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Journals >Chinese Optics Letters >Volume 21 >Issue 10 >Page 102601 > Article

Chinese Optics Letters
Vol. 21, Issue 10, 102601 (2023)

Generation of shaping nondiffracting structured caustic beams on the basis of stationary phase principle

Rijian Chen, Yile Shi, Ning Gong, Yefeng Liu, and Zhijun Ren^*

Author Affiliations

Key Laboratory of Optical Information Detecting and Display Technology, Zhejiang Normal University, Jinhua 321004, China

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DOI: 10.3788/COL202321.102601 Cite this Article Set citation alerts

Rijian Chen, Yile Shi, Ning Gong, Yefeng Liu, Zhijun Ren. Generation of shaping nondiffracting structured caustic beams on the basis of stationary phase principle[J]. Chinese Optics Letters, 2023, 21(10): 102601 Copy Citation Text

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Abstract

On the basis of the stationary phase principle, we construct a family of shaping nondiffracting structured caustic beams with the desired morphology. First, the analytical formula of a nondiffracting astroid caustic is derived theoretically using the stationary phase method. Then, several types of typical desired caustics with different shapes are numerically simulated using the obtained formulas. Hence, the key optical structure and propagation characteristics of nondiffracting caustic beams are investigated. Finally, a designed phase plate and an axicon are used to generate the target light field. The experimental results confirm the theoretical prediction. Compared with the classical method, the introduced method for generating nondiffracting caustic beams is high in light-energy utilization; hence, it is expected to be applied conveniently to scientific experiments.

Keywords

axicon nondiffracting structured caustic beams phase plate stationary phase method

r_{c} (φ) = \frac{1}{k_{⊥}} [Φ^{''} (φ) μ (φ) - Φ^{'} (φ) μ^{'} (φ)],

(1)

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r_{c} (φ) = \frac{1}{k_{⊥}} [\begin{matrix} (3 q - 2 q \sin^{2} φ) \sin φ \\ (3 q - 2 q \cos^{2} φ) \cos φ \end{matrix}] .

(2)

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Φ (φ) = - \frac{q}{2} \sin (2 φ),

(3)

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U (ρ, θ, z) = \frac{- i}{λ z} \exp (i k z) \int_{0}^{\infty} \int_{0}^{2 π} U_{0} (r, φ) \exp [\frac{i k}{2 z} (r^{2} + ρ^{2})] \times \exp [- \frac{i k}{z} ρ r \cos (φ - θ)] r d r d φ,

(4)

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U_{0} (r, φ) = A (φ) T (r),

(5)

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T (r) = {\begin{cases} \exp [- i k (n_{0} - 1) θ_{0} r], & r \leq R, \\ 0, & r \geq R, \end{cases}

(6)

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U (ρ, θ, z) = \frac{- i}{λ z} \exp (i k z) \exp (\frac{i k}{2 z} ρ^{2}) \times \int_{0}^{R} r d r \exp [- i k (n_{0} - 1) θ_{0} r] \exp (\frac{i k}{2 z} r^{2}) \times \int_{0}^{2 π} d φ \exp [i Φ (φ)] \exp [- \frac{i k}{z} ρ r \cos (φ - θ)] .

(7)

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\exp (i z \cos θ) = \sum_{m = - \infty}^{\infty} {(i)}^{m} J_{m} (z) \exp (i m θ),

(8)

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\exp [- \frac{i k}{z} ρ r \cos (φ - θ)] = \sum_{m = - \infty}^{\infty} {(- i)}^{m} J_{m} (\frac{k}{z} ρ r) \times \exp [- i m (φ - θ)] .

(9)

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U (ρ, θ, z) = \frac{- i}{λ z} \exp (i k z) \exp (\frac{i k}{2 z} ρ^{2}) \times \sum_{m = - \infty}^{\infty} {(- i)}^{m} \exp (i m θ) \int_{0}^{2 π} d φ \exp {i [Φ (φ) - m φ]} \times \int_{0}^{R} r d r J_{m} (\frac{k}{z} ρ r) \exp {i k [\frac{r^{2}}{2 z} - (n_{0} - 1) θ_{0} r]} .

(10)

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U (ρ, θ, z) \approx \frac{- i k_{r} \sqrt{λ z}}{λ k} \exp (i k z) \exp (\frac{i k}{2 z} ρ^{2}) \times \exp {i [\frac{- k {(n_{0} - 1)}^{2} θ_{0}^{2} z}{2} + \frac{π}{4}]} \times \sum_{m = - \infty}^{\infty} {(- i)}^{m} J_{m} (k_{r} ρ) \exp (i m θ) \times \int_{0}^{2 π} d φ \exp {i [Φ (φ) - m φ]},

(11)

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U (ρ, θ, z) = \frac{- i k_{r} \sqrt{λ z}}{λ k} \exp (i k z) \exp (\frac{i k}{2 z} ρ^{2}) \times \exp {i [\frac{- k {(n_{0} - 1)}^{2} θ_{0}^{2} z}{2} + \frac{π}{4}]} \times \sum_{m = - \infty}^{\infty} {(- i)}^{m} J_{m} (k_{r} ρ) \exp (i m θ) \times \int_{0}^{2 π} d φ \exp {i [- \frac{q}{2} \sin (2 φ) - m φ]} .

(12)

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\int_{0}^{π} d φ \exp [2 i m φ + i b \sin (2 φ)] = π {(- 1)}^{m} J_{m} (b) .

(13)

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\int_{0}^{2 π} d φ \exp {i [- \frac{q}{2} \sin (2 φ) - m φ]} = π [1 + \exp (- i m π)] \times {(- 1)}^{\frac{- m}{2}} J_{\frac{- m}{2}} (- \frac{q}{2}) .

(14)

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U (ρ, θ, z) = \frac{- i k_{r} \sqrt{λ z}}{2} \exp (i k z) \exp (\frac{i k}{2 z} ρ^{2}) \times \exp {i [\frac{- k {(n_{0} - 1)}^{2} θ_{0}^{2} z}{2} + \frac{π}{4}]} \times \sum_{n = - \infty}^{\infty} J_{2 n} (k_{r} ρ) \exp (2 i n θ) \times [1 + \exp (- 2 i n π)] J_{- n} (- \frac{q}{2}) .

(15)

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I (ρ, θ, z) = U (ρ, θ, z) U^{*} (ρ, θ, z),

(16)

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References (43)

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Rijian Chen, Yile Shi, Ning Gong, Yefeng Liu, Zhijun Ren. Generation of shaping nondiffracting structured caustic beams on the basis of stationary phase principle[J]. Chinese Optics Letters, 2023, 21(10): 102601

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