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.

[1] O. Svelto,

*Principles of Lasers*, (D. C. Hanna, trans. from Italian and ed.) 5th ed., New York: Springer, 2010.