In many laser optics applications, the laser beam is assumed Gaussian with an irradiance profile that follows an ideal Gaussian distribution. All actual laser beams will have some deviation from ideal Gaussian behavior. The M^{2} factor , also known as the beam quality factor, compares the performance of a real laser beam with that of a diffraction-limited Gaussian beam. Gaussian irradiance profiles are symmetric around the center of the beam and decrease as the distance from the center of the beam perpendicular to the direction of propagation increases (*Figure 1*).

**Figure 1: The waist of a Gaussian beam is
defined as the location where the irradiance is 1/e**^{2}** (13.5%) of its maximum value**

Equation 1 describes this distribution:

In *Equation 1*, I_{0} is the peak irradiance at the center of the beam,

r is the radial distance away from the axis,

w(z) is
the radius of the laser beam where the irradiance is 1/e^{2} (13.5%) of I_{0},

z is the distance propagated from the plane where the wavefront is flat,

and P is the total power of the beam.

However, this irradiance
profile does not stay constant as the beam propagates through space, hence the
dependence of w(z) on z. Due to diffraction, a Gaussian beam will converge and
diverge from an area called the beam waist (w_{0}), which is where the beam diameter reaches a minimum value.
The beam converges and diverges equally on both sides of the beam waist by the
divergence angle θ (*Figure 2*). The beam waist
and divergence angle are both measured from the axis and their relationship can
be seen in *Equation 2* and *Equation 3*:

**Figure 2: Gaussian beams are defined by
their beam waist (w**_{0}**), Rayleigh range (z**_{R}**), and divergence angle (θ)**

Variation of the beam diameter in the beam waist region is defined by:

The Rayleigh range of a
Gaussian beam is defined as the value of z where the cross-sectional area of
the beam is doubled. This occurs when w(z) has increased to √2 w_{0}. Using *Equation
5*, the Rayleigh range (z_{R}) can be expressed as:

This allows w(z) to also
be related to z_{R}:

The wavefront of the
laser is planar at the beam waist and approaches that shape again as the
distance from the beam waist region increases. This occurs because the radius
of curvature of the wavefront begins to approach infinity. The radius of
curvature of the wavefront decreases from infinity at the beam waist to a
minimum value at the Rayleigh range, and then returns to infinity when it is
far away from the laser (*Figure 3*); this is true for
both sides of the beam waist.

**Figure 3:** The curvature of the wavefront of a Gaussian beam is
near-zero when it is both very close and very far away from the beam waist

## Gaussian Beam Manipulation

Many laser optics systems require manipulation of a laser beam as opposed to simply using the “raw” beam. This may be done using optical components such as lenses, mirrors, prisms, etc. Below is a guide to some of the most common manipulations of Gaussian beams.

##### The Thin Lens Equation for Gaussian Beams

The behavior of an ideal thin lens can be described using the following equation:

In* Equation
8*, s’ is the distance from the lens to the image, s is the
distance from the lens to the object, and f is the focal length of the lens. If
the object and image are at opposite sides of the lens, s is a negative value
and s’ is a positive value. This equation ignores the thickness of a real lens
and is therefore only a simple approximation of real behavior (*Figure 4*). The thin lens
equation can also be written in a dimensionless form by multiplying both sides
of the equation by f:

**Figure 4:** The thin lens equation allows the position of an image
(s’) to be determined when the distance from the lens to the object (s) and the
focal length of the lens (f) are known

In addition to describing imaging applications, the thin lens equation is applicable to the focusing of a Gaussian beam by treating the waist of the input beam as the object and the waist of the output beam as the image. Gaussian beams remain Gaussian after passing through an ideal lens with no aberrations. In 1983, Sidney Self developed a version of the thin lens equation that took Gaussian propagation into account:

The total distance from the laser to the
focused spot is calculated by adding the absolute value of s to s’. *Equation 10* can also be
written in a dimensionless form by multiplying both sides by f:

This equation approaches the standard thin lens
equation as z_{R}/f approaches 0, allowing the standard thin lens equation to
be used for lenses with a long focal length. *Equations 10* and *11* can be used
to find the location of the beam waist after being imaged through the lens (*Figure 5*).

**Figure 5:** The ”object” when refocusing a Gaussian beam is the input
waist and the “image” is the output waist

A plot of the normalized image distance (s’/f)
versus the normalized object distance (s/f) shows the possible output waist
locations at a given normalized Rayleigh range (z_{R}/f) (*Figure 6*). This plot shows
that Gaussian beams focused through a lens have a few key differences when
compared to conventional thin lens imaging. Gaussian beam imaging has both
minimum and maximum possible image distances, while conventional thin lens
imaging does not. The maximum image distance of a refocused Gaussian beam
occurs at an object distance of -(f + z_{R}), as opposed to
–f. The point on the plot where s/f is equal to -1 and s’/f is equal to 1
indicates that the output waist will be at the back focal point of the lens if
the input is at the front focal point of a positive lens.

**Figure 6: The curve where z**_{R}**/f=0 corresponds to the conventional thin lens equation. The
curves where z**_{R}**/f>0 show that Gaussian imaging has minimum
and maximum image distances defined by the Rayleigh range**

In order to understand the beam waist and Rayleigh range after the beam travels through the lens, it is necessary to know the magnification of the system (α), given by:

Where w_{0} is beam waist
before the lens and w_{0}’ is the beam waist
after the lens. The thin lens equation for Gaussian beams can then be rewritten
to include the Rayleigh range of the beam after the lens (z_{R}‘):

The above equation will break down if the lens is at the beam waist (s=0). The inverse of the squared magnification constant can be used to relate the beam waist sizes and locations:

### Focusing a Gaussian Beam to a Spot

In many applications, such as laser materials processing or
surgery, it is highly important to focus a laser beam down to the smallest spot
possible to maximize intensity and minimize the heated area. In cases such as
these, the goal is to minimize w_{0}‘ (*Figure 7*). A modified version of *Equation
14* may be used to identify how to minimize the output beam
waist:

**Figure 7:** Focusing a laser beam down to the smallest possible size
is crucial for a wide range of applications including this laser cutting setup

After multiplying both sides by the denominator
from the left side of the equation and then multiplying both sides by (w_{0}‘)^^{2}, *Equation 15* becomes:

Solving for w_{0}‘ results in:

The focused beam waist can be minimized by
reducing the focal length of the lens and |s|-f. The terms next to w_{0} in *Equation 18* are defined
as another form of the magnification constant α in order to compare the values
of the input beam to the output beam after going through the lens (*Figure 8*).

**Figure 8:** For a magnification of 2, the output beam waist will be
twice the input beam waist and the output divergence will be half of the input
beam divergence

There are two limiting cases which further
simplify the calculations of the output beam waist size and location: when s is
much less than z_{R} or much greater than z_{R}. When
the lens is well within the laser’s Rayleigh range, then s << z_{R} and (|s|-f)^{2}<〖z_R〗^{2}. *Equation 19* simplifies to:

This also simplifies the calculations for the output beam’s waist, divergence, Rayleigh range, and waist location:

The other limiting situation where the lens is
far outside of the Rayleigh range and s >> z_{R}, simplifying *Equation 19* to:

Which makes the output beam waist diameter:

Similarly to when s << z_{R}, the calculations
for the output beam waist, divergence, Rayleigh range, and beam waist location
are also simplified:

When s >> z_{R}, the distance from the lens to the focused spot equals the focal
length of the lens.

Both of these results
intuitively make sense because the beam’s wavefront is approximately planar
both at and very far away from the beam waist. At these locations, the beam is
almost perfectly collimated (*Figure 9*). According to
the standard thin lens equation, a collimated input would have an image
distance equal to the focal length of the lens.

**Figure 9:** The focused spot of a Gaussian beam after it passes
through a lens will be located at the focal point of the lens if the input beam
waist is either very close or very far away from the lens. This is because the
input beam is approximately collimated at those points

### Gaussian Focal Shift

Counterintuitively, the intensity of a focused beam in a target
at a fixed distance (L) away from the lens is not maximized when the waist is
located at the target. The intensity on the target is actually maximized when
the waist occurs at a location before the target (*Figure 10*). This
phenomenon is known as Gaussian focal shift.

**Figure 10:** The minimum beam radius at a target occurs when the waist
of the focused beam occurs at a specific location before the target, not when
focused waist is located at the target

The lengthy derivation is not covered in this text, but the beam radius at the target can be described by the following expression:

Differentiating *Equation 34* with respect
to the focal length of the focusing lens (f) and solving for f when ^{d}*⁄*_{df}* *[*w** _{L}* (

*f*)]=0 reveals the lens focal length resulting in the minimum beam radius, and therefore highest intensity, at the target.

As |s| approaches either zero or infinity, ^{d}*⁄** _{df}* [

*w*

*(*

_{L}*f*)]=0 when f=L. In both of these cases, the input beam is approximately collimated, and it thereby follows that the smallest beam radius would occur at the focal point of the lens.

### Collimating a Gaussian Beam

Achieving a truly collimated beam where the
divergence is 0 is not possible, but achieving an approximately collimated beam
by either minimizing the divergence or maximizing the distance between the
point of observation and the nearest beam waist is possible. Since the output
divergence is inversely proportional to the magnification constant α, the
output divergence reaches a minimum value when |s|= f (*Figure 11*)

**Figure 11:** To collimate a Gaussian beam, the distance from the beam
waist to the collimating lens should be equal to the lens’ focal length

*References*

*“Gaussian Beam Optics.” CVI Laser Optics, IDEX Optics & Photonics.**O’Shea, Donald C. Elements of Modern Optical Design. Wiley, 1985.**Self, Sidney A. “Focusing of Spherical Gaussian Beams.” Applied Optics, vol. 22, no. 5, Jan. 1983.*