\section{Introduction}

Given a curve $B\subset C^{(2)}$, we define the \textbf{degree} of $B$ as the integer $d$ such that $C_P\cdot B=d$ where $C_P$ denotes the coordinate curve in $C^{(2)}$ with base point $P$. The curves of degree one in $C^{(2)}$ are completely characterized by the two results in \cite[Pg. 310, D-10]{ACGH} and \cite{Ciro}, where it is proven that a curve $B$ of degree one in $C^{(2)}$ different from a coordinate curve is smooth and it exists if and only if there exists a degree two morphism $f:C\rightarrow B$. Moreover, $B=\{f^{-1}(q)\ |\ q\in B\}\subset C^{(2)}$.

In \cite{Chan} a different proof of this result is given. From this proof we remark that considering the curve $B\subset C^{(2)}$ as before, then, the preimage of $B$ by $\pi_C: C\times C\rightarrow C^{(2)}$ is isomorphic to $C$ through the projection onto the first factor.

Let $\tilde{B}$ be an irreducible curve in $C^{(2)}$ different from a coordinate divisor. Let $B$ be its normalization and assume that there is no degree two morphism from $C$ to $B$. Then, since $C_P$ is ample in $C^{(2)}$, from the characterization of degree one curves we deduce that $\tilde{B}\cdot C_P\geq 2$. In this paper we present a characterization of curves with any degree. First of all we need the following definition:

\begin{defi}\label{defred}
We say that a diagram of morphisms of curves
\[
\xymatrix{
D \ar[d]_{(d:1)}\ar[r]^{(e:1)}& B\\
C &
}
\]
\textbf{reduces} if there exist curves $F$ and $H$ such that there exists a diagram
\[
\xymatrix{
D \ar@/_{10pt}/[dd]_{(d:1)}\ar[r]^{(e:1)} \ar@{.>}[d]^{(k:1)}& B \ar@{.>}[d]^{(k:1)}\\
F \ar@{.>}[d] \ar@{.>}[r]_{(e:1)} & H \\
C &
}
\]
with $k>1$, the upper square being a commutative diagram and the left vertical arrows giving a factorization of the original degree $d$ morphism.

When $k=d$ we will say that the diagram \textbf{completes}, and we will obtain a commutative diagram
\[
\xymatrix{
D \ar[d]_{(d:1)}\ar[r]^{(e:1)}& B \ar@{.>}[d]^{(d:1)}\\
C \ar@{.>}[r]_{(e:1)}& H.
}
\]
\end{defi}

Notice that when $d$ is a prime number both definitions are equivalent.

In this paper we prove

\begin{thm}\label{charcurves}
Let $B$ be an irreducible smooth curve such that there are no non-trivial morphisms $B\rightarrow C$. A morphism of degree one from the curve $B$ to the surface $C^{(2)}$ exists, with image $\tilde{B}$ of degree $d$ if, and only if, there exists a smooth irreducible curve $D$ and a diagram
\[
\xymatrix{
D \ar[d]_{(d:1)}\ar[r]^{(2:1)}& B\\
C &
}
\]
which does not reduce.
\end{thm}

If we consider the case $d=1$ we recover the results for degree one.

\vspace{5pt}
We prove the theorem in two steps, giving a separated proof for each implication(see Theorem \ref{const} and Theorem \ref{ext}). First, given a diagram
\[
\xymatrix{
D \ar[d]_{(d:1)}^{g}\ar[r]^{(2:1)}_{f}& B\\
C &
}
\]
which does not reduce we find a curve in $C^{(2)}$ defined by it as the image by $g^{(2)}$ of the immersion of $B$ in $D^{(2)}$ given by $f$. We prove that $g^{(2)}|_{B}$ has degree one, and hence the curve in $C^{(2)}$ has normalization $B$ and the normalization map is precisely $g^{(2)}|_{B}$. Second, given a curve lying in $C^{(2)}$ we find a diagram defined by the curve, its preimage by $\pi_C$ and the projection on one factor of $C\times C$. We compute the degrees of the different maps and prove that this diagram does not reduce.

In a following paper we are going to use this result to study and classify curves of degree two and some of degree three.

\textbf{Notation:} We work over the complex numbers. By curve we mean a complex projective reduced algebraic curve. Let $C$ be a smooth curve of genus $g\geq 2$, we put $C^{(2)}$ for its $2$nd symmetric product. We denote by $\pi_C:C \times C \rightarrow C^{(2)}$ the natural map, and $C_P \subset C^{(2)}$ a coordinate curve with base point $P\in C$. We put $\Delta_C$ for the main diagonal in $C^{(2)}$, and $\Delta_{C \times C}$ denotes the diagonal of the Cartesian product $C\times C$.

\section{Characterization}\label{Charsec}

We begin with a lemma that will simplify the rest of the exposition.

\begin{lemma}\label{digrdiag}
We consider a diagram of morphisms of smooth irreducible curves
\[
\xymatrix{
D \ar[d]_{(d:1)}^{g}\ar[r]_{(2:1)}^{f}& B\\
C &.
}
\]
The image of $B\subset D^{(2)}$ (with the immersion given by the fibers of $f$) by the morphism $g^{(2)}$ is the diagonal $\Delta_C\subset C^{(2)}$ if and only if the morphism $g$ factorizes through the curve $B$ by $f$.
\end{lemma}

\begin{proof}
Let $i$ be the involution on $D$ that defines $f$, that is, the change of sheet. Since $B=\{x+y\ |\ f(x)=f(y)\}=\{x+i(x)\}\subset D^{(2)}$, then $Im(g^{(2)}|_{B})=\{g(x)+g(i(x))\}$. It is contained in the diagonal $\Delta_C$ if and only if $g(x)=g(i(x))$ for all $x\in D$, that is, if and only if $g$ factorizes through $B$ by $f$.
\end{proof}

In the following theorem, given a diagram that does not reduce we deduce the existence of a curve in $C^{(2)}$ naturally attached to it.

\begin{thm}\label{const}
Assume that there exists a diagram of morphisms of smooth irreducible curves
\[
\xymatrix{
D \ar[d]_{(d:1)}^{g}\ar[r]_{(2:1)}^{f}& B\\
C &
}
\]
which does not reduce and such that the morphism $g$ does not factorize through $B$ by $f$. Then, $g^{(2)}$ gives a degree one map $B\rightarrow C^{(2)}$ with reduced image a curve $\tilde{B}$ of degree precisely $d$.

\end{thm}

\begin{proof}
Consider a diagram as above and look at the induced morphism $D^{(2)}\stackrel{g^{(2)}}{\rightarrow}C^{(2)}$. As we have seen in the Introduction, we have an immersion $B\subset D^{(2)}$ as the set of pairs of points in $D$ with the same image by $f$. Then, we consider $D$ inside $D \times D$ as $\pi^{-1}_D(B)\cong D$, that is, ordered pairs of points with the same image by $f$.

Let $\tilde{B}=g^{(2)}(B)_{red}$, the reduced image curve in $C^{(2)}$, and consider the map $B\stackrel{(k:1)}{\rightarrow}\tilde{B}$ induced by $g^{(2)}$. We want to see that $k=1$.

Notice that by Lemma \ref{digrdiag} we can assume that $\tilde{B}$ is not $\Delta_C$. We know that $B\cdot D_P=1$, hence,
\[
1={g^{(2)}}_*(B\cdot D_P)={g^{(2)}}_*(B)\cdot(\frac{1}{d}C_P) \Rightarrow {g^{(2)}}_*(B)\cdot C_P=d.
\]
In addition, since the map $B\stackrel{(k:1)}{\rightarrow} \tilde{B}$ is $g^{(2)}|_B$, we obtain that $d=(k\tilde{B})\cdot C_p$, and thus $\tilde{B}\cdot C_p=\frac{d}{k}$, that is, $k$ divides $d$.

Assume by contradiction that $k>1$.

Let $F$ be the preimage of $\tilde{B}$ by the morphism $\pi_C:C\times C \rightarrow C^{(2)}$. Then $F\rightarrow \tilde{B}$ has degree two and thus we obtain a diagram
\begin{equation}\label{diag}
\xymatrix@M+2pt{
D \times D \ar[r]^{\pi_D} \ar@/_20pt/[ddd]_{g\times g} & D^{(2)} \ar@/^20pt/[ddd]^{g^{(2)}}\\
D \ar@{_{(}->}[u] \ar[d] \ar[r]^{f}_{(2:1)}& B \ar[d]|(0.37){(k:1)} \ar@{_{(}->}[u] \\
F \ar@{_{(}->}[d] \ar[r]^{(2:1)}& \tilde{B}\ar@{_{(}->}[d]\\
C\times C \ar[r]^{\pi_C}& C^{(2)}.
}
\end{equation}
Observe that the exterior arrows form a commutative diagram, and hence, also the interior arrows give a commutative diagram. Thus, the morphism $D \rightarrow F$ has degree $k$. Now, the restriction to $D$ of $g\times g$ followed by the projection onto one factor of $C\times C$ is precisely $g:D\rightarrow C$ by construction. That is, we obtain the diagram
\[
\xymatrix{
D \ar@/_{10pt}/[dd]_{g} \ar[r]^{(2:1)} \ar[d]^{(k:1)}& B \ar[d]^{(k:1)}\\
F \ar[d] \ar[r]_{(2:1)} & \tilde{B} \\
C & .
}
\]
Hence, the original diagram reduces, contradicting our hypothesis.

Consequently, $k=1$ and thus we deduce that the curve $\tilde{B}$ has normalization $B$.

Moreover, looking at diagram (\ref{diag}) we deduce that $D \stackrel{(1:1)}{\rightarrow} F$, that is, the preimage of $\tilde{B}$ by $\pi_C$ has normalization $D$,  and we will denote it by $\tilde{D}$. So we have:
\begin{equation}\label{diagcompl}
\xymatrix@M+2pt{
D \times D \ar[r]^{\pi_D} \ar@/_20pt/[ddd]_{g\times g} & D^{(2)} \ar@/^20pt/[ddd]^{g^{(2)}}\\
D \ar@{-->}@/_20pt/[ddd]_{g} \ar@{_{(}->}[u] \ar[d]|(0.37){\approx} \ar@{-->}[r]^{f}& B \ar[d]|(0.37){\approx} \ar@{_{(}->}[u] \\
\tilde{D} \ar@{_{(}->}[d] \ar[r]& \tilde{B}\ar@{_{(}->}[d]\\
C\times C \ar[r]^{\pi_C} \ar[d]^{pr}& C^{(2)}\\
C&
}
\end{equation}
where the dashed arrows show the original diagram.
\end{proof}

Conversely, we have also a theorem in the opposite direction, from the existence of curves in $C^{(2)}$ we deduce the existence of diagrams which do not reduce.

\begin{thm}\label{ext}
Given an irreducible curve $\tilde{B}$ lying in $C^{(2)}$ with degree $d$, let $B$ be its normalization, and assume that there are no non trivial morphisms $B\rightarrow C$. Then, there exists a smooth irreducible curve $D$ and a diagram
\[
\xymatrix{
D \ar[d]_{(d:1)}\ar[r]^{(2:1)}& B\\
C &
}
\]
which does not reduce.
\end{thm}

\begin{proof}
First of all, we observe that $\tilde{B}$ is not the diagonal in $C^{(2)}$ because we are assuming that there are no morphisms from $B$ to $C$.

Let $\tilde{D}=\pi_C^*(\tilde{B})\in Div(C\times C)$ and $D$ its normalization. We notice that with our hypothesis $\tilde{D}$ is irreducible. Indeed, otherwise, one of its components would have as normalization the curve $B$, because we have a $(2:1)$ morphism from $\tilde{D}$ to $B$, and since $\tilde{D}\subset C\times C$ we would obtain a non trivial morphism from $B$ to $C$ contradicting our hypothesis.

Now, we are going to compute the degree of $\tilde{D}\rightarrow C$, given by the projection onto one factor:
\[
\begin{array}{c}
\tilde{D}\cdot (C\times P + P\times C)={\pi_C}_*({\pi_C}^*(\tilde{B})\cdot {\pi_C}^*(C_P))= \\
\tilde{B}\cdot {\pi_C}_*{\pi_C}^*(C_P)=2\tilde{B}\cdot C_P=2d.
\end{array}
\]
And therefore, since $\tilde{D}$ is symmetric with respect to the involution $(x,y) \rightarrow (y,x)$ by construction, $\tilde{D}\cdot (C\times P)=d$, and so, the degree of the morphism on $C$ is precisely $d$. In this way, we have a diagram \vspace{-10pt}
\[
\begin{array}{ccc}
{\xymatrix{
\tilde{D}\ar[r]^{(2:1)}\ar[d]_{(d:1)}&\tilde{B}\\
C&
}} & {\begin{array}{c} \\ \textrm{and taking their normalizations} \\ \textrm{we obtain a diagram of} \\ \textrm{morphisms of smooth curves} \end{array}}&
{\xymatrix{
D\ar[r]^{(2:1)}\ar[d]_{(d:1)}&B\\
C&. \vspace{-10pt}
}}
\end{array}
\]

We call $f:D \rightarrow B$ the map coming from $\pi_C|_{\tilde{D}}$ and $g:D \rightarrow C$ the map coming from the projection onto one factor of $C\times C$.

Let $\alpha$ be the degree one morphism induced in $B$ by the immersion of $\tilde{B}$ in $C^{(2)}$. Since we have $D \stackrel{(2:1)}{\longrightarrow}B$, as we have seen in the Introduction there exists an immersion of $B$ in $D^{(2)}$ as pairs of points with the same image by this morphism.

Since $D\stackrel{(1:1)}{\longrightarrow} \tilde{D} \subset C\times C$ we can consider that a general point in $D$ is a pair $(x,y)$ with $x,y\in C$. Moreover, since $D\rightarrow B$ is induced by $\pi_C|_{\tilde{D}}$, a general fiber of $D\rightarrow B$ will be two points $(x,y)$ and $(y,x)$. Hence, we can write a general point of $B\subset D^{(2)}$ as $(x,y)+(y,x)$.

Now, we consider the restriction to $B\subset D^{(2)}$ of $g^{(2)}$. By construction this morphism is precisely $\alpha$ and therefore the image is the original $\tilde{B}$. In particular, $g^{(2)}|_B$ is generically of degree one.

We are going to see that the diagram does not reduce by contradiction: Assume that there exist curves $F$ and $H$, and a diagram \vspace{-10pt}
\[
\xymatrix{
D \ar@/_{10pt}/[dd]_{g}\ar[r]^{f} \ar[d]^{h}& B \ar[d]^{r}\\
F \ar[d]_{l} \ar[r]^{s} & H \\
C & \vspace{-10pt}
}
\]
as in Definition \ref{defred}. Then, as we have seen in the Introduction, the fibers of $s$ give a curve isomorphic to $H$ inside $F^{(2)}$. Hence, we have
\[
\xymatrix@M+2pt{
& & \tilde{B}\ar@{_{(}->}[d] \\
B \ar@{^{(}->}[r] \ar[d] \ar@/^12pt/[rru]^{\alpha}&	D^{(2)} \ar[r]^{g^{(2)}} \ar[d]^{h^{(2)}} & C^{(2)}   \\
H	\ar@{^{(}->}[r] \ar@/_80pt/[rruu] & F^{(2)}\ar[ur]_{l^{(2)}} & \\
&&
}
\]

By definition, the image of $B\subset D^{(2)}$ by $h^{(2)}$ is $H\subset F^{(2)}$, that is, the embedding of $H$ in $F^{(2)}$ given by $s$, and we know that $l \circ h=g$ so $l^{(2)}\circ h^{(2)}=g^{(2)}$, hence
\[
{g^{(2)}}|_B:
\xymatrix@C+15pt{
 B \ar[r]^{h^{(2)}=r} \ar@/^20pt/[rr]^{(1:1)} & H \ar[r]^{l^{(2)}}& \tilde{B}
}
\]
thus $r$, as well as $h$, have degree one. Consequently, our diagram does not reduce (see Definition \ref{defred}).
\end{proof}

We observe that we could change the hypothesis of the non existence of morphisms from $B$ to $C$ by assuming that $\tilde{B}$ is not the diagonal and that $\pi^{-1}_C(\tilde{B})$ is irreducible.

Putting these two theorems together we find the characterization of curves in the symmetric square $C^{(2)}$ previously stated in Theorem \ref{charcurves}.

\bibliographystyle{alpha}


\section{Introduction \label{s:intro}}

Hot subdwarf B stars (sdBs) are evolved, core helium-burning objects with only thin hydrogen envelopes and masses around $0.5\,M_{\rm \odot}$
(Heber \cite{heber86}, see Heber \cite{heber09} for a review). To form such objects, the progenitor star has to lose almost its entire hydrogen envelope in the red-giant phase.

About half of the sdB stars are in close binaries with short periods from just a few hours to a few days (Maxted et al. \cite{maxted01}; Napiwotzki et al. \cite{napiwotzki04a}). Because the separation in these systems is much smaller than the size of the red-giant progenitor star, these binaries must have experienced a common-envelope and spiral-in phase (Han et al. \cite{han02,han03}). Although the common-envelope ejection channel is not properly understood (see Ivanova et al. \cite{ivanova13} for a review), it provides a reasonable explanation for the strong mass loss required to form sdB stars. However, for the other half of the known single-lined hot subdwarfs there is no evidence for close stellar companions as no radial velocity variations are found (Classen et al. \cite{classen11}).

Soker (\cite{soker98}) proposed that substellar objects like brown dwarfs (BDs) and planets, which enter the envelope of a red giant, might be able to trigger its ejection. Substellar objects with masses higher than $\simeq10\,M_{\rm J}$ were predicted to survive the common envelope phase and end up in a close orbit around the stellar remnant, while planets with lower masses would evaporate or merge with the stellar core. The stellar remnant is predicted to lose most of its envelope and settle on the extreme horizontal branch (EHB). Such a scenario has also been proposed to explain the formation of single low-mass white dwarfs (Nelemans \& Tauris \cite{nelemans98}).

\begin{figure*}[t!]
\centering
	\resizebox{9.15cm}{!}{\includegraphics{RV_phl457.eps}}
        \resizebox{9.15cm}{!}{\includegraphics{RV_cpdm64481.eps}}

\caption{Radial velocity plotted against orbital phase. The RV data were phase folded with the most likely orbital periods. The residuals are plotted below.}
\label{rv1}
\end{figure*}

The discovery of a brown dwarf ($M_{\rm BD}=0.053\pm0.006\,M_{\rm \odot}$) in close orbit ($0.08\,{\rm d}$) around such a white dwarf supports this scenario and shows that substellar companions can influence late stellar evolution (Maxted et al. \cite{maxted06}). With the discovery of the eclipsing sdB+BD binaries SDSS\,J0820+0008 and SDSSJ\,J1622+4730 in the course of the MUCHFUSS project (Geier et al. \cite{geier11a}), it was shown observationally that substellar companions are also able to form sdBs (Geier et al. \cite{geier11c}; Schaffenroth et al. \cite{schaffenroth14}).

We now have to address the question, how massive the companion must be to survive the CE-phase. It might also be possible to form an sdB and either evaporate the substellar companion or merge it with the red-giant core. Substellar companions of brown dwarf and even planetary mass in wide orbits have been detected around pulsating sdBs (Silvotti et al. \cite{silvotti07}; Lutz et al. \cite{lutz12}) and eclipsing sdB binaries using the timing method (see Zorotovic \& Schreiber \cite{zorotovic13} and references therein for a summary). Those results, although still under debate (see Wittenmyer et al. \cite{wittenmyer}, Horner et al. \cite{horner}), suggest that a high fraction of the sdB stars might be orbited by such objects. Substellar companions in close orbit might therefore be frequent as well.

Here we report the discovery of a reflection effect, but no eclipses, in the light curves of two close sdB binaries. CPD-64$^{\circ}$481 and PHL\,457 have been reported to be close sdB binaries with small RV shifts by Edelmann et al. (\cite{edelmann05}). Furthermore, PHL\,457 has been identified as long-period pulsator of V\,1093\,Her type (Blanchette et al. \cite{blanchette08}). Those two sdBs are among the best studied close sdB binaries. Detailed analyses showed that both are normal sdB binaries with typical atmospheric parameters (CPD-64$^{\circ}$481: $T_{\rm eff}= 27 500\pm 500\,\rm K,\, \log{g}=5.60 \pm 0.05$, Geier et al. \cite{geier10}; PHL\,457: $T_{\rm eff}= 26 500\pm 500\,\rm K,\, \log{g}=5.38\pm 0.05$, Geier et al. \cite{geier13a}).

Geier at al. (\cite{geier10}) constrained the companion mass of CPD-64$^{\circ}$481 to be as high as 0.62 $M_{\rm \odot}$ by measuring the projected rotational velocity of the sdB and assuming synchronised rotation. This assumption is reasonable, as the theoretical synchronisation timescales with stellar mass companions due to tidal interactions for binaries with periods of about 0.3 d are much smaller or comparable to the lifetime of the sdB on the EHB, depending on the theory (see Geier et al. \cite{geier10}).
The inclination angle was predicted to be as small as $7^{\rm \circ}$. Because no traces of the companion are seen in the spectrum, they concluded that the companion must be a WD, as main sequence stars would be visible in the optical spectra, if their masses are higher than $\sim0.45\,M_{\rm \odot}$ (Lisker et al. \cite{lisker05}). However, the detection of the reflection effect rules out such a compact companion.

Using the same method we constrained the companion mass of PHL\,457. Although the companion mass assuming synchronisation ($\sim0.26\,M_{\rm \odot}$) would still be consistent with observations, the derived inclination angle of $8^{\rm \circ}$ is very small and therefore unlikely.

Moreover, observational evidence, both from asteroseismic studies (Pablo et al. \cite{pablo11,pablo12}) and spectroscopic measurements (Schaffenroth et al. \cite{schaffenroth14}), indicates that synchronisation is not generally established in sdB binaries with low-mass companions (see also the discussion in Geier et al. \cite{geier10}). Therefore, the rotation of the sdBs in CPD-64$^{\circ}$481 and PHL\,457 is most likely not synchronised with their orbital motion and the method described in Geier et al. (\cite{geier10}) not applicable.

\section{Time-resolved spectroscopy and orbital parameters}\label{s:data}

In total, 45 spectra were taken with the FEROS spectrograph ($R\simeq48000,\lambda=3800-9200\,{\rm \AA}$) mounted at the ESO/MPG-2.2m telescope for studies of sdB stars at high resolution (Edelmann et al. \cite{edelmann05}; Geier et al. \cite{geier10}; Classen et al. \cite{classen11}). The spectra were reduced with the FEROS pipeline available in the \texttt{MIDAS} package. The FEROS pipeline, moreover, performs the barycentric correction.

To measure the RVs with high accuracy, we chose a set of sharp, unblended metal lines situated between $3600\,{\rm \AA}$ and $6600\,{\rm \AA}$. Accurate rest wavelengths were taken from the NIST database. Gaussian and Lorentzian profiles were fitted using the SPAS (Hirsch \cite{hirsch}) and FITSB2 routines (Napiwotzki et al. \cite{napiwotzki04b}, for details see Classen et al. \cite{classen11}). To check the wavelength calibration for systematic errors we used telluric features as well as night-sky emission lines. The FEROS instrument turned out to be very stable. Usually corrections of less than $0.5\,{\rm km\,s^{-1}}$ had to be applied. The RVs and formal $1\sigma$-errors are given in Appendix~\ref{app:RV}.

The orbital parameters and associated false-alarm probabilities are determined as described in Geier et al. (\cite{geier11c}). In order to estimate the significance of the orbital solutions and the contributions of systematic effects to the error budget, we normalised the $\chi^{2}$ of the most probable solution by adding systematic errors $e_{\rm norm}$ in quadrature until the reduced $\chi^{2}$ reached $\simeq1.0$. The hypothesis that both orbital periods are correct can be accepted with a high degree of confidence. The phased RV curves for the best solutions are of excellent quality (see Fig.~\ref{rv1}). The derived orbital parameters are given in Table~\ref{tab:orbits} and the orbital solution for CPD-64$^{\circ}$481 is perfectly consistent with the result presented in Edelmann et al. (\cite{edelmann05}).

\begin{table*}[t!]
\caption{Derived orbital solutions, mass functions and minimum companion masses}
\label{tab:orbits}
\centering
\begin{tabular}{lllrrlllll}
Object & $T_{0}^a$ & $\rm P^a$ &  $\gamma^a$ & $\rm K^a$ & $e_{\rm norm}$ & $f(M)$ & $M_{\rm 2}^b$ & $i_{\rm max}^c$\\
 & [$-$2\,450\,000] & [d] & [${\rm km\,s^{-1}}$] & [${\rm km\,s^{-1}}$] & [${\rm km\,s^{-1}}$] & [$M_{\rm \odot}$] & [$M_{\rm \odot}$] &$^\circ$\\
\hline
\\[-3mm]
CPD-64$^{\circ}$481 & $3431.5796\pm0.0002$ & $0.27726315\pm0.00000008$ & $93.54\pm0.06$ & $23.81\pm0.08$ & $0.16$ & $0.0004$ & $>0.048$ & 70\\
PHL\,457 & $5501.5961\pm0.0009$ & $0.3131\pm0.0002$   & $20.7\pm0.2$ & $13.0\pm0.2$ & $0.7$ & $0.00007$ & $>0.027$ & 75 \\
\hline \\[-3mm]
\end{tabular}
\begin{list}{}{}
\item[$^{\mathrm{a}}$] The systematic error adopted to normalise the reduced $\chi^{2}$ ($e_{\rm norm}$) is given for each case.
\item[$^{\mathrm{b}}$] The minimum companion masses take into account the highest possible inclination.
\item[$^{\mathrm{c}}$] $i=90$ is defined as an edge-on orbit
\end{list}

\end{table*}

\section{Photometry}

Time-resolved differential photometry in BVR-filters for CPD-64$^{\circ}$481 and VR filters for PHL\,457 was obtained with the SAAO STE4 CCD
on the 1.0m telescope at the Sutherland site of the South African Astronomical Observatory (SAAO). Photometric reductions
were performed using an automated version of DOPHOT (Schechter et al. \cite{schechter93}).

The differential light curves have been phased to the orbital periods derived from the RV-curves and
binned to achieve higher S/N. The light curves show sinusoidal variations ($\sim10\,{\rm mmag}$) with orbital phase characteristic for
a reflection effect (Fig.~\ref{lc}). It originates from the irradiation of a cool companion by the hot subdwarf primary.
The projected area of the companion's heated hemisphere changes while it orbits the primary. Compared to other reflection effect binaries the amplitude of the reflection effect in both systems is quite small. The amplitude of the reflection effect depends mostly on the separation of the system, the effective temperature of the subdwarf, and the visible irradiated area of the companion. Seen edge-on, the relative change of this area is the highest. However, for small inclinations the derived mass of the companion becomes higher and because there is a strong correlation between mass and radius on the lower main sequence (see Fig. \ref{m-r_cpd}), the radius of the companion and the absolute irradiated area becomes larger as well. Due to this degeneracy it is therefore not straight forward to claim that small reflection effects can simply be explained by small inclination angles.

We fitted models calculated with MORO, which is based on the Wilson-Devinney code (MOdified ROche model, Drechsel et al. \cite{drechsel95}), to the light curves as described in Schaffenroth et al. (\cite{schaffenroth13}). As no eclipses are present, the inclination is difficult to determine and we fitted light curve solutions for different fixed inclinations. The mass ratio, which can be calculated from the mass function for different inclinations, was also kept fixed, so that the mass of the sdB is equal to the canonical sdB mass $M_{\rm sdB}=0.47\,{\rm M_{\odot}}$ (see Fontaine et al. \cite{fontaine12} and references therein). Shape and amplitude of the variation mostly depends on the orbital inclination and the mass ratio, but also on the radius ratio of both components and the unknown albedo of the companion. Due to this high number of parameters, that are not independent from each other, we cannot find a unique solution.

Selecting only solutions for which the photometric radius is consistent with the spectroscopic radius derived from the surface gravity, we narrow down the number of solutions. Unfortunately, due to the degeneracy between the binary inclination and the radius of the companion we find equally good solutions for each inclination (see also \O stensen et al. \cite{oestensen13}). In the case of CPD-64$^{\circ}$481, see Fig.~\ref{m-r_cpd}, the derived mass and radius of the companion are in agreement with theoretical relations by Chabrier \& Baraffe (\cite{chabrier97}) for the whole range of inclinations. In the case of PHL\,457 the theoretical mass-radius relation is only consistent for an inclination of 50-70$^\circ$. For lower inclinations the measured radius would be larger than expected by the models. However, due to the many assumptions used in the analysis, it is difficult to estimate the significance of this result, as a smaller mass for the sdB could solve this issue.

In Fig. \ref{lc} we show model light curves for high and low inclination. Although small differences are present, a much better quality light curve is needed to resolve them. The sum of the deviations between the measurements and the models are somewhat, but not significantly, smaller for low inclinations. Therefore, we cannot draw firm conclusions from our photometric data at hand.

\begin{figure*}[t!]
\centering
	\resizebox{9.15cm}{!}{\includegraphics[angle=-90]{cpd_neu.eps}}
        \resizebox{9.15cm}{!}{\includegraphics[angle=-90]{phl.eps}}
	\caption{Phased and binned light curves in B-, V- and R-bands in the case of CPD-64$^{\circ}$481 (left panel) and V- and R-bands in the case of PHL\,457 (right panel).  Overplotted are two models for an inclination of 10$^\circ$ (solid) and 65$^\circ$ (dashed) for CPD-64$^{\circ}$481 and 10$^\circ$ (solid) and 70$^\circ$(dashed) for PHL\,457. The lightcurve models with higher inclinations (dashed) have broader minima and shallower maxima.}
	\label{lc}

\end{figure*}

\begin{figure*}[h!]
\centering
	\resizebox{9.15cm}{!}{\includegraphics{cpd_m-r}}
	\resizebox{9.15cm}{!}{\includegraphics{phl_m-r}}
		\caption{Mass-radius relation of the companion of CPD-64$^{\circ}$481 (left panel) and PHL\,457 (right panel) for different inclinations (filled rectangles), compared to theoretical relations by Chabrier \& Baraffe (\cite{chabrier97}) for an age of the system of 1Gyr (short dashed line) , 5 Gyr (long dashed line) and 10 Gyr (solid line). For all shown solutions mass and radius of the sdB are also consistent with the spectroscopic surface gravity measurement of $\log{g}=5.60 \pm 0.05$ for CPD-64$^{\circ}$481 (Geier et al. \cite{geier10}) and $\log{g}=5.38\pm 0.05$ (Geier et al. \cite{geier13a}) for PHL\,457.}
		\label{m-r_cpd}

\end{figure*}

\section{Brown dwarf nature of the unseen companions}

The two binaries are single-lined and their binary mass functions $f_{\rm m} = M_{\rm comp}^3 \sin^3i/(M_{\rm comp} + M_{\rm sdB})^2 = P K^3/2 \pi G$ can be determined.
The RV semi-amplitude and the orbital period can be derived from the RV curve, but the sdB mass $M_{\rm sdB}$, the companion mass $M_{\rm comp}$ and the inclination angle $i$ remain free parameters. Adopting the canonical sdB mass $M_{\rm sdB}=0.47\,{\rm M_{\odot}}$ and the $i_{\rm max}$, that can be constrained, because no eclipses are present in the lightcurve, we derive lower limits for the companion masses (see Table\,\ref{tab:orbits}).

Those minimum masses of $0.048\,M_{\rm \odot}$ for CPD-64$^{\circ}$481 and $0.027\,M_{\rm \odot}$ for PHL\,457 - the smallest minimum companion mass measured in any sdB binary so far - are significantly below the hydrogen-burning limit ($\sim0.07-0.08\,M_{\rm \odot}$, Chabrier et al. \cite{chabrier00}). As no features of the companion are found in the spectrum, we also derive an upper mass limit of $\sim0.45\,M_{\rm \odot}$ (Lisker et al. \cite{lisker05})

The initial sample of Edelmann et al. (\cite{edelmann05}) consisted of known, bright hot subdwarf stars. Because no additional selection criteria were applied, it can be assumed that the inclination angles of the binaries found in this survey are randomly distributed. Due to the projection effect it is much more likely to find binary systems at high rather than low  inclinations. The probability, that a binary has an inclination higher than a certain angle, can be calculated as described in Gray (\cite{gray92}), $P_{i>i_0}=1-(1-\cos i_0)$ . Since the companion mass scales with the inclination angle, we can derive the probability that the mass of the companion is smaller than the hydrogen-burning limit of $\sim0.08\,M_{\rm \odot}$, which separates stars from brown dwarfs.

For CPD-64$^{\circ}$481, the inclination angle must be higher than $38^{\rm \circ}$, which translates into a probability of $79\%$. In the case of PHL\,457, an inclination higher than $21^{\rm \circ}$ is required and the probability for the companion to be a brown dwarf is as high as $94\%$.
We therefore conclude that the cool companions in those two binary systems are likely brown dwarfs.

The only chance to constrain the inclination better would be very high signal-to-noise lightcurves.
Moreover, high resolution, high S/N spectra could help to constrain the mass ratio of the system. They could allow to discover emission lines from the irradiated hemisphere of the companion, as done for the sdOB+dM system AA Dor (Vu{\v c}kovi\'c et al. 2008). The strength of the emission lines should be independent of the inclination, depending only on the size of the companion, the separation of the system and the effective temperature of the primary. As the systems are very bright, it might be possible to find these emission lines despite the larger separation and lower effective temperature of our systems.

\begin{figure}[t!]
\centering
	\resizebox{9.15cm}{!}{\includegraphics{periodK_hwvir.eps}}
        \caption{The RV semiamplitudes of all known sdB binaries with reflection effects and spectroscopic solutions plotted against their orbital periods (Kupfer et al. \cite{kupfer13}). Diamonds mark eclipsing sdB binaries of HW\,Vir type where the companion mass is well constrained, triangles systems without eclipses, where only lower limit can be derived for the companion masses. Squares mark CPD-64$^{\circ}$481, PHL\,457, KBS\,13 and BPS\,CS\,22169$-$0001. Open symbols mark systems that have been discovered based on photometry, filled symbols have been discovered based on spectroscopy. The dashed lines mark the regions to the right where the minimum companion masses derived from the binary mass function (assuming $0.47\,M_{\rm \odot}$ for the sdBs) exceed $0.01\,M_{\rm \odot}$ (lower curve) and $0.08\,M_{\rm \odot}$ (upper curve).}
        \label{periodK}

\end{figure}
\section{Discussion}

Figure~\ref{periodK} gives an overview of the 29 sdB binaries with reflection effect and known orbital parameters (Kupfer et al. \cite{kupfer13}). While most companions are late M-dwarfs with masses close to $\sim0.1\,M_{\rm \odot}$, there is no sharp drop below the hydrogen-burning limit. The fraction of close substellar companions is substantial. An obvious feature in Fig.~\ref{periodK} is the lack of binaries with periods shorter than $\sim0.2\,{\rm d}$ and $K<50\,{\rm km\,s^{-1}}$ corresponding to companion masses of less than $\sim0.06\,M_{\rm \odot}$.

This feature could not be due to selection effects. About half of the known reflection effect binaries have been found based on RV-shifts detected in time-resolved spectra. As has been shown in this work, RV-semiamplitudes of a few tens of ${\rm km\,s^{-1}}$ are easily measurable. Furthermore, short-period binaries are found and solved easier than long-period systems.

The other half of the sample has been discovered based on variations in their light curves. Shape and amplitude of the light curves depend mostly on the radius of the companion for similar orbital periods and separations. Since the radii of late M-dwarfs, brown dwarfs and also Jupiter-size planets are very similar ($\sim0.1\,R_{\rm \odot}$), their light curves are expected to be very similar as well.

The most likely reason for this gap is the merger or evaporation of low-mass companions either before or after the CE-ejection corresponding to a population of single sdB stars. Other recent discoveries are perfectly consistent with this scenario. Charpinet et al. (\cite{charpinet11}) reported the discovery of two Earth-sized bodies orbiting a single pulsating sdB within a few hours. These might be the remnants of a more massive companion evaporated in the CE-phase (Bear \& Soker \cite{bear12}). Geier et al. (\cite{geier11b,geier13b}) found two fast rotating single sdBs, which might have formed in a CE-merger. Those discoveries provide further evidence that substellar companions play an important role in the formation of close binary and likely also single sdB stars.

We therefore conclude that the lack of short period systems with small RV variations $\rm K<50\,{\rm km\,s^{-1}}$ is real. However, the probability that substellar companions are present in systems with longer periods ($>0.2\,{\rm d}$) is quite high. In addition to the two binaries discussed here, two more systems with similar orbital parameters and reflection effects have been found (KBS\,13, For et al. \cite{for}; BPS\,CS\,22169$-$0001, Geier et al. \cite{geier12}). Following the line of arguments outlined above, we calculate the probability for those two systems to host a stellar companion to be $9\%$ for BPS\,CS\,22169$-$0001 and $20\%$ for KBS\,13. Multiplying those numbers  for all four binaries, we conclude that the probability that none of them has a substellar companions is less than $0.02\%$.

\begin{acknowledgements}
Based on observations at the La Silla Observatory of the European Southern Observatory for programmes number 073.D-0495(A), 074.B-455(A) and 086.D-0714(A). This paper uses observations made at the South African Astronomical Observatory (SAAO). V.S. acknowledges funding by the Deutsches Zentrum f\"ur Luft- und Raumfahrt (grant 50 OR 1110) and by the Erika-Giehrl-Stiftung.

\end{acknowledgements}


\section{Introduction }

Circular scanning is commonly used in high-resolution timing devices
such as streak cameras and the first experiments in this technique
were started in the 1950's. Much effort has gone into the development
of deflection systems, which would give a reasonable deflection of
a beam of electrons using moderate RF voltages applied to the deflectors.
Different types of deflectors have been proposed, developed and applied.
Rudenberg \cite{key-1} showed that loss in dynamic sensitivity, due
to transit time effects in parallel plate deflectors, could be reduced
by substituting parallel wire deflectors. The analysing system consisted
of two crossed pairs of shorted Lecher-wires, resonant at the frequency
(3000~MHz) of the beam modulating system under investigation \cite{key-2}.
It was used to measure ps electron bunches \cite{key-3}, where the
time dispersion of secondary electron emission process was determined
to be less than 6 ps \cite{key-4}.

However analysing systems based on parallel plates have also been
developed \cite{key-5} and applied \cite{key-6,key-7}. In these
circular-scan systems deflection frequencies up to 300~MHz were employed.
Later, circular-scan streak tubes operating at frequencies of 200~MHz
\cite{key-8} and 320~MHz \cite{key-9} have been developed for use
in a laser ranging systems. By applying a 320~MHz, 13~W RF signal,
which had a phase shift applied between the deflection plates, a 6~mm
circular sweep was obtained \cite{key-9}. A circular sweep speed
(which essentially defines the achievable time resolution of the device)
of $6.44\times10^{8}$~cm/s was achieved, providing a temporal resolution
of 30-35~ps and a range difference jitter of less then 6~ps. With
further development, parallel-plate deflection systems have achieved
even faster circular sweeps. A speed of $2.85\times10^{9}$~cm/sec
was achieved by Sibbet \emph{et al.} by applying 300~MHz, 15~W RF
power to a Photochron IIC streak tube \cite{key-10}. A 30~mm diameter
scan was obtained, enabling an instrumental time resolution better
then 6~ps. $6.25\times10^{9}$~cm/s was achieved by Bryukhnevitch
\emph{et al}. by applying 500~MHz, 17~W RF power to a PV006S streak
tube \cite{key-11}. A 40~mm diameter circular scan and time resolution
of less than 5~ps were achieved.

At higher frequencies other types of deflection systems have been
investigated in the 1970's. A travelling wave RF analysing system
was used by Kalibjian \emph{et al.} \cite{key-12}, giving a time
resolution of about 0.7~ps when a 3.6~GHz RF field was applied.
Alternatively a 10~GHz slit-aperture-resonator deflection system
was employed by Butslov \emph{et al}., giving a time resolution of
0.5~ps \cite{key-13} and a similar RF deflector has been investigated
by the Yerevan group for 3~GHz frequency applications \cite{key-14}.
More recently a cylindrical RF cavity was used to perform circular
sweeps, using a transverse magnetic field in TM110 mode, with circular
polarization and a frequency of 2.856 GHz \cite{key-15}. The time
resolution was about 0.6 ps.

In this article we present a sensitive and compact circular sweep
deflection system for keV electrons, capable of operating in the 500-1000~MHz
frequency range. The theory of the deflector and calculations based
on simulation studies are described in Sec.\ref{sec:The-Ultra-High-Frequency}.
Results of experimental studies of the deflector are presented in
Sec.\ref{sec:Experimental-studies}.

\selectlanguage{english}%

\section{\label{sec:The-Ultra-High-Frequency}\foreignlanguage{british}{The
Circular Scanning Deflector For keV Electrons }}

\selectlanguage{british}%
When an electron beam passes through a parallel-plate deflection system
that has an axial length $\Lambda$, a separation $d$ and a potential
difference $U_{d}$ between the plates (Fig. 1), it is deflected vertically
by an angle $\tan\theta=\dfrac{eU_{d}\varLambda}{mdv_{z}^{2}}$, where
$e,m,v_{z}$ are respectively the charge, mass and velocity of the
electron. The accelerating potential $U_{a}$ of the electron ``gun''
of the system produces an energy $eU_{a}=\dfrac{mv_{z}^{2}}{2}$ and
hence $\tan\theta=\dfrac{U_{d}\varLambda}{2U_{a}d}$. This produces
a deflection $\Delta Y$ at a distance $D$ from the plates:

\begin{equation}
\Delta Y=D\tan\theta=\dfrac{DU_{d}\varLambda}{2U_{a}d}\label{eq:DeltaY}
\end{equation}

The static sensitivity of the $Y$ deflection, i.e. the sensitivity
of the deflector at low frequencies, is defined by $\varepsilon_{s}=\Delta Y/U_{d}$.
The main problem for transverse deflection of keV electrons at high
frequencies is connected to the finite time that electrons take to
travel through the deflector. If the RF field changes according to
$\sin\omega t$, where $\omega=2\pi/T$ and $T$ is the period of
the RF Voltage, the dynamic sensitivity of the parallel plate RF deflector
$\varepsilon_{d}$, can be expressed as\foreignlanguage{english}{
\begin{equation}
\varepsilon_{d}=\varepsilon_{s}\dfrac{\sin\left(\omega T_{0}/2\right)}{\left(\omega T_{0}/2\right)}\label{eq:Eps_d}
\end{equation}
}\\
where $T_{0}=\Lambda/v_{z}$ is the transit time of the electron through
the deflector and $\omega T_{0}$ is the so called transit angle.
The dynamic sensitivity is a function of $\omega,\Lambda,v_{z}$ and
at very high frequencies the transit time of the electrons through
the deflecting field reduces the magnitude of the deflection angle,
i.e. $\varepsilon_{d}<\varepsilon_{s}$ unless the transit time is
smaller than a half-period of the highest frequency to be used (see
Ref.\cite{key-1,key-16}).

\selectlanguage{english}%
\begin{figure}
\begin{center}\includegraphics[width=0.6\columnwidth]{Fig_1}\end{center}

\protect\caption{\label{fig:Parallel-plate}\foreignlanguage{british}{Schematic of
a parallel-plate deflection system.}}
\end{figure}

\selectlanguage{british}%
To avoid transit time effects at high frequencies a dedicated RF deflector
was proposed by Shamaev%
\footnote{The Ph.D. thesis of Y.M. Shamaev is referred to in Ref.\cite{key-17},
but we were unable to access this unpublished document.%
} in the 1960's (the theory is described in Ref.\cite{key-16}). The
Shamaev deflector is based on a pair of helical electrodes (Fig. \ref{fig:Shamaev})
of periodic length $\Lambda$ and separation $d$. Electrons move
in the direction of the Z axis, which coincides with the axis of the
deflection system, with a constant velocity $v_{z}$. An applied sinusoidal
RF Voltage across the deflection plates produces a transverse ($E_{z}=0$)
electric field so that the equation of motion of electrons in this
field can be written as: $\dfrac{dv_{\perp}}{dt}=-\dfrac{eE_{\perp}}{m}$,
$\dfrac{dv_{z}}{dt}=0$, where $v_{\perp}=v_{x}+iv_{y}$ and $E_{\perp}=E_{x}+iE_{y}$
are complex variables.

\selectlanguage{english}%
\begin{figure}
\begin{center}\includegraphics[width=0.7\columnwidth]{Fig_2}\end{center}

\selectlanguage{british}%
\protect\caption{\selectlanguage{english}%
\label{fig:Shamaev}Schematic of the Shamaev helical RF deflector.\selectlanguage{british}%
}
\selectlanguage{english}%
\end{figure}

\selectlanguage{british}%
When the wavelength of the applied RF field $\lambda\gg\Lambda$,
one can to first approximation ignore the variation of the field along
the length of the electrode, in which case $E_{\perp}\thickapprox E\left(t\right)\exp\left(i2\pi z/\Lambda\right)$,
where $E\left(t\right)=E_{d}\sin\left(\omega t+\phi\right)$, $E_{d}=U_{d}/d$
is the amplitude of the electric field in the deflector for an applied
RF voltage of $U_{d}$ and $\phi$ is the RF phase at entry of an
electron into the deflector. The oscillating field can be represented
by a sum of two fields which sweep back and forth, by means of the
Euler formula:

\begin{equation}
\sin\left(\omega t+\phi\right)=\dfrac{1}{2i}\left\{ \exp\left[i\left(\omega t+\phi\right)\right]-\exp\left[-i\left(\omega t+\phi\right)\right]\right\} \label{eq:euler}
\end{equation}
\\
Assuming that electrons with constant velocity $v_{z}$ enter the
RF deflector at $t=0$, the equation of motion in the $X-Y$ plane
is given by:

\begin{equation}
\dfrac{dv_{\perp}}{dt}=\tfrac{eE_{d}}{2im}\left\{ \exp\left[i\left(\omega_{c}-\omega\right)t-i\phi\right]-\exp\left[i\left(\omega_{c}+\omega\right)t+i\phi\right]\right\} \label{eq:motion-XY}
\end{equation}
\\
where $\omega_{c}=\frac{2\pi z}{\Lambda t}=\frac{2\pi v_{z}}{\Lambda}=\frac{2\pi}{T_{c}}$.
The transverse velocity of the electron after the deflector $v_{\perp}$
can be determined by integration of Eq.\ref{eq:motion-XY} over the
range $t=0-\tau$, where $\tau=l/v_{z}$ and $l$ is the length of
the deflector. $T_{c}$ is the transit time over one periodic length
of the deflector ($l=\Lambda$).

{\small{}
\begin{eqnarray}
v_{\perp} & = & \dfrac{eE_{d}}{2m}\left\{ \dfrac{\exp\left[i\left(\omega_{c}+\omega\right)\right]\tau-1}{\omega_{c}+\omega}\exp\left(i\phi\right)-\dfrac{\exp\left[i\left(\omega_{c}-\omega\right)\right]\tau-1}{\omega_{c}-\omega}\exp\left(-i\phi\right)\right\} \label{eq:v-perp}\\
 & = & i\dfrac{eE_{d}\tau}{2m}\left\{ \dfrac{\sin x_{2}}{x_{2}}\exp i\left(x_{2}+\phi\right)-\dfrac{\sin x_{1}}{x_{1}}\exp i\left(x_{1}-\phi\right)\right\} \nonumber
\end{eqnarray}
}\\
where $x_{1}=\frac{\omega_{c}-\omega}{2}\tau=\frac{\omega_{c}-\omega}{2v_{z}}l$
and $x_{2}=\frac{\omega_{c}+\omega}{2}\tau=\frac{\omega_{c}+\omega}{2v_{z}}l$.
The $x,y$ components of the transverse velocity are:

\begin{eqnarray}
v_{x} & = & -\dfrac{eE_{d}\tau}{2m}\left\{ \dfrac{\sin x_{2}}{x_{2}}\sin\left(x_{2}+\phi\right)-\dfrac{\sin x_{1}}{x_{1}}\sin\left(x_{1}-\phi\right)\right\} \label{eq:v-perp-x}\\
v_{y} & = & -\dfrac{eE_{d}\tau}{2m}\left\{ -\dfrac{\sin x_{2}}{x_{2}}\cos\left(x_{2}+\phi\right)+\dfrac{\sin x_{1}}{x_{1}}\cos\left(x_{1}-\phi\right)\right\} \nonumber
\end{eqnarray}

\selectlanguage{english}%
\begin{figure}
\selectlanguage{british}%
\includegraphics[width=1\columnwidth]{Fig_3}

\selectlanguage{english}%
\protect\caption{\selectlanguage{british}%
\label{fig:Transverse-velocity}Transverse velocity components $v_{x}$
vs. $v_{y}$ in units of $v_{0}$.\selectlanguage{english}%
}

\selectlanguage{british}%
\selectlanguage{english}%
\end{figure}

\selectlanguage{british}%
Simulations were carried out to investigate the behaviour of the helical
RF deflector at different values of $k=\omega/\omega_{c}$ and $l=n\Lambda/4$
($n=1,2,3,4)$. Plots of $v_{x}$ vs. $v_{y}$ in units of $v_{0}=eE_{d}\Lambda/8mv_{z}$,
for different values of $k$ and $n$, are presented in Fig.3. These
studies show that the $x-y$ locus is circular only for the $k=1$
and $n=2,4$ cases. In the following we concentrate on the case $k=1,$
$n=4$ where the transit time of electrons through the deflector with
$l=\Lambda$ is equal to the period of the applied high frequency
sinusoidal voltage ($\omega=\omega_{c}$). We refer to this as the
Shamaev resonance condition, where one has maximum transverse velocity
$v_{\perp}=4v_{0}$, and Eq.\ref{eq:v-perp-x}, takes a simple form.
In the case of $k=1,$ $n=4$ then $x_{1}=0,$ $x_{2}=2\pi$, leading
to $\sin x_{1}/x_{1}=1,$ $\sin x_{2}=0$ and thus $v_{x}=-4v_{0}\sin\phi,$
$v_{y}=-4v_{0}\cos\phi$. The deflection angle is given by:

\selectlanguage{english}%
\begin{equation}
\tan\theta=\dfrac{v_{\perp}}{v_{z}}=\dfrac{eE_{d}\tau}{2mv_{z}}\sqrt{\left(\dfrac{\sin x_{1}}{x_{1}}-\dfrac{\sin x_{2}}{x_{2}}\right)^{2}+4\dfrac{\sin x_{1}\sin x_{2}}{x_{1}x_{2}}\sin^{2}\left(\dfrac{\omega\tau}{2}+\phi\right)}\label{eq:defl-angle}
\end{equation}
\\
\foreignlanguage{british}{which for $k=1$ and $n=4$ reduces to }$\tan\theta=\frac{4v_{0}}{v_{z}}=\frac{eE_{d}\Lambda}{2mv_{z}^{2}}=\frac{U_{d}\Lambda}{4dU_{a}}$.
\foreignlanguage{british}{When the Shamaev resonance condition is
applied to a helical deflector, transit-time effects are avoided and
$\varepsilon_{d}=\varepsilon_{s}/2$ (the factor 2 in the denominator
is related to the helical shape of the deflector electrodes). Electron
trajectories can be analysed in two parts: those inside and outside
the deflector. The electron's transverse coordinates at the end of
the RF deflector, $r_{x}=\varint v_{x}dt$ and $r_{y}=\varint v_{y}dt$,
are obtained by integrating over the range $t=0-l/v_{z}$, resulting}
\foreignlanguage{british}{in:}

{\small{}
\begin{eqnarray}
r_{x} & = & \dfrac{eE_{d}l^{2}}{4mv_{z}^{2}}\left\{ \cos\phi\left[\dfrac{1}{x_{1}}\left(1-\dfrac{\sin2x_{1}}{2x_{1}}\right)+\dfrac{1}{x_{2}}\left(1-\dfrac{\sin2x_{2}}{x_{2}}\right)\right]+\sin\phi\left[\dfrac{\sin^{2}x_{1}}{x_{1}^{2}}+\dfrac{\sin^{2}x_{2}}{x_{2}^{2}}\right]\right\} \label{eq:rxy}\\
r_{y} & = & \dfrac{eE_{d}l^{2}}{4mv_{z}^{2}}\left\{ -\sin\phi\left[\dfrac{1}{x_{1}}\left(1-\dfrac{\sin2x_{1}}{2x_{1}}\right)-\dfrac{1}{x_{2}}\left(1-\dfrac{\sin2x_{2}}{x_{2}}\right)\right]+\cos\phi\left[\dfrac{\sin^{2}x_{1}}{x_{1}^{2}}+\dfrac{\sin^{2}x_{2}}{x_{2}^{2}}\right]\right\} \nonumber
\end{eqnarray}
}\\
\foreignlanguage{british}{For $k=1$ and $n=4$ these formulae simplify
to $r_{x}=R_{0}\left[\sin\phi+\cos\phi/2\pi\right]$ and $r_{y}=R_{0}\left[\cos\phi+\sin\phi/2\pi\right]$,
where $R_{0}=\frac{eE_{d}\Lambda^{2}}{4mv_{z}^{2}}=\frac{U_{d}\Lambda^{2}}{8dU_{a}}$.
At the end of the deflector the electrons will scan through an ellipse
of half axes $a=R_{0}\left[1+1/2\pi\right]$ and $b=R_{0}\left[1-1/2\pi\right]$.
If the electrons are detected on a screen at a distance $D$ from
the deflector, the transverse coordinates will be $R_{x}=r_{x}+D\tan\theta_{x}$
and $R_{y}=r_{y}+D\tan\theta_{y}$, where $\tan\theta_{x}=-4v_{0}\sin\phi/v_{z}$
and $\tan\theta_{y}=-4v_{0}\cos\phi/v_{z}$, leading to: }

\selectlanguage{british}%
\begin{eqnarray}
R_{x} & = & r_{x}-D\dfrac{4v_{0}\sin\phi}{v_{z}}\label{eq:Rxy}\\
R_{y} & = & r_{y}-D\dfrac{4v_{0}\cos\phi}{v_{z}}\nonumber
\end{eqnarray}
\\
 For $U_{a}=2.5$~kV and $\Lambda=6\:(3)$~cm, corresponding to
Shamaev resonance conditions at 500 (1000)~MHz, one obtains, for
$U_{d}=20$~V and $d=1$~cm, $R_{0}=0.036$ (0.009)~cm and $\tan\theta=0.012$
(0.006). At distances $D=0,\:12,\:24$~cm, circles of radius 1, 5,
9 (1, 9, 17) $R_{0}$ respectively are produced. In Sec.\ref{sec:Experimental-studies}
it is demonstrated that the deflector forms a resonant circuit with
a Q-factor in excess of 100 at these frequencies. This results in
an observed deflection more than an order of magnitude larger than
has been calculated here.

\selectlanguage{english}%

\section{\label{sec:Experimental-studies}Experimental studies }

\begin{figure}
\selectlanguage{british}%
\begin{center}\includegraphics[width=0.8\columnwidth]{Fig_4}\end{center}

\selectlanguage{english}%
\protect\caption{\selectlanguage{british}%
\label{fig:Exp-schematic}A schematic representation of the experimental
set up: 1 thermionic cathode, 2 electron transparent accelerating
electrode, 3 electrostatic lens, 4 electrodes of the RF deflector,
5 phosphor screen.\selectlanguage{english}%
}
\end{figure}

\selectlanguage{british}%
The experimental set up for investigation of the characteristics of
the RF deflector is presented schematically in Fig.\ref{fig:Exp-schematic}.
It consists of a pumped vacuum vessel with various electrical and
mechanical feed throughs. A thermionic cathode (1) emits electrons,
which are accelerated by a voltage applied between the cathode and
an electron transparent electrode (2). An electrostatic lens (3) then
focuses the accelerated electrons on to the phosphor screen (5) at
the far end of the tube. The RF deflection system, consists of electrodes
(4), a RF source and tuning circuit. The experimental commissioning
followed the procedure: i. mechanical and vacuum testing; ii. electron
beam optics tuning; iii. RF deflector tuning.

\begin{figure}
\includegraphics[width=1\columnwidth]{Fig_5}

\protect\caption{\label{fig:Images-deflected}Images of the deflected 2.5 keV continuous
electron beam at the phosphor screen. The parameters of the deflector
and test set up are: $\Lambda=3$~cm, $d=1$~cm, $U_{d}=20$~V,
$U_{a}=2.5$~kV, $D=12$~cm. Applied RF frequencies are: a) 1000
MHz; b) 750 MHz. The radius of the scanned circle is 2 cm. The length
of the major axis of the ellipse is also around 2~cm.}
\end{figure}

The size of the focused thermionic electron beam spot on the phosphor
screen, mounted 12~cm from the end of the deflector, was less than
1~mm diameter. The RF deflection system consisted of one pair of
helical electrodes (Fig. \ref{fig:Shamaev}), which served as a capacitive
element, containing the applied RF power within the deflector. For
a given set of electrodes we observed that at some frequencies the
deflection angles, or radii of the scanned circles, were about an
order magnitude higher than anticipated from theory. Indeed for $U_{d}=20$~V,
$\omega=1000$~MHz, $\Lambda=3$~cm, $d=1$~cm and $U_{a}=2.5\, k$V,
the radius of the scanning circle at a distance $D=12$~cm was predicted
to be 0.81~mm, while it was measured at 20~mm (Fig. \ref{fig:Images-deflected}a).
For a tuned deflector at $\omega=750$~MHz frequency, electrons scan
on to an ellipse (Fig. \ref{fig:Images-deflected}b). These two cases
correspond to the theoretical predictions presented in Fig.\ref{fig:Transverse-velocity}-IId
(n = 4, k = 1) and Fig.\ref{fig:Transverse-velocity}-Va (n = 4, k
= 0.75). The observed factor $\sim25$ larger deflection than predicted
by Eq.\ref{eq:Rxy} can be explained if the deflection electrodes
form a resonant circuit at these frequencies with a high Q-factor.
The resonant frequency can be fixed at a desired value, i.e. near
the Shamaev resonant frequency, by tuning the capacitance of the deflection
system. This may be realized by suitable design of the electrodes
or, for given electrodes, by tuning of the deflector capacitance using
a coaxial cavity or an additional variable capacitor.

\begin{figure}
\begin{center}\includegraphics[width=0.6\columnwidth]{Fig_6}\end{center}

\protect\caption{\label{fig:equivalent-circuit}An equivalent circuit of the RF deflector
and resonant tuning system. $U_{d}$: applied RF Voltage; $C_{v}$:
variable capacitance; $C_{d}$: deflector capacitance; $R_{d}$ deflector
resistance; $L_{d}$: deflector inductance. }
\end{figure}

The equivalent circuit of the helical deflector and tuning system
is presented in Fig.\ref{fig:equivalent-circuit}. The Q-factor of
the resonant circuit was determined experimentally, by measuring the
diameter of the scanning circle as a function of the applied RF frequency.
An example of the resonant behaviour at frequencies around 500 MHz
for a deflector with parameters $\Lambda=6$~cm, $d=1$~cm, $U_{d}=10$~V,
$U_{a}=2.5$~kV, $D=12$~cm. is presented in Fig.\ref{fig:DiameterQ}.
From this it follows that the Q-factor of the formed resonance is
about 130. Thanks to this resonance, about 1~W (into $50\Omega$)
of RF power is enough to scan 2.5 keV electrons circularly at a radius
of 2 cm. The sensitivity of this new and compact RF deflector is about
1~mm/V or 0.1~rad/$W^{1/2}$ which is an order of magnitude higher
than the sensitivities of other RF deflectors used previously. Potentially
it has a number of applications in fixed-frequency cathode-ray-tube
based instruments, for example opto-electronic devices such as RF
Streak Cameras or the RF Photomultipler Tube \cite{key-18,key-19}.
The latter has been proposed for ps resolution timing measurements
in nuclear physics experiments.

\begin{figure}
\begin{center}\includegraphics[width=0.7\columnwidth]{Fig_7}\end{center}

\protect\caption{\label{fig:DiameterQ}Diameter of the scanning circle as a function
of RF frequency for the 500 MHz helical deflector. The parameters
of the deflector and test set up are: $\Lambda=6$~cm, $d=1$~cm,
$U_{d}=10$~V, $U_{a}=2.5$~kV, $D=12$~cm.}
\end{figure}

\selectlanguage{english}%

\section{\label{sec:Summary}Summary }

\selectlanguage{british}%
This paper describes a RF deflector of helical geometry to perform
circular sweeps of keV energy electrons using high frequency electromagnetic
fields. The sweep can be used convert the temporal dependence of a
pulse of electrons passing through the deflector to a position dependence
on a circle. By employing a position sensitive electron detector very
precise timing information at the ps level may then be obtained. A
major advantage of the helical electrodes is that the system can be
optimised to the velocity of the transiting electrons, so that loss
of deflection sensitivity due to finite transit time effects is avoided.

The theory of operation of the deflector is derived and optimum deflector
dimensions calculated for operational frequencies of 500 and 1000~MHz.
After fine tuning the effective capacitance of the system, the deflection
electrodes form a resonant circuit, with a quality factor Q in excess
of 100. On resonance, the sensitivity of the deflection system is
about 1~mm/V or 0.1~rad/$W^{1/2}$and around 1~W (into $50\Omega$)
of RF power is sufficient to scan 2.5 keV electrons circularly at
a radius of 2 cm radius. This makes the deflector suitable for ultra-high
resolution timing devices such as the Radio Frequency Photomultiplier
Tube \cite{key-18,key-19}, where development continues. Future developments
include a dual helical deflection system to produce a spiral rather
than circular scanning pattern.

\paragraph{Acknowledgements}

This work is supported in part by Grant Number 9-8/AB of the State
Committee of Science, Republic of Armenia and by the UK Science and
Technology Facilities Council (STFC 57071/1, 50727/1).