Abstract

We report on the interference between the double rings generated by the Fourier transform of a binary diffractive axicon. These two rings have the same size and correspond to the ± 1 diffracted order beams. The interference condition between both rings can be easily changed by adding a constant phase bias, resulting in a central ring that is either dark or bright. Additionally, this interference condition can be changed along the ring and can be easily tuned, thus allowing greater flexibility. We present experimental results obtained with a binary π-phase liquid-crystal spatial light modulator. These patterns might find applications in optical trapping systems, where the bright or dark regions could trap particles whose refractive index is either higher or lower than the medium.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2019 (2)

2018 (2)

S. Hasegawa, H. Ito, H. Toyoda, and Y. Hayasaki, “Diffraction-limited ring beam generated by radial grating,” OSA Continuum 1(2), 283–294 (2018).
[Crossref]

S. N. Khonina and A. P. Porfirev, “3D transformations of light fields in the focal region implemented by diffractive axicons,” Appl. Phys. B 124(9), 191 (2018).
[Crossref]

2016 (3)

J. A. Rodrigo and T. Alieva, “Polymorphic beams and Nature inspired circuits for optical current,” Sci. Rep. 6(1), 35341 (2016).
[Crossref]

J. A. Rodrigo and T. Alieva, “Light-driven transport of plasmonic nanoparticles on demand,” Sci. Rep. 6(1), 33729 (2016).
[Crossref]

D. Deng, Y. Li, Y. Han, X. Su, J. Ye, J. Gao, Q. Sun, and S. Qu, “Perfect vortex in three-dimensional multifocal array,” Opt. Express 24(25), 28270–28278 (2016).
[Crossref]

2015 (3)

2014 (2)

2013 (1)

2009 (2)

2008 (1)

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. 100(1), 013602 (2008).
[Crossref]

2006 (1)

2005 (1)

T. Cižmár, V. Garcés-Chávez, K. Dholakia, and P. Zemánek, “Optical conveyor belt for delivery of submicron objects,” Appl. Phys. Lett. 86(17), 174101 (2005).
[Crossref]

2003 (1)

J. E. Curtis and D. G. Grier, “Structure of optical vortices,” Phys. Rev. Lett. 90(13), 133901 (2003).
[Crossref]

1998 (1)

1996 (2)

1995 (1)

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995).
[Crossref]

1993 (1)

1992 (2)

G. Scott and M. McArdle, “Efficient generation of nearly diffraction-free beams using an axicon,” Opt. Eng. 31(12), 2640–2643 (1992).
[Crossref]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

1987 (1)

1986 (1)

Alieva, T.

J. A. Rodrigo and T. Alieva, “Polymorphic beams and Nature inspired circuits for optical current,” Sci. Rep. 6(1), 35341 (2016).
[Crossref]

J. A. Rodrigo and T. Alieva, “Light-driven transport of plasmonic nanoparticles on demand,” Sci. Rep. 6(1), 33729 (2016).
[Crossref]

Allen, L.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

Amato-Grill, J.

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. 100(1), 013602 (2008).
[Crossref]

Arrizon, V.

Arrizón, V.

Artl, J.

Ashkin, A.

Badham, K.

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

Berry, M.

M. Berry, “Optical currents,” J. Opt. A: Pure Appl. Opt. 11(9), 094001 (2009).
[Crossref]

Bhattacharya, S.

A. Vijayakumar and S. Bhattacharya, “Compact generation of superposed higher-order Bessel beams via composite diffractive optical elements,” Opt. Eng. 54(11), 111310 (2015).
[Crossref]

Bjorkholm, J. E.

Cai, Y.

Carcolé, E.

Chu, S.

Cižmár, T.

T. Cižmár, V. Garcés-Chávez, K. Dholakia, and P. Zemánek, “Optical conveyor belt for delivery of submicron objects,” Appl. Phys. Lett. 86(17), 174101 (2005).
[Crossref]

Cottrell, D. M.

Curtis, J. E.

J. E. Curtis and D. G. Grier, “Structure of optical vortices,” Phys. Rev. Lett. 90(13), 133901 (2003).
[Crossref]

Davis, J. A.

Deng, D.

Dholakia, K.

T. Cižmár, V. Garcés-Chávez, K. Dholakia, and P. Zemánek, “Optical conveyor belt for delivery of submicron objects,” Appl. Phys. Lett. 86(17), 174101 (2005).
[Crossref]

Durnin, J.

Dziedzic, J. M.

Forbes, A.

Friese, M. E. J.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995).
[Crossref]

Gahagan, K. T.

Gao, J.

Garcés-Chávez, V.

T. Cižmár, V. Garcés-Chávez, K. Dholakia, and P. Zemánek, “Optical conveyor belt for delivery of submicron objects,” Appl. Phys. Lett. 86(17), 174101 (2005).
[Crossref]

García-García, J.

Grier, D. G.

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. 100(1), 013602 (2008).
[Crossref]

J. E. Curtis and D. G. Grier, “Structure of optical vortices,” Phys. Rev. Lett. 90(13), 133901 (2003).
[Crossref]

Guenther, R.

R. Guenther, Modern Optics, Chapter 6, John Wiley and Sons, New York (1990).

Guertin, J.

Han, Y.

Hasegawa, S.

Hayasaki, Y.

He, H.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995).
[Crossref]

He, M.

Heckenberg, N. R.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995).
[Crossref]

Hossack, W. J.

Ismail, Y.

Ito, H.

Khonina, S. N.

S. N. Khonina and A. P. Porfirev, “3D transformations of light fields in the focal region implemented by diffractive axicons,” Appl. Phys. B 124(9), 191 (2018).
[Crossref]

Lafong, A.

Lei, M.

Li, M.

Li, Y.

Liang, Y.

McArdle, M.

G. Scott and M. McArdle, “Efficient generation of nearly diffraction-free beams using an axicon,” Opt. Eng. 31(12), 2640–2643 (1992).
[Crossref]

McLaren, M.

Méndez, G.

Mhlanga, T.

Moreno, I.

Nowakowski, T. J.

Ostrovsky, A. S.

Pinnell, J.

Porfirev, A. P.

S. N. Khonina and A. P. Porfirev, “3D transformations of light fields in the focal region implemented by diffractive axicons,” Appl. Phys. B 124(9), 191 (2018).
[Crossref]

Qu, S.

Ramos-García, R.

Read, N. D.

Rickenstorff-Parrao, C.

Rodrigo, J. A.

J. A. Rodrigo and T. Alieva, “Polymorphic beams and Nature inspired circuits for optical current,” Sci. Rep. 6(1), 35341 (2016).
[Crossref]

J. A. Rodrigo and T. Alieva, “Light-driven transport of plasmonic nanoparticles on demand,” Sci. Rep. 6(1), 33729 (2016).
[Crossref]

Rodríguez-Fajardo, V.

Roichman, Y.

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. 100(1), 013602 (2008).
[Crossref]

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. 100(1), 013602 (2008).
[Crossref]

Roux, F. S.

Rubinsztein-Dunlop, H.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995).
[Crossref]

Ruiz, U.

Rusch, L.

Sánchez-de-la-Llave, D.

Sánchez-López, M. M.

Scott, G.

G. Scott and M. McArdle, “Efficient generation of nearly diffraction-free beams using an axicon,” Opt. Eng. 31(12), 2640–2643 (1992).
[Crossref]

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

Su, X.

Sun, B.

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. 100(1), 013602 (2008).
[Crossref]

Sun, Q.

Swartzlander, G. A.

Toyoda, H.

Trichili, A.

Tsai, P.

Vaity, P.

Vijayakumar, A.

A. Vijayakumar and S. Bhattacharya, “Compact generation of superposed higher-order Bessel beams via composite diffractive optical elements,” Opt. Eng. 54(11), 111310 (2015).
[Crossref]

Wang, Z.

Woerdman, J. P.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

Yan, S.

Yao, V.

Ye, J.

Zemánek, P.

T. Cižmár, V. Garcés-Chávez, K. Dholakia, and P. Zemánek, “Optical conveyor belt for delivery of submicron objects,” Appl. Phys. Lett. 86(17), 174101 (2005).
[Crossref]

Zghal, M.

Appl. Opt. (3)

Appl. Phys. B (1)

S. N. Khonina and A. P. Porfirev, “3D transformations of light fields in the focal region implemented by diffractive axicons,” Appl. Phys. B 124(9), 191 (2018).
[Crossref]

Appl. Phys. Lett. (1)

T. Cižmár, V. Garcés-Chávez, K. Dholakia, and P. Zemánek, “Optical conveyor belt for delivery of submicron objects,” Appl. Phys. Lett. 86(17), 174101 (2005).
[Crossref]

J. Opt. A: Pure Appl. Opt. (1)

M. Berry, “Optical currents,” J. Opt. A: Pure Appl. Opt. 11(9), 094001 (2009).
[Crossref]

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

Opt. Eng. (2)

G. Scott and M. McArdle, “Efficient generation of nearly diffraction-free beams using an axicon,” Opt. Eng. 31(12), 2640–2643 (1992).
[Crossref]

A. Vijayakumar and S. Bhattacharya, “Compact generation of superposed higher-order Bessel beams via composite diffractive optical elements,” Opt. Eng. 54(11), 111310 (2015).
[Crossref]

Opt. Express (3)

Opt. Lett. (9)

J. García-García, C. Rickenstorff-Parrao, R. Ramos-García, V. Arrizón, and A. S. Ostrovsky, “Simple technique for generating the perfect optical vortex,” Opt. Lett. 39(18), 5305–5308 (2014).
[Crossref]

P. Vaity and L. Rusch, “Perfect vortex beam: Fourier transformation of a Bessel beam,” Opt. Lett. 40(4), 597–600 (2015).
[Crossref]

I. Moreno, J. A. Davis, M. M. Sánchez-López, K. Badham, and D. M. Cottrell, “Nondiffracting Bessel beams with polarization state that varies with propagation distance,” Opt. Lett. 40(23), 5451–5454 (2015).
[Crossref]

K. T. Gahagan and G. A. Swartzlander, “Optical vortex trapping of particles,” Opt. Lett. 21(11), 827–829 (1996).
[Crossref]

Y. Liang, S. Yan, M. He, M. Li, Y. Cai, Z. Wang, M. Lei, and V. Yao, “Generation of a double-ring perfect optical vortex by the Fourier transform of azimuthally polarized Bessel beams,” Opt. Lett. 44(6), 1504–1507 (2019).
[Crossref]

J. Pinnell, V. Rodríguez-Fajardo, and A. Forbes, “How perfect are perfect vortex beams?” Opt. Lett. 44(22), 5614–5617 (2019).
[Crossref]

V. Arrizón, D. Sánchez-de-la-Llave, U. Ruiz, and G. Méndez, “Efficient generation of an arbitrary nondiffracting Bessel beam employing its phase modulation,” Opt. Lett. 34(9), 1456–1458 (2009).
[Crossref]

A. S. Ostrovsky, C. Rickenstorff-Parrao, and V. Arrizon, “Generation of the “perfect” optical vortex using a liquid-crystal spatial light modulator,” Opt. Lett. 38(4), 534–536 (2013).
[Crossref]

A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11(5), 288–290 (1986).
[Crossref]

OSA Continuum (1)

Phys. Rev. A (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

Phys. Rev. Lett. (3)

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995).
[Crossref]

J. E. Curtis and D. G. Grier, “Structure of optical vortices,” Phys. Rev. Lett. 90(13), 133901 (2003).
[Crossref]

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. 100(1), 013602 (2008).
[Crossref]

Sci. Rep. (2)

J. A. Rodrigo and T. Alieva, “Polymorphic beams and Nature inspired circuits for optical current,” Sci. Rep. 6(1), 35341 (2016).
[Crossref]

J. A. Rodrigo and T. Alieva, “Light-driven transport of plasmonic nanoparticles on demand,” Sci. Rep. 6(1), 33729 (2016).
[Crossref]

Other (2)

R. Guenther, Modern Optics, Chapter 6, John Wiley and Sons, New York (1990).

ForthD displays, https://www.forthdd.com/products/spatial-light-modulators/

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Figures (7)

Fig. 1.
Fig. 1. (a) Regular continuous-phase diffractive axicon. (b) Binary-phase version of the axicon. (c) Ray diagrams of the double-ring formation in the Fourier plane with the positive axicon (red rays) and negative axicon (yellow rays). Rays 1-4 correspond to the converging axicon (+1 order). Rays 5-8 correspond to the diverging axicon (-1 order).
Fig. 2.
Fig. 2. Experimental results of the two-ring interference obtained by Fourier transform of a binary-phase diffractive axicon with a period of d = 10 pixels and (a) in phase, (b) with a relative phase of π/2, (c) with a relative phase of π and (d) with a relative phase of 3π/2. The corresponding intensity profile is shown on the right for each case.
Fig. 3.
Fig. 3. Experimental results of the two-ring interference obtained by Fourier transform of a binary-phase diffractive axicon with period (a) d = 15 pixels, (b) d = 20 pixels, (c) d = 30 pixels.
Fig. 4.
Fig. 4. Binary masks for displaying a binary axicon with parameters (a) $\ell = 0$ and ϕ0=π/2, (b) $\ell = 2$ and ϕ0=0.
Fig. 5.
Fig. 5. Experimental results of the Fourier transform of the binary-phase diffractive axicon with added spiral phase of charge $\ell = 1$ and constant phase ϕ0. (a) $\ell = 1$ and ϕ0=0, (b) $\ell = 1$ and ϕ0=π/2, (c) $\ell = 2$ and ϕ0=0, (d) $\ell = 2$ and ϕ0=π/2, (e) $\ell = 3$ and ϕ0=0 and (f) $\ell = 8$ and ϕ0=0. Red and yellow arrows indicate angular locations where the interference pattern shows darkness or brightness in the center of the ring, respectively.
Fig. 6.
Fig. 6. Computer simulations comparing the phase patterns, the intensity at the Fourier transform plane, and the phase along the maximum of the output for (a) a spiral phase having a charge of 10 and corresponding to a vortex lens, (b) a continuous phase axicon multiplied by a spiral phase having a charge of 10, (c) a binary phase axicon without a spiral phase and (d) a binary phase axicon with a spiral phase having a charge of 8.
Fig. 7.
Fig. 7. Experimental results of the Fourier transform of the binary phase diffractive axicon, showing a circular optical conveyor belt.

Equations (6)

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

t ( r ) = exp ( i 2 π r d ) exp ( i ϕ 0 )
R = f λ d
t ( r ) = n = c n exp ( i n 2 π r d ) exp ( i n ϕ 0 )
c n = 1 d 0 d t ( r ) exp ( i 2 π n r d ) d r
t ( r ) = c 1 { exp ( + i 2 π r d ) exp ( i ϕ 0 ) + exp ( i 2 π r d ) exp ( i ϕ 0 ) }
t ( r ) = n = c n exp ( i n 2 π r d ) exp ( i n ϕ 0 ) exp ( i n θ ) c 1 { exp ( + i 2 π r d ) exp ( i ϕ 0 ) exp ( i θ ) + exp ( i 2 π r d ) exp ( i ϕ 0 ) exp ( i θ ) } ,