Lasers, resonators, Gaussian beams

1. Lasers, resonators, Gaussian beams

2. 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

3. w0 2 2 e-r /w 0 w0 2 2 e-r /w w 2 e-ikr /2R 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 After an infinite number of round-trips: Intensity distribution: Field distribution Field distribution

4. 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)

5. k w0 k-vector distribution: 2 2 e-r /w 0 w0 2 2 e-r /w w 2 e-ikr /2R Let us look at a simple fundamental stable laser cavity Stable = shorter than concentric F C C F After an infinite number of round-trips: Intensity distribution: Field distribution Field distribution “Divergence” = width of that distribution. Uncertainty principle: the Gaussian is the least divergent beam.

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. Transverse modes F

13. 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”?

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

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

16. An unstable cavity can generate a “Super-Gaussian”

17. 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!

18. 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

19. The Concentric cavity and the point source