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Generation of bottle beam by focusing a super-Gaussian beam using a lens and an axicon
Optics Communications 277 (2007) 19–23 www.elsevier.com/locate/optcom
Generation of bottle beam by focusing a super-Gaussian beam using a lens and an axicon Ming-Dar Wei Department of Photonics, Feng Chia University, 100, Wenhwa Road, Seatwen, Taichung 407, Taiwan, ROC Received 30 November 2006; received in revised form 20 April 2007; accepted 27 April 2007
Abstract This study explores the characteristics of the bottle beams that are formed by super-Gaussian beams that impinge through an axicon and a positive lens. Analytical solutions for the on-axial intensity of Gaussian and apertured-plane beams were obtained. The barrier around the dark region has a larger variation of intensity for higher-order super-Gaussian profiles. Flattening the tops of the profiles increases the bottle lengths for a fixed second-order moment width or distance between the axicon and the lens. Ó 2007 Elsevier B.V. All rights reserved. PACS: 42.25.Bs; 42.60.Jf; 42.25.Fx Keywords: Focusing; Super-Gaussian beams; Bottle beams; Beam propagation
1. Introduction A dark focus, surrounded by regions of higher intensity, was initially created by using a holographic phase plate to trap atoms [1]. This beam is then called a bottle beam, which is constructed using a computer-generated hologram [2]. Bottle beams have recently attracted attention because of their applications in optical trapping, in which the bluedetuned intensity distribution acts as a repulsive wall that traps atoms in the dark region [1,3–5]. Various approaches have been demonstrated to generate bottle beams, using such elements as a special phase mask [1], amplitude masks [2] and a bimorph adaptive mirror [6], by focusing two identical Gaussian beams with opposing radii of curvature using an interferometer [7], and by utilizing the characteristics of a focused nondiffraction beam [8–13]. Recently, a bottle beam was directly produced from a bare laser by exploiting the constructive interference of Laguerre–Gaussian modes with a cavity configuration close to g1g2 = 1/4 [14]. A
E-mail address: [email protected] 0030-4018/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2007.04.044
self-imaged bottle beam with an embedded sequence of three-dimensional intensity voids was obtained by the interference of two Bessel beams with different radial wave vectors [15]. A novel element, called as a double-axicon, was fabricated to generate a self-imaged bottle beam [4]. Imaging different portions of a self-imaged bottle beam can trap high and low indices microparticles [5]. The axicon is useful for generating bottle beams, because it provides two main advantages. The first is its high energy efficiency, and the other is the ease of adjusting the layout of experiments to vary the beam geometry. Both a lens–axicon doublet and a combination of divergent and convergent axicons were adopted to produce hollow and bottle beams [16]. Additionally, focusing the nondiffraction beam, which can be generated using an axicon, can transform it into a bottle beam [8–13]. The nondiffraction beams focus to create a ring image in the focal plane of the lens, and the dark region in the three-dimensional view is formed from two cones, which share a single base on the focal plane. This phenomenon has been experimentally verified [13], using a dynamically one-parameter-tunable method to generate bottle beams and hollow beams. As
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M.-D. Wei / Optics Communications 277 (2007) 19–23
an apertured plane wave, associated with a plane wave existing within a finite transverse region, impinges through an axicon and a positive lens, the distance between the axicon and the focused lens is tuned to transform the beams from bottle beams to hollow beams. A tightly assembled axicon and lens can theoretically transform a Gaussian beam into a super-Gaussian profile, a ring-shaped profile or a doughnut profile [17]. However, the tunable parameter, which is the distance between the axicon and the lens changes, has not been comprehensively considered. This work investigates the characteristics of bottle beams that were generated by a super-Gaussian beam that impinges through an axicon and a positive lens. The effects of the beam profiles are of particular interest. The super-Gaussian function is popularly used to describe flat-topped profiles. The distribution of the super-Gaussian beam is given by h r n i ESG ðrÞ ¼ exp ; ð1Þ w where r is the radial coordinate; w is the radius of the beam at 1/e2, and n is order of the super-Gaussian beam. Increasing the order from n = 2 to infinity transforms the profiles from Gaussian to flat-topped. Therefore, the effect on the generation of bottle beams of gradually flattening the top of the profile can be examined. 2. Modeling Fig. 1 depicts the proposed scheme. The axicon, having the angle between the conical surface and the flat surface is c and the index is ng, has been successfully used to convert a plane wave into a Bessel beam. The transmittance of the axicon yields a linear phase shift to be sðrÞ ¼ expðibrÞ;
ð2Þ
where b = 2p(ng 1)tan(c)/k and k is the wavelength of the incident beam. When the lens with a focal length of f is located at a distance z0 beyond the axicon, the diffraction field E(q, z) at a distance z from the lens is given by the generalized Huygens formula [18] as, Z 1 2pi kDq2 krq expðikLÞ exp i Eðq; zÞ ¼ ESG ðrÞJ 0 kB B 2B 0 2 kAr exp i ibr r dr; ð3Þ 2B
super-Gaussian beam
γ
Z0
Z Z=0
Fig. 1. Schematic layout for a focused super-Gaussian beam.
where q is the radial coordinate; k is the wave number; J0 is the zero-order Bessel function, and L = z0 + z. The transformed ABCD matrix from the axicon through #the lens " 1 fz z0 þ z zf0 z A B to a distance z is ¼ . Since f1 1 zf0 C D the elements of the ABCD matrices are functions of the distance between the axicon and lens, z0, this dynamically tunable parameter controls the characteristics of the bottle beams. For a focused nondiffraction beam, the dark region in the three-dimensional view is formed from two cones, which share a base in the focal plane. The maximal dark region is in the focus plane given by z = f, yielding A = 0 and B = f; the transverse distribution in the focal plane is independent of z0 according to Eq. (3). Furthermore, this radius is almost independent of n, which fact is numerically confirmed in Section 3. Therefore, computing the on-axis intensity and the bottle length, associated with the on-axis dark region, determines the main characteristics of the bottle beam. The following discussion concerns two analytic solutions, relating to the Gaussian profile with n = 2 and the apertured plane with n = 1, which are the most common distributions in practical applications. (1) Gaussian beam When a Gaussian beam impinges on an axicon, a Bessel–Gaussian beam is produced. Thus, the following discussion equivalently analyzes the characteristics of the focused Bessel–Gaussian beam. At on-axis points, J0(0) = 1. Eq. (3) can be simplified to 2 Z 1 2pi r expðikLÞ exp 2 Eð0; zÞ ¼ kB w 0 2 kAr ibr r dr: ð4Þ exp i 2B Z 0
Based on the formula 3.462.5 in Ref. [19], the integral is rffiffiffi 1 m p 2 xelx 2mx dx ¼ 2l 2l l 2 m m exp ð5Þ 1 U pffiffiffi ; l l
1
kA and where U is the error function. Setting l ¼ w12 i 2B b m ¼ i 2, yields the on-axis field, rffiffiffi 2pi 1 ib p b2 expðikLÞ exp Eð0; zÞ ¼ kB 2l 4l l 4l
ib ð6Þ 1 U pffiffiffi 2 l
(2) Apertured-plane beam The focusing characteristics of the apertured-plane beam have been studied using a geometrical method [13]. The on-axis field distribution is expressed using the paraxial approximation, from Eq. (3), as Z w 2pi kAr2 Eð0; zÞ ¼ expðikLÞ ibr r dr: ð7Þ exp i kB 2B 0
M.-D. Wei / Optics Communications 277 (2007) 19–23
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pffiffiffi kA Setting a ¼ 2B and n ¼ a r 2ab into Eq. (7), yields Z x2 n b 2 expðin Þ þ 3=2 dn; Eð0; zÞ ¼ c1 ð8Þ a 2a x1 pffiffiffi b2 exp ikL þ i 4a where c1 ¼ 2pi , x1 ¼ 2pb ffiffia and x2 ¼ a kB w 2ab . An analytical form is c1 1 1 2 2 i expðix2 Þ þ i expðix1 Þ Eð0; zÞ ¼ 2 2 a c1 b 1 3=4 pffiffiffi 1=4 þ 3=2 ð1Þ pUi ½ð1Þ x2 2a 2
1 3=4 pffiffiffi 1=4 ð1Þ pUi ½ð1Þ x1 ; ð9Þ 2 where Ui is the imaginary error function. 3. Numerical results In the description of flat-topped profiles, the superGaussian function has the advantage of being explicitly expressible. However, propagations of super-Gaussian beams must be examined numerically, except in the aforementioned cases with n = 2 and n = 1. Moreover, since the distributions of bottle beams depend strongly on the width of the incident beams, the following set of field distributions is employed to maintain beam widths, which are calculated based on the second-order moment definition [20], for super-Gaussian beams as n N nr ESG ðrÞ ¼ exp ; W SG
Fig. 2. The evolution of a bottle beam with n = 2, c = 5°, ng = 1.5, w = 4 mm, z0 = 7.6 cm and f = 3.5 cm.
where Nn represents the ratio between the second-order moment width, WSG, and w for n such that 1=2 W SG Cð4=nÞ 12=n 2 ¼ Nn ¼ ; Cð2=nÞ w where C is the Gamma function. In the numerical computations, the parameters are set to c = 5°, ng = 1.5, k = 632 nm and f = 3.5 cm, as in Fig. 2. Fig. 2 displays the profile of a bottle beam with n = 2, WSG = 4 mm and z0 = 7.6 cm. A Bessel-like distribution is observed near the lens. Increasing z causes the side lobe to grow, and reduces the central intensity. The transverse pattern is then transferred to the doughnut ring and the size of the light region is minimal in the focal plane. Beyond the focal plane, the reverse process is performed to reconstruct the Bessel-like beam. A bottle beam is formed. For different orders of the super-Gaussian beam, similar bottle evolutions are obtained, but the bottle lengths vary. Fig. 3a plots the on-axis intensity as a function of z for z0 = 7.6 cm and WSG = 4 mm. Firstly, we concentrate on an incident beam that is an apertured plane wave, such that n = 1. Fig. 3b compares the numerical simulation with the analytical solution to Eq. (9). The results agree closely. The oscillation originates from the hard edge of the aperturedplane beam. Further, the values of z1,n and z2,n are set to
Fig. 3. (a) The on-axis intensity as a function of z with n = 2, 3, 5, 10, and 1 (labeled as n = inf). (b) The numerical and analytical results for an incident aperture-plane beam (n = 1).
represent the beginning and end points of the dark on-axis region of the order n, respectively; then, the bottle length is Dzn = z2,n z1,n. The bottle length decreases as n declines
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in Fig. 3a. The geometrical analysis in the previous work indicated that the point z = z1,1 is determined by the boundary (or transverse-edge) rays of the apertured plane wave focused by the lens, and z2,1 corresponds to the onaxis image of the apex of the axicon that is formed by the lens [13]. Thus, z2,n is expected to be independent of the shapes of the incident beams, as was confirmed in Fig. 3a.
Fig. 4. The transverse profile with (a) z = 32 mm, (b) z = 35 mm and (c) z = 38 mm.
However, as the super-Gaussian beams are gradually transformed from top-hat (n = 1) to Gaussian (n = 2) profiles with the drop in n, the decaying of the tail of the transverse field distribution is equivalent to increasing the distance between the boundary ray and the axis. Therefore, z1,n, associated with the point of intersection of the axis and the focused boundary ray, increases, reducing the bottle length by decreasing n. Not only do the bottle lengths vary, but also the maximum intensity and the variation of the on-axis intensity near dark region increase with n. Fig. 4 plots the transverse distributions in front of, in and beyond the focal plane at z = 32 mm, z = 35 mm and z = 38 mm, respectively. The transverse profiles in Fig. 4a reveal asymmetric distributions, in which the near z-axis side varies more steeply than the other side. Moreover, the intensity on the near z-axis side increases more steeply for larger n. A reverse distribution was observed when z exceeded the focal length f = 3.5 cm, as presented in Fig. 4c, and the variations of the slope on the near z-axis side were similar. Fig. 4b plots the symmetric transverse distribution in the focal plane for various values of n. All of the radii of the dark rings approach R = f(n 1)c = 1.527 mm, determined by geometric analysis [13]. Based on analyzing the both longitudinal and transverse distributions of the bottle beam, the barrier around the dark region has a larger gradient of intensity for focusing higher-order super-Gaussian profile. Fig. 5 plots the bottle length obtained by varying the second-order moment width. As in the preceding analysis, the boundary ray far from the z-axis focuses at greater z1,n as the width increases, shortening the bottle length. Apparently, the width is an important parameter in determining the bottle length, such that the bottle length with n = 2 declines from 5.128 cm to 2.322 cm as the width is varied from 2 mm to 10 mm. However, the bottle length is also influenced by the shape of the beam profile, such that, for example, the bottle length changes from 3.686 cm to 5.276 cm as n varies from 2 to 1 at width WSG = 4 mm. This result also demonstrates that the ray tracing method is unsuitable for analyzing quantitatively
Fig. 5. The bottle length against the second-order moment width.
M.-D. Wei / Optics Communications 277 (2007) 19–23
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propagate through an axicon and a positive lens. The distance between the axicon and the lens and the beam width still dominate the bottle lengths as in previous studies, but the effect of the beam profiles is not negligible. Since the longitudinal and transverse distributions, the barrier around the dark region of the bottle beam supports a larger variation of intensity for higher-order super-Gaussian beams. Flattening the tops of the profiles substantially increases the bottle lengths for a fixed beam width or a fixed distance between the axicon and the lens. The analytical on-axial intensities were obtained for Gaussian and apertured-plane beams. Acknowledgement
Fig. 6. The bottle length as a function of the distance between the axicon and the focused lens.
the focusing characteristics of a lower-order super-Gaussian beam. The relationship between the bottle length Dz and z0 is studied, as shown in Fig. 6. Since z2,1 is associated with the on-axis image of the apex of the axicon that is formed through the lens, reducing the distance between the axicon and the lens, z0, increases z2,n and the bottle length. The value of z2,n becomes infinity at z0 = f = 3.5 cm. The focused beam is transferred into a cylindrical-hollow beam with identical dark circles in all cross-sections beyond the focus plane of the lens. Further, hollow beams are formed for z0 < f, as in Ref. [13]. In summarizing the numerical results, the bottle lengths vary by several tenths of a percentage point as the incident super-Gaussian beam converted from n = 2 to n = 1. This effect of the shape cannot be neglected, although the distance between the axicon and the lens, z0, and the width remain dominant parameters. At fixed width, this scheme is a dynamically one-parameter-tunable to generate bottle beams, in which the parameter of z0 is adjusted to vary the volumes of the dark regions. Since the blue-detuned intensity distribution acts as a repulsive wall that traps atoms in the dark region of a bottle beam, we suggest that this scheme may be used to dilute or condense dynamically the trapped atoms. 4. Conclusions This work investigates the characteristics of bottle beams that were formed by super-Gaussian beams that
The authors would like to thank the National Science Council of the Republic of China for financially supporting this research under Contract No. NSC95-2112-035-001. References [1] R. Ozeri, L. Khaykovich, N. Davidson, Phys. Rev. A 59 (1999) R1750. [2] J. Arlt, M.J. Padgett, Opt. Lett. 25 (2000) 191. [3] L. Cacciapuoti, M. de Angelis, G. Pierattini, G.M. Tino, Eur. Phys. J. D 14 (2001) 373. [4] B.P.S. Ahluwalia, W.C. Cheong, X.-C. Yuan, L.-S. Zhang, S.-H. Tao, J. Bu, H. Wang, Opt. Lett. 31 (2006) 987. [5] B.P.S. Ahluwalia, X.-C. Yuan, S.H. Tao, W.C. Cheong, L.S. Zhang, H. Wang, J. Appl. Phys. 99 (2006) 113104. [6] O. Boyko, Th. A. Planchon, P. Merce‘re, C. Valentin, Ph. Balcou, Opt. Commun. 246 (2005) 131. [7] D. Yelin, B.E. Bouma, G.J. Tearney, Opt. Lett. 29 (2004) 661. [8] B. Lu, W. Huang, B. Zhang, F. Kong, Q. Zhai, Opt. Commun. 131 (1996) 223. [9] I. Manek, Y.B. Ovchinnikov, R. Grimm, Opt. Commun. 147 (1998) 67. [10] S. Chavez-Cerda, G.H.C. New, Opt. Commun. 181 (2000) 369. [11] J.C. Gutierrez-Vega, R. Rodriguez-Masegosa, S. Chavez-Cerda, J. Opt. A: Pure Appl. Opt. 5 (2003) 276. [12] M. de Angelisa, L. Cacciapuotib, G. Pierattinia, G.M. Tino, Opt. Laser Eng. 39 (2003) 283. [13] M.-D. Wei, W.-L. Shiao, Y.-T. Lin, Opt. Commun. 248 (2005) 7. [14] Ching-Hsu Chen, Po-Tse Tai, Wen-Feng Hsieh, Appl. Opt. 43 (2004) 6001. [15] B.P.S. Ahluwalia, X.-C. Yuan, S.-H. Tao, Opt. Commun. 238 (2004) 177. [16] M. de Angelisa, L. Cacciapuoti, G. Pierattini, G.M. Tino, Opt. Laser Eng. 39 (2003) 283. [17] K. Ait-Ameur, F. Sanchez, J. Mod. Opt. 46 (1999) 1537. [18] A.E. Siegman, Lasers, University Science, 1986. [19] I.S. Gradshteyn, I.M. Ryzhik, Table of Integral, Series, and Products, Academic Press, New York, 1980. [20] A.E. Siegman, SPIE 1224 (1990) 2.
Generation of bottle beam by focusing a super-Gaussian beam using a lens and an axicon Ming-Dar Wei Department of Photonics, Feng Chia University, 100, Wenhwa Road, Seatwen, Taichung 407, Taiwan, ROC Received 30 November 2006; received in revised form 20 April 2007; accepted 27 April 2007
Abstract This study explores the characteristics of the bottle beams that are formed by super-Gaussian beams that impinge through an axicon and a positive lens. Analytical solutions for the on-axial intensity of Gaussian and apertured-plane beams were obtained. The barrier around the dark region has a larger variation of intensity for higher-order super-Gaussian profiles. Flattening the tops of the profiles increases the bottle lengths for a fixed second-order moment width or distance between the axicon and the lens. Ó 2007 Elsevier B.V. All rights reserved. PACS: 42.25.Bs; 42.60.Jf; 42.25.Fx Keywords: Focusing; Super-Gaussian beams; Bottle beams; Beam propagation
1. Introduction A dark focus, surrounded by regions of higher intensity, was initially created by using a holographic phase plate to trap atoms [1]. This beam is then called a bottle beam, which is constructed using a computer-generated hologram [2]. Bottle beams have recently attracted attention because of their applications in optical trapping, in which the bluedetuned intensity distribution acts as a repulsive wall that traps atoms in the dark region [1,3–5]. Various approaches have been demonstrated to generate bottle beams, using such elements as a special phase mask [1], amplitude masks [2] and a bimorph adaptive mirror [6], by focusing two identical Gaussian beams with opposing radii of curvature using an interferometer [7], and by utilizing the characteristics of a focused nondiffraction beam [8–13]. Recently, a bottle beam was directly produced from a bare laser by exploiting the constructive interference of Laguerre–Gaussian modes with a cavity configuration close to g1g2 = 1/4 [14]. A
E-mail address: [email protected] 0030-4018/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2007.04.044
self-imaged bottle beam with an embedded sequence of three-dimensional intensity voids was obtained by the interference of two Bessel beams with different radial wave vectors [15]. A novel element, called as a double-axicon, was fabricated to generate a self-imaged bottle beam [4]. Imaging different portions of a self-imaged bottle beam can trap high and low indices microparticles [5]. The axicon is useful for generating bottle beams, because it provides two main advantages. The first is its high energy efficiency, and the other is the ease of adjusting the layout of experiments to vary the beam geometry. Both a lens–axicon doublet and a combination of divergent and convergent axicons were adopted to produce hollow and bottle beams [16]. Additionally, focusing the nondiffraction beam, which can be generated using an axicon, can transform it into a bottle beam [8–13]. The nondiffraction beams focus to create a ring image in the focal plane of the lens, and the dark region in the three-dimensional view is formed from two cones, which share a single base on the focal plane. This phenomenon has been experimentally verified [13], using a dynamically one-parameter-tunable method to generate bottle beams and hollow beams. As
20
M.-D. Wei / Optics Communications 277 (2007) 19–23
an apertured plane wave, associated with a plane wave existing within a finite transverse region, impinges through an axicon and a positive lens, the distance between the axicon and the focused lens is tuned to transform the beams from bottle beams to hollow beams. A tightly assembled axicon and lens can theoretically transform a Gaussian beam into a super-Gaussian profile, a ring-shaped profile or a doughnut profile [17]. However, the tunable parameter, which is the distance between the axicon and the lens changes, has not been comprehensively considered. This work investigates the characteristics of bottle beams that were generated by a super-Gaussian beam that impinges through an axicon and a positive lens. The effects of the beam profiles are of particular interest. The super-Gaussian function is popularly used to describe flat-topped profiles. The distribution of the super-Gaussian beam is given by h r n i ESG ðrÞ ¼ exp ; ð1Þ w where r is the radial coordinate; w is the radius of the beam at 1/e2, and n is order of the super-Gaussian beam. Increasing the order from n = 2 to infinity transforms the profiles from Gaussian to flat-topped. Therefore, the effect on the generation of bottle beams of gradually flattening the top of the profile can be examined. 2. Modeling Fig. 1 depicts the proposed scheme. The axicon, having the angle between the conical surface and the flat surface is c and the index is ng, has been successfully used to convert a plane wave into a Bessel beam. The transmittance of the axicon yields a linear phase shift to be sðrÞ ¼ expðibrÞ;
ð2Þ
where b = 2p(ng 1)tan(c)/k and k is the wavelength of the incident beam. When the lens with a focal length of f is located at a distance z0 beyond the axicon, the diffraction field E(q, z) at a distance z from the lens is given by the generalized Huygens formula [18] as, Z 1 2pi kDq2 krq expðikLÞ exp i Eðq; zÞ ¼ ESG ðrÞJ 0 kB B 2B 0 2 kAr exp i ibr r dr; ð3Þ 2B
super-Gaussian beam
γ
Z0
Z Z=0
Fig. 1. Schematic layout for a focused super-Gaussian beam.
where q is the radial coordinate; k is the wave number; J0 is the zero-order Bessel function, and L = z0 + z. The transformed ABCD matrix from the axicon through #the lens " 1 fz z0 þ z zf0 z A B to a distance z is ¼ . Since f1 1 zf0 C D the elements of the ABCD matrices are functions of the distance between the axicon and lens, z0, this dynamically tunable parameter controls the characteristics of the bottle beams. For a focused nondiffraction beam, the dark region in the three-dimensional view is formed from two cones, which share a base in the focal plane. The maximal dark region is in the focus plane given by z = f, yielding A = 0 and B = f; the transverse distribution in the focal plane is independent of z0 according to Eq. (3). Furthermore, this radius is almost independent of n, which fact is numerically confirmed in Section 3. Therefore, computing the on-axis intensity and the bottle length, associated with the on-axis dark region, determines the main characteristics of the bottle beam. The following discussion concerns two analytic solutions, relating to the Gaussian profile with n = 2 and the apertured plane with n = 1, which are the most common distributions in practical applications. (1) Gaussian beam When a Gaussian beam impinges on an axicon, a Bessel–Gaussian beam is produced. Thus, the following discussion equivalently analyzes the characteristics of the focused Bessel–Gaussian beam. At on-axis points, J0(0) = 1. Eq. (3) can be simplified to 2 Z 1 2pi r expðikLÞ exp 2 Eð0; zÞ ¼ kB w 0 2 kAr ibr r dr: ð4Þ exp i 2B Z 0
Based on the formula 3.462.5 in Ref. [19], the integral is rffiffiffi 1 m p 2 xelx 2mx dx ¼ 2l 2l l 2 m m exp ð5Þ 1 U pffiffiffi ; l l
1
kA and where U is the error function. Setting l ¼ w12 i 2B b m ¼ i 2, yields the on-axis field, rffiffiffi 2pi 1 ib p b2 expðikLÞ exp Eð0; zÞ ¼ kB 2l 4l l 4l
ib ð6Þ 1 U pffiffiffi 2 l
(2) Apertured-plane beam The focusing characteristics of the apertured-plane beam have been studied using a geometrical method [13]. The on-axis field distribution is expressed using the paraxial approximation, from Eq. (3), as Z w 2pi kAr2 Eð0; zÞ ¼ expðikLÞ ibr r dr: ð7Þ exp i kB 2B 0
M.-D. Wei / Optics Communications 277 (2007) 19–23
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pffiffiffi kA Setting a ¼ 2B and n ¼ a r 2ab into Eq. (7), yields Z x2 n b 2 expðin Þ þ 3=2 dn; Eð0; zÞ ¼ c1 ð8Þ a 2a x1 pffiffiffi b2 exp ikL þ i 4a where c1 ¼ 2pi , x1 ¼ 2pb ffiffia and x2 ¼ a kB w 2ab . An analytical form is c1 1 1 2 2 i expðix2 Þ þ i expðix1 Þ Eð0; zÞ ¼ 2 2 a c1 b 1 3=4 pffiffiffi 1=4 þ 3=2 ð1Þ pUi ½ð1Þ x2 2a 2
1 3=4 pffiffiffi 1=4 ð1Þ pUi ½ð1Þ x1 ; ð9Þ 2 where Ui is the imaginary error function. 3. Numerical results In the description of flat-topped profiles, the superGaussian function has the advantage of being explicitly expressible. However, propagations of super-Gaussian beams must be examined numerically, except in the aforementioned cases with n = 2 and n = 1. Moreover, since the distributions of bottle beams depend strongly on the width of the incident beams, the following set of field distributions is employed to maintain beam widths, which are calculated based on the second-order moment definition [20], for super-Gaussian beams as n N nr ESG ðrÞ ¼ exp ; W SG
Fig. 2. The evolution of a bottle beam with n = 2, c = 5°, ng = 1.5, w = 4 mm, z0 = 7.6 cm and f = 3.5 cm.
where Nn represents the ratio between the second-order moment width, WSG, and w for n such that 1=2 W SG Cð4=nÞ 12=n 2 ¼ Nn ¼ ; Cð2=nÞ w where C is the Gamma function. In the numerical computations, the parameters are set to c = 5°, ng = 1.5, k = 632 nm and f = 3.5 cm, as in Fig. 2. Fig. 2 displays the profile of a bottle beam with n = 2, WSG = 4 mm and z0 = 7.6 cm. A Bessel-like distribution is observed near the lens. Increasing z causes the side lobe to grow, and reduces the central intensity. The transverse pattern is then transferred to the doughnut ring and the size of the light region is minimal in the focal plane. Beyond the focal plane, the reverse process is performed to reconstruct the Bessel-like beam. A bottle beam is formed. For different orders of the super-Gaussian beam, similar bottle evolutions are obtained, but the bottle lengths vary. Fig. 3a plots the on-axis intensity as a function of z for z0 = 7.6 cm and WSG = 4 mm. Firstly, we concentrate on an incident beam that is an apertured plane wave, such that n = 1. Fig. 3b compares the numerical simulation with the analytical solution to Eq. (9). The results agree closely. The oscillation originates from the hard edge of the aperturedplane beam. Further, the values of z1,n and z2,n are set to
Fig. 3. (a) The on-axis intensity as a function of z with n = 2, 3, 5, 10, and 1 (labeled as n = inf). (b) The numerical and analytical results for an incident aperture-plane beam (n = 1).
represent the beginning and end points of the dark on-axis region of the order n, respectively; then, the bottle length is Dzn = z2,n z1,n. The bottle length decreases as n declines
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M.-D. Wei / Optics Communications 277 (2007) 19–23
in Fig. 3a. The geometrical analysis in the previous work indicated that the point z = z1,1 is determined by the boundary (or transverse-edge) rays of the apertured plane wave focused by the lens, and z2,1 corresponds to the onaxis image of the apex of the axicon that is formed by the lens [13]. Thus, z2,n is expected to be independent of the shapes of the incident beams, as was confirmed in Fig. 3a.
Fig. 4. The transverse profile with (a) z = 32 mm, (b) z = 35 mm and (c) z = 38 mm.
However, as the super-Gaussian beams are gradually transformed from top-hat (n = 1) to Gaussian (n = 2) profiles with the drop in n, the decaying of the tail of the transverse field distribution is equivalent to increasing the distance between the boundary ray and the axis. Therefore, z1,n, associated with the point of intersection of the axis and the focused boundary ray, increases, reducing the bottle length by decreasing n. Not only do the bottle lengths vary, but also the maximum intensity and the variation of the on-axis intensity near dark region increase with n. Fig. 4 plots the transverse distributions in front of, in and beyond the focal plane at z = 32 mm, z = 35 mm and z = 38 mm, respectively. The transverse profiles in Fig. 4a reveal asymmetric distributions, in which the near z-axis side varies more steeply than the other side. Moreover, the intensity on the near z-axis side increases more steeply for larger n. A reverse distribution was observed when z exceeded the focal length f = 3.5 cm, as presented in Fig. 4c, and the variations of the slope on the near z-axis side were similar. Fig. 4b plots the symmetric transverse distribution in the focal plane for various values of n. All of the radii of the dark rings approach R = f(n 1)c = 1.527 mm, determined by geometric analysis [13]. Based on analyzing the both longitudinal and transverse distributions of the bottle beam, the barrier around the dark region has a larger gradient of intensity for focusing higher-order super-Gaussian profile. Fig. 5 plots the bottle length obtained by varying the second-order moment width. As in the preceding analysis, the boundary ray far from the z-axis focuses at greater z1,n as the width increases, shortening the bottle length. Apparently, the width is an important parameter in determining the bottle length, such that the bottle length with n = 2 declines from 5.128 cm to 2.322 cm as the width is varied from 2 mm to 10 mm. However, the bottle length is also influenced by the shape of the beam profile, such that, for example, the bottle length changes from 3.686 cm to 5.276 cm as n varies from 2 to 1 at width WSG = 4 mm. This result also demonstrates that the ray tracing method is unsuitable for analyzing quantitatively
Fig. 5. The bottle length against the second-order moment width.
M.-D. Wei / Optics Communications 277 (2007) 19–23
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propagate through an axicon and a positive lens. The distance between the axicon and the lens and the beam width still dominate the bottle lengths as in previous studies, but the effect of the beam profiles is not negligible. Since the longitudinal and transverse distributions, the barrier around the dark region of the bottle beam supports a larger variation of intensity for higher-order super-Gaussian beams. Flattening the tops of the profiles substantially increases the bottle lengths for a fixed beam width or a fixed distance between the axicon and the lens. The analytical on-axial intensities were obtained for Gaussian and apertured-plane beams. Acknowledgement
Fig. 6. The bottle length as a function of the distance between the axicon and the focused lens.
the focusing characteristics of a lower-order super-Gaussian beam. The relationship between the bottle length Dz and z0 is studied, as shown in Fig. 6. Since z2,1 is associated with the on-axis image of the apex of the axicon that is formed through the lens, reducing the distance between the axicon and the lens, z0, increases z2,n and the bottle length. The value of z2,n becomes infinity at z0 = f = 3.5 cm. The focused beam is transferred into a cylindrical-hollow beam with identical dark circles in all cross-sections beyond the focus plane of the lens. Further, hollow beams are formed for z0 < f, as in Ref. [13]. In summarizing the numerical results, the bottle lengths vary by several tenths of a percentage point as the incident super-Gaussian beam converted from n = 2 to n = 1. This effect of the shape cannot be neglected, although the distance between the axicon and the lens, z0, and the width remain dominant parameters. At fixed width, this scheme is a dynamically one-parameter-tunable to generate bottle beams, in which the parameter of z0 is adjusted to vary the volumes of the dark regions. Since the blue-detuned intensity distribution acts as a repulsive wall that traps atoms in the dark region of a bottle beam, we suggest that this scheme may be used to dilute or condense dynamically the trapped atoms. 4. Conclusions This work investigates the characteristics of bottle beams that were formed by super-Gaussian beams that
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