Differentiation of a noninteger order and its opticalimplementation

Henryk Kasprzak

This paper gives the definition and examples of an arbitrary order differentiation based on some propertiesof the Fourier transform. To physically implement the generalized differentiation a coherent processingsystem with binary synthetic filter was used. The experimental part of the paper shows models of the 1-Dand 2-D binary filters and results of the filtering process.

1. Introduction

The generalization of differentiation to a nonintegerorder defined in this paper is based on the derivativetheorem of Fourier transform theory. Such generalizeddifferentiation of a function performed by appropriatemanipulation in its Fourier spectrum seems to be anatural, single-valued, and continuous transferencefrom a function to its derivatives of an integer order.Bracewelll mentions the possibility of fractional orderderivative calculations by using Fourier or Abel typetransforms. Abel2 was one of the first to use fractionalcalculus. Liouville3 presented the first major study ofa fractional order differentiation. Post4 defined anarbitrary order derivative as a generalization of differ-ence quotients. Such an approach was continued byOldham and Spanier5 who gave interesting examplesand applications.

The paper is basically divided into two parts. Thefirst part deals with definitions and examples of thegeneralized differentiation operator, while the secondis mainly oriented toward optical implementation ofsuch an operator by coherent optical filtering.

11. Basic Assumptions

From the Fourier transform pair of one independentvariable, defined by

F(u) f(x) exp(-i27rux)dx, (1)

The author is with Technical University of Wroclaw, Institute ofPhysics, Wroclaw, Wybrzeze Wyspianskiego 27, Poland.

Received 8 September 1981.0003-6935/82/183287-05$01.00/0.© 1982 Optical Society of America.

f(x) = 5 F(u) exp(i27rux)du, (2)

the Fourier integral theorem

f(x) = 5 f(t) exp(-27riut)dtj exp(i27rux)du (3)

may be derived. From Bracewell's' point of view, eachfunction describing a real optical field has its Fouriertransform.

III. Definitions

Let us assume that a generalization of the powerfactor n of the 2-7rui expression in derivative or integraltheorem to a noninteger positive number correspondsto an operation of real order integro-differential of theFourier transformed functions. Thus, a derivative ofreal order r of the Fourier transformed function f(x) hasits Fourier transform of the form (27rui)rF(u).Therefore a generalized derivative of a complex functionf (x) can be defined as

fr(x) = 3' (27rui)rI J f(x) exp(-2lriux)dxl

X exp(i27rux)du. (4)

Let us first restrict our attention to the class of real-valued functions f(x) and the order of differentiationsatisfying the inequality 0 < r < 1. Each real functionf(x) can be presented as a sum of its even part e (x) andits odd part o(x), respectively, i.e.,

1 1f (x) = e(x) + o(x) = - [f (x) + f (-X)] + -[f (x) -f (-x)]. (5)

2 2

By decomposing the Fourier transform F(u) of thefunction f (x) into a cosine transform of the even partE(u) and a sine transform of the odd part 0(u), re-spectively, one can write

F(u) = E(u) - iO(u).

Substitution of this result to definition (4) yields

(6)

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fr (x) = 2(27r)r cos -J ur[E(u) cos2irux

-O(u) sin2-ruxidu - sin r 3' ur[E(u) sin2,rux

+ O(u) cos2lrux]du}. (7)

This function is real.Note that operator (4) is single-valued only for the

principal value of expression r; therefore we restrict ourattention to this principal value. The linearity andhomogeneity of the differential operator (4) is an im-mediate consequence of the definition. Moreover, ifthere are no infinite discontinuities of the fr(x), deriv-atives are continuous with respect to the order of de-rivative r.

Figures 1(a), (b), and (c) show a graph of the samefunction (11) of two independent variables x and r forthree different regions of the variable x. The brokencurve in Fig. 1(b) denotes a locus of points in the x, rplane where fr(x) is zero. This curve is described by theformula

1x =-- 2 -r- 1

(14)

Another example which gives some interesting resultsbut requires the use of the 6 function is the generalizedderivative of a harmonic function. Let us consider afunction cos(px), where p is an arbitrary constant. TheFourier transform is given by

9(cos(px)) =2 - 2P) + 6(u + (15)

IV. Examples

Let us consider the unit area, unit amplitude trianglefunction A(x). Using the evenness property of thefunction and applying its cosine transform E(u) =(sinc7ru)2 , operator (7) gives

fr(x) = ( 12) cos 2Jr ur-2 sinru(1 + 2x) sinrudu

1 2

+ 3' ur-2 sinru(1 - 2x) sinrudu

+ sin -[3ri' ur-2 cos7ru(1 + 2x) sinirudu2o- 3 ur- 2 cosru(1 - 2x). sinrudull. (8)

These four integrals are, respectively, equal to

1 = C(r)(|l + XI -r - XI -r),

2 = C(r)(I1 - xI 1-r - lxI 1-r), ()3 = C(r)[|xll-rsgn(x) - 1 +x -r],

4 = C(r)[IxI 1-r sgn (-x) - 1 -xI 1-r],

where

CH= 2 1-r~r 2-rC(r) = -r2r

4cosr (2-r)2

(10)

Then by substituting the above transform in the oper-ator (7) and making use of the sifting property of the 6function, the derivative of real order r of cos(px) is seento be

cosr(px) = pr cos x + r 2); (16)

similarly,

sinr(px) = pr sin x + r2) (17)

These simple results may provide the basis for term-by-term differentiation of any periodic function forwhich the Fourier series expansion exists.

V. One-Dimensional Synthetic Spatial Filter

The ability to perform integro-differential operatorsof a noninteger order using electrical circuits has beenshown by Ichise et al. 6 and Oldham.7

To implement optically the differentiation operatorof a noninteger order a typical coherent processingsystem with synthetic spatial filter has been used.8 9

The filter was produced by the Vander Lugt method.'0The amplitude transmittance of the filter should beproportional to the intensity distribution in an inter-ference pattern produced by a plane wave and a wavewith complex amplitude distribution equal to (2rui) r,respectively, i.e.,

Substituting the above four integrals in (8) we obtain

(0 for x -1,fr(x) G(r)(1 + )l-r for -1 < x < 0,

G(r)[(1 + x)l-r - 2xl-r] for 0 < x 1,G(r)[(1 + x)-r - 2xl-r + (x - 1)1-r] for x > 1,

(11)

where

G(r) = (r) sin*r (12)G 1) 1-r

By using the respective properties of the gamma func-tion and the de l'Hopital theorem it can be shownthat

limG(r) = limG(r) = 1. (13)r-1 r-O

I(u) = IA exp(-i27rau) + (2irui)r 2, (18)

where a = sinO/X, 0 being the tilt angle of a plane wavewith respect to the recording plane.

By applying the simplification proposed by Burch"'and assuming that intensity 1(u) is equal to zero at theborder of the filter (for u = umax), the amplitudetransmittance distribution of such a filter is given by

(19)

where C is a constant depending on normalizationconditions. Figure 2 shows a cross section of the am-plitude transmittance distribution of the 1-D differ-entiation filter for an order of derivative equal to 1/2.This figure can be treated as a mask to prepare theproper halftone filter by using the photographicmasking method.

3288 APPLIED OPTICS / Vol. 21, No. 18 / 15 September 1982

t(u = C 1 + � U �r cos(27rau + r 7r�j ,I Umax 2

Fig. 1. Arbitrary order derivative of the triangle function.

VI. One-Dimensional Binary Filter

Halftone synthetic filter technology is rather cum-bersome and can be successfully employed only whenspecial programming write devices are accessible.Replacing the halftone filter by a binary filter consid-erably simplifies the method of recording. Amplitudetransmittance of a binary filter is defined as a local ratioof an amplitude of the wave, deflected in the first dif-fraction order to the amplitude of the incident wave.

Diffraction efficiency of such a binary grating is givenby the Kogelnik formula' 2

r r l(u) 2sin

I(u) d Id r 1 I(u)

d

Fig. 2. Cross section of the 1-D synthetic filter amplitude trans-mittance for the order of differentiation equal to 1/2.

0,6

0

(20)

UUr.s1

where 1(u) is a local width of a black stripe, and d marksthe distance between centers of two neighboring blackstripes. Thus, amplitude transmittance of the binaryfilter becomes equal to A%/jT(u). The maximum valueof N/-2?u is 0.318 for [l(u)]/d = 0.5.

To utilize the complete range of diffraction efficiencyin the area of the filter, normalization has been carriedout so that [l(u)]/d = 0.5 on the border of the filter. Itfollows that

(U) arcsin --- (21)d =r /1max

Referring to Eq. (19), the modulation factor amountsto (U/Umax) r, yielding finally

l(u) 1 1 r X=- arcsin .(22)

d x rUmj

Figure 3 is a graph of the 2-D function (22). Accordingto (19), if appropriate phase changes are to be realized,centers of black stripes for u > 0 and for u < 0 must beshifted in relation to the center of the filter by a distance

Fig. 3. Graph of the 2-D function (22).

equal to r/4 of the distance between the stripes. Figure4 shows a model of the binary filter for realizing theoperation of differentiation of the fractional order equalto 0.5.

VII. Two-Dimensional Binary Filter

As an example of 2-D generalized differential oper-ators, let us consider the transform

arl+r2dxldf2(x'y) <=> (27rui)l * (27rui)r2. F(UV), (23)

where r, and r2 are arbitrary orders of differentiationin directions x and y, respectively.

The same method of binary filter recording used forthe 1-D operator may be employed for the 2-D case.According to this model the local width of the blackstripes is described by the formula

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t (U)

III

Fig. 4. Model of the -- D -order differentiati binar fil.Fig. 4. Model of the 1-D /2-order differentiation binary filter.

1(uv) 1 arcin u I rlV Ir2arcsin - ~~~~(24)d -r maxI maxI

Note that black stripes are shifted in the positive di-rection of the u axis for the first quarter and in thenegative direction of the u axis for the third quarter ofthe filter by the value of (r2 + rl)/4 times the distancebetween the centers of black stripes. For the secondand fourth quarters the shift is equal to (r1 - r2)/4 in therespective directions.

Figure 5 shows a model of the 2-D binary filter to re-alize the operation of the type (23) for r, = r2 = /2-

Vil. Experimental Results

For coherent processing using binary filters the im-ages in neighboring diffraction orders must be separatedfrom each other. Therefore, the models of binary filtersshown in Figs. 4 and 5 have been demagnified thirtytimes to assure appropriate spatial frequency of thefilter.8 Aperture functions of the type rect(x), rect(y)or circ(r) were used as the functions to be differen-tiated.

Figure 6 shows the results of the filtration afterapplying the filter presented in Fig. 4. It may be seenthat the images in +1st and -1st diffraction orderspresent the square of the modulus of the 1-D derivativeof order r = 1/2 of a 2-D object produced in zero-orderdiffraction.

Each cross section of such a differentiated image inthe direction of differentiation is a function of therect(x) type. Thus each cross section of the image inthe direction of image deflection is a 1-D function pro-portional to I rect0 5 (x) 2. To test the exactness of theoptical system performance the two comparative graphsshown in Fig. 7 were constructed. The broken-linecurve in Fig. 7 is a plot of the theoretically calculatedfunction

0

while the continuous line has been obtained by mea-suring the aerial image of the first diffraction ordershown in Fig. 6 in the direction of diffraction.

Figures 8(a) and (b) show the effect of implementa-tion of the 2-D operation (23) in the first diffractionorder performed on the image, the latter being in thezero-order diffraction. The symmetry characteristicsindicate the correctness of the filtering operation.

IX. Conclusions

The proposed method of generalized differentiationpermits both an analytical calculation of derivatives ofan arbitrary order and physical implementation of suchoperators by coherent optical processing. The differ-entiation presented in this paper is concerned only witha simple real function or an amplitude object. Inter-esting results may be obtained by applying this methodto more complicated complex functions or to phaseobjects.

Fig. 5. Model of the 2-D binary filter to perform the operation(Orl+r2)/(dXrliyr2).

Fig. 6. Result of the 1/2-order differentiation produced by the filtershown in Fig. 5.

forx <-1/2,

_ _r(r) sin_7rr I_1 2 for -/2 < x < 1/2,

Irect0-5(X)12 = L i n + 2x)11[2rr(r) sin7rr 1 1 1 2

I( + 1)' for x > 1/2,

3290 APPLIED OPTICS / Vol. 21, No. 18 / 15 September 1982

(25)

Ired (0.5)(X)12

11

X

Fig. 7. Curves depicting the theoretical and experimental results.

Fig. 8. Images differentiated by use of the filter shown in Fig. 6.

References1. R. N. Bracewell, The Fourier Transform and Its Applications

(McGraw-Hill, New York, 1965).2. N. A. Abel, "Solution de quelques problemes a l'aide d'integrales

definies," Mag. Naturvidenkaberne, 1823, Norway.3. J. Liouville, J. Ec. Polytech. 13, Sec. 21, p. 1 (1832).4. E. L. Post, Trans. Am. Math. Soc. 32, p. 723 (1930).5. K. B. Oldham and J. Spanier, Fractional Calculus (Academic,

New York, 1974).6. M. Ichise, Y. Nagayanagi, and T. Kojima, J. Electroanal. Chem.

Interfacial Electrochem. 33, p. 253 (1971).7. K. B. Oldham, Anal. Chem. 45, p. 39 (1973).8. H. Kasprzak, "Realizowalnosc optyczna operacji rozniczkowania

rzedu rzeczywistego," "An Optical Implementation of the RealOrder Differentiation," Ph.D. Dissertation, Technical Universityof Wroclaw, Report 105/1980 (1980), in Polish.

9. H. Kasprzak, Opt. Appl. 10, p. 283 (1980).10. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill,

New York, 1968).11. J. Burch, Proc. IEEE 55, p. 599 (1967).12. R. Sirohi and V. Ram Mohan, Opt. Acta 22, p. 207 (1975).

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