www.iap.uni-jena.de

Wellenoptische Propagation zwischen

verkippten Ebenen

Statusmeeting

2016-11-25

Herbert Gross

Beam Propagation in Optical Systems

Segmentation of the system into diffraction-relevant sub-systems

Optimization of the propagator in every segment to get good numerical performance

considering:

1. Physical effects (diffraction, vectorial aspects,…)

2. Sampling requirements

3. Numerical efficiency

At the interface planes the field is re-interpolated or converted to raytrace data

The cascaded effect of diffraction needs several calculation steps

Fraunhofer

diffraction

corrected lens

segment 1

Raytrace

lens with

aberrations

segment 2

Finite differences

inhomogeneous

medium

segment 3

Mode

expansion

free space

segment 4

Fresnel

diffraction

corrected lens

segment 6

High-NA

diffraction

segment 7

stop

n(x,y,z)focus

Thin

phase

mask

segment 5

25

26

General simulation of optical systems

Simulation of diffraction effects

Calculation algorithms of wave

propagation

Partial coherent imaging and beam

propagation

Point spread function engineering and

Fourier optics

Segmented surfaces and multi-apertures

Optical System Simulation

ray

trace

interface plane

conversion

ray wave

field

propagationdiffraction

effects

lens

aberrations

Fresnel

lens

source

point

27

Tilted Plane Propagation

Usual setup for diffraction calculation of beam propagation:

initial and final plane perpendicular to optical axis

Calculation scheme for tilted planes with fast algorithms is a need in many applications

Idea for geometrical approach:

general rotation described by two special rotations:

1. rotate q azimuthal around z1 axis to match the y-axis with the plane of incidence

2. rotate j around x2 to match the new coordinates of the final plane

Matrix and geometry

1

1

1

1

1

1

3

3

3

coscossinsinsin

sincoscossincos

0sincos

z

y

x

z

y

x

MM

z

y

x

jqjqj

jqjqj

qq

qj

x1

y1

x3

y3

x2

y2

q

j

original

planeplane rotated

in the azimuthplane tilted

around the x-axis

z1 z2

z3

central axis

ray

28

Tilted Plane Propagation

Fast calculation of an azimuthal rotation:

sequence of 3 shear transforms

Fast calculation of a shear with Fourier

transforms:

1. calculate spectrum

2. shift by xo: phase factor in spectral function

3. height dependent shift

4. Total operator

x

y

x

y

rotation

x

y

x

y

shearing

+x

shearing

+x

shearing -y

y

x

yo

a

xo

original

sheared

dxexfvF vxi2)()(

dveevFdvevFxxf vxivixvxxi 22)(2

000 )()()(

atan0 oyx

)(ˆˆ)( 021

0 xfFeFxxfvix

29

Tilted Plane Propagation

Numerical check of azimuthal transform

by shear:

1. rotation of a profile

2. rotation and back-rotation:

residual error maschine precision

3. decompose a complete rotation of

360°by 12 successive rotations with

30° each:

nearly perfect reproduction of original

signal

30

Tilted Plane Propagation

Transition from plane 2

into plane 3:

line by line 1D Fourier transform

along x for every y

Projection relation:

Distance for height y:

Algorithm:

1. calculate spectrum

2. Calculate Fresnel propagation for every y/z by spectrum of plane waves and

back-transform

y3y2

j

adjusted

plane

plane tilted

around the x2-

axis

z2yo

zo

zo(0)jcos

23

yy

dydxeyxEvvE yx yvxvi

yx

2)0,,(0,,

xy

vyzivyiyikz

yx

vyzixvidvdveeevvEeezyxE yyxx

22 )(2)()(20,,,,

tanz y j

31

Tilted Plane Propagation

y

x

-8 -6 -4 -2 0 2 4 6 80

0.2

0.4

0.6

0.8

1

intensity I(x)

x-8 -6 -4 -2 0 2 4 6 8

0

0.2

0.4

0.6

0.8

1

intensity I(y)

y

y

intensity I(x)

x-1.5 -1 -0.5 0 0.5 1 1.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8x

-100 -50 0 50 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

intensity I(y)

y

a) nearly collimated beam b) curved in x-section c) curved in y-section d) curved in x-and y-sectiony

x

y

x

y

x

y

x

Example 1:

Gaussian beam cross section in a

70°tilted plane

Example 2:

Fresnel diffraction

of a tophat profile

in a tilted plane by

89.4°notice the scaling

and the asymmetry

Phase of a beam in

a slightly tilted plane

0.3°for different

curvatures

32

Tilted Plane Propagation

Performance of numerical calculations

Time contributions of the various

parts of the algorithm,

last column gives total time in sec

Total time of the propagation

algorithm for different sizes of the

sample points in x and y

incidence plane is y-z

Nx = Ny

time fft2

time czt2

time rotation

totaltime

64 0.0001 0.0040 0.050 0.063128 0.0003 0.0080 0.036 0.026256 0.0013 0.030 0.083 0.14512 0.0075 0.094 0.265 1.07

1024 0.031 0.361 0.870 7.02048 0.167 1.69 3.42 56.2

64 128 25610

-2

10-1

100

101

Log t [sec]

512 1024 2048

N

Nx = 256Ny varied

Ny = 256Nx varied

Propagation with Gain and Saturation

Change of amplitude

Gain function (homogeneous

saturation)

Small signal gain : go

Saturation intensity : Isat

Beam profile is changed,

if large amplification takes place

a) large gain

20 40 60 80 1000

2000

4000

6000

8000

-6 -4 -2 0 2 4 60

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100-6

-4

-2

0

2

4

6

20 40 60 80 1000

50

100

150

200

250

300

0 20 40 60 80 100-6

-4

-2

0

2

4

6

-6 -4 -2 0 2 4 60

0.2

0.4

0.6

0.8

1

b) small gain

amplification

amplification

z z

z z

r

r

start

profile

final

profile

final

gauss fit

zzyxgeyxEzyxE ),,()0,,(),,(

satsat

I

zyxI

zyxg

I

zyxE

zyxgzyxg

),,(1

),,(

),,(1

),,(),,( 0

2

0

33

34

Second Harmonic Generation

SHG in a birefringend crystal with

lateral walk-off in x

Pump beam / second harmonic:

- cross section at the end oif the crystal

- saturation of the pump beam along z

- development of light conversion

and efficiency

Ref.: S. Schmidt

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,

these are the 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 yg

RR1 2

A BC D

En,m(x,y)

round tripstop

35

Resonator with circular symmetry

Internal lens

Mode selection by internal stop

Convergence after 200 iterations

Higher order mode content

Numerical Mode Calculation

D

36

Resonator Mode Calculation after Fox-Li

Calculation of higher modes

Transform of the eigenvalue by a complex number m

Three higher modes are obtained

Re[m

Im[m

Ite

g m

g m

2

1

1

1

gIm

gRe

g

g

g

m

fundamental mode 1

g1(corr)

2

3

g2(corr)

calculated after transformation

37

Mode profiles with loss values per round trip

Losses as a function

of the stop size

Diagram of complex

Eigenvalues:

- distance from origin

loss

- azimuthal location:

phase

0 0.5-0.5

1

0

0.5

x

I(x)

00.

5

-

0.5

1

0

0.

5

x

I(x

)

0 0.5-0.5

1

0

0.5

x

I(x)

0 0.5-0.5

1

0

0.5

x

I(x)

0.04 % 2.20 % 19.03 % 44.64 %

loss factor

stop diameter

1

0.5

00 0.5 1 1.5 2

G

A

A2

1

A 3

Re

Im

A

A

A

G

1

2

3

-0.5-1 0.5 1

-0.5

-1

0.5

1complex

eigenvalues

Numerical Resonator Calculation

38

Resonator with Littrow Setup

Setup with Littrow arrangement for line narrowing

Spectral selection of output power

Outcoupling

mirror Gain medium

internal

stop

prism

train

Littrow

grating

1 2 3

4

5

67

8

round trip segments

193.4377 193.4378 193.4379 193.438 193.4381 193.4382 193.43830

0.1

0.2

0.3

0.4Power P(wl)

39

Resonator with Littrow Setup

Intensity distributions

Divergence distributions

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1Intensity I(y)

0.00000

193.43772

193.43780

193.43789

193.43797

193.43805

193.43814

193.43822

-5 -4 -3 -2 -1 0 1 2 3 4 50

0.2

0.4

0.6

0.8

1Intensity I(x)

-20 -15 -10 -5 0 5 10 15 200

0.2

0.4

0.6

0.8

1Intensity I(wy)

0.00000

193.43772

193.43780

193.43789

193.43797

193.43805

193.43814

193.43822

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1Intensity I(wx)

-1.5 -1 -0.5 0 0.5 1 1.5-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

y [mm]

cen

-o[pm]

40