Gaussian-apodized Bessel beams can be used to create a Bessel-like axial line focus at a distance from the focusing lens. For many applications it is desirable to create an axial intensity profile that is uniform along the Bessel zone. In this article, we show that this can be accomplished through phase-only shaping of the wavefront in the far field where the beam has an annular ring structure with a Gaussian cross section. We use a one-dimensional transform to map the radial input field to the axial Bessel field and then optimized the axial intensity with a Gerchberg-Saxton algorithm. By separating out the quadratic portion of the shaping phase the algorithm converges more rapidly.
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Bessel beams have been explored for a number of years as a means to localize the intensity of a laser beam over a distance much longer than the Rayleigh diffraction range corresponding to the focal spot size. Since an ideal conical wave with a constant approach angle to the optical axis will have the same spot size independent of propagation distance, these beams are often called diffraction-free. Moreover, in principle, linear superpositions of Bessel beams can be used to engineer beams with desired profiles . However, ideal Bessel beams require an infinite amount of energy and thus in practice Bessel beams are realized through truncation of the ideal wave. These apodized Bessel-beams accurately approximate the ideal Bessel beam in a “Bessel-zone” of finite width  and transition to a ring-beam in the far field [3, 4].
There are numerous applications of Bessel beams in a wide range of fields. On low energy systems, Bessel beams have been used for large depth of focus microscopic imaging systems [5, 6], optical tweezers , and optical coherence tomography . They also have applications in laser micro-machining , laser electron acceleration , and filament formation . In addition, to the best of our knowledge, the Bessel beam is the most efficient means of producing a high-intensity line focus, much more efficient than a cylindrical lens focus. The Bessel-Gauss beams in particular are useful for projecting a line focus to a position at some distance from the optic, especially for intense beams.
The two common methods for creating apodized Bessel beams are diffractive optical elements and axicons. The advantage of diffractive elements is that there is more flexibility in engineering the beam properties, but they are lossy and exhibit strong chromatic dispersion. Refractive and reflective axicons are more energy efficient but are difficult to fabricate and the Bessel-zone of such beams is localized near the tip of the axicon. In particular, with uniform input, the axial intensity profile from an axicon grows linearly with distance near the tip; with Gaussian beam input, the output axial profile has a nominal profile of zexp (−z2/w2). As will be shown below, the axial profile of a Bessel-Gauss focus has a Gaussian shape. For many applications it is advantageous to create an axial intensity profile that is nearly uniform in the Bessel-zone. The nonlinear material interactions necessary for micro-machining or plasma filament formation in air can take place within a sharp threshold and a uniform intensity profile will lead to well-controlled uniform plasma and the associated material modification. A number of methods have been proposed to engineer the radial phase profile of the optic to produce a uniform line focus. One of these is the logarithmic axicon which has been used to generate super-Gaussian profiles on the optical axis [12, 13]. Improvements in aspheric optical element production techniques have allowed the fabrication of refractive versions of the logarithmic axicon for constant intensity profile  and also for a profile that produces constant axial intensity in the presence of linear absorption . A two-step method to first convert a Gaussian beam to a ring beam with a controlled amplitude profile, then to redirect that beam to a flat top axial line focus was proposed by Honkanen and Turunen . Fresnel refractive axicons have also been attempted . A second technique to control the axial profile is with holographic diffractive elements. The first demonstration of this type was by Sochacki et al ; the holographic technique has recently been improved by using a liquid-crystal spatial light modulator (SLM) to create the desired axial profiles . While the holographic method can produce high-fidelity profiles, they exhibit low efficiency (∼5–7%), since the fidelity in the target region is achieved by allowing power to scatter away from the area of interest . It is also important to note that in the logarithmic axicon design the profile of the beam on the optical axis has the same functional form as the radial cross-section at the front of the axicon. That is, ring-beams with a Gaussian or super-Gaussian cross section passed through the axicon will form Gaussian or super-Gaussian profiles on the optical axis .
In this paper we present an efficient method for controlling the axial intensity profile of apodized Bessel-beams through a single step of phase-only shaping. The method we present in this paper is more general than the logarithmic axicon approach in that it allows different initial beam radial profiles to be shaped into a desired on-axis profile. Specifically, we consider starting with a Gaussian ring-beam and adjusting its wavefront to produce a desired axial intensity profile. Such a ring beam can be formed, for example, by passing a Gaussian beam through a diffraction grating with concentric grooves . While both phase and amplitude shaping will perform the task, it is either lossy or requires two separate shaping steps. Our phase shaping scheme is inspired by work that has been performed to shape the focal spot of conventional Gaussian beams. An example is shaping the wavefront of a Gaussian beam to create a super-Gaussian beam profile  at the focal plane of a lens. The technique we use in this paper to determine the optimum wavefront is the iterative Gerchberg-Saxton (GS) algorithm . The GS algorithm was originally introduced for phase retrieval in electron microscopes  and has since found applications in astronomy , radial beam shaping , pulse shaping  and image reconstruction . While it is well-known that the focal plane of a lens is the Fourier transform of the input field, it is less appreciated that there is a Fourier-like mapping of the input radial profile to the axial profile. This allows us to reduce the two dimensional problem to one. In this work we employ a modified version of the GS algorithm with rapid FFT iteration between two domains to optimize the axial profile to a super-Gaussian. We find that the phase shaping can be simplified by controlling the initial divergence of the input beam before the ring Gaussian is formed.
2. Bessel-Gauss beams
As mentioned in the Introduction, all physical Bessel beams must be of finite extent. One attractive representation of a finite conical wave is the Gaussian-apodized Bessel beam, also commonly called the Bessel-Gauss (BG) beam . By starting with an expression for a Gaussian beam tilted with an angle γ0 to the z-axis it has been shown within the paraxial approximation that these beams can be expressed analytically by:3], we have expressed the Gaussian beamlet profile in terms of the familiar Gouy phase Φ(z) and complex beam parameter q(z), which in turn is defined as below in terms of the z–dependent beam radius w(z) and radius of curvature R(z): Eq. (1)) intensity profile takes the form Eq. (3) if we set w(z) = w0.
Bagini et al extended the analytic formulation of the BG beams to create “generalized” Bessel-Gauss beams . In particular, it was shown that the profile given by Eq. (1) can be created remotely by propagating a ring-beam of the formFig. 1).
3. Phase-only shaping of axial intensity profile
3.1. Mapping E(r, 0) to E(0, z)
As seen above, a Gaussian ring-beam focused with a lens or axicon will produce a BG beam with a Gaussian axial intensity profile. To control the shape of the axial intensity profile we will make modifications to the radial input field. Rather than computing the full Fresnel transform, the input field E(r, 0) can be transformed directly to the axial field E(0, z) . (Note that from this point forward we will treat z = 0 as the input plane). Under the assumption of azimuthal symmetry the Fresnel propagation integral reduces toEquation 7 is in the form of a Fourier transform with Ωz taking on the role of the frequency variable which allows for rapid numerical evaluation using the fast Fourier transform (FFT).
While the mapping from the radial input field to the axial output field described above does not make any assumptions about the propagation distance z except that the Fresnel transform may be used, we are especially interested in the regime where the distance between the Bessel-zone is large compared to the (effective) Rayleigh range of the envelope on the Bessel function in the center of the Bessel-zone. Therefore, we will consider an input ring-beam of modulus E0f(r) with a modified radial phase factor ϕ(r2) focused by a lens of focal length zd:Eq. (5) adequately approximates Eq. (4) when a ≫ win.
Note that the lens phase can also be written in Ωz-space as exp [−iΩzds], where Ωzd = k/2zd. Therefore we can separate the lens phase from the input function, g(s) and the transform of the ring-beam with the shaper phase ϕ(s) will be a function of the shifted conjugate variable
3.2. Optimization of the input radial phase with the Gerchberg-Saxton algorithm
To construct a phase ϕ(r2) that will generate a desired on-axis intensity |HT(Ω′z)|2 we consider the following variational problemEq. (13) to be approximately zero is that
Similar to what has been used in pulse shaping applications, for example Rundquist et al , to optimize J[ϕ(s)] we use a modified GS algorithm. In this algorithm, the functions h(s) and HT (Ωz′) are uniformly discretized at the points si and (Ω′z)i respectively and following Fienup’s notation  the GS routine is
Since we are using the GS algorithm as an error reduction algorithm, i.e. ηj+1 < ηj, it follows that the sequence ηj will converge to a fixed value . However, this is not the same as convergence of the algorithm and in particular the sequence ϕ(j) may not be converging to a phase that is the global minimum of Eq. (13). This lack of convergence manifests itself in the form of stagnation of the algorithm at local minima. If the algorithm gets stuck before getting close to the target function, we use a modification of the GS algorithm called the input-output algorithm . Here a stronger adjustment is applied: |H(j+1) (Ωs′)| = HT (Ωs′) − α (|H(j) (Ωs′)| − |HT (Ωs′)|). A typical choice for α is 0.15. We find best convergence if we set α = 0 after the axial function approaches the target. If the optimization converges, ϕ(s) is the required shaper phase.
3.3. Optimization for a super-Gaussian axial profile in the Bessel zone
To create a profile with a uniform intensity we choose the following target field function as a test of our algorithm:Eq. (15) that in order for the the minimum value of J to be as small as possible that
Starting with choices of a, w0, k, and zd, we pick a value for the target width wT and run the GS algorithm. Figure 2 shows the results of such an optimization for input conditions zd = 1m, win = 5mm, and a = 50mm and a target width of wT = 7mm. For this set of parameters, γ0 ≈ 2.9° and the axial width without shaping would be 1.2mm. The optimized axial profile [Fig. 2(a)] is very close to the target super-Gaussian and the error as defined in Eq. (17) is approximately 10−4. Even more accurate optimization can be obtained with larger wT (see below in Fig. 4) The shape of the optimized radial phase for this run is shown in the blue curve of Fig. 2(b). There is a strong quadratic component that will be discussed below.
To see how the wavefront shaping affects the beam away from the z-axis, we numerically propagated the input Gaussian ring beam with the calculated optimum phase profile with Fresnel integration. Figure 3 shows the intensity profile in the Bessel zone. The Bessel zone radial profile looks as expected for a Gaussian-apodized Bessel beam, except that there is a taper to the diffraction rings. This taper indicates that the effective approach angle is z-dependent, γ(z). For these conditions, the radius to the first zero varies from 5μm to 6μm across the line focus. Figure 5(a) shows a cross-sectional lineout of the intensity profile rI(r) at a position that is approximately 17mm after the focal plane. To illustrate that the strong outwardly-decreasing taper is a profile that is geometrically consistent with a flat top line focus we also plot the function rI(r). Resulting from the initial wavefront shaping, the radial cross-section of the beam evolves towards a profile resembling a flat top sloped away from the optical axis. Without any shaping, the half width of the axial profile for these conditions would be 1.2mm, substantially smaller than this choice of target width, wT = 7mm. Note that the optimized phase profile is strongly quadratic in the quantity r − r0 (Fig. 2). This indicates a focusing phase, an idea supported by the image in Fig. 5(b). The first-order phase change to the beam is to defocus the ring so that the beamlets are larger where they cross the z-axis. This change in divergence can work with either sign: from the point of view of intense beam propagation, it is desirable to have the ring focus after the Bessel zone. We took advantage of this effect to improve the optimization convergence by estimating the quadratic component of the focusing phase. A straightforward calculation using the Gaussian beam propagation formula for w(z) (Eq. (2)), projected onto the z-axis is used to predict the effective divergence required to spread the beam to the desired size. After re-optimization, the phase correction excursion is considerably smaller as seen in the black dashed curve in Fig. 2.
3.4. Scaling of shaped axial profiles with wavelength and focal conditions
To explore the limits of what shaped widths could be found, we chose to scan through a range of values for wT, using the predicted quadratic phase as the starting phase for the beam. In Fig. 4, we show a contour map of the optimized axial profile as a function of the target super-Gaussian width wT for the same test conditions as above, zd = 1m, win = 5mm, and a = 50mm. The log of the convergence error is shown in the inset. Provided the target width of the superGaussian is sufficiently large, phase-only shaping yields very good results. For small wT, there is a overshoot seen at the edge that may be important for some applications. The 3D image on the right illustrates how the decrease in intensity with target width follows the expected inverse dependence.
Having determined an optimized shaper phase function, how does this function change when the system parameters are changed? The radial to axial transform in Eq. (6) transforms between the radial variable s = r2 and its shifted conjugate variable Ωz′ (see Eq. (12)). For our case we have optimized for an axial profile that is super-Gaussian axial profile in the variable δz = z − zd. Provided that wT/zd ≪ 1, Eq. (12) can be written to show that . This means that if we have optimized the radial phase profile for a width ΔΩs′, the corresponding length of the Bessel zone in z–space can be calculated as . Thus the same shaper phase profile will result in a similar axial profile if the wavelength or lens focal length are changed. For example, if we change the lens focal length in our example from 1m to 5m, the target width in Fig. 2(b) would change from 20mm to 500mm.
For broadband pulses this scaling leads to the conclusion that the length of the optimized axial region is proportional to the wavelength. With an axial length of the profile as a function of wavelength, the bandwidth will vary along the Bessel zone. This effect will be important for pulses that are sufficiently short that the fractional bandwidth is appreciable. It has been shown that Bessel beams can be made achromatic provided that the central transverse wavenumber kT = (ω/c)n(ω)sinγ0 (ω) can be made independent of frequency . This condition is possible if the ring-beam is formed by normal incidence on a diffraction grating, where the diffracted angle follows kT = 2π/d, where d is the groove spacing. Concentric gratings of this form have been developed .
4. Summary and Discussion
In this paper, we have shown that wavefront shaping of an annular ring-beam with a Gaussian cross-section can result in a line focus along the optical axis of nearly constant intensity. Starting from a Fourier transform integral mapping of the radial input profile to the axial profile, we used a modified Gerchberg-Saxton algorithm to optimize the focused profile using fast one-dimensional FFT operations. Across the input radial profile, the optimized phase excursions for our example are approximately 4π, well within the phase range of liquid crystal spatial light modulators. As mentioned in the introduction, the ring beam can be formed with a concentric diffraction grating, which also allows the radial spectral dispersion to be exploited to produce an achromatic beam. With the circular diffraction grating, a simple lens or curved mirror beam expander can be used to control the input quadratic phase of the beamlet before ring formation. This removes the quadratic component from the shaper, reducing further the phase excursions required over most of the beam profile. In fact, the residual phase seen in the black dashed curve of Fig. 2(b) is mostly quartic. A well-controlled spherical aberration term could remove that component as well. It may even be possible to design a multi element lens with the correct phase profile, as in the lens axicon designed by Burvall et al . At the expense of added complexity, the ring beam can be formed with another phase plate, in which case a two step shaping process can be used .
We anticipate this approach will find applications ranging from micro-machining to long-range filament formation in air. In the former case, the goal might be to deliver pulses at an intensity that is a prescribed amount over the damage threshold. In the second application, the line focus can be used to efficiently create an ionized column of desired length projected at a long range. We are currently working on the experimental demonstration of the methods described in this paper. The experimental work as well as the transition to applications will be addressed in subsequent publications.
The authors acknowledge funding support under the MURI AFOSR grant FA9550-10-0561 and C.D. acknowledges funding support from AFOSR under the grant FA9550-10-1-0394. The authors wish to thank Ewan Wright for many useful discussions and bringing to our attention reference . C. D. also wishes to thank Daniel Adams for useful input on the convergence of the G. S. algorithms.
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