Abstract

Two implementations are identified to create a Bessel beam directly, i.e. without the spatial filtering of an initially Gaussian beam. The first implementation is based on a resonator configuration whose lowest-loss transverse mode is a Bessel beam. Numerical simulation to corroborate the geometrical optical arguments is presented. The second implementation is based on the theorem that the angular-plane wave spectrum of a Bessel beam is composed of a cone of wave vectors. This cone is also generated through a phase-matching condition in a four-wave mixing process. This leads to the conclusion that anti-Stokes radiation generated in a nonlinear material will leave the substrate under the form of a Bessel beam.

© 2002 Optical Society of America

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References

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  1. N. Hodgson, H. Weber, Optical Resonators (Springer, New York, 1997).
  2. A. Onae, T. Kurosawa, Y. Miki, E. Sakuma, “Nearly diffraction-free CO2 laser beam,” J. Appl. Phys. 72, 4529–4532 (1992).
    [CrossRef]
  3. P. Paakkonen, J. Turunen, “Resonators with Bessel-Gauss modes,” Optics Commun. 156, 359–366 (1998).
    [CrossRef]
  4. G. Scott, N. McArdle, “Efficient generation of nearly diffraction-free beams using an axicon,” Opt. Eng. 31, 2640–2643 (1992).
    [CrossRef]
  5. W. Lauterborn, T. Kurz, M. Wiesenfeldt, Coherent Optics, 1st ed. (Springer-Verlag, Berlin, 1993).
  6. P. W. Milonni, J. H. Eberly, Lasers (Wiley, New York, 1988).
  7. R. Bracewell, The Fourier Transform and Its Applications, 3rd ed. (McGraw-Hill, New York, 2001).
  8. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980).
  9. R. Sutherland, Handbook of Nonlinear Optics (Marcel Dekker, New York, 1996).
  10. J. E. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [CrossRef] [PubMed]
  11. J. Reintjes, M. Bashkansky, “Stimulated Raman and Brillouin Scattering,” in Handbook of Optics IV, M. Bass, ed. (McGraw-Hill, New York, 2001), pp. 18.1–18.61.
  12. C. Pavé, P. Bélanger, “Custom laser resonators using graded phase mirrors: circular geometry,” IEEE J. Quantum Electron. 30, 1141–1148 (1994).
    [CrossRef]
  13. P. Bélanger, R. Lachance, C. Pavé, “Super Gaussian output from a CO2 laser by using a graded-phase mirror resonator,” Opt. Lett. 17, 739–741 (1992).
    [CrossRef] [PubMed]

1998 (1)

P. Paakkonen, J. Turunen, “Resonators with Bessel-Gauss modes,” Optics Commun. 156, 359–366 (1998).
[CrossRef]

1994 (1)

C. Pavé, P. Bélanger, “Custom laser resonators using graded phase mirrors: circular geometry,” IEEE J. Quantum Electron. 30, 1141–1148 (1994).
[CrossRef]

1992 (3)

P. Bélanger, R. Lachance, C. Pavé, “Super Gaussian output from a CO2 laser by using a graded-phase mirror resonator,” Opt. Lett. 17, 739–741 (1992).
[CrossRef] [PubMed]

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

A. Onae, T. Kurosawa, Y. Miki, E. Sakuma, “Nearly diffraction-free CO2 laser beam,” J. Appl. Phys. 72, 4529–4532 (1992).
[CrossRef]

1987 (1)

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

Bashkansky, M.

J. Reintjes, M. Bashkansky, “Stimulated Raman and Brillouin Scattering,” in Handbook of Optics IV, M. Bass, ed. (McGraw-Hill, New York, 2001), pp. 18.1–18.61.

Bélanger, P.

C. Pavé, P. Bélanger, “Custom laser resonators using graded phase mirrors: circular geometry,” IEEE J. Quantum Electron. 30, 1141–1148 (1994).
[CrossRef]

P. Bélanger, R. Lachance, C. Pavé, “Super Gaussian output from a CO2 laser by using a graded-phase mirror resonator,” Opt. Lett. 17, 739–741 (1992).
[CrossRef] [PubMed]

Bracewell, R.

R. Bracewell, The Fourier Transform and Its Applications, 3rd ed. (McGraw-Hill, New York, 2001).

Durnin, J. E.

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

Eberly, J. H.

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

P. W. Milonni, J. H. Eberly, Lasers (Wiley, New York, 1988).

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980).

Hodgson, N.

N. Hodgson, H. Weber, Optical Resonators (Springer, New York, 1997).

Kurosawa, T.

A. Onae, T. Kurosawa, Y. Miki, E. Sakuma, “Nearly diffraction-free CO2 laser beam,” J. Appl. Phys. 72, 4529–4532 (1992).
[CrossRef]

Kurz, T.

W. Lauterborn, T. Kurz, M. Wiesenfeldt, Coherent Optics, 1st ed. (Springer-Verlag, Berlin, 1993).

Lachance, R.

Lauterborn, W.

W. Lauterborn, T. Kurz, M. Wiesenfeldt, Coherent Optics, 1st ed. (Springer-Verlag, Berlin, 1993).

McArdle, N.

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

Miceli, J. J.

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

Miki, Y.

A. Onae, T. Kurosawa, Y. Miki, E. Sakuma, “Nearly diffraction-free CO2 laser beam,” J. Appl. Phys. 72, 4529–4532 (1992).
[CrossRef]

Milonni, P. W.

P. W. Milonni, J. H. Eberly, Lasers (Wiley, New York, 1988).

Onae, A.

A. Onae, T. Kurosawa, Y. Miki, E. Sakuma, “Nearly diffraction-free CO2 laser beam,” J. Appl. Phys. 72, 4529–4532 (1992).
[CrossRef]

Paakkonen, P.

P. Paakkonen, J. Turunen, “Resonators with Bessel-Gauss modes,” Optics Commun. 156, 359–366 (1998).
[CrossRef]

Pavé, C.

C. Pavé, P. Bélanger, “Custom laser resonators using graded phase mirrors: circular geometry,” IEEE J. Quantum Electron. 30, 1141–1148 (1994).
[CrossRef]

P. Bélanger, R. Lachance, C. Pavé, “Super Gaussian output from a CO2 laser by using a graded-phase mirror resonator,” Opt. Lett. 17, 739–741 (1992).
[CrossRef] [PubMed]

Reintjes, J.

J. Reintjes, M. Bashkansky, “Stimulated Raman and Brillouin Scattering,” in Handbook of Optics IV, M. Bass, ed. (McGraw-Hill, New York, 2001), pp. 18.1–18.61.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980).

Sakuma, E.

A. Onae, T. Kurosawa, Y. Miki, E. Sakuma, “Nearly diffraction-free CO2 laser beam,” J. Appl. Phys. 72, 4529–4532 (1992).
[CrossRef]

Scott, G.

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

Sutherland, R.

R. Sutherland, Handbook of Nonlinear Optics (Marcel Dekker, New York, 1996).

Turunen, J.

P. Paakkonen, J. Turunen, “Resonators with Bessel-Gauss modes,” Optics Commun. 156, 359–366 (1998).
[CrossRef]

Weber, H.

N. Hodgson, H. Weber, Optical Resonators (Springer, New York, 1997).

Wiesenfeldt, M.

W. Lauterborn, T. Kurz, M. Wiesenfeldt, Coherent Optics, 1st ed. (Springer-Verlag, Berlin, 1993).

IEEE J. Quantum Electron. (1)

C. Pavé, P. Bélanger, “Custom laser resonators using graded phase mirrors: circular geometry,” IEEE J. Quantum Electron. 30, 1141–1148 (1994).
[CrossRef]

J. Appl. Phys. (1)

A. Onae, T. Kurosawa, Y. Miki, E. Sakuma, “Nearly diffraction-free CO2 laser beam,” J. Appl. Phys. 72, 4529–4532 (1992).
[CrossRef]

Opt. Eng. (1)

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

Opt. Lett. (1)

Optics Commun. (1)

P. Paakkonen, J. Turunen, “Resonators with Bessel-Gauss modes,” Optics Commun. 156, 359–366 (1998).
[CrossRef]

Phys. Rev. Lett. (1)

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

Other (7)

J. Reintjes, M. Bashkansky, “Stimulated Raman and Brillouin Scattering,” in Handbook of Optics IV, M. Bass, ed. (McGraw-Hill, New York, 2001), pp. 18.1–18.61.

W. Lauterborn, T. Kurz, M. Wiesenfeldt, Coherent Optics, 1st ed. (Springer-Verlag, Berlin, 1993).

P. W. Milonni, J. H. Eberly, Lasers (Wiley, New York, 1988).

R. Bracewell, The Fourier Transform and Its Applications, 3rd ed. (McGraw-Hill, New York, 2001).

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980).

R. Sutherland, Handbook of Nonlinear Optics (Marcel Dekker, New York, 1996).

N. Hodgson, H. Weber, Optical Resonators (Springer, New York, 1997).

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Figures (7)

Fig. 1
Fig. 1

(a), Transformation of a plane wave in a conical wave by transmission through air axicon; (b), Resonator generating a Bessel beam.

Fig. 2
Fig. 2

Convergence rate of the Fox-Li iteration process.

Fig. 3
Fig. 3

Comparison between theoretical and numerical solutions.

Fig. 4
Fig. 4

Phase of the Bessel mode along the x axis.

Fig. 5
Fig. 5

Radial part of the spatial-frequency distribution.

Fig. 6
Fig. 6

Energy level diagram for Coherent Anti-Stokes Raman Scattering.

Fig. 7
Fig. 7

Wave vector diagrams for Coherent Anti-Stokes Raman Scattering.

Equations (15)

Equations on this page are rendered with MathJax. Learn more.

L=r2n-1γ,
Ex, y, z=E0J0αρexp-iωt-βz, k=2πλ=ωc, ρ2=x2+y2,α=k2-β21/2=k1-β2/k21/2.
β/k=2L/4L2+r21/2
α=kr4L2+r21/2.
α=n-1γk1+n-12γ21/2,
λ=10.6 μm, n=2.4 zinc selenide, γ=0.5°=8.7 mrad, r=0.75/2=9.53 mm,
Eρ=J0αρexp-ρ2w2,
E˜s= w22exp- w2α24exp- w2s24I0αw24 s,
sI0αw24 s=αI1αw24 s,
Imz= ez2πz m=0, 1.
ωa=2ωp-ωs.
ka=2kp-ks.
ka=qx, qy, qz=α, β, qz,
cos θ=4kp2+4ka2+4ks24kpka.
kj=ωjnjc j=p, s, a.

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