Three-dimensional analysis of optical forces generated by an active tractor beam using radial polarization Luis Carretero,1,∗ Pablo Acebal,1 and Salvador Blaya1 ´ Miguel Hern´andez, Dpto. de Ciencia de Materiales Optica y Tecnolog´ıa electr´onica, Avda. de la Universidad 3202 Elche Alicante, Spain
1 Universidad
∗ [email protected]
Abstract: We theoretically study the three-dimensional behavior of nanoparticles in an active optical conveyor. To do this, we solved the Langevin equation when the forces are generated by a focusing system at the near field. Analytical expressions for the optical forces generated by the optical conveyor were obtained by solving the Richards and Wolf vectorial diffraction integrals in an approximated form when a mask of two annular pupils is illuminated by a radially polarized Hermite-Gauss beam. Trajectories, in both the transverse plane and the longitudinal direction, are analyzed showing that the behavior of the optical conveyor can be optimized by conveniently choosing the configuration of the mask of the two annular pupils (inner and outer radius of the two rings) in order to trap and transport all particles at the focal plane. © 2014 Optical Society of America OCIS codes: (260.2110) Electromagnetic optics; (260.5430) Polarization; (140.7010) Laser trapping; (350.4855) Optical tweezers or optical manipulation.
References and links 1. A. Ashkin, “Acceleration and trapping of particles by radiation presure,” Phys. Rev. Lett. 24, 156–159 (1970). 2. T. Cizmar, V. Garces-Chavez, K. Dholakia, and P. Zemanek, “Optical conveyor belt for delivery submicron objects,” Appl. Phys. Lett. 86, 174101 (2005). 3. T. Cizmar, M. Siler, and P. Zemanek, “An optical nanotrap array movable over a milimetre range,” Appl. Phys. B 84, 197–203 (2006). 4. G. Milne, H. Dholakia, D. McGloin, K. Volke-Sepulveda, and P. Zemanek, “Transverse particle dynamics in a Bessel beam,” Opt. Express 15, 13972–13986 (2007). 5. M. Siler, P. Jakl, O. Brzobohaty, and P. Zemanek, “Optical forces induced behavior of a particle in a nondiffracting vortex beam,” Opt. Express 20, 24304–24318 (2012). 6. N. Wang, J. Chen, S. Liu, and Z. Lin, “Dynamical and phase study on stable optical pulling force in Bessel beams,” Phys. Rev. A 87, 063812 (2013). 7. D. B. Ruffner and D. G. Grier, “Optical conveyors: a class of active tractor beams,” Phys. Rev. Lett. 109, 163903 (2012). 8. T. A. Nieminen, N. R. Heckenberg, and H. Rubinstein-Dunlop, “Forces in optical tweezers with radially and azimuthally polarized trapping beams,” Opt. Lett. 33, 122–124 (2008). 9. S. E. Skelton, M. Sergides, R. Saija, M. Iati, O. Marago, and P. H. Jones, “Trapping volume control in optical tweezers using cylindrical vector beams,” Opt. Lett. 38, 28–30 (2013). 10. C. J. R. Sheppard and A. Choudhury, “Annular pupils, radial polarization and superresolution,” Appl. Opt. 43, 4322–4327 (2004). 11. Z. Chen and D. Zhao, “4 π focusing of spatially modulated radially polarized vortex beams,” Opt. Lett. 37, 1286–1288 (2012). 12. B. Richards and E. Wolf, “Electomagnetic diffraction of optical system II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A 253, 358–379 (1959).
#202540 - $15.00 USD Received 5 Dec 2013; revised 4 Jan 2014; accepted 14 Jan 2014; published 4 Feb 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.003284 | OPTICS EXPRESS 3284
13. K. S. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7, 77–87 (2000). 14. P. C. Chaumet and M. Nieto-Vesperinas, “Time-averaged total force on a dipolar sphere in an electromagentic fiel,” Opt. Lett. 25, 1065–1067 (2000). 15. M. Nieto-Vesperinas, J. S´aenz, R. G´omez-Medina, and L. Chantada, “Optical forces on small magnetodielectric particles,” Opt. Express 18, 11428–11443 (2010). 16. V. Kajorndenukul, W. Ding, S. Sukhov, C.-W. Qiu, and A. Dogariu, “Linear momentum increase and negative optical forces at dielectric interface,” Nat. Photonics 7, 787–790 (2013). 17. F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, 1965). 18. M. Borromeo and F. Marchesoni, “Brownian surfers,” Phys. Lett. A 249, 199–203 (1998). 19. P. Reimann, “Brownian motors: noisy transport far from equilibrium,” Phys. Rep. 361, 57–265 (2002). 20. M. Siler, T. Cizmar, A. Jonas, and P. Zemanek, “Surface delivery of a single nanoparticle under moving evanescent standing-wave illumination,” New J. Phys. 10, 113010 (2008).
1.
Introduction
In the 1970’s, Ashkin [1] demonstrated the optical trapping of particles using the radiation force generated by a focusing Gaussian beam. Since its first demonstration, optical manipulation by using focusing beams has been converted into a powerful tool for microscopic manipulation in different research fields like physics, biology, colloid science, or microfluidics. Among the different optical systems developed, the non diffracting Bessel beams have been one of the most used in optical nanotrap technology. Theoretical and experimental studies of non-paraxial Bessel beams and the resulting optical forces acting on a nanoparticle have been reported, such as the cases of a single Bessel beam or a standing Bessel beam obtained by illumination of an axicon with a linearly polarized beam [2, 3]. Using the same type of electric and magnetic field of Bessel beams given in [3], different computational models have been employed for analyzing the dynamic of particles produced by the optical forces generated by Bessel beams comparing the scattering Mie theory and geometrical ray optics [4]. Recently, Bessel beams, when the axicon is illuminated by a linearly polarized plane wave with different topological charge, have been used to study the behavior of microparticles near the center of an optical vortex beam [5]. Moreover, the influence of the orbital angular-momentum, using linear stability analysis on a spherical particle, has also been studied, [6] showing that a particle cannot be stably confined at the region of negative longitudinal optical force originated by Bessel beams with topological charge in the absence of ambient damping. Recently, Ruffner and Grier [7] experimentally demonstrated and analyzed the properties of a class of tractor beam obtained by the interference of two coaxial Bessel Beams that differ in their axial wave numbers. For this, it was employed linearly polarized light illuminating a computer designed phase profile which was focused using a high numerical aperture objective. In this paper, we theoretically examine in detail the particle dynamics for different configurations of the tractor beam type described in [7] when radially polarized Bessel beams are used in order to improve the trapping of spherical particles [8, 9]. These radially polarized Bessel beams will be obtained by focusing a radially polarized beam using a high-aperture system that illuminates a two-ringed phase-only transmission function. The choice of this polarization state, together with two annular pupils, gives a sharp focal spot [10, 11]. Analytical expressions for the electric field and optical forces generated by the optical conveyor will be obtained by solving the Richards and Wolf vectorial diffraction integrals in an approximated form. Our theoretical study will be carried out in the near field and for high aperture system using a vectorial diffraction analysis which differs to the scalar diffraction approximation used by Ruffner and Grier [7].
#202540 - $15.00 USD Received 5 Dec 2013; revised 4 Jan 2014; accepted 14 Jan 2014; published 4 Feb 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.003284 | OPTICS EXPRESS 3285
2.
Theoretical background
The electric field components (using cylindrical coordinates) in the vicinity of the focus of a radially polarized beam can be obtained by using vectorial diffraction theory as follows [12, 13]: er = A
α 0
ez = i A
hr (θ )l0 (θ )T (θ )J1 (krSin(θ ))Exp(i k zCos(θ ))d θ
α 0
eψ = 0 hz (θ )l0 (θ )T (θ )J0 (krSin(θ ))Exp(ikzCos(θ ))d θ (1)
where k is the wavenumber, A is an amplitude constant, and l0 (θ ) is the apodization function that we have assumed that is an order one Hermite-Gauss mode: β 2 Sin(θ )2 β Sin(θ ) (2) l0 (θ ) = Exp − Sin(α )2 Sin(α ) where hr (θ ) = Cos(θ )Sin(2θ ), hz (θ ) = 2 Cos(θ )Sin2 (θ ), α is the angular semi-aperture of the focusing system given by α = sin−1 (NA/n). NA is the numerical aperture and β is the ratio of the pupil radius and the beam waist, n is the refractive index between the high numerical optical system and the sample. Following the definitions given in reference [13], the main parameters used in Eqs. (1)–(3) are shown in Fig. 1. The apodization function is modified by a mask complex function T (θ ) given by: ⎧ g1 θ1 − δ21 ≤ θ ≤ θ1 + δ21 ⎪ ⎪ ⎪ ⎪ ⎨ T (θ ) = g2 θ2 − δ22 ≤ θ ≤ θ2 + δ22 ⎪ ⎪ ⎪ ⎪ ⎩ 0 otherwise
(3)
where we have assumed that g1 = 1 and g2 = Exp(iξ t), so emergent fields from the rings described in transmission Eq. (3) differ in their relative phase. This linear relative phase ξ t difference makes the conveyor work [7]. The inset in Fig. 1, shows the transmittance of the mask illuminated by a radially polarized beam considered (continuous line) compared to the same mask illuminated by a linearly Gaussian beam as apodization function (dashed lines). Values of the employed parameters are given in Table 1, the blue lines correspond to θ2 = 0.6, the red lines to θ2 = 0.7 and finally green lines correspond to θ2 = 0.8. As can be observed, the first ring is the same in all cases. If δ21 δ1Cos(θ1 ). Then, according to Eq. (19), the maximum theoretical axial range of the optical conveyor will be given by: λ (22) Δz = δ2e Sin(θ2 ) Since, in order to optimize the tractor beam, we have two degrees of freedom, δ1 and θ2 . Table 1. Values of the parameters used in the numerical simulations and theoretical axial range
Tractor T1 T2 T3
3.1.
θ2 0.6 0.7 0.8
δ2e 0.115 0.086 0.071
δ1 0.06 0.06 0.06
θ1 0.942 0.942 0.942
Δz 15.34 λ 17.86 λ 19.59 λ
Numerical results
Taking into account previous points, we analyze three configurations of the same tractor beam (see Table 1), taking in all cases δ1 = 0.06. For √ this, the NA system was 1.1, and the refractive index between the lens and the sample is n = ε = 1.33 (water). Using these numerical values, according to Eqs. (20) and (21), we have that θ1 = 0.942. Then, δ2e values for each conveyor configuration can be obtained by introducing in Eq. (17) Eq. (2) together with the numerical values of δ1 and θ1 . In Table 1, the obtained results for θ2
#202540 - $15.00 USD Received 5 Dec 2013; revised 4 Jan 2014; accepted 14 Jan 2014; published 4 Feb 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.003284 | OPTICS EXPRESS 3290
T1
T3
T2
Fig. 2. Normalized Intensity I = ir + iz at time t = 0 as a function of coordinates r and z for the three conveyors configurations (T1,T2 and T3) described in Table 1
0.4
100IA^2
0.3
0.2
0.1
0.0 20
10
0
10
20
zΛ
Fig. 3. Normalized Axial Irradiance I(r = 0) = iz at time t = 0s versus z-coordinate for the three conveyors configurations (T1 (blue),T2 (red) and T3 (green)) described in Table 1.
values between 0.6 to 0.8 are shown. This analysis is limited to this θ2 -range because at lower values than 0.6, δ2e increases and condition δ2
1 Universidad
∗ [email protected]
Abstract: We theoretically study the three-dimensional behavior of nanoparticles in an active optical conveyor. To do this, we solved the Langevin equation when the forces are generated by a focusing system at the near field. Analytical expressions for the optical forces generated by the optical conveyor were obtained by solving the Richards and Wolf vectorial diffraction integrals in an approximated form when a mask of two annular pupils is illuminated by a radially polarized Hermite-Gauss beam. Trajectories, in both the transverse plane and the longitudinal direction, are analyzed showing that the behavior of the optical conveyor can be optimized by conveniently choosing the configuration of the mask of the two annular pupils (inner and outer radius of the two rings) in order to trap and transport all particles at the focal plane. © 2014 Optical Society of America OCIS codes: (260.2110) Electromagnetic optics; (260.5430) Polarization; (140.7010) Laser trapping; (350.4855) Optical tweezers or optical manipulation.
References and links 1. A. Ashkin, “Acceleration and trapping of particles by radiation presure,” Phys. Rev. Lett. 24, 156–159 (1970). 2. T. Cizmar, V. Garces-Chavez, K. Dholakia, and P. Zemanek, “Optical conveyor belt for delivery submicron objects,” Appl. Phys. Lett. 86, 174101 (2005). 3. T. Cizmar, M. Siler, and P. Zemanek, “An optical nanotrap array movable over a milimetre range,” Appl. Phys. B 84, 197–203 (2006). 4. G. Milne, H. Dholakia, D. McGloin, K. Volke-Sepulveda, and P. Zemanek, “Transverse particle dynamics in a Bessel beam,” Opt. Express 15, 13972–13986 (2007). 5. M. Siler, P. Jakl, O. Brzobohaty, and P. Zemanek, “Optical forces induced behavior of a particle in a nondiffracting vortex beam,” Opt. Express 20, 24304–24318 (2012). 6. N. Wang, J. Chen, S. Liu, and Z. Lin, “Dynamical and phase study on stable optical pulling force in Bessel beams,” Phys. Rev. A 87, 063812 (2013). 7. D. B. Ruffner and D. G. Grier, “Optical conveyors: a class of active tractor beams,” Phys. Rev. Lett. 109, 163903 (2012). 8. T. A. Nieminen, N. R. Heckenberg, and H. Rubinstein-Dunlop, “Forces in optical tweezers with radially and azimuthally polarized trapping beams,” Opt. Lett. 33, 122–124 (2008). 9. S. E. Skelton, M. Sergides, R. Saija, M. Iati, O. Marago, and P. H. Jones, “Trapping volume control in optical tweezers using cylindrical vector beams,” Opt. Lett. 38, 28–30 (2013). 10. C. J. R. Sheppard and A. Choudhury, “Annular pupils, radial polarization and superresolution,” Appl. Opt. 43, 4322–4327 (2004). 11. Z. Chen and D. Zhao, “4 π focusing of spatially modulated radially polarized vortex beams,” Opt. Lett. 37, 1286–1288 (2012). 12. B. Richards and E. Wolf, “Electomagnetic diffraction of optical system II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A 253, 358–379 (1959).
#202540 - $15.00 USD Received 5 Dec 2013; revised 4 Jan 2014; accepted 14 Jan 2014; published 4 Feb 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.003284 | OPTICS EXPRESS 3284
13. K. S. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7, 77–87 (2000). 14. P. C. Chaumet and M. Nieto-Vesperinas, “Time-averaged total force on a dipolar sphere in an electromagentic fiel,” Opt. Lett. 25, 1065–1067 (2000). 15. M. Nieto-Vesperinas, J. S´aenz, R. G´omez-Medina, and L. Chantada, “Optical forces on small magnetodielectric particles,” Opt. Express 18, 11428–11443 (2010). 16. V. Kajorndenukul, W. Ding, S. Sukhov, C.-W. Qiu, and A. Dogariu, “Linear momentum increase and negative optical forces at dielectric interface,” Nat. Photonics 7, 787–790 (2013). 17. F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, 1965). 18. M. Borromeo and F. Marchesoni, “Brownian surfers,” Phys. Lett. A 249, 199–203 (1998). 19. P. Reimann, “Brownian motors: noisy transport far from equilibrium,” Phys. Rep. 361, 57–265 (2002). 20. M. Siler, T. Cizmar, A. Jonas, and P. Zemanek, “Surface delivery of a single nanoparticle under moving evanescent standing-wave illumination,” New J. Phys. 10, 113010 (2008).
1.
Introduction
In the 1970’s, Ashkin [1] demonstrated the optical trapping of particles using the radiation force generated by a focusing Gaussian beam. Since its first demonstration, optical manipulation by using focusing beams has been converted into a powerful tool for microscopic manipulation in different research fields like physics, biology, colloid science, or microfluidics. Among the different optical systems developed, the non diffracting Bessel beams have been one of the most used in optical nanotrap technology. Theoretical and experimental studies of non-paraxial Bessel beams and the resulting optical forces acting on a nanoparticle have been reported, such as the cases of a single Bessel beam or a standing Bessel beam obtained by illumination of an axicon with a linearly polarized beam [2, 3]. Using the same type of electric and magnetic field of Bessel beams given in [3], different computational models have been employed for analyzing the dynamic of particles produced by the optical forces generated by Bessel beams comparing the scattering Mie theory and geometrical ray optics [4]. Recently, Bessel beams, when the axicon is illuminated by a linearly polarized plane wave with different topological charge, have been used to study the behavior of microparticles near the center of an optical vortex beam [5]. Moreover, the influence of the orbital angular-momentum, using linear stability analysis on a spherical particle, has also been studied, [6] showing that a particle cannot be stably confined at the region of negative longitudinal optical force originated by Bessel beams with topological charge in the absence of ambient damping. Recently, Ruffner and Grier [7] experimentally demonstrated and analyzed the properties of a class of tractor beam obtained by the interference of two coaxial Bessel Beams that differ in their axial wave numbers. For this, it was employed linearly polarized light illuminating a computer designed phase profile which was focused using a high numerical aperture objective. In this paper, we theoretically examine in detail the particle dynamics for different configurations of the tractor beam type described in [7] when radially polarized Bessel beams are used in order to improve the trapping of spherical particles [8, 9]. These radially polarized Bessel beams will be obtained by focusing a radially polarized beam using a high-aperture system that illuminates a two-ringed phase-only transmission function. The choice of this polarization state, together with two annular pupils, gives a sharp focal spot [10, 11]. Analytical expressions for the electric field and optical forces generated by the optical conveyor will be obtained by solving the Richards and Wolf vectorial diffraction integrals in an approximated form. Our theoretical study will be carried out in the near field and for high aperture system using a vectorial diffraction analysis which differs to the scalar diffraction approximation used by Ruffner and Grier [7].
#202540 - $15.00 USD Received 5 Dec 2013; revised 4 Jan 2014; accepted 14 Jan 2014; published 4 Feb 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.003284 | OPTICS EXPRESS 3285
2.
Theoretical background
The electric field components (using cylindrical coordinates) in the vicinity of the focus of a radially polarized beam can be obtained by using vectorial diffraction theory as follows [12, 13]: er = A
α 0
ez = i A
hr (θ )l0 (θ )T (θ )J1 (krSin(θ ))Exp(i k zCos(θ ))d θ
α 0
eψ = 0 hz (θ )l0 (θ )T (θ )J0 (krSin(θ ))Exp(ikzCos(θ ))d θ (1)
where k is the wavenumber, A is an amplitude constant, and l0 (θ ) is the apodization function that we have assumed that is an order one Hermite-Gauss mode: β 2 Sin(θ )2 β Sin(θ ) (2) l0 (θ ) = Exp − Sin(α )2 Sin(α ) where hr (θ ) = Cos(θ )Sin(2θ ), hz (θ ) = 2 Cos(θ )Sin2 (θ ), α is the angular semi-aperture of the focusing system given by α = sin−1 (NA/n). NA is the numerical aperture and β is the ratio of the pupil radius and the beam waist, n is the refractive index between the high numerical optical system and the sample. Following the definitions given in reference [13], the main parameters used in Eqs. (1)–(3) are shown in Fig. 1. The apodization function is modified by a mask complex function T (θ ) given by: ⎧ g1 θ1 − δ21 ≤ θ ≤ θ1 + δ21 ⎪ ⎪ ⎪ ⎪ ⎨ T (θ ) = g2 θ2 − δ22 ≤ θ ≤ θ2 + δ22 ⎪ ⎪ ⎪ ⎪ ⎩ 0 otherwise
(3)
where we have assumed that g1 = 1 and g2 = Exp(iξ t), so emergent fields from the rings described in transmission Eq. (3) differ in their relative phase. This linear relative phase ξ t difference makes the conveyor work [7]. The inset in Fig. 1, shows the transmittance of the mask illuminated by a radially polarized beam considered (continuous line) compared to the same mask illuminated by a linearly Gaussian beam as apodization function (dashed lines). Values of the employed parameters are given in Table 1, the blue lines correspond to θ2 = 0.6, the red lines to θ2 = 0.7 and finally green lines correspond to θ2 = 0.8. As can be observed, the first ring is the same in all cases. If δ21 δ1Cos(θ1 ). Then, according to Eq. (19), the maximum theoretical axial range of the optical conveyor will be given by: λ (22) Δz = δ2e Sin(θ2 ) Since, in order to optimize the tractor beam, we have two degrees of freedom, δ1 and θ2 . Table 1. Values of the parameters used in the numerical simulations and theoretical axial range
Tractor T1 T2 T3
3.1.
θ2 0.6 0.7 0.8
δ2e 0.115 0.086 0.071
δ1 0.06 0.06 0.06
θ1 0.942 0.942 0.942
Δz 15.34 λ 17.86 λ 19.59 λ
Numerical results
Taking into account previous points, we analyze three configurations of the same tractor beam (see Table 1), taking in all cases δ1 = 0.06. For √ this, the NA system was 1.1, and the refractive index between the lens and the sample is n = ε = 1.33 (water). Using these numerical values, according to Eqs. (20) and (21), we have that θ1 = 0.942. Then, δ2e values for each conveyor configuration can be obtained by introducing in Eq. (17) Eq. (2) together with the numerical values of δ1 and θ1 . In Table 1, the obtained results for θ2
#202540 - $15.00 USD Received 5 Dec 2013; revised 4 Jan 2014; accepted 14 Jan 2014; published 4 Feb 2014 (C) 2014 OSA 10 February 2014 | Vol. 22, No. 3 | DOI:10.1364/OE.22.003284 | OPTICS EXPRESS 3290
T1
T3
T2
Fig. 2. Normalized Intensity I = ir + iz at time t = 0 as a function of coordinates r and z for the three conveyors configurations (T1,T2 and T3) described in Table 1
0.4
100IA^2
0.3
0.2
0.1
0.0 20
10
0
10
20
zΛ
Fig. 3. Normalized Axial Irradiance I(r = 0) = iz at time t = 0s versus z-coordinate for the three conveyors configurations (T1 (blue),T2 (red) and T3 (green)) described in Table 1.
values between 0.6 to 0.8 are shown. This analysis is limited to this θ2 -range because at lower values than 0.6, δ2e increases and condition δ2
