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\section{Introduction}
The interaction between nonlinearity and dispersion leads to various localized solutions to the nonlinear Schr$\ddot{o}$dinger equation (NLSE). Solitons are one of the most well-known solution \cite{Zakharov_JETP1972} and have advanced various research arenas such as Bose-Einstein condensates (BEC) \cite{Phillips_Science2000} and optics \cite{Dudley_NP2010Ten}. Notably, soliton mode-locking has enabled the coherent Kerr comb generation from optical microresonators recently \cite{Kippenberg_Nature2007,Kippenberg_Science2011,
Kippenberg_NP2014temporal} (a Kerr comb is an array of equally spaced spectral lines generated by cascaded four-wave-mixing in high-$Q$ microresonators, see ref. \cite{Kippenberg_Science2011} for a review). Breathers are another type of solution to the NLSE that occur widely in physical systems \cite{Kuznetsov_PR1986soliton,Ma1979,Akhmediev1986,Dudley_NP2010,Smerzi_PRL2001}. Different from solitons which preserve their shape as they propagate, breathers feature periodic variation in time or space. Systems in which breathers may exist include BEC \cite{Smerzi_PRL2001}, granular lattices \cite{Willis1998}, hydrodynamics \cite{Akhmediev_PRL2011Rogue}, optics \cite{Dudley_NP2010,Dudley_NP2014}. Breathers are attracting growing attention recently due to their close relationship to the growth and decay of the extreme events known as rogue waves \cite{Jalali_Natue2007,Dudley_NP2010,Akhmediev_PRL2011Rogue,Akhmediev_PRX2012,Dudley_NP2014,Dudley_NC2016}.

Optical systems are widely used to study breathers, due to the ease to tailor the linear and nonlinear properties of the systems. Optical breathers were first studied in conservative systems, namely optical fibres \cite{Dudley_NP2010,Akhmediev_PRX2012,Dudley_NP2014,Dudley_NC2016}. Recently optical breathers have also been observed in dissipative systems such as externally pumped fiber cavities \cite{Coen_OE2013} and microresonators \cite{Weiner_PRL2016,Gaeta_NC2017,Kippenberg_arXiv2016Breather,Kippenberg_arXiv2017inter}. Kerr comb generation in optical microresonators \cite{Kippenberg_Nature2007,Kippenberg_Science2011} constitutes an excellent experimental platform to explore breather dynamics, as microresonators provide essentially infinite propagation distance where breathers are densely sampled every round-trip. Furthermore, the wide frequency spacing for Kerr combs generated from microresonator \cite{Kippenberg_Science2011} makes it easy to separate individual lines and study their breathing dynamics \cite{Weiner_PRL2016}. To date, optical breathers have been mainly explored in the anomalous dispersion regime (for which nonlinearity and dispersion have the same sign). Nonlinear dynamics in the normal dispersion regime (opposite sign for nonlinearity and dispersion) are also important; for instance they help to improve pulse energy of mode-locked lasers \cite{Wise_LPR2008}. Dark solitons, which have been reported in optics \cite{Weiner_PRL1988}, BEC \cite{Lewenstein_PRL1999} and hydrodynamics \cite{Akhmediev_PRL2013} may exist in this regime. However, experimental observation of optical breathers in the normal dispersion regime remains elusive, to our knowledge. Discrete dark breathers have been shown to exist in granular lattices (e.g., Fermi-Pasta-Ulam lattice and Klein-Gordon lattice) \cite{Daraio_PRE2014,Romero_NJP2002}. Nevertheless, many questions about dark breathers, such as their dynamics in highly driven cases, remain open. Kerr comb generation in normal dispersion microresonators \cite{Weiner_NP2015mode,Weiner_Optica2014,Matsko_OL2014,Gorodetsky_OE2015,Wong_PRL2015,
Gaeta_OE2016dynamics,Gelens_PRA2016,Chembo_PRA2014} can provide a unique opportunity to study optical dark breathers. Numerical simulations \cite{Chembo_PRA2014,Gelens_PRA2016,Weiner_NP2015mode} and preliminary experiments \cite{Weiner_NP2015mode} suggest that optical dark breathers may exist in normal dispersion microresonators. In   this Article, we present the first clear observation and detailed characterization of optical dark breathers in the normal dispersion regime, which we obtain by destabilizing a dark pulse Kerr comb in a silicon-nitride (SiN) microresonator. The dark breather breathes relatively gently and features an energy exchange between comb lines. Furthermore, the breathing behaviour changes abruptly from line to line. The observed dark breather dynamics in dissipative cavities is governed by the Lugiato-Lefever equation (LLE), i.e., driven-damped NLSE \cite{Lugiato_PRL1987spatial,Coen_OL2013modeling} and differs substantially from the breathing ``double periodic" solution in the NLSE with normal dispersion \cite{Akhmediev_PRA1993}. In the highly driven regime, we observe that the dark breather in our dissipative cavity undergoes a period-tripling bifurcation and then a transition into a chaotic state. This behavior is prohibited in systems governed by the NLSE. The experimental observation of dark breathers is confirmed by rigorous comparison with numerical simulations.

\begin{figure*}[t]
\centering
\includegraphics[width=1.7\columnwidth]{Fig1.pdf}
\caption{\textbf{Generation dynamics of the dark breather.} When tuning the laser into cavity resonance, the comb is initiated by mode-interaction. \textbf{a,~b,} In the modulation instability regime, some coarsely spaced comb lines are generated and there are narrow RF peaks, whose frequencies are within the half-width-half-maximum (HWHM) of the cavity resonance. \textbf{c,~d}, With further tuning into resonance, we can get a broadband comb and a dark breather state. The simulated optical spectrum is in close agreement with the measured spectrum. The dark breather state has narrow RF peaks on the RF spectrum, whose frequencies are outside the HWHM of the resonance. The inset in \textbf{d} shows the measured comb power change of the dark breather. The dark breather only breathes weakly with a depth $\sim$3\%. \textbf{e,~f,} By lowering the pump power at the same wavelength, we can get a coherent dark pulse comb with low RF noise. The simulated optical spectrum also closely matches the measured spectrum. There is no significant difference in the optical spectra in the dark breather state and the dark pulse state.}
\label{Fig1Generation}
\end{figure*}

From a practical perspective, studying the dark breather gives insights into dark pulse Kerr comb generation in normal dispersion microresonators. The soliton Kerr combs generated in anomalous dispersion microresonators have been used in many applications, including communications \cite{Koos_Nature2017}, dual-comb spectroscopies \cite{Vahala_Science2016}, and low noise microwave generation \cite{Matsko_NC2015}. However, the conversion efficiency from the pump into the generated comb lines is usually less than a few percent for single-soliton combs \cite{Willner_OL2014,Weiner_OE2016}. In contrast, dark pulse Kerr combs can reach much higher conversion efficiency (above 30\%) \cite{Weiner_LPR2017}, providing an important tool for high power Kerr comb generation. Dark breathers represent an intrinsic source of instability for dark pulses; characterization and modeling of such breathers may  offer better opportunities for their control, and trigger the search for dark breathers in other physical systems.

\section{Results}
\subsection{Dark breather generation}

\begin{figure*}[t]
\centering
\includegraphics[width=1.85\columnwidth]{Fig2.pdf}
\caption{\textbf{Characterization of the dark breather.} \textbf{a,} The dark breather exhibits energy exchange between different comb lines. When choosing the comb lines with mode numbers in $[-3, 3]$ as the center group and comb lines in $[-10, -4]\cup[4, 10]$ as the wing group, we observe a phase difference in the fast evolving spectrum, but not an anti-phase exchange in (\lowercase\expandafter{\romannumeral1}). When eliminating modes in $[-2, 1]$ for the center group (keeping mode $-3$, 2, 3 only), there is a nearly out of phase energy exchange between the wing group and the modified center group in (\lowercase\expandafter{\romannumeral2}). \textbf{b,} The absolute and fractional breathing amplitudes of individual comb lines. The absolute breathing amplitude is stronger at the center, while the breathing depth is larger at the wing. \textbf{c,} Different comb lines breathe with different phase; the change of the breathing phase varies abruptly and asymmetric with respect to the pump. The colour density shows the normalized breathing of each individual line (normalized to the breathing amplitude of the same individual lines) and the circles on line mark the peaks of the oscillation for different lines, illustrating the breathing phase. The arrows illustrate the peaks for the modes $\pm$1. \textbf{d,} The dark breather retains a dark-pulse-like waveform. By line-by-line pulse shaping, dark pulses can be shaped into transform-limited pulses. Using the same phase on the pulse-shaper, the dark breather comb can also be shaped into a transform-limited pulse.}
\label{Fig2Characterization}
\end{figure*}

The dark breather is generated in a SiN microresonator with a radius of 100 $\mu$m, a loaded $Q$-factor of 0.8$\times$10$^6$, geometry of 2000 nm$\times$600 nm, and a normal dispersion of $\beta_2$=190 ps$^2$/km. The same microresonator pumped at the same resonance was previously used to generate a mode-locked dark pulse \cite{Weiner_NP2015mode}. By tuning the cw pump laser into resonance from blue to red, we are able to initialize the comb generation via mode-interaction \cite{Weiner_NP2015mode,Weiner_Optica2014} at the pump power of 1.7 W (off-chip power, the fiber-to-chip coupling loss is $\sim$2 dB per facet). At the pump wavelength of 1549.19 nm, we get some coarsely spaced comb lines as shown in
Fig. \ref{Fig1Generation}a, which is dominated by the mode-interaction assisted modulation instability dynamics. The RF spectrum of this comb shows some narrow sidebands at low frequencies (60 MHz) (Fig. \ref{Fig1Generation}b). These RF sidebands in the modulation instability regime can be attributed to the beat note between the main comb lines and the subcomb lines (see the inset of Fig. \ref{Fig1Generation}b) as reported in ref. \cite{Kippenberg_NP2012}. Hence, the sidebands are accommodated within the halfwidth-half-maximum (HWHM) of the cavity resonance (125 MHz). Thus, the comb in Figs. \ref{Fig1Generation}a, b is not a dark breather, despite the narrow RF sidebands. When further tuning the pump into the resonance, we can get a broadband, single free-spectral-range (FSR) comb at pump wavelength of 1549.26 nm (Fig. \ref{Fig1Generation}c). The comb in this state also has narrow RF sidebands as shown in Fig. 1d. However, the breathing frequency (740 MHz, $>$12 times of the frequency in Fig. 1b) is outside the HWHM of the resonance. Thus, these sidebands are attributed to a different generation mechanism from the main-sub comb beat. Supplemented with further evidence described below, we confirm that the comb in Figs. \ref{Fig1Generation}c, d corresponds to a dark breather and attribute the sidebands to periodic breathing (modulation) of the comb lines. The averaged simulated optical spectrum (see the section on numerical simulations and Methods below) of the dark breather is in close agreement with the measured optical spectrum. An important feature of the dark breather is that it only breathes weakly. We define the breathing depth as $(P_{max}-P_{min})/(P_{max}+P_{min})$, where $P_{max(min)}$ is the maximum (minimum) power. The comb power including (excluding) the pump breathes with a depth of $\sim$3$\%$ ($\sim$5$\%$). This is significantly different from bright soliton breathers, which can breathe with a depth of $\sim$50$\%$ (excluding the pump) \cite{Weiner_PRL2016}. We can get the coherent dark pulse comb by reducing the pump power to 1.6 W at the same wavelength. Now the comb shows low-noise operation (Figs. \ref{Fig1Generation}e, f). Again, the simulated dark pulse has an optical spectrum which closely matches the measurement. There is no significant difference between the dark breather and the mode-locked dark pulse in the optical spectrum analyzer averaged spectra, as the dark breather only breathes weakly; whereas the bright soliton and bright soliton breather combs show distinct averaged spectra \cite{Weiner_PRL2016,Kippenberg_arXiv2016Breather}.

\subsection{Characterization of dark breathers}

The high frequency RF sidebands, which were also reported in \cite{Weiner_NP2015mode}, are not sufficient by themselves to confirm the observation of dark breathers. For example, a comb with a constant spectrum shape but varying power can show similar peaks. Hence, we first need to confirm there is internal energy exchange between the comb lines. To do that, we use a pulse-shaper \cite{Weiner_RSI2000} to programmably select groups of comb line and record their fast evolution \cite{Weiner_PRL2016} (see Methods and Supplementary Information Sec. 1). We first choose the comb lines whose mode number with respect to the pump are within $[-3,~3]$ (pump line mode 0 included and larger number means higher frequency) as the center group, while the comb lines within $[-10,~-4] \bigcup [4,~10]$ as the wing group. The oscillation of the recorded power change of the center group and the wing group show a phase difference of 0.21$\pi$ in Fig. \ref{Fig2Characterization}a\lowercase\expandafter{\romannumeral1}. When further suppressing the comb line within $[-2, 1]$ for the center group (i.e., only keeping lines $-$3, 2 and 3), the center group breathes out of phase (0.93$\pi$) with the wing group as shown in Fig. \ref{Fig2Characterization}a\lowercase\expandafter{\romannumeral2}. Thus, it is confirmed that there is energy exchange within the Kerr comb in Fig. \ref{Fig1Generation}c. To gain more insight into the breathing dynamics of the dark breather, the pulse shaper is programmed to select single comb lines to record their fast breathing. The absolute breathing amplitude of the central comb lines within $[-2,~2]$ is much higher compared to other comb lines. However, the average power is also highest in the central comb lines. The modulation depth of individual lines tends to be largest in the wing of the spectrum (Fig. \ref{Fig2Characterization}b). Moreover, different lines breathe with different phases as shown in Fig. \ref{Fig2Characterization}c, which differs from the breathers in the NLSE with normal dispersion \cite{Akhmediev_PRA1993} (Supplementary Section 4). Abrupt changes in the phase and amplitude of the breathing can be seen between some adjacent lines, whereas bright soliton breathers have relatively smooth changes in breathing phase \cite{Weiner_PRL2016}. We also note that the breathing phase is not symmetric with respect to the pump (e.g., see the modes $\pm$1, illustrated by the arrows). Mode-interactions may be responsible for this asymmetry, as the dominant mode-interaction only exists on one side of the pump, breaking the symmetry (see numerical simulations below).

To confirm the observation of dark breathers, we need to verify that the comb in Figs. \ref{Fig1Generation}(c, d) has a dark-pulse like waveform. We use line-by-line pulse shaping (see Methods) to probe the corresponding waveform \cite{Weiner_RSI2000,Weiner_NP2015mode,Weiner_NP2011}. We first adjust the pump power to reach the mode-locked dark pulse state and apply spectral phase shaping to obtain a transform-limited pulse at the output (Fig. \ref{Fig2Characterization}d). We then adjust the pump power to transition into the dark breather state, while keeping the same phase profile on the pulse-shaper. The output is still shaped into a transform-limited pulse. This comparison suggests that the dark breather has a spectral phase profile close to that of the mode-locked dark pulse. Since the dark breather exhibits only a gentle modulation and the power spectrum remains close to that of the dark-pulse comb, we deduce that the dark breather retains a dark-pulse-like waveform (Supplementary Section 2). These findings confirm that we are observing a dark breather in our normal dispersion microresonator.

\subsection{Transition to chaotic breathing}
\begin{figure*}[t]
\centering
\includegraphics[width=1.8\columnwidth]{Fig3.pdf}
\caption{\textbf{Power dependence of the breathing properties.} \textbf{a,} The breathing depth of the comb power increases with increasing pump. \textbf{b,} The breathing rate decreases slightly with pump power. \textbf{c,} When the pump power is increased to 1.9 W, the system transitions to a period-tripling state, as evidenced by the additional sidebands spaced by $\sim$238 MHz. \textbf{d,} Broadband RF noise is observed when the pump power is further increased to 2.1 W. \textbf{e,} The phase diagram of the output comb power (pump included), which is obtained by plotting the first-order time derivative of the comb power (d$P_{\text{comb}}$/d$t$) versus $P_{\text{comb}}$, also shows chaotic variation. \textbf{f,} The optical spectrum in this chaotic breathing state is similar to the dark pulse state; the averaged simulated spectrum in the chaotic breathing state is in close agreement with the measured spectrum. \textbf{g,} The autocorrelation of the Kerr comb in the chaotic breathing state after pulse shaping (the same amplitude and phase adjustment used in Fig. \ref{Fig2Characterization}d) is close to the calculated autocorrelation trace of the transform limited pulse. The auto-correlation of the Kerr comb in the chaotic breathing state after pulse shaping (the same amplitude and phase adjustment used in Fig. \ref{Fig2Characterization}d) is close to the calculated auto-correlation trace of the transform limited pulse.}
\label{Fig3Chaotic}
\end{figure*}

\begin{figure*}[t]
\centering
\includegraphics[width=1.9\columnwidth]{Fig4.pdf}
\caption{\textbf{Numerical simulation on the dark breather dynamics.} \textbf{a,} Simulated breathing dynamics in the frequency and time domain. Mode-interaction is included in the simulation for modes in $[-57,~-54]$, leading to enhanced intensity for the perturbed modes. The dashed boxes show the breathing of the spectrum and the waveform. \textbf{b,}  Simulated spectral and temporal dynamics of the coherent dark pulse comb when decreasing the pump power to 628 mW. \textbf{c,} Simulated RF spectrum of the dark breather state, showing a breathing rate of 782 MHz, close to the experiments. The inset shows the change of the total comb power including pump; the breathing depth is about 3$\%$. \textbf{d,} The averaged simulated autocorrelation trace of the pulse-shaped dark breather comb (see Methods for numerical shaping method) is close to the autocorrelation of the pulse-shaped dark pulse comb. The inset is an example of the breathing dark-pulse-like waveform. \textbf{e,} The simulated energy exchange between comb lines at the center and the wing when selecting the comb line groups in the same manner with experiments in Fig. 2a. \textbf{f,} The simulated absolute breathing amplitude and fractional breathing of individual lines. \textbf{g,} The breathing phase of different comb lines. The colour density shows the normalized breathing of individual lines and the circles on the line mark the peaks of the breathing for different comb lines. The arrows illustrate that modes $\pm$1 breathe at different phases.
}
\label{Fig4simulations}
\end{figure*}

\begin{figure*}[t]
\centering
\includegraphics[width=1.9\columnwidth]{Fig5.pdf}
\caption{\textbf{Numerical simulations on chaotic breathing.} \textbf{a,} When the pump power increases, the breathing depth of the total comb power increases slightly and the breathing rate decreases slightly. \textbf{b,} When the pump power is increased to 752 mW, the dark breather goes through a tripling of the breathing period and the RF spectrum shows two equidistantly spaced weak tones between 0 and the strongest tone at 765 MHz. \textbf{c,} When further increasing the pump power to 785 mW, the breathing becomes chaotic and the RF spectrum becomes broadband, losing the original narrow sidebands. \textbf{d,} An example of the total comb power (including the pump line) plotted against its time derivative in the chaotic breathing state, showing stochastic variation. \textbf{e,} The spectral and temporal breathing dynamics. The arrows highlight the nonperiodic spectrum and the waveform evolution. \textbf{f,} When numerically shaping the chaotic breathing comb into a pulse (see Methods), the averaged auto-correlation trace is close to the auto-correlation of the shaped coherent comb. The inset depicts an example of the waveform of the chaotic breathing state, showing the dark-pulse-like waveform is retained.}
\label{Fig5simulations}
\end{figure*}

Fig. \ref{Fig1Generation} shows the breathing instability can be eliminated by decreasing the pump power. On the other hand, the absolute breathing depth increases slightly when increasing the pump power (Fig. \ref{Fig3Chaotic}a). At the same time, the breathing rate decreases slightly (Fig. \ref{Fig3Chaotic}b). The breathing rate of bright soliton breathers in the anomalous dispersion regime also decreases with increasing pump power \cite{Kippenberg_arXiv2016Breather}. The decrease in the breathing rate is an indication of approaching critical transitions for dynamical systems \cite{Kippenberg_arXiv2016Breather,Sugihara_Nature2009}. When increasing the pump power further to 1.9 W, the breathing of the dark breather goes through a period-tripling bifurcation (Fig. \ref{Fig3Chaotic}c). The period-doubling bifurcation occurs more commonly in dynamical systems \cite{Ott_2002} including bright soliton breathers in microresonators \cite{Gaeta_NC2017,Kippenberg_arXiv2016Breather}; however, it is not observed for the studied dark breather. The perturbation from mode-interaction is found to be important to this period-tripling, without which it does not occur (see Supplementary Section 5). Such a direct period-tripling transition was also demonstrated in passively mode-locked fiber lasers \cite{Akhmediev_PRE2004}.

By increasing the pump power further to 2.1 W, the breathing becomes chaotic. The measured RF spectrum shows broadband noise (Fig. \ref{Fig3Chaotic}d), and the phase diagram (plotting first-order time derivative of comb power d$P_{comb}$/d$t$ versus $P_{comb}$) shows the comb power changes chaotically (Fig. \ref{Fig3Chaotic}e). Chaotic breathing was predicted and observed (in fibre cavities) to exist in the LLE with anomalous dispersion \cite{Bekki1985,Coen_OE2013,Coen_Optica2016}; it has also been predicted to exist for dark breathers but not demonstrated yet \cite{Gelens_PRA2016}. Our results constitute the first observation of such a chaotic transition for dark breathers in all physical systems, to our knowledge. Note this chaotic breathing state is distinct from the chaotic state prior to the mode-locking transition \cite{Weiner_NP2015mode,Kippenberg_NP2014temporal} (see Supplementary Sections 1, 3). The comb in the chaotic breathing state retains the optical spectral features of the dark pulse (Fig. \ref{Fig3Chaotic}f), and the averaged spectrum in simulations is also in good agreement with the measurement. When we apply the same phase profile used in Fig. \ref{Fig2Characterization}d on the pulse shaper, the auto-correlation (AC) trace obtained under the chaotic breathing state still shows a clean pulse which is close to the transform-limit (as calculated from the spectrometer averaged spectrum). The visibility of the AC trace exceeds 97\% of the expected value; this differs dramatically from what is observed with the chaotic comb in the absence of a mode-locking transition \cite{Weiner_NP2011}. The AC results suggest the chaotic breathing state has a dark-pulse-like waveform (see also the simulation results in Fig. \ref{Fig5simulations}).  This is consistent with the fact that excitation of the chaotic breathing is reversible, i.e., the dark breather and the coherent comb can be obtained by simply decreasing the pump power from this chaotic state.

\subsection{Numerical simulations}

Kerr comb generation dynamics can be modeled by the LLE, i.e., the driven-damped NLSE \cite{Lugiato_PRL1987spatial,Coen_OE2013,Weiner_PRL2016}. For the NLSE, breather-like solutions also exist with normal dispersion \cite{Akhmediev_PRA1993} (Supplementary Section 4). However, dissipative effects in externally pumped microresonators change the dark breather dynamics dramatically. Here, we use the LLE to model the dark breather dynamics. When choosing the pump power as 700 mW and detuning as 0.0629, a dark breather can be generated (see Methods and Supplementary Section 3). As shown above, the simulated spectrum (averaged over slow time) is in close agreement of the measured spectrum (Fig. \ref{Fig1Generation}c). The inclusion of the mode-interaction gives the long wavelength peak around 1657 nm. From Fig. \ref{Fig4simulations}a, we can see the periodic variation of the spectral and temporal shape of the dark breather (see the dashed boxes). From the temporal breathing, we can see the waveform-background (high power part of the dark-pulse-like waveform) stays nearly unchanged in the breathing and the waveform-hole (low power part of the dark-pulse-like waveform) breathes more strongly (Supplementary Animation 1). This helps to elucidate the low breathing depth of the comb power, as the waveform-top carries most of the power. With the same detuning, a coherent dark pulse comb can be obtained by decreasing the pump power to 628 mW (Fig. \ref{Fig4simulations}b), consistent with experiments. As shown in Fig. \ref{Fig4simulations}c, the simulated dark breather has a breathing rate of 782 MHz, and the breathing depth of the comb power including (excluding) the pump is 3\% (6\%), both in agreement with the experiments. The simulation retains the dark-pulse-like waveform in the breathing state; the simulated AC (averaged over slow time, see Methods) remains close to the AC of the coherent comb after numerical pulse shaping (Fig. \ref{Fig4simulations}d).

The simulated dark breather also exhibits energy exchange between comb lines. When choosing the comb lines in the same manner as used in Fig. \ref{Fig2Characterization}a, the simulated power change is shown in Fig. \ref{Fig4simulations}e. The center group in $[-3, 3]$ has a different breathing phase with the wing group in $[-10, -4]\cup[4, 10]$, but is not completely out of phase (0.70$\pi$). Similar to experiments, when eliminating modes in $[-2, 1]$ from the center group, the center group and the wing group exhibit nearly out of phase energy exchange (1.11$\pi$). The breathing behaviour also differs greatly from line to line. The absolute breathing amplitude is stronger near the spectrum center, while the breathing depth tends to be stronger at the wing (Fig. \ref{Fig4simulations}f). The discrepancy of the absolute breathing amplitude from experiments (Fig. \ref{Fig2Characterization}b) may be attributed to the difficulty in modeling the mode-interaction accurately. The simulated breathing amplitude is also not symmetric with respect to the pump. The breathing phase of different lines changes asymmetrically with respect to the pump (see especially modes $\pm1$ in Fig. \ref{Fig4simulations}(g)). Note that the breathing becomes symmetric with respect to the pump when mode-interaction is excluded (Supplementary Section 5). This simulation together with the experimental result in Fig. \ref{Fig2Characterization}c suggests that the comb lines in the dark breathers are highly coupled and the dark pulse instability can be affected by a perturbation over 12 THz away. Furthermore, the breathing phase can change abruptly between adjacent lines, in agreement with experiments.

The simulated dark dynamics also depend strongly on the pump power. In agreement with experiments, the simulated breathing depth increases slightly while the breathing rate decreases slightly with increasing pump power, as shown in Fig. \ref{Fig5simulations}a. Moreover, when the pump power is further increased, the simulated dark breather undergoes a transition into chaotic breathing via period-tripling. When increasing the pump power to 752 mW, the simulated RF spectrum shows a period-tripling feature (Fig. \ref{Fig5simulations}b). Note that this period-tripling does not occur in the absence of mode-interaction (see Supplementary Information Sec. 5). When further increasing the pump power to 785 mW, the RF spectrum becomes broadband (Fig. \ref{Fig5simulations}c). In addition, the phase diagram of the total comb power shows chaotic features (Fig. \ref{Fig5simulations}d). The simulated spectral and temporal evolution are shown in Fig. \ref{Fig5simulations}e. From the arrows therein, we can see the breathing periodicity is lost (Supplementary Animation 2); however, the dark-pulse-like waveform and the dark-pulse-featured spectrum are retained. Furthermore, the averaged AC trace (after phase compensation) of the chaotic breathing state remains close to the AC of the coherent state. Breathers in the frame of NLSE cannot undergo period-bifurcation or chaotic transition, as the NLSE is fully integrable \cite{Manakov1974}. Here, the dissipative effects of the microresonators enable the observation of this chaotic transition. These transitions should also be possible for dark breathers in other systems, e.g., spatial solitons \cite{Lugiato_PRL1987spatial} and plasmas \cite{Bekki1985}.

\section{Discussion and conclusion}
Optical dark breathers in normal dispersion microresonators are clearly observed and comprehensively modeled for the first time. The dark breather features a high frequency modulation, a weak breathing depth, and an energy exchange between different lines. The breathing dynamics of comb lines around the pump are shown to be affected a mode-interaction over 12 THz away, which reveals the influence of mode-interaction on the instability of mode-locked dark pulses. A transition from periodic breathing into a chaotic breathing state where the comb retains a dark-pulse-like waveform is also observed, which highlights the dissipative features of the breathers in cavities. Since dissipation is unavoidable in the real world, we anticipate more features imprinted on breathers by dissipation can be revealed in optics and other physical systems. Our work constitutes an important addition to the study of breathers and may have implications to a variety of physical systems, including hydrodynamics, plasmas, BEC and granular lattices.

\section{Method}
\small\textbf{Fast recording of the spectral evolution.} The method to record the fast evolution of the spectral breathing is the same as that used in ref. \cite{Weiner_PRL2016} (see the Supplementary Information Sec. 1 for the experimental setup). After a pulse-shaper is used to select specific comb lines, the output is recorded by a fast photodiode. A portion of the output is used to trigger the oscilloscope in order to provide a timing reference for measurements performed on different comb lines. The recorded traces are Fourier transformed and numerically filtered by a 20 MHz bandpass filter to isolate the strongest breathing tone; the filtered spectra are inverse Fourier transformed to yield the spectral breathing traces presented in Figs. \ref{Fig2Characterization}a, c.

\textbf{Line-by-line pulse shaping.} The method for shaping the output comb into a transform-limited pulse is described in ref. \cite{Weiner_NP2011}. Comb lines with mode number in $[-7,~11]$ are used for pulse-shaping. The intensity of the comb lines is adjusted to prevent the strongest comb lines from dominating the pulse shaping. In this intensity adjustment, the pump line (mode 0) is attenuated by 20 dB; modes $\pm$1 are attenuated by 10 dB; modes $\pm$2 are attenuated by 5 dB, and the other modes are not attenuated.

\textbf{Numerical simulations.} The simulations are based on the Lugiato-Lefever equation \cite{Lugiato_PRL1987spatial,Coen_OL2013modeling}, which can be written as
\begin{equation}
\begin{aligned}
\left( {{t}_{R}}\frac{\partial }{\partial t}+\frac{\alpha +\kappa }{2}+i\frac{{{\beta }_{2}}L}{2}\frac{{{\partial }^{2}}}{\partial {{\tau }^{2}}}+i{{\delta }_{0}}-i\gamma L|E|^2 \right)E=\sqrt{\kappa P_{in}}
\end{aligned}
\label{eq:LLE}
\end{equation}
where $E$, $t_R$, $L$, $\beta_2$, $\gamma$, $\alpha$, $\kappa$, $\delta_0$, and $P_{in}$ are the envelope of the intracavity field, round-trip time, cavity length, group-velocity dispersion, nonlinear coefficient, intrinsic loss, coupling loss, pump phase detuning and pump power, respectively. In the simulations, the parameters used are $t_R$=4.3 ps, $\beta_2$=187 ps$^2$/km, $L$=628 $\mu$m, $\gamma$=0.9 (Wm)$^{-1}$, $\alpha$=0.0014, $\kappa$=0.0054, $P_{in}$=700 mW, which are representative of the studied device. Mode-interaction is included in the simulation by adding a phase shift $\Delta \phi$ per round-trip to the perturbed modes. In simulations, 4 modes within $[-57,~-54]$ are perturbed with an identical $\Delta \phi$. In the first step (0-100 ns), for which $\delta_0$=0.0102 and $\Delta \phi=-$0.815, the comb generation is initiated by the mode-interaction. In the second step (100-200 ns), for which $\delta_0$=0.0272, $\Delta \phi=-$0.815, the comb grows and becomes chaotic. In the third step (200-300 ns), for which $\delta_0$=0.0629, $\Delta \phi=-$0.326, a dark breather can be obtained (see Supplementary Information Sec. 3 for the comb generation dynamics).

The spectral breathing is numerically filtered in the same way as experiments to obtain the traces in Figs. \ref{Fig4simulations}e, g. For the traces presented in Fig. \ref{Fig4simulations}d and Fig. \ref{Fig5simulations}f, we extract the simulated phase of the coherent dark pulse, which we denote $\phi_{\text{dp}}(\omega)$, and then add $-\phi_{\text{dp}}(\omega)$ to compensate the phase and obtain a transform-limited pulse.  The AC is then calculated.  For the dark breather and chaotic breathing comb, the same $-\phi_{\text{dp}}(\omega)$is added to the complex spectral amplitude. Then a series of AC traces are computed based on snapshots of the intensity profile. 10$^4$ individual AC traces are calculated in a 1 $\mu$s time window (equivalent to $\sim$782 periods for the dark breather) and averaged to yield the traces in Fig. \ref{Fig4simulations}d and Fig. \ref{Fig5simulations}f. All the simulated comb modes are used in the simulated pulse shaping; the intensity of the comb lines within $[-2, ~2]$ are adjusted in the same way used in the line-by-line pulse shaping in experiments.
~~~

\textbf{Acknowledgements.} This work was supported in part by the Air Force Office of Scientific Research (AFOSR) (Grant No. FA9550-15-1-0211), by the DARPA PULSE program (Grant No. W31P40-13-1-0018) from AMRDEC, and by the National Science Foundation (NSF) (Grant No. ECCS-1509578). We gratefully acknowledges fruitful discussions with Xiaoxiao Xue and Nail Akhmediev.

\section*{1. Experimental setup and comb states}

\begin{figure}[h]
\includegraphics[width=0.75\columnwidth]{FigS1Setup3.pdf}
\caption{\textbf{Experimental setup and comb states.} \textbf{a,} The Kerr comb used for dark breather study is generated from a SiN microresonator, pumped by a cw laser. To record the fast evolution associated with the comb breathing, either individual comb lines or groups of lines are selected by a pulse-shaper; their power is recorded as a function of time by PD2. A fraction of the full comb is sent to a second photodiode, PD1, which is used for synchronization between different traces. An autocorrelator and the pulse shaper are also used for the time domain characterization. The comb states presented in the main text and their corresponding pump power and wavelength are also summarized. PD: photodiode, MI: modulation instability. LNA: low noise amplifier. There is a chaotic state prior to dark breather formation, i.e., state \uppercase\expandafter{\romannumeral2}, which is distinct from the chaotic breathing state \uppercase\expandafter{\romannumeral6}. The optical spectrum and RF spectrum of this state are shown in \textbf{b} and \textbf{c} respectively. \textbf{d,}  When using the phase compensation that is used to shape the coherent comb into transform limited pulses, the comb in chaotic state II cannot be shaped into a pulse. The autocorrelation (AC) trace (green line) differs markedly from the AC of the calculated transform limited pulse.}
\label{Fig1Setup}
\end{figure}

The experimental setup and the comb states described in the main text and the Methods section are illustrated in Fig. \ref{Fig1Setup}a. The pulse shaper is used for recording of the fast spectral breathing as well as for time domain characterization. The five comb states described in the main text are also summarized: \uppercase\expandafter{\romannumeral1}) a modulation instability dynamics dominated comb with low frequency RF sidebands; \uppercase\expandafter{\romannumeral3}) tuning the pump into the resonance further generates the dark breather; \uppercase\expandafter{\romannumeral4}) by decreasing the pump power from the dark breather state, a coherent dark pulse state can be obtained; \uppercase\expandafter{\romannumeral5}) by increasing the pump power from the dark breather state, a period-tripling breathing state is obtained; \uppercase\expandafter{\romannumeral6}) by increasing the pump power further, the breathing becomes chaotic. In addition to the comb states shown in the main text, there is another chaotic state (i.e., state \uppercase\expandafter{\romannumeral2}) that is observed  prior to dark breather formation (see Fig. \ref{Fig1Setup}c for its RF spectrum), which is distinct from the chaotic state (\uppercase\expandafter{\romannumeral6}) which is reached from the dark breather state. The optical spectrum of state \uppercase\expandafter{\romannumeral2} only shows some clustered comb lines (Fig. S1b). Furthermore, this chaotic comb state cannot be shaped into transform-limited pulses \cite{Weiner_NP2011}. Here, we select 11 lines (including the pump line whose intensity is attenuated by 10 dB) around the pump for pulse shaping and apply the phase profile that is used to shape the coherent dark pulse comb into transform-limited pulse, the AC trace deviates from the calculated AC of the transform-limited pulse greatly (Fig. \ref{Fig1Setup}d).

\section*{2. Intracavity waveform retrieval}

\begin{figure}[h]
\includegraphics[width=0.75\columnwidth]{FigS2Waveform.pdf}
\caption{\textbf{Intracavity waveform retrieval.} \textbf{a,} The retrieved comb phase via line-by-line pulse shaping. The intracavity pump phase is estimated based on the through-port data (see ref. \cite{Weiner_NP2015mode} and the Supplementary Information therein). \textbf{b,}  Using the averaged power spectra of the coherent dark pulse and the breather together with the retrieved phase, the intracavity waveform can be reconstructed, showing dark-pulse-like waveforms. Due to the difference in the averaged power spectra between the breather and the coherent state, the corresponding waveforms slightly differ.}
\label{Fig2Waveform}
\end{figure}

Using the comb phase retrieved in line-by-line pulse shaping, the intracavity waveform can be reconstructed \cite{Weiner_NP2015mode}. Note that the pump line in the through-port contains both the transmitted uncoupled pump and the coupled-out component from the cavity. Hence, the intensity and phase in the through-port are corrected to yield the pump line in the cavity, based on the method described in ref. \cite{Weiner_NP2015mode} and the Supplementary Information therein. By applying the retrieved phase in Fig. \ref{Fig2Waveform}(a) to the spectra of the dark breather and the coherent dark pulse, we can calculate dark-pulse-like waveforms. Since the dark breather breathes weakly, this further confirms that the dark breather retains the dark-pulse-like waveform.

\section*{3. Generation dynamics of the dark breather in simulations}

\begin{figure}[h]
\includegraphics[width=0.75\columnwidth]{FigS3Generation.pdf}
\caption{\textbf{Tuning the detuning to generate the dark breather.} \textbf{a,}  The spectral dynamics while detuning and mode interaction parameter $\Delta \phi$ are varied. \textbf{b,}  Typical spectra at 50 ns, 150 ns and 250 ns; at 250 ns, a dark-pulse-comb featured spectrum is obtained when the dark breather is generated. \textbf{c,}  Typical waveforms at 50 ns, 150 ns and 250 ns; this further confirms that the dark breather retains the dark-pulse-like waveform.}
\label{Fig3Generation}
\end{figure}

As described in the Methods, the dark breather is generated by increasing the detuning in a step-like way in simulations. In Fig. \ref{Fig3Generation}a, we show the generation dynamics in the 3 steps. In 0$-$100 ns, $\delta_0$=0.0102, $\Delta \phi=-$0.815, and the comb is initiated by the mode-interaction and coarsely spaced. In 100$-$200 ns, $\delta_0$=0.0272, $\Delta \phi=-$0.815, and the comb grows and becomes chaotic. In 200$-$300 ns, $\delta_0$=0.0629, $\Delta \phi=-$0.326, and a dark breather can be obtained. Since the full evolution over 200$-$300 ns are plotted in Fig. \ref{Fig3Generation}a, it is hard to resolve the breathing dynamics. For visualization of the breathing on a smaller time scale, see Fig. 4 in the main text. In Figs. \ref{Fig3Generation}b, c, we show the typical spectra and waveforms (at 50 ns, 150 ns, 250 ns) in each step. Note that the chaotic comb in 100$-$200 ns (prior to dark breather formation) is different from the chaotic breathing in Fig. 3 of the main text, and the time-domain waveform and spectrum do not resemble that which occurs in the coherent dark pulse state. This state is similar to the state \uppercase\expandafter{\romannumeral2} in Fig. \ref{Fig1Setup}.

\section*{4. Breathing solution in the nonlinear Schr\"{o}dinger equation}

\begin{figure}[h]
\includegraphics[width=0.65\columnwidth]{FigS5DoublePeriod.pdf}
\caption{\textbf{Breathing dynamics of the double period solution at $\epsilon$=0.5.} \textbf{a,} Temporal dynamics of the solution, \textbf{b,} two typical examples of the waveform at $t$=0.18 and $t$=1.98. \textbf{c,} Spectral breathing of the solution, \textbf{d,} all the frequency components (aside from the 0-frequency) breathe at the same phase, which is out phase of the breathing phase of the 0-frequency component.}
\label{Fig5DSP}
\end{figure}

An analytical breather solution to the nonlinear Schr\"{o}dinger equation (NLSE) also exists in the normal dispersion regime \cite{Akhmediev_PRA1993}. The breathing solution is also referred as the ``double period" solution, which can be written as \cite{Akhmediev_PRA1993}
\begin{equation}
\begin{aligned}
\Psi (\tau ,t)=\epsilon \frac{cn(t,\epsilon )-i\sqrt{1+\epsilon }sn(t,\epsilon )dn\left[ \sqrt{1+\epsilon }\tau ,\sqrt{{2\epsilon }/{(1+\epsilon )}\;} \right]}{\sqrt{1+\epsilon }dn\left[ \sqrt{1+\epsilon }\tau ,\sqrt{{2\epsilon }/{(1+\epsilon )}\;} \right]+dn(t,\epsilon )}\exp (it),
\end{aligned}
\label{eq:DPS}
\end{equation}
where $\Psi$, $\tau$, $t$ are the envelope of the waveform, fast time and slow time respectively; $sn$, $cn$, $dn$ are the Jacobi elliptic functions; $\epsilon$ is a free parameter of the solution. $\epsilon$ affects the amplitude of the waveform and the breathing period, but does not affect the waveform shape significantly. Here, we choose $\epsilon$=0.5 as an example of this solution. The corresponding temporal and spectral breathing can be found in Fig. \ref{Fig5DSP}. The spectral dynamics clearly shows the breathing of the solution. However, the features of the``double period" solution differ significantly from the dark breather we have investigated in microresonators. This breathing solution does not resemble the dark-pulse-like waveform in the microresonator (Fig. \ref{Fig5DSP}c). In the frequency domain, there is also energy exchange between the 0-frequency and the frequency components in the wings ($\pi$ phase difference). However, all the frequency components in the wing (i.e., aside from the 0-frequency) breathe with the same phase (Fig. \ref{Fig5DSP}d), which differs from the breathing of dark breather presented in Fig. 2c of the main text.

The NLSE is usually used to describe conservative systems, such as the propagation in optical fibers. The governing equation of the Kerr comb generation in microresonators is the Lugiato-Lefever equation (LLE) \cite{Lugiato_PRL1987spatial,Coen_OL2013modeling}, which is also regarded as an externally driven damped NLSE \cite{Smirnov_PRE1996}. The distinct breather dynamics in the NLSE and the LLE in the normal dispersion regime highlight the importance of dissipation for Kerr comb generation in microresonators.

\section*{5. Dark breather at the absence of mode-interaction}

\begin{figure}[h]
\includegraphics[width=0.75\columnwidth]{FigS4NoMI3.pdf}
\caption{\textbf{Dark breather at the absence of mode-interaction.} \textbf{a,} Simulated (averaged over slow time) spectrum in comparison with the experimental measured spectrum of the dark breather. \textbf{b,} Absolute breathing amplitude and breathing depth of different comb lines, both of which are symmetric with respect to the pump.  \textbf{c,} Spectral breathing dynamics for different comb lines, which are also symmetric with respect to the pump (see the arrows pointed to modes $\pm$1). The circles on the red line mark the peaks of the spectral breathing. \textbf{d,} In the absence of mode-interaction, the comb simulated at 752 mW pump power and $\delta_0$=0.0629 shows an oscillation period 23T$_1$ (where T$_1$ is the ``period" for a period-1 breather). This is in contrast to the period-3 bifurcation simulated in the presence of the mode interaction. \textbf{e,} Corresponding RF spectrum of \textbf{d}; the inset reveals low frequency RF tones at 1/(23T$_1$) and 2/(23T$_1$). \textbf{f,} If the mode-interaction strength is increased to 1.25 times the value used in the main text (i.e., $\Delta\phi=-$0.408), the simulated RF spectrum becomes broadband (using 752 mW pump power and $\delta_0$=0.0629, same as parts \textbf{d-e}).}
\label{Fig4NoMI}
\end{figure}

The simulated spectral breathing becomes symmetric with respect to the pump, when the mode-interaction is excluded (i.e., setting $\Delta \phi$=0). Note that this simulation uses the dark breather described in Fig. 4a of the main text as the seed. The simulated spectrum of the dark breather (averaged over slow time) no longer shows spikes around 1657 nm (Fig. \ref{Fig4NoMI}a). The breathing amplitude of different comb lines becomes symmetric with respect to the pump (Fig. \ref{Fig4NoMI}b). Furthermore, the breathing phase of different lines becomes symmetric with respect to the pump. For instance, modes $\pm$1 (see the arrows in (Fig. \ref{Fig4NoMI}c)) breathe with the same phase, whereas modes $\pm$1 breathe with different phase in the presence of mode-interaction (see Fig. 2c and Fig. 4g in the main text).

Furthermore, mode-interaction strongly affects the occurrence of the period-3 bifurcation reported in the main text. For instance, in the absence of mode-interaction, at the pump power of 752 mW, $\delta_0$=0.0629 where the simulated dark breather exhibits the period-3 bifurcation with mode-interaction in the main text, the simulated dark breather no longer shows the period-3 bifurcation (Figs. \ref{Fig4NoMI}d, e). Moreover, no period-3 bifurcation is found in simulation even at other pump powers in the absence of mode-interaction. In addition, if we increase the mode-interaction strength to 1.25 times that used in the main text, the dark breather will now be in the chaotic breathing state at 752 mW pump power, $\delta_0$=0.0629. We repeat simulations at other pump powers but do not find a period-3 bifurcation in the presence of this increased mode-interaction strength either. These results suggest that mode-interactions strongly affect the period bifurcation of the dark breather and are critical to the observation of the period-3 bifurcation reported in the main text.

\section*{6. Illustration of animations}
\textbf{Animation 1,} spectral and temporal breathing of the regular breather (corresponds to Fig. 4a in the main text). The breather repeats itself around 1.2$\sim$1.3 ns and $\sim$2.5 ns (end of the animation), as the period is 1.27 ns.

\textbf{Animation 2,} spectral and temporal breathing of the chaotic breather (corresponds to Fig. 5e in the main text). The chaotic breather does not repeat itself in the full 2.5 ns.

redpajama_set_name: RedPajamaArXiv


'EVERY cloud has a silver lining' described perfectly the situation in our house during the period March/April 1959.
The cloud was that dad had been laid low by a mysterious virus, and after a spell in hospital where he had been seriously ill, he was back home, but still very poorly upstairs.
The silver lining involved Crook Town AFC, who the family supported, and who had reached the semi-finals of the FA Amateur Cup.
Crook's success would later prove to be the turning point in dad's battle to get well, but the first problem was, with dad out of action, "Who was going to take me to the semi-final at Roker Park, Sunderland to watch Crook play Leytonstone?"
The issue had reached 'Brexit' proportions when a family friend Tommy Dixon agreed to take me. The plan was, that on my own, I would board one of the 'Football Special' trains at our railway station in Howden-Le-Wear, and meet up with Tommy, who would join the train at Shildon railway station.
Easier said than done, because the Football Special from Crook was practically full when it arrived in Howden. It then picked up more fans at the Etherley and Bishop Auckland stations before it arrived at Shildon where there were scores of people on the platform. Somehow, through the window of my carriage, and through the crowds of people I spotted Tommy, who waved and then joined me in my carriage, and we were on our way.
That was the first problem over with, the second problem was Leytonstone, our opponents at Roker Park in the afternoon. I had already had my heart broken the year before in the semi-final of the Amateur Cup at Roker Park when Crook lost 1-0 to Ilford in 1958. To add insult to injury my hero and good friend Bert Steward missed a penalty that afternoon, so it was a real bad day. Thoughts of the same thing happening again made it 'squeeky bum' time.
I needn't have worried, because Crook, who were not at their best, were too strong for Leytonstone. I recall Jimmy McMillan, the Crook outside left, once telling me that the sporting press reported that he put in a 'Finney-like' performance against Leytonstone. Jim was one of the few Crook players who was on form, the other was Colin Bainbridge, the Crook centre half. Some players were right off their game, including Brian Keating, the scorer of both Crook goals.
I had my eyes on reserve George Masters who played at right half in place of the injured Bill Jeffs to see how he would perform. I thought he did well even though the Crook Football Club committee later left him out of the team to play Barnet in the final at Wembley. With Bill Jeffs still unfit, they picked Derek Carr who had played only a handful of games for Crook, George had been at Crook all season, I don't think he took it well.
I always remember Crook's first goal at Roker Park, a left wing cross from Jimmy Mac on 28 minutes and a bullet header from Brian 'Buster' Keating. After that there were a series of thrilling non-stop attacks by Crook with Keating and Coates firing wide. On 82 minutes Crook's inside left MikeTracey headed on to the bar and Keating headed in the rebound to make the final score 2-0 in Crook's favour. I left Roker Park a lot happier than I did in 1958, now the question was, "Who was going to take me to the final at Wembley in April?"
We hope to have a Special Edition of Sport Archives available on Saturday April 20th to commemorate the 60th anniversary of the FA Amateur Cup Final between Crook Town and Barnet at Wembley Stadium on Saturday April 18th 1959.
Sport Archives is sad to hear of the passing of Derek Lewin, a fine footballer, and a gentleman on, and off the field. Derek was a very friendly person and a big supporter of the Durham Amateur Football Trust. He also contributed many times to the Sport Archives column as well as providing lots of information for my book Hopes and Heroes.


I am so lucky to meet and work with such lovely couples...Helen & Graham were just so friendly and fun and it was a pleasure to be part of their day! They married at The Carriage Hall in Plumtree and chose some gorgeous colours for their flowers...hot pink, lilac and peach...with lots of lovely foliage and succulents for a laid back, rustic vibe. Here's a few images from Firsthand Photography...to see the full gallery take a look at their blog! Thanks for stopping by!


Compiled By Jill Jusko The semiconductor equipment industry had been anticipating poor third-quarter market figures, and just-released data show that their expectations were more than met. Worldwide semiconductor manufacturing equipment shipments were $5.6 billion in the third quarter of 2001, according to the Semiconductor Equipment and Materials International (SEMI), San Jose, an international trade organization of companies that supply manufacturing technology and materials to the world's chip makers. That $5.6 billion figure is 55.6% below the same quarter a year ago and 20.2% behind the shipment figure for the second quarter of 2001. Even worse, worldwide equipment orders of $3.92 billion in the third quarter of this year are 75% below the same quarter a year ago and 12% below the orders figure for the second quarter of 2001. By region, North America shipped $1.66 billion in product during the third quarter of 2001, a drop of 52.7% from the same quarter a year ago. Japan, the next largest producer, saw total shipments of $1.47 billion in the third quarter of 2001, down 43.8% from the same quarter one year ago.


text:
Half year results for 6 months ended 30 June 2016
27 Sep 2016 aecadmin Regulatory
This document is posted @ London Stock Exchange and can be viewed and downloaded here
Document contents:
AIM: MLVN
(Formerly known as AEC Education PLC)
("MLVN" or the "Company" and together with its subsidiaries, the" Group")
Half year results for the six months ended 30 June 2016
· Revenues on continuing activities of £2.07m (2015: £2.3m)
· Operating loss of £0.46m (2015: loss of £0.62m)
· Loss before tax of £0.46m (2015: loss of £0.57m)
· Loss after tax from continuing activities of £0.48m (2015: loss of £0.57m)
· Loss per share on continuing activities of 0.77p (2015: 0.94p)
· Malaysia continues to contribute profits but London and Singapore continue to report losses for the year to date.
· The Irish operation is reported as a discontinued business for the current year as it has been sold on July 15, 2016. The details are documented in Note No:7.
· Initiatives have been implemented in London and Singapore to assist recovery in the second half of 2016.
· On June 15, 2016, loans amounting to £853,951 from two major shareholders of the Group were converted into 17,079,020 new ordinary shares at a conversion price of 5p per share.
This announcement includes inside information as defined in Article 7 of the Market Abuse Regulation No. 596/2014 and is disclosed in accordance with the Company's obligations under Article 17 of those Regulations.
William J Swords, Chairman, stated: "The Group's results for the six months ended 30 June 2016 are disappointing although better than the first six months of 2015. New arrangements in London and Singapore, through improvements in the management team and a focused strategic plan during the second half of 2016, are intended to shift the Group towards a better and positive 2017. A new deputy CEO from a successful education background has been appointed and will bring more leadership to initiate growing strategies that match today's needs. The improved structure and team in place will help the organisation to work on its new growth strategies which is being backed up with more funds injected into the Group's operations. Further announcements on the new plans will be made as the work progresses during the second half of 2016 and first half of 2017."
Tel: +44 (0) 207520 0470
William J Swords
WH Ireland Limited (NOMAD)
Mike Coe
CHAIRMAN'S STATEMENT
The Group's results for the six months ended 30 June 2016 are disappointing although better than the first six months of 2015. New arrangements in London and Singapore, through improvements in the management team and a focused strategic plan during the second half of 2016, are planned to shift the Group towards a better and positive 2017. A new deputy CEO from a successful education background has been appointed and will bring more leadership to initiate growing strategies that matches today's needs. The improved structure and the team in place will help the organisation to work on its new growth strategies which are backed up with more funds injected into the Group's operation, through new capital injections and the disposal of the Irish operations. Further announcements on the new plans will be made as the work progresses during the second half of 2016 and first half of 2017.
The period under review can be summarised as follows:
· London continued to record a low revenue performance but had a better management of its cost structure to reduce its loss.
· Ireland recorded a profit of £33K for the first half but as a result of its disposal in July 2016 , it has been reported as a discontinued business.
· The Cyprus joint venture continues to feel the effect of the slowing Russian economy and had recorded a break even result for the first half.
South East Asia/Middle East
· Singapore continued to record operating losses, although at a lower level than previously reported, due to lower revenue intakes, but we have put in place new management and new strategies that are intended to to kick start this market into profitability in early 2017.
· Malaysia continued its strong performance and showed a good operating profit.
The Group's revenues on continuing activities for the six months in 2016 reduced by 12% to £2.1m (2015: £2.3m). London and Singapore were responsible for the lower volumes as their jurisdictions continued to be affected by the lack of Edutrust accreditation and the uncertainty in the European markets. However, the Group's efforts to better manage its operating costs since the second half of 2015 has resulted in the Group's loss before tax for the six months from continuing operations decreasing to £0.46m (2015: £0.57m). A lack of marketing support and new programme initiatives has contributed to the struggles in both London and Singapore. While the operations are becoming smarter in running at a lower cost, the marketing side needs to improve the selling of all the programmes on offer. Recent initiatives to improve on local marketing in Singapore are showing signs of success.
No impairment was considered necessary for the first six month as we had already undertaken an extensive review and impairment charge during 2015 and we are confident that the plans and strategies we have in place for the second half of 2016 and 2017 will drive the Group's business positively.
For the first half reporting in 2016, we have reported the financial results of Malvern House Ireland as a discontinued business. Accordingly, we have re-stated the comparative number for 2015 financial results. More details can be found in Note No: 7 for the full impact of these comparative re-statements.
· The London operation recorded an operating loss of £133k which after, finance charges, resulted in a loss of £164k. For the same period last year, London recorded £539k of losses. The improvement over the prior year is mainly a result of costs management and higher gross margins in revenue earned. Revenue was down year on year by 54% to £0.7m (2015: £1.2m). Efforts are in place to strengthen the marketing team in London and a variety of educational offerings are being considered.
· Ireland recorded turnover of £1.3m in 2016 (2015 – £1.2m) and a profit after tax of £33k. The results of Ireland are reported as a discontinued operation.
· The Singapore operations recorded an operating loss of £96k (2015: £11k profit) mainly due to a lower revenue volume of £124k (2015: £227k). New marketing and programmes initiatives will be key in raising the profile of the Singapore operations within the Group.
· Malaysia returned an operating profit of £97k (2015: £26K loss). In Malaysia, revenue increased on the previous year by 29%. Malaysia continues to be on a healthy road and we expect this to continue throughout the financial year and into 2017.
The basic and diluted loss per share on the continuing business was (0.77p) (2015: 0.94p).
Net cash at the end of the year stood at £0.45m (2014: £0.36m).
In February 2016 the Group concluded an unsecured, interest free shareholder loan of £500,000 from KSP Investments Pte Limited ("KSP"). In June 2016, the Group agreed to the conversion of £646,000 of its outstanding loan balance with KSP and £207,951, being the remainder of its outstanding loan balance with CG Corp (a Cinnovation group company). These loans were converted at 5p into 17,079,020 new ordinary shares
Operational Review
In Europe, the London revenue has been impacted by a lack of marketing resources. This issue has been reviewed by the board and there are plans in place to support the team with more resources in the second half of 2016 and early 2017. Management has also taken the initiative to introduce some robust programmes that will widen of the revenue base by catering for students outside the visa-working requirements. The new funds injected to the business will be invested in new programmes and marketing resources. With the disposal of Dublin, our focus is on securing the future of our flagship in London.
The Asian units are also experiencing mixed results. Singapore started gaining more from its new initiatives in local markets and short courses and gradually moving towards a reduced loss position. There are plans in place to reapply for EduTrust by early 2017 when all the systems and processes are in place. This should boost the revenue in Singapore.
Malaysia remains, the most diversified of all the business units in terms of its programme range. Its business has been growing and accordingly the unit has reported higher profit. Moreover, market conditions, especially with the weak Ringgit should give Malaysia a stronger position in the Asia market and assist in its drive to increase student numbers in the next period although the weak currency will have a negative impact on reported results when converted into sterling. There are also plans in place to further grow Malaysia and to move to new premises in a better location.
The disposal of Malvern House Ireland was completed in July 2016 and a license agreement is being processed for the utilization of the Malvern Brand. The proceeds of the disposal amounting to €660,000 will be used to supplement the Group's cash resources. The disposal will have no impact on the other operations of the Group.
We look forward positively as we restructure the revenue generation models of all our operating units to include a wider range of products. We have also begun discussions with various parties on the development of online teaching platforms as a supplement to the traditional teaching methods. Further, we have started to re-introduce licensing models to offer our educational content and brand to non-traditional markets in Asia and the Middle East. We are confident that by mid 2017, all our new initiatives and programmes will begin to show an overall Group return.
UNAUDITED CONSOLIDATED INCOME STATEMENT
Six months to 30 June 2016
Twelve months to 31 December 2015
Audited
(restated)
Sales of services and other revenue
Cost of services sold & operating expenses
(6,403))
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(Loss) / profit for the period / year
(Loss) / profit attributable to equity holders
(Loss) / earnings per share on total activities
(Loss) / earnings per share on continuing activities
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UNAUDITED STATEMENT OF FINANCIAL POSITION
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Investment in joint venture
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UNAUDITED CONSOLIDATED STATEMENT OF CHANGES IN EQUITY
Translation Reserve
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Balance at 1 January 2015
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Unclaimed Dividends Returned
Balance at 31 December 2015/1 January 2016 (restated)
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New Share Issue
Balance at 30 June 2016
UNAUDITED CONSOLIDATED STATEMENT OF CASH FLOWS
(Loss) / profit before income tax from continuing activities
(Loss) / profit before income tax from discontinued activities
Adjustments for:
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Purchase of property, plant and equipment
Purchase of intangible fixed assets
Dividend paid to shareholders-unclaimed
(Decrease) / increase in finance lease liabilities
Repayment of term loan
Effect of foreign exchange rate changes on consolidation
Net increase in cash and cash equivalents
Cash and cash equivalents at beginning of period / year
Cash and cash equivalents at end of period / year
NOTES TO THE UNAUDITED INTERIM FINANCIAL INFORMATION FOR THE 6 MONTHS ENDED
Malvern International plc (formerly known as AEC Education plc) (the "Company") is a public limited liability company incorporated in England and Wales on 8 July 2004. The Company was admitted to AIM on 10 December 2004. Its registered office is Witan Gate House, 500-600 Witan Gate West, Milton Keynes MK9 1SH and its principal place of business is in Singapore. The registration number of the Company is 05174452.
The principal activities of the Company are that of investment holding and provision of educational consultancy services. The principal activity of the group is to provide an educational offering that is broad and geared principally towards preparing students to meet the demands of business and management. There have been no significant changes in the nature of these activities during the period
2. Adoption of new and revised International Financial Reporting Standards
No new IFRS standards, amendments or interpretations became effective in the six months to 30 June 2016 which had a material effect on this interim consolidated financial information.
3. Significant accounting policies
The accounting policies adopted are consistent with those of the previous financial year.
This interim consolidated financial information for the six months ended 30 June 2016 has been prepared in accordance with IAS 34, 'Interim financial reporting'. This interim consolidated financial information is unaudited and is not the Group's statutory financial statements and should be read in conjunction with the annual financial statements for the year ended 31 December 2015, which have been prepared in accordance with International Financial Reporting Standards (IFRS) and have been delivered to the Registrar of Companies. The auditors have reported on those accounts; their report was unqualified, but did include, without qualifying their report, references to which the auditors drew attention by way of emphasis of matter in respect of the preparation of the financial statements on a going concern basis.
The interim consolidated financial information for the six months ended 30 June 2016 is unaudited. In the opinion of the Directors, the interim consolidated financial information presents fairly the financial position, and results from operations and cash flows for the period. Comparative numbers for the six months ended 30 June 2015 are unaudited.
This interim consolidated financial information is presented in £ sterling, rounded to the nearest thousand.
Critical Accounting Judgements and Key Sources of Estimation Uncertainty
In the process of applying the Group's accounting policies above, management necessarily make judgements and estimates that have a significant effect on the amounts recognised in the financial statements. Changes in the assumptions underlying the estimates could result in a significant impact to the financial statements. The most critical of these accounting judgement and estimation areas are as follows:
Estimated Impairment of Brands, Licenses and Trademarks
The Group evaluates whether there is any indication that their brands, licenses and trademarks have suffered any impairment, in accordance with their stated accounting policy. The recoverable amount of brands, licenses and trademarks is determined from value in use calculations. The key assumptions for the value in use calculation are those regarding expected discounted future cash flows.
Estimated Impairment of Goodwill
The Group tests annually whether goodwill has suffered any impairment, in accordance with their stated accounting policy. The recoverable amount of goodwill is determined from value in use calculations. The key assumptions for the value in use calculation are those regarding expected discounted future cash flows.
Impairment of Assets other than Brands, Licenses, Trademarks and Goodwill
The Group reviews the carrying amounts of assets as at each net asset statement date to determine whether there is any indication of impairment in accordance with their stated accounting policy. If any such indication exists, the assets' recoverable amount or value in use is estimated. Determining the value in use of property, plant and equipment, which requires the determination of future cash flows expected to be generated from the continued use and ultimate disposal of the asset, requires the Company to make estimates and assumptions that can materially affect the financial statements. Any resulting impairment loss could have a material adverse impact on the Group's financial position and results of operations.
The Group is subject to income taxes in numerous jurisdictions. Significant judgement is required in determining the capital allowance, deductibility of certain expenses and taxability of certain income during the estimation of the provision for income taxes. There are many transactions and calculations for which the ultimate tax determination is uncertain during the ordinary course of business. The Group recognises liabilities based on estimates of whether additional taxes will be due. Where the final tax outcome is different from the amounts that were initially recorded, such differences will impact the income tax and deferred income tax provisions in the period in which such determination is made.
Evaluation of deferred income
The Group reviews the fees raised at the end of relevant periods to evaluate those amounts that cover the future provision of education not yet delivered to evaluate the amount of deferred income to be recognised in a future period
Revenue is recognised on the following basis:
· Course fees and examination fees are recognised as income based on classes or examinations conducted during the year.
· Accommodation fees are recognised as income based upon occupancy of act a point in time.
· Publication sales are recognised upon sale of study guides.
· Registration fees are recognised upon approval of respective applications.
· Revenues from support services are recognised when services are rendered.
· All other course fees in respect of courses offered with no obligation to impart lessons are recognised when the students register for the course and collect the study materials.
Deferred income relates to course and accommodation fees received in advance and is recognised in the income statement based on classes conducted and accommodation provided.
Intangible fixed assets
An intangible asset with indefinite useful life is tested for impairment annually and whenever there is an indication that the asset may be impaired.
License fees with a definite life are amortised using a straight line method over a period of 2 to 5 years. Brands with a definite life are amortised using a straight line method over a period of 25 years.
Impairment of tangible and intangible assets excluding goodwill
An assessment is made at each net asset statement date of whether there is any indication of impairment of any asset, or whether there is any indication that an impairment loss previously recognised for an asset in prior years may no longer exist or may have decreased. If any such indication exists, the asset's recoverable amount is estimated. An asset's recoverable amount is calculated as the higher of the asset's value in use or its fair value less costs to sell. Value in use is the present value of estimated future cash flows expected to arise from the continuing use of an asset and from its disposal at the end of its useful life.
An impairment loss is recognised only if the carrying amount of an asset exceeds its recoverable amount. An impairment loss is charged to the income statement in the period in which it arises unless the relevant asset is carried at a revalued amount in which case the impairment loss is treated as a revaluation decrease.
A previously recognised impairment loss is reversed only if there has been a change in the estimates used to determine the recoverable amount of an asset, however not to an amount higher than the carrying amount that would have been determined (net of any depreciation) had no impairment loss been recognised for the asset in prior years.
A reversal of an impairment loss is credited to the income statement in the period in which it arises unless the relevant asset is carried at a revalued amount in which case the impairment loss is treated as a revaluation increase.
Goodwill arising on acquisition is recognised as an asset and initially measured at cost, being the excess of the cost of the business combination over the Group's interest in the net fair value of the identifiable assets, liabilities and contingent liabilities recognised. After initial recognition, goodwill is measured at cost less accumulated impairment losses, if any.
For the purpose of impairment testing, goodwill is allocated to each of the Group's cash-generating sub-groups expected to benefit from the synergies of the combination. Cash-generating units to which goodwill has been allocated are tested for impairment annually, or more frequently when there is an indication that the unit may be impaired. If the recoverable amount of the cash-generating unit is less than the carrying amount of the unit, the impairment loss is allocated first to reduce the carrying amount of any goodwill allocated to the unit and then to the other assets of the unit pro-rata on the basis of the carrying amount of each asset in the unit. An impairment loss recognised for goodwill is not reversed in subsequent periods.
4. Dividend
No interim dividend for this financial year is proposed.
5. (Loss)/ earnings per share
The basic (loss)/earnings per share is calculated by dividing the (loss)/profit attributable to ordinary shareholders by the weighted average number of ordinary shares in issue during the relevant period. The weighted average number of shares in issue during the period was 64,450,963 (2015: 63,051,043).
The diluted (loss)/earnings per share is calculated by dividing the (loss)/profit attributable to ordinary shareholders by the weighted average number of ordinary shares in issue during the relevant period diluted for the effect of share options and warrants in existence at the relevant period. The weighted average number of shares in issue diluted for the effect of share options and warrants in existence during the period was 64,450,963 (2015: 63,051,043).
6. Share capital
On the 15th of June 2016, it was announced that the Company had agreed with certain shareholders that loans from them amounting in aggregate to £853,951 would be converted into ordinary shares in the Company at a value of 5 pence per share. 17,079,020 Ordinary Shares were issued at 5p each increasing the total number of Ordinary Shares held in the Company to 80,130,063 (previously: 63,051,043).
7. Subsequent events – discontinued activities
As reported in the 2015 Annual Report, the Group had undertaken the following activities to date in 2016.
On the 7th of July, 2016, the Company announced that it had disposed of one of its subsidiary companies, Malvern House Ireland Limited, in which it had a 55% interest in the share capital, to Oscar World Education Limited, Ireland. The consideration for the disposal was €660,000 (approximately £560,000) payable in cash on completion. The Purchaser will also sign a franchise agreement for one year (renewable yearly) enabling it to use the Malvern House brand name on payment of a royalty of 2.5% of tuition fee revenue per year. The proceeds of the disposal have been used to supplement the Group's cash resources. The disposal has no impact on the other operations of the Group.
This disposal has been reported as a discontinued activity in 2016 and the 2015 Financial Results have been re-stated on the same basis for comparative reasons.
The profit attributed to the discontinuance of the Malvern operations in Ireland is as follows:
June 2016 June 2015 December 2015
Half Year Half Year Full Year
(unaudited) (unaudited) (audited)
Revenue 1,267,354 1,231,189 2,905,301
Costs of Service Sold ( 718,035) (737,830) (1,446,089)
Personnel Costs ( 218,973) (175,023) (539,091)
Other Operating Expenses (264,497) (238,274) (619,280)
Depreciation & Finance Costs (33,192) (27,792) (55,561)
Tax Expense – – 17,150
Profit on Discontinued Activity 32,656 52,271 262,431
The Net Assets of the discontinued operation in Ireland are as follows:
Non-Current Assets 99,167 120,001 111,615
Current Assets 731,039 616,974 525,628
Total Assets 830,206 736,975 637,243
Current Liabilities 1,076,412 946,882 877,024
Non-Current Liabilities – – –
Total Liabilities 1,076,412 946,882 877,024
Net Assets (246,206) (209,907) (239,781)
Share Capital 725 725 725
Total Reserves (246,931) (210,632) (240,506)
Net Equity (246,206) (209,907) (239,781)
8. Subsequent events – Change of Name of the Company from AEC Education PLC to Malvern International PLC.
At the recently concluded AGM for the Group on the 12th of September, 2016, a resolution to change the name of the Group company from AEC Education PLC to Malvern International PLC was duly passed and came into effect on the 14th of September, 2016.
Result of AGM and Change ...

redpajama_set_name: RedPajamaCommonCrawl


text:
Boat takes an 80th anniversary spin
Joanna Hartman
Ryan Salm/ Sierra SunBill Lindemann drives 1926 Mercury all-aluminum boat with passengers from Sugar Pine Point State Park on Lake Tahoe on Tuesday. This summer marks the 80th anniversary of the Mercury.
The 1926 aluminum race boat formerly known as the Cigarette IV, now called the Mercury, celebrated her 80th birthday in style speeding across Lake Tahoe's early morning glass-like waters Tuesday.
The Mercury was built with precision; as the first all-aluminum speedster she was constructed with 979 pieces of duraluminum fastened with 14,250 rivets, 7,087 bolts and 238 screws. The original engine was a Curtiss V-12 aircraft engine that produced 625 horsepower, said Bill Lindemann, a Sierra District interpretive specialist and Mercury project manager.
After sporadic racing victories and a recorded top speed of 58.868 miles per hour as the "World's Fastest Boat" in 1926, the Mercury went into hiding.
Over a decade later she reappeared, re-christened as the Mercury, racing the waters of Lake Tahoe. A fuel fire and heavy damage sent her back to the boathouse in 1940, and was only occasionally released for leisurely cruises around Carnelian Bay.
The boat was left to the League to Save Lake Tahoe, re-gifted to the Sierra State Parks Foundation, and finally found her way to the Sugar Pine Point State Park Boathouse for storage.
Seventy-five years after her first victory in the President's Cup on the Potomac River in 1926, the Mercury was restored and re-re-powered with a V-8 to go back in the water under her own power.
With the new engine, even at 6,226 feet above sea level, the Mercury runs as efficiently now as she did 80 years ago with the V-12 engineered to speed to victory on Manhasset Bay, N.Y.
California State Park employees and volunteers were invited Tuesday to take a boatride on the Mercury down around Rubicon Bay.
"This is [the superintendent's] way of paying back seasonal workers, or volunteers, who work hard and don't get paid much. Or at all," said Lindemann of the Mercury's 80th anniversary.
"This is a treat," said Park Ranger Mike Rominger alongside his son, Andy. "[You] don't get to go on the Mercury very often."
The boat and the publicity are good for the park, said Lewis Grove, a volunteer docent. Restoration and maintenance relies greatly on public donations.
"People like to support things they enjoy doing, so this is good for that," Grove said.
Learn more about the history of the Mercury on the state parks Web site at http://www.parks.ca.gov, or visit the lustrous speedster at the Sugar Pine Point State Park Boathouse.


Taiwan is doing final upgrade b158 for all Taiwan devices. so try it: create a new Hicare account with Taiwan is the region!!
How you find this out?
This update adds a 10x zoom and optimizes the drawer homescreen display.
[What's new] Added a 10x zoom, which enables you to take clearer photos from a distance.
[Optimizations] Optimized the drawer homescreen display for a better experience. Decreased the CPU load for several core processes to achieve better system performance. Improved device security with Google security patches.
[Update notes] 1. This update will not erase your personal data, but we recommend that you back up any important data before updating. 2. The system will restart after the update. This will take about 5 minutes. 3. If you experience any issues during the update, please call the Huawei customer service hotline or visit an authorized Huawei service center for assistance.
Yes, it was put on the Huawei server within the last 24 hours.
It's the same as the Russian C10B156 update.
It's still not possible to install, but will be soon.
How about middle east C185B138 any idea when will be the new uodate?
It's now possible to install it. I'll be rebranding and checking it out tonight.
Yes, but it's not on Huawei's servers yet, and online update/OTA update is not out. Maybe this week. Strangely, the European is yet to be seen, even though European was first for other updates.
Maybe they are hiding the updates on different servers?
Seems they are going to change updates so c636 be the first like they use to be on mate8.
If it was on any server, I would know about it. It's not online yet.