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. 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

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 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.

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. 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

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. 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

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. 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

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. The rays through the center reproduce themselves
“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. 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!

18. 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

19. The Concentric cavity and the point source