Teoria elementare del microscopio a contrasto di fase

Riassunto

Dopo aver esposta un'analisi matematica semplificata della teoria fisica diAbbe sulla visione microscopica, si studia, sulla guida di essa, la teoria della formazione di immagini di oggetti perfettamente trasparenti, introducendo il principio della microscopia in contrasto di fase ideato daF. Zernike. Si conclude il lavoro ricavando le espressioni dei valori da assegnaro ai parametri che permettono la formazione di immagini a contrasto massimo.

Summary

A quantitative theory which should describe in detail the working of the phase contrast microscope in unrestricted conditions offers mathematical difficulties which hamper the treatment of such subjects. It is therefore preferable to work out some elementary theories for ideal cases, where such conditions are chosen that results can reasonably be expected to be valid in more complicated experimental circumstances.

The simplifying conditions introduced into the hypotheses are:

• a) The object has a periodical structure, extends to infinity and must be considered as absolutely flat.

• b) The light source is a monochromatic radiating point on the optical axis.

The paper gives at first an abridged account ofAbbe's theory on the image formation of an illuminated object. The object being an amplitude grating, the interference of all the diffracted waves in the image planeP' can be calculated following the optical diagram in fig. 1 and using LordRayleigh's formula (3) for the amplitude of the diffracted wave of orderk. The result of the calculation is given by equation (4) for the amplitudeA _x in the image plane, which represents the amplified image of the object grating. Coefficientc which appears in the formula is the breadth ratio of the opaque to the transparent strips of the grating.

Following the same procedure equation (7a) represents the image of a pure-phase grating similar to the preceding one except that the opaque strips are substitued by transparent strips of phase retardation ϕ (fig. 4). Formula (7a) shows that the vibration in the image plane of a pure-phase grating is given by the sum of two vectors of different phase: one representing the contribution of the zero-order wave and the other representing the resulting action of all the diffracted waves except the zero-order one. For small values of ϕ, as in the majority of practical cases, the two vectors differ in phase alternately π/2 or −π/2 (fig. 6) and therefore no visible contrasted image of the object occurs.

ApplyingZernike device for contrast phase microscopy, one introduces in the zero-order wave both an amplitude and a phase modification, in such way as to obtain the two vectors of equal amplitude and alternately in phase concordance and opposition. Under such conditions a full-contrast image of the phase-object is obtained.

The phase-shift ψ and the energy absorption coefficientt to be introduced in the zero-order wave is given by (8) and (9). Formula (11) and (12) give the approximations for small values of ϕ.

Fig. 9 shows the dependance of the absorption coefficientt on the object phase modulation ϕ for the different values of thec ratio.

Fig. 10 shows instead how the absorption coefficient depends (for small ϕ) on the numerical aperture of a finite lens, each curve referring to a lens accepting only up to the 1st, 2nd...kth-order diffracted wave. The ascissa is again thec-ratio.