Abstract

We introduce a powerful but simple methodology for numerical modeling of the far field of a hollow searchlight laser beam that is produced by passing a laser beam through a reflaxicon. Such a beam can be used in remote sensing as a space beacon. The far field is described by a Fourier–Bessel transform over an aperture function that includes a conical phase term introduced by the reflaxicon. Computations of the far field of the reflaxicon are difficult. The conventional approach for calculating the far field of such ideal aperture distributions as a plane wave or a Gaussian beam is to find exact solutions in the form of hypergeometric series and determine their asymptotic approximations for large values of some parameter. This approach does not extend to more complicated aperture distributions. We modify the transform by using the asymptotic form for the Bessel function as well as by limiting this form to include only the low-frequency (difference frequency) term. This approach is easily related to a one-dimensional Fourier transform of the aperture distribution, and thus numerical evaluations that make use of this approach can use the fast Fourier transform.

© 1997 Optical Society of America

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References

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  1. W. R. Edmonds, “The reflaxicon, a new reflective optical element, and some applications,” Appl. Opt. 12, 1940–1945 (1973).
    [CrossRef]
  2. J. H. Mcleod, “The axicon: a new type of optical element,” J. Opt. Soc. Am. 44, 592–597 (1954).
    [CrossRef]
  3. P. A. Belanger, M. Rioux, “Diffraction ring pattern at the focal plane of a spherical lens–axicon doublet,” Can. J. Phys. 54, 1774–1780 (1976).
    [CrossRef]
  4. G. Indebetouw, “Nondiffracting optical fields: some remarks on their analysis and synthesis,” J. Opt. Soc. Am. 6, 150–152 (1989).
    [CrossRef]
  5. M. Born, E. Wolf, Principles of Optics (Pergamon, Toronto, 1975).
  6. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1981).
  7. P. A. Belanger, M. Rioux, “Ring pattern of a lens–axicon doublet illuminated by a Gaussian beam,” Appl. Opt. 17, 1080–1086 (1978).
    [CrossRef]
  8. M. Abramowitz, I. Stigan, eds., Handbook of Mathematical Functions with Formulas (National Bureau of Standards, Washington, D.C., 1964).
  9. Y. L. Luke, Mathematical Functions and Their Approximations (Academic, New York, 1975).
  10. O. I. Smokty, A. V. Fabrikov, “Modeling the diffraction field in axicon–lens system,” Izv. Vyssh. Uchebn. Zaved. Fiz. 12, 36–41 (1987) (in Russian).
  11. M. V. Perez, C. Gomez-Rieno, J. M. Cuadrado, “Diffraction patterns and zone plates produced by thin linear axicon,” Opt. Acta 33, 1161–1176 (1986).
    [CrossRef]
  12. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), No. 7.340.
  13. O. I. Smokty, A. V. Fabrikov, “Modelling the diffraction field in axicon–lens system illuminated by Gaussian beam,” in Problems of Data Processing and Integral Automation of the Production in Transactions of Leningrad Institute of Information Science, Academy of Sciences of the USSR (Nauka, Leningrad, 1990), pp. 186–191 (in Russian).

1989 (1)

G. Indebetouw, “Nondiffracting optical fields: some remarks on their analysis and synthesis,” J. Opt. Soc. Am. 6, 150–152 (1989).
[CrossRef]

1987 (1)

O. I. Smokty, A. V. Fabrikov, “Modeling the diffraction field in axicon–lens system,” Izv. Vyssh. Uchebn. Zaved. Fiz. 12, 36–41 (1987) (in Russian).

1986 (1)

M. V. Perez, C. Gomez-Rieno, J. M. Cuadrado, “Diffraction patterns and zone plates produced by thin linear axicon,” Opt. Acta 33, 1161–1176 (1986).
[CrossRef]

1978 (1)

1976 (1)

P. A. Belanger, M. Rioux, “Diffraction ring pattern at the focal plane of a spherical lens–axicon doublet,” Can. J. Phys. 54, 1774–1780 (1976).
[CrossRef]

1973 (1)

1954 (1)

Belanger, P. A.

P. A. Belanger, M. Rioux, “Ring pattern of a lens–axicon doublet illuminated by a Gaussian beam,” Appl. Opt. 17, 1080–1086 (1978).
[CrossRef]

P. A. Belanger, M. Rioux, “Diffraction ring pattern at the focal plane of a spherical lens–axicon doublet,” Can. J. Phys. 54, 1774–1780 (1976).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Toronto, 1975).

Cuadrado, J. M.

M. V. Perez, C. Gomez-Rieno, J. M. Cuadrado, “Diffraction patterns and zone plates produced by thin linear axicon,” Opt. Acta 33, 1161–1176 (1986).
[CrossRef]

Edmonds, W. R.

Fabrikov, A. V.

O. I. Smokty, A. V. Fabrikov, “Modeling the diffraction field in axicon–lens system,” Izv. Vyssh. Uchebn. Zaved. Fiz. 12, 36–41 (1987) (in Russian).

O. I. Smokty, A. V. Fabrikov, “Modelling the diffraction field in axicon–lens system illuminated by Gaussian beam,” in Problems of Data Processing and Integral Automation of the Production in Transactions of Leningrad Institute of Information Science, Academy of Sciences of the USSR (Nauka, Leningrad, 1990), pp. 186–191 (in Russian).

Gomez-Rieno, C.

M. V. Perez, C. Gomez-Rieno, J. M. Cuadrado, “Diffraction patterns and zone plates produced by thin linear axicon,” Opt. Acta 33, 1161–1176 (1986).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), No. 7.340.

Indebetouw, G.

G. Indebetouw, “Nondiffracting optical fields: some remarks on their analysis and synthesis,” J. Opt. Soc. Am. 6, 150–152 (1989).
[CrossRef]

Luke, Y. L.

Y. L. Luke, Mathematical Functions and Their Approximations (Academic, New York, 1975).

Mcleod, J. H.

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1981).

Perez, M. V.

M. V. Perez, C. Gomez-Rieno, J. M. Cuadrado, “Diffraction patterns and zone plates produced by thin linear axicon,” Opt. Acta 33, 1161–1176 (1986).
[CrossRef]

Rioux, M.

P. A. Belanger, M. Rioux, “Ring pattern of a lens–axicon doublet illuminated by a Gaussian beam,” Appl. Opt. 17, 1080–1086 (1978).
[CrossRef]

P. A. Belanger, M. Rioux, “Diffraction ring pattern at the focal plane of a spherical lens–axicon doublet,” Can. J. Phys. 54, 1774–1780 (1976).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), No. 7.340.

Smokty, O. I.

O. I. Smokty, A. V. Fabrikov, “Modeling the diffraction field in axicon–lens system,” Izv. Vyssh. Uchebn. Zaved. Fiz. 12, 36–41 (1987) (in Russian).

O. I. Smokty, A. V. Fabrikov, “Modelling the diffraction field in axicon–lens system illuminated by Gaussian beam,” in Problems of Data Processing and Integral Automation of the Production in Transactions of Leningrad Institute of Information Science, Academy of Sciences of the USSR (Nauka, Leningrad, 1990), pp. 186–191 (in Russian).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Toronto, 1975).

Appl. Opt. (2)

Can. J. Phys. (1)

P. A. Belanger, M. Rioux, “Diffraction ring pattern at the focal plane of a spherical lens–axicon doublet,” Can. J. Phys. 54, 1774–1780 (1976).
[CrossRef]

Izv. Vyssh. Uchebn. Zaved. Fiz. (1)

O. I. Smokty, A. V. Fabrikov, “Modeling the diffraction field in axicon–lens system,” Izv. Vyssh. Uchebn. Zaved. Fiz. 12, 36–41 (1987) (in Russian).

J. Opt. Soc. Am. (2)

G. Indebetouw, “Nondiffracting optical fields: some remarks on their analysis and synthesis,” J. Opt. Soc. Am. 6, 150–152 (1989).
[CrossRef]

J. H. Mcleod, “The axicon: a new type of optical element,” J. Opt. Soc. Am. 44, 592–597 (1954).
[CrossRef]

Opt. Acta (1)

M. V. Perez, C. Gomez-Rieno, J. M. Cuadrado, “Diffraction patterns and zone plates produced by thin linear axicon,” Opt. Acta 33, 1161–1176 (1986).
[CrossRef]

Other (6)

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), No. 7.340.

O. I. Smokty, A. V. Fabrikov, “Modelling the diffraction field in axicon–lens system illuminated by Gaussian beam,” in Problems of Data Processing and Integral Automation of the Production in Transactions of Leningrad Institute of Information Science, Academy of Sciences of the USSR (Nauka, Leningrad, 1990), pp. 186–191 (in Russian).

M. Born, E. Wolf, Principles of Optics (Pergamon, Toronto, 1975).

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1981).

M. Abramowitz, I. Stigan, eds., Handbook of Mathematical Functions with Formulas (National Bureau of Standards, Washington, D.C., 1964).

Y. L. Luke, Mathematical Functions and Their Approximations (Academic, New York, 1975).

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Figures (5)

Fig. 1
Fig. 1

Hollow searchlight laser-beam-forming arrangement: 1, input laser beam; 2, reflaxicon; 3, beam cross section at a great distance z from the input.

Fig. 2
Fig. 2

Intensity I(r) versus radial coordinate r of hollow searchlight laser beam. Calculations are performed with Eq. (10) for curve 1 and Eq. (9) for curve 2, for ψ1(r′) ≡ 1 and R1 = 25 cm, R2 = 30 cm, β = 1 × 10-3 rad, λ = 0.5 µm, z = 200 km; λ/R2β ≈ 1.7 × 10-3.

Fig. 3
Fig. 3

Intensity I(r) versus radial coordinate r of hollow searchlight laser beam. Calculations are performed with Eq. (10) for curve 1, Eq. (12) for curve 2, Eq. (13) for curve 3, and Eq. (9) for curve 4; z = 2000 km.

Fig. 4
Fig. 4

Far-field intensity distribution for uniform aperture function calculated in the LFA approximation for three cases, where I0 = 1, R1 = 0, λ = 0.6 µm, z = 1 × 106 m, and β, R2 are variable. (a) R2 = 10 cm, β = 1 × 10-4, λ/βR2 = 0.06 ≪1. Values for exact integral, Eq. (7): I(0) = 2.4795 × 10-7; I(2.10-4) = 3.71709 × 10-6; I(2.10-5) = 5.03139 × 10-10. (b) R2 = 10 cm, β = 1 × 10-5, λ/βR2 = 0.6 ≈ 1. Values for exact integral, Eq. (7): I(0) = 2.31035 × 10-5; I(2.10 -5) = 3.84864 × 10-5; I(2.10-6) = 4.53274 × 10-7. (c) R2 = 1000 cm, β = 1 × 10-6, λ/βR2 = 0.06 ≪ 1. Values for exact integral, Eq. (7): I(0) = 24.795; I(2.10-6) = 371.709; I(2.10-7) = 0.0503139.

Fig. 5
Fig. 5

Far-field intensity distribution for Gaussian amplitude aperture function calculated in the LFA approximation for three cases, where I0. = 1, R1 = 0, R2 = ∞, λ = 0.6 µm, z = 1 × 106 m, and β, W are variable. (a) W2 = 10 cm, β = 1 × 10-4, λ/βW = 0.06 ≪1. Values for exact integral, Eq. (7): I(0) = 5.70088 × 10-12; I(2.10 -4) = 3.128119202 × 10-6; I(1.10-5) = 5.870995151 × 10-12. (b) W = 10 cm, β = 1 × 10-5, λ/βW = 0.6 ≈ 1. Values for exact integral, Eq. (7): I(0) = 5.86002 × 10-8; I(2.10 -5) = 3.1330269845873 × 10-5; I(2.10-6) = 6.04928655687 × 10-8. (c) W = 100 cm, β = 1 × 10-5, λ/βW = 0.06 ≪ 1. Values for exact integral, Eq. (7): I(0) = 5.70088 × 10-8; I(2.10 -5) = 1.136475 × 10-3; I(2.10-6) = 5.87544 × 10-8. (The general form of the sequence could not be determined and the result may be incorrect.)

Equations (20)

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Γr=expiω0rcircr/R2-circr/R1,  ω0=2kβ=4πβ/λ,
circr/R2-circr/R1=1,if R1rR20,if r<R1 or r>R2,
ψ2r=Γrψ1r,
ψr=C0ψ2rexpik/2zr2rJ0ωrdr,  C=-ik/zexpikzexpik/2zr2,
ω=k/zr.
Ir=Sr2,
Sr=kzR1R2ψ1rexpik/2zr2×expiω0rrJ0ωrdr.
Sr=k/zR1R2ψrexpiω0rrJ0ωrdr.
J0ωr12πωr1/2expiωr-π/4+exp-iωr-π/4,
Sr=1zr0λ1/2R1R2ψ1rexpik2zr2r×exp-iΩrdr,
Sr=1zr0λ1/2R1R2ψ1rr exp-iΩrdr,
Ωω0-ω=2πλzr0-r,  r0=2βz.
Sr=A2  1F13/2, 5/2; iρ2-A1  1F13/2, 5/2; iρ1,  Ai=4I0Ri3/9zλr01/2,  ρi=ω-ω0Ri,  i=1, 2,
 1F13/2, 5/2; iρ=3201expiρtt dt.
R2R1x exp-iΩxdx=i exp-iΩR2R2-i exp-iΩR1R1Ω--11/4π erf-11/4Ω R2+-11/4π erf-11/4Ω R12Ω3/2.
Ir=Csin2ΩΔΩ2Δ2,  C=I0R2Δ2λzr0,  r0=2βz.
Sr=2-3/4Γ3/2B exp-ρ02/2D-3/2-i2 ρ0,
B=I0W2r0λz1/2Γ3/42,  ρ0=πWλzr-r0,
Sr=B 1F13/4, 1/2; -ρ02-2iΓ5/4Γ3/4ρ0  1F15/4, 3/2; -ρ02.
0x exp-x2/W2-iΩxdx=18iΩW23/2 exp-18Ω2W2×K3/4-18Ω2W2-K1/4-18Ω2W2.

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