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The beam waist-to-waist transformation of Gaussian beams between input and output reference planes described by the scaled fractional Fourier transform is analyzed in this paper. We obtain the transfer matrix of ABCD optical system that corresponds to the scaled fractional Fourier transform. The results show that the beam waist-to-waist transformation of Gaussian beams can be described by the scaled fractional Fourier transform when the ABCD optical system has a suitable transfer matrix. The relationship between the input and output waist planes and some particular cases when a Gaussian beam passes through a thin lens is also discussed.

The propagation of Ince-Gaussian beams in a thermal lens medium is studied in this paper. Based on the ABCD matrix for Gaussian beams passing through a thermal lens medium, distinct expressions for the beam transverse intensity distributions and the longitudinal phase shift are deduced and discussed. Similar to Laguerre and Hermite-Gaussian beams, Ince-Gaussian beams, which constitute the third complete family of exact and orthogonal solutions of the paraxial wave equation, can also be used in other inhomogeneous media such as lenslike media and saturated absorption media.

Based on the Collins diffraction integral formula and irradiance moments, the propagation properties and kurtosis parametric characteristics of Hermite-cosh-Gaussian (HChG) beams passing through fractional Fourier transformation systems with spherically aberrated lenses are analyzed in detail. The closed-form expressions for the propagation equation and kurtosis parameter of HChG beams passing ideal fractional Fourier transformation systems are also obtained. By using an efficient algorithm some numerical calculations are also performed in this paper. The results show that the spherically aberrated lenses greatly influence on the intensity distribution and kurtosis parameter of HChG beams passage through fractional Fourier transformation systems. The intensity distribution and kurtosis parameter of HChG beams passage through two types of Lohmann's systems with spherically aberrated lenses, respectively, are no longer equal even in the case of the same fractional order and the same spherical aberration coefficients. The fundamental periods of the two types of Lohmann's systems with spherically aberrated lenses are different, 4 for the type I and 2 for the type II. Therefore, the two optical setups for implementing the fractional Fourier transformation are no longer equivalent in the presence of spherically aberrated lenses.

Based on the treatment that a rectangular function can be expanded as an approximate sum of complex Gaussian functions with finite numbers, the analytical expression of the Wigner distribution function for a Gaussian beam passing through a paraxial ABCD optical system with hard-edged aperture is obtained. By numerical simulation, it is shown that the effect of the aperture on the Wigner distribution function is prominent. By comparing the analytical results with the numerical integral results, it is shown that this method of expanding hard-edged aperture into Gaussian functions with finite numbers is proper and ascendant. This method could be extended to study the Wigner distribution functions of other light beams passing through a paraxial ABCD optical system with hard-edged aperture.

Under the condition of rotational symmetry fractional Fourier transforms become fractional Hankel transforms. The relationships between the fractional Hankel transform and the diffraction of a misaligned optical system are established in this paper. By settling and discussing the diffraction integral formulae of a misaligned ABCD optical system, the fractional Hankel transform of an input function can be obtained by selecting a suitable reference plane under given conditions.

A decentered flattened Gaussian beam (DEFGB), is defined by a tensor method. The propagation formula for a DEFGB passing through an axially nonsymmetrical paraxial optical system is derived through vector integration. The derived formula can be reduced to the formula for a generalized decentered elliptical flattened Gaussian beam under certain condition. As an example application of the derived formula, the propagation characteristics of a DEFGB in free space are calculated and discussed. As another example we have studied the properties of superposition with radial array consisted by DEFGB.

The apertured fractional Fourier transform (FRFT) system is introduced and applied to treat the propagation of Hermite-cosh-Gaussian beams. Based on the fact that a hard aperture function can be expanded into a finite sum of complex Gaussian functions, the approximate analytical expressions for the output field distribution of a Hermite-cosh-Gaussian beam through apertured FRFT systems are derived and compared with those obtained from diffraction integral formulae by numerical simulations. It is shown that our method can significantly improve the numerical calculation efficiency.

Summary The propagation of flattened Gaussian beams (FGBs) passing though misaligned optical systems with aperture has been studied in detail. The corresponding approximate equation has been derived, where the rectangular aperture function is expanded into a finite sum of complex Gaussian functions. Numerical examples are given to illustrate the application of the propagation equation and the properties of flattened Gaussian beams passing through a misaligned optical system with aperture are discussed.

Based on the Rayleigh–Sommerfeld diffraction integral equation, the on-axis phase and group velocities of the pulsed plane wave diffracted by a small circular aperture have been investigated, respectively. It is shown that the diffraction results in the superluminality of the phase and group velocities at some certain on-axis points in vacuum, and the group velocity is even infinity.

The phase shifts of a spherical light wave passing through an aperture are simply analysed on the basis of the integral theorem of Helmholtz and Kirchhoff in terms of Kirchhoff's boundary conditions and the physical meaning of the integral. It is shown that the phase shifts of spherical light wave passing through an aperture can be divided into two categories: the additional phase shift and the phase anomaly. The additional phase shift is related to the position of the field point, but the phase anomaly is not related to the position and is a phase jump at a certain point in the field space. The results obtained can explain some cases investigated previously.

Summary It is shown that, under some conditions, the generalized diffraction integral formula for any misaligned optical system in spatial-domain could be described by the fractional Fourier transform. On the other hand, the generalized diffraction integral formula for angular spectra through any misaligned optical system in spatial-frequency domain could also be described by the fractional Fourier transform. A practical optical system with a misaligned lens is given to illustrate its application.

Ganci's experiment is analysed by means of matrix optics and numerical simulation. It is shown that the boundary diffraction wave with phase jump π is a physical reality.

It is shown that fractional Fourier transform could be viewed as a tool for analysing the diffraction of misaligned optical systems, just as the fractional Fourier transforms are adapted to the mathematical expression of Fresnel diffraction. Some special cases are also discussed.

The effect of misalignment on optical fractional Fourier transforming systems is analysed in this paper. Meanwhile, a concept of perturbed fractional Fourier transform is put forward. By using the generalized misaligned diffraction integral formulae for misaligned optical systems, it is shown that the diffraction characteristics for misaligned Lohmann's fractional Fourier transforming systems and quasi-asymmetric inhomogeneous media are able to perform the perturbed fractional Fourier transform and are essentially equivalent. And also, a relative discussion is given.

The central spot of COS beam (CS-COS) can be obtained from COS beam by using the zero field truncation technique. The transversal intensity distributions of CS-COS through three different kinds of aperture functions are simulated numerically from the near field to the far field. It is shown that the CS-COS has the property of eliminating the diffraction fringes in the whole propagation process if a suitable aperture is introduced.

Relationships between the generalized diffraction integral formula for angular spectrum through any misaligned optical system in spatial-frequency domain and the fractional Fourier transform are bridged in this paper, which provides a new description of the diffraction of any misaligned optical system in spatial-frequency domain. Some special cases are also analyzed.

Airy pattern is formed by diffraction of a plane wave through a circular aperture in the far field with certain side lobes. It is shown that Airy pattern can be truncated at the first dark ring by a circular aperture to pick up the center spot (called Airy disc) without causing additional diffraction fringes, even in the very far field. This may be called zero field truncation technique. And is very useful in practice, for example, as a good quality optical source.

A novel model of beam formed by coherent superposition of two correlated Gaussian spherical beams with different centers is presented. This kind of beam has a transverse spreading smaller than a fundamental Gaussian beam. The beam quality factor is smaller than one.

By means of Collins' diffraction integral formula in the paraxial approximation, the analytical expressions for linearly polarized and azimuthally polarized Bessel-Gauss beams through any axisymmetric optical system that can be described by an ABCD ray transfer matrix are given. For example, the transformation characteristics of these two kinds of Bessel-Gauss beams through an optical system with a lens are compared by numerical calculation. The results are analyzed and discussed.

In the present paper, the ray path in a solid state Eckige-Schraube laser is analyzed by the tensor method. The properties of self compensation for thermal distortion and of symmetry in both transverse directions are proved.

The coupling between spatial and temporal parameters of ultrashort pulsed beams propagating through a dispersive system is treated by a 6×6 matrix formalism. A typical grating-pair system which is commonly used in pulsed beam compression is analysed in detail. Numerical examples show that the spatial-temporal coupling becomes severe under some conditions.

Summary A closed-form expression for the one-dimensional off-axial cosine-Gaussian beams passing through an apertured and misaligned paraxially ABCD optical system is derived. As special cases, the corresponding closed-forms for the off-axial or non off-axial cosine-Gaussian beams passing through apertured or unapertured and misaligned or aligned optical systems are also given. The obtained results could be straightforward to the two-dimensional case or sine-Gaussian beams.

Summary Analytical expressions for the one-dimensional off-axial Hermite-cosine-Gaussian beams and one-dimensional off-axial cosine-Gaussian beams passing through an apertured paraxial ABCD optical system have been respectively derived. Our results provide more convenient for studying their propagation and transformation than the usual way by using diffraction integral directly.

Summary When a Cos beam is apertured, it undergoes a phase shift as well as a focal shift. The propagation of relative phase shift of apertured Cos beam through free space is analysed. Based on numerical calculation it is shown that the variation rule of relative phase shift of Cos beam with Gaussian phase term is completely different from that of Cos beam without Gaussian phase term and means that the Gaussian phase term plays the main role in the description of relative phase shift.

Based on Fourier transforms the generalized diffraction integral formula for a beam through any misaligned optical system in spatial-frequency domain is derived in this paper. It is shown that the diffraction integral formula in spatial-frequency domain is related with the ray transfer matrix elements and the misaligned parameters. And some special cases are discussed, an example is also given to illustrate its application.

The concept of Fresnel number is discussed and expressions are derived for misaligned optical systems. For the case of perfectly aligned optical elements, the usual number,N, based on a Fresnel zone concept is found to be given byN = (a 2/λ)(D 1/B 1 +A 2/B 2), whereB 1,D 1 andB 2,A 2 are the transfer matrix elements of the optical systems before and after a circular aperture of radiusa respectively. A modified definition of the Fresnel number is proposed for Gaussian beam propagation. This parameterN′ G, is related to the complex beam parameter and may be represented by the angleθ = tan−1 N′ G, found in the familiar Collins chart and its dual representation. A general relation for the transformation of this Fresnel number is found. The expressions for Gaussian beam transformation are thus simplified, since the waist-waist transform is given byN′ G1 =N′ G2 = 0. Finally, these two kinds of Fresnel numbers are written as tensors when applied to cases involving elliptical apertures, astigmatic beams and nonsymmetrical systems.

The focal shift in a gaussian beam focussed by an optical system with an aperture is generalized by applying Collin's diffraction integral. By this implementation, the field distribution on axis is given. The method may be useful to determine the focal shift of a thin lens and of an optical focusing system as well.

A tensor method is presented to treat the transformation of pulsed nonideal beams in a four-dimension spatiotemporal domain. The equivalent beamwidth, mean pulse duration, divergence, spectrum width, radius of curvature, and pulsed beam quality factor for arbitrary-amplitude profile pulsed beams are introduced. The transformation rules of these parameters are derived by means of the four-dimension Huygens integral.

Two ABCD matrices for describing weakly aperturbed gaussian beams are discussed. Employing these matrices the mode changes of a typical resonator with an aperture are given analytically in a simple way.

By means of Collins' diffraction integral formula in the paraxial approximation, the analytical expressions for an azimuthally polarized Bessel - Gauss beam through any axisymmetric optical system that can be described by an ABCD ray transfer matrix are given in this paper. Free space propagation and the Fourier transfer system are also analysed and discussed as particular cases. En utilisant la forme integrale de la diffraction de Collins nous donnons dans cet article l'expression analytique de la propagation de faisceaux de Bessel - Gauss azimutalement polarises, a travers des systemes optiques de symetrie axiale qui peuvent etre decrits par une matrice ABCD dans l'approximation paraxiale. La propagation dans l'espace libre et a travers des systemes de Fourier sont egalement analyses et discutes comme des cas particuliers.

Based on the nonparaxial moment theory of light beam propagation, the propagation characteristics of nonparaxial scalar Gaussian beam, nonparaxial TEM and TE vector Gaussian beams have been investigated. The results reveal that both the transversal beam widths follow a simple hyperbolic law upon propagation. The analytical expressions of the beam propagation factor, beam waist and far field divergence angle are presented, respectively. Furthermore, the formulae can be very concise for highly nonparaxial cases. TE or TM polarization will result in different propagating features in the two transversal directions. The maximum transverse divergence angles of nonparaxial scalar and vector Gaussain beams are different, which indicates that nonparaxial scalar Gaussian beam is no longer approximate at subwavelength scales. When extending to the paraxial case, the results obtained are slightly different from the formerly paraxial ones. Moreover, in this case the beam propagation factor will always be greater than unity. This research also denotes some properties of subwavelength optics.

The laser mode properties of a corner-cube resonator, in which the corner cube is the key element and a flat mirror is used as the output mirror, are analyzed by a numerical simulating method. Examples of numerical calculations are given to illustrate mode propagation through an optical system. The simulated results agree with the experimental ones.

There are four kinds of reflectors: 1, flat mirror; 2, common retroreflectors; 3, phase conjugate mirror (PCM) and 4, the fourth one. "Corner cube" belongs to 2. However, if the dihedral angle lines and their images of the corner cube are taken into account, it is recognized as super-conjugator. The terminal mirror of laser cavity is replaced by such an element, it has ability to compensate distortions inside the cavity and misalignments insensitive and has being rapidly used. How to perform nonlinear phenomena by a well known linear element? Here we show the essentials by way of special array concept.

The variations of group velocities for Gaussian beams under different hypotheses were studied, and the rationality of the results were discussed and analysed. Then we show that the beam waist radius should be proportional to the square root ofwavelength. Under this condition, the group velocities are all equal to c on the beam axis and smaller than c off the axis.

An expression for the complex radius of curvature tensor describing every kind of gaussian beam is proposed. This expression can be used in the tensor ABCD law. The circular, orthogonal astigmatic and non orthogonal astigmatic beams are expressed by means of this tensor.

A novel method for laser beam smoothing by rotating random phase plate is presented in this paper. A random phase plate for CO 2 laser was made, and satisfactory results were obtained. The experimental results coincide very well with theoretical analysis.

According to the more complete diffraction theory, a series of new beams was invented both outside and inside laser cavity by using (lambda) /2 phase plate. A new beam with equivalent beam quality factor Me2 less than 1 is achieved on ten-watt CO2 laser and on hundred-Watt and thousand-Watt CO2 laser with axial express flux. It may be named as the first kind of new beam laser. From the point of view of practical applications, we designed and produced a new structure of laser cavity whose output mirror was a special mirror with step reflectivity. We have realized 1000 W new beam on CO2 laser with transverse express flux, which is close to the fundamental mode. It may be named as the second kind new laser beam. Its structure is simpler than others, but it brings excellent results.

A new suggestion that there is a phase jump of (pi) in the boundary wave is put forward in this paper. This suggestion may be a supplement of Huygens-Fresnel principle. Based on this new suggestion, a series of new beam was invented, both outside and inside laser cavity. Especially, a new CO2 laser with equivalent beam quality factor M2e < 1 is achieved. It can be considered as the result of some controllable nonlinear self-focusing, and the physical background of deformed quantum mechanics.© (1996) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

A new suggestion that there is a phase jump of (pi) in the boundary wave is put forward in this paper. This suggestion may be a supplement of Huygens-Fresnel principle. Based on this new suggestion, a series of new beam was invented, both outside and inside laser cavity. Especially, a new CO2 laser with equivalent beam quality factor M2 < 1 is achieved. it can be considered as the result of some controllable nonlinear self-focusing, and the physical background of deformed quantum mechanics.© (1996) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Focal shift of the zone plate is completely different from the corresponding circular and annular apertures. Focal powers of the zone plate are constant under some conditions but for the latter they are always variables. Astigmatism for a zone plate is also investigated. The sagittal focal surfaces are unchanged but the tangential focal surfaces are changed as a function: cos2(Theta) , besides, it has neither spherical aberration nor coma. They are also completely different from conventional imaging devices.© (1993) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

One kind of constant deviation prisms which can invariably transform the propagating direction of a ray or a beam is analyzed in this paper. Its transfer characteristics could be described by 2×2 matrix and its misaligned transfer characteristics could be described by 3×3 augmented matrix. Under the condition of manufacturing tolerance, its transfer matrices are also given. And it is shown that the constant deviation prism is a two-dimensional insensitive component.

The authors report on experimental demonstrations of the diffraction and dispersion phenomena of part Fresnel phase zone plates (FPZPs) by employing a He–Ne laser beam and a white point source. The experimental results have shown that a part square FPZP can behave like a complete circular FPZP, and a part circular FPZP has a focus at 1∕3 f and f with a shift. Some numerical simulations are performed. The simulations and the experiments have shown that the diffraction and dispersion properties of part FPZPs can be well explained by the different diffraction orders.

The concept of the Fresnel number is generalized to make it possible to describe the regular polygon and slit illuminated by a homogeneous plane wave. The generalization is based on nonlinear regression of the axial intensity distribution curve. A useful analytical expression for the Fresnel number is presented. A simple experiment to show the different Fresnel numbers in one observation plane is illustrated.

An ABCD matrix for describing the hard aperture under a large Fresnel number is defined in this Technical Note based on Li and Wolf's formula. It is useful for analyzing focal shifts of complicated optical systems with hard apertures.

When a Gaussian beam is weakly diffracted by a circular aperture it can be approximated by another Gaussian beam with slightly different properties in the far field. In this Technical Note, using the relationship between these two Gaussian beams, before and after the aperture, an ABCD matrix is defined for cascaded laser systems.

In this paper we study quantitatively the effects of atmospheric turbulence on the measurement sensitivity of a quadrant detecting array, along with the measurement variance induced by atmospheric turbulence. Our results show that the stronger the turbulence strength, the lower the measurement sensitivity, and the larger the measurement variance. This agrees with the experimental phenomena we have observed up to now. In addition, the measurement sensitivity in the X direction could be affected by the deviation of light spots in the Y direction due to the existence of gaps between the quadrant elements. Such effects still exist for the measurement variance even when there are no gaps at all. This effect is analyzed quantitatively.