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Physical Optics

Lecture 12: Gaussian beams

2018-06-27

Herbert Gross

Physical Optics: Content

2

No Date Subject Ref Detailed Content

1 11.04. Wave optics GComplex fields, wave equation, k-vectors, interference, light propagation,

interferometry

2 18.04. Diffraction GSlit, grating, diffraction integral, diffraction in optical systems, point spread

function, aberrations

3 25.04. Fourier optics GPlane wave expansion, resolution, image formation, transfer function,

phase imaging

4 02.05.Quality criteria and

resolutionG

Rayleigh and Marechal criteria, Strehl ratio, coherence effects, two-point

resolution, criteria, contrast, axial resolution, CTF

5 09.05. Photon optics KEnergy, momentum, time-energy uncertainty, photon statistics,

fluorescence, Jablonski diagram, lifetime, quantum yield, FRET

6 16.05. Coherence KTemporal and spatial coherence, Young setup, propagation of coherence,

speckle, OCT-principle

7 23.05. Polarization GIntroduction, Jones formalism, Fresnel formulas, birefringence,

components

8 30.05. Laser KAtomic transitions, principle, resonators, modes, laser types, Q-switch,

pulses, power

9 06.06. Nonlinear optics KBasics of nonlinear optics, optical susceptibility, 2nd and 3rd order effects,

CARS microscopy, 2 photon imaging

10 13.06. PSF engineering GApodization, superresolution, extended depth of focus, particle trapping,

confocal PSF

11 20.06. Scattering LIntroduction, surface scattering in systems, volume scattering models,

calculation schemes, tissue models, Mie Scattering

12 27.06. Gaussian beams G Basic description, propagation through optical systems, aberrations

13 04.07. Generalized beams GLaguerre-Gaussian beams, phase singularities, Bessel beams, Airy

beams, applications in superresolution microscopy

14 11.07. Miscellaneous G Coatings, diffractive optics, fibers

K = Kempe G = Gross L = Lu

Solution of the wave equation

Paraxial matrix calculus

Properties of Gaussian beams

Beam transformation

Mode matching

Generation of Gaussian beams in resonators

Ray equivalent auf gaussian beams

Generalized gaussian beams

Truncation of gaussian beams

Gaussian beams with aberrations

3

Contents

Helmholtz wave equation

Fast z-oscillation separated

Slowly varying envelope approximation

Paraxial approximation

paraxial wave equation

Conditions for scalar approximation:

1. Decoupling of field components,

wavelength small in comparison to free diameter

2. No large angles due to geometry,

Computation of field in large distances z

4

Solutions of the Wave Equation

02 EkE

0),,(

2

22

2

2

2

2

2

2

Ec

zyxn

z

E

y

E

x

E

ikzezyxEzyxE ),,(),,(

2

2

2 22

0

2

22E

zik

E

zE k

n x y

nn Eo

( , )

a

z

022

2

2

2

z

Eki

y

E

x

E

2

2

E

zk

E

z

5

Solutions of the Wave Equation

rapidly varying

envelope

slowly varying envelope

Slowly varying envelope approximation

Solution of the paraxial wave equation:

Gaussian beam

Interpretation:

- physical solution (diffraction included)

- only valid for small angles (paraxial)

- eigensolution (only scaling during propagation, no change of the gaussian shape)

- full orthogonal mode system

ikzzR

ikr

zw

r

zio eeeezw

wEzrE

)(2)()(

0

2

2

2

)(),(

Linear relation of ray transport

Simple case: free space

propagation

Advantages of matrix calculus:

1. simple calculation of component

combinations

2. Automatic correct signs of

properties

3. Easy to implement

General case:

paraxial segment with matrix

ABCD-matrix :

u

xM

u

x

DC

BA

u

x

'

'

z

x x'

ray

x'

u'

u

x

B

Matrix Formulation of Paraxial Optics

A B

C D

z

x x'

ray x'

u'u

x

Matrix Calculus

Paraxial raytrace transfer

Matrix formulation

Matrix formalism for finite angles

Paraxial raytrace refraction

Inserted

Matrix formulation

111 jjjj Udyy

1 jjjj Uyi in

nij

j

j

j''

1' jj UU

1 jj yy

1

'

''

j

j

j

j

j

jjj

j Un

ny

n

nnU

'' 1 jjjj iiUU

j

jj

j

j

U

yd

U

y

10

1

'

'1

j

j

j

j

j

jjj

j

j

U

y

n

n

n

nnU

y

'

'01

'

'

j

j

j

j

u

y

DC

BA

u

y

tan'tan

'

Linear transfer of spation coordinate x

and angle u

Matrix representation

Lateral magnification for u=0

Angle magnification of conjugated planes

Refractive power for u=0

Composition of systems

Determinant, only 3 variables

uDxCu

uBxAx

'

'

u

xM

u

x

DC

BA

u

x

'

'

mxxA /'

uuD /'

xuC /'

121 ... MMMMM kk

'det

n

nCBDAM


System inversion

Transition over distance L

Thin lens with focal length f

Dielectric plane interface

Afocal telescope

AC

BDM

1

10

1 LM

11

01

f

M

'0

01

n

nM

0

1L

M


Calculation of intersection length

Magnifications:

1. lateral

2. angle

3. axial, depth

Principal planes

Focal points


DsC

BsAs

'

DsC

BCADm

2'

DsC

BCAD

ds

ds

'sCA

BCADDsC

C

DBCADaH

C

AaH

1'

C

AaF '

C

DaF

2

2

)(

w

r

oeIrI

Gaussian Beams, Transverse Beam Profile

I(r) / I0

r / w

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-2 -1 0 1 2

0.135

0.0111.5

0.589

1.0

Transverse beam profile is gaussian

Beam radius w at 13.5% intensity

Radius r Diameter Amplitude A Intensity I Energy E P-truncation

0.58871 w 1.17741 w 0.7071 0.5000 0.5000 0.5000

w 2 w 0.3679 0.1353 0.8647 0.1353

1.0730 w 2.146 w 0.3162 0.1000 0.9000 0.1000

1.5 w 3 w 0.1054 0.0111 0.9889 0.0111

1.571 w w 0.0848 0.0072 0.9928 0.0072

2 w 4 w 0.0183 0.0003 0.9997 0.0003

Gaussian Beams: Parameters of the Profile

Typical parameters of a gaussian beam profile

2

2

2

1

2

2

2 1

2),(

o

To

z

zzw

r

o

To

e

z

zzw

PzrI

2

0 1)(

o

T

z

zzwzw

00000

zzw

o

o

Gaussian Beams, Definitions and Parameter

Paraxial TEM00 fundamental mode

Transverse intensity is gaussian

Axial isophotes are hyperbolic

Beam radius at 13.5% intensity

Only 2 independent beam parameter of the set:

1. waist radius wo

2. far field divergence angle o

3. Rayleigh range zo

4. Wavelength o

Relations

Expansion of the intensity distribution around the waist I(r,z)

Gaussian Beams

z

asymptotic

lines

x

hyperbolic

caustic curve

wo

w(z)

R(z)

o

zo

-4 -3 -2 -1 1 2 3 4

-4

-3

-2

-1

1

2

3

4

z / z

r / w

o

o

asymptotic

far field

waist

w(z)

o

intensity

13.5 %

Geometry of Gaussian Beams

-2

0

2-8

-6

-4

-2

0

2

4

6

8

0

0.5

1

z

intensity I

[a.u.]

x

z/zo

x/wo

I(x,z)

+ 4

- 4

+ 2

0

- 2- 4

- 8

+ 8

+ 4

00

1

Caustic of a Gaussian Beam

Intensity I(x,z)


Parameter:

= 0.5 mm

wo = 0.3 / 0.6 / 0.9 mm

Normalization of intensity


Parameter:

= 0.5 mm

wo = 0.3 / 0.6 / 0.9 mm

Normalization of power


Parameter:

= 0.5 mm

wo = 0.3 / 0.6 / 0.9 mm

Normalization of intensity


Parameter:

= 0.5 mm

wo = 0.3 / 0.6 / 0.9 mm

Normalization of power

21

Phase of a Gaussian Beam

Phase terms collected:

z

y

phase

plane wave Gouy phasewavefront bending

Ref: M. Dienerowitz

)(

)(2)(

0

2

2

2

)(),(

zzR

ikrkzi

zw

r

o eezw

wEzrE

)()(2

),(2

zzR

ikrkzzr

z

zzzR o

2

)( oz

zz arctan)(

2

2

0

0 1

''2)'(

''2)'(

)(

w

r

r

e

drrrI

drrrI

rE

2

02

2)( wIdrrrIP o

E(r) / E 0

r / w0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Gaussian Beam, Energy Function

Encircled energy function

Total power in the beam

222

1

12

)(

2),0(

o

To

z

zzw

P

zw

PzrI

2

0

0

0

0

2

wwz o

o

I(z) / I0(z)

z / zo

-6 -4 -2 0 2 4 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Gaussian Beam, Axial Intensity

Axial intensity: Lorentz function

Characteristic depth of the waist region:

Rayleigh range

T

oT

zz

zzzzR

2

)(

R / z0

z / zo

0 1 2 3 4 5 60

1

2

3

4

5

6

Gaussian Beams, Radius of Curvature

Gaussian beams have a spherical wavefront

Radius of curvature:

In the waist the phase is plane

The spherical wave is not concentric

to the waist center

In the distance of the Rayleigh range, the

curvature of the phase has ist maximum

Phase of a Gaussian beam:

Guoy phase

0

arctan)(z

zzz T

z

zo

-zo

-45°

-90°

+90°

+45°

Guoy Phase

f

zfz

zzz

TT

oTT

1

1

1112'

Transform of Gaussian Beams

Diffraction effects are taken into account

Geometrical prediction corrected in the waist region

No singulare focal point: waist with finite width

Focal shift: waist located towards the system, intra focal shift

Transform of paraxial beam

propagation

z'T / f

zT / f

-6 -5 -4 -3 -2 -1 0 1 2 3 4-4

-3

-2

-1

0

1

2

3

4

5

6

zo / f = 0.1

zo / f = 0.2

zo / f = 0.5

zo / f = 1

zo / f = 2

geometrical

limit zo / f = 0

Focussing Gaussbeam by a Lens

Two possibilities:

1. Start in waist

2. Start just before lens

waist position -z

Result

Comparison with lens

makers formula

(change of sign convention):

z

lens

focal length f'

z'

z

zo zo'

initial

plane

waist

final

plane

A BC D

1 0-1/f 1

Matrix :

z

z'

z

zo zo'

initial

plane

waist

final

planeA BC D

1 z0 1

Matrix :

1 0-1/f 1

waist

waist

lens

focal length f'

zf

zfz

zzfz

1

1

111

'

12

0

22

2

'ozzf

zfffz

nced

Wav

e

Optic

s

Focussing a Gaussbeam by a Lens

Setup:

Complex beam parameter

Transformation of Parameter

gives

geometrical model

z

focal length

f'

zT'zT

gaussian beam

waist

plane image plane

z

waist

plane

DCq

BAqq

'

0izzq

222

''o

oo

o

oo

zCDCz

DizzCBizzA

DizzC

BizzAizz

nced

Wav

e

Optic

s

-3 -2 -1 1 2 3 4

1

2

3

4

5

6

7

8

9

0

0.5

0.25

0.125

wo' / w

o

zT' / z'

o

1f_zo

=

22

'

oTo

o

zzf

f

w

w

0

max,2

1'z

ffz T

Transform of Gaussian Beams

Change of waist size / ‚magnification‘

The waist can not be located into an

arbitrary distance

largest possible distance:

-2

0

2-8

-6

-4

-2

0

2

4

6

8

0

0.5

1

z

intensity I

[a.u.]

x

Gaussian Beam Propagation

Paraxial transform of

a beam

Intensity I(x,z)2

)(2

2 )(

2),(

zw

r

ezw

PzrI

Propagation of Gaussian Beams

Paraxial transport through

ABCD-system

Astigmatic beam transfer

thin lensesx

y

z

Gauss Beam in Quadratic Grin-Medium

I(z)

I(r,z)

I(r)

z

r

Transform of Gaussian Beam

R

w

w'

R'

starting

plane

receiving

plane

paraxial

segment

A B

C D

incoming

Gaussian beam

z

outgoing

Gaussian beam

Transfer of a Gaussian beam by a paraxial ABCD system

w wB

wA

B

R'

2

2 2

R

AB

R

B

w

AB

RC

D

RD

B

w

'

2

2

2

2

2

given input

beam required output

beam

f

z

f1 2

z zL1L2

ww'

00

zTT '

2

00

2

'

2

0000

2

00'00'2

00

2/1 ''''2'2'

1zzzzzzzzzzzzz

zzf TTTTTT

Gaussian Beam Mode Matching

Given beam at before and after the system: search for transfer optic

In general two solutions

Location of lens at the intersection points of the two hyperbolic curves

0'

ww o

0' o

f

z

2

Lz

w0

f1

T

z'T

w'0

z'TT'

Beam Transformation by a Telescope

Afocal telescope of the Kepler or Galilean type

Application: adaptation of beam width

Beam radius enlarged: divergence reduced and vice versa

Kepler system:

Internal waist, can make problems with energy density

Gaussian beam with minimal diameter in a given distance L

Waist size

located in the half distance

Maximum value of beam radius

Mode Matching

wmax

w0

L

f

2

Lwo

Lw max

37

Laser Resonator Types

Ref: B. Böhme

Description of curved mirrors as combination of thin lens and plane mirror

RR1 2

A B

C D

f

f 1

2

A B

C D

221

1

22121

1

11222112

1G

B

GG

BG

R

BD

RR

B

R

A

R

DC

BR

BA

dc

baLMLmm

T

Equivalent Plan-plan-Resonator

Resonator Mode Calculation after Fox-Li

From the physical point of view the field inside a stable resonator is given as the

eigensolution of the electromagnetic field, that is reproduced for one round trip

through the resonator

Typically a system of eigensolutions is found by this boundary value problem, t

these are he modes of the resonator

The transverse limitations of the field due to a stop governs the modes

The eigenvalues g determine the losses of the modes

, , ,( , ) ( , , ', ') ( ', ') dx'dy'n m n m n mE x y K x y x y E x y

RR1 2

A BC D

En,m(x,y)

round tripstop

Condition of stability for a reproduced wave

Mirrors are phase surfaces

g1-g2 diagram of stability

yellow region delivers stable

operation

10 21 gg

A Do o

21

g

-3 -2

1

-1

2

3

g1

-3

-2

1

-1

2 3

2

stable regimes

symmetric

confocal

plane-plane

symmetric

concentric

Stability of a Gaussian Resonator

To

o

To zz

z

zzwzw

)(

Gaussian Beams: Asymptotic Limiting Case

Limiting case for large distances from waist z >> zo

Linear geometrical propagation

Geometrical aperture angle corresponds to divergence angle

Geometrical prediction are approximately valid

Equivalent description of the gaussian beams by two paraxial rays possible:

1. Waist ray corresponds to chief ray

2. divergence ray corresponds to marginal ray

Therefore description of propagation by identical ABCDS matrices

x

z

waist ray

divergence rayo

wo

z

beam hyperbola

General case:

Two rays with heights y and ybar and anglers u and ubar

Corresponds to marginal (MR) and chief (CR) ray

Rays must fulfill the Lagrange condition

x

zy ray MRy ray CR

o

beam hyperbola

o

oo

uyuyL

Gaussian Beam Ray Equivalent

Astigmatic Gaussian beams:

different values for wx, wy and Rx, Ry in both cross sections

Simple case of decoupled sections without twist

The intensity profile is

elliptical with changing axes

)()(

2

)()()(

2

)(

22

2

2

22

2

),,(zR

y

zR

xy

zR

xi

zw

y

zw

xy

zw

x

o

yxyxyxyx eeEzyxE

x / y

z

y-z cross

section

x-z cross

section

woxwoy

zTy

zTx

T

y

x

y

x

y

x

y

x

y

x

Astigmatic Gaussian Beams

Beams with twist:

ellipse of intensity

ellipse of curvature

with different phase

Beam rotates during

propagation in the azimuth

)()()( 2

2

22

2

intzw

y

zw

xy

zw

xE

yxyx

)()()( 2

2

22

2

zR

y

zR

xy

zR

xE

yxyx

phas

22

2

11

2

2tan

yx

xy

Int

ww

w

22

2

11

2

2tan

yx

xy

Phas

RR

R

isophotes of

intensity

isophase

surfaces

z

z

Beams with Twist

a/wo = 1

0 1 2 3 4 5 610

-12

10-10

10-8

10-6

10-4

10-2

100

a/wo = 2

a/wo = 3

a/wo = 4

x

Log |A|

Truncated Gaussian Beams

Untruncated gaussian beam: theoretical infinite extension

Real world: diameter D = 2a = 3w with 1% energy loss acceptable

Truncation: diffraction ripple occur, depending on ratio x = a / wo

Gaussian beam with width w

Stop with radius a

Truncation/transmission for a chaning lateral offset v in the range

v / w = 0.05 ...2.5

a

w

t

w

v

dtw

vtIete

wT

0

20

22

2

44 2

2

2

2

Gaussian Beam: Truncation by Offset

T

a/w

v/w = 0.05

v/w = 2.5

Transmission of a gaussian beam

with width w by a ring stop

inner radius b

outer radius a

Optimal beam width for maximal

transmission

2 2

2 2a a

w wT e e

w / aopt

E / E

1a

b

ln

12 awopt

Gaussian Bema: Truncation by Ring Stop

ri

Ii wi

circular

stop

gaussian

profil

z

a

b

Transmission of a gaussian beam by a ring stop with

for stop ratios = 0.05 ...2.5

Gaussian Beam: Truncation by Circular Ring Stop

Transmission of an elliptical gaussian beam with aspect ratio

by a circular stop withg radius a

Change of ratio a/w:

a / w = 6 , 5 , 4.5 , 4 , 3.5

3.0 , 2.5 , 2.0 , 1.75

1.5 , 1.25 , 1.0 , 0.9

0.8 , 0.7 , 0.6 , 0.5

0.4 , 0.3 , 0.2 , 0.1

Elliptical Gaussian Beam: Truncation by Stop

x

y

w

w

Transmission of a gaussian beam by a

centered slit of width b

Change of ratio a/w:

a / w = 2 , 1.5 , 1.4 , 1.3 , 1.2 , 1.1 , 1.0 0.95 ,

0.90 , 0.85 , 0.80 , 0.75 , 0.70 , 0.65 , 0.60 ,

0.55 , 0.50 , 0.45 , 0.40 , 0.35 , 0.30 ,

0.25 , 0.20 , 0.15 , 0.10 , 0.05

Gaussian Beam: Truncation by a Slit

x

y

bslit

w

gaussian

beam2

Slit

bT erf

w

Focussed Gaussian beam with spherical aberration

Asymmetry intra - extra focal

depending on sign of spherical aberration

Gaussian profile perturbed

Gaussian Beam with Spherical Aberration

c9 = -0.25

c9 = 0.25

c9 = 0

Gaussian beam with spherical aberration

The intensity becomes asymmetrical

Position and height of the peak are changing

0

11

2

1

0

24

1

2

2)( drree

zi

EzE

rRzw

rSik

w

r

Gaussian Bemas with Spherical Aberration

unperturbed

kS1 = 0.05 0.07 0.085 -0.07

I(0,z)

z

I(0,z)

z

I(0,z)

z

I(0,z)

z


Focal length : f = 200 mm , beam radius: w = 3 mm,

Spherical aberration: Zernike coefficient c9 = 0.10 (normalization at r = 1.5 w, PV is 2c9)

Colored plot: logarithmic, cross sections linear at z = 180, 200, 210 mm





Colored plot: logarithmic, cross sections linear at z = 180, 200, 210 mm





Colored plot: logarithmic, cross sections linear at z = 160, 180, 200, 210 mm





Colored plot: logarithmic, cross sections linear at z = 160,180, 200, 210 mm


Astigmatism

c = 0.3

20 21 22 23 24 25 26 27 28 29 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

I(z) I(y,z)I(x,z)

Coma

c = 0.3

20 21 22 23 24 25 26 27 28 29 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

I(z) I(y,z)I(x,z)

Gaussian Beam with Coma and Astigmatism

physical optics - uni-jena.de · 2018. 6. 4. · solution of the paraxial wave equation: gaussian...

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