Abstract

Simple asymptotic expressions based on diffraction and the method of stationary phase are derived for the axial line images produced by generalized axicons. It is shown that many of the previously numerically established results concerning the logarithmic uniform-intensity axicons readily follow from these expressions. The role of the radial and azimuthal hologram variations in axicon image formation is elucidated. In particular, the generation of hollow uniform-intensity axicon lines of finite length is demonstrated. Linear holographic axicons are shown to produce a uniform axial line of high transverse definition if the illumination intensity is suitably varied along the radial direction. The connection of axicon images with nondiffracting wave fields is also discussed.

© 1996 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  14. J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, 1986), Part III.
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    [CrossRef]
  24. S. De Silvestri, V. Magni, O. Svelto, G. Valentini, “Lasers with super-Gaussian mirrors,” IEEE J. Quantum Electron. 26, 1500–1509 (1990).
    [CrossRef]
  25. F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
    [CrossRef]
  26. A. Vasara, J. Turunen, A. T. Friberg, “General diffraction-free fields produced by computer-generated holograms,” in Holographic Systems, Components, and Applications II, IEE Conference Publication No. 311 (Institution of Electrical Engineers, London, 1989), pp. 85–89.
  27. Z. Jaroszewicz, J. F. Roman Dopazo, “Polychromatic illumination of logarithmic annular-aperture diffractive axicons,” in Diffractive Optics: Design, Fabrication, and Applications, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 228–231.

1995

S. Yu. Popov, A. T. Friberg, “Linear axicons in partially coherent light,” Opt. Eng. 34, 2567–2573 (1995).
[CrossRef]

1994

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
[CrossRef]

1993

1992

1991

1990

S. De Silvestri, V. Magni, O. Svelto, G. Valentini, “Lasers with super-Gaussian mirrors,” IEEE J. Quantum Electron. 26, 1500–1509 (1990).
[CrossRef]

1989

1988

1986

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

1982

Arfken, G.

G. Arfken, Mathematical Methods for Physicists (Academic, San Diego, Calif., 1985) Sec. 11.1.

Bara, S.

Born, M.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), Appendix III.

Caudrado, J. M.

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

Chew, W. C.

Cox, A. J.

D’Anna, J.

Davidson, N.

De Silvestri, S.

S. De Silvestri, V. Magni, O. Svelto, G. Valentini, “Lasers with super-Gaussian mirrors,” IEEE J. Quantum Electron. 26, 1500–1509 (1990).
[CrossRef]

Erdelyi, A.

A. Erdelyi, Asymptotic Expansions (Dover, New York, 1956), Sec. 2.9.

Fairchild, R. C.

R. C. Fairchild, J. R. Fienup, “Computer-originated aspheric holographic optical elements,” Opt. Eng. 21, 133–140 (1982).
[CrossRef]

Fienup, J. R.

R. C. Fairchild, J. R. Fienup, “Computer-originated aspheric holographic optical elements,” Opt. Eng. 21, 133–140 (1982).
[CrossRef]

Friberg, A. T.

S. Yu. Popov, A. T. Friberg, “Linear axicons in partially coherent light,” Opt. Eng. 34, 2567–2573 (1995).
[CrossRef]

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]

J. Turunen, A. Vasara, A. T. Friberg, “Holographic generation of diffraction-free beams,” Appl. Opt. 27, 3959–3962 (1988).
[CrossRef] [PubMed]

A. T. Friberg, S. Yu. Popov, “Partially coherently illuminated holographic axicons,” in Diffractive Optics: Design, Fabrication, and Applications, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 224–227.

A. Vasara, J. Turunen, A. T. Friberg, “General diffraction-free fields produced by computer-generated holograms,” in Holographic Systems, Components, and Applications II, IEE Conference Publication No. 311 (Institution of Electrical Engineers, London, 1989), pp. 85–89.

Friesem, A. A.

Gomez-Reino, C.

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

Gori, F.

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), formula 3.691, item 1.

Hasman, E.

Herman, R. M.

Jaroszewicz, Z.

Kolodziejczyk, A.

Magni, V.

S. De Silvestri, V. Magni, O. Svelto, G. Valentini, “Lasers with super-Gaussian mirrors,” IEEE J. Quantum Electron. 26, 1500–1509 (1990).
[CrossRef]

Marcuse, D.

D. Marcuse, Principles of Optical Fiber Measurement (Academic, New York, 1981), Sec. 4.7.

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), Sec. 7.3.

Perez, M. V.

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

Popov, S. Yu.

S. Yu. Popov, A. T. Friberg, “Linear axicons in partially coherent light,” Opt. Eng. 34, 2567–2573 (1995).
[CrossRef]

A. T. Friberg, S. Yu. Popov, “Partially coherently illuminated holographic axicons,” in Diffractive Optics: Design, Fabrication, and Applications, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 224–227.

Roman Dopazo, J. F.

Z. Jaroszewicz, J. F. Roman Dopazo, “Polychromatic illumination of logarithmic annular-aperture diffractive axicons,” in Diffractive Optics: Design, Fabrication, and Applications, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 228–231.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), formula 3.691, item 1.

Sherman, G. C.

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), Sec. 20.2.

Sochacki, J.

Stamnes, J. J.

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, 1986), Part III.

Staronski, L. R.

Svelto, O.

S. De Silvestri, V. Magni, O. Svelto, G. Valentini, “Lasers with super-Gaussian mirrors,” IEEE J. Quantum Electron. 26, 1500–1509 (1990).
[CrossRef]

Turunen, J.

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]

J. Turunen, A. Vasara, A. T. Friberg, “Holographic generation of diffraction-free beams,” Appl. Opt. 27, 3959–3962 (1988).
[CrossRef] [PubMed]

A. Vasara, J. Turunen, A. T. Friberg, “General diffraction-free fields produced by computer-generated holograms,” in Holographic Systems, Components, and Applications II, IEE Conference Publication No. 311 (Institution of Electrical Engineers, London, 1989), pp. 85–89.

Valentini, G.

S. De Silvestri, V. Magni, O. Svelto, G. Valentini, “Lasers with super-Gaussian mirrors,” IEEE J. Quantum Electron. 26, 1500–1509 (1990).
[CrossRef]

Vasara, A.

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]

J. Turunen, A. Vasara, A. T. Friberg, “Holographic generation of diffraction-free beams,” Appl. Opt. 27, 3959–3962 (1988).
[CrossRef] [PubMed]

A. Vasara, J. Turunen, A. T. Friberg, “General diffraction-free fields produced by computer-generated holograms,” in Holographic Systems, Components, and Applications II, IEE Conference Publication No. 311 (Institution of Electrical Engineers, London, 1989), pp. 85–89.

Wiggins, T. A.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), Appendix III.

Appl. Opt.

IEEE J. Quantum Electron.

S. De Silvestri, V. Magni, O. Svelto, G. Valentini, “Lasers with super-Gaussian mirrors,” IEEE J. Quantum Electron. 26, 1500–1509 (1990).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Acta

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

Opt. Commun.

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
[CrossRef]

Opt. Eng.

S. Yu. Popov, A. T. Friberg, “Linear axicons in partially coherent light,” Opt. Eng. 34, 2567–2573 (1995).
[CrossRef]

R. C. Fairchild, J. R. Fienup, “Computer-originated aspheric holographic optical elements,” Opt. Eng. 21, 133–140 (1982).
[CrossRef]

Opt. Lett.

Other

A. Vasara, J. Turunen, A. T. Friberg, “General diffraction-free fields produced by computer-generated holograms,” in Holographic Systems, Components, and Applications II, IEE Conference Publication No. 311 (Institution of Electrical Engineers, London, 1989), pp. 85–89.

Z. Jaroszewicz, J. F. Roman Dopazo, “Polychromatic illumination of logarithmic annular-aperture diffractive axicons,” in Diffractive Optics: Design, Fabrication, and Applications, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 228–231.

D. Marcuse, Principles of Optical Fiber Measurement (Academic, New York, 1981), Sec. 4.7.

A. Erdelyi, Asymptotic Expansions (Dover, New York, 1956), Sec. 2.9.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), Sec. 7.3.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975), Appendix III.

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), Sec. 20.2.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), formula 3.691, item 1.

A. T. Friberg, S. Yu. Popov, “Partially coherently illuminated holographic axicons,” in Diffractive Optics: Design, Fabrication, and Applications, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 224–227.

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, 1986), Part III.

G. Arfken, Mathematical Methods for Physicists (Academic, San Diego, Calif., 1985) Sec. 11.1.

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Figures (5)

Fig. 1
Fig. 1

Illustration of the geometry and notation relating to a uniformly illuminated, rotationally symmetric, generalized holographic axicon; the designed axial intensity profile is shown in the insert.

Fig. 2
Fig. 2

Variation of the asymptotically obtained image radius within the focal region for three representative apodized annular-aperture logarithmic axicons. In all cases R = 5 mm, d1 = 100 mm, and λ = 0.6328 μm.

Fig. 3
Fig. 3

Uniform-intensity axial line image produced by an apodized annular-aperture logarithmic axicon. The intensity profile is the leading asymptotic contribution, calculated from Eqs. (24) and (26), with d1 = 100 mm, d2 = 200 mm, r = 8 mm, R = 16 mm, and Δ = 7.5 mm−1. The wavelength is λ = 0.6328 μm, and I0 = 1.

Fig. 4
Fig. 4

Uniform-intensity axially hollow line image generated by an apodized annular-aperture logarithmic axicon. The intensity profile is the stationary-point contribution given by relation (17) with n = 1. Other parameters are the same as in Fig. 3.

Fig. 5
Fig. 5

On-axis intensity distribution generated by an apodized annular-aperture linear axicon. The solid curve corresponds to the stationary-point result of Eq. (32), while the dashed curve accounts in addition for the modified hologram amplitude transmittance of Eq. (33). The system parameters are as in Fig. 3.

Equations (33)

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t ( ρ ) = h ( ρ ) exp [ i k φ ( ρ ) ] ,
E ( ρ , ϕ , z ) = - i k E 0 2 π z exp [ i k ( z + ρ 2 / 2 z ) ] × 0 d ρ ρ T ( ρ ) exp { i k [ ρ 2 / 2 z + φ ( ρ ) ] } × 0 2 π d ϕ A ( ϕ ) exp [ - i k ρ ρ cos ( ϕ - ϕ ) / z ] .
A ( ϕ ) = n = - a n exp ( i n ϕ ) ,
a n = 1 2 π 0 2 π A ( ϕ ) exp ( - i n ϕ ) d ϕ .
exp ( i z cos θ ) = n = - i n J n ( z ) exp ( i n θ ) ,
0 2 π A ( ϕ ) exp [ - i k ρ ρ cos ( ϕ - ϕ ) / z ] d ϕ = 2 π n = - a n ( - i ) n exp ( i n ϕ ) J n ( k ρ ρ / z ) .
E ( ρ , ϕ , z ) = exp [ i k ( z + ρ 2 / 2 z ) ] × n = - a n ( - i ) n exp ( i n ϕ ) V n ( ρ , z ) ,
V n ( ρ , z ) = - i k E 0 z 0 ρ T ( ρ ) J n ( k ρ ρ / z ) × exp { i k [ ρ 2 / 2 z + φ ( ρ ) ] } d ρ .
V n ( ρ , z ) ~ - i k E 0 z V sp ( ρ , z ) ,
V sp ( ρ , z ) = [ 2 π k ψ ( 2 ) ( ρ c , z ) ] 1 / 2 exp [ i ( π / 4 ) ] H n ( ρ c , ρ , z ) × exp [ i k ψ ( ρ c , z ) ] ,
H n ( ρ , ρ , z ) = ρ T ( ρ ) J n ( k ρ ρ / z ) ,
ψ ( ρ , z ) = ρ 2 / 2 z + φ ( ρ ) ,
E ( ρ , ϕ , z ) ~ exp [ i k ( z + ρ 2 / 2 ) ] ( - i E 0 z ) [ 2 π k ψ ( 2 ) ( ρ c , z ) ] 1 / 2 × exp ( i π / 4 ) ρ c T ( ρ c ) exp [ i k ψ ( ρ c , z ) ] × n = - a n ( - i ) n exp ( i n ϕ ) J n ( k ρ ρ c / z ) ,
E ( ρ , ϕ , z ) ~ E 0 z exp [ i k ( z + ρ 2 / 2 ) ] [ k 2 π ψ ( 2 ) ( ρ c , z ) ] 1 / 2 × ρ c T ( ρ c ) exp { i [ k ψ ( ρ c , z ) - π / 4 ] } × 0 2 π A ( ϕ ) exp [ - i k ρ ρ c cos ( ϕ - ϕ ) / z ] d ϕ .
I ( ρ , ϕ , z ) ~ I 0 [ k ρ c 2 T 2 ( ρ c ) 2 π z 2 ψ ( 2 ) ( ρ c , z ) ] | 0 2 π A ( ϕ ) × exp [ - i k ρ ρ c cos ( ϕ - ϕ ) / z ] d ϕ | 2 ,
I n ( ρ , z ) = I 0 ( k / z ) 2 | 0 ρ T ( ρ ) J n ( k ρ ρ c / z ) × exp { i k [ ρ 2 / 2 z + φ ( ρ ) ] } d ρ | 2 ,
I sp , n ( ρ , z ) ~ I 0 [ 2 π k ρ c 2 T 2 ( ρ c ) z 2 ψ ( 2 ) ( ρ c , z ) ] J n 2 ( k ρ ρ c z ) .
φ ( ρ ) = - 1 2 a log { 2 a [ a 2 ρ 4 + ( 1 + 2 a d 1 - 2 a 2 r 2 ) ρ 2 + d 1 2 - 2 a d 1 r 2 + a 2 r 4 ] 1 / 2 + 2 a 2 ρ 2 + 1 + 2 a d 1 - 2 a 2 r 2 } ,
φ ( ρ ) = - 1 2 a log [ d 1 + a ( ρ 2 - r 2 ) ] .
ψ ( ρ , z ) = ρ 2 / 2 z - ( 2 a ) - 1 log [ d 1 + a ( ρ 2 - r 2 ) ] ,
ρ c = [ r 2 + ( z - d 1 ) / a ] 1 / 2 .
ψ ( 2 ) ( ρ , z ) = 1 z - d 1 - a ( ρ 2 + r 2 ) [ d 1 + a ( ρ 2 - r 2 ) ] 2 ,
ψ ( 2 ) ( ρ c , z ) = 2 ( z - d 1 + a r 2 ) / z 2 = 2 a ρ c 2 / z 2 .
I sp , 0 ( ρ , z ) = I 0 ( π k ) ( R 2 - r 2 ) ( d 2 - d 1 ) - 1 × T 2 ( r 2 + z - d 1 d 2 - d 1 ( R 2 - r 2 ) ) × J 0 2 ( k ρ z r 2 + z - d 1 d 2 - d 1 ( R 2 - r 2 ) ) ,
I sp , 0 ( 0 , z ) I 0 ( π k ) ( R 2 - r 2 d 2 - d 1 ) ,
T ( ρ ) = { 0.5 + arctan [ Δ ( ρ - r ] / π } × { 0.5 + arctan [ Δ ( R - ρ ) / π } ,
φ ( ρ ) = - 1 1 + a [ ( 1 + a ) ρ 2 + d 1 2 - a r 2 ] 1 / 2 ,
φ ( ρ ) = - 1 a [ d 1 2 + a ( ρ 2 - r 2 ) ] 1 / 2 .
ρ c = [ r 2 + ( z 2 - d 1 2 ) / a ] 1 / 2
ψ ( 2 ) ( ρ , z ) = 1 z - d 1 2 - a r 2 [ d 1 2 + a ( ρ 2 - r 2 ) ] 3 / 2 ,
ψ ( 2 ) ( ρ c , z ) = ( z 2 - d 1 2 + a r 2 ) / z 3 = a ρ c 2 / z 3 .
I sp , 0 ( ρ , z ) = I 0 ( 2 π k ) ( R 2 - r 2 ) ( d 2 2 - d 1 2 ) - 1 z × T 2 ( r 2 + z 2 - d 1 2 d 2 2 - d 1 2 ( R 2 - r 2 ) ) × J 0 2 ( k ρ z r 2 + z 2 - d 1 2 d 2 2 - d 1 2 ( R 2 - r 2 ) ) ,
T ( ρ ) = d 1 / z T ( ρ ) = [ 1 + a ( ρ 2 - r 2 ) / d 1 2 ] - 1 / 4 T ( ρ ) ,

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