Abstract

The performance of two-dimensional Talbot array illuminators is discussed in terms of compression ratio, fabrication cost, and illumination efficiency. By comparing two array-illuminator families, we try to answer the question, “Which Talbot array illuminator provides the best illumination performance and requires the least expenditure in fabrication for a given compression ratio?” We further present experimental results obtained with a quartz-glass four-level surface-relief array illuminator designed for a two-dimensional compression ratio of 16 and fabricated with only two lithographic masks.

© 1998 Optical Society of America

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References

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  1. A. W. Lohmann, “An array illuminator based on the Talbot effect,” Optik (Stuttgart) 79, 41–45 (1988).
  2. A. W. Lohmann, J. A. Thomas, “Making an array illuminator based on the Talbot effect,” Appl. Opt. 29, 4337–4340 (1990).
    [Crossref] [PubMed]
  3. P. Latimer, “Talbot plane patterns: grating images or interference effects?” Appl. Opt. 32, 1078–1083 (1993).
    [Crossref] [PubMed]
  4. K. Patorski, “The self-imaging phenomenon and its application,” in Vol. 28 of Progress in Optics Series (Elsevier, North Holland, Amsterdam, 1989), pp. 1–108.
  5. J. R. Leger, G. J. Swanson, “Efficient array illuminator using binary-optics phase plates at fractional Talbot planes,” Opt. Lett. 15, 288–290 (1990).
    [Crossref] [PubMed]
  6. V. Arrizón, J. Ojeda-Castañeda, “Irradiance at Fresnel planes of a phase grating,” J. Opt. Soc. Am. A 9, 1801–1806 (1992).
    [Crossref]
  7. V. Arrizón, J. Ojeda-Castañeda, “Talbot array illuminators with binary phase gratings,” Opt. Lett. 18, 1–3 (1993).
    [Crossref]
  8. P. Szwaykowski, V. Arrizón, “Talbot array illuminator with multilevel phase gratings,” Appl. Opt. 32, 1109–1114 (1993).
    [Crossref] [PubMed]
  9. C. Zhou, L. Liu, “Simple equations for the calculation of a multilevel phase grating for Talbot array illumination,” Opt. Commun. 115, 40–44 (1995).
    [Crossref]
  10. V. Arrizón, E. López-Olazagasti, A. Serrano-Heredia, “Talbot array illuminators with optimum compression ratio,” Opt. Lett. 21, 233–235 (1996).
    [Crossref]
  11. X.-Y. Da, Q.-Q. Wang, X.-J. Xue, “Two-dimensional Talbot array illuminators with single-step phase grating,” J. Opt. Soc. Am. A 13, 126–130 (1996).
    [Crossref]
  12. H. Hamam, “Design of Talbot array illuminators,” Opt. Commun. 131, 359–370 (1996).
    [Crossref]
  13. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 48–53.
  14. W. Klaus, Y. Arimoto, K. Kodate, “Talbot array illuminators providing spatial intensity and phase modulation,” J. Opt. Soc. Am. A 14, 1092–1102 (1997).
    [Crossref]
  15. M. B. Stern, “Binary optics fabrication,” in Micro-optics: Elements, Systems and Applications, H. P. Herzig, ed. (Taylor & Francis, London, 1997), pp. 63–64.
  16. 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]
  17. J. A. Cox, T. Werner, J. Lee, S. Nelson, B. Fritz, J. Bergstrom, “Diffraction efficiency of binary optical elements,” in Computer and Optically Formed Holographic Optics, I. Cindrich, S. H. Lee, eds., Proc. SPIE1211, 116–124 (1990).
    [Crossref]
  18. J. M. Miller, M. R. Taghizadeh, J. Turunen, N. Ross, “Multilevel-grating array generators: fabrication error analysis and experiments,” Appl. Opt. 32, 2519–2525 (1993).
    [Crossref] [PubMed]
  19. V. Arrizón, E. López-Olazagasti, “Binary phase grating for array generation at 1/16 of Talbot length,” J. Opt. Soc. Am. A 12, 801–804 (1995).
    [Crossref]

1997 (1)

1996 (3)

1995 (2)

V. Arrizón, E. López-Olazagasti, “Binary phase grating for array generation at 1/16 of Talbot length,” J. Opt. Soc. Am. A 12, 801–804 (1995).
[Crossref]

C. Zhou, L. Liu, “Simple equations for the calculation of a multilevel phase grating for Talbot array illumination,” Opt. Commun. 115, 40–44 (1995).
[Crossref]

1994 (1)

1993 (4)

1992 (1)

1990 (2)

1988 (1)

A. W. Lohmann, “An array illuminator based on the Talbot effect,” Optik (Stuttgart) 79, 41–45 (1988).

Arimoto, Y.

Arrizón, V.

Bergstrom, J.

J. A. Cox, T. Werner, J. Lee, S. Nelson, B. Fritz, J. Bergstrom, “Diffraction efficiency of binary optical elements,” in Computer and Optically Formed Holographic Optics, I. Cindrich, S. H. Lee, eds., Proc. SPIE1211, 116–124 (1990).
[Crossref]

Cox, J. A.

J. A. Cox, T. Werner, J. Lee, S. Nelson, B. Fritz, J. Bergstrom, “Diffraction efficiency of binary optical elements,” in Computer and Optically Formed Holographic Optics, I. Cindrich, S. H. Lee, eds., Proc. SPIE1211, 116–124 (1990).
[Crossref]

Da, X.-Y.

Fritz, B.

J. A. Cox, T. Werner, J. Lee, S. Nelson, B. Fritz, J. Bergstrom, “Diffraction efficiency of binary optical elements,” in Computer and Optically Formed Holographic Optics, I. Cindrich, S. H. Lee, eds., Proc. SPIE1211, 116–124 (1990).
[Crossref]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 48–53.

Grann, E. B.

Hamam, H.

H. Hamam, “Design of Talbot array illuminators,” Opt. Commun. 131, 359–370 (1996).
[Crossref]

Klaus, W.

Kodate, K.

Latimer, P.

Lee, J.

J. A. Cox, T. Werner, J. Lee, S. Nelson, B. Fritz, J. Bergstrom, “Diffraction efficiency of binary optical elements,” in Computer and Optically Formed Holographic Optics, I. Cindrich, S. H. Lee, eds., Proc. SPIE1211, 116–124 (1990).
[Crossref]

Leger, J. R.

Liu, L.

C. Zhou, L. Liu, “Simple equations for the calculation of a multilevel phase grating for Talbot array illumination,” Opt. Commun. 115, 40–44 (1995).
[Crossref]

Lohmann, A. W.

A. W. Lohmann, J. A. Thomas, “Making an array illuminator based on the Talbot effect,” Appl. Opt. 29, 4337–4340 (1990).
[Crossref] [PubMed]

A. W. Lohmann, “An array illuminator based on the Talbot effect,” Optik (Stuttgart) 79, 41–45 (1988).

López-Olazagasti, E.

Miller, J. M.

Moharam, M. G.

Nelson, S.

J. A. Cox, T. Werner, J. Lee, S. Nelson, B. Fritz, J. Bergstrom, “Diffraction efficiency of binary optical elements,” in Computer and Optically Formed Holographic Optics, I. Cindrich, S. H. Lee, eds., Proc. SPIE1211, 116–124 (1990).
[Crossref]

Ojeda-Castañeda, J.

Patorski, K.

K. Patorski, “The self-imaging phenomenon and its application,” in Vol. 28 of Progress in Optics Series (Elsevier, North Holland, Amsterdam, 1989), pp. 1–108.

Pommet, D. A.

Ross, N.

Serrano-Heredia, A.

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. 63–64.

Swanson, G. J.

Szwaykowski, P.

Taghizadeh, M. R.

Thomas, J. A.

Turunen, J.

Wang, Q.-Q.

Werner, T.

J. A. Cox, T. Werner, J. Lee, S. Nelson, B. Fritz, J. Bergstrom, “Diffraction efficiency of binary optical elements,” in Computer and Optically Formed Holographic Optics, I. Cindrich, S. H. Lee, eds., Proc. SPIE1211, 116–124 (1990).
[Crossref]

Xue, X.-J.

Zhou, C.

C. Zhou, L. Liu, “Simple equations for the calculation of a multilevel phase grating for Talbot array illumination,” Opt. Commun. 115, 40–44 (1995).
[Crossref]

Appl. Opt. (4)

J. Opt. Soc. Am. A (5)

Opt. Commun. (2)

C. Zhou, L. Liu, “Simple equations for the calculation of a multilevel phase grating for Talbot array illumination,” Opt. Commun. 115, 40–44 (1995).
[Crossref]

H. Hamam, “Design of Talbot array illuminators,” Opt. Commun. 131, 359–370 (1996).
[Crossref]

Opt. Lett. (3)

Optik (Stuttgart) (1)

A. W. Lohmann, “An array illuminator based on the Talbot effect,” Optik (Stuttgart) 79, 41–45 (1988).

Other (4)

K. Patorski, “The self-imaging phenomenon and its application,” in Vol. 28 of Progress in Optics Series (Elsevier, North Holland, Amsterdam, 1989), pp. 1–108.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 48–53.

M. B. Stern, “Binary optics fabrication,” in Micro-optics: Elements, Systems and Applications, H. P. Herzig, ed. (Taylor & Francis, London, 1997), pp. 63–64.

J. A. Cox, T. Werner, J. Lee, S. Nelson, B. Fritz, J. Bergstrom, “Diffraction efficiency of binary optical elements,” in Computer and Optically Formed Holographic Optics, I. Cindrich, S. H. Lee, eds., Proc. SPIE1211, 116–124 (1990).
[Crossref]

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Figures (8)

Fig. 1
Fig. 1

(a) Two-dimensional phase distribution of four neighboring elementary cells of the conventional TAIL for N 2-D = 16 [case (a), Section 2]. The phase levels are 0°, 90°, 135°, 180°, and 225°. (b) Two-dimensional phase distribution of a single elementary cell of the new TAIL for N 2-D = 16. The phase levels are 0°, 90°, 180°, and 270°.

Fig. 2
Fig. 2

Phase profile of the new TAIL for N 2-D = 1024 (N 1-D = 32), demonstrating the similarity to binary zone plates: (a) Three-dimensional view. (b) Top view. The scale along the right-hand side gives the phase in degrees.

Fig. 3
Fig. 3

Mask patterns of three different new TAIL’s for the compression-ratio series N 2-D = 22m ( N 1-D = 2 m ), where m represents the number of masks. The values in parentheses represent the etch depths. (a) m = 1. (b) m = 2. (c) m = 3.

Fig. 4
Fig. 4

Logarithmic intensity distributions: (a) Behind the conventional TAIL, according to Fig. 1(a), at z = z t /8. (b) Behind the new TAIL, according to Fig. 1(b), at z = z t /32. (c) Behind the conventional TAIL, according to Fig. 1(a), under the assumption of an etch-depth error of 10%. (d) Behind the new TAIL, according to Fig. 1(b), under the assumption of an etch-depth error of 10%. (e) Behind the conventional TAIL for N 2-D = 9, under the assumption of an alignment error of 2 μm along both TAIL axes. (f) Behind the new TAIL, according to Fig. 1(b), under the assumption of an alignment error of 2 μm along both TAIL axes (to the right and down). The minimum feature size of the ideal TAIL profile is 50 μm. The scale on the right-hand side shows the intensity in decibels [-40, 2]. The intensity is normalized to the value of N 2-D.

Fig. 5
Fig. 5

Illumination-efficiency reduction as a function of the etch-depth error for TAIL’s with (a) N 2-D = 16 and (b) N 2-D = 64. The solid curve represents the conventional TAIL, and the dotted and dashed curves represent the new TAIL’s. Behind the new TAIL’s, neighboring spots exhibit different illumination efficiencies with an increasing etch-depth error.

Fig. 6
Fig. 6

Illumination-efficiency reduction as a function of the alignment errors that occur simultaneously along both TAIL axes: (a) The new TAIL for N 2-D = 16 and the conventional TAIL for N 2-D = 19, each TAIL requiring two masks for their fabrication. The abbreviation fs represents the minimum feature size of the ideal TAIL profile. The solid curve represents the conventional TAIL, and the dotted curve the new TAIL. (b) The new TAIL for N 2-D = 64, requiring three masks for fabrication. Curve 1: error owing to mask 3 only. Curve 2: error owing to mask 2 only. Curve 3: errors owing to both masks that occur in the same direction. Curve 4: errors owing to both masks that occur in opposite directions.

Fig. 7
Fig. 7

Change in the illumination efficiency caused by variations in the observation distance near z mchc. The curves represent both the conventional and the new TAIL’s. The dotted, solid, and dashed curves represent minimum feature sizes of fs = 250, 50, 10 μm, respectively.

Fig. 8
Fig. 8

Experimental measurements: (a) The linear intensity distribution. (b) The interference fringes (representing the phase distribution) generated at z = z t /32 behind a quartz-glass phase grating designed according to Fig. 1(b). (c) The linear intensity distribution with an increased intensity level showing the distribution of the optical background noise caused by fabrication errors, i.e., an alignment error of 3 μm along both TAIL axes and an average etch-depth error of 15%. The fringes’ displacements shown in (b) indicate a phase shift of approximately 180° between neighboring spots. The numbers refer to the phase difference.

Tables (1)

Tables Icon

Table 1 Number of Phase Levels and Mask Numbers for the New and the Conventional 2-D TAIL’s Generating Spot Arrays with a Range of 2-D Compression Ratios of N2-D = 4–256a

Equations (6)

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ϕ k ,   n = -   π k 2 n / 2 .
ϕ k ,   n = - π k 2 2 n + 1 4 - 1 k .
ϕ k ,   n = -   π 2 k - 1 2 2 n .
ϕ k ,   n = - π k 2 2 n - 1 4 - 1 k .
ϕ k ,   N 1 - D = π k - 1 N 1 - D - k / N 1 - D ,
ϕ k + N 1 - D ,   N 1 - D = π + ϕ k ,   N 1 - D = π k N 1 - D - k + 1 / N 1 - D ,

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