Method of and device for measuring the refractive index of wafers of vitreous material

A light beam is sent onto a wafer, at different angles of incidence, thus giving rise to fluctuations in the transmittance of the wafer, as the angle of incidence varies, because of interference due to multiple reflections of the beam inside the wafer. The transmittance of the wafer is measured as the angle of incidence varies. The angular positions of transmittance maxima and minima are determined with respect to a maximum or minimum corresponding to normal incidence. The refractive index is obtained from these positions and from the number of maxima and minima in the different angles.

FIELD OF THE INVENTION 
My present invention relates to a method of measuring the refractive index 
and, more particularly to a method and a device for determining the 
refractive index of a wafer of vitreous material. 
Preferably, but not exclusively, the invention is employed in measuring the 
refractive index in a limited area (about 1 mm.sup.2) of a thin sample of 
an optical fiber preform of fluoride glass. 
BACKGROUND OF THE INVENTION 
It is well known that the refractive index of a body of a fluoride glass, 
such as an optical fiber preform, depends not only on the composition of 
the glass but also on its thermal history. In effect, during the casting 
and drawing processes stresses or inhomogeneities may arise which give 
rise to local fluctuations in the refractive index, which fluctuations 
must be detected and eliminated. 
The conventional devices for measuring the refractive index of samples of 
transparent material, which are based on measuring the limit angle (for 
instance Pulfrich refractometers) require sending a grazing beam onto the 
sample and analyzing the beam refracted by the sample. These devices 
cannot be employed to measure directly the refractive index of the core 
and the cladding of an optical fiber, since the size of the sample is too 
small to obtain a significant amount of refracted light. Using a Pulfrich 
refractometer for measurements on a glass sample whose composition is 
analogous to that of the preform, on the other hand, does not allow taking 
into account the thermal history of the glass, and in particular the fact 
that the cooling conditions of a wafer are different from those of a 
cylindrical body like the preform. 
European patent EP-B 0 085 978, describes a method of determining the 
refractive index, which method can also be employed in the case of small 
samples, such as those obtained by cutting an optical fiber preform. 
According to the known method, the sample is placed on a support which can 
be rotated. A light beam comprising two monochromatic radiations is sent 
towards the sample with a first angle of incidence and the two radiations 
are caused to interfere upon leaving the sample, thereby creating a first 
beat. Subsequently, the support is rotated, the beam is directed against 
on the sample with a second angle of incidence and the two radiations are 
again caused to interfere upon leaving the sample, thereby creating a 
second beat. The refractive index is obtained from the phase differences 
between the two beats and a reference beat obtained by making the two 
radiations of the beam interfere at the end of a path external to the 
sample. 
However, radiations which pass through the sample inevitably undergo 
multiple reflections inside it, and these bring about interference 
phenomena which result in a phase error limiting the accuracy of the 
measurement to such an extent that it is no longer possible to appreciate 
said fluctuations in the refractive index. Moreover, the measurement is 
quite sensitive to thermal expansion and to electrical drift. 
OBJECT OF THE INVENTION 
A more detailed analysis of the disturbance caused by multiple reflections, 
carried out by the inventor in order to correct or in any case to take 
into account errors stemming from such disturbance, has yielded the 
surprising result that it is possible to exploit the disturbance itself to 
obtain an accurate measurement of the refractive index. The object of the 
invention is thus to provide a method and a device which exploit 
interference phenomena due to multiple reflections inside the sample. 
SUMMARY OF THE INVENTION 
According to the invention a method is provided in which a source generates 
a light beam which is sent towards the wafer at different angles of 
incidence and the intensity of a beam transmitted by the wafer is measured 
as the angle of incidence varies, and in which: the beam generated by the 
source is a coherent monochromatic beam whose coherence length exceeds the 
thickness of the wafer; said beam, before being sent towards the wafer, is 
transformed into a collimated beam with plane wave front, in order to give 
rise to fluctuations of the wafer transmittance as the angle of incidence 
varies, because of the interference due to multiple reflections of the 
beam inside the wafer; the values of the wafer transmittance are obtained 
from the measured values of the intensity; the angular positions of the 
transmittance maxima and minima are determined, within a preset range of 
angles of incidence, with respect to a maximum or a minimum corresponding 
to normal incidence, and the refractive index is obtained from these 
positions and from the number of maxima and minima in the different angles 
.

SPECIFIC DESCRIPTION 
In FIG. 1, a source 1, e.g. a He-Ne laser, generates a beam of coherent 
monochromatic light. A spatial filter 2, comprising a pinhole diaphragm 2a 
placed between a first lens 2b, which focuses the beam emitted by the 
source onto the pinhole, and a second lens 2c, which collimates and 
expands the beam exiting the diaphragm, generates a beam with a planar 
wave front. The collimated beam is split into two fractions by means of a 
beam splitter 3. A first fraction is sent towards a sample 4, which is a 
wafer with plane and parallel faces whose thickness is smaller than the 
coherence length of source 1. This beam fraction passes through a 
polarizer 5 which orients the polarization plane of the beam fraction in 
such a way that the electrical field is parallel to the angle of incidence 
(S wave). 
Inside the sample, the beam undergoes multiple reflections and, due to the 
characteristics of the beam, there are noticeable fluctuations in the 
transmittance of the sample as an effect of interference between the 
various reflected beams. These fluctuations are exploited to determine the 
refractive index. The particular choice of the polarization makes the 
measurements easy since the amplitude of the fluctuations is larger in the 
case of an S wave. 
The second beam fraction is sent directly to a reference detector 6 whose 
output signals are provided to an analog-to-digital converter 7 connected 
to a processor 8. 
Sample 4 is mounted on a motorised support 9 which can be rotated under the 
control of processor 8 to vary the angle of incidence of the beam on 
sample 4. The support is associated with an extremely accurate angular 
position detector, in particular, an encoder with the ability to provide 
the position of the support with an accuracy on the order of tenths of a 
second of a degree. For the sake of simplicity, in the drawing the motor 
and the encoder are incorporated in support 9. The beam exiting sample 4 
passes through a movable diaphragm 10 which allows selection of the area 
of the sample on which the measurement is to be performed, and it is 
collected by a second detector 11 followed by an analog-to-digital 
converter 12 connected to processor 8. 
Processor 8 receives, from converters 7 and 12, current or voltage values 
representing the intensities of the reference beam and of the beam 
transmitted by wafer 4 (which is proportional to transmittance) as the 
angle of incidence varies, and computes the ratio of said values, 
associating the values of that ratio to the angular position of the 
support. Note that the intensity of the transmitted beam only could be 
used for processing. However, as shall be better shown below, the actual 
value of transmittance is not of interest for the invention and using the 
ratio between the two intensities allows the effects of noise to be 
reduced. Assuming that support 9 is rotated in a range from +45.degree. to 
-45.degree. with respect to normal incidence, sufficient data for 
subsequent processing are obtained by rotating support 9 in steps of a few 
tens of seconds of a degree. 
FIG. 2 shows the transmittance versus the angle of incidence (in degrees) 
for a small range of angles on the two sides of normal incidence. The 
values of the ordinates are not indicated, since they depend on the 
measured quantity (intensity of the beam transmitted by the sample or 
ratio between transmitted beam and reference beam intensities). The Figure 
clearly depicts transmittance fluctuations due to interference phenomena 
caused by multiple reflections inside wafer 4, and shows that oscillation 
frequency increases as the angle of incidence increases. According to the 
invention, the refractive index is obtained by identifying the positions 
of maxima and minima of the curve in FIG. 2 with respect to the position 
of normal incidence (0 in the Figure) and the number of maxima and minima 
corresponding to the various rotation steps. This number, as will be shown 
below, can be expressed, for a given thickness of the sample and a given 
wavelength of the radiation used, as a function of refractive index and 
angle of incidence. The position of normal incidence is in turn accurately 
determined by identifying first the approximate point around which the 
curve is symmetrical, then two maxima or minima which are symmetrical with 
respect to that point and finally by choosing as angle 0 the intermediate 
value between the two considered maxima and minima. The operation is made 
easier by the oscillation frequency being relatively low near normal 
incidence. 
For processing, the positions of transmittance maxima and minima are used 
in place of the actual values because the position is much less sensitive 
to errors due to drifts or to the state of cleanliness or finish of the 
faces of the sample. Greater processing complexity is counterbalanced by 
the improvement in accuracy. 
To determine the positions of maxima and minima, intervals containing each 
one maximum and one minimum are looked for in the curve. For this purpose, 
a function P(.theta.) (for instance, a straight line or a curve 
corresponding to a second degree polynomial function with a very small 
coefficient of the second degree term) is determined from the data, which 
function intersects the peaks in FIG. 2 essentially at mid height, and the 
values of .theta. corresponding to the intersections between the two 
curves are identified. Operations are simplified if both T and P are 
expressed as a function of cos.theta., since peaks in function 
T(cos.theta.) are essentially equally spaced, as FIG. 3 shows. For the 
sake of simplicity, a function of the type P(cos.theta.)=constant was 
considered for curve P in the drawing. Intersection points between P and T 
are those for which, in the points of the two curves P, T corresponding to 
two successive positions of sample 4, relation 
EQU (T.sub.i -P.sub.i).(T.sub.i-1 -P.sub.i-1) (1) 
applies, and moreover the distance between the previous point in which 
relation (1) has been met and the current point exceeds a given value 
(e.g. the distance between successive peaks) so as to eliminate spurious 
intersection points due to noise. Once the intervals have been determined, 
it is sufficient to approximate the experimental curve in each interval 
with a polynomial of at least the 3.sup.rd degree and to determine 
analytically the maximum and minimum thereof. The refractive index is then 
obtained from the angular positions .theta..sub.K of the maxima and minima 
of T and from the overall number V.sub.K of the maxima and minima present 
in the interval 0-.theta..sub.K. 
The following description provides a brief outline of the theory on which 
the method according to the invention is based. 
Considering sample 5 as a multi-layer, transmittance T can be expressed as 
a function of angle of incidence .theta., wavelength .lambda. of the 
incident radiation, refractive index n and thickness d according to the 
following relation: 
##EQU1## 
The relations above are obtained by applying the principles described for 
example in "Theory and calculations of optical thin films", by P. H. 
Berning, Physics of thin films, Vol. 1 pages 69 and fol. To identify the 
positions of the maxima and minima of T as a function of .theta. it will 
be enough to compute the derivative of T with respect to .theta. and to 
set it to 0. The result is a relation of the type: 
##EQU2## 
It can be verified that, for refractive index values typical of vitreous 
materials (in particular &lt;2), thicknesses of a few millimeters and 
wavelengths in the visible spectrum, in the range of angles under 
consideration the second member of relation (6) is very small (&lt;0.001) and 
therefore the sine of the first member can be considered equal to its 
argument. Therefore, relation (6) is equal to 0 when the argument of the 
sine is equal to m.pi., i.e. when 
##EQU3## 
Now, let us consider the function: 
##EQU4## 
This function is equal to 0 for .theta.=0 and can be interpreted, when it 
has an integer value, as the overall number of maxima and minima of T. 
Moreover, its trend is very similar to that of the curve obtained by 
plotting number V of maxima and minima of T, determined in the way 
described above, versus .theta.. This similarity can clearly be seen in 
FIG. 4, where the solid line corresponds to the aforesaid plot and the 
dashed line to curve (8). The trend of curve (8) varies abruptly even for 
small variations of n: however, if for each value of n one of the two 
curves is translated vertically so that it intersects the other in one 
point, the difference between the two curves, in the range of angles under 
consideration, is negligible (&lt;10.sup.-9). The value of n that minimizes 
the differences between curve (8) and the experimental data in the range 
of angles under consideration shall be the value of the refractive index 
of the sample. 
To obtain satisfactory results it is necessary to take some measures, which 
allow minimising the effects of error due to the inaccuracy in the 
knowledge of thickness d and taking into account only the shape of the 
curve F and not the position. 
To take into account the shape of the curve, one can consider the 
difference between the value of F and the experimental value for a first 
angle .theta..sub.R which can be the angle of normal incidence or the 
angle corresponding to one of the first peaks of the curve in FIG. 2. The 
effect of thickness d can be minimized by normalizing both V and function 
F (already corrected to take into account the shape difference) with 
respect to the value corresponding to a second angle .theta..sub.F, for 
example an angle near the last peak. In practice, denoting by V.sub.K, 
V.sub.R, V.sub.F the number of maxima and minima in correspondence with a 
generic angle .theta..sub.K and respectively with angles .theta..sub.R and 
.theta..sub.F, and by N the total number of maxima and minima determined 
experimentally, the refractive index can be determined by minimizing 
function 
##EQU5## 
The accuracy in measuring n can be further improved by using in relation 
(6), instead of V.sub.R, V.sub.F, values obtained through interpolation, 
with a second degree curve, of a preset number of previous and subsequent 
values, for example 50. 
The operations described above are also reported in the flow chart in FIG. 
5. 
The system just described does not require the creation of a beat between 
beams following different paths, so clearly it is immune to disturbances, 
such as thermal expansion in the components of the device, which cause 
variations in the optical path (aside from possible expansions of the 
wafer, which in any case are negligible with respect to those of the 
external components). Exploiting the number of transmittance maxima and 
minima and not their values, moreover, renders the system immune to 
electrical drifts.