TECHNICAL NOTE
Fractional Fourier transform:
simulations and experimental results
Yigal Bitran, David Mendlovic, Rainer G. Dorsch, Adolf W. Lohmann,
and Haldun M. Ozaktas
Recently two optical interpretations of the fractional Fourier transform operator were introduced. We
address implementation issues of the fractional-Fourier-transform operation. We show that the original
bulk-optics configuration for performing the fractional-Fourier-transform operation 3J. Opt. Soc. Am. A
10, 2181 1199324 provides a scaled output using a fixed lens. For obtaining a non-scaled output, an
asymmetrical setup is suggested and tested. For comparison, computer simulations were performed.
A good agreement between computer simulations and experimental results was obtained.
Key words: Fourier optics, optical information processing, fractional Fourier transforms.
Recently the fractional-Fourier-transform 1FRT2 opera-
tor was described in terms of physical optics opera-
tions.
1–3
In this Note we address some experimental
aspects of the FRT operator.
The first FRT definition
1,2
is modeled by the varia-
tion of the field during propagation along a quadratic
graded-index 1GRIN2 medium by a length proportional
to the FRT order a. The eigenmodes of quadratic
GRIN media are the Hermite–Gaussian 1HG2 func-
tions, which form an orthogonal and complete basis
set.
4
The mth member of this set is expressed as
C
m
1x2 5 H
m
1
Œ
2x@v2exp12x
2
@v
2
2, 112
where H
m
is a Hermite polynomial of order m and v is
a constant that is connected with the GRIN-medium
parameters. Each function u1x2 can be expressed as a
linear combination of C
m
1x2, where the coefficient of
each HG mode is denoted by A
m
. With the above
decomposition the FRT of order a is defined as
F
a
3u1x24 5
o
m
A
m
C
m
1x2exp1ib
m
aL2, 122
where L is the GRIN length that results in the
conventional Fourier transform and b
m
is the propaga-
tion constant for each HG mode.
In Ref. 3 the FRT operation is defined alternatively
as what happens to the signal u1x2 when its Wigner-
distribution function 1WDF2 is rotated by an angle
f5ap@2. Because the WDF of a function can be
rotated with bulk optics, Lohmann suggested
3
use of
the bulk-optics system of Fig. 1 for implementing the
FRT operator. In his paper,
3
Lohmann character-
ized this optical system using two parameters, Q and
R:
f 5 f
1
@Q, z 5 f
1
R, 132
where f
1
is an arbitrary fixed length, f is a variable
focal length of the lens, and z is the distance between
the lens and the input 1or the output2 plane. For a
fractional Fourier transform of order a, Q and R
should be chosen as
R 5 tan1f@22, Q 5 sin1f2, f5a1p@22. 142
By analyzing the optical configuration of Fig. 1, one
may write
F
a
3u1x24 5 C
1
e
u1x
0
2exp
1
ip
x
0
2
1 x
2
lf
1
tan f
2
3 exp
1
2i2p
xx
0
lf
1
sin f
2
dx
0
, 152
where l is the wavelength and C
1
is a constant.
Equation 152 defines the FRT for one-dimensional
Y. Bitran and D. Mendlovic are with the Faculty of Engineering,
Tel Aviv University, Tel Aviv 69978, Israel. R. G. Dorsch and
A. W. Lohmann is with the Angewandte Optik, Erlangen Univer-
sity, Erlangen 8520, Germany. H. M. Ozaktas is with the Depart-
ment of Electrical Engineering, Bilkent University, Bilkent 06533,
Ankara, Turkey.
Received 16 March 1994; revised manuscript received 11 October
1994.
0003-6935@95@081329-04$06.00@0.
r
1995 Optical Society of America.
10 March 1995 @ Vol. 34, No. 8 @ APPLIED OPTICS 1329
functions. Generalization for two-dimensional func-
tions is straightforward.
3
This FRT integral defini-
tion is fully equivalent to the modal definition given in
Eq. 122, as shown in Ref. 5.
Unlike the conventional Fourier-transform opera-
tion, which is scale invariant 1scaling the input object
results in a reciprocal scaling of the output2, the
generalized FRT is scale variant.
1,2
Thus if one uses
the wrong scaling factor at the input plane, it will be
impossible to get the correct FRT output simply by
scaling the output plane. In other words it is manda-
tory to use the correct scaling factor at the input
plane.
The influence of changing the scale factor on the
HG representation can be found by substitution of the
first HG mode C
0
1x2 in Eq. 152. The result is the
following relation between f
1
and the HG parameter v:
v
2
5lf
1
@p. 162
As is mentioned below, computer simulations were
done based on the HG representation. Thus the
above relation is important to adjust the scale factor
of the computer simulations and the bulk-optics ex-
periments.
Equation 152 implies a scale factor of 1l f
1
2
1@2
for both
the input and the output. However, f
1
5 f sin31ap2@24 is
a function of the FRT order. Thus to keep the scale
constant, one needs a zoom lens, which is inconve-
nient.
To avoid the scale changes when performing the
experiments, with the same lens for all fractional
orders, we suggest an asymmetrical configuration.
Figure 2 shows the optical setup including the two
free-space propagation lengths Af
1
and Bf
1
and the
fixed-lens focal length f 5 f
1
@Q. After some deriva-
tion, the output field due to the input u
0
1x
0
2 is
u1x2 5
e
u
0
1x
0
2exp
3
ip
lf
1
1
x
0
2
1 2 QB
B 1 A 2 QAB
1 x
2
1 2 QA
B 1 A 2 QAB
2
2xx
0
B 1 A 2 QAB
24
dx
0
.
172
We want this output equation to match the FRT
definition. By inspecting the coefficients of x
2
and
x
0
2
, one notes that the output of the asymmetrical
setup is multiplied by a quadratic phase term. Thus
only the absolute value can be matched to the FRT
definition,
0u1x20 5 0 F
a
3u
0
1x
0
240, 182
by choice of
A 5
sin f2112cos f2@Q
cos f
,
B 5 11 2 cos f2@Q.
192
Fig. 3. Input function 1a2 cross section and 1b2 two-dimensional,
separable function obtained with a pair of different one-dimen-
sional functions in outer-product form.
Fig. 2. Setup for performing the fixed-scale asymmetrical FRT.
Fig. 1. Setup for performing a two-dimensional fractional Fourier
transform according to the WDF definition.
1330 APPLIED OPTICS @ Vol. 34, No. 8 @ 10 March 1995
A convenient selection for Q is Q 5 1, which implies
f 5 f
1
.
To acquaint ourselves with the FRT operator, we
performed computer simulations. Based on the above
two definitions, two approaches were suggested: one
is based on the HG definition 3Eq. 1224, and the second
is based on the WDF definition 3Eq. 1524. Eq. 152
suggests the performance of a Fourier transformation
on the input function multiplied by a quadratic phase
term 1chirp term2. Because of the high frequencies
necessary to represent truly the chirp term, the
resolution necessary to represent the quadratic phase
term is much higher than that of the input, requiring
a high number of sampling points.
The other FRT computer-implementation approach
is based on the HG definition. With the modal
notation,
2
Eq. 122 can be written as
u
a
5 F
a
3u
0
4 5 Cb
a
C
21
u
0
, 1102
where b
a
is a diagonal matrix having exp12ipam@22 as
its mth diagonal element, and the C matrix is con-
structed from the HG functions C
m
as its rows. Each
HG function is normalized by 1h
m
2
1@2
to ensure that C
is orthonormal. We found the HG approach to be the
faster simulation procedure, and thus it was used
here for performing the computer experiments.
A
MATLAB subroutine was written based on Eq. 1102.
First, the HG modes C were calculated and stored as
a random-access-memory variable. Now each FRT
of a vector consists of the product of the input vector
by Cb
a
C
21
. C
21
5 C
T
, and b
a
is diagonal. Thus
the number of operations is 2N
2
for the two matrix–
vector multiplications, whereas b
a
1C
21
u
0
2 is a vector–
vector multiplication 1only N operations2. C is com-
puted only once. Consequently each FRT calculation
consist mainly of two matrix–vector multiplications,
which are performed relatively fast. For example,
the number of operations for a 256-element input
vector 1N 5 2562 is roughly 2N
2
5 132,000 operations.
The input vector length is always N; thus the required
number of operations is independent of the fractional
order.
The experimental demonstration of the FRT was
performed with bulk-optics setups, both for the sym-
metrical one 1Fig. 12 and the asymmetrical one 1Fig. 22.
Although the GRIN approach could be useful for
laboratory experiments, the superiority of the bulk-
optics system is apparent because of its much higher
SW 1space bandwidth product2 performance and flex-
ibility. The GRIN approach did serve for the com-
puter simulations.
Figure 3 shows the input function that was used for
the simulations and for the experiments. For sym-
metrical-set-up laboratory experiments a single lens
with f 5 250 mm was used. The control of the
fractional order was done by a change in the param-
eter z; thus, as mentioned above, the scale of the input
and the output depends on the FRT order. The input
mask size was 3.75 mm. The distance z to obtain
different fractional orders a is given by a combination
of Eqs. 132 and 142:
z 5 f tan f@2 sin f, 1112
where f5a1p@22.
Figures 41a2–41d2 show the experimental results of
the intensity distribution for fractional orders a equal
to 0.25, 0.5, 0.75, and 1, respectively. The optical
setup of Fig. 1 was used with the parameters of Table
1. A fixed lens was used. Thus the various order
experiments were done with a different scale factor of
Fig. 4. FRT experimental result obtained with the optical symmetrical setup of Fig. 1 with the parameters of Table 1. The FRT order is
1a2 a 5 0.25, 1b2 a 5 0.5, 1c2 a 5 0.75, and 1d2 a 5 1.
Table 1. Parameters
a
Used for Performing the Symmetrical
Setup Experiments
FRT
Order a f° RQf
1
1mm2 Z 1mm2
0.25 22.5 0.199 0.383 95.75 19.05
0.50 45 0.414 0.707 176.75 73.17
0.75 67.5 0.668 0.924 231.0 154.31
1 90 1 1 250 250
a
These parameters take into account different scale factors for
the various FRT orders.
Table 2. Parameters
a
Used for Performing the Asymmetrical
Setup Experiments
FRT Order a f° Af
1
1mm2 Bf
1
1mm2
0.25 22.5 83.0 19.0
0.50 45 146.5 73.25
0.75 67.5 200.25 154.25
1 90 250 250
a
f is 250 mm.
10 March 1995 @ Vol. 34, No. 8 @ APPLIED OPTICS 1331
the input and the output. This was taken into
account when the computer simulations were per-
formed, and an excellent agreement of the experimen-
tal and computer-simulation results was obtained.
For brevity we omitted the computer-simulation plots.
An additional set of experiments was performed in
order to explore the asymmetrical configuration.
According to Fig. 2, the parameters of the setup were
calculated to provide the FRT orders 0.25, 0.5, 0.75,
and 1. These parameters are presented in Table 2; f
and l are the same as in the symmetrical experiment.
Figure 5 shows the obtained results. The results
were compared with the computer-simulation results,
and again an excellent agreement was obtained.
To conclude, we have introduced the various possi-
bilities for implementing the fractional-Fourier-
transform operation. For computer simulations the
GRIN interpretation was used. A bulk-optics optical
implementation was suggested in a configuration
very similar to the conventional 2-f Fourier-transform-
ing system according to the WDF interpretation.
Two optical setups were introduced, a symmetrical
and an asymmetrical one. It was shown that, with
the symmetrical configuration with a fixed lens, the
input and the output objects should be scaled. The
asymmetrical setup avoids this scale factor but intro-
duces a quadratic phase distribution to the output
plane. Both setups were implemented optically, and
experimental results were demonstrated.
References
1. D. Mendlovic and H. M. Ozaktas, ‘‘Fractional Fourier transfor-
mations and their optical implementation. Part I,’’ J. Opt. Soc.
Am. A 10, 1875–1881 119932.
2. H. M. Ozaktas and D. Mendlovic, ‘‘Fractional Fourier transfor-
mations and their optical implementation. Part II,’’ J. Opt.
Soc. Am. A 10, 2522–2531 119932.
3. A. W. Lohmann, ‘‘Image rotation, Wigner rotation, and the
fractional Fourier transform,’’ J. Opt. Soc. Am. A 10, 2181–2186
119932.
4. A. Yariv, Optical Electronics, 3rd ed. 1Holt, New York, 19852.
5. D. Mendlovic, H. M. Ozaktas, and A. W. Lohmann, ‘‘Graded-
index fibers, Wigner distribution functions, and the fractional
Fourier transform,’’ Appl. Opt. 33, 6188–6193 119942.
Fig. 5. FRT experimental result obtained with the asymmetrical optical setup of Fig. 2 with the parameters of Table 2. The FRT order is
1a2 a 5 0.25, 1b2 a 5 0.5, 1c2 a 5 0.75 and 1d2 a 5 1.
1332 APPLIED OPTICS @ Vol. 34, No. 8 @ 10 March 1995
