Abstract

We analyzed the direct sampling (DS) method for diffractive lens encoding, using exact electromagnetic diffraction theory. In addition to previously published research [Pure Appl. Opt. 7, 565 (1998)] we present what we believe to be new results for TM polarization. We found that the validity of the scalar-based DS method is even more extended for TM than for TE polarization. Additionally, we fabricated and characterized DS-encoded blazed gratings and found good agreement between the experimental and theoretical diffraction efficiencies. We analyzed quantitatively the influence of the encoding schemes DS and analytic quantization (AQ) on the quality of the focal spot. We also investigated the focal spot sizes (FWHM) and the Strehl ratios of the DS- and the AQ-encoded cylindrical lenses.

© 2000 Optical Society of America

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References

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  1. G. P. Behrmann, J. N. Mait, “Hybrid (Refractive/Diffractive) Optics,” in Micro-Optics: Elements, Systems and Applications, H. P. Herzig, ed. (Taylor & Francis, London, 1997), pp. 259–292.
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    [CrossRef]
  3. P. Blattner, H. P. Herzig, K. J. Weible, J. M. Teijido, H. J. Heimbeck, E. Langenbach, J. Rogers, “Diffractive optics for compact space communication terminals,” J. Mod. Opt. 43, 1473–1484 (1996).
    [CrossRef]
  4. M. B. Stern, “Binary optics fabrication” in Micro-Optics: Elements, Systems and Applications, H. P. Herzig, ed. (Taylor & Francis, London, 1997), pp. 53–85.
  5. M. Kuittinen, H. P. Herzig, “Encoding of efficient diffractive microlenses,” Opt. Lett. 20, 2156–2158 (1995).
    [CrossRef] [PubMed]
  6. W. H. Welch, J. E. Morris, M. R. Feldmann, “Iterative discrete on-axis encoding of radially symmetric computer-generated holograms,” J. Opt. Soc. Am. A 10, 1729–1738 (1993).
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    [CrossRef]
  8. G. J. Swanson, (MIT Lincoln Laboratory, Lexington, Mass., 1989).
  9. U. Levy, N. Cohen, D. Mendlovic, “Analytic approach for optimal quantization of diffractive optical elements,” Appl. Opt. 38, 5527–5532 (1999).
    [CrossRef]
  10. J. Turunen, “Diffraction theory of microrelief gratings,” in Micro-Optics: Elements, Systems and Applications, H. P. Herzig, ed. (Taylor & Francis, London, 1997), pp. 31–52.
  11. M. G. Moharam, D. A. Pommet, E. B. Grann, T. K. Gaylord, “Stable implementation of the rigorous-coupled-wave analysis for surface relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A 12, 1077–1086 (1995).
    [CrossRef]
  12. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), pp.361–390.
  13. D. A. Pommet, M. G. Moharam, E. B. Grann, “Limits of scalar diffraction theory for diffractive phase elements,” J. Opt. Soc. Am. A 11, 1827–1834 (1994).
    [CrossRef]
  14. M. Born, E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, 1997), Chap. 9.

1999

1998

A. Schilling, P. Blattner, H. P. Herzig, “Direct sampling for diffractive microlens encoding from a rigorous point of view,” Pure Appl. Opt. 7, 565–574 (1998).
[CrossRef]

1996

W. Singer, H. P. Herzig, M. Kuittinen, E. Piper, J. Wangler, “Diffractive beamshaping elements at the fabrication limit,” Opt. Eng. 35, 2779–2787 (1996).
[CrossRef]

P. Blattner, H. P. Herzig, K. J. Weible, J. M. Teijido, H. J. Heimbeck, E. Langenbach, J. Rogers, “Diffractive optics for compact space communication terminals,” J. Mod. Opt. 43, 1473–1484 (1996).
[CrossRef]

1995

1994

1993

Behrmann, G. P.

G. P. Behrmann, J. N. Mait, “Hybrid (Refractive/Diffractive) Optics,” in Micro-Optics: Elements, Systems and Applications, H. P. Herzig, ed. (Taylor & Francis, London, 1997), pp. 259–292.

Blattner, P.

A. Schilling, P. Blattner, H. P. Herzig, “Direct sampling for diffractive microlens encoding from a rigorous point of view,” Pure Appl. Opt. 7, 565–574 (1998).
[CrossRef]

P. Blattner, H. P. Herzig, K. J. Weible, J. M. Teijido, H. J. Heimbeck, E. Langenbach, J. Rogers, “Diffractive optics for compact space communication terminals,” J. Mod. Opt. 43, 1473–1484 (1996).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, 1997), Chap. 9.

Cohen, N.

Feldmann, M. R.

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), pp.361–390.

Gaylord, T. K.

Grann, E. B.

Heimbeck, H. J.

P. Blattner, H. P. Herzig, K. J. Weible, J. M. Teijido, H. J. Heimbeck, E. Langenbach, J. Rogers, “Diffractive optics for compact space communication terminals,” J. Mod. Opt. 43, 1473–1484 (1996).
[CrossRef]

Herzig, H. P.

A. Schilling, P. Blattner, H. P. Herzig, “Direct sampling for diffractive microlens encoding from a rigorous point of view,” Pure Appl. Opt. 7, 565–574 (1998).
[CrossRef]

P. Blattner, H. P. Herzig, K. J. Weible, J. M. Teijido, H. J. Heimbeck, E. Langenbach, J. Rogers, “Diffractive optics for compact space communication terminals,” J. Mod. Opt. 43, 1473–1484 (1996).
[CrossRef]

W. Singer, H. P. Herzig, M. Kuittinen, E. Piper, J. Wangler, “Diffractive beamshaping elements at the fabrication limit,” Opt. Eng. 35, 2779–2787 (1996).
[CrossRef]

M. Kuittinen, H. P. Herzig, “Encoding of efficient diffractive microlenses,” Opt. Lett. 20, 2156–2158 (1995).
[CrossRef] [PubMed]

G. P. Behrmann, J. N. Mait, “Hybrid (Refractive/Diffractive) Optics,” in Micro-Optics: Elements, Systems and Applications, H. P. Herzig, ed. (Taylor & Francis, London, 1997), pp. 259–292.

Kuittinen, M.

W. Singer, H. P. Herzig, M. Kuittinen, E. Piper, J. Wangler, “Diffractive beamshaping elements at the fabrication limit,” Opt. Eng. 35, 2779–2787 (1996).
[CrossRef]

M. Kuittinen, H. P. Herzig, “Encoding of efficient diffractive microlenses,” Opt. Lett. 20, 2156–2158 (1995).
[CrossRef] [PubMed]

Langenbach, E.

P. Blattner, H. P. Herzig, K. J. Weible, J. M. Teijido, H. J. Heimbeck, E. Langenbach, J. Rogers, “Diffractive optics for compact space communication terminals,” J. Mod. Opt. 43, 1473–1484 (1996).
[CrossRef]

Levy, U.

Mait, J. N.

G. P. Behrmann, J. N. Mait, “Hybrid (Refractive/Diffractive) Optics,” in Micro-Optics: Elements, Systems and Applications, H. P. Herzig, ed. (Taylor & Francis, London, 1997), pp. 259–292.

Mendlovic, D.

Moharam, M. G.

Morris, J. E.

Piper, E.

W. Singer, H. P. Herzig, M. Kuittinen, E. Piper, J. Wangler, “Diffractive beamshaping elements at the fabrication limit,” Opt. Eng. 35, 2779–2787 (1996).
[CrossRef]

Pommet, D. A.

Rogers, J.

P. Blattner, H. P. Herzig, K. J. Weible, J. M. Teijido, H. J. Heimbeck, E. Langenbach, J. Rogers, “Diffractive optics for compact space communication terminals,” J. Mod. Opt. 43, 1473–1484 (1996).
[CrossRef]

Schilling, A.

A. Schilling, P. Blattner, H. P. Herzig, “Direct sampling for diffractive microlens encoding from a rigorous point of view,” Pure Appl. Opt. 7, 565–574 (1998).
[CrossRef]

Singer, W.

W. Singer, H. P. Herzig, M. Kuittinen, E. Piper, J. Wangler, “Diffractive beamshaping elements at the fabrication limit,” Opt. Eng. 35, 2779–2787 (1996).
[CrossRef]

Stern, M. B.

M. B. Stern, “Binary optics fabrication” in Micro-Optics: Elements, Systems and Applications, H. P. Herzig, ed. (Taylor & Francis, London, 1997), pp. 53–85.

Swanson, G. J.

G. J. Swanson, (MIT Lincoln Laboratory, Lexington, Mass., 1989).

Teijido, J. M.

P. Blattner, H. P. Herzig, K. J. Weible, J. M. Teijido, H. J. Heimbeck, E. Langenbach, J. Rogers, “Diffractive optics for compact space communication terminals,” J. Mod. Opt. 43, 1473–1484 (1996).
[CrossRef]

Turunen, J.

J. Turunen, “Diffraction theory of microrelief gratings,” in Micro-Optics: Elements, Systems and Applications, H. P. Herzig, ed. (Taylor & Francis, London, 1997), pp. 31–52.

Wangler, J.

W. Singer, H. P. Herzig, M. Kuittinen, E. Piper, J. Wangler, “Diffractive beamshaping elements at the fabrication limit,” Opt. Eng. 35, 2779–2787 (1996).
[CrossRef]

Weible, K. J.

P. Blattner, H. P. Herzig, K. J. Weible, J. M. Teijido, H. J. Heimbeck, E. Langenbach, J. Rogers, “Diffractive optics for compact space communication terminals,” J. Mod. Opt. 43, 1473–1484 (1996).
[CrossRef]

Welch, W. H.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, 1997), Chap. 9.

Appl. Opt.

J. Mod. Opt.

P. Blattner, H. P. Herzig, K. J. Weible, J. M. Teijido, H. J. Heimbeck, E. Langenbach, J. Rogers, “Diffractive optics for compact space communication terminals,” J. Mod. Opt. 43, 1473–1484 (1996).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Eng.

W. Singer, H. P. Herzig, M. Kuittinen, E. Piper, J. Wangler, “Diffractive beamshaping elements at the fabrication limit,” Opt. Eng. 35, 2779–2787 (1996).
[CrossRef]

Opt. Lett.

Pure Appl. Opt.

A. Schilling, P. Blattner, H. P. Herzig, “Direct sampling for diffractive microlens encoding from a rigorous point of view,” Pure Appl. Opt. 7, 565–574 (1998).
[CrossRef]

Other

G. J. Swanson, (MIT Lincoln Laboratory, Lexington, Mass., 1989).

J. Turunen, “Diffraction theory of microrelief gratings,” in Micro-Optics: Elements, Systems and Applications, H. P. Herzig, ed. (Taylor & Francis, London, 1997), pp. 31–52.

G. P. Behrmann, J. N. Mait, “Hybrid (Refractive/Diffractive) Optics,” in Micro-Optics: Elements, Systems and Applications, H. P. Herzig, ed. (Taylor & Francis, London, 1997), pp. 259–292.

M. B. Stern, “Binary optics fabrication” in Micro-Optics: Elements, Systems and Applications, H. P. Herzig, ed. (Taylor & Francis, London, 1997), pp. 53–85.

M. Born, E. Wolf, Principles of Optics (Cambridge University Press, Cambridge, 1997), Chap. 9.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), pp.361–390.

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Figures (8)

Fig. 1
Fig. 1

Dashed line, ideal phase function of a blazed grating (Λ b /MFS = 2.75); solid line, multilevel profile obtained by DS.

Fig. 2
Fig. 2

Calculated diffraction efficiencies (TE polarization) as a function of Λ b /λ with constant Λ b for blazed gratings with (a) Λ b /MFS = 2.25 and (b) Λ b /MFS = 2.75. For the phase depth of 2π, eight equally spaced phase levels were available. Dotted–dashed curves, scalar diffraction efficiencies; solid curves, rigorously calculated diffraction efficiencies; dashed curves, rigorous diffraction efficiencies for the phase-level distributions that were obtained by rigorous optimization (RSGO).

Fig. 3
Fig. 3

Calculated diffraction efficiencies (TM polarization) as a function of Λ b /λ with constant Λ b for blazed gratings with (a) Λ b /MFS = 2.25 and (b) Λ b /MFS = 2.75. For the phase depth of 2π, eight equally spaced phase levels were available. Dotted–dashed curves, scalar diffraction efficiencies; solid curves, rigorously calculated diffraction efficiencies; dashed curves, rigorous diffraction efficiencies for the phase-level distributions that were obtained by rigorous optimization (RSGO).

Fig. 4
Fig. 4

SEM pictures of the fabricated eight-level DS-encoded blazed gratings in fused silica: (a) for a grating period of 2.25 MFS, (b) for a grating period of 2.75 MFS. Numbers 0–7 indicate the eight different phase levels.

Fig. 5
Fig. 5

Experimental and theoretical diffraction efficiencies for (a) TE and (b) TM polarization. The gratings are eight-level DS-encoded fused-silica gratings with Λ b = 2.25 MFS.

Fig. 6
Fig. 6

Experimental and theoretical diffraction efficiencies for (a) TE and (b) TM polarization. The gratings are eight-level DS-encoded fused-silica gratings with Λ b = 2.75 MFS.

Fig. 7
Fig. 7

Focal spot sizes (FWHM) and Strehl ratios for DS- and AQ-encoded cylindrical lenses compared with the ideal case: (a) FWHM as a function of MFS, (b) Strehl ratio as a function of MFS.

Fig. 8
Fig. 8

Efficiencies in the various diffraction orders for DS- and AQ-encoded blazed gratings: (a) for Λ b = 2.25 MFS, (b) for Λ b = 2.75 MFS.

Tables (2)

Tables Icon

Table 1b/λ)0 Values for Different Λb/MFS Ratios of the DS Blazed Grating in the Binary Region: TE Polarizationa

Tables Icon

Table 2b/λ)0 Values for Different Λb/MFS Ratios of the DS Blazed Grating in the Binary Region: TE Polarizationa

Equations (4)

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

ηm=|Tm|2=1d0dtxexp-i2πmx/ddx2,
u2x, z2=-+ u1x, z1τx, z1×1kr12-iz12λr122expikr12dx,
ϕρ=(2π/λ)[f-f2+ρ21/2],
Ds=1-4π2Wrms/λ2,

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