Download Seminar2

COMPUTER MODELING OF
LASER SYSTEMS
Dejan Škrabelj
Advisers: Prof. Dr. Irena Drevenšek - Olenik
Dr. Marko Marinček
__________________________________________
OUTLINE
1.
2.
3.
4.
5.
Tatoo removal application - motivation
Revision of a basic laser physics
Laser model for a Q – switched solid state laser
Simulations of a ruby laser system
Conclusion
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Motivation – tatoo removal application
•
•
•
Non-destructive method for tatoo removal is a laser treatment.
High power pulses P ~ 10 MW are required to achieve pigment
break down.
Particular pigment color requires particular wavelength of the light.
Blue pigment – ruby laser wavelength, 694 nm.
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Motivation – tatoo removal application
•
•
desirable property of a laser beam: “top hat” profile.
We would like to construct a Q-switched ruby laser with a
supergaussian mirror output coupler. Good computer simulation model
can enormously reduce development time and development
expenses.
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Basic laser physics – laser system
• Laser is an optical oscillator.
• Front mirror is partially
transmitive.
• Resonator losses are
compensated by
amplification process based on
the stimulated emission, which
takes place in the laser rod.
For the stimulated emission we
have to attain the population
inversion with an external
pump source.
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Basic laser physics – QS technique
• Additional element is put in the
resonator, which mediates the
resonator losses.
•QS element is constituted from a
polarizer and an electrooptic
modulator. The generated laser light
is consequently linearly polarized.
• Produced pulses: Nd –YAG system:
E ~ 1J
t ~ 10ns
 P ~ 100MW
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Basic laser physics – unstable resonator
• In the stable resonator the light rays are
confined between the resonator mirrors. In
the unstable configuration, the light rays
are no longer confined between mirrors.
• Radii of curvatures of the resonator
mirrors and its length determine the type
of the resonator.
• Energy extraction is greater with use of
the unstable cavities.
• With an unstable cavity output coupled
with a supergaussian mirror we can obtain
“top–hat” profile of a laser beam.
• Supergaussian mirror has a non–
uniform reflectance profile.
R  R0e
2 ( wr ) ord
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Model – introduction



i ( r ,t )
E (r , t )  AE (r , t )e



i ( r , t )
(r , t )  A (r , t )e
| AE | | A |
•
For a given cavity configuration a good model should predict several
parameters as: pulse energy, pulse width, intensity distribution in a plane
perpendicular to the propagation direction, effective beam radii at different
distances, etc.
•
Resonator is divided into effective planes, which present resonator
elements.
•
We have to determine:
1. how the flux plane is propagated between resonator elements,
2. the influence of the particular element inside of the resonator on the
flux plane.
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Model – free space propagation
• For the propagation a method based on the 2D FT is used.
 
et ( s x , s y ; z ) 
  E ( x, y , z ) e
2i ( s x x  s y y )
dxdy,
  
 
E ( x, y , z ) 
  e ( s , s ; z )e
t
  
sx, y 
kx,y
2
x
y
 2i ( s x x  s y y )
ds x ds y ,
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Model – free space propagation
• Propagating field must obey the wave equation.


2
2
 E (r )  k E (r )  0,
d 2 et ( s x , s y ; z )
dz 2
 ( 2 ) 2 (1  2 s x2  2 s y2 )et ( s x , s y ; z )  0,
et ( s x , s y ; z 2 )  et ( s x , s y ; z1 )e
i 2 1 2 s x2  2 s 2y d
.
• The EM field transveral spectrum propagation is performed by a
simple multiplication with the phase factor!
real space:
Fourier space:
d
E( x, y, z1 )
FT
et ( s x , s y ; z1 )
e
E ( x, y, z2 )
IFT
i 2 12 s x2 2 s 2y d
et ( s x , s y ; z 2 )
• In the numerical calculation we cover the EM field with n x n
mesh points.
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Model – propagation through a lens
• Lens is characterized with its focal length f:
1
f
 (n 1)( r11  r12 )
• Curved partially transmitting mirror acts as a lens on the transmitted
part of the wavefront. Laser rod acts like a lens, too.
• If the optical wave passes through the slice of a medium with
refractive index n and a thickness d(x,y) its phase is changed:
 ( x, y)  nk0 d ( x, y)  k0 [d 0  d ( x, y)]
 nk0 d0  (n  1)k0 d ( x, y).
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Model – propagation through a lens
d ( x, y )  d 0  {R  [ R 2  ( x 2  y 2 )]1/ 2 }
x 2  y 2  R 2 
d ( x, y )  d 0 
x2  y2
2R
.
f  R /( n  1),
 ( x, y )  const. 
k0 ( x 2  y 2 )
2f
Transmitivity of a lens is equal to:
ik0 ( x2  y 2 )
Etransmitted

tL 
 ei ( x , y )  e 2 f .
Eincident
In the model the transmitivity of a lens is present with n x n matrix.
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Model – thermally induced laser rod lensing
• The heat produced by a flashlamp is absorbed inside the laser
rod with a cylindrical shape. The radial temperature distribution in a
cylinder with thermal conductivity D can be obtained from the heat
conduction equation.
T (r )  T (r0 )  4QD (r02  r 2 ),
dn
n(r )  [T (r )  T (r  0)] dT

  4QD
dn
dT
r 2.
 (r )  k0 [n0  n(r )]l
• The transmitivity factor of the rod introduced by the heating is
thr  e
ik ( x2  y 2 )
 2f
rod
,
f rod 
2K 1
dn
Q dT
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Model – effect of a mirror
1. Back resonator mirror R  100%
ik0 ( x2  y 2 )
only the phase of the EM field plane is modified: rM  e rb .
2. Front resonator mirror R  100%
* Reflected part of the EM field
rreflected  e
ik0 ( x2  y 2 )
rf
,
Areflected  R Aincident.
* Transmitted part of the EM field:
ttransmitted  e
ik0 ( x 2  y 2 )
2 f fron t
,
Atransmitted  1  R Aincident.
Reflectance R, r, and t – factors are present in the model with n x n
matrices.
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Model – QS element, gain
QS element is approximated with a nearly step
function.
Laser flux density addition represents gain
mechanism. QS technique:
population
inversion density
 photon flux density
n
n
t
 cn

t
 (cn ll'  tmin
)
R
 ruby  2
• The initial population inversion density is the
simulation input parameter.
• Spontaneous emission is the origin of the
lasing process.
|  i | a  niu (1  e  t / ),
'
niu  n0 ,
t' 
l'
cro d
.
  n / 21,  n / 21



  n / 2,  n / 21



  n / 21,n / 2 



 n / 2,n / 2 
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Model – simulation course
  n / 21,  n / 21



  n / 2,  n / 21
• At each resonator plane the photon flux matrix
  n / 21,n / 2 



 n / 2,n / 2 



 is modified.
• At each roundtrip time t part of the photon flux is transmitted
F
through the outcoupling mirror. The pulse intensity is a sum of all
individual contributions.
 j0 I ( t j ) t F
t 
,

I
(
t
)
 j 0 j
t j  j t F ,

 
effective
radius
  I ( x , y ) x dxdy
2
rx  2
  
 
  I ( x , y ) dxdy
  
effective
pulse width
j  1,2,3, 
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Simulations – ruby laser
• Ruby laser system is planed to become a new product of Fotona.
• We constructed a test QS ruby laser with a stable cavity in order to
estimate some model parameters. We estimated f rod  12m and begin
simulations.
• After many simulations we chose a cavity and a mirror, for which the
simulation gave the present NF intensity profile
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Simulations – ruby laser
and the pulse shape:
We ordered the optics and
waited ...
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Simulations – ruby laser
• Finally we mounted the supergaussian mirror in
the resonator.
•It turned out that the lensing parameter of the rod
was wrongly estimated. With the help of the model
we found that f rod  40m instead of f rod  12m.
f rod  40m
f rod  30m
f rod  20m
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Simulations – ruby laser
• We tested our system with higher repetition rate pulses. During the
initial tests an optical damage in the laser rod occured.
f rod  1 / Q
• The simulation predicted spiking behaviour at f rod  20m.
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Conclusion
• We have present a laser model for a Q – switched solid state
laser
• The model is used in the development of a new Fotona laser
planned to be used in the tatoo removal application.
• We have compared simulations and experiment and have
seen a good agreement.
• All obtained results will be considered in the next development
iteration.