2.1.1 Linear conversion

The refractive indices of the natural waves become almost equal in the region of quasi-longitudinal propagation with respect to the ambient magnetic field. In general, ray propagation in the open field line tube is quasi-transverse, ${\it\theta}\gg 1/{\it\gamma}$ (where ${\it\theta}$ is the wave vector tilt to the magnetic field, ${\it\gamma}$ the plasma Lorentz factor). However, due to refraction of radio waves in the pulsar plasma, in the inner magnetosphere the ray trajectory may contain a short segment almost parallel to the magnetic field. It is the place where the natural waves suffer linear conversion (Petrova Reference Petrova2001). The original ordinary waves become an incoherent mixture of the ordinary and extraordinary ones, which further on propagate independently. The total intensity is conserved, and the mode intensity ratio is determined by the characteristics of the plasma in the region of conversion. The anticorrelation of polarization mode intensities found in PSR B0329+54 (Edwards Reference Edwards2004) strongly suggests that it is the linear conversion that underlies the phenomenon of orthogonally polarized modes in pulsar emission. Being interpreted in terms of linear conversion, the mode intensities and their longitudinal and pulse-to-pulse fluctuations can be regarded as a probe of the pulsar plasma distribution and its temporal variations.

2.1.2 Polarization-limiting effect

As the open field line tube widens with distance from the neutron star, the plasma number density decreases and the refractive index of the ordinary waves tends to unity. Correspondingly, in the outer magnetosphere, the geometrical optics approximation is again broken and the waves are subject to the polarization-limiting effect (Cheng & Ruderman Reference Cheng and Ruderman1979; Barnard Reference Barnard1986; Lyubarskii & Petrova Reference Lyubarskii and Petrova1998a ; Petrova & Lyubarskii Reference Petrova and Lyubarskii2000). In the polarization-limiting region, the electric vector of the waves has no time to follow the ambient magnetic field direction and wave mode coupling starts. As a result, each of the incident natural waves becomes a coherent mixture of the two natural modes of the ambient plasma, acquiring some circular polarization and position angle shift. As the two types of waves evolve identically, the outgoing radiation presents a superposition of the two purely orthogonal elliptically polarized modes. This is in line with the empirical model of pulsar polarization (e.g. McKinnon & Stinebring Reference McKinnon and Stinebring2000).

2.1.3 Diagnostics of pulsar plasma

The ellipticity and position angle shift resulting from the wave mode coupling are related to each other, both being determined by the plasma characteristics in the polarization-limiting region. Based on this relation, a technique of number density diagnostics of a pulsar plasma was developed (Petrova Reference Petrova2003a ). It allows us to obtain the plasma number density distribution in the open field line tube proceeding from the observed polarization radio profile of a pulsar.

At a fixed altitude $r_{0}$ in the tube, the number density $N_{0}$ changes as function of the polar angle ${\it\chi}$ . Besides that, because of continuity, the plasma flow widens with altitude together with the open field line tube in such a way that

(2.1) $$\begin{eqnarray}N\left(r,\frac{{\it\chi}}{{\it\chi}_{f}(r)}\right)=N_{0}\left(\frac{{\it\chi}}{{\it\chi}_{f}(r_{0})}\right)\left(\frac{r}{r_{0}}\right)^{-3},\end{eqnarray}$$

where $N$ is the number density at the altitude $r$ and ${\it\chi}_{f}$ is the polar angle of the tube boundary at the corresponding altitude. Furthermore, it is convenient to normalize $N$ to the Goldreich–Julian number density (Goldreich & Julian Reference Goldreich and Julian1969),

(2.2) $$\begin{eqnarray}{\it\kappa}=\frac{NPce}{B},\end{eqnarray}$$

where $B$ is the magnetic field strength at the same altitude $r$ , $P$ the pulsar period, $c$ the speed of light and $e$ the electron charge. The resultant multiplicity ${\it\kappa}$ is independent of altitude and generally characterizes the number of secondary particles per primary particle (the theories of the polar gap pair creation cascade yield the multiplicity values in the range from less than unity up to ${\sim}10^{5}$ (Hibschman & Arons Reference Hibschman and Arons2001b ; Arendt & Eilek Reference Arendt and Eilek2002; Medin & Lai Reference Medin and Lai2010; Timokhin & Harding Reference Timokhin and Harding2015); in the outer gap scenario, the multiplicities may also be as high as ${\sim}10^{4}{-}10^{5}$ (e.g. Takata, Wang & Cheng Reference Takata, Wang and Cheng2010); in the wind zone, even higher values, ${\sim}10^{6}{-}10^{8}$ , are expected (e.g. Tanaka & Takahara Reference Tanaka and Takahara2013)). Thus, the plasma number density profile presented in terms of ${\it\kappa}({\it\chi}/{\it\chi}_{f})$ is a universal characteristic of the plasma distribution inside the open field line tube of a pulsar. It should be noted that since the radio beam coming into the polarization-limiting region usually occupies a small part of the tube cross-section, only a part of the plasma density distribution can be reconstructed from the observed polarization profile. Making use of the multifrequency polarization data allows to extend the region available for the plasma diagnostics substantially.

Figure 1 illustrates the relevant geometry. Figure 2 exhibits examples of application of the technique to the average polarization radio profiles of several gamma-ray pulsars. The resultant plasma density profiles show an exponential decrease towards the edge of the open field line tube, with the exponent being well fitted by a second-order polynomial. (Such a character of the plasma density distribution is similar to that found for the longer-period pulsars PSR B0355+54 and PSR B0628-28 in Petrova (Reference Petrova2003a ).)

Figure 1. Pulsar geometry; the magnetic moment and the line of sight are inclined to the rotation axis at the angles ${\it\alpha}$ and ${\it\zeta}$ , respectively, ${\it\beta}\equiv {\it\zeta}-{\it\alpha}$ is the impact angle and ${\it\psi}$ the geometrically defined position angle.

Figure 2. Plasma density profiles for the Vela (a) and the millisecond gamma-ray pulsars J1446-4701, J1744-1134 and J1939+2134 (b); ${\it\kappa}$ is the plasma multiplicity defined as the number density normalized to the Goldreich–Julian number density, ${\it\chi}$ the polar angle of a point inside the open field line tube, ${\it\chi}_{f}$ the polar angle of the tube boundary at the corresponding altitude. The circles, squares and triangles in the upper panel mark the points obtained from the polarization data of Johnston, Karastergiou & Willett (Reference Johnston, Karastergiou and Willett2006), Karastergiou & Johnston (Reference Karastergiou and Johnston2006) at 1.4, 3.1 and 8.4 GHz, respectively. The points in the lower panel are obtained from the polarization profiles of Dai et al. (Reference Dai, Hobbs, Manchester, Kerr, Shannon, van Straten, Mata, Bailes, Bhat and Burke-Spolaor2015) at 1.4 GHz; the squares and circles correspond to the main pulse and interpulse data, respectively. The polarization profiles are available via the EPN Database of Pulsar Profiles at http://www.epta.eu.org/epndb. The geometry assumed is as follows: ${\it\alpha}=35.6^{\circ }$ , ${\it\beta}=4^{\circ }$ for the Vela, ${\it\alpha}=60^{\circ }$ , ${\it\beta}=5^{\circ }$ for J1446-4701, ${\it\alpha}=40^{\circ }$ , ${\it\beta}=1.5^{\circ }$ for J1744-1134 and ${\it\alpha}=79^{\circ }$ , ${\it\beta}=4^{\circ }$ for J1939+2134, where ${\it\alpha}$ is the pulsar inclination and ${\it\beta}$ the impact angle.

The class of young energetic gamma-ray pulsars is presented by the Vela pulsar (see figure 2 a). As can be seen, the points resulting from the polarization data at different frequencies well match the same curve, confirming the validity of the technique used. The plasma multiplicity values appear in line with the results of numerical simulations of pair cascades in the polar gap (Hibschman & Arons Reference Hibschman and Arons2001b ; Arendt & Eilek Reference Arendt and Eilek2002; Medin & Lai Reference Medin and Lai2010), though they depend appreciably on the pulsar geometry assumed.

Figure 2(b) shows the plasma density profiles for three gamma-ray millisecond pulsars, PSRs J1446-4701, J1939+2134 and J1744-1134, which are the representatives of the three classes, with the radio pulse components leading, coincident with and lagging the main gamma-ray peak (cf. Abdo et al. Reference Abdo, Ajello, Allafort, Baldini, Ballet, Barbiellini, Baring, Bastieri, Belfiore and Bellazzini2013). As is evident from the figure, the plasma multiplicities for the three pulsars differ substantially, being the highest for the leading and the lowest for the trailing position of the radio pulse. In the former case, the plasma multiplicities are more or less similar to those of the Vela pulsar, where the radio profile also leads the high-energy peak. The latter is in general agreement with an interpretation in terms of a partially screened gap (e.g. Johnson et al. Reference Johnson, Venter, Harding, Guillemot, Smith, Kramer, Çelik, den Hartog, Ferrara and Hou2014), in which case the open field line tube is deficient of pair plasma.

Our result, however, should be taken with caution, since the scaling of the plasma density profiles along both axes noticeably depends upon the poorly known pulsar geometry. To the best of our knowledge, the only constraints on the geometrical parameters of the pulsars considered are obtained from gamma-ray light-curve modelling (e.g. Johnson et al. Reference Johnson, Venter, Harding, Guillemot, Smith, Kramer, Çelik, den Hartog, Ferrara and Hou2014). Keeping in mind these results, we have taken the values which provide ${\it\chi}/{\it\chi}_{f}\leqslant 1$ for the points of the plasma density profile. In fact, the values ${\it\chi}/{\it\chi}_{f}>1$ may be quite realistic, since for the millisecond pulsars the assumption of a purely dipolar magnetic field structure involved in our technique can hardly be regarded as appropriate. Note that the plasma density profiles appear to lie within ${\it\chi}/{\it\chi}_{f}\leqslant 1$ for quite special geometries, whereas over a broad range of geometrical parameters ${\it\chi}/{\it\chi}_{f}$ may be a few times larger, in which case the multiplicities are typically much higher as well.

The only advantage of our choice of the pulsar geometry and of our approach on the whole is definiteness. At present, the force-free model (e.g. Spitkovsky Reference Spitkovsky2006; Philippov, Spitkovsky & Cerutti Reference Philippov, Spitkovsky and Cerutti2015), its dissipative versions (e.g. Kalapotharakos et al. Reference Kalapotharakos, Kazanas, Harding and Contopoulos2012; Li, Spitkovsky & Tchekhovskoy Reference Li, Spitkovsky and Tchekhovskoy2012) and MHD extension (e.g. Tchekhovskoy, Spitkovsky & Li Reference Tchekhovskoy, Spitkovsky and Li2013) provide a more realistic representation of the pulsar magnetosphere. These models, being much more advanced as compared to the classical vacuum dipole model, contain a number of additional ingredients, such as the global electric current, the accelerating electric field, etc. One can expect that incorporating all this into the theory of polarization transfer would complicate it substantially, and the transparent unambiguous relation between the observed quantities, which underlies the above discussed technique of the plasma diagnostics would no longer be the case. It is yet to be understood what the main features are of the polarization evolution of radio waves in the force-free magnetosphere, and what it may imply for the plasma diagnostics. It should also be kept in mind that, since the particle motion in the force-free fields is principally distinct from the motion of a probe particle in fixed fields of the same strength, the polarization transfer in the force-free magnetosphere should be considered anew, e.g. based on the self-consistent two-fluid model (Beskin & Rafikov Reference Beskin and Rafikov2000; Kojima & Oogi Reference Kojima and Oogi2009; Petrova Reference Petrova2015).

One more important point can be noticed in figure 2(b). For the pulsar PSR J1939+2134, the points of the plasma density profile obtained from the main pulse and interpulse polarization data follow exactly the same curve. (Note that the uncertainty of the pulsar geometry discussed above does not affect the validity of this result.) Given that the main pulse and interpulse originate on opposite sides of the neutron star, such an exact match would seem improbable. This improbable coincidence disappears if the main pulse and interpulse were to come from the same region, as in the bidirectional model of pulsar radio emission (Dyks, Zhang & Gil Reference Dyks, Zhang and Gil2005; Petrova Reference Petrova2008a ), where the two components are emitted from the same pole in the outward and backward directions, respectively (for details on the physical grounds of the model see § 2.2.2 below). General considerations as to the origin of the radio components coincident with and lagging the high-energy peaks are presented in § 3.2. As was pointed out by the referee, the bidirectional model for the gamma-ray emission might be physically realizable in the current sheet inside the light cylinder, in which case the space-like current should be supplied by the counter-streaming particle beams.

The technique of plasma density diagnostics developed in Petrova (Reference Petrova2003a ) and discussed above is based on a simplistic assumption of a cold non-gyrotropic plasma embedded in a superstrong dipolar magnetic field. In this case, the ellipticity and position angle shift of outgoing waves exhibit one-to-one correspondence, both being determined by the plasma characteristics in the polarization-limiting region. The position angle shift resulting from polarization evolution in the gyrotropic plasma is suggested as a potential probe of the global current of a pulsar (Hibschman & Arons Reference Hibschman and Arons2001a ). Further generalizations of the theory of polarization transfer in a pulsar plasma by including realistic energy distributions of the plasma particles as well as finiteness and non-dipolar character of the magnetic field (Petrova Reference Petrova2006a ,Reference Petrova b ; Wang, Lai & Han Reference Wang, Lai and Han2010; Beskin & Philippov Reference Beskin and Philippov2012) lead to a complicated picture of observational consequences. Because of its numerous ingredients, the picture is no longer transparent and can hardly allow us to solve an inverse problem of plasma diagnostics. Nevertheless, the elaborated theory can explain a number of observed polarization peculiarities and seems appropriate for direct modelling of the pulsar polarization profiles.

2.1.4 The role of cyclotron absorption in polarization evolution

In the vicinity of the radio emission region, the magnetic field is so strong that the electron gyrofrequency is much higher than the radio wave frequency in the particle rest frame, ${\it\omega}_{H}\equiv eB/mc\gg {\it\omega}^{\prime }$ . As the magnetic field strength decreases with distance from the neutron star, in the outer magnetosphere the radio wave frequencies meet the cyclotron resonance condition, ${\it\omega}^{\prime }={\it\omega}_{H}$ . As a rule, the cyclotron resonance occurs above the polarization-limiting region. If the two regions lie close to each other, the plasma number density in the resonance region is still high enough to cause further evolution of radio wave polarization (Petrova Reference Petrova2006a ). The cyclotron absorption coefficients of the ordinary and extraordinary natural modes are slightly different: the extraordinary waves are suppressed somewhat stronger. The waves coming into the resonance region are already a coherent mixture of the two natural modes, and since these components are absorbed differently, the outgoing polarization changes. In this case, the evolution of the two orthogonally polarized modes is distinct and they finally become non-orthogonal.

If the waves pass through the resonance before the polarization-limiting region, the extraordinary ones are strongly suppressed, whereas the ordinary ones suffer weaker polarization evolution, with the resultant circular polarization having the opposite sense (Petrova Reference Petrova2006a ).

The pulse-to-pulse variations of the radio pulse polarization can be attributed to the temporal fluctuations of the plasma characteristics. In particular, the fine peculiarities of single-pulse polarization statistics observed in PSR B0329+54 (Edwards & Stappers Reference Edwards and Stappers2004) and PSR B0818-13 (Edwards Reference Edwards2004) can be reproduced by numerical modelling of polarization transfer in the plasma with weak pulse-to-pulse fluctuations of the number density (Petrova Reference Petrova2006b ).

2.2.1 The case of superstrong magnetic field

In the inner magnetosphere, where the number density of the scattering particles is the largest, the induced scattering is expected to be most efficient. In this region, the radio frequency in the particle rest frame is much less than the electron gyrofrequency, ${\it\omega}^{\prime }\equiv {\it\omega}{\it\gamma}(1-{\it\beta}\cos {\it\theta})\ll {\it\omega}_{H}$ (where ${\it\omega}$ and ${\it\theta}$ are the photon frequency and tilt to the ambient magnetic field in the laboratory frame, ${\it\beta}$ and ${\it\gamma}$ are the particle velocity in units of $c$ and Lorentz factor). Then the magnetic field affects the scattering process, changing both the scattering cross-section and the geometry of particle recoil (Canuto, Lodenquai & Ruderman Reference Canuto, Lodenquai and Ruderman1971; Blandford & Scharlemann Reference Blandford and Scharlemann1976). In the regime of superstrong magnetic field, the scattering particles slide along the magnetic field lines and the transverse component of their perturbed motion in the field of an incident wave is suppressed as well. Correspondingly, only the photons with the ordinary polarization are involved in this type of scattering.

As the pulsar radio emission is related to the processes in the ultrarelativistic secondary plasma, at each point the radiation is concentrated into a narrow beam with opening angle ${\sim}1/{\it\gamma}$ . However, beams of substantially differing frequencies may have different orientations as a result of refraction and ray propagation in the rotating magnetosphere. The problem of induced scattering of radio photons between two beams of substantially different frequencies, ${\it\omega}_{1}$ and ${\it\omega}_{2}$ , and orientations with respect to the ambient magnetic field, ${\it\theta}_{1}$ and ${\it\theta}_{2}$ , is considered in Petrova (Reference Petrova2004b ). Of course, in the rest frame of the scattering particles, the beam frequencies are equal, ${\it\omega}_{1}^{\prime }={\it\omega}_{2}^{\prime }$ , and hence, in the laboratory frame they are related by ${\it\omega}_{1}(1-{\it\beta}\cos {\it\theta}_{1})={\it\omega}_{2}(1-{\it\beta}\cos {\it\theta}_{2})$ .

In case of a superstrong magnetic field, the induced scattering between the two beams of intensities $I_{1}$ and $I_{2}$ is shown to result in the net intensity transfer from the lower-frequency beam to the higher-frequency one, with the total intensity of the two beams remaining approximately constant, $I_{1}+I_{2}=\text{const}$ . For the decreasing spectrum of pulsar radio emission, $I\propto {\it\omega}^{-{\it\xi}}$ (where $I$ is the original intensity of the pulsar radiation and ${\it\xi}$ the spectral index) this may imply significant amplification of the higher-frequency intensity. For example, if ${\it\theta}_{1}/{\it\theta}_{2}\approx 3$ , then ${\it\omega}_{2}/{\it\omega}_{1}\approx 10$ and for ${\it\xi}=3$ the original intensity ratio of the beams is $I_{1}/I_{2}=({\it\omega}_{1}/{\it\omega}_{2})^{-{\it\xi}}\sim 10^{3}$ . Correspondingly, in the case of efficient scattering, the higher-frequency intensity may increase by three orders of magnitude. Thus, the induced scattering can account for the observed energetics of giant pulses.

It is evident that the amplification is stronger for pulsars with steeper radio spectra. Steepness of radio spectrum was suggested as a criterion for the search for new giant pulse sources (Petrova Reference Petrova2004b ). The three sources discovered subsequently do meet this criterion (Knight et al. Reference Knight, Bailes, Manchester, Ord and Jacoby2006b ,Reference Knight, Bailes, Manchester and Ord a ).

Propagation origin of giant pulses implies that their intensity statistics are determined by pulse-to-pulse fluctuations in the pulsar plasma. Given that the plasma number density and the original radio intensity are Gaussian random quantities, the distribution of strongly amplified pulses has a power-law form and may fit the intensity distributions of the observed giant pulses (Petrova Reference Petrova2004b ).

Induced scattering inside the narrow beam leads to photon focusing in the direction closest to the ambient magnetic field (Petrova Reference Petrova2004a ). As a result, the beam width may decrease by approximately two orders of magnitude, becoming compatible with the characteristic scale of the observed microstructure. In giant pulses, the focusing effect is believed to be much stronger and can be responsible for their substructure at nanosecond time scales. Note that the focusing effect resolves the difficulty of too high brightness temperatures of the nanopulses observed in the Crab pulsar (Hankins et al. Reference Hankins, Kern, Weatherall and Eilek2003).

Generally speaking, the power-law statistics and extremely short-scale structures are characteristic of nonlinear processes. In particular, the power-law intensity distribution of giant pulses and their nanosecond substructure can also be interpreted in terms of the wave collapse resulting from the nonlinear modulation instability (Weatherall Reference Weatherall1998; Cairns Reference Cairns2004).

2.2.2 Induced scattering into background

Because of the strong directivity of pulsar radio emission, the photon direction corresponding to the maximum scattering probability lies outside the photon beam. Hence, induced scattering in this direction is possible only in the presence of background radiation. The latter may arise, e.g. as a result of spontaneous scattering of the beam photons. Although the intensity of the background radiation is negligible compared to the beam intensity, $\log I_{\mathit{bg}}/I_{\mathit{beam}}\approx -10$ , in the case of strong enough induced scattering, with the scattering depth ${\it\Gamma}\approx 30$ , a substantial part of the original beam intensity can be transferred into the background, the photons being predominantly scattered in the direction of the maximum scattering probability (Lyubarskii & Petrova Reference Lyubarskii and Petrova1996).

In the regime of superstrong magnetic fields, the scattered component is directed approximately along the scattering particle velocity and the induced scattering is most efficient at the upper boundary of the scattering region (Petrova Reference Petrova2008c ). Because of rotational aberration in that region, in the pulse profile observed the scattered component should lead the main pulse by ${\leqslant}30^{\circ }$ and therefore can be identified with the precursor. Within the framework of this mechanism, the complete linear polarization of the precursor emission and its spectral peculiarities are explained naturally.

In the regime of a moderately strong magnetic field, ( $1/\sqrt{{\it\gamma}}\ll {\it\omega}^{\prime }/{\it\omega}_{H}\ll 1$ ), the induced scattering of pulsar radio emission into the background can also be efficient (Petrova Reference Petrova2008a ). In this case, the perturbed motion of the scattering particle presents a drift in the crossed electric field of the incident wave and the ambient magnetic field (Blandford & Scharlemann Reference Blandford and Scharlemann1976; Ochelkov & Usov Reference Ochelkov and Usov1983). Correspondingly, the scattering geometry and the properties of the scattered radiation are essentially distinct. In particular, photons of both polarizations are involved in the scattering, whereas the scattered component is antiparallel to the particle velocity and appears in the pulse profile as an interpulse. A comprehensive picture of the observed interpulse properties may well be explained in terms of induced scattering (Petrova Reference Petrova2008a ).

It is important to note that the scattering nature of the precursor and the interpulse implies a physical relation between these components and the main pulse. The observational manifestations of such a relation are already known. The first example concerns the pulsar PSR B1822-09 (Fowler, Morris & Wright Reference Fowler, Morris and Wright1981; Gil et al. Reference Gil, Jessner, Kijak, Kramer, Malofeev, Malov, Seiradakis, Sieber and Wielebinski1994). In the weak emission mode its radio profile exhibits the main pulse and interpulse, whereas in the bright mode it contains the precursor, the main pulse and a much weaker interpulse. Thus, the precursor and interpulse are anticorrelated, and their appearance is controlled by the main pulse intensity. The second example is the pulsar PSR B1702-19 (Weltevrede, Wright & Stappers Reference Weltevrede, Wright and Stappers2007), in which case the subpulse structure of the main pulse and interpulse is modulated with exactly the same periodicity, testifying to the correlation of the instantaneous intensities of the two components.

2.2.3 Induced scattering by relativistically gyrating particles

The radio emission structure of the Crab pulsar presents an exceptional case. The radio profile contains a total of seven components spread out over the whole pulse period and characterized by distinct spectral and polarization properties (Moffett & Hankins Reference Moffett and Hankins1996, Reference Moffett and Hankins1999). At the lowest frequencies, the profile consists of the precursor, the main pulse and the interpulse, which have very steep spectra and disappear at higher frequencies. As the frequency increases, the precursor is replaced by the so-called low-frequency component (LFC) located at the earlier pulse phase, and instead of the interpulse there appears the high-frequency interpulse (IP $^{\prime }$ ) also shifted toward earlier phases as well as the pair of trailing high-frequency components (HFC1 and HFC2).

In order to explain this complicated picture, the theory of induced scattering is generalized to the case of relativistically gyrating particles (Petrova Reference Petrova2008b ,Reference Petrova d , Reference Petrova2009b ). In the inner magnetosphere, the magnetic field of a pulsar is so strong that any transverse momentum of the plasma particles is almost immediately lost via synchrotron re-emission. At higher altitudes, however, the particles acquire relativistic gyration as a result of resonant absorption of radio emission (see § 2.3 below). As the pulsar radio beam is broadband, ${\it\nu}\sim 10^{7}{-}10^{10}$ Hz, the resonance region appears quite extended in altitude. Therefore the lower-frequency radio waves are subject to strong-field scattering by the particles performing relativistic gyration because of resonance absorption of the higher-frequency radio emission.

At the conditions relevant to a pulsar magnetosphere, the induced scattering by relativistically gyrating particles may be efficient for several lowest harmonics of the gyrofrequency. In particular, the precursor and LFC of the Crab pulsar can be attributed to the main pulse scattering at the zeroth harmonic and from the first to the zeroth one, respectively (Petrova Reference Petrova2008d ). In this case, the location of the two components on the pulse profile, their spectral behaviour and high linear polarization are explained naturally.

Furthermore, the interpulse, IP $^{\prime }$ , HFC1 and HFC2 can be interpreted as the consequence of subsequent induced scattering of the main pulse, precursor and LFC, respectively, in the regime of a moderately strong magnetic field (Petrova Reference Petrova2009b ). Then HFC1 and HFC2 are actually a single component split by rotational aberration close to the light cylinder. Note that approximate constancy of the position angle of linear polarization across this component (Moffett & Hankins Reference Moffett and Hankins1999) testifies to its origin in the outer magnetosphere.

As is shown in Petrova (Reference Petrova2009b ), the fine structure in the observed dynamic spectrum of giant IP $^{\prime }$ s (Hankins & Eilek Reference Hankins and Eilek2007; Hankins, Jones & Eilek Reference Hankins, Jones and Eilek2015) can be identified with the structure of the giant main pulses modified by induced scattering in the channel ‘main pulse $\rightarrow$ precursor $\rightarrow$ IP $^{\prime }$ ’. The fine structure of the giant main pulses itself has also recently been interpreted in terms of induced scattering (Tanaka, Asano & Terasawa Reference Tanaka, Asano and Terasawa2015).

The implications of induced scattering of pulsar radio emission are not exhausted by those outlined above. The scattered components of the millisecond pulsars are addressed in § 3.2.

2.3.1 Resonant absorption of radio emission

In the inner magnetosphere, the plasma particles are at the ground Landau orbital due to extremely strong synchrotron re-emission. In the outer magnetosphere, in the region of cyclotron resonance of the radio waves, the particles can absorb and emit resonant photons, performing transitions between the Landau orbitals. In total, the incident radio emission is partially absorbed, whereas the particle pitch angle ${\it\psi}\equiv p_{\bot }/p_{\Vert }$ (where $p_{\bot }$ and $p_{\Vert }$ are the transverse and longitudinal components of the particle momentum, respectively) increases monotonically. Note that because of much weaker magnetic field synchrotron re-emission no longer prevents the momentum growth. In pulsar magnetospheres, the process of resonant absorption can be very efficient (Blandford & Scharlemann Reference Blandford and Scharlemann1976; Mikhailovskii et al. Reference Mikhailovskii, Onishchenko, Suramlishvili and Sharapov1982; Lyubarskii & Petrova Reference Lyubarskii and Petrova1998b ). On the one hand, this poses the problem of radio wave escape from the pulsar magnetosphere. On the other hand, as soon as the particles acquire substantial pitch angles, the very character of the absorption changes (Lyubarskii & Petrova Reference Lyubarskii and Petrova1998b ).

At the initial stage, the particle pitch angle grows rapidly up to values ${\it\psi}\approx {\it\theta}\sim 0.1$ (where ${\it\theta}$ is the incident angle of the resonant photons). Later on, the total particle momentum $p$ increases so as to satisfy the condition $p({\it\theta}-{\it\psi})=\text{const}$ . The estimates show that in pulsars, only the highest radio frequencies, ${\it\nu}>10$ GHz suffer resonant absorption in the regime ${\it\psi}\ll {\it\theta}$ , whereas over the rest of the radio range the radiation is absorbed in the regime ${\it\theta}-{\it\psi}\ll {\it\theta}$ , in which case the particle momenta can increase by 2–3 orders of magnitude (Petrova Reference Petrova2002).

The optical depth to resonant absorption in the regime ${\it\theta}-{\it\psi}\ll {\it\theta}$ is much less than that in the regime ${\it\psi}\ll {\it\theta}$ , and it is the former quantity that determines the total radio luminosity of a pulsar. For the long-period and millisecond pulsars, the optical depth is small, and their radio emission freely escapes from the magnetosphere. In the short-period normal pulsars, the optical depth is roughly of order unity, so that their radio emission may be substantially suppressed. As is argued in § 3.1 below, the resonant absorption can account for the radio quietness of young gamma-ray pulsars.

2.3.2 Spontaneous synchrotron re-emission

The synchrotron radiation of the particles with the evolved momenta falls into the optical and soft X-ray ranges. Spontaneous re-emission of the particles taking part in the resonant absorption of radio waves was suggested as a mechanism of the pulsar high-energy emission (Lyubarskii & Petrova Reference Lyubarskii and Petrova1998b ; Gil, Khechinashvili & Melikidze Reference Gil, Khechinashvili and Melikidze2001; Petrova Reference Petrova2003b ). The model was subsequently elaborated by including the accelerating electric field and applied to the cases of partially screened polar gap and slot gap (Harding, Usov & Muslimov Reference Harding, Usov and Muslimov2005; Harding et al. Reference Harding, Stern, Dyks and Frackowiak2008; Harding & Kalapotharakos Reference Harding and Kalapotharakos2015). It is important to note that the mechanism based on synchrotron re-emission for the first time implies the physical connection between the radio and high-energy emissions of a pulsar (Petrova Reference Petrova2003b ). In particular, the observed excess of optical emission during giant radio pulses of the Crab pulsar (Shearer et al. Reference Shearer, Stappers, O’Connor, Golden, Strom, Redfern and Ryan2003; Oosterbroek et al. Reference Oosterbroek, Cognard, Golden, Verhoeve, Martin, Erd, Schulz, Stüwe, Stankov and Ho2008; Strader et al. Reference Strader, Johnson, Mazin, Spiro Jaeger, Gwinn, Meeker, Szypryt, van Eyken, Marsden and O’Brien2013) can be understood as follows. As a giant pulse comes into the resonance region, the momentum evolution of the particles becomes more pronounced and their synchrotron emission is stronger.

2.3.3 Spontaneous scattering by relativistically gyrating particles

At the conditions relevant to pulsar magnetospheres, radio photons can also be reprocessed to high energies in another way. Over most of the resonance region there is a significant amount of under-resonance photons, with frequencies ${\it\omega}^{\prime }\ll {\it\omega}_{H}$ , which can be scattered off the relativistically gyrating particles. This process differs essentially from the common magnetized scattering by straight moving particles (Petrova Reference Petrova2008b ). The under-resonance photons are chiefly scattered to high harmonics of the particle gyrofrequency, ${\it\omega}_{sc}^{\prime }={\it\omega}^{\prime }+s{\it\omega}_{H}$ with $s\sim {\it\gamma}_{0}^{3}$ (where ${\it\gamma}_{0}$ is the Lorentz factor of the particle gyration), and the total scattering cross-section is much larger. Furthermore, the peak of the scattered radiation is markedly shifted in frequency beyond the synchrotron maximum. Correspondingly, although the total power scattered is always less than the synchrotron power, in the region beyond the synchrotron maximum the contribution of the scattered radiation can be substantial.

As is shown in Petrova (Reference Petrova2009a ), the spontaneous scattering of under-resonance radio photons off the relativistically gyrating particles can explain the radio–X-ray correlation observed in the Vela pulsar (Lommen et al. Reference Lommen, Donovan, Gwinn, Arzoumanian, Harding, Strickman, Dodson, McCulloch and Moffett2007). The scattered power appears to be a very strong function of the particle gyration energy, $L_{\text{sc}}\propto {\it\gamma}_{0}^{6}$ , and the quantity ${\it\gamma}_{0}$ is determined by the radio intensity coming into the resonance region and causing the particle momentum evolution. Thus, it is the scattered power that is tightly connected with the radio emission characteristics, and this connection should be most prominent beyond the synchrotron maximum. Recall that in the Vela pulsar the trough component is indeed noticeably connected with the radio emission beyond its spectral maximum. The numerical estimates as applied to this pulsar give reasonable results. In the observer frame, the peaks of the synchrotron and scattered radiation, $\hbar {\it\omega}_{syn}=0.2$ keV and $\hbar {\it\omega}_{sc}=1.5$ keV, agree well with the spectral maximum of the trough component and the range of its pronounced correlation with radio emission, while the total synchrotron power, $L_{syn}=10^{31}$ erg s $^{-1}$ , is compatible with the observed luminosity of this component.

The radio beam is also subject to induced scattering off the gyrating particles. In contrast to the spontaneous scattering considered above, the induced scattering is most efficient between the states below the resonance, ${\it\omega}_{sc}^{\prime }={\it\omega}^{\prime }$ , and can modify the radio profile shape (for details see § 2.2 above). Note that the efficiency of induced scattering is also a function of the particle gyration energy ${\it\gamma}_{0}$ . Hence, the variations of ${\it\gamma}_{0}$ should lead to joint fluctuations in the radio and high-energy emission, and the radio–high-energy connection in the Vela pulsar can be understood as the interplay between the processes of spontaneous and induced scattering by the gyrating particles (Petrova Reference Petrova2009a ).