Propagation of Gaussian and Non-Gaussian Laser Beams through Thin Lenses

​
wavelength
633 nm
1064 nm
1550 nm
beam waist radius at position z = 0
ω
o
(mm)
0.85
beam propagation factor
2
M
5
number of lenses
one lens
two lenses
focal length of lens 1
f
1
(m)
-1
position of lens 1
z
1
(m)
0.59
focal length of lens 2
f
2
(m)
0.6
position of lens 2
z
2
(m)
1.35
plot range z = 0 to
z
max
z
max
(m)
4.2
observation point
z (m)
3.46
​​
beam parameters at the observation point
z (m)
ω (mm)
R (m)
3.46
10.8
3.29
This Demonstration shows how the laser-beam characteristics (beam radius
ω
and wavefront radius of curvature
R
) change as the beam travels through one or two thin lenses. The beam caustic (
ω
versus position
z
along the propagation direction) and
R
versus
z
depend on the incident beam parameters (wavelength
λ
, waist radius
ω
0
and propagation factor
2
M
) and the optical system's design parameters (focal length
f
and position
z
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.).

Details

The incident beam is defined with three parameters: wavelength
λ
, waist radius
ω
0
and propagation factor
2
M
. The waist is the smallest beam radius in free space propagation (without any optical elements). The waist is set at position
z=0
. The propagation factor
2
M
(also known as beam quality factor) has a value of 1 for diffraction-limited Gaussian beams and
2
M
>1
for non-Gaussian beams. The three beam parameters fully define the beam divergence
θ=
2
M
λ
πω
0
(half-angle) and the Rayleigh range
z
R
=
2
πω
0
2
M
λ
. 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 (
f
1
and
f
2
) of the lenses and their positions with respect to the incident beam's waist (
z
1
and
z
2
). The lenses can be converging (
f>0
) or diverging (
f<0
). 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
z
1
<
z
2
to ensure that lens 1 is always to the left of lens 2. Therefore, if
z
1
appears to be limited in range you should simply increase
z
2
.
The beam radius
ω
versus distance
z
from the initial waist (known as a caustic) and the radius of curvature of the wavefront
R
versus
z
are shown graphically. The wavefront is planar (
R∞
) 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 (
R>0
) for a diverging beam and convex (
R<0
) for a converging beam. To zoom in on both graphs, you can adjust the upper limit
z
max
. Furthermore, you can obtain the values of
ω
and
R
in a table below the graphs by specifying the position
z
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
f
1
or
z
1
. 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
z
2
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.

References

[1] O. Svelto, Principles of Lasers, (D. C. Hanna, trans. from Italian and ed.) 5th ed., New York: Springer, 2010.
[2] R. Paschotta. "Gaussian Beams." RP Photonics Encyclopedia. (Jul 7, 2021) www.rp-photonics.com/gaussian_beams.html.
[3] 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.

External Links

Laser (ScienceWorld)
Gaussian Beam Propagation through Two Lenses

Permanent Citation

Nicholas Barnett, , Anna Petrova-Mayor
​
​"Propagation of Gaussian and Non-Gaussian Laser Beams through Thin Lenses"​
​http://demonstrations.wolfram.com/PropagationOfGaussianAndNonGaussianLaserBeamsThroughThinLens/​
​Wolfram Demonstrations Project​
​Published: July 15, 2021