Geometric phase Doppler effect: when structured light meets rotating structured materials

Zhenxing Liu; Yuanyuan Liu; Yougang Ke; Junxiao Zhou; Yachao Liu; Hailu Luo; Shuangchun Wen

doi:10.1364/OE.25.011564

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) M^{- 1} (α) .

Here, r is the radial coordinate, φ is the azimuthal coordinate. J(δ) is the operator of anisotropy with certain retardation δ:

J = [\begin{matrix} exp (i δ / 2) & 0 \\ 0 & exp (- i δ / 2) \end{matrix}] .

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) = (\begin{matrix} cos α & sin α \\ sin α & - cos α \end{matrix}) .

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) = cos \frac{δ}{2} (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}) + i sin \frac{δ}{2} (\begin{matrix} cos 2 α & sin 2 α \\ sin 2 α & - cos 2 α \end{matrix}) .

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) = q φ + α_{0},

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

| ψ_{I} 〉 = \frac{\sqrt{2}}{2} ({\hat{e}}_{x} - i σ {\hat{e}}_{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 |ψ_II〉 = T(r, φ)|ψ_I〉 can be written as

| ψ_{II} 〉 = \frac{\sqrt{2}}{2} ({\hat{e}}_{x} + i σ {\hat{e}}_{y}) exp [i (l φ - 2 σ α_{0})],

where l = −2σq. The particular phase lφ − 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 lħ 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 (φ - β) + (α_{0} + β),

where β is the angle of rotation. The output beam in the rotating q plate |ψ_III〉 = T(r, φ)|ψ_II〉 can be written as

| ψ_{III} 〉 = \frac{\sqrt{2}}{2} ({\hat{e}}_{x} - i σ {\hat{e}}_{y}) exp [i (m φ - 2 σ q β + 2 σ β)],

where m = l + 2σq. The Pancharatnam-Berry phase can be written as

γ = - 2 σ (q - 1) 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

Δ ω = \frac{d γ}{d t},

Substituting Eq. (10) into Eq. (11), the frequency shift can be obtained as

Δ ω = - 2 σ ω^{'} (q - 1),

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) 〉 = cos \frac{θ}{2} | N_{l} 〉 + sin \frac{θ}{2} | S_{m} 〉 exp (- i σ Φ),

and

| N_{l} 〉 = ({\hat{e}}_{x} + i σ {\hat{e}}_{y}) exp (i l φ) / \sqrt{2}

,

| S_{m} 〉 = ({\hat{e}}_{x} - i σ e_{y}) exp (i m φ) / \sqrt{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. |N_l〉 and |S_m〉 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) = - \int \int_{C} d S \cdot V (R),

where dS = ρ² sin θdθdΦρ̂ [47]. The Berry curvature is given by

V (R) = \nabla_{R} \times A .

Here,

\nabla_{R} = \frac{d}{d ρ} \hat{ρ} + \frac{1}{ρ} \frac{d}{d θ} \hat{θ} + \frac{1}{ρ sin θ} \frac{d}{d Φ} \hat{Φ}

is a gradient in spherical coordinates. The Berry connection can be showed as A = i〈ψ(R)|∇_R|ψ(R)〉 [47]. Then, we can obtain the Pancharatnam-Berry phase

γ (C) = - \frac{m - l - 2 σ}{4} Ω .

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: t₀ → t₁ → t₂ → t₃ 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′₀ → t′₁ → t′₂ → t′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) = - \frac{σ (q - 1)}{2} Ω .

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

Δ ω = - 2 σ ω^{'} (q - 1) .

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 z_t. 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 + \frac{k_{+} r^{2}}{R_{+} (z)} + m_{+} φ + Φ_{+} = const,

k_{-} z + \frac{k_{-} r^{2}}{R_{-} (z)} + m_{-} φ + 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

z_{p} = \frac{π (m_{+} - m_{-}) c}{| Δ ω |},

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 t_p = z_p/c, the period is given by

t_{p} = \frac{2 π | q |}{| (1 - q) 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 t_p can be easily realized, the typically enormous scale of z_p 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 z_p 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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Geometric phase Doppler effect: when structured light meets rotating structured materials

Abstract

1. Introduction

2. Evolution of geometric phase in inhomogeneous anisotropic plates

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

4. Structured field in the geometric phase Doppler effect

5. Conclusion

Funding

References and links

Cited By

Figures (5)

Equations (23)

Optics Express