Abstract

We propose an analytical solution of the focal ring generated at the focus of a toric lens. The analytical field of the focal ring is used with a Fourier transform lens to generate a Bessel beam. A comparative analysis between the use of an illuminated annular aperture, an axicon, and a toric lens to generate a Bessel beam is performed, and the benefits and drawbacks of each are discussed. This highlights the advantages of using a toric lens with a Gaussian beam to produce a focal line of increasing intensity, which is advantageous for applications such as high depth-of-field microscopy.

© 2020 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
  11. P.-A. Belanger and M. Rioux, “Diffraction ring pattern at the focal plane of a spherical lens–axicon doublet,” Can. J. Phys. 54, 1774–1780 (1976).
    [Crossref]
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    [Crossref]
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    [Crossref]
  21. M. Dong and J. Pu, “On-axis irradiance distribution of axicons illuminated by spherical wave,” Opt. Laser Technol. 39, 1258–1261 (2008).
    [Crossref]
  22. M. V. Perez, C. Gomez-Reino, and J. Cuadrado, “Diffraction patterns and zone plates produced by thin linear axicons,” Opt. Laser Technol. 33, 1161–1176 (1986).
    [Crossref]
  23. K.-S. Lee and J. P. Rolland, “Bessel beam spectral-domain high-resolution optical coherence tomography with micro-optic axicon providing extended focusing range,” Opt. Lett. 33, 1696–1698 (2008).
    [Crossref]
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    [Crossref]
  25. R. Dharmavarapu, S. Bhattacharya, and S. Juodkazis, “Diffractive optics for axial intensity shaping of Bessel beams,” J. Opt. 20, 085606 (2018).
    [Crossref]

2020 (1)

D. R. Beaulieu, I. G. Davison, K. Kılıc, T. G. Bifano, and J. Mertz, “Simultaneous multiplane imaging with reverberation two-photon microscopy,” Nat. Methods 17, 283–286 (2020).
[Crossref]

2018 (1)

R. Dharmavarapu, S. Bhattacharya, and S. Juodkazis, “Diffractive optics for axial intensity shaping of Bessel beams,” J. Opt. 20, 085606 (2018).
[Crossref]

2013 (1)

2010 (1)

2009 (1)

2008 (3)

2006 (1)

2004 (1)

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46, 15–28 (2004).
[Crossref]

2001 (1)

J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197, 239–245 (2001).
[Crossref]

1998 (1)

1997 (1)

B. Hafizi, E. Esarey, and P. Sprangle, “Laser-driven acceleration with Bessel beams,” Phys. Rev. E 55, 3539 (1997).
[Crossref]

1995 (1)

1992 (1)

N. Davidson, A. A. Friesem, and E. Hasman, “Efficient formation of nondiffracting beams with uniform intensity along the propagation direction,” Opt. Commun. 88, 326–330 (1992).
[Crossref]

1987 (1)

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

1986 (1)

M. V. Perez, C. Gomez-Reino, and J. Cuadrado, “Diffraction patterns and zone plates produced by thin linear axicons,” Opt. Laser Technol. 33, 1161–1176 (1986).
[Crossref]

1978 (1)

1976 (1)

P.-A. Belanger and M. Rioux, “Diffraction ring pattern at the focal plane of a spherical lens–axicon doublet,” Can. J. Phys. 54, 1774–1780 (1976).
[Crossref]

1969 (1)

1960 (1)

Arlt, J.

J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197, 239–245 (2001).
[Crossref]

Beaulieu, D. R.

D. R. Beaulieu, I. G. Davison, K. Kılıc, T. G. Bifano, and J. Mertz, “Simultaneous multiplane imaging with reverberation two-photon microscopy,” Nat. Methods 17, 283–286 (2020).
[Crossref]

Belanger, P. A.

Belanger, P.-A.

P.-A. Belanger and M. Rioux, “Diffraction ring pattern at the focal plane of a spherical lens–axicon doublet,” Can. J. Phys. 54, 1774–1780 (1976).
[Crossref]

Bhattacharya, S.

R. Dharmavarapu, S. Bhattacharya, and S. Juodkazis, “Diffractive optics for axial intensity shaping of Bessel beams,” J. Opt. 20, 085606 (2018).
[Crossref]

Bifano, T. G.

D. R. Beaulieu, I. G. Davison, K. Kılıc, T. G. Bifano, and J. Mertz, “Simultaneous multiplane imaging with reverberation two-photon microscopy,” Nat. Methods 17, 283–286 (2020).
[Crossref]

Brzobohaty, O.

Cao, Z.

Chebbi, B.

Cizmar, T.

Cuadrado, J.

M. V. Perez, C. Gomez-Reino, and J. Cuadrado, “Diffraction patterns and zone plates produced by thin linear axicons,” Opt. Laser Technol. 33, 1161–1176 (1986).
[Crossref]

Davidson, N.

N. Davidson, A. A. Friesem, and E. Hasman, “Efficient formation of nondiffracting beams with uniform intensity along the propagation direction,” Opt. Commun. 88, 326–330 (1992).
[Crossref]

Davison, I. G.

D. R. Beaulieu, I. G. Davison, K. Kılıc, T. G. Bifano, and J. Mertz, “Simultaneous multiplane imaging with reverberation two-photon microscopy,” Nat. Methods 17, 283–286 (2020).
[Crossref]

De Koninck, Y.

Dharmavarapu, R.

R. Dharmavarapu, S. Bhattacharya, and S. Juodkazis, “Diffractive optics for axial intensity shaping of Bessel beams,” J. Opt. 20, 085606 (2018).
[Crossref]

Dholakia, K.

T. Cizmar and K. Dholakia, “Tunable Bessel light modes: engineering the axial propagation,” Opt. Express 17, 15558–15570 (2009).
[Crossref]

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46, 15–28 (2004).
[Crossref]

J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197, 239–245 (2001).
[Crossref]

Dong, M.

M. Dong and J. Pu, “On-axis irradiance distribution of axicons illuminated by spherical wave,” Opt. Laser Technol. 39, 1258–1261 (2008).
[Crossref]

Dufour, P.

Durnin, J.

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

Eberly, J. H.

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

Esarey, E.

B. Hafizi, E. Esarey, and P. Sprangle, “Laser-driven acceleration with Bessel beams,” Phys. Rev. E 55, 3539 (1997).
[Crossref]

Friesem, A. A.

N. Davidson, A. A. Friesem, and E. Hasman, “Efficient formation of nondiffracting beams with uniform intensity along the propagation direction,” Opt. Commun. 88, 326–330 (1992).
[Crossref]

Garces-Chavez, V.

J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197, 239–245 (2001).
[Crossref]

Golub, I.

Gomez-Reino, C.

M. V. Perez, C. Gomez-Reino, and J. Cuadrado, “Diffraction patterns and zone plates produced by thin linear axicons,” Opt. Laser Technol. 33, 1161–1176 (1986).
[Crossref]

Goodell, J. B.

Gradshteyn, I.

I. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Elsevier/Academic, 2007).

Hafizi, B.

B. Hafizi, E. Esarey, and P. Sprangle, “Laser-driven acceleration with Bessel beams,” Phys. Rev. E 55, 3539 (1997).
[Crossref]

Hasman, E.

N. Davidson, A. A. Friesem, and E. Hasman, “Efficient formation of nondiffracting beams with uniform intensity along the propagation direction,” Opt. Commun. 88, 326–330 (1992).
[Crossref]

Hecht, E.

E. Hecht, Optics, 4th ed. (Pearson, 2002).

Jiang, Z.

Juodkazis, S.

R. Dharmavarapu, S. Bhattacharya, and S. Juodkazis, “Diffractive optics for axial intensity shaping of Bessel beams,” J. Opt. 20, 085606 (2018).
[Crossref]

Kilic, K.

D. R. Beaulieu, I. G. Davison, K. Kılıc, T. G. Bifano, and J. Mertz, “Simultaneous multiplane imaging with reverberation two-photon microscopy,” Nat. Methods 17, 283–286 (2020).
[Crossref]

Lee, K.-S.

Liu, Z.

Lu, Q.

McCarthy, N.

McGloin, D.

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46, 15–28 (2004).
[Crossref]

Mertz, J.

D. R. Beaulieu, I. G. Davison, K. Kılıc, T. G. Bifano, and J. Mertz, “Simultaneous multiplane imaging with reverberation two-photon microscopy,” Nat. Methods 17, 283–286 (2020).
[Crossref]

Miceli, J.

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

Nowacki, D.

Perez, M. V.

M. V. Perez, C. Gomez-Reino, and J. Cuadrado, “Diffraction patterns and zone plates produced by thin linear axicons,” Opt. Laser Technol. 33, 1161–1176 (1986).
[Crossref]

Piche, M.

Pu, J.

M. Dong and J. Pu, “On-axis irradiance distribution of axicons illuminated by spherical wave,” Opt. Laser Technol. 39, 1258–1261 (2008).
[Crossref]

Rioux, M.

P. A. Belanger and M. Rioux, “Ring pattern of a lens–axicon doublet illuminated by a Gaussian beam,” Appl. Opt. 17, 1080–1088 (1978).
[Crossref]

P.-A. Belanger and M. Rioux, “Diffraction ring pattern at the focal plane of a spherical lens–axicon doublet,” Can. J. Phys. 54, 1774–1780 (1976).
[Crossref]

Rolland, J. P.

Ryzhik, I. M.

I. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Elsevier/Academic, 2007).

Sedukhin, A. G.

Shaw, D.

Sibbett, W.

J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197, 239–245 (2001).
[Crossref]

Sprangle, P.

B. Hafizi, E. Esarey, and P. Sprangle, “Laser-driven acceleration with Bessel beams,” Phys. Rev. E 55, 3539 (1997).
[Crossref]

Wang, K.

Welford, W. T.

Wu, Q.

Zemanek, P.

Appl. Opt. (4)

Can. J. Phys. (1)

P.-A. Belanger and M. Rioux, “Diffraction ring pattern at the focal plane of a spherical lens–axicon doublet,” Can. J. Phys. 54, 1774–1780 (1976).
[Crossref]

Contemp. Phys. (1)

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46, 15–28 (2004).
[Crossref]

J. Opt. (1)

R. Dharmavarapu, S. Bhattacharya, and S. Juodkazis, “Diffractive optics for axial intensity shaping of Bessel beams,” J. Opt. 20, 085606 (2018).
[Crossref]

J. Opt. Soc. Am. (1)

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

Nat. Methods (1)

D. R. Beaulieu, I. G. Davison, K. Kılıc, T. G. Bifano, and J. Mertz, “Simultaneous multiplane imaging with reverberation two-photon microscopy,” Nat. Methods 17, 283–286 (2020).
[Crossref]

Opt. Commun. (2)

J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun. 197, 239–245 (2001).
[Crossref]

N. Davidson, A. A. Friesem, and E. Hasman, “Efficient formation of nondiffracting beams with uniform intensity along the propagation direction,” Opt. Commun. 88, 326–330 (1992).
[Crossref]

Opt. Express (2)

Opt. Laser Technol. (2)

M. Dong and J. Pu, “On-axis irradiance distribution of axicons illuminated by spherical wave,” Opt. Laser Technol. 39, 1258–1261 (2008).
[Crossref]

M. V. Perez, C. Gomez-Reino, and J. Cuadrado, “Diffraction patterns and zone plates produced by thin linear axicons,” Opt. Laser Technol. 33, 1161–1176 (1986).
[Crossref]

Opt. Lett. (3)

Phys. Rev. E (1)

B. Hafizi, E. Esarey, and P. Sprangle, “Laser-driven acceleration with Bessel beams,” Phys. Rev. E 55, 3539 (1997).
[Crossref]

Phys. Rev. Lett. (1)

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

Other (4)

Synopsys, “CODE V, Version 11.2.12760707” (Mountain View, California, 2019).

E. Hecht, Optics, 4th ed. (Pearson, 2002).

I. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Elsevier/Academic, 2007).

Wolfram Research Inc., “Mathematica, Version 12.0,” (2019).

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Figures (4)

Fig. 1.
Fig. 1. Toric lens and the coordinate system used for the diffractional analysis.
Fig. 2.
Fig. 2. Field solution at the focus of a toric lens with $a = 2\;{\rm mm} $, $f = 10\;{\rm mm} $, and $\lambda = 587\;{\rm nm} $: (a) intensity and (b) phase. The maximum relative error between the numerical and analytical solutions for the toric lens is 1.2%.
Fig. 3.
Fig. 3. (a) Radial intensity profiles at a distance ${f_{\textit{FT}}}$ after the Fourier transform lens and (b) axial intensity profiles after the Fourier transform lens. The maximum relative error between the numerical solutions and the equation ${J_0}(kaq^\prime \!/\!{f_{\textit{FT}}})$ is 2.5%.
Fig. 4.
Fig. 4. Axial intensity profile from CODE V simulations for a toric lens with parameters $a = 2\;{\rm mm} $ and $f = 10\;{\rm mm} $ coupled to a single lens ${f_{\textit{FT}}} = 10\;{\rm mm} $. Each curve represents the profile for a given Gaussian beam diameter ($w$). One can notice the displacement of the curve and a maximum intensity to farther axial distances.

Tables (1)

Tables Icon

Table 1. General Comportment of the Axial Intensity Profiles of the Bessel Beams Generated by Three Different Components with Their Advantages and Disadvantagesa

Equations (21)

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d E = ϵ A r E 0 ( x , y ) e i ( ω t k r + ϕ ( x , y ) ) d s ,
r = z 2 + ( X x ) 2 + ( Y y ) 2 .
r z ( 1 + 1 2 ( X x z ) 2 + 1 2 ( Y y z ) 2 ) .
ϕ ( x , y ) = k [ ( x 2 + y 2 ) a ] 2 f 2 ,
d E = ϵ A f E 0 ( ρ , θ ) e i ( ω t k f ) e i k ( X 2 + Y 2 ) 2 f e i k ( x 2 + y 2 ) 2 f e i k ( X x Y y ) f e i k ( x 2 + y 2 a ) 2 2 f d s .
x = ρ cos θ ; y = ρ sin θ ; X = q cos Θ , Y = q sin Θ ; d s = ρ d ρ d θ ,
d E = ϵ A f E 0 ( ρ , θ ) e i ( ω t k f ) e i k q 2 2 f e i k ρ 2 2 f × e i k ρ q f cos ( θ Θ ) e i k ( ρ a ) 2 2 f ρ d ρ d θ .
E ( q ) = ϵ A f e i ( ω t k f + i k a 2 2 f ) e i k q 2 2 f ρ = 0 2 a θ = 0 2 π E 0 ( ρ , θ ) e i k ρ q f cos ( θ Θ ) e i k ρ a f ρ d ρ d θ .
θ = 0 2 π e i k ρ q f cos ( θ Θ ) = 2 π J 0 ( k ρ q f ) ,
E ( q ) = 2 π ϵ A f e i ( ω t k f + i k a 2 2 f ) e i k q 2 2 f × ρ = 0 2 a E 0 ( ρ ) J 0 ( k ρ q f ) e i k ρ a f ρ d ρ .
t = ξ r 2 a ; ξ = k 2 a 2 f .
J 0 ( λ t ) = n = 0 ( 1 λ 2 ) n n ! ( t 2 ) n J n ( t ) ,
g ( ξ ) n + 1 , n = 0 ξ e i t J n ( t ) t n + 1 d t ,
= e i ξ ξ n + 1 [ J n + 1 ( ξ ) + ξ 2 n + 3 { J n + 2 ( ξ ) i J n + 1 ( ξ ) } ] ,
E ( q ) = 2 π ϵ A f e i ( ω t k f + k a 2 2 f ) e i k q 2 f 4 a 2 ξ e i ξ n = 0 ( ξ 2 ( 1 [ q a ] 2 ) ) n n ! [ J n + 1 ( ξ ) + ξ ( 2 n + 3 ) { J n + 2 ( ξ ) i J n + 1 ( ξ ) } ] ,
J n ( ξ ) 2 π ξ cos ( ξ π 4 n π 2 ) .
E ( q ) = 2 π ϵ A f 2 e i ( ω t k f + k a 2 2 f ) e i k q 2 f 8 k a 4 e i 3 π 4 n = 0 ( i k a 2 f ( 1 [ q a ] 2 ) ) n n ! ( 2 n + 3 ) .
e i 3 π 4 n = 0 ( i k a 2 f ( 1 [ q a ] 2 ) ) n n ! ( 2 n + 3 ) = 1 F 1 ( 3 2 5 2 | i k a 2 f [ ( q a ) 2 1 ] ) .
E ~ ( q ) = i λ f FT e i k 2 f T F { E ( q ) } .
E ~ ( q ) = i λ f FT e i k 2 f 0 J 0 ( 2 π q q λ f FT ) E ( q ) q d q .
E ~ ( 0 , z ) = i λ f e i k ( f + z ) 0 e i π λ f ( 1 z f ) q 2 E ( q ) q d q .