Abstract

The particle picture presented by the author in the paper “A particle picture of the optical resonator” [K. Altmann, ASSL 2014 Conference Paper ATu2A.29], which shows that the probability density of a photon propagating with a Gaussian wave can be computed by the use of a Schrödinger equation, is generalized to the case of a wave with arbitrary shape of the phase front. Based on a consideration of the changing propagation direction of the relativistic mass density propagating with the electromagnetic wave, 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 that in combination with a Schrödinger equation allows to describe the transverse quantum mechanical motion of the photon by the use of matter wave theory, even though the photon has no rest mass. The obtained results are verified for the plane, the spherical, and the Gaussian wave. Additional verification could be provided also by the fact that the mathematical equation describing the Guoy phase shift could be derived from this particle picture in full agreement with wave optics. One more verification could be obtained by the fact that within the range of the validity of paraxial wave optics, Snell's law could also be derived from this particle picture. Numerical validation of the obtained results for the case of the general wave is under development.

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

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References

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  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]

2015 (1)

2001 (1)

1996 (1)

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

1976 (1)

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

1926 (1)

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

1890 (1)

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

Altmann, K.

Feng, S.

Gouy, L. G.

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

Harihan, P.

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

Robinson, P. A.

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

Schrödinger, S.

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

Thomas, G.

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

Tsai, T.

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

Winful, H. G.

Am. J. Phys. (1)

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

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

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

J. Mod. Opt. (1)

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

Naturwissenschaften (1)

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

Opt. Express (1)

Opt. Lett. (1)

Other (9)

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

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

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

K. Altmann, to be submitted for publication.

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

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

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

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

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

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Figures (5)

Fig. 1
Fig. 1 Resonant Gaussian mode between two spherical mirrors. The green lines visualize the propagating energy density.
Fig. 2
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.
Fig. 3
Fig. 3 Visualization of two phase fronts Φ1 and Φ2 intersecting the z axis at z1 and z2.
Fig. 4
Fig. 4 Visualization of relation Δr/ΔΦ = r/R(z) as used in Eq. (23)
Fig. 5
Fig. 5 Refraction at the interface between vacuum and medium.

Equations (53)

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K (r,z) lim r 1 r 2 ,Δt0 1 Δt [ S N ( r 2 , z 2 ) S N ( r 1 , z 1 ) ].
K (r,z)= M ˜ c* lim r 1 r 2 ,Δt0 1 Δt [ S N ( r 2 , z 2 ) S N ( r 1 , z 1 ) ].
Δt= Φ( r 2 ,z ) 2 Φ( r 1 , z 1 ) c ,
Δt= z 2 z 1 c = Δz c .
K (r,z)= M ˜ c 2 lim r 1 r 2 ,Δz0 1 Δz [ S N ( r 2 , z 2 ) S N ( r 1 , z 1 ) ],
K (r,z)= E ph * lim r 1 r 2 ,Δz0 1 Δz [ S N ( r 2 , z 2 ) S N ( r 1 , z 1 ) ].
E ph =M c 2 = hc λ .
M= h cλ .
K (r,z)= E ph * lim r 1 r 2 ,Δz0 1 Δz [ dΦ( r 2 , z 2 ) dr dΦ( r 1 , z 1 ) dr ].
V(r,z)= E ph * 0 r lim r 1 r 2 ,Δz0 1 Δz [ dΦ( r 1 , z 1 ) dr dΦ( r 2 , z 2 ) dr ]dr .
[ 2 2M Δ +E(z)V(r,z) ]χ(r,z)=0.
V(x,y,z)= E ph * x || x y || y lim x 1 , y 1 , z 1 x 2 , y 2 , z 2 1 Δ || [ S N ( x 2 , y 2 , z 2 ) S N ( x 1 , y 1 , z 1 ) ]dxdy ,
Δ || = ( x 2|| x 1|| ) 2 + ( y 2|| y 1|| ) 2
V(x,y,z)= E ph * 0 x 0 y lim x 1 , y 1 , z 1 x 2 , y 2 , z 2 1 z 2 z 1 [ S N ( x 2 , y 2 , z 2 ) S N ( x 1 , y 1 , z 1 ) ]dxdy
Φ ˜ (r,z)= r 2 2R(z) z.
Φ(r,z)=z r 2 2R(z) .
R(z)=z+ z R 2 z .
z R = π w 0 2 λ .
K (r,z)= E ph * lim r 1 r 2 ,Δz0 1 Δz [ r 1 R( z 1 ) r 2 R( z 2 ) ]
K (r,z)= E ph r R 2 (z) ( ΔR Δz R(z) r Δr Δz )
ΔR Δz = R( z 2 )R( z 1 ) Δz = dR dz =1 z R 2 z 2 .
ΔΦ(r,z)=Φ( r 2 , z 2 )Φ( r 1 , z 1 )= Δr R(z) +ΔzΔz
lim Δz0 Δr ΔΦ = r R(z)
lim Δz0 Δr Δz = r R(z) .
K(r,z)= E ph r R 2 (z) ( 1 z R 2 z 2 R(z) r r R(z) )= E ph r R 2 (z) z R 2 z 2
K(r,z)= E ph r ( z R z 2 + z R 2 ) 2 .
V(r,z)= 1 2 M ω 2 (z) r 2 .
ω (z)= c z R z 2 + z R 2 .
[ 2M Δ +E 1 2 M ω 2 (z)( x 2 + y 2 ) ] χ nm (x,y,z)=0
χ nm (x,y,z)= 2 π 1 w p (z) 2 n+m n! m! H n ( 2 x w p (z) ) H m ( 2 y w p (z) )exp( x 2 + y 2 w p 2 (z) ).
w p 2 (z)= 2 M ω (z) .
w p 2 (z)= 2 z R Mc [ 1+ ( z z R ) 2 ].
w p 2 (0)= 2 Mc z R = λ π z R .
w p (z)= w 0 1+ ( z z R ) 2
K(r,z)= E ph r ( z R z 2 + z R 2 ) 2 = E ph w 0 4 z R 2 r w 4 (z)
K(w(z),z)= E ph w 0 4 z R 2 1 w 3 (z)
< χ nm | p ^ 2 (z)| χ nm >=M ω (z)(n+m+1).
M ω = h cλ c z R z 2 + z R 2 = h λ z R w 0 2 w 2 (z) = hλ λπ w 0 2 w 0 2 w 2 (z) = 2 w 2 (z) .
< χ nm | p ^ 2 (z)| χ nm >= 2 2 w 2 (z) (n+m+1).
< k x 2 >+< k y 2 >= < p ^ 2 (z)> 2 = 2(n+m+1) w 2 (z) .
Φ G = 1 k z { < k x 2 >+< k y 2 > } dz .
Φ G = 2(n+m+1) k z 1 w 2 (z) dz.
Φ G =(n+m+1)arctan(z/ z R )
Φ G =(n+1/2 )arctan(z/ z R ),
V(x,y,z)=V(x,z)+V(y,z)= 1 2 M[ ω x 2 (z) x 2 + ω x 2 (z) y 2 ]
< k x 2 >+< k y 2 >= 1 2 < χ nm | p ^ 2 (z)| χ nm >= 2n+1 w x 2 (z) + 2m+1 w y 2 (z)
Φ G (z)= 1 k 2 z <χ(x,y,z)| p ^ 2 (z)|χ(x,y,z)> dz ,
r=Acos( ω (z)t)
V(r,z)= E ph * 0 r lim r 1 r 2 ,Δz0 1 Δz [ S N ( r 2 , z 2 ) S N ( r 1 , z 1 ) ]dr .
V(r,z)= E ph * 0 r lim r 1 r 2 ,Δz0 1 Δz [ S N ( r 2 , z 2 ) S N ( r 1 , z 1 ) ]dr .
V(r,z)= E ph * 0 r lim r 1 r 2 ,Δz0 n m Δz [ S N ( r 2 , z 2 ) S N ( r 1 , z 1 ) ]dr .
S N S N / n m = sin( θ 1 ) sin( θ 2 ) = n m .
V(r,z)= E ph * 0 r lim r 1 r 2 ,Δz0 1 Δz [ r 1 R( z 1 ) r 2 R( z 2 ) ]dr .

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