Abstract

We propose an effective method for fabricating dual-periodic structures using the combination of multi-beam interference lithography and evanescent wave exposure. Four-beam evanescent wave interference lithography (EWIL) is used as a prototype to demonstrate the fabrication feasibility of one-dimensional (1D) micro-grating structures covered with nanodots and two-dimensional microdot structures filled with subwavelength fringes by designing reciprocal lattice vectors of interference fringes. We experimentally fabricated 1D nano-/micro-grating structures with periods of 140 nm and 12.5 µm and microdots filled with subwavelength gratings of 450 nm period by four-beam EWIL. These structures are applicable to superlattice photonic crystals and subwavelength structured surfaces.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2019 (1)

O. Sale, S. Hassan, D. Lowell, N. Hurley, and Y. Lin, “Holographic fabrication of graded photonic super-crystals through pixel-by-pixel phase coding of laser beams in a spatial light modulator,” Proc. SPIE 10930, 1093011 (2019).
[Crossref]

2018 (1)

Y. Suzuki, K. Suzuki, M. Michihata, K. Takamasu, and S. Takahashi, “One-shot stereolithography for biomimetic micro hemisphere covered with relief structure,” Precis. Eng. 54, 353–360 (2018).
[Crossref]

2017 (4)

2013 (1)

2012 (2)

H. Lü, Q. Zhao, Q. Zhang, D. Niu, and X. Wang, “Fabrication of two-dimensional superposed microstructure by interference lithography,” Appl. Opt. 51(3), 302–305 (2012).
[Crossref]

M. Röhrig, M. Thiel, M. Worgull, and H. Hölscher, “3D direct laser writing of nano- and microstructured hierarchical gecko-mimicking surfaces,” Small 8(19), 3009–3015 (2012).
[Crossref]

2011 (1)

E. A. Bezus, L. L. Doskolovich, and N. L. Kazanskiy, “Evanescent-wave interferometric nanoscale photolithography using guided-mode resonant gratings,” Microelectron. Eng. 88(2), 170–174 (2011).
[Crossref]

2010 (1)

C. Lu and R. H. Lipson, “Interference lithography: A powerful tool for fabricating periodic structures,” Laser Photonics Rev. 4(4), 568–580 (2010).
[Crossref]

2009 (1)

C. Yuan, X. Zhang, L. Su, B. Gao, and L. Shen, “Facile synthesis and self-assembly of hierarchical porous NiO nano/micro spherical superstructures for high performance supercapacitors,” J. Mater. Chem. 19(32), 5772–5777 (2009).
[Crossref]

2008 (1)

2007 (1)

2006 (3)

W. Yu, A. Mizutani, H. Kikuta, and T. Konishi, “Reduced wavelength-dependent quarter-wave plate fabricated by a multilayered subwavelength structure,” Appl. Opt. 45(12), 2601–2606 (2006).
[Crossref]

J. C. Martinez-Anton, “Surface relief subwavelength gratings by means of total internal reflection evanescent wave interference lithography,” J. Opt. A: Pure Appl. Opt. 8(4), S213–S218 (2006).
[Crossref]

B. W. Smith, Y. Fan, J. Zhou, N. Lafferty, and A. Estroff, “Evanescent wave imaging in optical lithography,” Proc. SPIE 6154, 61540A (2006).
[Crossref]

2003 (1)

H. Kikuta, H. Toyota, and W. Yu, “Optical elements with subwavelength structured surfaces,” Opt. Rev. 10(2), 63–73 (2003).
[Crossref]

1997 (2)

1996 (2)

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
[Crossref]

H. Hirayama, T. Hamano, and Y. Aoyagi, “Novel surface emitting laser diode using photonic band-gap crystal cavity,” Appl. Phys. Lett. 69(6), 791–793 (1996).
[Crossref]

1992 (1)

R. Magnusson and S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61(9), 1022–1024 (1992).
[Crossref]

1991 (1)

T. L. Koch and U. Koren, “Semiconductor photonic integrated circuits,” IEEE J. Quantum Electron. 27(3), 641–653 (1991).
[Crossref]

1982 (1)

S. J. Wilson and M. C. Hutley, “The optical properties of ‘moth eye’ antireflection surfaces,” Opt. Acta 29(7), 993–1009 (1982).
[Crossref]

1978 (1)

K. E. Bean, “Anisotropic etching of silicon,” IEEE Trans. Electron Devices 25(10), 1185–1193 (1978).
[Crossref]

Aoyagi, Y.

H. Hirayama, T. Hamano, and Y. Aoyagi, “Novel surface emitting laser diode using photonic band-gap crystal cavity,” Appl. Phys. Lett. 69(6), 791–793 (1996).
[Crossref]

Bean, K. E.

K. E. Bean, “Anisotropic etching of silicon,” IEEE Trans. Electron Devices 25(10), 1185–1193 (1978).
[Crossref]

Benisty, H.

J. M. Lourtioz, H. Benisty, V. Berger, J. M. Gerard, D. Maystre, and A. Tchelnokov, Photonic Crystals: Towards Nanoscale Photonic Devices (Springer, 2008).

Berger, V.

J. M. Lourtioz, H. Benisty, V. Berger, J. M. Gerard, D. Maystre, and A. Tchelnokov, Photonic Crystals: Towards Nanoscale Photonic Devices (Springer, 2008).

Bezus, E. A.

E. A. Bezus, L. L. Doskolovich, and N. L. Kazanskiy, “Evanescent-wave interferometric nanoscale photolithography using guided-mode resonant gratings,” Microelectron. Eng. 88(2), 170–174 (2011).
[Crossref]

Blaikie, R. J.

Borel, P. I.

Chen, B.

Chen, J. C.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
[Crossref]

Chen, X.

Chen, Y.

Chua, J. K.

Ding, Y.

Dong, L.

Doskolovich, L. L.

E. A. Bezus, L. L. Doskolovich, and N. L. Kazanskiy, “Evanescent-wave interferometric nanoscale photolithography using guided-mode resonant gratings,” Microelectron. Eng. 88(2), 170–174 (2011).
[Crossref]

Estroff, A.

B. W. Smith, Y. Fan, J. Zhou, N. Lafferty, and A. Estroff, “Evanescent wave imaging in optical lithography,” Proc. SPIE 6154, 61540A (2006).
[Crossref]

Fan, S.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
[Crossref]

Fan, Y.

B. W. Smith, Y. Fan, J. Zhou, N. Lafferty, and A. Estroff, “Evanescent wave imaging in optical lithography,” Proc. SPIE 6154, 61540A (2006).
[Crossref]

Frandsen, L. H.

Gao, B.

C. Yuan, X. Zhang, L. Su, B. Gao, and L. Shen, “Facile synthesis and self-assembly of hierarchical porous NiO nano/micro spherical superstructures for high performance supercapacitors,” J. Mater. Chem. 19(32), 5772–5777 (2009).
[Crossref]

Gao, W.

George, D.

Gerard, J. M.

J. M. Lourtioz, H. Benisty, V. Berger, J. M. Gerard, D. Maystre, and A. Tchelnokov, Photonic Crystals: Towards Nanoscale Photonic Devices (Springer, 2008).

Hamano, T.

H. Hirayama, T. Hamano, and Y. Aoyagi, “Novel surface emitting laser diode using photonic band-gap crystal cavity,” Appl. Phys. Lett. 69(6), 791–793 (1996).
[Crossref]

Hassan, S.

O. Sale, S. Hassan, D. Lowell, N. Hurley, and Y. Lin, “Holographic fabrication of graded photonic super-crystals through pixel-by-pixel phase coding of laser beams in a spatial light modulator,” Proc. SPIE 10930, 1093011 (2019).
[Crossref]

Hirayama, H.

H. Hirayama, T. Hamano, and Y. Aoyagi, “Novel surface emitting laser diode using photonic band-gap crystal cavity,” Appl. Phys. Lett. 69(6), 791–793 (1996).
[Crossref]

Hölscher, H.

M. Röhrig, M. Thiel, M. Worgull, and H. Hölscher, “3D direct laser writing of nano- and microstructured hierarchical gecko-mimicking surfaces,” Small 8(19), 3009–3015 (2012).
[Crossref]

Hurley, N.

O. Sale, S. Hassan, D. Lowell, N. Hurley, and Y. Lin, “Holographic fabrication of graded photonic super-crystals through pixel-by-pixel phase coding of laser beams in a spatial light modulator,” Proc. SPIE 10930, 1093011 (2019).
[Crossref]

Hutley, M. C.

S. J. Wilson and M. C. Hutley, “The optical properties of ‘moth eye’ antireflection surfaces,” Opt. Acta 29(7), 993–1009 (1982).
[Crossref]

Iwata, K.

Joannopoulos, J. D.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
[Crossref]

Kawata, S.

Kazanskiy, N. L.

E. A. Bezus, L. L. Doskolovich, and N. L. Kazanskiy, “Evanescent-wave interferometric nanoscale photolithography using guided-mode resonant gratings,” Microelectron. Eng. 88(2), 170–174 (2011).
[Crossref]

Kikuta, H.

Kjems, J.

Koch, T. L.

T. L. Koch and U. Koren, “Semiconductor photonic integrated circuits,” IEEE J. Quantum Electron. 27(3), 641–653 (1991).
[Crossref]

Konishi, T.

Koren, U.

T. L. Koch and U. Koren, “Semiconductor photonic integrated circuits,” IEEE J. Quantum Electron. 27(3), 641–653 (1991).
[Crossref]

Kristensen, M.

Kurland, I.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
[Crossref]

Lafferty, N.

B. W. Smith, Y. Fan, J. Zhou, N. Lafferty, and A. Estroff, “Evanescent wave imaging in optical lithography,” Proc. SPIE 6154, 61540A (2006).
[Crossref]

Li, L.

Lin, Q. Y.

Lin, Y.

Lipson, R. H.

C. Lu and R. H. Lipson, “Interference lithography: A powerful tool for fabricating periodic structures,” Laser Photonics Rev. 4(4), 568–580 (2010).
[Crossref]

Lourtioz, J. M.

J. M. Lourtioz, H. Benisty, V. Berger, J. M. Gerard, D. Maystre, and A. Tchelnokov, Photonic Crystals: Towards Nanoscale Photonic Devices (Springer, 2008).

Lowell, D.

Lu, C.

C. Lu and R. H. Lipson, “Interference lithography: A powerful tool for fabricating periodic structures,” Laser Photonics Rev. 4(4), 568–580 (2010).
[Crossref]

Lü, H.

Lutkenhaus, J.

Mack, C. A.

Magnusson, R.

R. Magnusson and S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61(9), 1022–1024 (1992).
[Crossref]

Martinez-Anton, J. C.

J. C. Martinez-Anton, “Surface relief subwavelength gratings by means of total internal reflection evanescent wave interference lithography,” J. Opt. A: Pure Appl. Opt. 8(4), S213–S218 (2006).
[Crossref]

Maruo, S.

Maystre, D.

J. M. Lourtioz, H. Benisty, V. Berger, J. M. Gerard, D. Maystre, and A. Tchelnokov, Photonic Crystals: Towards Nanoscale Photonic Devices (Springer, 2008).

Mehrotra, P.

Mekis, A.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
[Crossref]

Michihata, M.

Y. Suzuki, K. Suzuki, M. Michihata, K. Takamasu, and S. Takahashi, “One-shot stereolithography for biomimetic micro hemisphere covered with relief structure,” Precis. Eng. 54, 353–360 (2018).
[Crossref]

Mizutani, A.

Murukeshan, V. M.

Nakamura, O.

Niu, D.

Ohira, Y.

Philipose, U.

Ren, Z.

Röhrig, M.

M. Röhrig, M. Thiel, M. Worgull, and H. Hölscher, “3D direct laser writing of nano- and microstructured hierarchical gecko-mimicking surfaces,” Small 8(19), 3009–3015 (2012).
[Crossref]

Sale, O.

O. Sale, S. Hassan, D. Lowell, N. Hurley, and Y. Lin, “Holographic fabrication of graded photonic super-crystals through pixel-by-pixel phase coding of laser beams in a spatial light modulator,” Proc. SPIE 10930, 1093011 (2019).
[Crossref]

Shen, L.

C. Yuan, X. Zhang, L. Su, B. Gao, and L. Shen, “Facile synthesis and self-assembly of hierarchical porous NiO nano/micro spherical superstructures for high performance supercapacitors,” J. Mater. Chem. 19(32), 5772–5777 (2009).
[Crossref]

Shimizu, Y.

Skivesen, N.

Smith, B. W.

B. W. Smith, Y. Fan, J. Zhou, N. Lafferty, and A. Estroff, “Evanescent wave imaging in optical lithography,” Proc. SPIE 6154, 61540A (2006).
[Crossref]

Su, L.

C. Yuan, X. Zhang, L. Su, B. Gao, and L. Shen, “Facile synthesis and self-assembly of hierarchical porous NiO nano/micro spherical superstructures for high performance supercapacitors,” J. Mater. Chem. 19(32), 5772–5777 (2009).
[Crossref]

Suzuki, K.

Y. Suzuki, K. Suzuki, M. Michihata, K. Takamasu, and S. Takahashi, “One-shot stereolithography for biomimetic micro hemisphere covered with relief structure,” Precis. Eng. 54, 353–360 (2018).
[Crossref]

Suzuki, Y.

Y. Suzuki, K. Suzuki, M. Michihata, K. Takamasu, and S. Takahashi, “One-shot stereolithography for biomimetic micro hemisphere covered with relief structure,” Precis. Eng. 54, 353–360 (2018).
[Crossref]

Takahashi, S.

Y. Suzuki, K. Suzuki, M. Michihata, K. Takamasu, and S. Takahashi, “One-shot stereolithography for biomimetic micro hemisphere covered with relief structure,” Precis. Eng. 54, 353–360 (2018).
[Crossref]

Takamasu, K.

Y. Suzuki, K. Suzuki, M. Michihata, K. Takamasu, and S. Takahashi, “One-shot stereolithography for biomimetic micro hemisphere covered with relief structure,” Precis. Eng. 54, 353–360 (2018).
[Crossref]

Tan, S. K.

Tchelnokov, A.

J. M. Lourtioz, H. Benisty, V. Berger, J. M. Gerard, D. Maystre, and A. Tchelnokov, Photonic Crystals: Towards Nanoscale Photonic Devices (Springer, 2008).

Têtu, A.

Thiel, M.

M. Röhrig, M. Thiel, M. Worgull, and H. Hölscher, “3D direct laser writing of nano- and microstructured hierarchical gecko-mimicking surfaces,” Small 8(19), 3009–3015 (2012).
[Crossref]

Toyota, H.

H. Kikuta, H. Toyota, and W. Yu, “Optical elements with subwavelength structured surfaces,” Opt. Rev. 10(2), 63–73 (2003).
[Crossref]

Villeneuve, P. R.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77(18), 3787–3790 (1996).
[Crossref]

Wang, S. S.

R. Magnusson and S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61(9), 1022–1024 (1992).
[Crossref]

Wang, X.

Wang, Z.

Weng, Z.

Wilson, S. J.

S. J. Wilson and M. C. Hutley, “The optical properties of ‘moth eye’ antireflection surfaces,” Opt. Acta 29(7), 993–1009 (1982).
[Crossref]

Worgull, M.

M. Röhrig, M. Thiel, M. Worgull, and H. Hölscher, “3D direct laser writing of nano- and microstructured hierarchical gecko-mimicking surfaces,” Small 8(19), 3009–3015 (2012).
[Crossref]

Yu, W.

Yuan, C.

C. Yuan, X. Zhang, L. Su, B. Gao, and L. Shen, “Facile synthesis and self-assembly of hierarchical porous NiO nano/micro spherical superstructures for high performance supercapacitors,” J. Mater. Chem. 19(32), 5772–5777 (2009).
[Crossref]

Zhang, Q.

Zhang, X.

C. Yuan, X. Zhang, L. Su, B. Gao, and L. Shen, “Facile synthesis and self-assembly of hierarchical porous NiO nano/micro spherical superstructures for high performance supercapacitors,” J. Mater. Chem. 19(32), 5772–5777 (2009).
[Crossref]

Zhang, Z.

Zhao, Q.

Zhou, J.

B. W. Smith, Y. Fan, J. Zhou, N. Lafferty, and A. Estroff, “Evanescent wave imaging in optical lithography,” Proc. SPIE 6154, 61540A (2006).
[Crossref]

Appl. Opt. (3)

Appl. Phys. Lett. (2)

R. Magnusson and S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61(9), 1022–1024 (1992).
[Crossref]

H. Hirayama, T. Hamano, and Y. Aoyagi, “Novel surface emitting laser diode using photonic band-gap crystal cavity,” Appl. Phys. Lett. 69(6), 791–793 (1996).
[Crossref]

IEEE J. Quantum Electron. (1)

T. L. Koch and U. Koren, “Semiconductor photonic integrated circuits,” IEEE J. Quantum Electron. 27(3), 641–653 (1991).
[Crossref]

IEEE Trans. Electron Devices (1)

K. E. Bean, “Anisotropic etching of silicon,” IEEE Trans. Electron Devices 25(10), 1185–1193 (1978).
[Crossref]

J. Mater. Chem. (1)

C. Yuan, X. Zhang, L. Su, B. Gao, and L. Shen, “Facile synthesis and self-assembly of hierarchical porous NiO nano/micro spherical superstructures for high performance supercapacitors,” J. Mater. Chem. 19(32), 5772–5777 (2009).
[Crossref]

J. Opt. A: Pure Appl. Opt. (1)

J. C. Martinez-Anton, “Surface relief subwavelength gratings by means of total internal reflection evanescent wave interference lithography,” J. Opt. A: Pure Appl. Opt. 8(4), S213–S218 (2006).
[Crossref]

Laser Photonics Rev. (1)

C. Lu and R. H. Lipson, “Interference lithography: A powerful tool for fabricating periodic structures,” Laser Photonics Rev. 4(4), 568–580 (2010).
[Crossref]

Microelectron. Eng. (1)

E. A. Bezus, L. L. Doskolovich, and N. L. Kazanskiy, “Evanescent-wave interferometric nanoscale photolithography using guided-mode resonant gratings,” Microelectron. Eng. 88(2), 170–174 (2011).
[Crossref]

Opt. Acta (1)

S. J. Wilson and M. C. Hutley, “The optical properties of ‘moth eye’ antireflection surfaces,” Opt. Acta 29(7), 993–1009 (1982).
[Crossref]

Opt. Express (7)

D. Lowell, J. Lutkenhaus, D. George, U. Philipose, B. Chen, and Y. Lin, “Simultaneous direct holographic fabrication of photonic cavity and graded photonic lattice with dual periodicity, dual basis, and dual symmetry,” Opt. Express 25(13), 14444–14452 (2017).
[Crossref]

N. Skivesen, A. Têtu, M. Kristensen, J. Kjems, L. H. Frandsen, and P. I. Borel, “Photonic–crystal waveguide biosensor,” Opt. Express 15(6), 3169–3176 (2007).
[Crossref]

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Figures (9)

Fig. 1.
Fig. 1. Schematic diagram of multiple evanescent wave interference. (a) The nth incident beam in the coordinate system. (b) The nth evanescent wave and the corresponding wavevector.
Fig. 2.
Fig. 2. Calculated visibility factor $|{\hat{{\textbf P}}_a} \cdot {\hat{{\textbf P}}_b}|$ for two-beam interference as a function of incident angle θ and azimuth angle difference $\delta \phi $; (a) P polarization (b) S polarization.
Fig. 3.
Fig. 3. Four-beam EWIL. (a) Configuration of four incident beams and corresponding wavevectors. (b) Total intensity distribution of four-beam interference, which corresponds to $|{\hat{{\textbf E}}_1} + {\hat{{\textbf E}}_2} + {\hat{{\textbf E}}_3} + {\hat{{\textbf E}}_4}{|^2}$
Fig. 4.
Fig. 4. Intensity distribution of each two-beam interference. (a) $|{\hat{{\textbf E}}_1} + {\hat{{\textbf E}}_2}{|^2}$ (b) $|{\hat{{\textbf E}}_1} + {\hat{{\textbf E}}_3}{|^2}$, $|{\hat{{\textbf E}}_2} + {\hat{{\textbf E}}_4}{|^2}$ (c) $|{\hat{{\textbf E}}_1} + {\hat{{\textbf E}}_4}{|^2}$, $|{\hat{{\textbf E}}_3} + {\hat{{\textbf E}}_4}{|^2}$ (d) $|{\hat{{\textbf E}}_3} + {\hat{{\textbf E}}_4}{|^2}$.
Fig. 5.
Fig. 5. Calculated 2D interference pattern of four-beam EWIL in the xy-plane (z = 0). Insets show the schematic of the four incident wavevectors. (a) One-dimensional micro-grating and nanodot pattern, where the incident conditions are $({A_{1 - 4}},{\theta_{1 - 4}},{\phi_{1 - 4}},{\psi_{1 - 4}}) = $(1, 77°, 185°, 0°), (1, 77°, 265°, 0°), (1, 77°, 275°, 0°), and (1, 77°, 355°, 0°). (b) Microdots filled with nano-grating pattern, where the incident conditions are $({A_{1 - 4}},{\theta_{1 - 4}},{\phi_{1 - 4}},{\psi_{1 - 4}}) = $(1, 71°, 315°, 0°), (1, 71°, 225°, 0°), (1, 87°, 315°, 0°), and (1, 87°, 225°, 0°).
Fig. 6.
Fig. 6. Experimental setup of the four-beam EWIL (left) and the schematic of evanescent wave interference (right).
Fig. 7.
Fig. 7. Structures fabricated by two-beam EWIL. (a) Interference fringes of $|{\hat{{\textbf E}}_1} + {\hat{{\textbf E}}_2}{|^2}$ with a pitch of 143 nm. (b) Interference fringes of $|{\hat{{\textbf E}}_3} + {\hat{{\textbf E}}_4}{|^2}$ with a pitch of 139 nm. The insets show the four incident wavevectors.
Fig. 8.
Fig. 8. Dual-scale 1D grating structures fabricated by four-beam EWIL. (a) Microscopy image. Inset shows the schematic of four incident wavevectors. (b) AFM image of the scan area shown in (a). (c) Cross-sectional plot of A-A’ section.
Fig. 9.
Fig. 9. Fabricated microdots filled with subwavelength 1D grating structures. (a) Calculated intensity distribution at z = 0 plane. (b) Microscopy image. Inset shows the schematic of four incident wavevectors. (c) AFM image. (d) Cross-sectional image.

Equations (6)

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E ^ n = A n P ^ n exp ( i ( k ^ n r ^ ω t ) ) exp ( β n z ) ,
r ^ = x i ^ + y j ^ ,
k ^ n = k 0 n p sin θ n cos ϕ n i ^ + sin θ n sin ϕ n j ^ ,
P ^ n = ( cos θ n cos ϕ n cos ψ n + sin ϕ n sin ψ n ) i ^ + ( cos θ n sin ϕ n cos ψ n cos ϕ n sin ψ n ) j ^ + sin θ n cos ψ n k ^ ,
I ( r ^ , z ) = a n | E ^ a | 2 + a b n A a A b P ^ a P ^ b cos [ ( k ^ b k ^ a ) r ^ ] exp [ ( β a + β b ) z ] .
d a b = λ 2 n p sin θ sin ( δ ϕ / 2 ) = 2 π | k a k b | .

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