Abstract

Bessel beams have been increasingly used for their advantages of non-diffraction and long focal depth. In this paper, we studied the propagation of on-axis and off-axis Bessel beams in a gradient-index medium. By expressing a Bessel beam in integral form, the analytical expression of an on-axis, decentered, and tilted Bessel beam through a paraxial optical system is derived with the ABCD matrix method and Collins diffraction integral formula. Main lobe size and trajectory of the zeroth- and second-order Bessel beam are obtained, demonstrating that the Bessel beam is focused by the gradient-index medium and its main lobe trajectory is exactly the same as the corresponding geometrical ray for both the decentered and tilted Bessel beam. Effects of beam apodization are finally studied by the Fourier beam propagation method, showing that the side lobes of the Bessel beam vanish when the beam is focused inside the medium as only part of the beam enters the lens.

© 2018 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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2017 (1)

2016 (5)

2015 (4)

2014 (1)

2013 (3)

2011 (1)

T. A. 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. Methods 8, 417–423 (2011).
[Crossref]

2010 (1)

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photonics 4, 780–785 (2010).
[Crossref]

2003 (1)

2002 (1)

V. Garces-Chavez, D. Mcgloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
[Crossref]

1990 (1)

1988 (1)

Akturk, S.

Alda, J.

Altingöz, C.

Arrizon, V.

Badham, K.

Betzig, E.

T. A. 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. Methods 8, 417–423 (2011).
[Crossref]

Birch, P.

Boreman, G. D.

Chatwin, C.

Cheng, H.

Cheng, M.

Chu, X. X.

C. H. Qiao, X. X. Feng, and X. X. Chu, “Propagation and self-healing ability of a Bessel-Gaussian beam modulated by Bessel gratings,” Opt. Commun. 365, 24–28 (2016).
[Crossref]

Cottrell, D. M.

Davidson, M. W.

T. A. 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. Methods 8, 417–423 (2011).
[Crossref]

Davis, C. C.

Davis, J. A.

Deng, D.

Deng, F.

Dennis, M. R.

Dholakia, K.

D. McGloin, V. Garces-Chavez, and K. Dholakia, “Interfering Bessel beams for optical micromanipulation,” Opt. Lett. 28, 657–659 (2003).
[Crossref]

V. Garces-Chavez, D. Mcgloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
[Crossref]

Doster, T.

Durnin, J.

Eberly, J. H.

Eyyuboglu, H. T.

Fahrbach, F. O.

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photonics 4, 780–785 (2010).
[Crossref]

Feng, X. X.

C. H. Qiao, X. X. Feng, and X. X. Chu, “Propagation and self-healing ability of a Bessel-Gaussian beam modulated by Bessel gratings,” Opt. Commun. 365, 24–28 (2016).
[Crossref]

Galbraith, C. G.

T. A. 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. Methods 8, 417–423 (2011).
[Crossref]

Galbraith, J. A.

T. A. 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. Methods 8, 417–423 (2011).
[Crossref]

Gao, L.

T. A. 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. Methods 8, 417–423 (2011).
[Crossref]

Garces-Chavez, V.

D. McGloin, V. Garces-Chavez, and K. Dholakia, “Interfering Bessel beams for optical micromanipulation,” Opt. Lett. 28, 657–659 (2003).
[Crossref]

V. Garces-Chavez, D. Mcgloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
[Crossref]

Guo, L.

Han, L.

Huang, J.

Huang, Q.

Ituen, I.

Khalil, D.

Li, J.

Li, P.

Liu, S.

Loui, H.

H. Loui, “Fourier beam propagation,” (University of Colorado, 2004).

Mahmoud, M. A.

McGloin, D.

D. McGloin, V. Garces-Chavez, and K. Dholakia, “Interfering Bessel beams for optical micromanipulation,” Opt. Lett. 28, 657–659 (2003).
[Crossref]

V. Garces-Chavez, D. Mcgloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
[Crossref]

Melville, H.

V. Garces-Chavez, D. Mcgloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
[Crossref]

Miceli, J. J.

Milkie, D. E.

T. A. 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. Methods 8, 417–423 (2011).
[Crossref]

Moreno, I.

Moya-Cessa, H. M.

Nelson, W.

Palastro, J. P.

Peng, P.

Planchon, T. A.

T. A. 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. Methods 8, 417–423 (2011).
[Crossref]

Qiao, C. H.

C. H. Qiao, X. X. Feng, and X. X. Chu, “Propagation and self-healing ability of a Bessel-Gaussian beam modulated by Bessel gratings,” Opt. Commun. 365, 24–28 (2016).
[Crossref]

Ring, J. D.

Rohrbach, A.

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photonics 4, 780–785 (2010).
[Crossref]

Sánchez-López, M. M.

Shalaby, M. Y.

Sibbett, W.

V. Garces-Chavez, D. Mcgloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
[Crossref]

Simon, P.

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photonics 4, 780–785 (2010).
[Crossref]

Soto-Eguibar, F.

Sprangle, P.

Tong, C.

Voelz, D.

Watnik, A. T.

Wu, D.

Wu, G.

Xiao, X.

Yalizay, B.

Young, R.

Yu, W.

Zhang, Y.

Zhao, J.

Zhao, R.

Zuñiga-Segundo, A.

Appl. Opt. (4)

Chin. Opt. Lett. (1)

J. Opt. Soc. Am. A (6)

Nat. Methods (1)

T. A. 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. Methods 8, 417–423 (2011).
[Crossref]

Nat. Photonics (1)

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photonics 4, 780–785 (2010).
[Crossref]

Nature (1)

V. Garces-Chavez, D. Mcgloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
[Crossref]

Opt. Commun. (1)

C. H. Qiao, X. X. Feng, and X. X. Chu, “Propagation and self-healing ability of a Bessel-Gaussian beam modulated by Bessel gratings,” Opt. Commun. 365, 24–28 (2016).
[Crossref]

Opt. Express (1)

Opt. Lett. (4)

Other (1)

H. Loui, “Fourier beam propagation,” (University of Colorado, 2004).

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Figures (8)

Fig. 1.
Fig. 1.

Illustration of propagation of a Bessel beam in a gradient-index lens.

Fig. 2.
Fig. 2.

On-axis propagation of the zeroth-order Bessel beam. (a) Normalized amplitude distribution in the yOz plane in the logarithmic scale. (b) Normalized amplitude distribution at z=0.0005L in the linear scale. (c) Normalized amplitude distribution at z=0.2505L in the linear scale.

Fig. 3.
Fig. 3.

Same as Fig. 2, but for on-axis propagation of the second-order Bessel beam.

Fig. 4.
Fig. 4.

Same as Fig. 2, but for off-axis propagation of the decentered zeroth-order Bessel beam.

Fig. 5.
Fig. 5.

Same as Fig. 2, but for off-axis propagation of the decentered second-order Bessel beam.

Fig. 6.
Fig. 6.

Same as Fig. 2, but for off-axis propagation of the tilted zeroth-order Bessel beam.

Fig. 7.
Fig. 7.

Same as Fig. 2, but for off-axis propagation of the tilted second-order Bessel beam.

Fig. 8.
Fig. 8.

On-axis propagation of the zeroth-order Bessel beam calculated by the Fourier beam propagation method.

Equations (20)

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

E(r,ϕ,z)=A0exp(ikzz)Jn(krr)exp(inϕ),
E(x,y,z)=A0exp(ikzz)2πin02πexp(inϕ)×exp[ikr(xcosϕ+ysinϕ)]dϕ.
n=n0(1A1r22),
(ABCD)=(cos(z/β)βsin(z/β)n0n0sin(z/β)βcos(z/β)),
Eout(x,y,z)=ik2πBEin(x0,y0,0)×exp{ik2B[A(x02+y02)2(x0x+y0y)+D(x2+y2)]}dx0dy0.
Eout(x,y,z)=Cexp{ik2BD(x2+y2)}02πexp(inϕ)F1F2dϕ,
C=A0k4π2Bin1,
F1=exp[ikrx0cosϕ]exp{ik2B[Ax022x0x]}dx0,
F2=exp[ikry0sinϕ]exp{ik2B[Ay022y0y]}dy0.
exp[i(sx2+rx)]dx=iπsexp(ir24s).
F1=i2πBkAexp[ikB2A(xB+sinαcosϕ)2],
F2=i2πBkAexp[ikB2A(yB+sinαsinϕ)2],
Eout(x,y,z)=i2πBkACexp{ik2BD(x2+y2)}×exp{ikB2A[x2+y2B2+sin2α]}Jn(krrA).
E(x,y,z)=A0exp(ikzz)2πin02πexp(inϕ)×exp[ikr(xcosϕ+ysinϕycsinϕ)]dϕ,
E(x,y,z)=A0exp(ikzz)2πin02πexp(inϕ)×exp[ikr(xcosϕ+ysinϕ)+ikysinθc]dϕ,
|Eout(x,y,z)||02πexp(inϕ1)exp{ikrA[xcosϕ+(yAyc)sinϕ]}dϕ|.
y=Ayc+Bθc.
|Eout(x,y,z)||02πexp(inϕ1)exp{ikrA[xcosϕ+(yBsinθc)sinϕ]}dϕ|.
Eout(x,y,z+Δz)=IFFT[FFT[Ein(x,y,z)]A˜(kx,ky)],
Eout(x,y,z+Δz)=Eout(x,y,z+Δz)Δϕ(x,y),

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