Abstract

We report on the fabrication and the experimental demonstration of Moiré diffractive spiral phase plates with adjustable helical charge. The proposed optical unit consists of two axially stacked diffractive elements of conjugate structure. The joint transmission function of the compound system corresponds to that of a spiral phase plate where the angle of mutual rotation about the central axis enables continuous adjustment of the helical charge. The diffractive elements are fabricated by gray-scale photolithography with a pixel size of 200 nm and 128 phase step levels in fused silica. We experimentally demonstrate the conversion of a TEM00 beam into approximated Laguerre-Gauss (LG) beams of variable helical charge, with a correspondingly variable radius of their ring-shaped intensity distribution.

© 2015 Optical Society of America

1. Introduction

Spiral phase plates are elements with a helically increasing optical thickness. Depending on the design wavelength λd the step height is engineered such that the phase distribution of incident radiation is azimuthally shifted in the interval [0, 2πℓ], where is the helical charge, a signed nonzero integer. Along the central axis the vortex phase distribution of an incident plane wave of wavelength λd contains all shifted phase values ranging from 0 to 2πℓ, giving rise to destructive interference resulting in on-axis cancellation of the intensity distribution and the formation of a ring-shaped beam in the far-field with a radius proportional to [1]. These so called doughnut beams approximate a Laguerre-Gauss mode of helical index (or helical charge) , although, due to the lack of the corresponding amplitude modulation, they are actually a superposition of various LG-modes, but dominated by the mode with helical index . In the following we call these beams ”LG-like” beams. An extended discussion can be found in [2]. An LG-like beam with helical index carries an orbital angular momentum L = ℓħ per photon, exerting a torque proportional to the intensity on illuminated objects [35].

In the visible regime, spiral phase or vortex plates are used for micro manipulation of trapped particles by orbital angular momentum exchange [6]. In stimulated emission depletion (STED) microscopy spiral phase plates are regularly used for the creation of a ring-shaped de-excitation beam [7,8]. A different application of spiral phase plates is Fourier filtering in imaging systems for contrast enhancement. Spiral phase contrast offers an easy to implement contrast channel in an optical microscope where a vortex structure is inserted in the back focal plane of the microscope objective to achieve isotropic [9] or anisotropic edge enhancement [10,11]. In extrasolar planet detection vortex plates with a helicity of = 2 can serve as a filter to cancel the image of a star in a coronagraph [12, 13]. Vortex structures, as present in mica or some crystallized organic substances, can be located and highlighted by taking the intensity difference of two images spiral phase filtered with vortices of opposite handedness, e.g. = ±1 [14]. A combination of the two main benefits of spiral phase plates was reported in STED microscopy where the same vortex plate for generation of the doughnut-shaped de-excitation beam is also utilized for contrast enhancement of the probe image [15]. In optical communication systems light in multiple orbital angular momentum states created with optical vortices of different helicities can be used for data multiplexing at the same frequency [16]. A recent interest of vortex beams arises in ultra-intense laser field physics. For example, intense ultra-short vortex pulses are expected to excite otherwise forbidden molecular transitions [17], to produce relativistic twisted light pulses which exert a strong torque on materials [18], and to produce terawatt vortex laser beams for filamentation studies in gases [19].

A common alternative to spiral phase plates for the creation of vortex phase distributions is the use of a spatial light modulator (SLM), where any desired phase distribution can be generated. However, high resolution SLMs are expensive and are only available as reflective devices resulting in a less compact experimental setup. Furthermore, liquid crystal SLMs are polarization sensitive, which reduces their total efficiency and may be an issue in low signal applications. Other methods to generate vortex beams use computer generated holograms [20], or Dammann gratings [21, 22], which, however have a limited efficiency when generating a vortex beam with a defined helical index.

Here we report on the design and fabrication of high efficiency spiral phase plates in fused silica with an adjustable helicity for a fixed wavelength of λd = 532 nm. The proposed optical unit consists of two axially stacked diffractive elements with conjugate diffractive patterns exploiting the optical Moiré effect [23]. The combined transmission function of the superposed elements corresponds to that of a plain glass substrate, while mutual rotation about the central axis by an amount θ is designed to transform the homogeneous optical thickness to a spiral phase plate with a helicity depending on the rotation angle. This enables the creation of LG-like beams with an orbital angular momentum L = ℓħ, tunable from negative to positive values. We present experimental results of the transformation of a focused Gaussian beam to focused LG-like beams with helicities of = +1, +2, +3, +4, +5 and +10.

2. Moiré diffractive spiral phase plates

Rotational Moiré diffractive optical elements (MDOEs) were proposed in an earlier publication [23] and experimentally demonstrated as focus tunable Fresnel lenses [24]. The working principle is based on the optical Moiré effect, where two superposed gratings of a similar spacing can lead to a structure with a much larger beating period. The joint transmission function of two stacked optical phase elements is given by Tjoint = exp{i[Φ1 + Φ2]}, where Φ1,2 are the spatially modulated phase profiles of the two elements, respectively.

The desired transmission function of a spiral phase plate is Tspiral = exp{iℓϕ}, where ϕ is the angle in polar coordinates ranging from 0 to 2π. This transmission function can be obtained by stacking two phase elements of the form

Φ1(ϕ)=+aϕ2/2πΦ2(ϕ)=aϕ2/2π,
where a is a design parameter. If the second element is rotated by an angle θ with respect to the first one, its phase becomes
Φ2(ϕ)={a(ϕθ)2/2πforθϕ<2πa(ϕθ+2π)2/2πfor0ϕ<θ.
Distinguishing between two angular sectors is necessary because of the rotational ambiguity (i.e. a rotation of θ in one direction corresponds to a rotation of 2πθ into the other direction), and assures that the angular argument (in parentheses) is limited to the interval between 0 and 2π, where it is defined.

The joint transmission function of the compound optical unit Tjoint(ϕ) = exp{i[Φ1(ϕ) + Φ2(ϕ)]} is therefore:

Tjoint(ϕ)={exp{i[2θϕθ2]a/2π}forθϕ<2πexp{i[(2θ4π)ϕθ24π2+4πθ]a/2π}for0ϕ<θ.

All phase contributions in the exponent of Eq. (3) which are multiplied with ϕ correspond to the transmission functions of a spiral phase plate, whereas the other contributions are just uniform phase offsets (for a fixed rotation angle θ) which are of no significance for most applications. Thus two different spiral phase plates are obtained in the two sectors, with helicities of

={aπθforθϕ<2πaπθ2afor0ϕ<θ,
where the design parameter a determines the amount of mutual rotation θ required to change the vortex MDOE’s helicity by ±1. Note that also rotation angles, which produce non-integer values of , can be adjusted, which result in the generation of vortex beams with a fractional charge [25].

In order to produce the phase elements as diffractive optics components, the actual phase functions Φ1 and Φ2 in Eq. (1) are wrapped modulo 2π, i.e. they become

Φ1(ϕ)=mod2π{aϕ2/2π}Φ2(ϕ)=mod2π{aϕ2/2π},
respectively. At the designated wavelength, this wrapping procedure does not influence the corresponding transmission functions, but allows to produce the optical elements as DOE plates with a reduced thickness.

Figure 1(a) and 1(b) show the phase functions mod2π1(ϕ)} and mod2π2(ϕ)}, respectively, with a design parameter a = 90. For θ = 0° the vortex MDOE only introduces a uniform phase shift related to the optical thickness of the glass substrates, while a mutual rotation by θ = 2° leads to the phase profile of a first order spiral phase plate, i.e. with a helical index of = 1. In Fig. 1(c) the simulated combined phase function of the two phase plates is shown for an exemplary angle of mutual rotation of θ = 8°, which results in a vortex with a helicity of = 4. As expected, a second sector which contains a spiral phase functions with a different helicity appears in a small angular range corresponding to the rotation angle θ. As the sector formation is undesired for practical applications the design parameter a should be chosen sufficiently large to enable a wide tuning range of the dominant helical phase front with only a small sector containing the undesired phase contribution.

 figure: Fig. 1

Fig. 1 Phase functions for a design parameter a = 90 of (a) Φ1(ϕ), (b) Φ2(ϕ) and (c) of the compound unit for an exemplary angle of mutual rotation ϕ = 8° resulting in a helicity of = 4. The grey levels in the figure correspond to phase shifts in an interval between −π and π.

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3. Experiments

Two phase plates with transmission functions according to Eq. (5) (with parameter a = 90 and a maximal radius of 0.5 cm), were fabricated by grey scale lithography and subsequent selective reactive ion etching. The parameter a is limited by the condition that the steepest phase gradient in the phase functions of Eq. (5) has to be lower than the lithographic resolution (i.e. the resolvable pixel size p) divided by π. But actually the phase gradient along a circle with radius R around the center of the phase plates becomes infinite with decreasing radius, i.e. the phase pattern can only be resolved outside of a circular disk with a radius of Rmin. This leads to the condition:

a<Rminπ2p,
where p is the lithographic pixel resolution, and Rmin is the radius of the central disk outside of which the spiral phase pattern is correctly displayed. In our case we chose a=90, which corresponds to a negligiblunresolved region in the center with a radius of Rmin = 15μm, assuming an optimal lithographic resolution of p = λ/2 = 270 nm.

The transmission functions were realized as two grey scale patterns in a pixel raster with 200 nm resolution, and 128 grey values corresponding to 128 different exposure dose values. Each pattern was exposed four times to average out the effect of dose fluctuations during the exposure.

A fused silica plate was coated with diluted photoresist (AZ 4562) resulting in a resist layer of 3.2 μm. The photoresist was exposed by a direct writing lithography system based on an Ar-ion laser with wavelength λ = 364 nm. After development with slightly diluted developer (AZ 351B : H2O = 1 : 2) the desired pattern was reproduced in the height profile of the resist. In a preceding calibration procedure the height of the resist (after development) was determined as a function of the exposure dose. Applying this calibration during the exposure, a linear dependence between the grey values in the pattern and the final resist height was achieved.

The resist pattern was afterwards transferred into the fused silica substrate by selective reactive ion etching (rf power 150 W, DC bias 600 V, pressure 15 mtorr). CF4 with a flow rate of 45 sccm and O2 with a flow rate of 2.5 sccm were used to etch the fused silica and the resist, respectively. The ratio of the etch rates between fused silica and resist was 0.452. Height measurements of the finished phase plates are shown in Fig. 2.

 figure: Fig. 2

Fig. 2 (a): Photograph of one of the fabricated MDOEs. (b): Height measurement of its phase profile in the central region by white light interferometry. (c): Phase profile along the vertical line indicated in B.

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The helicity of the phase distribution of a plane wave passing the assembled vortex MDOE was measured with phase stepping interferometry [26]. The vortex MDOE was placed in the probe beam of a self assembled Mach-Zehnder interferometer operated with a frequency-doubled Nd:Yag laser at a wavelength of λ = 532 nm, as indicated in the upper row of Fig. 3.

 figure: Fig. 3

Fig. 3 Experimentally measured phase distribution of a plane wave (λ = 532 nm) after passing the vortex MDOE at different angles of mutual rotation θ. Upper row: Sketch of the interferometric setup, and 3 exemplary interferograms measured at different phase shifts α (indicated in the figure). Below: Phase behind the vortex plate, reconstructed from the corresponding interferograms, as a function of the rotation angle θ. The corresponding helicity of the combined vortex plate corresponds to the number of radially appearing 2π-phase jumps. Note that only positive helicities are shown, but the opposite sign can be obtained by reversing the mutual rotation direction of the two DOEs in the combined MDOE.

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Phase stepping was done with a mirror mounted on a piezo translation stage, and the corresponding interferograms were captured on a 14 bit digital sensor with a pixel size of 9×9 μm2 (pco.4000s). An example of 3 phase-stepped interferograms (with mutual phase offsets of 2π/3) is shown in the figure. From a set of 3 phase shifted interferograms the complex amplitude is computed and then numerically back-propagated by a distance of d (indicated in the Figure) into the plane of the vortex plate. The corresponding phase distributions directly behind the vortex plate are shown in Figure 3 for different angles of mutual rotation θ. Gray-scale values between black and white correspond to a phase ranging from 0 to 2π, respectively. The obtained phase patterns agree with those shown in Fig. 1. Particularly, the two sectors containing different helical phase profiles are reconstructed as expected. The observable deviations from the perfect ”star-like” spiral phase patterns correspond to an spherical phase curvature on the order of a quarter wavelength over the extension of the DOEs, which is probably due to the production process. For applications, as e.g. the generation of vortex beams, this effect is in most practical cases negligible.

Vortex MDOEs can be utilized to transform a Gaussian beam into a LG-like beam with adjustable helical charge , where the radius of the ring-shaped transverse intensity distribution is proportional to . We demonstrate the mode converting performance by imaging the Fourier transform of an extended laser beam at a wavelength λ = 532 nm passing the vortex MDOE. A digital sensor (mvBlueFOX-224G) with a pixel size of 4.4 μm was placed in the focal plane of the Fourier transforming lens of focal length f = 0.7 m. Figure 4 shows the normalized transverse intensity distributions of 4(a) the TEM00 mode for θ = 0° and 4(b)–4(g) LG-like beams for helicities ranging from = 1 to 10 corresponding to an angle θ ranging from 0° to 20°. By further increasing the angle θ, the influence of the complementary sector becomes notable, resulting in a gap in the ring-shaped intensity distribution and a non-zero intensity at the center. In Fig. 5(a) a quantitative comparison of the intensity distribution of a focused unmodified TEM00 beam (θ = 0°) and a focused Laguerre-Gaussian beam with = 1 (θ = 2°) is given. The peak intensity of the TEM00 beam is normalized to 1 and the doughnut beam intensity distribution (captured with the same exposure time) is proportionally scaled (see inset). The ratio of the two-dimensional integration over the LG-like beam and TEM00 beam intensity yields a 94% efficiency for the transformation of a TEM00 beam to an LG-like beam with = 1. For a perfect pair of Moiré spiral phase plates, an intensity distribution of 99.5% and 0.5% in the = 1 and = −179 LG-like beams would be expected, respectively, which corresponds to the relative areas of the two sectors within the combined DOE. The missing beam intensity in the experiment is probably due to tolerances in the DOE production and produces a marginal diffuse background by scattering. The green and blue curves show the corresponding cross-sections. From the zeros of the Airy disk (green curve) and the focal length of the Fourier transformation lens, the effective radius Σ of the aperture of the imaging system can be determined to be Σ = 3 mm.

 figure: Fig. 4

Fig. 4 Normalized intensity distribution of a TEM00 beam at λ = 532 nm, passing an MDOE spiral phase plate set at different angles of mutual rotation, focused on a digital sensor (mvBlueFOX-224G). (a) Unmodified TEM00 beam (θ = 0°). (b)–(g) TEM00 beam converted to LG-like beams of various helical charges .

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 figure: Fig. 5

Fig. 5 (a) Line profiles, as indicated in the insets, comparison of an unmodified TEM00 beam with θ = 0° (green curve) and a LG-like beam with helical charge = 1 (blue curve). The transformation efficiency is 94%. (b) Experimental results (blue circles) for the linear scaling of the radius of the ring-shaped intensity distribution with .

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The experimentally determined radii R of the ring-shaped intensity distributions for = 1 to = 10 are plotted in Fig. 5(b) (blue circles). Here we define the radius of a LG-like beam according to [1] as the distance from the center to the intensity maximum within the ring of light, which does, however, not correspond to the dark core width of the LG-like beam [2]. The linear scaling in can be approximated by [1]

R=bλfπΣ(1+0),
with b = 1.09 and 0 = 1.27 as free fit parameters (green line).

We also qualitatively investigated the dependence of the vortex beams on a mutual misalignment (decentering) of the two DOE plates. It turned out that small misalignments on the order of 10 μm still allow to produce almost undistorted vortex beams, if the relative rotation angle of the two plates is also adapted. Since the structure of the vortex plates is finer in their center, misalignments affect particularly the central area of the combined element. Therefore it turns out that higher vortex modes, which have a lower intensity in the central area, are less affected from a misalignment, than modes with lower helicities.

4. Conclusion

Using two DOEs with specially designed, conjugate phase profiles, an optical element is realized which produces vortex phase profiles with a helical index adjustable by a mutual rotation of the two plates. The experimental results agree with the theoretical expectations, and show a good quality of the obtained phase structures, which makes them practically employable in the typical field of application of vortex plates, as e.g. for the creation of donut beams with adjustable helical charge, or for Fourier filtering in microscopy. The vortex plates are not sensitive to polarization, yield a high conversion efficiency (typically more than 90%), and can potentially be used in high power applications.

Acknowledgments

This work was supported by the ERC Advanced Grant 247024 catchIT.

References and links

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

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

3. A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601 (2002). [CrossRef]  

4. 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 458185–8189 (1992). [CrossRef]   [PubMed]  

5. A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. 3161–204 (2011). [CrossRef]  

6. W. M. Lee, X.-C. Yuan, and W. C. Cheong, “Optical vortex beam shaping by use of highly efficient irregular spiral phase plates for optical micromanipulation,” Opt. Lett. 29, 1796–1798 (2004). [CrossRef]   [PubMed]  

7. S. W. Hell and J. Wichmann, “Breaking the diffraction resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy,” Opt. Lett. 19, 780–782 (1994). [CrossRef]   [PubMed]  

8. X. Hao, C. Kuang, T. Wang, and X. Liu, “Effects of polarization on the de-excitation dark focal spot in STED microscopy,” J. Opt. 12, 115707 (2010). [CrossRef]  

9. C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Upgrading a microscope with a spiral phase plate,” J. Microsc. 230, 134–142 (2008) [CrossRef]   [PubMed]  

10. A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Shadow effects in spiral phase contrast microscopy,” Phys. Rev. Lett. 94, 233902 (2005). [CrossRef]   [PubMed]  

11. G. Situ, G. Pedrini, and W. Osten, “Spiral phase filtering and orientation-selective edge detection/enhancement,” J. Opt. Soc. Am. A 261788–1797 (2009). [CrossRef]  

12. G. Foo, D. M. Palacios, and G. A. Swartzlander Jr., “Optical vortex coronagraph,” Opt. Lett. 30, 3308–3310 (2005). [CrossRef]  

13. D. Mawet, E. Serabyn, K. Liewer, R. Burruss, J. Hickey, and D. Shemo, “The vector vortex coronograph: Laboratory results and first light at Palomar observatory,” Astrophys. J. 709, 53 (2010). [CrossRef]  

14. R. Steiger, S. Bernet, and M. Ritsch-Marte, “Mapping of phase singularities with spiral phase contrast microscopy,” Opt. Exp. 21, 16282–16289 (2013). [CrossRef]  

15. M. A. Lauterbach, M. Guillon, A. Soltani, and V. Emiliani, “STED microscope with spiral phase contrast,” Sci. Rep. 3, 2050 (2013) [CrossRef]   [PubMed]  

16. H. Huang, G. Xie, Y. Yan, N. Ahmed, Y. Ren, Y. Yue, D. Rogawski, M. J. Willner, B. I. Erkmen, K. M. Birnbaum, S. J. Dolinar, M. P. J. Lavery, M. J. Padgett, M. Tur, and A. E. Willner, “100Tbit/s free-space data link enabled by three-dimensional multiplexing of orbital angular momentum, polarization, and wavelength,” Opt. Lett. 39, 197–200 (2014). [CrossRef]   [PubMed]  

17. M. Zürch, C. Kern, P. Hansinger, A. Dreischuh, and Ch. Spielmann, “Strong-field physics with singular light beams,” Nat. Phys. 8, 743–746 (2012). [CrossRef]  

18. Y. Shi, B. Shen, L. Zhang, X. Zhang, W. Wang, and Z. Xu, “Light fan driven by a relativistic laser pulse,” Phys. Rev. Lett. 112, 235001 (2014). [CrossRef]   [PubMed]  

19. C. Ament, L. Johnson, A. Schmitt-Sody, A. Lucero, T. Milster, and P. Polynkin, “Generation of multiterawatt vortex laser beams,” Appl. Opt. 53, 3355–3360 (2014). [CrossRef]   [PubMed]  

20. N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221–223 (1992). [CrossRef]   [PubMed]  

21. N. Zhang, X. C. Yuan, and R. E. Burge, “Extending the detection range of optical vortices by Dammann vortex gratings,” Opt. Lett. 35, 3495–3497 (2010). [CrossRef]   [PubMed]  

22. J. Yu, C. Zhou, W. Jia, A. Hu, W. Cao, J. Wu, and S. Wang, “Three-dimensional Dammann vortex array with tunable topological charge,” Appl. Opt. 51, 2485–2490 (2012). [CrossRef]   [PubMed]  

23. S. Bernet and M. Ritsch-Marte, “Adjustable refractive power from diffractive Moire elements,” Appl. Opt. 47, 3722–3730 (2008). [CrossRef]   [PubMed]  

24. S. Bernet, W. Harm, and M. Ritsch-Marte, “Demonstration of focus-tunable diffractive Moiré-lenses,” Opt. Exp. 21, 6955–6966 (2013). [CrossRef]  

25. J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71–78 (2004). [CrossRef]  

26. D. Malacara, Optical Shop Testing (Wiley, 2007). [CrossRef]  

References

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  1. J. E. Curtis and D. G. Grier, “Structure of optical vortices,” Phys. Rev. Lett. 90, 133901 (2003).
    [Crossref] [PubMed]
  2. Z. S. Sacks, D. Rozas, and G. A. Swartzlander, “Holographic formation of optical-vortex filaments,” JOSA B 15, 2226–2234 (1998).
    [Crossref]
  3. A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601 (2002).
    [Crossref]
  4. 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 458185–8189 (1992).
    [Crossref] [PubMed]
  5. A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. 3161–204 (2011).
    [Crossref]
  6. W. M. Lee, X.-C. Yuan, and W. C. Cheong, “Optical vortex beam shaping by use of highly efficient irregular spiral phase plates for optical micromanipulation,” Opt. Lett. 29, 1796–1798 (2004).
    [Crossref] [PubMed]
  7. S. W. Hell and J. Wichmann, “Breaking the diffraction resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy,” Opt. Lett. 19, 780–782 (1994).
    [Crossref] [PubMed]
  8. X. Hao, C. Kuang, T. Wang, and X. Liu, “Effects of polarization on the de-excitation dark focal spot in STED microscopy,” J. Opt. 12, 115707 (2010).
    [Crossref]
  9. C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Upgrading a microscope with a spiral phase plate,” J. Microsc. 230, 134–142 (2008)
    [Crossref] [PubMed]
  10. A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Shadow effects in spiral phase contrast microscopy,” Phys. Rev. Lett. 94, 233902 (2005).
    [Crossref] [PubMed]
  11. G. Situ, G. Pedrini, and W. Osten, “Spiral phase filtering and orientation-selective edge detection/enhancement,” J. Opt. Soc. Am. A 261788–1797 (2009).
    [Crossref]
  12. G. Foo, D. M. Palacios, and G. A. Swartzlander, “Optical vortex coronagraph,” Opt. Lett. 30, 3308–3310 (2005).
    [Crossref]
  13. D. Mawet, E. Serabyn, K. Liewer, R. Burruss, J. Hickey, and D. Shemo, “The vector vortex coronograph: Laboratory results and first light at Palomar observatory,” Astrophys. J. 709, 53 (2010).
    [Crossref]
  14. R. Steiger, S. Bernet, and M. Ritsch-Marte, “Mapping of phase singularities with spiral phase contrast microscopy,” Opt. Exp. 21, 16282–16289 (2013).
    [Crossref]
  15. M. A. Lauterbach, M. Guillon, A. Soltani, and V. Emiliani, “STED microscope with spiral phase contrast,” Sci. Rep. 3, 2050 (2013)
    [Crossref] [PubMed]
  16. H. Huang, G. Xie, Y. Yan, N. Ahmed, Y. Ren, Y. Yue, D. Rogawski, M. J. Willner, B. I. Erkmen, K. M. Birnbaum, S. J. Dolinar, M. P. J. Lavery, M. J. Padgett, M. Tur, and A. E. Willner, “100Tbit/s free-space data link enabled by three-dimensional multiplexing of orbital angular momentum, polarization, and wavelength,” Opt. Lett. 39, 197–200 (2014).
    [Crossref] [PubMed]
  17. M. Zürch, C. Kern, P. Hansinger, A. Dreischuh, and Ch. Spielmann, “Strong-field physics with singular light beams,” Nat. Phys. 8, 743–746 (2012).
    [Crossref]
  18. Y. Shi, B. Shen, L. Zhang, X. Zhang, W. Wang, and Z. Xu, “Light fan driven by a relativistic laser pulse,” Phys. Rev. Lett. 112, 235001 (2014).
    [Crossref] [PubMed]
  19. C. Ament, L. Johnson, A. Schmitt-Sody, A. Lucero, T. Milster, and P. Polynkin, “Generation of multiterawatt vortex laser beams,” Appl. Opt. 53, 3355–3360 (2014).
    [Crossref] [PubMed]
  20. N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221–223 (1992).
    [Crossref] [PubMed]
  21. N. Zhang, X. C. Yuan, and R. E. Burge, “Extending the detection range of optical vortices by Dammann vortex gratings,” Opt. Lett. 35, 3495–3497 (2010).
    [Crossref] [PubMed]
  22. J. Yu, C. Zhou, W. Jia, A. Hu, W. Cao, J. Wu, and S. Wang, “Three-dimensional Dammann vortex array with tunable topological charge,” Appl. Opt. 51, 2485–2490 (2012).
    [Crossref] [PubMed]
  23. S. Bernet and M. Ritsch-Marte, “Adjustable refractive power from diffractive Moire elements,” Appl. Opt. 47, 3722–3730 (2008).
    [Crossref] [PubMed]
  24. S. Bernet, W. Harm, and M. Ritsch-Marte, “Demonstration of focus-tunable diffractive Moiré-lenses,” Opt. Exp. 21, 6955–6966 (2013).
    [Crossref]
  25. J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71–78 (2004).
    [Crossref]
  26. D. Malacara, Optical Shop Testing (Wiley, 2007).
    [Crossref]

2014 (3)

2013 (3)

R. Steiger, S. Bernet, and M. Ritsch-Marte, “Mapping of phase singularities with spiral phase contrast microscopy,” Opt. Exp. 21, 16282–16289 (2013).
[Crossref]

M. A. Lauterbach, M. Guillon, A. Soltani, and V. Emiliani, “STED microscope with spiral phase contrast,” Sci. Rep. 3, 2050 (2013)
[Crossref] [PubMed]

S. Bernet, W. Harm, and M. Ritsch-Marte, “Demonstration of focus-tunable diffractive Moiré-lenses,” Opt. Exp. 21, 6955–6966 (2013).
[Crossref]

2012 (2)

J. Yu, C. Zhou, W. Jia, A. Hu, W. Cao, J. Wu, and S. Wang, “Three-dimensional Dammann vortex array with tunable topological charge,” Appl. Opt. 51, 2485–2490 (2012).
[Crossref] [PubMed]

M. Zürch, C. Kern, P. Hansinger, A. Dreischuh, and Ch. Spielmann, “Strong-field physics with singular light beams,” Nat. Phys. 8, 743–746 (2012).
[Crossref]

2011 (1)

2010 (3)

X. Hao, C. Kuang, T. Wang, and X. Liu, “Effects of polarization on the de-excitation dark focal spot in STED microscopy,” J. Opt. 12, 115707 (2010).
[Crossref]

D. Mawet, E. Serabyn, K. Liewer, R. Burruss, J. Hickey, and D. Shemo, “The vector vortex coronograph: Laboratory results and first light at Palomar observatory,” Astrophys. J. 709, 53 (2010).
[Crossref]

N. Zhang, X. C. Yuan, and R. E. Burge, “Extending the detection range of optical vortices by Dammann vortex gratings,” Opt. Lett. 35, 3495–3497 (2010).
[Crossref] [PubMed]

2009 (1)

2008 (2)

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Upgrading a microscope with a spiral phase plate,” J. Microsc. 230, 134–142 (2008)
[Crossref] [PubMed]

S. Bernet and M. Ritsch-Marte, “Adjustable refractive power from diffractive Moire elements,” Appl. Opt. 47, 3722–3730 (2008).
[Crossref] [PubMed]

2005 (2)

A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Shadow effects in spiral phase contrast microscopy,” Phys. Rev. Lett. 94, 233902 (2005).
[Crossref] [PubMed]

G. Foo, D. M. Palacios, and G. A. Swartzlander, “Optical vortex coronagraph,” Opt. Lett. 30, 3308–3310 (2005).
[Crossref]

2004 (2)

2003 (1)

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

2002 (1)

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601 (2002).
[Crossref]

1998 (1)

Z. S. Sacks, D. Rozas, and G. A. Swartzlander, “Holographic formation of optical-vortex filaments,” JOSA B 15, 2226–2234 (1998).
[Crossref]

1994 (1)

1992 (2)

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 458185–8189 (1992).
[Crossref] [PubMed]

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221–223 (1992).
[Crossref] [PubMed]

Ahmed, N.

Allen, L.

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601 (2002).
[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 458185–8189 (1992).
[Crossref] [PubMed]

Ament, C.

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 458185–8189 (1992).
[Crossref] [PubMed]

Bernet, S.

R. Steiger, S. Bernet, and M. Ritsch-Marte, “Mapping of phase singularities with spiral phase contrast microscopy,” Opt. Exp. 21, 16282–16289 (2013).
[Crossref]

S. Bernet, W. Harm, and M. Ritsch-Marte, “Demonstration of focus-tunable diffractive Moiré-lenses,” Opt. Exp. 21, 6955–6966 (2013).
[Crossref]

S. Bernet and M. Ritsch-Marte, “Adjustable refractive power from diffractive Moire elements,” Appl. Opt. 47, 3722–3730 (2008).
[Crossref] [PubMed]

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Upgrading a microscope with a spiral phase plate,” J. Microsc. 230, 134–142 (2008)
[Crossref] [PubMed]

A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Shadow effects in spiral phase contrast microscopy,” Phys. Rev. Lett. 94, 233902 (2005).
[Crossref] [PubMed]

Birnbaum, K. M.

Burge, R. E.

Burruss, R.

D. Mawet, E. Serabyn, K. Liewer, R. Burruss, J. Hickey, and D. Shemo, “The vector vortex coronograph: Laboratory results and first light at Palomar observatory,” Astrophys. J. 709, 53 (2010).
[Crossref]

Cao, W.

Cheong, W. C.

Curtis, J. E.

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

Dolinar, S. J.

Dreischuh, A.

M. Zürch, C. Kern, P. Hansinger, A. Dreischuh, and Ch. Spielmann, “Strong-field physics with singular light beams,” Nat. Phys. 8, 743–746 (2012).
[Crossref]

Emiliani, V.

M. A. Lauterbach, M. Guillon, A. Soltani, and V. Emiliani, “STED microscope with spiral phase contrast,” Sci. Rep. 3, 2050 (2013)
[Crossref] [PubMed]

Erkmen, B. I.

Foo, G.

Fürhapter, S.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Upgrading a microscope with a spiral phase plate,” J. Microsc. 230, 134–142 (2008)
[Crossref] [PubMed]

A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Shadow effects in spiral phase contrast microscopy,” Phys. Rev. Lett. 94, 233902 (2005).
[Crossref] [PubMed]

Grier, D. G.

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

Guillon, M.

M. A. Lauterbach, M. Guillon, A. Soltani, and V. Emiliani, “STED microscope with spiral phase contrast,” Sci. Rep. 3, 2050 (2013)
[Crossref] [PubMed]

Hansinger, P.

M. Zürch, C. Kern, P. Hansinger, A. Dreischuh, and Ch. Spielmann, “Strong-field physics with singular light beams,” Nat. Phys. 8, 743–746 (2012).
[Crossref]

Hao, X.

X. Hao, C. Kuang, T. Wang, and X. Liu, “Effects of polarization on the de-excitation dark focal spot in STED microscopy,” J. Opt. 12, 115707 (2010).
[Crossref]

Harm, W.

S. Bernet, W. Harm, and M. Ritsch-Marte, “Demonstration of focus-tunable diffractive Moiré-lenses,” Opt. Exp. 21, 6955–6966 (2013).
[Crossref]

Heckenberg, N. R.

Hell, S. W.

Hickey, J.

D. Mawet, E. Serabyn, K. Liewer, R. Burruss, J. Hickey, and D. Shemo, “The vector vortex coronograph: Laboratory results and first light at Palomar observatory,” Astrophys. J. 709, 53 (2010).
[Crossref]

Hu, A.

Huang, H.

Jesacher, A.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Upgrading a microscope with a spiral phase plate,” J. Microsc. 230, 134–142 (2008)
[Crossref] [PubMed]

A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Shadow effects in spiral phase contrast microscopy,” Phys. Rev. Lett. 94, 233902 (2005).
[Crossref] [PubMed]

Jia, W.

Johnson, L.

Kern, C.

M. Zürch, C. Kern, P. Hansinger, A. Dreischuh, and Ch. Spielmann, “Strong-field physics with singular light beams,” Nat. Phys. 8, 743–746 (2012).
[Crossref]

Kuang, C.

X. Hao, C. Kuang, T. Wang, and X. Liu, “Effects of polarization on the de-excitation dark focal spot in STED microscopy,” J. Opt. 12, 115707 (2010).
[Crossref]

Lauterbach, M. A.

M. A. Lauterbach, M. Guillon, A. Soltani, and V. Emiliani, “STED microscope with spiral phase contrast,” Sci. Rep. 3, 2050 (2013)
[Crossref] [PubMed]

Lavery, M. P. J.

Leach, J.

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71–78 (2004).
[Crossref]

Lee, W. M.

Liewer, K.

D. Mawet, E. Serabyn, K. Liewer, R. Burruss, J. Hickey, and D. Shemo, “The vector vortex coronograph: Laboratory results and first light at Palomar observatory,” Astrophys. J. 709, 53 (2010).
[Crossref]

Liu, X.

X. Hao, C. Kuang, T. Wang, and X. Liu, “Effects of polarization on the de-excitation dark focal spot in STED microscopy,” J. Opt. 12, 115707 (2010).
[Crossref]

Lucero, A.

MacVicar, I.

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601 (2002).
[Crossref]

Malacara, D.

D. Malacara, Optical Shop Testing (Wiley, 2007).
[Crossref]

Maurer, C.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Upgrading a microscope with a spiral phase plate,” J. Microsc. 230, 134–142 (2008)
[Crossref] [PubMed]

Mawet, D.

D. Mawet, E. Serabyn, K. Liewer, R. Burruss, J. Hickey, and D. Shemo, “The vector vortex coronograph: Laboratory results and first light at Palomar observatory,” Astrophys. J. 709, 53 (2010).
[Crossref]

McDuff, R.

Milster, T.

O’Neil, A. T.

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601 (2002).
[Crossref]

Osten, W.

Padgett, M. J.

Palacios, D. M.

Pedrini, G.

Polynkin, P.

Ren, Y.

Ritsch-Marte, M.

R. Steiger, S. Bernet, and M. Ritsch-Marte, “Mapping of phase singularities with spiral phase contrast microscopy,” Opt. Exp. 21, 16282–16289 (2013).
[Crossref]

S. Bernet, W. Harm, and M. Ritsch-Marte, “Demonstration of focus-tunable diffractive Moiré-lenses,” Opt. Exp. 21, 6955–6966 (2013).
[Crossref]

S. Bernet and M. Ritsch-Marte, “Adjustable refractive power from diffractive Moire elements,” Appl. Opt. 47, 3722–3730 (2008).
[Crossref] [PubMed]

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Upgrading a microscope with a spiral phase plate,” J. Microsc. 230, 134–142 (2008)
[Crossref] [PubMed]

A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Shadow effects in spiral phase contrast microscopy,” Phys. Rev. Lett. 94, 233902 (2005).
[Crossref] [PubMed]

Rogawski, D.

Rozas, D.

Z. S. Sacks, D. Rozas, and G. A. Swartzlander, “Holographic formation of optical-vortex filaments,” JOSA B 15, 2226–2234 (1998).
[Crossref]

Sacks, Z. S.

Z. S. Sacks, D. Rozas, and G. A. Swartzlander, “Holographic formation of optical-vortex filaments,” JOSA B 15, 2226–2234 (1998).
[Crossref]

Schmitt-Sody, A.

Serabyn, E.

D. Mawet, E. Serabyn, K. Liewer, R. Burruss, J. Hickey, and D. Shemo, “The vector vortex coronograph: Laboratory results and first light at Palomar observatory,” Astrophys. J. 709, 53 (2010).
[Crossref]

Shemo, D.

D. Mawet, E. Serabyn, K. Liewer, R. Burruss, J. Hickey, and D. Shemo, “The vector vortex coronograph: Laboratory results and first light at Palomar observatory,” Astrophys. J. 709, 53 (2010).
[Crossref]

Shen, B.

Y. Shi, B. Shen, L. Zhang, X. Zhang, W. Wang, and Z. Xu, “Light fan driven by a relativistic laser pulse,” Phys. Rev. Lett. 112, 235001 (2014).
[Crossref] [PubMed]

Shi, Y.

Y. Shi, B. Shen, L. Zhang, X. Zhang, W. Wang, and Z. Xu, “Light fan driven by a relativistic laser pulse,” Phys. Rev. Lett. 112, 235001 (2014).
[Crossref] [PubMed]

Situ, G.

Smith, C. P.

Soltani, A.

M. A. Lauterbach, M. Guillon, A. Soltani, and V. Emiliani, “STED microscope with spiral phase contrast,” Sci. Rep. 3, 2050 (2013)
[Crossref] [PubMed]

Spielmann, Ch.

M. Zürch, C. Kern, P. Hansinger, A. Dreischuh, and Ch. Spielmann, “Strong-field physics with singular light beams,” Nat. Phys. 8, 743–746 (2012).
[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 458185–8189 (1992).
[Crossref] [PubMed]

Steiger, R.

R. Steiger, S. Bernet, and M. Ritsch-Marte, “Mapping of phase singularities with spiral phase contrast microscopy,” Opt. Exp. 21, 16282–16289 (2013).
[Crossref]

Swartzlander, G. A.

G. Foo, D. M. Palacios, and G. A. Swartzlander, “Optical vortex coronagraph,” Opt. Lett. 30, 3308–3310 (2005).
[Crossref]

Z. S. Sacks, D. Rozas, and G. A. Swartzlander, “Holographic formation of optical-vortex filaments,” JOSA B 15, 2226–2234 (1998).
[Crossref]

Tur, M.

Wang, S.

Wang, T.

X. Hao, C. Kuang, T. Wang, and X. Liu, “Effects of polarization on the de-excitation dark focal spot in STED microscopy,” J. Opt. 12, 115707 (2010).
[Crossref]

Wang, W.

Y. Shi, B. Shen, L. Zhang, X. Zhang, W. Wang, and Z. Xu, “Light fan driven by a relativistic laser pulse,” Phys. Rev. Lett. 112, 235001 (2014).
[Crossref] [PubMed]

White, A. G.

Wichmann, J.

Willner, A. E.

Willner, M. J.

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 458185–8189 (1992).
[Crossref] [PubMed]

Wu, J.

Xie, G.

Xu, Z.

Y. Shi, B. Shen, L. Zhang, X. Zhang, W. Wang, and Z. Xu, “Light fan driven by a relativistic laser pulse,” Phys. Rev. Lett. 112, 235001 (2014).
[Crossref] [PubMed]

Yan, Y.

Yao, A. M.

Yao, E.

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71–78 (2004).
[Crossref]

Yu, J.

Yuan, X. C.

Yuan, X.-C.

Yue, Y.

Zhang, L.

Y. Shi, B. Shen, L. Zhang, X. Zhang, W. Wang, and Z. Xu, “Light fan driven by a relativistic laser pulse,” Phys. Rev. Lett. 112, 235001 (2014).
[Crossref] [PubMed]

Zhang, N.

Zhang, X.

Y. Shi, B. Shen, L. Zhang, X. Zhang, W. Wang, and Z. Xu, “Light fan driven by a relativistic laser pulse,” Phys. Rev. Lett. 112, 235001 (2014).
[Crossref] [PubMed]

Zhou, C.

Zürch, M.

M. Zürch, C. Kern, P. Hansinger, A. Dreischuh, and Ch. Spielmann, “Strong-field physics with singular light beams,” Nat. Phys. 8, 743–746 (2012).
[Crossref]

Adv. Opt. Photon. (1)

Appl. Opt. (3)

Astrophys. J. (1)

D. Mawet, E. Serabyn, K. Liewer, R. Burruss, J. Hickey, and D. Shemo, “The vector vortex coronograph: Laboratory results and first light at Palomar observatory,” Astrophys. J. 709, 53 (2010).
[Crossref]

J. Microsc. (1)

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Upgrading a microscope with a spiral phase plate,” J. Microsc. 230, 134–142 (2008)
[Crossref] [PubMed]

J. Opt. (1)

X. Hao, C. Kuang, T. Wang, and X. Liu, “Effects of polarization on the de-excitation dark focal spot in STED microscopy,” J. Opt. 12, 115707 (2010).
[Crossref]

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

JOSA B (1)

Z. S. Sacks, D. Rozas, and G. A. Swartzlander, “Holographic formation of optical-vortex filaments,” JOSA B 15, 2226–2234 (1998).
[Crossref]

Nat. Phys. (1)

M. Zürch, C. Kern, P. Hansinger, A. Dreischuh, and Ch. Spielmann, “Strong-field physics with singular light beams,” Nat. Phys. 8, 743–746 (2012).
[Crossref]

New J. Phys. (1)

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71–78 (2004).
[Crossref]

Opt. Exp. (2)

S. Bernet, W. Harm, and M. Ritsch-Marte, “Demonstration of focus-tunable diffractive Moiré-lenses,” Opt. Exp. 21, 6955–6966 (2013).
[Crossref]

R. Steiger, S. Bernet, and M. Ritsch-Marte, “Mapping of phase singularities with spiral phase contrast microscopy,” Opt. Exp. 21, 16282–16289 (2013).
[Crossref]

Opt. Lett. (6)

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 458185–8189 (1992).
[Crossref] [PubMed]

Phys. Rev. Lett. (4)

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

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601 (2002).
[Crossref]

A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Shadow effects in spiral phase contrast microscopy,” Phys. Rev. Lett. 94, 233902 (2005).
[Crossref] [PubMed]

Y. Shi, B. Shen, L. Zhang, X. Zhang, W. Wang, and Z. Xu, “Light fan driven by a relativistic laser pulse,” Phys. Rev. Lett. 112, 235001 (2014).
[Crossref] [PubMed]

Sci. Rep. (1)

M. A. Lauterbach, M. Guillon, A. Soltani, and V. Emiliani, “STED microscope with spiral phase contrast,” Sci. Rep. 3, 2050 (2013)
[Crossref] [PubMed]

Other (1)

D. Malacara, Optical Shop Testing (Wiley, 2007).
[Crossref]

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Figures (5)

Fig. 1
Fig. 1 Phase functions for a design parameter a = 90 of (a) Φ1(ϕ), (b) Φ2(ϕ) and (c) of the compound unit for an exemplary angle of mutual rotation ϕ = 8° resulting in a helicity of = 4. The grey levels in the figure correspond to phase shifts in an interval between −π and π.
Fig. 2
Fig. 2 (a): Photograph of one of the fabricated MDOEs. (b): Height measurement of its phase profile in the central region by white light interferometry. (c): Phase profile along the vertical line indicated in B.
Fig. 3
Fig. 3 Experimentally measured phase distribution of a plane wave (λ = 532 nm) after passing the vortex MDOE at different angles of mutual rotation θ. Upper row: Sketch of the interferometric setup, and 3 exemplary interferograms measured at different phase shifts α (indicated in the figure). Below: Phase behind the vortex plate, reconstructed from the corresponding interferograms, as a function of the rotation angle θ. The corresponding helicity of the combined vortex plate corresponds to the number of radially appearing 2π-phase jumps. Note that only positive helicities are shown, but the opposite sign can be obtained by reversing the mutual rotation direction of the two DOEs in the combined MDOE.
Fig. 4
Fig. 4 Normalized intensity distribution of a TEM00 beam at λ = 532 nm, passing an MDOE spiral phase plate set at different angles of mutual rotation, focused on a digital sensor (mvBlueFOX-224G). (a) Unmodified TEM00 beam (θ = 0°). (b)–(g) TEM00 beam converted to LG-like beams of various helical charges .
Fig. 5
Fig. 5 (a) Line profiles, as indicated in the insets, comparison of an unmodified TEM00 beam with θ = 0° (green curve) and a LG-like beam with helical charge = 1 (blue curve). The transformation efficiency is 94%. (b) Experimental results (blue circles) for the linear scaling of the radius of the ring-shaped intensity distribution with .

Equations (7)

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

Φ 1 ( ϕ ) = + a ϕ 2 / 2 π Φ 2 ( ϕ ) = a ϕ 2 / 2 π ,
Φ 2 ( ϕ ) = { a ( ϕ θ ) 2 / 2 π for θ ϕ < 2 π a ( ϕ θ + 2 π ) 2 / 2 π for 0 ϕ < θ .
T joint ( ϕ ) = { exp { i [ 2 θ ϕ θ 2 ] a / 2 π } for θ ϕ < 2 π exp { i [ ( 2 θ 4 π ) ϕ θ 2 4 π 2 + 4 π θ ] a / 2 π } for 0 ϕ < θ .
= { a π θ for θ ϕ < 2 π a π θ 2 a for 0 ϕ < θ ,
Φ 1 ( ϕ ) = mod 2 π { a ϕ 2 / 2 π } Φ 2 ( ϕ ) = mod 2 π { a ϕ 2 / 2 π } ,
a < R min π 2 p ,
R = b λ f π Σ ( 1 + 0 ) ,

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