1 / 19

Class question: Why do we use Gaussian beams?

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. w 0. 2. 2. e -r /w. 0. w 0. 2. 2. e -r /w. w. 2. e -ikr /2R.

rasha
Download Presentation

Class question: Why do we use Gaussian beams?

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 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

  2. 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

  3. 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)

  4. 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.

  5. 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

  6. 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

  7. 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

  8. 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 (page 74) Cylindrical symmetry: leads to the Gaussian beam Pages 66-69

  9. 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

  10. 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

  11. Transverse modes F

  12. 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 Answer 6: How good an approximation to the exact solution (non paraxial) to the wave equation? It has to do with the wavelength being considerably smaller than the resonator. Therefore the waist is also small, and the paraxial approximation holds

  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. w p w(z)2 = constant I(z) Power = 2 z w0 w Question: Why the phase factor on axis (the “P” factor)? If I(z) is the intensity on axis: Phase shift on axis Therefore the field on axis varies as P = atan(z/z0) Self trapping

  18. 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!

  19. 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

More Related