# Beam Expanders

A laser beam expander is designed to increase the diameter of a collimated input beam to a larger collimated output beam. Beam expanders are used in applications such as laser scanning, interferometry, and remote sensing. Contemporary laser beam expander designs are afocal systems that developed from well-established optical telescope fundamentals. In such systems, the object rays, located at infinity, enter parallel to the optical axis of the internal optics and exit parallel to them as well. This means that there is no focal length to the entire system.

## Theory: Telescopes

Optical telescopes, which have classically been 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.

Refracting telescopes fall into two categories: Keplerian and Galilean. A Keplerian telescope consists of positive focal length lenses that are separated by the sum of their focal lengths (Figure 1). The lens closest to the source image, or the object being viewed, 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 length (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. Please note that using the Effective Focal Length of the two lenses will give a good approximation of the total length, while using the Back Focal Length will give the most accurate answer.

**Figure 2:** Galilean Telescope

The Magnifying Power or the inverse of the magnification of the telescope is based upon the focal lengths of the objective and eye lenses.

**(1)**

**(2)**

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

## Theory: Laser Beam Expanders

In a laser beam expander design, the placement of the objective and image lenses is reversed. In the Keplerian beam expander design, 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). The focused spot heats the air between the lenses, deflecting light rays from their optical path, which can potentially lead to wavefront errors. For this reason, most beam expanders utilize the Galilean beam expander design or a variant of it (Figure 4).

**Figure 3:** Keplerian Beam Expander

**Figure 4:** Galilean Beam Expander

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

**(3)**

The Magnifying Power (MP) can now be expressed in terms of the Beam Divergences or Beam Diameters.

**(4)**

**(5)**

Interpreting the above equations, one sees that while the Output Beam Diameter (D_{O}) increases, the Output Beam Divergence (θ_{O}) decreases and vice versa. Therefore, if you use the beam expander as a beam minimizer, the beam 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 to the above, 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:** Calculating the Output Beam Diameter at a Specific Working Distance

**(6)**

Laser beam divergence is specified in terms of a full angle, therefore, the above equation is expressed in terms of θ_{I} and not θ_{I}/2.

Since a beam expander will increase the input beam by the Magnifying Power and decrease the input divergence by it as well, substituting equations (4) and (5) into (6) results in the following.

**(7)**

**(8)**

## Application Examples

### Example 1

Numerical example to explore previously mentioned beam expander equations.

**Initial Parameters**

Beam Expander Magnifying Power = MP = 10X

Input Beam Diameter = 1mm

Input Beam Divergence = 1mrad

Working Distance = L = 100m

**Calculated Parameter**

Output Beam Diameter

**(9)**

Compare this to the Beam Diameter without using a beam expander by using equation (6).

**(10)**

Although a beam expander will increase the input laser beam by a specific expansion power, it will also decrease the divergence by the same expansion power, resulting in a smaller collimated beam at a large distance.

### Example 2

Theoretical example for reducing a laser beam's divergence at a long working distance using a beam expander.

In addition to improving beam collimation, beam expanders can be used to focus laser beams. The following table shows simulated focusing performance for the 5X, 10X and 20X beam expanders. The spot sizes are given in units of microns and are calculated using a 0.63mm diameter laser beam at 632.8nm assuming M^{2}=1 and a perfectly collimated input beam.

Distance | Beam Expander Power | ||
---|---|---|---|

5X | 10X | 20X | |

1.2m | 439.19μm | 219.63μm | 111.04μm |

1.5m | 559.62μm | 279.84μm | 141.47μm |

2.5m | 961.07μm | 480.54μm | 242.89μm |

5.0m | 1964.86μm | 982.26μm | 496.36μm |

10m | 3973.17μm | 1985.49μm | 1002.87μm |

**Note:** The 1/e^{2} spot diameters listed were calculated from the equation: 2 * f/# * wavelength, where f/# is the working f/#.

## Edmund Optics Products

The TECHSPEC^{®} LC Fixed YAG Beam Expanders are available for applications where cost is the driving factor. Featuring 2-element Galilean designs with diffraction limited performance at YAG wavelengths, the LC beam expanders offer a variety of magnification ranges from 2X to 10X ideal for prototyping and OEM integration.

The TECHSPEC^{®} DA Fixed YAG Beam Expanders provide excellent value with λ/10 performance at design wavelength for apertures up to 4 mm. Featuring laser line V-coats for Nd:YAG harmonics down to 266 nm, these Galilean Designs use Fused Silica elements and provide divergence adjustability.

Examples of the application of the Galilean telescope design to laser beam expanders can be found in several Edmund Optics products, all of which can be used to collimate and focus laser beams. Our TECHSPEC^{®} DA Fixed HeNe Beam Expanders is a simple two-lens design, consisting of a negative lens and achromatic lens. Drawing of the internal optical elements is shown for reference.

Our TECHSPEC^{®} UDA Fixed Broadband Beam Expanders improves upon the simple two-lens design with a proprietary multi-element lens design that enhances its ability to create a collimated or focused laser beam diameter at a long working distance.

The TECHSPEC^{®} Research-Grade Variable Beam Expanders, ideal for high power laser applications where magnification changes may be required, feature λ/4 transmitted wavefront, Galilean designs, and high laser damage threshold anti-reflection coatings to ensure maximum transmittance while minimizing ghost reflections.

### References

- Greivenkamp, John E.
*Field Guide to Geometrical Optics*. Vol. FG01. Bellingham, WA: SPIE—The International Society for Optical Engineers, 2004. - Smith, Warren J.
*Modern Optical Engineering*. 3rd ed. New York, NY: McGraw-Hill Education, 2000.