Abstract

Gaussian-apodized Bessel beams can be used to create a Bessel-like axial line focus at a distance from the focusing lens. For many applications it is desirable to create an axial intensity profile that is uniform along the Bessel zone. In this article, we show that this can be accomplished through phase-only shaping of the wavefront in the far field where the beam has an annular ring structure with a Gaussian cross section. We use a one-dimensional transform to map the radial input field to the axial Bessel field and then optimized the axial intensity with a Gerchberg-Saxton algorithm. By separating out the quadratic portion of the shaping phase the algorithm converges more rapidly.

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References

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2012 (4)

2011 (1)

T. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods8, 417–423 (2011).
[CrossRef] [PubMed]

2010 (1)

2009 (1)

T. Čižmár and K. Dholakia, “Tunable Bessel light modes: engineering the axial propagation,” Opt. Express17(18), 15558–15570 (2009).
[CrossRef]

2008 (3)

2006 (1)

2004 (2)

2002 (3)

2001 (1)

J. Arlt, V. Garcés-Chávez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun.1974–6), 239–245 (2001).
[CrossRef]

2000 (1)

1998 (1)

M. Honkanen and J. Turunen, “Tandem systems for efficient generation of uniform-axial-intensity Bessel fields,” Opt. Commun.154(5), 368375 (1998).
[CrossRef]

1997 (2)

B. Hafizi, E. Esarey, and P. Sprangle, “Laser-driven acceleration with Bessel beams,” Phys. Rev. E55(3), 3539–3545 (1997).
[CrossRef]

L. Niggl, T. Lanzl, and M. Maier, “Properties of Bessel beams generated by periodic gratings of circular symmetry,” J. Opt. Soc. Am. A14(1), 27–33 (1997).
[CrossRef]

1996 (2)

A. T. Friberg, “Stationary-phase analysis of generalized axicons,” J. Opt. Soc. Am. A13(4), 743–750 (1996).
[CrossRef]

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Spagnolo, “Generalized bessel-gauss beams,” J. Mod. Opt.43(6), 1155–1166 (1996).

1992 (1)

1987 (2)

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun.64(6), 491–495 (1987).
[CrossRef]

J. R. Fienup, “Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint,” J. Opt. Soc. Am. A (1987).
[CrossRef]

1982 (2)

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt.21(15), 2758–2769 (1982).
[CrossRef] [PubMed]

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng.21, 829–832 (1982).
[CrossRef]

1972 (1)

R. W. Gerchberg and W. O. Saxton, “Phase determination from image and diffraction plane pictures in the electron microscope,” Optik35, 237–246 (1972).

Arlt, J.

J. Arlt, V. Garcés-Chávez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun.1974–6), 239–245 (2001).
[CrossRef]

Bagini, V.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Spagnolo, “Generalized bessel-gauss beams,” J. Mod. Opt.43(6), 1155–1166 (1996).

Bara, S.

Betzig, E.

T. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods8, 417–423 (2011).
[CrossRef] [PubMed]

Bhuyan, M. K.

F. Courvoisier, J. Zhang, M. K. Bhuyan, M. Jacquot, and J. M. Dudley, “Applications of femtosecond Bessel beams to laser ablation,” Appl. Phys. A (2012).

Burvall, A.

A. Burvall, K. Kolacz, Z. Jaroszewicz, and A. Friberg, “Simple lens axicon,” Appl. Opt.43(25), 4838–4844 (2004).
[CrossRef] [PubMed]

Chebbi, B.

B. Chebbi, I. Golub, and K. Gourley, “Homogenization of on-axis intensity distribution produced by a Fresnel refractive axicon,” Opt. Commun.285(7), 1636–1641 (2012).
[CrossRef]

I. Golub, B. Chebbi, D. Shaw, and D. Nowacki, “Characterization of a refractive logarithmic axicon,” Opt. Lett.35(16), 2828–2830 (2010).
[CrossRef] [PubMed]

Chen, Z.

Christodoulides, D.

Cižmár, T.

T. Čižmár and K. Dholakia, “Tunable Bessel light modes: engineering the axial propagation,” Opt. Express17(18), 15558–15570 (2009).
[CrossRef]

Courvoisier, F.

F. Courvoisier, J. Zhang, M. K. Bhuyan, M. Jacquot, and J. M. Dudley, “Applications of femtosecond Bessel beams to laser ablation,” Appl. Phys. A (2012).

Davidson, M. W.

T. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods8, 417–423 (2011).
[CrossRef] [PubMed]

De Koninck, Y.

DeMarco, B.

Dholakia, K.

T. Čižmár and K. Dholakia, “Tunable Bessel light modes: engineering the axial propagation,” Opt. Express17(18), 15558–15570 (2009).
[CrossRef]

J. Arlt, V. Garcés-Chávez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun.1974–6), 239–245 (2001).
[CrossRef]

Di Trapani, P.

Ding, Z.

Dudley, J. M.

F. Courvoisier, J. Zhang, M. K. Bhuyan, M. Jacquot, and J. M. Dudley, “Applications of femtosecond Bessel beams to laser ablation,” Appl. Phys. A (2012).

Dufour, P.

Efimov, A.

Esarey, E.

B. Hafizi, E. Esarey, and P. Sprangle, “Laser-driven acceleration with Bessel beams,” Phys. Rev. E55(3), 3539–3545 (1997).
[CrossRef]

Faccio, D.

Fienup, J. R.

J. R. Fienup, “Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint,” J. Opt. Soc. Am. A (1987).
[CrossRef]

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt.21(15), 2758–2769 (1982).
[CrossRef] [PubMed]

Frezza, F.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Spagnolo, “Generalized bessel-gauss beams,” J. Mod. Opt.43(6), 1155–1166 (1996).

Friberg, A.

A. Burvall, K. Kolacz, Z. Jaroszewicz, and A. Friberg, “Simple lens axicon,” Appl. Opt.43(25), 4838–4844 (2004).
[CrossRef] [PubMed]

Friberg, A. T.

Galbraith, C. G.

T. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods8, 417–423 (2011).
[CrossRef] [PubMed]

Galbraith, J. A.

T. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods8, 417–423 (2011).
[CrossRef] [PubMed]

Gao, L.

T. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods8, 417–423 (2011).
[CrossRef] [PubMed]

Garcés-Chávez, V.

J. Arlt, V. Garcés-Chávez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun.1974–6), 239–245 (2001).
[CrossRef]

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “Phase determination from image and diffraction plane pictures in the electron microscope,” Optik35, 237–246 (1972).

Golub, I.

Gonsalves, R. A.

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng.21, 829–832 (1982).
[CrossRef]

Gori, F.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun.64(6), 491–495 (1987).
[CrossRef]

Gourley, K.

B. Chebbi, I. Golub, and K. Gourley, “Homogenization of on-axis intensity distribution produced by a Fresnel refractive axicon,” Opt. Commun.285(7), 1636–1641 (2012).
[CrossRef]

Graf, T.

Guattari, G.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun.64(6), 491–495 (1987).
[CrossRef]

Hafizi, B.

B. Hafizi, E. Esarey, and P. Sprangle, “Laser-driven acceleration with Bessel beams,” Phys. Rev. E55(3), 3539–3545 (1997).
[CrossRef]

Hoffnagle, J. A.

Honkanen, M.

M. Honkanen and J. Turunen, “Tandem systems for efficient generation of uniform-axial-intensity Bessel fields,” Opt. Commun.154(5), 368375 (1998).
[CrossRef]

Jacquot, M.

F. Courvoisier, J. Zhang, M. K. Bhuyan, M. Jacquot, and J. M. Dudley, “Applications of femtosecond Bessel beams to laser ablation,” Appl. Phys. A (2012).

Jaroszewicz, Z.

Jefferson, C. M.

Kolacz, K.

A. Burvall, K. Kolacz, Z. Jaroszewicz, and A. Friberg, “Simple lens axicon,” Appl. Opt.43(25), 4838–4844 (2004).
[CrossRef] [PubMed]

Kolesik, M.

Kolodziejczyk, A.

Lanzl, T.

Lipson, A.

A. Lipson, S. G. Lipson, and H. Lipson, Optical Physics, 4th ed. (Cambridge University Press, 2010).
[CrossRef]

Lipson, H.

A. Lipson, S. G. Lipson, and H. Lipson, Optical Physics, 4th ed. (Cambridge University Press, 2010).
[CrossRef]

Lipson, S. G.

A. Lipson, S. G. Lipson, and H. Lipson, Optical Physics, 4th ed. (Cambridge University Press, 2010).
[CrossRef]

Liu, J. S.

Maier, M.

McCarthy, N.

Milkie, D. E.

T. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods8, 417–423 (2011).
[CrossRef] [PubMed]

Mills, M. S.

Mirtchev, T.

Moloney, J.

Nelson, J. S.

Niggl, L.

Nowacki, D.

Nuttall, J.

Padovani, C.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun.64(6), 491–495 (1987).
[CrossRef]

Pasienski, M.

Piché, M.

Planchon, T.

T. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods8, 417–423 (2011).
[CrossRef] [PubMed]

Polynkin, P.

Reitze, D. H.

Ren, H.

Roberts, A.

Rundquist, A.

Santarsiero, M.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Spagnolo, “Generalized bessel-gauss beams,” J. Mod. Opt.43(6), 1155–1166 (1996).

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “Phase determination from image and diffraction plane pictures in the electron microscope,” Optik35, 237–246 (1972).

Schettini, G.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Spagnolo, “Generalized bessel-gauss beams,” J. Mod. Opt.43(6), 1155–1166 (1996).

Shaw, D.

Sibbett, W.

J. Arlt, V. Garcés-Chávez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun.1974–6), 239–245 (2001).
[CrossRef]

Sochacki, J.

Spagnolo, G.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Spagnolo, “Generalized bessel-gauss beams,” J. Mod. Opt.43(6), 1155–1166 (1996).

Sprangle, P.

B. Hafizi, E. Esarey, and P. Sprangle, “Laser-driven acceleration with Bessel beams,” Phys. Rev. E55(3), 3539–3545 (1997).
[CrossRef]

Taghizadeh, M. R.

Turunen, J.

M. Honkanen and J. Turunen, “Tandem systems for efficient generation of uniform-axial-intensity Bessel fields,” Opt. Commun.154(5), 368375 (1998).
[CrossRef]

Venkataramani, S. C.

Wright, E. M.

Zamboni-Rached, M.

Zapata-Rodríguez, C.

Zhang, J.

F. Courvoisier, J. Zhang, M. K. Bhuyan, M. Jacquot, and J. M. Dudley, “Applications of femtosecond Bessel beams to laser ablation,” Appl. Phys. A (2012).

Zhao, Y.

Appl. Opt. (1)

A. Burvall, K. Kolacz, Z. Jaroszewicz, and A. Friberg, “Simple lens axicon,” Appl. Opt.43(25), 4838–4844 (2004).
[CrossRef] [PubMed]

Appl. Opt. (4)

Appl. Phys. A (1)

F. Courvoisier, J. Zhang, M. K. Bhuyan, M. Jacquot, and J. M. Dudley, “Applications of femtosecond Bessel beams to laser ablation,” Appl. Phys. A (2012).

J. Mod. Opt. (1)

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Spagnolo, “Generalized bessel-gauss beams,” J. Mod. Opt.43(6), 1155–1166 (1996).

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

J. R. Fienup, “Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint,” J. Opt. Soc. Am. A (1987).
[CrossRef]

J. Opt. Soc. Am. A (3)

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

Nat. Methods (1)

T. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods8, 417–423 (2011).
[CrossRef] [PubMed]

Opt. Commun. (1)

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun.64(6), 491–495 (1987).
[CrossRef]

Opt. Commun. (3)

J. Arlt, V. Garcés-Chávez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun.1974–6), 239–245 (2001).
[CrossRef]

M. Honkanen and J. Turunen, “Tandem systems for efficient generation of uniform-axial-intensity Bessel fields,” Opt. Commun.154(5), 368375 (1998).
[CrossRef]

B. Chebbi, I. Golub, and K. Gourley, “Homogenization of on-axis intensity distribution produced by a Fresnel refractive axicon,” Opt. Commun.285(7), 1636–1641 (2012).
[CrossRef]

Opt. Eng. (1)

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng.21, 829–832 (1982).
[CrossRef]

Opt. Express (4)

Opt. Lett. (4)

Optik (1)

R. W. Gerchberg and W. O. Saxton, “Phase determination from image and diffraction plane pictures in the electron microscope,” Optik35, 237–246 (1972).

Phys. Rev. E (1)

B. Hafizi, E. Esarey, and P. Sprangle, “Laser-driven acceleration with Bessel beams,” Phys. Rev. E55(3), 3539–3545 (1997).
[CrossRef]

Other (1)

A. Lipson, S. G. Lipson, and H. Lipson, Optical Physics, 4th ed. (Cambridge University Press, 2010).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Illustration of the propagation of the generalized Bessel-Gauss beam with color used to represent intensity. To highlight the structure of the low intensity rings, the cube root of the intensity is plotted. The black lines show the geometric path of the rings ztanγ0, and the red lines mark the full width at half maximum of the axial intensity profile.

Fig. 2
Fig. 2

(a) Lineouts of the target (red, dashed) and optimized (blue, slight vertical offset) axial profiles for wT = 7mm. The narrow black profile in (a) represents the axial profile of the unshaped Bessel zone. (b) Cross-sectional plots in the input plane. Red: input intensity profile; Blue: optimized radial phase profile for wT = 7 mm; Black/dashed: optimized phase profile after an initial guess of the focusing phase is added to the input beam.

Fig. 3
Fig. 3

Image of the calculated intensity vs r and z for the optimized axial profile for wT = 7 mm.

Fig. 4
Fig. 4

Left: Contour plot showing optimized axial intensity profiles as a function of the target width wT. The inset shows the log of the error in the optimization, calculated with Eq. (17). Right: Same data, but with the intensity scaled with the target width.

Fig. 5
Fig. 5

(a) Red: Radial section of the shaped ring-beam intensity I(r) at approximately 17mm past the focal plane. Blue: same radial section weighted with the radius, rI(r). (b) Image of the calculated shaped field on both sides of the Bessel zone.

Equations (19)

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

E ( r , z ) = E 0 w 0 w ( z ) exp [ i ( k z Φ ( z ) ) i β 2 2 k z ] exp [ i k 2 q ( z ) ( r 2 + ( β z k ) 2 ) ] J 0 [ β r ( 1 z q ( z ) ) ] ,
q ( z ) = 1 R ( z ) + i 2 k w 2 ( z ) , w ( z ) = w 0 1 + z 2 / z R 2 , R ( z ) = z ( 1 + z R 2 / z 2 ) , Φ ( z ) = tan 1 ( z / z R ) .
I ( z ) = w 0 2 w 2 ( z ) exp [ 2 z 2 w 2 ( z ) / sin 2 γ 0 ] .
E ( r , 0 ) = A 0 exp ( ( r 2 + a 2 ) w 0 2 ) I 0 ( 2 a r w 0 2 i β r ) ,
| E ( r , 0 ) | 2 A 0 2 w 0 2 4 π a r exp ( 2 ( r a ) 2 w 0 2 ) .
F ( z ) = E ( 0 , z ) = i k z 0 E ( r , 0 ) exp [ i k r 2 2 z ] r d r .
G ( Ω z ) = 2 i Ω z g ( s ) exp [ i Ω z s ] d s ,
g ( s ) = { E ( s , 0 ) s 0 0 s < 0 ,
E ( r , 0 ) = E 0 f ( r ) exp [ i ϕ ( r 2 ) ] exp [ i k r 2 2 z d ]
h ( s ) = { f ( s ) s 0 0 s < 0 .
f ( r ) = exp [ ( r a ) 2 w 0 2 ] ,
Ω z = Ω z Ω z d = k 2 ( 1 z 1 z d ) .
min ϕ , E 0 J [ ϕ ] = ( | H ( Ω z ) | | H T ( Ω z ) | ) 2 d Ω z ,
[ ( | H T ( Ω z ) | 2 d Ω z ) 1 / 2 2 π E 0 ( | h ( s ) | 2 d s ) 1 / 2 ] 2 J [ ϕ ] .
E 0 | H T ( Ω z ) | 2 d Ω z 2 π | h ( s ) | 2 d s .
H ( j ) ( Ω s ) = { h ( s ) exp [ ϕ ( j 1 ) ( s ) ] } h ( j ) ( s ) = 1 { H T ( Ω z ) exp [ Φ ( j ) ( Ω s ) ] } ,
η j = i ( | H ( j ) ( ( Ω z ) i ) | | H T ( ( Ω z ) i ) | ) 2 i | H T ( j ) ( ( Ω z ) i ) | 2 .
F T ( z ) = E T exp [ ( z z d ) 8 w T 8 ] .
E T 2 4.56 E 0 2 k 0 r 0 w 0 w T ,

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