Enhanced Second Harmonic Generation Efficiency via Wavefront

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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 eciency 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 eciency is required.

Nonlinear Crystal

However, many techniques used to

achieve high eciency are limited to fabrication methods and optical energy requirements. Here, we investigate and demonstrate substantial enhancement in the conversion eciency 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 eciency 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 eciency 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 eciency upon the transverse phase

conversion eciency, nonlinear generation

prole 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 specic 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

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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 eciency 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 prole of

plications.

the incident fundamental beam.

Over the years, multiple methods

40

have been used to improve the eciency 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 prole (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 eciency 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 eciency. These advances will likely

vided ecient and accurate tools to modify the

come from our through better understanding

phase prole 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-

prole 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 eciency

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 eciency 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 articial gauge

show enhancements in second harmonic conver-

elds in photon uids,

SHG from photonic

sion eciency 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 prole, 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-


Page 3 of 12

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 modied 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 coecient

(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 eciency 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 inuence the intensity or conversion eciency 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 eciency 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 conguration (magnication 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 (reec-

eciency 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 eciency in the

large area photodiodes (Thorlabs, DET100A).

backward direction from 0.01% to 0.1%.

The conversion eciency 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 reection

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 eciency (see Eqs. 7, 8).

ward spiral conguration.

42

This behavior, with large jumps in

tion (large macropixels) and higher resolution

The transverse intensity prole 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 eciency in

tal and second harmonic proles is caused by

both the forward and backward directions as

the wavefront shaping.

a function of algorithm iteration are shown in

positions in the prole 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 eciency 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 eciency was en-

gation distances, the energy in the second har-

hanced from 4.4% to 15.2% for a 5 mm prop-


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 prole 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)


Propagation of Second Harmonic

SHG Power (kW)

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Page 6 of 12

0

-1

0

1


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 prole 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 reection 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


This

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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 articial gauge elds

essarily directly or linearly proportional to the

in photon uids.

34

crystal length. More specically, Fig. 6e shows

Because there is little to no scattering in-

that the enhancement depends upon the loca-

volved in the problem, certain identiable 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 specic 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 reection 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 prole 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 prole 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 prole 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

eciencies.

crystal using lenses in a 4-f imaging congura-

Conclusion

tion. The magnication 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 eciency upon the phase prole 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 eciency of second har-

pulse energy incident upon the crystal was ap-


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 eciency 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 reections 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 reection 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 reections

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 eciency 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 conguration 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 sucient to produce results comparable to

cause the spatial resolution of the spatial light

established theory for at wavefronts.

modulator does have inuence 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 coecient to be

tude of the results, two dierent pixel sizes were

(60

× 72) or (24 × 24) pixels. With an individµm and a magnication 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.


14

It is also important to note

Page 9 of 12

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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


Enhanced Second Harmonic Generation Efficiency via Wavefront

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