APPLIED PHYSICS
NUMERICAL ANALYSIS OF SPATIAL DISTORTIONS EFFECT
ON FEMTOSECOND LASER INTERFERENCE PATTERNING
L. IONEL
National Institute for Laser, Plasma and Radiation Physics, Laser Department, 409 Atomistilor Street,
P.O. Box MG-36, 077125 Magurele-Bucharest, Romania, E-mail: laura.ionel@inflpr.ro
Received January 15, 2015
A numerical method is presented to study the laser beam distortions and their
influence on the interference patterning after propagation through an optical chirp
pulse amplification (CPA) system. This study is based on numerical simulation using
the ray-tracing model from Optica module of MATHEMATICA and it relates the
behavior of the distorted beam in terms of spatial distortions in case of user-induced
misalignments in single-grating stretcher-compressor system. It was shown that this
effect can be used both to control the femtosecond laser interference pattern design
and to improve the quality of the CPA beam profile for ultra-fast micro and nanopatterning experiments using multiple CPA laser beams.
Key words: Chirped Pulse Amplification, spatio-temporal distortions, lithographical
laser processing, multiple beam interferometry.
1. INTRODUCTION
Laser interference lithography is a flexible and potentially method to produce
periodic structures using multiple interfering highly-coherent light beams. In last
decades, the interference of laser was used to create periodic structures such as
distributed feedback laser, field emission displays, liquid crystal displays, optical
gratings, fiber gratings, directed nano-photonics, large area membrane reflectors
and anti-reflectors [1–3]. Thus, the lithographical laser processing became a very
important topic of interest offering the possibility to achieve micro and nanostructures, fast generated with a high processing control. Lately, ultra-fast
lithography method using an interfering femtosecond laser had been implemented
progressively as a complex direct processing technique for 1D, 2D and 3D nanosized and periodic structures generation [4–7]. It was shown that a design
interference pattern can be generated by controlling the number of beams [8],
phase shift and variation of amplitude between the interfering laser beams [9].
Rom. Journ. Phys., Vol. 60, Nos. 9–10, P. 1508–1514, Bucharest, 2015
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Numerical analysis of spatial distortions effect on femtosecond laser interference patterning
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The femtosecond laser systems are usually based on CPA method [10], in
order to generate ultra-short (few-cycle) intense laser pulses [11] with various
applications such as free-electron laser experiments [12], femtosecond
synchronization systems [13], nuclear photonics [14], the formation of self-guided
beam, without Kerr self-focusing [15], micro and nano-structuring [16] which
needs a high quality laser beam profile [17] and simultaneous processing. For this,
the CPA method often implements adaptive optics employing diffractive optical
elements [18–20] or holographic masks [21]. In this paper we investigate the
interference pattern of two and three countering CPA beams respectively by
simulation, in the presence of user-induced misalignments. Our optical system was
designed with Optica software, a ray-tracing package of MATHEMATICA,
following [22]. This optical system was designed to work with a single diffraction
grating for both optical stretcher and compressor [23]. This aspect increases the
difficulty of the alignment generated by the incidence angle on the diffraction
grating. Considering these, we made a study for several distinct incident angle
values on the diffraction grating in the CPA optical system, monitoring the effect
of the user-induced misalignments on the interference pattern. The interference
figure at the output of CPA chain were generated using MATHEMATICA
software taking into account the optical path from the CPA laser system. The
results are relevant for future interference processing experiments using CPA
optical systems.
2. INTERFERENCE PATTERNING
USING MULTIPLE CPA LASER BEAMS
The CPA system contains an optical stretcher, an amplifier and a compressor
designed to work with a laser system with 200 femtosecond pulses based on the
CPA principle, that was used for example in [24–25]. Previous works relate spatiotemporal distortions analysis after propagation through the designed optical CPA
system, in case of user-induced misalignments [26–27]. It was shown that
uncompensated spatial chirp after the CPA chain results in a high order spectral
modulation which not only lengthens the pulse duration but also modifies the laser
beam profile. Also, related to the interference pattern distribution, other work was
reported in [28], where a complementary study of electromagnetic field distribution
in the focal region using the coherent combination of two laser beams had been
performed.
According to these previous studies, in this paper we investigate the
interference pattern evolution after the CPA optical system in case of user-induced
misalignments using two and three laser beams respectively. The sketch of the
CPA optical system together with the adjacent interferometer module are shown in
figure 1. As it can be seen both the stretcher and the compressor use the same
diffraction grating. The double passing through the optical system is ensured by the
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fixed top roof mirror from the stretcher. The optical path differences introduced by
the stretcher are compensated by the two roof mirrors from the compressor. The
optical stretcher and the compressor use the same incident angle on the diffraction
grating. After the CPA chain, the output beam is guided through the interferometer
setup, illustrated on the inset picture. The countering beams are generated by the
two beam splitters included in the interferometer scheme, with an angle of
41 degree between the beams axes and normal direction which coincides with the
third countering CPA beam. The simulated interference patterns are obtained using
a numerical ray-tracing model and it is investigated for each case of user-induced
misalignments in the CPA chain, taking into account the uncompensated spatiotemporal distortions after the compressor.
Fig. 1 – Optical ray-tracing of the CPA interferometric laser system.
3. NUMERICAL SIMULATIONS AND RESULT DISCUSSION
Initially, the optical CPA system was projected to present no spatial
distortions (i.e. pulse front tilt or angular dispersion) using the same incident angle
of 23° both in optical stretcher and compressor. This was assessed as the optimum
value of the incident angle on the single diffraction grating according to the
multiple distortions studies elaborated for our CPA laser system. By using a single
diffraction grating for stretcher and compressor the compensation of the distortions
introduced by the optical stretcher was simplified. For this, it was considered the
previous studies which demonstrated that the pulse front tilt was completely
compensated after the compressor in same user-induced misalignment conditions
while the angular dispersion presents a negligible rate variation (~0.004 rad/nm)
[26] during the grating incident angle variation in range of 15–31º with 4º step.
In this work, first we analyzed the structures generated by two interfering
CPA beams. In our simulation, the user-induced misalignments consist of variation
of the incident angle on the diffraction grating around the ideal value of 23º in
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Numerical analysis of spatial distortions effect on femtosecond laser interference patterning
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range of 22.9–23.1º with 0.05º step in the CPA system. For each case, we obtained
different interference patterns, following the ray-tracing numerical model, taking
into account the rays path and the normal incidence of the laser beam. The position
of the central wavelength of the CPA beam on each optical component varies for
each value of the grating incident angle. The interference pattern variation as a
function of incident angle on the diffraction grating, in case of two CPA beams, is
illustrated in figure 2. It is shown how the slightly misalign in the stretchercompressor system by varying the incident angle on the diffraction grating in the
range of 22.9–23.1º with 0.05º step influences the fringes periodicity, in case of
two laser beams interference. A periodically increase of the interfringe was
observed, in range of 141–242 μm, during the user-induced misalignments as
shown in Figure 2.
Fig. 2 – Comparison of the two beams interference patterns in case of the variation of the incident
angle on the diffraction grating in the range of 22.9–23.1º with 0.05º step and calculated values
of fringes periodicity in the presence of user-induced misalignments.
This aspect can be explained both in terms of the interfering beams phase
shift introduced by the user-induced misalignments and also from the
uncompensated spatial distortions from the CPA chain which are still present after
the compressor and they lead to a spatial deformation of the laser beam profile. The
second mentioned approach was investigated before for grating incident angle
variation in range of 15–31º with 4º step and it had been shown that during the
user-induced misalignments the CPA beam intensity profile has similar aspects and
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dimensions on both planes while spatial deformations are still present due to the
uncompensated spatial distortions in the CPA system. The beam phase shift is
calculated from the ray-tracing model of the laser system at the output of the
compressor for each case of user-induced misalignments in the CPA chain.
The effect of these two approaches on the laser beams interference can also
be observed in case of three overlapping beams, when uniform interference
patterns in lattice are formed, as shown in Figure 3. This set of five pictures
presents the evolution of interference pattern in terms of shapes rotational angle (α)
and shapes spacing (d) in presence of grating incident angle variation in the same
range, previously mentioned. The lowest incident angle value (22.9º) induces very
elongated ellipsoid shapes with a spacing of about 141 µm with the corresponding
shapes rotational angle of 88.5º and phase shift of (5/7)π. In the case of (6/7)π
phase shift corresponding to 23.1º incident angle, the α parameter encounters a
decreasing of about 30º and the shapes spacing is significantly expanded to
242 µm.
Fig. 3 – Three beams interference pattern for regular structures formed by asymmetric interference
in presence of grating incident angle variation.
It is interesting that the shapes size obtained for each case uniformly increases,
generating in this way a uniform lattice, with a well-defined shape spacing. In the
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Numerical analysis of spatial distortions effect on femtosecond laser interference patterning
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case of 23.05º grating incident angle the shape of the ellipsoid is less sharper
compared to the four other cases. The α parameter decreases in the incident angle
variation range of 23–23.1º with almost double rate comparing to the range of
22.9–23º while the shape spacing is uniformly increasing during the user-induced
misalignments, as shown in Figure 3. The shape spacing variation behavior is
related to the output CPA beam phase shift, which presents a uniform increase in
the presence of incident angle variation. The highest shape spacing value
corresponds to the highest beam phase shift for the case of 23.1° grating incident
angle. At (4/5)π phase shift, corresponding to the 23° grating incident angle, the
shapes present a linear distribution oriented closer to the horizontal comparing to
the other four cases. In the cases of (3/4)π and (5/6)π phase shift which correspond
to the grating incident angle of 22.95° and 23.05° respectively, the linear shapes
distribution present an opposite orientation related to the vertical axis. This is the
result of the phase shift influence on the period and unit size of the shapes offering
the possibility of controlling the design of micro and nano-patterning structures.
4. CONCLUSIONS
We demonstrated that uncompensated spatio-temporal distortions after the
single-grating CPA chain in presence of user-induced misalignments (i.e. grating
incident angle variation) result in a high order spectral modulation on the laser
beam profile which influences the interference pattern distribution in case of
multiple CPA beam overlapping. Thus, it can be used as an alternative on-site
method of controlling the interference pattern distribution based on multiple CPA
laser beams. The system can be easily configured to a larger number of beams.
The method presented here opens the perspective of performing a variety of
interference patterns design by controlling only one parameter in the CPA optical
system. The major advantage of this interference patterning method comparing to
the classical lithographical femtosecond laser structuring using defined masks is
the high-flexibility of controlling the structures details such as: shapes rotation
angle, shapes spacing and shape structures.
This effect relates that stretcher-compressor alignment has certain limitations
and it can be used to control the spatial distortions in order to improve the quality
of the CPA beam profile for ultra-fast micro and nano-lithography experiments.
Acknowledgments. This work was supported by UEFISCDI, Code Project PN-II-ID-PCE2012-4-0539, Contract No. 33/2013 and by National Authority for Scientific Research, Project
LAPLAS3 (2015).
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