570 LETTERS TO THE EDITOR Vol. 55 Plane of Best Focus for a Sine-Wave Target in the Presence of Spherical Aberration RICHARD BARAKAT AND ELENA KITROSSER Optics Department, Itek Corporation, Lexington, Massachusetts 02173 (Received 11 December 1964) T HE usual theories 1-3 for determining the plane of best focus for an optical system with spherical aberration are defined in terms of the influence of the system on a point image. These theories are essentially equivalent to maximizing the Strehl criterion t(0), the ratio of the central illuminance of the system under consideration to the central illuminance of an aberration- free Airy objective for the same wavelength and numerical aper- ture; thus where W(ρ) is the aberration function. Note that t(0) always lies between zero and one. It can be shown that where T(ω) is the transfer function of the optical system in question. The cutoff frequency of the system in dimensionless units is ω c =2; this is related to the cutoff frequency in dimensional units by where Ω c is measured in lines/mm, λ in mm, and F is the ƒ/number of the system.
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570 L E T T E R S TO T H E E D I T O R Vol. 55
Plane of Best Focus for a Sine-Wave Target in the Presence of Spherical Aberration
RICHARD BARAKAT AND ELENA KITROSSER Optics Department, Itek Corporation, Lexington, Massachusetts 02173
(Received 11 December 1964)
THE usual theories1-3 for determining the plane of best focus for an optical system with spherical aberration are defined
in terms of the influence of the system on a point image. These theories are essentially equivalent to maximizing the Strehl criterion t(0), the ratio of the central illuminance of the system under consideration to the central illuminance of an aberration-free Airy objective for the same wavelength and numerical aperture; thus
where W(ρ) is the aberration function. Note that t(0) always lies between zero and one. I t can be shown that
where T(ω) is the transfer function of the optical system in question. The cutoff frequency of the system in dimensionless units is ω c =2; this is related to the cutoff frequency in dimensional units by
where Ωc is measured in lines/mm, λ in mm, and F is the ƒ/number of the system.
May 1965 L E T T E R S T O T H E E D I T O R 571
FIG. 1. Best focus for third-order spherical aberration for: (A) W4=0.7λ, (B) W4=0.9λ, (C) W 4 =l . l λ . The small circles indicate the computed values.
However, the process of maximizing the Strehl criterion is a global process in the sense that only the area under the transfer function is maximized and nothing is said concerning the values of T(ω) at any specified frequency. In many instances, it is necessary to determine the receiving plane for which the transfer function at a specified frequency is a maximum. The problem is of common occurrence in experimental measurements employing sine-wave targets. This suggests that we define the plane of best focus for a sine-wave target as the receiving plane W2 for which the transfer function is a maximum at a given frequency. Unfortunately, there does not seem to be any theory which allows this type of analysis to be carried out except by detailed calculation. The numerical analysis was performed for both third- and fifth-order spherical aberration.
For third-order spherical aberration, the wavefront is given by
where W4 is the third-order spherical-aberration coefficient measured in wavelength units and W2 is the defocusing parameter also in wavelength units. The optimum balancing condition in the context of the above theories is
That is, we must focus at half the distance between the paraxial plane (W2 = 0) and the marginal plane (W2 = —2W4).
The evaluation of the transfer function was performed using the program of Barakat and Morello.4 The results of the calculations are presented in Fig. 1 where the plane of best focus (W2) is plotted vs spatial frequency for values of W4 = 0.7, 0.9, 1.1. The computations were performed in the following manner: The trans-
FIG. 2. Best focus for fifth-order spherical aberration with ratio W4=-3W6/2 for: (A) W6=3λ, (B) W6=4λ, (C) W6=5λ, (D) W6=6λ. The small circles indicate the computed values.
fer function for W4 = 0.7 (say) was evaluated at W2=— 0.6 to —1.1 in steps of 0.025 and the maximum contrast was determined at ω = 0.2, 0.3, etc. Figure 1 was obtained in this manner. The three values of W4 were chosen so that when (5) was satisfied, t(0) = 0.9, 0.8, 0.7. The shift in focus is quite significant over the range of frequencies. In order to emphasize this shift, let us consider an ƒ/5 system operating in helium d light (5875.5 Å) which has a cutoff frequency of approximately 340 lines/mm and has an aberration function given by W4 = 0.7 [curve (A) in Fig. 1]. The plane of best focus corresponding to 51 lines/mm (ω = 0.3) is W2= -0 .94 , while that for 102 lines/mm (ω = 0.6) is W2= - 0 . 6 8 . This difference amounts to 0.28 wavelength units or approximately 1.2×10 -3 in., certainly a measurable amount.
As a second example, consider an optical system possessing fifth-order spherical aberration W6:
and in particular let
This ratio along with W2=3W6/5 determines the plane of best focus in the sense of maximum Strehl criterion. We set W6 = 3, 4, 5, 6 and let W2 vary as before; the results are recorded in Fig. 2. The curves labeled (A), (B), (C), (D) correspond to values of the Strehl criterion: t(0)=0.9, 0.8, 0.7, 0.6 when taken in the plane given by W2=3W6 /5 . These results are substantially different from those in Fig. 1. Note the pronounced dip in the medium-frequency region (ω~0.5) as the amount of spherical aberration is increased. As before, note the variations in the optimum focal position as the frequency is varied.
These results are sufficient to indicate that we must be careful in experimental studies where the plane of best focus for periodic detail is required.
1 R. Richter, Z. Instrumentenk. 45, 1 (1925). 2 W. Ta-Hang, Proc. Phys. Soc. (London) 53, 157 (1941). 3 A. Maréchal, Rev. Opt. 26, 257 (1947). 4 R. Barakat and M. V. Morello, J. Opt. Soc. Am. 52, 992 (1962).
