# Laser Beams And Resonators

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Information about Laser Beams And Resonators
Technology

Published on February 13, 2009

Author: mehmetdeveci

Source: slideshare.net

## Description

graduate degree project of laser beams and resonators

Mode Matching of Lasers to External Resonators Mehmet Deveci Review of the theory of laser beams and resonators

Mehmet Deveci

Introduction Fabry-perot Interferometer as a Laser Resonator The modes in a optical structure Resonators with Spherical Mirrors

Fabry-perot Interferometer as a Laser Resonator

The modes in a optical structure

Resonators with Spherical Mirrors

Paraxial Rays N ear the axis of an optical system

N ear the axis of an optical system

Ray Transfer Matrices

Wave Analysis of Beams and Resonators Is it plane wave? The Scalar Wave Equation : For light traveling in the z-direction : Solving them gives : (Similar to time-dependent Schrödinger equation) P(z) : complex phase shift q(z) : complex beam parameter (Gaussian variation in the beam intensity) Solution of above equation :

Is it plane wave?

Propogation Laws For Fundamental Mode The general equation : Two real beam parameters are introduced; R and w R: radius of the field w: measure of decrease of the field amplitude Fundamental mode

The general equation :

Amplitude distribution of the fundamental beam Distance at which 1/e times amplitude on the axis w: beam radius or spot size 2w: beam diameter

Distance at which 1/e times amplitude on the axis

w: beam radius or spot size

2w: beam diameter

Contour of a Gaussian Beam Minimum diameter at the beam waist A distance z away from the waist

Minimum diameter at the beam waist

Expansion of the beam and Equating the real and imaginary parts of : we get ;

and

Gaussian Beams waist 2w 0 2√2 w 0 z R

Higher Order Modes There are other solutions of A solution for general wave equation : Inserting above equation to general equation we get ; Hermite Polynomial of order m g: function of x and z h: function of y and z

There are other solutions of

Transverse mode numbers m and n Hermite Polynomials Phase shift

Transverse mode numbers m and n

Higher-Order Modes - HG H n (x) 2 1 2 3 4 H n (x) e -x /2 2 H n (x) e -x /2 2

Beam Transformation by a Lens Focusing a Laser Beam Producing a beam of suitable diameter and phase front curvature Ideal Lens leaves unchanged However a lens does change the parameters R(z) and w(z) What is the relationship between incoming and outgoing parameters?

Focusing a Laser Beam

Producing a beam of suitable diameter and phase front curvature

Ideal Lens leaves unchanged

However

a lens does change the parameters R(z) and w(z)

What is the relationship between incoming and outgoing parameters?

If q’s are measured at distance d 1 and d 2 Beam Transformation by a Lens

Beam Transformation by a Lens

Application

Seperating the real and imaginary part the condition is, obviously, f >z 0 .

Seperating the real and imaginary part

Laser Resonators Self consistency requires q 1 =q 2 =q

Self consistency requires

q 1 =q 2 =q

Laser Resonators R is equal to the radius of curvature of the mirrors The widt of the fundamental mode is; Beam radius w 0 in the center of the resonator, z=d/2

R 1 q: number of nodes m and n: rectangular mode numbers R 2 Resonance occurs when the phase shift fron one mirror to other is a multiple of π the frequency spacing between successive longitudinal resonance:

R 1

Mode Matching Modes of Laser Resonators can be characterised by light beams These beams are often injected to other optical structures with different sets of beam parameters These optical structures can assume various physical forms To match the modes of one structure to those of another we need to transform a gicen Gaussian beam

Modes of Laser Resonators can be characterised by light beams

These beams are often injected to other optical structures with different sets of beam parameters

These optical structures can assume various physical forms

To match the modes of one structure to those of another we need to transform a gicen Gaussian beam

where

Formulas for the confocal parameter and the location of beam waist

The confocal parameter b 2 as a function of the lens waist

Conclusion It was necessary for length measurement in metrology and calibration to concentrate the discussion of this work on the basic aspects of laser beams and resonators. A review of the theories from 1960’s and our contribution is done eachother Thank you For Your Interest

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