Abstract

Seshadri’s claim [J. Opt. Soc. A 23, 3238 (2006) ] that the singularity in the simple complex-source point model is not nonphysical is disputed. The branch cut behavior is further discussed. Two different forms for the complex-source point model for a Gaussian beam are identified, corresponding to different choices for the sign of a square root or in terms of oblate spheroidal coordinates to two different branch cuts. One of these solutions corresponds to a beamlike solution, while the second gives a wave that radiates outward from the source. Both exhibit nonphysical singularities. These singularities are avoided by using a complex-source/sink model. The choice of branch cut is also important in defining the oblate spheroidal beams, based on spheroidal harmonics.

© 2007 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  5. G. A. Deschamps, "Gaussian beam as a bundle of complex rays," Electron. Lett. 7, 684-685 (1971).
    [CrossRef]
  6. J. B. Keller and S. I. Rubinow, "Asymptotic solution of eigenvalue problems," Ann. Phys. (Leipzig) 9, 24-75 (1960).
    [CrossRef]
  7. L. A. Weinstein, Open Resonators and Open Waveguides (Golen Press, 1969).
  8. G. Toraldo di Francia, "Optical resonators," Opt. Acta 13, 323-342 (1966).
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    [CrossRef]
  10. G. Rodriguez-Morales and S. Chavez-Cerda, "Exact nonparaxial beams of the scalar Helmholtz equation," Opt. Lett. 29, 430-432 (2004).
    [CrossRef]
  11. A. L. Cullen, P. Nagenthiram, and A. D. Williams, "A variational approach to the theory of the open resonator," Proc. R. Soc. London, Ser. A 329, 153-169 (1972).
  12. C. J. R. Sheppard and S. Saghafi, "Beam modes beyond the paraxial approximation: a scalar treatment," Phys. Rev. A 57, 2971-2979 (1998).
    [CrossRef]

2006 (1)

2004 (1)

2001 (1)

2000 (1)

1998 (1)

C. J. R. Sheppard and S. Saghafi, "Beam modes beyond the paraxial approximation: a scalar treatment," Phys. Rev. A 57, 2971-2979 (1998).
[CrossRef]

1993 (1)

C. J. R. Sheppard and C. J. Cogswell, "Reflection and transmission confocal microscopy," in Proceedings of the International Conference on Optics within Life Sciences (1990), G.von Bally and S.Khanna, eds. (Elsevier, 1993), pp. 310-315.

1992 (1)

S. Hell and E. H. K. Stelzer, "Fundamental improvement of resolution with a 4Pi-confocal fluorescence microscope using two-photon excitation," Opt. Commun. 93, 277-282 (1992).
[CrossRef]

1972 (1)

A. L. Cullen, P. Nagenthiram, and A. D. Williams, "A variational approach to the theory of the open resonator," Proc. R. Soc. London, Ser. A 329, 153-169 (1972).

1971 (1)

G. A. Deschamps, "Gaussian beam as a bundle of complex rays," Electron. Lett. 7, 684-685 (1971).
[CrossRef]

1969 (1)

L. A. Weinstein, Open Resonators and Open Waveguides (Golen Press, 1969).

1966 (1)

G. Toraldo di Francia, "Optical resonators," Opt. Acta 13, 323-342 (1966).

1960 (1)

J. B. Keller and S. I. Rubinow, "Asymptotic solution of eigenvalue problems," Ann. Phys. (Leipzig) 9, 24-75 (1960).
[CrossRef]

Chavez-Cerda, S.

Cogswell, C. J.

C. J. R. Sheppard and C. J. Cogswell, "Reflection and transmission confocal microscopy," in Proceedings of the International Conference on Optics within Life Sciences (1990), G.von Bally and S.Khanna, eds. (Elsevier, 1993), pp. 310-315.

Cullen, A. L.

A. L. Cullen, P. Nagenthiram, and A. D. Williams, "A variational approach to the theory of the open resonator," Proc. R. Soc. London, Ser. A 329, 153-169 (1972).

Deschamps, G. A.

G. A. Deschamps, "Gaussian beam as a bundle of complex rays," Electron. Lett. 7, 684-685 (1971).
[CrossRef]

Hell, S.

S. Hell and E. H. K. Stelzer, "Fundamental improvement of resolution with a 4Pi-confocal fluorescence microscope using two-photon excitation," Opt. Commun. 93, 277-282 (1992).
[CrossRef]

Keller, J. B.

J. B. Keller and S. I. Rubinow, "Asymptotic solution of eigenvalue problems," Ann. Phys. (Leipzig) 9, 24-75 (1960).
[CrossRef]

Ludlow, I. K.

Nagenthiram, P.

A. L. Cullen, P. Nagenthiram, and A. D. Williams, "A variational approach to the theory of the open resonator," Proc. R. Soc. London, Ser. A 329, 153-169 (1972).

Rodriguez-Morales, G.

Rubinow, S. I.

J. B. Keller and S. I. Rubinow, "Asymptotic solution of eigenvalue problems," Ann. Phys. (Leipzig) 9, 24-75 (1960).
[CrossRef]

Saghafi, S.

C. J. R. Sheppard and S. Saghafi, "Beam modes beyond the paraxial approximation: a scalar treatment," Phys. Rev. A 57, 2971-2979 (1998).
[CrossRef]

Seshadri, S. R.

Sheppard, C. J.

C. J. R. Sheppard and S. Saghafi, "Beam modes beyond the paraxial approximation: a scalar treatment," Phys. Rev. A 57, 2971-2979 (1998).
[CrossRef]

Sheppard, C. J. R.

C. J. R. Sheppard, "High-aperture beams," J. Opt. Soc. Am. A 18, 1579-1587 (2001).
[CrossRef]

C. J. R. Sheppard and C. J. Cogswell, "Reflection and transmission confocal microscopy," in Proceedings of the International Conference on Optics within Life Sciences (1990), G.von Bally and S.Khanna, eds. (Elsevier, 1993), pp. 310-315.

Stelzer, E. H. K.

S. Hell and E. H. K. Stelzer, "Fundamental improvement of resolution with a 4Pi-confocal fluorescence microscope using two-photon excitation," Opt. Commun. 93, 277-282 (1992).
[CrossRef]

Toraldo di Francia, G.

G. Toraldo di Francia, "Optical resonators," Opt. Acta 13, 323-342 (1966).

Ulanowski, Z.

Weinstein, L. A.

L. A. Weinstein, Open Resonators and Open Waveguides (Golen Press, 1969).

Williams, A. D.

A. L. Cullen, P. Nagenthiram, and A. D. Williams, "A variational approach to the theory of the open resonator," Proc. R. Soc. London, Ser. A 329, 153-169 (1972).

Ann. Phys. (1)

J. B. Keller and S. I. Rubinow, "Asymptotic solution of eigenvalue problems," Ann. Phys. (Leipzig) 9, 24-75 (1960).
[CrossRef]

Electron. Lett. (1)

G. A. Deschamps, "Gaussian beam as a bundle of complex rays," Electron. Lett. 7, 684-685 (1971).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Acta (1)

G. Toraldo di Francia, "Optical resonators," Opt. Acta 13, 323-342 (1966).

Opt. Commun. (1)

S. Hell and E. H. K. Stelzer, "Fundamental improvement of resolution with a 4Pi-confocal fluorescence microscope using two-photon excitation," Opt. Commun. 93, 277-282 (1992).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. A (1)

C. J. R. Sheppard and S. Saghafi, "Beam modes beyond the paraxial approximation: a scalar treatment," Phys. Rev. A 57, 2971-2979 (1998).
[CrossRef]

Proc. R. Soc. London, Ser. A (1)

A. L. Cullen, P. Nagenthiram, and A. D. Williams, "A variational approach to the theory of the open resonator," Proc. R. Soc. London, Ser. A 329, 153-169 (1972).

Other (2)

C. J. R. Sheppard and C. J. Cogswell, "Reflection and transmission confocal microscopy," in Proceedings of the International Conference on Optics within Life Sciences (1990), G.von Bally and S.Khanna, eds. (Elsevier, 1993), pp. 310-315.

L. A. Weinstein, Open Resonators and Open Waveguides (Golen Press, 1969).

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Figures (2)

Fig. 1
Fig. 1

Intensity in a meridional plane for k d = 2 : (a) a complex-source wave for domain I, (b) a complex-source wave for domain II, (c) a complex-source/sink wave.

Fig. 2
Fig. 2

The contours of constant phase in a meridional plane for k d = 2 : (a) a complex-source wave for domain I, (b) a complex-source wave for domain II, (c) a complex-source/sink wave. Contours are plotted every 60 ° . In (a) the wave propagates outward. On traveling along the axis between two points on the axis on either side of the origin, there is a phase jump at z = 0 . In (b) the wave propagates beamlike from left to right. On traveling between two points on the axis on either side of the origin, there is a Gouy phase anomaly when the wave is traveling along the axis. If it is traveling along a line of constant phase, there is a phase jump at z = 0 . The phase is identical for positive z for (a) and (b). In (c) the wave also propagates beamlike from left to right. On traveling between two points on the axis on either side of the origin, there is a Gouy phase anomaly when the wave is traveling along the axis. If it is traveling along a line of constant phase, there is a phase singularity at z = 0 .

Equations (9)

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0 ξ < , 1 η 1 ( I ) ,
ξ < , 0 η 1 ( II ) .
U ( x , y , z ) = exp ( i k R ) R ,
R = [ x 2 + y 2 + ( z i d ) 2 ] 1 2 .
i R = ( d 2 + 2 i d z x 2 y 2 z 2 ) 1 2
R I = z i d , z > 0 , = z + i d , z < 0 ,
U I = exp ( i k z ) exp ( k d ) z i d , z > 0 , = exp ( i k z ) exp ( k d ) z i d , z < 0 .
i R II = d i z , z , d < 0 , = d + i z , z , d > 0 ,
U II = exp ( i k z ) exp ( k d ) z i d , z , d < 0 , = exp ( i k z ) exp ( k d ) z i d , z , d > 0 .

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