Abstract

We present a new feasible way to flatten the axial intensity oscillations for diffraction of a finite-sized Bessel beam, through designing a cardioid-like hole. The boundary formula of the cardioid-like hole is given analytically. Numerical results by the complete Rayleigh-Sommerfeld method reveal that the Bessel beam propagates stably in a considerably long axial range, after passing through the cardioid-like hole. Compared with the gradually absorbing apodization technique in previous papers, in this paper a hard truncation of the incident Bessel beam is employed at the cardioid-like hole edges. The proposed hard apodization technique takes two advantages in suppressing the axial intensity oscillations, i.e., easier implementation and higher accuracy. It is expected to have practical applications in laser machining, light sectioning, or optical trapping.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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2013 (2)

2010 (2)

T. Čižmár, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photon. 4, 388–394 (2010).
[Crossref]

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

2009 (1)

2005 (1)

2004 (2)

G. D. Gillen and S. Guha, “Modeling and propagation of near-field diffraction pattern: A more complete approach,” Am. J. Phys. 72, 1195–1201 (2004).
[Crossref]

S. H. Tao, W. M. Lee, and X. C. Yuan, “Experimental study of holographic generation of fractional Bessel beams,” Appl. Opt. 43, 122–126 (2004).
[Crossref] [PubMed]

2003 (1)

Z. Bouchal, “Nondiffracting optical beams: physical properties, experiments, and applications,” Czech. J. Phys. 53, 537–578 (2003).
[Crossref]

1999 (2)

1998 (2)

S. Y. Popov and A. T. Friberg, “Apodization of generalized axicons to produce uniform axial line images,” Pure Appl. Opt. 7, 537–548 (1998).
[Crossref]

S. Y. Popov, A. T. Friberg, M. Honkanen, J. Lautanen, J. Turunen, and B. Schnabel, “Apodized annular-aperture diffractive axicons fabricated by continuous-path-control electron beam lithography,” Opt. Commun. 154, 359–367 (1998).
[Crossref]

1997 (2)

1996 (1)

S. Brinkmann, T. Dresel, R. Schreiner, and J. Schwider, “Axicon-type test interferometer for cylindrical surfaces,” Optik 102, 106–110 (1996).

1995 (1)

1993 (1)

1992 (3)

1989 (1)

1988 (2)

1987 (2)

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

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

1978 (1)

Aguayo, S. L.

Ambriz, A. O.

Arlt, J.

C. A. McQueen, J. Arlt, and K. Dholakia, “An experiment to study a ‘nondiffracting’ light beam,” Am. J. Phys. 67, 912–915 (1999).
[Crossref]

Arrizón, V.

Bélanger, P. A.

Borghi, R.

Bouchal, Z.

Z. Bouchal, “Nondiffracting optical beams: physical properties, experiments, and applications,” Czech. J. Phys. 53, 537–578 (2003).
[Crossref]

Brinkmann, S.

S. Brinkmann, T. Dresel, R. Schreiner, and J. Schwider, “Axicon-type test interferometer for cylindrical surfaces,” Optik 102, 106–110 (1996).

Campos, J.

Choi, K. H.

H. Song, H. S. Lee, G. Y. Sung, K. H. Won, and K. H. Choi, “Spatial light modulator and holographic 3D image display including the same,” US patent, US20130335795A1.

Cižmár, T.

T. Čižmár, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photon. 4, 388–394 (2010).
[Crossref]

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

Cox, A. J.

D’Anna, J.

Davis, J. A.

DelaLlave, D. S.

DelaTocnaye, J. L. D.

Dholakia, K.

T. Čižmár, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photon. 4, 388–394 (2010).
[Crossref]

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

C. A. McQueen, J. Arlt, and K. Dholakia, “An experiment to study a ‘nondiffracting’ light beam,” Am. J. Phys. 67, 912–915 (1999).
[Crossref]

Dresel, T.

S. Brinkmann, T. Dresel, R. Schreiner, and J. Schwider, “Axicon-type test interferometer for cylindrical surfaces,” Optik 102, 106–110 (1996).

Dupont, L.

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] [PubMed]

Eberly, J. H.

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

Escalera, J. C.

Fahrbach, F. O.

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

Friberg, A. T.

S. Y. Popov, A. T. Friberg, M. Honkanen, J. Lautanen, J. Turunen, and B. Schnabel, “Apodized annular-aperture diffractive axicons fabricated by continuous-path-control electron beam lithography,” Opt. Commun. 154, 359–367 (1998).
[Crossref]

S. Y. Popov and A. T. Friberg, “Apodization of generalized axicons to produce uniform axial line images,” Pure Appl. Opt. 7, 537–548 (1998).
[Crossref]

A. Vasara, J. Turunen, and A. T. Friberg, “Realization of general nondiffracting beams with computer-generated holograms,” J. Opt. Soc. Am. A 6, 1748–1754 (1989).
[Crossref] [PubMed]

J. Turunen, A. Vasara, and A. T. Friberg, “Holographic generation of diffraction-free beams,” Appl. Opt. 27, 3959–3962 (1988).
[Crossref] [PubMed]

Gillen, G. D.

G. D. Gillen and S. Guha, “Modeling and propagation of near-field diffraction pattern: A more complete approach,” Am. J. Phys. 72, 1195–1201 (2004).
[Crossref]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, (McGraw-Hill Companies, United States, 1996), 2nd edition, pp. 184.

Gori, F.

Gracia, H. G.

Guha, S.

G. D. Gillen and S. Guha, “Modeling and propagation of near-field diffraction pattern: A more complete approach,” Am. J. Phys. 72, 1195–1201 (2004).
[Crossref]

HaÜusler, G.

Heckel, W.

Herman, R. M.

Honkanen, M.

S. Y. Popov, A. T. Friberg, M. Honkanen, J. Lautanen, J. Turunen, and B. Schnabel, “Apodized annular-aperture diffractive axicons fabricated by continuous-path-control electron beam lithography,” Opt. Commun. 154, 359–367 (1998).
[Crossref]

Iemmi, C.

Jaroszewicz, Z.

Jiang, Z. P.

Kartashov, Y. V.

Khalil, D.

Kolodziejczyk, A.

Lapointe, M. R.

M. R. Lapointe, “Review of non-diffracting Bessel beam experiments,” Opt. Laser Technol. 24, 315–321 (1992).
[Crossref]

Lautanen, J.

S. Y. Popov, A. T. Friberg, M. Honkanen, J. Lautanen, J. Turunen, and B. Schnabel, “Apodized annular-aperture diffractive axicons fabricated by continuous-path-control electron beam lithography,” Opt. Commun. 154, 359–367 (1998).
[Crossref]

Lee, H. S.

H. Song, H. S. Lee, G. Y. Sung, K. H. Won, and K. H. Choi, “Spatial light modulator and holographic 3D image display including the same,” US patent, US20130335795A1.

Lee, W. M.

Liu, Z. J.

Lu, Q. S.

Mahmoud, M. A.

Marquez, A.

Mazilu, M.

T. Čižmár, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photon. 4, 388–394 (2010).
[Crossref]

McQueen, C. A.

C. A. McQueen, J. Arlt, and K. Dholakia, “An experiment to study a ‘nondiffracting’ light beam,” Am. J. Phys. 67, 912–915 (1999).
[Crossref]

Méndez, G.

Miceli, J. J.

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

Petrov, D.

Popov, S. Y.

S. Y. Popov and A. T. Friberg, “Apodization of generalized axicons to produce uniform axial line images,” Pure Appl. Opt. 7, 537–548 (1998).
[Crossref]

S. Y. Popov, A. T. Friberg, M. Honkanen, J. Lautanen, J. Turunen, and B. Schnabel, “Apodized annular-aperture diffractive axicons fabricated by continuous-path-control electron beam lithography,” Opt. Commun. 154, 359–367 (1998).
[Crossref]

Rioux, M.

Rohrbach, A.

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

Santarsiero, M.

Schnabel, B.

S. Y. Popov, A. T. Friberg, M. Honkanen, J. Lautanen, J. Turunen, and B. Schnabel, “Apodized annular-aperture diffractive axicons fabricated by continuous-path-control electron beam lithography,” Opt. Commun. 154, 359–367 (1998).
[Crossref]

Schreiner, R.

S. Brinkmann, T. Dresel, R. Schreiner, and J. Schwider, “Axicon-type test interferometer for cylindrical surfaces,” Optik 102, 106–110 (1996).

Schwider, J.

S. Brinkmann, T. Dresel, R. Schreiner, and J. Schwider, “Axicon-type test interferometer for cylindrical surfaces,” Optik 102, 106–110 (1996).

Shalaby, M. Y.

Simon, P.

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

Sochacki, J.

Song, H.

H. Song, H. S. Lee, G. Y. Sung, K. H. Won, and K. H. Choi, “Spatial light modulator and holographic 3D image display including the same,” US patent, US20130335795A1.

Staronski, L. R.

Sung, G. Y.

H. Song, H. S. Lee, G. Y. Sung, K. H. Won, and K. H. Choi, “Spatial light modulator and holographic 3D image display including the same,” US patent, US20130335795A1.

Tao, S. H.

Torner, L.

Tremblay, R.

Turunen, J.

S. Y. Popov, A. T. Friberg, M. Honkanen, J. Lautanen, J. Turunen, and B. Schnabel, “Apodized annular-aperture diffractive axicons fabricated by continuous-path-control electron beam lithography,” Opt. Commun. 154, 359–367 (1998).
[Crossref]

A. Vasara, J. Turunen, and A. T. Friberg, “Realization of general nondiffracting beams with computer-generated holograms,” J. Opt. Soc. Am. A 6, 1748–1754 (1989).
[Crossref] [PubMed]

J. Turunen, A. Vasara, and A. T. Friberg, “Holographic generation of diffraction-free beams,” Appl. Opt. 27, 3959–3962 (1988).
[Crossref] [PubMed]

Vasara, A.

Vega, J. C. G.

Vysloukh, V. A.

Wiggins, T. A.

Won, K. H.

H. Song, H. S. Lee, G. Y. Sung, K. H. Won, and K. H. Choi, “Spatial light modulator and holographic 3D image display including the same,” US patent, US20130335795A1.

Yuan, X. C.

Yzuel, M. J.

Am. J. Phys. (2)

C. A. McQueen, J. Arlt, and K. Dholakia, “An experiment to study a ‘nondiffracting’ light beam,” Am. J. Phys. 67, 912–915 (1999).
[Crossref]

G. D. Gillen and S. Guha, “Modeling and propagation of near-field diffraction pattern: A more complete approach,” Am. J. Phys. 72, 1195–1201 (2004).
[Crossref]

Appl. Opt. (8)

Czech. J. Phys. (1)

Z. Bouchal, “Nondiffracting optical beams: physical properties, experiments, and applications,” Czech. J. Phys. 53, 537–578 (2003).
[Crossref]

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

Nat. Photon. (2)

T. Čižmár, M. Mazilu, and K. Dholakia, “In situ wavefront correction and its application to micromanipulation,” Nat. Photon. 4, 388–394 (2010).
[Crossref]

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

Opt. Commun. (1)

S. Y. Popov, A. T. Friberg, M. Honkanen, J. Lautanen, J. Turunen, and B. Schnabel, “Apodized annular-aperture diffractive axicons fabricated by continuous-path-control electron beam lithography,” Opt. Commun. 154, 359–367 (1998).
[Crossref]

Opt. Express (3)

Opt. Laser Technol. (1)

M. R. Lapointe, “Review of non-diffracting Bessel beam experiments,” Opt. Laser Technol. 24, 315–321 (1992).
[Crossref]

Opt. Lett. (3)

Optik (1)

S. Brinkmann, T. Dresel, R. Schreiner, and J. Schwider, “Axicon-type test interferometer for cylindrical surfaces,” Optik 102, 106–110 (1996).

Phys. Rev. Lett. (1)

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

Pure Appl. Opt. (1)

S. Y. Popov and A. T. Friberg, “Apodization of generalized axicons to produce uniform axial line images,” Pure Appl. Opt. 7, 537–548 (1998).
[Crossref]

Other (2)

H. Song, H. S. Lee, G. Y. Sung, K. H. Won, and K. H. Choi, “Spatial light modulator and holographic 3D image display including the same,” US patent, US20130335795A1.

J. W. Goodman, Introduction to Fourier Optics, (McGraw-Hill Companies, United States, 1996), 2nd edition, pp. 184.

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Figures (5)

Fig. 1
Fig. 1 (a) The transmittance distribution on the input plane z0 = 0 m under the soft apodization (SA) condition. The green and pink dashed curves depict two circles with radii of ϵR and R, respectively. (b) The shadowed annular district represents the equivalent transmitted region at a definite radius r, which occupies 2|φ| in the angle direction. (c) The transmittance distribution on the input plane under the hard apodization (HA) condition. The cyan solid curve encloses a cardioid-like hole, with transmittance being 1 and 0 inside and outside the hole, respectively.
Fig. 2
Fig. 2 (a) The axial intensity distribution of the Bessel beam diffracted from a circular hole without apodization. (b) The blue solid and red dashed curves represent the axial intensity distributions of the diffracted Bessel beam by the SA and HA techniques, corresponding to Figs. 1(a) and 1(c), respectively. In both cases, we choose the same smoothing parameter as ϵ0 = 0.5. (c) Axial intensity distributions of the diffracted Bessel beam by the HA technique, when the smoothing parameter ϵ varies from 0.7 to 0.9 with an interval of 0.05. The blue, green, red, cyan, and black curves correspond to different smoothing parameters of 0.7, 0.75, 0.8, 0.85, and 0.9, respectively.
Fig. 3
Fig. 3 Intensity deviation on three cross-sectional planes for the hard apodized Bessel beam. The smoothing parameter ϵ is 0.8, and other parameters are the same as above. (a) The intensity deviation ΔI1 = I1(x1, y1) − I0(x, y), where I1(x1, y1) represents the intensity distribution on the lateral plane z1 = 10 m and I 0 ( x , y ) = | J 0 ( β x 2 + y 2 ) | 2 is the intensity of the incident Bessel beam on the input plane z0 = 0 m. (b) and (c) are the same as (a) except for z2 = 20 m and z3=30 m, respectively.
Fig. 4
Fig. 4 The intensity distributions on three cross-sectional planes with longitudinal coordinates of (a) z1=10 m, (b) z2=20 m, and (c) z3=30 m, respectively. The smoothing parameter ϵ is assumed to be 0.8, and other parameters are the same as above. The blue solid and red dashed curves represent the intensity profiles along the x-axis and the y-axis, respectively. (d) Intensity deviation ΔIx = Ii(xi, 0) − |J0(β|x|)|2 and ΔIy = Ii(0, yi) − |J0(β|y|)|2 on the above-mentioned three cross-sectional planes are displayed from top to bottom, where Ii(xi, 0) and Ii(0, yi) (i=1,2,3) represent the intensities along the x-axis and the y-axis, respectively.
Fig. 5
Fig. 5 Side-view intensity distributions of the hard apodized Bessel beam on the (a) xz-plane and the (b) yz-plane, respectively. The smoothing parameter ϵ is selected as 0.8.

Equations (5)

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E ( x , y , z ) = k z i 2 π Ω E 0 ( x , y , 0 ) exp ( i k ρ ) ρ 2 ( 1 1 i k ρ ) d x d y ,
E ( 0 , 0 , z ) = ( i k z ) 0 R E 0 ( r , 0 ) exp ( i k r 2 + z 2 ) r 2 + z 2 [ 1 1 i k r 2 + z 2 ] r d r ,
T ( r ) = { 1 , r < ϵ R , 1 + cos [ π ( r ϵ R ) / ( R ϵ R ) ] 2 , ϵ R r R , 0 , r > R ,
F ( r ) = 2 | φ ( r ) | 2 π = T ( r ) = 1 + cos [ π ( r ϵ R ) / ( R ϵ R ) ] 2 , π φ π .
r ( φ ) = ϵ R + ( R ϵ R ) arccos ( 2 | φ | / π 1 ) π , π φ π .

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