Abstract

We investigate the diffraction effects of focused Gaussian beams yielding a double optical vortex by a nano-step structure fabricated in a transparent media. When approaching such a step the double vortex splits into single ones which move in a characteristic way. By observing this movement we can determine the position of the step with high resolution. Our theoretical predictions were verified experimentally.

© 2009 OSA

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2009 (3)

E. Frączek and G. Budzyń, “An analysis of an optical vortices interferometer with focused beam,” Opt. Appl. 39, 91–99 (2009).

R. K. Singh, P. Senthilkumaran, and K. Singh, “Structure of a tightly focused vortex beam in the presence of primary coma,” Opt. Commun. 282(8), 1501–1510 (2009).
[CrossRef]

R. K. Singh, P. Senthilkumaran, and K. Singh, “Tight focusing of vortex beams in presence of primary astigmatism,” J. Opt. Soc. Am. A 26(3), 576–588 (2009).
[CrossRef]

2008 (4)

B. Spektor, A. Normatov, and J. Shamir, “Singular Beam Microscopy,” Appl. Opt. 47(4), A78–A87 (2008).
[CrossRef] [PubMed]

J. Masajada, M. Leniec, E. Jankowska, H. Thienpont, H. Ottevaere, and V. Gomez, “Deep microstructure topography characterization with optical vortex interferometer,” Opt. Express 16(23), 19179–19191 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-23-19179 .
[CrossRef]

G. D’Aguanno, N. Mattiucci, M. Bloemer, and A. Desyatnikov, “Optical vortices during a superresolution process in a metamaterial,” Phys. Rev. A 77(4), 043825 (2008).
[CrossRef]

L. C. Thomson, Y. Boissel, G. Whyte, E. Yao, and J. Courtial, “Simulation of superresolution holography for optical tweezers,” N. J. Phys. 10(2), 023015 (2008).
[CrossRef]

2007 (4)

B. Sektor, A. Normatov, and J. Shamir, “Experimental validation of 20nm sensitivity of Singular Beam Microscopy,” Proc. SPIE 6616, 661622 (2007).
[CrossRef]

J. Masajada, “The interferometry based on the regular lattice of optical vortices,” Opt. Appl. 37, 167–185 (2007).

P. Kurzynowski and M. Borwińska, “Generation of vortex-type markers in a one-wave setup,” Appl. Opt. 46(5), 676–679 (2007).
[CrossRef] [PubMed]

S. Vyas and P. Senthilkumaran, “Interferometric optical vortex array generator,” Appl. Opt. 46(15), 2893–2898 (2007).
[CrossRef] [PubMed]

2006 (7)

2005 (1)

G. Gbur, H. F. Schouten, and T. D. Visser, “Achieving superresolution in near-field optical data readout systems using surface plasmons,” Appl. Phys. Lett. 87(19), 191109 (2005).
[CrossRef]

2004 (3)

S. Hell, “Strategy for far-field optical imaging and writing without diffraction limit,” Phys. Lett. A 326(1-2), 140–145 (2004).
[CrossRef]

J J. Masajada, “Small-angle rotations measurement using optical vortex interferometer,” Opt. Commun. 239(4-6), 373–381 (2004).
[CrossRef]

S. R. Oemrawsingh, E. R. Eliel, G. Nienhuis, and J. P. Woerdman, “Intrinsic orbital angular momentum of paraxial beams with off-axis imprinted vortices,” J. Opt. Soc. Am. A 21(11), 2089–2096 (2004).
[CrossRef]

2001 (1)

J. Masajada and B. Dubik, “Optical vortex generation by three plane wave interference,” Opt. Commun. 198(1-3), 21–27 (2001).
[CrossRef]

1998 (1)

1997 (1)

M. Totzeck and H. J. Tiziani, “Phase-singularities in 2D diffraction fields and interference microscopy,” Opt. Commun. 138(4-6), 365–382 (1997).
[CrossRef]

1991 (1)

1983 (1)

Bandres, M. A.

Bentley, J. B.

Berry, M. V.

M. V. Berry and S. Popescu, “Evolution of quantum superoscillations, and optical superresolution without evanescent waves,” J. Phys. A 39(22), 6965–6977 (2006).
[CrossRef]

Bloemer, M.

G. D’Aguanno, N. Mattiucci, M. Bloemer, and A. Desyatnikov, “Optical vortices during a superresolution process in a metamaterial,” Phys. Rev. A 77(4), 043825 (2008).
[CrossRef]

Boissel, Y.

L. C. Thomson, Y. Boissel, G. Whyte, E. Yao, and J. Courtial, “Simulation of superresolution holography for optical tweezers,” N. J. Phys. 10(2), 023015 (2008).
[CrossRef]

Borwinska, M.

P. Kurzynowski and M. Borwińska, “Generation of vortex-type markers in a one-wave setup,” Appl. Opt. 46(5), 676–679 (2007).
[CrossRef] [PubMed]

A. Popiołek-Masajada, M. Borwińska, and W. Frączek, “Testing a new method for small-angle rotation measurements with the optical vortex interferometer,” Meas. Sci. Technol. 17(4), 653–658 (2006).
[CrossRef]

Brunfeld, A.

Budzyn, G.

E. Frączek and G. Budzyń, “An analysis of an optical vortices interferometer with focused beam,” Opt. Appl. 39, 91–99 (2009).

Courtial, J.

L. C. Thomson, Y. Boissel, G. Whyte, E. Yao, and J. Courtial, “Simulation of superresolution holography for optical tweezers,” N. J. Phys. 10(2), 023015 (2008).
[CrossRef]

D’Aguanno, G.

G. D’Aguanno, N. Mattiucci, M. Bloemer, and A. Desyatnikov, “Optical vortices during a superresolution process in a metamaterial,” Phys. Rev. A 77(4), 043825 (2008).
[CrossRef]

Davis, J. A.

Desyatnikov, A.

G. D’Aguanno, N. Mattiucci, M. Bloemer, and A. Desyatnikov, “Optical vortices during a superresolution process in a metamaterial,” Phys. Rev. A 77(4), 043825 (2008).
[CrossRef]

Dubik, B.

J. Masajada and B. Dubik, “Optical vortex generation by three plane wave interference,” Opt. Commun. 198(1-3), 21–27 (2001).
[CrossRef]

Eliel, E. R.

Esposito, E.

Fra Czek, E.

Fraczek, E.

E. Frączek and G. Budzyń, “An analysis of an optical vortices interferometer with focused beam,” Opt. Appl. 39, 91–99 (2009).

Fraczek, W.

A. Popiołek-Masajada, M. Borwińska, and W. Frączek, “Testing a new method for small-angle rotation measurements with the optical vortex interferometer,” Meas. Sci. Technol. 17(4), 653–658 (2006).
[CrossRef]

Gbur, G.

G. Gbur, H. F. Schouten, and T. D. Visser, “Achieving superresolution in near-field optical data readout systems using surface plasmons,” Appl. Phys. Lett. 87(19), 191109 (2005).
[CrossRef]

Gibson, G. M.

Girkin, J. M.

Gomez, V.

Gutiérrez-Vega, J. C.

Hanson, S. G.

Hell, S.

S. Hell, “Strategy for far-field optical imaging and writing without diffraction limit,” Phys. Lett. A 326(1-2), 140–145 (2004).
[CrossRef]

Ishijima, R.

Jankowska, E.

Kurzynowski, P.

Leach, J.

Leniec, M.

Masajada, J J.

J J. Masajada, “Small-angle rotations measurement using optical vortex interferometer,” Opt. Commun. 239(4-6), 373–381 (2004).
[CrossRef]

Masajada, J.

J. Masajada, M. Leniec, E. Jankowska, H. Thienpont, H. Ottevaere, and V. Gomez, “Deep microstructure topography characterization with optical vortex interferometer,” Opt. Express 16(23), 19179–19191 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-23-19179 .
[CrossRef]

J. Masajada, “The interferometry based on the regular lattice of optical vortices,” Opt. Appl. 37, 167–185 (2007).

J. Masajada and B. Dubik, “Optical vortex generation by three plane wave interference,” Opt. Commun. 198(1-3), 21–27 (2001).
[CrossRef]

Mattiucci, N.

G. D’Aguanno, N. Mattiucci, M. Bloemer, and A. Desyatnikov, “Optical vortices during a superresolution process in a metamaterial,” Phys. Rev. A 77(4), 043825 (2008).
[CrossRef]

McConnell, G.

Miyamoto, Y.

Nienhuis, G.

Normatov, A.

B. Spektor, A. Normatov, and J. Shamir, “Singular Beam Microscopy,” Appl. Opt. 47(4), A78–A87 (2008).
[CrossRef] [PubMed]

B. Sektor, A. Normatov, and J. Shamir, “Experimental validation of 20nm sensitivity of Singular Beam Microscopy,” Proc. SPIE 6616, 661622 (2007).
[CrossRef]

Oemrawsingh, S. R.

Ottevaere, H.

Padgett, M. J.

Popescu, S.

M. V. Berry and S. Popescu, “Evolution of quantum superoscillations, and optical superresolution without evanescent waves,” J. Phys. A 39(22), 6965–6977 (2006).
[CrossRef]

Popiolek-Masajada, A.

A. Popiołek-Masajada, M. Borwińska, and W. Frączek, “Testing a new method for small-angle rotation measurements with the optical vortex interferometer,” Meas. Sci. Technol. 17(4), 653–658 (2006).
[CrossRef]

Rozas, D.

Sacks, Z. S.

Schouten, H. F.

G. Gbur, H. F. Schouten, and T. D. Visser, “Achieving superresolution in near-field optical data readout systems using surface plasmons,” Appl. Phys. Lett. 87(19), 191109 (2005).
[CrossRef]

Sektor, B.

B. Sektor, A. Normatov, and J. Shamir, “Experimental validation of 20nm sensitivity of Singular Beam Microscopy,” Proc. SPIE 6616, 661622 (2007).
[CrossRef]

Self, S. A.

Senthilkumaran, P.

Shamir, J.

Singh, K.

R. K. Singh, P. Senthilkumaran, and K. Singh, “Structure of a tightly focused vortex beam in the presence of primary coma,” Opt. Commun. 282(8), 1501–1510 (2009).
[CrossRef]

R. K. Singh, P. Senthilkumaran, and K. Singh, “Tight focusing of vortex beams in presence of primary astigmatism,” J. Opt. Soc. Am. A 26(3), 576–588 (2009).
[CrossRef]

Singh, R. K.

R. K. Singh, P. Senthilkumaran, and K. Singh, “Structure of a tightly focused vortex beam in the presence of primary coma,” Opt. Commun. 282(8), 1501–1510 (2009).
[CrossRef]

R. K. Singh, P. Senthilkumaran, and K. Singh, “Tight focusing of vortex beams in presence of primary astigmatism,” J. Opt. Soc. Am. A 26(3), 576–588 (2009).
[CrossRef]

Singher, L.

Spektor, B.

Swartzlander, G. A.

Takeda, M.

Thienpont, H.

Thomson, L. C.

L. C. Thomson, Y. Boissel, G. Whyte, E. Yao, and J. Courtial, “Simulation of superresolution holography for optical tweezers,” N. J. Phys. 10(2), 023015 (2008).
[CrossRef]

Tiziani, H. J.

M. Totzeck and H. J. Tiziani, “Phase-singularities in 2D diffraction fields and interference microscopy,” Opt. Commun. 138(4-6), 365–382 (1997).
[CrossRef]

Totzeck, M.

M. Totzeck and H. J. Tiziani, “Phase-singularities in 2D diffraction fields and interference microscopy,” Opt. Commun. 138(4-6), 365–382 (1997).
[CrossRef]

Visser, T. D.

G. Gbur, H. F. Schouten, and T. D. Visser, “Achieving superresolution in near-field optical data readout systems using surface plasmons,” Appl. Phys. Lett. 87(19), 191109 (2005).
[CrossRef]

Vyas, S.

Wada, A.

Wang, W.

Whyte, G.

L. C. Thomson, Y. Boissel, G. Whyte, E. Yao, and J. Courtial, “Simulation of superresolution holography for optical tweezers,” N. J. Phys. 10(2), 023015 (2008).
[CrossRef]

Woerdman, J. P.

Wozniak, W. A.

Wright, A. J.

Yao, E.

L. C. Thomson, Y. Boissel, G. Whyte, E. Yao, and J. Courtial, “Simulation of superresolution holography for optical tweezers,” N. J. Phys. 10(2), 023015 (2008).
[CrossRef]

Yokozeki, T.

Appl. Opt. (5)

Appl. Phys. Lett. (1)

G. Gbur, H. F. Schouten, and T. D. Visser, “Achieving superresolution in near-field optical data readout systems using surface plasmons,” Appl. Phys. Lett. 87(19), 191109 (2005).
[CrossRef]

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

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

J. Phys. A (1)

M. V. Berry and S. Popescu, “Evolution of quantum superoscillations, and optical superresolution without evanescent waves,” J. Phys. A 39(22), 6965–6977 (2006).
[CrossRef]

Meas. Sci. Technol. (1)

A. Popiołek-Masajada, M. Borwińska, and W. Frączek, “Testing a new method for small-angle rotation measurements with the optical vortex interferometer,” Meas. Sci. Technol. 17(4), 653–658 (2006).
[CrossRef]

N. J. Phys. (1)

L. C. Thomson, Y. Boissel, G. Whyte, E. Yao, and J. Courtial, “Simulation of superresolution holography for optical tweezers,” N. J. Phys. 10(2), 023015 (2008).
[CrossRef]

Opt. Appl. (2)

J. Masajada, “The interferometry based on the regular lattice of optical vortices,” Opt. Appl. 37, 167–185 (2007).

E. Frączek and G. Budzyń, “An analysis of an optical vortices interferometer with focused beam,” Opt. Appl. 39, 91–99 (2009).

Opt. Commun. (4)

J. Masajada and B. Dubik, “Optical vortex generation by three plane wave interference,” Opt. Commun. 198(1-3), 21–27 (2001).
[CrossRef]

J J. Masajada, “Small-angle rotations measurement using optical vortex interferometer,” Opt. Commun. 239(4-6), 373–381 (2004).
[CrossRef]

R. K. Singh, P. Senthilkumaran, and K. Singh, “Structure of a tightly focused vortex beam in the presence of primary coma,” Opt. Commun. 282(8), 1501–1510 (2009).
[CrossRef]

M. Totzeck and H. J. Tiziani, “Phase-singularities in 2D diffraction fields and interference microscopy,” Opt. Commun. 138(4-6), 365–382 (1997).
[CrossRef]

Opt. Express (3)

Opt. Lett. (3)

Phys. Lett. A (1)

S. Hell, “Strategy for far-field optical imaging and writing without diffraction limit,” Phys. Lett. A 326(1-2), 140–145 (2004).
[CrossRef]

Phys. Rev. A (1)

G. D’Aguanno, N. Mattiucci, M. Bloemer, and A. Desyatnikov, “Optical vortices during a superresolution process in a metamaterial,” Phys. Rev. A 77(4), 043825 (2008).
[CrossRef]

Proc. SPIE (1)

B. Sektor, A. Normatov, and J. Shamir, “Experimental validation of 20nm sensitivity of Singular Beam Microscopy,” Proc. SPIE 6616, 661622 (2007).
[CrossRef]

Other (5)

A. Erdèlyi, ed., Tables of integral transforms, McGraw-Hill, New York 1953.

B. Kress, and P. Meyrueis, Applied Digital Optics: from micro-optics to nano-photonics, Edited by John Wiley and Sons, Chichester, UK, April 2009.

V. P. Tychinsky, and C. H. F. Velzel, Super-resolution in Microscopy; in: Current trends in optics, (Academic Press, 1994), Chap. 18.

M. S. Soskin, and M. V. Vasnetsov, “Singular Optics,” Prog Opt. Amsterdam Elsevier, Vol 42, pp. 219–276 (2001)

A. S. Desyatnikov, L. Tornel, and Y. S. Kivshar, “Optical vortices and vortex solitons,” Progress in Optics, vol.47 (Amsterdam Elsevier) chapter 5, 2005.

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Figures (14)

Fig. 2
Fig. 2

The plot of the x-position of the vortex embedded in term f 1 versus q-parameter. The red line is for ideal focusing z=0, the blue and yellow lines represent the case when the focused spot goes beyond the upper surface of the sample at z=0.01 and z=0.1, respectively. The plot was made for objective focal length f=30mm.

Fig. 3
Fig. 3

Phase (a) and intensity (b) representations of term f1 . In the case referenced by (A),) the beam centre is located at the right side of the edge (q= −0.001mm), in case (B) – the beam centre is located on the edge (q=0), and in case (C) the beam centre is located at the left side of the edge (q= 0.001mm). The objective lens had a focal length of 30 mm, the step height was h=0.08µm and the refractive index of the sample was n=1.3. The y-axis is perpendicular to the edge and shows the scanning direction. The area of the image has a diameter of 2mm.

Fig. 4
Fig. 4

Phase (a) and intensity (b) representations of term f 3. in case (A) the beam centre is located on the right side on the edge (q= −0.001mm), in case (B) the beam centre is located on the edge (q=0), and in case (C) the beam centre is located at the left side of the edge (q= 0.001mm). It is easy to notice that the intensity and phase patterns are almost independent of the parameter q. The objective focal length was f=30mm and step height h=0,08μm.

Fig. 5
Fig. 5

((a) phase distribution for sum f1+f2 (b) The lines of equal amplitudes of term f3 are concentric circles (drawn as dotted lines) and the lines of equal amplitudes of term f 1 are elongated circles (drawn as solid lines). While changing the value of q the pattern representing f 1 moves and as a result the vortices change positions. In this figure vortices are referenced by black points. While crossing the edge (q=0) the distance between vortices reaches its local minimum, which in turn indicates the position of the edge.

Fig. 6
Fig. 6

Trajectory of optical vortices. The blue arrows show the pathway of one optical vortex and the red arrows show other one. The blue highlighted numbers show the value of parameter q. When the centre beam is far from the edge, one double charge optical vortex exists in the centre of the plot. When the beam center approaches the edges, the vortex splits into two optical vortices with m=1. When the centre of the beam is exactly located on the edge one vortex returns to centre of the plot, and the distance between the two vortices becomes minimal. To plot this figure the following values were used: step height is h=0.08µm and refractive index of the sample n=1.3, focal length of the objective is f=30mm.

Fig. 7
Fig. 7

Trajectory of optical vortices calculated for the step height h=0,08µm and refractive index of the sample n=1.3, the red line is for f=30mm, the black f=10m and the blue one is obtained by rescaling the black track by factor 3.

Fig. 8
Fig. 8

(a) The measurement system with lcos working as vortex generator; (b) the measurement system with special DOE (vortex lens) M are mirrors, L is laser, BS are beamsplitters, O is a sample on the scanning stage, OB – objectives, DOE – vortex lens.

Fig. 9
Fig. 9

Photo and topological scans of a 4 phase levels vortex diffractive lens of charge 2

Fig. 10
Fig. 10

Longitudinal and lateral numerical reconstruction windows of a Gaussian beam diffracted by a diffractive vortex lens of topological charge 2.

Fig. 11
Fig. 11

Interferogram examples while scanning the submicron step (h=300nm) with a objective focal length of 750mm. Optical vortices are marked by red circles. (A) The beam touches the edge from its left side (see Fig. 1) the vortices are slightly separated, (B) the beam gets closer to edge and vortices get more apart, (C) the beam center is close to the edge and vortices are closed to each other, (D,E and F), and then the beam center moves away from the edge. First, the distance between the vortices increases (D), then they become closer (E) and finally they are joined into one single vortex of charge 2 (F). These interferograms were chosen from a series of interferograms taken with scanning step value about 1μm. The first interferogram (A) was taken about 40μm before step and last about 40μm after step. Only the central part of the image is shown.

Fig. 1
Fig. 1

Description of the parameters used for calculations.

Fig. 12
Fig. 12

When the scanning beam is far from the edges, both vortices are located almost at the same plane (A). When the focused beam touches the edge, the vortices move apart. (B,C). When the beam center is near to the edge, the vortices get closer and the distance is minimal when the beam center is located exactly at the edge (D). When moving beyond the edge, the vortices get apart and again closer to each other (E,F)

Fig. 13
Fig. 13

The measured trajectory of optical vortices (see, Fig. 6 to compare with calculated trajectory). While the beam center approaches the edge the distance between two vortices become minimal (this is local minimum). The red dots indicates the two vortices at such minimal distance. The scanning step was 1 micron. The small figure (lower right hand corner) shows the part of the large Fig. for the scanning beam center being close to the edge. For this part the scanning step was 60nm. By inspecting the distance between vortices we can detect the edge position with an accuracy of ±60nm. The asymmetry of the picture is due to system aberrations and misalignment and errors introduced by the vortex lens. The object was scanned at a range ±15μm from the edge.

Fig. 14
Fig. 14

(A) The phase distribution of the erf function in formula (5c) along the line perpendicular to the step (y-axis) for x i=0. Focal length of the objective is 30mm, step height h=0.001mm (blue line), h=0.01mm (yellow line), h=0.1mm (red line). In general the π-jump moves away from the y i=0 line when h increases. (B) the same picture plotted for h=0.0003mm – no phase jump occurs at the plotted area, (C) h=0.00002mm the phase distribution is almost linear. (D) In case of h much bigger than λ, the position of the π-jump introduced by erf function depends on the focal length of the objective. Blue plot is for f=30mm, red for f=60mm and yellow for f=120mm. (E) the plot of phase distribution of the f3 f=60mm, (F) the plot of phase distribution of the f3 f=30mm.

Equations (12)

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UG(x,y)=(x+jsgnσy)mexp{x2+y2b}.
Ω  exp{jknd}(In1+In2),
Ω  =exp(jkzet)jλzetexp[jkλz(xi2+yi2)],
In1=exp{jkh}qUG(x,y)exp{jk2zet(x2+y22xxi2yyi)}dx dy,
In2=exp{jknh}qUG(x,y)exp{jk2zet(x2+y22xxi2yyi)}dx dy.
U(xi,yi)=14β2(2βexp(q(βq+2ikzyo)πsgnt12(βqsgnjikzx0jikzvo)+Ξyπβ(tt+t12)(sgn21)+2Ξyπikz2vo2(tt+t12erf[βqikzy0β]))
f1=2βexp(q(βq+2ikzyo)πsgnt12(βqsgnjikzx0jikzvo);
f2=Ξyπβ(tt+t12)(sgn21);
f3=2Ξyπikz2zo2(tt+t12erf[βqikzy0β]).
f1'=(qsgnb+kzx0)+j(ksgnq2z+ksgnq2zy0).
x0=sgnzbkqandy0=q.
f3=2Ξyπikz2vo2(tt+t12exp[ikzy0β](1+qβ)).

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