Abstract

Orienting two identical or complementary diffractive gratings with a small angle between the grating grooves allows a new crossed-grating device to be constructed. This device has an effective profile that varies locally. For understanding the effects of this variation and the diffraction efficiency of the gratings, the local profiles were correlated with the moiré period of the crossed-grating system by use of various techniques. Asymmetric intensity behavior in the first order of the crossed gratings was seen. Effectively, the diffraction efficiency of the crossed gratings yielded a response equivalent to that of a grating with variable blaze that could be useful in optical computing as a passive optical switching device. One of several models is described that creates greater asymmetric behavior.

© 1998 Optical Society of America

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References

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  1. M. E. Motamedi, “Merging microoptics with micromechanics: micro-opto-electro-mechanical (MOEM) devices,” Vol. CR49 of the SPIE Critical Review Series (SPIE Press, Bellingham, Wash., 1994), pp. 302–328.
  2. J. Guild, The Interference Systems of Crossed Diffraction Gratings (Clarendon, Oxford, 1956).
  3. G. H. Derrick, R. C. McPhedran, D. Maystre, M. Neviére, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
    [CrossRef]
  4. Rayleigh, “On the manufacture and theory of diffraction gratings,” Philos. Mag. 47, 81–93, 193–205 (1874).
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  6. MATHEMATICA is a registered trademark of Wolfram Research, Inc., Champaign, Il.
  7. GSOLVER is a registered trademark of the Grating Solver Development Company, Allen, Tex.
  8. MATLAB is a registered trademark of The Math Works, Inc., Natick, Mass.
  9. W. S. Rockward, “Crossed phase gratings using diffractive optical elements,” Ph.D. dissertation (Georgia Institute of Technology, Atlanta, Ga., 1997).
  10. D. C. O’Shea, J. Belectic, M. Poutous, “Binary-mask generation for diffractive optical elements using microcomputers,” Appl. Opt. 32, 2566–2572 (1993).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  13. FREEHAND is a trademark of the Macromedia Corporation, San Francisco, Calif.
  14. R. L. Morrison, S. L. Walker, F. B. McCormick, T. J. Cloonan, “Practical applications of diffractive optics in free-space photonic switching networks,” Vol. CR49 of the SPIE Critical Review Series (SPIE Press, Bellingham, Wash., 1994), pp. 265–289.
  15. LABVIEW is a registered trademark of the National Instruments Corporation, Austin, Tex.
  16. IBM-PC-based real-time interference fringe-analysis program from Wyko, Inc., Tucson, Ariz.
  17. D. A. Pommet, E. B. Grann, M. G. Moharam, “Effects of process errors on the diffraction characteristics of binary dielectric gratings,” Appl. Opt. 34, 2430–2435 (1995).
    [CrossRef] [PubMed]
  18. D. C. O’Shea, W. S. Rockward, “Light modulation using crossed phase gratings,” Opt. Lett. 23, 491–493 (1998).
    [CrossRef]

Belectic, J.

Cloonan, T. J.

R. L. Morrison, S. L. Walker, F. B. McCormick, T. J. Cloonan, “Practical applications of diffractive optics in free-space photonic switching networks,” Vol. CR49 of the SPIE Critical Review Series (SPIE Press, Bellingham, Wash., 1994), pp. 265–289.

Derrick, G. H.

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Neviére, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

Grann, E. B.

Guild, J.

J. Guild, The Interference Systems of Crossed Diffraction Gratings (Clarendon, Oxford, 1956).

Maystre, D.

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Neviére, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

McCormick, F. B.

R. L. Morrison, S. L. Walker, F. B. McCormick, T. J. Cloonan, “Practical applications of diffractive optics in free-space photonic switching networks,” Vol. CR49 of the SPIE Critical Review Series (SPIE Press, Bellingham, Wash., 1994), pp. 265–289.

McPhedran, R. C.

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Neviére, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

Moharam, M. G.

Morrison, R. L.

R. L. Morrison, S. L. Walker, F. B. McCormick, T. J. Cloonan, “Practical applications of diffractive optics in free-space photonic switching networks,” Vol. CR49 of the SPIE Critical Review Series (SPIE Press, Bellingham, Wash., 1994), pp. 265–289.

Motamedi, M. E.

M. E. Motamedi, “Merging microoptics with micromechanics: micro-opto-electro-mechanical (MOEM) devices,” Vol. CR49 of the SPIE Critical Review Series (SPIE Press, Bellingham, Wash., 1994), pp. 302–328.

Neviére, M.

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Neviére, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

O’Shea, D. C.

Pommet, D. A.

Poutous, M.

Rayleigh,

Rayleigh, “On the manufacture and theory of diffraction gratings,” Philos. Mag. 47, 81–93, 193–205 (1874).

Rockward, W. S.

Suleski, T. J.

Swanson, G. J.

G. J. Swanson, “Binary optics technology: the theory and design of multilevel diffractive optical elements,” Tech. Rep.854 (1-15 Lincoln Laboratory, MIT, Lexington, Mass., 1989).

Walker, S. L.

R. L. Morrison, S. L. Walker, F. B. McCormick, T. J. Cloonan, “Practical applications of diffractive optics in free-space photonic switching networks,” Vol. CR49 of the SPIE Critical Review Series (SPIE Press, Bellingham, Wash., 1994), pp. 265–289.

Appl. Opt. (4)

Appl. Phys. (1)

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Neviére, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

Opt. Lett. (1)

Philos. Mag. (1)

Rayleigh, “On the manufacture and theory of diffraction gratings,” Philos. Mag. 47, 81–93, 193–205 (1874).

Other (11)

G. J. Swanson, “Binary optics technology: the theory and design of multilevel diffractive optical elements,” Tech. Rep.854 (1-15 Lincoln Laboratory, MIT, Lexington, Mass., 1989).

MATHEMATICA is a registered trademark of Wolfram Research, Inc., Champaign, Il.

GSOLVER is a registered trademark of the Grating Solver Development Company, Allen, Tex.

MATLAB is a registered trademark of The Math Works, Inc., Natick, Mass.

W. S. Rockward, “Crossed phase gratings using diffractive optical elements,” Ph.D. dissertation (Georgia Institute of Technology, Atlanta, Ga., 1997).

FREEHAND is a trademark of the Macromedia Corporation, San Francisco, Calif.

R. L. Morrison, S. L. Walker, F. B. McCormick, T. J. Cloonan, “Practical applications of diffractive optics in free-space photonic switching networks,” Vol. CR49 of the SPIE Critical Review Series (SPIE Press, Bellingham, Wash., 1994), pp. 265–289.

LABVIEW is a registered trademark of the National Instruments Corporation, Austin, Tex.

IBM-PC-based real-time interference fringe-analysis program from Wyko, Inc., Tucson, Ariz.

M. E. Motamedi, “Merging microoptics with micromechanics: micro-opto-electro-mechanical (MOEM) devices,” Vol. CR49 of the SPIE Critical Review Series (SPIE Press, Bellingham, Wash., 1994), pp. 302–328.

J. Guild, The Interference Systems of Crossed Diffraction Gratings (Clarendon, Oxford, 1956).

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Figures (14)

Fig. 1
Fig. 1

Profile of the trapezoidal model with a 50/50 duty cycle. The width of the feature measured at the midpoint value along the sloped sidewalls is L + ∊.

Fig. 2
Fig. 2

Diagram of two identical binary, square gratings with a 50/50 duty cycle arranged facing each other. The grating grooves are parallel and sheared by a distance relative to each other. The shear distance δ is given as a percentage of the grating period.

Fig. 3
Fig. 3

Formation of moiré fringes by the slight rotation of two nearly identical phase gratings with a 50/50 duty cycle and a π-phase depth.

Fig. 4
Fig. 4

(a) Sketch of a magnified region of the moiré fringes created by the crossed phase gratings in Fig. 3. The prescribed line segments XX, YY, and ZZ from (a) are the equivalent shear locations of (b) δ = +50%, (c) δ = 0%, and (d) δ = -50%, respectively.

Fig. 5
Fig. 5

Schematic of the diffraction-efficiency instrumentation. The sample travels along both the x axis and the z axis (optic axis), whereas the photodetector travels along only the x axis. Also, the sample can be a single grating or a grating pair mounted in the shear-cell apparatus.

Fig. 6
Fig. 6

Image of two gratings, WR61 and WR65, mounted face to face inside the shear-cell device with a solid black line in the background. The various regions show the effects of the grating combinations on the solid line. Regions labeled W are unetched. Regions labeled X or Z are areas where one of the substrates is etched, and region Y is where both gratings overlap.

Fig. 7
Fig. 7

Normalized intensity distribution of grating sample WR65. The two dominant peaks represent the -1 and the +1 orders.

Fig. 8
Fig. 8

Plots of the diffraction efficiency as a function of shear for the 0, -2, and +2 orders. The data for these plots were taken by use of the crossed-grating combination of WR01–WR02. The diffraction-efficiency distributions are actually those of the ±1 orders of the effective grating with a 5-μm period but are labeled according to the orders of the single gratings, as explained in the text.

Fig. 9
Fig. 9

Plots of the diffraction efficiency as a function of shear for the 0, -1, and +1 orders. The data for these plots were taken by use of the crossed-grating combination of WR01–WR02. The diffraction-efficiency distributions for the +1 and the -1 orders should not exist in the shear region.

Fig. 10
Fig. 10

Plots of the diffraction efficiency as a function of shear for the 0, -2, and +2 orders. The data for these plots were taken by use of the crossed-grating combination of WR61–WR65. The diffraction-efficiency distributions for the ±2 orders are actually those of the ±1 orders of the effective grating with a 10-μm period.

Fig. 11
Fig. 11

Plots of the diffraction efficiency as a function of shear for the 0, -1, and +1 orders. The data for these plots were taken by use of the crossed-grating combination of WR61–WR65. The diffraction-efficiency distributions for the +1 and the -1 orders should not exist in the shear region.

Fig. 12
Fig. 12

Trapezoidal model of WR65–WR61 (exact): Simulation of the trapezoidal profile, duty-cycle, and phase-depth values from WR65–WR61. Plots of the diffraction efficiency for the zero and first orders are shown as a function of shear.

Fig. 13
Fig. 13

Diagram of the sheared gratings of model A. The constituent phase gratings are arranged at 0% shear to produce the strongest effective blazed grating. One grating has a period of w g and a phase depth of π, whereas the other has a period of 0.5w g and a phase depth of 0.5π.

Fig. 14
Fig. 14

Plots of the diffraction efficiency of the first orders as functions of shear for the sheared gratings of model A. There is no zero order for this combination.

Tables (1)

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Table 1 Characteristics of Gratings Used in This Study

Equations (7)

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f sq x = E 0   exp i ϕ x ,
ϕ x = 2 π d n - 1 λ 0 < x < L = 0.5 w g 2 π d λ 0.5 w g < x < w g ,
d = λ 2 n - 1 .
η m sq = sin π n - 1 d λ - m π n - 1 d λ - m 2 sin π n - 1 d 2 λ π n - 1 d 2 λ 2 ,
w m = w g cos θ / 2 sin   θ ,
w m w g θ .
Δ = d 0 2 tan   θ m = d 0 2 w g w m ,

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