In this paper, a new set of orthogonal functions convenient for the aberration analysis of the laser TEM00 mode is derived. This set of functions is a hypergeometric function. In terms of it, we can expand the aberration function of laser beam optics, and evaluate laser beam optical systems. Furthermore we can calculate the diffraction pattern.
As examples, the diffraction patterns of images of 1st coma and astigmatism aberrations are calculated. The application of this function to Strehl Definition (S.D.) and the comparison of behaviours between laser beam and constant amplitude beams in optical systems are discussed.

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Optimum discharge and cavity parameters for output power, amplification of small signal, output power curve versus cavity tuning and CH4 absorption are investigated experimentally for the 3.39μm He-Ne laser under various conditions.
Dependence of output power on discharge current, pressure, tube diameter and mirror transmittance is shown graphically. Output power curves versus cavity tuning shows a Lamb dip in a pressure rang lower than 1 torr and an inverted Lamb dip with a CH4 absorption cell in a cavity. Height, width and profile of the inverted Lamb dip are examined. Absorption coefficient of CH₄ at room temperature is also measured.

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An in-line X-ray hologram is made by using Alkα (8.34Å) or CKα (44.8Å) radiation to test their fitness for recording X-ray hologram. The result shows that AlKα is superior to CKα in its coherence and recording time. The X-ray source size is estimated to be several microns by Young's experiment. The effect of source size on X-ray holograms is demonstrated by the reconstructed real image of a fiber or a fine mesh. It is shown that a semitransparent object of a chemical fiber works as a lens for the AlKα line. A preliminary experiment of lensless Fourier-transform hologram is given.

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The theory and experimental results of diffraction-limit of moire contouring using coherent light from a point source described in Part (1) are extended to the moire fringes using an imperfect light source.
It is undesirable to use an ideal source such as a laser in a practical equipment from the view point of handling and cost. And it is desirable that the source which is easy to handle and is of low cost, for instance, as an incandescent lamp is able to be use for the equipment.
The diffraction-limit of moire contouring using an imperfect source, which has some spatial and spectral extent, and an observing method of the moire fringes are presented in this paper.

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Spatial filtering is shown to be an effective method for inspecting integrated circuit photomask which is the periodic array of a unit pattern. Detectable defect size and requirements for optical system and photomask period are discussed. The requirement for Fourier transforming lens is most severe, especially for masks of large period.

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The moire gauging technique proposed by Duffy is followed up and critical factors associated with the experiment are investigated. The technique is extended further to simultaneous measurement of two orthogonal components of in-plane displacement by the use of two pairs of small apertures located on apexes of a rhombus. The effect of out-of-plane displacement of the object under test is also studied.

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