There are two important conclusions that can be drawn from the radio emission properties of pulsars. Firstly the radio emission is thought to arise at an altitude ${r_{\rm em}}$ from the center of the neutron star, where ${r_{\rm em}}$ is of the order of several stellar radii, R=10⁶ cm (e.g. Kijak & Gil 1997, 1998, and references therein). Secondly the regions from where the radio emission arises are consistent with a purely dipolar magnetic field (Radhakrishnan & Cooke 1969). However the structure of the magnetic field at the surface of the neutron star is largely unknown. Strong non-dipolar surface magnetic fields have long been thought to play an important role in the radio emission of pulsars. For example, in order to sustain pair production in vacuum gaps, the Ruderman & Sutherland (RS75) model implicitly assumed that the radius of curvature of field lines above the polar cap should be about 10⁶ cm, which is 100 times smaller than that expected from a global dipolar magnetic field, thus indicating the presence of non-dipolar components. It is believed that thermal X-rays from pulsars are a good diagnostic tool to infer the structure of the surface magnetic field. Soft X-ray observations of pulsars show non-uniform surface temperatures which can be attributed to small scale magnetic anomalies on the pulsar polar cap (e.g. Page & Sarmiento 1990; Bulik et al. 1992, 1995). Several similar arguments in favour of the non-dipolar nature of the surface magnetic field can also be found in Becker & Trümper (1997); Cheng et al. (1998); Rudak & Dyks (1999); Cheng & Zhang (1999); Thompson & Duncan (1995, 1996), Murakami et al. (1999), and Tauris & Konar (2001). Also several theoretical studies concerning the formation and evolution of non-dipolar magnetic fields in neutron stars are found in the literature (e.g. Blandford et al. 1983; Krolik 1991; Ruderman 1991; Arons 1993; Chen & Ruderman 1993; Geppert & Urpin 1994; Mitra et al. 1999).

Woltjer (1964) proposed that the magnetic field in neutron stars results from the fossil field of the progenitor star which is amplified during the collapse stage and remains anchored in the superfluid core of the neutron star. It was also noted by several authors that shortly after or during the collapse of the neutron star magnetic fields can be generated in the outermost crust (e.g. by a mechanism like thermomagnetic instabilities; Blandford et al. 1983). Urpin et al. (1986) showed that in the crustal model it is possible to form small-scale surface field anomalies with typical sizes of the order of 100 meters. Further, Gil & Mitra (2001) demonstrated that such "sunspot''-like magnetic field structures on the polar cap surface help to sustain VG-driven radio emission of pulsars. In this paper we consider the scenario where the magnetic field on the neutron stars' surface is non-dipolar in nature which is due to superposition of the fossil field in the core and the crustal field structures. The crust gives rise to small-scale anomalies which can be modelled by a number of crust-anchored dipoles oriented in different directions (e.g. Blandford et al. 1983; Arons 1993). The superposition of global dipole and local anomaly is illustrated in Fig. 1, where for clarity of presentation only one local, crust-associated dipole is marked.

Formation of dense electron-positron pair plasma is essential for pulsar radiation, especially (but not only) at radio wavelengths. A purely quantum process for magnetic pair production $\gamma\rightarrow {\rm e}^-{\rm e}^+$ is commonly invoked as a source of this plasma (e.g. Sturrock 1971; Ruderman & Sutherland 1975). For superstrong magnetic fields close to the so-called quantum field $B_{\rm q}=4.4\times 10^{13}$ G, the process of free e^-e⁺ pair production can be dominated by the phenomenon of photon splitting (Adler et al. 1970; Bialynicka & Bialynicki et al. 1970; Baring & Harding 1998) and/or bound positronium formation (Usov & Melrose 1995, 1996). While the latter process can reduce the number of free pairs at magnetic fields $B\ga 0.1B_{\rm q}$ (e.g. Baring & Harding 2001), the former can suppress the magnetic pair production at $B\ga 10^{13}$ G entirely, provided that photons which are polarized both parallel and perpendicular to the local magnetic field direction can split (e.g. Baring 2001; Baring & Harding 2001). This assumption will be implicitly made throughout this paper. Under these circumstances one can roughly define a photon splitting critical line $B_{\rm cr}\sim B_{\rm q}$ and expect that there should be no radio pulsar above this line on the $B_{\rm p}-P$ diagram, where $B_{\rm p}=6.4\times 10^{19}(P\dot{P})^{1/2}$ G is the dipole surface magnetic field estimated at the pole from the pulsar period Pand its derivative $\dot{P}$ (Shapiro & Teukolsky 1983; Usov & Melrose 1996). This death-line is more illustrative than quantitative. In fact, a number of specific model-dependent death-lines separating radio-loud from radio-quiet pulsars are available in the literature (Baring & Harding 1998, 2001; Zhang & Harding 2000a, 2001). All these slightly period-dependent death-lines cluster around $B_{\rm q}$ on the $B_{\rm p}-P$ diagram, and hence the quantum critical field is conventionally treated as a threshold magnetic field above which pulsar radio emission ceases. In this paper we also use this terminology, bearing in mind that the photon splitting threshold realistically means a narrow range of magnetic fields around the critical quantum field $B_{\rm q}\sim 4\times 10^{13}$ G, certainly above 10¹³ G (see review by Baring 2001). For convenience, in all numerical examples presented in Figs. 2-6 and subsequent discussions we assume that the threshold magnetic field $B_{\rm cr}= B_{\rm q}$ .

In order to produce the necessary dense electron-positron plasma, a high-voltage accelerating region has to exist near the polar cap of pulsars. Two models of such acceleration regions are available in the literature, namely: the stationary space charge limited flow (SCLF) model (Sharleman et al. 1978; Arons & Sharleman 1979; Arons 1981) in which charged particles flow freely from the polar cap, and the highly non-stationary vacuum gap (VG) models (Ruderman & Sutherland 1975; Cheng & Ruderman 1977, 1980; Gil & Mitra 2001) in which the free outflow of charged particles from the polar cap surface is strongly impeded. In the VG models the charged particles accelerate within a height scale of about the polar cap radius (i.e. $\sim$ 10⁴ cm), due to a high potential drop across the gap, while in the SCLF models particles accelerate within a height scale of a stellar radius $\sim$ 10⁶ cm, due to the potential drop resulting from the curvature of field lines and/or the inertia of outstreaming particles. In both models the free e^-e⁺ pairs are created if the kinematic threshold $\varepsilon_\gamma\cdot\sin\theta_{\rm t}=2~mc^2$ is reached or exceeded and the local magnetic field is lower than the photon splitting threshold $B\sim B_{\rm cr}$ , where $\varepsilon_\gamma=\hbar\omega$ is the photon energy and $\theta_{\rm t}$ is the propagation angle with respect to the direction of the local magnetic field.

Recent discovery of high-magnetic-field pulsars (HBPs) however has challenged the existing pair creation theories. Few HBPs are seen to have the inferred surface dipolar fields above the photon splitting level: PSRs J1119-6127, J1814-1744 and J1726-3530 (Table 1). Yet another strong-field neutron star PSR J1846-0258 with $B_{\rm d}\sim 5\times 10^{13}$ G was discovered (Gotthelf et al. 2000), which seems to be radio-quiet (Kaspi et al. 1996), although its X-ray emission is apparently driven by dense ${\rm e}^-{\rm e}^+$ pair plasma (e.g. Cordes 2001). Bearing in mind that the actual threshold due to photon splitting and/or bound positronium formation can be well below the critical field $B_{\rm cr}\sim 4\times 10^{13}$ G, all high-magnetic-field radio pulsars with $B_{\rm d}>10^{13}$ G pose a challenge. To evade the photon splitting problem for these pulsars Zhang & Harding (Zhang & Harding 2000a, ZH00 hereafter) proposed "a unified picture for HBPs and magnetars''. They argued that radio-quiet magnetars cannot have active inner accelerators (thus no ${\rm e}^-{\rm e}^+$ pair production), while the HBPs can, with a difference attributed to the relative orientations of rotation and magnetic axes (neutron stars can be either parallel rotators (PRs) with $\vec{\Omega B}_{\rm p}>0$ or antiparallel rotators (APRs) with $\vec{\Omega B}_{\rm p}<0$ , where $\vec{\Omega}$ is the pulsar spin axis and $\vec{B}_{\rm p}$ is the magnetic field at the pole). If the photon splitting suppresses completely the pair production at the polar cap surface, then the VG inner accelerator cannot form, since the high potential drop cannot be screened at the top of the acceleration region. Hence, ZH00 argued that in the high magnetic field regime $(B_{\rm d}>B_{\rm cr})$ the pair production process is possible only if the SCLF accelerator forms. In fact, such SCLF accelerators are typically quite long and their pair formation front (PFF) can occur at high altitudes r, where the dipolar magnetic field $B_{\rm d}\propto r^{-3}$ has degraded below $B_{\rm cr}$ . Furthermore, ZH00 demonstrated that such a lengthened SCLF accelerator in a magnetar environment can form only for PRs and not for APRs. Consequently they concluded that the radio-loud HBPs are PRs with developed lengthened SCLF accelerator, while the radio-quiet magnetars (AXPs and SGRs) are APRs. It is worth emphasizing here that ZH00 developed their model under the assumption that the magnetic field at the surface of HBPs is purely dipolar.

Table 1: Radio-loud HBPs with inferred magnetic field $B_{\rm p}=6.4\times 10^{19}(P\dot{P})^{1/2}$ G higher than the critical quantum field $B_{\rm cr}=4.4\times 10^{13}$ G (after Table 1 in Zhang & Harding 2000b).
source P (s) $\dot{P}$ (s/s) B_p (G)

PSR J1814-1744 3.98 $7.43\times 10^{-13}$ $1.1\times 10^{14}$

PSR J1119-6127 0.41 $4.02\times 10^{-12}$ $8.2\times 10^{13}$

PSR J1726-3530 1.11 $1.22\times 10^{-12}$ $7.4\times 10^{13}$

**Table 1:** Radio-loud HBPs with inferred magnetic field $B_{\rm p}=6.4\times 10^{19}(P\dot{P})^{1/2}$ G higher than the critical quantum field $B_{\rm cr}=4.4\times 10^{13}$ G (after Table 1 in Zhang & Harding 2000b).
source	P (s)	$\dot{P}$ (s/s)	B_p (G)
PSR J1814-1744	3.98	$7.43\times 10^{-13}$	$1.1\times 10^{14}$
PSR J1119-6127	0.41	$4.02\times 10^{-12}$	$8.2\times 10^{13}$
PSR J1726-3530	1.11	$1.22\times 10^{-12}$	$7.4\times 10^{13}$

In this paper we propose an alternative model for radio-loud HBPs based on a highly non-dipolar surface magnetic field, in which the photon splitting within the VG inner acceleration region does not operate even if the dipole magnetic field exceeds the critical value $B_{\rm cr}$ at the polar cap. Thus our model requires that HBPs are APRs, which is a consequence of the VG scenario (e.g. Ruderman & Sutherland 1975; Gil & Mitra 2001). This model is a follow-up of Gil & Mitra (2001), who argued that the VG can form if the actual surface magnetic field is about 10¹³ Gauss. In other words, they assumed that all VG-driven radio pulsars require strong highly non-dipolar surface magnetic fields, with strength $B_{\rm s}$ being more or less independent of the value of the global dipole field inferred from the magnetic braking law. Thus, if $B_{\rm d}\ll 10^{13}$ G then $B_{\rm s}\gg B_{\rm d}$ and if $B_{\rm d}\ga 10^{13}$ G then $B_{\rm s}\sim B_{\rm d}$ . However, for radio pulsars to operate, in any case $B_{\rm s}<B_{\rm cr}\sim B_{\rm q}$ . We argue that such strong surface field anomalies can increase the dipolar field $B_{\rm d}\ll B_{\rm cr}$ in normal pulsars to values exceeding 10¹³ G (required by the conditions for VG formation - see Gil & Mitra 2001) if the global and local surface fields have the same polarities, or reduce the very high dipolar field $B_{\rm d}\ga B_{\rm cr}$ in HBPs, if both these components are of comparable values and have opposite polarities.

1 Introduction