SPHERICAL ABERRATION Beam expander with spherical lenses Experiment

SPHERICAL ABERRATION Beam expander with spherical lenses Experiment to measure: Michelson interferometer

RAINBOW Ray tracing

THE HALO Around moon (Albuquerque)

THE GLORY Around the airplane shadow

THE GLORY At sunrise

THE GLORY

THE GLORY

Gaussian beams

MAXWELL BEAM PROPAGATION

(1) One can try a Gaussian: Its Fourier transform is: Insert in (1): w 02/4 has been replaced by a complex number. The inverse Fourier transform is still a Gaussian, . but with w 0 replaced by a complex number z dependent

Let us look at a simple fundamental stable laser cavity Stable = shorter than concentric Can one talk of a “plane wave at the waist? F C C F w 0 After an infinite number of round-trips: Intensity distribution: Field distribution 2 2 -r e /w 0 Field distribution w 0 -r 2/w 2 -ikr 2/2 R e we

Let us look at a simple fundamental stable laser cavity Stable = shorter than concentric F C C k F w 0 After an infinite number of round-trips: Intensity distribution: 2 2 -r Field distribution e /w 0 k-vector distribution: Field distribution w 0 -r 2/w 2 -ikr 2/2 R e we “Divergence” = width of that distribution. Uncertainty principle: the Gaussian is the least divergent beam.

(1) One can try a Gaussian: Its Fourier transform is: Insert in (1): w 02/4 has been replaced by a complex number. The inverse Fourier transform is still a Gaussian, . but with w 0 replaced by a complex number z dependent

Other approach: trial solution in he space domain, inserting the Gaussian in:

Other approach: trial solution in the space domain, inserting the Gaussian in: That is the approach chosen by Kogelnik

Substitute in Maxwell’s equation to find:

Why do we use Gaussian beams? One clue: there was no Gaussian beam before 1960. Answer 1: It is the simplest fundamental solution that matches a laser cavity Answer 2: It is the least divergent beam (uncertainty principle) Answer 3: It is the self-preserving paraxial solution to Maxwell’s wave equation

What if we do not make the paraxial approximation? Confocal hyperboloids: envelope of rays Orthogonal Ellipsoids = phase fronts Laser cavity

The ABCD matrix apply to the q parameter!

PROPAGATION 1 L 0 1 STRAIGHT INTERFACE 1 0 n n 2 n 1 2 POSITIVE LENS 1 0 - 1 1 f CURVED INTERFACE 1 n 1 - n 2 Rn 2 n 1 0 n 1 n 2 R

Geometric versus Gaussian y+da = 1 d 0 1 a y a 0 Or

Geometric versus Gaussian Finding the focus 1 d 0 1 = 1 -1/f 0 1 y 0 0 a 1 – d/f 1 -1/f 0 -1/f d 1 y

He-Ne laser

1 d 0 1 Beam expander – reducer: use ABCD Matrices 1 -1/f 0 1 y a h 1 h 2 f 1 f 1

Beam expander – reducer: use ABCD Matrices 1 d 0 1 1 -1/f 0 1 y a qi f 2 f 1 qf

Principal plane(s) 1 d 0 1 1 -1/f ? F 0 1 y a

Nodal points 1 d 0 1 1 -1/f ? 0 1 y a