We derive solutions for radially polarized Bessel-Gauss beams in free-space by superimposing decentered Gaussian beams with differing polarization states. We numerically show that the analytical result is applicable even for large semi-aperture angles, and we experimentally confirm the analytical expression by employing a fiber-based mode-converter.
© 2013 Optical Society of America
Since their introduction several decades ago, Bessel-Gauss beams have sparked tremendous interest. As an approximation to the non-diffracting Bessel beam, Bessel-Gauss beams offer an extended depth of focus as well as other intriguing features such as an annular intensity distribution in the far-field [1–3]. Such features have fueled wide-ranging interest in potential applications ranging from materials processing  and micro-manipulation  to strong-field physics .
Interest has also grown in Bessel-Gauss beams with vector polarization states, in particular cylindrical polarization, i.e. radial or azimuthal polarization. Such cylindrical Bessel-Gauss beams have been explored in the context of laser electron acceleration  as well as in studies concerning tight-focusing of optical beams for microscopy and lithography applications [8–10]. Additionally, emission patterns from surface-emitting semiconductor lasers have been described with radially and azimuthally polarized Bessel-Gauss beam expressions formulated via paraxial vector wave equations [11–13].
In this contribution, we derive expressions for radially polarized Bessel-Gauss beams through a novel scalar-wave approach. The approach involves superposing multiple decentered scalar Gaussian beams with differing polarization states. The expressions we derive for radially polarized Bessel-Gauss beams provide new insight into the behavior of such beams in first-order, ABCD, optical systems. Moreover, they allow for straightforward numerical calculations. Additionally, we confirm our radially polarized Bessel-Gauss beam expressions through an experiment based on a fiber mode-converter.
The outline of the paper is as follows: In section 2, we define linearly polarized decentered Gaussian beams, and we superpose these component beams to formulate expressions for cylindrical, i.e. radially and azimuthally, polarized Bessel-Gauss beams in a way hitherto not demonstrated in the literature, to the best or our knowledge. In section 3, we derive the electromagnetic fields of the cylindrical Bessel-Gauss beams from the vector potential description developed in the preceding section, and we numerically confirm our analytical expressions through non-paraxial numerical simulations. In section 4, we describe an experimental verification of our radially polarized Bessel-Gauss expression. In section 5, we conclude this work.
2. Cylindrical Bessel-Gauss beams as superpositions of decentered Gaussian beams
In the following section, we first define the decentered Gaussian beam and propagate this beam in free-space using an adapted paraxial approximation. Next, we superpose many such decentered Gaussian component beams to generate cylindrical, i.e. radially and azimuthally polarized, Bessel-Gauss beams.
2.1. Decentered Gaussian beams
Throughout this section and the following ones, we will work with the vector potential A in the Lorenz gauge. With this gauge condition, the vector potential satisfies the wave equation, and we can write A in a familiar form as a superposition of plane-wave solutions:
Let us consider a Gaussian distribution of plane-wave amplitudes in the vicinity of a principal wavevector of fixed orientation given by K = (Kx, Ky, Kz). We denote deviations from this principal orientation in the x and y direction as α and β, respectively, such that:Eq. (4) into Eq. (1), setting z = 0, and integrating, we find an expression for the decentered Gaussian beam at its focus (z = 0) [14, 15]: Eq. (5), we see that the expression for AdG(x, y, z = 0, t) resembles that of a conventional Gaussian beam at its focus with a decentered focal spot and a tilted propagation direction. The decenterd focal spot is described by the parameters xd0 and yd0. They represent the coordinates of the beam focus in the x–y plane, which may also be written in terms of the vector rd0 = (xd0, yd0). The tilted propagation direction is characterized by the parameters εx and εy which we may likewise write in vector form as ε = (εx, εy). The vector ε is the projection of the principal wavevector K onto the x–y plane (divided by Kz). Thus, the tilt angle with respect to the z direction amounts to φ = tan−1ε where ε = |ε|. The geometry is illustrated in Fig. 1(a).
Having established the form of the decentered Gaussian beam at the focal plane (z = 0), we can find the full form of the beam by invoking a paraxial approximation and integrating for all z in Eq. (1). We perform the paraxial approximation as follows
Since decentered Gaussian beams are tilted beams, the ramifications of the tilt on the validity of the paraxial approximation is of particular interest. Specifically, the approximation error is given by δ = (1 − χ)(1/2) − (1 − χ/2), where χ stands for the term in the brackets in Eq. (7). Using  and , we can make a conservative estimate for the approximation error as χ ≤ 4tanφ/(cosφ · (2πw0/λ))+4/(cosφ ·(2πw0/λ))2 where φ is the tilt angle and w0/λ is the Gaussian waist parameter (in units of wavelength). As an example, consider a decentered beam with a tilt angle φ = 5°, a waist w0 ≈ 10λ, and a wavelength λ = 1μm. For these parameters, δ = 10−5, and we find that the residual phase Kzδz reaches π at z ≈ 50 mm. Since the above assessment is conservative, the preceding analysis helps identify regimes in which the paraxial approximation is strictly valid, however, it is not necessarily limited to these regimes.
Inserting our paraxial approximation, i.e. Eq. (8), into the plane-wave decomposition of the vector potential, i.e. Eq. (1), and integrating, we find the complete propagated form of the decentered Gaussian beam:Eq. (9) takes on a very intuitive form. It resembles a conventional Gaussian beam with a decentered focal spot and a propagation direction tilted with respect to the z-axis as illustrated in Fig. 1(a).
Before proceeding, let us make a brief note concerning polarization. We have described the decentered Gaussian beam with vector potential AdG pointing in the direction of a cartesian unit vector. If AdG points in some direction in the x–y plane, then we expect the corresponding electromagnetic fields to resemble a linearly polarized decentered Gaussian beam by analogy with the familiar Gaussian beam [16, 17]. However with AdG pointing in the z-direction, we expect a radially polarized decentered Gaussian beam now by analogy with the radially polarized Gaussian beam .
2.2. Superposition to cylindrical Bessel-Gauss beams
Bessel-Gauss beams are formed by superposing decentered Gaussian beams . As shown in Fig. 1(a), the centers of the decentered Gaussian beams are placed on a circle with a constant radius rd0 = |rd0| in the x–y plane, and the tilt angles of the beam directions point to the apex of a single cone. Specifically, this corresponds to a superposition of beams whose vectors rd0 and ε are collinear and where the parameters rd0 and ε are constant.
To carry out the superposition, we introduce cylindrical coordinates (r, θ, z). The focus of each decentered Gaussian beam in the superposition is given by coordinates (rd0, γ, 0). The angle between the vectors r and rd0 is (θ − γ). Consequently, we can rewrite Eq. (9) in cylindrical coordinates asEq. (10) and our preceding expression for the decentered Gaussian beam, Eq. (9), is that in Eq. (10) we allow the direction of the vector potential n̂ be a function of the beam focal position, specifically the parameter γ. A variation of n̂ throughout the superposition enables us to generate Bessel-Gauss beams with cylindrical polarization states.
Finally, let us carry out the superposition of the component beams as defined in Eq. (10). The superposition constitutes an integration of γ from 0 to 2π and yields a solution for the vector potential of the cylindrical Bessel-Gauss beam:Eq. (11), for two special cases. First, we consider the scenario with n̂ oriented along the radial or azimuthal direction. As we will see in section 3, this will produce electromagnetic fields representing radially and azimuthally polarized Bessel-Gauss beams. Next, we consider n̂ oriented along a cartesian unit vector. Here, we generate expressions for the well-known linearly polarized Bessel-Gauss beam as well as an alternative formulation for a radially polarized Bessel-Gauss beam. For these integrations, we apply the definition of the Bessel-function, e.g. as given in :
2.2.1. n̂ with radial or azimuthal orientation direction
Consider the decentered Gaussian component beam with its focal spot given by coordinates (rd0, γ, 0). In the superposition rd0 is fixed and γ is variable. With local linear polarization n̂ in the radial or azimuthal directions, n̂ = cosγx̂ + sinγŷ or n̂ = −sinγx̂ + cosγŷ, respectively. Expanding the trigonometric functions into exponential functions and inserting these expressions into the cylindrical Bessel-Gauss beam integral, Eq. (11), the integration can easily be performed by applying Eq. (12). The superposition is illustrated in Fig. 1(b). The result is given by:11, 12]; however, one advantage of superposing decentered Gaussian beams to radially and azimuthally polarized Bessel-Gauss beams is that the propagation of these component beams through first-order optical systems, i.e. ABCD optical systems, is known. Thus, a solution for radially or azimuthally polarized Bessel-Gauss beams in ABCD optical systems can be derived as well. This will be discussed in the appendix.
Figs. 2 (a) and (b) show this beam’s vector potential amplitude in the focus and far-field, respectively. The parameters are w0 = 1 mm, ε = 0.45°, rd0 = 0, and wavelength of λ = 1050 nm (corresponding to the experimental configuration described in section 4). The propagation of the amplitude through the focus is shown in Fig. 2 (c).
2.2.2. n̂ along the Cartesian-unit vector
Consider the case for which n̂ is orientated along any cartesian unit vector (x̂, ŷ, z ̂) or any linear combination thereof. For this case, the superposition is illustrated in Fig. 1(c), and the integral of Eq. (11) is given by:Figs. 2 (d) and (e) show this beam’s vector potential amplitude in the focus and far-field, respectively. Fig. 2 (f) displays the complete beam propagation through the focus. In the focus this beam solution shows an on-axis amplitude. The beam parameters are the same as those for the preceding subsection.
Before closing this section, it is worth noting that this concept of a ‘ray’ bundle of decentered Gaussian beams, i.e. our superposition procedure, can be extended to describe more general beam-shapes. For example, the component beams do not need to be continuously placed over the circle but can be situated discretely . In this way, the output of recently demonstrated of fiber-laser arrays can be described .
3. Electromagnetic fields of cylindrical Bessel-Gauss beams
In this section we obtain the electromagnetic fields of cylindrical Bessel-Gauss beams from the vector potential description of the previous section. We highlight the utility of the z-oriented solution in describing radially polarized Bessel-Gauss beams. Finally, we compare the fields derived from our analytical results to non-paraxial numerical simulations.
3.1. Electromagnetic fields and the vector potential
The electric and magnetic field are obtained from the vector potential A(x,y,z,t) and scalar potential φ(x,y,z,t) by:16, 17], and we note that the electric field E(x,y,z,t) is expressed by a term proportional to the vector potential, as well as a gradient term. The latter is required to ensure that the divergence of the electric field vanishes.
3.2. Fields of cylindrical beams
In section 2, we formulated cylindrical Bessel-Gauss beams in terms of the vector potential. Specifically, the vector potential descriptions took the form:Eq. (19), we can tabulate expressions for the field components in these cases:
Inspecting the expressions in Tables 1 and 2, we see the two cases correspond to transverse electric and transverse magnetic beams. Considering the azimuthal orientation θ̂ of the vector potential, we see that the resulting electric field in Table 1 is purely azimuthally polarized while the magnetic field possesses transverse as well as longitudinal components. Such beams are thus referred to as transverse electric beams. Orientation of the vector potential in the radial r̂ and ẑ directions produces radially polarized electric fields, as shown in Table 2. For this case, the magnetic field does not have a longitudinal component but the electric field does. This case corresponds to transverse magnetic beams.
Considering further the field expressions in Table 2, it is intriguing to note that there are two cylindrical Bessel-Gauss solutions that offer radial polarized fields. Specifically, the radial polarization can either arise from a radially oriented vector potential with amplitude ψ(r) or it can be derived from a z-oriented vector potential with amplitude ψ(z). In section 2, we have derived two different explicit paraxial expressions for ψ(r) and ψ(z). It is then reasonable to inquire how similar or different the resulting radially polarized electromagnetic fields are. Considering the magnetic field of the beam arising from the radially oriented vector potential, i.e. the first row of Table 2, and utilizing ψ(r) in the form of Eq. (14), we find Bθ ≈ iKzψ(r) (the primary z variation in ψ(r) follows from the fast-varying exp(iKzz) term). For the second radially polarized beam solution arising from the z-oriented vector potential, we employ ψ(z) in the form of Eq. (16), and noting that the primary radial variation of ψ(z) should come from the Bessel function term, it is straightforward to show that near the focus i.e. z ≪ zR and with small decenter parameter i.e. rd0 ≪ zR the magnetic field takes on the form Bθ ≈ Kzεψ(r). So, for this region of interest these two radially polarized solutions appear virtually identical. Although, these two solutions look nearly identical in this region, we have not thoroughly explored their properties outside this region. Such an investigation merits further study.
We should also emphasize that the amplitudes ψ(r) and ψ(z), which are used for the general field expressions in Tables 1 and 2, obey two different wave equations. As mentioned, in the Lorenz gauge and in a current-free medium, the vector potential A(x,y,z,t) satisfies the vector wave equation:Eq. (20) with ψ = ψ(z) and n̂ = ẑ, the vector wave equation reduces simply to: Eq. (20) with ψ = ψ(r) and n̂ = r̂ or θ̂, the vector wave equation reduces to: Eq. (23) has been used previously to study azimuthally and radially polarized Bessel-Gauss beams [11, 12]; however, it does not share the familiar form of Eq. (22) and does not allow straightforward plane-wave solutions. This is a key point; we have formulated expressions for radially polarized Bessel-Gauss beams by superposing component beams following the familiar wave equation for Cartesian unit vector orientation. In this way, we have indirectly solved for the more complicated wave equation in the form of Eq. (23). The developed radially polarized Bessel-Gauss beam expressions (at z = 0) serve as an input for non-paraxial, numerical propagation through Fourier transform techniques, as described in the following.
3.3. Non-paraxial numerical propagation
In this section we show that the vector potential can also be applied to numerically calculate the propagation of radially polarized beams. The z-oriented vector potential is advantageous for numerical calculations as the corresponding wave equation, Eq. (22), can be solved with a superposition of plane waves as given by Eq. (1). The electromagnetic fields are calculated from the numerical solution for the vector potential by using Eq. (19).
In a numerical implementation, we do not need to invoke a paraxial approximation, i.e. kz does not need to be approximated by a parabolic function, as performed in Eq. (8). This approximation was required to obtain an analytical solution for the Bessel-Gauss beam in form of Eq. (15) and Eq. (16). Thus, the numerical solution is directly calculated by using Eq. (1), which we can rewrite in terms of Fourier-transforms (FT) and inverse Fourier-transforms (IFT):Eq. (19). Additionally, we should note that since the cylindrical beams are azimuthally symmetric, the Fourier-transform in Eq. (24) is advantageously calculated by using a quasi-discrete Hankel-transform . For radially polarized light, this is a fast alternative to the commonly employed integral representation of the field in the focus .
For the example, we simulate the evolution of a radially polarized Bessel-Gauss beam with semi-aperture angle φ = 60° (= 1.05 rad), a Gaussian waist parameter of w0 = 10 μm, and a wavelength of λ = 1 μm. Fig. 3 (a) shows the initial state, and the numerical simulation of the propagation is shown in Fig. 3 (b). The radial and z-component of the electric fields are calculated by applying Eq. (19). Close-ups of the instantaneous fields are shown in Figs. 3(c) and (d). It can be seen that the radial component shows zero amplitude in the beam center, while the z-component is the strongest there. For the same beam parameters, in Fig. 4(a) we compare the numerically calculated z-component of the electric field to the analytically derived field at the beam center (r = 0), and in Fig. 4(b) we show a comparison of the corresponding radial electric fields at the off-axis maximum (at about r = 0.75 μm). Note that the z-component of the electric field exceeds the radial component for the present semi-aperture angle of φ = 60°. This effect has previously been described [8–10].
We additionally numerically calculate the ratio of the maximum z-component of the electric field to the maximum radial component of the electric field for radially polarized Bessel-Gauss beams as the semi-aperture angle of the beam is varied. The results of this computation are given in Fig. 4(c). We see that the z-component exceeds the radial component for large semi-aperture angles φ. The beam parameters are w0 = 10 μm and λ = 1 μm for this simulation; however, the results look virtually identical for other beam waists. Furthermore, we find that the results from the analytical solution parallel the numerical results. This demonstrates that the analytical model based on Eq. (16), which is a result of the approximation Eq. (8), holds even for large semi-aperture angles. Consequently, the error of the specially developed paraxial approximation, Eq. (8), is minimal.
For experimental investigation, we have designed an optical system that permits the production of radially polarized Bessel-Gauss beams. The beam generation is based on a specialty optical fiber in which a long period grating efficiently converts a linearly polarized waveguide mode to a radially polarized higher-order waveguide mode . Compared to free-space optics approaches, such as liquid crystal polarization converters or space-variant waveplates [24,25], the main advantage of the fiber-based approach is that it permits a fully integrated solution to the transformation from a linearly polarized beam to a radially polarized donut beam. Beam pointing stability and remote delivery of the generated beams are additional beneficial attributes.
A schematic of the setup is shown in Fig. 5(a). We use a tunable grating-stabilized external cavity diode laser with a wavelength of 1050 nm and a line width of 100 kHz. The laser is attached to a single-mode, polarization-maintaining fiber (PM980-XP, mode-field diameter of approximately 6.6 μm); at the fiber output, we obtain 30 mW of power. The light is then free-space coupled (using lenses L1 and L2 with focal lengths f = 4 mm) through a half-wave plate (HWP) and a quarter-wave plate (QWP) for polarization control and into a fiber module comprising a single mode fiber (SM980) spliced to a specialty optical fiber . The refractive index profile of this fiber is shown in Fig. 5(b). The linearly polarized fundamental HE11 fiber mode is mainly located in the inner core of the fiber, and a radially polarized TM01 fiber mode is dominantly located in the outer index ring. The calculated effective area of this TM01 mode at a wavelength of 1050 nm is Aeff = 51 μm2. The long period mircobend grating (LPG) converts the modes by matching the effective indices of the fundamental HE11 fiber mode to the radially polarized TM01 fiber mode . The LPG is inscribed in this specialty fiber by using a LDS3 fusion splicer . The LPG has a grating period of 762 μm and a length of about 23 mm (30 periods). After the mode conversion using the LPG, the power ratio PHE11/PTM01 is measured to be −19 dB at a wavelength of 1050 nm for the employed device. Thus, only approximately 1% of the power is left in the fundamental HE11 fiber mode after conversion. These values are obtained with an independent measurement of the LPG spectrum, as shown in Fig. 5(c). Specifically, this measurement is obtained by splicing the output of the specialty optical fiber to a single mode optical fiber.
We magnify the TM01 mode coming out of the fiber by about 60 times with a single lens L3 with f = 4 mm. Fig. 6(a) shows an image of the magnified mode that has been recorded in plane P1. The mode clearly shows an annular mode shape with some unconverted, residual HE11 power still visible in the center. With a polarizer situated directly before plane P1, the parts of the mode aligned with the polarizer are transmitted. This projection leads to a double crescent-shaped intensity profile rotating with the orientation of the polarizer as illustrated in Fig. 6(b). This confirms the radial polarization of the TM01 mode. Removing the polarizer, the mode is further magnified 10 times with a telescope (lenses L4 and L5 with focal lengths of 10 mm and 100 mm, respectively). The magnified mode is then sent through an axicon to produce a radially polarized Bessel-Gauss beam. Fig. 7(a) shows the partial beam profile at the position of the axicon, plane P2. Note that the whole beam cannot be recorded simultaneously due to limited size of the camera chip. The beam exhibits a ring radius of rd = 2350 μm and a Gaussian waist parameter of w ≈ 1050 μm. These parameters are determined by fitting the intensity profile to a Bessel-Gauss beam in the far-field . The imposed semi-aperture angle by the axicon is given by 0.45°. The on-axis crossing point of the Bessel-Gauss beam occurs at a distance rd/ tan(0.45°) ≈ 300 mm behind the axicon (plane P3). Fig. 8(a) shows the image acquired in plane P3, and Fig. 8(b) shows a comparison between the cross-section through the experimental data and our analytical expressions for radially polarized Bessel-Gauss beams. The analytical form is based on the electric field of the z-oriented vector potential Bessel-Gauss solution, where the fields are calculated with Eq. (16) and Eq. (19). The analytical expression agrees very well with the measured beam profile.
As a final note concerning our experimental work, we should point out that axicons are a common means to produce Bessel-Gauss beams; however, fabrication tolerances limit the quality of axicon-produced Bessel-Gauss beams . We have measured the relative optical path length of the axicon with a wavefront sensor (imagine optics) permitting a λ/100 rms absolute measurement accuracy while having a high dynamic range of hundreds of wavelengths. It can be seen in Fig. 7(b) that the axicon shows a round tip instead of a conic apex due to fabrication tolerances of diamond turning. The optical path impressed by the real axicon can be well approximated by a hyperbolic function , where n is the refractive index of axicon substrate (fused silica, n = 1.45) and δ is the base angle of the axicon, δ = 1°. In contrast to commonly used Gaussian illumination of axicons, the beam here possesses a ring shape, and thus, it avoids the round center of the axicon. This is why our measured beam profiles so precisely match our analytical Bessel-Gauss expressions. This technique may be useful for producing cleaner Bessel-Gauss beams in the future.
In conclusion, we have obtained expressions for radially and azimuthally polarized Bessel-Gauss beams by superposing decentered Gaussian beams with differing polarization states. This description can be applied to ABCD optical systems, which will be useful for the design of resonators for such beams. Additionally, we have developed a nonparaxial, fully numerical beam propagator for radially polarized, azimuthally symmetric beams. We have employed this to investigate radially polarized Bessel-Gauss beams and demonstrate the accuracy of our analytical expressions even for large semi-aperture angles. Finally, the form of the analytical solution has been experimentally verified by employing a fiber-based mode-converter generating an annular mode to produce a radially polarized Bessel-Gauss beam. Our radially polarized Bessel-Gauss beam generation scheme avoids the imperfect center of the axicon, and thus, results in a high-quality, radially polarized Bessel-Gauss beam.
Propagation of cylindrical Bessel-Gauss beams in ABCD optical systems
In the following, the solutions for radially and azimuthally polarized Bessel-Gauss beams are generalized for propagation in ABCD optical systems. The analytical description of decentered Gaussian beams in ABCD optical systems is known [14,15], and their essential form is identical to Eq. (9). A superposition of these beams to radially and azimuthally polarized Bessel-Gauss beams is then analogous to section 2.2. For an orientation of n̂ along a Cartesian unit vector, the results agree with previously derived expressions [15, 28]. If the orientation of n̂ is in the radial or azimuthal direction, we find the following expression:Eq. (28). The Gaussian beam parameters, q, w and ϕ transform like that of conventional on-axis Gausian beams, as demonstrated by Eq. (27). The formulation of cylindrical Bessel-Gauss beams in terms of ABCD-matrix parameters allows us to study their transformation under the influence of optical components.
It is worth noting that by comparing Eq. (26) with Eq. (14) one sees that Kz is replaced by k0. This change is required for consistency with the Collins integral . Additionally, Kz depends on the inclination angle which changes during propagation through an ABCD optical system. In fact, for propagation in a uniform medium, the phase-term Σ in Eq. (26) can be regarded as a correction of k0 to approximate Kz. However, for such propagation the description based on Kz in form of Eq. (16) is more accurate.
The authors acknowledge the contributions of Gilberto Abram to the implementation of the quasi-discrete Hankel-transform, as well as Jan Schulte for helpful discussions. We thank Imagine Optic, Inc. for the loan of the wavefront sensor (model HASO3). This work is supported by AFOSR grant FA9550-12-1-0499 and the Center for Free-Electron Laser Science. William P. Putnam acknowledges support from the NSF graduate research fellowships program.
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