Abstract

In this work, we present a method to generate a 3D lattice of vortex beams. We apply phase look-up tables (LUTs) designed to generate gratings having an arbitrary content of diffraction orders. This phase LUT can be applied to a variety of diffraction optical elements, such as linear phase gratings, blazed diffractive lenses, and spiral phase patterns. We concentrate on combinations of all of these to create 3D structures of vortex beams. In particular, we generate all of these elements in the first output quadrant and eliminate the zero-order diffraction that often unavoidably accompanies these patterns. We discuss different ways of producing these 3D vortex gratings, and how the various output beams are related to the arithmetic of the 3D distribution of topological charges. Experimental results are provided by means of a liquid crystal spatial light modulator.

© 2014 Optical Society of America

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2013 (3)

2012 (5)

2011 (1)

2010 (2)

I. Moreno, J. A. Davis, D. M. Cottrell, N. Zhang, and X. C. Yuan, “Encoding generalized phase functions on Dammann gratings,” Opt. Lett. 35, 1536–1538 (2010).
[CrossRef]

L. A. Romero and F. M. Dickey, “The mathematical theory of laser beam-splitting gratings,” Prog. Opt. 54, 319–386 (2010).
[CrossRef]

2009 (1)

2007 (1)

2006 (3)

S. Hasegawa, Y. Hayasaki, and N. Nishida, “Holographic femtosecond laser processing with multiplexed phase Fresnel lenses,” Opt. Lett. 31, 1705–1707 (2006).
[CrossRef]

S. H. Tao, X.-C. Yuan, J. Lin, and R. E. Burge, “Sequence of focused optical vortices generated by a spiral fractal zone plate,” Appl. Phys. Lett. 89, 031105 (2006).
[CrossRef]

S. Wong, M. Deubel, F. Pérez-Willard, S. John, G. A. Ozin, M. Wegener, and G. von Freymann, “Direct laser writing of three-dimensional photonic crystals with a complete photonic bandgap in chalcogenide glasses,” Adv. Mater. 18, 265–269 (2006).
[CrossRef]

2005 (3)

J.-I. Kato, N. Takeyasu, Y. Adachi, H.-B. Sun, and S. Kawata, “Multiple-spot parallel processing for laser micronanofabrication,” Appl. Phys. Lett. 86, 044102 (2005).
[CrossRef]

E. Schonbrun, R. Piestun, P. Jordan, J. Cooper, K. D. Wulff, J. Courtial, and M. Padgett, “3D interferometric optical tweezers using a single spatial light modulator,” Opt. Express 13, 3777–3786 (2005).
[CrossRef]

Y. Hayasaki, T. Sugimoto, A. Takita, and N. Nishida, “Variable holographic femtosecond laser processing by use of a spatial light modulator,” Appl. Phys. Lett. 87, 031101 (2005).
[CrossRef]

2004 (3)

S. Jeon, J.-U. Park, R. Cirelli, S. Yang, C. E. Heitzman, P. V. Braun, P. J. A. Kenis, and J. A. Rogers, “Fabricating complex three-dimensional nanostructures with high-resolution conformable phase masks,” Proc. Natl. Acad. Sci. USA 101, 12428–12433 (2004).
[CrossRef]

Y. Kuroiwa, N. Takeshima, Y. Narita, S. Tanaka, and K. Hirao, “Arbitrary micropatterning method in femtosecond laser microprocessing using diffractive optical elements,” Opt. Express 12, 1908–1915 (2004).
[CrossRef]

K. Crabtree, J. A. Davis, and I. Moreno, “Optical processing with vortex producing lenses,” Appl. Opt. 43, 1360–1367 (2004).
[CrossRef]

2003 (1)

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[CrossRef]

2001 (1)

T. Kondo, S. Matsuo, S. Juodkazis, and J. Misawa, “Femtosecond laser interference technique with diffractive beam splitter for fabrication of three-dimensional photonic crystals,” Appl. Phys. Lett. 79, 725–727 (2001).
[CrossRef]

1999 (1)

J. A. Davis, P. Tsai, D. M. Cottrell, T. Sonehara, and J. Amako, “Transmission variations in liquid crystal spatial light modulators caused by interference and diffraction effects,” Opt. Eng. 38, 1051–1057 (1999).
[CrossRef]

1992 (2)

1990 (1)

V. Yu. Bazhenov, V. Vasnetsov, and M. S. Soskin, “Laser-beams with screw dislocations in their wave-fronts,” JETP Lett. 52, 429–431 (1990).

1977 (1)

H. Dammann and E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
[CrossRef]

Adachi, Y.

J.-I. Kato, N. Takeyasu, Y. Adachi, H.-B. Sun, and S. Kawata, “Multiple-spot parallel processing for laser micronanofabrication,” Appl. Phys. Lett. 86, 044102 (2005).
[CrossRef]

Albero, J.

Amako, J.

J. A. Davis, P. Tsai, D. M. Cottrell, T. Sonehara, and J. Amako, “Transmission variations in liquid crystal spatial light modulators caused by interference and diffraction effects,” Opt. Eng. 38, 1051–1057 (1999).
[CrossRef]

Bazhenov, V. Yu.

V. Yu. Bazhenov, V. Vasnetsov, and M. S. Soskin, “Laser-beams with screw dislocations in their wave-fronts,” JETP Lett. 52, 429–431 (1990).

Braun, P. V.

S. Jeon, J.-U. Park, R. Cirelli, S. Yang, C. E. Heitzman, P. V. Braun, P. J. A. Kenis, and J. A. Rogers, “Fabricating complex three-dimensional nanostructures with high-resolution conformable phase masks,” Proc. Natl. Acad. Sci. USA 101, 12428–12433 (2004).
[CrossRef]

Burge, R. E.

S. H. Tao, X.-C. Yuan, J. Lin, and R. E. Burge, “Sequence of focused optical vortices generated by a spiral fractal zone plate,” Appl. Phys. Lett. 89, 031105 (2006).
[CrossRef]

Calabuig, A.

Cao, H.

Cao, W.

Cirelli, R.

S. Jeon, J.-U. Park, R. Cirelli, S. Yang, C. E. Heitzman, P. V. Braun, P. J. A. Kenis, and J. A. Rogers, “Fabricating complex three-dimensional nanostructures with high-resolution conformable phase masks,” Proc. Natl. Acad. Sci. USA 101, 12428–12433 (2004).
[CrossRef]

Cooper, J.

Cottrell, D. M.

Courtial, J.

Crabtree, K.

Dammann, H.

H. Dammann and E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
[CrossRef]

Dändliker, R.

Davis, J. A.

Deubel, M.

S. Wong, M. Deubel, F. Pérez-Willard, S. John, G. A. Ozin, M. Wegener, and G. von Freymann, “Direct laser writing of three-dimensional photonic crystals with a complete photonic bandgap in chalcogenide glasses,” Adv. Mater. 18, 265–269 (2006).
[CrossRef]

Dickey, F. M.

L. A. Romero and F. M. Dickey, “The mathematical theory of laser beam-splitting gratings,” Prog. Opt. 54, 319–386 (2010).
[CrossRef]

L. A. Romero and F. M. Dickey, “Theory of optimal beam splitting by phase gratings. I. One-dimensional gratings,” J. Opt. Soc. Am. A 24, 2280–2295 (2007).
[CrossRef]

Fernández-Alonso, M.

Furlan, W. D.

Gale, M. T.

García-Martínez, P.

J. Albero, P. García-Martínez, J. L. Martínez, and I. Moreno, “Second order diffractive optical elements in a spatial light modulator with large phase dynamic range,” Opt. Lasers Eng. 51, 111–115 (2013).
[CrossRef]

Granger, C. E.

Grier, D. G.

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[CrossRef]

Hasegawa, S.

Hayasaki, Y.

S. Hasegawa, Y. Hayasaki, and N. Nishida, “Holographic femtosecond laser processing with multiplexed phase Fresnel lenses,” Opt. Lett. 31, 1705–1707 (2006).
[CrossRef]

Y. Hayasaki, T. Sugimoto, A. Takita, and N. Nishida, “Variable holographic femtosecond laser processing by use of a spatial light modulator,” Appl. Phys. Lett. 87, 031101 (2005).
[CrossRef]

Heckenberg, N. R.

Heitzman, C. E.

S. Jeon, J.-U. Park, R. Cirelli, S. Yang, C. E. Heitzman, P. V. Braun, P. J. A. Kenis, and J. A. Rogers, “Fabricating complex three-dimensional nanostructures with high-resolution conformable phase masks,” Proc. Natl. Acad. Sci. USA 101, 12428–12433 (2004).
[CrossRef]

Hernandez, T. J.

Herzig, H. P.

Hirao, K.

Hnatovsky, C.

Hu, A.

Jeon, S.

S. Jeon, J.-U. Park, R. Cirelli, S. Yang, C. E. Heitzman, P. V. Braun, P. J. A. Kenis, and J. A. Rogers, “Fabricating complex three-dimensional nanostructures with high-resolution conformable phase masks,” Proc. Natl. Acad. Sci. USA 101, 12428–12433 (2004).
[CrossRef]

Jia, W.

John, S.

S. Wong, M. Deubel, F. Pérez-Willard, S. John, G. A. Ozin, M. Wegener, and G. von Freymann, “Direct laser writing of three-dimensional photonic crystals with a complete photonic bandgap in chalcogenide glasses,” Adv. Mater. 18, 265–269 (2006).
[CrossRef]

Jordan, P.

Juodkazis, S.

T. Kondo, S. Matsuo, S. Juodkazis, and J. Misawa, “Femtosecond laser interference technique with diffractive beam splitter for fabrication of three-dimensional photonic crystals,” Appl. Phys. Lett. 79, 725–727 (2001).
[CrossRef]

Kato, J.-I.

J.-I. Kato, N. Takeyasu, Y. Adachi, H.-B. Sun, and S. Kawata, “Multiple-spot parallel processing for laser micronanofabrication,” Appl. Phys. Lett. 86, 044102 (2005).
[CrossRef]

Kawata, S.

J.-I. Kato, N. Takeyasu, Y. Adachi, H.-B. Sun, and S. Kawata, “Multiple-spot parallel processing for laser micronanofabrication,” Appl. Phys. Lett. 86, 044102 (2005).
[CrossRef]

Kenis, P. J. A.

S. Jeon, J.-U. Park, R. Cirelli, S. Yang, C. E. Heitzman, P. V. Braun, P. J. A. Kenis, and J. A. Rogers, “Fabricating complex three-dimensional nanostructures with high-resolution conformable phase masks,” Proc. Natl. Acad. Sci. USA 101, 12428–12433 (2004).
[CrossRef]

Klotz, E.

H. Dammann and E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
[CrossRef]

Kondo, T.

T. Kondo, S. Matsuo, S. Juodkazis, and J. Misawa, “Femtosecond laser interference technique with diffractive beam splitter for fabrication of three-dimensional photonic crystals,” Appl. Phys. Lett. 79, 725–727 (2001).
[CrossRef]

Krolikowski, W.

Kuroiwa, Y.

Lin, J.

S. H. Tao, X.-C. Yuan, J. Lin, and R. E. Burge, “Sequence of focused optical vortices generated by a spiral fractal zone plate,” Appl. Phys. Lett. 89, 031105 (2006).
[CrossRef]

Ma, J.

Martínez, J. L.

J. Albero, P. García-Martínez, J. L. Martínez, and I. Moreno, “Second order diffractive optical elements in a spatial light modulator with large phase dynamic range,” Opt. Lasers Eng. 51, 111–115 (2013).
[CrossRef]

J. A. Davis, I. Moreno, J. L. Martínez, T. J. Hernandez, and D. M. Cottrell, “Creating 3D lattice patterns using programmable Dammann gratings,” Appl. Opt. 50, 3653–3657 (2011).
[CrossRef]

Martínez-León, L.

Matsuo, S.

T. Kondo, S. Matsuo, S. Juodkazis, and J. Misawa, “Femtosecond laser interference technique with diffractive beam splitter for fabrication of three-dimensional photonic crystals,” Appl. Phys. Lett. 79, 725–727 (2001).
[CrossRef]

McCormick, K. R.

McDuff, R.

Misawa, J.

T. Kondo, S. Matsuo, S. Juodkazis, and J. Misawa, “Femtosecond laser interference technique with diffractive beam splitter for fabrication of three-dimensional photonic crystals,” Appl. Phys. Lett. 79, 725–727 (2001).
[CrossRef]

Mitry, M. J.

Monsoriu, J. A.

Moreno, I.

Narita, Y.

Nishida, N.

S. Hasegawa, Y. Hayasaki, and N. Nishida, “Holographic femtosecond laser processing with multiplexed phase Fresnel lenses,” Opt. Lett. 31, 1705–1707 (2006).
[CrossRef]

Y. Hayasaki, T. Sugimoto, A. Takita, and N. Nishida, “Variable holographic femtosecond laser processing by use of a spatial light modulator,” Appl. Phys. Lett. 87, 031101 (2005).
[CrossRef]

Ozin, G. A.

S. Wong, M. Deubel, F. Pérez-Willard, S. John, G. A. Ozin, M. Wegener, and G. von Freymann, “Direct laser writing of three-dimensional photonic crystals with a complete photonic bandgap in chalcogenide glasses,” Adv. Mater. 18, 265–269 (2006).
[CrossRef]

Padgett, M.

Park, J.-U.

S. Jeon, J.-U. Park, R. Cirelli, S. Yang, C. E. Heitzman, P. V. Braun, P. J. A. Kenis, and J. A. Rogers, “Fabricating complex three-dimensional nanostructures with high-resolution conformable phase masks,” Proc. Natl. Acad. Sci. USA 101, 12428–12433 (2004).
[CrossRef]

Pascoguin, B. M. L.

Pérez-Willard, F.

S. Wong, M. Deubel, F. Pérez-Willard, S. John, G. A. Ozin, M. Wegener, and G. von Freymann, “Direct laser writing of three-dimensional photonic crystals with a complete photonic bandgap in chalcogenide glasses,” Adv. Mater. 18, 265–269 (2006).
[CrossRef]

Piestun, R.

Pons-Martí, A.

Prongué, D.

Rode, A. V.

Rogers, J. A.

S. Jeon, J.-U. Park, R. Cirelli, S. Yang, C. E. Heitzman, P. V. Braun, P. J. A. Kenis, and J. A. Rogers, “Fabricating complex three-dimensional nanostructures with high-resolution conformable phase masks,” Proc. Natl. Acad. Sci. USA 101, 12428–12433 (2004).
[CrossRef]

Romero, L. A.

L. A. Romero and F. M. Dickey, “The mathematical theory of laser beam-splitting gratings,” Prog. Opt. 54, 319–386 (2010).
[CrossRef]

L. A. Romero and F. M. Dickey, “Theory of optimal beam splitting by phase gratings. I. One-dimensional gratings,” J. Opt. Soc. Am. A 24, 2280–2295 (2007).
[CrossRef]

Sánchez-Ruiz, S.

Sand, D.

Schonbrun, E.

Shostka, N.

Shvedov, V. G.

Smith, C. P.

Sonehara, T.

J. A. Davis, P. Tsai, D. M. Cottrell, T. Sonehara, and J. Amako, “Transmission variations in liquid crystal spatial light modulators caused by interference and diffraction effects,” Opt. Eng. 38, 1051–1057 (1999).
[CrossRef]

Soskin, M. S.

V. Yu. Bazhenov, V. Vasnetsov, and M. S. Soskin, “Laser-beams with screw dislocations in their wave-fronts,” JETP Lett. 52, 429–431 (1990).

Sugimoto, T.

Y. Hayasaki, T. Sugimoto, A. Takita, and N. Nishida, “Variable holographic femtosecond laser processing by use of a spatial light modulator,” Appl. Phys. Lett. 87, 031101 (2005).
[CrossRef]

Sun, H.-B.

J.-I. Kato, N. Takeyasu, Y. Adachi, H.-B. Sun, and S. Kawata, “Multiple-spot parallel processing for laser micronanofabrication,” Appl. Phys. Lett. 86, 044102 (2005).
[CrossRef]

Tajahuerce, E.

Takeshima, N.

Takeyasu, N.

J.-I. Kato, N. Takeyasu, Y. Adachi, H.-B. Sun, and S. Kawata, “Multiple-spot parallel processing for laser micronanofabrication,” Appl. Phys. Lett. 86, 044102 (2005).
[CrossRef]

Takita, A.

Y. Hayasaki, T. Sugimoto, A. Takita, and N. Nishida, “Variable holographic femtosecond laser processing by use of a spatial light modulator,” Appl. Phys. Lett. 87, 031101 (2005).
[CrossRef]

Tanaka, S.

Tao, S. H.

S. H. Tao, X.-C. Yuan, J. Lin, and R. E. Burge, “Sequence of focused optical vortices generated by a spiral fractal zone plate,” Appl. Phys. Lett. 89, 031105 (2006).
[CrossRef]

Tsai, P.

J. A. Davis, P. Tsai, D. M. Cottrell, T. Sonehara, and J. Amako, “Transmission variations in liquid crystal spatial light modulators caused by interference and diffraction effects,” Opt. Eng. 38, 1051–1057 (1999).
[CrossRef]

Vasnetsov, V.

V. Yu. Bazhenov, V. Vasnetsov, and M. S. Soskin, “Laser-beams with screw dislocations in their wave-fronts,” JETP Lett. 52, 429–431 (1990).

von Freymann, G.

S. Wong, M. Deubel, F. Pérez-Willard, S. John, G. A. Ozin, M. Wegener, and G. von Freymann, “Direct laser writing of three-dimensional photonic crystals with a complete photonic bandgap in chalcogenide glasses,” Adv. Mater. 18, 265–269 (2006).
[CrossRef]

Wang, S.

Wegener, M.

S. Wong, M. Deubel, F. Pérez-Willard, S. John, G. A. Ozin, M. Wegener, and G. von Freymann, “Direct laser writing of three-dimensional photonic crystals with a complete photonic bandgap in chalcogenide glasses,” Adv. Mater. 18, 265–269 (2006).
[CrossRef]

White, A. G.

Wong, S.

S. Wong, M. Deubel, F. Pérez-Willard, S. John, G. A. Ozin, M. Wegener, and G. von Freymann, “Direct laser writing of three-dimensional photonic crystals with a complete photonic bandgap in chalcogenide glasses,” Adv. Mater. 18, 265–269 (2006).
[CrossRef]

Wu, J.

Wulff, K. D.

Yang, S.

S. Jeon, J.-U. Park, R. Cirelli, S. Yang, C. E. Heitzman, P. V. Braun, P. J. A. Kenis, and J. A. Rogers, “Fabricating complex three-dimensional nanostructures with high-resolution conformable phase masks,” Proc. Natl. Acad. Sci. USA 101, 12428–12433 (2004).
[CrossRef]

Yu, J.

Yuan, X. C.

Yuan, X.-C.

S. H. Tao, X.-C. Yuan, J. Lin, and R. E. Burge, “Sequence of focused optical vortices generated by a spiral fractal zone plate,” Appl. Phys. Lett. 89, 031105 (2006).
[CrossRef]

Zhang, N.

Zhou, C.

Adv. Mater. (1)

S. Wong, M. Deubel, F. Pérez-Willard, S. John, G. A. Ozin, M. Wegener, and G. von Freymann, “Direct laser writing of three-dimensional photonic crystals with a complete photonic bandgap in chalcogenide glasses,” Adv. Mater. 18, 265–269 (2006).
[CrossRef]

Appl. Opt. (8)

D. Prongué, H. P. Herzig, R. Dändliker, and M. T. Gale, “Optimized kinoform structures for highly efficient fan-out elements,” Appl. Opt. 31, 5706–5711 (1992).
[CrossRef]

J. A. Davis, I. Moreno, J. L. Martínez, T. J. Hernandez, and D. M. Cottrell, “Creating 3D lattice patterns using programmable Dammann gratings,” Appl. Opt. 50, 3653–3657 (2011).
[CrossRef]

J. Yu, C. Zhou, W. Jia, W. Cao, S. Wang, J. Ma, and H. Cao, “Three-dimensional Dammann array,” Appl. Opt. 51, 1619–1630 (2012).
[CrossRef]

J. Yu, C. Zhou, W. Jia, A. Hu, W. Cao, J. Wu, and S. Wang, “Three-dimensional Dammann vortex array with tunable topological charge,” Appl. Opt. 51, 2485–2490 (2012).
[CrossRef]

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http://support.microsoft.com/kb/214115 .

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Figures (8)

Fig. 1.
Fig. 1.

(a) Phase LUT12 to produce equally intense +1 and +2 orders. (b) Initial linear phase grating (three periods are shown). (c) Phase grating resulting from the application of LUT12 to the linear phase grating.

Fig. 2.
Fig. 2.

Phase gratings design: (a) linear phase, (b) phase after LUT12 is applied to the linear phase in (a) to create the (+1, +2) duplicator, (c) spiral phase (q=2), (d) addition of the linear phase in (a) and the spiral phase in (c), (e) phase after LUT12 is applied to the phase grating in (d), and (f) addition of the (+1, +2) duplicator grating in (b) with the spiral phase in (c).

Fig. 3.
Fig. 3.

Experimental diffraction patterns generated with 1D optimal phase gratings: (a) DC order (no grating is displayed), (b) (+1, +2) duplicator grating with equal intensity [mask in Fig. 2(b)], (c) grating with qX=+1, (d) grating with qX=+2 [mask in Fig. 2(e)], (e) grating with qG=2 [mask in Fig. 2(f)], and (f) grating with qG=2 qX=+2. The generated charge qlX is indicated at each diffraction order lX.

Fig. 4.
Fig. 4.

Design of optical lattice arrays: (a) addition of the (+1, +2) duplicator grating in Fig. 1(c) with a 90 deg rotated version of itself to create a 2D array and (b) addition of this previous phase with the spiral phase in Fig. 2(c).

Fig. 5.
Fig. 5.

Experimental diffraction patterns generated with 2D optimal phase gratings: (a) DC order (no grating is displayed), (b) (+1,+2)×(+1,+2) grating with equal intensity [mask in Fig. 4(a)], (c) grating with qG=+2 [mask in Fig. 4(b)], (d) grating with qX=+2, (e) grating with qX=qY=+1, (f) grating with qX=+1 and qY=1, (g) grating with qG=+2 and qX=qY=+1, and (h) grating with qG=2 and qX=qY=+1. The generated charge qlX,lY is indicated at each diffraction order (lX, lY).

Fig. 6.
Fig. 6.

Design of optical lens arrays: (a) the phase after LUT12 is applied to a quadratic lens phase and (b) addition of the phase grating in Fig. 4(a) and the lens phase in (a) to create a 3D focusing array.

Fig. 7.
Fig. 7.

Experimental diffraction patterns generated with 3D optimal phase gratings. Left and right columns show images at axial planes lZ=+1 and lZ=+2, respectively: (a) and (e) (+1, +2) lens [mask in Fig. 6(a)], (b) and (f) (+1, +2) lens multiplied by grating with four equal intensity orders [mask in Fig. 6(b)], (c) and (g) (+1, +2) lens multiplied by grating with qG=+2, and (d) and (h) (+1, +2) lens multiplied by grating with qX=+1 and qY=1. The generated charge qlX,lY,lZ is indicated at each diffraction order (lX, lY, lZ).

Fig. 8.
Fig. 8.

Experimental diffraction patterns generated with 3D optimal phase gratings. Left and right columns show images at axial planes lZ=+1 and lZ=+2, respectively. (+1, +2) lens multiplied by: (a) and (d) grating with qZ=+2; (b) and (e) grating with qX=+1, qY=1, and qZ=+2; and (c) and (f) grating with qX=+1, qY=1, qZ=+2, and qG=+1. The generated charge qlX,lY,lZ is indicated at each diffraction order (lX, lY, lZ).

Equations (10)

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exp[iφ(x)]=f(x)|f(x)|,
f(x)=lμlexp[iαl]exp[ilx].
t(x,y)=exp(iqGθ)·[exp(iqXθ)·exp(iγx)].
t(x,y)=exp(iqGθ)·LUT12[exp(iqXθ)·exp(iγx)]=exp(iqGθ)·[l=1,2clexp(ilqXθ)exp(ilγx)],
ql=qG+lqX.
t(x,y)=exp(iqGθ)·[exp(iqXθ)·exp(iγXx)]·[exp(iqYθ)·exp(iγYy)],
t(x,y)=eiqGθLUT12[eiqXθeiγXx]·LUT12[eiqYθeiγYy]=eiqGθ[lx=1,2clXeilXqXθeilXγXx]·[ly=1,2clYeilYqYθeilYγYy].
t(x,y,z)=exp(iqGθ)·[exp(iqXθ)·exp(iγXx)]·[exp(iqYθ)·exp(iγYy)]·[exp(iqZθ)·exp(iπr2λf)].
t(x,y,z)=eiqGθ·[lX=1,2clXeilXqXθeilXγXx]·[lY=1,2clYeilYqYθeilYγYy]·[lZ=1,2clZeilZqZθeilZπr2λf].
exp(iφ(x,y))=LUT12[exp(iπr2λf)].

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