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Beam Quality and Determination of M^² 

Background

The design of optical delivery systems for laser systems is highly dependent on the laser's beam quality. The beam quality, given by the parameter M², ranges from 1 for a diffraction-limited TEM₀₀ laser beam, to several hundred for a distorted, poor quality beam. The National Institute for Standards and Technology (NIST) and the International Standards Organization (ISO) are in the process of establishing standards for laser beam quality measurement. The proposed standards are based upon calculating M² from a set of beam diameter measurements. Several diameter measurement techniques have been endorsed by NIST. One of these methods, which has been adopted by U.S. Laser, involves the use of a CCD camera near the focus of an imaging lens. The beam waist diameter and Rayleigh range are measured, and are then used to calculate M². The technique is similar to focal plane divergence measurement technique with two major differences: (1) two diameters (instead of one) are taken at different locations along the beam; (2) the measurement of the actual beam waist and its position, not the diameter at the ideal waist position is used.

Definition of M²

The beam quality M² is the ratio of the laser beam's multimode diameter-divergence product to the ideal diffraction limited (TEM₀₀) beam diameter-divergence product. It can also be given by the square of the ratio of the multimode beam diameter to the diffraction-limited beam diameter.

In the equation above, D_m is the measured beam waist diameter, Θ_m is the measured full-angle divergence, d₀ is the theoretical "imbedded Gaussian" beam diameter, and θ₀ is the theoretical diffraction-limited divergence. The diameter-divergence products are given in units of mm•mrad. In addition to the quantities above we will use a quantity known as the Rayleigh range, denoted by z_R. The Rayleigh range is the distance a beam must propagate for its diameter to grow by a factor of factor . See Figure 1 for an illustration.

Selection of an Imaging Lens

It is important that two conditions are met by the lens:

1. The f-number, defined as the lens focal length in mm divided by the beam diameter at the lens in mm is at least 10, and preferably over 20. Alternatively, aberration corrected optics can be used.

2. The lens should be selected such that the focused beam diameter (times the square root of 2) on the CCD is as large as possible without overfilling. Overfilling the CCD array will result in false diameter readings. Underfilling the CCD will result in a loss of image resolution. A good rule of thumb is to have the CCD approximately ½ to ⅔ filled.

Note: For M² measurements the longest focal length lens that does not overfill the CCD camera should be used. Longer focal length lenses afford more precise Rayleigh range measurement. Better Rayleigh range measurements, in turn, enhance the quality of M² measurements.

Setup and Procedure

A schematic of the setup for M² measurements is shown below in Figure 1. The procedure for measuring M² is as follows:

1. Locate and measure the beam focus using the following steps:

C Move the CCD array along the axis and estimate the smallest spot size found (measure only the x-diameter now, y will be done later). Write down the estimated waist size.

C Find the positions on either side of the estimated focus where spot size is twice the estimated waist size.

C Locate the position of the actual focus, which lies exactly halfway between these two points.

C Move the position of the camera to the actual focus position. Record the focus diameter and scale position as D_m and Z₁.

2. Find the Rayleigh Range, Z_R. This is done by finding the place where the spot size grows to 1.414 times the focused spot size. Move the CCD camera toward the lens until this spot size is found. Record the new scale position (Z₂). The Rayleigh range Z_R is given by Z_R=|Z₁-Z₂|.

3. Calculate the imbedded Gaussian beam diameter using the relationship.

4. Find M² using the relationship

5. If the beam divergence has been previously measured, the diameter of the spot at the laser output can be calculated. If λ is the laser wavelength in microns and θ is the full-angle beam divergence in milliradians, the output diameter is given by the relation

Figure1. Setup for laboratory measurement of beam quality

References

1. Laser Far-Field Beam Profile Measurements by the Focal Plane Technique, National Bureau of Standards, March 1978.

2. Simple Beam Propagation Measurements on Ion Lasers, SPIE Vol. 1414 Laser Beam Diagnostics, 1991.
3. Beam Characterization and Measurement of Propagation Attributes, SPIE Vol. 1414 Laser Beam Diagnostics, 1991.
4. Standard for the Measurement of Beam Widths, Beam Divergence, and Propagation Factors, Proposal for a Working Draft, ISO, September, 1991.
5. Tech Note - Measuring Laser Beam Divergence, U.S. Laser Corporation, 1995.

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