On problem of the rigorous diffraction quantitative description
New data showing an inaccuracy of Kirchhoff's description for the diffraction of the limited aperture light beams are presented. A series of the known experimental facts, which did not have an unequivocal interpretation within the framework of this theory, gains a simple explanation with passag...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
1999
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irk-123456789-1202442017-06-12T03:05:09Z On problem of the rigorous diffraction quantitative description Anokhov, S. New data showing an inaccuracy of Kirchhoff's description for the diffraction of the limited aperture light beams are presented. A series of the known experimental facts, which did not have an unequivocal interpretation within the framework of this theory, gains a simple explanation with passage to conceptions of the rigorous diffraction theory. The difficulties hindering the wide application of this theory for solving practical problems are considered as well. 1999 Article On problem of the rigorous diffraction quantitative description / S. Anokhov // Semiconductor Physics Quantum Electronics & Optoelectronics. — 1999. — Т. 2, № 4. — С. 66-69. — Бібліогр.: 15 назв. — англ. 1560-8034 PACS: 42.25.K http://dspace.nbuv.gov.ua/handle/123456789/120244 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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New data showing an inaccuracy of Kirchhoff's description for the diffraction of the limited aperture light beams are presented. A series of the known experimental facts, which did not have an unequivocal interpretation within the framework of this theory, gains a simple explanation with passage to conceptions of the rigorous diffraction theory. The difficulties hindering the wide application of this theory for solving practical problems are considered as well. |
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Anokhov, S. |
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Anokhov, S. On problem of the rigorous diffraction quantitative description Semiconductor Physics Quantum Electronics & Optoelectronics |
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Anokhov, S. |
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Anokhov, S. |
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On problem of the rigorous diffraction quantitative description |
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On problem of the rigorous diffraction quantitative description |
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On problem of the rigorous diffraction quantitative description |
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On problem of the rigorous diffraction quantitative description |
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On problem of the rigorous diffraction quantitative description |
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on problem of the rigorous diffraction quantitative description |
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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1999 |
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On problem of the rigorous diffraction quantitative description / S. Anokhov // Semiconductor Physics Quantum Electronics & Optoelectronics. — 1999. — Т. 2, № 4. — С. 66-69. — Бібліогр.: 15 назв. — англ. |
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Semiconductor Physics Quantum Electronics & Optoelectronics |
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AT anokhovs onproblemoftherigorousdiffractionquantitativedescription |
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66 © 1999, Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
Semiconductor Physics, Quantum Electronics & Optoelectronics. 1999. V. 2, N 4. P. 66-69.
1. Introduction
The present article is initiated by the Wang’s publication
«On principles of diffraction» [1], in which the author, pro-
ceeding from the existing experimental facts and results of
his own investigation, comes to a conclusion about the ne-
cessity to correct the currently formed conception of dif-
fraction. His generalization concerning a set of known ef-
fects (illogical position of axial intensity maxima in the Fresnel
zone, focal spot displacement at convergent wave diffrac-
tion, etc.), quantitative explanation of which required to as-
cribe half-wave phase shift to diffracting wave edges rela-
tively to a remaining wave, was the ground for this. Accord-
ing to Wang, to correct the situation it is enough to improve
the former wave principle by including, once for all the indi-
cated phase shift for secondary waves touching with an
obstacle.
In this connection it is pertinently to remind about a
natural phase shift of a diffracted wave that figures in
the rigorous theory of diffraction and is not connected
with the action of Huygens-Fresnel principle. The fol-
lowing section of the article is devoted to discussion of
this problem. Besides this, the extra experimental facts,
which do not fit to the Kirchhoff diffraction theory, are
presented in the article. The problems hampering wide
practical distribution of the rigorous theory of the dif-
fraction are considered as well.
2. The phase shift in the diffracted wave
As is known, the formation of the modern conception of
diffraction occurred in rivalry conditions of two alternate
approaches to this phenomenon bound, accordingly, to
names of Young and Fresnel. The rigorous solution of a
problem of plane wave diffraction on an ideally conductive
half-plane obtained by Sommerfeld in 1896 [2,3] became the
decisive argument for the benefit of Young. According to it,
the resulting field can be represented by a superposition of
two components (Fig.1), namely, geometrical component AG
(the remaining part of the initial wave, that suffers disconti-
nuity and propagating further by rules of geometrical op-
tics) and diffracting or boundary wave AB – divergent wave,
geometrical center of which coincides with the edge of a
half-plane.
One of the most typical distinctive feature of the boun-
dary wave is the presence of the half-wave sudden change
in phase in it that coincides with the boundary of the geo-
metrical shadow. Thereof, these halves of the wavefront si-
tuated on different sides from the indicated boundary are
counter-phase (Fig. 1). A simple explanation of this circum-
PACS: 42.25.K
On problem of the rigorous diffraction quantitative
description
S. Anokhov
International Center «Institute of Applied Optics» of NAS of Ukraine
10-G vul. Kudryavska, 254053, Kyiv, Ukraine, phone: (38044) 212-21-58; fax: (38044) 212-48-12, e-mail: khizh@lomp.ip.kiev.ua
Abstract. New data showing an inaccuracy of Kirchhoff’s description for the diffraction of the limited
aperture light beams are presented. A series of the known experimental facts, which did not have an
unequivocal interpretation within the framework of this theory, gains a simple explanation with passage
to conceptions of the rigorous diffraction theory. The difficulties hindering the wide application of this
theory for solving practical problems are considered as well.
Keywords: diffraction, boundary wave, light beem, aperture diaphragm.
Paper received 28.09.99; revised manuscript received 30.11.99; accepted for publication 17.12.99.
Fig. 1. Schematic explanation of diffraction field structure behind the
screen based on rigorous diffraction theory.
A
A
AB G
x
Z
S.Anokhov: On problem of the rigorous diffraction quantitative...
67SQO, 2(4), 1999
stance made by Rubinovicz [4] consists in the following.
Sommerfeld solved the problem as a boundary one for
the Maxwell equations. By virtue of this, the solution ob-
tained by him, naturally, does not contain any discontinui-
ties of the field on the boundary of a geometrical shadow.
Otherwise, they should be connected to the presence of
actual charges or currents here, for occurrence of which
there are no physical grounds. As the incident wave under-
goes the discontinuity on edge of the screen, the boundary
wave should also have here the same effect, but of such
kind, at which the discontinuity of the composite field dis-
appears. The latter actually takes place due to the geometri-
cal component, and the boundary wave has opposite phase
when approaching to the boundary of a shadow from the
illuminated side and the identical phase when approaching
to it from the shadow one (Fig. 2).
The presence of such structure in the boundary wave
completely corresponds to the results of experimental ob-
servations [3-9]. Thus, at the electromagnetic wave diffrac-
tion on an arbitrary screen in the illuminated part of space
there always exists the divergent wave propagating from an
edge of the screen and π phase shifted relatively to the basic
wave. The explanation of the effects described in the paper
[1] becomes obvious with accounting it.
However, the discussed problem is not exhausted, as
there is no general quantitative description of the described
phenomenon so far. Effectivelly, it is talked about the ab-
sence of the rigorous theory of aperture diffraction that would
be capable to replace Kirchhoff’s theory, which is not so
reliable in this situation. The improvement of the wave prin-
ciple by including of half-wave phase shift for secondary
waves touching the obstacle, proposed in [1], can hardly
solve the problem.
It is enough only to pay attention to the fact that the
energy of selected counter-phase components in this case
are many orders of magnitude lower than the remaining wave
field, which excludes a possibility of their noticeable effect
on its common structure. Let’s remind, that in due time Fresnel
has met the similar problem, trying to calculate the simplified
Young’s model [10]. This circumstance, as known, has pushed
him to search other approach towards the diffraction.
3. Far field anomalies
According to the Kirchhoff theory, during the diffraction of
an arbitrary wave on an aperture, the distribution of the
field in the Fraunhofer zone is described by the Fourier trans-
forms of the initial field distribution on the aperture. This
transforms, in particular, lies in the basis of the modern theory
of aperture antennas [11,12]. However, as careful measure-
ments show, the indicated correspondence is fulfilled pre-
cisely only for the plane wave diffraction. With the devia-
tion of the amplitude distribution of a plane (quasi-plane)
wave at the aperture from homogeneous one, the resulting
Fourier-transform more and more differs from the real far
field pattern [13].
To estimate the scale of this discrepancy, let us turn to
the results of the experiment on single-mode laser beam dif-
fraction on a slit aperture, beam-waist of which was super-
posed with the plane of the slit when using a telescopic
system. The latter enables to consider the wave front of the
diffracting wave plane. Thus, we deal with the aperture dif-
fraction of a non-uniform plane wave with the Gaussian pro-
file of amplitude. The typical pattern of the far field intensity
distribution observed in this case is shown in Fig. 3. For
definiteness we shall consider such particular characteristic
of this field as angular distribution of its minima. If one ac-
cepts as the initial reference point, the equidistant sequence
of diffraction minima formed by the plane wave, and calcu-
lates the individual shifts of these minima after the replace-
ment of the plane wave by an arbitrary non-uniform wave
with the plane front, the common regularity arises [13]: the
absolute shift of minima always decreases with moving from
the center of the distribution (Fig. 4).
This rule, however, completely refutes the above men-
tioned experiment, demonstrating the opposite behavior of
actual minima of the far field, of real shift which practically
b
0 x
1
A G
c
x
A B
-0.5
0.5
à
1
0.5
0
A
x
Fig. 2. Addition (a) of amplitudes of geometrical (b) and boundary (c)
waves at edge diffraction of a plane wave.
S.Anokhov: On problem of the rigorous diffraction quantitative...
68 SQO, 2(4), 1999
always increases to the periphery of the distribution. The
objective character of such field evolution was also con-
firmed by the author’s analysis of accumulated in literature
facts that concerns the diffraction of non-uniform waves,
for several recent decades. Such deep discrepancy between
the expected and actual structures of the far field specifies
the presence of unknown physical factor at the diffraction
process, which formerly escaped from the attention of the
investigators.
To clarify its character, the computer simulation of
functions, Fourier-transforms’ shape of which would
coincide with the observed pattern of the far field, was
undertaken. As it was eventually found, the required field
distribution at the slit can be composed as the sum of two
components (Fig. 5), namely the diffracting wave itself and
π phase shifted wave, the most part of the power of which is
concentrated in a pair of separate peaks, which are clasped
to the edges of a slit [13]. For the shape and height of the
peaks shown in Fig. 5, the highest resemblance between the
calculated and observed structures of the final distribution,
is practically reached for all quantitative indexes.
At the same time, the actual existence of a similar wave
was not experimentally confirmed: in the field structure re-
corded using CCD-camera just behind the slit, any differ-
ences from initial distribution of the diffracting wave were
not found. Thus, the introduction of a new wave in this
case should be consider, only as formal technique for the
solution of the particular problem which makes it possible
to remove the arisen quantitative contradiction. The final
clearness into the situation can be brought only by the
sequential solution of this problem within the framework of
the rigorous theory of diffraction, which persist on half-wave
phase shift for a part of the wave field, as the logic link of the
diffraction process.
4. Fundamental problems of the rigorous theory
The preference that is given to the Kirchhoff’s theory, when
the practical case is under the consideration, can be explained
not only through its clearness and simplicity. Although be-
ing rehabilitated by Sommerfeld, the Young’s concept corre-
sponds to currently most rigorous quantitative approach to
the diffraction, and it has lead also to some unexpected prob-
lems. In particular, this concept is in a contradiction with the
so-called condition on a rim («a rim does not radiate»), the
idea that has been accepted for the rigorous solution of the
electrodynamics boundary-value problems [14]. As a result,
this concept has not yet found a complete understanding.
No less serious questions are caused by the analytical rep-
resentation of the diffraction field that is apparent from this
conception.
The given representation of the Sommerfeld solution in
the form of two components did not receive the complete
physical interpretation. In the area behind the screen both
geometrical and boundary waves possess an amplitude break
on the boundary of geometrical shadow. Moreover, the
boundary wave has broken derivative of the transversal
amplitude distribution and non-smooth wavefront. Being
intuitively understandable, these waves can not exist and
sin x
x
xπ
x
xa
b
3 32 21 1> >
-1 1 2 3 4 5 6 7 8-2-3-4-5-6-7-8
0
1
-d/2 d/2
x
A
Fig. 3. Distribution of a real far field at aperture (slit) diffraction of gaussian beam.
Fig. 4. The far field minima distribution for diffraction of an uniform
plane wave (a) and an arbitrary light beam (b).
Fig. 5. The amplitude profile of the combined wave on the slit estab-
lished by computer simulation (d designates the width of the slit).
S.Anokhov: On problem of the rigorous diffraction quantitative...
69SQO, 2(4), 1999
propagate as real waves. Separately they do not satisfy
wave equation, and the only their combination gives a proper
solution. However, there exists a set of reports [3-9] de-
scribing an isolation and study of the boundary wave itself,
but even up to now there is not any reasonable explanation
of the physical reality of geometrical and boundary waves,
thus putting questions about the validity of the solution.
Essentially, all the above listed moments represent al-
most insurmountable psychological barrier that severs a
rigorous diffraction theory from the practical optics, the lat-
ter always being bearing up against the physically clear
conceptions and laconic mathematical means. It is reason-
able to notice that all given above and difficult for interpre-
tation facts usually are omitted in literature on diffraction.
To the best knowledge of the author, there are few publica-
tions where this problem has been mentioned and its princi-
ple importance was acknowledged [8,15]. However, the fur-
ther serious advancement in this domain without its funda-
mental solution is hardly possible.
Conclusion
As it is clear, there are several enough serious aspects of
the diffraction, explanation of which within the frame-
work of Kirchhoff’s theory remains difficult. Besides,
the phenomena mentioned in the paper [1] and observed
in the case of the Fresnel diffraction, are necessary now
to be included to the same series of the structural peculi-
arities when considering the field formed in the
Fraunhofer zone at the aperture diffraction of a non-
uniform quasi-plane wave. The differences between the
observed field pattern and the structure calculated by
the traditional way are so significant that even require
the improvements of the model lying in the basement of
the conventional theory.
In particular, in this case it is not sufficient to have
the usual expansions of the initial wave at the slit aper-
ture onto plane waves ( Fourier transforms): for receiv-
ing the exact result it is necessary to supplement this wave
by one more π phase shifted, field maximums of which have
to correspond to minima of the diffracting wave (Fig. 5). The
Fourier-transform of such wave construction does repre-
sent the empirically found pre-image of rigorous solution of
this problem that is absent up to date.
The author is very grateful to Prof. A. Khizhiyak for fruit-
ful discussions and for his constructive remarks.
References
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(1995).
2. A. Sommerfeld, Mathematische Theorie der Diffraction// Math.
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3. M. Born, E. Volf, Principles of optics // Pergamon Press, N.Y.,
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4. A. Rubinowicz, Thomas Young and the theory of diffrac-
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5. R.W. Wood, Physical optics // MacMillan Company, N.Y. (1934)
6. S. Ganchi, An experiment on the physical reality of edge-dif-
fracted waves// Amer. J. Phys. 57, 370-373 (1989).
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