Abstract

Light beams carrying orbital angular momentum are key resources in modern photonics. In many applications, the ability to measure the complex spectrum of structured light beams in terms of these fundamental modes is crucial. Here we propose and experimentally validate a simple method that achieves this goal by digital analysis of the interference pattern formed by the light beam and a reference field. Our approach allows one to also characterize the beam radial distribution, hence retrieving the entire information contained in the optical field. Setup simplicity and reduced number of measurements could make this approach practical and convenient for the characterization of structured light fields.

© 2017 Optical Society of America

Full Article  |  PDF Article
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References

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    [Crossref]
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    [Crossref]
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2017 (6)

F. Bouchard, R. Fickler, R. W. Boyd, and E. Karimi, “High-dimensional quantum cloning and applications to quantum hacking,” Sci. Adv. 3, e1601915 (2017).
[Crossref]

F. Cardano, A. D’Errico, A. Dauphin, M. Maffei, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, L. Marrucci, M. Lewenstein, and P. Massignan, “Detection of Zak phases and topological invariants in a chiral quantum walk of twisted photons,” Nat. Commun. 8, 15516 (2017).
[Crossref]

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2017).
[Crossref]

R. Fickler, M. Ginoya, and R. W. Boyd, “Custom-tailored spatial mode sorting by controlled random scattering,” Phys. Rev. B 95, 161108 (2017).
[Crossref]

P. Zhao, S. Li, X. Feng, K. Cui, F. Liu, W. Zhang, and Y. Huang, “Measuring the complex orbital angular momentum spectrum of light with a mode-matching method,” Opt. Lett. 42, 1080–1083 (2017).
[Crossref]

G. Vallone, “Role of beam waist in Laguerre–Gauss expansion of vortex beams,” Opt. Lett. 42, 1097–1100 (2017).
[Crossref]

2016 (6)

A. Forbes, A. Dudley, and M. McLaren, “Creation and detection of optical modes with spatial light modulators,” Adv. Opt. Photon. 8, 200–227 (2016).
[Crossref]

M. Mirhosseini, O. S. Magaña-Loaiza, S. Omar, C. Chen, S. M. Hashemi Rafsanjani, and R. W. Boyd, “Wigner distribution of twisted photons,” Phys. Rev. Lett. 116, 1–6 (2016).
[Crossref]

H.-L. Zhou, D.-Z. Fu, J.-J. Dong, P. Zhang, D.-X. Chen, X.-L. Cai, F.-L. Li, and X.-L. Zhang, “Orbital angular momentum complex spectrum analyzer for vortex light based on the rotational Doppler effect,” Light Sci. Appl. 6, e16251 (2016).
[Crossref]

M. Malik, M. Erhard, M. Huber, M. Krenn, R. Fickler, and A. Zeilinger, “Multi-photon entanglement in high dimensions,” Nat. Photonics 10, 248–252 (2016).
[Crossref]

B. Hiesmayr, M. de Dood, and W. Löffler, “Observation of four-photon orbital angular momentum entanglement,” Phys. Rev. Lett. 116, 073601 (2016).
[Crossref]

F. Cardano, M. Maffei, F. Massa, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, and L. Marrucci, “Statistical moments of quantum-walk dynamics reveal topological quantum transitions,” Nat. Commun. 7, 11439 (2016).
[Crossref]

2015 (7)

X.-L. Wang, X.-D. Cai, Z.-E. Su, M.-C. Chen, D. Wu, L. Li, N.-L. Liu, C.-Y. Lu, and J.-W. Pan, “Quantum teleportation of multiple degrees of freedom of a single photon,” Nature 518, 516–519 (2015).
[Crossref]

K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin-orbit interactions of light,” Nat. Photonics 9, 796–808 (2015).
[Crossref]

F. Cardano and L. Marrucci, “Spin-orbit photonics,” Nat. Photonics 9, 776–778 (2015).
[Crossref]

M. Mirhosseini, O. S. Magaña-Loaiza, M. N. O’sullivan, B. Rodenburg, M. Malik, M. P. J. Lavery, M. J. Padgett, D. J. Gauthier, and R. W. Boyd, “High-dimensional quantum cryptography with twisted light,” New J. Phys. 17, 033033 (2015).
[Crossref]

V. Grillo, G. C. Gazzadi, E. Mafakheri, S. Frabboni, E. Karimi, and R. W. Boyd, “Holographic generation of highly twisted electron beams,” Phys. Rev. Lett. 114, 034801 (2015).
[Crossref]

B. Piccirillo, S. Slussarenko, L. Marrucci, and E. Santamato, “Directly measuring mean and variance of infinite-spectrum observables such as the photon orbital angular momentum,” Nat. Commun. 6, 8606 (2015).
[Crossref]

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photon. 7, 66–106 (2015).
[Crossref]

2014 (2)

M. Malik, M. Mirhosseini, M. P. J. Lavery, J. Leach, M. J. Padgett, and R. W. Boyd, “Direct measurement of a 27-dimensional orbital-angular-momentum state vector,” Nat. Commun. 5, 3115 (2014).
[Crossref]

G. Vallone, V. D’Ambrosio, A. Sponselli, S. Slussarenko, L. Marrucci, F. Sciarrino, and P. Villoresi, “Free-space quantum key distribution by rotation-invariant twisted photons,” Phys. Rev. Lett. 113, 060503 (2014).
[Crossref]

2013 (3)

M. Mirhosseini, M. Malik, Z. Shi, and R. W. Boyd, “Efficient separation of the orbital angular momentum eigenstates of light,” Nat. Commun. 4, 1–6 (2013).
[Crossref]

P. Bierdz, M. Kwon, C. Roncaioli, and H. Deng, “High fidelity detection of the orbital angular momentum of light by time mapping,” New J. Phys. 15, 113062 (2013).
[Crossref]

C. Schulze, A. Dudley, D. Flamm, M. Duparré, and A. Forbes, “Measurement of the orbital angular momentum density of light by modal decomposition,” New J. Phys. 15, 073025 (2013).
[Crossref]

2012 (3)

M. Mazilu, A. Mourka, T. Vettenburg, E. M. Wright, and K. Dholakia, “Simultaneous determination of the constituent azimuthal and radial mode indices for light fields possessing orbital angular momentum,” Appl. Phys. Lett. 100, 231115 (2012).
[Crossref]

E. Karimi, L. Marrucci, C. de Lisio, and E. Santamato, “Time-division multiplexing of the orbital angular momentum of light,” Opt. Lett. 37, 127–129 (2012).
[Crossref]

C. Schulze, S. Ngcobo, M. Duparré, and A. Forbes, “Modal decomposition without a priori scale information,” Opt. Express 20, 27866–27873 (2012).
[Crossref]

2011 (8)

A. Mourka, J. Baumgartl, C. Shanor, K. Dholakia, and E. M. Wright, “Visualization of the birth of an optical vortex using diffraction from a triangular aperture,” Opt. Express 19, 5760–5771 (2011).
[Crossref]

G. C. G. Berkhout, M. P. J. Lavery, M. J. Padgett, and M. W. Beijersbergen, “Measuring orbital angular momentum superpositions of light by mode transformation,” Opt. Lett. 36, 1863–1865 (2011).
[Crossref]

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

Q. S. Ferreira, A. J. Jesus-Silva, E. J. S. Fonseca, and J. M. Hickmann, “Fraunhofer diffraction of light with orbital angular momentum by a slit,” Opt. Lett. 36, 3106–3108 (2011).
[Crossref]

A. F. Abouraddy, T. M. Yarnall, and B. E. A. Saleh, “Angular and radial mode analyzer for optical beams,” Opt. Lett. 36, 4683–4685 (2011).
[Crossref]

M. P. J. Lavery, A. Dudley, A. Forbes, J. Courtial, and M. J. Padgett, “Robust interferometer for the routing of light beams carrying orbital angular momentum,” New J. Phys. 13, 093014 (2011).
[Crossref]

M. T. Gruneisen, R. C. Dymale, K. E. Stoltenberg, and N. Steinhoff, “Optical vortex discrimination with a transmission volume hologram,” New J. Phys. 13, 083030 (2011).
[Crossref]

L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo, E. Santamato, E. Nagali, and F. Sciarrino, “Spin-to-orbital conversion of the angular momentum of light and its classical and quantum applications,” J. Opt. 13, 064001 (2011).
[Crossref]

2010 (4)

B. Piccirillo, V. D’Ambrosio, S. Slussarenko, L. Marrucci, and E. Santamato, “Photon spin-to-orbital angular momentum conversion via an electrically tunable q-plate,” Appl. Phys. Lett. 97, 241104 (2010).
[Crossref]

G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. 105, 153601 (2010).
[Crossref]

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chávez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using light’s orbital angular momentum,” Phys. Rev. Lett. 105, 053904 (2010).
[Crossref]

S. Slussarenko, V. D’Ambrosio, B. Piccirillo, L. Marrucci, and E. Santamato, “The Polarizing Sagnac Interferometer: a tool for light orbital angular momentum sorting and spin-orbit photon processing,” Opt. Express 18, 27205–27216 (2010).
[Crossref]

2009 (3)

2008 (1)

G. C. G. Berkhout and M. W. Beijersbergen, “Method for probing the orbital angular momentum of optical vortices in electromagnetic waves from astronomical objects,” Phys. Rev. Lett. 101, 100801 (2008).
[Crossref]

2007 (1)

2006 (2)

H. I. Sztul and R. R. Alfano, “Double-slit interference with Laguerre–Gaussian beams,” Opt. Lett. 31, 999–1001 (2006).
[Crossref]

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[Crossref]

2003 (1)

2002 (2)

L. Gatteschi, “Asymptotics and bounds for the zeros of Laguerre polynomials: a survey,” J. Comput. Appl. Math. 144, 7–27 (2002).
[Crossref]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
[Crossref]

2001 (1)

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

1998 (1)

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, “Rotational frequency shift of a light beam,” Phys. Rev. Lett. 81, 4828–4830 (1998).
[Crossref]

1997 (1)

1996 (1)

M. Padgett, J. Arlt, N. Simpson, and L. Allen, “An experiment to observe the intensity and phase structure of Laguerre–Gaussian laser modes,” Am. J. Phys. 64, 77–82 (1996).
[Crossref]

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]

1992 (2)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref]

M. E. H. Ismail and X. Li, “Bound on the extreme zeros of orthogonal polynomials,” Proc. Am. Math. Soc. 115, 131–140 (1992).
[Crossref]

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F. Bouchard, R. Fickler, R. W. Boyd, and E. Karimi, “High-dimensional quantum cloning and applications to quantum hacking,” Sci. Adv. 3, e1601915 (2017).
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Figures (6)

Fig. 1.
Fig. 1.

Sketch of the experimental apparatus. (a) A He–Ne laser beam passes through a polarizer (P) and is spatially cleaned and collimated by means of an objective (Ob), a pinhole (ph) and a lens (L). A half-wave plate (HWP) and a polarizing beam splitter (PBS) are used in order to split the beam into the signal and reference arms, whose relative intensities can be controlled by HWP rotation. Fields resulting from a complex superposition of multiple helical modes were obtained by using q-plates and quarter-wave plates (QWPs), as shown in panels (b) and (c). After preparing the signal field, we place a further sequence of a QWP and a q-plate in case we need to shift the entire OAM spectrum. The reference field is a TEM0,0 Gaussian mode. In the upper arm of the interferometer, by orienting the QWP at 0 or 90° with respect to the beam polarization we can introduce a α=0 or π/2 phase delay between the signal and the reference field, respectively. The two beams are superimposed at the exit of a beam splitter (BS) and filtered through a polarizer, so that they share the same polarization state. The emerging intensity pattern is recorded on a CCD camera (with resolution 576×668). (b) A QWP oriented at 45° or 0, followed by q-plate with q=4 and δ=π or δ=π/2, is used for the generation of a light beam containing a single mode (m=8) or three modes (m=8,0,8), respectively. (c) two q-plates with q=1 and q=1/2 are aligned to generate spectra with m[3,3]. (d) A set of more complex distributions was obtained by displacing laterally the centre of a q-plate (q=1 and δ=π) with respect to the axis of the impinging Gaussian beam.

Fig. 2.
Fig. 2.

Experimental reconstruction of light OAM spectrum. We report the experimental characterization of optical fields containing one (a-b-c) and three (d-e-f) helical modes, generated using a q-plate with q=4 and δ=π or π/2, respectively. In panels (g)–(j) we report the experimental intensity patterns Iref, Is, I0 and Iπ/2, respectively, obtained when investigating the field generated by the q-plate with δ=π/2. Panels (a) and (d) show the OAM distributions in the two cases. Error bars are calculated as three times the standard error. Panels (b), (c) and (e), (f) show the measured amplitude and phase profiles of the non-vanishing helical modes that are present in the beam, where blue, red and green colored points are associated with modes with m=0,8,8, respectively. These results are compared with theoretical simulations, represented as continuous curves with the same color scheme adopted for the experimental results. For each value of m, we plot normalized coefficients c˜m=cm/Sm, where Sm is the total power associated with the helical mode. As expected from theory, a fraction of the beam is left in the fundamental Gaussian state, while an equal amount of light is converted into helical modes with m=±8, both having the radial profile of a HyGG8,8 mode. Simulated profiles of Gaussian and HyGG modes correspond to w0=1.45  mm and z=30  cm, the latter being the distance between the q-plate and the camera. Error bars are smaller than experimental points.

Fig. 3.
Fig. 3.

Complete spatial mode decomposition in terms of LG beams. We consider the light beam emerging from a q-plate with (q=4, δ=π), described by a HyGG8,8 mode [50]. We evaluate the overlap integral between the radial envelope c8(r) measured in our experiment at z=30  cm and LGp,8 modes at the same value of z and characterized by the optimal beam waist w˜0=w0/3 [51], where w0 is the input beam waist. In panels (a) and (b) we plot the squared modulus and the phase of the resulting coefficients (blue markers), respectively, showing a good agreement with the values obtained from numerical simulations (red markers). The phases of bp,8 with p equal to odd integers are absent in the plot since the corresponding amplitudes are consistent with zero, that is smaller than the associated statistical uncertainties.

Fig. 4.
Fig. 4.

Measure of shifted OAM power spectrum. OAM probability distributions are measured for two different optical fields, obtained when shining a sequence of two q-plates with q1=1 and q2=1/2 with horizontally polarized light. A further q-plate with q=4 shifts the final spectrum by M=8 units. (a) OAM spectrum for the case δ1=π and δ2=π. (b) The same data are reported for a different field, obtained when δ1=π and δ2=π/2. Error bars represent the standard error multiplied by three.

Fig. 5.
Fig. 5.

OAM spectrum for a shifted q-plate. We measure the OAM power spectrum at the exit of a q-plate (q=1, δ=π) shifted with respect to the axis of the impinging Gaussian beam, which is left-circularly polarized. The overall spectrum is shifted by M=8 units since we are using a further q-plate with q=4 and δ=π. However, we plot the original OAM distribution associated with the signal field. (a)–(f) Experimental (green) and simulated (red) OAM power spectra when the lateral shift is equal to aΔx, with a=1, 3, 6, 9, 12, 15 and Δx=0.125mm, respectively. Error bars represent the standard error multiplied by three. (g)–(i) Examples of the experimental intensity pattern I0 used for determining the power spectra reported in panels (a), (c), (e). The number of azimuthal fringes reveals that the OAM spectrum has been shifted. (j) and (k) First and second moment (m and m2) measured as a function of the lateral displacement. Error bars are not visible because smaller than the experimental points.

Fig. 6.
Fig. 6.

Detectable LG modes. Different colors, as reported in the legend, indicate whether a specific LGp,m, in a transverse plane z=30  cm and with a beam waist w0=0.16  mm [w(z=30  cm)0.4  mm] can be resolved in our setup. These parameters correspond to the ones used for the complete spatial decomposition of the HyGG beam generated by a q-plate (q=4, δ=π) in terms of LG beams.

Equations (20)

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Es(r,ϕ,z,t)=As(r,ϕ,z)ei(ωtkz),
As(r,ϕ,z)=m=K1K2cm(r,z)eimϕ,
cm(r,z)=12π02πdϕeimϕAs(r,ϕ,z).
P(m)=1S0drr|cm(r,z)|2,
As(r,ϕ,z)=p=P1P2m=K1K2bp,mLGp,m(r,ϕ,z),
bp,m=0rdrLGp,m*(r,z)cm(r,z),
I=Is+Iref+I˜α.
I˜α=2m|Aref||cm|cos[mϕ+α+βm],
I˜0iI˜π/2=2m|Aref||cm|ei[mϕ+βm].
cm(r,z)=14πAref*(r,z)02πdϕ(I˜0iI˜π/2)eimϕ,
LGp,m(ρ,ζ,ϕ)=2|m|+1p!π(p+|m|)!(1+ζ2)(ρ1+ζ2)|m|×eρ21+ζ2Lp|m|(2ρ2/(1+ζ2))×eiρ2ζ+1/ζeimϕi(2p+|m|+1)arctan(ζ),
HyGGp,m(ρ,ζ,ϕ)=21+|m|+pπΓ(1+|m|+p)Γ(1+|m|+p/2)Γ(1+|m|)×i|m|+1ζp/2(ζ+i)(1+|m|+p/2)×ρ|m|eiρ2/(ζ+i)+imϕ×F11(p/2;|m|+1;ρ2/(ζ(ζ+i))),
Q^(δ)=cos(δ2)(1001)+isin(δ2)(0eimϕeimϕ0),
rmax=d*N/2,
rmin=md/π,
r1=w(z){2p+m2[1+4(p1)(p+m1)]1/22}1/2,
rp=w(z){2p+m2+[1+4(p1)(p+m1)]1/22}1/2,
r˜p=w(z){2p+m+1}1/2.
{rmin<r1(i)rp<rmax(ii)Λ>2d(iii).
p<(N2d24w2(z)m1)/2.