# 光束质量和斯特列尔比

This is Sections 6.2, 6.3, 6.4, and 6.5 of the Laser Optics Resource Guide.

In order to accurately predict the real-world performance and quality of a laser, it is necessary to understand the laser’s M2 factor, which describes the quality of the beam. 。一旦了解了激光器的性能，定义与激光器一起使用的任何光学系统的真实性能将有助于了解最终系统的性能。将光学系统的实际性能与其理想性能进行比较，利用斯特列尔比使性能达到衍射极限。

## M2 因子

(1)$$M^2 = \frac{\pi w_0 \theta}{\lambda}$$

In Equation 1, w0 is the beam waist, θ is the divergence angle of the laser, and λ is the lasing wavelength (Figure 1). As defined in our Gaussian Beam Propagation application note, the divergence angle of a Gaussian beam is determined by the following equation:

(2)$$\theta = \frac{\lambda}{\pi w_0}$$

(3)$$M^2 = \frac{\pi w_0}{\lambda} \times \frac{\lambda}{\pi w_0} = 1$$
##### 图 1: 激光束发散角和束腰图解

Along with a laser beam’s optical power, the M2 factor determines the radiance of the beam. The M2 factor can also be used to approximate the radius of a beam as it propagates by replacing the wavelength of a laser with the wavelength multiplied by the M2 factor found in all of the equations in the Gaussian Beam Propagation application note.3

M2 因子很重要，因为它表示激光束在给定发散度下的聚焦能力。M2 因子越低，激光器的聚焦越精密，光束内功率的利用率越高，潜在有效功率越高。

(4)$$w \! \left( z \right) ^2 = w^2 _0 \left[ 1 + \left(z - z_0 \right)^2 \left( \frac{M^2 \lambda}{\pi w_0 ^2} \right)^2 \right]$$

## 光束参数乘积

(5)$$\text{BPP} = \frac{M^2 \lambda}{\pi}$$

## 桶中功率

(6)$$\text{Vertical Beam Quality} = \sqrt{\frac{\text{Ideal Power in Bucket}}{\text{Actual Power in Same Bucket}}}$$
(7)$$\text{Horizontal Beam Quality} = \frac{\text{Actual Beam Radius at a Given Power}}{\text{Ideal Beam Radius at the Same Power}}$$

## 斯特列尔比

##### 图 6: 该透镜的斯特列尔比为 0.826，由于大于 0.8，因此被认为是衍射极限

(8)$$S = \exp{\left[ - \left( 2 \pi \sigma \right)^2 \right]}$$

1. International Organization for Standardization. (2005). Lasers and laser-related equipment – Test methods for laser beam widths, divergence angles and beam propagation ratios (ISO 11146).
2. A. E. Siegman, “New developments in laser resonators”, Proc. SPIE 1224, 2 (1990)
3. Paschotta, Rüdiger. Encyclopedia of Laser Physics and Technology, RP Photonics, October 2017, www.rp-photonics.com/encyclopedia.html.
4. International Organization for Standardization. (2005). Lasers and laser-related equipment — Test methods for laser beam widths, divergence angles and beam propagation ratios — Part 1: Stigmatic and simple astigmatic beams (ISO 11146-1:2005).
5. A. Siegman, “’Non-Gaussian’ Beam”, OSA Annual Meeting, Long Beach, CA (1997)
6. Hofer, Lucas. “M² Measurement.” DataRay Inc., 12 Apr. 2016, www.dataray.com/blog-m2-measurement.html.
7. Strehl, Karl W. A. “Theory of the telescope due to the diffraction of light,” Leipzig, 1894.
8. Mahajan, Virendra N. "Strehl ratio for primary aberrations in terms of their aberration variance." JOSA 73.6 (1983): 860-861.

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