Abstract

Using the Rayleigh-Sommerfeld theory of diffraction we obtain an exact expression for the axial irradiance of a focused annular laser beam valid for all axial points. Conditions for the validity of the Fresnel theory are obtained. We discuss why and how the depth of focus and asymmetry of focused fields about the focal plane depend on the Fresnel number of the beam aperture as observed from the geometric focus. When a beam is focused on a distant target so that the Fresnel number is small (≲5), the principal maximum of axial irradiance occurs at a point which is significantly away from the geometric focus in the direction of the aperture. We discuss how to optimally focus a beam to illuminate a moving distant target in terms of the encircled energy on it. We show that, to obtain the maximum possible concentration of energy on a target, the beam must be focused on it, thus requiring active focusing for a moving target. However, if energy concentration is adequate for a beam focused on a target at a certain distance, it is more than adequate for a considerable range of the distance of a moving target without active focusing. In a shared-aperture optical system the aperture used for focusing a beam on the target is also used for imaging the target. Hence, in such a system the optical transfer function is also more than adequate over a wide range of the target distance without active focusing.

© 1983 Optical Society of America

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References

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  1. Y. Li, E. Wolf, Opt. Commun. 39, 211 (1981).
    [CrossRef]
  2. M. P. Givens, Opt. Commun. 41, 145 (1982).
    [CrossRef]
  3. Y. Li, E. Wolf, Opt. Commun. 42, 151 (1982).
    [CrossRef]
  4. W. H. Carter, Appl. Opt. 21, 1989 (1982).
    [CrossRef] [PubMed]
  5. Y. Li, J. Opt. Soc. Am. 72, 770 (1982).
    [CrossRef]
  6. H. Osterberg, L. W. Smith, J. Opt. Soc. Am. 51, 1050 (1961).
    [CrossRef]
  7. D. A. Holmes, J. E. Korka, P. V. Avizonis, Appl. Opt. 11, 565 (1972).
    [CrossRef] [PubMed]
  8. A. Arimoto, Opt. Acta 23, 245 (1976).
    [CrossRef]
  9. M. A. Gusinow, M. E. Riley, M. A. Palmer, Opt. Quantum Electron. 9, 465 (1977).
    [CrossRef]
  10. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980). Sec. 8.8.
  11. E. Collett, E. Wolf, Opt. Lett. 5, 264 (1980).See E. Wolf, Y. Li, Opt. Commun. 39, 205 (1981).
    [CrossRef] [PubMed]
  12. A. Sommerfeld, Optics, Vol. 4 (Academic, New York, 1972), pp. 199–201;substitute Eq. (8) on p. 201 into Eq. (6) on p. 199.
  13. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 43–44;substitute Eq. (3–23) on p. 44 into Eq. (3–15) on p. 43.
  14. Ref. 13, p. 59.
  15. The sag of a surface is defined as its deviation along its axis from a plane normal to the axis and passing through its vertex.
  16. V. N. Mahajan, J. Opt. Soc. Am. 72, 1258 (1982).
    [CrossRef]
  17. J. E. Harvey, R. V. Shack, Appl. Opt. 17, 3003 (1978);also J. E. Harvey, Am. J. Phys. 47, 974 (1979).
    [CrossRef] [PubMed]
  18. V. N. Mahajan, Appl. Opt. same issue 22, 0000 (1983).
  19. P. A. Stokseth, J. Opt. Soc. Am. 59, 1313 (1969).
    [CrossRef]
  20. R. Barakat, J. Opt. Soc. Am. 54, 920 (1963).J. Opt. Soc. Am. 55, 538 (1965);Appl. Opt. 5, 1850 (1966).
    [CrossRef] [PubMed]
  21. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge, U.P., New York, 1966), p. 576.

1983

V. N. Mahajan, Appl. Opt. same issue 22, 0000 (1983).

1982

1981

Y. Li, E. Wolf, Opt. Commun. 39, 211 (1981).
[CrossRef]

1980

1978

1977

M. A. Gusinow, M. E. Riley, M. A. Palmer, Opt. Quantum Electron. 9, 465 (1977).
[CrossRef]

1976

A. Arimoto, Opt. Acta 23, 245 (1976).
[CrossRef]

1972

1969

P. A. Stokseth, J. Opt. Soc. Am. 59, 1313 (1969).
[CrossRef]

1963

1961

Arimoto, A.

A. Arimoto, Opt. Acta 23, 245 (1976).
[CrossRef]

Avizonis, P. V.

Barakat, R.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980). Sec. 8.8.

Carter, W. H.

Collett, E.

Givens, M. P.

M. P. Givens, Opt. Commun. 41, 145 (1982).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 43–44;substitute Eq. (3–23) on p. 44 into Eq. (3–15) on p. 43.

Gusinow, M. A.

M. A. Gusinow, M. E. Riley, M. A. Palmer, Opt. Quantum Electron. 9, 465 (1977).
[CrossRef]

Harvey, J. E.

Holmes, D. A.

Korka, J. E.

Li, Y.

Y. Li, E. Wolf, Opt. Commun. 42, 151 (1982).
[CrossRef]

Y. Li, J. Opt. Soc. Am. 72, 770 (1982).
[CrossRef]

Y. Li, E. Wolf, Opt. Commun. 39, 211 (1981).
[CrossRef]

Mahajan, V. N.

V. N. Mahajan, Appl. Opt. same issue 22, 0000 (1983).

V. N. Mahajan, J. Opt. Soc. Am. 72, 1258 (1982).
[CrossRef]

Osterberg, H.

Palmer, M. A.

M. A. Gusinow, M. E. Riley, M. A. Palmer, Opt. Quantum Electron. 9, 465 (1977).
[CrossRef]

Riley, M. E.

M. A. Gusinow, M. E. Riley, M. A. Palmer, Opt. Quantum Electron. 9, 465 (1977).
[CrossRef]

Shack, R. V.

Smith, L. W.

Sommerfeld, A.

A. Sommerfeld, Optics, Vol. 4 (Academic, New York, 1972), pp. 199–201;substitute Eq. (8) on p. 201 into Eq. (6) on p. 199.

Stokseth, P. A.

P. A. Stokseth, J. Opt. Soc. Am. 59, 1313 (1969).
[CrossRef]

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge, U.P., New York, 1966), p. 576.

Wolf, E.

Y. Li, E. Wolf, Opt. Commun. 42, 151 (1982).
[CrossRef]

Y. Li, E. Wolf, Opt. Commun. 39, 211 (1981).
[CrossRef]

E. Collett, E. Wolf, Opt. Lett. 5, 264 (1980).See E. Wolf, Y. Li, Opt. Commun. 39, 205 (1981).
[CrossRef] [PubMed]

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980). Sec. 8.8.

Appl. Opt.

J. Opt. Soc. Am.

Opt. Acta

A. Arimoto, Opt. Acta 23, 245 (1976).
[CrossRef]

Opt. Commun.

Y. Li, E. Wolf, Opt. Commun. 39, 211 (1981).
[CrossRef]

M. P. Givens, Opt. Commun. 41, 145 (1982).
[CrossRef]

Y. Li, E. Wolf, Opt. Commun. 42, 151 (1982).
[CrossRef]

Opt. Lett.

Opt. Quantum Electron.

M. A. Gusinow, M. E. Riley, M. A. Palmer, Opt. Quantum Electron. 9, 465 (1977).
[CrossRef]

Other

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980). Sec. 8.8.

A. Sommerfeld, Optics, Vol. 4 (Academic, New York, 1972), pp. 199–201;substitute Eq. (8) on p. 201 into Eq. (6) on p. 199.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 43–44;substitute Eq. (3–23) on p. 44 into Eq. (3–15) on p. 43.

Ref. 13, p. 59.

The sag of a surface is defined as its deviation along its axis from a plane normal to the axis and passing through its vertex.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge, U.P., New York, 1966), p. 576.

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Figures (12)

Fig. 1
Fig. 1

Geometry of diffraction problem. ρ(ξ,η) is a point in the aperture plane. r(x,y) is a point in the observation plane. The z axis is perpendicular to the two planes.

Fig. 2
Fig. 2

(a) and (b): Diffraction of a spherical wave by an annular aperture of inner and outer radii of εa and a, respectively. F is the geometric focus of the wave, and P is an axial point of observation at distance z from the aperture.

Fig. 3
Fig. 3

Axial irradiance of circular and annular beams of radius a = 5λ focused at distance R = 5a, normalized by the irradiance I0 at a point on the wave front passing through the center of the annulus. N = 1 and 0.75 for ε = 0 and 0.5, respectively.

Fig. 4
Fig. 4

Variation of z0, zp, and I(zp) with N. z0 is the distance of an axial point from the aperture so that I(z0) = I(R) = π2N2I0. zp is the distance of the location of the principal maximum. I(zp) is the irradiance at this point.

Fig. 5
Fig. 5

Axial irradiance of a focused circular beam with N = 100.

Fig. 6
Fig. 6

Comparison of focal-point irradiance obtained by Rayleigh-Sommerfeld and Fresnel theories of diffraction.

Fig. 7
Fig. 7

Axial irradiance of a circular focused beam. D = 2a = 25 cm, λ = 10.6 μm, R = 1.47 km, so that N = a2R = 1. The solid curve gives the axial irradiance of a circular beam focused at distance R normalized by the focal-point irradiance. The dotted curve represents the focal-point irradiance of the beam when focused at distance z. The dashed curve gives the ratio of the solid and dotted curves. Wm is the peak defocus aberration of a beam focused at distance R with respect to a reference sphere centered at distance z.

Fig. 8
Fig. 8

Focused and defocused irradiance distributions and corresponding encircled energies for a circular focused beam considered in Fig. 7. The irradiance is normalized by the focal-point irradiance, and encircled-energy curves are normalized by the total power in the beam. The units of r and r0 are λR/D.

Fig. 9
Fig. 9

Encircled energy in a circle of fixed radius r0 for a circular focused beam considered in Fig. 7 as a function of the axial distance z from the aperture. The encircled energy is normalized by the total power in the beam. The units of r0 are λR/D.

Fig. 10
Fig. 10

OTF of a shared-aperture imaging system focused for a target at distance R. The OTF for z = R gives the imaging properties of the system for a focused target. The OTFs for z = (1 ± 0.4)R are the defocused OTFs for targets in two planes symmetric about the focal plane, one of them (z = 0.6R) being the plane containing the principal maximum of axial irradiance. The spatial frequency γ is in units of DR.

Fig. 11
Fig. 11

Axial irradiance of collimated circular and annular beams, normalized by aperture irradiance I0. As in Fig. 3, the aperture-to-wavelength ratio is a/λ = 5.

Fig. 12
Fig. 12

Axial irradiance of a collimated circular or annular beam normalized by the aperture irradiance I0. The axial distance z is in units of the far-field distance D2(1 − ε2)/λ.

Equations (77)

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U ( r ; z ) = 1 2 π U ( ρ ) z l l [ exp ( ikl ) / l ] d ρ ,
l = ( | ρ r | 2 + z 2 ) 1 / 2 = z + 1 2 z | ρ r | 2 1 8 z 3 | ρ r | 4 +
U ( r ; z ) = 1 2 π U ( ρ ) ( z / l 2 ) exp ( ikl ) ( ik 1 / l ) d ρ .
kl 1 ,
l z
l z + 1 2 z | ρ r | 2
U ( r ; z ) = exp [ ik ( z + r 2 / 2 z ) ] i λ z U ( ρ ) exp ( ik ρ 2 / 2 z ) × exp ( ik ρ · r / z ) d ρ ,
U ( ρ ) = ( A 0 / s ) exp ( iks ) ,
s = ( ρ 2 + R 2 ) 1 / 2
U ( z ) = z ε a a U ( ρ ) 1 l l [ exp ( ikl ) / l ] ρ d ρ ,
l = ( ρ 2 + z 2 ) 1 / 2
[ exp ( iks ) / s ] 1 l l [ exp ( ikl ) / l ] ρ d ρ = 1 l exp [ ik ( l s ) ] l s ,
I ( z ) = | U ( z ) | 2 = A 0 2 { ( cos α 1 d 1 ) 2 + ( cos α 2 d 2 ) 2 2 cos α 1 cos α 2 d 1 d 2 cos [ 2 π λ ( d 1 d 2 ) ] } ,
cos α 1 = z / s 1 , s 1 = ( z 2 + ε 2 a 2 ) 1 / 2 ,
cos α 2 = z / s 2 , s 2 = ( z 2 + a 2 ) 1 / 2 ,
d 1 = s 1 s 1 , s 1 = ( R 2 + ε 2 a 2 ) 1 / 2 ,
d 2 = s 2 s 2 , s 2 = ( R 2 + a 2 ) 1 / 2 .
I ( R ) = A 0 2 R 2 [ k 2 ( 1 s 1 1 s 2 ) 2 + 1 4 ( 1 s 1 2 1 s 2 2 ) 2 ] .
N = a 2 ( 1 ε 2 ) / λ R .
z a ,
R a
d 1 d 2 z R .
d 1 d 2 = 1 2 a 2 ( 1 ε 2 ) ( 1 z 1 R ) 1 8 a 4 ( 1 ε 4 ) ( 1 z 3 1 R 3 ) +
1 z 3 1 R 3 + λ a 4 ( 1 ε 4 ) ,
d 1 d 2 1 2 a 2 ( 1 ε 2 ) ( 1 z 1 R ) .
z 3 a 4 ( 1 ε 4 ) / λ .
I ( z ) = ( PA / λ 2 z 2 ) S ,
P = I 0 A
A = π a 2 ( 1 ε 2 )
S = [ sin ( 3 σ Φ ) / 3 σ Φ ] 2
σ Φ = π N 2 3 | R z 1 |
σ Φ = | n | π / 3 ,
R / z = 1 + 2 n / N ,
tan ( 3 σ Φ ) = ( R / z ) 3 σ Φ , z R .
I ( z ) = ( R / z ) 2 I ( R ) S ,
I ( R ) = PA / λ 2 R 2
S = I ( z ) / I ( R ) = ( R / z ) 2 S .
F = R / 2 a .
F = a ( 1 ε 2 ) / 2 λ N .
σ Φ = π ( 1 ε 2 ) | R z | / 8 3 λ F 2 .
I ( z ) = I ( R ) S ,
I e ( R ) = 4 π 2 ( R / λ ) 2 I 0 ( α 1 / 2 β 1 / 2 ) 2 ,
α = 1 + 1 / 4 F 2 ,
β = 1 + ε 2 / 4 F 2 .
I ( ρ ) = | u ( ρ ) | 2 = A 0 2 / ( ρ 2 + R 2 ) .
P = 2 π ε a a I ( ρ ) ρ d ρ = π I 0 R 2 ln ( α / β ) .
I a ( R ) = ( π 2 a 2 I 0 / λ 2 ) ( 1 ε 2 ) ln ( α / β ) .
I e ( R ) I a ( R ) = 16 F 2 ( α 1 / 2 β 1 / 2 ) 2 / ( 1 ε 2 ) ln ( α / β ) .
s R
s R + ρ 2 / 2 R
U ( ρ ) = ( A 0 / R ) exp [ ik ( R + ρ 2 / 2 R ) ] .
I ( r ; z ) = | U ( r ; z ) | 2 = ( 2 π / λ z ) 2 I 0 | ε a a exp [ i Φ ( ρ ) ] J 0 ( 2 π r ρ / λ z ) ρ d ρ | 2 ,
Φ ( ρ ) = π λ ( 1 z 1 R ) ρ 2
Φ ( ρ ; z = R + Δ ) Φ ( ρ ; z = R Δ ) ,
Φ ( ρ ) π λ R 2 ( R z ) ρ 2 .
E ( r 0 ; z ) = 2 π 0 r 0 I ( r ; z ) rdr .
I ( r ; R ) = ( PA / λ 2 R 2 ) [ 2 J 1 ( π r ) / π r ] 2 ,
E ( r 0 ; R ) = P [ 1 J 0 2 ( π r 0 ) J 1 2 ( π r 0 ) ] ,
I ( r ; z ) = [ 2 R / z ( 1 ε 2 ) ] 2 | ε 1 exp [ i Φ ( ρ ) ] J 0 ( π r ρ ) ρ d ρ | 2 .
I ( r ; z ) = [ 8 / ( 1 ε 2 ) ] 0 1 τ ( γ ; z ) J 0 ( 2 π r γ ) γ d γ ,
τ ( γ ; z ) = ( π 2 / 2 ) 0 I ( r ; z ) J 0 ( 2 π r γ ) rdr .
τ ( γ ; z ) = n = 1 A n ( z ) J 0 ( π r n γ ) ,
A n ( z ) = [ 2 / J 1 2 ( π r n ) ] 0 1 τ ( γ ; z ) J 0 ( π r n γ ) γ d γ .
A n ( z ) = ( 1 ε 2 ) I ( r n / 2 ; z ) / 4 J 1 2 ( π r n ) .
τ ( γ ; z ) = [ ( 1 ε 2 ) / 4 ] n = 1 I ( r n / 2 ; z ) J 0 ( π r n γ ) / J 1 2 ( π r n ) .
z 2.84 m ,
z / R 1.9 × 10 3 .
W m ( z ) = ( λ / 2 π ) Φ ( a ) = ( N / 2 ) ( R / z 1 ) λ .
I ( z ) = I 0 { cos 2 α 1 + cos 2 α 2 2 cos α 1 cos α 2 cos [ 2 π ( s 1 s 2 ) / λ ] } .
I ( z ) = 4 I 0 sin 2 [ π a 2 ( 1 ε 2 ) / 2 λ z ) ] .
z = a 2 ( 1 ε 2 ) / λ ( 2 n + 1 ) , n = 0 , 1 , 2 , .
z = a 2 ( 1 ε 2 ) / 2 λ ( n + 1 ) , n = 0 , 1 , 2 , .
z D 2 ( 1 ε 2 ) / λ ,
I ( r ; z ) = ( 2 π / λ z ) 2 I 0 | ε a a exp ( π i ρ 2 / λ z ) J 0 ( 2 π r ρ / λ z ) ρ d ρ | 2 .
Φ ( ρ ) = π ρ 2 / λ z .
I ( r ; z ) ( PA / λ 2 z 2 ) [ 2 J 1 ( π r ) / π r ] 2 , z D 2 / λ ,
E ( r 0 ) SP [ 1 J 0 2 ( π r 0 ) J 1 2 ( π r 0 ) ] .

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