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 . 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 . 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 . As He et al. showed in , 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 , 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 . 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  and classical  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 , with nonlinear Raman-active crystals . 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 = 2lp − ls. Hence, by applying the same logic to n sidebands, we may derive a simple equation that clearly predicts the TC of each sideband: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)ϕp − nϕs (where ϕn is the relative phase of the nth field, in the transverse plane) . 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.  utilizing two identical chirped femtosecond laser pulses with lp = ls = ±1 and lp = −ls = ±1. Further, our group has previously  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) , 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, (for polar angle of the phase map θ = 0 to ), 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).
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  or by using a cylindrical lens . One popular method involves building an interferometer and studying the interference pattern produced between a Gaussian beam and a vortex beam . 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. . 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  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 .
3. Setup and methods
Our experimental setup is shown in Fig. 2. As is described in , 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 . 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.
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 , 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). 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.
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  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. 3–5, 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).
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.
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 : , 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 |lp − ls| equal to 3. To confirm that the topological charge transfer took place, even for such a relatively large value of |lp − ls|, we measured the resultant TC of AS1. From Eq. (1), we predict the results shown in Table 2.
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.
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 . However, we obtained our results independently from Strohaber et al. and were unaware of their work prior to this publication.
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.
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