Abstract

We propose a design method of diffractive slit patterns for focusing surface plasmon polaritons. A scalar model of surface plasmon polariton excitation and interference is adopted, based on which the design method of diffractive slit patterns is built up. The validity of the proposed scalar model-based design is discussed through the comparison of the simulation results of the scalar model and the rigorous three-dimensional vectorial electromagnetic model using the rigorous coupled wave analysis.

©2008 Optical Society of America

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References

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  1. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
    [Crossref] [PubMed]
  2. E. Ozbay, “Plasmonics: Merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006).
    [Crossref] [PubMed]
  3. P. Berini, R. Charbonneau, and N. Lahoud, “Long-range surface plasmons on ultrathin membranes,” Nano Lett. 7, 1376–1380 (2007).
    [Crossref] [PubMed]
  4. L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Miller, J. Hua, U. Welp, D. E. Brown, and C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. 5, 1399–1402 (2005).
    [Crossref] [PubMed]
  5. I. P. Radko, S. I. Bozhevolnyi, A. B. Evlyukhin, and A. Boltasseva, “Surface plasmon polariton beam focusing with parabolic nanoparticle chains,” Opt. Express 15, 6576–6582 (2007).
    [Crossref] [PubMed]
  6. H. L. Offerhaus, B. van den Bergen, M. Escalante, F. B. Segerink, J. P. Korterik, and N. F. van Hulst, “Creating focused plasmons by noncollinear phasematching on functional gratings,” Nano Lett. 5, 2144–2148 (2005).
    [Crossref] [PubMed]
  7. Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, “Focusing surface plasmons with plasmonic lens,” Nano Lett. 5, 1726–1729 (2005).
    [Crossref] [PubMed]
  8. Z. Liu, J. M. Steele, H. Lee, and X. Zhang, “Tuning the focus of a plasmonic lens by the incident angle,” Appl. Phys. Lett. 88, 121108 (2006).
  9. H. Kim, J. Hahn, and B. Lee,“Focusing properties of surface plasmon polariton floatig dielectric lenses,” Opt. Express 16, 3049–3057 (2008).
    [Crossref] [PubMed]
  10. L. Feng, K. A. Tetz, B. Slutsky, V. Lomakin, and Y. Fainman, “Fourier plasmonics: Diffractive focusing of in-plane surface plasmon polariton waves,” Appl. Phys. Lett. 91, 081101 (2007).
    [Crossref]
  11. H. Kim, J. Hahn, and B. Lee, “Mathematical modeling of triangle-mesh-modeled three-dimensional surface objects for digital holography,” Appl. Opt. 47, D117–D127 (2008).
    [Crossref] [PubMed]
  12. R. Zia and M. L. Brongersma, “Surface plasmon polariton analogue to Young’s double-slit experiment,” Nat. Nanotechnol. 2, 426–429 (2007).
    [Crossref]
  13. H. Kim, I.-M. Lee, and B. Lee, “Extended scattering-matrix method for efficient full parallel implementation of rigorous coupled-wave analysis,” J. Opt. Soc. Am. A 24, 2313–2327 (2007).
    [Crossref]
  14. H. Kim and B. Lee, “Mathematical modeling of crossed nanophotonic structures with generalized scattering-matrix method and local Fourier modal analysis,” J. Opt. Soc. Am. B 25, 518–544 (2008).
    [Crossref]

2008 (2)

2007 (5)

R. Zia and M. L. Brongersma, “Surface plasmon polariton analogue to Young’s double-slit experiment,” Nat. Nanotechnol. 2, 426–429 (2007).
[Crossref]

H. Kim, I.-M. Lee, and B. Lee, “Extended scattering-matrix method for efficient full parallel implementation of rigorous coupled-wave analysis,” J. Opt. Soc. Am. A 24, 2313–2327 (2007).
[Crossref]

P. Berini, R. Charbonneau, and N. Lahoud, “Long-range surface plasmons on ultrathin membranes,” Nano Lett. 7, 1376–1380 (2007).
[Crossref] [PubMed]

I. P. Radko, S. I. Bozhevolnyi, A. B. Evlyukhin, and A. Boltasseva, “Surface plasmon polariton beam focusing with parabolic nanoparticle chains,” Opt. Express 15, 6576–6582 (2007).
[Crossref] [PubMed]

L. Feng, K. A. Tetz, B. Slutsky, V. Lomakin, and Y. Fainman, “Fourier plasmonics: Diffractive focusing of in-plane surface plasmon polariton waves,” Appl. Phys. Lett. 91, 081101 (2007).
[Crossref]

2006 (2)

Z. Liu, J. M. Steele, H. Lee, and X. Zhang, “Tuning the focus of a plasmonic lens by the incident angle,” Appl. Phys. Lett. 88, 121108 (2006).

E. Ozbay, “Plasmonics: Merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006).
[Crossref] [PubMed]

2005 (3)

L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Miller, J. Hua, U. Welp, D. E. Brown, and C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. 5, 1399–1402 (2005).
[Crossref] [PubMed]

H. L. Offerhaus, B. van den Bergen, M. Escalante, F. B. Segerink, J. P. Korterik, and N. F. van Hulst, “Creating focused plasmons by noncollinear phasematching on functional gratings,” Nano Lett. 5, 2144–2148 (2005).
[Crossref] [PubMed]

Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, “Focusing surface plasmons with plasmonic lens,” Nano Lett. 5, 1726–1729 (2005).
[Crossref] [PubMed]

2003 (1)

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[Crossref] [PubMed]

Barnes, W. L.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[Crossref] [PubMed]

Berini, P.

P. Berini, R. Charbonneau, and N. Lahoud, “Long-range surface plasmons on ultrathin membranes,” Nano Lett. 7, 1376–1380 (2007).
[Crossref] [PubMed]

Boltasseva, A.

Bozhevolnyi, S. I.

Brongersma, M. L.

R. Zia and M. L. Brongersma, “Surface plasmon polariton analogue to Young’s double-slit experiment,” Nat. Nanotechnol. 2, 426–429 (2007).
[Crossref]

Brown, D. E.

L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Miller, J. Hua, U. Welp, D. E. Brown, and C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. 5, 1399–1402 (2005).
[Crossref] [PubMed]

Charbonneau, R.

P. Berini, R. Charbonneau, and N. Lahoud, “Long-range surface plasmons on ultrathin membranes,” Nano Lett. 7, 1376–1380 (2007).
[Crossref] [PubMed]

Dereux, A.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[Crossref] [PubMed]

Ebbesen, T. W.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[Crossref] [PubMed]

Escalante, M.

H. L. Offerhaus, B. van den Bergen, M. Escalante, F. B. Segerink, J. P. Korterik, and N. F. van Hulst, “Creating focused plasmons by noncollinear phasematching on functional gratings,” Nano Lett. 5, 2144–2148 (2005).
[Crossref] [PubMed]

Evlyukhin, A. B.

Fainman, Y.

L. Feng, K. A. Tetz, B. Slutsky, V. Lomakin, and Y. Fainman, “Fourier plasmonics: Diffractive focusing of in-plane surface plasmon polariton waves,” Appl. Phys. Lett. 91, 081101 (2007).
[Crossref]

Feng, L.

L. Feng, K. A. Tetz, B. Slutsky, V. Lomakin, and Y. Fainman, “Fourier plasmonics: Diffractive focusing of in-plane surface plasmon polariton waves,” Appl. Phys. Lett. 91, 081101 (2007).
[Crossref]

Hahn, J.

H. Kim, J. Hahn, and B. Lee, “Mathematical modeling of triangle-mesh-modeled three-dimensional surface objects for digital holography,” Appl. Opt. 47, D117–D127 (2008).
[Crossref] [PubMed]

H. Kim, J. Hahn, and B. Lee,“Focusing properties of surface plasmon polariton floatig dielectric lenses,” Opt. Express 16, 3049–3057 (2008).
[Crossref] [PubMed]

Hua, J.

L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Miller, J. Hua, U. Welp, D. E. Brown, and C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. 5, 1399–1402 (2005).
[Crossref] [PubMed]

Kim, H.

Kimball, C. W.

L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Miller, J. Hua, U. Welp, D. E. Brown, and C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. 5, 1399–1402 (2005).
[Crossref] [PubMed]

Korterik, J. P.

H. L. Offerhaus, B. van den Bergen, M. Escalante, F. B. Segerink, J. P. Korterik, and N. F. van Hulst, “Creating focused plasmons by noncollinear phasematching on functional gratings,” Nano Lett. 5, 2144–2148 (2005).
[Crossref] [PubMed]

Lahoud, N.

P. Berini, R. Charbonneau, and N. Lahoud, “Long-range surface plasmons on ultrathin membranes,” Nano Lett. 7, 1376–1380 (2007).
[Crossref] [PubMed]

Lee, B.

Lee, H.

Z. Liu, J. M. Steele, H. Lee, and X. Zhang, “Tuning the focus of a plasmonic lens by the incident angle,” Appl. Phys. Lett. 88, 121108 (2006).

Lee, I.-M.

Liu, Z.

Z. Liu, J. M. Steele, H. Lee, and X. Zhang, “Tuning the focus of a plasmonic lens by the incident angle,” Appl. Phys. Lett. 88, 121108 (2006).

Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, “Focusing surface plasmons with plasmonic lens,” Nano Lett. 5, 1726–1729 (2005).
[Crossref] [PubMed]

Lomakin, V.

L. Feng, K. A. Tetz, B. Slutsky, V. Lomakin, and Y. Fainman, “Fourier plasmonics: Diffractive focusing of in-plane surface plasmon polariton waves,” Appl. Phys. Lett. 91, 081101 (2007).
[Crossref]

Miller, J. M.

L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Miller, J. Hua, U. Welp, D. E. Brown, and C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. 5, 1399–1402 (2005).
[Crossref] [PubMed]

Offerhaus, H. L.

H. L. Offerhaus, B. van den Bergen, M. Escalante, F. B. Segerink, J. P. Korterik, and N. F. van Hulst, “Creating focused plasmons by noncollinear phasematching on functional gratings,” Nano Lett. 5, 2144–2148 (2005).
[Crossref] [PubMed]

Ozbay, E.

E. Ozbay, “Plasmonics: Merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006).
[Crossref] [PubMed]

Pearson, J.

L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Miller, J. Hua, U. Welp, D. E. Brown, and C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. 5, 1399–1402 (2005).
[Crossref] [PubMed]

Pikus, Y.

Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, “Focusing surface plasmons with plasmonic lens,” Nano Lett. 5, 1726–1729 (2005).
[Crossref] [PubMed]

Radko, I. P.

Segerink, F. B.

H. L. Offerhaus, B. van den Bergen, M. Escalante, F. B. Segerink, J. P. Korterik, and N. F. van Hulst, “Creating focused plasmons by noncollinear phasematching on functional gratings,” Nano Lett. 5, 2144–2148 (2005).
[Crossref] [PubMed]

Slutsky, B.

L. Feng, K. A. Tetz, B. Slutsky, V. Lomakin, and Y. Fainman, “Fourier plasmonics: Diffractive focusing of in-plane surface plasmon polariton waves,” Appl. Phys. Lett. 91, 081101 (2007).
[Crossref]

Srituravanich, W.

Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, “Focusing surface plasmons with plasmonic lens,” Nano Lett. 5, 1726–1729 (2005).
[Crossref] [PubMed]

Steele, J. M.

Z. Liu, J. M. Steele, H. Lee, and X. Zhang, “Tuning the focus of a plasmonic lens by the incident angle,” Appl. Phys. Lett. 88, 121108 (2006).

Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, “Focusing surface plasmons with plasmonic lens,” Nano Lett. 5, 1726–1729 (2005).
[Crossref] [PubMed]

Sun, C.

Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, “Focusing surface plasmons with plasmonic lens,” Nano Lett. 5, 1726–1729 (2005).
[Crossref] [PubMed]

Tetz, K. A.

L. Feng, K. A. Tetz, B. Slutsky, V. Lomakin, and Y. Fainman, “Fourier plasmonics: Diffractive focusing of in-plane surface plasmon polariton waves,” Appl. Phys. Lett. 91, 081101 (2007).
[Crossref]

van den Bergen, B.

H. L. Offerhaus, B. van den Bergen, M. Escalante, F. B. Segerink, J. P. Korterik, and N. F. van Hulst, “Creating focused plasmons by noncollinear phasematching on functional gratings,” Nano Lett. 5, 2144–2148 (2005).
[Crossref] [PubMed]

van Hulst, N. F.

H. L. Offerhaus, B. van den Bergen, M. Escalante, F. B. Segerink, J. P. Korterik, and N. F. van Hulst, “Creating focused plasmons by noncollinear phasematching on functional gratings,” Nano Lett. 5, 2144–2148 (2005).
[Crossref] [PubMed]

Vlasko-Vlasov, V. K.

L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Miller, J. Hua, U. Welp, D. E. Brown, and C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. 5, 1399–1402 (2005).
[Crossref] [PubMed]

Welp, U.

L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Miller, J. Hua, U. Welp, D. E. Brown, and C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. 5, 1399–1402 (2005).
[Crossref] [PubMed]

Yin, L.

L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Miller, J. Hua, U. Welp, D. E. Brown, and C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. 5, 1399–1402 (2005).
[Crossref] [PubMed]

Zhang, X.

Z. Liu, J. M. Steele, H. Lee, and X. Zhang, “Tuning the focus of a plasmonic lens by the incident angle,” Appl. Phys. Lett. 88, 121108 (2006).

Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, “Focusing surface plasmons with plasmonic lens,” Nano Lett. 5, 1726–1729 (2005).
[Crossref] [PubMed]

Zia, R.

R. Zia and M. L. Brongersma, “Surface plasmon polariton analogue to Young’s double-slit experiment,” Nat. Nanotechnol. 2, 426–429 (2007).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (2)

L. Feng, K. A. Tetz, B. Slutsky, V. Lomakin, and Y. Fainman, “Fourier plasmonics: Diffractive focusing of in-plane surface plasmon polariton waves,” Appl. Phys. Lett. 91, 081101 (2007).
[Crossref]

Z. Liu, J. M. Steele, H. Lee, and X. Zhang, “Tuning the focus of a plasmonic lens by the incident angle,” Appl. Phys. Lett. 88, 121108 (2006).

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

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

Nano Lett. (4)

H. L. Offerhaus, B. van den Bergen, M. Escalante, F. B. Segerink, J. P. Korterik, and N. F. van Hulst, “Creating focused plasmons by noncollinear phasematching on functional gratings,” Nano Lett. 5, 2144–2148 (2005).
[Crossref] [PubMed]

Z. Liu, J. M. Steele, W. Srituravanich, Y. Pikus, C. Sun, and X. Zhang, “Focusing surface plasmons with plasmonic lens,” Nano Lett. 5, 1726–1729 (2005).
[Crossref] [PubMed]

P. Berini, R. Charbonneau, and N. Lahoud, “Long-range surface plasmons on ultrathin membranes,” Nano Lett. 7, 1376–1380 (2007).
[Crossref] [PubMed]

L. Yin, V. K. Vlasko-Vlasov, J. Pearson, J. M. Miller, J. Hua, U. Welp, D. E. Brown, and C. W. Kimball, “Subwavelength focusing and guiding of surface plasmons,” Nano Lett. 5, 1399–1402 (2005).
[Crossref] [PubMed]

Nat. Nanotechnol. (1)

R. Zia and M. L. Brongersma, “Surface plasmon polariton analogue to Young’s double-slit experiment,” Nat. Nanotechnol. 2, 426–429 (2007).
[Crossref]

Nature (1)

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[Crossref] [PubMed]

Opt. Express (1)

Science (1)

E. Ozbay, “Plasmonics: Merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006).
[Crossref] [PubMed]

Other (1)

H. Kim, J. Hahn, and B. Lee,“Focusing properties of surface plasmon polariton floatig dielectric lenses,” Opt. Express 16, 3049–3057 (2008).
[Crossref] [PubMed]

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Figures (10)

Fig. 1.
Fig. 1.

Phase distributions of complementary single point sources with the centers (a) at (xc , yc )=(0, 0) and (b) at (xc , yc )=(1.5µm, 1.5µm). (c) Phase distribution of a composite field of two point sources with the centers at (1.5µm, 1.5µm) and (-1.5µm, -1.5µm). Amplitude distributions of complementary single point sources with the centers (d) at (xc , yc )=(0, 0) and (e) at (xc , yc )=(1.5µm, 1.5µm). (f) Amplitude distribution of a composite field of two point sources with the centers at (1.5µm, 1.5µm) and (-1.5µm, -1.5µm).

Download Full Size | PPT Slide | PDF

Fig. 2.
Fig. 2.

(a). Ω+ of a point source with the center at the origin, (b) slit pattern extracted from Ω+ and SPP polarity, (c) Ω- of the point source with the center at the origin, (d) slit pattern extracted from Ω- and SPP polarity.

Download Full Size | PPT Slide | PDF

Fig. 3.
Fig. 3.

Single SPP spot generation: (a) slit pattern without the compensation of the π-phase difference, (b) SPP interference pattern, (c) slit pattern with the compensation of the π-phase difference, (d) SPP interference pattern

Download Full Size | PPT Slide | PDF

Fig. 4.
Fig. 4.

(a). Ω+ of a point source with the center at the off-origin (1.5µm, 1.5µm) (indicated by red lines), (b) slit pattern extracted from Ω+ and SPP polarity (Ω+ p is while and Ω+ n is black), (c) A+; amplitude profile of Ω+, (d) Ω- of the point source with the center at the off-origin (indicated by red lines), (e) slit pattern extracted from Ω- and SPP polarity (Ω- p is black and Ω- n is white), (f) A-; amplitude profile of Ω-.

Download Full Size | PPT Slide | PDF

Fig. 5.
Fig. 5.

Single SPP spot generation: (a) slit pattern without the compensation of the π -phase difference, (b) SPP interference pattern, (c) slit pattern with the compensation of the π -phase difference, (d) SPP interference pattern.

Download Full Size | PPT Slide | PDF

Fig. 6.
Fig. 6.

Two SPP spot generation: (a) slit pattern without the compensation of the π -phase difference, (b) SPP interference pattern, (c) slit pattern with the compensation of the π -phase difference, (d) SPP interference pattern..

Download Full Size | PPT Slide | PDF

Fig. 7.
Fig. 7.

RCWA results of single SPP focal spot generation: (a) slit pattern without the π -phase difference compensation, (b) x-polarization electric field intensity distribution, (c) y-polarization electric field intensity distribution, (d) z-polarization electric field intensity distribution

Download Full Size | PPT Slide | PDF

Fig. 8.
Fig. 8.

RCWA results of single SPP focal spot generation: (a) slit pattern with the π -phase difference compensation, (b) x-polarization electric field intensity distribution, (c) y-polarization electric field intensity distribution, (d) z-polarization electric field intensity distribution

Download Full Size | PPT Slide | PDF

Fig. 9.
Fig. 9.

RCWA results of off-origin single SPP focal spot generation: (a) slit pattern with the π -phase difference compensation, (b) x-polarization electric field intensity distribution, (c) y-polarization electric field intensity distribution, (d) z-polarization electric field intensity distribution

Download Full Size | PPT Slide | PDF

Fig. 10.
Fig. 10.

RCWA results of two SPP focal spot generation: (a) slit pattern with the π -phase difference compensation, (b) x-polarization electric field intensity distribution, (c) y-polarization electric field intensity distribution, (d) z-polarization electric field intensity distribution

Download Full Size | PPT Slide | PDF

Equations (22)

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

F ( x , y ) = ∑ m exp ( jk SPP ( x − x m ) 2 + ( y − y m ) 2 )
= ∑ m exp ( ( jk SPP r − k SPP i ) ( x − x m ) 2 + ( y − y m ) 2 ) ,
k SPP = 2 π λ ε a ε m ε a + ε m = 2 π λ SPP ,
G ( x , y ) = ∑ m exp ( ( − j k SPP r + k SPP i ) ( x − x m ) 2 + ( y − y m ) 2 )
= a ( x , y ) exp ( j ϕ ( x , y ) ) ,
Ω + = { ( x , y ) ∣ ϕ ( x , y ) = 0 and R 2 ≤ x 2 + y 2 ≤ R 1 } ,
Ω − = { ( x , y ) ∣ ϕ ( x , y ) = − π and R 2 ≤ x 2 + y 2 ≤ R 1 } .
A + = { a ( x , y ) ∣ ( x , y ) ∈ Ω + } ,
A − = { a ( x , y ) ∣ ( x , y ) ∈ Ω − } ,
G ( x , y ) = exp ( − jk SPP ( x − x c ) 2 + ( y − y c ) 2 ) .
U ( x , y ) = ∫ C exp ( jk SPP ( x − x ′ ) 2 + ( y − y ′ ) 2 ) p · n ds ,
∇ Ω ± = ( ∂ ϕ ( x , y ) ∂ x , ∂ ϕ ( x , y ) ∂ y ) ( x , y ) ∈ Ω ± ,
ϕ ( x , y ) = − tan − 1 [ ∑ m e k SPP i ( x − x m ) 2 + ( y − y m ) 2 sin ( k SPP r ( x − x m ) 2 + ( y − y m ) 2 ) ∑ m e k SPP i ( x − x m ) 2 + ( y − y m ) 2 cos ( k SPP r ( x − x m ) 2 + ( y − y m ) 2 ) ] ,
∂ ϕ ( x , y ) ∂ x = − ( cos ϕ ( x , y ) ) 2 ∂ ∂ x [ ∑ m e k SPP i ( x − x m ) 2 + ( y − y m ) 2 sin ( k SPP r ( x − x m ) 2 + ( y − y m ) 2 ) ∑ m e k SPP i ( x − x m ) 2 + ( y − y m ) 2 cos ( k SPP r ( x − x m ) 2 + ( y − y m ) 2 ) ] ,
∂ ϕ ( x , y ) ∂ y = − ( cos ϕ ( x , y ) ) 2 ∂ ∂ y [ ∑ m e k SPP i ( x − x m ) 2 + ( y − y m ) 2 sin ( k SPP r ( x − x m ) 2 + ( y − y m ) 2 ) ∑ m e k SPP i ( x − x m ) 2 + ( y − y m ) 2 cos ( k SPP r ( x − x m ) 2 + ( y − y m ) 2 ) ] .
Ω p + = { ( x , y ) ∣ ϕ ( x , y ) = 0 and ∇ Ω + · p ≥ 0 } ,
Ω n + = { ( x , y ) ∣ ϕ ( x , y ) = 0 and ∇ Ω + · p < 0 } ,
Ω p − = { ( x , y ) ∣ ϕ ( x , y ) = − π and ∇ Ω − · p ≥ 0 } ,
Ω n − = { ( x , y ) ∣ ϕ ( x , y ) = − π and ∇ Ω − · p < 0 } .
U ( x , y ) = ∫ C a ( x , y ) exp ( jk SPP ( x − x ′ ) 2 + ( y − y ′ ) 2 ) ds .
U ( x , y ) = ∫ C a ( x , y ) exp ( jk SPP ( x − x ′ ) 2 + ( y − y ′ ) 2 ) p · n ds .
E z ( x , y ) = ∫ C b ( x , y ) exp ( jk SPP ( x − x ′ ) 2 + ( y − y ′ ) 2 ) ds .

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