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

Full Article  |  PDF Article
OSA Recommended Articles
Contradiction within wave optics and its solution within a particle picture

Konrad Altmann
Opt. Express 23(3) 3731-3750 (2015)

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