Modeling superlattice patterns using the interference of sharp focused spherical waves

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1 Modeling superlattie patterns using the interferene of sharp foused spherial waves Fidirko N.S. Samara State Aerospae University Abstrat. In this paper, modelling of pseudonondiffrational beams forming superlattie strutures in a ross setion has been performed. To reate suh distributions we suggest using superposition of sharp foused spherial waves. Thus, we have done simulations for several spherial waves generated by oherent light soures loated on a ring with a ertain radius. It is shown that depending on the onfiguration of the soure field we an ahieve different superlattie patterns in a ross setion with a small amount of waves in the input field. Using more waves, we an obtain Bessel-like beams in the ross setion. Keywords: interferene, optial vorties, sharp fousing, polarization Citation: Fidirko NS. Modeling superlattie patterns using the interferene of sharp foused spherial waves. Proeedings of Information Tehnology and Nanotehnology (ITNT-2015), CEUR Workshop Proeedings, 2015; 1490: DOI: / Introdution A nondiffrating wave field is omprehended as a monohromati optial field whose transverse shape remains invariant in free-spae propagation. In 1987, Durnin proposed that nondiffrating wave fields are exat solutions to the homogeneous Helmholtz equation [1]; suh partiular solutions an be desribed as Bessel funtions and are alled nondiffrating Bessel beams. The realizable beams that propagate with relatively small divergene angles up to a ertain range have finite energy and are known as pseudonondiffrating optial beams. Along with his o-authors, Durnin first experimentally realized a pseudonondiffrating Bessel beam in a ylindrial oordinates system [2]. Sine then, nondiffrating Bessel beams have been extensively studied and applied in diverse fields, for example optial manipulation, the apture of miro partiles and optial oherene tomography [3-7]. In reent years, the attention of physiists and mathematiians has been drawn to two-dimensional nondiffrative superlattie patterns [8]. Besides, realization of suh distributions related to rystals, quasirystals and other periodi strutures has been atively researhed [9-12]. 45

2 A two dimensional distribution made by superposition of several plain latties is alled a superlattie [8, 12]. In this work, we show an approah to reate superlattie distributions using the interferene of sharp foused spherial waves. Model of sharp fousing A sharp foused eletromagneti field in the foal area in Cartesian oordinates an be desribed with the following equation: 2 PE (, ) E(,, z) if B(, ) T( ) 1 (,, z) H (, ) 0 0 PH (1) exp ik sin os( ) z os sin dd, where (,, z) ylindrial oordinates in the foal area, (, ) spherial angular oordinates of the output pupil of the fousing system, - maximal value of the azimuth angle, related to the numerial aperture, B(, ) transmission funtion, T( ) apodization funtion (for aplanati systems it is T( ) os ), k 2 / wavenumber, wavelength, f foal distane. (, ) P (, ) polarization matrixes for eletri and magneti fields respetively: P P H 2 1 os (os 1) sin os (os 1) ( ) sin os sin sin 2 x E (, ) sin os (os 1) 1 sin (os 1) ; ( ) y 2 x H (, ) 1 sin os 1 sin os (os 1). ( ) y 2 sin os (os 1) 1 os os 1 ( ) sin sin sin os is the polarization oeffiients of the soure field. where ( ), ( ) x For vortex fields y B(, ) R exp( im ), so formula (1) an be redued to an equation with one-time integration: Q (,, ) E E(,, z) ( ) ( ) 1 sin exp( os )d. H(,, ) ikf R T Q (,, ) ikz (4) z 0 H Q matries an be expliitly written for ertain types of where, (,, ) EH polarizations and onsist of superposition of Bessel funtions of different orders [13-14]. If all beams are generated by different zones of the optial element supplementing a lens with a high numerial aperture, the resulting field in the foal area will be a superposition of the fields established by different zones of the optial element: E (2) (3) 46

The diaphragm has several small holes, loated evenly on a ertain radius from the entre of the diaphragm. Table 1.

Every soure generates a spherial wave that is being foused on and interferes with waves from other point soures.

In figure 1, the shapes of the diaphragms generating different number of waves is evident.

3 E(,, z) Ei (,, z). (5) i The interferene of spherial waves The next step is to review an opaque diaphragm, imposed on the pupil of a fousing system with a high numerial aperture. The diaphragm has several small holes, loated evenly on a ertain radius from the entre of the diaphragm. Table 1. The resulting eletri field for the entral radius r 25 N Soure field Thereby, we gain a system of point light soures. Every soure generates a spherial wave that is being foused on and interferes with waves from other point soures. Herewith, we an vary the number of point soures and their distane from the entre of the aperture. In figure 1, the shapes of the diaphragms generating different number of waves is evident. In tables 1-5 results of modelling with different parameters are listed. From the tables above, with the interferene of three and four spherial waves in the ross setion of the foal area there are bright light spots loated in the lattie sites; furthermore, the onfigurations of the latties an be different. In addition, longitudinal plane long light hannels are formed. The nondiffrational nature of the beams, the spetrum of whih is loalized on the ring, has been evident for a long time. It has been suessfully used to reate different strutures that remain invariant 47

N Soure field 2 3 4 5 With the interferene of five or

see tables 3 and 4) the entral spot is dereased, whih

With the inrease in the numerial aperture (radius of

and different symmetries appear. а) b) ) Fig. 1.

4 in the longitudinal diretion [16-18]. The obtained distributions an be used to reate photoni rystals and plasma hannels. Table 2. The resulting eletri field for the entral radius r 50 N Soure field With the interferene of five or more waves in the ross setion, we an see a superlattie pattern. If the radius of the ring is inreased (for omparison see tables 3 and 4) the entral spot is dereased, whih orresponds to the inrease in the numerial aperture. With the inrease in the numerial aperture (radius of the ring), the interferene pattern beomes more omplex and different symmetries appear. а) b) ) Fig. 1. Shape of the diaphragm for а) 5 point soures, b) 10 point soures, ) 15 point soures with a entral radius r 50, the radius of every point is - rd 2 48

point soures so that the ring aperture is tightly filled, Bessel-like beams begin to form in

In these situations, the size of the entral spot of the Bessel beam depends on the radius of

It is worth noting that this approah to generating a Bessel beam is more onvenient than reating

wide range of superlattie patterns in the ross setion, whih will keep their struture at a long

In this ase, the radius of the holes in the diaphragm and the radius of the ring determine the

5 Table 3. The resulting eletri field for the entral radius r 25 N 10 Soure field If we use many point soures so that the ring aperture is tightly filled, Bessel-like beams begin to form in the ross setion. In these situations, the size of the entral spot of the Bessel beam depends on the radius of the ring aperture. It is worth noting that this approah to generating a Bessel beam is more onvenient than reating the ring aperture. Conslusion By varying the number of waves and distane between them one an therefore obtain a wide range of superlattie patterns in the ross setion, whih will keep their struture at a long distane. In this ase, the radius of the holes in the diaphragm and the radius of the ring determine the length of the longitudinal setion. Inreasing the size of the holes and the numerial aperture leads to a redution of the fous depth. More omplex superlattie patterns an be added by inreasing the number of phases of the point soures with speial phase elements and by adding polarization to the foused beam [13, 19-21]. Aknowledgements This work was finanially supported by the Russian Ministry of Eduation and Siene. 49

10 Table 4. The resulting eletri field for the entral radius r 50 N Soure field 2 12 15 Referenes 1. Durnin J.

Journal of the Optial Soiety of Ameria A, 1987; 4(4): 651-654. 2. Durnin J, Mieli JJ Jr, Eberly JH.

Garés-Chávez V, MGloin D, Melville H, Sibbett W, Dholakia K.

Nature, 2002; 419(6903): 145-147. 4. MGloin D, Garés-Chávez V, Dholakia K.

Arlt J, Garés-Chávez V, Sibbett W, Dholakia K. Optial miromanipulation using a Bessel light beam.

High-resolution optial oherene tomography over a large depth range with an axion lens.

6 10 Table 4. The resulting eletri field for the entral radius r 50 N Soure field Referenes 1. Durnin J. Exat solutions for nondiffrating beams. I. The salar theory. Journal of the Optial Soiety of Ameria A, 1987; 4(4): Durnin J, Mieli JJ Jr, Eberly JH. Diffration-free beams. Physial Review Letters, 1987; 58(15): Garés-Chávez V, MGloin D, Melville H, Sibbett W, Dholakia K. Simultaneous miromanipulation in multiple planes using a self-reonstruting light beam. Nature, 2002; 419(6903): MGloin D, Garés-Chávez V, Dholakia K. Interfering Bessel beams for optial miromanipulation. Optis Letters, 2003; 28(8): Arlt J, Garés-Chávez V, Sibbett W, Dholakia K. Optial miromanipulation using a Bessel light beam. Optis Communiations, 2001; 197(4-6): Ding Z, Ren H, Zhao Y, Nelson JS, Chen Z. High-resolution optial oherene tomography over a large depth range with an axion lens. Optis Letters, 2002; 27(4): Yu C, Wang MR, Varela AJ, Chen B. High-density non-diffrating beam array for optial interonnetion. Optis Letters, 2000; 177(1-6): Bouhal Z. Nondiffrating optial beams: physial properties, experiments, and appliations. Czehoslovak Journal of Physis, 2003; 53(7): Boguslawski M, Rose P, Denz C. Inreasing the strutural variety of disrete nondiffrating wave fields. Physial Review A, 2011; 84(1):

10 Table 5. The resulting eletri field for the entral radius r 75 N Soure field 2 12 15 10. Boguslawski M, Rose P, Denz C.

Chen YF, Liang HC, Lin YC, Tzeng YS, Su KW, Huang KF.

Physial Review A, 2011; 83(5): 053813. 12. Tsou CH, Wu TW, Tung JC, Liang HC, Tuan PH, Chen YF.

Computer Optis, 2010; 34(1): 58-68. [in Russian] 14. Khonina SN, Kazanskiy NL, Volotovsky SG.

Journal of Modern Optis, 2011; 58(9): 748-760. 15. Khonina SN, Kazanskiy NL, Volotovsky SG.

7 10 Table 5. The resulting eletri field for the entral radius r 75 N Soure field Boguslawski M, Rose P, Denz C. Nondiffrating kagome lattie. Applied Physis Letters, 2011; 98(6): Chen YF, Liang HC, Lin YC, Tzeng YS, Su KW, Huang KF. Generation of optial rystals and quasirystal beams: Kaleidosopi patterns and phase singularity. Physial Review A, 2011; 83(5): Tsou CH, Wu TW, Tung JC, Liang HC, Tuan PH, Chen YF. Generation of pseudonondiffrating optial beams with superlattie strutures. Optis Express, 2013; 21(20): Khonina SN, Volotovsky SG. Managing the ontribution of various vetor field omponents in the foal area of a high-aperture lens with binary phase strutures. Computer Optis, 2010; 34(1): [in Russian] 14. Khonina SN, Kazanskiy NL, Volotovsky SG. Vortex phase transmission funtion as a fator to redue the foal spot of high-aperture fousing system. Journal of Modern Optis, 2011; 58(9): Khonina SN, Kazanskiy NL, Volotovsky SG. Influene of vortex transmission phase funtion on intensity distribution in the foal area of high-aperture fousing system. Optial Memory and Neural Networks (Information Optis), 2011; 20(1): Ziolkowski RW, Besieris IM, Shaarawi AM. Aperture realizations of exat solutions to homogeneous-wave equations. Journal of the Optial Soiety of Ameria A, 1993; 10(1): Gutierrez-Vega JC, Iturbe-Castillo MD, Chavez-Cerda S. Alternative formulation for invariant optial fields: Mathieu beams. Optis Letters, 2000; 25(20):

8 18. Khonina SN. An easy and effiient way to reate different nondiffrational laser beams. Computer Optis, 2009; 33(1): [in Russian] 19. Gao X, Wang J, Gu H, Xu W. Fousing properties of onentri pieewise ylindrial vetor beam. Optik, 2007; 118: Khonina SN, Golub I. Optimization of fousing of linearly polarized light. Optis Letters, 2011; 36(3): Khonina SN, Fidirko NS. Researh of the interferene of ounterpropagating sharp foused beams with different polarizations. News of the Samara Siene Center of the RAS, 2014; 16(4): [in Russian] 22. Khonina SN, Ustinov AV. Analysis of interferene of ylindrial laser beams generated by ring optial elements with a vortex phase at sharp fousing. Computer Optis, 2015; 39(1): [in Russian] 52

Orbital angular momentum of mixed vortex beams

Orbital angular momentum of mixed vortex beams Z. Bouhal *, V. Kollárová, P. Zemánek, ** T. ižmár ** Department of Optis, Palaký University, 17. listopadu 5, 77 7 Olomou, Ceh Republi ** Institute of Sientifi