium with its surroundings, the fluctuation intensity as well as friction rate gets higher. Energy exchanges between the microlever and the thermal bath increase, but the balance remains the same: the overall flux is nil. The equipartition theorem provides therefore the average energy of the system 1 2 k < x 2 o >= 1 2 k B T . This experimental example shows how it is possible to control a dissipative coupling between a micromechanical oscillator and a thermal bath. Here an electro-mechanical coupling is established and dissipation takes place as currents induced by oscillator vibrations are dissipated in the resistance R also at the origin of Johnson noise shown in equation [START_REF] Ekinci | Ultrasensitive nanoelectromechanical mass detection[END_REF]. The coupling between the thermal bath and the oscillator is then truly tuned. Finally, it is important to note that the whole system remains in thermodynamic equilibrium with the thermal bath: the energy exchanges is balanced. This is in contrast with experimental strategy explored in next section.

AFM and cold damping

As indicated by the title of the pioneering article by Milatz, Van Zolingen and Van Iperen [START_REF] Milatz | The reduction in the brownian motion of electrometers[END_REF] in 1953, "The reduction in the Brownian motion of electrometers", it is possible to damp thermal fluctuations. However this cannot be done through a simple increase of the friction coefficient as this would immediately increase the thermal noise spectral density of force (equation ( 1)) with a cancellation of both effects to maintain fluctuations on position fixed by the equipartition principle of energy at thermal equilibrium.

It has been demonstrated in several circumstances that using an external feedback loop that acts against thermal fluctuations with an applied force proportional to the oscillator speed, it is really possible to damp these fluctuations. This is the so called cold damping procedure [START_REF] Cohadon | Cooling of a mirror by radiation pressure[END_REF][START_REF] Pinard | Full mechanical characterization of a cold damped mirror[END_REF][START_REF] Hopkins | Feedback cooling of a nanomechanical resonator[END_REF].

In the case of cold damping experiment based on an AFM set up, an active cooling can be implemented using an external laser applied onto the lever [START_REF] Mertz | Regulation of a microcantilever response by force feedback[END_REF], and either using a capacitive coupling or a bimorph piezo actuator, that is normally used to excite the lever at resonance, to rapidly react against the thermal fluctuations. We have successfully used the two latter strategies. We shall here describe the third one usable on any AFM. Compared to capacitive coupling it seems to be less intuitive. The experimental set-up is schematically described in figure 4.

The measured lever position x o is directly used through a feedback loop to apply a damping force F D proportional to the lever velocity, i.e. the x o time derivative, that counteracts the displacement. The ratio between the damping force F D and the lever speed is Γ r . The measured spectral density of the lever displacement is shown in figure 5 as Γ r is increased. As it is well known when cold damping is applied to an oscillator, the peak width is increased, and therefore the apparent friction coefficient is also increased. However the peak area decreases gradually which is classically described as a temperature reduction for the vibration mode. These new characteristics can be compared to the friction coefficient Γ 0 and the temperature T when no active loop is present. The characteristic time τ 0 = m/Γ 0 defined with no active loop becomes τ r = m/(Γ 0 + Γ r ) with cold damping operating. As the spectral density of force remains the same with and without cold damping operating, neither the thermal bath nor the system coupling to thermal bath are affected, and the spectral density of force remains the same whatever the value of Γ r :

S = 4k B T Γ 0 = 4k B T ef f (Γ 0 + Γ r ) (11)
This equality [START_REF] Jarzynski | Nonequilibrium equality for free energy differences[END_REF] leads to the following expression for the resonance mode temperature:

T eff = T Γ 0 Γ 0 + Γ r (12)
The thermal fluctuations measured at frequencies around this resonance are then progressively decreased as the feedback loop gain is increased. A quantitative and detailed analysis of this cold damping procedure shows that as the loop gain is increased, the peak integral is strongly reduced and the apparent resonant mode temperature is strongly decreased from 300 K down to 30 K. This is an apparent temperature as this is by no means a thermal bath temperature but the result of the stationary oscillator energy exchange with the thermal and with the external applied force. This we shall analyze below.

Conclusion: comparison of these two experiments

In both experiments presented here, the resonance curve is broadened which is immediately analyzed as an increase of the effective friction coefficient. This is a common point between the two experiments. However a discussion of the thermal energy transfer in both cases shows how much these two procedures differ.

In the first case, adding a capacitive coupling in the presence of a large resistance does not introduce a qualitative change. The friction coefficient is increased and the (a) Displacement spectral density: the thermal fluctuations are progressively cancelled as the loop gain G is increased from G0 to G7 (figure 4). The large thermal displacement at resonance has almost vanished. The small resonance frequency shift is due to a remaining proportional gain in the feedback loop device. (b) The effective temperature is proportional to the area under each lorentzian curve, namely the motion dispersion < x 2 o >. As the feed back loop system is implemented, it decreases from room temperature down to about 30 K.

oscillator coupling to thermal bath is made more efficient with a shorter time τ . This added coupling induces a broadening of the resonant peak but the integral of the resonance peak is essentially preserved and fixed by the thermal bath temperature. The thermal noise spectral density of force increases accordingly. The major change is then an increased coupling of the oscillator to the thermal bath with consequently faster heat exchanges but also a larger thermal noise spectral density of force applied to the oscillator. Originality here relies in the fact that the increased dissipation is due to the charge fluctuation at the sphere/surface due to the Johnson noise of the resistance. In the second case, the resonant peak also broadens. Although this is an apparent increase of the friction coefficient, this is not due to an increased direct coupling between the oscillator and the thermal bath. This coupling to the thermal bath does not change as the feedback loop is activated to induce cold damping. A new coupling is introduced with an external "intelligent" system which very rapidly and efficiently extracts thermal energy from the oscillator. This real time precise and detailed action directly on the thermal fluctuations is very much reminiscent of the way a Maxwell demon is acting as noticed in [START_REF] Hyuk | Fluctuation theorems of brownian particles controlled by a maxwell's demon[END_REF] that provides a theoretical description of this type of experiment related to general theorems [START_REF] Jarzynski | Nonequilibrium equality for free energy differences[END_REF][START_REF] Crooks | Nonequilibrium measurements of free energy differences for microscopically reversible markovian systems[END_REF]. In practice this extraction is so fast and so efficient that there is no time for the heat bath to impose its own temperature of 300 K. Thermal fluctuations are damped in detail and the temperature subsequently decreases. As this is a fast process of thermal energy extraction, the measured characteristic time τ r on the resonance curve is now much shorter than τ 0 the characteristic time of heat exchange when there is no cold damping.

It is then interesting to notice that it is the high quality factor of the oscillator i.e. the low friction coefficient that prevents the heat bath to efficiently maintain the energy oscillator at the value of k B T when the cold damping is acting. In stationary regime, the temperature determined from the oscillator fluctuation distribution is much smaller than the heat bath temperature, i.e. the cold damping process finally extracts energy from the heat bath through the oscillator. The whole process can then be described as a microscopic fridge as quoted in [START_REF] Hyuk | Fluctuation theorems of brownian particles controlled by a maxwell's demon[END_REF]. However an estimate of involved heat fluxes shows that it is not easy to make this fridge efficient enough to really cool down a thermal bath, even if it is a non realistic isolated nanosystem.

As a final remark, regarding NEMS sensor resolution, two different outcomes can be drawn from these two damping processes. In both cases, the mechanical response function of the lever is tuned as a result of the introduction of a new force on the system. However, force measurement accuracy is at best set by the thermal bath fluctuation level. The Johnson noise coupling is then undesirable for applications such as an accelerometer, a tiny force sensor etc. Therefore one has to pay attention to this additional damping process when implementing a capacitive coupling in a nanosystem. In contrast, the cold damping technique does not affect the force measurement noise. However, this comment remains rigorously true if the outside control system can be considered as perfect. Detection noise indeed limits cold damping efficiency and will be studied in a further article.

Figure 1 .

 1 Figure 1. (a) The metallic sphere radius R s is about 40 µm. The sphere surface distance d is between 50 nm and 500 nm. A voltage is established between the sample surface and the sphere: both form a distance dependent capacitance C(d). The resistance R is close to 30 MΩ and is due to a poor metallic coating of the AFM lever. (b) Schematic of the polarization circuit: it consists of a resistance and capacitance device. The capacitance depends on the lever motion through C(d) = C 0 + C ′ 0 x o . The inset above illustrates the Johnson model of a resistance: a perfect resistance associated with a noisy voltage source V J .

Figure 2 .

 2 Figure 2. Lever resonance curves (fundamental mode of vibration) are shown for two different coupling voltages.(a) no voltage is applied V = 0.0 V. (b)a voltage is applied V = 0.5 V. Sphere surface distance is here about 90 nm. Conclusion: Applying 0.5 V induces a width increase close to a factor 2. This is a fully reversible and controllable process. The electrostatic force gradient yields the resonance frequency decrease of the second curve.

Figure 3 .

 3 Figure 3. Widths deduced from resonance curves like ones in figure 2 are reported for different sphere plane distances. Curves are shown for two voltages: blue dots, V=0.0 volt; red squares V=0.5 volt. The fitting curve (1) determines the average value of the mechanical friction coefficient γ 0 = 27.7 ± 0.7 s -1 . The fitting curve (2) relies on equation (7): it is found that 4π 2 ǫ 2 0 R 2 s V 2 0 /m = 1, 4 10 -13 SI. It means that a 30 MΩ resistance can take into account this phenomenon.

Figure 4 .

 4 Figure 4. Schematic description of the cold damping feedback loop. The microlever is damped by the thermal bath with a rate Γ 0 and by the feedback loop through the bimorph actuator with a rate Γ r : The overall friction coefficient is Γ = Γ 0 + Γ r . As the loop gain G is increased, the damping becomes more and more efficient as it is shown in the figure 5.

  Figure 5.(a) Displacement spectral density: the thermal fluctuations are progressively cancelled as the loop gain G is increased from G0 to G7 (figure4). The large thermal displacement at resonance has almost vanished. The small resonance frequency shift is due to a remaining proportional gain in the feedback loop device. (b) The effective temperature is proportional to the area under each lorentzian curve, namely the motion dispersion < x 2 o >. As the feed back loop system is implemented, it decreases from room temperature down to about 30 K.

Figure 6 .

 6 Figure 6. (a) The capacitive coupling adds a classical dissipative channel due to an electromechanical coupling that couples the Johnson noise to the oscillator. The damping is increased but thermal fluctuations are accordingly increased. (b) The cold damping feedback loop extracts thermal energy from the oscillator by direct damping of thermal fluctuations. The damping is increased but no energy is randomly by the feedback loop to the oscillator. The temperature is decreased as thermal energy is extracted much more rapidly than what can be provided by the thermal bath.

  Experimentally an increase of the

	square of lever amplitude [a.u.]	0.0 0.5 1.0	31200 res 31437 rad.s -1 width 30 rad.s -1	31400	31600	(a)	square of lever amplitude [a.u.]	29800 0.0 0.1 0.2 0.3 0.4 0.5 0.6 width 52 rad.s 30000 res 30194 rad.s -1	-1	30200	30400 (b)
					-1						-1
			Angular frequency	[rad.s	]			Angular frequency	[rad.s	]