(3)

where w(l) is the radius of the beam entering the lens M3, l is the distance between the lens M3 and the beam waist w0 from the laser generator, and f3 is the focal length of the lens M3.

Since w0¢ lies on the back focus plane of the lens M4 with a longer focal length, f4, the Gaussian beam with a beam waist w¢0 will be collimated by the beam expander. The collimation ratio of the beam expander for a Gaussian beam is as follows

(4)

where T1 = f4/f3. The beam waist w²0 and divergence angle q² after the beam expander are

(5)

and

(6)

Substituting Equation (1) into Equation (5), the following expression can be obtained (7)

From Equations (4)-(7), it is concluded that the beam expansion ratio and the collimation ratio for a Gaussian beam depend not only on the specifications of the beam expander, but also on the laser beam parameters as well as the positions of the optical lenses.

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