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Superposition of monochromatic Bessel beams in (kρ, kz)-plane to obtain wave focusing: Spatial localized waves
Optics Communications 249 (2005) 407–413 www.elsevier.com/locate/optcom
Superposition of monochromatic Bessel beams in (kq, kz)-plane to obtain wave focusing: Spatial localized waves C.A. Dartora
b
a,*
, K.Z. Nobrega a,b, Alexandre Dartora H.E. Hernandez-Figueroa b
a,b
,
a DFMC, Institute of Physics, University of Campinas, C.P. 6165, 13015301 Campinas-SP, Brazil DMO, FEEC, University of Campinas, Albert Einstein Av. 400, CP 6101, 13083-970 Campinas-SP, Brazil
Received 31 August 2004; received in revised form 18 January 2005; accepted 21 January 2005
Abstract In this work we analyze the effect of superimposing monochromatic Bessel beams with different wave-vectors through the use of a multi-annular slit (or holographic methods). An analytical procedure based on the scalar diffraction theory allows one to obtain spatially localized waves. 2005 Elsevier B.V. All rights reserved. PACS: 42.79.D; 42.25 Keywords: Non-diffracting beams; Bessel beams; Wave equation; Scalar diffraction theory
1. Introduction After the pioneering work on Bessel beams by Durnin et al. in 1987 [1], non-diffracting beams have been drawing growing interest due to the fact that under ideal conditions they propagate without spreading [2–6]. These waves have potential applications in technology, like wireless communica* Corresponding author. Tel.: +55 1932374533; fax: +55 1937883704. E-mail addresses: dartora@ifi.unicamp.br (C.A. Dartora), [email protected] (K.Z. Nobrega).
tions, metrology, laser surgery, non-linear optics, optical tweezers, conduits in atom optics and so on [7–12]. Several solutions for the wave equation that are localized in spacetime have been obtained in the current literature and are best known as localized pulses [13–18], which propagate in free space without suffering diffraction and dispersion. Such solutions can be obtained by superposing diffraction-free beams in the frequency domain. As an example, one may invoke the X-shaped solutions, which are synthesized by an adequate superposition of Bessel beams in frequency domain.
0030-4018/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2005.01.030
408
C.A. Dartora et al. / Optics Communications 249 (2005) 407–413
While non-diffracting beams are not localized axially, localized pulses propagate forward along the z-axis, as a function of f = z vt. Our purpose here is to obtain a well-localized wave in both the transverse and axial directions using a monochromatic source and standard optical elements of the kind employed by Durnin [1]. The possibility of transverse shape modeling using a superposition of Bessel beams was previously examined by Bouchal et al. while the control of longitudinal intensity pattern has been analyzed in Refs. [19] and [20] using numerical optimization techniques. In Ref. [21] the superposition of Bessel beams to obtain a stationary localized wave field of arbitrary longitudinal shape is demonstrated analytically using the Fourier series. The main difference between the localized pulses and our solutions is that in the former the peakÕs intensity travels along the z-axis with the time, while in our solution the peak remains static along the z-axis, i.e., the axial and transverse patterns are time independent. In other words, the X-shaped waves are obtained through a frequency superposition of Bessel beams with a defined unitary ^k-vector (with ^k ¼ k=k), which means that independently of the frequency x = ck the k-vectors belong to a cone with a fixed axicon angle h, being kz = k cos h and kq = k sin h. Our aim is to make an adequate superposition of Bessel beams with a fixed frequency x0 and wave number k = x0/c but with wave vectors k belonging to different cones and consequently, different angles h. To do this a multi-annular slit with different radius of the rings can be used in an apparatus similar to that proposed in [1]. In fact, here we discuss the logarithmic spacing of the rings in the annular slit. The remainder of this paper is described as follows: In the next section we develop the theoretical framework based on the scalar diffraction theory to analyze the multi-annular structure. In Section 3 we present and discuss a few results for the ideal case of a lens of infinite radius and in the last section, the conclusions and remarks are added.
experiment [22]. Here, the scalar wave function W is a possible representation of the electric field, Ex, or the magnetic field, Hy, being the PoyntingÕs vector proportional to |W|2. The KirchhoffÕs integral in the paraxial approximation (distant field or Fraunhoffer region) reads [22–25] h i 2 k exp ik z þ q2z Wðq;u;zÞ ¼ 2piz Z 1 Z 2p . d. da sð.; aÞWð.; a;z0 ¼ 0Þ 0 0 2 k. kq. cosða uÞ ; exp i exp i z 2z ð1Þ where k = 2p/k is the vacuum wave number, s(., a) is the transmittance function (TF) of the aperture, W(., a, z 0 = 0) is the incident wave, (., a, z 0 ) and (q, u, z) are the diffractive aperture and observation points in cylindric coordinates, respectively. Consider an ideal multi-annular slit with n concentric rings as shown in Fig. 1. The experimental setup is the same as that one proposed by Durnin [1] except for the slit. We assume that the condition daj
kf R
holds, where aj is the radius and daj is the thickness of the jth ring, k is the wavelength of the laser illuminating the rings, f is the focal distance and R is
2. Theoretical framework: multi-annular structures At optical frequencies the scalar diffraction theory can be used to obtain excellent agreement with
ð2Þ
Fig. 1. Experimental setup with a multi-annular slit.
C.A. Dartora et al. / Optics Communications 249 (2005) 407–413
the radius of a thin lens. In this way, each ring in the slit has the following transfer function [25] sj ð.; aÞ ¼ dð. aj Þ
ð3Þ
and considering a forward z-propagating wave incident at the slit in (1) it can be shown that the wave generated by the jth ring at the lens surface is h i a2 q2 A0 aj k exp ik f þ 2f þ 2fj kaj q Wj ðq;u; f Þ ¼ J0 : f 2pif ð4Þ The thin lens transfer function is given by [22] kq2 slens ðq; uÞ ¼ exp i ð5Þ 2f and using expression (1) again, with (4) as the incident wave allow us to obtain the resultant field, generated by the n rings, after the lens P ðW ¼ nj¼0 Wj Þ h i ! 2 n Af k exp ik z þ q2z X a2j Wðq;u;zÞ ¼ aj exp ik iz 2f j¼0 Z R kaj q0 kqq0 q0 dq0 J 0 J0 f z q0 ¼0 02 kq exp i ; ð6Þ 2z where Af is some constant factor. We must observe that the plane where the lens is located corresponds to z = 0. The above equation is the general one describing the multi-annular structure shown in Fig. 1. If we make R ! 1, which means an infinite lens, Eq. (6) can be integrated analytically and following solution is easily attainable [26] Wðq; u; z; tÞ ¼ A exp ðix0 tÞ n X Aj exp ik jz z J 0 ðk jq qÞ
ð7Þ
j¼0
with A any common amplitude factor, x0 is the laser beam frequency, ! ka2j Aj ¼ aj exp i ð8Þ 2f and
k jq ¼
409
!
kaj ; f
k jz k 1
a2j : 2f 2
ð9Þ
The solution given in (7) is an ideal solution, valid for distances z < zmax, where zmax R/tan hmax is the shortest field depth, R is the true radius of the lens and tan hmax max (aj/f). In fact, Eq. (7) corresponds to the field generated by a finite aperture(with R remaining finite) until the Bessel beam with shortest field depth decays. After that distance, the behavior must be determined using the expression (6). However, in practice the field depths of all the Bessel beams are very close and after the distance corresponding to the shortest field depth zmax the diffraction effects cannot be compensated and the resulting field intensity is negligible. Notice that (7) is simply the superposition of Bessel beams generated by each one of the rings, as expected, and the dispersion relation (9) is valid only for kq kz, which is the paraxial approximation for the wave equation. For each ring we have a corresponding Bessel beam with transverse kq and longitudinal kz propagation numbers corresponding to the dimensions of the ring and the focal distance of the lens, in the case of multi-annular structures. Indeed we have obtained a superposition of Bessel beams in the (kq, kz)-plane with dispersion relation fixed to be k 2 ¼ k 2q þ k 2z . The solution (7) is obtained for an incident plane wave illuminating the slit, but if the wave arriving at the slit differs from a plane wave, we have more degrees of freedom and we can construct any desired spatial longitudinal pattern. Optical elements can be used to control the phase and amplitude of each part of the wave incident on the multi-annular slit. Another way to obtain the desired wave pattern is to manipulate the transmittance function of an aperture illuminated by a plane wave, by holographic methods [4,11], so we can have a solution of the kind n X Wðq; u; zÞ ¼ Aj exp ik jz z J 0 ðk jq qÞ; ð10Þ j¼0
which is of the same form as (7), with the aperture transmittance function given by n X Aj J 0 ðk jq .Þ sð.; aÞ ¼ j¼0
410
C.A. Dartora et al. / Optics Communications 249 (2005) 407–413
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and k jz ¼ k 2 k 2jq . In the non-ideal case the radius of the lens remains finite and we have h i 2 n k exp ik z þ q2z X Wðq; u;zÞ ¼ Aj z j¼0 2 Z R kq. k. exp i . d. J 0 k jq . J 0 ; z 2z .¼0 ð11Þ
which is similar to the expression Eq. (6). If we make q = 0 in (10) we have WðzÞ ¼
n X
Aj exp ik jz z ;
ð12Þ
j¼0
which is simply a sum of sinusoidal terms with different spatial frequencies kjz. In order to obtain the desired longitudinal intensity pattern we refer to a simple heuristic method, described as follows: (i) a reference function Wref(z) corresponding to the desired longitudinal pattern is chosen. We are interested only in the on-axis field pattern, i.e., q = 0; (ii) in both situations, multi-annular slit or hologram, the parameters of the experimental setup to be determined are Aj, kjq and kjz, related by the dispersion relation k 2 ¼ k 2jz þ k 2jq , and the number of Bessel beams to be superimposed n (j = 1,2, . . ., n); (iii) initial values for Aj, kjq and kjz, n are chosen. The resulting intensity pattern W is calculated and compared point to point in z-axis with the desired one. If the resulting pattern do not fit the desired function, the parameters are modified and another comparison between resulting pattern and desired one is made, until the fitting is considered satisfactory. A genetic algorithm can be used to generate the new parameters Aj, kjq and kjz from the old values, with n fixed. An analytical way to obtain the radius of the rings in the multi-annular structure is to choose a desired point z = zp where the field intensity is a maximum and the following procedure can be used:
o
jWðzÞj2
¼ 0; z¼zp oz X ðk nz k mz ÞAm An exp½iðk nz k mz Þzp ¼ 0;
ð13Þ
m;n
being kjz and Aj functions of the radius of the rings and obtained through (9) and (8). The expression (13) supply a system of equations that must be solved to obtain aj. For N rings we have a N · N system. However, this procedure assures that the maximum intensity will be at z = zp but it does not guarantee that the peak is narrow. A more rigorous procedure based on Fourier series is discussed in Ref. [21]. Using the main idea described here and expressions (7)–(9), in the next section we will analyze a multi-annular slit with logarithmic spacing of the rings and discuss relevant points related to it.
3. Logarithmic spacing of the rings Firstly, consider the expression (12) with only two rings for which the field is written as follows: A2 WðzÞ ¼ A1 exp ðik 1z zÞ 1 þ exp ½iðk 2z k 1z Þz : A1 It is clear that in this case the field intensity pattern |W|2 will be of sinusoidal form, with periodicity determined by the spatial frequency Dkz = (k2z k1z); the larger the difference, the smaller the period. The number kjz is a function of the focal distance f as well as the radius of the ring aj. The factor Dkz, in the paraxial approximation is written below Dk z ¼ k
a21 a22 2f 2
and we can see that the larger the focal distance f, the larger the spatial period, while the larger the difference between the radius of the rings, the smaller the spatial period. When the number of rings is greater than 2 the analysis is very complicated because there is more than one frequency difference Dkz, and the periodicity can even be absent. It is important to notice that the number of the rings controls the longitudinal spatial width
C.A. Dartora et al. / Optics Communications 249 (2005) 407–413
while the spacing of the rings controls the periodicity of the peaks. The number of peaks Np in a given region zmax will be determined by the periodicity p = 2p/Dkz, Np = zmax/p. For instance, an interesting case of a multiannular structure occurs when the spacing between the successive rings is logarithmic. We have used an heuristic method similar to that described in the previous section to obtain the following string: V ¼ f2; 2:2; 2:4; 2:6; 2:8; 3; 3:2; 3:4; 3:6g with the radius of the jth ring, in millimeters, given by aj ¼ log ðV j Þ; where Vj is the jth component of the string V. In the ideal case (R ! 1) one can use expression (7) to compute the spatial distribution of the field. Fig. 2 was obtained through this procedure. We have used f = 360 mm and k = 632.8 nm. Easily one can see peaks with maximum intensity and the wave pattern in the (q, z)-plane resembles the shape of an X remaining static in space. The general behavior of the field is that it creates a station-
411
ary intensity pattern that gives rise to constructive interference in only a small region of space. In order to analyze the influence of the various parameters such as the number of the rings, the separation of the rings and the focal distance, we have considered the on-axis intensity field (q = 0). Continuing to use the logarithmic spacing, we define the following: V jþ1 ¼ V j þ q;
ð14Þ
where q is a constant. In this way the vector V is a linear progression, and aj=log (Vj) mm. Fig. 3, obtained through expression (12) illustrates the influence of the various parameters. In Fig. 3(a) we have considered f = 360 mm, q = 0.2, varying the number of the rings from n = 5 to 8, being the vectors V5 = {2.4;2.6;2.8;3.0;3.2} and V8 = {2;2.2;2.4; 2.6;2.8;3.0;3.2;3.4;3.6}. It is clear from that figure that as greater the number the rings, smaller the longitudinal width of the peak is. This is due to the factthat with more rings the constructive interference becomes more difficult. Indeed, with the number of the rings going to infinity we would reproduce a Dirac delta function, but the periodicity
Fig. 2. Intensity pattern |W|2 (in arbitrary units) for the logarithmic spacing of the rings. f = 360 mm and k = 632.8 nm.
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C.A. Dartora et al. / Optics Communications 249 (2005) 407–413
Fig. 3. On-axis (q = 0) intensity pattern |W(z)|2 (in normalized arbitrary units) for the logarithmic spacing of the rings: (a) with f = 360 mm and q = 0.2, the influence of the number of rings n is analyzed, the dotted line corresponds to n = 5 and the solid one to n = 8; (b) f = 360 mm, n = 6 and we vary the rings separation parameter q, the dotted line corresponding to q = 0.4 and the solid one to q = 0.2; (c) with q = 0.2 and n = 8 the focal distance f is varied. k = 632.8 nm everywhere.
would disappear completely. Looking at Fig. 3(b), where we have analyzed the influence of the spacing varying the parameter q, we conclude that when q is increased the spatial period is decreased. For a greater q the spacing of the rings is larger and the spatial frequency differences become larger, with consequent reduction of the spatial period. We have considered n = 6 and for q = 0.2 and 0.4, we have V = {2.2;2.4;2.6;2.8;3.0;3.2} and {2.2;2.6;3.0;3.4;3.8;4.2}, respectively. Finally, in Fig. 3(c) we have varied the focal distance from 360 to 560 mm. As said before, the spatial period
grows with the increase of the focal distance. Also the peak moves forward along the z-axis. These fields could be used as a laser electromagnetic wall in two or more well localized points in the space, the number of points depending on the spatial periodicity and the shortest field depth zmax. After z = zmax the field intensity decays fast, becoming negligible. If by some special mechanism (piezoelectric materials) we appropriately change the radius of the rings, we can move the peak region along the z-axis. Changing the focal distance of the lens, the number of peaks in a given maxi-
C.A. Dartora et al. / Optics Communications 249 (2005) 407–413
mum distance is modified, being the spacing between peaks dependent of the focal distance f.
4. Conclusion and remarks In this paper we have proposed the superposition of Bessel beams to obtain field spatial localization. One can use an apparatus similar to DurninÕs experimental setup or holographic methods to do it. Both longitudinal and transverse localization are attainable. We have shown the field distribution for a simple logarithmic spacing of the rings in a slit with high localization of the field along the z-axis. This kind of fields can be used in laser tweezers or trappers and other applications. If these beams were modulated in time domain we could expect that the pulse would appear with a great amplitude only in the region where the composition of Bessel beams have its maximum.
Acknowledgement We thank Fundac¸a˜o de Amparo a´ Pesquisa do Estado de Sa˜o Paulo (FAPESP) for financial support.
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[7] J.C. Gutierrez-Vega, M.D. Iturbe-Castillo, E. Tepichin, G. Ramirez, R.M. Rodriguez-Dagnino, S. Chavez-Cerda, Opt. Photon. News (2000) 37; J.C. Gutierrez-Vega, M.D. Iturbe-Castillo, S. ChavezCerda, Opt. Lett. (25) (2000) 1493; J.C. Gutierrez-Vega, M.D. Iturbe-Castillo, G.A. Ramirez, E. Tepichin, R.M. Rodriguez-Dagnino, S. Chavez-Cerda, G.H.C. New, Opt. Commun. (195) (2001) 35. [8] V. Garce´z-Cha´vez, D. McGloin et al., Nature 419 (2002) 145. [9] S. Chavez-Cerda, G.S. McDonald, G.H. New, Opt. Commun. 123 (1996) 225; S. Chavez-Cerda, M.A. Meneses-Nava, J.M. Hickmann, Opt. Lett. 23 (24) (1998) 1871. [10] J. Roger-Salazar, G.H.C. New, S. Chavez-Cerda, Opt. Commun. (190) (2001) 117. [11] S. Chavez-Cerda, M.J. Padgett, I. Allison, G.H.C. New, J. Opt. B: Quant. Semiclass. Opt. 4 (2002) 52. [12] M. Erdelyi, Z.L. Horvath, G. Szabo, Zs Bor, F.K. Tittel, J.R. Carvalho, M.C. Smayling, J. Vac. Sci. Technol. B 15 (2) (1997) 287. [13] E. Recami, Physica A (252) (1998) 586; M. Zamboni-Rached, E. Recami, F. Fontana, Phys. Rev. E 64 (2001) 1; M. Zamboni-Rached, H.E. Hernandez-Figueroa, Opt. Commun. 191 (2001) 49. [14] J. yu Lu, J.F. Greenleaf, IEEE Trans. Ultrason. Ferroelect. Frequency Control (39) (1992) 19. [15] R.W. Ziolkowsky, Phys. Rev. A 39 (4) (1989) 2005. [16] R.W. Ziolkowski, I.M. Besieris, A.M. Shaarawi, J. Opt. Soc. Am. A 10 (1) (1993) 75. [17] D. Mugnai, A. Ranfagni, R. Ruggeri, Phys. Rev. Lett. 84 (21) (2000) 4830. [18] C.A. Dartora, K.Z. Nobrega, H.E. Hernandez-Figueroa, E. Recami, Opt. Commun. 229 (2004) 99. [19] J. Rosen, A. Yariv, Opt. Lett. 19 (1994) 843. [20] R. Piestun, B. Spektor, J. Shamir, J. Mod. Opt. 43 (1996) 1495. [21] M. Zamboni-Rached, Opt. Express 12 (2004) 4001. [22] M. Born, E. Wolf, Principles of Optics, sixth ed., Pergamon Press, New York, 1989, See also J.D. Jackson, Classical Electrodynamics, Wiley, 1998. [23] Pamela L. Greene, Dennis G. Hall, J. Opt. Soc. Am. A 15 (12) (1998) 3020. [24] P.L. Overleft, C.S. Kenney, J. Opt. Soc. Am. A 8 (5) (1991) 732. [25] C.A. Dartora, M. Zamboni-Rached, K.Z. No´brega, E. Recami, H.E. Herna´ndez-Figueroa, Opt. Commun. 222 (2003) 75. [26] I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, New York and London, 1965.
Superposition of monochromatic Bessel beams in (kq, kz)-plane to obtain wave focusing: Spatial localized waves C.A. Dartora
b
a,*
, K.Z. Nobrega a,b, Alexandre Dartora H.E. Hernandez-Figueroa b
a,b
,
a DFMC, Institute of Physics, University of Campinas, C.P. 6165, 13015301 Campinas-SP, Brazil DMO, FEEC, University of Campinas, Albert Einstein Av. 400, CP 6101, 13083-970 Campinas-SP, Brazil
Received 31 August 2004; received in revised form 18 January 2005; accepted 21 January 2005
Abstract In this work we analyze the effect of superimposing monochromatic Bessel beams with different wave-vectors through the use of a multi-annular slit (or holographic methods). An analytical procedure based on the scalar diffraction theory allows one to obtain spatially localized waves. 2005 Elsevier B.V. All rights reserved. PACS: 42.79.D; 42.25 Keywords: Non-diffracting beams; Bessel beams; Wave equation; Scalar diffraction theory
1. Introduction After the pioneering work on Bessel beams by Durnin et al. in 1987 [1], non-diffracting beams have been drawing growing interest due to the fact that under ideal conditions they propagate without spreading [2–6]. These waves have potential applications in technology, like wireless communica* Corresponding author. Tel.: +55 1932374533; fax: +55 1937883704. E-mail addresses: dartora@ifi.unicamp.br (C.A. Dartora), [email protected] (K.Z. Nobrega).
tions, metrology, laser surgery, non-linear optics, optical tweezers, conduits in atom optics and so on [7–12]. Several solutions for the wave equation that are localized in spacetime have been obtained in the current literature and are best known as localized pulses [13–18], which propagate in free space without suffering diffraction and dispersion. Such solutions can be obtained by superposing diffraction-free beams in the frequency domain. As an example, one may invoke the X-shaped solutions, which are synthesized by an adequate superposition of Bessel beams in frequency domain.
0030-4018/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2005.01.030
408
C.A. Dartora et al. / Optics Communications 249 (2005) 407–413
While non-diffracting beams are not localized axially, localized pulses propagate forward along the z-axis, as a function of f = z vt. Our purpose here is to obtain a well-localized wave in both the transverse and axial directions using a monochromatic source and standard optical elements of the kind employed by Durnin [1]. The possibility of transverse shape modeling using a superposition of Bessel beams was previously examined by Bouchal et al. while the control of longitudinal intensity pattern has been analyzed in Refs. [19] and [20] using numerical optimization techniques. In Ref. [21] the superposition of Bessel beams to obtain a stationary localized wave field of arbitrary longitudinal shape is demonstrated analytically using the Fourier series. The main difference between the localized pulses and our solutions is that in the former the peakÕs intensity travels along the z-axis with the time, while in our solution the peak remains static along the z-axis, i.e., the axial and transverse patterns are time independent. In other words, the X-shaped waves are obtained through a frequency superposition of Bessel beams with a defined unitary ^k-vector (with ^k ¼ k=k), which means that independently of the frequency x = ck the k-vectors belong to a cone with a fixed axicon angle h, being kz = k cos h and kq = k sin h. Our aim is to make an adequate superposition of Bessel beams with a fixed frequency x0 and wave number k = x0/c but with wave vectors k belonging to different cones and consequently, different angles h. To do this a multi-annular slit with different radius of the rings can be used in an apparatus similar to that proposed in [1]. In fact, here we discuss the logarithmic spacing of the rings in the annular slit. The remainder of this paper is described as follows: In the next section we develop the theoretical framework based on the scalar diffraction theory to analyze the multi-annular structure. In Section 3 we present and discuss a few results for the ideal case of a lens of infinite radius and in the last section, the conclusions and remarks are added.
experiment [22]. Here, the scalar wave function W is a possible representation of the electric field, Ex, or the magnetic field, Hy, being the PoyntingÕs vector proportional to |W|2. The KirchhoffÕs integral in the paraxial approximation (distant field or Fraunhoffer region) reads [22–25] h i 2 k exp ik z þ q2z Wðq;u;zÞ ¼ 2piz Z 1 Z 2p . d. da sð.; aÞWð.; a;z0 ¼ 0Þ 0 0 2 k. kq. cosða uÞ ; exp i exp i z 2z ð1Þ where k = 2p/k is the vacuum wave number, s(., a) is the transmittance function (TF) of the aperture, W(., a, z 0 = 0) is the incident wave, (., a, z 0 ) and (q, u, z) are the diffractive aperture and observation points in cylindric coordinates, respectively. Consider an ideal multi-annular slit with n concentric rings as shown in Fig. 1. The experimental setup is the same as that one proposed by Durnin [1] except for the slit. We assume that the condition daj
kf R
holds, where aj is the radius and daj is the thickness of the jth ring, k is the wavelength of the laser illuminating the rings, f is the focal distance and R is
2. Theoretical framework: multi-annular structures At optical frequencies the scalar diffraction theory can be used to obtain excellent agreement with
ð2Þ
Fig. 1. Experimental setup with a multi-annular slit.
C.A. Dartora et al. / Optics Communications 249 (2005) 407–413
the radius of a thin lens. In this way, each ring in the slit has the following transfer function [25] sj ð.; aÞ ¼ dð. aj Þ
ð3Þ
and considering a forward z-propagating wave incident at the slit in (1) it can be shown that the wave generated by the jth ring at the lens surface is h i a2 q2 A0 aj k exp ik f þ 2f þ 2fj kaj q Wj ðq;u; f Þ ¼ J0 : f 2pif ð4Þ The thin lens transfer function is given by [22] kq2 slens ðq; uÞ ¼ exp i ð5Þ 2f and using expression (1) again, with (4) as the incident wave allow us to obtain the resultant field, generated by the n rings, after the lens P ðW ¼ nj¼0 Wj Þ h i ! 2 n Af k exp ik z þ q2z X a2j Wðq;u;zÞ ¼ aj exp ik iz 2f j¼0 Z R kaj q0 kqq0 q0 dq0 J 0 J0 f z q0 ¼0 02 kq exp i ; ð6Þ 2z where Af is some constant factor. We must observe that the plane where the lens is located corresponds to z = 0. The above equation is the general one describing the multi-annular structure shown in Fig. 1. If we make R ! 1, which means an infinite lens, Eq. (6) can be integrated analytically and following solution is easily attainable [26] Wðq; u; z; tÞ ¼ A exp ðix0 tÞ n X Aj exp ik jz z J 0 ðk jq qÞ
ð7Þ
j¼0
with A any common amplitude factor, x0 is the laser beam frequency, ! ka2j Aj ¼ aj exp i ð8Þ 2f and
k jq ¼
409
!
kaj ; f
k jz k 1
a2j : 2f 2
ð9Þ
The solution given in (7) is an ideal solution, valid for distances z < zmax, where zmax R/tan hmax is the shortest field depth, R is the true radius of the lens and tan hmax max (aj/f). In fact, Eq. (7) corresponds to the field generated by a finite aperture(with R remaining finite) until the Bessel beam with shortest field depth decays. After that distance, the behavior must be determined using the expression (6). However, in practice the field depths of all the Bessel beams are very close and after the distance corresponding to the shortest field depth zmax the diffraction effects cannot be compensated and the resulting field intensity is negligible. Notice that (7) is simply the superposition of Bessel beams generated by each one of the rings, as expected, and the dispersion relation (9) is valid only for kq kz, which is the paraxial approximation for the wave equation. For each ring we have a corresponding Bessel beam with transverse kq and longitudinal kz propagation numbers corresponding to the dimensions of the ring and the focal distance of the lens, in the case of multi-annular structures. Indeed we have obtained a superposition of Bessel beams in the (kq, kz)-plane with dispersion relation fixed to be k 2 ¼ k 2q þ k 2z . The solution (7) is obtained for an incident plane wave illuminating the slit, but if the wave arriving at the slit differs from a plane wave, we have more degrees of freedom and we can construct any desired spatial longitudinal pattern. Optical elements can be used to control the phase and amplitude of each part of the wave incident on the multi-annular slit. Another way to obtain the desired wave pattern is to manipulate the transmittance function of an aperture illuminated by a plane wave, by holographic methods [4,11], so we can have a solution of the kind n X Wðq; u; zÞ ¼ Aj exp ik jz z J 0 ðk jq qÞ; ð10Þ j¼0
which is of the same form as (7), with the aperture transmittance function given by n X Aj J 0 ðk jq .Þ sð.; aÞ ¼ j¼0
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qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and k jz ¼ k 2 k 2jq . In the non-ideal case the radius of the lens remains finite and we have h i 2 n k exp ik z þ q2z X Wðq; u;zÞ ¼ Aj z j¼0 2 Z R kq. k. exp i . d. J 0 k jq . J 0 ; z 2z .¼0 ð11Þ
which is similar to the expression Eq. (6). If we make q = 0 in (10) we have WðzÞ ¼
n X
Aj exp ik jz z ;
ð12Þ
j¼0
which is simply a sum of sinusoidal terms with different spatial frequencies kjz. In order to obtain the desired longitudinal intensity pattern we refer to a simple heuristic method, described as follows: (i) a reference function Wref(z) corresponding to the desired longitudinal pattern is chosen. We are interested only in the on-axis field pattern, i.e., q = 0; (ii) in both situations, multi-annular slit or hologram, the parameters of the experimental setup to be determined are Aj, kjq and kjz, related by the dispersion relation k 2 ¼ k 2jz þ k 2jq , and the number of Bessel beams to be superimposed n (j = 1,2, . . ., n); (iii) initial values for Aj, kjq and kjz, n are chosen. The resulting intensity pattern W is calculated and compared point to point in z-axis with the desired one. If the resulting pattern do not fit the desired function, the parameters are modified and another comparison between resulting pattern and desired one is made, until the fitting is considered satisfactory. A genetic algorithm can be used to generate the new parameters Aj, kjq and kjz from the old values, with n fixed. An analytical way to obtain the radius of the rings in the multi-annular structure is to choose a desired point z = zp where the field intensity is a maximum and the following procedure can be used:
o
jWðzÞj2
¼ 0; z¼zp oz X ðk nz k mz ÞAm An exp½iðk nz k mz Þzp ¼ 0;
ð13Þ
m;n
being kjz and Aj functions of the radius of the rings and obtained through (9) and (8). The expression (13) supply a system of equations that must be solved to obtain aj. For N rings we have a N · N system. However, this procedure assures that the maximum intensity will be at z = zp but it does not guarantee that the peak is narrow. A more rigorous procedure based on Fourier series is discussed in Ref. [21]. Using the main idea described here and expressions (7)–(9), in the next section we will analyze a multi-annular slit with logarithmic spacing of the rings and discuss relevant points related to it.
3. Logarithmic spacing of the rings Firstly, consider the expression (12) with only two rings for which the field is written as follows: A2 WðzÞ ¼ A1 exp ðik 1z zÞ 1 þ exp ½iðk 2z k 1z Þz : A1 It is clear that in this case the field intensity pattern |W|2 will be of sinusoidal form, with periodicity determined by the spatial frequency Dkz = (k2z k1z); the larger the difference, the smaller the period. The number kjz is a function of the focal distance f as well as the radius of the ring aj. The factor Dkz, in the paraxial approximation is written below Dk z ¼ k
a21 a22 2f 2
and we can see that the larger the focal distance f, the larger the spatial period, while the larger the difference between the radius of the rings, the smaller the spatial period. When the number of rings is greater than 2 the analysis is very complicated because there is more than one frequency difference Dkz, and the periodicity can even be absent. It is important to notice that the number of the rings controls the longitudinal spatial width
C.A. Dartora et al. / Optics Communications 249 (2005) 407–413
while the spacing of the rings controls the periodicity of the peaks. The number of peaks Np in a given region zmax will be determined by the periodicity p = 2p/Dkz, Np = zmax/p. For instance, an interesting case of a multiannular structure occurs when the spacing between the successive rings is logarithmic. We have used an heuristic method similar to that described in the previous section to obtain the following string: V ¼ f2; 2:2; 2:4; 2:6; 2:8; 3; 3:2; 3:4; 3:6g with the radius of the jth ring, in millimeters, given by aj ¼ log ðV j Þ; where Vj is the jth component of the string V. In the ideal case (R ! 1) one can use expression (7) to compute the spatial distribution of the field. Fig. 2 was obtained through this procedure. We have used f = 360 mm and k = 632.8 nm. Easily one can see peaks with maximum intensity and the wave pattern in the (q, z)-plane resembles the shape of an X remaining static in space. The general behavior of the field is that it creates a station-
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ary intensity pattern that gives rise to constructive interference in only a small region of space. In order to analyze the influence of the various parameters such as the number of the rings, the separation of the rings and the focal distance, we have considered the on-axis intensity field (q = 0). Continuing to use the logarithmic spacing, we define the following: V jþ1 ¼ V j þ q;
ð14Þ
where q is a constant. In this way the vector V is a linear progression, and aj=log (Vj) mm. Fig. 3, obtained through expression (12) illustrates the influence of the various parameters. In Fig. 3(a) we have considered f = 360 mm, q = 0.2, varying the number of the rings from n = 5 to 8, being the vectors V5 = {2.4;2.6;2.8;3.0;3.2} and V8 = {2;2.2;2.4; 2.6;2.8;3.0;3.2;3.4;3.6}. It is clear from that figure that as greater the number the rings, smaller the longitudinal width of the peak is. This is due to the factthat with more rings the constructive interference becomes more difficult. Indeed, with the number of the rings going to infinity we would reproduce a Dirac delta function, but the periodicity
Fig. 2. Intensity pattern |W|2 (in arbitrary units) for the logarithmic spacing of the rings. f = 360 mm and k = 632.8 nm.
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Fig. 3. On-axis (q = 0) intensity pattern |W(z)|2 (in normalized arbitrary units) for the logarithmic spacing of the rings: (a) with f = 360 mm and q = 0.2, the influence of the number of rings n is analyzed, the dotted line corresponds to n = 5 and the solid one to n = 8; (b) f = 360 mm, n = 6 and we vary the rings separation parameter q, the dotted line corresponding to q = 0.4 and the solid one to q = 0.2; (c) with q = 0.2 and n = 8 the focal distance f is varied. k = 632.8 nm everywhere.
would disappear completely. Looking at Fig. 3(b), where we have analyzed the influence of the spacing varying the parameter q, we conclude that when q is increased the spatial period is decreased. For a greater q the spacing of the rings is larger and the spatial frequency differences become larger, with consequent reduction of the spatial period. We have considered n = 6 and for q = 0.2 and 0.4, we have V = {2.2;2.4;2.6;2.8;3.0;3.2} and {2.2;2.6;3.0;3.4;3.8;4.2}, respectively. Finally, in Fig. 3(c) we have varied the focal distance from 360 to 560 mm. As said before, the spatial period
grows with the increase of the focal distance. Also the peak moves forward along the z-axis. These fields could be used as a laser electromagnetic wall in two or more well localized points in the space, the number of points depending on the spatial periodicity and the shortest field depth zmax. After z = zmax the field intensity decays fast, becoming negligible. If by some special mechanism (piezoelectric materials) we appropriately change the radius of the rings, we can move the peak region along the z-axis. Changing the focal distance of the lens, the number of peaks in a given maxi-
C.A. Dartora et al. / Optics Communications 249 (2005) 407–413
mum distance is modified, being the spacing between peaks dependent of the focal distance f.
4. Conclusion and remarks In this paper we have proposed the superposition of Bessel beams to obtain field spatial localization. One can use an apparatus similar to DurninÕs experimental setup or holographic methods to do it. Both longitudinal and transverse localization are attainable. We have shown the field distribution for a simple logarithmic spacing of the rings in a slit with high localization of the field along the z-axis. This kind of fields can be used in laser tweezers or trappers and other applications. If these beams were modulated in time domain we could expect that the pulse would appear with a great amplitude only in the region where the composition of Bessel beams have its maximum.
Acknowledgement We thank Fundac¸a˜o de Amparo a´ Pesquisa do Estado de Sa˜o Paulo (FAPESP) for financial support.
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