Introduction

Wavefront manipulation, a powerful method of light field manipulations, has enabled to generate various structured light fields, such as vortex beams1,2, Airy beams3 and Bessel beams4. In particular, through spatiotemporal coupling, many kinds of interesting light fields have been developed, including spatiotemporal vortex beams5, vortex rings6, light springs7, and photon conchs8, greatly enriching the light field family. In addition to the creation of novel structured light fields, many efforts have also made on dynamic light manipulation, e.g. by electrically controlling q-plate system to tune the topological charges of light fields9; by moving one of two diffractive optical elements to adjust optical properties of optical elements, e.g. lensing and vortex phase modulation10,11. Light manipulation can not only provide more degrees of freedom for various applications, but also paly important role in aberrational corrections12. Consequently, wavefront manipulation finds wide applications ranging from biomedical imaging13, information security and communication14,15, to micro/nano fabrication16 and particle manipulation17.

To date, most of the manipulations have been developed in visible or NIR regions. In the past decade, however, THz wavefront manipulation has attracted increasing attention due to the unique merits of THz fields: low photon energy, strong water absorption and high penetration18, predicting promising applications in particle acceleration, medical imaging, free-space communication, material inspection, security screening, and so on18,19,20,21,22.

As is well known, ultrafast THz field can be generated by optical rectification in nonlinear optical crystals, photoexcitation of photoconductive antennas and laser −plasma interactions23,24. Regretfully, none of these methods can directly transfer the structured phases of the pumps directly to the generated THz fields. Accordingly, most of the THz wavefront manipulations can only be realized by using special THz phase-modulators25,26,27,28, which are primarily static, such as zone plates25, diffractive optical elements26, and metasurfaces27. These devices inherently lack the flexibility, tunability, and versatility due to their fixed structural design. Recently, more attention has turned to develop the flexible THz phase modulators, e.g. graphene metasurface29, Moiré device30 and spatial light modulator31. Nevertheless, significant challenge is still there. Compared with their NIR counterparts, both THz materials and components are much more underdeveloped, which usually results in not only large inserted loss, low efficiency and narrow bandwidth, but also expensive, difficult fabrication hindering practical implementation28. Very recently, a THz spatial light modulator was reported to offer extensive control with quick response, opening possibilities for arbitrary THz phase shaping31. However, it suffers from limited modulation resolution, unwanted diffraction orders and considerable power loss. Anyway, to our knowledge, there is no report on the multiple-wavefront THz phase manipulation with a single scheme until now.

In this work, we propose an all-optical wavefront manipulation mechanism to realize THz wavefront manipulation by modulating the wavefronts of NIR pump beams using mature and readily available NIR components, instead of the THz counterparts. As such, the manipulation process can be performed entirely in the NIR domain, which also mitigates effectively the impact of THz diffraction during the manipulation. The NIR manipulation works on an innovative phase manipulation mechanism enabling independent controls of two kinds of wavefront-phases with a 2D phase modulator along two orthogonal directions. Its corresponding implementation strategy based on a dislocation scheme using two conjugated phase plates, which are two PB plates here. This scheme can generate two kinds of phase-manipulations plus their assemble by adjusting independently the 2D dislocations of the phase plates as long as the two wavefront functions have equal partial derivatives with respect to each other’s spatial coordinates. Furthermore, the modulated wavefront phases out of the scheme are then efficiently transferred into THz region via difference-frequency generation (DFG), thereby realize THz field manipulation.

A proof-of-principle experiment is presented to validate the all-optical manipulation mechanism and its implementation strategy, where two liquid crystal PB plates are used as the 2D phase modulator (2D-PM). By changing the relative dislocations of the PB plates along two orthogonal directions, the vortex, Bessel or vortex-Bessel NIR pump fields are generated. The modulated wavefront-phases are then converted into corresponding THz fields, including THz vortex, Bessel or vortex-Bessel fields. These THz structured fields hold promising applications in THz communication, imaging, sensing and others. The proposed mechanism and implementation strategy provides a flexible and effectual method for universal and versatile light field manipulation, particularly in the spectral regions like THz, where direct phase modulation is technically restricted.

Results

Modulation of vortex phase and logarithmic-axicon phase

Figure 1 illustrates the dislocation scheme of the THz wavefront modulation based 2D-PM and DFG. Here, 2D-PM includes a polarizer (P), two PB plates (PB1, 2) made of liquid crystal polymer, and a quarter wave-plate (QW). The PB1 and PB2 own the same modulation phase which is designed for the incidence with right-circular polarization. More details of PB1 and PB2 can be seen in the Supplementary Note 1. According to Fig. 1, one can move one of the PB plates (e.g. PB1) to get Bessel beam along x direction, vortex beam along y direction, or Bessel-vortex beam along both x and y directions.

Fig. 1: The dislocation scheme of THz wavefront manipulation.
figure 1

2D-PM 2D phase modulator, P Polarizer, PB1, 2 PB Plates, QW quarter wave-plate, NC nonlinear DFG crystal, H Horizontal polarization, V Vertical polarization.

The dislocation scheme outputs two pulses with orthogonal polarizations, which is exactly required for pumping type-II phase-matched DFG to generate THz pulse. The pumps propagate collinearly with conjugated phases, which ensures to have good phase stability and exact spatial mode-matching for high-efficient DFG28. Due to the pumps around 800 nm, a 1 mm <110> ZnTe crystal is used as THz generator32. Here, the pumps are chirped by a parallel grating pair to suppress the THz generation by optical rectification for high purity of the generated THz pulse by DFG.

To ensure the availability of liquid crystal 2D-PM, we check experimentally the introduced vortex phases (Fig. 2) and logarithmic-axicon phases (Fig. 3) by moving PB1 relative to PB2 along 2D orthogonal directions under the illumination of 800 nm laser with right-hand circularly polarization. According to Fig. 2a1−a10, as the vertical displacement ΔLy of PB1 increases from −218 μm to 211 μm, the topological charge of the generated vortex phase changes from 6 to −6. Here, no logarithmic-axicon phase is added to the vortex phase, indicating that 2D-PM can work for independently modulating the vortex phase along y direction.

Fig. 2: Vortex phase induced by 2D-PM.
figure 2

a1a10 The vortex phase vs. the vertical displacement ΔLy of the PB plates. b The dependence of the introduced topological charge vs. ΔLy.

Fig. 3: The logarithmic-axicon phase induced by 2D-PM at different horizontal displacements ΔLx of PB1.
figure 3

a1a8 The space distributions and b1b8 the horizontal axial distributions of the logarithmic-axicon phases.

Figure 2b describes the relationship between the topological charge of the vortex phase and the PB1 displacement ΔLy with the blue marks for the experimental results and a red line for the theoretical fitting, respectively. It shows that the topological charge decreases linearly with ΔLy. The change of topological charge by 1 corresponds to the change of ΔLy by −36 μm, which is consistent with the theoretical value (−35.4 μm). The results in Fig. 2 indicate that this design can be used to generate vortex phase with a tunable topological charge by simply moving one PB plate (PB1) along y direction.

Figure 3a1−a8 shows the measured spatial logarithmic−axicon phases within the region [−π, π] with the displacement ΔLx by moving PB1 along x (horizontal) direction. As ΔLx is adjusted from −340 μm to 120 μm, the color ring numbers or the logarithmic-axicon phases increase with |ΔLx|. Figure 3b1–b8 are the radial phases of Fig. 3a1−a8 after cascaded within a radial range from −4.2 mm to 4.2 mm, showing the logarithmic-axicon phases have positive peaks for ΔLx < 0 but negative peaks for ΔLx > 0. As ΔLx changes from −340 μm to 120 μm, the peak of the logarithmic-axicon phase gradually decreases from 23.6 rad to −8.6 rad. As predicted, the absolute values of the peaks increase with |ΔLx|. The logarithmic-axicon phases do not carry any vortex phase, indicating that 2D-PM changes independently the logarithmic-axicon phase by ΔLx.

Extracted the peak values of logarithmic-axicon phases in Fig. 3b1–b8, Fig. 4 shows the dependence of the peak values on PB1 displacement ΔLx. There, the blue marks stand for the experimental results and the red line for the theoretical fitting. Obviously, the logarithmic-axicon phase peak has a linear relation to ΔLx, with a phase increase of 2π with ΔLx by −85.2 μm. Summarily, Fig. 3 confirms that this design can manipulate independently logarithmic-axicon phase by simply moving PB1 along x direction.

Fig. 4: The variation of the logarithmic-axicon phase vs. PB1 displacement.
figure 4

The blue marking and the red line for experimental and fitting results, respectively.

Generation of THz vortex beam and THz Bessel beam

As described above, for an incidence with linear polarization, the outputs of the 2D-PM can be regarded as a pair of pulses collinearly propagating with orthogonally linear polarizations and conjugated phases. Accordingly, the phase difference between the pulse pair, or the modulated phase by moving PB1, can be transferred when the pulses are used as the pumps of type-II phase-matching DFG. Figure 5 is the setup to generate and manipulate vortex or/and Bessel THz fields by DFG when choosing vortex and logarithmic-axicon phases as the target modulated functions. In the setup, the used laser source is a Ti: sapphire laser system, which outputs 1 kHz/50 fs/800 nm linearly polarized pulse train. The output pulse train is chirped to 2.1 ps by a parallel grating pair, then divided into two parts by a beam splitter (BS): one as the probe (80 mW) and the other as the pump (780 mW). For the probe, a pulse stretcher (PS, a 180° folding right-angle ZF52 prism pair) is used for chirp compensation to compress the pulse duration back to 50 fs for high temporal resolution. The pump is firstly phase-manipulated by 2D-PM to evolve into a collinear pulse pair with vortex or/and logarithmic-axicon phases and orthogonal polarizations. Between the pulse pair, a time delay of 0.42 ps is induced by a 1-mm-thick α-BBO crystal. The pulse pair then go through lens L1 (f = 500 mm) and L2 (f = 150 mm) to pump a 1 mm 〈110〉 ZnTe crystal (ZT1) for the generation of THz fields by DGF. Here, the orthogonal polarizations and some chirps of the pulse pair are required to suppress the optical rectification effect33, while the time delay between two pulses can also be used to change the central frequency of the THz fields34. The THz fields are imaged onto another 1 mm 〈110〉 ZnTe crystal (ZT2) by THz lenses TL1 (f = 100 mm) and TL2 (f = 100 mm) for electro-optic sampling to characterize the spatiotemporal distribution of the THz fields. Here, dynamic subtraction35 is used to improve the measured signal-to-noise ratio. In the setup, 2D-PM, ZT1, ZT2 and the camera (CM, Andor, ZYLA-5.5-CL3SCMOS, 2560 × 2200) are situated with object-image conjugation. The temporal synchronization among the mechanical switch (MS), camera shutter, and delay line (TDL) is realized by a microcontroller for fast and effective measurements of the THz fields.

Fig. 5: The experiment setup of the vortex or/and Bessel THz pulse based on 2D-PM.
figure 5

BS beam splitter, 2D-PM 2D phase modulator, P1-2 polarizer, QW quarter wave plate, α-BBO α-Ba(BO2)2 crystal, MS mechanical switch, L1-4 lenses, TL1-2 THz lenses, ZT1-2 〈110〉 ZnTe crystals, SW1-2 high resistance silicon wafers, TL1-2 THz lens, M1-3 mirrors, DSC dispersion compensator, TDL Delay line, CM sCMOS camera.

In this experiment, the generated THz pulses have typically a time duration of 6 ps and a central frequency of 2 THz (see the Supplementary Note 2). Figure 6 presents the THz field distributions measured by moving vertically PB1 at a distance of −31 μm. Here, Fig. 6a1–d1, a2–d2 and a3–d3 correspond to the electric field, the intensity, and the phase distributions of the THz field at different times, respectively. The recorded time interval is 133 fs. One can see the generated THz field exhibits a four-leaf petal-like amplitude distribution and a donut-shaped intensity profile. Its phase increases counterclockwise for an azimuthal angle from 0 to 4π, or the electric field and the phase distribution exhibit clockwise rotation with time. Figure 6a4–d4 are extracted from Fig. 6a3–d3 by the blue dots and the red lines for the experimental and the fitting results, respectively. Consequentially, the angular phase of the THz pulse increases linearly with azimuth at different times, which shows the generated THz vortex field has a topological charge of 2.

Fig. 6: The spatial distributions of a THz vortex pulse at different times.
figure 6

The measured electric field (a1d1), intensity (a2d2), phase distributions (a3d3) of THz vortex pulse at different times, and the dependence of the vortex phase on azimuthal angle (a4d4).

By moving PB1 vertically, the topological charge of the THz vortex pulse is modulated from −5 to 5. Figure 7a–c corresponds to the electric field, the intensity, and the phase of the measured THz pulses at the temporal points with peak intensities. In Fig. 7b, the diameter of the donut-shaped THz intensity gradually increases with the absolute value of topological charge. When the value is greater than 3, the THz ring tends to exceed the probe spot, resulting in a failed detection of the THz ring, thereby lower detected THz intensity and contrast. However, the THz vortex phase with different topological charges can still observed from the phase distribution, where the topological charge decreases linearly with the PB1 displacement. The topological charge changes 1 corresponds to a PB1 displacement of −16.9 μm, agreeing well with the theoretical value of −17.7 μm, as shown in Fig. 7d. It is worth noting that a PB1 displacement of 36 μm corresponds to change the topological charge by 1 in Fig. 2b, but by 2 in Fig. 7d, which is originated from the topological charge conservation during DFG for generating THz vortex pulse. So, the results in Fig. 7 confirms that this setup can generate THz vortex pulses with tunable topological charges.

Fig. 7: The spatial distributions of a THz vortex pulse with different topological charges.
figure 7

The measured electric field (a), intensity (b), and phase (c) of the THz vortex field with different dislocations ΔLy along vertical directions, as well as the corresponding topological charges (d) with the blue mark and the red line for the experimental and the fitting results.

By moving horizontally PB1, 2D-PM can impose a logarithmic-axicon phase into the pump pulse, thereby generate THz-Bessel pulse. Figure 8a1–d1 or a2–d2 are the results obtained by moving PB1 so that ΔLx = −6 μm or −56 μm. Figure 8a1 or a2 is the time-domain integration of the THz intensity, which has multiple ring structures with a central spot size of 415 μm or 463 μm, while Fig. 8b1 or b2 shows the 1D intensity distribution of the THz beam, extracted along the horizontal direction across the THz beam of Fig. 8a1 or a2. There, the blue dots represent the measured results, while the red lines are the theoretical fittings by the first-order Bessel function. The good agreements between the blue dots and the red lines imply that the generated THz fields are of THz Bessel beams. To check the diffraction of the generated THz Bessel beam, we move the THz crystal ZT2 along the propagation axis to detect the spot sizes of the THz Bessel beam at different propagation distances. Figure 8c1, c2 show the 1D intensity distributions of THz Bessel pulses measured at different axial positions, while Fig. 8d1, d2 are the corresponding the peak intensities of THz beams with the ZT2 displacement. From the figures, the THz Bessel beams generated at PB1 positions of −6 μm and −56 μm have different “diffraction-free” distances, ~9.5 mm and 15.3 mm, respectively, which are far longer than the Rayleigh length of the traditional Gaussian beams under the same conditions (1.43 mm for a Gaussian beam with a waist of 463 μm @ 2 THz). Based on the above measured results, this setup has successfully generated THz Bessel beams with adjustable diffraction-free distances by changing the PB1 horizontal position.

Fig. 8: The measured THz Bessel beams generated for different ΔLx: -6 μm (a1-d1) and -56 μm (a2-d2).
figure 8

a1, a2 THz transverse spatial intensity distributions, b1, b2 THz 1D transverse intensity distributions along horizontal direction, c1, c2 1D axial THz intensity, d1, d2 the peak intensity of THz rings in (c1, c2) with different propagation distances.

By adjusting simultaneously PB1 along both horizontal and vertical directions, both the vortex and logarithmic-axicon phases can be introduced simultaneously on the pump pulses. Figure 9 shows the generated THz pulse with the topological charges of −1 and −2, where Fig. 9a1-a2, b1-b2 and c1-c2 stand for the electric field, intensity and phase distributions of the THz fields, respectively. The generated THz pulses exhibit the electric field with multiple petal-shapes, the intensities with multiple ring-shapes, and spiral phase distributions, indicating the generated THz pulses are THz Bessel vortex pulses.

Fig. 9: The field distributions of the generated THz Bessel-vortex beams.
figure 9

The measured electric field (a1, a2), intensity (b1, b2), and phase distributions (c1, c2) of THz Bessel-vortex beams with topological charges −1 and −2, respectively.

Discussion

This paper introduces a manipulation mechanism, enabling universal and versatile wavefront-phase control in THz regime. This manipulation operates with a flexible NIR wavefront-phase manipulation combined with type-II phase-matched DFG. The NIR manipulation works on a universal manipulation mechanism, which manipulates independently a pair of wavefront-phases by changing two-dimensionally the dislocations of a couple of phase-elements with conjugated phase functions. The independence is ensured as long as the partial derivatives of the target phase pair with respect to each other’s spatial coordinates are equal, which can be satisfied by many important phase pairs, such as logarithmic-axicon versus vortex phases (for Bessel and vortex beams) and 1D versus 2D cubic phases (for 1D and 2D Airy beams). Notably, when only a single target wavefront is required, the method can accommodate arbitrary phase profiles. In principle, any phase modulator, including the graphene-based metasurface, can be used to act as the two phase-elements if they can induce desired phases. Here, to realize the NIR manipulation, a dislocation scheme is designed, where two PB plates are used as the conjugated phase elements due to their high efficiency and easy fabrication. Besides the NIR manipulation, the scheme also outputs a pair of collinear pulses with conjugated phases and orthogonal polarizations.

By using the resulting pulse pair as the pumps, the manipulated wavefront-phases allow to be transferred from NIR to THz region by type-II DFG. As a result, all the work for THz manipulation is performed at NIR, thus eliminates the need for THz components and mitigates strong THz diffraction during manipulation. Furthermore, the dislocation-based design allows for flexible and scalable wavefront control.

A subsequent proof-of-principle experiment validates this universal and versatile THz manipulation by utilizing a dual-phase modulator with a pair of PB plates and a < 110> ZnTe crystal-based type-II DFG. By moving one of the PB plates along one or two of the spatial directions, we achieved logarithmic-axicon, vortex, or combinations of these phases. This method generates THz vortex beams with tunable topological charges from +5 to −5 and produces Bessel beams with diffraction-free distances of 7.86 mm and 15.95 mm through horizontal adjustment of the PB plate. Additionally, THz Bessel-vortex beams with topological charges of 1 and 2 have been created. These THz structured beams hold considerable potential for applications in THz imaging, sensing, communication, and THz-matter interaction. Summarily, this paper provides a flexible way for dynamic and multifaceted manipulation of light fields, particularly in the THz domain, where direct phase modulation is typically challenging.

Methods

Principle of Terahertz wavefront modulation

It is well known that THz pulse can be generated by nonlinear frequency conversion, including optical rectification36 and difference frequency33. The conversion can be described by three-wave nonlinear coupling equation as

$$\frac{d{E}_{THz}}{dz}\propto -ik{E}_{p1}\cdot {E}_{p2}^{\ast }=-ik{A}_{p1}\cdot {A}_{p2}^{*}\cdot \exp (i{\varphi }_{p1}-i{\varphi }_{p2}).$$
(1)

Here, ETHz, Ep1 = Ap1·exp(p1) and Ep2 = Ap2·exp(p2) stands for THz field, as well as the two pump pulses with their amplitudes Ap1, Ap2 and phases φp1, φp2, while k is the coupling coefficient, respectively. From Eq. (1), it is obvious that the phase of the THz field is determined by the phase difference of the pumps. For optical rectification, THz field is originated from the difference frequency conversion between the spectral components of the same pulse, so some pump phases, e. g. vortex phase, fail to transform into the THz pulse during the conversion. However, for traditional difference frequency, it occurs between two separated pump pulses, so φp1 and φp2 can be manipulated separately. Consequentially, the generated THz phase can be manipulated by modulating the two pump phases separately. In other words, the manipulation in THz region can be replaced by the manipulation in near infrared (NIR) regime, which makes THz manipulation be much easier, simpler and more effective.

Obviously, the universality and versatility of the THz manipulation depends mainly on the NIR phase-manipulation which is described as following. Suppose that the modulation functions of the two phase-elements are ϕ (x, y) and −ϕ (x, y), they induce totally null phase if they are exactly aligned. When a dislocation between them occurs along x or y direction by Δx or Δy, the total phase variation, or the induced net-phase modulation of the phase-elements, shall be

$$\begin{array}{c}\Delta {\phi }_{x}(x,y)=\phi (x+\Delta x,y)-\phi (x,y)\\ \approx \frac{\partial \phi (x,y)}{\partial x}\Delta x={\varphi }_{1}(x,y)\Delta x,\end{array}$$
(2a)

or

$$\begin{array}{c}\Delta {\phi }_{y}(x,y)=\phi (x,y+\Delta y)-\phi (x,y)\\ \approx \frac{\partial \phi (x,y)}{\partial y}\Delta y={\varphi }_{2}(x,y)\Delta y.\end{array}$$
(2b)

Eq. (2a) and Eq. (2b) mean the net modulation phase Δϕx (x, y) or Δϕy (x, y) is ϕ (x, y) differential along the dislocation direction. Obviously, if

$$\frac{\partial {\varphi }_{2}(x,y)}{\partial x}=\frac{\partial {\varphi }_{1}(x,y)}{\partial y},$$
(3)

Δϕx (x, y) and Δϕy (x, y) can be changed independently. If Δϕx (x, y) and Δϕy (x, y) are nonzero simultaneously, the phase variation Δϕ (x, y) can be expressed

$$\begin{array}{c}\Delta \phi (x,y)=\phi (x+\Delta x,y+\Delta y)-\phi (x,y)\\ \approx {\varphi }_{1}(x,y)\Delta x+{\varphi }_{2}(x,y)\Delta y.\end{array}$$
(4)

The discussion above shows, as long as the target phases φ1 (x, y) and φ2 (x, y) fulfill Eq. (3), the function ϕ (x, y) can be designed for the phase plates by solving Eqs. (2a) and (2b). Accordingly, this design can implement the independent light manipulations with φ1(x, y), φ2(x, y) or φ1(x, y)+φ2(x, y) by moving one of the plates along x, y directions or both of them.

There are lots of interesting function pair for the phase modulation meeting Eq. (3), e.g. logarithmic-axicon vs. vortex phases for Bessel beam vs. vortex beams, 1D vs. 2D cubic phase for 1D vs. 2D Airy beams, and so on. These mean that this mechanism can work for versatile manipulations with up to three target wavefront-phases by only one setup for flexible light phase manipulations. However, if only one target required, it is universally applicable to any wavefront function. This NIR phase-manipulation can output a pair of pulses with conjugated modulated phases and orthogonal polarizations. The orthogonal polarizations allow the pulse pair owning perfect polarization matching to pump type-II phase-matched DFG, which makes the modulated phases be transferred to the generated THz fields. All above mean this universal mechanism can work for flexible manipulations with multiple-wavefront phases owing wide applications.

Any phase modulators, e.g. graphene-based metasurface, can be used to carry out this work if they can induce desired phases. Here, two PB phase plates are chosen as the mutually conjugated phase modulators due to the high efficiency and easy fabrication. According to Eq. (1)~(4), we implement the concept of the MIR wavefront-phase manipulation by a dislocation scheme with a pair of PB phase plates. The detail design is subsequently described for the 2D-PM.

Realization of Terahertz wavefront modulation

To realize the proposed phase manipulation, a pair of NIR pulses are required as the pumps to generate THz wave by DFG according to Eq. (1). Here, type-II phase-matching is preferred to suppress optical rectification generally used for THz generation, which means it is better that the pump pulses have orthogonal polarizations besides the controllable phases described in Eqs. (2) and (3). In order to get this kind of pump pair, a dislocation scheme have been designed. It includes mainly a 2D phase modulator (2D-PM), a pair of identical PB optical elements (PB) made of liquid crystal polymers, situated after a polarizer (P) but before a quarter wave-plate (QWP). 2D-PM imposes phase-modulations and changes the output polarizations to the orthogonal directions for different circularly polarized inputs, which means that, if a circularly polarized beam passes through 2D-PM, it is equivalent to be phase-modulated by the two plates with conjugated phases37, thereby can achieve net-phase modulation as Eqs. (2a) and (2b). If the incident beam is a linearly polarized laser, it can be regarded as the superposition of its two orthogonal polarization components: one with right-circular polarization and the other with left-circular polarization. After 2D-PM, the outputs include two orthogonal circular-polarization beams with a conjugated net modulation phase. Finally, a quarter wave-plate (QW) with a fast axis orientation of 45° is used to convert the two circularly polarized beams into two with orthogonal linearly polarizations.

As an example, we choose the target phases φ1(x, y) and φ2(x, y) of 2D−PM to be logarithmic-axicon and vortex phases as followings

$${\varphi }_{1}(x,y)=\frac{\pi a}{\lambda }\,ln\left(\frac{{x}^{2}+{y}^{2}}{{b}^{2}}\right),$$
(5a)
$${\varphi }_{2}(x,y)=-\frac{2\pi a}{\lambda }\cdot arctan \frac{y}{x}.$$
(5b)

Here, a and b are size coefficients. It is easy to confirm that φ1 (x, y) of Eq. (5a) and φ2 (x, y) of Eq. (5b) meet Eq. (3). By substituting Eqs. (5a) and (5b) into Eqs. (2a) and (2b), the phase modulation ϕ (x, y) of the two PB-plates for right-circular polarized inputs shall be designed as

$$\phi (x,y)=\frac{2\pi a}{\lambda }\left[x\cdot \,ln\left(\frac{\sqrt{{x}^{2}+{y}^{2}}}{b}\right)-y\cdot arctan \left(\frac{y}{x}\right)-x\right].$$
(6)