1 September 1999
Optics Communications 168 Ž1999. 157–160
www.elsevier.comrlocateroptcom
Parity and the self-imaging phenomenon
B. Ruiz
a,b
, S. Granieri
a,b,)
, H. Rabal
a,b
a
b
Centro de InÕestigaciones Opticas (CIOp), P.O. Box 124, 1900 La Plata, Argentina
UID OPTIMO, Facultad de Ingenierıa,
´ UniÕersidad Nacional de La Plata, La Plata, Argentina
Received 9 March 1999; received in revised form 8 June 1999; accepted 20 June 1999
Abstract
The parity of self-images obtained at a half of Talbot distance is investigated. Conditions for inversion are found. A
subset of the functions fulfilling Montgomery’s condition, namely those composed of sinusoids with frequencies in the ratio
of the square root of odd numbers, replicate in inverted form. Some examples are commented, numerically simulated and
verified by experiment. q 1999 Published by Elsevier Science B.V. All rights reserved.
Keywords: Diffraction; Self-imaging
1. Introduction
Self-images were discovered by Talbot w1x and
extensively investigated thereafter w2,3x. The most
usual example used, both in theory and experiment,
is that of Ronchi rulings. It is, gratings with square
transmittance profile. However, gratings with rectangular profile and general non-binary profiles were
also analised w4–6x.
When considering non-binary profiles, the question arises about if self images are inverted or not.
Actually, proper self-images are not, by definition,
inverted. So, we must consider other cases, as are
the, so called, negatiÕe self-images, occurring half
way between Talbot images. The name negative
self-images is inaccurate as, in fact, these images are
not contrast reversed replicas of the original grating
but half period shifted versions. In the popular Ronchi
)
Corresponding author. Fax: q54 221 471 2771; e-mail:
granieri@topos.fisica.unlp.edu.as
ruling case, shifted and contrast reversed possibilities
coincide, this being, probably, the origin of the
confusing name. But when rectangular gratings of
different duty cycle are considered the correct result
is apparent. Something similar occurs with the parity
of the image, as it is still obscure even in rectangular
gratings. Periodic binary profiles must show even
parity. Odd parity requires negative parts and physical feasibility Žor easy feasibility. indicates that a
certain constant bias is needed to instrument gratings
with positive transmittance. But then, this leads to
gratings with multiple gray levels. So, to look for the
answer to the question of the inversion we must look
into non-binary profiles.
In this paper we find the conditions for optical
fields inversion by free propagation. In Section 2 the
theoretical aspects are considered and a one-dimensional case is analised. In Section 3 the obtained
conditions are tested with computer simulations and
the result is demonstrated by means of an experiment.
0030-4018r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved.
PII: S 0 0 3 0 - 4 0 1 8 Ž 9 9 . 0 0 3 2 9 - 6
B. Ruiz et al.r Optics Communications 168 (1999) 157–160
158
2. Theory
Taking into account Eq. Ž7., Eq. Ž3. can be rewritten
as
To look for conditions for inverted self-images we
are going to start by looking at the solutions of the
Helmholtz equation
u Ž x , y ; z . s Ý u m Ž x , y . exp i k 2 y k m2 z .
= 2 u Ž x , y ; z . q k 2 u Ž x , y ; z . s 0,
with u mŽ x, y . solution of Eq. Ž4.. If we restrict to the
paraxial approximation, the period p becomes
Ž 1.
where uŽ x, y; z . is the field distribution,
E
2
2
= '
E x2
E
q
2
E y2
E
q
ps
2
E z2
,
Ž 2.
is the Laplacian operator, k s 2prl is the wave
number and l is the wavelength of the light. In order
to satisfy the self-imaging condition uŽ x, y; z . should
be periodic in z. Thus, the complex field can be
expressed as
ž
u Ž x , y ; z . s Ý u m Ž x , y . exp i
m
2p
p
ž(
m
4p km
k m2
Ž 9.
/
,
Ž 10 .
then the allowed k m values are proportional to the
square root of integer numbers and Eq. Ž3. takes the
form
ž
u Ž x , y ; z . s exp Ž ikz . Ý u m Ž x , y . exp yi
m
k m2
2k
/
z .
Ž 11 .
/
mz ,
Ž 3.
In this approximation, due to the expŽ ikz . phase
factor, only the intensity of uŽ x, y; z . is a periodic
function of z.
If the boundary condition in the plane z s 0 is
where p is the z-period. Notice that p is the Talbot
distance when uŽ x, y;0. is a periodic function of the
transversal coordinates Ž x, y .. We are going to call it
the Talbot distance even for other more general
situations giving rise to self images.
When this expression is replaced in the Helmholtz
equation, Eq. Ž1., it results in
and we want to find the components that produce
inverted images under propagation, then we should
impose that
=t 2 u m Ž x , y . q k m2 u m Ž x , y . s 0,
u Ž x , y ; z 0 . s u 0 Ž yx ,y y . ,
Ž 4.
where
E x2
E2
q
E y2
Ž 5.
is the Laplacian operator in the transverse coordinates x, y, and
k m2 s k 2 y
4p 2
p2
m2 .
Ž 6.
2
2
m
.
Ž 7.
Eq. Ž7. should remain valid for all m such that
k m - k.
p
2
.
Ž 13 .
Ž 14 .
Using Eq. Ž3., the condition can be written as
Ý u m Ž x , y . exp Ž ip m . s Ý u m Ž yx ,y y . .
Ž 8.
Ž 15 .
m
From Eq. Ž15. it can be deduced that the components
with even index must be even functions or odd
functions with respect to both variables, it is
u 2 m Ž x , y . s u 2 m Ž yx ,y y .
2p m
(k y k
z0 s
m
From Eq. Ž6. the period p results
ps
Ž 12 .
with
E2
=t 2 '
u 0 Ž x , y . s u Ž x , y ;0 .
Ž 16 .
and the components with odd index must be odd
functions with respect to one of the variables and
even with respect to the other
u 2 mq1 Ž x , y . s yu 2 mq1 Ž yx ,y y .
Ž 17 .
B. Ruiz et al.r Optics Communications 168 (1999) 157–160
159
2.1. Example in one dimension
If the boundary condition at z s 0 is
u 0 Ž x . s Ý Ž a 2 my1 sin k 2 my1 x q b 2 m cos k 2 m x . ,
m
Ž 18 .
where every term is a general solution of Eq. Ž4. in
the x variable, then, in the paraxial approximation,
the propagated field at half the Talbot distance pr2
results
uŽ x ; z0 .
s Ý a 2 my1 sin k 2 my1 xexp Ž yip Ž 2 m y 1 . .
m
qb 2 m cos k 2 m xexp Ž yip 2 m . ,
Ž 19 .
with
k 2 my1 s '2 m y 1 k 1
Ž 20 .
and
k 2 m s 'm k 2 .
Ž 21 .
We obtain so, that those one-dimensional gratings,
the odd Fourier series development of which consists
only of components with frequencies related by the
square root of odd numbers, invert when propagate a
half of Talbot distance. When the component frequencies are between them as the square roots of
even numbers, the functions are even. Both kinds of
functions are subsets of the set that fulfills Montgomery’s conditions w7x. These functions are, in general, not periodic, as their frequencies are not commensurate. Only a subset of them are periodic,
namely those composed of harmonic numbers that
are themselves square numbers. But this particular
cases are slightly tricky, as their inverted images
coincide with their half Žfundamental. period shifted
versions. So, to look for unmistakable inverted selfimages, the non-periodic subset must be chosen.
3. Results
In order to demonstrate the proposed approach,
we select a function fulfilling the condition Eq. Ž18.
and Eq. Ž20.. It is
u 0 Ž x . s 2 q sin k 1 x q sin'3 k 1 x ,
Ž 22 .
Ž
.
with k 1 s 12.56 mm . In Fig. 1 a the intensity of
the propagated wave is numerically calculated and
y1
Fig. 1. Ža. numerically calculated intensity for the propagation of
the optical field from a pupil with profile u 0 Ž x .; Žb. and Žc.
intensity profiles obtained from slices of Ža. corresponding to
z s 0 and z s z 0 respectively.
shown as gray levels. The familiar formation of
self-images can be readily appreciated. Fig. 1Žb. and
Žc., show the intensity profile of the field at z s 0
and at half the Talbot distance respectively. Comparison between both curves evidences that the profiles
are one inverted with respect to the other, thus
confirming the predicted behavior. We performed
also an experiment to confirm the obtained theoretical results. A photographic reduction of a grating
with that profile was used in a Talbot set up Žfree
160
B. Ruiz et al.r Optics Communications 168 (1999) 157–160
Fig. 2. Ža. and Žc. intensity patterns obtained experimentally from a Talbot’s setup for a grating with profile u 0 Ž x . corresponding to z s 0
and z s z 0 respectively; Žb. and Žd. intensity profiles from Ža. and Žc..
propagation with collimated illumination. and the
planes at the pupil Ž z s 0. and half Talbot distance
Ž z s z 0 . were observed. Fig. 2Ža. and Žb. show both,
pupil plane and its intensity profile, and Fig. 2Žc. and
Žd. the same for the self-image obtained at half the
Talbot distance. The inversion of the image can be
appreciated, thus confirming the calculations. Both,
in the numerical calculations and in the experiment,
the effect of the finite aperture of the grating is
evident only near the edges. These regions have not
be included in the figures.
result could be used in the design of Talbot interferometers as well as opening perspectives for the
generation of new kinds of fields for optical engineering applications.
4. Conclusions
References
We have studied the conditions required by pupil
functions for their half Talbot distances self-images
to be inverted. The obtained condition was verified
both in numerical simulations and experimentally.
Only functions composed by sinusoids that have
certain incommensurate frequencies Žnamely in the
ratio of the square root of those odd numbers that are
not themselves squares. will produce unmistakable
inverted self-images in paraxial approximation. This
Acknowledgements
This work was supported by the Facultad de
Ingenierıa,
´ Universidad Nacional de La Plata and
Grant PMT PICT 0249 BID-CONICET, Argentina.
w1x H.F. Talbot, Philos. Mag. 9 Ž1836. 401.
w2x K. Patorski, The self-imaging phenomenon and its applications, in: E. Wolf ŽEd.., Progress in Optics, vol. 27, Chap. 1,
North-Holland, Amsterdam, 1989.
w3x http:rrw3.osa.orgrstandardrtalbib.htm.
w4x J.T. Winthrop, C.R. Worthington, J. Opt. Soc. Am. 55 Ž1965.
373.
w5x H. Konitz, Optik 68 Ž1984. 127.
w6x L. Wronkowski, J. Mod. Opt. 34 Ž1987. 1057.
w7x W.D. Montgomery, J. Opt. Soc. Am. 57 Ž1967. 772.