2094
Salvatore Ganci
J. Opt. Soc. Am. A/Vol. 3, No. 12/December 1986
Maggi-Rubinowicz transformation for phase apertures
Salvatore Ganci
Gruppo Nazionale Didattica della Fisica, ConsiglioNazionale delle Ricerche, Dipartimento di Fisica
dell'Universitb, Via Dodecaneso, 33, 16146 Genova, Italy
Received August 16, 1985; accepted July 28, 1986
The proof of the Maggi-Rubinowicztransformation is extended to a general phase aperture. General formulas are
established for spherical and plane waves.
JJ(an {IQexp[ik(rQ+ A)]I1iexp(ikr)
INTRODUCTION
+r |
As is known,", 2 the
Maggi-Rubinowicz
transformation
makes it possible to express the surface integral constituting
Kirchhoff's formulation of the problem of diffraction in a
line integral extended to the closed line constituting the
boundary of the aperture. The derivation of this transformation (in the case of a spherical or plane wave) has been
included in textbooks on optics for some time,3 -5 and the
problem is considered here only in the special case when one
has an aperture in an opaque screen.
-1
rQ
exp[ik(rQ + A)]
exp(ikr)]ldxdy.
LrJ
(2)
The integral extended to the aperture A can be considered in
all aspects as the Kirchhoff solution of the problem of diffraction through an aperture A in an opaque screen, and
thus Kirchhoff's boundary conditions are implicit:
As the Kirchhoff integral can easily be applied to problems referring to phase apertures, it seems obvious that such
problems
On
IA
exp(ikrQ),
-
can also be dealt with by using the Maggi-
UA(x, Y, 0) =
Rubinowicz transformation.
the plane x, y, let there be a constant phase jump. We can
then describe the expression of the electric field U of the
wave incident in the form
U(x, y, 0)
=
Aexp(ikrQ),
Qe A
reQ
-exp
Q
an
Similarly, the integral extended to aperture B may be considered in all aspects as the solution of the problem of diffraction through an aperture B (complementary to A) in an
opaque screen, and in this case also Kirchhoff's boundary
conditions are implicit:
- exp[ik(rQ + A)],
(1)
[ik (rQ + A)],
B
j
'P
UB(X, Y, 0) =
IJ
{_,an
Qe A
O.
47r
- U(xy,0)
=
|
n Ir
I{A
n
[f
- I exp(ikrQ)
rQ
a
an
r
1 exp(ikr)I}dxdy
exp(ikrQ)]
-
I exp(ikr)
I- exp(ikr)I dxdy
On r
J
0740-3232/86/122094-07$02.00
QA
OUB( o).o
(x, y, 0) -exp(ikr)
Qe B
Q
The diffracted wave field at the point P(x, y, z) is
U(x, y, ) =
QeB.
UA(xy,0)=0,
With reference to Fig. 1, a spherical wave (A/rQ)exp(ikrQ) is
incident upon the plane x, y. Between surfaces A and B of
A
Qe B
0,
MAGGI-RUBINOWICZ TRANSFORMATION
FOR PHASE APERTURES
Q
rQ
xlB 0) =0,
QEA.
To each of the previous cases one can apply the MaggiRubinowicz transformation, which, with reference to the
geometry and notation of Fig. 1, gives
4irJA
{[
OU (x, Y. 0)]
O-an
-UA(X,
r
exp(ikr)
Y, 0)W
© 1986 Optical Society of America
4
exp(ikr)]}dxdY
Vol. 3, No. 12/December 1986/J. Opt. Soc. Am. A
Salvatore Ganci
2095
S
X
J
z
ItI
I
/
/
II
I
I
\S
III
II
I
I
II
II
N
Fig. 1.
Diffraction geometry of the Maggi-Rubinowicz
transformation
applied to a phase aperture.
Case of the spherical wave.
Between
surfaces A and B let there be a constant phase jump. r is the closed-line boundary of A and B; dl is the length of an infinitesimal element of r.
The unit vectors nA and nB are orthogonal to rB and to dl.
-
IR
exp(ikR) + UB1,
A
P J
UB1,
4
IB
7rn
{[an
n
and cos(nA, s) = -cos(nB, s). Let
P J
(3)
f(n, s) =
4
I exp(ikr)
(x, Y, 0)
cos(nA,s)
( R
Jr
exp(ikR) +
1
- UB(X,Y
"'"
-
IR
) a'n
a [rr exp(ikr)I|dxdy
P J
exp[ik(R + A)]+ UB2,
UB2,
U(x,y,z) =
4ir
[1- exp(ikA)]
exp ik(rB + s)f(n, s)dl,
Pe J
r rBS
A-expik(R + A) + A [1 - exp(ikA)]
R r s exp
4r
PEJ
(4)
A
sin(rB, dl),
1 + cos(rB, s)
JrI
expik(rB+ s)f(n, s)dl,
Pe J
Pe J
(7)
where
U
A | 1 exp[ik(rB + s)]
A
sin(rB, dl)dl,
47r r rBs
1 + cos(rB, s)
(5)
UB2= A
X sin(rB, dl)dl,
The plane wave can be treated in the same way,6 and we
obtain similar relations.
More interesting (for extension to Fraunhofer diffraction)
is the special case of a plane monochromatic
1 exp[ik('r + s + )]cos(nB, s)
4T r rBs
where R is the length of the geometrical ray from S to P.
wave normally
incident upon a phase aperture. With reference to the geometry and notation of Fig. 2, the expressions of the bound-
1 + cos(rB, S)
(6)
ary waves UB1 and UB2 are
2096
J. Opt. Soc. Am. A/Vol. 3, No. 12/December 1986
Salvatore Ganci
X
J
I'.,
I
I
I/
P
I
'%
I
I
j
II
1
%
%
Fig. 2.
Diffraction geometry of the Maggi-Rubinowicz transformation
applied to a phase aperture.
Case of a plane monochromatic wave of
amplitude A traveling in the direction of the z axis (normal incidence). The unit vectors nA and nB are orthogonal to dl and to the z axis.
UB1 =
A
-
I
4w ir
EXAMPLES OF APPLICATIONS
IF
cos(nA,s)1
exp(iks)
_ dl,
-
sI1+
(8)
cos(s,z)J
The Single Phase Slit
UB2 =A
[-exp ik( + ) cos(nB,s)
sA
1 + cos(s z)
47r
dl.
(9)
First, we consider the classical problem of a half-plane in the
special case of a normally incident plane monochromatic
wave.
With reference to Fig. 3, let a,
Let f(n; s)
=
cos(nA, s)/1 + cos(s, z), which again is
3,and y be the direction
cosines of s and 1, 0, and 0 be the direction cosines of n. The
Maggi-Rubinowicz transformation gives
UB =
1[
)
UB=II exp (iks)
A exp(ikz) + A [1 - exp(ikA)]
4w
j[
U(X' ,yZ)
exp(iks)f(n, s)] dl,
P
J
fF1exp(iks)
eS
A7r
(10)
=A
A exp ik(z + A) + -
4 r Jr,
[1 - exp(ikA)]
4w
Jr[-exp(iks)f(n,
s)]dl,
PC J
A simple inspection of Eqs. (7) and (10) shows that one does
not have a diffracted field whenever the phase jump is 2mwr
(m = 0, 1, 2, .. .). On the other hand, we have the maximum
A
1
fs (1 + Cos
UB- 4r
l -exp(iks)
2
22
JrLXs
fX +f,
A
4w
from the same transformation applied to the corresponding aperture in an opaque screen because of the factor
[1 - exp(ikA)].
(11)
)1 dl.
32]-
frequencies), Eq. (11) gives
phase jump is (2m + 1)7r (m = 0, 1, 2, ... ). The Maggi-
Rubinowicz transformation applied to a phase aperture thus
2(1-
a2 +
If we consider the diffracted wave field at P(x, 0, z) in the
approximation so- 7r/2, and let f = IX and f = /X (spatial
contrast of the fringes in the observation plane whenever the
differs
1dl
a
a
+'Y
4wr .rS
dl
fx
f+7/2[
exp (ik So
(fx2 + fy 2 ) i- 7 /2l
cos /
1 + os l do.
cos
J
(12)
The method of stationary phase7 is applied to Eq. (11), and
we find for the boundary wave at P the remarkable result
Vol. 3, No. 12/December 1986/J. Opt. Soc. Am. A
Salvatore Ganci
Y
So
z
P
Fig. 3. Diffraction geometry of the Maggi-Rubinowicztransformation applied to the half-plane problem.
A X
B
I~'I2
.
-
A
I
_
_
_
So
_
_
_
1
--
_
-
S
-o.
-
so2
---
-I/2
B
Fig. 4.
Diffraction geometry of the Maggi-Rubinowicz transformation
applied to a single phase slit.
-I
2097
2098
J. Opt. Soc. Am. A/Vol. 3, No. 12/December 1986
Salvatore Ganci
A
y
%s
S
'A
I
x
t
I
Fig. 5.
A
1
UB 2-
S
A polygonal phase aperture with a constant phase jump between A and B.
exp[i(kso + r/4)].
(13)
Obviously, if the direction cosines of n are -1, 0, and 0, Eq.
(13) changes sign.
With reference to Fig. 4, the single phase slit has a transmission function
Polygonal Phase Aperture
In the special case of the Fraunhofer approximation, the
Maggi-Rubinowicz
transformation
is a powerful method for
calculation of the diffracted wave field through a polygonal
aperture in an opaque screen.8 The boundary wave from
the ith side is simply the unidimensional Fourier transform
of the x projection of this side [Eq. (4) of Ref. 8] or the
unidimensional Fourier transform of the side if this side is
t(x,
) = {exp(ikA),
orthogonal to the x axis [Eq. (6) of Ref. 8].
In the special case of a polygonal phase aperture (Fig. 5),
[x: > 1/2
Equation (10) and relation (13) directly applied to this case
give
UB = UBI + UB2 = A [1 2w
the factor [1 - exp(ikA)] is taken into account in the use of
Eqs. (4) and (6) of Ref. 8.
An interesting application is given in the following subsec-
tion.
exp(ikA)exp(ir/4)
Young's Double-Phase-Aperture Interference
x
[exp(iksol)- exp(ikO2)].
In the Fraunhofer approximation this is
SO
Sol
S02
With reference to Fig. 6, a double-square phase aperture of
width is represented. Between A and B and A' and B let
there be a constant phase jumps kAl and kA2.
Equation
exp(iks0 2 ) = exp i(so,
+
(10), in which the line-integral
calculation
is
performed, yields a set of Fourier-transform relations as
observed in the previous example. Let f and f, be the
spatial frequencies, defined by
Xlxfx).
Then
f = cos O/X xO/Xz,
UB = A
~Xs
0
[1 - exp(ikA)] exp [ik (0
+
2
Sin(wxf
1JfX
fx = Cos 0- tx,/Xz,
X
)
f +
4
y = coso/X
(14)
where xOis the abscissa of the observation point P.
The diffracted wave field at point P is the superposition of
Yo/Xz.
The boundary wave from each side of the phase aperture in
the direction r identified by giving it the spatial frequencies
fA,and fy following Eqs. (4) and (6) of Ref. 8. Let
K = - 2X exp(ikz)exp [ik xo2 +Y02
UB and the geometrical wave A exp(ikz), or A exp[ik(z + A)].
This conclusion is fully consistent with the results obtained
by using the Kirchhoff or the Rayleigh-Sommerfeld formulation of the problem.
UB1 a = - K
_______
/x2
+
exp(-iwf,1)exp(iwf.,d)1 sin(wf1),
7f.,
wy
Vol. 3, No. 12/December 1986/J. Opt. Soc. Am. A
Salvatore Ganci
UB2a = K
2
2099
U(P) = UG + [1 - exp(ikA)] UBA + [1 - exp(ikA2 )] UBA',
exp(iwf
1)exp(iwfd)lsin(fyl)
+ / 2fX2
7r~~fy
1
+ fy 2
(15)
where
UB3.= K 2 f exp(iwfxl)exp(irfxd)l f ,
wfXl
f/ + f2
a
Inwl)
+f2 exp(-iwf.l)exp(iwfd)l
UB4.= -K f2
Pc L
'A exp(ikz),
UG =
A exp[ik(z + Al)],
P E LA
A exp[ik(z + A 2 )],
P E LA'
and
srf
UB1 = -K 2 +/2 exp(-irfxl)exp(-irfxd)l
7rfxl~w/~
fX2 + f,2
UB
'K2
UB2a =
_____
K2+
f
2
UBA=K exp(-i7rfd)2 Sin(lf)
sin(f)
U~~exn(-Lwfd'fIlr~f
wf l
exp(iwx/1)exp(-iwfd)l sin(wfyl)
UBA'= K
w(i
fd)12 sin(rf)
sin(1/fy)
Simple inspection of Eq. (15) shows that one does not have a
diffracted field whenever exp(ikA,) = exp(ikA 2 ) = 1. If
exp (ikA,) = exp(ikA2 ) = exp(ikA), the boundary-wave con-
UB3.'= K 2
/2+ f 22exp(iwfx1)exp(-iwrfd)1 wrfl )
tribution to the diffracted wave field is
2
UB4 a.=-K
/X2+
exp(-irfx1)exp(-irfxd)l nfw/fy
4Y2
UB = K[1 - exp(ikA)]l 2
lfx
wl/X
s
fY cos(7rfxd). (16)
wl/y
For each aperture, the line integral in Eq. (10) is the sum
Aside from the amplitude and phase factors, this expression
of the boundary wave is the classical result of the Fourier-
UBA = UB1a + UB2a + UB3a + UB4a,
optics treatment of Young's double-slit experiment. 9 A
simple inspection of Eq. (16) shows a maximum of fringing
effects in the diffracted wave field when exp(ikA) = -1.
UBA' = UBla. + UB2a + UB3a'+ UB4a'-
The diffracted wave field at point P is
y
I
I
X
i
L
la
A1
2a
3a
I I
I
I -
L
I II ",
ZI-
LA
Fig. 6.
Diffraction geometry for Young's double-phase-aperture
constant phase jump beween A' and B.
interference.
kAl is the constant phase jump between A and B; kA2 is the
2100
J. Opt. Soc. Am. A/Vol. 3, No. 12/December 1986
Salvatore Ganci
REFERENCES AND NOTES
6. The corresponding explicit expression of the boundary wave can
1. G. A. Maggi, "Sulla propagazione libera e perturbata delle onde
luminose in un mezzo isotropo," Ann. Matematica 16, 21-48
(1888).
2. A. Rubinowicz, "Zur Kirchhoffschen Beugungstheorie," Ann.
Phys. 73, 339-364 (1924).
7. M. Born and E. Wolf,Principles of Optics, 6th ed. (Pergamon,
3. A.Rubinowicz,Die Beugunswelle in der Kirchhoffschen Theorie
der Beugung, 2nd ed. (Springer-Verlag, Berlin, 1966).
4. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon,
Oxford, 1980), pp. 449-453.
5. A. Sommerfeld, Optics (Academic, New York, 1954), pp. 311318.
be found in Ref. 3, Chap. III.
Oxford, 1980), pp. 752-753.
8. S. Ganci, "Simple derivation of formulas for Fraunhofer diffrac-
tion at polygonal apertures from Maggi-Rubinowicztransformation," J. Opt. Soc. Am. A 1, 559-561 (1984).
9. One of the referees suggested this example and kindly drew my
attention to the fact that this treatment may have application in
problems involvingsegmented optics or phased telescope arrays.