Analytical Expression of Gaussian Beam Converted by Axicon-lens System

Mengshi DAI; Chongguang LI; Yuli LOU; Xiaoxue CHEN

doi:10.3788/gzxb20215007.0726001

Journals >Acta Photonica Sinica >Volume 50 >Issue 7 >Page 298 > Article

Acta Photonica Sinica
Vol. 50, Issue 7, 298 (2021)

Analytical Expression of Gaussian Beam Converted by Axicon-lens System

Mengshi DAI, Chongguang LI, Yuli LOU^*, and Xiaoxue CHEN

Author Affiliations

Key Laboratory of Laser Information Processing Technology and Application， College of Science，Kunming University of Science and Technology， Kunming650500， China

show less

DOI: 10.3788/gzxb20215007.0726001 Cite this Article

Mengshi DAI, Chongguang LI, Yuli LOU, Xiaoxue CHEN. Analytical Expression of Gaussian Beam Converted by Axicon-lens System[J]. Acta Photonica Sinica, 2021, 50(7): 298 Copy Citation Text

show less

Abstract

Based on Fraunhofer diffraction integral formula， using convolution theorem and the property of zero-order Hank transform， the analytical mathematical formula of light intensity distribution on the back focal plane of the lens after the fundamental transverse mode Gaussian beam perpendicularly enters the axicon-lens combination optical system is derived， and the mathematical relationship between the bottle beam intensity distribution and the main optical parameters in the axicon-lens combination is established， which provides theoretical support for the rapid simulation of bottle beam intensity distribution and the design of optical system for generating bottle beam. According to the derived formula， the numerical simulation and experiment are carried out， obtaining a uniformly distributed bottle beam with similar ring size， respectively， which verifies the correctness of the theoretical results.

Keywords

Analytic expression Axicon-lens combination Bottle beam Fraunhofer diffraction Gaussian beam

u (x, y) = \sqrt[]{\frac{2 P_{0}}{π ω {}^{2}}} e x p (- \frac{x^{2} + y^{2}}{ω^{2}})

(1)

View in Article

t (x, y) = \{\begin{matrix} e x p [- j k (n - 1) β \sqrt[]{x^{2} + y^{2}}] \begin{matrix} \begin{matrix}  \end{matrix} & \sqrt[]{x^{2} + y^{2}} \leq R \end{matrix} \\ 0 \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} \end{matrix} & \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} \end{matrix} & \sqrt[]{x^{2} + y^{2}} > R \end{matrix} \end{matrix} \end{matrix}

(2)

View in Article

u (x, y, 0) = A_{0} e x p (- \frac{x^{2} + y^{2}}{ω^{2}}) e x p [- j k (n - 1) β \sqrt[]{x^{2} + y^{2}}]

(3)

View in Article

u (x_{f}, y_{f}, z) = \frac{e x p (j k z)}{j λ z} e x p (j k \frac{{x_{f}}^{2} + {y_{f}}^{2}}{2 z}) \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} u (x, y, 0) e x p [- j \frac{2 π}{λ z} (x x_{f} + y y_{f})] d x d y

(4)

View in Article

\begin{array}{l} u (x_{f}, y_{f}, f) = \frac{A_{0} e x p (j k f)}{j λ f} e x p (j k \frac{{x_{f}}^{2} + {y_{f}}^{2}}{2 f}) \int_{0}^{\infty} \int_{0}^{\infty} e x p (- \frac{x^{2} + y^{2}}{ω^{2}}) e x p [- j k (n - 1) β \sqrt[]{x^{2} + y^{2}}] \\ e x p [- j 2 π (\frac{x_{f}}{λ f} x + \frac{y_{f}}{λ f} y)] d x d y \end{array}

(5)

View in Article

u (x_{f}, y_{f}, f) = c \times F \{e x p (- \frac{x^{2} + y^{2}}{ω^{2}}) e x p [- j k (n - 1) β \sqrt[]{x^{2} + y^{2}}]\}

(6)

View in Article

u (x_{f}, y_{f}, f) = \frac{c π λ ω^{2}}{(n - 1) β} e x p \{- π^{2} ω^{2} {[\frac{\sqrt[]{{x_{f}}^{2} + {y_{f}}^{2}}}{λ f} - \frac{(n - 1) β}{λ}]}^{2}\}

(7)

View in Article

I (x_{f}, y_{f}, f) = c_{1}^{2} e x p \{- 2 π^{2} ω^{2} {[\frac{\sqrt[]{{x_{f}}^{2} + {y_{f}}^{2}}}{λ f} - \frac{(n - 1) β}{λ}]}^{2}\}

(8)

View in Article

Mengshi DAI, Chongguang LI, Yuli LOU, Xiaoxue CHEN. Analytical Expression of Gaussian Beam Converted by Axicon-lens System[J]. Acta Photonica Sinica, 2021, 50(7): 298

Download Citation

Tools

Save the article for my favorites

Paper Information