Chiro-optical fields with asymmetric orbital angular momentum and polarization

Rui Liu; Yan Li; Duo Deng; Yi Liu; Liang Tao Hou; Yan Hua Han

doi:10.1364/OE.449884

1. Introduction

Since the concept of the optical vortex was proposed [1], extensive attention has been attracted due to the special properties of its phase singularity [2]. Optical vortex has been widely used in many applications, such as optical communications [3,4], vector lasers [5,6], nanoscale microscopy [7], and nonlinear optics [8]. For the conventional circular optical vortex, there is a spiral wavefront with exp(ilφ), where l is called topological charge (TC) and φ is the azimuthal coordinate. The properties of the conventional circular optical vortex are almost completely determined by the TC [9], which also limits the potential application of the optical vortex beams in certain fields. In order to couple multiple OAM beams into an optical fiber with a fixed annular index profile, the concept of the perfect vortex is proposed [10–13]. The perfect vortex can modulate the spatial distribution characteristics of the conventional vortex beams while maintaining the orbital angular momentum carried by them, which greatly enhances their applications.

In recent years, chiro-optical fields have received extensive attention from researchers due to their superiority in optical tweezers [14,15] and the manipulation of particles [16,17]. In addition, the chirality of the optical vortex can be transferred to the chiro-optical field, by using phase modulation method. And the chiro-optical fields have characteristics of self-healing [18] and can be used for optical communication [19,20] and chiral micro/nanofabrication [21,22]. In addition, the OAM of the chiro-optical field generated by the vortex phase is maintained even if the circular fringe distribution is broken. However, most of the chiro-optical field generated by the vortex phase generally has an axial or cylindrical symmetry and the OAM magnitude on each side-lobe is the same [23,24]. Furthermore, the polarization of these chiro-optical fields is also uniform or rotational symmetry [25]. To improve the chiro-optical fields for practical applications, the continuous and adjustable OAM modes and polarization on each side-lobe need to be explored.

In this paper, we proposed a flexible method for generating asymmetric chiro-optical fields with locally controllable OAM and polarization states. For the single-lobed chiro-optical field, a phase plate was calculated by the coordinate transformation method of the perfect vortex and then loaded into the spatial light modulator to generated the single-loaded chiro-optical field. Then, the method of rotation matrix was employed to stitch several discrete chiro-optical fields and generate a multi-lobed chiro-optical field. Different from the traditional symmetrical distribution of chiro-optical fields, the local topological charge(TC) of each side-lobe of the multi-lobed chiro-optical field could be arbitrarily changed by adjusting the corresponding parameter ‘ l ‘ of HPPs. Furthermore, the polarization state of each side-lobe of the asymmetric chiro-optical field can also be modulated independently by using two cooperative spatial light modulators. The experimental setup composed of two spatial light modulators. One (SLM1) is used to modulate the radially polarized beam into arbitrarily linearly polarized beam, and the another spatial light modulator (SLM 2) is used to load the phase information of the chiro-optical field to generate a multi-lobed chiro-optical field with controllable polarization state of each side-lobe.

2. Chiral field of a locally controllable OAM

Generally, the phase of a perfect vortex beam can be expressed as:

(1)$$\exp [i\varphi (r,\theta )] = \exp[i(l\theta + \beta r)]$$

where φ denotes the phase of the perfect vortex beam. (r, θ) are the polar coordinates. l represents the TC of the perfect vortex and β represents the axicon parameter. In order to obtain a chiro-optical field in the shape of a spiral, the equiangular spiral is considered. The phase of perfect vortex in the Cartesian coordinate system could be expressed as:

(2)$$\exp[i\varphi (x,y)] = \exp[i(l \cdot \textrm{arctan}(\frac{y}{x}) + \beta \sqrt {{x^2} + {y^2}} )]$$

where (x, y) are the two dimensional coordinates. Then, a transformation matrix is employed to achieve the coordinate transformation for the equiangular spiral:

(3)$$\left( \begin{array}{l} {x_0}\\ {y_0} \end{array} \right) = \left( {\begin{array}{cc} {{e^{A\theta }}}&{ - {e^{B\theta }}}\\ {{e^{C\theta }}}&{{\textrm{e}^{D\theta }}} \end{array}} \right)\left( \begin{array}{l} x\\ y \end{array} \right)$$

where (x₀, y₀) represents the coordinate after transformation and it can be expressed as:

(4)$$\left\{ \begin{array}{l} {x_0}^2 = {e^{2A\theta }}{x^2} - 2{e^{(\textrm{A} + \textrm{B})\theta }}xy + {e^{2B\theta }}{y^2}\\ {y_0}^2 = {e^{\textrm{2\textrm{C}}\theta }}{x^2} + 2{e^{(\textrm{C} + \textrm{D})\theta }}xy + {e^{2\textrm{D}\theta }}{y^2} \end{array} \right.$$

Thus, the polar coordinates of chiro-optical field, ‘r₀’and ‘θ₀’ can be expressed as:

(5)$$\left\{ \begin{array}{l} {r_0} = \sqrt {{x_0}^2 + {y_0}^2} = \sqrt {({e^{2A\theta }} + {e^{2C\theta }}){x^2} + ({e^{2B\theta }} + {e^{2D\theta }}){y^2} - 2xy({e^{(\textrm{A} + \textrm{B})\theta }} - {e^{(\textrm{C} + \textrm{D})\theta }})} \\ {\theta_0} = {\tan^{ - 1}}(\frac{{{y_0}}}{{{x_0}}}) \end{array} \right.$$

When A, B, C, D in Eq. (4) take the same value A, the relationship between polar coordinate ‘r ‘ and ‘r₀’ can be expressed as:

(6)$${r_0} = r\sqrt 2 {e^{A\theta }}$$

Obviously, Eq. (6) is the function of an equiangular spiral. Therefore, the phase of the chiro-optical field can be expressed as:

(7)$$\exp [i\varphi ({r_0},{\theta _0})] = \exp[i(l{\theta _0} + \beta {r_0})]$$

In order to achieve the multi-lobed chiro-optical field, tight focusing system was employed to generate the far-field diffraction of a chiro-optical field. In tight focusing system, the electric field in the neighborhood of the focusing spot can be expressed as a Fourier transform:

(8)$$\begin{aligned} &\vec{E}(x,y,z) = \int\!\!\!\int {[U(\theta ,\varphi )\exp(i{k_z}z)/{\textrm{cos}}\theta ]} \exp [ - i({k_x}x + {k_y}y)]d{k_x}d{k_y}\\ &= FFT\{ U(\theta ,\varphi ){{\vec{E}}_t}(\theta ,\varphi )\exp(i{k_z}z)/{\textrm{cos}}\theta \} \end{aligned}$$

where k = 2π/λ; U(θ, φ) is the apodization function; E_t(θ, φ) is the transmitted field; F denotes the Fourier transform; θ=arcsin(r/R×NA/n_t) is the deflection angle, in which R is the maximum radius of the aperture. n_t= 1 is the air index behind high NA objective.

The Matlab software was used to calculate the HPPs and the checkerboard-segmentation method was employed for improving the quality of the chiro-optical field [13]. According to Eq. (6), the spatial distribution of the chiro-optical field depends on the parameter ‘A’. Therefore, the value of parameter ‘A’ could influence on the shape of the single-lobe chiro-optical field and the characteristic of OAM. As shown in Figs. 1(a)–1(c), the HPPs are calculated based on Eq. (6) with the value of ‘A’ are 0.2, 0.4, and 0.6, respectively. the corresponding simulation results of the chiro-optical fields can be seen in Figs. 1(d)–1(f), in which the blue lines are the equiangular spirals that are consistent with the helix-shaped optical field. Moreover, the TC of Chiro-optical fields could be different as shown in Figs. 1(g)–1(i). To identify the TC of chiro-optical fields, the interference patterns of the chiro-optical field and spherical wave are shown in Figs. 1(j)–1(l). The number of interference fringes is equal to the TC. It means that the chiro-optical fields retain the OAM characteristic and the TC are 1, 3, and 6 respectively.

Fig. 1. (a)-(c) The HPPs with A = 0.2, 0.4, 0.6. (d)-(f) The simulation results with A =0.2, 0.4, 0.6. (g)-(i) The simulation results with A =0.4, l = 1, 3, 6. (j)-(l) The interferograms of the corresponding chiro-optical fields and spherical wave.

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In order to prove the controllability of chiro-optical filed, an experimental setup was built as shown in Fig. 2(a). A linearly polarized laser beam with a wavelength of 532 nm was used as the incident beam, which was generated by a diode pumping solid-state (MGL-III-532-100 mW). Then the laser beam was expanded and collimated by a collimator. The beam reflected from the Beam Splitter and Lens 1 was used as reference beam for interference. The beam passing through the Beam Splitter was used to generate spherical wave. An aperture was used to ensure the beam quality. Behind the aperture, a spatial light modulator (SLM) (DHC, RSLM-II, 1920×1080 pixels, pixel pitch 6.4μm, effective face reading size is 15.36 mm × 8.64 mm) was employed for loading the calculated HPP. Since the SLM can only modulate the horizontally polarized component of the incident beam, a polarizer was used to generate a horizontal polarized beam. Finally, the modulated beam reflected by the SLM was focused by a Lens2, and a chiro-optical field was captured by a CCD camera. The experimental results of Figs. 1(d)–1(l) are shown in Figs. 2(b)–2(g), that can identify the parameter ‘A’ can modulate the shape of chiro-optical field and the interference results can prove that the optical vortex has not been destroyed.

Fig. 2. (a) The diagram of the experimental setup. (b)-(c)The experimental results with A = 0.2, 0.6. (d)-(e) The experimental results with l = 3, 6. (f)-(g) The experimental results of detecting TC.

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In order to achieve the asymmetric chiro-optical fields that have locally controllable OAM and rotation angle, a rotation matrix is employed to calculate the HPP. In the Cartesian coordinate system, the transformed coordinates with Rotation matrix can be expressed as:

(9)$$\left( \begin{array}{l} {x_n}\\ {y_n} \end{array} \right) = \left( {\begin{array}{cc} {\cos (\alpha )}&{ - \sin (\alpha )}\\ {\sin (\alpha )}&{\cos (\alpha )} \end{array}} \right)\left( \begin{array}{l} {x_0}\\ {y_0} \end{array} \right)$$

where (x_n, y_n) represents the coordinate of the chiro-optical field after rotation. When the azimuth angle α is given by different values, the chiro-optical field will also rotate at the same angle. Consequently, Eq. (6) can be rewritten as:

(10)$$\exp [i{\varphi _n}({x_n},{x_n})] = \exp[i({l_n} \cdot \textrm{arctan}(\frac{{{y_n}}}{{{x_n}}}) + \beta \sqrt {{x_n} + {y_n}} )]$$

As shown in Figs. 3(a)–3(b), the value of α is set as π/3, 2π/3 respectively and the corresponding simulation optical field results have also been rotated counterclockwise π/3 and 2π/3. In order to conveniently observe the changes of the azimuth angle, the dashed auxiliary line is drawn in each chiro-optical field.

Fig. 3. (a)-(b)The simulation optical field with rotation angles of π/3, 2π/3. (c) The HPP of multi-lobed chiro-optical field (d) The encoding method of Checkerboard. The simulation and experimental results of multi-lobed chiro-optical fields with two side-lobes (e) and (f), three side-lobes (i) and (j). The interference simulation and experimental results two side-lobes (g) and (h), three side-lobes (k) and (l).

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By superimposing basic phase distribution of chiro-optical fields with different rotation angles and OAM states, the multi-lobed chiro-optical field with a controllable number of lobes can be obtained. Moreover, the TC on each lobe can also be arbitrarily modulated. The phase function of the target optical field can be expressed as:

(11)$$\exp [i\phi ({x_n},{x_n})] = \sum\limits_{n = 1}^c {\exp [i({l_n} \cdot \textrm{arctan}(\frac{{{y_n}}}{{{x_n}}}) + \beta \sqrt {{x_n} + {y_n}} )](c = 1,2,3,4\ldots \ldots )}$$

As shown in Fig. 3(c), the whole back aperture plane is considered as a square checkerboard with length of 1000 pixels, which also could be considered as part of SLM. When encoding the HPP, the 1000 pixels will be divided into different groups, that depending on the number of chiro-optical field lobes. As shown in Fig. 3(d), a encoding method with a group of two pixels was used to generate a two-lobed chiro-optical field. By superimposing the HPPs with α=0, TC = 1, and α=π, TC = 2, a two-lobed chiro-optical field can be generated and the simulation and experimental result are shown in Figs. 3(e) and 3(f) respectively. The number of interference fringes of the generated two-lobed chiro-optical field are 1 and 2 respectively, that means the TC could be controlled separately. Furthermore, a three-lobed chiro-optical field was generated by HPPs with α=0, TC = 2, α=2π/3, TC = 3 and α=4π/3, TC = 4. The corresponding simulation and experimental results are shown in Figs. 3(i) and 2(j). Similarly, the simulation and experimental interference fringes of the three-lobed optical field are shown in Fig. 3(k) and 3(l) respectively. Thus, by modulating the parameter of HPP, asymmetrical chiro-optical fields with controllable side-lobe numbers and arbitrary OAM states are generated.

3. Chiro-optical field with a locally controllable polarization state

When the chiro-optical field has a continuously adjustable polarization state, it is conducive to expanding its practical applications in the fields of optical communication and manipulation. To modulate the polarization state of each side-lobe of the multi-lobed chiro-optical field, a radially polarized beam is selected as the incident beam. As the incident beam is a non-uniform polarized beam, the polarization state of each position on the cross-section of the beam is different. This non-uniform polarization is helpful to modulate the polarization state of each side-lobe of the chiro-optical field. There have been reported a method by adding a phase retarder to a radially polarized beam to achieve polarization state modulation [26]. However, those research are limited to conventional radially polarized beam [27]. In general, these methods cannot achieve arbitrary polarization modulation of chiro-optical fields.

Therefore, we proposed a method to achieve locally controllable polarization of the perfect vortex and the chiral optical beam. The method for controlling the polarization state of the beam is shown in Fig. 4(a) that means the radially polarized beam was used as the incident beam. To generated the radially polarized beam, the Gaussian-laser beam with the wavelength 532 nm passed through the collimator. Behind the collimator, a radial-polarization converter (RPC) was employed to generate a radially polarized beam as shown in Fig. 4(b). After the radially polarized beam passes through the QWP1, HWP, SLM and QWP2 successively, the corresponding linearly polarized beam can be obtained, where the angle between the fast axis of QWP1 and the x axis is 0, the angle between the fast axis of HWP and the x axis is π/8, the angle between the fast axis of QWP2 and the x axis is π/4 respectively. When the modulated beam passed through the second SLM (SLM2), the transmitted beam can be modulated by the the second HPP that Loading the phase of multi-lobed chiro-optical field. Finally, the far-field diffraction pattern was captured by the CCD camera placed at the focal plane of L3. To verify the polarization, a polarizer (analyzer) was employed in front of the lens L3.

Fig. 4. (a) The principle of converting radial polarization to linear polarization. (b) The Optical structure design for the generation of vector chiro-optical field..

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The phase screen loaded on the two SLMs is important to modulate the polarization state. The radially polarized beam can be expressed in the form of Jones vector as

(12)$$\left( \begin{array}{l} {E_x}\\ {E_y} \end{array} \right) = c\left( \begin{array}{l} \cos \theta \\ \sin \theta \end{array} \right)$$

where c represents the complex amplitude distribution of the beam. θ represents the polar angle of incident perfect vortex beam on the cross section. The Jones vector of the transmitted beam after passing through the optical path in Fig. 4(a) can be expressed as [28]:

(13)$$\begin{aligned} &{J_{out}} = \frac{1}{2}\left( {\begin{array}{cc} 1&0\\ 0&{{e^{ - i\frac{\pi }{2}}}} \end{array}} \right)\left( {\begin{array}{cc} 1&{ - 1}\\ 1&1 \end{array}} \right)\left( {\begin{array}{cc} 1&0\\ 0&{{e^{i\phi }}} \end{array}} \right)\left( {\begin{array}{cc} 1&1\\ 1&{ - 1} \end{array}} \right)\left( {\begin{array}{cc} 1&0\\ 0&{{e^{ - i\frac{\pi }{2}}}} \end{array}} \right)\left( \begin{array}{l} \cos \theta \\ \sin \theta \end{array} \right)\\ &= {e^{ - i\frac{\pi }{2}}}{e^{i\frac{\phi }{2}}}\left( \begin{array}{l} \cos \frac{\phi }{2}\cos \theta - \sin \theta \sin \frac{\phi }{2}\\ \sin \frac{\phi }{2}\cos \theta + \cos \frac{\phi }{2}\sin \theta \end{array} \right) = {e^{ - i\frac{\pi }{2}}}{e^{i\frac{\phi }{2}}}\left( \begin{array}{l} \sin (\frac{\phi }{2} + \theta )\\ \cos (\frac{\phi }{2} + \theta ) \end{array} \right) \end{aligned}$$

where ϕ represents the phase that loaded on the SLM1. The radially polarized beam can be converted into linearly polarized beam only by taking an appropriate ϕ, and the Jones vector of the linearly polarized beam can be expressed as:

(14)$${E_L} = \left( \begin{array}{l} \cos {\alpha_n}\\ \sin {\alpha_n} \end{array} \right)$$

where α_n indicates the polarization direction of linearly polarized beam. Therefore, when the value of ϕ is (π−2(α_n+θ)), Eq. (13) can be written as:

(15)$${J_{out = e}}{e^{ - i\frac{\pi }{2}}}{e^{i\frac{{\pi - 2({\alpha _n} + \theta )}}{2}}}\left( \begin{array}{l} \cos {\alpha_n}\\ \sin {\alpha_n} \end{array} \right)$$

Combining the Eq. (8) and Eq. (13), the HPPs could be calculated with controllable polarization state of each side-lobe:

(16)$$\begin{array}{l} {E_{out}} = {e^{ - i\frac{\pi }{2}}}{e^{i\phi 2}}{e^{i\frac{{{\phi _1}}}{2}}}\left( \begin{array}{l} \sin (\frac{{{\phi_1}}}{2} + \theta )\\ \cos (\frac{{{\phi_1}}}{2} + \theta ) \end{array} \right)\\ \left\{ \begin{array}{l} {\phi_1} = \pi - 2({\alpha_n} + \theta )\\ {e^{i{\phi_2}}} = \sum\limits_{n = 1}^c {\exp [i({l_n} \cdot \textrm{arctan}(\frac{{{y_n}}}{{{x_n}}}) + \beta \sqrt {{x_n} + {y_n}} )](c = 1,2,3,4\ldots \ldots )} \end{array} \right. \end{array}$$

The HPPs loaded on the two SLMs are shown in Figs. 5(a) and 5(c). The simulation result is shown in Fig. 5(b). As the value of α_n is set to π/4, π/2, 3π/4, and π respectively, each side-lobe of the four-lobed chiro-optical field has independent and different polarization state. By rotating the analyzer that is placed behind SLM2, the polarization state on each side-lobe can be verified. The verification results are shown in Figs. 5(d)–5(g). The corresponding experimental results are shown in Figs. 5(h)–5(k). The arrow in the lower right corner indicates the rotation angle of the analyzer. The experimental results clearly show that the polarization of each lob could be controlled. Table 1 summarizes different method to compare the results of our proposed scheme with that of the other methods. From the view of number of lobs, modulation of polarization state, pattern of optical field, vortex properties and the method of research.

Fig. 5. (a) and (c)The HPPs that loaded on SLM1 and SLM2. (b)The Simulation Results of multi-lobed chiro-optical field that can be modulated in any polarization. (d)-(g)The simulation results multi-lobed chiro-optical fields after rotating the analyzer at different angles. (h)-(k)The experimental results multi-lobed chiro-optical fields after rotating the analyzer at different angles.

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Table 1. Comparison with previous research results

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4. Conclusion

In this paper, we achieved the asymmetric chiro-optical fields that has locally controllable orbital angular momentum (OAM) and polarization states. The HPP that calculated based on coordinates transformation of the perfect vortex, is employed to achieve the OAM controllability of a single chiro-optical field. Furthermore, a multi-lobed chiro-optical field with asymmetric OAM on each side-lobe is generated by combining several discrete chiro-optical fields with different rotation angles and TCs. The polarization state of each side-lobe of the asymmetric chiro-optical field can also be modulated independently by an optical system that using two special HPPs and wave-plane. The proposed asymmetric chiro-optical fields with locally controllable OAM and polarization states may have widely applications in optical communications, particle manipulation, and enantiomer-selective sensing.

Funding

National Natural Science Foundation of China (11874133); Natural Science Foundation of Shandong Province (ZR2021MF111).

Disclosures

There are no conflicts of interest related to this article.

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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References	Modulate of polarization state	Number of focus	The pattern of optical field	Vortex properties	Method
[15]	No	1	Chiro-optical field	Less variability in the chiro-optical field	Exp.
[19]	No	Multiple	Chiro-optical field	Less variability in the chiro-optical field	Exp.
[24]	No	Multiple	Chiro-optical field	Less variability in the chiro-optical field	Sim.
[25]	No	Multiple	Chiro-optical field	Topological charge is destroyed	Exp.
[27]	Yes	1	Gaussian beam	No vortex properties	Sim.
[28]	Yes	Multiple	Gaussian beam	No vortex properties	Exp.
Our Research	Yes	Multiple	Chiro-optical field	The vortex properties are almost preserved and adjustable	Exp.

Chiro-optical fields with asymmetric orbital angular momentum and polarization

Abstract

1. Introduction

2. Chiral field of a locally controllable OAM

3. Chiro-optical field with a locally controllable polarization state

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (5)

Tables (1)

Equations (16)

Optics Express