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An exact decomposition of the diffracted field into a direct wave and a boundary diffraction wave is obtained for an incident inhomogeneous wave, namely, the complex-source-point spherical wave. Our result, in the paraxial approximation, is consistent with already published results on the diffraction of a Gaussian beam

Bélanger, Pierre-André

Couture, Marc

446 J. Opt. Soc. Am./Vol. 73, No. 4/April 1983
Boundary diffraction of an inhomogeneous wave
Pierre-A. Belanger
Laboratoire de Recherches en Optique et Laser, D6partement de Physique, Universit6 Laval, Qu6bec GlK 7P4,
Canada
Marc Couture
Coll6ge d'Enseignement Gen6ral et Professionnel de Rimouski 60, de 1'Evecher Ouest, Rimouski, Canada
Received August 24, 1982; revised manuscript received November 20, 1982
An exact decomposition of the diffracted field into a direct wave and a boundary diffraction wave is obtained for
an incident inhomogeneous wave, namely, the complex-source-point spherical wave. Our result, in the paraxial
approximation, is consistent with already published results on the diffraction of a Gaussian beam.
1. INTRODUCTION
The idea of the boundary-diffraction wave is primarily due
to Sir Thomas Young, who suggested in 1801' that diffraction
caused by the presence of an obstacle in the path of light could
be considered interference between the direct light and light
reflected at each point of the boundary of the diffracting body.
Young's ideas about diffraction were expressed only in a
qualitative manner and were soon forgotten, as the more-
quantitative Huygens-Fresnel model of secondary waves
began to dominate the field.
Nevertheless, the strong appeal of Young's model led some
authors to investigate further in that direction. Maggi2 and
Rubinowicz3 independently demonstrated that, in the case
of diffraction by an aperture in an opaque screen, the Kirch-
hoff integral (a mathematical formulation of the Huygens-
Fresnel physical model) leads, for plane or spherical waves
impinging upon the aperture, to a decomposition of the type
suggested by Young, called the boundary-diffraction-wave
(BDW) theory..
Some years ago, Miyamoto and Wolf4'5 obtained a gener-
alization of that theory. They noted that the integrand ap-
pearing in the Kirchhoff diffraction formula can be written
as the curl of a potential vector W(P, Q), where P is the ob-
servation point and Q is an (unspecified) point in the aperture.
Thereafter they showed that the diffracted field at point P,
U(P), may be expressed as the sum of two terms. The first,
UB(P), is the contour integral of W along the boundary of the
aperture. The second, US(P), is shown to reduce to the sum
of line integrals on infinitesimal paths surrounding the
discontinuities of W in the aperture. They were able to obtain
a general expression for the vector potential W and to show
that US(P) is indeed equal to the direct (unperturbed) inci-
dent field for the cases already discussed by Maggi and Ru-
binowicz as well as in the case of a general incident wave in the
geometrical-optics approximation.
It is important to realize, as these authors suggest else-
where,6 that this alternative approach is not only of academic
interest. Replacing double integrals with contour integrals
can be quite time saving, especially if one considers today's
complicated cascaded optical systems or related ones, such
as optical resonators.
The purpose of the present paper is to present an exact
decomposition of the diffracted field, for an inhomogeneous
wave this time: the complex-source-point spherical wave
(CSPSW). The CSPSW is a spherical wave whose origin is
a point with complex coordinates rather than real ones. Its
expression is easily shown to be an exact solution of the
Helmholtz equation. This solution was first mentioned by
Deschamps,7 and its properties were extensively analyzed later
by Felsen.8 It was shown recently 9 that the CSPSW can be
viewed as the result of the summation of all the corrections
to the paraxial Gaussian beam. This implies that the present
work is directly related to the problem of diffraction of a laser
beam by an aperture.
In Section 2 we derive an expression for the vector potential
W associated with the CSPSW. Then, in Section 3, we find
the locations of the singularities of W in the plane of the ap-
erture and discuss the question of the lit and shadowed regions
in the diffraction plane. In Section 4, the contribution from
the integrals around these singularities is shown to give exactly
the unperturbed incident field. In Section 5, we compare our
exact result with those of Otis'0 and Takenaka et al. ,11,12 who
studied the diffraction of a Gaussian beam in the geometri-
cal-optics approximation.
2. VECTOR POTENTIAL W(P, Q)
As was explained earlier, according to the BDW theory the
diffracted field U(P) at an observation point P (Fig. 1) can be
split into two parts:
U(P) = UB(P) + US(P). (2.1)
The first term on the right-hand side of Eq. (2.1), UB(P),
is a contribution from the edge of the diffracting aperture P.
Its complete expression [see Ref. 5, Eq. (2.6)] is
UB(P) = 9 W(PQ) *dI, (2.2)
0030-3941/83/040446-05$01.00 © 1983 Optical Society of America
P.-A. Belanger and M. Couture
Vol. 73, No. 4/April 1983/J. Opt. Soc. Am. 447
r
Fig. 1. Illustrating notation relating to the diffracted field at point
P by an aperture r located in the plane z' from the origin 0.
where the integration is made counterclockwise (viewed from
P) and where the vector potential W [see Ref. 4, Eq. (4.15)]
is
W(P, Q) = W(r, r')
W_ eiks o
= Wt + S r eiksvUi(r' + p&)dA, (2.3)
4 7rs f-
Ui being the incident field and f a unit vector in the direction
of s = r'- r. The term W_ is discussed later.
The other term Us(P) arises from the singularities of W(P,
Q) in the aperture and is discussed in Section 3. Here we take
the incident field Ui to be the CSPSW of Ref. 7,
Ui(r) = exp[ikR, (r)]
R,, (r) (2.4)
where RC is the complex length of the vector R, that is,' 3
RC = (RC RC)1.2,
R,(r) = r - izoi.
This expression is a solution of the Helmholtz equation
obeying the Sommerfeld radiation condition. According to
Miyamoto and Wolf,4 for such a wave the term W_ in the ex-
pression for W is equal to zero. The remaining part of that
expression is evaluated by first calculating v Ui:
vUi(r' + g) = R, (i- ) epiR,) (2.5)
where
R, = Rj (r' + pf) = r' + g& - izoi.
As g X s vanishes, the expression for W(r, r') becomes
W(r, r') = s & X R, (r')
47rs
x fj (k _) exp (B, 2 dA, (2.6)
where RC under the integral stands again for RC (r + pA).
In complete analogy [see Ref. 4, Eq. (5.7)] with the regular
spherical wave, the integrand in Eq. (2.6) is shown to be the
total derivative of
and Wolf [see Ref. 4, Eq. (5.10)] with the complex vector
R, (r') replacing the real vector r'. However, as the complex
nature of the radius RC changes the characteristics of the
spherical wave profoundly, making it nonuniform, the
singularities of W will be considerably different from those
of the regular spherical wave. This point is examined in
Section 3.
3. SINGULARITIES OF W IN THE APERTURE
The second term US(P) contributing to the total diffracted
field is made of the sum of line integrals on paths encircling
the points of discontinuity of W [see Ref. 5, Eq. (2.7)] in the
aperture
Us(P) = U1S(r) = Z W(r, r') -dl,
i rj
(3.1)
where the rj's are infinitesimal circular paths around the
points of discontinuity of W, the integration being made
clockwise when viewed from P.
The locations of these discontinuities are obtained by
making the denominator of the expression of W(r, r') equal
zero, that is,
R,(r') = 0
or
Rc (r') + s * R, (r') = 0.
(3.2)
(3.3)
The singularity resulting from Eq. (3.2) corresponds to the
singularity in the incident field itself and is excluded from the
present analysis by putting the plane of the aperture on the
right-hand side of the plane z = 0, where this singularity
lies. 14
The other singularity, described by Eq. (3.3), is completely
different from that of the regular spherical wave. In the latter
case, the location of the singularity is the point where a
straight line going from the origin to the observation point,
called the direct ray, intersects the plane of the aperture. This
is not true in the present case. To find the new location of the
singularity, one has to realize that Eq. (3.3), containing com-
plex quantities, represents in fact two equations:
Re[Rjr')] =-r' *
Im[R, (r')] = z0& -i.
(3.4)
(3.5)
Written in terms of the cylindrical coordinates of points
P(p, z, i) and Q(p', z', 0') (Fig. 2) and then squared, Eqs. (3.4)
exp[ik(R, + p)]
RCJR + I * Rj]
This enables us to integrate directly, obtaining
W(r r') = eik [& X R, (r')]exp[ikR,, (r')]
4wzs Rc (r')[RB,(r') + § * RB(r')]
(2.7)
(2.8)
This expression is identical with that obtained by Miyamoto
-- constant 0 line on hyperboloid Z plane
(observation plane)
Fig. 2. Illustrating the two lines of singularity Q5 P and Q2P lying
on the hyperboloid.
P.-A. Belanger and M. Couture
448 J. Opt. Soc. Am./Vol. 73, No. 4/April 1983
and (3.5) become, respectively,
p'2Z + p2z' - pp'(z ±z')cos(4) - ) = 0, (3.6)
(pp')2 sin'N4 - 4/) + (pZ')2 + (p'Z)2 - 2zz'pp' cos(o - 0/)
= Zr 21p' 2 + p2 - 2pp' cos(0 - 0')]. (3.7)
These two equations can be combined and, after tedious but
straightforward calculation, shown to define two lines of
singularity, analogous to the direct rays mentioned earlier,
for each observation point P. The singularities are again lo-
cated at the intersection between these lines and the plane of
the aperture.
These two lines lie on the same hyperboloid of revolution
whose annular focus is a circle of radius zo located in the plane
z = 0. This hyperboloid is obtained by eliminating the trig-
onometric functions in Eq. (3.7) with the help of Eq. (3.6).
The resulting equation is the equation of the hyperboloid but
with its parameters evaluated in the plane z'. The same hy-
perboloid is more simply written with the aid of Eqs. (3.8) and
(3.9):
p2 z 2  = 1.
a2 z0
2
-a 2
(3.8)
The specification of the observation point determines the
parameter a; in turn, the distance p' from the axis to the point
where the lines of singularity intersect the plane of the aper-
ture located at z' is evaluated with'5
Pe2 z' 2
- = 1.
a2 Z0 2 -a 2
(3.9)
Furthermore, these lines do not follow constant-angle paths
on the hyperboloid. For any position along z axis, they are
symmetrically located on each side of the constant -0 line
going through P; their angle O' in the plane of the aperture is
determined by Eq. (3.6), namely,
P'2z + p2z'
cos~k - 4/) - pp'(z + Z')I (3.10)
which gives two values that are equal but in opposite sign for
the relative angle a = p-4'. This angle may be shown always
to be less than or equal to 7r/2, as the observation point recedes
to infinity.
Figure 2 illustrates the two lines of singularity (PQ, and
PQ2) associated with an observation point as well as the two
points of singularity of W in the aperture (Qj and Q2)-
With these results it is now easy to identify the lit and
shadowed regions in the diffraction area. The shadowed re-
x
Fig. 3. Illustrating notation for the integration around infinitesimal
circular contours rP and r2 around the singularity Q, and Q2.
gion is defined as the region in space where an observation
point defines no singularity of W inside the aperture; other-
wise it is called the lit region. For an aperture with circular
symmetry, the boundary of the two regions is well defined and
corresponds to a hyperboloid of the type mentioned earlier,
which coincides with the edge of the aperture in the z' plane.
Note that this boundary does not exactly coincide, as pre-
dicted by geometrical optics, 4 with a normal to the surfaces
of constant phase of the wave. However, it may be shown
that, as soon as one considers points some wavelengths away
from the annulus of singularity of the CSPSW (located at z
= 0, p = zo), the two nearly coincide.
4. CONTRIBUTION FROM THE
SINGULARITIES
We now calculate the contribution US(P) from the singular-
ities to the total diffracted field in the lit region. We must use
Eq. (3.1), which reads now
US(r) = W(r, r') * dl + A W(r, r') * dl, (4.1)
where F, and F2 are infinitesimal circular contours of radius
a around points of singularity Qi and Q2 (Fig. 3).
If one recalls expression (2.8) for W, one finds that one has
to evaluate, to first order in a, the expressions
N1, 2 - dl * [s%,2 X R,(r;,2)] (numerator) (4.2)
and
D1,2 - s,2R, (r, 2) [s1,2R& (r , 2) + sj,2 -R. (r;,2)]
(denominator). (4.3)
Note that both expressions have been multiplied by sK,2.
After some calculation, one finds to first order that
NI,2 = ad0j[(z - izo)x' - (z' - izo)p]cos 0
+ (z - izo)y' sin 0} + 0(cr2)dO (4.4)
and
D1,2 = uR,(r)I[(x' - p)R,(r) + splcos 0
+ y'Rj(r)sin 01 + o(U2), (4.5)
where in both cases + (-) is for 2 (1) and where the following
relation has been used:
Rc(r) = R,(r') + s. (4.6)
Each of the two contributions to US(r) can now be written
as
lim 1 W(r, rD) . dl = exp [ikR. (r)]
a-0 f r12 4 -4R, (r)
x r A cos O + B sin 0
XJo CcosO+DsinoOM
where
A = (x' - p)(z - izo) + (z - z')p,
B = y'(z -izo),
C = (x' - p)R,(r) + sp,
D = y'R,(r),
and + (-) corresponds to 2 (1) again.
(4.7)
P.-A. Belanger and M. Couture
Vol. 73, No. 4/April 1983/J. Opt. Soc. Am. 449
Both integrals are readily evaluated by means of the residue
theorem (see Appendix A) and using the relations A = iD and
B = -iC; each of them is shown to equal 2wr. We thus finally
obtain the expected result:
US(r) = exp[ikRc(r) (4.8)
R, (r)
We therefore conclude that, when the observation point is
in the lit region, the contributions from the two points of
singularity of W in the aperture to the diffracted field add up
exactly to yield the unperturbed incident field.
5. DISCUSSION
We have thus extended the results of BDW to a more-general
case (which encompasses the spherical wave as a special case,
e.g., when zo equals zero), the inhomogeneous CSPSW. This
wave is also asymptotically equivalent, in the paraxial domain,
to the well-known Gaussian beam. 9
Otis 10 considered the application of BDW theory to such
a beam and obtained a result equivalent to ours in the paraxial
domain. In particular, he found that the shadow boundary
is given by a surface that coincides with the edge of the aper-
ture and remains always perpendicular to the constant-phase
surfaces of the wave, following hyperbolas. However, Otis 9
seems to have found only one singularity in the aperture, in-
stead of two as we did.
In two others papers, Takenaka et al.,11,1 2 examining the
same problem, indeed found two singularities in the aperture.
The locations of these singularities exactly match those that
we obtained if we use the approximation p << Iz - izd, clearly
of a paraxial nature, in Eqs. (3.8)-(3.10).
The result of Otis10 is easily explained if one notes that the
complex singularity around which he makes a contour inte-
gration is equivalent to two integrations in real space.
For the case of a spherical wave (or a plane wave) and also
in the geometrical-optics approximation,4' 5 the singularities
in the aperture of the vector potential W are exactly those
points that lie on the rays that pass through P. For the
CSPSW the interpretation is not straightforward. Even in
the paraxial approximation, in which the CSPSW becomes
the regular Gaussian beam for which the geometrical rays are
the mentioned hyperbolas, it cannot be said that the singu-
larities in the aperture lie on the rays that pass through P
unless one defines the rays as our lines of singularity that ro-
tate while propagating. This double singularity of the vector
potential W renders the separation between the lit and the
shadowed boundary less drastic. For instance, a noncircu-
lar-symmetric aperture is such that for a certain region in
space only one singularity is seen at point P. For this region
only half of the direct wave contributes to the total field. The
shadow boundary is somewhat ambiguous in real space for the
inhomogeneous waves, such as the CSPSW.
6. CONCLUSION
Using the generalization of the boundary-diffraction theory
of Miyamoto and Wolf,4' 5 we have obtained an exact decom-
position of the diffracted field in a direct wave and a boundary
wave when the incident field is the complex-source-point
spherical wave. Our result contains all the known exact de-
composition in boundary-diffraction-wave theory and could
be generalized to another incident field, such as a complex-
source-point sinusoidal wave, sin kR0 /R,. This last field is
also paraxially equivalent to the usual Gaussian beam and
does not contain the discontinuity of the complex-source-
point spherical wave.
APPENDIX A
In this appendix we evaluate the integral of Eq. (4.7),
namely,
F= 2 A cos :i B sin dM,
Jo C cosO +D sinS ' (Al)
where A = iD and B = -iC.
When the usual substitution z = ei0 is made, the integral
becomes a contour integral along unit circle in the complex
plane. It may be written as
(A2)1 (TC +)z2 C+ iDdzi f (C =FiD)Z2 +C± iD z
The integrand possesses three simple poles located at z I =
0 and at the two values z2,3 of( C i iD'1/2
= C =F iD (A3)
However, it can be shown that the following relation:
IC+iDI >IC-iDI (A4)
always holds for z > z' so that, for the upper signs, the poles
Z 2 and z 3 lie outside the unit circle. The only pole contrib-
uting to the integral is thus z 1, and the residue at that point
is easily evaluated:
Resz2 0 = 1.
On the other hand, all three poles contribute to the integral
when the lower signs are considered. The residues are found
to be
Resz=o = -1,
Resz2 = Res,~ 3 = 1.
The integral being equal to 27ri times the sum of the resi-
dues located inside the contour of integration, one sees that
its value is 27r for both choices of signs.
ACKNOWLEDGMENT
This work was made possible by the financial support of the
Ministbre de l'Education du Qu6bec and of the National Re-
search Council of Canada.
REFERENCES
1. T. Young, "Lectures on natural philosophy, diffraction," Phil.
Trans. R. Soc. London 92, 12-49 (1802).
2. G. A. Maggi, Ann. Mat. Ila, 16, 21 (1888).
3. A. Rubinowicz, "Die Beugungswelle in der Kirchhoffschen
Theorie des Beugungsercheinungen," Ann. Phys. 53, 257-278
(1917).
4. K. Miyamoto and E. Wolf, "Generalization of the Maggi-Rubi-
nowicz theory of the boundary diffraction wave-Part I," J. Opt.
Soc. Am. 52, 615-625 (1962).
P.-A. Belanger and M. Couture
450 J. Opt. Soc. Am./Vol. 73, No. 4/April 1983
5. K. Miyamoto and E. Wolf, "Generalization of the Maggi-Rubi-
nowicz theory of the boundary diffraction wave-Part II," J. Opt.
Soc. Am. 52, 626-637 (1962).
6. E. Wolf, "Some recent research on diffraction of light," in Modern
Optics (Polytechnic Institute of Brooklyn, New York, 1967), pp.
433-453.
7. G. A. Deschamps, "Gaussian beam as a bundle of complex rays,"
Electron. Lett. 7, 684-685 (1971).
8. L. B. Felsen, "Complex-source-point solutions of the field
equations and their relation to the propagation and scattering
of Gaussian beams," in Symposia Mathematica (Academic, New
York, 1976), Vol. XVIII.
9. M. Couture and P. A. Belanger, "From Gaussian beam to com-
plex-source-point-spherical wave," Phys. Rev. A 24, 355-359
(1981).
10. G. Otis, "Application of the boundary-diffraction wave theory
P.-A. Belanger and M. Couture
to Gaussian beams," J. Opt. Soc. Am. 64, 1545-1550 (1974).
11. T. Takenaka, M. Kakeya, and 0. Fukumitsu, "Asymptotic rep-
resentation of the boundary-diffraction wave for a Gaussian beam
incident on a circular aperture," J. Opt. Soc. Am. 70, 1323-1328
(1980).
12. T. Takenaka and 0. Fukumitsu, "Asymptotic representation of
the boundary-diffraction wave for a three-dimensional Gaussian
beam incident upon a Kirchhoff half-screen," J. Opt. Soc. Am.
72, 331-336 (1982).
13. z0 is the Rayleigh range 7rWo2/X of the corresponding paraxial
Gaussian beam of beam waist Wo.
14. The case in which the plane of the aperture is on the left-hand side
of the plane z = 0 could also be analyzed in a way similar to that
for a regular converging spherical wave.
15. Note that, in order that Eqs. (3.4) and (3.5) be satisfied, a must
always be smaller than z0.


Boundary diffraction of an inhomogeneous wave

Bleton, Paul

Une vérité viscérale (à partir de Mort à Crédit de L.-F. Céline)

It is shown that the paraxial Gaussian beam becomes the complex-source-point spherical wave when all-order corrections are made according to the method of Lax, Louisell, and McKnight. Apparent contradictions between previously published first-order corrections are also discussed

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Boundary diffraction of an inhomogeneous wave

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From Gaussian beam to complex-source-point spherical wave

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