Abstract

This paper gives a treatment for finding 3rd order aberrations in solid-immersion lenses (SILs) using spherical-aberration as the basis for a polynomial power expansion of the wavefront. Unlike previous work, the treatment is general for any incident and lens media, for any lens thickness, and for any chief-ray specification. Using this treatment, a tolerance analysis is given with emphasis on thickness tolerance and limitations on field of view. Major findings include tight thickness tolerance for high-index hyperhemispheres and a tolerance window for hemispheres centered about a thickness less than the radius of curvature of the lens.

© 2008 Optical Society of America

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References

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  1. S. M. Mansfield and G. S. Kino, "Solid immersion microscope," Appl. Phys. Lett. 57, 2615-2616 (1990).
    [CrossRef]
  2. B. D. Terris, H. J. Mamin, D. Rugar, W. R. Studenmunt, and G. S. Kino, "Near-field optical data storage using a solid immersion lens," Appl. Phys. Lett. 65, 388-390 (1994).
    [CrossRef]
  3. R. Shack, Introduction to Abberations, (personal communication).
  4. M. Baba, T. Sasaki, M. Yoshita, and H. Akiyama, "Aberrations and allowances for errors in a hemisphere solid immersion lens for submicron-resolution photoluminescence microscopy," Appl. Phys. 85, 6923-6925 (1999).
  5. S. M. Mansfield, "Solid Immersion Microscopy," Ph. D. Dissertation, Standford University (1992)
  6. J. Jo, The Vector Behavior of Aberrations in High Numerical Aperture (0.9<NA<3.1) Laser Focusing Systems, Ph. D. Dissertation (2001)
  7. W. T. Welford, Aberrations of Optical Systems, (Adam Hilger, 1986) pp. 130-161.
  8. Zemax: commercially available software from the Zemax development corporation

1999

M. Baba, T. Sasaki, M. Yoshita, and H. Akiyama, "Aberrations and allowances for errors in a hemisphere solid immersion lens for submicron-resolution photoluminescence microscopy," Appl. Phys. 85, 6923-6925 (1999).

1994

B. D. Terris, H. J. Mamin, D. Rugar, W. R. Studenmunt, and G. S. Kino, "Near-field optical data storage using a solid immersion lens," Appl. Phys. Lett. 65, 388-390 (1994).
[CrossRef]

1990

S. M. Mansfield and G. S. Kino, "Solid immersion microscope," Appl. Phys. Lett. 57, 2615-2616 (1990).
[CrossRef]

Akiyama, H.

M. Baba, T. Sasaki, M. Yoshita, and H. Akiyama, "Aberrations and allowances for errors in a hemisphere solid immersion lens for submicron-resolution photoluminescence microscopy," Appl. Phys. 85, 6923-6925 (1999).

Baba, M.

M. Baba, T. Sasaki, M. Yoshita, and H. Akiyama, "Aberrations and allowances for errors in a hemisphere solid immersion lens for submicron-resolution photoluminescence microscopy," Appl. Phys. 85, 6923-6925 (1999).

Kino, G. S.

B. D. Terris, H. J. Mamin, D. Rugar, W. R. Studenmunt, and G. S. Kino, "Near-field optical data storage using a solid immersion lens," Appl. Phys. Lett. 65, 388-390 (1994).
[CrossRef]

S. M. Mansfield and G. S. Kino, "Solid immersion microscope," Appl. Phys. Lett. 57, 2615-2616 (1990).
[CrossRef]

Mamin, H. J.

B. D. Terris, H. J. Mamin, D. Rugar, W. R. Studenmunt, and G. S. Kino, "Near-field optical data storage using a solid immersion lens," Appl. Phys. Lett. 65, 388-390 (1994).
[CrossRef]

Mansfield, S. M.

S. M. Mansfield and G. S. Kino, "Solid immersion microscope," Appl. Phys. Lett. 57, 2615-2616 (1990).
[CrossRef]

Rugar, D.

B. D. Terris, H. J. Mamin, D. Rugar, W. R. Studenmunt, and G. S. Kino, "Near-field optical data storage using a solid immersion lens," Appl. Phys. Lett. 65, 388-390 (1994).
[CrossRef]

Sasaki, T.

M. Baba, T. Sasaki, M. Yoshita, and H. Akiyama, "Aberrations and allowances for errors in a hemisphere solid immersion lens for submicron-resolution photoluminescence microscopy," Appl. Phys. 85, 6923-6925 (1999).

Studenmunt, W. R.

B. D. Terris, H. J. Mamin, D. Rugar, W. R. Studenmunt, and G. S. Kino, "Near-field optical data storage using a solid immersion lens," Appl. Phys. Lett. 65, 388-390 (1994).
[CrossRef]

Terris, B. D.

B. D. Terris, H. J. Mamin, D. Rugar, W. R. Studenmunt, and G. S. Kino, "Near-field optical data storage using a solid immersion lens," Appl. Phys. Lett. 65, 388-390 (1994).
[CrossRef]

Yoshita, M.

M. Baba, T. Sasaki, M. Yoshita, and H. Akiyama, "Aberrations and allowances for errors in a hemisphere solid immersion lens for submicron-resolution photoluminescence microscopy," Appl. Phys. 85, 6923-6925 (1999).

Appl. Phys.

M. Baba, T. Sasaki, M. Yoshita, and H. Akiyama, "Aberrations and allowances for errors in a hemisphere solid immersion lens for submicron-resolution photoluminescence microscopy," Appl. Phys. 85, 6923-6925 (1999).

Appl. Phys. Lett.

S. M. Mansfield and G. S. Kino, "Solid immersion microscope," Appl. Phys. Lett. 57, 2615-2616 (1990).
[CrossRef]

B. D. Terris, H. J. Mamin, D. Rugar, W. R. Studenmunt, and G. S. Kino, "Near-field optical data storage using a solid immersion lens," Appl. Phys. Lett. 65, 388-390 (1994).
[CrossRef]

Other

R. Shack, Introduction to Abberations, (personal communication).

S. M. Mansfield, "Solid Immersion Microscopy," Ph. D. Dissertation, Standford University (1992)

J. Jo, The Vector Behavior of Aberrations in High Numerical Aperture (0.9<NA<3.1) Laser Focusing Systems, Ph. D. Dissertation (2001)

W. T. Welford, Aberrations of Optical Systems, (Adam Hilger, 1986) pp. 130-161.

Zemax: commercially available software from the Zemax development corporation

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Figures (17)

Fig. 1.
Fig. 1.

Geometry of chief and marginal ray bending at a single refracting surface.

Fig. 2.
Fig. 2.

Maximum NA′ conditions for hemisphere and hyperhemisphere SIL configurations.

Fig. 3.
Fig. 3.

Graph of W 040 generated using raytrace methods and the analytical approach described as a function of SIL thickness for n=1.0, n′=1.5, R=1.5mm, NA=0.1.

Fig. 4.
Fig. 4.

Graph of analytical and raytrace generated W 040 at a larger NA than Fig. 3, showing deviation due to paraxial approximations made for the analytical calculations.

Fig. 5.
Fig. 5.

Graph of W040 and u′ for a high input NA that results in a non-physical hyperhemisphere location. Only when -1<u′<0 are the results from the analytical solution valid.

Fig. 6.
Fig. 6.

Comparison of spherical aberration curves for the current 3rd order treatment vs. the previous treatments by Baba and Mansfield.

Fig. 7.
Fig. 7.

Graph of W 040 for fixed u, showing λ/20 thickness tolerance windows.

Fig. 8.
Fig. 8.

Graph of analytical and raytrace generated spherical aberration and input marginal ray angle u for fixed u′.

Fig. 9.
Fig. 9.

Graph of W 040 for fixed u′ showing λ/20 thickness tolerance windows.

Fig. 10.
Fig. 10.

A graph of W 040, δλW 020 and δλW 111 as a function of SIL thickness for a SIL with R=0.5mm, NA=0.2, and ȳ=10µm.

Fig. 11.
Fig. 11.

Illustration of two methods to achieve non-zero image height in a SIL: a) lateral chief ray displacement in a telecentric system; and b) changing chief ray angle for a system with the stop at the SIL vertex.

Fig. 12.
Fig. 12.

Third order aberrations vs. image coordinate for a hemisphere. W 040 and W 131 are both zero, and the W 222 curves for the telecentric and stop at vertex conditions are degenerate.

Fig. 13.
Fig. 13.

Third order aberrations vs. image coordinate for l′=0.9R.

Fig. 14.
Fig. 14.

Third order aberrations vs. image coordinate for l′=1.1R.

Fig. 15.
Fig. 15.

Third order aberrations vs. image coordinate for a hyperhemisphere.

Fig. 16.
Fig. 16.

Third order aberrations as a function of SIL thickness with n′=1.5, R=1.5mm, NA=0.36, and ȳ=80µm.

Fig. 17.
Fig. 17.

Schematic of aspheric plate at the center of curvature with field and pupil footprints showing the relationship between pupil and field vectors and the wavefront aberration expansion variable, .

Tables (2)

Tables Icon

Table 1. Third order and raytrace derived tolerance data for several R=500µm radius SIL configurations and fixed u.

Tables Icon

Table 2. 3rd order and raytrace derived tolerance data for several R=1.5mm SIL configurations and fixed u.

Equations (83)

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W 040 = 1 8 A 2 y Δ ( u n )
W 131 = 4 γ W 040 ,
W 222 = 4 γ 2 W 040 ,
W 220 S = 2 γ 2 W 040 ,
W 311 = 4 γ 3 W 040 ,
W 400 = γ 4 W 040 ,
γ = A ¯ A ,
A = n i = n i ,
A ¯ = n i ¯ = n i ¯ ,
Δ ( u n ) = u n u n .
NA = nu
α = y R
y = lu
y = l u
y ¯ y ¯ = l u ¯
i = u α
i = u α
i ¯ = u ¯ α ¯
A = n ( u α ) .
A = n ( u + y R ) .
A ¯ = n ( u ¯ + y ¯ R ) .
n u ¯ = n u ¯ y ¯ ϕ ,
ϕ = ( n n ) R .
u ¯ ' = n n u ¯ y ¯ ( n n ) n R .
u = i + α .
ni = n i ,
u = n n ( u α ) + α .
u = nu n y n ' R ( n n ) ,
u = NA · R n R l ( n n ) .
n u = nu y ( n n ) R
Δ ( u n ) = u n u n = nu n 2 y ( n n ) n 2 R nu n 2
Δ ( u n ) = NA ( 1 n ' 2 1 n 2 ) y ( n ' n ) n ' 2 R
l = y u .
l = y u ( n n l ( n n ) nR ) .
l ( nR + l ( n n ) nR ) = l n n
l = l n R n R l ( n n ) ,
l = l n R n R l ( n n ) ,
n ( u + y R ) = 0
y = u R
l = R .
u n = n u n 2 + y ( n n ) n 2 R = u n
y u ( n n ) n 2 R = n n 2 1 n
l = ( n + n ) n R .
l = ( n + n ) n R .
s = λ NA ,
s = λ NA = λ n u ,
s = λ n n R l ( n n ) NA · R .
s = λ n n R R ( n n ) NA · R
s = λ NA n n = s n n
s = λ n n R R ( n + n ) ( n n ) n NA · R
s = λ NA 1 n ( n + n 2 n 2 n )
s = λ NA ( n n ) 2 = s ( n n ) 2
W 40 = ( l R ) 2 R n ( n 1 ) ( 1 cos ( θ ) 1 2 sin 2 θ )
u = u n R l ( n n ) nR .
l = n ( n n ) R
W 040 = 1 8 n 2 n 2 u 4 R ( n n ) 3
δ λ W 020 = 1 2 Ay Δ ( δ n n ) ,
δ λ W 111 = A ¯ y Δ ( δ n n ) ,
( δ n n ) = n 1 n ν ,
ν = n M 1 n s n L ,
n 1 n ν = n 1 n ν .
n = n ν ν ( 1 n ) + n ν .
u ¯ = y ¯ R .
W 220 S = W 220 P + 1 2 W 222 ,
W 220 M = W 220 P + W 222 ,
W 220 T = W 220 P + 3 2 W 222 ,
W 220 P = 1 4 א 2 P ,
א = n u ¯ y n u y ¯ ,
P = C Δ ( 1 n ) = n n ' n ' n R .
W = j , m , n W klm H k ρ l cos m φ ,
k = 2 j + m ,
l = 2 n + m .
G = ρ + γ H ,
W = W 040 G 4
= W 040 ( G · G ) 2 .
G · G = ( ρ + γ H ) · ( ρ + γ H )
= ρ · ρ + 2 γ H · ρ + γ 2 H · H .
( G · G ) 2 = ( ρ · ρ ) 2 ( W 040 , Spherical )
+ 4 γ ( H · ρ ) ( ρ · ρ ) ( W 131 , Coma )
+ 4 γ 2 ( H · ρ ) 2 ( W 222 , Astigmatism )
+ 2 γ 2 ( H · H ) ( ρ · ρ ) ( W 220 , Field Curvature )
+ 4 γ 2 ( H · H ) ( H · ρ ) ( W 311 , Distortion )
+ γ 2 ( H · H ) 2 . ( W 400 , Piston )

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