Issues in Optical Diffraction Theory.

This paper focuses on unresolved or poorly documented issues pertaining to Fresnel's scalar diffraction theory and its modifications. In Sec. 2 it is pointed out that all thermal sources used in practice are finite in size and errors can result from insufficient coherence of the optical field. A quarter-wave criterion is applied to show how such errors can be avoided by placing the source at a large distance from the aperture plane, and it is found that in many cases it may be necessary to use collimated light as on the source side of a Fraunhofer experiment. If these precautions are not taken the theory of partial coherence may have to be used for the computations. In Sec. 3 it is recalled that for near-zone computations the Kirchhoff or Rayleigh-Sommerfeld integrals are applicable, but fail to correctly describe the energy flux across the aperture plane because they are not continuously differentiable with respect to the assumed geometrical field on the source side. This is remedied by formulating an improved theory in which the field on either side of a semi-reflecting screen is expressed as the superposition of mutually incoherent components which propagate in the opposite directions of the incident and reflected light. These components are defined as linear combinations of the Rayleigh-Sommerfeld integrals, so that they are rigorous solutions of the wave equation as well as continuously differentiable in the aperture plane. Algorithms for using the new theory for computing the diffraction patterns of circular apertures and slits at arbitrary distances z from either side of the aperture (down to z = ± 0.0003 λ) are presented, and numerical examples of the results are given. These results show that the incident geometrical field is modulated by diffraction before it reaches the aperture plane while the reflected field is spilled into the dark space. At distances from the aperture which are large compared to the wavelength λ these field expressions are reduced to the usual ones specified by Fresnel's theory. In the specific case of a diffracting half plane the numerical results obtained were practically the same as those given by Sommerfeld's rigorous theory. The modified theory developed in this paper is based on the explicit assumption that the scalar theory of light cannot explain plolarization effects. This premise is justified in Sec. 4, where it is shown that previous attempts to do so have produced dubious results.

Optical Diffraction in Close Proximity to Plane Apertures. IV. Test of a Pseudo-Vectorial Theory.

Rayleigh's pseudo-vectorial theory of the diffraction of polarized light by apertures which are small compared to the wavelength of light is analyzed with respect to its mathematical rigor and physical significance. It is found that the results published by Rayleigh and Bouwkamp for s-polarized incident do not obey the conditions assumed in their derivation and must therefore be dismissed. It is also found that the theory leads to paradoxical predictions concerning the polarization of the diffracted field, so that the pseudo-vectorial approach is intrinsically incapable of describing polarization effects.

Optical Diffraction in Close Proximity to Plane Apertures. III. Modified, Self-Consistent Theory.

The classical theory of diffraction at plane apertures illuminated by normally incident light is modified so that diffraction on the source side of the screen is taken into consideration and the energy transport across the aperture plane is described by continuous functions. The modified field expressions involve the sums and differences of the Rayleigh-Sommerfeld diffraction integrals as descriptors of a bidirectional flow of energy in the near zones on either side of the aperture. The theory is valid for unpolarized fields, and a pragmatic argument is presented that it is applicable to metallic as well as black screens. The modified field expressions are used for numerical near-field computations of the diffraction profiles and transmission coefficients of circular apertures and slits. In the mid zone the modified theory is reduced to the Fresnel approximation, and here the latter may be used with confidence.

Optical Diffraction in Close Proximity to Plane Apertures. II. Comparison of Half-Plane Diffraction Theories.

The accuracy and physical significance of the classical Rayleigh-Sommerfeld and Kirchhoff diffraction integrals are assessed in the context of Sommerfeld's rigorous theory of half-plane diffraction and Maxwell's equations. It is shown that the Rayleigh-Sommerfeld integrals are in satisfactory agreement with Sommerfeld's theory in most of the positive near zone, except at sub-wavelength distances from the screen. On account of the bidirectional nature of diffraction by metallic screens the Rayleigh-Sommerfeld integrals themselves cannot be used for irradiance calculations, but must first be resolved into their forward and reverse components and it is found that Kirchhoff's integral is the appropriate measure of the forward irradiance. Because of the inadequate boundary conditions assumed in their derivation the Rayleigh-Sommerfeld and Kirchhoff integrals do not correctly describe the flow of energy through the aperture.

Optical Diffraction in Close Proximity to Plane Apertures. I. Boundary-Value Solutions for Circular Apertures and Slits.

In this paper the classical Rayleigh-Sommerfeld and Kirchhoff boundary-value diffraction integrals are solved in closed form for circular apertures and slits illuminated by normally incident plane waves. The mathematical expressions obtained involve no simplifying approximations and are free of singularities, except in the aperture plane itself. Their use for numerical computations was straightforward and provided new insight into the nature of diffraction in the near zone where the Fresnel approximation does not apply. The Rayleigh-Sommerfeld integrals were found to be very similar to each other, so that polarization effects appear to be negligibly small. On the other hand, they differ substantially at sub-wavelength differences from the aperture plane and do not correctly describe the diffracted field as an analytical continuation of the incident geometrical field.

Algorithms for Fresnel Diffraction at Rectangular and Circular Apertures.

This paper summarizes the theory of Fresnel diffraction by plane rectangular and circular apertures with a view toward numerical computations. Approximations found in the earlier literature, and now obsolete, have been eliminated and replaced by algorithms suitable for use on a personal computer.

Computation of Fresnel Integrals.

This paper describes a method for spreadsheet computations of Fresnel integrals to six significant figures, based on successive improvements of known rational approximations which are accurate to only three figures. Outside the range of validity of the improved approximations, known series expansions are used to obtain the Fresnel integrals to six figures.

"Wolf Shifts" and Their Physical Interpretation Under Laboratory Conditions.

This paper attempts to reconcile conflicting points of view of laboratory physicists and coherence theorists on correlation-induced spectral changes arising from the partial coherence of primary and secondary light sources. It is shown that, under normal laboratory conditions and in the Fraunhofer approximation, the directional spectrum of light does not change on propagation in free space, and that each frequency component of the total spectrum is preserved in accordance with the principle of energy conservation. It is demonstrated, and illustrated by examples, that descriptions of diffraction by the theory of partial coherence and by classical wave optics are fully equivalent for incoherent primary sources. A statistical approach is essential, and coherence theory is required, for partially coherent primary sources.

Results of a CCPR Intercomparison of Spectral Irradiance Measurements by National Laboratories.

An intercomparison of spectral irradiance measurements by 12 national laboratories has been carried out between 1987 and 1990. The intercomparison was conducted under the auspices of the Comité Consultatif de Photometrie et Radiometrie (CCPR) of the Comité International des Poids et Mesures, and the National Institute of Standards and Technology (NIST) served as the pilot laboratory. The spectral range of the intercomparison was 250 to 2400 nm and the transfer standards used were commercial tungsten-halogen lamps of two types. The world-wide consistency of the results (one standard deviation) was on the order of 1% in the visible spectral region and 2 to 4% in the ultraviolet and infrared portions of the spectrum. The intercomparison revealed no statistically significant differences between spectral-irradiance scales based on blackbody physics and absolute detector radiometry.

Spectroradiometric Determination of the Freezing Temperature of Gold.

A direct spectroradiometric determination of the temperature of freezing gold was performed by measuring the spectral radiances of a gold blackbody relative to those of a laser-irradiated integrating sphere which was calibrated with absolute silicon detectors and an electrically calibrated radiometer. The measurements were performed at three laser wavelengths near 600 nm, and the temperature of the blackbody was calculated by substituting the measured spectral radiances into Planck's radiation formula. The result obtained, TAu=(1337.33± 0.34) K, is 0.25 K below the gold-point assignment in the International Practical Temperature Scale of 1968 (IPTS-68) and has been adopted in September 1990 as the new gold-point value in the International Temperature Scale of 1990 (ITS-90). The effect of this change in the gold-point assignment on pyrometric, radiometric, and photometric measurement services provided by the National Institute of Standards and Technology is assessed.

Effects of the International Temperature Scale of 1990 (ITS-90) on CIE Documentary Standards for Radiometry, Photometry, and Colorimetry.

The differences between ITS-90 and IPTS-68 above 1235 K are described. It is shown that none of the following CIE definitions or recommendations require revision because of the introduction of the ITS-90: International Lighting Vocabulary definitions; CIE Standard Illuminants A, D65, other illuminants; and sources for realizing CIE Illuminants. The effect of the ITS-90 on previously calibrated sources for realizing CIE illuminants is negligibly small.

The 1990 NIST Scales of Thermal Radiometry.

Following an absolute NIST measurement of the freezing temperature of gold and the adoption of the International Temperature Scale of 1990 (ITS-90), NIST has adopted new measurement scales for the calibration services based on thermal radiometry. In this paper, the new scales are defined and compared to the ITS-90, and the effects of the scale changes on NIST measurement services in optical pyrometry, radiometry, and photometry are assessed quantitatively. The changes in reported calibration values are within quoted uncertainties, and have resulted in small improvements in accuracy and better consistency with other radiometric scales.

Spectral Transmittance Characteristics of Holmium Oxide in Perchloric Acid Solution.

The work describes the methods and procedures used to determine the wavelengths of minimum transmittance of holmium oxide in perchloric acid solution. Measurements of spectral transmittance of the solutions were made by means of a high precision spectrophotometer over the wavelength range 200 nm to 680 nm. The wavelength scale accuracy of this instrument was verified by extensive measurements of mercury and deuterium emission lines. The measurements of spectral transmittance of the holmium oxide solutions were made as a function of temperature, purity, concentration, and spectral bandwidth. Analysis of the uncertainties associated with these parameters and the uncertainties associated with the calibration of the instrument wavelength scale and the data analysis have resulted in an estimated uncertainty of ±0.1 nm for the determination of the wavelengths of minimum transmittance of the holmium oxide solution.