## Appendix A. Phase extraction

The extraction of the phase from our experimental data involves several steps which will be explained here. We will highlight the advantages of using the finite-fringe method in conjunction with Fourier filtering to reduce influences from experimental noise.

The first processing step is the decomposition of the interferogram in its (wrapped) phase and magnitude components using a Fourier transform technique (figure 10), which eliminates low frequency background variations and high frequency noise from the interference phase (Takeda, Ina & Kobayashi 1982; Kreis 1986). For a uniform refractive index field, the finite-fringe-width interferogram consists of evenly spaced interference fringes with a Gaussian intensity modulation (figure 10
*a*). Additional phase variations result in distortion of this fringe pattern (figure 10
*b*). To extract the phase, the two-dimensional Fourier spectrum is computed (figure 10
*c*) and a rectangular window is applied (the green shaded area). The window only keeps a part of one of the half-planes corresponding with the modulation of carrier fringes. The carrier phase is then removed by recentring the windowed spectrum to the carrier phase frequency (figure 10
*d*). The same spectrum recentring translation has to be used for the reference and test interferograms. Therefore, we recentre the spectrum for the reference interferogram first and use the same translation for subsequent interferograms. Finally, an inverse Fourier transformation yields the interference phase modulo
$2\unicode[STIX]{x03C0}$
(figure 10
*e*) and magnitude (figure 10
*f*) corresponding with the interferogram of figure 10(*b*).

Figure 10. Demonstration of the phase extraction process with a computer generated interferogram.

## Appendix B. Calibration of
$\text{d}n/\text{d}T$
and refraction correction

Since the conversion from the optical path-length difference into temperature fields involves careful calibration, here we will describe the steps taken to ensure the reliability of our experimental results.

The temperature at the top and bottom of the substrate were measured with a surface probe (Anritsu Meter Co., Ltd. N-141K-02-1-TC1-ANP) and a thermocouple inside the heater, respectively. First, a reference interferogram was recorded at room temperature
$T_{0}=22\,^{\circ }\text{C}$
to extract phase and magnitude corresponding to an isothermal substrate. Next, for every increase of
$10\,^{\circ }\text{C}$
in heater temperature set point
$T_{imp}$
, an interferogram was recorded and the temperature was measured at the top of the quartz top obtain
$\unicode[STIX]{x0394}T=T_{imp}-T_{s\unicode[STIX]{x1D6F4}}$
, which is plotted in figure 11(*a*). With this increasing temperature gradient the apparent translation due to refraction increases too (figure 11
*b*), since apart from changing the optical path length, index gradients also cause refraction of light. This effect manifests itself by an apparent translation of the interferogram when the substrate is heated to high temperatures due to the index gradient from high to low values (top to bottom). When phase subtraction is performed, the two interferograms therefore no longer correspond to the same location on the camera, causing errors in the extracted phase. The apparent translation relative to the reference interferogram is found using two-dimensional cross-correlation of the magnitude extracted from the interferograms and the phase and magnitude are corrected accordingly. This translation is corrected for before phase subtraction and magnitude division.

Figure 11. Temperature difference
$\unicode[STIX]{x0394}T=T_{imp}-T_{s\unicode[STIX]{x1D6F4}}$
across the quartz substrate in the absence of a drop (*a*) and the apparent translation relative to the reference interferogram (*b*).

Figure 12 shows phases extraction of interferograms recorded at room temperature and with the heater at
$400\,^{\circ }\text{C}$
. The phase map in figure 12(*c*) shows that the index gradient in the air above the quartz is negligible compared to the index gradient inside the quartz. This supports the assumption that the refractive index change due to the heating of air is negligible compared to the refractive index change inside the quartz. Next, the phase is unwrapped (figure 12
*d*). Figure 12(*e*) shows the vertical profiles corresponding to the region indicated by the rectangle in figure 12(*d*). An offset
$\unicode[STIX]{x1D719}_{imp}$
is added in figure 12(*e*) such that the phase
$\unicode[STIX]{x1D719}_{imp}$
at the heater surface correspond to the temperature difference between the heater temperature
$T_{imp}$
and room temperature
$T_{0}$
,

where
$\text{d}n/\text{d}T$
is the later determined value.

Unwrapping errors are encountered near the top and bottom edge of the quartz plate. Figure 12(*e*) shows that the mean value of the measured phase follows a linear behaviour near the edges, but at the very edge the phase cannot be unwrapped reliably. This is a result of the quartz near the edges being of lesser optical quality as a result of the polishing process and the occurrence of diffraction occurring close to the top and bottom surfaces. To determine the phase at the very edge, the phase profiles are therefore extrapolated towards the edges of the plate using a linear fit of the phase profile. For both edges of the plate, the fit was based on the phase field between 0.5 and 1 mm distance from the edge (the blue shaded area of figure 12
*e*). The phase difference
$\unicode[STIX]{x0394}\unicode[STIX]{x1D719}=\unicode[STIX]{x1D719}_{imp}-\unicode[STIX]{x1D719}_{s\unicode[STIX]{x1D6F4}}$
is then computed from the extrapolated values,
$\unicode[STIX]{x1D719}_{imp}$
at the heater surface and
$\unicode[STIX]{x1D719}_{s\unicode[STIX]{x1D6F4}}$
at the quartz surface. The refractive index difference
$\unicode[STIX]{x0394}n$
across the quartz plate is then

Now,
$\text{d}n/\text{d}T$
can be determined from a linear fit of
$\unicode[STIX]{x0394}n$
as a function of
$\unicode[STIX]{x0394}T$
(figure 12
*f*), assuming that the temperature field in the absence of a drop essentially varies only across the quartz plate. The result,
$\text{d}n/\text{d}T=(1.20\pm 0.01)\times 10^{-5}~\text{K}^{-1}$
, is consistent with the values reported in the literature (Malitson 1965; Toyoda & Yabe 1983).

For the Leidenfrost experiments, the heater can now be set to any temperature
$T_{imp}$
to record an interferogram with and without a drop. The phase difference between these two then gives the amount of cooling due to the drop.

## Appendix C. Abel inversion

Presented here are the details on our novel approach to Abel inversion. As already outlined in § 2.2, our method is based on fitting the data using the projection on a basis function expansion. The experimental data are thus fitted in the Abel space while the thereby obtained coefficients are then used to reconstruct the temperature field.

The interferometric phase
$\unicode[STIX]{x1D719}(y,z)$
is a two-dimensional projection of the refractive index field
$\unicode[STIX]{x0394}n(x,y,z)$
, The underlying field must therefore be determined from the projection by tomographic reconstruction. If the underlying field is axisymmetric (as approximately assumed here), a single projection
$\unicode[STIX]{x1D719}(y,z)$
is sufficient to reconstruct
$\unicode[STIX]{x0394}n(r,z)$
(in cylindrical coordinates). The projection is then related to the field by the Abel transform:

where in our case

The analytic Abel inversion is given by

However, this integral is known to be sensitive to noise, especially near the axis of symmetry (Ma, Gao & Wu 2008). In order to reliably extract the local heat flux from the temperature field, we propose an inversion method based on the basis function expansion method (BASEX) which imposes that the reconstructed temperature field satisfies the Laplace equation for steady heat conduction. With BASEX, the unknown distribution
$f(r,z)$
is approximated by a complete set of functions with linear coefficients (Dribinski *et al.*
2002). The phase projection of this expansion solution (the forward Abel transform) is computed by numerical integration. The coefficients are then obtained by matching coefficients of the projection to the measured projection data
$F(y,z)$
using linear regression. Based on physical arguments given in § 2.1, our domain is restricted to the width
$W$
of the quartz plate, thus (C 1) becomes a definite integral:

The axisymmetric solution to the Laplace equation
$\unicode[STIX]{x1D6FB}^{2}f=0$
in cylindrical coordinates is given by the family of functions (Jackson 1998)

where
$A_{p}$
and
$B_{p}$
are constants,
$k$
is a real or imaginary number and
$\text{J}_{0}(r)$
and
$\text{Y}_{0}(r)$
are Bessel functions of first and second kind, respectively. Since
$\text{Y}_{0}(kr)$
diverges for
$r\rightarrow 0$
, we must have
$B=0$
. The remaining functions
$f_{k}(r,z)=\sum _{p=1}^{N}A_{p}\text{J}_{0}(kr)\text{e}^{\pm kz}$
can be divided in two families of solutions. The first family provides a complete orthogonal set of functions on a disc of radius
$a$
in the form of a Fourier–Bessel series:

where
$k=\unicode[STIX]{x1D701}_{p}/a$
is real, with
$\unicode[STIX]{x1D701}_{p}$
the
$p$
th root of the zeroth-order Bessel function of the first kind. The constant
$a$
represents the largest distance from
$r=0$
and is given by
$\sqrt{(l_{y}/2)^{2}+(W/2)^{2}}$
where
$l_{y}$
is the field of view of the camera. The second family provides a complete orthogonal set of functions on a surface at constant
$r$
with
$0\leqslant z\leqslant c$
in the form of a Fourier series

where
$k=\text{i}p\unicode[STIX]{x03C0}r/c$
is imaginary with
$p=1,2,\ldots$
and
$I_{0}(kr)=\text{J}_{0}(\text{i}kr)$
is the zeroth-order modified Bessel function of the first kind. In our case, we take
$c=H_{s}$
, and add a translation in the
$z$
-direction such that
$z=0$
at the top of the substrate for the measured data.

To approximate the three-dimensional field
$\unicode[STIX]{x0394}n(r,z)$
, we use a linear combination of both families plus an offset term
$f_{0}$
:

Next, we compute the phase projection by numerical integration of (C 1), taking into account that
$f$
is zero for
$x<-W/2$
and
$x>W/2$
:

The integration is carried out numerically for every point
$(y,z)=(y_{k},z_{k})$
, with
$r_{k}=\sqrt{x^{2}+y_{k}^{2}}$
, corresponding with the measured
$F_{exp}(y_{k},z_{k})$
. The parameters
$A_{p}$
,
$B_{p}$
,
$C_{p}$
,
$D_{p}$
and
$f_{0}$
are determined by a least squares fit to a measured projection
$F_{exp}(y_{k},z_{k})$
by means of linear regression. The reconstructed field then follows by substitution of the parameter values in (C 10). In our analysis we used
$M=N=10$
to obtain high accuracy, without amplifying high-frequency noise. Feature scaling is applied to improve the numerical stability of the linear regression. Since the integrals for the forward Abel transform only depend on the geometry of the problem, the method excels for large data sets with an identical coordinate system: the integrals have to be computed only once in that case and the inversion from any experimentally observed projection reduces to a few matrix multiplications to solve the regression problem.

## Appendix D. Reynolds and Péclet number estimation in the vapour film

The values of the Reynolds number, comparing the inertial and viscous effects in the vapour film, and of the Péclet number, comparing heat transfer by convection and conduction, are evaluated *a posteriori*. It is reasonable to assess them above all in the neck region, where the vapour flow velocity attains its maximum (Sobac *et al.*
2015*a*
). The Reynolds number is then defined as
$Re=(\unicode[STIX]{x1D70C}_{v}U_{neck}h_{neck}^{2})/(\unicode[STIX]{x1D707}_{v}\ell _{neck})$
, where
$U_{neck}$
is the velocity in the middle of the vapour film cross-section at the neck location,
$h_{neck}$
is the neck thickness (cf. § 4.4) and
$\ell _{neck}$
is the length of the neck region as measured at the height
$2h_{neck}$
of the vapour film profile. The maximum
$Re$
value for the parameters of the present study is thereby found to be of the order of 0.5, justifying the dominant role of viscous forces in the vapour film assumed here (Stokes flow). Indeed, for an ethanol drop of
$R=3.56$
mm with
$T_{s\unicode[STIX]{x1D6F4}}=330\,^{\circ }\text{C}$
, one can obtain
$U_{neck}=1.75~\text{m}~\text{s}^{-1}$
,
$h_{neck}=58~\unicode[STIX]{x03BC}$
m and
$\ell _{neck}=906~\unicode[STIX]{x03BC}$
m giving rise to the mentioned
$Re$
value. The Péclet number
$Pe=Pr\,Re$
is then estimated to be of the order of 0.4, as well, supporting *a posteriori* the hypothesis of a mostly conductive heat transfer in (across) the vapour film. Here
$Pr=\unicode[STIX]{x1D708}_{v}/\unicode[STIX]{x1D6FC}_{v}\sim 0.8$
is the Prandtl number, and
$\unicode[STIX]{x1D708}_{v}=\unicode[STIX]{x1D707}_{v}/\unicode[STIX]{x1D70C}_{v}$
and
$\unicode[STIX]{x1D6FC}_{v}$
are the kinematic viscosity and the thermal diffusivity of the vapour, respectively.

## Appendix E. Influence of the needle

While in the experiments the Leidenfrost drop is kept attached to a needle, the theory assumes a free Leidenfrost drop. In the present appendix, we numerically assess the actual influence of the needle on the Leidenfrost drop shape as well as on the temperature profile in the substrate. Mathematically, incorporating the presence of a needle just consists in imposing as a boundary condition for the upper shape of the drop, governed by (3.1), a vertical slope at the external needle radius (assuming complete wetting) rather than merely no singularity at the symmetry axis. Figure 13 reports the results for an ethanol Leidenfrost drop over the quartz plate used in the experiments, as described in § 2.1, with an imposed temperature of
$330\,^{\circ }\text{C}$
. As one can see, it is just the upper shape that is mostly affected by the presence of a needle. At the same time, the effect of the needle on the vapour film thickness and on the substrate temperature field is minor, although it proves to be stronger for smaller drops as could be expected. However, even for
$R=0.87\ell _{c}$
, the smallest drop dealt with in the present set-up, the neck thickness is affected only by 0.5 %, while the maximum substrate cooling only by
$0.5~\text{K}$
. Thus, we conclude that the presence of a needle is not really essential here.

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