Image Formation and Nonlinear Transfer E. MENZEL AND I. WEINGΛRTNER

Instilut A für Physik der Technischen Hochschule, 33 Braunschweig, Germany

(Recieved 10 January 1967) INDEX HEADINGS: Microscopy; Fourier transforms; Modulation transfer;

Image formation.

LINEAR Fourier formalism can describe the distribution of energy in the image only when the object is illuminated

incoherently.1 In many other cases, and these include examples in photography and vision, the transfer of high modulations involves nonlinear processes. Image formation in the microscope can be treated as linear only when the object has small amplitude and phase modulation.2,3 Limits for the maximum modulation as a function of the tolerable nonlinear perturbation have been discussed.4 For some special techniques in microscopy, such as the phase-contrast method invented by Zernike, the distributions of energy in the images of certain simple objects can be calculated exactly.5'6 The Fourier transform of any such distribution can be divided by the Fourier transform of the object to obtain a transfer function. But, in the case of nonlinear transfer, the result de­pends not only on the optical system but also on the object. For such cases we call the transfer function a "pseudo transfer function"; it is related to the "describing function."7 By dis­cussing simple examples, we can learn the properties of these transfer functions.

Figure 1 shows the calculated I (x) for the phase-contrast image of a phase edge with a phase retardation φ. For simplification, coherent illumination and a symmetrical pupil without an aper­ture stop are assumed. The Zernike plate is described by a phase shift 7Γ/2, an amplitude transmittance t, and a width corresponding to the spatial frequency R = 2M. Such a phase microscope is a symmetrical channel of transfer and does not alter the symmetry of the object so long as linear transfer can describe the image. As φ increases, the image of an antisymmetrical phase edge becomes asymmetrical. This is seen in Fig. 1. In the formula for the image I(x), the term A (x) is antisymmetrical in x and the term S(x)

FIG. 1. Phase-contrast images of phase edges with the phase retarda­tion φ. t: amplitude transmittance of the Zernike plate. Zero of I(x) is suppressed.

is symmetrical

A and S are form factors independent of the edge modulation φ. For small φ, I (x) is proportional to φ and is characterized by linear transfer. For larger φ both terms contribute to the nonlinearity.

Functions Bn(R), which describe the nonlinear transfer, were obtained from a Fourier transform of the calculated I(x) and division by the Fourier transform of the object. Bn then has the form

R is the spatial frequency. A third term resulting from t2 in Eq. (1) is without interest. The real and imaginary terms in Eq. (2) correspond to A and S, respectively. Both terms contain modula­tion factors and a form factor E or G. E(R) is readily calculated and G(R) may be obtained with a computer. Figure 2 shows E and G for different amplitude transmittances t of the Zernike plate. When t = 0, E becomes zero and a dark-field image results. Different means are used to normalize G when t = 0 and when t≠O.

FIG. 2. Form factors of the pseudo-transfer-functions for Fig. 1.

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When φ is small, Bn (R) = E (R). E (R) is thus the phase-contrast function previously obtained from linear considerations.2,3 Equa­tion (2) shows that this function can be measured not only when the modulation φ of a phase edge is small but also when it is larger and provides a greater output signal and better accuracy. These new results agree well with previous measurements.8

A similar calculation was made for an object consisting of a phase strip. This produces no symmetry change in the image and therefore no imaginary term in the corresponding Bn(R). Two modulation functions appear again as in Eq. (2) and the term sinφ/φ combines with the same E(R). The term containing (1 – cosφ)/φ is real and the form factor differs from that in Eq. (2); it approaches infinity for values of R corresponding to the width of the object strip, because these values of R make the frequency spectrum of the object zero.

1 H. H. Hopkins, Proc. Roy. Soc. (London) A217, 408 (1953). 2 E. Menzel, Optik 15, 460 (1958). 3 E. Menzel, in Optics in Metrology, P. Mollet, Ed. (Pergamon Press, Oxford, 1960). 4 H. Hauser, Opt. Acta 9, 121 (1962). 5 E. Menzel, Optik 5, 385 (1949). 6 H. Wolter, Ann. Physik, Leipzig 7, 33 (1950). 7 E. Ingelstam, Japan J. Appl. Phys. Suppl. I, 4, 15 (1965). 8 D. Haina and E. Menzel, Optik 21, 1 (1964).

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