m-lines technique: prism coupling measurement and discussion of accuracy for homogeneous
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2000 J. Opt. A: Pure Appl. Opt. 2 188
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J. Opt. A: Pure Appl. Opt. 2(2000) 188–195. Printed in the UK PII: S1464-4258(00)04726-7
-lines technique: prism coupling
measurement and discussion of
accuracy for homogeneous
S Monneret†, P Huguet-Chantˆ
ome and F Flory
Laboratoire d’Optique des Surfaces et des Couches Minces, Ecole Nationale Sup´
Physique de Marseille, Domaine Universitaire de Saint-J´
ome, 13397 Marseille Cedex 20,
Received 2 June 1999, in ﬁnal form 4 February 2000
Abstract. A method is proposed to measure the thickness of the air layer between the prism
and the waveguide in a totally reﬂecting prism coupler. The coupling efﬁciency of a Gaussian
beam from the prism into the waveguide can be calculated when the air-layer thickness (ALT)
To perform measurements of the indices and thicknesses of planar waveguides using the
m-lines technique, it is necessary to have a good knowledge of the prism’s characteristics and
to accurately measure the angles. However, we show by means of an example that the small
distance between the prism and the guide (i.e. the ALT) should be taken into account in order
to achieve accurate measurements.
Keywords: Thin ﬁlms, m-lines, prism coupling, optical waveguides, refractive index
Thewellknownprismcoupler [1–5] can be consideredasone
of the best ways to couple large amounts of light in planar
optical waveguides. The totally reﬂecting prism coupler
(TRPC) technique, also referred to as the m-lines technique,
is commonly used to determine the optical properties of
thin ﬁlms [6, 7]. The refractive index and the thickness as
well as the anisotropy of dielectric planar waveguides can be
determined in this way [8–11].
The m-lines appear for the different incident directions
corresponding to the coupling of light in the waveguide. As
the coupling between the prism and the waveguide increases,
the m-lines are broadened and shifted . For m-lines
is neglected as soon as the thickness of the air layer between
the prism and the waveguide is greater than about half the
wavelength of the coupled beam [8,9].
To the best of our knowledge, the study of the prism’s
inﬂuence on the m-lines technique has never been treated
because of the difﬁculty in determining the thickness of the
coupling air layer between the prism and the guide (air layer
Thus this paper is devoted ﬁrst to the method we propose
to measure the ALT, then to the study of the prism’s inﬂuence
† Present address: ENSIC, DCPR (UMR 7630 CNRS-INPL), BP 451,
54001 Nancy Cedex 01, France.
on the measurements obtained with the m-lines technique.
After a short review of the principle of the TRPC, we
present the experimental method implemented to determine
the ALT. It is shown that this method gives the ALT with
a difference of 1 nm or less for two sets of independent
measurements performed at two different wavelengths.
Numerical results concerning the consequences of prism
coupling on measurements are given afterwards. A general
nand the thickness tare the parameters to be determined.
The absorption can be taken into account but we concern
ourselves mainly with weakly absorbing materials.
2. The totally reﬂecting prism coupler
We shall ﬁrst brieﬂy review the main properties of the
common TRPC. In this paper, we consider prisms that are
rectangular and isosceles. A schematic representation of the
device is given in ﬁgure 1.
The coupling of an incident laser beam by a prism into a
planar waveguide [1–5] is governed by the incident angle
θpof the beam on the prism base. Under total internal
reﬂection conditions on this base, strong coupling of light
into the waveguide can occur via resonant frustrated total
1464-4258/00/030188+08$30.00 © 2000 IOP Publishing Ltd
Discussion of the accuracy of the m-lines technique
Figure 1. Schematic representation of the TRPC, and CCD
recording of a m-line.
reﬂection, i.e. via evanescent waves in the air layer (ﬁgure 1).
Such coupling occurs only when resonant conditions inside
the waveguide are met. This leads to a ﬁnite number of
discrete incidences of the laser beam, for which the light can
be strongly coupled into the guide. We call these incidences
synchronism angles θsync (ﬁgure 1).
In the experiments, the resonant coupling of the laser
beam into the waveguide is observed through the appearance
of a dark line in the reﬂected beam. The dark line can
be associated with a bright line (ﬁgure 1, CCD recording).
According to , we call such lines m-lines.
Consequently, the method referred to as the m-lines
technique consists in measuring the synchronism angles
corresponding to the m-lines. The optical parameters nand
tare calculated from the measured θsync [6–9]. Under such
conditions, the propagation constants determined from the
θsync are assumed to be those of the free guided modes.
However, this hypothesis is not perfectly true due to prism
coupling. The coupling is, of course, directly dependent on
the ALT: we shall now consider the measurement of the ALT.
3. Measurement of the ALT
The determination of prism coupling efﬁciency has been the
subject of several studies. Chilwell  proposed a method
the prism’s base. This method is not easy to implement when
it comes to determining the absolute value of the ALT.
A second way to assess the efﬁciency of prism coupling
is to use m-lines. Midwinter  has ﬁrst shown that there
is a relationship between the intensity proﬁle of a m-line and
the coupling of the incident beam in the waveguide. Falco
et al  have then shown experimentally that the near-ﬁeld
intensity proﬁle of the reﬂected beam directly depends on the
coupling of the incident beam.
After these studies, this principle was developed to
study nonlinear changes in the refractive index of thin-ﬁlm
materials[15,16]. This methodoffersseveraladvantages: the
beam itself, coupled in the waveguide, is used; the ALT is
determined at the exact point where light is actually coupled;
such determination is possible in real time; we will see that
the precision of the method is quite high.
Figure 2. Modelling of the reﬂected beam. Coordinate frames.
Solid arrow: mean incidence of the Gaussian beam. Broken
arrow: one of the incidences of the Gaussian beam.
First we present the modelling of the reﬂected beam
needed for ALT determination, which has not been presented
in [15, 16]. We then describe the measurement method,
validate it experimentally, and ﬁnally give the precision of
3.1. Modelling of the reﬂected beam
Let us consider the prism coupler schematically drawn in
ﬁgure 2. All the media are assumed to be isotropic and
homogeneous. The waveguide and the substrate can be
dissipative, so that refractive indices are complex numbers.
In the coupling zone, interfaces are assumed to be plane and
parallel. A ﬁrst reference (O,X,Y,Z) is chosen in order
that the boundaries between the media may be parallel to the
(XO Y ) plane. The plane of incidence is the (XOZ) plane.
The system is assumed to be inﬁnite along the Ydirection. E
harmonic waves with an exp(−iωt) temporal dependence
which will be omitted in the presentation.
This modelling concerns only the particular case of a TE
or TM linearly polarized incident light beam.
Let our system be illuminated by a Gaussian laser beam
focused on the prism’s base and centred on the point O
(ﬁgure2). WedenoteOiandOrthe geometrical images of O
through the entrance and exit faces of the prism, respectively.
Two Cartesian coordinate frames (xi,y
are centred on Oiand Or(ﬁgure 2). These coordinate frames
are such that they are directly associated with the path of the
beam in the air.
Let us ﬁrst consider the beam propagating in the air
without any prism. In this case, the amplitude of the incident
electric ﬁeld in the plane (xiOiyi)is given by
where w0is the half width at 1/e2in intensity of the Gaussian
beam’s waist in the air and E0the maximum amplitude of Ei.
For zi=0 one can express E0exp(−x2
0)as a sum of
plane waves. As the plane of incidence is the (XOZ) plane,
S Monneret et al
and because the structure is invariant in Ywe obtain
Ei(σ ) exp(2jπσxi)dσ
is the transverse spatial frequency of each plane wave, θ
the angle between the incidence of this plane wave and the
mean incidence of the Gaussian beam, and ˆ
Ei(σ ) the one-
dimensional Fourier transform of Ei(xi,0,0). Its analytical
expression is given by:
Ei(σ ) =E0√πw0exp(−π2w2
Let us now consider the beam propagating in the real
system, i.e. with the prism. After propagating through the
material, the electric ﬁeld in the air is given by
Ei(σ )f (σ )
where the longitudinal spatial frequency of each plane wave
Thefunction f(σ)isdeﬁnedby f(σ)=t
1(σ )t2(σ )r (σ )
exp(j1φp(σ )). Values t1(σ ),t2(σ ) and r(σ) are the
transmission and reﬂection coefﬁcients  in amplitude on
the prism’s faces (ﬁgure 2) for a plane wave characterized
by the spatial frequency σ.r(σ) is obtained from a classical
plane wave matrix method , and is strongly dependent on
σnear a resonance. 1φpis the change of phase of a plane
wave of spatial frequency σas a result of its path inside the
prism. Its expression is given by
1φp(σ ) =k0Lp
where θiis the mean angle of incidence of the beam on
the prism and Lpthe mean total length of the beam’s path
inside the prism. Lpis easily determined by geometrical
considerations, and obviously depends on θiand on the size
of the prism.
Let us now assume that the incident Gaussian beam has
a small aperture. Hence we can neglect the dependence of
t1and t2on σand assume t1(σ )t2(σ ) =t1(0)t2(0). The
small aperture of the beam implies that σλ−1
carried into equations (3) and (6), leads to an expression of
the longitudinal spatial frequency µversus the transverse
spatial frequency σ:
2π µ(σ ) ≈k01−λ2
The electric ﬁeld distribution of the reﬂected beam is now
Figure 3. Typical calculated evolution of the transverse intensity
proﬁle Ir(xr,zr=1 m) of the reﬂected light beam. The given
proﬁles correspond to: (a) No coupling (inﬁnite ALT), (b) weak
coupling (ALT =250 nm) and (c) stronger coupling
(ALT =150 nm). Calculated for λ=514.5 nm with a beam waist
w0=15 µm and a SrTiO3prism, in the case of a waveguide of
refractive index 1.500 and of thickness 1046 nm. Substrate index
i(σ )f (σ ) exp(jk0zr)exp(−jπσ2zr). (10)
r)be the transverse intensity proﬁle of the
reﬂected light beam. This proﬁle is deﬁned as
For a given value zr0of zr,Ir(xr,z
0)is therefore obtained
directly from the one-dimensional inverse Fourier transform
transform algorithm is used to compute Ir. The sampling
must be chosen carefully because of the steep slope of the
function exp(−πλ0σ2zr)versus σwhen zrincreases.
In ﬁgure 3 we provide an example of the dependence
r)on the ALT. For an inﬁnite ALT, the reﬂected
beamis Gaussian (ﬁgure 3(a)). When the coupling increases,
the m-line appears and becomes more and more contrasted
(ﬁgure 3(b), ALT ta=250 nm and ﬁgure 3(c), ALT
ta=150 nm). Oscillations appearing in the reﬂected beam
can also be observed experimentally, especially for low-loss
waveguides. They can be regarded as interferences between
the part of the beam that is almost directly reﬂected on the
prism’s base and that which has passed through the guide.
3.2. Measurement method
Let us now study the characteristics of the reﬂected near-ﬁeld
intensity distribution INF, deﬁned as
INF corresponds to the transverse intensity distribution of
light. It can be directly recorded on a screen by imaging the
prism’s base around Owith the use of a lens.
Figure 4 gives typical experimental recordings of INF
obtained with a CCD camera in various coupling conditions.
The appearance of such reﬂected spots can be understood as
follows: the ﬁrst spot represents the part of the incident beam
that is directly reﬂected on the prism’s base; the second one
representsthe light leaking out afterhavingbeen coupled into
Discussion of the accuracy of the m-lines technique
Figure 4. Typical experimental recordings of reﬂected near-ﬁeld
images, obtained with a CCD camera. (a) no coupling, (b) weak
coupling, and (c) strong coupling.
the waveguide. Therefore only the ﬁrst spot is visible when
no coupling occurs (ﬁgure 4(a)), and for strong coupling
(ﬁgure 4(c)) the second spot is much brighter than the ﬁrst
one. Because the coupling efﬁciency cannot be more than
81% [3,13], the ﬁrst spot is always visible. We denote by I1
the ﬁrst relative maximum of the INF(xr,0)proﬁle and by
I2the second one, according to the propagation direction of
Values of I2depend on the coupling, on the waveguide
absorption, and on the beam waist. Dependence on the
beam’s waist is due to the ﬁltering effect of the function ˆ
on the spatial frequencies, which depends on the ALT .
Figure 5 represents typical changes of the near-ﬁeld
ratio (NFR) I1/I2with the ALT tafor a single-layer
waveguide, showing a one-to-one relationship between these
parameters for each conﬁguration (i.e. prism and waveguide
characteristics, beam wavelength, waist and polarization,
resonance order). Thus, this one-to-one relationship implies
that, for a given conﬁguration, the measurement of the NFR
I1/I2gives the ALT. Moreover, our calculations show that
the value of I1/I2corresponding to the optimal coupling
efﬁciency depends neither on the resonance order nor on the
3.3. Validation of the method
In order to validate the method described above, we have
determined the values of the ALT tasimultaneously by
performing two independent sets of measurements with two
different laser beams (ﬁgure 6).
The waveguide we have used is a single thin ﬁlm of
SiO2deposited by ion-assisted deposition on a fused silica
substrate. The two incident beam spots are superimposed on
thebaseof the SrTiO3prism. Theexperimentalconﬁguration
is deﬁned by
green beam: λ=514.5nm,
red beam: λ=632.8nm,
Thegreen and thered images ofthe corresponding near ﬁelds
Figure 5. Typical evolutions of the NFR versus the ALT and
inﬂuences of the beam waist w0and of the imaginary part κof the
waveguide refractive index. (a)κ=0; w0=15 µm.
(b)κ=7.8×10−5;w0=15 µm. (c)κ=7.8×10−5;
w0=30 µm. Calculated for λ=514.5 nm and a SrTiO3prism, in
the case of a waveguide of refractive index 1.500 and of thickness
1046 nm. Substrate index 1.4616. TE0resonance.
Figure 6. Experimental two-beam set-up used for validating the
determination of the prism coupling efﬁciency.
L2(ﬁgure 6) because with this waveguide the two resonances
are very close together. Values of the green and the red NFRs
are measured simultaneously with a CCD camera connected
to a digital image processing system. Curves such as those
given in ﬁgure 5 are used to obtain the value of the ALT as
measured with the green and the red beams. The extinction
coefﬁcientκof the waveguide, determined by a photothermal
deﬂection technique  is considered:
λ=514.5 nm: κ≈7.8×10−5,
λ=632.8 nm: κ≈3.4×10−5.
In order to verify our results, ﬁve thorough independent
measurements with various coupling efﬁciencies have been
usedtopresstheguideagainsttheprism. Theresultsare given
in ﬁgure 7. Values of the ALT obtained either from the red
or the green beam are the same, within approximately 1 nm.
As the ALT can be measured, the coupling efﬁciency
of the incident beam into the waveguide can be calculated as
wellas the electricﬁeld distribution in thewhole system .
S Monneret et al
Figure 7. Simultaneous determination of the ALT with two laser
beams. Five independent measurements with various coupling
efﬁciencies are reported. Waveguide: thin ﬁlm of SiO2deposited
by ion assisted deposition (deposition conditions: deposition
−1, ion energy ≈250 eV, current
density ≈200 µAcm
−2. Film properties: n=1.494 at
λ=632.8 nm, n=1.500 at λ=514.5 nm, κ=7.8×10−5at
λ=514.5 nm, κ=3.4×10−5at λ=632.8 nm, t=1046 nm).
Substrate: fused silica (ns=1.4570 at λ=632.8 nm,
ns=1.4616 at λ=514.5 nm). Prism: SrTiO3. Green beam:
λ=514.5 nm, w0=46.4±0.5µm, TE0resonance. Red beam:
λ=632.8 nm, w0=34.7±0.5µm, TM0resonance.
Because of the limited dynamics of the CCD camera, the
coupling efﬁciencies that can be measured with this method
are restricted to values greater than 30% of the maximum
coupling efﬁciency. This corresponds to NFRs typically
between 0.1 and 10.
Such a range is not convenient for determining the ALT
in the case of weak coupling as is usually considered for
typical m-lines measurements. A two-beam set-up (like that
presented in ﬁgure 6) can be used to remove this constraint.
A ﬁrst beam is weakly coupled for the measurement of
each m-line by searching a dark line as thin as can be visible
to the naked eye. A second beam, at a different wavelength,
is used at the same time to measure the NFR. By combining
the two wavelengths and the mode orders, both beams can be
coupled with the same ALT, each into its own mode, which
make weak coupling of the ﬁrst beam possible and permits
us to measure the NFR of the second one.
4. High-accuracy m-lines measurements
As mentioned previously, the m-lines can be broadened and
shifted when the coupling efﬁciency increases [1,2]. This
means that the synchronism angles θsync depend on the ALT.
The weak coupling hypothesis is commonly used to neglect
thisdependencebecause of the former difﬁcultyinmeasuring
the ALT. However this hypothesis limits the accuracy of the
waveguide parameter measurements . The aim of what
is presented below is to assess such a limitation and also to
compare it with the other sources of inaccuracy inherent in
the m-lines technique.
First we recall the differences in the conditions of
light propagation between the free waveguide (i.e. the weak
coupling hypothesis) and the waveguide coupled to a prism.
Finally, an example shows that for accurate measurements,
the effect of the very small distance between the prism and
the guide could lead to non-negligible errors in the recovery
of the waveguide parameters.
4.1. The weak coupling regime
4.1.1. Theory. Let us recall some general results concern-
ing guided optics, in order to clarify the main changes occur-
ring in the light propagation whether the waveguide is free
or coupled to a prism.
We ﬁrst consider the free waveguide. Guided modes
occur when the ﬁelds are evanescent in the air and the
substrate [18, 21]. Then, we need to ﬁnd the solutions
to Maxwell’s equations, which can satisfy the boundary
conditions at the waveguide–substrate and waveguide–air
interfaces. We know that there exists only a ﬁnite number of
discreteguidedmodesthat can propagate in a free waveguide.
Each guided mode is then characterized by its polarization
each mode can be associated with the propagation of one
single wave inside the structure, of propagation constant
βm;βmis invariant inside the waveguide and represents
the projected part of the wavevector on the guided wave
propagation axis . For more convenience, the effective
index αm=βm/k0can also be used, where k0is the modulus
of the wave vector of the light beam in vacuum.
In a prism–ﬁlm coupler, the resonant modes are not
the modes of the free guide. In this case, a continuum of
propagating waves exists in the guide; and the propagation
constant β depends only on the incidence of the input
beam on the prism, independently of any of the guide’s
parameters. Because of this fundamental change, for a
Gaussian incident beam, one can no longer associate the
beam propagating inside the waveguide with a single wave
but with a continuous set of waves . The aperture of such
a wavepacket strongly depends on the ALT .
Nevertheless, the different resonances of the waves
correspond to the modes of the waveguide coupled to the
prism . When increasing the ALT, these resonances
grow sharper and change notably. For an inﬁnite ALT, they
correspond, of course, to the free guided modes. Therefore,
each resonance is labelled with the same mode order mas
the corresponding free guided mode. We call βmr the mean
propagationconstantof the resonant beam propagating inside
the coupled waveguide, and αmr the corresponding effective
Such dependence of βmr on the ALT is directly
responsible for the intrinsic limit on the accuracy of the m-
lines technique. In order to suppress such a limit and because
of the difﬁculty in determining the ALT in the TRPC, it
is usually assumed that the measurements are made in a
weak coupling regime. Calculations have shown that the
ALT should be greater than half the wavelength in order
not to disturb the guided modes [8,9]. The weak coupling
approximation then allows us to consider the free guided
modes as corresponding to the m-lines. In this condition
it is easy to recover nand tfrom the measured propagation
Discussion of the accuracy of the m-lines technique
4.1.2. Accuracy of the m-lines measurements. Let us
deﬁne several parameters concerning the evaluation of the
errors made during m-lines experiments and calculations:
=error due to the coupling to the prism.
1αexp =uncertainty on the measurement of αmr .
On the one hand, the uncertainty 1αexp stems from the
uncertainty on both the refractive index npand the angle
Apof the prism (see ﬁgure 2). On the other hand, it results
fromtheuncertaintyonthemeasured synchronism angle. Let
1np,1Apand 1θsync be these uncertainties. The maximum
absolute error 1αexp of αmr is then given by
The refractive index of the prism can generally be
determined from the common minimum deviation method,
giving in our case 1np=2×10−5†. The angle Apof the
prism is measured with 1Ap=1×10−3degrees and the
synchronism angles with 1θsync =5×10−3degrees.
angle θsync for realistic values of npand Ap, the values of the
multiplicative coefﬁcients of equation (11) ∂αmr
∂θsync imply that 1αexp varies only weakly around 1×10−4.
One can therefore assume that the weak coupling hypothesis
is satisﬁed as long as 1αth is negligible in comparison with
1αexp. We assume this condition is met when 1αth <
We must now calculate the error 1αth versus the
coupling in order to deﬁne experimental conditions in which
the coupling has no signiﬁcant bearing on the measurements.
For this purpose, we shall ﬁrst study the inﬂuence of the
coupling of the prism.
4.2. Numerical simulations of the prism’s inﬂuence
The ALT that corresponds to a given coupling efﬁciency
depends a lot on the characteristics of the coupled beam
(i.e. wavelength, polarization, mode order, beam waist) .
Consequently, the ALT is not the most suitable parameter
to deﬁne the weak coupling regime. On the other hand, for
a given beam waist, the NFR associated with the maximal
coupling efﬁciency depends neither on the mode’s order nor
on the polarization state. The NFR can therefore be used as
a typical parameter of the coupling regime.
relative to the NFR. First, 1αth clearly tends to zero for an
inﬁnite NFR (that is for an inﬁnite ALT). Additionally, the
TM resonances are much more sensitive to the coupling than
the TE ones in all the calculations we performed with various
waveguide parameters, wavelengths, beam waists or prism
indices (an example is shown in ﬁgure 8). To satisfy the
criterion 1αth <1α
lim, the NFR should be greater than
several hundreds, corresponding to extremely weak coupling
† This implies corrections 8 and 10 indicated below.
Figure 8. Evolution of 1αth versus NFR for the TE2and TM2
resonances. Calculated for λ=514.5 nm with a beam waist
w0=15 µm and a SrTiO3prism, in the case of a waveguide of
refractive index 2.27 and of thickness 427 nm, with an extinction
coefﬁcient κ=5×10−5. Substrate index 1.4616.
4.3. High accuracy determination of optical properties
of light guiding thin ﬁlms
4.3.1. m-line detection. With the m-lines technique, the
purpose is to measure the angles for which dark lines appear
in the reﬂected beam. These dark lines, when thin, can be
efﬁciently detected by the eye, whereas for a coherent laser
beam, the speckle makes the detection of these lines difﬁcult
to perform with a photodetector. To be in a weak coupling
regime, it is necessary to measure the angles for lines as thin
as possible, i.e. on the limit of visibility.
Measurements performed on the limit of visibility yield
a contrast and a width of the line one metre from the prism
of 20% and 0.025◦, respectively. The contrast is deﬁned as
Imax+Imin with Imax andImin the maximum and theminimum of
the m-line intensity proﬁle respectively; the width of the line
is deﬁned as the apparent angle between Imax and Imin as seen
from the prism. This is in good agreement with the reference
results  about human eye mean contrast sensitivity. It
can be considered that the width of the line on the limit
of visibility, 1 m from the prism, does not change with the
polarizationor the mode’sorder. Nordoesitchange in a wide
range of waveguide, prism or beam characteristics. Hence,
with this contrast, the ALT in the m-lines conditions can be
evaluated without measuring the NFR.
4.3.2. Accuracy of the m-lines technique. Considering
that the approximate values of the refractive index and of the
thickness of the guide are known, the contrast on the limit of
visibility (i.e. 20%) allows the calculation of the ALT, then
For example, ﬁgure 9 shows the differences 1αth
between the effective indices corresponding to the guided
modes of the free waveguide and those corresponding to the
associated resonances of the coupled waveguide. The layer
considered has a refractive index of 2.27 and a thickness of
427 nm. One can notice that non-negligible errors (1αth >
1αlim =1×10−5)occur in the TM resonances.
On the other hand, our experimental uncertainties on
the prism’s angles or index and on the synchronism angle
measurements prove more signiﬁcant than the effect of non-
weak coupling. The accuracy we currently obtain in the
S Monneret et al
Figure 9. Example of calculation of 1αth for the resonances of a
guide with the conditions of m-lines measurements. Calculated for
λ=514.5 nm with a beam waist w0=15 µm and a SrTiO3prism,
in the case of a waveguide of refractive index 2.27, of extinction
coefﬁcient 5 ×10−5and of thickness 427 nm. Substrate index
index and thickness measurements of thin-ﬁlm dielectric
waveguides is of 1 ×10−3for the index and 1 nm for the
thickness. In order to reach higher accuracy, the ﬁrst step
would be to improve the accuracy of the measurements of the
prismparameters and ofthe synchronism angles. Thesecond
stepwouldthenbetocorrectαmr with 1αth by taking account
of the coupling with the prism. In this way, an accuracy
of 1 ×10−4for the refractive index and of 0.1 nm for the
thickness of a homogeneous and isotropic waveguide could
Onecandeterminethe indexand the thickness of a waveguide
using the m-lines technique on the assumption of weak
couplingwith the prism. This hypothesis allowsonetomatch
the measured effective indices to the effective indices of the
guided modes of the free waveguide. The necessity for this
hypothesis is warranted mostly by the difﬁculty in measuring
the coupling of the incident beam in the guide.
We propose a general experimental method to measure
the thickness of the air layer between the prism and the
guidewhich determines the couplingefﬁciency. This method
is based on an electromagnetic model of the TRPC with
a Gaussian beam. It offers several advantages: ﬁrst the
precision of the ALT measurement is of the order of 1 nm;
then the beam actually coupled in the guide is used; lastly,
the measurement can be made in real time.
As one can measure the ALT, it then becomes possible
to test the validity of the hypothesis of weak coupling
used by the m-lines technique. Using the model of the
TRPC mentioned above, we have studied the behaviour of
the difference 1αth between the effective indices of the
resonances of the guide coupled to the prism on the one hand
and the effective indices of the associated guided modes of
the free waveguide on the other.
We have shown by means of an example that in m-lines
conditions, the ALT can be such that 1αth becomes non-
negligible by contrast to the other sources of uncertainty,
as regards TM resonances especially. However, in our
experiment, these sources of uncertainty are conducive to
an accuracy of no better than 1×10−3for the index and 1 nm
for the thickness. Such an accuracy is immune to the effect
of non-weak coupling.
However, if the angles were measured with an accuracy
better than 1 ×10−3degrees, by taking into account the
for the refractive index and of 0.1 nm for the thickness,
respectively, for homogeneous and isotropic waveguides.
The method proposed here for single layer waveguides
can easily be extended to multilayer waveguides and
consequently to inhomogeneous waveguides which can be
approximated by multilayers.
We are very grateful to Thierry Kakouridis for his precious
help in improving the English.
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