Abstract

An apodized, multilevel diffractive lens that can produce a desired diffraction-limited focal spot is proposed for many applications, such as focusing, imaging, optical storage, and optical trapping. The three key points for the design are the innovative idea of complex conjugate subzones, the use of Babinet’s principle, and the equivalent-pupil (or aperture) function theory of diffractive focusing elements composed of concentric transparent rings. As a concrete example, we numerically design a mixed multilevel diffractive lens (the highest phase level is 8) to produce a diffraction-limited Gaussian focal spot. Some related problems, such as the validity range and the combination with high-numerical-aperture refractive lenses, are also discussed.

© 2006 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975), Subsect. 8.6.2.
  2. S. Sinzinger and J. Jahns, Microoptics, 2nd ed. (Wiley-VCH, 2003), Subsect. 6.3.6.
    [CrossRef]
  3. R. Brunner, M. Burkhardt, A. Pesch, O. Sandfuchs, M. Ferstl, S. Hohng, and J. O. White, "Diffraction-based solid immersion lens," J. Opt. Soc. Am. A 21, 1186-1191 (2004).
    [CrossRef]
  4. M. Kuittinen and H. P. Herzig, "Encoding of efficient diffractive microlenses," Opt. Lett. 20, 2156-2158 (1995).
    [CrossRef] [PubMed]
  5. U. Levy, D. Mendlovic, and E. Marom, "Efficiency analysis of diffractive lenses," J. Opt. Soc. Am. A 18, 86-93 (2001).
    [CrossRef]
  6. A. V. Baez, "Fresnel zone plate for optical image formation using extreme ultraviolet and soft x radiation," J. Opt. Soc. Am. 51, 405-412 (1961).
    [CrossRef]
  7. C. D. Pfeifer, L. D. Ferris, and W. M. Yen, "Optical image formation with a Fresnel zone plate using vacuum-ultraviolet radiation," J. Opt. Soc. Am. 63, 91-95 (1973).
    [CrossRef]
  8. G. Schmahl, D. Rudolph, P. Guttmann, and O. Christ, "Zone plates for x-ray microscopy," in X-ray Microscopy, G.Schmahl and D.Rudolph, eds. (Springer-Verlag, 1984), Vol. 43, pp. 63-74.
  9. D. J. Stigliani, R. Mittra, and R. G. Semonin, "Resolving power of a zone plate," J. Opt. Soc. Am. 57, 610-613 (1967).
    [CrossRef]
  10. H. Arsenault, "Diffraction theory of Fresnel zone plates," J. Opt. Soc. Am. 58, 1536 (1968).
    [CrossRef]
  11. J. A. Sun and A. Cai, "Archaic focusing properties of Fresnel zone plates," J. Opt. Soc. Am. A 8, 33-35 (1991).
    [CrossRef]
  12. E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, and D. Attwood, "Nanofabrication and diffractive optics for high-resolution x-ray applications," J. Vac. Sci. Technol. B 18, 2970-2975 (2000).
    [CrossRef]
  13. W. Chao, E. Anderson, G. P. Denbeaux, B. Harteneck, J. A. Liddle, D. L. Olynick, A. L. Pearson, F. Salmassi, C. Y. Song, and D. T. Attwood, "20-nm-resolution soft x-ray microscopy demonstrated by use of multilayer test structures," Opt. Lett. 28, 2019-2021 (2003).
    [CrossRef] [PubMed]
  14. L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, and R. Seemann, "Sharper images by focusing soft x-rays with photon sieves," Nature (London) 414, 184-188 (2001).
    [CrossRef]
  15. Q. Cao and J. Jahns, "Focusing analysis of the pinhole photon sieve: individual far-field model," J. Opt. Soc. Am. A 19, 2387-2393 (2002).
    [CrossRef]
  16. Q. Cao and J. Jahns, "Nonparaxial model for the focusing of high-numerical-aperture photon sieves," J. Opt. Soc. Am. A 20, 1005-1012 (2003).
    [CrossRef]
  17. R. Menon, D. Gil, G. Barbastathis, and H. I. Smith, "Photon-sieve lithography," J. Opt. Soc. Am. A 22, 342-345 (2005).
    [CrossRef]
  18. G. E. Artzner, J. P. Delaboudinière, and X. Y. Song, "Photon sieves as EUV telescopes for solar orbiter," in Innovative Telescopes and Instrumentations for Solar Astrophysics, S.L.Keil, S.V.Avakyan, and S.I.Vavilov, eds., Proc. SPIE 4853, 158-161 (2003).
  19. M. Howells, http://www-esg.lbl.gov/esg/personnel/howells/Xraysieves.pdf.
  20. M. J. Simpson and A. G. Michette, "Imaging properties of modified Fresnel zone plates," Opt. Acta 31, 403-413 (1984).
    [CrossRef]
  21. Q. Cao and J. Jahns, "Modified Fresnel zone plates that produce sharp Gaussian focal spots," J. Opt. Soc. Am. A 20, 1576-1581 (2003).
    [CrossRef]
  22. Q. Cao and J. Jahns, "Comprehensive focusing analysis of various Fresnel zone plates," J. Opt. Soc. Am. A 21, 561-571 (2004).
    [CrossRef]
  23. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), Subsect. 2.1.5.
  24. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1975), Section 9.11.
  25. R. L. Morrison, "Symmetries that simplify the design of spot array phase gratings," J. Opt. Soc. Am. A 9, 464-471 (1992).
    [CrossRef]
  26. Y. Komma, S. Kadowaki, S. Nishino, Y. Hori, and M. Kato, "A holographic optical element for an integrated optical head," Opt. Rev. 3, 251-257 (1996).
    [CrossRef]
  27. R. Brunner, R. Steiner, K. Rudolf, and H.-J. Dobschal, "Diffractive-refractive hybrid microscope objective for193-nm inspection systems," in Gradient Index, Miniature, and Diffractive Optical Systems III, T.J.Suleski, Proc. SPIE 5177, 9-15 (2003).
  28. T. Stone and N. George, "Hybrid diffractive-refractive lenses and achromats," Appl. Opt. 27, 2960-2971 (1988).
    [CrossRef] [PubMed]
  29. N. Davidson, A. A. Friesem, and E. Hasman, "Analytic design of hybrid diffractive-refractive achromats," Appl. Opt. 32, 4770-4774 (1993).
    [CrossRef] [PubMed]

2005 (1)

2004 (2)

2003 (3)

2002 (1)

2001 (2)

U. Levy, D. Mendlovic, and E. Marom, "Efficiency analysis of diffractive lenses," J. Opt. Soc. Am. A 18, 86-93 (2001).
[CrossRef]

L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, and R. Seemann, "Sharper images by focusing soft x-rays with photon sieves," Nature (London) 414, 184-188 (2001).
[CrossRef]

2000 (1)

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, and D. Attwood, "Nanofabrication and diffractive optics for high-resolution x-ray applications," J. Vac. Sci. Technol. B 18, 2970-2975 (2000).
[CrossRef]

1996 (1)

Y. Komma, S. Kadowaki, S. Nishino, Y. Hori, and M. Kato, "A holographic optical element for an integrated optical head," Opt. Rev. 3, 251-257 (1996).
[CrossRef]

1995 (1)

1993 (1)

1992 (1)

1991 (1)

1988 (1)

1984 (1)

M. J. Simpson and A. G. Michette, "Imaging properties of modified Fresnel zone plates," Opt. Acta 31, 403-413 (1984).
[CrossRef]

1973 (1)

1968 (1)

1967 (1)

1961 (1)

Adelung, R.

L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, and R. Seemann, "Sharper images by focusing soft x-rays with photon sieves," Nature (London) 414, 184-188 (2001).
[CrossRef]

Anderson, E.

Anderson, E. H.

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, and D. Attwood, "Nanofabrication and diffractive optics for high-resolution x-ray applications," J. Vac. Sci. Technol. B 18, 2970-2975 (2000).
[CrossRef]

Arsenault, H.

Artzner, G. E.

G. E. Artzner, J. P. Delaboudinière, and X. Y. Song, "Photon sieves as EUV telescopes for solar orbiter," in Innovative Telescopes and Instrumentations for Solar Astrophysics, S.L.Keil, S.V.Avakyan, and S.I.Vavilov, eds., Proc. SPIE 4853, 158-161 (2003).

Attwood, D.

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, and D. Attwood, "Nanofabrication and diffractive optics for high-resolution x-ray applications," J. Vac. Sci. Technol. B 18, 2970-2975 (2000).
[CrossRef]

Attwood, D. T.

Baez, A. V.

Barbastathis, G.

Berndt, R.

L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, and R. Seemann, "Sharper images by focusing soft x-rays with photon sieves," Nature (London) 414, 184-188 (2001).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975), Subsect. 8.6.2.

Brunner, R.

R. Brunner, M. Burkhardt, A. Pesch, O. Sandfuchs, M. Ferstl, S. Hohng, and J. O. White, "Diffraction-based solid immersion lens," J. Opt. Soc. Am. A 21, 1186-1191 (2004).
[CrossRef]

R. Brunner, R. Steiner, K. Rudolf, and H.-J. Dobschal, "Diffractive-refractive hybrid microscope objective for193-nm inspection systems," in Gradient Index, Miniature, and Diffractive Optical Systems III, T.J.Suleski, Proc. SPIE 5177, 9-15 (2003).

Burkhardt, M.

Cai, A.

Cao, Q.

Chao, W.

W. Chao, E. Anderson, G. P. Denbeaux, B. Harteneck, J. A. Liddle, D. L. Olynick, A. L. Pearson, F. Salmassi, C. Y. Song, and D. T. Attwood, "20-nm-resolution soft x-ray microscopy demonstrated by use of multilayer test structures," Opt. Lett. 28, 2019-2021 (2003).
[CrossRef] [PubMed]

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, and D. Attwood, "Nanofabrication and diffractive optics for high-resolution x-ray applications," J. Vac. Sci. Technol. B 18, 2970-2975 (2000).
[CrossRef]

Christ, O.

G. Schmahl, D. Rudolph, P. Guttmann, and O. Christ, "Zone plates for x-ray microscopy," in X-ray Microscopy, G.Schmahl and D.Rudolph, eds. (Springer-Verlag, 1984), Vol. 43, pp. 63-74.

Davidson, N.

Delaboudinière, J. P.

G. E. Artzner, J. P. Delaboudinière, and X. Y. Song, "Photon sieves as EUV telescopes for solar orbiter," in Innovative Telescopes and Instrumentations for Solar Astrophysics, S.L.Keil, S.V.Avakyan, and S.I.Vavilov, eds., Proc. SPIE 4853, 158-161 (2003).

Denbeaux, G.

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, and D. Attwood, "Nanofabrication and diffractive optics for high-resolution x-ray applications," J. Vac. Sci. Technol. B 18, 2970-2975 (2000).
[CrossRef]

Denbeaux, G. P.

Dobschal, H.-J.

R. Brunner, R. Steiner, K. Rudolf, and H.-J. Dobschal, "Diffractive-refractive hybrid microscope objective for193-nm inspection systems," in Gradient Index, Miniature, and Diffractive Optical Systems III, T.J.Suleski, Proc. SPIE 5177, 9-15 (2003).

Ferris, L. D.

Ferstl, M.

Friesem, A. A.

George, N.

Gil, D.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), Subsect. 2.1.5.

Guttmann, P.

G. Schmahl, D. Rudolph, P. Guttmann, and O. Christ, "Zone plates for x-ray microscopy," in X-ray Microscopy, G.Schmahl and D.Rudolph, eds. (Springer-Verlag, 1984), Vol. 43, pp. 63-74.

Harm, S.

L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, and R. Seemann, "Sharper images by focusing soft x-rays with photon sieves," Nature (London) 414, 184-188 (2001).
[CrossRef]

Harteneck, B.

W. Chao, E. Anderson, G. P. Denbeaux, B. Harteneck, J. A. Liddle, D. L. Olynick, A. L. Pearson, F. Salmassi, C. Y. Song, and D. T. Attwood, "20-nm-resolution soft x-ray microscopy demonstrated by use of multilayer test structures," Opt. Lett. 28, 2019-2021 (2003).
[CrossRef] [PubMed]

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, and D. Attwood, "Nanofabrication and diffractive optics for high-resolution x-ray applications," J. Vac. Sci. Technol. B 18, 2970-2975 (2000).
[CrossRef]

Hasman, E.

Herzig, H. P.

Hohng, S.

Hori, Y.

Y. Komma, S. Kadowaki, S. Nishino, Y. Hori, and M. Kato, "A holographic optical element for an integrated optical head," Opt. Rev. 3, 251-257 (1996).
[CrossRef]

Howells, M.

M. Howells, http://www-esg.lbl.gov/esg/personnel/howells/Xraysieves.pdf.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1975), Section 9.11.

Jahns, J.

Johnson, L.

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, and D. Attwood, "Nanofabrication and diffractive optics for high-resolution x-ray applications," J. Vac. Sci. Technol. B 18, 2970-2975 (2000).
[CrossRef]

Johnson, R. L.

L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, and R. Seemann, "Sharper images by focusing soft x-rays with photon sieves," Nature (London) 414, 184-188 (2001).
[CrossRef]

Kadowaki, S.

Y. Komma, S. Kadowaki, S. Nishino, Y. Hori, and M. Kato, "A holographic optical element for an integrated optical head," Opt. Rev. 3, 251-257 (1996).
[CrossRef]

Kato, M.

Y. Komma, S. Kadowaki, S. Nishino, Y. Hori, and M. Kato, "A holographic optical element for an integrated optical head," Opt. Rev. 3, 251-257 (1996).
[CrossRef]

Kipp, L.

L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, and R. Seemann, "Sharper images by focusing soft x-rays with photon sieves," Nature (London) 414, 184-188 (2001).
[CrossRef]

Komma, Y.

Y. Komma, S. Kadowaki, S. Nishino, Y. Hori, and M. Kato, "A holographic optical element for an integrated optical head," Opt. Rev. 3, 251-257 (1996).
[CrossRef]

Kuittinen, M.

Levy, U.

Liddle, J. A.

Lucero, A.

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, and D. Attwood, "Nanofabrication and diffractive optics for high-resolution x-ray applications," J. Vac. Sci. Technol. B 18, 2970-2975 (2000).
[CrossRef]

Marom, E.

Mendlovic, D.

Menon, R.

Michette, A. G.

M. J. Simpson and A. G. Michette, "Imaging properties of modified Fresnel zone plates," Opt. Acta 31, 403-413 (1984).
[CrossRef]

Mittra, R.

Morrison, R. L.

Nishino, S.

Y. Komma, S. Kadowaki, S. Nishino, Y. Hori, and M. Kato, "A holographic optical element for an integrated optical head," Opt. Rev. 3, 251-257 (1996).
[CrossRef]

Olynick, D. L.

W. Chao, E. Anderson, G. P. Denbeaux, B. Harteneck, J. A. Liddle, D. L. Olynick, A. L. Pearson, F. Salmassi, C. Y. Song, and D. T. Attwood, "20-nm-resolution soft x-ray microscopy demonstrated by use of multilayer test structures," Opt. Lett. 28, 2019-2021 (2003).
[CrossRef] [PubMed]

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, and D. Attwood, "Nanofabrication and diffractive optics for high-resolution x-ray applications," J. Vac. Sci. Technol. B 18, 2970-2975 (2000).
[CrossRef]

Pearson, A. L.

Pesch, A.

Pfeifer, C. D.

Rudolf, K.

R. Brunner, R. Steiner, K. Rudolf, and H.-J. Dobschal, "Diffractive-refractive hybrid microscope objective for193-nm inspection systems," in Gradient Index, Miniature, and Diffractive Optical Systems III, T.J.Suleski, Proc. SPIE 5177, 9-15 (2003).

Rudolph, D.

G. Schmahl, D. Rudolph, P. Guttmann, and O. Christ, "Zone plates for x-ray microscopy," in X-ray Microscopy, G.Schmahl and D.Rudolph, eds. (Springer-Verlag, 1984), Vol. 43, pp. 63-74.

Salmassi, F.

Sandfuchs, O.

Schmahl, G.

G. Schmahl, D. Rudolph, P. Guttmann, and O. Christ, "Zone plates for x-ray microscopy," in X-ray Microscopy, G.Schmahl and D.Rudolph, eds. (Springer-Verlag, 1984), Vol. 43, pp. 63-74.

Seemann, R.

L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, and R. Seemann, "Sharper images by focusing soft x-rays with photon sieves," Nature (London) 414, 184-188 (2001).
[CrossRef]

Semonin, R. G.

Simpson, M. J.

M. J. Simpson and A. G. Michette, "Imaging properties of modified Fresnel zone plates," Opt. Acta 31, 403-413 (1984).
[CrossRef]

Sinzinger, S.

S. Sinzinger and J. Jahns, Microoptics, 2nd ed. (Wiley-VCH, 2003), Subsect. 6.3.6.
[CrossRef]

Skibowski, M.

L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, and R. Seemann, "Sharper images by focusing soft x-rays with photon sieves," Nature (London) 414, 184-188 (2001).
[CrossRef]

Smith, H. I.

Song, C. Y.

Song, X. Y.

G. E. Artzner, J. P. Delaboudinière, and X. Y. Song, "Photon sieves as EUV telescopes for solar orbiter," in Innovative Telescopes and Instrumentations for Solar Astrophysics, S.L.Keil, S.V.Avakyan, and S.I.Vavilov, eds., Proc. SPIE 4853, 158-161 (2003).

Steiner, R.

R. Brunner, R. Steiner, K. Rudolf, and H.-J. Dobschal, "Diffractive-refractive hybrid microscope objective for193-nm inspection systems," in Gradient Index, Miniature, and Diffractive Optical Systems III, T.J.Suleski, Proc. SPIE 5177, 9-15 (2003).

Stigliani, D. J.

Stone, T.

Sun, J. A.

Veklerov, E.

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, and D. Attwood, "Nanofabrication and diffractive optics for high-resolution x-ray applications," J. Vac. Sci. Technol. B 18, 2970-2975 (2000).
[CrossRef]

White, J. O.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975), Subsect. 8.6.2.

Yen, W. M.

Appl. Opt. (2)

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (9)

J. Vac. Sci. Technol. B (1)

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, and D. Attwood, "Nanofabrication and diffractive optics for high-resolution x-ray applications," J. Vac. Sci. Technol. B 18, 2970-2975 (2000).
[CrossRef]

Nature (London) (1)

L. Kipp, M. Skibowski, R. L. Johnson, R. Berndt, R. Adelung, S. Harm, and R. Seemann, "Sharper images by focusing soft x-rays with photon sieves," Nature (London) 414, 184-188 (2001).
[CrossRef]

Opt. Acta (1)

M. J. Simpson and A. G. Michette, "Imaging properties of modified Fresnel zone plates," Opt. Acta 31, 403-413 (1984).
[CrossRef]

Opt. Lett. (2)

Opt. Rev. (1)

Y. Komma, S. Kadowaki, S. Nishino, Y. Hori, and M. Kato, "A holographic optical element for an integrated optical head," Opt. Rev. 3, 251-257 (1996).
[CrossRef]

Other (8)

R. Brunner, R. Steiner, K. Rudolf, and H.-J. Dobschal, "Diffractive-refractive hybrid microscope objective for193-nm inspection systems," in Gradient Index, Miniature, and Diffractive Optical Systems III, T.J.Suleski, Proc. SPIE 5177, 9-15 (2003).

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), Subsect. 2.1.5.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1975), Section 9.11.

G. E. Artzner, J. P. Delaboudinière, and X. Y. Song, "Photon sieves as EUV telescopes for solar orbiter," in Innovative Telescopes and Instrumentations for Solar Astrophysics, S.L.Keil, S.V.Avakyan, and S.I.Vavilov, eds., Proc. SPIE 4853, 158-161 (2003).

M. Howells, http://www-esg.lbl.gov/esg/personnel/howells/Xraysieves.pdf.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975), Subsect. 8.6.2.

S. Sinzinger and J. Jahns, Microoptics, 2nd ed. (Wiley-VCH, 2003), Subsect. 6.3.6.
[CrossRef]

G. Schmahl, D. Rudolph, P. Guttmann, and O. Christ, "Zone plates for x-ray microscopy," in X-ray Microscopy, G.Schmahl and D.Rudolph, eds. (Springer-Verlag, 1984), Vol. 43, pp. 63-74.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Schematic view of a complex conjugate subzone for the case of (a) 4-phase-level, (b) 8-phase-level.

Fig. 2
Fig. 2

Function h n ( M , α ) for the case of (a) M = 2 , (b) M = 4 .

Fig. 3
Fig. 3

Local structures at the three boundaries between (a) region 1 and region 2, (b) region 2 and region 3, (c) region 3 and region 4. For simplicity, we do not draw the additional phase Δ M .

Fig. 4
Fig. 4

Calculated intensity distribution (solid curve) and the desired diffraction-limited Gaussian intensity distribution (dashed curve) at the focal region. (a) Linear plots, (b) logarithmic plots.

Fig. 5
Fig. 5

Schematic view of the relation between t n i m and S n i m . For simplicity, we show only the case of 4-phase-level. The upper part is for a certain α value, the lower part is for the traditional layout.

Fig. 6
Fig. 6

Schematic view of the traditional layout for the case of (a) 4-phase-level, (b) 8-phase-level.

Tables (1)

Tables Icon

Table 1 Related Parameters in the Four Regions

Equations (31)

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

C n i = S n λ f n + ( i 0.5 ) D n L ,
α + β = 2 M .
α = β = 1 M .
U n i m ( R ) 2 f f n i m exp [ j k ( f n i m + R 2 2 f n i m ) + j φ n i m ] sin ( k d n i m 2 f n i m ) J 0 ( k r n i m f n i m R ) ,
U n i m ( R ) 2 f f n i m exp [ j k ( f n i m f ) + j φ n i m ] sin ( k d n i m 2 f n i m ) J 0 ( k r n i m f n i m R ) ,
U n i m ( R ) 2 f f n exp [ j k ( f n i m f ) + j φ n i m ] sin ( k d n i m 2 f n ) J 0 ( k r n f n R ) .
exp [ j k ( f n i m f ) + j φ n i m ] = exp ( j Δ M ) exp ( j k δ n i m ) ,
sin ( k d n i m 2 f n ) = sin γ n i m ,
U n i m ( R ) 2 f f n exp ( j k δ n i m ) sin ( γ n i m ) J 0 ( k r n f n R ) ,
U n ( R ) = i = 1 L m = 2 2 U n i m ( R ) ,
U n ( R ) 2 π f f n h n ( M , α ) J 0 ( k r n f n R ) ,
h n ( M , α ) = M π cos [ π 2 M ( M α 1 ) ] [ sin ( α π 2 ) + sin ( β π 2 ) ] .
U ( R ) = n = 1 N U n ( R ) D n D n = n = 1 N U n ( R ) 2 λ f n D n ,
W ( r n ) = f F h n ( M , α ) f n 2 .
U ( R ) 2 π λ F 0 A r W ( r ) J 0 ( 2 π R λ Q r ) d r ,
U ( R ) 2 π λ F 0 A r W ( r ) J 0 ( 2 π R λ F r ) d r .
Λ t = Λ a + Λ b .
ϕ t = ϕ a + ϕ b ,
ϕ b ϕ a .
exp ( j π ) ϕ b ϕ a ,
exp [ j k ( f n i m f ) + j φ n i m ] = exp [ j k ( g n i m f ) + j φ n i m ] exp ( j k δ n i m ) .
f n i m = f 2 + S n + ( S n i m S n ) = f n + S n i m S n 2 f n ,
g n i m = f 2 + S n + ( t n i m S n ) = f n + t n i m S n 2 f n .
δ n i m = f n i m g n i m = S n i m t n i m 2 f n .
δ n i m = sgn ( m ) M α 1 4 M λ ,
g q = f + ( n 1 ) λ + q 0.5 2 M λ ,
φ q = q 1 M π .
exp [ j k ( g q f ) + j φ q ] = exp ( j Δ M ) ,
U n i ± m ( R ) 4 f f n cos [ π 2 M ( M α 1 ) ] sin ( γ n i m ) J 0 ( k r n f n R ) .
U n ( R ) = i = 1 L m = 1 2 U n i ± m ( R ) .
m = 1 2 U n i ± m ( R ) = 4 f f n cos [ π 2 M ( M α 1 ) ] [ sin ( α π 2 ) + sin ( β π 2 ) ] J 0 ( k r n f n R )

Metrics