Abstract

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

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

1. Introduction

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

  • • Theory of relativity (E = mc2),
  • • Newton's first law,
  • • Maxwell's equations.

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

K(r,z)=Eph*limr1r2,z1z21Δz[SN(r2,z2)SN(r1,z1)]
where for the sake of simplicity rotational symmetry of the wave has been assumed. Δz is the distance of the points z1 and z2 where two wave fronts Φ1 and Φ2 with infinitesimal distance intersect the propagation axis of the wave, as shown by Fig. 3 in [1]. The SN(ri,zi) are normalized Poynting vectors erected in the points r1 and r2 on the wave fronts Φ1 and Φ2 as also shown by Fig. 3 in [1]. In the infinitesimal limit the points ri as well as zi are assumed to move into each other. Eph is the energy of the photon given by
Eph=Mc2=hcλ
where λ and c are the wavelength and the speed of light in a vacuum, respectively. M = h/(cλ) is the relativistic mass of the photon.

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

[22MΔ+E(z)V(r,z)]χ(r,z)=0,
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. In Eq. (3) 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.

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

[2MΔ+Enm12Mω2(z)(x2+y2)]χnm(x,y,z)=0
as also shown in [1]. ω describes the frequency of the transverse quantum mechanical oscillation of the photon, and is given by
ω(z)=czRz2+zR2.
Here zR = πw02/λ is the Rayleigh range, and w0 is the spot size at the beam waist. In [1] the Eqs. (4) and (5) have been verified by comparing the probability density of the photon, which is represented by the squared absolute values |χnm(x,y,z)|2 of the eigenfunctions χnm of the Schrödinger equation, with the result obtained by the use of wave optics for the normalized local intensity of a Gaussian wave of order n,m as described by Eq. (16.60) in [2]. This comparison provided full agreement between both quantities.

As well known, the energy levels Enm of the Schrödinger equation of the 2-dimensional harmonic oscillator are given by

Enm(z)=ω(z)(n+m+1).
In the following it will be shown that a direct relation exists between the energy levels Enm(z) and the Gouy phase shift [3] which seems to allow for explaining the Guoy phase shift as an energetic effect.

2. Is the Gouy phase shift an energetic effect?

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

k¯z,nm(z)=<kz,nm2>k=k<kx,nm2>+<ky,nm2>k
and establish a relationship between the expectation value <kx,nm2>+<ky,nm2> of the transverse part of the square of the propagation constant k, and the expectation value of the square of the photon's transverse quantum mechanical momentum given by
<χnm|p^2(z)|χnm>=Mω(z)(n+m+1)
which can by the use of Eq. (6) be transformed into
<χnm|p^2(z)|χnm>=MEnm(z)
In order to establish a relationship between <kx,nm2>+<ky,nm2> and <χnm|p^2(z)|χnm>we take into account that the momentum of a freely propagating photon is given by p = ћk. This shows that the expression for the wave number k is obtained, if p is divided by ћ. It can therefore be concluded that division of the expectation value <χnm|p^2(z)|χnm>by ћ2 delivers <kx,nm2>+<ky,nm2>. In this way, we obtain
<kx,nm2>+<ky,nm2>=<χnm|p^2(z)|χnm>2=MEnm(z)2
This delivers after insertion into Eq. (7)
k¯z,nm(z)=k(1MEnm(z)k22)
If we replace in this equation the relativistic mass M of the photon according to Eq. (2) by M = Eph/c2, we obtain
k¯z,nm(z)=k(1EphEnm(z)c2k22)
If we take into account now that Eph can be expressed as
Eph=ck
we obtain
k¯z,nm(z)=k(1Enm(z)Eph)=kEph[EphEnm(z)]=1c[EphEnm(z)]
This equation can be rewritten as
ck¯z,nm(z)=EphEnm(z)
This shows that ck¯z,nm(z)is equal to the difference between the total energy Eph of the photon and the energy eigenvalues Emn(z) of the Schrödinger equation which increase with decreasing z, since ω increases according to the Eq. (5). It can be therefore concluded that part of the total energy of the photon is transformed into the energy of the transverse quantum mechanical motion of the photon, and another part is transformed into ck¯z,nm(z). Since according to this result ck¯z,nm(z) represents a real energy, it must be assumed that also the effective axial propagation constant k¯z,nm(z) represents a real propagation constant. This leads to the conclusion that
λnm(z)=2πk¯z,nm(z)=hcck¯z,nm(z)=hcEphEnm(z)
represents the real local wave length of the photon at the position z. According to this conclusion λnm(z) increases inversely proportional to k¯z,nm(z), and therefore, describes the Gouy phase shift in agreement with wave optics. But this conclusion also shows that λnm(z) increases inversely proportional to energy difference EphEnm(z) which decreases with decreasing z. This seems to show that the deeper physical reason for Gouy phase shift consists in the fact that the energy of the photon is increasingly converted into its transverse quantum mechanical motion when the photon approaches the focus. This allows to conclude that the Gouy phase shift is an energetic effect. Therefore, it seems to be reasonable to designate the energy ck¯z,nm(z) as Gouy energy, and to introduce for this energy the term EG,nm(z). Based on this result Eq. (15) can be rewritten as
Eph=EG,nm(z)+Enm(z),
which shows that the total energy Eph is the sum of an axial part described by EG,nm(z), and the energy eigenvalues Emn(z) of the Schrödinger equation. However the particle picture not only explains the Gouy phase shift as an energetic effect, it also allows to compute the mathematical expression describing the Gouy phase shift in agreement with wave optics as already shown in [1].

The upper part of Fig. 1 shows for λ=1[µm] and w0=1.12√2/k [µm] how the energies EG,00(z) and E00(z) change with the propagating wave. The lower part shows how the spot size changes. Under the same conditions, the upper part of Fig. 2 shows how the local wave length λ00(z) changes as a function of z [µm]. The lower part shows how the Gouy phase shift changes.

 

Fig. 1 The upper part of this figure shows for λ=1[µm] and w0=1.12√2/k [µm] how the energies EG,00(z) and E00(z) change with the propagating wave. The lower part shows how the spot size changes´as a function of z [µm].

Download Full Size | PPT Slide | PDF

 

Fig. 2 Under the same conditions as used for Fig. 1. the upper part of this figure shows how the local wave length λ00(z) changes as a function of z. The lower part shows how the Gouy phase shift changes.

Download Full Size | PPT Slide | PDF

Since EGnm(z) vanishes for Enm(z) = Eph, λnm(z) goes to infinity under this condition. Therefore, Enm(z) < Eph must be considered to be a limiting condition for Enm. The condition Enm(z) < Eph transforms, after replacing Enm(z) according to Eq. (6) and Eph according to Eq. (2), into

ω(z)(n+m+1)<hcλ
which after replacing ω(z) according to Eq. (5) transforms into
zRz2+zR2(n+m+1)<2πλ.
This equation transforms for z = 0 into
n+m+1<2πλzR=2(πw0λ)2.
This delivers for the spot size w0 at the beam waist the condition
w0>λπn+m+12.
If we replace in this equation the “>” sign by the “=” sign we obtain for n = m = 0
w0=λ2π=2k.
This delivers the relation
ω(0)=czR=2πcλ=ω.
This result shows that under the condition Enm(z) = Eph the frequency ω of transverse quantum mechanical oscillation of the photon is becoming equal to the frequency ω of the initially incoming electromagnetic wave. Therefore, in this case the oscillation of the wave is in the focus totally transformed into the transverse quantum mechanical oscillation of the photon. Simultaneously, the effective axial propagation constant k¯z,00 vanishes according to Eq. (14), and the local wave length λnm(z) goes to infinity according to Eq. (16).

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

The fact that effective axial propagation constant k¯z,00(0) vanishes under the condition w0=λ/2π also can be derived from the wave optics description of the Gouy effect by differentiating the phase of the fundamental mode that is given by Φ(z)=kz+ΦG(z) with ΦG(z)=arctan(z/zR). Differentiation of Φ(z) versus z delivers for z=0

k¯z,00(0)=Φ(0)z=k+ΦG(0)z=k+1zR=0
This result shows that the phase Φ(z) of the fundamental mode stops to change at the beam waist under the condition w0=√2/k. Therefore, the question arises what is happening with a wave whose phase stops to oscillate with the propagating wave. Can it still be con­sidered to be a wave which transports energy? The presented particle picture says that the total energy of the photon is totally transformed into its transverse quantum mechanical motion under the above mentioned condition, and the wave itself does not transport energy anymore. An other aspect of this problem arises, if one considers the expectation value of the square of the z component of the wave-vector which is according to Eq. (7) given by <kz2>=k¯z,00k. This expression shows that <kz²> also vanishes for w0=√2/k according to Eq. (24), and is becoming even negative for w0<√2/k. Therefore, the question arises, how is a wave with negative <kz²> propagating. These problems deserve further consideration.

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

To answer this question we take into account that the quantity Δz used in Eq. (1) can be replaced by Δt = Δz/c which represents the infinitesimal time step which the phase front Φ1 takes to move in the phase front Φ2 as shown by Fig. 3 in [1]. This transforms Eq. (1) into

K(r,z)=Mc*limr1r2,z1z21Δt[SN(r2,z2)SN(r1,z1)]
which shows that the force K(r,z) exerted on the photon is equal to the momentum of the photon multiplied with the infinitesimal directional change of the normalized Poynting vector versus the infinitesimal time step Δt. Therefore, since an infinitesimal momentum change versus an infinitesimal time step represents a force, the right side of Eq. (25) physically represents a force, though it may not be possible to conclude that a photon, which is found at a certain position r,z, changes its momentum at this position. However, it seems to be necessary to assume that an overall continuous change of the momentum of the photons propagating with an electromagnetic wave takes place, since otherwise the photons would not follow the propagating wave, and would spread out from the wave due to their relativistic mass. Therefore, it seems to be possible to conclude that the K(r,z) represents a force which causes a directional change of the momentum of the photons, and in this way, forces the photons to follow the electromagnetic wave.

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

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

4. Summary and conclusions

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

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

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

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

References

1. K. Altmann, “Embedding the photon with its relativistic mass as a particle into the electromagnetic wave,” Opt. Express 26(2), 1375–1389 (2018). [CrossRef]   [PubMed]  

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

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

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

5. M. Eichhorn and M. Pollnau, “Spectrosopic foundations of lasers: Spontaneous emission into a resonator mode,” IEEE J. Sel. Top. Quantum Electron. 21(1), 9000216 (2015). [CrossRef]  

6. A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24(4), 156–159 (1970). [CrossRef]  

7. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11(5), 288–290 (1986). [CrossRef]   [PubMed]  

8. Y. Harada and T. Asakura, “Radiation Forces on a dielectric sphere in the Rayleigh Scattering Regime,” Opt. Commun. 124(5–6), 529–541 (1996). [CrossRef]  

References

  • View by:
  • |
  • |
  • |

  1. K. Altmann, “Embedding the photon with its relativistic mass as a particle into the electromagnetic wave,” Opt. Express 26(2), 1375–1389 (2018).
    [Crossref] [PubMed]
  2. A. E. Siegman, Lasers (University Science Books, 1986).
  3. L. G. Gouy, “Sur une propriété nouvelle des ondes lumineuses,” C. R. Acad. Sci. Paris 110, 1251 (1890).
  4. S. Feng and H. G. Winful, “Physical origin of the Gouy phase shift,” Opt. Lett. 26(8), 485–487 (2001).
    [Crossref] [PubMed]
  5. M. Eichhorn and M. Pollnau, “Spectrosopic foundations of lasers: Spontaneous emission into a resonator mode,” IEEE J. Sel. Top. Quantum Electron. 21(1), 9000216 (2015).
    [Crossref]
  6. A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24(4), 156–159 (1970).
    [Crossref]
  7. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11(5), 288–290 (1986).
    [Crossref] [PubMed]
  8. Y. Harada and T. Asakura, “Radiation Forces on a dielectric sphere in the Rayleigh Scattering Regime,” Opt. Commun. 124(5–6), 529–541 (1996).
    [Crossref]

2018 (1)

2015 (1)

M. Eichhorn and M. Pollnau, “Spectrosopic foundations of lasers: Spontaneous emission into a resonator mode,” IEEE J. Sel. Top. Quantum Electron. 21(1), 9000216 (2015).
[Crossref]

2001 (1)

1996 (1)

Y. Harada and T. Asakura, “Radiation Forces on a dielectric sphere in the Rayleigh Scattering Regime,” Opt. Commun. 124(5–6), 529–541 (1996).
[Crossref]

1986 (1)

1970 (1)

A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24(4), 156–159 (1970).
[Crossref]

1890 (1)

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

Altmann, K.

Asakura, T.

Y. Harada and T. Asakura, “Radiation Forces on a dielectric sphere in the Rayleigh Scattering Regime,” Opt. Commun. 124(5–6), 529–541 (1996).
[Crossref]

Ashkin, A.

Bjorkholm, J. E.

Chu, S.

Dziedzic, J. M.

Eichhorn, M.

M. Eichhorn and M. Pollnau, “Spectrosopic foundations of lasers: Spontaneous emission into a resonator mode,” IEEE J. Sel. Top. Quantum Electron. 21(1), 9000216 (2015).
[Crossref]

Feng, S.

Gouy, L. G.

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

Harada, Y.

Y. Harada and T. Asakura, “Radiation Forces on a dielectric sphere in the Rayleigh Scattering Regime,” Opt. Commun. 124(5–6), 529–541 (1996).
[Crossref]

Pollnau, M.

M. Eichhorn and M. Pollnau, “Spectrosopic foundations of lasers: Spontaneous emission into a resonator mode,” IEEE J. Sel. Top. Quantum Electron. 21(1), 9000216 (2015).
[Crossref]

Winful, H. G.

C. R. Acad. Sci. Paris (1)

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

IEEE J. Sel. Top. Quantum Electron. (1)

M. Eichhorn and M. Pollnau, “Spectrosopic foundations of lasers: Spontaneous emission into a resonator mode,” IEEE J. Sel. Top. Quantum Electron. 21(1), 9000216 (2015).
[Crossref]

Opt. Commun. (1)

Y. Harada and T. Asakura, “Radiation Forces on a dielectric sphere in the Rayleigh Scattering Regime,” Opt. Commun. 124(5–6), 529–541 (1996).
[Crossref]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. Lett. (1)

A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24(4), 156–159 (1970).
[Crossref]

Other (1)

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

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (2)

Fig. 1
Fig. 1 The upper part of this figure shows for λ=1[µm] and w0=1.12√2/k [µm] how the energies EG,00(z) and E00(z) change with the propagating wave. The lower part shows how the spot size changes´as a function of z [µm].
Fig. 2
Fig. 2 Under the same conditions as used for Fig. 1. the upper part of this figure shows how the local wave length λ00(z) changes as a function of z. The lower part shows how the Gouy phase shift changes.

Equations (25)

Equations on this page are rendered with MathJax. Learn more.

K (r,z)= E ph * lim r 1 r 2 , z 1 z 2 1 Δz [ S N ( r 2 , z 2 ) S N ( r 1 , z 1 ) ]
E ph =M c 2 = hc λ
[ 2 2M Δ +E(z)V(r,z) ]χ(r,z)=0,
[ 2M Δ + E nm 1 2 M ω 2 (z)( x 2 + y 2 ) ] χ nm (x,y,z)=0
ω (z)= c z R z 2 + z R 2 .
E nm (z)= ω (z)(n+m+1).
k ¯ z,nm (z)= < k z,nm 2 > k =k < k x,nm 2 >+< k y,nm 2 > k
< χ nm | p ^ 2 (z)| χ nm >=M ω (z)(n+m+1)
< χ nm | p ^ 2 (z)| χ nm >=M E nm (z)
< k x,nm 2 >+< k y,nm 2 >= < χ nm | p ^ 2 (z)| χ nm > 2 = M E nm (z) 2
k ¯ z,nm (z)=k( 1 M E nm (z) k 2 2 )
k ¯ z,nm (z)=k( 1 E ph E nm (z) c 2 k 2 2 )
E ph =ck
k ¯ z,nm (z)=k( 1 E nm (z) E ph )= k E ph [ E ph E nm (z) ]= 1 c [ E ph E nm (z) ]
c k ¯ z,nm (z)= E ph E nm (z)
λ nm (z)= 2π k ¯ z,nm (z) = hc c k ¯ z,nm (z) = hc E ph E nm (z)
E ph = E G,nm (z)+ E nm (z),
ω (z)(n+m+1)< hc λ
z R z 2 + z R 2 (n+m+1)< 2π λ .
n+m+1< 2π λ z R =2 ( π w 0 λ ) 2 .
w 0 > λ π n+m+1 2 .
w 0 = λ 2 π = 2 k .
ω (0)= c z R = 2πc λ =ω.
k ¯ z,00 (0)= Φ(0) z =k+ Φ G (0) z =k+ 1 z R =0
K (r,z)=Mc* lim r 1 r 2 , z 1 z 2 1 Δt [ S N ( r 2 , z 2 ) S N ( r 1 , z 1 ) ]

Metrics