Abstract

The influence of mask-alignment errors on the fabrication of four-level binary lenses has been investigated. The pupil function was derived as a function of the amount of misalignment between the two mask patterns based on scalar diffraction theory. It was found that the phase term in the pupil function affects only the location of the point-spread function, whereas the uneven light transmittance degrades the image quality in the direction of the misalignment. The Strehl intensity was found to decrease almost linearly with respect to the amount of error.

© 1998 Optical Society of America

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References

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  1. G. J. Swanson, “Binary optics technology: the theory and design of multi-level diffractive optical elements,” Tech. Rep. 854 (Massachusetts Institute of Technology Lincoln Laboratory, Lexington, Mass., 1989).
  2. H. Dammann, “Spectral characteristics of stepped-phase gratings,” Optik 53, 409–417 (1979).
  3. K. M. Flood, J. M. Finlan, “Multiple phase level computer-generated hologram etched in fused silica,” in Holographic Optics: Optically and Computer Generated, I. N. Cindrich, S. H. Lee, eds., Proc. SPIE1052, 91–96 (1989).
    [CrossRef]
  4. 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. N. Cindrich, S. H. Lee, eds., Proc. SPIE1211, 116–124 (1990).
    [CrossRef]
  5. M. W. Farn, J. W. Goodman, “Effect of VLSI fabrication errors on kinoform efficiency,” in Computer and Optically Formed Holographic Optics, I. N. Cindrich, S. H. Lee, eds., Proc. SPIE1211, 125–136 (1990).
    [CrossRef]
  6. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Chap. 9.
  7. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  8. Ref. 6, Chap. 8.
  9. B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
    [CrossRef]

1979

H. Dammann, “Spectral characteristics of stepped-phase gratings,” Optik 53, 409–417 (1979).

1959

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

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. N. Cindrich, S. H. Lee, eds., Proc. SPIE1211, 116–124 (1990).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Chap. 9.

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. N. Cindrich, S. H. Lee, eds., Proc. SPIE1211, 116–124 (1990).
[CrossRef]

Dammann, H.

H. Dammann, “Spectral characteristics of stepped-phase gratings,” Optik 53, 409–417 (1979).

Farn, M. W.

M. W. Farn, J. W. Goodman, “Effect of VLSI fabrication errors on kinoform efficiency,” in Computer and Optically Formed Holographic Optics, I. N. Cindrich, S. H. Lee, eds., Proc. SPIE1211, 125–136 (1990).
[CrossRef]

Finlan, J. M.

K. M. Flood, J. M. Finlan, “Multiple phase level computer-generated hologram etched in fused silica,” in Holographic Optics: Optically and Computer Generated, I. N. Cindrich, S. H. Lee, eds., Proc. SPIE1052, 91–96 (1989).
[CrossRef]

Flood, K. M.

K. M. Flood, J. M. Finlan, “Multiple phase level computer-generated hologram etched in fused silica,” in Holographic Optics: Optically and Computer Generated, I. N. Cindrich, S. H. Lee, eds., Proc. SPIE1052, 91–96 (1989).
[CrossRef]

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. N. Cindrich, S. H. Lee, eds., Proc. SPIE1211, 116–124 (1990).
[CrossRef]

Goodman, J. W.

M. W. Farn, J. W. Goodman, “Effect of VLSI fabrication errors on kinoform efficiency,” in Computer and Optically Formed Holographic Optics, I. N. Cindrich, S. H. Lee, eds., Proc. SPIE1211, 125–136 (1990).
[CrossRef]

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

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. N. Cindrich, S. H. Lee, eds., Proc. SPIE1211, 116–124 (1990).
[CrossRef]

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. N. Cindrich, S. H. Lee, eds., Proc. SPIE1211, 116–124 (1990).
[CrossRef]

Richards, B.

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Swanson, G. J.

G. J. Swanson, “Binary optics technology: the theory and design of multi-level diffractive optical elements,” Tech. Rep. 854 (Massachusetts Institute of Technology Lincoln Laboratory, Lexington, Mass., 1989).

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. N. Cindrich, S. H. Lee, eds., Proc. SPIE1211, 116–124 (1990).
[CrossRef]

Wolf, E.

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Chap. 9.

Optik

H. Dammann, “Spectral characteristics of stepped-phase gratings,” Optik 53, 409–417 (1979).

Proc. R. Soc. London Ser. A

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Other

G. J. Swanson, “Binary optics technology: the theory and design of multi-level diffractive optical elements,” Tech. Rep. 854 (Massachusetts Institute of Technology Lincoln Laboratory, Lexington, Mass., 1989).

K. M. Flood, J. M. Finlan, “Multiple phase level computer-generated hologram etched in fused silica,” in Holographic Optics: Optically and Computer Generated, I. N. Cindrich, S. H. Lee, eds., Proc. SPIE1052, 91–96 (1989).
[CrossRef]

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. N. Cindrich, S. H. Lee, eds., Proc. SPIE1211, 116–124 (1990).
[CrossRef]

M. W. Farn, J. W. Goodman, “Effect of VLSI fabrication errors on kinoform efficiency,” in Computer and Optically Formed Holographic Optics, I. N. Cindrich, S. H. Lee, eds., Proc. SPIE1211, 125–136 (1990).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Chap. 9.

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

Ref. 6, Chap. 8.

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Figures (13)

Fig. 1
Fig. 1

Schematic description of a binary Fresnel lens with four-level surface structures.

Fig. 2
Fig. 2

Fabrication processes for the four-level structure for two different mask patterns. In (a) the two-level structure is created, and it is converted into a four-level structure in (b). The portion removed in each etching process is represented by hatching.

Fig. 3
Fig. 3

Radial distribution of the Fresnel lens described in relation to the two mask patterns, which are ideally matched. The ranges r 2 < r < r j as well as r j+1 < rR (R is the lens’s radius) are omitted.

Fig. 4
Fig. 4

Relationship between the two mask patterns when a translational error l is assumed for mask B while mask A is fixed.

Fig. 5
Fig. 5

Surface profiles at r = r j fabricated with the misaligned masks. (a) a(r j , θ) < 0, (b) a(r j , θ) > 0. According to the sign of a(r j , θ), the profile shows quite distinct characteristics.

Fig. 6
Fig. 6

Phase-modulation functions representing the surface profiles shown in Fig. 5.

Fig. 7
Fig. 7

Optical configurations used for the derivation of the point-spread function.

Fig. 8
Fig. 8

Amplitude transmittance obtained by the Fresnel lens. Because the distribution is independent of s y , the results are shown as one-dimensional functions of s x .

Fig. 9
Fig. 9

Three-dimensional distribution of the point-spread function calculated for ρ = 0.6.

Fig. 10
Fig. 10

Cross section of the point-spread function in the x direction. The y coordinate is fixed at y = 0 because the plane containing the peak point always coincides with y = 0.

Fig. 11
Fig. 11

Cross section of the point-spread function in the y direction. The x coordinate is changed according to x = ρλ/(8NA), so the peak point is always contained.

Fig. 12
Fig. 12

Strehl intensity calculated as a function of |ρ|.

Fig. 13
Fig. 13

Widths of the point-spread function calculated as a function of |ρ|. W x , width in the x direction; W y , that in the y direction.

Equations (21)

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r j = 2 jf λ + j λ 2 1 / 2     j > 0 ,   integer .
d A = λ 2 n - 1 ,     d B = λ 4 n - 1 ,
a r j ,   θ = s / Δ j B ,
a r j ,   θ = 4 lr j cos   θ λ f 2 + r j 2 ,
| l | λ 4 NA ,
NA = R f 2 + R 2 .
a r ,   θ = 4 lr   cos   θ λ f 2 + r 2 ,
C m = 1 T 0 T   p u exp - i 2 π mu / T d u ,
C - 1 = 1 T exp i   7 π 4 0 1 + a T / 4 exp i   2 π u T d u + exp i   5 π 4 1 + a T / 4 2 + a T / 4 exp i   2 π u T d u + exp i   7 π 4 2 + a T / 4 T / 2 exp i   2 π u T d u + exp i   3 π 4 T / 2 3 + a T / 4 exp i   2 π u T d u + exp i   π 4 3 + a T / 4 4 + a T / 4 exp i   2 π u T d u + exp i   3 π 4 4 + a T / 4 T exp i   2 π u T d u = = 2 2 π 1 + sin π a 2 1 / 2 exp i   π a 4 .
C - 1 = 1 T exp i   5 π 4 0 aT / 4 exp i   2 π u T d u + exp i   7 π 4 aT / 4 1 + a T / 4 exp i   2 π u T d u + exp i   5 π 4 1 + a T / 4 T / 2 exp i   2 π u T d u + exp i   π 4 T / 2 2 + a T / 4 exp i   2 π u T d u + exp i   3 π 4 2 + a T / 4 3 + a T / 4 exp i   2 π u T d u + exp i   π 4 3 + a T / 4 T exp i   2 π u T d u = = 2 2 π 1 - sin π a 2 1 / 2 exp i   π a 4 .
C - 1 = 2 2 π 1 - sin π | a | 2 1 / 2     exp i   π a 4 ,
Φ r ,   θ = 2 2 π 1 - sin kr | l   cos   θ | f 2 + r 2 1 / 2 × exp i   klr   cos   θ 2 f 2 + r 2 ,
Φ ˆ ξ ,   η = 2 2 π 1 - sin k | l ξ | f 2 + ξ 2 + η 2 1 / 2 × exp i   kl ξ 2 f 2 + ξ 2 + η 2 1 / 2 1 / 2 .
U x ,   y = - i λ   E   Ω   Θ s x ,   s y exp - ik s x x + s y y d Ω ,
s x = ξ f 2 + ξ 2 + η 2 1 / 2 , s y = η f 2 + ξ 2 + η 2 1 / 2 , s z = - 1 - s x 2 - s y 2 1 / 2 ,
Θ s x ,   s y = 2 2 π 1 - sin k | ls x | 1 / 2 exp i   kls x 2 .
d Ω = d s x d s y
U x ,   y = D   NA 1 - sin k | ls x | 1 / 2 × exp - ik s x x - l 2 + s y y d s x d s y ,
I x ,   y = | U x ,   y | 2 | U 0 ,   0 l = 0 | 2 ,
l = ρ   λ 4 NA ,
Θ ˆ s x ,   s y = 1 - sin π 2 | ρ s x | NA 1 / 2

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