Abstract

Based on the complementary V-shaped antenna structure, ultrathin vortex phase plates are designed to achieve the terahertz (THz) optical vortices with different topological charges. Utilizing a THz holographic imaging system, the two dimensional complex field information of the generated THz vortex beam with the topological number l=1 is directly obtained. Its far field propagation properties are analyzed in detail, including the rotation, the twist direction, and the Gouy phase shift of the vortex phase. An analytic Laguerre-Gaussian mode is used to simulate and explain the measured phenomena. The experimental and simulation results overlap each other very well.

©2013 Optical Society of America

1. Introduction

As special light beams, vortex beams are always interesting members in optics, which possess helical wavefronts, an on-axis phase singularity, and quantization orbital angular momentums [13]. Owing to the unique properties, vortex beams have been applied in many industrial and research fields, such as optical micro-manipulation [4, 5], bio-medical [68], optical information transmission [911], and so on. Researchers have paid more and more attention on investigations about optical vortices. Currently, techniques for generating optical vortices mainly include mode converters [12, 13], computer generated holograms [14, 15], spiral phase plates [16], and so on. Many methods have been proposed to detect optical vortices, such as the measurement of the mechanical torque arising from orbital angular momentum [2, 3], Stokes parameters method [17], dispersion and phase modulation method [18], and the traditional interferometric method [19].

As a novel far infrared radiation, the terahertz (THz) ray has become a hot topic of optical researches in the past decades and the THz technology has exhibited strong application potentials in many imaging and sensing fields [2023]. However, there are only considerably less works about the THz vortex beam [24]. In the radio frequency community, a discrete sub-wavelength two dimensional structure concept has been proposed to design reflectarray and lens-array elements for modulating the phase distributions of electro-magnetic waves. Currently, the technique has been rapidly developed for advanced sensing or communications applications [2530]. In 2011, N. F. Yu et al. proposed a V-shaped antenna meta-surface structure to achieve the phase modulation for cross polarized scattered fields [31]. In 2012, P. Genevet et al. utilized the design to build a vortex phase plate (VPP) and generate an optical vortex in the infrared wave band [32]. In 2013, we extended the structure into the THz wave band and fabricated ultrathin THz lenses and phase holograms [33].

In this paper, the complementary V-shaped antenna structure is applied to mold ultrathin VPPs in the THz wave band. Utilizing a THz holographic imaging system, the complex field information of the generated THz vortex beam with the topological number l=1 is coherently measured and its far field propagation properties are investigated in detail. A basis Laguerre-Gaussian mode is used to simulate and explain the evolutions of the intensity and phase of the THz vortex beam. This work prompts the development of ultrathin THz elements and investigations on the THz vortex beams.

2. Designs

According to [33], based on the surface plasmonic resonance effect, eight kinds of complementary V-shaped slit antennas are designed to realize the various phase shifts for the transmitted cross polarized lights. Figure 1(a) shows the design sketch of the antenna on the X-Y plane. Each antenna unit consists of two equivalent rectangular slits connected at one end in a square region with a length p = 200 μm. The slit width (w=5μm) is fixed. The slit length (h), the angle (θ) between two slits and the angle (β) between the bisector line of the V-shaped antenna and the Y-axis can be adjusted to achieve the phase modulation of the scattered field. It should be noted that the selection of the complementary structure is to ensure the enough diffraction efficiency of the expected THz spectral component. Figure 1(b) presents eight antenna designs which correspond to the phase distributions of the scattered fields from 3π/4 to π with a π/4 interval. The first four antennas have θ=130°,120°,100°,60° with corresponding lengths of h=78,82,90,150μm, while the β is fixed as 45°. The other four units are the mirror images of the first four ones.

 figure: Fig. 1

Fig. 1 (a) A complementary V-shaped antenna phase modulation unit. (b) Eight kinds of complementary V-shaped antenna structures corresponding to phase shifts from −3π/4 to π with a π/4 interval. (c) Photography of the central region of the designed vortex phase plate (VPP) for l = 1.

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To built a VPP with the topological number l=1, the required phase distribution in polar coordinates (r,α) can be easily calculated by φ=lα. The phase values are quantized to eight values. A series of complementary V-shaped antennas are picked in terms of the phase distribution and are filled in the corresponding positions. The designed VPP consists of 40×40 units in the 8×8mm2 area. In the experiment, the VPP is fabricated in a gold film (with a 100 nm thickness) deposited on a double-side polished high resistivity silicon substrate (with a 500 μm thickness) using the conventional photolithography and metallization process. The central region of the VPP is shown in Fig. 1(c). The central wavelength of the VPP is 400 μm (corresponding to 0.75 THz), so the thickness of the effective layer of the VPP is only 1/4000 of the wavelength. When the incident THz beam with a horizontal polarization passes through the VPP, the transmitted vertical polarized THz beam has same transmission intensity and the corresponding phase modulation on each antenna unit. Then, a vortex THz field is formed.

3. Results and discussions

3.1 Complex field information of the THz vortex beam

To check the function of the VPP, a THz holographic imaging system [34, 35] is utilized to measure the intensity and phase information of the transmitted cross-polarized THz field. Figure 2(a) shows the experimental scheme. A laser beam with a 800 nm central wavelength, a 100 fs pulse duration and a 1 kHz repetition ratio illuminates a <110> ZnTe crystal with a 3 mm thickness (is not shown in Fig. 2(a)) to radiate the horizontal polarized THz wave with a 15 mm diameter due to the optical rectification effect. After the THz wave passing through the VPP, the transmitted THz vortex beam with a vertical polarization impinges on the sensor crystal (another <110> ZnTe with a 3 mm thickness). The probe beam with a vertical polarization is reflected onto the sensor crystal by a 50/50 non-polarization beam splitter (BS). In the crystal, the probe polarization is modulated by the THz field to carry the two dimensional THz information. To measure the THz vertical polarization component, the <001> axis of the sensor crystal is perpendicular with the vertical direction [34, 36]. The reflected probe beam is incident into the imaging unit of the system and the THz complex field is extracted by the balanced electro-optic detection technique. The detailed principle about the imaging system has been published in [34, 35]. By changing the optical path difference between the THz beam and the probe beam, 128 THz temporal images are obtained and the corresponding time window is 17 ps. Performing the Fourier transformation on the temporal signal at each pixel, the intensity and the phase information of the 0.75 THz component is exactly extracted. It should be noted that the refractive index of the silicon substrate is about 3.4 at 0.75 THz [37], so its optical thickness reaches to 1.7 mm and the time difference between the main pulse and echo pulse is about 11 ps. Utilizing zero-padding, the interference effects between the main pulse and echo pulse is removed.

 figure: Fig. 2

Fig. 2 (a) Terahertz (THz) holographic imaging system. (b) and (c) display the measured intensity and phase distribution of the generated THz vortex beam with l = 1 at 0.75THz, respectively. (d) The phase curves with the azimuthal angle α and the radial distance r = 1.5 mm.

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Figures 2(b) and 2(c) show the intensity and phase distributions of the 0.75 THz vortex beam. It should be noted that when the value is less than 0.2 on the normalized THz intensity image, the color of the corresponding pixel is set as gray on the phase map to filter the uncertain noise. It can be seen that the intensity distribution is mainly uniform except for two regions with higher transmissivity. The difference may be caused by the fabrication error. The phase map exhibits the expected variation. The distance between the VPP and the sensor crystal is about 4 mm. Owing to the diffraction, the measured phase presents a smoothly monotonically increase from 3π/4 to π with the azimuth angle α. To more clearly observe the phase distribution, the phase data with various α and fixed r = 1.5 mm are extracted and plotted in Fig. 2(d). It shows the good linear relationship between the phase and the azimuth angle, which demonstrates that the designed VPP can be applied to form a THz vortex beam well.

3.2 Evolution properties of the THz vortex beam in the far field

To investigate the evolution properties of the THz vortex beam in the far field, a silicon lens with a 25 mm focal length and a 25.4 mm diameter is used to focus the THz field and a Z-scan measurement is performed by moving the lens and the VPP together around the focal spot, as shown in Fig. 3(a). The distance between the VPP and the lens is about 4 mm. The position of the focal spot is set as the initial point. The scan range along the Z-axis is from −20 mm to 20 mm with the 1 mm scan resolution. On each scan point, the 0.75 THz spectral component is extracted to build the focusing process of the THz vortex beam. The intensity and the phase evolutions of the 0.75 THz vortex beam around the focal spot are recorded in Media 1 and Media 2 of Figs. 3(b) and 3(c). Figures 3(b) and 3(c) show that the intensity and the phase maps with Z = −20 mm, −10 mm, 0 mm, 10 mm, 20 mm on the X-Y plane. The intensity of the converging THz wave shows a doughnut shape due to the central phase singularity. On the focal plane, the radius of the THz ring beam is about 1.1 mm. The non-uniformity on the ring is attributed to the transmission discrepance on the VPP (as shown in Fig. 2(b)). In the phase evolution, the phase profile presents a spiral distribution and always rotates in a clockwise sense when the vortex beam approaches or departs the focal spot. Meanwhile, twist directions of the phase before and after the focal spot are opposite.

 figure: Fig. 3

Fig. 3 (a) Experimental setup for observing the intensity and phase evolutions of the THz vortex beam in the focusing process. (b) Media 1 and (c) Media 2 are the intensity and phase maps of the measured THz vortex beam with Z = −20 mm, −10 mm, 0 mm, 10 mm, and 20 mm. (d) Phase distributions of the Laguerre-Gaussian (LG) mode with l=1,p=0at Z = −20 mm, −10 mm, 0 mm, 10 mm, and 20 mm. (e) Correlation coefficients of the basis LG modes in the measured THz vortex beam.

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To explain these phenomena in the phase evolution, the vortex beam is decomposed by a series of basis Laguerre-Gaussian (LG) modes Epl [38], which is given by

Epl(r,α,z)[2rw(z)]Lp|l|(2r2w(z)2)exp[r2w(z)2]exp[ikzr22(zR2+z2)+ilα+iΦG(z)],
where k is the wave number in vacuum, p is the radial index which is 0 for a linear polarized LG mode, Lp|l|(x) is the generalized Laguerre polynomial. w(z) is the beam radius at a propagation distance z, as given by
w(z)=w01+z2/zR2.
Parameter zR is the Rayleigh range and is expressed as zR=kw02/2. w0 is the radius of the beam waist. In addition, ΦG(z) is the Gouy phase shift, which is an additional phase shift for a beam passing through the focal region. It is given by [24]

ΦG(z)=(2p+|l|+1)arctan(z/zR).

In our experiment, correlation coefficients of the LG modes in the measured THz vortex beam Ev are calculated by

Cp,l=Ev(Epl)*rdrdα,
where the asterisk denotes the complex conjugate, z, w0 and k are set as 0 mm, 1.1 mm and 157 cm−1(corresponding to 0.75 THz). Figure 3(e) presents the relative charge distribution for the measured vortex beam. It can be seen that the generated vortex field is 87% correlated with the LG mode of l=1,p=0. It means that the main phase properties of the vortex field can be explained by analyzing the analytic expression of the LG mode. According to Eq. (1), the phase distributions of the LG mode with l=1,p=0 at Z = −20 mm, −10 mm, 0 mm, 10 mm, 20 mm are calculated, as shown in Fig. 3(d). It can be seen that the simulation results are in well agreement with the experimental ones.

The phase of the LG mode includes three terms kzr22(zR2+z2), lα and ΦG(z). To identify their functions, the phase distributions of lα+ΦG(z) and kzr22(zR2+z2)+lα are calculated at Z = −20 mm, −10 mm, 0 mm, 10 mm, 20 mm, respectively. Figures 4(a) and 4(b) give the theoretical results. In Fig. 4(a), the vortex phase exhibits a clockwise rotation as the propagation distance, which denotes that the lα forms a vortex phase and the term ΦG(z) determines the rotation of the vortex phase. In Fig. 4(b), the vortex phase presents a spiral distribution and its twist has an inverse direction after passing through the focal spot, which demonstrates that the term kzr22(zR2+z2) converts the vortex phase into a spiral profile and causes the variation of its twist direction.

 figure: Fig. 4

Fig. 4 (a) and (b) are the phase distributions of lα+ΦG(z) and kzr22(zR2+z2)+lα with Z = −20 mm, −10 mm, 0 mm, 10 mm, 20 mm, respectively.

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3.3 Gouy phase shift of the THz vortex beam

To further observe the propagation properties of the THz vortex beam in the focusing process, its Gouy phase shift is also investigated. On different positions of the Z-axis, the central lines (X = 0 mm) of each intensity and phase maps are extracted to exhibit the longitudinal distributions of the THz field, as shown in Figs. 5(a) and 5(c). In Fig. 5(a), the cross section of the ring intensity distribution of the focused THz vortex beam is presented, which is symmetrical along the Z-axis. In Fig. 5(c), the phase evolution of the THz vortex beam in the focusing process is clearly displayed. The phase rotation and the variation of the twist direction can be observed in Fig. 5(c). The longitudinal phase shift around the optical axis only reaches about 1.5π due to the limitation of the measurement range.

 figure: Fig. 5

Fig. 5 (a) and (b) represent the experimental and theoretical longitudinal intensity distributions of the THz vortex beam on the Y-Z plane, respectively. (c) and (d) are the corresponding longitudinal phase distributions. (e) is the paraxial phase shifts extracted from (c) and (d). The red solid curve and blue squares correspond to the theoretical and experimental results, respectively.

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To compare with the experimental results, propagation of the LG mode with l=1,p=0 is also calculated from Z = −20 mm to Z = 20 mm with the 1 mm interval. The intensity and phase maps are exhibited in Figs. 5(b) and 5(d), respectively. It is clear that the experimental results are in excellent agreement with the simulation ones. To obtain the Gouy phase shift, the paraxial phase values (Y = 0.25 mm) in Figs. 5(c) and 5(d) are extracted and plotted in Fig. 5(e). The data on the optical axis (Y = 0 mm) are not selected for avoiding its phase singularity and noise. In Fig. 5(e), the red solid curve and the blue squares represent the theoretical and experimental results, respectively. Both of them present a 1.5π phase change and match each other very well. The phenomena nicely manifest the Gouy phase shift of the THz vortex beam between the front and the back of the focal point.

3.4 THz vortex beams with other topological numbers

To further demonstrate the validity of the VPP design method, other two VPPs with l=2 and l=3 are designed and fabricated. The photographs of their central regions are shown in Figs. 6(a) and 6(b). The THz imaging system is used to check effects of the two VPPs. The experimental results are presented in Figs. 6(c) and 6(d). It is evident that the expected vortex phase distributions for the 0.75 THz component are formed, which exhibited the linear phase variation in two and three cycles around the optical axis. To check the qualities of generated vortex beams, the vortex phase distributions with l=2 and l=3 are simulated by using the phase term lα. The simulation results are shown in Figs. 6(e) and 6(f), which are consistent with the experimental results. It indicates that THz vortex beams with higher topological numbers can be generated using this technique.

 figure: Fig. 6

Fig. 6 (a) and (b) are the photographs of the central parts of two designed VPPs with l = 2 and l = 3. (c) and (d) are the measured vortex phase distributions with l = 2, and l = 3. (e) and (f) are the simulated vortex phase distributions with l = 2 and l = 3.

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4. Conclusions

In a conclusion, the ultrathin planar THz VPP is designed based on the complementary V-shaped antenna structures and the THz vortex beam with the topological number l=1 is generated utilizing the VPP. By utilizing the THz holographic imaging system, the vortex phase distribution of the THz beam is observed and the propagation properties of the THz vortex beam in the far field are investigated. Taking advantage of the LG mode with l=1,p=0, the phase evolution of the THz vortex beam in the focusing process are systematically analyzed. In addition, the experimental results also demonstrated that the THz vortex beams with higher topological numbers can be generated based on the method. We believed that the work is valuable for investigations on special light beams, the exploitation of planar THz elements and the THz information transmission.

Acknowledgments

This work was supported by the 973 Program of China (No. 2013CBA01702), the National Natural Science Foundation of China (Nos. 91233202, 61205097, 11204188, and 11174211), the National High Technology Research and Development Program of China (No. 2012AA101608-6), the Beijing Natural Science Foundation (No. KZ201110028035 and 1132011), and the Program for New Century Excellent Talents in University (No. NCET-12-0607).

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5. N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22(1), 52–54 (1997). [CrossRef]   [PubMed]  

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References

  • View by:

  1. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A Math. Phys. Sci. 336(1605), 165–190 (1974).
    [Crossref]
  2. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
    [Crossref] [PubMed]
  3. S. M. Barnett and L. Allen, “Orbital angular momentum and nonparaxial light beams,” Opt. Commun. 110(5–6), 670–678 (1994).
    [Crossref]
  4. H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995).
    [Crossref] [PubMed]
  5. N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22(1), 52–54 (1997).
    [Crossref] [PubMed]
  6. R. W. Steubing, S. Cheng, W. H. Wright, Y. Numajiri, and M. W. Berns, “Laser induced cell fusion in combination with optical tweezers: the laser cell fusion trap,” Cytometry 12(6), 505–510 (1991).
    [Crossref] [PubMed]
  7. J. T. Finer, R. M. Simmons, and J. A. Spudich, “Single myosin molecule mechanics: piconewton forces and nanometre steps,” Nature 368(6467), 113–119 (1994).
    [Crossref] [PubMed]
  8. S. Seeger, S. Monajembashi, K. J. Hutter, G. Futterman, J. Wolfrum, and K. O. Greulich, “Application of laser optical tweezers in immunology and molecular genetics,” Cytometry 12(6), 497–504 (1991).
    [Crossref] [PubMed]
  9. G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12(22), 5448–5456 (2004).
    [Crossref] [PubMed]
  10. A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental two-photon, three-dimensional entanglement for quantum communication,” Phys. Rev. Lett. 89(24), 240401 (2002).
    [Crossref] [PubMed]
  11. G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: Preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88(1), 013601 (2001).
    [Crossref] [PubMed]
  12. C. Tamm and C. O. Weiss, “Bistability and optical switching of spatial patterns in a laser,” J. Opt. Soc. Am. B 7(6), 1037–7038 (1990).
    [Crossref]
  13. M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96(1–3), 123–132 (1993).
    [Crossref]
  14. N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17(3), 221–223 (1992).
    [Crossref] [PubMed]
  15. V. Y. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” JETP Lett. 52(8), 429–431 (1990).
  16. M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate,” Opt. Commun. 112(5–6), 321–327 (1994).
    [Crossref]
  17. I. Freund, “Poincaré vortices,” Opt. Lett. 26(24), 1996–1998 (2001).
    [Crossref] [PubMed]
  18. M. E. Grein, H. A. Haus, L. A. Jiang, and E. P. Ippen, “Action on pulse position and momentum using dispersion and phase modulation,” Opt. Express 8(12), 664–669 (2001).
    [Crossref] [PubMed]
  19. J. M. Vaughan and D. V. Willetts, “Interference properties of a light beam having a helical wave surface,” Opt. Commun. 30(3), 263–267 (1979).
    [Crossref]
  20. M. Tonouchi, “Cutting-edge terahertz technology,” Nat. Photonics 1(2), 97–105 (2007).
    [Crossref]
  21. C. Jansen, S. Wietzke, O. Peters, M. Scheller, N. Vieweg, M. Salhi, N. Krumbholz, C. Jördens, T. Hochrein, and M. Koch, “Terahertz imaging: applications and perspectives,” Appl. Opt. 49(19), E48–E57 (2010).
    [Crossref] [PubMed]
  22. A. Redo-Sanchez and X. C. Zhang, “Terahertz science and technology trends,” IEEE J. Sel. Top. Quant. 14(2), 1–10 (2008).
    [Crossref]
  23. A. Bitzera and M. Waltherb, “Terahertz near-field imaging of metallic subwavelength holes and hole arrays,” Appl. Phys. Lett. 92(23), 231101 (2008).
    [Crossref]
  24. J. Hamazaki, Y. Mineta, K. Oka, and R. Morita, “Direct observation of Gouy phase shift in a propagating optical vortex,” Opt. Express 14(18), 8382–8392 (2006).
    [Crossref] [PubMed]
  25. D. M. Pozar, S. D. Targonski, and H. D. Syrigos, “Design of millimeter wave microstrip reflectarrays,” IEEE Trans. Antenn. Propag. 45(2), 287–296 (1997).
    [Crossref]
  26. J. Perruisseau-Carrier, F. Bongard, R. Golubovic-Niciforovic, R. Torres-Sánchez, and J. R. Mosig, “Contributions to the modeling and design of reconfigurable reflecting cells embedding discrete control elements,” IEEE Trans. Microw. Theory Tech. 58(6), 1621–1628 (2010).
    [Crossref]
  27. J. Perruisseau-Carrier, “Dual-polarized and polarization-flexible reflective cells with dynamic phase control,” IEEE Trans. Antenn. Propag. 58(5), 1494–1502 (2010).
    [Crossref]
  28. J. A. Encinar, L. S. Datashvili, J. A. Zornoza, M. Arrebola, M. Sierra-Castañer, J. L. Besada-Sanmartín, H. Baier, and H. Legay, “Dual-polarization dual-coverage reflectarray for space applications,” IEEE Trans. Antenn. Propag. 54(10), 2827–2837 (2006).
    [Crossref]
  29. C. Cheng, B. Lakshminarayanan, and A. Abbaspour-Tamijani, “A programmable lens-array antenna with monolithically integrated MEMS switches,” IEEE Trans. Microw. Theory Tech. 57(8), 1874–1884 (2009).
    [Crossref]
  30. J. Y. Lau and S. V. Hum, “A planar reconfigurable aperture with lens and reflectarray modes of operation,” IEEE Trans. Microw. Theory Tech. 58(12), 3547–3555 (2010).
  31. N. F. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
    [Crossref] [PubMed]
  32. P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100(1), 013101 (2012).
    [Crossref]
  33. D. Hu, X. K. Wang, S. F. Feng, J. S. Ye, W. F. Sun, Q. Kan, P. J. Klar, and Y. Zhang, “Ultrathin terahertz planar elements,” Adv. Opt. Mater. 1(2), 186–191 (2013).
    [Crossref]
  34. X. K. Wang, Y. Cui, W. F. Sun, J. S. Ye, and Y. Zhang, “Terahertz polarization real-time imaging based on balanced electro-optic detection,” J. Opt. Soc. Am. A 27(11), 2387–2393 (2010).
    [Crossref] [PubMed]
  35. X. K. Wang, Y. Cui, W. F. Sun, J. S. Ye, and Y. Zhang, “Terahertz real-time imaging with balanced electro-optic detection,” Opt. Commun. 283(23), 4626–4632 (2010).
    [Crossref] [PubMed]
  36. R. X. Zhang, Y. Cui, W. Sun, and Y. Zhang, “Polarization information for terahertz imaging,” Appl. Opt. 47(34), 6422–6427 (2008).
    [Crossref] [PubMed]
  37. C. M. Randall and R. D. Rawcliffe, “Refractive indices of germanium, silicon, and fused quartz in the far infrared,” Appl. Opt. 6(11), 1889–1895 (1967).
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  38. A. E. Siegman, Lasers (University Science Books, 1986, Chap. 16).

2013 (1)

D. Hu, X. K. Wang, S. F. Feng, J. S. Ye, W. F. Sun, Q. Kan, P. J. Klar, and Y. Zhang, “Ultrathin terahertz planar elements,” Adv. Opt. Mater. 1(2), 186–191 (2013).
[Crossref]

2012 (1)

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100(1), 013101 (2012).
[Crossref]

2011 (1)

N. F. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

2010 (6)

J. Y. Lau and S. V. Hum, “A planar reconfigurable aperture with lens and reflectarray modes of operation,” IEEE Trans. Microw. Theory Tech. 58(12), 3547–3555 (2010).

X. K. Wang, Y. Cui, W. F. Sun, J. S. Ye, and Y. Zhang, “Terahertz polarization real-time imaging based on balanced electro-optic detection,” J. Opt. Soc. Am. A 27(11), 2387–2393 (2010).
[Crossref] [PubMed]

X. K. Wang, Y. Cui, W. F. Sun, J. S. Ye, and Y. Zhang, “Terahertz real-time imaging with balanced electro-optic detection,” Opt. Commun. 283(23), 4626–4632 (2010).
[Crossref] [PubMed]

C. Jansen, S. Wietzke, O. Peters, M. Scheller, N. Vieweg, M. Salhi, N. Krumbholz, C. Jördens, T. Hochrein, and M. Koch, “Terahertz imaging: applications and perspectives,” Appl. Opt. 49(19), E48–E57 (2010).
[Crossref] [PubMed]

J. Perruisseau-Carrier, F. Bongard, R. Golubovic-Niciforovic, R. Torres-Sánchez, and J. R. Mosig, “Contributions to the modeling and design of reconfigurable reflecting cells embedding discrete control elements,” IEEE Trans. Microw. Theory Tech. 58(6), 1621–1628 (2010).
[Crossref]

J. Perruisseau-Carrier, “Dual-polarized and polarization-flexible reflective cells with dynamic phase control,” IEEE Trans. Antenn. Propag. 58(5), 1494–1502 (2010).
[Crossref]

2009 (1)

C. Cheng, B. Lakshminarayanan, and A. Abbaspour-Tamijani, “A programmable lens-array antenna with monolithically integrated MEMS switches,” IEEE Trans. Microw. Theory Tech. 57(8), 1874–1884 (2009).
[Crossref]

2008 (3)

R. X. Zhang, Y. Cui, W. Sun, and Y. Zhang, “Polarization information for terahertz imaging,” Appl. Opt. 47(34), 6422–6427 (2008).
[Crossref] [PubMed]

A. Redo-Sanchez and X. C. Zhang, “Terahertz science and technology trends,” IEEE J. Sel. Top. Quant. 14(2), 1–10 (2008).
[Crossref]

A. Bitzera and M. Waltherb, “Terahertz near-field imaging of metallic subwavelength holes and hole arrays,” Appl. Phys. Lett. 92(23), 231101 (2008).
[Crossref]

2007 (1)

M. Tonouchi, “Cutting-edge terahertz technology,” Nat. Photonics 1(2), 97–105 (2007).
[Crossref]

2006 (2)

J. A. Encinar, L. S. Datashvili, J. A. Zornoza, M. Arrebola, M. Sierra-Castañer, J. L. Besada-Sanmartín, H. Baier, and H. Legay, “Dual-polarization dual-coverage reflectarray for space applications,” IEEE Trans. Antenn. Propag. 54(10), 2827–2837 (2006).
[Crossref]

J. Hamazaki, Y. Mineta, K. Oka, and R. Morita, “Direct observation of Gouy phase shift in a propagating optical vortex,” Opt. Express 14(18), 8382–8392 (2006).
[Crossref] [PubMed]

2004 (1)

2002 (1)

A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental two-photon, three-dimensional entanglement for quantum communication,” Phys. Rev. Lett. 89(24), 240401 (2002).
[Crossref] [PubMed]

2001 (3)

G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: Preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88(1), 013601 (2001).
[Crossref] [PubMed]

I. Freund, “Poincaré vortices,” Opt. Lett. 26(24), 1996–1998 (2001).
[Crossref] [PubMed]

M. E. Grein, H. A. Haus, L. A. Jiang, and E. P. Ippen, “Action on pulse position and momentum using dispersion and phase modulation,” Opt. Express 8(12), 664–669 (2001).
[Crossref] [PubMed]

1997 (2)

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22(1), 52–54 (1997).
[Crossref] [PubMed]

D. M. Pozar, S. D. Targonski, and H. D. Syrigos, “Design of millimeter wave microstrip reflectarrays,” IEEE Trans. Antenn. Propag. 45(2), 287–296 (1997).
[Crossref]

1995 (1)

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995).
[Crossref] [PubMed]

1994 (3)

S. M. Barnett and L. Allen, “Orbital angular momentum and nonparaxial light beams,” Opt. Commun. 110(5–6), 670–678 (1994).
[Crossref]

J. T. Finer, R. M. Simmons, and J. A. Spudich, “Single myosin molecule mechanics: piconewton forces and nanometre steps,” Nature 368(6467), 113–119 (1994).
[Crossref] [PubMed]

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate,” Opt. Commun. 112(5–6), 321–327 (1994).
[Crossref]

1993 (1)

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96(1–3), 123–132 (1993).
[Crossref]

1992 (2)

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17(3), 221–223 (1992).
[Crossref] [PubMed]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

1991 (2)

S. Seeger, S. Monajembashi, K. J. Hutter, G. Futterman, J. Wolfrum, and K. O. Greulich, “Application of laser optical tweezers in immunology and molecular genetics,” Cytometry 12(6), 497–504 (1991).
[Crossref] [PubMed]

R. W. Steubing, S. Cheng, W. H. Wright, Y. Numajiri, and M. W. Berns, “Laser induced cell fusion in combination with optical tweezers: the laser cell fusion trap,” Cytometry 12(6), 505–510 (1991).
[Crossref] [PubMed]

1990 (2)

V. Y. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” JETP Lett. 52(8), 429–431 (1990).

C. Tamm and C. O. Weiss, “Bistability and optical switching of spatial patterns in a laser,” J. Opt. Soc. Am. B 7(6), 1037–7038 (1990).
[Crossref]

1979 (1)

J. M. Vaughan and D. V. Willetts, “Interference properties of a light beam having a helical wave surface,” Opt. Commun. 30(3), 263–267 (1979).
[Crossref]

1974 (1)

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A Math. Phys. Sci. 336(1605), 165–190 (1974).
[Crossref]

1967 (1)

Abbaspour-Tamijani, A.

C. Cheng, B. Lakshminarayanan, and A. Abbaspour-Tamijani, “A programmable lens-array antenna with monolithically integrated MEMS switches,” IEEE Trans. Microw. Theory Tech. 57(8), 1874–1884 (2009).
[Crossref]

Aieta, F.

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100(1), 013101 (2012).
[Crossref]

N. F. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

Allen, L.

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22(1), 52–54 (1997).
[Crossref] [PubMed]

S. M. Barnett and L. Allen, “Orbital angular momentum and nonparaxial light beams,” Opt. Commun. 110(5–6), 670–678 (1994).
[Crossref]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96(1–3), 123–132 (1993).
[Crossref]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Arrebola, M.

J. A. Encinar, L. S. Datashvili, J. A. Zornoza, M. Arrebola, M. Sierra-Castañer, J. L. Besada-Sanmartín, H. Baier, and H. Legay, “Dual-polarization dual-coverage reflectarray for space applications,” IEEE Trans. Antenn. Propag. 54(10), 2827–2837 (2006).
[Crossref]

Baier, H.

J. A. Encinar, L. S. Datashvili, J. A. Zornoza, M. Arrebola, M. Sierra-Castañer, J. L. Besada-Sanmartín, H. Baier, and H. Legay, “Dual-polarization dual-coverage reflectarray for space applications,” IEEE Trans. Antenn. Propag. 54(10), 2827–2837 (2006).
[Crossref]

Barnett, S. M.

Bazhenov, V. Y.

V. Y. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” JETP Lett. 52(8), 429–431 (1990).

Beijersbergen, M. W.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate,” Opt. Commun. 112(5–6), 321–327 (1994).
[Crossref]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96(1–3), 123–132 (1993).
[Crossref]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Berns, M. W.

R. W. Steubing, S. Cheng, W. H. Wright, Y. Numajiri, and M. W. Berns, “Laser induced cell fusion in combination with optical tweezers: the laser cell fusion trap,” Cytometry 12(6), 505–510 (1991).
[Crossref] [PubMed]

Berry, M. V.

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A Math. Phys. Sci. 336(1605), 165–190 (1974).
[Crossref]

Besada-Sanmartín, J. L.

J. A. Encinar, L. S. Datashvili, J. A. Zornoza, M. Arrebola, M. Sierra-Castañer, J. L. Besada-Sanmartín, H. Baier, and H. Legay, “Dual-polarization dual-coverage reflectarray for space applications,” IEEE Trans. Antenn. Propag. 54(10), 2827–2837 (2006).
[Crossref]

Bitzera, A.

A. Bitzera and M. Waltherb, “Terahertz near-field imaging of metallic subwavelength holes and hole arrays,” Appl. Phys. Lett. 92(23), 231101 (2008).
[Crossref]

Blanchard, R.

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100(1), 013101 (2012).
[Crossref]

Bongard, F.

J. Perruisseau-Carrier, F. Bongard, R. Golubovic-Niciforovic, R. Torres-Sánchez, and J. R. Mosig, “Contributions to the modeling and design of reconfigurable reflecting cells embedding discrete control elements,” IEEE Trans. Microw. Theory Tech. 58(6), 1621–1628 (2010).
[Crossref]

Capasso, F.

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100(1), 013101 (2012).
[Crossref]

N. F. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

Cheng, C.

C. Cheng, B. Lakshminarayanan, and A. Abbaspour-Tamijani, “A programmable lens-array antenna with monolithically integrated MEMS switches,” IEEE Trans. Microw. Theory Tech. 57(8), 1874–1884 (2009).
[Crossref]

Cheng, S.

R. W. Steubing, S. Cheng, W. H. Wright, Y. Numajiri, and M. W. Berns, “Laser induced cell fusion in combination with optical tweezers: the laser cell fusion trap,” Cytometry 12(6), 505–510 (1991).
[Crossref] [PubMed]

Coerwinkel, R. P. C.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate,” Opt. Commun. 112(5–6), 321–327 (1994).
[Crossref]

Courtial, J.

Cui, Y.

Datashvili, L. S.

J. A. Encinar, L. S. Datashvili, J. A. Zornoza, M. Arrebola, M. Sierra-Castañer, J. L. Besada-Sanmartín, H. Baier, and H. Legay, “Dual-polarization dual-coverage reflectarray for space applications,” IEEE Trans. Antenn. Propag. 54(10), 2827–2837 (2006).
[Crossref]

Dholakia, K.

Encinar, J. A.

J. A. Encinar, L. S. Datashvili, J. A. Zornoza, M. Arrebola, M. Sierra-Castañer, J. L. Besada-Sanmartín, H. Baier, and H. Legay, “Dual-polarization dual-coverage reflectarray for space applications,” IEEE Trans. Antenn. Propag. 54(10), 2827–2837 (2006).
[Crossref]

Feng, S. F.

D. Hu, X. K. Wang, S. F. Feng, J. S. Ye, W. F. Sun, Q. Kan, P. J. Klar, and Y. Zhang, “Ultrathin terahertz planar elements,” Adv. Opt. Mater. 1(2), 186–191 (2013).
[Crossref]

Finer, J. T.

J. T. Finer, R. M. Simmons, and J. A. Spudich, “Single myosin molecule mechanics: piconewton forces and nanometre steps,” Nature 368(6467), 113–119 (1994).
[Crossref] [PubMed]

Franke-Arnold, S.

Freund, I.

Friese, M. E. J.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995).
[Crossref] [PubMed]

Futterman, G.

S. Seeger, S. Monajembashi, K. J. Hutter, G. Futterman, J. Wolfrum, and K. O. Greulich, “Application of laser optical tweezers in immunology and molecular genetics,” Cytometry 12(6), 497–504 (1991).
[Crossref] [PubMed]

Gaburro, Z.

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100(1), 013101 (2012).
[Crossref]

N. F. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

Genevet, P.

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100(1), 013101 (2012).
[Crossref]

N. F. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

Gibson, G.

Golubovic-Niciforovic, R.

J. Perruisseau-Carrier, F. Bongard, R. Golubovic-Niciforovic, R. Torres-Sánchez, and J. R. Mosig, “Contributions to the modeling and design of reconfigurable reflecting cells embedding discrete control elements,” IEEE Trans. Microw. Theory Tech. 58(6), 1621–1628 (2010).
[Crossref]

Grein, M. E.

Greulich, K. O.

S. Seeger, S. Monajembashi, K. J. Hutter, G. Futterman, J. Wolfrum, and K. O. Greulich, “Application of laser optical tweezers in immunology and molecular genetics,” Cytometry 12(6), 497–504 (1991).
[Crossref] [PubMed]

Hamazaki, J.

Haus, H. A.

He, H.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995).
[Crossref] [PubMed]

Heckenberg, N. R.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995).
[Crossref] [PubMed]

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17(3), 221–223 (1992).
[Crossref] [PubMed]

Hochrein, T.

Hu, D.

D. Hu, X. K. Wang, S. F. Feng, J. S. Ye, W. F. Sun, Q. Kan, P. J. Klar, and Y. Zhang, “Ultrathin terahertz planar elements,” Adv. Opt. Mater. 1(2), 186–191 (2013).
[Crossref]

Hum, S. V.

J. Y. Lau and S. V. Hum, “A planar reconfigurable aperture with lens and reflectarray modes of operation,” IEEE Trans. Microw. Theory Tech. 58(12), 3547–3555 (2010).

Hutter, K. J.

S. Seeger, S. Monajembashi, K. J. Hutter, G. Futterman, J. Wolfrum, and K. O. Greulich, “Application of laser optical tweezers in immunology and molecular genetics,” Cytometry 12(6), 497–504 (1991).
[Crossref] [PubMed]

Ippen, E. P.

Jansen, C.

Jiang, L. A.

Jördens, C.

Kan, Q.

D. Hu, X. K. Wang, S. F. Feng, J. S. Ye, W. F. Sun, Q. Kan, P. J. Klar, and Y. Zhang, “Ultrathin terahertz planar elements,” Adv. Opt. Mater. 1(2), 186–191 (2013).
[Crossref]

Kats, M. A.

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100(1), 013101 (2012).
[Crossref]

N. F. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

Klar, P. J.

D. Hu, X. K. Wang, S. F. Feng, J. S. Ye, W. F. Sun, Q. Kan, P. J. Klar, and Y. Zhang, “Ultrathin terahertz planar elements,” Adv. Opt. Mater. 1(2), 186–191 (2013).
[Crossref]

Koch, M.

Kristensen, M.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate,” Opt. Commun. 112(5–6), 321–327 (1994).
[Crossref]

Krumbholz, N.

Lakshminarayanan, B.

C. Cheng, B. Lakshminarayanan, and A. Abbaspour-Tamijani, “A programmable lens-array antenna with monolithically integrated MEMS switches,” IEEE Trans. Microw. Theory Tech. 57(8), 1874–1884 (2009).
[Crossref]

Lau, J. Y.

J. Y. Lau and S. V. Hum, “A planar reconfigurable aperture with lens and reflectarray modes of operation,” IEEE Trans. Microw. Theory Tech. 58(12), 3547–3555 (2010).

Legay, H.

J. A. Encinar, L. S. Datashvili, J. A. Zornoza, M. Arrebola, M. Sierra-Castañer, J. L. Besada-Sanmartín, H. Baier, and H. Legay, “Dual-polarization dual-coverage reflectarray for space applications,” IEEE Trans. Antenn. Propag. 54(10), 2827–2837 (2006).
[Crossref]

Lin, J.

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100(1), 013101 (2012).
[Crossref]

McDuff, R.

Mineta, Y.

Molina-Terriza, G.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: Preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88(1), 013601 (2001).
[Crossref] [PubMed]

Monajembashi, S.

S. Seeger, S. Monajembashi, K. J. Hutter, G. Futterman, J. Wolfrum, and K. O. Greulich, “Application of laser optical tweezers in immunology and molecular genetics,” Cytometry 12(6), 497–504 (1991).
[Crossref] [PubMed]

Morita, R.

Mosig, J. R.

J. Perruisseau-Carrier, F. Bongard, R. Golubovic-Niciforovic, R. Torres-Sánchez, and J. R. Mosig, “Contributions to the modeling and design of reconfigurable reflecting cells embedding discrete control elements,” IEEE Trans. Microw. Theory Tech. 58(6), 1621–1628 (2010).
[Crossref]

Numajiri, Y.

R. W. Steubing, S. Cheng, W. H. Wright, Y. Numajiri, and M. W. Berns, “Laser induced cell fusion in combination with optical tweezers: the laser cell fusion trap,” Cytometry 12(6), 505–510 (1991).
[Crossref] [PubMed]

Nye, J. F.

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A Math. Phys. Sci. 336(1605), 165–190 (1974).
[Crossref]

Oka, K.

Padgett, M. J.

Pas’ko, V.

Perruisseau-Carrier, J.

J. Perruisseau-Carrier, “Dual-polarized and polarization-flexible reflective cells with dynamic phase control,” IEEE Trans. Antenn. Propag. 58(5), 1494–1502 (2010).
[Crossref]

J. Perruisseau-Carrier, F. Bongard, R. Golubovic-Niciforovic, R. Torres-Sánchez, and J. R. Mosig, “Contributions to the modeling and design of reconfigurable reflecting cells embedding discrete control elements,” IEEE Trans. Microw. Theory Tech. 58(6), 1621–1628 (2010).
[Crossref]

Peters, O.

Pozar, D. M.

D. M. Pozar, S. D. Targonski, and H. D. Syrigos, “Design of millimeter wave microstrip reflectarrays,” IEEE Trans. Antenn. Propag. 45(2), 287–296 (1997).
[Crossref]

Randall, C. M.

Rawcliffe, R. D.

Redo-Sanchez, A.

A. Redo-Sanchez and X. C. Zhang, “Terahertz science and technology trends,” IEEE J. Sel. Top. Quant. 14(2), 1–10 (2008).
[Crossref]

Rubinsztein-Dunlop, H.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995).
[Crossref] [PubMed]

Salhi, M.

Scheller, M.

Scully, M. O.

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100(1), 013101 (2012).
[Crossref]

Seeger, S.

S. Seeger, S. Monajembashi, K. J. Hutter, G. Futterman, J. Wolfrum, and K. O. Greulich, “Application of laser optical tweezers in immunology and molecular genetics,” Cytometry 12(6), 497–504 (1991).
[Crossref] [PubMed]

Sierra-Castañer, M.

J. A. Encinar, L. S. Datashvili, J. A. Zornoza, M. Arrebola, M. Sierra-Castañer, J. L. Besada-Sanmartín, H. Baier, and H. Legay, “Dual-polarization dual-coverage reflectarray for space applications,” IEEE Trans. Antenn. Propag. 54(10), 2827–2837 (2006).
[Crossref]

Simmons, R. M.

J. T. Finer, R. M. Simmons, and J. A. Spudich, “Single myosin molecule mechanics: piconewton forces and nanometre steps,” Nature 368(6467), 113–119 (1994).
[Crossref] [PubMed]

Simpson, N. B.

Smith, C. P.

Soskin, M. S.

V. Y. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” JETP Lett. 52(8), 429–431 (1990).

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Spudich, J. A.

J. T. Finer, R. M. Simmons, and J. A. Spudich, “Single myosin molecule mechanics: piconewton forces and nanometre steps,” Nature 368(6467), 113–119 (1994).
[Crossref] [PubMed]

Steubing, R. W.

R. W. Steubing, S. Cheng, W. H. Wright, Y. Numajiri, and M. W. Berns, “Laser induced cell fusion in combination with optical tweezers: the laser cell fusion trap,” Cytometry 12(6), 505–510 (1991).
[Crossref] [PubMed]

Sun, W.

Sun, W. F.

D. Hu, X. K. Wang, S. F. Feng, J. S. Ye, W. F. Sun, Q. Kan, P. J. Klar, and Y. Zhang, “Ultrathin terahertz planar elements,” Adv. Opt. Mater. 1(2), 186–191 (2013).
[Crossref]

X. K. Wang, Y. Cui, W. F. Sun, J. S. Ye, and Y. Zhang, “Terahertz real-time imaging with balanced electro-optic detection,” Opt. Commun. 283(23), 4626–4632 (2010).
[Crossref] [PubMed]

X. K. Wang, Y. Cui, W. F. Sun, J. S. Ye, and Y. Zhang, “Terahertz polarization real-time imaging based on balanced electro-optic detection,” J. Opt. Soc. Am. A 27(11), 2387–2393 (2010).
[Crossref] [PubMed]

Syrigos, H. D.

D. M. Pozar, S. D. Targonski, and H. D. Syrigos, “Design of millimeter wave microstrip reflectarrays,” IEEE Trans. Antenn. Propag. 45(2), 287–296 (1997).
[Crossref]

Tamm, C.

C. Tamm and C. O. Weiss, “Bistability and optical switching of spatial patterns in a laser,” J. Opt. Soc. Am. B 7(6), 1037–7038 (1990).
[Crossref]

Targonski, S. D.

D. M. Pozar, S. D. Targonski, and H. D. Syrigos, “Design of millimeter wave microstrip reflectarrays,” IEEE Trans. Antenn. Propag. 45(2), 287–296 (1997).
[Crossref]

Tetienne, J. P.

N. F. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

Tonouchi, M.

M. Tonouchi, “Cutting-edge terahertz technology,” Nat. Photonics 1(2), 97–105 (2007).
[Crossref]

Torner, L.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: Preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88(1), 013601 (2001).
[Crossref] [PubMed]

Torres, J. P.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: Preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88(1), 013601 (2001).
[Crossref] [PubMed]

Torres-Sánchez, R.

J. Perruisseau-Carrier, F. Bongard, R. Golubovic-Niciforovic, R. Torres-Sánchez, and J. R. Mosig, “Contributions to the modeling and design of reconfigurable reflecting cells embedding discrete control elements,” IEEE Trans. Microw. Theory Tech. 58(6), 1621–1628 (2010).
[Crossref]

van der Veen, H. E. L. O.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96(1–3), 123–132 (1993).
[Crossref]

Vasnetsov, M.

Vasnetsov, M. V.

V. Y. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” JETP Lett. 52(8), 429–431 (1990).

Vaughan, J. M.

J. M. Vaughan and D. V. Willetts, “Interference properties of a light beam having a helical wave surface,” Opt. Commun. 30(3), 263–267 (1979).
[Crossref]

Vaziri, A.

A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental two-photon, three-dimensional entanglement for quantum communication,” Phys. Rev. Lett. 89(24), 240401 (2002).
[Crossref] [PubMed]

Vieweg, N.

Waltherb, M.

A. Bitzera and M. Waltherb, “Terahertz near-field imaging of metallic subwavelength holes and hole arrays,” Appl. Phys. Lett. 92(23), 231101 (2008).
[Crossref]

Wang, X. K.

D. Hu, X. K. Wang, S. F. Feng, J. S. Ye, W. F. Sun, Q. Kan, P. J. Klar, and Y. Zhang, “Ultrathin terahertz planar elements,” Adv. Opt. Mater. 1(2), 186–191 (2013).
[Crossref]

X. K. Wang, Y. Cui, W. F. Sun, J. S. Ye, and Y. Zhang, “Terahertz polarization real-time imaging based on balanced electro-optic detection,” J. Opt. Soc. Am. A 27(11), 2387–2393 (2010).
[Crossref] [PubMed]

X. K. Wang, Y. Cui, W. F. Sun, J. S. Ye, and Y. Zhang, “Terahertz real-time imaging with balanced electro-optic detection,” Opt. Commun. 283(23), 4626–4632 (2010).
[Crossref] [PubMed]

Weihs, G.

A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental two-photon, three-dimensional entanglement for quantum communication,” Phys. Rev. Lett. 89(24), 240401 (2002).
[Crossref] [PubMed]

Weiss, C. O.

C. Tamm and C. O. Weiss, “Bistability and optical switching of spatial patterns in a laser,” J. Opt. Soc. Am. B 7(6), 1037–7038 (1990).
[Crossref]

White, A. G.

Wietzke, S.

Willetts, D. V.

J. M. Vaughan and D. V. Willetts, “Interference properties of a light beam having a helical wave surface,” Opt. Commun. 30(3), 263–267 (1979).
[Crossref]

Woerdman, J. P.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate,” Opt. Commun. 112(5–6), 321–327 (1994).
[Crossref]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96(1–3), 123–132 (1993).
[Crossref]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Wolfrum, J.

S. Seeger, S. Monajembashi, K. J. Hutter, G. Futterman, J. Wolfrum, and K. O. Greulich, “Application of laser optical tweezers in immunology and molecular genetics,” Cytometry 12(6), 497–504 (1991).
[Crossref] [PubMed]

Wright, W. H.

R. W. Steubing, S. Cheng, W. H. Wright, Y. Numajiri, and M. W. Berns, “Laser induced cell fusion in combination with optical tweezers: the laser cell fusion trap,” Cytometry 12(6), 505–510 (1991).
[Crossref] [PubMed]

Ye, J. S.

D. Hu, X. K. Wang, S. F. Feng, J. S. Ye, W. F. Sun, Q. Kan, P. J. Klar, and Y. Zhang, “Ultrathin terahertz planar elements,” Adv. Opt. Mater. 1(2), 186–191 (2013).
[Crossref]

X. K. Wang, Y. Cui, W. F. Sun, J. S. Ye, and Y. Zhang, “Terahertz polarization real-time imaging based on balanced electro-optic detection,” J. Opt. Soc. Am. A 27(11), 2387–2393 (2010).
[Crossref] [PubMed]

X. K. Wang, Y. Cui, W. F. Sun, J. S. Ye, and Y. Zhang, “Terahertz real-time imaging with balanced electro-optic detection,” Opt. Commun. 283(23), 4626–4632 (2010).
[Crossref] [PubMed]

Yu, N.

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100(1), 013101 (2012).
[Crossref]

Yu, N. F.

N. F. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

Zeilinger, A.

A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental two-photon, three-dimensional entanglement for quantum communication,” Phys. Rev. Lett. 89(24), 240401 (2002).
[Crossref] [PubMed]

Zhang, R. X.

Zhang, X. C.

A. Redo-Sanchez and X. C. Zhang, “Terahertz science and technology trends,” IEEE J. Sel. Top. Quant. 14(2), 1–10 (2008).
[Crossref]

Zhang, Y.

D. Hu, X. K. Wang, S. F. Feng, J. S. Ye, W. F. Sun, Q. Kan, P. J. Klar, and Y. Zhang, “Ultrathin terahertz planar elements,” Adv. Opt. Mater. 1(2), 186–191 (2013).
[Crossref]

X. K. Wang, Y. Cui, W. F. Sun, J. S. Ye, and Y. Zhang, “Terahertz real-time imaging with balanced electro-optic detection,” Opt. Commun. 283(23), 4626–4632 (2010).
[Crossref] [PubMed]

X. K. Wang, Y. Cui, W. F. Sun, J. S. Ye, and Y. Zhang, “Terahertz polarization real-time imaging based on balanced electro-optic detection,” J. Opt. Soc. Am. A 27(11), 2387–2393 (2010).
[Crossref] [PubMed]

R. X. Zhang, Y. Cui, W. Sun, and Y. Zhang, “Polarization information for terahertz imaging,” Appl. Opt. 47(34), 6422–6427 (2008).
[Crossref] [PubMed]

Zornoza, J. A.

J. A. Encinar, L. S. Datashvili, J. A. Zornoza, M. Arrebola, M. Sierra-Castañer, J. L. Besada-Sanmartín, H. Baier, and H. Legay, “Dual-polarization dual-coverage reflectarray for space applications,” IEEE Trans. Antenn. Propag. 54(10), 2827–2837 (2006).
[Crossref]

Adv. Opt. Mater. (1)

D. Hu, X. K. Wang, S. F. Feng, J. S. Ye, W. F. Sun, Q. Kan, P. J. Klar, and Y. Zhang, “Ultrathin terahertz planar elements,” Adv. Opt. Mater. 1(2), 186–191 (2013).
[Crossref]

Appl. Opt. (3)

Appl. Phys. Lett. (2)

A. Bitzera and M. Waltherb, “Terahertz near-field imaging of metallic subwavelength holes and hole arrays,” Appl. Phys. Lett. 92(23), 231101 (2008).
[Crossref]

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100(1), 013101 (2012).
[Crossref]

Cytometry (2)

S. Seeger, S. Monajembashi, K. J. Hutter, G. Futterman, J. Wolfrum, and K. O. Greulich, “Application of laser optical tweezers in immunology and molecular genetics,” Cytometry 12(6), 497–504 (1991).
[Crossref] [PubMed]

R. W. Steubing, S. Cheng, W. H. Wright, Y. Numajiri, and M. W. Berns, “Laser induced cell fusion in combination with optical tweezers: the laser cell fusion trap,” Cytometry 12(6), 505–510 (1991).
[Crossref] [PubMed]

IEEE J. Sel. Top. Quant. (1)

A. Redo-Sanchez and X. C. Zhang, “Terahertz science and technology trends,” IEEE J. Sel. Top. Quant. 14(2), 1–10 (2008).
[Crossref]

IEEE Trans. Antenn. Propag. (3)

D. M. Pozar, S. D. Targonski, and H. D. Syrigos, “Design of millimeter wave microstrip reflectarrays,” IEEE Trans. Antenn. Propag. 45(2), 287–296 (1997).
[Crossref]

J. Perruisseau-Carrier, “Dual-polarized and polarization-flexible reflective cells with dynamic phase control,” IEEE Trans. Antenn. Propag. 58(5), 1494–1502 (2010).
[Crossref]

J. A. Encinar, L. S. Datashvili, J. A. Zornoza, M. Arrebola, M. Sierra-Castañer, J. L. Besada-Sanmartín, H. Baier, and H. Legay, “Dual-polarization dual-coverage reflectarray for space applications,” IEEE Trans. Antenn. Propag. 54(10), 2827–2837 (2006).
[Crossref]

IEEE Trans. Microw. Theory Tech. (3)

C. Cheng, B. Lakshminarayanan, and A. Abbaspour-Tamijani, “A programmable lens-array antenna with monolithically integrated MEMS switches,” IEEE Trans. Microw. Theory Tech. 57(8), 1874–1884 (2009).
[Crossref]

J. Y. Lau and S. V. Hum, “A planar reconfigurable aperture with lens and reflectarray modes of operation,” IEEE Trans. Microw. Theory Tech. 58(12), 3547–3555 (2010).

J. Perruisseau-Carrier, F. Bongard, R. Golubovic-Niciforovic, R. Torres-Sánchez, and J. R. Mosig, “Contributions to the modeling and design of reconfigurable reflecting cells embedding discrete control elements,” IEEE Trans. Microw. Theory Tech. 58(6), 1621–1628 (2010).
[Crossref]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

C. Tamm and C. O. Weiss, “Bistability and optical switching of spatial patterns in a laser,” J. Opt. Soc. Am. B 7(6), 1037–7038 (1990).
[Crossref]

JETP Lett. (1)

V. Y. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” JETP Lett. 52(8), 429–431 (1990).

Nat. Photonics (1)

M. Tonouchi, “Cutting-edge terahertz technology,” Nat. Photonics 1(2), 97–105 (2007).
[Crossref]

Nature (1)

J. T. Finer, R. M. Simmons, and J. A. Spudich, “Single myosin molecule mechanics: piconewton forces and nanometre steps,” Nature 368(6467), 113–119 (1994).
[Crossref] [PubMed]

Opt. Commun. (5)

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96(1–3), 123–132 (1993).
[Crossref]

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate,” Opt. Commun. 112(5–6), 321–327 (1994).
[Crossref]

S. M. Barnett and L. Allen, “Orbital angular momentum and nonparaxial light beams,” Opt. Commun. 110(5–6), 670–678 (1994).
[Crossref]

J. M. Vaughan and D. V. Willetts, “Interference properties of a light beam having a helical wave surface,” Opt. Commun. 30(3), 263–267 (1979).
[Crossref]

X. K. Wang, Y. Cui, W. F. Sun, J. S. Ye, and Y. Zhang, “Terahertz real-time imaging with balanced electro-optic detection,” Opt. Commun. 283(23), 4626–4632 (2010).
[Crossref] [PubMed]

Opt. Express (3)

Opt. Lett. (3)

Phys. Rev. A (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Phys. Rev. Lett. (3)

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995).
[Crossref] [PubMed]

A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental two-photon, three-dimensional entanglement for quantum communication,” Phys. Rev. Lett. 89(24), 240401 (2002).
[Crossref] [PubMed]

G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: Preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88(1), 013601 (2001).
[Crossref] [PubMed]

Proc. R. Soc. Lond. A Math. Phys. Sci. (1)

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A Math. Phys. Sci. 336(1605), 165–190 (1974).
[Crossref]

Science (1)

N. F. Yu, P. Genevet, M. A. Kats, F. Aieta, J. P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

Other (1)

A. E. Siegman, Lasers (University Science Books, 1986, Chap. 16).

Supplementary Material (2)

Media 1: MOV (1414 KB)     
Media 2: MOV (1270 KB)     

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Figures (6)

Fig. 1
Fig. 1 (a) A complementary V-shaped antenna phase modulation unit. (b) Eight kinds of complementary V-shaped antenna structures corresponding to phase shifts from −3π/4 to π with a π/4 interval. (c) Photography of the central region of the designed vortex phase plate (VPP) for l = 1.
Fig. 2
Fig. 2 (a) Terahertz (THz) holographic imaging system. (b) and (c) display the measured intensity and phase distribution of the generated THz vortex beam with l = 1 at 0.75THz, respectively. (d) The phase curves with the azimuthal angle α and the radial distance r = 1.5 mm.
Fig. 3
Fig. 3 (a) Experimental setup for observing the intensity and phase evolutions of the THz vortex beam in the focusing process. (b) Media 1 and (c) Media 2 are the intensity and phase maps of the measured THz vortex beam with Z = −20 mm, −10 mm, 0 mm, 10 mm, and 20 mm. (d) Phase distributions of the Laguerre-Gaussian (LG) mode with l=1, p=0 at Z = −20 mm, −10 mm, 0 mm, 10 mm, and 20 mm. (e) Correlation coefficients of the basis LG modes in the measured THz vortex beam.
Fig. 4
Fig. 4 (a) and (b) are the phase distributions of lα+ Φ G ( z ) and kz r 2 2( z R 2 + z 2 ) +lα with Z = −20 mm, −10 mm, 0 mm, 10 mm, 20 mm, respectively.
Fig. 5
Fig. 5 (a) and (b) represent the experimental and theoretical longitudinal intensity distributions of the THz vortex beam on the Y-Z plane, respectively. (c) and (d) are the corresponding longitudinal phase distributions. (e) is the paraxial phase shifts extracted from (c) and (d). The red solid curve and blue squares correspond to the theoretical and experimental results, respectively.
Fig. 6
Fig. 6 (a) and (b) are the photographs of the central parts of two designed VPPs with l = 2 and l = 3. (c) and (d) are the measured vortex phase distributions with l = 2, and l = 3. (e) and (f) are the simulated vortex phase distributions with l = 2 and l = 3.

Equations (4)

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E p l ( r,α,z )[ 2 r w( z ) ] L p | l | ( 2 r 2 w ( z ) 2 )exp[ r 2 w ( z ) 2 ]exp[ i kz r 2 2( z R 2 + z 2 ) +ilα+i Φ G ( z ) ],
w( z )= w 0 1+ z 2 / z R 2 .
Φ G ( z )=( 2p+| l |+1 )arctan( z/ z R ).
C p,l = E v ( E p l ) * rdrdα ,

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