## Abstract

In the paper “Embedding the photon with its relativistic mass as a particle into the electromagnetic wave” [Opt. Express **26**, 1375 (2018). [CrossRef] [PubMed] ], it has been shown that the problem why the energy and the mass density of an electromagnetic wave are propagating in the same direction can be solved by the assumption that a transverse force is exerted on the photons. This leads to the result that the photon is moving within a transverse potential, which allows the description of the transverse quantum mechanical motion of the photon by a Schrödinger equation. These results are used to show that, in the case of a Gaussian wave, the effective axial propagation constant ${\overline{k}}_{z,nm}(z)$ can be expressed as ${\overline{k}}_{z,nm}(z)=\left[{E}_{ph}-{E}_{nm}(z)\right]/\hslash c$, where E_{ph} is the total energy of the photon and E_{nm}(z) are the energy eigenvalues of the transverse quantum mechanical motion of the photon. Since, according to this result, $\hslash c{\overline{k}}_{z,nm}(z)$ represents a real energy, it has also been concluded that the effective axial propagation constant represents a real propagation constant. This leads to the conclusion that ${\lambda}_{nm}(z)=2\pi /{\overline{k}}_{z,nm}(z)=hc/\left({E}_{ph}-{E}_{nm}(z)\right)$ represents the real local wave length of the photon at the position z. According to this conclusion, λ_{nm}(z) increases inversely proportionally to the energy difference E_{ph}-E_{nm}(z), which decreases with decreasing z, and therefore describes the Gouy phase shift in agreement with wave optics. This shows that the deeper physical reason for Gouy phase shift consists in the fact that the energy of the photon is increasingly converted into its transverse quantum mechanical motion when the photon approaches the focus. This explains the Gouy phase shift as an energetic effect.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

In [1] the question has been considered, why the energy density and the mass density of an electromagnetic wave propagate in the same direction, even if the energy density, whose propagation direction is described by the Poynting vector, is propagating along a curved line as shown by Fig. 1 in [1] which displays a resonant Gaussian wave between two spherical mirrors of an optical resonator. Therefore, if a particle of relativistic mass propagating with the electromagnetic wave is assumed to represent a quantum of energy, its propagation should be described by the Poynting vector. On the other hand, if the particle is considered to represent a real particle of mass, its propagation should be described by Newton's first law, if no force is exerted on it. Therefore, this consideration leads to a contradiction between fundamental physical laws:

In this context it shall be stressed that this contradiction not only concerns the modes in an optical resonator but furthermore concerns any electromagnetic wave. To solve this problem, in [1] the assumption has been made that a transverse force is exerted on the mass density and in consequence on the photons which forces them to follow the propagating energy density. Based on a consideration of the directional change of the Poynting vector during an infinitesimal propagation step of the wave, in [1] the following expression has been derived for this force

_{1}and z

_{2}where two wave fronts Φ

_{1}and Φ

_{2}with infinitesimal distance intersect the propagation axis of the wave, as shown by Fig. 3 in [1]. The S

_{N}(r

_{i},z

_{i}) are normalized Poynting vectors erected in the points r

_{1}and r

_{2}on the wave fronts Φ

_{1}and Φ

_{2}as also shown by Fig. 3 in [1]. In the infinitesimal limit the points r

_{i}as well as z

_{i}are assumed to move into each other. E

_{ph}is the energy of the photon given bywhere λ and c are the wavelength and the speed of light in a vacuum, respectively. M = h/(cλ) is the relativistic mass of the photon.

From the above results it has furthermore been concluded in [1] that the photon is moving within a transverse potential V(r,z) which is obtained by integrating the negative value of the force **K**(r,z) given by Eq. (1) over r along the curvature of the phase front Φ(r,z). Based on this conclusion in [1] the assumption has been made that the transverse motion of the photon is described by the following Schrödinger equation

In case of a Gaussian wave Eq. (3) transforms into the Schrödinger equation of the 2-dimensional harmonic oscillator given by

_{┴}describes the frequency of the transverse quantum mechanical oscillation of the photon, and is given byHere z

_{R}= πw

_{0}

^{2}/λ is the Rayleigh range, and w

_{0}is the spot size at the beam waist. In [1] the Eqs. (4) and (5) have been verified by comparing the probability density of the photon, which is represented by the squared absolute values |χ

_{nm}(x,y,z)|

^{2}of the eigenfunctions χ

_{nm}of the Schrödinger equation, with the result obtained by the use of wave optics for the normalized local intensity of a Gaussian wave of order n,m as described by Eq. (16.60) in [2]. This comparison provided full agreement between both quantities.

As well known, the energy levels E_{nm} of the Schrödinger equation of the 2-dimensional harmonic oscillator are given by

_{nm}(z) and the Gouy phase shift [3] which seems to allow for explaining the Guoy phase shift as an energetic effect.

## 2. Is the Gouy phase shift an energetic effect?

To answer this question we consider the effective axial propagation constant for a finite beam which according to the Eqs. (3) and (19) in [4] can be expressed as

^{2}delivers $<{k}_{x,nm}^{2}>+<{k}_{y,nm}^{2}>$. In this way, we obtain

_{ph}/c

^{2}, we obtain

_{ph}can be expressed aswe obtain

_{ph}of the photon and the energy eigenvalues E

_{mn}(z) of the Schrödinger equation which increase with decreasing z, since ω

_{┴}increases according to the Eq. (5). It can be therefore concluded that part of the total energy of the photon is transformed into the energy of the transverse quantum mechanical motion of the photon, and another part is transformed into $\hslash c{\overline{k}}_{z,nm}(z)$. Since according to this result $\hslash c{\overline{k}}_{z,nm}(z)$ represents a real energy, it must be assumed that also the effective axial propagation constant ${\overline{k}}_{z,nm}(z)$ represents a real propagation constant. This leads to the conclusion that

_{nm}(z) increases inversely proportional to ${\overline{k}}_{z,nm}(z)$, and therefore, describes the Gouy phase shift in agreement with wave optics. But this conclusion also shows that λ

_{nm}(z) increases inversely proportional to energy difference ${E}_{ph}-{E}_{nm}(z)$ which decreases with decreasing z. This seems to show that the deeper physical reason for Gouy phase shift consists in the fact that the energy of the photon is increasingly converted into its transverse quantum mechanical motion when the photon approaches the focus. This allows to conclude that the Gouy phase shift is an energetic effect. Therefore, it seems to be reasonable to designate the energy $\hslash c{\overline{k}}_{z,nm}(z)$ as Gouy energy, and to introduce for this energy the term E

_{G,nm}(z). Based on this result Eq. (15) can be rewritten aswhich shows that the total energy E

_{ph}is the sum of an axial part described by E

_{G,nm}(z), and the energy eigenvalues E

_{mn}(z) of the Schrödinger equation. However the particle picture not only explains the Gouy phase shift as an energetic effect, it also allows to compute the mathematical expression describing the Gouy phase shift in agreement with wave optics as already shown in [1].

The upper part of Fig. 1 shows for λ=1[µm] and w_{0}=1.12√2/k [µm] how the energies E_{G,00}(z) and E_{00}(z) change with the propagating wave. The lower part shows how the spot size changes. Under the same conditions, the upper part of Fig. 2 shows how the local wave length λ_{00}(z) changes as a function of z [µm]. The lower part shows how the Gouy phase shift changes.

Since E_{Gnm}(z) vanishes for E_{nm}(z) = E_{ph}, λ_{nm}(z) goes to infinity under this condition. Therefore, E_{nm}(z) < E_{ph} must be considered to be a limiting condition for E_{nm}. The condition E_{nm}(z) < E_{ph} transforms, after replacing E_{nm}(z) according to Eq. (6) and E_{ph} according to Eq. (2), into

_{┴}(z) according to Eq. (5) transforms intoThis equation transforms for z = 0 intoThis delivers for the spot size w

_{0}at the beam waist the conditionIf we replace in this equation the “>” sign by the “=” sign we obtain for n = m = 0This delivers the relationThis result shows that under the condition E

_{nm}(z) = E

_{ph}the frequency ω

_{┴}of transverse quantum mechanical oscillation of the photon is becoming equal to the frequency ω of the initially incoming electromagnetic wave. Therefore, in this case the oscillation of the wave is in the focus totally transformed into the transverse quantum mechanical oscillation of the photon. Simultaneously, the effective axial propagation constant ${\overline{k}}_{z,00}$ vanishes according to Eq. (14), and the local wave length λ

_{nm}(z) goes to infinity according to Eq. (16).

Concerning the Eqs. (20)–(22) it shall be mentioned that there is mathematical agreement between Eq. (22) and Eq. (30) given in [5] which according to the considerations given in [5] describes the fundamental mode radius. Equation (21) can therefore be considered as a generalization of Eq. (30) in [5] for higher order Gaussian modes. This shows once more that the particle picture provides results in agreement with wave optics as has been used in [5].

The fact that effective axial propagation constant ${\overline{k}}_{z,00}(0)$ vanishes under the condition ${w}_{0}=\lambda /\sqrt{2}\pi $ also can be derived from the wave optics description of the Gouy effect by differentiating the phase of the fundamental mode that is given by Φ(z)=kz+Φ_{G}(z) with Φ_{G}(z)=arctan(z/z_{R}). Differentiation of Φ(z) versus z delivers for z=0

_{0}=√2/k. Therefore, the question arises what is happening with a wave whose phase stops to oscillate with the propagating wave. Can it still be considered to be a wave which transports energy? The presented particle picture says that the total energy of the photon is totally transformed into its transverse quantum mechanical motion under the above mentioned condition, and the wave itself does not transport energy anymore. An other aspect of this problem arises, if one considers the expectation value of the square of the z component of the wave-vector which is according to Eq. (7) given by $<{k}_{z}^{2}>={\overline{k}}_{z,00}k$. This expression shows that <k

_{z}²> also vanishes for w

_{0}=√2/k according to Eq. (24), and is becoming even negative for w

_{0}<√2/k. Therefore, the question arises, how is a wave with negative <k

_{z}²> propagating. These problems deserve further consideration.

## 3. Is the force exerted on the photons a real force?

To answer this question we take into account that the quantity Δz used in Eq. (1) can be replaced by Δt = Δz/c which represents the infinitesimal time step which the phase front Φ_{1} takes to move in the phase front Φ_{2} as shown by Fig. 3 in [1]. This transforms Eq. (1) into

**K**(r,z) exerted on the photon is equal to the momentum of the photon multiplied with the infinitesimal directional change of the normalized Poynting vector versus the infinitesimal time step Δt. Therefore, since an infinitesimal momentum change versus an infinitesimal time step represents a force, the right side of Eq. (25) physically represents a force, though it may not be possible to conclude that a photon, which is found at a certain position r,z, changes its momentum at this position. However, it seems to be necessary to assume that an overall continuous change of the momentum of the photons propagating with an electromagnetic wave takes place, since otherwise the photons would not follow the propagating wave, and would spread out from the wave due to their relativistic mass. Therefore, it seems to be possible to conclude that the

**K**(r,z) represents a force which causes a directional change of the momentum of the photons, and in this way, forces the photons to follow the electromagnetic wave.

An other interesting aspect of the force exerted on the photon consists in the fact that, in case of a Gaussian wave, a ray bouncing between two virtual phase fronts with infinitesimal distance carries through a sinusoidal motion which according to Eq. (48) in [1] is described by r = Acos(ω_{┴}t), and therefore oscillates with a frequency ω_{┴} identical with the frequency of the quantum mechanical transverse oscillation of the photon given by Eq. (5). Therefore, since a ray of infinitesimal length can be considered to represent the classical equivalent to the photon, it follows that the transverse motion of this classical equivalent particle is described by the same potential which describes the motion of the photon with the difference that the photon obeys the laws of quantum mechanics. It can be therefore concluded that the same force, which is exerted on the classical equivalent particle, is also exerted on the photon. This shows that the force derived above by considering of the momentum change of the photon, also can be derived from a geometric optics consideration.

Equation (25) also shows that this force does not change due to the splitting of E_{ph} into E_{G,nm} + E_{nm}, since the momentum Mc of the photon remains unchanged. The latter follows from the fact that the expectation value of the momentum of the harmonic oscillator vanishes with the consequence that the total momentum Mc of the photon remains unchanged. Therefore, also the radiation pressure in the focal region remains unchanged. The latter is important, since both quantities are used to describe the optical tweezers [6–8].

## 4. Summary and conclusions

Based on a consideration of the question, why the energy density and the relativistic mass density of an electromagnetic wave are propagating in the same direction, in [1] the assumption has been made that a transverse force is exerted on the mass density and in consequence on the mass of the photons which forces them to follow the propagating energy density. This assumption leads to the result that the photon is moving within a transverse potential which allows to describe the transverse quantum mechanical motion of the photon by a Schrödinger equation which is identical with the Schrödinger equation describing the motion of the electron, except that the mass of the electron is replaced by the relativistic mass of the photon [1]. In case of Gaussian waves this Schrödinger equation is identical with the Schrödinger equation of the 2-dimensional harmonic oscillator [1]. This Schrödinger equation is used to compute the expectation value of the square of the photon's transverse momentum which, after some transformations, allows to express the effective axial propagation constant ${\overline{k}}_{z,nm}(z)$ of the wave as given by Eq. (3) in [4] by ${\overline{k}}_{z,nm}(z)=\left[{E}_{ph}-{E}_{nm}(z)\right]/\hslash c$ where E_{ph} is the total energy of the photon, and the E_{nm}(z) are the energy eigenvalues of the transverse quantum mechanical motion of the photon. Since according to this result $\hslash c{\overline{k}}_{z,nm}(z)$ represents a real energy, also the effective axial propagation constant ${\overline{k}}_{z,nm}(z)$ must be assumed to represent a real propagation constant. This leads to the conclusion that ${\lambda}_{nm}(z)=2\pi /{\overline{k}}_{z,nm}(z)=hc/\left({E}_{ph}-{E}_{nm}(z)\right)$ represents the real local wave length of the photon at the position z. According to this conclusion, λ_{nm}(z) increases inversely proportional to ${\overline{k}}_{z,nm}(z)$ with decreasing z, and therefore, describes the z dependence of the Gouy phase shift in agreement with wave optics. But this conclusion also shows that λ_{nm}(z) increases inversely proportional to the energy difference E_{ph}-E_{nm}(z) which decreases with decreasing z. This shows that the deeper physical reason for Gouy phase shift seems to consist in the fact that the energy of the photon is increasingly converted into its transverse quantum mechanical motion when the photon approaches the focus. This explains the Gouy phase shift as an energetic effect. Based on this result the proposal has been made to designate the energy $\hslash c{\overline{k}}_{z,nm}(z)$ as Gouy energy, and to introduce for this energy the term E_{G,nm}(z). This delivers for the total energy of the photon the expression E_{ph} = E_{G,nm}(z) + E_{nm}(z). However the particle picture not only explains the Gouy phase shift as an energetic effect, it also allows to compute the mathematical expression describing the Gouy phase shift in agreement with wave optics as already shown in [1].

Since according to these results only E_{G,nm}(z) changes when photon approaches the focus but not its total energy E_{ph}, it follows that the wave length of the photon as measured by a spectroscopic device is λ = hc/E_{ph} and not λ_{nm}(z), because the measurement process changes the structure of the propagating wave with the consequence that the change of the wave length disappears together with the Gouy phase shift.

Since E_{G,nm}(z) decreases, when the photon approaches the focus, and is transformed into E_{nm}(z), which simultaneously increases by the same amount, it can be concluded that the wave structure of the photon is becoming less important compared with its particle property near the focus. This result may provide new aspects concerning the theoretical description of the optical tweezers.

The question, if the force exerted on the photons represents a real force, has been considered in Sect. 3. Equation (25) shows that the force described by **K**(r.z) is equal to the momentum of the photon multiplied with the infinitesimal directional change of the normalized Poynting vector versus a time step Δt describing an infinitesimal propagation step of the wave. Therefore, since an infinitesimal momentum change versus an infinitesimal time step represents a force, the right side of Eq. (25) physically represents a force, though it may not be possible to conclude that a photon, which is found at a certain position r,z, changes its momentum at this position. Since however nevertheless an overall momentum change of the photons must take place which forces them to follow the electromagnetic wave, **K**(r.z) seems to represent the force, which causes this momentum change.

## References

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