Second-order

nonlinear interaction of noncollinear propagating broadband laser pulses

Chi2D  

and  

chi3D  

use  

the  

Euler  

method

within  

a  

split-step  

algorithm  

to  

compute  

the  

evolution  

of  

the  

2  

or  

3  

dimensional  

electrical  

fields  

containing

pump,  

signal  

and  

idler  

pulses.  

The  

complex  

electrical  

fields  

after  

a  

linear  

propagation  

step  

are  

Fourier  

transformed  

into  

the  

spatiotemporal

domain.   

  

A   

novel   

set   

of   

coupled   

nonlinear   

equations   

calculate   

the   

change   

in   

the   

ordinary   

and   

extra-ordinary   

polarized   

electrical   

fields,

respectively.  

The  

different  

terms  

of  

the  

equations  

include  

all  

possible  

second-order  

nonlinear  

interaction  

types  

in  

one  

step.  

This  

includes  

sum

frequency  

generation  

(SPM),  

second  

harmonic  

generation  

(SHG),  

difference  

frequency  

generation  

(DFG),  

as  

well  

as  

all  

thinkable  

parasitic  

and

cascaded   

conversion   

processes.   

The   

phase   

of   

the   

newly   

generated   

frequency   

components   

defines   

the   

constructive   

or   

deconstructive

interference with the fundamental field and the propagation direction within the next linear propagation step.

Propagation of ultra-short laser pulses in isotropic and birefringent media

The   linear   evolution   of   the   arbitary   light   field   containing   a   super   position   of   all   involved   fs-laser   pulses   is   realized   in   Fourier   space   of   the   ordinary and   extraordinary   field.   Following   a   solution   of   the   wave   equation,   a   linear   phase   given   by   the   Sellmeier   formalism   is   added   to   the   optical   and spatial   frequency   spectrum.   The   result is       the       application       of       dispersion, diffraction    and    displacement    due    to walk-off   and   non-collinear   propagation within     each     propagation     step     in     z direction.

Treatment    

of    

all    

pulses    

in    

only    

two

orthogonal polarized fields

Second  

order  

nonlinear  

frequency  

conversion  

processes  

e.g.

the   

second   

harmonic   

frequency   

generation   

(SHG),   

the   

sum

frequency    

generation    

(SFG)    

or    

the    

difference    

frequency

generation   

(DFG,   

OPG,   

OPA,   

OPCPA)   

are   

the   

fundamental

concepts   

and   

part   

of   

almost   

any   

modern   

high   

energy   

and

ultra-short pulsed laser system.

Analytical   

methods   

to   

predict   

the   

efficiency   

of   

a   

certain

conversion     

process     

is     

only     

possible     

under     

defined

circumstances,   

e.g.   

a   

negligible   

pump  

wave.  

The   

traditional

way  

to  

compute  

the  

parametric  

conversion  

numerically  

is  

the

use of the three well known coupled equations,

.

Each  

equation  

describes  

the  

change  

of  

the  

specific  

complex

pump,  

signal  

and  

idler  

wave,  

respectively.  

Consequently  

each

phase-matched  

nonlinear  

process  

needs  

to  

be  

implemented

individually.   

In   

particular   

ultra-broadband   

optical   

parametric

amplifiers  

(OPCPA)  

are  

subject  

of  

a  

variety  

of  

parasitic  

and

cascaded nonlinear processes.

  

  

In  

contrast,

  

chi2D  

and  

chi3d  

rely  

on  

a  

new  

concept  

utilizing

the   

fact,   

that   

all   

second   

order   

conversion   

processes   

of

practical  

use  

are  

realized  

in  

birefringent  

nonlinear  

crystals.

Consequently,  

the  

superposition  

of  

all  

involved  

optical  

pulses

(signal,  

idler,  

pump,  

parasitic  

and  

cascaded  

signals,  

etc.)  

can

be  

described  

within  

only  

two  

orthogonal  

polarized  

complex

electrical  

fields.  

A  

novel  

set  

of  

only  

two  

nonlinear  

coupled

equations    

dedicated    

to    

the    

ordinary    

and    

extraordinary

electrical  

fields  

inherently  

include  

all  

possible  

C

²-nonlinear

processes.   

Combined   

in   

a   

split-step   

Fourier   

algorithm   

the

linear  

propagation  

effects,  

such  

as  

diffraction,  

dispersion  

and

walk-off,  

as  

well  

as  

the  

relevant  

C

³-nonlinear  

effects  

like  

self-

phase     

modulation     

and     

self-focusing     

can     

be     

easily

implemented. [

Lang, et. al., Optics Express

21

(01)  (2013)

]

Super fluorescence cone / cascaded frequency generation in chirped pulse

parametric amplifiers

The   

following   

example   

is   

taken

from      

an      

experimental      

and

theoretical    

study    

of    

a    

double

stage  

high  

power  

OPCPA  

system

published   

in   

Matyschok,   

et.   

al.,

Optics Express

21

(24)  (2013).

The  

colorful  

photograph  

shows  

a

screen   

illuminated   

by   

the   

well

known   

super   

fluorescence   

cone

visible    

in    

high    

power    

OPCPA

systems.      

Beside      

the      

visible

parasitic  

idler  

and  

signal  

SHG  

or  

the  

scattered  

green

pump  

light  

passing  

through  

the  

hole  

in  

the  

screen,  

a

variety    

of    

additional    

features    

emerge    

at    

different

angles.

The    

animation    

illustrates    

the    

generation    

of    

the

spectral,  

angular  

distributions  

of  

all  

visible  

and  

non-

visible  

mixing  

products  

in  

optimum  

temporal  

overlap

between   

pump   

and   

seed   

pulse   

as   

simulated   

with

chi2D.

The     

detailed     

information     

about     

the     

respective

polarization,    

spectrum,    

propagation    

direction    

and

phase  

allows  

to  

address  

each  

mixing  

product  

to  

its

specific  

cascading  

path.  

It  

can  

be  

seen  

that  

during  

the

amplification  

process  

the  

simulation  

reproduces  

each

and  

every  

feature  

visible  

in  

the  

photograph  

as  

taken  

of

the   

experiment.   

Since   

the   

complex   

electrical   

fields

contain   

the   

full   

information   

regarding   

the   

spectral

phase,  

it  

is  

possible  

to  

map  

the  

relative  

group  

delay  

of

each angular and spectral resolved mixing product.

Having  

a  

look  

on  

the  

quantitative  

agreement  

between  

experiment  

and  

simulation  

both  

subsequential  

amplification  

stages  

of  

the

OPCPA  

system  

were  

simulated  

using  

the  

chi2D  

code.  

The  

two  

plots  

show  

the  

comparison  

of  

simulated  

and  

experimental  

results.

Left:  

Measured  

spectra  

of  

the

amplified    

signal    

pulse    

after

the   

first   

stage   

(red   

shaded)

and   

for   

the   

further   

amplified

signal  

pulse  

after  

the  

second

stage  

(grey  

shaded);  

extracted

spectra   

from   

the   

simulation

(blue lines), respectively.

Right:          

Measured          

and

simulated  

pulse  

energies  

after

the first and the second stage.

The  

chi23D  

forum  

includes  

examples  

as  

well  

as  

a  

question  

&

answer section:

A     

simple     

starting     

guide     

named     

HowToChi2D     

can     

be

downloaded in pdf-format:

chi3D

  

is  

a  

Matlab  

class  

based  

on  

the  

same  

algorithms  

used  

in  

chi2D  

but  

can  

now  

handle  

  

2+1  

or  

3+1  

dimensions.  

In

contrast  

to  

chi2D  

it  

can  

be  

called  

directly  

from  

Matlab  

similar  

to  

a  

standard  

Matlab  

toolbox.  

This  

enables  

to  

write  

scripts  

and

simulated   

chains   

of   

nonlinear   

conversion   

processes,   

e.g.   

SHG,   

DFG,   

OPCPA.   

Furthermore,   

it   

is   

possible   

to   

implement

subsequent  

components  

like  

imaging  

systems,  

delay  

plates,  

windows  

or  

a  

gas  

between  

the  

stages.  

Since  

the  

complex

electrical  

fields  

can  

be  

passed  

over  

to  

each  

subsequent  

component  

in  

the  

script,  

parameter  

scans  

of  

the  

whole  

system

considering pulse energy, pointing, SPM, etc. are now possible.

In contrast to the free version chi3D is a freelance project and offered on commercial basis together with the support.

Dr. Tino Lang Quälkampsweg 100 D-22880 Wedel E-mail: info@chi23d.com Essential Dimensions tax-ID.: 26/026/53505

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