Simple Yet Efficient Content Based Video Copy Detection

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# Simultaneous microscopic imaging of thickness and refractive index of thin
layers by heterodyne interferometric reflectometry (HiRef)

Alexander Nahmad-Rohen<EMAIL_ADDRESS>Instituto de Ciencias
Aplicadas y Tecnología, Universidad Nacional Autónoma de México, Circuito
Exterior S/N, Ciudad Universitaria, Ciudad de México, 04510, México  David
Regan  School of Physics & Astronomy, Cardiff University, The Parade, Cardiff,
CF24 3AA, United Kingdom  Paola Borri  School of Biosciences, Cardiff
University, Museum Avenue, Cardiff, CF10 3AX, United Kingdom  Wolfgang
Langbein<EMAIL_ADDRESS>School of Physics & Astronomy, Cardiff
University, The Parade, Cardiff, CF24 3AA, United Kingdom

###### Abstract

The detection of spatial or temporal variations in very thin samples has
important applications in the biological sciences. For example, cellular
membranes exhibit changes in lipid composition and order, which in turn
modulate their function in space and time. Simultaneous measurement of
thickness and refractive index would be one way to observe these variations,
yet doing it noninvasively remains an elusive goal. Here we present a
microscopic-imaging technique to simultaneously measure the thickness and
refractive index of thin layers in a spatially resolved manner using
reflectometry. The heterodyne-detected interference between a light field
reflected by the sample and a reference field allows measurement of the
amplitude and phase of the reflected field and thus determination of the
complex reflection coefficient. Comparing the results with the simulated
reflection of a thin layer under coherent illumination of high numerical
aperture by the microscope objective, the refractive index and thickness of
the layer can be determined. We present results on a layer of polyvinylacetate
(PVA) with a thickness of approximately 80 nm. These results have a precision
better than 10% in the thickness and better than 1% in the refractive index
and are consistent within error with measurements by quantitative differential
interference contrast (qDIC) and literature values. We discuss the
significance of these results, and the possibility of performing accurate
measurements on nanometric layers. Notably, the shot-noise limit of the
technique is below 0.5 nm in thickness and 0.0005 in refractive index for
millisecond measurement times.

##  I Introduction

Reflectometry has long been used as a noninvasive optical probe for properties
of a sample. Interferometric reflectometry (iRef) is an enhancement of
reflectometry which uses the interference between two beams to infer the
complex amplitude of the reflected field rather than its intensity.
Alternatively, it can employ the interference of the reflected field with
itself (either different reflection orders or the reflections at different
interfaces).

The interference of reflections at different interfaces of a coated glass
capillary filled with a liquid has been used to measure the liquid’s
refractive index.ref-BornhopUS7130060B2  The same principle has been used on
planar layers to measure crystal etch rateref-SteinslandSAA86  and diamond
crystal growth;ref-CatledgeDRM9  in both cases, this has been done on layers
with thicknesses on the order of microns. Adsorption of polymer layers as thin
as 80 nm has also been studied with this type of technique.ref-MunchJCSFT86  A
commercial setup has been used to study biological samples by looking at the
interference between light reflected at the interface between air and the
sample, which sits on a substrate consisting of a SiO2 layer on top of a Si
layer, and light reflected at the SiO2-Si interface. This setup has been used
to noninvasively monitor DNA degradation, DNA-protein binding and single-virus
deposition on surfaces (using ebolavirus and Marburg virus, both of which have
a minor axis of about 80 nm).ref-AvciS15  It has also been used to observe
antibody-antigen and DNA-DNA interaction.ref-AvciS15 ; ref-DaaboulBB26  Its
accuracy is about 50 nm, although with the aid of nanoparticle labelling
(which makes it invasive) the accuracy improves to about 20 nm. On a much
larger scale, there is GPS interferometric reflectometry, which consists of a
receiver above ground detecting electromagnetic waves emitted by satellite,
containing the interference of the direct path with the reflection from the
ground; this has been used to measure soil moisture.ref-ChewGPSS20

Interferometric scattering microscopy (iSCAT) uses reflectometry of a planar
interface to measure the properties of small objects close to the interface.
It has been used to observe virus capsid self-assembly,ref-GarmannPNAS116  the
bending of 200-nm microtubules,ref-AmosJCSS14  the dynamics of coexisting
lipid domainsref-deWitPNAS112  and differences in the lateral diffusion of
labelled lipid molecules in supported lipid bilayers depending on the label
(fluorophores and metallic nanoparticles).ref-ReinaJPD51  Differential iSCAT
has been used to measure protein secretions by single cells.ref-
GemeinhardtJVE141

Ellipsometry uses reflectometry at inclined incidence and exploits the
differences in the Fresnel coefficients for the polarisation components
parallel and perpendicular to the plane of incidence. By measuring the
intensity and polarisation of the light reflected by a thin layer, the complex
reflection coefficients for the two polarisations are determined, and thus the
thickness and/or refractive index of the layer can be determined. Notably,
this technique uses a defined angle of incidence, requiring layers which are
laterally homogeneous over a size much larger than the light wavelength
employed, typically at least 100 µm. The technique has been used to measure
the thicknesses of uniform inorganicref-HenckJVSTA10  and organicref-
StrombergJRNBSAPC67A ; ref-HoeoekAC73  layers and long-range non-uniform
inorganic layersref-LiuAO33  with nanometre resolution, as well as the
refractive index of metallic substrates on which nanometric layers of water
are adsorbed.ref-McCrackinJRNBSAPC67A  Biological applications of ellipsometry
include the measurement of the thickness of macaque retinal nerve layers,ref-
WeinrebAO108  which was measured to be between about 20 µm and about 215 µm.
Spectroscopic ellipsometry, combining measurements at multiple wavelengths,
has been employed to determine the thickness and (by studying absorption
bands) the chemical composition of silane films several hundred nanometres
thick,ref-FranquetTSF441  as well as to measure the thickness and electric
permittivity of nanometric tellurium layers and decananometric polypyrrole
layers.ref-ArwinTSF113

Despite the high sensitivity offered by these techniques, full advantage of
the complex reflected field is typically not taken. Most of the aforementioned
measurements are only of the thickness or the refractive index (or some other
parameter, such as moisture) of the sample. Ellipsometry ref-ArwinTSF113  can
determine both refractive index and thickness, but it requires laterally
homogeneous layers over a large scale and does not have microscopic imaging
capabilities.

We present here a heterodyne interferometric reflectometry technique (HiRef)
which can measure the thickness and refractive index of a sample
simultaneously and with microscopic optical-diffraction-limited lateral
spatial resolution. We demonstrate accurate measurements on samples tens of
nanometres thick, and we discuss the possibility of improving this to a few
nanometres in the future, as well as potential applications.

##  II Heterodyne interferometric reflectometry (HiRef)

We develop here a theoretical description of HiRef, which will be used for
quantitative analysis of the thickness and refractive index of thin samples.

###  II.1 Reflection by a layer

Suppose a homogeneous, isotropic layer of a material with thickness $d$ and
refractive index $n_{1}$ is sandwiched between two homogeneous semi-infinite
media with refractive indices $n_{0}>n_{1}$ and $n_{2}<n_{1}$. Neglecting
absorption, a plane light wave incident on the layer (for example, from the
$n_{0}$ side, as shown in figure 1) can be either reflected or transmitted,
with the relative amplitude of the reflected and transmitted waves given by
the Fresnel coefficients:ref-HechtO

$\displaystyle
r_{jl}^{\,\scalebox{0.6}{\raisebox{2.41112pt}{\cancel{\vphantom{.}\phantom{\,}}\cancel{\vphantom{.}\phantom{\,}}}}}$
$\displaystyle=$
$\displaystyle\frac{n_{j}\text{cos}\\!\left({\theta_{j}}\right)-n_{l}\text{cos}\\!\left({\theta_{l}}\right)}{n_{j}\text{cos}\\!\left({\theta_{j}}\right)+n_{l}\text{cos}\\!\left({\theta_{l}}\right)},$
(1) $\displaystyle
t_{jl}^{\,\scalebox{0.6}{\raisebox{2.41112pt}{\cancel{\vphantom{.}\phantom{\,}}\cancel{\vphantom{.}\phantom{\,}}}}}$
$\displaystyle=$
$\displaystyle\frac{2n_{j}\text{cos}\\!\left({\theta_{j}}\right)}{n_{j}\text{cos}\\!\left({\theta_{j}}\right)+n_{l}\text{cos}\\!\left({\theta_{l}}\right)},$
(2) $\displaystyle r_{jl}^{\perp}$ $\displaystyle=$
$\displaystyle\frac{n_{l}\text{cos}\\!\left({\theta_{j}}\right)-n_{j}\text{cos}\\!\left({\theta_{l}}\right)}{n_{l}\text{cos}\\!\left({\theta_{j}}\right)+n_{j}\text{cos}\\!\left({\theta_{l}}\right)},$
(3) $\displaystyle t_{jl}^{\perp}$ $\displaystyle=$
$\displaystyle\frac{2n_{j}\text{cos}\\!\left({\theta_{j}}\right)}{n_{l}\text{cos}\\!\left({\theta_{j}}\right)+n_{j}\text{cos}\\!\left({\theta_{l}}\right)},$
(4)

where the subscript $jl$ indicates that the light is incident on the planar
boundary between media with refractive indices $n_{j}$ and $n_{l}$ from the
$n_{j}$ side, $\theta_{j}$ is the angle of incidence, $\theta_{l}$ is the
angle of transmission (given by Snell’s law,
$n_{j}\text{sin}\\!\left({\theta_{j}}\right)=n_{l}\text{sin}\\!\left({\theta_{l}}\right)$),
and the superscripts  ~~. ~~~~.~~ and $\perp$ denote the polarisation
components parallel and perpendicular, respectively, to the plane of
incidence.

Figure 1: Sketch of reflection by a homogeneous layer of refractive index
$n_{1}$ between two homogeneous semi-infinite media of refractive index
$n_{0}$ and $n_{2}$.

If we place a detector on the $n_{0}$ side, we can collect the light reflected
by the layer. The amplitude of the reflected field is given by interference
between all the reflection orders. Light incident from the $n_{0}$ side is
reflected at the first boundary with a relative amplitude $s_{0}=r_{01}$,
where the superscript is omitted for clarity in the following calculations.
Light transmitted into the second semi-infinite medium is not measured. Light
transmitted back into the first semi-infinite medium after having been
reflected a total of $j$ times at the $n_{1}$-$n_{2}$ interface and $j-1$
times at the $n_{0}$-$n_{1}$ interface has the relative complex amplitude

$\displaystyle s_{1}$ $\displaystyle=$ $\displaystyle
t_{01}t_{10}\sum_{j=1}^{\infty}r_{10}\,\\!^{j-1}r_{12}\,\\!^{j}e^{ij\Delta},$
(5)

where $\Delta=2kdn_{1}/\text{cos}\\!\left({\theta_{1}}\right)$ is the phase
due to propagation through the layer and back, $k=2\pi/\lambda$ is the wave
number of the light of wavelength $\lambda$, and $\theta_{1}$ is the
propagation angle in the layer. The relative complex amplitude of the detected
wave is then

$\displaystyle s$ $\displaystyle=$ $\displaystyle
s_{0}+s_{1}=r_{01}+t_{01}r_{12}t_{10}e^{i\Delta}\sum_{j=0}^{\infty}\left(r_{10}r_{12}e^{i\Delta}\right)^{j}$
(6) $\displaystyle=$ $\displaystyle
r_{01}+\frac{t_{01}r_{12}t_{10}e^{i\Delta}}{1-r_{10}r_{12}e^{i\Delta}}\,,$

where $|r_{10}r_{12}e^{i\Delta}|<1$ for $\theta_{0}$ outside the total
internal reflection (TIR) regime. Note that if there is no layer (i.e. $d=0$)
then the right side of equation 6 reduces to $r_{02}$, the reflection
coefficient at the boundary between the two semi-infinite media.

Figure 2: $|s/r_{02}|$ (top) and $\text{arg}\\!\left({s}\right)$ (bottom) as
functions of $d$ and $n_{1}$ for normal incidence with $n_{2}=1.333$,
$n_{0}=1.518$ and $\lambda=550$ nm.

For normal incidence ($\theta_{0}=0$), equation 6 can be written as

$\displaystyle s$ $\displaystyle=$
$\displaystyle\frac{(n_{0}+n_{1})(n_{1}-n_{2})e^{i\Delta}+(n_{0}-n_{1})(n_{1}+n_{2})}{(n_{0}-n_{1})(n_{1}-n_{2})e^{i\Delta}+(n_{0}+n_{1})(n_{1}+n_{2})}\,.$
(7)

Figure 2 shows the resulting amplitude and phase of $s$ for $n_{0}=1.518$,
$n_{2}=1.333$, $\lambda=550$ nm, layer thicknesses between $0$ and $600$ nm
and layer refractive indices between $n_{2}$ and $n_{0}$. The aforementioned
values of $n_{0}$ and $n_{2}$ have been chosen to match microscope-slide
glassref-OcheiMLS  and waterref-HechtO , respectively. It is evident that $s$
is periodic in $d$ with a period that is decreasing with $n_{1}$, as expected
from the term $e^{i\Delta}$ in equation 6; the thickness range was chosen to
show three such periods. It is also clear that $|s|\leqslant r_{02}$ for any
$d$ as long as $n_{2}\leqslant n_{1}\leqslant n_{0}$.

###  II.2 Interferometric detection of reflection focussed by a microscope
objective

HiRef makes use of the heterodyne-detected interference between two beams, one
of which (called the probe beam) interacts with the sample and one of which
(called the reference beam) acts as an external reference, to obtain the
reflected amplitude and phase. This allows the determination of two properties
of the sample, such as thickness and refractive index, simultaneously.

We consider here an aplanatic objective of numerical aperture NA. The light
field incident on the sample has a wide angular distribution over the polar
angle $\theta$, and the maximum angle of incidence is given by
$\theta_{\text{m}}=\text{arcsin}\\!\left({\text{NA}/n_{0}}\right)$. The probe
beam entering the objective back-focal plane is chosen to be circularly
polarised, so the probe field in cartesian polarisation basis is given by

$\displaystyle\mathbf{E}_{\text{p}}(\theta)$ $\displaystyle=$
$\displaystyle\frac{E(\theta)}{\sqrt{2}}{\begin{pmatrix}1\\\
i\end{pmatrix}}e^{-i\omega t}$ (8)

with

$\displaystyle E(\theta)$ $\displaystyle=$ $\displaystyle
E_{0}\sqrt{\text{cos}\\!\left({\theta}\right)}\,e^{-\left(\zeta\text{sin}\\!\left({\theta}\right)/\text{sin}\\!\left({\theta_{\text{m}}}\right)\vphantom{b^{2}}\right)^{2}},$
(9)

where we assume a gaussian beam profile in the back-focal plane of the
objective with inverse objective fill factor $\zeta$.ref-NovotnyPONO  The
fields parallel and perpendicular to the plane of incidence, which are not
mixed by the reflection, are then given by

$\displaystyle
E_{\text{p}}^{\,\scalebox{0.6}{\raisebox{2.41112pt}{\cancel{\vphantom{.}\phantom{\,}}\cancel{\vphantom{.}\phantom{\,}}}}}(\theta,\varphi)$
$\displaystyle=$ $\displaystyle\frac{1}{\sqrt{2}}E(\theta)e^{i(\varphi-\omega
t)},$ (10) $\displaystyle E_{\text{p}}^{\perp}(\theta,\varphi)$
$\displaystyle=$ $\displaystyle\frac{i}{\sqrt{2}}E(\theta)e^{i(\varphi-\omega
t)},$ (11)

where $\varphi$ is the azimuthal angle. The probe beam is reflected by the
sample and recollimated by the objective, and it then interferes with a
frequency-shifted reference beam, which does not interact with the sample, in
an image plane of the objective back-focal plane. The reference beam is also
circularly polarised and has the same beam profile as the probe beam. The
reflection inverts the circular polarisation relative to the propagation
direction, so we use a reference beam which has opposite circular polarisation
to the beam entering the objective. We can therefore assume that its field
distribution in the image plane of the back-focal plane of the objective is
equal to that of the probe beam, so, up to a constant phase shift,
$\mathbf{E}_{\text{r}}(\theta)=e^{-i\Omega t}\mathbf{E}_{\text{p}}(\theta)$,
where $\Omega$ is the heterodyne frequency shift (see section III.1). We thus
have, for each incidence direction $(\theta,\varphi)$, the interference term

$\displaystyle I(\theta,\varphi)$ $\displaystyle=$ $\displaystyle
2\text{Re}\\!\left({E_{\text{r}}^{{\,\scalebox{0.6}{\raisebox{2.41112pt}{\cancel{\vphantom{.}\phantom{\,}}\cancel{\vphantom{.}\phantom{\,}}}}}\ast}s^{\,\scalebox{0.6}{\raisebox{2.41112pt}{\cancel{\vphantom{.}\phantom{\,}}\cancel{\vphantom{.}\phantom{\,}}}}}E_{\text{p}}^{\,\scalebox{0.6}{\raisebox{2.41112pt}{\cancel{\vphantom{.}\phantom{\,}}\cancel{\vphantom{.}\phantom{\,}}}}}+E_{\text{r}}^{\perp\ast}s^{\perp}E_{\text{p}}^{\perp}}\right)$
(12) $\displaystyle=$
$\displaystyle-|E(\theta)|^{2}\text{Re}\\!\left({\\!\left(s^{\,\scalebox{0.6}{\raisebox{2.41112pt}{\cancel{\vphantom{.}\phantom{\,}}\cancel{\vphantom{.}\phantom{\,}}}}}(\theta)+s^{\perp}(\theta)\right)e^{i\Omega
t}\vphantom{\left(e^{i\Omega t}\right)^{\frac{b}{b}}}\\!}\right)\\!,$

where the star indicates the complex conjugate. The interference does not
depend on $\varphi$ because we use co-circular polarisations for the probe and
reference beams. A dual-channel lock-in detection recovers both the in-phase
and in-quadrature components at the frequency $\Omega$, so we can detect the
complex signal

$\displaystyle S$ $\displaystyle=$ $\displaystyle
S_{0}\int_{0}^{\theta_{\text{m}}}{|E(\theta)|^{2}\left(s^{\,\scalebox{0.6}{\raisebox{2.41112pt}{\cancel{\vphantom{.}\phantom{\,}}\cancel{\vphantom{.}\phantom{\,}}}}}(\theta)+s^{\perp}(\theta)\right)\text{sin}\\!\left({\theta}\right)}\,\text{d}{\theta},$
(13)

where $S_{0}$ is a constant which depends on the laser powers used, the
detection efficiency and other setup-specific constants. The parameters used
here are such that TIR does not occur, as the NA of the objective is below the
lowest refractive index of the layer structure considered. Therefore, the
expressions derived in section II.1 for $s$ are valid. Expressions for $s$
including the case of TIR can be derived in a similar fashion.

Figure 3: $|S/S_{02}|$ (top) and $\text{arg}\\!\left({s}\right)$ (bottom) as
functions of $d$ and $n_{1}$ for an aplanatic objective with numerical
aperture 1.27 and fill factor 1, materials with refractive indices
$n_{2}=1.333$ and $n_{0}=1.518$, and circularly polarised light with
wavelength $\lambda=550$ nm.

Figure 3 shows the amplitude and phase of $S/S_{02}$, where $S_{02}$ is the
signal detected in the absence of the thin layer, for which
$s^{\,\scalebox{0.6}{\raisebox{2.41112pt}{\cancel{\vphantom{.}\phantom{\,}}\cancel{\vphantom{.}\phantom{\,}}}}}=r_{02}^{\,\scalebox{0.6}{\raisebox{2.41112pt}{\cancel{\vphantom{.}\phantom{\,}}\cancel{\vphantom{.}\phantom{\,}}}}}$
and $s^{\perp}=r_{02}^{\perp}$. Note that $S_{02}$ is real. The wavelength and
refractive indices used are the same as in figure 2;
$\theta_{\text{m}}=56.8^{\circ}$, as determined by $\text{NA}=1.27$; and
$\zeta=1$. We find that the qualitative pattern seen with normal incidence
remains, but it is distorted, squeezed upwards and towards the left, with
increasing distortion as the thickness increases. The signal still satisfies
$|S|\leqslant S_{02}$.

###  II.3 Retrieving thickness and refractive index

Figure 4: $d$ (top) and $n_{1}$ (bottom) as functions of $s/r_{02}$ for normal
incidence with $n_{2}=1.333$, $n_{0}=1.518$ and $\lambda=550$ nm.

For normal incidence, equation 6 can be solved analytically for $d$, yielding

$\displaystyle d$ $\displaystyle=$
$\displaystyle\frac{i}{2kn_{1}}\,\text{log}\\!\left({\frac{n_{2}-n_{1}}{n_{2}+n_{1}}\,\frac{n_{0}+n_{1}-(n_{0}-n_{1})s}{n_{0}-n_{1}-(n_{0}+n_{1})s}}\right).$

Now, since $d$ is a real quantity, the argument of the logarithm in equation
LABEL:eq-d has a magnitude of 1. This condition can be used to obtain

$\displaystyle n_{1}$ $\displaystyle=$
$\displaystyle\sqrt{n_{0}n_{2}\,\frac{n_{0}-n_{2}-2n_{0}\text{Re}\\!\left({s}\right)+(n_{0}+n_{2})|s|^{2}}{n_{0}-n_{2}-2n_{2}\text{Re}\\!\left({s}\right)-(n_{0}+n_{2})|s|^{2}}}\,.$

Figure 4 shows the resulting thickness and refractive index of the layer as a
function of $s/r_{02}$, where we have chosen the solution (the branch of the
complex logarithm) having the smallest thickness range.

We can see that around the circumference, changing the phase of $s/r_{02}$
while keeping its magnitude close to 1, the thickness of the layer changes
while its refractive index remains close to $n_{0}$. Going instead radially,
changing the real part of $s/r_{02}$ starting from 1, the refractive index
increases from $n_{2}$ for $s=1$ to $\sqrt{n_{2}n_{0}}$ for $s=0$ to $n_{0}$
for $s=-1$.

Figure 5: $d$ (top) and $n_{1}$ (bottom) as functions of $S/S_{02}$ for an
aplanatic objective with numerical aperture 1.27 and fill factor 1, materials
with refractive indices $n_{2}=1.333$ and $n_{0}=1.518$, and circularly
polarised light with wavelength $\lambda=550$ nm.

In the case of reflection of a focussed beam, given by equation 13, and shown
in figure 3, the refractive index and thickness of the layer can be determined
numerically for the angular distribution we have considered. Figure 5 shows
the resulting $d$ and $n_{1}$ as functions of $S/S_{02}$.

We find that the qualitative behaviour is similar to the one at normal
incidence shown in figure 4, apart from a region with no solutions (white)
given by $|S/S_{02}|>R(\text{arg}\\!\left({S/S_{02}}\right))$. The radius $R$
of this region decreases with increasing angle
$\text{arg}\\!\left({S/S_{02}}\right)$, down to about 0.7 in the fourth
quadrant, before increasing back to 1, joining the $S/S_{02}=1$ point of zero
thickness. This behaviour reflects the angular averaging of the reflection
coefficient, which for a finite thicknesses $d$ reduces $|S/S_{02}|$ below its
maximum of 1.

Measuring $S/S_{02}$ therefore allows the determination of the refractive
index and the thickness with high spatial resolution given by the sub-micron
focus size created by the objective.

##  III Materials & methods

###  III.1 Experimental setup

Figure 6: Schematic diagram of the HiRef setup used. OPO: optical parametric
oscillator; BS: beam-splitter; $\lambda/4$: quarter-wave plate; $\lambda/2$:
half-wave plate.

The experimental setup is shown schematically in figure 6 and described below,
with more detail given elsewhere.ref-ZorinyantsPRX7  A titanium-sapphire laser
(Spectra-Physics Mai Tai) emits 100-fs pulses centred at a wavelength of 820
nm with 80-MHz repetition rate, which pump an optical parametric oscillator
(OPO, Inspire Radiantis). The OPO emits a beam of 150-fs pulses centred at 550
nm with horizontal (in the plane of the schematic) linear polarisation. This
beam enters an acousto-optic modulator (AOM, IntraAction ASM-802B67) driven at
82 MHz. The zeroth- and first-order deflected beams are used as reference and
probe beams, respectively, for HiRef. The probe beam is reflected by a 20R/80T
beam-splitter (Edmund Optics NT68-367) and transmitted through a quarter-wave
plate (Casix WPA400-550-4) and a half-wave plate (Casix WPA400-550-2), which
allow full control of its polarisation state. After the wave plates, the beam
is reflected by a dichroic mirror (Eksma Optics, HR415-550nm/HT630-1300nm),
transmitted through a 10-mm-thick SF2 glass block used to adjust the beam
centring in the back-focal plane of the objective by tip and tilt, and
focussed by a lens with a focal length of 100 mm (Edmund Optics T49-360) into
the image plane of an inverted microscope (Nikon Ti-U). The beam enters the
right port of the microscope, is reflected upwards by the port prism, is
collimated by a $1.5\times$ tube lens with a focal length of 300 mm, and is
focussed onto the sample by a $60\times$ 1.27-NA water-immersion objective
(Nikon MRD70650). The power at the sample is about 10 µW.

The beam reflected by the sample travels back through the optics and is
transmitted by the 20R/80T beam-splitter. The beam is then recombined with the
reference beam by a low-polarising 45R/45T beam-splitter (BS, Optosigma
039-0235). The reference beam’s optical path length is matched to the probe
beam’s path length by an optical delay line using a linear stage (Physik
Instrumente M-403.6DG) and a fused-silica corner cube (Eksma Optics
#340-1217M+3217), and its group velocity dispersion is matched to the probe
beam’s using 142 mm of SF2 glass (Schott). Note that the probe makes a double
pass through the microscope optics described above, accumulating the
corresponding amount of dispersion. A linear polariser at 45∘ creates a well-
defined polarisation state of the reference beam with fields of equal
amplitude and phase in the horizontal and vertical polarisations.

The two outputs of the BS are separated into horizontal and vertical
polarisations by a Wollaston prism and are focussed onto silicon diodes
(Hamamatsu S5973-02) to allow balanced detection of the horizontal and
vertical polarisation components separately. The reference beam’s power was
chosen to be around 0.5 mW per diode to ensure shot-noise-limited detection.
The differential diode current corresponding to each polarisation is amplified
with a transimpedance of 100 k and analysed by a dual-channel lock-in
amplifier (Zurich Instruments HF2) locked to the difference of 2 MHz between
the AOM upshift and the pulse repetition rate. The lock-in amplifier extracts
the interference between the probe and reference beams in amplitude and phase.

The dual-polarisation detection is used to adjust the polarisation at the
sample to be circular on axis as follows: A circular polarisation changes
helicity upon reflection on an in-plane-isotropic surface, resulting in a
cross-circularly-polarised reflected beam emerging from the sample, which
after returning through the wave plates is cross-polarised to the horizontal
input polarisation. Therefore, minimising the detected signal in the
horizontal polarisation using the wave plates ensures a circular polarisation
at the sample. Typically, a horizontal field amplitude below 1% of the
vertical field amplitude can be stably achieved.

The reflected probe beam $E_{\text{p}}$, with power $|E_{\text{p}}|^{2}$, and
the reference beam $E_{\text{r}}$, with power $|E_{\text{r}}|^{2}$, interfere
at the BS, yielding two beams with powers
$S_{1}=|E_{\text{p}}+E_{\text{r}}|^{2}/2$ and
$S_{2}=|E_{\text{p}}-E_{\text{r}}|^{2}/2$. The difference
$S_{1}-S_{2}=2\text{Re}\\!\left({E_{\text{p}}E_{\text{r}}\,\\!^{\ast}}\right)$
is measured by the balanced detector and analysed by the lock-in amplifier to
obtain the amplitude and phase of the combined field. Because the amplitude
and phase of $E_{\text{r}}$ are independent of the sample and ideally
constant, the amplitude and phase of the reflected probe beam can be
determined.

Due to the high NA used, the focal depth of the focussed probe beam is of the
order of 1 µm, much smaller than the thickness of the coverslip and the depth
of the well formed by the imaging spacer (see section III.2), so the glass and
oil layers enclosing the sample layer can be considered infinitely thick for
the purpose of interference of reflected beams. Furthermore, the short
coherence length of the 100-fs pules (around 30 µm) suppresses interference
with reflections occuring at other surfaces.

The sample is moved by an $xyz$ piezoelectric stage with nanometric position
accuracy (MadCityLabs NanoLP200) and scanned during image acquisition. Images
were acquired with a pixel size $\delta_{x}=108$ nm at 0.2 ms per pixel. Each
image was a square 80 µm on a side and took around 110 s to acquire.

Immediately before each field of view was imaged with HiRef, the same field of
view was imaged with quantitative differential interference contrast (qDIC),
which provides an accurate measurement of the thickness under the assumption
that the refractive index is known.ref-ReganL35

###  III.2 Sample preparation

Figure 7: qDIC phase (left; $m=-0.9212$, $M=0.5581$), HiRef amplitude (centre;
$m=9.8$ mV, $M=76.5$ mV) and HiRef phase (right; $m=-\pi$, $M=\pi$) images of
one of the studied fields of view of the PVA layer. In the left and centre
images, brighter regions are empty and darker regions are the PVA layer.
Considerable phase drift is visible in the right image. The red lines show
examples of line trace positions (see section III.3) taken along the shear for
qDIC and along the fast scan axis for HiRef. Note that the trace is centred on
the same point at a step in all three images.

Three different samples were used.

One sample consisted of a flat layer of polyvinylacetate (PVA) several tens of
nanometres thick on a #1 glass coverslip (figure 7). The coverslip was washed
with acetone and etched with a 3:1 solution of sulphuric acid to hydrogen
peroxide at 95 ∘C to remove contaminants. 0.26 g of PVA and 4.33 g of water
were mixed to form a 6% mass/mass PVA solution. The solution was spin-coated
on the coverslip at 3,000 rpm for 30 s with 6 s of constant acceleration and
deceleration before and after the 30-s constant-speed period. Cuts were made
with a sharp razor at different angles in the spin-coated PVA layer in order
to create gaps without material. A 120-µm-thick square imaging spacer (Grace
BioLabs, OR, USA) with a circular hole 13 mm in diameter was adhered to the
coverslip to form a shallow well. The well was filled with water-immersion oil
($n_{2}=1.3339$, Zeiss Immersol W 2010) and covered with a glass microscope
slide. Because the immersion oil reduced the adhesiveness of the imaging
spacer, nail varnish was used to create a tight seal around the chamber.

Figure 8: qDIC phase (a, d), HiRef amplitude (b, e) and HiRef phase (c, f)
images of two different fields of view of the ternary-SLB sample. The
greyscale values are as follows: a: $m=-0.0456$, $M=-0.0081$; b: $m=0.4371$ V,
$M=0.4769$ V; c: $m=-0.1235$, $M=0.1232$; d: $m=-0.0481$, $M=0.0106$; e:
$m=0.2497$ V, $M=0.2678$ V; f: $m=-0.1203$, $M=0.1107$. In all images,
brighter regions are empty and darker regions are the lipid bilayer. The DOPC
domains appear slightly brighter than the sphingomyelin domains in the left
and central images; domain boundaries where the change in brightness is
visible are indicated by red arrows.

Each of the other two samples consisted of a supported lipid bilayer (SLB) on
a #1 glass coverslip (figure 8). The coverslip was washed and etched as
described above. Either pure dipentadecanoylphosphatidylcholine (DC15PC) or a
ternary lipid solution consisting of dioleoylphosphatidylcholine (DOPC),
chicken-egg sphingomyelin and cholesterol in an 11:5:4 molar ratio
(DOPC:sphingomyelin:cholesterol) was diluted in isopropanol to a lipid
concentration of 1 mg/ml. 150 µl of this solution were deposited on the
coverslip and spin-coated as described above; the spin-coating parameters were
the same as for the PVA sample. In order to avoid rapid absorption of moisture
in the later steps of sample preparation, which would have destroyed the
structure of the bilayer, the coverslip was placed in a nitrogen-filled
centrifuge tube with a small piece of moist tissue paper and incubated at 37
∘C for 1 hr. Finally, an imaging spacer with the aformentioned dimensions was
adhered to the coverslip to form a shallow well. The well was filled with
degassed phosphate-buffered saline (PBS) and covered with a glass microscope
slide. Prior to imaging, the ternary SLB sample was stored at 4 ∘C for a few
hours to allow liquid-ordered (LO) and liquid-disordered (LD) lipid domains to
form.

###  III.3 Data analysis

The HiRef phase images of the SLB samples were filtered to remove low-
frequency noise, which obscured the structural features owing to the low
constrast from a single lipid bilayer. This filtering consisted of multiplying
the Fourier spectrum of the phase of the image data by $1-f(\nu)$, where $f$
is a sum of gaussian functions, each centred at a frequency $\nu$ and having a
frequency width of 1 Hz, to remove frequencies below 5 Hz (phase drift), a
collection of frequency peaks near 10 Hz (likely electronic noise from nearby
equipment) and 15 harmonics of the fast-axis scan frequency, given by
$1/(2N_{x}t_{\text{e}})$, where $t_{\text{e}}=200$ µs is the pixel dwell time
and $N_{x}$ is the number of pixels in one row of the image.

Because the position of the sample stage at any given time during image
acquisition is not exactly the commanded position, the position $x^{\prime}$,
$y^{\prime}$ measured by the stage sensor is recorded at each pixel together
with the signal $S^{\prime}$ at that pixel, forming a time trace of the
measurement with the time points $t^{\prime}$. This time trace is converted
into a regular grid of pixels in a post-processing procedure called
regularisation. For every pixel position $(x,y)$ on the grid, the signal $S$
is calculated as

$\displaystyle S(x,y)$ $\displaystyle=$
$\displaystyle\sum_{t^{\prime}}S^{\prime}(t^{\prime})\,W(t^{\prime})\left/\sum_{t^{\prime}}W(t^{\prime})\right.\,,$
(16)

where

$\displaystyle W(t^{\prime})$ $\displaystyle=$
$\displaystyle\exp\left(-\left(\frac{x-x^{\prime}}{\Delta_{x}}\right)^{2}-\left(\frac{y-y^{\prime}}{\Delta_{y}}\right)^{2}\right)$
(17)

and the sum is taken over all the time points $t^{\prime}$ for which
$|x-x^{\prime}|<2\Delta_{x}$ and $|y-y^{\prime}|<2\Delta_{y}$ for
computational efficiency. We used $\Delta_{x}=\Delta_{y}=\delta_{x}$.
Regularisation improves the quality of the images in terms of accurate
representation of the contents of the sample.

After regularisation, line traces with a length of 121 pixels (about $13$ µm)
for the PVA sample or 61 pixels (about $7.5$ µm) for the SLB samples were
taken along the fast scan axis, the $x$ axis. The positions of the traces were
chosen so they were centred along the edge of the PVA or lipid (depending on
the sample) and thus roughly half of the pixels on any given trace were on PVA
or lipid and the rest were on an empty region; the line traces thus had a step
in the middle. The step function

$\displaystyle\Theta(x)$ $\displaystyle=$
$\displaystyle\frac{a_{1}}{2}\,\text{tanh}\\!\left({\frac{x-a_{2}}{a_{3}}}\right)+\frac{a_{4}}{2}\,\text{sech}\\!\left({\frac{x-a_{5}}{a_{3}}}\right)$
(18) $\displaystyle+\ a_{6}(x-a_{2})+a_{7}\vphantom{\frac{b}{b}}$

was simultaneously fitted to the amplitude and phase traces, sharing the
spatial parameters ($a_{2}$, $a_{3}$ and $a_{5}$), since features are expected
to be at the same locations in the amplitude and phase data. The hyperbolic
tangent produces the step, while the hyperbolic secant describes
irregularities in the edges, caused for example by the nature of the cuts with
the razor (PVA sample) or folds in the lipid (SLB samples). The linear
component fits slow drift due to sample tilt and to thermal drifts in the beam
paths and in the axial ($z$) position of the sample stage.

02468$-0.1$$-0.05$$0$$0.05$$0.1$$0.15$$x$ (µm)$\phi$ (rad)

024681012$0.035$$0.04$$0.045$$0.05$$0.055$$0.06$$0.065$$0.07$$x$ (µm)$|S|$ (V)

024681012$\pi/2$$\pi$$3\pi/2$$2\pi$$5\pi/2$$x$
(µm)$\text{arg}\\!\left({S}\right)$ (rad)

Figure 9: Line traces marked in figure 7: qDIC phase (top) and HiRef (centre
and bottom). Each of the traces is shown with its step function fit $\Theta$.
The step function in the qDIC data does not include the hyperbolic secant.
Note that the spatial parameters are the same in the two HiRef fits.

Let us assume for the following discussion that with increasing $x$ the trace
moves from the glass-water interface onto the investigated layer. For the
phase traces, the constant term $a_{7}$ represents a phase offset, and the
fitted step height $a_{1}$ is given by

$\displaystyle\text{arg}\\!\left({\frac{S}{S_{02}}}\right)$ $\displaystyle=$
$\displaystyle a_{1}$ (19)

(for comparison with calculations, see also figure 5). For the amplitude
traces, we have

$\displaystyle\left|\frac{S}{S_{02}}\right|$ $\displaystyle=$
$\displaystyle\frac{a_{7}-a_{1}/2}{a_{7}+a_{1}/2}\,,$ (20)

fully determining $S/S_{02}$. Figure 9 shows an example of a line trace from
the PVA sample.

The thickness and refractive index were computed numerically from the measured
value of $S/S_{02}$ at each line trace. For the computation, the region of
$(d,n_{1})$ space given by $0\leqslant d\leqslant 120$ nm and $n_{2}\leqslant
n_{1}\leqslant n_{0}$ was partitioned finely and the difference between the
quantity $S/S_{02}$ calculated with equation 13 and the quantity calculated
from the line trace fit was minimised; the values of $d$ and $n_{1}$ assigned
to the layer at the line trace were those which minimised this difference.

##  IV Results & discussion

$0.5$$0.6$$0.7$$0.8$$0.9$$-\pi$$-\frac{3\pi}{4}$$-\frac{\pi}{2}$$|S/S_{02}|$$\text{arg}\\!\left({S}\right)$$60$$65$$70$$75$$80$$85$$90$$95$$100$1.481.491.501.511.52$d$
(nm)$n_{1}$

Figure 10: Thickness and refractive index of a PVA film measured
simultaneously by HiRef (coloured points). Different colours indicate
different fields of view. Uncertainties are $\Delta_{d}=0.5$ nm and
$\Delta_{n_{1}}=0.0005$. The empty square indicates the result of 472 qDIC
measurements assuming a refractive index of 1.502. Inset: Relative amplitude
and phase steps measured in PVA. Fit errors (sum of squares) are typically
below $10^{-3}$ for $|S/S_{02}|$ and below 0.6 rad2 for
$\text{arg}\\!\left({S}\right)$.

Our HiRef results (12 traces) yield a thickness of $(77.4\pm 4.5)$ nm and a
refractive index of $1.509\pm 0.005$. qDIC measurements (472 traces) yielded a
thickness of $(84.7\pm 6.3)$ nm assuming a refractive index of $1.502\pm
0.017$; this refractive index is the average of a number of literature
values.ref-SchnepfN9 ; ref-BodurovNN16 ; ref-MahendiaJAP113 ; ref-KumarJMO52 ;
ref-Kumar-O119  The uncertainties are the standard deviation in all cases.

Figure 10 shows the results of the thickness and refractive index
measurements. Different colours indicate measurements taken in different
fields of view; the empty square indicates the qDIC average for reference. The
uncertainty in each individual HiRef measurement, calculated as half of the
partition size used for the numerical retrieval of $d$ and $n_{1}$, are
$\Delta_{d}=0.5$ nm and $\Delta_{n_{1}}=0.0005$. The standard error of the
qDIC measurements is $\Delta_{d}=0.3$ nm. The inset shows the relative
amplitude and phase steps corresponding to the measurements shown in the main
graph.

$40$$50$$60$$70$$80$$90$$100$$110$$40$$50$$60$$70$$80$$90$$100$$110$$\stackrel{{\scriptstyle\displaystyle
d\text{ (nm),}}}{{\text{HiRef}}}$$\stackrel{{\scriptstyle\displaystyle d\text{
(nm),}}}{{\text{qDIC}}}$ Figure 11: Comparison of HiRef (horizontal axis) and
qDIC (vertical axis) results. Each point corresponds to a pair of
measurements, one made with HiRef and one with qDIC, at the same position of a
sample.

The HiRef measurements are consistent within error with our qDIC measurements.
Figure 11 shows a comparison of the layer thickness calculated with the two
techniques; each point corresponds to one pair of measurements made taking
HiRef and qDIC line traces at the same location. The Pearson correlation
coefficient is low, only 0.47, indicating that the fluctuations in both
techniques arise from different sources. The standard deviation of the HiRef
results is only 5.9% in thickness and 0.3% in refractive index even with
relatively few (12) individual measurements. To estimate the precision of the
HiRef measurements, we analysed three traces close to each other, probing
virtually the same structure. We found a maximum variation of 3 nm in
thickness and 0.007 in refractive index, which is about two thirds (thickness)
and half (refractive index) of the uncertainties mentioned above, across the
field of view.

The HiRef analysis presented can be used with thin layers of refractive index
$n_{1}$ satsifying $n_{2}\leqslant n_{1}\leqslant n_{0}$ if TIR is avoided at
any of the interfaces and both the $n_{0}$ material and the layer in question
are transparent. We note, however, that for cases where any of these
conditions is not met, equation 6 can be adapted, taking a different
refractive-index range, extinction and/or TIR into account.

Our measurements on the SLB samples explore the limit of the present
experimental setup for measuring the thickness and refractive index of a
nanometric layer. The HiRef measurements of the thicknesses and refractive
indices are $d=(12.7\pm 2.9)$ nm and $n_{1}=1.418\pm 0.032$ for DOPC (52
traces) and $d=(9.0\pm 3.2)$ nm and $n_{1}=1.427\pm 0.038$ for DC15PC (48
traces). While the refractive indices are within error of the literature
values of 1.445 for DOPCref-DevanathanFEBSJ273  and 1.440 for DC15PC,ref-
HowlandBJ92  the thicknesses are two to three times larger than those of
single bilayers of the aforementioned lipids (4.1 nm and 5.3 nm,
respectively).ref-ReganL35  However, we show below that the analysis using
simulated data with the experimental noise leads to a large systematic error.

Despite the aforementioned inaccuracies in the thickness measurements, it is
possible to see sub-nanometre thickness differences as brightness changes in
the HiRef images of the ternary-SLB sample (figure 8, red arrows).
Sphingomyelin enriched with between 60% and 100% as much cholesterol as there
is sphingomyelin forms bilayers 5.0 nm thick,ref-ReganL35  which makes the
thickness difference between the enriched sphingomyelin domains and the DOPC
domains 0.9 nm.

Figure 12: As figure 2, but for a range in $d$ reduced to 10 nm to show the
changes in $S/S_{02}$ relevant for single lipid bilayers.

While HiRef is applicable to very thin layers and sub-nanometre differences in
thickness are qualitatively visible in HiRef images of layers 4–5 nm thick
(such as cell membranes), accurate thickness and refractive index measurements
on such thin layers are hindered by the signal-to-noise ratio of the present
data. Figure 12 shows $|S/S_{02}|$ and $\text{arg}\\!\left({S}\right)$ for an
aplanatic objective with a numerical aperture of 1.27 and all parameters as in
figure 3, but with a reduced $d$ range suited for the bilayer thicknesses; it
can be seen that, for intermediate values of $n_{1}$ (around 1.45),
$|S/S_{02}|$ varies by about 2% over this range and
$\text{arg}\\!\left({S}\right)$ varies by about 0.2 rad. Notably, the present
measurements are not shot-noise-limited; the shot noise is about 53 µV for the
chose acquisition parameters. Given that signals are between 250 and 500 mV
for the lipid images studied, the shot noise is about 0.01% of the signal, or
0.05% of the $\text{arg}\\!\left({S}\right)$ range and 0.5% of the
$|S/S_{02}|$ range. For the nominal bilayer thickness of 5.0 nm and
$n_{1}=1.45$, this noise results in an uncertainty smaller than 0.5 nm in $d$
and smaller than 0.0005 in $n_{1}$. This is more than one order of magnitude
below the uncertainties determined from the measurements.

$0$$2$$4$$6$$8$$10$$12$$n_{2}$1.351.401.451.50$n_{0}$$d$
(nm)$n_{1}$$\sigma=5\times 10^{-4}$$\sigma=5\times 10^{-5}$$\sigma=0$

Figure 13: Retrieved thickness $d$ and refractive index $n_{1}$ from 50 line
traces with simulated data with different levels of noise. The empty square
indicates the thickness (4 nm) and refractive index (1.440) used in the
construction of the data.

To evaluate the performance of HiRef for the experimental setup if limited by
shot noise, we carried out the same data analysis procedure with simulated
data. This data consisted of an $N_{x}\times N_{y}$ array of complex values
with $N_{x}=N_{y}=40$ divided into two types of homogeneous regions. The value
at one region type was $r_{02}$ for the parameters ($n_{0}$, $n_{2}$ and
$\lambda$) used for figure 2 and normal incidence, and the value at the other
was the value of $s$ corresponding to $d=4$ nm and $n_{1}=1.440$ for the same
parameters and normal incidence. Phase noise was then added to the data by
generating a $1\times N_{x}N_{y}$ array of random numbers with a gaussian
distribution with standard deviation $\sigma$ and mean $\mu=0$ (see below).
This noise array was then low-pass-filtered, wrapped into an $N_{x}\times
N_{y}$ array and added to the phase of the simulated data. Versions of the
data with $\sigma=0$, $\sigma=5\times 10^{-5}$ (which is approximately the
shot noise) and $\sigma=5\times 10^{-4}$, but otherwise identical to each
other, were created and analysed; the version with $\sigma=5\times 10^{-4}$
was qualitatively similar to experimental data in terms of the relative
strengths of the phase noise and the phase data. The results of this analysis,
consisting of 50 line traces in each case, are shown in figure 13. The
$(d,n_{1})$ values found were $(4.1\pm 0.1,1.443\pm 0.002)$ with $\sigma=0$,
$(4.1\pm 0.1,1.442\pm 0.005)$ with $\sigma=5\times 10^{-5}$, and $(5.2\pm
3.482,1.435\pm 0.048)$ with $\sigma=5\times 10^{-4}$, where $d$ is in nm. We
note that both the error and the uncertainty increase with the phase noise,
but even a noise level similar to that seen in the experimental data is not
capable of causing an error of 200% or 300%, as seen in the experiment.
Therefore, the HiRef technique is capable of retrieving (within error) the
correct thickness and refractive index.

##  V Conclusions

We have presented a microscopy technique to measure thickness and refractive
index of a thin layer simultaneously and noninvasively. The technique detects
the interference between a beam which interacts with the layer and a beam
which does not, retrieves the amplitude and phase of the beam reflected by the
sample, and numerically computes the thickness and refractive index of the
layer from these values using exact expressions for the interference between
the beams.

We used a spin-coated PVA layer with a thickness of about 80 nm to test the
technique. The measured thickness and refractive index have a standard
deviation of only 5.9% and 0.3% (respectively) of the measured values and are
consistent with qDIC thickness measurements and PVA refractive index values
from the literature.

Other microscopy techniques are limited to measuring a single property of the
sample (e.g. qDIC) and/or are invasive (e.g. fluorescence microscopy,
nanoparticle labelling, atomic-force microscopy and electron microscopy).
Alternatively, ellipsometry can measure the thickness and the refractive index
of a sample at the same time but requires laterally homogeneous samples over
large areas and is not spatially resolved at a microscopic scale. Here we have
used an optical microscopy technique to measure a sample’s thickness and
refractive index at the same time in a noninvasive manner.

In the reported experimental data, the results are dominated by classical
fluctuations of the amplitude and phase of the detected signal, limiting the
technique’s ability to reliably measurable layer thicknesses below several
tens of nanometres. It is possible to image much smaller thicknesses and
changes in thickness and refractive index, such as phase transitions in lipid
bilayers, when reducing the classical noise below the shot-noise limit, which
for the presented measurements is below 0.5 nm in the thickness and 0.0005 in
the refractive index, more than an order of magnitude below the present
classical noise. Therefore, a suited referencing technique is required, which
could consist of a common-path referencing using a defocussed beam with a
different heterodyne frequency shift. This would allow us to address important
questions which are inaccessible with current techniques. For example, the
difference in thickness between lipid domains in different thermodynamic
phases is about 0.9 nm.ref-ReganL35  Such measurements, performed on live
neurons rather than artificial lipid bilayers, might then be able to probe
whether or not neural activity involves a phase transition,ref-
NahmadRohenPhDthesis  as proposed by Heimburg and Jackson in 2005.ref-
HeimburgPNAS102  Another potential application would be the observation of
lipid rafts in cell membranes.ref-MunroC115

The data presented in this work is available from the Cardiff University data
archive, http://doi.org/10.17035/xxx.

##  VI Acknowledgements

A.N.R. acknowledges support of his PhD studies by CONACYT (scholarship
recipient number 581516). The microscope setup used was developed within the
UK EPSRC Leadership fellowship award of P.B. (grant number EP/I005072/1 and
equipment grant number EP/M028313/1). The authors thank Iestyn Pope for
support in instrument development and data acquisition and George Zorinyants
and Joseph Williams for their contributions to the qDIC data analysis.

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