Measurement of the Gaussian laser beam divergence Yasuzi Suzaki and Atsushi Tachibana

Hitachi Ltd, Totsuka Works, Totsuka-ku, Yokohama, Japan. Received 21 December 1976. It is well known that the radius w(z) of the Gaussian laser

beam is given by1

where w0 is the radius of beam waist, λ the wavelength of the laser, and z the distance from the beam waist. The beam radius w is defined as the radial distance at which light in­tensity is 1/e2 of its value on the beam axis. Figure 1 shows these relations. The gradient dw/dz of the beam radius locus at the distance z, defined as θ(z), is derived from Eq. (1) and is given as follows:

So, the beam divergence θ(∞) is obtained from Eq. (2) by settings = ∞,

If we define z~* by

z* is given by

z* means the distance from the beam waist to the position where the gradient of the beam radius locus has the value 0.99 θ( ∞). The beam divergence is usually obtained by measuring the gradient of the beam radius along the beam axis. So, to obtain the correct beam divergence with this method, the beam radius at a position far from z* must be measured. Figure 2 shows the relation of z* vs beam waist radii in the case

Fig. 1. Profile of a Gaussian beam.

Fig. 2. Distance z * vs radius of beam waist w0.

of a 0.63-μm He-Ne and a 10.6-μm CO2 laser beam. In many conventional He-Ne laser oscillators, the beam waist radii w0 are about 0.5 mm, and the beam waist locations are near the output mirror of the oscillators. In this case, z* is about 10 m. In CO2 laser radar systems, the radii of the beams trans­mitted through the optical antennas ordinarily exceed tens of millimeters. If, for example, w0 is 20 mm, then z * is about 800 m. Thus, the measurement of the beam divergence from a measurement of the variation of the beam radius along the axis may be difficult.

Here, we propose an improved method for measuring beam divergences. When a Gaussian beam passes through a lens of known focal length, the beam radius in the focal planes Wf is independent of the lens position on the beam axis.2 So Wf is obtained from the following relations1:

If the beam to be measured passes through a lens of focal length / and the beam radius in the focal plane of that lens is Wf, the beam divergence 0(∞) follows from Eq. (6). The beam divergence of 0.63-μm He-Ne laser beams was measured with

June 1977 / Vol. 16, No. 6 / APPLIED OPTICS 1481

Fig. 3. Profile of the test beam.

Fig. 4. Measured results of beam divergence 0(∞).

this method. The beam profile is as shown in Fig. 3, where the solid line shows the theoretical curve and the dots show the values. It is found from this profile that the beam waist radius is 0.167 mm, and the beam divergence θ(∞) is 1.21 × 10 - 3 rad. Then, we set a planoconvex 65-mm focal length lens on the beam axis and measured the beam radius Wf at the focal plane. The measurements were made for several positions of the lens. The beam radius Wf was measured by the scan­ning knife edge method.2,3 The results are shown in Fig. 4. The values θ(∞) obtained from Wf at different lens positions are nearly independent of the lens position, as the theory predicts, and agree well with the theoretical value. The error is less than 2% for different set positions of lens.

Using this proposed method, accurate laser beam diver­gences are easily obtained. The most conspicuous advantage of this method is that it enables us to measure the beam div­ergences by inserting a lens anywhere along the axis of the beam under test.

References 1. H. Kogelnik, Bell. Syst. Tech. J. 44, 455 (1965). 2. J. A. Arnaud, W. M. Hubbard, G. D. Mandeville, B. de la Claviere,

E. A. Franke, and J. M. Franke, Appl. Opt. 10, 2275 (1971). 3. Y. Suzaki and A. Tachibana, Appl. Opt. 14, 2809 (1975).

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