Effective Fresnel number and the focal shifts of focused partially coherent beams

We define the effective Fresnel number of the cylindrical lens illuminated by a plane wave or Schell-model beams. On the basis of the concept of the effective Fresnel number, the focusing properties of the cylindrical lens illuminated by the Schell-model beam are investigated in a simple way. It is shown that the relative focal shift can be evaluated by an analytical formulation, which is expressed as a function of the effective Fresnel number. To evaluate our approach, we make the comparison between the results obtained by our method and the numerical calculation based on the diffraction integral. The results indicate that we can simply and exactly evaluate the focal shifts with our method.     

Predictions of Rayleigh's diffraction theory for the effect of focal shift in high-aperture systems

In their work on diffraction [J. Opt. Soc. Am.51, 1050 1961], Osterberg and Smith have computed in an exact manner from the Rayleigh-Sommerfeld diffraction integral of the first kind the irradiance distribution along the axis of a converging spherical wave, and they found that in a scalar optical system of high relative aperture and finite value of Fresnel number, the central peak value of the axial irradiance may occur inside, at, or outside the geometrical focal point as the angular semiaperture of the system is less than, equal to, or greater than, respectively, a particular angle that falls near 70 degrees . These findings are now reexamined using a different assumption that takes into account diffraction at the edge of the aperture. Different results are obtained that agree well with the predictions of other theories of diffraction of light and give confidence to the common conclusions drawn by investigators of the effect of focal shift, that the point of the principal maximum of axial irradiance is not at the geometrical focus but shifted toward the aperture in systems of different relative aperture and finite value of Fresnel number.     

Focal shift in small-Fresnel-number focusing systems of different relative aperture

Axial irradiance distribution arising from the diffraction of a uniform, converging, spherical wave at a circular aperture is studied on the basis of scalar boundary-diffraction wave theory. The combined effects of Fresnel number and angular aperture on the focal shift are evaluated, and the validity of the results is checked against the Kirchhoff boundary conditions.     

Focal shift in focused radially polarized ultrashort pulsed laser beams

Beginning with a beam coherence polarization (BCP) matrix, we obtain an analytical intensity expression for radially polarized ultrashort pulsed laser beams that pass through an apertureless aplanatic lens. We also investigate the intensity distribution of radially polarized beams in the vicinity of the focus. The focal shift of these beams is studied in detail. The focal shift depends strongly on Z(F) that coincides with pi times the Fresnel number.     

Focal swift for a Gaussian beam: an experimental study

The beam radius along a focused, unapertured Gaussian beam was measured and used to calculate the dependence of the geometrical Fresnel number on the effective Fresnel number of the beam as it emerged from the focusing lens. The resulting data clearly demonstrate a focal shift away from the focal plane given by geometrical optics. The data agree very well with a theory due to Carter. This theory indicates that a significant focal shift occurs in these beams if the focusing lens is placed within the near field of the focal plane.     

Possibility of an optical focal shift with polarization masks

The polarization phase shift (PPS) has emerged as an important analytical tool in optical metrology. The present study utilizes the concept of controlling the polarization phase in applications such as focal shift and automatic focusing. When elliptically polarized light, in general, is incident upon a circularly symmetric polarization mask consisting of circular and annular zones with each zone having a unique linear polarizability, the polarization-phase difference introduced between the polarization-masked zones is also circularly symmetric. With the mask at the lens aperture, the polarization phase introduced is multiplicative with the lens function and is shown to result in a shift of the Gaussian focus plane. Because the polarization phase can be controlled by variation of the polarization parameters, the effective focal length of the imaging system can be varied within a small range. A study of the point-spread functions at the shifted focal planes has shown that the quality of the focal patch in these planes is comparable with that produced by a diffraction-limited imaging system at Gaussian focus. The shift of focus can be achieved by control of the polarization of the input beam. It is anticipated that this technique may find application in areas for which dynamic focusing within a small range is required.     

球面波通过透镜-光阑分离系统的焦移

用Collins公式研究了均匀球面波通过透镜-光阑分离系统的轴上光强分布。经推导得出焦移解析表达式。在这类光学系统中,光束的焦移可能大于,等于或小于零。分析表明,焦移的大小与菲涅尔数、透镜-光阑的分离程度以及入射波波面曲率半径等因素有关,并得出了消除焦移的条件。     

Numerical calculation of a converging vector electromagnetic wave diffracted by an aperture using Borgnis potentials. II. Application to the study of focal shift

Focal shift of the converging spherical wavefront light diffracted by a circular aperture is numerically studied with the method of calculating the vector diffractive field by using Borgnis potentials given in Part I [J. Opt. Soc. Am. A23, 872 (2006)]. The quantitative dependence of the focal shift on the geometric parameters is discussed. The focal shift is mainly determined by the Fresnel number (N(f)) on the geometric focusing plane of the converging light, and an empirical formula between the fractional focal shift and the Fresnel number is deduced for N(f)<2. The focal shift of the same geometry is also studied on the basis of the scalar Rayleigh theory of diffraction, and its comparison with and difference from the result of our method are presented.     

Focal shifts in focused beams with an elliptical diffracting screen

The approximate analytical formula of the focal shift has been derived for an elliptical diffracting screen that consists of a circular lens and an elliptical aperture. It is found that the focal shift of the beam focused by this elliptical diffraction screen depends not only on the product of the Fresnel number of the focusing geometry and the standard deviation of a mapped function, but also on the ellipticity (the ratio between the minor and the major axes) of the elliptical aperture. The focal shift increases as the decrease in the ellipticity of elliptical aperture. By using an approximate analytical formula and the diffraction integral formula, some numerical simulation comparisons are done. The presented analyses show that the actual analytical expression for the focal shift of a rotationally asymmetric diffraction screen cannot generally be obtained by using the azimuthal average of the screen amplitude transmittance.     

Measuring the focal length of optical systems by grating shearing interferometry

Using grating shearing interferometry, a new and simple technique to measure the effective focal length of optical systems is described. The diffraction pattern of a phase grating positioned at the focal point of the lens under test is evaluated for this purpose. The relative lateral shift between the undiffracted zero order and the diffracted first orders caused by the grating is measured. By utilizing knowledge of the wavelength of light, the grating period, and the diameter of an aperture stop placed in front of the test lens, we can determine the effective focal length of the test lens. Results of measurements are presented and compared with calculated values. The dependence of the focal length on the wavelength of the light is shown by using two laser sources of different wavelengths. An analysis of the measurement accuracy is given.     

Focal shifts in diffracted converging electromagnetic waves. I. Kirchhoff theory

Starting with the vector formulation of the Kirchhoff diffraction theory, expressions for the total energy density distribution along the axis are presented without using any of the usual assumptions except the assumption made by Kirchhoff for the boundary conditions of a black screen. To make the Kirchhoff integral compatible with Maxwell's equations, a line integral around the edge of the aperture is added in the analysis. The consequence of ignoring the contribution of this line integral to the axial field distribution is examined numerically. The focal shift effect is investigated for both aplanatic systems and parabolic mirrors having an arbitrary numerical aperture (NA) and finite value of the Fresnel number. The combined effects of the Fresnel number and NA on the focal shift are evaluated, and the validity of the results is carefully checked by comparing the wavelength with the system dimensions.     

Focal shift and the axial optical coordinate for high-aperture systems of finite Fresnel number

Analytic expressions are given for the on-axis intensity predicted by the Rayleigh-Sommerfeld and Kirchhoff diffraction integrals for a scalar optical system of high numerical aperture and finite value of Fresnel number. A definition of the axial optical coordinate is introduced that is valid for finite values of Fresnel number, for high-aperture systems, and for observation points distant from the focus. The focal shift effect is reexamined. For the case when the focal shift is small, explicit expressions are given for the focal shift and the axial peak in intensity.     

Focal shift, optical transfer function, and phase-space representations

The focal shift for a lens of finite value of Fresnel number can be defined in terms of the second moment of the intensity distribution in transverse planes. The connection with the optical transfer function is described. The specification of the focused amplitude in terms of the fractional Fourier transform is discussed, and the connections among the fractional Fourier transform, the Wigner distribution, and the ambiguity function are described, leading to a model for effects of Fresnel number in terms of a rotation in phase space. The uncertainty principle is discussed, including the significance of the beam propagation factor M2 and the width of optical fiber beam modes. Calculation of the moments in terms of the modulus and the phase of the illuminating wave is presented, and the use of the Kaiser-Teager energy operator is also described.     

Subharmonic focal-length intensities formed by Fresnel lenses

Binary Fresnel lenses produce focused spots at subharmonics of the principal focal length of the lens. The intensities of these focal spots can be controlled by variation of the relative widths of the rings of the Fresnel lens compared with the spacings between the rings. Theory is presented and experimentalverification is provided with Fresnel lenses written onto the magneto-optic spatial light modulator.     

Focal shift in focused truncated pulsed-laser beam

The focal shift of a focused truncated pulsed-laser beam is investigated. In the case of the Fresnel approximation, the analytic expression of the time-averaged intensity distribution along the axis is derived based on the series expansion. It shows that the focal shift of the pulsed beam can be completely determined by a series of normalized spectrum moments and the central Fresnel number defined according to the central frequency of the pulse. The absolute value of the focal shift of the pulsed beam decreases monotonously and slowly with the normalized spectrum width increasing and the central Fresnel number fixed, and it increases monotonously with the central Fresnel number decreasing and the normalized spectrum width fixed. Besides the central Fresnel number and the normalized spectrum width, the shape of spectral intensity of the pulse affects the focal shift too.     

Modeling microlenses by use of vectorial field rays and diffraction integrals

A nonparaxial vector-field method is used to describe the behavior of low-f-number microlenses by use of ray propagation, Fresnel coefficients and the solution of Maxwell equations to determine the field propagating through the lens boundaries, followed by use of the Rayleigh-Sommerfeld method to find the diffracted field behind the lenses. This approach enables the phase, the amplitude, and the polarization of the diffracted fields to be determined. Numerical simulations for a convex-plano lens illustrate the effects of the radii of curvature, the lens apertures, the index of refraction, and the wavelength on the variations of the focal length, the focal plane field distribution, and the cross polarization of the field in the focal plane.     

Establishment of the Maximum Encircled Energy in the Geometrical Focal Plane

The classical theory of the focusing of light predicts that light energy is highly concentrated in the geometrical focal plane; in other words, the geometrical focal plane contains more energy per unit area than any other plane parallel to it. However, it has recently been shown that the classical theory is valid only for focusing systems of large Fresnel number. This paper examines how the maximum encircled energy is accumulated in the geometrical focal plane as the Fresnel number of the diffracting aperture increases.     

On-axis diffractional behavior of two-dimensional pupils

We show that, at any Fresnel number, a suitable one-dimensional Fourier transform relates the complex-amplitude distribution along the optical axis with the zero-order circular harmonic of the amplitude transmittance of a two-dimensional diffracting screen. First, our general result is applied to recognize that any rationally nonsymmetric screen generates an axial-irradiance distribution that exhibits focal shift. In this way we identify a wide set of two-dimensional screens that produce the same focal shift as that produced by the clear circular aperture. Second, we identify several apodizers for shaping the axial-amplitude distribution. We discuss some examples for achieving high-precision focusing, axial hyperresolution, or high focal depth.     

余弦-高斯光束通过光阑-透镜分离系统的焦移

利用Collins公式,研究了余弦-高斯光束通过光阑-透镜分离系统时的轴上光强分布和聚焦特性。研究表明轴上光强分布和相对焦移是光学系统参数、光束参数 、菲涅尔数和光阑尺寸的函数。通过选择适当的光学系统和光束参数,焦移现象消失或反转。     

Focal shifts in diffracted converging electromagnetic waves. II. Rayleigh theory

Part II of this study is an application of the Rayleigh vector diffraction integrals to an investigation of the effect of focal shifts in converging spherical waves diffracted in systems of arbitrary relative aperture. The results are compared numerically with those obtained in Part I [J. Opt. Soc. Am. A 22, 68 (2005)] from the Kirchhoff vector diffraction theory. The effect of the numerical aperture (NA) on focal shifts can be considered in two regions: When NA < or = 0.5 the system behaves like an paraxial system, and the Fresnel number is the dominant factor. When 0.5 < NA < or = 0.9 the absolute value of the relative focal shift decreases with increasing value of NA.