Research Article A Wavelet Algorithm for Fourier-Bessel Transform Arising in Optics

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1 Hindawi Publishing Cororation International Engineering Mathematics Volume 25, Article ID , 9 ages htt://dx.doi.org/.55/25/ Research Article A Wavelet Algorithm for Fourier-Bessel Transform Arising in Otics Nagma Irfan and A. H. Siddiqi Deartment of Mathematics, School of Engineering and Technology, Sharda University, Knowledge Park III, Greater Noida, Delhi, NCR 236, India Corresondence should be addressed to Nagma Irfan; nagmairfanmath@gmail.com Received May 25; Revised 2 August 25; Acceted 2 August 25 Academic Editor: Song Cen Coyright 25 N. Irfan and A. H. Siddiqi. This is an oen access article distributed under the Creative Commons Attribution License, which ermits unrestricted use, distribution, and reroduction in any medium, rovided the original work is roerly cited. The aim of the aer is to roose an efficient and stable algorithm that is quite accurate and fast for numerical evaluation of the Fourier-Bessel transform of order ], ] >, using wavelets. The hilosohy behind the roosed algorithm is to relace the art tf(t) of the integral by its wavelet decomosition obtained by using CAS wavelets thus reresenting F ] () as a Fourier-Bessel series with coefficients deending strongly on the inut function tf(t). The wavelet method indicates that the aroach is easy to imlement and thus comutationally very attractive.. Introduction The Fourier-Bessel transform (also designated as Hankel transform) is a very useful tool of mathematical hysics []. It is a very useful instrument in a wide range of hysical roblems which have an axial symmetry. It is articularly imortant in otics and two-dimensional image rocessing, it naturally occurs in image reconstruction from rojections or from reflected ulses, and it is a useful tool in the analysis and synthesis of three-dimensional wave fields. The resent develoment is essentially motivated by otics alication. The influence of the Lalacian on a function in cylindrical coordinates is equal to the roduct of the squared arameter of the transformation and the transform of the function [2] ( d2 dr 2 + d r dr )f(r) 2 F (), ( d2 dr 2 + d r dr r 2 )f(r) 2 F (). There are two tyes of the Hankel transform. The first one is defined on the semi-infinite interval. In this case the direct and inverse transforms of the ]th kind are reresented as a symmetric air. When we are dealing with roblems that show circular symmetry, Hankel transforms may be () very useful [3, 4]. Lalace s artial differential equation in cylindrical coordinates can be transformed into an ordinary differential equation by using the Hankel transform. Because the Hankel transform is the two-dimensional Fourier transform of a circularly symmetric function, it lays an imortant role in otical data rocessing [5 7]. In otics, the Hankel transform aears in many contexts, not the least of which is the roagation of cylindrically symmetric laser beams. Most classical otical systems like mirrors or lenses are axially symmetric devices. Hankel transform also roved to be extremely useful in roblems associated with seismology, geohysics [8, 9], electroscattering, acoustics, hydrodynamics, image rocessing [], time deendent Schrodinger equation, and so forth. Mathematical Background. The Fourier-Bessel transform may be defined by the following exression: F ] () χ ] {f (t)} tf (t) J ] (t) dt, (2) f (t) =χ ] {F ] ()} F ] () J ] (t) d, (3) where J ] is the ]th-order Bessel function of the first kind.

2 2 International Engineering Mathematics InthecaseofthefiniteHankeltransformonlyadirect transform has an integral form. Without loss of generality its exression is F ] () = tf (t) J ] (t) dt (4) (see []). Practical calculation of direct and inverse Hankel transform is connected with two roblems. The first roblem isbasedonthefactthatnoteverytransforminthereal hysical situation has analytical exression for result of inversehankeltransform.thesecondoneisthedetermination of functions as a set of their values for numerical calculations. The classical traezoidal rule, Cotes rule, and other rules connected with the relacement of the integrand by sequence of olynomials have high accuracy if integrand is a smooth function. But tf(t)j ] (t) (or F ] ()J ] (t)) isa quick oscillating function if t (or ) islarge.therearetwo general methods of the effective calculation in this area. The first is the fast Hankel transform [2]. The secification of that method is transforming the function to the logarithmical sace and fast Fourier transform in that sace. This method needs a smoothing of the function in log sace. The second method is based on the searation of the integrand into roduct of slowly varying comonent and a raidly oscillating Bessel function [3]. But it needs the smoothness of the slow comonent for its aroximation by lower-order olynomials. To overcome these difficulties, various different techniques are available in the literature. Several aers have been written on the numerical evaluation of the HT in general and the zeroth order in articular [4 24]. There are two general methodsoftheeffectivecalculationinthisarea.thefirstisthe fast Hankel transform [25]. The secification of that method is transforming the function to the logarithmical sace and fast Fourier transform in that sace. This method needs a smoothing of the function in log sace. The second method is based on the searation of the integrand into roduct of slowly varying comonent and a raidly oscillating Bessel function [26]. But it needs the smoothness of the slow comonent for its aroximation by lower-order olynomials. From variety of algorithm, a otential user would robably find it difficulttoselectanyonealgorithmthatmightbebestfora articular alication. For an overview of these algorithms and their numerical comlexity, the reader is referred to [27 3]. The organization of the aer is as follows: Section 2 gives a brief descrition of the CAS wavelets followed by the derivation of the algorithm in Section 3. The efficiency and stabilityofthealgorithmareshownbyalyingittofour test functions with known analytical transform in Section 4. At the end, a brief conclusion and future work are given in Section Proerties of CAS Wavelets 2.. Wavelets and CAS Wavelets. Wavelets constitute a family of functions constructed from dilation and translation of a single function φ(t) called the mother wavelets. When the dilation arameter is 2 and the translation arameter is wehavethefollowingfamilyofdiscretewavelets[32]: ψ kn (t) =2 k/2 φ (2 k t n), (5) where ψ kn form a wavelet orthonormal basis for L 2 (R). CAS wavelets ψ nm (t) = ψ(k,n,m,t) involve four arguments n, k, m,andt,wheren =,,...,2 k, k is assumed to be any nonnegative integer, m is any integer, and t is normalized time. CAS wavelets are defined as [33] where ψ nm (t) = { 2 k/2 CAS m (2 k t n), for n t<n+, { 2k 2 k {, otherwise, (6) CAS m (t) = cos (2mπt) + sin (2mπt). (7) An efficient algorithm has been resented for the Fourier- Bessel transform. 3. Outline of Algorithm The function f(t) reresenting hysical fields either are zero or have an infinitely long decaying tail outside a disk of finite radius R. Hence, in most ractical alications either the signal f(t) hasacomactsuortorforagivenε>there exists R>such that tf(t)j R ](t)dt < ε. Therefore, in either case, R F ] () = tf (t) J ] (t) dt = tf (t) J ] (t) dt, (by scaling) known as the finite Hankel transform (FHT), is a good aroximation of the HT as given by (2). Writing tf(t) = g(t) in (8), we get We may exand g(t) as follows: (8) F ] () = g (t) J ] (t) dt. (9) g (t) = n= c nm ψ nm (t), () m Ζ where c nm = g(t),ψ nm (t). By truncating infinite series () at levels n=2 k and m=m, we obtain an aroximate reresentation for g(t) as g (t) 2 k n= M c nm ψ nm (t) =C T ψ (t), () m= M

3 International Engineering Mathematics 3 where the matrices C and ψ are given by C=[c ( M),c ( M+),...,c,M,c ( M),...,c M,..., c (2 k )( M),...,c (2 k )M ]T, ψ (t) =[ψ ( M),ψ ( M+),...,ψ,M,ψ ( M),...,ψ M,...,ψ (2 k )( M),...,ψ (2 k )M ]T. Substituting () in (9), we get (2) F ] () C T ψ (t) J ] (t) dt. (3) Taking M=and k=,(3)reducesto F ] () C T [ ψ ( ) (t) J ] (t) dt, ψ, (t) J ] (t) dt, ψ, (t) J ] (t)dt,..., T ψ ( ) (t) J ] (t) dt,..., ψ, (t) J ] (t) dt]. Now, we relabel and write (4) as (4) F ] () [c,c,...,c 42 ][I ],I n,...,i n ]T, (5) where I l ] s are the lth-lace integral in (4). The integrals arising in (4) are evaluated by using the formulae a L J ] (t) dt = 2 lim J ]+2z+ (a), L z= (6) Re ] > (see [34]) and are calculated with the hel of Simson s onethird rule, Simson s three-eighth rule, comosite Simson s one-third rule, and comosite Simson s three-eighth rule, resectively. In numerical analysis, Simson s rule and comosite Simson s rule are method for numerical integration, the numerical aroximation of definite integrals. 4. Numerical Results In this section, we test the roosed algorithm (5) by evaluating the aroximate Hankel transforms of 4 well-known test functions with known analytical Hankel transforms. Note that in all the examles the truncation is done at level N=, k=,andl=6in(5).weobservedthattheaccuracyof the method is very high even at such a low level of truncation. Examle. Let f(r) = r ] sin(πr 2 /4), r<;then F ] () = 2 (π 2 ) ] ] [U ]+ ( π 2,) U ]+2 ( π 2,)] (7).2.. F () H () Figure : The exact transform, F ] () (solid line), and the aroximate transform, H ] () (dotted line), where ] = E() Figure 2: Comarison of the errors. (obtained from [34] by utting a = π/4, b = ), where U ] (w, ) is a Lommel function of two variables, = L 2 [ η= [( ) η ( π 2 ) 2η (J ]+2η+ () π 2 J ]+2η+2 ())]] as L. (8) The comarison of the aroximation H]() (dotted line) with the exact Hankel transform F]() (solid line) is shown infigures,3,and5andtheerrore() = H]() F]() in Figures2,4,and6. Simson s One-Third Rule.SeeFiguresand2. Simson s Three-Eighth Rule.SeeFigures3and4. Comosite Simson s One-Third Rule.SeeFigures5and6. Comosite Simson s Three-Eighth Rule.SeeFigures7and8. Examle 2. The following examle was solved numerically by [35].

4 4 International Engineering Mathematics F () H () Figure 3: The exact transform, F ] () (solid line), and the aroximate transform, H ] () (dotted line), where ] =. F () F() Figure 7: The exact transform, F ] () (solid line), and the aroximate transform, F() (dotted line), where ] = E() Figure 4: Comarison of the errors E() Figure 8: Comarison of the errors..2.. F () H () Figure 5: The exact transform, F ] () (solid line), and the aroximate transform, H ] () (dotted line), where ] =. 3 5 For f (r) =e r, its HT is F () = ( + 2 ). (9) 3/2 Wesolvetheaboveroblembytheroosedalgorithmand observe that our method gives a result comarable to [35]. Note that F () and F () are indicated by F () (solid line) and F() (dotted line) in Figures 9,, 3, and 5 and the error E() = F() F () is shown in Figures, 2, 4, and 6. Simson s One-Third Rule.SeeFigures9and. Simson s Three-Eighth Rule. See Figures and 2. Comosite Simson s One-Third Rule. See Figures 3 and 4. Comosite Simson s Three-Eighth Rule. See Figures 5 and E() Figure 6: Comarison of the errors. Examle 3 (sombrero function). A very imortant and often used function is the Circ function that can be defined as [22] Circ ( t a )={, t a, { (2), t > a. { This function is quite common in otical roblems where it is used, for instance, to reresent a circular uil of radius a.

5 International Engineering Mathematics F () F() F () F() Figure 9: The exact transform, F () (solid line), and the aroximate transform, F() (dotted line). Figure 3: The exact transform, F ] () (solid line), and the aroximate transform, H ] () (dotted line) E() Figure : Comarison of the errors. E() Figure 4: Comarison of the errors F () F() Figure : The exact transform, F () (solid line), and the aroximate transform, F() (dotted line). F () F() Figure 5: The exact transform, F () (solid line), and the aroximate transform, F() (dotted line) E() Figure 2: Comarison of the errors. E() Figure 6: Comarison of the errors.

6 6 International Engineering Mathematics F () F() F () F() Figure 7: The exact transform, F () (solid line), and the aroximate transform, F() (dotted line). Figure 9: The exact transform, F () (solid line), and the aroximate transform, F() (dotted line) E() Figure 8: Comarison of the errors. E() Figure 2: Comarison of the errors. The Fourier-Bessel transform of (2) is the well-known sombrero function. The zeroth-order Hankel transform of Circ(t/a) is the sombrero function [29], given by S () =a 2 J (a) a. (2) The exact and numerical transforms differ very slightly but the differences are hardly visible. Simson s One-Third Rule. See Figures 7 and 8. Simson s Three-Eighth Rule. See Figures 9 and 2. Comosite Simson s One-Third Rule. See Figures 2 and 22. Comosite Simson s Three-Eighth Rule. See Figures 23 and 24. Examle 4. Let f(t) = (2/π)[arccos(t) t( t 2 ) /2 ], t ; then, F () F() Figure 2: The exact transform, F () (solid line), and the aroximate transform, F() (dotted line) F () = 2 J2 (/2) 2,, (22) a well-known result. The air (f(t), F ()) arises in otical diffraction theory [36]. The function f(t) is the otical transfer function of an aberration-free otical system with acircularaerture,andf () is the corresonding sread function E() Figure 22: Comarison of the errors.

7 International Engineering Mathematics F () F() F () F() Figure 23: The exact transform, F () (solid line), and the aroximate transform, F() (dotted line). Figure 25: The exact transform, F () (solid line), and the aroximate transform, F() (dotted line) E() Figure 24: Comarison of the errors E() Figure 26: Comarison of the errors..5 Barakat and Sandler [26] evaluated F () numerically using Filon quadrature hilosohy but the associated error is areciable for <, whereas our method gives almost zero error in that range. Simson s One-Third Rule. See Figures 25 and 26. Simson s Three-Eighth Rule. See Figures 27 and 28. Comosite Simson s One-Third Rule. See Figures 29 and 3. Comosite Simson s Three-Eighth Rule. See Figures 3 and F () F() Figure 27: The exact transform, F () (solid line), and the aroximate transform, F() (dotted line). 5. Conclusion.2 Since the basis functions used to construct the wavelets are orthogonal and have comact suort, it makes them more useful and simle in actual comutations. Also, since the numbers of mother wavelet s comonents are restricted to one, they do not lead to the growth of comlexity of calculations. Our choice of wavelets makes them more attractive in their alications in the alied hysical roblems as they eliminate the roblems connected with the Gibbs henomenon taking lace in [3]. A good agreement between the obtained solution and some well-known results has been obtained. Four test examles are rovided to show the advantage of using wavelets. This method is caable of greatly E() Figure 28: Comarison of the errors.

8 8 International Engineering Mathematics F () F() Figure 29: The exact transform, F () (solid line), and the aroximate transform, F() (dotted line) reducing the size of calculations while still maintaining high accuracy of the numerical solution. Proosed wavelet method is very simle and attractive. The imlementation of the current aroach in analogy to existing methods is more convenient and the accuracy is high. The numerical examle and the comared results suort our claim. The difference between the exact and aroximate solutions for each examle was lotted grahically to determine the accuracy of numerical solutions. 5.. Future Work. Since comutational work is fully suortive of comatibility of the roosed algorithm, the same may be extended to other hysical roblems also. A very high level of accuracy exlicitly reflects the reliability of this scheme for such roblems. We would like to stress that the aroximate solution includes not only time information but also frequency information due to the localization roerty of wavelet basis; with some change we can aly this method with the hel of other wavelet bases Conflict of Interests The authors declare that there is no conflict of interests regarding the ublication of this aer E() Figure 3: Comarison of the errors. F () F() Figure 3: The exact transform, F () (solid line), and the aroximate transform, F() (dotted line)....2 E() Figure 32: Comarison of the errors. Acknowledgments The authors would like to exress their sincere gratitude to the editor and reviewers for their constructive comments and suggestion. References [] C. J. Tranter, Integral Transforms in Mathematical Physics, Methuen, 956. [2]E.B.Postnikov, AboutcalculationoftheHankeltransform using reliminary wavelet transform, JournalofAliedMathematics,vol.6, ,22. [3] J. C. Fernandes, M. T. Vilhena, B. E. Bodmann, and V. Borges, On the build-u factor from the multi-grou neutron diffusion equation with cylindrical symmetry, World Nuclear Science and Technology,vol.3,no.,. 5,23. [4] D.W.Zhang,X.-C.Yuan,N.Q.Ngo,andP.Shum, FastHankel transform and its alication for studying the roagation of cylindrical electromagnetic fields, Otics Exress, vol., no. 2, , 22. [5] S. Ueda, Alication of Hankel transform for solving a fracture roblem of a cracked iezoelectric stri under thermal loading, in Fourier Transform Materials Analysis, S. M. Salih, Ed., chater, InTech, Rijeka, Croatia, 22. [6] V.Magni,G.Cerullo,andS.DeSilvestri, Highaccuracyfast Hankel transform for otical beam roagation, the Otical Society of America A,vol.9,no., ,992. [7] R.Barakat,E.Parshall,andB.H.Sandler, Zero-orderHankel transformation algorithms based on Filon quadrature hilosohy for diffraction otics and beam roagation, the Otical Society of America A,vol.5,no.3, ,998.

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