Abstract

For one-dimensional binary-phase [(0, π) and (0, non-π)] fanout elements and for one-dimensional continuous or multilevel quantized phase fanout elements, an upper bound on diffraction efficiency is presented for fanouts ranging from 2 to 25. The upper bound is determined by optimizing with respect to the array phase the upper bound on diffraction efficiency for a coherent array. To determine the upper bound for binary-phase gratings, restrictions on the array phase are imposed. For fanouts that are >5, the upper bound on the diffraction efficiency for continuous phase fanouts ranges between 97 and 98%; for (0, π)-binary-phase fanouts the upper bound ranges between 83 and 84%; and for (0, non-π)-binary-phase, between 87 and 88%.

© 1992 Optical Society of America

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  1. N. Streibl, K.-H. Brenner, A. Huang, J. Jahns, J. Jewell, A. W. Lohmann, D. A. B. Miller, M. Murdocca, M. Prise, T. Sizer, “Digital optics,” Proc. IEEE 77, 1954–1969 (1989).
    [CrossRef]
  2. M. R. Feldman, C. C. Guest, “High efficiency hologram encoding for generation of spot arrays,” Opt. Lett. 14, 479–481 (1989).
    [CrossRef] [PubMed]
  3. J. Turunen, A. Vasara, J. Westerholm, “Kinoform phase relief synthesis: a stochastic method,” Opt. Eng. 28, 1162–1167 (1989).
    [CrossRef]
  4. J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).
    [CrossRef]
  5. F. Wyrowski, “Coding and quantization techniques in digital phase holography,” in Holographic Optics II: Principles and Applications, G. M. Morris, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1136, 215–219 (1989).
  6. H. P. Herzig, D. Prongué, R. Dändliker, “Design and fabrication of highly efficient fan-out elements,” Jpn. J. Appl. Phys. 29, L1307–L1309 (1990).
    [CrossRef]
  7. J. N. Mait, “Extensions to Dammann’s method of binary-phase grating design,” in Holographic Optics: Optically and Computer Generated, I. N. Cindrich, S. H. Lee, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1052, 41–46 (1989).
  8. A. Vasara, J. Turunen, J. Westerholm, M. R. Taghizadeh, “Stepped-phase kinoforms,” in Optics in Complex Systems, F. Lanzl, H.-J. Preuss, J. Weigelt, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1319, 298–299 (1990).
  9. F. Wyrowski, “Characteristics of diffractive optical elements/digital holograms,” in Computer and Optically Formed Holographic Optics, I. N. Cindrich, S. H. Lee, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1211, 2–10 (1990).
  10. R. L. Morrison, “Symmetries that simplify design of spot-array phase gratings,” in Annual Meeting Technical Digest, Vol. 15 of OSA 1990 Technical Digest Series (Optical Society of America, Washington, D.C., 1990), p. 86.
  11. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1986).
  12. A. Vasara, “Array generation with periodic Fourier-transform holograms,” licentiate thesis (Helsinki University of Technology, Espoo, Finland, 1989).
  13. H. Dammann, K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
    [CrossRef]
  14. U. Krackhardt, N. Streibl, “Design of Dammann-gratings for array generation,” Opt. Commun. 74, 31–34 (1989).
    [CrossRef]
  15. U. Killat, G. Rabe, W. Rave, “Binary phase gratings for star couplers with high splitting ratio,” Fiber Integr. Opt. 4, 159–167 (1982).
    [CrossRef]
  16. J. N. Mait, “Design of binary-phase and multiphase Fourier gratings for array generation,” J. Opt. Soc. Am. A 7, 1514–1528 (1990).
    [CrossRef]

1990 (2)

H. P. Herzig, D. Prongué, R. Dändliker, “Design and fabrication of highly efficient fan-out elements,” Jpn. J. Appl. Phys. 29, L1307–L1309 (1990).
[CrossRef]

J. N. Mait, “Design of binary-phase and multiphase Fourier gratings for array generation,” J. Opt. Soc. Am. A 7, 1514–1528 (1990).
[CrossRef]

1989 (5)

M. R. Feldman, C. C. Guest, “High efficiency hologram encoding for generation of spot arrays,” Opt. Lett. 14, 479–481 (1989).
[CrossRef] [PubMed]

U. Krackhardt, N. Streibl, “Design of Dammann-gratings for array generation,” Opt. Commun. 74, 31–34 (1989).
[CrossRef]

J. Turunen, A. Vasara, J. Westerholm, “Kinoform phase relief synthesis: a stochastic method,” Opt. Eng. 28, 1162–1167 (1989).
[CrossRef]

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).
[CrossRef]

N. Streibl, K.-H. Brenner, A. Huang, J. Jahns, J. Jewell, A. W. Lohmann, D. A. B. Miller, M. Murdocca, M. Prise, T. Sizer, “Digital optics,” Proc. IEEE 77, 1954–1969 (1989).
[CrossRef]

1982 (1)

U. Killat, G. Rabe, W. Rave, “Binary phase gratings for star couplers with high splitting ratio,” Fiber Integr. Opt. 4, 159–167 (1982).
[CrossRef]

1971 (1)

H. Dammann, K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
[CrossRef]

Brenner, K.-H.

N. Streibl, K.-H. Brenner, A. Huang, J. Jahns, J. Jewell, A. W. Lohmann, D. A. B. Miller, M. Murdocca, M. Prise, T. Sizer, “Digital optics,” Proc. IEEE 77, 1954–1969 (1989).
[CrossRef]

Dammann, H.

H. Dammann, K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
[CrossRef]

Dändliker, R.

H. P. Herzig, D. Prongué, R. Dändliker, “Design and fabrication of highly efficient fan-out elements,” Jpn. J. Appl. Phys. 29, L1307–L1309 (1990).
[CrossRef]

Downs, M. M.

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).
[CrossRef]

Feldman, M. R.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1986).

Görtler, K.

H. Dammann, K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
[CrossRef]

Guest, C. C.

Herzig, H. P.

H. P. Herzig, D. Prongué, R. Dändliker, “Design and fabrication of highly efficient fan-out elements,” Jpn. J. Appl. Phys. 29, L1307–L1309 (1990).
[CrossRef]

Huang, A.

N. Streibl, K.-H. Brenner, A. Huang, J. Jahns, J. Jewell, A. W. Lohmann, D. A. B. Miller, M. Murdocca, M. Prise, T. Sizer, “Digital optics,” Proc. IEEE 77, 1954–1969 (1989).
[CrossRef]

Jahns, J.

N. Streibl, K.-H. Brenner, A. Huang, J. Jahns, J. Jewell, A. W. Lohmann, D. A. B. Miller, M. Murdocca, M. Prise, T. Sizer, “Digital optics,” Proc. IEEE 77, 1954–1969 (1989).
[CrossRef]

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).
[CrossRef]

Jewell, J.

N. Streibl, K.-H. Brenner, A. Huang, J. Jahns, J. Jewell, A. W. Lohmann, D. A. B. Miller, M. Murdocca, M. Prise, T. Sizer, “Digital optics,” Proc. IEEE 77, 1954–1969 (1989).
[CrossRef]

Killat, U.

U. Killat, G. Rabe, W. Rave, “Binary phase gratings for star couplers with high splitting ratio,” Fiber Integr. Opt. 4, 159–167 (1982).
[CrossRef]

Krackhardt, U.

U. Krackhardt, N. Streibl, “Design of Dammann-gratings for array generation,” Opt. Commun. 74, 31–34 (1989).
[CrossRef]

Lohmann, A. W.

N. Streibl, K.-H. Brenner, A. Huang, J. Jahns, J. Jewell, A. W. Lohmann, D. A. B. Miller, M. Murdocca, M. Prise, T. Sizer, “Digital optics,” Proc. IEEE 77, 1954–1969 (1989).
[CrossRef]

Mait, J. N.

J. N. Mait, “Design of binary-phase and multiphase Fourier gratings for array generation,” J. Opt. Soc. Am. A 7, 1514–1528 (1990).
[CrossRef]

J. N. Mait, “Extensions to Dammann’s method of binary-phase grating design,” in Holographic Optics: Optically and Computer Generated, I. N. Cindrich, S. H. Lee, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1052, 41–46 (1989).

Miller, D. A. B.

N. Streibl, K.-H. Brenner, A. Huang, J. Jahns, J. Jewell, A. W. Lohmann, D. A. B. Miller, M. Murdocca, M. Prise, T. Sizer, “Digital optics,” Proc. IEEE 77, 1954–1969 (1989).
[CrossRef]

Morrison, R. L.

R. L. Morrison, “Symmetries that simplify design of spot-array phase gratings,” in Annual Meeting Technical Digest, Vol. 15 of OSA 1990 Technical Digest Series (Optical Society of America, Washington, D.C., 1990), p. 86.

Murdocca, M.

N. Streibl, K.-H. Brenner, A. Huang, J. Jahns, J. Jewell, A. W. Lohmann, D. A. B. Miller, M. Murdocca, M. Prise, T. Sizer, “Digital optics,” Proc. IEEE 77, 1954–1969 (1989).
[CrossRef]

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1986).

Prise, M.

N. Streibl, K.-H. Brenner, A. Huang, J. Jahns, J. Jewell, A. W. Lohmann, D. A. B. Miller, M. Murdocca, M. Prise, T. Sizer, “Digital optics,” Proc. IEEE 77, 1954–1969 (1989).
[CrossRef]

Prise, M. E.

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).
[CrossRef]

Prongué, D.

H. P. Herzig, D. Prongué, R. Dändliker, “Design and fabrication of highly efficient fan-out elements,” Jpn. J. Appl. Phys. 29, L1307–L1309 (1990).
[CrossRef]

Rabe, G.

U. Killat, G. Rabe, W. Rave, “Binary phase gratings for star couplers with high splitting ratio,” Fiber Integr. Opt. 4, 159–167 (1982).
[CrossRef]

Rave, W.

U. Killat, G. Rabe, W. Rave, “Binary phase gratings for star couplers with high splitting ratio,” Fiber Integr. Opt. 4, 159–167 (1982).
[CrossRef]

Sizer, T.

N. Streibl, K.-H. Brenner, A. Huang, J. Jahns, J. Jewell, A. W. Lohmann, D. A. B. Miller, M. Murdocca, M. Prise, T. Sizer, “Digital optics,” Proc. IEEE 77, 1954–1969 (1989).
[CrossRef]

Streibl, N.

N. Streibl, K.-H. Brenner, A. Huang, J. Jahns, J. Jewell, A. W. Lohmann, D. A. B. Miller, M. Murdocca, M. Prise, T. Sizer, “Digital optics,” Proc. IEEE 77, 1954–1969 (1989).
[CrossRef]

U. Krackhardt, N. Streibl, “Design of Dammann-gratings for array generation,” Opt. Commun. 74, 31–34 (1989).
[CrossRef]

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).
[CrossRef]

Taghizadeh, M. R.

A. Vasara, J. Turunen, J. Westerholm, M. R. Taghizadeh, “Stepped-phase kinoforms,” in Optics in Complex Systems, F. Lanzl, H.-J. Preuss, J. Weigelt, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1319, 298–299 (1990).

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1986).

Turunen, J.

J. Turunen, A. Vasara, J. Westerholm, “Kinoform phase relief synthesis: a stochastic method,” Opt. Eng. 28, 1162–1167 (1989).
[CrossRef]

A. Vasara, J. Turunen, J. Westerholm, M. R. Taghizadeh, “Stepped-phase kinoforms,” in Optics in Complex Systems, F. Lanzl, H.-J. Preuss, J. Weigelt, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1319, 298–299 (1990).

Vasara, A.

J. Turunen, A. Vasara, J. Westerholm, “Kinoform phase relief synthesis: a stochastic method,” Opt. Eng. 28, 1162–1167 (1989).
[CrossRef]

A. Vasara, J. Turunen, J. Westerholm, M. R. Taghizadeh, “Stepped-phase kinoforms,” in Optics in Complex Systems, F. Lanzl, H.-J. Preuss, J. Weigelt, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1319, 298–299 (1990).

A. Vasara, “Array generation with periodic Fourier-transform holograms,” licentiate thesis (Helsinki University of Technology, Espoo, Finland, 1989).

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1986).

Walker, S. J.

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).
[CrossRef]

Westerholm, J.

J. Turunen, A. Vasara, J. Westerholm, “Kinoform phase relief synthesis: a stochastic method,” Opt. Eng. 28, 1162–1167 (1989).
[CrossRef]

A. Vasara, J. Turunen, J. Westerholm, M. R. Taghizadeh, “Stepped-phase kinoforms,” in Optics in Complex Systems, F. Lanzl, H.-J. Preuss, J. Weigelt, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1319, 298–299 (1990).

Wyrowski, F.

F. Wyrowski, “Coding and quantization techniques in digital phase holography,” in Holographic Optics II: Principles and Applications, G. M. Morris, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1136, 215–219 (1989).

F. Wyrowski, “Characteristics of diffractive optical elements/digital holograms,” in Computer and Optically Formed Holographic Optics, I. N. Cindrich, S. H. Lee, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1211, 2–10 (1990).

Fiber Integr. Opt. (1)

U. Killat, G. Rabe, W. Rave, “Binary phase gratings for star couplers with high splitting ratio,” Fiber Integr. Opt. 4, 159–167 (1982).
[CrossRef]

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

Jpn. J. Appl. Phys. (1)

H. P. Herzig, D. Prongué, R. Dändliker, “Design and fabrication of highly efficient fan-out elements,” Jpn. J. Appl. Phys. 29, L1307–L1309 (1990).
[CrossRef]

Opt. Commun. (2)

H. Dammann, K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
[CrossRef]

U. Krackhardt, N. Streibl, “Design of Dammann-gratings for array generation,” Opt. Commun. 74, 31–34 (1989).
[CrossRef]

Opt. Eng. (2)

J. Turunen, A. Vasara, J. Westerholm, “Kinoform phase relief synthesis: a stochastic method,” Opt. Eng. 28, 1162–1167 (1989).
[CrossRef]

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).
[CrossRef]

Opt. Lett. (1)

Proc. IEEE (1)

N. Streibl, K.-H. Brenner, A. Huang, J. Jahns, J. Jewell, A. W. Lohmann, D. A. B. Miller, M. Murdocca, M. Prise, T. Sizer, “Digital optics,” Proc. IEEE 77, 1954–1969 (1989).
[CrossRef]

Other (7)

F. Wyrowski, “Coding and quantization techniques in digital phase holography,” in Holographic Optics II: Principles and Applications, G. M. Morris, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1136, 215–219 (1989).

J. N. Mait, “Extensions to Dammann’s method of binary-phase grating design,” in Holographic Optics: Optically and Computer Generated, I. N. Cindrich, S. H. Lee, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1052, 41–46 (1989).

A. Vasara, J. Turunen, J. Westerholm, M. R. Taghizadeh, “Stepped-phase kinoforms,” in Optics in Complex Systems, F. Lanzl, H.-J. Preuss, J. Weigelt, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1319, 298–299 (1990).

F. Wyrowski, “Characteristics of diffractive optical elements/digital holograms,” in Computer and Optically Formed Holographic Optics, I. N. Cindrich, S. H. Lee, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1211, 2–10 (1990).

R. L. Morrison, “Symmetries that simplify design of spot-array phase gratings,” in Annual Meeting Technical Digest, Vol. 15 of OSA 1990 Technical Digest Series (Optical Society of America, Washington, D.C., 1990), p. 86.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1986).

A. Vasara, “Array generation with periodic Fourier-transform holograms,” licentiate thesis (Helsinki University of Technology, Espoo, Finland, 1989).

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Figures (4)

Fig. 1
Fig. 1

Representation of (a) grating phase ϕ(u) and (b) response p(x) for an odd numbered spot array.

Fig. 2
Fig. 2

Representation of (a) grating phase ϕ(u) and (b) response p(x) for an even numbered spot array.

Fig. 3
Fig. 3

Calculated upper bound on diffraction efficiency for fanout elements assuming (a) continuous-phase gratings, (b) (0, non-π)-binary-phase gratings, and (c) (0, π)-binary-phase gratings. Circles indicate even numbered fanouts, and triangles indicate odd numbered fanouts.

Fig. 4
Fig. 4

Spot-array locations of free phases that are available to increase the diffraction efficiency for separable arrays assuming (a) continuous-phase gratings, (b) (0, non-π)-binary-phase gratings, (c) (0, π)-binary-phase gratings and nonseparable arrays, (d)–(f) similar phase structures. Open circles indicate phases that can be arbitrarily set to zero by removing constant phases and linear phase shifts.

Tables (5)

Tables Icon

Table I Array-Phase Values Assuming a (0, π) Binary-Phase Grating

Tables Icon

Table II Array-Phase Values Assuming a (0, non-π) Binary-Phase Grating

Tables Icon

Table III Array-Phase Values Assuming a Continuous-Phase Grating

Tables Icon

Table IV Diffraction Efficiency and Uniformity (in Decibels) for (0, π)-Binary Gratings a

Tables Icon

Table V Diffraction Efficiency and Uniformity (in Decibels) for (0, non-π)-Binary Gratings

Equations (36)

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P ( u ) = exp [ j ϕ ( u ) ] .
p ( x ) = α q ( x ) , x X ,
η = x X | p ( x ) | 2 d x ,
η [ | Q ( u ) | d u ] 2 | Q ( u ) | 2 d u .
P o ( u ) = P ˜ ( u ) * comb ( u ) ,
p o ( x ) = p ˜ ( x ) comb ( x ) = n = N N p ˜ ( n ) δ ( x n ) ,
p ˜ ( x ) = 1 / 2 1 / 2 P ˜ ( u ) exp ( j 2 π x u ) d u
q o ( x ) = n = N N a o , n exp ( j θ o , n ) δ ( x n ) ,
Q o ( u ) = n = N N a o , n exp ( j θ o , n ) exp ( j 2 π u n ) .
P e ( u ) = P ˜ ( u ) * { ( 1 / 2 ) comb ( u / 2 ) ( 1 / 2 ) comb [ ( u 1 ) / 2 ] } ,
p e ( x ) = 2 j sin π x exp ( j π x ) p ˜ ( x ) comb ( 2 x ) = p ˜ ( x ) comp ( x 1 / 2 ) = n = p ˜ ( n + 1 / 2 ) δ ( x n 1 / 2 ) .
q e ( x ) = n = N N 1 a e . n exp ( j θ e , n ) δ ( x n 1 / 2 ) ;
Q e ( u ) = n = N N 1 a e , n exp ( j θ e . n ) exp [ j 2 π u ( n + 1 / 2 ) ] = exp ( j π u ) n = N N 1 a e , n exp ( j θ e , n ) exp ( j 2 π u n ) ,
P ˜ ( u ) = rect ( u ) + [ exp ( j ϕ ) 1 ] Σ ( u ) ,
p o , ϕ ( x ) = n = { sinc ( n ) + [ exp ( j ϕ ) 1 ] σ ( n ) } δ ( x n ) .
p ˜ o , π ( n ) = 2 σ ( n )
p ˜ o , π ( 0 ) = 1 2 σ ( 0 ) .
σ ( 0 ) = 1 / 2 1 / 2 Σ ( u ) d u .
q o , π ( x ) = a o , 0 δ ( x ) + n = 1 N a o , n [ exp ( j θ o , n ) δ ( x n ) + exp ( j θ o , n ) δ ( x + n ) ] ;
Q o , π ( u ) = a o , 0 + 2 n = 1 N a o . n cos ( 2 π n u θ o , n ) ;
p ˜ o , ϕ ( n ) = [ exp ( j ϕ ) 1 ] σ ( n ) = 2 j sin ϕ / 2 exp ( j ϕ / 2 ) σ ( n ) ,
p ˜ o , ϕ ( 0 ) = 1 + [ exp ( j ϕ ) 1 ] σ ( 0 ) = [ 1 σ ( 0 ) ] + exp ( j ϕ ) σ ( 0 ) = { 1 2 σ ( 0 ) [ 1 σ ( 0 ) ] ( 1 cos ϕ ) } 1 / 2 exp ( j ψ ) ,
ψ = tan 1 [ σ ( 0 ) sin ϕ 1 σ ( 0 ) ( 1 cos ϕ ) ] .
q o , ϕ ( x ) = j exp ( j ϕ / 2 ) { a o , 0 exp ( j θ o , 0 ) δ ( x ) + n = 1 N a o , n [ exp ( j θ o , n ) δ ( x n ) + exp ( j θ o , n ) δ ( x + n ) ] } ,
Q o , ϕ ( u ) = j exp ( j ϕ / 2 ) [ a o , 0 exp ( j θ o , 0 ) + 2 n = 1 N a o , n cos ( 2 π u n θ o , n ) ] ,
θ o , 0 = ψ ϕ / 2 π / 2 .
P e , ϕ ( u ) = { rect ( u ) + [ exp ( j ϕ ) 1 ] Σ ( u ) } * ( 1 / 2 ) comb ( u / 2 ) + { exp ( j ϕ ) rect ( u ) + [ 1 exp ( j ϕ ) ] Σ ( u ) } * ( 1 / 2 ) comb [ ( u 1 ) / 2 ] ;
p e , ϕ ( x ) = ( sinc ( x ) { 1 + exp [ j ( 2 π x + ϕ ) ] } + [ exp ( j ϕ ) 1 ] [ 1 exp ( j 2 π x ) ] σ ( x ) ) × comb ( 2 x ) = { [ 1 exp ( j ϕ ) ] / 2 } n = [ sinc ( n + 1 / 2 ) 2 σ ( n + 1 / 2 ) ] δ ( x n 1 / 2 ) .
q e . ϕ ( x ) = n = 0 N 1 a e , n [ exp ( j θ e , n ) δ ( x n 1 / 2 ) + exp ( j θ e , n ) δ ( x + n + 1 / 2 ) ] ;
Q e , ϕ ( u ) = 2 n = 0 N 1 a e , n cos [ 2 π u ( n + 1 / 2 ) θ e . n ] ,
U = 10 log I max + I min I max I min ,
P o ( u , υ ) = P ˜ ( u , υ ) * * comb ( u , υ ) ;
p o ( x , y ) = p ˜ ( x , y ) comb ( x , y ) = m = n = p ˜ ( n , m ) δ ( x n , y m ) .
q o ( x , y ) = m = M M n = N N a n , m exp ( j θ n , m ) δ ( x n , y m ) ,
Q o ( u , υ ) = m = M M n = N N a n , m exp ( j θ n , m ) exp [ j 2 π ( n u + m υ ) ] .
Q o ( u , υ ) = m = M M n = N N a n a m exp [ j ( θ n + θ m ) ] exp [ j 2 π ( n u + m υ ) ] ,

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