Abstract

The imaging properties of three types of refractive axicons are examined by using them in an imaging system. A linear axicon, a logarithmic axicon, and a Fresnel axicon are characterized by determining their point spread functions (PSFs) experimentally and by numerical simulation. The PSFs, which vary along the depth of field for the cases considered in the present investigation, are used in digital filters to denoise the images. A comparison of the imaging performance of these three optical elements is presented.

© 2013 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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2013 (1)

2012 (2)

S. K. Tiwari, S. R. Mishra, S. P. Ram, and H. S. Rawat, “Generation of a Bessel beam of variable spot size,” Appl. Opt. 51, 3718–3725 (2012).
[CrossRef]

B. Chebbi, I. Golub, and K. Gourley, “Homogenization of on-axis intensity distribution produced by a Fresnel refractive axicon,” Opt. Commun. 285, 1636–1641 (2012).
[CrossRef]

2011 (4)

2010 (2)

I. Golub, B. Chebbi, D. Shaw, and D. Nowacki, “Characterization of a refractive logarithmic axicon,” Opt. Lett. 35, 2828–2830 (2010).
[CrossRef]

A. Saikaley, I. Pak, I. Golub, and B. Chebbi, “Imaging with a fiberoptic bundle/axicon telescope system,” Proc. SPIE 7750, 775010 (2010).
[CrossRef]

2009 (2)

Z. Zhai, S. Ding, Q. Lv, X. Wang, and Y. Zhong, “Extended depth of field through an axicon,” J. Mod. Opt. 56, 1304–1308 (2009).
[CrossRef]

K. Gourley, I. Golub, and B. Chebbi, “First experimental demonstration of a Fresnel axicon,” Proc. SPIE 7099, 70990D (2009).
[CrossRef]

2008 (2)

2007 (1)

2005 (1)

G. Mikuła, A. Kolodziejczyk, M. Makowski, C. Prokopowicz, and M. Sypek, “Diffractive elements for imaging with extended depth of focus,” Opt. Eng. 44, 058001 (2005).
[CrossRef]

1995 (1)

1992 (1)

1989 (1)

1980 (1)

G. Roy and R. Tremblay, “Influence of the divergence of a laser beam on the axial intensity distribution of an axicon,” Opt. Commun. 34, 1–3 (1980).
[CrossRef]

Ares García, J.

Bará, S.

Borra, E. F.

Brousseau, D.

Cathey, W.

Chebbi, B.

B. Chebbi, I. Golub, and K. Gourley, “Homogenization of on-axis intensity distribution produced by a Fresnel refractive axicon,” Opt. Commun. 285, 1636–1641 (2012).
[CrossRef]

A. Saikaley, S. Matoug, I. Golub, and B. Chebbi, “Imaging properties of different refractive axicons,” Proc. SPIE 8007, 80070Y (2011).
[CrossRef]

K. Gourley, I. Golub, and B. Chebbi, “Demonstration of a Fresnel axicon,” Appl. Opt. 50, 303–306 (2011).
[CrossRef]

I. Golub, B. Chebbi, D. Shaw, and D. Nowacki, “Characterization of a refractive logarithmic axicon,” Opt. Lett. 35, 2828–2830 (2010).
[CrossRef]

A. Saikaley, I. Pak, I. Golub, and B. Chebbi, “Imaging with a fiberoptic bundle/axicon telescope system,” Proc. SPIE 7750, 775010 (2010).
[CrossRef]

K. Gourley, I. Golub, and B. Chebbi, “First experimental demonstration of a Fresnel axicon,” Proc. SPIE 7099, 70990D (2009).
[CrossRef]

B. Chebbi and I. Golub, School of Engineering, Laurentian University, Sudbury, ON, Canada, are preparing a manuscript called “More on the Bessel beam approximation for the radial intensity distribution of logarithmic axicons,” (to be submitted).

Diaz, A.

Ding, S.

Z. Zhai, S. Ding, Q. Lv, X. Wang, and Y. Zhong, “Extended depth of field through an axicon,” J. Mod. Opt. 56, 1304–1308 (2009).
[CrossRef]

Dowski, E.

Drapeau, J.

Druart, G.

Eddins, S. L.

R. C. Gonzalez, R. E. Woods, and S. L. Eddins, Digital Image Processing using Matlab (Prentice-Hall, 2004).

Golub, I.

B. Chebbi, I. Golub, and K. Gourley, “Homogenization of on-axis intensity distribution produced by a Fresnel refractive axicon,” Opt. Commun. 285, 1636–1641 (2012).
[CrossRef]

A. Saikaley, S. Matoug, I. Golub, and B. Chebbi, “Imaging properties of different refractive axicons,” Proc. SPIE 8007, 80070Y (2011).
[CrossRef]

K. Gourley, I. Golub, and B. Chebbi, “Demonstration of a Fresnel axicon,” Appl. Opt. 50, 303–306 (2011).
[CrossRef]

I. Golub, B. Chebbi, D. Shaw, and D. Nowacki, “Characterization of a refractive logarithmic axicon,” Opt. Lett. 35, 2828–2830 (2010).
[CrossRef]

A. Saikaley, I. Pak, I. Golub, and B. Chebbi, “Imaging with a fiberoptic bundle/axicon telescope system,” Proc. SPIE 7750, 775010 (2010).
[CrossRef]

K. Gourley, I. Golub, and B. Chebbi, “First experimental demonstration of a Fresnel axicon,” Proc. SPIE 7099, 70990D (2009).
[CrossRef]

B. Chebbi and I. Golub, School of Engineering, Laurentian University, Sudbury, ON, Canada, are preparing a manuscript called “More on the Bessel beam approximation for the radial intensity distribution of logarithmic axicons,” (to be submitted).

Gomez García, M.

Gonzalez, R. C.

R. C. Gonzalez, R. E. Woods, and S. L. Eddins, Digital Image Processing using Matlab (Prentice-Hall, 2004).

Goodman, J.

J. Goodman, Introduction to Fourier Optics (Roberts and Company, 2005).

Gourley, K.

B. Chebbi, I. Golub, and K. Gourley, “Homogenization of on-axis intensity distribution produced by a Fresnel refractive axicon,” Opt. Commun. 285, 1636–1641 (2012).
[CrossRef]

K. Gourley, I. Golub, and B. Chebbi, “Demonstration of a Fresnel axicon,” Appl. Opt. 50, 303–306 (2011).
[CrossRef]

K. Gourley, I. Golub, and B. Chebbi, “First experimental demonstration of a Fresnel axicon,” Proc. SPIE 7099, 70990D (2009).
[CrossRef]

Guérineau, N.

Haïdar, R.

Jaroszewicz, Z.

Kattnig, A.

Kolodziejczyk, A.

Kotlyar, V.

Kovalev, A.

Lv, Q.

Z. Zhai, S. Ding, Q. Lv, X. Wang, and Y. Zhong, “Extended depth of field through an axicon,” J. Mod. Opt. 56, 1304–1308 (2009).
[CrossRef]

Makowski, M.

G. Mikuła, A. Kolodziejczyk, M. Makowski, C. Prokopowicz, and M. Sypek, “Diffractive elements for imaging with extended depth of focus,” Opt. Eng. 44, 058001 (2005).
[CrossRef]

Matoug, S.

A. Saikaley, S. Matoug, I. Golub, and B. Chebbi, “Imaging properties of different refractive axicons,” Proc. SPIE 8007, 80070Y (2011).
[CrossRef]

Mikula, G.

G. Mikuła, Z. Jaroszewicz, A. Kolodziejczyk, K. Petelczyc, and M. Sypek, “Imaging with extended focal depth by means of lenses with radial and angular modulation,” Opt. Express 15, 9184–9193 (2007).
[CrossRef]

G. Mikuła, A. Kolodziejczyk, M. Makowski, C. Prokopowicz, and M. Sypek, “Diffractive elements for imaging with extended depth of focus,” Opt. Eng. 44, 058001 (2005).
[CrossRef]

Mishra, S. R.

Nowacki, D.

Ojeda-Castaneda, J.

Pak, I.

A. Saikaley, I. Pak, I. Golub, and B. Chebbi, “Imaging with a fiberoptic bundle/axicon telescope system,” Proc. SPIE 7750, 775010 (2010).
[CrossRef]

Petelczyc, K.

Piché, M.

Primot, J.

Prokopowicz, C.

G. Mikuła, A. Kolodziejczyk, M. Makowski, C. Prokopowicz, and M. Sypek, “Diffractive elements for imaging with extended depth of focus,” Opt. Eng. 44, 058001 (2005).
[CrossRef]

Ram, S. P.

Rawat, H. S.

Roy, G.

G. Roy and R. Tremblay, “Influence of the divergence of a laser beam on the axial intensity distribution of an axicon,” Opt. Commun. 34, 1–3 (1980).
[CrossRef]

Saikaley, A.

A. Saikaley, S. Matoug, I. Golub, and B. Chebbi, “Imaging properties of different refractive axicons,” Proc. SPIE 8007, 80070Y (2011).
[CrossRef]

A. Saikaley, I. Pak, I. Golub, and B. Chebbi, “Imaging with a fiberoptic bundle/axicon telescope system,” Proc. SPIE 7750, 775010 (2010).
[CrossRef]

Sauer, H.

Shaw, D.

Snoeyink, C.

Sochacki, J.

Soifer, V. A.

Stafeev, S.

Sypek, M.

G. Mikuła, Z. Jaroszewicz, A. Kolodziejczyk, K. Petelczyc, and M. Sypek, “Imaging with extended focal depth by means of lenses with radial and angular modulation,” Opt. Express 15, 9184–9193 (2007).
[CrossRef]

G. Mikuła, A. Kolodziejczyk, M. Makowski, C. Prokopowicz, and M. Sypek, “Diffractive elements for imaging with extended depth of focus,” Opt. Eng. 44, 058001 (2005).
[CrossRef]

Taboury, J.

Tepichin, E.

Tiwari, S. K.

Tremblay, R.

G. Roy and R. Tremblay, “Influence of the divergence of a laser beam on the axial intensity distribution of an axicon,” Opt. Commun. 34, 1–3 (1980).
[CrossRef]

Wang, X.

Z. Zhai, S. Ding, Q. Lv, X. Wang, and Y. Zhong, “Extended depth of field through an axicon,” J. Mod. Opt. 56, 1304–1308 (2009).
[CrossRef]

Woods, R. E.

R. C. Gonzalez, R. E. Woods, and S. L. Eddins, Digital Image Processing using Matlab (Prentice-Hall, 2004).

Zhai, Z.

Z. Zhai, S. Ding, Q. Lv, X. Wang, and Y. Zhong, “Extended depth of field through an axicon,” J. Mod. Opt. 56, 1304–1308 (2009).
[CrossRef]

Zhong, Y.

Z. Zhai, S. Ding, Q. Lv, X. Wang, and Y. Zhong, “Extended depth of field through an axicon,” J. Mod. Opt. 56, 1304–1308 (2009).
[CrossRef]

Appl. Opt. (6)

J. Mod. Opt. (1)

Z. Zhai, S. Ding, Q. Lv, X. Wang, and Y. Zhong, “Extended depth of field through an axicon,” J. Mod. Opt. 56, 1304–1308 (2009).
[CrossRef]

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

Opt. Commun. (2)

B. Chebbi, I. Golub, and K. Gourley, “Homogenization of on-axis intensity distribution produced by a Fresnel refractive axicon,” Opt. Commun. 285, 1636–1641 (2012).
[CrossRef]

G. Roy and R. Tremblay, “Influence of the divergence of a laser beam on the axial intensity distribution of an axicon,” Opt. Commun. 34, 1–3 (1980).
[CrossRef]

Opt. Eng. (1)

G. Mikuła, A. Kolodziejczyk, M. Makowski, C. Prokopowicz, and M. Sypek, “Diffractive elements for imaging with extended depth of focus,” Opt. Eng. 44, 058001 (2005).
[CrossRef]

Opt. Express (2)

Opt. Lett. (3)

Proc. SPIE (3)

A. Saikaley, I. Pak, I. Golub, and B. Chebbi, “Imaging with a fiberoptic bundle/axicon telescope system,” Proc. SPIE 7750, 775010 (2010).
[CrossRef]

A. Saikaley, S. Matoug, I. Golub, and B. Chebbi, “Imaging properties of different refractive axicons,” Proc. SPIE 8007, 80070Y (2011).
[CrossRef]

K. Gourley, I. Golub, and B. Chebbi, “First experimental demonstration of a Fresnel axicon,” Proc. SPIE 7099, 70990D (2009).
[CrossRef]

Other (3)

B. Chebbi and I. Golub, School of Engineering, Laurentian University, Sudbury, ON, Canada, are preparing a manuscript called “More on the Bessel beam approximation for the radial intensity distribution of logarithmic axicons,” (to be submitted).

J. Goodman, Introduction to Fourier Optics (Roberts and Company, 2005).

R. C. Gonzalez, R. E. Woods, and S. L. Eddins, Digital Image Processing using Matlab (Prentice-Hall, 2004).

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Figures (9)

Fig. 1.
Fig. 1.

Setup to measure PSF.

Fig. 2.
Fig. 2.

PSF images of the point source recorded on a CMOS camera for regular axicon (α=5°). (a) p=55mm, q=145mm. (b) p=55mm, q=245mm. (c) p=55mm, q=345mm. (d) p=80mm, q=120mm. (e) p=80mm, q=220mm. (f) p=80mm, q=320mm. (g) p=125mm, q=75mm. (h) p=125mm, q=175mm. (i) p=125mm, q=275mm. The size of the images is (0.56mm×0.56mm).

Fig. 3.
Fig. 3.

PSF images of the point source recorded on a CMOS camera for Fresnel axicon (α=5°). (a) p=55mm, q=145mm. (b) p=55mm, q=245mm. (c) p=55mm, q=345mm. (d) p=80mm, q=120mm. (e) p=80mm, q=220mm. (f) p=80mm, q=320mm. (g) p=125mm, q=75mm. (h) p=125mm, q=175mm. (i) p=125mm, q=275mm. The size of the images is (0.56mm×0.56mm).

Fig. 4.
Fig. 4.

PSF images of the point source recorded on a CMOS camera for LA. (a) p=70mm, q=230mm. (b) p=70mm, q=330mm. (c) p=80mm, q=220mm. (d) p=80mm, q=320mm. (e) p=90mm, q=210mm. (f) p=90mm, q=310mm. (g) p=145mm, q=155mm. (h) p=145mm, q=255mm. The size of the images is (0.56mm×0.56mm).

Fig. 5.
Fig. 5.

Measured and predicted spot size obtained using a regular axicon for an incoming Gaussian beam.

Fig. 6.
Fig. 6.

Simulated and measured PSF profiles for regular axicon with phase function φ(r)=(n1)rtanα [13,16].

Fig. 7.
Fig. 7.

Simulated and measured PSF profiles for LA with phase function φ(r)=const(1/2a)ln(f1+ar2), where a=(f2f1)/b2, f1 and f2 mark the beginning and the end distances of the DOF, and b is the radius of the aperture [12,19].

Fig. 8.
Fig. 8.

Image processing procedure shown for LA at p=200mm, q=62mm. (a) Original grayscale image. (b) After Wiener filter is applied. (c) After application of Wiener filter and adaptive histogram equalization. The size of the images is (4.48mm×3.36mm).

Fig. 9.
Fig. 9.

Images obtained using different axicons after digital processing. Axicon (α=5°): (a) p=190mm, q=45mm, ε=13.2D. (b) p=80mm, q=20mm, ε=48.2D. (c) p=30mm, q=20mm., ε=69D. Fraxicon (α=5°): (d) p=190mm, q=45mm, ε=13.2D. (e) p=80mm, q=20mm, ε=48.2D. (f) p=30mm, q=20mm, ε=69D. Axicon (α=10°): (g) p=175mm, q=70mm, ε=9.13D. (h) p=80mm, q=20mm, ε=33.4D. (i) p=30mm, q=20mm. ε=54.2D. LA (f1=50mm, f2=150mm): (j) p=200mm, q=55mm, ε=13.2D. (k) p=200mm, q=62mm, ε=11.3D. (l) p=140mm, q=80mm. ε=9.6D. The size of the images is (4.48mm×3.36mm).

Tables (1)

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Table 1. Object and Image Distances for Axicon-type Optical Elements Covered in the Literature

Equations (3)

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r=2.4qk(n1)tanα(1p+1q+1Ri).
E(x,y,z)=1jλAE(x,y,0)exp[jkR(x,x,y,y,z)]R(x,x,y,y,z)cos(n⃗,R⃗(x,x,y,y,z))dxdy,
F(u,v)=[H*(u,v)|H(u,v)|2+NSR]G(u,v),

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