Embedding the photon with its relativistic mass as a particle into the electromagnetic wave

Konrad Altmann

doi:10.1364/OE.26.001375

1. Introduction

The relation between photon and electromagnetic wave has been considered from many aspects [1]; however, no explicit mathematical expression has been derived describing how the photon is embedded as a particle into the propagating electromagnetic wave. In contrary, David Bohm has argued in his quantum theory book [2] that there is no quantity for light equivalent to the electron probability density P_e(x) = |ψ(x)|². In particular, he is claiming [2]: “There is, strictly speaking, no function that represents the probability of finding a light quantum at a given point”. In the following, nevertheless, the attempt is made to embed the photon with its relativistic mass as a particle into the electromagnetic wave.

In previous publications of the author [3] and [4] it has been shown that by the use of a particle picture the transverse motion of a photon propagating with a Gaussian beam can be described by the Schrödinger equation of the 2-dimensional harmonic oscillator. This result has been derived by considering a photon reflected back and forth between totally reflecting equiphase surfaces of the propagating wave. It could be shown that the probability density of the photon obtained in this way describes the normalized intensity distribution of the propagating wave in full agreement with the result obtained by the use of wave optics. Since the consideration of a photon reflected back and forth between totally reflecting equiphase surfaces involves conceptual theoretical problems, it has, in the present paper, been replaced by a more straightforward approach by considering the infinitesimal directional change of the Poynting vector during an infinitesimal propagation step of the wave. In this way, the above results could be generalized to the case of a wave with arbitrary shape of the phase front. Based on a consideration of the relativistic mass density propagating with the electromagnetic wave, which can be expected to change its propagation direction analogous to the Poynting vector, a transverse force acting on the photon is derived. The expression obtained for this force makes it possible to show that the photon moves within a transverse potential which in combination with a Schrödinger equation allows to describe the transverse quantum mechanical motion of the photon by the use of matter wave theory like the motion of the electron, even though the photon has no rest mass. The obtained results are verified for the plane, the spherical and the Gaussian wave. An additional verification could be provided by the fact that also the mathematical equation describing the Guoy phase shift [5] could be derived from the quantum mechanical particle picture in full agreement with wave optics. In this way, wrong conclusions drawn in [4] from the results derived in this paper could be revised. One more verification could be obtained by the fact that within the range of validity of paraxial wave optics also Snell's law could be derived from this particle picture.

Numerical validation of the obtained results by the use of a beam propagation code for the case of the general wave is under development [6].

2. Derivation of a transverse force exerted on a photon propagating with an electromagnetic wave

According to Einstein's energy-mass relation E = mc² the energy propagating with an electromagnetic wave simultaneously represents a propagating relativistic mass. However, as well known from the propagation of laser beams, the energy density, whose propagation follows the Poynting vector, is usually not propagating along a straight line as shown by Fig. 1 which visualizes the propagating energy density of a resonant Gaussian wave between two spherical mirrors of an optical resonator. Therefore, the question arises, what is happening with a particle of mass propagating with the wave. Since it simultaneously represents a quantum of energy its propagation direction should change in agreement with the changing direction of the Poynting vector as shown by the green lines in Fig. 1. However, if the quantum of energy is considered to be a particle of mass its motion should be controlled by Newton's first law which claims that its motion, and therefore, also its propagation direction is remaining unchanged, if no force is exerted on it. A solution of this problem could be found, if one assumes that the mass of light does not follow the propagating energy. However, since it has been proven by many experiments that the direction of propagating light changes under the influence of gravity, which according to the theory of general relativity acts on the mass of light, it must be assumed that energy and mass are propagating together in the same direction. Therefore, the question arises, why does the propagating relativistic mass density follow the propagating energy density. Which interaction takes place between the propagating mass and the propagating energy?

Fig. 1 Resonant Gaussian mode between two spherical mirrors. The green lines visualize the propagating energy density.

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To investigate this interaction in more detail we consider that a propagating wave can be thought to be made up by a bundle of thin bent channels whose walls follow the lines of the propagating Poynting vector. In Fig. 2 these walls are shown by green lines, and the mass propagating within the channels is visualized by blue arrows. Due to the momentum change of the mass density propagating along these channels, the mass of the light is compressed like a compressible fluid, when the wave propagates from the right mirror to the waist, and is expanding again, after the wave has passed through the waist. It must therefore be assumed that a force opposite to the change of the momentum of the mass density is exerted on the propagating mass density as shown by the red arrows in Fig. 2. In principle, this force seems to be comparable with the force exerted on the body of a motorcyclist driving along a narrow curve. Therefore, if a small particle of the propagating mass is considered, it can be concluded that a force is exerted on this particle opposite to the infinitesimal momentum change of this particle during an infinitesimal propagation step. Since this momentum change must be proportional to the directional change of the propagating energy density, it can furthermore be concluded that the transverse force exerted on the particle must be proportional to the negative value of the directional change of the normalized Poynting vector versus an infinitesimal propagation step.

Fig. 2 Subdivision of a wave into a bundle of thin channels whose walls follow the green lines of the propagating Poynting vector. The blue arrows symbolize the propagating mass density. The red arrows symbolize the force exerted on the mass density. The distance between the red line and the topmost green line symbolizes, how this force changes along the propagation direction.

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To derive an expression for the infinitesimal directional change of the Poynting vector we consider two equiphase surfaces Φ_i(r,z_i) of a propagating wave with infinitesimal distance as shown in Fig. 3, where for the sake of simplicity rotational symmetry of the wave is assumed. The z_i are the points where the Φ_i(r,z_i) intersect the optical axis. The red arrows symbolize the normalized Poynting vectors erected in the points r_i located on the Φ_i. The points r_i as well as z_i are assumed to move into each other in the infinitesimal limit. The time, which the phase front Φ₁ takes to propagate into Φ₂, shall for further use be designated by Δt. Since according to the above arguments the direction of the Poynting vector must be assumed to be in alignment with the direction of the momentum of the propagating mass density, it can be concluded that the change of the momentum of a small particle of mass during the time Δt is proportional to the change of the normalized Poynting vector during Δt. In this way, it can furthermore be concluded that the transverse force exerted on a small particle of mass is proportional to the negative value of the differential quotient obtained after dividing the infinitesimal change of the normalized Poynting vector by Δt according to the following expression

\vec{K} (r, z) \propto - \lim_{r_{1} \to r_{2}, Δ t \to 0} \frac{1}{Δ t} [{\vec{S}}_{N} (r_{2}, z_{2}) - {\vec{S}}_{N} (r_{1}, z_{1})] .

Here S_N(r_i,z_i) designates the normalized Poynting vector erected on Φ(r,z_i) in the point r_i as visualized by the red arrows in Fig. 3. Since the particle of mass is propagating with the speed of light c, its momentum is given by

\tilde{M} c

, if

\tilde{M}

is its mass. Since according to Newton's second law a momentum change divided by a time interval describes a force, it can furthermore be concluded that the force exerted on the particle is given by

\vec{K} (r, z) = - \tilde{M} c * \lim_{r_{1} \to r_{2}, Δ t \to 0} \frac{1}{Δ t} [{\vec{S}}_{N} (r_{2}, z_{2}) - {\vec{S}}_{N} (r_{1}, z_{1})] .

In order to derive an expression for Δt, we look for two points r_i for which S(r₁) and S(r₂) are in alignment with each other. Under this condition the energy density of the electromagnetic wave propagates exactly perpendicularly to the Φ_i according to the orientation of the Poynting vector. Therefore, we obtain under this condition

Δ t = \frac{Φ (r_{2}, z {}_{2}) - Φ (r_{1}, z_{1})}{c},

keeping in mind that this relation is only valid, if the above condition is met. It may be argued that Eq. (3) may deliver different results for Δt, if the above condition is met by more than one pair of points r_i. However, this is not the case, since, due to the constant speed of light, Eq. (3) always delivers the same result for Δt under the above condition. Moreover, it be may argued that, assuming that Eq. (3) delivers different values for Δt dependent on whether or not the above condition is met, is in contradiction with the fact that light is propagating with constant speed. However, this is also not the case, since according to Eq. (2) only the limiting case of an infinitesimal distance between the phase fronts is physically relevant. For rotational symmetric waves, the above condition is met for r₁ = r₂ = 0. Therefore, we obtain in this case

Δ t = \frac{z_{2} - z_{1}}{c} = \frac{Δ z}{c} .

This delivers for the force exerted on a particle of mass

\tilde{M}

Fig. 3 Visualization of two phase fronts Φ₁ and Φ₂ intersecting the z axis at z₁ and z₂.

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\vec{K} (r, z) = - \tilde{M} c^{2} \lim_{r_{1} \to r_{2}, Δ z \to 0} \frac{1}{Δ z} [{\vec{S}}_{N} (r_{2}, z_{2}) - {\vec{S}}_{N} (r_{1}, z_{1})],

So far arguing is based on terms of classical physics combined with the theory of relativity. Therefore, the question arises what is the physical meaning of a small particle of mass propagating with an electromagnetic wave as considered above. To answer this question, we take into account that the propagating energy is subdivided into quanta of energy called photons. Therefore, since the above argumentation does not require defining the mass of the small particle exactly, it can be assumed that of the mass of the small particle is represented by the relativistic mass of the photon. In this way, the pressure exerted on the walls of the thin channels considered above can be interpreted as a radiation pressure exerted by the photons impinging on these walls under a small angle. Therefore, it should be possible to assume that the force exerted on the particles of mass can be interpreted as a force exerted on the photons, even though it may not be possible to assume that the photons exactly propagate within channels with infinitesimal cross section. Thus, after replacing the mass $\tilde{M}$ by the mass M of the photon, we obtain

\vec{K} (r, z) = - E_{p h} * \lim_{r_{1} \to r_{2}, Δ z \to 0} \frac{1}{Δ z} [{\vec{S}}_{N} (r_{2}, z_{2}) - {\vec{S}}_{N} (r_{1}, z_{1})] .

Here E_ph is the energy of the photon given by

E_{p h} = M c^{2} = \frac{h c}{λ} .

where λ and c are the wavelength and the speed of light in a vacuum, respectively. M is the relativistic mass of the photon given by

M = \frac{h}{c λ} .

Equation (6) shows that the force K(r,z) exerted on the photon is proportional to its energy E_ph. Equation (6) can be transformed furthermore, if we take into account that the directional change of the normalized Poynting vector S_N(r,z) versus the z coordinate is identical with the negative value of the change of the tangents to the phase fronts Φ(r,z) versus the r coordinate, as visualized by the green arrows in Fig. 3. This allows to transform Eq. (6) into

\vec{K} (r, z) = E_{p h} * \lim_{r_{1} \to r_{2}, Δ z \to 0} \frac{1}{Δ z} [\frac{d Φ (r_{2}, z_{2})}{d r} - \frac{d Φ (r_{1}, z_{1})}{d r}] .

The use of the coordinates r,z in the above equations does not mean that the photon is located at r,z, it only means that the force K is exerted on the photon, if the photon can be found at the position described by r,z. This is in agreement with the quantum mechanical assumption that a Coulomb force is exerted on an electron, if the latter can be found at a certain distance from the atomic nucleus. Without this assumption, no quantum mechanical potential, and therefore not even the Schrödinger equation describing the hydrogen atom could have been derived.

2. Derivation of a potential and a Schrödinger equation describing the transverse motion of the photon

Integration of the negative value of the force K(r,z) given by Eq. (9) over r along the curvature of the phase front Φ(r,z) shows that the photon is moving within a transverse potential given by

V (r, z) = E_{p h} * \int_{0}^{r} \lim_{r_{1} \to r_{2}, Δ z \to 0} \frac{1}{Δ z} [\frac{d Φ (r_{1}, z_{1})}{d r} - \frac{d Φ (r_{2}, z_{2})}{d r}] d r .

This equation shows that the transverse motion of the photon is described by a potential like the motion of the electron. It can be therefore assumed that its transverse motion is also described by a Schrödinger equation like the motion of the electron except for the difference that the mass of the electron is replaced by the relativistic mass M of the photon. Thus, it seems to be possible to conclude that the transverse motion of the photon is described by the following Schrödinger equation

[\frac{ℏ^{2}}{2 M} Δ_{⊥} + E (z) - V (r, z)] χ (r, z) = 0 .

where r and z are not Cartesian coordinates in the usual sense, since z describes the point where the phase front Φ(r,z) intersects the optical axis. From Eq. (11) it turns out that the Schrödinger equation not only describes the motion of a particle with rest mass, but also describes the motion of the photon which only has a relativistic mass. This furthermore shows that the transverse motion of the photon is described by matter wave theory as introduced by de Broglie and Schrödinger almost hundred years ago to describe the motion of the electron. Therefore, the photon exhibits wave-particle duality like the electron, but even more. On the one hand, its motion along propagation direction is described by wave optics, and on the other hand, its transverse quantum mechanical motion is described by matter wave theory.

For the general case of an arbitrarily unsymmetric wave the potential describing the quantum mechanical transverse motion of the photon can be derived from Eq. (6) in a similar way. This delivers

V (x, y, z) = E_{p h} * \int_{x_{| |}}^{x} \int_{y_{| |}}^{y} \lim_{x_{1}, y_{1}, z_{1} \to x_{2}, y_{2}, z_{2}} \frac{1}{Δ_{| |}} [S_{N} (x_{2}, y_{2}, z_{2}) - S_{N} (x_{1}, y_{1}, z_{1})] d x d y,

where the integration is done over the equiphase surface of the wave. As in the case of rotational symmetry, it is again assumed that the point x₁,y₁ located on Φ₁ moves in the limit into the corresponding point x₂,y₂ located on Φ₂. z₁ and z₂ are the points where the Φ_i intersect the z axis. Therefore, only x and y represent Cartesian coordinates, but not the triple x,y,z. To compute Δ_║, we look again for a pair of points x_i║, y_i║ for which S₁(x_1║,y_1║,z₁) and S₂(x_2║,y_2║,z₂) are in alignment with each other. The distance Δ_║ between these points is given by

Δ_{| |} = \sqrt{{(x_{2 | |} - x_{1 | |})}^{2} + {(y_{2 | |} - y_{1 | |})}^{2}}

It may be asked, why the integration only starts from one pair x_║ and y_║, even though Eq. (12) involves two pairs of corresponding points x_i║,y_i║. The latter follows from the fact that the integration is carried through in Eq. (12) after computing the infinitesimal quotient. However, since in the case of a general wave, the z axis does not represent a physical propagation direction, it can be replaced by any other choice of the z axis without limiting generality. A mathematically convenient choice would be to define the direction of the z axis in alignment with S₁(x_1║,y_1║,z₁) and S₂(x_2║,y_2║,z₂). Then Δ_|| again can be replaced by z₂-z₁ as shown above for the case of a rotational symmetric wave. This delivers

V (x, y, z) = E_{p h} * \int_{0}^{x} \int_{0}^{y} \lim_{x_{1}, y_{1}, z_{1} \to x_{2}, y_{2}, z_{2}} \frac{1}{z_{2} - z_{1}} [S_{N} (x_{2}, y_{2}, z_{2}) - S_{N} (x_{1}, y_{1}, z_{1})] d x d y

This equation provides a general relation between the electromagnetic field and the potential V(x,y,z) describing the quantum mechanical transverse motion of the photon. It can be used in combination with the above Schrödinger equation in the same way as Eq. (10).

The eigensolutions χ(x,y,z) of the above Schrödinger equation allow to compute the probability density |χ(x,y,z)|² of the photons propagating with an electromagnetic wave. This result shows how the photon can be assumed to be embedded into the propagating electromagnetic wave with its relativistic mass. This seems to be in contradiction with the statement of David Bohm mentioned in the introduction.

In the next section the relations given by the Eq. (9), (10) and (11) will be verified for the plane, the spherical, and the Gaussian wave.

3. Verification of the Eqs. (9), (10) and (11) for the case of the plane, the spherical, and the Gaussian wave

Since in case of a plane wave dΦ(r,z)/dr vanishes for all values of r and z, the latter obviously also holds for the potential V(r,z). Therefore, since for a vanishing potential the transverse motion of the photon is not confined, no quantum mechanical eigenfunctions with a confined transverse extension exist. This is in agreement with the wave optics result that plane waves are not described by eigenmodes with confined transverse extension. The same holds for the spherical wave. Since in this case the Poynting vectors S(r₁,z₁) and S(r₂.z₂) are in alignment with each other for all r_i and z_i, the relation dΦ(r₁,z₁)/dr = dΦ(r₂,z₂)/dr holds for all r_i, z_i with the consequence that also in this case the potential vanishes.

However, the Eqs. (9), (10) and (11) also prove true for the case of a Gaussian wave. To shows this, we take into account that the phase of a Gaussian wave is according to Eq. (4).7.7) in [7] given by

\tilde{Φ} (r, z) = - \frac{r^{2}}{2 R (z)} - z .

This delivers for phase front Φ(r,z) shown in Fig. 3

Φ (r, z) = z - \frac{r^{2}}{2 R (z)} .

Here R(z) is the radius of curvature of the phase front of a Gaussian wave which is according to Eq. (17).5) in [8] given by

R (z) = z + \frac{z_{R}^{2}}{z} .

Here z_R is the Rayleigh range which according to Eq. (17).4) in [8] is given by

z_{R} = \frac{π w_{0}^{2}}{λ_{}} .

where w₀ is the spot size at the beam waist of a Gaussian wave. Therefore, after inserting Eq. (16) into Eq. (9), and carrying through the differentiation versus r we obtain

\vec{K} (r, z) = E_{p h} * \lim_{r_{1} \to r_{2}, Δ z \to 0} \frac{1}{Δ z} [\frac{r_{1}}{R (z_{1})} - \frac{r_{2}}{R (z_{2})}]

As shown in [4], this expression can be transformed into

\vec{K} (r, z) = E_{p h} \frac{r}{R^{2} (z)} (\frac{Δ R}{Δ z} - \frac{R (z)}{r} \frac{Δ r}{Δ z})

with

\frac{Δ R}{Δ z} = \frac{R (z_{2}) - R (z_{1})}{Δ z} = \frac{d R}{d z} = 1 - \frac{z_{R}^{2}}{z^{2}} .

To compute Δr/Δz = (r₂ – r₁)/Δz we take into account that ΔΦ = Φ(r₂,z₂) - Φ(r₁,z₁), as shown in Fig. 4, can be transformed into

Δ Φ (r, z) = Φ (r_{2}, z_{2}) - Φ (r_{1}, z_{1}) = - \frac{Δ r}{R (z)} + Δ z \approx Δ z

Here the approximate transformation at the right side of this expression follows from the fact that in paraxial approximation the relation Δr <<Δz <<R holds which allows to neglect Δr/R. Therefore, since, as visualized by Fig. 4, in the limit the relation

\lim_{Δ z \to 0} \frac{Δ r}{Δ Φ} = \frac{r}{R (z)}

holds, we obtain

\lim_{Δ z \to 0} \frac{Δ r}{Δ z} = \frac{r}{R (z)} .

Thus, after inserting the Eqs. (21) and (24) into Eq. (20) we obtain

K (r, z) = E_{p h} \frac{r}{R^{2} (z)} (1 - \frac{z_{R}^{2}}{z^{2}} - \frac{R (z)}{r} \frac{r}{R (z)}) = - E_{p h} \frac{r}{R^{2} (z)} \frac{z_{R}^{2}}{z^{2}}

Therefore, after replacing R(z) according to Eq. (17), we obtain for the force exerted on the photon

K (r, z) = - E_{p h} r {(\frac{z_{R}}{z^{2} + z_{R}^{2}})}^{2} .

To compute the potential, -K(r,z) has to be integrated along the curvature of the phase front. However, in paraxial approximation this integration can be carried through directly along r. This delivers

V (r, z) = \frac{1}{2} M ω_{⊥}^{2} (z) r^{2} .

Here ω_┴ is the frequency of the transverse quantum mechanical oscillation of the photon, and is according to Eq. (26) given by

ω_{⊥} (z) = \frac{c z_{R}}{z^{2} + z_{R}^{2}} .

If the potential V(r,z), as given by Eq. (27), is inserted into the Schrödinger equation given by Eq. (11) the following expression

[\frac{ℏ}{2 M} Δ_{⊥} + E - \frac{1}{2} M ω_{⊥}^{2} (z) (x^{2} + y^{2})] χ_{n m} (x, y, z) = 0

is obtained, where the coordinates x,y and z are used in the same way as r and z in Eq. (16).

Fig. 4 Visualization of relation Δr/ΔΦ = r/R(z) as used in Eq. (23)

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Equation (29) is identical with the Schrödinger equation for the 2-dimensional harmonic oscillator. As well known, the eigensolutions of this equation are given by

χ_{n m} (x, y, z) = \sqrt{\frac{2}{π}} \frac{1}{w_{p} (z) \sqrt{2^{n + m} n!} m!} H_{n} (\frac{\sqrt{2} x}{w_{p} (z)}) H_{m} (\frac{\sqrt{2} y}{w_{p} (z)}) \exp (- \frac{x^{2} + y^{2}}{w_{p}^{2} (z)}) .

with w_p² given by

w_{p}^{2} (z) = \frac{2 ℏ}{M ω_{⊥} (z)} .

Here the subscript p refers to particle. According to Eq. (30), w_p represents the distance from the optical axis, where the value of the probability density of a photon in the ground level drops to |χ₀₀(x,y,z)|² = exp(−2)|χ₀₀(0,0,z)|². After inserting of ω_┴ according to Eq. (28) w_p² transforms into

w_{p}^{2} (z) = \frac{2 ℏ z_{R}}{M c} [1 + {(\frac{z}{z_{R}})}^{2}] .

Here the factor at the left side of the square bracket can be replaced by

w_{p}^{2} (0) = \frac{2 ℏ}{M c} z_{R} = \frac{λ}{π} z_{R} .

This shows that the expression obtained for w_p(0) is identical with the expression obtained by the use of paraxial wave optics for the Gaussian spot size w₀ at the beam waist as shown by Eq. (17).4) in [8]. Therefore, Eq. (32) can be transformed into

w_{p} (z) = w_{0} \sqrt{1 + {(\frac{z}{z_{R}})}^{2}}

which is according to Eq. (17).5) in [8] identical with the expression for w(z) describing the z dependence of the spot size of a Gaussian wave. By the use of the Eqs. (33) and (34) the expression for the force K(r,z) can be transformed as follows

K (r, z) = - E_{p h} r {(\frac{z_{R}}{z^{2} + z_{R}^{2}})}^{2} = - E_{p h} \frac{w_{0}^{4}}{z_{R}^{2}} \frac{r}{w^{4} (z)}

Thus, the force exerted on photons, which can be found at a distance w(z) from the axis, is given by

K (w (z), z) = - E_{p h} \frac{w_{0}^{4}}{z_{R}^{2}} \frac{1}{w^{3} (z)}

This expression shows that K(w(z),z) is strongest, when the mass density follows the narrow curve near the beam waist, as shown by the red arrows in Fig. 2 where the distance between the red line and the topmost green line visualizes the force K(w(z),z). This force is becoming smaller with increasing distance from the waist, and disappears, when the beam is going over into a spherical wave. Since in the latter case the mass density propagates along straight lines, no transverse force is exerted on the photons, as already discussed above.

Comparison of Eq. (30) with Eq. (16).60) in [8] shows that the probability density |χ_nm(x,y,z)|² of the photon is in full agreement with the normalized local intensity provided by paraxial wave optics for a Gaussian mode of order n,m. In this way, the Eqs. (9), (10) and (11) could be verified for the case of a Gaussian wave. In the next section, it will be shown that also the Gouy phase shift [5] can be computed on the basis of the above results. This provides an additional verification of the Eqs. (10) and (11).

4. Quantum mechanical computation of the Gouy Phase shift for the case of a Gaussian wave

Unfortunately, in [4] wrong conclusions have been drawn from the above results. It has been concluded that the above results lead to a contradiction with results obtained by the use of wave optics, especially concerning the Gouy phase shift [5]. In this section, it will be shown that, in contrary, the Gouy phase shift can be computed in full agreement with wave optics by the use of the above described particle picture.

An interpretation of the Gouy phase shift as a geometrical quantum effect has already been given in [9]. However, in this paper only a quantum mechanical interpretation of the Gouy effect is given, but not an explicit mathematical computation of the Gouy phase shift based on a quantum mechanical model as presented below. Moreover, it has been assumed that the photons exactly propagate in z direction, and that therefore their transverse momentum vanishes. This assumption seems to be not correct, since it must be assumed that the photons, which represent the propagating energy, propagate in the direction of the Poynting vector which has a transverse component. In the following, a computation of the Gouy phase shift is given based on the above presented quantum mechanical particle picture.

For this purpose, we consider the expression for the quantum mechanical expectation value of the square of the momentum in case of a 2-dimensional harmonic oscillator. This expression delivers for the expectation value of the square of the photon's transverse momentum

< χ_{n m} | {\hat{p}}_{⊥}^{2} (z) | χ_{n m} > = ℏ M ω_{⊥} (z) (n + m + 1) .

By the use of the Eqs. (8), (28) and (33) the expression Mω_┴, used in this equation, can be transformed as follows

M ω_{⊥} = \frac{h}{c λ} \frac{c z_{R}}{z^{2} + z_{R}^{2}} = \frac{h}{λ z_{R}} \frac{w_{0}^{2}}{w^{2} (z)} = \frac{h λ}{λ π w_{0}^{2}} \frac{w_{0}^{2}}{w^{2} (z)} = \frac{2 ℏ}{w^{2} (z)} .

Insertion of this result into Eq. (37) delivers

< χ_{n m} | {\hat{p}}_{⊥}^{2} (z) | χ_{n m} > = \frac{2 ℏ^{2}}{w^{2} (z)} (n + m + 1) .

For further consideration, we compare the momentum of a freely propagating photon, which is given by p = h/λ, with the wave vector of a plane wave, which is given by k = 2π/λ . This comparison shows that the expression for the wave vector k is obtained, if p is divided by ћ. Therefore, it can be concluded that division of the expectation value

< {\hat{p}}_{⊥}^{2} (z) >

by ћ² delivers the expectation value

< k_{x}^{2} > + < k_{y}^{2} >

of the transverse part of the square of the wave vector k. In this way, we obtain

< k_{x}^{2} > + < k_{y}^{2} > = \frac{< {\hat{p}}_{⊥}^{2} (z) >}{ℏ^{2}} = \frac{2 (n + m + 1)}{w^{2} (z)} .

Now we take into account that by the use of wave optics the following expression for the Gouy phase shift is obtained

Φ_{G} = - \frac{1}{k} \int_{}^{z} {< k_{x}^{2} > + < k_{y}^{2} >} d z .

as shown by Eq. (4) in [10]. This expression delivers after insertion of Eq. (40)

Φ_{G} = - \frac{2 (n + m + 1)}{k} \int_{}^{z} \frac{1}{w^{2} (z)} d z .

Thus, after carrying through the integration, we obtain

Φ_{G} = - (n + m + 1) arc \tan (z / z_{R})

in agreement with Eq. (20) in [10].

This result shows that the Gouy phase shift can be derived in full agreement with wave optics by the use of the quantum mechanical particle picture. Furthermore, it demonstrates that the Gouy effect can be equivalently understood as a wave optics as well as a quantum mechanical effect, even though the quantum mechanical description seems to provide a more straightforward understanding of some features of this effect. For instance the factor n + m + 1 in Eq. (43) immediately follows from the quantum mechanical description of the 2-dimensional harmonic oscillator. Also the fact that the Gouy phase shift for a cylindrical wave is given by

Φ_{G} = - (n + 1 / 2) arc \tan (z / z_{R}),

immediately follows from the fact that in this case the photon is moving within the potential of a 1-dimensional harmonic oscillator. Moreover, by the use of the particle picture the Gouy phase shift can easily be computed for an elliptic Gaussian beam. In this case the potential is given by

V (x, y, z) = V (x, z) + V (y, z) = \frac{1}{2} M [ω_{⊥ x}^{2} (z) x^{2} + ω_{⊥ x}^{2} (z) y^{2}]

where ω_┴x and ω_┴y are the transverse oscillation frequencies of the photon in x and y direction. This delivers

< k_{x}^{2} > + < k_{y}^{2} > = \frac{1}{ℏ^{2}} < χ_{n m} | {\hat{p}}_{⊥}^{2} (z) | χ_{n m} > = \frac{2 n + 1}{w_{x}^{2} (z)} + \frac{2 m + 1}{w_{y}^{2} (z)}

where w_x(z) and w_y(z), respectively, describe the z dependence of the spot size in the x-z and the y-z plane analogous to Eq. (34). For w_x = w_y Eq. (46) transforms into Eq. (40).

These results provide, for the case of a Gaussian wave, an additional verification of the particle picture of the photon described by the Eqs. (10) and (11). Moreover, numerical evaluation of Eq. (14) would make it possible to compute the Gouy shift for a wave with arbitrary phase front by the use of

Φ_{G} (z) = - \frac{1}{k ℏ^{2}} \int_{}^{z} < χ (x, y, z) | {\hat{p}}_{⊥}^{2} (z) | χ (x, y, z) > d z,

where χ(x,y,z) is an eigensolution of the Schrödinger equation with the potential given by Eq. (14). According to this equation, the Gouy phase shift vanishes for a spherical wave in the same way as for a plane wave, since for this type of waves the potential, and therefore, also the eigensolutions χ(x,y,z) are vanishing. This is in agreement with the mathematical description of the spherical wave for which the distance of the phase fronts is independent of z. Therefore, since Eq. (43) describes the Guoy phase shift for paraxial waves, it may be possible to observe that the Guoy phase shift decreases, when a paraxial wave changes into a spherical wave.

5. Ray optics confirmation of the frequency ω_┴ describing the transverse motion of a photon moving with a Gaussian beam

A surprising result is obtained, if one considers the transverse motion of a ray bouncing between two phase fronts of a Gaussian beam. In section 4 of [4] it has been shown that for an infinitesimal distance between the phase fronts the transverse motion of the ray is described by the following expression

r = A \cos (ω_{⊥} (z) t)

where A is the amplitude of the transverse motion of the ray. This equation shows that the ray is oscillating in transverse direction with the same frequency ω_┴ as obtained for the quantum mechanical transverse motion of the photon. Therefore, between the transverse motion of photon and ray the same correspondence relation seems to exist as has been described by Schrödinger for the correspondence between quantum mechanical and classical oscillator [11]. Differentiation of Eq. (48) shows that the transverse motion of the ray is periodically accelerating and slowing down. The ray is therefore moving like the mass of a classical harmonic oscillator. This provides an alternative aspect concerning the force acting on the mass of the photon according to Newton's second law.

6. Snell's law derived from a particle picture

In paraxial approximation also Snell's law can be derived from the above described particle picture. For this purpose we derive the potential V(r,z) directly from Eq. (6). This delivers

V (r, z) = E_{p h} * \int_{0}^{r} \lim_{r_{1} \to r_{2}, Δ z \to 0} \frac{1}{Δ z} [{\vec{S}}_{N} (r_{2}, z_{2}) - {\vec{S}}_{N} (r_{1}, z_{1})] d r .

Since in paraxial approximation S_N(r₂,z₂) - S_N(r₁,z₁) can be replaced by S_N┴(r₂,z₂)- S_N┴(r₁,z₁), where S_N┴ is the normalized transverse component S_┴/|S| of the Poynting vector S, this expression can be transformed into

V (r, z) = E_{p h} * \int_{0}^{r} \lim_{r_{1} \to r_{2}, Δ z \to 0} \frac{1}{Δ z} [S_{N ⊥} (r_{2}, z_{2}) - S_{N ⊥} (r_{1}, z_{1})] d r .

When the paraxial wave passes from a vacuum into a medium, this expression transforms into

V (r, z) = E_{p h} * \int_{0}^{r} \lim_{r_{1} \to r_{2}, Δ z \to 0} \frac{n_{m}}{Δ z} [S_{N ⊥} (r_{2}, z_{2}) - S_{N ⊥} (r_{1}, z_{1})] d r .

Here Δz is replaced by Δz/n_m = Δt*c/n_m according to Eq. (4), where n_m is the refractive index of the medium. If it is taken into account now that for paraxial waves the potential V(r,z), which describes the transverse quantum mechanical motion of the photon, must remain unchanged, when the wave perpendicularly passes into the medium, it must be concluded that also the integrand in Eq. (51) must remain unchanged. This can only be achieved, if the change of Δz into Δz/n_m is compensated by replacing S_N┴ by S_N┴/n_m in Eq. (51). Thus, if we designate the angle, which the Poynting vector forms with the normal to the interface between vacuum and medium before the wave enters the medium with θ₁, and the angle, which the Poynting vector forms with this normal after the wave has entered the medium with θ₂, we obtain

\frac{S_{N ⊥}}{S_{N ⊥} / n_{m}} = \frac{\sin (θ_{1})}{\sin (θ_{2})} = n_{m} .

as shown in Fig. 5. In this way, Snell's law could, within the range of validity of the paraxial approximation, be derived from a particle picture.

Fig. 5 Refraction at the interface between vacuum and medium.

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The transformation of S_N┴ into S_N┴/n_m demanded above can be validated by the use of Gaussian wave theory, if Eq. (19) is used to transform Eq. (49) into

V (r, z) = E_{p h} * \int_{0}^{r} \lim_{r_{1} \to r_{2}, Δ z \to 0} \frac{1}{Δ z} [\frac{r_{1}}{R (z_{1})} - \frac{r_{2}}{R (z_{2})}] d r .

Since the radius R(z), used in this expression, transforms according to the Gaussian mode algorithm into n_mR(z), when the wave passes from a vacuum into a medium, the expression in the square brackets in Eq. (53) transforms in the same way as the expression in the square brackets in Eq. (51). The transformation of R(z) into n_mR(z) can also easily be shown by the use of the software LASCAD [12].

7. The Schrödinger equation describing the transverse quantum mechanical motion of a photon in a medium

If the wave is propagating in a medium, the mass M of the photon, used in the Schrödinger equation, has to be replaced by an effective relativistic mass M_m = n_mM in agreement with the requirement that the momentum of the propagating mass density must remain unchanged. The use of M_m in the Schrödinger equation can be verified by the fact that, in case of a Gaussian wave, the eigensolutions of the Schrödinger equation modified in this way deliver a probability density of the photon in agreement the normalized intensity distribution obtained by the use of paraxial wave optics. The latter follows from the fact that in Eq. (33) Mc is replaced by M_mc/n_m = Mc. The modification of the effective relativistic mass of the photon can be explained by the fact that the stream of the propagating photons is compressed by a factor n_m in propagation direction due to the reduced speed of the propagating wave, even though a single photon is still propagating with the speed of light in vacuum. This strange behaviour of the photon has been explained by Richard Feynman in section 3 of [13], where he is giving a quantum electrodynamics explanation for the refractive index. Due the compression of the stream of the propagating photons the propagating mass density is increased by a factor n_m. This explains, why the increased effective relativistic mass M_m of the photon has to be used in the Schrödinger equation.

8. Summary and conclusions

In previous publications of the author [3] and [4] it has been shown that by the use of a particle picture the transverse motion of a photon propagating with a Gaussian beam can be described by the Schrödinger equation of the 2-dimensional harmonic oscillator. This particle picture has been generalized to the case of a wave with arbitrary shape of the phase front. Based on a consideration of the relativistic mass density propagating with the electromagnetic wave a transverse force acting on the photon has been derived. The expression obtained for this force makes it possible to show that the photon moves within a transverse potential which, in combination with a Schrödinger equation, allows to describe the transverse quantum mechanical motion of the photon. The obtained Schrödinger equation is identical with the Schrödinger equation describing the motion of the electron except for the difference that the mass of the electron is replaced by the relativistic mass of the photon. This shows that the transverse motion of the photon is described by matter wave theory, even though the photon has no rest mass. It shows furthermore that the photon exhibits wave-particle duality like the electron, but even more. On the one hand, its motion along propagation direction is described by wave optics, and on the other hand, its transverse motion is described by matter wave theory.

The eigensolutions χ(x,y,z) of the obtained Schrödinger equation allow to compute the probability density |χ(x,y,z)|² of a photon propagating with an electromagnetic wave. This result seems to be in contradiction to the statement of David Bohm mentioned in the introduction, and it seems to show, how the photon is embedded with its relativistic mass as a particle into the electromagnetic wave. This can be expected to open new perspectives concerning the quantum theory of radiation, and explains, how the relativistic mass density is forced to follow the propagating energy density. If this force can be considered as a real force, since it allows to compute a quantum mechanical potential, or as a virtual force, since it is not comparable with any forces acting between physical systems known so far, deserves further consideration. It shall however be mentioned that also gravity just exists, and can physically not be further explained. In addition, the obtained relation between the photon and the electromagnetic wave opens one more perspective, since it makes it possible to interpret the propagating electromagnetic wave as a macroscopic quantum mechanical system described by a Schrödinger equation like a microscopic system.

The obtained results have been verified for the plane, the spherical and the Gaussian wave. In the latter case the result obtained for the probability density |χ(x,y,z)_nm|² of the photon is in full agreement with the normalized local intensity of the Gaussian modes n,m obtained by the use of paraxial wave optics. An additional verification has been obtained by considering the Guoy phase shift [5]. It could be shown that in case of a Gaussian wave a mathematical expression describing the Guoy phase shift can be derived from the particle picture which is in full agreement with the expression derived by the use of wave optics. In this way, wrong conclusions drawn in [4] from the derived results could be revised. The result concerning the Gouy phase shift demonstrates that the Gouy effect can be equivalently understood as a wave optics as well as a quantum mechanical effect, even though the quantum mechanical description seems to provide a more straightforward understanding of some features of this effect. Moreover, the quantum mechanical description makes it possible to compute the Gouy phase shift for a general wave with arbitrary shape of the phase front. One more verification of the particle picture could be obtained by the fact that within the range of validity of paraxial wave optics also Snell's law could be derived from this particle picture.

The obtained results provide the physical reason for the analogies between optics and quantum mechanics described in previous work such as in [14] and [15].

In future work the obtained results shall be confirmed numerically by the use of a wave optics beam propagation code [6]. It is expected that application to the case of a wave truncated by an aperture may deliver a quantum mechanics interpretation of the diffraction of light. Moreover, the obtained result is expected to allow for efficient computation of the eigenmodes of an electromagnetic wave.

References and links

1. Ch. Roychoudhuri, The Nature of Light: What is a Photon? (CRC Press, 2008)

2. D. Bohm, Quantum Theory, (Prentice-Hall, 1951, republished by Dover Publications, 1989), pp. 98.

3. K. Altmann, “A Particle Picture of the Optical Resonator,” ASSL 2014 Conference Paper ATu2A.29, 16–21 November 2014, Shanghai, China. [CrossRef]

4. K. Altmann, “Contradiction within wave optics and its solution within a particle picture,” Opt. Express 23(3), 3731–3750 (2015). [CrossRef] [PubMed]

5. L. G. Gouy, “Sur une propriété nouvelle des ondes lumineuses,” C. R. Acad. Sci. Paris 110, 1251 (1890).

6. K. Altmann, to be submitted for publication.

7. O. Svelto, Principles of Lasers (Plenum Press, 1998).

8. A. E. Siegman, Lasers (University Science Books, 1986).

9. P. Harihan and P. A. Robinson, “The Gouy phase shift as a geometrical quantum effect,” J. Mod. Opt. 43(2), 219–221 (1996).

10. S. Feng and H. G. Winful, “Physical origin of the Gouy phase shift,” Opt. Lett. 26(8), 485–487 (2001). [CrossRef] [PubMed]

11. S. Schrödinger, “Der stetige Übergang von der Mikro- zur Makromencanik,” Naturwissenschaften 14, 664–666 (1926). [CrossRef]

12. LASCAD, “LASer Cavity Analysis and Design,” www.las-cad.com.

13. R. P. Feynman, QED - The Strange Theory of Light and Matter (Princeton University Press, 1985) Chap. 3.

14. W. van Haeringen and D. Lenstra, Analogies in Optics and Micro Electronics (Kluwer Academic Publishers, 1990).

15. T. Tsai and G. Thomas, “Analog between optical waveguide system and quantum-mechanical tunneling,” Am. J. Phys. 44(7), 636–638 (1976). [CrossRef]

Embedding the photon with its relativistic mass as a particle into the electromagnetic wave

Abstract

1. Introduction

2. Derivation of a transverse force exerted on a photon propagating with an electromagnetic wave

2. Derivation of a potential and a Schrödinger equation describing the transverse motion of the photon

3. Verification of the Eqs. (9), (10) and (11) for the case of the plane, the spherical, and the Gaussian wave

4. Quantum mechanical computation of the Gouy Phase shift for the case of a Gaussian wave

5. Ray optics confirmation of the frequency ω_┴ describing the transverse motion of a photon moving with a Gaussian beam

6. Snell's law derived from a particle picture

7. The Schrödinger equation describing the transverse quantum mechanical motion of a photon in a medium

8. Summary and conclusions

References and links

Cited By

Figures (5)

Equations (53)

Optics Express

Abstract

1. Introduction

2. Derivation of a transverse force exerted on a photon propagating with an electromagnetic wave

2. Derivation of a potential and a Schrödinger equation describing the transverse motion of the photon

3. Verification of the Eqs. (9), (10) and (11) for the case of the plane, the spherical, and the Gaussian wave

4. Quantum mechanical computation of the Gouy Phase shift for the case of a Gaussian wave

5. Ray optics confirmation of the frequency ω┴ describing the transverse motion of a photon moving with a Gaussian beam

6. Snell's law derived from a particle picture

7. The Schrödinger equation describing the transverse quantum mechanical motion of a photon in a medium

8. Summary and conclusions

References and links

Cited By

Figures (5)

Equations (53)

Optics Express

5. Ray optics confirmation of the frequency ω_┴ describing the transverse motion of a photon moving with a Gaussian beam