210 likes | 252 Views
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. w 0. 2. 2. e -r /w. 0. w 0. 2. 2. e -r /w. w. 2. e -ikr /2R.
E N D
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
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
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 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.
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 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
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
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
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”?
“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
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!
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