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Abstract

We propose a scheme for producing and magnifying a hollow beam, as might be desired for purposes of storing and guiding cold atoms, through the use of a simple spherically aberrating lens and a projection lens. The field is a superposition of J_0 Bessel fields, so that simple (linear, circular) polarizations can be utilized. We analyze some of the beam properties through analytical approximations. Some examples of field zeros along the optical axis are given, together with some of their characteristics. Numerical calculations largely confirm the validity of the analytical expressions. For the most important zero nearly all of the beam power is contained within the first two Bessel spacings, with a resulting highly efficient trapping. Isophotes are calculated and displayed for the region surrounding this null point. They have regular shapes, for which we give an approximate expression.

The propagation of Bessel-Gauss, generalized Bessel-Gauss, and modified Bessel-Gauss beams, for which the exact form of the optical fields is known, is analyzed according to the approximate theory developed previously by the authors [J. Opt. Soc. Am. 17 , 1021 (2000)]. Approximations are developed for the fields themselves that are highly accurate and yet are simple in their form and physical description. A set of simple equations is developed, which directly give the parameters describing an image beam following passage through a perfect lens of focal length f, starting with any of the above-mentioned object beams. Ray propagation for these types of beams is described, and it is specifically noted that the intensity maxima do not follow straight paths, while the auxiliary F (p, z) function in fact does follow straight paths.

The propagation of Bessel–Gauss, generalized Bessel–Gauss, and modified Bessel–Gauss beams, for which the exact form of the optical fields is known, is analyzed according to the approximate theory developed previously by the authors [J. Opt. Soc. Am.17, 1021 (2000)]. Approximations are developed for the fields themselves that are highly accurate and yet are simple in their form and physical description. A set of simple equations is developed, which directly give the parameters describing an image beam following passage through a perfect lens of focal length f, starting with any of the above-mentioned object beams. Ray propagation for these types of beams is described, and it is specifically noted that the intensity maxima do not follow straight paths, while the auxiliary F?(ρ,?z) function in fact does follow straight paths.

An approximate method for determining the radial and axial intensity of a Bessel-like beam is presented for the general case in which a radial Bessel distribution of any order is modulated by an arbitrary function. For Bessel–Gauss, generalized Bessel–Gauss, and Bessel–super-Gauss beams, this simple approximation yields results that are very close to the exact values, while they are exact for Bessel beams. A practical beam that can be generated with a combination of simple lenses is also analyzed and illustrated.

Optical electromagnetic fields whose transverse structure can be represented by a zero-order Bessel function propagate along the optical axis indefinitely. Methods for producing such beams are described, and some experimental results are shown. Particular emphasis is put on the fields produced as a result of spherical aberration. Possible applications of these optical fields are outlined.

Methods for the measurement of the waist size and position for Gaussian beams are summarized. An alternative method is given which would apply to pulsed systems. The general theory of diffraction of Gaussian beams is developed which provides a new method for the location of the beam waist. These methods which use hole gratings are employed to demonstrate their feasibility using a small cw source.

The use of hole gratings in small-spot laser damage testing is discussed. If the intensity loss due to the transmission of the grating and due to the production of several spots can be tolerated, a hole grating is shown to increase the ease of establishing a damage threshold by the production of spots with a wide range of intensities whose ratios are well understood. It allows the investigation of defect-related damage since several regions are illuminated with equal intensities, and it permits the investigation of the effects of several closely spaced simultaneous illuminations. Several types of arrays of circles and ellipses are investigated, and the effects of hole size, hole spacing, hole shape, and elliptical hole orientation are discussed. The effects of apertures of the grating are also discussed. Two methods of suppression of diffraction spots lying outside the Airy disk are described and illustrated which utilize distributions of either circular hole sizes or of elliptical hole orientations. Two arrays are used in damage tests of metal surfaces to illustrate their use.

It is proposed to obtain intense optical fields, whose form shows little change in size over long paths, through the use of either conical lenses or spherical lenses showing spherical aberration together with a single projecting lens. The conical lens is shown to produce fields whose transverse structure is given by a zero-order Bessel function J_0, while the spherical aberrating lens produces (real or virtual) J_0-like transverse structures, provided that the central portion of the aberrating lens is occluded. In all cases projection gives a J_0 real-image optical structure. Intensity, size of the transverse structure, and range considerations are developed, and some aspects of optimization are discussed. A negative aberrating lens gives a long range of nearly constant size in the image field, and a universal expression is presented to describe the image size as a function of image distance for this case. Projection with an aberrating projection lens is shown to improve the constancy of the final J_0 pattern size dramatically. Typical photographic results are included for beams generated by using a low-power He–Ne laser. Brief considerations of practical uses of diffractionless beams are presented.

Design considerations are presented for pairs of spherically aberrating elements that produce diffractionless beams of high efficiency and nearly constant size and intensity over specified ranges of axial position. An approximate design, assuming an aberration quadratic in radial ray positions, is followed by a final lens (mirror) specification that is verified through meridional ray tracing. This then provides an accurate determination of beam characteristics. Examples are presented that represent a variety of applications. As a design aid, a simple prescription is given for generating families of substantially different Bessel-like beams from any given pair of elements under small changes in element separation. Pattern sizes and ranges are compared with those of Gaussian beams, shadow lengths are examined for Bessel-type beams, and beam efficiencies are fully analyzed.

Thermal excitations on a germanium surface under simultaneous irradiation by two monochromatic optical beams, one strong and one weak, are predicted as functions of the angular separation and frequency difference between the beams, their relative polarization, their intensities, and pulse durations. Nonlinear optical reflection for Q-switched ruby laser pulses is then described. Weak reflected and diffracted beam intensities show tendencies in which the former is preferentially enhanced for a downshifted weak beam frequency, while the latter depends only on the shift magnitude. Both are suppressed for large shifts or large angular separations between input beams.

The M^2 factor of Bessel–Gauss beams derived by Borghi and Santarsiero [ Opt. Lett.22, 262–264 (1997)] is shown to predict the e^-2 axial position rather than the half-intensity position of the on-axis intensity as the Rayleigh range divided by M^2 for large values of k_tw_0. For small values of k_tw_0, the half-intensity axial position of the J_0 Bessel–Gauss beam is the Rayleigh range divided by M^2. Also, the ratio of the half-intensity lengths of J_0 Bessel–Gauss and comparable Gaussian beams having the same radial size of their central regions is shown to be M^2/1.3. For equal input powers and large k_tw_0, the values of peak intensity times effective range for J_0 Bessel–Gauss beams is a constant and is a factor of 1.3 larger than the corresponding product for the comparable simple Gaussian beam.