Abstract

Integrable spatial-phase-modulating surface-emitting lasers, which utilize the band edge mode of two-dimensional photonic-crystals as resonators, project static arbitrary two-dimensional beam patterns from on-chip size. In this device, holes shifting from the lattice point of a two-dimensional photonic crystal provide spatial phase modulation to light waves, which form standing waves in the resonator. Thus far, the origin of the beam patterns has not been studied, especially the formation of subsidiary beam patterns against the designed beam pattern. In this work, we clarify the origin of beam patterns in two types of spatial phase modulating method, which impose in-plane shifting of holes according to circular and linear shift methods. Based on a theoretical study of spatial phase modulation, we reveal that the circular shift method provides a symmetric beam pattern, while the linear shift method causes an asymmetric beam pattern. Consequently, we demonstrated the asymmetric two-dimensional beam pattern by the linear shift method for the first time.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. A. E. Siegman, Lasers (Univ Science Books, 1986).
  2. L. B. Lesem, P. M. Hirsch, and J. A. Jordan, “The Kinoform: A New Wavefront Reconstruction Device,” IBM J. Res. Develop. 13(2), 150–155 (1969).
    [Crossref]
  3. B. C. Kress and P. Meyrueis, Digital Diffractive Optics: An Introduction to Planar Diffractive Optics and Related Technology (Wiley, 2000).
  4. M. S. Seghilani, M. Myara, M. Sellahi, L. Legratiet, I. Sagnes, G. Beaudoin, P. Lalanne, and A. Garnache, “Vortex Laser based on III-V semiconductor metasurface: direct generation of coherent Laguerre-Gauss modes carrying controlled orbital angular momentum,” Sci. Rep. 6(1), 38156 (2016).
    [Crossref] [PubMed]
  5. K. Kitamura, “Generation of Optical Vortex Beams by Surface-emitting Lasers,” Kogaku 46(11), 439–443 (2017).
  6. Y. Kurosaka, K. Hirose, T. Sugiyama, Y. Takiguchi, and Y. Nomoto, “Phase-modulating lasers toward on-chip integration,” Sci. Rep. 6(1), 30138 (2016).
    [Crossref] [PubMed]
  7. Y. Takiguchi and Y. Nomoto, Japanese Patent Application JP2013–045826.
  8. J. Goodman, Introduction to Fourier Optics, 3rd ed, (Roberts & Co Publishers, 2005).
  9. M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, “Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure,” Appl. Phys. Lett. 75(3), 316–318 (1999).
    [Crossref]
  10. K. Hirose, Y. Liang, Y. Kurosaka, A. Watanabe, T. Sugiyama, and S. Noda, “Watt-class high-power, high-beam-quality photonic-crystal lasers,” Nat. Photonics 8(5), 406–411 (2014).
    [Crossref]
  11. T. Okino, K. Kitamura, D. Yasuda, Y. Liang, and S. Noda, “Position-modulated Photonic-crystal Lasers and Control of Beam Direction and Polarization,” in Proceedings of Conference on Lasers and Electro-Optics2015, Paper SW1F.1.
    [Crossref]
  12. S. Noda, K. Kitamura, T. Okino, D. Yasuda, and Y. Tanaka, “Photonic-Crystal Surface-Emitting Lasers: Review and Introduction of Modulated-Photonic Crystals,” IEEE J. Sel. Top. Quantum Electron. 23(6), 1–7 (2017).
    [Crossref]
  13. S. Noda, T. Okino, K. Kitamura, Y. Tanaka, and Y. Liang, Japanese Patent Application JP2013–046564.
  14. K. Sakoda, Optical Properties of Photonic Crystals (Springer-Verlag, 2005).
  15. A. Weber, Mathematical Methods for Physicists, 6th ed. (Elsevier Academic Press, 2005).

2017 (2)

K. Kitamura, “Generation of Optical Vortex Beams by Surface-emitting Lasers,” Kogaku 46(11), 439–443 (2017).

S. Noda, K. Kitamura, T. Okino, D. Yasuda, and Y. Tanaka, “Photonic-Crystal Surface-Emitting Lasers: Review and Introduction of Modulated-Photonic Crystals,” IEEE J. Sel. Top. Quantum Electron. 23(6), 1–7 (2017).
[Crossref]

2016 (2)

Y. Kurosaka, K. Hirose, T. Sugiyama, Y. Takiguchi, and Y. Nomoto, “Phase-modulating lasers toward on-chip integration,” Sci. Rep. 6(1), 30138 (2016).
[Crossref] [PubMed]

M. S. Seghilani, M. Myara, M. Sellahi, L. Legratiet, I. Sagnes, G. Beaudoin, P. Lalanne, and A. Garnache, “Vortex Laser based on III-V semiconductor metasurface: direct generation of coherent Laguerre-Gauss modes carrying controlled orbital angular momentum,” Sci. Rep. 6(1), 38156 (2016).
[Crossref] [PubMed]

2014 (1)

K. Hirose, Y. Liang, Y. Kurosaka, A. Watanabe, T. Sugiyama, and S. Noda, “Watt-class high-power, high-beam-quality photonic-crystal lasers,” Nat. Photonics 8(5), 406–411 (2014).
[Crossref]

1999 (1)

M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, “Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure,” Appl. Phys. Lett. 75(3), 316–318 (1999).
[Crossref]

1969 (1)

L. B. Lesem, P. M. Hirsch, and J. A. Jordan, “The Kinoform: A New Wavefront Reconstruction Device,” IBM J. Res. Develop. 13(2), 150–155 (1969).
[Crossref]

Beaudoin, G.

M. S. Seghilani, M. Myara, M. Sellahi, L. Legratiet, I. Sagnes, G. Beaudoin, P. Lalanne, and A. Garnache, “Vortex Laser based on III-V semiconductor metasurface: direct generation of coherent Laguerre-Gauss modes carrying controlled orbital angular momentum,” Sci. Rep. 6(1), 38156 (2016).
[Crossref] [PubMed]

Chutinan, A.

M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, “Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure,” Appl. Phys. Lett. 75(3), 316–318 (1999).
[Crossref]

Garnache, A.

M. S. Seghilani, M. Myara, M. Sellahi, L. Legratiet, I. Sagnes, G. Beaudoin, P. Lalanne, and A. Garnache, “Vortex Laser based on III-V semiconductor metasurface: direct generation of coherent Laguerre-Gauss modes carrying controlled orbital angular momentum,” Sci. Rep. 6(1), 38156 (2016).
[Crossref] [PubMed]

Hirose, K.

Y. Kurosaka, K. Hirose, T. Sugiyama, Y. Takiguchi, and Y. Nomoto, “Phase-modulating lasers toward on-chip integration,” Sci. Rep. 6(1), 30138 (2016).
[Crossref] [PubMed]

K. Hirose, Y. Liang, Y. Kurosaka, A. Watanabe, T. Sugiyama, and S. Noda, “Watt-class high-power, high-beam-quality photonic-crystal lasers,” Nat. Photonics 8(5), 406–411 (2014).
[Crossref]

Hirsch, P. M.

L. B. Lesem, P. M. Hirsch, and J. A. Jordan, “The Kinoform: A New Wavefront Reconstruction Device,” IBM J. Res. Develop. 13(2), 150–155 (1969).
[Crossref]

Imada, M.

M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, “Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure,” Appl. Phys. Lett. 75(3), 316–318 (1999).
[Crossref]

Jordan, J. A.

L. B. Lesem, P. M. Hirsch, and J. A. Jordan, “The Kinoform: A New Wavefront Reconstruction Device,” IBM J. Res. Develop. 13(2), 150–155 (1969).
[Crossref]

Kitamura, K.

K. Kitamura, “Generation of Optical Vortex Beams by Surface-emitting Lasers,” Kogaku 46(11), 439–443 (2017).

S. Noda, K. Kitamura, T. Okino, D. Yasuda, and Y. Tanaka, “Photonic-Crystal Surface-Emitting Lasers: Review and Introduction of Modulated-Photonic Crystals,” IEEE J. Sel. Top. Quantum Electron. 23(6), 1–7 (2017).
[Crossref]

T. Okino, K. Kitamura, D. Yasuda, Y. Liang, and S. Noda, “Position-modulated Photonic-crystal Lasers and Control of Beam Direction and Polarization,” in Proceedings of Conference on Lasers and Electro-Optics2015, Paper SW1F.1.
[Crossref]

Kurosaka, Y.

Y. Kurosaka, K. Hirose, T. Sugiyama, Y. Takiguchi, and Y. Nomoto, “Phase-modulating lasers toward on-chip integration,” Sci. Rep. 6(1), 30138 (2016).
[Crossref] [PubMed]

K. Hirose, Y. Liang, Y. Kurosaka, A. Watanabe, T. Sugiyama, and S. Noda, “Watt-class high-power, high-beam-quality photonic-crystal lasers,” Nat. Photonics 8(5), 406–411 (2014).
[Crossref]

Lalanne, P.

M. S. Seghilani, M. Myara, M. Sellahi, L. Legratiet, I. Sagnes, G. Beaudoin, P. Lalanne, and A. Garnache, “Vortex Laser based on III-V semiconductor metasurface: direct generation of coherent Laguerre-Gauss modes carrying controlled orbital angular momentum,” Sci. Rep. 6(1), 38156 (2016).
[Crossref] [PubMed]

Legratiet, L.

M. S. Seghilani, M. Myara, M. Sellahi, L. Legratiet, I. Sagnes, G. Beaudoin, P. Lalanne, and A. Garnache, “Vortex Laser based on III-V semiconductor metasurface: direct generation of coherent Laguerre-Gauss modes carrying controlled orbital angular momentum,” Sci. Rep. 6(1), 38156 (2016).
[Crossref] [PubMed]

Lesem, L. B.

L. B. Lesem, P. M. Hirsch, and J. A. Jordan, “The Kinoform: A New Wavefront Reconstruction Device,” IBM J. Res. Develop. 13(2), 150–155 (1969).
[Crossref]

Liang, Y.

K. Hirose, Y. Liang, Y. Kurosaka, A. Watanabe, T. Sugiyama, and S. Noda, “Watt-class high-power, high-beam-quality photonic-crystal lasers,” Nat. Photonics 8(5), 406–411 (2014).
[Crossref]

T. Okino, K. Kitamura, D. Yasuda, Y. Liang, and S. Noda, “Position-modulated Photonic-crystal Lasers and Control of Beam Direction and Polarization,” in Proceedings of Conference on Lasers and Electro-Optics2015, Paper SW1F.1.
[Crossref]

Murata, M.

M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, “Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure,” Appl. Phys. Lett. 75(3), 316–318 (1999).
[Crossref]

Myara, M.

M. S. Seghilani, M. Myara, M. Sellahi, L. Legratiet, I. Sagnes, G. Beaudoin, P. Lalanne, and A. Garnache, “Vortex Laser based on III-V semiconductor metasurface: direct generation of coherent Laguerre-Gauss modes carrying controlled orbital angular momentum,” Sci. Rep. 6(1), 38156 (2016).
[Crossref] [PubMed]

Noda, S.

S. Noda, K. Kitamura, T. Okino, D. Yasuda, and Y. Tanaka, “Photonic-Crystal Surface-Emitting Lasers: Review and Introduction of Modulated-Photonic Crystals,” IEEE J. Sel. Top. Quantum Electron. 23(6), 1–7 (2017).
[Crossref]

K. Hirose, Y. Liang, Y. Kurosaka, A. Watanabe, T. Sugiyama, and S. Noda, “Watt-class high-power, high-beam-quality photonic-crystal lasers,” Nat. Photonics 8(5), 406–411 (2014).
[Crossref]

M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, “Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure,” Appl. Phys. Lett. 75(3), 316–318 (1999).
[Crossref]

T. Okino, K. Kitamura, D. Yasuda, Y. Liang, and S. Noda, “Position-modulated Photonic-crystal Lasers and Control of Beam Direction and Polarization,” in Proceedings of Conference on Lasers and Electro-Optics2015, Paper SW1F.1.
[Crossref]

Nomoto, Y.

Y. Kurosaka, K. Hirose, T. Sugiyama, Y. Takiguchi, and Y. Nomoto, “Phase-modulating lasers toward on-chip integration,” Sci. Rep. 6(1), 30138 (2016).
[Crossref] [PubMed]

Okino, T.

S. Noda, K. Kitamura, T. Okino, D. Yasuda, and Y. Tanaka, “Photonic-Crystal Surface-Emitting Lasers: Review and Introduction of Modulated-Photonic Crystals,” IEEE J. Sel. Top. Quantum Electron. 23(6), 1–7 (2017).
[Crossref]

T. Okino, K. Kitamura, D. Yasuda, Y. Liang, and S. Noda, “Position-modulated Photonic-crystal Lasers and Control of Beam Direction and Polarization,” in Proceedings of Conference on Lasers and Electro-Optics2015, Paper SW1F.1.
[Crossref]

Sagnes, I.

M. S. Seghilani, M. Myara, M. Sellahi, L. Legratiet, I. Sagnes, G. Beaudoin, P. Lalanne, and A. Garnache, “Vortex Laser based on III-V semiconductor metasurface: direct generation of coherent Laguerre-Gauss modes carrying controlled orbital angular momentum,” Sci. Rep. 6(1), 38156 (2016).
[Crossref] [PubMed]

Sasaki, G.

M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, “Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure,” Appl. Phys. Lett. 75(3), 316–318 (1999).
[Crossref]

Seghilani, M. S.

M. S. Seghilani, M. Myara, M. Sellahi, L. Legratiet, I. Sagnes, G. Beaudoin, P. Lalanne, and A. Garnache, “Vortex Laser based on III-V semiconductor metasurface: direct generation of coherent Laguerre-Gauss modes carrying controlled orbital angular momentum,” Sci. Rep. 6(1), 38156 (2016).
[Crossref] [PubMed]

Sellahi, M.

M. S. Seghilani, M. Myara, M. Sellahi, L. Legratiet, I. Sagnes, G. Beaudoin, P. Lalanne, and A. Garnache, “Vortex Laser based on III-V semiconductor metasurface: direct generation of coherent Laguerre-Gauss modes carrying controlled orbital angular momentum,” Sci. Rep. 6(1), 38156 (2016).
[Crossref] [PubMed]

Sugiyama, T.

Y. Kurosaka, K. Hirose, T. Sugiyama, Y. Takiguchi, and Y. Nomoto, “Phase-modulating lasers toward on-chip integration,” Sci. Rep. 6(1), 30138 (2016).
[Crossref] [PubMed]

K. Hirose, Y. Liang, Y. Kurosaka, A. Watanabe, T. Sugiyama, and S. Noda, “Watt-class high-power, high-beam-quality photonic-crystal lasers,” Nat. Photonics 8(5), 406–411 (2014).
[Crossref]

Takiguchi, Y.

Y. Kurosaka, K. Hirose, T. Sugiyama, Y. Takiguchi, and Y. Nomoto, “Phase-modulating lasers toward on-chip integration,” Sci. Rep. 6(1), 30138 (2016).
[Crossref] [PubMed]

Tanaka, Y.

S. Noda, K. Kitamura, T. Okino, D. Yasuda, and Y. Tanaka, “Photonic-Crystal Surface-Emitting Lasers: Review and Introduction of Modulated-Photonic Crystals,” IEEE J. Sel. Top. Quantum Electron. 23(6), 1–7 (2017).
[Crossref]

Tokuda, T.

M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, “Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure,” Appl. Phys. Lett. 75(3), 316–318 (1999).
[Crossref]

Watanabe, A.

K. Hirose, Y. Liang, Y. Kurosaka, A. Watanabe, T. Sugiyama, and S. Noda, “Watt-class high-power, high-beam-quality photonic-crystal lasers,” Nat. Photonics 8(5), 406–411 (2014).
[Crossref]

Yasuda, D.

S. Noda, K. Kitamura, T. Okino, D. Yasuda, and Y. Tanaka, “Photonic-Crystal Surface-Emitting Lasers: Review and Introduction of Modulated-Photonic Crystals,” IEEE J. Sel. Top. Quantum Electron. 23(6), 1–7 (2017).
[Crossref]

T. Okino, K. Kitamura, D. Yasuda, Y. Liang, and S. Noda, “Position-modulated Photonic-crystal Lasers and Control of Beam Direction and Polarization,” in Proceedings of Conference on Lasers and Electro-Optics2015, Paper SW1F.1.
[Crossref]

Appl. Phys. Lett. (1)

M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, “Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure,” Appl. Phys. Lett. 75(3), 316–318 (1999).
[Crossref]

IBM J. Res. Develop. (1)

L. B. Lesem, P. M. Hirsch, and J. A. Jordan, “The Kinoform: A New Wavefront Reconstruction Device,” IBM J. Res. Develop. 13(2), 150–155 (1969).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

S. Noda, K. Kitamura, T. Okino, D. Yasuda, and Y. Tanaka, “Photonic-Crystal Surface-Emitting Lasers: Review and Introduction of Modulated-Photonic Crystals,” IEEE J. Sel. Top. Quantum Electron. 23(6), 1–7 (2017).
[Crossref]

Kogaku (1)

K. Kitamura, “Generation of Optical Vortex Beams by Surface-emitting Lasers,” Kogaku 46(11), 439–443 (2017).

Nat. Photonics (1)

K. Hirose, Y. Liang, Y. Kurosaka, A. Watanabe, T. Sugiyama, and S. Noda, “Watt-class high-power, high-beam-quality photonic-crystal lasers,” Nat. Photonics 8(5), 406–411 (2014).
[Crossref]

Sci. Rep. (2)

Y. Kurosaka, K. Hirose, T. Sugiyama, Y. Takiguchi, and Y. Nomoto, “Phase-modulating lasers toward on-chip integration,” Sci. Rep. 6(1), 30138 (2016).
[Crossref] [PubMed]

M. S. Seghilani, M. Myara, M. Sellahi, L. Legratiet, I. Sagnes, G. Beaudoin, P. Lalanne, and A. Garnache, “Vortex Laser based on III-V semiconductor metasurface: direct generation of coherent Laguerre-Gauss modes carrying controlled orbital angular momentum,” Sci. Rep. 6(1), 38156 (2016).
[Crossref] [PubMed]

Other (8)

A. E. Siegman, Lasers (Univ Science Books, 1986).

B. C. Kress and P. Meyrueis, Digital Diffractive Optics: An Introduction to Planar Diffractive Optics and Related Technology (Wiley, 2000).

Y. Takiguchi and Y. Nomoto, Japanese Patent Application JP2013–045826.

J. Goodman, Introduction to Fourier Optics, 3rd ed, (Roberts & Co Publishers, 2005).

T. Okino, K. Kitamura, D. Yasuda, Y. Liang, and S. Noda, “Position-modulated Photonic-crystal Lasers and Control of Beam Direction and Polarization,” in Proceedings of Conference on Lasers and Electro-Optics2015, Paper SW1F.1.
[Crossref]

S. Noda, T. Okino, K. Kitamura, Y. Tanaka, and Y. Liang, Japanese Patent Application JP2013–046564.

K. Sakoda, Optical Properties of Photonic Crystals (Springer-Verlag, 2005).

A. Weber, Mathematical Methods for Physicists, 6th ed. (Elsevier Academic Press, 2005).

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Figures (8)

Fig. 1
Fig. 1 Schematic drawing explaining wavefront modulation: (a) square lattice PCSEL, (b) iPMSEL.
Fig. 2
Fig. 2 Structure of iPMSELs.
Fig. 3
Fig. 3 Schematic of phase-modulating method based on (a) circular shift and (b) linear shift. In this study, the angle θ is 45°.
Fig. 4
Fig. 4 (a) Phase angle distribution of the beam pattern “ABCD”. (b) Corresponding beam pattern of the phase angle distribution in wavenumber space normalized by 2π/a.
Fig. 5
Fig. 5 Beam patterns of Fig. 4 in wavenumber space normalized by 2π/a. (a) 0th order and (b) −1st order.
Fig. 6
Fig. 6 Relationship between the ± 1st and 0th order beams for a single lightwave vs. positional shift r in the circular shift method. (a) Amplitude. (b) Intensity.
Fig. 7
Fig. 7 Relationship of the ± 1st and 0th order beam of a single lightwave vs. maximum distance L in the linear shift method. (a) Amplitude. (b) Intensity.
Fig. 8
Fig. 8 FFP of the iPMSEL based on axial displacement methods. (a) Measured FFP. (b) Line plot of the FFP.

Equations (34)

Equations on this page are rendered with MathJax. Learn more.

N(x,y)=AIFT[ F( k x , k y ) ].
N(x,y)=I(x,y)exp( iϕ( x,y ) ),
Δ ϕ Γ ( x,y )=exp{ iϕ( x,y ) }.
Δ ϕ Γn ( x,y )=exp{ inϕ( x,y ) },
Δ ϕ U ( x,y )=exp{ i 2πr a sinϕ( x,y ) },
Δ ϕ D ( x,y )=exp{ i 2πr a sinϕ( x,y ) },
Δ ϕ R ( x,y )=exp{ i 2πr a cosϕ( x,y ) },
Δ ϕ L ( x,y )=exp{ i 2πr a cosϕ( x,y ) },
Δ ϕ U ( x,y )= n= J n ( 2πr a )exp( inϕ( x,y ) ) ,
Δ ϕ D ( x,y )= n= J n ( 2πr a )exp( inϕ( x,y ) ) ,
Δ ϕ R ( x,y )= n= e j nπ 2 J n ( 2πr a )exp( inϕ( x,y ) ) ,
Δ ϕ L ( x,y )= n= e j nπ 2 J n ( 2πr a )exp( inϕ( x,y ) ) ,
Δ ϕ U ( x,y )=exp[ i 2Lsinθ a { ϕ( x,y ) ϕ 0 } ],
Δ ϕ D ( x,y )=exp[ i 2Lsinθ a { ϕ( x,y ) ϕ 0 } ],
Δ ϕ R ( x,y )=exp[ i 2Lcosθ a { ϕ( x,y ) ϕ 0 } ],
Δ ϕ L ( x,y )=exp[ i 2Lcosθ a { ϕ( x,y ) ϕ 0 } ],
Δ ϕ U ( x,y )=Δ ϕ R ( x,y )=exp[ i 2 Lϕ( x,y ) a ]
Δ ϕ D ( x,y )=Δ ϕ L ( x,y )=exp[ i 2 Lϕ( x,y ) a ]
Δ ϕ U ( x,y )=Δ ϕ R ( x,y )= n= A n exp{ inϕ( x,y ) } ,
Δ ϕ D ( x,y )=Δ ϕ L ( x,y )= n= A n exp{ inϕ( x,y ) } ,
A n =exp{ iπ( 2 L a n ) }sinc{ π( 2 L a n ) },
A 0 =sinc( 2 πL a ),
A 1 =exp{ iπ( 2 πL a 1 ) }sinc{ π( 2 πL a 1 ) },
A 1 =exp{ iπ( 2 πL a +1 ) }sinc{ π( 2 πL a +1 ) },
A 0 '=sinc( 2 πL a ),
A 1 '=sinc{ π( 2 L a 1 ) },
A 1 '=sinc{ π( 2 L a +1 ) },
F( X,Y,Z )= e ikZ e i k 2Z ( X 2 + Y 2 ) iλZ N( x,y,0 )exp[ i k Z ( xX+yY ) ]dxdy ,
F( k x , k y )=A N( x,y,0 )exp{ i( k x x+ k y y ) }dxdy , k x =k X Z =ksin θ tilt cos θ rot , k y =k Y Z =ksin θ tilt sin θ rot ,
e z 2 ( t 1 t ) = n= t n J n (z) ,
e izsinθ = n= e inθ J n (z) .
f( z )= z c = m= B m z m ,
B m = 1 2πi C f(t) t m+1 dt ,
B m = 1 2π 0 2π e i( c+1m1 )ϕ dϕ .

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