of the undulation at AE, to the point P, to be of equal intensity,
since the only rays, with respect to which this hypothesis would be
incorrect, are such as have no sensible influence on the quantity of
light which it receives. For the same reason, in order to simplify
the calculation of the result of all these elementary undulations,
we may consider their constituent motions as performed in the
same direction, the angles which they form with each other being
inconsiderable. The problem is thus reduced to that which has been
solved in the Memoir on Diffraction, already quoted: _To find the
result of any number of systems of parallel undulations of light, of
the same frequency, when their intensities and relative situations
are given_.—The intensities are here proportional to the length of
the small illuminating arcs, and the relative situations are given
from the differences of the paths described.
We have considered, correctly speaking, only the section of the
undulation made by a plane perpendicular to the margin of the screen
represented by A. We may now take into account the whole extent
of the undulation, and suppose it to be divided, by equidistant
meridians perpendicular to the plane of the figure, into infinitely
thin wedges or strata; and we may apply to all of these the reasoning
which has [p438] been employed for one section, and thus demonstrate
that the rays which have a marked obliquity must destroy each other.
These strata, in the case here considered, being all parallel to the
edge of the screen, and infinitely extended, while the undulation
is intercepted but on one side; the intensity of the result of all
the impressions, which they transmit to P, will be the same for each
of them: for the rays emanating from them must be considered as of
equal intensity, at least for the very small extent of the generating
undulation, which has a sensible influence on the light received at
P. Besides, each elementary result will evidently be retarded by the
same quantity, with respect to the ray derived from the point of the
stratum nearest to P, that is to say, to the point in which it cuts
the plane of the figure: consequently the intervals between these
elementary results will be equal to the differences of the paths
described by the rays AP, _m′_P, _m_P, and so forth, which are in
the plane of the figure, and their intensities will be proportional
to the arcs A_m′_, _m′_, _m_, _m_M, and so forth. We may therefore
consider the intensity of the general result as determined by the
calculation already mentioned, as belonging to the section of the
undulation made by a plane perpendicular to the margin of the screen.
While the outline of the screen remains rectilinear, it is
sufficient, in order to determine the situations of the dark and
light stripes, and their relative intensities, to consider the
section of the undulation made by a plane perpendicular to that
outline: but when it is curved, or composed of lines meeting at any
angles, it becomes necessary to obtain the integral effect for two
directions at right angles to each other, or for a circle surrounding
the point considered. This last method is the most simple in some
particular cases, as when we have to calculate, for example, the
intensity of the light in the projection of the centre of a circular
screen or opening: [a simplification which, though sufficiently
obvious, had perhaps not occurred to Mr. FRESNEL, until it was
pointed out to him by the Translator of this paper.]
It will now be easy to form a distinct idea of the method [p439]
which must be followed, in order to calculate the situation and
the intensity of the dark and bright stripes, in the different
circumstances under which it is proposed to compare the theory with
experiment. When the screen is infinitely extended on one side, or
is broad enough to allow us to neglect the rays which pass beyond
it, we are to determine, for any point P at the distance of the
place at which the fringes are to be observed, the result of all
the elementary undulations coming from the part AMF only of the
incident wave; and comparing the intensities at different collateral
points, P, P′, P″, we are to find the situation of the darkest and
the brightest points. In this manner we find, for a screen closed
on one side, 1st, that the intensity of the light decreases rapidly
within the [shadow] beginning from the tangent CAB, and _so much the
more rapidly as the undulation is smaller_; and this in a continuous
manner, without any alternations of maxima and minima; 2ndly, that
out of the shadow, the intensity of the light, after augmenting
considerably to a certain point, which may be called a maximum of
the first order, decreases to another point, which is the minimum
of the first order: that it increases again to a second maximum, to
which succeeds a second minimum, and so forth; 3rdly, that none of
these minimums completely vanish, as in the case of fringes produced
by the concourse of two luminous pencils of equal intensity, and
that the difference between the maxima and minima diminishes in
proportion as we go further from the shadow; whence we may understand
why the fringes which surround shadows in a homogeneous light, are
less marked and less numerous, than those which are obtained by
a combination of two mirrors, and those in white light much less
brilliant; 4thly, that the intervals been the maxima and minima are
unequal, and diminish, as we depart from the shadow, in proportions
which remain unaltered, whatever may be the distance from the screen
at which we measure them; and 5thly, that the same maxima and minima,
calculated for different distances from the screen, are situated in
hyperbolas of a sensible curvature, of which the foci are the edge
of the screen, and the luminous point. All these consequences of the
theory are precisely confirmed by experiment. [p440]
The general formula gives the position of the maxima and minima for
any distances whatever of the luminous point from the screen, and
from the screen to the micrometer, when the length of the undulation
of the light employed is known. In order to submit the theory to a
decisive test, instead of determining the length of the undulation
by measures of the external fringes, and then employing it in
calculations of the same kind, I deduced it from an experiment on
diffraction of a very different kind; and after having first verified
it by the fringes obtained from two mirrors, of which it represented
the breadth within a hundredth part of the truth, I introduced it
into the formula which I afterwards compared with 125 measurements of
exterior fringes, made under very different circumstances; for the
distance of the radiant point from the screen was varied from four
inches to six or seven yards, and the distance between the screen
and the micrometer was varied from 1/13th of an inch to more than
four yards: and the results of all these comparisons were perfectly
satisfactory, as maybe seen in the comparative table published in the
XIth volume of the Annales de Chimie et de Physique, p. 339, 343.
When the screen, instead of extending infinitely on one side, is
narrow enough to admit some light on that side, not too much weakened
by the rapid decrease of intensity produced by obliquity, we must
take into the calculation the light on both sides, and find, for
each point of the shadow, the general result of all the elementary
undulations derived from the points on the right and left. We thus
demonstrate that the interior parts of the shadow must be divided
by a series of dark and bright stripes, nearly equal in breadth,
of which the situations differ very little from those which would
be deduced from the approximative formula which has already been
given for the same purpose, when they are still separated from the
borders of the shadow by an interval of several of their breadths.
But when the opaque body is narrow enough, and the micrometer far
enough removed for the observed stripes to be very near the exterior
stripes, then the results of this more exact calculation, as well
as those of experiment, show that the approximation is no longer
accurate. The [p441] calculation determines also, with remarkable
precision, the singular alterations which the exterior fringes often
undergo, when the other series extends beyond the shadow, and mixes
its effects with those of the exterior.
I have also verified the theory by examining the fringes derived
from a narrow slit of indefinite length; and determining, for the
different points enlightened by the luminous pencil, the result of
all the elementary undulations derived from the part of the primitive
wave comprehended in the breadth of the slit; and I have found a
satisfactory agreement between the calculation and the observations,
even when the fringes thus obtained afforded the most capricious and
apparently irregular appearances.
In this mode of considering the problems relating to diffraction, we
have not taken into the calculation the greater or less thickness of
the edges of the screen, but merely the extent of the primitive wave
which is capable of sending elementary undulations to the points for
which we are to find the intensity of illumination; and the opaque
substance has no other effect than simply to intercept a part of the
wave: for this reason the result is necessarily independent of the
nature of the body, of its mass, and of the thickness of its edges.
Nevertheless, if the surface of the edges were very extensive, it
would be impossible to consider the portion of the wave as quitting
the slit without having received some previous modification, and it
would be necessary to take into the calculation the small fringes
derived from the effect of the remoter parts of the slit. But
while the thickness is moderate, or the edges rounded off into a
well marked curve, the small fringes derived from this cause may
be neglected, and the emerging wave may be considered as of equal
intensity throughout, at the moment of its quitting the screen,
especially if the intensity of the light is to be calculated for a
pretty considerable distance from the screen. We must not, indeed,
forget, that according to the reasoning which has been employed,
the formulas for diffraction are only sufficiently exact when this
distance is very considerable, in comparison with the breadth of an
undulation, since it is in this case only that we can neglect the
rays that are decidedly oblique, and [p442] can suppose all those,
which are essentially concerned in the effect, to be nearly of equal
intensity. It is not, however, surprising that the same formulas will
give the position of the fringes with sufficient accuracy at small
distances from the screen, when its edges are thin, since, the mean
breadth of an undulation being but about one fifty thousandth of an
inch, a tenth of an inch becomes comparatively a very considerable
distance.
These are the three principal kinds of phenomena presented to us by
diffraction, when the edges of the screen, or of the opening made in
it, are sufficiently extensive to afford fringes independent of any
effect from their terminations: and in such cases it is sufficient
to make the integral calculation for the plane perpendicular to
the edges of the screen only, in order to determine the position
of the dark and bright stripes, and their comparative intensities.
But when the screen or the opening are of small dimensions in every
direction, it becomes necessary to extend the integration to the
effects produced in two perpendicular planes: and the results of the
calculation agree perfectly with observation, as will appear from two
curious instances.
When the screen is circular, the calculation leads to this singular
result, that the centre of the shadow projected by it must be as
much enlightened as if the screen were not in existence. It was Mr.
POISSON that first pointed out this consequence of my formulas,
which I did not at first observe, though it is immediately deducible
from the theory by very simple geometrical considerations. Mr. ARAGO
made the experiment with the shadow of a screen 1/13th of an inch
in diameter, perfectly round, and fixed on a plate of glass. The
result confirmed the fact which had been announced by the theory.
It is only the centre itself that possesses this property, and the
same brightness is only extended to a sensible distance from this
mathematical point when the screen is of very small diameter, and
when its shadow is observed at a great distance: for the wider that
the screen becomes, the more the little bright circle is contracted;
and when the screen is four tenths of an inch in diameter, we only
see a single point of light, at the distance of a yard, even with a
powerful magnifier. It must be observed, that if the screen [p443]
were too large, the reasoning, from which the formulas have been
deduced, would no longer be rigorously applicable to the rays
inflected into the shadow, because of their too great obliquity,
which would render it impossible to consider their effects as equal
in intensity to those of the direct rays.
When we calculate, by the same formulas, the intensity of the light
in the centre of the projection of a small circular aperture, made
in a large screen, we find that this centre will exhibit alternately
a bright and a dark appearance, according to the distance at which
the shadow is viewed; and that in homogeneous light this darkness
must be perfect. This new inference from the general formulas may be
deduced from the theory by very simple geometrical considerations.
Thus we find that the values of the successive distances, at
which the centre of the shadow becomes completly dark, are _b_ =
_a__r_^2/(2_a__d_−_r_^2), _b_ = _a__r_^2/(4_a__d_−_r_^2), _b_ =
_a__r_^2/(8_a__d_−_r_^2); and so forth; _r_ being the semidiameter
of the aperture, _a_ and _b_ its respective distances from the
luminous point and from the micrometer, and _d_ the length of the
undulation of the light employed. Now, if we place the micrometer at
the distances indicated by these formulas, we observe, in fact, that
the centre of the projection of the opening is so completely deprived
of light, that it appears like a spot of ink in the middle of the
illuminated part, at least with respect to the minimums of the first
three orders, as indicated by the formulas here inserted: those of
the subsequent orders, which are nearer to the screen, exhibiting
no longer the same degree of darkness, on account of the want of
homogeneity of the light employed.
There is still a multitude of other phenomena of diffraction, such
as those of multiplied and coloured images, reflected by striated
surfaces, as seen through a texture of fine fibres, as well as the
coloured rings, produced by an irregular collection of such fibres,
or of light powders, consisting of particles nearly equal, placed
between the eye of the spectator and a luminous object; all of which
may be explained and rigorously computed by means of the theory which
has been laid down. It would, however, occupy too much of our time
to describe them here, and to [p444] show how exactly they concur
in confirming the theory; which indeed appears to be abundantly
demonstrated by the numerous and diversified facts which have been
already adduced in support of it. It will be sufficient to conclude
this extract of the Memoir on Diffraction with a detailed description
of an important experiment of Mr. ARAGO, which furnishes us with a
method of determining the slightest differences of the refractive
powers of bodies, with a degree of accuracy almost unlimited.
We have seen that the fringes, produced by two very narrow slits, are
always placed symmetrically with regard to a plane passing through
the luminous point and the middle of the interval between the slits,
as long as the two pencils of light which interfere have passed
through the same medium, for instance, the air, as happens in the
ordinary arrangement of the apparatus. But the result is different
when one of the pencils continues to pass through the air, and the
other has to be transmitted by a more refractive body, a thin plate
of mica, for example, or a piece of glass blown very thin: the
fringes are then displaced, and carried towards the side on which
the transparent substance is placed: and if its thickness becomes
at all considerable, they are removed out of the enlightened space,
and disappear altogether. This important experiment, which was first
made by Mr. Arago, may also be performed with the apparatus of the
two mirrors, if the plate be placed in the way of one of the pencils,
either before or after its reflection.
Let us now see what inference may be drawn from this remarkable fact,
by the assistance of the principle of interferences. The light stripe
in the middle is always derived, as we have already seen, from the
simultaneous arrival of rays which have issued at the same moment
from the luminous point; consequently, in the common circumstances
of the experiment, they must have described paths exactly equal, in
order to arrive in the same time at the place of meeting: but it
is obvious that if they pass through mediums in which light is not
propagated with the same velocity, that pencil, which has travelled
the more slowly, will arrive at the given point later than the other,
and the point will [p445] therefore no longer be in the bright
stripe. The stripe must therefore necessarily change its place
towards the pencil which travels the more slowly, in order that
the shortness of its path may compensate for the delay during its
transmission through the solid: and the converse of the proposition
enables us to conclude, that where the stripes are displaced, the
pencil towards which they move has been retarded in its passage. The
natural inference, therefore, “from Mr. ARAGO’s experiment,” is, that
light is propagated more rapidly in the air than in mica or glass,
and generally in all bodies more refractive than the air; a result
directly opposite to the Newtonian theory of refraction, which,
supposes the particles of light to be strongly attracted by dense
substances, which would cause the velocity of light to be greater in
these bodies than in rarer mediums.
This experiment furnishes a method of comparing the velocity of the
propagation of light in different mediums, [or, in other words,
the refractive density, which is always supposed in this theory,
to be reciprocally proportional to it.] If, in fact, we measure
very accurately, by means of a spherometer, the thickness of the
thin plate of glass which has been placed in the way of one of
the luminous pencils, and if the displacement of the fringes has
been measured by the micrometer; since we know that, before the
interposition of the glass, the paths described were equal for the
middle of the central stripe, we may calculate how much difference
is occasioned by the change of position, and this difference will
give the retardation in the plate of glass, of which the thickness
is known: so that, by adding this thickness to the difference
calculated, we shall find the little path which the other pencil has
described in the air, while the former was transmitted by the plate
of glass; and this path, compared with the thickness of the plate of
glass, will give the proportion of the velocity of the light in the
air, to its velocity within the glass.
We may also consider this problem in another point of view, with
which it is convenient to make ourselves familiar. The duration of
each undulation, as we have seen, does not depend on the greater or
less velocity with which the [p446] agitation is propagated along
the fluid, but merely on the duration of the previous oscillation
which gave it birth; consequently, when the luminous waves pass
from one medium into another, in which they are propagated more
slowly, each undulation is performed in the same interval of time as
before, and the greater density of the medium has no other effect
than that of diminishing the length of the undulation, in the same
proportion as the velocity of light is diminished: for the length
of the undulation is equal to the space that the first agitation
describes during the time of a complete oscillation. We may therefore
calculate the relative velocities of light in different mediums, by
comparing the length of the undulations of the same kind of light in
those mediums. Now, the middle of the central stripe is formed by the
reunion of such rays of the two pencils as have performed the same
number of undulations, in their way from the luminous point, whatever
may be the nature of the mediums transmitting the light. If then the
central stripe is brought towards the side of the pencil which has
passed through the glass, it is because the undulations of light are
shorter within the glass than in the air; and it is necessary, in
consequence, that the path described on this side should be shorter
than the other, in order that the number of undulations may remain
the same. Let us suppose, then, that the central stripe has been
displaced to the extent of twenty breadths of fringes, for example,
or of twenty times the interval between the middle points of two
consecutive dark stripes; we must necessarily conclude that the
interposition of the plate of glass has retarded the progress of
the pencil passing through it to the extent of twenty undulations;
or that it has performed within the plate twenty undulations more
than the same pencil would have performed in an equal thickness of
air, since each breadth of a fringe answers to the difference of a
single undulation. If then we know the thickness of the plate, and
the length of an undulation of the light employed, which is easily
deduced from the measurement of the fringes, by the formula that has
been given, we can calculate the number of undulations comprehended
in the same thickness of air, and by adding twenty to the number, we
shall have that of the [p447] undulations performed in the thickness
of the glass; and the proportion of these two numbers will be that of
the velocities of light in the different mediums. Now this proportion
is found by experiment the same with that of the sines of incidence
and of refraction between air and glass; which agrees with the theory
of the refraction of undulations, as will be seen hereafter.
The same experiment may be employed, on the other hand, for
determining with extreme precision the thickness of a thin plate of a
substance of known refractive density; placing it in the way of one
of the two pencils of light, and measuring the displacement of the
fringes which it occasions.
This method of determining refractive densities is however liable to
some difficulties, when we wish to apply it to a body much more dense
than air, such as water, or glass, for example; since it is necessary
to employ a very thin plate only, in order that the fringes may not
be too much displaced for observation; and then it becomes difficult
to measure the thickness of such a plate with sufficient accuracy.
We may, indeed, place in the way of the other pencil a thick plate
of a transparent substance, of which the refractive density has been
ascertained by the ordinary methods, and we can then employ as thick
a plate of the new substance. But then it becomes simpler to measure
its refractive density by the common method: [unless we choose to
immerse the whole apparatus in a fluid very nearly approaching
to it in refractive density, which may sometimes be done without
inconvenience. TR.]
The case, in which Mr. Arago’s experiment has a decided advantage
over the direct method, is when we desire to determine very slight
differences of velocity in mediums of nearly equal refractive
density: for by lengthening the passage of the light in the two
mediums of which we wish to compare the refractive density, we can
increase the accuracy of the results almost without limit. In order
to form an idea of the extreme precision that may be attained by
these measurements, it is sufficient to observe that the length of
the yellow undulations in air being about .000021 E.I., there are two
millions of them in the length of about 42 inches. Now [p448] it is
very easy to observe the difference of one fifth of a fringe, which
corresponds to a retardation of one fifth of an undulation in one of
the pencils, that is, the ten millionth part of the whole length of
42 inches; we might therefore, by introducing any gas or vapour into
a tube of this length, terminated by two plane glasses, estimate very
accurately the variation of its refractive power.