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Journals >Acta Photonica Sinica >Volume 50 >Issue 10 >Page 1026002 > Article

Acta Photonica Sinica
Vol. 50, Issue 10, 1026002 (2021)

Local Optical Chirality-analysis of Tightly Focused Vortex Beams

Shenyan GUO, Zhiwei CUI^*, Ju WANG, and Fuping WU

Author Affiliations

School of Physics and Optoelectronic Engineering，Xidian University，Xi'an 710071，China

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DOI: 10.3788/gzxb20215010.1026002 Cite this Article

Shenyan GUO, Zhiwei CUI, Ju WANG, Fuping WU. Local Optical Chirality-analysis of Tightly Focused Vortex Beams[J]. Acta Photonica Sinica, 2021, 50(10): 1026002 Copy Citation Text

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Abstract

Based on the Richards-Wolf vectorial diffraction theory, the electric and magnetic field components of the focused vortex beams with different states of polarization are derived. The chiral density and superchiral factor are introduced. The effects of the waist radius, polarization state, and topological charge on the local chirality of tightly focused vortex beams are numerically simulated. The numerical results show that the chirality is remarkable when the waist radius of the beam is equal to the focal length of the high numerical aperture lens. The topological charge has a significant effect on the local chirality of tightly focused vortex beams. The radially polarized focused vortex beam has more obvious effect on the local chiral enhancement.

Keywords

Focusing Optical chirality Polarization Topological charges Vortex beams

E (r, φ, z = 0) = {(\sqrt[]{2} \frac{r}{w_{0}})}^{|l|} L_{p}^{|l|} (2 \frac{r^{2}}{w_{0}^{2}}) e x p (- \frac{r^{2}}{w_{0}^{2}}) e x p (i l φ)

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E (r, φ, z = 0) = {(\sqrt[]{2} \frac{r}{w_{0}})}^{|l|} e x p (- \frac{r^{2}}{{w_{0}}^{2}}) e x p (i l φ)

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E = \frac{- i k f}{2 π} \int_{0}^{θ_{m a x}} \int_{0}^{2 π} a (θ, φ) e^{i k \cdot r_{p}} s i n θ d θ d φ

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a (θ, φ) = T (θ) E (θ, φ) P_{e} (θ, φ)

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P_{e} (θ, φ) = [\begin{array}{l} p_{x} (c o s^{2} φ c o s θ + s i n^{2} φ) - p_{y} (1 - c o s θ) s i n φ c o s φ \\ p_{x} (c o s θ - 1) s i n φ c o s φ + p_{y} (s i n^{2} φ c o s θ + c o s^{2} φ) \\ - p_{x} s i n θ c o s φ - p_{y} s i n θ s i n φ \end{array}]

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A (θ) = {(\sqrt[]{2} \frac{f s i n θ}{w_{0}})}^{|l|} e x p [- \frac{{(f s i n θ)}^{2}}{{w_{0}}^{2}}]

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\hat{k} = s i n θ c o s φ \hat{x} + s i n θ s i n φ \hat{y} + c o s θ \hat{z}

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r_{p} = r_{p} c o s φ_{p} \hat{x} + r_{p} s i n φ_{p} \hat{y} + z_{p} \hat{z}

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E = E_{0} \int_{0}^{θ_{m a x}} \int_{0}^{2 π} F (θ) A (θ) e^{i l φ} P_{e} (θ, φ) e^{i k z_{p} c o s θ} e^{i k r_{p} s i n θ c o s (φ - φ_{p})} d φ d θ

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H = H_{0} \int_{0}^{θ_{m a x}} \int_{0}^{2 π} F (θ) A (θ) e^{i l φ} P_{m} (θ, φ) e^{i k z_{p} c o s θ} e^{i k r_{p} s i n θ c o s (φ - φ_{p})} d φ d θ

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P_{m} (θ, φ) = [\begin{array}{l} - p_{x} (1 - c o s θ) s i n φ c o s φ - p_{y} (c o s^{2} φ c o s θ + s i n^{2} φ) \\ p_{x} (s i n^{2} φ c o s θ + c o s^{2} φ) + p_{y} (1 - c o s θ) s i n φ c o s φ \\ - p_{x} s i n θ s i n φ + p_{y} s i n θ s i n φ \end{array}]

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\{\begin{array}{l} P_{e}^{x - L P} = [\begin{array}{l} c o s^{2} φ c o s θ + s i n^{2} φ \\ (c o s θ - 1) s i n φ c o s φ \\ - s i n θ c o s φ \end{array}] \\ P_{m}^{x - L P} = [\begin{array}{l} - (1 - c o s θ) s i n φ c o s φ \\ s i n^{2} φ c o s θ + c o s^{2} φ \\ - s i n θ c o s φ \end{array}] \end{array}

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[\begin{matrix} I_{l}^{n} (ρ) \\ J_{l}^{n} (ρ) \\ K_{l} (ρ) \end{matrix}] = \int_{0}^{2 π} e^{i l φ} [\begin{matrix} c o s (n φ) \\ s i n (n φ) \\ 1 \end{matrix}] e^{i ρ c o s (φ - φ_{p})} d φ = π [\begin{matrix} i^{l + n} e^{i (l + n) φ_{p}} J_{l + n} (ρ) + i^{l - n} e^{i (l - n) φ_{p}} J_{l - n} (ρ) \\ - i \cdot i^{l + n} e^{i (l + n) φ_{p}} J_{l + n} (ρ) + i \cdot i^{l - n} e^{i (l - n) φ_{p}} J_{l - n} (ρ) \\ 2 i^{l} e^{i l φ_{p}} J_{l} (ρ) \end{matrix}]

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[\begin{matrix} E_{x}^{x - L P} \\ E_{y}^{x - L P} \\ E_{z}^{x - L P} \end{matrix}] = \frac{E_{0}}{2} \int_{0}^{θ_{m a x}} F (θ) A (θ) e^{i k z_{p} c o s θ} [\begin{matrix} (c o s θ - 1) I_{l}^{2} (k r_{p} s i n θ) + (c o s θ + 1) K_{l} (k r_{p} s i n θ) \\ (c o s θ - 1) J_{l}^{2} (k r_{p} s i n θ) \\ - 2 s i n θ I_{l}^{1} (k r_{p} s i n θ) \end{matrix}] d θ

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[\begin{matrix} H_{x}^{x - L P} \\ H_{y}^{x - L P} \\ H_{z}^{x - L P} \end{matrix}] = \frac{H_{0}}{2} \int_{0}^{θ_{m a x}} F (θ) A (θ) e^{i k z_{p} c o s θ} [\begin{matrix} (c o s θ - 1) J_{l}^{2} (k r_{p} s i n θ) \\ (c o s θ + 1) K_{l} (k r_{p} s i n θ) - (c o s θ - 1) I_{l}^{2} (k r_{p} s i n θ) \\ - 2 s i n θ J_{l}^{1} (k r_{p} s i n θ) \end{matrix}] d θ

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\{\begin{array}{l} P_{e}^{y - L P} = P_{m}^{x - L P} \\ P_{m}^{y - L P} = - P_{e}^{x - L P} \end{array}

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\{\begin{array}{l} [\begin{matrix} E_{x}^{y - L P} \\ E_{y}^{y - L P} \\ E_{z}^{y - L P} \end{matrix}] = Z [\begin{matrix} H_{x}^{x - L P} \\ H_{y}^{x - L P} \\ H_{z}^{x - L P} \end{matrix}] \\ [\begin{matrix} H_{x}^{y - L P} \\ H_{y}^{y - L P} \\ H_{z}^{y - L P} \end{matrix}] = - \frac{1}{Z} [\begin{matrix} E_{x}^{x - L P} \\ E_{y}^{x - L P} \\ E_{z}^{x - L P} \end{matrix}] \end{array}

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\{\begin{array}{l} P_{e}^{C P} = (P_{e}^{x - L P} \pm i P_{m}^{x - L P}) / \sqrt[]{2} \\ P_{m}^{C P} = (P_{e}^{y - L P} \pm i P_{m}^{y - L P}) / \sqrt[]{2} \end{array}

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\{\begin{array}{l} [\begin{matrix} E_{x}^{C P} \\ E_{y}^{C P} \\ E_{z}^{C P} \end{matrix}] = \frac{1}{\sqrt[]{2}} [\begin{matrix} E_{x}^{x - L P} \pm i Z H_{x}^{x - L P} \\ E_{y}^{x - L P} \pm i Z H_{y}^{x - L P} \\ E_{z}^{x - L P} \pm i Z H_{z}^{x - L P} \end{matrix}] \\ [\begin{matrix} H_{x}^{C P} \\ H_{y}^{C P} \\ H_{z}^{C P} \end{matrix}] = \frac{1}{\sqrt[]{2}} [\begin{matrix} Z^{- 1} E_{x}^{y - L P} \pm i H_{x}^{y - L P} \\ Z^{- 1} E_{y}^{y - L P} \pm i H_{y}^{y - L P} \\ Z^{- 1} E_{z}^{y - L P} \pm i H_{z}^{y - L P} \end{matrix}] \end{array}

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\{\begin{array}{l} P_{e}^{R P} = {[\begin{matrix} c o s θ c o s φ \\ c o s θ s i n φ \\ - s i n θ \end{matrix}]}_{}^{} \\ P_{m}^{R P} = [\begin{matrix} - s i n φ \\ c o s φ \\ 0 \end{matrix}] \end{array}

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[\begin{matrix} E_{x}^{R P} \\ E_{y}^{R P} \\ E_{z}^{R P} \end{matrix}] = E_{0} \int_{0}^{θ_{m a x}} F (θ) A (θ) e^{i k z_{p} c o s θ} [\begin{matrix} c o s θ I_{l}^{1} (k r_{p} s i n θ) \\ c o s θ J_{l}^{1} (k r_{p} s i n θ) \\ - s i n θ K_{l} (k r_{p} s i n θ) \end{matrix}] d θ

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[\begin{matrix} H_{x}^{R P} \\ H_{y}^{R P} \\ H_{z}^{R P} \end{matrix}] = H_{0} \int_{0}^{θ_{m a x}} F (θ) A (θ) e^{i k z_{p} c o s θ} [\begin{matrix} - J_{l}^{1} (k r_{p} s i n θ) \\ I_{l}^{1} (k r_{p} s i n θ) \\ 0 \end{matrix}] d θ

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\{\begin{array}{l} P_{e}^{A P} = P_{m}^{R P} \\ P_{m}^{A P} = - P_{e}^{R P} \end{array}

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\{\begin{array}{l} [\begin{matrix} E_{x}^{A P} \\ E_{y}^{A P} \\ E_{z}^{A P} \end{matrix}] = Z [\begin{matrix} H_{x}^{R P} \\ H_{y}^{R P} \\ H_{z}^{R P} \end{matrix}] \\ [\begin{matrix} H_{x}^{A P} \\ H_{y}^{A P} \\ H_{z}^{A P} \end{matrix}] = - \frac{1}{Z} [\begin{matrix} E_{x}^{R P} \\ E_{y}^{R P} \\ E_{z}^{R P} \end{matrix}] \end{array}

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g = \frac{ω}{2 c^{2}} I m (E \cdot H^{*})

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g_{0} = \pm \frac{ε_{0} ω}{2 c} {|E_{C P L}|}^{2} s i n Δ ϕ

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g / g_{C P L} = Z_{0} \frac{I m (E \cdot H^{*})}{{|E|}^{2}}

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Shenyan GUO, Zhiwei CUI, Ju WANG, Fuping WU. Local Optical Chirality-analysis of Tightly Focused Vortex Beams[J]. Acta Photonica Sinica, 2021, 50(10): 1026002

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