Application of Hertz Vector Diffraction Theory to the Diffraction of Focused Gaussian Beams and Calculations of Focal Parameters

Hertz vector diffraction theory is applied to a focused TEM00 Gaussian light field passing through a circular aperture. The resulting theoretical vector field model reproduces plane-wave diffractive behavior for severely clipped beams, expected Gaussian beam behavior for unperturbed focused Gaussian beams as well as unique diffracted-Gaussian behavior between the two regimes. The maximum intensity obtainable and the width of the beam in the focal plane are investigated as a function of the clipping ratio between the aperture radius and the beam width in the aperture plane

Use of Michelson and Fabry-Perot interferometry for independent determination of the refractive index and physical thickness of wafers

We present a method to independently measure the refractive index and the thickness of materials having flat and parallel sides by using a combination of Michelson and Fabry-Perot interferometry techniques. The method has been used to determine refractive-index values in the infrared with uncertainties in the third decimal place and thicknesses accurate to within Ϯ5 m for materials at room and cryogenic temperatures. © 2005 Optical Society of America OCIS codes: 120.2230, 120.3180, 120.4290, 160.4760. The refractive index, n, and the thermo-optic coefficient, dn͞dT, of materials are often determined by one's interferometrically measuring the phase change that light undergoes in passing through a plane-parallel slab of the material. Because the phase change depends on the value of n as well as the slab thickness, d, to obtain accurate values of n and dn͞dT, it is important to know d accurately. FabryPerot etalon interferometry has been used to optically measure d, 5 but the precision of thickness measurements with this method is limited by the precision of the known refractive-index value. Recent research by Coppala et al. 6 demonstrated that independent values for n and d can be obtained with interferometry and a continuously tunable laser source. In this paper we demonstrate that the Michelson and the Fabry-Perot interferometric methods can be used sequentially to determine independent and absolute values of both the material's thickness and the material's refractive index over a wide range of temperatures of practical interest. The method does not require that either quantity be initially well known. With this method, both n and d can be determined by use of a fixed-wavelength laser source. First, by use of data from both experiments, the material's physical thickness is determined. Then the thickness value is used to determine the material's refractive index (and thermo-optic coefficient) with either of the interferometric methods. We present experimental verification of this method by measuring n and d for a range of common infrared materials at both room temperature and cryogenic temperatures. The intensity of a coherent collimated beam of light transmitted by a plane-parallel transparent plate is given by the Airy formula 7 : where I o is the incident intensity, r is the reflection coefficient for the electric field, and f is the phase difference accumulated by the light beam in a double traversal through the plate. As the sample is rotated in the path of the laser light, the net transmitted intensity will modulate owing to the changing phase, f . The angle-dependent phase difference between subsequent transmitted light paths through the sample is given by 7 f () ϭ 4nd cos t ϭ 4d ͙n 2 Ϫ sin 2 , where d is the sample thickness, is the laser wavelength, t is the angle of refraction, and is the angle of incidence of the laser path with respect to the normal of the sample surface

Projection of Diffraction Patterns for Use in Cold-Neutral-Atom Trapping

Scalar diffraction theory is combined with beam-propagation techniques to investigate the projection of near-field diffraction patterns to spatial locations away from the aperture for use in optically trapping cold neutral alkali-metal atoms. Calculations show that intensity distributions with localized bright and dark spots usually found within a millimeter of the diffracting aperture can be projected to a region free from optical components such as a cloud of cold atoms within a vacuum chamber. Calculations also predict that the critical properties of the optical dipole atom traps are not only maintained for the projected intensity patterns but also can be manipulated and improved by adjustment of the optical components outside the vacuum chamber

Vector Diffraction Theory of Light Propagation Through Nanostructures

It is well known that “vector” diffraction theory needs to be invoked to describe the propagation of light through apertures having dimensions on the order of the wavelength of light. For regions close to the aperture, use of Kirchhoff boundary conditions in the aperture plane is invalid. The Hertz vector formalism provides a way to describe the diffraction of light beams through apertures having sizes ranging from half the wavelength of light to larger values. Here we will present a summary of the method used to calculate the distribution of all of the electromagnetic field components and a Poynting vector component at and near the plane of a single elliptical aperture

Vector Diffraction Theory of Refraction of Light by a Spherical Surface

Focusing of light by a curved surface is described using the vector Kirchhoff diffraction theory. The electromagnetic fields of a light beam incident as a plane wave on a curved surface separating two transparent media having different refractive indices are expressed as dimensionless double integrals. The integrals are evaluated for a few specific cases, and the three-dimensional distribution of irradiance near the focus is determined. The role of aberration in limiting the maximum achievable irradiance is studied. The distribution of the longitudinal components of the electric field in the focal region is also studied, and the region where the longitudinal fields maximize is determined

Description of Light Focusing by a Lens using Vector Diffraction Theory

For a plane wave incident on a spherical lens at normal incidence, the field distributions in the focal region are calculated using vector diffraction theory showing the effects of aberration

Description of Light Propagation Through a Circular Aperture Using Nonparaxial Vector Diffraction Theory

Using nonparaxial vector diffraction theory derived using the Hertz vector formalism, integral expressions for the electric and magnetic field components of light within and beyond an apertured plane are obtained for an incident plane wave. For linearly polarized light incident on a circular aperture, the integrals for the field components and for the Poynting vector are numerically evaluated. By further two-dimensional integration of a Poynting vector component, the total transmission of a circular aperture is determined as a function of the aperture radius to wavelength ratio. The validity of using Kirchhoff boundary conditions in the aperture plane is also examined in detail