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Images of wedge-shaped objects under illumination of partially coherent light
Volume 37, number 1
OPTICS COMMUNICATIONS
1 April 1981
IMAGES OF WEDGE-SHAPED OBJECTS UNDER ILLUMINATION OF PARTIALLY COHERENT LIGHT N. NAKAJIMA and T. ASAKUBA Research Institute of Applied Electricity,
Hokkaido
University, Sappom, Hokkakio 060, Japan
Received 29 September 1980 Revised manuscript received 28 November 1980
The images of wedge-shaped objects under illumination of partially coherent light are investigated as a function of the vertical angle of objects and the spatial coherence condition of the ilhrminating light. The actual images of wedge-shaped objects are numerically evaluated by using an electronic computer and shown in contour lines of equal intensity.
In various optical instruments such as the microscope, microdensitometers, spectrographs, slide projectors, and enlargers, the object is illuminated by an illuminator whose numerical aperture is equal to or less than that of the following imaging systems. Under this condition, a certain degree of spatial coherence between any two neighbouring points appears in the illuminating light over the object plane and the image is consequently formed by partially coherent light. In addition, the study of coherent and partially coherent image formation has become important in recent years since the lasers are being used as an illuminator for various imaging systems. The image formation under illumination of partially coherent light was first treated by Hopkins [ 1,2 3 who introduced the ideas of an effective source and a projected source. Since the work of Hopkins, a large number of studies have been done on the partially coherent imaging properties of various objects such as the point, line, bar, edge, sinusoidal object, and circular disk. A detailed description on these past works can be found in an excellent review article [3] and, therefore, a further discussion on the past works is omitted here. Even though, as is stated above, the images of various extended objects under illumination of partially coherent light have already been performed, no study is found on the partially coherent image of a wedgeshaped extended object. In many cases, almost all the extended objects are constructed by wedge-shaped parts with various vertical angles. Therefore, the study for the images of wedge-shaped objects under illumina0 030-4018/81/0000-00000/$02.50
0 North-Holland
tion of partially coherent light is important and of great interest in various optical imaging systems, and becomes a problem to be performed in this paper. Kubota [4] only studied the images of wedge-shaped opening objects under the limited illumination of incoherent light. In this paper, the images of acute wedge-shaped opening objects (i.e. obtuse opague wedge objects) under illumination of partially coherent light are investigated as a function of the vertical angle of objects and the degree of spatial coherence in the illuminating light. When the object is illuminated with partially coherent light characterized by the mutual intensity function Q,52) sh owing the spatial coherence condition, its image is expressed by [3,5]
(1) where O(Q stands for the complex amplitude transmittance of the object, K(x -4) is the amplitude impulse response of an imaging system, and * indicates a complex conjugate. By assuming that the mutual intensity function is spatially stationary, i.e.
and if the replacement of coordinates e= (u, u) and x = (x, y) at the object and image planes by the Cartesian coordinates is taken, eq. (1) becomes
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5
OPTICS COMMUNICATIONS
1 April 1981
IMAGES OF WEDGE-SHAPED OBJECTS UNDER ILLUMINATION OF PARTIALLY COHERENT LIGHT N. NAKAJIMA and T. ASAKUBA Research Institute of Applied Electricity,
Hokkaido
University, Sappom, Hokkakio 060, Japan
Received 29 September 1980 Revised manuscript received 28 November 1980
The images of wedge-shaped objects under illumination of partially coherent light are investigated as a function of the vertical angle of objects and the spatial coherence condition of the ilhrminating light. The actual images of wedge-shaped objects are numerically evaluated by using an electronic computer and shown in contour lines of equal intensity.
In various optical instruments such as the microscope, microdensitometers, spectrographs, slide projectors, and enlargers, the object is illuminated by an illuminator whose numerical aperture is equal to or less than that of the following imaging systems. Under this condition, a certain degree of spatial coherence between any two neighbouring points appears in the illuminating light over the object plane and the image is consequently formed by partially coherent light. In addition, the study of coherent and partially coherent image formation has become important in recent years since the lasers are being used as an illuminator for various imaging systems. The image formation under illumination of partially coherent light was first treated by Hopkins [ 1,2 3 who introduced the ideas of an effective source and a projected source. Since the work of Hopkins, a large number of studies have been done on the partially coherent imaging properties of various objects such as the point, line, bar, edge, sinusoidal object, and circular disk. A detailed description on these past works can be found in an excellent review article [3] and, therefore, a further discussion on the past works is omitted here. Even though, as is stated above, the images of various extended objects under illumination of partially coherent light have already been performed, no study is found on the partially coherent image of a wedgeshaped extended object. In many cases, almost all the extended objects are constructed by wedge-shaped parts with various vertical angles. Therefore, the study for the images of wedge-shaped objects under illumina0 030-4018/81/0000-00000/$02.50
0 North-Holland
tion of partially coherent light is important and of great interest in various optical imaging systems, and becomes a problem to be performed in this paper. Kubota [4] only studied the images of wedge-shaped opening objects under the limited illumination of incoherent light. In this paper, the images of acute wedge-shaped opening objects (i.e. obtuse opague wedge objects) under illumination of partially coherent light are investigated as a function of the vertical angle of objects and the degree of spatial coherence in the illuminating light. When the object is illuminated with partially coherent light characterized by the mutual intensity function Q,52) sh owing the spatial coherence condition, its image is expressed by [3,5]
(1) where O(Q stands for the complex amplitude transmittance of the object, K(x -4) is the amplitude impulse response of an imaging system, and * indicates a complex conjugate. By assuming that the mutual intensity function is spatially stationary, i.e.
and if the replacement of coordinates e= (u, u) and x = (x, y) at the object and image planes by the Cartesian coordinates is taken, eq. (1) becomes
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5