1. Introduction

There was a century-long debate concerning the momentum of light and the electromagnetic energy-momentum tensor in a dielectric medium related to two famous German scientists: Max Abraham and Hermann Minkowski. Many excellent reviews have been devoted to this problem [1–4], but the controversy is still not fully resolved, as can be seen from recent publications [5–13]. Theoretical papers devoted to derivations of components of the energy-momentum tensor based on different formulations of the electromagnetic field theory continue to appear. For example, it has been shown that the Abraham energy-momentum tensor contradicts special relativity, whereas the Minkowski tensor is fully consistent with it [14–18].

The debate about the momentum of light has gained additional importance, following recent advances in the development of single- and few-photon sources and detectors for quantum technologies (see a recent review [19]). Investigations of the discrete nature of light, single photon experiments including temporal and spatial control, manipulation, and structuring of photons, all these achievements are opening a new era of quantum photonics. That is why it is important to know what happens to a single photon in a dielectric medium, where, due to the excitation of dipole oscillations of bound electrons and polarization wave, the photon (together with the excited dipoles) forms a relativistic quasiparticle (photon in medium, which will be called m-photon) propagating at the speed of light $v=c/n$ in nondispersive dielectrics. Concerning a full classification of the existing photonic quasiparticles, a recent review [20] can be recommended.

In many papers and textbooks, without mentioning the above controversy, people are simply using the Minkowski momentum for a photon in a dielectric medium $p_M =n \hbar \omega /c$, where $n$ is the refractive index, $\omega$ is the frequency of light, and $c$ is the speed of light in free space (vacuum). We all agree, at least many of us, that this expression is correct and is broadly used in photonics, laser physics, and nonlinear optics. However, to obtain and verify this expression from classical electrodynamics, the components of the energy-momentum tensor must be derived. Even for the case of plane monochromatic waves, this derivation is quite complex and can only be followed by experts in the field. Moreover, using different formulations of electrodynamics results in different expressions for the energy-momentum tensor and photon momentum – the reason for the ongoing Abraham/Minkowski debate.

In this paper, we present a simple derivation of the photon momentum, by considering photons in media as relativistic quasiparticles (m-photons). In contrast to the complex derivations and discussions of the energy-momentum tensor components, our derivations can be easily reproduced by a very broad scientific audience including students. In Sec. 2, we first introduce the well-known formulas for the momentum, mass, and energy of the m-photon and discuss arguments for and against the Abraham and Minkowski momenta. Afterwards in Sec. 3, we demonstrate that the Minkowski form of the photon momentum follows directly from the relativistic energy conservation law for the m-photon quasiparticle. In Sec. 4, the presented theory is directly applied to derive a new expression for the photon momentum in dispersive media. In the final part, we discuss the effect of light-induced optical stretching and its potential application for measurement of photon momenta in dispersive media.

2. Mass, momentum, and energy of the m-photon (Abraham vs. Minkowski)

In this section, we introduce the well-known formulas for the momentum, mass, and energy of the m-photon following Abraham and Minkowski. For simplicity, we consider first a uniform, non-dispersive, non-absorbing, linear dielectric medium with the refractive index $n\geq 1$. In such a medium the photon phase and group velocities coincide and are determined by $\textbf {v}=c/n\times \textbf {k}/{k}$, where $\textbf {k}$ is the wave vector. Both, the speed of light $c/n$ and its wavelength $\lambda /n$, where $\lambda$ is the wavelength of light in free space, are reduced. The frequency of light $\omega$ remains unchanged. In this paper, the refractive index and light frequency are defined for a medium at rest with anti-reflection coatings and the geometry shown in Fig. 1. The medium at rest coincides with the laboratory coordinate frame.

 

Fig. 1. Illustration of a photon entering a dielectric medium with antireflection coatings at normal incidence, exciting dipole oscillations, and generating a "photon in medium" quasiparticle (m-photon).

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The speed of light and its wavelength are directly proportional to each other. For the electrical field of a plane wave propagating along the $z$ axis, we can write

Below we provide expressions for the m-photon mass and energy, which follow from the Abraham and Minkowski momenta. A photon has a zero mass in vacuum. In a dielectric medium it becomes a quasiparticle (m-photon) with non-zero mass, which can be found by equating the relativistic momentum to that of Abraham or Minkowski. From $v=c/n$, the relativistic Lorentz factor $\gamma =1/\sqrt {1-v^2/c^2}=n/\sqrt {n^2-1}$, and the expression $\gamma \: m_{A,M}\: \textbf {v} =\textbf {p}_{A,M}$, we obtain two different rest masses for Abraham and Minkowski m-photons, which are related by the following expression:

Now we turn to the discussion of the m-photon energy. In Abraham’s case, the m-photon energy $E_A=\hbar \omega$ is equal to the photon energy in vacuum and at first glance everything appears perfect, since the energy conservation is fulfilled. In case of Minkowski, the m-photon energy $E_M=n^2\: \hbar \omega$ is $n^2$ times larger than the energy of the photon in vacuum. To fulfil the energy conservation law for the Minkowski m-photon, an additional term $U=(n^2-1) \hbar \omega$ must be introduced:

In the next section, we will use relativistic energy conservation principles to present a direct derivation of Eq. (5), without the reliance on additional assumptions. Based on this, we will explain the physical origin of the $U$ term and arrive at the Minkowski form for the momentum of the m-photon.

3. Energy conservation for a photon crossing boundary with a dielectric medium

In this section, we demonstrate for the first time to the best of our knowledge that the Minkowski form of the photon momentum follows directly from the relativistic energy conservation law for the m-photon quasiparticle. The first statement that we make, is that a low energy (infrared, visible, ultraviolet) photon with a zero-mass in vacuum is not creating a new mass when entering into the dielectric medium. The low energy photon is not able to start a nuclear reaction and/or to convert its energy to the birth of a new particle. In a non-absorbing medium the photon energy can be transferred only to kinetic energy, defined as the total relativistic energy minus the rest energy:

 

Fig. 2. Illustration of $\chi ^{(1)}$ processes responsible for refraction and absorption.

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The excited dipole oscillations of bound electrons and moving polarization wave are the integral part of the m-photon quasiparticle. The masses $m_M$ and $m_d$ are connected to each other by $m_M=m_d\gamma$, i.e. the so-called relativistic mass of oscillating dipoles $m_d\gamma$ is equal to the rest mass of the m-photon. Taking this into account, Eq. (6) can be rewritten in the form equivalent to Eq. (5) :

Now we turn to the discussion of the derived energy conservation Eq. (9) in its final, most important form:

The Abraham mass satisfies the following equation $\hbar \omega = \gamma m_Ac^2$, which assumes that the Abraham photon is generating a new mass in the dielectric medium, since in this case the photon energy is spent not only on the kinetic energy, but also on the rest energy, which is not physically justified. In contrast, the presented derivation of Eq. (10) and above discussions provide a straightforward physical justification for the Minkowski momentum $n\hbar \omega /c$, which in this form is broadly used in scientific literature as the photon momentum in a dielectric medium.

4. Dispersive medium

A single photon radiated by an atom or a quantum dot has a certain time duration and frequency bandwidth satisfying the Heisenberg uncertainty relation $\Delta \omega \Delta \: t \sim 2\pi$. Therefore, it is reasonable to expect that the behaviour of such a photon in a dispersive medium with a frequency dependent refractive index will change.

In a dispersive medium, due to the difference between phase and group velocities, the planes of constant phase and constant amplitude do not in general coincide and can be tilted against each other. In such a medium, the propagation of light is determined by the group velocity $v_g=d\omega /dk=c/n_g$, where $n_g=n+\omega \: dn/d\omega$ is the refractive index for the group velocity (the group index) calculated at the carrier frequency $\omega _0$. The group index is constrained by causality to be greater than unity regardless of the sign of the medium refractive index [14,31]. In a uniform isotropic medium the following relation between the group and phase velocities is fulfilled, $\textbf {v}_g n_g=\textbf {v}_p n=c\: \textrm {sign}(n)\textbf {k}/{k}$, where the phase velocity, wavevector, and the refractive index are defined at the carrier frequency. This relation is valid also in case of $n<0$ when the phase and group velocities are antiparallel with the phase and group fronts propagating in the opposite directions [14,31].

It has been experimentally demonstrated that single photons propagate in different media with the group velocity [32–34]. For the m-photon propagating with the group velocity, we can use the energy conservation Eq. (6) with the Lorentz factor determined by the group velocity $\gamma =1/\sqrt {1-v_g^2/c^2}=n_g/\sqrt {n_g^2-1}$. We will get the same Eqs. (7)–(10), with $n$ replaced by $n_g$ and the m-photon momentum determined by $p=p_M=n_g\hbar \omega /c$. This expression corresponds to the m-photon propagating with the group velocity as relativistic quasiparticle with the rest mass $m_M= (\hbar \omega /c^2)\:n_g\sqrt {n_g^2-1}$ and the total energy $E_M=n_g^2\hbar \omega$.

Using $\hbar \omega = \gamma m_Ac^2$, it is straightforward to show that the Abraham momentum in a dispersive medium is determined by $p_A=p_M/n_g^2=\hbar \omega /cn_g$. All papers agree with this expression for the Abraham momentum, whereas several different expressions have been proposed for the Minkowski momentum in a dispersive medium. The first one is $p_{M1}=n^2\hbar \omega /cn_g$ [21,22,26] and another $p_{M2}=n\hbar \omega /c$ [2,5]. Recall that according to the relativistic quasiparticle approach, as it is shown above, $p_M=n_g \hbar \omega /c$. Thus, there are three different expressions for the m-photon momentum. This could be a good motivation for experimentalists to look at what happens to the light momentum in a dispersive medium and which expression is correct. There is only one relatively old experiment [35], where an attempt to measure the light momentum in a dispersive medium has been performed with sufficient accuracy, with the result that this momentum is equal to $p=n\hbar \omega /c$. However, recent theoretical simulations of this experiment leave a lot of questions and uncertainties [36].

Let us briefly discuss the case of $0<n<1$, which naturally occurs at high frequencies, when the refractive index is determined by $n=\sqrt {1-\omega _p^2/\omega ^2}$, where $\omega _p$ is a model parameter (plasma frequency). In this case the phase velocity $v=c/n$ is larger than the speed of light and cannot be used in the relativistic energy conservation Eqs. (6), (10). These equations can be used only with the group velocity. It is easy to verify that the group velocity refractive index is given by $n_g=1/n$, corresponding to the group velocity $v_g=cn$. According to the expressions derived above, we get in such a medium for the m-photon Minkowski momentum $p_M=n_g\hbar \omega /c=\hbar \omega /cn$. Note, that the Abraham momentum will be defined by $p_A=\hbar \omega /cn_g=n\hbar \omega /c$.

In a medium with negative refraction index, the light momentum defined by $p=n\hbar \omega /c$ is oppositely directed to the group velocity and corresponds in case of total reflection to the attractive (negative) light pressure [14,31]. This scenario is difficult to combine with the behaviour of single photon quasiparticles. According to the above discussions, their momentum is determined by $p=n_g\hbar \omega /c$ and they can generate only the positive light pressure. This conclusion is in agreement with recent numerical simulations examining optical forces in negative refractive index materials and demonstrating the positive light pressure [37].

5. Optical stretching

The effect of light-induced optical stretching can be used for experimental verification of the existing expressions for the photon momentum. According to Minkowski, for a photon entering a dielectric medium at normal incidence from free space (see Fig. 1), the momentum increases and the change of momentum $\Delta p=(n-1)\hbar \omega /c = F \Delta \: t$ produces a stretching force $F$ acting at the interface in the opposite direction to the propagation of the photon. If we have a laser pulse with the pulse duration $\Delta \: t=\tau$ and the total number of photons $N$, the stretching force acting at the interface will be given by $F=(n-1)\hbar \omega N/c\tau =(n-1)P/c$, where $P$ is the laser pulse power. When the laser pulse leaves the medium at the opposite end, the same stretching force is produced at the interface acting now in the direction of the laser pulse. The total stretching force acting at the dielectric medium (assuming that the medium length fulfils the condition $l<<c\tau$) $F=2(n-1)P/c$. If the dielectric material is elastic, as for example polydimethylsiloxan (PDMS), the light-induced changes in the material length can be measured by interferometric techniques. Choosing a medium with a strong dispersion, one should be able to verify the value of the momentum $p=n_d\hbar \omega /c$ in the dispersive medium and the corresponding expression for the stretching force $F=2(n_d-1)P/c$, where there are 3 possibilities for the $n_d$ value: $n$, $n^2/n_g$, and $n_g$.

6. Conclusion

In this paper the origin of differences between the Abraham and Minkowski expressions for the photon mass, momentum, and energy in a dielectric medium has been clarified by considering photons in media as relativistic quasiparticles. The Abraham expressions can be easily derived from the equation $\hbar \omega =\gamma mc^2$. For the first time, to the best of our knowledge, we have shown how the Minkowski expressions can be derived from the following energy conservation equation $\hbar \omega =\gamma mc^2-mc^2=\gamma m_Mc^2 - m_Mc^2/\gamma$. In this case, it is considered that the photon energy is transferred into the kinetic energy of a "photon in medium" quasiparticle (m-photon) and is not used for the creation of a new mass (as in the Abraham case). Therefore, from the relativistic energy conservation law the Minkowski expressions for the photon mass, given by Eq. (2), momentum $p=n\hbar \omega /c$, and energy $E=n^2\hbar \omega$ in a dielectric medium are correct and fully justified.

Using the same relativistic approach for quasiparticles moving with the group velocity, a new expression for the photon momentum in a dispersive dielectric medium $p=n_g\hbar \omega /c$, where $n_g$ is the group index, has been derived. Additionally, we have discussed the effect of light induced optical stretching, which may be exploited for an experimental verification of the existing expressions for the photon momentum in dispersive media.