Single-Molecule Microscopy Group

Gaussian Beams

Physical Optics

31-05-2017

Maria Dienerowitz

maria.dienerowitz@med.uni-jena.de

www.single-molecule-microscopy.uniklinikum-jena.de

Revision

• Wave equation to Helmholtz equation

• Plane & spherical waves, Gaussian beams

Gaussian Beams

• Analytical expression

• Beam parameters

Transmission through optical components

• Thin lens, focussing, collimating

• ABCD law

Contents

Fundamentals of Photonics – B.Saleh, M.Teich 2

Introducing the Gaussian beam

3

Where to start?

Gaussian beam

simplifications Laser output

electromagnetic theory

concise solution find mathematical model

4

wave equation

-> complex wave function

amplitude of wave = magnitude

Wave equation to Helmholtz equation

Helmholtz equation => time independent!!

optical intensity

5

Solutions of the Helmholtz equation

Helmholtz equation

plane wave spherical wave

6

Solutions of the Helmholtz equation

Helmholtz equation

plane wave spherical wave

7

rapidly varying envelope

within a distance

Helmholtz equation to paraxial Helmholtz equation

slowly varying envelope

8

Helmholtz equation to paraxial Helmholtz equation

wavefront normals

within a distance

slowly varying envelope

9

Helmholtz equation to paraxial Helmholtz equation

paraxial Helmholtz equation

solution: Gaussian beam

10

Revision

• Wave equation to Helmholtz equation

• Plane & spherical waves, Gaussian beams

Gaussian Beams

• Analytical expression

• Beam parameters

Transmission through optical components

• Thin lens, focussing, collimating

• ABCD law

Contents

Fundamentals of Photonics – B.Saleh, M.Teich 11

Gaussian beam: analytical expression

beam intensity?

beam power? beam diameter? focal depth?

12

Gaussian beam: analytical expression

plane wave wavenumber

Gaussian beam

13

beam intensity

Gaussian beam: intensity

14

beam power

Gaussian beam: power

beam area

!!! continuous and pulsed Gaussian beams

15

Gaussian beam: width

intensity drops by factor 1/e2

Full Width Half Maximum -> half intensity

16

The Rayleigh range

Gaussian

Rayleigh range

17

Beam waist

beam diameter

18

Depth of focus

beam area

19

beam divergence

Beam divergence

20

The phase of a wave

21

The phase of a Gaussian beam

z

y

phase

plane wave Gouy phase wavefront bending

22

Summary - part I

Rayleigh range = ½ peak intensity

Rayleigh range = beam area twice as large than at beam waist

beam width = intensity drops by factor 1/e2

23

Summary - part II

spherical wave

Gaussian beam

wavelength

plane wave

24

Revision

• Wave equation to Helmholtz equation

• Plane & spherical waves, Gaussian beams

Gaussian Beams

• Analytical expression

• Beam parameters

Transmission through optical components

• Thin lens, focussing, collimating

• ABCD law

Contents

Fundamentals of Photonics – B.Saleh, M.Teich 25

phase shift

Transmission through a transparent plate

phase

26

Transmission through a thin lens

phase

27

magnification

beam waist

depth of focus (=2zR) beam divergence

waist location

radius of curvature

Transmission through a thin lens

28

waist location

Ray optics

magnification

29

Matrix Optics

paraxial approximation

thin lens refraction planar boundary

Snell 30

Transmission through arbitrary systems: ABCD law

complex beam parameter

free space

31

Beam shaping: focussing

32

Beam shaping: expansion/narrowing

increasing beam width decreasing beam width

double the beam diameter: f1=50mm, f2=100mm, d=150mm f1=150mm, f2=300mm, d=450mm

33

Conjugate planes in a microscope

© Nikon MicroscopyU

illumination light path

image forming light path

34

Beam relaying: conjugate planes

beam steering acousto-optical deflector

35