(44-6) 02 * << * >> * Russian * English * Content * All Issues

Optical force acting on a particle in the presence of a backward energy flow near the focus of a gradient lens
A.G. Nalimov 1,2

IPSI RAS – Branch of the FSRC "Crystallography and Photonics" RAS,
443001, Samara, Russia, Molodogvardeyskaya 151,
Samara National Research University, 443086, Samara, Russia, Moskovskoye Shosse 34

 PDF, 1269 kB

DOI: 10.18287/2412-6179-CO-744

Pages: 871-875.

Full text of article: Russian language.

We show that a 70-nm dielectric nanoparticle placed on the optical axis near the surface (at a distance less than 100 nm) of a high-NA gradient microlens made of silicon, which is illuminated by a laser beam of 1.55 μm wavelength, is attracted to the lens surface with a piconewton force. The profile of the lens refractive index is described by a hyperbolic secant function. If a cut-out is made in the lens output surface, then the nanoparticle will be pulled into this cut-out, producing a kind of 'optical magnet'. If a reverse energy flow is to be generated on the optical axis near the output surface of such a gradient lens, this will lead to an absorbing dielectric nanoparticle being pulled toward the surface with a greater force than a similar non-absorbing particle. In the absence of a reverse flow, both absorbing and non-absorbing particles will be attracted to the surface with an equal force. The electromagnetic fields involved are calculated using a finite difference time domain (FDTD) method and the acting forces are calculated using a Maxwell stress tensor.

force, backward force, moment of force, optical tweezers, Maxwell stress tensor, rotation, gradient lens.

Nalimov AG. Optical force acting on a particle in the presence of a backward energy flow near the focus of a gradient lens. Computer Optics 2020; 44(6): 871-875. DOI: 10.18287/2412-6179-CO-744.

The work was partly funded by the Russian Science Foundation under grant #18-19-00595 (Sections "Incident fields with left-handed circular polarization and a phase vortex m=2" and “Circularly polarized light without a phase vortex”), the Russian Foundation for Basic Research under grant #18-29-20003 (Section “Dependence of the force on the particle size”), and the RF Ministry of Science and Higher Education within a state contract with the "Crystallography and Photonics" Research Center of the RAS (in part "Introduction", “Problem formulation”, “Conclusion”).


  1. Li H, Cao Y, Zhou L, Xu X, Zhu T, Shi Y, Qiu C, Ding W. Optical pulling forces and their applications. Adv Opt Photon 2020; 12: 288-366.
  2. Sraj I, Szatmary AC, Marr DWM, Eggleton CD. Dynamic ray tracing for modeling optical cell manipulation. Opt Express 2010; 18: 16702-16714.
  3. Zhong M, Xue G, Zhou J, Wang Z, Li Y. Measurement of interaction force between RGD-peptide and Hela cell surface by optical tweezers. Chin Opt Lett 2012; 10: 101701-101701.
  4. Zhou JH, Zhong MC, Wang ZQ, Li YM. Calculation of optical forces on an ellipsoid using vectorial ray tracing method. Opt Express 2012; 20: 14928-14937.
  5. Liu S, Li Z, Weng Z, Li Y, Shui L, Jiao Z, Chen Y, Luo A, Xing X, He S. Miniaturized optical fiber tweezers for cell separation by optical force. Opt Lett 2019; 44: 1868-1871.
  6. Drobczyński S, Duś-szachniewicz K. Real-time force measurement in double wavelength optical tweezers. J Opt Soc Am B 2017; 34: 38-43.
  7. Yu Y, Zhang Z, Li Z, Wang X. Methods of calibration to optical trapping force upon non-spherical cells. Chin Opt Lett 2006; 4: 722-724.
  8. Muradoglu M, Chiu WSY, Ng TW. Optical force lateral push–pulling using focus positioning. J Opt Soc Am B 2012; 29: 874-880.
  9. Wang D, Wang Z. Optical pulling force in periodic backward-wave waveguides. Conference on Lasers and Electro-Optics, OSA Technical Digest (online) 2017: FTh1H.4.
  10. Jing P, Liu Y, Keeler EG, Cruz NM, Freedman BS, Lin LY. Optical tweezers system for live stem cell organization at the single-cell level. Biomed Opt Express 2018; 9: 771-779.
  11. Liu H, Panmai M, Peng Y, Lan S. Optical pulling and pushing forces exerted on silicon nanospheres with strong coherent interaction between electric and magnetic resonances. Opt Express 2017; 25(11): 12357-12371.
  12. Kuznetsov AI, Miroshnichenko AE, Fu YH, Zhang J, Luk´yanchuk B. Magnetic light. Sci Rep 2012; 2: 492.
  13. Evlyukhin AB, Novikov SM, Zywietz U, Eriksen RL, Reinhardt C, Bozhevolnyi SI, Chichkov BN. Demonstration of magnetic dipole resonances of dielectric nanospheres in the visible region. Nano Lett 2012; 12(7): 3749-3755.
  14. Shi L, Tuzer TU, Fenollosa R, Meseguer F. A new dielectric metamaterial building block with a strong magnetic response in the sub-1.5-micrometer region: silicon colloid nanocavities. Adv Mater 2012; 24(44): 5934-5938.
  15. Geffrin JM, García-Cámara B, Gómez-Medina R, Albella P, Froufe-Pérez LS, Eyraud C, Litman A, Vaillon R, González F, Nieto-Vesperinas M, Sáenz JJ, Moreno F. Magnetic and electric coherence in forward- and back-scattered electromagnetic waves by a single dielectric subwavelength sphere. Nat Commun 2012; 3: 1171.
  16. Fu YH, Kuznetsov AI, Miroshnichenko AE, Yu YF, Luk’yanchuk B. Directional visible light scattering by silicon nanoparticles. Nat Commun 2013; 4: 1527.
  17. Harada Y, Asakura T. Radiation forces on a dielectric sphere in the Rayleigh scattering regime. Opt Commun 1996; 124: 529-541.
  18. Bekshaev AYa. Subwavelength particles in an inhomogeneous light field: optical forces associated with the spin and orbital energy flows. J Opt 2013; 15: 044004.
  19. Biener G, Vrotsos E, Sugaya K, Dogariu A. Optical torques guiding cell motility. Opt Express 2009; 17: 9724-9732.
  20. Nieto-Vesperinas M. Optical torque on small bi-isotropic particles. Opt Lett 2015; 40: 3021-3024.
  21. Chen J, Ng J, Lin ZF, Chan CT. Optical pulling force. Nat Photonics 2011; 5(9): 531-534.
  22. Novitsky AV, Novitsky DV. Negative propagation of vector Bessel beams. J Opt Soc Am A 2007; 24: 2844-2849.
  23. Stafeev SS, Nalimov AG. Longitudinal component of the Poynting vector of a tightly focused optical vortex with circular polarization. Computer Optics 2018; 42(2): 190-196. DOI: 10.18287/2412-6179-2018-42-2-190-196.
  24. Kotlyar VV, Kovalev AA, Nalimov AG. Energy density and energy flux in the focus of an optical vortex: reverse flux of light energy. Opt Lett 2018; 43(2): 2921-2924. DOI: 10.1364/OL.43.002921.
  25. Brzobohatý O, Karásek V, Šiler M, Chvátal L, Čižmár T, Zemánek P. Experimental demonstration of optical transport, sorting and self-arrangement using a ‘tractor beam’. Nat Photonics 2013; 7(2): 123-127.
  26. Nalimov AG, Stafeev SS. Energy flux of a vortex field focused using a secant gradient lens. Computer Optics 2020; 44(5): 707-711. DOI: 10.18287/2412-6179-CO-688.
  27. Nalimov AG, Kotlyar VV. Calculation of the moment of force acting by a cylindrical Gaussian beam on a cylindrical microparticle. Computer Optics 2007; 31(2): 16-20.

© 2009, IPSI RAS
151, Molodogvardeiskaya str., Samara, 443001, Russia; E-mail: ko@smr.ru ; Tel: +7 (846) 242-41-24 (Executive secretary), +7 (846) 332-56-22 (Issuing editor), Fax: +7 (846) 332-56-20