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Focal shift of apodized truncated hyperbolic-cosine-Gaussian beam
Optics Communications 273 (2007) 21–27 www.elsevier.com/locate/optcom
Focal shift of apodized truncated hyperbolic-cosine-Gaussian beam Xiumin Gao a b
a,*
, Jinsong Li
b,c
Electronics and Information College, Hangzhou Dianzi University, Xiasha Campus, Hangzhou 310018, PR China College of Optics and Electronics, University of Shanghai for Science and Technology, Shanghai 20093, PR China c Institute of Optoelectronics Technology, China Jiliang University, Hangzhou 310018, PR China Received 6 July 2006; received in revised form 7 December 2006; accepted 12 December 2006
Abstract Focal shift of hyperbolic-cosine-Gaussian beams induced by pure phase apodizer was investigated theoretically. The pure phase apodizer consists of three concentric zones: center circle zone, inner annular zone and outer annular zone. And the phase variance of the inner annular zone is adjustable. Results show that intensity peak moves far from optical aperture and then shrinks sharply for certain radii of zones with increasing phase variance of the inner annular zone. Simultaneously, one new intensity peak occurs near optical aperture, moves far from the optical aperture, and then becomes intensity maximum peak, and repeats the evolution process of the former intensity peak. Tunable focal shift occurs with focal switch. Decreasing the phase variance can change the move direction of the intensity peaks. In addition, the maximum distance between the two intensity peaks can be altered by beam parameters of cosh parts, and the distance value increases and then decreases with increasing beam parameters of cosh parts for certain radii of zones. Tunable focal shift is also discussed to construct optical tweezers. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Focal shift; Hyperbolic-cosine-Gaussian beam; Pure phase apodizer; Optical tweezers
1. Introduction In many optical systems Gaussian beams are very useful and common beams. In fact, Gaussian beams are only one solution of the Helmholtz equation. The more general solutions of the Helmholtz equation are Hermite-sinusoidalGaussian beams, which were introduced by Casperson and coworkers [1,2] and have attracted much attention recently. These beams can be obtained by using phase mirrors [3]. The hyperbolic-cosine-Gaussian (ChG) beams, which are regarded as the special cases of sinusoidal-Gaussian beams or Hermite-sinusoidal-Gaussian beams, are of practical interest because of their efficience in extracting energy from conventional laser amplifiers, and the flat-top field distribution can be obtained by choosing suitable beam parameters of cosh parts [4]. So propagation and focusing properties of ChG beams have become object of some works [5–8]. *
Corresponding author. Tel.: +86 57186791561; fax: +86 57186791505. E-mail address: [email protected] (X. Gao).
0030-4018/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2006.12.021
In the investigation of the focusing properties of optical beams, tracing the movement of the point of absolute maximum intensity along optical axis has attracted many researchers for several decades [9–18]. It was found that the point of absolute maximum intensity does not coincide with the geometrical focus but shifts along optical axis [14–16], this phenomenon is referred to as focal shift. More interesting, the focal shift may be incontinuous in certain optical focusing systems. Focal shift may be accompanied by an effective permutation of the focal point, namely, intensity maximum can jump from one position to other position, and this effect is called focal switch [17,18]. In this article, focal shift of focused hyperbolic-cosineGaussian beams induced by a pure concentric phase apodizer was investigated through scalar diffractive theory. Dependence of tunable focal shift on phase variance of the apodizer is studied, and the maximum distance between two intensity peaks, which illustrates focal split, is also discussed. Section 2 shows the principle of focusing optical system, in which ChG beam is apodized. And Section 3
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X. Gao, J. Li / Optics Communications 273 (2007) 21–27
presents results and discussions. Conclusions are summarized in Section 4. 2. Principle of focusing system In the focusing system we investigated, ChG beam passes through a phase apodizer, and then converges through an objective lens. It should be noted that the apodizer may be not only phase plate that works in transmittance mode, but also a reflector with tunable phase distribution in reflective mode, as shown in Fig. 1. The phase apodizer consists of three zones, center circle zone, inner annular zone and outer annular zone. The motive for choosing this kind of apodizer is that this kind of apodizer may be easy to produce and duplicate for mass production, and the analysis method of the three-portion apodizer can be also conveniently extended to analyze the generalized apodizer. The phase variance of inner annular zone is adjustable. Parameters a, b, and 1 are the normalized convergence angles for each zone, respectively, as Fig. 2 shows. Parameters a and b are the ratios of convergence angle of center circle zone and inner annular zone to the aperture angle of the focusing optical system, respectively, b ¼ a1 =a and a ¼ a2 =a, where a1 and a2 are the genuine values of convergence angle corresponding to center circle zone and inner annular zone, and a is the aperture angle of the optical system. Electric field of the ChG beam at z = 0 is, 2 x þ y2 Eðx; y; z ¼ 0Þ ¼ A0 cosh ðXx xÞ cosh ðXy yÞ exp x20 ð1Þ where x0 is waist width, cosh is hyperbolic-cosine function. Xx and Xy are beam parameters of cosh parts. Now the
Fig. 1. Schematic of apodized focusing system. (a) Transmitted mode apodizer, (b) reflective apodizer.
Fig. 2. Parameters illustration of focused modulated beams. Parameters a, b, and 1 are the normalized convergence angles for each zone, respectively. / is the phase variance of the inner annular zone.
radial coordinate in z ¼ 0 plane is introduced, x ¼ r0 cos ðuÞ, y ¼ r0 sin ðuÞ, so Eq. (1) can be rewritten as Eðx; y; z ¼ 0Þ ¼ Eðr0 ; uÞ ¼ A0 cosh ½Xx r0 cos ðuÞ cosh ½Xy r0 sin ðuÞ r20 exp 2 x0
ð2Þ
where r0 is the radial coordinate. By simple deviation, ChG beam can be expressed as Eðh; uÞ ¼ A0 cosh ½NA1 bx sin ðhÞ cos ðuÞ cosh ½NA1 by sin ðhÞ sin ðuÞ sin2 ðhÞ exp NA2 w2
ð3Þ
where NA is the numerical aperture of the focusing system, and rp is the outer radius of the truncated ChG beam. Here, w ¼ x0 =rp . Parameters bx ¼ rp Xx , and by ¼ rp Xy indicate beam parameters of cosh parts and are called decentered parameters below. The phase variances of center zone and outer annular zone are zero, and the phase variance of inner annular zone is tunable and symbolized as /. According to scalar theory, electric amplitude distribution in focal region of the focusing optical system can be written as [19], Z Z i b 2p U ðr; w; zÞ ¼ Eðh; uÞ exp ½ikr sinðhÞ cosðu wÞ k 0 0 Z Z i 1 2p Eðh; uÞ ikz cosðhÞ sinðhÞdhdu þ k a 0 exp ½ikr sinðhÞ cosðu wÞ ikz cosðhÞ Z Z i a 2p sinðhÞdhdu þ Eðh; uÞ expði/Þ k b 0 exp ½ikr sinðhÞ cosðu wÞ ikz cosðhÞ sinðhÞdhdu ð4Þ where, u 2 ½0; 2p. It should be noted that focal shift refers to the distance between the genuine focal plane and geometrical focal plane. And here genuine focal plane is the plane that passes through the intensity maximum peak and is perpendicular to optical axis. The light intensity in the focal region is proportional to the modulus square of Eq. (4).
X. Gao, J. Li / Optics Communications 273 (2007) 21–27
3. Results and discussions The optical distribution in focal region is calculated according to the Eq. (4). Without losing generality and validity, the intensity is normalized by optical intensity maximum and it is supposed that NA = 0.6, w = 3 in computation. Fig. 3 illustrates the evolution of axial intensity distribution for bx ¼ by ¼ 10; b ¼ 0:1; a ¼ 0:6 with changing /. Arrows show the move direction of the intensity peaks. When / increases from 0 to p, it can be seen that intensity peak moves away from geometric focal plane in the direction far from optical aperture, which shows that focal shift occurs. In this evolution process, one new intensity peak comes into being in the near side of the maximum intensity peak, becomes stronger and stronger with increasing /, and also shifts in the same move direction of the intensity maximum peak. However, the former peak moves more quickly than the new smaller intensity peak, so the distance between them broadens with increasing /. When the two peaks appear simultaneously, there is one local axial
Fig. 3. Axial intensity distribution for bx ¼ by ¼ 10, b = 0.1, a = 0.6, / ranges from (a) 0 to p, (b) p to 0. Arrows shows the move direction of the intensity peaks.
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intensity minimum. From the figure, we can see that the local intensity minimum also shifts and its intensity value decreases smoothly with increasing /. When / increases up to p, the intensity of the two intensity peaks equals, the distance between the two peaks comes to maximum value, and there are two maximum intensity peaks. Increase / continuously, the evolution of axial intensity distribution is shown in Fig. 3b. The intensity peak far from optical aperture shrinks. Therefore, the intensity peak near optical aperture becomes only one maximum peak, focal switch occurs. With increasing /, the new intensity maximum peak goes on shifting. At the same time, the intensity peak far from optical aperture also shifts continuously, however its move velocity is smaller than that of intensity maximum peak, so the distance between the two intensity peaks decreases. And the local axial intensity minimum shifts and then disappears when / approaches 2p. It is found that the evolution process in Fig. 3a and b is symmetric. And / ¼ p is critical value where there are two intensity maximum peaks with maximum distance between them. Now we trace the focal shift. Dependence of focal shift on / is illustrated in Fig. 4. It is shown that the value of focal shift is positive and increases with increasing / considerably. There is one critical position with / ¼ p, at which there are two focal shift values, and focal switch also happens. With increasing /, the negative value of focal shift increases when / changes from p to 1.5p. And the focal shift changing process repeats with increasing /. When / decreases, the focal shift and focal switch can also occur, the difference is the move direction of intensity peaks. By altering /, focal shift can occurs with focal switch, and the move direction can also be changed. Now promising application of the above changing process is discussed. Optical tweezers offers a very convenient, noninvasive access to process at the microscopic scale, and has become valuable tool and accelerated many major advances in numerous areas of science since Ashkin and co-workers accomplished optical tweezers experimentally
Fig. 4. Dependence of focal shift on / for bx ¼ by ¼ 10, b = 0.1, a = 0.6.
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[20–25]. Many approaches have been proposed to construct optical trap, including generalized phase-contrast technique [26], diffractive optical element [27], holograms [28], self-reconstruct light beam [29], and interferometer pattern [30]. In optical trapping system, it is usually deemed that the forces exerted on the particles in light field include two kinds of forces, one is the optical gradient force, which is proportional to the optical intensity gradient; the other is the scattering force, which is proportional to the optical intensity [20,31]. Therefore, the tunable focal shift predicts that the position of optical trap may be controllable. If one intensity peak acts as one optical trap, the focal shift means that the optical trapped particle in the intensity peak can be moved. So in the proposed apodized system we investigated, the particles may be trapped in maximum intensity peak and moved far from optical aperture by increasing phase shift / from 0 to p. At the same time, one new maximum intensity peak occurs. With phase shift / increasing from p to 2p, the intensity peak far from aperture shrinks, which means that the trapped particle is deposited there. The peak near optical aperture becomes only one maximum intensity peak and can trap new one particle, then carries the trapped particle far away from the aperture with increasing /. Therefore, the particles may be transported from one place to another far from aperture by increasing / continuously, as shown in Fig. 5. Decrease of the phase shift may induce the optical trap move towards optical aperture. Therefore, the focused apodized ChG beam can be used to construct tunable optical tweezers for dynamical manipulation of particles. Here, typical three-dimensional optical intensity distributions in w ¼ p=4 plane are illustrated in Fig. 6 due to the geometry properties of the ChG beam and focal intensity distribution [5]. From the transverse and axial optical distributions, we can see that focal shift and focal switch can occur and the evolution principle is similar to that in Fig. 3. Arrows denote the motion direction of the optical peaks. It should be noted that the transverse optical distributions are very steep which shows that the optical gradient force is big, therefore the optical trap formed by these optical peaks are stable. In order to understand the focal shift and focal switch further, axial intensity distributions for b = 0.1, a = 0.6, and different decentered parameters are also computed. Fig. 7 shows that axial intensities for bx ¼ by ¼ 5,
Fig. 5. Schematic of optical tweezers application of focused beam. In collecting zone, new intensity peak occurs, and traps particles in the region, then move them far from optical aperture to release zone. And then release those trapped particles. Arrows show the move direction of trapped particles.
bx ¼ by ¼ 3, respectively. It can be seen that focal shift principle is similar to that in Fig. 3, however, the maximum distance between that the two intensity peaks decreases with decreasing decentered parameters. And for bx ¼ by ¼ 5, the value of the axial local intensity minimum
Fig. 6. Intensity distributions for bx ¼ by ¼ 10, b = 0.1, a = 0.6, and (a) / ¼ 0, (b) / ¼ 0:6p, (c) / ¼ p, and (d) / ¼ 1:6p.
X. Gao, J. Li / Optics Communications 273 (2007) 21–27
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Fig. 7. Axial intensity distribution for b = 0.1, a = 0.6, / ranges from 0 to p, and (a) bx ¼ by ¼ 5, (b) bx ¼ by ¼ 3. Arrows shows the move direction of the intensity peaks. Unit of number in this figure is p.
increases when / increases from 0 to p, which is inverse to that in Fig. 3. For bx ¼ by ¼ 3, the axial local intensity minimum occurs when / approaches p. Therefore, the maximum distance between the two intensity peaks can be altered by decentered parameters, and there is critical value of / at which the axial local intensity minimum comes into being. In order to confirm the principle of the axial intensity evolution, we also computed axial intensity distributions for bigger decentered parameters under the condition that / increases from 0 to p. Fig. 8 shows that axial intensity distributions for b = 0.1, a = 0.6, bx ¼ by ¼ 15, 20, and 30, respectively. Arrows show the move direction of the intensity peaks. With increasing /, intensity peak moves in the direction far from optical aperture. In the evolution process, one new intensity peak comes into being in the near side of the maximum intensity peak, becomes stronger and stronger with increasing /, and also shifts. The distance between the two peaks increases with increasing /. When the two peaks appear simultaneously, there is one local axial intensity minimum.
Fig. 8. Axial intensity distribution for b = 0.1, a = 0.6, / ranges from 0 to p, and (a) bx ¼ by ¼ 15, (b) bx ¼ by ¼ 20, (c) bx ¼ by ¼ 30. Arrows shows the move direction of the intensity peaks. Unit of number in this figure is p.
We can see that the local intensity minimum also shifts and its intensity value increases with increasing /, and change rate increases with decentered parameters of the beam, as shown in Fig. 8a and b. In addition, for certain decentered
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Fig. 10. Dependence of maximum focal shift on decentered parameters of the beam for b = 0.1, a = 0.6.
Fig. 9. Axial intensity distribution for b = 0.1, a = 0.6, / ¼ p, and bx ¼ by ¼ b ranges from (a) 0 to 10, (b) 30 to 45.
increases to its maximum. Then the distance decreases with increasing b, and when value of b equals 38, the distance becomes zero, which shows that the focal shift phenomenon disappears. Fig. 10 shows the dependence of maximum focal shift on decentered parameters of the beam for b = 0.1, a = 0.6. From above process, it can be seen that for certain geometrical parameters of the apodizer, the maximum distance between the two intensity peaks can be altered by decentered parameters, which illustrates that if the apodized beam is used to construct tunable optical tweezers, the distance between collecting zone and release zone, shown in Fig. 4, can be adjusted. 4. Conclusions
parameters, there is a critical /, at which axial local intensity minimum begin to occur, and the critical / increases with increasing decentered parameters. In the whole evolution process, it can be seen that the critical /, at which axial local intensity minimum begin occurs, decreases, and then increases with increasing decentered parameters of the beam. From above simulation, it can be seen that the maximum distance between the two intensity peaks can be altered by decentered parameters, and occurs at / ¼ p. Now we focus on adjustable degree of the maximum distance. Fig. 9 illustrates axial intensity distributions for b = 0.1, a = 0.6, / ¼ p, and different decentered parameters b ¼ bx ¼ by of the ChG beams. For small b, there is only one intensity peak in changing process, focal shift does not occur, there are not two intensity peaks at all, so the maximum distance is zero. Increase b continuously and b value is bigger than about 4, the focal shift and focal split occur, the maximum distance increases sharply, then the increase speed of the distance decreases with increasing b. When b equals about 18, the distance
Focal shift of hyperbolic-cosine-Gaussian beams induced by a phase apodizer was investigated. The pure phase apodizer consists of three concentric zones: center circle zone, inner annular zone and outer annular zone. Phase variance of the inner annular zone can be altered. Simulation results show that for certain radii of zones, intensity peak moves far from optical aperture and then shrinks sharply with increasing phase variance of the inner zone. Simultaneously, one new intensity peak occurs near optical aperture and shifts in the same direction as that of former intensity peak, its intensity increases sharply and then becomes intensity maximum peak, which shows that tunable focal shift companied with focal switch. Decrease the phase variance may change the move direction of the two intensity peaks. In addition, the maximum distance between the two intensity peaks can be altered by decentered parameter of the beam, the value of the maximum distance increases and then decreases with increasing decentered parameter. The tunable focal shift and focal switch may be used to construct controllable optical tweezers.
X. Gao, J. Li / Optics Communications 273 (2007) 21–27
Acknowledgements This work was supported by Science Research Project of Hangzhou Dianzi University. And the equation derivation and codes used in this article can be obtained conveniently through [email protected].
[14] [15] [16] [17] [18] [19] [20]
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J.J. Stamnes, B. Spjelkavik, Opt. Commun. 40 (1981) 81. M.P. Givens, Opt. Commun. 42 (1982) 145. Y. Li, J. Opt. Soc. Am. 72 (1982) 770. M. Martines-Corral, V. Climent, Appl. Opt. 35 (1996) 24. Y. Li, J. Opt. Soc. Am. A 14 (1997) 1297. M. Gu, Advanced Optical Imaging Theory, Springer, Heidelberg, 2000, p. 163. A. Ashkin, J.M. Dziedzic, J.E. Bjorkholm, S. Chu, Opt. Lett. 11 (1986) 288. M.P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, K. Dholakia, Science 296 (2002) 1101. D.G. Grier, Nature 424 (2003) 810. M.P. MacDonald, G.C. Spalding, K. Dholakia, Nature 426 (2003) 421. S. Kawata, Y. Inouye, T. Sugiura, Jpn. J. Appl. Phys. 33 (1994) L1725. A. Ashkin, J.M. Dziedzic, T. Yamane, Nature 330 (1989) 7691. R.L. Eriksen, P.C. Mogensen, J. Glu¨ckstad, Opt. Lett. 27 (2002) 267. E.R. Dufresne, D.G. Grier, Rev. Sci. Instrum. 69 (1998) 1974. J.E. Curtis, B.A. Koss, D.G. Grier, Opt. Commun. 207 (2002) 169. V. Garces-Chaves, D. McGloin, H. Melville, W. Sibbett, K. Dholakia, Nature 419 (2002) 145. L. Paterson, M.P. MacDonald, J. Arlt, W. Sibbett, P.E. Bryant, K. Dholakia, Science 292 (2001) 912. K. Visscher, G.J. Brakenhoff, Optik 89 (1992) 174.
Focal shift of apodized truncated hyperbolic-cosine-Gaussian beam Xiumin Gao a b
a,*
, Jinsong Li
b,c
Electronics and Information College, Hangzhou Dianzi University, Xiasha Campus, Hangzhou 310018, PR China College of Optics and Electronics, University of Shanghai for Science and Technology, Shanghai 20093, PR China c Institute of Optoelectronics Technology, China Jiliang University, Hangzhou 310018, PR China Received 6 July 2006; received in revised form 7 December 2006; accepted 12 December 2006
Abstract Focal shift of hyperbolic-cosine-Gaussian beams induced by pure phase apodizer was investigated theoretically. The pure phase apodizer consists of three concentric zones: center circle zone, inner annular zone and outer annular zone. And the phase variance of the inner annular zone is adjustable. Results show that intensity peak moves far from optical aperture and then shrinks sharply for certain radii of zones with increasing phase variance of the inner annular zone. Simultaneously, one new intensity peak occurs near optical aperture, moves far from the optical aperture, and then becomes intensity maximum peak, and repeats the evolution process of the former intensity peak. Tunable focal shift occurs with focal switch. Decreasing the phase variance can change the move direction of the intensity peaks. In addition, the maximum distance between the two intensity peaks can be altered by beam parameters of cosh parts, and the distance value increases and then decreases with increasing beam parameters of cosh parts for certain radii of zones. Tunable focal shift is also discussed to construct optical tweezers. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Focal shift; Hyperbolic-cosine-Gaussian beam; Pure phase apodizer; Optical tweezers
1. Introduction In many optical systems Gaussian beams are very useful and common beams. In fact, Gaussian beams are only one solution of the Helmholtz equation. The more general solutions of the Helmholtz equation are Hermite-sinusoidalGaussian beams, which were introduced by Casperson and coworkers [1,2] and have attracted much attention recently. These beams can be obtained by using phase mirrors [3]. The hyperbolic-cosine-Gaussian (ChG) beams, which are regarded as the special cases of sinusoidal-Gaussian beams or Hermite-sinusoidal-Gaussian beams, are of practical interest because of their efficience in extracting energy from conventional laser amplifiers, and the flat-top field distribution can be obtained by choosing suitable beam parameters of cosh parts [4]. So propagation and focusing properties of ChG beams have become object of some works [5–8]. *
Corresponding author. Tel.: +86 57186791561; fax: +86 57186791505. E-mail address: [email protected] (X. Gao).
0030-4018/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2006.12.021
In the investigation of the focusing properties of optical beams, tracing the movement of the point of absolute maximum intensity along optical axis has attracted many researchers for several decades [9–18]. It was found that the point of absolute maximum intensity does not coincide with the geometrical focus but shifts along optical axis [14–16], this phenomenon is referred to as focal shift. More interesting, the focal shift may be incontinuous in certain optical focusing systems. Focal shift may be accompanied by an effective permutation of the focal point, namely, intensity maximum can jump from one position to other position, and this effect is called focal switch [17,18]. In this article, focal shift of focused hyperbolic-cosineGaussian beams induced by a pure concentric phase apodizer was investigated through scalar diffractive theory. Dependence of tunable focal shift on phase variance of the apodizer is studied, and the maximum distance between two intensity peaks, which illustrates focal split, is also discussed. Section 2 shows the principle of focusing optical system, in which ChG beam is apodized. And Section 3
22
X. Gao, J. Li / Optics Communications 273 (2007) 21–27
presents results and discussions. Conclusions are summarized in Section 4. 2. Principle of focusing system In the focusing system we investigated, ChG beam passes through a phase apodizer, and then converges through an objective lens. It should be noted that the apodizer may be not only phase plate that works in transmittance mode, but also a reflector with tunable phase distribution in reflective mode, as shown in Fig. 1. The phase apodizer consists of three zones, center circle zone, inner annular zone and outer annular zone. The motive for choosing this kind of apodizer is that this kind of apodizer may be easy to produce and duplicate for mass production, and the analysis method of the three-portion apodizer can be also conveniently extended to analyze the generalized apodizer. The phase variance of inner annular zone is adjustable. Parameters a, b, and 1 are the normalized convergence angles for each zone, respectively, as Fig. 2 shows. Parameters a and b are the ratios of convergence angle of center circle zone and inner annular zone to the aperture angle of the focusing optical system, respectively, b ¼ a1 =a and a ¼ a2 =a, where a1 and a2 are the genuine values of convergence angle corresponding to center circle zone and inner annular zone, and a is the aperture angle of the optical system. Electric field of the ChG beam at z = 0 is, 2 x þ y2 Eðx; y; z ¼ 0Þ ¼ A0 cosh ðXx xÞ cosh ðXy yÞ exp x20 ð1Þ where x0 is waist width, cosh is hyperbolic-cosine function. Xx and Xy are beam parameters of cosh parts. Now the
Fig. 1. Schematic of apodized focusing system. (a) Transmitted mode apodizer, (b) reflective apodizer.
Fig. 2. Parameters illustration of focused modulated beams. Parameters a, b, and 1 are the normalized convergence angles for each zone, respectively. / is the phase variance of the inner annular zone.
radial coordinate in z ¼ 0 plane is introduced, x ¼ r0 cos ðuÞ, y ¼ r0 sin ðuÞ, so Eq. (1) can be rewritten as Eðx; y; z ¼ 0Þ ¼ Eðr0 ; uÞ ¼ A0 cosh ½Xx r0 cos ðuÞ cosh ½Xy r0 sin ðuÞ r20 exp 2 x0
ð2Þ
where r0 is the radial coordinate. By simple deviation, ChG beam can be expressed as Eðh; uÞ ¼ A0 cosh ½NA1 bx sin ðhÞ cos ðuÞ cosh ½NA1 by sin ðhÞ sin ðuÞ sin2 ðhÞ exp NA2 w2
ð3Þ
where NA is the numerical aperture of the focusing system, and rp is the outer radius of the truncated ChG beam. Here, w ¼ x0 =rp . Parameters bx ¼ rp Xx , and by ¼ rp Xy indicate beam parameters of cosh parts and are called decentered parameters below. The phase variances of center zone and outer annular zone are zero, and the phase variance of inner annular zone is tunable and symbolized as /. According to scalar theory, electric amplitude distribution in focal region of the focusing optical system can be written as [19], Z Z i b 2p U ðr; w; zÞ ¼ Eðh; uÞ exp ½ikr sinðhÞ cosðu wÞ k 0 0 Z Z i 1 2p Eðh; uÞ ikz cosðhÞ sinðhÞdhdu þ k a 0 exp ½ikr sinðhÞ cosðu wÞ ikz cosðhÞ Z Z i a 2p sinðhÞdhdu þ Eðh; uÞ expði/Þ k b 0 exp ½ikr sinðhÞ cosðu wÞ ikz cosðhÞ sinðhÞdhdu ð4Þ where, u 2 ½0; 2p. It should be noted that focal shift refers to the distance between the genuine focal plane and geometrical focal plane. And here genuine focal plane is the plane that passes through the intensity maximum peak and is perpendicular to optical axis. The light intensity in the focal region is proportional to the modulus square of Eq. (4).
X. Gao, J. Li / Optics Communications 273 (2007) 21–27
3. Results and discussions The optical distribution in focal region is calculated according to the Eq. (4). Without losing generality and validity, the intensity is normalized by optical intensity maximum and it is supposed that NA = 0.6, w = 3 in computation. Fig. 3 illustrates the evolution of axial intensity distribution for bx ¼ by ¼ 10; b ¼ 0:1; a ¼ 0:6 with changing /. Arrows show the move direction of the intensity peaks. When / increases from 0 to p, it can be seen that intensity peak moves away from geometric focal plane in the direction far from optical aperture, which shows that focal shift occurs. In this evolution process, one new intensity peak comes into being in the near side of the maximum intensity peak, becomes stronger and stronger with increasing /, and also shifts in the same move direction of the intensity maximum peak. However, the former peak moves more quickly than the new smaller intensity peak, so the distance between them broadens with increasing /. When the two peaks appear simultaneously, there is one local axial
Fig. 3. Axial intensity distribution for bx ¼ by ¼ 10, b = 0.1, a = 0.6, / ranges from (a) 0 to p, (b) p to 0. Arrows shows the move direction of the intensity peaks.
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intensity minimum. From the figure, we can see that the local intensity minimum also shifts and its intensity value decreases smoothly with increasing /. When / increases up to p, the intensity of the two intensity peaks equals, the distance between the two peaks comes to maximum value, and there are two maximum intensity peaks. Increase / continuously, the evolution of axial intensity distribution is shown in Fig. 3b. The intensity peak far from optical aperture shrinks. Therefore, the intensity peak near optical aperture becomes only one maximum peak, focal switch occurs. With increasing /, the new intensity maximum peak goes on shifting. At the same time, the intensity peak far from optical aperture also shifts continuously, however its move velocity is smaller than that of intensity maximum peak, so the distance between the two intensity peaks decreases. And the local axial intensity minimum shifts and then disappears when / approaches 2p. It is found that the evolution process in Fig. 3a and b is symmetric. And / ¼ p is critical value where there are two intensity maximum peaks with maximum distance between them. Now we trace the focal shift. Dependence of focal shift on / is illustrated in Fig. 4. It is shown that the value of focal shift is positive and increases with increasing / considerably. There is one critical position with / ¼ p, at which there are two focal shift values, and focal switch also happens. With increasing /, the negative value of focal shift increases when / changes from p to 1.5p. And the focal shift changing process repeats with increasing /. When / decreases, the focal shift and focal switch can also occur, the difference is the move direction of intensity peaks. By altering /, focal shift can occurs with focal switch, and the move direction can also be changed. Now promising application of the above changing process is discussed. Optical tweezers offers a very convenient, noninvasive access to process at the microscopic scale, and has become valuable tool and accelerated many major advances in numerous areas of science since Ashkin and co-workers accomplished optical tweezers experimentally
Fig. 4. Dependence of focal shift on / for bx ¼ by ¼ 10, b = 0.1, a = 0.6.
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[20–25]. Many approaches have been proposed to construct optical trap, including generalized phase-contrast technique [26], diffractive optical element [27], holograms [28], self-reconstruct light beam [29], and interferometer pattern [30]. In optical trapping system, it is usually deemed that the forces exerted on the particles in light field include two kinds of forces, one is the optical gradient force, which is proportional to the optical intensity gradient; the other is the scattering force, which is proportional to the optical intensity [20,31]. Therefore, the tunable focal shift predicts that the position of optical trap may be controllable. If one intensity peak acts as one optical trap, the focal shift means that the optical trapped particle in the intensity peak can be moved. So in the proposed apodized system we investigated, the particles may be trapped in maximum intensity peak and moved far from optical aperture by increasing phase shift / from 0 to p. At the same time, one new maximum intensity peak occurs. With phase shift / increasing from p to 2p, the intensity peak far from aperture shrinks, which means that the trapped particle is deposited there. The peak near optical aperture becomes only one maximum intensity peak and can trap new one particle, then carries the trapped particle far away from the aperture with increasing /. Therefore, the particles may be transported from one place to another far from aperture by increasing / continuously, as shown in Fig. 5. Decrease of the phase shift may induce the optical trap move towards optical aperture. Therefore, the focused apodized ChG beam can be used to construct tunable optical tweezers for dynamical manipulation of particles. Here, typical three-dimensional optical intensity distributions in w ¼ p=4 plane are illustrated in Fig. 6 due to the geometry properties of the ChG beam and focal intensity distribution [5]. From the transverse and axial optical distributions, we can see that focal shift and focal switch can occur and the evolution principle is similar to that in Fig. 3. Arrows denote the motion direction of the optical peaks. It should be noted that the transverse optical distributions are very steep which shows that the optical gradient force is big, therefore the optical trap formed by these optical peaks are stable. In order to understand the focal shift and focal switch further, axial intensity distributions for b = 0.1, a = 0.6, and different decentered parameters are also computed. Fig. 7 shows that axial intensities for bx ¼ by ¼ 5,
Fig. 5. Schematic of optical tweezers application of focused beam. In collecting zone, new intensity peak occurs, and traps particles in the region, then move them far from optical aperture to release zone. And then release those trapped particles. Arrows show the move direction of trapped particles.
bx ¼ by ¼ 3, respectively. It can be seen that focal shift principle is similar to that in Fig. 3, however, the maximum distance between that the two intensity peaks decreases with decreasing decentered parameters. And for bx ¼ by ¼ 5, the value of the axial local intensity minimum
Fig. 6. Intensity distributions for bx ¼ by ¼ 10, b = 0.1, a = 0.6, and (a) / ¼ 0, (b) / ¼ 0:6p, (c) / ¼ p, and (d) / ¼ 1:6p.
X. Gao, J. Li / Optics Communications 273 (2007) 21–27
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Fig. 7. Axial intensity distribution for b = 0.1, a = 0.6, / ranges from 0 to p, and (a) bx ¼ by ¼ 5, (b) bx ¼ by ¼ 3. Arrows shows the move direction of the intensity peaks. Unit of number in this figure is p.
increases when / increases from 0 to p, which is inverse to that in Fig. 3. For bx ¼ by ¼ 3, the axial local intensity minimum occurs when / approaches p. Therefore, the maximum distance between the two intensity peaks can be altered by decentered parameters, and there is critical value of / at which the axial local intensity minimum comes into being. In order to confirm the principle of the axial intensity evolution, we also computed axial intensity distributions for bigger decentered parameters under the condition that / increases from 0 to p. Fig. 8 shows that axial intensity distributions for b = 0.1, a = 0.6, bx ¼ by ¼ 15, 20, and 30, respectively. Arrows show the move direction of the intensity peaks. With increasing /, intensity peak moves in the direction far from optical aperture. In the evolution process, one new intensity peak comes into being in the near side of the maximum intensity peak, becomes stronger and stronger with increasing /, and also shifts. The distance between the two peaks increases with increasing /. When the two peaks appear simultaneously, there is one local axial intensity minimum.
Fig. 8. Axial intensity distribution for b = 0.1, a = 0.6, / ranges from 0 to p, and (a) bx ¼ by ¼ 15, (b) bx ¼ by ¼ 20, (c) bx ¼ by ¼ 30. Arrows shows the move direction of the intensity peaks. Unit of number in this figure is p.
We can see that the local intensity minimum also shifts and its intensity value increases with increasing /, and change rate increases with decentered parameters of the beam, as shown in Fig. 8a and b. In addition, for certain decentered
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Fig. 10. Dependence of maximum focal shift on decentered parameters of the beam for b = 0.1, a = 0.6.
Fig. 9. Axial intensity distribution for b = 0.1, a = 0.6, / ¼ p, and bx ¼ by ¼ b ranges from (a) 0 to 10, (b) 30 to 45.
increases to its maximum. Then the distance decreases with increasing b, and when value of b equals 38, the distance becomes zero, which shows that the focal shift phenomenon disappears. Fig. 10 shows the dependence of maximum focal shift on decentered parameters of the beam for b = 0.1, a = 0.6. From above process, it can be seen that for certain geometrical parameters of the apodizer, the maximum distance between the two intensity peaks can be altered by decentered parameters, which illustrates that if the apodized beam is used to construct tunable optical tweezers, the distance between collecting zone and release zone, shown in Fig. 4, can be adjusted. 4. Conclusions
parameters, there is a critical /, at which axial local intensity minimum begin to occur, and the critical / increases with increasing decentered parameters. In the whole evolution process, it can be seen that the critical /, at which axial local intensity minimum begin occurs, decreases, and then increases with increasing decentered parameters of the beam. From above simulation, it can be seen that the maximum distance between the two intensity peaks can be altered by decentered parameters, and occurs at / ¼ p. Now we focus on adjustable degree of the maximum distance. Fig. 9 illustrates axial intensity distributions for b = 0.1, a = 0.6, / ¼ p, and different decentered parameters b ¼ bx ¼ by of the ChG beams. For small b, there is only one intensity peak in changing process, focal shift does not occur, there are not two intensity peaks at all, so the maximum distance is zero. Increase b continuously and b value is bigger than about 4, the focal shift and focal split occur, the maximum distance increases sharply, then the increase speed of the distance decreases with increasing b. When b equals about 18, the distance
Focal shift of hyperbolic-cosine-Gaussian beams induced by a phase apodizer was investigated. The pure phase apodizer consists of three concentric zones: center circle zone, inner annular zone and outer annular zone. Phase variance of the inner annular zone can be altered. Simulation results show that for certain radii of zones, intensity peak moves far from optical aperture and then shrinks sharply with increasing phase variance of the inner zone. Simultaneously, one new intensity peak occurs near optical aperture and shifts in the same direction as that of former intensity peak, its intensity increases sharply and then becomes intensity maximum peak, which shows that tunable focal shift companied with focal switch. Decrease the phase variance may change the move direction of the two intensity peaks. In addition, the maximum distance between the two intensity peaks can be altered by decentered parameter of the beam, the value of the maximum distance increases and then decreases with increasing decentered parameter. The tunable focal shift and focal switch may be used to construct controllable optical tweezers.
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Acknowledgements This work was supported by Science Research Project of Hangzhou Dianzi University. And the equation derivation and codes used in this article can be obtained conveniently through [email protected].
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