Lasers, resonators, Gaussian beams

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

e-r /w e-r /w e-ikr /2R Can one talk of a “plane wave at the waist? w0 Let us look at a simple fundamental stable laser cavity Stable = shorter than concentric Can one talk of a “plane wave at the waist? w0 C C F F After an infinite number of round-trips: Intensity distribution: e-r /w 2 Field distribution w0 w 2 e-r /w e-ikr /2R 2 Field distribution

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)

k e-r /w e-r /w e-ikr /2R Field distribution k-vector distribution: w0 Let us look at a simple fundamental stable laser cavity Stable = shorter than concentric w0 k-vector distribution: k C C F F After an infinite number of round-trips: Intensity distribution: e-r /w 2 Field distribution w0 w 2 e-r /w e-ikr /2R 2 Field distribution “Divergence” = width of that distribution. Uncertainty principle: the Gaussian is the least divergent beam.

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

Fraunhofer approximation: the far-field is the Fourier Transform Answer 3: It is the self-preserving paraxial solution to Maxwell’s wave equation Fraunhofer approximation: the far-field is the Fourier Transform of the field at z=0 What are the choices? Sech Gaussian Bessel beam

Class question: 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 Answer 4: any resonator mode can be a made by a superposition of Gaussians

Leads to the Hermite Gaussian modes = linear superposition Answer 4: any resonator mode can be a made by a superposition of Gaussians MAXWELL General equation: Leads to the Hermite Gaussian modes = linear superposition of Gaussians Cylindrical symmetry: leads to the Gaussian beam

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 Answer 4: any resonator mode can be a made by a superposition of Gaussians Answer 5: Resonator mode with the smallest mode volume

The mode volume is defined by: Answer 5: Resonator mode with the smallest mode volume The mode volume is defined by: This is why an aperture is used to ensure TEM00 mode

Transverse modes F

Class question: 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 Answer 4: any resonator mode can be a made by a superposition of Gaussians Answer 5: Resonator mode with the smallest mode volume Comment: What about “Super-Gaussians”?

The rays through the center reproduce themselves “Concentric” configuration: The rays through the center reproduce themselves

If the distance between mirrors is larger than twice the radius, The beams “spill over” the mirrors

An unstable cavity can generate a “Super-Gaussian”

What about N round-trips? Question: about the 1/q parameter equal to itself after 1 RT. What about N round-trips? Simple answer: instead of using the ABCD matrix for one round-trip, use the one for N round-trips. There is a difference!

yes How to calculate the location of a beam waist? More Questions… How to calculate the location of a beam waist? Location where R(z) is infinite Too much math – do we have to…??? Tip one: use 1/q rather than q parameter yes “Algebraic manipulation” sofwares –good for matrix multiplicatons Still simplifications by hand required

The Concentric cavity and the point source