Reconfigurable generation of chiral optical fields with multiple selective degrees of freedom

Duo Deng; Duo Deng; Xing Liu; Zhenjun Yang; Yan Li

doi:10.1364/OE.506660

1. Introduction

Chirality refers to the property of an object that is nonsuperimposable with its mirror image and is almost omnipresent in nature [1]. Chiral optical fields have attracted much attention because their chirality can interplay with that of chiral molecules [2] or nanostructures [3]. As well-known circular dichroism, circularly polarized light is crucial for the characterization of the optical activity that could be used to detect biomolecules [4,5] or enantiomers [6,7]. The handedness of circularly polarized light is related to the photonic spin angular momentum (SAM), which has two states defined as S=±ħ [8]. Most recently, a new phenomenon named vortical dichroism has come into researchers’ sight, in which vortex beams were employed as the probe beams to discriminate the optical activity of chiral structures and have demonstrated gigantic vortical differential scattering [9,10]. Such beams, which are characterized by an azimuthal phase exp(ilϕ), carry the orbital angular momentum (OAM) of lћ per photon, where l is the topological charge (TC) and ϕ is the azimuthal angle [11]. The handedness of vortex beams can indeed be exhibited in local chiral effects, being dependent on the sign of the TC [12]. The number of OAM modes is theoretically unlimited, far more than that of SAM with only two modes. As the vortical differential scattering effect depends on both the sign and magnitude of the OAM, high-order vortex beams have been applied to enhance the characteristic signals for detecting chiral molecules [13]. The OAM of the light field can be arbitrarily manipulated with the help of metasurfaces [14–16], which can promote the popularization of vortical dichroism. Furthermore, chiral optical fields can provide an optical force on chiral particles [17], which can be applied to guide or rotate particles [18–20]. However, the effects of moving particles using chiral optical fields carrying SAM and OAM are quite different, as SAM leads to the particles rotating around their own axis while OAM makes them orbit around the beam axis [21]. In addition, researchers have confirmed that chiral nanostructures, which can be fabricated by chiral optical fields possessing OAM [22–27], have advanced applications such as the detection of chiral chemical composites [28] and the control of chiral chemical reactions [29]. Indeed, it is worth noting that the parameters of the chiral optical field have a direct influence on the structural parameters when it is used to fabricate chiral structures [30].

Responding to its unique characteristics and extensive applications, the manufacture of chiral optical fields has become the focus of researchers. Specifically, various schemes have been proposed to generate chiral optical fields based on the helical phase. For example, the methods of vortex phase splicing and vortex phase modulation can be used to generate chiral optical fields. Yang and coworkers proposed to generate chiral optical fields with a controllable even number of spiral lobes by grafting the phase of several OAM beams [31]. Chiral optical fields with controllable OAM and polarization on each spiral lobe were fabricated based on the coordinate transformation of the perfect vortex phase [32]. A multi-twisted beam with a controlled twisted shape was generated by superimposing modulated vortex phases [33]. In addition, the interference method is also an important method for generating chiral optical fields. For example, chiral optical fields with controllable chirality and number of fringes could be generated by coaxial interference of a vortex beam and a plane wave [25] or two vortex beams [27]. These studies motivated researchers to control the parameters of the generated chiral optical fields by shaping the vortex optical fields. Therefore, methods such as generating elliptical perfect vortices [34], remainder-phase optical vortices [35], and multi-fractional-order optical vortices [36] have the potential for application in chiral field modulation. Nevertheless, previous schemes, which can only control the chirality and number of fringes but not the size, local intensity distribution, ellipticity, and orientation of the chiral optical fields, greatly limit the flexible applications of chiral optical fields. Thus, it is necessary to explore an alternative scheme for effectually generating a flexible chiral optical field.

In this work, we proposed both theoretically and experimentally a simple method to generate chiral optical fields with five separately controllable degrees of freedom, including the local intensity distribution, number of fringes, size, orientation, and ellipticity. The chiral optical field is generated by superimposing two anisotropic vortices whose wavefronts have a nonlinear phase variation concerning the azimuthal angle. The nonlinear coefficient, TCs, axicon parameter, rotation angle, and stretching factor of the anisotropic vortices can determine the local intensity distribution, number of fringes, size, orientation, and ellipticity of the chiral optical field, respectively. Furthermore, the OAM density distribution of the chiral optical field was theoretically calculated and the magnitude of the local OAM density could be continually enhanced with the variation in the local intensity distribution. The generated chiral optical field with five separately controllable degrees of freedom may be applied in chiral structure fabrication to increase the types of chiral nanostructures or applied in optical tweezers to provide appropriate optical force and pull different kinds of particles.

2. Theoretical overview

As chiral optical fields can be obtained by superimposing two vortex beams, we propose to fabricate chiral optical fields with multiple selective degrees of freedom by adjusting the characteristics of the vortex beam. An anisotropic vortex is characterized by the nonlinear phase variation in the phase profile. The phase variation is a function of the azimuthal angle, and the function should be periodic as φ(0)= φ(2π). Therefore, the phase of an ideal anisotropic vortex can be expressed as:

(1)$$\varphi (\phi )= l\phi + \alpha \cos (l\phi ),$$

where l is the TC of the anisotropic vortex and α is the nonlinear coefficient controlling the degree of nonlinear phase variation. As shown in Fig. 1(a), compared with the isotropic vortex whose phase varies linearly with the azimuthal angle, the anisotropic vortex has a nonlinear variation in the phase profile. The phase profile of an anisotropic vortex with l = 3 and α=1 is shown in Fig. 1(b). The normalized intensities of anisotropic vortices with the same TC l = 3 and different nonlinear coefficient α=0, 1, and 2 are shown in Figs. 1(c)–1(e). We can see that as the nonlinear coefficient α increases, the intensity distribution on the vortex ring will change and show a trend of moving towards several points.

Fig. 1. (a) A plot of the phase as a function of azimuthal angle for an isotropic vortex with l = 3 (black line) and two anisotropic vortices with l = 3, α=1 (red dashed line) and l = 3, α=2 (blue dotted line). (b) The phase profile of an anisotropic vortex with l = 3 and α=1.

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The chiral optical field can be obtained by superimposing a pair of conjugate anisotropic vortices with opposite phases. The transmittance function used to generate the coaxial anisotropic vortices can be expressed in the polar coordinates (r, ϕ) as:

(2)$$T(r,\phi ) = \textrm{exp} \{{i[{l\phi + \alpha \cos (l\phi )} ]} \}+ \textrm{exp} \{{ - i[{l\phi + \alpha \cos (l\phi )} ]} \},$$

where l = l₁=-l₂, and l₁ and l₂ are the TCs of the first and second anisotropic vortices.

This pair of anisotropic vortices passes through a lens and then produces a chiral optical field. Factoring out the parts that are different in the ϕ dependences of the field, we can obtain

(3)$$\begin{aligned} E &\propto \textrm{exp} [{i\varphi (\phi )} ]+ \textrm{exp} [{ - i\varphi (\phi )} ]\\ & = 2\cos [{\varphi (\phi )} ]. \end{aligned}$$

According to Eq. (3), we can see that the chiral optical field is closely related to the phase of the first anisotropic vortex. Then the stationary point of the chiral optical field is marked with the phase

(4)$$\varphi (\phi )= l\phi + \alpha \cos (l\phi ) = 0.$$

The phase ϕ satisfying Eq. (4) could make the cosine function in Eq. (3) obtain the maximum value. Thus, the stationary point also describes the extreme point with the maximum intensity of the chiral optical field. As shown in Fig. 1(a) with the green solid line, when the nonlinear coefficient α increases, the corresponding solution of ϕ satisfying Eq. (4) decreases. Furthermore, for an anisotropic vortex with l = 3 and a different nonlinear coefficient α, the parameter ϕ satisfying Eq. (4) is shown in Table 1. Therefore, along with the increasing nonlinear coefficient α, the stationary point of the chiral optical field moves clockwise. Based on the above theoretical derivation, the nonlinear coefficient of the anisotropic vortex affects the local intensity distribution of the chiral optical field, which will be the first controllable degree of freedom of the proposed chiral optical field.

Table 1. The stationary point of the chiral optical field consisting of the anisotropic vortex with l = 3 and different nonlinear coefficients α.

View Table

Then an axicon phase is added to adjust the radius of the anisotropic vortex, and the phase is designed as:

(5)$$\varphi ({r,\phi } )= l\phi + \alpha \cos (l\phi ) + \eta r,$$

where η is the axicon parameter.

A quasi-Bessel beam can be generated with the phase, which can be expressed as:

(6)$$E(r,\phi ,z) = {J_l}({{k_r}r} )\textrm{exp}[{i\varphi (\phi )+ i{k_z}z} ],$$

in which

(7)$$\begin{aligned} {k_r} &= k\,\sin ({{\eta / k}} )\\ {k_z} &= k\, \cos({{\eta / k}} ). \end{aligned}$$

Here J_l is an l-th order Bessel function of the first kind. k_r and k_z are the radial and longitudinal wavenumbers, with the wavenumber k = 2π/λ, and λ is the wavelength in vacuum.

A simple lens can transform the quasi-Bessel beam into the form of a quasi-perfect vortex whose field can be expressed as

(8)$$E({r,\phi } )= \frac{{{i^{l - 1}}}}{{{k_r}}}\delta ({r - R} )\textrm{exp} [{i\varphi (\phi )} ],$$

where δ(x) is the Dirac delta function, R = k_rf/k is the radius of the quasi-perfect vortex, and f is the focal length of the lens.

When two quasi-perfect vortices are employed to form the chiral optical field, the size of the chiral optical field can be adjusted with the axicon parameter. Thus, the size becomes the second degree of freedom of the multi-parameter controllable chiral optical field.

The number of fringes could be the third degree of freedom, because the number of fringes of the chiral optical field is closely related to the TCs of the anisotropic vortices. If only the spiral phases of the anisotropic vortices are considered, the field can be simplified as:

(9)$$u \propto \textrm{exp} [{i{l_1}\phi } ]+ \textrm{exp} [{i{l_2}\phi } ].$$

Then the intensity could be calculated:

(10)$$I = 2 + 2\cos [{({{l_1} - {l_2}} )\phi } ].$$

Based on Eq. (10), we can conclude that the number of fringes on the field’s intensity profile is the absolute value of the difference between the TCs of the anisotropic vortices. Thus, a chiral optical field with arbitrary number of fringes could be generated by adjusting the TCs of the pair of anisotropic vortices.

In addition, the orientation of the chiral optical field can rotate by rotating the azimuthal angle,

(11)$$\phi' = \phi + {\phi _m},$$

where ϕ_m is the rotation angle. By using the corrected azimuth angle ϕ′ in the phase calculation of anisotropic vortices, the chiral optical field generated by superimposing a pair of conjugate anisotropic vortices rotated clockwise. Therefore, the orientation of the chiral optical field can serve as the fourth controllable degree of freedom.

The elliptic of the chiral optical field can be generated by stretching the anisotropic vortex into an elliptic anisotropic vortex,

(12)$$\left\{ {\begin{array}{{c}} {\rho = \sqrt {{{({mx} )}^2} + {y^2}} }\\ {\theta = \arctan \frac{y}{{mx}}} \end{array}} \right.$$

where (ρ, θ) is the stretched polar coordinates and m is the stretching factor that determines the eccentricity. The eccentricity satisfies the following relationship,

(13)$$e = \left\{ \begin{array}{cc} \sqrt {1 - {m^2}} &\mathrm{\ 0\ < s} \le \textrm{1 }\\ {\sqrt {{m^2} - 1}}& \mathrm{\ s\ > 1\ } \end{array} \right.$$

When the stretched coordinates are used to calculate the phases of elliptic anisotropic vortices, an elliptic chiral optical field can be obtained by superimposing two elliptic anisotropic vortices. The chiral optical field can be freely changed from a circle to an ellipse by adjusting the stretching factor, which is considered the fifth controllable degree of freedom.

3. Results and discussion

To verify the theoretical analysis of the proposed multi-parameter controllable chiral optical fields, optical devices were assembled as shown in Fig. 2. A linearly polarized laser beam generated by a diode-pumped solid-state laser (532 nm) is expanded and collimated by a beam expander. Then the beam is filtered by an aperture and modulated by a polarizer. A phase-only spatial light modulator (SLM, UPOLabs, HDSLM80R-PLUS, 1920 × 1200 pixels, pixel pitch 8 µm, 60 Hz) is illuminated by the Gaussian beam. The phase uploaded on the SLM is used to generate coaxial anisotropic vortex beams. To avoid invalid pixels in the phase plate uploaded to the SLM, the phase is calculated by the pixel checkboard method reported in our previous work [37]. The beams reflected by the SLM are Fourier transformed by a Fourier lens (f = 160 mm). The chiral optical field is recorded by a charge-coupled device (Basler, acA4112, 4096 × 3000 pixels, pixel size 3.45 µm, 30 Hz) that is placed at the focal plane of the lens.

Fig. 2. Experimental setup. BE, beam expander; A, aperture; P, polarizer; BS, beam splitter; SLM, spatial light modulator; L, lens; CCD, charge-coupled device.

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As the first controllable dimension, chiral optical fields with different local intensity distributions are generated by anisotropic vortices with the same TCs by different nonlinear coefficients, as shown in Fig. 3. As the conjugate anisotropic vortices have opposite phases, the parameters of the conjugate anisotropic vortices are abbreviated as the parameters (l, α, η) of the first anisotropic vortex. As the simulation results show in Figs. 3(a1)–3(e1), when the nonlinear coefficients of the anisotropic vortices increase, the local intensity distribution on the fringes of the chiral optical field generated by the conjugate anisotropic vortices changes. Specifically, with the increase in the nonlinear coefficient α, the energy of the chiral optical field moves from the center of each fringe to the edge in a clockwise direction, which is consistent with the theoretical derivation. At the same time, the number of fringes, size and other parameters of these fields do not change. The experimental results agree well with the simulation results, as shown in Figs. 3(a2)–3(e2). Furthermore, the chirality of the optical field could be controlled by the modes of the anisotropic vortices. As shown in Figs. 3(a1)–3(e1) and Figs. 3(a3)–3(e3), when the TCs of the anisotropic vortices change to their conjugate states, the optical field undergoes a mirror transformation, and the chirality changes from left-handed to right-handed. The experimental results of the right-handed chiral optical field with different local intensity distributions are shown in Figs. 3(a4)–3(e4). When the optical field has right-handedness, the intensity of the field changes counterclockwise with increasing nonlinear coefficient α. Furthermore, it is worth mentioning that the number of maximum intensity points in the intensity distribution is consistent with the number of fringes of the chiral optical field. In general, the local intensity distribution on the fringes of the proposed chiral optical field can be freely controlled.

Fig. 3. The local intensity distribution of the chiral optical fields is affected by the nonlinear coefficient of the anisotropic vortex. The chiral optical fields are generated by conjugate anisotropic vortices with the parameters (a1-a2) l = 3, η=3, α=0.5, (b1-b2) l = 3, η=3, α=1, (c1-c2) l = 3, η=3, α=1.5, (d1-d2) l = 3, η=3, α=2, (e1-e2) l = 3, η=3, α=2.5, (a3-a4) l = −3, η=3, α=0.5, (b3-b4) l = −3, η=3, α=1, (c3-c4) l = −3, η=3, α=1.5, (d3-d4) l = −3, η=3, α=2, and (e3-e4) l = −3, η=3, α=2.5, respectively. The first and third lines are the simulation results, and the second and fourth lines are the experimental results.

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The OAM density changes with the energy shift on the fringes of the chiral optical field. The OAM distribution has significant relevance in the mechanical transfer of OAM to particles in optical trapping and tweezing [38,39]. Therefore, the OAM density distribution of the field is calculated to prove that by changing the local intensity distribution of the chiral optical field, the OAM density can be continuously variable, supporting different application scenarios. The OAM density of the field is calculated based on the Poynting vector, which can be calculated as [40]

(14)$$\boldsymbol{S} = \frac{{{\varepsilon _0}\omega }}{4}[{i({u\nabla {u^ \ast } - {u^ \ast }\nabla u} )+ 2k{{|u |}^2}{\boldsymbol{e}_z}} ],$$

where ω is the angular frequency, ɛ₀ is the permittivity of vacuum, and e_z is the z-direction. Then the OAM density is calculated from the Poynting vector

(15)$${j_z} = r{S_\phi },$$

where S_ϕ is the ϕ component of S.

Figures 4(a) show the OAM density of chiral optical fields generated by conjugate anisotropic vortices with l = 3, η=3, and α=1.1 to 1.9. The OAM density oscillates from positive to negative, making the chiral field an ideal tool in the field of optical trapping and tweezing. The increase in the nonlinear coefficient α continually enhances the magnitude of local OAM density of the chiral optical field, as shown in Fig. 4(b). The optical force is proportional to the OAM density [41]. Therefore, when manipulating particles with the proposed chiral optical field, we can select the appropriate OAM density by changing the nonlinear coefficient α.

Fig. 4. (a) The OAM density of the chiral optical field consisted of conjugate anisotropic vortices with l = 3, η=3, and α=1.1 to 1.9. (b) The maximum of the field’s OAM density continuously enhances with the variation of the nonlinear coefficient α.

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The controllable dimensions of the chiral optical field include not only its local intensity distribution but also its number of fringes and size. As shown in Fig. 5, chiral optical fields with 1 to 5 fringes are generated by simulation and experiment. Here, the TCs of anisotropic vortices used to generate the left-handed chiral optical field are Fig. 5(a1) l₁= 0, l₂=−1, Fig. 5(b1) l₁= 1, l₂=−1, Fig. 5(c1) l₁= 1, l₂=−2, Fig. 5(d1) l₁= 2, l₂=−2, and Fig. 5(e1) l₁= 2, l₂=−3. The axicon parameter η and nonlinear coefficient α are 3 and 1 for these anisotropic vortices. The corresponding simulation and experimental results are shown in Figs. 5(a1)–5(e1) and Figs. 5(a2)–5(e2), respectively. When the TCs of anisotropic vortices are transformed into conjugate modes, the chirality of the optical fields changes to right-handed as shown in Figs. 5(a3)–5(e3) and Figs. 5(a4)–5(e4). The chirality of the optical field changes with the sign of TC, but the number of fringes does not change. In addition, chiral optical fields with the same number of fringes and intensity distribution but different sizes are generated by conjugate anisotropic vortex beams with different degrees of axicon parameters. As shown in Fig. 6, with increasing η, the size of the field is gradually amplified on the premise of constant number of fringes and local intensity distribution.

Fig. 5. The number of fringes of the chiral optical fields is affected by the modes of anisotropic vortices. The chiral optical fields are generated by conjugate anisotropic vortices with α=1, η=3 and (a1), (a2) l₁= 0 and l₂=−1, (a3), (a4) l₁= 0 and l₂= 1, (b1), (b2) l₁= 1 and l₂=−1, (b3), (b4) l₁=−1 and l₂= 1, (c1), (c2) l₁= 1 and l₂=−2, (c3), (c4) l₁=−1 and l₂= 2, (d1), (d2) l₁= 2 and l₂=−2, (d3), (d4) l₁=−2 and l₂= 2, (e1), (e2) l₁= 2 and l₂=−3, (e3), (e4) l₁=−2 and l₂= 3. The first and third lines are the simulation results, and the second and fourth lines are the experimental results.

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Fig. 6. The size of the chiral optical fields is affected by the degree of the axicon parameter. The chiral optical fields are generated by conjugate anisotropic vortices with (a1-a2) l = 3, α=1, η=1, (b1-b2) l = 3, α=1, η=2, (c1-c2) l = 3, α=1, η=3, (d1-d2) l = 3, α=1, η=4, and (e1-e2) l = 3, α=1, η=5, (a3-a4) l = −3, α=1, η=1, (b3-b4) l = −3, α=1, η=2, (c3-c4) l = −3, α=1, η=3, (d3-d4) l = −3, α=1, η=4, and (e3-e4) l = −3, α=1, η=5. The first and third lines are the simulation results, and the second and fourth lines are the experimental results.

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Furthermore, the orientation and eccentricity are also controllable degrees of freedom of the proposed chiral optical fields. As shown in Figs. 7(a1)–7(e1) and Figs. 7(a2)–7(e2), chiral optical fields with different orientations are generated by simulation and experiment. The rotation angles of the chiral optical fields are ϕ_m= 0°, ϕ_m= 15°, ϕ_m= 30°, ϕ_m= 45°, and ϕ_m= 60°. Based on the simulation and experimental results, we can see that the chiral optical fields generated can rotate freely by rotating the azimuthal angle. As shown in Figs. 7(f1)–7(j1) and Figs. 7(f2)–7(j2), the eccentricity of the chiral optical fields can be freely controlled. By utilizing the rotation angle ϕ_m and the stretching factor m, we can stretch or compress the chiral optical fields in any direction, which makes the generated chiral optical field more flexible.

Fig. 7. The orientation and eccentricity of the chiral optical fields are affected by the rotation angle and stretching factor, respectively. The chiral optical fields are generated by conjugate anisotropic vortices with l = 3, α=1, η=3, and (a1-a2) ϕ_m= 0°, (b1-b2) ϕ_m= 15°, (c1-c2) ϕ_m= 30°, (d1-d2) ϕ_m= 45°, (e1-e2) ϕ_m= 60°, (f1-f2) m = 0.5, (g1-g2) m = 0.7, (h1-h2) m = 0.9, (i1-i2) m = 1.1, and (j1-j2) m = 1.3. The first and third lines are the simulation results, and the second and fourth lines are the experimental results.

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Figure 8 illustrates two five-dimensional chiral optical fields as examples. The two chiral fields are different in number of fringes, size, local intensity distribution, orientation, and eccentricity. The results prove the effectiveness of the proposed method and inspire the potential applications in chiral structure fabrications and optical tweezers.

Fig. 8. Five-dimensional controllable chiral optical fields. The chiral optical fields are generated by conjugate anisotropic vortices with (a) l₁= 1, l₂=−2, α=0.7, η=2.5, ϕ_m= 30°, and m = 0.8, (b) l₁= 2, l₂=−3, α=1.7, η=3.5, ϕ_m= 45°, and m = 1.2.

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4. Conclusions

In summary, we propose a simple method to fabricate chiral optical fields with five separately controllable degrees of freedom by superimposing a pair of anisotropic vortices. The local intensity distribution, number of fringes, size, orientation, and eccentricity of the proposed chiral optical field could be freely controlled by adjusting the nonlinear coefficient, TC, axicon parameter, rotation angle, and stretching factor of anisotropic vortices, respectively. The OAM density of the field is investigated and proven to continuously vary with the variation of the field’s local intensity distribution, which can be tuned by adjusting one independent parameter. This method can find potential applications as chiral optical fields with separately controllable multiply parameters could be used to fabricate various chiral nanostructures or provide appropriate optical forces to guide and trap different kinds of particles.

Funding

National Natural Science Foundation of China (12304320); Science and Technology Project of Hebei Education Department (QN2023038); Science Foundation of Hebei Normal University (L2023B07).

Disclosures

The authors declare no conflicts of interests.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Reconfigurable generation of chiral optical fields with multiple selective degrees of freedom

Abstract

1. Introduction

2. Theoretical overview

3. Results and discussion

4. Conclusions

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (8)

Tables (1)

Equations (15)

Optics Express