of the 2p time delay. Indeed, in this energy region,
ionization toward the s continuum is much lower than toward the d continuum, which may
justify the approximation of t1(2p) as the Wigner
delay for the d channel but not on the level of a
few attoseconds. Figure 4B shows the same
quantities for 2s ionization. Here, the difference
between t1(2s) and t W(2s) is not visible, which
justifies the interpretation of t1(2s) as Wigner
delay. The colored dots and square have been
obtained by subtracting from the experimental data (see Fig. 1A) the calculated tA(2p) for
the angle-integrated case and the continuum-continuum contribution tcc(2s), thus extracting
the absolute Wigner delays for 2s ionization for
the first time. The 2s Wigner time delay does
not vary with energy in this region and is approximately equal to –3 as. Other calculations
of the 2s Wigner delays (20, 34) agree well with
our experimental result. We stress that the increase with excitation energy observed in Fig. 1A
reflects essentially the energy dependence of
tcc(2s) − tcc(2p), which itself is dominated by the
variation of tcc(2s).
Solving a 7-year-old puzzle
We compared our theoretical results with calculations in the conditions of an attosecond
streak-camera experiment—i.e., with a stronger
IR field, a single attosecond pulse, and angle-resolved detection (20)—obtaining agreement
for the Wigner delays, the tcc, and time delay
differences within a few attoseconds. Then, we
carried out an energy-integrated instead of energy-resolved analysis of the sideband oscillations and
obtained time delay differences that were, in general, below those indicated in Fig. 1A, close to the
value retrieved in (7). This leads us to suggest that
the discrepancy of the experimental result (7) with
theory (19–21) might be due to the influence of
shake-up processes, not resolved and accounted
for in the analysis of the experiment (35).
The good agreement obtained between our
experimental data and numerical calculations
of the photoionization time delays from the 2s
and 2p shells in neon for photon energies ranging
from 65 up to 100 eV gives us confidence in this
type of measurement for other, more complex,
atomic or molecular systems (36, 37) and points
out the necessity for keeping high-frequency resolution in addition to high temporal resolution.
Our method can be improved by using longer laser pulses, allowing for the generation of trains
with reproducible attosecond pulses (high temporal resolution) and narrow harmonic bandwidth (high spectral resolution), in combination
with a highly stable optical interferometer and
an electron spectrometer with high resolving
power. The use of lasers in the mid-IR region for
harmonic generation (38) will have several other
advantages: broader XUV spectra (39), better energy sampling, and access to time scales of a few
femtoseconds, which will be of great interest for
photochemistry applications. Our approach provides the means to study and control photo-induced processes both in the time and frequency
domain from the visible to the x-ray range.
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A.L. thanks J. Ye for fruitful discussions. This research was
supported by the European Research Council (Advanced Grant
PALP 339253), the Swedish Research Council (grant no. 2013-
8185), and the Knut and Alice Wallenberg Foundation. J.M.D. was
funded by the Swedish Research Council, grant no. 2014-3724.
E.L. was funded by the Swedish Research Council, grant no. 2016-
03789. The authors declare that they have no competing interests.
All data and analysis details presented in this work are available
upon request to M.I.
Materials and Methods
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22 August 2017; accepted 19 October 2017
Published online 2 November 2017
Arbitrary spin-to–orbital angular
momentum conversion of light
Robert C. Devlin,1 Antonio Ambrosio,1,2,3,4 Noah A. Rubin,1*
J. P. Balthasar Mueller,1 Federico Capasso1†
Optical elements that convert the spin angular momentum (SAM) of light into vortex
beams have found applications in classical and quantum optics. These elements—SAM-to–
orbital angular momentum (OAM) converters—are based on the geometric phase and
only permit the conversion of left- and right-circular polarizations (spin states) into states
with opposite OAM. We present a method for converting arbitrary SAM states into total
angular momentum states characterized by a superposition of independent OAM. We
designed a metasurface that converts left- and right-circular polarizations into states with
independent values of OAM and designed another device that performs this operation
for elliptically polarized states. These results illustrate a general material-mediated
connection between SAM and OAM of light and may find applications in producing complex
structured light and in optical communication.
Circularly polarized light carries spin angu- lar momentum (SAM) (1) of ±ℏ per pho- ton depending on the handedness (where ℏ is Planck’s constant h divided by 2p), whereas the most general state of polar-
ization (elliptical) is a superposition of the two.
Electromagnetic fields with an azimuthal phase
dependenceexpði‘fÞ, where ‘ is any integer, also
carry orbital angular momentum (OAM) (2). Such
beams (optical vortices) exhibit ‘ multiples of 2p
in phase around the azimuth with an undefined
phase at the center, resulting in a “doughnut”
intensity profile (3). In contrast to SAM, the OAM
can take any integer value in [–1, +1] correspond-
ing to ‘ℏ OAM per photon (4). A paraxial, circularly
polarized helical beam has been shown to carry