Nonparaxial propagation of radially polarized chirped Airy beams in uniaxial crystal orthogonal to the optical axis*

Yaohui Chen(陈耀辉), Lixun Wu(吴理汛), Zhixiong Mo(莫智雄),Lican Wu(吴利灿), and Dongmei Deng(邓冬梅)

Guangdong Provincial Key Laboratory of Nanophotonic Functional Materials and Devices,South China Normal University,Guangzhou 510631,China

Keywords: radial polarization,nonparaxial propagation,Airy beam,uniaxial crystal,chirp

1. Introduction

In 1979, as the fact that the Airy function is a solution to the Schr¨odinger equation was demonstrated by Berry and Balazs, an Airy wave packet was predicted theoretically.[1]Although the Airy wave also has the characteristics of selfhealing and self-accelerating, it is not physically realizable because of its infinite energy. Since the intensive research and experimental demonstration on the Airy beams with finite energy were conducted by Siviloglou and Christodoulides in 2007,[2,3]the Airy beam has attracted tremendous attention because of its special characteristics such as self acceleration,[2,3]nondiffraction[4]and self-healing.[5]In the last decades,the Airy beam has been applied in many areas[6]such as the optical cleaning of the micro particles,[7]the lightsheet microscopy,[8]the light bullet generation[9,10]and the curved plasma channel generation.[11,12]Recently, some new types of the auto-focusing beam have been reported[13,14]and a kind of novel property of the Airy-like beam has been investigated.[15,16]

So far, researchers have diffusely studied the paraxial propagation of Airy beams in media with external potentials,[17–19]inhomogeneous medium,[20–24]linear medium,[4,5]nonlinear medium[25,26]and apertured optical system.[27]Researches on the propagation of laser beams in uniaxial crystals have important effects on the design of compensators, polarizers, amplitude and phase modulation devices.[28–30]However,the theory of paraxial propagation is no longer applicable when the beam waist radius is equivalent to the wavelength in a uniaxial crystal or the far-field divergence angle is large. In this case, the nonparaxial effects of the beam should be concentrated on. Varieties of nonparaxial methods have been developed to deal with beam propagation beyond the paraxial approximation.[31–34]In particular,a practical method is to solve the nonparaxial propagation of laser beams in uniaxial crystals by solving the boundary conditions of Maxwell’s equations.[29,30,35,36]

Since 2007, the first realization of the finite-energy Airy beams has been given,[2]an analytical vectorial structure of radially polarized light beams has been investigated[37]and a review on cylindrical vector beams has been written.[38]For their interesting properties, the radially polarized beams have been applied in different aspects such as optical microfabrication,[39]high-resolution microscopy,[40]acceleration techniques,[41,42]particularly material processing[43,44]and particles guiding or trapping.[45–47]Also,some researches on the propagation of radially polarized Airy beams have been reported in recent years.[48–51]Although the way of strengthening the longitudinal component of the electric field to reduce the size of the light spot does not work in crystals,[52–54]it is still valuable to study the radial polarization in uniaxial crystals due to the interesting effects given in the works.[55,56]On the other hand, the chirp parameter is regarded as a powerful tool in the generation and the manipulation of optical beams.[57,58]Zhang et al. studied the impacts of the first and the second order chirp on initial finite energy Airy beams.[18]Therefore,some novel phenomena may be discovered by combining the radial polarization, nonparaxial propagation and chirped Airy beams in uniaxial crystals.

In this paper,we investigate the nonparaxial propagation properties of Airy beams with the first and the second order chirps in uniaxial crystal orthogonal to the optical axis,respectively. The intensity evolutions and transverse intensity of RPCAiBs are analyzed in detail. Based on the results obtained,[59,60]we pay more attention to the different behaviors of the x- and the y-components and some interesting results are found.

The organization of the paper is as follows. In Section 2,the theoretical model for the nonparaxial propagation in uniaxial crystal orthogonal to the optical axis is presented and we derive analytical solutions of RPCAiBs. In Section 3,the nonparaxial evolutions and transverse intensity of RPCAiBs with the first and the second order chirp factors and the ratios of the extraordinary refractive index to the ordinary refractive index which can be adjusted by using the electro-optical crystal are analyzed. Finally, some useful results are summarized in Section 4.

2. Analytic solution of RPCAiBs in uniaxial crystals orthogonal to the optical axis

In the Cartesian coordinate system, the propagation axis is set as the z-axis and the optical axis of the uniaxial crystal is the x-axis. The relative dielectric tensors ε of the uniaxial crystal can be expressed as[30]

where noand neare the ordinary and the extraordinary refractive indices of the uniaxial crystal. We assume that RPCAiBs in the x-direction is incident on the uniaxial crystal at the initial plane z=0. Therefore, the electric field distribution of RPCAiBs in the input plane can be expressed as

where w0denotes the initial length size; a is the truncation factor of the RPCAiBs;Ai(···)represents the first kind of the Airy function;β1xand β1yare the first order chirp parameters,while β2xand β2yare the second order chirp parameters.

Under the theory of the nonparaxial propagation in uniaxial crystal orthogonal to the optical axis,the electric field of a beam in uniaxial crystal orthogonal to the optical axis can be expressed as[30]

with kezand kozdefined by

Utilizing the properties of Fourier transforms,[61]the electric field of the beams can be written as the three components:[30]

where

with

and Ai′(···) indicates the first-order derivative of the Airy function.

3. The numerical calculations and analysis

Based on the analytical expressions for RPCAiBs propagating in uniaxial crystals orthogonal to the optical axis, the propagation properties of RPCAiBs including the influences of the first and the second order chirp factors are further investigated. In the following simulations, the calculation parameters are chosen as λ =633 nm, w0=100 μm, a=0.1,no=2.616,and Zr=kw20/2 is the Rayleigh range.

Comparing the maximum intensity of three components of the beams in Fig. 1, we can find that the intensity of the z-component of the beam is much smaller than other components. Therefore, in the following analysis, we ignore the z-component and only consider the x-component and the ycomponent of the beams. Comparing the maximum intensity of Ixand Iy, we can find that before focusing, the minimum value of Ixappears before that of Iy,and the minimum value of Ixis larger than that of Iy.

To investigate the nonparaxial propagation of the xcomponent of RPCAiBs in uniaxial crystals orthogonal to the optical axis, the intensity evolutions of the beam in different observation planes are depicted in Fig.2. As shown in Fig.2,the propagation trajectory of the beam in the x-direction is further than that in the y-direction.The maximum intensity of the beam in the x-direction is smaller than that in the y-direction.Comparing Figs.2(a1)–2(a4)and Figs.2(b1)–2(b4),it is obvious that as β1x,yincreases,the beam is tilted towards the positive x-axis and the positive y-axis in the x-direction and the y-direction,respectively,and when β1x,y＜0,the beam is tilted towards the negative direction. In addition,as β1x,yincreases,the inclination of the beam in the x-direction becomes larger than that in the y-direction.

Figure 3 represents the nonparaxial propagation of the ycomponent of RPCAiBs in different observation planes with different first order chirp factors in uniaxial crystal orthogonal to optical axis.It is interesting that the intensity of the beam in the x-direction mainly distributes on the main lobe while the intensity of the beam in the y-direction mainly distributes on the side lobe.The propagation of the beam in the x-direction is much further than that in the y-direction.When β1x,yincreases,the focus position of the beam in the x-direction is advanced and the length of the focus of the beam is smaller, as shown in Figs.3(a1)–3(a4)and Figs.3(b1)–3(b4). In addition,larger β1x,ymakes the distribution of the beam more concentrated.Combining Figs. 3(a1)–3(b4) and Figs. 3(a5)–3(b5), it is obvious that the initial slope of the beam in the x-direction varies more with β1x,ythan that in the y-direction.

The transverse intensity distributions are plotted to analyze the influence of the first order chirp factor on RPCAiBs. Taking the beams with β1x,y=5 as an example, at the start, the intensity mainly distributes on the side lobe as shown in Fig. 4(e1). With the increase of the propagation distance it can be seen that the x-component of the beams moves faster than the y-component in the x-direction, while the y-component moves faster than the x-component in the ydirection in Figs. 4(e2)–4(e4). In addition, a change of β1x,ycauses the x-component of the beams to move more than the ycomponent in the x-direction and the y-component move more than the x-component in the y-direction when the propagation distance is equal,as shown in Figs.4(a2)–4(e4).

Fig.1. The maximum intensity of radially polarized Airy beams in uniaxial crystals orthogonal to the optical axis with ne=1.5no.

Fig. 2. The nonparaxial propagation trajectories and intensity evolutions of the x-component of RPCAiBs in different observation planes in uniaxial crystals orthogonal to the optical axis with ne=1.5no and β2x,y=0: (a1)and(b1)β1x=β1y=-1;(a2)and(b2)β1x=β1y=0;(a3)and(b3)β1x=β1y=1;(a4)and(b4)β1x=β1y=2.

Fig. 3. The nonparaxial propagation trajectories and intensity evolutions of the y-component of RPCAiBs in different observation planes in uniaxial crystals orthogonal to the optical axis with ne=1.5no and β2x,y=0: (a1)and(b1)β1x=β1y=-1;(a2)and(b2)β1x=β1y=0;(a3)and(b3)β1x=β1y=1;(a4)and(b4)β1x=β1y=2.

Fig.4. Intensity distributions of RPCAiBs in uniaxial crystals orthogonal to the optical axis with β2x,y =0 and ne =1.5no: (a1)–(a4)β1x =β1y=-5;(b1)–(b4)β1x=β1y=-3;(c1)–(c4)β1x=β1y=0;(d1)–(d4)β1x=β1y=3;(e1)–(e4)β1x=β1y=5.

To investigate the nonparaxial propagation of the xcomponent of RPCAiBs in uniaxial crystals orthogonal to the optical axis,the intensity evolutions and the trajectories of the beam in different observation planes are depicted in Fig. 5.Comparing Figs. 5(a2)–5(a4) and Figs. 5(b2)–5(b4), we can find that the maximum intensity of the x-component in the xdirection varies more with β2x,ythan that in the y-direction.At the same time, with the increase of β2x,y, the intensity of the side lobe of the beam in the x-direction becomes more obvious compared to the main lobe while the intensity of the beam in the y-direction has no such phenomenon. Comparing Figs. 5(a5) and 5(b5), one can see that as the β2x,yincreases,the trajectory of the beam in the x-direction changes more than that in the y-direction.

Fig. 5. The nonparaxial propagation trajectories and intensity evolutions of the x-component of RPCAiBs in different observation planes in uniaxial crystals orthogonal to the optical axis with ne =1.5no and β1x,y =0: (a1)and(b1)β2x =β2y =-0.01;(a2)and(b2)β2x =β2y =0;(a3)and(b3)β2x=β2y=0.01;(a4)and(b4)β2x=β2y=0.02.

Fig. 6. The nonparaxial propagation trajectories and intensity evolutions of the y-component of RPCAiBs in different observation planes in uniaxial crystals orthogonal to the optical axis with ne=1.5no and β1x,y=0: (a1)β2x=β2y=-0.01,(b1)β2x=β2y=-0.05;(a2)and(b2)β2x=β2y=0;(a3)β2x=β2y=0.01,(b3)β2x=β2y=0.05;(a4)β2x=β2y=0.02,(b4)β2x=β2y=0.1.

To further learn about the variation of the intensity during propagation, the intensity evolutions of the y-component of RPCAiBs with different second order chirp factors are depicted. Comparing Figs. 6(a1) and 6(b1), we can derive that the propagation of the beam in the x-direction is much further than that in the y-direction,and the intensity in the x-direction has a focused performance in the main lobe while in the ydirection it always distributes on the side lobes. With β2x,ybecoming larger, the depth of the focus of the beam in the xdirection decreases and the length of the focus increases but the maximum intensity of the beam in the y-direction is unchanged,as shown in Figs.6(a1)–6(b4).

In Fig. 7, transverse intensity distributions of RPCAiBs with different second order chirp factors are plotted. Comparing Figs.7(a2)–7(e2),we can conclude that when z=9Zr,larger β2x,ymakes the intensity tend to be distributed in the x-direction. Moreover, comparing Figs. 7(a2)–7(e4), we can derive that the distance between the x-and the y-components of RPCAiBs with larger β2x,yincreases more slowly with the propagation distance increasing.

Fig.7. Intensity distributions of RPCAiBs in uniaxial crystals orthogonal to the optical axis with β1x,y =0 and ne =1.5no: (a1)–(a4)β2x =β2y=-0.02;(b1)–(b4)β2x=β2y=-0.01;(c1)–(c4)β2x=β2y=0;(d1)–(d4)β2x=β2y=0.01;(e1)–(e4)β2x=β2y=0.02.

Fig. 8. Intensity distributions of RPCAiBs in uniaxial crystals orthogonal to the optical axis with β1x,y =3 and β2x,y =0.01: (a1)–(a4)ne=0.8no;(b1)–(b4)ne=no;(c1)–(c4)ne=1.2no;(d1)–(d4)ne=1.5no;(e1)–(e4)ne=1.8no.

To further learn about the effect of the ratio of neto noon RPCAiBs, the intensity distributions of the beams are depicted in Fig.8.Comparing Figs.8(a1)–8(e4),we can find that the distance between the x-component and the y-component will increase with the propagation distance increasing when ne／=no. It can be seen in Figs. 8(a1)–8(e2) that only the velocity of the x-component in the x–y plane is adjusted by the ratio of neto no. In addition, in Figs. 8(a3)–8(e3), when the ratio of neto noincreases, the intensity of the focus of the y-component does not change while that of the x-component becomes larger.

4. Conclusion

In summary, we have analyzed the propagation evolutions, the trajectories in different observation planes and the transverse intensity of RPCAiBs by deriving analytical expressions of three components of the nonparaxial propagation of RPCAiBs in a uniaxial crystal orthogonal to the optical axis.Although the initial slope of RPCAiBs in both the x-direction and the y-direction increases as β1xand β1ybecome larger,the initial slope of the beams in the x-direction increases more.In addition,with β1xand β1yincreasing,the focus position of the y-component of RPCAiBs is advanced, the length of the focus becomes smaller and the intensity distribution becomes more concentrated while the x-component of RPCAiBs has no difference in these three aspects with larger β1xand β1y. The acceleration of RPCAiBs in both the x-direction and the ydirection becomes larger with β2xand β2yincreasing, but the acceleration of the beams in the x-direction increases more. In addition,the ratio of neto nocan adjust not only the difference of the depth of the focus between the x-and the y-components but also the difference of the velocity between the x- and the y-components. We believe this investigation of RPCAiBs will be available for applications of laser processing. It can be expected that the desired shape of the focus and its depth can be achieved by adjusting the first and the second order chirp factors,and the ratio of neto no.

Acknowledgment

The authors thank Yujun Liu for valuable advice.