Nondiffractive three-dimensional polarization features of optical vortex beams

Andrei Afanasev; Jack J. Kingsley-Smith; Francisco J. Rodríguez-Fortuño; Anatoly V. Zayats

doi:10.1117/1.APN.2.2.026001

Journals >Advanced Photonics Nexus >Volume 2 >Issue 2 >Page 026001 > Article

Advanced Photonics Nexus
Vol. 2, Issue 2, 026001 (2023)

Nondiffractive three-dimensional polarization features of optical vortex beams

Andrei Afanasev^1、*, Jack J. Kingsley-Smith², Francisco J. Rodríguez-Fortuño², and Anatoly V. Zayats²

Author Affiliations

¹The George Washington University, Department of Physics, Washington, DC, United States

²King’s College London, Department of Physics and London Centre for Nanotechnology, London, United Kingdom

show less

DOI: 10.1117/1.APN.2.2.026001 Cite this Article Set citation alerts

Andrei Afanasev, Jack J. Kingsley-Smith, Francisco J. Rodríguez-Fortuño, Anatoly V. Zayats. Nondiffractive three-dimensional polarization features of optical vortex beams[J]. Advanced Photonics Nexus, 2023, 2(2): 026001 Copy Citation Text

EndNote(RIS)

BibTex

Plain Text

show less

$Polarization parameters for a focused Laguerre–Gaussian vortex beam with l=1, linearly polarized along the x direction and propagating along the z axis. The beam waist is w0=λ. (a) The intensity distribution of the beam in the (xz) plane. (b–d) Colormaps of the (a) py, (b) pzz, and (d) pxx-pyy polarization parameters with polarization ellipses overlaid on top, indicating the polarization state at each point in space. The polarization structure around the phase singularity is nondiffractive and invariant with respect to the beam waist. The inserts in (c) show the cross-sections at different z positions. The insert in (d) shows the zoom near the beam centre.$

Fig. 1. Polarization parameters for a focused Laguerre–Gaussian vortex beam with l=1, linearly polarized along the

x

direction and propagating along the

z

axis. The beam waist is

w_{0} = λ

. (a) The intensity distribution of the beam in the (

x z

) plane. (b–d) Colormaps of the (a)

p_{y}

, (b)

p_{z z}

, and (d)

p_{x x} - p_{y y}

polarization parameters with polarization ellipses overlaid on top, indicating the polarization state at each point in space. The polarization structure around the phase singularity is nondiffractive and invariant with respect to the beam waist. The inserts in (c) show the cross-sections at different

z

positions. The insert in (d) shows the zoom near the beam centre.

Download full size | View in the Article

$Polarization parameters for a focused Laguerre–Gaussian vortex beam with l=1and σ=−1 (left-hand circularly polarized), propagating along the positive z axis. The beam waist is (a–d) w0=λ and (e–h) w0=2λ. (a) and (e) The intensity distribution of the beam in the (xz) plane. (b–d) and (f–h) Colormaps of the (b,f) py, (c,g) pz, and (d,h) pzz polarization parameters with polarization ellipses overlaid on top, indicating the polarization state at each point in space. The polarization structure around the phase singularity is nondiffractive and invariant with respect to the beam waist.$

Fig. 2. Polarization parameters for a focused Laguerre–Gaussian vortex beam with l=1and

σ = - 1

(left-hand circularly polarized), propagating along the positive

z

axis. The beam waist is (a–d)

w_{0} = λ

and (e–h)

w_{0} = 2 λ

. (a) and (e) The intensity distribution of the beam in the (

x z

) plane. (b–d) and (f–h) Colormaps of the (b,f)

p_{y}

, (c,g)

p_{z}

, and (d,h)

p_{z z}

Download full size | View in the Article

Fig. 3. Cross sections of the linearly polarized vortex beams with

w_{0} = λ

depicted in Fig. 1 showing (a)

p_{z z}

and (b)

| E |^{2}

at the focal plane and after propagating a distance of

5 λ

. The vertical dashed lines show the points at which

p_{z z} = 0

and the corresponding field intensity for each cross-section plane.

Download full size | View in the Article

$(a) The angular spectrum of a linearly polarized (x direction) vortex beam with l=1. The green line indicates the light line and the red dotted line indicates the inner limit of the cropped region dictated by the objective with NA=0.5. (b) The intensity distribution of the corresponding vortex beam with NA=0.5 generated by integrating the field distribution in (a). The beam waist is w0=λ. (c) Colormap of the py polarization parameter with polarization ellipses overlaid on top, indicating the polarization state at each point in space. The nondiffractive behavior near the phase singularity is maintained and only peripheral fields are affected by the NA reduction. (d–f) The same quantities as in (a–c) but with part of the angular spectrum removed and astigmatism applied along x direction.$

Fig. 4. (a) The angular spectrum of a linearly polarized (

x

direction) vortex beam with l=1. The green line indicates the light line and the red dotted line indicates the inner limit of the cropped region dictated by the objective with

NA = 0.5

. (b) The intensity distribution of the corresponding vortex beam with

NA = 0.5

generated by integrating the field distribution in (a). The beam waist is

w_{0} = λ

. (c) Colormap of the

p_{y}

polarization parameter with polarization ellipses overlaid on top, indicating the polarization state at each point in space. The nondiffractive behavior near the phase singularity is maintained and only peripheral fields are affected by the NA reduction. (d–f) The same quantities as in (a–c) but with part of the angular spectrum removed and astigmatism applied along

x

direction.

Download full size | View in the Article

Fig. 5. (a–d) The phase of

E_{x}

in the focal plane of a nonparaxial l=3 vortex beam linearly polarized along

x

direction with

w_{0} = λ

and (e–g) the

p_{z z}

polarization parameter in a

z = 5 λ

plane (to avoid focal plane features) simulated for different finite numbers of constituent plane waves

N

(indicated in the panels). The l=3 vortex splits in three l=1 vortices with an increased splitting for a decreased

N

Download full size | View in the Article


	Circular $σ \cdot l < 0$	Circular $σ \cdot l > 0$	Linear	Radial	Azimuthal
$η_{⊥}$	$\frac{1}{\sqrt{2}} (\hat{x} - i \hat{y})$	$- \frac{1}{\sqrt{2}} (\hat{x} + i \hat{y})$	$\hat{x}$	$\hat{ρ}$	$\hat{ϕ}$
$p_{z}$	$- \frac{1}{1 + 2 ζ^{2}}$	1	0	0	0
$p_{y}$	$\frac{\sqrt{2} \| ζ \| \cos ϕ}{1 + 2 ζ^{2}}$	0	$- \frac{2 \| ζ \| \cos ϕ}{1 + ζ^{2}}$	$- \frac{4 \| ζ \| \cos ϕ}{1 + 4 ζ^{2}}$	0
$p_{x}$	$- \frac{\sqrt{2} \| ζ \| \sin ϕ}{1 + 2 ζ^{2}}$	0	0	$\frac{4 \| ζ \| \sin ϕ}{1 + 4 ζ^{2}}$	0
$p_{x x} - p_{y y}$	0	0	$\frac{3}{1 + ζ^{2}}$	$- \frac{3 \cos 2 ϕ}{1 + 4 ζ^{2}}$	$3 \cos 2 ϕ$
$p_{x y}$	0	0	0	$- \frac{3}{2} \frac{\sin 2 ϕ}{1 + 4 ζ^{2}}$	$\frac{3}{2} \sin 2 ϕ$
$p_{x z}$	$- \frac{3}{\sqrt{2}} \frac{\| ζ \| \sin ϕ}{1 + 2 ζ^{2}}$	0	$\frac{3 \| ζ \| \sin ϕ}{1 + ζ^{2}}$	0	0
$p_{y z}$	$\frac{3}{\sqrt{2}} \frac{\| ζ \| \cos ϕ}{1 + 2 ζ^{2}}$	0	0	0	0
$p_{z z}$	$\frac{1 - 4 ζ^{2}}{1 + 2 ζ^{2}}$	1	$\frac{1 - 2 ζ^{2}}{1 + ζ^{2}}$	$\frac{1 - 8 ζ^{2}}{1 + 4 ζ^{2}}$	1