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Nonlinear mixing of optical vortices with fractional topological charge in Raman sideband

generation

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2017 J. Opt. 19 015607

(http://iopscience.iop.org/2040-8986/19/1/015607)

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Nonlinear mixing of optical vortices withfractional topological charge in Ramansideband generation

J Strohaber1, Y Boran2, M Sayrac2, L Johnson1, F Zhu2,A A Kolomenskii2 and H A Schuessler2

1Department of Physics and Astronomy, Florida A&M University, Tallahassee, FL-32307, USA2Department of Physics and Astronomy, Texas A&M University, College Station, TX-77843-4242, USA

E-mail: jstroha1@gmail.com

Received 6 October 2016Accepted for publication 17 October 2016Published 8 December 2016

AbstractWe studied the nonlinear parametric interaction of femtosecond fractionally-charged opticalvortices in a Raman-active medium. Propagation of such beams was described using theKirchhoff–Fresnel integrals by embedding a non-integer 2π phase step in a Gaussian beamprofile. When using fractionally-charged pump or Stokes beams, we observed the production ofnew topological charge and phase discontinuities in the Raman field. These newly generatedfractionally-charged Raman vortex beams were found to follow the same orbital angularmomentum algebra derived by Strohaber et al (2012 Opt. Lett. 37 3411) for integer vortexbeams.

Keywords: femtosecond, Raman sideband generation, optical vorices

(Some figures may appear in colour only in the online journal)

1. Introduction

It is well-known that light possesses spin angular momentumassociated with its polarization state; however, it is lessknown that it can also possess orbital angular momentum(OAM). Spin angular momentum of light has been studied byBeth [1] in a well-known experiment designed to investigatethe transfer of optical angular momentum to matter. Beth’sexperiment used birefringent wave plates suspended by a finequartz wire. A similar experiment using Laguerre–Gaussian(LG) beams was proposed by Allen [2], who suggested theuse of a cylindrical lens as a mode converter. Beams of lightwith OAM such as LG beams are characterized by a centralphase singularity, in which an enclosed line integral of thegradient of the phase around the singularity results in aninteger ℓ number of 2π phase changes [2, 3]. This is known asthe Berry phase and indicates the amount of topologicalcharge (TC) enclosed within the given line integral. In theseexperiments, the transfer of angular momentum was due to

the intrinsic spin or the external degrees of freedom of thefield as described by linear optics.

Recently a number of investigations involving the non-linear interaction of optical vortices with matter haveappeared in the literature [4–6]. The first two experimentswere carried out by Strohaber et al [4], and Hassinger et al[5]. In [4], two time-delayed chirped pulses embedded with anazimuthal phase from a spiral phase plate were crossed in aPbWO4 crystal, and, in [5], broadband radiation was spec-trally separated into two portions, where the differencebetween the central frequencies of each portion was used ina wave mixing process. In these two original experiments,the authors demonstrated, for the first time, the coherenttransfer of optical angular momentum in the generationof cascaded Raman sidebands. The authors of bothgroups derived an OAM-algebra, or vortex or TC-algebra,which was verified by phase measurements. Later experi-ments showed that this same algebra = + -ℓ n ℓ nℓ1n P S

AS[ ( )and = + -ℓ n ℓ nℓ1n

SS P( ) ] also held when Ince–Gaussian

Journal of Optics

J. Opt. 19 (2017) 015607 (10pp) doi:10.1088/2040-8986/19/1/015607

2040-8978/17/015607+10$33.00 © 2016 IOP Publishing Ltd Printed in the UK1

modes were used [7]. Figure 1 shows an energy level diagramillustrating how pump and Stokes photons mix to generatenew photons with new energy and momentum. To producethe first anti-Stokes (AS) order, a pump photon is absorbedleaving the system in a virtual state. The Stokes beam sti-mulates the emission of a Stokes photon bringing the systemto an excited state. This is followed by the absorption of asecond pump photon and the spontaneous emission of a newAS photon. For high pumping rates, the newly generated ASradiation participates in pumping the system to higher virtualstates, which results in higher frequency AS radiation. Similararguments can be made for the lower frequency Stokesradiation. These processes are known as cascaded Ramansideband generation.

In the present work, we advance upon our earlier work[4, 7] by investigating the production of Raman sidebandswith fractionally charged phase singularities. A fundamentaldifference between previous work, involving optical beammodes with integer charges, and the present work with frac-tionally charged vortices is that the latter are not self-similarbeams and the propagation dynamics are therefore morecomplex. Fractionally-charged vortices are of interest in laser-matter interactions because OAM may influence selectionrules. A question that naturally arises is how can a fractionalamount of angular momentum be quantum mechanicallytransferred to matter [8, 9]? As will be shown later, a phase-discontinuity due to a non-integer phase step with a fractionof 2π produces a beam consisting of an infinite number ofinteger-charged vortices. In our analysis, imaging of crosssectional distributions of laser beams plays an important role;therefore, we first consider changes in the beam profile withdifferent values of the TC.

2. Propagation of a Gaussian beam with a fractionalazimuthal phase

For comparison with the experiment, we theoreticallydescribe the propagation of a Gaussian beam with a frac-tionally charged azimuthal phase using the Kirchhoff–Fresnel

integral. In addition to comparison with the measured data,results obtained in the Fresnel approximation provide aninsight into the effects of the phase on the diffracted beam.The theoretical results also elucidate diffraction phenomenatypically encountered during the production of optical vor-tices. For example, it may be surmised that the multi-ringedpattern, often observed in the diffracted beam profile of aGaussian beam embedded with an azimuthal phase, is due toimproper spatial amplitude modulation. One is tempted toexpand the distribution in the LG basis as

å åY = r r=¥

=-¥¥

c LG ;ℓ ℓ p

ℓ0 , however, this superposition does

not produce the observed diffraction effects. Our analysisindicates that the multi-ringed diffraction phenomena typi-cally encountered in vortex generation in phase-only pro-duction is the result of the phase singularity at the center ofthe beam. The expression for the electric field, in the Fresnelapproximation, of a fractionally charged vortex beam with anembedded azimuthal phase variation of exp(iaθ) is (seeappendix A),

⎜ ⎟ ⎜ ⎟

⎧⎨⎩⎡⎣⎢

⎛⎝

⎞⎠

⎛⎝

⎞⎠

⎤⎦⎥

⎫⎬⎭/ /

åp dpr

r r

= +

´ -

q r

=-¥

¥-

- +

E r z Ap

c

I I

,1

i e8

e

1

4

1

4.

1n

nn

nn

n n

2i

,014

1 22

1 22

2( )

( )( ) ( )

Here p p= - -c a a n1 sin ,nn( ) ( ) [ ( )] = +p w z1 ik 2 ,2

02

/ /r = =b p b r z2 , ik ,2 2 2 with k being the wavenumber, z—the propagation distance, w0—the beam waist, a—the TC,and Iα(x) is the modified Bessel function.

Equation (1) is the propagation equation for the electricfield amplitude produced by the diffraction of a Gaussianbeam through a phase plate having an arbitrary step height. Asimilar result has been obtained by Berry for the diffraction ofa plane electromagnetic wave by a phase plate of arbitraryheight [11]. Equation (1) is useful to us as it provides insightinto the evolution of fractionally-charged optical vortices.One drawback of using a plane wave is that for small frac-tional step heights, edge diffraction from the phase mismatchdominates the diffraction pattern. A comparison between auniform plane wave and a Gaussian beam is shown in figure 2

Figure 1. Time-delayed, chirped-pulse cascaded Raman sideband generation: (a) frequency versus time of two time-delayed chirped pulsesemployed to produce Raman transitions. The frequency difference between the pulses is Δω=btd. In order to access the Raman transitions,this difference must be close to the Raman frequency ωR=ωtd. (b) Diagram of cascade generation of sideband frequencies, where ωP is thefrequency of the pump radiation and ωs is that of the Stokes radiation.

2

J. Opt. 19 (2017) 015607 J Strohaber et al

for the sequence of TCs of a=(0.01, 0.1, 0.5, 1.0).Equation (1) will be further used to calculate the inter-ferograms of fractionally-charged vortices. Experimentalresults are expected to show the appearance of lines of lowerintensity due to a phase discontinuity, which are ubiquitousfeatures of fractional vortices.

3. Experimental setup

In our experiments, we used amplified femtosecond radiationfrom a Ti:sapphire chirped-pulse amplification system(Spectra Physics, Spitfire). Our system produces ∼50 fs laserpulses at a repetition rate of 1 kHz. The central wavelengthwas 800 nm, and the energy per pulse was ∼1 mJ. Outputradiation from the laser was sent through a variable iris (notshown) used to control mode quality [7] and beam power.Following the iris, the radiation was sent through the sym-metric beam crossing setup shown on the left in figure 3, withthe outputs being the time-delayed pump and Stokes beams.In contrast to our previous setups [4, 7], the Michelsoninterferometer was placed last with one of the mirrors beingreplaced by a parallel aligned liquid crystal on silicon spatiallight modulator (Hamamatsu LCOS-SLM 104683). Thisconfiguration allowed us to independently modulate theradiation in both the pump and the Stokes beams [12, 13] andproduced two sets of sidebands (solid and doted lines infigure 3). One arm of the Michelson was used to produce anindependent set of Raman sidebands with Gaussian profilesserving as a reference (dotted lines in figure 3) to investigatethe phase content of the generated sidebands produced by theother arm (solid lines in figure 3) [4, 5, 7]. This is thesimultaneous Young’s double slit experiment [4], and allowsfor each Raman sideband of different frequency to simulta-neously have a reference beam of the same frequency.Figure 3 illustrates this by the solid and dotted beams exitingthe PbWO4 crystal. As can be seen in the figure, the two setsof sidebands are being generate at two different but closelyspaced locations in the crystal. By displaying a sequence ofgrayscale images on the SLM, the phase modulation as afunction of normalized grayscale value (0 to 1) was found to

Figure 2. Comparison of calculated near-field intensity profiles of fractionally-charged vortices produced using a plane electromagnetic wave(row 1), and a Gaussian beam (row 2). The topological charges from left to right are 0.01, 0.1, 0.5 and 1.0.

Figure 3. Illustration of the beam crossing setup with the followingelements: beam splitter (BS), mirror (M), D-shaped mirror (M-D),lens (L), translation stage (T), spatial light modulator (SLM) andlead tungsten crystal (PbWO4). The SLM, used to produce thedesired beam mode, is positioned in the stationary arm of theMichelson interferometer. Radiation from the variable arm was usedas the reference. The solid and dotted beams exiting the crystal showthat the reference and the signal sidebands are generated simulta-neously at different locations.

3

J. Opt. 19 (2017) 015607 J Strohaber et al

have a linear dependence y=10.5x, which for a grayscalevalue of 0.6 corresponds to a phase shift of 2π [12]. Toproduce Raman sidebands, we employed time-delayed pulsesin the beam crossing setup (figure 3). Images of the pump andStokes beams, and the generated Raman orders were capturedusing an iSight CCD (Apple, iPhone 5s) consisting of 8megapixels with 1.5 μm per pixel, and a charge-coupleddevice (Spiricon, SP503U) having a resolution of ´640 480pixels.

To produce Raman sidebands, the pump and Stokesbeams were focused into the Raman-active crystal using alens having a nominal focal length of 40 cm. The separationdistance between the beams at the position of the lens(figure 3) was ∼2.25 cm and resulted in a full-angle of 3.2°between the two beams. The nonlinear crystal was a 0.5 mmthick lead tungstate crystal (MTI Corporation). Lead tungstate(PbWO4) is a uniaxial crystal belonging to the tetragonalcrystal group (space group -C aI4h4

61 ) [14]. Because lead

tungstate is a stable, non-hygroscopic and low cost material, itis a popular Raman crystal that has found applications in theconstruction of Raman lasers [15] and in the generation ofsidebands [4, 5]. Lead tungstate has a broad optical trans-parency spanning the wavelength range from 0.33 to 5.5 μm[14]. Of particular relevance to the intense femtosecondapplications used in this work, PbWO4 has a relatively highdamage threshold. Stimulated Raman scattering experimentshave been used to obtain spectra over a large wavelengthrange. The spectrum of PbWO4 is dominated by the totallysymmetric (breathing) Ag optical modes of its tetrahedral

-WO24 ions at 900 cm−1 [15] and a relatively strong Raman

line at 325 cm−1 from the internal mode of WO4 [15]. Wehave previously shown [7] that the frequency difference of thedelayed chirped pulses is capable of being tuned to the Ramantransitions in a nonlinear medium (figure 1). For linear

chirped pulses, the phase content has a quadratic timedependence, so that the instantaneous frequency is givenby ω=ω0+bt. Adding a delay td between the pulses resultsin a frequency difference of Δω=btd (figure 1). When thisΔω is equal to the Raman frequency ΔωR i.e., the differencebetween the two lowest states in figure 1, we get the pro-duction of sidebands. The optimal conditions for sidebandgeneration were found by experimentally scanning the chirp(b) and delay (td) between the two pulses.

4. Experimental results

Using the setup described in the last section, we producedfractionally-charged vortices from a computer generatedhologram of an azimuthal phase ramp having an azimuthalphase mismatch. When the phase step was greater than theavailable phase modulation of the SLM, we used phasewrapping similar to that found in Fresnel lenses. Figure 4(a)shows experimental near field (z≈0.5 m) images of theintensity and phase of fractionally charged vortices rangingwithin a=0−3.4. We took images in phase steps of 0.1 rad,but present images with phase steps of 0.5 rad. The beamswith integer values of =a ℓ in columns 1 and 3 are nearlycylindrically symmetric, and exhibit a central multifurcationor region where a fringe branches into multiple fringes. Thebeams with half-integer values of /=a ℓ 2 in columns 2 and 4exhibit a line of zero intensity. From the interferograms, it canbe seen that the line of zero intensity is associated with aphase discontinuity i.e., where the maxima of the interferencefringes abruptly shift by half the period of the interferencepattern. These near-field beams were projected onto a whitescreen and images were taken with an iSight camera. Usingequation (1), we simulated the near-field intensity profiles and

Figure 4. (a) Experimentally measured near-field interferograms of fractionally-charged vortices produced by an SLM. The fractional chargesare indicated in the upper corner of each sub-panel. (b) Theoretical interferograms of near-field fractional vortices calculated using theKirchhoff–Fresnel integral (equation (1)).

4

J. Opt. 19 (2017) 015607 J Strohaber et al

interferograms, figure 4(b). To determine the TCs of thevortices in the far-field at the location immediately before thecrystal, we use a 40 cm focal length lens to focus the beamsfrom the SLM. In the focus, we investigated the TC by usingthe tilted lens method. As with the near-field beams, we usedour theoretical calculations (equation (1)) to simulate theintensity profile of the far-field beams shown in rows 1 and 3,figure 5(b). When the TC is an integer value =a ℓ, the tiltedlens pattern shows a number of nodes equal to the TC. Forhalf-integer vortices, an additional partial lobe appears.

For the integer vortices, we were able to derive exacttheoretical expression describing the modal conversion of thevortices to Hermite–Gaussian modes, and for the fractionalvortices we found an approximate expression. These solutionsare derived in appendix B, and provided us with guidance forfinding the optimal experimental configuration. This optimalwas achieved by approaching the focal region from down-stream with our CCD camera while rotating the lens, so as toimprove the separation of the modal lobes in the spatial imageand to avoid possible damage to our CCD. The measuringposition was inside the focal region, but at a position thatresulted in an elliptical shape of the beam as seen in figure 5for the lowest order Gaussian mode.

To show how the tilted lens generates Hermite Gaussianbeams, we use an expression we previously derived in [7] forthe modal decomposition of LG beams in the HG basis,

åpp

= --

´=

-

Nk ℓ k

ℓ

LG 12

2

1

i sgn HG .

2ℓ

ℓ ℓℓ

k

ℓ

kℓ k k

0, 00

,

( )! (∣ ∣ )!

( ( ))( )

∣ ∣∣ ∣

∣ ∣

∣ ∣

In equation (2), the LG beam is at the position of the lens z=0. For the beam at z= z±, we included only the Gouy

phases for each -HG ℓ k k,∣ ∣ in the sum

å

pp

Y = -

´-

´

F¢ F¢ +F

=

-- F¢ -F

N

k ℓ kℓ

1 e e2

2

1i sgn

HG e .

3

ℓℓ ℓ ℓ

ℓ

k

ℓk

ℓ k kk

0,far field

0i i 1

2

0

,i

G G G

G G

( )

! (∣ ∣ )!( ( )) ( )

∣ ∣ ∣ ∣ ( )∣ ∣

∣ ∣

∣ ∣( )

Equation (3), gives the field distributions for the HG beams infigure 5(b), rows (2) and (4). For the non-integer opticalvortices, the analytical solution is somewhat difficult to find,but we made an approximation to zero order

å= Y=-¥

¥

F c . 4ℓ

ℓ ℓ0,farfield ( )

Here the cℓ are those given in equation (1). The results ofequation (4) are plotted in figure 5 and show good qualitativeagreement with the experimental results.

Before investigating the production of Raman sidebandswith fractionally-charged optical vortices, we produced aseries of integer vortices in the pump and Stokes beams to testour new setup. Interferograms of generated sidebands in asimultaneous Young’s double slit experiment definitivelyverified that the sidebands followed the expected OAM-algebra derived by us [4]. In figure 6(a), the pump beam wasembedded with =ℓ 0p units () of OAM and the Stokes beamwith =ℓ 1.S Based on the OAM-algebra = +ℓ N ℓ1 PASN ( )-Nℓ ,S the AS orders are expected to possess TC of

= -ℓ Nℓ .SASN The interferograms in column 2 of figure 6(a)show agreement with the expected OAM content. Thisagreement can be seen by counting the number of bifurcationsin the recorded interferograms. To aid the reader, we haveplace red dots at the location of the bifurcations. For example,in the AS3 order in figure 6(a), the expected magnitude of theTC is 3 for which we count three bifurcations. The data in

Figure 5. (a) Rows 1 and 2 are far-field images of fractional vortices in the focus of a 40 cm focal length lens. The topological charges areindicated in each image. Rows 3 and 4 are the result of tilting the focusing lens by an angle of about 20 degrees. (b) Rows 1 and 3 aretheoretical far-field diffraction patterns based on the Kirchhoff–Fresnel diffraction integral. Rows 2 and 4 were calculated by expanding thebeams in the Hermite–Gaussian basis set, see text for details.

5

J. Opt. 19 (2017) 015607 J Strohaber et al

figure 6(b) were taken using beams with =ℓ 1p and =ℓ 0.S Inthis case, the OAM-algebra predicts that the AS orders willhave TC of = +ℓ ℓN 1 ,PASN ( ) or more explicitly, =ℓ 2,AS1

=ℓ 3AS2 and =ℓ 4.AS3 This is in agreement with the mea-sured data as seen by the number of observed bifurcationsindicated by the red dots in the interferograms. The negativevalues of = -ℓ NℓSASN can be seen by the inversion of themultifurcations relative to those in figure 6(a). Lastly, wegenerated Raman sidebands by spatially modulating both thepump and the Stokes beams. Figure 7 shows the first two ASorders generated using = -ℓ 1p and =ℓ 1,S whichgives = - +ℓ N2 1ASN ( ).

In our first experiments producing sidebands with frac-tional vortices, we used a pump beam embedded witha fractional charge of /=ℓ 3 4p and a Stokes beam with

=ℓ 1.S From the OAM-algebra, we expect to get

/= - =ℓ ℓ ℓ2 1 2PAS1 S in the first AS order. Panels (a) and(b) in figure 8 show the intensity profile and interferogram inthis order. Visual inspection of the interference fringes infigure 8(b) indicates that the AS1 order has a TC of =ℓ 0.5 asseen by the line phase dislocation. For comparison, theintensity profile and interferogram of an =ℓ 0.5 vortex isshown in figures 8(e) and (f).

Since this is the first reported production of a fraction-ally-charged vortex in Raman sideband generation, definiteverification is required. Lineouts (figure 8(b)) were taken onboth sides of the phase discontinuity (dashed lines infigure 8(b)), and a running correlation was calculatedfigure 8(d). This correlation shows that the fringes to theleft of the beam in figure 9(b) are in phase with each other(+1), while fringes to the right are out of phase (−1). Thisresult is expected from a beam with =ℓ 0.5 units of OAM.

Figure 6. (a) Images of Raman sidebands in the first three anti-Stokes orders. The pump was encoded with =ℓ 0P units of OAM and theStokes with =ℓ 1.S The second column in panel (a) are interferograms verifying the phase content of the beams. (b) same as (a) except with

=ℓ 1P and =ℓ 0.S The location of the bifurcations is indicated by the red dots.

Figure 7. Images of Raman sidebands in the first two anti-Stokes orders. The pump was encoded with = -ℓ 1P units of OAM and the Stokeswith =ℓ 1.S The interferograms verify the OAM algebra = + -ℓ ℓ ℓN 1 NPASN S( ) .

6

J. Opt. 19 (2017) 015607 J Strohaber et al

Figure 8. Panels (a) and (b): Intensity profile and interferogram in the first anti-Stokes order. The pump and Stokes beams contained /=ℓ 3 4p

and =ℓ 1S units of OAM respectively, and the OAM algebra predicts an OAM content of /=ℓ 1 2.AS1 (c) Lineouts of fringes on either side ofthe phase dislocation. (d) Running correlation of the line-outs shown in (c). The correlation indicates that the fringes begin in phase (+1) inthe first half of the image and are out of phase (−1) in the second half. Panels (e)–(h) are simulated of the experimental data shown in panels(a)–(d).

Figure 9. Production of new topological charge from phase discontinuities. The value in the lower left corner of the images is the topologicalcharge of the Stokes beam. For values of ℓS near 0.5 and 1.5 a phase discontinuity can be observed followed by the appearance of abifurcation. Row (5), columns 2 and 3 are an observed phase discontinuity circled in the panels of row 1, column 5 and row 4, column 2. Thelast two images were obtained using the tilted lens method.

7

J. Opt. 19 (2017) 015607 J Strohaber et al

Panels (e)–(h) are simulations of those shown in panels(a)–(d).

As the simplest example, we have observed and verifiedthe generation of a fractionally-charged Raman vortex withTC of 1/2. Next we demonstrate how new TC is produced inthe sidebands from phase discontinuities similar to thoseshown in figure 4. To do this, we produced higher orderfractionally-charged vortices by embedding the pump radia-tion with =ℓ 1P and varying the OAM of the Stokes radiationfrom =ℓ 0S to = -ℓ 2S in steps of 0.1 radian. From theOAM-algebra, we expect the radiation in the AS1 order topossess TCs ranging from =ℓ 2AS1 to =ℓ 4.AS1 Figure 9shows the experimental interferograms for this sequence ofpump and Stokes beams. Our first observation is that forinteger values of ℓS no multifurcations (typically associatedwith beam having >ℓ 1) survive, but instead spatially sepa-rated bifurcations appear, which can be seen in rows 1–5,columns 1–5; and row 5, column 1. When the Stokes beamhas an OAM content of around = -ℓ 0.4 0.6,S we observe aline dislocation spanning roughly four fringes (as shownwithin the red circle in row 1, column 5, and its close-up inrow 5, column 2) followed by the appearance of a newbifurcation that moves in from the right of the image. Thearea near the new bifurcation shows a decrease in intensityalong a line (most noticeable for =ℓ 0.5) and is a featuretypically observed in beams near the area where there is astepwise change in phase. A similar occurrence appears whenthe OAM of the Stokes beam is =ℓ 1.6S (row 4, column 2,and row 5, column 3). We have traced the formation of newTC from other combinations of pump and Stokes beams, andfor which similar results were obtained. Two observationswere made: first, the multifurcations in the Raman sidebandswere broken-up into single bifurcations, and second, a ‘short’phase dislocation was usually observed before the appearanceof a new bifurcation. We speculate that the absence of a‘long’ phase discontinuity in the Raman sidebands (as seen infigure 4) with higher order OAM is due to coherent back-ground instabilities [18], which spatially splits the OAMcharge into separate bifurcations.

5. Discussion and summary

In conclusion, we have for the first time produced fractionallycharged vortices in Raman sidebands. The effect of diffrac-tion on fractionally charged vortices was investigated, and theTCs were determined by the simultaneous Young’s doubleslit experiment and the tilted lens methods. The observedfractional charges in the sidebands were found to follow thesame OAM algebra that was earlier established for helical-LGwith integer charges. The field of future applications forfractionally-charged vortex beams is wide open, possiblyspanning novel strong field physics to new pathways inoptical excitations.

Acknowledgments

This work was funded by the Robert A Welch Foundation,Grant No. A1546 and the Qatar Foundation under Grants No.NPRP 6-465-1-091. Y Boran and M Sayrac contributedequally to this work.

Appendix A

An analytical solution using the Kirchhoff–Fresneldiffraction integral is found by embedding an azimuthal phasevariation exp(iaθ) that is in general not an integer multiple of2π, but an arbitrary real number, into a Gaussian beamamplitude

/y = q-x y N, , 0 e e . A.1r w a00 i2

02( ) ( )

Here /p=N w200

02 normalizes the intensity of the trans-

verse beam, w0 is the beam waist and a is any real number.This beam can be propagated using the Fresnel integral incylindrical coordinates,

⎛⎝⎜

⎞⎠⎟

/

ò òlq= - - ¢ ¢ ¢

´

p

q q q

¥

- ¢ ¢ - ¢ + ¢ - ¢

E r zz

kr

zr r

N

,i

e exp i2

d d

e e e e A.2

kz

r w a krz

kr rz

i2

0 0

2

00 i i

2i cos .2

02

2

( )

( ) ( )( )

The angular integral can be evaluated by first making aFourier series expansion of the fractional phase factor,

å åpp

= = --

q q q¢

=-¥

¥¢

=-¥

¥¢c a

a ne e sin

11

1e ,

A.3

a

nn

n

n

n ni i i( ) ( )

( )

where n is any integer number. To compress the equationsand make them more transparent, we make the followingsubstitutions: cn=(−1)nsin(πa)/[π(a−n)], /= +p w12

02

/k zi 2 , and b=ikr/z. With these substitutions and the Fourierexpansion in equation (A.3), the integral in equation (A.2)reduces to,

ò òå h= ¢ ¢

´

qp

h h=-¥

¥ ¥- ¢

- ¢

E r z A c r r, e d d e

e e .

A.4nn

n p r

n br

i

0 0

2

i i cos

2 2

( )( )

( )

Here the angular variable has changed since we have madethe substitution η=θ − θ′. The complex exponential havinga cosine argument can be expanded using the well-knownJacobi–Anger expansion,

å= ¢h h¢

=-¥

¥

J bre i e . A.5br

m

mm

mi cos i( ) ( )( )

8

J. Opt. 19 (2017) 015607 J Strohaber et al

Using equation (A.5), the angular integral can be per-formed

òå åp d= ¢ ¢ ¢q

=-¥

¥

=-¥

¥ ¥- ¢E r z A c r r J br, 2 i e d e

A.6n m

mn

nn m m

p ri,

0

2 2

( ) ( )

( )

In order to carry out the integration over the radial spatialcoordinate, we integrate by parts one time,

ò

ò

¢ ¢ ¢ = - ¢ +

´ ¢¢

¢

¥- - ¢

¥

¥- ¢

r r J brp

J brp

rr

J br

d e1

2e

1

2

d ed

d.

A.7

np r

np r

p rn

0 20

2

0

2 2 2 2

2 2

( ) ( )

( )( )

The first term on the right is non-zero when m=0 otherwiseit is equal to δm,0/2p

2 after evaluating the integration bounds.Using the recurrence relation 2dJα(x)/dx=Jα−1(x)−Jα+1(x), the last integral reduces to

ò

òd

¢ ¢ ¢

= + ¢ ¢ - ¢

¥-

¥- ¢

- +

r r J br

p

b

pr J br J br

d e

2 4d e .

A.8

np r

n p rn n

0

,02 2 0

1 1

2 2

2 2

( )

[ ( ) ( )]( )

The remaining integral over the radial coordinate can becarried out using a modified version of Hankel’s more generalintegral [10]

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟/ò

p= -

¥-J bt t

p

b

pI

b

pe d

2exp

8 8. A.9m

p tm

0

2

2 2

2

2

2 2( ) ( )

Using equation (A.9) we arrive at

⎜ ⎟

⎜ ⎟

⎡⎣⎢

⎛⎝

⎞⎠

⎛⎝

⎞⎠

⎤⎦⎥

/

/

òd r p

r

r

¢ ¢ ¢

= +

-

r

¥-

--

+

r r J br

p pI

I

d e

2 2 8e

1

4

1

4. A.10

np r

nn

n

0

,02 2

14 1 2

2

1 22

2 2

2

( )

( )

( )

( )

Here Iα(x) are the modified Bessel functions and ρ2=b2/2p2. By combining equations (A.6) and (A.10), the finalexpression for the electric field in the Fresnel approximationfor a fractionally charged vortex is

⎜ ⎟ ⎜ ⎟

⎧⎨⎩⎡⎣⎢

⎛⎝

⎞⎠

⎛⎝

⎞⎠

⎤⎦⎥

⎫⎬⎭/ /

åp dpr

r r

= +

´ -

q r

=-¥

¥-

- +

E r z Ap

c

I I

,1

i e8

e

1

4

1

4. A.11

n

nn

nn

n n

2i

,014

1 22

1 22

2( )

( )( ) ( )

Notice from equation (3) that when n=0 the expansioncoefficient is zero c0=0, and because δn,0=0 whenn≠0, we can neglect the first term δn,0 in the bracketedexpression.

Appendix B

When a lens is tilted, the focal length of the beam changesalong the axis about which the lens is being rotated and alongthe axis perpendicular to the rotation axis and the beam [16].If the lens is rotated about the y-axis, the diameter of the lensalong that axis is unchanged y′=y, but the diameter of thelens in the x-direction effectively shrinks as x′=x cos(θ),where θ is the tilt angle. In both cases, the thickness of thelens at its ‘center’ increases as d′/cos(θ). From these rela-tionships and using the Lensmaker’s equation for a plano-convex lens, we found, in the approximation that d/R=1,the two focal lengths

q q» »f f f fcos , cos . B.1x l y lens3

ens( ) ( ) ( )

Plots of the effective focal lengths in the x and y direc-tions are shown in figure B.1(a). In our experiments, we use alens with a focal length of 40 cm. This graph was used toguide us in adjusting the tilt angle of the lens. To determinethe required focal length difference, we used Gaussian optics.

When a Gaussian beam is focused, the effectivefocal length / /= +l f f z1 2

0L2( ) and the waist =w0

/ +w f z f0L 0L2 2 are given in terms of the focal length f of

the lens, and of the waist w0L and Rayleigh range z0L of theinput beam [17]. Upon tilting the lens, the focal lengths alongthe x direction fx and along the y direction fy are notequal, therefore the beam parameters are not equal. TheRayleigh ranges z0x and z0y of the beam in the x and ydirections are found by setting the beam spot sizes

/= + -w w z l z1x x x x02

02( ) and /= + -w w z l z1y y y y0

202( )

equal to w01 at the position of the lens z=0. In this way, theRayleigh ranges in the x and y directions are found to be

= -z l w w 1x x x0 012

02 and = -z l w w 1 .y y y0 01

202 We

wish to find the position in the focus where wx=wy, and theGouy phase difference is equal to pF¢ - F = 2.G G This canbe achieved by setting the derivative of the Gouy phase dif-ference equal to zero

⎡⎣⎢⎢

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥⎥

--

-=

z

z l

z

z l

z

d

darctan arctan 0. B.2x

x

y

y0 0( )

After some algebra, the z position of the maximum of theGouy phase difference can be expressed by either of thefollowing two solutions,

⎡

⎣⎢⎢

⎛⎝⎜

⎞⎠⎟

⎤

⎦⎥⎥

⎡

⎣⎢⎢

⎛⎝⎜

⎞⎠⎟

⎤

⎦⎥⎥

= +-

-- + +

-

-

= --

-

´ - + +-

-

+

-

z l zl l

z z

z

z

z z

l l

z l zl l

z z

z

z

z z

l l

1 1

1 1 .

B.3

y yx y

x y

x

y

x y

x y

x xx y

y x

y

x

y x

x y

00 0

0

0

0 02

2

00 0

0

0

0 02

2

( )( )

( )( )

( )

By setting the operand in equation (B.2) equal to π/2 and bytaking the z positions to be those of equation (B.3), the dif-ference in effective focal lengths between the x and y

9

J. Opt. 19 (2017) 015607 J Strohaber et al

directions is given by the simple expression

- =l l z z2 . B.4x y x y0 0 ( )

In figure B.1(b) we have plotted the beam spot sizes inthe x (solid blue curve) and y (solid red curve) directions, theassociated Gouy phases and the Gouy phase difference (solidblack curve). Using equation (B.4), we find that the differencein the focal lengths for a π/2 phase difference is fx−fy≈10cm. From figure B.1(a) this corresponds to a lens tilt angle ofabout ∼35°.

References

[1] Beth R A 1936 Mechanical detection and measurement of theangular momentum of light Phys. Rev. 50 115–25

[2] Heckenberg N R, Friese M E J, Nieminen T A andRubinsztein-Dunlop H 1999 Mechanical effects of opticalvortices Optical Vortices (Horizons in World Physics) edM Vasnetsov and K Staliunas vol 228 (New York: NovaScience) pp 75–105

[3] Strohaber J 2013 Frame dragging with optical vortices Gen.Relat. Gravit. 45 2457

[4] Strohaber J, Zhi M, Sokolov A V, Kolomenskii A A,Paulus G G and Schuessler H A 2012 Coherent transfer ofoptical orbital angular momentum in multi-order Ramansideband generation Opt. Lett. 37 3411

[5] Hansinger P, Maleshkov G, Garanovich I L, Skryabin D V,Neshev D N, Dreischuh A and Paulus G G 2014 Vortexalgebra by multiply cascaded four-wave mixing offemtosecond optical beams Opt. Express 22 11079

[6] Gariepy G, Leach J, Kim K T, Hammond T J, Frumker E,Boyd R W and Corkum P B 2014 Creating high-harmonicbeams with controlled orbital angular momentum Phys. Rev.Lett. 113 153901

[7] Strohaber J, Abul J, Richardson M, Zhu F,Kolomenskii A A and Schuessler H A 2015 Cascade Ramansideband generation and orbital angular momentum relationsfor paraxial beam modes Opt. Express 23 22463

[8] Ballantine K E, Donegan J F and Eastham P R 2016 There aremany ways to spin a photon: Half-quantization of a totaloptical angular momentum Sci. Adv. 2 e1501748

[9] Schmiegelow C T, Schulz J, Kaufmann H, Ruster T,Poschinger U G and Schmidt-Kaler F 2016 Transfer ofoptical orbital angular momentum to a bound electronNature Comm. 7 12998

[10] Watson G N 2008 A Treatise On The Theory of BesselFunctions (Merchant Books)

[11] Berry M V 2004 Optical vortices evolving from helicoidalinteger and fractional phase steps J. Opt. A: Pure Appl. Opt.6 259

[12] Strohaber J, Kaya G, Kaya N, Hart N, Kolomenskii A A,Paulus G G and Schuessler H A 2011 In situ tomography offemtosecond optical beams with a holographic knife-edgeOpt. Express 19 143321

[13] Forbes A, Dudley A and McLaren M 2016 Creation anddetection of optical modes with spatial light modulators Adv.Opt. Photonics 8 200

[14] Kaminskii A A et al 1999 Properties of Nd3+-doped andundoped tetragonal PbWO4, NaY(WO4)2, CaWO4, andundoped monoclinic ZnWO4 and CdWO4 as laser-active andstimulated Raman scattering-active crystals Appl. Opt.38 4533

[15] Kaminskiia A A et al 2000 High efficiency nanosecondRaman lasers based on tetragonal PbWO4 crystals 183277–87

[16] Massey G A and Siegmann A E 1969 Reflection and refractionof Gaussian light beams at tilted elipsoidal surfaces.’ Appl.Opt. 5 975

[17] Yariv A 1989 Quantum Electronics (New York: Wiley)[18] Ricci F, Löffler W and van Exter M P 2012 Instability of

higher-order optical vortices analyzed with a multipinholeinterferometer Opt. Express 20 22961–75

Figure B.1. (a) The effective focal lengths along the x (solid blue line) and y (solid red line) axis from a lens tilted by an angle of q around they-axis. (b) Graph showing the beam spot size and Gouy phase along the x (solid blue line) and y (solid red line) directions. The solid blackline is the difference between the Gouy phases.

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J. Opt. 19 (2017) 015607 J Strohaber et al