Abstract

Ray tracing for a planar microlens by solving a ray equation is performed. Off-axial dependence of spot size and wave aberration of fabricated planar microlens samples are estimated. A typical spherical aberration of the present planar microlenses at N.A. = 0.3 was 1.1λ. The curvature field was −0.7λ, and comatic aberration was 0.3λ for a tilted angle of 5° at 0.3 N.A. It was estimated that the field curvature and comatic aberration were dominant next to the spherical aberration.

© 1988 Optical Society of America

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References

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  1. S. Misawa, M. Oikawa, K. Iga, “Ray Tracing in a Distributed-Index Planar Microlens,” Jpn. J. Appl. Phys. 21, L589 (1982).
    [CrossRef]
  2. S. Misawa, M. Oikawa, K. Iga, “Ray Tracing in a Planar Microlens,” Optics 12, 467 (1983) (in Japanese).
  3. S. Misawa, M. Oikawa, K. Iga, “Measurement of Numerical Aperture of Planar Microlens by Far Field Pattern Method,” in Technical Digest, Topical Meeting on Gradient-Index Optical Imaging Systems (Optical Society of America, Washington, DC, 1984), paper ThE-D4.
  4. N. Arai, “A New Calculation Method of Optical Path Length Through Gradient Index Media,” in Technical Digest, Fourth Topical Meeting on Gradient Index Optical Imaging Systems (Japanese Society of Applied Physics, Tokyo, 1983), paper B5.
  5. W. H. Southwell, “Ray Tracing in Gradient-Index Media,” J. Opt. Soc. Am. 72, 908 (1982).
    [CrossRef]
  6. E. W. Marchand, Gradient-Index Optics (Academic, New York, 1978), pp. 100.
  7. S. Misawa, M. Oikawa, K. Iga, “Maximum and Effective N.A. of Planar Microlens,” Appl. Opt. 23, 1784 (1984).
    [CrossRef] [PubMed]
  8. K. Iga, S. Misawa, “Distributed-Index Planar Microlens and Stacked Planar Optics: a Review of Progress,” Appl. Opt. 25, 3388 (1986).
    [CrossRef] [PubMed]
  9. Y. Matsui, Method of Lens Design (Kyo-Ritsu Syuppan Press, Tokyo, 1972) (in Japanese).
  10. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), Sec. 5.1.

1986 (1)

1984 (1)

1983 (1)

S. Misawa, M. Oikawa, K. Iga, “Ray Tracing in a Planar Microlens,” Optics 12, 467 (1983) (in Japanese).

1982 (2)

S. Misawa, M. Oikawa, K. Iga, “Ray Tracing in a Distributed-Index Planar Microlens,” Jpn. J. Appl. Phys. 21, L589 (1982).
[CrossRef]

W. H. Southwell, “Ray Tracing in Gradient-Index Media,” J. Opt. Soc. Am. 72, 908 (1982).
[CrossRef]

Arai, N.

N. Arai, “A New Calculation Method of Optical Path Length Through Gradient Index Media,” in Technical Digest, Fourth Topical Meeting on Gradient Index Optical Imaging Systems (Japanese Society of Applied Physics, Tokyo, 1983), paper B5.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), Sec. 5.1.

Iga, K.

K. Iga, S. Misawa, “Distributed-Index Planar Microlens and Stacked Planar Optics: a Review of Progress,” Appl. Opt. 25, 3388 (1986).
[CrossRef] [PubMed]

S. Misawa, M. Oikawa, K. Iga, “Maximum and Effective N.A. of Planar Microlens,” Appl. Opt. 23, 1784 (1984).
[CrossRef] [PubMed]

S. Misawa, M. Oikawa, K. Iga, “Ray Tracing in a Planar Microlens,” Optics 12, 467 (1983) (in Japanese).

S. Misawa, M. Oikawa, K. Iga, “Ray Tracing in a Distributed-Index Planar Microlens,” Jpn. J. Appl. Phys. 21, L589 (1982).
[CrossRef]

S. Misawa, M. Oikawa, K. Iga, “Measurement of Numerical Aperture of Planar Microlens by Far Field Pattern Method,” in Technical Digest, Topical Meeting on Gradient-Index Optical Imaging Systems (Optical Society of America, Washington, DC, 1984), paper ThE-D4.

Marchand, E. W.

E. W. Marchand, Gradient-Index Optics (Academic, New York, 1978), pp. 100.

Matsui, Y.

Y. Matsui, Method of Lens Design (Kyo-Ritsu Syuppan Press, Tokyo, 1972) (in Japanese).

Misawa, S.

K. Iga, S. Misawa, “Distributed-Index Planar Microlens and Stacked Planar Optics: a Review of Progress,” Appl. Opt. 25, 3388 (1986).
[CrossRef] [PubMed]

S. Misawa, M. Oikawa, K. Iga, “Maximum and Effective N.A. of Planar Microlens,” Appl. Opt. 23, 1784 (1984).
[CrossRef] [PubMed]

S. Misawa, M. Oikawa, K. Iga, “Ray Tracing in a Planar Microlens,” Optics 12, 467 (1983) (in Japanese).

S. Misawa, M. Oikawa, K. Iga, “Ray Tracing in a Distributed-Index Planar Microlens,” Jpn. J. Appl. Phys. 21, L589 (1982).
[CrossRef]

S. Misawa, M. Oikawa, K. Iga, “Measurement of Numerical Aperture of Planar Microlens by Far Field Pattern Method,” in Technical Digest, Topical Meeting on Gradient-Index Optical Imaging Systems (Optical Society of America, Washington, DC, 1984), paper ThE-D4.

Oikawa, M.

S. Misawa, M. Oikawa, K. Iga, “Maximum and Effective N.A. of Planar Microlens,” Appl. Opt. 23, 1784 (1984).
[CrossRef] [PubMed]

S. Misawa, M. Oikawa, K. Iga, “Ray Tracing in a Planar Microlens,” Optics 12, 467 (1983) (in Japanese).

S. Misawa, M. Oikawa, K. Iga, “Ray Tracing in a Distributed-Index Planar Microlens,” Jpn. J. Appl. Phys. 21, L589 (1982).
[CrossRef]

S. Misawa, M. Oikawa, K. Iga, “Measurement of Numerical Aperture of Planar Microlens by Far Field Pattern Method,” in Technical Digest, Topical Meeting on Gradient-Index Optical Imaging Systems (Optical Society of America, Washington, DC, 1984), paper ThE-D4.

Southwell, W. H.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), Sec. 5.1.

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

Jpn. J. Appl. Phys. (1)

S. Misawa, M. Oikawa, K. Iga, “Ray Tracing in a Distributed-Index Planar Microlens,” Jpn. J. Appl. Phys. 21, L589 (1982).
[CrossRef]

Optics (1)

S. Misawa, M. Oikawa, K. Iga, “Ray Tracing in a Planar Microlens,” Optics 12, 467 (1983) (in Japanese).

Other (5)

S. Misawa, M. Oikawa, K. Iga, “Measurement of Numerical Aperture of Planar Microlens by Far Field Pattern Method,” in Technical Digest, Topical Meeting on Gradient-Index Optical Imaging Systems (Optical Society of America, Washington, DC, 1984), paper ThE-D4.

N. Arai, “A New Calculation Method of Optical Path Length Through Gradient Index Media,” in Technical Digest, Fourth Topical Meeting on Gradient Index Optical Imaging Systems (Japanese Society of Applied Physics, Tokyo, 1983), paper B5.

E. W. Marchand, Gradient-Index Optics (Academic, New York, 1978), pp. 100.

Y. Matsui, Method of Lens Design (Kyo-Ritsu Syuppan Press, Tokyo, 1972) (in Japanese).

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), Sec. 5.1.

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Figures (7)

Fig. 1
Fig. 1

Coordinate system for ray tracing.

Fig. 2
Fig. 2

Index distribution of a fabricated planar microlens sample.

Fig. 3
Fig. 3

Schematic explanation of the incident condition of rays and calculation of the wave aberration.

Fig. 4
Fig. 4

Change of focused spot in the axial direction for parallel incidence. The numbers in this figure show the distance from the end of the distributed index region to the observed plane.

Fig. 5
Fig. 5

Change of focused spot for a tangential fan of rays. The incident rays are tilted in the Y-direction.

Fig. 6
Fig. 6

(a) Wave aberration at the meridional plane. Here the N.A. shows the height of each ray from the axis at the emergent plane divided by the distance from this plane to the focal plane and multiplied by the substrate index 1.54. (b) Wave aberration at the sagittal plane. This graph is for incident angle ϕ = 0°. The graphs for the other incident angles are nearly the same as this.

Fig. 7
Fig. 7

(a) Spherical aberration; (b) comatic aberration; (c) astigmatic aberration; (d) curvature of field; (e) distortion; (f) arrowlike aberration.

Tables (1)

Tables Icon

Table I Characteristic of Fabricated Planar Microlens Sample

Equations (22)

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d d s ( n d r d s ) = n ,
T = ( p , r q , l ) ,
p = n d r / d s , q = n d θ / d s , l = n d z / d s .
d p d s = n r + r n q 2 ;
r d q d s = 2 n p q + 1 r n θ ;
d l d s = n z .
| T | 2 = p 2 + r 2 q 2 + l 2 = n 2 ( C r 2 + C θ 2 + C z 2 ) = n 2 .
d r d z = p n 2 p 2 r 2 q 2 ,
d θ d z = q n 2 p 2 r 2 q 2 ,
d p d z = n n r + r q 2 n 2 p 2 r 2 q 2 ,
r d q d z = n r n θ 2 p q n 2 p 2 r 2 q 2 .
L = z 0 z 1 n 1 + ( d r / d z ) 2 + r 2 ( d θ / d z ) 2 d z = z 0 z 1 n 2 n 2 p 2 r 2 q 2 d z ,
d L d z = n 2 n 2 p 2 r 2 q 2 ,
d r d z = p n 2 p 2 C 2 / r 2 ,
d θ d z = C / r 2 n 2 p 2 C 2 / r 2 ,
d p d z = 1 2 n 2 r + C 2 r 3 n 2 p 2 C 2 / r 2 ,
d L d z = n 2 n 2 p 2 C 2 / r 2 .
d r d z = p n 2 p 2 ,
d p d z = 1 2 n 2 2 r n 2 p 2 ,
d L d z = n 2 n 2 p 2 .
n ( 0 ) = 1 . 81 , g = 0 . 849 ( mm 1 ) , n 2 ( r , z ) = n 2 ( 0 ) { 1 , ( g z ) 2 , ( g z ) 4 , ( g z ) 6 } × [ 1 1 3 . 24 9 . 55 1 . 04 8 . 42 51 . 4 12 . 5 3 . 75 39 . 2 30 600 9 . 82 5 . 7 220 2000 ] [ 1 ( g r ) 2 ( g r ) 4 ( g r ) 6 ] , ( in the distributed index region ) = n 2 2 ( in the substrate ) ,
W dif = W 11 r cos θ + W 20 r 2 + W 22 r 2 cos 2 θ + W 31 r 3 cos θ + W 33 r 3 cos 3 θ + W 40 r 4 + W 42 r 4 cos 2 θ + W 51 r 5 cos θ + W 60 r 6 ,

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