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Photonics Research
Vol. 12, Issue 3, 431 (2024)

Chiral forces in longitudinally invariant dielectric photonic waveguides

Josep Martínez-Romeu^1、†, Iago Diez^1、†, Sebastian Golat², Francisco J. Rodríguez-Fortuño², and Alejandro Martínez^1、*

Author Affiliations

¹Nanophotonics Technology Center, Universitat Politècnica de València, Valencia 46022, Spain

²Department of Physics, King’s College London, London WC2R 2LS, UK

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DOI: 10.1364/PRJ.509634 Cite this Article Set citation alerts

Josep Martínez-Romeu, Iago Diez, Sebastian Golat, Francisco J. Rodríguez-Fortuño, Alejandro Martínez. Chiral forces in longitudinally invariant dielectric photonic waveguides[J]. Photonics Research, 2024, 12(3): 431 Copy Citation Text

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Abstract

We calculate numerically the optical chiral forces in rectangular cross-section dielectric waveguides for potential enantiomer separation. Our study considers force strength and time needed for separating chiral nanoparticles, mainly via quasi-TE guided modes at short wavelengths (405 nm) and the 90°-phase-shifted combination of quasi-TE and quasi-TM modes at longer wavelengths (1310 nm). Particle tracking simulations show successful enantiomer separation within two seconds. These results suggest the feasibility of enantiomeric separation of nanoparticles displaying sufficient chirality using simple silicon photonic integrated circuits, with wavelength selection based on the nanoparticle size.

F = \frac{1}{2} ℜ [\underset{interaction}{\underset{⏟}{(\nabla \otimes E) p^{*} + μ (\nabla \otimes H) m^{*}}} \underset{recoil}{\underset{⏟}{- \frac{k^{4} η}{6 π} (p^{*} \times m)}}] .

(1)

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p = α_{e} ε E + i \frac{1}{c} α_{c} H, m = α_{m} H - i \frac{1}{η} α_{c} E,

(2)

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α_{0 e} = 4 π r^{3} \frac{(ε_{p} - ε_{m}) (μ_{p} + 2 μ_{m}) - κ^{2}}{(ε_{p} + 2 ε_{m}) (μ_{p} + 2 μ_{m}) - κ^{2}}, α_{0 m} = 4 π r^{3} \frac{(ε_{p} + 2 ε_{m}) (μ_{p} - μ_{m}) - κ^{2}}{(ε_{p} + 2 ε_{m}) (μ_{p} + 2 μ_{m}) - κ^{2}}, α_{0 c} = 12 π r^{3} \frac{κ}{(ε_{p} + 2 ε_{m}) (μ_{p} + 2 μ_{m}) - κ^{2}},

(3)

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α_{e} = \frac{α_{0 e} - i \frac{k^{3}}{6 π} (α_{0 c}^{2} - α_{0 e} α_{0 m})}{1 + {(\frac{k^{3}}{6 π})}^{2} (α_{0 c}^{2} - α_{0 e} α_{0 m}) - i \frac{k^{3}}{6 π} (α_{0 e} + α_{0 m})}, α_{m} = \frac{α_{0 m} - i \frac{k^{3}}{6 π} (α_{0 c}^{2} - α_{0 e} α_{0 m})}{1 + {(\frac{k^{3}}{6 π})}^{2} (α_{0 c}^{2} - α_{0 e} α_{0 m}) - i \frac{k^{3}}{6 π} (α_{0 e} + α_{0 m})}, α_{c} = \frac{α_{0 c}}{1 + {(\frac{k^{3}}{6 π})}^{2} (α_{0 c}^{2} - α_{0 e} α_{0 m}) - i \frac{k^{3}}{6 π} (α_{0 e} + α_{0 m})} .

(4)

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W_{e} = \frac{1}{4} ε {| E |}^{2} [\frac{J}{m^{3}}], W_{m} = \frac{1}{4} μ {| H |}^{2} [\frac{J}{m^{3}}], G = \frac{1}{2 ω c} J (E \cdot H^{*}) [\frac{J \cdot s}{m^{3}}], S_{e} = \frac{1}{4 ω} J (ε E^{*} \times E) [\frac{J \cdot s}{m^{3}}], S_{m} = \frac{1}{4 ω} J (μ H^{*} \times H) [\frac{J \cdot s}{m^{3}}], Π = \frac{1}{2} E \times H^{*} [\frac{W}{m^{2}}] .

(5)

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F_{chiral} = \underset{helicity gradient}{\underset{⏟}{ω ℜ (α_{c}) \nabla G}} - \underset{vortex}{\underset{⏟}{\frac{1}{c} J (α_{c}) \nabla \times ℜ Π}} + \underset{electric spin}{\underset{⏟}{(2 k J (α_{c}) - \frac{k^{4}}{3 π} ℜ (α_{e}^{*} α_{c})) ω S_{e}}} + \underset{magnetic spin}{\underset{⏟}{(2 k J (α_{c}) - \frac{k^{4}}{3 π} ℜ (α_{c}^{*} α_{m})) ω S_{m}}}, F_{achiral} = \underset{electric gradient}{\underset{⏟}{ℜ (α_{e}) \nabla W_{e}}} + \underset{magnetic gradient}{\underset{⏟}{ℜ (α_{m}) \nabla W_{m}}} - \underset{spin - curl}{\underset{⏟}{ω \nabla \times (J (α_{e}) S_{e} + J (α_{m}) S_{m})}} + \underset{radiation pressure}{\underset{⏟}{(\frac{k}{c} J (α_{e} + α_{m}) - \frac{k^{4}}{6 π} \frac{1}{c} (ℜ (α_{e}^{*} α_{m}) + {| α_{c} |}^{2})) ℜ Π}} - \underset{flow}{\underset{⏟}{\frac{k^{4}}{6 π} \frac{1}{c} J (α_{e}^{*} α_{m}) J Π}},

(6)

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0 = - \underset{friction}{\underset{⏟}{γ \frac{d x}{d t}}} + \underset{optical}{\underset{⏟}{F}} + \underset{stochastic}{\underset{⏟}{γ \sqrt{2 D} ξ (t)}},

(7)

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x^{(m + 1)} = x^{(m)} + M_{x, ⊥}^{(m)} F_{x}^{(m)} Δ t + \sqrt{2 M_{x, ⊥}^{(m)} k_{B} T Δ t} N_{x} (0,1), y^{(m + 1)} = y^{(m)} + M_{y, ⊥}^{(m)} F_{y}^{(m)} Δ t + \sqrt{2 M_{y, ⊥}^{(m)} k_{B} T Δ t} N_{y} (0,1), z^{(m + 1)} = z^{(m)} + M_{z, | |}^{(m)} F_{z}^{(m)} Δ t + \sqrt{2 M_{z, | |}^{(m)} k_{B} T Δ t} N_{z} (0,1),

(8)

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t_{sort} = 3 π η k_{B} T \frac{r}{F^{2}} {(1 + \sqrt{1 + \frac{F (L_{0} + Δ x)}{k_{B} T}})}^{2} .

(9)

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m \frac{d^{2} x}{d t^{2}} = - \underset{friction}{\underset{⏟}{γ \frac{d x}{d t}}} + \underset{optical}{\underset{⏟}{F}} + \underset{stochastic}{\underset{⏟}{γ \sqrt{2 D} ξ (t)}} .

(A1)

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0 = - γ \frac{d x}{d t} + F + γ \sqrt{2 D} ξ (t),

(A2)

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x^{(m + 1)} = x^{(m)} + M_{x, ⊥}^{(m)} F_{x}^{(m)} Δ t + \sqrt{2 M_{x, ⊥}^{(m)} k_{B} T Δ t} N_{x} (0,1), y^{(m + 1)} = y^{(m)} + M_{y, ⊥}^{(m)} F_{y}^{(m)} Δ t + \sqrt{2 M_{y, ⊥}^{(m)} k_{B} T Δ t} N_{y} (0,1), z^{(m + 1)} = z^{(m)} + M_{z, | |}^{(m)} F_{z}^{(m)} Δ t + \sqrt{2 M_{z, | |}^{(m)} k_{B} T Δ t} N_{z} (0,1),

(A3)

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Δ t ≫ τ = \frac{m}{γ} = \frac{ρ_{{SiO}_{2}} \frac{4}{3} π r^{3}}{6 π η r} = \frac{2 ρ_{{SiO}_{2}} r^{2}}{9 η} \sim 2.4 ns .

(A4)

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Δ t ≪ \frac{Δ x_{\min} γ}{\max | F |} \sim 162 μs,

(A5)

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M_{⊥} (h) = M (1 - \frac{9}{8} (\frac{r}{h}) + \frac{1}{2} {(\frac{r}{h})}^{3} - \frac{1}{8} {(\frac{r}{h})}^{5}), M_{| |} (h) = M (1 - \frac{9}{16} (\frac{r}{h}) + \frac{1}{8} {(\frac{r}{h})}^{3} - \frac{1}{16} {(\frac{r}{h})}^{5}),

(A6)

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(+) - EF = \frac{N_{+}}{N_{+} + N_{-}}, (-) - EF = \frac{N_{-}}{N_{+} + N_{-}} .

(A7)

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d_{opt} = \frac{F}{γ} t,

(B1)

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d_{B} = \sqrt{2 D t},

(B2)

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2 d_{opt} = L_{0} + 2 d_{B} + Δ x,

(B3)

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2 \frac{F}{γ} t = L_{0} + 2 \sqrt{2 D} \sqrt{t} + Δ x .

(B4)

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\sqrt{t_{sort}} = \frac{γ}{F} \sqrt{\frac{D}{2}} (1 + \sqrt{1 + \frac{F (L_{0} + Δ x)}{D γ}}),

(B5)

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t_{sort} = 3 π η k_{B} T \frac{r}{F^{2}} {(1 + \sqrt{1 + \frac{F (L_{0} + Δ x)}{k_{B} T}})}^{2} .

(B6)

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Josep Martínez-Romeu, Iago Diez, Sebastian Golat, Francisco J. Rodríguez-Fortuño, Alejandro Martínez. Chiral forces in longitudinally invariant dielectric photonic waveguides[J]. Photonics Research, 2024, 12(3): 431

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