Abstract
Diffraction is a phenomenon that is caused by medium features that are small compared to the
wavelength of the wave. Sommerfeld defines diffraction as “any deviation of light rays from
rectilinear paths which cannot be interpreted as reflection or refraction”. Landau and Lifshitz state that
diffraction refers to phenomena which are the “consequence of deviations from geometrical optics”.
In this paper we present a brief historical sketch of diffraction theory. We present original material
from the key works of Grimaldi, Huygens, Young, Fresnel, Kirchhoff and others representing
milestones in the development of the theory. The examples range from the early experiments by
Grimaldi in the 1600’s to the seismic diffraction modeling of edge and tip waves.
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A brief historical sketch of diffraction theory
Grimaldi. Leonardo da Vinci is generally credited for the earliest descriptions of diffraction
phenomena. The first written mention of the phenomenon of diffraction is due to Francisco Maria
Grimaldi (1618-1663), a Jesuit and professor of mathematics at
Bologna. In his two-volume work, published posthumously in
1665, Grimaldi relates a set of experiments which lead him to
conclude that other than rectilinear, refracted and reflected Figure 1: From the title page of
Grimaldi’s book (1665).
propagation of light, there exists a fourth mode of light which
he terms diffraction (figure 1). In particular, two experiments in the first part of the book are of
importance; the second is shown in figure 2. A cone of light is let pass through
two small apertures (CD) and (GH). The outcome of the experiment, in
contradiction to geometrical optics, is described by Grimaldi as follows:
When the light is incident on a smooth white surface it will show an
illuminated base IK notably greater than the rays would make which are
transmitted in straight lines through the two holes. This is proved as often as
the experiment is tried by observing how great the base IK is in fact and
deducing by calculation how great the base NO ought to be which is formed
by the direct rays. Further it should not be omitted that the illuminated base
IK appears in the middle suffused with pure light, and at either extremity its Figure 2: Grimaldi's
light is coloured.
drawing of his second
experiment (1665).
The word diffraction stems from the latin verb diffringere, composed of dis
and frangere, and means “to break in different directions”. Although Hooke and Newton later used the
term “inflexion” for the same phenomenon, Grimaldi’s “diffraction” has survived.
Huygens and the Wave Theory of Light. Grimaldi’s observation of diffraction clearly led him to
reject a corpuscular theory of light. He believed light to be a fluid and provides the analogy to waves
generated by throwing a stone into water. Much of the history of diffraction theory in the following
two centuries is closely interwoven with the debate about the nature of light. Robert Hooke (16351703) in his Micrographia (1665) proposed that light has a wave character, describing it as a rapid
vibrational (longitudinal) motion of the medium, an idea he deduced from the observation of coloured
interference patterns. Hooke’s famous contemporary, Isaac Newton (1642-1727), remained much
more ambivalent about the nature of light. At first, his dispersion experiments made him conclude that
the colours he observed are related to corpuscles of light exciting the ether in characteristic vibrations.
Later in life, having failed to explain the rectilinear propagation of light in terms of waves
(geometrical optics), he seemed to embrace the particle theory more than the wave theory. However,
at the same time as Newton was at work in England, Christiaan Huygens (1629-1695), in the
Netherlands, was developing a form of wave-theory now known as Huygens’ Principle (or
construction). Huygens correctly predicted that light slows down on entering more dense media
(which Descartes, Hooke and Newton had been unable to explain). Using Huygens’ construction he
derived the laws of reflection and refraction, and went as far as explaining double refraction in calcite.
(He also discovered polarisation).
Huygens' original construction is more reminiscent of scattering theory and mechanical vibrations
than of light propagation as we know it today. In other words the wave disturbances used by Huygens
are single longitudinal pulses or shock-waves such as in an elastic medium. In particular, Huygens
makes no reference to the phase of wave motion - which explains why his theory is unable to explain
interference and diffraction effects. On this evidence, then, Huygens did not regard light as a train of
waves (or even a transverse wave):
But although we do not understand the cause of the elasticity, we cannot fail to observe that most
bodies possess this property: it is not unnatural, therefore, to suppose that it is a characteristic also
of the small, invisible particles of the ether. If, indeed, one looks for some other mode of
accounting for the gradual propagation of light, he will have difficulty in finding one better adapted
than elasticity to explain the fact of uniform speed. (Huygens, p. 18)
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On page 19, next to the figure of the candle, Huygens turns to the issue of propagation, and the idea of
secondary wavelets begins to emerge:
…it follows from what has already been said concerning the production of light
that each point of a luminous body, such as the sun, a candle, or a piece of
burning carbon, gives rise to its own waves, and is the centre of these waves.
Thus if A, B and C represent different points in a candle flame, concentric circles
described about each of these points will represent the waves to which they give
rise. And the same is true for all the points on the surface and within the flame.
(Huygens, p. 19)
More clearly a little later:
Figure 3: From
In considering the propagation of waves, we must remember that each particle of Huygens, p.19
the medium through which the wave spreads does not communicate its motion only to that
neighbour which lies in the straight line drawn from the luminous point, but shares it with all the
particles which touch it and resist its motion. Each particle is thus to be considered as the centre of
a wave. (Huygens, p. 21)
Without mentioning the word interference, this phenomenon is explicitly forbidden between the
“enormous number of waves” that are “crossing one another without confusion and without disturbing
one another” (Hugyens, p. 19). Although one might expect the secondary
waves to spread out in all directions, giving rise to diffractions, this is
denied by Huygens:
For instance, that part of the wave BG, whose centre is the luminous
point A, develops into the arc CE, limited by the straight lines, ABC and
AGE. For although the secondary waves produced by the particles lying
within the space CAE may spread to the region outside, nevertheless
they do not combine at the same instant to produce one single wave
limiting the motion and lying in their circumference CE which is their Figure 4: From Huygens,
common tangent. This explains the fact that light, provided its rays are p.21
not reflected or refracted, always travels in straight lines, so that no body is illuminated by it unless
the straight-line patch from the source to this body is unobstructed. (Huygens, p. 21)
Oddly enough, this explanation uses a concept which can be described as lack of constructive
interference. Another problem of Huygens’ theory is that it does not prevent the “backward”
propagation of the secondary wavelets, as Huygens was well aware (in the modern theory based on
Fresnel and Kirchhoff this is prevented by the obliquity factor).
The combination of Huygens’ Principle with the phenomenon of interference allows for a theory that
can explain rectilinear propagation of light, the laws of reflection and refraction, as well as the laws of
diffraction. Thus, the earliest scientific solution to the diffraction problem is profoundly based on
Christiaan Huygens’ work from 1678 (published in 1690).
Young and Interference. Thomas Young (1773-1829) made two significant contributions to the
development of diffraction theory. The best known is his experiment, now generally called Young's
experiment, in which he gave “a proof of the general law of the interference of two portions of light''
(all quotes from Young, 1804). Young gave this demonstration on November 24, 1803, to the Royal
Society of London in a lecture entitled “Experiments and Calculations relative to Physical Optics''.
This lecture was published in 1804 in the Philosophical Transactions. It is perhaps less well known
that Young did not, in fact, use the historic “double-slit'' experiment to show that light has wave-like
behaviour and that it can interfere and be diffracted. Young's experiment was much simpler and “may
be repeated with great ease whenever the sun shines, and without any other apparatus than is at hand
to every one''. The “apparatus'' used by Young was a “slip of card, about one-thirtieth of an inch in
breadth'' which Young held edgewise into the sunbeam which was made to enter the room via a
“looking-glass'', in other words, a mirror, and a small hole in a “piece of thick paper'' that Young had
put in front of the window shutter that was also “perforated with a fine needle''. If the beam of light
has a diameter slightly larger than the thickness of the card then the beam is split into two. Young
observed that “the shadow itself was divided by similar parallel fringes'' but in such a way as to leave
the “middle of the shadow always white''. The 1803 lecture was the culmination of a series of
investigations that Young had started much earlier. In 1800 he published “Experiments on Sound and
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Light'' in the Philosophical Transactions of the Royal Society and presented a detailed account of his
theory of interference in the Bakerian Lecture On the Theory of Light and Colours in 1801. The twoslit experiment appears in his Lectures on Natural Philosophy of 1807:
In order that the effects of two portions of light may be thus combined, it is necessary that they be
derived from the same origin, and that they arrive at the same point by different paths, in directions
not much deviating from one another…the simplest case appears to be, when a beam of
homogeneous light falls on a screen in which there are two very small holes or slits, which may be
considered as centres of divergence, from whence the light
is diffracted in every direction. In this case, when the two
newly formed beams are received on a surface placed so as
to intercept them, their light is divided by dark stripes into
portions nearly equal, but becoming wider as the surface is
more remote from the apertures… (Young 1807)
Figure 5: From Young’s 1807 lecture.
For illustration, Young refers to a figure from his hydraulics
lecture in the same book (fig. 5) that shows the pattern of interacting waves produced by throwing two
stones simultaneously into a pond. The importance of Young's work was not fully recognized by his
contemporaries and he was even rudely attacked in several anonymously published reviews. Young’s
second significant contribution to diffraction theory is the idea of boundary diffraction waves, the precursor to the modern geometrical theory of diffraction; we shall discuss it in more detail below.
Fresnel: Synthesis of Wave Theory and Interference Principle. Huygens’ work forms the basis of
the first wave treatment of the phenomenon of diffraction by Augustin Jean Fresnel (1788-1827).
Fresnel modified the classical principle of Huygens and proposed that the secondary spherical waves
interfere amongst themselves and in fact he did so not knowing of the work that Young had done
earlier. Having published some of his results in 1816, Fresnel’s work came to fruition when, a little
later, the French Academie of Sciences announced that the Grand Prix for 1819 would be awarded for
the best work on diffraction, apparently in the hope that the submissions would provide the final proof
for a corpuscular theory of light. Fresnel won the prize with ease (there being only one other
submission); however, one of the judges, Poisson, was convinced that Fresnel’s wave theoretical
calculations could not be right. Spurred on to perform his own calculations, Poisson then predicted the
seemingly ridiculous result that a bright spot of light should appear in the centre of the shadow of a
small illuminated disk. Much to Poisson’s surprise this spot was indeed observed when Arago
performed the experiment for Fresnel and the spot is now known as “Poisson’s spot”. However, the
question of the distribution of the intensity of elementary waves upon diffraction from an aperture in
an opaque screen remained unclear to Fresnel because he only considered the contribution of those
elementary sources which belonged to the aperture.
Helmholtz, Kirchhoff, Rayleigh, Sommerfeld. Fresnel’s idea influenced all
subsequent developments of the theory of diffraction. The major step in its
development was carried out by Gustav Kirchhoff (1824-1887) in 1882 based on
mathematical developments by Green and Helmholtz. Kirchhoff showed that the
Young-Fresnel principle is an approximate form of Helmholtz’s (1859) rigorous
integral theorem and based Fresnel’s source theory on this (Kirchhoff diffraction
theory). In the case of diffraction from the aperture of a plane screen, Kirchhoff
described the field on the screen in the form of an integral over the aperture
(Kirchhoff approximation), that is, as a result of the interference of Huygen’s
secondary sources. Rayleigh (1842-1919) then used Kirchhoff’s integral to show
that the wavefield can be satisfactorily determined by the boundary conditions on
the screen which can be written in integral form (1897). With this equation, the
intensity of the elementary sources reduces to solving an integral equation. From
the work of Sommerfeld (1896) and Rayleigh (1897) it became clear that the Figure 6: From
study of diffraction of waves could formally be reduced to an ordinary boundary Young's 1807 lecture
value problem of mathematical physics, that is, to the integration of differential on diffraction.
equations with the partial derivatives or to the solution of the integral equations.
Young and Boundary Diffraction Waves. Although Fresnel’s synthesis of Huygens’ Principle with
Young’s interference came to dominate the development of diffraction theory, in fact the very first
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attempt to explain the phenomenon of the diffraction of light for a plane screen was given by Young
(1802) before Fresnel’s memoir and using a completely different mechanism. He considered the
phenomenon of diffraction to be the result of two waves – the direct, unobstructed portion of the
primary wave (now called “geometrical wave”) from the source, existing only in the illuminated zone
of propagation, and reflected waves (now called “boundary diffraction waves”) radiating from the
edge of the screen in the illuminated region and in the shadow region. The superposition of these 2
types of waves gives the diffraction pattern (fig. 6).
Modern synthesis: Maggy, Rubinovicz, Keller, Klem-Musatov. Following the success of Fresnel’s
synthesis of Huygens’ Principle with the interference phenomenon, Young rejected his alternative
explanation and it did not attract serious attention until the 20th century, when it was formulated
mathematically by Maggi (1888), and Rubinovicz (1917, 1953). They showed that by transforming
the (Kirchhoff) surface integral to a contour integral by means of Stokes’
theorem, this result can be represented as the interference of direct waves from
the sources and diffracted waves from the edge of the screen. In this way,
Kirchhoff’s diffraction theory validated Young’s attempt to explain diffraction
as the phenomenon of deviation from the laws of geometric optics, which
manifests itself in the form of diffracted waves. The modern formulation of this
theory, known as “geometrical theory of diffraction” is largely due to the
seminal work of Keller. His landmark paper from 1953 on the representation of
diffraction by rays is one of the most cited in the modern diffraction literature.
Keller (1962) notes that the theory was in part motivated by Sommerfeld's work
on diffraction by a half plane (1896) as the
Figure 7: Illustration of
the cone of diffracted
rays from a ray hitting
the edge of thin screen
obliquely (Keller, 1962)
cylindrical
wavefronts
for
normal
incidence on the edge lend themselves to a
representation by rays. Generalizing
Sommerfeld's results to oblique incidence
the wavefronts become conical, so that a single ray incident on the
edge forms a cone of diffracted rays. In analogy to reflected and
refracted rays, Keller represents the diffraction energy generated by
an incident ray by a diffracted ray (fig. 7). Diffraction by an edge is
characterized by an edge diffracted ray, whereas diffraction from a
vertex is characterized by a vertex diffracted ray. Keller generalizes
Fermat's principle to formulate the law of edge, vertex and surface
diffraction. Keller’s geometrical theory, was extended by KlemMusatov (1984) to the seismic problem, and forms the cornerstone
of the forward problem of seismic diffraction (fig. 8).
Figure 8: Theoretical seismogram for
a model of a half-plane: reflected
wave (top), edge wave and total field
(bottom). From Klem-Musatov and
Aizenberg (1985).
References
Fresnel, M. A. [1819] Mémoire sur la Diffraction de la Lumière, translated into English by Henry Crew, “The Wave Theory
of Light, Memoirs of Huygens, Young and Fresnel”, American Book Company, 1900, New York, Cincinnati, Chicago.
Grimaldi, F.M. [1665] Physicomathesis de lumine, coloribus et iride, Bononiae. Translation from Latin by Klaus Helbig.
Helmholtz, H. [1859] Theorie der Luftschwingungen in Röhren mit offenen Enden: Journal für die reine und angewandte
Mathematik (Crelle's Journal), 57, 1-72.
Huygens, C. [1690] Traité de la Lumière. First three chapters translated into English by Henry Crew, op.cit.
Keller, J. B. [1962]. A geometrical theory of diffraction, J. Opt. Soc. Am., 52, 116-130.
Kirchhoff, G. [1882] Berliner Sitzungsbericht, p.641. Ann. der Physik 18 p.663. Vorlesungen uber mathematische Physik 2
Klem-Musatov, K. D. and Aizenberg, A. M. [1984]. Ray method and the theory of edge waves. Geophys. J. R. Astron. Soc.,
79, 35-50.
Maggi, G. A. [1888] Sulla propagazione libera e perturbata delle onde luminose in un mezzo isotropo, Ann. Mat. 16 , 21–48.
Rayleigh, Lord (J. W. Strutt) [1897] On the passage of waves through apertures in plane screens: Philosophical Magazine,
43, 259-271, reprinted in Rayleigh’s Scientific Papers, volume 4, 283-296.
Rubinowicz A., Ann. d. Physik 53, 257 (1917); 73 (1924); 81, 153 (1926); Acta Phys. Polon. 12, 225 (1953).
Sommerfeld, A. [1896] Mathematische Theorie der Diffraction: Mathematische Annalen, 47, 317-374.
Young T. [1802] On the Theory of Light and Colours, Bakerian Lecture 1801, Phil. Trans. Roy. Soc. 20, 26.
Young, T. [1804] The Bakerian Lecture: Experiments and Calculations relative to physical Optics, Philosophical
Transactions of the Royal Society of London.
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