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# Laser beam expanders

Laser beam expanders increase the diameter of a collimated input beam to a larger collimated output beam for applications such as laser scanning, interferometry, and remote sensing. Contemporary laser beam expanders are afocal systems developed from well-established optical telescope fundamentals. In such systems, the object rays enter parallel to the optical axis of the internal optics and exit parallel to them. This means that the entire system does not have a focal length.

## Telescope Theory

Optical telescopes, traditionally used to view distant objects such as celestial bodies in outer space, are divided into two types: refracting and reflecting. Refracting telescopes utilize lenses to refract, or bend, light, while reflecting telescopes utilize mirrors to reflect light.

There are two categories of refracting telescopes: Keplerian and Galilean. A Keplerian telescope consists of lenses with positive focal lengths separated by the sum of their focal lengths (Figure 1). The lens closest to the object being viewed, or source image, is called the objective lens, while the lens closest to the eye, or image created, is called the image lens.

##### Figure 1: Keplerian telescope

A Galilean telescope consists of a positive lens and a negative lens that are also separated by the sum of their focal lengths (Figure 2). However, since one of the lenses is negative, the separation distance between the two lenses is much shorter than in the Keplerian design. While using the effective focal length of the two lenses will provide a good approximation of the total length, using the back focal length will provide the most accurate length.

##### Figure 2: Galilean telescope

The magnifying power, or the inverse of the magnification, of a telescope is based upon the focal lengths of the objective and eye lenses:

(1)$\text{Magnifying Power}\left(\text{MP}\right)=\frac{1}{\text{Magnification}\left[\text{m}\right]}$

(2)$\text{MP}=-\frac{{\text{Focal Length}}_{\text{Objective Lens}}}{{\text{Focal Length}}_{\text{Image Lens}}}$

If the magnifying power is greater than 1, the telescope magnifies. When the magnifying power is less than 1, the telescope minifies.

## Beam Expander Theory

In a laser beam expander, the placement of the objective and image lenses is reversed. Keplerian beam expanders consist of two lenses with positive focal lengths separated by the sum of their focal lengths. They offer high expansion rations and allow for spatial filtering because the collimated input beam focuses to a spot between the objective and image lenses, producing a point within the system where the laser's energy is concentrated (Figure 3). However, this heats the air between the lenses, deflecting light rays from their optical path and potentially leading to wavefront errors especially in high-power laser applications.

##### Figure 3: Keplerian beam expanders have an internal focus which is detrimental to high power applications, but useful for spatial filtering in lower power applications

Galilean beam expanders, in which an objective lens with a negative focal length and an image lens with a positive focal length are separated by the sum of their focal lengths, are simple, lower-cost designs that also avoid the internal focus of Keplerian beam expanders (Figure 4). The lack of an internal focus makes Galilean beam expanders better suited for high-power laser applications than Keplerian designs.

##### Figure 4: Galilean beam expanders have no internal foci and are ideally suited for high power lasers applications

When using the Keplerian or Galilean designs in laser beam expander applications, it is important to be able to calculate the output beam divergence. This determines the deviation from a perfectly collimated source. The beam divergence is dependent on the diameters of the input and output laser beams.

(3)$\frac{\text{Input Beam Divergence}\left({\theta }_{I}\right)}{\text{Output Beam Divergence}\left({\theta }_{O}\right)}=\frac{\text{Output Beam Diameter}\left({D}_{O}\right)}{\text{Input Beam Diameter}\left({D}_{I}\right)}$

The magnifying power (MP) can now be expressed in terms of the beam divergences or beam diameters.

(4)$\text{MP}=\frac{{\theta }_{I}}{{\theta }_{O}}$

(5)$\text{MP}=\frac{{D}_{O}}{{D}_{I}}$

When interpreting Equation 4 and Equation 5, one can see that while the output beam diameter (D0) increases, the output beam divergence (θO) decreases and vice versa. Therefore, when using a beam expander to minimize the beam, its diameter will decrease but the divergence of the laser will increase. The price to pay for a small beam is a large divergence angle.

In addition, it is important to be able to calculate the output beam diameter at a specific working distance (L). The output beam diameter is a function of the input beam diameter and the beam divergence after a specific working distance (L) (Figure 5).

##### Figure 5: A laser's input beam diameter and divergence can be used to calculate the output beam diameter at a specific working distance

(6)${D}_{L}={D}_{O}+L\cdot \mathrm{tan}\left(2{\theta }_{O}\right)$

Laser beam divergence is specified in terms of a half angle, which is why a factor of 2 is required in the second term in Equation 6.

A beam expander will increase the input beam and decrease the input divergence by the Magnifying Power. Substituting Equations 4 and 5 with Equation 6 results in the following:

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