2. Model

Figure 1. Overall geometry of the system.

The primary assumption of the model is that the broadband radio emission source of HFIP is located behind the pulsar, as illustrated in figure 1. The precise origin of the source remains uncertain. Nevertheless, it is reasonable to assume that the mechanism is analogous to that observed in the main pulse and the low-frequency interpulse (Philippov et al. Reference Philippov, Uzdensky, Spitkovsky and Cerutti2019). In principle, although less likely, this could be a reflected or refracted radio emission originating from a distinct location, such as a polar cap.

Electromagnetic waves, propagating in magnetised plasma quasi-perpendicular to the magnetic field, have two distinct polarisations, the ‘ordinary modes’ and ‘extraordinary modes’ (O-modes and X-modes, for short). These two polarisations possess different propagation characteristics.

The X-mode possesses a component of the wave electric field perpendicular to the magnetic field. Consequently, it can resonate with gyrating electrons and positrons, resulting in significant absorption. It is widely recognised that cyclotron resonant absorption severely restricts wave propagation through a magnetosphere (Blandford & Scharlemann Reference Blandford and Scharlemann1976; Mikhailovskii et al. Reference Mikhailovskii, Onishchenko, Suramlishvili and Sharapov1982). Absorption typically occurs near the light cylinder, where the wave frequency and the cyclotron frequency coincide. Although the resonance region is typically confined to a small volume, the absorption cross-section is very high, resulting in a substantial overall optical depth. The optical depth in the Crab magnetosphere is approximately $\tau _{\textrm {abs}} \sim 0.1\mathcal{M}(\nu /30\,\textrm {GHz})^{-1/3}$ (cf. equation (8) in Rafikov & Goldreich Reference Rafikov and Goldreich2005). Consequently, the X-mode is strongly absorbed ( $\tau _{\textrm {abs}} \gtrsim 1$ ) in a plasma with multiplicity $\mathcal{M} \gtrsim 10$ , irrespective of whether the propagation is perpendicular or oblique. In the Crab pulsar environment, $\mathcal{M} \sim 10^4 - 10^5$ on open field lines and possibly lower on closed field lines (Philippov & Kramer Reference Philippov and Kramer2022).

It is crucial to emphasise that the nearly 100 % polarisation of HFIP is naturally explained by the strong absorption of one of the two modes. The constancy of the position angle can be attributed to pure geometry: the radio source is consistently located behind the magnetosphere at the same rotation phase. Variations in the position angle and circular polarisation, if any, can be affected as plasma normal modes propagate through the dynamical magnetospheric plasma (Petrova & Lyubarskii Reference Petrova and Lyubarskii2000; Beskin & Philippov Reference Beskin and Philippov2012), as well as caused by variability of the source.

In contrast, the O-mode propagating perpendicular to the magnetic field has its wave electric field aligned with the ambient magnetic field. Consequently, it perceives plasma but not the magnetic field. Consequently, it does not undergo cyclotron absorption and cannot propagate at a frequency below the plasma frequency (it is said to have a cutoff at the plasma frequency). The plasma index of refraction for the perpendicular-propagating O-mode is identical to that of the unmagnetised plasma

(2.1) \begin{equation} n_{\textrm {pl}}^{2}(r,\omega )\equiv \left (\frac {kc}{\omega }\right )^2=1-\frac {\omega _p^2}{\omega ^2}=1-\frac {4\pi e^2 n_e(r)/m_e}{\omega ^2}, \end{equation}

where $\omega$ and $k$ are the frequency and wave number of the wave, $c$ is the speed of light, $\omega _p$ is called the non-relativistic plasma frequency, $e$ is the elementary charge and $n_{e}$ is the lepton (electron and positron, if present) number density. In relativistic electron–positron plasmas, one should replace the electron mass $m_{e}$ with the ensemble-averaged quantity $m_{e}\langle \gamma _{e}^{-3}\rangle ^{-1}$ , with $\gamma _{e}$ being the particles’ Lorenz factor (Gedalin, Melrose & Gruman Reference Gedalin, Melrose and Gruman1998). This factor can be heuristically understood as the effect accounting for the relativistic parallel inertia. This effect arises when the acceleration due to the transverse-propagating ordinary-wave electric field is parallel (or antiparallel) to the particle velocity, which in turn is along the magnetic field. Plasma on the open field lines is highly relativistic, $\langle \gamma _{e}^{-3}\rangle ^{-1}\gg 1$ , as it has a large bulk Lorentz factor and also can have relativistically large thermal spread at large distances. On the closed field lines, however, the plasma is non-relativistic because of cooling and magnetic confinement, $\gamma _{e}\sim 1$ . Consequently, we can disregard this factor henceforth. It can always be accounted for later, if necessary.

It is crucial to emphasise that we assume the propagation of O-mode waves is strictly perpendicular to the magnetospheric field. This assumption holds true for the waves propagating in the plane of the magnetic equator of a magnetic dipole, which serves as a model for the magnetic field of a neutron star. Consequently, this assumption imposes a constraint on the location of the source, yet it remains fully consistent with the source being situated at the current sheet. Furthermore, propagation in the equatorial plane inherently leads to the disappearance of Faraday rotation, irrespective of the plasma’s composition. (Note: generally, Faraday rotation exists even in electron–positron plasma, provided it is non-neutral. The charge imbalance induced by the spinning magnetosphere is equal to the Goldreich–Julian density).

3. Propagation

In this section, we employ the natural system of units, $\hbar =c=G=1$ . Propagation of light through plasma in a strong gravitational field has been studied extensively, see for example de Felice (Reference de Felice1971), Broderick & Blandford (Reference Broderick and Blandford2003a,b ) Tsupko & Bisnovatyi-Kogan (Reference Tsupko and Bisnovatyi-Kogan2013), Atamurotov, Ahmedov & Abdujabbarov (Reference Atamurotov, Ahmedov and Abdujabbarov2015), Perlick & Tsupko (Reference Perlick and Tsupko2017) and Balek, Bezděková & Bičák (Reference Balek, Bezděková and Bičák2024). Here, we present relevant theoretical calculations of the ray trajectories, which will be used in the next section.

The plasma dispersion of the O-mode, $\omega ^{2}=\omega _p^2+k^{2}$ , in the covariant form reads

(3.1) \begin{equation} g^{\mu \nu }p_\mu p_\nu +\omega _p^2=0, \end{equation}

where the zeroth component of the 4-momentum is $p_{t}=-\omega$ and the spatial components are $p_{i}=k_{i},\ i=1,2,3$ . Here, $\omega =\omega (\boldsymbol{x})$ and $k=k(\boldsymbol{x})$ are the local frequency and wavenumber of the electromagnetic wave, and $g^{\mu\nu}$ is the spacetime metric tensor.

The dispersion relation induces a Hamiltonian for our system

(3.2) \begin{equation} H(\boldsymbol{x},\boldsymbol{p})=\frac {1}{2}\left [g^{\mu \nu }(\boldsymbol{x})p_\mu p_\nu +\omega _p^2(\boldsymbol{x})\right ]\!. \end{equation}

Then, geodesic equations for light rays are

(3.3) \begin{align} &\frac {{\rm d} x^\mu }{{\rm d} \lambda }=\frac {\partial H}{\partial p_\mu }, \end{align}

(3.4) \begin{align} &\frac {{\rm d} p_\mu }{{\rm d} \lambda }=-\frac {\partial H}{\partial x^\mu }, \end{align}

(3.5) \begin{align} &H=0, \end{align}

where $\lambda$ is the affine parameter along the ray trajectory.

We assume Schwarzschild geometry

(3.6) \begin{align} {\rm d}s^2&=g_{\mu \nu }{\rm d}x^\mu {\rm d}x^\nu \end{align}

(3.7) \begin{align} &=-\left (1-\frac {2m}{r}\right ){\rm d}t^2+\left (1-\frac {2m}{r}\right )^{-1}{\rm d}r^2+r^2{\rm d}\Omega ^2. \end{align}

Here ${\rm d}\Omega ^2 = ({{\rm d}{\theta}^2 + \sin^{2}{\theta}\, {\rm d}{\varphi}^2})$ is the metric on the two-sphere, $\theta$ and $\varphi$ are the azimuthal and polar angles. Thereby, the rotation of the neutron star is neglected. In our units, the gravitational (Schwarzschild) radius is given by $r_g = 2m$ , where $m$ is the mass of the neutron star. The Hamiltonian, (3.2), in Kerr space–time reads

(3.8) \begin{align} H(\boldsymbol{x},\boldsymbol{p})&=\frac {1}{2}\left [\left (1-\frac {2m}{r}\right )p_{r}^{2} +\frac {p_{\theta }^{2}}{r^{2}}+\frac {p_{\phi }^{2}}{r^{2}\sin ^{2}\theta } \right .\nonumber \\ &\qquad \left .-\left (1-\frac {2m}{r}\right )^{-1}p_{t}^{2}+\omega _{p}^{2}(\boldsymbol{x})\right ]. \end{align}

Let $S$ be action, then $p_{\mu }=\partial S/\partial x^{\mu }$ . Consequently, (3.1) transforms into the Hamilton–Jacobi equation, which can be solved using separation of variables, $S=-E t+L \phi +S_{r}(r)+ S_{\theta }(\theta )$ . Here, the two constants of motion represent conserved quantities, specifically energy and angular momentum

(3.9) \begin{align} &E=-\partial S/\partial t=-p_{t}=\omega _{\infty }, \end{align}

(3.10) \begin{align} &L=\partial S/\partial \phi =p_{\phi }=k b= \omega _{\infty }b, \end{align}

where $b$ is the impact parameter, the constant $\omega _{\infty }=\omega (r\to \infty )$ is the frequency of radiation at infinity and the last equalities in (3.9), (3.10) hold true because $\omega _p(r)=0$ as $r\to \infty$ . Hereafter, we assume that $\omega _p(r)$ solely depends on the radial coordinate, thereby maintaining spherical symmetry.

The third constant of motion that emerges from the separation of variables in the Hamilton–Jacobi equation, referred to as Carter’s constant, appears as an identity and is expressed as

(3.11) \begin{align} \mathcal{C}&=p_{\theta }^{2}+L\cot ^{2}\theta \nonumber \\ &=-r^{2}\left [\left (1-\frac {2m}{r}\right )p_{r}^{2}+\frac {L^{2}}{r^{2}} -\frac {E^{2}}{\left (1-({2m}/{r})\right )}+\omega _{p}(r)\right]\!. \end{align}

By employing spherical symmetry, we can assume, for convenience and without loss of generality, that motion starts in the equatorial plane, $\theta =\pi /2$ , with vanishing $\theta$ -momentum at infinity, $p_{\theta ,\infty }=0$ . Then, Carter’s constant vanishes identically. Note that (3.11) with $\mathcal{C}=0$ readily determines the radial momentum

(3.12) \begin{align} p_{r}^{2}=\left (1-\frac {2m}{r}\right )^{-2}\left [E^{2}-\left (1-\frac {2m}{r}\right )\left (\frac {L^{2}}{r^{2}}+\omega _{p}^{2}(r)\right )\right]\!. \end{align}

Substituting (3.9)–(3.12) into the Hamilton equations, (3.3)–(3.5), the light ray equations can be written as

(3.13) \begin{align} &\frac {{\rm d} r}{{\rm d} \lambda }=\sqrt {\omega _\infty ^2-\left (1-\frac {2m}{r}\right )\left (\frac {\omega _\infty ^2b^2}{r^2}+\omega _p^2\right )}, \end{align}

(3.14) \begin{align} &\frac {{\rm d}\theta }{{\rm d} \lambda }=0, \end{align}

(3.15) \begin{align} &\frac {{\rm d} \phi }{{\rm d} \lambda }=\frac {\omega _\infty b}{r^{2}}, \end{align}

(3.16) \begin{align} &\frac {{\rm d} t}{{\rm d} \lambda }={\omega _\infty }\left /{\left (1-\frac {2m}{r}\right )}\right . . \end{align}

These equations contain three constants of motion that separate the variables in the Hamilton–Jacobi equation. Two of them, $\omega _{\infty }$ and $b$ (in the combination $\omega _{\infty }b$ ), represent the conservation of energy and angular momentum, respectively. The third one (Carter’s constant), together with (3.14), yields that motion is restricted to a plane, chosen to be the equatorial plane, $\theta (\lambda )=\theta (0)=\pi /2$ , and $p_{\theta }=0$ everywhere, not just at infinity.

To proceed further, we need to employ a model of the electron–positron plasma density distribution. Motivated by the recent finding suggesting a power-law distribution in the magnetosphere of the Crab pulsar (Medvedev Reference Medvedev2024), we assume $\omega _p^2\propto n_{e}(r)\propto r^{-\kappa }$ with $\kappa$ being a constant to be determined. This yields the following parameterisation of the plasma index of refraction, (2.1):

(3.17) \begin{equation} n_{\textrm {pl}}^{2}(r,\omega _{\infty })=1-\left (\frac {r_{0}(\omega _{\infty })}{r}\right )^{\kappa }. \end{equation}

Here, $r_0(\omega _{\infty })$ is the frequency-dependent plasma ‘reflection radius’ in the absence of gravity. It is defined as the radius at which the plasma frequency equals the wave frequency at infinity, $\omega _{p}(r_0) = \omega _{\infty }$ . Thus, in the absence of gravitational blueshift, this is the point where the wave would be reflected, as indicated by the condition $n_{\textrm {pl}}(r_0, \omega _{\infty }) = 0$ . Consequently,

(3.18) \begin{equation} r_0^\kappa (\omega _{\infty })=r_*^\kappa \,\frac {\omega _{p*}^2}{\omega _\infty ^2}, \end{equation}

where we chose to normalise quantities by their values at the neutron star surface, namely $r_*$ is the neutron star radius and $\omega _{p*}\equiv \omega _p(r_*)$ is the plasma frequency at the neutron star’s surface.

Figure 2. The effective index of refraction as a function of distance, for various values of the so-called ‘reflection radius’ parameter $r_{0}=3,\ldots , 7$ and $\kappa =3$ . All distances are in units of mass $m$ , so that $r_{g}=2$ and the Crab pulsar’s radius is $r_{*}\simeq 4.8$ (indicated by the vertical dashed line). Bright colours denote $n_{\textrm {eff}}\gt 1$ and dark colours correspond to $n_{\textrm {eff}}\lt 1$ .

The equation that describes the geometric path of a light ray in a medium, here referred to as the ‘crystal ball equation’ (see Appendix A for discussion), is straightforwardly obtained from (3.13) and (3.15)

(3.19) \begin{equation} \frac {{\rm d} \phi }{{\rm d} r}=\frac {b}{r\sqrt {1-{2m}/{r}}\sqrt {r^2 n_{\textrm {eff}}^2-b^2}}, \end{equation}

where the effective index of refraction is

(3.20) \begin{equation} n_{\textrm {eff}}=\sqrt {\left (1-\frac {2m}{r}\right )^{-1}-\left (\frac {r_0}{r}\right )^\kappa }. \end{equation}

Figure 2 illustrates the behaviour of the effective index of refraction, $n_{\textrm {eff}}$ , as a function of distance. The figure depicts a range of values for the reflection radius parameter $r_{0}$ , which varies between 3 and 7. The figure considers a representative value of $\kappa =3$ , which corresponds to the Goldreich–Julian density in the dipolar magnetic field. All distances are measured in units of mass, $m$ . Consequently, $r_{g}=2$ , and the Crab pulsar’s radius (for a mass of $M_{*}\sim 1.4 M_{\odot }$ and a radius of $R_{*}\simeq 10$ km) is approximately $r_{*}\simeq 4.8$ . This value is roughly equivalent to 2.4 Schwarzschild radii. Bright colours of the curves indicate regions where $n_{\textrm {eff}}\gt 1$ , signifying the dominance of gravitational lensing. Conversely, dark colours correspond to regions where $n_{\textrm {eff}}\lt 1$ , indicating the dominance of defocusing by plasma.

We observe an intriguing interplay between gravity and plasma dispersion, especially at short distances when $r\sim r_{0}\sim r_{g}$ . This interplay results in a non-monotonic index of refraction. This phenomenon may hold significance for the study of light propagation near a black hole. Further exploration of this effect is warranted in a separate context. In contrast, for our neutron star system, which has a substantially larger size, the refraction index behaviour is relatively straightforward. It remains below unity in the vicinity of $r_{0}$ , indicating the predominance of plasma dispersion, and increases above unity at greater distances, where gravitational lensing effect dominates.

The geometrical path of the light ray, $\phi (r)$ , is determined by the straightforward integration of (3.19). The minimum radius, or the distance of the closest approach, $r_{m}$ , is identified by the vanishing denominator. Consequently, it is implicitly determined, as a function of $b$ , as a positive root of the equation

(3.21) \begin{equation} r_{m}n_{\textrm {eff}}(r_{m})=b. \end{equation}

Note that this equation is polynomial for rational values of $\kappa$ and transcendental for irrational values. By symmetry of the trajectory with respect to the $r=r_{m}$ point, the total change in the angle $\phi$ is

(3.22) \begin{equation} \phi _{\textrm {tot}}=2\int ^{r_{m}}_{\infty }\frac {b\, {\rm d}r}{r\sqrt {1-{2m}/{r}}\sqrt {r^2 n_{\textrm {eff}}^2-b^2}}. \end{equation}

A straight, non-deflected path has a total angle change $\phi _{\textrm {tot}} = \pi$ . Consequently, the deflection angle $\hat \alpha$ is determined as $\hat \alpha \equiv \phi _{\textrm {tot}} - \pi$ . Positive values of $\hat \alpha$ indicate focusing, while negative values correspond to defocusing.

Figure 3. Deflection of parallel light rays propagating from right to left for various impact parameters, $b=4,\ldots ,10$ , and for $r_{0}=5.5$ and $\kappa =3$ . Brighter colours indicate weak deflection, while darker tones represent strongly deflected rays. The $(r,\phi )$ coordinates of the innermost ray, along with its impact parameter, $b$ , and the distance of the closest approach, $r_{m}$ , are depicted. Also, plotted are: the Schwarzschild radius, $r_{g}=2$ (smallest dashed circle), the non-transparent region when gravitational blueshift of $\omega$ is accounted for (grey disk), the neutron star size, $r_{*}\simeq 4.8$ (solid black circle) and the reflection radius, $r_{0}$ , defined by (3.18) (large dashed grey circle). All distances are measured in units of the neutron star mass $m$ .

Figure 3 illustrates the deflection of light rays from a light source located at infinity on the right and propagating leftward. The figure depicts a range of values for the impact parameter $b$ , which varies between 4 and 10. The light rays are coloured by their deflection angle with brighter colours indicating weaker deflections, whereas darker colours indicate stronger deflections. Other parameters are fixed at $r_{0}=5.5$ and $\kappa =3$ . For the innermost ray (thick black curve), the coordinate system, $(r,\phi )$ , its impact parameter, $b$ , and the distance of the closest approach, $r_{m}$ , are depicted. Additionally shown with circles are the gravitational radius $r_{g}=2$ (dashed black), the neutron star radius $r_{*}\simeq 4.8$ (solid black) and the reflection radius $r_{0}$ (dashed grey). We also depict the region of non-transparency (grey disk), which is different from $r_{0}$ (which is defined for $\omega _{\infty }$ ), because of the gravitational blueshift of the wave frequency as it enters deeper into the gravitational field, $\omega \gt \omega _{\infty }$ , and becomes maximum at $r_{m}$ . Consequently, the distinction between the actual reflection radius, which corresponds to the edge of the grey disk, and the reflection radius $r_{0}$ , for which gravitational effects are disregarded, elucidates the magnitude and significance of the gravitational blueshift’s impact.

As observed, rays with minimal deflections, $|\hat \alpha | \ll 1$ , are present for sufficiently large values of $b$ , i.e. at considerable distances. Consequently, the analytical treatment can be further simplified by assuming $r \gg 2m$ and $r \gg r_0$ . We evaluate the applicability of the latter assumption in § 5. These assumptions correspond to the limit of weak gravitational field and weak plasma refraction. In this limit, referred to as the ‘far-away approximation,’ the crystal ball equation simplifies to a classical form

(3.23) \begin{equation} \frac {{\rm d} \phi }{{\rm d} r}=\frac {b}{r\sqrt {r^2 n^2(r)-b^2}}. \end{equation}

This equation describes the trajectory of a light ray’s path through a transparent medium with a radially dependent index of refraction, $n(r)$ , similar to a crystal ball. Consequently, this equation is named accordingly. Under the assumptions made, the effective refractive index squared is obtained to the next to the leading order as follows:

(3.24) \begin{equation} n^{2}(r)=n^2_{\textrm {weak}}(r)\approx 1+\frac {2m}{r}-\left (\frac {r_0}{r}\right )^\kappa\!, \end{equation}

where the second term represents gravitational lensing, while the third term represents the plasma de-lensing effect.

Figure 4. A schematic representation of the pulsar system, illustrating its equivalence to Young’s slits. Variables drawn with blue ink describe light ray propagation, while those in red ink describe two-slit interference.

Next, we note an important point that the two nearly non-deflected rays, travelling on two opposite sides of the pulsar as in figure 4, can interfere, even at considerable distances. Such an interference process has been proposed (Medvedev Reference Medvedev2024) as an explanation for the multiple spectral bands (the so-called ‘zebra pattern’) observed in the dynamic spectra of the Crab pulsar HFIP (Hankins & Eilek Reference Hankins and Eilek2007; Eilek & Hankins Reference Eilek and Hankins2016; Hankins et al. Reference Hankins, Eilek and Jones2016). The system under consideration is therefore analogous to Young’s two slits, as illustrated in figure 4. The separation between the ‘slits’ (referred to as ‘hot spots’ in Medvedev Reference Medvedev2024) is twice the impact parameter distance at which the deflection angle vanishes, $a(\omega )=2b_{0}$ .

Strictly speaking, the interference phenomenon does not necessitate the two rays to undergo minimal deflection. Large deflection angles are permissible for a generic Young’s slit system. Nevertheless, observations of the Crab pulsar impose certain constraints on the possible deflection angle. We note that the deflection angle determines the difference in optical paths, $\Delta l$ , and consequently the fringe order, $m_i$ , as illustrated in figure 4. Specifically, constructive interference occurs when $a(\omega )\sin |\hat \alpha | = \Delta l = m_i\lambda$ (see discussion in the next section). For concreteness, let us assume that $a \sim 10^7$ cm (i.e. ten stellar radii) and $\lambda \sim 1$ cm (corresponding to a frequency of 30 GHz). Firstly, Medvedev (Reference Medvedev2024) shows that $m_{i}\gtrsim 10^{2}$ is favoured by observations, resulting in $|\hat \alpha |\gtrsim m_{i}\lambda /a\sim 10^{-5}$ . Secondly, the duration of the HFIP pulses, $\tau$ , is typically a few milliseconds. For interference to occur, the difference in light travel time along the rays should be much smaller than the pulse duration, $\Delta l \ll \tau c$ . This yields the second constraint on the possible fringe order $m_{i}\sim \Delta l/\lambda \ll \tau c/\lambda \sim 10^{4}$ and the deflection angle $|\hat \alpha |\sim \Delta l/a\ll 10^{-3}$ . Consequently, the interfering rays must undergo a tiny, although non-zero, deflection.

By considering (3.23), we observe that ray deflection vanishes identically when $n=1$ . Indeed, direct integration with $n=1$ yields $\phi _{\textrm {tot}}= \pi$ for any value of $b$ . This is approximately the case for (3.19) and (3.22), as well, provided $r_{m}\gg 2m,\, r_{0}$ . We remind the reader that the distance of the closed approach, $r_{m}$ , is implicitly given by (3.21). Furthermore, since the integral in (3.22) takes the major contribution near $r\sim r_{m}$ and noting that in the ‘far-away approximation,’

(3.25) \begin{equation} r_{m}\simeq b_{0}, \end{equation}

we can determine $b_{0}$ from

(3.26) \begin{equation} n_{\textrm {weak}}(b_{0})\simeq 1. \end{equation}

We remind the reader that $b_{0}$ is the impact parameter of a special ray for which the defocussing effect of the plasma and the focussing effect due to gravity cancel each other. Upon solving (3.26), we obtain

(3.27) \begin{equation} b_{0}^{\kappa -1}\simeq \frac {r_0^\kappa }{2m}=\frac {r_0^\kappa }{r_{\textrm {g}}}. \end{equation}

Consequently, the slit (or hotspot) separation is

(3.28) \begin{equation} a(\omega )=2b_{0}\simeq 2\left (\frac {r_0^\kappa }{r_{g}}\right )^{{1}/({\kappa -1})}\propto \omega ^{-{2}/({\kappa -1})}. \end{equation}

Hereafter, we omit the frequency subscript ‘ $\infty$ ’ as redundant.

4. Interference

The interference pattern generated by a monochromatic wave passing through two slits consists of bright fringes at locations where the phase difference along two paths is a multiple of $2\pi$ , resulting in a path length difference that is a multiple of the wavelength, $\lambda$ . The same ‘striped’ pattern is observed in the frequency domain by an observer located at a specific position, provided that the source possesses a broadband spectrum.

For each frequency, the condition for a bright $i$ -th fringe (interference maximum) is

(4.1) \begin{equation} a(\omega _i)\sin \theta _{\textrm {obs}}=m_{i}\lambda _{i}=[m_1\pm (i-1)](2\pi c/\omega _i), \end{equation}

where $1\le i\le N$ and observations of the Crab pulsar’s HFIP indicate the presence of approximately $N\sim 30$ fringes between approximately 5 and 30 GHz. Here, $m_{1}$ represents the fringe order (i.e. the number of the fringe) corresponding to the lowest frequency $\omega _{1}$ . Observationally, the fringes exhibit a proportional separation with the ‘6 % rule,’ i.e. $\Delta \omega =(\omega _{i+1}-\omega _{i})\simeq 0.057\,\omega _{i}$ . Consequently, we define a parameter $\delta =0.057$ that characterises the spectral band separation as follows:

(4.2) \begin{equation} \omega _{i+1}/\omega _{i}=(1+\delta ). \end{equation}

Finally, $\theta _{\textrm {obs}}$ denotes the location of an observer relative to the source-slits direction, as illustrated in figure 4. It is clear that $\theta _{\textrm {obs}}=|\hat \alpha |$ , with $\hat \alpha$ being the ray deflection angle.

Combining (4.1) and (4.2), we obtain the relation first derived in Medvedev (Reference Medvedev2024)

(4.3) \begin{equation} a(\omega )=a(\omega _{1})\,\frac {\omega _1}{\omega }\left (1\pm \frac {\log _{1+\delta }(\omega /\omega _1)}{m_1}\right)\!. \end{equation}

It is reasonable to assume that the total number of fringes produced in space significantly exceeds the quantity observed, $N_{\textrm {tot}} \gg N$ . Consequently, the order of any observed fringe, $m_{i}$ , should also be substantially larger than $N$ , as statistically $m_{i} \sim N_{\textrm {tot}}/2$ . This implies that there is a greater likelihood of observing higher-order fringes from the ‘bulk’ of the pattern compared with those near its edge, $m_{i} \lesssim N$ . Therefore, assuming $m_{1} \delta \gg 1$ , the above equation can be expressed in the following form:

(4.4) \begin{equation} a(\omega )=a(\omega _{1})\left (\frac {\omega }{\omega _1}\right )^{-\beta },\quad \beta =1\mp \frac {1}{m_1\delta }\approx 1. \end{equation}

Direct comparison of (3.28) and (4.4) yields

(4.5) \begin{align} &\kappa =1+\frac {2}{1\mp ({1}/({m_1\delta }))}\approx 3,\\[-10pt]\nonumber \end{align}

(4.6) \begin{align} &a(\omega _{1}) \approx 2 \Big(\frac{r_{0}^{3}(\omega _{1})}{r_{g}}\Big)^{1/2} =2\Big(\frac{r_{*}^{3}}{r_{g}}\Big)^{1/2}\frac{\omega_{p*}}{\omega_{1}}.\end{align}

The plasma density profile derived from observations is, therefore,

(4.7) \begin{equation} n_{e}(r)\propto r^{-3}. \end{equation}

This result is in remarkable agreement with theoretical expectations for a dipolar magnetic field.

Indeed, pulsar theory and simulations predict that the minimum density needed to be present in the magnetosphere for it to co-rotate with the pulsar as a rigid body up to the light cylinder is the so-called Goldreigh–Julian density

(4.8) \begin{equation} n_{\textrm {GJ}}(r)=\frac {\Omega B}{2ce}\simeq (0.07\,\textrm {cm}^{-3})\frac {B}{P}, \end{equation}

where $B$ is the magnetic field strength in gauss and $P=2\pi /\Omega$ is the spin period in seconds. The actual density of plasma in the magnetosphere can be larger than $n_{\textrm {GJ}}$ by a factor of multiplicity $\mathcal{M}\gt 1$ . For the Crab pulsar ( $P\simeq 0.0335$ s) with a dipolar field $B\simeq (4\times 10^{12}\textrm {G})(r/r_*)^{-3}$ ,

(4.9) \begin{equation} n_{e}(r)\simeq (8.5 \times 10^{12}\,\textrm {cm}^{-3})\ \mathcal{M} \left (\frac {r}{r_*}\right )^{-3} \propto r^{-3}. \end{equation}

For our analytical results obtained above, we employed the ‘far-away approximation,’ which implies

(4.10) \begin{align} &\frac {b_{0}}{r_{g}}=\left (\frac {r_{*}}{r_{g}}\right )^{3/2}\frac {\omega _{p*}}{\omega }\gg 1, \end{align}

(4.11) \begin{align} &\frac {b_{0}}{r_{0}}=\left (\frac {r_{*}}{r_{g}}\right )^{1/2}\left (\frac {\omega _{p*}}{\omega }\right )^{1/3}\gg 1. \end{align}

For the Crab pulsar, the pulsar radius over the gravitational radius is $r_{*}/r_{g}\sim 2.4$ , and the plasma frequency at the star surface is

(4.12) \begin{equation} \nu _{p*}=\omega _{p*}/2\pi \simeq 26\mathcal{M}^{1/2}\,\textrm {GHz}. \end{equation}

Consequently, both constraints are reasonably well satisfied, with the exception of a small multiplicity, $\mathcal{M}\sim 1$ , when $\omega _{p*}\sim \omega$ near the upper end of the observational frequency range.

Now, we validate our findings directly from (3.22) without resorting to approximations. For this purpose, we numerically solve equation $\phi _{\textrm {tot}}=\pi$ for $b_{0}$ across various values of $\kappa$ . Notably, the obtained numerical solution, $b_{0}(r_{0})$ , appears to be remarkably well approximated by the simplified analytical expression given by (3.27). Subsequently, employing $a=2b_{0}$ , we compute the interference pattern and determine the positions of the fringes. The parameters of the model (e.g. the value of $m_{1}$ ) are properly adjusted to ensure that the number of fringes within the observational frequency range is approximately 30, as observed for the Crab pulsar. Finally, we present the $\Delta \omega$ vs. $\omega$ dependence in figure 5. In this figure, the results for $\kappa =2.8,\ 2.9,\ 3.1,\ 3.2$ are presented. Approximate data points from Hankins et al. (Reference Hankins, Eilek and Jones2016) are also depicted. The straight black dashed line indicates the anticipated dependence for the analytically obtained value of $\kappa =3$ (given by (4.5)), which also coincides with the best fit to the data. It is evident that the numerical points converge to the observed linear trend $\Delta \omega =0.057\omega$ as $\kappa \to 3$ . Clearly, $\kappa \approx 3$ provides the most accurate approximation to the observed data. In contrast, other values of $\kappa$ tend to deviate from the observed ‘6 % rule’.

Figure 5. The frequency fringe pattern obtained in the double-slit set-up with the slit separation, $a(\omega )=2b_{0}$ , obtained from (3.22) for $\phi _{\textrm {tot}}=\pi$ . Four different values of the power-law index $\kappa =2.8,\ 2.9,\ 3.1,\ 3.2$ are presented. The black dashed line indicates the anticipated dependence for the analytically obtained value of $\kappa =3$ , see (4.5). Other numerical parameters are adjusted so that there approximately $N\simeq 30$ fringes in the 5–30 frequency domain (arbitrary units). The approximate data points from Hankins et al. (Reference Hankins, Eilek and Jones2016) are also shown.