Abstract

A microscope objective with (small) central obstruction of aperture and small wave-front deformation has a loss of amplitude at the center of the Airy disk which may be expressed in terms of losses due to spherical aberration, coma, and central obstruction. These losses are essentially independent; the net loss is the sum of the separate losses. Data are presented from which these losses for Schwarzschild spherical mirror systems of high initial magnification may be computed. It is shown that for visible or ultraviolet light and for numerical apertures greater than about 0.5, the net loss of central amplitude becomes excessive unless the object field is restricted to an extraordinarily small diameter. The numerical aperture at which such restriction of the field is necessary may be increased if the initial magnification is made small, e.g., about 8×, as is possible and convenient for infrared microspectroscopy.

© 1951 Optical Society of America

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References

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  1. A. E. Conrady, Applied Optics and Optical Design (Oxford University Press, London, 1929).
  2. D. S. Grey, J. Opt. Soc. Am. 40, 283 (1950), Fig. 11.
    [CrossRef]
  3. See reference 2, especially Fig. 4, which shows such an objective adapted for immersion use.
  4. Norris, Seeds, and Wilkins, J. Opt. Soc. Am. 41, 111 (1951).
    [CrossRef]
  5. R. Kingslake, Proc. Phys. Soc. (London) 61, 147 (1948).
    [CrossRef]
  6. Blout, Bird, and Grey, J. Opt. Soc. Am. 40, 304 (1950).
    [CrossRef]
  7. D. S. Grey, U. S. Pat.2,520,634.

1951 (1)

1950 (2)

1948 (1)

R. Kingslake, Proc. Phys. Soc. (London) 61, 147 (1948).
[CrossRef]

J. Opt. Soc. Am. (3)

Proc. Phys. Soc. (London) (1)

R. Kingslake, Proc. Phys. Soc. (London) 61, 147 (1948).
[CrossRef]

Other (3)

D. S. Grey, U. S. Pat.2,520,634.

A. E. Conrady, Applied Optics and Optical Design (Oxford University Press, London, 1929).

See reference 2, especially Fig. 4, which shows such an objective adapted for immersion use.

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Figures (8)

Fig. 1
Fig. 1

The Schwarzschild mirror arrangement consists of a concave mirror and a convex mirror. Light from the short conjugate focus is incident first on the concave mirror, then on the convex mirror.

Fig. 2
Fig. 2

The effect of obscuration on the distribution of energy in the Airy disk is shown here. The abscissa is radial distance z from the center of the Airy disk; the ordinate is integrated intensity within the annular region z to zz.

Fig. 3
Fig. 3

Coma increases as area obscuration decreases. F is the coefficient of third-order coma for zero spherical aberration, plotted here as a function of Q. G is a correction term for higher order coma, etc., the amplitude loss due to coma is approximately equal to 0.274y2(1−Q)−1λ−2[(NA)3F+(NA)5G]2.

Fig. 4
Fig. 4

The third-order estimate of coma yields an estimate of the semidiameter y of the object field divided by wavelength of light λ. The above curve is obtained by permitting a total loss of intensity at the center of the Airy disk of 22.5 percent, and assuming that spherical aberration causes one-third of this loss. Then, when (y/λ)NA3 assumes the values above, obscuration and the coma predicted by third-order theory absorb the remaining tolerance for intensity loss. Field diameters indicated by this figure are overestimated by a factor equal to the quantity computed/predicted, given in Table I.

Fig. 5
Fig. 5

The wave-front deformation s due to spherical aberration is plotted against numerical aperture R for a system of numerical aperture 0.705, area obscuration Q=0.1, and focal length 1 mm. The range of s varies with Q but only by a trivial amount; it is proportional to focal length, and it is approximately proportional to (NA)7 for 0.50≦NA≦0.75.

Fig. 6
Fig. 6

The loss of central amplitude due to the wave-front deformation s is computed by plotting 2π2(s/λ)2 vs r2 where r is numerical aperture divided by maximum numerical aperture. The loss of central amplitude is equal to the average value of this curve divided by the fraction of aperture which is not obscured. The curve above applies to the deformation s shown in Fig. 5. The amplitude loss is found to be 0.124 giving an intensity loss of 23 percent from spherical aberration alone.

Fig. 7
Fig. 7

Object-field size plotted against Q, computed according to the criterion: (δs+δc+δQ) gives an amplitude loss not greater than 12 percent at the center of the Airy disk. Curve I: NA 0.705, focal length 0.6 mm, no coverslip. Curve II: NA 0.65, f=0.6 mm with 0.2-mm coverslip, or NA 0.65, f=1.2 mm, no coverslip. Curve III: NA 0.65, f=0.9 mm, no coverslip. Curve IV: NA 0.65, focal length 0.6 mm without coverslip. Curve V: NA 0.50, focal length=3 mm. Curve VI: NA 0.40, focal length=15 mm. The loss δc from coma is independent of focal length, but the loss δs from spherical aberration is inversely proportional to focal length squared; hence, when focal length is decreased, a larger portion of the total tolerance remains for coma (compare curves II, III, and IV). When a coverslip is introduced, zonal spherical aberration increases, and the portion of the tolerance consumed by δs increases (compare curves IV and II). The objectives whose performance is depicted by V and VI have virtually no spherical aberration for the focal length and NA specified. All curves are for wavelength 0.25 micron.

Fig. 8
Fig. 8

When the area obscuration Q is made small, the linear dimensions for a given focal length increase. The ratio of diameter D of the concave mirror divided by diameter d of the convex mirror is plotted above vs Q. This ratio is also equal to the distance from the concave reflecting surface to the object, divided by the focal length of the system.

Tables (1)

Tables Icon

Table I Third-order estimates of offense against the sine condition are less than computed values. The ratio by which actual comatic wave-front deformation is greater than the predicted value is equal to the ratio of the computed value of [ ( B 1 + 1.2 B 2 ) 2 + 0.06 B 2 2 ] 1 2 divided by the predicted value of B1 This ratio is tabulated as computed/predicted. For a given value of Q, the ratio should approximately fit the form: computed/predicted=1+D(NA)2, where D is a constant dependent on Q, and increasing as Q increases.

Equations (28)

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A = ( K ) 1 2 n a N A - π π cos [ 2 π ( + 0 ) λ ] R d R d θ .
A 1 A 0 = n a N A - π π cos [ 2 π ( + 0 ) / λ ] R d R d θ π N A 2 [ ( N A 2 - n a 2 ) / N A 2 ] 1 2 .
A 1 A 0 = [ 1 - ( n a N A ) 2 ] 1 2 × { 1 - n a N A - π π { 1 - cos [ 2 π ( + 0 ) / λ ] } R d R d θ π N A 2 [ 1 - ( n a / N A ) 2 ] } .
A 1 A 0 = ( 1 - Q ) 1 2 × { 1 - r 1 1 - π π { 1 - cos 2 π [ ( + 0 ) / λ ] } r d r d θ π ( 1 - Q ) } .
A 1 A 0 = ( 1 - δ Q ) ( 1 - δ ) .
δ Q = 1 - ( 1 - Q ) 1 2 = 1 2 Q + 1 8 Q 2 + 1 16 Q 3 + .
δ = r 1 1 - π π [ 1 - cos [ 2 π ( + 0 ) / λ ] r d r d θ π ( 1 - Q ) .
+ 0 = s + c = F 1 ( r 2 ) + r y cos θ F 2 ( r 2 ) .
cos 2 π ( s + c λ ) = 1 - 1 2 [ 2 π ( s + c ) λ ] 2 + 1 24 [ 2 π ( s + c ) λ ] 4 + .
δ = r 1 1 - π π 1 2 [ 2 π ( s + c ) / λ ] 2 r d r d θ π ( 1 - Q ) .
δ = δ s + δ c ,
δ s = r 1 1 - π π ( 2 π 2 / λ 2 ) s 2 r d r d θ π ( 1 - Q ) = 2 π 2 λ 2 ( 1 - Q ) r = r 1 1 s 2 d ( r 2 ) ,
δ c = r 1 1 - π π ( 2 π 2 / λ 2 ) c 2 r d r d θ π ( 1 - Q ) = 2 π 2 λ 2 ( 1 - Q ) r 1 1 c 2 r d r .
A 1 / A 0 = ( 1 - δ Q ) ( 1 - δ s - δ c ) = 1 - δ Q - δ s - δ c ( nearly ) .
O S C = b 1 R 2 + b 2 R 4 .
c = y R cos θ ( b 0 + b 1 R 2 + b 2 R 4 ) .
c = y r cos θ ( B 0 + B 1 r 2 + B 2 r 4 ) ,
δ c = 2 ( y λ ) 2 π 2 1 - Q r 1 1 ( B 0 + B 1 r 2 + B 2 r 4 ) 2 r 3 d r ,
δ c = 2 ( y λ ) 2 π 2 1 - Q { B 0 2 4 ( 1 - Q 2 ) + B 0 B 1 3 ( 1 - Q 3 ) + B 0 B 2 4 ( 1 - Q 4 ) + B 1 2 8 ( 1 - Q 4 ) + B 1 B 2 5 ( 1 - Q 5 ) + B 2 2 12 ( 1 - Q 6 ) } .
- B 0 = 2 [ B 1 3 ( 1 + Q + Q 2 ) 1 + Q + B 2 4 ( 1 + Q 2 ) ] .
δ c = 2 π 2 1 - Q ( y λ ) 2 { B 1 2 [ 1 - Q 4 8 - 1 + Q 2 9 ] + B 1 B 2 [ 1 - Q 5 5 - 1 + Q 2 + 6 ] + B 2 2 [ 1 - Q 6 12 - 1 + Q 2 + 16 ] } .
δ c = 2 π 2 1 - Q ˙ ( y λ ) 2 { B 1 2 ( 1 8 - 1 9 ) + B 1 B 2 ( 1 5 - 1 6 ) + B 2 2 ( 1 12 - 1 16 ) } .
δ c = 0.274 1 - Q ( y λ ) 2 { ( B 1 + 1.2 B 2 ) 2 + 0.06 B 2 2 } .
y λ ( N A ) 3 = 1.91 ( 1 - Q ) 1 2 ( 0.08 - 1 2 Q - 1 8 Q 2 ) 1 2 F .
C = [ ( B 1 + 1.2 B 2 ) 2 + 0.06 B 2 2 ] 1 2 estimated B 1 = B estimated B 1 .
y λ = 1.91 ( 1 - Q ) 1 2 ( 0.12 - δ s - 1 2 Q - 1 8 Q 2 ) 1 2 ( N A ) 3 F + ( N A ) 5 G .
δ s = 2 π 2 r 1 1 s 2 d ( r 2 ) λ 2 ( 1 - Q ) .
s = H 0 + H 1 r 2 + H 2 r 4 + H 3 r 6 + .