Fig. 1. Normalized momentum density and wavevector at the waist plane. The (a) r, (b) φ, (c) z components of Poynting momentum p can be decomposed into the (d) r, (e) φ, (f) z components of po and (g) the z components of ps. Here, the horizontal components of ps are zero. (h) The normalized φ component of k/W indicates off-axis vortexes and (i) the normalized wavenumber |k|/W is larger than one, which is a necessary condition for super-oscillations. Here, l1=−1, l2=−3, A1=A2 throughout the paper.

Fig. 2. The distributions of optical force as the optical azimuthal backflow acts on gold dipolar nanoparticles [50]. The (a) r, (b) φ, (c) z components of total optical force F. The total optical force can be divided into the (d) r, (e) φ, (f) z components of radiation Frad related to Poynting momentum, the (g) r, (h) φ components of gradient force Fgrad related to energy density, and (i) the z component of Fcurl related to the spin momentum. Here, εp=−11.74+1.2611i and μp=1.0; l1=−1 and l2=−3. All the force is normalized by the maximal value of total optical force throughout the paper.

Fig. 3. The optimization of material’s permittivity for eliminating the misalignment between force and Poynting momentum and achieving stable optical trapping. Since the Frad/Fgrad is related to the ξ1=Im{αee}/Re{αee}, we scan the permittivity of nanoparticles for maximal (a) log10|ξ1| to eliminate the misalignment between force and Poynting momentum. The trapping potential well is related to Re{αee} and we scan the permittivity of nanoparticles for maximal (b) log10|ξ2|. Here, μp=1.0.

Fig. 4. The total optical force and the trapping potential well V/kBT. When εp=−1.586+1.653i, the (a) r, (b) φ, (c) z components of F match well with the Poynting momentum given by Figs. 1(a)–1(c). However, the (d) potential well is much smaller than one and the particle cannot be trapped stably. When εp=−2.609−0.339i, the (d) r, (e) φ, (f) z components of F are shown and the (h) potential well is much larger than one. When εp=−2.7458+0.23171i, the (i) r, (j) φ, (k) z components of F are shown and the (l) potential well is much larger than one. Here, l1=−1,l2=−3.

Fig. 5. The azimuthal optical force Fφ for different TCs of OAM states. The appearance of optical azimuthal backflow is determined by |l1−l2|, which can be verified in the Fφ when (a) l1=−1 and l2=−4, (b) l1=−1 and l2=−5, (c) l1=−1 and l2=−6, (d) l1=−2 and l2=−5, (e) l1=−2 and l2=−6, (f) l1=−3 and l2=−6. Here, εp=−2.7458+0.23171i and μp=1.0.

Fig. 6. Normalized momentum density at the focal plane. (a) The z components of Poynting momentum p can be decomposed into the (b) z components of po and (c) the z components of ps. The horizontal components of p, po, and ps are zero. Here, ηx=ηy=1, numerical aperture NA=0.95, the beam waist w0=0.26λ, and A=1 throughout the paper.

Fig. 7. The distributions of optical force as the optical axial backflow acts on dipolar nanoparticles. The (a) r, (b) φ, (c) z components of total optical force F. The total optical force can be divided into the (d) r, (e) φ components of gradient force Fgrad, (f) the z component of radiation force Frad related to Poynting momentum, the (g) r, (h) φ components of flow force Fflow related to imaginary Poynting vector, and (i) the z component of Fcurl related to the spin momentum. Here ,εp=−11.74+1.2611i and μp=1.0.

Fig. 8. The optimization of material’s nanoparticle properties to ensure stable optical trapping and stably generate a reverse optical force. Since the trapping potential well is related to the ξ2 and the Fz is related to ξ3=|(σe+σm)/max(σe,σm)| and ξ4=σ′/max(|σe|,|σm|,|σe+σm|), we scan the permittivity of nanoparticles for maximal (a), (d) log10|ξ2| to ensure stable optical trapping and scan the permittivity of nanoparticles for minimal (b), (e) log10ξ3 and maximal (c), (f) log10ξ4 to stably generate a reverse optical force. Here, in (a)–(c) μp=1.0+5.0i and in (d)–(f), μp=1.0+10.0i.

Fig. 9. The axial optical force and the trapping potential well V/kBT. When εp=−7.795−2.7825i and μp=1.0+5.0i, a significant reverse axial optical force emerges in the (a) z components of F. But Fz have a markedly different distribution from the (b) z components of Frad, due to the influence of the (c) z components of Fcurl and (d) potential well is larger than one. When εp=0.535–3.715i and μp=1.0+5.0i, the distribution of the (e) z components of F aligns with the (f) z components of Frad with tiny (g) z components of Fcurl and (h) potential well is also larger than one. When εp=0.085–3.6525i and μp=1.0+10.0i, the z components of (i) F, (j) Frad, (k) Fcurl and (l) the potential well have similar distributions and results to (e)–(h).