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handling high current proton beams is a challenge for beam diagnostics in future proton accelerators . in order to achieve a high beam power ,
it is mandatory to understand the beam dynamics and to determine the beam size and profile with precision . in high power proton linacs
, such understanding would allow a full characterization of the beam after each linac section . with a beam transverse size of 2 to ,
intercepting profile diagnostics such as otr screens or cherenkov monitors can be used only during tuning and specific beam operation modes . to determine the longitudinal profile , feschenko - type bunch shape monitors ( bsm @xcite )
can be used in the low energy sections of the linac , however , bsm may not be sensitive to short bunch lengths . in the higher energy sections ,
it is highly advisable to use non interceptive devices to avoid radiations or damage to the equipment .
wall current monitors or other methods based on detecting the fields at the vacuum chamber boundary could be used , but they are intrinsically limited in resolution due to the rather low relativistic @xmath0 @xcite and do not have an adapted temporal resolution .
therefore , a diagnostic based on smith - purcell radiation could be of high interest .
we will investigate here the possibility of using a smith - purcell diagnostic for the measurement of proton beams longitudinal profiles , using a numerical simulation code presented hereafter .
smith - purcell radiation can be described by the `` surface current '' theory @xcite : when charged particles travel over a grating , they induce an image charge on its surface and keep pace with it .
the corrugations of the grating cause the created current to accelerate , which in turn leads to the emission of radiation . in our coordinate system
, we assume that the protons travel perpendicularly to the grooves of a blazed grating .
due to the periodic structure of the grating , the emitted wavelength @xmath1 depends on the observation angle @xmath2 defined in fig .
[ fig : coordinates ] .
the grating shape and pitch defines its efficiency , that varies depending on @xmath2 and on the second observation angle called @xmath3 .
our simulation describes the smith - purcell intensity radiated per unit solid angle for a single particle , as described in the equation ( 2a ) of @xcite for a grating of period _
l _ and length z positioned at a distance @xmath4 from the beam center :    @xmath5\ ] ]    the term @xmath6 represents the grating efficiency that contains the contribution from each period of the grating , and is different for each grating shape @xcite . here
we use echelette gratings whose efficiency will be shown in the results section .    for a given bunch of @xmath7 particles ,
the emitted radiation has a high degree of coherence when the radiation wavelength is comparable to the bunch length . in the equation (
[ equ : coherence ] ) , @xmath8 and @xmath9 represent the coherent and incoherent components of the emitted radiation respectively . assuming that the transverse and longitudinal profiles are not correlated , @xmath8 can be expressed as the product two functions _
t _ and _ g _ encoding the longitudinal ( temporal ) and transverse profile respectively . @xmath10 and @xmath11 are the beam dimensions in x and y directions , and @xmath12 is the frequency .
@xmath13    @xmath14    in the case where the coherent component prevails , the emitted radiation encodes the form factor of the bunch and its measurement allows to recover information about the bunch longitudinal profile .
if we assume that the beam is gaussian in the transverse plane and _ g _ is normalized to 1 , then the emitted energy is proportional to the form factor as shown in [ equ : approxcoherence ] .
@xmath15    this form factor corresponds to the modulus of the fourier transform of the temporal profile . in order to recover the full profile the phase needs to be reconstructed .
several techniques are possible , and are documented in @xcite .    in this work
, we use a code package written by g. doucas @xcite , that takes several beam parameters as an input and generates the a smith - purcell spectrum , which is a calculation made from the fourier transform of the temporal profile and chosen grating parameters .
the parameters used for this study on proton beams are summarized in table [ tab : beamparameters ] .
we assume that the beam transverse dimensions would allow approaching a grating at a distance @xmath16 from the beam center .
.used beam parameters for the simulation [ cols="<,^,^",options="header " , ]     [ tab : gratingparameters ]
knowing the expected signal level , it is now necessary to verify that the device is sensitive to a longitudinal profile change , which is a crucial feature for a profile measurement tool .
the sensitivity to a variation in the bunch fwhm length is shown in fig . [
fig : bunchlength ] for the chosen grating and optics parameters .
the coherent emission becomes progressively predominant as the bunch length decreases , which leads to a significant and measurable signal increase .
this is shown in fig .
[ fig : bunchlength ] , where the wavelength at @xmath2 = 90varies by more than 20% for a bunch length varying from 3 to , but a similar change is also observed at other wavelengths showing a significant change of the power spectrum distribution .        the signal level for the simulated case , as shown in fig . [ fig : bunchlength ] , is of the order of per bunch , and needs to be compared with the performances of the detectors used . for example ,
if the detectors measure an averaged signal over several ms , which is the case of the pyroelectric sensors often used in experiments involving electrons @xcite , they will detect the smith - purcell power produced by a whole train ( around bunches ) , thus a higher signal by several orders of magnitude , up to mj .
however , these values do not take into account the transmission of the different optical elements that would need to be installed in front of the detectors .
also , as seen in fig .
[ fig : theta ] , the expected smith - purcell emission would occur in the millimeter - wave range where the background level needs to be evaluated to be able to distinguish it from the actual signal .
if we use detectors that are sensitive only to signal variations , they will not impacted by the effect of the temperature . however
, electromagnetic noise and other sources of radiations related to the beam can create background .
since the smith - purcell radiation is linearly polarized , a way to decorrelate the signal and the background could then be to measure the two polarization components .
angle , for different @xmath2 values .
the emission source is the center of the figure , the detection optics would be placed on the right half of the figure.,width=340 ]    the efficiency of the grating is given in fig .
[ fig : phi ] depending on the @xmath3 angle .
this indicates that most of the radiation would be emitted along the x axis in a cone of approximately 40opening , allowing all signal to enter the optics .
our simulations indicate that with the parameters used for our simulation a longitudinal bunch profile monitor based on coherent smith - purcell radiation installed at near a high intensity proton beam would be sensitive to changes in the pulse length and could be useful in the tuning phase of such accelerator .
however such device has never been tested at low @xmath0 proton beams and preliminary tests on a high intensity proton source would be useful .
the authors acknowledge george doucas from john adams institute in oxford for his useful advice .
the use of smith - purcell radiation to measure electrons longitudinal profiles has been demonstrated at several facilities in the picosecond and sub - picosecond range . there is a strong interest for the development of non intercepting longitudinal profile diagnostics for high current proton beams . 
 we present here results of simulations on the expected yield of longitudinal profile monitors using smith - purcell radiation for such proton beams .