The DeLLight experiment to observe an optically-induced change of the vacuum index

As mentioned in Sec. II, when the weak probe pulse collides with the pump pulse, it engenders a vacuum index gradient which tends to bend the rays of the probe towards regions of higher index/intensity. This is a direct consequence of Fermat’s principle: taking $z$ as the propagation direction of the probe, and $x$ and $y$ as the transverse directions, the optical path length associated with a ray of the probe is

where the prime denotes differentiation with respect to $z$ along the ray trajectory, and where in the second line we have assumed both $\delta n\ll 1$ and $\left(x^{\prime}\right)^{2}+\left(y^{\prime}\right)^{2}\ll 1$ so that higher-order terms can be neglected. This integral is minimized by the actual trajectory, and the problem is formally equivalent to a variational problem in classical mechanics, with $z$ playing the role of time, $\left(\left(x^{\prime}\right)^{2}+\left(y^{\prime}\right)^{2}\right)/2$ the kinetic energy, and $-\delta n$ the potential energy. Since we assume the deflections to be small, $x^{\prime}$ and $y^{\prime}$ are the angles $\theta_{x}$ and $\theta_{y}$ between the tangent vector of the trajectory and the $z$-axis. Therefore, the corresponding Euler-Lagrange equations can be written as

This allows us to determine the expected scaling of the accumulated deflection angle with the parameters characterizing the pump pulse. We assume an axisymmetric pulse characterized by two scales: the transverse width $W_{0}$ and the duration $T_{0}$. The index perturbation $\delta n=n_{2}\,I_{\rm pump}$ will scale as $n_{2}\,\mathcal{E}/T_{0}/\left(\pi W_{0}^{2}\right)$, where $\mathcal{E}$ is the energy of the pump pulse. The transverse gradient of $\delta n$, equal to $\partial\left(\delta n\right)/\partial y$, will scale as $n_{2}\,I_{\rm pump}/W_{0}\sim n_{2}\,\mathcal{E}/T_{0}/(\pi W_{0}^{3})$, and Eq. (4) tells us that this is equal to the deflection rate in the propagation direction $z$. Integrating over $z$, and noting that the longitudinal size of the pump pulse is of order $c\,T_{0}$, we find

This is a rough estimate; it does not account for the non-uniformity of $\partial\left(\delta n\right)/\partial y$ within the pump pulse, nor the spatial extent of the probe, which will itself be a beam rather than an individual ray. But it does allow us to extract an expected order of magnitude for the deflection angle: letting $n_{2}=n_{2,{\rm max}}=1.55\times 10^{-33}\,{\rm cm}^{2}/{\rm W}$, $\mathcal{E}=2.5\,{\rm J}$, and $W_{0}=5\,\mu{\rm m}$, we find $\delta\theta\sim 0.3\times 10^{-12}\,{\rm rad}=0.3\,{\rm prad}$. Since this approximates the probe as a ray of infinitesimal width, we can expect the average deflection of a spatially extended probe pulse to be smaller, so that this value must be an overestimate. Indeed, we shall see below that taking the finite extension of the probe into account tends to reduce this estimate.

The finite spatial extent of both the pump and the probe are taken into account to get a more accurate prediction. For simplicity here, we assume the pulses to be exactly counter-propagating (i.e., $\theta_{\rm tilt}=0$). The general case, in which the tilt angle $\theta_{\rm tilt}$ is non-zero, is presented in Appendix A.

For definiteness and simplicity, let us assume that the probe and pump intensity profiles are separable into transverse $F_{\perp}(x,y)$ and longitudinal $G_{z}(z)$ Gaussian profiles:

with $G_{z}(z)=e^{-\frac{4\ln 2}{T_{0}^{2}}z^{2}}$ and

Here $T_{0}$ is the duration of the pulse (FWHM), while the length scales $W_{0}$ and $w_{0}$ are the standardly defined waists at focus of the pump and probe, respectively. We have included an impact parameter, $b$, between the trajectories of the pump and probe pulses. As mentioned in Sec. II, this is required because a non-zero deflection of the barycenter requires some asymmetry in the response of the probe to the presence of the pump, whereas it will be exactly symmetrical if $b=0$. We can also justify a non-zero impact parameter by noting (see Eq. (4)) that the rate of deflection is equal to the gradient of $\delta n$ in the transverse directions, so we wish for the probe to be centered on a region where the pump intensity is strongly varying.

We can also consider the pulses to be collimated in the interaction region. Indeed, for a Gaussian beam propagation, the beam width varies longitudinally from the focus point as $w(z)=w_{0}\sqrt{1+(z/z_{R})^{2}}$, where $z$ is the longitudinal position along the beam measured from the focus, and $z_{R}=\pi w_{0}^{2}/\lambda$ is the Rayleigh length. For a pulse duration $T_{0}=30$ fs, the longitudinal size of the interaction region is of order $z_{l}=c\,T_{0}=9\,\mu{\rm m}$. For a waist at focus $w_{0}=5\,\mu{\rm m}$ and a wavelength $\lambda=800\,{\rm nm}$, the Rayleigh length is $z_{R}\simeq 100\ \mu{\rm m}$. Thus, the relative variation of the beam width during the interaction is $(w(z=c\,T_{0})-w_{0})/w_{0}\simeq(z_{l}/z_{R})^{2}/2=4\times 10^{-3}$, so the approximation of collimated pulses in the interaction region is valid.

The fact that the total deflection is an integrated effect means that $g_{z}(z)$ drops out of the result, and to each transverse position $(x,y)$ we may assign a local deflection angle $\delta\theta_{y}(x,y)$. Note that, since the focus axes of pump and probe are considered to be shifted along $y$, the probe “sees” a symmetric $\delta n$ in $x$, so the mean of $\delta\theta_{x}$ will vanish; it is for this reason we restrict our attention to $\delta\theta_{y}$. As per Eq. (4), the deflection is found by integrating $\partial_{y}\left(\delta n\right)$ over $z$: $\delta\theta_{y}=\partial_{y}\left(\int{\rm d}z\,\delta n\right)$. Writing $\delta n=n_{2,{\rm max}}\times I_{\rm pump}=n_{2,{\rm max}}\times I_{\perp}\,G_{z}$ and integrating over $z$, we find the local deflection angle $\delta\theta_{y}(x,y)$ to be:

Here, $\mathcal{E}$ is the total energy of the pump pulse.

The mean deflection $\langle\delta\theta_{y}\rangle$ is found by averaging $\delta\theta_{y}(x,y)$ over $x$ and $y$, the transverse form of the probe intensity acting as a weight function:

Substituting (8) into (9), we find:

where we have defined

and

$b_{\rm opt}$ is the value of the impact parameter $b$ at which $\langle\delta\theta_{y}\rangle$ is optimized. This optimal value is $\langle\delta\theta_{y}\rangle_{\rm max}$, the maximum possible barycenter shift given the pump energy and the pulse widths. For achievable pump energy $\mathcal{E}=2.5$ J with the LASERIX facility, and with a minimum waist at focus $w_{0}=W_{0}=5\ \mu$m for both the probe and the pump pulses, the average deflection angle is $\langle\delta\theta_{y}\rangle_{\rm max}=0.13$ prad. Note that its functional form corresponds closely to that predicted in Eq. (5) from purely dimensional considerations. In this analysis, only the value $A$ is model-dependent; the value given here is for the Euler-Heisenberg model derived from QED (with orthogonal polarizations for the pump and probe).

An example of the mean deflection $\langle\delta\theta_{y}\rangle$ is shown, in solid black, in the left panel of Fig. 4, as a function of $b$ at fixed $\mathcal{E}=2.5\,{\rm J}$ and $W_{0}=w_{0}=5\,\mu{\rm m}$.

Experimental constraints dictate that the tilt angle $\theta_{\rm tilt}$ cannot be exactly zero. Analytical calculations (see Appendix A) show that the effect of the tilt angle is to introduce an overall correction factor $r_{\rm tilt}$ to the expression (10) for the deflection angle:

with

Here, $W_{z}$ ($w_{z}$) is the longitudinal size of the pump (probe) pulse, defined analogously to the waist (i.e., the electric field amplitude is proportional to $e^{-z^{2}/w_{z}^{2}}$). The parameter $R$ measures the ratio of the longitudinal to transverse sizes of the pulses. It is the only parameter that depends on their longitudinal size. To understand the form of the numerator of $r_{\rm tilt}$, consider two spherically symmetric pulses, so that $R=1$ and the denominator is also just $1$. In this case, the rotation of the pulse profiles is trivial, and the value of $r_{\rm tilt}$ comes from a combination of two factors: the refractive index change $\delta n$ is proportional to ${\rm cos}^{4}\left(\theta_{\rm tilt}/2\right)$ (see Eq. (1)), while the interaction time during which the refractive index profile is able to act on the pulse is proportional to $1/{\rm cos}\left(\theta_{\rm tilt}/2\right)$. Turning to values achievable with the LASERIX facility, taking a duration $T_{0}=30\,{\rm fs}$, which corresponds to a longitudinal size $w_{z}=W_{z}=7.6$ $\mu$m, and a minimum waist at focus $w_{0}=W_{0}=5\,\mu{\rm m}$, we find $R=1.5$. Examples of $r_{\rm tilt}$, as a function of $R$ for several fixed values of $\theta_{\rm tilt}$, are shown in the right panel of Fig. 4. Also shown in the left panel, in solid red, is the theoretical prediction for the mean deflection $\langle\delta\theta_{y}\rangle$ as a function of $b$ with a tilt angle of $30\degree$ and a pulse duration (FWHM) of $30\,{\rm fs}$. For the optimal impact parameter $b_{\rm opt}$, the deflection is $\langle\delta\theta_{y}\rangle_{\rm max}=0.11\ \mathrm{prad}$, representing only a slight reduction with respect to the case of a “head-on” collision.

As well as a deflection, the perturbed probe will also be characterized by a delay, on account of the slower wave speed due to the $\delta n$ induced by the pump. For $\theta_{\rm tilt}=0$, the delay accumulated at any point within the probe is $\delta t=\left(\int{\rm d}z\,\delta n\right)/c$, and is thus proportional to the bracketed expression in Eq. (8). The mean delay $\langle\delta t\rangle$ is defined analogously to the mean deflection of Eq. (9), and is found to be:

This leads to a phase delay $\langle\delta\psi\rangle=\omega_{0}\,\langle\delta t\rangle$, where $\omega_{0}$ is the carrier frequency of the probe. At a wavelength of $800\,{\rm nm}$, and for $b=b_{\rm opt}$ (so that the deflection $\langle\delta\theta_{y}\rangle$ is optimized), this yields an average phase delay of:

Therefore, with $\mathcal{E}$ on the order of 1 J and $b_{\rm opt}$ on the order of a few $\mu{\rm m}$, the mean phase delay is on the order of a few picoradians, and is thus of the same order as the mean deflection angle.

In this section, we shall illustrate how the Sagnac interferometer yields an amplification of the barycenter shift in the measured intensity profile, considering a true imperfect interferometer.

To account for inevitable low-frequency variations of the measured position of the unperturbed probe, we propose to alternate shots with and without interaction between the pump and probe pulses, which we respectively label as “ON” and “OFF” measurements. We extract the barycenters of the intensity profile measured in the dark output of the interferometer for successive ON and OFF measurements, which we name $\bar{y}^{\rm ON}(i)$ and $\bar{y}^{\rm OFF}(i)$, respectively. The signal $\delta y(i)$ of the $i^{\rm th}$ “ON-OFF” measurement, corresponding to a shift of the barycenter due to the interaction with the pump, is then defined as

The measured $\delta y(i)$ will have a certain distribution characterized by its mean value $\langle{\delta y}\rangle$ and its standard deviation $\sigma_{y}$ (which we shall henceforth refer to as the spatial resolution). Collecting $N$ such measurements, the statistical error (one standard deviation) of the observed mean $\langle{\delta y}\rangle$ is equal to $\sigma_{y}/\sqrt{N}$.

Since classical electromagnetism predicts no light-by-light refraction (and hence $\delta y=0$), we wish to measure a signal whose difference from zero is statistically significant, i.e., for which the measured mean $\langle\delta y\rangle$ is a few standard deviations $\sigma_{y}/\sqrt{N}$ away from zero. Assuming that the average signal $\langle{\delta y}\rangle$ is equal to $\langle\delta y\rangle_{\rm Sagnac}=\langle\delta y\rangle_{\rm direct}/\left(2\sqrt{\mathcal{F}}\right)$ of Eq. (29), the sensitivity of the experiment, in terms of the number of standard deviations $N_{\mathrm{sd}}$, is

Using Eqs. (11), (12) and (23), the number of standard deviation of the signal, in terms of the experimental parameters, is

where $f$ is the focal length of the lenses used to focus the probe and reference pulses inside the Sagnac interferometer, and $r_{\rm tilt}$ is the correction factor defined in Eq. (14). The number of “ON-OFF” measurements is related to the repetition rate $f_{\mathrm{rep}}$ of the laser shots and to the the total duration of the experiment, $T_{\mathrm{obs}}(\mathrm{days})$, given in number of days: $N=86400\times f_{\mathrm{rep}}({\rm Hz})\times T_{\mathrm{obs}}(\mathrm{days})$. With experimental parameters given in local units, and for the expected QED signal ($n_{2,\mathrm{max}}=n_{2,\mathrm{QED}}$), we get

As detailed in Sec. IV, preliminary experimental tests show that an extinction factor of the Sagnac interferometer $\mathcal{F}=0.4\times 10^{-5}$ (corresponding to an amplification factor $1/\left(2\sqrt{\mathcal{F}}\right)$ = 250) and a spatial resolution $\sigma_{y}=10\leavevmode\nobreak\ {\rm nm}$ can be achieved. A waist at focus $W_{0}=5\ \mathrm{\mu}$m for the pump beam is a standard value reachable with intense pulses. A same waist at focus $w_{0}=5\ \mathrm{\mu}$m for the probe beam can be obtained by requiring a waist $w\simeq 25\leavevmode\nobreak\ {\rm mm}$ for the collimated probe beam in the interferometer (before focus) and by using optical lenses with focal lengths $f=500$ mm. A conservative tilt angle of $30\degree$ yields to a correction factor $r_{\rm tilt}\approx 0.9$. With an energy of the pump pulse of 2.5 J as delivered by the LASERIX facility, the expected signal in the dark output of the interferometer²²2The direct signal, using a direct pointing method at a distance $f=500$ mm without interferometer, would be only $\langle\delta y\rangle_{\rm direct}=60$ fm. is $\langle\delta y\rangle_{\rm Sagnac}=15$ pm. With the relatively high repetition rate of LASERIX $f_{\mathrm{rep}}=10$ Hz, we get $N_{\mathrm{sd}}=0.95\times\sqrt{T_{\mathrm{obs}}(\mathrm{days})}$, which means that the sensitivity of the expected QED signal at 1 standard deviation (1-sigma) is obtained after 1.1 days of collected data, or a 5-sigma observation after about one month.

The optical Kerr effect in the residual gas is at first sight a nonlinear effect much stronger than the corresponding effect in vacuum. For instance, the Kerr index in air at atmospheric pressure ($n_{2}\simeq 3\times 10^{-19}\,{\rm cm}^{2}/{\rm W}$) is $2\times 10^{14}$ times larger than the expected nonlinear index $n_{2,{\rm QED}}$ of vacuum. Since this value is proportional to the pressure, setting $n_{2}$ to lie an order of magnitude below $n_{2,{\rm QED}}$ requires a residual pressure of $\sim 10^{-12}$ mbar. However this results is true only if the pump-probe interaction volume $V$ is large enough to contain a few atoms. For the DeLLight experiment, the interaction volume is very small, of the order of $V\simeq w_{0}^{2}\times\Delta t\times c\simeq 225\ \mu\mathrm{m}^{3}$. The vacuum chamber of the DeLLight experiment is designed to ensure a nominal pressure for the full chamber below $10^{-6}$ mbar and a local pressure in the interaction area below $10^{-8}$ mbar. At this local pressure, there is on average only 0.1 residual atom inside the pump-probe interaction volume $V$. Therefore, at achievable pressures, we are already in a regime far below that of the coherence required for a possible refraction effect, and where the notion of refractive index for air can no longer be defined.

Moreover, in our setup using the LASERIX facility, the intensity of the pump in the interaction area is of the order of $10^{20}$ W/cm². At this intensity, the residual gas is completely ionized by the pump, all the electrons being removed from the atoms. We thus have an electromagnetic wave crossing a pure relativistic plasma. Investigating the dynamics of this system will require numerical simulations, but its effect on the DeLLight signal is expected to be negligible. Indeed the plasma density and the plasma index is transversally uniform in the pump-probe interaction area, so that the perturbation of the probe phase is symmetric and the mean deflection (the signal) is zero

Experimental tests are feasible that could distinguish between a possible artefact induced by the residual gas and the vacuum signal. First, by decreasing the intensity of the pump by a factor of 10, the vacuum signal should be reduced by the same factor, while any artefact due to the plasma should be constant (the intensity still being high enough to completely ionize the atoms). Also, by inverting the propagation directions of the probe and reference pulses (i.e., moving from a counter-propagating pump-probe test to a co-propagating test), any plasma signal should remain while the vacuum signal must be suppressed according to the $r_{\rm tilt}$ factor of Eq. (14).

Finally we add that the Kerr effect in air will be used to calibrate and validate the DeLLight method. The measurement will be carried out with a relatively low intensity ($\sim 10^{11}$ W/cm²) in order to avoid generating a plasma via ionization of air molecules, and the Kerr effect will be measured as a function of the pressure. More generally, the use of the Kerr or plasma signals can be used to monitor and control the spatial and temporal overlap of the pump and probe pulses in the interaction area.

Two different Sagnac interferometers have been developed and tested: first, a rectangular Sagnac interferometer composed of three mirrors, without any focus inside the interferometer; then a triangular configuration composed of two mirrors and two optical lenses so as to focus the laser pulses at the midpoint of the interferometer. Both geometries are insensitive to beam pointing fluctuations in angle and transverse position.

The optical setups are shown in Figure 8. The incident laser pulses are delivered by the LASERIX beam with a repetition rate of 10 Hz. The energy per pulse is about 20 $\mu$J, and the pulse duration is 50 fs. The central wavelength is $\lambda_{0}=815$ nm with a spectral width $\Delta\lambda=40$ nm (FWHM). A spatial filter composed of two spherical lenses and a pinhole located in the focal point of the telescope produces a smooth transverse intensity profile that is close to Gaussian. An initial beamsplitter (BS-1) provides two distinct beams: the probe beam, used here only to study the performance of the interferometer, and the pump beam which will be used in a second step to validate the DeLLight technique by measuring the Kerr effect in a medium. For the measurements presented in this article, the pump beam is stopped. Results of the measurements of the Kerr effect in a medium will be presented in a future article.

Before entering inside the interferometer, a polarizer selects the horizontally polarized (p-pol.) component and a neutral density filter sets the suitable intensity of the incident probe pulse. The width $w$ of the intensity profile of the probe pulse is $w\approx 1$ mm (FWHM).

The beamsplitter BS-2 of the Sagnac interferometer is a 50/50 commercial femtosecond p-pol beamsplitter (Semrock FS01-BSTiS-5050P-25.5). Its coating is produced by the Ion Beam Sputtering technique which delivers uniform atomic layers. The expected theoretical values of the reflection and transmission coefficients $r^{2}$ and $t^{2}$ vary in a range $r^{2}=t^{2}=50\pm 0.2\%$ over a broad spectrum between 650 and 1100 nm. Using Eq. (22), a deviation of $0.2\%$ (equivalent to $\delta a=4\times 10^{-3}$) corresponds to an extinction factor $\mathcal{F}=(\delta a)^{2}=1.6\times 10^{-5}$. The phase factor $\phi$ of the beamsplitter (Eq. (20)) is not characterized by the producer. The thickness of the beamsplitter is 3 mm and the group delay dispersion (GDD) is less than 30 fs² for both reflection and transmission. An anti-reflection coating has been deposited on the rear side of the beamsplitter with a reflectivity coefficient $r_{\mathrm{AR}}^{2}=0.1\ \%$ at 800 nm. The mirrors inside the interferometer are standard femtosecond dielectric mirrors with a low GDD value (typically less than 50 fs²) and standard laser grade surface qualities: a flatness peak-to-valley $<\lambda/10$ at 633 nm, a quality 20-10 Scratch-Dig, and a roughness with typical $\mathrm{RMS}<5\ \mathring{\mathrm{A}}$. The beamsplitter and one of the mirrors are controlled by kinematic mirror mounts with two static piezoelectric adjusters for horizontal and vertical alignment with an angular resolution of $0.5\ \mu$rad for a 0.1 V step. The lateral position of one mirror is controlled by a micrometric translation stage. For the triangular configuration of the Sagnac interferometer, the beam is focused using two best form optical lenses L-1 and L-2, placed between the two mirrors M-1 and M-2. The lenses have equal focal length $f=100$ mm, and are separated from each other by a distance $2f$. One lens is mounted on a 3-axis piezoelectric adjuster. The length of the longest interferometer arm between the two mirrors M1 and M2 is about 40 cm.

The dark output of the interferometer is read by a CCD camera (Basler acA1300-60gm) containing $1024\times 1280$ pixels. The pixel dimension is $5.3\times 5.3\ \mathrm{\mu m}^{2}$ and the maximum charge storage capacity before saturation (full well capacity) is $10^{4}$ electrons per pixel. An interferential multilayer dielectric filter $\Delta\lambda=3$ nm centered at 808 nm is placed in front of the CCD camera. Rotation of the incident angle of the filter allows us to select a wavelength from 808 to 800 nm and thereby optimize the extinction factor by minimizing the deviation coefficient $\delta a$ of the beamsplitter. Since $r^{2}$ and $t^{2}$ depend also on the polarization of the incident beam, the deviation coefficient $\delta a$ is also minimized by rotating the incident polarization with a half-wave plate WP installed just after the polarizer, and before the Sagnac interferometer.

The extinction has been measured first for the rectangular Sagnac interferometer with three mirrors and no focusing. A typical transverse intensity profile recorded by the CCD camera in the dark output of the interferometer, after optimization of the extinction factor, is presented in Fig. 9. The two spots due to back-reflections on the rear side of the beamsplitter are clearly observed on opposite lateral sides of the main signal, with approximately equal intensities $I_{\rm AR}$. As illustrated in Fig. 7, ³³3While Fig. 7 shows the case of a triangular two-mirror geometry with focusing, the contributions of the back-reflected beams to the output intensity are the same as in the case of a rectangular three-mirror geometry without focusing. one spot $I_{\rm AR,1}$ corresponds to the direct image of the incident intensity after a single reflection on the rear side of the beamsplitter, while the second spot $I_{\rm AR,2}$ is a superposition of four distinct reflected waves :

where $I_{\rm in}$ is the incident intensity at the entrance of the Sagnac interferometer. At lowest order, we have $r^{2}=t^{2}=1/2$ and $\delta\phi=0$, which gives $I_{\rm AR,1}=I_{\rm AR,2}=I_{\rm in}\times r_{\rm AR}^{2}/2$. The attenuation factor $\mathcal{F}_{\rm AR}$ of the back-reflections is $\mathcal{F}_{\rm AR}=I_{\rm AR}/I_{0}=r^{2}_{\rm AR}/2$. It has been measured with a photodiode and a set of calibrated neutral density filters, and is $\mathcal{F}_{\rm AR}=4\times 10^{-4}$. It corresponds to $r^{2}_{\rm AR}=8\times 10^{-4}$ at 800 nm, in agreement with the typical value given by the producer. For an incidence angle of $45\degree$, the transverse distance between the back reflections and the interference signal is $\delta_{\rm AR}=e/\sqrt{n^{2}-1/2}$, where $e$ is the thickness of the beamsplitter and $n$ the refractive index of its substrate. Here, $e=3$ mm, $n\simeq 1.45$, and $\delta_{\rm AR}\simeq 2.37$ mm.

The interference signal is located in the central part, delimited by the dotted white circle in Fig. 9. The residual phase noise of the interference signal is shown in the right image of Fig. 9 by increasing the sensitivity scale of the display (the intensity of the back-reflections cannot be directly attenuated using a neutral density filter because of the too small thickness of the beamsplitter and therefore a too small transverse distance between the signal area and the back-reflections). Two types of noise pattern can be distinguished: hot spots in the central area of the expected intensity signal and interference rings with large transverse size (low spatial frequency). The extinction factor measured in the central area is at worst $\mathcal{F}\simeq 2\times 10^{-5}$.

The noise pattern is due to a difference in phase between the two counter-rotating pulses when they interfere; we thus refer to it as the phase noise pattern. It is induced by surface defects on the mirrors and on the substrate of the beamsplitter. From Eq. (22), the observed phase noise at a level $\mathcal{F}=2\times 10^{-5}$ corresponds to a difference of phase $\delta\phi=4$ mrad. It is equivalent to a difference in the optical path lengths $\delta l=\delta\phi\times\lambda_{0}/(2\pi)\simeq 5\ \mathring{\mathrm{A}}$. This value is in agreement with the current tolerance of the surface quality of the mirrors and beamsplitter, indicating that the residual hot spots seems to be produced by roughness defects on the surface of the mirrors or the substrate of the beamsplitter. This is experimentally confirmed by the fact that the pattern of the noisy hot spots is horizontally translated when the position of the incident beam is slightly horizontally translated (by the translation stage TS-1 shown in Fig. 8), allowing to scan different part of the surface of the mirrors and beamsplitter. The probable origin of the interference rings, spread over a large transverse size, is the relatively limited quality of the flatness of the mirrors and the beamsplitter, and a modest parallelism of the beamspitter ($<5$ arcminutes). Another possible origin of the residual noise is a deviation from a symmetric $50/50$ beamsplitter. However the dielectric coating is expected to be very uniform at the atomic scale. Therefore the asymmetry should be also uniform and should not exhibit local hot spots.

An additional limitation of the extinction is the presence of a double back-reflection which interferes with the interference signal, as discussed in Sec. III.3.4 (see Eqs. (31) and (32)). For the beamsplitter used in the prototype, we have $\delta a\leq 4\times 10^{-3}$ and $r^{2}_{\rm AR}\simeq 10^{-3}$. Therefore, the contribution from the additional term is small.

In order to reduce the phase noise, and thereby reach an extinction factor on the order of $10^{-6}$, a new beamsplitter and mirrors are under development with superpolished surface qualities: surface flatness peak-to-valley $<\lambda/100$ at 633 nm, surface roughness $\mathrm{RMS}<1\ \mathring{\mathrm{A}}$, parallelism of the beamsplitter $<1$ arcsecond, and an anti-reflection coating on the rear side of the beamsplitter with a reflectivity coefficient $r^{2}_{\mathrm{AR}}\sim 10^{-4}$ at 800 nm.

The extinction has also been measured for a triangular Sagnac interferometer with a focusing of the probe beam by two optical lenses. The minimum waist measured at focus is $w_{0}\approx 25\ \mu$m. A similar quality of the global extinction has been measured in the dark output of the interferometer. Degradation of the extinction by a factor about 2 can be observed locally due to additional defects of lens surfaces. Focusing with off-axis parabolic mirrors (rather than with lenses) is also envisaged.