Obliquity Factor Research Articles

Coherence theory of a radio telescope

A new theory is developed to describe the operation of a radio telescope consisting of an array of two-element interferometers. The theory is based on recent developments in the theory of partial coherence, which employs an expansion of the scalar field amplitude into an angular spectrum of plane waves. Expressions are found describing the correlation of the field amplitudes in the antenna plane which are more general than usually obtained, and which suggest a simplified data processing procedure which might yield improved images. The usual quasi-monochromatic approximation is made and the conditions under which it is valid are discussed in detail. Finally the theory is connected with the usual procedure for producing images of the radio sources and the existence of an obliquity factor is shown which could affect images over a large field of view.

Some uses of the Huygens' obliquity factor

It is shown how the effect of the mounting surface for a flush‐mounted aperture antenna can be considered as a modification of the Huygens' obliquity factor, which represents the pattern of the radiation of an element of the aperture. The same method, in conjunction with Babinet's principle, applied to a small slit, gives the corresponding element pattern for an elementary dipole and lends credence to the method even when applied to very small apertures. An application is made to a solid‐state laser configuration, and a result obtained by a more involved method is shown to correspond to the obliquity factor for an air‐dielectric interface. A correction to the formula is also found, and an (apparently new) expression is found for the obliquity factor when there is a change of medium at the aperture.

The propagation of gaussian beams very near the source

The field of a gaussian beam is examined in detail. It is shown that the usual formula for the field does not satisfy the wave equation within the Rayleigh distance except in the limit of large waist diameter. For small waist sizes the departure from the results of the usual formula can be substantial within a distance of about 1.3 wavelengths from the source. An exact formula is found for the on‐axis radiation. In order to make this study, certain features such as the obliquity factor, which is commonly ignored, have to be considered in detail. This leads to the need for a physical characterization of the source, which is not capable of a unique specification. The choice is probably not critical—the usual formulas for the transformation of gaussian beams do not need to refer to it at all—and the one used here pertains closely to the condition of radiation from the dielectric‐air interface of a solid state laser.

Obliquity-factor correction to solid-state radiation patterns

The Huygens obliquity factor, in a special form appropriate to the dielectric-air interface of the laser, provides an additional factor to the existing radiation pattern formula, and leads to a beam sharpening which gives good agreement with experimental measurements. A simple approximate formula relates the half-power angles as calculated with and without the correction, though it is inadequate to take into account finer structure-dependent features.

Calculation of the far-field halfpower width and mirror reflection coefficients of double-heterostructure lasers

A comparison is made between theoretical and experimental values of the beamwidth of a d.h. injection laser. The importance of the obliquity factor is pointed out. The good agreement between theory and experiment permits the determination of material properties from laser-diode measurements. Further, mirror reflection coefficients for lasers immersed in different liquids are calculated. Marked differences in the values of the Fresnel reflection coefficients, for normal incidence, are found.

Obliquity factor for radiation from solid-state laser

The radiated field consists of forward radiation at an angle θ and backward radiation at π – θ reflected from the dielectric-air interface into an additional component of field at angle θ. The total radiation contains a factor that simplifies into an expression recently obtained by more sophisticated methods, and which is here interpreted as the Huygens obliquity factor for the arrangement. A small correction to the expression is also obtained.

A Theoretical Study of the Axial Field Amplitude of Microwave Paraboloidal, Spherical and Plane Zone Plate Antennas

A theoretical investigation of the spherical, the plane and the paraboloidal microwave zone plate antennas has been made on the basis of the Fresnel-Huyghen's diffraction theory. Rigorous formulae for the field amplitude at an axial point for the three zone plates have been derived. Some approximate expressions have also been given by assuming that the obliquity factor remains almost a constant over a zone.

Obliquity Factors of Huygens and Kirchhoff Theories Applied to Experimental Diffraction by Two Long, Coplanar, Parallel Strips

Fraunhofer diffraction by two long, thin, parallel conducting strips in a plane is investigated experimentally using microwaves whose wavelength is of the order of the strip width. The radiation is polarized with the electric field vector parallel to the strip axes. The results are compared with scalar theory, employing either the obliquity factor suggested by Stokes or that obtained in a Kirchhoff theory development. It is found for this polarization that, while both scalar theories give fairly good agreement with experimental results, the one employing Kirchhoff's obliquity factor more closely predicts the results obtained in the laboratory.

Particle Sizing by Means of the Forward Scattering Lobe

The angular distribution of scattered light in the main lobe of the Fraunhofer diffraction pattern of a particle changes rapidly with its size, but is largely independent of its refractive index. Even when the particle is so small (diameter less than a few wavelengths) that its forward lobe must be calculated from the Mie theory, the relative angular distribution within the central part is found to be given approximately by the Fraunhofer formula with obliquity factor, even for particles smaller than one wavelength, except near a minimum in the particle extinction efficiency factor calculated from the Mie theory. A measurement of the ratio between the scattered intensities at a pair of convenient angles within the lobe can thus give a useful estimate of the size of spherical and nonspherical particles without knowing their refractive index. The sizing error caused by the narrowing of the lobe near an extinction minimum is acceptable for many practical purposes, and can in any case be detected by comparing the size estimates derived from measurements at two different wavelengths. Families of curves for sizing by this simple procedure have been prepared. Sideways scattering instruments have been preferred hitherto because of their easier construction. A realization of their possibilities for sizing unknown particles should, however, stimulate the development of suitable forward scattering instruments.

Convolution Formulation of Fresnel Diffraction*

A convolution formulation of Fresnel diffraction is presented. The diffracted amplitude is expressed as a convolution in either direct or reciprocal space. Approximations involve the use of parabolic wavefronts and the omission of the obliquity factor. The formulation is readily applied to many optical phenomena.A theory of Fresnel transforms is given. The Fresnel transforms are expressed as a convolution in either direct or reciprocal space. Two examples of their use are given.

On Plane Blazed Gratings

The Fraunhofer patterns of blazed gratings are derived on the basis of a scalar theory which includes the non-linear dependence of the obliquity factor and the phase modulation on the spatial frequencies defining the positions of the source and of the observer. The solution based on the usual ‘linear communications’ theory is compared with one based on the more general non-linear theory; it is shown that the former is meaningful only in the neighborhood of the blaze wavelength. The behavior of blazed gratings is examined in the light of non-linear theory in the region away from the blaze wavelength. It is shown that the envelope function describing the amplitude distribution due to a single groove depends on the single parameter defining half the phase difference between the two edges of a single diffracting facet. It is also shown that certain wavelengths are missing from the zero order and that ‘dark’ lines exist into which no light of any order is transmitted. A useful maximum for the aspect ratio is derived. The Littrow and spectrograph configurations are examined in some detail.

The effect of flanges on the radiation patterns of waveguide and sectoral horns

Measurements are given of radiation patterns of open-ended rectangular waveguide with conducting flanges attached to the long edges of the aperture. In particular, the considerable effect of different flange lengths and included angles on the E-plane patterns is examined; these measurements are made at three frequencies in order to supplement data, already published, describing measurements made at a single frequency. A suggestion that the E-plane radiation pattern can be approximated by the radiation pattern of a primary source and two secondary sources near the end of the flanges has been substantiated if a suitable obliquity factor is assumed. Some E-plane patterns of flanged H-plane sectoral horns are also given. It is found that there are differences between these patterns and those of similarly flanged waveguide.The effect of flanges on the H-plane patterns of an E-plane sectoral horn is comparatively small, as was expected.

A New Formulation of Scalar Diffraction Theory for Restricted Aperture

A formulation for scalar problems in physical optics is presented. Approximations involve the neglecting of the obliquity factor and the dependence of amplitude on distance and the replacement of spherical by paraboloidal wavefronts. Expressions are obtained for wave function in a multi-component system in terms of convolution integrals, the components of the system being described by functions of either direct or Fourier space. The formulation is equally applicable to light optics, electron microscopy or electron diffraction and permits interchange in the techniques of these disciplines. Some elementary examples of the application to thin lenses, and to periodic objects are given.

An ionospheric attenuation equivalence theorem

The phenomenon of radio-wave attenuation by way of ionospheric partial reflections and scattering is examined theoretically, and an equivalence theorem, relating oblique incidence to vertical-incidence phenomena, is derived. For radio frequencies well in excess of the equivalent critical frequency of the medium this may be written [ρ] ƒ i 0 = λ [ρ] ƒ cos i 0 0 where ρ is the fractional attenuation, f is the frequency, and i 0 the angle of incidence in the oblique-incidence case. Where the radio-wave electric vector is at right angles to the plane of propagation, λ is unity; when the electric vector lies in the plane of propagation, λ is equal to the usual obliquity factor cos 2 i 0.

Ionospheric Measurements at Oblique Incidence over Eastern Australia

An account is given of night observations of oblique-incidence pulse transmissions on 5.8 Mc/s. over a baseline of 763 km., using a responder technique. The experiment aimed at measuring apparent path lengths as a guide to the identification of the reflecting layer in the ionosphere. The characteristics of beacon triggering are discussed in relation to the echo system received after ionospheric reflection. A correlation of Es occurrences as observed at oblique incidence and vertical incidence near the midpoint of the trajectory was effected, and certain characteristics of the Es layer are presented. Unusual records of Pedersen rays are reproduced and sudden height increases and diffuseness of F2 echoes are discussed. A check on the oblique-incidence theory using a Millington transmission curve on vertical-incidence h'f records yielded reasonable agreement between measured and deduced reflection heights, although there is evidence of a tendency for the predicted obliquity factor to be too low. A rough analysis of oblique-incidence penetrations yielded an average value for the frequency separation of the magneto-ionic M.U.F.'s close to half the gyromagnetic frequency.

A Correction to the Treatment of Fresnel Diffraction

Measurements of the diffraction pattern of microwaves near circular apertures reveal the quantitative correctness of Kirchhoff's obliquity factor. It has been customary in textbooks of physical optics to assume with Fresnel that the amplitude from zones an infinite distance from the center of the aperture is zero. However, according to Kirchhoff's equation, the amplitude from a zone as the zone number increases without bound is not zero but half the amplitude from the central zone. It is noted that a marginal correction in the textbooks will give results that agree not only with the observed behavior of light, but of microwave diffraction as well, up to the plane of the aperture. The more daring instructor may wish to make a more complete break with the past and explain the diffraction pattern on the basis of Thomas Young's theory. By Young's simple theory that the diffraction pattern of the aperture is an interference pattern between the incident wave and secondary waves from the edge of the aperture, the positions of maximum and minimum intensity may be predicted not only in the Fresnel region but also behind the aperture and in the plane of the aperture itself.

On the radiation from the boundary of diffracting apertures and obstacles

Neglecting the obliquity factor, which is justified when one is considering only small angle diffraction, it is shown that the surface integral which is usually employed for the determination of the disturbance at any point can be easily converted into a line integral along the boundary of the diffracting screen. The formulae thus obtained show that with either an aperture or an obstacle the illumination in the region of shadow can be completely represented as the effect of radiations arising from the boundary, while in the region of light the disturbance due to the direct light is superposed on this. The phase of the boundary radiation is determined by the region (of light or shadow) to which the ray towards the point of observation proceeds from the boundary, being opposite to that of the incident light in the former case, and being the same in the latter case. It is however shown that this leads to no discontinuity in the illumination as the point of observation passes from the region of light into the region of shadow. The boundary radiation can again be effectively replaced by the radiations arising from a finite number of point-sources situated on the boundary called ‘poles’, for which the path to the observation pointvia the boundary is a maximum or a minimum. The phase of the resultant disturbance due to regions of the boundary including and lying on either side of a pole is shown to lead over or lag behind that of the wave from the pole by the quantity π/4, according as the pole is one of maximum or minimum path. Applying these ideas to the diffraction pattern of a circular disc, it is shown that the calculated radii of the rings in the region of shadow agree well with those deduced from Lommel’s theory.

LXXI. The photometric measurement of the obliquity factor of diffraction

LXXI. <i>The photometric measurement of the obliquity factor of diffraction</i>

The Photometric Measurement of the Obliquity Factor of Diffraction

IN vol. lxxviii. of NATURE (May 21, 1908, p. 55) was published a note on “Secondary Waves of Light,” in which I described the diffraction effects produced by an obliquely held rectangular aperture or reflecting surface, and pointed out that the observed distribution of illumination in the pattern was not in accordance with that deduced in the ordinary way. I indicated an explanation of the discrepancy, that it was due to the variation of the obliquity factor of diffraction within the limits of the pattern.

Secondary Waves of Light

IT has hitherto been held that, so long as the diffraction apertures used (cut in perfectly opaque or perfectly reflecting screens) are large compared with the wave-length of light, Fresnel's expression for the amplitude of the disturbance due to a surface-element gives us a close approximation to the observed diffraction effects, and that the exact value for the obliquity factor is of little importance (e.g. see Schuster's “Optics”, sec. 48). That this is true only in the special case in which the apertures are held normal to the, waves of light, and not in other cases, is shown by some new diffraction phenomena that I have made the subject of study.