Abstract

In a previous paper [Phys. Rev. 138, B1561 (1965)] some new results were presented relating to the structure of the electromagnetic field near the focus of a coherent beam emerging from an aplanatic optical system. The present paper supplements the previous one by providing detailed analysis of the behavior of the Poynting vector in the focal region of such a beam. In particular, diagrams showing the flow lines and the contours of constant amplitude of the time-averaged Poynting vector are given. The energy flow is found to have vortices near certain points of the focal plane. Two diagrams showing the behavior of the flow near a typical vortex are also included.

© 1967 Optical Society of America

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  1. B. Richards and E. Wolf, Proc. Roy. Soc. (London) A253, 358 (1959).
  2. A. Boivin and E. Wolf, Phys. Rev. 138, B1561 (1965).
  3. A. I. Carswell, Phys. Rev. Letters 15, 647 (1965).
  4. (a) A. L. Bloom: Paper 9A-6 presented at the International Quantum Electronics Conference, Phoenix, Ariz., April, 1966. [Abstract published in IEEE QE-2, 87 lxiii (1966)]. (b) D. J. Innes and A. L. Bloom, Spectra-Physics Laser Technical Bulletin, # 5 (1966) (Published by Spectra-Physics, Inc., Mountain View, Calif.).
  5. A. Boivin and M. Gravel, J. Opt. Soc. Am. 56, 1438A (1966).
  6. A misprint in the formula (3.22) for |〈S〉| given in Ref. corrected here.
  7. See, for example, R. P. Agnew, Differential Equations (McGraw-Hill Book Company, New York, 1960), p. 306.
  8. The Poynting-vector flow lines, away from the singularities, resemble closely the contour lines for the fraction of total illumination within circles centered on axis in receiving planes u = const [E. Wolf, Proc. Roy. Soc. (London) A204, 533 (1951), Fig. 3a on p. 542]. The qualitative resemblance of these contour lines to flow lines appears to be a consequence of the principle of conservation of energy, applied to the region generated by the rotation of a portion of any particular contour line about the optical axis.
  9. V. S. Ignatowsky, Trans. Opt. Inst. Petrograd, Vol. 1, paper IV (1919).
  10. In Ref. 1, the designations of the vertical axis in Figs. 4 and 5 have been interchanged. The designation is corrected in our reproduction of Fig. 5(b) (shown here as Fig. 4).
  11. This implies that the numbers on the contour in Figs. 5 and 6 represent the quantity 100|〈S(u,υ)〉|/|〈S(0,0)〉|, for an aplanatic system with α = 45°. The conversion of these dimensionless quantities to gaussian units or to the rationalized mks units may be effected by the relations [cf. Eq. (3.23) of Ref. 1]: |S(0,0)| = (cA2/8π)|I0(0,0)|2 (gaussian units), |S(0,0)|=(A2/2Z0)|I0(0,0)|2 (mks units). Here A is given by Eq. (4), I0(0,0) = 0.5021 (with α = 45°) and Z0 = cµ0 = 376.727 ohms, µ0, being the magnetic permeability of free space. There are errors in the corresponding equations in footnote 9 of Ref. 2, relating to the values of the time-averaged electric energy density. The correct equations are 〈we〉 = (1/8π)〈E2〉 = (AG2/16π) |I0(0,0) |2 (gaussian units), 〈we〉 = ½∊0E2〉 = ¼∊0Amks2|I0(0,0)|2 (mks units), AG = 141.20 statV/cm, Amks = 1.339×107 V/m, where ∊0 is the permittivity of free space. The caption to Fig. 5 of Ref. 2 does not specify the normalization of ez. The figure displays (for an aplanatic syst with α = 45°) the quantity 70.898ez(0,υ,ϕ)/|e(0,0,0)| which is identical with ez(0,υ,ϕ) when A is taken to be equal to AG = 141.20 stat V/cm.

Agnew, R. P.

See, for example, R. P. Agnew, Differential Equations (McGraw-Hill Book Company, New York, 1960), p. 306.

Bloom, A. L.

(a) A. L. Bloom: Paper 9A-6 presented at the International Quantum Electronics Conference, Phoenix, Ariz., April, 1966. [Abstract published in IEEE QE-2, 87 lxiii (1966)]. (b) D. J. Innes and A. L. Bloom, Spectra-Physics Laser Technical Bulletin, # 5 (1966) (Published by Spectra-Physics, Inc., Mountain View, Calif.).

Boivin, A.

A. Boivin and M. Gravel, J. Opt. Soc. Am. 56, 1438A (1966).

A. Boivin and E. Wolf, Phys. Rev. 138, B1561 (1965).

Carswell, A. I.

A. I. Carswell, Phys. Rev. Letters 15, 647 (1965).

Gravel, M.

A. Boivin and M. Gravel, J. Opt. Soc. Am. 56, 1438A (1966).

Ignatowsky, V. S.

V. S. Ignatowsky, Trans. Opt. Inst. Petrograd, Vol. 1, paper IV (1919).

Richards, B.

B. Richards and E. Wolf, Proc. Roy. Soc. (London) A253, 358 (1959).

Wolf, E.

B. Richards and E. Wolf, Proc. Roy. Soc. (London) A253, 358 (1959).

A. Boivin and E. Wolf, Phys. Rev. 138, B1561 (1965).

The Poynting-vector flow lines, away from the singularities, resemble closely the contour lines for the fraction of total illumination within circles centered on axis in receiving planes u = const [E. Wolf, Proc. Roy. Soc. (London) A204, 533 (1951), Fig. 3a on p. 542]. The qualitative resemblance of these contour lines to flow lines appears to be a consequence of the principle of conservation of energy, applied to the region generated by the rotation of a portion of any particular contour line about the optical axis.

Other (11)

B. Richards and E. Wolf, Proc. Roy. Soc. (London) A253, 358 (1959).

A. Boivin and E. Wolf, Phys. Rev. 138, B1561 (1965).

A. I. Carswell, Phys. Rev. Letters 15, 647 (1965).

(a) A. L. Bloom: Paper 9A-6 presented at the International Quantum Electronics Conference, Phoenix, Ariz., April, 1966. [Abstract published in IEEE QE-2, 87 lxiii (1966)]. (b) D. J. Innes and A. L. Bloom, Spectra-Physics Laser Technical Bulletin, # 5 (1966) (Published by Spectra-Physics, Inc., Mountain View, Calif.).

A. Boivin and M. Gravel, J. Opt. Soc. Am. 56, 1438A (1966).

A misprint in the formula (3.22) for |〈S〉| given in Ref. corrected here.

See, for example, R. P. Agnew, Differential Equations (McGraw-Hill Book Company, New York, 1960), p. 306.

The Poynting-vector flow lines, away from the singularities, resemble closely the contour lines for the fraction of total illumination within circles centered on axis in receiving planes u = const [E. Wolf, Proc. Roy. Soc. (London) A204, 533 (1951), Fig. 3a on p. 542]. The qualitative resemblance of these contour lines to flow lines appears to be a consequence of the principle of conservation of energy, applied to the region generated by the rotation of a portion of any particular contour line about the optical axis.

V. S. Ignatowsky, Trans. Opt. Inst. Petrograd, Vol. 1, paper IV (1919).

In Ref. 1, the designations of the vertical axis in Figs. 4 and 5 have been interchanged. The designation is corrected in our reproduction of Fig. 5(b) (shown here as Fig. 4).

This implies that the numbers on the contour in Figs. 5 and 6 represent the quantity 100|〈S(u,υ)〉|/|〈S(0,0)〉|, for an aplanatic system with α = 45°. The conversion of these dimensionless quantities to gaussian units or to the rationalized mks units may be effected by the relations [cf. Eq. (3.23) of Ref. 1]: |S(0,0)| = (cA2/8π)|I0(0,0)|2 (gaussian units), |S(0,0)|=(A2/2Z0)|I0(0,0)|2 (mks units). Here A is given by Eq. (4), I0(0,0) = 0.5021 (with α = 45°) and Z0 = cµ0 = 376.727 ohms, µ0, being the magnetic permeability of free space. There are errors in the corresponding equations in footnote 9 of Ref. 2, relating to the values of the time-averaged electric energy density. The correct equations are 〈we〉 = (1/8π)〈E2〉 = (AG2/16π) |I0(0,0) |2 (gaussian units), 〈we〉 = ½∊0E2〉 = ¼∊0Amks2|I0(0,0)|2 (mks units), AG = 141.20 statV/cm, Amks = 1.339×107 V/m, where ∊0 is the permittivity of free space. The caption to Fig. 5 of Ref. 2 does not specify the normalization of ez. The figure displays (for an aplanatic syst with α = 45°) the quantity 70.898ez(0,υ,ϕ)/|e(0,0,0)| which is identical with ez(0,υ,ϕ) when A is taken to be equal to AG = 141.20 stat V/cm.

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