The Angular Momentum of Topologically Structured Darkness

Samuel N. Alperin and Mark E. Siemens∗

Department of Physics & Astronomy, University of Denver, Denver, Colorado, USA(Dated: August 12, 2018)

We theoretically analyze and experimentally measure the extrinsic angular momentum contribu-tion of topologically structured darkness found within fractional vortex beams, and show that thisstructured darkness can be explained by evanescent waves at phase discontinuities in the generatingoptic. We also demonstrate the first direct measurement of the intrinsic orbital angular momen-tum of light with both intrinsic and extrinsic angular momentum, and explain why the total orbitalangular momenta of fractional vortices do not match the winding number of their generating phases.

Though it is well known that the orbital angular mo-mentum (OAM) of light is a physical momentum thatcan be used to apply mechanical torque to small par-ticles [1, 2], its classification as an intrinsic property oflight remains a subject of discussion [3–5]. A clearly in-trinsic property such as photon spin is independent of thetransverse position of the calculation axis, but the photonOAM is more complicated. Zambrini et al. showed thatwhile the average OAM of a beam is invariant under ar-bitrary linear translations, its OAM spectrum is not [4].This led those authors to deem the OAM a quasi-intrinsicproperty. It has also been stated that the angular mo-mentum of a beam is intrinsic if and only if there existsa choice of axis such that the mode has no net transversemomentum [3, 5]. An extension of these conclusions isthat to understand the natures of the intrinsic and ex-trinsic OAM, we must study a system in which both arepresent. In this letter we consider what is arguably themost natural of such systems: fractional vortex beams.

It is well known that passing a Gaussian beam througha 2πm helical phase optic imparts the beam with anOAM of m~ for any m ∈ Z. However it has only recentlycome to light that the OAM of fractional vortex beams ismore subtle: the mode formed by Gaussian illuminationof a phase ramp of topological charge µ ∈ R does not ingeneral have an average OAM of µ~ per photon, as shownin Fig. 1.a. Though the expectation value of the OAM ofthese fractional vortex modes has been calculated [6] andexperimentally verified [7, 8], there has been no physicalexplanation for the discrepancy between the topologicalcharge of the generating phase structure and the OAMof the resulting mode.

In this letter we show that the intrinsic OAM of a frac-tional vortex beam can be directly measured, and that itmatches the topological charge of the generating phase,even for non-integer µ. This suggests that the fundamen-tal action of the 2πµ spiral phase applied to a beam isnot to impart the beam with µ~ of total OAM, but toimpart the beam with µ~ of intrinsic OAM. This leads usto investigate the form and physical meaning of the otherOAM component, that of extrinsic angular momentum.We do this by separating fractional vortex modes into co-

∗ msiemens@du.edu

FIG. 1. (a) Plot of average OAM as a function of generatingspiral phase step height µ, as drawn in (b). Three coloredpoints along the curve in (a) correspond to the modes labeledin matching colors in (c). (c) Shows various theoretical frac-tional vortices for three different µ. Intensity is representedby brightness, and phase is represented by color. The arrowsshow the flow of transverse Poynting vectors, which are withina scaling factor of local OAM per photon.

herent sums of intrinsic and extrinsic components, whichin the case of fractional OAM modes means separatinginto light and dark parts.

We start with a description of our direct optical mea-surement of the intrinsic OAM of light. In a prior workthe authors demonstrated a technique for the quantita-tive measurement of total OAM by transforming the lightwith a cylindrical lens, recording an image with a cam-era at its focus, and calculating high-order moments [8].This OAM measurement technique exploits the fact thatat the focal plane of a cylindrical lens, a camera recordsthe intensity of the 1D Fourier transform of a mode in-cident on the lens. From that image the OAM can becalculated as

`meas =4π

fλ

∫∫∞−∞ I(x′, y)`x

′ y dx′ dy∫∫∞−∞ I(x′, y)` dx′ dy

. (1)

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where I(x′, y) is the spatially-resolved light intensity atthe focal plane of the lens [8]. However, for modes with-out cylindrical symmetry in the transverse plane, thecylindrical lens orientation with respect to the mode un-der test affects which OAM components are measured.For fractional modes, there is mirror symmetry in thetransverse plane, along the line of the lateral discontinu-ity, and to measure the total OAM, the cylindrical lensmust be oriented at π

4 with respect to the line of symme-try [8].

Any extrinsic component to the OAM can be writ-ten as a net linear transverse momentum, and thus theremust be exactly one orientation of the cylindrical lens forwhich all of the extrinsic component is not counted in themeasurement. As the intrinsic OAM is all that is left tobe measured and is by definition invariant on measure-ment orientation, there exists exactly one orientation ofthe cylindrical lens for which the intrinsic OAM can bemeasured directly. As it is known that upon propagation

the line of symmetry moves in the φ direction, we choosethe orientation of the cylindrical lens to be oriented π

2radians from the orientation of the discontinuity, so asto zero the measurement to extrinsic angular momentumby mapping the transverse extrinsic momentum contri-bution onto the optical axis in the Fourier domain.

The result of our intrinsic OAM measurement is shownin blue in Fig. 2. This is the first direct measurement ofintrinsic OAM in the presence of extrinsic OAM, and itconfirms the nontrivial claim that the intrinsic OAM offractional vortex modes goes as µ~. The extrinsic OAMcan be calculated as the difference between the total andthe intrinsic OAM, the result of which is shown in blackin Fig. 2.

Having shown that the intrinsic OAM of fractional vor-tex beams matches the topological charges of their gen-erating phases, we now set out to understand the opticalstructure of these beams in connection with intrinsic andextrinsic OAM contributions.

We start by constructing the form of the fractional vor-tex mode. We can write the general form of a fractionalvortex in terms of LG modes

ψµ =

∞∑l=−∞

C`

∞∑p=0

Cp(`)LG`p (2)

such that [6]

C` =i(1− e2iπ(µ−bµc)

)eiθ(µ−l)−iθ(µ−bµc)

2π(µ− l)(3)

in which θ represents the orientation angle of φ = 0, andthus specifies the direction of the lateral phase disconti-nuity in the final structure. For each ` in the sum, thereis a hypergeometric response in the radial structure, asin Karimi et al [9]. Restricting the mathematical expres-sion for the hypergeometric response to physical cases, itcan be simplified so that

-1 0 1μ

-1

0

1

OAM

Total OAM measurement

Intrinsic OAM measurement

Extrinsic OAM

FIG. 2. Blue: Direct experimental measurement of the in-trinsic orbital angular momentum of fractional vortex beamsgenerated by non-integer spiral phases of topological chargeµ. The blue inset shows the experimental alignment: a frac-tional vortex beam is input into a cylindrical lens such thatthe vortex chain (green dots) is perpendicular to the axis ofthe lens; in this case, the vortices in the chain are all focusedonto the optical axis in the Fourier plane (green rectangle) andextrinsic OAM contributions are eliminated. The measuredtotal OAM of the same fractional vortex modes is shown inred, achieved by a 45◦ rotation of the input mode, as shownin the red inset. The extrinsic OAM (black) is then calculatedas the difference between the total OAM and intrinsic OAM.

Cp(`) =`Γ(`2 + p

)2√p!(`+ p)!

. (4)

As can be seen in Fig. 1.c, the transverse momentaof the modes have a great deal of interesting structure.Gbur elegantly showed that as µ goes continuously fromone integer to the next, a radial string of unit vortices ofalternating charge forms until µ is a half integer and thestring extends infinitely from the center of the mode [10].As µ continues to increase, the chain collapses towardsinfinity, leaving a single additional vortex which movesinto the center. The vortex chain can be seen in Fig. 3,and the movement of the ‘additional’ vortex can be seenin Fig. 1.c: At µ = 1.25 there is a clearly defined, local2π phase wrap near the mode center, while at µ = 1.50the core opens up to accept the ‘new’ vortex left by theinfinite chain, and at µ = 1.75 there is a well defined corephase wrap of 4π.

Fig. 3 compares theoretical and experimental fields forψ4.4, demonstrating that our model matches experimen-tal observations of both the intensity and phase of frac-

3

FIG. 3. Intensity and phase of theoretical (top) and exper-imental (bottom) realizations of fractional vortex mode ψ4.4

at a distance of 120ZR from the generating optic. Notably, the

vortex chain (a string of unit vortices of alternating sign) isresolved in both theoretical and experimental phase images.

tional vortex modes. In our measurements, phase was ex-perimentally measured using phase stepping holography[11, 12]. To experimentally generate fractional vortexbeams, we pass a laser with a Gaussian profile through aspatial light modulator encoded with a fractional spiralphase hologram, which takes the form of a forked grat-ing with a lateral discontinuity, and take images with aCCD a distance of 1

20ZR beyond the modulator. The ex-perimental results accurately resolve the dominant phasestructure and the lateral stripe from the linear phase dis-continuity in the generating phase, which is composedof a line of unit vortices of alternating sign that extendsoutward from near the center of the mode [10].

Having built and experimentally confirmed the struc-ture of fractional vortex beams ψµ, we now separate theminto their intrinsic and extrinsic parts, in keeping withour measurements in Fig. 2. We define the intrinsiccomponent of ψµ as

ψiµ(r, φ) = |ψµ(r, θ + π)|eiµφ. (5)

This intrinsic component has a phase defined by the non-integer phase wrap of the generating optic, and its am-plitude is a circularly symmetric wrap of the radial func-tion of ψµ at the angle opposite the discontinuity, wherethere are no effects of the discontinuity. We are carefulnot to call this theoretical object a mode, as it is not asolution to the wave equation and cannot propagate in-dependently. However we find it conceptually useful andit is essential for the development of our understanding

FIG. 4. From left to right, theoretical (top row) and experi-

mental (bottom row) amplitude and phase of ψ1.5, ψi1.5, and

ψd1.5. Local transverse Poynting vectors are also shown as

white arrows in plots of ψd1.5, which demonstrate that thedominant transverse momentum is made of off axis linearmomenta pointing in opposite directions, suggesting no netextrinsic contribution, which is consistent with observationsin Fig. 1.

of its extrinsic counterpart. The intrinsic component ψiµpossesses uniform average OAM of µ~ per photon. Inorder to describe the complete optical mode ψµ, we in-troduce a second component, that of structured darkness,as

ψdµ = ψµ − ψiµ. (6)

We can evaluate the complex form of ψdµ numericallyfor any µ. We can also perform the same operation onexperimentally measured vortex beams via phase recon-struction. Fig. 4 demonstrates the match between thetheory and experiment in the experimentally viable caseof non-zero propagation from the generating phase optic.Both the numerical and experimental plots of the struc-tured darkness associated with ψ1.5 demonstrate thatnearly all measurable transverse momentum takes theform of off axis, linear momentum which cancel and thusform a net zero extrinsic contribution. This is consistentwith our result shown in Fig. 2: the extrinsic angularmomentum is zero for half-integer µ. Similar analysis ofmodes with other values of µ were also consistent withFig. 2.

We now discuss a physical interpretation of structureddarkness. It has been shown that the dark singularityat the center of an integer vortex beam, generated bythe Gaussian excitation of a spiral phase, is the result ofdepletion of the Gaussian into non-propagating evanes-cent waves wherever the propagation angle of component

4

FIG. 5. (Top) Far field amplitude and phase of fractional vor-tex beams, calculated as the propagating part of beams withGaussian amplitude and fractional 2πµ spiral phase. Thisshows that the dark stripe seen in any fractional vortex beamrepresents the region where the field was non-propagating.(Bottom) Near field amplitude of beam centers of integer andfractional modes generated as the Gaussian excitations of spi-ral phases, showing that the lateral discontinuity of fractionalbeams comes from depletion into evanescent waves at the sur-face of the generating phase optic. The orientation of the darkstripe rotates by π

2over propagation to the far field.

plane waves is too high to be physical [13]. Similarly, weclaim that the form of the structured darkness of a frac-tional vortex beam at z = 0 represents that of evanescent

waves excited along the lateral phase discontinuity of thegenerating phase optic.

To prove that the structured darkness of fractional vor-tex modes represents the regions depleted by the exci-tation of evanescent waves at the generating phase op-tic, we demonstrate that we can numerically generatea fractional mode by calculating the propagating, non-evanescent part of a mode with Gaussian amplitude andfractional 2πµ spiral phase. This is done by taking theFourier transform of the helically phased Gaussian, ap-plying a circular aperture, and inverse Fourier transform-ing back. In the case that the aperture blocks only non-propagating evanescent waves, we can calculate the nearfield of the propagating field. By selecting a much smalleraperture in k-space, the far field of the mode can be calcu-lated because this is mathematically equivalent to spatialfiltering. Fig. 5.a-b shows the result of such far field cal-culations for µ = 1 and µ = 1.5. Both cases match theexpectation in the far field in both phase and amplitude,without any a priori knowledge of the Laguerre-Gaussiancompositions, or even of the form of any mode other thanthe input Gaussian. Fig. 5.c-d shows the amplitudes ofthe near field results for the same values of µ, zoomedin to the beam center. Even in the near field, there is aclearly defined lateral singularity along the discontinuityof the generating phase. Given that this behavior is ap-parent with only the assumption that evanescent wavesdo not propagate, we conclude that the region of struc-tured darkness seen beyond the near field represents theeffect of evanescent structure at the surface of the phaseoptic on the propagating mode. While this is somewhatcounterintuitive as it is well known that evanescent wavesdo not propagate, it is clear that their structure can in-fluence that of a surrounding, propagating field.

The authors thank A.A. Voitiv for assisting in data ac-quisition, J.M. Knudsen, J.T. Gopinath and R.D. Nieder-riter for helpful discussions, and acknowledge financialsupport from the National Science Foundation (1509733,1553905).

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