Investigation of the Abraham–Minkowski dilemma in Smith–Purcell radiation from photonic crystals

Free electron emission has fueled many scientific and technological advances over the last few decades.^1–10 When a free electron propagates near or through a periodic structure (a grating, for example), free electron emission occurs. This phenomenon, known as the Smith–Purcell effect, was first observed by Smith and Purcell in 1953.¹¹ For over 70 years of development, the formula (conventional formula) used to predict the photon frequency emitted via Smith–Purcell radiation is based on Minkowski’s momentum.^12–17 In Minkowski’s momentum, the momentum of a photon in a medium is assumed to be p_Min = nℏk,¹⁸ where n is the refractive index of the medium, ℏ is the reduced Planck constant, and k is the wave vector of the emitted photons in free space. Tsesses et al. conducted a theoretical investigation into Smith–Purcell radiation in the infrared and visible spectra, employing Minkowski’s momentum formulation.¹⁹ Similarly, Huang et al. performed both theoretical and experimental studies on quantum recoil effects during interactions between free electrons and atomic lattices, also utilizing Minkowski’s momentum concept.²⁰ Conversely, there exists a contrasting momentum, e.g., Abraham’s momentum, which contends that the momentum of a photon in a medium is given by p_Abr = ℏk/n.¹⁸ 

The debate over whether light’s momentum within a medium is accurately described by the Abraham or Minkowski formulation has persisted for over a century.^12,18 It is remarkable that the Abraham–Minkowski dilemma continues to be a focal point of significant debate to this day.^{12,18,21–24} Recently, both the Abraham and Minkowski formulations of momentum are now recognized as correct within their respective contexts. The Abraham momentum represents the kinetic momentum of light and aligns well with classical mechanical scenarios, particularly those involving the center of mass motion. In contrast, the Minkowski momentum, identified as the canonical momentum, proves especially useful in quantum mechanical treatments, such as those involving photon recoil and field translations.

To our knowledge, this dilemma has not been explored within the context of Smith–Purcell radiation. This is because most of the Smith–Purcell radiation was measured in free space, where the value of the refractive index is one [Fig. 1(a)].^12–17 Recent studies have explored the atomic version of Smith–Purcell radiation in van der Waals materials [Fig. 1(b)].^20,25–31 However, the output photon energies are in the x-ray regime, where the refractive index of van der Waals materials is remarkably close to one. As a result, both the Abraham and Minkowski-based theories predict identical photon energies. We would like to point out that the Abraham–Minkowski dilemma generally applies to all forms of free electron radiation, including Cherenkov radiation and Smith–Purcell radiation. Notably, Cherenkov radiation is characterized by its broadband nature, which means that the observed Cherenkov radiation spectrum in any specific direction exhibits a wide range of frequencies. Given the current state of experimental technology, it is exceptionally challenging to definitively determine whether the photon energy in Cherenkov radiation is governed by Abraham momentum or Minkowski momentum. As a result, despite extensive theoretical and experimental studies on Cherenkov radiation, its implications for the Abraham–Minkowski dilemma remain largely unexplored.

Here, we investigate the Abraham–Minkowski dilemma within the realm of Smith–Purcell radiation from photonic crystals, where the refractive index deviates from one. We develop a formula to predict the photon frequency of Smith–Purcell radiation based on Abraham’s momentum. We employ formulas derived from both Abraham’s and Minkowski’s theories to investigate Smith–Purcell radiation, which is produced by 3 MeV free electrons impinging a photonic crystal with a grating length of 2.1 μm. Our analysis reveals that the emission is red light when evaluated through the formula based on Abraham’s momentum, whereas the emission shifts to blue light when analyzed using the formula based on Minkowski’s momentum. Furthermore, our results demonstrate that the variation in wavelength, as predicted by the Abraham and Minkowski momentum-based formulas, is influenced by both the grating length and the refractive index. The investigation of the Abraham–Minkowski dilemma in the context of Smith–Purcell radiation can enrich our understanding of both phenomena (e.g., the Abraham–Minkowski dilemma and Smith–Purcell radiation). Smith–Purcell radiation has shown promising applications in tunable light sources. To accurately determine the output of Smith–Purcell-based light sources, a comprehensive understanding of the Abraham–Minkowski dilemma in this specific realm is essential. Furthermore, understanding the role of the refractive index of materials in Smith–Purcell radiation provides an additional degree of control in manipulating free-electron light sources.

Recently, an atomic version of Smith–Purcell radiation has been intensively investigated, where electrons impinge on a van der Waals structure (sub-nanometer grating), resulting in Smith–Purcell radiation. In this scenario, Smith–Purcell radiation is produced within a material, yet the wavelength of the radiation falls within the x-ray spectrum due to the sub-nanometer grating, a range in which the refractive index of all known materials is approximately one. Therefore, theories based on Abraham’s momentum and Minkowski’s momentum yield identical results, i.e., Eq. (12). The atomic version of Smith–Purcell radiation is also known as parametric x-ray radiation.^35–40 

To address the Abraham–Minkowski dilemma within the realm of Smith–Purcell radiation, we study Smith–Purcell radiation from photonic crystals, where the refractive index deviates from one [Fig. 1(c)]. The effective refractive index of a multilayer grating can be determined using effective medium theory.⁴¹ Our key findings demonstrate considerable robustness with respect to variations in the refractive index. In particular, the predicted output photon energies of Smith–Purcell radiation based on Abraham and Minkowski momenta exhibit substantial differences for any refractive index value other than unity. To illustrate this phenomenon, we have chosen a representative refractive index of 1.9 for our analysis, without loss of generality.

It is important to note that Cherenkov radiation can occur when a free electron traverses a photonic crystal,⁴² as illustrated in Fig. 1(c). One of the main distinctions between Cherenkov and Smith–Purcell radiation lies in the emission process: Cherenkov radiation is generated and propagated within the same crystal, whereas Smith–Purcell radiation is emitted via periodic grating.⁴² Furthermore, Smith–Purcell radiation exhibits narrowband photon energies at specific observation angles, in contrast to the broadband nature of Cherenkov radiation.

In Fig. 1(c), the predicted wavelength of Smith–Purcell radiation is determined by Eq. (6) for Minkowski’s momentum and Eq. (11) for Abraham’s momentum, respectively. Furthermore, it is no longer possible to simplify these two equations to yield an identical equation. Now, we consider Smith–Purcell radiation generated by 3 MeV free electrons impinging a photonic crystal with a grating length of 2.1 μm. The emission appears as red light when analyzed based on Abraham’s momentum [Fig. 2(a)] and as blue light when analyzed based on Minkowski’s momentum [Fig. 2(b)]. Figure 2(c) compares the predicted wavelengths of Smith–Purcell radiation based on Abraham’s momentum and Minkowski’s momentum at various grating lengths, where the electron energy is 3 MeV and the refractive index is 1.9. The results indicate that the disparity in wavelength predicted by theories based on Abraham’s momentum and Minkowski’s momentum depends on the grating length. We clarify that the incident electrons penetrate the photonic-crystal grating shown in Figs. 1(c), 2(a), and 2(b), leading to the emission of both Smith–Purcell radiation and Cherenkov radiation. In this work, we focus on Smith–Purcell radiation, as it provides a valuable framework for studying the Abraham–Minkowski dilemma.

Figure 3 compares the predicted wavelengths of Smith–Purcell radiation based on Abraham’s momentum and Minkowski’s momentum at various refractive indexes, where the electron energy is 3 MeV and the grating length is 2.1 μm. The findings suggest that the variation in wavelength predicted by Abraham’s momentum and Minkowski’s momentum changes in relation to the refractive index. In addition, Fig. 3 demonstrates that both Abraham’s momentum and Minkowski’s momentum produced identical outcomes in previous studies where the refractive index (n) equals one. We build upon previous research by venturing into an unexplored regime where n ≠ 1. This exploration reveals that the predicted wavelengths of Smith–Purcell radiation, based on Abraham’s momentum and Minkowski’s momentum, significantly diverge from one another.

In summary, we have demonstrated that the Abraham–Minkowski dilemma can be potentially resolved within the context of Smith–Purcell radiation emanating from photonic crystals. Our findings show that the predicted wavelengths of Smith–Purcell radiation, based on Abraham’s momentum and Minkowski’s momentum, significantly diverge from one another. In particular, we find that 3 MeV free electrons impinge a photonic crystal with a grating length of 2.1 μm, resulting in the emission of red light when analyzed based on Abraham’s momentum, and blue light when analyzed based on Minkowski’s momentum. The discrepancy in the predicted wavelengths, when based on Abraham’s momentum vs Minkowski’s momentum, is substantial enough to be readily distinguishable with existing spectrometers. Thus, our findings offer a method to address the Abraham–Minkowski dilemma within the context of Smith–Purcell radiation, thereby enhancing our understanding of both the Abraham–Minkowski dilemma and Smith–Purcell radiation.

This research was supported by the National Natural Science Foundation of China (Grant Nos. 62371391 and 11704310), the Youth Innovation Team of Shaanxi Universities (Grant No. 21JP084), and by the Key Core Technology Research Project for Strategic Industry Chains of the Xi’an Science and Technology Bureau (Grant No. 23LLRH0057).

The authors have no conflicts to disclose.

Suguo Chen: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal). Pengtao Wang: Data curation (equal); Investigation (equal). Yue Wang: Data curation (equal); Formal analysis (equal). Sunchao Huang: Data curation (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). Lei Hou: Data curation (equal); Funding acquisition (equal); Methodology (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.