The propagation problem of light can be solved by the Fresnel or Fraunhofer diffraction integral[
Chinese Optics Letters, Volume. 13, Issue Suppl., S21408(2015)
Analysis of laser beam propagation properties with the Collins formula in a focused laser system
A Collins formula method with a scaling factor between the target and source plane is proposed for laser propagation in optical system design, which can be used to evaluate laser optical system performance and tolerance analysis. The laser propagation in optical systems can be calculated by the Collins integral formula, and an angular spectrum method is derived by coordinate substitution. It is introduced a scaling factor
The propagation problem of light can be solved by the Fresnel or Fraunhofer diffraction integral[
The Collins formula is as follows[
From the convolution theorem, the solution of Eq. (
Sign up for Chinese Optics Letters TOC Get the latest issue of Advanced Photonics delivered right to you!Sign up now
The coordinate after scaling is
Therefore,
Providing a real optical system, after a distance propagation, the matrix
The optical intensity distribution is shown in Fig.
Figure 1.Coma wavefront error and the resulting laser spot distribution with different scaling factors.
Figure 2.Astigmatism wavefront error and the resulting laser spot distribution with different scaling factors.
The optical system design software Zemax is used to design the laser focus system. However, the software is based on ray tracing, such that it cannot analyze the optical intensity distribution after laser beam propagation through an aberration optical system. If we take the aforementioned Collins formula [Eq. (
Figure 3.Ensemble framework of our laser optical system.
Figure
Figure 4.Wavefront with tolerance and the resulting laser spot distribution and PIB curve.
The distance of the lens can be adjusted to compensate the focus shift and calibrated by a 1000 Ω electric displacement meter. The calibrating accuracy is 5 μm and the range is from
Figure 5.Laser spot radius with compensation error.
When the RMS wavefront errors are equal, the spot radius with negative focus compensation is larger than the radius with positive focus compensation. This can be explained by Fig.
Figure 6.PIB curve with positive compensation error and negative compensation error.
The experimental setup is shown in Fig.
Figure 7.Experimental setup.
The light spot is focused at 90 m from the laser source through adjusting the distance between the front lens and the back lens. The recording device is a CCD detector with pixel size
Figure 8.Measured laser spot radius at the distance of 90 m.
In laser propagation, the optical design software Zemax cannot fully evaluate the optical intensity distribution because of the diffraction effects and the aberration. Furthermore, the laser beam is like a Gaussian profile, not a plane wave. A method that uses the Collins formula with a scaling factor is presented in this Letter to compensate this shortcoming of optical design. The optical aberration is readily added into the integral to obtain the focused light spot intensity distribution over an arbitrary distance. An example of a focus system is provided to focus a laser beam from 30 to 100 m. The theoretical results show that the spot radius is 1.05 mm at the distance 90 m, which agrees well with the experimental results. This verifies the effectiveness of our algorithm, which can be applied in the design of a laser propagation system.
[1] X. H. Zhong. Foundation of Modern Optics, 60(2011).
[2] A. E. Siegman. Lasers, 30(1986).
[4] B. D. Lu. Laser Optics: Laser Beam Propagation and Beam Quality Control, 54(1992).
[5] X. L. Ji, X. Y. Tao, B. D. Lu. Acta Phys. Sin., 53, 952(2004).
[6] R. K. Singh, P. Senthilkumaran, K. Singh. J. Opt. Soc. Am. A, 25, 1307(2008).
[7] C. J. R. Sheppard, P. Torok. J. Opt. Soc. Am. A, 20, 2156(2003).
Get Citation
Copy Citation Text
RuHai Guo, "Analysis of laser beam propagation properties with the Collins formula in a focused laser system," Chin. Opt. Lett. 13, S21408 (2015)
Category: Lasers and Laser Optics
Received: Jan. 5, 2015
Accepted: Mar. 10, 2015
Published Online: Aug. 8, 2018
The Author Email: RuHai Guo (hitgrh@163.com)