As a first step towards a generalization of the Maggi-Rubinowicz theory of the boundary diffraction wave, a new vector potential W(Q,P) is associated with any monochromatic scalar wavefield U(P). This potential has the property that the normal component of its curl, taken with respect to the coordinates of any point Q on a closed surface S surrounding a field point P, is equal to the integrand of the Helmholtz-Kirchhoff integral; that is, curlQW(Q,P)·n=14π{U(Q)∂∂n(exp(iks)s)−exp(iks)s∂∂nU(Q)}.where s is the distance QP and ∂/∂n denotes the differentiation along the inward unit normal n to S.Further it is shown that the vector potential always has singularities at some points Q on S and that the field at P may be rigorously expressed as the sum of disturbances propagated from these points alone.A closed expression for the vector potential associated with any given monochromatic wavefield that obeys the Sommerfeld radiation condition at infinity is derived and it is shown that in the special case when U is a spherical wave, this expression reduces to that found by G. A. Maggi and A. Rubinowicz in their researches on the boundary diffraction wave.

Generalization of the maggi-rubinowicz theory of the boundary diffraction wave part I

Evaluation of photographic optical systems through use of the optical transfer function or sine wave response has become commonplace. The fundamental physical concepts which lie behind the theory are not, however, widely known and are outlined for use by the optical engineer. The significance, limitations, and advantages of this tool for image evaluation are discussed along with some suggestions of direction along which future work may be carried. The discussion is from the point of view of an individual interested in the design, fabrication, evaluation, and use of large systems of very high precision.

Optical Image Evaluation and the Transfer Function

Theoretisch untersucht wird der durch Aberrationen verursachte Abfall der (‘;inkoharenten’;) Ubertragungsfunktion optischer Systeme, und zwar bei ‘kleinen7rsquo; Orts-Frequenzen. In diesem Orts-Fre...

Der Einfluß der Aberrationen auf die Optische Übertragungsfunktion bei Kleinen Orts-Frequenzen

The treatment of the problem of deducing the phase of an optical wavefield from image- and diffraction-plane intensity measurements recently presented by Robinson [ J. Opt. Soc. Am68, 87– 92 ( 1968)] is shown to be valid only in the limit of geometrical optics. A simple proof and example of the nonuniqueness of the phase retrieval problem in this limit is presented. It is argued that although the phase retrieval is not unique in the geometrical optics limit it does appear to possess a unique solution within the framework of the diffraction theory of image formation.

On the uniqueness question in the problem of phase retrieval from intensity measurements

Spot diagrams give much information about the image formation of an optical system. In order to evaluate the image qualities of lenses quantitatively, it is shown that the geometric optical response function and the intensity distribution can be calculated directly from the coordinates of the spots. Image evaluation using the geometric optical “Strehl definition” is also discussed.Furthermore, the relation between wave optics and geometrical optics is considered in the case of partial coherence, and it is deduced that, when the wave aberration function is large compared with the wavelength (more than 2λ), the degree of coherence in the object plane scarcely affects the intensity distribution in the image plane.

On a Comparison between Wave Optics and Geometrical Optics by Using Fourier Analysis. III. Image Evaluation by Spot Diagram

In this paper the relation between wave optics and geometrical optics is studied by using Fourier analysis The geometric optical intensity distribution of light in the point image Ig(x,y) is expressed by Ig(x,y)=a−1|∂2ϕ∂u2∂2ϕ∂υ2−(∂2ϕ∂u∂υ)2|−1,where ϕ(u,υ) is the aberration function and (x,y), (u,υ) are the normalized coordinates in the image plane and the exit pupil, respectively, a being the area of the (u,υ) region Hereupon the geometric optical response function Rg(s,t) is given by the Fourier transform of Ig(x,y), that is; Rg(s,t)=a−1∫∫Aexp[−2πi(s∂∂u+t∂∂υ)ϕ]∂u∂υ,where (s,t) are the normalized line frequenciesThen the foregoing formula for Rg(s,t) is compared with the wave optical response function which H H Hopkins has deduced, and it is shown that, when the magnitude of the aberration function ϕ(u,ϕ) is small compared with the wavelength, the wave optical intensity distribution Iw(x,y) can be expressed approximately by the convolution of Ig(x,y) and the intensity distribution of the aberration-free diffraction image, and when the magnitude of ϕ(u,ϕ) becomes large compared with the wavelength, Iw(x,y) is given approximately by Ig(x,y), if Ig(x,y) does not become infinite at any point In the case where Ig(x,y) is infinite at certain points, if we neglect the high-frequency Fourier component of the intensity distribution, Iw(x,y) is nearly equal to Ig(x,y)

On a Comparison between Wave Optics and Geometrical Optics by Using Fourier Analysis. I. General Theory

In the previous paper (part I), the relation between wave optics and geometrical optics has been studied generally by using the Fourier analysis. In this paper the general theory is applied to the concrete examples of the astigmatism, coma, and spherical aberration, and the following results are obtained.In case of astigmatism, if the magnitude of the wave aberration |ωλ| is larger than 2λ (λ being the wavelength), the geometric optical response function Rg(s) approximates to the wave optical response function Rw(s) within the region of the normalized frequency s where |λs/2| 1.2λ and |λs/2| 1.6λ and |λt/2| 1.3λ and |λs/2|<0.4.

Comparison between Wave Optics and Geometrical Optics Using Fourier Analysis. II. Astigmatism, Coma, Spherical Aberration

The problem of how the aberration of a lens should be balanced is studied in order to get the best result in combination with its receiver (photographic emulsion, television pick-up tube, etc.).The intensity distribution of light in the line image formed by the lens is assumed to be expressed approximately by the sum of two error functions corresponding to the image core and halo, respectively, and the diffusing characteristics of the light receiver are represented by a third error function. A formula is deduced which expresses the resultant response of the total system (lens plus receiver) with parameters used in the three error functions described above. As a measure of the over-all image quality of the total system the maximum value in the resultant response (S.D.)—the generalized Strehl definition—is chosen.This method is applied to the error balancing of a lens having primary and secondary spherical aberration, the intensity distribution being obtained by spot diagrams. As a result, the over-corrected type is better when the receiver has a small degree of diffusion, and the under-corrected type is preferred when it has a large one.

Kenrō Miyamoto

Papers

On a Comparison between Wave Optics and Geometrical Optics by Using Fourier Analysis. I. General Theory

Comparison between Wave Optics and Geometrical Optics Using Fourier Analysis. II. Astigmatism, Coma, Spherical Aberration

On the error balancing of an optical system in combination with its receiver.