Propagation of Gaussian and Non-Gaussian Laser Beams through Thin Lenses
This Demonstration shows how the laser-beam characteristics (beam radius and wavefront radius of curvature ) change as the beam travels through one or two thin lenses. The beam caustic ( versus position along the propagation direction) and versus depend on the incident beam parameters (wavelength , waist radius and propagation factor ) and the optical system's design parameters (focal length and position of the lenses with respect to the incident beam waist). You can study the effect of each variable and manually optimize the optical system to achieve a desired output (i.e. particular focal spot size at a given distance from the lens, collimated beam, etc.).
The incident beam is defined with three parameters: wavelength , waist radius and propagation factor . The waist is the smallest beam radius in free space propagation (without any optical elements). The waist is set at position . The propagation factor (also known as beam quality factor) has a value of 1 for diffraction-limited Gaussian beams and for non-Gaussian beams. The three beam parameters fully define the beam divergence (half-angle) and the Rayleigh range . The beam propagation can be studied after a single thin lens (lens 1) or a combination of two thin lenses (lens 1 followed by lens 2). You can control the focal lengths ( and ) of the lenses and their positions with respect to the incident beam's waist ( and ). The lenses can be converging () or diverging (). The sign of the focal length (and the shape of the lens) changes automatically as the respective slider is adjusted. Alternatively, you can type the focal length (click the "+" to the right of the slider) with a "-" sign for diverging lens and no sign for converging lens. The positions of the lenses is preset to to ensure that lens 1 is always to the left of lens 2. Therefore, if appears to be limited in range you should simply increase .
The beam radius versus distance from the initial waist (known as a caustic) and the radius of curvature of the wavefront versus are shown graphically. The wavefront is planar () for a collimated beam and at the beam's waist (waist of the incident beam and any waist, or focal spot, formed with lenses). The wavefront is concave () for a diverging beam and convex () for a converging beam. To zoom in on both graphs, you can adjust the upper limit . Furthermore, you can obtain the values of and in a table below the graphs by specifying the position of an observation point.
All dynamic variables have the option for animation (click the "+" to the right of the slider and select the "Play" button). The "Play" button can be used for a quick observation of the effect of one design parameter on the beam propagation. For example, if the goal is to focus the beam down to a specific spot size with a single lens (snapshot 1), you can animate either or . You can alter the spot size with the addition of a second lens (snapshot 2). Similarly, if the goal is to collimate the beam with a given set of lenses (snapshot 3) you could vary the lens separation (set in play mode) and observe the caustic. Animating the "observation point" allows you to quickly check the position where a desired beam size is achieved (i.e. to place a detector) or simply to scan the beam as it travels through the optical system.
 O. Svelto, Principles of Lasers, (D. C. Hanna, trans. from Italian and ed.) 5th ed., New York: Springer, 2010.
 R. Paschotta. "Gaussian Beams." RP Photonics Encyclopedia. (Jul 7, 2021) www.rp-photonics.com/gaussian_beams.html.
 R. Hinton. "Laser Beam Quality: Beam Propagation and Quality Factors: A Primer." Laser Focus World. (Jul 7, 2021) www.laserfocusworld.com/lasers-sources/article/14036821/beam-propagation-and-quality-factors-a-primer.