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Propagation properties of Bessel and Bessel–Gaussian beams in a fractional Fourier transform optical system
ARTICLE IN PRESS Optics & Laser Technology 42 (2010) 280–284
Contents lists available at ScienceDirect
Optics & Laser Technology journal homepage: www.elsevier.com/locate/optlastec
Propagation properties of Bessel and Bessel–Gaussian beams in a fractional Fourier transform optical system Chengliang Zhao, Kaikai Huang , Xuanhui Lu Department of Physics, Zhejiang University, Hangzhou 310027, PR China
a r t i c l e in f o
a b s t r a c t
Article history: Received 12 June 2009 Accepted 1 July 2009 Available online 22 July 2009
The properties of Bessel–Gaussian beams (BGBs) and Bessel beams (BBs) propagating through a fractional Fourier transform (FRT) optical system have been investigated. The analytical transformation formulae for BBs and BGBs propagation through a FRT optical system are derived based on definition of the FRT in the cylindrical coordinate system. By using the derived formula, numerical examples are illustrated. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Bessel–Gaussian beams Fractional Fourier transform Propagation properties
1. Introduction In recent years, optical beams with an intensity minimum (zero in ideal case) along the beam axis, called dark-hollow beams (DHBs), have attracted much attention because of their increasing applications in atom optics. Atoms can be guided in DHBs by dipole potentials [1–3]. Many studies of the DHBs application in Bose–Einstein condensation [4], and in creating optical traps and tweezers for manipulating of micro-particles and biological species [5–8] have been reported. Several modes have been presented to describe DHBs. The well-known example is a TEM01* beam [9] (also known as a doughnut beam). Cai et al. also introduce a new model termed the hollow Gaussian beam [10]. An alternative model that may be used to describe DHBs is the high-order Bessel–Gaussian beams (BGBs), which was introduced by Bagini et al. [11] and Palma et al. [12]. It has been shown that the high-order Bessel–Gaussian mode is a convenient model to describe dark-hollow beams. It can be used for guiding, focusing and trapping ultracold atoms [13]. In a previous paper, we reported the properties of BGBs propagating through a misaligned optical system [14] and a theoretical analysis of the radiation force on a dielectric sphere produced by a highly focused BGB [15]. The fractional Fourier transform (FRT) is a generalization of the ordinary Fourier transform. In 1980, Namias introduce the FRT as a mathematical way for solving theoretical physics problems [16].
Corresponding author. Tel.: +86 135 888 83747.
E-mail address: [email protected] (K. Huang). 0030-3992/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2009.07.003
However, it had not attracted much attention until Ozaktas, Mendlovic and Lohmann introduced FRT into optics in 1993 [17,18]. Since then many studies of its properties, optical implementations and applications have been reported [19–25]. The propagation properties of many beams, Hermite–Gaussian beams [26], Hermite–cosh-Gaussian beams [27], hollow Gaussian beam [28], elliptical Gaussian beams [29], in an FRT optical system have been studied. However, to the best of our knowledge, Bessel beams (BBs) and Bessel–Gaussian beams have never considered. Therefore, for the properties and the wide application of the BBs and BGBs, the study on the behavior of BBs and BGBs propagating through FRT optical systems would be of practical interest. In this paper, by using the connection between the Collins formula and the FRT, an analytical formula for the BBs and BGBs is derived. The propagation properties of BBs and BGBs in the FRT plane are illustrated numerically.
2. Analytical formula for BBs and BGBs through FRT optical system The two kinds of optical setups introduced by Lohmann for implementing FRT are shown in Fig. 1. Generally, the Fresnel diffraction and the Collins formula have been used for studying the propagation of any beam through a paraxial optical system. However, for discussing the FRT optical system, the connections between the Fresnel diffraction and the FRT and between the Collins formula and FRT have been obtained [19,22]. Within the framework of the connection between the Collins formula and the FRT, the propagation of the BBs and BGBs through the FRT
ARTICLE IN PRESS C. Zhao et al. / Optics & Laser Technology 42 (2010) 280–284
281
Fig. 1. Optical system for performing the FRT: (a) one lens system and (b) two lens system.
of p-order optical system can be treated as following form [17]: Z 1 Z 1 ipðx21 þ y21 þ x22 þ y22 Þ 1 Ep ðx2 ; y2 Þ ¼ Eðx1 ; y1 Þexp lf tan f ilf sin f 1 1 2piðx1 x2 þ y1 y2 Þ dx1 dy1 ; exp lf sin f ð1Þ where f is the standard focal length, l the wavelength of the input beam and f ¼ pp/2. Eq. (1) is the expression of the FRT in the rectangular coordinate system. And the expression also can be transformed in the cylindrical coordinate system as follows [28]: Z 1 Z 2p ipðr12 þ r22 Þ 1 Ep ðr2 ; y2 Þ ¼ Eðr1 ; y1 Þexp lf tan f ilf sin f 0 0 2pir1 r2 cos ðy1 y2 Þ r1 dr1 dy1 ; exp ð2Þ lf sin f where r1, y1 and r2, y2 are the radial and the azimuthal coordinate in the input and output planes, respectively. The electric field of BGBs in the cylindrical coordinate system at z ¼ 0 can be defined as follows: Ep ðr1 ; y1 ; z ¼ 0Þ ¼ E0 Jn ðar1 Þexpðr12 =w20 Þexpðiny1 Þ; n ¼ 1; 2; 3; . . .
ð3Þ
where r1 is the radial and y1 the azimuthal coordinate, Jn the Bessel function of the first kind of n-order, a and E0 constants, w0 the waist of Gaussian beam. When w0-N, the HBGBs can reduce to a pure Bessel beam, and when a-0, we obtain a Gaussian beam. Fig. 2(a) shows the intensity distribution of BGBs for various n values. It is found from Fig. 2(a) that the dark region increases with the n. We also plot the BGBs with the w0-N and a ¼ 0 as shown in Fig. 2(b). Substituting Eq. (3) into (2) and using the following integral formula: Z 2p 1 exp½inðp=2 y2 ÞJn ðkr1 r2 =LÞ ¼ exp½iðkr1 r2 =LÞcos ðy1 2p 0 y2 Þ iny1 dy1 ;
ð5Þ
The following integral formulae are used to resolve Eq. (5): 1 a2 þ g2 ag Im ð6:1Þ x expðbx2 ÞJm ðaxÞJm ðgxÞ ¼ exp 4b 2b 2b 0 1
Im ðxÞ ¼ expðimp=2ÞJm ðixÞ ¼ ð1Þm expðimp=2ÞJm ðixÞ;
where Jn(x) and In(x) denote the n-order Bessel polynomial and modified Bessel polynomial, respectively. After tedious but straightforward integration, we obtained a result as follows: Ep ðr2 ; y2 Þ ¼
pE0 1 expðiny2 Þ ilf sin f 1=w20 þ ip=ðlf tan fÞ ( ) ipr22 a2 þ ½2pr2 =ðlf sin fÞ2 exp exp lf tan f 4½1=w20 þ ip=ðlf tan fÞ ! ipar2 : Jn lf sin f½1=w20 þ ip=ðlf tan fÞ
ð7Þ
Eq. (7) is the analytical formula of the BGBs through the FRT optical system.
3. Numerical simulations and discussion
ð4Þ
We can transform Eq. (2) into h p i ipr22 2pE0 Ep ðr2 ; y2 Þ ¼ y2 exp exp in ilf sin f 2 lf tan f ! Z 1 r2 ipr12 2pr1 r2 r dr : Jn ðar1 ÞJn exp 12 lf sin f 1 1 w0 lf tan f 0
Z
Fig. 2. Normalized intensity distribution of BGBs for different orders.
ð6:2Þ
The propagation properties of the BGBs through FRT optical systems are studied by using Eq. (7). In the following calculation, we assume that the wavelength l ¼ 632.8 nm. In Fig. 3, the normalized intensity distribution of the BGBs through the FRT optical system is plotted against the different fractional order with n ¼ 5, a ¼ 1, f ¼ 100 mm, w0 ¼ 1 mm. It is found that the intensity distribution of the BGBs in the FRT plane varies as the fractional order. When the fractional order 0opo1, with increasing of the fractional orders the intensity distribution of the output beam becomes more and more convergent, the beam waist of 1 mm may be easily converged into 0.1 mm. From Fig. 3, it can be seen that, when 1opo2, with increasing the fractional orders, the intensity distribution of the output beam becomes more and more divergent. The normalized intensity distribution of the BGBs against the beam order is shown in Fig. 4. The parameters are f ¼ 100 mm, a ¼ 1, w0 ¼ 1 mm, p ¼ 1.02. When the
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Fig. 3. Normalized intensity distribution of BGBs in the FRT plane against different fractional order.
Fig. 4. Normalized intensity distribution of BGBs in the FRT plane for different beam order.
order n ¼ 0, the intensity distribution shows that HBGBs become zero-order Bessel–Gaussian beams. It can be shown that the radius of the bright ring of the intensity distribution of the HBGBs increases when n increases. This also means that the dark region of the HBGBs increases as n increases. This result is similar to Fig. 3(a). Fig. 5 shows the normalized intensity of BGBs with different w0 in the FRT plane with f ¼ 100 mm, a ¼ 1, n ¼ 5,
p ¼ 1.02. From Fig. 5, it can be found that the intensity distribution related with the input beam waist. The intensity distribution is plotted with w0-N, such that the BGBs become Bessel beams. From Figs. 4 and 5, the results show that with the certain fractional order, the intensity distribution is related with the input beam waist and the order. The dark region increases with the input beam waist and the order. The intensity
ARTICLE IN PRESS C. Zhao et al. / Optics & Laser Technology 42 (2010) 280–284
283
Fig. 5. Normalized intensity distribution of BGBs in the FRT plane against different beam waist.
Fig. 6. Normalized intensity distribution of BGBs in the FRT plane against different focal length.
distribution of BGBs for various focal lengths f is also studied in Fig. 6 with n ¼ 5, a ¼ 1, p ¼ 1.02, w0 ¼ 1 mm. The dark region also increases as f increases. From the above discussion, we should choose the appropriate beam parameters, fractional order and the focal length in order to obtain a required intensity distribution.
4. Conclusions To summarize, the analytical formula of BGBs through the fractional Fourier transform by the definition of the FRT in the cylindrical coordinate system has been derived. The effects caused
ARTICLE IN PRESS 284
C. Zhao et al. / Optics & Laser Technology 42 (2010) 280–284
by fractional order p, the beam order n, beam waist w0 and the focal length f are studied. The results show that BGBs can induce to zero-order Bessel–Gaussian beams and Bessel beams with n ¼ 0 and w0-N, respectively, and the dark region is related to the above parameters.
Acknowledgement This work is supported by National Nature Science Foundation of China (Grant no. 10874012), China Academy of Space Technology Innovation Fund (Grant no. CAST200825), and by the Hi-Tech Research and Development Program of China (863 Project, Grant no. 2007AA12Z130). References [1] Kuga T, Torii Y, Shiokawa N, et al. Novel optical trip of atoms with a doughnut beam. Phys Rev Lett 1997;78:4713–16. [2] Ovchinnikov YB, Manek I, Grimm R. Cs atoms based on evanescent-wave cooling. Phys Rev Lett 1997;79:2225–8. [3] Song Y, Milam D, H Jr WT. Long, narrow all-light atom guide. Opt Lett 1999;24:1805–7. [4] Bongs K, Burger S, Dettmer S, et al. Waveguide for Bose–Einstein condensates. Phys Rev A 2001;63:031602. [5] Christou J, Tikhonenko V, Kivshar YS, et al. Vortex soliton motion and steering. Opt Lett 1996;21:1649–51. [6] Mehta D, Rief M, Spudich JA, et al. Single-molecule biomechanics with optical methods. Science 1999;283:1689–95. [7] Paterson L, Macdonald M, Arlt J, et al. Controlled rotation of optically trapped microscopic particles. Science 2001;292:912–14. [8] Simpson NB, Dholakia K, Allen L, et al. Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner. Opt Lett 1997;22: 52–54. [9] Siegmen AE. Lasers. Mill Valley, Calif: University Science; 1986. [10] Cai YJ, Lu XH, Lin Q. Hollow Gaussian beams and their propagation properties. Opt Lett 2003;28:1084–6.
[11] Bagini V, Frezza F, Santarsiero M, et al. Generalized Bessel–Gauss beams. J Mod Opt 1996;43:1155–66. [12] Palma C, Borghi R, Cincotti G. Beams originated by J0-correlated Schellmodel planar sources. Opt Commun 1996;125:113–21. [13] Lu XH, Chen XM, Zhang L, et al. High-order Bessel–Gaussian beam and its propagation properties. Chin Phys Lett 2003;20:2155–7. [14] Zhao CL, Wang LG, Chen H, et al. Propagation of high-order Bessel–Gaussian beam through a misaligned first-order optical system. Opt Laser Technol 2007;39:1199–203. [15] Zhao CL, Wang LG, Lu XH. Radiation forces of highly focused Bessel–Gaussian beams on a dielectric sphere. Optik 2008;119:477–80. [16] Namias V. The fractional order Fourier transform and its applications to quantum mechanics. J Inst Math Appl 1980;25:241–65. [17] Lohmann AW. Image rotation, Wigner rotaion, and the Fractional Fourier transform. J Opt Soc Am A 1993;10:2181–6. [18] Mendlovic D, Ozaktas HM. Fractional Fourier transforms and their optical implementation: I. J Opt Soc Am A 1993;10:1875–81. [19] Liu ZY, Wu XY, Fan DY. Collins formula in frequency-domain and fractional Fourier transforms. Opt Commun 1998;155:7–11. [20] Lohmann AW, Mendlovic D, Zalevsky Z. Fractional Hilbert transform. Opt Lett 1996;21:281–3. [21] Mendlovic D, Zalevsky Z, Dorsch RG, et al. New signal representation based on the fractional Fourier transform: definitions. J Opt Soc Am A 1995;12: 2424–2431. [22] Pellat-Finet P. Fresnel diffraction and the fractional-order Fourier transforms. Opt Lett 1994;19:1388–90. [23] Yu L, Lu X, Zeng Y, et al. Deriving the integral representation of a fractional Hankel transform from a fractional Fourier transform. Opt Lett 1998; 23:1158–60. [24] Zhang Y, Dong B Z, Gu B Y, et al. Beam shaping in the fractional Fourier transform domain. J Opt Soc Am A 1998;15:1114–20. [25] Zhu BH, Liu ST. Multifractional correlation. Opt Lett 2001;26:578–80. [26] Zhao DM, Mao HD, Liu HJ, et al. Propagation of Hermite–Gaussian beams in apertured fractional Fourier transforming systems. Optik 2003;114:504–8. [27] Zhao DM, Mao HD, Zheng CW, et al. The propagation properties and kurtosis parametric characteristics of Hermite–cosh-Gaussian beams passing through fractional Fourier transformation systems. Optik 2005;116: 461–468. [28] Zheng CW. Fractional Fourier transform for a hollow Gaussian beam. Phys Lett A 2006;355:156–61. [29] Du XY, Zhao D M. Fractional Fourier transform of truncated elliptical Gaussian beams. Appl Opt 2006;45:9049–52.
Contents lists available at ScienceDirect
Optics & Laser Technology journal homepage: www.elsevier.com/locate/optlastec
Propagation properties of Bessel and Bessel–Gaussian beams in a fractional Fourier transform optical system Chengliang Zhao, Kaikai Huang , Xuanhui Lu Department of Physics, Zhejiang University, Hangzhou 310027, PR China
a r t i c l e in f o
a b s t r a c t
Article history: Received 12 June 2009 Accepted 1 July 2009 Available online 22 July 2009
The properties of Bessel–Gaussian beams (BGBs) and Bessel beams (BBs) propagating through a fractional Fourier transform (FRT) optical system have been investigated. The analytical transformation formulae for BBs and BGBs propagation through a FRT optical system are derived based on definition of the FRT in the cylindrical coordinate system. By using the derived formula, numerical examples are illustrated. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Bessel–Gaussian beams Fractional Fourier transform Propagation properties
1. Introduction In recent years, optical beams with an intensity minimum (zero in ideal case) along the beam axis, called dark-hollow beams (DHBs), have attracted much attention because of their increasing applications in atom optics. Atoms can be guided in DHBs by dipole potentials [1–3]. Many studies of the DHBs application in Bose–Einstein condensation [4], and in creating optical traps and tweezers for manipulating of micro-particles and biological species [5–8] have been reported. Several modes have been presented to describe DHBs. The well-known example is a TEM01* beam [9] (also known as a doughnut beam). Cai et al. also introduce a new model termed the hollow Gaussian beam [10]. An alternative model that may be used to describe DHBs is the high-order Bessel–Gaussian beams (BGBs), which was introduced by Bagini et al. [11] and Palma et al. [12]. It has been shown that the high-order Bessel–Gaussian mode is a convenient model to describe dark-hollow beams. It can be used for guiding, focusing and trapping ultracold atoms [13]. In a previous paper, we reported the properties of BGBs propagating through a misaligned optical system [14] and a theoretical analysis of the radiation force on a dielectric sphere produced by a highly focused BGB [15]. The fractional Fourier transform (FRT) is a generalization of the ordinary Fourier transform. In 1980, Namias introduce the FRT as a mathematical way for solving theoretical physics problems [16].
Corresponding author. Tel.: +86 135 888 83747.
E-mail address: [email protected] (K. Huang). 0030-3992/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2009.07.003
However, it had not attracted much attention until Ozaktas, Mendlovic and Lohmann introduced FRT into optics in 1993 [17,18]. Since then many studies of its properties, optical implementations and applications have been reported [19–25]. The propagation properties of many beams, Hermite–Gaussian beams [26], Hermite–cosh-Gaussian beams [27], hollow Gaussian beam [28], elliptical Gaussian beams [29], in an FRT optical system have been studied. However, to the best of our knowledge, Bessel beams (BBs) and Bessel–Gaussian beams have never considered. Therefore, for the properties and the wide application of the BBs and BGBs, the study on the behavior of BBs and BGBs propagating through FRT optical systems would be of practical interest. In this paper, by using the connection between the Collins formula and the FRT, an analytical formula for the BBs and BGBs is derived. The propagation properties of BBs and BGBs in the FRT plane are illustrated numerically.
2. Analytical formula for BBs and BGBs through FRT optical system The two kinds of optical setups introduced by Lohmann for implementing FRT are shown in Fig. 1. Generally, the Fresnel diffraction and the Collins formula have been used for studying the propagation of any beam through a paraxial optical system. However, for discussing the FRT optical system, the connections between the Fresnel diffraction and the FRT and between the Collins formula and FRT have been obtained [19,22]. Within the framework of the connection between the Collins formula and the FRT, the propagation of the BBs and BGBs through the FRT
ARTICLE IN PRESS C. Zhao et al. / Optics & Laser Technology 42 (2010) 280–284
281
Fig. 1. Optical system for performing the FRT: (a) one lens system and (b) two lens system.
of p-order optical system can be treated as following form [17]: Z 1 Z 1 ipðx21 þ y21 þ x22 þ y22 Þ 1 Ep ðx2 ; y2 Þ ¼ Eðx1 ; y1 Þexp lf tan f ilf sin f 1 1 2piðx1 x2 þ y1 y2 Þ dx1 dy1 ; exp lf sin f ð1Þ where f is the standard focal length, l the wavelength of the input beam and f ¼ pp/2. Eq. (1) is the expression of the FRT in the rectangular coordinate system. And the expression also can be transformed in the cylindrical coordinate system as follows [28]: Z 1 Z 2p ipðr12 þ r22 Þ 1 Ep ðr2 ; y2 Þ ¼ Eðr1 ; y1 Þexp lf tan f ilf sin f 0 0 2pir1 r2 cos ðy1 y2 Þ r1 dr1 dy1 ; exp ð2Þ lf sin f where r1, y1 and r2, y2 are the radial and the azimuthal coordinate in the input and output planes, respectively. The electric field of BGBs in the cylindrical coordinate system at z ¼ 0 can be defined as follows: Ep ðr1 ; y1 ; z ¼ 0Þ ¼ E0 Jn ðar1 Þexpðr12 =w20 Þexpðiny1 Þ; n ¼ 1; 2; 3; . . .
ð3Þ
where r1 is the radial and y1 the azimuthal coordinate, Jn the Bessel function of the first kind of n-order, a and E0 constants, w0 the waist of Gaussian beam. When w0-N, the HBGBs can reduce to a pure Bessel beam, and when a-0, we obtain a Gaussian beam. Fig. 2(a) shows the intensity distribution of BGBs for various n values. It is found from Fig. 2(a) that the dark region increases with the n. We also plot the BGBs with the w0-N and a ¼ 0 as shown in Fig. 2(b). Substituting Eq. (3) into (2) and using the following integral formula: Z 2p 1 exp½inðp=2 y2 ÞJn ðkr1 r2 =LÞ ¼ exp½iðkr1 r2 =LÞcos ðy1 2p 0 y2 Þ iny1 dy1 ;
ð5Þ
The following integral formulae are used to resolve Eq. (5): 1 a2 þ g2 ag Im ð6:1Þ x expðbx2 ÞJm ðaxÞJm ðgxÞ ¼ exp 4b 2b 2b 0 1
Im ðxÞ ¼ expðimp=2ÞJm ðixÞ ¼ ð1Þm expðimp=2ÞJm ðixÞ;
where Jn(x) and In(x) denote the n-order Bessel polynomial and modified Bessel polynomial, respectively. After tedious but straightforward integration, we obtained a result as follows: Ep ðr2 ; y2 Þ ¼
pE0 1 expðiny2 Þ ilf sin f 1=w20 þ ip=ðlf tan fÞ ( ) ipr22 a2 þ ½2pr2 =ðlf sin fÞ2 exp exp lf tan f 4½1=w20 þ ip=ðlf tan fÞ ! ipar2 : Jn lf sin f½1=w20 þ ip=ðlf tan fÞ
ð7Þ
Eq. (7) is the analytical formula of the BGBs through the FRT optical system.
3. Numerical simulations and discussion
ð4Þ
We can transform Eq. (2) into h p i ipr22 2pE0 Ep ðr2 ; y2 Þ ¼ y2 exp exp in ilf sin f 2 lf tan f ! Z 1 r2 ipr12 2pr1 r2 r dr : Jn ðar1 ÞJn exp 12 lf sin f 1 1 w0 lf tan f 0
Z
Fig. 2. Normalized intensity distribution of BGBs for different orders.
ð6:2Þ
The propagation properties of the BGBs through FRT optical systems are studied by using Eq. (7). In the following calculation, we assume that the wavelength l ¼ 632.8 nm. In Fig. 3, the normalized intensity distribution of the BGBs through the FRT optical system is plotted against the different fractional order with n ¼ 5, a ¼ 1, f ¼ 100 mm, w0 ¼ 1 mm. It is found that the intensity distribution of the BGBs in the FRT plane varies as the fractional order. When the fractional order 0opo1, with increasing of the fractional orders the intensity distribution of the output beam becomes more and more convergent, the beam waist of 1 mm may be easily converged into 0.1 mm. From Fig. 3, it can be seen that, when 1opo2, with increasing the fractional orders, the intensity distribution of the output beam becomes more and more divergent. The normalized intensity distribution of the BGBs against the beam order is shown in Fig. 4. The parameters are f ¼ 100 mm, a ¼ 1, w0 ¼ 1 mm, p ¼ 1.02. When the
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Fig. 3. Normalized intensity distribution of BGBs in the FRT plane against different fractional order.
Fig. 4. Normalized intensity distribution of BGBs in the FRT plane for different beam order.
order n ¼ 0, the intensity distribution shows that HBGBs become zero-order Bessel–Gaussian beams. It can be shown that the radius of the bright ring of the intensity distribution of the HBGBs increases when n increases. This also means that the dark region of the HBGBs increases as n increases. This result is similar to Fig. 3(a). Fig. 5 shows the normalized intensity of BGBs with different w0 in the FRT plane with f ¼ 100 mm, a ¼ 1, n ¼ 5,
p ¼ 1.02. From Fig. 5, it can be found that the intensity distribution related with the input beam waist. The intensity distribution is plotted with w0-N, such that the BGBs become Bessel beams. From Figs. 4 and 5, the results show that with the certain fractional order, the intensity distribution is related with the input beam waist and the order. The dark region increases with the input beam waist and the order. The intensity
ARTICLE IN PRESS C. Zhao et al. / Optics & Laser Technology 42 (2010) 280–284
283
Fig. 5. Normalized intensity distribution of BGBs in the FRT plane against different beam waist.
Fig. 6. Normalized intensity distribution of BGBs in the FRT plane against different focal length.
distribution of BGBs for various focal lengths f is also studied in Fig. 6 with n ¼ 5, a ¼ 1, p ¼ 1.02, w0 ¼ 1 mm. The dark region also increases as f increases. From the above discussion, we should choose the appropriate beam parameters, fractional order and the focal length in order to obtain a required intensity distribution.
4. Conclusions To summarize, the analytical formula of BGBs through the fractional Fourier transform by the definition of the FRT in the cylindrical coordinate system has been derived. The effects caused
ARTICLE IN PRESS 284
C. Zhao et al. / Optics & Laser Technology 42 (2010) 280–284
by fractional order p, the beam order n, beam waist w0 and the focal length f are studied. The results show that BGBs can induce to zero-order Bessel–Gaussian beams and Bessel beams with n ¼ 0 and w0-N, respectively, and the dark region is related to the above parameters.
Acknowledgement This work is supported by National Nature Science Foundation of China (Grant no. 10874012), China Academy of Space Technology Innovation Fund (Grant no. CAST200825), and by the Hi-Tech Research and Development Program of China (863 Project, Grant no. 2007AA12Z130). References [1] Kuga T, Torii Y, Shiokawa N, et al. Novel optical trip of atoms with a doughnut beam. Phys Rev Lett 1997;78:4713–16. [2] Ovchinnikov YB, Manek I, Grimm R. Cs atoms based on evanescent-wave cooling. Phys Rev Lett 1997;79:2225–8. [3] Song Y, Milam D, H Jr WT. Long, narrow all-light atom guide. Opt Lett 1999;24:1805–7. [4] Bongs K, Burger S, Dettmer S, et al. Waveguide for Bose–Einstein condensates. Phys Rev A 2001;63:031602. [5] Christou J, Tikhonenko V, Kivshar YS, et al. Vortex soliton motion and steering. Opt Lett 1996;21:1649–51. [6] Mehta D, Rief M, Spudich JA, et al. Single-molecule biomechanics with optical methods. Science 1999;283:1689–95. [7] Paterson L, Macdonald M, Arlt J, et al. Controlled rotation of optically trapped microscopic particles. Science 2001;292:912–14. [8] Simpson NB, Dholakia K, Allen L, et al. Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner. Opt Lett 1997;22: 52–54. [9] Siegmen AE. Lasers. Mill Valley, Calif: University Science; 1986. [10] Cai YJ, Lu XH, Lin Q. Hollow Gaussian beams and their propagation properties. Opt Lett 2003;28:1084–6.
[11] Bagini V, Frezza F, Santarsiero M, et al. Generalized Bessel–Gauss beams. J Mod Opt 1996;43:1155–66. [12] Palma C, Borghi R, Cincotti G. Beams originated by J0-correlated Schellmodel planar sources. Opt Commun 1996;125:113–21. [13] Lu XH, Chen XM, Zhang L, et al. High-order Bessel–Gaussian beam and its propagation properties. Chin Phys Lett 2003;20:2155–7. [14] Zhao CL, Wang LG, Chen H, et al. Propagation of high-order Bessel–Gaussian beam through a misaligned first-order optical system. Opt Laser Technol 2007;39:1199–203. [15] Zhao CL, Wang LG, Lu XH. Radiation forces of highly focused Bessel–Gaussian beams on a dielectric sphere. Optik 2008;119:477–80. [16] Namias V. The fractional order Fourier transform and its applications to quantum mechanics. J Inst Math Appl 1980;25:241–65. [17] Lohmann AW. Image rotation, Wigner rotaion, and the Fractional Fourier transform. J Opt Soc Am A 1993;10:2181–6. [18] Mendlovic D, Ozaktas HM. Fractional Fourier transforms and their optical implementation: I. J Opt Soc Am A 1993;10:1875–81. [19] Liu ZY, Wu XY, Fan DY. Collins formula in frequency-domain and fractional Fourier transforms. Opt Commun 1998;155:7–11. [20] Lohmann AW, Mendlovic D, Zalevsky Z. Fractional Hilbert transform. Opt Lett 1996;21:281–3. [21] Mendlovic D, Zalevsky Z, Dorsch RG, et al. New signal representation based on the fractional Fourier transform: definitions. J Opt Soc Am A 1995;12: 2424–2431. [22] Pellat-Finet P. Fresnel diffraction and the fractional-order Fourier transforms. Opt Lett 1994;19:1388–90. [23] Yu L, Lu X, Zeng Y, et al. Deriving the integral representation of a fractional Hankel transform from a fractional Fourier transform. Opt Lett 1998; 23:1158–60. [24] Zhang Y, Dong B Z, Gu B Y, et al. Beam shaping in the fractional Fourier transform domain. J Opt Soc Am A 1998;15:1114–20. [25] Zhu BH, Liu ST. Multifractional correlation. Opt Lett 2001;26:578–80. [26] Zhao DM, Mao HD, Liu HJ, et al. Propagation of Hermite–Gaussian beams in apertured fractional Fourier transforming systems. Optik 2003;114:504–8. [27] Zhao DM, Mao HD, Zheng CW, et al. The propagation properties and kurtosis parametric characteristics of Hermite–cosh-Gaussian beams passing through fractional Fourier transformation systems. Optik 2005;116: 461–468. [28] Zheng CW. Fractional Fourier transform for a hollow Gaussian beam. Phys Lett A 2006;355:156–61. [29] Du XY, Zhao D M. Fractional Fourier transform of truncated elliptical Gaussian beams. Appl Opt 2006;45:9049–52.