Abstract

A three-dimensional (3-D) optical-scanning technique is proposed based on spatial optical phase code activation on an input beam. This code-multiplexed optical scanner (C-MOS) relies on holographically stored 3-D beam-forming information. Proof-of-concept C-MOS experimental results by use of a photorefractive crystal as a holographic medium generates eight beams representing a basic 3-D voxel element generated via a binary-code matrix of the Hadamard type. The experiment demonstrates the C-MOS features of no moving parts, beam-forming flexibility, and large centimeter-size apertures. A novel application of the C-MOS as an optical security lock is highlighted.

© 2003 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. Gottlieb, C. L. M. Ireland, J. M. Ley, “Electro-Optic and Acousto-Optic Scanning and Deflection,” (Marcel Dekker, New York, 1983).
  2. V. J. Fowler, J. Schlafer, “A survey of laser beam deflection techniques,” in Proceedings of IEEE Conference, 54.
  3. Q. W. Song, X. Wang, R. Bussjager, J. Osman, “Electro-optic beam-steering device based on a lanthanum-modified lead zirconate titanate ceramic wafer,” Appl. Opt. 35, 3155–3162 (1996).
    [CrossRef] [PubMed]
  4. N. Asada, “Silicon micromachined two dimensional galvano optical scanner,” IEEE Trans. Magn. 306, 4647–4648 (1994).
    [CrossRef]
  5. N. A. Riza, “MOST: multiplexed optical scanner technology,” in Proceedings of IEEE Annual Meeting of Lasers and Electro-Optic Society, (Institute of Electrical and Electronics Engineers, 2000), 2, pp. 828–829.
  6. N. A. Riza, Y. Huang, “High speed optical scanner for multi-dimensional beam pointing and acquisition,” in Proceedings of IEEE 12th Annual Meeting of Lasers and Electro-Optic Society, (Institute of Electrical and Electronics Engineers, 1999), pp. 184–185.
  7. N. A. Riza, Z. Yaqoob, “Ultra high speed scanner for optical data handling,” in Proceedings of IEEE 13th Annual Meeting of Lasers and Electro-Optic Society, (Institute of Electrical and Electronics Engineers, Puerto Rico, 2000), pp. 822–823.
  8. N. A. Riza, Z. Yaqoob, “Agile optical beam scanners using wavelength and space manipulations,” in Algorithms and Systems for Optical Information Processing V, B. Javidi, D. Psaltis, eds., Proc. SPIE4471, 262–271 (2001).
    [CrossRef]
  9. Z. Yaqoob, N. A. Riza, “Free-space wavelength-multiplexed optical scanner demonstration,” Appl. Opt. 41, 5568–5573 (2002).
    [CrossRef] [PubMed]
  10. Z. Yaqoob, A. A. Rizvi, N. A. Riza, “Free-space wavelength multiplexed optical scanner,” App. Opt. 40, 6425–6438 (2001).
    [CrossRef]
  11. Z. Yaqoob, J. Steedle, N. A. Riza, “Wide angle high speed large aperture optical scanner,” in Proceedings of IEEE 14th Annual Meeting of Lasers and Electro-Optic Society, (Institute of Electrical and Electronics Engineers, San Diego, California, 2001), 2, pp. 616–617.
  12. N. A. Riza, “BOPSCAN Technology: A methodology and implementation of the billion point optical scanner,” in International Optical Design Conference 1998, L. R. Gardner, K. P. Thompson, eds., Proc. SPIE3482, 572–578 (1998).
    [CrossRef]
  13. N. A. Riza, “Digital control polarization-based optical scanner,” U.S. Patent6,031,658, Feb.29, 2000.
  14. N. A. Riza, S. A. Khan, “P-MOS: polarization multiplexed optical scanner,” OSA Annual Meeting, Orlando, USA, 2002.
  15. R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971).
  16. P. K. Rastogi, Holographic Interferometry (Springer Series in Optical Sciences, Vol. 68, Springer, New York, 1994).
    [CrossRef]
  17. P. Günter, Electro-optic and Photorefractive Materials (Springer Proceedings in Physics, Vol. 18, Springer-Verlag, New York1987).
    [CrossRef]
  18. S. A. Benton, “Photographic Holography,” in Optics in Entertainment, C. S. Outwater, ed., Proc. SPIE391, pp. 2–9. (1983).
    [CrossRef]
  19. J. J. Amodei, D. R. Bosomworth, “Hologram storage and retrieval in photochromic strontium titanate crystals,” Appl. Opt. 8, 2473–2477 (1969).
    [CrossRef] [PubMed]
  20. F. H. Mok, M. C. Tackitt, H. M. Stoll, “Storage of 500 high resolution holograms in LiNbO3 crystal,” Opt. Lett. 16, 605–607 (1991).
    [CrossRef] [PubMed]
  21. X. Yang, Z. Gu, “Three dimensional optical data storage and retrieval system based on phase-code and space multiplexing,” Opt. Eng. 35, 452–456 (1996).
    [CrossRef]
  22. H. Kogelink, “Coupled wave theory for thick hologram gratings,” J. Bell Syst. Tech. 48, 2909–2947 (1969).
    [CrossRef]
  23. R. K. Yarlagadda, J. E. Hershey, Hadamard Matrix Analysis and Synthesis With Applications to Communications and Signal/Image Processings (Kluwer Academic, Boston, Mass., 1997).
  24. J. F. Heanue, M. C. Bashaw, L. Hesselink, “Recall of linear combinations of stored data pages based on phase-code multiplexing in volume holography,” Opt. Lett. 19, 1079–1081 (1994).
    [CrossRef] [PubMed]
  25. C. Denz, G. Roosen, G. Pauliat, T. Tschudi, “Potentialities and limitations of hologram multiplexing by using the phase encoding technique,” App. Opt. 31, 5700–5706 (1992).
    [CrossRef]
  26. Z. Wen, Y. Tao, “Orthogonal codes and cross-talk in phase-code multiplexed volume holographic data storage,” Optics Commun. 148, 11–17 (1998).
    [CrossRef]
  27. O. M. Efimov, L. B. Glebov, L. N. Glebova, K. C. Richardson, V. I. Smirnov, “High-efficiency Bragg gratings in photothermorefractive glass,” Appl. Opt. 38, 619–627 (1999).
    [CrossRef]
  28. O. M. Efimov, L. B. Glebov, V. I. Smirnov, “High-frequency Bragg gratings in a photothermorefractive glass,” Opt. Lett. 25, 1693–1695 (2000).
    [CrossRef]
  29. N. A. Riza, “Reconfigurable optical wireless,” in Proceedings of IEEE 12th Annual Meeting of Lasers and Electro-Optic Society, (Institute of Electrical and Electronics Engineers, San Francisco, 1999), 1, pp. 70–71.

2002 (1)

2001 (1)

Z. Yaqoob, A. A. Rizvi, N. A. Riza, “Free-space wavelength multiplexed optical scanner,” App. Opt. 40, 6425–6438 (2001).
[CrossRef]

2000 (1)

1999 (1)

1998 (1)

Z. Wen, Y. Tao, “Orthogonal codes and cross-talk in phase-code multiplexed volume holographic data storage,” Optics Commun. 148, 11–17 (1998).
[CrossRef]

1996 (2)

X. Yang, Z. Gu, “Three dimensional optical data storage and retrieval system based on phase-code and space multiplexing,” Opt. Eng. 35, 452–456 (1996).
[CrossRef]

Q. W. Song, X. Wang, R. Bussjager, J. Osman, “Electro-optic beam-steering device based on a lanthanum-modified lead zirconate titanate ceramic wafer,” Appl. Opt. 35, 3155–3162 (1996).
[CrossRef] [PubMed]

1994 (2)

1992 (1)

C. Denz, G. Roosen, G. Pauliat, T. Tschudi, “Potentialities and limitations of hologram multiplexing by using the phase encoding technique,” App. Opt. 31, 5700–5706 (1992).
[CrossRef]

1991 (1)

1969 (2)

Amodei, J. J.

Asada, N.

N. Asada, “Silicon micromachined two dimensional galvano optical scanner,” IEEE Trans. Magn. 306, 4647–4648 (1994).
[CrossRef]

Bashaw, M. C.

Benton, S. A.

S. A. Benton, “Photographic Holography,” in Optics in Entertainment, C. S. Outwater, ed., Proc. SPIE391, pp. 2–9. (1983).
[CrossRef]

Bosomworth, D. R.

Burckhardt, C. B.

R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971).

Bussjager, R.

Collier, R. J.

R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971).

Denz, C.

C. Denz, G. Roosen, G. Pauliat, T. Tschudi, “Potentialities and limitations of hologram multiplexing by using the phase encoding technique,” App. Opt. 31, 5700–5706 (1992).
[CrossRef]

Efimov, O. M.

Fowler, V. J.

V. J. Fowler, J. Schlafer, “A survey of laser beam deflection techniques,” in Proceedings of IEEE Conference, 54.

Glebov, L. B.

Glebova, L. N.

Gottlieb, M.

M. Gottlieb, C. L. M. Ireland, J. M. Ley, “Electro-Optic and Acousto-Optic Scanning and Deflection,” (Marcel Dekker, New York, 1983).

Gu, Z.

X. Yang, Z. Gu, “Three dimensional optical data storage and retrieval system based on phase-code and space multiplexing,” Opt. Eng. 35, 452–456 (1996).
[CrossRef]

Günter, P.

P. Günter, Electro-optic and Photorefractive Materials (Springer Proceedings in Physics, Vol. 18, Springer-Verlag, New York1987).
[CrossRef]

Heanue, J. F.

Hershey, J. E.

R. K. Yarlagadda, J. E. Hershey, Hadamard Matrix Analysis and Synthesis With Applications to Communications and Signal/Image Processings (Kluwer Academic, Boston, Mass., 1997).

Hesselink, L.

Huang, Y.

N. A. Riza, Y. Huang, “High speed optical scanner for multi-dimensional beam pointing and acquisition,” in Proceedings of IEEE 12th Annual Meeting of Lasers and Electro-Optic Society, (Institute of Electrical and Electronics Engineers, 1999), pp. 184–185.

Ireland, C. L. M.

M. Gottlieb, C. L. M. Ireland, J. M. Ley, “Electro-Optic and Acousto-Optic Scanning and Deflection,” (Marcel Dekker, New York, 1983).

Khan, S. A.

N. A. Riza, S. A. Khan, “P-MOS: polarization multiplexed optical scanner,” OSA Annual Meeting, Orlando, USA, 2002.

Kogelink, H.

H. Kogelink, “Coupled wave theory for thick hologram gratings,” J. Bell Syst. Tech. 48, 2909–2947 (1969).
[CrossRef]

Ley, J. M.

M. Gottlieb, C. L. M. Ireland, J. M. Ley, “Electro-Optic and Acousto-Optic Scanning and Deflection,” (Marcel Dekker, New York, 1983).

Lin, L. H.

R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971).

Mok, F. H.

Osman, J.

Pauliat, G.

C. Denz, G. Roosen, G. Pauliat, T. Tschudi, “Potentialities and limitations of hologram multiplexing by using the phase encoding technique,” App. Opt. 31, 5700–5706 (1992).
[CrossRef]

Rastogi, P. K.

P. K. Rastogi, Holographic Interferometry (Springer Series in Optical Sciences, Vol. 68, Springer, New York, 1994).
[CrossRef]

Richardson, K. C.

Riza, N. A.

Z. Yaqoob, N. A. Riza, “Free-space wavelength-multiplexed optical scanner demonstration,” Appl. Opt. 41, 5568–5573 (2002).
[CrossRef] [PubMed]

Z. Yaqoob, A. A. Rizvi, N. A. Riza, “Free-space wavelength multiplexed optical scanner,” App. Opt. 40, 6425–6438 (2001).
[CrossRef]

N. A. Riza, Z. Yaqoob, “Ultra high speed scanner for optical data handling,” in Proceedings of IEEE 13th Annual Meeting of Lasers and Electro-Optic Society, (Institute of Electrical and Electronics Engineers, Puerto Rico, 2000), pp. 822–823.

N. A. Riza, “MOST: multiplexed optical scanner technology,” in Proceedings of IEEE Annual Meeting of Lasers and Electro-Optic Society, (Institute of Electrical and Electronics Engineers, 2000), 2, pp. 828–829.

N. A. Riza, Y. Huang, “High speed optical scanner for multi-dimensional beam pointing and acquisition,” in Proceedings of IEEE 12th Annual Meeting of Lasers and Electro-Optic Society, (Institute of Electrical and Electronics Engineers, 1999), pp. 184–185.

N. A. Riza, Z. Yaqoob, “Agile optical beam scanners using wavelength and space manipulations,” in Algorithms and Systems for Optical Information Processing V, B. Javidi, D. Psaltis, eds., Proc. SPIE4471, 262–271 (2001).
[CrossRef]

Z. Yaqoob, J. Steedle, N. A. Riza, “Wide angle high speed large aperture optical scanner,” in Proceedings of IEEE 14th Annual Meeting of Lasers and Electro-Optic Society, (Institute of Electrical and Electronics Engineers, San Diego, California, 2001), 2, pp. 616–617.

N. A. Riza, “BOPSCAN Technology: A methodology and implementation of the billion point optical scanner,” in International Optical Design Conference 1998, L. R. Gardner, K. P. Thompson, eds., Proc. SPIE3482, 572–578 (1998).
[CrossRef]

N. A. Riza, “Digital control polarization-based optical scanner,” U.S. Patent6,031,658, Feb.29, 2000.

N. A. Riza, S. A. Khan, “P-MOS: polarization multiplexed optical scanner,” OSA Annual Meeting, Orlando, USA, 2002.

N. A. Riza, “Reconfigurable optical wireless,” in Proceedings of IEEE 12th Annual Meeting of Lasers and Electro-Optic Society, (Institute of Electrical and Electronics Engineers, San Francisco, 1999), 1, pp. 70–71.

Rizvi, A. A.

Z. Yaqoob, A. A. Rizvi, N. A. Riza, “Free-space wavelength multiplexed optical scanner,” App. Opt. 40, 6425–6438 (2001).
[CrossRef]

Roosen, G.

C. Denz, G. Roosen, G. Pauliat, T. Tschudi, “Potentialities and limitations of hologram multiplexing by using the phase encoding technique,” App. Opt. 31, 5700–5706 (1992).
[CrossRef]

Schlafer, J.

V. J. Fowler, J. Schlafer, “A survey of laser beam deflection techniques,” in Proceedings of IEEE Conference, 54.

Smirnov, V. I.

Song, Q. W.

Steedle, J.

Z. Yaqoob, J. Steedle, N. A. Riza, “Wide angle high speed large aperture optical scanner,” in Proceedings of IEEE 14th Annual Meeting of Lasers and Electro-Optic Society, (Institute of Electrical and Electronics Engineers, San Diego, California, 2001), 2, pp. 616–617.

Stoll, H. M.

Tackitt, M. C.

Tao, Y.

Z. Wen, Y. Tao, “Orthogonal codes and cross-talk in phase-code multiplexed volume holographic data storage,” Optics Commun. 148, 11–17 (1998).
[CrossRef]

Tschudi, T.

C. Denz, G. Roosen, G. Pauliat, T. Tschudi, “Potentialities and limitations of hologram multiplexing by using the phase encoding technique,” App. Opt. 31, 5700–5706 (1992).
[CrossRef]

Wang, X.

Wen, Z.

Z. Wen, Y. Tao, “Orthogonal codes and cross-talk in phase-code multiplexed volume holographic data storage,” Optics Commun. 148, 11–17 (1998).
[CrossRef]

Yang, X.

X. Yang, Z. Gu, “Three dimensional optical data storage and retrieval system based on phase-code and space multiplexing,” Opt. Eng. 35, 452–456 (1996).
[CrossRef]

Yaqoob, Z.

Z. Yaqoob, N. A. Riza, “Free-space wavelength-multiplexed optical scanner demonstration,” Appl. Opt. 41, 5568–5573 (2002).
[CrossRef] [PubMed]

Z. Yaqoob, A. A. Rizvi, N. A. Riza, “Free-space wavelength multiplexed optical scanner,” App. Opt. 40, 6425–6438 (2001).
[CrossRef]

N. A. Riza, Z. Yaqoob, “Agile optical beam scanners using wavelength and space manipulations,” in Algorithms and Systems for Optical Information Processing V, B. Javidi, D. Psaltis, eds., Proc. SPIE4471, 262–271 (2001).
[CrossRef]

N. A. Riza, Z. Yaqoob, “Ultra high speed scanner for optical data handling,” in Proceedings of IEEE 13th Annual Meeting of Lasers and Electro-Optic Society, (Institute of Electrical and Electronics Engineers, Puerto Rico, 2000), pp. 822–823.

Z. Yaqoob, J. Steedle, N. A. Riza, “Wide angle high speed large aperture optical scanner,” in Proceedings of IEEE 14th Annual Meeting of Lasers and Electro-Optic Society, (Institute of Electrical and Electronics Engineers, San Diego, California, 2001), 2, pp. 616–617.

Yarlagadda, R. K.

R. K. Yarlagadda, J. E. Hershey, Hadamard Matrix Analysis and Synthesis With Applications to Communications and Signal/Image Processings (Kluwer Academic, Boston, Mass., 1997).

App. Opt. (2)

Z. Yaqoob, A. A. Rizvi, N. A. Riza, “Free-space wavelength multiplexed optical scanner,” App. Opt. 40, 6425–6438 (2001).
[CrossRef]

C. Denz, G. Roosen, G. Pauliat, T. Tschudi, “Potentialities and limitations of hologram multiplexing by using the phase encoding technique,” App. Opt. 31, 5700–5706 (1992).
[CrossRef]

Appl. Opt. (4)

IEEE Trans. Magn. (1)

N. Asada, “Silicon micromachined two dimensional galvano optical scanner,” IEEE Trans. Magn. 306, 4647–4648 (1994).
[CrossRef]

J. Bell Syst. Tech. (1)

H. Kogelink, “Coupled wave theory for thick hologram gratings,” J. Bell Syst. Tech. 48, 2909–2947 (1969).
[CrossRef]

Opt. Eng. (1)

X. Yang, Z. Gu, “Three dimensional optical data storage and retrieval system based on phase-code and space multiplexing,” Opt. Eng. 35, 452–456 (1996).
[CrossRef]

Opt. Lett. (3)

Optics Commun. (1)

Z. Wen, Y. Tao, “Orthogonal codes and cross-talk in phase-code multiplexed volume holographic data storage,” Optics Commun. 148, 11–17 (1998).
[CrossRef]

Other (16)

M. Gottlieb, C. L. M. Ireland, J. M. Ley, “Electro-Optic and Acousto-Optic Scanning and Deflection,” (Marcel Dekker, New York, 1983).

V. J. Fowler, J. Schlafer, “A survey of laser beam deflection techniques,” in Proceedings of IEEE Conference, 54.

N. A. Riza, “Reconfigurable optical wireless,” in Proceedings of IEEE 12th Annual Meeting of Lasers and Electro-Optic Society, (Institute of Electrical and Electronics Engineers, San Francisco, 1999), 1, pp. 70–71.

N. A. Riza, “MOST: multiplexed optical scanner technology,” in Proceedings of IEEE Annual Meeting of Lasers and Electro-Optic Society, (Institute of Electrical and Electronics Engineers, 2000), 2, pp. 828–829.

N. A. Riza, Y. Huang, “High speed optical scanner for multi-dimensional beam pointing and acquisition,” in Proceedings of IEEE 12th Annual Meeting of Lasers and Electro-Optic Society, (Institute of Electrical and Electronics Engineers, 1999), pp. 184–185.

N. A. Riza, Z. Yaqoob, “Ultra high speed scanner for optical data handling,” in Proceedings of IEEE 13th Annual Meeting of Lasers and Electro-Optic Society, (Institute of Electrical and Electronics Engineers, Puerto Rico, 2000), pp. 822–823.

N. A. Riza, Z. Yaqoob, “Agile optical beam scanners using wavelength and space manipulations,” in Algorithms and Systems for Optical Information Processing V, B. Javidi, D. Psaltis, eds., Proc. SPIE4471, 262–271 (2001).
[CrossRef]

Z. Yaqoob, J. Steedle, N. A. Riza, “Wide angle high speed large aperture optical scanner,” in Proceedings of IEEE 14th Annual Meeting of Lasers and Electro-Optic Society, (Institute of Electrical and Electronics Engineers, San Diego, California, 2001), 2, pp. 616–617.

N. A. Riza, “BOPSCAN Technology: A methodology and implementation of the billion point optical scanner,” in International Optical Design Conference 1998, L. R. Gardner, K. P. Thompson, eds., Proc. SPIE3482, 572–578 (1998).
[CrossRef]

N. A. Riza, “Digital control polarization-based optical scanner,” U.S. Patent6,031,658, Feb.29, 2000.

N. A. Riza, S. A. Khan, “P-MOS: polarization multiplexed optical scanner,” OSA Annual Meeting, Orlando, USA, 2002.

R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971).

P. K. Rastogi, Holographic Interferometry (Springer Series in Optical Sciences, Vol. 68, Springer, New York, 1994).
[CrossRef]

P. Günter, Electro-optic and Photorefractive Materials (Springer Proceedings in Physics, Vol. 18, Springer-Verlag, New York1987).
[CrossRef]

S. A. Benton, “Photographic Holography,” in Optics in Entertainment, C. S. Outwater, ed., Proc. SPIE391, pp. 2–9. (1983).
[CrossRef]

R. K. Yarlagadda, J. E. Hershey, Hadamard Matrix Analysis and Synthesis With Applications to Communications and Signal/Image Processings (Kluwer Academic, Boston, Mass., 1997).

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 (7)

Fig. 1
Fig. 1

C-MOS creation and operation: (a) Shows the hologram recording with changing optical phase codes in a transmissive assembly, (b) shows a hologram reading for a C-MOS attained in transmissive assembly.

Fig. 2
Fig. 2

C-MOS creation and operation: (a) shows the hologram recording with changing optical phase codes in a 90° assembly, (b) shows a hologram reading for a C-MOS attained in 90° assembly.

Fig. 3
Fig. 3

Experimental setup for C-MOS demonstration by use of orthogonal phase codes and photorefractive crystal as the holographic material for 3-D beam scanning.

Fig. 4
Fig. 4

Geometry to calculate the maximum scan angle in the x direction for C-MOS in 90° assembly. The y direction has the same scan angle because of the same dimensions in x and y directions. Here, angle AOC represents the maximum scan angle in one direction. The total scan angle is 2 × Angle AOC = Angle AOB.

Fig. 5
Fig. 5

Generation of a voxel scan element in 3-D space. The circles represent the location of the generated beams in a cube volume element. Xo, Yo, and Zo are the scanning dimensions in the x, y, and z directions, respectively. θx and θy are corresponding scan angles in x and y directions.

Fig. 6
Fig. 6

Eight 3-D beams generated from the proposed experimental C-MOS voxel: (a)–(d) beams are on the front face of the voxel, while (e)–(h) beams (appearing more focused) are at the back plane of the voxel.

Fig. 7
Fig. 7

Proposed C-MOS-based optical security lock system.

Tables (1)

Tables Icon

Table 1 Hadamard Matrix of order sixteen where each row is orthogonal to all other rows representing a phase code for code multiplexing. Here each row corresponds to a phase code. Dark boxes represents a phase of 0°, while white boxes represents a phase shift of 180°. The first eight rows have been used for forming a voxel through the C-MOS.

Equations (10)

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

Sm=Smx, yexpjkm · r,
P=n=1N Pn expjkn · rexpjϕmn,
Im=|Sm+Pn|2 =Smx, yS*mx, y+n=1N PnP*n+n=1N Smx, yP*mx, yexpjk-kn · rexp-jϕnm+n=1N Pnl=1,lnN P*l expjkn-kl · r×expjϕnm-ϕlm.
Rpx, y=l=1N Pl expjk · rexpjϕlm ×m=1mn=1N Smx, yP*mx, yexpjk-kn · rexp-jϕnm.
Rpx, y=m=1mn=1N Smx, yPmP*mx, y ×expjkn · rexpjϕnp-ϕnm.
n=1Nexpjϕnp-ϕnm=0,  for pm,
n=1Nexpjϕnp-ϕnm=N,  for p=m.
θimax=tan-1x/2-y · tanθrd,
1f=1f3+1f4-tf3×f4,
BFD=f-t×ff3.

Metrics