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Enhanced second harmonic generation efficiency via wavefront shaping Jonathan V. Thompson, Brett H. Hokr, Graham A. Throckmorton, Dawei Wang, Marlan O Scully, and Vladislav V. Yakovlev ACS Photonics, Just Accepted Manuscript • Publication Date (Web): 06 Jun 2017 Downloaded from http://pubs.acs.org on June 9, 2017
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ACS Photonics
Enhanced second harmonic generation e ciency via wavefront shaping Jonathan V. Thompson, Wang,
†
†
Brett H. Hokr,
Marlan O. Scully,
†, ‡
†
Graham A. Throckmorton,
and Vladislav V. Yakovlev
‡
Dawei
∗, †
Texas A&M University, College Station, TX 77843, and Baylor University, Waco, TX 76798 E-mail:
[email protected] Abstract
Nonlinear Crystal
Optical second harmonic generation is a fundamental nonlinear eect with a large impact on laser technology and optical imaging/sensing. For most practical applications of second harmonic generation, high conversion e ciency is required.
Nonlinear Crystal
However, many techniques used to
achieve high e ciency are limited to fabrication methods and optical energy requirements. Here, we investigate and demonstrate substantial enhancement in the conversion e ciency of second harmonic generation in a nonlinear crystal via the application of wavefront shaping. In
Figure 1: Control of the second harmonic gen-
a one-dimensional understanding of second har-
eration conversion e ciency is achieved by ap-
monic generation, a phase oset applied to the
plication of a phase mask to the incident fun-
fundamental wave has no eect on the intensity
damental wave.
of the generated light. We show that when the
Keywords
pump eld is not a plane wave, enhanced conversion e ciency can be controlled by the application of a phase mask to the fundamental
wavefront shaping,
beam. This investigation of the dependence of
second harmonic genera-
tion, optimization algorithm, nonlinear crystal,
conversion e ciency upon the transverse phase
conversion e ciency, nonlinear generation
pro le of the incident pump laser yields the promise of new ways to enhance nonlinear gen-
Optical second harmonic generation (SHG)
eration with limited optical energy and without
was the rst discovered nonlinear optical ef-
the need for speci c fabrication techniques.
fect,
1
and is now widely used for wavelength
∗
conversion in green
†
sensing and chemical structure assessment,
To whom correspondence should be addressed 1 ‡ 2
laser pointers,
and biomedical imaging.
79
optical
26
For example, sec-
ond harmonic generation imaging microscopy has been used to image molecular structures such as collagen
ACS Paragon Plus Environment 1
10
7
in tissues. This is useful for
ACS Photonics
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the detection and research of cancer, and connective tissue disorders.
11
10,12
brosis,
Page 2 of 12
ing of the wavefront does have an eect on the nonlinear process. For example, the transverse
The conversion e ciency of a nonlinear pro-
mode (beam state) of second harmonic light can
cess is a critical parameter for practical ap-
be controlled by changing the phase pro le of
plications.
the incident fundamental beam.
Over the years, multiple methods
40
have been used to improve the e ciency of
In this article, we present theoretical and ex-
second harmonic generation processes including
perimental results which indicate that shaping
the selection of the nonlinear material,
in-
of the phase pro le (i.e. wavefront shaping) of
the use of
the fundamental beam not only controls the sec-
teraction length of the crystal, higher intensities, riodic poling,
18
16
13,15,16
photonic crystals,
13,14
17
or pe-
ond harmonic beam shape, but also substan-
to name a few. When it comes
tially impacts the intensity or conversion e -
to nonlinear biomedical imaging and sensing,
ciency of second harmonic generation, as con-
the above methods are not always applicable;
ceptually illustrated in Fig. 1.
however, the strength of the signal generated
demonstrated using second harmonic conver-
through a nonlinear optical interaction will de-
sion e ciency from a nonlinear crystal (with
termine the ultimate sensitivity and imaging
no scattering present) as a feedback param-
speed and depth ultimate sensitivity, imaging
eter for wavefront shaping optimization algo-
speed, and depth .
rithms. Recent advances in wavefront shaping
19
It is therefore imperative
4143
This eect is
44,45
to explore new ways to increase the nonlinear
techniques
conversion e ciency. These advances will likely
vided e cient and accurate tools to modify the
come from our through better understanding
phase pro le of a laser beam.
of the nonlinear optical process in all three di-
clude search algorithms for feedback-based op-
mensions, and, more particularly, from better
timization
understanding the role of the transverse phase
utilization of these search algorithms with sec-
pro le on of the incident fundamental beam.
ond harmonic generation as feedback produces
A one dimensional theory
20,21
41,46
and technologies
have pro-
These tools in-
of the applied phase mask. The
is often su -
a phase mask for which the generation e ciency
cient to describe the generation of second har-
is enhanced. These techniques have been suc-
monic (SH) light.
Three-dimensional treat-
cessfully demonstrated in highly scattering en-
ment, however, is necessary when the incident
vironments with both second harmonic genera-
beam is either naturally disturbed by dirac-
tion
tion
back.
22,23
and scattering,
24
or through active
47
and two-photon uorescence
48
as feed-
Here, we show enhancements in sec-
preparation of particular beam states rang-
ond harmonic conversion e ciency as high as
ing
and
an order of magnitude when no scattering is
Other
present by implementing similar techniques and
from
Gaussian
other sophisticated
22,25,26
3033
to
Bessel
2729
beams states.
instances that require three-dimensional treat-
iterative search algorithms.
ment include the treatment of arti cial gauge
show enhancements in second harmonic conver-
elds in photon uids,
SHG from photonic
sion e ciency as high as an order of magnitude
intensity-dependent phase shifts due
by implementing an iterative search algorithm.
crystals,
35
34
to pump elds that are not plane-waves,
22,36
In particular, we
as
These results may impact laser technologies in
well as shaping of the second harmonic beam
the form of lower input powers required for fre-
through facet engraving,
quency doubling, as well as increasing the sen-
phy,
38,39
37
nonlinear hologra-
and plasmonic metasurfaces.
33
sitivity of second harmonic biomedical imaging
Of particular interest here is the treatment
and other detection/sensing techniques. These
of nonlinear generation when the wavefront, or
results should also translate to many other non-
phase pro le, of the pump beam is non-uniform.
linear optical conversion methods.
In the one dimensional case (plane wave), a phase shift in the fundamental beam has no impact upon nonlinear generation.
However,
when the pump beam is not a plane wave, shap-
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Photodiodes
6ps Laser
SHGF
Lens Beamsplitter
CCD
Half-wave plate
Lens
Polarizing beam cube
Lens
I0
Glass Slide
Mirror
LCOS SLM
SHG Crytal
SHGB
Mirror
a)
b)
Figure 2: Experimental setup. (a) The spatial phase of a 6 picosecond laser pulse was modi ed via spatial light modulator (SLM). The phase mask was imaged onto the sample with a 4-f imaging setup. For enhanced generation from a nonlinear crystal (b), the forward and backward generated signals (SHG F , SHGB ) as well as a reference signal (I 0 ) were collected with photodiodes. These signals were used to create metrics that drove an optimization algorithm to enhance the second harmonic generation by wavefront shaping.
Numerical Simulations Normalized Conversion Efficiency (SHG/I0)
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ACS Photonics
where
2π
10
represent the fundamental and
= 2ω1 ), respectively, nj is the refractive index for the j th eld, o is the permittivity of free space, and c is the
3π/2
8
j = 1, 2
second harmonic elds ( ω2
speed of light. The electric eld and nonlinear polarization are given by
z = 5 mm
6
π
Phase Mask
E˜j (r, t) = A˜j (r)ei(kj z−ωj t) + c.c. P˜j (r, t) = P˜j (r)e−iωj t + c.c.
4 π/2 2
z = 5 mm z =10 mm 0
200 400 600 800
The tilde indicates complex variables,
z =10 mm
0
Phase Mask
monic generation,
P˜j
is written as
is the
21
P˜1 (r) = 4o deff A˜2 A˜∗1 ei(k2 −k1 )z P˜2 (r) = 2o deff A˜2 e2ik1 z
Simulated enhancement of second
harmonic generation via wavefront shaping for dierent interaction lengths (left).
A˜
complex eld amplitude, and for second har-
Algorithm Iteration Figure 4:
(2)
(3)
1
The opti-
14
mized phase masks are shown on the right.
Here,
Results
of the medium. Solution of the wave equation
is the eective nonlinear coe cient
(Eq. 1) yields the intensity of the two frequencies of light via the relation
Theoretical Model
2 Ij = 2nj 0 c A˜j (z) .
Second harmonic generation is modeled via the three-dimensional wave equation
∇2 E˜j −
deff
(4)
21,22,49
Substitution of Eq. 2 into Eq. 1, and applica-
tion of the slowly n2j ∂ 2 E˜j 1 ∂ 2 P˜j = (1) 2 Paragon Plus Environment ACS c2 ∂t2 o c2 ∂t 3
varying approximation yields
ACS Photonics
2.2
Enhancement in Backward Direction 2π
a)
Normalized Conversion Efficiency (SHGB/I0)
Enhancement in Forward Direction Normalized Conversion Efficiency (SHGF/I0)
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11
2π
b)
10
2 3π/2 1.8 Phase Mask: Large
1.6
π
1.4 π/2 1.2 1
Large Pixels Small Pixels 0
0
200 400 600 800 Phase Mask: Small
9
3π/2
8 7 Phase Mask: Large
6
π
5 4 π/2
3 Large Pixels Small Pixels
2 1
Algorithm Iteration
0
200 400 600 800 Phase Mask: Small
0
Algorithm Iteration
Figure 3: Enhanced conversion e ciency of second harmonic generation in in the forward (a) and backward (b) direction from a nonlinear crystal due to wavefront shaping of the fundamental. For both cases, the nal phase masks as determined by the search algorithm are shown for large (top) and small (bottom) pixel sizes.
a dierential equation for the complex amplitude,
where
22
F1 =
∂ A˜j (r) i 2 ˜ i(ωj )2 ˜ = Pj (r)e−ikj z ∇⊥ Aj (r) + ∂z 2kj 2kj o c2
F2 =
ω22 deff A21 . k 2 c2
(9)
the amplitude has a dependence upon the trans-
The dependence of the eld amplitude on the
verse gradient of the phase. As opposed to the
transverse phase is more apparent by explicitly
one-dimensional case (plane wave), these terms
writing the amplitude and phase as
A˜j (r) = Aj (r)e
2ω12 deff A2 A1 , k 1 c2
and
From the equations in this form, we see that
(5)
iφj (r)
∆k = k2 − 2k1 , ∆φ = φ2 − 2φ1 ,
will survive the absolute value in Eq. 4. Thus, a phase mask applied to the fundamental beam
(6)
will in uence the intensity or conversion e ciency of the generated light.
and separating the real and imaginary parts of Eq. 5. This yields the following system of equa-
Experimental Results
tions.
−1 2 (∂x Aj )(∂x φj ) ∂z Aj (r) = 2kj 2 + (∂y Aj )(∂y φj ) + Aj ∇⊥ φj + (−1)j Fj sin ∆kz + ∆φ −1 2 2 Aj ∂z φj (r) = Aj (∂x φj ) + (∂y φj ) 2kj 2 − ∇⊥ Aj + Fj cos ∆kz + ∆φ
The
enhancement
of
second
harmonic
con-
version e ciency was experimentally demonstrated (see Methods for details) by applying a
(7)
phase mask to a 6 picosecond laser pulse with a wavelength of 1064 nm (Attodyne, APLX-1064532). As depicted in Fig. 2, the phase mask was applied to the fundamental pulse with a spatial light modulator (LCOS-SLM, Hamamatsu X10468-08).
(8)
The applied phase mask was
then imaged with a 4-f telescope con guration (magni cation of 1/8) to a Potassium Titanyl Phosphate (KTiOPO 4 , KTP) nonlinear crystal
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ACS Photonics
(type II, rotated 45
◦
with respect to the beam
agation length, and from 2.3% to 23% for a
polarization). The second harmonic light gener-
10 mm propagation length.
ated in both the forward and backward (re ec-
e ciency from the back side of the crystal is
tion from back surface) directions was measured
taken into account, these values correspond to
with respect to the incident fundamental using
an enhancement in conversion e ciency in the
large area photodiodes (Thorlabs, DET100A).
backward direction from 0.01% to 0.1%.
The conversion e ciency was used as feedback
optimum phase masks calculated by the simu-
for a search algorithm, which determined an op-
lation are also shown in Fig. 4. Here, the phase
timum phase mask for enhancement.
pattern more clearly shows a checkerboard-like
In this
case, we used both a genetic algorithm a continuous sequential algorithm
41
46
and
behavior.
If the re ection
The
While this pattern is not as clearly
in an out-
visible in the experimental results, possibly due
Both algorithms
to experimental factors such as noise and slight
produced similar results; therefore, only results
aberration eects, large phase jumps are clearly
from the genetic algorithm are shown here. Be-
observed.
cause enhancement may also depend upon the
the phase, gives sharp jumps in the transverse
resolution of the phase mask applied, we also
gradient of the phase which in turn eects the
show results for phase masks with low resolu-
conversion e ciency (see Eqs. 7, 8).
ward spiral con guration.
42
This behavior, with large jumps in
tion (large macropixels) and higher resolution
The transverse intensity pro le before and af-
(small macropixels). At the plane of the crystal,
ter wavefront shaping of the fundamental and
these macropixels had sizes of (150 ×180)
second harmonic pulses are shown in Fig. 5.
and (60×60)
µm,
µm
respectively.
Here, a grid-like pattern in both the fundamen-
The second harmonic conversion e ciency in
tal and second harmonic pro les is caused by
both the forward and backward directions as
the wavefront shaping.
a function of algorithm iteration are shown in
positions in the pro le with intensities much
Fig. 3.
brighter than those without shaping.
The second harmonic conversion e -
This causes localized Numer-
ciency, calculated from the second harmonic di-
ical simulation results showing similar behavior
vided by the incident fundamental ( ISHG /I0 ),
are shown in Fig. 6a-b. Here, a grid-like pattern
is shown in Fig. 3 as a function of algorithm
caused by the shaped wavefront is also observed
iteration for both the forward and backward
in the fundamental and generated light at the
directions. For comparison, the conversion ef-
output of the interaction region.
ciencies were normalized by the initial ( at
like patterns are likely due to the phase jumps
wavefront) value, which was 9.7% for the for-
in the transverse direction of the applied phase
ward generated light, and 0.02% for the back-
mask.
ward generated light. After 800 iterations, the
the enhancement of second harmonic conversion
conversion e ciency was enhanced by a factor
is sensitive to the transverse phase gradients.
of two (19.5%) in the forward direction and
Therefore, we expect to see enhancement local-
as much as an order of magnitude (0.2%) in
ized to the regions where the phase gradient is
the backward direction. The nal phase masks
non-zero.
(both large and small macropixels) as deter-
Examination of Eqs. 7 & 8 shows that
Discussion
mined by the algorithm are also shown in the gures. Similar results from numerical calculations are shown in Fig. 4. As depicted in the g-
Further analysis of the theoretical model pro-
ure, numerical simulations were performed for
vides more insight into the eects of wave-
dierent crystal (propagation) thicknesses. Dif-
front shaping upon second harmonic genera-
ferent enhancements were achieved based upon
tion. Taking a cross-section of the second har-
the crystal lengths, but these enhancements
monic intensity as is propagates through the
are not necessarily linear with propagation distance.
These grid-
crystal (Fig. 6c-e) shows that after long propa-
Here, the conversion e ciency was en-
gation distances, the energy in the second har-
hanced from 4.4% to 15.2% for a 5 mm prop-
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ACS Photonics Fundamental Profile
Numerical Simulation Profile
1
1
a)
0.35
b)
0.3
0.8
0.8 0.25
0.6 Flat wavefront, Large pixels
0.6
Shaped wavefront, Large pixels
Fundamental
0.2
Second Harmonic
0.4
0.15
0.4
0.2
0.1 0.2
Flat wavefront, Small pixels
Shaped wavefront, Small pixels
0.05
0 Shaped Fundamental
Second Harmonic Profile
Shaped Second Harmonic
0
1
0
0.8 0.6 Shaped wavefront, Large pixels
0.4
c)
1 0.5 0 -0.5 -1 -1.5
Flat wavefront
0
0.2 Flat wavefront, Small pixels
Shaped wavefront, Small pixels
2.5
5
7.5
1.5
d)
1
1x106
0 -0.5
5x105
-1 -1.5
Shaped wavefront
0
Propagation (mm)
0
Figure 5: Transverse pro le of the fundamental (top) and second harmonic (bottom) pulse af-
2.4 2 1.6 1.2 0.8 0.4 0
Flat Shaped
2.5
5
7.5
1.5x106
5
0
7.5
f)
1.0x106 5.0x105
10
Propagation (mm)
ter the nonlinear crystal with (right side) and
2.5
Propagation (mm)
e)
0
2x106
0.5
Intensity (W/cm2)
Flat wavefront, Large pixels
1.5
Transverse Position (mm)
Transverse Position (mm)
Propagation of Second Harmonic
SHG Power (kW)
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0
-1
0
1
Transverse Position (mm)
without (left side) wavefront shaping applied. Both large and small pixel sizes where used for
Figure 6:
wavefront shaping.
fundamental (a) and second harmonic (b) pulse
Transverse pro le of the simulated
after the nonlinear crystal with and without wavefront shaping applied.
monic eld pumps back into the fundamental eld.
the simulated second harmonic intensity inside
At this point, it is important to recall
the interaction region is also shown for at (c)
that the simulation assumes the wave-vector mismatch is zero ( k2
= 2k1 ).
and shaped (d) incident wavefronts.
This behavior
was also observed by Sheng et al
22
tion gives the power as a function of propagation distance (e). A slice of the intensity at the
tween second harmonic generation and prop-
output of the crystal is also shown (f ).
agation eects when the pump beam is not a
36
Integra-
tion of the intensity over the transverse direc-
and is due
to diraction and the complex interplay be-
plane-wave.
A cross section of
When wavefront shaping is ap-
plied, this parametric pumping of energy back
the turnover in harmonic power occurs with re-
into the fundamental eld is counteracted, and
spect to the back surface of the crystal.
the harmonic eld continues to grow. Integra-
length dependence explains the dierence in en-
tion over the transverse direction shows this ef-
hancement between the forward and backward
fect in the average power as a function of prop-
direction experimental results. Because the sec-
agation distance (Fig. 6e). This indicates that
ond harmonic light detected in the backward
the enhancement attainable via wavefront shap-
direction was likely due to a re ection from the
ing is dependent upon the length of the crys-
back surface of the crystal, this light propa-
tal.
gated through an eectively thicker nonlinear
This dependence is not necessarily linear
with crystal length, but it depends upon where
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ACS Photonics
medium. This indicates that while the enhance-
monic generation in nonlinear crystals. These
ment attainable via wavefront shaping is depen-
results should apply to other nonlinear eects,
dent upon the length of the crystal it is not nec-
and may also be related to arti cial gauge elds
essarily directly or linearly proportional to the
in photon uids.
34
crystal length. More speci cally, Fig. 6e shows
Because there is little to no scattering in-
that the enhancement depends upon the loca-
volved in the problem, certain identi able pat-
tion of the turnover in harmonic power with re-
terns in the phase mask may be responsible for
spect to the exit aperture of the crystal. Thus,
the majority of the enhancement. This has been
the enhancement of generated signal is depen-
observed, with the genetic algorithm converg-
dent upon the crystal length; however, a factor
ing to a phase mask with patterns consisting
of two longer crystal will not necessarily yield
of large (close to
a factor of two larger enhancement. For exam-
ing of this phase pattern for speci c applica-
ple, according to Fig. 6e, signal from a crystal
tions will remove the need for a feedback fed
that is 5 mm long will be enhanced by a factor
search algorithm. Furthermore, more sophisti-
of 3.4 times, whereas signal from a 10 mm long
cated feedback metrics may also provide for the
crystal will be enhanced by a factor of 10. This
extension of this technique to biological applica-
dependence explains the dierence observed in
tions including the enhancement of second har-
the enhancement between the experimental re-
monic images of collagen structures in a mostly
sults in the forward and backward direction.
transparent sample.
Because the second harmonic light detected in
phase jumps. Understand-
Methods
the backward direction was likely due to a re ection from the back surface of the crystal, this light propagated through an eectively thicker
Experimental Setup
nonlinear medium. A slice through the brightest points of the in-
The experimental setup is illustrated in Fig. 2.
tensity pro le at the output of the crystal is
Here, a phase mask was applied to a 1064 nm, 6
shown in Fig. 6f for both shaped and at input wavefronts.
π)
picosecond laser pulse (Attodyne APLX-1064-
This gives better perspective
532) by a liquid crystal on silicon spatial light
to see that not only are the bright intensity
modulator (LCOS-SLM, Hamamatsu X10468-
spikes enhanced, but the darker regions are at
08). Prior to phase shaping, the beam was ex-
least as bright or brighter than generation from
panded by a telescope to ll more of the active
the unshaped fundamental. It also shows how
region of the spatial light modulator.
drastic the intensity spikes may be. While this
A half-
wave plate and polarizing beam cube were used
spatial pro le may not be ideal for all appli-
to control the power and set the polarization
cations, there are instances, including optical
to the phase-only modulation orientation of the
detection, when the spatial pro le is less impor-
LCOS-SLM. The phase mask applied to the
tant than the attainment of greater conversion
beam was imaged onto the face of a nonlinear
e ciencies.
crystal using lenses in a 4-f imaging con gura-
Conclusion
tion. The magni cation of the lenses (Thorlabs,
In conclusion, we have theoretically and experi-
in diameter (0.7 mm, FWHM) than that of the
mentally demonstrated a dependence of second
nonlinear crystal (5 mm).
AC254-400-C-ML, AC254-050-C-ML) was chosen to ensure that the shaped pulse was smaller
The second harmonic light was generated
harmonic conversion e ciency upon the phase pro le of the incident fundamental wave.
by a Potassium Titanyl Phosphate (KTiOPO 4 , ◦
We
have shown that the utilization of wavefront
KTP) nonlinear crystal (type II, rotated 45
shaping techniques can yield a substantial in-
with respect to the beam polarization).
crease to the conversion e ciency of second har-
pulse energy incident upon the crystal was ap-
ACS Paragon Plus Environment 7
The
ACS Photonics
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proximately 2.3
µJ ( uence of 100 µJ/cm2 ) with
a repetition rate of 10 kHz.
Page 8 of 12
els were (150 ×180)
The crystal was
and (60×60)
µm
plane of the crystal.
µm
at the
For the large or small
×
phase matched by hand and the orientation was
macropixels, there were a total of (10
rotated about the optical axis to optimize the
(25
conversion e ciency of the unshaped pulses.
respectively. The genetic algorithm was allowed
The second harmonic light generated (532 nm)
to select a value between 0 and 2 π as the phase
was detected in both the forward and backward
delay for each of the macropixels.
×
11) or
33) macropixels in the modulator array,
directions using large area photodiodes (Thor-
After wavefront optimization via the search
labs DET100A) with glass color lters (Thor-
algorithm, an image of the output from the non-
labs FGB39) to reject the fundamental.
The
linear crystal in the forward direction was ac-
light detected in the backward direction was
quired using imaging lenses (Thorlabs, AC254-
likely due to re ections from the back surface
035-A-ML, AC254-045-A-ML) and a charge-
of the crystal, and was separated from the inci-
coupled device (CCD) (Mightex CGE-B013-U).
dent fundamental for detection by re ection o
The lenses were placed such that the back sur-
a glass slide (1 mm thick). This same glass slide
face of the crystal was imaged to the CCD. A
was also used to direct a small proportion of
glass color lter (Thorlabs FGB39) was used to
the incident beam into an additional large area
reject the fundamental light when imaging the
photodiode (Thorlabs DET100A) to be used as
generated second harmonic (532 nm).
a reference (I0 ). Both the reference beam and
mirror (Thorlabs M254C00) was used to reject
the forward generated beam were further at-
the second harmonic light when imaging the
tenuated by neutral density lters with 2 OD
fundamental.
(Thorlabs ND20B, ND10B) placed directly be-
without disruption to the alignment. Further-
fore the photodiodes.
more, neutral density lters were used to avoid
A cold
These lters were interchanged
The signal from the photodiodes was collected
saturation of the CCD in both cases. Care was
and averaged 128 times by an oscilloscope (Ag-
also taken at each stage to prevent re ections
ilent DSO6034A). This signal was used as feed-
from optics in the forward direction from prop-
back (I2 /I1 ) (ISHG /I0 ) for an optimization al-
agating back into the nonlinear crystal.
gorithm which searched for a favorable phase
Numerical Simulations
mask to enhance the feedback metric; conversion e ciency in this case.
46
A genetic algo-
with a population of 20 individuals cy-
Comparison of the experimental results with
cled over 800 generations was used as the op-
second harmonic theory were performed via nu-
timization algorithm.
merical simulation of Eq. 5.
rithm
The optimization algo-
rithm used was a genetic algorithm
46
Here, a spectral
method in the transverse direction,
with a
×
22
with a
population of 20 individuals per generation, cy-
grid of 128
cled over 800 generations.
The results from
tion with a nite dierence step in the propa-
this algorithm were compared with those from
gation (z ) direction. Enhancements for a total
a continuous sequential algorithm
in an out-
propagation distance of both 5 mm and 10 mm
and found to be
were computed with 128 and 256 total steps
ward spiral con guration similar.
42
41
in the
Therefore, only results from the ge-
z
128 points, was used in conjunc-
direction, respectively. This grid size
netic algorithm have been presented here. Be-
was su cient to produce results comparable to
cause the spatial resolution of the spatial light
established theory for at wavefronts.
modulator does have in uence on the magniused by grouping the pixels of the spatial light
we used an initial fundamental beam waist of 6 2 0.7 mm and intensity of 2 × 10 W/cm (9.8 × 103 W total power). We assumed the eective
modulator into macropixels composed of either
nonlinear coe cient to be
tude of the results, two dierent pixel sizes were
(60
× 72) or (24 × 24) pixels. With an individµm and a magni cation of
the index of refraction,
22
Here
deff = 1.4 cm/V, and n = 1.8, for both the
ual pixel size of 20
fundamental (1064 nm) and second harmonic
1/8 from the 4-f imaging setup, these macropix-
(532 nm) waves.
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It is also important to note
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ACS Photonics
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Graphical TOC Entry
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2.2
Nonlinear Crystal
Normalized Conversion Efficiency
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Nonlinear Crystal
2 1.8 1.6 1.4 1.2 1
0
200 400 600 800 Algorithm Iteration
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