# High-Power, High-Beam-Quality, Long-Pulse-Width 532 nm Laser Based on a 4f Optical System
## 2 Theoretical Analysis  
A basic 4f optical system is shown in Figure 1. Its basic structure consists of the following five parts: (1) an input plane where the optical information to be processed (such as an image or beam) is placed; (2) a lens L\({}_{1}\) with a focal length of f\({}_{1}\), which performs a Fourier transform of the light field distribution on the input plane; (3) a Fourier plane, located at the focal plane of the first lens, where the Fourier transform of the input light field is formed; (4) a lens L\({}_{2}\) with a focal length of f\({}_{2}\), which performs a Fourier transform of the light field that has been Fourier-transformed, recovering it to the spatial domain; (5) an output plane located at the focal plane of the second lens, where the processed optical information is displayed. In a 4f optical system, after passing through the first lens, the input light field forms its Fourier transform at the Fourier plane. Optical filters placed at the Fourier plane can modulate specific spatial frequency components. After passing through the second lens, the modulated light field at the Fourier plane is Fourier-transformed again, forming the output light field. The name \"4f system\" comes from its structure. If the focal length f1 of L1 is equal to the focal length f2 of L2, and the distance between them is exactly twice their focal length, the entire system's distance is 4f, which is a standard 4f system where the image and object field regions are at a 1:1 ratio. This configuration can be used for spatial filtering [14], digital image processing [15], and amplification of laser beams [16]. In such a system, it is crucial to correctly choose the focal lengths and positions of the lenses, while ensuring the lenses are stable and the optical axis is aligned, to avoid aberrations and beam distortions. This ensures that the 4f system functions as an image transfer system without introducing degradation in beam quality, and maintains the q-parameter of the input and output beams unchanged [17]. When the number of particles in the upper energy level exceeds the threshold, laser oscillation occurs. As particles in the upper level are consumed, the inversion number decreases, and oscillation stops when it falls below the threshold. Continuous pumping accumulates particles in the upper level, and once the threshold is exceeded again, a second pulse is generated, repeating this process until the pumping stops. Each pulse peak is low because it's generated near the threshold, and increasing the pumping energy only increases the number of small peaks. To enhance the peak power of the pulse, it's necessary to accumulate a large number of particles in the upper level. This can be achieved by setting the laser's threshold very high to suppress oscillation, allowing particle accumulation. When the inversion number reaches its maximum, suddenly lowering the threshold causes the accumulated particles to transition rapidly to the lower level, releasing energy in a very short time and producing a high peak power giant pulse laser output. The oscillation threshold condition of the laser can be expressed as:  
\[\Delta n_{\text{th}}\geq\frac{\text{g}}{\text{A}_{21}}.\frac{1}{\tau_{\text{c }}} \tag{1}\]  
where,  
\[\tau_{\text{c}}=\frac{\text{Q}}{2\pi\text{v}} \tag{2}\]  
Therefore,  
\[\Delta n_{\text{th}}\geq\frac{\text{g}}{\text{A}_{21}}.\frac{2\pi\text{v}}{ \text{Q}} \tag{3}\]  
In the equation, g represents the number of modes, A21 is the spontaneous emission probability, and \(\tau_{\text{c}}\) is the photon lifetime in the cavity. Q is called the quality factor, which is defined as:  
\[\text{Q}=2\pi\text{v}_{0}\bigg{(}\frac{\text{Stored energy in the cavity}}{\text{Energy loss per second}}\bigg{)} \tag{4}\]  
In the equation, \(\
u_{0}\) is the central frequency of the laser. Figure 1: 4f optical system diagram.