Abstract

Design considerations are presented for pairs of spherically aberrating elements that produce diffractionless beams of high efficiency and nearly constant size and intensity over specified ranges of axial position. An approximate design, assuming an aberration quadratic in radial ray positions, is followed by a final lens (mirror) specification that is verified through meridional ray tracing. This then provides an accurate determination of beam characteristics. Examples are presented that represent a variety of applications. As a design aid, a simple prescription is given for generating families of substantially different Bessel-like beams from any given pair of elements under small changes in element separation. Pattern sizes and ranges are compared with those of Gaussian beams, shadow lengths are examined for Bessel-type beams, and beam efficiencies are fully analyzed.

© 1994 Optical Society of America

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References

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  1. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
    [Crossref]
  2. G. Indebetouw, “Nondiffracting optical fields: some remarks on their analysis and synthesis,” J. Opt. Soc. Am. A 6, 150–152 (1989); A. J. Cox, D. C. Dibble, “Nondiffracting beam from a spatially filtered Fabry–Perot resonator,” J. Opt. Soc. Am. A 9, 282–286 (1992).
    [Crossref]
  3. A. Vasara, J. Turunen, A. T. Friberg, “Realization of general nondiffracting beams with computer-generated holograms,” J. Opt. Soc. Am. A 6, 1748–1754 (1989).
    [Crossref] [PubMed]
  4. J. H. McLeod, “The axicon: a new type of optical element,” J. Opt. Soc. Am. 44, 592–597 (1954); M. V. Perez, C. Gomez-Reino, J. M. Cuadrado, “Diffraction patterns and zone plates produced by thin linear axicons,” Opt. Acta 33, 1161–1177 (1986).
    [Crossref]
  5. R. M. Herman, T. A. Wiggins, “Production and uses of diffractionless beams,” J. Opt. Soc. Am. A 8, 932–942 (1991).
    [Crossref]
  6. R. M. Herman, T. A. Wiggins, “Apodization of diffractionless beams,” Appl. Opt. 31, 5913–5915 (1992).
    [Crossref] [PubMed]
  7. A. J. Cox, J. D’Anna, “Constant-axial-intensity nondiffracting beams,” Opt. Lett. 17, 232–234 (1992).
    [Crossref] [PubMed]
  8. P. L. Overfelt, C. S. Kenney, “Comparison of the propagation characteristics of Bessel, Bessel-Gauss, and Gaussian beams diffracted by a circular aperture,” J. Opt. Soc. Am. A 8, 732–745 (1991).
    [Crossref]
  9. F. A. Jenkins, H. E. White, Fundamentals of Optics, 3rd ed. (McGraw-Hill, New York, 1957), Chap. 9, p. 135.
  10. P. Sprangle, B. Hafizi, “Comment on nondiffracting beams,” Phys. Rev. Lett. 66, 837 (1991); B. Hafizi, P. Sprangle, “Diffraction effects in directed radiation beams,” J. Opt. Soc. Am. A 8, 706–717 (1991).
    [Crossref] [PubMed]
  11. M. R. Lapointe, “Review of non-diffracting Bessel beam experiments,” Opt. Laser Technol. 24, 315–321 (1992).
    [Crossref]
  12. G. Scott, N. McArdle, “Efficient generation of nearly diffraction-free beams using an axicon,” Opt. Eng. 31, 2640–2643 (1992).
    [Crossref]
  13. S. A. Self, “Focusing of spherical Gaussian beams,” Appl. Opt. 22, 658–661 (1983); R. M. Herman, J. Pardo, T. A. Wiggins, “Diffraction and focusing of Gaussian beams,” Appl. Opt. 24, 1346–1354 (1985).
    [Crossref] [PubMed]
  14. L. R. Staronski, J. Sochacki, Z. Jaroszewicz, A. Kolodziejczyk, “Design of uniform-intensity refractive axicons,” Opt. Eng. 31, 516–521 (1992).
    [Crossref]

1992 (5)

R. M. Herman, T. A. Wiggins, “Apodization of diffractionless beams,” Appl. Opt. 31, 5913–5915 (1992).
[Crossref] [PubMed]

A. J. Cox, J. D’Anna, “Constant-axial-intensity nondiffracting beams,” Opt. Lett. 17, 232–234 (1992).
[Crossref] [PubMed]

M. R. Lapointe, “Review of non-diffracting Bessel beam experiments,” Opt. Laser Technol. 24, 315–321 (1992).
[Crossref]

G. Scott, N. McArdle, “Efficient generation of nearly diffraction-free beams using an axicon,” Opt. Eng. 31, 2640–2643 (1992).
[Crossref]

L. R. Staronski, J. Sochacki, Z. Jaroszewicz, A. Kolodziejczyk, “Design of uniform-intensity refractive axicons,” Opt. Eng. 31, 516–521 (1992).
[Crossref]

1991 (3)

1989 (2)

1987 (1)

1983 (1)

1954 (1)

Cox, A. J.

D’Anna, J.

Durnin, J.

Friberg, A. T.

Hafizi, B.

P. Sprangle, B. Hafizi, “Comment on nondiffracting beams,” Phys. Rev. Lett. 66, 837 (1991); B. Hafizi, P. Sprangle, “Diffraction effects in directed radiation beams,” J. Opt. Soc. Am. A 8, 706–717 (1991).
[Crossref] [PubMed]

Herman, R. M.

Indebetouw, G.

Jaroszewicz, Z.

L. R. Staronski, J. Sochacki, Z. Jaroszewicz, A. Kolodziejczyk, “Design of uniform-intensity refractive axicons,” Opt. Eng. 31, 516–521 (1992).
[Crossref]

Jenkins, F. A.

F. A. Jenkins, H. E. White, Fundamentals of Optics, 3rd ed. (McGraw-Hill, New York, 1957), Chap. 9, p. 135.

Kenney, C. S.

Kolodziejczyk, A.

L. R. Staronski, J. Sochacki, Z. Jaroszewicz, A. Kolodziejczyk, “Design of uniform-intensity refractive axicons,” Opt. Eng. 31, 516–521 (1992).
[Crossref]

Lapointe, M. R.

M. R. Lapointe, “Review of non-diffracting Bessel beam experiments,” Opt. Laser Technol. 24, 315–321 (1992).
[Crossref]

McArdle, N.

G. Scott, N. McArdle, “Efficient generation of nearly diffraction-free beams using an axicon,” Opt. Eng. 31, 2640–2643 (1992).
[Crossref]

McLeod, J. H.

Overfelt, P. L.

Scott, G.

G. Scott, N. McArdle, “Efficient generation of nearly diffraction-free beams using an axicon,” Opt. Eng. 31, 2640–2643 (1992).
[Crossref]

Self, S. A.

Sochacki, J.

L. R. Staronski, J. Sochacki, Z. Jaroszewicz, A. Kolodziejczyk, “Design of uniform-intensity refractive axicons,” Opt. Eng. 31, 516–521 (1992).
[Crossref]

Sprangle, P.

P. Sprangle, B. Hafizi, “Comment on nondiffracting beams,” Phys. Rev. Lett. 66, 837 (1991); B. Hafizi, P. Sprangle, “Diffraction effects in directed radiation beams,” J. Opt. Soc. Am. A 8, 706–717 (1991).
[Crossref] [PubMed]

Staronski, L. R.

L. R. Staronski, J. Sochacki, Z. Jaroszewicz, A. Kolodziejczyk, “Design of uniform-intensity refractive axicons,” Opt. Eng. 31, 516–521 (1992).
[Crossref]

Turunen, J.

Vasara, A.

White, H. E.

F. A. Jenkins, H. E. White, Fundamentals of Optics, 3rd ed. (McGraw-Hill, New York, 1957), Chap. 9, p. 135.

Wiggins, T. A.

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

Opt. Eng. (2)

L. R. Staronski, J. Sochacki, Z. Jaroszewicz, A. Kolodziejczyk, “Design of uniform-intensity refractive axicons,” Opt. Eng. 31, 516–521 (1992).
[Crossref]

G. Scott, N. McArdle, “Efficient generation of nearly diffraction-free beams using an axicon,” Opt. Eng. 31, 2640–2643 (1992).
[Crossref]

Opt. Laser Technol. (1)

M. R. Lapointe, “Review of non-diffracting Bessel beam experiments,” Opt. Laser Technol. 24, 315–321 (1992).
[Crossref]

Opt. Lett. (1)

Phys. Rev. Lett. (1)

P. Sprangle, B. Hafizi, “Comment on nondiffracting beams,” Phys. Rev. Lett. 66, 837 (1991); B. Hafizi, P. Sprangle, “Diffraction effects in directed radiation beams,” J. Opt. Soc. Am. A 8, 706–717 (1991).
[Crossref] [PubMed]

Other (1)

F. A. Jenkins, H. E. White, Fundamentals of Optics, 3rd ed. (McGraw-Hill, New York, 1957), Chap. 9, p. 135.

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Figures (3)

Fig. 1
Fig. 1

Shape factor q for lenses of index equal to 1.517 and minimum thickness as a function of BF c 2. Shown are values for the lens’ f-number for which the maximum incidence angle of refraction or the emergent refraction angle is 1 rad.

Fig. 2
Fig. 2

Beam size S as a function of position Z from the second lens. The central curve is from the data of example 3, for which Q = 28.95 cm. The other curves are for changes in Q as indicated, with all other parameters held fixed.

Fig. 3
Fig. 3

Ray construction for determining output Bessel-type beam intensities from the second lens.

Equations (54)

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S = π / ( k sin β ) .
F = F c ( 1 - B ρ 2 ) ,
ρ 2 = ρ 1 ( F 1 - Q ) / F 1 ,
q = ( r 2 + r 1 ) / ( r 2 - r 1 ) ,
q 0.7 ± 0.9 ( B F c 2 - 1.0 ) 1 / 2 ,
B F c 2 1.0 + 1.2 ( q - 0.7 ) 2 .
η = - ( ρ 2 / f ) ( d f / d ρ ) [ 1 - ( ρ / f ) ( d f / d ρ ) ] ,
ξ = f [ 1 - ( ρ / f ) ( d f / d ρ ) ] .
ξ 1 + ξ 2 = Q .
η 2 / η 1 = ξ 2 / ( Q - ξ 1 ) .
η 2 B ρ 3 / ( 1 + B ρ 2 ) ,
ξ F c ( 1 - B ρ 2 ) 2 / ( 1 + B ρ 2 ) .
ξ F c ( 1 - B ρ 2 ) 3 .
S - π F 1 k ρ 1 F 2 x 2 ,
x 2 = Q - F 1 - F 2 ,
Z = F 2 2 / x 2 + F 2 ,
Q = F 1 ( 1 - π k ρ 1 Z S ) ,
ρ 2 = ( π / k ) ( Z / S ) .
F 2 c = [ ( ρ 22 2 ξ 21 3 - ρ 21 2 ξ 22 3 ) / ( ρ 22 2 - ρ 21 2 ) ] 3 ,
B 2 = ( ξ 21 3 - ξ 22 3 ) / ( ρ 22 2 ξ 21 3 - ρ 21 2 ξ 22 3 ) .
B 2 / B 1 ( - F 1 c / F 2 c ) p
F 1 c / F 1 c = B 1 / B 1 = Z i / ( Z i + F 1 c ) ,
f 1 c / f 1 c = ( q 1 / q 1 ) 2 / 3 = Z i / ( Z i + f 1 c ) .
( - 2.50 , 2.50 , 0.30 , 0.34 ) 0.93 ( 2.944 , 0.52 ) ( - 2.36 , - 2.835 ) 1.18 ( 2.918 , - 0.720 ) ( 1.330 , 0.20 , 0.636 , 0.42 ) 1.18 ( 1.754 , 0.41 , - 10.776 , 0.78 ) .
( 30.00 , 0.00122 , 3.50 , 4.10 ) 25.75 ( - 5.253 , 0.927 ) ( 30.334 , 0.496 ) 26.60 ( - 4.584 , - 5.45 ) ( 20.966 , 0.935 , - 62.233 , 4.70 ) 26.60 ( - 0.735 , 0.20 , - 1.065 , 0.42 ) .
( - 9.20 , 0.2222 , 0.50 , 0.60 ) 28.87 ( 37.792 , 0.002197 ) ( - 9.02 , - 3.29 ) 28.95 ( 37.799 , - 0.6425 ) ( 4.073 , 0.20 , 2.174 , 0.80 ) 28.95 ( 23.796 , 0.568 , - 109.326 , 3.60 ) .
( - 10.00 , 0.0288 , 0.60 , 0.80 ) 200 ( 210 , 2.835 × 10 - 6 ) ( - 10.00 , - 0.50 ) 190 ( mirror ' s radius of curvature = 400 ) ( - 20.68 , 0.20 , 6.893 , 1.10 ) 190 ( radius of curvature = 400 , ρ m = 22.6 ) .
S = π Z k ρ 2 .
W B = 1.75 Z k ρ 2 .
w = w 0 [ 1 + ( Z / Z R ) 2 ] 1 / 2 2 Z k w 0 ,
W G = w 0 = 2 Z k w .
Z B = k N S 2 π ,
Z B π N Z 2 k ρ 2 2 .
Z R = k w 0 2 2 2 Z 2 k w 2 .
L R ( β + λ / 4 R ) .
S λ 2 ( B 1 F 1 c 3 Q - F 1 c - F 2 c ) 1 / 2 F 2 c ( Q - F 1 c - F 2 c ) ( ζ 3 ζ - 1 ) 1 / 2 ,
ζ = z ( Q - F 1 c - F 2 c ) F 2 c 2 .
( z + δ z ) = f 2 ( ρ 2 ) 2 x 2 + δ Q .
( z + δ z ) z [ 1 + z δ Q f 2 ( ρ 2 ) 2 ] - 1 .
( S + δ S ) S [ 1 + z δ Q f 2 ( ρ 2 ) 2 ] - 1 .
I 1 ( ρ 1 ) = 2 P exp [ - 2 ( ρ 1 / w 1 ) 2 ] π w 1 2 ,
P ring = 4 π 2 Z I 2 ( ρ 2 ) k ( d ρ 2 d r 2 ) .
d ρ 2 d r 2 = Z ρ 2 d ρ 2 d Z .
Z ρ 2 d ρ 2 d Z = S d d Z ( Z S ) = 1 - Z S d S d Z ,
P ring = 4 π 2 Z I 2 ( ρ 2 ) k ( 1 - Z S d S d Z ) .
I 2 ( ρ 2 ) 2 P exp [ - 2 ( ρ 2 / w 2 ) 2 ] π w 2 2 ,
w 2 | f 2 c f 1 c | w 1 .
ɛ = 8 π Z k w 2 2 exp [ - 2 ( ρ 2 / w 2 ) 2 ] ( 1 - Z S d S d Z ) .
ɛ = 8 π Z k w 2 2 exp [ - 2 ( π Z / k S w 2 ) 2 ] ( 1 - Z S d S d Z ) .
ɛ ( Z ; Z w ) = 8 k S w 2 π Z w 2 exp [ - 2 ( Z S w / S Z w ) 2 ] Z ( 1 - Z S d S d Z ) .
ɛ 8 k S w 2 π Z w ( Z Z w ) exp [ - 2 ( Z / Z w ) 2 ] .
Z S d S d Z = ζ - / 2 3 ζ - 1 ,
ɛ ( Z ; Z w ) ( 4 k S w 2 π Z w ) Z Z w exp [ - 2 ( Z S w / S Z w ) 2 ] 1 ( ζ - 1 ) .
ɛ ( Z ) max = 2 k S 2 π e Z ( ζ - 1 ) .

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