Generation of the periodically polarized structured light beams

Xibo Sun; Lanqin Liu; Wanqing Huang; Ying Zhang; Wenyi Wang; Tianran Zheng; Xi Feng; Yuanchao Geng; Qihua Zhu

doi:10.1364/OE.25.021460

1. Introduction

Besides scalability of high power, the flexibility of laser systems is another important development direction, in the sense that it is relatively easy to adapt parameters such as beam phasefront or beam polarization to specific requirements [1]. In recent years, structured light beam, with phase and polarization singularities, open the door to numerous opportunities in research and industrial applications. The former, provided with twisted phasefronts, are called vortex beams. They are associated with orbital angular momentum [2, 3], which has effect on the laser particles interactions [4, 5] and can potentially increase the communication system capacity [6]. The latter are known as vector beams. The states of the polarization can affect the speed and quality of laser processing [7]. Therefore, the researches on the generation methods of structured light beams have attracted much attention. Most available techniques to generate the vortex beam rely on transform elements such as forked holograms [8], astigmatic mode converters [9] and q-plates [10] on account of the high conversion efficiency and flexibility. And current generation methods of vector beam include segmented waveplates [11], interferometric methods [12], fibers [13], mode superposition [14] and conical intra-cavity prisms [15].

Conical refraction (CR) is one of the methods to generate structured light beams, which was first proposed by William Hamilton in 1832 as an intrinsic property of biaxial crystals [16]. The circularly polarized light passes through an appropriately cut crystal and comes into a beam of light that propagates on a dual-cone [17]. The characteristics of the conical refraction beam such as far-field structure [18, 19], phase structure [20] and polarization structure [21, 22] have been widely studied in recent years. This phenomenon can be regarded as a transformation in polarization and wavefront structures [16] applied in polarimeter [23, 24], communication [25] and optical manipulation [26]. The theoretical model of most previous studies predicting this phenomenon ocurring in biaxial crystals is based on the Belsky-Khapalyuk-Berry integrals [27]. And Turpin et al. demonstrated experimental results on polarization singularities by means of the Stokes vector formalism [22]. The intensity distributions of CR beams and the states of polarization at different transverse planes along the beam propagation direction are determined. Some fundamentals and applications of conical refraction are investigated in ref. [28].The physics behind the phenomenon of CR is very rich and that the singular properties of CR can be really useful in a wide variety of situations.

In traditional communications, the intensity, frequency and polarization of beams have been regarded as the information carrier [25]. Structured light beams with polarization states dependent on the transverse position have been generated to enlarge the channel capacity. In this study, we demonstrate the generation of structured light beams with periodical polarization and phase singularities. The period of the spatial distribution can be used as a parameter to describe the polarization state. The beams are promising in free-space optical polarization demultiplexing and multiplexing. Our technique is based on the application of the conical refraction phenomenon and the 4f-system. The biaxial crystal placed at the transformation plane plays the role of a phase and polarization element. We employ a simple Fourier transform methods to express the electric field distribution of the CR beam without the Belsky-Khapalyuk-Berry integrals [27]. As a result, the input beams with homogeneous polarization states are transformed into periodically structured light beams. The properties of the periodically structured beams via this method for left-handed circularly and linearly polarized Gaussian inputs are analyzed theoretically and experimentally. The states of polarization and the phase distributions can be converted to periodical structures while the intensity profiles stay the same. It is easy and flexible to control the states of polarization and the phase distribution of the periodically structured beam by adjusting the states of polarization of the input beam, the length of crystal and the focal length.

2. Theoretical analysis

2.1. The theoretical model

The illustration of the generation method instituted in our study is shown in Fig. 1. L1 and L2 are same lens to compose the 4f-system. Plane 1 is the input plane of the 4f-system. The waist of the input beam is at plane 1. Plane 2 is the waist of the beam focused by L1. Plane 3 is the focal image plane (FIP) of crystal. Plane 4 is the output plane of 4f-system at the back focus of L2. There is a longitudinal shift of the focus of L1 caused by the inserted crystal. The shift Δ=l(1−1/n₂) is determined using the theory of geometrical optics. For simplicity, the constant phase multipliers are omitted. And because the input plane is on the front focus, the quadratic phase terms introduced by the lenses can be offset. The lens apertures are considered to be much larger as compared with the spatial extent of the beam at the respective planes. We can use the integral relation between the field distributions in the front E_l and back E_f focal planes of a thin lens,

E_{f} (r^{″}) = \frac{1}{λ f} \iint E_{l} (r^{'}) exp [- i \frac{k}{f} (r^{″} \cdot r^{'})] d r^{'} = F {E_{l}}

where F{} denotes Fourier transform, r′ is the position at front focal plane and r″ is the position at back focal plane.

Fig. 1 Illustration of the generation method. Plane 1 is the input plane of 4f-system, plane 2 is the incident beam waist, plane 3 is the focal image plane, plane 4 is the output plane of 4f-system, and plane A is arbitrary plane after crystal.

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In this 4f-system, the electric fields at each plane shown in Fig. 1 have the following relation,

\begin{array}{l} E_{2} = F {E_{1}} \\ E_{4} = F {E_{3}} \end{array}

The input beam of this setup can be written as

E_{1} = E_{1} (\begin{matrix} d_{x} \\ d_{y} \end{matrix})

in which, the component

(\begin{matrix} d_{x} \\ d_{y} \end{matrix})

represents the polarization of input beam, for instance,

(\begin{matrix} 1 \\ - i \end{matrix})

and

(\begin{matrix} 0 \\ 1 \end{matrix})

denote the left-handed circular and y-linear polarization (the y-axis is in the vertical direction), respectively. E₁ is the transverse profile of the input beam.

According to the integral relation shown in Eq. (2) and on the assumption that the polarization is homogeneous independent with the position, the beam waist of the incident beam at the plane 2, can be represented as

E_{2} = F {E_{1}} = F {E_{1}} (\begin{matrix} d_{x} \\ d_{y} \end{matrix}) = E_{2} (\begin{matrix} d_{x} \\ d_{y} \end{matrix})

We consider a slab of transparent biaxial crystal with thickness l and three main refraction indices n₁, n₂, n₃, (n₁ < n₂ < n₃). The semi-angle of the cone in crystal can be given as $A = \sqrt{(n_{2} - n_{1}) (n_{3} - n_{2})} / n_{2}$ .

According to the previous studies by Berry [27] and Phelan [29], the electric field E_A (r, z) transformed by conical diffraction at the plane A can be expressed as

E_{A} (r, z) = \frac{k}{2 π} \iint exp [i k (P \cdot r - \frac{1}{2} Z P^{2})] [cos (k R_{0} P) \cdot I - i \cdot sin (k R_{0} P) M (ϕ)] a_{2} (P) d P

where r = (r, ϕ) is the position at transverse plane in cylinder coordinates,

M (ϕ) = (\begin{matrix} cos ϕ & sin ϕ \\ sin ϕ & - cos ϕ \end{matrix})

, R₀ = A · l is the radius of the refracted ring beam beyond the crystal when the condition R₀/w′ ≫ 1 (w′ is the beam waist at plane 2) is fulfilled, z is the distance from incident beam waist to arbitrary plane (plane A), Z = (z−Δ)n₂ = l +(z−l)n₂ is the propagation distance from the focal image plane (plane 3) as showed in Fig. 1, a₂(P) = F{E₂}=E₁ is the Fourier transform of the electric field at plane 2, and kP is the transverse wavevectors satisfying P = r/f under the paraxial approximation.

Equation (5) represent the transform relation between the electric fields at plane 2 and plane A. This transform induced by conical diffraction has the similar form with Fourier transform. Therefor we have not used the Belsky-Khapalyuk-Berry integrals [27] to reach the solution of CR beam but a Fourier transform methods to express the electric field distribution. Replacing a₂(P) with a₂(P) $(\begin{matrix} d_{x} \\ d_{y} \end{matrix})$ , the associated Fourier transform of E_A therefore can be derived as

a_{A} (P, z) = exp (- \frac{1}{2} i k P^{2} Z) [cos (k P R_{0}) I - i sin (k P R_{0}) M (ϕ)] a_{2} (P) (\begin{matrix} d_{x} \\ d_{y} \end{matrix}) .

When Z = 0, E_A describes the electric field at the focal image plane. So the associated Fourier transform of E₃ at image focal plane (plane 3) can be derives as

a_{3} (P) = F {E_{3}} = a_{2} (P) [cos (k P R_{0}) I - i sin (k P R_{0}) M (ϕ)] \cdot (\begin{matrix} d_{x} \\ d_{y} \end{matrix})

We can derive the expression of the electric field in the output plane (plane 4) via standard Fourier transform shown in Eq. (2) and the relation P = r/f.

E_{4} (r) = E_{1} [cos (k R_{0} r / f) I - i sin (k R_{0} r / f) M (ϕ)] \cdot (\begin{matrix} d_{x} \\ d_{y} \end{matrix}) = (\begin{matrix} E_{x} \\ E_{y} \end{matrix})

The intensity of the output beam can be derived as

I_{4} = E_{4}^{*} E_{4} = {E_{1}}^{2} .

These expressions of Eq. (8) and Eq. (9) show us that the intensity profile of the output beam remains unchanged, while the polarization distribution is no longer homogeneous. The states of polarization of the output beams will be discussed in the next section in details.

2.2. The methods to represent the structures of beam

Stokes parameter is a standard tool to analyze the states of polarization of a light beam with electric field E = (E_x, E_y), which can be calculated by the relation shown as follows [21]

\begin{array}{l} S_{0} = & {| E_{x} |}^{2} + {| E_{y} |}^{2} = I_{0 °} + I_{90 °} \\ S_{1} = & {| E_{x} |}^{2} - {| E_{y} |}^{2} = I_{0 °} - I_{90 °} \\ S_{2} = & 2 Re [{E_{x}}^{*} E_{y}] = I_{45 °} - I_{- 45 °} \\ S_{3} = & 2 Im [{E_{x}}^{*} E_{y}] = I_{R} - I_{L} \end{array}

where I_φ (φ = 0°, 45°, 90°, −45°), I_R and I_L indicate the intensity of linearly polarized lights with azimuth φ (φ = 0° corresponds to x-linear polarization), right-handed and left-handed circularly polarized lights, respectively, which can be measured in experiments directly.

With the following equations, which show the relations between Stokes parameters and the azimuth α and ellipticity β of the polarization ellipse, the states of polarization can be confirmed uniquely by means of measuring the Stokes parameters.

\begin{array}{l} α = \frac{1}{2} arctan (\frac{S_{2}}{S_{1}}) \\ β = \frac{1}{2} arctan (\frac{S_{3}}{\sqrt{{S_{1}}^{2} + {S_{2}}^{2}}}) \end{array}

By substituting Eq. (8) with the polarization state $(\begin{array}{r} 1 \\ - i \end{array})$ of the input beam into formulas shown in Eq. (10), the Stokes parameters of the output beam can be derived. By substituting the results into the Eq. (11), the polarization ellipses of the output beam for left-handed circularly polarized input will be expressed as

\begin{array}{l} α_{lc} = - \frac{1}{2} arctan (\frac{cos ϕ}{sin ϕ}) \\ β_{lc} = \frac{1}{2} arctan [\frac{cos (2 k R_{0} r / f)}{\sqrt{{sin}^{2} (2 k R_{0} r / f)}}] . \end{array}

The azimuth α_lc and ellipticity β_lc become azimuthal and radial periodical, respectively. The radial profile period T satisfies T = λf/(2Al). In a similar way, the polarization ellipses with the linearly polarized input beam could be calculated as

\begin{array}{l} α_{lp} = - \frac{1}{2} arctan \frac{{sin}^{2} (k R_{0} r / f) sin (2 φ)}{\sqrt{{(2 \sin^{2} (k R_{0} r / f) {sin}^{2} (φ) - 1)}^{2}}} \\ β_{lp} = \frac{1}{2} arctan \frac{2 cos (k R_{0} r / f) sin (k R_{0} r / f) sin φ}{\sqrt{{sin}^{2} (2 φ) \sin^{4} (k R_{0} r / f) + {(2 \sin^{2} (k R_{0} r / f) {sin}^{2} (φ) - 1)}^{2}}} . \end{array}

The polarization distribution has been changed into periodical structure according to the trigonometric function in Eq. (13) and Eq. (13). By means of changing the focal length f and length of crystal l, the period of the polarization structure can be regulated.

3. Numerical calculations and discussions

In this section, based on the analytical expressions derived in the last section, numerical examples are performed to illustrate the transformation progress of circularly and linearly polarized input beams. In the numerical examples, the parameters are chosen as follows.

The wavelength of incident beam is 633nm, with radius w = 1mm in left-handed circular and linear polarization. A 10mm long KGd(WO₄)₂ crystal with main refraction indices n₁ = 2.01348, n₂ = 2.04580, n₃ = 2.08608 is adopted as the biaxial crystal yielding CR ring radius of R₀ = 0.18mm. The focal lengths of thin lenses L1 and L2 are both 200mm. The waist w′ of the beam after L1 can be calculated by Fourier optics and w′ = 20μm (The condition R₀/w′ ≫ 1 is fulfilled).

Figure 2(a) shows the intensity and polarization distributions of CR beam in FIP obtained according to the Fourier transform of the expressions Eq. (7). And Fig. 2(b) shows the results gained from the theoretical model predicting the transformation based on the Belsky-Khapalyuk-Berry integrals [27] which denote the exact solution of the paraxial model. The similar intensity and polarization distributions of numerical results from two solutions can be observed, which demonstrates the correctness of the analysis provided in last section.

Fig. 2 The intensity and polarization distributions of CR beam. (a) The result obtained by means of Fourier transform expression. (b) The result of Belsky-Khapalyuk-Berry solution with radial integrals containing Bessel functions. The green ellipses denotes the left-handed states while the blue ellipses represent the right-handed states and linearly states.

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Then intensity distributions of the output beams can be represented in Fig. 3(a) for the left-handed circularly and linearly polarized input beams. There seems no difference in the intensity distributions among inputs and outputs. To obtain the Stokes parameters of the output beam, which can uniquely confirm the states of polarization, the intensity distributions I_φ (φ = 0°, 45°, 90°, −45°), I_R and I_L of the output beams are represented in Figs. 3(b)–(c). When the states of polarization of the input beams are adjusted, the intensity profiles of the output beams stay the same, while the properties such as the states of polarization and the phase distribution are changed.

Fig. 3 The intensity profile distributions obtained from numerical simulations. (a) The intensity profiles of inputs and outputs. (b,c) The intensity distributions of each components I_φ (φ = 0°, 45°, 90°, −45°), I_R and I_L of the outputs for left-handed circularly and y-linearly polarized Gaussian inputs.

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As for the left-handed circularly polarized input beam, the linearly polarized compositions of the output shown in Fig. 3(b) represent ’ripple-like’ distributions. While the circularly polarized compositions are turned into a kind of structures with central symmetry, consisting of a series of concentric rings with hollow center for right-handed circularly polarized input and solid center for left-handed circularly polarized input. For the y-linearly polarized input beam shown in Fig. 3(c), the circularly polarized compositions come into the ’ripple-like’ distribution with null-intensity points lying on the y-axis, while the null-intensity points of linearly polarized (φ = 0°, 45°, 90°, −45°) compositions lie on the perpendicular directions to polarizations. To find out the reason for this phenomenon, the phase distributions are considered, and the y-linearly polarized composition is taken as an example represented in Fig. 4.

Fig. 4 The phase distribution. (a)The phase distributions of y-linearly polarized composition of output beam for left-handed circularly polarized input beam. (b) The details of phase distributions marked by a solid square in (a). The dashed lines represent the x-axis and solid annular arrows represent the handedness of the phase singularities.

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As we can see, there are phase singularities at the points where the intensities are null shown in Fig. 3(b). And phase distributions for y-linear polarization possess phase singularities along the y-axis. The phase singularities lying sides of x-axial have the opposite handedness, such as right-handedness for y > 0 and left-handedness for y < 0.

According to relations shown in Eq. (10) and the simulated results represented in Fig. 3, the numerical transverse pattern of Stokes parameters of the outputs are illustrated in Fig. 5 for left-handed circularly polarized and y-linearly polarized Gaussian inputs.

Fig. 5 Transverse pattern of Stokes parameters obtained from numerical simulations for left-handed circularly polarized and y-linearly polarized Gaussian inputs

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By means of the relations between the Stokes parameters and the polarization ellipse shown in Eq. (11), the polarization distributions of the Gaussian output beams for different polarized input beams are confirmed uniquely and represented in Fig. 6. The states of polarization distributions are indicated by polarization ellipses in two kinds of colors. The green ellipses denote the left-handed states while the red ellipses represent the right-handed states.

Fig. 6 The states of polarization distributions of outputs from 4f-system for (a) left-handed circularly polarized and (b) linearly polarized input beams. The green ellipses denote the left-handed states while the red ellipses represent the right-handed states.

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As shown in Fig. 3 and Fig. 6, the intensity profiles of the electric fields still satisfy Gaussian distribution, while the states of polarization distributions are no longer homogeneous (linearly or circularly polarized). In Fig. 6(a), as for the left-handed circularly polarized input beam, the output beam has periodical annular structures of polarization composed of left-handed states and right-handed states in adjacent rings. The azimuths and the ellipticities of the ellipses change continuously along the ring, such that every two diametrically opposite points have orthogonal azimuth of ellipses. In Fig. 6(b), as for the y-linearly polarized input beam, there are periodical semi-annular structures with the opposite signs of polarizations at points symmetrical about x-axis. The polarization distribution along x-axis is y-linearly polarized same with the polarization of input beams.

In order to show the effects of the crystal length and focal length on the period, different parameters are used to calculate the stokes parameters of the output beam. Figure 7 represents density plots of the numerically calculated Stokes parameters for (a) crystal length l = 5mm, 10mm, 15mm and focal length f = 200mm, and (b) focal length f = 150mm, 200mm, 250mm and crystal length l = 10mm. By means of changing the focal length f and length of crystal l, the period of the polarization structure can be regulated. The profile period is in direct proportion to the focal length f and in inverse proportion to the crystal length l.

Fig. 7 Transverse pattern of the Stokes parameters obtained from numerical simulations for different (a) crystal length l = 5mm, 10mm, 15mm and (b)focal length f = 150mm, 200mm, 250mm.

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4. Experimental results

To test the validity of the obtained theoretical results, we have performed corresponding experimental measurements. The diagrams of the experimental setups are illustrated in Fig. 8.

Fig. 8 Schematic diagrams of the experimental setups for (a) the measurement of the state of polarization and (b)the interference experiment. QWP, quarter-wave plate; P, polarizer; L, lens; BS, beam splitter; M, mirror.

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A collimated random polarized He-Ne laser at wavelength of 632.8nm is used as the light source. The input beam is coupled to a beam expander, yielding a beam waist radius 1mm at input plane. Polarizer (P1) and quarter-wave plate (QWP1) are adopted to generate the left-handed circularly polarized beams and y-linearly polarized beams. Then the beam is focused by a focusing lens (L1) along one of the optic axes of a biaxial crystal. The experiments are carried out using a pair of lenses with 200mm focal length and a 10mm long KGd(WO₄)₂ crystal with n₁ = 2.01348, n₂ = 2.04580, n₃ = 2.08608. After that, the FIP of CR beam is set at the front focal plane of L2 to project the image of FIP into the CCD camera at back focal plane. Quarter-wave plate (QWP2) and polarizer (P2) before CCD are used as polarization state detectors to measure the stokes parameters of the output beams. In Fig. 8(b), beam splitters (BS1 and BS2) and mirrors are applied to derive the Gaussian beam as a reference beam when the interference experiment is carried out.

We have performed experiments in order to prove theoretical results analyzed above. Figure 9 shows the obtained experimental input and output beams. The experimental results agree with the theoretical results represented in Fig. 3 in last section. According to the relations shown in Eq. (10), the measured Stokes parameters are represented in Fig. 10. Minor discrepancy has been observed for the directions of axis of symmetry for the case of a linearly polarized input beam. This can be explained in terms of the experimental error induced by the polarizer used. The axis of the polarizer slightly deviate from the direction required by the theoretical analysis, which disturbs the axis of symmetry of intensity distributions. On account of the relations between the polarization ellipse and Stokes parameters denoted in Eq. (11), the periodical polarization structures of outputs can be confirmed as shown in Fig. 6.

Fig. 9 The intensity profile distributions obtained from experiments. (a) The intensity profiles of inputs and outputs. (b,c) The intensity distributions of each component I_φ (φ = 0°, 45°, 90°, −45°), I_R and I_L of the outputs for left-handed circularly and y-linearly polarized Gaussian inputs.

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Fig. 10 Transverse pattern of Stokes parameters obtained from experiments for left-handed circularly polarized and y-linearly polarized Gaussian inputs.

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The phase distribution of the periodically structured beam generated is measured by superimposing it on a co-propagating coherent plane wave. To observe the interference, a fraction of the Gaussian beam, derived from the beam splitter placed behind the quarter-wave plate (QWP1), is aligned by means of the mirrors (M1 and M2) so that it can be superimposed on the output beam. The Gaussian beam acts as a plane wave coherent with the y-polarized output beam. The interference pattern is recorded by the CCD. The theoretically expected and experimentally recorded intensity patterns are shown in Fig. 11. The theoretical analysis and numerical simulation employed here have shown consistency. A series of clear forklike pattern can be observed at the null-intensity points as marked by red circle in Fig. 11. The forks appear along the y-axis. And the forks lying sides of x-axial are in the opposite directions. The direction of the forks represents the handedness of phase singularities. The patterns validate the properties of the phase singularities show in Fig. 4.

Fig. 11 The intensity interference patterns. (a) The theoretically expected result. (b) The experimentally recorded result. Solid circles are used to mark the forks. Dashed lines represent the x-axis.

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Based on the analysis above, we can construct periodical polarization structures into arbitrary homogeneously polarized beams without changing the intensity distributions. The period and the polarization structures can be regulated respectively by changing the system parameters (focal length and crystal length) and the polarization states of input beams.

5. Conclusion

In summary, a kind of structured light beam with periodical polarization and phase singularities is generated by a transformation setup consisting of conical refraction transformation and 4f-system. Theoretical analysis indicates that the periodical polarization structures and phase singularities can be formed without changing the intensity distributions of the input beam. And the period of the structure is dependent on the length of crystal and the focal length. It is in direct proportion to the focal length and in inverse proportion to the length of crystal. The states of polarization and phase distributions of outputs are studied theoretically and experimentally in detail for different polarized inputs. It is shown that the polarization of the input beam can be used to control the polarization and phase structures of the output beam. The periodical polarization structures can be constructed into arbitrary homogeneously polarized beams without changing the intensity distributions. In traditional communications, the intensity, frequency and polarization of beams have been regarded as the information carrier. These periodically structured light beams have the polarization states dependent on the transverse position.

Funding

Science Foundation of China Academy of Engineering Physics, China (Grant No. 2014A0401018); Foundation of State Key Laboratory for Plasma Physics, China (Grant No. 9140C680604150C68299).

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Generation of the periodically polarized structured light beams

Abstract

1. Introduction

2. Theoretical analysis

2.1. The theoretical model

2.2. The methods to represent the structures of beam

3. Numerical calculations and discussions

4. Experimental results

5. Conclusion

Funding

References and links

Cited By

Figures (11)

Equations (13)

Optics Express