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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

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C F F C Let us look at a simple fundamental stable laser cavity Stable = shorter than concentric After an infinite number of round-trips: Intensity distribution: w0w0 e -r /w 0 2 2 Field distribution w0w0 w e -ikr /2R 2 2 e -r /w 2 Can one talk of a “plane wave at the waist?

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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)

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C F F C Let us look at a simple fundamental stable laser cavity Stable = shorter than concentric After an infinite number of round-trips: Intensity distribution: w0w0 e -r /w 0 2 2 Field distribution w0w0 w e -ikr /2R 2 2 e -r /w 2 k-vector distribution: k “Divergence” = width of that distribution. Uncertainty principle: the Gaussian is the least divergent beam.

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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

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Fraunhofer approximation: the far-field is the Fourier Transform of the field at z=0 What are the choices? Sech Gaussian Bessel beam

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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

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MAXWELL General equation: Cylindrical symmetry: leads to the Gaussian beamPages 66-69 Leads to the Hermite Gaussian modes = linear superposition of Gaussians (page 74)

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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

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The mode volume is defined by: This is why an aperture is used to ensure TEM 00 mode

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Transverse modes F

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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

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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”?

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

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

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An unstable cavity can generate a “Super-Gaussian”

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z w Phase shift on axis P = atan(z/z 0 ) If I(z) is the intensity on axis: Power = w(z) 2 2 I(z) = constant Therefore the field on axis varies as w0w0 w Self trapping Question: Why the phase factor on axis (the “P” factor)?

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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!

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

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Stephen Benson U.S. Particle Accelerator School January 17, 2011 The FEL as a Diagnostic* * This work was supported by U.S. DOE Contract No. DE-AC05-84-ER40150,

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