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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 18 — Aug. 31, 2009
  • pp: 16144–16159
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Micro-step localization using double charge optical vortex interferometer

Jan Masajada, Monika Leniec, Sławomir Drobczyński, Hugo Thienpont, and Bernard Kress  »View Author Affiliations


Optics Express, Vol. 17, Issue 18, pp. 16144-16159 (2009)
http://dx.doi.org/10.1364/OE.17.016144


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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

1. Introduction

2. Theoretical backgroung

The theory is based on solving scalar diffraction integrals in the near field approximation. We made the same assumption in paper [1

1. 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]

]. Let us consider here a micro-step fabricated in a transparent material.

When scanning the micro-step with a focused Gaussian beam yielding an optical vortex of topological charge 2 we observe the vortex (or vortices) behavior(s) at the image plane produced by an objective lens.

The focused Gaussian beam yielding optical vortices is described by the following equation:

UG(x,y)=(x+jsgnσy)mexp{x2+y2b}.
(1)

In this equation m is the optical vortex topological charge, sgn is the sign of the charge, and b is given by b=ω02(1+zet2R2),where zet is the distance between the sample plane and the waist position of the beam, ω0 is the Gaussian beam waist radius and R is the Rayleigth distance. We set m to = 2. We use the relation ω0=2λπfD for determining ω0 for given objective focal length f. D is diameter of laser beam illuminating the objective. In our calculations D=3mm. Formula (1) is a simplified version of the general formula describing fundamental Gaussian mode. Here we neglect the phase factor representing the Gouy phase. When the incident plane is a waist plane (or nearly waist plane) (zet = 0), the formula (1) is exact; since we assume that the waist of the beam coincides with the upper part of our sample, we can use this simplification. We also assume that our focusing system is aberration free, so the focused beam can be described just by setting the proper value for parameters in formula 1. These steps are typical in paraxial Gaussian optics (see, for example [22

22. L. Singher, J. Shamir, and A. Brunfeld, “Focused-beam interaction with a phase step,” Opt. Lett. 16(2), 61–63 (1991). [CrossRef] [PubMed]

,23

23. S. A. Self, “Focusing of spherical Gaussian beams,” Appl. Opt. 22(5), 658–661 (1983). [CrossRef] [PubMed]

]).

As shown in [1

1. 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]

], the solution of our diffraction problem can be written as
Ω  exp{jknd}(In1+In2),
(2)
where Ω is the constant term of Fresnel integrals.

Ω  =exp(jkzet)jλzetexp[jkλz(xi2+yi2)],
In1=exp{jkh}qUG(x,y)exp{jk2zet(x2+y22xxi2yyi)}dx dy,
(3a)
In2=exp{jknh}qUG(x,y)exp{jk2zet(x2+y22xxi2yyi)}dx dy.
(3b)

We can scan through the sample by changing the value of parameter q; for example, when q=0, the beam center is located exactly at the sidewall edge.

We now insert the expression for U G with m=2 and find the sum of integrals (2). After calculations the resulting expression of the diffracted field can be written in the following form:
U(xi,yi)=14β2(2βexp(q(βq+2ikzyo)πsgnt12(βqsgnjikzx0jikzvo)+Ξyπβ(tt+t12)(sgn21)+2Ξyπikz2vo2(tt+t12erf[βqikzy0β]))
(4)
where erf is the error function [24

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

], ikz=jk2z;     β=1b+ikz;     Ξy=     exp{ikz2βyi2}; t12=exp{jkh}exp{jknh};tt=exp{jkh}+exp{jknh}; vo=(x0+jsgny0).

At the object plane we have the referential system based on (x o, y o), and at the image plane we have the referential system based on (x i, y i.)

This formula can be split into the three following distinctive members:

f1=2βexp(q(βq+2ikzyo)πsgnt12(βqsgnjikzx0jikzvo);
(5a)
f2=Ξyπβ(tt+t12)(sgn21);
(5b)
f3=2Ξyπikz2zo2(tt+t12erf[βqikzy0β]).
(5c)

The first member f1 shows an interesting behavior when the beam crosses the edges of the micro-step. It contains an optical vortex (marked as vo in the above formulas), which is located at the center of the beam for q=0. For other values of q the position of the vortex changes drastically. In order to locate this vortex, we calculate the positions of the zeros in the complex field amplitude. After some calculation, and after removing terms having the form e and real terms independent of q, x o and y o, we get:

f1'=(qsgnb+kzx0)+j(ksgnq2z+ksgnq2zy0).
(6)

This expression is equal to zero when both the real and imaginary parts are equal to zero. Thus, we have two equations describing the position of vortex point. The solution of this equation is:

x0=sgnzbkqandy0=q.
(7)

Due to the large value of the multiplying coefficient q in the Eq. (7) , the zero intensity point moves rapidly along the x o axis (see 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.
and 3b
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.
). Its movement in the y direction is slower and equal in magnitude to the distance between the edge and the beam center. It is interesting to notice that the position of the vortex (vortex seeded in term f1) does not depend on the step height h. For small changes of q, the amplitude pattern of the f1 term preserves its general geometry. More visible changes occur in the phase distribution (Fig. 3a), however when applying smaller q the phase distribution remains almost constant.

Let us assume now that our vortex is symmetrical i.e. sgn=± 1. In such a case f2=0. The third term f3 contains a vortex with a charge magnitude of 2. The vortex stays at the beam center while scanning the sample. f3 depends also on the erf function. For a small q we can decomposed the erf function into a power series and reject all terms of power 2 or higher. We get thus the expression:

f3=2Ξyπikz2vo2(tt+t12exp[ikzy0β](1+qβ)).
(8)

The argument of expression 1+qβ is very small and changes linearly with q, which means that for a small q the influence of the erf function on the phase distribution in f3 is negligible (Fig. 4a
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.
). Also the amplitude of f3 change slightly for small q changes (Fig. 4b).

Therefore, the whole diffraction pattern can be considered as a sum of almost constant (independent on q) term f3 and fast moving amplitude and phase patterns described by term f1 (when q changes). For the specific focusing objectives and sample parameters described here, this general pattern will contain two optical vortices. In the image plane vortices emerge at the point where the amplitudes of terms f1 and f3 are the same, and the phases are shifted by π. When q become different than zero the first terms moves quickly along the x-axis. As a result, the two vortices move apart. The line joining both vortex points will rotate in a specific way when crossing the edge of the step. These dynamics are illustrated in 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.
.

For larger values of q, we must take into account the rapid decrease of the f1 amplitude due to factor exp(-βq2). Due to this factor, the influence of the term f 1 decreases and the two vortices get closer again to finally form the input pattern; i.e. optical vortex with charge two is reconstructed. When the value of q gets large enough, the focused light spot looses contact with the edge. We can summarize our discussion as follows: while approaching the edge of the step, the input vortex splits into two vortices which move apart, up to some maximal distance. They then get again closer, and when the scanning beam center is located exactly at the edge location, the distance between vortex points is minimal (local minimum). This distance can then again increase to a maximal value and decrease until the entire geometry of the input beam is reconstructed. Figure 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.
and 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.
shows an example of a vortex trajectory. Due to the high sensitivity of the term f1 over the value of q, - we can observe these dynamics with a high precision, which enables a superresolving detection of the edge location. The location resulting in a minimal distance between the vortex points gives the position of the step edge .

In practical measurements the parameter sgn may differ from ± 1. In this case we get vortices with asymmetrical distributions of the phase lines [25

25. Z. S. Sacks, D. Rozas, and G. A. Swartzlander Jr., “Holographic formation of optical-vortex filaments,” J. Opt. Soc. Am. B 15(8), 2226–2234 (1998). [CrossRef]

,26

26. 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]

], and the factor f2 becomes different than zero. The incident vortex is split into two single vortices due to vortex asymmetry itself. In result the sensitivity of our method is somehow lower. Fortunately small deviations from ± 1 are not harmful for the presented method and modern technology supports us with vortex generators of sufficient quality. It is interesting to notice that f 2 does not depends on q or x o,y o. The additional separation of incident vortex may be also caused by aberrations of the optical system or its misalignment.

Up to now we have assumed that the incident beam is focused at the upper sample surface. Here we want to comment on the case when the focusing is not ideal. Inspecting our formula leads to the conclusion that small defocusing has negligible influence on the measurement results. The influences of small and large defocusing on the investigated parameters are shown in Fig. 2.

3. Experiment

In order to generate a Gaussian beam with optical vortex we have used two different methods. In the first method the vortex beam was generated using a spatial light modulator and an objective with a long focal length of 750mm [27

27. J. Leach, G. M. Gibson, M. J. Padgett, E. Esposito, G. McConnell, A. J. Wright, and J. M. Girkin, “Generation of achromatic Bessel beams using a compensated spatial light modulator,” Opt. Express 14(12), 5581–5587 (2006). [CrossRef] [PubMed]

,28

28. J. B. Bentley, J. A. Davis, M. A. Bandres, and J. C. Gutiérrez-Vega, “Generation of helical Ince-Gaussian beams with a liquid-crystal display,” Opt. Lett. 31(5), 649–651 (2006). [CrossRef] [PubMed]

]. Here, the collimated laser beam is incident onto the spatial light modulator on which fork-like fringe pattern was written. We extract from the diffraction pattern the first and zero order diffraction beam, simply by applying a small aperture. The first order diffraction beam contains an optical vortex of m=2. The zero order beam serves as a reference beam (Fig. 8a
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.
).

In the second method a special diffractive structure, a diffractive vortex lens, with short focal length (30mm) [29

29. G. A. Swartzlander Jr., “Achromatic optical vortex lens,” Opt. Lett. 31(13), 2042–2044 (2006). [CrossRef] [PubMed]

] was used. The reference beam was generated in a standard Mach-Zender interferometric configuration (Fig. 8b).

For the purpose of our experiments we have designed, modeled and fabricated diffractive vortex lenses of topological charge 2. In order to achieve at least 80% diffraction efficiency, we have opted for 4 levels phase elements, to be fabricated by conventional multilevel lithography and subsequent reactive ion beam etching (RIE), in quartz wafers. Figure 9
Fig. 9 Photo and topological scans of a 4 phase levels vortex diffractive lens of charge 2
shows one of the diffractive lenses etched under 4 levels and diced out into individual dies. The 3D topological scans of the lens have been performed by a white light confocal interferometric microscope (Wyko). The longitudinal and lateral numerical reconstructions are shown in Fig. 10
Fig. 10 Longitudinal and lateral numerical reconstruction windows of a Gaussian beam diffracted by a diffractive vortex lens of topological charge 2.
, as intensity and phase windows. They have been computed by using extended scalar theory of diffraction [30

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

] in the Fresnel regime. These numerical reconstructions show the typical toroidal section of the diffracted vortex beams, as well as the phase uniformity along the radial dimension, owing to the topological charge 2 of the vortex lens. The lateral reconstruction windows show how the phase vortex changes direction at the focal plane (window d) as well as the toroidal beam waist at this location.

The object we have measured was a microstep of height 300nm manufactured in our laboratory. This same object was introduced in both measurement systems in the focus plane of the objective lens. As for illumination, we used a He-Ne laser beam at λ=633nm.

Figure 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 Description of the parameters used for calculations.
and 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)
show measurements examples for the system described respectively in Fig. 8a and 8b. In Fig. 11 the six interferograms were taken respectively close to the edge, in the vicinity of the edge and just behind the edge. The scanning step was about 1μm. The focal length of the objective was 750 mm. This example shows that in principle, even for a long focal length of the objective lens, the vortex dynamics in case of submicron step heights differ from that discussed in paper [1

1. 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]

], for HARMS samples.

Figure 12 shows six interferograms produced by the system referred to in Fig. 8b. By inspecting the fork dynamics we can see that when moving towards the edges, both vortices get apart while when approaching close to the edge they get closer to each other, and when the beam crosses the edge they drift apart again. Figure 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.
shows these dynamics in a more general view. In the image plane, both vortices were always slightly apart even if we put them far from the edge. This separation was small (1/7th of the distance between vortices when the beam was located at the edge). This is probably due to errors in the optical system (such as misalignments), and in smaller way to errors in our vortex lens.

With our experimental setup we can measure the position of the edge with accuracy of ±60nm which apparently exceed the resolution limit for objective having a focal length 30mm. The numerical calculations shows that for smaller focal lengths we could expect a much better resolution. To verify this conclusion experimentally we would need a better setup for generating and focusing the optical vortex. Since it is hard to manufacture vortex lenses with shorter focal length, we plan for future measurements to use a separate high quality element for vortex generation (without focusing) and work with microscopic objectives with high numerical apertures.

4. HARMS sample measurement

By using a focused Gaussian beam with optical vortices having a charge of m=2, we can perform a high quality inspection of the HARMS structures. The behavior of vortices in the diffracted field is similar to those reported in [1

1. 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]

] for an optical vortex with m=1. For the sidewalls perpendicular to the underlying substrate, the vortex dynamics are simple – none or one additional vortex appear when the scanning beam center is close to the edge. When the sidewalls are tilted, a number of new vortices appear and move chaotically when the beam crosses the edges. The system sensitivity is a function of the power of the objective lens. Although similar in their behavior, the dynamics of higher charge vortices are more complicated than for fundamental vortexes. Therefore, higher charge vortices are not interesting for HARMS elements inspection.

In this paper we discuss the diffraction effects produced by nano-step with small heights h (h< λ) rather than by HARMS structures having step heights measured in hundreds of microns. As discussed in [1

1. 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]

] the erf function introduces a π-jump in the phase distribution of the term f3. This π-jump was responsible for the characteristic behavior of the vortices in case of a perfect micro-step geometry (for HARMS samples). The position of the π-jump depends on the balance between the factors t12 and tt in f3 (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.
). For small step heights, this π-jump is shifted below the area where vortices can appear and has no influence on their behavior (Fig. 14 B-C). The other factor which is important here is the focusing power of the objective. Smaller focal lengths objectives result in smaller areas where vortices can appear and the π-jump moves away (Fig. 14 D-F). We can conclude that with a short focus objective lens (such as microscopic objectives), we could precisely measure the position of an edge having heights smaller than λ. When the step height is greater than λ, measurements are still possible but the vortex behavior becomes more complicated. This is because the π-jump is at the area where vortices are created and more than two vortices can be observed during measurements. On the other hand, for HARMS elements inspection, objectives with short focal length may produce diffraction images with no additional vortex (for perfect sidewall), whereas for non perfect tilted sidewalls a higher number of vortices is generated.

In case of HARMS like sample inspection the main focus was measuring the slope of the edge which characterizes the quality of the pillar sidewall. The behavior of optical vortices is strictly dependant on the angle of that edge. In the present considerations this factor is negligible. We have positioned the plate with a sample normal to the incident beam. For a small step height of less than λ, slope angles of 2-3 degrees of arc (due technology errors or misalignment) cannot be measured, when using f=30mm objective lens.

5. Conclusions

A new method for submicron lateral step detection is presented. The tracking of the relative positions of two optical vortices has been introduced, which enables edge detection with resolutions exceeding the diffraction limit of classical optical microscopic imaging systems. The resolution of our method depends on three factors: the quality of the optical element generating the optical vortex, the numerical aperture of the objective lens and the method used for the vortex points localization. It is difficult to estimate at the present time what would be the final resolution of the method; to answer this question more measurements and more numerical and analytical modeling is necessary.

The resolution of our system working with an f=30mm (NA~0.07) objective and red laser light exceeds the diffraction limit for a microscopes working with objective having numerical aperture 0.95. Our analytical and numerical studies show that when using objectives with such high numerical aperture the resolution of our system can be improved by factor of 15. On the other hand we have neglected polarization effects which may have remarkable influence on the vortex pattern, for objectives with NA>0.4 [31

31. 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]

,32

32. 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]

].

The constituting elements of our microscope are very simple. In its basic configuration, the system is composed of a laser, an optical element generating the optical vortex, a focusing objective, a translation table, a camera and a computer. Besides, the analysis of the resulting fringe patterns can be performed by commercially available fringe processing packages and performed by a computer for a fully computerized system. The simplicity of the measurement system gives space for its improvement and adaptation for various needs.

It is hard to judge what the future of this method is. What can we do for sure is to localize (with vortex lens f=30mm) the 100nm nano-step with accuracy ±25nm (as resulting from careful extrapolation of experimental results). To make a progress the following steps are necessary: our system was mounted as separate elements on the optical table. To avoid alignment errors a better system would be made of dedicated opto-mechanical mounts producing much tighter alignment tolerances, in an experimental setup the vortex lens must be split into two elements – one is vortex lens without focusing power and the second is classical objective with high numerical aperture, the study on the polarization effects are necessary to answer if better resolution is possible by applying focusing objectives with NA>0.4. Nevertheless, in the past the successful superresolution step detection was the first step for developing most of superresolution method in microscopy.

Acknowledgments

We gratefully acknowledge financial support of the grant KBN (N N505255635).

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

18.

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]

19.

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]

20.

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]

21.

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]

22.

L. Singher, J. Shamir, and A. Brunfeld, “Focused-beam interaction with a phase step,” Opt. Lett. 16(2), 61–63 (1991). [CrossRef] [PubMed]

23.

S. A. Self, “Focusing of spherical Gaussian beams,” Appl. Opt. 22(5), 658–661 (1983). [CrossRef] [PubMed]

24.

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

25.

Z. S. Sacks, D. Rozas, and G. A. Swartzlander Jr., “Holographic formation of optical-vortex filaments,” J. Opt. Soc. Am. B 15(8), 2226–2234 (1998). [CrossRef]

26.

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]

27.

J. Leach, G. M. Gibson, M. J. Padgett, E. Esposito, G. McConnell, A. J. Wright, and J. M. Girkin, “Generation of achromatic Bessel beams using a compensated spatial light modulator,” Opt. Express 14(12), 5581–5587 (2006). [CrossRef] [PubMed]

28.

J. B. Bentley, J. A. Davis, M. A. Bandres, and J. C. Gutiérrez-Vega, “Generation of helical Ince-Gaussian beams with a liquid-crystal display,” Opt. Lett. 31(5), 649–651 (2006). [CrossRef] [PubMed]

29.

G. A. Swartzlander Jr., “Achromatic optical vortex lens,” Opt. Lett. 31(13), 2042–2044 (2006). [CrossRef] [PubMed]

30.

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

31.

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]

32.

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]

OCIS Codes
(110.0180) Imaging systems : Microscopy
(120.3180) Instrumentation, measurement, and metrology : Interferometry
(260.1960) Physical optics : Diffraction theory

ToC Category:
Imaging Systems

History
Original Manuscript: June 22, 2009
Revised Manuscript: July 31, 2009
Manuscript Accepted: August 11, 2009
Published: August 26, 2009

Citation
Jan Masajada, Monika Leniec, Sławomir Drobczyński, Hugo Thienpont, and Bernard Kress, "Micro-step localization using double charge optical vortex interferometer," Opt. Express 17, 16144-16159 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-18-16144


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References

  1. 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]
  2. J. Masajada and B. Dubik, “Optical vortex generation by three plane wave interference,” Opt. Commun. 198(1-3), 21–27 (2001). [CrossRef]
  3. J J. Masajada, “Small-angle rotations measurement using optical vortex interferometer,” Opt. Commun. 239(4-6), 373–381 (2004). [CrossRef]
  4. 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]
  5. P. Kurzynowski, W. A. Woźniak, and E. Frą Czek, “Optical vortices generation using the Wollaston prism,” Appl. Opt. 45(30), 7898–7903 (2006). [CrossRef] [PubMed]
  6. P. Kurzynowski and M. Borwińska, “Generation of vortex-type markers in a one-wave setup,” Appl. Opt. 46(5), 676–679 (2007). [CrossRef] [PubMed]
  7. S. Vyas and P. Senthilkumaran, “Interferometric optical vortex array generator,” Appl. Opt. 46(15), 2893–2898 (2007). [CrossRef] [PubMed]
  8. J. Masajada, “The interferometry based on the regular lattice of optical vortices,” Opt. Appl. 37, 167–185 (2007).
  9. M. S. Soskin, and M. V. Vasnetsov, “Singular Optics,” Prog Opt. Amsterdam Elsevier, Vol 42, pp. 219–276 (2001)
  10. A. S. Desyatnikov, L. Tornel, and Y. S. Kivshar, “Optical vortices and vortex solitons,” Progress in Optics, vol.47 (Amsterdam Elsevier) chapter 5, 2005.
  11. W. Wang, T. Yokozeki, R. Ishijima, A. Wada, Y. Miyamoto, M. Takeda, and S. G. Hanson, “Optical vortex metrology for nanometric speckle displacement measurement,” Opt. Express 14(1), 120–127 (2006). [CrossRef] [PubMed]
  12. V. P. Tychinsky, and C. H. F. Velzel, Super-resolution in Microscopy; in: Current trends in optics, (Academic Press, 1994), Chap. 18.
  13. M. Totzeck and H. J. Tiziani, “Phase-singularities in 2D diffraction fields and interference microscopy,” Opt. Commun. 138(4-6), 365–382 (1997). [CrossRef]
  14. B. Sektor, A. Normatov, and J. Shamir, “Experimental validation of 20nm sensitivity of Singular Beam Microscopy,” Proc. SPIE 6616, 661622 (2007). [CrossRef]
  15. B. Spektor, A. Normatov, and J. Shamir, “Singular Beam Microscopy,” Appl. Opt. 47(4), A78–A87 (2008). [CrossRef] [PubMed]
  16. E. Frączek and G. Budzyń, “An analysis of an optical vortices interferometer with focused beam,” Opt. Appl. 39, 91–99 (2009).
  17. S. Hell, “Strategy for far-field optical imaging and writing without diffraction limit,” Phys. Lett. A 326(1-2), 140–145 (2004). [CrossRef]
  18. 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]
  19. 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]
  20. 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]
  21. 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]
  22. L. Singher, J. Shamir, and A. Brunfeld, “Focused-beam interaction with a phase step,” Opt. Lett. 16(2), 61–63 (1991). [CrossRef] [PubMed]
  23. S. A. Self, “Focusing of spherical Gaussian beams,” Appl. Opt. 22(5), 658–661 (1983). [CrossRef] [PubMed]
  24. A. Erdèlyi, ed., Tables of integral transforms, McGraw-Hill, New York 1953.
  25. Z. S. Sacks, D. Rozas, and G. A. Swartzlander., “Holographic formation of optical-vortex filaments,” J. Opt. Soc. Am. B 15(8), 2226–2234 (1998). [CrossRef]
  26. 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]
  27. J. Leach, G. M. Gibson, M. J. Padgett, E. Esposito, G. McConnell, A. J. Wright, and J. M. Girkin, “Generation of achromatic Bessel beams using a compensated spatial light modulator,” Opt. Express 14(12), 5581–5587 (2006). [CrossRef] [PubMed]
  28. J. B. Bentley, J. A. Davis, M. A. Bandres, and J. C. Gutiérrez-Vega, “Generation of helical Ince-Gaussian beams with a liquid-crystal display,” Opt. Lett. 31(5), 649–651 (2006). [CrossRef] [PubMed]
  29. G. A. Swartzlander., “Achromatic optical vortex lens,” Opt. Lett. 31(13), 2042–2044 (2006). [CrossRef] [PubMed]
  30. B. Kress, and P. Meyrueis, Applied Digital Optics: from micro-optics to nano-photonics, Edited by John Wiley and Sons, Chichester, UK, April 2009.
  31. 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]
  32. 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]

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