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the emission of radio frequency ( rf ) radiation from the extended air showers ( eas ) produced by ultra high energy cosmic rays ( uhecr , i.e. , particles with energy @xmath0ev ) impinging on the earth atmosphere has been investigated since the 60s , after askaryan @xcite proposed a coherent mechanism for the production of cherenkov radiation at mhz frequencies ( the so called askaryan effect ) . the first detection of rf radiation from eas was proved in 1965 by jelley et al . @xcite .
the idea of askaryan relies on the excess of negative charges in the shower , since positrons tend to disappear through their annihilation into photons , and further electrons are created via compton scattering by photons .
if the number of electrons and positrons is the same , emission would not occur because the electric field generated by opposite charges is equal ( in module ) and opposite ( in sign ) . upon certain conditions
, the cherenkov yield is not proportional to the number @xmath1 of charged particles in the shower , but to @xmath2 , where @xmath3 is the fractional excess of charge .
the quadratic dependence on @xmath1 ( and therefore on @xmath4 ) makes the askaryan effect attractive as mechanism for the detection of uhecrs .    in the following years
, other mechanisms were proposed , which predict a coherent emission of rf radiation , and invoke the interaction of the shower electrons and positrons with the earth magnetic field .
negative and positive charges are separated by the magnetic field , creating an electric dipole . during the motion of the dipole ,
rf radiation is emitted in coherent way ( geomagnetic effect ) .
alternatively , charged particles are bent because of the lorentz force and emit synchrotron radiation ( geosynchrotron effect ) .
the bunch of particles in the eas arriving to the detector can be schematically depicted as a dish , with thickness @xmath5 of a few meters and diameter depending of the energy of the primary .
as will see in the followings , the requirement that the wavelength @xmath6 is comparable with the dish thickness puts a frequency cutoff above which coherence is lost ( and , consequently , the emission strongly hampered ) : @xmath7  mhz , where @xmath8 is the speed of light .
the coherence condition explains why the experimental investigation is mostly concentrated in the frequency range @xmath9  mhz .
this picture is not complete .
only very recently , it has been shown , with monte carlo simulations @xcite , that geosynchrotron radiation can be coherent above @xmath10100  mhz under certain conditions : since the emission occurs in air , which has a ( altitude dependent ) refractive index @xmath11 , a time compression of rf wavefront is produced , which causes the coherence to extend to higher frequency , even beyond the ghz .
the compression takes place only inside a cone with aperture @xmath12 equal to the cherenkov angle , and angular spread @xmath13 . outside this small angular region ,
coherence is lost and the rf output power falls dramatically .
in this coherence regime , sometimes named 
geomagnetic cherenkov radiation " , the emitted electric field amplitude @xmath14 depends on the frequency and can be modelled , as shown by refs .
, as @xmath15 with @xmath16 300  mhz and @xmath17 500  mhz . here we will not deal with such particular mechanism .
the first experimental proof of the coherent cherenkov emission in laboratory controlled conditions was performed in 2001 by directing picosecond pulses of gev photons against a silica sand target @xcite . successively other tests were conducted using rock salt @xcite or ice @xcite as targets . because of the higher density of those materials , the shower dimensions are different than in air , and askaryan effect can extend beyond ghz frequencies .
the coherent rf emission by charged particles in the presence of a magnetic field has been measured in controlled conditions and successfully checked versus electrodynamics simulations in ref . .
beside coherent emission , a rf incoherent emission mechanism exists , that is , bremsstralhung radiation , the emission of photons during the deflection of electrons and positrons under the coulomb field of the nuclei .
the emission is always present but is weak since it scales linearly with @xmath4 and the cross section is low .    in this paper we do not discuss the various mechanisms of production of rf radiation and their relative importance .
our purpose is to explicitly show how and when coherent mechanisms appear , and to derive a general formula which involves the coherent and the incoherent regimes .
we suppose that the eas is composed by @xmath1 charged particles divided into @xmath18 electrons and @xmath19 positrons ( @xmath20 ) , and indicate with @xmath3 the charge excess : @xmath21 ( @xmath22 ) .
depending of the rf radiation mechanism , the phase of the electromagnetic wave from a single emitter ( electron or positron ) can change randomly with time or can not . for bremsstrahlung , for example ,
the phase changes randomly and therefore the superposition of @xmath1 sources is incoherent and the emitted energy is proportional to @xmath1 . for synchrotron , geomagnetic , and cherenkov emission the phase of a single emitter does not change randomly with time .
the superposition of @xmath1 emitters can be coherent or incoherent depending on the random relative position of the individual sources .    to simplify the calculations
, we suppose that the electric field amplitude is unity and the fields are polarized in the same direction .    for the case of cherenkov radiation , the electron and positron electric fields have opposite sign .
therefore the total electric field @xmath14 ( at the detector ) is @xmath23 where @xmath24 and @xmath25 are the phases of the rf wave emitted by the @xmath26-th electron and positron , respectively . for any @xmath26 , @xmath25 and @xmath24
are independent random variables with the same distribution @xmath27 , that is , @xmath28
. therefore their mean values , for any @xmath26 , are @xmath29 .
the energy @xmath30 is @xmath31 we are interested to the mean value @xmath32 : @xmath33 since @xmath24 is independent of @xmath34 ( @xmath35 ) , @xmath36 = @xmath37 = @xmath38 = @xmath39 .
it is reasonable to assume that the probability density @xmath40 is an even function between @xmath41 and @xmath42 , so @xmath43 . in conclusion
, @xmath44 analogously @xmath45 @xmath46 the energy @xmath47 is therefore @xmath48\mean{\cos\phi}^2\ ] ]    the factor which determines the presence of the coherence is precisely @xmath49 .
it can be easily seen in the case where the phases @xmath50 are uniformly distributed between @xmath41 and @xmath42 : @xmath51 two extreme regimes can be identified :    1 .
@xmath52 : @xmath53 , which is the condition for complete incoherence .
the total energy is @xmath54 .
this is the case , for example , of the optical and uv cherenkov light emission .
@xmath55 : total coherence .
@xmath56 , depends on the excess @xmath3 of negative charges in the shower .
the cherenkov angle . ]    .
[ fg : systemref ]    we treat now the generic case for the cherenkov radiation where the phase @xmath50 is not uniformely distributed .
the results can be easily adapted to the geosynchronous and the geomagnetic radiation . with reference to fig.[fg : systemref ] , we introduce a reference system with versors @xmath57 , where the eas axis coincides the @xmath58 axis .
@xmath59 is a single emitter with cylindrical coordinates @xmath60 , @xmath61 is the cherenkov cone aperture , @xmath62 the angular coordinate along the emission cone .
the detector d is placed on the axis @xmath58 at the distance @xmath63 : @xmath64 .
p is the projection of s on the x - y plane .
then : @xmath65 , @xmath66 .
the wave vector @xmath67 is @xmath68 and the phase @xmath50 of the electric field in the point d is @xmath69\ ] ] the phase @xmath50 depends also on @xmath63 ( fixed value ) , but since in the calculation of the energy only the phase differences enter , the term with @xmath63 always eliminates , so we can put @xmath70 to simplify the computation : @xmath71\ ] ] we schematize the shower as a flat cylinder , with thickness @xmath5 and charge distribution @xmath72 along the shower axis ( @xmath73 ) , and radial charge distribution @xmath74 such as @xmath75 = 1 .
the angular variables @xmath76 and @xmath62 are distributed uniformily between 0 and @xmath77
. then @xmath78\big]\ ] ] the computation is straightforward in the case @xmath79 ( uniform distribution ) : @xmath80 where @xmath81 is the bessel function of the first kind .
calling : @xmath82 and @xmath83 then @xmath84    the same procedure can be applied to the geosynchrotron emission .
the electric field of the synchrotron radiation is @xcite : @xmath85}{r~(1-\vec{\beta}\cdot\hat{n})^3 } \bigg]_{retarded}\ ] ] where @xmath86 is the versor of the direction of observation , @xmath87 is the particle acceleration , and the electron electric charge @xmath88 must be considered with the sign . since positrons and electrons have opposite acceleration under magnetic field , the product @xmath89 is the same for the two particles , so the electric fields sum up .
therefore , for geosynchrotron radiation , the eq.([eq : efield ] ) becomes @xmath90 and the final result is : @xmath91\mean{\cos\phi}^2\ ] ] in case of total coherence , @xmath92 .
( [ eq : wmedio ] ) and ( [ eq : wmedio2 ] ) can be summarized stating that the total power emitted by @xmath1 sources is obtained multiplying the power from a single source by a factor @xmath93 : @xmath94 where @xmath95 represents the spatial coherence factor , function of the radiation frequency @xmath96 and of the shower shape and dimension , and the sign @xmath97 depends on the emission mechanism (  for cherenkov , + for synchrotron and geomagnetic ) .
the factor @xmath98 in eq.([eq : m ] ) determines the coherent or incoherent regime . if @xmath99 , the incoherence is total , and @xmath100 , i.e. the total emitted power is the some of the @xmath1 single powers . on the opposite
, complete coherence occurs as @xmath101 , and @xmath102 for cherenkov emission , @xmath103 for geosynchrotron and geomagnetic emission , i.e. , the power is proportional to the squared number of emitters .
the factorization ( @xmath104 ) has the physical meaning that the longitudinal and radial developments of the showers are independent of each other , which is a reasonable approximation that can be extended to the general case .
@xmath105 ( @xmath106 ) , takes into account the radial emitter distribution @xmath74 , in the plane perpendicular to the shower axis , while @xmath107 ( @xmath108 ) takes into account the distribution along the longitudinal axis .    as a check ,
note that when @xmath109 , @xmath110 and @xmath111 ( total spatial incoherence ) and when @xmath112 , @xmath113 and @xmath114 ( total spatial coherence ) .
as an example of the above calculations , we apply them to the case of eas produced by cosmic rays in the atmosphere .    it is well known that a relativistic particle with velocity @xmath115 such that @xmath116 , where @xmath117 is the refractive index of air , radiates electromagnetic waves at the angle @xmath118 at the energy rate ( in si units ) : @xmath119 since @xmath120 , @xmath121 , and @xmath122 , for an eas with @xmath1 secondary particles the radiated power @xmath123 in the bandwidth @xmath124 centred around the frequency @xmath96 , neglecting coherence effets , is : @xmath125     for the coherent cherenkov effect assuming the nkg radial charge distribution . ]    where @xmath126 is the fine - structure constant and @xmath127 the planck constant .
when coherence occurs , the emitted power is obtained from the above equation substituting the factor @xmath1 with @xmath2 and taking into consideration the amplitude @xmath98 ( eq.([eq : m ] ) ) .
then @xmath128 to calculate @xmath129 from eq.([eq : ar ] ) we must know the radial charge distribution @xmath130 .
some examples can be found in the original work of askaryan @xcite .
here we treat the case of the nishima , kamata , greisen ( nkg ) distribution , which describes adequately the charge distribution in an eas@xcite .
the normalized density function @xmath74 is : @xmath131 where @xmath132 is the factorial gamma function , @xmath133 , and @xmath134 m. @xmath135 has been evaluated numerically and the result shown in fig.[fg : fignkg ] .
the frequency cutoff , where the power is reduced by a factor of 2 , is @xmath136 mhz .    regarding the longitudinal coherence , in the simple case of an uniform charge distribution , we get , as shown before in eq.([eq : az ] ) , a @xmath137 function with the first zero occurring at @xmath138 mhz for @xmath139 m. beyond that frequency , the rf emitted power scales as @xmath140 .
more realistic cases are discussed in ref .  .
we have derived a general formula for the emission of radio frequency from particle showers , which describes also the transition from the incoherent regime to the coherent one .
a coherence factor is introduced , which depends on the spatial charge distribution of the radio frequency emitters .
the formula applies also to dense materials , such as ice , for example , which has been proposed as target medium ( precisely , antartic ice sheet ) the for the detection of very high energy neutrinos @xcite .
extended air showers produced by cosmic rays impinging on the earth atmosphere irradiate radio frequency radiation through different mechanisms . upon certain conditions , 
 the emission has a coherent nature , with the consequence that the emitted power is not proportional to the energy of the primary cosmic rays , but to the energy squared . 
 the effect was predicted in 1962 by askaryan and it is nowadays experimentally well established and exploited for the detection of ultra high energy cosmic rays .    in this paper 
 we discuss in details the conditions for coherence , which in literature have been too often taken for granted , and calculate them analytically , finding a formulation which comprehends both the coherent and the incoherent emissions . 
 we apply the result to the cherenkov effect , obtaining the same conclusions derived by askaryan , and to the geosynchrotron radiation .    
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