Abstract

A theory of open laser resonators is formulated within the framework of the electromagnetic coherence theory. It is shown that if only one Fox–Li mode contributes to the field at a given frequency, then the field at that frequency is necessarily completely coherent in view of the space–frequency counterpart of the recently introduced degree of coherence of electromagnetic fields [Opt. Express 11, 1137 (2003)]. It is also shown that the relation between the number of Fox–Li modes and the new degree of coherence is analogous to the relation established in the scalar theory of laser resonator modes. Difficulties that arise with the formerly introduced visibility-based definition of the electromagnetic degree of coherence are briefly discussed.

© 2005 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, Cambridge, UK, 1999), historical introduction.
  2. E. Wolf, G. S. Agarwal, “Coherence theory of laser resonator modes,” J. Opt. Soc. Am. A 1, 541–546 (1984).
    [CrossRef]
  3. D. Pohl, “Operation of a ruby laser in the purely transverse electric mode TE01,” Appl. Phys. Lett. 20, 266–267 (1972).
    [CrossRef]
  4. R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77, 3322–3324 (2000).
    [CrossRef]
  5. A. V. Nesterov, V. G. Niziev, V. P. Yakunin, “Generation of high-power radially polarized beam,” J. Phys. D Appl. Phys. 32, 2871–2875 (1999).
    [CrossRef]
  6. A. V. Nesterov, V. G. Niziev, “Laser beams with axially symmetric polarization,” J. Phys. D Appl. Phys. 33, 1817–1822 (2000).
    [CrossRef]
  7. I. Moshe, S. Jackel, A. Meir, “Production of radially or azimuthally polarized beams in solid-state lasers and the elimination of thermally induced birefringence effects,” Opt. Lett. 28, 807–809 (2003).
    [CrossRef] [PubMed]
  8. J. R. Leger, D. Chen, G. Mowry, “Design and performance of diffractive optics for custom laser resonators,” Appl. Opt. 34, 2498–2509 (1995).
    [CrossRef] [PubMed]
  9. J. Tervo, J. Turunen, “Paraxial-domain diffractive elements with 100% efficiency based on polarization gratings,” Opt. Lett. 25, 785–786 (2000).
    [CrossRef]
  10. J. Tervo, V. Kettunen, M. Honkanen, J. Turunen, “Design of space-variant diffractive polarization elements,” J. Opt. Soc. Am. A 20, 282–289 (2003).
    [CrossRef]
  11. F. Zernike, “The concept of degree of coherence and its application to optical problems,” Physica 5, 785–795 (1938).
    [CrossRef]
  12. J. Tervo, T. Setälä, A. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11, 1137–1143 (2003); www.opticsexpress.org .
    [CrossRef] [PubMed]
  13. T. Setälä, J. Tervo, A. T. Friberg, “Complete electromagnetic coherence in the space-frequency domain,” Opt. Lett. 29, 328–330 (2004).
    [CrossRef] [PubMed]
  14. T. Setälä, J. Tervo, A. T. Friberg, “Theorems on complete electromagnetic coherence in the space–time domain,” Opt. Commun. 238, 229–236 (2004).
  15. R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130, 2529–2539 (1963).
    [CrossRef]
  16. J. Tervo, T. Setälä, A. T. Friberg, “Theory of partially coherent electromagnetic fields in the space-frequency domain,” J. Opt. Soc. Am. A 21, 2205–2215 (2004).
    [CrossRef]
  17. P. Roman, E. Wolf, “Correlation theory of stationary electromagnetic fields. Part I—the basic field equations,” Nuovo Cimento 17, 462–476 (1960).
    [CrossRef]
  18. P. Roman, E. Wolf, “Correlation theory of stationary electromagnetic fields. Part II—conservation laws,” Nuovo Cimento 17, 477–490 (1960).
    [CrossRef]
  19. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).
  20. E. Wolf, “New theory of partial coherence in the space–frequency domain. Part I: spectra and cross-spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343–351 (1982).
    [CrossRef]
  21. F. Gori, M. Santarsiero, R. Simon, G. Piquero, R. Borghi, G. Guattari, “Coherent-mode decomposition of partially polarized, partially coherent sources,” J. Opt. Soc. Am. A 20, 78–84 (2003).
    [CrossRef]
  22. H. Kogelnik, T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1567 (1966).
    [CrossRef] [PubMed]
  23. A. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986).
  24. A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
    [CrossRef]
  25. P. M. Morse, H. Feshbach, Methods of Mathematical Physics (McGraw-Hill, New York, 1953).
  26. F. Smithies, Integral Equations (Cambridge U. Press, Cambridge, UK, 1970).
  27. The exact conditions for the existence and validity of the biothogonal expansion in Eq. (22) do not appear to be known. However, it is readily shown that if the Fox–Li modes are complete in the sense that the field Fj+1(ρ, ω)on the left-hand side of Eq. (15) is expressible as their linear combination, then the resonator kernel admits the biorthogonal series representation as given in Eq. (22). A similar assumption (though stated slightly differently) was also invoked in Footnote 15 of Ref. 2.
  28. The steps are essentially identical to those made in Ref. 2 but with the scalar functions replaced by appropriate vector- or tensor-valued functions. The vectorial formulation is given in detail in Ref. 29.
  29. J. Tervo, T. Saastamoinen, J. Turunen, T. Setälä, A. T. Friberg, “Degree of coherence and electromagnetic resonators,” in Photon Management, F. Wyrowski, ed., Proc. SPIE5456, 28–35 (2004).
    [CrossRef]
  30. The same result holds also within the scalar theory, although it is not explicitly mentioned in Refs. 2 and 19.

2004

2003

2000

A. V. Nesterov, V. G. Niziev, “Laser beams with axially symmetric polarization,” J. Phys. D Appl. Phys. 33, 1817–1822 (2000).
[CrossRef]

J. Tervo, J. Turunen, “Paraxial-domain diffractive elements with 100% efficiency based on polarization gratings,” Opt. Lett. 25, 785–786 (2000).
[CrossRef]

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77, 3322–3324 (2000).
[CrossRef]

1999

A. V. Nesterov, V. G. Niziev, V. P. Yakunin, “Generation of high-power radially polarized beam,” J. Phys. D Appl. Phys. 32, 2871–2875 (1999).
[CrossRef]

1995

1984

1982

1972

D. Pohl, “Operation of a ruby laser in the purely transverse electric mode TE01,” Appl. Phys. Lett. 20, 266–267 (1972).
[CrossRef]

1966

1963

R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130, 2529–2539 (1963).
[CrossRef]

1961

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
[CrossRef]

1960

P. Roman, E. Wolf, “Correlation theory of stationary electromagnetic fields. Part I—the basic field equations,” Nuovo Cimento 17, 462–476 (1960).
[CrossRef]

P. Roman, E. Wolf, “Correlation theory of stationary electromagnetic fields. Part II—conservation laws,” Nuovo Cimento 17, 477–490 (1960).
[CrossRef]

1938

F. Zernike, “The concept of degree of coherence and its application to optical problems,” Physica 5, 785–795 (1938).
[CrossRef]

Agarwal, G. S.

Blit, S.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77, 3322–3324 (2000).
[CrossRef]

Bomzon, Z.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77, 3322–3324 (2000).
[CrossRef]

Borghi, R.

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, Cambridge, UK, 1999), historical introduction.

Chen, D.

Davidson, N.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77, 3322–3324 (2000).
[CrossRef]

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Mathematical Physics (McGraw-Hill, New York, 1953).

Fox, A. G.

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
[CrossRef]

Friberg, A. T.

T. Setälä, J. Tervo, A. T. Friberg, “Theorems on complete electromagnetic coherence in the space–time domain,” Opt. Commun. 238, 229–236 (2004).

J. Tervo, T. Setälä, A. T. Friberg, “Theory of partially coherent electromagnetic fields in the space-frequency domain,” J. Opt. Soc. Am. A 21, 2205–2215 (2004).
[CrossRef]

T. Setälä, J. Tervo, A. T. Friberg, “Complete electromagnetic coherence in the space-frequency domain,” Opt. Lett. 29, 328–330 (2004).
[CrossRef] [PubMed]

J. Tervo, T. Setälä, A. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11, 1137–1143 (2003); www.opticsexpress.org .
[CrossRef] [PubMed]

J. Tervo, T. Saastamoinen, J. Turunen, T. Setälä, A. T. Friberg, “Degree of coherence and electromagnetic resonators,” in Photon Management, F. Wyrowski, ed., Proc. SPIE5456, 28–35 (2004).
[CrossRef]

Friesem, A. A.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77, 3322–3324 (2000).
[CrossRef]

Glauber, R. J.

R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130, 2529–2539 (1963).
[CrossRef]

Gori, F.

Guattari, G.

Hasman, E.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77, 3322–3324 (2000).
[CrossRef]

Honkanen, M.

Jackel, S.

Kettunen, V.

Kogelnik, H.

Leger, J. R.

Li, T.

H. Kogelnik, T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1567 (1966).
[CrossRef] [PubMed]

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
[CrossRef]

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).

Meir, A.

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Mathematical Physics (McGraw-Hill, New York, 1953).

Moshe, I.

Mowry, G.

Nesterov, A. V.

A. V. Nesterov, V. G. Niziev, “Laser beams with axially symmetric polarization,” J. Phys. D Appl. Phys. 33, 1817–1822 (2000).
[CrossRef]

A. V. Nesterov, V. G. Niziev, V. P. Yakunin, “Generation of high-power radially polarized beam,” J. Phys. D Appl. Phys. 32, 2871–2875 (1999).
[CrossRef]

Niziev, V. G.

A. V. Nesterov, V. G. Niziev, “Laser beams with axially symmetric polarization,” J. Phys. D Appl. Phys. 33, 1817–1822 (2000).
[CrossRef]

A. V. Nesterov, V. G. Niziev, V. P. Yakunin, “Generation of high-power radially polarized beam,” J. Phys. D Appl. Phys. 32, 2871–2875 (1999).
[CrossRef]

Oron, R.

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77, 3322–3324 (2000).
[CrossRef]

Piquero, G.

Pohl, D.

D. Pohl, “Operation of a ruby laser in the purely transverse electric mode TE01,” Appl. Phys. Lett. 20, 266–267 (1972).
[CrossRef]

Roman, P.

P. Roman, E. Wolf, “Correlation theory of stationary electromagnetic fields. Part II—conservation laws,” Nuovo Cimento 17, 477–490 (1960).
[CrossRef]

P. Roman, E. Wolf, “Correlation theory of stationary electromagnetic fields. Part I—the basic field equations,” Nuovo Cimento 17, 462–476 (1960).
[CrossRef]

Saastamoinen, T.

J. Tervo, T. Saastamoinen, J. Turunen, T. Setälä, A. T. Friberg, “Degree of coherence and electromagnetic resonators,” in Photon Management, F. Wyrowski, ed., Proc. SPIE5456, 28–35 (2004).
[CrossRef]

Santarsiero, M.

Setälä, T.

J. Tervo, T. Setälä, A. T. Friberg, “Theory of partially coherent electromagnetic fields in the space-frequency domain,” J. Opt. Soc. Am. A 21, 2205–2215 (2004).
[CrossRef]

T. Setälä, J. Tervo, A. T. Friberg, “Complete electromagnetic coherence in the space-frequency domain,” Opt. Lett. 29, 328–330 (2004).
[CrossRef] [PubMed]

T. Setälä, J. Tervo, A. T. Friberg, “Theorems on complete electromagnetic coherence in the space–time domain,” Opt. Commun. 238, 229–236 (2004).

J. Tervo, T. Setälä, A. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11, 1137–1143 (2003); www.opticsexpress.org .
[CrossRef] [PubMed]

J. Tervo, T. Saastamoinen, J. Turunen, T. Setälä, A. T. Friberg, “Degree of coherence and electromagnetic resonators,” in Photon Management, F. Wyrowski, ed., Proc. SPIE5456, 28–35 (2004).
[CrossRef]

Siegman, A.

A. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986).

Simon, R.

Smithies, F.

F. Smithies, Integral Equations (Cambridge U. Press, Cambridge, UK, 1970).

Tervo, J.

Turunen, J.

Wolf, E.

E. Wolf, G. S. Agarwal, “Coherence theory of laser resonator modes,” J. Opt. Soc. Am. A 1, 541–546 (1984).
[CrossRef]

E. Wolf, “New theory of partial coherence in the space–frequency domain. Part I: spectra and cross-spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343–351 (1982).
[CrossRef]

P. Roman, E. Wolf, “Correlation theory of stationary electromagnetic fields. Part I—the basic field equations,” Nuovo Cimento 17, 462–476 (1960).
[CrossRef]

P. Roman, E. Wolf, “Correlation theory of stationary electromagnetic fields. Part II—conservation laws,” Nuovo Cimento 17, 477–490 (1960).
[CrossRef]

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).

M. Born, E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, Cambridge, UK, 1999), historical introduction.

Yakunin, V. P.

A. V. Nesterov, V. G. Niziev, V. P. Yakunin, “Generation of high-power radially polarized beam,” J. Phys. D Appl. Phys. 32, 2871–2875 (1999).
[CrossRef]

Zernike, F.

F. Zernike, “The concept of degree of coherence and its application to optical problems,” Physica 5, 785–795 (1938).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

D. Pohl, “Operation of a ruby laser in the purely transverse electric mode TE01,” Appl. Phys. Lett. 20, 266–267 (1972).
[CrossRef]

R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. 77, 3322–3324 (2000).
[CrossRef]

Bell Syst. Tech. J.

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys. D Appl. Phys.

A. V. Nesterov, V. G. Niziev, V. P. Yakunin, “Generation of high-power radially polarized beam,” J. Phys. D Appl. Phys. 32, 2871–2875 (1999).
[CrossRef]

A. V. Nesterov, V. G. Niziev, “Laser beams with axially symmetric polarization,” J. Phys. D Appl. Phys. 33, 1817–1822 (2000).
[CrossRef]

Nuovo Cimento

P. Roman, E. Wolf, “Correlation theory of stationary electromagnetic fields. Part I—the basic field equations,” Nuovo Cimento 17, 462–476 (1960).
[CrossRef]

P. Roman, E. Wolf, “Correlation theory of stationary electromagnetic fields. Part II—conservation laws,” Nuovo Cimento 17, 477–490 (1960).
[CrossRef]

Opt. Commun.

T. Setälä, J. Tervo, A. T. Friberg, “Theorems on complete electromagnetic coherence in the space–time domain,” Opt. Commun. 238, 229–236 (2004).

Opt. Express

Opt. Lett.

Phys. Rev.

R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130, 2529–2539 (1963).
[CrossRef]

Physica

F. Zernike, “The concept of degree of coherence and its application to optical problems,” Physica 5, 785–795 (1938).
[CrossRef]

Other

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).

M. Born, E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, Cambridge, UK, 1999), historical introduction.

P. M. Morse, H. Feshbach, Methods of Mathematical Physics (McGraw-Hill, New York, 1953).

F. Smithies, Integral Equations (Cambridge U. Press, Cambridge, UK, 1970).

The exact conditions for the existence and validity of the biothogonal expansion in Eq. (22) do not appear to be known. However, it is readily shown that if the Fox–Li modes are complete in the sense that the field Fj+1(ρ, ω)on the left-hand side of Eq. (15) is expressible as their linear combination, then the resonator kernel admits the biorthogonal series representation as given in Eq. (22). A similar assumption (though stated slightly differently) was also invoked in Footnote 15 of Ref. 2.

The steps are essentially identical to those made in Ref. 2 but with the scalar functions replaced by appropriate vector- or tensor-valued functions. The vectorial formulation is given in detail in Ref. 29.

J. Tervo, T. Saastamoinen, J. Turunen, T. Setälä, A. T. Friberg, “Degree of coherence and electromagnetic resonators,” in Photon Management, F. Wyrowski, ed., Proc. SPIE5456, 28–35 (2004).
[CrossRef]

The same result holds also within the scalar theory, although it is not explicitly mentioned in Refs. 2 and 19.

A. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986).

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Equations (29)

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Γ(r1, r2, τ)=E*(r1, t)E(r2, t+τ),
W(r1, r2, ω)=12π -Γ(r1, r2, τ)exp(iωτ)dτ,
W(r1, r2, ω)=W(r2, r1, ω),
DDf*(r1)·W(r1, r2, ω)·f(r2)d3r1d3r20,
DDW(r1, r2, ω)F2d3r1d3r2<,
W(r1, r2, ω)F=tr[W(r1, r2, ω)·W(r1, r2, ω)]1/2=i,j|Wij(r1, r2, ω)|21/2,
W(r1, r2, ω)=nλn(ω)ϕn*(r1, ω)ϕn(r2, ω),
Dϕn(r1, ω)·W(r1, r2, ω)d3r1=λn(ω)ϕn(r2, ω).
Dϕn*(r, ω)·ϕm(r, ω)d3r=δnm,
Wn(r1, r2, ω)=λn(ω)ϕn*(r1, ω)ϕn(r2, ω)
μ(r1, r2, ω)=W(r1, r2, ω)F[tr W(r1, r1, ω)]1/2[tr W(r2, r2, ω)]1/2.
W(r1, r2, ω)=F*(r1, ω)F(r2, ω),
2F(r, ω)+(ω/c)2F(r, ω)=0,
·F(r, ω)=0,
Fj+1(ρ, ω)=AL(ρ, ρ, ω)·Fj(ρ, ω)d2ρ,
Wj+1(ρ1, ρ2, ω)=AAL*(ρ1, ρ1, ω)·Wj(ρ1, ρ2, ω)·LT(ρ2, ρ2, ω)d2ρ1d2ρ2,
Wj+1(ρ1, ρ2, ω)=σ(ω)Wj(ρ1, ρ2, ω),
AAL*(ρ1, ρ1, ω)·W(ρ1, ρ2, ω)·LT(ρ2, ρ2, ω)d2ρ1d2ρ2 =σ(ω)W(ρ1, ρ2, ω).
AL(ρ1, ρ2, ω)·ψn(ρ2, ω)d2ρ2=αn(ω)ψn(ρ1, ω)
AL(ρ2, ρ1, ω)·χn(ρ2, ω)d2ρ2=βn(ω)χn(ρ1, ω).
Aψn*(ρ, ω)·χm(ρ, ω)d2ρ=δnm,
L(ρ1, ρ2, ω)=nαn(ω)ψn(ρ1, ω)χn*(ρ2, ω).
βn(ω)=αn*(ω)
W(ρ1, ρ2, ω)=mncmn(ω)ψm*(ρ1, ω)ψn(ρ2, ω),
W(ρ1, ρ2, ω)=cmm(ω)ψm*(ρ1, ω)ψm(ρ2, ω).
cmn(ω)=pΛp(ω)ξp,m*(ω)ξp,n(ω),
W(ρ1, ρ2, ω)=pΛp(ω)ζp*(ρ1, ω)ζp(ρ2, ω),
ζp(ρ, ω)=mξp,m(ω)ψm(ρ, ω).
W(ρ1, ρ2, ω)=d(ω)mnψm*(ρ1, ω)ψn(ρ2, ω)=d(ω)mψm(ρ1, ω)*mψm(ρ2, ω).

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