1. Introduction

The phenomenon of focal shift, wherein the point of maximum intensity of a diffracted field is located not at the geometric focus, but rather somewhat closer to the diffraction plane, is well known. In general, the magnitude of this shift depends strongly on the Fresnel number of the aperture, or on the effective Fresnel number of the field in unapertured cases [1–4]. The shift is significant only when the Fresnel number of the system is sufficiently small; for most imaging applications, therefore, it may safely be neglected. However, in laser cavities and other laser systems the Fresnel number is often on the order of unity or smaller [5], in which case the focal shift becomes important. A focal shift has been observed for a variety of systems, including a converging spherical wave diffracted by a circular aperture [2,3,6,7], a uniformly illuminated aperture with a central obscuration [8], and more general diffracting screens [4,9]. It has likewise been noted and thoroughly studied for Gaussian beams focused by an aperturing [10,11] or nonaperturing [1,12,13] lens with a low Fresnel number, as well as for other types of focused beams, including Laguerre–Gauss [14], Bessel [15], J ₀-Bessel–Gauss [16,17], and flattened Gaussian beams [18]. However, all these diverse systems involve only fields that are linearly polarized and satisfy a scalar wave equation.

In the present work we examine the focal shift of a new family of beams, found by solving the vector wave equation in the paraxial limit. These vector beams have general polarization; recently reported linearly [19] and azimuthally [20] polarized Bessel–Gauss beams may be identified as members of the set [21]. An attractive property of these more general vector Bessel–Gauss beam solutions is their correspondence to modes of important cylindrical waveguides and resonators, in particular the modes emitted by a concentric-circle-grating surface-emitting (CCGSE) semiconductor laser [22]. A correspondence can also be found with the modes supported within an optical fiber, in the limit of large Gaussian beam size [23], and with the free-space modes excited by light exiting the end of such a fiber [21].

We begin with a summary of the set of general vector beam solutions to be considered, including a description of their form after passing through a nonaperturing thin lens. We then define the waist and focal plane of such beams and present images and animated figures exploring their qualitative features near the focus of a long-focal-length lens, as well as numerical solutions showing the dependence of the focal shift on the mode number and various beam parameters. We find that the focal shift depends strongly on the beam and system parameters; it is largest for a narrow beam, long focal length, and large wavelength.

2. Calculations

The system under consideration is shown schematically in Fig. 1. The beam is a member of the set of paraxial vector Bessel–Gauss beams, which are most easily expressed in cylindrical coordinates. We can write its electric field in terms of a vector amplitude U(ρ,ϕ,z):

The exp(-iωt) harmonic time dependence will be taken as implicit throughout the following work. The field amplitude is assumed to include a Gaussian envelope g(ρ,z) to provide the necessary transverse localization in free space,

 

Fig. 1. Schematic diagram of the geometry under consideration, showing the notation used. The blue dashed lines represent the geometric focus of the lens, and the red curves the profile of the focused beam.

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Here g(ρ,z) is the familiar Gaussian solution,

where w ₀ is the Gaussian beam waist, L = ${\mathit{\text{kw}}}_0^2$/2 is the Rayleigh range, k = ω/c, w(z) = w ₀√1 + (z/L)² and Φ(z) = tan^-1 (z/L). Considering only transversely polarized fields for concreteness, we seek a vector function f(ρ,ϕ,z) with only ρ and ϕ components, which may be described by [21,22]

Here the notation {ρ, ϕ} denotes either the radial or the azimuthal component,

m is the azimuthal mode number, β is an arbitrary constant, and the constant a_m is the mode amplitude. The subscript m has been suppressed from f_ρ, f_ϕ , Θ_ρ and Θ_ϕ here for economy of notation. The radial component f_ρ uses the upper operator (+) in Eq. (4) and either of the upper two trigonometric functions in Eq. (7), and the azimuthal component f_ϕ uses the lower operator (-) in Eq. (4) and the corresponding lower function in Eq. (7). Note that the sinusoidal functions in Eq. (7) provide a characteristic 2m-fold azimuthal symmetry for most parameter sets [22].

The familiar scalar Fresnel diffraction integral [24] can be converted to cylindrical coordinates and used to obtain a diffraction integral for each component of a general transverse vector beam affected by a circularly symmetric pupil (e.g., a lens or circular aperture). For the case of a nonaperturing thin lens, with a pupil function modeled by a quadratic phase object, the vector field amplitude following the lens is found to be [22]

where

I_v is the modified Bessel function of order v, f is the geometric focal length of the lens, and the notation is the same as that used in Eq. (4). We further define a Fresnel number N_w for the Gaussian portion of the beam,

as well as a separate Fresnel number N_β for the Bessel portion:

Because we are considering higher-order Bessel functions, this definition for N_β differs by a factor of (m+2)² from that used in the discussion of linearly polarized J ₀-Bessel–Gauss beams [16,17]; note that the spot radius of J_m (βρ) is approximately (m + 2)/β.

It is worthwhile to note that the focal shift seen in scalar beams is embodied in this more general vector analysis. The familiar Gaussian beam, as well as other linearly polarized beams, may be found as members of the family of vector Bessel–Gauss beams [21], and thus the origins of their focal shift must be the same as for the more general beams. In particular, the focal shift appears when the distance from the wavefront at the lens to the point of observation is not approximated as the focal length of the lens [7].

In usual treatments of focal shift in linearly polarized, scalar beams, the location of the actual focus is defined to be the point at which the beam reaches its maximum on-axis intensity. However, this definition is not applicable to all types of beams. In the case of a general vector beam, the presence of the higher-order Bessel functions in Eq. (4) results in an on-axis null for all mode numbers except m = 1, since all Bessel functions J_v (u) except v = 0 vanish as u goes to zero. We therefore use a more general definition for the focal plane of these beams. For a standard linearly polarized Gaussian beam, slightly more than 80% of the energy propagates within a radius w ₀ of the axis, so we define the width ρ ₀ of a vector beam as that radius within which 80% of the beam’s power is enclosed, and the focal plane z = z ₀ as the plane in which that width is at its smallest value. Note that the plane containing the 2m points of maximum intensity of a vector beam is not this focal plane, but rather is in general between the focal plane defined here and the geometric focus.

The intensity of the beam is the time average of the z component of the Poynting vector,

The magnetic field H is easily found using Maxwell’s equations, and the asterisk (*) denotes the complex conjugate. The power carried may be found by integrating this intensity over the ρ-ϕ plane, so we seek the propagation distance z ₀ at which the smallest value of ρ ₀ satisfies

The width of a vector beam at a given propagation distance z must be calculated from Eq. (13) numerically, and therefore the focal shift ∆z = f - z ₀ must also be found numerically.

3. Numerical results

The three-dimensional structure of these vector beams in the region of the geometric focus is complicated, as shown in Fig. 2. In the animation associated with each of these snapshots the viewer is traveling along the direction of propagation of a vector beam, from z = 0.9 f to z = 1.1 f Animations (a)–(d) show modes m = 0, 1, 2, and 7, respectively, for λ = 632.8 nm,f = 80 mm, w ₀ = 1 mm, and β = √0.00004 μm^-1 ≃ 0.006 μm^-1 . These values correspond to N_w ≃ 20 and N_β ≃ 0.5(m + 2)². As described in previous work, this relatively large value of the inverse Bessel width β ensures that the beams show the characteristic 2m-fold azimuthal symmetry and that the m = 0 and m = 2 beams differ [22]. The white lines in these snapshots have been added to aid in the interpretation of Fig. 3. The most noteworthy features are the change in the shape of the beam and the contrast reversal that occur in the vicinity of the geometric focus; the distinction between the plane at which each beam is narrowest and the plane containing its highest intensity is also seen in these sequences.

 

Fig. 2. (1 MB each) Snapshots from animations showing vector beam intensity patterns as the viewer travels along the direction of propagation, near the geometric focus. Images (a)–(d) show modes m = 0, 1, 2, and 7, respectively. The beam parameters f = 80 mm and β ≃ 0.006 μm^-1 correspond to Fresnel numbers N_w ≃ 20 and N_β ≃ 0.5 (m + 2)². Each mode has been independently normalized, but the individual images within each animation are not separately scaled. The white lines have been added to aid in the interpretation of Fig. 3. [Media 1] [Media 2] [Media 3] [Media 4]

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Fig. 3. Intensity plotted in an ρ-z plane slicing longitudinally through the three-dimensional beam, for (a) m = 0, ϕ = 0 (circularly symmetric beam), (b) m = 1, ϕ = π/2, (c) m = 2, ϕ = π/4, and (d) m = 7, ϕ = π/14. Beam parameters are as in Fig. 2. In images (b) and (c), note that the beam does not vanish entirely in the region surrounding z = f, but the energy is no longer found at this particular azimuth ϕ.

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The structure of the beams is shown further in Fig. 3, which presents the intensity of each mode in a longitudinal cross-section containing the beam axis. Images (a)–(d) show m = 0, 1, 2, and 7, respectively. A specific choice of azimuthal coordinate is necessary to fully describe the ρ-z plane; in each case we have selected ϕ = π/(2m), indicated by a white line added to the snapshots of Fig. 2. The beam parameters are the same as those in Fig. 2. In images (b) and (c), note that the beam does not vanish entirely in the region surrounding z = f, but because of the contrast reversal noted above the energy is no longer found at the particular azimuth ϕ corresponding to this plane. The exact focal shift is not evident from Fig. 3, but a distinct asymmetry in the intensity of the field on either side of the geometric focus may be seen, indicating that the energy is somewhat more concentrated just before z = f, and therefore that the beam is likely to be narrower there.

This focal shift is confirmed in Fig. 4, a plot of the beam width in micrometers, calculated from Eq. (13), against the propagation distance, for modes m = 0 through 7, with the same beam parameters as used in Fig. 2. The asymmetry about z = f shown in Fig. 3 is also apparent here. Each of the modes reaches a minimum waist just before the geometric focal length, showing a small focal shift ∆z toward the lens that decreases (moves toward the geometric focus) slowly with mode number for m ≥ 1. The shift depends more strongly on the other beam parameters, however, as demonstrated in the following two figures. Fig. 5 shows its dependence on the Gaussian Fresnel number N_w for the modes m = 0 and m = 1. Both beams have N_β ≃ 2, corresponding to β ≃ 0.006 μm^-1 and β ≃ 0.0095 μm^-1 , respectively, and to change N_w the Gaussian width w ₀ is varied from 300 μm to 1900 μm. Similarly, Fig. 6 plots the focal shift against the Bessel Fresnel number N_β for these two modes, with N_w ≃ 20 and the change in N_β accomplished by varyingβ from 0.001 μm^-1 (0.0015 μm^-1) to 0.010 μm^-1 for m = 0 (m = 1). As these two figures show, the focal shift decreases as either N_w or N_β increases, and is slightly larger for m = 1 than for m = 0, an effect which becomes more evident for smaller N_w or N_β . Note that differences between the two modes in Fig. 6 arise primarily from the (m + 2) factor in N_β , as defined in Eq. (11); when the focal shift is plotted against β itself, the lines are nearly identical.

 

Fig. 4. Beam width in micrometers plotted against normalized propagation distance z/f for several mode numbers m. The beam parameters are as in Fig. 2. The minimum width for each mode occurs slightly before z/f = 1, indicating a positive focal shift ∆z that decreases slowly with mode number for m ≥ 1.

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Fig. 5. Relative focal shift ∆z/f plotted against the Gaussian Fresnel number N_w , varied by changing w ₀, for modes m = 0 (blue line) and 1 (red line). The Bessel Fresnel number is N_β ≃ 2 for each beam, corresponding to β ≃ 0.006 μm^-1 and β ≃ 0.0095 μm^-1, respectively.

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Fig. 6. Relative focal shift ∆z/f plotted against the Bessel Fresnel number N_β , varied by changing β, for modes m = 0 (blue line) and 1 (red line). The Gaussian Fresnel number is N_w ≃ 20. Differences between the two modes arise primarily from the (m + 2)² factor in N_β , as defined in Eq. (11); when the focal shift is plotted against β itself, the lines are nearly identical.

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The interaction between the Bessel and Gaussian Fresnel numbers is shown in Fig. 7, which plots the relative focal shift against N_w for m = 0 and several values of N_β . The largest focal shifts are seen for small N_w and N_β , which may be achieved either with small w ₀ and large β, which together produce a very narrow beam, or with a large wavelength λ or long focal length f.

 

Fig. 7. Relative focal shift plotted against the Gaussian Fresnel number N_w , for m = 0 and several values of the Bessel Fresnel number N_β . For a single mode, the largest focal shifts are seen for small N_w and N_β . This situation corresponds to either small w ₀ and large β, which together produce a very narrow beam, or a large wavelength λ or focal length f

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4. Conclusions

We have explored the behavior of vector, paraxial Bessel–Gauss beams near the focus of a long-focal-length, nonaperturing thin lens. The three-dimensional structure of these beams near the focus is complicated, with a 2m-fold azimuthally symmetric intensity pattern that shows both a shape change and contrast reversal near the geometric focal plane. The phenomenon of focal shift familiar from Gaussian and other scalar beams has also been found for this more general family of beams: the beam width, defined here as the radius encircling 80% of the total power, reaches a minimum slightly closer to the lens than the geometric focal length predicts. However, in general vector Bessel-Gauss beams vanish on the axis, and so for these beams the point of maximum intensity is not located at the true focus, but rather in a plane slightly closer to the geometric focus. The magnitude of the focal shift is typically on the order of a few percent of the lens focal length, though its exact value depends strongly on the parameters of the beam and the optical system. The focal shift is largest when the Bessel and Gaussian Fresnel numbers N_β and N_w are small; this corresponds to a narrow beam, long wavelength, and long lens focal length.