Fractional Fourier transform description with use of differential operators

erators, and a large number of optical systems associated with it are found. At the same time, the output of ampies. Mathematical definitions for the P-order convolution and correlation are proposed as generalizations of the classical ones such that, when the P-order FRT is applied to them, theorems that generalize the classical convolution and correlation are verified. © 1997 Optical Society of America [S0740-3232(97)01511-l]


INTRODUCTION
The fractional Fourier transform (FRT) and its optical implementation have been the subject of considerable at tention in recent times: The concept emerged in quan tum mechanics and is a generalization o f the Fourier transform (FT) with respect to an order P .Fractionalorder convolution and correlation have also been defined and optically generated.1"14 The FRT is usually defined through integrals that very much resemble the integral representation o f scalar dif fraction in the Fresnel region.We present here a de scription of the same transform, that is, one that is equivalent to it, by using differential operators, namely, the propagation operator and the thin-lens phase delay operator (or function).
The use of operators constitutes a powerful symbolic in strument to describe in a comprehensive way the relevant properties of optical systems without being distracted by many details.Very general properties can be high lighted in this way, and adimensional variables are not required.
Differential exponential operators are recognized to be difficult to treat mathematically,15" 19 and this is indeed ered.If limiting apertures can be separately treated,19 considerable insight c m be gained by using operators.
stract use of operators15" 19 me that, although one is re quired to work with infinite series, the explicit physical representation of every mathematical expression is readily present.This is not so when using abstract op erators, such as the scaling operator, which is easily de fined through its properties but cannot be straightfor wardly implemented optically.When it is necessary to use the scaling operator, we put it in at the final step.
In the appendices there are complementary calcula tions.There we have found explicitly equivalent systems to one given in Ref. 20.In the main text we use the con cept of equivalence to find an infinite number o f optical systems giving rise to the FRT, which are equivalent to the simplest one.We have used this concept earlier to associate an infinite set o f cylindrical systems with the FT.21 We begin in Section 2 by reviewing an auxiliary useful concept, that of the g-index FT.22This is not a new con -and frequencyscaled version o f the ordinary FT.20 Its introduction makes the following mathematical steps easier.
Different equivalent ways to express the FRT, some of them in terms o f operators, are proposed in Section 3. The amplitude distribution on an image is then calculated in terms of the FRT in Section 4. In Section 5 some other mathematical definitions o f usual operations are given, namely, P-order convolution and correlation.Both in tend to be generalizations o f classical definitions such that, when the P-order FRT is applied to them, theorems that generalize the classical convolution and correlation are verified.
We also have, from Eq. (1), = J ^" 1 = 4T * When q = 2 tt, the classical FT is obtained and for q > 0 it coincides with the q -index transform defined by23 (2) 0740-3232/97/112905-09$10.00 © 1997 Optical Society of America lows we are going to work with distributions.24 For ease, sometimes we use the following notation to describe optical systems: We call Pz( g ) the propagation operator applied t o g when it is calculated at a distance z, and L f the function (or operator) that describes the phase tive).That is, x expf e ~ " )2 + ~dudo if we take into account Eq. (B3) in Appendix B with s = = The effect of the finite size o f the pupil o f the lens is not considered here in order to keep the results in a closed form.
If we rewrite Eq. ( 1) with q =■ k/f and use Eqs.
(3), we obtain the identification of the q-FT with the simple sys tem o f Fig. 1 and parameter f: j r q{ g } = Pf L f P f { g }.
Expression (1) is also equivalent to22 _2\ (4) F q i g }(*> y ) = j exp -j q r ' and, with the use of Eqs.(3) and q = k/f, can be written as jr q {g } = Lf P f L f { g ) , ( 6) which identifies the q-FT with the simple system of Fig. 2 and parameter f.
The general setup shown in Fig. 1 is represented by and the setup shown in Fig. 2 is

FRACTIONAL TRANSFORM: EQUIVALENT DEFINITIONS
We are going to adopt one expression for the FRT, given by Eq. ( 9), and we shall find different ways to express Eq. ( 9), which are given by Eqs.(10)|1 = By using the q-FT with q = k/F sin and Eq. ( 2), we can write Eq. ( 9) as j r p{ g } ( x , y ) = exp ~(x, y ) = --It can be seen from the last result and Eq. ( 9) that is periodic with period 4. It can be proved that it is enough to work with 0 < P < 1, that is, 0 < <f < it 12.

A. Optical Systems Description with the Fractional Fourier Transform
We are going to describe the systems in Figs. 1 and 2 by means of the FRT.Equation (11) lets us write the q -FT in terms o f the P-order FRT (P-FRT): Equations ( 12) and ( 13) represent the two simple systems giving the FRT (Figs. 1 and 2).Equation ( 9) can also be written as with q = k lF sin (f>. For the system in Fig. 1, with U f U zl + l/z 2 and Eqs.(A l) and (A2) in Appendix A, we have with q = kf/A, where A = f ( z 1 + z 2) -ZiZ2.
In Eq. ( 18) we put as in Eq. ( 17) and obtain = If, in Eq. ( 19), we take F tan < f> = A / (f-z2), we obtain the following for the FRT: It can be seen from Eq. (20a) that not all output distri butions correspond to a FRT.Besides, when only inten sity is measured, so that the phase factor produces no ef fect, the result is always proportional to a squared FRT of a certEiin order given by Eqs.(20b).= -Zi), we obtain with cos < f> = ( f -z{)/f and F Al\zx( 2 f -z{)'\V2. ( Then Eqs. ( 20) and ( 21) are descriptions o f the optical system in Fig. 1 by means o f the FRT.

B. Equivalent Systems Representing the Fractional Fourier Transform
The FRT can be represented by many optical systems (ac tually, an infinite number o f systems), some o f which we are going to describe.
From Eqs. (20) we see that the system in Fig. 1, with a -A /( z 2 -^i), represents &~p.It is shown in Fig. 4.
Conversely, given 0 and F , there are infinite systems associated with &'p (P = 20/7t).They are the systems described in Fig. 4, and ing <P'p .In general, as the FRT is determined by 0 and F and the system in Fig. 1 has three free parameters (z1, z 2, and f ) , it is possible to choose one o f them arbitrarily to represent the FRT.If, for example, we take z-^=z2 = z, then the output lens collapses and Eq.(20a) be comes z = F sin 0 , f = F sin 0 /(1 -cos 0 ), (23) which coincides with the FRT description in Eq. ( 13).Then, from Fig. 4, we can say that a new expression for the P-FRT using operators is f p{ g } (x .y ) = exp with any /*; this includes as particular cases Eq. ( 12) (z x = =

C. Cascading Systems Description
We can describe the system shown in Fig. 5 as being com posed of three FRT cascading systems such as that in Fig. 1; that is, by using Eq. ( 22 As the system in Figs. 1 and 2, with the conditions given in Appendix A, are equivalent, the system in Fig. 2, with = represents the FRT.But this means that f i = f 2 = the system in Fig. 2 represents the FRT with or, alternatively, the FRT can be represented by the sys tem in Fig. 2 with rameters given in Subsection 4.B.

Optical system described with the FRT in Subsection
To be able to apply order additivity, we have to impose = F i .This implies that f i = -, P Zi_z^2p-pi{ g } = h.(24) In addition, it holds that 2P -P x = 2, and then which implies that the complex system in Fig. 5 with the input g has the same behavior as a propagation of dis tance (z 1 --z) with the input function --.
-z in Eq. ( 25), we have h ---, with f x = f 2/2(z -f ) and /"and z arbitrary.Then we can freely take any two parameters (of the four parameters o f the system) to obtain --as the image through the system in Fig. 5.That is so because in Eq. ( 24) with z -Z\ there is only one FRT.
I f we want to find an alternative system that gives, for --, we put IF2-1 = -successive application o f the system in Fig. 2 with the pa rameters given by Eqs. ( 23) and then the system in Fig. 2 with parameters given by Eq. ( 23) but with 0 changed to 7T -0.This procedure is shown in Fig. 6, where the pa rameters are In each particular case, the most convenient parameter, s, is chosen so as to make the calculation easy.Then the expression generally used is z = F sin 0, l = F sin 0 /(1 -cos 0 ), for any F and <f>(=f=rmr).

CONVOLUTION AND CORRELATION
We propose here definitions o f convolution and correlation associated with the FRT.In the first place, it is required that they generalize the usual definitions o f convolution and correlation in the following sense: For 0 = W2, the usual definitions should be obtained.For example, the P convolution must verify a convolution theorem for 3^p so that for 0 = 7r/2 the theorem associated with the FT should result: A. Definition o f Convolution We define the P-order convolution between the functions p g and h, by using the notation g * h, as , (/ , ( ~jk r 2 g*hl(x,y) = exP 2 F t a n ^, Jkr2

, g e x p \ W T ^] * h (27)
with P = 2 0 /tt.In this definition it can be recognized that there is an external phase correction and that the main operation is a classical convolution between two complex functions, which are those to be fractionally con volved, each with the same quadratic phase modification.This phase modification depends on the operation frac tional order and the parameter F .
The external phase correction does not affect intensity measurements but must be taken into account if the field distribution obtained in fractional convolution is used as input to other processing operations.
The expression of the P convolution with use o f the propagation operator is obtained from Eq. ( 27) and Eq.= ig * hj(x, y ) X 6Xp^ 2F tan 0 J with r 2 = -a )2 + (y -/J)2.For example, if h = exp (jar2), we choose s -a + k/(2F tan 0) =£ 0 and obtain, by applying Eq. ( 13), with P i determined by the equations given in Subsection 4.B.Expression (29b) shows that the P convolution between a function g and a quadratic phase delay is the product o f a lens and the P r FRT o f g .^tás convolution can be op tically obtained by using a lens against the input function transmittance, followed by a free propagation and by a lens with focal length opposite to that o f the first one.
From another point of view, this system is shift invari ant in a wide sense because the output is the P convolu tion o f the input and the impulse response o f the system.
Expression (29b) is valid when P = 1, in which case the system reduces to a propagation o f distance z = k/2a.

B. Convolution Theorem
The P-FRT applied to the P convolution verifies that = To prove this theorem, we start with as defined in Eq. ( 11), P convolution as defined in Eq. ( 27), and q -k/F sin 0:

C. Definition o f Correlation
We define the P-order correlation between the functions g P and h , by using the notation g * h as (31) The asterisk as a superscript indicates conjugation.

D. Correlation Theorem
The P-FRT applied to the P correlation verifies that To prove this theorem, we use the P-FRT defined by Eq. ( 11) and the convolution theorem (26), with q = klF sin < f> : When <p = 7t/2 , this correlation theorem is the classical FT correlation theorem, which is what we wanted.
The optical implementation o f the correlation can be performed by taking g as input, using a complex holo-given in Eq. (32), and obtaining the result in the (4 -P)-Fourier plane.

E. Autocorrelation Theorem
If, in Eq. (31), we take g = h, we obtain the P-order au tocorrelation.From the correlation theorem results the autocorrelation theorem:

CONCLUSIONS
We have used a description o f the Fourier transform (FT) named the q -index FT to analyze the fractional Fourier transform (FRT).When it is used, in operator descrip tion, abstract scaling operators do not appear in the middle of system descriptions but, if required, appear only in the final results.
The concept of equivalent optical systems was then used to find an infinite set o f systems that represent the FRT.That concept, when applied with the use of opera tors, results in a set of relations among the parameters (focal lengths and propagation distances), which must be satisfied for the systems to be equivalent.The relations between these parameters and the two parameters that determine the FRT, P and F , were found.
These relations simplify the calculations o f the output o f any system considered as a cascade and composed of any number o f lenses.By considering that both the < f> and F values can be chosen independently, we can use dif ferent focal distances for the lenses in the system.Two application examples were given to illustrate the simplic ity of the approach.
The output of a simple optical system with three inde pendent parameters was found in terms of the FRT ap plied to the input.The output intensity distribution of the system always corresponds to that o f a certain FRT.Some calculations on the FRT and the convolution were given in the appendices.
Mathematical definitions of P convolution and P corre lation that generalize the usual ones were proposed and expressed in terms of operators.A physical way to gen erate a certain P convolution in terms o f lenses and propagations was described.More general cases require the use of complex filters, as for example, holographic el ements.
B. Ruiz is also with the Faculty of Astronomic and Geo physic Sciences, Universidad Nacional de La Plata, Paseo del Bosque, 1900-La Plata, Argentina.H. Rabal is also with the Faculty of Engineering, Universidad Nacional de - Pz(g ) = e x p f ê ^l ( g ) , L f = exP| ~i j r * (3a) (3b) with k the wave number o f the light.Equation (3a) is equal to the integral expression25 p*(g)= ¿ J 7 * (bv)

Fig. 3 .
Fig. 3. Optical system equivalent to the systems in Figs. 1 and 2, with the conditions deduced in Appendix A.
P{g}{x> y ) = TT-exp with q = k/F sin (f>.Equation (14) expresses a convolution.If h (x, y ) = exp(jkr¿/2F tan <f>), then it can be written as ^F{g}(x, y) -jq I -j k r 2 \ I x y 2tt eX^\2P cot <f>!^ ^\ co s <f>' cos < f> (15) (20a) with If we use Eq.(B3) from Appendix B, this last equation be comes cos 4>=(f~ z 2)/f, F = A/[z2( 2 f -z 2) ] m .(20b) (16) == i = describes the FRT as a propagation followed by a lens and a change of scale in the output.Equations (14)-FT, that is, when < f> = -ttI2.Equation (16) can be used to define the FRT when < j> = -rrnr and shows continuity with the above definition.An important property o f the FRT is order additivity.= = ~ 1 tained by changing F to Equations (20) Eire not valid, in the real field, for z 2 = 0 and z 2 5= 2 f .For those values it is convenient to work with the classical FT.But if we want to use the FRT, we can sepEirate the last propagation into two suc cessive propagations z 2 = z + (z 2 ^z ) such that the disteince z satisfies Eqs.(20) Eind the propagation o f distance (z 2 ■ -z) can be expressed by means of Eq. (13).

1 i y 2
with any P, whose elements are analyzed in Subsection 4.C.