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Optics Communications 266 (2006) 558–561

Prevailing effects of interference or diffraction by multiple apertures

Orlando Quintero, John Fredy Barrera *, Rodrigo Henao, Francisco F. Medina

Grupo de Optica y Fotonica, Instituto de Fısica, Universidad de Antioquia, A.A. 1226 Medellın, Colombia

Received 13 December 2005; received in revised form 15 March 2006; accepted 3 May 2006

Abstract

The effects of interference and diffraction produced by a bidimensional structure composed of multiple and identical apertures arestudied. Such effects are competitive during the propagation, that is, in some planes the interference effects are more visible, while inothers prevail the diffraction ones. Using the number of Fresnel zones, the spatial domains where the interference or diffraction effectsprevail are defined and parameterized. Simulation and experimental results are further introduced to verify the proposed method.� 2006 Elsevier B.V. All rights reserved.

PACS: 42.25.Hz; 42.25.Fx

Keywords: Interference; Fraunhofer diffraction; Fresnel diffraction; Fresnel zones

1. Introduction

Let us consider a diffracting structure composed by mul-tiple identical apertures, arranged in a bidimensional way,either periodic or nonperiodically. The effects of interfer-ence and diffraction produced by the structure in the differ-ent planes during its propagation are competitive, this is, insome planes the interference effects are more visible, whilein others prevail the diffraction ones. In the diffraction, theindividual effects become more evident for each aperture;while, in the interference the collective effects are quitenotorious.

For an appropriate analysis of these effects and, ofcourse of the diffracting structure, is necessary to determineeach of these effects in a separate form. Thus, it is conve-nient to establish, by means of an appropriate and simplecriterion, the regions where the interference prevails overthe diffraction and vice versa; as well as, those regionswhere there is not a clear prevalence of an effect over theother one.

0030-4018/$ - see front matter � 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.optcom.2006.05.005

* Corresponding author. Tel.: +57 42105630; fax: +57 42105666.E-mail address: jbarrera@fisica.udea.edu.co (J.F. Barrera).

The objective of this work is to establish a criterion thatallows the delimitation of the regions previously men-tioned. This criterion will be established from the numberof Fresnel zones [1–3]. These zones are determined froman axial point in the observation plane.

This criterion is useful in the analysis of details ordefects in the optoelectronics structures, where selectivelythe details or defects of the main structure can be analyzein separate form, and also, it can be made in much com-fortable observation planes that the direct contact plane,where they are usually carried out [4].

Computer simulations and experimental results areshown to illustrate the different situations.

2. Basic theory

The diffracting structure is inscribed in a circle of radioR, this structure is illuminated by a spherical wave thatcomes from a coherent punctual source S located at a dis-tance r0 of the plane that contains such structure. The num-ber of Fresnel zones is calculated from the point P; wherethe line SP goes through the center of the circle that cir-cumscribes the structure and is perpendicular to it, the cor-responding setup is shown in Fig. 1.

Fig. 1. Setup for diffraction by a diffracting structure.

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Considering that r ^ r0� R and R� k, where k is thewavelength, for calculating the number of Fresnel zones,the parabolic approach can be used. Therefore, the numberof Fresnel zones N will be given by [3,5]

N ¼ R2

k1

r0

þ 1

r

� �ð1Þ

Let us denote by N0 the number of Fresnel zones for anindividual aperture and by N the number of Fresnel zonessubtended by the diffracting structure from that samepoint. Note that number of Fresnel zones assembles allthe geometrical parameters involved in the opticalarrangement.

In this case, a plane wave illumination will be employed,it means r0!1; therefore the number of Fresnel zonesturns into:

N ¼ R2

krð2Þ

and

N 0 ¼R2

0

krð3Þ

where R0 is the radio of an individual aperture.

3. Experimental and simulation processes

In the optical arrangement shown in Fig. 2, the collima-tion system converts a laser beam in a plane wave that

Fig. 2. Experimental setup (CS: collimation system; CCD: camera).

impinges on the diffracting structure, and then the diffrac-tion pattern generated at a r distance of the structure is reg-istered with the CCD camera. The numerical calculationshave been performed using a simulation package thatworks according to the modified convolution approach[6] on a matrix of 512 · 512 points with the sample intervalequivalent to 10 lm.

In the experimental and simulation process, the diffrac-tion pattern generated at a r distance of the structure wasregister and calculated, respectively, when it is illuminatedby a monochromatic plane wave of wavelength equivalentto k = 632.8 nm. In both cases triangular and square struc-tures were used; these structures are shown in Fig. 3(a) and(b). Although for easiness circular apertures are consideredto illustrate the validity of the proposal, the number ofFresnel zones criterion can be applied when the diffractingstructure is composed by apertures with asymmetricalshapes. When irregular apertures are employed, the exter-nal and the internal circumferences that circumscribe theseapertures must have an appropriate relation [7].

Three regions are defined for the analysis of the results,according to the different values that N and N0 take:

1. Region of Fraunhofer: N < 1, and therefore N0 < 1.2. Region of transition: N > 1, but, N0 < 1.3. Region of Fresnel: N0 > 1, that implies N > 1.

In Tables 1 and 2 are shown the experimental and sim-ulations results in the three regions of interest. In all cases,the computer simulations were obtained by using the sameparameters as those employed in the experimental results.Furthermore, computer simulations performed with two(Young’s experiment) and three identical aligned circularapertures are presented in Table 3. In all experiments andsimulations the radio of the circular apertures was r0 =0.175 mm.

The experimental and simulation results show theobtained patterns with different structures in the regionsof Fresnel, transition, and Fresnel. In the Fresnel regionis possible to know the shape of the individual apertures

Fig. 3. Diffracting structures used in the simulations and experiments:(a) triangular structure, formed by three identical circular apertures,placed in the vertexes of an equilateral triangle and (b) square structure,formed by four identical circular apertures, placed in the vertexes of asquare. In both structures, the radio of the each circular aperture wasR0 = 0.175 mm.

Table 1Experimental and simulation results

Diffraction patterns generated by the triangular structure for differentnumber of Fresnel zones N and N0.

Table 3Computer simulations

Diffraction patterns generated by structures with two (Young’s experi-ment) and three aligned apertures for the same number of Fresnel zones N

and N0.

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and the structure, due to the diffraction patterns generatedby each aperture are not overleaping. In the transitionregion the interference and diffraction effects generated bythe diffracting structure are combined, there is a weak rec-ognition on shape of the apertures and on the shape of

Table 2Experimental and simulation results

Diffraction patterns generated by the square structure for different numberof Fresnel zones N and N0.

the diffracting structure, furthermore, the interferenceeffects can be observed. The prevailing of interference or dif-fraction effect depends on the proximity of the observationplane with the Fresnel or Fraunhofer domain. When intothe transition region the observation plane is proximalto the Fresnel region, the recognition on the aperturesand the structure increases and the interference effectdecreases, while it is near the Fraunhofer region the interfer-ence effect increases and the weak recognition of the struc-ture and the apertures decreases until that it is not possiblededuce the shape of the structure and the individual aper-tures. Starting from the patterns obtained in Fraunhoferregion is practically impossible to deduce the shape of theindividual apertures, as well as, infer the shape of the dif-fracting structure. In this region the diffraction patternscoming from the apertures are completely overlapped.

The obtained results using the different structures in thethree established domains present a similar behavior.Where, the simulation and experimental results for thetriangular and rectangular structures present a goodmatching. Also the obtained patterns in the computer sim-ulations by two (Young’s experiment) and three alignedapertures are according to the established criterion. Thediffraction pattern in the Fraunhofer region for Young’sexperiment presents the usual fringe pattern, where theinterference effect is predominant. In Fresnel region the dif-fraction pattern of the individual apertures is obtained,

O. Quintero et al. / Optics Communications 266 (2006) 558–561 561

therefore the diffraction effect prevails. In all cases, for theFraunhofer region the periodicities are present by means offringe patterns. The orientation of these fringe patterns isrelated with the spatial distribution of the apertures intothe structure, this effect is evident in the fringe patterns gen-erated by two and three apertures. Additionally, as the dif-fracting structure is composed by multiple apertures it canbe considered as an arrangement of Young’s pairs [8].

Therefore, the number of Fresnel zones is the parame-ter that determines the regions where the interference orthe diffraction effects are predominant, as well as, theregion in which both effects are significant. This parame-ter depends on the size and spatial arrangement of theindividual apertures that contains the diffracting structure.Therefore in this case, the number of Fresnel zones isused to parameterize the overlapping of the intensitiescoming from the individual diffraction patterns and thephase delay generated by the structure. For example, iftwo structures contain the same number of identical aper-tures, but the spatial position of the apertures in eachstructure is different, so the radio of the circumferencethat circumscribes each diffracting structure also is differ-ent. Then the distances among each structure and theregions of Fresnel, transition, and Fraunhofer are differ-ent for each structure. Besides, the same effect is presentedfor two diffracting structures composed by identical aper-tures, where the radio that circumscribes it is the same butthe size of the individual apertures is different for eachstructure.

4. Conclusions

The experimental and simulations results allow conclud-ing the following:

The number of Fresnel zones N, is the main parameterthat determines when the effect of interference prevails overthe diffraction one and the kind of diffraction generated bythe diffracting structure. Also, the recognition of the shapeof the apertures that conform the structure can be obtainedfrom the number of Fresnel zones N that subtend thestructure.

If N > 1, the effect of interference is tenuous and func-tion, either for the number of apertures or their separation.In this case, there is little information about the periodicityand the effect of diffraction is described by means of theFresnel transform. When N < 1, the effects of interferenceare visible and predominate in relation to the diffractionones and essentially independent from the separation andthe number of the apertures. In the case N 6 1, the effectsof diffraction are described by means of the Fourier trans-form (Fraunhofer diffraction) and prevails the informationof the periodicity.

For N 6 0.33 is practically impossible to deduce theshape of the apertures. When N > 0.33 the recognition ofthe shape of the apertures increases proportionally to thegrowth of N, being important when N > 1.

Starting from the properties of these domains it is possi-ble to establish the forms and sizes of the details or defects,independently of the forms and sizes of the complex struc-tures, this due to the properties of the Fraunhofer andFresnel domains.

Acknowledgments

John Fredy Barrera acknowledges the financial supportfrom COLCIENCIAS – Colombia. The Grants from CODI,Universidad de Antioquia, is gratefully acknowledged.

References

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