\part*{Supplementary information\label{part:Supplementary-information}}


\section{Comparison with other sources}

\begin{table}[h]
\begin{tabular}{|c|c|c|c|c|}
\hline 
\multicolumn{1}{|c|}{Reference} & Emission HWHM & Pattern HWHM & Geometry limited maximum angle & Coherence angle\tabularnewline

\hline 
\cite{golzhauser1998holographic} & - & 1.6$^{\circ}$ & $\sim$8$^{\circ}$ & $\sim$3$^{\circ}$\tabularnewline

\hline 
\cite{de2004brightness} & - & - & 1.6$^{\circ}$& 1.4$^{\circ}$ \tabularnewline
\hline
\cite{morin1996low,prigent2000charge,prigent2001charge} & - & - & 1.4$^{\circ}$ & -\tabularnewline
\hline 
\cite{oshima2005highly} & 3.3$^{\circ}$ & 3.3$^{\circ}$ & 4.3$^{\circ}$ & too large for detector\tabularnewline
\hline 
\cite{chang2009fully,tsong2013} & 2.3$^{\circ}$ & 2.3$^{\circ}$ & 12.8$^{\circ}$ & -\tabularnewline
\hline 
\cite{longchamp2012low} & - & - & NA=0.48, 30$^{\circ}$ & 5.5$^{\circ}$\tabularnewline
\hline 
This work & 7.25$^{\circ}$ & 7.25$^{\circ}$ & variable & 14.5$^{\circ}$\tabularnewline
\hline 

\end{tabular}


\caption{\label{tab:comp}A comparison of coherence angle from other works}
\end{table}

There are many previous measurements of coherence angle, a broad selection are shown in Table \ref{tab:comp}. Some of these quantities are quoted directly from the referenced papers, where in other cases they had to be calculated from information provided about the experimental apparatus. This is compiled in order to compare our work using the same figures of merit presented in the literature.

\section{Tip etching}

\begin{suppfigure}[h]
\includegraphics[width=2.8cm,height=2.8cm]{002}\includegraphics[width=2.8cm,height=2.8cm]{004}\includegraphics[width=2.8cm,height=2.8cm]{005}

\includegraphics[width=2.8cm,height=2.8cm]{007}\includegraphics[width=2.8cm,height=2.8cm]{014}\includegraphics[width=2.8cm,height=2.8cm]{032}

\caption{FIM images showing the etching of the tip. Imaging voltages are 13.0
kV, 6.50 kV, 6.10 kV, 5.35kV, 5.05kV for each image, respectively}
\end{suppfigure}


To create such the tip used in the experiment, nitrogen gas is leaked
into the system under field-ion microscope (FIM) observation. Nitrogen
reacts with the tungsten, causing small protrusions. The field at
these protrusions is high enough to cause field-desorption - removing
the material. The high-fields used during the etching process inhibit
nitrogen from migrating near the apex. As a result the reaction and
field-desorption to occur preferentially at the shank of the tip.
What remains is a sharp apex with a higher aspect-ratio than single-atoms
tips fabricated using previous methods \cite{unnanotip,unnanotip3}\cite{pitters2012creation}. 

The tips in this work were etched using a specific recipe. Before
applying the field-assisted nitrogen technique, a relatively sharp
tip with a radius of curvature $>$10nm was first obtained by electrochemically
etching a polycrystalline tungsten wire in NaOH. The tip was then
loaded into a ultra-high vacuum (UHV) chamber with a base pressure
of 1x$10^{-10}$ Torr and placed in a LN$_{2}$ cooled FIM setup.
A high voltage was applied on the tip and a background pressure of
1x$10^{-5}$ Torr of helium was leaked into the chamber. A bias voltage
between 10-20 kV was necessary to obtain a clean, stable FIM image
of the tip. A background pressure of 2x$10^{-6}$ Torr of nitrogen
was then introduced into the chamber. The bias voltage was held constant
for 45 minutes. This allowed the material along the shank of the tip
outside of the FIM image, to be etched away. The voltage was then
decreased automatically at a rate of 1 V/s, until the bias reached
5.5 kV. When it was evident from the FIM image that etching had reached
a standstill, the voltage was lowered in 100 V increments at 1 V/s.
When the voltage reached 5 kV, the N$_{2}$ pressure was lowered to
8x$10^{-7}$ Torr and the voltage was lowered in 10 V increments until
the tip was sufficiently sharp, typically ending in a single atom. 





\section{Interpretation of fringe patterns}

The electron trajectories are bent towards the optical axis by the
electric field resulting in an interference pattern analogous to that
of the double slit, where the slit separation is tuned by varying
the voltage between tip and bundle \cite{hasselbach2010progress,lichte1998gottfried}.
This is shown schematically in Figure \ref{fig:regimes}g. The
angular intensity of the interference pattern can be expressed as:
\begin{equation}
I(\alpha)=4I_{o}\cos^{2}\left(\frac{4\pi a\gamma_{o}U_{f}}{\lambda}\alpha\right),\label{eq:pattern}
\end{equation}
 where $a$ is the tip-bundle distance, $U_{f}$ is the relative bias
between tip and sample, $\lambda$ is the deBroglie wavelength of
the electron and $\gamma_{o}$ is the angular deflection per Volt
characteristic of the bi-prism. The bi-prism can be expressed as a
double slit with an effective slit separation of $D=a\gamma_{o}U_{f}$.
The pitch of the fringe pattern can be changed by tuning the voltage
of the tip relative to the bundle, as is shown in Supplementary Figure \ref{fig:biprism_graph}.
Practically, the tip-sample distance, extraction voltage and current
are related by the Fowler-Nordheim equation \cite{fowler1928electron}.
The tip can only be operated safely in a narrow window for beam current
ranging between 0.03 - 2 nA. This limits the amount we can explore
the dependence of the pattern on these variables. These finer fringe
patterns are only revealed at higher magnifications and are evident
in Figures \ref{fig:leeps_images}, \ref{fig:regimes}, and \ref{fig:coh_angle}.

\begin{suppfigure}
\includegraphics[height=.4\columnwidth]{dvsu.pdf}
\includegraphics[height=.4\columnwidth]{vis}

\caption{The fine fringe pattern is generated by electrons that do not scatter off the nanotube bundle but instead are deflected by the
electric field surrounding it. This results in a bi-prism mode, analogous
to that of the double slit, where the effective slit separation can
be tuned by changing the bias between the tip and the bundle. This
is illustrated in \ref{fig:regimes}(g). As the field increases, so does the separation
of the virtual sources, resulting in a finer fringe pattern. (a):
demonstrates the dependence of the effective source separation on
tip-bundle bias $U_{f}$ and separation $a$. (b)The visibility of the fringe patterns falls off linearly
with tip-sample distance. This fits the conjecture that the reduced
visibility is simply to due obstruction of the coherence electron
wave by the nanotube bundle. }
\label{fig:biprism_graph}

\end{suppfigure}


It also must be noted that the visibility of the fringe patterns falls
off linearly as magnification increases as seen in  Supplementary Figure \ref{fig:biprism_graph}b.
While increased visibility has been previously seen with reduced temperature
of the emitter \cite{cho2004quantitative}, we doubt that the drop
off in visibility is due to a change in the coherence properties of
the source itself. This can be more simply explained by considering
the geometry of the experiment. As the magnification increases, the
tip is brought closer to the nanotube bundle, resulting in an increased
fraction of the beam being blocked by the tube. This results in
a greater fraction of electrons being absorbed or incoherently scattered
by the metallic nanotube bundle. This yields a smaller fraction of
coherent electron irradiation to contribute to the pattern, hence
the reduced visibility. This conjecture fits with the linear fall
off of visibility demonstrated in Supplementary Figure \ref{fig:biprism_graph}b: the solid angle (and therefore the number of electrons)
eclipsed by the bundle would vary linearly with tip-sample distance.
This lack of visibility due to the gross size of the sample also limits
the application of holographic reconstruction to this sample.








\end{document}
