1. Introduction

As a typical anisotropic medium, uniaxial crystals play an important role in light modulation and control. Recently, the investigation of the propagation of light beams in uniaxial crystals has attracted intensive attention [1–14]. In 2001, A. Ciattoni et al. [1] proposed a paraxial vector theory of light beams propagating in uniaxial crystals. Benefit from their theory, numerous works have investigated the propagation of many kinds of light beams in uniaxial crystals, such as Hermite–Gauss beams [2], Laguerre–Gauss beams and Bessel–Gauss beam [3], circular Airy beams (CABs) [4], circular Airy Vortex Beams (CAVBs) [5], etc. Through these investigations, not only the scientific curiosities of researchers have been satisfied, but also many great applications have been found. For example, Ref. [6] found that an optical vortex possessing reversed circular polarization with a topological charge number change of 2 will be generated when a particular circularly polarized beam propagating along the optical axis of a uniaxial crystal. It can be used to realize the generation of vortex beams [7] and the change of the Bessel beam order [8]. W. Zhu et al. [9] showed that the uniaxial crystals can be used to generate and control the right-handed circularly polarized (RHCP) and left-handed circularly polarized (LHCP) multiring modes of light beams. F. Tang et al. [10] pointed out that the theory of light beams propagating in uniaxial crystals can be utilized to study the transmission of an image through the uniaxial crystals. W. Zhu et al. [11] and our previous work [12] demonstrated that a uniaxial crystal can generated huge photon spin Hall effect. And in our previous work [13], we found that the uniaxial crystals could be used to tailor the orbital angular momentum (OAM) spectrum of the quadratic-power-exponent-phase vortex beam to obtain flat superimposed OAM states.

Thanks to these extensive studies, researchers have gained great insight into the transmission of light beams in uniaxial crystals and the use of uniaxial crystals to regulate the light field to achieve many novel applications. However, almost all these investigations focus on the evolution of two circularly polarized components, LHCP and RHCP components. During the propagation, the LHCP and RHCP components are coupled and will convert to each other. To ensure the conservation of angular momentum, spin-orbit coupling must exist in the process of the circular polarization conversion. To date, the inherent physics of the origin and efficiency of OAM beam generation via the spin-orbit coupling in the circular polarization conversion of a light beam propagating inside a uniaxial crystal have been well explained [14].

In this paper, from another perspective rather than the circular polarization conversion, we focus on the evolution of the radial- and azimuthal-polarization components. For the first time we find that when the incident light beam is CAVBs, i.e., CABs possessing a spiral phase, the radial- and azimuthal-polarization components will be separated during the propagation and finally focus on two separated focus points. CABs are a kind of novel circular symmetric beams with an airy radial profile. The CABs were first proposed theoretically in 2010 [15] and demonstrated experimentally in 2011 [16]. Due to their “abruptly autofocusing” property, the CABs have important potential application values in biomedical treatment, material processing and nonlinear optical processes, and have received great attention recently [17]. Our results can provide CABs with unique control capabilities, which is interesting and can facilitate the flexible applications of CABs.

2. Qualitative discussion

In this section, we first give a qualitative discussion about the uniaxial crystals’ ability of separating the radial- and azimuthal-polarization components of CAVBs. To show this visually, we give a schematic diagram shown in Fig. 1, in which the green areas represent uniaxial crystals of length L, the blue dotted lines and red lines represent the energy transport trajectory of the radial polarization (R-pol) component and the azimuthal polarization (A-pol) component, respectively. Figure 1(a) and Fig. 1(b) correspond to that the crystal is positive uniaxial crystal (n_e> n_o) and negative uniaxial crystal (n_e< n_o), respectively. Here n_e and n_o represent the refractive indices of extraordinary wave and ordinary wave, respectively. It shows that due to the presence of uniaxial crystals, the radial- and azimuthal-polarization components gradually separate from each other during the propagation, and finally abruptly focus on two separate focal points.

 

Fig. 1. The schematic diagram of controlling separation of radial polarization (R-pol) and azimuthal polarization (A-pol) by using uniaxial crystals. (a) and (b) correspond to that the crystal is positive uniaxial crystal (n_e> n_o) and negative uniaxial crystal (n_e< n_o), respectively. The green areas are the uniaxial crystals of length L. The blue lines represent the energy transport trajectory of the radial polarization component; the red lines represent the energy transport trajectory of the azimuthal polarization component.

Download Full Size | PDF

To understand the underlying physical mechanism of separating the radial polarization and azimuthal polarization components in uniaxial crystal, we give the following explanation. Since the CAVB focuses radially towards the beam axis, it can be known that in the crystal, the radial polarization component is the extraordinary wave of the beam and the azimuthal polarization component is the ordinary wave. According to our previous work [18], for arbitrary angular spectrum component with transverse wave vector k of the input beam, the longitudinal wave vector of the ordinary and extraordinary waves are, respectively, ${k_{oz}} = {(k_0^2n_o^2 - {k^2})^{{1 / 2}}}$ and ${k_{ez}} = {(k_0^2n_e^2 - {k^2})^{{1 / 2}}}{{{n_o}} / {{n_e}}}$ with ${k_0}$ being the wave vector of the light beam in vacuum. Hence, in the crystal, the polar angle of the wave vector of the ordinary wave is ${\theta _o} = \arctan ({k / {{k_{oz}}}})$ and the polar angle of the wave vector of the extraordinary wave is ${\theta _e} = \arctan ({k / {{k_{ez}}}})$. Since ${k_{oz}}$ and ${k_{ez}}$ are different, ${\theta _o}$ and ${\theta _e}$ are different. This leads to that the ordinary and extraordinary waves propagate along different directions, hence will be separated gradually during the propagation. When the crystal is positive uniaxial crystal (n_e> n_o), it can be obtained that ${k_{oz}} < {k_{ez}}$. In this case, ${\theta _o} > {\theta _e}$, hence the ordinary wave (i.e., the azimuthal polarization component) will auto-focus before the extraordinary wave (i.e., the radial polarization component) as shown in Fig. 1(a). While when the crystal is negative uniaxial crystal (n_e< n_o), it can be obtained that ${k_{oz}} > {k_{ez}}$. In this case, ${\theta _o} < {\theta _e}$, hence the extraordinary wave (i.e., the radial polarization component) will auto-focus before the ordinary wave (i.e., the azimuthal polarization component) as shown in Fig. 1(b).

3. Theory and quantitative investigation

In the above section, a qualitative discussion about the uniaxial crystals’ ability of separating the radial- and azimuthal-polarization components of CAVBs has been given. In this section, we investigate this phenomenon quantitatively. For this, we should first have the evolution expressions of the radial- and azimuthal-polarization components of the CAVBs propagating in and after the crystals. Since the expressions of two circularly polarized components in uniaxial crystals have been well derived [18], the use of them is strongly encouraged, because to address the evolution of the radial- and azimuthal-polarization components, we just need to perform simple coordinate transformation according to relations ${\hat{e}_r} = {{({e^{ - i\varphi }}{{\hat{e}}_ + } + {e^{i\varphi }}{{\hat{e}}_ - })} / {\sqrt 2 }}$ and ${\hat{e}_\varphi } ={-} i{{({e^{ - i\varphi }}{{\hat{e}}_ + } - {e^{i\varphi }}{{\hat{e}}_ - })} / {\sqrt 2 }}$, where ${\hat{e}_ + } = {{({{\hat{e}}_x} + i{{\hat{e}}_y})} / {\sqrt 2 }}$ and ${\hat{e}_ - } = {{({{\hat{e}}_x} - i{{\hat{e}}_y})} / {\sqrt 2 }}$ are unit vectors employed to describe LHCP and RHCP, respectively; ${\hat{e}_r}$ and ${\hat{e}_\varphi }$ are unit vectors employed to describe the radial polarization and azimuthal polarization in cylindrical coordinate system $(r,\varphi ,z)$; and ${\hat{e}_x}$ and ${\hat{e}_y}$ are unit vectors of x and y axes, respectively.

According to our previous work [18], the expression of the light field in the uniaxial crystal is

Performing simple coordinate transformation, it is straightforward to obtain the radial- and azimuthal-polarization components to be

If considering the incident light is LHCP lth-order CAVB, its initial electric field can be expressed as

Now, according to Eqs. (1), (4), (6) and (8), we can calculate the evolution of the radial- and azimuthal-polarization components during the propagation.

We first consider the case that the crystal is positive uniaxial crystal (n_e> n_o) as shown in Fig. 1(a). For this we choose the crystal to be TiO₂. Figure 2 (a) is the calculated square root of normalized intensity distribution of the total light field, in which the four white dotted lines show the position of z = 20 mm, 38.88 mm, 41.01 mm and 43 mm, respectively. Figures 2(b)-(e) are the normalized intensity distribution of the radial polarization (R-pol) and azimuthal polarization (A-pol) components on x-axis at z = 20 mm, 38.88 mm, 41.01 mm and 43 mm, respectively. In the calculation, the parameters are chosen to be r₀ = 0.25 mm, w = 0.014 mm, a = 0.01, l = -1 and C = 1 V∕mm; the light wavelength is λ = 532 nm; the refractive indices of the crystal is chosen to be n_o= 2.6678 and n_e= 2.978 [19], and the crystal length is L = 30 mm. From Fig. 2(a), it can be seen clearly that during the propagation in the uniaxial crystal, the light energy will split into two parts, each of which will travel along a different energy transport trajectory and eventually focus on a different focus, just as shown in Fig. 1. It can also be seen that a bottle like beam can be generated between the two focal planes. Figure 2(b) can demonstrate that the two parts are respective the energy of the radial polarization and azimuthal polarization components of the beam. From Fig. 2(c), it can be seen that the plane of z = 38.88 mm is the focus plane of azimuthal polarization component. On this plane, the energy of the azimuthal polarization component is concentrated in the central region (the pink region of Fig. 2(c)), while the radial polarization component is not focused. Then from Fig. 2(d), it can be seen that the plane of z = 42.01 mm is the focus plane of the radial polarization component. On this plane, the energy of the radial polarization component is concentrated in the central region (the blue region of Fig. 2(d)), while the energy of the azimuthal polarization component is relatively low. Hence, when the crystal is positive uniaxial crystal, the radial- and azimuthal-polarization components can be separated from each other, and the azimuthal polarization component will focus before the radial polarization component. This is exactly as predicted in Fig. 1(a).

 

Fig. 2. (a) The square root of normalized intensity distribution of total light field. The normalized intensity distribution of the radial polarization (R-pol) and azimuthal polarization (A-pol) components on x-axis at (b) z = 20 mm, (c) z = 38.88 mm, (d) z = 40.01 mm and (d) z = 43 mm. The four positions are shown by white dotted lines in (a).

Download Full Size | PDF

To further confirm the separation phenomenon of the azimuthal polarization and radial polarization components, we also calculate the normalized intensity distribution and the polarization distribution of the total light field on z = 38.88 mm and z = 42.01 mm planes, i.e., the focal planes of the azimuthal polarization and radial polarization components. The calculated results are show in Fig. 3, in which Fig. 3(a) is the normalized intensity distribution on z = 38.88 mm plane, Fig. 3(b) is the polarization distribution on z = 38.88 mm plane, Fig. 3(c) is the normalized intensity distribution on z = 42.01 mm plane, and Fig. 3(d) is the polarization distribution on z = 42.01 mm plane. In the polarization distribution, the blue polarization ellipse represents left-handed elliptic polarization, and the red polarization ellipse represents left-handed elliptic polarization. It can be seen clearly, on z = 38.88 mm plane, the polarization is mainly azimuthal polarization, and on z = 42.01 mm plane, the polarization is mainly radial polarization. This confirms that the radial and azimuthal polarization components are focused in different focal planes. Here it should be noted that on z = 38.88 mm plane (the focal plane of the azimuthal polarization component), the intensity of the radial polarization component is very low (see Fig. 2(c)), so that the polarization in Fig. 3(b) is nearly pure azimuthal polarization. However, on z = 42.01 mm plane (the focal plane of the radial polarization component), although the intensity of the azimuthal polarization component is much lower than that of the radial polarization component, it still cannot be ignored (see Fig. 2(d)), so that the polarization in Fig. 3(d) is not pure radial polarization, but polarization ellipse with relatively large eccentricity.

 

Fig. 3. (a) is the normalized intensity distribution on z = 38.88 mm plane, (b) is the polarization distribution on z = 38.88 mm plane, (c) is the normalized intensity distribution on z = 42.01 mm plane, and (d) is the polarization distribution on z = 42.01 mm plane. In (b) and (d), the blue polarization ellipse represents left-handed elliptic polarization, and the red polarization ellipse represents left-handed elliptic polarization.

Download Full Size | PDF

We also calculate the normalized intensity distributions of the radial polarization and azimuthal polarization components, and their corresponding transverse energy flux density distributions on the planes before and after the focal plane. The results are show in Fig. 4, in which Figs. 4(a1) and 4(b1) are respective the normalized intensity distributions of the radial and azimuthal polarization components on z = 37 mm plane (i.e., a plane before the focal plane of azimuthal polarization component), Figs. 4(a2) and 4(b2) are respective the transverse energy flux density distributions of the radial and azimuthal polarization components on z = 37 mm plane, Figs. 4(a3) and 4(b3) are respective the partial enlarged details of Figs. 4(a2) and 4(b2); Figs. 4(c1) and 4(d1) are respective the normalized intensity distributions of the radial and azimuthal polarization components on z = 39.5 mm plane (i.e., a plane after the focal plane of azimuthal polarization component but before the focal plane of the radial polarization component), Figs. 4(c2) and 4(d2) are respective the transverse energy flux density distributions of the radial and azimuthal polarization components on z = 39.5 mm plane, Figs. 4(c3) and 4(d3) are respective the partial enlarged details of Figs. 4(c2) and 4(d2). From Figs. 4(a2) and 4(b2), it can be seen that before the focal plane of azimuthal polarization component, both the energy of the radial and azimuthal polarization components flows toward the beam center. This is the fundamental reason of the autofocusing property of the CAVB. It can also be seen that in the process of the energy flows towards the beam center, the transverse energy flow of the azimuthal polarization component of is closer to the beam center than that of the radial polarization component, hence the azimuthal polarization component can focus before the radial polarization component. This agrees well with the conclusion above. Then from Figs. 4(c2) and 4(d2), it can be seen that on z = 39.5 mm plane, the energy of the radial polarization component still flows toward the beam center, while the energy of the azimuthal polarization component flows away from the beam center. This is due to that z = 39.5 mm plane is a plane after the focal plane of azimuthal polarization component but before the focal plane of the radial polarization component, hence the radial polarization component is still in the focusing process, while the azimuthal polarization component is in the diverging process after focusing.

 

Fig. 4. (a1) and (c1) are the normalized intensity distributions of the radial polarization component; (b1) and (d1) are the normalized intensity distributions of the azimuthal polarization component; (a2) and (c2) are the transverse energy flux density distributions of the radial polarization component; (b2) and (d2) are the transverse energy flux density distributions of the azimuthal polarization component; (a3), (b3), (c3) and (d3) are respective the partial enlarged details of (a2), (b2), (c2) and (d2). The results are calculated on different z planes, which are labeled in the figure.

Download Full Size | PDF

In above, the case that the crystal is positive uniaxial crystal has been investigated in details, then in the following, we consider the case that the crystal is negative uniaxial crystal (n_e< n_o) as shown in Fig. 1(b). For this, we choose the crystal to be NaNO₃, whose refractive indices are n_o= 1.5897 and n_e= 1.3346 at light wavelength λ = 532 nm [20]. Figure 5 (a) is the calculated distribution of the square root of normalized intensity of the total light field, in which the four white dotted lines show the position of z = 15 mm, 22.3 mm, 27.47 mm and 33 mm, respectively. Figures 5(b)-(e) are the normalized intensity distribution of the radial polarization (R-pol) and azimuthal polarization (A-pol) components on x-axis at z = 15 mm, 22.3 mm, 27.47 mm and 33 mm, respectively. In the calculation, the crystal length is L = 20 mm, and the other parameters are the same as those used in Fig. 2. From Fig. 5(a), it can be seen clearly when the crystal is negative uniaxial crystal, the light energy will also split into two parts during the propagation, and each of them will travel along a different energy transport trajectory and eventually focus on a different focus, just as shown in Fig. 1. Similar to Fig. 2(b), Fig. 5(b) demonstrates that the two parts are respective the energy of the radial polarization and azimuthal polarization components of the beam. From Fig. 5(c), it can be seen that the plane of z = 22.3 mm is the focus plane of the radial polarization component. On this plane, the energy of the radial polarization component is concentrated in the central region (the blue region of Fig. 5(c)), while the azimuthal polarization component is not focused. Then from Fig. 5(d), it can be seen that the plane of z = 27.47 mm is the focus plane of the azimuthal polarization component. On this plane, the energy of the azimuthal polarization component is concentrated in the central region (the pink region of Fig. 5(d)), while the energy of the radial polarization component is relatively low. Hence, when the crystal is negative uniaxial crystal, the radial- and azimuthal-polarization components can be separated from each other, and the radial polarization component will focus before the azimuthal polarization component. This is exactly as predicted in Fig. 1(b). Moreover, it can also be seen from Fig. 5(a) that a chain of bottle like beam is generated between the two focuses. This is because the sidelobes of the Airy intensity envelope of the radial polarization component are focused between the two focal points. Since a bottle like beam is formed between two adjacent sidelobe focal planes, a chain of bottle like beam is obtained finally. Since the light evolutions in both positive and negative uniaxial crystals are very similar (the only difference is that in the case of positive uniaxial crystal, the focus of the azimuthal polarization component is before the focus of the radial polarization component, while in the case of negative uniaxial crystal, the focus of the radial polarization component is before the focus of the azimuthal polarization component), it can be expected that the results of the intensity and polarization distributions on the two focal planes, and the transverse energy flux density distributions on the planes before and after the focal plane in the case of negative uniaxial crystal should be very similar to the case of positive uniaxial crystal. Hence, we do not repeatedly calculate and discuss these results for the case of negative uniaxial crystal.

 

Fig. 5. (a) The normalized intensity distribution of the total light field. The normalized intensity distribution of the radial polarization (R-pol) and azimuthal polarization (A-pol) components on x-axis at (b) z = 15 mm, (c) z = 22.3 mm, (d) z = 27.47 mm and (d) z = 33 mm. The four positions are shown by the white dotted lines in (a).

Download Full Size | PDF

Until now, the separation of the radial- and azimuthal-polarization components of CAVBs propagating in uniaxial crystal has been demonstrated numerically both when the crystal is positive and negative. To give deeper understanding of this phenomenon, we further investigate the influence of the crystal length L on the separation distance of the two focal points. For this, we calculate and compare the distribution of the square root of normalized intensity of the total light field under different crystal length L. Figure 6(a)-(c) give the calculated normalized intensity distribution of the total light field under crystal length L = 10 mm, 15 mm and 20 mm, respectively (in the calculation, the other parameters are same as those used in Fig. 5). In the figure, the positions of the focal points are shown by the white dotted lines. It can be seen clearly that the focal length of the autofocusing is prolonged as the increase of the crystal length. This phenomenon is due to that for arbitrary angular spectrum component with transverse wave vector k of the input beam, both the longitudinal wave vector of the ordinary and extraordinary waves, i.e., ${k_{oz}} = {(k_0^2n_o^2 - {k^2})^{{1 / 2}}}$ and ${k_{ez}} = {(k_0^2n_e^2 - {k^2})^{{1 / 2}}}{{{n_o}} / {{n_e}}}$, are larger than the vacuum longitudinal wave vector ${k_z} = {(k_0^2 - {k^2})^{{1 / 2}}}$, so that both the polar angles of the wave vectors of ordinary wave and extraordinary wave in the crystal, i.e., ${\theta _o} = \arctan ({k / {{k_{oz}}}})$ and ${\theta _e} = \arctan ({k / {{k_{ez}}}})$, are smaller than the polar angle of the vacuum wave vector $\theta = \arctan ({k / {{k_z}}})$, implying that the CAVB focuses slower in crystal than in free space. Hence the focal length of the autofocusing is prolonged as the increase of the crystal length. Here, it should be noted that the above analysis does not limit the type of the uniaxial crystal and is therefore valid for both positive and negative uniaxial crystals. It can be further found that the separation distance of the two focal points increases as the increase of the crystal length. This is due to that the propagating directions of the ordinary and extraordinary waves are different in the crystal, so that the separation of them becomes larger as the increase of the crystal length. This analysis does also not limit the type of the uniaxial crystal and is therefore valid for both positive and negative uniaxial crystals.

 

Fig. 6. The normalized intensity distribution of the total light field under crystal length (a) L = 10 mm, (b) L = 15 mm and (c) L = 20 mm. The positions of the focal points are shown by the white dotted lines. In the calculation, the other parameters are same as those used in Fig. 3.

Download Full Size | PDF

Besides crystal length, the refractive indices n_e and n_o will also affect the separation of the radial- and azimuthal-polarization components of the CAVBs. Since the separation is due to that the polar angle of the wave vector of the ordinary wave and extraordinary wave, i.e., ${\theta _o} = \arctan ({k / {{k_{oz}}}})$ and ${\theta _e} = \arctan ({k / {{k_{ez}}}})$, are different. Its physical root is that the anisotropy of the crystal leads to ${k_{oz}} \ne {k_{ez}}$. Hence, it can be expected that the separation of the radial- and azimuthal-polarization components will become more significantly as the increase of the anisotropy, i.e., $|{{n_e} - {n_o}} |$. Therefore, to observe a large separation distance between the two focal points, it is suggested that the crystal should have large birefringence.

4. Extended discussion

In the separation of the radial- and azimuthal-polarization components of the CAVBs, the autofocusing property of the CAVB plays an important role. On one hand, the autofocusing makes the transverse wave vector k of the angular spectrum of the beam to be nonzero, leading to that the radial- and azimuthal-polarization components can propagate along different directions. On the other hand, the autofocusing reduces the transverse size of the beam at focus plane drastically, makes it become possible to separate the radial- and azimuthal-polarization components when the interval between their intensity patterns is larger than their size. For this reason, the separation of the radial- and azimuthal-polarization components in uniaxial crystal is a unique phenomenon of CAVBs, which will not occur when the incident beam is other kind of beams, such as Gaussian beam, Bessel beam and Laguerre Gaussian beam, due to that these beams do not have autofocusing property. Of course, focusing a light beam through a lens can achieve the same effect as autofocusing. Therefore, if the beam is other kind of vortex beams followed by a lens, we think it is possible to obtain the same result, i.e., the radial- and azimuthal-polarization components are gradually separated during the propagation.

Theoretically, to discuss the case of an arbitrary vortex followed by a lens, we can consider the input light beam to be

Inserting Eq. (10) into Eq. (3), it can be obtained that

5. Conclusion

In summary, the propagation of the radial- and azimuthal-polarization components of CAVBs in uniaxial crystal is studied. It is found that the two polarization components will be gradually separated during the propagation and finally focus on two separated focus points. When the crystal is positive uniaxial crystal, it is found that the azimuthal polarization component will focus before the radial polarization component. While when the crystal is negative uniaxial crystal, it is found that the radial polarization component will focus before the azimuthal polarization component. The influences of the crystal length and birefringence on the separation of the radial- and azimuthal-polarization components are also investigated and discussed. Our results can provide CAVBs with unique control capabilities and could offer deeper understanding of the propagation of light beam in uniaxial crystal.