Formation of quasi-solitons in transverse confined ferromagnetic
film media
A.A. Serga 1
Technische Universität Kaiserslautern, Department of Physics and
Forschungsschwerpunkt MINAS, D - 67663 Kaiserslautern, Germany
M. Kostylev 2
School of Physics, The University of Western Australia, 35 Stirling
Highway, Crawley WA 6009, Australia
St.Petersburg Electrotechnical University, 197376, St.Petersburg, Russia
B. Hillebrands
Technische Universität Kaiserslautern, Department of Physics and
Forschungsschwerpunkt MINAS, D - 67663 Kaiserslautern, Germany
Abstract The formation of quasi-2D spin-wave waveforms in longitudinally
magnetized stripes of ferrimagnetic film was observed by using time- and space-
resolved Brillouin light scattering technique. In the linear regime it was found
that the confinement decreases the amplitude of dynamic magnetization near the
lateral stripe edges. Thus, the so-called effective dipolar pinning of dynamic mag-
netization takes place at the edges.
In the nonlinear regime a new stable spin wave packet propagating along a
waveguide structure, for which both transversal instability and interaction with
the side walls of the waveguide are important was observed. The experiments and
a numerical simulation of the pulse evolution show that the shape of the formed
waveforms and their behavior are strongly influenced by the confinement.
We report on the observation of a new type of a stable, two-dimensional
nonlinear spin wave packet propagating in a magnetic waveguide structure
and suggest a theoretical description of our experimental findings. Stable
two-dimensional spin wave packets, so-called spin wave bullets, were previ-
ously observed, however solely in long and wide samples of a thin ferrimag-
netic film of yttrium-iron-garnet (YIG) [1, 2, 3], that were practically un-
1Email address: serha@rhrk.uni-kl.de
2Email address: kostylev@cyllene.uwa.edu.au
bounded in both in-plane directions compared to the lateral size of the spin
wave packets and the wavelength of the carrier spin wave. In a waveguide
structure, where the transverse dimension is comparable to the wavelength,
up to day only quasi one-dimensional nonlinear spin wave objects were ob-
served, which are spin wave envelope solitons. Here a typical system is a
narrow (' 1-2mm) stripe of a YIG ferrite film [4, 5]. Both for solitons and bul-
lets the spreading in dispersion is compensated by the longitudinal nonlinear
compression. Concerning the transverse dimension, solitons have a cosine-
like amplitude distribution due to the lateral confinement in the waveguide,
whereas bullets show a transverse nonlinear instability compensating pulse
widening due to diffraction and leading to transverse confinement.
Here we report on the observation of a new stable spin wave packet prop-
agating along a waveguide structure, for which both transversal instability
and interaction with the side walls of the waveguide are important.
The experiments were carried out using a longitudinally magnetized long
YIG film stripe of 2.5mm width and 7µm thickness. The magnetizing field
was 1831Oe. The spin waves were excited by a microwave magnetic field
created with a microstrip antenna of 25µm width placed across the stripe
and driven by electromagnetic pulses of 20ns duration at a carrier frequency
of 7.125GHz. As is well known the backward volume magnetostatic spin
wave (BVMSW) [6] excited in the given experimental configuration is able to
form both envelope solitons and bullets [4], depending on the geometry. The
spatio-temporal behavior of the traveling BVMSW packets was investigated
by means of space- and time-resolved Brillouin light scattering spectroscopy
The obtained results are demonstrated in Fig. 1 where the spatial distri-
butions of the intensity of the spin wave packets are shown for given moments
of time. The spin wave packets propagate here from left to right and decay
in the course of their propagation along the waveguide because of magnetic
loss. The left set of diagrams corresponds to the linear case. The power of
the driving electromagnetic wave is 20mW. The right set of diagrams corre-
sponding to the nonlinear case was collected for a driving power of 376mW.
Differences between these two cases are clearly observed. First of all the
linear spin wave packet is characterized by a cosine-like lateral profile while
the cross section of the nonlinear pulse is sharply modified relative to the
linear case and has a pronounced bell-like shape. Second, the intensity of the
linear packet decays monotonically with time while the intensity of the non-
linear packet initially increases because of its strong transversal compression
(see the second diagram from the top in Fig. 1).
Both of these nonlinear features provide clear evidence for the develop-
Figure 1: Bullet formation in the transversally confined yttrium-iron-garnet
film.
ment of a transversal instability and bullet formation. It is interesting that
the bell-like cross-section shape survives even at the end of the propaga-
tion distance when the pulse intensity decreases more than ten times and
the nonlinear contribution to the spin wave dynamics should considerably
diminish.
In order to interpret the experimental result we have assumed that the de-
velopment of nonlinear instabilities in a laterally confined medium is strongly
modified by a quantization of the spin wave spectrum. That is why we have
transformed the two-dimensional Nonlinear Schrödinger Equation tradition-
ally used for the analysis of bullet dynamics [4] into a system of coupled
equations for amplitudes of the spin wave width modes. The specific form of
the discrete set of these orthogonal modes is defined by the actual boundary
conditions at the lateral edges of the stripe. We developed a two-dimensional
theory of linear spin-wave dynamics in magnetic stripes. As an important
outcome we found that the Guslienko-Slavins effective boundary condition [8]
for dynamic magnetization at the stripe lateral edges, being initially derived
for spin waves with vanishing longitudinal wavenumbers, is also valid in the
case of propagating width modes with non-vanishing longitudinal wavenum-
bers [9] . The effective boundary condition shows that the magnetization
vector at the lateral stripe edges is highly pinned, that means that the am-
plitude of dynamic magnetization practically vanishes at the edges. For
simplicity it is even possible to consider the stripe width modes to be totally
pinned at the stripe lateral edges. As seen from Fig. 1 this conclusion is in a
good agreement with the experiment.
The analysis of the system of nonlinear equations derived from the Non-
linear Schrödinger Equation shows that the formation of the two-dimensional
waveform can be considered as an enrichment of the spectrum of the width
modes. The partial waveforms carried by the modes have the same carrier
frequencies equal to that of the initial signal and the carrier wave numbers
which satisfy the dispersion relations for the modes. In the linear regime all
the modes are orthogonal to each other and do not interact. In the nonlinear
(high amplitude) regime the width modes become intercoupled by the four-
wave nonlinear interaction, resulting in an intermodal energy transfer and
the mode spectrum enrichment.
As the spin wave input antenna effectively generates only the lowest width
mode, the initial waveform launched in the stripe is determined by it solely.
Therefore to understand the underlaying physics of quasi-bullet formation it
is necessary to consider the nonlinear interaction of higherorder width modes
with it.
Our theoretical analysis shows that the interaction of the lowest width
mode (n = 1) with higherorder modes is different for odd and even higher or-
der modes. While interacting with even modes, the lowest width mode plays
the role of the pumping wave. This parametrically transfers its energy to the
higher width modes. The interaction is purely parametric and therefore a
threshold process. It needs an initial signal to start the process. This signal
usually is a thermally excited mode. Therefore the amplified waveform needs
a large distance of propagation and a group velocity equal to the velocity of
the lowest width mode in order to reach the soliton amplitude level. If there
is a damping of the pumped wave, even modes will never reach an amplitude
comparable with that of the lowest mode. As a result they can contribute to
the nonlinear waveform profile only, if the amplitude of the initial waveform
is far beyond the threshold of soliton formation.
Interaction of modes of the same type of symmetry are described by a
parametric term as well as by an additional pseudo-linear (tri-linear) exci-
tation term, playing the role of an external source of excitation. Such a
pseudo-linear excitation is a threshold-free process. In contrast to paramet-
ric processes it does not need an initial amplitude value to start the the
process. The pseudo-linear excitation is possible only due to the effective
dipolar pinning of the magnetization at the stripe edges. If the edge spins
were unpinned, the interaction of all the width modes would be purely para-
metric.
The purely parametric mechanism of developing a transversal instability
is typical for the process of bullet formation from a plane-wave waveform in
an unconfined medium, which distinguishes it from the process of soliton and
bullet formation in the waveguide structures.
In contrast, the transverse instability of a wave packet in a confined
medium starts as a pseudolinear excitation of higher-order width modes.
This mechanism ensures a rapid growth of the symmetric n = 3 mode up
to the level where the parametric mechanism starts to work. After that the
main mode together with the n = 3 mode are capable to rapidly generate
a large set of yet higher modes through both pseudo-linear and parametric
mechanisms.
Our theory shows that the efficiency of both nonlinear interaction mech-
anisms (parametric and tri-linear) strongly depends on the group velocity
difference of modes and the initial length of the nonlinear pulse. In larger
stripes the group velocities of modes are closer to each other. As a result
the nonlinearly generated higher-order modes longer remain within the pump
pulse. If the pulse is long enough, they reach significant amplitudes and a
bullet-like waveform is formed. In narrower stripes the group velocity differ-
ence is larger, and consequently the nonlinearly generated highorder wave-
forms leave faster the pumping area. As a result, for the same pulse length,
they do not reach significant amplitudes. The nonlinear steepening results
Figure 2: Lateral shapes of the nonlinear SW packets. 1 and 2 – theoretical
results calculated for the ferrite stripes of width of 2.5mm and 1mm , respec-
tively. 3 and 4 – experimental profiles observed in YIG waveguides of width
of 2.5mm and 1mm, respectively. 1 and 3: bullets. 2 and 4: solitons.
in the transformation of the lowest mode into a soliton.
The results of our calculations of the lateral shapes of the nonlinear spin
wave packets in wide (2.5mm) and narrow 1mm ferrite stripes are shown in
Fig. 2. The excellent correspondence with the experimental data provides
good evidence for the validity of the developed theory.
Support by the Deutsche Forschungsgemeinschaft, the Australian Re-
search Council, and Russian Foundation for Basic Research is gratefully ac-
knowledged.
References
[1] O. Büttner, M. Bauer, S.O. Demokritov, B. Hillebrands, Yu.S. Kivshar,
V. Grimalsky, Yu. Rapoport, A.N. Slavin, 61, 11576 (2000).
[2] A.A. Serga, B. Hillebrands, S.O. Demokritov, A.N. Slavin, 92, 117203
(2004).
[3] A.A. Serga, B. Hillebrands, S.O. Demokritov, A.N. Slavin, P. Wierzbicki,
V. Vasyuchka, O. Dzyapko, A. Chumak, 94, 167202 (2005).
[4] A.N. Slavin, O. Büttner, M. Bauer, S.O. Demokritov, B. Hillebrands,
M.P. Kostylev, B.A. Kalinikos, V. Grimalsky, Yu. Rapoport, Chaos 13,
693 (2003).
[5] M. Chen, M.A. Tsankov, J.M. Nash, C.E. Patton, 49, 12773 (1994).
[6] F.R. Morgenthaler, Proceedings of the IEEE 76, 138 (1988).
[7] S.O. Demokritov, B. Hillebrands, A.N. Slavin, Phys. Rep. 348, 441
(2001).
[8] K.Y.Guslienko, S.O.Demokritov, B.Hillebrands, and A.N.Slavin, 66,
132402 (2002).
[9] M.Kostylev, J.-G. Hu, and R.L.Stamps, 90, 012507 (2007).

  The formation of quasi-2D spin-wave waveforms in longitudinally magnetized
stripes of ferrimagnetic film was observed by using time- and space-resolved
Brillouin light scattering technique. In the linear regime it was found that
the confinement decreases the amplitude of dynamic magnetization near the
lateral stripe edges. Thus, the so-called effective dipolar pinning of dynamic
magnetization takes place at the edges.
  In the nonlinear regime a new stable spin wave packet propagating along a
waveguide structure, for which both transversal instability and interaction
with the side walls of the waveguide are important was observed. The
experiments and a numerical simulation of the pulse evolution show that the
shape of the formed waveforms and their behavior are strongly influenced by the
confinement.

Formation of quasi-solitons in transverse confined ferromagnetic
film media
A.A. Serga 1
Technische Universität Kaiserslautern, Department of Physics and
Forschungsschwerpunkt MINAS, D - 67663 Kaiserslautern, Germany
M. Kostylev 2
School of Physics, The University of Western Australia, 35 Stirling
Highway, Crawley WA 6009, Australia
St.Petersburg Electrotechnical University, 197376, St.Petersburg, Russia
B. Hillebrands
Technische Universität Kaiserslautern, Department of Physics and
Forschungsschwerpunkt MINAS, D - 67663 Kaiserslautern, Germany
Abstract The formation of quasi-2D spin-wave waveforms in longitudinally
magnetized stripes of ferrimagnetic film was observed by using time- and space-
resolved Brillouin light scattering technique. In the linear regime it was found
that the confinement decreases the amplitude of dynamic magnetization near the
lateral stripe edges. Thus, the so-called effective dipolar pinning of dynamic mag-
netization takes place at the edges.
In the nonlinear regime a new stable spin wave packet propagating along a
waveguide structure, for which both transversal instability and interaction with
the side walls of the waveguide are important was observed. The experiments and
a numerical simulation of the pulse evolution show that the shape of the formed
waveforms and their behavior are strongly influenced by the confinement.
We report on the observation of a new type of a stable, two-dimensional
nonlinear spin wave packet propagating in a magnetic waveguide structure
and suggest a theoretical description of our experimental findings. Stable
two-dimensional spin wave packets, so-called spin wave bullets, were previ-
ously observed, however solely in long and wide samples of a thin ferrimag-
netic film of yttrium-iron-garnet (YIG) [1, 2, 3], that were practically un-
1Email address: serha@rhrk.uni-kl.de
2Email address: kostylev@cyllene.uwa.edu.au
bounded in both in-plane directions compared to the lateral size of the spin
wave packets and the wavelength of the carrier spin wave. In a waveguide
structure, where the transverse dimension is comparable to the wavelength,
up to day only quasi one-dimensional nonlinear spin wave objects were ob-
served, which are spin wave envelope solitons. Here a typical system is a
narrow (' 1-2mm) stripe of a YIG ferrite film [4, 5]. Both for solitons and bul-
lets the spreading in dispersion is compensated by the longitudinal nonlinear
compression. Concerning the transverse dimension, solitons have a cosine-
like amplitude distribution due to the lateral confinement in the waveguide,
whereas bullets show a transverse nonlinear instability compensating pulse
widening due to diffraction and leading to transverse confinement.
Here we report on the observation of a new stable spin wave packet prop-
agating along a waveguide structure, for which both transversal instability
and interaction with the side walls of the waveguide are important.
The experiments were carried out using a longitudinally magnetized long
YIG film stripe of 2.5mm width and 7µm thickness. The magnetizing field
was 1831Oe. The spin waves were excited by a microwave magnetic field
created with a microstrip antenna of 25µm width placed across the stripe
and driven by electromagnetic pulses of 20ns duration at a carrier frequency
of 7.125GHz. As is well known the backward volume magnetostatic spin
wave (BVMSW) [6] excited in the given experimental configuration is able to
form both envelope solitons and bullets [4], depending on the geometry. The
spatio-temporal behavior of the traveling BVMSW packets was investigated
by means of space- and time-resolved Brillouin light scattering spectroscopy
The obtained results are demonstrated in Fig. 1 where the spatial distri-
butions of the intensity of the spin wave packets are shown for given moments
of time. The spin wave packets propagate here from left to right and decay
in the course of their propagation along the waveguide because of magnetic
loss. The left set of diagrams corresponds to the linear case. The power of
the driving electromagnetic wave is 20mW. The right set of diagrams corre-
sponding to the nonlinear case was collected for a driving power of 376mW.
Differences between these two cases are clearly observed. First of all the
linear spin wave packet is characterized by a cosine-like lateral profile while
the cross section of the nonlinear pulse is sharply modified relative to the
linear case and has a pronounced bell-like shape. Second, the intensity of the
linear packet decays monotonically with time while the intensity of the non-
linear packet initially increases because of its strong transversal compression
(see the second diagram from the top in Fig. 1).
Both of these nonlinear features provide clear evidence for the develop-
Figure 1: Bullet formation in the transversally confined yttrium-iron-garnet
film.
ment of a transversal instability and bullet formation. It is interesting that
the bell-like cross-section shape survives even at the end of the propaga-
tion distance when the pulse intensity decreases more than ten times and
the nonlinear contribution to the spin wave dynamics should considerably
diminish.
In order to interpret the experimental result we have assumed that the de-
velopment of nonlinear instabilities in a laterally confined medium is strongly
modified by a quantization of the spin wave spectrum. That is why we have
transformed the two-dimensional Nonlinear Schrödinger Equation tradition-
ally used for the analysis of bullet dynamics [4] into a system of coupled
equations for amplitudes of the spin wave width modes. The specific form of
the discrete set of these orthogonal modes is defined by the actual boundary
conditions at the lateral edges of the stripe. We developed a two-dimensional
theory of linear spin-wave dynamics in magnetic stripes. As an important
outcome we found that the Guslienko-Slavins effective boundary condition [8]
for dynamic magnetization at the stripe lateral edges, being initially derived
for spin waves with vanishing longitudinal wavenumbers, is also valid in the
case of propagating width modes with non-vanishing longitudinal wavenum-
bers [9] . The effective boundary condition shows that the magnetization
vector at the lateral stripe edges is highly pinned, that means that the am-
plitude of dynamic magnetization practically vanishes at the edges. For
simplicity it is even possible to consider the stripe width modes to be totally
pinned at the stripe lateral edges. As seen from Fig. 1 this conclusion is in a
good agreement with the experiment.
The analysis of the system of nonlinear equations derived from the Non-
linear Schrödinger Equation shows that the formation of the two-dimensional
waveform can be considered as an enrichment of the spectrum of the width
modes. The partial waveforms carried by the modes have the same carrier
frequencies equal to that of the initial signal and the carrier wave numbers
which satisfy the dispersion relations for the modes. In the linear regime all
the modes are orthogonal to each other and do not interact. In the nonlinear
(high amplitude) regime the width modes become intercoupled by the four-
wave nonlinear interaction, resulting in an intermodal energy transfer and
the mode spectrum enrichment.
As the spin wave input antenna effectively generates only the lowest width
mode, the initial waveform launched in the stripe is determined by it solely.
Therefore to understand the underlaying physics of quasi-bullet formation it
is necessary to consider the nonlinear interaction of higherorder width modes
with it.
Our theoretical analysis shows that the interaction of the lowest width
mode (n = 1) with higherorder modes is different for odd and even higher or-
der modes. While interacting with even modes, the lowest width mode plays
the role of the pumping wave. This parametrically transfers its energy to the
higher width modes. The interaction is purely parametric and therefore a
threshold process. It needs an initial signal to start the process. This signal
usually is a thermally excited mode. Therefore the amplified waveform needs
a large distance of propagation and a group velocity equal to the velocity of
the lowest width mode in order to reach the soliton amplitude level. If there
is a damping of the pumped wave, even modes will never reach an amplitude
comparable with that of the lowest mode. As a result they can contribute to
the nonlinear waveform profile only, if the amplitude of the initial waveform
is far beyond the threshold of soliton formation.
Interaction of modes of the same type of symmetry are described by a
parametric term as well as by an additional pseudo-linear (tri-linear) exci-
tation term, playing the role of an external source of excitation. Such a
pseudo-linear excitation is a threshold-free process. In contrast to paramet-
ric processes it does not need an initial amplitude value to start the the
process. The pseudo-linear excitation is possible only due to the effective
dipolar pinning of the magnetization at the stripe edges. If the edge spins
were unpinned, the interaction of all the width modes would be purely para-
metric.
The purely parametric mechanism of developing a transversal instability
is typical for the process of bullet formation from a plane-wave waveform in
an unconfined medium, which distinguishes it from the process of soliton and
bullet formation in the waveguide structures.
In contrast, the transverse instability of a wave packet in a confined
medium starts as a pseudolinear excitation of higher-order width modes.
This mechanism ensures a rapid growth of the symmetric n = 3 mode up
to the level where the parametric mechanism starts to work. After that the
main mode together with the n = 3 mode are capable to rapidly generate
a large set of yet higher modes through both pseudo-linear and parametric
mechanisms.
Our theory shows that the efficiency of both nonlinear interaction mech-
anisms (parametric and tri-linear) strongly depends on the group velocity
difference of modes and the initial length of the nonlinear pulse. In larger
stripes the group velocities of modes are closer to each other. As a result
the nonlinearly generated higher-order modes longer remain within the pump
pulse. If the pulse is long enough, they reach significant amplitudes and a
bullet-like waveform is formed. In narrower stripes the group velocity differ-
ence is larger, and consequently the nonlinearly generated highorder wave-
forms leave faster the pumping area. As a result, for the same pulse length,
they do not reach significant amplitudes. The nonlinear steepening results
Figure 2: Lateral shapes of the nonlinear SW packets. 1 and 2 – theoretical
results calculated for the ferrite stripes of width of 2.5mm and 1mm , respec-
tively. 3 and 4 – experimental profiles observed in YIG waveguides of width
of 2.5mm and 1mm, respectively. 1 and 3: bullets. 2 and 4: solitons.
in the transformation of the lowest mode into a soliton.
The results of our calculations of the lateral shapes of the nonlinear spin
wave packets in wide (2.5mm) and narrow 1mm ferrite stripes are shown in
Fig. 2. The excellent correspondence with the experimental data provides
good evidence for the validity of the developed theory.
Support by the Deutsche Forschungsgemeinschaft, the Australian Re-
search Council, and Russian Foundation for Basic Research is gratefully ac-
knowledged.
References
[1] O. Büttner, M. Bauer, S.O. Demokritov, B. Hillebrands, Yu.S. Kivshar,
V. Grimalsky, Yu. Rapoport, A.N. Slavin, 61, 11576 (2000).
[2] A.A. Serga, B. Hillebrands, S.O. Demokritov, A.N. Slavin, 92, 117203
(2004).
[3] A.A. Serga, B. Hillebrands, S.O. Demokritov, A.N. Slavin, P. Wierzbicki,
V. Vasyuchka, O. Dzyapko, A. Chumak, 94, 167202 (2005).
[4] A.N. Slavin, O. Büttner, M. Bauer, S.O. Demokritov, B. Hillebrands,
M.P. Kostylev, B.A. Kalinikos, V. Grimalsky, Yu. Rapoport, Chaos 13,
693 (2003).
[5] M. Chen, M.A. Tsankov, J.M. Nash, C.E. Patton, 49, 12773 (1994).
[6] F.R. Morgenthaler, Proceedings of the IEEE 76, 138 (1988).
[7] S.O. Demokritov, B. Hillebrands, A.N. Slavin, Phys. Rep. 348, 441
(2001).
[8] K.Y.Guslienko, S.O.Demokritov, B.Hillebrands, and A.N.Slavin, 66,
132402 (2002).
[9] M.Kostylev, J.-G. Hu, and R.L.Stamps, 90, 012507 (2007).
