Sébastien M Popoff
email: sebastien.popoff@espci.psl.eu

Yaron Bromberg

Maxime W Matthès

Rodrigo Gutiérrez-Cuevas
email: rodrigo.gutierrez-cuevas@espci.psl.eu

A practical guide to Digital Micro-mirror Devices (DMDs) for wavefront shaping

Digital micromirror devices have gained popularity in wavefront shaping, offering a high frame rate alternative to liquid crystal spatial light modulators. They are relatively inexpensive, offer high resolution, are easy to operate, and a single device can be used in a broad optical bandwidth. However, some technical drawbacks must be considered to achieve optimal performance. These issues, often undocumented by manufacturers, mostly stem from the device's original design for video projection applications. Herein, we present a guide to characterize and mitigate these effects. Our focus is on providing simple and practical solutions that can be easily incorporated into a typical wavefront shaping setup.

Introduction

Since the advent of adaptive optics, various technologies have been employed to modulate the amplitude and/or phase of light. Early adaptive optics devices, utilized in fields like microscopy and astronomy, offer rapid modulation capable of compensating for the aberrations of optical systems in real-time. However, these devices are constrained by a limited number of actuators, restricting their utility in complex media where a large number of degrees of freedom is essential. Liquid Crystal Spatial Light Modulators (LC-SLMs), which allow for the control of light phase across typically more than a million pixels, have emerged as powerful tools for wavefront shaping in complex media since the seminal work of A. Most and I. Vellekoop in the mid-2000s [START_REF] Vellekoop | Focusing coherent light through opaque strongly scattering media[END_REF]. Nonetheless, LC-SLMs are hampered by their slow response time, permitting only a modulation speed ranging from a few Hz to about 100 Hz.

Digital Micromirror Devices (DMDs) have emerged as a technology bridging the gap between these two types of systems; they offer a large number of pixels (similar to LC-SLMs) and fast modulation speeds (typically up to several tens of kHz). Their high speed capabilities made them attractive for real-time applications, in particular for high-resolution imaging microscopy requiring fast scanning or illumination shaping [START_REF] Cha | Nontranslational three-dimensional profilometry by chromatic confocal microscopy with dynamically configurable micromirror scanning[END_REF][START_REF] Zhuang | Application of digital micromirror devices (DMD) in biomedical instruments[END_REF], biolithography [START_REF] Yoon | Emerging applications of digital micromirror devices in biophotonic fields[END_REF], and optical tweezers [START_REF] Gauthier | Direct imaging of a digital-micromirror device for configurable microscopic optical potentials[END_REF]. However, DMDs are restricted to hardware binary amplitude modulation and are not optimized for coherent light applications. Utilizing DMDs for coherent control of light in complex media is therefore non-trivial and necessitates specific adaptations for efficient use.

To comprehend both the capabilities and limitations of DMD technology for coherent wavefront shaping, it is crucial to understand the device's operating principles and its original design intentions. Investigated and developed by Texas Instruments since the 1980s, DMDs gained prominence in the 1990s for video projection applications since the 1990s under the commercial name of Digital Light Processing (DLP) [START_REF] Larry | Digital light processing for high-brightness high-resolution applications[END_REF][START_REF] Dudley | Emerging digital micromirror device (dmd) applications[END_REF]. The technology enables high-resolution, high-speed, and high-contrast-ratio modulation of light. DMDs operate by toggling the state of small mirrors between two distinct angles, denoted as ±θ DMD . The device is originally engineered for amplitude modulation in video projection applications. In this configuration, one mirror angle directs light into the projection lens, while the alternate angle results in the light path being blocked (see Fig. 1). Given that projectors utilize incoherent light and that the DMD plane is optically conjugated with the projection screen, aberrations within the DMD plane are generally not problematic. Similarly, phase fluctuations induced by temperature variations, as well as minor vibrations from the cooling hardware, are inconsequential in this context. The DMD is designed to produce binary on/off modulation, which is then leveraged to generate grayscale images via pulse-width modulation. Color modulation is accomplished through the use of a color wheel in conjunction with a bright white light source.

Third-party companies have developed kits tailored for research applications, which include a DMD, a control board, and a software interface. Specifically, Vialux devices [START_REF]Home vialux gmbh[END_REF] offer an FPGA board that enables high-speed modulation by allowing frames to be stored in the device's memory [START_REF] Hofling | ALP: universal DMD controller for metrology and testing[END_REF]. However, standard Texas Instruments video projector evaluation modules can also be repurposed into wavefront shaping devices [START_REF] Cox | Converting a texas instruments DLP4710 DLP evaluation module into a spatial light modulator[END_REF], though at a compromised modulation speed. These systems can further be converted into phase or complex field modulators. This is typically achieved by encoding the optical phase into the spatial displacement of binary spatial fringes displayed on the DMD, followed Figure 1: Principle of operation of a DMD in a digital projector. Left, incident light can be reflected towards the projection lens (state on), or onto a beam dump (state off). Right, zoom on the pixels. Image adapted from [START_REF] Jackson | Visual analysis of a texas instruments digital micromirror device[END_REF].

by filtering the high spatial frequencies in the Fourier plane [START_REF] Lee | Binary computer-generated holograms[END_REF]. Such a configuration permits multi-level complex modulation but sacrifices spatial resolution. The implementation and performance of such systems are explored in a separate tutorial [START_REF] Gutiérrez-Cuevas | Binary holograms for shaping light with digital micromirror devices[END_REF] and will not be elaborated upon here. For the remainder of this paper, it will be assumed that the DMD is used for complex modulation via such a method.

While other articles exist describing the various aspects of DMDs [START_REF] Park | Properties of dmds for holographic displays[END_REF][START_REF] Scholes | Structured light with digital micromirror devices: a guide to best practice[END_REF][START_REF] Cox | Converting a texas instruments DLP4710 DLP evaluation module into a spatial light modulator[END_REF][START_REF] Wang | Diffraction characteristics of a digital micromirror device for computer holography based on an accurate three-dimensional phase model[END_REF], this tutorial aims to provide a guide for easily setting up a DMD for wavefront shaping applications in complex media. In particular, we provide characterization and validation procedures that requires minimal changes compared to typical wavefront shaping setups. We first introduce the diffraction properties of a DMD and elaborate on how these could impact the system's efficiency. We also furnish a straightforward criterion for selecting the appropriate DMD parameters for a specified excitation wavelength. In the next section, we delve into the aberration impacts brought about by the non-flatness of the DMD surface. We demonstrate a simple process to characterize this effect and provide compensation solutions. In the third segment, we detail the influence of mechanical vibrations that are induced by the DMD's cooling system. Lastly, we discuss how the thermalization of the DMD chip can potentially result in variations to the DMD response over time.

Choosing the right DMD: Diffraction effects

A 1D model

A significant distinction between liquid crystal modulators and DMDs lies in the geometry of the pixel surface. This difference gives rise to diffraction effects that can adversely affect both the modulation quality and system efficiency. The impact of these diffraction effects is highly contingent on several factors: the wavelength of the illumination, the pixel pitch, and both the incident and outgoing angles. Therefore, alongside selecting an appropriate anti-reflection coating, it is crucial to ensure that the pixel pitch is compatible with the specific configuration being used. Texas Instruments offers chips with a variety of pixel pitches d, ranging approximately from 5 to ∼ 25 µm [START_REF]Texas instruments DLP display and projection chipset selection guide[END_REF].

To achieve a qualitative understanding of this issue, we consider a 1D array of pixels as illustrated in Fig. 2. Initially, let's assume that all pixels are in the same state and are illuminated by a plane wave originating from the far field. Under these conditions, the pixelated modulator essentially functions as a grating, with a period d that is equivalent to the pixel pitch. It is important to underscore that these modulators possess a hardware-limited fill factor, typically around 90%. This translates to an effective active pixel size of d ′ < d.

In general, a grating gives rise to various diffraction orders with differing intensities and angles θ p , as dictated by the grating equation: sin(θ p ) = pλ/d = p sin(θ D ), where λ is the wavelength of the light, θ D is the angle of the first-order diffraction, and p is an integer value denoting the orders of diffraction. The intensity of the individual diffraction orders is influenced by the response of a single pixel, constituting the unit cell of the grating, and that is governed by d, d ′ , and its rotation angle in the case of a DMD. Importantly, we can decouple the effects of the grating's periodicity, which influences the angles of the diffraction orders, from the effects of the response of a single pixel, which shapes the envelope of the angular response.

For a case of normal incidence with an LC-SLM (Liquid Crystal Spatial Light Modulator), we can assume that the response of a single pixel is uniform over its surface. Similarly, in the scenario of a blazed grating, such as a DMD, a linear phase slope is present on each pixel. This is due to the tilt angle θ B of the mirrors. For an arbitrarily selected incidence angle α, a global phase slope is introduced. This results in a trivial shift of the angular diffraction pattern by an angle α. In essence, the incident angle α serves to shift the entire angular diffraction pattern relative to the case of normal incidence, while the blazed angle θ B -the tilt angle of the mirror in the DMD projected onto the axis we consider-only shifts the envelope by an angle of 2θ B . Whenever the fill factor approaches 100%, i.e. when d ≈ d ′ , the envelope for a flat grating achieves its maximum at θ 0 = -α; this corresponds to the angle of the zeroth-order diffraction, and the intensity nears zero for all other orders. In these specific conditions, a singu-flat grating (LC-SLM) blazed grating (DMD) a. b. lar diffraction order is visually perceived, corresponding to the optimum scenario. The addition of a blazed angle θ B results in both a shift in the envelope and in the position of its maximum, now indicated by θ 0 = 2θ B -α. In the general case, this position may not align with a diffraction order anymore [START_REF] Park | Properties of dmds for holographic displays[END_REF]. We provide in Appendix A. a 1D computation of this effect. A more accurate computation of the far field can be found in [START_REF] Wang | Diffraction characteristics of a digital micromirror device for computer holography based on an accurate three-dimensional phase model[END_REF].

We represent in Fig. 3 the angular response of a flat grating and a blazed grating for a 1D filling fraction of 95% (correponding to a 2D filling fraction of ≈ 90%). For a flat grating, the zero-th order contains most of the intensity, the other orders being negligible in comparison. For the blazed grating example shown, we are in a situation close to the worst case scenario: Two diffraction orders have a significant and comparable intensity, and other orders also have a non-negligible contributions. In the optimal scenario, where the peak of the envelope corresponds to a diffraction order, it results in a single diffraction order carrying the majority of the energy. This state is achieved when the conditions of the blazed grating equation are fulfilled [START_REF] Casini | On the intensity distribution function of blazed reflective diffraction gratings[END_REF]:

sin(2θ B -α) + sin(α) = 2 sin(θ B ) cos(θ B -α) = p λ d . (1)
We note that the incident angle α olso affects the diffraction efficiency.

The 2D case

To analyze more precisely the effect of diffraction in a DMD, one needs to consider the 2D surface of the modulator. We can establish a Cartesian coordinate system on the plane of the DMD, with axes x and y aligned with the pixel sides (refer to Fig. 4a). Pixels are uniformly repeated along these axes. However, a technical challenge arises in that the axis of rotation of the pixels aligns with the pixel diagonals, resulting in a rotation by 45 degrees with respect to the x and y axes. For the convenience of alignment and manipulation of the optical setup, it is preferable to work with the incident and outgoing beams which have the optical axis contained in the horizontal plane, i.e. a plane parallel to the table surface. A straightforward and prevalent solution is to rotate the chip by 45 degrees relative to the horizontal plane, which aligns the pixel's axis of rotation to be vertical. This configuration is depicted in Fig. 4b.

The 2D array can be seen as 2 orthogonal 1D gratings in the x and y directions, both having the same properties. A more comprehensive depiction of the 2D system can be found in [START_REF] Scholes | Structured light with digital micromirror devices: a guide to best practice[END_REF]. In order to utilize Eq.( 1), one first needs to project the different angles of the problem onto the incident planes of the two 1D gratings. This yields α = arctan tan(α ′ )/ √ 2 , θ = arctan tan(θ ′ )/ √ 2 , and θ B = arctan tan(θ DMD )/ √ 2 , where θ DMD represents the true rotation angle of the mirrors relative to the diagonal of the pixels, and α ′ and θ ′ denote the incident and outgoing angles in the horizontal plane. We can quantify how close we are to the ideal case, i.e., when satisfying the blazed equation outlined by Eq.( 1), by defining a blazed number µ as introduced in [START_REF] Popoff | Setting up a dmd: Diffraction effects[END_REF]:

µ = ⌊4 d λ [sin(θ B )cos(θ B -α)]⌋ mod 2 -1 , (2)
with ⌊.⌋ representing the floor function, and mod 2 the modulo 2 operation. µ is maximal and equals 1 when the blazing equation is satisfied, i.e. when one order of diffraction contains most of the energy, and minimal when we are in the worst-case scenario, i.e. when four orders of diffraction have a significant and equal intensity.

To demonstrate the effect, we conduct a simulation of a DMD using Python (refer to tutorial and code in [START_REF] Popoff | Setting up a dmd: Diffraction effects[END_REF]) with two pixel pitches of d = 7.6µm and d = 10.8µm, under a coherent excitation at λ = 633nm. Fig. 5 shows the estimated blazed number µ as a function of the angle of incidence α ′ in the horizontal plane, along with the far field diffraction pattern for two distinct incident angles. It should be noted that the efficiency of diffraction can be altered by adjusting the angle of incidence. However, its impact is relatively confined within an acceptable angular range that aligns with experimental limitations (i.e. for angles far from ±90 • ). Far field patterns are centered around the maximum of the envelope θ max = 2θ B -α (marked by a yellow cross). We see that for small values of α, the pixel pitch of d = 10.8µm leads to a blazed number µ close 1. It corresponds in the far field to having one bright order of diffraction close to the maximum of the envelope.

Modulation cross-talk

In practice, we place a pinhole or iris to select one order of diffraction, corresponding to the on state. Having a small value for the blaze number µ not only restricts the amount of light modulation due to the diminished diffraction efficiency, it also influences the modulation quality by inducing cross-talk between the two states of the DMD pixels. Until now, we have assumed that all the pixels are in the same state. In actual usage of the DMD, it becomes necessary to modulate the state of each pixel individually. When µ approaches zero, higher orders of diffraction still possess a significant intensity, as demonstrated in the 1D case in Fig 3 . One adverse implication is that pixels in the off state may contain orders of diffraction that are not blocked by the pinhole, and therefore, will contribute as an interference to the modulated wavefront. We show in Fig. 6 the normalized amplitude of the diffraction patterns corresponding to all the pixels in the on (blue curve) and off (orange curve) states for the same experimental conditions but with two different pixel pitches leading to situations close to the worst (a) and best (b) case scenarios. We observe the presence of a non-negligible contribution of the off state at the main diffraction order of the on state in the first case. While this contribution might appear weak, it does affect the quality of the modulation since the modulation scheme typically necessitates about half the pixels to be in the off state for phase modulation [START_REF] Lee | Binary computer-generated holograms[END_REF], and even more so for elaborate modulation schemes [START_REF] Goorden | Superpixel-based spatial amplitude and phase modulation using a digital micromirror device[END_REF][START_REF] Gutiérrez-Cuevas | Binary holograms for shaping light with digital micromirror devices[END_REF].

Dispersion

DMDs are composed of metallic small mirrors, the response of which is minimally affected by wavelength changes. This is particularly advantageous for broadband applications requiring amplitude modulation and operating on a plane conjugated to the DMD's surface. This is the case for the originally intended application of video projection. However, for wavefront-shaping applications, it is typically required to select a specific diffraction order to acheive phase or complex modulation [START_REF] Lee | Binary computer-generated holograms[END_REF][START_REF] Gutiérrez-Cuevas | Binary holograms for shaping light with digital micromirror devices[END_REF]. Under such circumstances, the wavelength-dependency of the diffraction effect becomes important. The blazed number, denoted by µ (according to Eq.( 2)), scales inversely with the wavelength. Fig. 7 shows the blazed number µ as a function of the wavelength for two pixel pitches d = 7.6µm and d = 10.8µm, with an incident angle of α ′ = 20 • . Within the visible spectrum, µ fluctuates between 0 to 1 over a typical range of roughly 100 nm. ) with the same experimental conditions. In the first case, µ is close to 0, we observe a non-negligible contribution of the off state at the main diffraction order of the on state, thus creating unwanted cross-talk. In the second case, µ is close to 1, the contribution of the off state is negligible. TL;DR: DMDs act as blazed gratings. For a given operation wavelength, we need to find the right pixel pitch to have a good modulation quality and diffraction efficiency. It can be done by estimating the blazed number µ introduced in eq. ( 2) or directly using our custom online tool [START_REF] Popoff | DMD Diffraction Tool[END_REF].

Python code example

We provide in the paper repository [START_REF] Popoff | Respoitory for the paper "a practical guide to digital micro-mirror devices (dmds) for wavefront shaping[END_REF] a Python code to simulate the diffraction effect of a DMD by computing the far field pattern for a set of realistic parameters. It provides a simple way to estimate the blazed number µ introduced in eq. ( 2) to assess the quality of the modulation at the desired wavelength for a given pixel pitch. We also propose an online tool accessible at https://www.wavefrontshaping.net/post/id/49.

Characterizing aberration effects

Presentation of the problem

While only capable of providing hardware binary amplitude modulation, DMDs serve as a potent tool for wavefront shaping and sensing. These applications critically require characterization and correction of aberrations caused by the non-flatness of the DMD surface. For LC-SLMs, the manufacturer typically characterizes the surface's inhomogeneities within the plane of the modulator and provides a spatial phase profile of the introduced aberrations based on the operational wavelength. It is noteworthy that when utilized for intensity modulation on a plane conjugated to the DMD plane, as it is done in digital projectors, the system becomes insensitive to the aberrations caused by the DMD surface. Consequently, these effects are commonly overlooked and rarely documented in the information provided by the manufacturers. Nonetheless, they are non-negligible, with deviations from a flat surface that typically spans several wavelengths [START_REF] Brown | Multicolor structured illumination microscopy and quantitative control of polychromatic light with a digital micromirror device[END_REF].

Finding the correction pattern

In the literature, various methods have been proposed to characterize the phase pattern of the DMD aberrations in the plane of the modulator. Typically, this involves using a model for the aberrations, tweaking the parameters to align with the measurements [START_REF] Matthès | Optical complex media as universal reconfigurable linear operators[END_REF][START_REF] Scholes | Structured light with digital micromirror devices: a guide to best practice[END_REF][START_REF] Brown | Multicolor structured illumination microscopy and quantitative control of polychromatic light with a digital micromirror device[END_REF]]. An alternative approach entails direct measurement of the distorted wavefront, either employing a wavefront sensor such as a Shack-Hartmann [START_REF] Lee | Compensation of wavefront aberration introduced by dmds' operation principle[END_REF], or via interferometry. While such methods yield accurate results, they necessitate adaptation to the particular setup conditions, and frequently require supplementary optical components, meticulous alignment, and custom software. Furthermore, they also demand a meticulously calibrated and stable setup. In this section, we introduce a straightforward method to characterize the aberrations using a lens and a camera. This technique can be employed for any system that offers phase modulation, such as LC-SLMs or deformable mirrors.

We assume the DMD is configured to deliver phase modulation [START_REF] Lee | Binary computer-generated holograms[END_REF][START_REF] Gutiérrez-Cuevas | Binary holograms for shaping light with digital micromirror devices[END_REF]. This implies that the modulator can be divided into N sections, which we designate as macropixels, where the phase can be controlled independently. We use a lens and a camera in its Fourier plane, illuminating the modulator with a collimated beam that extends over the entire area of the modulator intended for use. In scenarios where there are minimal or no aberrations, the intensity pattern observed would mimic the PSF of the lens, such as an Airy disk depicted in Fig. 8.c. However, in practice, we encounter a substantially distorted pattern, like the one represented in Fig. 8.d. A more detailed depiction of the setup for aberration characterization, in the context of a wavefront shaping application in complex media is presented in Fig. 9.a. In the case where the medium is a multimode fiber, one could leverage the reflection from the input surface to directly visualize the intensity pattern of the input plane, as shown in Fig. 9.b.

We hypothesize that the aberrations brought about by the DMD are smooth, and can be depicted by a phase pattern ϕ aber in the plane of the DMD array. This could be feasibly approximated by a finite number of Zernike polynomials Z n (r, θ) [START_REF] Zernike | Beugungstheorie des schneidenver-fahrens und seiner verbesserten form, der phasenkontrastmethode[END_REF], as follows: The goal is to find and display the phase value ϕ corr i on each macropixel i that best compensates for the aberrations, i.e. ϕ corr i = -ϕ aber (r i , θ i ). We create this pattern in the basis of Zernike polynomials

ϕ aberr (r, θ) ≈ N n=0 a n Z n (r, θ) . (3)
ϕ corr i = N n=0 a ′ n Z n (r, θ) , (4)
the best correction is obtained for

a ′ n = -a n ∀n ∈ [0..N ] . (5)
We perform a sequential optimization of parameters a n to maximize a specific function designed to be maximal for the ideal correction of optical aberrations. We first generate a mask, represented by a disk, centered around the point of maximum intensity of the original image (refer to Fig. 8.d) with a radius equivalent to a single speckle grain. The exact size of this radius for a successful optimization is not critical and can be determined by approximating the dimensions of the ideal point spread function, expressed as r 0 ≈ M λ 2N A , where N A is the numerical aperture of the optical system and M refers to its magnification. For a given output intensity pattern, we compute an element-wise product between this image and the created mask, followed by a summation. This calculated sum is then divided by an analogous product, but with the complementary mask substituted in place of the original in order to minimize the side lobes of the point spread function. We set initial parameter values as a ′ n = 0 ∀n ∈ [0..N ]. For each parameter, we test different values of a ′ n , construct the phases for every micropixel according to output is retained. To mitigate potential noise or instability, this complete process is reiterated 3 times for each parameter.

ϕ corr i = N n=0 a ′ n Z n (r i , θ i ),
To estimate the quality of the correction, we compute the Strehl ratio of the point spread function. It is defined as the maximum of the measured PSF divided by the maximum of the ideal one. This optimal PSF is the squared modulus of the Fourier transform of a circular aperture [START_REF] Biddell | On the diffraction of an object-glass with circular aperture[END_REF]:

P SF ideal ∝ J 1 ka R R 2 + f 2 2 , (6)
with J 1 the Bessel function of the first kind of order 1, k = 2π/λ the wavenumber, a the radius of the aperture, R the radial coordinate in the Fourier plane, and f the focal length of the lens.

As an illustration, we conduct an optimization procedure using 11 Zernike polynomials. We use a V-9501 Vialux DMD with a DLP9500 TI chip of resolution 1920 by 1200 pixels and a pixel pitch of 10.8 µm. The optimization is performed on a disk of radius 340 pixels, The illumination is done using an expanded laser beam at 633 nm, corresponding to the aperture of our optical setup. We exclude the first three Zernike polynomials in the optimization process, starting from the radial degree 2. Indeed, the initial one, known as the piston, does not influence the PSF quality. The subsequent ones, the tip and the tilt, cause the PSF to shift. Our procedure relies on optimizing the maximum of the PSF, wherever that is, rendering us indifferent to these two parameters. After optimization, it is possible to generate the correction pattern using a selected number of Zernike polynomials in order to investigate their impact on the PSF quality. We see here that using about 10 Zernike polynomials is sufficient to obtain a Strehl ratio > 0.99. Fig. 11 demonstrates the Strehl ratio and the intensity profiles of the PSF for different counts of the utilized Zernike polynomials. It is important to note that we do not use the full surface of the DMD. Using a larger area may lead to stronger deformations of the PSF, requiring a larger number of Zernike polynomials to be corrected accurately. The experimental data, in addition to the Python code used to generate the figures, can be accessed in the dedicated repository [START_REF] Popoff | Respoitory for the paper "a practical guide to digital micro-mirror devices (dmds) for wavefront shaping[END_REF].

Python code example

We provide in the paper repository [START_REF] Popoff | Respoitory for the paper "a practical guide to digital micro-mirror devices (dmds) for wavefront shaping[END_REF] a Python code example to simulate the effect of aberration on a DMD and then perform a sequential optimization as previously proposed to learn the characterize pattern. We make use of the aotools package [START_REF] Townson | Aotools: a python package for adaptive optics modelling and analysis[END_REF] for generating Zernike polynomials.

TL;DR:

DMDs are not flat and can introduce aberrations; these are typically much stronger than those commonly observed with liquid crystal SLMs. This can be counteracted in situ using a standard setup and a straightforward optimization procedure to maximize the intensity at the central position of the point spread function.

Mechanical and thermal stability

Unlike the original purpose of the DMD, i.e. amplitude modulation for video projectors, typical scientific applications require a high stability of the generated wavefront. This is particularly true for applications in complex media, such as strongly scattering media or multimode fiber, where a small change in the phase front can lead to a large change in the output intensity profile. While LC-SLMs design has been improved and adapted to scientific applications over the last decades, DMD are still relatively new tools for wavefront shaping and sensing and are prone to instabilities that need to be addressed by the user. In this section, we present the effect of mechanical and thermal instabilities and how to limit their impact on the wavefront quality with simple solutions.

Mechanical stability

Most DMD kits consist of two primary components, the chip itself and the electronic board that controls it. This could be the standard electronics board typically used for video projectors, as seen in TI evaluation kits, or an FPGA specifically designed for rapid scientific usage, as offered by Vialux [START_REF]Home vialux gmbh[END_REF] for instance. Integral to these electronics is a fan designed to cool both the chip and the electronic board. However, due to the use of a rigid flat cable for connection between the chip and the electronics board, these parts are not mechanically independent. As such, vibrations originating from the board are partially transmitted to the chip, resulting in minute rotations of the mirror surface. Although this perturbation is inconsequential for video projection, they can have significant impacts on applications involving complex media given their high sensitivity to phase front variations.

Due to its high sensitivity in complex media, it is convenient to characterize this effect directly on the system's response, rather than constructing a distinct setup to analyze the wavefront itself. An example of such a setup is demonstrated in Fig. 12, although a similar approach can be employed with a scattering medium. We enlarge a laser beam onto the DMD and transmit the incoming light through a multimode fiber. Additional elements are required in the setup to fulfill the requirement for complex modulation [START_REF] Gutiérrez-Cuevas | Binary holograms for shaping light with digital micromirror devices[END_REF]. For the sake of clarity, we present a simplified version of the setup where those elements are not present. The output from the fiber is then made to interfere with a reference arm in an off-axis configuration [START_REF] Cuche | Spatial filtering for zero-order and twin-image elimination in digital off-axis holography[END_REF], allowing us to detect changes in the output complex field by recording the interference pattern using a camera.

Laser DMD reference arm In the supplementary materials [START_REF] Popoff | Supplemetary Material for 'A practical guide to Digital Micro-mirror Devices (DMDs) for wavefront shaping[END_REF], we present an animation illustrating the dynamic pattern. In off-axis holography, the local transverse displacement of the fringes is directly proportional to the phase, with a displacement equivalent to the period of the fringes corresponding to 2π [START_REF] Cuche | Spatial filtering for zero-order and twin-image elimination in digital off-axis holography[END_REF]. This permits us to estimate the fluctuation in phase over time at a given position of the output plane. As illustrated in Fig. 13 (depicted by the red curve), the phase varies rapidly over time, a fluctuation attributed to the rapid mechanical vibrations transmitted by the board.

BS Camera BS

A simple yet effective solution consists in dampening the vibrations at the flat ca-time (s) Phase (rad) We employ off-axis holography using the setup depicted in Fig. 12. The red and blue curves correspond to the phase fluctuations measured respectively without and with vibration damping by securing the flat cable with foam, as illustrated in Fig. 14. ble's level by clamping it with a soft material, as depicted in Fig. 14. This can be achieved using commonly available materials. In this context, we utilize simple foam, typically used for packaging, and secure it to the cable with two metallic plates, screws, and nuts. We observe a significant decrease in the phase fluctuations, as demonstrated in Fig. 13 (blue curve).

TL;DR:

The functioning of DMDs can be perturbed by vibrations transmitted from the electronics board via the rigid flat cable that links it to the DMD chip. This adverse effect can be minimized by securing the cable with a soft material that serves to dampen these vibrations.

Thermal stability

Electronics utilized to control the DMD chip experience thermal variations during operation. The dynamics of this effect are dependent on the frame rate. Specifically, the chips heats up more quickly when increasing the frame rate. The increase of temperature can reach more than 15 degrees Celsius when running a sequence at maximum speed (20-30 kHz) [START_REF] Rudolf | Thermal stability of wavefront shaping using a dmd as a spatial light modulator[END_REF]. Notably, this effect is less pronounced when the device is on but not running a sequence. Temperature fluctuations can cause deformations on the chip's surface and can modify the phase response of the glass protective window. This creates low order aberrations, which degrade the quality of the wavefront. While this issue is comparatively less critical than static aberrations and mechanical instabilities previously detailed, it nonetheless has a substantial impact on the complex medium's response when a DMD is employed to modulate the input wavefront.

Before initiating a wavefront shaping experiment, it is important to characterize the influence of temperature to assess the extent to which it impacts the results. Although the exact effect on the wavefront distortion can be directly quantified [START_REF] Rudolf | Thermal stability of wavefront shaping using a dmd as a spatial light modulator[END_REF], it is typically more convenient to directly measure the effect on the studied system's response. To do so, we use a setup similar to the one presented in the previous section and depicted in Fig. 12. We can then estimate the field or intensity decorrelation over time. We present here results with intensity correlation, as it does not require a reference arm. The measured output pattern typically take the form of a seemingly random speckle pattern, that is sensitive to minute changes in the input wavefront. The correlation estimation is obtained by comparing the output intensity pattern I(⃗ r, t) at a given time t to the one at t = 0. We use the following expression for the correlation:

C(t) = Ī(⃗ r, t) Ī(⃗ r, t = 0) ⃗ r Ī(⃗ r, t) 2 ⃗ r Ī(⃗ r, t = 0) 2 ⃗ r , (7)
with Ī(⃗ r, t) = I(⃗ r, t) -⟨I(⃗ r, t)⟩ t and ⟨.⟩ ⃗ r . ⟨.⟩ t represent the spatial averaging over the region of interest of the output plane (i.e. the plane of the camera sensor) and the temporal averaging over the measured frames. Fig. 15 shows the measured decorrelation over time when running an sequence in an infinite loop. t = 0 correspond in the first case (a) to the start of a sequence and is set in the second one to 4.2 hours post the after of the sequence (b). For the scenario (a), it is to note that before the sequence commenced, the DMD was active but remained in the idle state, meaning that it was not executing a sequence. We observe a non-negligible decorrelation over time after the sequence started that slows down after about 1 hour. When running the same experiment 4.2 hours after starting the sequence, the correlation is now stable. Temporal decorrelation of a speckle pattern measured at the output of a multimode fiber when running a sequence in an infinite loop. We use Eq. [START_REF] Dudley | Emerging digital micromirror device (dmd) applications[END_REF] to estimate the correlation between the intensity pattern at a given time t and the one at t = 0. t = 0 corresponds respectively to the start of the sequence (a) and to 4.2 hours after the start of the sequence (b).

The more efficient way to counteract this effect is to use a closed-loop system to stabilize the temperature of the DMD chip. This can be achieved by using a thermoelectric cooler as demonstrated in Ref. [START_REF] Rudolf | Thermal stability of wavefront shaping using a dmd as a spatial light modulator[END_REF] and depicted in Fig. 16.

TL;DR:

DMDs take about an hour to thermaly stabilize when a sequence is running. It can countered by using a thermoelectric cooler and a temperature sensor. It can also be simply mitigated by letting a sequence run for few hours before starting the experiment.

Conclusion

DMDs are powerful tools for wavefront shaping applications in complex media due to their high pixel count, relative low cost, and high refresh rate. However, mostly due to their original purpose of amplitude modulation of incoherent light for video projection, Figure 16: Thermal stabilization of the DMD chip. Schematic representation of the setup utilized to regulate the temperature of the DMD chip. A thermoelectric cooler and a heat sink is employed to control the temperature, while a thermistor is used to measure deviations from the temperature setpoint. A closed-loop system is used to regulate the temperature. Image adapted from [START_REF] Rudolf | Thermal stability of wavefront shaping using a dmd as a spatial light modulator[END_REF].

several effects need to be taken into account when using DMDs for wavefront shaping. First, the choice of its pixel pitch must be made carefully by considering the wavelength of operation to ensure a good modulation quality and diffraction efficiency. Furhtermore, the DMD surface is not flat and can introduce aberrations which can be corrected using a simple optimization procedure. Finally, the DMD is sensitive to mechanical vibrations and thermal variations, which can be mitigated ensuring a good mechanical and thermal isolation of the device. with Π(x) the rectangular function, representing the finite size of the pixel, defined as:

Π(x) = 1, if -1 2 < x < 1 2 , 0, otherwise. (9)
The intensity as a function of the angle in the far-field is given, up to a homotetic transformation, by the absolute value squared of The Fourier transform of Eq. 8 can be written as:

I flat (θ) ∝
We observe that the envelope (right hand term) is maximal for sin(θ max ) = sin(α -2θ B ) while the effect of the periodicity (left hand term) is maximal for sin(θ p ) + sin(α) = pλ/d, representing the orders of diffraction.

Figure 2 :

 2 Figure 2: 1D grating geometry. Schematic representation of the geometry of two types of modulators: (a) the liquid crystal modulator, equivalent to a flat grating, and (b) the DMD geometry, equivalent to a blazed grating. α denotes the incident angle relative to the normal of the array plane, θ 0 refers to the angle of the zeroth diffraction order, and θ B is the tilt angle of the mirrors.

Figure 3 :

 3 Figure 3: Flat grating vs blazed grating. Far field diffraction patterns for a 1D flat grating (left) and a 1D blazed grating (right) for an input angle of α = -20 • , a filling fraction of 95% (corresponding to a 2D filling fraction of ≈ 90%), a pixel tilt angle of θ B = 5 • , and a wavelength to pixel pitch ratio 0.05. Vertical lines represent the angles of the diffraction orders and the black dashed curve represent the amplitude of the field. a.b.

Figure 4 :

 4 Figure 4: 2D grating geometry. (a) Schematic representation of the geometry of the DMD. The incident and reflection angles α ′ and θ ′ 0 are situated within the horizontally plane, illustrated in yellow. (b) Photograph of the DMD chip oriented so that the rotation axis of the pixels is aligned vertically.

Figure 5 :

 5 Figure 5: Blazed number and far field diffraction patterns, extreme cases. Blazing number µ (Eq.(2)) as a function of the incident angle α ′ (top) for a pixel pitch of d = 7.6µm (left) and d = 10.8µm (right). Corresponding far field diffraction pattern (bottom) for two incident angles α ′ = 0 • and α ′ = 25 • . The yellow cross indicates the maximum of the envelope θ max = 2θ B -α.

Figure 6 :

 6 Figure 6: Cross-talk between on and off states. We show the computed normalized amplitude of the diffraction patterns corresponding to all the pixels in the on state (blue) and off state (orange) for two different pixel pitches, d = 7.6µm (a.) and d = 10.8µm (b.) with the same experimental conditions. In the first case, µ is close to 0, we observe a non-negligible contribution of the off state at the main diffraction order of the on state, thus creating unwanted cross-talk. In the second case, µ is close to 1, the contribution of the off state is negligible.

Figure 7 :

 7 Figure 7: Dispersion of the diffraction effect. Blazing number µ (Eq.(2)) as a function of the incident wavelength for an incident angle α ′ = 20 • and a pixel pitch of d = 7.6µm (blue curve) and d = 10.8µm (dashed black curve).

Figure 8 :

 8 Figure 8: Setup for aberration correction. (a) A schematic representation of the DMD array divided into macropixels, of which the phase can be controlled. The precise shape of the macropixels and the area controlled are not crucial for the characterization of aberrations to be effective, provided that the spatial sampling is sufficient to capture the highest spatial frequencies of the aberrations. (b) A schematic representation of the setup employed to characterize the aberrations. A camera is used to image the far field on the DMD surface using a lens. (c) shows the ideal intensity pattern corresponding to the numerical aperture of the illumination setup, i.e. its theoretical ideal point spread function. (d) show the actual recorded intensity pattern corresponding to a distorted point spread function due to the DMD aberrations.

Figure 9 :

 9 Figure9: Detailed aberration characterization setup. (a) A laser beam is expanded and collimated using a telescope (lenses L and L'). The DMD is represented in transmission for simplicity. Light reflected from the DMD is focused by a lens L 1 onto the complex medium to study, namely a scattering medium (SC) or a multimode fiber (MMF). A beamsplitter is used to image the far field of the DMD onto a camera using a lens (L 2 ). The intensity pattern, up to a homotetic transformation, represent the input excitation on the medium which also corresponds to the PSF of the illumination setup. (b) In the case of a multimode fiber, one can take advantage of the reflection of the input facet to directly image the intensity pattern of the input plane.

  record the resulting intensity profile, and evaluate the corresponding cost function. For each parameter, the value that results in the maximum

Figure 10 :

 10 Figure 10: Correction phase map. (a) Schematic representation of the function we want to optimize. We both want to maximize the intensity of the main lobe of the point spread function and minimize its side lobes. (b) Values of the experimentally obtained optimal coefficients in the Zernike polynomials basis. (c) Resulting optimal phase pattern in the chosen illumination area.

Figure 11 :

 11 Figure 11: Effect of Zernike polynomials on the optimization of the PSF. (a) Strehl ratio of the point spread function as a function of the number of Zernike polynomials used for the compensation of the aberrations. (b) Intensity profile of the point spread function for different number of Zernike polynomials in linear and logarithmic scale. To generate this data, we take the final solution of the optimization experimental procedure, i.e. the optimal coefficients a ′ n , and generate the corrected PSF with an increasing number of Zernike polynomials.

Figure 12 :

 12 Figure 12: Measuring stability at the output of a multimode fiber. Schematic representation of the setup used to measure the phase fluctuations through a multimode fiber. An equivalent setup can be used with a scattering medium.

Figure 13 :

 13 Figure 13: Vibration induced phase fluctuations. Phase fluctuations measured over time at a specific position on a plane located at the distal end of a multimode fiber.We employ off-axis holography using the setup depicted in Fig.12. The red and blue curves correspond to the phase fluctuations measured respectively without and with vibration damping by securing the flat cable with foam, as illustrated in Fig.14.

Figure 14 :

 14 Figure 14: Dumping mechanical vibrations. Pictures of experimental setups where a damping of the vibrations is implemented. (a) The flat cable is secured with foam and metallic plates. (b) The flat cable is in tension against a foam material supported by a metallic plate.

Figure 15 :

 15 Figure15: Effect of temperature. Temporal decorrelation of a speckle pattern measured at the output of a multimode fiber when running a sequence in an infinite loop. We use Eq.[START_REF] Dudley | Emerging digital micromirror device (dmd) applications[END_REF] to estimate the correlation between the intensity pattern at a given time t and the one at t = 0. t = 0 corresponds respectively to the start of the sequence (a) and to 4.2 hours after the start of the sequence (b).

R

  flat (x) ∝ Π (x/d ′ ) ⊗ x k δ(x -kd) e -j 2π λ sin(α)x R blazed (x)

p

  δ(sin(θ) + sin(α)) × sinc 2 [sin(θ) + sin(α)] λ d ′ I blazed (θ) ∝ p δ(sin(θ) + sin(α) -p θ D ) orders of diffraction × sinc 2 [sin(θ) + sin(α)] λ d ′ envelope .

Data and code availability

Data and code examples are available in the dedicated repository [START_REF] Popoff | Respoitory for the paper "a practical guide to digital micro-mirror devices (dmds) for wavefront shaping[END_REF].

Appendix A: 1D calculation of the DMD diffraction effect

Assuming the effect of the device's finite size and illumination to be negligible, and situating ourselves within the context of the small-angle approximation, we can represent the field reflected from the device under the influence of plane wave illumination across two systems as follows:


Marion Nickum

Pascale Desrumaux

Psychological Health at Work Among Lawyers: Burnout, Demands, and Engagement

Lawyers are facing a real crisis of psychological health at work

o High levels of psychological distress à depression, suicide, anxiety, addiction [START_REF] Brough | Improving criminal justice work-places: Translating theory and research into evidenced-based practice[END_REF] o Most at risk of suffering from BURNOUT [START_REF] Morgillo | Do not make their trauma your trauma: Coping with burnout as a family law attorney[END_REF] Task-related variables

Step 1: All significant Spearman's rank correlations

Step 2: Mediations -Hayes and Preacher procedure ( 2014)

Figure 1 .Figure 2 .

 12 Figure1. Tested model with decision latitude among French lawyers in 2020[START_REF] Nickum | Burnout among lawyers: Effects of workload, latitude and mediation via engagement and over-engagement[END_REF]


Raphaëlle Chaix

Maria Jesús Alvarez-López

Maud Fagny

Laure Lemee

Béatrice Regnault

Richard J Davidson

Antoine Lutz

Perla Kaliman

Epigenetic clock analysis in long-term meditators

Keywords: Stress, Mindfulness, Epigenetics, Methylation, Biological aging, Blood

In this paper, we examined whether meditation practice influences the epigenetic clock, a strong and reproducible biomarker of biological aging, which is accelerated by cumulative lifetime stress and with age-related chronic diseases. Using the Illumina 450 K array platform, we analyzed the DNA methylome from blood cells of long-term meditators and meditation-naïve controls to estimate their Intrinsic Epigenetic Age Acceleration (IEAA), using Horvath's calculator. IEAA was similar in both groups. However, controls showed a different IEAA trajectory with aging than meditators: older controls (age ≥ 52) had significantly higher IEAAs compared with younger controls (age < 52), while meditators were protected from this epigenetic aging effect. Notably, in the meditation group, we found a significant negative correlation between IEAA and the number of years of regular meditation practice. From our results, we hypothesize that the cumulative

Introduction

Recently, DNA methylation has emerged as a robust biomarker of biological aging. The methylation level of a large number of genomic sites are highly associated with age, to such a point that methods based on a small number of these sites can predict the chronological age with an average error of only 4 years [START_REF] Hannum | Genome-wide methylation profiles reveal quantitative views of human aging rates[END_REF][START_REF] Horvath | DNA methylation age of human tissues and cell types[END_REF]. More precisely, the DNA methylation (DNAm) age can be estimated from the methylation levels of 353 CpG sites of the genome and has been found to highly correlate to chronological age and to be stable across most tissues and cell types (i.e. whole blood, peripheral mononuclear cells, brain cells, breast, kidney, liver, lung and saliva) [START_REF] Horvath | DNA methylation age of human tissues and cell types[END_REF]. The deviation between the DNAm age and the chronological age of an individual provides information regarding his epigenetic aging rate [START_REF] Chen | DNA methylation-based measures of biological age: meta-analysis predicting time to death[END_REF]. Faster-running epigenetic clocks have been negatively associated with longevity (Horvath et al., 2015), and positively associated with chronic diseases [START_REF] Horvath | Obesity accelerates epigenetic aging of human liver[END_REF][START_REF] Horvath | Increased epigenetic age and granulocyte counts in the blood of Parkinson ' s disease patients[END_REF]Levine et al., 2015aLevine et al., , 2015b;;[START_REF] Perna | Epigenetic age acceleration predicts cancer, cardiovascular, and all-cause mortality in a German case cohort[END_REF], frailty [START_REF] Breitling | Frailty is associated with the epigenetic clock but not with telomere length in a German cohort[END_REF], cognitive and physical fitness in the elderly (Marioni et al., 2015b) and all cause mortality even after adjusting for chronological age and a variety of known risk factors [START_REF] Christiansen | DNA methylation age is associated with mortality in a longitudinal Danish twin study[END_REF]Marioni et al., 2015a;[START_REF] Perna | Epigenetic age acceleration predicts cancer, cardiovascular, and all-cause mortality in a German case cohort[END_REF]. The epigenetic clock therefore may represent an accurate tool to measure the effectiveness of lifestyle-based interventions for the prevention of age-related diseases. This is a health priority area today as revealed by data from the Centers for Disease Control and Prevention indicating that 78% of adults aged 55 years and over present one or more chronic diseases (www.cdc.gov).

Consistent with previous work describing the noxious impact of psychological stress on health span [START_REF] Juster | Allostatic load biomarkers of chronic stress and impact on health and cognition[END_REF], it has recently been shown that cumulative lifetime chronic stress [START_REF] Zannas | Lifetime stress accelerates epigenetic aging in an urban, African American cohort: relevance of glucocorticoid signaling[END_REF] and trauma [START_REF] Boks | Longitudinal changes of telomere length and epigenetic age related to traumatic stress and post-traumatic stress disorder[END_REF] predict accelerated epigenetic aging. Meditation-based stress reduction interventions have increasingly become a focus of scientific interest to promote healthy aging. Recent research suggests that meditation has beneficial effects in stress and age-related neuroplastic changes, and mood and cognitive disorders (Davidson and McEwen, 2012;[START_REF] Acevedo | The neural mechanisms of meditative practices: novel approaches for healthy aging[END_REF][START_REF] Luders | Exploring age-related brain degeneration in meditation practitioners[END_REF]. Molecular mechanisms involved in the aging process, such as inflammation, immune and epigenetic pathways [START_REF] Black | Mindfulness meditation and the immune system: a systematic review of randomized controlled trials[END_REF][START_REF] Kaliman | Rapid changes in histone deacetylases and inflammatory gene expression in expert meditators[END_REF], as well as telomere maintenance (Epel et al., 2009;[START_REF] Alda | Zen meditation, length of telomeres, and the role of experiential avoidance and compassion[END_REF], are also sensitive to contemplative practices. However, it remains unexplored whether meditation experience modulates the rate of the epigenetic aging. Given the well-characterized effect of meditation practice on stress reduction [START_REF] Mcewen | In pursuit of resilience: stress, epigenetics, and brain plasticity[END_REF], we tested the hypothesis that long-term practitioners would show slower rates of epigenetic aging than age and sex matched controls. To do so, we analyzed DNA methylome data from blood cells of long-term meditators and meditation-naïve controls.

Material and methods

Participants

Long-term meditators (n = 18) and meditation-naïve controls (n = 20) were studied. The recruitment strategy, and inclusion and exclusion criteria, are described in detail in supplementary information S1. The two groups had similar group distribution of age, sex, body mass index and ethnicity (S2). Participant's written informed consent was approved by the UW-Madison. Meditators had more than 3 years of practice (with a minimum of 30 min of daily sitting meditation and at least 3 intensive retreats), including both mindfulness and compassion-related meditations. The control group comprised of individuals with no prior meditation experience.

These participants were studied in the context of our previous work [START_REF] Kaliman | Rapid changes in histone deacetylases and inflammatory gene expression in expert meditators[END_REF], before (8 am, Time 1) and after (4 pm, Time 2) an intensive day of mindfulness meditation or leisure time in the same environment for meditators and controls, respectively. Detailed information about the interventions is provided in S3. As no impact of daytime or interventions was found on epigenetic aging measures (S3), results presented in this paper correspond to Time 1 samples.

Isolation of peripheral blood mononuclear cells (PBMC)

Blood samples were obtained from each participant and PBMCs were isolated, as described [START_REF] Kaliman | Rapid changes in histone deacetylases and inflammatory gene expression in expert meditators[END_REF]. DNA was isolated using QIAamp DNA Blood Mini Kit, and stored at -80 °C until processing. Sample processing is described in detail in the supplementary information (S1).

Genome-wide DNA methylation profiling

Methylation analysis was performed using Illumina Infinium HumanMethylation450 BeadChip. DNA methylation data from all participants were generated at the same time. DNA samples from both groups were randomized across the chips. For each participant, DNA methylation data at more than 485,512 sites was obtained. Four participants (3 controls and 1 meditator) with poor methylome quality were removed, leaving a data set of 17 meditators and 17 controls.

DNA methylation (DNAm) age and intrinsic epigenetic age acceleration (IEAA)

DNAm age was calculated from raw β-values using an elastic net regression model based on 353 CpGs. IEAA is defined as the residual resulting from a multivariate regression model of DNAm age on chronological age and blood cell counts [START_REF] Chen | DNA methylation-based measures of biological age: meta-analysis predicting time to death[END_REF]. DNAm age and IEAA were estimated using the DNAm age calculator [START_REF] Horvath | DNA methylation age of human tissues and cell types[END_REF]. All values for DNAm age and IEAA are provided in S4.

We tested whether IEAA was significantly different between meditators and controls using a linear model including age and sex as covariates. To explore IEAA trajectory as a function of age in meditators and controls, we generated a total of four subgroups including 9 subjects with ages < 52 and 8 subjects with ages ≥ 52, both for meditators and controls. All tests were corrected for sex. The cut-off age was set to 52 following two criteria: (i) to generate balanced subgroups in terms of sample size (ii) that the older group includes ages at which most individuals present one or several chronic diseases (www.cdc.gov). We checked that BMI, ethnicity and sex showed similar distributions between the subgroups (t-test pvalue for BMI and chi-square p-values with Yates's correction for sex and ethnicity > 0.5 when comparing control subgroups and p-values =0.85, 0.08 and 0.25 respectively for these tests when comparing meditator subgroups). In addition, the two subgroups of expert meditators showed similar distribution of years of meditation experience (p-value = 0.59).

Regression analysis between IEAA and the number of years of practice, taking into account age and sex as covariates, were carried out within the meditators group without including controls, and in the two age subgroups including controls as basal ("zero years meditation experience"). We confirmed that IEAA and years of meditation followed a normal distribution with the Shapiro-Wilk test (p-value=0.59 and 0.17 respectively), and we did not identify any outlier in terms of IEAA (p-value=0.46) or years of meditation (p-value=0.09) using the Grubbs test.

Results

DNA methylation (DNAm) age and Intrinsic epigenetic age acceleration (IEAA) estimation

We measured the DNA methylation (DNAm) ages in long-term meditators and meditationnaïve controls. DNAm and chronological ages were highly correlated to each other (Pearson r = 0.88 in controls, r = 0.79 in meditators, r = 0.83 overall, p-value < 10 -3 in all cases) (Fig. 1A), confirming the accuracy of the epigenetic clock model [START_REF] Horvath | DNA methylation age of human tissues and cell types[END_REF].

We calculated the Intrinsic Epigenetic Age Aceleration (IEAA) which adjusts the epigenetic aging rate for blood cell count estimates, leading to a measure unaffected by both variation in chronological age and blood cell composition [START_REF] Chen | DNA methylation-based measures of biological age: meta-analysis predicting time to death[END_REF]. Correcting for age and sex, we found that meditators and controls exhibited similar IEAAs (p-value = 0.70) (Fig. 1B, Table 1A), indicating no difference in the epigenetic aging rate between the two groups.

Age-related IEAA trajectories in controls and meditators

We next examined how aging influenced IEAA in controls and long term meditators. We compared IEAA between ages < 52 and ≥ 52 in the control group, and we performed the same analysis in long term meditators.

We found that controls presented a different IEAA trajectory of aging compared with meditators: while controls aged 52 and over showed significantly higher IEAAs than younger controls (p-value = 0.04, Fig. 1C and Table 1A), meditators were protected from this effect, showing similar IEAAs in both age subgroups (p-value = 0.47, Fig. 1D and Table 1A).

IEAA and years of meditation practice

To further explore the influence of meditation practice on IEAA, we regressed IEAA on years of regular meditation practice including age and sex as covariates. Expert meditators showed a mean decrease in IEAA of 0.24 [0.06, 0.42] (p-value = 0.02) for each additional year of regular meditation practice (Fig. 1E and Table 1B). We also performed this analysis considering the control group as "zero meditation years" (Fig. 1F andG, Table 1B). We found a significant negative correlation between meditation years of practice and IEAA in subjects aged ≥ 52 (p-value = 0.03) while such an effect was not found in younger subjects (ages < 52) (p-value =0.7).

Discussion

The current study was designed to determine if the rate of epigenetic aging in long-term meditators was different than in meditationnaïve controls (the meditators recruited for this study had between 5 and 30 years of continuous regular practice). We found no difference in the intrinsic epigenetic aging acceleration (IEAA) between long-term meditators and meditation-naïve controls. However, we found a different IEAA trajectory in controls and meditators as a function of age, with an increase of IEAA with age in controls but not in meditators, suggesting that integrating meditation practice into daily routine may have a protecting effect in terms of epigenetic aging in the long run. Cumulative lifetime stress is one of the factors described to accelerate epigenetic aging [START_REF] Zannas | Lifetime stress accelerates epigenetic aging in an urban, African American cohort: relevance of glucocorticoid signaling[END_REF]. Our study suggests that psychological stress reduction in meditators as a result of their regular practice may contribute to the stability of their IEAA across ages compared with controls. Consistent with this hypothesis, we found that in long-term meditators the epigenetic aging rate decreased significantly with the number of years of regular meditation practice. When controls were included in this analysis as "zero meditation experience", the significant effect of years of meditation on IEAA was only found in subjects aged ≥ 52, indicating that this protective effect of meditation practice is observed mainly in older subjects. As a whole, these findings suggest that the protective effect of meditation on epigenetic age acceleration may be progressive and cumulative.

Data presented here extend previous evidence showing the influence of meditation training on aging biomarkers, notably the presence of longer telomeres in long-term meditators [START_REF] Alda | Zen meditation, length of telomeres, and the role of experiential avoidance and compassion[END_REF], the increase in telomerase activity after a 3 months meditation retreat [START_REF] Jacobs | Intensive meditation training, immune cell telomerase activity, and psychological mediators[END_REF], a yogic meditation training [START_REF] Lavretsky | A pilot study of yogic meditation for family dementia caregivers with depressive symptoms: effects on mental health, cognition, and telomerase activity[END_REF] and a mindfulness-based stress reduction program [START_REF] Lengacher | Influence of mindfulness-based stress reduction (MBSR) on telomerase activity in women with breast cancer (BC)[END_REF]. Telomere length and epigenetic aging rate, however, have been shown to be mutually uncorrelated and independently linked to disease and mortality risk, probably through different aging signaling pathways (Marioni et al., 2016). Previous reports on telomere biology, together with our findings on epigenetic aging, suggest that meditation practice may modulate different molecular mechanisms involved in cell aging and may represent a useful preventive strategy for age-related chronic diseases.

Our data may contribute to reveal the benefits of long-term meditation on healthy aging although they present several limitations that warrant further investigation. In particular, longitudinal randomized and controlled trials involving larger samples are required to help disentangle the effect of meditation from pre-existing individual differences and/or from other meditation-associated lifestyle factors on epigenetic aging. It is also critical to explore in future studies the type of practices (i.e. mindfulness-related meditations, compassionrelated meditations, breath work, visualizations, body scanning or other mind-body techniques) and the frequency and duration of the trainings that could efficiently decrease the epigenetic aging rate in otherwise meditation-naïve healthy subjects or clinical populations. 1. In Figs. 1G andH, the years of meditation practice was set to zero for controls. In Fig. 1H, the regression line for men is not traced because of their limited number. In the violin plots (Figures B-D), the white dot is the median, the thick vertical line corresponds to the interquartile range and the thin vertical line to the 95% confidence interval. The bulge corresponds to the kernel density estimation to show the distribution shape of the data. Wider sections of the violin plot represent a higher probability that the samples will have a given IEAA value, the skinnier sections represent a lower probability.

Fig. 1 .

 1 Fig. 1. (A) DNA methylation age (DNAm) correlation with chronological age by group and sex (B-D) Violin plots showing Intrinsic epigenetic age acceleration (IEAA) measured in (C) meditators and controls, (D) controls aged less than 52 and controls aged 52 and over and (E) meditators aged less than 52 and meditators aged 52 and over. (E-G) IEAA regressed on years of meditation practice in (F) the meditators' group, (G) subjects aged less than 52, (H) subjects aged 52 and over. P-value for figure A indicates the significance level of the correlation and p-values for figures B-D are extracted from the linear models presented in Table1. In Figs.1G and H, the years of meditation practice was set to zero for controls. In

Psychoneuroendocrinology. Author manuscript; available in PMC 2018 March 22.

Acknowledgments

Funding

This work was supported by CNRS PEPS INEE 2015 (RC), NCCAM (NIH) (P01-AT004952 (RJD and AL), Fetzer Institute, John Templeton Foundation, anonymous donor (RJD), LABEX CORTEX ANR-11-LABX-0042, Université de Lyon ANR-11-IDEX-0007 and ERC-Consolidator 617739-BRAINandMINDFULNESS (AL). We thank Claire Dandine, Hervé Perdry and Alice Urvoy for helpful discussions.

Authors' contributions RC, RJD, AL and PK designed the study; MJA and PK generated the samples; BR and LL produced the methylation data; RC, MF and BR analyzed the data; RC and PK wrote the manuscript.

All authors declare that they have no conflicts of interest.

Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.psyneuen. 2017.08.016.

Supplementary Material

Refer to Web version on PubMed Central for supplementary material.

Author Manuscript

Chaix et al.

Page 11

Table 1 Linear models fitted to test A) if IEAA differs between i) controls and meditators, ii) controls aged less than 52 and controls aged 52 and over, iii) meditators aged less than 52 and meditators aged 52 and over, and B) to test is IEAA depends on years of meditation practice (years_med) i) within the meditators group, ii) in all subjects younger than 52 years old and iii) in all subjects 52 years old and over. These models integrate age of participants in years and sex (0 females, 1 males) as covariates. In the two latter models, years_med was set to zero for controls. Beta coefficients, their 95% confidence intervals (95% CI) and P-values are listed for each explanatory variable. P-values below the 5% cutoff are in bold. Table S5 (in Supplementary information) provides the standardized beta coefficients from these linear models.

A -Difference in IEAA between groups Controls versus meditators, all ages (Fig. 1B) Controls < 52 versus Controls ≥ 52 (Fig. 1C) Meditators < 52 versus Meditators ≥ 52 (Fig. 1D)