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m-lines technique: prism coupling measurement and discussion of accuracy for homogeneous

waveguides

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2000 J. Opt. A: Pure Appl. Opt. 2 188

(http://iopscience.iop.org/1464-4258/2/3/304)

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J. Opt. A: Pure Appl. Opt. 2(2000) 188–195. Printed in the UK PII: S1464-4258(00)04726-7

m

-lines technique: prism coupling

measurement and discussion of

accuracy for homogeneous

waveguides

S Monneret†, P Huguet-Chantˆ

ome and F Flory

Laboratoire d’Optique des Surfaces et des Couches Minces, Ecole Nationale Sup´

erieure de

Physique de Marseille, Domaine Universitaire de Saint-J´

erˆ

ome, 13397 Marseille Cedex 20,

France

E-mail: francois.flory@enspm.u-3mrs.fr

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)

is known.

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

1. Introduction

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 [1]. For m-lines

measurements,thecouplingisgenerallyconsideredweakand

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

thickness (ALT)).

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

discussiononhigh-accuracythin-ﬁlmcharacterizationisthen

presented. Thispaperonlydealswithhomogeneousisotropic

single-layeropticalwaveguidesforwhichtherefractiveindex

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 [1], 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 [12] proposed a method

basedontheobservationofinterferencefringeslocalized near

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 [13] 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 [14] 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

the measurement.

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

istheelectricvectoroftheelectromagneticﬁeld. Weconsider

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

i,z

i)and (xr,y

r,z

r)

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

Ei(xi,y

i,z

i=0)=E

0exp −x2

i

w2

0exp −y2

i

w2

0(1)

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

i

w2

0)as a sum of

plane waves. As the plane of incidence is the (XOZ) plane,

189

S Monneret et al

and because the structure is invariant in Ywe obtain

Ei(xi,y

i,0)=exp −y2

i

w2

0Z+∞

−∞ ˆ

Ei(σ ) exp(2jπσxi)dσ

(2)

where

σ=sin(θ )/λ0(3)

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:

ˆ

Ei(σ ) =E0√πw0exp(−π2w2

0σ2). (4)

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

Er(xr,y

r,z

r)=exp −y2

r

w2

0Z+∞

−∞ ˆ

Ei(σ )f (σ )

×exp(2jπσxr)exp(2jπµzr)dσ(5)

where the longitudinal spatial frequency of each plane wave

is

µ=cos(θ)/λ0.(6)

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 [17] 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 [18], 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

np

(n2

p−1) 1+λ

2

0

2sin2(θi)

n2

p−sin2(θi)σ2!(7)

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

0, which,

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

0σ2

2.(8)

The electric ﬁeld distribution of the reﬂected beam is now

given by

Er(xr,y

r,z

r)=exp −y2

r

w2

0Z+∞

−∞

F(σ,x

r)

×exp(2jπσxr)dσ(9)

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

1.4616. TE0resonance.

with

F(σ,z

r)=ˆ

E

i(σ )f (σ ) exp(jk0zr)exp(−jπσ2zr). (10)

Let Ir(xr,z

r)be the transverse intensity proﬁle of the

reﬂected light beam. This proﬁle is deﬁned as

Ir(xr,z

r)=|E

r

(xr,0,z

r)|2.(11)

For a given value zr0of zr,Ir(xr,z

r

0)is therefore obtained

directly from the one-dimensional inverse Fourier transform

versusσofthefunctionF(σ,z

r

0). Numerically,afastFourier

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

of Ir(xr,z

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(xr,y

r)=|E

r

(xr,y

r,0)|2.(12)

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

190

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

the light.

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

Ei

on the spatial frequencies, which depends on the ALT [16].

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

polarization state.

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,

w

0=46.4±0.5µm,TE0resonance,

red beam: λ=632.8nm,

w

0=34.7±0.5µm,TM0resonance.

Thegreen and thered images ofthe corresponding near ﬁelds

ofthereﬂectedbeamsareobtainedbymeansofthesinglelens

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 [19] 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

performed. TheALThasbeentunedbychangingthepressure

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 [16].

191

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

rate ≈1.0nms

−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.

3.4. Discussion

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 [20]. 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.

Numericalsimulationsoftheprism’sinﬂuencearethengiven.

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

stateandanintegermwhichistheorderofthemode. Besides,

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 [21]. 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[22] 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 [23]. The aperture of such

a wavepacket strongly depends on the ALT [16].

Nevertheless, the different resonances of the waves

correspond to the modes of the waveguide coupled to the

prism [3]. 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

index.

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

constants [24].

192

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:

1αth =|α

mr −αm|

=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

1αexp =∂αmr

∂Ap

1Ap+∂αmr

∂np

1np+∂αmr

∂θsync 1θsync.(13)

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.

Calculatingthedependenceof1αexp onthesynchronism

angle θsync for realistic values of npand Ap, the values of the

multiplicative coefﬁcients of equation (11) ∂αmr

∂Ap,∂αmr

∂npand

∂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 <

1αlim =1×10−5.

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) [16].

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.

Severalremarkscanbemadeabouttheevolutionof1αth

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

conditions.

† 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

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 [25] 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,

thecontrastaccountsdirectlyforthevisibility. Consequently,

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

of 1αth.

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

193

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

1.4616.

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

be expected.

5. Conclusion

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

presenceoftheprism,onecouldreachanaccuracyof1×10−4

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.

Acknowledgment

We are very grateful to Thierry Kakouridis for his precious

help in improving the English.

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