1 Introduction

A highly focused beam, named nanojet (NJ), is generated in case that a mesoscale scatterer is irradiated by an electromagnetic wave [1]. As an analogue of the NJ in the tera-hertz (THz) region, the concept of terajet (TJ) is first introduced in 2014 [2]. It is generated by a dielectric scatterer under illumination of a THz wave. Quite different absorption losses in respective frequency bands give rise to different localized field characteristics of the two jets. The NJ/TJ can be produced in either a transmission mode or a reflection mode. In the transmission mode, it is formed by the interference of scattered waves and is usually located behind the scatterer. In a reflection mode, it is formed by interference of reflected incident and scattered waves, and is located in front of scatterer, and relevant previous studies concentrate on generation and applications of NJs [3,4,5,6,7,8,9,10].

The TJ/NJs are featured by high intensity, sub-wavelength waist and wavelength-scale propagation distance, which are characterized by wave intensity enhancement factor, full width at half-maximum (FWHM) and focal length (FL) in order. Among these parameters, the FL is the most important one that reflects axial position of a TJ/NJ and determines its application fields [1, 11]. A TJ/NJ with longer FL may find its use in many fields such as manipulation and detection of single nanoparticle and biomolecule [12, 13], control of on/off of an optical path [14, 15], color splitter of dielectric double-material for image sensor [16], measurement of rotation speed for a motor [17], non-destructive metrology for semiconductor devices [18], imaging of reflective fluorescence/confocal microscopy [19,20,21], power enhancement of antennas radiation [22], and on-chip bendable wave communications [14, 23, 25]. On the other hand, a TJ/NJ with a short FL may find its use in high-resolution microscope imaging [11, 22, 24]. Due to the great application potential of TJ/NJs with different FLs, some methods have been proposed to tailor the axial position of FL of TJ/NJ [11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]. The traditional method is either to tailor the refractive index contrast of dielectric scatterer and surrounding medium [4, 5, 9] or to change the shape and geometry of scatterer [8, 11, 26,27,28,29]. Recently, Minin and his co-workers have proposed a method of placing metal screens on two sides of scatterer to tailor the axial position of FL of NJ [25]. However, all these methods aimed at the FL tailoring via adjusting the optical and geometrical parameters of scattering system. From the viewpoint of application, the change of dielectric system is unrealistic and unfeasible for dynamical control of FL in realtime. It is essential to seek other simpler, more convenient and flexible methods to tailor the axial position of a TJ/NJ for various potential applications mentioned above. Here, we report such a method. It is proposed on the basis of adjustment of polarization state of incident wave. First, our study focuses on effect of polarization state of an incident microwave on the performance of TJ generated in the reflection mode, which is realized by putting a cuboid scatterer onto a reflection screen. The results show that the FL can be effectively tailored by the polarization state. Then, the study focuses on relationship between the polarization effect and asymmetry of scatterer structure. Asymmetries caused by geometries of scatterer and reflection screen and their relative position are exemplified.

2 Structural model, principle and numerical method

Figure 1 shows schematic of generation of TJ in a reflection mode, which is realized by a scattering system consisted of a reflection screen and a cuboid put onto it. An xyz Cartesian coordinate system is fixed at the center of the top surface of the cuboid scatterer. The cuboid has a width Wc along the x-axis, a length Lc along the y-axis and a height Hc along the z-axis. It adopts usually a dielectric material. Here, Teflon material is adopted, which is widely used as a lens material due to its weak absorption at THz frequency region (~0.05 cm−1 @ 0.1 THz [30]) and has a refractive index nc = 1.46 at the wavelength λ = 3 mm studied here [31, 32]. Due to quite low absorption coefficient, the wave power dissipation in the Teflon scatterer can be neglected. The reflection screen has a width Wrs, a length Lrs and a height Hrs, and adopts the aluminum metal with an electric conductivity of 3.56 × 107 S/m at λ = 3 mm [3]. Assume that the thickness of reflection screen Hrs has a value of 0.33λ, which is taken from Ref. [3]. The cuboid has the same height as that of the reflection screen (Hc = Hrs), if without other specified.

Fig. 1
figure 1

Schematic of generation of TJ in a reflection mode realized by putting a cuboid on a reflection screen

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A plane THz wave (λ = 3 mm) polarized along either x- or y-axis is incident onto the structure along -z direction as shown in Fig. 1. Assume that it has an intensity of I0. Waves scattered by the cuboid and reflected by the screen interfere with each other, leading to the generation of a TJ in the space z > 0. The projections of the TJ onto xz (left) and xy (right) planes are schematically indicated in Fig. 1. The TJ is characterized by the following three parameters [2, 3]:

  1. (i)

    Focal length (FL), defined as the distance from the z coordinate of scatterer’s top surface and the z position where the wave intensity maximizes [also called focal point as indicated];

  2. (ii)

    FWHMx(y), defined as full width at half-maximum intensity along the transversal x/y-axis at focal point;

  3. (iii)

    wave intensity enhancement factor, defined as the ratio of maximum intensity Imax to incident intensity I0.

Based on the structural model shown in Fig. 1, we have theoretically studied the effect of wave polarization state on the performance of TJ produced by an asymmetrical scatterer structure in the reflection mode. The study focuses on the polarization effect on the characteristic parameters of TJ, especially the FL. The study was accomplished by aid of a commercially available program (CST Microwave Studio). A transient solver in time domain was adopted, and a fine hexahedral mesh with an element size of one-forty-fifth wavelength (λ/45) was employed for the whole computational domain. Open boundary condition was adopted. Adaptive mesh refinement and strict iterative termination condition were adopted to guarantee the convergence of numerical method and the accuracy of numerical calculation results.

3 Numerical results

3.1 Polarization effect of TJ generated by a symmetrical structure

The study first concentrates on the polarization effect of the scattering system having a symmetrical structure. Figure 2a and b shows the TJs projected onto xz plane and generated by a symmetrical structure with Lrs = Wrs = 2λ, Lc = Wc = λ and Hc = Hrs = 0.33λ under the illumination of x- and y-polarized waves and under the assumption that the cuboid is placed at the center of the top surface of the reflection screen (in this way, the whole scattering system is transversally symmetrical in structure). The white dashed line marks the focal position of TJ. One can see that the two TJs have the same focal positions and thereby the same FL. Thus, for symmetrical structure, the change of polarization state does not affect the FL of TJ. However, it affects considerably the shape of wave spot. As shown in Fig. 2a and b, the two TJs have considerably different widths along the x-axis at the focal position. This feature is further verified by the projections onto the xy plane in Fig. 2c and d. One can see that the TJ generated in the case of x-polarized incident wave has larger x-width, while that generated in the case of y-polarized wave has larger y-width. In specific, the TJ generated in the case of the x-polarized incident wave has an FWHMx of 0.453λ and an FWHMy of 0.448λ, while the TJ generated in the case of y-polarized wave has an FWHMx of 0.448λ and an FWHMy of 0.453λ. In a word, for symmetric structure, the polarization state affects mainly the FWHM while hardly the FL of TJ.

Fig. 2
figure 2

TJ generated by symmetrical structure with Lrs = Wrs = 2λ, Lc = Wc = λ and Hc = Hrs = 0.33λ, and projected onto (a, b) xz and (c, d) xy planes

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3.2 Polarization effect of TJ generated by an asymmetrical structure

Attention is next paid to the polarization effect on the performance of the TJ generated by an asymmetric structure. Here, the asymmetries arising from the geometries of scatterer and reflection screen and their relative spatial position are exemplified to demonstrate the effect.

3.2.1 Polarization effect arising from geometric asymmetry of reflection screen

The study first focuses on the case that the asymmetry originates from the geometry of reflection screen. In this case, the cuboid is assumed to be symmetrical with Lc = Wc, while the reflection screen is asymmetrical with Lrs ≠ Wrs. Consider that the two centers of cuboid and reflection screen are all located on the z-axis. Figure 3(a) shows the Wrs dependence of the FL of TJ generated in the case that the incident wave is y- or x-polarized and other geometric parameters are Hc = Hrs = 0.33λ, Lc = Wc = λ and Lrs = 2λ. Note that the change of Wrs is equivalent to the change of asymmetry. The red and blue asterisks represent the y- and x-polarized cases, respectively. The intersection of two curves happens at Wrs = Lrs = 2λ, which implies that the whole structure of cuboid plus reflection screen is symmetrical. This means that the two TJs generated by a symmetrical structure in case of y- or x-polarized wave irradiation have the equal FL, i.e., polarization effect on the FL is null for a symmetrical scatterer, consistent with the preceding result shown in Fig. 2. As the Wrs is larger than the Lrs, the structure becomes asymmetrical and the FL of the TJ generated in the case of y-polarized incident wave has a value definitely larger than that in the case of x-polarized incident wave, indicating that the FL shows definite polarization effect. The effect maximizes at Wrs = 3λ. Aiming at this maximal effect, we have simulated the TJs projected onto xz plane and generated by an irradiation of a y- or x-polarized incident wave. The results are shown in Fig. 3b and c. The white dashed lines mark the focal positions of two TJs, and the red dot and arrow symbols represent the polarization directions along y and x-axes, respectively. We note that the FL difference of two TJs is evident.

Fig. 3
figure 3

a Wrs dependence of FL of TJ generated in the case of Lc = Wc = λ, Lrs = 2λ and Hc = Hrs = 0.33λ. b, c: TJs projected onto the xz plane and generated in the case of Hc = Hrs = 0.33λ, Lc = Wc = λ, Lrs = 2λ, Wrs = 3λ and an irradiation by a y- or x-polarized incident wave. d Lrs dependence of FL of TJ generated in the case of Lc = Wc = λ, Wrs = 3λ and Hc = Hrs = 0.33λ. e, f: TJs projected onto the xz plane and generated in the case of an irradiation by a y- or x-polarized incident wave and Lc = Wc = λ, Lrs = 5λ, Wrs = 3λ and Hc = Hrs = 0.33λ

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By fixing Wrs = 3λ, at which the polarization effect on FL maximizes as demonstrated above, we have further investigated the effect by changing the Lrs, which is equivalent to the change of asymmetry too. The results are illustrated in Fig. 3d. For convenience, Table 1 brings together the FL values of x- and y- polarization states and the polarization effect ΔFL for different Lrs values. As Lrs increases from 2λ to 11λ, the TJ generated in the case of an illumination by a y-polarized plane wave has always a longer FL than that generated in the case of an illumination by an x-polarized plane wave. As expected, two plots intersect as Lrs = Wrs = 3λ. The intersection means a null polarization effect on the FL due to a symmetrical structure. As Lrs = 5λ, the FL difference has a maximum of 0.65λ (see Table 1) and the FL changes by as much as ninefold. Aiming at the maximum polarization effect, we have calculated the TJs projected onto xz plane and generated by an irradiation of a y- or x-polarized incident wave. The results are shown in Fig. 3e and f. One can see that the polarization effect on the FL is more evident than that observed in Fig. 3b and c.

Table 1 FL values of x- and y-polarization states and the polarization effect ΔFL for different Lrs values in the range of 2λ-11λ. Other parameters are Lc = Wc = λ, Wrs = 3λ and Hc = Hrs = 0.33λ
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In summary, for a given symmetrical cuboid, the polarization effect changes considerably with the alteration of asymmetry of the reflection screen. The maximum polarization effect takes place for Wrs = 3λ and Lrs = 5λ in case of Lc = Wc = λ and Hc = Hrs = 0.33λ. It is essential to further study the wave intensity distribution features of the TJ generated by the scatterer structure in case that the maximal polarization effect takes place. Figure 4a–c shows the wave intensity distribution along the z-, y- and x-axes of the TJ generated in the case of Wrs = 3λ and Lrs = 5λ (other parameters are Lc = Wc = λ and Hc = Hrs = 0.33λ), at which the maximal polarization effect happens as stated above. The black dashed lines in Fig. 4a indicate the focal position, the red/blue curve in each diagram corresponds to the y-/x-polarized wave illumination, and the curves in Fig. 4b and c are taken from the focal positions. We first note from Fig. 4a that the TJ generated in the case of y-polarized incident plane wave has a larger FL than that generated in the case of x-polarized incident plane wave. We also note from Fig. 4a–c that the TJ generated in the case of y-polarized wave illumination has larger FWHMx(y) and higher wave intensity at the focal point than that generated in the case of x-polarized wave illumination.

Fig. 4
figure 4

Wave intensity distribution along a z-, b y- and c x-axis of the TJ generated in the case of Wrs = 3λ, Lrs = 5λ, Lc = Wc = λ and Hc = Hrs = 0.33λ

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Regarding the polarization effect on the FL, there is an argument to be clarified. One can see from Fig. 4a that the difference in maximum intensities between the two states of polarization is quite small. An evaluation shows that the difference is about 3%. Therefore, small a difference may affect accuracy of the FL determination and hence the claimed polarization effect. One may query that the difference may be within the uncertainty of the numerical method, which is closely related to the grid size of the computation window adopted in the simulation, i.e., the grid number per wavelength λ, named N. To reduce the uncertainty and increase the numerical accuracy, the grid size should be adopted as small as possible, i.e., the N should be large as soon as possible. We have repeated the simulations by adopting different grid sizes ranging from λ/30 to λ/100. As too large N increases significantly the computation, the simulation considers a grid size down to λ/100. We have monitored the normalized maximum intensity of the TJ, Imax/ I0, as a function of the grid number N per wavelength. The results in the two cases of x- and y-polarizations are illustrated in Fig. 5. One can see that the grid size affects definitely the Imax/I0 factor. As the grid becomes fine, the Imax/ I0 decreases. This is because the iterative procedure changes as the grid size is changed. An asymptotic tendency is discernible as the grid number N is greater than 80. The Imax/I0 value at N = 100 is assumed to be the asymptotic one, i.e., the accurate value. Based on the assumption, we can evaluate the uncertainty of Imax/I0 atN = 45, which is adopted for all of the simulation results in this work. It is smaller than 2%. One can see that the simulation uncertainty is affirmatively smaller than the difference of the maximum intensities between the two polarizations, 3% mentioned above. Thus, the polarization effect on FL claimed above is sound.

Fig. 5
figure 5

Mesh grid number per wavelength dependence of maximum intensity Imax/I0 of TJ generated by a structure with Wrs = 3λ, Lrs = 5λ, Lc = Wc = λ and Hc = Hrs = 0.33λ under the illumination of a x- or b y-polarized plane wave

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Aiming at the above-mentioned geometric parameters for which the polarization effect maximizes, i.e., Wrs = 3λ and Lrs = 5λ, Lc = Wc = λ and Hc = Hrs = 0.33λ, we have further investigated this effect by changing the polarization angle θ from 0° to 90° (θ is defined as the intersection angle between the polarization direction of the incident wave and the x-axis). Figure 6 shows the varying relations of the characteristic parameters of TJ to the polarization angle θ. These characteristic parameters include the FL, I/I0, FWHMx and FWHMy. One can see that the FL changes largely within 0.08λ to 0.73λ, Imax/I0 changes weakly, and the FWHMx and FWHMy change moderately within 0.46λ–0.66λ and 0.40λ–0.60λ in order as the θ increases from 0° to 90°. Table 2 brings together the values of these characteristic parameters for each polarization angle θ. It appears that there exists a threshold polarization angle θc around 45°. As θ < θc, the characteristic parameters show a small polarization effect. As θ > θc, the effect becomes significant for the FL and considerable for the FWHMx and FWHMy, while remains small still for the I/I0. The observations imply that not only the FL but also the FWHMx(y) can be tailored by the polarization state of incident wave.

Fig. 6
figure 6

Polarization angle θ dependence of FL, Imax/I0, FWHMx and FWHMy of the TJ generated in the case of Lrs = 5λ, Wrs = 3λ, Lc = Wc = λ and Hc = Hrs = 0.33λ

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Table 2 Characteristic parameters of TJ generated in the case of Lrs = 5λ, Wrs = 3λ, Hc = Hrs = 0.33λ and Lc = Wc = λ for different polarization angles θ
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In addition, aiming at the maximum polarization effect, which takes place for Wrs = 3λ and Lrs = 5λ, Lc = Wc = λ and Hc = Hrs = 0.33λ, we have investigated the influence of cuboid height Hc on the polarization effect on FL of TJ. In Fig. 7, we show the dependence of the FL on the Hc in the range of 0.3λ–λ. The red and blue curves represent the y- and x-polarization cases, respectively. We first note that the FL changes dramatically as the Hc is changed. This is also true for the polarization effect on FL. In addition, the effect decreases with a rise in Hc. It is significant for Hc < 0.6λ while quite weak as Hc > 0.6λ. The feature is associated with the change of asymmetry with the Hc: the larger the Hc is, the higher the symmetry, i.e., the lower the asymmetry.

Fig. 7
figure 7

Cuboid height Hc dependence of FL of TJ generated in the case of Lc = Wc = λ, Hrs = 0.33λ, Wrs = 3λ and Lrs = 5λ

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3.2.2 Polarization effect arising from geometric asymmetry of cuboid

Attention is next paid to the polarization effect arising from geometric asymmetry of the cuboid. In this case, the reflection screen is assumed to be symmetrical with Lrs = Wrs, while the cuboid is asymmetrical with Lc ≠ Wc. It is still assumed that the centers of cuboid and reflection screen all are located on the z-axis.

First, the study is carried out in the case that the Lc varies within 0.2λ–2λ while the other parameters are fixed at Lrs = Wrs = 2λ, Wc = λ and Hc = Hrs = 0.33λ. Fig. 8a shows the Lc dependence of FL of the TJ generated in the two cases that the incident wave is either x- or y-polarized. One can see that the polarization effect changes dramatically within 0.0-0.5λ as the Lc changes from 0.2λ to 2λ, i.e., the polarization effect changes dramatically with the alteration of the asymmetry of cuboid for a given symmetrical reflection screen. As expected, the two plots intersect as Lc = Wc = λ. The intersection implies a null polarization effect on the FL due to a symmetrical structure. As Lc = 0.2λ, the polarization effect maximizes with a value of ~0.5λ.

Fig. 8
figure 8

Lc dependence of FL of TJ generated in the case that the cuboid is asymmetrical with fixed Wc = λ while varied Lc and the reflection screen is symmetrical with a Lrs = Wrs = 2λ and b Lrs = Wrs = 3λ. The other parameters are Hc = Hrs = 0.33λ

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Second, the further study is motivated by the question: if overall size change of the symmetrical reflection screen influences the polarization effect. To make the argument clear, we have repeated the above study by changing the size of symmetrical reflection screen from Lrs = Wrs = 2λ to Lrs = Wrs = 3λ. Fig. 8b shows the FL calculated as a function of Lc in the range of 0.8λ–3λ. The other parameters remain at Wc = λ and Hc = Hrs = 0.33λ. It is observed again that the change of the asymmetry of cuboid influences dramatically the polarization effect. Moreover, the overall size change of the symmetrical reflection screen results in considerable weakening of the polarization effect. As shown in Fig. 8b, the polarization effect maximizes at Lc = 2.2λ with a value of 0.24λ, which is about half of that in Fig. 8a, ~0.5λ as given above. In addition, the overall size change of the symmetrical reflection screen also results in alteration of location of the maximum effect, which is located at Lc = 0.2λ as shown in Fig. 8a and Lc = 2.2λ in Fig. 8b.

It is worthwhile to point out that the intersection of two plots takes place also at other Lc values, for example Lc = 1.425λ in Fig. 8a and several Lc values in Fig. 8b, for which the relevant cuboids are not symmetrical as Lc ≠ Wc (= λ). This means that the FL polarization effect is absent not only for a symmetrical structure, but also for some specific asymmetrical structures.

In words, the asymmetry of cuboid influences dramatically the polarization effect on the FL. The polarization effect on the FL does not exist only in case that the cuboid has a specific size. The polarization still takes an effect as the symmetrical reflection screen changes overall in size, and the overall size change of the symmetrical reflection screen results in both considerable change in extent of polarization effect and remarkable alteration of location of its maximum value. The polarization effect on the FL is absent not only for a symmetrical structure, but also for some specific asymmetrical structures due to the use of reflection mode.

Aiming at the maximum polarization effect observed in Fig. 8a, which takes place for Lc = 0.2λ, we have further studied the polarization angle θ dependence of characteristic parameters FL, Imax/I0, FWHMx and FWHMy of the TJ generated in case of Wc = λ, Lc = 0.2λ, Lrs = Wrs = 2λ and Hc = Hrs = 0.33λ. The calculated results are illustrated in Fig. 9. Table 3 brings together the values of these characteristic parameters for each θ. As the θ increases from 0° to 90°, the FL increases from 0.28λ to 0.77λ. The increase is by 275% as the polarization direction changes from x-axis to y-axis. This implies that, for an asymmetric structure consisting of a symmetric reflection screen and an asymmetric cuboid, the FL of the TJ can still be effectively tailored by the polarization state. In contrast, the Imax/I0, FWHMx and FWHMy all reveal a relatively moderate θ dependence. Similar to Fig. 6, the plots in Fig. 9 also reveal a feature of threshold polarization angle θc ≈ 60°.

Fig. 9
figure 9

Polarization angle θ dependence of FL, Imax/I0, FWHMx and FWHMy of the TJ generated in case of Wc = λ, Lc = 0.2λ, Lrs = Wrs = 2λ and Hc = Hrs = 0.33λ

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Table 3 Characteristic parameters of TJ generated in case of Wc = λ, Lc = 0.2λ, Lrs = Wrs = 2λ and Hc = Hrs = 0.33λ
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3.2.3 Polarization effect due to asymmetry of relative position of cuboid and reflection screen

As demonstrated above, the FL of the TJ generated in reflection mode by either an asymmetrical reflection screen or cuboid can be effectively tailored by the polarization state. In addition to the asymmetry arising from the cuboid or reflection screen geometry, their relative position can also cause an asymmetry. The FL of the TJ generated shows also the polarization effect as demonstrated below. As examples, in Fig. 10, we show the projections onto xz plane of TJs generated by a structure with the asymmetry induced by relative position of the reflection screen and cuboid. Assume that both the reflection screen and cuboid themselves are symmetrical with Lrs = Wrs = 2λ, Lc = Wc = λ, and Hc = Hrs = 0.33λ, and the geometric center of cuboid is no longer located on the z-axis while that of the reflection screen keeps on the z-axis (note that the whole structure in this case is no longer symmetrical with respect to the y-axis but still symmetrical with respect to x-axis). Figure 10a and b shows the case that the cuboid has an offset of 0.3λ along with the x-axis, and Fig. 10c and d shows the case that the cuboid has a larger offset of 0.5λ along with the x-axis. For convenience, the geometric shapes (projected onto xz plane) of the cuboid and reflection screen have been indicated by black boxes at the bottom of each panel. The horizontal white dashed lines mark the focal positions in the two cases of x- and y-polarized incident waves. The FL is similar to 0.18λ in Fig. 10a and 0.12λ in Fig. 10b, yielding a polarization effect of 0.06λ, and is about 0.18λ in Fig. 10c and 0.06λ in Fig. 10d, yielding a polarization effect of 0.12λ. One can see that the polarization effect arising from asymmetry of relative position of cuboid and reflection screen is much smaller than that arising from the geometric asymmetry of the reflection screen or cuboid, 0.65λ for the former and 0.5λ for the latter as given above

Fig. 10
figure 10

TJ projected onto xz plane and generated by a structure with an asymmetry induced by relative position of reflection screen and cuboid in case of illumination by a, c: y- and b, d: x-polarized wave. Both reflection screen and cuboid themselves are symmetrical with Lrs = Wrs = 2λ, Lc = Wc = λ, and Hc = Hrs = 0.33λ, and the geometric center of reflection screen keeps always on the z axis while that of the cuboid is no longer located on the z axis with an x-offset of 0.3λ for panels (a) and (b) and of 0.5λ for panels (c) and (d)

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Here, there are two queries to be clarified. In one hand, one may query that edge diffraction phenomenon [34, 35] may play a role in the generation of the TJs shown in Fig. 10. We think that this possibility is small because a jet beam generated by the edge diffraction distributes predominantly in a higher refractive index medium instead of a lower refractive index medium as observed in Fig. 10, where all the jets are generated mainly in lower refractive index medium (air). Moreover, if the edge diffraction really takes effect, the scattered field mainly distribute near the edge region inside the scatterer. Instead, one can see from Fig. 10 that all of the jets are mainly generated in the traditional region near the optical axis z, only weak scattered field can be observed inside the scatterer.

One may also query that if the scatterer boundary takes effect. As pointed out above, the open boundary condition has been adopted in our simulation. This implies that the scatterer boundary should not take an effect on the results. This conclusion should have nothing to do with whether the right boundary of the scatterer is aligned with that of reflection screen or not [the right-justified boundary corresponds to Fig. 10c and d, and the non-justified boundary to Fig. 10a and b]. Indeed, one can see from Fig. 10 that the scatterer boundary has less effect on the results whether it is right-justified with the boundary of reflection screen or not.

4 Discussion

The polarization effect of FL of the TJ generated in the reflection mode reveals some interesting features as follows.

  1. 1)

    The polarization effect of FL of the TJ generated in the reflection mode displays an asymmetry dependence completely different from that generated in the transmission mode. In the case of transmission mode, a larger scatterer has always a weaker confinement to incident waves, and hence results in generation of a TJ with a larger FL [33]. For example, for an asymmetrical scatterer that has a size along the x-axis larger than along the y-axis, the FL of TJ generated in the transmission mode in case of illumination of an x-polarized incident wave is always larger than that generated in case of illumination of a y-polarized incident wave. In the reflection mode, however, a completely reversed feature is observed as shown in Fig. 3a and d, i.e., the TJ generated in case of illumination of a y-polarized incident wave has a larger FL than that generated in case of illumination of an x-polarized incident wave instead, whether the Wrs is greater or smaller than Lrs. The situation becomes complicated in Fig. 9a and 9b. We can see that the asymmetry dependence of the FL polarization effect cannot be clearly resolved. The TJ generated by illumination of a y-polarized incident wave has a larger FL in some cases, while a smaller FL in the other cases.

  2. 2)

    The polarization effect on the FL is absent not only for a symmetrical structure, but also for some specific asymmetrical structures.

  3. 3)

    There exists a threshold polarization angle effect on the characteristic parameters of TJ.

Some features may be explained as follows.

For a TJ, whether it is generated in a transmission or reflection mode, the focusing position is determined by a comprehensive interaction between scatterer structure and incident wave’s two components of electric field along x- and y-axes. It is, therefore, comprehensible that the polarization effect on the FL is closely related to the asymmetry of scatterer structure as shown by the geometric asymmetry dependence of FL in Fig. 3a and d, and Fig. 8a and b.

The varying feature of the FL with the polarization angle θ observed in Figs. 6 and 9 may be interpreted as follows. Attention is first paid to the feature of FL values at θ = 0° and 90°, which correspond to the x- and y-polarizations, respectively. In Fig. 6, the FL values at θ = 0° and 90° correspond to those values in the case that the FL has a maximal polarization effect, i.e., those FL values at Lrs = 5λ in Fig. 3d. Similarly, the FL values at θ = 0° and 90° in Fig. 6 correspond to those FL values at Lc = 0.2λ in Fig. 8a, where the FL polarization effect maximizes. We can see from these figures that, as Lrs = 5λ or Lc = 0.2λ, the TJ generated in case of illumination of a y-polarized incident wave has the maximal FL while that generated in case of illumination of an x-polarized incident wave has the minimal FL. For a wave that is polarized along an arbitrary direction in the xy plane with a polarization angle θ and is incident onto an asymmetric structure, its electric field can be decomposed into x- and y-components and the FL of TJ is correlated with the proportion of two electric field components. Since the TJs with the maximal and minimal FLs are generated in case of irradiation of the y- and x-polarized waves, respectively, a larger proportion of y-component of electric field leads, therefore, to a larger FL, while a larger proportion of x-component results in a smaller FL. As the FL is a comprehensive result concerning with both x- and y-components of electric field of incident wave, it is thus reasonable that the FL has a median value as the incident wave is polarized with an arbitrary θ. As the polarization changes, the proportion of x- and y-components and hence the FL value changes as shown in Figs. 6 and 9.

In the reflection mode, the use of reflection screen results in a different formation principle of the TJ from the case of transmission mode. The introduction of reflection screen increases complexity of interference of the scattered waves and thereby results in other features of polarization effect, such as dramatic asymmetry dependence of the FL polarization effect, absence of the FL polarization effect for some specific asymmetrical structures, and threshold polarization angle effect on the characteristic parameters of TJ.

5 Conclusion

Study shows that polarization state of incident wave is one key factor in tailoring the performance of TJ generated by symmetric or asymmetric scatterer structures in reflection mode. For a symmetrical structure, it affects mainly FWHMx(y) of the TJ generated. For an asymmetrical structure, however, it affects more the FL as a result of interaction between electric field of incident wave and the dielectric scatterer structure. Further study shows that the effect of polarization state on FL of TJ is closely related to the geometric asymmetry of scatterer structure and both have a non-monotonic relationship. The geometric asymmetry of reflection screen or cuboid leads to a remarkable polarization effect on the FL, and the FL may change by as much as nine times as the polarization state is changed. In contrast, the effect has a weak dependence on the asymmetry caused by relative spatial position of reflection screen and cuboid.

The use of reflection screen results in a different formation principle of the TJ from the case of transmission mode. The introduction of reflection screen increases complexity of interference of scattered waves and induces some interesting FL polarization effect features, such as an asymmetry dependence of FL polarization effect quite different from the case in the transmission mode, absence of FL polarization effect for some specific asymmetrical structures, and threshold polarization angle effect on the characteristic parameters of TJ.

In addition, the characteristic parameters of TJ all exhibit a monotonic dependence on polarization angle θ, and the FL, FWHMx(y) and I/I0 change with the polarization angle θ strongly, moderately and weakly in order. It is concluded that polarization state of incident wave is a simpler, more flexible, effective, and convenient factor in tailoring the FL of a PNJ generated by an asymmetric scatterer than others, such as the geometry of scatterer. Meanwhile, the FWHMx(y) can be also tailored in some extents. With properly designed polarization state and asymmetrical scattering structure, a TJ with desired FL and FWHMx(y) is expected for various applications.