Abstract

Complex-amplitude modulation is necessary if we wish to suppress the noise that surrounds the signal in diffractive optics. This is desirable in, e.g., space-invariant optical interconnection. We present a mathematical scheme to synthesize high-carrier-frequency diffractive elements that perform phase and amplitude modulation of the first carrier-grating order without the use of absorption. The amplitude information is encoded in grating-depth variations and in the phase information in pulse-position modulation. Use of rigorous diffraction theory in the design of the local grating structure ensures virtually noise-free reconstruction with efficiencies that are significantly higher than the values one typically obtains, e.g., by conventional holographic recording.

© 1996 Optical Society of America

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References

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  1. F. Wyrowski, O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
    [CrossRef]
  2. M. R. Taghizadeh, J. Turunen, “Synthetic diffractive elements for optical interconnection,” Opt. Comput. Process. 2, 221–242 (1992).
  3. F. Wyrowski, “Consideration on convolutions and phase factors,” Opt. Commun. 81, 353–358 (1991).
    [CrossRef]
  4. D. C. Chu, J. R. Fienup, J. W. Goodman, “Multi-emulsion on-axis computer-generated hologram,” Appl. Opt. 12, 1386–1388 (1973).
    [CrossRef] [PubMed]
  5. A. W. Lohmann, D. B. Paris, “Binary Fraunhofer holograms generated by computer,” Appl. Opt. 6, 1739–1748 (1967).
    [CrossRef] [PubMed]
  6. J. P. Kirk, A. L. Jones, “Phase-only complex-valued spatial filter,”J. Opt. Soc. Am. 61, 1023–1028 (1971).
    [CrossRef]
  7. E. V. Jull, J. W. Heath, G. R. Ebbeson, “Gratings that diffract all incident energy,”J. Opt. Soc. Am. 67, 557–560 (1977).
    [CrossRef]
  8. E. G. Loewen, M. Nevière, D. Maystre, “Efficiency optimization of rectangular groove gratings for use in the visible and IR regions (TE),” Appl. Opt. 18, 2262–2266 (1979).
    [CrossRef] [PubMed]
  9. J. Turunen, P. Blair, J. M. Miller, M. R. Taghizadeh, E. Noponen, “Bragg holograms with binary surface-relief profile,” Opt. Lett. 18, 1022–1024 (1993).
    [CrossRef] [PubMed]
  10. E. Noponen, J. Turunen, “Binary high-frequency-carrier diffractive optical elements: electromagnetic theory,” J. Opt. Soc. Am. A 11, 1097–1109 (1994).
    [CrossRef]
  11. E. Tervonen, J. Turunen, J. Pekola, “Pulse-frequency-modulated high-frequency-carrier diffractive elements for pattern projection,” Opt. Eng. 33, 2579–2587 (1994).
    [CrossRef]
  12. P. Blair, M. R. Taghizadeh, W. Parkes, C. D. W. Wilkinson, “High-efficiency binary fan-out gratings by modulation of a high-frequency carrier grating,” Appl. Opt. 34, 2406–2413 (1995).
    [CrossRef] [PubMed]
  13. P. Ehbets, H. P. Herzig, P. Nussbaum, P. Blattner, R. Dändliker, “Interferometric fabrication of modulated submicron gratings in photoresist,” Appl. Opt. 34, 2540–2547 (1995).
    [CrossRef] [PubMed]
  14. P. Ehbets, H. P. Herzig, M. Kuittinen, F. S. M. Clube, Y. Darbellay, “High-carrier-frequency fan-out gratings fabricated by TIR holographic lithography,” Opt. Eng. 34, 2377–2383 (1995).
    [CrossRef]
  15. F. Wyrowski, “Upper bound of the efficiency of diffractive phase elements,” Opt. Lett. 16, 1915–1917 (1991).
    [CrossRef] [PubMed]
  16. E. Noponen, A. Vasara, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Synthetic diffractive optics in the resonance domain,” J. Opt. Soc. Am. A 9, 1206–1213 (1992).
    [CrossRef]
  17. E. Noponen, J. Turunen, F. Wyrowski, “Synthesis of paraxial-domain diffractive elements by rigorous electromagnetic theory,” J. Opt. Soc. Am. A 12, 1128–1133 (1995).
    [CrossRef]
  18. R. Waldhäusl, P. Dannberg, E. B. Kley, A. Bräuer, W. Karthe, “Highly efficient blazed grating couplers in planar polymer waveguides,” Int. J. Optoelectron. 8, 529–536 (1993).
  19. E. Noponen, J. Turunen, A. Vasara, “Electromagnetic theory and design of diffractive-lens arrays” J. Opt. Soc. Am. A 10, 434–443 (1993).
    [CrossRef]
  20. R. A. Bartolini, “Characteristics of relief phase holograms recorded in photoresist,” Appl. Opt. 13, 129–139 (1974).
    [CrossRef] [PubMed]
  21. H. Bartelt, S. K. Case, “High-efficiency hybrid computer-generated holograms,” Appl. Opt. 21, 2886–2890 (1982).
    [CrossRef] [PubMed]
  22. B. Robertson, J. Turunen, H. Ichikawa, J. M. Miller, M. R. Taghizadeh, A. Vasara, “Hybrid kinoform fanout holograms in dichromated gelatin,” Appl. Opt. 30, 3711–3720 (1991).
    [CrossRef] [PubMed]

1995 (4)

1994 (2)

E. Noponen, J. Turunen, “Binary high-frequency-carrier diffractive optical elements: electromagnetic theory,” J. Opt. Soc. Am. A 11, 1097–1109 (1994).
[CrossRef]

E. Tervonen, J. Turunen, J. Pekola, “Pulse-frequency-modulated high-frequency-carrier diffractive elements for pattern projection,” Opt. Eng. 33, 2579–2587 (1994).
[CrossRef]

1993 (3)

1992 (2)

E. Noponen, A. Vasara, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Synthetic diffractive optics in the resonance domain,” J. Opt. Soc. Am. A 9, 1206–1213 (1992).
[CrossRef]

M. R. Taghizadeh, J. Turunen, “Synthetic diffractive elements for optical interconnection,” Opt. Comput. Process. 2, 221–242 (1992).

1991 (4)

1982 (1)

1979 (1)

1977 (1)

1974 (1)

1973 (1)

1971 (1)

1967 (1)

Bartelt, H.

Bartolini, R. A.

Blair, P.

Blattner, P.

Bräuer, A.

R. Waldhäusl, P. Dannberg, E. B. Kley, A. Bräuer, W. Karthe, “Highly efficient blazed grating couplers in planar polymer waveguides,” Int. J. Optoelectron. 8, 529–536 (1993).

Bryngdahl, O.

F. Wyrowski, O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
[CrossRef]

Case, S. K.

Chu, D. C.

Clube, F. S. M.

P. Ehbets, H. P. Herzig, M. Kuittinen, F. S. M. Clube, Y. Darbellay, “High-carrier-frequency fan-out gratings fabricated by TIR holographic lithography,” Opt. Eng. 34, 2377–2383 (1995).
[CrossRef]

Dändliker, R.

Dannberg, P.

R. Waldhäusl, P. Dannberg, E. B. Kley, A. Bräuer, W. Karthe, “Highly efficient blazed grating couplers in planar polymer waveguides,” Int. J. Optoelectron. 8, 529–536 (1993).

Darbellay, Y.

P. Ehbets, H. P. Herzig, M. Kuittinen, F. S. M. Clube, Y. Darbellay, “High-carrier-frequency fan-out gratings fabricated by TIR holographic lithography,” Opt. Eng. 34, 2377–2383 (1995).
[CrossRef]

Ebbeson, G. R.

Ehbets, P.

P. Ehbets, H. P. Herzig, M. Kuittinen, F. S. M. Clube, Y. Darbellay, “High-carrier-frequency fan-out gratings fabricated by TIR holographic lithography,” Opt. Eng. 34, 2377–2383 (1995).
[CrossRef]

P. Ehbets, H. P. Herzig, P. Nussbaum, P. Blattner, R. Dändliker, “Interferometric fabrication of modulated submicron gratings in photoresist,” Appl. Opt. 34, 2540–2547 (1995).
[CrossRef] [PubMed]

Fienup, J. R.

Goodman, J. W.

Heath, J. W.

Herzig, H. P.

P. Ehbets, H. P. Herzig, M. Kuittinen, F. S. M. Clube, Y. Darbellay, “High-carrier-frequency fan-out gratings fabricated by TIR holographic lithography,” Opt. Eng. 34, 2377–2383 (1995).
[CrossRef]

P. Ehbets, H. P. Herzig, P. Nussbaum, P. Blattner, R. Dändliker, “Interferometric fabrication of modulated submicron gratings in photoresist,” Appl. Opt. 34, 2540–2547 (1995).
[CrossRef] [PubMed]

Ichikawa, H.

Jones, A. L.

Jull, E. V.

Karthe, W.

R. Waldhäusl, P. Dannberg, E. B. Kley, A. Bräuer, W. Karthe, “Highly efficient blazed grating couplers in planar polymer waveguides,” Int. J. Optoelectron. 8, 529–536 (1993).

Kirk, J. P.

Kley, E. B.

R. Waldhäusl, P. Dannberg, E. B. Kley, A. Bräuer, W. Karthe, “Highly efficient blazed grating couplers in planar polymer waveguides,” Int. J. Optoelectron. 8, 529–536 (1993).

Kuittinen, M.

P. Ehbets, H. P. Herzig, M. Kuittinen, F. S. M. Clube, Y. Darbellay, “High-carrier-frequency fan-out gratings fabricated by TIR holographic lithography,” Opt. Eng. 34, 2377–2383 (1995).
[CrossRef]

Loewen, E. G.

Lohmann, A. W.

Maystre, D.

Miller, J. M.

Nevière, M.

Noponen, E.

Nussbaum, P.

Paris, D. B.

Parkes, W.

Pekola, J.

E. Tervonen, J. Turunen, J. Pekola, “Pulse-frequency-modulated high-frequency-carrier diffractive elements for pattern projection,” Opt. Eng. 33, 2579–2587 (1994).
[CrossRef]

Robertson, B.

Taghizadeh, M. R.

Tervonen, E.

E. Tervonen, J. Turunen, J. Pekola, “Pulse-frequency-modulated high-frequency-carrier diffractive elements for pattern projection,” Opt. Eng. 33, 2579–2587 (1994).
[CrossRef]

Turunen, J.

Vasara, A.

Waldhäusl, R.

R. Waldhäusl, P. Dannberg, E. B. Kley, A. Bräuer, W. Karthe, “Highly efficient blazed grating couplers in planar polymer waveguides,” Int. J. Optoelectron. 8, 529–536 (1993).

Wilkinson, C. D. W.

Wyrowski, F.

E. Noponen, J. Turunen, F. Wyrowski, “Synthesis of paraxial-domain diffractive elements by rigorous electromagnetic theory,” J. Opt. Soc. Am. A 12, 1128–1133 (1995).
[CrossRef]

F. Wyrowski, “Upper bound of the efficiency of diffractive phase elements,” Opt. Lett. 16, 1915–1917 (1991).
[CrossRef] [PubMed]

F. Wyrowski, “Consideration on convolutions and phase factors,” Opt. Commun. 81, 353–358 (1991).
[CrossRef]

F. Wyrowski, O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
[CrossRef]

Appl. Opt. (8)

Int. J. Optoelectron. (1)

R. Waldhäusl, P. Dannberg, E. B. Kley, A. Bräuer, W. Karthe, “Highly efficient blazed grating couplers in planar polymer waveguides,” Int. J. Optoelectron. 8, 529–536 (1993).

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (4)

Opt. Commun. (1)

F. Wyrowski, “Consideration on convolutions and phase factors,” Opt. Commun. 81, 353–358 (1991).
[CrossRef]

Opt. Comput. Process. (1)

M. R. Taghizadeh, J. Turunen, “Synthetic diffractive elements for optical interconnection,” Opt. Comput. Process. 2, 221–242 (1992).

Opt. Eng. (2)

E. Tervonen, J. Turunen, J. Pekola, “Pulse-frequency-modulated high-frequency-carrier diffractive elements for pattern projection,” Opt. Eng. 33, 2579–2587 (1994).
[CrossRef]

P. Ehbets, H. P. Herzig, M. Kuittinen, F. S. M. Clube, Y. Darbellay, “High-carrier-frequency fan-out gratings fabricated by TIR holographic lithography,” Opt. Eng. 34, 2377–2383 (1995).
[CrossRef]

Opt. Lett. (2)

Rep. Prog. Phys. (1)

F. Wyrowski, O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Diffraction of a plane wave by a grating with an arbitrary periodic (complex) refractive-index distribution nII(x, z).

Fig. 2
Fig. 2

(a) Amplitude and (b) phase of the complex-amplitude transmission function of a grating that splits a normally incident plane wave into three equal-efficiency orders.

Fig. 3
Fig. 3

(a) Amplitude and (b) phase of the complex-amplitude transmission function of a grating that splits a normally incident plane wave into four equal-efficiency orders.

Fig. 4
Fig. 4

(a) Carrier-grating structure: h = h0 is the groove depth that gives the highest efficiency η = ηc,max. (b) Dependence of ηc on groove depth h with fill factor c/dc = 1/2 and dc = 0.8λ at Bragg incidence. (c) Dependence of the phase ϕc of the minus first order on groove depth h.

Tables (3)

Tables Icon

Table 1 Efficiency η, Uniformity Error E, and Noise Measure N for a High-Carrier-Frequency-Encoded Three-Beam Splitter

Tables Icon

Table 2 Efficiency η, Uniformity Error E, and Noise Measure N for a High-Carrier-Frequency-Encoded Four-Beam Splitter

Tables Icon

Table 3 Efficiencies ηn (%) of Some Propagating Orders of Three-Beam and Four Beam Elements when Q = 70

Equations (27)

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E ( x , z ) = m = - T m exp [ i ( α m x + t m z ) ] ,
α m = k n I sin θ + 2 π m / d ,
t m = [ ( k n III ) 2 - α m 2 ] 1 / 2 ,
η m = n III cos θ m n I cos θ T m 2 ,
η = m W η m .
m W η ¯ m = 1.
E s ( x , 0 ) = m W T ¯ m exp ( i 2 π m x / d ) .
1 d 0 d E s ( x , 0 ) 2 d x = 1 ,
η l = [ 1 d 0 d E s ( x , 0 ) d x ] 2 ,
t ( x ) = I max - 1 / 2 E s ( x , 0 ) ,
I max = max 0 x d E s ( x , 0 ) 2 .
η m = I max - 1 η ¯ m ,
η = I max - 1 .
ϕ m = arg { T ¯ m } .
E s ( x , 0 ) = 2 cos ( 2 π x / d ) .
E s ( x , 0 ) = 1 3 [ 1 + 2 cos ( 2 π x / d ) exp ( i ϕ 1 ) ] .
I max = { ( 5 + 4 cos ϕ 1 ) / 3 when 0 ϕ 1 < π / 2 or 3 π / 2 ϕ 1 < 2 π ( 5 - 4 cos ϕ 1 ) / 3 when π / 2 ϕ 1 < 3 π / 2 .
η l = 1 3 { 0 d [ 1 + 4 cos 2 ( 2 π x / d ) ] 1 / 2 d x } 0.9381
t ( x ) = 1 / 5 [ 1 - i 2 cos ( 2 π x / d ) ]
E s ( x , 0 ) = cos ( 2 π x / d ) + cos ( 6 π x / d ) exp ( i ϕ 3 ) .
E ( x , 0 ) exp ( - i 2 π Q x / d ) = m W T m exp ( i α m - Q x ) = n + Q W T n + Q exp ( i α n x ) ,
d c = d / Q ,
sin θ n = 2 n + Q 2 n III λ d = 1 + 2 n / Q 2 n III λ d c .
η c ( x ) = η c , max t ( x ) 2 ,
Δ x ( h ) = - d c [ ϕ c ( h ) - ϕ c ( h 0 ) ] / 2 π .
E = ( η max - η min ) / ( η max + η min ) ,
N = n < - Q / 2 η n - η .

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