Approximation of Fresnel Integrals with Applications to Diffraction Problems

Home , Fresnel diffraction, Fresnel integral

Hindawi Mathematical Problems in Engineering Volume 2018, Article ID 4031793, 13 pages https://doi.org/10.1155/2018/4031793

Research Article Approximation of Fresnel Integrals with Applications to Diffraction Problems

M. Sandoval-Hernandez ,1,2 H. Vazquez-Leal ,3 L. Hernandez-Martinez,1 U. A. Filobello-Nino,3 V. M. Jimenez-Fernandez,3 A. L. Herrera-May,4,5 R. Castaneda-Sheissa,3 R. C. Ambrosio-Lazaro,6 and G. Diaz-Arango 1

1 Instituto Nacional de Astrof´ısica, Optica´ y Electronica,´ Luis Enrique Erro No. 1, Sta. Mar´ıa, 72840 Tonantzintla, PUE, Mexico 2DGETI-CBTis No. 268, Av. La Bamba S/N Esq. Boulevard Jarocho, Fracc. Geo Villas del Puerto, 91777 Veracruz, VER, Mexico 3Facultad de Instrumentacion´ Electronica,UniversidadVeracruzana,Cto.GonzaloAguirreBeltr´ an´ S/N, 91000 Xalapa, VER, Mexico 4Centro de Investigacion´ de Micro y Nanotecnolog´ıa, Universidad Veracruzana, Calzada Ru´ız Cortines 455, 94292 Boca del R´ıo,VER,Mexico 5Facultad de Ingenier´ıa de la Construccion´ y el Habitat,´ Universidad Veracruzana, Calzada Ru´ız Cortines 455, 94294 Boca del R´ıo,VER,Mexico 6Facultad de Electronica,´ Benemerita Universidad AutonomadePuebla,Av.SanClaudioS/N,72000Puebla,PUE,Mexico´

Correspondence should be addressed to M. Sandoval-Hernandez; [email protected]

Received 3 April 2018; Accepted 5 June 2018; Published 4 July 2018

Academic Editor: Gisele Mophou

Copyright © 2018 M. Sandoval-Hernandez et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Tis article introduces two approximations that allow the evaluation of Fresnel integrals without the need for using numerical algorithms. Tese equations accomplish the characteristic of being continuous in the same interval as Fresnel. Both expressions have been determined applying the least squares method to suitable expressions. Accuracy of equations improves as � increases; −5 as for small values of �, it is possible to achieve an absolute error less than 8×10 . To probe the efciency of the equations, two case studies are presented, both applied in the optics feld. Te frst case is related to the semi-infnite opaque screen for Fresnel difraction. In this case study Fresnel integrals are evaluated with the proposed equations to calculate the irradiance distribution and the Cornu spiral for difraction computations of the Fresnel difraction; obtained results show a good accuracy. Te second case is related to the double aperture problem for Fresnel difraction.

1. Introduction that the propagation rate of truncated, nondifracting, and accelerating beams could be accurately calculated using a Fresnel integrals are two transcendental functions named method that represents the beams by a fnite superposition of afer Augustin-Jean Fresnel; these are widely used in physics, Gaussian[1]tobeinturnpropagatedbymeansoftheFresnel specially in optics and electromagnetic theory [1, 2]. For instance, in optics, solution of the Fresnel difraction for difractionintegral.In[9],theiterativeFresnelintegrals a rectangular aperture is calculated, usually, in terms of method (IFIM) has been applied, for the frst time, to the integrals that cannot be obtained by analytical methods [3– simulation and generation of the complete near-feld Fresnel � 6]. Tese integrals are known as Fresnel integrals and contain difraction images crated by -apertures. Te article [2] arguments in terms of the rectangular coordinate system provides a development for the double Fresnel integral based (Cartesian coordinate system). on adequate variable changes of the problem under study to In [7] the iterative Fresnel integral method is introduced predict electromagnetic reduction of the double razor edge for numerical calculation of the Fresnel difraction patterns with refection on the earth surface. In [10], Fresnel integrals from rectangular apertures tilted at an arbitrary angle to were applied to obtain the derivation of linear FM Pulse the optical axis. Te work presented in [8] demonstrated Compression Spectra. 2 Mathematical Problems in Engineering

For some years proposals to evaluate these integrals have been provided. Te work from [11] introduces polynomial 0.6 approximations to evaluate Fresnel integrals. Tese approxi- 0.4 mations are applied in difraction analysis [12]. An algorithm 0.2 to evaluate Fresnel integrals based on the continued fractions −4 −3 −2 −1 0 1234 method is provided in [13]. On the other hand, to evaluate −0.2 Fresnel integrals for high values of |�|, the solution is resort x to the expansion in terms of Bessel functions [14, 15]. Besides, −0.4 in [16], one of the performed approximations was made by −0.6 the sum of the frst term of the Fresnel integrals asymptotic expansion and an exponential function which approaches C(x) infnity at the zero of Fresnel function argument. S(x) Likewise, several diferent numerical alternatives have been proposed to evaluate Fresnel integrals. [17] presented Figure 1: Fresnel integrals. the Chebyshev approximations for the Fresnel integrals which employ diferent values of the argument at diferent evaluation intervals, depending on its magnitude. In [18], a 0.6 method for spreadsheet computations of Fresnel integrals to six signifcant fgures is introduced; it is based on successive improvements of known relational approximations which are 0.4 accurate to just three fgures. S Tis work presents two expressions to calculate, analyt- 0.2 ically, the Fresnel sine integral and Fresnel cosine integral; with our approximations it is possible to evaluate Fresnel integrals in a simple way without the need for using numerical algorithms like Simpson’s rule, among others. Tese −0.8 −0.6 −0.4 −0.2 0 0.2 0.60.4 0.8 C expressions can be evaluated using any calculus sofware −0.2 or standard scientifc calculator. In this article we assume that exact Fresnel functions are the ones provided by Maple 17. −0.4 Te paper is organized as follows. Section 2 provides an introduction to the Fresnel integrals. In Section 3, the −0.6 least squares method is presented. Deduction of equations that approximate Fresnel integrals is performed in Section 4. Study cases are introduced in Section 5. Section 6 presents Cornu Spiral thediscussionabouttheobtainedresultsandefciencyofthe equations. Finally, conclusions of this work are provided in Figure 2: Cornu spiral. Section 7.

thus, it is possible to obtain analytic functions for one variable 2. Fresnel Integrals [20]. Te Fresnel sine and cosine integrals are represented by √� � (�) = (√� (√��) + √−� (√−��)) , (3) �(�) and �(�), respectively; these are two integral functions 4 erf erf that originate by applying the analysis of Fresnel difraction phenomena, which is defned by the following inte- √� � (�) = (√−� erf (√��) + √� erf (√−��)) . (4) grals: 4 Figure 1 shows the Fresnel integrals graph. � ��2 Te Cornu spiral, also known as clothoid or Euler spiral, � (�) = ∫ cos ( )��, (1) �(�) 0 2 is the parametric curve generated by the Fresnel integrals and �(�) from (1) and (2). Te spiral is a tangent curve to the � ��2 abscissa axis at the origin and its radius of curvature decreases � (�) = ∫ sin ( )��, (2) 0 2 inversely in proportion to the distance travelled over it (see Figure 2). In the optics feld, the Cornu spiral can be employed to where �≥0is a real number and � is a real variable. When obtain a quantitative image from a difraction pattern. In � tends to infnite, �=�=1/2[19]. this way, when the difraction is observed from a position Fresnel integrals (�(�) and �(�))areoddfunctionsof close to the difractor obstacle, it is referred to as near-feld �. Tey can be extended to the complex numbers domain; difraction or Fresnel zone. Terefore, in order to perform the Mathematical Problems in Engineering 3 difraction pattern analysis, the semi-periodic Fresnel zones are calculated from the observation point. In consequence, the contribution from each zone is a phasor that adds up. At the limit of the infnitesimal width zones the Cornu spiral is obtained. TeCornuspiral[1]hasthepropertythatitscurvatureat (x2,y2) any point is proportional to the distance along the curvature from the origin. Tis property allows using it as transition curve for tracing highways and railroads, given the fact that a (x1,y1) (xn,yn) mobile that has this displacement at constant speed will have a constant angular acceleration [21, 22]. Te transition curve y=f(x,a1,a2,. . .,ai) has infnite radius at the tangent point to the straight part of the path, radius � at the tangency point to the uniform circular curve; therefore, the most common type of curves Figure 3: Least squares method. on highways is straight stretch-clothoid-circular-clothoid- straight stretch [21, 22].

=0 3. Least Squares Approximations . . Let (�0,�0), (�1,�1), (�2,�3), ...,(��,��)bethecoordinates � = 0,1,2,...,� of a data set, such that ,andtheadjustment n �� 2 curve be �=�(�,�0,�1,�2,...,��),where�� (� = 0, 1, 2, . . .) = (∑ ((��)−�(��,�0,�1,�2,...,��)) ) areadjustmentconstants.Teleastsquaresapproximations �� =1 attempt to minimize the sum of the squares from the vertical =0. distances of �� values to the ideal model �(�) and obtain �(� ,� ,� ,...,� ) the model function 0 1 2 � , which minimizes the (6) square error defned by

Solvingsystem(6)foradjustmentconstants��,itis �(�0,�1,�2,...,��) possible to obtain a model that provides the best ft to the data

� set. Tis method allows creating a continuous function in the 2 (5) space with a minimized error with respect to the samples = ∑ ((��)−�(��,�0,�1,�2,...,��)) . �=1 [23]. In consequence, it is possible to test various models and select the one that provides the best ft. Figure 3 depicts the methodology. Notice that in Figure 3 the asterisk symbol Terefore, (5) is designed to minimize the error with represents the data set. For every data element a value (��,��) respect to the sample set. To do this, partial derivatives are is given, where � = 1, 2, 3, . . ..Itispossibletoobtainthe defned with respect to every adjustment constant, creating a best function that fts the data set, represented by the blue system of nonlinear equations given as line.

� �� 2 4. Deduction of Approximations = (∑ ((� )−�(�,� ,� ,� ,...,� )) ) �� 0 1 2 � 0 0 �=1 Tis article proposes two equations which allow the numerical evaluation of the sine and cosine integrals (�(�) and �(�)) =0 given by (1) and (2), respectively. � �� 2 = (∑ ((��)−�(��,�0,�1,�2,...,��)) ) 2 �� sin (�� /2) 0 0 �=1 � (�) =(− =0 �(|�| +20�exp (−200�√|�|))

� 8 3 �� 2 + (1 − exp (−�� )) = (∑ ((��)−�(��,�0,�1,�2,...,��)) ) 25 ��1 ��1 (7) �=1 2 9 + (1 − (− ��2)) =0 25 exp 2

� �� 2 1 = (∑ ((� )−�(�,� ,� ,� ,...,� )) ) + (1 − (−��))) (�) � � 0 1 2 � 10 exp sgn ��2 ��2 �=1 4 Mathematical Problems in Engineering

2 cos (�� /2) � (�) =(− 0.025 �(|�| +��exp (��√|�|)) 0.020 0.015 8 3 2 2 0.010 + (1 − exp (−�� )) + (1 − exp (−�� )) (8) 25 25 Absolut error 0.005 0 1 9 0 1234567 + (1 − (− ��))) (�) 10 exp 2 sgn x Figure 4: Absolute error for �(�) in the interval 0≤�≤7.

tends to 1/2. Using as adjustment functions (7) and (8) and Notice that in (7) and (8) the frst term decreases at the NonlinearFit command in Maple 17, model parameters therateof1/�� [13]. In consequence, the frst term in areadjustedtothedatasetreducingtheerrorbyleastsquares. these equations dictates the behaviour for each one since it In the procedure of obtaining the approximate models, the acts as the characteristic term of a solution just like the data set (��, ��) consists of 300 elements, obtained from the solution of a diferential equation [5]. Te following terms exact solutions �(�) and �(�) for the 0≤�≤3interval; represent the stationary state, when approaching infnity therefore, the resultant equations are

(� |�|2 /2) sin 8 69 3 2 9 2 � (�) =(− + (1 − exp (− � |�| )) + (1 − exp (− � |�| )) �(|�| + 20� exp (−200�√|�|)) 25 100 25 2 (9) 1 + (1 − (−1.55294068198794� |�|))) (�) , 10 exp sgn

2 cos (� |�| /2) � (�) =(− �(|�| + 16.7312774552827� exp (−1.57638860756614�√|�|)) 8 2 + (1 − (−0.608707749430681� |�|3)) + (1 − (−1.71402838165388� |�|2)) 25 exp 25 exp (10) 1 9 + (1 − (− � |�|))) (�) . 10 exp 10 sgn

1 1 + 0.926� In Figures 4 and 5 the absolute error is presented for (9) � (�) = −( +�(�)) 2 and (10), respectively. Te maximum error for (9) is about 2 2 + 1.792� + 3.104� 0.0297629 � = 1.4 when .ItcanbenoticedinFigure4that ��2 � ⋅ ( ) accuracy improves as value increases. cos 2 For (10), the maximum absolute error occurs when �= (12) 1.62 and is 0.0207910487.Likewise,inFigure5itcanbe 1 � −( +�(�)) noticedthattheerrorfor(10)tendstodecreaseas value 2 + 4.142� + 3.492�2 +6.67�3 increases. ��2 In [18] two approximations are proposed to evaluate the ⋅ ( ) sine and cosine integrals; there are sin 2

1 1 + 0.926� −3 � (�) = +( +�(�)) |�(�)| ≤ 2 × 10 �≥0 2 where and . 2 2 + 1.792� + 3.104� As a way to compare the accuracy of our approximations, ��2 Figure 6 depicts the behaviour of the absolute error between ⋅ ( ) sin 2 (9) and (11), which approximate the cosine integral. Initially the absolute error in (9) is higher than that in (11), (11) 1 but tends to decrease considerably. At �<6.5,theerroris −( +�(�)) 2 + 4.142� + 3.492�2 + 6.67�3 lower than (11) (see Figure 6(a)). By �=8,theabsoluteerror 10−3 2 for (11) is about , while that for (9) is lower. For instance, �� at �=9, the absolute error for (11) has kept constant, but for ⋅ cos ( ) −4 2 (9) the absolute error has decreased to 10 (see Figure 6(b)). Mathematical Problems in Engineering 5

0.020 10−2 −3 0.015 10 10−4 0.010 10−5 0.005 Absolut error 10−6 0 −7 0 1234567 10 x 0 12345678 Figure 5: Absolute error for �(�) in the interval 0≤�≤7. x Eq. (10) Eq. (12) (a) Absolute error comparison in the interval 0≤�≤8

−3 −2 1. × 10 10 5. × 10−4 −3 10 1. × 10−4 5. × 10−5 10−4 1. × 10−5 −5 10 5. × 10−6 −6 10 1. × 10−6 5. × 10−7 10−7

0 12345678 8 9101112 x x

Eq. (9) Eq. (10) Eq. (11) Eq. (12) (a) Absolute error comparison in the interval 0≤�≤8 (b) Absolute error comparison in the interval 8≤�≤12

10−3 10−4 10−4 10−5 10−5 10−6 −6 10 10−7 10−7 10−8 −9 10−8 10

8 9 101112 30.0 30.1 30.2 30.3 30.4 30.5 x x

Eq. (9) Eq. (10) Eq. (11) Eq. (12) (b) Absolute error comparison in the interval 8≤�≤12 (c) Absolute error comparison in the interval 30 ≤ � ≤ 30.5

Figure 7: Absolute error comparison between (10) and (12). 10−4 10−5 −6 10 For higher � values the accuracy improves for (11) and (9); −3 −5 10−7 for example, at � = 30.5,theabsoluteerroris10 and 10 , 10−8 respectively (see Figure 6(c)). From this analysis, it can be concluded that (9) provides better accuracy than (11). Figure 7 30.0 30.1 30.2 30.3 30.4 30.5 depicts the absolute error of approximations for the sine x integral (10) and (12). When � < 3.5,absoluteerrorfor(10)is −3 Eq. (9) higher than that for (12), with a maximum error of 10 (see Eq. (11) Figure 7(a)). � > 3.5 (c) Absolute error comparison in the interval 30 ≤ � ≤ 30.5 When , the accuracy improves notoriously for (10). For instance, in Figure 7(b), notice that at �=11the −4 Figure 6: Absolute error comparison between (9) and (11). absolute error is about 1×10 , while (12) keeps its absolute 6 Mathematical Problems in Engineering

0.6 0.6 S S 0.4 0.4 0.2 0.2

−4 −2 −0.2 2 46−4 −2 −0.20 2 46 −0.4 C − C − 0.4 0.6 −0.6

C(x) Exact S(x) C(x) Tis work Tis work (a) Comparison between (1) and (9) (b) Comparison between (2) and (10)

Figure 8: Comparison between exact equations and approximate equations (9) and (10).

z 0.6

S 0.4

0.2 (０5)

(０4)

(０3) −0.6 −0.4 −0.2 0 0.40.2 0.6 C (０2) −0.2 (０1)

−0.4 Σ −0.6

Figure 10: Semi-infnite plane. Exact Tis work Figure 9: Comparison between Cornu graph and exact equations. 5.1. Te Semi-Infnite Opaque Screen for Fresnel Difraction. For this study case we will consider the semi-infnite opaque screen defned by the Σ plane (see Figure 10), where �� is 2 2 the intensity at point � in units �� /� or �/� and �0 is −3 error at about 10 .Teaccuracyof(10)continuestoimprove the intensity of these incident parallel wavefronts. From [1], for larger � values. At � = 30.4, the absolute error approaches intensity at � is proportional to 10−6 1×10−3 , while (12) keeps its error magnitude close to (see 2 2 �� 1 1 Figure 7(c)). Nevertheless, although (9) and (10) exhibit an �� = =[( −�(�)) +( −�(�)) ]. (13) improved accuracy as � increases, they just handle one term 2 2 2 that describes the behaviour of the Fresnel integrals while (11) and (12) handle two terms. For simplicity, we assume that the semi-infnite opaque Figure 8 shows a comparison between approximations, screen for Fresnel difraction is illuminated by a plane wave (9) and (10), and exact Fresnel integrals (1) and (2). On the given by other hand, Figure 9 shows the Cornu graphic generated 2 by the exact equations (9) and (10). It can be noticed that V (�) =�√ , (14) inside the spiral, as � value increases, accuracy improves �� notoriously. where � is the source, � the observation point, and � the wavelength of the projected light beam. By substituting (14) 5. Study Cases in (9) and (10), equations are now in terms of �;thus,itis possible to calculate �(V(�)) and �(V(�)).Now,substitutingin Tis section provides two study cases applied to the optics (13), we obtain feld, specifcally, the difraction phenomenon. Te frst 2 2 addresses the semi-infnite opaque screen subject; the second �� 1 1 � = =[( −�(V (�))) +( −�(V (�))) ]. (15) is about the double aperture problem for Fresnel difraction. � 2 2 2 Mathematical Problems in Engineering 7

(P4) (P4) 1.2 1.2 1.0 1.0 0.8 0.8 (P5) 0.6 (P5) 0.6 Edge, I/Io Edge, 0.4 I/Io Edge, 0.4 (P3) (P2) P3 P2 0.2 (P1) ( )0.2 ( ) (P1)

−0.004 −0.003 −0.002 −0.001 0 0.001 0.002 −0.004 −0.003 −0.002 −0.001 0 0.001 0.002 S S Exact Tis work Exact Tis work Point exact Point approximate Point exact Point approximate (a) Te corresponding calculated irradiance distribution with �= (a) Te corresponding calculated irradiance distribution with �= 4.0� and � = 632.8�� 4.0� and � = 540�� (P1) (P1)

0.6 0.6 S 0.4 S 0.4

0.2 0.2 (P2) (P2) (P3) (P3) −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 C C −0.2 −0.2

(P5) (P5) −0.4 −0.4

−0.6 −0.6 (P4) (P4)

Cornus Spiral Exact Cornus Spiral TW Cornus Spiral Exact Cornus Spiral TW Point Exact Point T W Point Exact Point T W (b) Te Cornu spiral for a semi-infnite opaque screen with �= (b) Te Cornu spiral for a semi-infnite screen with � = 4.0� 4.0� and � = 632.8�� and � = 540�� Figure 11: Te semi-infnite opaque screen with � = 4.0� and �= Figure 12: Te semi-infnite opaque screen with � = 4.0� and �= 632.8��. 540��.

the corresponding calculated irradiance distribution and the Proposed numerical values are � = 4.0� and �= Cornu spiral in Figures 11–14. 632.8��, 540��, 390��, 100��.Toobtainthequantitative Observation points �1 and �2 are on the shade of the values of the intensities from the Cornu spiral, the length Σ plane. Te arc cords decrease monotonically within the from point (0.5,0.5)topoints�1, �2, �3, �4,and�5 is region; in consequence, the irradiance abruptly declines. In measured, calculated using (9) and (10). fact, as � decreases, the bands move closer to the edge Likewise, Figures 11–14 depict the diagrams of the irra- and become thinner [1]. Observation points were calculated diance distribution and Cornu spiral for a semi-infnite using (9) and (10) and compared with the ones obtained by opaque screen, obtained using (9) and (10), against the exact exact equations. Tese points are depicted in the magnitude equations for diferent values of �.TeoriginoftheCornu diagrams for � = 632.8��, � = 540��, � = 390��,and�= spiral diagram corresponds to the edge of the semi-infnite 100�� (see Figures 11(a), 12(a), 13(a), and 14(a), respectively). opaque screen, point (�3), where the intensity has decreased Te points obtained by the approximate method in these a quarter for negative values of V;innumericalterms,we diagrams keep a relationship with the approximated Cornu have that �(0) = �(0) = 0,and�� =�0/4 because spiral in Figures 11(b), 12(b), 13(b), and 14(b), respectively. half of the waveform is obstructed. Te amplitude of the perturbation has been reduced by half and the irradiance drops by a quarter. As already mentioned, this occurs at 5.2. Te Double Aperture Problem for Fresnel Difraction. point �3 of the plane (see Figure 10), which can be located at Light difraction coming through a rectangular aperture is 8 Mathematical Problems in Engineering

(P4) (P4) 1.2 1.2 1.0 1.0 0.8 0.8 (P5) 0.6 (P5) 0.6 Edge, I/Io Edge,

Edge, I/Io Edge, 0.4 0.4 (P3) P2 (P3) P2 0.2 ( ) (P1) 0.2 ( ) (P1)

−0.003 −0.002 −0.001 0 0.001 0.002 −0.003 −0.002 −0.001 0 0.001 0.002 S S Exact Tis work Exact Tis work Point exact Point approximate Point exact Point approximate (a) Te corresponding calculated irradiance distribution with (a) Te corresponding calculated irradiance distribution with � = 4.0� and � = 390�� = 4.0� and � = 100��

0.6 0.6 (P2) (P1) S 0.4 S 0.4 (P2) (P1)

0.2 0.2 (P3) (P3) −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 C −0.2 C −0.2 P5 ( ) (P5) −0.4 −0.4

−0.6 − (P4) 0.6 (P4) Cornus Spiral Exact Cornus Spiral TW Point Exact Point T W Cornus Spiral Exact Cornus Spiral TW Point Exact Point T W (b) Te Cornu spiral for a semi-infnite screen with � = 4.0� and � = 390�� (b) Te Cornu spiral for a semi-infnite screen with � = 4.0� and � = 100�� Figure 13: Te semi-infnite opaque screen with � = 4.0� and �= 390��. Figure 14: Te semi-infnite opaque screen with � = 4.0� and �= 100��.

analysed in [1, 25]. Tis study case considers the double To generate the simulation shown in Figure 16(b), built- aperture for Fresnel difraction presented and studied in in functions of Matlab were replaced by the proposed [24] (see Figure 15). We assume that the light source with � � � functions (9) and (10) (see the Appendix). Te Pearson wavelength is ,atadistance 0,andtheapertureiscentred � �, � correlation coefcient, ,iswildlyemployedinstatistical on the Cartesian system ; the origin of the coordinate analysis, pattern recognition, and image processing [26, 27]. system is located at the exact centre of the double aperture For the monochromatic digital images, the Person correlation (see Figure 15). Upper and lower edges of the aperture are coefcient is defned as located at �=�and �=−�, respectively. For this system, �−� the width for each aperture is ( )andseparationbetween ∑ (� −� )(� −� ) centresis(�+�). Besides, it is considered that difracted light �= �=1 � � � � , � √ 2√ 2 (16) is observed on a screen located at � away from it. Reference ∑�=1 (�� −��) ∑�=1 (�� −��) [24] can be consulted for further detailed analysis of this study case. �ℎ �� Simulations performed in Matlab for this study case where �� and �� are intensity values of � pixel in 1 �� (based on [24]) are presented in Figure 16(a). Tis work and 2 image, respectively; �� and �� are mean intensity considers the same aperture proportions given in [24], these values of frst and second image, respectively. Te correlation are 2� = 2��, 2� = 4��, 2� = 6��, �0 = 400��,image coefcient is �=1when both images are completely identical. area 10�� × 10��,andstepsizeof0.01��. Additionally, For this study case, correlation value is very high between the the illumination wavelength is � = 632��. images generated with a value of 0.999964736. Mathematical Problems in Engineering 9

Figure 15: Confguration of the double aperture problem for Fresnel difraction.

−10 −10 −8 −8 −6 −6 −4 −4 −2 −2 0 0 2 2 4 4 6 6 8 8 10 10 −10 −8 −6 −4 −2 0 2 4 6 8 10 −10 −8−6−4−20246810 (a) Computer simulated Fresnel difraction using approximate equations (b) Computer simulated Fresnel difraction using approximate equations and iterative Fresnel integrals method [24] (9) and (10)

Figure 16: Computer simulated Fresnel difraction from double rectangular apertures.

6. Discussion Notice how the error decreases on observation points �1 and �2 (see Tables 1–4), because they are internally Tis section presents numerical results of the previous study moving within the spiral since its geometry is contracting cases. For the frst case, the comparison is performed using and are in function of the difracted light wavelength. For the Cornus spiral and observation points of the irradiance observation points �4 and �5,theerrorhasincreasedtoabout −7 distribution. We calculated the absolute error diference 8×10 because they have moved to the right, closer to from (0.5, 0.5) to every observation points of the difraction the origin. Tis happens for two reasons: the wavelength is pattern. Table 1 shows data for the case when � = 4.0� and getting smaller and we approach the region where the error � = 632.8��;theerroris0.0185625807 and 0.006802459 at value is higher for (9) and (10). Nevertheless, in Table 4, the observation points �1 and �2,respectively. absolute error at �5 has diminished due to the accuracy of our 10 Mathematical Problems in Engineering

Table 1: Observation points comparison for � = 4.0� and � = 632.8��.

Observation point Exact Approximate Absolute error 1 0.227105905 0.208543324 0.018562581 2 0.375123732 0.381926192 0.006802459 3 0.707106781 0.707106781 0.000000000 4 1.655561529 1.643873584 0.011690945 5 1.247629737 1.245639720 0.001990017

Table 2: Observation points comparison for � = 4.0� and � = 540��.

Observation point Exact Approximate Absolute error 1 0.212072842 0.199156973 0.012915869 2 0.358007985 0.364436149 0.006428163 3 0.707106781 0.707106781 0.000000000 4 1.655525630 1.643873930 0.011651701 5 1.247648595 1.245609700 0.002038895

Table 3: Observation point comparison for � = 4.0� and � = 390��.

Observation point Exact Approximate Absolute error 1 0.183070676 0.187027639 0.003956963 2 0.322264740 0.326617422 0.004352681 3 0.707106781 0.707106781 0.000000000 4 1.655510405 1.643827879 0.011682526 5 1.247659193 1.245607630 0.002051563

Table 4: Observation point comparison for � = 4.0� and � = 100��.

Observation point Exact Approximate Absolute error 1 0.094716894 0.094879587 0.000162693 2 0.185219606 0.188826768 0.003607161 3 0.707106781 0.707106781 0.000000000 4 1.655561120 1.643873413 0.011687707 5 1.247601422 1.245601596 0.001999826 proposed expressions (see Figures 4 and 5). For observation shading zone, and the zero are kept at the origin of the Cornu point �3, the error is zero becuase the intersection with � axis diagram. Te proposed equations in this work are continuous is always the same. Tis error is in good agreement with the throughout the range of real numbers and are evaluated at exact spiral because (9) and (10) evaluated at �=0are equal the origin, with an exact value of zero. As for the second to zero. study case, simulations showed a correlation close to the unit. For the second study case, the Pearson correlation coef- Terefore, the simulations of the double aperture problem for cient value was 0.999964736, which indicates that properties Fresnel difraction produce near identical fgures; thus, the of difraction patterns from the reported simulation in simulation in Figure 16(a) presented in [24] keeps an almost [24] and the ones obtained from our approximations (see perfect correlation against the simulation in Figure 16(b). Figure 16) are identical. Tis result shows the accuracy that With the obtained results, the proposed equations can be can be accomplished by using our proposal on applications employed in diverse applications, specially for values of for Fresnel difraction. � > 1.5;asthevalueof� increases, accuracy improves considerably. Applying the experience of approximate Fresnel 7. Conclusions functions on this paper, as a future work, we will attempt to improve precision of the results. One possible path to Approximations proposed on this article allow evaluating followwouldbetheuseofadiferenttechnique,besidesleast Fresnel sine and Fresnel cosine integrals with high accuracy. squares method, that allows improving precision closer to In the frst study case, we built the Cornu spiral and the the origin region; among other techniques to be applied is irradiance distribution with good results since the lower the power series extender method [28, 29]. Tis method may wavelength of the difraction, the higher accuracy on the be useful because it provides highly accurate approximations Mathematical Problems in Engineering 11 at the origin, while maintaining good prediction for large %t, domains. format long

Appendix l=lnm*1e-6; a=amm*sqrt(2/(l*q0mm)); Modified Matlab/Octave Program for b=bmm*sqrt(2/(l*q0mm)); the Double Aperture [24] c=cmm*sqrt(2/(l*q0mm)); W=Wmm*sqrt(2/(l*q0mm)); s=smm*sqrt(2/(l*q0mm)); %The original program of Matlab was publicated on dimenU=b:s:W+b; %Computer simulation of Fresnel diffraction longiU=length(dimenU); from double rectangular Cu2=zeros(1,longiU); %apertures in one and two dimensions using Su2=zeros(1,longiU); the iterative Fresnel %integrals method. dimenU=a:s:W+a; % longiU=length(dimenU); %Abedin, Kazi Monowar and Rahman, SM Mujibur. Cu1=zeros(1,longiU); % Optics & Laser Technology, vol.44(2), Su1=zeros(1,longiU); pp.394--402. %2012, Elsevier dimenU=-a:s:W-a; longiU=length(dimenU); Cu3=zeros(1,longiU); %Program modified by Mario Sandoval H. Su3=zeros(1,longiU); %New name of funtion Matlab/Octave function D=dbslitMprueba(amm,bmm,cmm,Wmm, dimenU=-b:s:W-b; smm,q0mm,lnm,t) longiU=length(dimenU); %Replace to------Cu4=zeros(1,longiU); ------Su4=zeros(1,longiU); %function D=dbslit(amm,bmm,cmm,Wmm,smm, %------q0mm,lnm,t) dimenV=c:s:W+c; ------longiV=length(dimenV); Cv2=zeros(1,longiV); %Date of modification Sv2=zeros(1,longiV); %01/12/2017 dimenV=-c:s:W-c; %Using the program dbslit, the values of the longiV=length(dimenV); inputs, i.e., aperture Cv1=zeros(1,longiV); %dimension Sv1=zeros(1,longiV); %Arguments %half-widths: a, b and c in [mm], %amm, %Functions modified %bmm, %FFRESNELC and FFRESNELS replace to function %cmm, inter-build of matlab %Image area half-width W in [mm], %FresnelC and FresnelS %Wmm, %Step size s in [mm], %You must be write the equations (9) and (10) %smm, on file .M each one. %The aperture-image plane distance i=0; q0 in [mm], for contador=b:s:W+b %q0mm, i=i+1; %The illuminating wavelength l in [nm] Cu2(1,i)=FFRESNELC(contador); %lnm, Su2(1,i)=FFRESNELS(contador); %and the intensity factor t are input to the end program. The intensity %factor t is used to control the apparent i=0; visual intensity in the for contador=a:s:W+a %generated image. For most of the images, i=i+1; a factor of unity is good Cu1(1,i)=FFRESNELC(contador); %enough. 12 Mathematical Problems in Engineering

Su1(1,i)=FFRESNELS(contador); E(-p+2+m,q)=E(p,q); end end end i=0; y=-W:s:W; for contador=-a:s:W-a ymm=y*sqrt((l*q0mm)/2); i=i+1; imagesc(ymm,ymm,E,[01]);colormap(gray); Cu3(1,i)=FFRESNELC(contador); Su3(1,i)=FFRESNELS(contador); end Data Availability Tedatausedtosupportthefndingsofthisstudyare i=0; available from the corresponding author upon request. for contador=-b:s:W-b i=i+1; Cu4(1,i)=FFRESNELC(contador); Conflicts of Interest Su4(1,i)=FFRESNELS(contador); end Te authors declare that they have no conficts of interest.

∧ A=(Cu2+Cu3-Cu1-Cu4). 2+(Su2+Su3-Su1-Su4) ∧ Acknowledgments . 2; Te frst author is currently doing a postdoctoral stay at i=0; the Instituto Nacional de Astrofsica, Optica y Electronica in for contador=c:s:W+c Puebla, Mexico, and wishes to thank M. Eng. Roberto Ru´ız i=i+1; Gomez and M.TN. Cesar´ Colocia Garc´ıa for their support Cv2(1,i)=FFRESNELC(contador); andtoacknowledgethefnancialsupportfromtheSecretary Sv2(1,i)=FFRESNELS(contador); of Public Education of Mexico´ (SEP) through Grant 2703 end E476300.0275652.

i=0; References for contador=-c:s:W-c i=i+1; [1] E. Hecht and A. Zajac, “Optics, massachusetts,” 1987. Cv1(1,i)=FFRESNELC(contador); [2] V. Balassone and H. Romero, “Determination of the solution Sv1(1,i)=FFRESNELS(contador); of the integral double of fresnel to predict the electromagnetic end attenuation in the case of double knife edges with ground refex- ion,” Revista Tecnica´ de la Facultad de Ingeniera Universidad del ∧ ∧ B=(Cv1-Cv2). 2+(Sv1-Sv2). 2; Zulia,vol.38,pp.83–91,2015. � B=B ; [3]E.J.Purcell,S.E.Rigdon,andD.E.Varberg,“Calculo,”´ Pearson n=size(B); Educacion´ , 2007. B=repmat(B(:,1),1,n); [4]C.H.EdwardsandD.E.Penney,“Calculo´ con trascendentes � A=A ; tempranas,” Pearson Educacion,´ 2008. A=repmat(A(:,1),1,n); [5] D. G. Zill, “Calculo´ con geometra analtica,” Grupo Editorial � A=(A) ; Iberoamerica,´ 1987. D=B*A; [6] A. A. Goloborodko, “Computer simulation of Talbot phe- D=t*D/max(max(D)); nomenon using the Fresnel integrals approach,” Optik - Inter- m=2*fix(((2*W)/s)/2); national Journal for Light and Electron Optics,vol.127,no.10, E=zeros(m+1,m+1); pp. 4478–4482, 2016. for q=1:1:m/2+1 [7] K.M.AbedinandS.M.MujiburRahman,“TeiterativeFresnel for p=1:1:m/2+1 integrals method for Fresnel difraction from tilted rectangular E(m/2+2-p,q+m/2)=D(p,q); apertures: Teory and simulations,” Optics & Laser Technology, end vol. 44, no. 4, pp. 939–947, 2012. end [8]M.Cywiak,D.Cywiak,andE.Ya´nez,˜ “Finite Gaussian wavelet superposition and Fresnel difraction integral for calculating for q=m+1:-1:m/2+2 the propagation of truncated, non-difracting and accelerating for p=1:1:m/2+1 beams,” Optics Communications,vol.405,pp.132–142,2017. E(p,-q+2+m)=E(p,q); [9] K. M. Abedin and S. M. M. Rahman, “Fresnel difraction from end N-apertures: Computer simulation by iterative Fresnel integrals end method,” Optik - International Journal for Light and Electron Optics,vol.126,no.23,pp.3743–3751,2015. [10] C. E. Cook, “Pulse Compression—Key to More Efcient Radar for q=1:1:m+1 Transmission,” Proceedings of the IRE,vol.48,no.3,pp.310–316, for p=1:1:m/2 1960. Mathematical Problems in Engineering 13

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