Abstract

This Letter presents a theoretical and experimental study of an axicon illuminated by an off-axis paraxial point source. The Fresnel diffraction integral is applied to show that a paraxial point source produces a Bessel beam. A simple analytical relationship is demonstrated between the location of the point source and the spatial frequency and the center of the resulting Bessel beam in the image plane of a camera. Finally, experimental verification is given by translating a point source of light along the optical axis of an axicon and comparing the resulting predicted and recorded beam intensity profiles. The resulting images are then analyzed to predict the location of the point source with excellent accuracy.

© 2012 Optical Society of America

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References

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2011

2007

M. Dong and J. Pu, Opt. Laser Technol. 39, 1258 (2007).
[CrossRef]

2005

G. Mikula, A. Kolodziejczyk, M. Makowski, C. Prokopowicz, and M. Sypek, Opt. Eng. 44, 058001 (2005).
[CrossRef]

Z. Jaroszewicz, A. Burvall, and A. T. Friberg, Opt. Photon. News 16, 34 (2005).
[CrossRef]

2002

2001

1998

1987

J. Durnin and J. J. Miceli, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef]

1970

1954

Bin, Z.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed.(Cambridge University, 1999), p. 952.

Burvall, A.

Z. Jaroszewicz, A. Burvall, and A. T. Friberg, Opt. Photon. News 16, 34 (2005).
[CrossRef]

Chen, Z.

Chi, W.

Chou, K. C.

Collins, J.

Ding, Z.

Dong, M.

M. Dong and J. Pu, Opt. Laser Technol. 39, 1258 (2007).
[CrossRef]

Durnin, J.

J. Durnin and J. J. Miceli, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef]

Friberg, A. T.

Z. Jaroszewicz, A. Burvall, and A. T. Friberg, Opt. Photon. News 16, 34 (2005).
[CrossRef]

George, N.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Co., 2004), p. 491.

Jaroszewicz, Z.

Z. Jaroszewicz, A. Burvall, and A. T. Friberg, Opt. Photon. News 16, 34 (2005).
[CrossRef]

Kolodziejczyk, A.

G. Mikula, A. Kolodziejczyk, M. Makowski, C. Prokopowicz, and M. Sypek, Opt. Eng. 44, 058001 (2005).
[CrossRef]

Leung, B. O.

Makowski, M.

G. Mikula, A. Kolodziejczyk, M. Makowski, C. Prokopowicz, and M. Sypek, Opt. Eng. 44, 058001 (2005).
[CrossRef]

McLeod, J. H.

Miceli, J. J.

J. Durnin and J. J. Miceli, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef]

Mikula, G.

G. Mikula, A. Kolodziejczyk, M. Makowski, C. Prokopowicz, and M. Sypek, Opt. Eng. 44, 058001 (2005).
[CrossRef]

Nelson, J. S.

Prokopowicz, C.

G. Mikula, A. Kolodziejczyk, M. Makowski, C. Prokopowicz, and M. Sypek, Opt. Eng. 44, 058001 (2005).
[CrossRef]

Pu, J.

M. Dong and J. Pu, Opt. Laser Technol. 39, 1258 (2007).
[CrossRef]

Ren, H.

Sypek, M.

G. Mikula, A. Kolodziejczyk, M. Makowski, C. Prokopowicz, and M. Sypek, Opt. Eng. 44, 058001 (2005).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed.(Cambridge University, 1999), p. 952.

Zhao, Y.

Zhu, L.

Appl. Opt.

Appl. Spectrosc.

J. Opt. Soc. Am.

Opt. Eng.

G. Mikula, A. Kolodziejczyk, M. Makowski, C. Prokopowicz, and M. Sypek, Opt. Eng. 44, 058001 (2005).
[CrossRef]

Opt. Laser Technol.

M. Dong and J. Pu, Opt. Laser Technol. 39, 1258 (2007).
[CrossRef]

Opt. Lett.

Opt. Photon. News

Z. Jaroszewicz, A. Burvall, and A. T. Friberg, Opt. Photon. News 16, 34 (2005).
[CrossRef]

Phys. Rev. Lett.

J. Durnin and J. J. Miceli, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef]

Other

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Co., 2004), p. 491.

M. Born and E. Wolf, Principles of Optics, 7th ed.(Cambridge University, 1999), p. 952.

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Figures (3)

Fig. 1.
Fig. 1.

Schematic showing location of point source and resulting Bessel beam in the image plane.

Fig. 2.
Fig. 2.

Normalized intensity profiles for predicted Bessel pattern (dashed) and measured Bessel pattern (solid) for a point source located 335mm from the axicon.

Fig. 3.
Fig. 3.

Calculated axial position of a point source of light plotted against the actual axial position. Solid line is a linear best fit with a slope of 1.02.

Equations (17)

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E0(r)=|1R|eikR,
Eax(rax,θax)=eikzp|zp|×eik[rp2+rax22rpraxcos(θaxθp)2zp+(n1)raxα],
Ei(ri,θi)=ikeikC2πCEax(rax,θax)×eikC[Drax2+Ari22raxricos(θiθax)]rdθaxdrax,
Ei(ri,θi)=ikeikC2πCeikzp|zp|×eik[rp2+rax22rpraxcos(θaxθp)2zp+(n1)raxα]×eikC[Drax2+Ari22raxricos(θiθax)]rdθaxdrax=ikeik(zp+C)2πCzp×0D2r{02πeikg(rp,θp:rax,θax:ri,θi)×eik[(n1)raxα+Ari2+Drax22C+rp2+rax22zp]dθax}drax,
g(rp,θp:rax,θax:ri,θi)=2rpraxcos(θaxθp)2zp+2raxricos(θiθax)2C,
g(rp,θp:rax,θax:ri,θi)=csin(ϕ1+γ),
ϕ1=π2θax+θp,
ϕ2=θi+θp,
c=(rpraxzp)2+(riraxC)2+2rprirax2Crpcos(ϕ2),
γ=tan1(rirax2Csin(α)rprax2zp+rirax2Ccos(α)).
2πJ0(x)=02πeixcos(ϕϕ0)dϕ
Ei(ri,θi)=ikeik(zp+C)2πCzp0D2J0(kc)×raxeik[(n1)raxα+Ari2+Drax22C+rp2+rax22zp]drax.
Ei(ri,θi)=ikeik(zp+C)2πCzp2πi(DC+1zp)×J0(kc(rs))eiCzpα(n1)Dzp+C,
rs=Czpα(n1)Dzp+C.
rc=Crpzp,
θc=θp+π,
zp=CββDkα(n1),

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