Abstract

We have studied the increase of the power contained in Bessel beams generated using the Durnin ring technique, which is compatible with microelectromechanical systems technology. Increasing the ring width to increase the output power will lead to deviation from the Bessel beam profile and its diffraction properties. In this work, the effect of the ring width on the generated beam is investigated. An analytical expression for the generated beam depth of focus (DOF) is obtained. A Fourier optics model is also developed to estimate the transverse field profile. The theoretical predictions are assisted by numerical simulations and experimental measurements. The developed models allow engineering the beam diffraction properties to make the necessary compromise between the DOF and the amount of energy carried by the beam depending on the targeted application.

© 2013 Optical Society of America

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References

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2012 (1)

I. Abdo, N. Ashry, M. Sadek, M. A. Hakim, and D. Khalil, “Effect of ring width on ring generated Bessel beam,” Proc. SPIE 8236, 823608 (2012).
[CrossRef]

2011 (1)

2009 (1)

2002 (1)

1997 (2)

M. Erdely, Z. L. Horvath, G. Zsabó, Zs. Bor, F. K. Tittel, J. R. Cavallaro, and M. C. Smayling, “Generation of diffraction-free beams for applications in optical microlithography,” J. Vac. Sci. Technol. B 15, 287–292 (1997).
[CrossRef]

L. Niggel, T. Lanz, and M. Maier, “Properties of Bessel beams generated by periodic grating of circular symmetry,” J. Opt. Soc. Am. A 14, 27–33 (1997).
[CrossRef]

1993 (1)

T. Wulle and S. Herminghaus, “Nonlinear optics of Bessel beams,” Phys. Rev. Lett. 71, 209–213 (1993).
[CrossRef]

1992 (1)

1988 (1)

1987 (2)

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[CrossRef]

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

1986 (1)

1962 (1)

Abdo, I.

I. Abdo, N. Ashry, M. Sadek, M. A. Hakim, and D. Khalil, “Effect of ring width on ring generated Bessel beam,” Proc. SPIE 8236, 823608 (2012).
[CrossRef]

Ashkin, A.

Ashry, N.

I. Abdo, N. Ashry, M. Sadek, M. A. Hakim, and D. Khalil, “Effect of ring width on ring generated Bessel beam,” Proc. SPIE 8236, 823608 (2012).
[CrossRef]

Bjorkholm, J. E.

Bor, Zs.

M. Erdely, Z. L. Horvath, G. Zsabó, Zs. Bor, F. K. Tittel, J. R. Cavallaro, and M. C. Smayling, “Generation of diffraction-free beams for applications in optical microlithography,” J. Vac. Sci. Technol. B 15, 287–292 (1997).
[CrossRef]

Brown, D.

Cavallaro, J. R.

M. Erdely, Z. L. Horvath, G. Zsabó, Zs. Bor, F. K. Tittel, J. R. Cavallaro, and M. C. Smayling, “Generation of diffraction-free beams for applications in optical microlithography,” J. Vac. Sci. Technol. B 15, 287–292 (1997).
[CrossRef]

Chen, Z.

Chu, S.

Ding, Z.

Dudley, A.

Durnin, J.

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[CrossRef]

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

Dziedzic, J. M.

Eberly, J.

Eberly, J. H.

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

Erdely, M.

M. Erdely, Z. L. Horvath, G. Zsabó, Zs. Bor, F. K. Tittel, J. R. Cavallaro, and M. C. Smayling, “Generation of diffraction-free beams for applications in optical microlithography,” J. Vac. Sci. Technol. B 15, 287–292 (1997).
[CrossRef]

Forbes, A.

Friberg, A. T.

Fujiwara, S.

Goto, M.

Hakim, M. A.

I. Abdo, N. Ashry, M. Sadek, M. A. Hakim, and D. Khalil, “Effect of ring width on ring generated Bessel beam,” Proc. SPIE 8236, 823608 (2012).
[CrossRef]

Herminghaus, S.

T. Wulle and S. Herminghaus, “Nonlinear optics of Bessel beams,” Phys. Rev. Lett. 71, 209–213 (1993).
[CrossRef]

Horvath, Z. L.

M. Erdely, Z. L. Horvath, G. Zsabó, Zs. Bor, F. K. Tittel, J. R. Cavallaro, and M. C. Smayling, “Generation of diffraction-free beams for applications in optical microlithography,” J. Vac. Sci. Technol. B 15, 287–292 (1997).
[CrossRef]

Huang, H.

Khalil, D.

I. Abdo, N. Ashry, M. Sadek, M. A. Hakim, and D. Khalil, “Effect of ring width on ring generated Bessel beam,” Proc. SPIE 8236, 823608 (2012).
[CrossRef]

Khilo, N.

Lanz, T.

Lin, Y.

Maier, M.

Marcuse, D.

D. Marcuse, Light Transmission Optics, 2nd ed. (Van Nostrand Reinhold, 1982).

Miceli, J. J.

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

Nelson, J. S.

Niggel, L.

Ramachandran, S.

Ren, H.

Sadek, M.

I. Abdo, N. Ashry, M. Sadek, M. A. Hakim, and D. Khalil, “Effect of ring width on ring generated Bessel beam,” Proc. SPIE 8236, 823608 (2012).
[CrossRef]

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 2007).

Seka, W.

Smayling, M. C.

M. Erdely, Z. L. Horvath, G. Zsabó, Zs. Bor, F. K. Tittel, J. R. Cavallaro, and M. C. Smayling, “Generation of diffraction-free beams for applications in optical microlithography,” J. Vac. Sci. Technol. B 15, 287–292 (1997).
[CrossRef]

Steinvurzel, P.

Tantiwanichapan, K.

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 2007).

Tittel, F. K.

M. Erdely, Z. L. Horvath, G. Zsabó, Zs. Bor, F. K. Tittel, J. R. Cavallaro, and M. C. Smayling, “Generation of diffraction-free beams for applications in optical microlithography,” J. Vac. Sci. Technol. B 15, 287–292 (1997).
[CrossRef]

Turunen, J.

Vasara, A.

Vasilyeu, R.

Wulle, T.

T. Wulle and S. Herminghaus, “Nonlinear optics of Bessel beams,” Phys. Rev. Lett. 71, 209–213 (1993).
[CrossRef]

Zhao, Y.

Zsabó, G.

M. Erdely, Z. L. Horvath, G. Zsabó, Zs. Bor, F. K. Tittel, J. R. Cavallaro, and M. C. Smayling, “Generation of diffraction-free beams for applications in optical microlithography,” J. Vac. Sci. Technol. B 15, 287–292 (1997).
[CrossRef]

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

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

J. Vac. Sci. Technol. B (1)

M. Erdely, Z. L. Horvath, G. Zsabó, Zs. Bor, F. K. Tittel, J. R. Cavallaro, and M. C. Smayling, “Generation of diffraction-free beams for applications in optical microlithography,” J. Vac. Sci. Technol. B 15, 287–292 (1997).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Phys. Rev. Lett. (2)

T. Wulle and S. Herminghaus, “Nonlinear optics of Bessel beams,” Phys. Rev. Lett. 71, 209–213 (1993).
[CrossRef]

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

Proc. SPIE (1)

I. Abdo, N. Ashry, M. Sadek, M. A. Hakim, and D. Khalil, “Effect of ring width on ring generated Bessel beam,” Proc. SPIE 8236, 823608 (2012).
[CrossRef]

Other (2)

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 2007).

D. Marcuse, Light Transmission Optics, 2nd ed. (Van Nostrand Reinhold, 1982).

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Figures (13)

Fig. 1.
Fig. 1.

Geometrical parameters of optical elements used for Bessel beam generation with Durnin ring.

Fig. 2.
Fig. 2.

Large-width ring representation as subtraction of two concentric circular apertures.

Fig. 3.
Fig. 3.

|Jo|2 Bessel intensity compared to plots of the Fourier transform of the large-width ring different values for ΔR/Rav.

Fig. 4.
Fig. 4.

Evaluated overlap integral between Jo Bessel profile and large-width ring Fourier transform as a function of the normalized ring width.

Fig. 5.
Fig. 5.

Procedure for the numerical algorithm applied.

Fig. 6.
Fig. 6.

Analytical axial intensity compared with numerical axial intensity for different spot size. Rav=600μm and ΔR=200μm.

Fig. 7.
Fig. 7.

Axial intensity as a function of the propagation distance for different values of normalized width ΔR/Rav.

Fig. 8.
Fig. 8.

Analytical and numerical DOF as a function of the normalized width for Rav=500μm and Rav=750μm.

Fig. 9.
Fig. 9.

Efficiency as a function of normalized width for Rav=500μm and Rav=750μm.

Fig. 10.
Fig. 10.

Experimental setup used for Bessel beam generation.

Fig. 11.
Fig. 11.

Captured beam using CMOS camera.

Fig. 12.
Fig. 12.

Axial intensity as a function of the propagation distance for a ring with Rav=600μm and ΔR=200μm.

Fig. 13.
Fig. 13.

Axial intensity as a function of the propagation distance for a ring with Rav=700μm and ΔR=400μm.

Equations (34)

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ktr=ksinθ,
tanθ=Rf,
Zmax=D2tanθ,
F(νx,νy)=f(x,y)exp(j2π(νxx+νyy))dxdy,
g(x,y)=jλfexp(j2kf)F(xλf,yλf).
I(x,y)=|g(x,y)|2=1(λf)2|F(xλf,yλf)|2.
E(ρ,z)=i=1NEo(kzi)Jo(ktriρ)ejkziz,
E(0,z)=i=1NEo(kzi)ejkziz.
E(0,z)=kz1kz2Eo(kz)ejkzzdkz,
E(0,z)=Δkzejkzavzsinc(Δkzz2),
I(0,z)=sinc2(Δkzz2).
Zs=2πΔkz.
Δkz=Ravf2kΔR,
Zs=2πf2RavkΔR.
DOF=2.8f2RavkΔR.
Pin=2πEo24wo2,
Pout=2πEo24wo2(exp(2R12wo2)exp(2R22wo2)).
Efficiency=PoutPin=(exp(2R12wo2)exp(2R22wo2)).
F(νx,νy)=F2(νx,νy)F1(νx,νy),
g(ρ)=jλfIiexp(j2kf)[R2J1(2πR2ρλf)ρλfR1J1(2πR1ρλf)ρλf],
I(ρ)=Ii(λf)2[R2J1(2πR2ρλf)ρλfR1J1(2πR1ρλf)ρλf]2.
J1(x±Δx/2)x±Δx/2J1(x)x±Δx2ddx(J1(x)x)
ddx[J1(x)x]=J2(x)x.
J1(x±Δx/2)x±Δx/2J1(x)xΔx2J2(x)x.
g(ρ)jλfIiexp(j2kf)(2π)[J1(2πλRavfρ)2πλRavfρ(R22R12)ΔR(R22+R12)2RavJ2(2πλRavfρ)].
R22R12=2RavΔR,
R22+R12=2Rav2+(ΔR)22.
g(ρ)jλfIiexp(j2kf)(2π)(ΔRRav)[2J1(ktravρ)ktravρJ2(ktravρ)(ΔR2Rav)2J2(ktravρ)].
J0(x)+J2(x)=2xJ1(x),
g(ρ)jλfIiexp(j2kf)(2π)(ΔRRav)[Jo(ktravρ)(ΔR2Rav)2J2(ktravρ)].
g(ρ)jλfIiexp(j2kf)(Ring Area)[Jo(ktravρ)(ΔR2Rav)2J2(ktravρ)].
T=|2π0g(ρ)Jo(ktravρ)ρdρ|22π0|g(ρ)|2ρdρ2π0|Jo(ktravρ)|2ρdρ.
H(ktr)=exp(jdk2ktr2),
t(ρ)=exp(jkρ22f).

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