Abstract

It is shown that the condition provided by paraxial wave optics 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 for nearly plane waves, paraxial wave optics becomes exact within wave optics, this contradiction also concerns wave optics. A solution for this problem is proposed within a particle picture as presented recently by the author. It is based on a consideration of the change of momentum of a photon bouncing between two equiphase surfaces with vanishing distance. This leads to a transverse force exerted on the photon. Assigning a relativistic mass to the photon leads to a Schrödinger equation describing a transverse motion of the photon. In this way the transverse modes of an optical resonator can be understood as the quantum mechanical eigenfunctions of a single photon.

© 2015 Optical Society of America

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References

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  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).

2013 (1)

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]

1993 (1)

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

1982 (1)

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

1964 (1)

1890 (1)

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

Allen, L.

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

Altmann, K.

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

Goldsborough, J. P.

Gouy, L. G.

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

Herzig, H. P.

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]

Kim, M. S.

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]

Naqavi, A.

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]

Nienhuis, G.

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

Rockstuhl, C.

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]

Scharf, T.

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]

Völkel, R.

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]

Weible, K. J.

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]

Appl. Opt. (1)

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. Opt. (1)

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]

Phys. Lett. (1)

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

Phys. Rev. A (1)

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

Other (7)

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

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

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]

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

K. Altmann, to be submitted for publication.

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

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

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Figures (7)

Fig. 1
Fig. 1 Planar-spherical resonator subdivided by a totally reflecting equiphase surface S.
Fig. 2
Fig. 2 Dependence of the axial phase shift ψ(ζ,β) on the normalized parameters ζ = z/λq and β = w0q.
Fig. 3
Fig. 3 Transverse components of the momentum changes of a photon reflected between two equiphase surfaces.
Fig. 4
Fig. 4 Dependence of the local wavelength λl on the normalized distance s = z/zR from the beam waist according to the particle picture and to the PWO approach
Fig. 5
Fig. 5 Dependence of the local wavelength λl on the normalized distance ζ = z/λq from the beam waist according to the particle picture and to the PWO approach.
Fig. 6
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.
Fig. 7
Fig. 7 Schematic illustration of the quantum mechanics and the ray tracing approach.

Equations (75)

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ω qnm = 2πcq 2L +(1+n+m) c L arccos ( 1 L R 1 )( 1 L R 2 )
ψ(z)=arctan z z R
[ 2M Δ+( ω ω q )V(x,y) ]χ(x,y)=0.
ω q =πqc/L.
λ q = 2L q = 2πc ω q .
M= ω q / c 2 = πq cL = 2π c λ q .
w(z)= w 0 1+ ( z z R ) 2 ,
R(z)=z+ z R 2 z ,
ω qnm = 2πc λ q +(n+m+1) c L arccos 1 L R E = 2πc λ q +(n+m+1) c L arcsin L R E ,
ω qnm,SC = 2πc λ q +(n+m+1) c λ q arccos ( 1+ λ q R S )( 1 λ q R E ) .
ω qnm,SC 2πc λ q +(n+m+1) c λ q arcsin λ q R E 2πc λ q +(n+m+1) c R E .
z R = π w 0 2 λ q ,
ψ(z)=arctan z λ q π w 0 2 .
z=ς λ q ,
w 0 =β λ q ,
ψ(ς,β)=arctan ς π β 2
R i ( z i )= z i + z R 2 z i ,i| 1,2 |.
| Δ P |=2Mc.
Δ P ( r 1 , z 1 )= 2Mc R 1 r 1 ,
Δ P ( r 2 , z 2 )= 2Mc R 2 r 2 .
Δ P ( r 1 , z 1 )+Δ P ( r 2 , z 2 )=2Mc( r 1 R 1 r 2 R 2 )= 2Mc R 1 R 2 ( r 1 R 2 r 2 R 1 ) = 2Mc R 1 R 2 [ r 2 ( R 2 R 1 ) R 2 ( r 2 r 1 ) ].
lim R 1 R 2 Δ P ( r 1 , z 1 )+Δ P ( r 2 , z 2 )= 2Mc R 2 2 ( r 2 ΔR R 2 Δr ).
t round = 2 c ( z 2 z 1 )= 2 c Δz.
K(r,z)= M c 2 R 2 (z) ( r ΔR Δz R Δr Δz ).
dR dz (z)=1 z R 2 z 2 .
dr dz (z)= r R(z)
lim Δz0 K(r,z)= M c 2 r R 2 (z) ( 1 z R 2 z 2 R(z) r r R(z) )= M c 2 r R 2 (z) z R 2 z 2
K(r,z)=M ( c z R z 2 + z R 2 ) 2 r.
V(r,z)= 1 2 M ω t 2 (z) r 2
ω t (z)= c z R z 2 + z R 2 .
[ 2M Δ t +E 1 2 M ω t 2 (z)( x 2 + y 2 ) ]χ(x,y,z)=0
E(z)=[ ω(z) ω q ].
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 t (z) .
w p 2 (z)= 2 z R Mc [ 1+ ( z z R ) 2 ].
w p 2 (0)= 2 Mc z R = λ q π z R .
w p (z)= w p (0) 1+ ( z z R ) 2
E nm (z)= ω t (z)( n+m+1 ).
ω qnm (z)= 2πc λ q +(n+m+1) c z R z 2 + z R 2 .
ω qnm (z)= 2πc λ q +(n+m+1) cz z 2 + z R 2 .
ω qnm (z) 2πc λ q +(n+m+1) c LR E .
z R 2 = R E L( 1 L R E ),
lim L/R0 z R = L R E .
λ l (z)= [ 1 λ q (n+m+1) z R 2π[ z 2 + z R 2 ] ] 1 .
λ l (z) λ q λ q = [ 1 (n+m+1) λ q 2π[ (z/ z R ) 2 +1 ] z R ] 1 1.
s= z z R ,
λ l (s) λ q λ q = [ 1 (n+m+1) 2 π 2 ( s 2 +1) β 2 ] 1 1.
β 2 λ l (s) λ q λ q n+m+1 2 π 2 ( s 2 +1) ,
β 2 λ l (0) λ q λ q n+m+1 2 π 2 .
exp[ i( k q ψ(z) z )z ]
k lPWO (z)= k q ψ(z) z
λ lPWO (z)= 2π k lPWO (z) = 2π k q ψ(z) z = 2π 2π λ q ψ(z) z = λ q 1 λ q ψ(z) 2πz
λ lPWO λ q λ q = 1 1 λ q ψ(z) 2πz 1
λ lPWO λ q λ q = 1 1 arctan(s) 2 π 2 s β 2 1
λ lPWO λ q λ q = 2 π 2 β 2 2 π 2 β 2 arctan(s) s 1.
lim s0 λ lPWO λ q λ q = 2 π 2 β 2 2 π 2 β 2 1 1.
lim s0 λ lPWO λ q λ q = 2 π 2 β 2 2 π 2 β 2 (n+m+1) 1
λ l (z=0) λ q = [ 1 1 2 π 2 β 2 ] 1 = 2 π 2 β 2 2 π 2 β 2 1 .
β= 1 2 π .
ω nm (z)=(n+m+1) ω t (z)=(n+m+1) c z R z 2 + z R 2 .
cos[ Δ ω n 1 m 1 , n 2 m 2 (z)t ]=cos[ ( n + 2 m 2 n 1 m 1 ) c z R z 2 + z R 2 t ]
Δ ω 00,10 (s)= c z R 1 s 2 +1 = c π β 2 λ q 1 s 2 +1 .
Δ ω 00,10 (s=0)= c π β 2 λ q .
H= p x 2 + p y 2 2m + 1 2 m ω 2 ( x 2 + y 2 ).
x n = x 0 cosnθ+ s 0 sinnθ.
x n =Ccos(nθ+γ)
C= x 0 2 + s 0 2 ,
tanγ= x 0 s 0 .
x n =Ccos( θ t round n t round +γ ).
θ=arccos( S= A+D 2 )=arcsin 1 S 2 .
lim Δz0 θ= 1 S 2 .
[ A B C D ]=[ 1 Δz 0 1 ][ 1 0 2 /R 1 1 ][ 1 Δz 0 1 ][ 1 0 2 / ( R 1 +R'Δz) 1 ]
lim t round 0 θ t round = lim Δz0 cθ 2Δz = lim Δz0 c 1 S 2 4Δ z 2 =c 1R' R 2 = c z R zR = c z R z 2 + z R 2 .
lim n t round t x=Acos( ω ray (z)t+γ ).
ω ray (z)= c z R z 2 + z R 2 .

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