Contradiction within wave optics and its solution within a particle picture

Konrad Altmann

doi:10.1364/OE.23.003731

1. Introduction

It is well known that the computation of the resonance frequencies ω_qnm of the eigenmodes of a resonator is based on the condition that the total phase shift of an eigenmode wave must be an integer multiple of 2π after one round-trip. This condition delivers, in case of an empty resonator with spherical end mirrors, according to paraxial wave optics (PWO) the following relation, see Eq. (19.23) in [1]

ω_{q n m} = \frac{2 π c q}{2 L} + (1 + n + m) \frac{c}{L} arc \cos \sqrt{(1 - \frac{L}{R_{1}}) (1 - \frac{L}{R_{2}})}

In this equation q is the axial mode index, L is the length of the resonator, R₁ and R₂ are the radii of curvature of the mirrors, and n and m are the transverse modal numbers. It is shown that Eq. (1) leads to a contradiction, if the resonator is divided into subcavities by inserting a totally reflecting equiphase surface into the resonator, and moreover, that Eq. (1) implies a violation of the law energy conservation. Since the second term in Eq. (1) accounts for the axial phase shift ψ of the Gaussian mode, it is analyzed, whether a relation exists between these problems and the mathematical expression for ψ which is given by

ψ (z) = arc \tan \frac{z}{z_{R}}

for the Gaussian fundamental mode. Here z is the distance from the beam waist and z_R the Rayleigh range. Although the derivation of Eq. (2) is exact within PWO, it could be shown that it leads to a contradiction, if z_R goes to infinity. Since PWO transforms into an exact representation of wave optics for nearly plane waves, i. e. for z_R→∞, these problems also concern wave optics. Therefore a solution of these problems does not seem to be possible within the wave optics approach.

In this paper a solution is proposed within a particle picture as recently presented by the author [2, 3], which is worked out in more detail in the present paper.

To develop a particle picture of a propagating wave, it is considered that, according to the theory of relativity, a propagating electromagnetic wave represents a propagating mass, whose quantum is the relativistic mass of the photon. This aspect has not been really integrated into the wave optics theory so far. It seems that the particle aspect of the photon, proven by Einsteins' photoelectric effect and the Compton effect, stands aside of wave theory originally initiated by Huygens more than 350 years ago. The theory of the laser is an example for this missing conjunction between wave and particle theory. The computation of the modes of a laser is exclusively based on the wave theory, the description of stimulated emission, which basically enables the function of the laser, is based on the particle theory. Even though the theory of coherence has established a relation between stimulated emission and wave theory, there is still no direct relation between the photon and the modes of a laser resonator. To establish this relation is the intention of the present paper. If one neglects the aspects of wave theory, and considers the propagating mass only, the question arises, as to why this mass follows the propagating wave, and why it is not dissipating in the transverse direction, since it is not kept together by interatomic forces. Is there a force acting on the mass which is compelling it to follow the propagating wave? To investigate this question, the change of momentum is considered when the mass, propagating with a wave, is bouncing between two reflecting equiphase surfaces (RES) with small distance. In case of vanishing distance between the RES, this leads to the requested transverse force exerted on the quantum of the propagating mass or photon. In this way it can be shown that the photon is moving within a transverse quantum mechanical potential. This leads to a Schrödinger equation describing a transverse quantum mechanical motion of the photon in a propagating wave. Applied to an optical resonator, the solutions of this Schrödinger equation show that the transverse modes of an optical resonator can be understood as the quantum mechanical eigenfunctions of a single photon. In addition, this result provides a physical explanation for the analogy between paraxial wave optics and the harmonic oscillator already known for a long time [4].

The possibility to describe the transverse motion of a photon in a resonator by the use of a Schrödinger equation has already been outlined by the author earlier [5] in a sligthly different context. In [5] the following Schrödinger equation for the photon has been derived

[\frac{ℏ}{2 M} Δ + ℏ (ω - ω_{q}) - V (x, y)] χ (x, y) = 0 .

Here V(x,y) is the transverse potential within which the photon is moving. In the context of [5], V(x,y) was only dependent on the transverse coordinates x and y. In the present context it also depends on z. |χ|² describes the position probability density of the photon, ω is the angular frequency of the photon. ω_q is defined by

ω_{q} = π ​ q c / L .

The corresponding axial wave length is given by

λ_{q} = \frac{2 L}{q} = \frac{2 π c}{ω_{q}} .

In case of an empty resonator with two end mirrors, ћω_q is the associated plane wave energy of the photon in the limiting case of plane end mirrors. From the above equations the corresponding relativistic mass M of the photon is obtained as

M = ℏ ω_{q} / c^{2} = ℏ \frac{π q}{c L} = \frac{2 ℏ π}{c λ_{q}} .

2. Contradiction within wave optics

It is well known that all important properties, such as spot size w(z), curvature R(z) of the phase front, and axial phase shift ψ(z) of a Gaussian beam, are determined by the Rayleigh range z_R and the beam waist spot size w₀ according to the following two equations

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

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

and Eq. (2), see for instance Eqs. (17.5) in [1]. The Eqs. (2), (7) and (8) show that all empty resonators with spherical end mirrors and with given w₀ and z_R have identical transverse mode structures independent of the position z of the end mirrors along the resonator axis as long as the curvatures of the mirrors are given by the Eq. (8). To look at this in more detail, we consider a resonator with one planar and one spherical end mirror as shown in Fig. 1.

Fig. 1 Planar-spherical resonator subdivided by a totally reflecting equiphase surface S.

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We now assume that a certain transverse eigenmode propagates from the planar to the spherical end mirror E with curvature R_E. After it is reflected, it propagates back, but is now reflected at a totally reflecting equiphase surface S inserted between the two mirrors. Since there is no loss, the wave continues to be reflected back and forth between S and E. If S is positioned at a nodal phase front of the wave, it seems to be obvious that this wave represents a standing wave in the subcavity SE, made up by S and E. (Since S is a totally reflecting surface but not a physical mirror, this condition is not really necessary. The wave makes a phase jump of π at S anyway, and a standing wave would be created nevertheless.) Therefore, it can be expected that the local phase and intensity distribution of this wave are not changing during a round trip through SE. Consequently, according to the generally accepted definition of an eigenmode, see for instance [1] Chapt. 14, the wave propagating in SE can be considered to be an eigenmode of SE. This is confirmed by the fact that the intensity distribution of any transverse mode does not change, when it propagates through the surface S, and also does not change, when S is made totally reflecting. Basically, this is comparable with any other vibrating system. For instance, if one considers a string vibrating in an overtone, there are nodes where the string does not move. Thus, its behavior is not changed, if its transverse motion is fixed at one of these nodes. Therefore, a vibration, representing an eigenmode of the full string, automatically represents eigenmodes of the subresonators created by the fixation of the string at a node. However, according to PWO, the mode, generated in this way in the subcavity SE, is not an eigenmode of SE, since the frequencies belonging to the eigenmodes of SE according to Eq. (1) are different from the frequency of the wave propagating from the full resonator into the subcavity SE. This turns out as follows.

According to Eq. (1) the resonance frequencies of the full cavity are given by

ω_{q n m} = \frac{2 π c}{λ_{q}} + (n + m + 1) \frac{c}{L} arc \cos \sqrt{1 - \frac{L}{R_{E}}} = \frac{2 π c}{λ_{q}} + (n + m + 1) \frac{c}{L} arc \sin \sqrt{\frac{L}{R_{E}}},

where the relation

\arcsin x = \arccos \sqrt{1 - x^{2}}

has been used. If we now assume that the distance between S and E is λ_q, we obtain for the resonance frequencies of the subcavity SE from Eq. (1)

ω_{q n m, S C} = \frac{2 π c}{λ_{q}} + (n + m + 1) \frac{c}{λ_{q}} arc \cos \sqrt{(1 + \frac{λ_{q}}{R_{S}}) (1 - \frac{λ_{q}}{R_{E}})} .

For λ_q << R_S, R_E and R_S ≈R_E this transforms into

ω_{q n m, S C} \approx \frac{2 π c}{λ_{q}} + (n + m + 1) \frac{c}{λ_{q}} arc \sin \frac{λ_{q}}{R_{E}} \approx \frac{2 π c}{λ_{q}} + (n + m + 1) \frac{c}{R_{E}} .

Thus, it turns out that the resonance frequencies of the full cavity and the subcavity are different. This contradicts the above made assumption that the wave passes unchanged through the surface S from the full cavity into the subcavity SE, and then, according to the above arguments, generates an eigenmode in the subcavity SE.

But there is one more problem. If it is assumed that Eq. (11) is correct, it follows that the resonance frequencies diminish with increasing radius of curvature R_E, or equivalently with increasing L. Since the subcavity can be positioned at any distance L from the waist, this results in a violation of the law of energy conservation, because the energy of a photon cannot change when it propagates away from the waist in a vacuum.

Since the second term in Eq. (1) accounts for the axial phase shift of the Gaussian modes, it shall now be analyzed, whether there is a relation between these problems and the mathematical expression for ψ, as given by Eq. (2). Although the derivation of Eq. (2) is exact within PWO, it delivers results which seem to be physically incorrect. This can be seen when ψ(z) is plotted over a logarithmic scale. For this purpose z_R in Eq. (2) is replaced according to the relation (see Eq. (17.4) in [1])

z_{R} = \frac{π w_{0}^{2}}{λ_{q}},

which delivers

ψ (z) = arc \tan \frac{z λ_{q}}{π w_{0}^{2}} .

If we now express z and w₀ as multiples of the wavelength λ_q according to

z = ς λ_{q},

and

w_{0} = β λ_{q},

we obtain

ψ (ς, β) = arc \tan \frac{ς}{π β^{2}}

Figure 2 shows ψ(ζ,β) for different values of β. The diagram shows ψ(ζ,β) for strongly focused waves that means waves with small w₀ i.e. small β up to nearly plane waves with large β. It can be seen that for a nearly plane wave ψ(ζ,β) almost disappears for finite values of ζ, but nevertheless jumps from ~0 to π/2 for ζ→∞ . This contradicts the generally accepted assumption that the properties of a wave are not changing anymore at long distances from the beam waist. This contradiction is confirmed by the fact that this jump of the phase shift disappears, if ζ and β simultaneously go to infinity. This turns out from Eq. (16), since β appears squared in the denominator, whereas ζ in the numerator is not squared. Thus, if both quantities increase simultaneously, ζ/πβ², and therefore, also ψ drops to zero. This leads to the additional contradiction that ψ(ζ,β) is undefined for ζ→∞ and/or β→∞ .

Fig. 2 Dependence of the axial phase shift ψ(ζ,β) on the normalized parameters ζ = z/λ_q and β = w₀/λ_q.

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Since PWO becomes an exact representation of wave optics for nearly plane waves, these problems cannot be attributed to the approximation made to derive the paraxial wave equation from the Helmholtz equation. They seem to relate to the Helmholtz equation itself, and therefore to wave optics generally. Therefore, it cannot be seen, how the above discussed problems can be solved within wave optics theory. In the following a solution is proposed within a particle picture.

3. A particle picture of a propagating electromagnetic wave

3.1. Derivation of a Schrödinger equation for a photon bouncing between equiphase surfaces

To develop the particle picture we consider a photon bouncing between two RES positioned at z₁ and z₂ as shown in Fig. 3. Though the following consideration is not restricted to Gaussian beams, we confine it to the latter, in a first step. Since the curvatures of the RES of a Gaussian beam are given by Eq. (8), R₁ an R₂ in Fig. 3 shall be given by

R_{i} (z_{i}) = z_{i} + \frac{z_{R}^{2}}{z_{i}}, i \in | 1, 2 | .

It shall now be considered how the momentum of a photon changes when it bounces between the two RES. Though a single photon cannot be localized, it can be claimed that the movement of the photon follows the Poynting vector. It describes the movement of the energy of an electromagnetic wave, and the photon is the quantum of this propagating energy. Since the Poynting vector is oriented perpendicular to the RES, the photon also impinges at a right angle on the RES. Therefore, the change of momentum of a photon reflected on a RES is given by

| Δ \vec{P} | = 2 M c .

To derive the particle picture we need the components of the momentum change perpendicular to the optical axis as shown by the red arrows in Fig. 3. For the mirror at z₁ this component is given by

Δ P_{↑} (r_{1}, z_{1}) = \frac{2 M c}{R_{1}} r_{1},

and for the mirror at z₂ by

Δ P_{↓} (r_{2}, z_{2}) = - \frac{2 M c}{R_{2}} r_{2} .

Here r₁ is the distance from the optical axis of the point where the photon impinges on the mirror at z₁, and r₂ is the corresponding distance where the photon impinges on the mirror at z₂. It shall be stressed that these coordinates are not understood as measurable positions of a photon, they are understood in the same way as in quantum mechanics the displacement coordinates of particles are used such as to describe the vibrations of the atomic nuclei of a molecule, though in the end only a position probability of the nuclei can be computed. This issue is discussed furthermore below.

Fig. 3 Transverse components of the momentum changes of a photon reflected between two equiphase surfaces.

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From Eqs. (19a) and (19b) we obtain for the overall momentum change perpendicular to the optical axis, which a photon experiences during one round trip

\begin{array}{l} Δ P_{↑} (r_{1}, z_{1}) + Δ P_{↓} (r_{2}, z_{2}) = 2 M c (\frac{r_{1}}{R_{1}} - \frac{r_{2}}{R_{2}}) = \frac{2 M c}{R_{1} R_{2}} (r_{1} R_{2} - r_{2} R_{1}) \\ = \frac{2 M c}{R_{1} R_{2}} [r_{2} (R_{2} - R_{1}) - R_{2} (r_{2} - r_{1})] . \end{array}

This delivers

\lim_{R_{1} \to R_{2}} Δ P_{↑} (r_{1}, z_{1}) + Δ P_{↓} (r_{2}, z_{2}) = \frac{2 M c}{R_{2}^{2}} (r_{2} Δ R - R_{2} Δ r) .

Since a reflected wave exerts a radiation pressure on the RES, it must be assumed that a photon bouncing between two RES exerts a time-averaged quantized force on the RES, and that, in consequence, the RES must exert a time averaged quantized force on the photon to compensate for the radiation pressure. Though it is easy to derive a relation between the momentum change of a continuously acting quantity like an electromagnetic wave and a force, this is not straightforward in the case of a single photon. To establish a relation between the change of a momentum and a force, it is necessary to define a time span during which the momentum change takes place. Though it is not possible to establish a fixation in time for the change of the momentum of a single photon, it can be stated that the photon undergoes one change of momentum at the RES at z₁ and a second change of momentum at the RES at z₂ during the time t_round which a photon takes for a round trip. The latter is given by

t_{r o u n d} = \frac{2}{c} (z_{2} - z_{1}) = \frac{2}{c} Δ z .

Thus, it follows from Eqs. (21) and (22) that, after replacing R₂ by R(z), the time averaged quantized force exerted on the photon during one round trip is given by

K (r, z) = \frac{M c^{2}}{R^{2} (z)} (r \frac{Δ R}{Δ z} - R \frac{Δ r}{Δ z}) .

In the limit Δz→0 the differential quotient dR/dz is obtained from Eq. (8) as

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

For dr/dz a simple geometrical consideration delivers

\frac{d r}{d z} (z) = \frac{r}{R (z)}

Since for Δz →0, the problem of a fixation in time of the change of the momentum becomes irrelevant, the force given by Eq. (23) can be considered to be a real force in this limit. After inserting Eqs. (24) and (25) into Eq. (23), this force is obtained as

\lim_{Δ z \to 0} K (r, z) = \frac{M c^{2} r}{R^{2} (z)} (1 - \frac{z_{R}^{2}}{z^{2}} - \frac{R (z)}{r} \frac{r}{R (z)}) = - \frac{M c^{2} r}{R^{2} (z)} \frac{z_{R}^{2}}{z^{2}}

Replacing here R(z), according to Eq. (8), delivers

K (r, z) = - M {(\frac{c z_{R}}{z^{2} + z_{R}^{2}})}^{2} r .

which is valid for any z. Thus, we obtain that the photon is moving within a parabolic potential well given by

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

with

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

Inserting Eq. (28) into Eq. (3) delivers

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

with

E (z) = ℏ [ω (z) - ω_{q}] .

Except for the dependence of ω_t and consequently also of χ on the z-coordinate, Eq. (30) is identical with the Schrödinger equation of the two-dimensional harmonic oscillator. The only difference between Eq. (30) and the Schrödinger equation for a particle with rest mass m > 0 is that the potential given by Eqs. (28) and (29) only depends on the parameters λ_q and z_R, and not on the rest mass of a particle, which vanishes in the case of a photon. The solutions of Eq. (30) are well known.

With respect to Eq. (29) it shall be pointed out that ω_t does not depend on the mass M of the photon. It only depends on z_R, which according to Eq. (12), is proportional to w₀²/λ . But, since according to Eq. (19.6) in [1], the latter expression only depends on the geometrical properties of a resonator, this also holds for z_R. It can therefore be expected that ω_t can be also derived by a geometric optics approach. This is shown in Sect. 4.

It may be argued that the above given derivation of a Schrödinger equation describing the transverse motion of a photon is only valid, if the RES represent real mirrors, since otherwise no real force is exerted on the photon. Concerning this argument it shall be recalled that the propagating wave represents a propagating mass. This delivers two arguments. First, the properties of the propagating mass are not changed when this mass is bouncing between RES. Therefore, the motion of a photon propagating with this mass also cannot be changed, if the photon is bouncing in the same way. However, if the photon is bouncing in this manner, a transverse force must be exerted on the photon, otherwise it must be expected that it is moving away to infinity in the transverse direction like any other particle. Second, a real transverse force is obviously exerted on the propagating mass by the end mirrors of a real cavity, since a force compensating for the radiation pressure is exerted on the reflected wave by the mirrors, and this force has a transverse component whose magnitude depends locally on the distance from the cavity axis. But this force cannot be assumed to be exerted on the end mirrors only, since, under this assumption, no smooth propagating wave would be generated. The above derived result, describing a transverse force acting continuously between the real mirrors, solves this problem. If it cannot be considered to be a real force, this force may be considered as a virtual force. As already mentioned in the introduction, the requirement of a transverse force acting on the mass of a propagating wave does not fit into the wave optics picture, since it cannot be considered within this approach. It must however be taken into account, if the electromagnetic wave is being considered as a propagating mass.

A second argument may be that, due to the uncertainty principle, the momentum of the photon becomes undefined, if the distance between the RES vanishes. However, since the RES are not considered to be real mirrors, they do not fix the position of the photon like in an experiment. Therefore, in the limit Δz→0, the coordinates z₁₍₂₎ and r₁₍₂₎ are, strictly speaking, only used to describe the virtual momentum changes of the mass of the photon but not to localize it, as already pointed out above.

There is however one more problem. Since in case of a photon the mass represented by the energy of the transverse motion is not small compared with the total mass, as in the case of a particle with non-vanishing rest mass, the question arises, whether the total mass still can be used in the Schrödinger equation describing the transverse motion of the photon. Though this question deserves more careful consideration, the preliminary assumption is made that Eq. (30) delivers a good approximation for the description of the transverse motion of the photon as long as z_R is not too small. This is confirmed by the fact that, as already mentioned above, the result obtained for ω_t can be reproduced by a geometric optics approach which does not involve the mass of the photon.

Since the above given derivation of a potential describing the transverse motion of the photon is not restricted to spherical RES, but can be applied to phase fronts of any shape, it can be concluded that a transverse quantum mechanical motion of the photon described by a Schrödinger equation can be generally attributed to any photon moving with a wave having distinct phase fronts. Therefore, instead of the spherical surfaces introduced above parabolic surfaces could also have been used, which represent the mathematically exact description of the equiphase surfaces of Gaussian beams. But it is interesting that the spherical surfaces deliver results in full agreement with PWO as shown below.

3.2. Computation of the transverse eigenfunctions

The eigenfunctions of Eq. (30) 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)}) .

Here w_p², where p refers to particle, is given by

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

According to Eq. (32), w_p represents the value of r, where the position probability density of a photon in the ground level drops to exp(−2). After insertion of Eq. (29) we obtain from Eq. (33)

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

which delivers

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

This is in agreement with Eq. (12) obtained by the use of PWO. Insertion of Eq. (35) into Eq. (34) delivers

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

This is additionally in agreement with PWO, as Eq. (7) shows.

If we moreover compare Eq. (32) with Eq. (16.60) in [1], we obtain the important result that, in the whole resonator, the photons' position probability density |χ_nm(x,y,z)|², computed by the use of Eqs. (32), (35) and (36), is for any modal numbers n and m in full agreement with the normalized intensity distribution provided by PWO for a transverse Gaussian mode.

3.3. Computation of the eigenvalues

The eigenvalues of Eq. (30) are given by

E_{n m} (z) = ℏ ω_{t} (z) (n + m + 1) .

After inserting Eq. (31) this delivers

ω_{q n m} (z) = \frac{2 π c}{λ_{q}} + (n + m + 1) \frac{c z_{R}}{z^{2} + z_{R}^{2}} .

This result is not in agreement with Eq. (11), as can be seen, if R_E in Eq. (11) is replaced according to Eq. (8), which delivers

ω_{q n m} (z) = \frac{2 π c}{λ_{q}} + (n + m + 1) \frac{c z}{z^{2} + z_{R}^{2}} .

However, the Eqs. (37) and (39) are not generally different. They agree for z→0. The latter cannot be directly derived from Eq. (39), because the derivation of Eq. (11) from Eq. (10) does not hold in this case, since R_S becomes infinite, if S approaches the beam waist. Therefore, it is more convenient to consider Eq. (9) for L<<R_E, which results in a planar-spherical resonator with very small distance between the mirrors. For this case Eq. (9) delivers

ω_{q n m} (z) \approx \frac{2 π c}{λ_{q}} + (n + m + 1) \frac{c}{\sqrt{L R {}_{E}}} .

Since according to Eq. (19.4) in [1], the Rayleigh range z_R, in case of a planar-spherical resonator, is given by

z_{R}^{2} = R_{E} L (1 - \frac{L}{R_{E}}),

we obtain

\lim_{L / R \to 0} z_{R} = \sqrt{L R_{E}} .

Thus, it turns out that for z→0, Eq. (38) is in agreement with the result obtained by PWO for a planar-spherical resonator with vanishing distance between the mirrors.

However Eq. (38) shows that also in the particle picture, the problem arises that the law of energy conservation is violated, since according to this equation the energy of the propagating photon depends on z. However, within the particle picture, a solution seems to be possible, as proposed in the following.

3.4. The energy balance of a photon in a propagating beam

To solve the problem concerning energy conservation arising from Eq. (38), the following solution is proposed. If a propagating beam wave is focused by a mirror or a lens, the energy of a photon propagating with the beam remains unchanged, due to energy conservation. But part of its energy is converted into the energy of a quantum mechanical transverse motion of the photon, dependent on its distance from the beam waist and on the spot size at the beam waist. Due to this energy conversion, the wave optics property of a propagating photon is changed, and its wavelength increases or decreases, when it approaches the beam waist or propagates away from it. However, the total energy of the photon propagating in this way remains unchanged. Depending on the amount of energy that is converted into the quantum mechanical transverse motion, the photon can be found in different eigenstates described by Eq. (32) in case of Gaussian beams. However, the total energy of the photon does not depend on the mode order.

With respect to this proposal, it seems to be useful to introduce for the modified wavelength of the photon the term local wavelength λ_l(z) which shall be defined as twice the z-dependent distance of two points where the vector of the electrical field passes through zero. For λ_l(z) we obtain, according to the proposed solution of the energy problem, from Eq. (38) the expression

λ_{l} (z) = {[\frac{1}{λ_{q}} - \frac{(n + m + 1) z_{R}}{2 π [z^{2} + z_{R}^{2}]}]}^{- 1} .

This delivers λ_l = λ_q for z →∞. Therefore, for z →∞, energy and wavelength of the photon are equal to the values initially defined in Eqs. (4) and (5), describing a photon in the associated plane wave in the limiting case of plane end mirrors. If we divide Eq. (43) by λ_q we obtain

\frac{λ_{l} (z) - λ_{q}}{λ_{q}} = {[1 - \frac{(n + m + 1) λ_{q}}{2 π [{(z / z_{R})}^{2} + 1] z_{R}}]}^{- 1} - 1.

Introducing the relation

s = \frac{z}{z_{R}},

and replacing z_R, according to the Eqs. (12) and (15), delivers

\frac{λ_{l} (s) - λ_{q}}{λ_{q}} = {[1 - \frac{(n + m + 1)}{2 π^{2} (s^{2} + 1) β^{2}}]}^{- 1} - 1.

Since in many cases β² >>1 holds, this equation can be simplified to

β^{2} \frac{λ_{l} (s) - λ_{q}}{λ_{q}} \approx \frac{n + m + 1}{2 π^{2} (s^{2} + 1)},

under this condition. For s→0 this equation simplifies to

β^{2} \frac{λ_{l} (0) - λ_{q}}{λ_{q}} \approx \frac{n + m + 1}{2 π^{2}} .

Although wave optics theory does not allow to speak about a change of the wavelength of a photon propagating in a vacuum, the axial phase shift ψ described by Eq. (2) de facto also results in an increased local wavelength, as shown in Fig. 17.16 in [1]. Therefore, the effect of the axial phase shift on the local wavelength shall now be analyzed and compared with the results obtained above.

According to Eq. (17.1) in [1] the phase of the Gaussian fundamental wave is given by the expression

\exp [- i (k_{q} - \frac{ψ (z)}{z}) z]

Here k_q = 2π/λ_q is the propagation vector of the associated plane wave. Defining

k_{l P W O} (z) = k_{q} - \frac{ψ (z)}{z}

as the local propagation vector according to PWO, we obtain for the local wavelength, caused by the axial phase shift, according to PWO

λ_{l P W O} (z) = \frac{2 π}{k_{l P W O} (z)} = \frac{2 π}{k_{q} - \frac{ψ (z)}{z}} = \frac{2 π}{\frac{2 π}{λ_{q}} - \frac{ψ (z)}{z}} = \frac{λ_{q}}{1 - λ_{q} \frac{ψ (z)}{2 π z}}

From this equation we obtain

\frac{λ_{l P W O} - λ_{q}}{λ_{q}} = \frac{1}{1 - λ_{q} \frac{ψ (z)}{2 π z}} - 1

Replacing here z by sz_R and then z_R according to Eqs. (12) and (15) delivers

\frac{λ_{l P W O} - λ_{q}}{λ_{q}} = \frac{1}{1 - \frac{a r c \tan (s)}{2 π^{2} s β^{2}}} - 1

which transforms into

\frac{λ_{l P W O} - λ_{q}}{λ_{q}} = \frac{2 π^{2} β^{2}}{2 π^{2} β^{2} - \frac{a r c \tan (s)}{s}} - 1.

For s→0 this delivers

\lim_{s \to 0} \frac{λ_{l P W O} - λ_{q}}{λ_{q}} = \frac{2 π^{2} β^{2}}{2 π^{2} β^{2} - 1} - 1.

Taking additionally into account that, according to Eq. (16.60) in [1], the axial phase shift of any Gaussian mode with modal numbers n and m is given by (n + m + 1)ψ(z/z_R), we obtain

\lim_{s \to 0} \frac{λ_{l P W O} - λ_{q}}{λ_{q}} = \frac{2 π^{2} β^{2}}{2 π^{2} β^{2} - (n + m + 1)} - 1

Exactly the same result is obtained, if s = 0 is inserted into Eq. (46). Thus, it turns out that for s = 0 the local wavelength at the beam waist computed by the use of the particle picture is in full agreement with the local wavelength computed by the use of PWO, and that this holds for all transverse modes. This result is indeed surprising, since the particle picture and the above made proposal concerning the energy balance of the photon represent a physical approach which is completely different from the wave optics methods used with PWO. But beyond, this result confirms the above made proposal.

The deeper reason for this agreement between the particle picture and PWO for the case s→0 is the fact that, for this case, the energy levels of the photon obtained within the particle picture agree with the resonant frequencies of the modes obtained within PWO as shown in Sect. 3.3. Consequently this agreement concerning the local wavelength is canceled for s>0. This is shown in Figs. 4(a) and 4(b). For n = m = 0 and β = 1, the blue line in Fig. 4(a) shows (λ_l - λ_q)/λ_q according to Eq. (46) i.e. according to the particle picture, the red line shows (λ_lPWO - λ_q)/λ_q according to Eq. (54) i.e. according to PWO. It can be seen that the blue line drops much faster with increasing s than the red line. This is a consequence of the fact that the axial phase shift continues to jump from 0 to π/2 even for large values of ζ, as Fig. 2 shows. The latter is also the deeper reason for the discrepancy between Eqs. (38) and (39) describing the frequency levels. Figure 4(b) shows the relative increment (λ_l-λ_q)/λ_q of the local wavelength multiplied by β² i.e. β²(λ_l-λ_q)/λ_q for β = 5 and n = m = 0. As can be seen, there is only a very small difference between the two Figs. If β increases even more almost nothing is changing. This is expected, since the right hand side of Eq. (47) does not depend on β.

Fig. 4 Dependence of the local wavelength λ_l on the normalized distance s = z/z_R from the beam waist according to the particle picture and to the PWO approach

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For s = n = m = 0 Eq. (46) delivers

\frac{λ_{l} (z = 0)}{λ_{q}} = {[1 - \frac{1}{2 π^{2} β^{2}}]}^{- 1} = \frac{2 π^{2} β^{2}}{2 π^{2} β^{2} - 1} .

This result shows that the local wavelength λ_l goes to infinity for

β = \frac{1}{\sqrt{2} π} .

According to the proposal concerning the energy balance of the photon, this means that the energy of the photon is completely converted into the energy of the transverse quantum mechanical motion. Thus, it seems that a photon originating from a quasi point source like an atom initially does not have the wave optics property represented by the wavelength, and that the latter only develops with increasing distance from the point source. This may be important for the understanding of the emission of light from an atom. For

β < 1 / (\sqrt{2} π)

, the local wavelength jumps to a negative value and then goes to -∞ for β→0. Exactly the same result is obtained from Eq. (56) for n = m = 0. The reason for this strange behavior may be that Eq. (8) is not correct for a nearly spherical wave that means for β→0 i.e. z_R→0, and needs to be revised. However, it also may be necessary to revise Eq. (30) for this limiting case, as indicated above.

The result λ_l →λ_q for z→∞ obtained from Eq. (43) does not necessarily indicate that all transverse modes in a real cavity have the same output frequency. Since in a real cavity the mirrors usually represent nodal phase fronts of the mode, and since the position of these nodal phase fronts depends on the phase shift, λ_q is expected to assume different values to adjust the position of the nodal surfaces to the mirrors surfaces. Due this fact Eq. (2) is only approximately correct dependent on the magnitude of q. It would be therefore more convenient to replace λ_q by λ_∞ that means the value of the wavelength which the photon assumes in a divergent beam for z→∞.

3.5. Confirmation by measurements

3.5.1 Measurement of the local wavelength

It is well known that the physical understanding of the axial phase shift is based on the Guoy effect, which has been detected by L. G. Guoy in 1890 by the use of a very simple apparatus. Using a pinhole light source, he overlapped a focused beam with an unfocused beam and observed a “bulls-eye” interference pattern [1,6] in the focus. It has been shown [7,8,] that this experiment can be carried through today with much more accuracy using laser technology and modern measurement methods. In this way, it should be possible to check, if the description of the local wavelength according to Eq. (46) based on the particle picture is correct, or if the description according to Eq. (54), based on PWO is correct. In Fig. 5 the relative increment of the local wavelength is plotted once more for β = 1 but using for the abscissa the scaling ζ = z/λ_q . Figure 5 shows that, in this case, the relative increment of the local wavelength amounts to ~5% in the focus. According to the particle picture it decreases to almost zero at a distance of ~15*λ_q from the focus, whereas it still amounts to ~2% at this distance according to PWO.

Fig. 5 Dependence of the local wavelength λ_l on the normalized distance ζ = z/λ_q from the beam waist according to the particle picture and to the PWO approach.

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3.5.2 Measurement of the transverse mode beat frequency

An additional experimental confirmation seems to be possible by measuring the beat frequency of the power output generated by a superposition of more than one eigenfunction χ_nm of the photon.

If the time-independent Eq. (30) is replaced by the corresponding time-dependent Schrödinger equation, the eigenfunctions χ_nm are multiplied by the time factor exp[-iω_nm(z)t] with

ω_{n m} (z) = (n + m + 1) ω_{t} (z) = (n + m + 1) \frac{c z_{R}}{z^{2} + z_{R}^{2}} .

Therefore, if the state function of the photon is represented by a superposition of several eigenfunctions, the photons' position probability oscillates around a mean value with time-dependencies given by

\cos [Δ ω_{n_{1} m_{1}, n_{2} m_{2}} (z) t] = \cos [(n {}_{2}+ m_{2} - n {}_{1}- m_{1}) \frac{c z_{R}}{z^{2} + z_{R}^{2}} t]

dependent on the modal numbers n and m of the eigenfunctions contributing to the state function of the photon. Due to this oscillation of the photons' position probability density, a locally oscillating increment of the density distribution of the photons is generated by stimulated emission, and in consequence, a beating power output. According to Eq. (60), the frequency of this oscillating increment locally depends on the distance z from the beam waist. If the state function is represented by a combination of the eigenfunctions χ₀₀ and χ₁₀, we obtain

Δ ω_{00, 10} (s) = \frac{c}{z {}_{R}} \frac{1}{s^{2} + 1} = \frac{c}{π β^{2} λ {}_{q}} \frac{1}{s^{2} + 1} .

Figure 6 displays the oscillation frequency of the local position probability of the photon as a function of the normalized distance s = z/z_R from the beam waist. It shows that this oscillation frequency decreases, compared with the value at the beam waist, by a factor 50 at a distance of z = 7*z_R from the beam waist. It is therefore expected that the beat frequency of the power output strongly depends on the position of the active medium, if the latter extends over a short section along the beam axis. This is in contradiction with the results presented in [9]. According to these results the transverse mode beats should only depend on the resonance frequencies of the transverse modes described by Eq. (1). The latter is also claimed in Sect. 19.3 in [1]. The beat frequency, computed according to PWO, cannot be shown in Fig. 6, since it depends, according to Eq. (1), in addition to β on the curvatures and the distance of the end mirrors.

Fig. 6 Variation of the beat frequency between TEM₀₀ and TEM₀₁ mode dependent on the distance of a short active medium from the beam waist.

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If for the proposed experiment a thin plate of solid state laser material is used as active medium, the absorbed pump power density should be equally distributed transverse to the beam axis. Otherwise, a thermal lensing effect is generated with the consequence that the beam waist follows the position of the plate, when it is shifted along the beam axis. The latter can be demonstrated by the use of the program LASCAD [10]. This also seems to be the reason, why this effect has not been detected so far, since in solid state laser configurations the beam waist usually coincides with the focus of the pump beam.

For s = 0 we obtain from Eq. (61)

Δ ω_{00, 10} (s = 0) = \frac{c}{π β^{2} λ {}_{q}} .

This result shows that for s = 0 the oscillation frequency of the photons' position probability density only depends on the parameters β and λ_q . According to this equation, the oscillation frequency of the photons' position probability decreases by a factor 100, if the waist spot size w₀ increases from λ_q to 10* λ_q . Consequently, also the beat frequency of the power output is expected to decrease similarly, if the active medium only extends over a small range around the beam waist. This is once more in contradiction with the currently accepted theory of transverse mode beating, as described in Sect. 19.3 in [1]. It claims that the beat frequency according to Eq. (1) also depends on the curvatures and the distance L of the mirrors, and not only on w₀ and λ_q . This is hard to believe, since according to PWO a Gaussian mode is completely described by the Eqs. (2), (7), (8), (12), β and the axial wavelength λ_q which does not change as long as the distance L between the mirrors is equal to any integer multiple of λ_q . For β→0 the problem arises that Δω_00,10 becomes infinite. This problem is closely connected with the above discussed question, if the Schrödinger equation for the transverse motion of the photon, represented by Eq. (30), remains a good approximation in this case or must be revised.

Measurements to verify Eqs. (60), (61) and (62) are in progress [11].

4. Geometric optics description of the transverse motion of a ray reflected between two equiphase surfaces with vanishing distance

As is well known, the harmonic motion of a classical particle with mass m is described by the Hamiltonian

H = \frac{p_{x}^{2} + p_{y}^{2}}{2 m} + \frac{1}{2} m ω^{2} (x^{2} + y^{2}) .

The solution of this equation describes a periodic sinusoidal motion of a particle with angular frequency ω. By the use of Schrödingers' correspondence rules Eq. (63) can be converted into the corresponding quantum mechanical Hamiltonian of the harmonic oscillator.

Since a similar correspondence principle holds for the relationship between geometric and wave optics, it should be possible to establish a description of the transverse motion of the photon using a ray tracing approach. Based on the correspondence principle, this would establish a confirmation of the results obtained above within a quantum mechanics particle picture.

For this purpose we again consider the cavity proposed in Fig. 3, with RES at the positions z₁ and z_2, and with radii equal to the radii of the local Gaussian phase front, and assume that a ray is bouncing back and forth between these RES. According to Eq. (15.48) in [1], the series of points where such a ray consecutively hits the RES at z₂ is given by

x_{n} = x_{0} \cos n θ + s_{0} \sin n θ .

For the sake of simplicity, the consideration has been restricted to one dimension. In Eq. (64) x₀ is the initial ray position and s₀ the initial ray slope. The two terms in Eq. (64) can be summarized into

x_{n} = C \cos (n θ + γ)

with

C = \sqrt{x_{0}^{2} + s_{0}^{2}},

and

\tan γ = \frac{x_{0}}{s_{0}} .

The time a ray takes for a round-trip is again given by Eq. (22). We can therefore express the reflection points over a time axis as follows

x_{n} = C \cos (\frac{θ}{t_{r o u n d}^{}} n t_{r o u n d} + γ) .

as shown by Fig. (15.15) in [1]. According to Eq. (15.45) in [1], θ is given by

θ = arc \cos (S = \frac{A + D}{2}) = arc \sin \sqrt{1 - S^{2}} .

which delivers

\lim_{Δ z \to 0} θ = \sqrt{1 - S^{2}} .

In Eq. (69) A and D are elements of the round trip matrix of the propagating ray. The latter is given by

[\begin{matrix} A & B \\ C & D \end{matrix}] = [\begin{matrix} 1 & Δ z \\ 0 & 1 \end{matrix}] [\begin{matrix} - 1 & 0 \\ {- 2 / R}_{1} & - 1 \end{matrix}] [\begin{matrix} 1 & Δ z \\ 0 & 1 \end{matrix}] [\begin{matrix} 1 & 0 \\ - 2 / (R_{1} + R' Δ z) & 1 \end{matrix}]

with R' = dR/dz, according to Eq. (24). From Eqs. (68), (70) and (71) we obtain, after a small amount of algebra

\lim_{t_{r o u n d} \to 0} \frac{θ}{t_{r o u n d}} = \lim_{Δ z \to 0} \frac{c θ}{2 Δ z} = \lim_{Δ z \to 0} c \sqrt{\frac{1 - S^{2}}{4 Δ z^{2}}} = c \sqrt{\frac{1 - R'}{R^{2}}} = \frac{c z_{R}}{z R} = \frac{c z_{R}}{z^{2} + z_{R}^{2}} .

Inserting this result into Eq. (68) delivers

\lim_{n t_{r o u n d} \to t} x = A \cos (ω_{r a y} (z) t + γ) .

with

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

Thus, we obtain the surprising result that ω_t(z), given by Eq. (29), is identical with the geometric optics quantity ω_ray(z) as already indicated in Sect. 3.1. To state this in words, the geometric optics photon, described by a bouncing ray, moves with the same transverse oscillation frequency as obtained for the quantum mechanical photon based on a consideration of the change of momentum at the RES. Since a mass M, according to Eq. (6), can be also attributed to the geometric optics photon, this additionally confirms that a force according to Eq. (27) must be exerted on this photon.

In this way the assumption of a transverse quantum mechanical motion of the photon is confirmed by a geometric optics approach. A schematic illustration of both approaches is shown in Fig. 7. The left picture shows the photon as a particle squeezed between two RES with vanishing distance. The right picture shows a zig-zag ray bouncing back and forth between two RES.

Fig. 7 Schematic illustration of the quantum mechanics and the ray tracing approach.

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5. Summary and conclusions

It has been shown that Eq. (1) describing the condition provided by paraxial wave optics (PWO) for the resonance frequencies of the eigenmodes of an optical resonator, leads to a contradiction, if the resonator is divided into subcavities. Moreover, it is shown that the results obtained in this way imply a violation of energy conservation.

Since the second term in Eq. (1) accounts for the axial phase shift ψ of the Gaussian modes, it has been analyzed whether a relation exists between these problems and the mathematical expression for ψ as given by Eq. (2). Though the derivation of Eq. (2) is exact within paraxial wave optics (PWO), it could be shown that it leads to a contradiction, if the Rayleigh range z_R goes to infinity. This turns out, when ψ(z) is plotted over a logarithmic scale (see Fig. 2), which shows that for large values of z_R the phase shift ψ(z) almost disappears for finite values of z, but nevertheless jumps from ~0 to π/2 for z→∞ . This contradicts the generally accepted assumption that the properties of a wave are not changing anymore at long distances from the beam waist.

Since PWO transforms into an exact representation of wave optics for nearly plane waves i. e. for z_R→∞, these problems even concern wave optics. Therefore a solution of these problems does not seem to be possible within wave optics theory.

In this paper a solution is proposed within a particle picture, as presented recently by the author [2,3], which has been worked out in more detail in the present paper. This particle picture shows that the transverse modes can be understood as the transverse quantum mechanical eigenfunctions of a single photon. Additionally, it provides a physical explanation for the analogy between paraxial wave optics and the harmonic oscillator as described previously [4].

To develop the particle picture of a propagating wave, it has been recalled that according to the theory of relativity a propagating electromagnetic wave represents a propagating mass, whose quantum is the relativistic mass of the photon. Therefore, the change of momentum is considered when the mass, propagating with a wave, is bouncing between two reflecting equiphase surfaces (RES) with small distance. In case of vanishing distance between the RES, this leads to a transverse force exerted on the quantum of the propagating mass or photon. In this way it could be shown that the photon is moving within a transverse quantum mechanical potential. This leads to a Schrödinger equation describing the transverse motion of the photon in a propagating wave.

However, it turned out that the obtained particle picture also violates of the law of energy conservation, as shown in Sect. 3.3. But in the particle picture a solution of this problem seems to be possible and has been proposed as follows. The total energy of the photon is constant in the whole cavity, but due to the focusing of a beam, part of the energy of the photon is converted into a quantum mechanical transverse motion of the photon. Due to this energy conversion, the wave-optics property of the propagating photon is changed, and its local wavelength increases or decreases, when it approaches the beam waist or propagates away from it. Depending on the amount of energy converted into the quantum mechanical transverse motion, the photon can be found in different eigenstates as described by Eq. (32) for the case of a resonator with spherical end mirrors.

Since the axial phase shift ψ de facto also results in an increased local wavelength, as shown in Fig. 17.16 in [1], the effect of the axial phase shift on the local wavelength has been analyzed and compared with the results obtained for the local wavelength based on the above proposal. In this way it turned out that both approaches deliver identical results for the local wavelength at the beam waist. This is surprising, since the particle picture together with the above made proposal concerning the energy balance of the photon represent a physical approach which is completely different from the wave optics methods used with PWO. But beyond, this result confirms the above made proposal.

However, with increasing distance from the beam waist the results obtained for the local wavelength by both approaches differ considerably. Therefore it is proposed to check experimentally which approach is physically correct. This could be done by repeating the measurements, which L.G. Gouy made in 1890, when he detected the Gouy effect [6]. It has been shown [7,8,] that this experiment can be carried through today with much more accuracy using laser technology and modern measurement methods. In this way it should be possible to check, whether the description of the local wavelength, according to Eq. (46) based on the particle picture, is correct, or if the description according to Eq. (54) based on PWO, is correct.

Measurement of the transverse mode beats provides an other possibility to verify the particle picture. According to the latter a beating power output should be observed if the state function of the photon is represented by a superposition of more than one eigenfunction. Due to the different time dependency of the eigenfunctions this leads to an oscillating position probability of the photon, with the consequence that a locally oscillating increment of the density distribution of the photons is generated by stimulated emission. As shown in Sect. 3.5.2 the oscillation frequency of this increment depends locally on the distance from the beam waist. This is in contradiction to the currently accepted opinion, see [1] Sect. 19.3, that the frequency of the transverse mode beats only depends on the resonance frequencies of the transverse modes described by Eq. (1).

An important property of the particle picture is that only a single photon is involved. This leads to the conclusion that the reflection at the mirrors enables a single spontaneously emitted photon to build up a resonant state in a cavity with macroscopic dimensions. From this result it can be further concluded that this single resonant photon generates a second resonant photon by stimulated emission, and so forth, which finally leads to a coherent state. Therefore only a single photon in a resonant state seems to be necessary to generate a laser mode. Hence the particle picture seems to explain why it is possible that a laser mode can develop in a cavity by spontaneous emission combined with stimulated emission.

However, the particle picture provides a further important result. Since it shows that the intensity distribution in the cavity can be interpreted as the position probability density of a particle represented by the photon, it turns out that the laser resonator represents a macroscopic quantum mechanical system. In this way the particle picture shows a new facet of the particle wave dualism introduced by Schrödinger into physics.

Through further research, it shall be investigated how the particle picture can be extended to resonators with aspheric mirrors, and/or with internal elements. In case of mirrors with limited transverse extension, it should be possible to show that the effect of diffraction corresponds in the particle picture to the quantum mechanical tunnel effect. It may be further expected that the particle picture will generally be important to model photonic systems, since analogies between micro optics and quantum mechanics have already been reported by other authors [12]. For instance, it is expected that the transverse quantum mechanical motion of a photon in a step index fiber can be modeled by the use of a rectangular potential well of finite depth.

Acknowledgments

I thank my son Georg Altmann for helping me to create the figures and plots, and for reviewing the algebraic transformations.

References and links

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

2. K. Altmann, “A Particle Picture of the Optical Resonator,” in OSA OpticsInfoBase, Conference Paper ATu2A.29, Advanced Solid State Lasers (ASSL) 2014. [CrossRef]

3. K. Altmann, “A Particle Picture of the Optical Resonator,” in OSA OpticsInfoBase, Summary of Conference Paper ATu2A.29–1, Advanced Solid State Lasers (ASSL) 2014. [CrossRef]

4. G. Nienhuis and L. Allen, “Paraxial wave optics and harmonic oscillators,” Phys. Rev. A 48(1), 656–665 (1993). [CrossRef] [PubMed]

5. K. Altmann, “Derivation of the transverse mode structure of an optical resonator by a Schrödinger equation,” Phys. Lett. 91(1), 1–4 (1982). [CrossRef]

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

7. M. S. Kim, A. Naqavi, T. Scharf, K. J. Weible, R. Völkel, C. Rockstuhl, and H. P. Herzig, “Experimental and theoretical study of the Gouy phase anomaly of light in the focus of microlenses,” J. Opt. 15(10), 105708 (2013). [CrossRef]

8. Th. Videbaek, “Visualizing the Gouy phase of a laser beam,” Laser Teaching Center, Stony Brook University.

9. J. P. Goldsborough, “Beat frequencies between modes of a concave-mirror optical resonator,” Appl. Opt. 3(2), 267–275 (1964). [CrossRef]

10. LAS-CAD GmbH Munich, Germany, http://www.las-cad.com

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

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

Contradiction within wave optics and its solution within a particle picture

Abstract

1. Introduction

2. Contradiction within wave optics

3. A particle picture of a propagating electromagnetic wave

3.1. Derivation of a Schrödinger equation for a photon bouncing between equiphase surfaces

3.2. Computation of the transverse eigenfunctions

3.3. Computation of the eigenvalues

3.4. The energy balance of a photon in a propagating beam

3.5. Confirmation by measurements

3.5.1 Measurement of the local wavelength

3.5.2 Measurement of the transverse mode beat frequency

4. Geometric optics description of the transverse motion of a ray reflected between two equiphase surfaces with vanishing distance

5. Summary and conclusions

Acknowledgments

References and links

Cited By

Figures (7)

Equations (75)

Optics Express