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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References

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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).
    [Crossref]
  3. A. I. Carswell, Phys. Rev. Letters 15, 647 (1965).
    [Crossref]
  4. (a)A. L. Bloom: Paper 9A-6 presented at the International Quantum Electronics Conference, Phoenix, Ariz., April, 1966. [Abstract published in IEEE QE-2,87lxiii (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. 1 is 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,v)〉|/|〈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〉=12∊0〈E2〉=14∊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 system with α=45°) the quantity 70.898ez(0,v,ϕ)/|e(0,0,0)| which is identical with ez(0,v,ϕ) when A is taken to be equal to AG=141.20 statV/cm.

1966 (1)

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

1965 (2)

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

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

1959 (1)

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

1951 (1)

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.

1919 (1)

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

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-2,87lxiii (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).
[Crossref]

Carswell, A. I.

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

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.

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

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

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.

J. Opt. Soc. Am. (1)

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

Phys. Rev. (1)

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

Phys. Rev. Letters (1)

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

Proc. Roy. Soc. (London) (2)

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.

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

Trans. Opt. Inst. Petrograd (1)

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

Other (5)

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,v)〉|/|〈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〉=12∊0〈E2〉=14∊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 system with α=45°) the quantity 70.898ez(0,v,ϕ)/|e(0,0,0)| which is identical with ez(0,v,ϕ) when A is taken to be equal to AG=141.20 statV/cm.

(a)A. L. Bloom: Paper 9A-6 presented at the International Quantum Electronics Conference, Phoenix, Ariz., April, 1966. [Abstract published in IEEE QE-2,87lxiii (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 misprint in the formula (3.22) for |〈S〉| given in Ref. 1 is corrected here.

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

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Figures (6)

Fig. 1
Fig. 1

Illustrating the notation.

Fig. 2
Fig. 2

Flow lines of the time-averaged Poynting vector in the neighborhood of the focus of an aplanatic system with angular semi-aperture α=45° on the image side. u and v denote the normalized longitudinal and transversal coordinates defined by Eq. (1).

Fig. 3
Fig. 3

Flow lines of the time-averaged Poynting vector, showing vortex behavior, in the neighborhood of the two singularities of the flow closest to the focus (u=0, v=3.67 and u=0, v=3.99). (Aplanatic system, α=45°.)

Fig. 4
Fig. 4

The variation of the only nonvanishing component of the time-averaged energy flow |〈Sz〉| along any meridional section ϕ=const of the focal plane in an aplanatic system of angular semi-aperture α on the image side. The values are normalized to 100 at the focus. [After B. Richards and E. Wolf, Proc. Roy. Soc. (London) A253, 358 (1959).]

Fig. 5
Fig. 5

Contours of constant magnitude of the time-averaged Poynting vector (constant |〈S〉|) in the focal region of an aplanatic system with angular semi-aperture α=45°. The values are normalized to 100 at the focus.

Fig. 6
Fig. 6

Contours of constant magnitude of the time-averaged Poynting vector near the two singularities closest to the focus (u=0, v=3.67 and u=0, v=3.99). The values correspond to normalization 100 at the focus. (Aplanatic system, α=45°.)

Equations (13)

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u = k r cos θ sin 2 α = k z sin 2 α , v = k r sin θ sin α = k ( x 2 + y 2 ) 1 2 sin α ,
S x = ( c A 2 / 4 π ) cos ϕ Im { I 1 ( I 2 * - I 0 * ) } , S y = ( c A 2 / 4 π ) sin ϕ Im { I 1 ( I 2 * - I 0 * ) } , S z = ( c A 2 / 8 π ) { I 0 2 - I 2 2 } ,
I 0 ( u , v ) = 0 α cos 1 2 θ sin θ ( 1 + cos θ ) J 0 ( v sin θ / sin α ) e i u cos θ / sin 2 α d θ , I 1 ( u , v ) = 0 α cos 1 2 θ sin 2 θ J 1 ( v sin θ / sin α ) e i u cos θ / sin 2 α d θ , I 2 ( u , v ) = 0 α cos 1 2 θ sin θ ( 1 - cos θ ) J 2 ( v sin θ / sin α ) e i u cos θ / sin 2 α d θ .
A = π f l / λ ,
S = ( c A 2 / 8 π ) ( ( I 0 2 - I 2 2 ) 2 + 4 { Im [ I 1 ( I 2 * - I 0 * ) ] } 2 ) 1 2 .
sin γ = 2 Im [ I 1 ( I 2 * - I 0 * ) ] [ ( I 0 2 - I 2 2 ) 2 + 4 { Im [ I 1 ( I 2 * - I 0 * ) ] } 2 ] 1 2 , cos γ = I 0 2 - I 2 2 [ ( I 0 2 - I 2 2 ) 2 + 4 { Im [ I 1 ( I 2 * - I 0 * ) ] } 2 ] 1 2 .
S ( - u , v , ϕ ) = S ( u , v , ϕ ) ,
γ ( - u , v , ϕ ) = 2 π - γ ( u , v , ϕ ) ,             ( u 0 ) .
d v d u = 1 sin α d d z [ ( x 2 + y 2 ) 1 2 ] .
sin α d v d u = [ S x 2 + S y 2 ] 1 2 S z = 2 Im [ I 1 ( I 2 * - I 0 * ) ] I 0 2 - I 2 2 .
I 0 - I 2 = 0             when             v = 3.67 , 6.92 , 10.11 , 13.27 , 16.42 , 19.57 , I 0 + I 2 = 0             when             v = 3.99 , 7.13 , 10.26 , 13.39 , 16.52 , 19.66 , .
S ( 0 , 0 ) = ( c A 2 / 8 π ) I 0 ( 0 , 0 ) 2 ( gaussian units ) , S ( 0 , 0 ) = ( A 2 / 2 Z 0 ) I 0 ( 0 , 0 ) 2 ( mks units ) .
w e = ( 1 / 8 π ) E 2 = ( A G 2 / 16 π ) I 0 ( 0 , 0 ) 2 ( gaussian units ) , w e = 1 2 0 E 2 = 1 4 0 A mks 2 I 0 ( 0 , 0 ) 2 ( mks units ) , A G = 141.20 statV / cm , A mks = 1.339 × 10 7 V / m ,