53 EXAMPLE GAUSSIAN BEAM FOCUSED THROUGH LENS z1 z2 d 01 0 1 cm 0 7000 n 1 f 20 cm f A Gaussian beam passes through a lens of focal length f at the waist of the beam The lens focuses the beam into a second Gaussian beam Where is the location of the second waist The transformation matrix for z1 z2 is 0 1 d d 1 f 1 1 1 1 f f free space lens 1 ABCD 0 where z2 z1 d Setting z1 0 z2 d The initial Gaussian beam at z1 has 201 n W1 0 R1 1 0 R1 R2 1 0 R2 The focused Gaussian beam with waist at z2 has W2 202 n 0 d 1 54 The ABCD law for Gaussian beams has 1 R2 C A A D R1 B R1 B R1 2 B W1 BD W12 2 where R1 0 1 0 B BD D BD A 2 CA 2 0 Therefore C R R W1 1 1 W1 where A 1 d f C B d 1 f D 1 1f 1 df d2 0 W1 d f 1 f2 W12 f 1 f 2 20 4 2 2 01 n The beam focuses at a point slightly shorter than the focal length d 20 cm 1 20 20 7000 10 8 2 0 1 4 2 1 2 20 cm 1 002 What is the spot size From the ABCD formulation 1 1 W2 W1 1 W1 DA BC A B R1 2 B W1 2 1 DA BC W 2 1 A B2 W1 1 df df 1 1 2 2 2 2 W 1 df Wd 1 1 df Wd 1 1 Since d f we can approximate 2 1 1 W1 W1 W2 W1 f 2 f2 202 n f 2 2 0 0 01 n 02 f2 W2 W 1 f 02 0 n 01 20 cm 7000 10 8 cm 4 45 10 cm 45 m 0 1 cm 1 55 The lens focused 01 02 100 m 45 m To focus to a small spot size 1 0 small 2 f small 3 01 large The maximum possible spot size fills the lens If the diameter of the lens is D then 01 D 2 f 0 F 2 0 02 D 2 n n f is the f number of lens where F D low Divergence Large F 02 large large Divergence Small F 02 small