Researching | New kind of Hermite–Gaussian-like optical vortex generated by cross phase

Journals >Chinese Optics Letters >Volume 18 >Issue 10 >Page 100501 > Article

Chinese Optics Letters
Vol. 18, Issue 10, 100501 (2020)

New kind of Hermite–Gaussian-like optical vortex generated by cross phase

Chen Wang¹, Yuan Ren^1、2、*, Tong Liu¹, Linlin Chen¹, and Song Qiu¹

Author Affiliations

¹Department of Aerospace Science and Technology, Space Engineering University, Beijing 101416, China

²State Key Laboratory of Laser Propulsion & Application, Space Engineering University, Beijing 101416, China

show less

DOI: 10.3788/COL202018.100501 Cite this Article Set citation alerts

Chen Wang, Yuan Ren, Tong Liu, Linlin Chen, Song Qiu. New kind of Hermite–Gaussian-like optical vortex generated by cross phase[J]. Chinese Optics Letters, 2020, 18(10): 100501 Copy Citation Text

show less

Abstract

We propose a new kind of optical vortex called the Hermite–Gaussian-like optical vortex (HGOV) inspired by the cross phase (CP). Theoretically, we investigate how the CP is decoupled from the phase of a cylindrical lens. We also investigate the propagation characteristics of an HGOV, which has a Hermite–Gaussian-like intensity distribution but still retains the orbital angular momentum. Furthermore, we derive the Fresnel diffraction integral of an HGOV and study the purity at infinity. Besides, we show a novel function of the self-measurement of the HGOV. Finally, we show that we can change the relative positions of singularities and the direction of an HGOV precisely, which facilitates applications in optical micro-manipulation.

Keywords

cross-phase Hermite-Gaussian-like optical vortex orbital angular momentum

ψ (x, y) = u (x \cos θ - y \sin θ) (x \sin θ + y \cos θ),

(1)

View in Article

ψ_{p}^{90 °} = \exp (i x^{2} \frac{k}{2 f}),

(2)

View in Article

ψ_{p}^{90 °} = \exp {\frac{ik}{2 f} [\frac{1}{2} (x^{2} + y^{2}) + \frac{1}{2} (x^{2} - y^{2})]} = \exp [\frac{ik}{2 f} \frac{1}{2} (x^{2} + y^{2})] \exp [\frac{ik}{2 f} (\frac{\sqrt{2}}{2} x - \frac{\sqrt{2}}{2} y) (\frac{\sqrt{2}}{2} x + \frac{\sqrt{2}}{2} y)],

(3)

View in Article

ψ_{p}^{90 °} = \exp [\frac{ik}{2 f} \frac{1}{2} (x^{2} + y^{2})] \exp [\frac{ik}{2 f} (x \cos \frac{π}{4} - y \sin \frac{π}{4}) (x \sin \frac{π}{4} + y \cos \frac{π}{4})] .

(4)

View in Article

ω = \sqrt{\frac{(1 + \sqrt{2}) f λ}{\sqrt{2} π}} .

(5)

View in Article

u = \frac{1 + 1 / \sqrt{2}}{ω^{2}} .

(6)

View in Article

U (x, y, 0) = \exp (- \frac{x^{2} + y^{2}}{ω^{2}}) \exp (x - i y) ψ^{'} (x, y) .

(7)

View in Article

E (x, y, z) = \frac{1}{i λ z} \exp (i k z) \exp [\frac{i k}{2 z} (x^{2} + y^{2})] F {U (x_{0}, y_{0}, 0) \exp [\frac{i k}{2 z} ({x_{0}}^{2} + {y_{0}}^{2})]},

(8)

View in Article

P_{m} = \frac{C_{m}}{\sum_{- \infty}^{+ \infty} C_{n}},

(9)

View in Article

C_{m} = \int_{0}^{\infty} 〈 a_{m} (r, φ, z) \cdot | a_{m} (r, φ, z) 〉 r d r,

(10)

View in Article

a_{m} (r, z) = 1 / {(2 π)}^{1 / 2} \int_{0}^{2 π} Ψ (r, φ, z) \exp (- i m φ) d φ,

(11)

View in Article

E (x, y, z) = \frac{1}{i λ z} \exp (i k z) \exp [\frac{i k}{2 z} (x^{2} + y^{2})] F [\exp (- \frac{x_{0}^{2} + y_{0}^{2}}{ω^{2}}) (x_{0} - i y_{0})] * F {u x_{0} y_{0} \exp [\frac{i k}{2 z} (x_{0}^{2} + y_{0}^{2})]},

(12)

View in Article

\int_{- \infty}^{\infty} x^{n} \exp (- p x^{2} + 2 q x) d x = n! \exp (\frac{q^{2}}{p}) \sqrt{\frac{π}{p}} {(\frac{q}{p})}^{n} \sum_{s = 0}^{[\frac{n}{2}]} \frac{1}{(n - 2 s)! s!} {(\frac{p}{4 q^{2}})}^{s} = \sqrt{\frac{π}{p}} \exp (\frac{q^{2}}{p}) \sum_{s = 0}^{[\frac{n}{2}]} \frac{n!}{(n - 2 s)! s!} p^{s - n} 4^{- s} q^{n - 2 s},

(13)

View in Article

E (x, y, z) = \frac{π}{2 λ z^{2} {(β α)}^{3 / 2}} (- k x α + U γ_{1} z) \exp (i k z + i k \frac{x^{2} + y^{2}}{2 z} + \frac{γ_{1}^{2}}{α} - \frac{k^{2} x^{2}}{4 β z^{2}}) + i \frac{π}{2 λ z^{2} {(β α)}^{3 / 2}} (k y α - U γ_{2} z) \exp (i k z + i k \frac{x^{2} + y^{2}}{2 z} + \frac{γ_{2}^{2}}{α} - \frac{k^{2} y^{2}}{4 β z^{2}}),

(14)

View in Article

{\begin{matrix} α = \frac{1}{ω^{2}} + k \frac{1}{2 i z} + \frac{U^{2}}{4 β} \\ β = \frac{1}{ω^{2}} + k \frac{1}{2 i z} \\ γ_{1} = \frac{1}{2} (k \frac{y}{i z} + k \frac{x U}{2 β z}) \\ γ_{2} = \frac{1}{2} (k \frac{x}{i z} + k \frac{y U}{2 β z}) \end{matrix} .

(15)

View in Article

Λ (0) = \frac{Φ}{c^{2} k} m,

(16)

View in Article

Λ (z) = \frac{Φ m}{c^{2} k} \frac{{(\frac{z}{z_{R}})}^{2} + (1 - \frac{z}{f_{x}})}{{(\frac{z}{z_{R}})}^{2} + \frac{1}{2} {(1 - \frac{z}{f_{x}})}^{2} + \frac{1}{2}},

(17)

View in Article

Figures&Tables (5)

References (19)

Copy Citation Text

Chen Wang, Yuan Ren, Tong Liu, Linlin Chen, Song Qiu. New kind of Hermite–Gaussian-like optical vortex generated by cross phase[J]. Chinese Optics Letters, 2020, 18(10): 100501

Download Citation

Set citation alerts for the article

Set citation alerts for the article

Save the article for my favorites

Paper Information

Recommended Topics

laser devices and laser physics

Lasers and Laser Optics

laser manufacturing

Instrumentation, Measurement and Metrology

Set citation alerts for the article

Please enter your email address