Abstract

We investigate the transfer of orbital angular momentum among multiple beams involved in a coherent Raman interaction. We use a liquid crystal light modulator to shape pump and Stokes beams into optical vortices with various integer values of topological charge, and cross them in a Raman-active crystal to produce multiple Stokes and anti-Stokes sidebands. We measure the resultant vortex charges using a tilted-lens technique. We verify that in every case the generated beams’ topological charges obey a simple relationship, resulting from angular momentum conservation for created and annihilated photons, or equivalently, from phase-matching considerations for multiple interacting beams.

© 2015 Optical Society of America

1. Introduction and background

An optical vortex is a simple yet intriguing object which finds its use in a multitude of areas of research and technology [1]. An optical vortex beam exhibits a characteristic donut-shaped transverse profile (a ring of light) with a spiral wavefront. The zero-intensity center axis of such a beam is a basic light-wave phase singularity [2]. An integer number of 2π phase accumulation around one turn of the wavefront spiral corresponds to integer topological charge (TC), or, in the ideal case, an integer amount of orbital angular momentum (OAM) that the vortex carries [3]. As He et al. showed in [4], a focused optical vortex can impart its orbital angular momentum onto a trapped microparticle and make it spin in a direction determined by the helicity of the beam. Of particular pertinence to our current work is a study of optical vortices interacting nonlinearly in atomic vapors [5], where TC transfer allows identification of nonlinear pathways. A variety of vortex applications, however, simply utilize its stable and reproducible donut shape, with a perfectly dark center, resulting from destructive interference at the point of phase anomaly. In optical tweezers, vortex beams are used for studying proteins as well as for micro- (and nano-) manipulation of absorbing or scattering particles such as biological cells [6–8]. In stimulated-emission-depletion (STED) microscopy, an optical vortex serves as a perfect depletion beam [9]. In astronomy, a vortex coronagraph allows for the detection of faint extrasolar planets near their very bright host stars [10, 11]. Other interesting applications of optical vortices include quantum [12] and classical [13] communication systems based on information encryption via OAM states of photons.

Our work focuses on the interaction of optical vortices, and femtosecond optical vortices in particular [14], with nonlinear Raman-active crystals [15]. Coherent multi-sideband Raman generation offers opportunities for the production of ultrashort (sub-femtosecond) optical pulses of adjustable shapes, and for non-sinusoidal field synthesis [16–18]. The possibility of adjusting the transverse beam profile and producing coherent Raman sidebands of various vortex shapes adds another dimension to light-field shaping. The goal of our present work is to explore the process of transferring topological charge (otherwise known as the TC algebra) from input femtosecond beams into Raman-generated vortex sidebands. Topological charge is related to the OAM, but is generally not the same [19, 20]. However, the equation that governs TC transfer may still be derived from either orbital angular momentum conservation for created and annihilated photons or, equivalently, from considerations of phase-matching between the applied and generated beams. OAM, as a rule, is only conserved in cylindrically symmetric systems (i.e. in collinear setups), but is approximately conserved at sufficiently small angles (on the order of 10°, which is significantly larger than the value used in our experiment), as [12, 21–23] have shown in spontaneous parametric down conversion, second harmonic generation, and four wave mixing.

Returning to our derivation of the TC algebra, we note that one photon is added to the Stokes pulse and two photons are removed (annihilated) from the pump pulse to make a photon of the first anti-Stokes (AS) sideband, so that the resulting OAM conservation equation becomes lAS1 = 2lpls. Hence, by applying the same logic to n sidebands, we may derive a simple equation that clearly predicts the TC of each sideband:

ln=(n+1)lpnls
where n is the order of the sideband (i.e. n = 1, 2,... correspond to AS orders, and n = −1, −2,... correspond to Stokes orders), ln is the TC of a sideband of order n, lp is the TC of the pump, and ls = l − 1 is the TC of Stokes [24]. In the language of phasematching, this conservation of orbital angular momentum comes from a phase relationship among the applied and generated beams: ϕn = (n + 1)ϕps (where ϕn is the relative phase of the nth field, in the transverse plane) [24]. This is analogous to how photon momentum conservation is equivalent to wavevector (k-vector) matching.

In this work we verify the TC algebra by using a computer-controlled spatial light modulator (SLM) to shape one or both beams incident on the Raman-active crystal (similar SLM functionality has been demonstrated in [25, 26]). Prior work on TC algebra in the context of coherent Raman interactions includes J. Strohaber et al. [24] utilizing two identical chirped femtosecond laser pulses with lp = ls = ±1 and lp = −ls = ±1. Further, our group has previously [27] studied the generation of multi-color optical vortices in a PbWO4 crystal using two-color, Fourier-transform limited, femtosecond input pulses (with a vortex shape applied by spiral phase plate). We then checked the TC algebra with the TC of pump and Stokes equal, respectively, to lp = 1 and ls = 0. Here, we extend upon our previous work by observing Raman vortices which were generated from pump and Stokes pulses with arbitrary lp and ls (up to ±3). Our results provide a further test of the TC algebra.

2. Vortex production and measurement

There are several methods to convert ordinary Gaussian beams into vortex beams. This can be done with spiral phase plates [24,27], computer-generated holograms (or gratings with defects) [28], or SLMs [29–32]. In our present work, we use an SLM-based method. Our particular SLM modulates the phase of incoming light as it reflects off of a mirror covered by a programmable liquid crystal layer. We program this liquid crystal layer using computer-generated phase masks, an example of which is shown in Fig. 1. These phase masks are bitmaps where the pixel value varies based on basic equations for the phase that should be added to a Gaussian beam in order to produce a vortex beam. For example, for a first order vortex beam generated from a Gaussian beam, ϕ=arctan(yx) (for polar angle of the phase map θ = 0 to π2), where ϕ is the added phase, x is one spatial direction in the cross-section of the beam, and y is the perpendicular spatial direction (again, in the cross-section of the beam). An analogous relationship can be defined for a 3rd order vortex, as shown by Fig. 1(a).

 

Fig. 1 Production and measurement of optical vortices. (a) Computer-generated phase mask applied to the SLM. (b) Donut-shaped beam profile (taken approximately 76 cm after the SLM) resulting from Gaussian beam reflection off of the SLM. (c) Schematic of an astigmatic focusing method for TC measurement. (d) Resultant intensity distribution in the focal plane. Data in (a), (b), and (d) are shown for a vortex beam with TC = 3, hence we see 4 distinguishable spots in part (d).

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Several methods have been proposed for measuring the topological charge of optical vortices. These include using the diffraction pattern after propagating an optical vortex through an annular aperture [33] or by using a cylindrical lens [34]. One popular method involves building an interferometer and studying the interference pattern produced between a Gaussian beam and a vortex beam [35]. Our group has employed this method in the past, generating two sets of coherent Raman sidebands by two sets of input beams, one set of vortices and another set of reference Gaussian sidebands [24, 27], and then interfering them. Another, simpler method was proposed by P. Vaity et al. [36]. The essence of this method is to use a tilted lens to determine the topological charge of an optical vortex by taking advantage of astigmatic focusing produced by this lens, as demonstrated schematically in Fig. 1(c) and (d). The main advantages of this method are its simplicity and clarity; in addition, because there is no need for a reference beam, we have more power available to generate higher order optical vortices. This method was successfully used by Buono and collaborators [37] for measuring the transfer of TC in second harmonic generation. Previously, we have employed this method and showed that it performs equally well to the standard interferometric method (albeit without giving detailed phase information) in determining the topological charge of each sideband [27].

3. Setup and methods

Our experimental setup is shown in Fig. 2. As is described in [38], we used a Ti:Sapphire regenerative amplifier (Coherent, Legend) to produce infrared (λ = 802 nm) 35 fs pulses with a 1 kHz repetition rate and 1W average power. We then chirped these pulses by changing the grating distance within the compressor unit of the amplifier, producing pulses of around 200 fs. We retrieved the precise chirp using a second harmonic (SH) frequency-resolved optical gating (FROG) setup, with software provided by R. Trebino [39]. We determined the sign of the chirp by performing a SH-FROG before and after riding a slab of glass to the beampath of our chirped pulse.

 

Fig. 2 (a) Our experimental setup. Dashed lines correspond to Stokes beam, while solid lines correspond to pump. The blue lines correspond to the one-beam modulation case and the red lines correspond to the two-beam modulation case. The angle of the SLM is greatly exaggerated. Typical sidebands produced from this arrangement are also shown. The inset schematically depicts our two chirped pulses and the delay between them. (b) Computer generated phase masks (left), optical vortices obtained with these phase masks just before the focusing lens, approximately 76 cm after the SLM (middle), and vortices focused with a tilted lens (right).

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The beam was then split into pump and Stokes beams by a non-polarizing beamsplitter. We adjusted the power in each beam with separate neutral density filters and observed the behavior of the generated coherent Raman sidebands. By tuning the power we were able to reduce nonlinear parasitic effects and thus optimize the quality of the sidebands. Finally, after either one or both beams were spatially modulated, they are focused (by separate 50 cm lenses) and recombined at a small angle (3.16 degrees, to satisfy phase matching conditions) in a lead tungstate (PbWO4) crystal placed 2.5 cm before the focus, resulting in a beam diameter of about 1 mm. This distance ensures that the intensity was low enough that parasitic effects (such as self-phase modulation) did not dominate over the Raman generation.

The relative pulse delay, phase matching, and chirp chosen excited the PbWO4 Raman mode of 325 cm−1 [38], as we confirmed with a spectrometer. As is shown in the inset of Fig. 2, for positive chirp (which is what we used for all results given here), the leading pulse acted as pump and the delayed pulse acted as Stokes. In this configuration, the Raman mode is driven by the instantaneous frequency difference Δω = bt, where t is the relative delay of the two pulses (in femtoseconds) and b is their chirp rate (fs − cm−1)[38]. A standard set of sidebands thus generated is shown in Fig. 2(a). We label the sidebands as anti-Stokes One (AS1), anti-Stokes Two (AS2), and so on.

A spatial light modulator (Hamamatsu x10468 − 02) was used for all spatial shaping of the beam. We optimized the phase masks (i.e. by adding a constant phase or multiplying by a constant) to produce beams which appear darkest in the center in the far field, as measured by a beam profiler (Spiricon SP620U). We confirmed independently, by using an interferometric setup, that these beams correspond to vortex beams that exhibit the proper behavior when interfered with Gaussian beams. These phase masks and examples of the beams thus produced are shown in Fig. 2. We performed two sets of experiments; for the first, we only spatially shaped one beam (so that the topological charge did not equal 0) and left the other as a Gaussian. Therefore, only one beam was input to the SLM and we were able to use the full size of the beam. For the second setup, we reduced the size of both beams by 25% with a telescope placed before the beamsplitter, so that the diffraction that results from the clipping of one beam on the SLM was avoided. The telescope was built with thin negative miniscus and plano-convex lenses to produce minimal spherical aberrations.

4. Results

As mentioned above, after generating a set of Raman vortex sidebands, the TC for each sideband was checked by focusing with a lens tilted by 6 degrees [36] and counting the number of spots in the focal plane. The number of observed spots is equal to ln + 1. The spots form a tilted row and the slope of this row corresponds to the sign of TC, such that positive slope corresponds to positive TC and vice versa. As shown by Figs. 35, our results confirm that higher order optical vortices (i.e. vortices generated by lp and ls ≠ 1 or 0) follow the algebra given by Eq. (1).

 

Fig. 3 TC measurement of Raman sidebands using a tilted lens. For each block: columns 1 and 3 are the sidebands before the lens, Columns 2 and 4 are the sidebands after the lens. From top to bottom – AS1, AS2, AS3. Results are summarized in Table 1. (a) Left (right) two columns: sidebands generated with lp = 0 and ls = −1 (ls = 1). (b) Left (right) two columns: sidebands generated with lp = 0 and ls = −2 (ls = 2).

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Fig. 4 TC measurement of Raman sidebands using a tilted lens. Columns 1 and 3 are the sidebands before the lens, Columns 2 and 4 are the sidebands after the lens. Left (right) two columns: sidebands generated with lp = 0 and ls = −3 (ls = 3). From top to bottom – AS1 (TC=±3; 4 spots), AS2 (TC=±6; 6 spots).

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Fig. 5 TC measurement of Raman sidebands using a tilted lens. For each block: digital phase maps for generating pump and Stokes beams (left), AS1 generated when these phase maps are applied (middle), AS1 focused with tilted lens (right).

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4.1. Results for one modulated beam

This experiment was performed with six sets of TC values. We set lp equal to zero for the whole experiment and ls = ±1; ±2; ±3. We present our calculations for the topological charge of each set in Table 1.

Tables Icon

Table 1. Predicted, and measured, TC for (from top to bottom) ls = +1(−1), ls = +2(−2), and ls = +3(−3). In all cases, lp = 0.

As depicted in Fig. 3, the theory matches our experimental results.

We were able to generate up to 7 high quality vortices with ls = ±1, without any background nonlinear processes and noise, and 15 vortices (up to blue wavelengths) at the maximum intensity below the burning point of the crystal. However, the intensity of the vortices generated with ls = ±2 and ±3 is lower than for ls = ±1. Accordingly, only 6 AS sidebands were generated for ls = ±2 and 5 AS sidebands for ls = ±3. We hypothesize that this is due to the increase in the bright area of each vortex, as was shown in [40, 41] and as seen in Fig. 2. Therefore, as the vortex order increases, the peak intensity (at a fixed point) becomes smaller, impacting the Raman generation negatively. We can calculate the bright area Al of a Laguerre-Gauss beam of order l generated from a Gaussian beam of radius w0, using a formula from [40]: Al=πw02(l+1.3)2l2e1.4/l, so the intensity of an l = 2 beam is 75% of that for an l = 1 beam, and the intensity of an l = 3 beam is 83% of that for an l = 2 beam. However, it is important to note that the question of conversion efficiency is quite complicated, and a reduced efficiency can be due to a wide variety of other factors, such as spatial overlap and the increased divergence of higher order beams. Reduction in peak intensity is only the most obvious and most dramatic cause.

Finally, we checked the TC of the first three AS sidebands for ls = ±1, ±2 and the first two AS sidebands for ls = ±3, as is shown in Figs. 3 and 4. Our measurements were limited by the resolution of the intensity distribution at the focus for each vortex; despite our use of loosely focusing lenses, we were unable to resolve the 10 spots theoretically predicted for AS3 of ls = ±3.

4.2. Results for two modulated beams

In this experiment, we tested 4 different combinations of TC of input beams but kept the difference |lpls| equal to 3. To confirm that the topological charge transfer took place, even for such a relatively large value of |lpls|, we measured the resultant TC of AS1. From Eq. (1), we predict the results shown in Table 2.

Tables Icon

Table 2. Predicted, and measured, TC for 4 different cases of mixed lp and ls.

As depicted in Fig. 5, Eq. (1) correctly predicts the results in all 4 cases. The sidebands generated in this experiment are worse in quality (have diffraction fringes and are not complete donut shapes) and efficiency (they are barely seen by eye) than sidebands generated with one modulated beam. Our experience has shown that small changes (such as any coma introduced by lenses) from ideal spatial alignment can add a significant amount of distortion to Raman-generated beams. This issue is especially relevant for Raman-generated vortices, as any distortions in the beam profile reflect distortions in the carried OAM.

5. Summary

We have produced multi-color optical vortices in Raman sideband generation with two femtosecond linearly chirped pulses, verified the TC algebra, and used a single SLM to shape two beams.

When finalizing our work, we became aware of [42]. However, we obtained our results independently from Strohaber et al. and were unaware of their work prior to this publication.

Acknowledgments

A. Zhdanova and M. Shutova contributed equally to this paper. This work is supported by the National Science Foundation (grant No. PHY-1307153) and the Welch Foundation (grant No. A1547). We thank Peter Zhokhov and Anton Shutov for valuable help and providing experimental equipment. M. S. is supported by the Herman F. Heep and Minnie Belle Heep Texas A&M University Endowed Fund held/administered by the Texas A&M Foundation. A. Z. gratefully acknowledges her Diversity Fellowship from Texas A&M University.

References and links

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2. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A. 336, 165–190 (1974). [CrossRef]  

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5. A. M. Akulshin, R. J. Mclean, E. E. Mikhailov, and I. Novikova, “Distinguishing nonlinear processes in atomic media via orbital angular momentum transfer,” Opt. Lett. 40, 1109–1112 (2015). [CrossRef]   [PubMed]  

6. F. M. Fazal and S. M. Block, “Optical tweezers study life under tension,” Nature Phot. 5, 318–321 (2011). [CrossRef]  

7. M. J. Padgett, “Light in a twist: optical angular momentum,” Proc. SPIE 8637, 863702 (2013). [CrossRef]  

8. D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003). [CrossRef]   [PubMed]  

9. T. A. Klar, E. Engel, and S. W. Hell, “Breaking Abbe’s diffraction resolution limit in fluorescence microscopy with stimulated emission depletion beams of various shapes,” Phys. Rev. E 64, 066613 (2001). [CrossRef]  

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

11. G. A. Swartzlander, “Peering into darkness with a vortex spatial filter,” Opt. Lett. 26, 497–499 (2001). [CrossRef]  

12. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001). [CrossRef]   [PubMed]  

13. G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Freespace information transfer using light beams carrying orbital angular momentum,” Opt. Express 12, 5448–5456 (2004). [CrossRef]   [PubMed]  

16. I. Mariyenko, J. Strohaber, and C. Uiterwaal, “Creation of optical vortices in femtosecond pulses,” Opt. Express 13, 7599–7608 (2005). [CrossRef]   [PubMed]  

17. F. Lenzini, S. Residori, F. Arecchi, and U. Bortolozzo, “Optical vortex interaction and generation via nonlinear wave mixing,” Phys. Rev. A 84, 1–4 (2011). [CrossRef]  

14. A. V. Sokolov and S. E. Harris, “Ultrashort pulse generation by molecular modulation,” J. Opt. B: Quantum S. O. 5, R1–R26 (2003). [CrossRef]  

15. M. Zhi and A. V. Sokolov, “Broadband coherent light generation in a Raman-active crystal driven by two-color femtosecond laser pulses,” Opt. Lett. 32, 2251–2253 (2007). [CrossRef]   [PubMed]  

18. A. V. Sokolov, M. Y. Shverdin, D. R. Walker, D. D. Yavuz, A. M. Burzo, G. Y. Yin, and S. E. Harris, “Generation and control of femtosecond pulses by molecular modulation,” J. Mod. Opt. 52, 285–304 (2005). [CrossRef]  

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References

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  1. M. V. Vasnetsov and K. Staliunas, Optical Vortices Horizons in World Physics Volume 228 (Nova Science Publishers, Inc, 1999).
  2. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A. 336, 165–190 (1974).
    [Crossref]
  3. A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. 3, 161–204 (2011).
    [Crossref]
  4. H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995).
    [Crossref] [PubMed]
  5. A. M. Akulshin, R. J. Mclean, E. E. Mikhailov, and I. Novikova, “Distinguishing nonlinear processes in atomic media via orbital angular momentum transfer,” Opt. Lett. 40, 1109–1112 (2015).
    [Crossref] [PubMed]
  6. F. M. Fazal and S. M. Block, “Optical tweezers study life under tension,” Nature Phot. 5, 318–321 (2011).
    [Crossref]
  7. M. J. Padgett, “Light in a twist: optical angular momentum,” Proc. SPIE 8637, 863702 (2013).
    [Crossref]
  8. D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
    [Crossref] [PubMed]
  9. T. A. Klar, E. Engel, and S. W. Hell, “Breaking Abbe’s diffraction resolution limit in fluorescence microscopy with stimulated emission depletion beams of various shapes,” Phys. Rev. E 64, 066613 (2001).
    [Crossref]
  10. G. Foo, D. M. Palacios, and G. A. Swartzlander, “Optical vortex coronagraph,” Opt. Lett. 30, 3308–3310 (2005).
    [Crossref]
  11. G. A. Swartzlander, “Peering into darkness with a vortex spatial filter,” Opt. Lett. 26, 497–499 (2001).
    [Crossref]
  12. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
    [Crossref] [PubMed]
  13. G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Freespace information transfer using light beams carrying orbital angular momentum,” Opt. Express 12, 5448–5456 (2004).
    [Crossref] [PubMed]
  14. I. Mariyenko, J. Strohaber, and C. Uiterwaal, “Creation of optical vortices in femtosecond pulses,” Opt. Express 13, 7599–7608 (2005).
    [Crossref] [PubMed]
  15. F. Lenzini, S. Residori, F. Arecchi, and U. Bortolozzo, “Optical vortex interaction and generation via nonlinear wave mixing,” Phys. Rev. A 84, 1–4 (2011).
    [Crossref]
  16. A. V. Sokolov and S. E. Harris, “Ultrashort pulse generation by molecular modulation,” J. Opt. B: Quantum S. O. 5, R1–R26 (2003).
    [Crossref]
  17. M. Zhi and A. V. Sokolov, “Broadband coherent light generation in a Raman-active crystal driven by two-color femtosecond laser pulses,” Opt. Lett. 32, 2251–2253 (2007).
    [Crossref] [PubMed]
  18. A. V. Sokolov, M. Y. Shverdin, D. R. Walker, D. D. Yavuz, A. M. Burzo, G. Y. Yin, and S. E. Harris, “Generation and control of femtosecond pulses by molecular modulation,” J. Mod. Opt. 52, 285–304 (2005).
    [Crossref]
  19. N. Bozinovic, S. Golowich, P. Kristensen, and S. Ramachandran, “Control of orbital angular momentum of light with optical fiber,” Opt. Lett. 37(13), 2451–2453 (2014).
    [Crossref]
  20. A. M. Amaral, E. L. Falco-Filho, and C. B. de Arajo, “Characterization of topological charge and orbital angular momentum of shaped optical vortices,” Opt. Express 22(24), 30315–30324 (2014).
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  21. D. Persuy, M. Ziegler, O. Crégut, K. Kheng, M. Gallart, B. Hönerlage, and P. Gilliot, “Four-wave mixing in quantum wells using femtosecond pulses with Laguerre-Gauss modes,” Phys. Rev. B 92, 115312 (2015).
    [Crossref]
  22. T. Roger, J. J. Heitz, E. M. Wright, and D. Faccio, “Non-collinear interaction of photons with orbital angular momentum,” Sci. Rep. 3, 3491 (2013).
    [Crossref] [PubMed]
  23. G. Molina-Terriza, J. P. Torres, and L. Torner, “Orbital angular momentum of photons in noncollinear parametric downconversion,” Opt. Commun. 228, 155–160 (2003).
    [Crossref]
  24. J. Strohaber, M. Zhi, A. V. Sokolov, A. A. Kolomenskii, G. G. Paulus, and H. A. Schuessler, “Coherent transfer of optical orbital angular momentum in multi-order Raman sideband generation,” Opt. Lett. 37, 3411–3413 (2012).
    [Crossref]
  25. N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
    [Crossref] [PubMed]
  26. R. Vasilyeu, A. Dudley, N. Khilo, and A. Forbes, “Generating superpositions of higher-order Bessel beams,” Opt. Express 17(26), 23389–23395 (2009).
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  27. M. Zhi, K. Wang, X. Hua, H. Schuessler, J. Strohaber, and A. V. Sokolov, “Generation of femtosecond optical vortices by molecular modulation in a Raman-active crystal,” Opt. Express 21, 27750–27758 (2013).
    [Crossref]
  28. K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, “Second-harmonic generation and the orbital angular momentum of light,” Phys. Rev. A 54, R3742–R3745 (1996).
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  29. J. Liao, X. Wang, W. Sun, Y. Tan, D. Kong, Y. Nie, J. Qi, H. Jia, J. Liu, J. Yang, J. Tan, and X. Li, “Analysis of femtosecond optical vortex beam generated by direct wave-front modulation,” Opt. Eng. 52, 106102 (2013).
    [Crossref]
  30. H. Ma, Z. Liu, H. Wu, X. Xu, and J. Chen, “Adaptive correction of vortex laser beam in a closed-loop system with phase only liquid crystal spatial light modulator,” Opt. Commun. 285, 859–863 (2012).
    [Crossref]
  31. W. M. Lee, X. C. Yuan, and K. Dholakia, “Experimental observation of optical vortex evolution in a Gaussian beam with an embedded fractional phase step,” Opt. Commun. 239, 129–135 (2004).
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  32. M. Padgett, J. Courtial, and L. Allen, “Light’s orbital angular momentum,” Phys. Today 57(5), 35–40 (2004).
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  33. C.-S. Guo, L.-L. Lu, and H.-T. Wang, “Characterizing topological charge of optical vortices by using an annular aperture,” Opt. Lett. 34, 3686–3688 (2009).
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  34. Y. Peng, X.-T. Gan, P. Ju, Y.-D. Wang, and J.-L. Zhao, “Measuring topological charges of optical vortices with multi-singularity using a cylindrical lens,” Chinese Phys. Lett. 32, 024201 (2015).
    [Crossref]
  35. M. Harris, C. A. Hill, P. R. Tapster, and J. M. Vaughan, “Laser modes with helical wave fronts,” Phys. Rev. A 49, 3119–3122 (1994).
    [Crossref] [PubMed]
  36. P. Vaity, J. Banerji, and R. Singh, “Measuring the topological charge of an optical vortex by using a tilted convex lens,” Phys. Lett. A 377, 1154–1156 (2013).
    [Crossref]
  37. W. Buono, L. Moraes, J. Huguenin, C. Souza, and A. Khoury, “Arbitrary orbital angular momentum addition in second harmonic generation,” New J. Phys. 16, 093041 (2014).
    [Crossref]
  38. M. Zhi and A. V. Sokolov, “Broadband generation in a Raman crystal driven by a pair of time-delayed linearly chirped pulses,” New J. Phys. 10, 025032 (2008).
    [Crossref]
  39. R. Trebino, K. W. Delong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbugel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
    [Crossref]
  40. S. G. Reddy, S. Prabhakar, A. Kumar, J. Banerji, and R. Singh, “Higher order optical vortices and formation of speckles,” Opt. Lett. 39, 4364–4367 (2014).
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  41. R. L. Phillips and L. C. Andrews, “Spot size and divergence for Laguerre Gaussian beams of any order,” Appl. Opt. 22, 643–644 (1983).
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  42. J. Strohaber, J. Abul, F. Zhu, A. A. Kolomenskii, and H. A. Schuessler, “Cascade Raman sideband generation and orbital angular momentum relations for paraxial beam modes,” Opt. Express 23, 22463–22476 (2015).
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2015 (4)

A. M. Akulshin, R. J. Mclean, E. E. Mikhailov, and I. Novikova, “Distinguishing nonlinear processes in atomic media via orbital angular momentum transfer,” Opt. Lett. 40, 1109–1112 (2015).
[Crossref] [PubMed]

D. Persuy, M. Ziegler, O. Crégut, K. Kheng, M. Gallart, B. Hönerlage, and P. Gilliot, “Four-wave mixing in quantum wells using femtosecond pulses with Laguerre-Gauss modes,” Phys. Rev. B 92, 115312 (2015).
[Crossref]

Y. Peng, X.-T. Gan, P. Ju, Y.-D. Wang, and J.-L. Zhao, “Measuring topological charges of optical vortices with multi-singularity using a cylindrical lens,” Chinese Phys. Lett. 32, 024201 (2015).
[Crossref]

J. Strohaber, J. Abul, F. Zhu, A. A. Kolomenskii, and H. A. Schuessler, “Cascade Raman sideband generation and orbital angular momentum relations for paraxial beam modes,” Opt. Express 23, 22463–22476 (2015).
[Crossref] [PubMed]

2014 (4)

2013 (6)

M. J. Padgett, “Light in a twist: optical angular momentum,” Proc. SPIE 8637, 863702 (2013).
[Crossref]

P. Vaity, J. Banerji, and R. Singh, “Measuring the topological charge of an optical vortex by using a tilted convex lens,” Phys. Lett. A 377, 1154–1156 (2013).
[Crossref]

J. Liao, X. Wang, W. Sun, Y. Tan, D. Kong, Y. Nie, J. Qi, H. Jia, J. Liu, J. Yang, J. Tan, and X. Li, “Analysis of femtosecond optical vortex beam generated by direct wave-front modulation,” Opt. Eng. 52, 106102 (2013).
[Crossref]

T. Roger, J. J. Heitz, E. M. Wright, and D. Faccio, “Non-collinear interaction of photons with orbital angular momentum,” Sci. Rep. 3, 3491 (2013).
[Crossref] [PubMed]

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref] [PubMed]

M. Zhi, K. Wang, X. Hua, H. Schuessler, J. Strohaber, and A. V. Sokolov, “Generation of femtosecond optical vortices by molecular modulation in a Raman-active crystal,” Opt. Express 21, 27750–27758 (2013).
[Crossref]

2012 (2)

J. Strohaber, M. Zhi, A. V. Sokolov, A. A. Kolomenskii, G. G. Paulus, and H. A. Schuessler, “Coherent transfer of optical orbital angular momentum in multi-order Raman sideband generation,” Opt. Lett. 37, 3411–3413 (2012).
[Crossref]

H. Ma, Z. Liu, H. Wu, X. Xu, and J. Chen, “Adaptive correction of vortex laser beam in a closed-loop system with phase only liquid crystal spatial light modulator,” Opt. Commun. 285, 859–863 (2012).
[Crossref]

2011 (3)

F. M. Fazal and S. M. Block, “Optical tweezers study life under tension,” Nature Phot. 5, 318–321 (2011).
[Crossref]

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. 3, 161–204 (2011).
[Crossref]

F. Lenzini, S. Residori, F. Arecchi, and U. Bortolozzo, “Optical vortex interaction and generation via nonlinear wave mixing,” Phys. Rev. A 84, 1–4 (2011).
[Crossref]

2009 (2)

2008 (1)

M. Zhi and A. V. Sokolov, “Broadband generation in a Raman crystal driven by a pair of time-delayed linearly chirped pulses,” New J. Phys. 10, 025032 (2008).
[Crossref]

2007 (1)

2005 (3)

A. V. Sokolov, M. Y. Shverdin, D. R. Walker, D. D. Yavuz, A. M. Burzo, G. Y. Yin, and S. E. Harris, “Generation and control of femtosecond pulses by molecular modulation,” J. Mod. Opt. 52, 285–304 (2005).
[Crossref]

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

I. Mariyenko, J. Strohaber, and C. Uiterwaal, “Creation of optical vortices in femtosecond pulses,” Opt. Express 13, 7599–7608 (2005).
[Crossref] [PubMed]

2004 (3)

W. M. Lee, X. C. Yuan, and K. Dholakia, “Experimental observation of optical vortex evolution in a Gaussian beam with an embedded fractional phase step,” Opt. Commun. 239, 129–135 (2004).
[Crossref]

M. Padgett, J. Courtial, and L. Allen, “Light’s orbital angular momentum,” Phys. Today 57(5), 35–40 (2004).
[Crossref]

G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Freespace information transfer using light beams carrying orbital angular momentum,” Opt. Express 12, 5448–5456 (2004).
[Crossref] [PubMed]

2003 (3)

A. V. Sokolov and S. E. Harris, “Ultrashort pulse generation by molecular modulation,” J. Opt. B: Quantum S. O. 5, R1–R26 (2003).
[Crossref]

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[Crossref] [PubMed]

G. Molina-Terriza, J. P. Torres, and L. Torner, “Orbital angular momentum of photons in noncollinear parametric downconversion,” Opt. Commun. 228, 155–160 (2003).
[Crossref]

2001 (3)

T. A. Klar, E. Engel, and S. W. Hell, “Breaking Abbe’s diffraction resolution limit in fluorescence microscopy with stimulated emission depletion beams of various shapes,” Phys. Rev. E 64, 066613 (2001).
[Crossref]

G. A. Swartzlander, “Peering into darkness with a vortex spatial filter,” Opt. Lett. 26, 497–499 (2001).
[Crossref]

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[Crossref] [PubMed]

1997 (1)

R. Trebino, K. W. Delong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbugel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[Crossref]

1996 (1)

K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, “Second-harmonic generation and the orbital angular momentum of light,” Phys. Rev. A 54, R3742–R3745 (1996).
[Crossref] [PubMed]

1995 (1)

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

1994 (1)

M. Harris, C. A. Hill, P. R. Tapster, and J. M. Vaughan, “Laser modes with helical wave fronts,” Phys. Rev. A 49, 3119–3122 (1994).
[Crossref] [PubMed]

1983 (1)

1974 (1)

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A. 336, 165–190 (1974).
[Crossref]

Abul, J.

Akulshin, A. M.

Allen, L.

M. Padgett, J. Courtial, and L. Allen, “Light’s orbital angular momentum,” Phys. Today 57(5), 35–40 (2004).
[Crossref]

K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, “Second-harmonic generation and the orbital angular momentum of light,” Phys. Rev. A 54, R3742–R3745 (1996).
[Crossref] [PubMed]

Amaral, A. M.

Andrews, L. C.

Arecchi, F.

F. Lenzini, S. Residori, F. Arecchi, and U. Bortolozzo, “Optical vortex interaction and generation via nonlinear wave mixing,” Phys. Rev. A 84, 1–4 (2011).
[Crossref]

Banerji, J.

S. G. Reddy, S. Prabhakar, A. Kumar, J. Banerji, and R. Singh, “Higher order optical vortices and formation of speckles,” Opt. Lett. 39, 4364–4367 (2014).
[Crossref] [PubMed]

P. Vaity, J. Banerji, and R. Singh, “Measuring the topological charge of an optical vortex by using a tilted convex lens,” Phys. Lett. A 377, 1154–1156 (2013).
[Crossref]

Barnett, S. M.

Berry, M. V.

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A. 336, 165–190 (1974).
[Crossref]

Block, S. M.

F. M. Fazal and S. M. Block, “Optical tweezers study life under tension,” Nature Phot. 5, 318–321 (2011).
[Crossref]

Bortolozzo, U.

F. Lenzini, S. Residori, F. Arecchi, and U. Bortolozzo, “Optical vortex interaction and generation via nonlinear wave mixing,” Phys. Rev. A 84, 1–4 (2011).
[Crossref]

Bozinovic, N.

N. Bozinovic, S. Golowich, P. Kristensen, and S. Ramachandran, “Control of orbital angular momentum of light with optical fiber,” Opt. Lett. 37(13), 2451–2453 (2014).
[Crossref]

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref] [PubMed]

Buono, W.

W. Buono, L. Moraes, J. Huguenin, C. Souza, and A. Khoury, “Arbitrary orbital angular momentum addition in second harmonic generation,” New J. Phys. 16, 093041 (2014).
[Crossref]

Burzo, A. M.

A. V. Sokolov, M. Y. Shverdin, D. R. Walker, D. D. Yavuz, A. M. Burzo, G. Y. Yin, and S. E. Harris, “Generation and control of femtosecond pulses by molecular modulation,” J. Mod. Opt. 52, 285–304 (2005).
[Crossref]

Chen, J.

H. Ma, Z. Liu, H. Wu, X. Xu, and J. Chen, “Adaptive correction of vortex laser beam in a closed-loop system with phase only liquid crystal spatial light modulator,” Opt. Commun. 285, 859–863 (2012).
[Crossref]

Courtial, J.

Crégut, O.

D. Persuy, M. Ziegler, O. Crégut, K. Kheng, M. Gallart, B. Hönerlage, and P. Gilliot, “Four-wave mixing in quantum wells using femtosecond pulses with Laguerre-Gauss modes,” Phys. Rev. B 92, 115312 (2015).
[Crossref]

de Arajo, C. B.

Delong, K. W.

R. Trebino, K. W. Delong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbugel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[Crossref]

Dholakia, K.

W. M. Lee, X. C. Yuan, and K. Dholakia, “Experimental observation of optical vortex evolution in a Gaussian beam with an embedded fractional phase step,” Opt. Commun. 239, 129–135 (2004).
[Crossref]

K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, “Second-harmonic generation and the orbital angular momentum of light,” Phys. Rev. A 54, R3742–R3745 (1996).
[Crossref] [PubMed]

Dudley, A.

Engel, E.

T. A. Klar, E. Engel, and S. W. Hell, “Breaking Abbe’s diffraction resolution limit in fluorescence microscopy with stimulated emission depletion beams of various shapes,” Phys. Rev. E 64, 066613 (2001).
[Crossref]

Faccio, D.

T. Roger, J. J. Heitz, E. M. Wright, and D. Faccio, “Non-collinear interaction of photons with orbital angular momentum,” Sci. Rep. 3, 3491 (2013).
[Crossref] [PubMed]

Falco-Filho, E. L.

Fazal, F. M.

F. M. Fazal and S. M. Block, “Optical tweezers study life under tension,” Nature Phot. 5, 318–321 (2011).
[Crossref]

Fittinghoff, D. N.

R. Trebino, K. W. Delong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbugel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[Crossref]

Foo, G.

Forbes, A.

Franke-Arnold, S.

Friese, M. E. J.

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

Gallart, M.

D. Persuy, M. Ziegler, O. Crégut, K. Kheng, M. Gallart, B. Hönerlage, and P. Gilliot, “Four-wave mixing in quantum wells using femtosecond pulses with Laguerre-Gauss modes,” Phys. Rev. B 92, 115312 (2015).
[Crossref]

Gan, X.-T.

Y. Peng, X.-T. Gan, P. Ju, Y.-D. Wang, and J.-L. Zhao, “Measuring topological charges of optical vortices with multi-singularity using a cylindrical lens,” Chinese Phys. Lett. 32, 024201 (2015).
[Crossref]

Gibson, G.

Gilliot, P.

D. Persuy, M. Ziegler, O. Crégut, K. Kheng, M. Gallart, B. Hönerlage, and P. Gilliot, “Four-wave mixing in quantum wells using femtosecond pulses with Laguerre-Gauss modes,” Phys. Rev. B 92, 115312 (2015).
[Crossref]

Golowich, S.

Grier, D. G.

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[Crossref] [PubMed]

Guo, C.-S.

Harris, M.

M. Harris, C. A. Hill, P. R. Tapster, and J. M. Vaughan, “Laser modes with helical wave fronts,” Phys. Rev. A 49, 3119–3122 (1994).
[Crossref] [PubMed]

Harris, S. E.

A. V. Sokolov, M. Y. Shverdin, D. R. Walker, D. D. Yavuz, A. M. Burzo, G. Y. Yin, and S. E. Harris, “Generation and control of femtosecond pulses by molecular modulation,” J. Mod. Opt. 52, 285–304 (2005).
[Crossref]

A. V. Sokolov and S. E. Harris, “Ultrashort pulse generation by molecular modulation,” J. Opt. B: Quantum S. O. 5, R1–R26 (2003).
[Crossref]

He, H.

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

Heckenberg, N. R.

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

Heitz, J. J.

T. Roger, J. J. Heitz, E. M. Wright, and D. Faccio, “Non-collinear interaction of photons with orbital angular momentum,” Sci. Rep. 3, 3491 (2013).
[Crossref] [PubMed]

Hell, S. W.

T. A. Klar, E. Engel, and S. W. Hell, “Breaking Abbe’s diffraction resolution limit in fluorescence microscopy with stimulated emission depletion beams of various shapes,” Phys. Rev. E 64, 066613 (2001).
[Crossref]

Hill, C. A.

M. Harris, C. A. Hill, P. R. Tapster, and J. M. Vaughan, “Laser modes with helical wave fronts,” Phys. Rev. A 49, 3119–3122 (1994).
[Crossref] [PubMed]

Hönerlage, B.

D. Persuy, M. Ziegler, O. Crégut, K. Kheng, M. Gallart, B. Hönerlage, and P. Gilliot, “Four-wave mixing in quantum wells using femtosecond pulses with Laguerre-Gauss modes,” Phys. Rev. B 92, 115312 (2015).
[Crossref]

Hua, X.

Huang, H.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref] [PubMed]

Huguenin, J.

W. Buono, L. Moraes, J. Huguenin, C. Souza, and A. Khoury, “Arbitrary orbital angular momentum addition in second harmonic generation,” New J. Phys. 16, 093041 (2014).
[Crossref]

Jia, H.

J. Liao, X. Wang, W. Sun, Y. Tan, D. Kong, Y. Nie, J. Qi, H. Jia, J. Liu, J. Yang, J. Tan, and X. Li, “Analysis of femtosecond optical vortex beam generated by direct wave-front modulation,” Opt. Eng. 52, 106102 (2013).
[Crossref]

Ju, P.

Y. Peng, X.-T. Gan, P. Ju, Y.-D. Wang, and J.-L. Zhao, “Measuring topological charges of optical vortices with multi-singularity using a cylindrical lens,” Chinese Phys. Lett. 32, 024201 (2015).
[Crossref]

Kane, D. J.

R. Trebino, K. W. Delong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbugel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[Crossref]

Kheng, K.

D. Persuy, M. Ziegler, O. Crégut, K. Kheng, M. Gallart, B. Hönerlage, and P. Gilliot, “Four-wave mixing in quantum wells using femtosecond pulses with Laguerre-Gauss modes,” Phys. Rev. B 92, 115312 (2015).
[Crossref]

Khilo, N.

Khoury, A.

W. Buono, L. Moraes, J. Huguenin, C. Souza, and A. Khoury, “Arbitrary orbital angular momentum addition in second harmonic generation,” New J. Phys. 16, 093041 (2014).
[Crossref]

Klar, T. A.

T. A. Klar, E. Engel, and S. W. Hell, “Breaking Abbe’s diffraction resolution limit in fluorescence microscopy with stimulated emission depletion beams of various shapes,” Phys. Rev. E 64, 066613 (2001).
[Crossref]

Kolomenskii, A. A.

Kong, D.

J. Liao, X. Wang, W. Sun, Y. Tan, D. Kong, Y. Nie, J. Qi, H. Jia, J. Liu, J. Yang, J. Tan, and X. Li, “Analysis of femtosecond optical vortex beam generated by direct wave-front modulation,” Opt. Eng. 52, 106102 (2013).
[Crossref]

Kristensen, P.

N. Bozinovic, S. Golowich, P. Kristensen, and S. Ramachandran, “Control of orbital angular momentum of light with optical fiber,” Opt. Lett. 37(13), 2451–2453 (2014).
[Crossref]

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref] [PubMed]

Krumbugel, M. A.

R. Trebino, K. W. Delong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbugel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[Crossref]

Kumar, A.

Lee, W. M.

W. M. Lee, X. C. Yuan, and K. Dholakia, “Experimental observation of optical vortex evolution in a Gaussian beam with an embedded fractional phase step,” Opt. Commun. 239, 129–135 (2004).
[Crossref]

Lenzini, F.

F. Lenzini, S. Residori, F. Arecchi, and U. Bortolozzo, “Optical vortex interaction and generation via nonlinear wave mixing,” Phys. Rev. A 84, 1–4 (2011).
[Crossref]

Li, X.

J. Liao, X. Wang, W. Sun, Y. Tan, D. Kong, Y. Nie, J. Qi, H. Jia, J. Liu, J. Yang, J. Tan, and X. Li, “Analysis of femtosecond optical vortex beam generated by direct wave-front modulation,” Opt. Eng. 52, 106102 (2013).
[Crossref]

Liao, J.

J. Liao, X. Wang, W. Sun, Y. Tan, D. Kong, Y. Nie, J. Qi, H. Jia, J. Liu, J. Yang, J. Tan, and X. Li, “Analysis of femtosecond optical vortex beam generated by direct wave-front modulation,” Opt. Eng. 52, 106102 (2013).
[Crossref]

Liu, J.

J. Liao, X. Wang, W. Sun, Y. Tan, D. Kong, Y. Nie, J. Qi, H. Jia, J. Liu, J. Yang, J. Tan, and X. Li, “Analysis of femtosecond optical vortex beam generated by direct wave-front modulation,” Opt. Eng. 52, 106102 (2013).
[Crossref]

Liu, Z.

H. Ma, Z. Liu, H. Wu, X. Xu, and J. Chen, “Adaptive correction of vortex laser beam in a closed-loop system with phase only liquid crystal spatial light modulator,” Opt. Commun. 285, 859–863 (2012).
[Crossref]

Lu, L.-L.

Ma, H.

H. Ma, Z. Liu, H. Wu, X. Xu, and J. Chen, “Adaptive correction of vortex laser beam in a closed-loop system with phase only liquid crystal spatial light modulator,” Opt. Commun. 285, 859–863 (2012).
[Crossref]

Mair, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[Crossref] [PubMed]

Mariyenko, I.

Mclean, R. J.

Mikhailov, E. E.

Molina-Terriza, G.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Orbital angular momentum of photons in noncollinear parametric downconversion,” Opt. Commun. 228, 155–160 (2003).
[Crossref]

Moraes, L.

W. Buono, L. Moraes, J. Huguenin, C. Souza, and A. Khoury, “Arbitrary orbital angular momentum addition in second harmonic generation,” New J. Phys. 16, 093041 (2014).
[Crossref]

Nie, Y.

J. Liao, X. Wang, W. Sun, Y. Tan, D. Kong, Y. Nie, J. Qi, H. Jia, J. Liu, J. Yang, J. Tan, and X. Li, “Analysis of femtosecond optical vortex beam generated by direct wave-front modulation,” Opt. Eng. 52, 106102 (2013).
[Crossref]

Novikova, I.

Nye, J. F.

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A. 336, 165–190 (1974).
[Crossref]

Padgett, M.

M. Padgett, J. Courtial, and L. Allen, “Light’s orbital angular momentum,” Phys. Today 57(5), 35–40 (2004).
[Crossref]

Padgett, M. J.

Palacios, D. M.

Pas’ko, V.

Paulus, G. G.

Peng, Y.

Y. Peng, X.-T. Gan, P. Ju, Y.-D. Wang, and J.-L. Zhao, “Measuring topological charges of optical vortices with multi-singularity using a cylindrical lens,” Chinese Phys. Lett. 32, 024201 (2015).
[Crossref]

Persuy, D.

D. Persuy, M. Ziegler, O. Crégut, K. Kheng, M. Gallart, B. Hönerlage, and P. Gilliot, “Four-wave mixing in quantum wells using femtosecond pulses with Laguerre-Gauss modes,” Phys. Rev. B 92, 115312 (2015).
[Crossref]

Phillips, R. L.

Prabhakar, S.

Qi, J.

J. Liao, X. Wang, W. Sun, Y. Tan, D. Kong, Y. Nie, J. Qi, H. Jia, J. Liu, J. Yang, J. Tan, and X. Li, “Analysis of femtosecond optical vortex beam generated by direct wave-front modulation,” Opt. Eng. 52, 106102 (2013).
[Crossref]

Ramachandran, S.

N. Bozinovic, S. Golowich, P. Kristensen, and S. Ramachandran, “Control of orbital angular momentum of light with optical fiber,” Opt. Lett. 37(13), 2451–2453 (2014).
[Crossref]

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref] [PubMed]

Reddy, S. G.

Ren, Y.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref] [PubMed]

Residori, S.

F. Lenzini, S. Residori, F. Arecchi, and U. Bortolozzo, “Optical vortex interaction and generation via nonlinear wave mixing,” Phys. Rev. A 84, 1–4 (2011).
[Crossref]

Richman, B. A.

R. Trebino, K. W. Delong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbugel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[Crossref]

Roger, T.

T. Roger, J. J. Heitz, E. M. Wright, and D. Faccio, “Non-collinear interaction of photons with orbital angular momentum,” Sci. Rep. 3, 3491 (2013).
[Crossref] [PubMed]

Rubinsztein-Dunlop, H.

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

Schuessler, H.

Schuessler, H. A.

Shverdin, M. Y.

A. V. Sokolov, M. Y. Shverdin, D. R. Walker, D. D. Yavuz, A. M. Burzo, G. Y. Yin, and S. E. Harris, “Generation and control of femtosecond pulses by molecular modulation,” J. Mod. Opt. 52, 285–304 (2005).
[Crossref]

Simpson, N. B.

K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, “Second-harmonic generation and the orbital angular momentum of light,” Phys. Rev. A 54, R3742–R3745 (1996).
[Crossref] [PubMed]

Singh, R.

S. G. Reddy, S. Prabhakar, A. Kumar, J. Banerji, and R. Singh, “Higher order optical vortices and formation of speckles,” Opt. Lett. 39, 4364–4367 (2014).
[Crossref] [PubMed]

P. Vaity, J. Banerji, and R. Singh, “Measuring the topological charge of an optical vortex by using a tilted convex lens,” Phys. Lett. A 377, 1154–1156 (2013).
[Crossref]

Sokolov, A. V.

M. Zhi, K. Wang, X. Hua, H. Schuessler, J. Strohaber, and A. V. Sokolov, “Generation of femtosecond optical vortices by molecular modulation in a Raman-active crystal,” Opt. Express 21, 27750–27758 (2013).
[Crossref]

J. Strohaber, M. Zhi, A. V. Sokolov, A. A. Kolomenskii, G. G. Paulus, and H. A. Schuessler, “Coherent transfer of optical orbital angular momentum in multi-order Raman sideband generation,” Opt. Lett. 37, 3411–3413 (2012).
[Crossref]

M. Zhi and A. V. Sokolov, “Broadband generation in a Raman crystal driven by a pair of time-delayed linearly chirped pulses,” New J. Phys. 10, 025032 (2008).
[Crossref]

M. Zhi and A. V. Sokolov, “Broadband coherent light generation in a Raman-active crystal driven by two-color femtosecond laser pulses,” Opt. Lett. 32, 2251–2253 (2007).
[Crossref] [PubMed]

A. V. Sokolov, M. Y. Shverdin, D. R. Walker, D. D. Yavuz, A. M. Burzo, G. Y. Yin, and S. E. Harris, “Generation and control of femtosecond pulses by molecular modulation,” J. Mod. Opt. 52, 285–304 (2005).
[Crossref]

A. V. Sokolov and S. E. Harris, “Ultrashort pulse generation by molecular modulation,” J. Opt. B: Quantum S. O. 5, R1–R26 (2003).
[Crossref]

Souza, C.

W. Buono, L. Moraes, J. Huguenin, C. Souza, and A. Khoury, “Arbitrary orbital angular momentum addition in second harmonic generation,” New J. Phys. 16, 093041 (2014).
[Crossref]

Staliunas, K.

M. V. Vasnetsov and K. Staliunas, Optical Vortices Horizons in World Physics Volume 228 (Nova Science Publishers, Inc, 1999).

Strohaber, J.

Sun, W.

J. Liao, X. Wang, W. Sun, Y. Tan, D. Kong, Y. Nie, J. Qi, H. Jia, J. Liu, J. Yang, J. Tan, and X. Li, “Analysis of femtosecond optical vortex beam generated by direct wave-front modulation,” Opt. Eng. 52, 106102 (2013).
[Crossref]

Swartzlander, G. A.

Sweetser, J. N.

R. Trebino, K. W. Delong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbugel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[Crossref]

Tan, J.

J. Liao, X. Wang, W. Sun, Y. Tan, D. Kong, Y. Nie, J. Qi, H. Jia, J. Liu, J. Yang, J. Tan, and X. Li, “Analysis of femtosecond optical vortex beam generated by direct wave-front modulation,” Opt. Eng. 52, 106102 (2013).
[Crossref]

Tan, Y.

J. Liao, X. Wang, W. Sun, Y. Tan, D. Kong, Y. Nie, J. Qi, H. Jia, J. Liu, J. Yang, J. Tan, and X. Li, “Analysis of femtosecond optical vortex beam generated by direct wave-front modulation,” Opt. Eng. 52, 106102 (2013).
[Crossref]

Tapster, P. R.

M. Harris, C. A. Hill, P. R. Tapster, and J. M. Vaughan, “Laser modes with helical wave fronts,” Phys. Rev. A 49, 3119–3122 (1994).
[Crossref] [PubMed]

Torner, L.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Orbital angular momentum of photons in noncollinear parametric downconversion,” Opt. Commun. 228, 155–160 (2003).
[Crossref]

Torres, J. P.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Orbital angular momentum of photons in noncollinear parametric downconversion,” Opt. Commun. 228, 155–160 (2003).
[Crossref]

Trebino, R.

R. Trebino, K. W. Delong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbugel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[Crossref]

Tur, M.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref] [PubMed]

Uiterwaal, C.

Vaity, P.

P. Vaity, J. Banerji, and R. Singh, “Measuring the topological charge of an optical vortex by using a tilted convex lens,” Phys. Lett. A 377, 1154–1156 (2013).
[Crossref]

Vasilyeu, R.

Vasnetsov, M.

Vasnetsov, M. V.

M. V. Vasnetsov and K. Staliunas, Optical Vortices Horizons in World Physics Volume 228 (Nova Science Publishers, Inc, 1999).

Vaughan, J. M.

M. Harris, C. A. Hill, P. R. Tapster, and J. M. Vaughan, “Laser modes with helical wave fronts,” Phys. Rev. A 49, 3119–3122 (1994).
[Crossref] [PubMed]

Vaziri, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[Crossref] [PubMed]

Walker, D. R.

A. V. Sokolov, M. Y. Shverdin, D. R. Walker, D. D. Yavuz, A. M. Burzo, G. Y. Yin, and S. E. Harris, “Generation and control of femtosecond pulses by molecular modulation,” J. Mod. Opt. 52, 285–304 (2005).
[Crossref]

Wang, H.-T.

Wang, K.

Wang, X.

J. Liao, X. Wang, W. Sun, Y. Tan, D. Kong, Y. Nie, J. Qi, H. Jia, J. Liu, J. Yang, J. Tan, and X. Li, “Analysis of femtosecond optical vortex beam generated by direct wave-front modulation,” Opt. Eng. 52, 106102 (2013).
[Crossref]

Wang, Y.-D.

Y. Peng, X.-T. Gan, P. Ju, Y.-D. Wang, and J.-L. Zhao, “Measuring topological charges of optical vortices with multi-singularity using a cylindrical lens,” Chinese Phys. Lett. 32, 024201 (2015).
[Crossref]

Weihs, G.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[Crossref] [PubMed]

Willner, A.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref] [PubMed]

Wright, E. M.

T. Roger, J. J. Heitz, E. M. Wright, and D. Faccio, “Non-collinear interaction of photons with orbital angular momentum,” Sci. Rep. 3, 3491 (2013).
[Crossref] [PubMed]

Wu, H.

H. Ma, Z. Liu, H. Wu, X. Xu, and J. Chen, “Adaptive correction of vortex laser beam in a closed-loop system with phase only liquid crystal spatial light modulator,” Opt. Commun. 285, 859–863 (2012).
[Crossref]

Xu, X.

H. Ma, Z. Liu, H. Wu, X. Xu, and J. Chen, “Adaptive correction of vortex laser beam in a closed-loop system with phase only liquid crystal spatial light modulator,” Opt. Commun. 285, 859–863 (2012).
[Crossref]

Yang, J.

J. Liao, X. Wang, W. Sun, Y. Tan, D. Kong, Y. Nie, J. Qi, H. Jia, J. Liu, J. Yang, J. Tan, and X. Li, “Analysis of femtosecond optical vortex beam generated by direct wave-front modulation,” Opt. Eng. 52, 106102 (2013).
[Crossref]

Yao, A. M.

Yavuz, D. D.

A. V. Sokolov, M. Y. Shverdin, D. R. Walker, D. D. Yavuz, A. M. Burzo, G. Y. Yin, and S. E. Harris, “Generation and control of femtosecond pulses by molecular modulation,” J. Mod. Opt. 52, 285–304 (2005).
[Crossref]

Yin, G. Y.

A. V. Sokolov, M. Y. Shverdin, D. R. Walker, D. D. Yavuz, A. M. Burzo, G. Y. Yin, and S. E. Harris, “Generation and control of femtosecond pulses by molecular modulation,” J. Mod. Opt. 52, 285–304 (2005).
[Crossref]

Yuan, X. C.

W. M. Lee, X. C. Yuan, and K. Dholakia, “Experimental observation of optical vortex evolution in a Gaussian beam with an embedded fractional phase step,” Opt. Commun. 239, 129–135 (2004).
[Crossref]

Yue, Y.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref] [PubMed]

Zeilinger, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[Crossref] [PubMed]

Zhao, J.-L.

Y. Peng, X.-T. Gan, P. Ju, Y.-D. Wang, and J.-L. Zhao, “Measuring topological charges of optical vortices with multi-singularity using a cylindrical lens,” Chinese Phys. Lett. 32, 024201 (2015).
[Crossref]

Zhi, M.

Zhu, F.

Ziegler, M.

D. Persuy, M. Ziegler, O. Crégut, K. Kheng, M. Gallart, B. Hönerlage, and P. Gilliot, “Four-wave mixing in quantum wells using femtosecond pulses with Laguerre-Gauss modes,” Phys. Rev. B 92, 115312 (2015).
[Crossref]

Adv. Opt. Photon. (1)

Appl. Opt. (1)

Chinese Phys. Lett. (1)

Y. Peng, X.-T. Gan, P. Ju, Y.-D. Wang, and J.-L. Zhao, “Measuring topological charges of optical vortices with multi-singularity using a cylindrical lens,” Chinese Phys. Lett. 32, 024201 (2015).
[Crossref]

J. Mod. Opt. (1)

A. V. Sokolov, M. Y. Shverdin, D. R. Walker, D. D. Yavuz, A. M. Burzo, G. Y. Yin, and S. E. Harris, “Generation and control of femtosecond pulses by molecular modulation,” J. Mod. Opt. 52, 285–304 (2005).
[Crossref]

J. Opt. B: Quantum S. O. (1)

A. V. Sokolov and S. E. Harris, “Ultrashort pulse generation by molecular modulation,” J. Opt. B: Quantum S. O. 5, R1–R26 (2003).
[Crossref]

Nature (2)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[Crossref] [PubMed]

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[Crossref] [PubMed]

Nature Phot. (1)

F. M. Fazal and S. M. Block, “Optical tweezers study life under tension,” Nature Phot. 5, 318–321 (2011).
[Crossref]

New J. Phys. (2)

W. Buono, L. Moraes, J. Huguenin, C. Souza, and A. Khoury, “Arbitrary orbital angular momentum addition in second harmonic generation,” New J. Phys. 16, 093041 (2014).
[Crossref]

M. Zhi and A. V. Sokolov, “Broadband generation in a Raman crystal driven by a pair of time-delayed linearly chirped pulses,” New J. Phys. 10, 025032 (2008).
[Crossref]

Opt. Commun. (3)

G. Molina-Terriza, J. P. Torres, and L. Torner, “Orbital angular momentum of photons in noncollinear parametric downconversion,” Opt. Commun. 228, 155–160 (2003).
[Crossref]

H. Ma, Z. Liu, H. Wu, X. Xu, and J. Chen, “Adaptive correction of vortex laser beam in a closed-loop system with phase only liquid crystal spatial light modulator,” Opt. Commun. 285, 859–863 (2012).
[Crossref]

W. M. Lee, X. C. Yuan, and K. Dholakia, “Experimental observation of optical vortex evolution in a Gaussian beam with an embedded fractional phase step,” Opt. Commun. 239, 129–135 (2004).
[Crossref]

Opt. Eng. (1)

J. Liao, X. Wang, W. Sun, Y. Tan, D. Kong, Y. Nie, J. Qi, H. Jia, J. Liu, J. Yang, J. Tan, and X. Li, “Analysis of femtosecond optical vortex beam generated by direct wave-front modulation,” Opt. Eng. 52, 106102 (2013).
[Crossref]

Opt. Express (6)

Opt. Lett. (8)

Phys. Lett. A (1)

P. Vaity, J. Banerji, and R. Singh, “Measuring the topological charge of an optical vortex by using a tilted convex lens,” Phys. Lett. A 377, 1154–1156 (2013).
[Crossref]

Phys. Rev. A (3)

M. Harris, C. A. Hill, P. R. Tapster, and J. M. Vaughan, “Laser modes with helical wave fronts,” Phys. Rev. A 49, 3119–3122 (1994).
[Crossref] [PubMed]

K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, “Second-harmonic generation and the orbital angular momentum of light,” Phys. Rev. A 54, R3742–R3745 (1996).
[Crossref] [PubMed]

F. Lenzini, S. Residori, F. Arecchi, and U. Bortolozzo, “Optical vortex interaction and generation via nonlinear wave mixing,” Phys. Rev. A 84, 1–4 (2011).
[Crossref]

Phys. Rev. B (1)

D. Persuy, M. Ziegler, O. Crégut, K. Kheng, M. Gallart, B. Hönerlage, and P. Gilliot, “Four-wave mixing in quantum wells using femtosecond pulses with Laguerre-Gauss modes,” Phys. Rev. B 92, 115312 (2015).
[Crossref]

Phys. Rev. E (1)

T. A. Klar, E. Engel, and S. W. Hell, “Breaking Abbe’s diffraction resolution limit in fluorescence microscopy with stimulated emission depletion beams of various shapes,” Phys. Rev. E 64, 066613 (2001).
[Crossref]

Phys. Rev. Lett. (1)

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

Phys. Today (1)

M. Padgett, J. Courtial, and L. Allen, “Light’s orbital angular momentum,” Phys. Today 57(5), 35–40 (2004).
[Crossref]

Proc. R. Soc. Lond. A. (1)

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A. 336, 165–190 (1974).
[Crossref]

Proc. SPIE (1)

M. J. Padgett, “Light in a twist: optical angular momentum,” Proc. SPIE 8637, 863702 (2013).
[Crossref]

Rev. Sci. Instrum. (1)

R. Trebino, K. W. Delong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbugel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[Crossref]

Sci. Rep. (1)

T. Roger, J. J. Heitz, E. M. Wright, and D. Faccio, “Non-collinear interaction of photons with orbital angular momentum,” Sci. Rep. 3, 3491 (2013).
[Crossref] [PubMed]

Science (1)

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref] [PubMed]

Other (1)

M. V. Vasnetsov and K. Staliunas, Optical Vortices Horizons in World Physics Volume 228 (Nova Science Publishers, Inc, 1999).

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Figures (5)

Fig. 1
Fig. 1 Production and measurement of optical vortices. (a) Computer-generated phase mask applied to the SLM. (b) Donut-shaped beam profile (taken approximately 76 cm after the SLM) resulting from Gaussian beam reflection off of the SLM. (c) Schematic of an astigmatic focusing method for TC measurement. (d) Resultant intensity distribution in the focal plane. Data in (a), (b), and (d) are shown for a vortex beam with TC = 3, hence we see 4 distinguishable spots in part (d).
Fig. 2
Fig. 2 (a) Our experimental setup. Dashed lines correspond to Stokes beam, while solid lines correspond to pump. The blue lines correspond to the one-beam modulation case and the red lines correspond to the two-beam modulation case. The angle of the SLM is greatly exaggerated. Typical sidebands produced from this arrangement are also shown. The inset schematically depicts our two chirped pulses and the delay between them. (b) Computer generated phase masks (left), optical vortices obtained with these phase masks just before the focusing lens, approximately 76 cm after the SLM (middle), and vortices focused with a tilted lens (right).
Fig. 3
Fig. 3 TC measurement of Raman sidebands using a tilted lens. For each block: columns 1 and 3 are the sidebands before the lens, Columns 2 and 4 are the sidebands after the lens. From top to bottom – AS1, AS2, AS3. Results are summarized in Table 1. (a) Left (right) two columns: sidebands generated with lp = 0 and ls = −1 (ls = 1). (b) Left (right) two columns: sidebands generated with lp = 0 and ls = −2 (ls = 2).
Fig. 4
Fig. 4 TC measurement of Raman sidebands using a tilted lens. Columns 1 and 3 are the sidebands before the lens, Columns 2 and 4 are the sidebands after the lens. Left (right) two columns: sidebands generated with lp = 0 and ls = −3 (ls = 3). From top to bottom – AS1 (TC=±3; 4 spots), AS2 (TC=±6; 6 spots).
Fig. 5
Fig. 5 TC measurement of Raman sidebands using a tilted lens. For each block: digital phase maps for generating pump and Stokes beams (left), AS1 generated when these phase maps are applied (middle), AS1 focused with tilted lens (right).

Tables (2)

Tables Icon

Table 1 Predicted, and measured, TC for (from top to bottom) ls = +1(−1), ls = +2(−2), and ls = +3(−3). In all cases, lp = 0.

Tables Icon

Table 2 Predicted, and measured, TC for 4 different cases of mixed lp and ls.

Equations (1)

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l n = ( n + 1 ) l p n l s

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