Generation of a sub-half-wavelength focal spot with purely transverse spin angular momentum

Abstract

We theoretically demonstrate that optical focus fields with purely transverse spin angular momentum (SAM) can be obtained when a kind of special incident fields is focused by a high numerical aperture (NA) aplanatic lens (AL). When the incident pupil fields are refracted by an AL, two transverse Cartesian components of the electric fields at the exit pupil plane do not have the same order of sinusoidal or cosinoidal components, resulting in zero longitudinal SAMs of the focal fields. An incident field satisfying above conditions is then proposed. Using the Richard–Wolf vectorial diffraction theory, the energy density and SAM density distributions of the tightly focused beam are calculated and the results clearly validate the proposed theory. In addition, a sub-half-wavelength focal spot with purely transverse SAM can be achieved and a flattop energy density distribution parallel to z-axis can be observed around the maximum energy density point.

Appendix

• 1. Detailed expression of det ($A|\overrightarrow {{E_{o} }} (\theta ,\varphi )$) = 0:

  $$\left\{ \begin{aligned} \varPhi_{1,2a} = \varPsi_{2,2b} = \varPhi_{3,2c + 1} (\forall a,b,c \in N) \hfill \\ \varPhi_{1,2a + 1} = \varPsi_{2,2b + 1} = \varPhi_{3,2c} (\forall a,b,c \in N) \hfill \\ \varPsi_{1,2a} = \varPhi_{2,2b} = \varPsi_{3,2c + 1} (\forall a,b,c \in N) \hfill \\ \varPsi_{1,2a + 1} = \varPhi_{2,2b + 1} = \varPsi_{3,2c} (\forall a,b,c \in N) \hfill \\ \sin \theta (f_{11} + g_{21} ) = 2\cos \theta f_{30} \hfill \\ \sin \theta (2f_{10} + f_{12} + g_{22} ) = 2\cos \theta f_{31} \hfill \\ \sin \theta (g_{12} + 2f_{20} - f_{22} ) = 2\cos \theta g_{31} \hfill \\ for\;n = 2,3,4 \cdots \hfill \\ \sin \theta (f_{1,n - 1} + f_{1,n + 1} - g_{2,n - 1} + g_{2,n + 1} ) = 2\cos \theta f_{3,n} \hfill \\ \sin \theta (g_{1,n - 1} + g_{1,n + 1} + f_{2,n - 1} - f_{2,n + 1} ) = 2\cos \theta g_{3,n} \hfill \\ \end{aligned} \right..$$

  (A1)

• 2. Electric field and SAM density corresponding to $\vec{E}_{i} (\theta ,\varphi )$ in Eq. (12):

$$\begin{aligned} E_{x} (r,\phi ,z) &= - ik\int_{0}^{{\theta_{\hbox{max} } }} {(\cos \phi J_{1} + \cos 2\phi J_{2} )} \hfill \\ &\quad \times\sin \theta \cos \theta \exp (ikz\cos \theta ){\text{d}}\theta , \hfill \\ \end{aligned}$$

(A2)

$$\begin{aligned} E_{y} (r,\phi ,z) &= - ik\int_{0}^{{\theta_{\hbox{max} } }} {(\sin \phi J_{1} + \sin 2\phi J_{2} )} \hfill \\ &\quad\times \sin \theta \cos \theta \exp (ikz\cos \theta ){\text{d}}\theta , \hfill \\ \end{aligned}$$

(A3)

$$\begin{aligned} E_{z} (r,\phi ,z) &= - k\int_{0}^{{\theta_{\hbox{max} } }} {(J_{0} + \cos \phi J_{1} )} \hfill \\ &\quad\times \sin^{2} \theta \exp (ikz\cos \theta ){\text{d}}\theta , \hfill \\ \end{aligned}$$

(A4)

$$\begin{aligned} &s_{x} \propto \int_{0}^{{\theta_{\hbox{max} } }} {\int_{0}^{{\theta_{\hbox{max} } }} {\sin \theta \cos \theta \sin^{2} \theta^{'} (\sin \phi J_{1} + \sin 2\phi J_{2} )} } \hfill \\ &\quad \times (J_{0}^{'} + \cos \phi J_{1}^{'} )\cos [kz(\cos \theta^{'} - \cos \theta )]{\text{d}}\theta^{'} {\text{d}}\theta , \hfill \\ \end{aligned}$$

(A5)

$$\begin{aligned} s_{y} \propto \int_{0}^{{\theta_{\hbox{max} } }} {\int_{0}^{{\theta_{\hbox{max} } }} {\sin^{2} \theta \sin \theta^{'} \cos \theta^{'} (J_{0} + \cos \phi J_{1} )} } \hfill \\ \times\;\;\;(\cos \phi J_{1}^{'} + \cos 2\phi J_{2}^{'} )\cos [kz(\cos \theta^{'} - \cos \theta )]{\text{d}}\theta^{'} {\text{d}}\theta , \hfill \\ \end{aligned}$$

(A6)

$$\begin{aligned} &s_{z} \propto \int_{0}^{{\theta_{\hbox{max} } }} {\int_{0}^{{\theta_{\hbox{max} } }} {\sin \theta \cos \theta \sin \theta^{'} \cos \theta^{'} (\cos \phi J_{1} + \cos 2\phi J_{2} )} } \hfill \\ &\times\, (\sin \phi J_{1}^{'} + \sin 2\phi J_{2}^{'} )\sin [kz(\cos \theta^{'} - \cos \theta )]{\text{d}}\theta^{'} {\text{d}}\theta . \hfill \\ \end{aligned}$$

(A7)