Abstract

We examine the geometric phase Doppler effect that appears when a structured light interacts with a rotating structured material. In our scheme the structured light possesses a vortex phase and the structured material works as an inhomogeneous anisotropic plate. We show that the Doppler effect manifests itself as a frequency shift which can be interpreted in terms of a dynamic evolution of Pancharatnam-Berry phase on the hybrid-order Poincaré sphere. The frequency shift induced by the change rate of Pancharatnam-Berry phase with time is derived from both the Jones matrix calculations and the theory of the hybrid-order Poincaré sphere. Unlike the conventional rotational Doppler effect, the frequency shift is proportional to the variation of total angular momentum of light beam, irrespective of the orbital angular momentum of input beams.

© 2017 Optical Society of America

1. Introduction

The rotational Doppler effect refers to the angular velocity between source and observer leading to a frequency shift which is proportional to the total angular momentum per photon [1–5]. In recent years, the rotational Doppler effect has been studied in various contexts such as optical manipulation and detection [6–11]. More recently, a reversed optical force has been detected using a rotational Doppler frequency shift experiment by rotating the inhomogeneous anisotropic waveplate at controlled angular velocity [12, 13]. And the rotational Doppler effect manifests itself as a frequency shift which arises from the nonzero optical radiation torque and can be derived from the law of energy conservation.

In another hand, structured materials have attracted much attention due to the fascinating abilities of controlling light [14]. Numerous works to explore the transmission properties have also been proposed in recent years [15–18]. Structured beams such as the vortex beam with helical wavefronts and the vector beam with a spatially inhomogeneous polarization state [19] have various applications including optical manipulation [20], free-space data transmission [21,22], and high-resolution imaging [23]. Numerous impressive methods to generate the structured beams have been proposed by structured materials, such as nematic liquid crystals [24,25], plasmonic nanostructrues [26, 27], and dielectric metasurfaces [28–31]. In these methods the structured materials works as an inhomogeneous anisotropic waveplate. By designing and tailoring the specified geometry of the nanostructures, one can achieve desired functionalities to control the phase and polarization of light.

In this paper, we investigate the geometric phase Doppler effect when a structured light is incident on a rotating structured material, where the structured light possesses a vortex phase and the structured material works as an inhomogeneous anisotropic plate, as shown in Fig. 1. In the geometric Doppler effect or spatial Doppler effect, it is the spatial rotation rate of a fixed q plate that leads to a spatial frequency shift [32–35]. However, in the geometric phase Doppler effect, not only does the q plate have a spatial rotation rate, but it also rotates with the time, thus resulting in a frequency shift in the time domain. Additionally, note that in conventional rotational Doppler effect, the frequency shifts are observed by rotating homogeneous anisotropic waveplate or/and Dove prism. While in our scheme, a rotating inhomogeneous anisotropic waveplate illuminated by a circularly polarized vortex wave acts as a rotating source. The Jones matrix calculations are employed to obtain the generated frequency shift that can be interpreted in terms of a dynamic evolution of Pancharatnam-Berry phase on the hybrid-order Poincaré sphere [36].

 

Fig. 1 Schematic illustration of the propagation of circular polarized light beams passing through two half-wave q plates, wherein the first one is fixed while the second one rotates with a angular velocity. The red and blue arrows represent the left- and right-handed circularly polarized waves, respectively. And the relative displacements of arrows between the input and the output are resulting from the induced geometric phase.

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2. Evolution of geometric phase in inhomogeneous anisotropic plates

We now consider the evolution of polarization and phase in the structured material. One can use the Jones formalism to fully characterize the optical propagation through the inhomogeneous media [37]:

T(r,φ)=M(α)J(δ)M1(α).
Here, r is the radial coordinate, φ is the azimuthal coordinate. J(δ) is the operator of anisotropy with certain retardation δ:
J=[exp(iδ/2)00exp(iδ/2)].
Let α(x, y) be the angle between optical axis and a fixed reference axis x. The matrix M describing the rotation of local optical axis can be represented as
M(α)=(cosαsinαsinαcosα).
It can be easily proved that the Jones matrix T(r, φ) describing the optical field at each transverse position (r, φ) is the following:
T(r,φ)=cosδ2(1001)+isinδ2(cos2αsin2αsin2αcos2α).

The inhomogeneous birefringent elements having specified geometry can be designated as q plates [24]. The optical axis direction is specified by a space-variant angle

α(r,φ)=qφ+α0,
where q is the topological charge characterizing the spatial rotation rate of optical axis and α0 is a constant angle specifying the initial orientation on the axis x.

As the q plate is an inhomogeneous media, we can thus obtain the manipulation of polarization state and phase by using its effective birefringent nature. Without loss of the generality, we assume that the inhomogeneous anisotropic waveplate is illuminated by a circularly polarized wave

|ψI=22(e^xiσe^y).
The spin angular momentum is σħ with σ = −1 and σ = +1 representing the left- and right-handed circular polarization, respectively [38]. The evolution of the vortex beam in the q plate |ψII〉 = T(r, φ)|ψI〉 can be written as
|ψII=22(e^x+iσe^y)exp[i(lφ2σα0)],
where l = −2σq. The particular phase − 2σα0 is purely geometric in origin, and thereby is the so-called Pancharatnam-Berry geometric phase. The azimuthal phase factor exp(ilφ) is the vortex phase, associated with orbital angular momentum per photon [39]. Compared with the input field, the output field obtains a reversed spin angular momentum and a modified orbit angular momentum given by 2σqħ.

 

Fig. 2 Schematic illustration of the evolution of optical field in the rotational Doppler effect. The q plate rotating uniformly with an angular velocity ω′ is illuminated by a circularly polarized vortex wave with an angular frequency ω. Here, σ = −1 and σ = +1 represent the left- and right-handed circular polarization, respectively. And the output beams in (a) and (b) possess two opposite helical wave fronts with different angular frequencies, ω + Δω and ω − Δω.

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We next consider the field passing through the second rotating q plate, whose structure can be written as

α(r,φ)=q(φβ)+(α0+β),
where β is the angle of rotation. The output beam in the rotating q plate |ψIII〉 = T(r, φ)|ψII〉 can be written as
|ψIII=22(e^xiσe^y)exp[i(mφ2σqβ+2σβ)],
where m = l + 2σq. The Pancharatnam-Berry phase can be written as
γ=2σ(q1)β.
Here, it follows from Eq. (10) that the resulting geometric phase depends on the topological charge of the q plate. However, if the q plate is replaced by a rotating spiral phase plate, the effect would not arise because the spiral phase plate is essentially not a geometric phase element, and it shall not induce the geometric phase of the Doppler effect.

When the Pancharatnam-Berry phase γ evolves linearly in time, we can acquire the frequency shift [see Figs. 2(a) and 2(b)]. The frequency shift of Doppler effect can be written as

Δω=dγdt,
Substituting Eq. (10) into Eq. (11), the frequency shift can be obtained as
Δω=2σω(q1),
where ω′ = dβ/dt is the angular frequency of the rotating q plate. This relationship shows that the frequency shift is proportional to both the variation of total angular momentum of light −2σ(q − 1) and the angular velocity ω′.

3. Evolution of geometric phase on hybrid-order Poincaré sphere

The Doppler effect manifests itself as a frequency shift which can be interpreted in terms of a dynamic evolution of Pancharatnam-Berry phase on the hybrid-order Poincaré sphere. The state evolution in the q plate represented by the hybrid-order Poincaré sphere can be expressed as a two-dimensional Jones vector [36,40]

|ψ(θ,Φ)=cosθ2|Nl+sinθ2|Smexp(iσΦ),
and |Nl=(e^x+iσe^y)exp(ilφ)/2, |Sm=(e^xiσey)exp(imφ)/2. Here, Φ = π/2 − 2α0 for σ = +1 and Φ = π/2 + 2α0 for σ = −1; (θ, Φ) is the latitude and longitude on the sphere. |Nl〉 and |Sm〉 represent two orthogonal circular polarization bases. Indeed, the hybrid-order Poincaré sphere extends the orbital Poincaré sphere [41–43] and high-order Poincaré sphere [44,45] to a more general form. It should be noted that the diffraction inside the q plate has been neglected, so that only the evolution of polarization and phase are taking place, which is accessible when the thickness of the plate is less than the Rayleigh range of the incident beam [46].

Then let us consider the resulting Pancharatnam-Berry phase. This additional phase arising from the cyclic transformation of ψ(R) over a circuit C in parameter space R is specified by

γ(C)=CdSV(R),
where dS = ρ2 sin θdθdΦρ̂ [47]. The Berry curvature is given by
V(R)=R×A.
Here, R=ddρρ^+1ρddθθ^+1ρsinθddΦΦ^ is a gradient in spherical coordinates. The Berry connection can be showed as A = iψ(R)|∇R|ψ(R)〉 [47]. Then, we can obtain the Pancharatnam-Berry phase
γ(C)=ml2σ4Ω.
Here, m represents the orbital angular momentum of the output beam. Ω is the solid angle on the hybrid-order Poincaré sphere enclosed by the circuit C (see the green area in Figs. 3 and 4). Such a phase shift is reminiscent of the one in system of plane wave that equals to half of the solid angle subtended by the closed circuit on the original Poincaré sphere.

 

Fig. 3 Schematic illustration showing realization of the evolution along different longitude lines on the hybrid-order Poincaré sphere. The sphere is assumed with state σ = +1 and l = 0 in the north pole, and the state σ = −1 and m = +1 in the south pole. Insets (t0)–(t3) show the rotating q plate (q = 1/2) with different initial angle α0. Here the sense of the positive rotating angle is chosen as anticlockwise which means ω′ < 0, α0 < 0 corresponding to the longitude line’s moving direction: t0t1t2t3 and polarization states moving from north pole to south pole.

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Fig. 4 Schematic illustration showing realization of the evolution along different longitude lines on the hybrid-order Poincaré sphere. The state on the pole is assumed as the same as Fig. 3. Insets (t0)–(t3) show the rotating q plate (q = 1/2) with different initial angle α0. Here, ω′ > 0, α0 > 0 corresponding to the longitude line’s moving direction: t′0t′1t′2t′3 and polarization states moving from north pole to south pole.

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Substituting m = l + 2σq into Eq. (16), the geometric phase can be obtained as

γ(C)=σ(q1)2Ω.
The vector beam continuous transformations following a closed circuit on the hybrid-order Poincaré sphere. The solid angle Ω is double the longitude Φ of the evolving route, and we have Ω = 4β. Substituting Eq. (17) into Eq. (11), the frequency shift resulting from Pancharatnam-Berry phase can be written as
Δω=2σω(q1).
This result coincides well with the result obtained by the Jones Matrix as shown in Eq. (12). It should be clear that the frequency shift arising in the inhomogeneous anisotropic waveplate is obviously different from conventional Doppler frequency shift, because it is proportional to the variation of total angular momentum of light beam, irrespective of the orbital angular momentum of input beams. The arising frequency shift is spin dependent and thus considered as a “geometric phase Doppler effect”.

Moreover, compared with previous frequency shifts in Refs. [12] and [13], which also arise in the inhomogeneous anisotropic waveplate and are derived from the law of energy conservation, we derived the frequency shift from the theory of the hybrid-order Poincaré sphere. Thus, the final frequency shift can be understood more intuitively via the dynamic evolution of Pancharatnam-Berry phase on the hybrid-order Poincaré sphere.

The evolution of polarization states in the q plate can be described as the transformations of point on the surface along its longitude and latitude. It is known that a homogeneous half waveplate can transform a left-handed circularly polarized light to a right-handed one which can be described as evolution of state from the north pole to south pole along the longitude line on the plane-wave Poincaré sphere, whose longitude depends on the orientation of the optical axis [48]. While on the hybrid-order Poincaré sphere, the incident light beam is evolving in the q plate with corresponding point moving from the north pole to the south pole. The evolving longitude depends on the orientation of the initial angle α0 of q plate. A rotation of α0 would rotate the longitude by an angle 2α0 as shown in Figs. 3 and 4.

In our case, we choose a special value with q = 1/2 for the purpose of illustration. In Fig. 3, we assume the state α0 = 0 at time t0 as the initial state corresponding to the longitude intersecting the axis S1, a 45-degree rotation of α0 would move the longitude to the one intersecting the axis S2 corresponding to the time t3. And there should also exist two unique states corresponding to the time t1 and t2, respectively. A circular motion of q plate would exactly result in a complete circular rotation of the longitude line and they finally return to the initial state at the same time. In this case, the q plate and the longitude achieve the synchronization of rotation control and one-to-one correspondence. Therefore, we can select the longitude by rigidly rotating the q plate. Further, when the q plate is rotating in the opposite direction, the longitude also rotates oppositely corresponding to the time t′1, t′2 and t′3 in Fig. 4. Specially, in the case of q = 1, both the initial angle and the evolving route remain unchanged whatever the rotation of q plate for its rotational invariant feature.

4. Structured field in the geometric phase Doppler effect

To understand the geometric phase Doppler effect, we now consider a Gauss beam with linear polarization passing through a rotating q plate. In fact, the instant field distribution of the out-put beam is a “double spot” pattern rotating around axis z as shown in Fig. 5. The propagation characteristic of output light beam is attributed to the frequency shift. Due to the different frequencies, these two spin components obtain different phase shifts while passing the distance between the initial plane z0 and the current plane zt. Various phase differences lead to various transverse orientations of the double-spot pattern. At any instant of time, the 3D spatial intensity distribution of the rotating beam is helical which is determined by

k+z+k+r2R+(z)+m+φ+Φ+=const,
kz+kr2R(z)+mφ+Φ=const,
where R+(z) and R(z) are the radius of curvature of the wave front; Φ+ and Φ are the Gouy phases; k+ and k are the wavevectors; m+ and m are the orbital angular momentum. As we would neglect the phase difference caused by radial and Gouy phases, Eqs. (19) and (20) can be written as
(k+k)z+φ(m+m)=0.
Here, φ = 2π in a pitch. The spatial structure exhibits a screwing type along the +z axis with the pitch
zp=π(m+m)c|Δω|,
where Δk = k+k = 2Δω/c and c is the speed of light in a vacuum. In addition, the pattern rotates with an angular velocity ϖ = σω′(q − 1)/q which means the sense of rotation is specified by the chirality of the incident beam, the parameter q and the rotational direction of the q plate. These two “double spot” patterns are shown to screw in the opposite direction in the case of q = 1 and σ = +1 [see Figs. 5(a) and 5(b)] depending on the rotation direction of q plate. After substituting Eq. (18) into Eq. (22), we can obtain the relationship between the output beam and the rotating q plate; then by using tp = zp/c, the period is given by
tp=2π|q||(1q)ω|.
This relationship specifies the period of rotation of the output beam which is directly associated with the rotational angular velocity and the structure of q plate. Though the magnitude of tp can be easily realized, the typically enormous scale of zp makes its observation hard to implement under the magnitude limitation of the parameter q and the angular velocity ω′.

 

Fig. 5 3D spatial structure of output beam screws with pitch zp after the rotating q plate. The rotational Doppler effect induces rotation of intensity pattern with the rotational angular velocity ϖ = σω′(q − 1)/q. Here, q = 1/2, σ = +1. (a) the positive rotation (ω′ > 0, ϖ < 0), (b) the opposite rotation (ω′ < 0, ϖ > 0).

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5. Conclusion

In conclusion, we have demonstrated that the rotational Doppler effect can be interpreted in terms of a dynamic evolution of Pancharatnam-Berry phase on the hybrid-order Poincaré sphere, and that the frequency shift depends on the variation of total angular momentum but is independent of the orbit angular momentum of the input beam, which is significantly different from the conventional rotational Doppler effect [1–5]. The geometric phase Doppler effect has been examined by the calculations with Jones matrix and the theory of hybrid-order Poincaré sphere, and the two methods coincide well with each other. In addition, this interesting effect is also different from the geometric Doppler effect [32–35] in which the temporal frequency shift is replaced by a spatial frequency shift.

Funding

National Natural Science Foundation of China (No. 11474089).

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3. G. Nienhuis, “Doppler effect induced by rotating lenses,” Opt. Commun. 132(1), 8–14 (1996). [CrossRef]  

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5. J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, “Rotational frequency shift of a light beam,” Phys. Rev. Lett. 81(22), 4828–4830 (1998). [CrossRef]  

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References

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  1. B. A. Garetz and S. Arnold, “Variable frequency shifting of circularly polarized laser radiation via rotating halfwave retardation plate,” Opt. Commun. 31(1), 1–3 (1979).
    [Crossref]
  2. R. Simon, H. J. kimble, and E. C. G. Sudarshan, “Evolving geometric phase and its dynamical manifestation as a frequency shift: an optical experiment,” Phys. Rev. Lett. 61(1), 19–22 (1988).
    [Crossref] [PubMed]
  3. G. Nienhuis, “Doppler effect induced by rotating lenses,” Opt. Commun. 132(1), 8–14 (1996).
    [Crossref]
  4. J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, “Measurement of the rotational frequency shift imparted to a rotating light beam possessing orbital angular momentum,” Phys. Rev. Lett. 80(15), 3217–3219 (1998).
    [Crossref]
  5. J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, “Rotational frequency shift of a light beam,” Phys. Rev. Lett. 81(22), 4828–4830 (1998).
    [Crossref]
  6. M. Michalski, W Hütner, and H. Schimming, “Experimental demonstration of the rotational frequency shift in a molecular system,” Phys. Rev. Lett. 95(20), 203005 (2005).
    [Crossref] [PubMed]
  7. S. Barreiro, J.W. R. Tabosa, H. Failache, and A. Lezama, “Spectroscopic observation of the rotational Doppler effect,” Phys. Rev. Lett. 97(11), 113601 (2006).
    [Crossref] [PubMed]
  8. T. D. Thomas, E. Kukk, K. Ueda, T. Ouchi, K. Sakai, T. X. Carroll, C. Nicolas, O. Travnikova, and C. Miron, “Experimental observation of rotational Doppler broadening in a molecular system,” Phys. Rev. Lett. 106(19), 193009 (2011).
    [Crossref] [PubMed]
  9. O. Korech, U. Steinitz, R. J. Gordon, I. S. Averbukh, and Y. Prior, “Observing molecular spinning via the rotational Doppler effect,” Nat. Photonics 7(9), 711–714 (2013).
    [Crossref]
  10. M. P. J. Lavery, F. C. Speirits, S. M. Barbett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341(6145), 537–540 (2013).
    [Crossref] [PubMed]
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  33. A. Niv, Y. Gorodetski, V. Kleiner, and E. Hasman, “Topological spin-orbit interaction of light in anisotropic inhomogeneous subwavelength structures,” Opt. Lett. 33(24), 2910–2912 (2008).
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  43. G. F. Calvo, “Wigner representation and geometric transformations of optical orbital angular momentum spatial modes,” Opt. Lett. 30(10), 1207–1209 (2005).
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  44. A. Holleczek, A. Aiello, C. Gabriel, C. Marquardt, and G. Leuchs, “Classical and quantum properties of cylindrically polarized states of light,” Opt. Express 19(10), 9714–9736 (2011).
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  45. G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107(5), 053601 (2011).
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  46. W. Shu, Y. Liu, Y. Ke, X. Ling, Z. Liu, B. Huang, H. Luo, and X. Yin, “Propagation model for vector beams generated by metasurfaces,” Opt. Express 24(18), 21177–21189 (2016).
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2017 (1)

2016 (1)

2015 (7)

Y. Liu, Y. Ke, J. Zhou, H. Luo, and S. Wen, “Manipulating the spin-dependent splitting by geometric Doppler effect,” Opt. Express 23(13), 16682–16692 (2015).
[Crossref] [PubMed]

X. Yi, Y. Liu, X. Ling, X. Zhou, Y. Ke, H. Luo, and S. Wen, “hybrid-order Poincaré sphere,” Phys. Rev. A 91(2), 023801 (2015).
[Crossref]

M. Seghilani, M. Myara, I. Sagnes, B. Chomet, R. Bendoula, and A. Garnache, “Self-mixing in low-noise semi-conductor vortex laser: detection of a rotational Doppler shift in backscattered light,” Opt. Lett. 40(24), 5778–5781 (2015).
[Crossref] [PubMed]

D. Hakobyan and E. Brasselet, “Optical torque reversal and spin-orbit rotational Doppler shift experiments,” Opt. Express 23(24), 31230–31239 (2015).
[Crossref] [PubMed]

A. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. Lavery, M. Tur, S. Ramachandran, A. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photon. 7(1), 66–106 (2015).
[Crossref]

D. Hakobyan, H. Magallanes, G. Seniutinas, S. Juodkazis, and E. Brasselet, “Tailoring orbital angular momentum of light in the visible domain with metallic metasurfaces,” Adv. Opt. Mater. 4(2), 306 (2015).
[Crossref]

M. I. Shalaev, J. Sun, A. Tsukernik, A. Pandey, K. Nikolskiy, and N. M. Litchinitser, “High-efficiency all-dielectric metasurfaces for ultracompact beam manipulation in transmission mode,” Nano Lett. 15(9), 6261–6266 (2015).
[Crossref] [PubMed]

2014 (4)

E. Karimi, S. A. Schulz, I. D. Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light: Sci. Appl. 3, e167 (2014).
[Crossref]

Y. Liu, X. Ling, X. Yi, X. Zhou, H. Luo, and S. Wen, “Realization of polarization evolution on higher-order Poincaré sphere with metasurface,” Appl. Phys. Lett. 104(19), 191110 (2014).
[Crossref]

X. Yi, X. Ling, Z. Zhang, Y. Li, X. Zhou, Y. Liu, S. Chen, H. Luo, and S. Wen, “Generation of cylindrical vector vortex beams by two cascaded metasurfaces,” Opt. Express 22(14), 17207–17215 (2014).
[Crossref] [PubMed]

D. Hakobyan and E. Brasselet, “Left-handed optical radiation torque,” Nat. Photonics 8(8), 610–614 (2014).
[Crossref]

2013 (2)

O. Korech, U. Steinitz, R. J. Gordon, I. S. Averbukh, and Y. Prior, “Observing molecular spinning via the rotational Doppler effect,” Nat. Photonics 7(9), 711–714 (2013).
[Crossref]

M. P. J. Lavery, F. C. Speirits, S. M. Barbett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341(6145), 537–540 (2013).
[Crossref] [PubMed]

2012 (2)

N. M. Litchinitser, “Structured light meets structured matter,” Science,  337(6098), 1054–1055 (2012).
[Crossref] [PubMed]

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics,  6(7), 488–496 (2012).
[Crossref]

2011 (4)

A. Picón, A. Benseny, J. Mompart, and G. F. Calvo, “Spin and orbital angular momentum propagation in anisotropic media: theory,” J. Opt. 13(6), 383–385 (2011).
[Crossref]

T. D. Thomas, E. Kukk, K. Ueda, T. Ouchi, K. Sakai, T. X. Carroll, C. Nicolas, O. Travnikova, and C. Miron, “Experimental observation of rotational Doppler broadening in a molecular system,” Phys. Rev. Lett. 106(19), 193009 (2011).
[Crossref] [PubMed]

A. Holleczek, A. Aiello, C. Gabriel, C. Marquardt, and G. Leuchs, “Classical and quantum properties of cylindrically polarized states of light,” Opt. Express 19(10), 9714–9736 (2011).
[Crossref] [PubMed]

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107(5), 053601 (2011).
[Crossref] [PubMed]

2010 (1)

N. Dahan, Y. Gorodetski, K. Frischwasser, V. Kleiner, and E. Hasman, “Geometric Doppler effect: spin-split dispersion of thermal radiation,” Phys. Rev. Lett. 105(13), 136402 (2010).
[Crossref]

2009 (2)

2008 (4)

S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photon. Rev. 2(4), 299–313 (2008).
[Crossref]

E. Karimi, B. Piccirillo, L. Marrucci, and E. Santamato, “Light propagation in a birefringent plate with topological charge,” Opt. Lett. 34(8), 1225–1227 (2008).
[Crossref]

K. Y. Bliokh, Y. Gorodetski, V. Kleiner, and E. Hasman, “Coriolis effect in optics: unified geometric phase and spin-Hall effect,” Phys. Rev. Lett. 101(3), 030404 (2008).
[Crossref] [PubMed]

A. Niv, Y. Gorodetski, V. Kleiner, and E. Hasman, “Topological spin-orbit interaction of light in anisotropic inhomogeneous subwavelength structures,” Opt. Lett. 33(24), 2910–2912 (2008).
[Crossref] [PubMed]

2007 (1)

2006 (2)

S. Barreiro, J.W. R. Tabosa, H. Failache, and A. Lezama, “Spectroscopic observation of the rotational Doppler effect,” Phys. Rev. Lett. 97(11), 113601 (2006).
[Crossref] [PubMed]

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
[Crossref] [PubMed]

2005 (2)

M. Michalski, W Hütner, and H. Schimming, “Experimental demonstration of the rotational frequency shift in a molecular system,” Phys. Rev. Lett. 95(20), 203005 (2005).
[Crossref] [PubMed]

G. F. Calvo, “Wigner representation and geometric transformations of optical orbital angular momentum spatial modes,” Opt. Lett. 30(10), 1207–1209 (2005).
[Crossref] [PubMed]

2003 (1)

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90(20), 203901 (2003).
[Crossref] [PubMed]

2002 (1)

2001 (1)

1999 (1)

1998 (2)

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, “Measurement of the rotational frequency shift imparted to a rotating light beam possessing orbital angular momentum,” Phys. Rev. Lett. 80(15), 3217–3219 (1998).
[Crossref]

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, “Rotational frequency shift of a light beam,” Phys. Rev. Lett. 81(22), 4828–4830 (1998).
[Crossref]

1996 (2)

1992 (1)

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(11), 8185 (1992).
[Crossref] [PubMed]

1988 (2)

T. H. Chyba, L. J. Wang, L. Mandel, and R. Simon, “Measurement of the Pancharatnam phase for a light beam,” Opt. Lett. 13(7), 562–564 (1988).
[Crossref] [PubMed]

R. Simon, H. J. kimble, and E. C. G. Sudarshan, “Evolving geometric phase and its dynamical manifestation as a frequency shift: an optical experiment,” Phys. Rev. Lett. 61(1), 19–22 (1988).
[Crossref] [PubMed]

1984 (1)

M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. Lond. A 392(1802), 45–57 (1984).
[Crossref]

1979 (1)

B. A. Garetz and S. Arnold, “Variable frequency shifting of circularly polarized laser radiation via rotating halfwave retardation plate,” Opt. Commun. 31(1), 1–3 (1979).
[Crossref]

1936 (1)

R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50(2), 115 (1936).
[Crossref]

Ahmed, N.

A. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. Lavery, M. Tur, S. Ramachandran, A. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photon. 7(1), 66–106 (2015).
[Crossref]

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics,  6(7), 488–496 (2012).
[Crossref]

Aiello, A.

Alfano, R. R.

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107(5), 053601 (2011).
[Crossref] [PubMed]

Allen, L.

S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photon. Rev. 2(4), 299–313 (2008).
[Crossref]

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, “Measurement of the rotational frequency shift imparted to a rotating light beam possessing orbital angular momentum,” Phys. Rev. Lett. 80(15), 3217–3219 (1998).
[Crossref]

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, “Rotational frequency shift of a light beam,” Phys. Rev. Lett. 81(22), 4828–4830 (1998).
[Crossref]

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(11), 8185 (1992).
[Crossref] [PubMed]

Arnold, S.

B. A. Garetz and S. Arnold, “Variable frequency shifting of circularly polarized laser radiation via rotating halfwave retardation plate,” Opt. Commun. 31(1), 1–3 (1979).
[Crossref]

Ashrafi, N.

Ashrafi, S.

Averbukh, I. S.

O. Korech, U. Steinitz, R. J. Gordon, I. S. Averbukh, and Y. Prior, “Observing molecular spinning via the rotational Doppler effect,” Nat. Photonics 7(9), 711–714 (2013).
[Crossref]

Bao, C.

Barbett, S. M.

M. P. J. Lavery, F. C. Speirits, S. M. Barbett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341(6145), 537–540 (2013).
[Crossref] [PubMed]

Barreiro, S.

S. Barreiro, J.W. R. Tabosa, H. Failache, and A. Lezama, “Spectroscopic observation of the rotational Doppler effect,” Phys. Rev. Lett. 97(11), 113601 (2006).
[Crossref] [PubMed]

Beijersbergen, M. W.

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(11), 8185 (1992).
[Crossref] [PubMed]

Bendoula, R.

Benseny, A.

A. Picón, A. Benseny, J. Mompart, and G. F. Calvo, “Spin and orbital angular momentum propagation in anisotropic media: theory,” J. Opt. 13(6), 383–385 (2011).
[Crossref]

Berry, M. V.

M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. Lond. A 392(1802), 45–57 (1984).
[Crossref]

Beth, R. A.

R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50(2), 115 (1936).
[Crossref]

Bliokh, K. Y.

K. Y. Bliokh, Y. Gorodetski, V. Kleiner, and E. Hasman, “Coriolis effect in optics: unified geometric phase and spin-Hall effect,” Phys. Rev. Lett. 101(3), 030404 (2008).
[Crossref] [PubMed]

Bomzon, Z.

Boyd, R. W.

E. Karimi, S. A. Schulz, I. D. Leon, H. Qassim, J. Upham, and R. W. Boyd, “Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface,” Light: Sci. Appl. 3, e167 (2014).
[Crossref]

Brasselet, E.

D. Hakobyan, H. Magallanes, G. Seniutinas, S. Juodkazis, and E. Brasselet, “Tailoring orbital angular momentum of light in the visible domain with metallic metasurfaces,” Adv. Opt. Mater. 4(2), 306 (2015).
[Crossref]

D. Hakobyan and E. Brasselet, “Optical torque reversal and spin-orbit rotational Doppler shift experiments,” Opt. Express 23(24), 31230–31239 (2015).
[Crossref] [PubMed]

D. Hakobyan and E. Brasselet, “Left-handed optical radiation torque,” Nat. Photonics 8(8), 610–614 (2014).
[Crossref]

Calvo, G. F.

Cao, Y.

Carroll, T. X.

T. D. Thomas, E. Kukk, K. Ueda, T. Ouchi, K. Sakai, T. X. Carroll, C. Nicolas, O. Travnikova, and C. Miron, “Experimental observation of rotational Doppler broadening in a molecular system,” Phys. Rev. Lett. 106(19), 193009 (2011).
[Crossref] [PubMed]

Chen, S.

Chomet, B.

Chyba, T. H.

Courtial, J.

M. J. Padgett and J. Courtial, “Poincaré sphere equivalent for light beams containing orbital angular momentum,” Opt. Lett. 24(7), 430–432 (1999).
[Crossref]

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, “Measurement of the rotational frequency shift imparted to a rotating light beam possessing orbital angular momentum,” Phys. Rev. Lett. 80(15), 3217–3219 (1998).
[Crossref]

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, “Rotational frequency shift of a light beam,” Phys. Rev. Lett. 81(22), 4828–4830 (1998).
[Crossref]

Crawford, P. R.

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90(20), 203901 (2003).
[Crossref] [PubMed]

Dahan, N.

N. Dahan, Y. Gorodetski, K. Frischwasser, V. Kleiner, and E. Hasman, “Geometric Doppler effect: spin-split dispersion of thermal radiation,” Phys. Rev. Lett. 105(13), 136402 (2010).
[Crossref]

Dholakia, K.

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, “Rotational frequency shift of a light beam,” Phys. Rev. Lett. 81(22), 4828–4830 (1998).
[Crossref]

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, “Measurement of the rotational frequency shift imparted to a rotating light beam possessing orbital angular momentum,” Phys. Rev. Lett. 80(15), 3217–3219 (1998).
[Crossref]

Dolinar, S.

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics,  6(7), 488–496 (2012).
[Crossref]

Failache, H.

S. Barreiro, J.W. R. Tabosa, H. Failache, and A. Lezama, “Spectroscopic observation of the rotational Doppler effect,” Phys. Rev. Lett. 97(11), 113601 (2006).
[Crossref] [PubMed]

Fazal, I. M.

J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics,  6(7), 488–496 (2012).
[Crossref]

Franke-Arnold, S.

S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photon. Rev. 2(4), 299–313 (2008).
[Crossref]

Frischwasser, K.

N. Dahan, Y. Gorodetski, K. Frischwasser, V. Kleiner, and E. Hasman, “Geometric Doppler effect: spin-split dispersion of thermal radiation,” Phys. Rev. Lett. 105(13), 136402 (2010).
[Crossref]

Gabriel, C.

Galvez, E. J.

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90(20), 203901 (2003).
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Figures (5)

Fig. 1
Fig. 1 Schematic illustration of the propagation of circular polarized light beams passing through two half-wave q plates, wherein the first one is fixed while the second one rotates with a angular velocity. The red and blue arrows represent the left- and right-handed circularly polarized waves, respectively. And the relative displacements of arrows between the input and the output are resulting from the induced geometric phase.
Fig. 2
Fig. 2 Schematic illustration of the evolution of optical field in the rotational Doppler effect. The q plate rotating uniformly with an angular velocity ω′ is illuminated by a circularly polarized vortex wave with an angular frequency ω. Here, σ = −1 and σ = +1 represent the left- and right-handed circular polarization, respectively. And the output beams in (a) and (b) possess two opposite helical wave fronts with different angular frequencies, ω + Δω and ω − Δω.
Fig. 3
Fig. 3 Schematic illustration showing realization of the evolution along different longitude lines on the hybrid-order Poincaré sphere. The sphere is assumed with state σ = +1 and l = 0 in the north pole, and the state σ = −1 and m = +1 in the south pole. Insets (t0)–(t3) show the rotating q plate (q = 1/2) with different initial angle α0. Here the sense of the positive rotating angle is chosen as anticlockwise which means ω′ < 0, α0 < 0 corresponding to the longitude line’s moving direction: t0t1t2t3 and polarization states moving from north pole to south pole.
Fig. 4
Fig. 4 Schematic illustration showing realization of the evolution along different longitude lines on the hybrid-order Poincaré sphere. The state on the pole is assumed as the same as Fig. 3. Insets (t0)–(t3) show the rotating q plate (q = 1/2) with different initial angle α0. Here, ω′ > 0, α0 > 0 corresponding to the longitude line’s moving direction: t′0t′1t′2t′3 and polarization states moving from north pole to south pole.
Fig. 5
Fig. 5 3D spatial structure of output beam screws with pitch zp after the rotating q plate. The rotational Doppler effect induces rotation of intensity pattern with the rotational angular velocity ϖ = σω′(q − 1)/q. Here, q = 1/2, σ = +1. (a) the positive rotation (ω′ > 0, ϖ < 0), (b) the opposite rotation (ω′ < 0, ϖ > 0).

Equations (23)

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T ( r , φ ) = M ( α ) J ( δ ) M 1 ( α ) .
J = [ exp ( i δ / 2 ) 0 0 exp ( i δ / 2 ) ] .
M ( α ) = ( cos α sin α sin α cos α ) .
T ( r , φ ) = cos δ 2 ( 1 0 0 1 ) + i sin δ 2 ( cos 2 α sin 2 α sin 2 α cos 2 α ) .
α ( r , φ ) = q φ + α 0 ,
| ψ I = 2 2 ( e ^ x i σ e ^ y ) .
| ψ II = 2 2 ( e ^ x + i σ e ^ y ) exp [ i ( l φ 2 σ α 0 ) ] ,
α ( r , φ ) = q ( φ β ) + ( α 0 + β ) ,
| ψ III = 2 2 ( e ^ x i σ e ^ y ) exp [ i ( m φ 2 σ q β + 2 σ β ) ] ,
γ = 2 σ ( q 1 ) β .
Δ ω = d γ d t ,
Δ ω = 2 σ ω ( q 1 ) ,
| ψ ( θ , Φ ) = cos θ 2 | N l + sin θ 2 | S m exp ( i σ Φ ) ,
γ ( C ) = C d S V ( R ) ,
V ( R ) = R × A .
γ ( C ) = m l 2 σ 4 Ω .
γ ( C ) = σ ( q 1 ) 2 Ω .
Δ ω = 2 σ ω ( q 1 ) .
k + z + k + r 2 R + ( z ) + m + φ + Φ + = const ,
k z + k r 2 R ( z ) + m φ + Φ = const ,
( k + k ) z + φ ( m + m ) = 0 .
z p = π ( m + m ) c | Δ ω | ,
t p = 2 π | q | | ( 1 q ) ω | .

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