1. Introduction

Light beams focused into a subwavelength area are of considerable interest owing to their applications in various fields such as microscopy, lasers, optical information storage, and optical trapping. Analyses considering different aspects of Gaussian light beams including the focusing have been in the literature for many years [1–8]. The propagation properties of such a beam are generally studied in the paraxial approximation. However for a strongly focused beam when the spot size of the beam is less than the wavelength it is necessary to go beyond the paraxial approximation.

It is well known, that Gaussian beam changes its form at the focusing into a subwavelength area due to the effects of nonparaxiality. Various methods for consideration of nonparaxial propagation of light beams have been proposed. In [9] the expansion procedure of the electric field into a power series in terms of small parameter was introduced. Many papers were devoted to nonparaxial corrections of fundamental and higher-order Gaussian beams [10–17]. The angular spectrum approach was used in [10] to obtain an exact solution for the electromagnetic field. In [11] non-paraxial propagation of Laguerre-Gaussian and Hermite- Gaussian beams was investigated by the expansion theory introduced in [9]. In [17] all nonparaxial corrections for standard Hermite-Gaussian beams were evaluated. Another approaches were proposed and used for investigation of high-aperture beams free-space propagation based on the complex-source-sink solution of Maxwell equations in [18, 19]. Similar solutions for the scalar field of strongly focused Gaussian beams are given in [20]. In [21] all nonparaxial corrections for coherent Bessel-Gauss beams are obtained using formalism suggested by Wunsche [15].

Analytical methods may also be developed for investigation of paraxial light beams propagation in an inhomogeneous medium [22–28]. The ABCD matrix approach for beam expansion and compression analysis in quadratic-index waveguides was developed in papers [23, 26]. Note, that in paraxial approximation the shape of the Gaussian beam is preserved after passing through the graded-index waveguide, even if the beam is decentered [8].

A quantum-theoretical method of coherent states was used in [29, 30] for consideration of beam propagation in a graded-index medium beyond the paraxial approximation. The first order corrections were obtained using this method. In [31] the explicit expressions for the beam trajectory and width were obtained and mode structure forming in a graded-index waveguide have been investigated.

Note that the number of corrections must be increased as the beam divergence angle grows, in other case such models became inaccurate. Besides the nonparaxial effects are accumulated with distance, so the small corrections became significant at long distances. Below we will show, that the revival effect of the beam trajectory at long distances predicted in [29] is not physical, but is only the result of the use of the expression which is not valid for these distances. Therefore the developing of explicit methods for consideration of nonparaxial propagation is much interesting.

In this paper we present a rigorous description of the propagation and focusing of a Gaussian beams with spherical wavefront in a graded-index medium using the quantum-theoretical method of generalized coherent states which allows us to calculate the average values with the help of an operator approach. The whole dynamics of the system is transferred to the operators in this approach. This allows us to investigate the evolution of the characteristics of the propagating beam with the help of pure algebraic procedures, that is without using explicit expressions for field wavefunctions and without the calculation of any integrals. The concept of choice of the quantum formalism in the waveguide theory is the following. It is well known that the Maxwell equations for the scalar wave paraxial beams may be reduced with high accuracy to a parabolic type equation. This approximation enables us to use well-developed quantum-mechanical methods for the investigation of wave propagation in inhomogeneous media, because the parabolic equation formally coincides with the Schrodinger equation in quantum mechanics for particles moving in a time-dependent potential well. All that is required is to redefine the parameters in the Schrodinger equation. The role of time is now taken by the longitudinal coordinate, and Planck’s constant is superseded by the radiation wavelength in vacuum. The potential is defined as a function of the refractive index of the medium. The intimate relationship between the wave mechanics of particles and the optics of light beams has been discussed in detail in many papers (see, e.g., [32, 33]).

2. Formulation of the problem

The equation describing the propagation of radiation in an inhomogeneous medium from the Maxwell equations for the electric field $\overrightarrow{E}$ exp(’iνt) may be obtained as

where k=2π/λ is the wave number, n(x,y,z) is the refractive index of the medium.

In the case of slowly inhomogeneous media ((λ/2π)∇n ²/n ² << 1) the third term corresponding to the polarization is small and the electric field is described by the Helmholtz equation:

If the medium is homogeneous in the longitudinal direction z, equation (2) may be reduced to the equivalent Schrodinger equation for the reduced field ψ(x,y):

where ε and ψ(x, y) are the eigenvalue and the eigenfunction of the Hamiltonian

The evolution of the field

E(x, y, z)=exp(iβz)ψ (x, y)

is determined by the propagation constant

$\beta \left(\epsilon \right)=k{n}_0{\left[1-\frac{2\epsilon }{n_0^2}\right]}^{\frac{1}{2}}.$.

The evolution of any operator acting on the solution of Eqn (2) is specified by an equation analogous to the Heisenberg equation:

where $\dot{Â}$=dÂ/dz; [Â, Â]=Â$\widehat{\beta }$-$\widehat{\beta }$ Âis the commutator of the operators  and $\widehat{\beta }$.

The propagation-constant operator assumes the form

$\hat{\beta }=k{n}_0{\left[1-\frac{2\hat{H}}{n_0^2}\right]}^{\frac{1}{2}},$,

and its eigenvalues determine the propagation constants β(ε).

The evolution operator for the field has the form Û=exp(i$\widehat{\beta }$ z), and the evolution of any operator  is given by Â(z)=Û ⁺ ÂÛ.

Thus, the solution of the Helmholtz equation (2) reduces in this instance to the solution of the Heisenberg equation for operators the average values of which determine the parameters of the investigated beam, for example, the coordinate of its gravity center and its width.

Consider the propagation of radiation in a medium homogeneous in the direction z with a parabolic variation of the refractive index in the transverse directions x and y:

where n ₀ is the refractive index along the axis and ω is the gradient parameter of the medium. The expression (6) describes the parabolic distribution of the refractive index of the medium, which is infinite in transverse directions. However the modes of the medium (6) as the modes of graded-index waveguide (fiber) may also be considered. This is valid as a first approximation for lowest-order waveguide modes when the waveguide carries a sufficiently large number of modes.

Consider an incident beam in the form of coherent states which are Gaussian wave packets and represent the eigenfunctions of the annihilation operator Â:

${\hat{a}}_{\mathrm{1,2}}\mid {\alpha }_1{\alpha }_2〉={\alpha }_{\mathrm{1,2}}\mid {\alpha }_1{\alpha }_2〉,$${\hat{a}}_1=\frac{1}{\sqrt{2}}\left[\sqrt{k\omega }\hat{x}+i\sqrt{\frac{k}{\omega }}{\hat{p}}_x\right],{\hat{a}}_2=\frac{1}{\sqrt{2}}\left[\sqrt{k\omega }\hat{y}+i\sqrt{\frac{k}{\omega }}{\hat{p}}_y\right],$${\hat{p}}_x=-\frac{i}{k}\frac{\mathit{\partial }}{\mathit{\partial }x},{\hat{p}}_y=-\frac{i}{k}\frac{\mathit{\partial }}{\mathit{\partial }y}.$,

The wavefunction |α ₁ α ₂〉 defines the space distribution of the electric field and has the form

$\mid {\alpha }_1{\alpha }_2〉={\left(\frac{k\omega }{\pi }\right)}^{\frac{1}{2}}\exp \left\{\begin{array}{cc}-\frac{k\omega }{2}\left(x^2+y^2\right)& +\sqrt{2k\omega }\left({\alpha }_{1}x+{\alpha }_{2}y\right)\\ -\frac{1}{2}& \left({\alpha }_1^2+{\alpha }_2^2+{\mid {\alpha }_1\mid }^2+{\mid {\alpha }_2\mid }^2\right)\end{array}\right\}.$.

The complex eigenvalues ${\alpha }_1=\frac{1}{\sqrt{2}}\left(\sqrt{k\omega }{x}_0+i\sqrt{\frac{k}{\omega }}{p}_{x0}\right)$ and ${\alpha }_2=\frac{1}{\sqrt{2}}\left(\sqrt{k\omega }{y}_0+i\sqrt{\frac{k}{\omega }}{p}_{y0}\right)$ determine the initial coordinates x ₀, y ₀ of the gravity center of the beam and the angle of its inclination p_{x 0}=n ₀sinφ _x0, p_{y 0}=n ₀sinφ _y0 to the axis of the medium. The term “coherent states” was introduced by Glauber [34] for a one-dimensional steady-state quantum oscillator in connection with problems in quantum optics. Such states were constructed and investigated already by Schrodinger [35] in order to establish a relationship between the classical and quantum approaches.

In the paraxial approximation, when only the first term in the expansion series of the propagation-constant function $\widehat{\beta }$ in terms of Ĥ/$n_0^2$ is considered,

the coherent states have the minimum possible width and an angular diffraction divergence on propagation in a medium the refractive index of which is defined by expression (6). According to beam optics, the center of gravity of such wave packets moves along a geometrical path and the packet width does not change during the propagation process. Furthermore, the coherent states are the generating functions for the modes of the medium:

$\mid {\alpha }_1{\alpha }_2〉=e^{-\frac{{\mid {\alpha }_1\mid }^2}{2}-\frac{{\mid {\alpha }_2\mid }^2}{2}}\sum _{m_1,m_2=0}^{\infty }\frac{{\alpha }_1^{m_1}{\alpha }_2^{m_2}}{\sqrt{m_1!m_2!}}\mid m_1{m}_2〉.$.

3. Evolution of the beam path and width

Consider the evolution of the beam path and width in a medium (6). For simplicity, we confine ourselves to one component of the field. The beam path is found by calculating the average value of the coordinate operator:

where x̂(z) is found from the solution of the Heisenberg equation (5). The evolution of the width of the wave packet is found similarly:

In the paraxial approximation, the expression for the beam path has the following form:

〈x〉_α=x ₀ cos(ωz-θ),

where θ is determined from the relationship α=|α|e^iθ . The beam width ${〈x〉}_{\alpha }=x_0\cos \left(\omega z-\theta \right),$ is not changed at the propagation in this approximation.

When account was taken of the next term in expansion (7), the expression for the beam path assumed the form [30]

Substitution of the solution of eqn (5) for the coordinate operator in formula (8) and calculation of the matrix elements gives the following expression for the coordinate of the center of gravity of the beam taking into account all terms of expansion (7):

The evolution of the wave-packet width is described by the expression

We may note that, in contrast to the paraxial approximation, the trajectory of gravity center of wave beam depends on the radiation wavelength and differs from the trajectory of geometrical ray. Moreover the amplitude of the trajectory decreases with distance (Fig.1), i.e. the sum of periodical functions gives the damping oscillations though there are no energy losses at propagation. Beams with the small initial inclination angle (Fig.1a) or small axis displacement (Fig.1b) have almost periodical trajectories and invariable diameters up to sufficient distances. It means that paraxial approximation may be used successfully in these cases. For large inclination angles and axis displacements visible deviations from paraxial approximation are appeared already at short distances.

 

Fig. 1. Trajectories of wave beams (dashed curves) of wavelength λ=0.65 µm accompanied by its width (solid curves) for gradient parameter ω=5·10^-2 µm^-1 with initial inclination angles 15° and 60° (a) and initial axis displacements 7.76 and 26 µm (b).

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The expressions (11) and (12) are the exact solutions including all correction terms to the paraxial solution and were obtained to our knowledge in the first time. Analysis of these expressions shows that the relaxation length of the unaxial ray on the axis or the mode structure establishment length decreases at the increase of initial amplitude of a ray or its inclination angle to the axis of medium, wavelength of radiation and gradient parameter of medium:

$z_0\approx \frac{1}{\mid \alpha \mid }·\frac{L}{\lambda }·L,\mathrm{where}L=2\pi n_0/\omega .$.

At a distance of the order of z ₀, the mode structure of the field is established and the use of the localized beam representation becomes ineffective. Hence, it follows that the distance corresponding to the establishment of the mode structure is most sensitive to changes in the gradient parameter, so that it is essential to employ fibres with the smallest gradient parameter for the transmission of an image in graded-index optic fibres. In typical graded-index fibres with the core radius r ₀=50µm and the gradient parameterω=5·10^-3 µm ^-1, the mode structure is established at distances z ₀≈30-40 cm.

Thus, an off-axial wave packet spreads on propagation in a graded-index medium and the center of gravity of the wave packet relaxes towards the axis of the medium, while its width increases to a value determined by the initial beam amplitude |α|. The limits of applicability of the ray optics are determined by the initial beam coordinate, the radiation wavelength, and the gradient parameter of the medium.

The explicit expressions (11) and (12) take into account all terms in the series expansion of the propagation constant function in terms of small parameter f∝ω/${\mathit{\text{kn}}}_0^2$. Since the gradient parameter determines the width of fundamental mode ω=2/${\mathit{\text{ka}}}_0^2$, then the parameter f is analogical to the small parameter introduced by Lax et al. [9]. The use of approximated expressions which take into account only the first terms of expansion can lead to nonphysical results. Particularly, the revival effect at long distances predicted in [29] is only the result of the use of the expressions which are not valid for these distances.

4. The focusing of Gaussian beam

The propagation of Gaussian beams, matching the width of the fundamental mode, was examined above. Below we consider the evolution of an axial Gaussian beam, of width greater or less than the width of the fundamental mode of the medium. The expression for the field of the incident Gaussian beam in the z=0 plane assumes the form

where R is the radius of wavefront curvature, a ₀≅0.85w ₀, w ₀ is the full width at half maximum (FWHM).

Focusing of spherical Gaussian beams of low-aperture was considered in [5]. Here we consider also the case when significant aperturing effect occurs. In this limit the waist has no Gaussian profile.

Note, that the arbitrary Gaussian beam with a spherical wave front can be described by the generalized coherent-state representation [36]. Variation of the distribution of the field with distance in a graded-index medium has the following form:

Here N is the maximum number of propagating modes determined by the relationship N≤${\mathit{\text{kn}}}_0^2$/2ω-1/2.

The beam width evolution is described by the expression

For beams with initially planar wave fronts χ=0 and $\mu =\frac{1}{2}k{a}_0^2.$. When account is taken in expansion (7) only of the terms quadratic in the Hamiltonian, the beam width is described by the expression

$\Delta x_0^2\left(z\right)=\frac{1}{2k\omega }\left\{1+2v^2-\frac{2v}{u^2}{\left[1+\frac{v^4}{u^4}-2\frac{v^2}{u^2}\cos \frac{4{\omega }^2}{k{n}_0^3}z\right]}^{-\frac{3}{4}}\cos \left[\frac{3}{2}\varphi +\left(\frac{2\omega }{n_0}+\frac{3{\omega }^2}{k{n}_0^3}\right)z\right]\right\},$,

where $\varphi =\arctan [\left(\frac{v^2}{u^2}\sin \frac{{4\omega }^2}{k{n}_0^3}z\right)/\left(1-\frac{v^2}{u^2}\cos \frac{{4\omega }^2}{k{n}_0^3}z\right)].$.

The minimum beam width is attained at a distance

i.e. the length of graded-index lens depends on the parameters of incident beam.

In the paraxial approximation, expression (15) has the form

$\Delta x_0^2=\frac{1}{2k\omega }\left[1+2v^2-2uv\cos \frac{2\omega }{n_0}z\right].$.

The minimum beam width is attained in this approximation at a distance z=n ₀ π/2ω and is determined by the expression:

$\Delta x_{min}^2=\frac{1}{k^2{\omega }^2{a}_0^2}.$.

In the case of paraxial approximation the focusing distance in graded-index medium depends only on the parameters of medium. When account is taken of nonparaxiality, the focusing distance (16) depends also on the parameters of radiation. The focusing distance or the graded-index lens length for nonparaxial beam is less than the focusing distance for paraxial beam.

The distribution of the field intensity I(x, z)=ψ(x, z)² may be found from formula (14). We may note that the total energy is normalized with respect to unity P=∫I (x, z)dx=1 and remains constant for all values of z. Fig.2 represents the distribution of the field intensity along the axial line of the medium.

 

Fig. 2. Variation with distance of the field intensity along the axial line of a graded-index medium withω=5·10^-2 µm^-1 and n ₀=1.5; half-width of the incident beam a ₀=15 µm.

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The field intensity in the focal plane z _f=44.9 µm increases by a factor of 30 compared with the intensity in the initial plane. The focal planes of nonparaxial and paraxial beams, where the highest intensity on the axis and smallest waist of the beam are attained, are not the same. The focusing distance for nonparaxial beam is shorter than for paraxial beam. There is a significant difference in the intensity distribution along the medium axis at the front of focus and behind it. This asymmetry is caused by the effect of nonparaxiality. Note, that these effects have also been shown to appear for a free-space beam focusing [37]. Fig.3 presents the distributions of the field intensity in the transverse plane. As can seen from Fig.3, the distributions of the field intensity differ significantly from the initial Gaussian distribution.

Note that taking into account only guided modes we obtain usual classical value for minimal spot size determined by the diffraction limit (a _min≈λ/2n).

 

Fig. 3. Distribution of the field intensity in a transverse plane of a waveguide at various distances from the initial plane z=0 (a), z=40 (b) and z=44.9 µm (c) (the parameters of the beam and the medium correspond to those in Fig.2).

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5. Diffraction of strongly focused Gaussian beam

It is well known that when focusing light onto a small object then this object may lead to an energy concentration below the diffraction limit due to the presence of evanescent waves. Below we consider the strongly focused spot of subwavelength size at initial plane z=0. The electric field of such spot may be represented as the sum of propagating and evanescent modes: E=E _p+E _ev. In near-field zone the contribution of evanescent modes may become significant to compare with propagating modes. Inversely, in far-field zone only the propagating modes may be considered.

The evanescent fields become visible if the spots beyond the diffraction limit are considered. In Fig.4 the distributions of the evanescent and propagating fields are presented for the Gaussian beam of the width a ₀=50 nm in a graded-index medium with n ₀=1.5 and ω=10^-2 µm ^-1. The intensity of evanescent field on the axis in 2.4 times greater than the intensity of the propagating field. The amplitude of evanescent field on the axis increases with the decrease of beam width, whereas the amplitude of propagating field decreases. The region where the evanescent field dominates is only within a fraction of the wavelength of the light from the initial plane, i.e. in near-field zone.

 

Fig. 4. Distribution of the evanescent (red curve) and propagating (blue curve) fields in a transverse direction at the initial plane.

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Note, that the oscillations in lateral parts of propagating and evanescent fields (Fig.4) take place in anti-phase to each other, so the total intensity I=|E_p +E_ev |² distribution has not any oscillating tails (Fig.5).

 

Fig. 5. Distribution of the total field intensity at the initial plane.

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In far-field zone the evanescent fields do not contribute to the total field, so the propagating modes may only be considered. Distribution of propagating field does not have a truly Gaussian profile (Fig.6) owing to the absence of evanescent field at far-field zone.

 

Fig. 6. Distribution of the propagating field intensity at the initial plane.

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The ratio of energy guided by propagating modes to the total input energy η=P_g /P ₀ is rapidly decreased if the focused spot became less than the value of λ/2n owing to increase of evanescent part of energy (Fig.7). For a given aperture size the energy guided by propagating modes grows and energy of evanescent modes decreases if the refractive index of medium increases.

 

Fig. 7. The ratio of energy guided by propagating modes P _g (solid curves) and evanescent modes P _ev (dashed curves) to the total input energy P ₀ as function of spot size a ₀ for different values of refractive index: 1-n ₀=1.5; 2-n ₀=2; 3-n ₀=3.87.

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Beam width evolution with distance is investigated for different values of aperture size at initial plane. Variation in the angle θ ₀ at which the far-field intensity drops to one half of the intensity on the axis is calculated for different values of the initial spot size. It is obtained, that for the Gaussian beam the maximum value of θ ₀ is about 37°. As is shown in [18] for the electromagnetic mixed-dipole wave the maximum value of θ ₀ is 37.8°, which is practically the same. It was obtained in [16], that the upper limit on the beam divergence angle is 65.5°, which is cannot be reached in our case. The maximum angle, where the intensity in a transverse plane in the far-field drops to 1/e of the on-axis value is 45° (Fig.8).

 

Fig. 8. Variation in the angle θ ₀ at which the far-field intensity drops to one half and 1/e of the intensity on the axis for different values of the initial spot size.

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6. Discussion

It is known that the threshold for the maximum focusing of the light beam can be estimated from the uncertainty relationship

where the angular divergence of the beam is Δp ²<<$n_0^2$.

The uncertainty relationship (17) is obtained for the beam width and divergence, which are the averaged parameters. For beam focusing the electric field distribution is more important. Calculations show that the significant part of incident beam power may be focused into the central spot with the subwavelength size beyond the diffraction limit. The recent experimental and theoretical studies [38, 39] of near-field emission from uncoated tapered fiber probes prove that subwavelength resolution can be obtained. In [38] the λ/3 spatial resolution using solid immersion lens microscopy was obtained.

The relationship (17) corresponds to the beams with plane wave front. In the case of beams with spherical wave front one can be obtained the following uncertainty relationship [36]

and due to Δp≤1, R _min≥Δx=a ₀/0, i.e. the minimal value of radius of curvature is a diffraction limited. Note that R may be negative, that corresponds to the light sources with a converging spherical wave front. Such a spherical wave front is formed, particularly, by the solid immerse lens.

To diminish the spot size to subwavelength area the medium with high refractive index may be used. However such materials have a high absorption. Another method to decrease the spot size is to use the longitudinally inhomogeneous medium. Multimode graded-index waveguide tapers may be used as focusing or defocusing elements. Strictly adiabatic tapers that do not radiate along their length are proposed and investigated in [40–42]. It was shown that impractically long length of taper is often required.

Choosing the optimal change with distance z of gradient parameter ω(z) we can reduce the size of initial spot. The size of the focused spot may be decreased when series-distributed graded-index lenses of length Δz_i =n ₀ π/2ω_i and with different gradient parameters ω ₁<ω ₂<…<ω _n are used. For example, the perfect matching between two waveguides with different gradient parameters ω(z)=ω- and ω(z)=ω ₊ may be achieved if the transparent barrier $\omega \left(z\right)={\omega }_0=\sqrt{{\omega }_{-}{\omega }_{+}}$ with length of L=(πn ₀/2ω ₀)(2p+1), p=0, 1, 2, … (the analog of the Ramsauer effect in quantum mechanics [43]) or the barrier ω(z)=ω ²-{1+2a sin[(2+b)ω-z/n ₀]}, where |a|, |b|<<1, are used.

The size of the focused spot may be reduced by an increase of the gradient parameter of the medium. There is a cutoff value of the gradient parameter determining the possibility for the propagation of the fundamental mode: ω _max=${\mathit{\text{kn}}}_0^2$. At ω>ω _max only the evanescent waves are possible. On the other hand, the width of the fundamental mode is determined by the gradient parameter: $a_0=\sqrt{2/\left(k\omega \right)}.$. Thus, for the minimal value of the beam width we have: a _min=√2/(kn ₀), i.e. FWHM w ₀≅λ/4n ₀. However, the condition of a weak inhomogeneity of the medium is infringed for high values of ω, which necessitates allowance for the polarization effects, i.e. the vector wave equation (1) should be considered instead of Helmholtz equation (2). These effects for paraxial beam in a graded-index medium were considered by us on the basis of a treatment of coherent states [44, 45]. It was shown in [44], that the polarization is limiting factor to focus a light beam into a subwavelength spot. However, the radially polarized radiation may be used to avoid this limitation [46].

7. Summary

We have analyzed propagation and focusing of Gaussian beams in graded-index medium beyond the paraxial approximation.

The expressions for the beam trajectory and width taking into account all correction terms of a series of expansion of powers of a small dimensionless parameter f∝ω/${\mathit{\text{kn}}}_0^2$ are obtained explicitly. Exact analytical expressions (11) and (12) for the trajectory and width of decentered Gaussian beams in graded-index medium were not obtained earlier to our knowledge. It is obtained, that the gravity center drops on the axis of medium even the lossless medium is considered. It is shown, that the effect of the beam trajectory revival predicted earlier in [29], is not physical, but due to the use of approximated expression at large distances where the validity of this approximation is disappeared.

The localized wave packets (coherent states), which have minimum width and angular diffraction spreading during propagation in a medium with a quadratic index profile may be used for description of an off-axial Gaussian beams in paraxial approximation. The center of gravity of such wave packets moves along a geometrical path, i.e. obeys geometrical optics and the packet width does not change during the propagation process. In nonparaxial case the beam center does not follow along a geometrical ray trajectory. The limits of applicability of the ray optics are determined by the initial beam coordinate, the radiation wavelength, and the gradient parameter of the medium.

Nonparaxiality causes a change of initially Gaussian profile at the propagation and to an asymmetric distribution of the field intensity in the longitudinal direction. The focal planes of nonparaxial and paraxial beams are not the same. The focusing distance for nonparaxial beam is shorter than for paraxial beam.

Diffraction limited minimal value for radius of wavefront curvature of a spherical beam has been determined from the uncertainty relation.

It is shown, that the nonparaxial focusing in longitudinally homogeneous graded-index medium and the paraxial focusing in longitudinally inhomogeneous medium give the same minimal spot size limited by diffraction.

The minimal spot size using only the propagating modes is diffraction limited. Further decrease of spot can be carried out if the evanescent modes are included. For a given aperture size the energy guided by propagating modes grows and energy of evanescent modes decreases if the refractive index of medium increases. So far as evanescent modes cannot be registered at far-field zone, far-field image of the spot will be greater than the real spot size.

Far-field radiation of high-aperture Gaussian beam has not a truly Gaussian profile, but has oscillating tails outside the central region. In near-field region these oscillations are compensated by anti-phase oscillations of the same amplitude of evanescent field.

It is obtained, that for the Gaussian beam the maximum value of θ ₀ at which the far-field intensity drops to one half of the intensity on the axis approximately is 37°. The maximum angle, where the intensity in a transverse plane in the far-field drops to 1/e of the on-axis value is 45°.