Abstract

A simple interferometer for the characterization of axicon lenses is presented. The phase cone acquired by a wave propagating through an axicon, when interfered with a collinear reference wave, produces a nearly cylindrically symmetric self-referenced interference pattern from which the distortions of the axicon surface may be readily obtained. Comparison with two-dimensional off-axis interferometry is used to validate the self-referenced technique. The measurements are based on retrieval of the accrued spatial phase distribution from interference fringes with on- and off-axis reference beams and are found to be equivalent. We use the ellipticity of the phase maps to qualify axicon lenses, which are expected to exhibit radial symmetry and engage the self-referential capability of the on-axis method to derive deviation maps that characterize the surface quality of the axicons.

© 2008 Optical Society of America

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2007 (3)

2006 (1)

2003 (1)

M. de Angelis, S. De Nicola, P. Ferraro, A. Finizio, and G. Pierattini, “Test of a conical lens using a two-beam shearing interferometer,” Opt. Lasers Eng. 39, 155-163 (2003).
[CrossRef]

2002 (3)

T. Kololuoma, K. Kataja, S. Juuso, J. Aikio, and J. T. Rantala, “Fabrication and characterization of hybrid-glass-based axicons,” Opt. Eng. 41, 3136-3140 (2002).
[CrossRef]

Z. H. Ding, H. W. Ren, Y. H. Zhao, J. S. Nelson, and Z. P. Chen, “High-resolution optical coherence tomography over a large depth range with an axicon lens,” Opt. Lett. 27, 243-245(2002).
[CrossRef]

G. Indebetouw, “Properties of a scanning holographic microscope: improved resolution, extended depth-of-focus, and/or optical sectioning,” J. Mod. Opt. 49, 1479-1500 (2002).
[CrossRef]

2001 (1)

2000 (2)

J. Fan, E. Parra, I. Alexeev, K. Y. Kim, H. M. Milchberg, L. Y. Margolin, and L. N. Pyatnitskii, “Tubular plasma generation with a high-power hollow Bessel beam,” Phys. Rev. E 62, R7603-R7606 (2000).
[CrossRef]

J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297-301(2000).
[CrossRef]

1999 (1)

K. Ait-Ameur and F. Sanchez, “Gaussian beam conversion using an axicon,” J. Mod. Opt. 46, 1537-1548 (1999).

1998 (3)

J. Garcia-Marquez, D. Malacara-Hernandez, and M. Servin, “Analysis of interferograms with a spatial radial carrier or closed fringes and its holographic analogy,” Appl. Opt. 37, 7977-7982 (1998).
[CrossRef]

I. Manek, Y. B. Ovchinnikov, and R. Grimm, “Generation of a hollow laser beam for atom trapping using an axicon,” Opt. Commun. 147, 67-70 (1998).
[CrossRef]

J. Fan, T. R. Clark, and H. M. Milchberg, “Generation of a plasma waveguide in an elongated, high repetition rate gas jet,” Appl. Phys. Lett. 73, 3064-3066 (1998).
[CrossRef]

1997 (1)

1995 (1)

Z. Bouchal and M. Olivik, “Non-diffractive vector Bessel beams,” Opt. Eng. 42, 1555-1566 (1995).

1992 (2)

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

S. C. Tidwell, D. H. Ford, and W. D. Kimura, “Transporting and focusing radially polarized laser beams,” Opt. Eng. 31, 1527-1531 (1992).
[CrossRef]

1990 (2)

I. Golub and R. Tremblay, “Light focusing and guiding by an axicon-pair-generated tubular light beam,” J. Opt. Soc. Am. B 7, 1264-1267 (1990).
[CrossRef]

O. Ren and R. Birngruber, “Axicon: a new laser beam delivery system for corneal surgery,” IEEE J. Quantum Electron. 26, 2305-2308 (1990).
[CrossRef]

1988 (1)

1985 (1)

M. Couture and M. Piche, “Optical-pumping of infrared-lasers using an axicon geometry,” J. Opt. Soc. Am. A 2, P38 (1985).

1982 (1)

1981 (2)

1980 (1)

1978 (2)

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

M. Blanchard, Y. Dastous, G. Roy, and R. Tremblay, “Axicon lenses to produce very long plasmas for lasers,” J. Opt. Soc. Am. 68, 1389 (1978).

1965 (1)

1962 (1)

1954 (1)

Appl. Opt. (7)

Appl. Phys. Lett. (1)

J. Fan, T. R. Clark, and H. M. Milchberg, “Generation of a plasma waveguide in an elongated, high repetition rate gas jet,” Appl. Phys. Lett. 73, 3064-3066 (1998).
[CrossRef]

IEEE J. Quantum Electron. (1)

O. Ren and R. Birngruber, “Axicon: a new laser beam delivery system for corneal surgery,” IEEE J. Quantum Electron. 26, 2305-2308 (1990).
[CrossRef]

J. Mod. Opt. (2)

K. Ait-Ameur and F. Sanchez, “Gaussian beam conversion using an axicon,” J. Mod. Opt. 46, 1537-1548 (1999).

G. Indebetouw, “Properties of a scanning holographic microscope: improved resolution, extended depth-of-focus, and/or optical sectioning,” J. Mod. Opt. 49, 1479-1500 (2002).
[CrossRef]

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (1)

Opt. Commun. (3)

I. Manek, Y. B. Ovchinnikov, and R. Grimm, “Generation of a hollow laser beam for atom trapping using an axicon,” Opt. Commun. 147, 67-70 (1998).
[CrossRef]

J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177, 297-301(2000).
[CrossRef]

M. D. Wei, “Generation of bottle beam by focusing a super-Gaussian beam using a lens and an axicon,” Opt. Commun. 277, 19-23 (2007).
[CrossRef]

Opt. Eng. (4)

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

Z. Bouchal and M. Olivik, “Non-diffractive vector Bessel beams,” Opt. Eng. 42, 1555-1566 (1995).

T. Kololuoma, K. Kataja, S. Juuso, J. Aikio, and J. T. Rantala, “Fabrication and characterization of hybrid-glass-based axicons,” Opt. Eng. 41, 3136-3140 (2002).
[CrossRef]

S. C. Tidwell, D. H. Ford, and W. D. Kimura, “Transporting and focusing radially polarized laser beams,” Opt. Eng. 31, 1527-1531 (1992).
[CrossRef]

Opt. Express (1)

Opt. Lasers Eng. (1)

M. de Angelis, S. De Nicola, P. Ferraro, A. Finizio, and G. Pierattini, “Test of a conical lens using a two-beam shearing interferometer,” Opt. Lasers Eng. 39, 155-163 (2003).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. E (1)

J. Fan, E. Parra, I. Alexeev, K. Y. Kim, H. M. Milchberg, L. Y. Margolin, and L. N. Pyatnitskii, “Tubular plasma generation with a high-power hollow Bessel beam,” Phys. Rev. E 62, R7603-R7606 (2000).
[CrossRef]

Other (2)

J. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Co., 2005).

R. Halif and J. Flusser, “Numerically stable direct least-squares fitting of ellipses,” (Department of Software Engineering, Charles University,2000).

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Figures (7)

Fig. 1
Fig. 1

(a) Experimental setup used to measure axicon phase maps: BS, nonpolarizing beam splitter. (b) Geometry of a plane wave transmitted through an axicon.

Fig. 2
Fig. 2

Interference distributions measured for an axicon with an on-axis reference beam. (a) Source interferogram in Cartesian coordinates. (b) Pattern mapped to cylindrical coordinates. The arches result from clipping by the edge of the source interferogram.

Fig. 3
Fig. 3

Processing of the on-axis interferogram shown in Fig. 2b by use of a 1-D Fourier transform. (a) Log of the magnitude of the Fourier transform of the cylindrically mapped interferogram. The sideband to be isolated is indicated by the dashed line. The solid line represents f Ax ( θ ) ; see text. (b) Reconstructed phase map after returning to Cartesian coordinates. The contours of the measured phase cone are projected onto a plane below the phase surface.

Fig. 4
Fig. 4

Interference distribution measured for an axicon with an off-axis reference beam: (a) source interferogram and (b) 2-D Fourier transform. The isolated ring of frequencies is indicated by the dashed circle.

Fig. 5
Fig. 5

Contour plot of the deviation map δ Φ Ax ( x , y ) . The isophase contour lines span a range of π rad .

Fig. 6
Fig. 6

Variation of ellipticity with a contour level for one axicon derived from three independent phase map measurements.

Fig. 7
Fig. 7

Axicon parameters as a function of rotation angle. (a) Ellipse angles measured for two axicons using the on-axis (+) and off-axis (×) methods. The vertical offset was shifted to serve as a visual separation of the measurements. (b) Ellipticities for the two axicons. The ellipticity measured for a spherical lens using the off-axis method is shown for comparison.

Equations (11)

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E ( x , y , z , t ) = E 0 ( x , y ) e i φ 0 e i ( k z ω t ) e i 2 π ( x f x + y f y ) ,
Φ Ax ( r , θ ) 2 π r f Ax ( θ ) + δ Φ Ax ( r , θ ) .
E r e f * E Probe = | E r e f | | E p r o b e | e i [ 2 π ( x Δ f x + y Δ f y ) + δ 0 + Φ Ax ( x , y ) ] .
F r { | E int | 2 } | θ = D θ ± R θ [ f r ± f Ax ( θ ) ] ,
D θ = F r { | E 0 , Ref ( r , θ ) | 2 + | E 0 , Pr ( r , θ ) | 2 } ,
R θ ( f r ) = F r { | E 0 , Ref ( r , θ ) | | E 0 , Pr ( r , θ ) | e i δ Φ Ax ( r , θ ) } .
Φ Ax ( r , θ ) = Imag { ln [ F r 1 { R θ ( f r ) } ] } ,
F { x , y } { | E int | 2 } = D ˜ { x , y } ± R ˜ ( f x ± Δ f x , f y ± Δ f y ) ,
D ˜ { x , y } = F { x , y } { | E 0 , r e f ( x , y ) | 2 + | E 0 , p r o b e ( x , y ) | 2 } ,
R ˜ ( f x , f y ) = F { x , y } { | E 0 , r e f ( x , y ) | | E 0 , p r o b e ( x , y ) | e i Φ Ax ( x , y ) } .
Φ Ax ( x , y ) = Imag { ln [ F { x , y } 1 { R ˜ ( f x , f y ) } ] } ,

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