Redistributing the energy flow of a tightly focused radially polarized optical field by designing phase masks

Zhongsheng Man; Zhidong Bai; Shuoshuo Zhang; Xiaoyu Li; Jinjian Li; Xiaolu Ge; Yuquan Zhang; Shenggui Fu

doi:10.1364/OE.26.023935

1. Introduction

Because of their peculiar properties, vector beams with cylindrically symmetric states of polarization (SoPs) have emerged as a rather appealing research topic [1]. Two extreme forms of such beams are the radially polarized (RP) and azimuthally polarized (AP) beams. A RP plane beam focused by a high numerical aperture (NA) objective can create a strong longitudinal electric field in the focal region, resulting in a tighter focal spot with perfect circular symmetry [2,3]. The AP plane beam generates an azimuthally polarized electric field with a hollow-shaped pattern under a high NA focusing condition [2]. Further, an arbitrary three-dimensional polarization orientation can be obtained based on the superposition of the RP and AP beams [4]. Additional amplitude or phase modulations have also been considered in redistributing the focal electric field distributions of input cylindrical vector beams, resulting in more-enriched focal profiles including the transversely or longitudinally polarized optical needle [5–14], the transversely or longitudinally polarized optical channel [15,16], sharper focus profiles [17–19], optical cage or chain [20–22], flat-top focus profiles [23,24], optical bottle-hollow profiles [25], and multiple-foci profiles [16,26,27]. These unique features hint at their great potential in a variety of applications, such as electron acceleration [28], optical imaging [29], data storage [30,31], and optical trapping of particles [32–35].

For the cylindrical vector beams, most focus has so far been mainly on the tightly focused electric field distributions. The studies of energy flow in the focal plane of optical fields are also important and useful, as they indicate the wide use of such beams in manipulating and transporting absorptive particles [36,37]. In an optical tweezers system, the absorptive particles can be trapped by the strongly focused optical field when absorbing some of the beam energy. Simultaneously, they obtain a fraction of the energy flow, which causes the trapped absorptive particles to move so that the trajectory aligns with the Poynting vector with a velocity that is proportional to the modulus of the Poynting vector. There are a few reports on the energy flow of tightly focused optical fields with amplitude or phase modulations [38–40]. The redistributions of the transverse energy flow of a tightly focused AP beam modulated by rotationally symmetric sector-shaped obstacles and an amplitude-modulated spiral phase hologram have been studied [38,39]. The energy flow in the focal plane of one specific kind of vector optical field, which has a space-invariant orientation along the long axis and a space-varying ellipticity, has also been explored [40]. Moreover, Wu and colleagues have examined the energy flow of a RP beam after passing through a sector-shaped opaque obstacle and a thin lens [41]. As a counterpart, a systematic study is needed of the energy flow of the RP beam under high-NA focusing conditions, taking full advantage of the perfect cylindrical symmetry of the radial SoP. Furthermore, the impact of the modulation masks in designing the energy flow of the RP beam also needs investigating.

In this paper, we study the redistribution of the energy flow in the focal plane of a tightly focused RP beam. Using the Richards and Wolf vector diffraction theory, expressions for the electromagnetic fields and the Poynting vector in the focal region are presented. Based on this analytical model, we show there is only a longitudinal energy flow with a toroidal pattern in the focal plane for the incident RP plane field. When adding the traditional vortex phase to the RP input field, a transverse energy flow ring with perfect rotational symmetry is obtained in the focal plane. Furthermore, breaking of this rotational symmetry is also achieved when introducing an annular optical vortex with carefully tailored inner and outer radii, resulting in a transverse energy flow distribution exhibiting a regular polygon shape. As a result, the absorptive particles can be transported along more flexible paths. Moreover, the rotational symmetry of the electromagnetic field is also broken with the appearance of the regular polygon- shaped focal spot. These properties might be helpful in applications such as optical trapping and manipulation of particles and laser processing

2. Principle

Mathematically, a RP laser beam with finite aperture can be described in the cylindrical coordinate system (r, φ, z) as [1,42–45]

\begin{array}{l} E_{i} (r, φ, z) = A_{0} c i r c (r / r_{0}) (\cos φ {\hat{e}}_{x} + \sin φ {\hat{e}}_{y}) \\ = A_{0} c i r c (r / r_{0}) [\cos φ (\cos φ {\hat{e}}_{r} - \sin φ {\hat{e}}_{φ}) + \sin φ (\sin φ {\hat{e}}_{r} + \cos φ {\hat{e}}_{φ})] . \\ = A_{0} c i r c (r / r_{0}) {\hat{e}}_{r} \end{array}

Here, A₀ is a constant determining the relative amplitude distribution of the input field, r and φ are the polar radius and azimuthal angle, respectively, circ(r/r₀) is the circular function with r₀ the radius of the optical field,

{\hat{e}}_{x}

and

{\hat{e}}_{y}

are the unit vectors along the x and y axes, respectively, and

{\hat{e}}_{r}

and

{\hat{e}}_{φ}

are the respective unit vectors along the radial and azimuthal directions in the polar system.

With regard to the setup of the focus system (Fig. 1), a RP laser beam described by Eq. (1) passes initially through an annular spiral phase plate (inner and outer radii are labeled r₁ and r₂, respectively) and is subsequently focused by a high NA objective obeying the sine condition. An annular spiral phase plate is placed at the front focal plane to load the optical vortex on the incident optical field. Based on the Richards and Wolf vectorial diffraction theory, an expression for the electric field in the focal region of this tightly focused RP beam superposed with the optical vortex can be derived in the cylindrical coordinate system (ρ_S, ϕ_S, z_S) [1,2,46,47],

E_{o} (ρ_{S}, ϕ_{S}, z_{S}) = \frac{- i k f}{2 π} \int_{0}^{2 π} \int_{0}^{α} \sin θ \sqrt{\cos θ} K c i r c (\sin θ / \sin α) l_{0} (θ, φ) M_{e} d φ d θ,

\begin{array}{l} K = \exp {i k [z_{S} \cos θ + ρ_{S} \sin θ \cos (ϕ - ϕ_{S})]} \\ = \exp {i k [z_{S} \cos θ - ρ_{S} \sin θ \cos (φ - ϕ_{S})]} . \end{array}

where k and f denote respectively the wave number in the focal field and focal length; φ and θ are the azimuthal and tangential angles with respect to the z and x₀ axes (Fig. 1); α is the maximum tangential angle, given by α = arcsin(NA/n) with NA the numerical aperture of the focusing objective and n the refractive index in the focal space; ϕ is the azimuthal angle with respect to the x axis of the wave vector k in the image space (Fig. 1). The function l₀(θ, φ) denotes the complex amplitude distribution of the Bessel-Gaussian beam, which is given by [2]

l_{0} (θ, φ) = \exp [- β^{2} {(\frac{\sin θ}{\sin α})}^{2} + i Φ (θ, φ)] J_{1} (2 β \frac{\sin θ}{\sin α}),

where β is the ratio of the pupil radius and beam waist, J₁ is the Bessel function of the first kind of order one, Φ(θ, φ) denotes the additional phase shift stemming from the introduction of the annular spiral phase plate and written

Φ (θ, φ) = {\begin{cases} l φ f o r θ \in [θ_{1}, θ_{2}] \\ 0 o t h e r w i s e \end{cases}

where l is the topological charge of the optical vortex phase, θ₁ and θ₂ are the tangential angles with respect to the z axis and determined by the inner and outer radii of the annular spiral phase plate r₁ and r₂. They satisfy relations

Fig. 1 Schematic diagram for the focusing system. A Radially polarized beam is firstly passing through spiral phase plate for loading optical vortex with topological l and subsequently focused by a high NA objective.

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{\begin{cases} r_{1} = f \sin θ_{1} \\ r_{2} = f \sin θ_{2} \end{cases},

We introduce annular factors defined as the ratios between the inner and outer radii of the annular spiral phase plate and the radius of the optical field,

{\begin{cases} τ_{1} = \frac{r_{1}}{r_{0}} = \frac{f \sin θ_{1}}{f \sin α} = \frac{\sin θ_{1}}{\sin α} \\ τ_{2} = \frac{r_{2}}{r_{0}} = \frac{f \sin θ_{2}}{f \sin α} = \frac{\sin θ_{2}}{\sin α} \end{cases} .

In Eq. (2), M_e is the electric polarization vector in the tightly focused field with general expression

M_{e} = [\begin{array}{l} M_{e x} \\ M_{e y} \\ M_{e z} \end{array}] = [\begin{array}{l} \cos θ \cos φ \\ \cos θ \sin φ \\ \sin θ \end{array}] .

Similarly, the magnetic field in the focal region of the vortex-superposed optical beam is written in the cylindrical coordinate system (ρ_S, ϕ_S, z_S) as [46]

H_{o} (ρ_{S}, ϕ_{S}, z_{S}) = \frac{- i k f}{2 π} \int_{0}^{2 π} \int_{0}^{α} \sin θ \sqrt{\cos θ} K c i r c (\sin θ / \sin α) l_{0} (θ, φ) M_{m} d φ d θ,

where M_m is the magnetic polarization vector in the image space. In an isotropic dielectric, the electric and magnetic vectors are known to be orthogonal and satisfy the relation [40,46]

M_{m} = k \times M_{e},

where k = (−sinθcosφ, −sinθsinφ, cosθ) is the unit vector of the wave vector in the focal field. This then gives the magnetic polarization vector M_m in the form

M_{m} = [\begin{array}{l} M_{m x} \\ M_{m y} \\ M_{m z} \end{array}] = [\begin{array}{l} - \sin φ \\ \cos φ \\ 0 \end{array}] .

Evidently, this magnetic polarization vector is much different than the electric polarization vector in Eq. (8). In terms of the full time-dependent three-dimensional magnetic and electric fields, the energy current is given by the time-averaged Poynting vector [46],

P \propto \frac{c}{8 π} Re (E \times H^{*}),

where E and H are the electric and magnetic fields in the image space, respectively; the asterisk denotes complex conjugation. We can calculate the electromagnetic field intensity distributions and the energy flow of the tightly focused RP beam superposed with optical vortex using Eqs. (2)–(12). In the following calculations, all length measurements are in units of the wavelength; furthermore, the NA of the focusing lens is 0.95 and the refractive index n is assumed to be 1.

3. Energy flow in the focal plane of RP beam without any phase modulations

For purpose of comparison, we first investigate the distribution of the energy flow for a tightly focused RP beam without any phase modulation. We do this especially for the transverse energy flow in the focal plane, as it is useful in manipulating absorptive particles. The focal plane corresponds to setting τ₁ = τ₂ = 0 in Eq. (7). Figure 2 depicts the transverse and longitudinal components of the normalized Poynting vectors in the focal plane, both of which are normalized to the maximum of the total Poynting vector in the focal plane. Also, the ratio of the pupil radius and the beam waist β is set to 1.25 in the above calculations. Nearly no energy flow can be found for the transverse component whereas, for the longitudinal component, it exhibits a doughnut-shaped pattern with perfect circular symmetry due to the perfect cylindrically symmetric SoP of the input optical field. As a result, the absorptive particles cannot be transported in the focal plane of the tightly focused RP plane beam.

Fig. 2 Transverse (a) and longitudinal (b) components of the normalized Poynting vectors in the focal plane of the tightly focused RP beam without any phase modulations. Both images have dimensions 3λ × 3λ.

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4. Redistribution of the energy flow with traditional optical vortex

Recently, because of the increasing interest in the study of the transverse energy flow of tightly focused beams modulated by phase masks [39,40], the traditional vortex phase mask is applied to the RP input field to investigate the energy flow in the focal plane, which corresponds to instance τ₁ = 0 and τ₂ = 1 in Eq. (7). Figure 3 shows the normalized energy flow in the focal plane of the RP optical field modulated by traditional vortex phase masks with topological charge l = 1, 3, and 5, respectively. All images are normalized to the maximum of the total energy flow in the focal plane for each input beam mode. Comparing with the longitudinal energy flow, it can be seen from the corresponding transverse energy flow that a transverse energy flow ring with perfect rotational symmetry plays a non-negligible role in the total energy flow. This flow can be used to transport absorptive particles along a circular trajectory. Also, the radius of the circle is controlled by l; it increases when the topological charge grows. In addition, the magnitude of the transverse component of the Poynting vector, compared with the longitudinal component, gradually increases as l increases, and therefore it becomes easier to manipulate absorptive particles in the transverse plane.

Fig. 3 Normalized Poynting vectors in the focal plane of the tightly focused RP beam with traditional vortex phase modulations when l = 1, 3, and 5, respectively. The transverse and longitudinal components are shown in the first and second rows. The direction of the transverse energy flow is indicated by black arrows. All images have dimensions 4λ × 4λ.

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Figure 4 shows the corresponding normalized electromagnetic field intensity distributions in the focal plane for the above three input beam modes. All the images for the electric fields are normalized to the maximum total electric field strength in the focal plane; similarly, all images for the magnetic fields are normalized to the maximum total magnetic field strength in the focal plane. The electromagnetic fields all exhibit perfect rotationally symmetric intensity distributions for the traditional optical vortex phase modulations that stems from the rotational symmetry of the transverse and longitudinal energy flow in the focal plane (Fig. 2). To be specific, both fields are hot spot distributions when l = 1. The transverse component dominates the total electric field, whereas for the longitudinal component, the field takes on a doughnut-shaped pattern. These focusing properties are much different from that of a RP plane input field [2], because the energy flows change greatly when a traditional optical vortex is applied to the RP input field (see Figs. 2 and 3). Indeed, with a larger value of l, we can always obtain a doughnut-shaped pattern for both the electric and magnetic total fields (for l = 3 and 5, see second and third row panels in Fig. 4). Furthermore, it is obvious that the magnitude of the transverse component of the electric field, compared with the z component, will gradually decrease as l increases, and thus a strong longitudinally polarized electric field can be obtained with large topological charge of optical vortex.

Fig. 4 Normalized electromagnetic field intensity distributions in the focal plane of a tightly focused RP laser beam with traditional vortex phase modulations when l = 1, 3, and 5, corresponding to the first, second, and third rows, respectively. The transverse, longitudinal, and total field intensity distributions are shown in the first, second, and third columns, respectively. The fourth column gives the normalized total magnetic field intensity distributions in the focal plane. All images have dimensions of 4λ × 4λ.

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5. Breaking the rotationally symmetry of the transverse energy flow

Recalling from Fig. 3, a transverse energy flow ring with rotational symmetry and controllable radius can be obtained by introducing a traditional vortex phase mask modulation. As a result, absorptive particles can be transported along a circular trajectory. To meet the diverse demands for flexible manipulation of particles, breaking the rotationally symmetry of the transverse energy flow is critical and important. We find that polygonal profiles for the transverse component of the Poynting vector can be achieved simply by tailoring the values τ₁ and τ₂ in Eq. (7). The first and second rows in Fig. 5 display the transverse and longitudinal components, respectively, of the Poynting vectors of the tightly focused optical fields with (l, β) = (3, 1.3), (4, 1.25), (5, 1,2), and (6, 1) when (τ₁, τ₂) = (0.3865, 0.9775), thus an annular optical vortex is introduced; the value β is also carefully tailored to obtain a standard and uniform polygonal pattern for the transverse energy flow. All images are normalized to the maximum of the total energy flow in the focal plane for each input beam mode. Obviously, the number of sides of the polygon is equal to the topological charge of the annular optical vortex. Furthermore, the proportion of the transverse energy flow gradually increases when l increases, and the size of the polygon of the transverse energy flow also increases. The longitudinal components of the Poynting vectors (second row in Fig. 5) also exhibit polygon-shaped patterns but with non-uniform profiles, the size of which depends on l. Although the absorptive particles cannot be transported exactly along the path of the polygon, it is still useful in designing a uniform transverse energy to manipulate these particles along a certain path at an approximate uniform speed.

Fig. 5 Normalized Poynting vectors in the focal plane of the tightly focused RP beam with annular vortex phase modulations when l = 3, 4, 5, and 6, respectively. The transverse and longitudinal components are presented in the first and second rows. The direction of the transverse energy flow is indicated by black arrows. All images have dimensions of 5λ × 5λ.

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Figure 6 shows the corresponding normalized electromagnetic fields intensity distributions in the focal plane of the above four input beam modes. All images for the electric fields are normalized to the maximum of the total electric field in the focal plane, and likewise for all images of the magnetic field distributions. Noting that the rotationally symmetric focus for the input RP optical field is also broken, arriving from the rotationally non-symmetric energy flow distributions (Fig. 5). Specifically, it can be seen from the first and second columns in Fig. 6 that the profiles of the transverse and longitudinal components of the electric fields are polygons, i.e., triangle, quadrangle, pentagon, and hexagon. The number of sides of the polygon is equal to the topological charge l of the annular optical vortex, and the size of the polygon gradually increases as l increases. As a result, the total electric field, which is of course the vector sum of the transverse and longitudinal components, has a polygonal-symmetric intensity distribution, (third column in Fig. 6). The total magnetic field intensity distributions (fourth column in Fig. 6) exhibit nearly the same profiles as the corresponding longitudinal Poynting vector, (second row in Fig. 5). Overall, it should be emphasized that the proposed method does not only achieve a redistribution of the energy flow but also provides a new opportunity for focus shaping of tightly focused optical fields.

Fig. 6 Normalized electromagnetic field intensity distributions in the focal plane of the tightly focused RP beam with annular vortex phase modulations when l = 3, 4, 5, and 6, respectively (first, second, third, and fourth rows, respectively). The transverse, longitudinal, and total field intensity distributions are shown in the first, second, and third columns, respectively. The fourth column gives the normalized total magnetic field intensity distributions in the focal plane. All images have dimensions 5λ × 5λ.

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Based on the rotational-symmetry-broken energy flow profiles of the tightly focused RP input optical field presented above, a detailed procedure for controlling the energy flow needs to be outlined. By adjusting the phase distribution, the energy flow distribution can be controlled flexibly and conveniently. From theory, the energy flow should be calculated using Eqs. (2)–(12), and it is essential to consider the values of for example the annular factors and topological charge of vortex phase. Indeed, there are mature methods for experimentally generating RP optical field with arbitrary phase distributions [48,49]. With theories and experimental methods, we believed that various types of energy flow profiles with broken rotational symmetry of tightly focused RP input optical field can be achieved experimentally.

6. Conclusion

We studied the redistribution of the energy flow in the focal plane of tightly focused RP beam. Expressions for calculating the electromagnetic fields and Poynting vector of tightly focused RP input fields were obtained using the Richards and Wolf vector diffraction theory. Based on this analytical model, there is only a longitudinal energy flow with a doughnut-shaped pattern in the focal plane for the incident RP plane field. When adding the traditional vortex phase to the RP input field, a transverse energy flow ring with perfect rotational symmetry was obtained in the focal plane. Furthermore, breaking of this rotational symmetry was also achieved by introducing an annular optical vortex with carefully tailored inner and outer radii, resulting in a transverse energy flow in the shape of a regular polygon. As a consequence, the absorptive particles may be transported along more flexible paths. Moreover, the rotational symmetry of the electromagnetic field is also broken because a polygonal focal spot is generated. These properties may assist in laser fabrication and optical manipulation. We believe that this work not only provides a new method for manipulating energy flows of a tightly focused RP input optical field, but also offers a new opportunity for focus shaping of tightly focused vector beams.

Funding

National Natural Science Foundation of China (NSFC) (11604182, 11704226, 61605117); Natural Science Foundation of Shandong Province (ZR2016AB05, ZR2017MA051); Key Laboratory of Optoelectronic Devices and Systems of Ministry of Education and Guangdong Province (GD201704).

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Redistributing the energy flow of a tightly focused radially polarized optical field by designing phase masks

Abstract

1. Introduction

2. Principle

3. Energy flow in the focal plane of RP beam without any phase modulations

4. Redistribution of the energy flow with traditional optical vortex

5. Breaking the rotationally symmetry of the transverse energy flow

6. Conclusion

Funding

References

Cited By

Figures (6)

Equations (12)

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