Abstract

We study the field that is produced by a paraxial refractive axicon lens. The results from geometrical optics, scalar wave optics, and electromagnetic diffraction theory are compared. In particular, the axial intensity, the on-axis effective wavelength, the transverse intensity, and the far-zone field are examined. A rigorous electromagnetic diffraction analysis shows that the state of polarization of the incident beam strongly affects the transverse intensity distribution, but not the intensity distribution in the far zone.

© 2017 Optical Society of America

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References

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  1. J. H. McLeod, “The axicon: a new type of optical element,” J. Opt. Soc. Am. 44, 592–597 (1954).
    [Crossref]
  2. J. Sochacki, A. Kolodziejczyk, Z. Jaroszewicz, and S. Bara, “Nonparaxial design of generalized axicons,” Appl. Opt. 31, 5326–5330 (1992).
    [Crossref]
  3. S. Brinkmann, T. Dresel, R. Schreiner, and J. Schwider, “Axicon-type test interferometer for cylindrical surfaces,” in Simulation and Experiment in Laser Metrology, Z. Füzessy, W. Jüptner, and W. Osten, eds. (Akademie Verlag, 1996), pp. 82–87.
  4. I. Velchev and W. Ubachs, “Higher-order stimulated Brillouin scattering with nondiffracting beams,” Opt. Lett. 26, 530–532 (2001).
    [Crossref]
  5. F. P. Schafer, “On some properties of axicons,” Appl. Phys. B 39, 1–8 (1986).
    [Crossref]
  6. M. Rioux, R. Tremblay, and P. A. Bélanger, “Linear, annular, and radial focusing with axicons and applications to laser machining,” Appl. Opt. 17, 1532–1536 (1978).
    [Crossref]
  7. B. Shao, S. C. Esener, J. M. Nascimento, E. L. Botvinick, and M. W. Berns, “Dynamically adjustable annular laser trapping based on axicons,” Appl. Opt. 45, 6421–6428 (2006).
    [Crossref]
  8. T. Wulle and S. Herminghaus, “Nonlinear optics of Bessel beams,” Phys. Rev. Lett. 70, 1401–1404 (1993).
    [Crossref]
  9. G. Bickel, G. Hausler, and M. Haul, “Triangulation with expanded range of depth,” Opt. Eng. 24, 975–977 (1985).
    [Crossref]
  10. Z. Ding, H. Ren, Y. Zhao, J. S. Nelson, and Z. Chen, “High-resolution optical coherence tomography over a large depth range with an axicon lens,” Opt. Lett. 27, 243–245 (2002).
    [Crossref]
  11. Q. Ren and R. Birngruber, “Axicon: a new laser beam delivery system for corneal surgery,” IEEE J. Quantum Electron. 26, 2305–2308 (1990).
    [Crossref]
  12. H. E. Hernandez-Figueroa, E. Recami, and M. Zamboni-Rached, eds., Non-Diffracting Waves (Wiley, 2014).
  13. L. M. Soroko, “Axicons and meso-optical imaging devices,” Vol. 27 of Progress in Optics, E. Wolf, ed. (Elsevier, 1989), pp. 109–160.
  14. Z. Jaroszewicz, Axicons-Design and Propagation Properties (SPIE, 1997).
  15. Z. Jaroszewicz, A. Burvall, and A. T. Friberg, “Axicon-the most important optical element,” Opt. Photon. News 16(4), 34–39 (2005).
    [Crossref]
  16. W. R. Edmonds, “Imaging properties of a conic axicon,” Appl. Opt. 13, 1762–1764 (1974).
    [Crossref]
  17. J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [Crossref]
  18. J. Tervo and J. Turunen, “Generation of vectorial propagation-invariant fields by polarization-grating axicons,” Opt. Commun. 192, 13–18 (2001).
    [Crossref]
  19. S. Fujiwara, “Optical properties of conic surfaces. I. Reflecting cone,” J. Opt. Soc. Am. 52, 287–292 (1962).
    [Crossref]
  20. J. Turunen and A. T. Friberg, “Electromagnetic theory of reflaxicon fields,” Pure Appl. Opt. 2, 539–547 (1993).
    [Crossref]
  21. R. Dutta, K. Saastamoinen, J. Turunen, and A. T. Friberg, “Broadband spatiotemporal axicon fields,” Opt. Express 22, 25015–25026 (2014).
    [Crossref]
  22. N. Radwell, R. D. Hawley, J. B. Gotte, and S. Franke-Arnold, “Achromatic vector vortex beams from a glass cone,” Nat. Commun. 7, 10564 (2016).
    [Crossref]
  23. J. H. McLeod, “Axicons and their uses,” J. Opt. Soc. Am. 50, 166–169 (1960).
    [Crossref]
  24. C. J. Zapata-Rodriguez and A. Sanchez-Losa, “Three-dimensional field distribution in the focal region of low-Fresnel-number axicons,” J. Opt. Soc. Am. A 23, 3016–3026 (2005).
    [Crossref]
  25. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London A 253, 358–379 (1959).
    [Crossref]
  26. T. G. Brown, “Unconventional polarization states: beam propagation, focusing, and imaging,” Vol. 56 of Progress in Optics, E. Wolf, ed. (Elsevier, 2011), pp. 81–129.
  27. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1995).
  28. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  29. Z. L. Horvath and Z. Bor, “Diffraction of short pulses with boundary diffraction wave theory,” Phys. Rev. E 63, 026601 (2001).
    [Crossref]
  30. J. Turunen, A. Vasara, and A. T. Friberg, “Holographic generation of diffraction-free beams,” Appl. Opt. 27, 3959–3962 (1988).
    [Crossref]
  31. A. Vasara, J. Turunen, and A. T. Friberg, “Realization of general nondiffracting beams with computer-generated holograms,” J. Opt. Soc. Am. A 6, 1748–1754 (1989).
    [Crossref]
  32. J. Turunen and A. T. Friberg, “Self-imaging and propagation-invariance in electromagnetic fields,” Pure Appl. Opt. 2, 51–60 (1993).
    [Crossref]
  33. J. Turunen and A. T. Friberg, “Propagation-invariant optical fields,” Vol. 54 of Progress in Optics, E. Wolf, ed. (Elsevier, 2009), pp. 1–88.
  34. U. Levy, S. Derevyanko, and Y. Silberberg, “Light modes of free space,” Vol. 61 of Progress in Optics, T. D. Visser, ed. (Elsevier, 2016), pp. 237–281.
  35. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1998).
  36. G. Toraldo di Francia, Electromagnetic Waves (Interscience, 1955).
  37. J. T. Foley and E. Wolf, “Wave-front spacing in the focal region of high-numerical-aperture systems,” Opt. Lett. 30, 1312–1314 (2005).
    [Crossref]
  38. T. D. Visser and J. T. Foley, “On the wavefront spacing of focused, radially polarized beams,” J. Opt. Soc. Am. A 22, 2527–2531 (2005).
    [Crossref]
  39. A. T. Friberg, “Stationary phase analysis of generalized axicons,” J. Opt. Soc. Am. A 13, 743–750 (1996).
    [Crossref]

2016 (1)

N. Radwell, R. D. Hawley, J. B. Gotte, and S. Franke-Arnold, “Achromatic vector vortex beams from a glass cone,” Nat. Commun. 7, 10564 (2016).
[Crossref]

2014 (1)

2006 (1)

2005 (4)

2002 (1)

2001 (3)

I. Velchev and W. Ubachs, “Higher-order stimulated Brillouin scattering with nondiffracting beams,” Opt. Lett. 26, 530–532 (2001).
[Crossref]

J. Tervo and J. Turunen, “Generation of vectorial propagation-invariant fields by polarization-grating axicons,” Opt. Commun. 192, 13–18 (2001).
[Crossref]

Z. L. Horvath and Z. Bor, “Diffraction of short pulses with boundary diffraction wave theory,” Phys. Rev. E 63, 026601 (2001).
[Crossref]

1996 (1)

1993 (3)

J. Turunen and A. T. Friberg, “Self-imaging and propagation-invariance in electromagnetic fields,” Pure Appl. Opt. 2, 51–60 (1993).
[Crossref]

J. Turunen and A. T. Friberg, “Electromagnetic theory of reflaxicon fields,” Pure Appl. Opt. 2, 539–547 (1993).
[Crossref]

T. Wulle and S. Herminghaus, “Nonlinear optics of Bessel beams,” Phys. Rev. Lett. 70, 1401–1404 (1993).
[Crossref]

1992 (1)

1990 (1)

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

1989 (1)

1988 (1)

1987 (1)

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref]

1986 (1)

F. P. Schafer, “On some properties of axicons,” Appl. Phys. B 39, 1–8 (1986).
[Crossref]

1985 (1)

G. Bickel, G. Hausler, and M. Haul, “Triangulation with expanded range of depth,” Opt. Eng. 24, 975–977 (1985).
[Crossref]

1978 (1)

1974 (1)

1962 (1)

1960 (1)

1959 (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London A 253, 358–379 (1959).
[Crossref]

1954 (1)

Bara, S.

Bélanger, P. A.

Berns, M. W.

Bickel, G.

G. Bickel, G. Hausler, and M. Haul, “Triangulation with expanded range of depth,” Opt. Eng. 24, 975–977 (1985).
[Crossref]

Birngruber, R.

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

Bor, Z.

Z. L. Horvath and Z. Bor, “Diffraction of short pulses with boundary diffraction wave theory,” Phys. Rev. E 63, 026601 (2001).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1995).

Botvinick, E. L.

Brinkmann, S.

S. Brinkmann, T. Dresel, R. Schreiner, and J. Schwider, “Axicon-type test interferometer for cylindrical surfaces,” in Simulation and Experiment in Laser Metrology, Z. Füzessy, W. Jüptner, and W. Osten, eds. (Akademie Verlag, 1996), pp. 82–87.

Brown, T. G.

T. G. Brown, “Unconventional polarization states: beam propagation, focusing, and imaging,” Vol. 56 of Progress in Optics, E. Wolf, ed. (Elsevier, 2011), pp. 81–129.

Burvall, A.

Z. Jaroszewicz, A. Burvall, and A. T. Friberg, “Axicon-the most important optical element,” Opt. Photon. News 16(4), 34–39 (2005).
[Crossref]

Chen, Z.

Derevyanko, S.

U. Levy, S. Derevyanko, and Y. Silberberg, “Light modes of free space,” Vol. 61 of Progress in Optics, T. D. Visser, ed. (Elsevier, 2016), pp. 237–281.

Ding, Z.

Dresel, T.

S. Brinkmann, T. Dresel, R. Schreiner, and J. Schwider, “Axicon-type test interferometer for cylindrical surfaces,” in Simulation and Experiment in Laser Metrology, Z. Füzessy, W. Jüptner, and W. Osten, eds. (Akademie Verlag, 1996), pp. 82–87.

Durnin, J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref]

Dutta, R.

Eberly, J. H.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref]

Edmonds, W. R.

Esener, S. C.

Foley, J. T.

Franke-Arnold, S.

N. Radwell, R. D. Hawley, J. B. Gotte, and S. Franke-Arnold, “Achromatic vector vortex beams from a glass cone,” Nat. Commun. 7, 10564 (2016).
[Crossref]

Friberg, A. T.

R. Dutta, K. Saastamoinen, J. Turunen, and A. T. Friberg, “Broadband spatiotemporal axicon fields,” Opt. Express 22, 25015–25026 (2014).
[Crossref]

Z. Jaroszewicz, A. Burvall, and A. T. Friberg, “Axicon-the most important optical element,” Opt. Photon. News 16(4), 34–39 (2005).
[Crossref]

A. T. Friberg, “Stationary phase analysis of generalized axicons,” J. Opt. Soc. Am. A 13, 743–750 (1996).
[Crossref]

J. Turunen and A. T. Friberg, “Self-imaging and propagation-invariance in electromagnetic fields,” Pure Appl. Opt. 2, 51–60 (1993).
[Crossref]

J. Turunen and A. T. Friberg, “Electromagnetic theory of reflaxicon fields,” Pure Appl. Opt. 2, 539–547 (1993).
[Crossref]

A. Vasara, J. Turunen, and A. T. Friberg, “Realization of general nondiffracting beams with computer-generated holograms,” J. Opt. Soc. Am. A 6, 1748–1754 (1989).
[Crossref]

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

J. Turunen and A. T. Friberg, “Propagation-invariant optical fields,” Vol. 54 of Progress in Optics, E. Wolf, ed. (Elsevier, 2009), pp. 1–88.

Fujiwara, S.

Gotte, J. B.

N. Radwell, R. D. Hawley, J. B. Gotte, and S. Franke-Arnold, “Achromatic vector vortex beams from a glass cone,” Nat. Commun. 7, 10564 (2016).
[Crossref]

Haul, M.

G. Bickel, G. Hausler, and M. Haul, “Triangulation with expanded range of depth,” Opt. Eng. 24, 975–977 (1985).
[Crossref]

Hausler, G.

G. Bickel, G. Hausler, and M. Haul, “Triangulation with expanded range of depth,” Opt. Eng. 24, 975–977 (1985).
[Crossref]

Hawley, R. D.

N. Radwell, R. D. Hawley, J. B. Gotte, and S. Franke-Arnold, “Achromatic vector vortex beams from a glass cone,” Nat. Commun. 7, 10564 (2016).
[Crossref]

Herminghaus, S.

T. Wulle and S. Herminghaus, “Nonlinear optics of Bessel beams,” Phys. Rev. Lett. 70, 1401–1404 (1993).
[Crossref]

Horvath, Z. L.

Z. L. Horvath and Z. Bor, “Diffraction of short pulses with boundary diffraction wave theory,” Phys. Rev. E 63, 026601 (2001).
[Crossref]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1998).

Jaroszewicz, Z.

Z. Jaroszewicz, A. Burvall, and A. T. Friberg, “Axicon-the most important optical element,” Opt. Photon. News 16(4), 34–39 (2005).
[Crossref]

J. Sochacki, A. Kolodziejczyk, Z. Jaroszewicz, and S. Bara, “Nonparaxial design of generalized axicons,” Appl. Opt. 31, 5326–5330 (1992).
[Crossref]

Z. Jaroszewicz, Axicons-Design and Propagation Properties (SPIE, 1997).

Kolodziejczyk, A.

Levy, U.

U. Levy, S. Derevyanko, and Y. Silberberg, “Light modes of free space,” Vol. 61 of Progress in Optics, T. D. Visser, ed. (Elsevier, 2016), pp. 237–281.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

McLeod, J. H.

Miceli, J. J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref]

Nascimento, J. M.

Nelson, J. S.

Radwell, N.

N. Radwell, R. D. Hawley, J. B. Gotte, and S. Franke-Arnold, “Achromatic vector vortex beams from a glass cone,” Nat. Commun. 7, 10564 (2016).
[Crossref]

Ren, H.

Ren, Q.

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

Richards, B.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London A 253, 358–379 (1959).
[Crossref]

Rioux, M.

Saastamoinen, K.

Sanchez-Losa, A.

Schafer, F. P.

F. P. Schafer, “On some properties of axicons,” Appl. Phys. B 39, 1–8 (1986).
[Crossref]

Schreiner, R.

S. Brinkmann, T. Dresel, R. Schreiner, and J. Schwider, “Axicon-type test interferometer for cylindrical surfaces,” in Simulation and Experiment in Laser Metrology, Z. Füzessy, W. Jüptner, and W. Osten, eds. (Akademie Verlag, 1996), pp. 82–87.

Schwider, J.

S. Brinkmann, T. Dresel, R. Schreiner, and J. Schwider, “Axicon-type test interferometer for cylindrical surfaces,” in Simulation and Experiment in Laser Metrology, Z. Füzessy, W. Jüptner, and W. Osten, eds. (Akademie Verlag, 1996), pp. 82–87.

Shao, B.

Silberberg, Y.

U. Levy, S. Derevyanko, and Y. Silberberg, “Light modes of free space,” Vol. 61 of Progress in Optics, T. D. Visser, ed. (Elsevier, 2016), pp. 237–281.

Sochacki, J.

Soroko, L. M.

L. M. Soroko, “Axicons and meso-optical imaging devices,” Vol. 27 of Progress in Optics, E. Wolf, ed. (Elsevier, 1989), pp. 109–160.

Tervo, J.

J. Tervo and J. Turunen, “Generation of vectorial propagation-invariant fields by polarization-grating axicons,” Opt. Commun. 192, 13–18 (2001).
[Crossref]

Toraldo di Francia, G.

G. Toraldo di Francia, Electromagnetic Waves (Interscience, 1955).

Tremblay, R.

Turunen, J.

R. Dutta, K. Saastamoinen, J. Turunen, and A. T. Friberg, “Broadband spatiotemporal axicon fields,” Opt. Express 22, 25015–25026 (2014).
[Crossref]

J. Tervo and J. Turunen, “Generation of vectorial propagation-invariant fields by polarization-grating axicons,” Opt. Commun. 192, 13–18 (2001).
[Crossref]

J. Turunen and A. T. Friberg, “Electromagnetic theory of reflaxicon fields,” Pure Appl. Opt. 2, 539–547 (1993).
[Crossref]

J. Turunen and A. T. Friberg, “Self-imaging and propagation-invariance in electromagnetic fields,” Pure Appl. Opt. 2, 51–60 (1993).
[Crossref]

A. Vasara, J. Turunen, and A. T. Friberg, “Realization of general nondiffracting beams with computer-generated holograms,” J. Opt. Soc. Am. A 6, 1748–1754 (1989).
[Crossref]

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

J. Turunen and A. T. Friberg, “Propagation-invariant optical fields,” Vol. 54 of Progress in Optics, E. Wolf, ed. (Elsevier, 2009), pp. 1–88.

Ubachs, W.

Vasara, A.

Velchev, I.

Visser, T. D.

Wolf, E.

J. T. Foley and E. Wolf, “Wave-front spacing in the focal region of high-numerical-aperture systems,” Opt. Lett. 30, 1312–1314 (2005).
[Crossref]

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London A 253, 358–379 (1959).
[Crossref]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1995).

Wulle, T.

T. Wulle and S. Herminghaus, “Nonlinear optics of Bessel beams,” Phys. Rev. Lett. 70, 1401–1404 (1993).
[Crossref]

Zapata-Rodriguez, C. J.

Zhao, Y.

Appl. Opt. (5)

Appl. Phys. B (1)

F. P. Schafer, “On some properties of axicons,” Appl. Phys. B 39, 1–8 (1986).
[Crossref]

IEEE J. Quantum Electron. (1)

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

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (4)

Nat. Commun. (1)

N. Radwell, R. D. Hawley, J. B. Gotte, and S. Franke-Arnold, “Achromatic vector vortex beams from a glass cone,” Nat. Commun. 7, 10564 (2016).
[Crossref]

Opt. Commun. (1)

J. Tervo and J. Turunen, “Generation of vectorial propagation-invariant fields by polarization-grating axicons,” Opt. Commun. 192, 13–18 (2001).
[Crossref]

Opt. Eng. (1)

G. Bickel, G. Hausler, and M. Haul, “Triangulation with expanded range of depth,” Opt. Eng. 24, 975–977 (1985).
[Crossref]

Opt. Express (1)

Opt. Lett. (3)

Opt. Photon. News (1)

Z. Jaroszewicz, A. Burvall, and A. T. Friberg, “Axicon-the most important optical element,” Opt. Photon. News 16(4), 34–39 (2005).
[Crossref]

Phys. Rev. E (1)

Z. L. Horvath and Z. Bor, “Diffraction of short pulses with boundary diffraction wave theory,” Phys. Rev. E 63, 026601 (2001).
[Crossref]

Phys. Rev. Lett. (2)

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref]

T. Wulle and S. Herminghaus, “Nonlinear optics of Bessel beams,” Phys. Rev. Lett. 70, 1401–1404 (1993).
[Crossref]

Proc. R. Soc. London A (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London A 253, 358–379 (1959).
[Crossref]

Pure Appl. Opt. (2)

J. Turunen and A. T. Friberg, “Electromagnetic theory of reflaxicon fields,” Pure Appl. Opt. 2, 539–547 (1993).
[Crossref]

J. Turunen and A. T. Friberg, “Self-imaging and propagation-invariance in electromagnetic fields,” Pure Appl. Opt. 2, 51–60 (1993).
[Crossref]

Other (11)

J. Turunen and A. T. Friberg, “Propagation-invariant optical fields,” Vol. 54 of Progress in Optics, E. Wolf, ed. (Elsevier, 2009), pp. 1–88.

U. Levy, S. Derevyanko, and Y. Silberberg, “Light modes of free space,” Vol. 61 of Progress in Optics, T. D. Visser, ed. (Elsevier, 2016), pp. 237–281.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1998).

G. Toraldo di Francia, Electromagnetic Waves (Interscience, 1955).

T. G. Brown, “Unconventional polarization states: beam propagation, focusing, and imaging,” Vol. 56 of Progress in Optics, E. Wolf, ed. (Elsevier, 2011), pp. 81–129.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1995).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

S. Brinkmann, T. Dresel, R. Schreiner, and J. Schwider, “Axicon-type test interferometer for cylindrical surfaces,” in Simulation and Experiment in Laser Metrology, Z. Füzessy, W. Jüptner, and W. Osten, eds. (Akademie Verlag, 1996), pp. 82–87.

H. E. Hernandez-Figueroa, E. Recami, and M. Zamboni-Rached, eds., Non-Diffracting Waves (Wiley, 2014).

L. M. Soroko, “Axicons and meso-optical imaging devices,” Vol. 27 of Progress in Optics, E. Wolf, ed. (Elsevier, 1989), pp. 109–160.

Z. Jaroszewicz, Axicons-Design and Propagation Properties (SPIE, 1997).

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Figures (11)

Fig. 1.
Fig. 1. Refractive axicon with radius a , base angle α , and refractive index n . Rays are normally incident on the front face A . A marginal ray crosses the z axis at a distance L from the apex, which is taken to be in the plane z = 0 .
Fig. 2.
Fig. 2. Length L of the focal line as a function of the base angle α . In this example the refractive index n = 1.5 , and the axicon radius a = 2    cm .
Fig. 3.
Fig. 3. Paraxial refractive axicon with radius a , base angle α , and refractive index n . A plane wave with a Gaussian amplitude distribution is normally incident on the front face A . The thickness of the cylindrical base is denoted by t , and z = 0 indicates the output plane.
Fig. 4.
Fig. 4. Normalized intensity distribution along the z axis as given by geometrical optics [Eq. (8)] (blue), wave optics using the full diffraction integral [Eq. (21)] (green), and wave optics employing the method of stationary phase [Eq. (24)] (red). In panel (a) the beam waist w 0 = 0.5    cm , which is smaller than the axicon radius. In panel (b)  w 0 = 1    cm , which is equal to the axicon radius. In both these examples, the refractive index n = 1.5 , the base angle α = 1 ° , the axicon radius a = 1    cm , and the wavelength λ = 632.8    nm .
Fig. 5.
Fig. 5. Normalized transverse intensity distribution according to Eq. (21) in different planes. From left to right, z = 1    m , 2 m, 3 m, and 4 m. In these examples λ = 632.8    nm , α = 1 ° , w 0 = 1    cm , a = 1    cm , and n = 1.5 .
Fig. 6.
Fig. 6. Transverse intensity distribution according to Eq. (21), normalized to unity at θ = 0 , as a function of the polar angle, in different cross sections. From left to right, z = 1.4    m , 1.6 m, 1.8 m, 2.0 m, and 2.2 m. All other parameters are the same as in Fig. 5.
Fig. 7.
Fig. 7. Transverse intensity distribution according to Eq. (21), normalized to unity at θ = 0 , as a function of the polar angle, in different cross sections. From left to right, z = 1.4 m, 1.6 m, 1.8 m, 2.0 m, and 2.2 m. The beam waist w 0 is now increased to 1 m, from 1 cm in Figs. 5 and 6. All the other parameters are the same as in Fig. 5.
Fig. 8.
Fig. 8. Transverse intensity distribution according to Eq. (21), normalized to unity at θ = 0 , as a function of the polar angle, in different cross sections. From left to right, z = 10    m , 15 m, and 20 m. The beam waist w 0 = 1    m . All other parameters are the same as in Fig. 5.
Fig. 9.
Fig. 9. Axial intensity distribution I ( z ) = | E x ( 0 , 0 , z ) | 2 for an incident beam that x -polarized, as given by Eq. (79). In this example n = 1.5 , a = 1    cm , α = 1 ° , and λ = 632.8    nm .
Fig. 10.
Fig. 10. Transverse intensity distribution for an incident beam that is x -polarized [Eq. (83)] (purple and green curves), and for a radially polarized [Eqs. (85) and (86)] or azimuthally polarized beam [Eqs. (87) and (88)] (red and blue curves), for z = 0.5    m and z = 1.5    m . The other parameters are n = 1.5 , a = 1    cm , α = 1 ° , and λ = 632.8    nm .
Fig. 11.
Fig. 11. Far-zone intensity distribution I ( θ ) as given by Eq. (95) for an incident beam that is x -polarized. In this example, n = 1.5 , a = 1    cm , α = 1 ° , and λ = 632.8    nm .

Tables (1)

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Table 1. Effective Wavelength on Axis for an Incident Beam with λ = 632.80    nm

Equations (123)

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L = a tan γ a tan α ,
γ = 90 ° β + α .
I ( in ) ( ρ ) = I 0 exp ( 2 ρ 2 / w 0 2 ) ,
P ( ρ ) = I 0 exp ( 2 ρ 2 / w 0 2 ) 2 π ρ δ ρ .
L 1 = ρ ( tan γ tan α ) ,
L 2 = ( ρ + δ ρ ) ( tan γ tan α ) ,
P ( ρ ) δ L T 1 2 ( ω ) T 2 2 ( ω ) cos ( β α ) = 2 π I 0 T 1 2 ( ω ) T 2 2 ( ω ) cos ( β α ) tan γ tan α ρ exp ( 2 ρ 2 / w 0 2 ) .
I ( z ) = D 1 z exp [ 2 z 2 / w 0 2 ( tan γ tan α ) 2 ] ,
D 1 = 2 π I 0 T 1 2 ( ω ) T 2 2 ( ω ) cos ( β α ) ( tan γ tan α ) 2 .
U ( in ) ( ρ , ω ) = U 0 ( ω ) exp ( ρ 2 / w 0 2 ) ,
d ρ tan α ρ α .
Δ = ( 1 n ) k d = ( 1 n ) k ρ α ,
U ( out ) ( ρ , ω ) = T ( ρ , ω ) U ( in ) ( ρ , ω ) ,
T ( ρ , ω ) = C ( ω ) exp [ i k ( 1 n ) ρ α ]
C ( ω ) = T 1 ( ω ) T 2 ( ω ) exp ( i k n t ) .
U ( r , ω ) = i λ z = 0 U ( out ) ( ρ , ω ) e i k R R d 2 ρ ,
U ( x , y , z ) = i C U 0 λ z exp ( i k z ) × z = 0 exp [ i k ( 1 n ) ξ 2 + η 2 α ] × exp [ ( ξ 2 + η 2 ) / w 0 2 ] × exp { i k 2 z [ ( x ξ ) 2 + ( y η ) 2 ] } d ξ d η ,
ρ = ( ξ , η ) = ρ ( cos μ , sin μ ) ,
ρ = ( x , y ) = ρ ( cos δ , sin δ ) ,
U ( ρ , z ) = i C U 0 λ z exp ( i k z ) exp ( i k 2 z ρ 2 ) × 0 2 π 0 a exp [ i k ( 1 n ) ρ α ] exp ( ρ 2 / w 0 2 ) × exp ( i k 2 z ρ 2 ) × exp [ i k ρ ρ z cos ( μ δ ) ] ρ d ρ d μ .
U ( ρ , z ) = i 2 π C U 0 λ z exp ( i k z ) exp ( i k 2 z ρ 2 ) × 0 a exp [ i k ( 1 n ) ρ α ] exp ( i k 2 z ρ 2 ) × exp ( ρ 2 / w 0 2 ) J 0 ( k ρ ρ z ) ρ d ρ ,
I ( ρ , z ) = | U ( ρ , z ) | 2 ,
U ( ρ , z ) = i C U 0 ( 2 π k z ) 1 / 2 ( n 1 ) α exp ( i π / 4 ) exp ( i k z ) × exp ( i k ρ 2 / 2 z ) exp [ i k z ( n 1 ) 2 α 2 / 2 ] × exp [ z 2 ( 1 n ) 2 α 2 / w 0 2 ] J 0 [ ( n 1 ) k ρ α ] .
I ( ρ , z ) = D 2 z exp [ 2 z 2 ( 1 n ) 2 α 2 / w 0 2 ] × { J 0 [ ( n 1 ) k ρ α ] } 2 ,
D 2 = C 2 I 0 2 π k ( n 1 ) 2 α 2 .
I ( ρ ) = { J 0 [ ( n 1 ) k ρ α ] } 2 , ( z < L ) ,
E ( in ) ( r ) = E 0 x ^ e i k ( z ε ) = E 0 ( 1 , 0 , 0 ) T e i k ( z ε ) ,
T 1 = 2 n + 1 .
E ( ) ( r ) = E 0 T 1 e i k ( z ε ) e i k n [ t + α ( a ρ ) ] ( 1 , 0 , 0 ) T .
n ^ = ( sin α cos ϕ , sin α sin ϕ , cos α ) T ,
s = z ^ × n ^ ,
= ( sin α sin ϕ , sin α cos ϕ , 0 ) T .
p ^ = z ^ × s ^ ,
= ( cos ϕ , sin ϕ , 0 ) T .
E ( ) ( ρ , ϕ ) = E s ( ) ( ρ , ϕ ) + E p ( ) ( ρ , ϕ ) ,
E s ( ) ( ρ , ϕ ) = [ E ( ) ( ρ , ϕ ) · s ^ ] s ^ ,
= Λ ( ρ ) ( sin 2 ϕ , cos ϕ sin ϕ , 0 ) T ,
E p ( ) ( ρ , ϕ ) = [ E ( ) ( ρ , ϕ ) · p ^ ] p ^ ,
= Λ ( ρ ) ( cos 2 ϕ , cos ϕ sin ϕ , 0 ) T ,
Λ ( ρ ) = E 0 T 1 e i k ( z ε ) e i k n [ t + α ( a ρ ) ] .
T s = 2 n cos α n cos α + 1 n 2 sin 2 α ,
T p = 2 n cos α cos α + n 1 n 2 sin 2 α .
sin β = n sin α .
q ^ = ( cos ( β α ) cos ϕ , cos ( β α ) sin ϕ , sin ( β α ) ) T ,
q ^ · s ^ = 0
q ^ · p ^ = cos ( β α ) .
E ( + ) ( ρ , ϕ ) = T s [ E ( ) ( ρ , ϕ ) · s ^ ] s ^ + T p [ E ( ) ( ρ , ϕ ) · p ^ ] q ^ ,
= T s Λ ( ρ ) ( sin 2 ϕ cos ϕ sin ϕ 0 ) + T p Λ ( ρ ) ( cos ( β α ) cos 2 ϕ cos ( β α ) cos ϕ sin ϕ sin ( β α ) cos ϕ ) .
T p cos ( β α ) = T s .
E ( + ) ( ρ , ϕ ) = T s Λ ( ρ ) ( 1 , 0 , 0 ) T .
E ( out ) ( ρ , ϕ ) = exp ( i k ρ α ) E ( + ) ( ρ , ϕ ) ,
= exp ( i k ρ α ) T s Λ ( ρ ) ( 1 , 0 , 0 ) T .
E ( r ) = 1 2 π × z = 0 [ z ^ × E ( out ) ( r ) ] e i k R R d 2 r ,
E ( r ) = 1 2 π z = 0 ( E x ( out ) ( r ) z 0 E x ( out ) ( r ) x ) e i k R R d 2 r .
E ( r ) = T s 2 π x ^ z = 0 e i k ρ α Λ ( ρ ) z e i k R R d 2 r .
E ( in ) ( r ) = E 0 u ^ e i k ( z ε ) ,
u ^ = A x x ^ + A y y ^ ,
E ( r ) = 1 2 π A x T s x ^ z = 0 e i k ρ α Λ ( ρ ) z e i k R R d 2 r 1 2 π A y T s y ^ z = 0 e i k ρ α Λ ( ρ ) z e i k R R d 2 r .
E ( in ) ( r ) = E 0 ρ ^ e i k ( z ε ) ,
= E 0 ( cos ϕ , sin ϕ , 0 ) T e i k ( z ε ) .
E ( ) ( r ) = E 0 T 1 e i k ( z ε ) e i k n [ t + α ( a ρ ) ] ρ ^ ,
= Λ ( ρ ) ( cos ϕ , sin ϕ , 0 ) T ,
E ( ) ( ρ , ϕ ) · s ^ = 0 ,
E ( ) ( ρ , ϕ ) · p ^ = Λ ( ρ ) .
E ( + ) ( ρ , ϕ ) = T p [ E ( ) ( ρ , ϕ ) · p ^ ] q ^ ,
= T p Λ ( ρ ) ( cos ( β α ) cos ϕ cos ( β α ) sin ϕ sin ( β α ) ) .
E ( out ) ( ρ , ϕ ) = exp ( i k ρ α ) E ( + ) ( ρ , ϕ ) ,
= T s Λ ( ρ ) exp ( i k ρ α ) ( cos ϕ , sin ϕ , 0 ) T .
E ( in ) ( r ) = E 0 ϕ ^ e i k ( z ε ) ,
= E 0 ( sin ϕ , cos ϕ , 0 ) T e i k ( z ε ) .
E ( ) ( r ) = E 0 T 1 e i k ( z ε ) e i k n [ t + α ( a ρ ) ] ϕ ^ = Λ ( ρ ) ( sin ϕ , cos ϕ , 0 ) T .
E ( ) ( ρ , ϕ ) · s ^ = Λ ( ρ ) ,
E ( ) ( ρ , ϕ ) · p ^ = 0 .
E ( + ) ( ρ , ϕ ) = T s [ E ( ) ( ρ , ϕ ) · s ^ ] s ^ ,
= T s Λ ( ρ ) ( sin ϕ , cos ϕ , 0 ) T ,
E ( out ) ( ρ , ϕ ) = exp ( i k ρ α ) E ( + ) ( ρ , ϕ ) ,
= T s Λ ( ρ ) exp ( i k ρ α ) ( sin ϕ , cos ϕ , 0 ) T .
E x ( 0 , 0 , z ) = z 2 π T s 0 a 0 2 π exp [ i k ρ α ] Λ ( ρ ) × e i k R R 2 [ 1 R i k ] ρ d ϕ d ρ ,
= z T s E 0 T 1 e i k n ( t + α a ) e i k ε 0 a e i k ( 1 n ) ρ α × e i k R R 2 [ 1 R i k ] ρ d ρ ,
λ eff = λ cos ( β α ) ,
I ( ρ ) = { J 0 [ ( n 1 ) k ρ α ] } 2 , ( z < L ) ,
E x ( r ) = 1 2 π z = 0 E x ( out ) ( r ) z e i k R R d 2 r ,
= z T s E 0 T 1 2 π e i k n ( t + α a ) e i k ε × 0 a 0 2 π e i k ρ α ( 1 n ) e i k R R 2 [ i k 1 / R ] ρ d ϕ d ρ ,
R = ( x ρ cos ϕ ) 2 + ( y ρ sin ϕ ) 2 + z 2 .
E x ( r ) = z T s 2 π 0 a 0 2 π Λ ( ρ ) cos ϕ e i k ρ α ( i k 1 / R ) × e i k R R 2 ρ d ϕ d ρ ,
E y ( r ) = z T s 2 π 0 a 0 2 π Λ ( ρ ) sin ϕ e i k ρ α ( i k 1 / R ) × e i k R R 2 ρ d ϕ d ρ .
E x ( r ) = z T s 2 π 0 a 0 2 π Λ ( ρ ) sin ϕ e i k ρ α ( i k 1 / R ) × e i k R R 2 ρ d ϕ d ρ ,
E y ( r ) = z T s 2 π 0 a 0 2 π Λ ( ρ ) cos ϕ e i k ρ α ( i k 1 / R ) × e i k R R 2 ρ d ϕ d ρ .
E ( r r ^ ) = i k e i k r 2 π r r ^ × z = 0 z ^ × E ( out ) ( r ) e i k r ^ · r d 2 r .
E i ( r r ^ ) = i k e i k r 2 π r z = 0 E i ( out ) ( r ) e i k r ^ · r d 2 r ( i = x , y , z ) ,
r ^ = ( sin θ cos ϕ , sin θ sin ϕ , cos θ ) T ,
E ( r r ^ ) = ( cos θ E x ( r r ^ ) 0 sin θ cos ϕ E x ( r r ^ ) ) .
E x ( r r ^ ) = i k e i k r 2 π r T s 0 a e i k ρ α Λ ( ρ ) ρ × { 0 2 π e i k ρ sin θ cos ( ϕ ϕ ) d ϕ } d ρ ,
= i k e i k r r T s T 1 E 0 e i k n ( t + α a ) e i k ε × 0 a e i k ρ α ( 1 n ) ρ J 0 ( k ρ sin θ ) d ρ .
I ( θ ) = | E x ( r r ^ ) | 2 = cos 2 θ | E x ( r r ^ ) | 2 ,
E ( r r ^ ) = ( cos θ E x ( r r ^ ) cos θ E y ( r r ^ ) 0 ) ,
E x ( r r ^ ) = k e i k r r T s T 1 E 0 e i k n ( t + α a ) e i k ϵ cos ϕ × 0 a e i k ρ α ( 1 n ) ρ J 1 ( k ρ sin θ ) d ρ ,
E y ( r r ^ ) = k e i k r r T s T 1 E 0 e i k n ( t + α a ) e i k ε sin ϕ × 0 a e i k ρ α ( 1 n ) ρ J 1 ( k ρ sin θ ) d ρ .
E ( r r ^ ) = ( cos θ E x ( r r ^ ) cos θ E y ( r r ^ ) 0 ) ,
E x ( r r ^ ) = k e i k r r T s T 1 E 0 e i k n ( t + α a ) e i k ϵ sin ϕ × 0 a e i k ρ α ( 1 n ) ρ J 1 ( k ρ sin θ ) d ρ ,
E y ( r r ^ ) = k e i k r r T s T 1 E 0 e i k n ( t + α a ) e i k ε cos ϕ × 0 a e i k ρ α ( 1 n ) ρ J 1 ( k ρ sin θ ) d ρ .
F ( k ) = 0 a f ( ρ ) exp [ i k g ( ρ ) ] d ρ .
F ( k ) ( 2 π k ) 1 / 2 exp ( ± i π / 4 ) | g ( ρ c ) | 1 / 2 f ( ρ c ) exp [ i k g ( ρ c ) ] , ( k ) ,
f ( ρ ) = exp ( ρ 2 / w 0 2 ) J 0 ( k ρ ρ z ) ρ ,
g ( ρ ) = ( 1 n ) ρ α + ρ 2 2 z .
g ( ρ ) = ( 1 n ) α + ρ z ,
g ( ρ ) = 1 z .
ρ c = z ( n 1 ) α .
0 ρ c a .
z > a α ( n 1 ) .
× ( A × B ) = A ( · B ) ( A · ) B + ( B · ) A B ( · A ) .
· B = · ( E ( out ) ( r ) e i k R R ) ,
= E x ( out ) ( r ) x e i k R R + E y ( out ) ( r ) y e i k R R + E z ( out ) ( r ) z e i k R R ,
= E x ( out ) ( r ) e i k R [ i k ( x x ) ( x x ) / R ] R 2 + E y ( out ) ( r ) e i k R [ i k ( y y ) ( y y ) / R ] R 2 + E z ( out ) ( r ) e i k R [ i k z z / R ] R 2 .
· B = E x ( out ) ( r ) i k ( x x ) e i k R R 2 + E y ( out ) ( r ) i k ( y y ) e i k R R 2 + E z ( out ) ( r ) i k z e i k R R 2 .
A ( · B ) = z ^ [ E x ( out ) ( r ) i k ( x x ) e i k R R 2 + E y ( out ) ( r ) i k ( y y ) e i k R R 2 + E z ( out ) ( r ) i k z e i k R R 2 ] .
( A · ) B = E ( out ) ( r ) z e i k R R .
( A · ) B = E ( out ) ( r ) i k z e i k R R 2 .
E ( r r ^ ) = i k 2 π z = 0 { z ^ [ E x ( out ) ( r ) ( x x ) + E y ( out ) ( r ) ( y y ) + E z ( out ) ( r ) z ] E ( out ) ( r ) ( z ^ · r ) } e i k R R 2 d 2 r .
E ( r r ^ ) = i k e i k r 2 π r 2 z = 0 { z ^ [ E x ( out ) ( r ) x + E y ( out ) ( r ) y + E z ( out ) ( r ) z ] E ( out ) ( r ) ( z ^ · r ) } e i k r ^ · r d 2 r ,
= i k e i k r 2 π r z = 0 [ z ^ ( E ( out ) ( r ) · r ^ ) E ( out ) ( r ) ( z ^ · r ^ ) ] e i k r ^ · r d 2 r .
A × ( B × C ) = B ( A · C ) C ( A · B ) ,
E ( r r ^ ) = i k e i k r 2 π r r ^ × z = 0 z ^ × E ( out ) ( r ) e i k r ^ · r d 2 r ,

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