Spin-Wave Modes in Transition from a Thin Film to a Full Magnonic Crystal
M.\ Langer
R.\ A.\ Gallardo
T.\ Schneider
S.\ Stienen
A.\ Rold\'an-Molina
Y.\ Yuan
K.\ Lenz
J.\ Lindner
P.\ Landeros
J.\ Fassbender
% Surface-modulated magnonic crystals are the natural link between continuous films with sinusoidal spin-wave eigenmodes and one-dimensional magnonic crystals composed of individual nanowires. Nevertheless, the transformation process of the spin-wave modes in this transition remains yet unclear. Here, spin-wave modes in their entire transition from a flat film to a `full' (one-dimensional) magnonic crystal are studied by ferromagnetic resonance (FMR) and micromagnetic simulations. For this purpose, the surface of a pre-patterned thin permalloy film was sequentially ion milled resulting in hybrid structures, referred to as surface-modulated magnonic crystals, with increasing modulation depth. After each step, FMR measurements were carried out in backward-volume and Damon-Eshbach geometry. The evolution of each spin-wave resonance is studied together with the corresponding mode profile obtained by micromagnetic simulations. Simple rules describing the transition of the modes from the film to the modes of the full magnonic crystal are provided unraveling the complexity of spin-wave states in these hybrid systems. % %Magnonic crystals, such as bi-component systems and periodically patterned magnetic thin films, form the most relevant group of systems that today's magnonic devices are based on. %This work contributes to the fundamental understanding of the dynamics of magnonic crystals by investigating the transition from a thin film to an array of free standing wires (one dimensional magnonic crystal). A systematic investigation of the frequency-dependence and the mode profiles of spin waves in such surface-modulated magnonic crystals is carried out. In case of magnonic crystals with a pronounced modulation of internal magnetic properties, spin waves are strongly dynamically coupled leading to hybrid spin-wave states, which together with increasing edge-effects hinder a straight-forward understanding of their physical origin. In this case, the approach to follow the evolution of spin waves from their quasi-harmonic limit, the planar film, up to the limit of a magnonic crystal with strong property-modulation, is specifically suited to unravel this complexity. %Ferromagnetic resonance measurements of surface-modulated magnonic crystals with gradually increasing modulation heights are corroborated by numerical simulations as well as semi-analytical theory. %Intermediate states of thin films with a pronounced surface modulation show complex spin-wave spectra with an even larger amount of spin-wave states accessible compared to the array of free standing wires (one-dimensional magnonic crystal). An increasing level of mode hybridization with the modulation height is reported as well as the analysis of the optimal modulation height to achieve similar mode amplitudes. Furthermore, the physical nature of such hybrid spin-wave modes was unraveled. It was found that the spin-wave spectra split up in two fundamental branches: a wire mode and a trench mode. These branches itself couple to higher ($n>3$) spin-wave modes and evolve gradually from different harmonic spin-wave modes in the continuous film limit, i.e.\ the wire mode arises from the $n=1$ ($1^{\mathrm{st$) backward-volume spin-wave mode and the trench mode originates from the uniform mode. Moreover it was observed, that the $n=2$ backward-volume mode changes with increasing modulation size into an edge mode. %By multistep ion-milling of a single permalloy (Ni$_{80$Fe$_{20$) film, such a transition is performed experimentally. Initially, the surface of the 37 nm thin film is lithographically patterned into a regular stripe lattice of 300 nm periodicity. For each ion-milling step, subsequent frequency-dependent ferromagnetic resonance (FMR) measurements are carried out and corroborated by analytical theory as well as numerical simulations. Both give insight into the spin-wave dispersion and can be employed to map out the spin-wave mode profiles. Effects of dipolar coupling and mode localization are observed to become more and more prominent and are related to the evolution of the internal demagnetizing field distribution. %
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\begin{document}

\title{Spin-Wave Modes in Transition from a Thin Film to a Full Magnonic Crystal} %Title of paper

\author{M.\ Langer}
\affiliation{Helmholtz-Zentrum Dresden -- Rossendorf, Institute of Ion Beam Physics and Materials Research, Bautzner Landstr.\ 400, 01328 Dresden, Germany}
\affiliation{Institute for Physics of Solids, Technische Universit\"at Dresden, Zellescher Weg 16, 01069 Dresden, Germany}
\affiliation{Swiss Light Source, Paul Scherrer Institute, 5232 Villigen, Switzerland}

\author{R.\ A.\ Gallardo}
\affiliation{Departamento de F\'isica, Universidad T\'ecnica Federico Santa Mar\'ia, Avenida Espa$\tilde{n}$a 1680,
2390123 Valpara\'iso, Chile}
\affiliation{Center for the Development of Nanoscience and Nanotechnology (CEDENNA), 917-0124 Santiago, Chile}

\author{T.\ Schneider}
\affiliation{Helmholtz-Zentrum Dresden -- Rossendorf, Institute of Ion Beam Physics and Materials Research, Bautzner Landstr.\ 400, 01328 Dresden, Germany}
\affiliation{Department of Physics, Technische Universit\"at Chemnitz, Reichenhainer Str.\ 70, 09126 Chemnitz, Germany}

\author{S.\ Stienen}
\affiliation{Helmholtz-Zentrum Dresden -- Rossendorf, Institute of Ion Beam Physics and Materials Research, Bautzner Landstr.\ 400, 01328 Dresden, Germany}

\author{A.\ Rold\'an-Molina}
%\affiliation{Instituto de F\'isica, Pontificia Universidad Cat\'olica de Valpara\'iso, Avenida Brasil 2950,
%2390123 Valpara\'iso, Chile}
\affiliation{Universidad de Ays\'en, Calle Obispo Vielmo 62, Coyhaique, Chile}

\author{Y.\ Yuan}
\affiliation{Helmholtz-Zentrum Dresden -- Rossendorf, Institute of Ion Beam Physics and Materials Research, Bautzner Landstr.\ 400, 01328 Dresden, Germany}

\author{K.\ Lenz}
\affiliation{Helmholtz-Zentrum Dresden -- Rossendorf, Institute of Ion Beam Physics and Materials Research, Bautzner Landstr.\ 400, 01328 Dresden, Germany}

\author{J.\ Lindner}
\affiliation{Helmholtz-Zentrum Dresden -- Rossendorf, Institute of Ion Beam Physics and Materials Research, Bautzner Landstr.\ 400, 01328 Dresden, Germany}

\author{P.\ Landeros}
\affiliation{Departamento de F\'isica, Universidad T\'ecnica Federico Santa Mar\'ia, Avenida Espa$\tilde{n}$a 1680,
2390123 Valpara\'iso, Chile}
\affiliation{Center for the Development of Nanoscience and Nanotechnology (CEDENNA), 917-0124 Santiago, Chile}

\author{J.\ Fassbender}
\affiliation{Helmholtz-Zentrum Dresden -- Rossendorf, Institute of Ion Beam Physics and Materials Research, Bautzner Landstr.\ 400, 01328 Dresden, Germany}
\affiliation{Institute for Physics of Solids, Technische Universit\"at Dresden, Zellescher Weg 16, 01069 Dresden, Germany}

\date{\today}

\begin{abstract}
%
Surface-modulated magnonic crystals are the natural link between continuous films with sinusoidal spin-wave eigenmodes and one-dimensional magnonic crystals composed of individual nanowires. Nevertheless, the transformation process of the spin-wave modes in this transition remains yet unclear. Here, spin-wave modes in their entire transition from a flat film to a `full' (one-dimensional) magnonic crystal are studied by ferromagnetic resonance (FMR) and micromagnetic simulations. For this purpose, the surface of a pre-patterned thin permalloy film was sequentially ion milled resulting in hybrid structures, referred to as surface-modulated magnonic crystals, with increasing modulation depth. After each step, FMR measurements were carried out in backward-volume and Damon-Eshbach geometry. The evolution of each spin-wave resonance is studied together with the corresponding mode profile obtained by micromagnetic simulations. Simple rules describing the transition of the modes from the film to the modes of the full magnonic crystal are provided unraveling the complexity of spin-wave states in these hybrid systems.

%
%Magnonic crystals, such as bi-component systems and periodically patterned magnetic thin films, form the most relevant group of systems that today's magnonic devices are based on. 
%This work contributes to the fundamental understanding of the dynamics of magnonic crystals by investigating the transition from a thin film to an array of free standing wires (one dimensional magnonic crystal). A systematic investigation of the frequency-dependence and the mode profiles of spin waves in such surface-modulated magnonic crystals is carried out. In case of magnonic crystals with a pronounced modulation of internal magnetic properties, spin waves are strongly dynamically coupled leading to hybrid spin-wave states, which together with increasing edge-effects hinder a straight-forward understanding of their physical origin. In this case, the approach to follow the evolution of spin waves from their quasi-harmonic limit, the planar film, up to the limit of a magnonic crystal with strong property-modulation, is specifically suited to unravel this complexity.
%Ferromagnetic resonance measurements of surface-modulated magnonic crystals with gradually increasing modulation heights are corroborated by numerical simulations as well as semi-analytical theory. 
%Intermediate states of thin films with a pronounced surface modulation show complex spin-wave spectra with an even larger amount of spin-wave states accessible compared to the array of free standing wires (one-dimensional magnonic crystal). An increasing level of mode hybridization with the modulation height is reported as well as the analysis of the optimal modulation height to achieve similar mode amplitudes. Furthermore, the physical nature of such hybrid spin-wave modes was unraveled. It was found that the spin-wave spectra split up in two fundamental branches: a \textit{wire} mode and a \textit{trench} mode. These branches itself couple to higher ($n>3$) spin-wave modes and evolve gradually from different harmonic spin-wave modes in the continuous film limit, i.e.\ the \textit{wire} mode arises from the $n=1$ ($1^{\mathrm{st}}$) backward-volume spin-wave mode and the \textit{trench} mode originates from the uniform mode. Moreover it was observed, that the $n=2$ backward-volume mode changes with increasing modulation size into an edge mode.
%By multistep ion-milling of a single permalloy (Ni$_{80}$Fe$_{20}$) film, such a transition is performed experimentally. Initially, the surface of the 37 nm thin film is lithographically patterned into a regular stripe lattice of 300 nm periodicity. For each ion-milling step, subsequent frequency-dependent ferromagnetic resonance (FMR) measurements are carried out and corroborated by analytical theory as well as numerical simulations. Both give insight into the spin-wave dispersion and can be employed to map out the spin-wave mode profiles. Effects of dipolar coupling and mode localization are observed to become more and more prominent and are related to the evolution of the internal demagnetizing field distribution. 
%
\end{abstract}
%

\pacs{}

\maketitle 

\section{Introduction}
\label{Int}
%
Periodically patterned magnetic materials with periodicities ranging from micrometers down to several tens of nanometers are referred to as magnonic crystals (MCs).\cite{Vasseur1996,Nikitov2001,Neusser2009,Kruglyak2010,Gubbiotti2010,Lenk2011,Krawczyk2014,Chumak2015,Mansurova2017} In the last decade, this group of meta-materials, such as bi-component systems,\cite{Ma2012,Tacchi2012,Yu2013,Saha2013,Mruczkiewicz2013,Obry2013,Rychly2017} free standing structures,\cite{Topp2010,Ding2011,Duerr2012,Schwarze2013,Huber2013,Montoncello2013} and continuous films with periodic structures on top, also known as surface-modulated magnonic crystals (SMMCs),\cite{Barsukov2011,Landeros2012,Liu2013,Gallardo2014,Kakazei2014,Langer2016,Khanal2017,Heimbach2018} experienced a growing scientific interest. As spin waves (SWs) offer unique properties such as charge-less propagation and high group velocities, there are multiple applications conceivable since industry is in need for higher efficiencies as well as high performances in information technology including the transport and processing of data.\cite{Cuykendall1987,Schneider2008,Lee2008,Khitun2010,Khitun2011,Berut2012,Khitun2012,Sato2013,Jamali2013,Klingler2014,Chumak2015} The possibility to manipulate the band gaps\cite{Kostylev2008,Lee2009,Wang2009,Wang2010,Ma2012,Di2013,Krawczyk2013,Kumar2014} and to tailor the SW properties paves the way for many applications based on magnonic devices,\cite{Inoue2011,Rana2018} such as magnonic filters,\cite{Kim2009} switches\cite{Khitun2010,Vogt2014,Balinskiy2018}, grating couplers\cite{Yu2013} and transistors.\cite{Chumak2014} Moreover, fast developments in spintronics and spin-caloritronics\cite{Bauer2012} hold out the perspective of novel promising hybrid-topics\cite{Chumak2015} in the future, where the unique properties of MCs are combined with functional entities, such as recently shown for spin-torque oscillators.\cite{Urazhdin2014a}

%For the application as information carriers and logic devices,\cite{Kostylev2005,Lee2008,Khitun2010} the channelization\cite{Wagner2015} of spin waves with long propagation distance is crucial. As Demidov \emph{et al.} recently demonstrated, a magnetic film with ridge-like thickness profile, similar to the surface of SMMCs, enables the guiding of different channelized Damon-Eshbach-type spin-wave modes.\cite{Demidov2015} For carriage of information and performing logical operations, it may also be desired to use an extended periodic system capable of guiding spin-waves in parallel channels. Studying modulation-dependent spin-wave properties in SMMCs is promising in terms of optimizing the spin-wave channelization in extended magnetic systems with a pronounced periodic internal field variation.

In an ongoing miniaturization process, future devices will require small SW wavelengths in an exchange-dominated regime.\cite{Liu2018} For this reason, a small base periodicity of a few hundred nanometers or less is desired with characteristic SW mode wavelengths of several tens to hundreds of nanometers. There are several advanced methods to investigate the transition from a thin film to a MC with a pronounced modulation, such as e.g.\ thermal landscape modulation\cite{Vogel2015} or periodic Oersted-fields of current-carrying meander structures.\cite{Chumak2009-JPD,Chumak2010,Bai2011,Wang2017} However, these approaches are so far non-applicable to periodicities of 500~nm and below.

This work complements these studies in focusing on transitional systems which offer even richer SW spectra with adjustable amplitudes. The transition of these modes from the film limit to full MC limit is studied by incrementally introducing periodic trenches into the film surface. The resulting structures [sketched in Fig.~\ref{experiment}(a)] can be seen as a periodic array of wires on top of a thin film and are referred to as surface-modulated magnonic crystals.

In table~\ref{tab1}, the literature is summarized in which SMMCs are employed under a systematic variation of the modulation height. This summary reveals the diverging structural properties and the variety of scientific objectives for which these structures were used. As most of the previous investigations cover only a fraction of the full transition (column 4 in table~\ref{tab1}), this work is intended to close the gap between MCs in the surface-perturbation regime\cite{Landeros2012,Gallardo2014} and strongly modulated systems.\cite{Liu2013,Kakazei2014} It is important to furthermore note that most previous work aimed at different objectives, such as the SW transmission of surface-patterned waveguides,\cite{Chumak2008,Chumak2009-JAP,Chumak2009-APL} the optimization of the reconfigurability of the magnonic properties\cite{Liu2013,Kakazei2014}, and the study of SWs in SMMCs in the perturbation regime.\cite{Landeros2012,Gallardo2014} Nonetheless, the work that is closer related to this topic concerns either the numerical analysis of SW resonances together with the respective mode profiles\cite{Aranda2014} or the theoretical and numerical calculations of the locally varying internal demagnetizing fields which act on the SWs opening band gaps and showing flat bands around the backward-volume geometry.\cite{Gallardo2018} This experimental study is meant to complement these theoretical works and shall provide a comprehensible interpretation of the evolution of SW modes during different levels of surface modulation.

\begin{figure}[t]
\includegraphics[width=0.95\columnwidth]{experiment0.eps}
\caption{\label{experiment} (Color online) (a) Sketch of the sequential fabrication of several SMMCs by ion-milling of a single sample. (b) TEM image of a sample milled to $\Delta d = 10$~nm surface modulation.} %(c)--(d) SEM images of the top view of the same sample before the removal of the resist mask.}
\end{figure}

%The transitional SMMCs are fabricated by sequential ion milling of a lithographically surface-patterned permalloy (Ni$_{80}$Fe$_{20}$) thin film offering the benefit of studying the transition gradually on the same sample. Subsequent to each ion-milling step, broadband ferromagnetic resonance (FMR) measurements are performed and corroborated by micromagnetic simulations.
Up to now mainly the two limits---the thin film limit\cite{Barsukov2011, Landeros2012, Gallardo2014, Langer2016} and the `full' MC\cite{Demokritov2001,Roussigne2001,Gubbiotti2007,Kostylev2008,Wang2009,Tacchi2010,Wang2010,Saha2015,Belmeguenai2016}---were intensively studied. This work is focused on the dynamic properties of transitional systems---SMMCs. For this purpose, a stripe-patterned thin permalloy (Ni$_{80}$Fe$_{20}$) film was sequentially ion-milled and characterized after each milling step using broadband ferromagnetic resonance (FMR). This allows to study SMMCs with different modulation heights using the same sample. The SW resonances and SW mode profiles are compared to the results of micromagnetic simulations in order to study the transition of SW modes from film modes to the modes in a one-dimensional MC. The huge variety of modes in SMMCs, the strong coupling between them, and the distortion of the mode profiles due to the hybridization and inhomogeneous internal demagnetizing fields hinder a straightforward interpretation. To disentangle these effects, the approach to follow the modes under an incremental increase of the modulation height becomes particularly favorable.

%\todo{Here a new paragraph should be added suggesting the possibility to use standing BV modes to channelize DE-type spin-wave propagation (similar to the papers [Park2002-PRL] and [Demidov2015])This is important because it is a good way to motivate this research for applications and we already know, that this can be done. Also, it will be a good connection to the STXM paper that will follow. ??? The Park-Paper only observed edge modes in BV not in DE - why should it be cited here? Und m.E. ist es nicht richtig, dass die BV mode den channel bildet, sondern die statischen Dipolfelder der Geometrie?!? }
%One possibility is to use standing backward volume (BV) spin wave modes to channel propagating Damon-Eshbach (DE) type spin waves. Demidov \emph{et al.} demonstrated that magnonic waveguides based on a magnetic film with a ridge-like thickness profile enable guiding of both the edge and the center spin wave modes.\cite{Demidov2015}
%
%The article addresses the complex evolution of hybrid spin-wave modes in the gradual transition from a thin magnetic film to an array of periodical wires (here, referred to as `full magnonic crystal') carried out by sequential ion-milling of a patterned permalloy (Ni$_{80}$Fe$_{20}$) film with subsequent spin-wave detection. Here, the term `hybrid mode' refers to dipolarly coupled standing spin-wave modes of different $k$-vectors. The approach implies, that all structural geometries obtained during this transition are settled in the group of SMMCs with the same periodicity and a different modulation height. SMMCs reveal several advantages compared to bi-component MCs. In the thin film limit, both types of MCs show very similar dynamic properties, however in the case of an SMMC, the alternating internal properties are tailorable by the modulation height $\Delta d$, which provides the possibility to study the aforementioned gradual transition from (thin) film physics to magnonic crystals with large band gaps using a \emph{single} sample. Note, that the variety of modes that can be excited is much larger in an intermediate regime with $\Delta d\cong d/2$ compared to the array of wires and moreover, intensities are of a comparable size. Even though, there are other advanced methods to realize such transition, as e.g.\ thermal landscape modulation\cite{Vogel2015} or periodic Oersted-fields of current-carrying meander structures\cite{Chumak2010}, these approaches are so far non-applicable to periodicities of 500~nm and below.
%
%The study takes advantage of micromagnetic simulations\cite{Vansteenkiste2014,Wagner2015} and quasi-analytical theory\cite{Sokolovskyy2011,Klos2012,Krawczyk2013} based on the plane-wave method (PWM) to understand and describe the measured spin-wave dynamics and moreover, to obtain the spin-wave mode profiles revealing wavelengths in the range of 50--300~nm right at the edge of the exchange-dominated regime. The evolution of spin-wave modes is gradually followed from the quasi-harmonic thin film limit up to the limit of an array of wires (`full' magnonic crystal). 
%
%A high complexity of mode-coupling was found for an intermediate level of surface-modulation as well as a transformation of the spin-wave character with the modulation height $\Delta d$.
% 
%The study reveals a transformation of the mode character from the film limit to the array of free-standing wires. 
%In the backward-volume geometry, the uniform mode undergoes a transition to the fundamental mode in the thin part (termed `trench'). At the same time, the $n=1$ (1$^{\mathrm{st}}$) and $n=2$ %(2$^{\mathrm{nd}}$) SW modes change towards the center- and the edge mode of the thick part (termed `wire'). In the Damon-Eshbach geometry, the uniform mode is observed to transition similarly towards a %\textit{trench} mode whereas the 1$^{\mathrm{st}}$ and the 2$^{\mathrm{nd}}$ Damon-Eshbach mode converge towards the center mode within the thick part of the MC. In principle, the results of this study %shall be transferable to bi-component MCs as well.

The manuscript is organized as follows. Details about the sample fabrication and the measurement technique are given in Sec.~\ref{Exp} followed by Secs.~\ref{BV} and \ref{DE} containing the results for the backward-volume (BV) and the Damon-Eshbach (DE) geometry, respectively. Section~\ref{Con} summarizes the results.

\section{Experiment}
\label{Exp}

The experiments are based on a polycrystalline $d = 36.8$~nm thin permalloy (Ni$_{80}$Fe$_{20}$) film deposited by electron beam deposition on surface-oxidized Si(001) substrate. The surface of the film was lithographically stripe patterned using ma-N~2401 negative resist with a wire width (resist covered wires) of $w = 140$~nm and a periodicity of $a_0=300$~nm. In order to incrementally remove the magnetic material between the resist-covered stripes, sequential Ar-ion milling was employed. The procedure is schematically depicted in Fig.~\ref{experiment}(a). Figure~\ref{experiment}(b) shows the corresponding cross-section of a patterned permalloy film after milling $\Delta d = 10$~nm into the film. Altogether, the film was milled five times until an array of separate wires, i.e.\ a `full' MC, was achieved. 
%
\begin{figure}[t]
\includegraphics[width=0.95\columnwidth]{FilmLimit0.eps}
\caption{\label{FilmLimit} (Color online) Measurement and simulation of the frequency-field dependence of (a),(c) a flat film and (b),(d) a surface-modulated magnonic crystal with tiny modulations ($\Delta d=2$~nm). Orange lines represent the spin waves' frequency-field dependences calculated from Eq.~(\ref{BVdisp}).}
\end{figure}

After each ion-milling step, the frequency-field-dependence $f(H_0)$ was measured using a broadband ferromagnetic resonance (FMR) setup, as described in reference~[\onlinecite{Langer2016}]. 
An FMR pre-characterization of the thin film properties was carried out prior to the sequential ion milling yielding the saturation magnetization $\mu_{0} M_{\mathrm{s}}=0.9236$~T, the $g$-factor $g=2.11$ and the exchange stiffness $D=23.6$~Tnm$^2$.\cite{Langer2016, Langer2017} It shall be noted that the investigations concentrate on symmetric modes only, since the dynamic measurements are carried out using a symmetric excitation scheme.
%
\begin{table*}[th]
\caption{Overview of previous investigations of systematic modulation $\Delta d/d$ variations on the spin-wave properties of SMMCs. The following abbreviations are used: SWT---spinwave transducer, TMS---two-magnon scattering, PWM---plane-wave method.}
\label{tab1}
\centering
\begin{tabular}{ccccccccccccccc}
\hline\hline
\multirow{2}{*}{authors / year} & ~ & 1D / 2D & ~ & thickness & ~ & modulation & ~ & \multirow{2}{*}{material} & ~ & period & ~ & main & ~ & studied \\
& ~ & SMMC & ~ & $d$ (nm) & ~ & $\Delta d/d$ (\%) & ~ & & ~ & $a_0$ (\textmu m) & ~ & method & ~ & quantity \\
\hline
Chumak \textit{et al.} 2008 [\onlinecite{Chumak2008}] & ~ & 1D & ~ & 5\,500 & ~ & 0--16.4 & ~ & YIG & ~ & 300 & ~ & SWT & ~ & SW transmission\\
\multirow{2}{*}{Chumak \textit{et al.} 2009 [\onlinecite{Chumak2009-JAP}]} & \multirow{2}{*}{~} & \multirow{2}{*}{1D} & \multirow{2}{*}{~} & 5\,500 & \multirow{2}{*}{~} & 0--16.4 & \multirow{2}{*}{~} & \multirow{2}{*}{YIG} & \multirow{2}{*}{~} & \multirow{2}{*}{300} & \multirow{2}{*}{~} & \multirow{2}{*}{SWT} & \multirow{2}{*}{~} & \multirow{2}{*}{SW transmission}\\
& & & & 14\,000 & & 0--12.9 & & & & & & & &\\
Chumak \textit{et al.} 2009 [\onlinecite{Chumak2009-APL}] & ~ & 1D & ~ & 5\,500 & ~ & 0--41.8 & ~ & YIG & ~ & 300 & ~ & SWT & ~ & SW transmission\\
%Chumak \textit{et al.} 2009 \left[\onlinecite{Chumak2009-JPD}\right] & MT & $d$ & 5.5--3.2 \textmu m (42\%) & YIG & 300 & NA & T, R\\
%Bai \textit{et al.} 2011 \left[\onlinecite{Bai2011}\right] & MT & $H_{\mathrm{Oe}}$ & 5.5--3.2 \textmu m (42\%) & YIG & 300 & NA & T, R & - \\
Landeros \textit{et al.} 2012 [\onlinecite{Landeros2012}] & ~ & 1D & ~ & 30 & ~ & 0--6.7 & ~ & Permalloy & ~ & 0.25 & ~ & TMS Theory & ~ & FMR response\\
Liu \textit{et al.} 2013 [\onlinecite{Liu2013}] & ~ & 2D & ~ & 60--120 & ~ & 50--100 & ~ & Permalloy & ~ & 0.62 & ~ & FMR & ~ & FMR response\\ %rev. Approach
Kakazei \textit{et al.} 2014 [\onlinecite{Kakazei2014}] & ~ & 1D & ~ & 30--60 & ~ & 50--100 & ~ & Permalloy & ~ & 0.4 & ~ & FMR & ~ & FMR response \\ %rev. Approach
Gallardo \textit{et al.} 2014 [\onlinecite{Gallardo2014}] & ~ & 1D / 2D & ~ & 30 & ~ & 0--10 & ~ & Permalloy & ~ & 0.3 & ~ & TMS Theory & ~ & FMR response\\
Aranda \textit{et al.} 2014 [\onlinecite{Aranda2014}] & ~ & 1D & ~ & 20 & ~ & 0--100 & ~ & Permalloy & ~ & 0.5 & ~ & Simulation & ~ & FMR response\\
Gallardo \textit{et al.} 2018 [\onlinecite{Gallardo2018}] & ~ & 1D & ~ & 10 & ~ & 0--50 & ~ & Permalloy & ~ & 0.3 & ~ & PWM Theory & ~ & FMR response\\
\hline\hline
\end{tabular}
\end{table*}

The etching depths of each SMMC were determined by fitting of the $f(H_0)$ dependence of SW modes standing vertically in the film, also referred to as perpendicular standing spin-wave (PSSW) modes.\cite{Tannenwald1957,Seavey1959} For larger modulation heights $\Delta d > 10$~nm, vibrating sample magnetometry measurements were employed to determine the magnetic volume of an etched reference film before ($M_1$) and after ($M_2$) the ion milling step. This approach estimates the modulation height $\Delta d=d\times\left(1-M_2/M_1\right)$ by the ratio of the measured magnetic volume $M_2/M_1$. 

The following discussion of SW modes in SMMCs is subdivided by the in-plane orientation of the external field with respect to the modulation axis of the SMMC, i.e.\ backward-volume (BV) and Damon-Eshbach (DE) geometry.
%
%\section{Theory}
%
%\todo{`Short' version of PWM.}

%\section{Results}
%\label{Res}
%It is noted that both symmetric and antisymmetric SW modes can be excited as eigenmodes of SMMCs. Since the experimental work is carried out using a symmetric excitation method, the investigations concentrate on the symmetric modes. 
%
%The material parameters for the fit functions and the micromagnetic simulations are all selected coherently due to the FMR characterization of the uniform mode of a flat film with the magnetic polarization $\mu_{0} M_{\mathrm{S}}=0.9236$~T, the $g$-factor $g=2.11$ and the exchange stiffness $D=23.6$~Tnm$^2$.\cite{Langer2016}
%
%In order to achieve an abstract description of the mode character, the following notation is used. In the film limit, standing spin-wave modes require an even number of $2\times n$ nodes within one period and hence, they are labeled with their `half node' number $n$. In the limit of the one-dimensional MC, the number $m$ is used in order to express the rank of the spin-wave modes according to the number of peaks within the width of a wire.
%

\section{Backward-volume geometry}
\label{BV}

SWs in an SMMC with field orientation parallel to the modulation axis, i.e.\ $\mathbf{k} \| \mathbf{M}$ (BV geometry), are particularly interesting due to the highly inhomogeneous demagnetizing fields $H_{\mathrm{d}}$, which are largest compared to all other in-plane field orientations. Before different transitional SMMCs are addressed, both, the planar film limit as well as the limit of a `full' MC of separate wires are addressed. 

In order to understand the complex mode structure of the SMMCs, micromagnetic simulations were carried out using the MuMax$^3$-code.\cite{Vansteenkiste2014} The FMR simulations were carried out employing both, pulsed\cite{McMichael2005} and continuous-wave\cite{Wagner2015} excitation. Both approaches are described in more detail in Ref.~\onlinecite{Langer2017}. For all simulations, the material parameters presented in Sec.~\ref{Exp} were used and several SMMCs were micromagnetically reconstructed using a fixed film thickness of $d=36$~nm and six modulation heights of $\Delta d=2, 4, 9, 13.5, 18$ and 36~nm with the latter representing the `full' MC. For the realization of an extended MC, periodic boundary conditions were applied to the in-plane axes.%To obtain the spin-wave spectra of each SMMC, a pulsed excitation\cite{McMichael2005} was realized and the integral dynamic magnetization was plotted as the FMR response.

\subsection{The Limits: Thin Film and Full Magnonic Crystal}

\vspace{3mm}
\textbf{Thin Film Limit.}\quad The limit of a thin film with perturbation-like modulations has previously been studied in experiment\cite{Barsukov2011,Koerner2013,Langer2016} and theory using two-magnon scattering perturbation theory.\cite{Landeros2012,Gallardo2014} In such systems, the demagnetizing field $H_{\mathrm{d}}$ acts as a small periodic perturbation introducing a base periodicity $a_0$ crucial for the presence of standing (in-plane) SW modes. While the dispersion remains unaffected, an introduction of periodic perturbations in the real space leads to a discretization of standing SW modes in the $k$-space according to $k=2\pi n/a_0$ with $k$ being the in-plane wave vector and $n=1, 2, 3,...$ the film mode ``quantum number''. As mentioned above, the SW dispersion follows the well known relation:\cite{Kalinikos1986}
%
%\begin{equation}
%\left(\frac{\omega}{\gamma}\right)^{\negmedspace2}\negthickspace=\!\left( \mu_{0} H_0 + D k^2 \right) \times \left[ \mu_{0} H_0 + D k^2 + \mu_{0} M_{\mathrm{S}} F(kd) \right]
%\label{BVdisp}
%\end{equation}
%
\begin{align}
\left(\frac{\omega}{\gamma}\right)^{\negmedspace2}\negthickspace&=\!\left[ \mu_{0} H_\mathrm{eff} + D k^2 \right]\label{BVdisp}\\\nonumber
&~\times \left[ \mu_{0} H_\mathrm{eff} + \mu_{0} M_{\mathrm{s}} \left( \frac{1-\mathrm{e}^{-kd}}{kd} \right) + D k^2 \right]
\end{align}
%
\noindent with the FMR frequency $f = \omega/(2\pi)$ and the effective field $H_\mathrm{eff}=H_\mathrm{0}+H_\mathrm{d}$ composed of the external field $H_{0}$ and the internal demagnetizing field $H_{\mathrm{d}}$.
%\noindent with $f = \omega/(2\pi)$ the FMR frequency, $D$ the exchange stiffness constant, $\mu_{0} M_{\mathrm{S}}$ the magnetic polarization and $d$ the film thickness.

Equation~(\ref{BVdisp}) describes a parabolic shape of the dispersion for dipole-exchange spin waves in BV geometry with minimum frequency at $k \neq 0$. The characteristic shape implies an energy degeneracy, which together with the scattering condition $k=2\pi n/a_0$ enables the two-magnon scattering channel\cite{Landeros2012,Gallardo2014} from the uniform mode to $k\neq 0$ standing SW modes. This is the reason for the occurrence of many high-intensity modes at the same time in contrast to the DE geometry (see Sec.~\ref{DE}).

In Figs.~\ref{FilmLimit}(a) and \ref{FilmLimit}(c), the measured and the simulated frequency-field dependences $f(H_0)$ are depicted for a flat film and in Figs.~\ref{FilmLimit}(b) and \ref{FilmLimit}(d) for an SMMC with 2~nm modulation height. Fig.~\ref{FilmLimit}(d) additionally provides the $f(H_0)$ dependence of the $n=1...4$ modes calculated from Eq.~(\ref{BVdisp}) and plotted as orange solid lines. Clearly, standing SW modes in an SMMC with tiny modulation follow the $f(H_0)$ dependence of propagating modes in a flat film of the same properties quite well. However, deviations from the film limit can be observed in the vicinity of crossing points [see crossing between dashed and solid lines in Fig.~\ref{FilmLimit}(b)] in the $f(H_0)$ dependence, where different standing SW modes couple to each other. 

In analogy to coupled oscillators in mechanics, both modes split up into an acoustical and an optical branch with in-phase and anti-phase oscillation, respectively. The evolving frequency gap between acoustical and optical mode is connected with the emergence of a magnonic band structure reflecting the transition from pure film physics to the physics of MCs.
%
%Important differences of backward-volume modes in a weakly modulated MC compared to the planar film are (i) the pinned character of the spin waves and (ii) the dipolar coupling of different backward-volume modes leading to a mode-splitting (also termed `anticrossing'\cite{Bauer2014}) at each point in the $f(H_0)$-dependence, where different modes meet. Such effects are very well described by perturbation theory.\cite{Arias1999,Landeros2012,Gallardo2014} The other limit would be a periodic array of separate wires. Here, exchange between neighboring wires is completely switched off leading to very similar spin-wave dynamics as compared to a single wire with an additional stray field contribution arising from neighboring wires. In addition, minor differences occur due to different dipolar boundary conditions.\cite{Duan2015}
%

\vspace{3mm}
\textbf{Full MC.}\quad The opposite limit is the periodic array of wires, also known as `full' or one-dimensional MC,\cite{Tacchi2012,Gubbiotti2010} with the $f(H_0)$ dependence shown in Fig.~\ref{WireLimit}. Figure~\ref{WireLimit}(a) is the measurement and \ref{WireLimit}(b) the simulation with orange lines indicating the simulated resonance modes of a single wire. If the stray field contribution of neighboring wires (approximately 30~mT) is considered to shift the modes to lower resonance fields, both the full MC [colorplot in Fig.~\ref{WireLimit}(b)] and the single wire [orange lines in Fig.~\ref{WireLimit}(b)] reveal very similar results for the given dimensions. 

Figure~\ref{wiremodes} contains the corresponding mode profiles of the simulated spin-wave resonances at 300~mT. For the full MC, the mode number is denoted as $n'$ %The existence of the differently localized modes is known to be connected to the strong variation of the internal demagnetizing field 
and the mode profiles of standing spin-waves are distorted by the strong locally varying internal demagnetizing fields [e.g., the \textit{center} mode ($n'=0$) and the \textit{edge} mode ($n'=2$) displayed in Fig.~\ref{wiremodes}(a) and (b)]. In particular the second mode ($n'=2$) is strongly distorted by the presence of two energetic minima (spin-wave wells\cite{Bayer2003, Bayer2005}) at the edges of the wire created by the strong demagnetizing fields at this location. It is important to note that the amount of \textit{localized modes} in the edge regions is given by the number of SW resonances below the quasi-uniform mode ($n'=0$), which is the first mode with the ability to overcome the maximum of the demagnetizing field in the center and, thus, to extend over the complete structure. It is only possible due to the characteristic parabolic dispersion in the backward-volume geometry that modes of a higher mode number $n'$ bear less energy than the quasi-uniform mode. %These modes are confined to certain areas of the structure in contrast to extended modes which spread their amplitude over the entire MC. The occurrence of these two categories of SW modes can be understood in a simple picture in which the characteristic shape of the demagnetizing field forms energy wells\cite{Bayer} at the edges of the structure. Spin-wave modes with an energy higher than the well bear sufficient energy to extend over the complete MC while those with a lower energy are confined inside the regions with reduced or negative demagnetizing fields.
%
\begin{figure}[t]
\includegraphics[width=0.95\columnwidth]{WireLimit0.eps}
\caption{\label{WireLimit} (Color online) (a) Measurement and (b) simulation of the frequency-field dependence of an array of wires (one-dimensional MC) with the orange lines indicating the simulated resonance branches of a single wire with the same properties. The edge mode is clearly suppressed in the measurement due to tapered side-walls and side-wall roughness of the measured sample. The vertical line marks the resonance positions at 300~mT shown in Fig.~\ref{wiremodes}.}
\end{figure}

Typical examples for extended modes are found at higher energies, and their mode profiles are illustrated in Figs.~\ref{wiremodes}(c) and \ref{wiremodes}(d). These modes are clearly of a higher order, which is reflected by the number of $n'$ nodes and $n'+1$ peaks inside the magnetic wires and they carry only higher intensities when crossing the center mode to which they can couple [see the inset in Fig.~\ref{WireLimit}(b)]. Note that a symmetric excitation only allows for the measurement of symmetric SW eigenmodes and, thus, the peak number $n'+1$ of SW modes in the `full' MC limit must be odd ($n'=0, 2, 4,...$).\cite{Demokritov2001, Demokritov2003} 
%The mode profiles corresponding to each resonance branch in Fig.~\ref{WireLimit}(b) are shown in Fig.~\ref{wiremodes} at an external field of 300~mT. The two fundamental branches can be identified as the \emph{edge} mode ($m=3$) in Fig.~\ref{wiremodes}(a) and the \emph{wire} mode ($m=1$, also termed `wire center mode') In the measurement [Fig.~\ref{wiremodes}(b)], the edge mode appears to be completely suppressed which is believed to be due to the beveled edge shape and edge roughness of the sample. Figures~\ref{wiremodes}(c),(d) reveal the mode profiles of higher modes ($m=5, 7$) standing within the boundaries of the wire. These modes only carry significant intensities when crossing the center mode to which they couple [see the inset in Fig.~\ref{WireLimit}(b)].
%

\subsection{Transitional SMMCs}
%\del{The sequential ion-milling was performed with very small etchings steps in the beginning ($\Delta d \longrightarrow 0$) and large etching steps in the end ($\Delta d \longrightarrow d$). This is due to the demagnetizing field increasing substantially in the range of $\Delta d=0--10$~nm (see inset Fig.~\ref{multistep}) and gradually saturating above that range. For SMMCs with larger modulation heights, the changes become less significant with $\Delta d$ and thus, the alteration of the mode character is uncritical.}

To explain the mode character in hybrid structures with a significant modulation height, two spin-wave ``quantum numbers'' need to be introduced. $l=1, 3, 5,...$ defines the number of peaks of a spin-wave mode inside the thin part of the SMMC and $m=1, 3, 5,...$ reflects the number of peaks within the thick part. In the film limit, the two quantum numbers are linked by the discrete film mode ``quantum number'' $n$ with $2n=m+l$ whereas in the full MC only the $m$ peaks inside the wire region remain connecting $m$ with the full magnonic crystal quantum number $n'$ by $n'=m-1$.
%\new{It is noted that this relation is only valid for extended modes, since the localized modes are no longer able to maintain all of their  }
%\todo{In the case of localized modes the.... WAS MEINST DU MIT: rule distorted due to the BCs in case of localized modes?}
\begin{figure}[t]
\includegraphics[width=1\columnwidth]{wiremodes0.eps}
\caption{\label{wiremodes} (Color online) Simulated (bottom row) and idealized (top row) mode profiles in an array of wires in backward-volume geometry [as shown in Fig.~\ref{WireLimit}(b)] at 300~mT. The modes are labeled according to their mode number $n'$.}
\end{figure}

Figure~\ref{multistep} shows the $f(H_0)$ plots of several SMMCs with a modulation height $\Delta d$ in the range of 2.0--13.5~nm. In Figs.~\ref{multistep}(a)--(d) and \ref{multistep}(e)--(h), the FMR measurements and the simulations are illustrated, respectively. In the inset, the level of the modulation is provided together with the $x$-component [axes defined according to Fig.~\ref{experiment}(a)] of the simulated internal demagnetizing field $H_{\mathrm{d},x}$ as colorplot. Inside the thick part (also termed `wire') of the MC, the demagnetizing field acts \textit{against} (negative sign) the local magnetization direction as usually. Interestingly, inside the thin part (`trench'), it acts \textit{with} the magnetization and is, thus, a \textit{magnetizing} field. The increasing contrast between the magnetizing- and demagnetizing fields with the modulation height is easily inferred from the insets in Figs.~\ref{multistep}(e)--(h). %This effect is important for the understanding of the evolution of SW modes in SMMCs with increasing modulation height $\Delta d$.

%It is again distinguished between new{fundamental modes} and higher order SW modes. The former can be directly excited uniformly whereas the latter gain intensity from coupling to a fundamental branch. In Figs.~\ref{multistep}(e)--(h), \new{fundamental modes} are marked by dashed colored lines. The color of each mode is chosen corresponding to the $H_{\mathrm{d},x}$ field where orange color represents \textit{magnetizing} fields and blue color demagnetizing fields.

The corresponding mode profiles for the marked resonance positions in Fig.~\ref{multistep} are provided in Fig.~\ref{profiles}. Before discussed in detail, the modes are distinguished into three categories: fundamental modes, localized modes and extended modes. The latter two categories are distinguished by whether or not the mode spreads over the complete structure whereas a fundamental mode defines the lowest (center) mode within one of the two different regions, i.e.\ wire or trench.

%Prior to a detailed discussion, a splitting into two fundamental modes, a SW excitation in the trench (\textit{trench} mode) and an excitation of the wire (\textit{wire} mode), can already be concluded. 
%Both modes are carrying already strong intensities at a small level of modulation (\ref{multistep}(a),(e)). 
%Now, due to the increasing contrast between the (de-)magnetizing fields in the trench and in the wire, both modes undergo a vast separation leading to the evolution of a large gap between them (see Figs.~\ref{multistep}(g) and \ref{multistep}(h)). 

%At first, the important role of the quasi-uniform mode (termed: trench mode) with $n=0$ and the 1$^\mathrm{st}$ SW mode (termed: wire mode) with $n=1$ shall be addressed.
%Both modes are carrying already strong intensities at a small level of modulation (\ref{multistep}(a),(e)). 
%Now, due to the increasing contrast between the (de-)magnetizing fields in the trench and in the wire, both modes undergo a vast separation leading to the evolution of a large gap between them (see Figs.~\ref{multistep}(g) and \ref{multistep}(h)). 
%
%Next, the transition of each SW mode from the film to the `full' MC shall be addressed.
%
%For the simulation, the following structural and magnetic properties were chosen as obtained in the measurement of the flat film: $M_{\mathrm{S}}=735$~kA/m (obtained by vibrating sample magnetometry), $g=2.11$ (obtained the out-of-plane $f(H_0)$-dependence measured by FMR), $D=23.6$~Tnm$^2$\cite{Langer2016}, $d=36$~nm [obtained from transmission electron microscopy, see Fig.~\ref{experiment}(b)], $a_0=300$~nm, $w=140$~nm (both, according to the lithography mask) and $\Delta d$ the modulation height (obtained according to the description provided in Sec.~\ref{app}) given in each plot. 
%
\begin{figure}[t]
\includegraphics[width=0.95\columnwidth]{multistep0.eps}
\caption{\label{multistep} (Color online) (a)--(d) FMR measurements of MCs with different modulation heights as sketched in the insets. (e)--(h) The dynamic response with the internal demagnetizing field in the inset both calculated using pulsed micromagnetic simulations. Colored dots represent the field-frequency position of the mode profiles illustrated in Fig.~\ref{profiles}.}
\end{figure}
%
%\begin{figure}[H]
%\includegraphics[width=1\linewidth]{numericalfH.eps}
%\caption{\label{numericalfH} (Color online) (a-f) FMR measurements of a multiple ion-milled permalloy film. (g-l) The dynamic response calculated using pulsed micromagnetic simulations with similar %properties. Orange lines in (h) indicate the position of non-interacting film modes in backward-volume geometry according to Eq.~\ref{BVdisp}. Similar lines in (l) indicate the modes of a single nanowire %obtained by pulsed micromagnetic simulations.}
%\end{figure}
%The other limit, the periodic array of wires, is shown in the Figs.~\ref{multistep}(f) and \ref{multistep}(l) together with the modes of a \emph{single} infinitely long wire with $w=140$~nm (orange lines). As described above, the dynamics of the single wire are very similar to the dynamics of the array, if the field offset of $\mu_0 H_0\cong 30$~mT generated by the stray field of neighboring wires is taken into account.

\vspace{3mm}
\textbf{Quasi-Uniform Mode.}\quad A classical uniform mode does no longer exist in an SMMC due to the inhomogeneous internal demagnetizing fields. However, from the mode profiles in Figs.~\ref{profiles}(a) and \ref{profiles}(f) it is evident that the uniform mode evolves to the fundamental mode of the trench region (termed ``trench mode''), i.e.\ a local quasi-uniform excitation of the trench region of the SMMC. In other words, it seems that the quasi-uniform ($n=0$) mode represents the first mode which is able to excite the complete SMMC. Therefore, the mode needs sufficient energy to overcome the magnetizing fields inside the trench region. One consequence is, that the main amplitude of this mode is found inside the trench region. Due to the higher energy necessary to excite the trench, the resonance condition does not match for the wire region anymore. It has been observed that the magnetizing fields inside the trench region increase with the level of surface modulation separating the quasi-uniform mode even stronger from the energetically lower modes which localize only in the wire region. This evolution is reflected by the increasing gap between the trench (orange line) and wire mode (bright blue line) in the $f(H_0)$-dependence illustrated in Figs.~\ref{multistep}(e)--(h). The concentration of the main amplitude of the mode inside the trench region can be seen in the corresponding mode profiles shown in Fig.~\ref{profiles}(a) and (f) at 75--100~mT. Throughout the mode's evolution, it maintains its main amplitude in the center of the trench region. Thus, in principle the trench mode can be understood as an fundamental $l=0$ excitation of the trench region. But as the coupling to higher SW resonances increases with the modulation height, the uniform ($l=0$) characteristics are gradually reduced and are replaced by a modulus due to the strong hybridization with higher BV modes crossing the $f(H_0)$-dependence of the mode. This leads to the effect that the quasi-uniform branch rather appears as an $l=1,3,5,...$ mode changing the peak number $l$ by $\pm 2$ at each interaction point with a higher SW resonance in the $f(H_0)$-dependence. The crossing point in Figs.~\ref{multistep}(g),(h) at approximately $f=15$~GHz and $\mu_{0} H_\mathrm{0}=200$~mT is such an example. Here the 4$^{\mathrm{th}}$ mode crosses the quasi-uniform branch revealing $l=1$ peaks below the gap and $l=3$ peaks above.

It can be concluded that the quasi-uniform mode evolves towards a fundamental trench mode ($l=0$) which gradually splits up into discreet standing BV modes with $l=1,3,5...$ peaks confined inside the trench region. Moreover, the trench mode gradually loses intensity due to the reduction of the film thickness in the trench region until the full MC limit is reached and the mode completely disappears. This transition is also accompanied by an increasingly strong magnetizing field in the trench region, shifting the trench mode to higher field and frequency values supporting the formation of a large gap with respect to 1$^{\mathrm{st}}$ SW mode [see orange dashed line in Figs.~\ref{multistep}(e)--(h)].%The origin of this effect is connected to the negative slope of the spin-wave dispersion close to $k=0$ for spin-waves in the dipole-exchange regime.\cite{Langer2017} If a small internal field alteration $\delta H_\mathrm{d}$ is considered at a constant frequency $f$, the modification of the wave vector $\delta k$ connected to this alteration can be calculated with the help of Eq.~(\ref{BVdisp}):
%
%\begin{equation}
%\omega_0(k=0, H_{\mathrm{eff}})=\omega(\delta k, H_{\mathrm{eff}}+\delta H_\mathrm{d})
%\label{BVuni}
%\end{equation}
%
%
%\begin{equation}
%\begin{align}
%&\omega_0(k=0, H_{\mathrm{eff}})=\omega(\delta k, H_{\mathrm{eff}}+\delta H_\mathrm{d})\\
%&\left[ \mu_{0} H_0 + \mu_{0} H_{\mathrm{eff}} \right] \times \left[ \mu_{0} H_0 + \mu_{0} H_{\mathrm{eff}} \right]=\\
%&\left[ \mu_{0} H_{\mathrm{eff}} + \mu_{0} \delta H_\mathrm{d} + D (\delta k)^2 \right]\\\nonumber
%&\times \left[ \mu_{0} H_{\mathrm{eff}} + \mu_{0} \delta H_\mathrm{d} + \mu_{0} M_{\mathrm{S}} \left( \frac{1-\mathrm{e}^{-\delta k\,d}}{\delta k\,d} \right) + D (\delta k)^2 \right]\\
%&\mathrm{The}~\mathrm{term}~F(kd)=[1-\mathrm{exp}(-\delta k\,d)](\delta k\,d)\\\nonumber
%&\mathrm{can}~\mathrm{be}~\mathrm{approximated}~\mathrm{for}~\delta k\,d<<1~\mathrm{by}~F\approx1-1/2(\delta k\,d)\,.\\\nonumber
%&\mathrm{Neglecting}~\mathrm{all}~\mathrm{non-linear}~\mathrm{terms}~\mathrm{in}\\\nonumber
%&\delta H~\mathrm{and}~\delta k~,~\mathrm{one}~\mathrm{finally}~\mathrm{yields}\\\nonumber
%&\delta k = \frac{4}{d}\frac{\mu_{0} \delta H_\mathrm{d}}{\mu_{0} M_{\mathrm{S}}}
%\end{align}
%\label{BVuni}
%\end{equation}
%
%The term $F(kd)=[1-\mathrm{exp}(-kd)]/(kd)$ can be approximated for $kd\ll 1$ by $F\approx 1-1/2(kd)$ and neglecting all non-linear terms finally yields
%
%\begin{equation}
%\delta k = \frac{1}{d}\left(\frac{2}{\mu_{0} H_\mathrm{eff}}+\frac{4}{\mu_{0} M_{\mathrm{s}}}\right)\mu_{0} \delta H_\mathrm{d}~.
%\label{BVinf}
%\end{equation}%
%
%Now keeping in mind that a quasi-uniform excitation requires a ``flat'' internal field landscape, only two regions are eligible: The center of the \textit{trench} and the center of the \textit{wire}. The internal demagnetizing field forms a local minimum in the former case and a local maximum in the latter case [as can be seen from the insets in Figs.~\ref{multistep}(e)--\ref{multistep}(h)]. If a quasi-uniform ($k=0$) excitation is assumed at a local field maximum, the neighboring spins experience a slightly lower internal field $H_{\mathrm{d}}$, which requires a small decrease of $k$ to satisfy Eq.~(\ref{BVinf}). Thus, there is no solution of the spin-wave dispersion anymore and the mode is suppressed. Only if the trench region with a local minimum in $H_{\mathrm{d}}$ is quasi-uniformly excited, neighboring spins can still respond with a slightly increased wave vector according to Eq.~(\ref{BVinf}). 

%Thus, quasi-uniform mode becomes the \new{fundamental mode} of the \textit{trench} of SMMCs in the transition from a thin film to a full magnonic crystal. That implies the gradual loss of amplitude and finally the vanishing of this mode in the case of fully separated wires (termed ``full magnonic crystal'').

%\del{The $f(H_0)$ dependence of this mode can be approximated by Kittel's law replacing the effective field $\mu_{0} H_{\mathrm{eff}}$ by an external field $\mu_{0} H_0$ and an additional demagnetizing field of positive sign $\mu_{0} H_{\mathrm{d}}$ as follows}
%
%
%\begin{equation}
%\left(\frac{\omega}{\gamma}\right)^{\negmedspace2}\negthickspace=\!\left( \mu_{0} H_0 + \mu_{0} H_\mathrm{d} \right)\times \left( \mu_{0} H_0 + \mu_{0} H_\mathrm{d} + \mu_{0} M_{\mathrm{S}} \right)
%\label{Trench}
%\end{equation}
%
%\del{The internal field $\mu_{0} H_{\mathrm{d}}$ can either be taken from simulations or approximated analytically by\cite{Gallardo2014,Gallardo2016}}
%
%\begin{equation}
%\begin{split}
%\mu_0 H_{\mathrm{d},x}(x)\negthickspace=\!&-\frac{\mu_0 M_{\mathrm{S}}}{2}\sum_q\frac{w}{a_0}\mathrm{sinc}\!\left(\frac{g_q w}{2}\right)\!\frac{\mathrm{e}^{-\left|g_q\right|d}}{g_q\!\left(d-\Delta d\right)}\\
%&\times\left[\mathrm{e}^{\left|g_q\right|\left(d-\Delta d\right)}-1\right]\left(\mathrm{e}^{\left|g_q\right|\Delta d}-1\right)\mathrm{e}^{ig_q x}
%\end{split}
%\label{Demag}
%\end{equation}
%
%\del{for $x=a_0/2$ where $g_q=2\pi q/a_0$ is the reciprocal lattice vector of order $q$.}


%A comparison between the uniform mode in the film limit (blue dotted line in Fig.~\ref{FilmLimit}(d)) and the $n=0$ mode in SMMCs (orange dashed line in Fig.~\ref{multistep}(e)) reveals the transition of the uniform mode to a localized quasi-uniform excitation of the thin part of the SMMC (here termed \textit{trench} mode). The transition of the mode character can be understood best with the help of Fig.~\ref{profiles} where the $\Delta d$-dependent mode profiles at the marked resonance positions in Fig.~\ref{multistep} are displayed. Figure~\ref{profiles}(a) shows the mode profiles of the \textit{trench} mode ($n=0$) within one period $a_0$. Moreover, a colorplot of the $\Delta d$-dependent mode amplitude within the trench region can be found in Fig.~\ref{profiles}(f). 
%
%A clear maximum in the center of the trench is observed, which becomes more confined with increasing modulation height. Moreover, the amplitude in the wire region vanishes as $\Delta d$ increases. These observations are evidences for a transition from the uniform mode towards a quasi-uniform ($k=0$) excitation of the trench center, which vanishes if the trench is completely removed.
%
%The confinement of the center peak of the \textit{trench} mode is explicitly explained in Ref.~[\onlinecite{Langer2016'}] and increasing with $\Delta d$. The effect is connected with the diverging internal $H_{\mathrm{d}}$ fields. For spins in neighboring regions to the trench center, the higher internal fields shift the solution of the SW dispersion in Eq.~(\ref{BVdisp}) to $k\neq0$. And due to an increasing internal field contrast, the local $k$-vector becomes larger with $\Delta d$ leading to a narrowing of the center peak. 
%
%The width of this central peak and the evolution of its confinement can be explained with the local adaptation of the wave vector further apart from the \textit{trench} center (see reference~[\onlinecite{Langer2016A}]). Since the space-dependent internal $H_{\mathrm{d}}(x)$ field is not constant, the $k$-vector adapts in order to accomplish the SW dispersion in Eq.~(\ref{BVdisp}). And therefore, spins between the \textit{trench} center and the boundaries of \textit{trench} oscillate with $k\neq0$. And due to an increasing internal field contrast, the local $k$-vector becomes larger with $\Delta d$. 

%
%The mode profiles (see Fig.~\ref{wiremodes}) obtained by continuous wave\cite{Wagner2015} FMR simulations reveal the character of each branch in the $f(H_0)$ dependence at $\mu_0 H_0=280$~mT: (a) the edge-mode,\cite{Bayer2005} (b) the 1$^{\mathrm{st}}$ (fundamental) center mode, (c) the 3$^{\mathrm{rd}}$ center mode and (d) the 5$^{\mathrm{th}}$ center mode. The mode-splitting (anticrossing) visible at $f=17.7$~GHz and $\mu_0 H_0=460$~mT is of a similar character as already seen for crossing spin-waves in the limit of small $\Delta d$. Here, the fundamental mode wire-external splits up in an optical and acoustical branch due to the coupling to the 3$^{\mathrm{rd}}$ center mode, as can be clearly seen from the mode profiles in Fig.~\ref{wiremodes} at $\mu_0 H_0=460$~mT. Thus, it is worth to mention that a more accurate description of the 3$^{\mathrm{rd}}$ (5$^{\mathrm{th}}$) mode, would be `the optical coupling mode of the fundamental mode and the 3$^{\mathrm{rd}}$ (5$^{\mathrm{th}}$) mode'. Note, that the unexpected minimum of the fundamental mode in the center of the wire at 280~mT is already an effect of the anticrossing with the 3$^{\mathrm{rd}}$ mode. Moreover, one can see that only spin waves with odd number were observed to couple to the fundamental mode, since a coupling requires a sufficient amplitude in the center of the structure, which is only given for spin waves of odd wavenumber.
%\todo{shorten text of individual modes}
\vspace{3mm}
$\mathbf{1^{\mathrm{st}}}$ \textbf{BV Mode.}\quad The bright blue dashed lines in the Figs.~\ref{multistep}(e)--(h) indicate the wire mode. Originating from the 1$^{\mathrm{st}}$ BV mode in the film limit, [see $n=1$ mode in the inset of Fig.~\ref{FilmLimit}(d)], this mode evolves from an extended BV mode with the wavelength $\lambda = a_0$ to a fundamental (center) mode of the wire with one peak ($m=1$) confined in this region. The mode profiles in Figs.~\ref{profiles}(b) and \ref{profiles}(g) demonstrate this transition. Note that the additional peaks visible for $\Delta d = 36.0$~nm are due to a strong coupling to the $n'=4$ mode [also visible in Figs.~\ref{WireLimit}(a) and \ref{wiremodes}(c)]. 

The reason for the mode's confinement in the wire region lies in the energy of the mode. Already in the film limit, the resonance frequency of the mode lies for all field values $>50$~mT below the uniform mode. With insufficient energy for an excitation of the complete structure, the mode localizes in those regions where the energy is internally reduced due to the presence of a demagnetizing field. Thus, the mode is strongly suppressed in regions with \textit{magnetizing} fields (i.e.\ the trench). Even at low modulation heights of only a few nanometers, the demagnetizing fields in the wire region are strong enough to shift the complete mode to higher field values [bright blue dashed line in Figs.~\ref{multistep}(e)--(h)] with the formation of a large frequency and field gap with respect to the quasi-uniform mode [orange dashed line in Figs.~\ref{multistep}(e)--(h)]. As the locally alternating demagnetizing fields increase with the modulation height, the gap between both fundamental modes of the wire and the trench becomes very large reaching nearly 5~GHz at $\Delta d=13.5$~nm.

Eventually, the 1$^{\mathrm{st}}$ BV mode ($n=1$) becomes therefore the wire mode with $m=2n-1=1$ peaks inside the wire. The mode profiles in Figs.~\ref{profiles}(b) and \ref{profiles}(g) support this transition of the mode character consistently up to the limit of the full MC [Fig.~\ref{wiremodes}(b)]. Here the rule $n'=m-1$ applies and the mode forms the 'new' quasi-uniform mode ($n'=0$) of the full MC.
%It shall be mentioned that the \textit{wire} mode is of a different nature than the \textit{trench} mode with respect to the local $k$-vector. Even though both modes exhibit a similar shape within the respective region, the \textit{wire} mode is localized at a local maximum of the internal $H_{\mathrm{d}}$ field, whereas the \textit{trench} mode is excited at a local minimum. As mentioned above, for small wavelength excitations, the $k$-vector locally rises with higher internal fields and in-turn, drops with lower internal fields due to Eq.~(\ref{BVdisp}).\cite{Langer2017} So, if the center of the wire was excited quasi-uniformly ($k=0$), like in case of the \textit{trench} mode, there would be no possible solution solving Eq.~(\ref{BVdisp}) for the neighboring spins, which are exposed to a reduced $H_{\mathrm{d}}$ field. This would immediately damp out this mode. Hence, the \new{fundamental \textit{wire} mode} must exhibit a $k\neq0$ excitation in the center, and hence, should be interpreted as an $m=1$ standing spin-wave mode within the wire region.
%
%Eventually, mode 1 is following the transition from an $n=1$ film mode to an $m=1$ mode in the `full' MC.

\begin{figure}[!ht]
\includegraphics[width=1\columnwidth]{profiles0.eps}
\caption{\label{profiles} (Color online) (a)--(e) Mode profiles at the marked resonance positions in Fig.~\ref{multistep} revealing the transitional states between the film and the full MC limit. (g)--(j) The $\Delta d$-dependent mode profiles inside the wire region being an important proof for the transition from the $n^{\mathrm{th}}$ film mode to the $m^{\mathrm{th}}$ mode in the full MC limit.}
\end{figure}

\vspace{3mm}
$\mathbf{2^{\mathrm{nd}}}$ \textbf{BV Mode.}\quad %With moderate modulation heights of $\Delta d\approx 10$~nm, the 2$^{\mathrm{nd}}$ SW mode in the film limit starts to gain intensity, as illustrated in the Figs.~\ref{multistep}(g) and \ref{multistep}(h), and evolves towards an \textit{edge} mode inside the wire region. This transition is again reflected by the mode profiles in the Figs.~\ref{profiles}(c) and \ref{profiles}(h). 
The transition of the $2^{\mathrm{nd}}$ BV mode is particularly interesting because the mode reveals with its $2n=4$ peaks distributed over a full period $a_0$ a wavenumber which is close to the minimum of the parabolic SW dispersion in backward-volume geometry [Eq.~(\ref{BVdisp})]. This means that the mode does not only have less energy than the quasi-uniform preventing the mode from extending over the full SMMC. The energy is even below the fundamental wire mode (with $m=1$), meaning the $2^{\mathrm{nd}}$ BV mode cannot fully extend over the wire region and is forced to mainly localize at the edges of the wire where the energy is lowest due to the high demagnetizing fields. 

This is the reason why the mode profile of this spin-wave mode is highly distorted and its evolution can be divided into two steps: (i) The localization of $m=2n-1=3$ peaks inside the wire and subsequently, (ii) the gradual suppression of the central peak and the localization at the two wire edges. The reason for (i) is equivalent to the explanation for the localization of the wire mode. In the film limit, the 2$^{\mathrm{nd}}$ SW mode lies even below the 1$^{\mathrm{st}}$ mode (at fields above 100~mT) with the same consequence that the mode cannot extend over the complete structure and instead localizes $m=3$ peaks in the wire region. The reason for (ii) is the formation of the distinct spin-wave wells,\cite{Bayer2005} i.e., minima of $H_{\mathrm{d}}(x)$, coinciding with the outermost peaks of this mode close to the edges of the wire and scaling with the degree of surface modulation. Thus, the mode profiles of the second mode in Fig.~\ref{profiles}(c) and \ref{profiles}(h) are interpreted as a transition from the $n=2$ mode to an $m=2n-1=3$ mode with increasing mode localization at the edges ending up as the edge mode in the full MC [see Fig.~\ref{wiremodes}(a)] with the mode number $n'=m-1=2$.
%
%The mode appears as a hybrid mode of a 3rd mode within the \textit{wire} and an edge mode. Vividly speaking, the mode exhibits similar characteristics like an oscillating string being excited at the two edges only. Analogous to a guitar string, it oscillates with a defined number of peaks between both boundaries and the amplitude towards the center is being reduced.
%
%Hence, the abstracted transition of the edge mode reads $n$=2 $\rightarrow$ $m$=3 keeping in mind that without following the modes' evolution the assigned of mode number $m=3$ would be doubtable.

\vspace{3mm}
\textbf{Higher BV Modes.}\quad Backward-volume modes of higher order ($n>2$) are also present throughout the transition from a film to a full MC. And as mentioned before, these modes only carry significant intensity when crossing one of the fundamental modes (see Fig.~\ref{multistep}) to which they can couple. Thus, these modes exhibit quite different mode profiles depending on whether they couple to the trench or to the wire mode. In the former case, the modes extend over the complete structure with the largest amplitude in the trench region. In contrast to localized modes, extended modes conserve their total peak number $2n=m+l$ under a gradual increase of the ratio $m/l$ due to the internal field contrast between trench and wire scaling with the surface modulation. In other words, the modes adopt their wavelength to the underlying energy landscape such that the wavelength is reduced where the internal field is high (trench) and vice versa where it is small (wire/edges). This is supported by the mode profiles of the 4$^{\mathrm{th}}$ mode [Fig.~\ref{profiles}(e) and \ref{profiles}(j)] where with increasing modulation height the peaks squeeze up inside the wire region whereas they get wider in the trench region. In the case of higher modes coupling to the wire mode, the modes localize such that they concentrate their amplitude inside the thick part [see profiles of mode 3 and 4 in Figs.~\ref{profiles}(d) and \ref{profiles}(e), respectively]. However, for the transition to the full MC, only the profile inside the wire region is crucial revealing the steady evolution of the mode character to the full MC. This steady process is depicted in Figs.~\ref{profiles}(i) and \ref{profiles}(j) revealing the concentration of $2n-1$ peaks inside the wire.

%Generally, it is observed, that film modes localize $m=2n-1$ peaks within the wire region during this transition process. Together with a single `silent' (damped) minimum in the center of the trench, the total number of $2n$ peaks within one period $a_0$ is conserved throughout this process. % In some cases, like for modes 3 and 4, the mode profiles in the `full' MC [see Figs.~\ref{wiremodes}(c),(d)] rather appear as $m=3$ and $m=5$ modes instead of $m=5$ and $m=7$. In both cases, the last peaks at the edges are incomplete raising the question whether they contribute to the peak number $m$ in the limit of the `full' MC or not. 
%
%To answer this question, it is useful to identify their role in the transition to the `full' MC. Take for example the 3$^{\mathrm{rd}}$ mode at $\Delta d=18$~nm [see Fig.~\ref{profiles}(d)]. Only when both edge peaks are fully taken into account, the total number of nodes within one period remains 6, making $n$ a conserved quantity. At this stage of the transition, the mode's profile inside the wire region is already set and since both peaks' amplitudes are clearly damped outside the wire, they should fully contribute to the mode number $m=5$ inside the wire.
%
%Certainly, modes of a high peak number $m$ tend to cut off their edge peaks, since in that way, the high cost of exchange energy $Dk^2 \propto (m/w)^2$ is minimized when condensing $2n-1$ peaks within the narrow wire region. Thus, the individual pinning of the $m^{\mathrm{th}}$ mode at the wire edges is likely depending on the width $w$ of the wire. This is supported by the mode profiles shown in Ref.~[\onlinecite{Langer2017}] where an SMMC of a different wire width $w=166$~nm, but with the same periodicity $a_0=300$~nm, thickness $d=36.8$~nm, and material parameters was analyzed. The significantly larger nominal wire width results in similar mode characteristics but with a different pinning condition at the wire edges.
%
In brief, the coherent pictures is that the $n^{\mathrm{th}}$ film mode transforms into the $m^{\mathrm{th}}$ mode with $m=2n-1$ during the transition from the film to a highly modulated SMMC. Together with an $l=1$ `silent' (damped) peak inside the etched trench region, this relation reflects the modes' momentum conservation fitting $2n$ peaks inside one period $a_0$ during this process. Arriving in the full MC limit, where only the wire region remains, the simple relation $n'=2(n-1)$ is found as the link to the SW modes in the film limit. Only the $n=0$ mode maintains a local quasi-uniform character inside the trench region and vanishes completely in the full MC.

%Even though, the pinning condition at the \textit{wire} can appear very different and e.g., in case of mode 3 and 4, the last peak at the edges is incomplete.
%
%\begin{figure}[t]
%\includegraphics[width=\linewidth]{suscept.eps}
%\caption{\label{suscept} (Color online) Susceptibility of the modes above the large frequency gap (blue) and below the gap (orange) dependent from the modulation height $\Delta d$. (a-b) The susceptibility at 150~mT and (c-d) the susceptibility at 400~mT. The measured data is shown on the left side accomplished with the simulation on the right side.}
%\end{figure}
%
%It is obvious that for this final state deviations from the measurement are somewhat larger due to (i) neglecting beveled edges leading also to (ii) an effective broadening of the wires, which together with (iii) the increasing impact of the parameter $w$ on the $f(H_0)$-dependence result in major differences. Moreover, the structural edges result in a very narrow minimum of the demagnetizing field, which in turn is hyper-sensitive to small individual changes in the edge-shape. Such effect causes a major increase of the linewidth of this mode. The strong impact of the simplified structure edge on the position of the edge-mode [see Fig.~\ref{wiremodes}(a)] and its linewidth is already evident from the Figs.~\ref{multistep}(e) and (k). However, in case of separated wires, the edge mode is not observed due the effects discussed above.
%
%In MCs with a strong property-modulation, the internal demagnetizing field becomes prominent and thus, other approaches, such as micromagnetic simulations and PWM, must be employed. In Fig.~\ref{numericalfH}(a-f) the results of frequency-dependent FMR measurements of an SMMC with gradually increasing modulation height are presented. The data is corroborated with (g-l) $f(H_0)$-plots obtained by pulsed micromagnetic simulations\cite{Mumax} of a perfect SMMC with similar structural and magnetic properties. The perturbation heights are given in the inset of each plot. 
%The situation becomes more complex in the case of an SMMC with a moderate modulation height, as shown in Figs.~\ref{multistep}(c)--(e) and \ref{multistep}(i)--(k). While for $\Delta d = 4$~nm the resonance branches are still governed by the anticrossing of the film modes [orange lines in Fig.~\ref{multistep}(h)], the character of the SW modes for 9 and 18~nm is not intuitively clear. 
%With the help of numerically obtained mode profiles the different coupling strengths and thus, the mode hybridization can be understood.
%With the help of numerical simulations (see Fig.~\ref{multistep}) as well as theoretical calculations, the evolution towards strongly modulated SMMCs can be understood.
%
%\begin{figure}[tb]
%\includegraphics[width=0.9\linewidth]{theoryfH.eps}
%\caption{\label{theoryfH} (Color online) FMR measurements (circles) of each etching step compared to the results obtained from theory (lines). %Note that for all steps the magnetic parameters were chosen according to the characterization of (a) the intact film: $M_{\mathrm{S}} = 735$~kA/m, $g = 2.11$, $D = 23.6$~Tnm$^2$, $d = 36.8$~nm, $a_0 = 300$~nm, $w = 156$~nm. (f) shows the $f(H_0)$ of an array of completely separate wires.
%The parameters are given in Sec.~\ref{BV}.}
%\end{figure}
%
%At first, some general interpretations of different branches in the $f(H_0)$ shall be explained. In all stages of surface modulation, the main mode at low field values originating from the uniform film mode, is a fundamental branch, where the resonant spins are located in the thin part of the MC, whereas the main high-field branch corresponds to a fundamental excitation of the thick part of the MC. It is already evident from the position of the film modes compared to the simulated ones that this branch evolves from the 1$^{\mathrm{st}}$ backward-volume mode in the film limit [see Fig.~\ref{multistep}(h)] to the fundamental wire mode in Fig.~\ref{multistep}(l). The physical explanation of this effect can be provided by the local distribution of the internal demagnetizing field $H_{\mathrm{d}}$. While in the \textit{thick} part of the structure $H_{\mathrm{d}}$ acts against the local direction of magnetization, it acts in the same direction of the local spins in the \textit{thin} part, as a `magnetizing field'. As the uniform mode as well as the 1$^{\mathrm{st}}$ backward-volume mode show a maximal amplitude in both, the thick and the thin part, they both are in principle able to transform gradually into central modes. However, it is the higher resonance field (at sufficient frequencies) of the 1$^{\mathrm{st}}$ film mode compared to the uniform one, that leads to a favoring of its transition to a central wire mode.
%
%\begin{figure}[tb]
%\includegraphics[width=0.9\linewidth]{2SW.eps}
%\caption{\label{2SW} (Color online) Mode profiles obtained by simulations at $\mu_0 H_0=70$~mT and 400~mT. The modes undergo a transition (a,d) from the $n=0$ mode to the fundamental trench mode (b,e) from the $n=1$ mode to the fundamental wire mode (c,f) from the $n=2$ mode to the edge mode. Note, that some transitional states do not show the pure character of the respective mode due to a pronounced coupling to other spin-wave modes. For $\Delta d = 2,4,36$~nm and $\mu_0 H_0=400$~mT, the wire mode shows for instance a pronounced minimum in the wire center which is due to the coupling to the $n=3$ spin-wave mode.}
%\end{figure}
%
%Furthermore, it is observed, that the 2$^{\mathrm{nd}}$ backward-volume mode in the film is developing into a SW mode with edge mode character.\cite{Bayer2005} An explanation for this transition can be provided employing the mode profiles as provided in Fig.~\ref{2SW}. The figure shows the profiles obtained by numerical calculations for two different field values of (a) 70~mT and (b) 400~mT and for various modulation heights. Starting at $\Delta d = 2$~nm, an almost harmonic (film-like) character of the mode is observed at 70~mT. At 400~mT this mode's amplitude is insignificantly low and, for that reason, not shown. With a higher level of modulation, the minima of the demagnetizing field at the edges of the thick part become prominent. Showing major amplitudes at the edges, the 2$^{\mathrm{nd}}$ mode tends to a confinement in this part with increasing modulation height. Thus, this SW mode undergoes a transition toward an edge mode, as can seen best for $\Delta d = 18$~nm and 36~nm in Fig.~\ref{2SW}(f).
%
%Higher backward-volume modes ($n>3$) are also present in this system. One example is the anticrossing of the trench mode (excitation in the thin part) with the 3$^{\mathrm{rd}}$ and 4$^{\mathrm{th}}$ mode shown in Fig.~\ref{4nmModulation}. While the coupling to the first is very strong for $\Delta d = 4$~nm modulation, it almost vanishes for $\Delta d = 9$~nm and remains very small for higher modulation depths. As can be seen from Fig.~\ref{4nmModulation}(a), for small $\Delta d$, the maxima of the amplitude of the fundamental mode can easily be shifted inside the thin part and thus, adopt to the shape of the coupling SW. With higher $\Delta d$ the location of the modes maximum becomes strongly pinned and the coupling to the 3$^{\mathrm{rd}}$ mode is suppressed [see Fig.~\ref{4nmModulation}(a)], as it requires a shift of the SW maximum towards the edges of the thin part. Such shift is more unlikely to happen due to the inhomogeneity of the internal $H_{\mathrm{d}}$-field.
%
%Moreover, it can be seen from Fig.~\ref{4nmModulation}, that the mode profiles generally adopt inside the respective region, where the excitation takes place. As lined out above, this can either be the thick part and corresponds to the coupling to the fundamental `high field' mode or the thin part --as discussed above-- for the fundamental branch at lower resonance fields. That means, that the coupling of SWs with different wave number should not be understood as a lateral coupling with e.g.\ a uniform excitation in one part of the structure and a higher SW mode in the other part. It rather means, that in either the thick or the thin part a fundamental mode forms a hybrid SW when coupling to higher SW in the same respective part of the MC. The existence of sufficiently large SW amplitude in the other part of the system, which is not excited, is only allowed for the fundamental mode in the thin part because its energy is higher than the fundamental mode of the thick part. In the other case, the amplitude of coupling SW modes will abruptly descend in the thinner region.
%
%It is concluded, that the coupling strength of each fundamental branch to a higher SW mode is strongly dependent on the ratio of the amplitudes of both, the higher SW and the fundamental mode at the locations where excitation takes place.
%
%Where one of the fundamental modes crosses the dispersion of higher BV-type SW modes, anticrossing (mode splitting) occurs. The grade of mode-splitting was found to differ a lot dependent on the level of surface modulation, and will be discussed later in the manuscript. Another fundamental observation 
%
%A comparison of the same measurement with the resonance positions obtained by the quasi-analytical theoretical model is presented in Fig.~\ref{theoryfH}. As the model is based on an intact film with altering magnetic properties, it resembles the dynamic properties of a BCMC. Hence, comparing both results, the theoretical ones with the micromagnetic simulations, allows to point out the differences and similarities of physics in SMMCs and BCMCs.
%
%\begin{figure}[tb]
%\includegraphics[width=0.9\linewidth]{4nmModulation.eps}
%\caption{\label{4nmModulation} (Color online) The $f(H_0)$ of an MC with 4~nm and 9~nm surface modulation shown in (a) and (b), respectively. Symbols mark the spin-wave resonance positions evolving from a state dominated by the $n = 3$ backward-volume mode up to the $n = 4$ one. Orange lines correspond to the virtual position of non-interacting backward-volume modes labeled with $n=a_0/\lambda$. Two surface plots in (c) and (d) depict the evolution of the spin-wave mode profiles in both cases. An increasing localization of the quasi-uniform mode towards the center of the thickness reduced part is identified as the reason for the suppression of the $3^{\mathrm{rd}}$ backward-volume mode with larger surface modulations.}
%\end{figure}

\section{Damon-Eshbach geometry}
\label{DE}
%\todo{shorten insgesamt. 1. DE Mode mit FMR gemessen. Charakteristika erwaehnen. Im Film Limit propagierende DE Moden sind harmonisch fuehrt zu Resonanz der stehenden Moden bei Modulation. 2. und 3 Mode nur allgemein als higher order modes diskutieren}
The DE orientation ($\mathbf{k} \bot \mathbf{M}$) with the field applied parallel to the surface edges offers different insights. Since the static internal demagnetizing fields vanish in this geometry, the transition of spin-wave modes is much more influenced by the dynamic fields and geometrical conditions.
%For the sake of completeness, the measurement results for this orientation are provided as well. However, the micromagnetic part of the study will reveal that major changes of the mode character occur at a very high level of surface modulation where the experimental data remains incomplete.
%The field orientation parallel to the stripe-like edges with the wave vector $k$ perpendicular to the magnetization $M$ is referred to as Damon-Eshbach geometry. 

\subsection{The Limits: Thin Film and Full Magnonic Crystal}

\vspace{3mm}
\textbf{Thin Film Limit.}\quad The measurements at lowest modulation heights of $\Delta d=$~2.3~nm, and 4.7~nm did not reveal any significant changes of the FMR spectra compared to the continuous thin film with the typical Kittel-like behavior of the uniform mode. %This means the film limit in DE geometry is governed by the unaltered quasi-uniform film mode.
Figure~\ref{multistepDE} displays the position of the uniform film mode (orange symbols) together with the calculated position of the 1$^{\mathrm{st}}$ DE mode in the film limit (orange dot-dashed line) according to Eq.~(\ref{DEdisp}). It is noted that the 1$^{\mathrm{st}}$ DE mode carries no intensity in the film limit and a prominent surface-modulation of approximately $d/4$ was necessary for the FMR detection of this mode. 

The appearance of higher non-uniform modes at much higher modulations $\Delta d$ compared to the BV geometry can be explained by the DE SW dispersion\cite{Kalinikos1986,Gallardo2014} reading
%
\begin{equation}
\begin{split}
\left(\frac{\omega}{\gamma}\right)^{\negmedspace2}\negthickspace=\!&\left[ \mu_{0} H_{\mathrm{eff}} + \mu_{0} M_{\mathrm{s}} \left( 1-\frac{1-\mathrm{e}^{-kd}}{kd} \right) + D k^2 \right]\\
&\times \left[ \mu_{0} H_{\mathrm{eff}} + \mu_{0} M_{\mathrm{s}} \left( \frac{1-\mathrm{e}^{-kd}}{kd} \right) + D k^2 \right]~.
\end{split}
\label{DEdisp}
\end{equation}
%
In contrast to the BV geometry, there is no energy degeneracy of the uniform mode with higher modes. Thus, no two-magnon scattering channel is present that could easily transfer intensity to higher $k\neq 0$ SW modes.
%
\begin{figure}[t]
\includegraphics[width=1\columnwidth]{multistepDE0.eps}
\caption{\label{multistepDE} (Color online) $f(H_0)$-dependence of SW modes in DE geometry at different surface modulations. The measured quasi-uniform mode (first DE mode) is indicated by full symbols ($\boxplus/\boxminus$ symbols). Black open symbols correspond to the measurement of the full MC. Grey semi-filled symbols represent the simulation of the full MC. Theoretical calculations of the uniform mode in the full MC based on Eq.~(\ref{uniform}) are shown as black solid/dashed lines. The orange dot-dashed line indicates the calculated 1$^{\mathrm{st}}$ SW mode in the film limit [Eq.~(\ref{DEdisp})].}
\end{figure}
%
%Hence, there are no scattering channels present to transfer intensity from the uniform mode to higher $k\neq 0$ SW modes. This is why DE modes are of no (or extremely small) intensities for small modulation heights $\Delta d$ in contrast to standing spin-wave modes in BV orientation. \new{Besides the quasi-uniform mode, the measurements at a modulation height of $\Delta d=$~2.3~nm, and 4.7~nm did not show a recognizable alteration of the FMR spectra of the continuous thin film.}
%A prominent level of modulation in the magnitude of $d/4$ was found to be necessary for the detection of at least a small intensity of the 1$^{\mathrm{st}}$ mode. Thus, the film limit in DE geometry is governed by the \new{almost unaltered quasi-uniform film mode}.
%Figure~\ref{multistepDE} displays the position of the uniform film mode (orange symbols) together with the calculated position of the 1$^{\mathrm{st}}$ DE mode (orange dot-dashed line) according to Eq.~(\ref{DEdisp}), which does not carry intensity.

\vspace{3mm}
\textbf{Full MC.}\quad In this limit, the mode profiles are much less distorted compared to the BV geometry due to the vanishing static demagnetizing fields. Thus, the modes can be more easily assigned to the corresponding idealized profile illustrated in the top row of Fig.~\ref{wiremodes}. The main (quasi-uniform) mode with $n'=0$ can be calculated using the demagnetizing factor of the wires\cite{Lindner2004,Belmeguenai2016}
%
%Again, the two limits, film and full MC, reveal major differences in this field orientation. For example, the uniform mode in the film limit expressed by 
%
%\begin{equation}
%\left(\frac{\omega}{\gamma}\right)^{\negmedspace2}\negthickspace=\!\mu_{0} H_0\times\left[ \mu_{0} H_0 + \mu_{0} M_{\mathrm{S}} \right]
%\label{unifilm}
%\end{equation}
%
%strongly differs from the quasi-uniform mode in a one-dimensional MC expressed by
%
\begin{equation}
\begin{split}
\left(\frac{\omega}{\gamma}\right)^{\negmedspace2}\negthickspace=&\left[ \mu_{0} H_{\mathrm{eff}} + N_{x}\mu_{0} M_{\mathrm{s}} \right]\\
&\times \left[ \mu_{0} H_{\mathrm{eff}} - N_x\mu_{0} M_{\mathrm{s}} + \mu_{0} M_{\mathrm{s}} \right]
\end{split}
\label{uniform}
\end{equation}
%
\noindent with $N_{x}$ being the in-plane demagnetizing factor along the short wire ($x$-) axis [as defined in Fig.~\ref{experiment}(a)] and with the shape-anisotropy $N_x\mu_{0} M_{\mathrm{s}}$. 
%
%Full symbols indicate the measurement of the quasi-uniform mode and open blue symbols correspond to the measured position of the 1$^{\mathrm{st}}$ and 3$^{\mathrm{rd}}$ SW mode. Black semi-filled symbols indicate the simulation of the the 1$^{\mathrm{st}}$ and 3$^{\mathrm{rd}}$ SW mode of the full magnonic crystal. The orange dot-dashed line is the 1$^{\mathrm{st}}$ mode in the thin film limit according to Eq.~(\ref{DEdisp}) and the dashed (solid) blue line indicates the 1$^{\mathrm{st}}$ mode of the full MC using Eq.~(\ref{uniform}) with $N_x=0.160$ ($N_x=0.115$).}% The green line highlights the frequency of 17~GHz, which corresponds to the data shown in Fig.~\ref{transitionDE}.}}
%Measurement of the frequency-field dependences of the 1$^{\mathrm{st}}$ Damon-Eshbach mode (open symbols) for the modulation heights $\Delta d=$~8.7~nm, 13.2~nm and 36.8~nm. Crossed symbols represent the results of micromagnetic simulations. The standard material parameters were used as fitting parameters yielding an effective thickness $d=27.6$~nm and 23.0~nm for $\Delta d=$~8.7~nm, and 13.2~nm. In case of the `full' magnonic crystal, the fit yields a demagnetizing factor of $N_x=0.115$.}
%
%The demagnetizing factors can be obtained analytically by\cite{Brown1962}
%
%\begin{equation}
%N_{x}=\frac{1}{\pi}\bigg[\frac{1-p^2}{2p}\mathrm{ln}\left(1+p^2\right)+p\mathrm{ln}p+\mathrm{arctan}\left(\frac{1}{p}\right)\bigg]~.
%\label{N}
%\end{equation}
%
%\noindent where $p$ denotes the thickness $d$ vs.\ width $w$ ratio $p=w/d$. Similar to Ref.~[\onlinecite{Belmeguenai2016}], this mode is referred to as $m=1$ mode or center mode to not be confused with the uniform mode. There are also higher order modes standing across the width of the wire. 
Again, higher $k\neq 0$ modes are also present with the wave vector quantized with $k=n'\pi/w$, $n'=2,4,6...$ due to the number of nodes fitting in the wire width $w$.\cite{Belmeguenai2016} It is noticed that the pinning of the mode at the edges of the structure has a strong influence on this estimation and the actual $f(H_0)$ dependence of the mode.%Therefore, a modified mode number $m \rightarrow m+\lambda$ can be defined with $\lambda$ being the pinning parameter leading to a higher accuracy of the resonance positions.\cite{Belmeguenai2016}

The measurement of the full magnonic crystal (black open symbols in Fig.~\ref{multistepDE}) reveals two modes---the quasi-uniform ($n'=0$) and the 1$^{\mathrm{st}}$ DE mode ($n'=2$). The positions of both modes obtained by micromagnetic simulations are plotted as semi-filled gray squares in Fig.~\ref{multistepDE}). Apart from a systematic reduction of the measured SW resonance frequency compared to the simulation, a qualitative agreement is achieved. It is demonstrated by the $f(H_0)$-dependences calculated with the help of Eq.~(\ref{uniform}) that the frequency of the uniform ($n'=0$) mode is reduced if the effective demagnetizing factor of the wire is smaller. When $N_{x}$ was fitted (solid black in Fig.~\ref{multistepDE}) to the measurement data, a value of $N_x=0.115$ is obtained. $N_{x}$ can also be calculated\cite{Brown1962} employing the ideal geometrical properties of the wires listed in Sec.~\ref{Exp} yielding $N_x=0.160$. As the latter approach (dashed black line in Fig.~\ref{multistepDE}) yields a reasonable agreement of theory and simulation, it can be concluded that imperfect (tapered) edges are likely effectively reducing the demagnetizing factor of the wires leading to the deviation between the simulated and measured $f(H_0)$-dependences.
%
%In case of the measurements carried out in this work, only the 1$^{\mathrm{st}}$ mode was detected (black open pentagons in Fig.~\ref{multistepDE}). However, micromagnetic simulations (black crossed pentagons and squares in Fig.~\ref{multistepDE}) reveal the existence of higher ($m>1$) modes with small intensity. The simulated resonance position of the second highest intensity mode with $m=3$ (termed `\textit{edge} mode') is also shown (crossed squares) in Fig.~\ref{multistepDE} where the crossed pentagons represent the $n=1$ (\textit{wire}) mode.
%
\subsection{Transitional SMMCs}
%
%\todo{Mention that apart from the quasi-uniform mode, no localized modes exist in DE geometry. These modes are, in general, extended modes revealing some partially propagating character due to the non-reciprocity of the system.\cite{Gladii2016,Gladii2017,Di2013,Di2015}}
%
%DE modes in SMMCs are known to reveal a dipole-dipole induced frequency non-reciprocity\cite{Mruczkiewicz2014,Lisenkov2015} leading to a propagation of higher SW modes even at $k=0$. Such effects were observed in this work but will be addressed elsewhere.
%
In Fig.~\ref{multistepDE}, the $f(H_0)$-dependence of the measured $\Delta d$-dependent SW resonances is illustrated for the quasi-uniform mode (full symbols) and the 1$^{\mathrm{st}}$ DE mode (open symbols with and without filling pattern). The measurement is corroborated with the results from theoretical calculations [according to Eqs.~(\ref{DEdisp}) and (\ref{uniform})] of the 1$^{\mathrm{st}}$ mode in the film limit (orange dot-dashed line) and in the full MC limit (black dashed and solid lines). The results from the micromagnetic simulations of the full MC are plotted as gray semi-filled symbols. The green line marks the frequency of 17~GHz at which the field-dependent dynamic response was simulated and compared to the measurement data (Fig.~\ref{transitionDE}) for different modulation heights $\Delta d$. The dynamic response is plotted in gray scale and colored full symbols mark the measured SW resonances. 
%results of the measurement and simulation are compared in Fig.~\ref{transitionDE} depending on $\Delta d$. 
The corresponding simulated mode profiles are presented in Fig.~\ref{ProfilesDE} for the quasi-uniform mode and the first three higher modes.

%During this study, different effects connected to this phenomenon, such as SW propagation were observed and are not further discussed since the 
%The calculation of the $\Delta d$-dependent position of the 1$^{\mathrm{st}}$ SW mode is depicted as bright blue dotted line Fig.~\ref{transitionDE}.} However, at a modulation height of about half the film thickness, applying Eq.~(\ref{DEdisp}) leads to major deviations from the measured [open blue squares in Fig.~\ref{transitionDE}] and simulated (colorplot) resonance positions.
%However, micromagnetic simulations [colorplot in Fig.~\ref{transitionDE}(a)] indicate that the 1$^{\mathrm{st}}$ DE mode (together with the 2$^{\mathrm{nd}}$ mode) evolves into a \new{fundamental (\textit{wire}) mode} in the full magnonic crystal. At the same time, the quasi-uniform mode [orange lines and symbols in Fig.~\ref{transitionDE}(a)] gradually disappears.

Before the transition process for each of these modes is addressed in detail, it is noted that DE modes in SMMCs show significant dipolar induced non-reciprocal properties.\cite{Lisenkov2015} As a consequence, the $n\neq0$ DE modes reveal a propagating character and their mode profiles are no longer fully symmetric with a tendency to show a non-uniform vertical mode profile in addition. In the following discussion this complexity will not be addressed and the main focus will be on the averaged transversal characteristics of the DE modes.

\begin{figure}[t]
\includegraphics[width=1\columnwidth]{transitionDE0.eps}
\caption{\label{transitionDE} (Color online) Micromagnetic simulation of the SW resonances at $f=17$~GHz (green line in Fig.~\ref{multistepDE}) depending on the modulation height $\Delta d$ in DE orientation. The measurement is represented by full symbols. The blue dashed line indicates the position of the 1$^{\mathrm{st}}$ calculated by Eq.~(\ref{DEdisp}) using an effective film thickness $d_{\mathrm{eff}}$.}%Note the fading uniform mode and the emerging of the \textit{wire} mode in the `full' magnonic crystal from the 1$^{\mathrm{st}}$ and the 2$^{\mathrm{nd}}$ spin-wave mode.%\todo{bleibt aber Side-by-side mit Fig9a/f,b/g. Mode DE1b raus. Mode 2,3 unwichtiger.}}
\end{figure}

%\begin{figure*}[t]
%\includegraphics[width=\textwidth]{TransitionProfiles.eps}
%\caption{\label{transitionDE} \new{(Color online) (a) Micromagnetic Simulation of the SW resonances at $f=17$~GHz depending on the modulation height $\Delta d$ in Damon-Eshbach orientation. Symbols represent the results from the measurement. Note the fading uniform mode and the emerging of the \textit{wire} mode in the `full' magnonic crystal from the 1$^{\mathrm{st}}$ and the 2$^{\mathrm{nd}}$ spin-wave mode. (b)--(e) Simulated spin-wave mode profiles for the uniform and first Demon-Eshbach mode. (d,e) Colorplots of the mode profiles in dependence of the relative modulation depth $\Delta d/d$. The amplitude is weighted by the local film thickness.}}
%\end{figure*}
%
%\vspace{3mm}
%\textbf{Quasi-Uniform Mode.}\quad In Figs.~\ref{transitionDE}(b,c) the $\Delta d$-dependent mode profiles of the quasi-uniform mode and the first DE-type standing SW mode are plotted. On the left side [Figs.~\ref{transitionDE}(b) and \ref{transitionDE}(c), the space-dependent spin-wave amplitude is plotted, whereas the colorplots on the right side [Figs.~\ref{transitionDE}(d) and \ref{transitionDE}(e)] reveal the effective amplitudes, i.e.\ the same profiles scaled with the local film thickness. In case of the $n=0$ (quasi-uniform) mode, the dynamic activity appears to be concentrated inside the thin part, which is analogous to the character of the \emph{trench} mode in the BV geometry. However, when scaled with the local film thickness, the response of both, thick and thin part remains equally high in the center of both regions. This observation is a major difference compared to the behavior of the $n=0$ mode in BV direction. Here, in the DE orientation, the effective behavior remains quasi-uniform in the \emph{full} structure. And as the modulation becomes dominant, the effective dynamic activity is reduced close to the edges of the wire, where dynamic demagnetizing fields become relevant altering the SW dispersion. This leads to the picture illustrated in Fig.~\ref{ProfilesDE}(d), where a constant effective SW amplitude is visible together with channels of reduced dynamic activity at the boundaries between both regions.
%\vspace{3mm}
%\textbf{Quasi-Uniform Mode.}\quad \new{From the colorplot in Fig.~\ref{multistepDE} as well as the $f(H_0)$-dependences (filled symbols) in Fig.~\ref{transitionDE}, it can be seen that the quasi-uniform mode maintains its approximate $f(H_0)$-position as it gradually loses intensity until it vanishes in the full MC limit. Besides that,} the dynamic activity appears to concentrate inside the thin part [see Fig.~\ref{profilesDE}(a)], which is analogous to the character of the \emph{trench} mode in the BV geometry. However, when scaled with the local film thickness [Fig.~\ref{profilesDE}(e)] the response of both, thick and thin part remains equal during the whole transition. This observation is a major difference compared to the behavior of the mode in BV direction. \new{Since in the DE orientation no static dipolar fields are present, the SW amplitude is only reduced close to the edges where the \textit{dynamic} stray fields maximize. Thus, similar to the BV direction, the main FMR mode of the full MC does not originate from the quasi-uniform mode of SMMCs.}
%\done{The discussion of each SW mode in DE geometry has been reduced}
\vspace{3mm}
\textbf{Quasi-Uniform Mode.}\quad 
In the $f(H_0)$-dependence, the uniform mode maintains its Kittel-like behavior up to significant modulations even above $\Delta d=d/2$, as can be seen from Fig.~\ref{multistep} and Fig.~\ref{transitionDE}, which is a main difference compared to the BV geometry, where the uniform mode evolves rapidly towards a trench mode with a strong up-shift of the resonance frequency due to the presence of magnetizing fields. The preservation of its original uniform character is also reflected in the mode profiles shown in Figs.~\ref{ProfilesDE}(a),(e). Apart from the wire edges, the amplitude of the mode is almost evenly spread over a full period even up to prominent modulation heights $\Delta d>d/2$. Interestingly, at very high modulations close to $\Delta d\rightarrow d$ the mode seems to evolve a stronger amplitude in the center of the trench [see mode profile at $\Delta d=31.5$~nm in Fig.~\ref{ProfilesDE}(a)] , similar to the observation in the BV orientation. In Fig.~\ref{multistepDE}, it can be seen that the quasi-uniform mode maintains its approximate $f(H_0)$-position and loses gradually intensity until it vanishes in the full MC limit. Thus, similar to the BV direction, the main FMR mode ($n'=0$) of the full MC does not originate from the quasi-uniform mode of the SMMC.

%In the mode profiles in Fig.~\ref{ProfilesDE}(a), the dynamic activity appears to concentrate inside the thin part, which is analogous to the character of the trench mode in the BV geometry. However, when scaled with the local film thickness [Fig.~\ref{ProfilesDE}(e)] the response of both, thick and thin part remains equal during the whole transition. This observation is a major difference compared to the behavior of the quasi-uniform mode in BV direction. Since no static dipolar fields are present in the DE orientation, the SW amplitude is equally spread and reduces only close to the edges where the \textit{dynamic} stray fields maximize. From the colorplot in Fig.~\ref{transitionDE} and from the $f(H_0)$-dependences (full symbols) in Fig.~\ref{multistepDE}, it can be seen that the quasi-uniform mode maintains its approximate $f(H_0)$-position as it gradually loses intensity until it vanishes in the full MC limit. Thus, similar to the BV direction, the main FMR mode of the full MC does not originate from the quasi-uniform mode of the SMMC.
%\done{The discussion of each SW mode in DE geometry has been reduced}

%\vspace{3mm}
%$\mathbf{1^{\mathrm{st}}}$ \textbf{SW Mode.}\quad \del{RAUSLASSEN, da DE1b nicht mehr markiert?: The colorplot in Fig.~\ref{transitionDE}(a) reveals a splitting of the 1$^{\mathrm{st}}$ SW mode into two modes. A high-intensity mode changing gradually into a \emph{wire} mode and, closer to the dotted line corresponding to Eq.~(\ref{DEdisp}), another low-intensity mode. The latter fades out for high $\Delta d$. The profiles of both modes are illustrated in Figs.~\ref{transitionDE}(c) and \ref{transitionDE}(e).} \new{The blue dashed line in the colorplot Fig.~\ref{transitionDE}(a) marks the $1^\mathrm{st}$ spin wave mode. This mode changes gradually into a \emph{wire} mode for high $\Delta d$. Figure \ref{transitionDE}(c) and \ref{transitionDE}(e) depict the corresponding mode profiles. Starting from the film limit, the character of the mode is dominated by the harmonic character of an $n=1$ film mode. It remains fully symmetric throughout the whole transition.} \del{Only for pronounced modulations ($\Delta d \geq 27$~nm), an asymmetry within the thin part occurs in case of the low-intensity mode (Figs.~\ref{ProfilesDE}(c),(h)), while the high-intensity mode (Figs.~\ref{ProfilesDE}(b),(g)) remains fully symmetric throughout the whole transition.} This mode keeps one peak within the thick part throughout the transition whereas the number of peaks within the thin region increases with $\Delta d$. Once the thin part of the SMMC is fully removed, only the peak within the thick part remains as the \emph{wire} mode of the full MC. \del{It shall be noted, that the origin of the low-intensity mode remains unclear but might be due to the breaking of the symmetry by the \textit{dynamic} demagnetizing fields.} 
\vspace{3mm}
$\mathbf{1^{\mathrm{st}}}$ \textbf{DE Mode.}\quad The measured $f(H_0)$-dependences in Fig.~\ref{multistepDE} indicate instead that the $1^{\mathrm{st}}$ DE mode ($\boxplus/\boxminus$ symbols in Fig.~\ref{multistepDE}) gradually evolves to the uniform mode ($n'=0$) of the full CM (open squares in Fig.~\ref{multistepDE}). Clearly, at the modulations of $\Delta d=8.7$~nm and $\Delta d=13.2$~nm the $1^{\mathrm{st}}$ mode gains intensity and is slightly shifted towards lower frequencies and higher field values closer to the position of the quasi-uniform mode of the full MC. This supposed transition is supported by the micromagnetic simulation (colorplot) in Fig.~\ref{transitionDE}. Here, a steady $\Delta d$-dependent development of the 1$^{\mathrm{st}}$ SMMC mode towards the $n'=0$ full MC mode is identified together with the experimentally obtained data (full squares) at 17~GHz. For small modulations, this behavior can be calculated by Eq.~(\ref{DEdisp}) employing a reduced effective film thickness of $d_{\mathrm{eff}}=a^{-1}_0[wd+(a_0-w)(d-\Delta d)]$. This relation is depicted as a dotted blue line in Fig.~\ref{transitionDE} revealing a reasonable agreement for modulations of $\Delta d\leq d/2$. For larger modulations, this estimation becomes systematically wrong and the simulated mode profiles in Figure \ref{ProfilesDE}(b),(f) start to reveal major deviations from the sinusoidal film-mode character and are dominated by a central peak inside the wire region, which is already characteristic for the wire mode in the full MC. 

Similar to the BV geometry, the full evolution of the $1^{\mathrm{st}}$ DE mode goes via a state of a pronounced peak inside the wire region ($m=1$) and a suppression of the dynamic response in the trench region ($l=0$). In the full MC limit, only this one peak inside the wire remains and the mode is identified as the quasi-uniform ($n'=0$) mode.

%the colorplot in Fig.~\ref{transitionDE} identifies the 1$^{\mathrm{st}}$ SW mode as the one gradually evolving to the 1$^{\mathrm{st}}$ mode of the full MC. As the simulations reveal an almost linear shift of the resonance position, this transition is also supported by the linear fit (blue dashed line) to the few data points of the measurement (blue open squares).
%%\new{During the experiment, the 1$^{\mathrm{st}}$ SW mode was first detected at $\Delta d=8.7$~nm (crossed blue squares in Fig.~\ref{multistepDE}).} At a modulation height of $\Delta d=13.2$~nm, the mode gains intensity and is slightly shifted towards lower frequencies and higher field values. \new{For small modulations, this behavior can be modeled by Eq.~(\ref{DEdisp}) employing a reduced effective film thickness of $d_{\mathrm{eff}}=a^{-1}_0[wd+(a_0-w)(d-\Delta d)]$. This is demonstrated in Fig.~\ref{transitionDE} where the $\Delta d$-dependent measurement (open symbols) and simulation (colorplot) is shown at $f=17$~GHz. The calculation based on Eq.~(\ref{DEdisp}) and the effective thickness $d_{\mathrm{eff}}$ is depicted as a dotted blue line revealing a good agreement for modulations of $\Delta d\leq d/2$.}
%
%The blue dashed line in the colorplot Fig.~\ref{transitionDE}(a) marks the $1^\mathrm{st}$ spin wave mode. This mode changes gradually into a \emph{wire} mode for high $\Delta d$. Figure \ref{transitionDE}(c) and \ref{transitionDE}(e) depict the corresponding mode profiles. Starting from the film limit, the character of the mode is dominated by the harmonic character of an $n=1$ film mode. It remains fully symmetric throughout the whole transition. This mode keeps one peak within the thick part throughout the transition whereas the number of peaks within the thin region increases with $\Delta d$. Once the thin part of the SMMC is fully removed, only the peak within the thick part remains as the \emph{wire} mode of the full MC.

\begin{figure}[t]
\includegraphics[width=0.9\columnwidth]{ProfilesDE0.eps}
\caption{\label{ProfilesDE} (Color online) (a)--(d) Simulated spin-wave mode profiles in Damon-Eshbach geometry in the transition from a film to a full MC. (e)--(h) The $\Delta d$-dependent colorplots of the mode profiles integrated over the total film thickness.}
\end{figure}
%
%
%\del{$\mathbf{2^{\mathrm{nd}}}$ \textbf{SW Mode.}\quad Due to the coupling to PSSW-type modes, the mode profiles of the 2$^{\mathrm{nd}}$ and 3$^{\mathrm{rd}}$ Damon-Eshbach modes are significantly asymmetrically distorted. Even though the distortion causes complex time-dependent SW amplitude oscillations, the main mode character is conserved. In Figs.~\ref{ProfilesDE}(d),(i) the $2n$ peaks of the 2$^{\mathrm{nd}}$ mode are clearly identified as well as the gradual transition of the mode profile within the thick part towards the \textit{wire} mode. In fact, the two mode profiles of the $n=1$ and the $n=2$ mode become less different as very high modulations are reached. For example, at $\Delta d = 35$~nm, both modes consist of a clear central peak within the thick part and another small-wavelength excitation (consisting of approximately 3 peaks) within the thin part. The only difference is that the phase of the small-wavelength excitation in the thin part is shifted around $\pi$. 
%From this observation, it can be concluded that close to the full MC the mode character of both the $n=1$ mode and the $n=2$ mode is dominated by a \textit{wire} mode character inside the thick part with an optical ($n=2$ mode) and acoustical ($n=1$ mode) lateral coupling low-wavelength excitation inside the thin part.}
%\vspace{3mm}
%\del{$\mathbf{3^{\mathrm{rd}}}$ \textbf{SW Mode.}\quad The general mode character of the $n=3$ mode with its $2n=6$ peaks in total is also preserved during the transition. However inside the thick part, this mode was observed to evolve into an \textit{edge} mode in the full MC limit (see Figs.~\ref{ProfilesDE}(e),(j)). The typical mode profile characteristics of this mode are already established at $\Delta d = 31.5$~nm inside the \textit{thick} part with pronounced peaks at both edges and a smaller anti-phase peak in the center. According to the above findings for the $n=1$ and $n=2$ modes, one would expect the presence of an analogous second mode close to the full MC limit with an anti-phase dynamic activity inside the \textit{thin} part. Such mode exists in the spin-wave spectra presented in Fig.~\ref{transitionDE}, but reveals a much smaller (negligible) resonance amplitude. The different behavior is due to the symmetry of the mode profile inside the thick part. The characteristic pronounced peaks at the edges of the thick part hinder the lateral coupling of an anti-phase excitation inside the thin part due to the high cost of exchange energy.}

\vspace{3mm}
$\mathbf{2^{\mathrm{nd}}}$ \textbf{DE Mode.}\quad The intensity plot of the simulated dynamic response in Fig.~\ref{transitionDE} reveals that the 2$^{\mathrm{nd}}$ DE mode appears at much higher modulation compared to the $1^{\mathrm{st}}$ mode. This might as well be the reason why this mode was not observed in the FMR measurement. Moreover, the simulations indicate an interesting detail, namely the merging of the $1^{\mathrm{st}}$ and $2^{\mathrm{nd}}$ DE mode in the full MC limit. Note that the two modes remain as individual resonances even up to a modulation of $\Delta d/d=0.97$. At these high modulations, the two resonances move gradually towards each other until they fully merge into the quasi-uniform mode in the full MC. 

To understand the evolution of this mode and its merging process, the mode profiles in Figs.~\ref{ProfilesDE}(c),(g) need to be considered. Besides the distortions due to the vertical profile, the mode shows its characteristic four peaks within one period for small modulations [see the profile for $\Delta d=9$~nm in Fig.~\ref{ProfilesDE}(c)]. In the regime of small modulations, two peaks of the mode are located at the two wire edge, which is natural for the $2^{\mathrm{nd}}$ DE mode since its wavelength is exactly twice the periodicity $a_0$. But as the modulation height increases, it comes to the critical point where these two peaks either need to be packed into the trench region or the wire region (as observed for the $2^{\mathrm{nd}}$ BV mode) in order to avoid the edges. In contrast to the BV orientation, there are no static demagnetizing fields present and, therefore, the mode is highly influenced by the geometrical properties. That means, that the width of the wire and the trench are likely to play an important role here. To be more precise, as the width of the wire is slightly smaller than the trench, an evolution towards a mode with $m=1$ and $l=3$ is energetically more favorable than a mode with $m=3$ and $l=1$. This is supported by the mode profile for $\Delta d=31.5$~nm in Fig.~\ref{ProfilesDE}(c) revealing that one peak inside the wire and three peaks in the trench region are established.
%Note that at high modulations the $1^{\mathrm{st}}$ and $2^{\mathrm{nd}}$ DE mode reveal a strong lateral coupling, i.e.\ both central peak inside the wire oscillate in phase whereas the peaks inside the trench perform an anti-phase oscillation with the peaks of the $1^{\mathrm{st}}$ mode.}
\\

\textbf{Higher DE Modes.}\quad gain, there are also higher modes, like the $3^{\mathrm{rd}}$ DE mode, which maintains its characteristic $2n=6$ peaks throughout the transition until the full MC is reached (see Figs.~\ref{ProfilesDE}(d),(h)). In this limit, only the three peaks inside the wire remain, forming the characteristic shape of the $n'=2$ mode with two peaks at the edges and one in the center. 

This seems to be a general transition pattern in the DE geometry. For moderate modulations, the modes maintain their sinusoidal behavior much longer compared to the BV geometry. As the modulation becomes prominent, the modes avoid steady peaks at the edges which are either placed inside the trench or inside the wire, depending on which of both possibilities
\begin{align}
m&=n-1,\quad l=n+1\nonumber \\
\text{or}\quad m&=n+1,\quad l=n-1\nonumber
\end{align}
is closest to the natural wavelength $\lambda=n/a_0$ of the mode. In MCs with almost equal size of wire and trench, this circumstance would be relevant for all even ($n=2,4,6,...$) modes which finally reach the full MC with $n'=m-1$ either as $n'=n-2$ mode or $n'=n$ mode. Modes of an odd number ($n=1,3,5,...$) reveal a node close to the edges such that the number of peaks is exactly cut in halves when the full MC is reached. Using the relation $n'=m-1$, this means that odd modes evolve to the $n'=n-1$ full MC mode.

%Starting from the 2$^{\mathrm{nd}}$ SW mode, higher modes were not detected in the FMR measurement, which is likely connected to the circumstance that these mode only gain significant amplitude at a relatively high degree of surface modulation ($\Delta d/d\geq0.8$), which was not covered by the measurement. This is supported by the late increase of intensities of the 2$^{\mathrm{nd}}$ and 3$^{\mathrm{rd}}$ mode in the colorplot in Fig.~\ref{transitionDE}. For the given film-thickness, higher DE-type SW modes resonate in a frequency range close to the $1^{\mathrm{st}}$ PSSW mode. This leads to  a characteristic vertical profile of the DE modes shown in Figs.~\ref{ProfilesDE}(c),(g). Nonetheless, the overall mode character of the 2$^{\mathrm{nd}}$ (3$^{\mathrm{rd}}$) mode revealing $2n=4$ (6) peaks inside one period is still easily identified.
%
%Interestingly, the micromagnetic simulations indicate that the 2$^{\mathrm{nd}}$ DE mode merges together with the $1^{\mathrm{st}}$ DE mode to the quasi-uniform mode of the full MC, whereas the 3$^{\mathrm{rd}}$ DE mode becomes the $n'=2$ MC mode. This behavior might be related to the geometrical conditions of the MC. If 
%
%The authors interpret the merging of the $1^{\mathrm{st}}$ and $2^{\mathrm{nd}}$ mode rather as a degeneration of these modes due to the flat demagnetizing field inside the wires. Since the 2$^{\mathrm{nd}}$ SW mode was observed to become the edge mode of the full MC in the BV geometry, the edge (3$^{\mathrm{rd}}$) mode of the full MC in DE geometry is expected to share the same energy as the center ($1^{\mathrm{st}}$) mode. 
%
%Due to the coupling to PSSW-type modes, the profiles of the 2$^{\mathrm{nd}}$ and 3$^{\mathrm{rd}}$ DE modes are asymmetrically distorted (not shown) but the main mode character is conserved.
%\todo{ kann man das beschreiben, warum die koppeln?} \del{Even though the distortion causes complex time-dependent SW amplitude oscillations, the}  In fact, the mode profiles of the $n=1$ and the $n=2$ mode become less different as very high modulations are reached. For example, at $\Delta d = 35$~nm, both modes have a high amplitude within the thick part and another small-wavelength excitation (consisting of approximately 3 peaks) within the thin part. The only difference is that the phase of the small-wavelength excitation in the thin part is shifted around $\pi$. 
%From this observation, it can be concluded that close to the full MC the mode character of both the $n=1$ mode and the $n=2$ mode is dominated by a \emph{wire} mode character inside the thick part with an optical ($n=2$ mode) and acoustical ($n=1$ mode) lateral coupling low-wavelength excitation inside the thin part. 
%Also mode $n=3$ is mainly preserved during the transition. In the final MC limit this mode turns into an \textit{edge} mode.} 
%\del{The typical mode profile characteristics of this mode are already established at $\Delta d = 31.5$~nm inside the \textit{thick} part with pronounced peaks at both edges and a smaller anti-phase peak in the center. According to the above findings for the $n=1$ and $n=2$ modes, one would expect the presence of an analogous second mode close to the full MC limit with an anti-phase dynamic activity inside the \textit{thin} part. Such mode exists in the spin-wave spectra presented in Fig.~\ref{transitionDE}, but reveals a much smaller (negligible) resonance amplitude. The different behavior is due to the symmetry of the mode profile inside the thick part. The characteristic pronounced peaks at the edges of the thick part hinder the lateral coupling of an anti-phase excitation inside the thin part due to the high cost of exchange energy.}
%
%\del{MOVED TO SUMMARY: In order to summarize the transition of spin-wave modes in DE geometry, it is noted that the quasi-uniform mode keeps its character throughout the transition but with a vanishing amplitude in the full MC limit where it, finally, disappears. The $n=1$ and $n=2$ modes are found to transform into laterally coupling acoustical and optical modes of the \emph{wire} mode with a low-wavelength excitation inside the thin part. Moreover, the \emph{egde} mode of the full MC gradually evolves from the $n=3$ DE mode.}


%In short, it is found that the $1^{\mathrm{st}}$ SW mode evolves to the wire mode in the full MC, similar to the BV geometry. The major difference is, that this mode originates from a merging of the $1^{\mathrm{st}}$ and the $2^{\mathrm{nd}}$ SW mode. The 3$^{\mathrm{rd}}$ SW mode, however, is found to evolve to the 3$^{\mathrm{rd}}$ mode of the full MC. Thus, in contrast to the BV geometry, no general expression for the mode transition can be formulated. Only if the wire ($1^{\mathrm{st}}$) mode and the edge ($2^{\mathrm{nd}}$) mode degenerate in the full MC limit, the transition rule of the $n^{\mathrm{th}}$ film mode becoming the $m=2n-1$ full MC mode would still hold. In this case, the $1^{\mathrm{st}}$ and the $3^{\mathrm{rd}}$ full MC mode share the same resonance position such that the $3^{\mathrm{rd}}$ mode of the full MC (open circle in Fig.~\ref{transitionDE}) would actually represent the $5^{\mathrm{th}}$ mode.


%In case of the uniform mode, the expression of the resonance equation must be modified such that the transition from the film limit expressed by Kittel's equation towards the wire limit with the modified shape anisotropy due to the demagnetizing factor of the wires is realized in a continuous way. 
%To address this mater, an effective demagnetizing factor $N_eff$ is introduced such that $N_eff\rightarrow0$ yields the Kittel equation of a thin film and $N_eff\rightarrow N_wire$ results in the typical resonance condition of for an array of separate wires.
%To address this mater, an effective demagnetizing factor $N_{\mathrm{y}}$ is introduced yielding the following resonance equation
%
%\begin{equation}
%\begin{split}
%\left(\frac{\omega}{\gamma}\right)^{\negmedspace2}\negthickspace=\!&\left[ \mu_{0} H_0 + N_{y}\mu_{0} M_{\mathrm{S}} \right]\\
%&\times \left[ \mu_{0} H_0 - N_{\mathrm{y}}\mu_{0} M_{\mathrm{S}} + \mu_{0} M_{\mathrm{S}} \right]~.
%\end{split}
%\label{uniform}
%\end{equation}
%
%Eq.~(\ref{uniform}) yields the familiar equations for the two limits, the thin film ($N_{\mathrm{y}}\rightarrow0$)
%
%\begin{equation}
%\left(\frac{\omega}{\gamma}\right)^{\negmedspace2}\negthickspace=\!\mu_{0} H_0\times\left[ \mu_{0} H_0 + \mu_{0} M_{\mathrm{S}} \right]
%\label{unifilm}
%\end{equation}
%
%and the array of wires (1D magnonic crystal) ($N_{\mathrm{y}}\rightarrow N_{\mathrm{wire}}$)
%
%\begin{equation}
%\begin{split}
%\left(\frac{\omega}{\gamma}\right)^{\negmedspace2}\negthickspace=\!&\left[ \mu_{0} H_0 + N_{\mathrm{wire}}\mu_{0} M_{\mathrm{S}} \right]\\
%&\times \left[ \mu_{0} H_0 - N_{\mathrm{wire}}\mu_{0} M_{\mathrm{S}} + \mu_{0} M_{\mathrm{S}} \right]~.
%\end{split}
%\label{uniwire}
%\end{equation}
%
%with the $N_{\mathrm{y}}$ being the demagnetizing factor of the wire along the $x$ axis\cite{Aharoni1997}
%
%As the crucial role of the distribution of the internal demagnetizing field is already lined out in Sec.~\ref{BV}, this chapter shall focus on the relation between the internal demagnetizing field and the dispersion and mode profiles of hybrid SWs in SMMCs. Therefore, a transmission electron microscopy (TEM)-holography analysis of an SMMC with 10~nm surface modulation was conducted employing the same permalloy film as used in Sec.~\ref{BV}. The method allows for a screening of the local external and internal stray field of the structure and yields at the same time high-quality images of the structural properties. Employing numerical simulations, both approaches can then be followed to calculate the FMR response: (i) Using the structural properties on the one hand and (ii) adding the measured internal $H_{\mathrm{d}}$-field distribution to a regular permalloy thin film, similar to the approach of the analytical theory, on the other hand.
%
%The result shows surprising similarities of both approaches, but also unravels the important differences of MCs with and without structural steps, such as non-reciprocal SW dispersion and major effects on the edge modes.

\section{Conclusion}
\label{Con}
%\done{Write one conclusion for both geometries, BV and DE with the clear focus on the BV geometry, since the important transitions (for DE geometry) are happening at way higher modulations. Take text from BV part}
The gradual evolution of SW modes in the transition from a thin film to a full MC has been investigated by FMR measurements and micromagnetic simulations in the BV and DE geometry. %This has been achieved by successively increasing the depth of the structural surface modulation of a permalloy film by ion milling. The experimentally by FMR obtained spin-wave spectra after each milling step were compared to analytical theory and micromagnetic simulations in order to identify the modulation-height-dependent mode character and profiles.
The uniform mode maintains as a trench mode in BV orientation and as a quasi-uniform mode in the DE geometry until it finally vanishes in the full MC limit for both orientations.

For non-uniform modes, simple transition rules are found. While in the BV orientation, the transition process is governed by the presence of strong magnetizing and demagnetizing fields, which lead to a confinement of $m=2n-1$ peaks inside the wire region, the transition process for DE modes is governed by the preservation of the modes natural wavelength leading to the transition to an $m=n$, $l=n$ mode for odd $n$ and either $m=n-1$, $l=n+1$ or $m=n+1$, $l=n-1$ for even $n$, depending on the geometry of the MC. For all modes, the transition ends in the full MC limit where only the peak number $m$ in the wire region pre-determines the final state with $n'=m-1$. Thus, in the BV geometry the $n^{\mathrm{th}}$ film mode evolves to the $n'=2(n-1)$ mode of the full MC. In the DE geometry, odd film modes with $n=1,3,5,...$ evolve to the $n'=n-1$ full MC mode whereas even modes with $n=2,4,6,...$ can either become the $n'=n-2$ mode or the $n'=n$ mode.

\section{Acknowledgment}

We thank B.\ Scheumann for the film deposition, A.\ Banholzer and C.\ Fowley for the technical support in the patterning process and P.\ C.\ Grubitz and A.\ Jansen for their help with the fitting of the measurement data. The support by the Structural Characterization and Nanofabrication Facilities Rossendorf at IBC and the HZDR Department of Information Services and Computing is gratefully acknowledged. 
This work was supported by the Centers of Excellence with Basal/CONICYT financing (grant no.\ FB0807), CONICYT PAI/ACADEMIA 79140033, FONDECYT 1161403, FONDE-CYT Iniciacion 11170736, CONICYT PCCI (grant no.\ 140051), DAAD PPP ALECHILE (grant no.\ 57136331) and from the Deutsche Forschungsgemeinschaft (grant no.\ {LE2443/5-1}). Funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sk\l{}odowska-Curie grant agreement No.\ 701647 is gratefully acknowledged.
%
%\begin{figure}[H]
%\centering
%\includegraphics[width=0.85\linewidth]{PWM.eps}
%\caption{\label{PWM} (Color online) (a)--(d) Fitted spin-wave resonance spectra (colored circles) for different modulation heights $\Delta d$ together with the quasi-analytical calculations (PWM) plotted as orange lines.}
%\end{figure}
% 

%\section{Appendix}
%\label{App}
%
%\vspace{3mm}
%\del{\textbf{(ii) Validation of the Quasi-Analytical Model}\quad Quasi-analytical theory based on the PWM\cite{Krawczyk2008,Sokolovskyy2011,Klos2012,Krawczyk2013,Gallardo2018} was used as well in order to understand the transition of the SW modes in backward-volume geometry. 
%A firm agreement between the three datasets of the measured, simulated and theoretically estimated SW resonances would justify the use of the PWM to test whether the formulated transition rules apply generally. Moreover, the theoretical model allows for fast calculation of the band structure of the MC, which is difficult to access experimentally, especially if the periodicity of the system is 300~nm or less.
%Figure~\ref{PWM} shows the results from PWM (orange lines) in comparison to the measurement (colored circles). The structural and magnetic parameters for the PWM-based calculations were used according to the ones selected for the micromagnetic simulations (see beginning of Sec.~\ref{BV}). The only minor difference is that the precise $\Delta d$ values listed in Fig.~\ref{PWM} were employed. In the simulations, these values were abstracted due to the necessity to model the SMMC by a natural number of cells.
%Up to the modulation height of approximately 9~nm the agreement of theory and experiment is striking. However, for higher modulation sizes significant differences between measurement and theory are visible and are likely due to thickness-related approximations within the theory.\cite{Gallardo2018} It shall be noted that up to now, such systems were theoretically handled using perturbation theory,\cite{Landeros2012,Gallardo2014} which cannot be employed for such high levels of surface modulation.
%The firm quantitative agreement between the numerically calculated SW resonances and the ones obtained by PWM also can be confirmed by a comparison of the mode profiles. This was already done and published elsewhere\cite{Langer2017} confirming a very high level of agreement up to a modulation height of $\Delta d=10$~nm.
%}
%
%\begin{figure}[t]
%\includegraphics[width=\linewidth]{suscept.eps}
%\caption{\label{suscept} (Color online) Susceptibility of the modes above the large frequency gap (blue) and below the gap (orange) dependent from the modulation height $\Delta d$. (a),(b) The susceptibility at 150~mT and (c),(d) the susceptibility at 400~mT. The measured data is shown on the left side accomplished with the simulation data on the right side.}
%\end{figure}
%
%\del{The proven accuracy of the theory for moderate modulation heights justifies its application to calculate the spin-wave dispersion and mode profiles in the $k$-space. For MCs of 300~nm periodicity or less, the measurement of the SW dispersion is challenging due to the limitations associated with the maximum $k$-values accessible in Brillouin light scattering (BLS). That makes PWM an ideal tool regarding band-gap and band-structure analyses and optimizations for such surface-modulated MCs.
%\vspace{3mm}
%\textbf{(i) Resonance Amplitude Analysis}\quad Figure~\ref{multistep} reveals an enriched spin-wave spectrum of SMMCs compared to the `full' MC presented in Fig.~\ref{WireLimit}. The dominance of the quasi-uniform \textit{trench} mode for small $\Delta d$ values and its vanishing for large values indicates the existence of a threshold where both \new{fundamental modes}, the \textit{trench} and the \textit{wire} mode, exhibit the same amplitude. In order to identify this threshold, an analysis of the measured and simulated mode amplitudes was carried out. Figure~\ref{suscept} depicts the resulting $\Delta d$-dependent amplitudes of both, measurement (\ref{suscept}(a),(c)) and simulation (\ref{suscept}(b),(d)) for two different field values, 150~mT (\ref{suscept}(a),(c)) and 400~mT (\ref{suscept}(a),(c)). 
%In the measurement a threshold was found at approximately $\Delta d/d=0.26$ for $H_0=150$~mT and $\Delta d/d=0.29$ for $H_0=400$~mT. The data differs quite significantly compared to the simulation. Here, the threshold was obtained at $\Delta d/d=0.33$ for both external field values.
%}
%
%\section{Appendix}
%\label{app}
%
%The approximated dispersion of PSSW modes reads\cite{Kalinikos1986}
%\begin{equation}
%\left(\frac{\omega}{\gamma}\right)^2 = \left( \mu_{0} H_0 + D k^2_{\bot} \right) \times \left( \mu_{0} H_0 + D k^2_{\bot} + \mu_{0} M_{\mathrm{eff}} \right)
%\label{PSSWdisp}
%\end{equation}
%with $f = \omega/(2\pi)$ the FMR frequency $\mu_{0} H_0$ the external magnetic field, $D$ the exchange stiffness constant, $k_{\bot}$ the SW wave number along the axis perpendicular to the film plane and $\mu_{0} M_{\mathrm{eff}}$ the effective magnetization of the film. The wave number $k_{\bot}$ can be estimated using the parameter $d$ the local film thickness $k_{\bot} = p\pi/d$ (p = 1, 2, 3, ...). 
%
% BV Dispersion
%\begin{equation}
%\begin{split}
%\left(\frac{\omega}{\gamma}\right)^{\negmedspace2}\negthickspace=\!&\left( \mu_{0} H_0 + \mu_{0} H_{\mathrm{d}} + D k^2 \right)\\
%&\times \left[ \mu_{0} H_0 + \mu_{0} H_{\mathrm{d}} + D k^2 + \mu_{0} M_{\mathrm{S}} F(kd) \right]
%\end{split}
%\label{BVdisp}
%\end{equation}
%
%\noindent with $f = \omega/(2\pi)$ the FMR frequency, $D$ the exchange stiffness constant, $\mu_{0} M_{\mathrm{S}}$ the magnetic polarization, $\mu_{0} H_{\mathrm{d}}$ the internal demagnetizing field and $k$ the in-plane wave vector. For standing spin-wave modes the discretization $k=2\pi n/a_0$ applies due to the modulation periodicity $a_0$. Here, $n$ is the spin-wave `quantum number' reflecting the number of nodes fitting in one period $a_0$. Furthermore, $F(kd)=\left[1-\exp(-kd)\right]/(kd)$ is derived from the matrix element of the dipolar interaction\cite{Kalinikos1986} of a film of thickness $d$.
%
%\todo{find new references, Pedro Papers etc.}
\bibliography{references}
\bibliographystyle{kl_ms_num}
\end{document}
MultistepEtching6.bbl
\providecommand{\noopsort}[1]{}\providecommand{\singleletter}[1]{#1}%
\begin{thebibliography}{10}
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