# Polarization-Controlled Nonlinear Computer-Generated Holography

Lisa Ackermann  Institute of Photonic Technologies, Friedrich-Alexander-
Universität Erlangen-Nürnberg, Konrad-Zuse-Straße 3/5, 91052 Erlangen, Germany
School of Advanced Optical Technologies (SAOT), Friedrich-Alexander-
Universität Erlangen-Nürnberg, Paul-Gordan-Straße 6, 91052 Erlangen, Germany
Corresponding author<EMAIL_ADDRESS>Clemens Roider
Institute of Photonic Technologies, Friedrich-Alexander-Universität Erlangen-
Nürnberg, Konrad-Zuse-Straße 3/5, 91052 Erlangen, Germany  Kristian Cvecek
Institute of Photonic Technologies, Friedrich-Alexander-Universität Erlangen-
Nürnberg, Konrad-Zuse-Straße 3/5, 91052 Erlangen, Germany  School of Advanced
Optical Technologies (SAOT), Friedrich-Alexander-Universität Erlangen-
Nürnberg, Paul-Gordan-Straße 6, 91052 Erlangen, Germany  Nicolas Barré
Institute of Photonic Technologies, Friedrich-Alexander-Universität Erlangen-
Nürnberg, Konrad-Zuse-Straße 3/5, 91052 Erlangen, Germany  School of Advanced
Optical Technologies (SAOT), Friedrich-Alexander-Universität Erlangen-
Nürnberg, Paul-Gordan-Straße 6, 91052 Erlangen, Germany  Christian Aigner
Institute of Photonic Technologies, Friedrich-Alexander-Universität Erlangen-
Nürnberg, Konrad-Zuse-Straße 3/5, 91052 Erlangen, Germany  Michael Schmidt
Institute of Photonic Technologies, Friedrich-Alexander-Universität Erlangen-
Nürnberg, Konrad-Zuse-Straße 3/5, 91052 Erlangen, Germany  School of Advanced
Optical Technologies (SAOT), Friedrich-Alexander-Universität Erlangen-
Nürnberg, Paul-Gordan-Straße 6, 91052 Erlangen, Germany

###### Abstract

Dynamic phase-only beam shaping with a liquid crystal spatial light modulator
is a powerful technique for tailoring the intensity profile or wave front of a
beam. While shaping and controlling the light field is a highly researched
topic, dynamic nonlinear beam shaping has hardly been explored so far. One
potential reason is that generating the second harmonic is a degenerate
process as it mixes two fields at the same frequency. To overcome this
problem, we propose the use of type II phase matching as a control mechanism
to distinguish between the two fields. Our experiments demonstrate that
distributions of arbitrary intensity can be shaped in the frequency-converted
field at the same quality as for linear beam shaping and with conversion
efficiencies similar to without beam shaping. We envision this method as a
milestone toward beam shaping beyond the physical limits of liquid crystal
displays by facilitating dynamic phase-only beam shaping in the ultraviolet
spectral range.

##  1 Introduction

The first operating laser in the early 1960s [1] was the dawn for many
research fields in modern optics although some of their fundamental effects
were already demonstrated or proposed in theory decades earlier. Holography
and nonlinear optics emerged independently from each other but both fields
benefited from the high coherence and high power of new light sources.
Holography is based on the interference of light waves and incorporates phase
and amplitude information to go beyond photography. Dynamic phase-only beam
shaping with a liquid crystal spatial light modulator (LC-SLM) is a method
emerging from holography for arbitrarily controlling the intensity
distribution of the beam with many applications in research [2, 3, 4] and
industry [5, 6, 7]. As this method only modulates the wave front, there are no
significant losses. As a drawback, liquid crystal displays are technically
limited to the visible, near-infrared and mid-infrared spectral ranges. This
is not an insurmountable problem, as frequency conversion processes such as
second harmonic generation or sum frequency generation are coherent processes
that preserve the phase of the impinging fundamental wave. Combining nonlinear
optics and holography allows for shaping the light field at the fundamental
while achieving the targeted outcome at the frequency-converted field. Even
though both research fields can be combined, the concept of nonlinear
holography is only currently emerging.
Yariv showed decades ago that four wave mixing can be interpreted as
holographic recording and reconstruction and proposed using it for the
realization of real time holography [8]. In this process, the interaction
between the fields can be interpreted as one field diffracted by the shaped
pattern of another field. Meanwhile many investigations followed on the
nonlinear conversion of structured light for the conservation of singularities
[9, 10] and orbital or spin angular momentum and vortex beams [11, 12, 13, 14,
15, 16, 17, 18, 19, 20, 21, 22, 23]. Here, the physical principles of the
nonlinear conversion of structured light have been well-explored and several
works use beam characteristics such as polarization [18], differing
wavelengths [15, 16, 17] or non-collinear geometries [19] as a control
mechanism for the nonlinear conversion of optical vortices. The recent review
paper published by Buono and Forbes gives an overview on nonlinear optics with
structured light [24].
Currently, there are two major approaches to nonlinear holography: directly
structuring the nonlinear crystal or imaging the plane of an LC-SLM into the
crystal. 3D structuring of the nonlinear crystal leads to a modulation of the
nonlinear susceptibility which shapes the wave front of the emerging light
field. Such elements are called nonlinear photonic crystals as the modulation
of the nonlinear susceptibility affects the beam generation and propagation
[25, 26, 27, 28, 29, 30, 31]. There are demonstrations of a binary hologram in
a nonlinear crystal [32], a structured element combined with structured light
[33, 34] or plasmonic metasurfaces [35]. Such 3D structured nonlinear crystals
act as volume holograms or phased arrays and in theory give more degrees of
freedom as a thin hologram. As their implementation is technically
challenging, the freedom of design is so far strongly limited and furthermore
only static solutions are possible. Such practical limitations motivate the
consideration of thin holograms which are easier to realize.
In a dynamic approach, the plane of a phase-only LC-SLM is directly imaged
into the nonlinear crystal [36, 37, 38, 39]. It thus acts as a thin hologram
and brings the SLM and the nonlinear crystal together in a common plane, where
the SLM dynamically shapes the wave front of the fundamental as if the
hologram was directly structured into the crystal. The phase mask on the SLM
imprints a locally varying phase on the incoming light field which leads to a
modification of the resulting wave front where the slope corresponds to wave
vectors forming the angular spectrum. During the nonlinear interaction, all
light fields which fulfill the phase matching condition interact and create a
new, frequency-converted outgoing wave. If the incoming fields are not
distinguishable, i.e. the fields are degenerate, all wave vectors will add up,
resulting in a new outgoing wave that has an angular spectrum, containing all
possible sums of incoming wave vectors. This complicates beam shaping of the
outgoing, frequency-converted wave as the initially applied angular spectrum
at the SLM is not conserved. Consequently, there needs to be a control
mechanism which makes the incoming fields distinguishable. Current approaches
for second harmonic generation [36, 37, 38, 39] are based on a degenerate type
I phase matching process and thus involve a non-collinear geometry to control
the beam-shaped outcome. This imposes severe limitations in the quality and
the efficiency of the achievable results. We utilize type II phase matching
and demonstrate high-quality and highly-efficient nonlinear beam shaping in
the second harmonic with a LC-SLM in a collinear geometry:

Figure 1: Basic concept of nonlinear beam shaping with linear beam shaping as
reference: The applied wave front generates an angular spectrum of wave
vectors which are centered around the optical axis. Their distribution results
in a target image in the far field. The plane of the SLM is imaged into the
nonlinear crystal. As for type II phase matching only crossed polarization
states mix efficiently, the shaped light field can only mix with the unshaped
one. The addition of the wave vectors results in the tailored light field with
only half the initial deflection angle $\alpha$.

Since the nonlinear conversion is only possible with phase matching, this
constraint can be exploited for beam shaping. In type II phase matching, only
crossed polarization states can mix and through this assignment, the two
fields at the fundamental are clearly distinguishable even when working in a
collinear geometry. This concept is sketched in Figure 1. By keeping one field
unshaped while the other polarization carries the full phase information,
there is an unambiguous assignment of the frequency-doubled wave front for
every wave vector. The experimental implementation is simple as the
polarization is set diagonally before it impinges on the SLM and as the SLM is
polarization-sensitive, only half of the light field is shaped. After imaging
this plane into the nonlinear crystal, only crossed polarization components
add up for the second harmonic field. The same concept can be transferred to
sum frequency generation where the differing frequencies ensure a non-
degenerate mixing process.
Based on those physical principles, we present a setup for nonlinear beam
shaping with high quality while the conversion efficiency is almost defined by
the crystal’s conversion efficiency without beam shaping. Experimental results
are shown in Section 2.1. In Section 2.2 we model the chosen KTP crystal to
estimate the relative conversion efficiency. In Section 3 we discuss
optimizing the conversion efficiency and give an outlook on the applicability
of the method for beam shaping in the ultraviolet spectral range. Furthermore,
Section 4 presents the experimental setup.

##  2 Results

For second harmonic generation, two fields at the fundamental frequency,
$\bm{E_{1}}^{(\omega)}$ and $\bm{E_{2}}^{(\omega)}$, add up to the frequency-
doubled field. From the three wave mixing process follows a quadratic relation
between the fields
$\bm{E}^{(2\omega)}\propto\bm{E_{1}}^{(\omega)}\cdot\bm{E_{2}}^{(\omega)}$.
The phase matching condition

$\displaystyle\bm{k}^{(2\omega)}\stackrel{{\scriptstyle!}}{{=}}\bm{k_{1}}^{(\omega)}+\bm{k_{2}}^{(\omega)},$
(1)

requires the fundamental and the second harmonic to maintain their phase
relationship during propagation through the nonlinear material to avoid
destructive interference and weak conversion. Here $\bm{k_{1/2}}^{(\omega)}$
are the wave vectors at the fundamental frequency and $\bm{k}^{(2\omega)}$ is
the resulting wave vector at the second harmonic. We first present the
experimental results before we model the nonlinear crystal to evaluate the
relative conversion efficiency.

###  2.1 Nonlinear Beam Shaping

Figure 2: Experimentally recorded results for nonlinear beam shaping for two
different crystal lengths and the initial result at the fundamental as
reference. The measured conversion efficiency $\eta$ is denoted above the
individual images.

With polarization as a control mechanism, nonlinear beam shaping can be
performed similar to linear beam shaping. The phase mask for wave front
shaping can be calculated with the same algorithms and tools as for linear
beam shaping. However, as wave front shaping generates an angular spectrum of
wave vectors which are centered around the optical axis, the nonlinear crystal
needs to equally support vectors which slightly deviate from the optimum phase
matching angle. This required angular tolerance or acceptance is one
characteristic of nonlinear crystals and it is defined for an angular
deviation where the conversion efficiency drops to half of its maximum value.
We chose KTP as the nonlinear crystal for our application as it exhibits a
high angular acceptance.
Figure 2 shows experimentally recorded profiles of the second harmonic at
$532\,\textrm{nm}$ and images of the fundamental at $1064\,\textrm{nm}$
(Figure 2 a-d) as reference. The corresponding phase masks are calculated with
the Gerchberg Saxton algorithm [40]. When tailoring the target image in the
far field, the propagation can be mathematically described by a Fourier
transform. Thus, the intensity profile of the target image effectively
reflects the generated angular spectrum. Its size directly shows the required
angles of the wave vectors which need to be supported during frequency
conversion. Consequently, the larger the targeted image size is, the higher
the angular acceptance of the nonlinear KTP crystal needs to be, to equally
convert the second harmonic signal for all wave vectors.
As a shorter nonlinear crystal exhibits a higher angular acceptance, second
harmonic generation is supported for a broad spectrum of wave vectors and
arbitrary target structures can be shaped over a large area. We demonstrate
this in Figure 2 e-h for four different target distributions with a
$2\,\textrm{mm}$ long KTP crystal. All structures are homogeneously converted
and the quality is comparable to the results at the fundamental. We calculate
the conversion efficiency by dividing the measured power of the shaped target
structure at the second harmonic by the power at the fundamental. The
conversion efficiency is around $6-8\,\%$ for all structures. Conversely, a
$9\,\textrm{mm}$ long KTP crystal promises higher conversion efficiency at
lower angular acceptance. The results of the two smaller target structures for
the $9\,\textrm{mm}$ long crystal (Figure 2 i,j) are of the same quality as
for the $2\,\textrm{mm}$ crystal. The conversion efficiency is $>30\,\%$ and
this is only a little less than the initial conversion efficiency of the
nonlinear crystal without beam shaping which is around $40\,\%$. For the
$2\,\textrm{mm}$ crystal the values are even closer with $8.5\,\%$ without and
values around $6-8\,\%$ with beam shaping. Those results demonstrate the
applicability of nonlinear beam shaping in a regime of high conversion
efficiency while maintaining high quality. The homogeneous conversion in the
range of the initial conversion efficiency of the nonlinear crystal is due to
a plateau in the conversion efficiency for small angular deflections. We
further investigate this favorable effect for beam shaping in Section 2.2.
Figure 2 k,l also shows the limitations of nonlinear beam shaping when working
beyond this plateau. The globe and snowflake are almost cut at the borders as
the required angles are not supported by phase matching. As parts of the light
field are not converted, the conversion efficiency decreases. These results
are shown to demonstrate the limitations outside the plateau of high
conversion efficiency. It is nonetheless possible to shape a smaller target
structure which is magnified with a telescope afterwards.

###  2.2 Modeling the Nonlinear Crystal

Figure 3: Normalized conversion efficiency $\eta$ for lateral deflection of
the ordinary/extraordinary beam and projection along the two axes
$\delta\theta$ and $\delta\phi$ for a $2\,\textrm{mm}$ KTP crystal.
Experimental measurements along these axes are in good agreement with the
simulated curves. The refractive index ellipsoid for the biaxial KTP crystal
(dimensions not to scale) shows the phase matching angles
$\theta_{m}=90\,^{\circ}$ and $\phi_{m}=24.8\,^{\circ}$ with the axes for beam
deflection along $\delta\theta$ and $\delta\phi$. The deflection of the
ordinary beam along $\delta\phi$ is indicated with two incoming fields where
the extraordinary beam is polarized along the xy plane of the crystal and the
deflected ordinary beam is polarized along the z-axis.

The wave vector is a function of the wavelength and the corresponding
refractive index, which depends on the wavelength. Consequently, the phase
matching condition in Equation 1 is generally not fulfilled if fields with
different wavelengths are propagating through a dispersive medium.
Birefringence is one effect which is used for phase matching: For a biaxial
crystal the refractive index is determined by three direction-dependent
indices $n_{x}$, $n_{y}$, and $n_{z}$ along the principal axes [41]. The
effective refractive index at any position in space is given by their
projection. The direction of the polarization determines the refractive index
which is perceived by the corresponding light field. Thus, phase matching can
occur under a specific angle of incidence depending on the phase matching
type. Type II phase matching is fulfilled, if two crossed polarization states
mix where their average value of the refractive indices corresponds to the
refractive index of the second harmonic:
$(n^{o/e}(\omega)+n^{e/o}(\omega))\cdot 0.5=n^{e/o}(2\omega)$.
KTP is a biaxial crystal which allows for type I and type II phase matching at
different angles of incidence $\theta$ and $\phi$. As KTP exhibits a higher
conversion efficiency for type II phase matching, this option is the typical
choice. The phase matching angles $\theta_{m}$ and $\phi_{m}$ are marked in
the refractive index ellipsoid in Figure 3. The two red curves along the three
mutually perpendicular planes of the coordinate system indicate the effective
refractive index at the fundamental resulting for a field either polarized
within the plane of incidence or perpendicular to that plane. From a
perspective of the highest symmetry, the beam oscillating perpendicular to the
plane of incidence is called the ordinary beam (dashed lines in Figure 3) as
the refractive index is the same for any angle while a beam with the
polarization axis within the plane of incidence is called the extraordinary
beam (solid lines in Figure 3) as the effective refractive index is given by
the corresponding ellipse defined by the two refractive indices of the axes
spanning that plane. Likewise, the two green curves indicate the refractive
index for the frequency-doubled field. As the blue curve marks the mean value
of the refractive indices at the fundamental for the two polarization states,
its crossing point with the green curve marks the angles for type II phase
matching.
Wave front shaping introduces a spectrum of wave vectors centered around the
optical axis. Even in a collinear geometry, the generated angular spectrum
deviates from the ideal phase matching condition. This deviation causes a
refractive index mismatch and affects the conversion efficiency. Due to the
high angular acceptance of the chosen KTP crystal, a deflection along
$\delta\theta$ and $\delta\phi$ only causes marginal contributions to the
phase mismatch. Our application for beam shaping is different from the general
definition of the angular acceptance which accounts for a deflection of the
total field of the fundamental: The two mixing fields have a slightly
different angle as the shaped field is deflected while the other field
propagates without any modifications along the phase matching angle. This
results in the phase mismatch $\Delta\bm{k}$:

$\displaystyle\begin{split}\Delta\bm{k}=\bm{k_{1}}^{\omega}(\theta_{m},\phi_{m})+\bm{k_{2}}^{\omega}(\theta_{m}+\delta\theta,\phi_{m}+\delta\phi)\\\
-\bm{k}^{(2\omega)}(\theta_{m}+\delta\theta/2,\phi_{m}+\delta\phi/2)\end{split}$
(2)

This scenario is sketched for two wave vectors in front of the refractive
index ellipsoid in Figure 3.
We calculate the refractive index mismatch for a deflection along
$\delta\theta$ and $\delta\phi$ by calculating the corresponding refractive
indices [42] for the wave vectors. In good approximation, the resulting wave
vector at the second harmonic is generated at half of the initial deflection
angle, as this is the mean value between the deflected and undeflected wave
vector at the fundamental. The phase mismatch is thus given by the difference
of the wave vectors projected in the direction of the frequency doubled field.
The relative conversion efficiency can be deduced from the projected mismatch
$\Delta k$ by solving the following equation:

$\displaystyle I_{2\omega}\propto
I^{2}_{\omega}L^{2}\operatorname{sinc}^{2}\left(\frac{\Delta kL}{2}\right)$
(3)

This analytical relation between the second harmonic intensity $I_{2\omega}$
and the intensity at the fundamental $I_{\omega}$ with respect to the crystal
length $L$ and the phase mismatch $\Delta k$ follows when solving the
nonlinear wave equation for the assumption of low depletion and the slowly
varying amplitude approximation [43]. On this basis, we calculate the relative
2D conversion efficiency for a deviation within the plane of $\delta\theta$
and $\delta\phi$ in Figure 3. According to Snell’s law and assuming the small
angle approximation, the internal angle corresponds to the external angle
connected via the effective refractive index which is perceived by the
corresponding wave vector. The graphs in Figure 3 show the external angle, as
this is the relevant value for beam shaping.
Besides calculating the relative conversion efficiency, we perform experiments
for an angular deviation introduced by the SLM for beam shaping. To evaluate
different angles, we apply a blazed grating either in horizontal or vertical
direction on the SLM and image this plane into the nonlinear crystal which has
a length of $2\,\textrm{mm}$. Similar to beam shaping, only one polarization
component is shaped. The corresponding setup is sketched at the bottom of
Figure 3. In the far field we measure the resulting power of the frequency-
doubled field and divide it by the measured power of the field at the
fundamental. This results in two curves for the relative conversion
efficiency, either in the direction of $\delta\theta$ or $\delta\phi$, with
respect to no deflection at $0\,^{\circ}$ at the optimum phase matching angle.
As either the ordinary or extraordinary polarization state can be deflected
within the $\delta\theta$ and $\delta\phi$ plane, two different scenarios are
possible and both are shown in Figure 3. Deflecting the ordinary beam causes a
highly symmetric profile of the relative conversion efficiency and is thus the
favorable option for our application. In addition to the experimentally
measured points, we add the analytically calculated curve of the relative
conversion efficiency.
Simulation and experiment are in good agreement and show that the conversion
efficiency exhibits a broad plateau. Within that range, the light is
homogeneously converted, independently of the exact deflection angle.
Moreover, the conversion efficiency is not significantly reduced with respect
to the actual conversion efficiency without beam shaping. Thus, arbitrary
intensity profiles can be shaped within that angular range without reduction
in homogeneity or efficiency. The acceptance angle approximately reaches $\pm
2\,^{\circ}$ for a $2\,\textrm{mm}$ crystal and $\pm 1\,^{\circ}$ for the
$9\,\textrm{mm}$ long crystal. It makes sense to compare this value with the
maximum deflection angle of the SLM which is determined by
$\sin^{-1}(\lambda/(2u))\cdot 1/M$, where $u$ is the pixel pitch and $M$ is
the magnification between the SLM and the crystal. The two smaller images in
Figure 2 approximately acquire $19\,\%$ of the SLM’s linear field of view,
while the two bigger images acquire $32\,\%$. This corresponds to deflection
angles of approximately $1.1\,^{\circ}$ and $1.9\,^{\circ}$ (bearing in mind
that the plane of the SLM is imaged with a factor of $M=0.25$ into the
nonlinear crystal and thus the maximum deflection angle is $\pm 6\,^{\circ}$).
At the phase matching angle $\theta_{m}=90\,^{\circ}$, the crystal is non-
critically phase-matched along $\delta\theta$. This means when expanding the
mismatch $\Delta k$ in a Taylor series, the first derivative $\partial\Delta
k/\partial\theta=0$ due to the crystal’s symmetry and thus only higher order
terms with a weaker effect on the phase mismatch contribute [44]. Besides the
control mechanism for the wave mixing process it is crucial to choose a
nonlinear crystal with a high angular acceptance. We consider temperature-
controlled noncritical phase matching or quasi-phase matching as another
option but do not discuss this further within the scope of this paper.

##  3 Discussion

Nonlinear beam shaping is a powerful tool as it enables dynamic phase-only
beam shaping in new spectral ranges. To optimize this process, two parameters
should be considered: While the conversion efficiency should be maximized, the
angular acceptance needs to be high enough to ensure homogeneous conversion of
the angular spectrum. As both parameters are mutually dependent, we discuss
the proper choice of the experimental parameters to optimize the outcome.
Moreover, we give an outlook on dynamic phase-only beam shaping in the
ultraviolet spectral range.

###  3.1 Optimizing the Conversion Efficiency with Respect to the Angular
Acceptance

When approaching a high conversion regime, it is more critical to maintain
proper phase matching as the phase mismatch gains an increasing impact on the
efficiency of the converted outcome. At high conversion, the angular range of
tolerance narrows and this decreases the plateau of homogeneous conversion for
the generated angular spectrum. Equation 3 shows the impact of the
experimental parameters on the conversion efficiency while keeping the laser
power constant. Both increasing the intensity and the crystal length promises
higher conversion efficiency. However, both parameters also affect the angular
acceptance of the nonlinear crystal.
The effect of the crystal length $L$ is directly stated in Equation 3. While a
longer crystal increases the relative conversion with $L^{2}$, it also
increases the contribution of the phase mismatch as a multiplier of $\Delta k$
in the argument of the $\operatorname{sinc}$ function. We use the model of the
nonlinear crystal to calculate the relative conversion of the target
structures for the $9\,\textrm{mm}$ crystal as they are shown in Figure 2.

Figure 4: Simulated beam conversion for the $9\,\textrm{mm}$ crystal. The
simulated efficiency is calculated from the integrated relative conversion
multiplied with the efficiency of the $9\,\textrm{mm}$ crystal without beam
shaping (compare Figure 5).

The beam-shaped structures directly reflect the generated angular spectrum and
thus the relative conversion can be simulated when multiplying the target
image with the calculated angular conversion. Figure 4 shows the simulated
results with the corresponding conversion efficiency. This value is calculated
as the total efficiency without beam shaping multiplied by the integrated
relative conversion. Working beyond the limits of phase matching results in
weak conversion and the simulated as well as the measured values for the
conversion efficiency drop significantly. The experimental parameters should
be chosen to work within the pleateau of homogeneous conversion as otherwise
not only the quality gets worse but also the efficiency drops significantly.
The comparison of the simulated and experimentally measured conversion
efficiency explains the strong decrease in efficiency for the two larger
structures with their measured conversion efficiency marked as gray crosses in
Figure 5. This decrease can be traced back to insufficient phase matching as
the structures are chosen larger than the angular acceptance. As simulation
and experiment are in good agreement, proper experimental parameters can be
derived on that basis even in a high conversion regime.

Figure 5: Measured output power vs. input power for second harmonic generation
with a $2\,\textrm{mm}$ and $9\,\textrm{mm}$ KTP crystal without beam shaping.
The crosses indicate the conversion efficiency of the structures in Figure 2
for beam shaping, with the two intentionally cropped structures in gray.

Similar to the crystal length $L$, an increase in intensity also improves the
conversion efficiency. It can either be increased by increasing the power of
the chosen light source or by reducing the illuminated area. As the laser
power is typically technically limited, we will focus on the second case. Here
the plane of the SLM is imaged with a certain demagnification into the
nonlinear crystal to decrease the illuminated area. It is also possible to
illuminate the SLM with a smaller beam but this reduces the number of
illuminated pixels and unnecessarily increases the intensity on the SLM with
respect to potential damage thresholds. It is thus beneficial to design the
telescope with the required demagnification. This demagnification increases
the intensity in the nonlinear crystal and scales the intensity with $1/M^{2}$
to gain higher conversion efficiencies. Likewise, the telescope increases the
angular spectrum as the deflection angles are proportional to $M$. Similar to
the crystal length $L$, this factor also affects the phase mismatch $\Delta k$
in the argument of the $\operatorname{sinc}$ function as it acts as a
multiplier of the initially induced angular spectrum. Consequently, the
intensity needs to be considered when modeling the phase mismatch to determine
optimum parameters.
In our experimental setup we work with a telescope that decreases the image
size by a factor of $0.25$ and thus increases the intensity by a factor of
$16$. This reduction affects the deflection angles and increases them by a
factor of $4$ as the effective pixel size of the SLM changes. We consider that
adjusting the intensity with respect to the required angular spectrum for a
given crystal length is reasonable. The resulting plateau should be exactly
within the required angular range - neither broader nor narrower - to ensure
homogeneous conversion with maximized efficiency.

###  3.2 Towards Dynamic Phase-only Beam Shaping in UV Range

Dynamic beam shaping in the ultraviolet (UV) spectral range is highly limited
as many devices absorb the UV light, including liquid crystal displays. This
paper demonstrates nonlinear beam shaping from the infrared to visible
spectral range. Nonetheless, other conversion processes are possible, reaching
from second harmonic generation at different fundamental frequencies up to
other nonlinear processes like sum frequency generation. Currently, there is
research on new materials with tunable birefringence in the UV spectral range
for spatial light modulators with results showing light modulation at
$303\,\textrm{nm}$ [45]. We see the potential of our method for approaching
even deeper UV spectral ranges. This section presents a few thoughts on proper
nonlinear crystals and conversion processes for dynamic phase-only beam
shaping in the UV range.
Ultrashort pulsed laser systems often feature nonlinear crystals for frequency
conversion, for example to the second or third harmonic. High-energetic short
wavelengths in the UV range of ultrashort pulsed laser systems are thus often
generated with nonlinear frequency conversion from the visible or infrared
range. Consequently, the initial conditions often enable a direct integration
of nonlinear beam shaping into the laser setup and ideally this can be
performed with conversion efficiencies close to the initial values.
To achieve high-quality results, the nonlinear crystal should have a broad
angular acceptance and the conversion process needs to be non-degenerate. Sum
frequency generation with an LBO crystal from $1064\,\textrm{nm}$ and
$532\,\textrm{nm}$ to $355\,\textrm{nm}$ is one option. The major benefit of
this crystal is the high angular acceptance. Similarly, sum frequency
generation with CLBO [46] from $1064\,\textrm{nm}$ and $266\,\textrm{nm}$ to
$213\,\textrm{nm}$ should be possible. In both options the process is
inherently non-degenerate as two different wavelengths mix. While the SLM can
either shape the light field in the infrared or visible spectral range, only
the infrared light can be shaped in the latter case as the SLM can not shape
light at $266\,\textrm{nm}$. With one of the options suggested here or other
nonlinear processes, we see the potential of using nonlinear beam shaping to
approach the UV spectral range. This enables dynamic phase-only beam shaping
beyond the physical limits of the liquid crystal display.

##  4 Materials and Methods

Figure 6: Experimental setup for nonlinear beam shaping: A half wave plate
introduces diagonal polarization on the beam impinging on the SLM. Thus, only
half of the field is shaped while the crossed polarization component remains
unshaped. A telescope images the plane of the SLM into the nonlinear KTP
crystal. Here, the two polarization components mix and shape the targeted
field. After a filter separates the second harmonic from the fundamental, a
lens images the target image, where the second harmonic signal is recorded. As
the shaped result is sensitive to the deflected polarization component, a half
wave plate after the SLM allows for changing the deflected and non-deflected
polarization component.

Nonlinear phase-only beam shaping with a liquid crystal display is possible as
the applied phase information is conserved during the frequency conversion.
However, as second harmonic generation is a degenerate process, the mixing
waves need to be controlled by making the interacting fields distinguishable.
We use the requirements of type II phase matching to unambiguously define the
outcome of the second harmonic. Under this condition, only fields of crossed
polarization can mix efficiently. A half wave plate, abbreviated with
$\lambda/2$ in Figure 6 is placed in front of the SLM to generate diagonal
polarization with respect to the SLM’s orientation. As the liquid crystals are
polarization-sensitive, only half of the field is shaped, while the other part
remains unshaped. Consequently, a shaped component can only mix with the
unshaped plane wave front. In Figure 6 the two polarization components are
indicated as the ordinary and the extraordinary beam. The polarization
configuration in the figure refers to where the ordinary beam is shaped. If
the extraordinary beam is shaped, an additional half wave plate after the SLM
helps to rotate the polarization states correspondingly. This might be
beneficial when working off-axis to separate non-diffracted light as the area
of quasi-homogeneous conversion is shifted away from the center of the optical
axis. A telescope images the plane of the SLM into the nonlinear crystal. This
ensures a homogeneous intensity distribution within the nonlinear crystal
which is required to convert all field components equally within a range of
proper phase matching. In good approximation, the resulting wave vectors at
the second harmonic reduce to half of the initially set angle of the shaped
field component, as this is the sum of the two involved wave vectors.
The SLM is the model LSH0701010 from Hamamatsu (Hamamatsu, Japan) and has a
pixel resolution of $800\,\textrm{px}\times 600\,\textrm{px}$ with a pixel
pitch of $20$ µm. The beam diameter on the LC-SLM is $14\,\textrm{mm}$. To
increase the intensity to obtain a higher conversion efficiency, we design our
telescope to demagnify the beam by $M=0.25$. The corresponding focal lengths
are $f_{1}=200\,\textrm{mm}$ and $f_{2}=50\,\textrm{mm}$. Before the target
distribution is imaged to the far field with a $f_{3}=200\,\textrm{mm}$ lens,
the remaining IR light is blocked with the filter FL532-10 from Thorlabs (NJ,
USA). The laser system Fuego from Time-Bandwidth Products (CA, USA) emits
$10\,\textrm{ps}$ pulses at a repetition frequency of $200\,\textrm{kHz}$ at a
wavelength of $1064\,\textrm{nm}$. The power measurements were done with the
power meter PM10 from Coherent (CA, USA) and PM160 from Thorlabs. We record
RGB images with the camera model UI-3000SE-C-HQ from IDS (Obersulm, Germany).

##  5 Conclusion

Nonlinear computer-generated holography is gaining increasing attention and
there is a lot of research on structuring nonlinear crystals but this process
is technically challenging and static. These deficiencies can be overcome by
thin holograms generated with dynamic liquid crystal displays. However,
research on nonlinear beam shaping is still in its infancy. We present a high-
quality and highly efficient method for nonlinear dynamic phase-only beam
shaping with a simple concept: The mixing process needs to be non-degenerate
by employing a control mechanism and the angular acceptance of the nonlinear
crystal needs to support the angular spectrum generated by wave front shaping.
Based on this, we demonstrate beam shaping at the second harmonic with
conversion efficiencies close to the ones without beam shaping. This is mainly
due to the fact that we work at a plateau of constant conversion efficiency
for the generated angular spectrum. While working within that range, the
conversion efficiency is mainly given by the conversion efficiency without
beam shaping and thus this method is highly efficient. Furthermore, the
quality of the frequency-converted result is of the same quality as for beam
shaping at the fundamental. We see high potential in this approach as it not
only enables highly-efficient and high-quality beam shaping with thin
holograms but due to its simplicity, it can be easily combined with other
elaborated processes. Based on nonlinear beam shaping, the spectral range of
dynamic liquid crystal displays can be extended beyond the physical limits as
we discuss here for the UV spectral range.

## Funding

The authors gratefully acknowledge funding from the Erlangen Graduate School
in Advanced Optical Technologies (SAOT) by the German Research Foundation
(DFG) in the framework of the German excellence initiative.

## Authors’ contributions

LA and CR conceived the method. LA recorded the experimental data and
performed simulations. LA, CR and KC analyzed the data. CA performed
provisional experiments. LA, CR, KC, NB and MS contributed to the manuscript
and MS acquired funding.

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