Abstract

The design of advanced diffraction gratings that produce spot arrays is highly influenced by the computational capabilities that are available to the designer. This is due to the increased pattern complexity that is required for larger spot arrays or higher efficiencies. Symmetries that lead to a significant reduction in the design complexity can be incorporated into the grating pattern. In addition, a translational symmetry leads to the highly desired result of even-numbered spot arrays. We examine the symmetries that can be applied to both general-and discrete-level design parameterization.

© 1992 Optical Society of America

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References

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  1. H. Dammann, K. Gortler, “High-efficiency in-line multiple imaging by multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
    [CrossRef]
  2. H. Dammann, E. Klotz, “Coherent-optical generation and inspection of two-dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
    [CrossRef]
  3. F. B. McCormick, F. A. P. Tooley, T. J. Cloonan, J. L. Brubaker, A. L. Lentine, R. L. Morrison, S. L. Walker, S. J. Hinterlong, M. J. Herron, “S-SEED-based photonic switching network demonstration,” in Optical Society of America 1990 Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990).
  4. M. E. Prise, N. C. Craft, R. E. LaMarche, M. M. Downs, S. J. Walker, L. A. D’Asaro, L. M. F. Chirovsky, “Module for optical logic circuits using symmetric self-electrooptic effect devices,” Appl. Opt. 29, 2164–2170 (1990).
    [CrossRef] [PubMed]
  5. J. Jahns, M. M. Downs, M. E. Prise, N. Stribl, S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).
    [CrossRef]
  6. J. N. Mait, “Design of binary-phase and multiphase Fourier gratings for array generation,” J. Opt. Soc. Am. A 7, 1514–1528 (1990).
    [CrossRef]
  7. J. Turunen, A. Vasara, J. Westerholm, A. Salin, “Optimization and fabrication of grating beamsplitters,”J. Phys. D 21, 102–105 (1990).
    [CrossRef]
  8. J. Turunen, A. Vasara, J. Westerholm, “Stripe-geometry two-dimensional Dammann gratings,” Opt. Commun. 74, 245–252 (1990).
    [CrossRef]
  9. M. R. Feldman, C. G. Guest, “Iterative encoding of high-efficiency holograms for generation of spot arrays,” Opt. Lett. 14, 479–481 (1990).
    [CrossRef]
  10. U. Krackhardt, N. Streibl, “Design of Dammann-gratings for array generation,” Opt. Commun. 74, 31–36 (1989).
    [CrossRef]
  11. R. L. Morrison, S. L. Walker, “Binary phase gratings generating even numbered spot arrays,” in Optical Society of America Annual 1989 Meeting, Vol. 18 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989).
  12. S. J. Walker, Jugen Jahns, “Array generation with multilevel phase gratings,” J. Opt. Soc. Am. A 7, 1509–1513 (1990).
    [CrossRef]
  13. Noted by the reviewer.
  14. U. Killat, G. Rabe, W. Rave, “Binary phase gratings for star couplers with high splitting ratios,” Fiber Integr. Opt. 4, 159–164 (1982).
    [CrossRef]

1990 (6)

1989 (2)

U. Krackhardt, N. Streibl, “Design of Dammann-gratings for array generation,” Opt. Commun. 74, 31–36 (1989).
[CrossRef]

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

1982 (1)

U. Killat, G. Rabe, W. Rave, “Binary phase gratings for star couplers with high splitting ratios,” Fiber Integr. Opt. 4, 159–164 (1982).
[CrossRef]

1977 (1)

H. Dammann, E. Klotz, “Coherent-optical generation and inspection of two-dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
[CrossRef]

1971 (1)

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

Brubaker, J. L.

F. B. McCormick, F. A. P. Tooley, T. J. Cloonan, J. L. Brubaker, A. L. Lentine, R. L. Morrison, S. L. Walker, S. J. Hinterlong, M. J. Herron, “S-SEED-based photonic switching network demonstration,” in Optical Society of America 1990 Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990).

Chirovsky, L. M. F.

Cloonan, T. J.

F. B. McCormick, F. A. P. Tooley, T. J. Cloonan, J. L. Brubaker, A. L. Lentine, R. L. Morrison, S. L. Walker, S. J. Hinterlong, M. J. Herron, “S-SEED-based photonic switching network demonstration,” in Optical Society of America 1990 Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990).

Craft, N. C.

D’Asaro, L. A.

Dammann, H.

H. Dammann, E. Klotz, “Coherent-optical generation and inspection of two-dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
[CrossRef]

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

Downs, M. M.

Feldman, M. R.

Gortler, K.

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

Guest, C. G.

Herron, M. J.

F. B. McCormick, F. A. P. Tooley, T. J. Cloonan, J. L. Brubaker, A. L. Lentine, R. L. Morrison, S. L. Walker, S. J. Hinterlong, M. J. Herron, “S-SEED-based photonic switching network demonstration,” in Optical Society of America 1990 Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990).

Hinterlong, S. J.

F. B. McCormick, F. A. P. Tooley, T. J. Cloonan, J. L. Brubaker, A. L. Lentine, R. L. Morrison, S. L. Walker, S. J. Hinterlong, M. J. Herron, “S-SEED-based photonic switching network demonstration,” in Optical Society of America 1990 Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990).

Jahns, J.

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

Jahns, Jugen

Killat, U.

U. Killat, G. Rabe, W. Rave, “Binary phase gratings for star couplers with high splitting ratios,” Fiber Integr. Opt. 4, 159–164 (1982).
[CrossRef]

Klotz, E.

H. Dammann, E. Klotz, “Coherent-optical generation and inspection of two-dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
[CrossRef]

Krackhardt, U.

U. Krackhardt, N. Streibl, “Design of Dammann-gratings for array generation,” Opt. Commun. 74, 31–36 (1989).
[CrossRef]

LaMarche, R. E.

Lentine, A. L.

F. B. McCormick, F. A. P. Tooley, T. J. Cloonan, J. L. Brubaker, A. L. Lentine, R. L. Morrison, S. L. Walker, S. J. Hinterlong, M. J. Herron, “S-SEED-based photonic switching network demonstration,” in Optical Society of America 1990 Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990).

Mait, J. N.

McCormick, F. B.

F. B. McCormick, F. A. P. Tooley, T. J. Cloonan, J. L. Brubaker, A. L. Lentine, R. L. Morrison, S. L. Walker, S. J. Hinterlong, M. J. Herron, “S-SEED-based photonic switching network demonstration,” in Optical Society of America 1990 Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990).

Morrison, R. L.

F. B. McCormick, F. A. P. Tooley, T. J. Cloonan, J. L. Brubaker, A. L. Lentine, R. L. Morrison, S. L. Walker, S. J. Hinterlong, M. J. Herron, “S-SEED-based photonic switching network demonstration,” in Optical Society of America 1990 Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990).

R. L. Morrison, S. L. Walker, “Binary phase gratings generating even numbered spot arrays,” in Optical Society of America Annual 1989 Meeting, Vol. 18 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989).

Prise, M. E.

Rabe, G.

U. Killat, G. Rabe, W. Rave, “Binary phase gratings for star couplers with high splitting ratios,” Fiber Integr. Opt. 4, 159–164 (1982).
[CrossRef]

Rave, W.

U. Killat, G. Rabe, W. Rave, “Binary phase gratings for star couplers with high splitting ratios,” Fiber Integr. Opt. 4, 159–164 (1982).
[CrossRef]

Salin, A.

J. Turunen, A. Vasara, J. Westerholm, A. Salin, “Optimization and fabrication of grating beamsplitters,”J. Phys. D 21, 102–105 (1990).
[CrossRef]

Streibl, N.

U. Krackhardt, N. Streibl, “Design of Dammann-gratings for array generation,” Opt. Commun. 74, 31–36 (1989).
[CrossRef]

Stribl, N.

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

Tooley, F. A. P.

F. B. McCormick, F. A. P. Tooley, T. J. Cloonan, J. L. Brubaker, A. L. Lentine, R. L. Morrison, S. L. Walker, S. J. Hinterlong, M. J. Herron, “S-SEED-based photonic switching network demonstration,” in Optical Society of America 1990 Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990).

Turunen, J.

J. Turunen, A. Vasara, J. Westerholm, A. Salin, “Optimization and fabrication of grating beamsplitters,”J. Phys. D 21, 102–105 (1990).
[CrossRef]

J. Turunen, A. Vasara, J. Westerholm, “Stripe-geometry two-dimensional Dammann gratings,” Opt. Commun. 74, 245–252 (1990).
[CrossRef]

Vasara, A.

J. Turunen, A. Vasara, J. Westerholm, “Stripe-geometry two-dimensional Dammann gratings,” Opt. Commun. 74, 245–252 (1990).
[CrossRef]

J. Turunen, A. Vasara, J. Westerholm, A. Salin, “Optimization and fabrication of grating beamsplitters,”J. Phys. D 21, 102–105 (1990).
[CrossRef]

Walker, S. J.

Walker, S. L.

R. L. Morrison, S. L. Walker, “Binary phase gratings generating even numbered spot arrays,” in Optical Society of America Annual 1989 Meeting, Vol. 18 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989).

F. B. McCormick, F. A. P. Tooley, T. J. Cloonan, J. L. Brubaker, A. L. Lentine, R. L. Morrison, S. L. Walker, S. J. Hinterlong, M. J. Herron, “S-SEED-based photonic switching network demonstration,” in Optical Society of America 1990 Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990).

Westerholm, J.

J. Turunen, A. Vasara, J. Westerholm, A. Salin, “Optimization and fabrication of grating beamsplitters,”J. Phys. D 21, 102–105 (1990).
[CrossRef]

J. Turunen, A. Vasara, J. Westerholm, “Stripe-geometry two-dimensional Dammann gratings,” Opt. Commun. 74, 245–252 (1990).
[CrossRef]

Appl. Opt. (1)

Fiber Integr. Opt. (1)

U. Killat, G. Rabe, W. Rave, “Binary phase gratings for star couplers with high splitting ratios,” Fiber Integr. Opt. 4, 159–164 (1982).
[CrossRef]

J. Opt. Soc. Am. A (2)

J. Phys. D (1)

J. Turunen, A. Vasara, J. Westerholm, A. Salin, “Optimization and fabrication of grating beamsplitters,”J. Phys. D 21, 102–105 (1990).
[CrossRef]

Opt. Acta (1)

H. Dammann, E. Klotz, “Coherent-optical generation and inspection of two-dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
[CrossRef]

Opt. Commun. (3)

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

J. Turunen, A. Vasara, J. Westerholm, “Stripe-geometry two-dimensional Dammann gratings,” Opt. Commun. 74, 245–252 (1990).
[CrossRef]

U. Krackhardt, N. Streibl, “Design of Dammann-gratings for array generation,” Opt. Commun. 74, 31–36 (1989).
[CrossRef]

Opt. Eng. (1)

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

Opt. Lett. (1)

Other (3)

F. B. McCormick, F. A. P. Tooley, T. J. Cloonan, J. L. Brubaker, A. L. Lentine, R. L. Morrison, S. L. Walker, S. J. Hinterlong, M. J. Herron, “S-SEED-based photonic switching network demonstration,” in Optical Society of America 1990 Annual Meeting, Vol. 15 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990).

R. L. Morrison, S. L. Walker, “Binary phase gratings generating even numbered spot arrays,” in Optical Society of America Annual 1989 Meeting, Vol. 18 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989).

Noted by the reviewer.

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Figures (9)

Fig. 1
Fig. 1

Production of spot arrays using a Fourier-transform setup.

Fig. 2
Fig. 2

Transitions and phase levels associated with one period of a multilevel grating. a.u., Arbitrary units.

Fig. 3
Fig. 3

Standard spot array design (top) and even-numbered spot array design (bottom). The small darkened circles in the even-numbered design are suppressed (zero-intensity) orders.

Fig. 4
Fig. 4

One period of a continuous-phase surface exhibiting reflection symmetry about the period midpoint.

Fig. 5
Fig. 5

One period of a continuous-phase surface exhibiting translation and reflection symmetry.

Fig. 6
Fig. 6

One period of a discrete multilevel phase grating with reflection symmetry about the period midpoint.

Fig. 7
Fig. 7

One period of an even-numbered multilevel phase grating with translation symmetry and reflection symmetry.

Fig. 8
Fig. 8

One period of a standard binary-phase grating.

Fig. 9
Fig. 9

One period of an even-numbered binary-phase grating with translation symmetry about the period midpoint.

Equations (33)

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h = λ Δ n ,
A ( n x , n y ) = 0 1 0 1 t ( x , y ) exp [ i θ ( x , y ) ] exp ( - 2 π i n x x ) × exp ( - 2 π i n y y ) d x d y ,
S = λ f P ,
A ( n ) = 0 1 t ( x ) exp [ i θ ( x ) ] exp ( - 2 π i n x ) d x ,
I ( n ) = A ( n ) · A * ( n ) = A ( - n ) · A * ( - n ) = I ( - n ) ,
t ( 1 - x ) = t ( x ) ,             0 x 1 / 2 , θ ( 1 - x ) = θ ( x ) + ϕ ,
A ( n ) = 0 1 / 2 t ( x ) exp [ i θ ( x ) ] × [ exp ( - 2 π i n x ) + exp ( i ϕ ) exp ( 2 π i n x ) ] d x .
A ( n ) = 2 0 1 / 2 exp [ i θ ( x ) ] cos ( 2 π n x ) d x
A ( n ) = - 2 i 0 1 / 2 exp [ i θ ( x ) ] sin ( 2 π n x ) d x .
t ( x ) = t ( x - 1 / 2 ) ,             1 / 2 x 1 , θ ( x ) = θ ( x - 1 / 2 ) + ϕ ,
A ( n ) = 0 1 / 2 t ( x ) exp [ i θ ( x ) ] exp ( - 2 π i n x ) × [ 1 + ( - 1 ) n exp ( i ϕ ) ] d x .
A ( n ) = 0 ,             n even , A ( n ) = 2 0 1 / 2 t ( x ) exp [ i θ ( x ) ] exp ( - 2 π i n x ) d x ,             n odd .
t ( x ) = t ( 1 / 2 - x ) ,             1 / 4 x 1 / 2 , θ ( x ) = θ ( 1 / 2 - x ) + ϕ ,
A ( n ) = 2 0 1 / 4 t ( x ) exp [ i θ ( x ) ] × [ exp ( - 2 π i n x ) - exp ( i ϕ ) exp ( 2 π i n x ) ] d x .
A ( n ) = - 1 2 π i n k = 1 N exp ( i θ k ) × [ exp ( - 2 π i n x k ) - exp ( - 2 π i n x k - 1 ) ] .
A ( n ) = 1 2 π i n k = 1 N exp ( - 2 π i n x k ) [ exp ( i θ k + 1 ) - exp ( i θ k ) ] .
A ( 0 ) = k = 1 N ( x k - x k - 1 ) exp ( i θ k )
A ( 0 ) = exp ( i θ 1 ) + k = 1 N x k [ exp ( i θ k ) - exp ( i θ k + 1 ) ] .
x k = 1 - x N - k ,             N / 2 + 1 k N , θ k = θ N + 1 - k + ϕ .
x k = 1 - x N + 1 - k ,             N / 2 + 1 k N , θ k = θ N + 2 - k + ϕ ,
A ( n ) = 1 2 π i n [ exp ( i θ 1 ) - ( - 1 ) n exp ( i θ N / 2 ) ] [ 1 - exp ( i ϕ ) ] + 1 2 π i n k = 1 N / 2 - 1 [ exp ( i θ k + 1 ) - exp ( i θ k ) ] × [ exp ( - 2 π i n x k ) - exp ( i ϕ ) exp ( 2 π i n x k ) ] .
A ( n ) = - 1 π n k = 1 N / 2 - 1 [ exp ( i θ k + 1 ) - exp ( i θ k ) ] sin ( 2 π n x k ) .
A ( 0 ) = 1 / 2 [ 1 + exp ( i ϕ ) ] exp ( i θ N / 2 ) + k = 1 N / 2 - 1 x k [ exp ( i θ k ) - exp ( i θ k + 1 ) ] [ exp ( i ϕ ) + 1 ] .
A ( 0 ) = exp ( i θ N / 2 ) + k = 1 N / 2 - 1 2 x k [ exp ( i θ k ) - exp ( i θ k + 1 ) ] ,
x k = x k - N / 2 + 1 / 2 ,             N / 2 + 1 k N , θ k = θ k - N / 2 + π .
x k = 1 / 2 - x N / 2 - k ,             N / 4 + 1 k N / 2 , θ k = θ N / 4 - k + ϕ .
A ( n ) = 0 ,             n even , A ( n ) = 1 - exp ( i ϕ ) π i n [ exp ( i θ 1 ) - ( i ) n exp ( i θ N / 4 ) ] + 1 π i n k = 1 N / 4 - 1 [ exp ( i θ k + 1 ) - exp ( i θ k ) ] [ exp ( - 2 π i n x k ) - exp ( i ϕ ) exp ( 2 π i n x k ) ] ,             n odd .
A ( n ) = - 2 π n k = 1 N / 4 - 1 [ exp ( i θ k + 1 ) - exp ( i θ k ) ] sin ( 2 π n x k ) ,
A ( n ) = 2 exp ( i θ 1 ) π i n - 2 ( i ) n exp ( i θ N / 4 ) π i n + 2 π i n k = 1 N / 4 - 1 [ exp ( i θ k + 1 ) - exp ( i θ k ) ] cos ( 2 π n x k ) ,
θ k = θ 0 + ( - 1 ) k Δ θ / 2 ,
A ( n ) = sin ( Δ θ / 2 ) π n k = 1 N ( - 1 ) k exp ( - 2 π i n x k ) ,
A ( 0 ) = exp ( i θ 1 ) + 2 sin ( Δ θ / 2 ) k = 1 N ( - 1 ) k x k .
A ( n ) = 2 π n k = 1 N / 2 ( - 1 ) k exp ( - 2 π i n x k ) ,             n odd , A ( n ) = 0 ,             n even ,

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